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Journal of Functional Analysis 257 (2009) 1261–1287 www.elsevier.com/locate/jfa Uniqueness results for nonlocal Hamilto...

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Journal of Functional Analysis 257 (2009) 1261–1287 www.elsevier.com/locate/jfa

Uniqueness results for nonlocal Hamilton–Jacobi equations ✩ Guy Barles a , Pierre Cardaliaguet b,∗ , Olivier Ley a , Aurélien Monteillet b a Laboratoire de Mathématiques et Physique Théorique, Fédération Denis Poisson, Université de Tours,

Parc de Grandmont, 37200 Tours, France b Laboratoire de Mathématiques, CNRS UMR 6205, Université de Brest, 6 Av. Le Gorgeu BP 809, 29285 Brest, France

Received 18 March 2008; accepted 19 April 2009 Available online 13 May 2009 Communicated by H. Brezis

Abstract We are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal, non-monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh–Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts. © 2009 Elsevier Inc. All rights reserved. Keywords: Nonlocal Hamilton–Jacobi equations; Dislocation dynamics; Fitzhugh–Nagumo system; Nonlocal front propagation; Level-set approach; Geometrical properties; Lower-bound gradient estimate; Viscosity solutions; Eikonal equation; L1 -dependence in time

✩ This work was partially supported by the ANR (Agence Nationale de la Recherche) through MICA project (ANR-06-BLAN-0082). * Corresponding author. E-mail addresses: [email protected] (G. Barles), [email protected] (P. Cardaliaguet), [email protected] (O. Ley), [email protected] (A. Monteillet).

0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.014

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1. Introduction In this article, we are interested in uniqueness results for different types of problems which can be written as nonlocal Hamilton–Jacobi equations of the following form: ut = c[1{u0} ](x, t)|Du| u(x, 0) = u0 (x)

in RN × (0, T ),

(1.1)

in R ,

(1.2)

N

where T > 0, the solution u is a real-valued function, ut and Du stand respectively for its time and space derivatives and 1A is the indicator function of a set A. Finally u0 is a bounded, Lipschitz continuous function. For any indicator function χ or more generally for any χ ∈ L∞ with 0 χ 1 a.e., the function c[χ] depends on χ in a nonlocal way and, in the main examples we have in mind, it is obtained from χ through a convolution type procedure (either only in space or in space and time). In particular, in our framework, despite the fact that χ has no regularity neither in x nor in t, c[χ] will always be Lipschitz continuous in x; on the contrary we impose no regularity with respect to t. More precisely we always assume in what follows that there exist constants C, c, c > 0 such that (H1) For any χ ∈ L∞ (RN × (0, T ), [0, 1]), the velocity c = c[χ] is (x, t)-measurable and, for all x, y ∈ RN and t ∈ [0, T ], c(x, t) − c(y, t) C|x − y|, 0 < c c(x, t) c.

(1.3)

The first consequence of assumption (H1) is that, for any given function χ ∈ L∞ (RN × (0, T ), [0, 1]), there exists a unique solution to

ut (x, t) = c[χ](x, t)Du(x, t) in RN × (0, T ), u(·, 0) = u0 in RN ,

(1.4)

for any bounded and Lipschitz continuous initial data u0 . To give a first flavor of our main uniqueness results, we can point out the following key facts: Eq. (1.1) can be seen as the “level-set approach”-equation associated to the motion of the front Γt := {x: u(x, t) = 0} with the nonlocal velocity c[1{u(·,t)0} ]. However, in the non-standard examples we consider, it is not only a nonlocal but also non-monotone “geometrical” equation; by non-monotone we mean that the inclusion principle, which plays a central role in the “levelset approach”, does not hold and, therefore, the uniqueness of solutions cannot be proved via standard viscosity solutions methods. In fact, the few uniqueness results which exist in the literature (see below) rely on L1 type estimates on the solution; this is natural since one has to connect the continuous function u and the indicator function 1{u0} . The main estimates concern measures of sets of the type {x: a u(x, t) b} for a, b close to 0. Whether or not the aforementioned estimate has to be uniform on time, or of integral type, strongly depends on the properties of the convolution kernel. In order to emphasize this fact, we are going to concentrate on two model cases: the first one is a dislocation type equation (see Section 3) in which the kernel belongs to L∞ while the second one is related


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to the Fitzhugh–Nagumo system arising in neural wave propagation or in chemical kinetics in which the kernel is essentially the kernel of the heat equation (see Section 4). In that case, it is not in L∞ . The fact that the convolution kernel is, or is not, bounded is indeed the key difference here. Before going further, let us give some references: for the first model case (dislocation type equations), we refer the reader to Barles, Cardaliaguet, Ley and Monneau [4] where general results are provided for these equations. We point out—and we will come back to this fact later—that uniqueness in the non-monotone case was first obtained by Alvarez, Cardaliaguet and Monneau [1] and then by Barles and Ley [6] using different arguments; we also refer to Rodney, Le Bouar and Finel [20] for the physical background on these equations. The Fitzhugh– Nagumo system has been investigated in particular by Giga, Goto and Ishii [13], and by Soravia and Souganidis [21]. They provided a notion of weak solution for this system (see (4.1) below) and proved existence of such weak solutions. They also study the connections with the phase field model (a reaction–diffusion system which leads to such a front propagation model). However the uniqueness question has been left open until now. Let us return to the key steps to prove uniqueness for (1.1)–(1.2). A major issue is the properties of the solutions of the eikonal equations of the form ut = c(x, t)|Du|

in RN × (0, T ),

(1.5)

where c is a continuous function, satisfying suitable assumptions. Of course, such partial differential equations appear naturally when considering 1{u0} as an a priori given function. We recall that this equation is related via the level-set approach to the motion of fronts with an (x, t)-dependent normal velocity c(x, t) and to deal with compact fronts and to simplify matter, we assume that the initial datum satisfies the following conditions: the subset {u0 > 0} is non-empty and there exists R0 > 0 such that u0 = −1 in RN \ B(0, R0 ).

(1.6)

This implies, in particular, that the initial front Γ0 = {u0 = 0} is a non-empty compact subset of B(0, R0 ). Assumption (H1) ensures existence and uniqueness of a solutions to (1.5) but we also assume that the function c = c[χ] is nonnegative (and even positive), together with (H2) There exists η0 > 0 such that −u0 (x) − Du0 (x) + η0 0

in RN in the viscosity sense.

The above assumption implies that the set {u = 0} has a zero Lebesgue measure (cf. Ley [15]) which is an important property for our arguments. Indeed [4] provides a counter-example (even in a (quasi) monotone case) where fattening phenomena leads to a non-uniqueness feature for a nonlocal equation. In addition to this non-fattening property, a key consequence of (H1)–(H2) is a lower bound on the gradient Du on a set {x: |u(x, t)| η} for a small enough η (cf. [15]). We now concentrate on the estimates of the measures of the volume of sets like {a u(·, t) b} where −η a < b η. We first note that such estimates are related with perimeter estimates of the α level-sets of u for α close to 0 (typically |α| < η): indeed, combining the co-area formula with the lower bound on the gradient of the solution, we obtain

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b

1{au(·,t)b} dx =

|Du|−1 dHn−1 ds

a {u(·,t)=s}

RN

b−a sup Per u(·, t) = s , η asb

(1.7)

where η is the lower bound on |Du| on the set {x: |u(x, t)| η}. In [1] and [6], perimeter estimates for the α level-sets of u were obtained by using bounds on the curvatures of these sets. Although this approach was powerful, it has the drawback to require strong assumptions on the dependence in x of c[χ] (typically a C 1,1 regularity). Unfortunately such strong regularity does not always hold: for instance it is not the case for the Fitzhugh– Nagumo system. The key contribution of this paper is to provide L1 ([0, T ]) or L∞ ([0, T ]) estimates of the volume of the set {a u(·, t) b} (or, almost equivalently, of the perimeter of the α level-sets of u) in situations where the velocity c[χ] is less regular in x. As a consequence we are able to prove new uniqueness results. For the dislocation dynamics model, our approach allows to relax the assumptions of [1] and [6] on the data. The proofs are also simpler, requiring only L1 ([0, T ]) estimates and a mild regularity (c[χ] is merely measurable in time and Lipschitz continuous in space). So the main conclusion here is that “soft” estimates are sufficient provided the convolution kernel is in L∞ . On the contrary, for the Fitzhugh–Nagumo system, where the convolution kernel is unbounded, these L1 -estimates are no more sufficient and the uniqueness proof rather requires heavy L∞ -estimates on the perimeter. These estimates are obtained by establishing, through optimal control type arguments, that the set {x: u(x, t) > 0} satisfies a uniform “interior cone property”, from which we deduce (explicit) estimates on the perimeter. The paper is organized as follows: in Section 2, we recall the notion of weak solution for (1.1) introduced in [4]. In Section 3 we prove uniqueness of the solution for the dislocation type equation, while we deal with the Fitzhugh–Nagumo case in Section 4. The main technical results of this paper are gathered in Section 5: we recall here some useful results for the eikonal equation (1.5), we show the interior cone property and deduce the uniform perimeter estimates. Notation. In the sequel, | · | denotes the standard euclidean norm in RN , B(x, R) (resp. B(x, R)) is the open (resp. closed) ball of radius R centered at x ∈ RN . We denote the essential supremum of f ∈ L∞ (RN ) or f ∈ L∞ (Rn × (0, T )) by |f |∞ . Finally, Ln and Hn denote, respectively, the n-dimensional Lebesgue and Hausdorff measures. 2. Definition of weak solutions to (1.1) We will use the following definition of weak solutions introduced in [4]. Definition 2.1. Let u : RN × [0, T ] :→ R be a continuous function. We say that u is a weak solution of (1.1)–(1.2) if there exists χ ∈ L∞ (RN × [0, T ]; [0, 1]) such that (1) u is an L1 -viscosity solution of (1.4). (2) For almost all t ∈ [0, T ], 1{u(·,t)>0} χ(·, t) 1{u(·,t)0}

in RN .


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Moreover, we say that u is a classical solution of (1.1) if in addition, for almost all t ∈ [0, T ] and almost everywhere in RN , 1{u(·,t)>0} = 1{u(·,t)0} . We refer to [4, Appendix] for the basic definition and properties of L1 -viscosity solutions and to [14,16,18,19,8,9] for a complete presentation of the theory. 3. Model problem 1: dislocation type equations In this section, we consider equations arising in dislocations theory (cf. [2,20]) where, for all χ ∈ L∞ (RN × (0, T )) or L1 (RN × (0, T )), c[χ] is defined by c[χ](x, t) = (c0 ∗ χ)(x, t) + c1 (x, t)

in RN × (0, T ),

(3.1)

where c0 , c1 are given functions, satisfying suitable assumptions which are described later on and “∗” stands for the usual convolution in RN with respect to the space variable x. Our main result below applies to slightly more general cases but the main interesting points appear on this model case. We refer to [4] for a complete description of the characteristics and difficulties connected to (1.1) in this case; as recalled in the introduction, this equation is not only nonlocal but it is also, in general, non-monotone, which means that the maximum principle (or, here, inclusion principle) does not hold and one cannot apply directly viscosity solutions’ theory. Roughly speaking, a (more or less) direct use of viscosity solutions’ theory requires that c0 0 in RN × (0, T ), an assumption which is not natural in the dislocations’ framework. We use the following assumptions on c0 and c1 . (H3) c0 , c1 ∈ C 0 (RN × [0, T ]) and there exists a constant C such that, for any x, y ∈ RN and t ∈ [0, T ], c0 (x, t) − c0 (y, t) + c1 (x, t) − c1 (y, t) C|x − y|. Moreover, c0 ∈ C 0 ([0, T ]; L1 (RN )) and there exist c, c > 0 such that, for any x ∈ RN and t ∈ [0, T ], c0 (x, t) c, 0 < c −c0 (·, t)L1 + c1 (x, t) c0 (·, t)L1 + c1 (x, t) c. This assumption ensures that the velocity c[χ] in (3.1) satisfies (H1) with constants independent of 0 χ 1 with compact support in some fixed ball (see Step 1 in the proof of Theorem 3.1). Note that, in contrast to [4], we do not assume that c0 , c1 are C 1,1 (or semiconvex). We provide a direct proof of uniqueness for the solution of the dislocation equation (1.1); we recall that the existence of weak solutions is obtained in [4,5] and that, in our case, the weak solutions are classical solutions since the set {u = 0} has a zero Lebesgue measure by the result of [7,15] since c[χ] 0 for all 0 χ 1.

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Theorem 3.1. Suppose that c0 , c1 satisfy (H3) and that u0 is a Lipschitz continuous function satisfying (H2) and such that (1.6) holds. Then (1.1)–(1.2) has a unique (Lipschitz) continuous viscosity solution in RN × [0, T ]. Proof. 1. Uniform bounds for the velocity. By (H3) and Lemma 5.3, the set {u(·, t) 0} remains in a fixed ball B(0, R0 + cT ) of RN . Then, for any subset A of B(0, R0 + cT ), c[1A ] satisfies (H1) with constants which are uniform in A. 2. L∞ -estimate. If u1 , u2 are two solutions of (1.1)–(1.2), for 0 < τ T , we set δτ :=

sup u1 (x, t) − u2 (x, t).

RN ×[0,τ ]

Since u0 is Lipschitz continuous and 0 c[1{ui 0} ] c (i = 1, 2), for τ small enough, we have δτ η/2 where η is obtained by applying Theorem 5.1 to the ui ’s. By Lemma 5.2, we have τ δτ |Du0 |∞ e

Cτ

c[1{u

1 0}

] − c[1{u2 0} ] (·, t)∞ dt

0

τ |Du0 |∞ eCτ

c0 (·, t) ∗ (1{u

1 (·,t)0}

− 1{u2 (·,t)0} )∞ dt

0

τ c|Du0 |∞ e

|1{u1 0} − 1{u2 0} | dx dt

CT 0 RN

by using the L∞ -bound |c0 |∞ c. 3. L1 -estimate. We have |1{u1 0} − 1{u2 0} | 1{−δτ u1 0} + 1{−δτ u2 0}

in RN × [0, τ ].

Using Proposition 5.5 we get τ |1{u1 0} − 1{u2 0} | dx dt

2δτ ψτ , ηc

0 RN

where we have set ψτ = L N

x: u0 (x) −δτ − c|Du0 |∞ τ − LN x: u0 (x) 0 .

4. Uniqueness on [0, τ ] for small τ . Using this information in (3.2) yields δτ

2c |Du0 |∞ eCT ψτ δτ , ηc

(3.2)


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namely δτ Lψτ δτ , where L = L(T , c, c, C, η, |Du0 |∞ ) is a constant. Since the 0-level set of u0 has a zero Lebesgue measure from assumption (H2), we have ψτ → 0 as τ → 0. Therefore, for τ small enough, Lψτ < 1 and necessarily δτ = 0. It follows that u1 = u2 on RN × [0, τ ]. 5. Uniqueness on the whole time interval. Step 4 gives the uniqueness for small times but then we can consider τ = sup τ > 0; u1 = u2 on RN × [0, τ ] . In fact, by continuity of u1 and u2 , τ is a maximum. If τ < T , then we can repeat the above proof from time τ instead of time 0. This is, in fact, rather straightforward since u(·, τ ) satisfies the same properties as u0 . Finally, τ = T and the proof is complete. 2 4. Model problem 2: a Fitzhugh–Nagumo type system We are now interested in the following system: ⎧ ⎨ ut = α(v)|Du| v − v = g + (v)1{u0} + g − (v)(1 − 1{u0} ) ⎩ t u(·, 0) = u0 , v(·, 0) = v0

in RN × (0, T ), in RN × (0, T ), in RN ,

(4.1)

which is obtained as the asymptotics as ε → 0 of the following Fitzhugh–Nagumo system arising in neural wave propagation or chemical kinetics (cf. [21]): ⎧ ⎨ uε − ε uε = 1 f uε , v ε in RN × (0, T ), t ε ⎩ ε vt − v ε = g uε , v ε in RN × (0, T ),

(4.2)

where

f (u, v) = u(1 − u)(u − a) − v g(u, v) = u − γ v

(0 < a < 1), (γ > 0).

The functions α, g + and g − appearing in (4.1) are Lipschitz continuous functions on R associated with f and g. The functions g − and g + are bounded and satisfy g − g + in R. The initial datum v0 is bounded and of class C 1 in RN with |Dv0 |∞ < +∞. System (4.1) corresponds to a front Γ (t) = {u(·, t) = 0} moving with normal velocity α(v), the function v being itself the solution of an interface reaction–diffusion equation depending on the regions separated by Γ (t). The u-equation in (4.1) can be written as (1.1)–(1.2) although the dependence of c in 1{u(·,t)0} is less explicit than in the first model case. More precisely, for

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χ ∈ L∞ (RN × [0, T ], [0, 1]), let v be the solution of vt − v = g + (v)χ + g − (v)(1 − χ) v(·, 0) = v0

in RN × [0, T ], in RN .

(4.3)

Then the problem (4.1) reduces to (1.1)–(1.2) with c[χ](x, t) = α(v(x, t)). Under the additional assumption that α > 0 in R, we prove uniqueness of solutions to the system (4.1) (or equivalently (1.1)). We suppose (H4) v0 is bounded and C 1 , g − , g + are Lipschitz continuous with |Dv0 |∞ < +∞ and g g − (r) g + (r) g

for all r ∈ R.

(H5) α is Lipschitz continuous and there exist c, c, C > 0 such that, for all r, r ∈ R, c α(r) c, α(r) − α(r ) C|r − r |. (H6) u0 is Lipschitz continuous and satisfies (1.6) with K0 := {u0 0} which is the closure of a non-empty bounded open subset of RN with C 2 boundary. Theorem 4.1. Under assumptions (H2), (H4), (H5), (H6), system (4.1) has a unique solution. We recall that the existence of weak solutions is obtained in [13,21]. Moreover, since α > 0 in R, weak solutions are classical thanks to the results of [15]. Before giving the uniqueness proof, we start by a preliminary result on the inhomogeneous heat equation. 4.1. Classical estimates for the inhomogeneous heat equation We first gather some regularity results for the solutions of the heat equation (4.3). The explicit resolution of the heat equation (4.3) shows that for any (x, t) ∈ RN × [0, T ], v(x, t) = G(x − y, t)v0 (y) dy RN

t +

G(x − y, t − s) g + (v)χ + g − (v)(1 − χ) (y, s) dy ds,

0 RN

where G is the Green function defined by G(y, s) =

2 1 − |y| 4s . e (4πs)N/2

(4.4)

Lemma 4.2. Assume that (H4) holds. For χ ∈ L∞ (RN × [0, T ]; [0, 1]), let v be the unique solution of (4.3). Set γ = max{|g|, |g|}. Then there exists a constant kN depending only on N such that


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(i) v is uniformly bounded: for all (x, t) ∈ RN × [0, T ], v(x, t) |v0 |∞ + γ t. (ii) v is continuous on RN × [0, T ]. (iii) For any t ∈ [0, T ], v(·, t) is of class C 1 in RN . (iv) For all t ∈ [0, T ], x, y ∈ RN , √ v(x, t) − v(y, t) |Dv0 |∞ + γ kN t |x − y|. (v) For all 0 s t T , x ∈ RN , √ √ v(x, t) − v(x, s) kN |Dv0 |∞ + γ kN s t − s + γ (t − s). In particular, under assumption (H5), the velocity c[χ] (given here by α(v)) is bounded, continuous on RN × [0, T ] and Lipschitz continuous in space, uniformly with respect to χ . It follows that (1.4) has a unique continuous (classical) viscosity solution for all χ ∈ L∞ (RN × [0, T ]; [0, 1]). Proof of Lemma 4.2. Properties (i), (ii) and (iii) are straightforward applications of assumption (H4) and of the representation formula for the solution. We also note that Dv(x, t) = G(x − y, t)Dv0 (y) dy RN

t +

+

− DG(x − y, t − s) g (v)χ + g (v)(1 − χ) (y, s) dy ds

0 RN

γ |Dv0 |∞ + 2(4π)N/2

t

|x−y|2

(t − s)−N/2−1 |x − y|e− 4(t−s) dy ds

0 RN

|Dv0 |∞ + γ kN t 1/2 , whence (iv). As for (v) we have, from the semi-group property, v(x, t) − v(x, s) =

G(x − y, t − s) v(y, s) − v(x, s) dy

RN

t +

G(x − y, t − τ ) g + (v)χ + g − (v)(1 − χ) (y, τ ) dy dτ.

s RN

The first integral can be estimated by using (iv) while the second one is bounded by γ (t − s).

2

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4.2. Proof of Theorem 4.1 1. Properties of the velocity. As explained above, for any measurable subset A of RN , the velocity c[1A ] in (1.1) satisfies (H1) with constants which are uniform in A: for all x, x ∈ RN , t ∈ [0, T ], c c[1A ] c c[1A ](x, t) − c[1A ](x , t) C|x ˜ − x |, √ with C˜ := C(|Dv0 |∞ + γ kN T ). By Lemma 5.3, it follows that the set {u(·, t) 0} remains in a fixed ball B(0, R0 + cT ) of RN . 2. First estimate (eikonal equation). We start as in the proof of Theorem 3.1. Let u1 , u2 be two solutions of (1.1) and v1 , v2 be the solutions of (4.3) associated with u1 , u2 respectively. For 0 τ T , we set δτ :=

sup u1 (x, t) − u2 (x, t)

RN ×[0,τ ]

and we choose τ small enough in order that δτ < η/2 where η is given by applying Theorem 5.1 to the ui ’s. By Lemma 5.2, we have

δτ |Du0 |∞ e

˜ Cτ

τ

c[1{u 0} ] − c[1{u 0} ] (·, t) dt 1 2 ∞

0

|Du0 |∞ e

˜ Cτ

τ

α(v1 ) − α(v2 ) (·, t) dt ∞

0

C|Du0 |∞ e

˜ CT

τ

(v1 − v2 )(·, t) dt. ∞

(4.5)

0

It remains to estimate |(v1 − v2 )(·, t)|∞ . 3. Second estimate (heat equation). The function v = v1 − v2 solves vt − v = (1{u1 0} − 1{u2 0} ) g + (v1 ) − g − (v1 ) + 1{u2 0} g + (v1 ) − g + (v2 ) − 1{u2 0} g − (v1 ) − g − (v2 ) + g − (v1 ) − g − (v2 ) in RN × [0, T ]. Since g + and g − are Lipschitz continuous, say with Lipschitz constant M, we have 1{u 0} g + (v1 ) − g + (v2 ) − 1{u 0} g − (v1 ) − g − (v2 ) + g − (v1 ) − g − (v2 ) 3M|v|. 2 2


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Moreover |1{u1 0} − 1{u2 0} |g + (v1 ) − g − (v1 ) (g − g)|1{u1 0} − 1{u2 0} |, by (H4). This implies that both v and −v are viscosity subsolutions of wt − w − 3M|w| = (g − g)|1{u1 0} − 1{u2 0} |

in RN × [0, T ],

whence |v| = max{v, −v} is also a subsolution as the maximum of two subsolutions. Therefore we have |v|t − |v| − 3M|v| (g − g)|1{u1 0} − 1{u2 0} |

in RN × [0, T ].

In particular the function w : (x, t) → e−3Mt |v(x, t)| satisfies wt − w (g − g)e−3Mt |1{u1 0} − 1{u2 0} | in RN × [0, T ]. By the comparison principle, since w(·, 0) = 0, we have for any (x, t) ∈ RN × [0, τ ], t w(x, t)

G(x − y, t − s)(g − g)e−3Ms |1{u1 0} − 1{u2 0} |(y, s) dy ds.

0 RN

Using the definition of δτ , we have |1{u1 0} − 1{u2 0} |(y, s) 1{−δτ u1 0, η = η(C, c, η0 ) > 0 such that the viscosity solution u of (1.5) satisfies in the viscosity sense eγ t Du(x, t)2 + η 0 −u(x, t) − 4

in RN × [0, T ].

(5.1)

We refer the reader to [11,15] for the proof of this result. Let us mention that (H1) implies that p ∈ RN → c(x, t)|p| is convex for every (x, t) ∈ RN × [0, T ] which is a key assumption to prove (ii). We remark that, in (ii), u is Lipschitz continuous because the assumptions of (i) are satisfied. Therefore u is differentiable a.e. in RN ×[0, T ] and (5.1) holds a.e. in RN ×[0, T ]. Part (ii) gives a lower-bound gradient estimate for u near the front {(x, t) ∈ RN × [0, T ]: u(x, t) = 0}. Indeed, if |u(x, t)| < η/2, then −Du(x, t) − 2ηe−γ T /2 := −η < 0 in |u| < η/2

(5.2)

in the viscosity sense (and almost everywhere in {|u| < η/2}). We continue by giving an upper-bound for the difference of two solutions with different velocities ci . Lemma 5.2. (See [6].) For i = 1, 2, let ui ∈ C 0 (RN × [0, T ]) be a solution of (ui )t = ci (x, t)|Dui | in RN × [0, T ], in RN , ui (x, 0) = u0 (x) where ci satisfies (H1) and u0 is Lipschitz continuous. Then, for any t ∈ [0, T ], (u1 − u2 )(·, t)

t ∞

|Du0 |∞ e

Ct 0

(c1 − c2 )(·, s) ds. ∞

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Finite speed of propagation implies a uniform bound for compact fronts governed by eikonal equations: Lemma 5.3. (See [6].) Suppose that (H1) holds and that u0 is Lipschitz continuous and satisfies (1.6). Let u be the viscosity solution of (1.5) with initial condition u0 . Then, for all t ∈ [0, T ], u(·, t) 0 ⊂ B(0, R0 + ct). Lemma 5.4 (Viscosity increase principle). (See [6].) Let w ∈ C 0 (RN ) satisfying (H2) and δ < η0 /2. If x ∈ {−δ w δ}, then sup

w w(x) + δ.

B(x,2δ/η0 )

We refer the reader to [6] for the proofs of these results. 5.2. Estimates on the measure of level-sets for solutions of (1.5) Now we turn to the key estimates on the measure of small level-sets of the solution of the eikonal equation (1.5). For every −η/2 a < b η/2, we consider the function ϕ : R → R+ , depending on a and b such that ϕ = 0 on (−∞, a), ϕ (t) = (b − a)−1 in (a, b) and ϕ = 1 on [b, +∞). In fact, ϕ is chosen in such a way that (b − a)ϕ is the indicator function of [a, b]. We omit to write the dependence of ϕ with respect to a, b for the sake of simplicity of notations. Proposition 5.5. Assume (H1), (H2) hold and suppose that {u0 0} is a compact subset of RN . Let −η/2 a < b η/2 where η is defined in (5.1) and let u be the unique Lipschitz continuous viscosity solution of (1.5). Then, for any 0 < τ T τ 1{aub} dx dt 0 RN

b−a ηc

ϕ u(x, τ ) − ϕ u(x, 0) dx,

(5.3)

RN

where η is defined in (5.2). It follows τ 1{aub} dx dt

b − a N L u(·, τ ) a − LN u(·, 0) b dx, ηc

(5.4)

0 RN

and τ 1{aub} dx dt 0 RN

b − a N L u(·, 0) a − c|Du0 |∞ τ − LN u(·, 0) b . ηc

(5.5)


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Remark 5.6. The above proposition is related with results obtained by the fourth author in [17] for the eikonal equation with a changing sign velocity. Proof of Proposition 5.5. By the definition of ϕ τ

τ 1{aub} dx dt =

0 RN

(b − a)ϕ u(x, t) dx dt.

0 RN

Using the fact that −η/2 a < b η/2 and the definition of η in (5.2), we can estimate the right-hand side by τ

c(x, t) |Du| dx dt, (b − a)ϕ u(x, t) c η

0 RN

since c c on RN × (0, T ) and |Du| η on the set {|u| η/2}. Therefore, by the equation, we have the following equality b−a cη

τ

(b − a) ϕ u(x, t) c(x, t)|Du| dx dt = cη

0 RN

τ

ϕ u(x, t) t dx dt,

0 RN

and (5.3) follows by applying Fubini’s theorem and integrating. Inequality (5.4) follows easily by taking into account the form of ϕ. To deduce (5.5), it is sufficient to note that, since u0 + c|Du0 |∞ t is a supersolution of (1.5), we have, by comparison, u(x, t) u0 (x) + c|Du0 |∞ t in RN × (0, T ). 2 5.3. Estimate of the perimeter of sets with the interior cone property Definition 5.7. Let K be a compact subset of RN . We say that K has the interior cone property of parameters ρ and θ if 0 < ρ < θ and if, for any x ∈ ∂K, there exists some ν ∈ SN −1 such that the set ρ,θ := x + [0, θ ]B(ν, ρ/θ ) Cν,x ρ = x + λν + λ ξ : λ ∈ [0, θ ], ξ ∈ B(0, 1) θ

is contained in K (see Fig. 1). Theorem 5.8. Let K be a compact subset of RN having the interior cone property of parameters ρ and θ . Then there exists a positive constant Λ = Λ(N, ρ, θ/ρ) such that for all R > 0, HN −1 ∂K ∩ B(0, R) ΛLN K ∩ B(0, R + ρ/4) .

(5.6)

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ρ,θ

Fig. 1. Cν,z : interior cone at z of parameters ρ, θ and axis ν.

Proof. 1. Restriction to a finite number of axes for the interior cones. We first observe that ρ,θ if z ∈ ∂K and Cν,z ⊂ K, then for all ν ∈ SN −1 verifying |ν − ν | ρ/(2θ ), we have ρ/2,θ Cν ,z ⊂ K. By compactness of SN −1 , we can cover SN −1 with the traces on SN −1 of at most p := β(N )/(ρ/(2θ ))N −1 balls of radius ρ/(2θ ) centered at νi , for some positive constant β(N ) ρ/2,θ and 1 i p. Therefore, for any z ∈ ∂K, there exists 1 i p such that Cνi ,z ⊂ K. 2. Local study of points of the boundary with the same interior cone axis. We fix 1 i p and set ρ/2,θ Ai = {z ∈ ∂K; Cνi ,z ⊂ K}. Up to a rotation of K, we can assume that νi = (0, . . . , 0, −1) =: ν. Let us fix z ∈ Ai , that we write z = (x, y) with x ∈ RN −1 and y ∈ R. Let us set V = BN −1 (x, ρ/4) × (y − θ/2, y + θ/2) and Di = V ∩

ρ/2,θ

Cνi ,(x ,y )

(x ,y )∈Ai ∩V

(see Fig. 2). Then Ai ∩ V ⊂ ∂Di ∩ V : indeed if (x , y ) ∈ Ai ∩ V , then (x , y ) ∈ Di ∩ V , and (x , y ) cannot lie in the interior of Di , otherwise for λ > 0 small enough, we would have (x , y ) − λν ∈ Di , which would imply that (x , y ) lies in the interior of one of the cones forming Di , and therefore in the interior of K, which is absurd since (x , y ) ∈ ∂K. 3. The set ∂Di ∩ V is a Lipschitz graph of constant (2θ/ρ)2 − 1. More precisely let us prove that ∂Di ∩ V is equal to Gi = (x , y ): x ∈ BN −1 (x, ρ/4) and y = max y : (x , y ) ∈ ∂C for one of the cones C forming Di . First of all, it is easy to show that Di is closed, and that the maximum in the definition of Gi exists and is not equal to y + θ2 ; otherwise there would exist a cone C in Di such that (x, y) ∈ int(C) ⊂ int(K), which is absurd. The inclusion Gi ⊂ ∂Di ∩ V follows from the same argument used for the inclusion Ai ∩ V ⊂ ∂Di ∩ V in Step 2. Conversely, let us fix (x , y ) ∈ ∂Di ∩ V .


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Fig. 2. Illustration of the proof of Theorem 5.8.

Then (x , y ) ∈ Di since Di is closed, so that (x , y ) is included in the trace on V of one of the cones forming Di , let us say (x , y ) ∈ C. But then (x , y ) cannot belong to int(C), otherwise we would have (x , y ) ∈ int(Di ), so we deduce that (x , y ) ∈ ∂C ∩ V . Moreover if there exists y > y such that (x , y ) ∈ ∂C for some other of the cones C forming Di , then we must have (x , y ) ∈ int(C ) ∩ V ⊂ int(Di ), which is absurd, and proves that y is equal to the maximum in the definition of Gi , and that ∂Di ∩ V ⊂ Gi . Therefore ∂Di ∩ V is a Lipschitz graph of constant μ = (2θ/ρ)2 − 1 as a supremum of graphs of cones of the same parameters ρ/2 and θ . 4. Estimate of the perimeter of Ai in V . It follows from Step 3 that ∂Di ∩ V is HN −1 measurable with HN −1 (∂Di ∩ V ) LN −1 BN −1 (x, ρ/4) 1 + μ2 , hence HN −1 (Ai ∩ V ) ωN −1

N −1 ρ 2θ , 4 ρ

where ωj denotes the volume of the unit ball of Rj . 5. Covering of Ai with balls of fixed radius. By Besicovitch’s covering theorem (see [12]), there exists a constant ξN depending only on N such that for any ε > 0 and R > 0, there exist numbers

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Γ1 , . . . , ΓξN and a finite family (xkj ) (for 1 k ξN and 1 j Γk ) of points of Ai ∩ B(0, R) such that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Ai ∩ B(0, R) ⊂

ξN Γk

B(xkj , ε),

k=1 j =1

for each k, the balls B(xkj , ε), 1 j Γk , are pairwise disjoint.

The family (xkj )j is a priori only countable, but has to be finite by boundedness of Ai and ξN Γk . Let us therefore because the radius of covering balls is fixed. We now want to estimate k=1 compute

ξN Γk

K∩B(0,R+ε)

k=1 j =1

1B(xkj ,ε) .

On the one hand, we have ξN

Γk

k=1 j =1 K∩B(0,R+ε)

1B(xkj ,ε) ξN LN K ∩ B(0, R + ε) ,

(5.7)

because for each k, the balls B(xkj , ε) are pairwise disjoint. On the other hand, for each k and j , ρ/2,θ the ball B(xkj , ε) contains a fixed portion of the cone Cνi ,xkj , portion which is included in K ∩ B(0, R + ε) by the interior cone property, since xkj ∈ Ai ∩ B(0, R). We call ρ/2,θ γ := LN B(xkj , ε) ∩ Cνi ,xkj the volume of this portion of cone, the computation of which is done in Step 7. Note that γ is independent of xkj . Therefore K∩B(0,R+ε)

ξN Γk k=1 j =1

1B(xkj ,ε) =

ξN Γk

k=1 j =1 K∩B(0,R+ε)

1B(xkj ,ε)

ξN

Γk γ .

(5.8)

k=1

From (5.7) and (5.8), we deduce ξN k=1

Γk

ξN N L K ∩ B(0, R + ε) . γ

Since B((x, y), ε) ⊂ V = BN −1 (x, ρ/4) × (y − θ/2, y + θ/2), as soon as ε < min{ρ/4, θ/2} = ξN Γk cylinders of the form of ρ/4, we deduce from this that Ai ∩ B(0, R) can be covered by k=1 V centered at points of Ai ∩ B(0, R), so that, from (5.7),


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N −1 ξN ρ 2θ HN −1 Ai ∩ B(0, R) Γk ωN −1 4 ρ k=1

N −1 ρ 2θ N ξN ωN −1 L K ∩ B(0, R + ε) . γ 4 ρ 6. Sum for all axes. What we have done does not depend on the fixed direction axis νi , and we β(N ) know, thanks to Step 1 that ∂K is the union of less than p = (ρ/2θ) N−1 sets of the form Ai , so that we finally have

H

N −1

∂K ∩ B(0, R)

N −1 ρ 2θ N β(N) ξN ωN −1 L K ∩ B(0, R + ε) N −1 γ 4 ρ (ρ/2θ )

which gives (5.6). 7. Computation of the value of γ . As soon as ε θ 2 − (ρ/2)2 (the length of the longest segρ/2,θ ρ/2,θ ment included in ∂Cνi ,xkj ), then B(xkj , ε) contains at least the straight portion of Cνi ,xkj of length l = ρμε/(2θ ), the volume of which equals N ρ ωN −1 l N ωN −1 μ ε . = N μN −1 N 2θ This gives a lower bound for γ . Moreover, we obtain a more precise estimate for Λ in (5.6): since ρ < θ , we see that ρ/4 θ 2 − (ρ/2)2 , so that sending ε to ρ/4, we get 1 (θ/ρ)2N LN K ∩ B(0, R + ρ/4) . HN −1 ∂K ∩ B(0, R) 4N +1 Nβ(N )ξN ρ (2θ/ρ)2 − 1

2

5.4. Propagation of the interior cone property We want to prove that the interior cone property is preserved for sets whose evolution is governed by the eikonal equation (1.5). We assume: (H7) The function c(·, t) is C-Lipschitz continuous with a constant C independent of t ∈ [0, T ] and, for all R > 0, there exists an increasing modulus of continuity ωR such that, for all x ∈ B(0, R), t, s ∈ [0, T ], then c(x, t) − c(x, s) ωR |t − s| . Theorem 5.9. Assume that c satisfies (H1) and (H7) and that u0 satisfies (H6). Let u be the unique uniformly continuous viscosity solution of (1.5). Then there exist ρ > 0 and θ > 0 depending only on K0 , T , c, c and C, such that K(t) = {x ∈ RN ; u(x, t) 0} has the interior cone property of parameters ρ and θ for all t ∈ [0, T ]. More precisely, let r > 0 be such that K0 has

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the interior ball property of radius r > 0, then we can choose 2 c −1 c θ = min , cωR ,r 6Cc 4

and ρ =

c θ 2c

where R > 0 is such that K0 + cT B(0, 1) ⊂ B(0, R). Proof. 1. Minimal time function. We first claim that t → u(x, t) is nondecreasing for any x ∈ RN and, if u(x, t) = 0, then u(x, s) > 0 for any s ∈ (t, T ]. Indeed, let (x, t) ∈ RN × [0, T ] such that u(x, t) = 0. Since u is Lipschitz continuous and the lower-gradient bound estimate (5.2) holds almost everywhere in {|u| < η/2}, there exists r0 > 0 small enough such that, for all 0 t s t + r0 and r r0 ,

u(y, s) − u(y, t) dy =

s ut (y, τ ) dy dτ t B(x,r)

B(x,r)

s =

c(y, τ )Du(y, τ ) dy dτ

t B(x,r)

LN B(x, r) cη(s − t). Dividing by LN (B(x, r)) and letting r → 0, we get that u(x, s) > 0 if t < s t + r0 . The proof that t → u(x, t) is nondecreasing for any x ∈ RN can be obtained in a similar way by simpler arguments (in particular we do not need (5.2)). Therefore, the minimal time function v(x) = min t ∈ [0, T ]; u(x, t) 0 is defined at points x ∈ K(T ), and for any t ∈ [0, T ], x ∈ RN ; u(x, t) 0 = x ∈ RN ; v(x) t , x ∈ RN ; u(x, t) = 0 = x ∈ RN ; v(x) = t . Moreover, v is (1/c)-Lipschitz on K(T ): let us fix x and y in K(T ) with v(x) v(y). The function u : (z, t) →

sup

|z −z|c|t−v(x)|

u z , v(x)

is the unique uniformly continuous viscosity solution (see [3]) of

ut (z, t) = cDu(z, t) u ·, v(x) = u ·, v(x)

in RN × v(x), T , in RN .


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The comparison principle for continuous viscosity solutions implies that u u in RN × [v(x), T ]. In particular 1 1 u y, |x − y| + v(x) u y, |x − y| + v(x) , c c which implies by definition of u and v that 1 1 0 = u x, v(x) u y, |x − y| + v(x) u y, |x − y| + v(x) , c c from which the Lipschitz property follows, since we deduce that 1 v(y) |x − y| + v(x). c 2. Interior cone property at time t ∈ [μ, T ] for some μ > 0. To prove the claim of the theorem, we will use arguments from control theory. For this we need the velocity c to be C 1 in space, additional condition that we can assume without loss of generality by replacing c by suitable space convolution cδ of c. Then we get the result for cδ , and, letting δ → 0+ , obtain the desired result since the constants θ and ρ do not depend on δ. It is well known that, for each time t, the set K(t) can be seen as the reachable set from K0 for the controlled system x (t) = c x(t), t a(t)

for t ∈ [0, T ],

(5.9)

where the control a takes its values in the unit closed ball. Let x be an extremal trajectory, i.e. a trajectory verifying x(T ) ∈ ∂K(T ). For such a trajectory, it is easy to see that t → u(x(t), t) is nondecreasing, from which we infer that x(t) ∈ ∂K(t) for any t ∈ [0, T ], that is to say, v(x(t)) = t. The Pontryagine maximum principle [10] implies the existence of a nonvanishing adjoint p such that the following system is satisfied on [0, T ]: ⎧ ⎨ x (t) = cx(t), t p(t) , |p(t)| ⎩ −p (t) = Dc x(t), t p(t).

(5.10)

In particular, from the regularity assumptions on c and since p(t) p(t) d p(t) = −Dc x(t), t + Dc x(t), t , , dt |p(t)| |p(t)| |p(t)| the map t → have

p(t) |p(t)|

is 2C-Lipschitz continuous. So, for any fixed t ∈ [0, T ] and any s ∈ [0, t], we x (s) − x (t) M(t − s) + ωR (t − s),

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where M = 3Cc and R := R0 + cT is given by Lemma 5.3. By integration on [t, t], we deduce that, for any t ∈ [0, t], x(t) − x(t) − x (t)(t − t) M (t − t)2 + ωR (t − t)(t − t). 2

(5.11)

Let x ∈ ∂K(t), and let x(·) be an extremal trajectory with x(t) = x. We are going to show that for any t ∈ [0, t], the ball B(t) of radius r(t) centered at x(t) − x (t)(t − t) is contained in K(t) for some r(t) to determine, i.e. that we have for any ξ ∈ B(0, r(t)), v x(t) − x (t)(t − t) + ξ t. We therefore estimate, using the Lipschitz continuity of v and (5.11), t − v x(t) − x (t)(t − t) + ξ 1 t − v x(t) − x (t)(t − t) − |ξ | c 1 M 1 2 t − v x(t) − (t − t) + ωR (t − t)(t − t) − r(t) c 2 c 1 M =t −t − (t − t)2 + ωR (t − t)(t − t) + r(t) . c 2 c

Thus if we set r(t) = 2 (t − t), the above quantity is nonnegative as soon as t −t

c 2M

c and ωR (t − t) . 4

For this choice, it follows B(t) = B x(t) − x (t)(t − t), r(t) x (t) c (t x (t) (t − t) + x = x(t) − (t) − t)ξ, ξ ∈ B(0, 1) |x (t)| 2|x (t)| ⊂ K(t). Since x(t) = x and c |x (t)| c, this proves the interior cone property at x as soon as t μ = c −1 c min( 2M , ωR ( 4 )), of parameters ρ1 =

c θ1 , 2c

2 c −1 , cωR with θ1 = min (c/4) . 2M

3. Interior cone property for small time t ∈ [0, μ]. With the previous notation, let x ∈ ∂K(t) and x(·) be an extremal trajectory of (5.9) with x(t) = x and p(·) be an associated adjoint. Let us recall that the regularity of K0 given by (H6) implies that it has the interior ball property, i.e.


1285

there exists r > 0 independent of y ∈ ∂K0 such that B y − ν(y)r, r ⊂ K0 , where ν(y) is the unit outer normal to K0 at y ∈ ∂K0 . In particular, since x(·) is extremal, ν(x(0)) = p(0)/|p(0)| and so p(0) r, r ⊂ K0 . B x(0) − |p(0)|

(5.12)

We will prove that, for t μ, K(t) has the interior cone property of parameters ρ = r/2 and r/2,r p(t) . We write y as θ = r. Let y ∈ Cν,x with ν = − |p(t)| y=x−

1 p(t) λ + λξ, |p(t)| 2

(5.13)

where 0 λ r and |ξ | 1. Let y(·) be the solution of ⎧ ⎨ y (t) = cy(t), t p(t) |p(t)| ⎩ y(t) = y,

for t ∈ [0, t],

where p(·) is the adjoint associated with x(·) by (5.10). It is enough to prove that y(0) ∈ K0 , since then y = y(t) ∈ K(t). Because of (5.12), we only have to show that y(0) − x(0) − p(0) λ λ. |p(0)| Moreover, we remark that (5.13) implies that y(t) − x(t) − p(t) λ = 1 λξ λ . |p(t)| 2 2 Let us therefore set p(t) 2 f (t) = y(t) − x(t) + λ , |p(t)| so that f (t)

λ2 4 .

It only remains to prove that f (0) λ2 . But

p(t) f (t) = 2 y(t) − x(t), y (t) − x (t) + 2λ y (t) − x (t), |p(t)| d p(t) + 2λ y(t) − x(t), dt |p(t)| p(t) = 2 y(t) − x(t), c y(t), t − c x(t), t |p(t)|

1286


p(t) p(t) , + 2λ c y(t), t − c x(t), t |p(t)| |p(t)| p(t)p(t), p (t) p (t) − + 2λ y(t) − x(t), |p(t)| |p(t)|3 p (t) 2 −2C y(t) − x(t) − 2λC y(t) − x(t) − 2λy(t) − x(t) |p(t)| p(t)p(t), p (t) . − 2λ y(t) − x(t) |p(t)|3 Thanks to (5.10), we know that p (t) |p(t)| C

p(t)p(t), p (t) C, and |p(t)|3

so that 2 f (t) −2C y(t) − x(t) − 6λC y(t) − x(t). But if we set g(t) = |y(t) − x(t)|2 , then 2 g (t) = 2 y(t) − x(t), y (t) − x (t) −2C y(t) − x(t) = −2Cg(t), which implies that for all t ∈ [0, t] g(t)e2Ct g(t)e2Ct , that is to say thanks to (5.13) y(t) − x(t) |y − x|eC(t−t) 3λ eCt . 2 We therefore obtain f (t) −2C

3λ Ct e 2

2 − 6λC

3λ Ct 9 e = − Ce2Ct + 9CeCt λ2 . 2 2

If we set k = 92 Ce2Ct + 9CeCt , we finally have f (0) f (t) + kλ2 t

λ2 + kλ2 t λ2 4

as soon as kt 34 . Thus if we set b to be the unique solution of 92 be2b + 9beb = get that f (0) λ2 as soon as t b/C. If we assume that c b −1 c μ = min , ωR , C 2M 4

3 4

(b > 0), we


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which is always possible by reducing c, we see that K(t) has the interior cone property of parameters ρ2 = r/2 and θ2 = r for all 0 t μ (note that the parameters ρ2 , θ2 depend only on K0 ). 4. End of the proof. We remark that 1 ρ2 ρ1 c = , = θ1 2c 2 θ2 whence we finally obtain that for any t 0, K(t) has the interior cone property of parameters c ρ = 2c θ with θ = min{θ1 , θ2 }. 2 References [1] O. Alvarez, P. Cardaliaguet, R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces Free Bound. 7 (2005) 415–434. [2] O. Alvarez, P. Hoch, Y. Le Bouar, R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal. 181 (3) (2006) 449–504. [3] G. Barles, Solutions de viscosité des équations de Hamilton–Jacobi, Springer-Verlag, Paris, 1994. [4] G. Barles, P. Cardaliaguet, O. Ley, R. Monneau, Global existence results and uniqueness for dislocation equations, SIAM J. Math. Anal. 40 (2008) 44–69. [5] G. Barles, P. Cardaliaguet, O. Ley, A. Monteillet, Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations, Nonlinear Anal. (2009), doi:10.1016/j.na.2009.01.156, in press. [6] G. Barles, O. Ley, Nonlocal first-order Hamilton–Jacobi equations modelling dislocations dynamics, Comm. Partial Differential Equations 31 (8) (2006) 1191–1208. [7] G. Barles, H.M. Soner, P.E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (2) (1993) 439–469. [8] M. Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with L1-time dependence and Neumann boundary conditions, Discrete Contin. Dyn. Syst. 21 (3) (2008) 763–800. [9] M. Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with L1-time dependence and Neumann boundary conditions. Existence and applications to the level-set approach, Discrete Contin. Dyn. Syst. 21 (4) (2008) 1047–1069. [10] F.H. Clarke, The maximum principle under minimal hypotheses, SIAM J. Control Optim. 14 (6) (1976) 1078–1091. [11] M.G. Crandall, P.-L. Lions, Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 277 (1) (1983) 1–42. [12] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992. [13] Y. Giga, S. Goto, H. Ishii, Global existence of weak solutions for interface equations coupled with diffusion equations, SIAM J. Math. Anal. 23 (4) (1992) 821–835. [14] H. Ishii, Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Eng. Chuo Univ. 28 (1985) 33–77. [15] O. Ley, Lower-bound gradient estimates for first-order Hamilton–Jacobi equations and applications to the regularity of propagating fronts, Adv. Differential Equations 6 (5) (2001) 547–576. [16] P.-L. Lions, B. Perthame, Remarks on Hamilton–Jacobi equations with measurable time-dependent Hamiltonians, Nonlinear Anal. 11 (5) (1987) 613–621. [17] A. Monteillet, Integral formulations of the geometric eikonal equation, Interfaces Free Bound. 9 (2) (2007) 253–283. [18] D. Nunziante, Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence, Differential Integral Equations 3 (1) (1990) 77–91. [19] D. Nunziante, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Anal. 18 (11) (1992) 1033–1062. [20] D. Rodney, Y. Le Bouar, A. Finel, Phase field methods and dislocations, Acta Materialia 51 (2003) 17–30. [21] P. Soravia, P.E. Souganidis, Phase-field theory for FitzHugh–Nagumo-type systems, SIAM J. Math. Anal. 27 (5) (1996) 1341–1359.


Rieffel deformation via crossed products ✩ P. Kasprzak Department of Mathematical Methods in Physics, Faculty of Physics, Warsaw University, Warsaw, Poland Received 23 May 2008; accepted 6 May 2009 Available online 2 June 2009 Communicated by Alain Connes

Abstract We start from Rieffel data (A, Ψ, ρ), where A is a C∗ -algebra, ρ is an action of an abelian group Γ on A and Ψ is a 2-cocycle on the dual group. Using Landstad theory of crossed product we get a deformed C∗ -algebra AΨ . In the case of Γ = Rn we obtain a very simple proof of invariance of K-groups under the deformation. In the general case we also get a very simple proof that nuclearity is preserved under the deformation. We show how our approach leads to quantum groups and investigate their duality. The general theory is illustrated by an example of the deformation of SL(2, C). A description of it, in terms of ˆ γˆ , δ, ˆ is given. noncommutative coordinates α, ˆ β, © 2009 Elsevier Inc. All rights reserved. Keywords: C∗ -algebras; Quantum groups

Contents 1. 2. 3.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Landstad theory of crossed products . . . . . . . . . . . . . . . . . . . . . . . . . . . Rieffel deformation of C∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Deformation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Functorial properties of the Rieffel deformation . . . . . . . . . . . . . . 3.3. Preservation of nuclearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. K-theory in the case of Γ = Rn . . . . . . . . . . . . . . . . . . . . . . . . . Rieffel deformation of locally compact groups . . . . . . . . . . . . . . . . . . . . 4.1. From an abelian subgroup with a dual 2-cocycle to a quantum group

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The research was supported by the KBN under grant 115/E-343/SPB/6.PR UE/DIE 50/2005–2008. E-mail address: [email protected].


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P. Kasprzak / Journal of Functional Analysis 257 (2009) 1288–1332

4.2. Dual quantum group . . . . . . . . . 4.3. Haar measure . . . . . . . . . . . . . . 5. An example of quantization of SL(2, C) . 5.1. C∗ -algebra structure . . . . . . . . . 5.2. Commutation relations . . . . . . . 5.3. Comultiplication . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction In [12] Rieffel described the method of deforming of C∗ -algebras known today as the Rieffel deformation. Having an action of Rd on a C∗ -algebra A and a skew symmetric operator J : Rd → Rd Rieffel defined a new product that gave rise to the deformed C∗ -algebra AJ . In [13] the Rieffel deformation was applied to the C∗ -algebra of continuous functions vanishing at infinity on a Lie group G. An action of Rn was constructed using the left and right shifts along a fixed abelian Lie subgroup Γ . Having the deformed C∗ -algebra Rieffel introduced a comultiplication, a coinverse and a counit, showing that it is a locally compact quantum group. M. Enock and L. Vainerman in [3] gave a method of deforming of the dual object associated ˆ where C∗r (G) is the reduced group C∗ with the locally compact group G that is (C∗r (G), ) ˆ algebra and is the canonical comultiplication on it. Using an abelian subgroup Γ ⊂ G and ˆ a 2-cocycle Ψ on the Pontryagin dual group Γˆ they twisted the canonical comultiplication ∗ ∗ on the reduced group C -algebra Cr (G) obtaining a new quantum group. They also presented a formula for a multiplicative unitary and described a Haar measure for this new quantum group. The existence of these two methods of deforming of objects related to a group G prompts the question about the relations between them. In this paper it is shown that they are dual versions of the same mathematical procedure. Let us note that the deformation framework of Enock and Vainerman is in a sense more general than the one of Rieffel: instead of a skew symmetric matrix on Rn they use a 2-cocycle Ψ on the abelian subgroup Γ . This suggests that it should be possible to perform the Rieffel deformation of a C∗ -algebra A acted on by an abelian group Γ with a 2cocycle Ψ on Γˆ . A formulation of Rieffel deformation in that context is one of the results of this paper. Let us briefly describe the contents of the whole paper. In the next section we revise the Landstad theory of crossed products. We prove a couple of useful results that we could not find in the literature. In Section 3 we use the Landstad’s theory to give a new approach to the Rieffel deformation of C∗ -algebras. In Section 4 we apply the Rieffel deformation to locally compact groups. We show that Enock–Vainerman’s and Rieffel’s approach give mutually dual, locally compact quantum groups. Moreover, a formula for a Haar measure on a quantized algebra of functions is given. In the last section we use our scheme to deform SL(2, C). The subgroup Γ consists of diagonal matrices. We show that the deformed C∗ -algebra A is generated in the sense of ˆ γˆ , δˆ and give a detailed description of the commuWoronowicz by four affiliated elements α, ˆ β, tation relations they satisfy. Moreover, we show that the comultiplication Ψ ∈ Mor(A; A ⊗ A) acts on the generators in the standard way: Ψ (α) ˆ = αˆ ⊗ αˆ βˆ ⊗ γˆ ,

ˆ = αˆ ⊗ βˆ βˆ ⊗ δ, ˆ Ψ (β)

Ψ (γˆ ) = γˆ ⊗ αˆ δˆ ⊗ γˆ ,

ˆ = δˆ ⊗ δˆ γˆ ⊗ β. ˆ Ψ (δ)

(1)

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Throughout the paper we will freely use the language of C∗ -algebras and the theory of locally compact quantum groups. For the notion of multipliers, affiliated elements, algebras generated by a family of affiliated elements and morphism of C∗ -algebras we refer the reader to [18] and [15]. For the theory of locally compact quantum groups we refer to [6] and [9]. For the theory of quantum groups given by a multiplicative unitary we refer to [19]. Some remarks about the notation. For a subset X of a Banach space B, X cls denotes the closed linear span of X. Let A be a C∗ -algebra and A∗ be its Banach dual. A∗ is an A-bimodule where for ω ∈ A∗ and b, b ∈ A we define b · ω · b by the formula: b · ω · b (a) = ω(b ab) for any a ∈ A. 2. Landstad theory of crossed products Let us start this section with a definition of Γ -product. For a detailed treatment of this notion see [10]. Definition 2.1. Let Γ be a locally compact abelian group, Γˆ its Pontryagin dual, B a C∗ -algebra, λ a homomorphism of Γ into the unitary group of M(B) continuous in the strict topology of M(B) and let ρˆ be a continuous action of Γˆ on B. The triple (B, λ, ρ) ˆ is called a Γ -product if: ρˆγˆ (λγ ) = γˆ , γ λγ for any γˆ ∈ Γˆ and γ ∈ Γ . The unitary representation λ : Γ → M(B) gives rise to a morphism of C∗ -algebras λ ∈ Mor(C∗ (Γ ); B). Identifying C∗ (Γ ) with C∞ (Γˆ ) via the Fourier transform, we get a morphism λ ∈ Mor(C∞ (Γˆ ); B). Let τγˆ ∈ Aut(C∞ (Γˆ )) denote the shift automorphism: τγˆ (f )(γˆ ) = f (γˆ + γˆ )

for all f ∈ C∞ (Γˆ ).

It is easy to see that λ intertwines the action ρˆ with τ : λ τγˆ (f ) = ρˆγˆ λ(f )

(2)

for any f ∈ C∞ (Γˆ ). The following lemma seems to be known but we could not find any reference. Lemma 2.2. Let (B, λ, ρ) ˆ be a Γ -product. Then the morphism λ ∈ Mor(C∞ (Γˆ ); B) is injective. Proof. The kernel of the morphism λ is an ideal in C∞ (Γˆ ) hence it is contained in a maximal ideal. Therefore there exists γˆ0 such that f (γˆ0 ) = 0 for all f ∈ ker λ. Eq. (2) implies that ker λ is τ invariant. Hence f (γˆ0 + γˆ ) = 0 for all γˆ . This shows that f = 0 and ker λ = {0}. 2 In what follows we usually treat a C∗ -algebra C∞ (Γˆ ) as a subalgebra of M(B) and we will not use the embedding λ explicitly.


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Definition 2.3. Let (B, λ, ρ) ˆ be a Γ -product and x ∈ M(B). We say that x satisfies the Landstad conditions if: ⎧ ⎪ ρˆγˆ (x) = x for all γˆ ∈ Γˆ ; ⎨ (i) (ii) the map Γ γ → λγ xλ∗γ ∈ M(B) is norm continuous; (3) ⎪ ⎩ (iii) f xg ∈ B for all f, g ∈ C∞ (Γˆ ). In computations it is useful to smear unitary elements λγ ∈ M(B) with a function h ∈ L1 (Γ ): λh =

h(γ )λγ dγ ∈ M(B). Γ

Note that λh ∈ M(B) coincides with the Fourier transform of h: F(h) ∈ C∞ (Γˆ ). In the original form of Definition 2.3 given by Landstad the third condition had the form: λf x, xλf ∈ B

for all f, g ∈ L1 (Γ ).

(4)

Our conditions are simpler to check, which turns out to be useful in the example considered at the end of the paper. As shown below both definitions of invariants are in fact equivalent. The argument is very similar to the one given in [11] which shows that the third Landstad condition can be also replaced by λf x ∈ B

for all f ∈ L1 (Γ ).

Assume that x ∈ M(B) satisfies the Landstad conditions (3). Choose ε 0 and a function f ∈ L1 (Γ ). By the second Landstad condition we can find a finite volume neighborhood O of the neutral element e ∈ Γ such that:

λγ x − xλγ ε

for all γ ∈ O.

(5)

Let χO denote the normalized characteristic function of O ⊂ Γ : χO (γ ) =

1 vol O

0

if γ ∈ O, if γ ∈ Γ \ O.

Then by (5) we have:

λχO x − xλχO ε.

(6)

If necessary, we can choose a smaller neighborhood and assume also that:

λf λχO − λf ε. The calculation below is self-explanatory λf x = λf x − λf λχO x + λf λχO x = (λf x − λf λχO x) + (λf λχO x − λf xλχO ) + λf xλχO

(7)

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and together with (6) and (7) shows that:

λf x − λf xλχO ε x + λf . Hence we can approximate λf x by elements of the form λf xλχO ∈ B. This shows that λf x ∈ B. A similar argument proves the second inclusion: xλf ∈ B. The set of elements satisfying Landstad’s conditions is a C∗ -algebra. We shall call it the Landstad algebra and denote it by A. It follows from Definition 2.3, that if a ∈ A then λγ aλ∗γ ∈ A and the map Γ ∈ γ → λγ aλ∗γ ∈ A is norm continuous. An action of Γ on A defined in this way will be denoted by ρ. It can be shown that the embedding of A into M(B) is a morphism of C∗ -algebras (cf. [8, Section 2]). Hence the multipliers algebra M(A) can also be embedded into M(B). Let x ∈ M(B). Then x ∈ M(A) if and only if it satisfies the following two conditions: ⎧ ⎪ (i) ⎪ ⎨ (ii) ⎪ ⎪ ⎩

ρˆγˆ (x) = x for all γˆ ∈ Γˆ ; for all a ∈ A, the map Γ γ → λγ xλ∗γ a ∈ M(B) is norm continuous.

(8)

Note that the first and the second condition of (3) imply conditions (8). Examples of Γ -products can be obtained via the crossed-product construction. Let A be a C∗ algebra with an action ρ of Γ on A. There exists the standard action ρˆ of the group Γˆ on A ρ Γ and a unitary representation λγ ∈ M(A ρ Γ ) such that the triple (A ρ Γ, λγ , ρ) ˆ is a Γ -product. It turns out that all Γ -products (B, λ, ρ) ˆ are crossed-products of the Landstad algebra A by the action ρ implemented by λ. The following theorem is due to Landstad [10, Theorem 7.8.8]: Theorem 2.4. A triple (B, λ, ρ) ˆ is a Γ -product if and only if there is a C∗ -dynamical system (A, Γ, ρ) such that B = A ρ Γ . This system is unique up to isomorphism and A consist of the elements in M(B) that satisfy Landstad conditions while ργ (a) = λγ aλ∗γ . Remark 2.5. The main problem in the proof of the above theorem is to show that the Landstad algebra is not small. It is solved by integrating the action ρˆ over the dual group. More precisely, we say that an element x ∈ M(B)+ is ρ-integrable ˆ if there exists y ∈ M(B)+ such that ω(y) =

d γˆ ω ρˆγˆ (x)

for any ω ∈ M(B)∗+ . We denote y by E(x). If x ∈ M(B) is not positive then we say that it is ρ-integrable ˆ if it can be written as a linear combination of positive ρ-integrable ˆ elements. The set of ρ-integrable ˆ elements will be denoted by D(E). The averaging procedure induces a map: E : D(E) → M(B). It can be shown that for a large class of x ∈ D(E), E(x) is an element of the Landstad algebra A. This is the case for f1 bf2 where b ∈ B and f1 , f2 ∈ C∞ (Γˆ ) are square integrable. Moreover the


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map: B b → E(f1 bf2 ) ∈ M(B)

(9)

is continuous with the following estimate for norms

E(f1 bf2 ) f1 2 b

f2 2

(10)

where · 2 is the L2 -norm. Furthermore, we have

cls E(f1 bf2 ): b ∈ B, f1 , f2 ∈ C∞ (Γˆ ) ∩ L2 (Γˆ ) = A.

(11)

The last equality was not proven in [10]. We shall need it at some point so let us give a proof here. Let a ∈ A and f1 , f2 , f3 , f4 be continuous, compactly supported functions on Γ . Consider an element x = λf1 λf2 aλf3 λf4 . Clearly x = F(f1 )F(f2 )aF(f3 )F(f4 ) and F(f1 ), . . . , F(f4 ) ∈ L2 (Γˆ ), hence by (10) x ∈ D(E). We compute: E(x) =

d γˆ ρˆγˆ (λf1 λf2 aλf3 λf4 )

=

d γˆ ρˆγˆ

=

d γˆ

dγ1 dγ2 dγ3 dγ4 f1 (γ1 )f2 (γ2 )λγ1 +γ2 aλγ3 +γ4 f3 (γ3 )f4 (γ4 )

dγ1 dγ2 dγ3 dγ4 γˆ , γ1 + γ2 + γ3 + γ4 f1 (γ1 )f2 (γ2 )

× λγ1 +γ2 aλγ3 +γ4 f3 (γ3 )f4 (γ4 ) . Using properties of the Fourier transform we obtain: E(x) =

dγ1 dγ2 dγ3 ργ1 +γ2 (a)f1 (γ1 )f2 (γ2 )f3 (γ3 )f4 (−γ1 − γ2 − γ3 ).

(12)

If f1 , f2 , f3 approximate the Dirac delta function and f4 (0) = 1 then using (12) we see that elements E(λf1 λf2 aλf3 λf4 ) approximate a in norm. This proves (11). The following lemma is simple but very useful: Lemma 2.6. Let (B, λ, ρ) ˆ be a Γ -product, A its Landstad algebra, ρ an action of Γ on A implemented by λ and V ⊂ A a subset of the Landstad algebra which is invariant under the action ρ and such that (C∞ (Γˆ )VC∞ (Γˆ ))cls = B. Then V cls = A. Proof. This proof is similar to the proof of formula (11). Let f1 , f2 , f3 , f4 be continuous, compactly supported functions on Γ . Using (10) and (11) we get: cls A = E λf1 (λf2 vλf3 )λf4 : v ∈ V .

(13)

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A simple calculation shows that: E λf1 (λf2 vλf3 )λf4 = dγ1 dγ2 dγ3 ργ1 +γ2 (v)f1 (γ1 )f2 (γ2 )f3 (γ3 )f4 (−γ1 − γ2 − γ3 ). Note that the integrand Γ × Γ (γ1 , γ2 , γ3 ) → ργ1 +γ2 (v)f1 (γ1 )f2 (γ2 )f3 (γ3 )f4 (−γ1 − γ2 − γ3 ) ∈ V is a norm continuous, compactly supported function, hence E(λf1 vλf2 λf3 ) ∈ V cls . From this and (13) we get V cls = A.

2

The next proposition shows that morphisms of Γ -products induce morphisms of their Landstad algebras. The below result can be, to some extent, deduced from the results of paper [4]. Proposition 2.7. Let (B, λ, ρ) ˆ and (B , λ , ρˆ ) be Γ -products and let A, A be Landstad algebras for B and B respectively. Assume that π ∈ Mor(B, B ) satisfies: • π(λγ ) = λγ ; • π(ρˆγˆ (b)) = ρˆγˆ (π(b)). Then π(A) ⊂ M(A ) and π|A ∈ Mor(A, A ). Moreover, if π(B) ⊂ B then π(A) ⊂ A . If π(B) = B then π(A) = A . Proof. We start by showing that π(A) ⊂ M(A ). Let a ∈ A. Then ρˆγˆ π(a) = π ρˆγˆ (a) = π(a). Hence π(a) is ρˆ invariant. Moreover the map: ∗ Γ γ → λ γ π(a)λ γ = π λγ aλ∗γ ∈ M(A ) is norm continuous. This shows that π(a) satisfies the first and the second Landstad condition of (3) which guaranties that π(a) ∈ M(A ). To prove that the homomorphism π restricted to A is in fact a morphism from A to A we have to check that the set π(A)A is linearly dense in A . We know that π(B)B is linearly dense in B . Using this fact in the last equality below, we get cls ∗ cls ∗ = π C (Γ )A A C∗ (Γ ) C (Γ )π(A)A C∗ (Γ ) cls = π(B)B = B . Moreover ∗ ∗ λγ π(a)a λ γ = π λγ aλ∗γ λγ a λ γ ,


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hence the set π(A)A is ρ -invariant (remember that ρ is the action of Γ implemented by λ ). This shows that π(A)A satisfies the assumptions of Lemma 2.6 and gives the density of π(A)A in A . Assume now that π(B) ⊂ B . Let a ∈ A satisfy Landstad conditions (3). Then as was shown at the beginning of the proof, π(a) satisfies the first and the second Landstad condition. Moreover f π(a)g = π(f ag) ∈ B for all f, g ∈ C∞ (Γˆ ), hence π(a) also satisfies the third Landstad condition. Therefore π(a) ∈ A . If π(B) = B then the equality E f1 π(b)f2 = π E(f1 bf2 )

(14)

and property (11) shows that π(A) = A . To prove (14) take ω ∈ M(B )∗ . Then ω π E(f1 bf2 ) = ω ◦ π E(f1 bf2 ) = =

d γˆ ω ◦ π ρˆγˆ (f1 bf2 )

d γˆ ω ρˆγˆ f1 π(b)f2 = ω

d γˆ ρˆγˆ f1 π(b)f2

= ω E f1 π(b)f2 . Hence π(E(f1 bf2 )) = E(f1 π(b)f2 ).

2

Let Γ be an abelian locally compact group and φ : Γ → Γ a continuous homomorphism. For γˆ ∈ Γˆ we set φ T (γˆ ) = γˆ ◦ φ ∈ Γˆ . The map φ T : Γˆ → Γˆ ,

φ T (γˆ ) = γˆ ◦ φ

is a continuous group homomorphism called the dual homomorphism. We have a version of Proposition 2.7 with two different groups. Proposition 2.8. Let (B, λ, ρ) ˆ be a Γ -product, (B , λ , ρˆ ) a Γ -product, φ : Γ → Γ a surjective continuous homomorphism and φ T : Γˆ → Γˆ the dual homomorphism. Assume that π ∈ Mor(B, B ) satisfies: • π(λγ ) = λφ(γ ) ; • ρˆγˆ (π(b)) = π(ρˆφ T (γˆ ) (b)). Then π(A) ⊂ M(A ) and π|A ∈ Mor(A, A ). Moreover, if π(B) ⊂ B then π(A) ⊂ A . If π(B) = B then π(A) = A . Let π ∈ Mor(B, B ) be a morphism of C∗ -algebras satisfying the assumptions of Proposition 2.7 such that π(B) = B . We have an exact sequence of C∗ -algebras: π

0 → ker π → B → B → 0.

(15)

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The C∗ -algebra ker π has a canonical Γ -product structure. Indeed, consider a morphism α ∈ Mor(B; ker π) associated with the ideal ker π ⊂ B: α(b)j = bj where b ∈ B and j ∈ ker π . Note that α(b) = b

for any b ∈ ker π ⊂ B.

(16)

For all γ ∈ Γ we set λ˜ γ = α(λγ ) ∈ M(ker π). The map Γ γ → λ˜ γ ∈ M(ker π) is a strictly continuous representation of Γ on ker π . Moreover ker π is invariant under the action ρ. ˆ The restriction of ρˆ to ker π will also be denoted by ρ. ˆ It is easy to check that ρˆγˆ (λ˜ γ ) = γˆ , γ λ˜ γ which shows that the triple (ker π, λ˜ , ρ) ˆ is a Γ -product. Let I, A, A be Landstad algebras for the Γ -products (ker π, λ˜ , ρ), ˆ (B, λ, ρ), ˆ (B , λ , ρˆ ) respectively. Our objective is to show that the exact sequence (15) induces an exact sequence of Landstad algebras: 0 → I → A → A → 0.

(17)

Let π¯ ∈ Mor(A; A ) denote a morphism of Landstad algebras induced by π . We assumed that π is surjective, hence by Proposition 2.7 π(A) ¯ = A and we have an exact sequence of C∗ -algebras: π¯

0 → ker π¯ → A → A → 0. It is easy to check that the morphism α ∈ Mor(B; ker π) satisfies the assumptions of Proposition 2.7, hence α(A) ⊂ M(I). If we show that α restricted to ker π¯ identifies it with the Landstad algebra I, then the existence of the exact sequence (17) will be proven. There are two conditions to be checked: (i) α(ker π) ¯ = I; (ii) if x ∈ ker π¯ and α(x) = 0 then x = 0. Ad(i) Let a ∈ ker π¯ and f ∈ C∞ (Γˆ ). Then af ∈ B ∩ ker π , hence α(a)f = α(af ) = af ∈ ker π,

(18)

where we used (16). This shows that α(a) satisfies the third Landstad condition for Γ -product (ker π, λ˜ , ρ). ˆ As in Proposition 2.7 we check that α(a) also satisfies the first and the second Landstad condition, hence α(ker π¯ ) ⊂ I. Furthermore, E(f1 bf2 ) ∈ ker π¯ for all b ∈ ker π , and we have α E(f1 bf2 ) = E f1 α(b)f2 = E(f1 bf2 ). Using (11) we see that α(ker π¯ ) = I. Ad(ii) Assume that a ∈ ker π¯ and α(a) = 0. Note that af ∈ ker π for any f ∈ C∞ (Γˆ ). Using (16) we get af = α(af ) = α(a)α(f ) = 0. Hence af = 0 for any f ∈ C∞ (Γˆ ), which implies that a = 0. We can summarize the above considerations in the following:


1297

Proposition 2.9. Let (B, λ, ρ), ˆ (B , λ , ρˆ ) be Γ -products with Landstad algebras A, A respectively, π ∈ Mor(B, B ) a surjective morphism intertwining ρˆ and ρˆ such that π(λγ ) = λγ . Let ˜ ρ) (ker π, λ, ˆ be the Γ -product described after Proposition 2.8 and let I ⊂ M(ker π ) be its Landstad algebra. Then I can be embedded into A and we have a Γ -equivariant exact sequence: π¯

0 → I → A → A → 0 where π¯ = π|A . 3. Rieffel deformation of C∗ -algebras 3.1. Deformation procedure Let (B, λ, ρ) ˆ be a Γ -product. A 2-cocycle on the group Γˆ is a continuous function Ψ : Γˆ × 1 ˆ Γ → T satisfying: (i) Ψ (e, γˆ ) = Ψ (γˆ , e) = 1 for all γˆ ∈ Γˆ ; (ii) Ψ (γˆ1 , γˆ2 + γˆ3 )Ψ (γˆ2 , γˆ3 ) = Ψ (γˆ1 + γˆ2 , γˆ3 )Ψ (γˆ1 , γˆ2 ) for all γˆ1 , γˆ2 , γˆ3 ∈ Γˆ . (For the theory of 2-cocycles we refer to [5].) For γˆ , γˆ1 we set Ψγˆ (γˆ1 ) = Ψ (γˆ1 , γˆ ). It defines a family of functions Ψγˆ : Γˆ → T1 . Using the embedding λ ∈ Mor(C∞ (Γˆ ); B) we get a strictly continuous family of unitary elements Uγˆ = λ(Ψγˆ ) ∈ M(B).

(19)

The 2-cocycle condition for Ψ gives: Uγˆ1 +γˆ2 = Ψ (γˆ1 , γˆ2 )Uγˆ1 ρˆγˆ1 (Uγˆ2 ).

(20)

Theorem 3.1. Let (B, λ, ρ) ˆ be a Γ -product and let Ψ be a 2-cocycle on Γˆ . For any γˆ ∈ Γˆ the map ρˆγΨˆ : B b → ρˆγΨˆ (b) = Uγ∗ˆ ρˆγˆ (b)Uγˆ ∈ B is an automorphism of C∗ -algebra B. Moreover, ρˆ Ψ : Γˆ γˆ → ρˆγΨˆ ∈ Aut(B) is a strongly continuous action of Γˆ on B and the triple (B, λγ , ρˆ Ψ ) is a Γ -product. Proof. Using Eq. (20) we get ρˆγΨˆ1 +γˆ2 (b) = Uγ∗ˆ1 +γˆ2 ρˆγˆ1 +γˆ2 (b)Uγˆ1 +γˆ2

= Ψ (γˆ1 , γˆ2 )Uγ∗ˆ1 ρˆγ1 (Uγˆ2 )∗ ρˆγˆ1 ρˆγˆ2 (b) ρˆγ1 (Uγˆ2 )Uγˆ1 Ψ (γˆ1 , γˆ2 ) = Uγ∗ˆ1 ρˆγ1 Uγ∗ˆ2 ρˆγˆ2 (b)Uγˆ2 Uγˆ1 = ρˆγΨˆ1 ρˆγΨˆ2 (b) .

1298


This shows that ρˆ Ψ is an action of Γˆ on B. Applying ρˆ Ψ to λγ we get: ρˆγΨˆ (λγ ) = Uγ∗ˆ ρˆγˆ (λγ )Uγˆ = γˆ , γ Uγ∗ˆ λγ Uγˆ = γˆ , γ λγ . The last equality follows from commutativity of Γ . Hence the triple (B, λγ , ρˆ Ψ ) is a Γ product. 2 The above theorem leads to the following procedure of deformation of C∗ -algebras. The data needed to perform the deformation is a triple (A, ρ, Ψ ) consisting of a C∗ -algebra A, an action ρ of a locally compact abelian group Γ and a 2-cocycle Ψ on Γˆ . Such a triple is called deformation data. The resulting C∗ -algebra will be denoted AΨ . The procedure is carried out in three steps: ˆ be the standard Γ -product struc(1) Construct the crossed product B = A ρ Γ . Let (B, λ, ρ) ture of the crossed product. (2) Introduce a Γ -product (B, λ, ρˆ Ψ ) as described in Theorem 3.1. (3) Let AΨ be the Landstad algebra of the Γ -product (B, λ, ρˆ Ψ ). Note that AΨ still carries an action ρ Ψ of Γ given by ργΨ (x) = λγ xλ∗γ . In this case it is not the formula defining the action itself, but its domain of definition that changes under deformation. The triple (AΨ , Γ, ρ Ψ ) will be called a twisted dynamical system. The procedure of deformation described above is called the Rieffel deformation. Using Theorem 2.4 we immediately get Proposition 3.2. Let (A, ρ, Ψ ) be deformation data and (AΨ , Γ, ρ Ψ ) be the twisted dynamical system considered above. Then A ρ Γ = AΨ ρ Ψ Γ. In what follows we investigate the dependence of the Rieffel deformation on the choice of a 2-cocycle. Let f : Γˆ → T1 be a continuous function such that f (e) = 1. For all γˆ1 , γˆ2 ∈ Γˆ we set ∂f (γˆ1 , γˆ2 ) =

f (γˆ1 + γˆ2 ) . f (γˆ1 )f (γˆ2 )

One can check that the map ∂f : Γˆ × Γˆ (γˆ1 , γˆ2 ) →

f (γˆ1 + γˆ2 ) ∈T f (γˆ1 )f (γˆ2 )

is a 2-cocycle. 2-cocycles of this form are considered to be trivial. We say that a pair of 2-cocycles Ψ1 , Ψ2 is in the same cohomology class if they differ by a trivial 2 cocycle: Ψ2 = Ψ1 ∂f .


1299

Theorem 3.3. Let (A, ρ, Ψ ) be deformation data, giving rise to a Landstad algebra AΨ . Then the isomorphism class of the Landstad algebra AΨ depends only on the cohomology class of Ψ . Theorem 3.3 easily follows from the next two lemmas. Lemma 3.4. Let (A, ρ, Ψ ) be deformation data with a trivial 2-cocycle Ψ = ∂f . Then A and AΨ are isomorphic. More precisely, treating f as an element of C∗ -algebra M(A ρ Γ ) we have AΨ = {f af ∗ : a ∈ A}. Proof. Fixing the second variable in Ψ we get a family Ψγˆ of the form Ψγˆ = f (γˆ )f ∗ τγˆ (f ),

(21)

where τγˆ (f )(γˆ ) = f (γˆ + γˆ ). Let Uγˆ ∈ M(A ρ Γ ) be the unitary element given by Ψγˆ (cf. (19)). The function f can be embedded into M(A ρ Γ ) and using (21) we get: Uγˆ = f (γˆ )f ∗ ρˆγˆ (f ).

(22)

Assume that a ∈ A. Then ρˆγΨˆ f af ∗ = Uγ∗ˆ ρˆγˆ f af ∗ Uγˆ = Uγ∗ˆ ρˆγˆ (f )a ρˆγˆ (f )∗ Uγˆ . Using Eq. (22) we see that ρˆγΨˆ f af ∗ = f (γˆ )f ρˆγˆ (f )∗ ρˆγˆ (f )a ρˆγˆ (f )∗ f (γˆ )f ∗ ρˆγˆ (f ) = f af ∗ which means that the element f af ∗ satisfies the first Landstad condition for the Γ -product (A ρ Γ, λ, ρˆ Ψ ). It is easy to check that it also satisfies the second and third Landstad condition, hence f Af ∗ ⊂ AΨ . An analogous reasoning proves the opposite inclusion f Af ∗ ⊃ AΨ . 2 Let Ψ1 , Ψ2 be a pair of 2-cocycles on Γˆ . Their product Ψ1 Ψ2 is also a 2-cocycle. Let (A, Γ, ρ) be a dynamical system. The deformation data (A, ρ, Ψ1 ) gives rise to the twisted dynamical system (AΨ1 , Γ, ρ Ψ1 ). Furthermore, the triple (AΨ1 , ρ Ψ1 , Ψ2 ) is deformation data which gives rise to the C∗ -algebra (AΨ1 )Ψ2 . At the same time, using the deformation data (A, ρ, Ψ1 Ψ2 ) we can introduce the C∗ -algebra AΨ1 Ψ2 . Lemma 3.5. Let (A, Γ, ρ) be a Γ -product and let Ψ1 , Ψ2 be 2-cocycles on the group Γˆ . Let (AΨ1 )Ψ2 be a C∗ -algebra constructed from the deformation data (AΨ1 , ρ Ψ1 , Ψ2 ) and let AΨ1 Ψ2 be a C∗ -algebra constructed from the deformation data (A, ρ, Ψ1 Ψ2 ). Then Ψ AΨ1 Ψ2 AΨ1 2 .

1300


Proof. The algebras AΨ1 Ψ2 and (AΨ1 )Ψ2 can be embedded into M(A ρ Γ ): they are Landstad algebras of the Γ -products (A ρ Γ, λ, ρˆ Ψ1 Ψ2 ) and (A ρ Γ, λ, (ρˆ Ψ1 )Ψ2 ) respectively. Note that UγΨˆ 1 Ψ2 = UγΨˆ 1 UγΨˆ 2 , hence ∗

ρˆγΨˆ 1 Ψ2 (b) = U Ψ1 Ψ2 γˆ ρˆγˆ (b)UγΨˆ 1 Ψ2

Ψ ∗ ∗ = U Ψ2 γˆ U Ψ1 γˆ ρˆγˆ (b)UγΨˆ 1 UγΨˆ 2 = ρˆ Ψ1 γˆ 2 (b).

This shows that ρˆ Ψ1 Ψ2 = (ρˆ Ψ1 )Ψ2 and implies that the (A ρ Γ, λ, ρˆ Ψ1 Ψ2 ) and (A ρ Γ, λ, (ρˆ Ψ1 )Ψ2 ) are in fact the same Γ -products. Therefore their Landstad algebras coincide. 2 3.2. Functorial properties of the Rieffel deformation Let (B, λ, ρ) ˆ be a Γ -product, Ψ a 2-cocycle on the dual group Γˆ and H a Hilbert space. Using Theorem 3.1 we introduce the twisted Γ -product (B, λ, ρˆ Ψ ). Let A, AΨ ⊂ M(B) be Landstad algebras of (B, λ, ρ) ˆ and (B, λ, ρˆ Ψ ) respectively and π ∈ Rep(B; H ) a representation of the ∗ C -algebra B. The representation of B extends to multipliers M(B) and can be restricted to A and AΨ . Theorem 3.6. Let (B, λ, ρ), ˆ (B, λ, ρˆ Ψ ) be Γ -products considered above, A, AΨ their Landstad algebras and π a representation of C∗ -algebra B on a Hilbert space H . Then π is faithful on A if and only if it is faithful on AΨ . Proof. Assume that π is faithful on A and let a ∈ AΨ be such that π(a) = 0. Invariance of a with respect to the action ρˆ Ψ implies that ρˆγˆ (a) = Uγˆ aUγ∗ˆ . Hence ρˆγˆ (f )Uγˆ aUγ∗ˆ ρˆγˆ gf ∗ λ−γ = ρˆγˆ f agf ∗ λ−γ

(23)

for all f, g ∈ C∞ (Γˆ ). The element a ∈ AΨ belongs to ker π therefore π ρˆγˆ (f )Uγˆ aUγ∗ˆ ρˆγˆ gf ∗ λ−γ = 0. Combining it with Eq. (23) we obtain π(ρˆγˆ (f agf ∗ λ−γ )) = 0. Assume now that f, g ∈ L2 (Γˆ ) ∩ C∞ (Γˆ ). Let E denote the averaging map with respect to undeformed action ρ. ˆ Then f agf ∗ λ−γ ∈ D(E) and E(f agf ∗ λ−γ ) = 0. Indeed, let ω ∈ B(H )∗ . Then = ω ◦ π E f agf ∗ λ−γ ω π E f agf ∗ λ−γ = 0. = d γˆ ω π ρˆ f agf ∗ λ−γ Hence ω(π(E(f agf ∗ λ−γ ))) = 0 for any ω ∈ B(H )∗ and π(E(f agf ∗ λ−γ )) = 0. But E(f agf ∗ λ−γ ) ∈ A and π is faithful on A hence E f agf ∗ λ−γ = 0

for all f, g ∈ L2 (Γˆ ) ∩ C∞ (Γˆ ).

(24)


1301

We will show that the above equation may be satisfied only if a = 0. Let f ∈ L1 (Γ ) be an approximation of the Dirac delta function as used in Theorem 7.8.7 of [10]. This theorem says that for any y of the form y = f agf ∗ we have the following norm convergence: lim E(yλ−γ )λγ λf dγ = y. ε→0

Using (24) we get E(yλ−γ ) = E(f agf ∗ λ−γ ) = 0 hence f agf ∗ = 0 for all f, g ∈ L2 (Γˆ ) ∩ C∞ (Γˆ ). This immediately implies that a = 0 and shows that π is faithful on AΨ . A similar argument shows that faithfulness of π on AΨ implies its faithfulness on A. 2 Definition 3.7. Let (A, ρ, Ψ ), (A , ρ , Ψ ) be deformation data with groups Γ and Γ respectively. Let φ : Γ → Γ be a surjective continuous homomorphism, φ T : Γˆ → Γˆ the dual homomorphism and π ∈ Mor(A, A ). We say that (π, φ) is a morphism of deformation data (A, ρ, Ψ ) and (A , ρ , Ψ ) if: • Ψ ◦ (φ T × φ T ) = Ψ ; • ρ φ(γ ) π(a) = π(ργ (a)). Using universal properties of crossed products, we see that a morphism (π, φ) of the deformation data induces the morphism π φ ∈ Mor(A Γ ; A Γ ) of crossed products. One can check ˆ and the that π φ satisfies the assumptions of Proposition 2.8 with the Γ -product (A ρ Γ, λ, ρ) Γ -product (A ρ Γ , λ , ρˆ ). This property is not spoiled by the deformation procedure. Applying Proposition 2.8 and Theorem 3.6 to the morphism π φ ∈ Mor(A ρ Γ ; A ρ Γ ), Γ -product (A ρ Γ, λ, ρˆ Ψ ) and Γ -product (A ρ Γ , λ , ρˆ Ψ ) we get Proposition 3.8. Let (π, φ) be a morphism of deformation data (A, ρ, Ψ ) and (A , ρ , Ψ ) and let π φ ∈ Mor(A ρ Γ ; A ρ Γ ) be the induced morphism of the crossed products considered above. Then π φ (AΨ ) ⊂ M(A Ψ ) and π φ |AΨ ∈ Mor(AΨ ; A Ψ ). Morphism π ∈ Mor(A; A ) is injective if and only if so is π φ |AΨ ∈ Mor(AΨ ; A Ψ ) and π(A) = A if and only if π φ (AΨ ) = A Ψ . Let (I, Γ, ρI ), (A, Γ, ρ), (A , Γ, ρ ) be dynamical systems and let π

0 → I → A → A → 0

(25)

be an exact sequence of C∗ -algebras which is Γ -equivariant. Morphism π induces a surjective morphism π ∈ Mor(A ρ Γ ; A ρ Γ ). It sends A to A by means of π and it is identity on C∗ (Γ ). Its kernel can be identified with I ρI Γ so we have an exact sequence of crossed product C∗ -algebras: π

0 → I ρI Γ → A ρ Γ → A ρ Γ → 0.

(26)

Note that π ∈ Mor(A ρ Γ ; A ρ Γ ) satisfies the assumptions of Proposition 2.9 with the ˆ and (A ρ Γ, λ , ρˆ ). This property is not spoiled by the deforΓ -products (A ρ Γ, λ, ρ) mation procedure. Hence applying Proposition 2.9 to the Γ -products (A ρ Γ, λ, ρˆ Ψ ) and (A ρ Γ, λ , ρˆ Ψ ) we obtain

1302


Theorem 3.9. Let (I, Γ, ρI ), (A, Γ, ρ), (A , Γ, ρ ) be dynamical systems. Let π

0 → I → A → A → 0 be an exact sequence of C∗ -algebras which is Γ -equivariant, Ψ a 2-cocycle on the dual group Γˆ and I Ψ , AΨ , A Ψ the Landstad algebras constructed from the deformation data (I, ρI , Ψ ), (A, ρ, Ψ ), (A , ρ , Ψ ). Then we have the Γ -equivariant exact sequence: πΨ

0 → I Ψ → AΨ −→ A Ψ → 0 where the morphism π Ψ ∈ Mor(AΨ ; A Ψ ) is the restriction of the morphism π ∈ Mor(A ρ Γ, A ρ Γ ) to the Landstad algebra AΨ ⊂ M(A ρ Γ ). 3.3. Preservation of nuclearity Theorem 3.10. Let (A, ρ, Ψ ) be the deformation data which gives rise to the Landstad algebra AΨ . C∗ -algebra A is nuclear if and only if AΨ is. The proof follows from the equality A ρ Γ = AΨ ρ Ψ Γ (Proposition 3.2) and the following: Theorem 3.11. Let A be a C∗ -algebra with an action ρ of an abelian group Γ . Then A is nuclear if and only if A ρ Γ is nuclear. The above theorem can be deduced from Theorem 3.3 and Theorem 3.16 of [14]. 3.4. K-theory in the case of Γ = Rn In this section we will prove the invariance of K-groups under the Rieffel deformation in the case of Γ = Rn . The tool we use is the analogue of the Thom isomorphism due to Connes [2]: Theorem 3.12. Let A be a C∗ -algebra, and ρ an action of Rn on A. Then Ki (A) Ki+n A ρ Rn . Theorem 3.13. Let (A, Rn , ρ) be a dynamical system and let (A, ρ, Ψ ) be the deformation data giving rise to the Landstad algebra AΨ . Then Ki (A) Ki AΨ . Proof. Proposition 3.2 asserts that A ρ Rn AΨ ρ Ψ Rn . Hence using Theorem 3.12 we get Ki (A) Ki+n A ρ Rn Ki+n AΨ ρ Ψ Rn Ki AΨ .

2


1303

4. Rieffel deformation of locally compact groups 4.1. From an abelian subgroup with a dual 2-cocycle to a quantum group In this section we shall apply our deformation procedure to the algebra of functions on a locally compact group G. First we shall fix a notation and introduce auxiliary objects. Let G g → Rg ∈ B(L2 (G)) be the right regular representation of G on Hilbert space L2 (G) of the right invariant Haar measure. Let C∞ (G) ⊂ B(L2 (G)) be the C∗ -algebra of continuous functions on G vanishing at infinity, C∗r (G) ⊂ B(L2 (G)) the reduced group C∗ -algebra generated by Rg and V ∈ B(L2 (G × G)) the Kac–Takesaki operator: Vf (g, g ) = f (gg , g ) for any f ∈ L2 (G × G). By G ∈ Mor(C∞ (G); C∞ (G) ⊗ C∞ (G)) we will denote the comultiplication on C∞ (G). It is known that the Kac–Takesaki operator V is an element of M(C∗r (G) ⊗ C∞ (G)) which implements comultiplication: G (f ) = V (f ⊗ 1)V ∗ for any f ∈ C∞ (G). Let Γ ⊂ G be an abelian subgroup of G, Γˆ its dual group and Γˆ ∈ Mor(C∞ (Γˆ ); C∞ (Γˆ )⊗C∞ (Γˆ )) the comultiplication on C∞ (Γˆ ). Let π R ∈ Mor(C∗ (Γ ); C∗r (G)) be a morphism induced by the following representation of the group Γ : Γ γ → Rγ ∈ M C∗r (G) . Identifying C∗ (Γ ) with C∞ (Γˆ ) we get π R ∈ Mor(C∞ (Γˆ ); C∗r (G)). Let us fix a 2-cocycle Ψ on the group Γˆ . Our objective is to show that an action of Γ 2 on the ∗ C -algebra C∞ (G) given by the left and right shifts and a 2-cocycle on Γˆ 2 determined by Ψ , give rise to a quantum group. We shall describe this construction step by step. Let ρ R be the action of Γ on C∞ (G) given by right shifts: ργR (f )(g) = f (gγ ) for any f ∈ C∞ (G). Let B R be the crossed product C∗ -algebra C∞ (G) ρ R Γ and (B R , λ, ρ) ˆ the standard ˆ Γ -product structure on it. The standard embeddings of C∞ (G) and C∞ (Γ ) into M(B R ) enable us to treat (π R ⊗ id)Ψ and V ∗ (1 ⊗ f )V (where f ∈ C∞ (Γˆ )) as elements of M(C∗r (G) ⊗ B R ). One can show that V ∗ (1 ⊗ λγ )V = Rγ ⊗ λγ for all γ ∈ Γ , which implies that V ∗ (1 ⊗ f )V = π R ⊗ id Γˆ (f )

(27)

for any f ∈ C∞ (Γˆ ). Using Ψ we deform the standard Γ -product structure on B R to (B R , λ, ρˆ Ψ ). Proposition 4.1. Let (B R , λ, ρˆ Ψ ) be the deformed Γ -product and V (π R ⊗ id)Ψ ∈ M(C∗r (G) ⊗ B R ) the unitary element considered above. Then V (π R ⊗ id)Ψ is invariant with respect to the action id ⊗ ρˆ Ψ . Proof. The 2-cocycle equation for Ψ implies that: id ⊗ ρˆγΨˆ Ψ = (id ⊗ ρˆγˆ )Ψ = (I ⊗ Uγˆ )∗ Γˆ (Uγˆ )Ψ.

1304


The second leg of V is invariant with respect to the action ρˆ hence id ⊗ ρˆγΨˆ V = I ⊗ Uγ∗ˆ (id ⊗ ρˆγˆ )V (I ⊗ Uγˆ ) = I ⊗ Uγ∗ˆ V (I ⊗ Uγˆ ) = V π R ⊗ id Γˆ (Uγˆ )∗ (I ⊗ Uγˆ ). The last equality follows from (27). Finally id ⊗ ρˆγΨˆ V π R ⊗ id Ψ = V π R ⊗ id Γˆ (Uγˆ )∗ (I ⊗ Uγˆ )(I ⊗ Uγˆ )∗ π R ⊗ id Γˆ (Uγˆ ) π R ⊗ id Ψ = V π R ⊗ id Ψ where in the last equality we used the fact that Uγˆ is unitary.

2

Let ρ L be the action of Γ on C∞ (G) given by left shifts: ργL (f )(g) = f (γ −1 g) for any f ∈ C∞ (G). Let B L be the crossed product C∗ -algebra C∞ (G) ρ L Γ and let (B L , λ, ρ) ˆ be the standard Γ -product structure on it. For any γˆ1 , γˆ2 ∈ Γˆ we set Ψ (γˆ1 , γˆ2 ) = Ψ (γˆ1 , −γˆ1 − γˆ2 ). This defines a function Ψ ∈ Cb (Γˆ 2 ). The standard embeddings of C∞ (G) and C∞ (Γˆ ) into M(B L ) enable us to treat (π R ⊗ id)Ψ V and V (1 ⊗ f )V ∗ (where f ∈ C∞ (Γˆ )) as elements of M(C∗r (G) ⊗ B R ). One can show that V (1 ⊗ λγ )V ∗ = Rγ ⊗ λγ for all γ ∈ Γ , which implies that V (1 ⊗ f )V ∗ = π R ⊗ id Γˆ (f )

(28)

denote a 2-cocycle defined by the formula: for any f ∈ C∞ (Γˆ ). Let Ψ (γˆ1 , γˆ2 ) ≡ Ψ (−γˆ1 , −γˆ2 ) Ψ

we deform the standard Γ -product structure on B L to (B L , λ, ρˆ Ψ ). for any γˆ1 , γˆ2 ∈ Γˆ . Using Ψ

Proposition 4.2. Let (B L , λ, ρˆ Ψ ) be the deformed Γ -product and (π R ⊗ id)Ψ V ∈ M(C∗r (G) ⊗ B L ) the unitary element considered above. Then (π R ⊗ id)Ψ V ∈ M(C∗r (G) ⊗ B L ) is invariant with respect to the action id ⊗ ρˆ Ψ . Proof. One can check that (γˆ1 + γˆ2 , γˆ )Ψ (γˆ1 , γˆ2 ). (γˆ2 , γˆ )Ψ Ψ (γˆ1 , γˆ2 + γˆ ) = Ψ (γˆ1 , −γˆ1 − γˆ2 − γˆ ) = Ψ Hence γˆ ) ˆ (U γˆ )∗ Ψ . id ⊗ ρˆγΨˆ Ψ = (I ⊗ U Γ


1305

Moreover γˆ )∗ V (I ⊗ U γˆ ) = (I ⊗ U γˆ )∗ π R ⊗ id ˆ (U γˆ )V . id ⊗ ρˆγΨˆ V = (I ⊗ U Γ Following the proof of Proposition 4.1 we get our assertion.

2

Let ρ denote the action of Γ 2 on C∞ (G) given by the left and right shifts, B the crossed product C∗ -algebra C∞ (G) ρ Γ 2 and (B, λ, ρ) ˆ the standard Γ 2 -product. The standard embedding of C∞ (G) into M(B) applied to the second leg of V ∈ M(C∗r (G) ⊗ C∞ (G)) embeds V into M(C∗r (G) ⊗ B). We have two embeddings λL and λR of C∞ (Γˆ ) into M(B) corresponding to the left and the right action of Γ . Moreover by Eqs. (27) and (28) we have: V 1 ⊗ λL (f ) V ∗ = π R ⊗ λL Γˆ (f ), V ∗ 1 ⊗ λR (f ) V = π R ⊗ λR Γˆ (f )

(29)

for any f ∈ C∞ (Γˆ ). Note also that: (id ⊗ λγ1 ,γ2 )V id ⊗ λ∗γ1 ,γ2 = (R−γ1 ⊗ I )V (Rγ2 ⊗ I ).

(30)

Let us introduce elements Ψ L and Ψ R : Ψ L = π R ⊗ λL Ψ , Ψ R = π R ⊗ λR (Ψ ) ∈ M C∗r (G) ⊗ B .

(31)

Multiplying Ψ L , V and Ψ R we get the unitary element: V Ψ = Ψ L V Ψ R ∈ M C∗r (G) ⊗ B .

(32)

⊗ Ψ on Γˆ 2 we deform the standard Γ 2 -product structure on B to Using the 2-cocycle Ψ ⊗Ψ Ψ ). (B, λ, ρˆ

Proposition 4.3. Let (B, λ, ρˆ Ψ ⊗Ψ ) be the deformed Γ 2 -product structure and V Ψ ∈ M(C∗r (G) ⊗ B) the unitary element given by (32). Then V Ψ is invariant with respect to the action id ⊗ ρˆ Ψ ⊗Ψ . Moreover, for any γ1 , γ2 ∈ Γ we have (id ⊗ λγ1 ,γ2 )V Ψ id ⊗ λ∗γ1 ,γ2 = (R−γ1 ⊗ I )V Ψ (Rγ2 ⊗ I ).

(33)

Proof. Invariance of V Ψ with respect to the action id ⊗ ρˆ Ψ ⊗Ψ follows easily from Propositions 4.1 and 4.2. The group Γ is abelian, hence (id ⊗ λγ1 ,γ2 )V Ψ id ⊗ λ∗γ1 ,γ2 = (id ⊗ λγ1 ,γ2 )Ψ L V Ψ R id ⊗ λ∗γ1 ,γ2 = Ψ L (id ⊗ λγ1 ,γ2 )V id ⊗ λ∗γ1 ,γ2 Ψ R . Using (30) we get

1306


(id ⊗ λγ1 ,γ2 )V Ψ id ⊗ λ∗γ1 ,γ2 = Ψ L (R−γ1 ⊗ I )V (Rγ2 ⊗ I )Ψ R = (R−γ1 ⊗ I )Ψ L V Ψ R (Rγ2 ⊗ I ) = (R−γ1 ⊗ I )V Ψ (Rγ2 ⊗ I ). This proves (33).

2

The first leg of V Ψ belongs to C∗r (G) so it acts on L2 (G). It is well known that slices of Kac– Takesaki operator V by normal functionals ω ∈ B(L2 (G))∗ give a dense subspace of C∞ (G) (see [1, Section 2]). We will show that slices of V Ψ give a dense subspace of C∞ (G)Ψ ⊗Ψ .

Theorem 4.4. Let (B, λ, ρˆ Ψ ⊗Ψ ) be the deformed Γ 2 -product structure and V Ψ ∈ M(C∗r (G)⊗B) the unitary operator given by (32). Then V = (ω ⊗ id)V Ψ : ω ∈ B L2 (G) ∗

is a norm dense subset of C∞ (G)Ψ ⊗Ψ . Proof. We need to check that for any ω ∈ B(L2 (G))∗ the element (ω ⊗ id)V Ψ ∈ M(B) satisfies Landstad conditions for Γ 2 -product (B, λ, ρˆ Ψ ⊗Ψ ). The first Landstad condition is equivalent to the invariance of the second leg of V Ψ with respect to the action ρˆ Ψ ⊗Ψ (Proposition 4.3). Using (33) we get λγ1 ,γ2 (ω ⊗ id)V Ψ λ∗γ1 ,γ2 = (Rγ2 · ω · R−γ1 ⊗ id)V Ψ

(34)

for any γ1 , γ2 ∈ Γ . The norm continuity of the map Γ 2 (γ1 , γ2 ) → Rγ2 · ω · R−γ1 ∈ B L2 (G) ∗ implies that (ω ⊗ id)V Ψ satisfies the second Landstad condition. To check the third Landstad condition we need to show that f1 (ω ⊗ id)V Ψ f2 ∈ B

(35)

for any f1 , f2 ∈ C∞ (Γˆ × Γˆ ). Let us consider the set cls W = f1 (ω ⊗ id)V Ψ f2 : f1 , f2 ∈ C∞ (Γˆ × Γˆ ), ω ∈ B L2 (G) ∗ .

(36)

We will prove that W = B which is a stronger property than (35). Taking for ω ∈ B(L2 (G))∗ elements of the form π R (h3 ) · μ · π R (h4 ), for f1 ∈ C∞ (Γˆ × Γˆ ) elements λR (h1 )λL (h2 ) where h1 , h2 ∈ C∞ (Γˆ ) and similarly for f2 we do not change the closed linear span. Thus we have: W = λR (h1 )λL (h2 ) π R (h3 ) · μ · π R (h4 ) ⊗ id V Ψ λR (h5 )λL (h6 ): cls h1 , h2 , . . . , h6 ∈ C∞ (Γˆ ), μ ∈ B L2 (G) ∗ .


1307

Note that λR (h1 )λL (h2 ) π R (h3 ) · μ · π R (h4 ) ⊗ id V Ψ λR (h5 )λL (h6 ) = λR (h1 ) (μ ⊗ id) π R ⊗ λL Ψ (h4 ⊗ h2 ) V π R ⊗ λR Ψ (h3 ⊗ h5 ) λL (h6 ), hence W coincides with the following set: R λ (h1 ) (μ ⊗ id) π R ⊗ λL Ψ (h4 ⊗ h2 ) V π R ⊗ λR Ψ (h3 ⊗ h5 ) λL (h6 ): cls h1 , h2 , . . . , h6 ∈ C∞ (Γˆ ), μ ∈ B L2 (G) ∗ . Using the fact that Ψ and Ψ are unitary we get W = λR (h1 ) (μ ⊗ id) π R ⊗ λL (h4 ⊗ h2 )V π R ⊗ λR (h3 ⊗ h5 ) λL (h6 ): cls h1 , h2 , . . . , h6 ∈ C∞ (Γˆ ), μ ∈ B L2 (G) ∗ = λR (h1 )λL (h2 ) π R (h3 ) · μ · π R (h4 ) ⊗ id (V ) λR (h5 )λL (h6 ): cls h1 , h2 , . . . , h6 ∈ C∞ (Γˆ ), μ ∈ B L2 (G) ∗ . Now again R λ (h1 )λL (h2 ) π R (h3 ) · μ · π R (h4 ) ⊗ id (V ) λR (h5 )λL (h6 ): cls h1 , h2 , . . . , h6 ∈ C∞ (Γˆ ), μ ∈ B L2 (G) ∗ cls = f1 (ω ⊗ id)V f2 : f1 , f2 ∈ C∞ (Γˆ × Γˆ ), ω ∈ B L2 (G) ∗ hence we get cls W = f1 (ω ⊗ id)V f2 : f1 , f2 ∈ C∞ (Γˆ × Γˆ ), ω ∈ B L2 (G) ∗ . The set {(ω ⊗ id)V : ω ∈ B(L2 (G))∗ } is dense in C∞ (G) which shows that W = B and proves formula (36). We see that the elements of the set V satisfy the Landstad conditions. To prove that V is dense in C∞ (G)Ψ ⊗Ψ we use Lemma 2.6. According to (34), V is a ρ Ψ ⊗Ψ -invariant subspace of C∞ (G)Ψ ⊗Ψ . Moreover we have that (C∗ (Γ 2 )VC∗ (Γ 2 ))cls = W = B. Hence the assumptions of Lemma 2.6 are satisfied and we get the required density. 2 Remark 4.5. The representation of C∞ (G) on L2 (G) is covariant. The action of Γ 2 is implemented by the left and right shifts: Lγ1 , Rγ2 ∈ B(L2 (G)), where by Lg ∈ B(L2 (G)) we understand the unitarized left shift. More precisely, let δ : G → R+ be the modular function for the right Haar measure. Then Lg ∈ B(L2 (G)) is a unitary given by: 1 (Lg f )(g ) = δ(g) 2 f g −1 g

1308


for any g, g ∈ G and f ∈ L2 (G). This covariant representation of C∞ (G) induces the representation of crossed product B = C∞ (G) ρ Γ 2 , which we denote by π can . Clearly it is faithful on C∞ (G), hence by Theorem 3.6 it is faithful on C∞ (G)Ψ ⊗Ψ . Let us introduce the unitary operator: W = id ⊗ π can V Ψ ∈ B L2 (G) ⊗ L2 (G) .

(37)

Theorem 4.6. The unitary operator W ∈ B(L2 (G) ⊗ L2 (G)) considered above satisfies the pentagonal equation: ∗ W12 W23 W12 = W13 W23 .

Remark 4.7. A similar construction of the operator W and the proof that it satisfies the pentagonal equation was given by Enock and Vainerman in [3] and independently by Landstad in [7]. We included the following proof for the completeness of the exposition. Proof. Let us introduce two unitary operators X, Y ∈ B(L2 (G) ⊗ L2 (G)): X = id ⊗ π can Ψ R ,

Y = id ⊗ π can Ψ L

(38)

where Ψ R , Ψ L ∈ M(C∗r (G) ⊗ B) are elements defined by (31). Note that X ∈ M C∗r (G) ⊗ C∗r (G) ,

Y ∈ M C∗r (G) ⊗ C∗l (G) ,

(39)

hence W = Y V X ∈ M(C∗r (G) ⊗ K) where K is the algebra of compact operators acting on L2 (G). Inserting γˆ3 → (−γˆ1 − γˆ2 − γˆ3 ) into the 2-cocycle condition Ψ (γˆ1 , γˆ2 + γˆ3 )Ψ (γˆ2 , γˆ3 ) = Ψ (γˆ1 + γˆ2 , γˆ3 )Ψ (γˆ1 , γˆ2 )

(40)

and taking the complex conjugate we get Ψ (γˆ1 , −γˆ1 − γˆ3 )Ψ (γˆ2 , −γˆ1 − γˆ3 − γˆ2 ) = Ψ (γˆ1 , γˆ2 )Ψ (γˆ1 + γˆ2 , −γˆ1 − γˆ2 − γˆ3 ). This implies that Ψ (γˆ1 , γˆ2 )Ψ (γˆ1 + γˆ2 , γˆ3 ) = Ψ (γˆ1 , γˆ3 )Ψ (γˆ2 , γˆ1 + γˆ3 )

(41)

where Ψ (γˆ1 , γˆ2 ) = Ψ (γˆ1 , −γˆ1 − γˆ2 ). Using Eqs. (38), (40), (41) and the fact that V implements the coproduct we obtain: ∗ ∗ ∗ X12 V12 Y23 V12 = Y13 V13 Y23 V13 , ∗ ∗ V12 X23 V12 X12 = V23 X13 V23 X23 .

Now we can check the pentagonal equation:


1309

∗ ∗ ∗ ∗ W12 W23 W12 = X12 V12 Y12 W23 Y12 V12 X12 ∗ ∗ = X12 V12 Y23 V23 X23 V12 X12 ∗ ∗ ∗ ∗ = X12 V12 Y23 V12 V12 V23 V12 V12 X23 V12 X12 ∗ ∗ (V13 V23 ) V23 X13 V23 X23 = Y13 V13 Y23 V13

= Y13 V13 Y23 X13 V23 X23 = (Y13 V13 X13 )(Y23 V23 X23 ) = W13 W23 . In the second equality we used the fact that the second leg of element Y commutes with the first leg of W (see (39)). 2 Our next aim is to show that W is manageable. For all γˆ ∈ Γˆ we set u(γˆ ) = Ψ (−γˆ , γˆ ). It defines a function u ∈ Cb (Γˆ ). Applying π R ∈ Mor(C∞ (Γˆ ); C∗r (G)) to u ∈ M(C∞ (Γˆ )) we get the unitary operator: J = π R (u) ∈ M C∗r (G) ⊂ B L2 (G) .

(42)

Theorem 4.8. Let W ∈ B(L2 (G) ⊗ L2 (G)) be the multiplicative unitary and J ∈ B(L2 (G)) be entering Definition 1.2 the unitary operator (42). Then W is manageable. Operators Q and W ∗ ∗ of [19] equal respectively 1 and (J ⊗ 1)W (J ⊗ 1). Remark 4.9. The presented proof seems to be simpler than the Landstad’s proof given in [7]. In what follows we shall use the bracket notation for the scalar product: let H be a Hilbert space, x, y ∈ H , and T ∈ B(H ). Then (x|T |y) denotes the scalar product (x|T y). Proof. Let x, y, z, t ∈ L2 (G), γ1 , γ2 , γ3 , γ4 ∈ Γ . The Kac–Takesaki operator is manageable, therefore x ⊗ t|(Rγ1 ⊗ Lγ2 )V |(Rγ3 ⊗ Rγ4 )|z ⊗ y = (R−γ1 x ⊗ L−γ2 t|V |Rγ3 z ⊗ Rγ4 y) = Rγ3 z ⊗ L−γ2 t|V ∗ |R−γ1 x ⊗ Rγ4 y = z ⊗ t|(R−γ3 ⊗ Lγ2 )V ∗ (R−γ1 ⊗ Rγ4 )|x ⊗ y . Using well-known equalities V ∗ (I ⊗ Rg )V = Rg ⊗ Rg , V (I ⊗ Lg )V ∗ = Rg ⊗ Lg and commutativity of Γ we get the following formula: z ⊗ t|(R−γ3 ⊗ Lγ2 )V ∗ (R−γ1 ⊗ Rγ4 )|x ⊗ y = z ⊗ t|(R−γ3 +γ4 ⊗ Rγ4 )V ∗ (R−γ1 +γ2 ⊗ Lγ2 )|x ⊗ y . Hence:

(43)

1310


x ⊗ t|(Rγ1 ⊗ Lγ2 )V (Rγ3 ⊗ Rγ4 )|z ⊗ y = z ⊗ t|(R−γ3 +γ4 ⊗ Rγ4 )V ∗ (R−γ1 +γ2 ⊗ Lγ2 )|x ⊗ y .

(44)

Using continuity arguments, this equality will be extended. We will repeatedly use the identifications C∗ (Γ 2 ) = C∞ (Γˆ 2 ) = C∞ (Γˆ ) ⊗ C∞ (Γˆ ), etc. Let uγ be a unitary generator of C∗ (Γ ). Let us define the following morphisms: Φ1R ∈ Mor C∗ (Γ ) ⊗ C∗ (Γ ); C∗r (G) ⊗ C∗r (G) : Φ1R (uγ1 ⊗ uγ2 ) = Rγ1 ⊗ Rγ2 , Φ1L ∈ Mor C∗ (Γ ) ⊗ C∗ (Γ ); C∗r (G) ⊗ C∗l (G) : Φ1L (uγ1 ⊗ uγ2 ) = Rγ1 ⊗ Lγ2 and automorphism Θ ∈ Aut(C∞ (Γˆ 2 )) given by the formula: Θ(f )(γˆ1 , γˆ2 ) = f (−γˆ1 , γˆ1 + γˆ2 ) for any f ∈ C∞ (Γˆ 2 ). One can check that Θ(uγ1 ⊗ uγ2 ) = u−γ1 +γ2 ⊗ uγ2 . Using the above morphisms we reformulate (44): x ⊗ t|Φ1L (uγ1 ⊗ uγ2 )V Φ1R (uγ3 ⊗ uγ4 )|z ⊗ y = z ⊗ t|Φ1R ◦ Θ(uγ3 ⊗ uγ4 )V ∗ ΦlL ◦ Θ(uγ1 ⊗ uγ2 )|x ⊗ y . By linearity and continuity we get x ⊗ t|Φ1L (f )V Φ1R (g)|z ⊗ y = z ⊗ t|Φ1R ◦ Θ(g)V ∗ Φ1L ◦ Θ(f )|x ⊗ y for any f, g ∈ M(C∞ (Γˆ ) ⊗ C∞ (Γˆ )). In particular x ⊗ t|Φ1L Ψ V Φ1R (Ψ )|z ⊗ y = z ⊗ t|Φ1R ◦ Θ(Ψ )V ∗ Φ1L ◦ Θ Ψ |x ⊗ y . It is easy to see that X = Φ1R (Ψ ), Y = Φ1L (Ψ ) and Θ(Ψ ) = Ψ (u ⊗ I ) where X and Y are given by (38). Therefore Φ1R ◦ Θ(Ψ ) = Φ1R Ψ (u ⊗ I ) = (J ⊗ I )X ∗ .

(45)

Φ1L ◦ Θ Ψ = Y ∗ J ∗ ⊗ I

(46)

Similarly we prove that

and finally we get


1311

x ⊗ t|Φ1L Ψ V Φ1R (Ψ )|z ⊗ y = z ⊗ t|(J ⊗ I )X ∗ V ∗ Y ∗ J ∗ ⊗ I |x ⊗ y . This shows that = (J ⊗ I )Y V X J ∗ ⊗ I ∗ = (J ⊗ I )W ∗ J ∗ ⊗ I W and Q = 1.

2

Proposition 4.10. Let W ∈ B(L2 (G) ⊗ L2 (G)) and J ∈ B(L2 (G)) be the unitaries defined in (37) and (42) respectively. Let x, y be vectors in L2 (G) and ωx,y ∈ B(L2 (G))∗ a functional given by ωx,y (T ) = (x|T |y) for any T ∈ B(L2 (G)). Then we have ∗ (ωx,y ⊗ id)W = (ωJ ∗ x,J ¯ ∗ y¯ ⊗ id)(W ).

(47)

Proof. Using manageability of W we get: ∗ ∗ ) = (ωy, (ωx,y ⊗ id)W = (ωy, ¯ x¯ ⊗ id)(W ¯ x¯ ⊗ id) (J ⊗ 1)W J ⊗ 1 ∗ ∗ = (ωJ ∗ x,J 2 = (ωJ ∗ y,J ¯ ∗ x¯ ⊗ id) W ¯ ∗ y¯ ⊗ id)(W ) . Let A be a C∗ -algebra obtained by slicing the first leg of a manageable multiplicative unitary W ∈ B(L2 (G) ⊗ L2 (G)): A = (ω ⊗ id)W : ω ∈ B L2 (G) ∗ · . Theorem 1.5 of [19] shows that A carry the structure of a quantum group. The comultiplication on A is given by the formula: A a → W (a ⊗ I )W ∗ ∈ M(A ⊗ A). At the same time, using the morphism π can ∈ Rep(B; L2 (G)) introduced in Remark 4.5 we can faithfully represent C∞ (G)Ψ ⊗Ψ on L2 (G). By Theorem 4.4 π can (C∞ (G)Ψ ⊗Ψ ) = A, hence we ⊗Ψ Ψ . Our next objective is to can transport the structure of a quantum group from A to C∞ (G) ⊗Ψ Ψ which does not use multiplicative present a useful formula for comultiplication on C∞ (G) unitary W . The construction is done in two steps. • Let ρ be the action of Γ 2 on C∞ (G) given by left and right shifts along the subgroup Γ ⊂ G. The comultiplication is covariant: G ργ1 ,γ2 (f ) = (ργ1 ,0 ⊗ ρ0,γ2 ) G (f ) for any f ∈ C∞ (G). Therefore, it induces a morphism of crossed products: ∈ Mor C∞ (G) Γ 2 ; C∞ (G) Γ 2 ⊗ C∞ (G) Γ 2 . restricted to C∞ (G) ⊂ M(C∞ (G) Γ 2 ) coincides with G and restricted to C∞ (Γˆ 2 ) ⊂ M(C∞ (G) Γ 2 ) is given by

1312


(h) = λL ⊗ λR h ∈ M C∞ (G) Γ 2 ⊗ C∞ (G) Γ 2 where λL , λR ∈ Mor(C∞ (Γˆ ), C∞ (G) Γ 2 ) are morphisms introduced after the proof of Proposition 4.2 and h ∈ C∞ (Γˆ 2 ). • Let Ψ be a 2-cocycle on Γˆ . Recall that Ψ ∈ M(C∞ (Γˆ 2 )) is defined by Ψ (γˆ1 , γˆ2 ) = Ψ (γˆ1 , −γˆ1 − γˆ2 ). Let us introduce the unitary element Υ ∈ M(C∞ (G) Γ 2 ⊗ C∞ (G) Γ 2 ): Υ = λR ⊗ λL Ψ and a morphism Ψ ∈ Mor(C∞ (G) Γ 2 ; C∞ (G) Γ 2 ⊗ C∞ (G) Γ 2 ) given by the formula Ψ (a) = Υ (a)Υ ∗

(48)

for any a ∈ C∞ (G) Γ 2 . Theorem 4.11. Let Ψ ∈ Mor(C∞ (G) Γ 2 ; C∞ (G) Γ 2 ⊗ C∞ (G) Γ 2 ) be the morphism defined by formula (48). For all a ∈ C∞ (G)Ψ ⊗Ψ we have Ψ (a) ∈ M C∞ (G)Ψ ⊗Ψ ⊗ C∞ (G)Ψ ⊗Ψ and Ψ |C∞ (G)Ψ⊗Ψ ∈ Mor C∞ (G)Ψ ⊗Ψ ; C∞ (G)Ψ ⊗Ψ ⊗ C∞ (G)Ψ ⊗Ψ . Moreover Ψ |C∞ (G)Ψ⊗Ψ coincides with the comultiplication implemented by W : C∞ (G)Ψ ⊗Ψ a → W (a ⊗ 1)W ∗ ∈ M C∞ (G)Ψ ⊗Ψ ⊗ C∞ (G)Ψ ⊗Ψ . Proof. By Theorem 1.5 of [19] it is enough to show that Ψ Ψ id ⊗ Ψ V Ψ = V12 V13 . From the definition of it follows that (id ⊗ )V Ψ = (id ⊗ ) π R ⊗ λL Ψ V π R ⊗ λR (Ψ ) = π R ⊗ λL Ψ V 12 V π R ⊗ λR Ψ 13 . Hence

id ⊗ Ψ V Ψ = (1 ⊗ Υ ) π R ⊗ λL Ψ V 12 V π R ⊗ λR Ψ 13 1 ⊗ Υ ∗ .


1313

By Eq. (29) we get (1 ⊗ Υ )V12 = V12 π R ⊗ λR ⊗ λL ◦ ( Γˆ ⊗ id) Ψ and V13 (1 ⊗ Υ ) =

R π ⊗ λR ⊗ λL ◦ (σ ⊗ id) ◦ (id ⊗ Γˆ ) Ψ V13

where σ is the flip operator. Therefore id ⊗ Ψ V Ψ = π R ⊗ λL Ψ V 12 π R ⊗ λR ⊗ λL ◦ ( Γˆ ⊗ id) Ψ ∗ R × π R ⊗ λR ⊗ λL ◦ (σ ⊗ id) ◦ (id ⊗ Γˆ ) Ψ V π ⊗ λR Ψ 13 . We compute Ψ (γˆ1 + γˆ2 , −γˆ1 − γˆ2 − γˆ3 )Ψ (γˆ2 , −γˆ1 − γˆ2 − γˆ3 ) = Ψ (γˆ1 , γˆ2 )Ψ (γˆ1 , −γˆ1 − γˆ3 )Ψ (γˆ2 , −γˆ1 − γˆ2 − γˆ3 )Ψ (γˆ2 , −γˆ1 − γˆ2 − γˆ3 ) = Ψ (γˆ1 , γˆ2 )Ψ (γˆ1 , −γˆ1 − γˆ3 ). The above equality implies that ∗ R π ⊗ λR ⊗ λL ◦ ( Γˆ ⊗ id) Ψ π R ⊗ λR ⊗ λL ◦ (σ ⊗ id) ◦ (id ⊗ Γˆ ) Ψ = π R ⊗ λR Ψ 12 π R ⊗ λL Ψ 13 . Hence id ⊗ Ψ V Ψ = π R ⊗ λL Ψ V π R ⊗ λR Ψ 12 π R ⊗ λL Ψ V π R ⊗ λR Ψ 13 Ψ Ψ = V12 V13 .

This ends the proof.

2

4.2. Dual quantum group Let G be a locally compact group, Γ an abelian subgroup of G and Ψ a 2-cocycle on Γˆ . Using the results of previous sections we can construct the quantum group (C∞ (G)Ψ ⊗Ψ , Ψ ) 2 2 and the multiplicative unitary W ∈ B(L (G) ⊗ L (G)). In this section we will investigate the dual quantum group in the sense of duality given by W . Our objective is to show that this is the twist, in the sense of M. Enock and L. Vainerman (see [3]), of the canonical quantum group structure on the reduced group C∗ -algebra C∗r (G). Theorem 4.12. Let W ∈ B(L2 (G) ⊗ L2 (G)) be a manageable multiplicative unitary (37) and ˆ ˆ ˆ ) a quantum group obtained by slicing the second leg of W : (A, A Aˆ = (id ⊗ ω) W ∗ : ω ∈ B L2 (G) · .

1314


Then 1. Aˆ = C∗r (G). 2. The comultiplication on Aˆ is given by ˆ ˆ ˆ ˆ (a) = ΣX ∗ Σ (a)ΣXΣ ∈ M(Aˆ ⊗ A) Aˆ a → A ˆ is the canonical comultiplication on C∗r (G) and X is given by (38). where 3. The coinverse on Aˆ is given by ∗ ˆ κˆ Aˆ (a) = J κ(a)J

where κˆ is the canonical coinverse on C∗r (G) and J is given by (42). The proof was communicated to the author by S.L. Woronowicz. Proof. Using Eq. (33) we get Rγ1 (id ⊗ ω)W Rγ2 = (id ⊗ R−γ2 Lγ1 · ω · L−γ1 Rγ2 )W

(49)

for any γ1 , γ2 ∈ Γ . Therefore Rγ ∈ B(L2 (G)) is a multiplier of Aˆ and representation: ˆ Γ γ → Rγ ∈ M(A) is strictly continuous. This representation induces a morphism which we denote by χ ∈ ˆ Applying it to Ψ and Ψ we obtain Mor(C∞ (Γˆ ), A). ˆ X = (χ ⊗ χ)(Ψ ) ∈ M(Aˆ ⊗ A), Y = χ ⊗ π L Ψ ∈ M(Aˆ ⊗ K). Recall that W ∈ M(Aˆ ⊗ A), hence V = Y ∗ W X ∗ ∈ M(Aˆ ⊗ K)

(50)

∗ V , V V ∗ ∈ M(A ˆ ⊗ K ⊗ C∞ (G)). The pentagonal equation which immediately implies that V12 23 12 23 for V together with (50) gives

∗ ∗ V23 V12 V23 ∈ M Aˆ ⊗ K ⊗ C∞ (G) , V13 = V12 therefore V ∈ M Aˆ ⊗ C∞ (G) .

(51)

W ∈ M C∗r (G) ⊗ A .

(52)

Similarly we prove that


1315

Formula (51) and point 6 of Theorem 1.6 of [19] imply that the natural representation of ˆ Similarly, (52) implies that the natC∗r (G) on L2 (G) is in fact an element of Mor(C∗r (G), A). 2 ˆ C∗r (G)). The general properties of ˆ ural representation of A on L (G) is an element of Mor(A, morphisms gives ˆ C∗r (G)Aˆ = A, ˆ ∗r (G) = C∗r (G). AC ˆ ∗r (G) = C∗r (G), which But C∗r (G) and Aˆ are closed under the star operation, hence Aˆ = Aˆ ∗ = AC proves point 1 of our theorem. To prove point 2 we recall that the comultiplication on Aˆ is implemented by ΣW ∗ Σ , hence ˆ ˆ (a) = ΣX ∗ V ∗ Y ∗ (I ⊗ a)Y V XΣ A = ΣX ∗ V ∗ (I ⊗ a)V XΣ ˆ = ΣX ∗ Σ (a)(ΣXΣ). Point three follows from Proposition 4.10.

2

4.3. Haar measure Let G be a locally compact group, Γ an abelian subgroup of G and Ψ a 2-cocycle on Γˆ . Throughout this section we shall assume that the modular function δ on G restricted to Γ is identically equal to 1. Let (C∞ (G)Ψ ⊗Ψ , Ψ ) be the quantum group that we considered previ ously. In what follows we will identify C∞ (G)Ψ ⊗Ψ with its image in B(L2 (G)). Definition 4.13. Let f ∈ C∞ (G) and Rg ∈ B(L2 (G)) be the right regular representation of group G. We say that f is quantizable if there exists ω ∈ B(L2 (G))∗ such that f (g) = ω(Rg ) for any g ∈ G. Given a quantizable function f we introduce an operator Q(f ) ∈ C∞ (G)Ψ ⊗Ψ ⊂ B(L2 (G)) given by:

Q(f ) = (ω ⊗ id)W ∈ C∞ (G)Ψ ⊗Ψ . Note that the equation f (g) = ω(Rg ) does not determine ω ∈ B(L2 (G))∗ . Nevertheless, the operator Q(f ) does not depend on the choice of the functional that gives rise to f . It is easy to see that the vector space of quantizable functions equipped with the pointwise multiplication forms an algebra which in the literature is called the Fourier Algebra. We use the term quantizable function to stress that with such an f we can associate the operator Q(f ) = (ω ⊗ id)W ∈ B(L2 (G)).

Theorem 4.14. Let (C∞ (G)Ψ ⊗Ψ , Ψ ) be the quantum group with multiplicative unitary W ∈ B(L2 (G) ⊗ L2 (G)) considered above. Let f, h ∈ C∞ (G) be quantizable functions given by functionals ω ∈ B(L2 (G))∗ and μ ∈ B(L2 (G))∗ respectively. They yield operators Q(f ), Q(h) ∈ B(L2 (G)). Assume that h ∈ L2 (G). Then Q(f )h ∈ L2 (G) is a quantizable function and Q Q(f )h = Q(f )Q(h).

(53)

1316


Proof. Note that Q(f )Q(h) = (ω ⊗ id)(W )(μ ⊗ id)W = (ω ⊗ μ ⊗ id)(W13 W23 ) ∗ W23 W12 . = (ω ⊗ μ ⊗ id) W12 The above calculation shows that Q(f )Q(h) is given by the quantization of the function k ∈ C∞ (G): k(g) = (ω ⊗ μ) W ∗ (I ⊗ Rg )W . Using the identity W ∗ (I ⊗ Rg )W = X ∗ (Rg ⊗ Rg )X we get k(g) = (ω ⊗ μ) X ∗ (Rg ⊗ Rg )X . Therefore, to prove formula (53) we need to show that (ω ⊗ μ) X ∗ (Rg ⊗ Rg )X = Q(f )h (g).

(54)

In order to do that we compute (ω ⊗ id) (Rγ1 ⊗ Lγ2 )V (Rγ3 ⊗ Rγ4 ) h(g) = Lγ2 f (γ1 · γ3 )Rγ4 h(g) 1 = δ 2 (γ2 )f (γ1 − γ2 )gγ3 h (−γ2 )gγ4 . Using the assumption that δ(γ ) = 1 for any γ ∈ Γ we get (ω ⊗ id) (Rγ1 ⊗ Lγ2 )V (Rγ3 ⊗ Rγ4 ) h(g) = f (γ1 − γ2 )gγ3 h (−γ2 )gγ4 . The equality h(g) = μ(Rg ) implies that (ω ⊗ id) (Rγ1 ⊗ Lγ2 )V (Rγ3 ⊗ Rγ4 ) h (g) = (ω ⊗ μ) (Rγ1 −γ2 ⊗ R−γ2 )(Rg ⊗ Rg )(Rγ3 ⊗ Rγ4 ) . Let ϑ ∈ Aut(C∞ (Γˆ × Γˆ )) be the automorphism given by ϑ(f )(γˆ1 , γˆ2 ) = f (γˆ1 , −γˆ1 − γˆ2 )

for all f ∈ C∞ (Γˆ × Γˆ ).

By continuity, (55) extends to (ω ⊗ id) π R ⊗ π L (f1 )V π R ⊗ π R (f2 ) h(g) = (ω ⊗ μ) π R ⊗ π R ϑ(f1 )(Rg ⊗ Rg ) π R ⊗ π R (f2 ) for any f1 , f2 ∈ Cb (Γˆ ⊗ Γˆ ). Taking f1 = Ψ and f2 = Ψ we obtain ϑ(Ψ ) = Ψ and

(55)


1317

(ω ⊗ id)(Y V X) h(g) = (ω ⊗ μ) X ∗ (Rg ⊗ Rg )X where X and Y were introduced in (38). Recall that W = Y V X, hence Q(f )h(g) = (ω ⊗ μ) X ∗ (Rg ⊗ Rg )X = k(g). This proves formula (54) and ends the proof of our theorem.

2

Let f ∈ C∞ (G) be a quantizable function i.e. f (g) = ω(Rg ) for some ω ∈ B(L2 (G))∗ . Suppose that Q(f ) = 0. This means that (ω ⊗ id)W = 0 which together with Theorem 4.12 shows that ω(Rg ) = 0 for all g ∈ G. Hence f (g) = 0 for any g ∈ G, which shows that the quantization map Q is injective and its inverse is well defined. We shall show that the closure of this inverse is the GNS map for a Haar measure of (C∞ (G)Ψ ⊗Ψ , Ψ ). Let us introduce N0 ⊂ C∞ (G)Ψ ⊗Ψ : N0 = Q(f ): f -quantizable and f ∈ L2 (G) . For all Q(f ) ∈ N0 we set η0 (Q(f )) = f . This defines a map η0 : N0 → L2 (G). Proposition 4.15. Let η0 be the map defined above. Then this is a densely defined, closable map from C∞ (G)Ψ ⊗Ψ to L2 (G). Proof. Let Ψ Σ be a 2-cocycle obtained from Ψ by a flip of variables: Ψ Σ (γˆ1 , γˆ2 ) = Ψ (γˆ2 , γˆ1 ). Let QΣ be the quantization map related to Ψ Σ . Using the equality (ω ⊗ μ) X ∗ (Rg ⊗ Rg )X = (μ ⊗ ω) ΣX ∗ Σ(Rg ⊗ Rg )ΣXΣ and Theorem 4.14 we see that for quantizable, square integrable functions h, h ∈ L2 (G) we have Q(h)h = QΣ (h )h.

(56)

Let us assume that limn→∞ Q(fn ) = 0 and limn→∞ η0 (Q(fn )) = f . Using Eq. (56) we see that: QΣ (h)f = lim QΣ (h)fn = lim Q(fn )h = 0 n→∞

n→∞

(57)

for all quantizable functions h ∈ L2 (G). To conclude that f is 0 we have to show that the set of operators

QΣ (h): h is quantizable and h ∈ L2 (G) ⊂ B L2 (G)

separates elements of L2 (G). In order to do that we introduce a multiplicative unitary W Σ related to the 2-cocycle Ψ Σ . The C∗ -algebra obtained by the slices of the first leg of W Σ will be denoted Σ Σ by AΨ . By point 1 of Theorem 1.5 of [19], AΨ separates elements of L2 (G), hence it is enough to note that:

1318


cls Σ AΨ = (ω ⊗ id)W Σ : ω ∈ B L2 (G) cls = (ωx,y ⊗ id)W Σ : x, y are of compact support cls ⊂ QΣ (h): h is quantizable and h ∈ L2 (G) . The last inclusion follows from the fact that, when x and y are of compact support, then the function f defined by f (g) = ωx,y (Rg ) is also of compact support. 2 The closure of the map η0 will be denoted by η and its domain will be denoted by N. Proposition 4.16. Let η : N → L2 (G) be the map introduced above. Then N is a left ideal in C∞ (G)Ψ ⊗Ψ and η(ab) = aη(b) for all a ∈ C∞ (G)Ψ ⊗Ψ and b ∈ N.

Proof. Let b ∈ N and a ∈ C∞ (G)Ψ ⊗Ψ . Let us fix a sequence of quantizable functions fn such that a = limn→∞ Q(fn ). Map η is the closure of η0 , therefore there exists a sequence hm ∈ C∞ (G) of quantizable functions such that: lim Q(hm ) = b

m→∞

and

lim η Q(hm ) = η(b).

m→∞

Using Theorem 4.14 we get Q(fn )η(Q(hm )) = η(Q(fn )Q(hm )) and aη Q(hm ) = lim Q(fn )η Q(hm ) = lim η Q(fn )Q(hm ) . n→∞

n→∞

The closedness of the map η implies that aQ(hm ) ∈ N

and η aQ(hm ) = aη Q(hm ) .

Taking limits with respect to m and using the closedness of η once again we conclude that ab ∈ N and η(ab) = aη(b). 2 The above proposition shows that the map η : N → L2 (G) is a GNS map. To show that η ˜ corresponds to the Haar measure of (C∞ (G)Ψ ⊗Ψ , Ψ ) we shall need the following Proposition 4.17. Let η : N → L2 (G) be the map introduced above. For a ∈ N and ϕ ∈ (C∞ (G)Ψ ⊗Ψ )∗ let us consider their convolution ϕ ∗ a = (id ⊗ ϕ) (a) ∈ C∞ (G)Ψ ⊗Ψ . Then ϕ ∗ a is an element of N and η(ϕ ∗ a) = (id ⊗ ϕ)W η(a).

(58)

Proof. Recall that with any normal functional ω ∈ B(L2 (G))∗ we can associate a function fω ∈ C∞ (G) where fω (g) = ω(Rg ). Assume that a = Q(fω ) for some fω ∈ L2 (G). In particular a ∈ N and η(a) = fω . We compute


1319

ϕ ∗ a = (id ⊗ ϕ) W ∗ (a ⊗ 1)W = (id ⊗ ϕ) W ∗ (ω ⊗ id)(W ) ⊗ 1 W ∗ W12 W23 = (ω ⊗ id ⊗ ϕ) W23 = (ω ⊗ id ⊗ ϕ)(W12 W13 ) = (b · ω ⊗ id)W

(59)

where b = (id ⊗ ϕ)W ∈ M(C∗r (G)). Therefore to prove that ϕ ∗ a is an element of N it is enough to show that fb ·ω ∈ L2 (G) for all b ∈ M(C∗r (G)). First we check it for b = Rg . Note that fb·ω (g ) = b · ω(Rg ) = ω Rg Rg = ω(Rg g ) = fω (g g) = (Rg fω )(g ) = (bfω )(g ) for any g, g ∈ G, therefore fb·ω = bfω . By linearity this equality is satisfied for any b ∈ span{Rg : g ∈ G}. We extend it using a continuity argument. There exists a net of operators bi ∈ lin-span{Rg : g ∈ G} strongly convergent to b ∈ M(C∗r (G)). Functional ω is strongly continuous hence limi bi · ω = b · ω in the norm sense. Therefore limi fbi ·ω = fb·ω where limi is taken in the uniform sense. At the same time limi bi fω = bfω in the L2 -norm, hence fb·ω (g) = lim fbi ·ω (g) = lim bi fω (g) = bfω (g) i

i

for almost all g ∈ G. This shows that fb ·ω ∈ L2 (G) and fb·ω = bfω

(60)

for any b ∈ M(C∗r (G)). Using (59) and (60) we get the following sequence of equalities: η(ϕ ∗ a) = fb·ω = bfω = bη(a) = (id ⊗ ϕ)W η(a) which proves (58) for a = Q(fω ). But the set

a = (ω ⊗ id)W : fω ∈ L2 (G)

is a core for η, hence Eq. (58) is satisfied for any a ∈ N.

2

Remark 4.18. Let π R , π L ∈ Rep(C∗ (Γ ), L2 (G)) be representations that send generators uγ ∈ M(C∗ (Γ )) to Rγ and Lγ ∈ B(L2 (G)) respectively. Let f, f˜ ∈ M(C∗ (Γ )) and ω ∈ B(L2 (G))∗ be such that fω ∈ L2 (G). Then using a method similar to the one used in the proof of Proposition 4.17 we can show that fπ R (f )·ω·π R (f˜) = π R (f )π L κ(f˜) (fω ) where κ is the coinverse on C∗ (Γ ).

(61)

1320


⊗Ψ With GNS-map η : N → L2 (G) we can associate a weight hΨ : C∞ (G)Ψ → R+ : + hΨ (a ∗ a) = (η(a)|η(a)).

Proposition 4.19. Let hΨ be the weight on C∞ (G)Ψ ⊗Ψ introduced above. Then it is a faithful trace. In particular it is strictly faithful. Proof. Let u ∈ M(C∗ (Γ )) be the unitary element which appears in formula (42). From the above remark and Proposition 4.10 it follows that ∗ R η (ωx,y ⊗ id)W = η π R (u) · ωx, ¯ y¯ · π (u) ⊗ id W = π R (u)π L κ(u) η (ωx, ¯ y¯ ⊗ id)W = π R (u)π L κ(u) η (ωx,y ⊗ id)W . The set a = (ω ⊗ id)W : fω ∈ L2 (G) is a core for η, hence we have η a ∗ = π R (u)π L κ(u) η(a) for any a ∈ N. Now we can prove the trace property: hΨ a ∗ a = η(a)|η(a) = η(a)|η(a) ∗ ∗ = π L κ(u) π R (u)∗ η a ∗ |π L κ(u) π R (u)∗ η a ∗ = η a ∗ |η a ∗ = hΨ aa ∗ . Let us prove the faithfulness of hΨ . Assume that hΨ (a ∗ a) = 0. Then hΨ a ∗ c∗ ca = 0 = hΨ caa ∗ c∗ hence η(a ∗ c∗ ) = a ∗ η(c∗ ) = 0. The set of elements of the form η(c∗ ) is dense in L2 (G), hence a = 0. The notion of strict faithfulness was introduced in [9]. It can be shown that a faithful trace is automatically strictly faithful. This ends our proof. 2 Using Propositions 4.17 and 4.19 one can check that the assumptions of Theorem 3.9 of [9] are satisfied. Hence we get

Theorem 4.20. Let (C∞ (G)Ψ ⊗Ψ , Ψ ) be the quantum group with the multiplicative unitary W and the weight hΨ considered above. Then hΨ is a Haar measure for (C∞ (G)Ψ ⊗Ψ , Ψ ) and W is the canonical multiplicative unitary.


1321

5. An example of quantization of SL(2, C) In this section we use the Rieffel deformation to quantize the special linear group: SL(2, C) =

α γ

β δ

: α, β, γ , δ ∈ C, αδ − βγ = 1 .

(In what follows SL(2, C) will be denoted by G.) The resulting quantum group is the C∗ algebraic version of one of the ∗-Hopf algebras introduced by S.L. Woronowicz and S. Zaˆ γˆ , δ, ˆ satisfying the krzewski in paper [17]. As a ∗-algebra it is generated by four elements α, ˆ β, following commutation relations: αˆ βˆ = βˆ α, ˆ αˆ δˆ = δˆα, ˆ αˆ γˆ = γˆ α, ˆ ˆ βˆ γˆ = γˆ β, ˆ βˆ δˆ = δˆβ, γˆ δˆ = δˆγˆ , αˆ δˆ = 1 + βˆ γˆ ,

ˆ αˆ αˆ ∗ = αˆ ∗ α, ˆ αˆ βˆ ∗ = t βˆ ∗ α,

ˆ βˆ βˆ ∗ = βˆ ∗ β,

ˆ αˆ γˆ ∗ = t −1 γˆ ∗ α,

ˆ βˆ γˆ ∗ = γˆ ∗ β,

γˆ γˆ ∗ = γˆ ∗ γˆ ,

ˆ αˆ δˆ∗ = δˆ∗ α,

ˆ βˆ δˆ∗ = t −1 δˆ∗ β,

γˆ δˆ∗ = t δˆ∗ γˆ ,

ˆ δˆδˆ∗ = δˆ∗ δ,

(62)

where t is a nonzero real parameter. The comultiplication, coinverse and counit act on them in the standard way: (α) ˆ = αˆ ⊗ αˆ + βˆ ⊗ γˆ , ˆ = αˆ ⊗ βˆ + βˆ ⊗ δ, ˆ (β) (γˆ ) = γˆ ⊗ αˆ + δˆ ⊗ γˆ , ˆ = γˆ ⊗ βˆ + δˆ ⊗ δ, ˆ (δ)

ˆ κ(α) ˆ = δ, ˆ = −β, ˆ κ(β) κ(γˆ ) = −γˆ , ˆ = α, κ(δ) ˆ

ε(α) ˆ = 1, ˆ = 0, ε(β) ε(γˆ ) = 0, ˆ = 1. ε(δ)

(63)

The deformation procedure in our example is based on the abelian subgroup Γ ⊂ G of diagonal matrices: Γ =

w 0

0 w −1

: w ∈ C∗ .

1322


To simplify some calculations we pull back the action of Γ 2 on C∞ (G) to the action of C2 on C∞ (G). The resulting action is denoted by ρ:

(ρz1 ,z2 f )(g) = f

e−z1 0

0 ez1

z e2 g 0

0

e−z2

.

(64)

Let us fix a 2-cocycle on the dual group. The additive group (C, +) is self-dual, with the duality given by: C2 (z1 , z2 ) → exp i Im(z1 z2 ) ∈ T. Let s ∈ R. For any z1 , z2 ∈ C we set Ψ (z1 , z2 ) = exp is Im(z1 z¯ 2 ) . It is clear, that Ψ ∈ Cb (C2 ) satisfies the 2-cocycle condition. Using results of Section 4 we deform the standard C2 -product structure on C∞ (G) ρ C2 to (C∞ (G) ρ C2 , λ, ρˆ Ψ ⊗Ψ ). In our is just the complex conjugate of Ψ and the deformed action of the dual group is given by case Ψ

⊗Ψ (b) = λ−s z¯ 1 ,s z¯ 2 ρˆz1 ,z2 (b)λ∗−s z¯ 1 ,s z¯ 2 ρˆzΨ1 ,z 2

for any b ∈ C∞ (G) ρ C2 . The Landstad algebra A of the deformed C2 -product carries the structure of a quantum group. Our aim is to show that this quantum group is the C∗ -algebraic version of the Hopf ∗-algebra described above. The relation between parameters s, t ∈ R is t = exp(−2s). 5.1. C∗ -algebra structure ˆ γˆ , δˆ η A and show that they In this section we will construct four affiliated elements α, ˆ β, ∗ generate C -algebra A. Let Tr , Tl ∈ C∗ (C2 )η ⊂ (C∞ (G) ρ C2 )η be infinitesimal generators of the left and right shifts. By definition Tl and Tr are normal elements satisfying: λz1 ,z2 = exp i Im(z1 Tl ) exp i Im(z2 Tr )

(65)

for any z1 , z2 ∈ C. Let α, β, γ , δ be coordinate functions on G: η η α, β, γ , δ ∈ C(G) = C∞ (G) ⊂ C∞ (G) ρ C2 . Consider also a unitary element: U = exp is Im Tr∗ Tl ∈ M C∗ C2 ⊂ M C∞ (G) ρ C2 . We use it to define four normal elements affiliated with C∞ (G) ρ C2 :

(66)


αˆ = U αU ∗ ,

βˆ = U ∗ βU,

γˆ = U ∗ γ U,

δˆ = U δU ∗ .

1323

(67)

ˆ γˆ , δˆ which will be needed later. In the next lemma we present different formulas for α, ˆ β, Lemma 5.1. Let α, β, γ , δ ∈ C∞ (G)η ⊂ (C∞ (G) ρ C2 )η be coordinate functions on G. Let Tl , Tr be infinitesimal generators defined by (65) and let ˆ γˆ , δˆ ∈ C∞ (G) ρ C2 η α, ˆ β, be normal elements (67). Then ⎧ 1. ⎪ ⎪ ⎪ ⎨ 2. ⎪ 3. ⎪ ⎪ ⎩ 4.

α and Tl + Tr strongly commute and αˆ = exp −s Tl∗ + Tr∗ α; β and Tl − Tr strongly commute and βˆ = exp s Tl∗ − Tr∗ β; ∗ γ and Tl − Tr strongly commute and γˆ = exp s Tr − Tl∗ γ ; δ and Tl + Tr strongly commute and δˆ = exp s T ∗ + Tr∗ δ.

(68)

l

Proof. The fact that Tl + Tr and α strongly commute follows from the identity exp i Im z(Tl + Tr ) α exp −i Im z(Tl + Tr ) = α. We check it below: exp i Im z(Tl + Tr ) α exp −i Im z(Tl + Tr ) = λz,z αλ∗z,z = exp(−z + z)α = α. To prove the equality αˆ = exp(−s(Tl∗ + Tr∗ ))α note that αˆ = exp is Im Tr∗ Tl α exp −is Im Tr∗ Tl = exp is Im (Tl + Tr )∗ Tl α exp −is Im (Tl + Tr )∗ Tl ,

(69)

where we used the fact that exp(is Im(Tl∗ Tl )) = 1. Using the strong commutativity of Tl + Tr and α and the following identity: exp is Im(wTl ) α exp −is Im(wTl ) = exp(−sw)α, we get αˆ = exp(−s(Tl∗ + Tr∗ ))α. This ends the proof of point 1 of (68). Using the same techniques we prove points 2, 3, 4. 2 ˆ γˆ , δˆ are generators of C∗ -algebra A. In particular we have Our objective is to show that α, ˆ β, to show that they are affiliated with A. The following proposition is the first step toward the proof of this fact.

Proposition 5.2. Let (C∞ (G) ρ C2 , λ, ρˆ Ψ ⊗Ψ ) be the deformed C2 -product, A its Landstad ˆ ∈ M(A) for algebra and αˆ ∈ (C∞ (G) ρ C2 )η the normal element defined in (67). Then f (α) any f ∈ C∞ (C).

1324


Proof. Let us first prove the invariance of f (α) ˆ under the action ρˆ Ψ ⊗Ψ . It is enough to check that αˆ is invariant. In order to do that we calculate λz1 ,z2 αλ∗z1 ,z2 = ρz1 ,z2 (α) = exp(−z1 + z2 )α.

(70)

Furthermore ⊗Ψ ⊗Ψ ρˆzΨ1 ,z U αU ∗ (α) ˆ = ρˆzΨ1 ,z 2 2

⊗Ψ ⊗Ψ ⊗Ψ = ρˆzΨ1 ,z (U )ρˆzΨ1 ,z (α)ρˆzΨ1 ,z (U )∗ . 2 2 2

⊗Ψ (U ) and ρˆ Ψ ⊗Ψ (α) separately: We compute ρˆzΨ1 ,z z1 ,z2 2

⊗Ψ (α) = λ−s z¯ 1 ,s z¯ 2 ρˆz1 ,z2 (α)λ∗−s z¯ 1 ,s z¯ 2 ρˆzΨ1 ,z 2

= λ−s z¯ 1 ,s z¯ 2 αλ∗−s z¯ 1 ,s z¯ 2 = exp(s z¯ 1 + s z¯ 2 )α, ⊗Ψ ⊗Ψ exp is Im Tr∗ Tl (U ) = ρˆzΨ1 ,z ρˆzΨ1 ,z 2 2 = ρˆz1 ,z2 exp is Im Tr∗ Tl = exp is Im Tr∗ + z¯ 2 (Tl + z1 ) = U λs z¯ 2 ,−s z¯ 1 Ψ (z1 , z2 ). Using (70) we get

⊗Ψ ρˆzΨ1 ,z (α) ˆ = exp(s z¯ 1 + s z¯ 2 )U λs z¯ 2 ,−s z¯ 1 αλ∗−s z¯ 2 ,s z¯ 1 U ∗ 2

= exp(s z¯ 1 + s z¯ 2 ) exp(−s z¯ 1 − s z¯ 2 )U αU ∗ = α. ˆ Let us now check that the map C2 (z1 , z2 ) → λz1 ,z2 f (α)λ ˆ ∗z1 ,z2 ∈ M C∞ (G) C2

(71)

is norm continuous. For this note that: λz1 ,z2 f (α)λ ˆ ∗z1 ,z2 = U λz1 ,z2 f (α)λ∗z1 ,z2 U ∗ = Uf e−z1 +z2 α U ∗ . Function f is continuous and vanishes at infinity, hence we get norm continuity (71). This shows that f (α) ˆ satisfies the first and second Landstad condition of (3) which is enough to be an element of M(A). 2 To prove that αˆ is affiliated to A we need one more


1325

Proposition 5.3. The set I = f (α)A: ˆ f ∈ C∞ (C) is linearly dense in A.

Proof. Recall that ρ Ψ ⊗Ψ is the action of C2 on A implemented by unitary elements λz1 ,z2 . It is easy to see that I is invariant under ρ Ψ ⊗Ψ . Let g ∈ C∞ (C) be a function given by the formula −1 g(z) = (1 + z¯ z) . Then g(α) ˆ = U (1 + α ∗ α)U ∗ and we have: 2 cls ∗ 2 cls −1 ∗ 2 ∗ ˆ C = C C 1 + α ∗ α U ∗ AC∗ C2 C C g(α)AC cls ⊂ C∗ C2 IC∗ C2

(72)

where we used the equality C∗ (C2 )U = C∗ (C2 ). Note that the set U ∗ AC∗ (C2 ) is linearly dense in C∞ (G) ρ C2 . Using the fact that α is affiliated with C∞ (G) ρ C2 we see that the set C∗ (C2 )(1 + α ∗ α)−1 U ∗ AC∗ (C2 ) is linearly dense in C∞ (G) ρ C2 . Hence by (72) the set C∗ (C2 )IC∗ (C2 ) is linearly dense in C∞ (G) ρ C2 . Using Lemma 2.6 we get the linear density of I in A. 2 Let us define the homomorphism of C∗ -algebras: C∞ (C) f → π(f ) = f (α) ˆ ∈ M(A). Theorem 5.4. Let π be the homomorphism defined above. π is a morphism of C∗ -algebras: π ∈ Mor(C∞ (C); A). In particular αˆ is the normal element affiliated with A. Proof. By Proposition 5.3 we have π(C∞ (C))A · = A which shows that π ∈ Mor(C∞ (C); A). Let id ∈ C∞ (C)η be the identity function: id(z) = z for all z ∈ C. Applying morphism π to id ∈ C∞ (C)η we get π(id) = id(α) ˆ = αˆ ∈ Aη . 2 ˆ γˆ , δˆ η A. In the next theorem we prove that they Using the same techniques we show that β, are in fact generators of A. ˆ γˆ , δˆ η A be affiliated elements introduced in (67). Let us consider the set: Theorem 5.5. Let α, ˆ β, ˆ 3 (γˆ )f4 (δ): ˆ f1 , f2 , f3 , f4 ∈ C∞ (C) ⊂ M(A). V = f1 (α)f ˆ 2 (β)f ˆ γˆ , δˆ ∈ Aη . Then V is a subset of A and V cls = A. In particular A is generated by elements α, ˆ β, Proof. Let us start with a proof that V ⊂ A. Mimicking the proof of Theorem 5.2 we show that elements of V satisfy the first and the second Landstad condition (3). To check that they also satisfy the third one, we need to show that ˆ 3 (γˆ )f4 (δ)y ˆ ∈ C∞ (G) ρ C2 xf1 (α)f ˆ 2 (β)f

(73)

1326


for any x, y ∈ C∗ (C2 ). Let us consider the set cls ˆ 3 (γˆ )f4 (δ)y: ˆ W = xf1 (α)f ˆ 2 (β)f f1 , f2 , f3 , f4 ∈ C∞ (C), x, y ∈ C∗ C2 . Note that W = (C∗ (C2 )VC∗ (C2 ))cls . We will show that: W = C∞ (G) ρ C2 which is a stronger property than (73). Using (67) we get cls W = xUf1 (α)U ∗ 2 f2 (β)f3 (γ )U 2 f4 (δ)U ∗ y: f1 , f2 , f3 , f4 ∈ C∞ (C), x, y ∈ C∗ C2 . By unitarity of U we can substitute x with xU ∗ and y with Uy not changing W: cls W = xf1 (α)U ∗ 2 f2 (β)f3 (γ )U 2 f4 (δ)y: f1 , f2 , f3 , f4 ∈ C∞ (C), x, y ∈ C∗ C2 . The map C2 (z1 , z2 ) → ρz1 ,z2 f (α) = f exp(−z1 + z2 )α is norm continuous, hence:

cls cls = xf (α): f ∈ C∞ (C), x ∈ C∗ C2 . f (α)x: f ∈ C∞ (C), x ∈ C∗ C2

In particular cls W = f1 (α)xU ∗ 2 f2 (β)f3 (γ )U 2 f4 (δ)y: f1 , f2 , f3 , f4 ∈ C∞ (C), x, y ∈ C∗ C2 . Similarly, we commute f4 (δ) and y: cls W = f1 (α)xU ∗ 2 f2 (β)f3 (γ )U 2 yf4 (δ): f1 , f2 , f3 , f4 ∈ C∞ (C), x, y ∈ C∗ C2 . Substituting x with xU 2 and y with U ∗2 y we get cls W = f1 (α)xf2 (β)f3 (γ )yf4 (δ): f1 , f2 , f3 , f4 ∈ C∞ (C), x, y ∈ C∗ C2 . Commuting back f1 (α) (f4 (δ) resp.) and x (y resp.) we obtain cls W = xf1 (α)f2 (β)f3 (γ )f4 (δ)y: f1 , f2 , f3 , f4 ∈ C∞ (C), x, y ∈ C∗ C2 . The last set is obviously the whole C∞ (G) ρ C2 . Therefore we conclude that elements of V satisfies the Landstad conditions and V ⊂ A. Moreover V is ρ Ψ ⊗Ψ -invariant and the set ∗ 2 ∗ 2 2 C (C )VC (C ) is linearly dense in C∞ (G) ρ C . Using Lemma 2.6 we see that V cls = A.


1327

ˆ γˆ , δˆ separate representations of A and In particular α, ˆ β, −1 −1 −1 −1 ∈ A. 1 + βˆ ∗ βˆ 1 + γˆ ∗ γˆ 1 + δˆ∗ δˆ 1 + αˆ ∗ αˆ ˆ γˆ , δ. ˆ By Theorem 3.3 of [18] we see that A is generated by α, ˆ β,

2

5.2. Commutation relations ˆ γˆ , δˆ satisfy relations (62). Note that The aim of this section is to show that generators α, ˆ β, in general it is impossible to multiply affiliated elements, so we have to give a precise meaning to (62). We start with considering a more general type of relations. Let p, q be real, strictly positive numbers and (R, S) a pair of normal operators acting on H . The precise meaning of the relations RS = pSR, RS ∗ = qS ∗ R was given in [16]: Definition 5.6. Let (R, S) be a pair of normal operators acting on a Hilbert space H . We say that (R, S) is a (p, q)-commuting pair if: 1. |R| and |S| strongly commute. 2. (Phase R)(Phase S) = (Phase S)(Phase R). 3. On ker R ⊥ we have (Phase R)|S|(Phase R)∗ =

√ pq |S|.

4. On ker S ⊥ we have (Phase S)|R|(Phase S)∗ =

q/p |R|.

The set of all (p, q)-commuting pairs of normal operators acting on a Hilbert space H is denoted by Dp,q (H ). Note that (1, 1)-commuting pair of normal operators is just a strongly commuting pair of operators. We need a version of the above definition which is suitable for a pair of normal elements affiliated with a C∗ -algebra. In what follows we shall use the symbol z(T ) to denote the z1 transform of an element T : z(T ) = T (1 + T ∗ T )− 2 . Definition 5.7. Let A be a C∗ -algebra and (R, S) a pair of normal elements affiliated with A. We say that (R, S) is a (p, q)-commuting pair if ∗ ) = z(√pq S ∗ )z(√q/p R), 1. z(R)z(S √ √ 2. z( q/p R)z(S) = z( pq S)z(R).

1328


The set of all (p, q)-commuting pairs of normal elements affiliated with a C∗ -algebra A is denoted by Dp,q (A). It turns out that Definitions 5.6 and 5.7 are in a sense equivalent. Namely we have: Proposition 5.8. Let (R, S) be a pair of normal operators acting on H . It is a (p, q)-commuting pair in the sense of Definition 5.6 if and only if

√ √ z(R)z S ∗ = z pq S ∗ z( q/p R), √ √ z( q/p R)z(S) = z( pq S)z(R).

(74)

Proof. It is easy to see that a pair (R, S) of (p, q)-commuting operators satisfies (74). We will prove the opposite implication. Using (74) we get: √ z( q/p R)z(S)z(S)∗ = z( pq S)z(R)z(S)∗ √ √ = z( pq S)z( pq S)∗ z( q/p R).

(75)

z( q/p R)∗ z( q/p R)z(S)z(S)∗ √ √ = z( q/p R)∗ z( pq S)z( pq S)∗ z( q/p R) = z(S)z(S)∗ z( q/p R)∗ z( q/p R).

(76)

Hence

2 and we get T is a normal operator, hence zT∗ zT = zT zT∗ = z|T |

2 2 2 2 z q/p |R| z |S| = z |S| z q/p |R| . This shows that (|R|, |S|) is a pair of strongly commuting operators. Using the polar decomposition of normal operators R and S we rewrite the second equation of (74): Phase(R)z

√ q/p |R| z |S| Phase(S) = Phase(S)z pq |S| z |R| Phase(R).

Strong commutativity of |R| and |S| and identities Phase(S) Phase(S)∗ z |S| = z |S| , Phase(R) Phase(R)∗ z |R| = z |R| gives Phase(R) Phase(S) Phase(S)∗ z |S| z q/p |R| Phase(S) √ = Phase(S) Phase(R) Phase(R)∗ z |R| z pq |S| Phase(R).


1329

Uniqueness of the polar decomposition implies that phases of R and S commute: Phase(R) Phase(S) = Phase(S) Phase(R). Using Eq. (75) we get Phase(R)z

√ q/p |R| z |S| = z pq |S| Phase(R)z q/p |R| .

We already know that |R| and |S| strongly commute, hence √ Phase(R)z |S| z q/p |R| = z pq |S| Phase(R)z q/p |R| . This shows that on ker R ⊥ we have Phase(R)|S| Phase(R)∗ = Similarly, one can prove that Phase(S)|R| Phase(S)∗ =

√ pq |S|.

√ q/p |R| on ker S ⊥ .

2

ˆ γˆ , δˆ η A satisfy relations (62) in the sense of Definition 5.7. The next theorem shows that α, ˆ β, ˆ γˆ , δˆ η A be elements given by (67). Then Theorem 5.9. Let α, ˆ β, ˆ (β, ˆ γˆ ) ∈ D1,1 (A), 1. (α, ˆ δ), ˆ (γˆ , δ) ˆ ∈ D1,t (A), 2. (α, ˆ β), ˆ ˆ 3. (α, ˆ γˆ ), (β, δ) ∈ D1,t −1 (A), ˆ βˆ γˆ η A (a product of two strongly commutwhere t = exp(−2s). Consider normal elements αˆ δ, ˆ βˆ γˆ ) ∈ D1,1 (A) and ing normal elements is well defined). Then (αˆ δ, αˆ δˆ − βˆ γˆ = 1. ˆ ∈ D1,1 (A) and (β, ˆ γˆ ) ∈ D1,1 (A). Note that the afProof. Directly from (67) it follows that (α, ˆ δ) filiated element αδ ηC∞ (G) is ρ-invariant: ρz1 ,z2 (αδ) = αδ where ρ is the action defined by (64). Therefore, at the level of the crossed product, αδ commutes with C∗ (C2 ). Using the fact that U ∈ M(C∗ (C2 )) we get αˆ δˆ = U αδU ∗ = αδ. ˆ βˆ γˆ ) ∈ D1,1 (A) and Similar reasoning shows that βˆ γˆ = βγ . Therefore (αˆ δ, αˆ δˆ − βˆ γˆ = αδ − βγ = 1. ˆ ∈ D1,t (A). Using the faithful representation π can of A on L2 (G) Now let us prove that (α, ˆ β) ˆ ∈ ˆ β) we can treat generators α, ˆ βˆ as normal operators acting on L2 (G). We will show that (α, 2 ˆ ˆ β) ∈ D1,t (A). D1,t (L (G)) which by Proposition 5.8 is equivalent with the containment (α, Using Lemma 5.1 we get:

1330


⎧ Phase(α) ˆ = exp is Im(Tl + Tr ) Phase(α), ⎪ ⎪ ⎪ ⎨ |α| ˆ = exp −s Re(Tl + Tr ) |α|, ˆ = exp −is Im(Tl − Tr ) Phase(β), ⎪ Phase(β) ⎪ ⎪ ⎩ ˆ = exp s Re(Tl − Tr ) |β|. |β| Moreover, it is easy to check that ⎧ exp is Im(Tl + Tr ) β exp −is Im(Tl + Tr ) = exp(−2s)β, ⎪ ⎪ ⎪ ⎨ expis Im(T − T )α exp−is Im(T − T ) = exp(−2s)α, l r l r ⎪ exp −is Re(T + T ) |β| exp is Re(T + Tr ) = exp(2is)β = |β|, l r l ⎪ ⎪ ⎩ exp −is Re(Tl − Tr ) |α| exp is Re(Tl − Tr ) = exp(2is)α = |α|.

(77)

(78)

Eqs. (77) and (78) show together that: 1.

ˆ = Phase(β) ˆ Phase(α), Phase(α) ˆ Phase(β) ˆ

2.

ˆ Phase(α) ˆ Phase(α)| ˆ β| ˆ ∗ = exp(−2s)|β|,

3.

ˆ α| ˆ ∗ = exp(−2s)|α|, Phase(β)| ˆ Phase(β) ˆ

4.

ˆ strongly commute. |α| ˆ and |β|

ˆ ∈ D1,t (L2 (G)). Using the same techniques we prove Note that ker αˆ = ker βˆ = {0} hence (α, ˆ β) all other assertions of our theorem. 2 5.3. Comultiplication Let Ψ ∈ Mor(A; A ⊗ A) be the comultiplication on A. As was shown in Theorem 4.11, it is given by: Ψ (a) = Υ (a)Υ ∗ ,

(79)

where ∈ Mor(C∞ (G) C2 ; C∞ (G) C2 ⊗ C∞ (G) C2 ) is uniquely characterized by two properties: • (Tl ) = Tl ⊗ I , (Tr ) = I ⊗ Tr ; • restricted to C∞ (G) coincides with the comultiplication on C∞ (G). In our case the unitary element Υ is of the following form: Υ = exp is Im Tr∗ ⊗ Tl .

(80)

ˆ γˆ , δˆ be the Theorem 5.10. Let (A, Ψ ) be the quantum group considered above and let α, ˆ β, Ψ generators of A given by (67). Comultiplication acts on generators in the standard way: ˆ = αˆ ⊗ αˆ + βˆ ⊗ γˆ , Ψ (α)

ˆ = αˆ ⊗ βˆ + βˆ ⊗ δ, ˆ Ψ (β)

Ψ (γˆ ) = γˆ ⊗ αˆ + δˆ ⊗ γˆ ,

ˆ = δˆ ⊗ δˆ + γˆ ⊗ β. ˆ Ψ (δ)

(81)


1331

Remark 5.11. The action of Ψ in the formula above is given by the sum of affiliated elements. In general it is not a well-defined operation. But in our case (as will be shown) this is a sum of two normal strongly commuting elements of (A ⊗ A)η . This operation is well defined and gives a normal element affiliated with (A ⊗ A)η . Proof. Applying morphism to U ∈ M(C∗ (Γ 2 )) (see (66)) we get: (U ) = exp is Im Tl ⊗ Tr∗ .

(82)

Let R be a unitary element given by the formula: R = exp is Im Tr∗ ⊗ Tl exp is Im Tl ⊗ Tr∗ . Using (79), (80) and (82) we get: Ψ (α) ˆ = R(α ⊗ α + β ⊗ γ )R ∗ = R(α ⊗ α)R ∗ + R(β ⊗ γ )R ∗ .

(83)

Note that ∗ U ⊗ U ∗ R = exp is Im Tr∗ ⊗ I − I ⊗ Tr∗ (I ⊗ Tl − Tl ⊗ I ) .

(84)

It is easy to check that elements (Tr ⊗ I − I ⊗ Tr ) and (I ⊗ Tl − Tl ⊗ I ) strongly commute with α ⊗ α. Hence by identity (84), (U ∗ ⊗ U ∗ )R commutes with α ⊗ α. Similarly, we check that the unitary element (U ⊗ U )R commutes with β ⊗ γ . Using these two facts we get R(α ⊗ α)R ∗ = (U ⊗ U ) U ∗ ⊗ U ∗ R(α ⊗ α)R ∗ (U ⊗ U ) U ∗ ⊗ U ∗ = (U ⊗ U )(α ⊗ α) U ∗ ⊗ U ∗ = αˆ ⊗ αˆ

(85)

R(β ⊗ γ )R ∗ = U ∗ ⊗ U ∗ (U ⊗ U )R(β ⊗ γ )R ∗ U ∗ ⊗ U ∗ (U ⊗ U ) = U ∗ ⊗ U ∗ (β ⊗ γ )(U ⊗ U ) = βˆ ⊗ γˆ .

(86)

and

Eqs. (83), (85), (86) give: Ψ (α) ˆ = αˆ ⊗ αˆ + βˆ ⊗ γˆ . All other assertions of our theorem are proven using the same techniques.

2

1332


References [1] S. Baaj, G. Skandalis, Unitaires mulitiplicatifs et dualite pour les produits croises de C∗ -algebres, Ann. Sci. Ecole Norm. Sup. (4) 26 (1993) 425–488. [2] A. Connes, An analogue of the Thom isomorphism for crossed products of a C∗ -algebra by an action of R, Adv. Math. 39 (1981) 31–55. [3] M. Enock, L. Vainerman, Twisted Kac algebras obtained from 2-cocycles, Comm. Math. Phys. 178 (1996). [4] S. Kaliszewski, J. Quigg, I. Raeburn, Proper actions, fixed-point algebras and naturality in nonabelian duality, J. Funct. Anal. 254 (12) (2008) 2949–2968. [5] A. Kleppner, Multipliers on Abelian groups, Math. Ann. 158 (1965) 11–34. [6] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole Norm. Sup. (4) 33 (2000) 837–934. [7] M.B. Landstad, Quantizations arising from abelian subgroups, Internat. J. Math. 5 (1994) 897–936. [8] M.B. Landstad, D. Olesen, G.K. Pedersen, Towards a Galois theory for crossed products of C∗ -algebras, Math. Scand. 43 (2) (1979) 311–321. [9] T. Masuda, Y. Nakagami, S.L. Woronowicz, A C∗ -algebraic framework for quantum groups, Internat. J. Math. 14 (9) (2003) 903–1001. [10] G.K. Pedersen, C∗ -Algebras and Their Automorphism Groups, Academic Press, 1979. [11] W. Pusz, P.M. Sołtan, Analysis on a homogeneous space of a quantum group, preprint, math.OA/0509610. [12] M.A. Rieffel, Deformation quantization for action of Rd , Mem. Amer. Math. Soc. 506 (1993). [13] M.A. Rieffel, Non-compact quantum groups associated with Abelian subgroups, Comm. Math. Phys. 171 (1995). [14] M. Takesaki, Theory of Operator Algebras III, in: Oper. Alg. Non-commut. Geom., vol. 8, in: Encyclopaedia Math. Sci., vol. 127, Springer-Verlag, Berlin, 2003. [15] S.L. Woronowicz, Unbounded elements affiliated with C∗ -algebras and non-compact quantum groups, Comm. Math. Phys. 136 (1991) 399–432. [16] S.L. Woronowicz, Operator equalities related to the quantum E(2) group, Comm. Math. Phys. 144 (1992). [17] S.L. Woronowicz, S. Zakrzewski, Quantum deformations of Lorentz group. Hopf ∗-algebra level, Compos. Math. 90 (2) (1994). [18] S.L. Woronowicz, C∗ -algebras generated by unbounded elements, Rev. Math. Phys. 7 (3) (1995) 481–521. [19] S.L. Woronowicz, From multiplicative unitaries to quantum groups, Internat. J. Math. 7 (1) (1996).


Null controllability for the parabolic equation with a complex principal part ✩ Xiaoyu Fu School of Mathematics, Sichuan University, Chengdu 610064, China Received 25 May 2008; accepted 29 May 2009 Available online 12 June 2009 Communicated by J. Coron

Abstract The paper is devoted to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators (α + iβ)∂t + nj,k=1 ∂k (a j k ∂j ) (with real functions α and β), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg–Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimates. © 2009 Elsevier Inc. All rights reserved. Keywords: Null controllability; Observability; Universal approach; Carleman estimate; Parabolic equation with a complex principal part

1. Introduction and main results Assume given T > 0 and a bounded domain Ω of Rn (n ∈ N) with C 2 boundary Γ . Fix an open non-empty subset ω of Ω and denote by χω the characteristic function of ω. Let ω0 be another non-empty open subset of Ω such that ω0 ⊂ ω. Put Q = (0, T ) × Ω,

Σ = (0, T ) × Γ,

Q0 = (0, T ) × ω0 .

✩ This work is partially supported by the NSF of China under grants 10525105, 10771149 and 10831007. The author gratefully acknowledges Professor Xu Zhang for his guidance, encouragement and suggestions. E-mail address: [email protected].


1334

X. Fu / Journal of Functional Analysis 257 (2009) 1333–1354

In the sequel, we will use the notation yj = yxj , where xj is the j -th coordinate of a generic point x = (x1 , . . . , xn ) in Rn . In a similar manner, we use the notation zj , vj , etc. for the partial derivatives of z and v with respect to xj . Throughout this paper, we will use C = C(T , Ω, ω) to denote generic positive constants which may vary from line to line (unless otherwise stated). For any c ∈ C, we denote its complex conjugate by c. Fix a j k (·) ∈ C 1,2 (Q; R) satisfying a j k (t, x) = a kj (t, x),

(t, x) ∈ Q, j, k = 1, 2, . . . , n,

(1.1)

and for some constant s0 > 0, n

a j k ξj ξk s0 |ξ |2 ,

(t, x, ξ ) ≡ (t, x, ξ1 , . . . , ξn ) ∈ Q × Cn .

(1.2)

j,k=1

Next, we fix a function f (·) ∈ C 1 ( C) satisfying f (0) = 0 and the following condition: lim

s→∞

|f (s)| |s| ln1/2 |s|

= 0.

(1.3)

Note that f (·) in the above can have a superlinear growth. We are interested in the following semilinear parabolic equation with a complex principal part: ⎧ n jk ⎪ ⎪ ⎪ (1 + ib)y a yj k = χω u + f (y) − t ⎨ ⎪ ⎪ ⎪ ⎩y = 0 y(0) = y0

in Q,

j,k=1

(1.4) on Σ, in Ω,

√ where i = −1 and b is a given real number. In (1.4), y = y(t, x) is the state and u = u(t, x) is the control. One of our main objectives in this paper is to study the null controllability of system (1.4), by which we mean that, for any given initial state y0 , find (if possible) a control u such that the weak solution of (1.4) satisfies y(T ) = 0. In order to derive the null controllability of (1.4), by means of the well-known duality argument (see [15, p. 282, Lemma 2.4], for example), one needs to consider the following dual system of the linearized system of (1.4) (which can be regarded as a linearized complex Ginzburg– Landau equation): ⎧ ⎨ Gz = q(t, x)z z=0 ⎩ z(T ) = zT

in Q, on Σ, in Ω,

(1.5)

where q(·) ∈ L∞ (0, T ; L∞ (Ω)) is a potential and Gz (1 + ib)zt +

n jk a zj k . j,k=1

(1.6)


1335

By means of global Carleman inequality, we shall establish the following explicit observability estimate for solutions of system (1.5). Theorem 1.1. Let a j k (·) ∈ C 1,2 (Q; R) satisfy (1.1)–(1.2), q(·) ∈ L∞ (0, T ; L∞ (Ω)) and b ∈ R. Then there is a constant C > 0 such that for all solutions of system (1.5), it holds

z(0)

L2 (Ω)

C(r)|z|L2 ((0,T )×ω) ,

∀zT ∈ L2 (Ω),

(1.7)

r |q|L∞ (0,T ;L∞ (Ω)) .

(1.8)

where 2

C(r) C0 eC0 r ,

C0 = C 1 + b 2 ,

Thanks to the dual argument and the fixed point technique, Theorem 1.1 implies the following controllability result for system (1.4). Theorem 1.2. Let a j k (·) ∈ C 1,2 (Q; R) satisfy (1.1)–(1.2), f (·) ∈ C 1 (C) satisfy f (0) = 0 and (1.3), and b ∈ R. Then for any given y0 ∈ L2 (Ω), there is a control u ∈ L2 ((0, T ) × ω) such that the weak solution y(·) ∈ C([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) of system (1.4) satisfies y(T ) = 0 in Ω. The controllability problem for system (1.4) with b = 0 (i.e., linear and semilinear parabolic equations) has been studied by many authors and it is now well understood. Among them, let us mention [5,6,13,9] on what concerns null controllability, [7–9,29] for approximate controllability, and especially [30] for recent survey in this respect. However, for the case b = 0, very little is know in the previous literature. To the best of our knowledge, [10] is the only paper addressing the global controllability for multidimensional system (1.4). We refer to [21] for a recent interesting result on local controllability of semilinear complex Ginzburg–Landau equation. We remark that condition (1.3) is not sharp. Indeed, following [5], one can establish the null controllability of system (1.4) when the nonlinearity f (y) is replaced by a more general term of the form f (y, ∇y) under the assumptions that f (·,·) ∈ C 1 (C1+n ), f (0, 0) = 0 and

|(s,p)|→∞

lim

1

0 fs (τ s, τp) dτ | 3/2

= 0, (1 + |s| + |p|) 1 1 |( 0 fp1 (τ s, τp) dτ, . . . , 0 fpn (τ s, τp) dτ )| lim

|(s,p)|→∞

|

ln

ln1/2 (1 + |s| + |p|)

= 0,

(1.9)

where p = (p1 , . . . , pn ). Moreover, following [6], one can show that the assumptions on the growth of the nonlinearity f (y, ∇y) in (1.9) are sharp in some sense. Since the techniques are very similar to [5,6], we shall not give the details here. Instead, as a byproduct of the fundamental identity established in this work (to show the observability inequality (1.7)), we shall develop a universal approach for controllability/observability problems governed by partial differential equations (PDEs for short), which is the second main object of this paper. The study of controllability/observability problem for PDEs began in the 1960s, for which various techniques have been developed in the last decades [1,4,13,17,30]. It is well known that the controllability of PDEs depends strongly on the nature of the system, say time reversibility or without. Typical examples are the wave and heat equations.

1336


It is clear now that there exist essential differences between the controllability/observability theories for these two equations. Naturally, one expects to know whether there are some relationship between these two systems of different nature. Especially, it would be quite interesting to establish a unified controllability/observability theory for parabolic and hyperbolic equations. This problem was posed by D.L. Russell in [22], where one can also find some preliminary result; further results are available in [18,27]. In [16], the authors analyzed the controllability/observability problems for PDEs from the point of view of methodology. It is well known that these problems may be reduced to the obtention of suitable observability inequalities for the underlying homogeneous systems. However, the techniques that have been developed to obtain such estimates depend heavily on the nature of the equations. In the context of the wave equation one may use multipliers [17] or microlocal analysis [1]; while, in the context of heat equations, one uses Carleman estimates [8,13]. Carleman estimates can also be used to obtain observability inequalities for the wave equation [26]. However, the Carleman estimate that has been developed up to now to establish observability inequalities of PDEs depend heavily on the nature of the equations, and therefore a unified Carleman estimate for those two equations has not been developed before. In this paper, we present a point-wise weighted identity for partial differential operators (α + iβ)∂t + nj,k=1 ∂k (a j k ∂j ) (with real functions α and β), by which we develop a unified approach, based on global Carleman estimate, to deduce not only the controllability/observability results for systems (1.4) and (1.5), but also all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimates (see Section 2 for more details). We point out that this identity has other interesting applications, say, in [20] it is applied to derive an asymptotic formula of reconstructing the initial state for a Kirchhoff plate equation with a logarithmical convergence rate for smooth data; while in [11] it is applied to establish sharp logarithmic decay rate for general hyperbolic equations with damping localized in arbitrarily small set by means of an approach which is different from that in [3]. The rest of this paper is organized as follows. In Section 2, we establish a fundamental pointwise weighted identity for partial differential operators of second order and give some of its applications. In Section 3, we derive a modified point-wise inequality for the parabolic operator. This estimate will play a key role when we establish in Section 4 a global Carleman estimate for the parabolic equation with a complex principal part. Finally, we will prove our main results in Section 5. 2. A weighted identity for partial differential operators and its applications In this section, we will establish a point-wise weighted identity for partial differential operators of second order with a complex principal part, which has an independent interest. First, we introduce the following second order operator: Pz (α + iβ)zt +

m jk a zj k ,

m ∈ N.

(2.1)

j,k=1

We have the following fundamental identity. Theorem 2.1. Let α, β ∈ C 2 (R1+m ; R) and a j k ∈ C 1,2 (R1+m ; R) satisfy a j k = a kj (j, k = 1, 2, . . . , m). Let z, v ∈ C 2 (R1+m ; C) and Ψ, ∈ C 2 (R1+m ; R). Set θ = e and v = θ z. Then


1337

θ (PzI1 + PzI1 ) + Mt + div V

m j k j k j k j k 1 jk 2 j k 2 a j k a − a a j k + αa t − a Ψ (vk v j + v k vj ) = 2|I1 | + 2 j,k,j ,k =1

+i

m

m j k βa j t + a j k (β t )j (v k v − vk v) − a j k αk (vj v t + v j vt )

j,k=1

+ i βΨ +

m

βa j k j

k

j,k=1

(vvt − vv t ) + B|v|2 ,

(2.2)

j,k=1

where ⎧ m m jk ⎪ ⎪ jk ⎪ A a j k − Ψ, a

− j k ⎪ ⎪ ⎨ j,k=1 j,k=1 m ⎪ jk ⎪ ⎪ ⎪ a vj k + Av, I1 iβvt − α t v + ⎪ ⎩

(2.3)

j,k=1

and ⎧ m m 2 2 ⎪ ⎪ 2 jk ⎪ M α

+ β − αA |v| + α a v v + iβ a j k j (v k v − vk v), ⎪ t j k ⎪ ⎪ ⎪ j,k=1 j,k=1 ⎪ ⎪ 1 ⎪ ⎪ k m ⎪ V V ,...,V ,...,V , ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ k ⎪ −iβ a j k j (v t v − vvt ) + a j k t (vj v − v j v) − αa j k (vj v t + v j vt ) V ⎪ ⎪ ⎪ ⎪ j,j ,k =1 ⎪ ⎨ (2.4) + 2a j k a j k − a j k a j k j (vj v k + v j vk ) − Ψ a j k (vj v + v j v) ⎪ ⎪ j k 2 ⎪ ⎪ + a (2A j + Ψj − 2α j t )|v| , ⎪ ⎪ m ⎪ ⎪ ⎪ 2 2 ⎪ j k ⎪ ⎪ αa j t k + αΨ t B α t t + β t t − (αA)t − 2 ⎪ ⎪ ⎪ ⎪ j,k=1 ⎪ ⎪ m ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ a j k Ψk j + 2 a j k j A k + AΨ . + ⎪ ⎩ j,k=1

j,k=1

Several remarks are in order. Remark 2.1. We see that only the symmetry condition of (a j k )m×m is assumed in the above. Therefore, Theorem 2.1 is applicable to ultra-hyperbolic or ultra-parabolic differential operators. jk Remark 2.2. Note that when Ψ = − m j,k=1 (a j )k and α(t, x) ≡ a, β(t, x) ≡ b with a, b ∈ R, Theorem 2.1 is reduced to [10, Theorem 1.1]. Here, we add an auxiliary function Ψ in the righthand side of the multiplier I1 so that the corresponding identity coincides with [12, Theorem 4.1] for the case of hyperbolic operators. Moreover, we will see that the modified identity is better than [10] in some sense.

1338


jk Remark 2.3. By choosing α(t, x) ≡ 1, β(t, x) ≡ 0 and Ψ = −2 m j,k=1 (a j )k in Theorem 2.1, one obtains a weighted identity for the parabolic operator. By this and following [23], one may recover all the controllability/observability results for the parabolic equations in [5] and [13]. Remark 2.4. By choosing a j k (t, x) ≡ a j k (x) and α(t, x) = β(t, x) ≡ 0 in Theorem 2.1, one obtains the identity derived in [12] for the controllability/observability results for the general hyperbolic equations. Due to the finite speed of propagation and its hyperbolic nature, to prove the observability result by using Carleman estimate, it is shown that triple (T , ω, Ω) should satisfy suitable geometric assumptions (say, the classical ones arising when applying multiplier methods [17]). Remark 2.5. By choosing (a j k )1j,km to be the identity matrix, α(t, x) ≡ 0, β(t, x) ≡ 1 and Ψ = − in Theorem 2.1, one obtains the point-wise identity derived in [14] for the observability results for the nonconservative Schrödinger equations. Also, this yields the controllability/observability results in [28] for the plate equations and the results for inverse problem for the Schrödinger equation in [2]. On the other hand, for the Schrödinger equation, the terms in the Carleman estimate that we can indeed bound depends strongly on the pseudoconvexity condition in question (weak, strong, etc.), and the size of the control region depends also of that pseudoconvexity region [19]. Remark 2.6. By choosing (a j k )1j,km to be the identity matrix, α(t, x) ≡ 0, β(t, x) ≡ p(x) and Ψ = − in Theorem 2.1, one obtains the point-wise identity for the Schrödinger operator: ip(x)∂t + . Further, by choosing

(t, x) = sϕ,

ϕ = eγ (|x−x0 |

2 −c|t−t |2 ) 0

with γ > 0, c > 0, x0 ∈ Rn \ Ω and assuming ∇ log p · (x − x0 ) > −2. One can recover the fundamental Carleman estimate for Schrödinger operator ip(x)∂t + derived in [25, Lemma 2.1]. Proof of Theorem 2.1. The proof is divided into several steps. Step 1. By θ = e , v = θ z, we have (recalling (2.4) for the definition of I1 ) θ Pz = (α + iβ)vt − (α + iβ) t v +

m jk a vj k j,k=1

+

m j,k=1

a j k j k v − 2

m

a j k j vk −

j,k=1

m jk a j k v j,k=1

= I1 + I 2 ,

(2.5)

where I2 αvt − iβ t v − 2

m j,k=1

a j k j vk + Ψ v.

(2.6)


1339

Hence, by recalling (2.3) for the definition of I1 , we have θ (PzI1 + PzI1 ) = 2|I1 |2 + (I1 I 2 + I2 I 1 ).

(2.7)

Step 2. Let us compute I1 I 2 + I2 I 1 . Denote the four terms in the right-hand side of I1 and I2 by j j I1 and I2 , respectively, j = 1, 2, 3, 4. Then I11 I21 + I22 + I11 I21 + I22 = − β 2 t |v|2 t + β 2 t t |v|2 .

(2.8)

Note that

2vv t = |v|2 t − (vvt − vv t ), 2vv k = |v|2 k − (vvk − vv k ).

Hence, we get I11 I23 + I24 + I11 I23 + I24 = −2i

m j k βa j vv k t − βa j k j t vv k j,k=1

m j k + 2i βa j vv t k − βa j k j k vv t − iβΨ (vv t − vt v) j,k=1 m

= −i

m jk jk βa j (vv k − vvk ) t + i βa j (vv t − vvt ) k

j,k=1

−i

m

j,k=1

m jk jk βa j t (vvk − vv k ) + i βΨ + βa j k (vvt − vv t ).

j,k=1

(2.9)

j,k=1

Next, I12 I 12 + I 22 + I 32 + I 42 + I 21 I21 + I22 + I23 + I24 m jk αa j t |v|2 k = − α 2 t |v|2 t + 2 j,k=1

+ α 2 t t |v|2 − 2

m jk αa j t k + αΨ t |v|2 .

j,k=1

Noting that a j k = a kj , we have

(2.10)

1340


I13 I21 + I22 + I13 I21 + I22 m m m jk jk αa (vj v t + v j vt ) k − αa vj v k t a j k αk (vj v t + v j vt ) −

=

j,k=1

+

j,k=1

m

jk αa t vj v k + i

j,k=1 m

=

m

j,k=1

m jk βa t (vj v − v j v) k + i a j k (β t )k (v j v − vj v)

j,k=1

j,k=1

jk αa (vj v t + v j vt ) + iβa j k t (vj v − v j v) k

j,k=1

−

m m jk αa vj v k t − a j k αk (vj v t + v j vt ) j,k=1

+

1 2

m

j,k=1 m jk αa t (vj v k + vk v j ) + i a j k (β t )k (v j v − vj v).

j,k=1

(2.11)

j,k=1

Using the symmetry condition of a j k again, we obtain m

2

a j k a j k j (vj v kk + v j vkk )

j,k,j ,k =1 m

=

j,k,j ,k =1

j k j k a a j (vj v k + v j vk ) k −

m

j,k,j ,k =1

a j k a j k j k (vj v k + v j vk ). (2.12)

By (2.12), we get I13 I23 + I13 I23 = −2

m j,k,j ,k =1

+

m j,k,j ,k =1

jk a j a j k (vj v k + v j vk ) k + 2

j k j k a a j (vj v k + v j vk ) k −

m j,k,j ,k =1 m

a j k a j k j k (vj v k + v j vk )

j k j k a a j k (vj v k + v j vk ).

j,k,j ,k =1

(2.13) Further, I13 I24 + I13 I24 =

m m jk Ψ a (vj v + v j v) k − a j k Ψ (vj v k + v j vk ) j,k=1

−

j,k=1

m m jk jk a Ψk |v|2 j + a Ψk j |v|2 . j,k=1

j,k=1

(2.14)


1341

Finally, I14 I 12 + I 22 + I 32 + I 42 + I 41 I21 + I22 + I23 + I24 m jk 2 2 = αA|v| t − (αA)t |v| − 2 a j A|v|2 k

+2

j,k=1

m

jk a j A k + AΨ |v|2 .

(2.15)

j,k=1

Step 3. By (2.8)–(2.15), combining all ‘ ∂t∂ -terms’, all ‘ ∂x∂ k -terms’, and (2.7), we arrive at the desired identity (2.2). 2 We have the following point-wise estimate for the complex parabolic operator Gz. Corollary 2.1. Let b ∈ R and a j k (t, x) ∈ C 1,2 (R1+n ; R) satisfy condition (1.1). Let z, v ∈ C 2 (R1+m ; C) and Ψ, ∈ C 2 (R1+m ; R). Set θ = e and v = θ z. Put n jk a j k . Ψ = −2

(2.16)

j,k=1

Then θ 2 |Gz|2 + Mt + div V

n j k j k j k 1 jk j k j k 2 j k 2 a j k a − ak a j + at + a a j k (vk v j + v k vj ) |I1 | + 2 j,k,j ,k =1

+ ib

n

n jk jk jk at j + 2a j t (v k v − vk v) − ib a j k (vvt − vv t ) + B|v|2 ,

j,k=1

(2.17)

j,k=1

where ⎧ n n 2 ⎪ 2 jk ⎪ ⎪ M = 1 + b

− A |v| + a v v + ib a j k j (v k v − vk v), t j k ⎪ ⎪ ⎪ ⎪ j,k=1 j,k=1 ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ −ib a j k j (v t v − vvt ) + a j k t (vj v − v j v) − a j k (vj v t + v j vt ) Vk = ⎪ ⎪ ⎪ ⎪ ⎪ j,j ,k =1 ⎪ ⎪ ⎪ ⎨ + 2a j k a j k − a j k a j k j (vj v k + v j vk ) − Ψ a j k (vj v + v j v) (2.18) ⎪ + a j k (2A j + Ψj − 2 j t )|v|2 , ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ ⎪ 2 jk ⎪ a j t k + Ψ t B = 1 + b tt − At − 2 ⎪ ⎪ ⎪ ⎪ j,k=1 ⎪ ⎪ n ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ a j k Ψk j + 2 a j k j A k + AΨ . + ⎪ ⎩ j,k=1

j,k=1

1342


Proof. Recalling (1.6) for the definition of Gz, taking m = n, α(x) ≡ 1, β(x) ≡ b in Theorem 2.1, by using Hölder inequality and simple computation, we immediately obtain the desired result. 2 3. A modified point-wise estimate Note that the term ib nj,k=1 (a j k j )k (vvt − vv t ) in the right-hand side of (2.17) is not good. In this section, we make some modification on this term and derive the following modified pointwise inequality for the parabolic operator with a complex principal part. Theorem 3.1. Let b ∈ R and a j k (t, x) ∈ C 1,2 (R1+n ; R) satisfy (1.1)–(1.2). Let z, v ∈ C 2 (R1+m ; C) and Ψ, ∈ C 2 (R1+m ; R) satisfy (2.16). Put θ = e ,

v = θ z.

(3.1)

Then 2θ 2 |Gz|2 + Mt + div V˜ |I1 |2 +

n

˜ 2 cj k (vk v j + v k vj ) + B|v|

j,k=1

+ ib

n

jk at j

j,k=1

n j k 1 j k a j k j a + 2a j t + (v k v − vk v), 1 + b2 jk

(3.2)

j ,k =1

where M, V k , B is given by (2.18) and ⎧ n n j k j k ⎪ ib ⎪ k k ˜ ⎪ V a j k a (v j v − vj v) = V + ⎪ 2 ⎪ 1+b ⎪ ⎪ j,j ,k =1 j,k,j ,k =1 ⎪ ⎪ ⎪ ⎪ 2 ⎪ jk b b2 2 j k j k 2 ⎪ 2 j k ⎪ ⎪ − a θ a j k j a |z| θ

a (z z + z z) + j j j ⎪ k ⎪ 1 + b2 1 + b2 ⎪ ⎪ ⎪ ⎪ ⎪ 2b2 j k j k ⎪ ⎪ a j k a j |v|2 , − ⎪ ⎪ 2 ⎪ 1 + b ⎪ ⎨

n j k j k j k 1 jk 1 j k j k j k j k ⎪ ⎪c = a a j k , 2 a j k a − ak a j + at + ⎪ ⎪ 2 2(1 + b2 ) ⎪ ⎪ j ,k =1 ⎪ ⎪ ⎪

n

2 ⎪ n ⎪ ⎪ j k j k b2

j k

2b2 ⎪ ˜ ⎪ ⎪ a j k a k j − a j k

B =B−

⎪ ⎪

1 + b2 1 + b2

⎪ ⎪ j,k=1 j,k,j ,k =1 ⎪ ⎪ ⎪ ⎪ n ⎪ j k j k j k j k ⎪ b2 ⎪ ⎪ + a 2a a a j k j k . +

k j ⎪ k j ⎩ 1 + b2 j,k,j ,k =1

(3.3)


1343

Proof. We divided the proof into several steps. Step 1. Note that v = θ z, it is easy to check that vvt − vv t = θ 2 (zzt − zzt ),

vvk − vv k = θ 2 (zzk − zzk ).

(3.4)

Recalling (1.6) for the definition of Gz and by (3.4), we have −ib

n n jk jk 2 a j k (vvt − vv t ) = −ib a j k θ (zzt − zzt ) j,k=1

j,k=1

=−

n ibθ 2 j k a j k (1 − ib)Gzz − (1 − ib)Gzz 2 1+b j,k=1

−

n n ibθ 2 j k j k a a zj k z − a j k zj k z

j k 1 + b2 j ,k =1

+

≡

b2 θ 2 1 + b2

3

n

j ,k =1

j,k=1

a j k j

k

n j k a zj k z + a j k zj k z j,k=1

(3.5)

Jk .

k=1

Step 2. Let us estimate Jk (k = 1, 2, 3) respectively. First, note that v = θ z, we have J1 = −

n ibθ 2 j k a j k (1 − ib)Gzz − (1 − ib)Gzz 1 + b2 j,k=1

2

n

(1 − ib)θ Gz 2

ibθ z j k

− √ a j k

−

√

1 + b2 1 + b2 j,k=1

n

2 b2

j k

a j k |v|2 . = −θ |Gz| −

1 + b2

2

2

j,k=1

Next, by using (3.4) again, we have J2 = −

n n ibθ 2 j k j k a a zj k z − a j k zj k z

j k 2 1+b j ,k =1

=−

ib 1 + b2

n

j,k=1

2 j k j k θ a j k a (zj z − zj z) k

j,k,j ,k =1

(3.6)

1344


+

=−

ib 1 + b2

j,k,j ,k =1 n

ib 1 + b2

+

n

j,k,j ,k =1

ib 1 + b2

θ 2 2 k a j k j k + a j k j k k a j k (zj z − zj z)

j k j k a j k a (v j v − vj v) k

n j,k,j ,k =1

j k 2 k a j k + a j k j k k a j k (v j v − vj v).

(3.7)

Next, noting that a j k satisfy (1.1), and recalling v = θ z, it follows

J3 =

n n b2 θ 2 j k j k a a zj k z + a j k zj k z

j k 2 1+b j ,k =1

=

b2 1 + b2 −

+

=

+

−

n j,k,j ,k =1

b2 θ 2 1 + b2 b2 1 + b2

b2 1 + b2

j,k,j ,k =1 n j,k,j ,k =1 n

j,k,j ,k =1

2b2 1 + b2 b2 1 + b2

2 j k j k θ a j k a (zj z + zj z) − θ 2 a j k j k j a j k |z|2 k

n

j,k,j ,k =1 n j,k,j ,k =1

n j,k,j ,k =1

j k j k a j k a (zj zk + zj zk ) 2 j k j k 2 θ a j k j a k |z|

2 j k j k θ a j k a (zj z + zj z) − θ 2 a j k j k j a j k |z|2 k

n

b2 2 + 1 + b2 +

j,k=1

j k j k a j k a k |v|2 j j k j k a j k a (v j vk + vj v k )

n j,k,j ,k =1

a j k j k a j k j k + 2

j,k,j ,k =1

a j k k a j k j k j

j k j k a a j k j k |v|2

where we have used the following fact:

n

(3.8)

X. Fu / Journal of Functional Analysis 257 (2009) 1333–1354 n

θ2

a j k j

j,k,j ,k =1 n

=

k

a j k (zj zk + zj zk )

j k j k a j k a (v j vk + vj v k ) + 2

j,k,j ,k =1

n

−2

1345

j k j k a j k a k |v|2 j + 2

j,k,j ,k =1

n

j k j k a j k a j k |v|2

j,k,j ,k =1 n

j,k,j ,k =1

j k j k a j k a k j |v|2

(3.9)

and n

2 j k j k 2 θ a j k j a k |z|

j,k,j ,k =1

n

=

j,k,j ,k =1

= 4

2 j k j k θ 2a j a j k + a j k a j k j k j k |z|2

n j,k,j ,k =1

a j k j k a j k j k + 4

2

n

a j k j k + + 2

j,k=1

n

n j,k,j ,k =1

a j k k a j k j k j

j k j k a a j k j k |v|2 .

(3.10)

j,k,j ,k =1

Step 3. Noting that a j k satisfy (1.1)–(1.2), we have the following fact. n n b 1 jk a (v j vk + vj v k ) + 2i a j k k (v j v − vj v) 2(1 + b2 ) 1 + b2 j,k=1

= =

1 1 + b2 1 1 + b2

−

n j,k=1 n

j,k=1

n 4b2 j k a j k vj v k + 2ib k (v j v − vj v) + 4b2 j k |v|2 − a j k |v|2 1 + b2 j,k=1

a j k (vj + 2ib j v)(vk + 2ib k v) −

j,k=1

4b2 1 + b2

n

4b2 1 + b2

n

a j k j k |v|2

j,k=1

a j k j k |v|2 .

(3.11)

j,k=1

Finally, combining (2.17), (3.4)–(3.8) and (3.11), we arrive at the desired result (3.2).

2

4. Global Carleman estimate for parabolic operators with complex principal part We begin with the following known result. Lemma 4.1. (See [13,24].) There is a real function ψ ∈ C 2 (Ω) such that ψ > 0 in Ω and ψ = 0 on ∂Ω and |∇ψ(x)| > 0 for all x ∈ Ω \ ω0 .

1346


For any (large) parameters λ > 1 and μ > 1, put θ = e ,

= λρ,

ϕ(t, x) =

eμψ(x) , t (T − t)

ρ(t, x) =

eμψ(x) − e2μ|ψ|C(Ω;R) . t (T − t)

(4.1)

For j, k = 1, . . . , n, it is easy to check that

t = λρt ,

j = λμϕψj ,

j k = λμ2 ϕψj ψk + λμϕψj k

(4.2)

and |ρt | Ce2μ|ψ|C(Ω) ϕ 2 ,

|ϕt | Cϕ 2 .

(4.3)

In the sequel, for k ∈ N, we denote by O(μk ) a function of order μk for large μ (which is independent of λ); by Oμ (λk ) a function of order λk for fixed μ and for large λ. We have the following Carleman estimate for the differential operator G defined in (1.6): Theorem 4.1. Let b ∈ R and a j k satisfy (1.1)–(1.2). Then there is a μ0 > 0 such that for all μ μ0 , one can find two constants C = C(μ) > 0 and λ1 = λ1 (μ), such that for all z ∈ C([0, T ]; L2 (Ω)) ∩ C((0, T ]; H01 (Ω)) and for all λ λ1 , it holds

(λϕ)−1 θ 2 |zt |2 + |z|2 dx dt + λ3 μ4

Q

C 1+b

2

ϕ 3 θ 2 |z|2 dt dx + λμ2

Q

ϕ 3 θ 2 |z|2 dt dx .

θ 2 |Gz|2 dt dx + λ3 μ4

Q

ϕθ 2 |∇z|2 dt dx Q

(4.4)

(0,T )×ω

Proof. The proof is long, we divided it into several steps. Step 1. Recalling (3.3) for the definition of cj k , by (4.2)–(4.3), we have n

cj k (vk v j + v k vj )

j,k=1

=

n j,k,j ,k =1

jk 2 a j k j k a j k − ak a j k j

1 jk 1 j k j k a a + at +

j k (vk v j + v k vj ) 2 2(1 + b2 )

2

n n

1

2

jk 2 λμ a ψj v k + ϕ a j k a j k ψj ψk (vk v j + v k vj ) = 4λμ ϕ

2

2(1 + b ) j,k=1

+ λϕO(μ)|∇v|2 .

j,k,j ,k =1

(4.5)


1347

On the other hand, by (4.2)–(4.3), recalling (2.16) and (2.3) for the definitions of Ψ and A, it is easy to check that n

Ψ = −2λμ ϕ 2

a j k ψj ψk + λϕO(μ),

j,k=1

A = λ2 μ 2 ϕ 2

n

a j k ψj ψk + ϕ 2 Oμ (λ).

(4.6)

j,k=1

Next, recall (2.3) and (2.18) for the definition of A and B, respectively. By (4.2)–(4.3) and combining (4.6), we have n

B =2

a j k j Ak − 2A

j,k=1

n jk a j k j,k=1

n n n jk jk 2 jk a j k + a Ψk j + 1 + b tt − At − 2 a j tk + 2 t j,k=1

j,k=1

n

2

= 2λ3 μ4 ϕ 3

a j k ψj ψk + λ3 ϕ 3 O μ3 + ϕ 3 Oμ λ2 .

j,k=1

(4.7)

j,k=1

˜ we have Hence, by recalling (3.3) for the definition of B,

n

2

2

3 4 3 jk λ μ ϕ a ψ ψ B˜ =

+ λ3 ϕ 3 O μ3 + ϕ 3 Oμ λ2 . j k 2

1+b

(4.8)

j,k=1

Step 2. By (4.2)–(4.3), we have

n n

jk

at j + 2a j k j t (v k v − vk v) Cλμ ϕ 2 a j k ψj (v k v − vk v)

ib

j,k=1

j,k=1

n

2

jk Cλμϕ

a ψj v k + Cλμϕ 3 |v|2 .

j,k=1

It is easy to see that (4.9) can be absorbed by (4.5) and (4.8). Similarly, by using (4.2)–(4.3) again, we have

ib

1 + b2

n

j k

j k a j k j a (v k v − vk v)

j,k,j ,k =1

n

1

3 2 jk 2 2 jk ψk + λϕ O μ b λμ ϕ a ψ ψ (v v − v v) a

j k k j

1 + b2

j ,k =1

(4.9)

1348


n

1

3 2 jk jk ψk a ψj (v k v − vk v)

bλμ ϕ a ψ

j 2

1+b j ,k =1

n

C

2 2 jk + bλμ ϕ a ψ (v v − v v)

j k k

1 + b2

j,k=1

2

n

2

n

1 b2

2

jk 4 3

jk λμ ϕ a ψ v + λμ ϕ a ψ ψ

|v|2

j k j k

1 + b2 1 + b2 j,k=1

j,k=1

2

n

C Cb2

jk + λμϕ a ψ v + λμ3 ϕ 3 |v|2 .

j k

1 + b2 1 + b2

(4.10)

j,k=1

Therefore (4.10) also can be absorbed by (4.5) and (4.8). Combining (4.5), (4.8)–(4.10), by (1.2), we end up with The right-hand side of (3.2)

n n 1 2 j k λμ ϕ a ψ ψ a j k vk v j + λϕO(μ)|∇v|2 j k 2(1 + b2 ) j ,k =1

j,k=1

n

2

2 3 2

3 4 3

jk 3 3 3 + λ μ ϕ

a ψj ψk + λ ϕ O μ + ϕ Oμ λ |v|2

1 + b2

j,k=1

2 2 1 s0 λμ ϕ|∇ψ|2 + λϕO(μ) |∇v|2 2 2(1 + b ) +

2 2 3 4 3 s0 λ μ ϕ |∇ψ|4 + λ3 ϕ 3 O μ3 + ϕ 3 Oμ λ2 |v|2 . 2 1+b

(4.11)

Step 3. Integrating (3.2) on Q, by (4.11), noting that θ (0) = θ (T ) ≡ 0, we have

|I1 |2 dx dt +

Q

+

ϕ λμ2 |∇ψ|2 + λO(μ) |∇v|2 dx dt

Q

ϕ 3 λ3 μ4 |∇ψ|4 + λ3 O μ3 + Oμ λ2 |v|2 dx dt

Q

C

|θ Gz|2L2 (Q)

+

˜ div V · ν dx dt .

(4.12)

Q

Next, recall (2.4) and (3.3) for the definition of V and V˜ . Noting that z|Σ = 0 and vi = (which follows from (1.5) and v|Σ = 0, respectively), by (4.2) and Lemma 4.1, we have

∂v ∂ν νi


div V˜ · ν dx dt =

Q

1349

div V · ν dx dt Q

2

n ∂ψ

∂v

2 j k ϕ a νj νk dt dx 0. ∂ν ∂ν

= λμ

(4.13)

j,k=1

Σ

On the other hand, T

+

Left side of (4.12) = 0

Ω\ω0

2 ϕ λμ |∇ψ|2 + λO(μ) |∇v|2

ω0

+ ϕ 3 λ3 μ4 |∇ψ|4 + λ3 O μ3 + ϕ 3 Oμ λ2 |v|2 dt dx

T

2 ϕ λμ |∇ψ|2 + λO(μ) |∇v|2

0 Ω\ω0

+ ϕ 3 λ3 μ4 |∇ψ|4 + λ3 O μ3 + ϕ 3 Oμ λ2 |v|2 dt dx 2 − Cλμ ϕ |∇v|2 + λ2 μ2 ϕ 2 |v|2 dt dx.

(4.14)

Q0

By the choice of ψ , we know that minx∈Ω\ω0 |∇ψ| > 0. Hence, there is a μ0 > 0 such that for all μ μ0 , one can find a constant λ1 = λ1 (μ) so that for any λ λ1 , it holds T

ϕ λμ2 |∇ψ|2 + λO(μ) |∇v|2 dt dx

0 Ω\ω0

+

ϕ 3 λ3 μ4 |∇ψ|4 + λ3 O μ3 + ϕ 3 Oμ λ2 |v|2 dt dx

Q

T c0 λμ

2

ϕ |∇v|2 + λ2 μ2 ϕ 2 |v|2 dt dx,

(4.15)

0 Ω\ω0

where c0 = min(minx∈Ω\ω0 |∇ψ|2 , minx∈Ω\ω0 |∇ψ|4 ) is a positive constant. Next, noting that v = θ z, by (4.2), we have zj = θ −1 (vj − j v) = θ −1 (vj − λμϕψj v), vj = θ (zj + j z) = θ (zj + λμϕψj z).

(4.16)

By (4.16), we get 1 2 θ |∇z|2 + λ2 μ2 ϕ 2 |z|2 |∇v|2 + λ2 μ2 ϕ 2 |v|2 Cθ 2 |∇z|2 + λ2 μ2 ϕ 2 |z|2 . C

(4.17)

1350


Therefore, it follows from (4.14)–(4.15) and (4.17) that λμ

ϕθ 2 |∇z|2 + λ2 μ2 ϕ 2 |z|2 dt dx

2 Q

T = λμ

+

2 Ω\ω0

0

C

ϕθ 2 |∇z|2 + λ2 μ2 ϕ 2 |z|2 dt dx

ω0

ϕ λμ2 |∇ψ|2 + λO(μ) |∇v|2 dt dx

Q

+

ϕ 3 λ3 μ4 |∇ψ|4 + λ3 O μ3 + ϕ 3 Oμ λ2 |v|2 dt dx

Q

+ λμ2

ϕθ 2 |∇z|2 + λ2 μ2 ϕ 2 |z|2 dt dx .

(4.18)

Q0

Now, combining (4.12)–(4.13) and (4.18), we end up with

|I1 |2 dx dt + λμ2

Q

ϕθ 2 |∇z|2 + λ2 μ2 ϕ 2 |z|2 dt dx

Q

C

θ |Gz| dt dx + λμ 2

2

Q

2

2 2 2 2 2 ϕθ |∇z| + λ μ ϕ |z| dt dx . 2

(4.19)

Q0

Step 4. We choose a cut-off function ζ ∈ C0∞ (ω; [0, 1]) so that ζ ≡ 1 on ω0 . Then

2

2

2 2

ζ ϕθ (1 + ib)z t = ζ 2 (1 + ib)z ϕθ 2 t + ζ 2 ϕθ 2 (1 + ib)(1 − ib)(zzt + zzt ). By (1.6), (4.20) and noting θ (0, x) = θ (T , x) ≡ 0, we find 0=

2

ζ 2 (1 + ib)z ϕθ 2 t + ϕθ 2 (1 + ib)(1 − ib)(zzt + zzt ) dt dx

Q0

=

ζ 2θ 2

Q0

n

2

jk

(1 + ib)z (ϕt + 2λϕρt ) + ϕ(1 − ib)z − a zj k + Gz j,k=1

+ ϕ(1 + ib)z − = Q0

n jk a zj k + Gz j,k=1

2

2 θ ζ 2 (1 + ib)z (ϕt + 2λϕρt )

(4.20)

X. Fu / Journal of Functional Analysis 257 (2009) 1333–1354 n

+ ζ 2ϕ

1351

a j k (1 − ib)zj zk + (1 + ib)zj zk

j,k=1 n

+ μζ 2 ϕ

a j k (1 − ib)zzj ψk + (1 + ib)zzj ψk

j,k=1 n

+ 2λμζ 2 ϕ 2


j,k=1 n

+ 2ζ ϕ

a j k (1 − ib)zzj ζk + (1 + ib)zzj ζk

j,k=1

+ ζ 2 ϕ (1 − ib)zGz + (1 + ib)zGz dt dx.

(4.21)

Hence, by (4.21) and (1.2), we conclude that, for some δ > 0, 2 ζ 2 ϕθ 2 |∇z|2 dx dt Q0

2

2

= θ ζ 2 (1 + ib)z (ϕt + 2λϕρt ) Q0

+ μζ 2 ϕ

j,k

+ 2λμζ 2 ϕ 2 + 2ζ ϕ



j,k

a j k (1 − ib)zzj ζk + (1 + ib)zzj ζk

j,k

+ ζ ϕ (1 − ib)zGz + (1 + ib)zGz dt dx

2

δ

ζ 2 ϕθ 2 |∇z|2 dt dx

Q0

C (1 + b2 ) 2 2 2 2 3 2 2 + θ |Gz| dt dx + λ μ ϕ θ |z| dt dx . δ λ2 μ2 Q0

(4.22)

Q0

Now, we choose δ = 1. By (4.22), we conclude that

1 2 2 2 2 2 2 2 3 2 2 ϕθ |∇z| dt dx C 1 + b θ |Gz| dt dx + λ μ ϕ θ |z| dt dx . (4.23) λ2 μ2 Q0

Combining (4.19) and (4.23), we arrive at

Q0

Q0

1352


|I1 | dx dt + λ μ 2

ϕ θ |z| dt dx + λμ

3 4

Q

C 1+b

2

3 2

2

Q

ϕθ 2 |∇z|2 dt dx

2

Q

θ |Gz| dt dx + λ μ 2

2

3 4

Q

ϕ θ |z| dt dx . 3 2

2

(4.24)

(0,T )×ω

Step 5. Finally, let us estimate “ Q (λϕ)−1 θ 2 (|zt |2 + |z|2 ) dx dt”. Noting that v = θ z and the definition of Gz, we have ⎧ n ⎪ jk ⎪ ⎪ θ Gz = θ (1 + ib)zt + a zj k = I1 + I2 , ⎪ ⎪ ⎪ ⎪ j,k=1 ⎪ ⎪ ⎪ ⎪ n ⎪ jk ⎪ ⎪ ⎪ a vj k + Av, = ibv −

v + I ⎪ 1 t t ⎪ ⎨ j,k=1

n ⎪ ⎪ ⎪ ⎪ a j k j vk + Ψ v, I2 = vt − ib t v − 2 ⎪ ⎪ ⎪ ⎪ j,k=1 ⎪ ⎪ ⎪ n n n ⎪ ⎪ jk ⎪ jk jk ⎪ ⎪ a a j k .

, A = a

+ Ψ = −2 j k j k ⎪ ⎩ j,k=1

j,k=1

(4.25)

j,k=1

Then, by (4.25), (4.2)–(4.3) and (4.6), we have

(λϕ)−1 θ 2 |zt |2 + |z|2 dx dt

Q

C

(λϕ)−1 θ 2

2

n

a j k zj k dx dt |zt |2 +

j,k=1

Q

=C

(λϕ)−1 |vt − t v|2 dx dt

Q

+C C

n

2 n n n

j k

jk jk jk a vj k − 2 a j k v dx dt a j vk + a j k v −

−1

(λϕ)

j,k=1

Q

C

−1

(λϕ) Q

j,k=1

j,k=1

(λϕ)−1 |I1 |2 + |I2 |2 + (λμϕ)4 |v|2 + (λμϕ)2 |∇v|2 dx dt

Q

j,k=1

2|I1 |2 + |Gz|2 +

3 4 3 2 λ μ ϕ |v| + λμ2 ϕ|∇v|2 dx dt.

(4.26)

Q

Finally, combining (4.24) and (4.26), we get the proof of Theorem 4.1.

2


1353

5. Proof of the main results In this section, we will give the proofs of Theorems 1.1 and 1.2. Thanks to the classical dual argument and the fixed point technique, proceeding as in [13,21], Theorem 1.2 is a consequence of the observability result (1.7). Therefore, we only give here a brief proof of Theorem 1.1. Proof of Theorem 1.1. We apply Theorem 4.1 to system (1.5). Recalling that q(·) ∈ L∞ (0, T ; L∞ (Ω)) and using (4.2), we obtain λ3 μ4 ϕ 3 θ 2 |z|2 dt dx + λμ2 ϕθ 2 |∇z|2 dt dx Q

C 1+b

2

Q

θ |qz| dt dx + λ μ 2

2

ϕ θ |z| dt dx

3 4

Q

3 2

(0,T )×ω

2 2 C 1 + b 1 + |r| θ 2 |z|2 dt dx + λ3 μ4 Q

2

ϕ θ |z| dt dx . 3 2

2

(5.1)

(0,T )×ω

Choosing λ C(1 + b2 )(1 + |r|2 ) and μ large enough, from (5.1), one deduces that

ϕ 3 θ 2 |z|2 dt dx C

Q

ϕ 3 θ 2 |z|2 dt dx.

(5.2)

(0,T )×ω

Finally, by (5.2) and applying the usual energy estimate to system (1.5), we conclude that inequality (1.7) holds, with the observability constant C given by (1.8), which completes the proof of Theorem 1.1. 2 References [1] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992) 1024–1065. [2] L. Baudouin, J.P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems 18 (2002) 1537–1554. [3] N. Burq, Décroissance de l’énergie locale de l’équation des ondes pour le probléme extérieur et absence de résonance au voisinagage du réel, Acta Math. 180 (1998) 1–29. [4] J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr., vol. 136, American Mathematical Society, Providence, RI, 2007. [5] A. Doubova, E. Fernández-Cara, M. Gonzälez-Burgos, E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim. 41 (2002) 798–819. [6] T. Duyckaerts, X. Zhang, E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 1–41. [7] C. Fabre, J.P. Puel, E. Zuazua, Approximate controllability of the semilinear heat equations, Proc. Roy. Soc. Edinburgh 125 (1995) 31–61. [8] E. Fernández-Cara, E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations 5 (2000) 465–514. [9] E. Fernández-Cara, E. Zuazua, Null and approximate controllability for weakly bolwing up heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (5) (2000) 583–616. [10] X. Fu, A weighted identity for partial differential operators of second order and its applications, C. R. Math. Acad. Sci. Paris 342 (2006) 579–584.

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[11] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping, preprint. [12] X. Fu, J. Yong, X. Zhang, Exact controllability for the multidimensional semilinear hyperbolic equations, SIAM J. Control Optim. 46 (2007) 1578–1614. [13] A.V. Fursikov, O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Research Institute of Mathematics, Seoul National University, Seoul, Korea, 1996. [14] I. Lasiecka, R. Triggiani, X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via point-wise Carleman estimates, Part I: H 1 (Ω)-estimates, J. Inverse Ill-Posed Probl. 12 (2004) 43–123. [15] X. Li, J. Yong, Optimal control theory for infinite-dimensional systems, in: Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. [16] W. Li, X. Zhang, Controllability of parabolic and hyperbolic equations: Towards a unified theory, in: Control Theory of Partial Differential Equations, in: Lect. Notes Pure Appl. Math., vol. 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, pp. 157–174. [17] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1, Contrôlabilité exacte, Rech. Math. Appl., vol. 8, Masson, Paris, 1988. [18] A. López, X. Zhang, E. Zuazua, Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations, J. Math. Pures Appl. 79 (2000) 741–808. [19] A. Mercado, A. Osses, L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems 24 (2008). [20] K. Phung, X. Zhang, Time reversal focusing of the initial state for Kirchoff plate, SIAM J. Appl. Math. 68 (2008) 1535–1556. [21] L. Rosier, B. Zhang, Null controllability of the complex Ginzburg–Landau equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 1535–1556. [22] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. Appl. Math. 52 (1973) 189–221. [23] S. Tang, X. Zhang, Carleman inequality for the backward stochastic parabolic equations with general coefficients, C. R. Acad. Sci. Paris Sér. I 339 (2004) 775–780. [24] G. Wang, L. Wang, The Carleman inequality and its application to periodic optimal control governed by semilinear parabolic differential equations, J. Optim. Theory Appl. 118 (2003) 249–461. [25] G. Yuan, M. Yamamoto, Lipschitz stability in inverse problems for a Kirchhoff plate equation, Asymptot. Anal. 53 (2007) 29–60. [26] X. Zhang, Explicit observability estimate for the wave equation with potential and its application, Proc. R. Soc. Lond. Ser. A 456 (2000) 1101–1115. [27] X. Zhang, A remark on null exact controllability of the heat equation, SIAM J. Control Optim. 40 (2001) 39–53. [28] X. Zhang, Exact controllability of the semilinear plate equations, Asymptot. Anal. 27 (2001) 95–125. [29] E. Zuazua, Approximate controllability for semilinear heat equations with global Lipschitz nonlinearities, Control Cybernet. 28 (1999) 665–683. [30] E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in: Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3, Elsevier Science, 2006, pp. 527– 621.


Universal Lp improving for averages along polynomial curves in low dimensions Spyridon Dendrinos a , Norberto Laghi b,1 , James Wright b,∗,1 a Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom b Maxwell Institute for Mathematical Sciences, The University of Edinburgh, JCMB, King’s Buildings,

Edinburgh EH9 3JZ, United Kingdom Received 28 May 2008; accepted 8 May 2009 Available online 9 June 2009 Communicated by C. Kenig

Abstract We prove sharp Lp → Lq estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree. Our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to M. Christ. Almost sharp Lorentz space estimates are obtained as well. © 2009 Elsevier Inc. All rights reserved. Keywords: Averaging operators; Polynomial curves; Universal bounds

1. Introduction and statement of results Recently there has been considerable attention given to certain euclidean harmonic analysis problems associated to a curve or surface where the underlying euclidean arclength or surface measure (which typically defines the classical problem) is replaced by the so-called affine arclength or surface measure. This has the effect of making the problem affine invariant as well as invariant under reparameterisations of the underlying variety. For this reason there have been * Corresponding author.

E-mail addresses: [email protected] (S. Dendrinos), [email protected] (N. Laghi), [email protected] (J. Wright). 1 Supported in part by an EPSRC grant. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.011

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S. Dendrinos et al. / Journal of Functional Analysis 257 (2009) 1355–1378

many attempts to obtain universal results, establishing uniform bounds over a large class of curves or surfaces. The affine arclength or surface measure also has the mitigating effect of dampening any curvature degeneracies of the curve or surface and therefore the expectation is that the universal bounds one seeks will be the same as those arising from the most nondegenerate situation. This line of research has been actively pursued for the problem of Fourier restriction, a central problem in euclidean harmonic analysis; see for example [1,2,5,6,14–17,21,23,28]. Drury initiated an investigation along these lines for the problem of achieving precise regularity results for averages along curves or surfaces, in particular determining sharp Lp → Lq estimates, and this has been followed up by several authors; see for example [7,8,15,19,20,22,24–27]. In this paper we continue an investigation by Oberlin to establish such a result for averaging operators along general polynomial curves in Rd when d = 2 or d = 3 (in [22], the d = 2 case was fully resolved and partially resolved for d = 3). More specifically, if γ : I → Rd parametrises a smooth curve in Rd on an interval I , set Lγ (t) = det γ (t) · · · γ (d) (t) ; this is the determinant of a d × d matrix whose j th column is given by the j th derivative of γ , γ (j ) (t). The affine arclength measure ν = νγ on γ is defined on a test function φ by ν(φ) =

2 φ γ (t) Lγ (t) d(d+1) dt;

I

one easily checks that this measure is invariant under reparameterisations of γ . A basic problem in the theory of averaging operators along curves (or more generally, for generalised Radon transforms) is to determine the exponents p and q so that the a priori estimate Tf Lq (Rd ) Cf Lp (Rd )

(1)

holds uniformly for a large class of curves γ where Tf (x) = f ∗ ν(x) =

2 f x − γ (t) Lγ (t) d(d+1) dt.

I

The use of the affine arclength measure allows us to think about global estimates, not only establishing (1) with a constant C uniform over a large class of curves but also possibly obtaining such a constant independent of the parametrising interval I . As discussed above, the exponents p and q in (1) that we expect, should come from the most non-degenerate situation which in this case is the curve γ (t) = (t, . . . , t d ) in Rd where Lγ ≡ constant. A simple scaling argument shows that necessarily we must have 1/q = 1/p −2/d(d +1) if (1) is to hold as a global estimate. Furthermore by testing (1) on f = χBδ where Bδ is the ball of radius δ with centre 0, we obtain the added necessary condition (d 2 + d)/(d 2 − d + 2) p (d + 1)/2. It is a remarkable result of Christ [9] that (up to the endpoints) these restrictions on p and q are in fact sufficient for (1) to hold in this non-degenerate situation. Stovall [30], building on an argument of Christ [10], has converted Christ’s restricted weak-type estimates at the endpoints into strong type estimates.


1357

To date, progress that has been made to establish universal bounds in (1) for curves γ where Lγ ≡ constant has not been as substantial as for the corresponding problem of Fourier restriction. The case for curves γ (t) = (t, φ(t)) given as the graph of a convex function φ has been considered by Choi, Drury, Oberlin and Pan and the best result here is due to Oberlin [20] where the additional hypothesis that φ is monotone increasing is imposed and then only a weak-type estimate is obtained at the endpoint (2/3, 1/3) (in [7] Choi obtained strong type estimates at (2/3, 1/3) but these estimates are not universal – the constant C in (1) depends on φ – and in fact the author needs to impose much more stringent conditions on φ). Compare this with the situation for the corresponding Fourier restriction problem in two dimensions where Sjölin [29] obtained uniform bounds over the class of all convex curves – see also [21]. The class of convex curves is a natural class to examine in light of simple counterexamples to (1) where Lγ changes sign too often (of course if γ is convex, Lγ does not change sign). By the above discussion on necessary conditions, we see that the endpoint estimate to aim for in (1) is (2/3, 1/3) in two dimensions. Consider the curve γ given by γ (t) = (t, t k sin(1/t)). By testing (1) on f = χDδ where Dδ = {(x, y): |x| δ, |y| δ k } one easily shows that if (1) were to hold for this example, then 1/q 1/p − (k − 1)/3(k + 1). Therefore if Lγ changes sign too often then (1) may not hold uniformly for all curves in the expected Lp range. In [22] Oberlin established (1) in two dimensions for the family of polynomial curves γ (t) = P(t) = (P1 (t), P2 (t)) where each P1 and P2 is a general real polynomial of bounded degree. Specifically he established (1) with a constant C only depending on the degrees of the polynomials defining P. This is a natural class of curves to consider as the number of sign changes of LP is controlled by the degree of the polynomials Pj . Furthermore Oberlin established (1) in three dimensions for polynomial curves of the form P(t) = (t, P2 (t), P3 (t)) but the estimates are not universal in the sense that the constant C can be taken to depend only on the degrees of the polynomials. For the corresponding Fourier restriction problem in the setting of polynomial curves, see [2] and [14]. In this paper we give an alternative approach to the results in [22] and strengthen the threedimensional result to general polynomial curves P(t) = (P1 (t), P2 (t), P3 (t)); furthermore all estimates will be uniform over the class of polynomials of bounded degree. Our hope is that this approach will generalise to general polynomials curves in all dimensions. From now on we shall focus on the operator 2 (2) Af (x) = f x − P(t) LP (t) d(d+1) dt. I

We are now ready to state our main result which is a global estimate. Theorem 1. Let d = 2, 3. Then for every > 0, Af

d 2 +d , d+1 + 2 (Rd )

L 2d−2

Cf

L

d+1 2 (Rd )

and Af

d+1 , d 2 +d + d 2 −d+2 (Rd )

L d−1

Cf

d 2 +d

,

L d 2 −d+2 (Rd )

where the constant C depends only on > 0, the degrees of the polynomials defining the curve P and in particular not on the parametrising interval I .

1358


When d = 2 there is just a single endpoint and the above two estimates agree. Here Lp,r (Rd ) denote the familiar Lorentz spaces. As discussed above, it follows from Christ [9] that the p, q exponents in the estimates A : Lp,r → Lq,s in Theorem 1 are best possible. Also, up to the factor, the r, s exponents are also best possible in general. In fact by considering the local operator Aloc (defined by restricting the integration to the unit interval) in the non-degenerate case P(t) = (t, t 2 , . . . , t d ) an Lp,r to Lq,s estimate would give rise to an Lp,r → Lp,s estimate for Aloc and a result of A. Blozinski [4] states that there are no nontrivial positive2 bounded linear translation invariant operators from Lp,r to Lp,s whenever s < r.3 Since C can be taken to be independent of I and A is a positive operator, Theorem 1 is equivalent to establishing the concluding estimates for the global analogue of A where the integration in (2) is replaced by the entire real line. The proof of Theorem 1 combines an elegant combinatorial argument of Christ in [9], together with a recent geometric inequality for vector polynomials which was established in [14]. Christ’s method is elementary but powerful and has seen applications outside the model curve case (t, . . . , t d ) (see [3,11,18]) as well as substantial generalisations (see [12] and [31]). We mention again that Christ has developed a method that may be used to deduce strong-type estimates (even Lorentz type estimates) from restricted weak-type estimates (see [10]) and we will follow this method to deduce the Lorentz bounds in Theorem 1. Finally, we wish to emphasise the fact that the result of Theorem 1 is obtained by using slightly different ingredients in different dimensions; whilst the basic techniques employed do not change, the relevant arguments need to be suitably adjusted. This is reflected in the structure of the paper: in the next section we recall the rudiments of Christ’s argument in [9] followed by a description in Section 3 of the key geometric inequality for polynomial curves established in [14], an essential fact in our arguments. In Section 4 we deal with the restricted weak-type estimates in three dimensions, and in Section 5 we show how these can be turned into strong-type and indeed Lorentz-space estimates, again in three dimensions. In the last section we produce the necessary arguments needed to deal with the two-dimensional case. Notation. Throughout this paper, whenever we write A B or A = O(B) for any two nonnegative quantities A and B, we mean that there exists a strictly positive constant c, possibly depending on the degree of the map P, so that A cB; this constant is subject to change from line to line and even from step to step. We also write A ∼ B if A B A. 2. Rudiments of Christ’s argument For a nonnegative finite measure μ supported on an interval I and a curve parametrised by γ : I → Rd , consider the averaging operator Af (x) = f x − γ (t) dμ(t). In this section we recall the basics of the combinatorial argument of Christ in [9] to prove a restricted weak-type estimate A : Lp,1 (Rd ) → Lq,∞ (Rd ). This is equivalent to proving 2 The assumption of positivity was removed by Cowling and Fournier in [13]. 3 We thank the referee for pointing out that in fact there is a general principle that if a bounded operator is translation and dilation invariant, then no Lp,r → Lq,s are possible when s < r.


AχE , χF |E|1/p |F |1/q

1359

(3)

for any two measurable sets E, F ⊂ Rd where | · | denotes the Lebesgue measure. Without loss of generality we may assume that |E|, |F | and AχE , χF are all positive quantities. Define two positive parameters α and β by the relations α :=

1

AχE , χF , |F |

β :=

1 ∗ A χF , χE |E|

so that α|F | = β|E|

where A∗ f (y) = f (y + γ (t)) dμ(t). Thus α is the average value of AχE on F and β is the average of A∗ χF on E. By passing to refinements of the sets E and F , without changing significantly the basic quantity K := AχE , χF = χE , A∗ χF to be estimated in (3), we will be able to bound pointwise AχE by α on F and bound pointwise A∗ χF by β on E. Precisely one defines the following refinements of E and F :

F1 = x ∈ F : AχE (x) α/2 ,

F2 = x ∈ F1 : AχE1 (x) α/8 , . . . ,

E1 = y ∈ E: A∗ χF1 (y) β/4 ,

En = y ∈ En−1 : A∗ χFn (y) β/22n ,

etc. It is a simple matter to check that AχEn , χFn K/22n and χEn , A∗ χFn+1 K/22n+1 for each n and so En , Fn = ∅. If d = 3, we fix an x0 ∈ F2 , set S = {s ∈ I : x0 − γ (s) ∈ E1 } and note μ(S) = AχE1 (x0 ) α/8.

(4)

Next observe that for every s ∈ S, if Ts = {t ∈ I : x0 − γ (s) + γ (t) ∈ F1 }, then μ(Ts ) = A∗ χF1 x0 − γ (s) β/4.

(5)

Finally we see that for every s ∈ S and t ∈ Ts , if Us,t = {u ∈ I : x0 − γ (s) + γ (t) − γ (u) ∈ E}, then μ(Us,t ) = AχE x0 − γ (s) + γ (t) α/2.

(6)

Hence we end up with a structured parameter domain P = {(s, t, u) ∈ I 3 : s ∈ S, t ∈ Ts , u ∈ Us,t } so that if Φγ (s, t, u) := x0 − γ (s) + γ (t) − γ (u), Φγ (P) ⊂ E. Therefore if Φγ is injective we have |E| P

JΦ (s, t, u) ds dt du = γ

JΦ (s, t, u) ds dt du γ

S Ts Us,t

where JΦγ (s, t, u) = det(γ (s)γ (t)γ (u)) is the determinant of the Jacobian matrix for the mapping Φγ , reducing matters to understanding the smallness of JΦγ (for instance, sublevel sets of JΦγ ) in order to bound from below the above integral over the structured set P. If γ (t) = (t, t 2 , t 3 ) (the non-degenerate example in three dimensions) and μ = | · | is the Lebesgue

1360


measure, then JΦγ (s, t, u) = 6(s − t)(t − u)(s − u) and so (4), (5) and (6) quickly imply |E| β 2 α 4 which gives (3) with p = 2 and q = 3, the desired endpoint estimate in this case. If d = 2, we fix a y0 ∈ E1 , set S = {s ∈ I : y0 + γ (s) ∈ F1 } and note μ(S) = A∗ χE1 (y0 ) β/4. Next observe that for every s ∈ S, if Ts = {t ∈ I : y0 + γ (s) − γ (t) ∈ E}, then μ(Ts ) = AχE y0 + γ (s) α/2.

(7)

(8)

Hence we end up with a structured parameter domain P = {(s, t) ∈ I 2 : s ∈ S, t ∈ Ts } so that if Φγ (s, t) := y0 + γ (s) − γ (t), Φγ (P) ⊂ E. Therefore if Φγ is injective we have JΦ (s, t) ds dt JΦγ (s, t) ds dt = |E| γ P

S Ts

where JΦγ (s, t) = − det(γ (s)γ (t)). If γ (t) = (t, t 2 ) (the non-degenerate example in two dimensions) and μ = | · | is the Lebesgue measure, then JΦγ (s, t) = 2(s − t) and so (7), (8) imply |E| βα 2 which gives (3) with p = 3/2 and q = 3, the desired endpoint estimate in this case. When we consider a general polynomial curve γ (t) = P(t) = (P1 (t), P2 (t)) in two dimensions with μ the affine arclength measure on P, we will only be able to prove JΦ (s, t) ds dt = JΦ (s, t) ds dt βα 2 (9) γ γ P

S Ts

in the range α β. In fact, without further refinements in the argument (see, for example, [9]), this integral bound can be false in the range β α. Nevertheless, due to the fact that the sharp endpoint estimate lies on the line of duality Lp → Lp , it will be the case that |E| βα 2 for all α, β. Knowing only (9) in the range β α leads to some further difficulties when establishing the Lorentz bounds and these difficulties do not present themselves in the three-dimensional case. This is why we choose to address the three-dimensional case first. 3. A geometric inequality As we have seen in the previous section, Christ’s argument in [9] is based in part on analysis of the map ΦP (t1 , . . . , td ) = (−1)d P(t1 ) + (−1)d+1 P(t2 ) + · · · − P(td ). In particular it would be desirable to have the following properties about ΦP : Key properties (a) ΦP is 1–1; 1 (b) |JΦP (t1 , . . . , td )| C dj =1 |LP (tj )| d j 0, bI ∈ / I and a nonnegative integer kI which is bounded above by a constant only depending on the degrees of the polynomials defining P; (P3) for (t1 , . . . , td ) ∈ I d , d

1 LΓ (tj ) d JΦ (t1 , . . . , td ) C |tj − tk | Γ j =1

j 0, let Bα = {u ∈ I : |u − b| δα 6/(k+6) } so that μ(Bα ) ck δ (k+6)/6 α. We will choose δ > 0 to be sufficiently small in each instance so that the following holds: if W ⊂ I is a set satisfying μ(W ) > c0 α for some c0 > 0, then μ(W \ Bα ) (c0 /2)α. • For δ > 0 and t, set Bt,α = {u ∈ I : |u − t| δα|t − b|−k/6 }. k/6 – If for all u ∈ W ⊂ I , |u − b| C0 |t − b|, then μ(W ∩ Bt,α ) 2C0 δα (in fact k/6 k/6 k/6 du C k/6 μ(B ) 2C t,α 0 |t − b| 0 δα) and therefore if μ(W ) c0 α, W ∩Bt,α |u − b| we have μ(W \ Bt,α ) (c0 /2)α if δ > 0 is chosen sufficiently small. – On the other hand, if we do not know a priori that |u − b| C0 |t − b| on W but we happen to know |t − b| C0 α 6/(k+6) , then automatically we have the control |u − b| |t − b| on Bt,α since |t − b| C0 α 6/(k+6) implies α|t − b|−k/6 |t − b| and thus |u − t| |t − b| on Bt,α . Case 1. On Ts , |t − b| (1/8)|s − b| holds; note then that |s − t| ∼ |s − b|. Case (1a). On Us,t , |u − b| (1/4)|t − b| holds; note then that |u − t| ∼ |t − b| and |u − s| ∼ |s − b|. Thus

|s − b|k/3

S

Ts

∼

|t − b|k/3 |s − t|

S

|u − b|k/3 du dt ds

Us,t

|s − b|k/6+k/6+2

|t − b|k/6+k/6+1

Ts \Bβ

S\Bα

|t − b|k/3+1 Ts

Us,t

|s − b|k/3+2

|u − b|k/3 |u − s||u − t| du dt ds

|u − b|k/6+k/6 du dt ds.

Us,t \Bα

Now choosing δ > 0 in each Bα , Bβ to ensure that the μ measures of the above sets have not 6

been altered significantly, and using the fact that on Us,t \ Bα we have |u − b| α k+6 (as well as analogous estimates on Ts \ Bβ and S \ Bα ), we see that the last iterated integral is bounded below by a constant multiple of 6

6

k

α k+6 (k/6+2) × α × β k+6 (k/6+1) × β × α k+6 × α = α 4 β 2 .


1365

Case (1b). On Us,t , (1/4)|t − b| |u − b| 4|t − b| holds; here then |u − s| ∼ |s − b| but now |u − t| may vanish. Then

|s − b|

|t − b|

k/3

S

k/3

Ts

∼

|s − t| Us,t

|s − b|k/3+2 S

|t − b|k/3 Ts

|s − b|

|t − b|

k/3+2

S

|u − b|k/3 |u − t| du dt ds

Us,t

|u − b|k/3 |u − s||u − t| du dt ds

|u − b|k/3 |u − t| du dt ds,

k/3 Us,t \Bt,α

Ts

and using that on Us,t \ Bt,α one has |u − t| α|t − b|−k/6 (together with the fact that |s − b| |t − b| and |u − b| ∼ |t − b| in this case) this last quantity is bounded below by

|s − b|k/3+2

α S

|t − b|k/6

Ts

α


Us,t \Bt,α

|s − b|

|t − b|

k/3+1 Ts \Bβ

S\Bα

|u − b|k/6 du dt ds.

k/3+1 (Us,t \Bt,α )\Bα

Since |u − b| 2|t − b| on Us,t , we see that we can choose δ > 0 in each Bα , Bβ and Bt,α so as not to change the μ measures much when we excise these intervals from S, Ts and Us,t . Therefore the last iterated integral above is bounded below by a constant times α × α 2 × β 2 × 4 2 (here we have used the fact, and will continue to do so, that for any set E ⊂ R, α = α β (k/3)+1 dt μ(E)2 ). E |t − b| Case (1c). On Us,t , |u − b| 4|t − b|; here |u − t| ∼ |u − b| but now |u − s| may vanish. Then

|s − b|k/3 S

|t − b|k/3 |s − t| Ts

Us,t

|s − b|

α

S\Bα

|t − b|k/3

|s − b|

k/6 Ts \Bβ

|u − b|k/3+1 du dt ds

Us,t \Bs,α

Ts

α

Us,t \Bs,α

|s − b|k/6+1

S\Bα

|u − b|k/3+1 |u − s| du dt ds

k/3

Ts

|t − b|

k/3+1

S\Bα

|u − b|k/3 |u − s||u − t| du dt ds

|t − b|

k/3+1

|u − b|k/3+1 du dt ds.

(Us,t \Bs,α )\Bα

Since we do have the control |u − b| |s − b| on Bs,α (since for s ∈ S \ Bα , |s − b| α 6/(k+6) ), we see that by appropriate choices of δ > 0 in Bα , Bβ and Bs,α , the above sets do not change in μ measure. Thus the final iterated integral is at least a constant multiple of α × α × β 2 × α 2 = α 4 β 2 .

1366


Case 2. On Ts , (1/8)|s − b| |t − b| 2|s − b| holds. Case (2a). On Us,t , |u − b| (1/4)|t − b| holds; here then |u − t| ∼ |t − b|, and we may also deduce |u − s| ∼ |s − b|. Since |t − b| ∼ |s − b|,

|s − b|

|t − b|

k/3

S

k/3

Ts

|s − t|

|u − b|k/3 |u − s||u − t| du dt ds

Us,t

|s − b|

|t − b|

k/3+1

k/3+1

|s − t|

Ts \Bs,β

S

β

|s − b|k/6+1

|t − b|k/3+1

Ts \Bs,β

S


|s − b|k/3+2

|t − b|k/6

Ts \Bs,β

S\Bα

Us,t

Us,t

β



Us,t \Bα

Again since |t − b| |s − b|, appropriate choices of δ > 0 can be made so as not to change the μ measures of S, Ts and Us,t when we excise from them the above intervals. Hence the last iterated integral is bounded below by a constant multiple of 6

6

β × α × α k+6 (k/6+2) × β × α × α k+6 (k/6) = α 4 β 2 . Case (2b). On Us,t , (1/4)|t − b| |u − b| 4|t − b| holds. Here we may compare all quantities containing b; namely |s − b| ∼ |t − b| ∼ |u − b|. Hence

|s − b|

|t − b|

k/3

S

Ts

k/3

|s − b|

βα

2 S\Bα

|u − b|k/3 |u − s||u − t| du dt ds

Us,t

|t − b|

k/2

S

|s − t|

k/3

Ts \Bs,β

|s − t| Us,t \(Bt,α ∪Bs,α )

|s − b|

k/6 Ts \Bs,β

|u − b|k/6 |u − t||u − s| du dt ds

|t − b|

k/6


Us,t \(Bt,α ∪Bs,α )

Again we see that the sets we are integrating over have not changed in μ measure much when we remove intervals and so the last iterated integral is at least a constant times βα 2 × α × β × α = α4 β 2 . Case (2c). On Us,t , |u − b| 4|t − b|; here |u − t| ∼ |u − b| but |u − s| and |t − s| may vanish. Since |u − b| |s − b| ∼ |t − b|,


|s − b|

|t − b|

k/3

S

k/3

Ts

|s − t|

|t − b|

k/3

S\Bα

αβ

k/3

Ts \Bs,β

|u − b|k/3 |u − s||u − t| du dt ds

Us,t

|s − b|

1367

|s − t| Us,t \Bs,α

|s − b|k/3+1

|t − b|k/6

Ts \Bs,β

S\Bα

|u − b|k/3+1 |u − s| du dt ds


Us,t \Bs,α

One checks that removing Bα , Bs,β and Bs,α has not changed the μ measures of our sets very much and so this last iterated integral is at least a constant times αβ × α 2 × β × α = α 4 β 2 . Case 3. On Ts , |t − b| 2|s − b| holds; in this case |t − s| ∼ |t − b|. Case (3a). On Us,t , |u − b| (1/4)|t − b| holds; here |t − u| ∼ |t − b| but |u − s| may vanish. Since |t − b| |s − b|,

|s − b|k/3

S

|t − b|k/3 |s − t| Ts

Us,t

|s − b|k/3

α

|t − b|k/3+2

Ts \Bβ

S\Bα

|u − b|k/3 |u − s||u − t| du dt ds

Us,t \Bs,α

|s − b|k/6+1

|u − b|k/3 |u − s| du dt ds

Ts \Bβ

S\Bα

|t − b|k/3+1


(Us,t \Bs,α )\Bα

Again the removal of intervals has not changed significantly the μ measures and so the last iterated integral is at least a constant multiple of α × α k/(k+6)+1 × β 2 × α 6/(k+6)+1 = α 4 β 2 . Case (3b). On Us,t , (1/4)|t − b| |u − b| 4|t − b| holds; here |s − u| ∼ |u − b| but |u − t| can vanish. Since |u − b| ∼ |t − b|,

|s − b|k/3 S

|t − b|k/3 |s − t| Ts

S

α S\Bα

Us,t

|s − b|k/3

|u − b|k/3 |u − s||u − t| du dt ds

|t − b|k/3+1 Ts

|s − b|

k/3 Ts \Bβ

|u − b|k/3+1 |u − t| du dt ds

Us,t \Bt,α

|t − b|

k/3+1

|u − b|k/6+1 du dt ds,

Us,t \Bt,α

and as before we see that the last iterated integral is bounded below by a constant times α × α k/(k+6)+1 × β 2 × α 6/(k+6)+1 = α 4 β 2 .

1368


Case (3c). On Us,t , |u − b| 4|t − b|; here we may deduce that |u − t| ∼ |u − b| and |u − s| ∼ |u − b|. Thus

|s − b|

|t − b|

k/3

S

k/3

Ts

|t − b|

k/3

S\Bα

|u − b|k/3 |u − s||u − t| du dt ds

Us,t

|s − b|

|s − t|

|u − b|k/3+2 du dt ds,

k/3+1

Ts \Bβ

Us,t \Bα 6

and as before this last iterated integral is at least a constant multiple of α × α k+6 (k/6) × β 2 × α × 6 α k+6 (k/6+2) = α 4 β 2 . This completes the bound for (15) and thus proves (14), completing the proof of Theorem 2. 2 5. Strong-type inequalities We now wish to complete the proof of Theorem 1 when d = 3. We shall suitably modify the arguments in [10] in order to achieve this goal. We will concentrate only on the first estimate stated in Theorem 1 and thanks to our geometric inequality and previous arguments, we just have to show that the operator AI : L2 (R3 ) → L3,2+ (R3 ), uniformly in I . This is equivalent to showing AI f, g C f 2 g3/2,2−

for any f ∈ L2 R3 , g ∈ L3/2,2− R3 .

(17)

Following [10], it suffices to select f, g of the form f=

∈Z

2 χE ,

g=

2m χFm ,

m∈Z

where the sets E are pairwise disjoint and so are the sets Fm . However, we shall specialise further, and pick the function g = g0 to be simply the characteristic function of a measurable set, g0 := χF . If we prove estimate (17) with g replaced by g0 , we then have an L2 → L3,∞ bound; one can then use Christ’s arguments toturn this into the claimed Lorentz space bound. We may normalise the L2 norm of f , so that 22 |E | = 1, and then the desired L2 → L3,∞ bound becomes

2 AI χE , χF |F |2/3 .

(18)

We decompose the sum above in order to stabilise certain quantities. For dyadic numbers , η ∈ (0, 1/2] we define L,η to be those where |E | ∼ η2−2

and AI χE , χF ∼ |E |1/2 |F |2/3 .


1369

The number M of indices in L,η is therefore finite and satisfies Mη 1. Our aim is then to prove

2 AI χE , χF min a , ηb |F |2/3

(19)

∈L,η

for some positive exponents a, b. By summing over the dyadic and η, we see that (19) implies (18). Next we may assume that |i − j | C log(1/) for any two distinct indices appearing in the sum over L,η where C > 0 will be an absolute constant.4 One now defines sets

G = x ∈ F : AI χE c0 |E |1/2 |F |2/3 |F |−1 , for a certain c0 > 0. If c0 is chosen sufficiently small, then AI χE , χF \G 1/2 AI χE , χF and so AI χE , χG ∼ AI χE , χF . By Theorem 2 we have AI χE , χG |E |1/2 |G |2/3 and so |G | 3/2 |F |.

(20)

By the Cauchy–Schwarz inequality,

−1

|F |

2 2 −1 |G | |F | χG

∈L,η

F

|F |−1

∈L,η

|G | + |F |−1

|Gk ∩ G |

k=

∈L,η

and therefore either (|F |−1 ∈L,η |G |)2 |F |−1 k= |Gk ∩ G | holds or we have ∈L,η |G | |F |. If the former holds, then by (20)

M 3/2

2

2 |G | M 2 |F |−1 max |Gk ∩ G |

∈L,η

k=

and the above dichotomy becomes either

|G | |F |

(21)

∈L,η

or there exist i = j so that |Gi ∩ Gj | 3 |F |.

(22)

4 By splitting the sum over L ,η into O(C log(1/)) sums, this assumption will cost us only a factor of O(C log(1/)) in the estimate (19).

1370


The key is now to show that (22) leads to a contradiction; this implies that (21) holds, and therefore

2 AI χE , χF ∼

∈L,η

2 AI χE , χG

∈L,η

2 |E | 3

3/2

1/3

∈L,η

2/3 |G | η1/6 |F |2/3 .

∈L,η

On the other hand,

2 AI χE , χF ∼

∈L,η

2 |E |1/2 |F |2/3

∈L,η

Mη1/2 |F |2/3 η1/2 |F |2/3 and these two estimates together imply (19). To disprove (22) we need the following result. Lemma 1. There exists a finite set of pairs (A, B) satisfying 1 A < 2, 2 < B 3 and A + B = 4 so that whenever E, E , G ⊂ R3 are measurable sets of finite measure satisfying AI χE (x) β

and AI χE (x) θ

for all x ∈ G,

there exists a pair (A, B) from our collection so that |E | β A β 2 θ B where β = β |G| |E| . Proof. Set ΦP (s, t, u) = −P(s) + P(t) − P(u) and define refinements

E 1 = y ∈ E: A∗I χG (y) β /2 ,

G1 = x ∈ G: AI χE 1 (x) β/4 . We have

β|G| A∗I χG1 , χE 1 = AI χE 1 , χG − AI χE 1 , χG\G1 AI χE 1 , χG − 4 ∗ ∗ β|G| 3β|G| β|G| AI χE , χG − . = AI χG , χE − AI χG , χE\E 1 − 4 4 4

Hence G1 = ∅. Now, pick x0 ∈ G1 and set

S = s ∈ I : x0 − P(s) ∈ E 1

⇒

μ(S) = AI χE 1 (x0 ) β/4.


1371

For s ∈ S, set

Ts = t ∈ I : x0 − P(s) + P(t) ∈ G

β μ(Ts ) = A∗I χG x0 − P(s) . 2

⇒

Finally for s ∈ S and t ∈ Ts , set

Us,t = u ∈ I : x0 + ΦP (s, t, u) ∈ E

μ(Us,t ) = AI χE x0 − P(s) + P(t) θ.

⇒

The idea is to estimate the measure of E by observing that if

P = (s, t, u) ∈ I 3 : s ∈ S, t ∈ Ts , u ∈ Us,t

then x0 + ΦP (P) ⊂ E .

Hence the arguments of Section 4 apply and we have

|E |

|s − b|k/3

S

|t − b|k/3 |s − t| Ts

|u − b|k/3 |t − u||s − u| du dt ds.

Us,t

To estimate the last iterated integral we can proceed in exactly the same manner as in the proof of Theorem 2 and split the analysis into 9 = 3 × 3 cases (1a)–(3c). In all cases, we obtain a bound from below equal to a constant multiple of β A β 2 θ B , for A and B belonging to a fixed finite set and always satisfying 1 A < 2, 2 < B 3 and A + B = 4. We explicitly present here a couple of cases to show that A and B can take different values, and leave the remaining cases to the interested reader. Let us suppose that for all t ∈ Ts , |t − b| (1/8)|s − b| and for all u ∈ Us,t , |u − b| (1/4)|t − b|. Then |s − t| ∼ |s − b|, |u − t| ∼ |t − b| and |u − s| ∼ |s − b|. Hence

|s − b|

|t − b|

k/3

S

k/3

Ts

∼

|s − b|k/6+k/6+2


Us,t

|t − b|k/6+k/6+1

Ts \Bβ

S\Bβ

|t − b|k/3+1 Ts

|u − b|k/3 |u − s||u − t| du dt ds

Us,t

|s − b|k/3+2 S

|s − t|

|u − b|k/6+k/6 du dt ds,

Us,t \Bθ

and the last iterated integral is bounded below by a constant times 6

6

k

β k+6 (k/6+2) × β × β k+6 (k/6+1) × β × θ k+6 × θ = β

2k+18 k+6

β 2θ

2k+6 k+6

.

Another case is one where, for all t ∈ Ts , (1/8)|s − b| |t − b| 2|s − b|, and for all Us,t , (1/4)|t − b| |u − b| 4|t − b|. Here |s − b| ∼ |t − b| ∼ |u − b|. Hence

1372


|s − b|

|t − b|

k/3

S

Ts

k/3

|s − b|

β θ 2

|t − b|

k/3

Ts \Bs,β

|s − t|

|u − b|k/6 |u − t||u − s| du dt ds

Us,t \(Bt,θ ∪Bs,θ )

|s − b|k/6

S\Bβ

|u − b|k/3 |u − s||u − t| du dt ds

Us,t

k/2

S

|s − t|

|t − b|k/6

Ts \Bs,β


Us,t \(Bt,θ ∪Bs,θ )

Again we see that the sets we are integrating over have not changed in μ measure much when we remove intervals and so the last iterated integral is at least a constant times β θ 2 × β × β × θ = ββ 2 θ 3 . The remaining seven cases can be treated in a similar way. 2 We can now conclude our argument; pick E = Ei , E = Ej , G = Gi ∩ Gj , and β = |E|1/2 |F |−1/3 , θ = |E |1/2 |F |−1/3 , β = β|G|/|E|. By Lemma 1 we have |E | A+B |F |(A+B)/3 |E|A/2 |F |B/2 β 2 |G|2 |E|−2 4 |F |−4/3 |E|A/2 |E |B/2 2 |E||F |−2/3 |G|2 |E|−2 12 |E|A/2−1 |E |B/2 , where we have used the fact that |G| 3 |F |. Using the relation A + B = 4 we deduce |E |1−A/2 −12 |E|1−A/2 , which is equivalent to 2−jp 24/(2−A) 2−ip , implying j i − C log(1/); since the roles of i, j can be exchanged one has |i − j | C log(1/), which contradicts our assumptions and therefore (22) cannot hold. This gives us the weak-type bound (18). As we have already mentioned, the arguments in [10] can now be reproduced verbatim to obtain the Lorentz bound (17), completing the proof of Theorem 1 for d = 3. 6. Two-dimensional estimates In this section we present the arguments necessary to prove Theorem 1 in the case d = 2, starting with the restricted weak type estimates. Theorem 3. Let d = 2. The operator (10) satisfies AR : L3/2,1 R2 → L3,∞ R2 .

(23)


1373

Proof. The preparatory statements of Sections 3 and 4 can obviously be applied also in this setting and we quickly reduce our analysis to the operators AI f (x) =

f x − P(t) |t − b|k/3 dt :=

I

f x − P(t) dμI (t),

I

for each fixed I . We set

AI χE , χF = α|F |,

AI χE , χF = β|E|,

with |E| = 0, |F | = 0, and observe it suffices to establish5

AI χE , χF |E|2/3 |F |2/3

⇔

|E| α 2 β

⇔

|F | β 2 α,

(24)

uniformly in I . As discussed in Section 2 we will apply Christ’s argument to prove |E| α 2 β

in the range α β

(25)

and similarly F | β 2 α in the range β α. But from the relation α|F | = β|E|, we see that (25) implies (24). This only works since we are proving an estimate on the line of duality. We shall concentrate on the estimate in (25) (the proof of the second estimate is similar) and so we assume from now on that α β. By the discussion in Section 2 we can find a point x0 ∈ E and S ⊂ I so that μ(S) β; for each s ∈ S there is Ts ⊂ I so that μ(Ts ) α;

if P = (s, t) ∈ I 2 : s ∈ S, t ∈ Ts , ⇒ x0 + ΦP (P) ⊂ E. Therefore (see Section 2) |E| P

JΦ (s, t) ds dt P

|s − b|k/2

S

|t − b|k/2 |s − t| dt ds.

(26)

Ts

As before we use a simple pigeonhole argument to reduce to various cases where the factors |s − b|, |t − b| and |s − t| in the integrand of the interated integral in (26) have a definite size relationship. We shall use similar dynamic notation as in Section 4: Bα = {t ∈ I : |t − b| δα 3/(k+3) } and Bs,α = {t ∈ I : |t − s| δα|s − b|−k/3 } with analogous conclusions as before if δ > 0 is chosen small enough in any particular situation. 5 We shall again abuse notation and relabel the measures μ as μ. I

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Case 1. On Ts , |t − b| (1/2)|s − b| holds; in this case |s − b| ∼ |t − s|. Thus

|s − b|

|t − b|

k/2

S

k/2

|s − t| dt ds

Ts

|s − b|

|t − b|k/2 dt ds

k/2+1 Ts \Bα

S\Bβ

|s − b|k/3

|t − b|2k/3+1 dt ds βα 2 .

Ts \Bα

S\Bβ

Here we have not used the relation α β. In addition, 3 k+4 3 k+2 |s − b|k/2+1 |t − b|k/2 dt ds β 2 k+3 α 2 k+3 . Ts \Bα

S\Bβ 3 k+4

3 k+2

Notice that β 2 k+3 α 2 k+3 α 2 β for α β. The former of these two estimates suffices for the proof of Theorem 3. However, both estimates will be required in order to obtain Lorentz space bounds. Case 2. On Ts , (1/2)|s − b| |t − b| 2|s − b| holds.

|s − b|k/2

S

|t − b|k/2 |s − t| dt ds Ts

|s − b|k/2

|t − b|k/2 |s − t| dt ds

Ts \Bs,α

S\Bβ

α

|s − b|

|t − b|k/2 dt ds.

k/6 Ts \Bs,α

S\Bβ

We make the important observation here that, in this case, |t − b| |s − b| on Bs,α and therefore μ(Ts \ Bs,α ) α if δ > 0 is chosen appropriately. Therefore the last iterated integral is bounded below by k/6+k/6 α |s − b| |t − b|k/3 dt ds α 2 β. Ts \Bs,α

S\Bβ

Case 3. On Ts , 2|s − b| |t − b| holds; in this case |t − s| ∼ |t − b|. Thus |s − b|

|t − b|

k/2

S

k/2

|s − t| dt ds

Ts

|s − b|

|t − b|k/2+1 dt ds

k/2

S

Ts

3 k+2 2 k+3

|t − b|k/2+1 dt ds

Ts \Bα

S\Bβ

β

|s − b|k/2

α

3 k+4 2 k+3

α2β

since α β. This completes the proof of (25) and hence the proof of Theorem 3.

2


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To prove the Lorentz estimates for the operator AI we put ourselves back in the setting of Section 5, with the (obvious) difference that we must consider the estimates just proven. Recall the appropriate setup: – there are 4 sets E(= Ei ), E (= Ej ), G(= Gi ∩ Gj ), F with |E| ∼ η2−3i/2 , |E | ∼ η2−3j/2 , and G ⊂ F , – four parameters > 0, β = |E|2/3 |F |−1/3 , δ = |E |2/3 |F |−1/3 , β = β|G|/|E|, – we may assume |G| > 3 |F |, AI χE β on G, AI χE δ on G, – we further assume β δ, which is equivalent to |E| |E |,6 and we wish to show that (22) leads to a contradiction; this will manifest itself in two possible forms, the inequality |E| c |E | or the inequality |G| K −1 3 |F |, for some c 0 and for a sufficiently large K. Clearly |G| K −1 3 |F | contradicts (22). The inequality |E| c |E | is equivalent to 23(i−j )/2 (1/)c which in turn is equivalent to 0 i − j c log(1/) which contradicts our basic assumptions on i and j . As indicated at the end of Section 2 the arguments in Section 5 break down in the two-dimensional setting and a slightly more elaborate argument is needed here. To carry out our arguments, we define two refinements

E 1 = y ∈ E: A∗I χG (y) β /2 ,

G1 = x ∈ G: AI χE 1 (x) β/4 .

The standard argument shows that G1 = ∅, thus we pick x0 ∈ G1 and set

Us,t

S = s ∈ I : x0 − P(s) ∈ E 1

Ts = t ∈ I : x0 − P(s) + P(t) ∈ G

= u ∈ I : x0 − P(s) + P(t) − P(u) ∈ E

⇒ ⇒ ⇒

μ(S) = AI χE 1 (x0 ) β/4, μ(Ts ) = A∗I χG x0 − P(s) β /2, μ(Us,t ) = AI χE x0 − P(s) + P(t) δ.

Case A. |G| p |E|, where p > 0 will be determined later. For fixed s ∈ S we have ψs (Ts × Us,t ) ⊂ E ,

where ψs (t, u) = x0 − P(s) + P(t) − P(u),

therefore

|E |

|t − b|

k/2

Ts

|u − b|k/2 |u − t| du dt δ C β

D

Us,t

thanks to Cases 1, 2 and 3 in this section; here (C, D) = (2, 1), (A, B) or (B, A), where k+4 3 k+2 (A, B) := ( 32 k+3 , 2 k+3 ), and in all instances C + D = 3. Hence 6 Since our arguments are completely symmetrical, this assumption does not pose any restrictions, as the roles of E and E can be interchanged.

1376


|E | δ C β = δ C β D |G|D |E|−D δ C β D p(D−1) |G||E|−1 D

C |E |2C/3 |F |−C/3 D |E|2D/3 |F |−D/3 p(D−1) |G||E|−1 , which is equivalent to |E|1−2D/3 3+p(D−1) |E |2C/3−1 |F |−1 |G| 6+p(D−1) |E |2C/3−1 , the contradiction we wished to find. Case B. |G| p |E|. This case is more involved and will be split into subcases. Let

Q = (s, t) ∈ I 2 : s ∈ S, t ∈ Ts , ΦP (s, t) = x0 − P(s) + P(t). Clearly ΦP (Q) ⊂ G, hence |G|

|s − b|

|t − b|k/2 |s − t| dt ds.

k/2

S

Ts

Let Ts = Ts1 ∪ Ts2 ∪ Ts3 , where

Ts1 = Ts ∩ t ∈ I : |t − b| (1/2)|s − b| ,

Ts2 = Ts ∩ t ∈ I : (1/2)|s − b| < |t − b| 2|s − b| ,

Ts3 = Ts ∩ t ∈ I : |t − b| 2|s − b| . Also let

S 2 = s ∈ S: μ Ts1 β /6 , S 1 = s ∈ S: μ Ts2 β /6 ,

S 3 = s ∈ S: μ Ts3 β /6 . Case B1. μ(S 1 ) β/12. Then either μ(S 2 ) β/12 or μ(S 3 ) β/12.

Case (B1a). μ(S 2 ) β/12. In this case, by Case 1, B B k/2 |G| |s − b| |t − b|k/2 |s − t| dt ds β A β = 3 |E|2 |F |−1 |G|/|E| . S2

Ts1

This implies |F | 3 |E|2−B |G|B−1 3 −p(2−B) |G|2−B+B−1

⇔

|G| p(2−B)−3 |F |,

contradicting |G| 3 |F | for p chosen sufficiently large (note B < 2).


1377

Case (B1b). μ(S 3 ) β/12. Here, by Case 3, A A |G| |s − b|k/2 |t − b|k/2 |s − t| dt ds β β B = 3 |E|2 |F |−1 |G|/|E| . S3

Ts3

This leads to |F | 3 |E|2−A |G|A−1 3 −p(2−A) |G|2−A+A−1

⇔

|G| p(2−A)−3 |F |,

contradicting |G| 3 |F | for sufficiently large p (note A < 2 if k = 07 ). 2,1

Case B2. μ(S 1 ) > β/12. To take care of this case we shall define subsets Ts

, Ts2,2 of Ts2 as

Ts2,1 = t ∈ Ts2 : μ {u ∈ Us,t : |u − b| 2|t − b|} δ/2 ,

Ts2,2 = t ∈ Ts2 : μ {u ∈ Us,t : |u − b| > 2|t − b|} δ/2 . Case (B2a). There exists s0 ∈ S 1 so that μ(Ts2,1 0 ) β /12. Hence, we bound the measure of E 2,1 by integrating over Ts0 . By Cases 1 and 2, we have |t − b|k/2 |u − b|k/2 |u − t| du dt β δ 2 = 3 |E|−1/3 |E |4/3 |G||F |−1 . |E | Ts2,1 0

Us0 ,t

This implies |E|1/3 3 |E |1/3 |G||F |−1 6 |E |1/3 , giving us the desired contradiction. Case (B2b). For every s ∈ S 1 we have μ(Ts2,1 ) < β /12. Thus, we must have that μ(Ts2,2 ) β /12. Now the integration occurs over Ts2,2 ; fixing an s ∈ S 1 , we have |t − b|k/3 |u − b|k/2 |u − t| du dt |s − b|k/6 δ A β . |E | |s − b|k/6 Us,t

Ts2,2

Now, if we choose S ⊂ S 1 , so that μ(S) = β/100 we have k 3 k/3 A |E | |s − b| ds δ β |s − b|k/3+k/6 ds δ A β β × β 6 k+3 = δ A β β B , S

S\Bβ

and this implies β|E | δ A β β B ⇔

|E | δ A β B |G||E|−1 = 3 |E |2A/3 |E|2B/3−1 |F |−1 |G| 6 |E |2A/3 |E|2B/3−1

⇔

|E|1−2B/3 6 |E |2A/3−1 ,

which is the required contradiction. This completes the proof of Theorem 1. 7 The case k = 0 is simpler and is dealt with in [10].

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Acknowledgment We are indebted to the referee for pointing out several errors and inconsistencies in an earlier version of this paper and we would like to thank the referee for making a number of very useful suggestions to improve the paper. References [1] F. Abi-Khuzam, B. Shayya, Fourier restriction to convex surfaces in R3 , Publ. Mat. 50 (1) (2006) 71–85. [2] J.-G. Bak, D. Oberlin, A. Seeger, Restriction of Fourier transforms to curves and related oscillatory integrals, Amer. J. Math. 131 (2) (2009) 277–311. [3] J. Bennett, A. Carbery, J. Wright, A non-linear generalisation of the Loomis–Whitney inequality and applications, Math. Res. Lett. 12 (4) (2005) 443–458. [4] A. Blozinski, Convolution of L(p, q) functions, Proc. Amer. Math. Soc. 32 (1) (1972) 237–240. [5] A. Carbery, C. Kenig, S. Ziesler, Restriction for flat surfaces of revolution in R3 , Proc. Amer. Math. Soc. 135 (6) (2007) 1905–1914. [6] A. Carbery, S. Ziesler, Restriction and decay for flat hypersurfaces, Publ. Mat. 46 (2) (2002) 405–434. [7] Y. Choi, Convolution operators with the affine arclength measure on plane curves, J. Korean Math. Soc. 36 (1) (1999) 193–207. [8] Y. Choi, The Lp –Lq mapping properties of convolution operators with the affine arclength measure on space curves, J. Aust. Math. Soc. 75 (2) (2003) 247–261. [9] M. Christ, Convolution, curvature and combinatorics: A case study, Int. Math. Res. Not. 19 (1998) 1033–1048. [10] M. Christ, Quasi-extremals for a Radon-like transform, preprint, 2006. [11] M. Christ, B. Erdogan, Mixed norm estimates for a restricted X-ray transform, J. Anal. Math. 87 (2002) 187–198. [12] M. Christ, B. Erdogan, Mixed norm estimates for certain generalised Radon transforms, Trans. Amer. Math. Soc. 360 (10) (2008) 5477–5488. [13] M. Cowling, J. Fournier, Inclusions and noninclusions of spaces of convolution operators, Trans. Amer. Math. Soc. 221 (1) (1976) 59–95. [14] S. Dendrinos, J. Wright, Fourier restriction to polynomial curves I: A geometric inequality, Amer. J. Math., in press. [15] S.W. Drury, Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc. 108 (1) (1990) 89–96. [16] S.W. Drury, B.P. Marshall, Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Cambridge Philos. Soc. 97 (1) (1985) 111–125. [17] S.W. Drury, B.P. Marshall, Fourier restriction theorems for degenerate curves, Math. Proc. Cambridge Philos. Soc. 101 (3) (1987) 541–553. [18] B. Erdogan, Mixed norm estimates for a restricted X-ray transform in R4 and R5 , Int. Math. Res. Not. 11 (2001) 575–600. [19] D. Oberlin, Oscillatory integrals with polynomial phase, Math. Scand. 69 (1) (1991) 45–56. [20] D. Oberlin, Convolution with affine arclength measures in the plane, Proc. Amer. Math. Soc. 127 (12) (1999) 3591– 3592. [21] D. Oberlin, Fourier restriction estimates for affine arclength measures in the plane, Proc. Amer. Math. Soc. 129 (11) (2001) 3303–3305. [22] D. Oberlin, Convolution with measures on polynomial curves, Math. Scand. 90 (1) (2002) 126–138. [23] D. Oberlin, A uniform Fourier restriction theorem for surfaces in R3 , Proc. Amer. Math. Soc. 132 (4) (2004) 1195– 1199. [24] D. Oberlin, Two estimates for curves in the plane, Proc. Amer. Math. Soc. 132 (11) (2004) 3195–3201. [25] Y. Pan, A remark on convolution with measures supported on curves, Canad. Math. Bull. 36 (2) (1993) 245–250. [26] Y. Pan, Convolution estimates for some degenerate curves, Math. Proc. Cambridge Philos. Soc. 116 (1) (1994) 143–146. [27] Y. Pan, Lp -improving properties for some measures supported on curves, Math. Scand. 78 (1) (1996) 121–132. [28] B. Shayya, An affine restriction estimate in R3 , Proc. Amer. Math. Soc. 135 (4) (2007) 1107–1113. [29] P. Sjölin, Fourier multipliers and estimates for the Fourier transform of measures carried by smooth curves in R2 , Studia Math. 51 (1974) 169–182. [30] E. Stovall, Endpoint bounds for a generalized Radon transform, J. London Math. Soc., arXiv at http://arxiv.org/abs/ 0905.3926. [31] T. Tao, J. Wright, Lp improving bounds for averages along curves, J. Amer. Math. Soc. 16 (3) (2003) 605–638.


Quasi-invariance of the Wiener measure on the path space over a complete Riemannian manifold Elton P. Hsu ∗,1 , Cheng Ouyang Department of Mathematics, Northwestern University, Evanston, IL 60208, United States Received 13 June 2008; accepted 14 May 2009 Available online 4 June 2009 Communicated by L. Gross

Abstract We prove a generalization of the Cameron–Martin theorem for a geometrically and stochastically complete Riemannian manifold; namely, the Wiener measure on the path space over such a manifold is quasiinvariant under the flow generated by a Cameron–Martin vector field. © 2009 Elsevier Inc. All rights reserved. Keywords: Wiener measure; Cameron–Martin theorem; Brownian motion; Quasi-invariance; Riemannian manifold; Completeness

1. Introduction The Cameron–Martin theorem is a fundamental result in stochastic analysis. Let Po (R) = Co ([0, 1]; R) be the (pinned) path space over R, i.e., the space of continuous functions w : [0, 1] → R such that w0 = o, the origin. Let μ be the Wiener measure on Po (R). Denote by w = {ws , 0 s 1} the canonical coordinate process on Po (R). Under μ, the process w is a Brownian motion starting from the origin. Now consider the shifted Brownian motion w h = w + h, where h ∈ Po (R) is a Cameron–Martin path, i.e., it has a distributional derivative h˙ such that

* Corresponding author.

E-mail addresses: [email protected] (E.P. Hsu), [email protected] (C. Ouyang). 1 The research of the first author on this work was supported in part by the NSF grant DMS-0407819.


1380

E.P. Hsu, C. Ouyang / Journal of Functional Analysis 257 (2009) 1379–1395

1 |h|2H

=

|h˙ s |2 ds < ∞.

(1.1)

0

The Cameron–Martin theorem (Cameron and Martin [1]) asserts that the law μh of w h and the Wiener measure μ are mutually absolutely continuous. Furthermore, the Radon–Nikodym derivative is given by dμh 1 = exp h, wH − |h|2H , dμ 2 where 1 h, wH =

h˙ s dws

0

is the Itô stochastic integral of h˙ with respect to the Brownian motion w. It is in this sense we say that the Wiener measure μ is quasi-invariant under a Cameron–Martin shift. A more general form of the Cameron–Martin theorem is the Girsanov theorem (Girsanov [4]), a simple formulation of which is as follows. Suppose that F∗ = {Fs , 0 s 1} is a filtration of σ -algebras on a probability space (Ω, F , P) and W an F∗ -Brownian motion. Let V = {Vs , 0 s 1} be an F∗ -adapted and progressively measurable process such that s e(s) = exp

1 Vu dWu − 2

0

s

|Vu |2 du ,

0 s 1,

0

is a martingale. Let Q be a new probability measure defined by dQ/dP = e(1). Then the process s Xs = W s −

Vu du,

0 s 1,

0

is a Brownian motion under Q. The Cameron–Martin theorem has a generalization to the Wiener measure on the path space Po (M) = Co ([0, 1], M) over a compact Riemannian manifold M. Driver [2] found the correct analogue of the euclidean Cameron–Martin shift w h = w + h under which the Wiener measure is quasi-invariant. The shift should be embedded in a flow generated by a vector field Dh defined geometrically on the path space. In the euclidean case Dh is simply the constant vector field Dh (γ ) = h and the flow is given by ζ t γ = γ + th. For a Riemannian manifold M we define the vector field Dh as follows. Fix a point o ∈ M and an orthonormal frame uo ∈ O(M) at o. Let U (γ ) be the horizontal lift from uo along a path γ ∈ Po (M). For each s ∈ [0, 1], U (γ )s : Rn → Tγs M


1381

is an isometry of the two indicated euclidean spaces. We define the vector field Dh on the path space Po (M) by Dh (γ ) = U (γ )h. More precisely, Dh (γ )s = U (γ )s hs ,

0 s 1.

In the above cited paper, Driver showed that when the manifold M is compact and h ∈ C 1 [0, 1] the vector field Dh indeed generates a flow {ζ t , t ∈ R} in the path space Po (M) and the Wiener measure on Po (M) (the law of a Riemannian Brownian motion on M starting from o) is quasiinvariant under the flow. Later in Hsu [5] and Enchev and Stroock [3], the existence of the flow and the quasi-invariance of the Wiener measure were extended to all Cameron–Martin vector fields Dh , h ∈ H . If we denote by μt the law of ζ t under the Wiener measure μ, then its Radon–Nikodym derivative with respect to μ has the form 1 1 t 1 t 2 dμt θ ds def = exp θs , dws − = et , s dμ 2 0

(1.2)

0

where θ t can be expressed more or less explicitly in terms of the flow and the curvature tensor of M and w ∈ Po (Rn ) is the anti-development of γ ∈ Po (M). We have dθst 1 = h˙ s + RicU (γ )s hs , dt t=0 2

(1.3)

where Ricu : Rn → Rn is the scalarized Ricci curvature tensor at an orthonormal frame u ∈ O(M). This relation and the flow equation

dζ t = Dh ζ t , dt

ζ 0 (γ ) = γ

(1.4)

gives an integration by parts formula for the vector field Dh

F (Dh G) dμ = Po (M)

G Dh∗ F dμ

(1.5)

Po (M)

on cylinder functions F and G on Po (M). The adjoint operator Dh∗ = −Dh + Dh∗ 1 of the vector field (first order differential operator) Dh with respect to the Wiener measure μ is given by

Dh∗ 1 =

1 0

1 h˙ s + RicU (γ )s hs , dws . 2

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The three objects crucial to our study of the Cameron–Martin theorem are the Radon– Nikodym derivative, the flow, and the integration by parts (through the divergence Dh∗ 1). They all appear in another representation of the Radon–Nikodym derivative t et = exp

Dh∗ 1

−s ds . ζ

(1.6)

0

This formula can be verified directly from (1.2) and (1.3). The question whether there is a complete generalization of the Cameron–Martin theorem for a general complete but possibly noncompact Riemannian manifold has been open for quite sometime. For this generalization we need to address two problems: the existence of the flow on the path space in an appropriate sense and the quasi-invariance of the Wiener measure under this flow. Since the vector field is Dh (γ ) = U (γ )h, the flow equation (1.4) involves the horizontal lift {U (ζ t )s , 0 s 1} of the process ζ t = {ζst , 0 s 1}, thus for each fixed t, the process ζ t should be a semimartingale. In Hsu [6] the first author showed that if M is geometrically complete and its Ricci curvature has at most a linear growth |Ricx | C 1 + r(x) , where r(x) = d(x, o), the Riemannian distance from o to x, then the flow {ζ t } generated by the vector field Dh exists and the Wiener measure is quasi-invariant. It was also pointed out there that the existence of the flow, if properly interpreted, can be proved for any geometrically and stochastically complete Riemannian manifold. Note that a linear bound on the Ricci curvature implies stochastic completeness. The question was left open whether the quasi-invariance is true solely under the condition of geometric and stochastic completeness. These two completeness conditions are natural for our problem: geometric completeness for the existence of the flow and stochastic completeness for the Wiener measure to be a probability measure on the path space Po (M). The purpose of this paper is to prove this quasi-invariance. The results of this work represent a complete generalization of the Cameron–Martin theorem to the Wiener measure on the path space of a Riemannian manifold. We will prove the existence of the flow in very much the same way as in Hsu [6], but with more care of the details in view of later applications. The expression for the would-be Radon– Nikodym derivative (1.2) or (1.6) still makes sense. The density formula (1.2) represents the terminal value of a local exponential martingale on the time interval [0, 1]. The Wiener measure is quasi-invariant under the flow if we can show that the local exponential martingale is uniformly integrable, or equivalently, 1 E exp 0

θst , dws

1 − 2

1

t 2 θ ds = 1. s

0

A sufficient condition for the uniform integrability is the Novikov criterion


1383

1 1 t 2 θs ds < ∞. E exp 2 0

In Hsu [6], this criterion was shown to hold for sufficiently small |t| under the above mentioned linear growth restriction on the Ricci curvature. In the current paper, we avoid verifying the uniform integrability; instead, we take advantage of the fact that by our construction, the flow {ζ t } is the limit of a sequence of flows for which the Wiener measure is quasi-invariant. For our argument to work, it is crucial that μ{0 < et < ∞} = 1, which indeed holds under the assumption of stochastic completeness. The rest of this paper has two sections. In Section 2 we show by an approximation argument that the flow equation has a unique solution. In Section 3, we show that the Wiener measure is quasi-invariant under the flow. 2. Existence of the flow In this section we show that the Cameron–Martin vector field Dh generates a flow {ζ t } in the path space Po (M). We assume that M is a geometrically and stochastically complete Riemannian manifold of dimension n. On a geometrically complete Riemannian manifold every bounded closed subset of M is compact, which ensures that the flow will be defined for all t ∈ R. When M is stochastically complete, we have pM (s, x, y) dy = 1, (x, s) ∈ M × R+ , M

where pM (s, x, y) is the (minimal) heat kernel of M (the transition density function of Brownian motion on M). Under this condition Brownian motion on M does not explode and the Wiener measure μ is a probability measure. In particular, with probability 1 every Brownian path γ [0, 1] is a compact subset of M. Remark 2.1. Let B∗ = {Bs , 0 s 1} be the standard filtration of σ -fields on the path space Po (M). Then B1 = B(Po (M)), the Borel σ -field on Po (M) viewed as a metric space in the usual way. Throughout this paper, we often speak of the composition F ◦ G (or F (G)) of two measurable maps F, G : Po (M) → Po (M). For this composition to make sense, a measurable map such as F is always meant to be so in the Borel sense F : (Po (M), B1 ) → (Po (M), B1 ), i.e., F −1 B1 ⊂ B1 . Remark 2.2. Throughout this paper, an almost sure statement always refers to the Wiener measure μ. Let O(M) be the orthonormal frame bundle of M and π : O(M) → M the canonical projection. Let Hi , i = 1, . . . , n, be the canonical horizontal vector fields on O(M). Fix an orthonormal frame uo ∈ Oo (M) at o. Let w be the euclidean Brownian motion given by the coordinate process on the flat path space (Po (Rn ), B∗ , ν). Consider the following stochastic differential equation dUs =

n i=1

Hi (Us ) ◦ dwsi ,

U0 = uo

(2.1)

1384


of a horizontal process U = U (w) on the orthonormal frame bundle O(M). The projection γ = πU of its solution is the stochastic development of w. By the pathwise uniqueness of the above stochastic differential equation, the relation J w = γ defines the so-called Itô map J : Po (Rn ) → Po (M). The process U is the horizontal lift U (γ ) of γ to O(M) and at the same time the stochastic development U (w) of w on the orthonormal frame bundle O(M). The resulting slight abuse of notation U (γ ) = U (w) should not cause any confusion. Note that U (γ ) is also the solution of a stochastic differential equation on O(M) driven by the Brownian motion γ on M. The line integral of the solder form θ on O(M) gives s ws =

θ (◦ dUs ). 0

This procedure gives the inverse map J −1 : Po (M) → Po (Rn ) (see Hsu [7]). Throughout the discussion we fix an Rn -valued Cameron–Martin path h. The Cameron– Martin vector field on Po (M) is defined by Dh (γ ) = U (γ )h (see Driver [2]), where U (γ ) is the horizontal lift of γ . The equation of the flow generated by Dh is

dζ t = U ζ t h, dt

ζ 0 (γ ) = γ .

(2.2)

Here we assume that ζ t = {ζst , 0 s 1} is an M-valued semimartingale under the Wiener measure μ and U (ζ t ) is the horizontal lift of ζ t . Note that the notation ζ t plays the dual role as a process {ζst , 0 s 1} and as a map ζ t : Po (M) → Po (M). In the latter capacity it is a Po (M)-valued random variable. To prove the existence of the flow generated by Dh , we first convert the flow equation from the curved path space Po (M) to the flat path space Po (Rn ) by the Itô map J : Po (Rn ) → Po (M). This step has two purposes. First, we will introduce a cut-off function in the flow equation on the flat path space to deal with possible unboundedness of the curvature tensor and its derivatives. Doing so directly on the flow equation on the curved path space Po (M) will result in a much more complicated vector field when the equation with the cut-off function is mapped to the flat path space by the Itô map. Second, after introducing a cut-off function, we see easily that the flow equation in the flat path space has globally Lipschitz coefficients so that Picard’s iteration can be applied. The formal calculation of the pullback vector field p = J∗−1 Dh is well known (see Driver [2] and Hsu [5]) and will not be repeated here. The result is s p(w)s = hs −

K(w)τ ◦ dwτ , 0

where s K(w)s =

ΩU (J w)τ (◦ dwτ , hτ ). 0


1385

Here Ω is the curvature form, which is by definition a o(n)-valued horizontal 2-form on O(M), n and ◦ dw denotes Stratonovich stochastic integration. To alleviate the notation, for a, b ∈ R we have written Ωu (a, b) instead of more precise Ωu (H a, H b) with H a = ni=1 Hi ai . Under the Wiener measure μ on Po (M), the anti-development w = J −1 γ is a euclidean Brownian motion starting from the origin whose law is the Wiener measure ν on Po (Rn ). Conversely, under ν on Po (Rn ), the development γ = J w is a Brownian motion on M from o whose law is μ. Therefore studying the flow equation (2.2) under the measure μ is equivalent to that of the flow equation

dξ t = p ξt , dt

ξ 0 (w) = w

(2.3)

on Po (Rn ) under the measure ν. Once {ξ t } is found, the desired flow on Po (M) is simply ζ t = J ◦ ξ t ◦ J −1 . In terms of Itô integrals the vector field on Po (Rn ) is given by 1 p(z)s = hs + 2

s

s RicU (J z)τ hτ dτ −

0

K(z)τ , dzτ ,

0

s K(z)s =

ΩU (J z)τ (dzτ , hτ ) + 0

1 2

s

Hi ΩU (J z)τ (ej , hτ ) d zi , zj τ .

(2.4)

0

Here {ei } is the canonical orthonormal basis of Rn . In order that the switch between the path spaces Po (M) and Po (Rn ) work properly it is crucial that all stochastic processes involved are semimartingales with respect to the Wiener measures ν or μ. It turns out sufficient to seek solutions in the space of semimartingales of the special form s zs =

s Aτ dτ +

0

Oτ dwτ ,

0 s 1,

(2.5)

0

where A and O are, respectively, Rn - and O(n)-valued processes, both being adapted to the canonical Borel filtration B∗ on Po (Rn ). Suppose that s ξst

=

s Atτ

0

dτ +

Oτt dwτ .

(2.6)

0

Then the flow equation (2.3) becomes ⎧ t ⎪ ⎪

⎪ t ⎪ O = I − K ξ λ O λ dλ, ⎪ ⎪ ⎪ ⎨ 0

⎪ t ⎪ ⎪ 1 ⎪ t t λ∗ ˙ ⎪ Ric A = O O h dλ. h + λ ⎪ U (J ξ ) ⎪ ⎩ 2 0

(2.7)

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In order to solve these equations by Picard’s iteration we need to introduce an appropriate norm on a semimartingale of the form (2.5) (see Hsu [5]): 1

z = E

|As |2 ds + E sup |Os |2 .

2

(2.8)

0s1

0

If the manifold M is compact, the components of the curvature tensor Ω and their derivatives are uniformly bounded, hence the coefficients of (2.7) are globally Lipschitz with respect to the above norm. In this case one can directly apply Picard’s iteration. For a geometrically and stochastically complete Riemannian manifold M, we will use a cutoff function defined on M to truncate the curvature tensor. Let φ : M → R be a smooth function with compact support. We define a new vector field on Po (Rn ):

1 p (z)s = hs + 2

s

φ

φ (J z)τ RicU (J z)τ hτ dτ −

0

s K φ (z)s =

s

K φ (z)τ , dzτ ,

0

1 φ (J z)τ ΩU (J z)τ (dzτ , hτ ) + 2

0

s

φ (J z)τ Hi ΩU (J z)τ (ej , hτ ) d zi , zj τ .

0

This definition should be compared with (2.4). The new vector field uses only the values of the curvature tensor components and their derivatives on a fixed compact region, namely, on the φ,t support of the cut-off function φ. We also note that Ks ∈ o(n). Consider the flow equation

dξ φ,t = p φ ξ φ,t , dt

ξ φ,0 (w) = w,

or equivalently, consider the integral equation t ξ

φ,t

=w+

p φ ξ φ,λ dλ.

(2.9)

0

We write as before s ξsφ,t

=

s Aφ,t u

0

du +

Ouφ,t dwu . 0

In terms of the pair {Aφ,t , O φ,t } the flow equation becomes

(2.10)


1387

⎧ t ⎪ ⎪

⎪ φ,t ⎪ O = I − K φ ξ φ,λ O φ,λ dλ, ⎪ ⎪ ⎪ ⎨ 0

⎪ t ⎪ ⎪ 1 φ,λ ⎪ φ,t φ,t φ,λ∗ ˙ ⎪ A Ric = O O h dλ. h + φ J ξ φ,λ ⎪ U (J ξ ) ⎪ ⎩ 2

(2.11)

0

These equations can be solved by Picard’s iteration as if the manifold is compact. The crucial step is to show that in the norm z defined in (2.8) the vector field is globally Lipschitz continuous. Proposition 2.3. There is a constant C such that for any semimartingales zi of the form (2.5) with the norm defined in (2.8), we have φ p (z1 ) − p φ (z2 ) C z1 − z2 . Let ξit and ηit be semimartingales of the form (2.5) such that t ξit

= ξi0

+

p φ ηiλ dλ.

0

Then t ξ − ξ t ξ 0 − ξ 0 + C 1 2 1 2

t

λ η − ηλ dλ. 1

2

0

Proof. Since the vector field p φ only uses the components of the curvature tensor and their derivatives on a compact subset of the manifold, the Lipschitz continuity of p φ can be proved in exactly the same way as in the case of a compact manifold, which involves nothing more than routine bounds of stochastic integrals with respect to dwτ by Doob’s inequality and those with respect to dτ by taking absolute value under the integrals. The details can be found in Hsu [5] and will not be repeated here, but we point out an important fact, namely, even after introducing the cut-off function, the new K φ still takes values in the space o(n) of anti-symmetric matrices and the corresponding O φ,t takes values in the space O(n) of orthogonal matrices O(n) (see the first equation in (2.11)). As a consequence, O φ,t is always uniformly bounded. 2 Proposition 2.4. There exists a unique family of semimartingales {ξ φ,t , t ∈ R} of the form (2.5) such that with probability 1: (a) (b) (c) (d)

ξ φ,0 (w) = w; p φ (ξ φ,t )s is jointly continuous in (t, s) ∈ R × [0, 1]; φ,t ξs is jointly C 1 in t ∈ R and continuous in s ∈ [0, 1]; φ,t (d/dt)ξs = p φ (ξ φ,t )s .

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Proof. We only outline the proof here, the technical details being mostly contained in Hsu [5]. The cut-off function φ is fixed in the course of this proof. For simplicity, we drop the superscript φ wherever no confusion is likely to occur. Let ξ t,0 (w) = w and t ξ

t,n

=ξ

t,0

+

p ξ λ,n−1 dλ.

0

From Proposition 2.3 we have t,n ξ − ξ t,n−1 C

t

λ,n−1 ξ − ξ λ,n−2 dλ.

0

This inequality implies that the limit ξ t = limn→∞ ξ t,n exists and is the solution to (2.11). The uniqueness is clear because we are dealing with a Volterra type integral equation. This shows the existence and uniqueness of the solution of the flow equation (2.9) in the class of semimartingales of the form (2.5). From (2.9) we have (a) immediately. The joint continuity claimed in (b) follows from Kolmogorov’s sample path continuity criterion (for processes with two time parameters). Using this we see that (c) and (d) follow again from (2.9). 2 Proposition 2.5. Let {ξ φ,t } be the flow in Proposition 2.4. (a) For each fixed t the law ν φ,t of ξ φ,t and the Wiener measure ν on Po (Rn ) are mutually absolutely continuous. The Radon–Nikodym derivative is given by 1 1 dν φ,t 1 φ,t 2 φ,t∗ φ,t As = exp Os As dws − ds . dν 2 0

(2.12)

0

We have dν φ,t = exp dν

t

φ

lh ξ φ,−λ dλ,

0

where φ lh (w) =

1

1

h˙ s + φ (J w)s RicU (J w)s hs , dws . 2

0

(b) For all (t1 , t2 ) ∈ R × R we have almost surely, ξ φ,t1 ◦ ξ φ,t2 = ξ φ,t1 +t2 .

(2.13)


1389

φ,t

Proof. From (2.6) and the fact that Os ∈ O(n) the assertion in (a) and (2.12) immediately from Girsanov’s theorem. The second formula for the Radon–Nikodym derivative can be obtained from differentiating (2.9) with respect to t and use the flow equation (2.11). For (b), both sides of (2.13) (with t1 as time variable) are the solution of the flow equation with initial value ξ φ,t2 at t1 = 0, hence the equality holds by the uniqueness of solutions of the flow equation (see Proposition 2.3). 2 For each fixed t we define the semimartingale ζ φ,t on the probability space (Po (M), B∗ , μ) by ζ φ,t = J ◦ ξ φ,t ◦ J −1 .

(2.14)

The maps J and J −1 send semimartingales to semimartingales, therefore ζ φ,t = {ζs , 0 s 1} is an M-valued semimartingale for each fixed t. φ,t

Proposition 2.6. Let ζ φ,t = J ◦ ξ φ,t ◦ J −1 . The following assertions hold. (a) For each fixed t the law μφ,t of ζ φ,t and the Wiener measure μ are mutually absolutely continuous and the Radon–Nikodym derivative is give by dν φ,t dμφ,t = ◦ J −1 . dμ dν (b) For all fixed (t1 , t2 ) ∈ R × R we have almost surely, ζ φ,t1 ◦ ζ φ,t2 = ζ φ,t1 +t2 .

(2.15)

Proof. These properties are inherited from the corresponding properties of ξ φ,t proved in Proposition 2.5. 2 Now we come to the main result of this section. Since the manifold is assumed to be geometrically complete, we can gradually remove the effect of the cut-off function and construct a flow for the Cameron–Martin vector field Dh . Theorem 2.7. Let M be a geometrically and stochastically complete Riemannian manifold. There exists a family of semimartingales {ζ t , t ∈ R} such that with probability 1: (a) (b) (c) (d) (e)

ζ 0 (γ ) = γ ; U (ζ t )s is jointly continuous in (t, s) ∈ R × [0, 1]; ζst is jointly C 1 in t ∈ R and continuous in s ∈ [0, 1]; dζst /dt = U (ζ t )s hs ; For all fixed (t1 , t2 ) ∈ R × R, we have almost surely ζ t1 ◦ ζ t2 = ζ t1 +t2 .

Proof. Without loss of generality we assume that h ∞ 1. We choose a sequence of cut-off functions {φN } as follows. Denote by BR the geodesic ball of radius R centered at o. For a

1390


positive integer N , we let φN be a bounded (uniformly in N ) smooth function on M such that φN (x) = 1 for x ∈ B2N and φN (x) = 0 for x ∈ / B3N (see Remark 2.8 below). Consider the flow ζ N,t = ζ φN ,t in Proposition 2.6. These flows will not feel the presence of the cut-off function as long as they stay within the geodesic ball B2N . More precisely, define an increasing sequence of stopping times σN = inf s 1: r(γs ) = N (with the convention that inf ∅ = 1) and let CN = {σN = 1}. Then CN = γ ∈ Po (M): γs ∈ BN for all s ∈ [0, 1] . If s σN , then the initial path ζ N,0 = {γs , 0 s 1} of the flow {ζ N,t } lies within the geodesic ball BN . Since φN = 1 on B2N , from (2.2) and λ U ζ hs |hs | 1 s we have d(ζsN,t , γs ) |t| for small time t and

r ζsN,t d ζsN,t , γs + r(γs ) |t| + N. By a routine open-closed argument with the above inequality we see that the above inequalities will hold for all s σN and |t| N . As a consequence, for M N , the flows ζsM,t and ζsN,t satisfy the same equation for all |t| N and s σN . By uniqueness we must have the consistency ζ M,t (γ )s = ζ N,t (γ )s ,

M N, s σN , |t| N.

Note that since CN = {σN = 1}, the above equality also holds for 0 s 1 if γ ∈ CN . We can now define ζ t (γ )s = ζ N,t (γ )s ,

0 s σN , |t| N.

(2.16)

This defines ζ t = {ζst , 0 s 1} for all γ ∈ CN . By stochastic completeness Brownian motion on M does not explode. With probability one the path γ [0, 1] is a compact subset of M, hence μ{CN } ↑ 1. This shows that (2.16) defines a family {ζ t } of semimartingales such that ζ t (γ ) = ζ N,t (γ ),

γ ∈ CN , |t| N.

We now prove the properties of the flow {ζ t } listed in the statement of the theorem. To start with, (a) follows from the fact that ζ N,0 (γ ) = γ . The joint continuity of U (ζ N,t )s can be proved again by Kolmogorov’s sample path continuity criterion for processes with two time parameters. The assumption of geometric completeness is needed for this step of the proof; see Remark 2.8 below. From ζst = J (ξ t )s and the definition of the Itô map J (see (2.1)) we have (d) as the equality of two semimartingales for each fixed t. This together with (b) implies (c) and also (d) for all s and t.


1391

For the composition property (e), by (2.15) we have first

ζ 2N,t1 ζ 2N,t2 γ = ζ 2N,t1 +t2 γ . If |t1 | + |t2 | N , then ζ 2N,t2 γ = ζ t2 γ

and ζ 2N,t1 +t2 γ = ζ t1 +t2 γ

for all γ ∈ CN . But it is clear that ζ t2 γ ∈ C2N , hence

ζ 2N,t1 ζ 2N,t2 γ = ζ t1 ζ t2 γ . It follows that ζ t1 (ζ t2 γ ) = ζ t1 +t2 γ for all γ ∈ CN , hence for almost all γ ∈ Po (M) because μ{CN } ↑ 1. 2 Remark 2.8. The cut-off functions used in the proof of Theorem 2.7 exist by the assumption that M is geometrically complete. However, even without the geometric completeness (but retaining the stochastic completeness), we can construct a sequence of cut-off functions {φN } such that the curvature tensor and its derivatives are uniformly bounded on {φN > 0} for each fixed N and that the sequence of increasing interiors {φN = 1}◦ exhausts M. Using such a sequence of the cut-off functions, we can show by properly modifying the proof given above that a jointly continuous flow {ζst (γ ), e− (γ , s) < t < e+ (γ , s)} can be constructed up to its natural lifetime in t. This means that limt→e± (γ ,s) ζ t (γ , s) = ∂M in the one-point compactification M∂ = M ∪ {∂M }. There are also manifolds not geometrically complete which nevertheless has a sequence of cut-off functions having all the properties used in the proof of Theorem 2.7. In this case we can construct a global measurable flow {ζ t , t ∈ R} as the solution of the corresponding integral equation in the space of Po (M)-valued random variables. However, this flow may not have a version jointly continuous in the two parameters t and s, for Kolmogorov’s sample path continuity theorem requires that the state space M be a complete metric space (see, e.g., Kallenberg [8, p. 313]). A case in point is M = R2 with a single point removed and hs = (0, s). While the removed point is not needed for the existence of the measurable flow ζ t (γ ) = γs + ts (for each fixed t), it is needed if we want to have a version of the flow that is jointly continuous in t and s. Finally we show that the flow {ζ t } is infinitely differentiable in the t-direction. Proposition 2.9. There is a version of the flow {ζ t , t ∈ R} such that with probability one the path t → ζst (γ ) is smooth for every s ∈ [0, 1]. More precisely,

P γ ∈ Po (M): ζ (γ ) ∈ C ∞,0 R × [0, 1]; M = 1. Proof. The proof involves repeatedly use of Kolmogorov’s sample path continuity criterion. To start with, we have shown in Theorem 2.7 that U (ζ t )s has a version jointly continuous in t and s, the flow ζst has a version jointly C 1 in t and continuous in s, and the identity dζst /dt = U (ζ t )s hs holds. Once we know that ζ t is C 1 in t, we can differentiate the stochastic differential equation (in the s-direction) satisfied by U (ζ t )s hs with respect to t and use Kolmogorov’s criterion to conclude that U (ζ t )s hs has a version jointly C 1 in t and continuous in s. The identity dζst /dt = U (ζ t )s hs then shows that ζst has a version jointly C 2 in t and continuous in s. This argument

1392


can be repeated indefinitely and shows that ζst has a version jointly C m in t and continuous in s for any positive integer m. For each m, there is an exceptional set Ωm of probability zero of the path space Po (M) on which the assertion does not hold. By excluding from the path space the union ∞ m=1 Ωm , which still has probability zero, we see that there is version of the flow jointly continuous in s and infinitely differentiable in t. 2 3. Quasi-invariance of the Wiener measure Let M be a geometrically and stochastic complete Riemannian manifold and {ζ t } the flow of the Cameron–Martin vector field Dh on the path space Po (M) constructed in the preceding section. In this section we show that for each t ∈ R, the law μt of the semimartingale ζ t is mutually absolutely continuous with respect to the Wiener measure μ. We have shown in Proposition 2.6 that the law μN,t of ζ N,t and μ are mutually absolutely continuous and the density function is t etN = exp

lhN ζ N,−λ dλ,

0

where 1 lhN (γ ) =

1 h˙ s + φN (γs )RicU (γ )s hs , dws . 2

0

Here w is the stochastic anti-development of γ . Recall that CN = γ ∈ Po (M): max r(γs ) N . 0s1

For γ ∈ CN and |λ| N we have lhN (γ ) =

1 1 def h˙ s + RicU (γ )s hs , dw = lh 2 0

and ζ N,−λ = ζ −λ , hence for |t| N t etN

= exp

def lh ζ −λ dλ = et

on CN .

0

Since M is geometrically and stochastically complete, the flow does not explode and μ{0 < et < ∞} = 1. Now we prove the main result of this section.

(3.1)


1393

Theorem 3.1. The laws μt of ζ t and the Wiener measure μ are mutually absolutely continuous and the Radon–Nikodym derivative is given by dμt = exp dμ

t

lh ζ −λ dλ.

0

Proof. Write X = Po (M) for simplicity. Fix N |t|. Then ζ t (γ ) = ζ N,t (γ ) and et (γ ) = etN (γ ) for γ ∈ CN . For a nonnegative bounded measurable function F on Po (M) we have

F ζ t dμ

X

F ζ N,t dμ

CN

F ζ N,t dμ − F ∞ μ(X\CN )

X

F etN dμ − F ∞ μ(X\CN )

= X

F et dμ − F ∞ μ(X\CN )

CN

→

F et dμ. X

Therefore we have

F et dμ X

F ζ t dμ.

(3.2)

X

By the monotone convergence theorem, the above inequality holds for all nonnegative measurable F . From Theorem 2.6 we have ζ N,t (ζ N,−t γ ) = γ . Let G = F (ζ N,t )et (ζ N,t ). Then

F et dμ =

X

G ζ N,−t dμ

X

=

N Ge−t dμ X

F ζ t et ζ t e−t dμ

CN

→ X

F ζ t et ζ t e−t dμ.

1394


Therefore

F et dμ

X

F ζ t et ζ t e−t dμ.

(3.3)

X

On the other hand, from the definition (3.1) of et and the identity ζ −λ ◦ ζ t = ζ t−λ it is easy to verify that

et ζ t e−t = 1. This identity together with (3.2) and (3.3) implies

F et dμ =

X

X

F ζ t dμ =

F dμt . X

In view of μ{0 < et < ∞} = 1 this shows that μ and μt are mutually absolutely continuous and dμt /dμ = et . 2 Remark 3.2. We have restricted our discussion in this work to the Wiener measure on the path space over a geometrically and stochastically complete Riemannian manifold equipped with the Levi-Civita connection. Our methods and the results remain valid for a general non-degenerate diffusion measure on a differentiable manifold provided that (1) the diffusion process is stochastically complete (i.e., with probability 1 it has infinite lifetime); (2) the manifold is geometrically complete under the Riemannian metric defined by the diffusion measure; (3) the connection (needed for defining the Cameron–Martin vector fields) is compatible with the Riemannian metric and has an anti-symmetric torsion. In particular, our results remain valid for canonical Brownian motions on Lie groups equipped with the Cartan connections. Acknowledgments This research was carried out in part by the first author during his visit to Bielefeld University, Germany. The first author would like to express his gratitude to Professor Alexander Grigoryan for his hospitality and financial support during the visit. The authors would also like to thank the anonymous referee for the careful reading of the paper and for many helpful suggestions for improvement. References [1] R.H. Cameron, W.T. Martin, Transformation of Wiener integrals under translations, Ann. Math. 45 (1944) 386–396. [2] B. Driver, A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal. 110 (2) (1992) 272–376. [3] O. Enchev, D.W. Strook, Towards a Riemannian geometry for Brownian motion on a compact Riemannian manifold, J. Funct. Anal. 134 (1995) 392–416. [4] I.V. Girsanov, On transformations of a class of random processes by an absolute change of measures, Theory Probab. Appl. 5 (3) (1960) 314–330. [5] Hsu, P. Elton, Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold, J. Funct. Anal. 134 (1995) 417–450.


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[6] Hsu, P. Elton, Quasi-invariance of the Wiener measure on path spaces: Noncompact case, J. Funct. Anal. 193 (2002) 278–290. [7] Hsu, P. Elton, Stochastic Analysis on Manifolds, Grad. Ser. Math., vol. 38, Amer. Math. Soc., Providence, RI, 2002. [8] Kallenberg, Olaf, Foundations of Modern Probability, second ed., Springer-Verlag, Berlin/Heidelberg/New York, 2002.


Rank and regularity for averages over submanifolds Philip T. Gressman 1 University of Pennsylvania, Mathematics Department, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104, United States Received 25 July 2008; accepted 17 April 2009 Available online 13 May 2009 Communicated by I. Rodnianski

Abstract This paper establishes endpoint Lp –Lq and Sobolev mapping properties of Radon-like operators which satisfy a homogeneity condition (similar to semiquasihomogeneity) and a condition on the rank of a matrix related to rotational curvature. For highly degenerate operators, the rank condition is generically satisfied for algebraic reasons, similar to an observation of Greenleaf, Pramanik and Tang [A. Greenleaf, M. Pramanik, W. Tang, Oscillatory integral operators with homogeneous polynomial phases in several variables, J. Funct. Anal. 244 (2) (2007) 444–487] concerning oscillatory integral operators. © 2009 Elsevier Inc. All rights reserved. Keywords: Radon transform; Oscillatory integral operator; Rotational curvature

1. Introduction The purpose of this paper is to establish endpoint Lp –Lq and Sobolev inequalities for a broad class of highly degenerate Radon-like averaging operators. The literature relating to this goal is both broad and deep, beginning with the groundbreaking work of Phong and Stein [15,16], and including but not limited to the works of Bak, Oberlin and Seeger [1], Cuccagna [4], Greenblatt [6], Greenleaf and Seeger [8,9], Lee [10,11], Phong and Stein [17], Phong, Stein and Sturm [18], Pramanik and Yang [19], Rychkov [20], Seeger [21], and Tao and Wright [24]. This literature provides a comprehensive theory of Radon transforms in the plane (optimal Lp –Lq and Sobolev bounds were established by Seeger [21] and others). Tao and Wright [24] have also E-mail address: [email protected]. 1 Partially supported by NSF grant DMS-0653755.


P.T. Gressman / Journal of Functional Analysis 257 (2009) 1396–1428

1397

established sharp (up to loss) Lp –Lq inequalities for completely general averaging operators over curves in any dimension. In the remaining cases, though, little has been proved regarding optimal inequalities for Radon-like operators. Among the reasons for this is that the rotational curvature (in the sense of Phong and Stein [13,14]) is essentially controlled by a scalar quantity for averaging operators in the plane, but is governed in higher dimensions (and higher codimension) by a matrix condition which is increasingly difficult to deal with using standard tools. While it is generally impossible for rotational curvature to be nonvanishing in this case, the corresponding matrix can be expected to have nontrivial rank. Under this assumption, works along the lines of Cuccagna [4] and Greenleaf, Pramanik and Tang [7] have been able to use this weaker information as a replacement for nonvanishing rotational curvature. In particular, Greenleaf, Pramanik and Tang showed that optimal L2 -decay inequalities for “generic” oscillatory integral operators can be established in the highly degenerate case with only the knowledge that the corresponding matrix quantity has rank one or higher at every point away from the origin. The purpose of this paper, then, is to explore and extend this phenomenon as it can be applied to the setting of Radon-like operators. Fix positive integers n and n , and let S be a smooth mapping into Rn which is defined on a neighborhood of the origin in Rn × Rn × Rn . The purpose of this paper is to prove a range of sharp Lp –Lq and Sobolev inequalities for the Radon-like operator defined by Tf (x , x ) := f y , x + S(x , x , y ) ψ(x , x , y ) dy , (1)

where x , y ∈ Rn and x ∈ Rn (n represents the dimension of the manifolds over which f is averaged, and n represents the codimension). When no confusion arises, the variable x will stand for the pair (x , x ), and n will refer to the sum n + n . The assumption to be made on S is that it exhibits a sort of approximate homogeneity (aka semiquasihomogeneity). The notation to be used to describe this scaling will be as follows: given any multiindex γ := (γ1 , . . . , γm ) of length m, any z := (z1 , . . . , zm ) ∈ Rm , and any integer j , let 2j γ z := (2j γ1 z1 , . . . , 2j γm zm ). The entries of a multiindex will always be integers, but they will be allowed to be negative in situations where negative entries make sense. The order of the multiindex γ (denoted by |γ |) is the sum of the entries, i.e., γ1 + · · · + γm , and may be negative in some cases. As for the mapping S, it will be assumed that there exist multiindices α and β of length n and α and β of length n such that the limit of lim 2jβ S 2−j α x , 2−j α x , 2−jβ y =: S P (x , x , y )

j →∞

(2)

as j → ∞ exists and is a smooth function of x , x , and y which does not vanish identically (note that, given a smooth mapping S, there is always at least one choice of multiindices so that this condition holds). Furthermore, it will be assumed that βi > αi for i = 1, . . . , n . The assumption on α and β will together with (2) be referred to as the homogeneity conditions. As with the variable x, the multiindices α and β of length n will represent (α , α ) and (β , β ) respectively. Although the mapping S exhibits a weak sort of homogeneity, the second of the homogeneity conditions guarantees, in fact, that the averaging operator (1) is not homogeneous. For this reason, it turns out that there is more than one family of dilations that come into

1398


play in the study of (1). To simplify the proofs somewhat, it will also be convenient to define α˜ to represent (α , β ). The main nondegeneracy condition to be used is stated as follows: for each pair (x, y ) in the support of the cutoff ψ in (1) and each η ∈ Rn \ {0}, consider the n × n mixed Hessian matrix H P whose (i, j )-entry is given by HijP (x , x , y , η ) :=

∂ 2 P η · S (x , x , y ) . ∂xi yj

(3)

Throughout the paper, it will be assumed that there is a positive integer r > 0 such that, at each point (x , x , y ) = (0, 0, 0) and for each η = 0, the matrix H P (x , x , y , η ) has rank at least r. This condition is very closely related to the condition of nonvanishing rotational curvature of Phong and Stein [13,14]; however, even when r is maximal, the operators (1) can and generally do have vanishing rotational curvature at the origin. When r < n the rotational curvature may actually vanish at every point. It should be pointed out that this rank condition is fairly easy to satisfy in cases of low codimension, but will not be satisfied for averages over curves in spaces of dimension three or greater (so the present situation is well outside the work of Seeger [21] and Tao and Wright [24], for example). The rank condition may, however, be satisfied by averages over of “relatively small” dimension (for example, submanifolds of dimension roughly √ manifolds n in Rn ). When this rank condition is satisfied, the following theorems hold: Theorem 1. Suppose that the operator (1) satisfies the homogeneity conditions and that the mixed Hessian (3) has rank at least r whenever (x , x , y ) = (0, 0, 0) and η = 0. If the support |+|β | of ψ is sufficiently near the origin and nr > |α |β then T maps Lp (Rn ) to Lq (Rn ) provided | that the following inequalities are satisfied: |β | + |β | |α | + |β | − < |β |, p q 1 + 1 − 1 < 1 − 2n + r 1 − 1 . p q r p q

(4) (5)

Additionally, T maps Lp to Lq if either one of the inequalities (4) or (5) is replaced with equality. If both inequalities are replaced with equality, then T is of restricted weak-type (p, q). The Riesz diagram corresponding to these estimates is shown in Fig. 1.

|+|β | In the event that nr < |α |β , the condition (5) excludes the possibility of equality in condi | tion (4). Both (4) and (5) are “optimal,” but in varying senses. In the case of the former, any oper | | − |α |+|β > ator satisfying the homogeneity condition is unbounded from Lp to Lq if |β |+|β p q |β |. For the latter, there exists an operator satisfying the rank condition which is unbounded for pairs ( p1 , q1 ) which lie outside the closure of the region defined by (5).

|+|β | Theorem 2. Suppose that T satisfies the rank and homogeneity conditions and nr > |α |β (and | the support of ψ is sufficiently near the origin). Then the operator T maps the space Lp (Rn ) to p the Sobolev space Ls (Rn ) (s 0) provided that the following two conditions are satisfied:


1399

|+|β | Fig. 1. Riesz diagram corresponding to the estimates proved for T (the shaded area) in the case when nr > |α |β | . Restricted weak-type inequalities are obtained at the nontrivial vertices (marked by circles).

|α | 1 + |β | 1 − , s max β1 , . . . , βn p p s 1 1 1 < − − . r 2 2 p

(6)

(7)

Just like the constraint (4), the inequality (6) is necessarily satisfied by any operator satisfying the hypotheses of Theorem 2. Another interesting feature of Theorems 1 and 2 is that the homogeneity condition and the rank condition on the Hessian are decoupled, in the sense that each of the constraints (4)–(7) depends (quantitatively, at least) on only one of the two assumptions made of T . A consequence of this is that when |β | is large, the Lp –Lq boundedness of T near p the line of duality p1 + q1 = 1 as well as the Lp –Ls boundedness for p near 2 are almost completely insensitive to the condition on the rank of the Hessian. This phenomenon was observed by Greenleaf, Pramanik and Tang [7] in the context of oscillatory integral operators (this is the “low-hanging fruit”). It is of particular interest, then, to make a statement quantifying the strength of the rank assumption on the mixed Hessian (3). For some particular combinations of α , α , β , and β , there may not, in fact, be any operators satisfying the homogeneity condition because S is assumed to be smooth. For this reason, it will not be possible to make a nonvacuous statement valid for every possible combination of multiindices. To rectify, the multiindices α , β and α will be considered fixed, and a “positive fraction” of the choices of β will be examined. There are a variety of ways to formulate this concept; here a set of multiindices E of length n will be said to have lower density provided that

lim inf N →∞

#{β ∈ E | βi N ∀i = 1, . . . , n } . Nn

1400


Let Λα,β be the space of all n -tuples of real polynomials (p1 , . . . , pn ) in the variables x , y , and x (for x , y ∈ Rn and x ∈ R n ) such that pl 2j α x , 2j α x , 2jβ y = 2βl pl (x , x , y ) for each integer j and l = 1, . . . , n ; suppose further that Λα,β is given the topology of a real, finite-dimensional vector space. Each element (p1 , . . . , pn ) naturally induces an operator of the form (1) which satisfies the homogeneity condition. The strength of the condition (3) can now be quantified as follows: Theorem 3. Fix α , α , and β . Let K1 be the least common multiple of the entries of α , α and β ; let K2 be the number of distinct values (modulo K1 ) taken by the sum αi + βj for i, j =

1, . . . , n . Then for any β in some set of lower density K1−n , the operators (1) corresponding to the polynomials Λα,β generically satisfy the rank condition provided r < n −

1 − K2−1 (n )2 + 2n.

In the context of averages over hypersurfaces with isotropic homogeneity (taking the entries of α , α , and β to equal one, corresponding to the case nX = nZ in the work of Greenleaf, Pramanik and Tang √ [7]), a generic mixed Hessian (3) has everywhere (except the origin) rank at least n − 1 − 2n + 2, and the hypotheses of Theorems 1 and 2 are satisfied for any choice of β 3 when n > 25. On the opposite extreme, a rank one condition holds provided that n < n (n2−4) (an extremely large codimension, similar to those encountered by Cuccagna [4] and well beyond the range of nonvanishing rotational curvature) and Theorems 1 and 2 hold with r = 1 for all multiindices β satisfying |β | > (n )2 (n − 4). 1.1. Examples When considering the class of averaging operators to which Theorems 1 and 2 apply, there are a few general points to bear in mind. The first is that, for each of these operators, there is a single, distinguished point in the incidence manifold

(x , x , y , y ) ∈ R2n +2n y = x + S(x , x , y ) ,

namely the origin (x , x , y ) = 0, at which the operator is at its most degenerate (when measured in terms of the rank condition). In particular, this means that the class of operators considered here does not include translation-invariant operators which are highly degenerate (since, aside from those operators already considered by Cuccagna [4], the rank would be expected to drop on a hypersurface or some even more complicated set). Likewise, the curvature condition considered here agrees with the corresponding conditions found in the study of the restriction phenomenon only in the nondegenerate case. For example, the nondegenerate surfaces of codimension 2 studied by Christ [2] trivially satisfy a rank condition like the one considered here (Christ’s hypothesis (4.1)), but the conjectured curvature conditions for degenerate codimension 2 submanifolds (hypothesis (3.6)) involve a more subtle interplay between rank and geometry than is considered here. See D. Oberlin [12] for an example of what would be considered a nondegenerate, low rank surface (analogous to the results of Cuccagna) in the context of restriction theory


1401

as well. While the particular structure assumed in this paper (namely, the existence of this distinguished point) may seem rather special, Theorem 3 guarantees that it is generic in many different settings (and so it is the behavior of the translation-invariant operators, for example, which is not generic). The second general point to keep in mind is that the class of operators considered in Theorems 1 and 2 cannot be expected to be expressed simply in terms of polynomial coefficients or Newton data (as was the case, for example, in the work of Phong and Stein [15,16]). When measured in terms of the rank condition, mappings S which can be expressed as a linear combination of a small number of monomials do not behave generically, nor do mappings S which have a simple tensor-product structure. To put it another way, the rank condition is a means of encoding subtle interactions between the various terms of S which are not properly encoded by the Newton polytope (even when expressed in adapted coordinates as in Varˇcenko [25]). For this reason, it is simplest to describe the examples below by grouping monomials into subcollections whose interactions can be easily understood. To that end, fix some positive integer r and choose the dimension of submanifolds n and the codimension n so that n is a multiple of r and n (2n + 1)r. An example of a mapping S which satisfies the rank r condition required by Theorems 1 and 2 is given (with components (S1 , . . . , Sn )) by

Sj (x , x , y ) :=

n /r

si,2j +i−2 x(i−1)r+1 , . . . , xir , y(2j +i−3)r+1 , . . . , y(2j +i−2)r

i=1

+

n /r

si,2j +i−1 y(2j +i−2)r+1 , . . . , y(2j +i−1)r , x(i−1)r+1 , . . . , xir

i=1

+ ϕj (x )

r

xj r+l y(j −1)r+l

l=1

(where there is a periodicity convention with indices so that xn +1 corresponds with x1 , etc.) whenever the mappings ϕj and sk,l (for k, l = 1, . . . , n /r) are quasihomogeneous (so that each Sj is quasihomogeneous with respect to a fixed dilation structure) and satisfy the properties ∂2s

k,l that ϕj (x ) = 0 only when x = 0 and ∂x∂y (x1 , . . . , xr , y1 , . . . , yr ) is a degenerate matrix only when (x1 , . . . , xr ) = 0. This structure guarantees that the mixed Hessians H p (x , x , y , η ) have a block structure (with individual blocks having size r × r) which guarantees that H P (x , x , y , η ) has rank r or greater unless η = 0 or (x , x , y ) = 0. That such building blocks s exist is not difficult to see; an example of a quasihomogeneous function satisfying the 2 constraint that ∂∂xs ∂y is nondegenerate away from x = 0 is given by

s(x1 , . . . , xr , y1 , . . . , yr ) :=

r j =1

cj xj yj

r

σj |xj |

τi

l=1

when the τi are positive, the σj nonnegative, and the constants cj are nonvanishing. With these building blocks, it is straightforward to build many non-translation-invariant averaging operators of high codimension which fall outside the scope of previous research (the main reason being that, in the microlocal sense, this class of examples is populated by highly degenerate operators).

1402


To give some slightly more explicit representatives, fix n 3 and n = 1; choose any α , β , α , and β . The scalar-valued mapping

S(x , x , y

) := cx (β −α1 −β2 )/α x1 yn −2

n (β −βj )/αj (β −αj +1 )/βj cj xj + yj + c˜j xj +1 yj j =1

(where all coefficients are nonzero) falls in this class with r = 1. This example arises from taking the si,j functions to have the form x s y. Likewise a codimension 1 averaging operator satisfying the rank 2 condition is given by S(x , x , y ) := cx k x1 y4 + +

k k cj xj k−1 xj +1 yj + xj xj k−1 +1 yj − xj yj +1 + xj +1 yj +1

j odd

k k c˜j xj +2 yj k−1 yj +1 + xj +2 yj yj k−1 +1 − xj +3 yj + xj +3 yj +1

j odd

when n is even and greater than or equal to six (and again, all coefficients are nonzero). Here the si,j are multiples of the function s(x1 , x2 , y1 , y2 ) := x1k−1 x2 y1 + x1 x2k−1 y1 − x1k y2 + x2k y2 . Note also that any such example is stable under small perturbations of the coefficients within the class of quasihomogeneous polynomials of the appropriate scaling. 1.2. Organization of the paper The following section (Section 2) is primarily concerned with a number of basic ideas and lemmas from analysis, the three major topics being the reduction of the analysis of semiquasihomogeneous function to the polynomial case, a stationary phase lemma, and a discussion of nonisotropic Sobolev spaces. This section also contains the main decomposition which will be used to prove both 1 and 2; the decomposition itself is inspired by the decomposition introduced by Christ, Nagel, Stein and Wainger [3]. The analysis of the individual terms of the main decomposition is carried out in Sections 3 and 4. The inequalities in Section 3 share the common feature that they are “trivial” in the sense that they primarily encode information about the size of supports of various cutoff functions (i.e., they do not involve any subtle orthogonality or curvature considerations). Section 4, in contrast, contains the analysis terms which incorporates the geometry of the problem (using Cotlar–Stein almost orthogonality and an operator van der Corput-type lemma). Finally, Sections 5 and 6 contain the proofs of Theorems 1, 2 and 3. Section 5 incorporates the estimates from Sections 3 and 4 to prove the sufficiency portions of Theorems 1 and 2 followed by Knapp-type examples which demonstrate the sharpness of the conclusions of those theorems. Section 6 is a largely self-contained proof of Theorem 3. 2. Preliminaries To begin, a few comments about homogeneity are necessary. Given multiindices α , α , and β , a smooth function f defined on a neighborhood of the origin in R2n +n (as a function of


1403

x , x , and y ) will be called nearly homogeneous of degree l 0 if lim 2lj f 2−j α x, 2−jβ y =: f P (x , x , y )

j →∞

(8)

exists as j → ∞ for every x , x , and y and is nonzero at some point (the limit function f P will be called the principal part of f ). In fact, given any smooth function f not vanishing to infinite order at the origin, there is a unique nonnegative integer l such that f is nearly homogeneous of degree l, the limit (8) must be uniform on compact sets, and the principal part must be a polynomial. Furthermore, for any multiindex γ of length n and any multiindex δ of length n ,

lim 2j l−j α·γ −jβ ·δ

j →∞

∂ |γ |+|δ| f −j α ∂ |γ |+|δ| f P −jβ 2 = x, 2 y (x, y ) (∂x)γ (∂y )δ (∂x)γ (∂y )δ

(9)

(where α · γ = ni=1 αi γi and likewise for δ · β ) with uniform convergence on compact sets. These assertions all follow directly from Taylor’s theorem with remainder by regrouping terms according to homogeneity degree (with respect to α , α , and β ); the proofs are straightforward and will not be given here. 2.1. Remarks on the dual operator T ∗ The next item to be explored is the nature of the operator T ∗ which is dual to (1). For fixed x and y , let Φx ,y (x ) := x +S(x , x , y ). To express T ∗ as an integral operator, it is necessary to invert the mapping Φx ,y . To that end, consider the derivative of the lth component of S (denoted by Sl ) with respect to xk . The function Sl is nearly homogeneous of degree βl by assumption; therefore (9) guarantees that the derivative ∂xk Sl (x, y ) vanishes at the origin whenever βl > αk . Since βi > αi for each i, it follows that the Jacobian matrix of Φ0,0 (x ) at x = 0 is upper triangular with ones along the diagonal (after a suitable permutation of the rows and columns). As a result, the inverse function theorem guarantees the existence of a smooth inverse to Φx ,y near x = 0 for all sufficiently small x and y . It follows that the dual operator T ∗ may be written as ∗ ˜ T g(y , y ) = g x , y − S x , Φx−1 ψ(x , y) dx ,y (y ), y for a new cutoff ψ˜ equal to the old cutoff ψ divided by the absolute value of the Jacobian determinant of Φx ,y . The next step is to compute the principal part of the dual mapping S ∗ defined by ∗ S (y , y , x ) := −S(x , Φx−1 ,y (y ), y ). To do so, consider yet another important consequence of the assumption βi > αi : lim 2j α Φ2−j α x ,2−jβ y 2−j α x = x

j →∞

(10)

with uniform convergence on compact sets. Furthermore, for any R > 0, the inverse function theorem provides a uniform constant CR such that |x | CR 2j α Φ2−j α x ,2−jβ y 2−j α x

(11)

1404


uniform in j , valid for all x such that the right-hand side is itself bounded by RCR . It therefore must be the case that

lim 2j α Φ −1 −j α

j →∞

2

x ,2−jβ y

−j α 2 y = y

(since (11) guarantees that the sequence on the left-hand side is bounded, and the uniformity of (10) shows that any convergent subsequence must have limit y ). Consequently, if S satisfies the homogeneity condition and has principal part S P (x , x , y ), then S ∗ (x , y , y ) satisfies the homogeneity condition with x scaled by α , y by β , and y by α with principal part −S P (x , y , y ). Thus if (1) has a mixed Hessian (3) with rank at least r near the origin, then so does T ∗ . This fact will be used later to simplify the proof of Theorem 2. 2.2. Main tools Now comes the time to prove the main tools which power the arguments necessary for Theorems 1 and 2. To simplify matters, fix, once and for all, a smooth function ϕ0 on the real line which is supported on [−2, 2], identically one on [−1, 1], and monotone on [0, ∞) and (−∞, 0]. The first order of business is the integration-by-parts lemma. The key idea of the method of stationary phase is that the main contributions to an oscillatory integral occur where the gradient of the phase is “small.” While there is an intrinsic way of stating that the gradient of the phase vanishes, there is (unfortunately) no coordinate-independent way of quantifying “smallness.” The answer, then, is to be explicit about the coordinates being used, and to change those coordinates whenever it is necessary and proper to do so. This changing of coordinate systems is captured here by what will be called scales. More precisely: a scale S on Rd will be any multiindex of length d with entries in Z. A vector v ∈ Rd will have length relative to S given by |v|S :=

d

1 22Si |vi |2

2

i=1

(the term “scale” was chosen because S implicitly induces a rescaling of the standard coordinate γ system via this formula). Likewise, the derivative ∂S (for some standard multiindex γ ) is meant to represent the derivative 2

d

i=1 γi Si

∂ ∂t1

γ1

∂ ··· ∂td

γd

(where the standard coordinates are here labeled t1 , . . . , td ). In this notation, the integration-byparts lemma is stated as follows: Lemma 1. Let Φ be any real-valued, C ∞ function defined on some open subset of Rd , and let ϕ be a C ∞ function compactly supported in the domain of Φ. Then for any positive integer N , there exists a constant CN such that


1405

+1 γ N γ N 2 N |γ |=0 |∂S ϕ(t)| |γ |=2 |∂S Φ(t)| eiΦ(t) ϕ(t) dt CN 1+ dt, (1 + |∇Φ(t)|S )N 1 + |∇Φ(t)|S

(12)

where S is any scale and 0 < 1. Proof. Consider the following integral on Rd+1 : Iα := ei(−2παt0 +Φ(t)) ϕ0 (t0 )ϕ(t) dt0 dt. By Fubini’s theorem and continuity of this integral near α = 0, there is at least one value of α ∈ (0, 1) (depending only on ϕ0 ) and a constant Cα = 0 (depending only on α and ϕ0 ) such that Cα−1 Iα is precisely the value of the integral to be computed. Let such an α be fixed once and for ˜ t˜ ) and ϕ( ˜ t˜ ) represent the phase and amplitude, all. Let t˜ := (t0 , t1 , . . . , td ), and likewise let Φ( respectively, appearing in the integral defining Iα . Let k be any nonnegative integer, let l be an integer such that 2−l < 2πα 2−l+1 , and let S˜ be the scale on Rd+1 given by (l, S1 − k, . . . , Sn − k). Now consider the following differential operator on Rd+1 : LS˜ f (t˜ ) :=

˜ t˜ ) · ∇ ˜ f (t˜ ) ∇S˜ Φ( S . ˜ ˜ i|∇ Φ(t )|2 S˜

Since α = 0, the operator LS˜ is well defined because the denominator is nonzero. The standard integration-by-parts argument dictates that N ˜ ˜ ˜ t˜ ) d t˜ = ei Φ(t˜ ) Lt ˜ ϕ( (LS˜ )N ei Φ(t˜ ) ϕ( ˜ t˜ ) d t˜ Iα = S

for each integer N 0, where Lt ˜ is the adjoint of LS˜ . Now an elementary induction on the S Leibnitz rule gives that, for each N , there is a constant CN depending on N (and the dimension d) such that N +1 γ

N N ˜ ˜ γ t N C |γ |=1 |∂S˜ Φ(t )| N ∂ ϕ( L (13) ϕ( ˜ t˜ ) ˜ t˜ ). ˜ N S˜ S ˜ t˜ )| ˜ t˜ )| ˜ |∇ Φ( |∇ Φ( ˜ S |γ |=0 S

˜ t˜ )|2 = 22l 4π 2 α 2 + At this point, several simplifications are in order. First, observe that |∇ Φ( ˜ S

2−2k |∇Φ(t)|2S > 1 + 2−2k |∇Φ(t)|2S . Next,

N N N γ γ −k|γ | γ 2−2k ∂ Φ( ∂ Φ(t) ˜ t˜ ) = ∂ 2 Φ(t) S S ˜ S

|γ |=2

|γ |=2

|γ |=2

(where the dummy multiindex γ is meant to have length d + 1 on the left-hand side and length d ˜ t˜ ) differs from Φ(t) by a linear term. Finally, in the two sums on the right-hand side) because Φ( N N γ γ ∂ ϕ( ∂ ϕ(t) ˜ S ˜ ˜ t ) CN χ[−2,2] (t0 )

|γ |=0

S

|γ |=0

1406


again by the Leibnitz rule and the compact support of ϕ0 . Combining these three observations with the inequality (13) and performing the (trivial) integral over t0 first gives (12) if k is chosen so that 2−k 2−k+1 . 2 The second idea to be used repeatedly throughout all that follows is contained in the proposition below. In simplest terms, the result is that the integral of certain simple ratios (which appear and will appear frequently) can be estimated by removing appropriate terms from the denominator and multiplying by an appropriate factor of two coming from the scale: Proposition 1. For any multiindex γ and any positive integer N sufficiently large (depending only on γ and the dimension), there is a constant CN,γ such that

2−|S |−γ ·S |t γ | dt C N,γ (|τ | + |t|S )N |τ |N −d−|γ |

(14)

for any scale S and any real τ . Proof. The inequality (14) follows immediately from a change of variables. Changing ti → |τ |2−Si for i = 1, . . . , d, the desired integral is equal to |t γ | dt, 2−|S |−γ ·S |τ ||γ |+d−N (1 + |t|)N and this new integral is clearly finite when N > |γ | + d.

2

2.3. Fractional differentiation Fractional differentiation is, of course, an essential component of Theorem 2. Here it will be useful to develop nonisotropic versions of the standard Bessel potentials (found, for example in Stein [22]). Since the operator (1) is not actually homogeneous, however, there will be more than one natural choice of scaling to use in defining the nonisotropic Bessel potentials; not only that, it will be necessary to make certain estimates of these Bessel potentials using conflicting families of dilations. For this reason, it is worthwhile to proceed in nearly complete generality and work with a large family of potentials. Recalling the fixed function ϕ0 on the real line, let ϕΠ (ξ ) := ni=1 ϕ0 (ξi ) (clearly ξ ∈ Rn ). Given any multiindex γ (with strictly positive entries) and any complex number s satisfying Re(s) 0, consider the tempered distribution Jγs whose Fourier transform is given by ∞ s ∧ 2sj ϕΠ 2−j γ ξ − ϕΠ 2−(j −1)γ ξ . Jγ (ξ ) := ϕΠ (ξ ) +

(15)

j =1

Note that when s is real, (Jγs )∧ (ξ ) is nonnegative for all ξ by the monotonicity conditions on ϕ0 . p A function f on Rn will be said to belong to the space Ls,γ (Rn ) provided that Jγs f p < ∞. When s is real, γ = 1 := (1, . . . , 1) and 1 < p < ∞, the Calderón–Zygmund theory of singular p integrals guarantees that the space Ls,γ (Rn ) is the usual Sobolev space. More generally, the p n space Ls,γ (R ) can be thought of as the space of functions which are differentiable to order s/γi in the ith coordinate direction. It also follows that ∂ l f ∈ Lp , 1 < p < ∞, provided that l · γ s.


1407

As there are various scalings to be exploited in the proofs to follow, it is necessary to record the behavior of the distribution Jγs when it is restricted to a box which has a potentially different scaling δ (that is, a box with side lengths approximately 2−δi for i = 1, . . . , n). For this reason, consider the distribution obtained from multiplying Jγs by the Schwartz function ϕΠ (2δ x). The resulting distribution will be called Jγs |δ ; its Fourier transform is given by the convolution s ∧ Jγ δ (ξ ) = 2−|δ|

∧ ϕˆ Π 2−δ (ξ − η) Jγs (η) dη.

(16)

Now ϕΠ (2−j γ ξ )−ϕΠ (2−(j −1)γ ξ ) = 0 when |2−j γi ξi | 1 for any value of the index i. It follows that on the support of ϕΠ (2−j γ ξ ) − ϕΠ (2−(j −1)γ ξ ), 2Re(s)j ( 12 |ξi |)Re(s)/γi . Hence it follows that n Re(s) s ∧ − Re(s) J (ξ ) 1 + 2 γi |ξi | γi . γ

i=1

Inserting this inequality into (16) gives that s ∧ J (ξ ) 2−|δ| γ δ

n Re(s) −δ − Re(s) γi γi ϕˆ Π 2 (ξ − η) 1 + 2 |ηi | dη. i=1

Now when Re(s) 0, 2− Re(s)/γi |ηi |Re(s)/γi |ξi |Re(s)/γi + |ηi − ξi |Re(s)/γi ; the result is that there exists a constant C independent of δ and Im(s) such that

n Re(s) s ∧ δ Re(s) J (ξ ) C 2 γ + |ξi | γi . γ δ i=1

The same procedure yields the more general family of inequalities

n Re(s) l s ∧ δ γi ∂ J (ξ ) C 2Re(s) γ + |ξi | δ γ δ

(17)

i=1

where, again, the constant does not depend on δ or Im(s). This inequality will be indispensable in applying the integration-by-parts lemma in the presence of a fractional differentiation which is not of the same sort of scaling as the rest of the integral. The standard arguments appearing in the theory of regular homogeneous distributions guarantee that Jγs − Jγs |δ is a C ∞ function which is, in fact, of rapid decay. Let j (x) be the inverse Fourier transform of the difference ϕΠ (2−j γ ξ ). The usual integration-by-parts arguments require that, for each positive integer N , there exists a constant C such that |0 (x)| CN,l (1 + |x|N )−1 . Rescaling, it follows that j |γ | j (x) CN 2 1 + |x|N jγ

1408

P.T. Gressman / Journal of Functional Analysis 257 (2009) 1396–1428 δ γ be defined for any two multiindices δ and γ of the same length of γδii as i ranges over all entries. Now for any multiindices δ and γ ,

for the same constant CN . Let to equal the maximum value |x|δ :=

n δ 2 2 i x i

1 2

=

i=1

n ( δi −j )γi j γ 2 2 γ i 2 i xi

1 2

2

( γδ −j ) mini γi

|x|j γ

i=1

provided j γδ . Therefore, taking the inverse Fourier transform of the right-hand side (15) and integrating over the set of x’s where |x|δ s J − J s C N γ γ δ 1

1 2

gives that

2Re(s)j +j γ ·l +

0j γδ

2

Re(s)j +j γ ·l+N ( γδ −j ) mini γi

.

j > γδ

Choosing N sufficiently large guarantees that s δ J − J s f Cγ ,Re(s) 2Re(s) γ f p γ γ δ p

(18)

for all 1 p ∞, uniform in Im(s) and δ. When no confusion will arise, the convolution operators corresponding to convolution with Jγs and Jγs |δ will simply be written as Jγs and Jγs |δ (i.e., the star will be suppressed). 2.4. Main decomposition The time has now come to describe the decomposition of the operator (1) which will be used to prove Theorems 1 and 2. The decomposition to be used here is inspired by the decomposition away from the singular set developed by Phong and Stein [15] and the later more elaborate decomposition of Christ, Nagel, Stein and Wainger [3]. The first step, as is easily imagined, is to decompose the support of the operator (1) away from the origin (x , x , y ) = (0, 0, 0) in a way that is consistent with the scalings of the homogeneity condition. Given an amplitude ψ supported near the origin, fix some smooth function ϕ on Rn × Rn × Rn which is identically one on the support of ψ and is itself compactly supported. Now let ψj (x, y ) := ψ(x, y ) ϕ 2j α x, 2jβ y − ϕ 2(j +1)α x, 2(j +1)β y and consider the following two families of operators: Tj f (x) := Uj f (x) :=

f y , x + S(x, y ) ψj (x, y ) dy , ∞ f y , x + S(x, y ) ψ(x, y )ϕ 2j α x, 2jβ y dy = Tl f (x). l=j

Clearly T = ∞ function on Rn j =0 Tj suitably defined. For example, if f (y , y ) is a Schwartz whose support is at a nonzero distance from the hyperplane y = 0, then Tf = ∞ j =0 Tj f with convergence in the Schwartz space topology (and, in fact, only finitely many terms of the sum


1409

are nonzero). This is because the supports of Tj f and Uj f are contained in a box of side lengths comparable to 2−j αi for i = 1, . . . , n, and the supports in y of the cutoffs for both operators are similarly restricted to a box of sides 2−jβi for i = 1, . . . , n . The operators Tj will be further decomposed (according to a new family of dilations which is potentially in conflict with the one already used). To that end, choose ϕ˜ to be a smooth function of compact support on Rn which is supported in the Euclidean ball of radius 1 and is identically one on the ball of radius 12 . Now for any nonnegative integers j, k, let (Pj k f )∧ (ξ , ξ ) := ϕ˜ 2−k−1 2−jβ ξ − ϕ˜ 2−k 2−jβ ξ fˆ(ξ , ξ ), (Qj f )∧ (ξ , ξ ) := ϕ˜ 2−jβ ξ fˆ(ξ , ξ ). Observe that for fixed k, the operators Pj k exhibit a scaling symmetry consistent with the homogeneity condition, but that for fixed j , the scaling is isotropic (and, hence, potentially conflicting). Observe that |ξ |−jβ 1 in the frequency support of Qj and 2k−1 |ξ |−jβ 2k+1 for Pj k , and that, for each j , the sum Qj +

∞

Pj k = I

k=0

where I is the identity operator. As with the Tj ’s, this equation can be interpreted as saying Qj f + ∞ j =0 Pj k f = f for any Schwartz function f supported a finite distance away from the hyperplane y = 0. In this case, the convergence is in the Schwartz topology, and every term Qj f and Pj k f retains the property that it is supported away from y = 0. The main decomposition of the operator T , then, will be the following sum over j and k: T=

∞

Tj Qj +

j =0

∞ ∞

Tj Pj k .

(19)

j =0 k=0

At one point, it will also be necessary to use the summation-by-parts equality ∞ j =0

Tj Qj = U0 Q0 +

∞

Uj (Qj − Qj −1 ).

j =1

In the a priori sense, this equality is valid because of the finite summation-by-parts formula N j =0

Tj Qj = U0 Q0 − UN +1 QN +

N

Uj (Qj − Qj −1 )

j =1

coupled with the fact that UN +1 QN f = 0 for N sufficiently large when f is supported away from y = 0. Lastly, each of these decompositions remains valid (i.e., is defined in the a priori sense) if a fractional differentiation operator is applied on one or both sides (though if JγsRR is applied on the right, the test function f must be chosen so that JγsRR f is supported away from y = 0 rather than f itself).

1410


3. “Trivial” inequalities This section contains the proofs of a variety of inequalities typically referred to as “size” or “trivial” inequalities, the reason being that the proofs of these inequalities typically do not depend on the geometry of S in any real way, only on the size of the support of the cutoffs involved. Of course, when fractional differentiations are added to the mix (as will be done shortly), oscillatory integral estimates and integration-by-parts arguments like Lemma 1 are necessary to establish even the trivial inequalities. Before making this addition, though, it is necessary and worthwhile to make a series of straightforward estimates which are not especially subtle in any way. In light of the decomposition (19), the indices j and k will be fixed from this point and through the next several sections to refer exclusively to the indices of summation in (19). Moreover, the following notation is adopted: the expression A B will mean that there exists a constant C such that, for all j, k 0, A CB (and so A B is only meaningful if one or both sides depend on either j or k). If the expression A or B includes a fractional integration, the expression A B means that A CB uniformly in j , k, and the imaginary parts of any fractional integration exponents. With this notational device in hand, the first and most basic set of inequalities to establish is the following:

Tj Qj 1→1 2−j |α | ,

(20)

Tj Pj k 1→1 2

−j |α |

,

(21)

Tj Qj ∞→∞ 2

−j |β |

,

(22)

Tj Pj k ∞→∞ 2

−j |β |

,

(23)

Tj Qj 1→∞ 2j |β | ,

Tj Pj k 1→∞ 2

j |β |+kn

(24) (25)

.

The unifying theme of these inequalities is that they are proved fairly directly from estimates of the size of the support of the amplitude ψj appearing in the definition of Tj . In fact, f y , x + S(x, y ) ψj (x, y ) dy dx Tj f (x) dx

f (y , x ) dx supψj (x, y ) dx dy 2−j |α | f 1 x

since 2−j |α | represents the size of the support of supx |ψj (x, y )| in x (for fixed y ). Similar reasoning gives that Tj ∞→∞ 2−j |β | . The Littlewood–Paley-type projections Qj and Pj k are uniformly bounded on Lp for all p (since each Qj can be appropriately rescaled to Q0 and each Pj k to P00 ); thus (20)–(23) follow. The main observation behind the L1 –L∞ inequality is that Fubini’s theorem guarantees that the following inequalities hold uniformly in j and k:

Qj L1 →L1 L∞ 2j |β | , y

y

Pj k L1 →L1 L∞ 2j |β y

y

|+kn

.


1411

The justification for these estimates is that both Qj and Pj k can be expressed as a convolution with a measure of smooth density on the hyperplane x = 0. The density is bounded by a constant times 2j |β | in the former case and 2j |β |+kn in the latter, which can be seen by simply rescaling the operators Qj and Pj k to coincide with Q0 and P00 as before. From these facts and the definition of Tj , however, Tj Pj k f (x) 2j |β |+kn ψj (x, y ) f (y , y ) dy dy 2jβ +kn f 1

Rn

(and likewise for Tj Qj ).

α 3.1. Fractional differentiation and L∞ –L∞ x BMOx bounds

In order to prove Theorem 2, it is absolutely essential to prove generalizations of (21) and (22) in the presence of fractional derivative operators (and (20) and (23) as well, but these are readily obtained from what is known about the dual operator T ∗ ). Moreover, to obtain a range of sharp results, it is necessary here just as in the work of Christ, Nagel, Stein and Wainger [3] to be able to sum the corresponding estimates in a critical case (here, when there is no decay in j of the norms of the individual terms). For this reason, stating the inequality as an L1 –L1 or L∞ –L∞ bound is unsatisfactory; even the Calderón–Zygmund weak-(1, 1) bound is unsuccessful here (unlike in [3]) because its proof requires that a separate Lp –Lp has already been established. In general, the operators here are expected to be bounded on Lp for a single value of p in the critical case (because the rate of decay varies as p varies, unlike the translation-invariant case in which it is constant). The solution is to prove a BMO-type inequality and to appeal to analytic interpolation. In this case, the operators in question may not even be bounded from L∞ to BMO, but they are bounded from L∞ to a mixed-norm space involving L∞ and a nonisotropic version of BMO. The space α will be designated L∞ x BMOx , and is defined to be the space of functions f for which there exists a constant Cf such that, for almost every x and any box B on Rn with side lengths 2sαi for i = 1, . . . , n (s ∈ R), 1 f (x) − f B,x dx Cf |B| B

where f B,x :=

1 |B| B

f (x , x ) dx . The inequality to be proved in this section, then, is that

when Re(sL ), Re(sR ) 0 and Re(sL ) γα˜L + Re(sR ) γβR = |β |, then for every fixed > 0, ∞

sL Tj Qj JγsRR JγL j =0

∞

sL Tj Pj k JγsRR JγL j =0

1,

(26)

L∞ →L∞ BMOαx x

L∞ →L∞ BMOαx x

uniformly in k, Im(sL ) and Im(sR ) (recall that

δ γ

2

:= maxi

k(Re(sL ) γ1 +Re(sR ) γ1 +) L

δi γi ).

R

,

(27)

1412


To prove (26) and (27), it is first necessary to revisit the “trivial” inequalities in the presence of fractional differentiation, as well as to introduce several new inequalities: s J L Tj Qj J sR

γL γR ∞→∞ 1, s k(Re(sL ) γ1 +Re(sR ) γ1 ) J L Tj Pj k J sR L R , γL γR ∞→∞ 2 s l J L ∂ Tj Qj J sR γL −j α γR ∞→∞ 1,

(28) (29) (30)

s l k(Re(sL ) γ1 +Re(sR ) γ1 ) sR J L ∂ L R , γL −j α −k1 Tj Pj k JγR ∞→∞ 2 s J L Tj Qj J sR ∞ ∞ 1 2−j |α | , γL γR L →L L s J L Tj Pj k J sR γL

x

(31) (32)

x

γR L∞ →L∞ L1 x x

2

−j |α |+k(Re(sL ) γ1 +Re(sR ) γ1 ) L

R

,

(33)

l where (∂−j α ) represents a scaled, mixed derivative in only the single-prime directions (i.e., not in the double-primed directions). The proofs of these inequalities are virtually identical because it will not be necessary to use the fact that Pj k is cutoff away from small frequencies, which is the main qualitative feature distinguishing it from Qj . For this reason, the attention will be focused primarily on (29), (31), and (33). In what follows, for the proofs of (28), (30), and (32), simply fix k = 0. By (18), it suffices to prove a modified form of (28)–(33). Specifically, it suffices to replace and JγsRR by JγsRR |jβ+k1 ; this is true by virtue of the identity JγsLL by JγsLL |j α+k1 ˜

0 JγL j α+k1 f + JγsLL − JγsLL j α+k1 f JγsLL f = JγsLL j α+k1 ˜ ˜ ˜

(34)

is the identity operator. Therefore, (and likewise for JγsRR ) which is itself true because Jγ0L |j α+k1 ˜ one may assume without loss of generality that differential inequalities of the form (17) hold (which will be necessary to apply Lemma 1). Tj Pj k JγsRR |jβ+k1 f (x). The function Vj k f (x) is For convenience, let Vj k f (x) := JγsLL |j α+k1 ˜ given by integration against a kernel Kj k (x, y), given by the expression (35) e2πi(ξ ·(x−w)+η ·(z −y )+η ·(w +S(w,z )−y )) ϕj k (ξ, η, w, z ) dξ dη dw dz , where the amplitude function ϕj k is equal to the product of several simpler pieces: the cutoff ψj (w, z ) from the definition of Tj ; the cutoff in η arising from Pj k , which happens to be )∧ (ξ ) supported on the set where |η |−jβ 2k+1 ; and finally, the Fourier transforms (JγsLL |j α+k1 ˜ sR ∧ and (JγR |jβ+k1 ) (η). In the case of (30) and (31), the amplitude that arises is slightly different. This time the amplitude is given by e−2πiη

·S(w,z )

l ∂−j α −k1 e2πiη ·S(w,z ) ϕj k (ξ, η, w, z )

(36)

with the derivative acting on the w variables only. This new amplitude can, of course, be expressed as a finite linear combination of scaled w -derivatives of ϕj k times a finite number of scaled w -derivatives of η · S(w, z ) (a simple integration-by-parts is all that is necessary to turn the derivative in x to a derivative in w ).


1413

At this point, the main piece of information needed to apply Lemma 1 is the scale S to be used. To that end, choose scale −j α − k1 in the w variable, j α + k1 in the ξ variable, and −jβ − k1 and jβ + k1, respectively, in z and η . In the remaining directions, the scales chosen are jβ + k1 in ξ and η and −jβ − k1 in w . With respect to the chosen scale, all scaled derivatives of degree at least two of the phase Φx,y (ξ, η, w, z ) := 2π ξ · (x − w) + η · (z − y ) + η · w + S(w, z ) − y are bounded uniformly in j and k; that is, for all j, k, |∂Sl Φx,y (ξ, η, w, z )| Cl when |l| 2. This fact is a direct consequence of the uniform convergence of the scaled derivatives of S, as in (9), coupled with the fact that βi > αi for all i. Likewise, the scaled derivatives of the cutoff ϕj k are all uniformly bounded in j and k (and the imaginary parts of sL and sR ) by a constant times

n n Re(sL ) Re(sR ) Re(sL )(j γα˜ +k γ1 ) Re(sR )(j γβ +k γ1 ) (γ ) (γ ) L L + R R + |ξi | L i |ηi | R i 2 (37) 2 i=1

i=1

and supported where |w |j α 1, |z |jβ 1 and |η |−jβ −k1 1. Note that this fact is also true of the scaled derivatives of the amplitude (36) since the scaled derivatives of the phase η · S(w, z ) are uniformly bounded. The magnitude of the scaled gradient of Φx,y , on the other hand, is greater than some fixed constant (independent of j and k) times 2k |x − w |j α + |x − w |jβ + |z − y |jβ + w + S(w, z) − y jβ + 2−k ξ − ∇w η · S(w, z )−j α + η + ∇z η · S(w, z )−jβ + 2−k ξ − η − ∇w η · S(w, z ) . −jβ

Again, since the scaled derivatives of η · S(w, z ) are uniformly bounded, there is a constant C0 independent of j and k such that the magnitude of the scaled gradient is greater than C0 times 2k |x − w |j α + |x − w |jβ + |z − y |jβ + w + S(w, z) − y jβ + 2−k |ξ |−j α + |η |−jβ + |ξ − η |−jβ − C0 . Choose < C0−1 and apply the integration-by-parts argument of Lemma 1. The result is that the kernel Kj k (x, y) (modulo a multiplicative constant independent of j and k) is bounded from above by the integral over ξ, η, w and z (suitably cutoff in w , z and η ) of a fraction whose numerator is (37) and whose denominator is N 2kN |x − w |j α + |x − w |jβ + |z − y |jβ + w + S(w, z) − y jβ N + 2−kN |ξ |−j α + |η |−jβ + |ξ − η |−jβ + 1

(38)

for any fixed positive integer N . To obtain the operator norm of Vj k on L∞ , the kernel Kj k (x, y) must be integrated over y and the supremum over all x is taken. This integral can be estimated by using Proposition 1 recursively: performing the y integral first, Lemma 1 dictates that the L∞

1414


operator norm of Vj k is less than the integral over ξ, η, w, z (suitably cutoff in w , z and η ) of a new fraction whose numerator is (37) times an additional factor of 2j |β| , but whose denominator is (modulo a multiplicative constant independent of j and k) N 2kN2 |x − w |j α + |x − w |jβ 2 N + 2−kN2 |ξ |−j α + |η |−jβ + |ξ − η |−jβ 2 + 1

(39)

for N2 := N − n. Proposition 1 is repeated for the integrals over w, ξ , and η (in the process, the triangle inequality |ξi | |ξi − ηi | + |ηi | is used when the ξ integral is encountered to make terms in the numerator match terms in the denominator). After these integrations are complete, the denominator is trivial (assuming that N was chosen sufficiently large). To conclude, the integrals over η and z are estimated using the size of the support of ϕj k in these directions. Collecting all the powers of 2 encountered in this way gives precisely the inequalities (28)–(31). 1 For the norm of Vj k as a mapping from L∞ to L∞ x Lx , the kernel Kj k is integrated in x and y and the supremum over x is taken. Just as before, Proposition 1 is applied recursively. This time the order of integration is y followed by x , then w , ξ and η . After these steps, the denominator is again trivial, and the remaining integrals over w , z and η are carried out by computing the size of the support of ϕj k in these directions. Collecting powers of 2 as before gives precisely the same result as above with the addition of another factor of 2−j |α | . In light of (28)–(33), the argument to establish (26) and (27) proceeds as follows. First observe that given any smooth function f on Rn and any box B ⊂ Rn of side lengths 2tαi for i = 1, . . . , n and some t ∈ R, the following inequality holds: 1 |B|

B

n 1 ∂f sα f (x) − f B,x dx 2 min 2 i .

f L∞ L1 , f ∞ , ∂x x x |B| i ∞ l=1

The first two terms on the right-hand side follow from fairly straightforward applications of the triangle inequality. The latter perhaps requires more explanation. The triangle inequality guarantees that 1 f (x) − f B,x dx 1 f (x , x ) − f (y , x ) dx dy , |B| |B|2 B

B B

and the fundamental theorem of calculus allows one to estimate the difference |f (x , x ) − f (y , x )| in terms of the gradient: 1 d f (x , x ) − f (y , x ) = f θ x + (1 − θ )y , x dθ dθ 0

n 1 x − y ∂f θ x + (1 − θ )y , x i i dθ ∂x i

i=1 0

2

n i=1

2

∂f ∂x .

tαi

i ∞


1415

sR For any bounded g, let f := JγsLL ( ∞ j =0 Tj Qj )JγR g. The inequalities (28), (30), and (32) give that 1 |B|

∞ n f (x) − f B,x dx min 2(j −t)|α | , 1, 2(t−j )αi g ∞ j =0

B

i=1

uniformly in t and g ∞ , of course. Summing in j and taking the supremum over B and x sL ∞ gives (26). If instead one takes f := JγL ( j =0 Tj Pj k )JγsRR g, the same reasoning gives that 1 |B|

f (x) − f B,x dx

B

∞

2

k(Re(sL ) γ1 +Re(sR ) γ1 ) L

R

min 2

(j −t)|α |

, 1,

j =0

n

2

(t−j )αi k

2

g ∞ ,

i=1

which yields (27) (in fact, it yields the slightly better inequality in which 2k is replaced by log(1 + k)). 4. L2 –L2 inequalities 4.1. Orthogonality inequalities The goal of this section is to prove the necessary orthogonality inequalities for the operators Tj Qj and Tj Pj k on L2 (Rn ). As in the previous section, a number of slightly different inequalities are necessary, but the proofs of these inequalities are nearly indistinguishable. The precise statement of these inequalities goes as follows: Fix sL , γL , sR , γR and a positive integer M. Then for any positive integers j1 , j, k, if |j − j1 | is sufficiently large (independent of the choices of j1 , j , and k) then Pj k J sL Tj J sR Pj k 2−(k+j +j1 )M , γL γR 1 2→2 (Qj − Qj −1 )J s1 Uj J s2 (Qj − Qj −1 ) 2−(j +j1 )M . 1 1 l1 l2 2→2

(40) (41)

If, in addition, j is sufficiently large, then it is also true that

k sL sR P0k JγL Tj JγR Pj k Q0 + k1 =0

2−(k+j )M ,

(42)

2→2

Q0 J sL Uj J sR (Qj − Qj −1 ) 2−j M . γL γR 2→2

(43)

Heuristically speaking, these inequalities assert that Pj k effectively commutes with Tj and Qj − Qj −1 likewise effectively commutes with Uj , so that, for fixed k, the terms of the decomposition (19) are effectively mutually orthogonal. The advantage of this, of course, is that it is precisely what is needed to apply the Cotlar–Stein almost-orthogonality lemma to conclude

1416


that the operator norm on L2 of the sum (for fixed k) is comparable to the supremum of the operator norms over j (which is an absolutely necessary element of the proof of Theorem 2). The proof to be given now is that of (40); all others are proved in a similar manner. Let Vj k := JγsLL Tj JγsRR Pj k . Conjugated by the Fourier transform, the operator Vj k has a kernel (on frequency space) given by Kj k (ξ, η) :=

e2πi(−ξ ·w+η ·z +η

·(w +S(w,z )))

ϕj k (ξ, η, w, z ) dw dz

where, as before, ϕj k is supported where |η |−jβ −k1 1, |w|j α 1, and |z |jβ 1; additionally,

n n Re(sL ) Re(sR ) (γ ) (γ ) ϕj k (ξ, η, w, z ) 1 + |ξi | L i |ηi | R i . 1+ i=1

i=1

Let jm := max{j1 , j }. To estimate the size of the kernel Kj k , a suitable scale S must be chosen. Choose scale −jm α − 12 k1 for w and −jm β − 12 k1 for z , then choose scale −jm α − 12 k1 for w . The family of phases Φξ,η (w, z ) satisfies ∇Φξ,η (w, z ) 2− k2 |ξ |−j α + |η |−j β + |ξ − η |−j α − C0 2 k2 m m m S for some constant C0 independent of j, j1 , and k (due to the uniform convergence of the scaled derivatives S as in (9)). The quantity 1 := mini βi − αi is strictly positive; clearly |ξ − η |−jm α 21 jm |ξ − η |−jm β . Now it must either be the case that |ξ |−jm β 2k or |η |−jm β 2k . If the former is true (which occurs when jm = j1 ), then |η |−jm β 2−2 |j −j1 | |η |−jβ 2−2 |j −j1 |+k , where 2 := mini βi . On the other hand, if jm = j , then |ξ |−jm β 2−2 |j −j1 |+k (this is the case which occurs in (42) and (43)). By the triangle inequality, then, |ξ − η |−jm α 21 jm +k when |j − j1 | is sufficiently large (for some bound uniform in j and k). It follows that, when |j − j1 | is sufficiently large, the scaled gradient of the phase satisfies the improved inequality ∇Φξ,η (w, z ) 2− k2 |ξ |−j α + |η |−j β + |ξ − η |−j α + 21 jm + k2 . m m m S To compute the operator norm on L1 associated to the kernel Kj k , apply Lemma 1 (and note that the scaled derivatives of ϕj k with respect to w and z are clearly bounded when the cutoff arises from the operators Tj or Uj ), then integrate over ξ and take the supremum over η. As in the previous section, Proposition 1 is applied to the integral in ξ . Next, the integrals in w and z are estimated using the size of the support of ϕj k . The fact that the scaled gradient of the phase has k magnitude no smaller than a constant times 21 jm + 2 gives that the L1 -operator norm is less than a constant times 2−Mjm −Mk for any fixed positive M by taking N sufficiently large in Lemma 1. The operator norm on L∞ is computed in a completely analogous way, integrating over η and taking the supremum over ξ ; the result is the same, i.e., the operator norm on L∞ can be made smaller than 2−Mjm −Mk provided |j −j1 | is sufficiently large. Finally, Riesz–Thorin interpolation gives (40).


1417

4.2. van der Corput inequalities In this section, the rank condition on the mixed Hessian (3) finally comes into play. Let r be the minimum value of the rank of (3) over (x , x , y ) = (0, 0, 0) and η = 0. The main inequalities to be proved in this section are that for j sufficiently large, for any sL , γL , sR , γR and any z | − Re(sL ) γα˜L − Re(sR ) γβR , it must be the case that satisfying Re(z) = |α |+|β 2 jz s 2 J L Tj J sR Qj jz s 2 J L Tj J sR Pj k

γL

γR

2→2

1,

γL

γR

2→2

2

(44)

−k 2r +k(Re(sL ) γ1 +Re(sR ) γ1 ) L

R

(45)

uniformly in j , k, Im(sL ), Im(sR ), and (of course) Im(z). As before, the inequality (18) and and JγsRR |jβ+k1 (in the identity (34) allow one to replace the fractional derivatives by JγsLL |j α+k1 ˜ the case of (44), take k = 0). Note that the condition that j be sufficiently large is the same as requiring that the cutoff ψ of the operator (1) is supported sufficiently near the origin, and so has no major effect on the potency of any of these inequalities. It is first necessary to further localize the cutoffs ψj (x, y ). To that end, let ϕ1 , . . . , ϕm be any finite partition of unity on the support of ψ0 . Define Tji f (x) := f y , x + S(x, y ) ψj (x, y )ϕi 2j α x, 2jβ y dy . The inequalities (44) and (45) will be proved with Tj replaced by Tji for i = 1, . . . , m, then summed over i to obtain the estimates originally desired. To simplify notation, the index i will be suppressed and it will simply be assumed that the cutoffs ψj are sufficiently localized around the points (2−j α x0 , 2−jβ y0 ). As is customary, the engine behind the proof is a T T ∗ argument; that is, the operator norm on 2 L of the operator 2 JγsLL j α+k1 Tj JγsRR jβ+k1 Pj k Tj∗ JγsLL j α+k1 ˜ ˜ (and likewise for Qj ) will be computed. Just as in the previous proofs, this operator is given by integration against a kernel Kj k (x, y) which is itself expressed as an oscillatory integral with phase Φx,y (ξ, η, ν, w, z, u , v ) given by 2π ξ · (x − w) + η · (z − y) + ν · u + ν · w − z + S(w, u + v ) − S(z, v ) and amplitude ϕj k (x, y, η, ξ, ν, w, z, u , v ) which is a product of these factors: ψj (w, u + v ) )∧ (ξ ) and (JγsLL |j α+k1 )∧ (η); finally and ψj (z, u ) from the definition of Tj ; (JγsLL |j α+k1 ˜ ˜ sR ∧ 2 |(JγR |jβ+k1 ) (ν)| and the modulus squared of the cutoff arising from Pj k . Once again, Lemma 1 will be the main computational tool once a suitable scale S is chosen. Choose scale −j α˜ − k1 for w and z and the dual scale j α˜ + k1 for ξ and η. Choose jβ + k1 for ν and jβ + k1 for ν , and choose −jβ − k1 for u and −jβ for v . The thing to notice about this choice of scale is that the derivatives in v do not have a factor of 2−k in the scale. In the language of Lemma 1, it is this special, asymmetric case which leads to operator van der Corput-type bounds for the kernel Kj k . Of course, this omission also means that one must take some extra care in analyzing the scaled derivatives of the phase.

1418


As before, one expects the scaled derivatives of order 2 or greater of the phase are uniformly bounded except for those various derivatives are taken exclusively in the ν and v directions (since all other derivatives have enough factors of k to balance the fact that |ν |−jβ ≈ 2k ). As for these exceptional derivatives, the relevant portion of the phase is examined by breaking it into two pieces. The first is the difference ν · (S(w, u + v ) − S(w, v )). The fundamental theorem of calculus provides the identity

n ν · S(w, u + v ) − S(w, v ) = 2k 2jβi ui

i=1

1 0

∂S 2−k ν · 2−jβi (w, θ u + v ) dθ ; ∂yi

it follows immediately from differentiating this equality that all scaled derivatives of ν · (S(w, u + v ) − S(w, v )) are bounded uniformly by a constant times |u |jβ +k1 since w, u , and v are restricted to be suitably small. The second piece to examine is ν · (w − z + S(w, v ) − S(z, v )). Again, the fundamental theorem of calculus gives that the scaled derivative of this piece with respect to νi is simply equal to

2

jβi +k

n wi − zi + 2j αl +k wl − zl l =1

+

n

1

2−j αl +jβi

0

2jβl +k wl − zl

1

l =1

2−jβl +jβi

0

∂Si θ w + (1 − θ )z, v dθ ∂xl

∂Si θ w + (1 − θ )z, v dθ ∂xl

where the integrals in the l sum are uniformly bounded and the integrals in the l sum tend to zero uniformly as j → ∞ (because of the uniform convergence (9) in both cases and the fact that the entries of β strictly dominate those of α ). Similarly, the scaled derivative of this second piece with respect to vi is equal to

n

2

j αl +k

wl

− zl

l =1

2−jβi −j αl −k ν ·

0

+

1

n

2

jβl +k

wl

l =1

− zl

1

∂ 2S θ w + (1 − θ )z, v dθ ∂xl yi

2−jβi −jβl −k ν ·

0

∂ 2S θ w + (1 − θ )z, v dθ. ∂xl yi

As before, the second integral tends uniformly to zero as j → ∞ by virtue of (9) and the domination of α by β . The first integral is, in the limit, an average of the (l , i)-entry of the mixed Hessian matrix H P over points near some fixed point (x0 , x0 , y0 , ν0 ) (without loss of generality, one may localize in ν with a finite partition of unity as was already done for the physical variables). Fixing Φ2 := ν · (w − z + S(w, v ) − S(z, v )), the information just given about the derivatives of this second term may be written in matrix form as

A 2jβ+k1 ∂ν Φ2 = 2j α ∂v Φ2 C

B D

2j α +k1 (w − z ) 2jβ +k1 (w − z )

(46)


1419

where A has uniformly bounded entries, B tends uniformly to an n × n identity matrix, C tends uniformly to an integral of the rescaled Hessian matrix (3), and D tends uniformly to zero. The rank condition on H P implies that there is an r × r submatrix of C which is invertible (with coefficients of the inverse bounded uniformly in j and k). For simplicity, assume that this submatrix lies in the first r rows and r columns of the full matrix C. For j sufficiently large, then, the matrix in (46) has an (r + n ) × (r + n )-invertible submatrix (which must contain B). It follows that for some uniform constant C, |∇ν Φx,y |jβ +k1 + |∇v Φx,y |−jβ

|w − z |

jβ +k1

+

r i=1

2

j αi +k

wi

− zi − C|u |jβ +k1

−C

n

2j αi +k wi − zi ;

i=r+1

furthermore, differentiating the identities for ∇ν Φx,y and ∇v Φx,y likewise gives that all scaled derivatives of the phase (of fixed order) are bounded uniformly by some constant times 1 + |u |jβ +k1 +|∇ν Φx,y |jβ +k1 +|∇v Φx,y |−jβ + ni=r+1 2j αi +k |wi −zi |. The full scaled gradient, however, has magnitude at least 2k |x − w |j α + |x − w |jβ + |z − y |j α + |z − y |jβ + 2−k |ξ |−j α + |ξ − ν |−jβ + |η |−j α + |η − ν |−jβ + 2k |u |jβ + 2−k |ν |−jβ + 2k w − z + S(w, v ) − S(z, v )jβ + ∇v ν · S(w, v ) − S(z, v ) −jβ − C.

Restrict attention for the moment to the situation in which |xi − yi | 2−k−j αi for i = r + 1, . . . , n . In this case 2j αi +k wi − zi 2j αi +k xi − wi + 2j αi +k yi − zi + 1 for i = r + 1, . . . , n . Thus if one decreases the scales of ν and v to equal jβ + (k − m)1 for ν and −jβ − m1, respectively, for some fixed m suitably large (independent of j and k and the imaginary parts of sL and sR ), it follows that all scaled derivatives of the phase have magnitude at most 1 + |∇Φx,y |S (up to a uniform multiple) and that the magnitude of the scaled gradient is at least 2k |x − w |j α + |x − w |jβ + |z − y |j α + |z − y |jβ + 2−k |ξ |−j α + |ξ − ν |−jβ + |η |−j α + |η − ν |−jβ + 2k |u |jβ + 2−k |ν |−jβ + 2k |w − z |jβ + 2k |w − z |j α − C. Now apply Lemma 1 and Proposition 1 recursively as before. Since the operator in question is self-adjoint, it suffices to compute its norm as a mapping on L1 , meaning that the kernel Kj k should be integrated over x and the supremum should be taken over y. The integration over x, ˜ performed first, gives a factor of 2−j |α|−kn and reduces the denominator to

1420


2k |z − y |j α + |z − y |jβ + 2−k |ξ |−j α + |ξ − ν |−jβ + |η |−j α + |η − ν |−jβ + 2k |u |jβ + 2−k |ν |−jβ + 2k |w − z |jβ + 2k |w − z |j α + 1 ˜ (taken to a suitably large power). Integration over w produces an additional factor of 2−j |α|−kn and eliminates the terms involving w on the last line. Integration over ξ then over η both give ˜ ˜ L +k1/γL ) (because of the growth of the cutoff ϕ ). Integration in factors of 2−j |α|−kn 2Re(sL )(j α/γ jk ˜ . Over u , one gets an additional 2−j |β |−kn . Integration z gives yet another factor of 2−j |α|−kn over ν (using the finite support in ν ) gives a factor of 2j |β|+kn 22 Re(sR )(jβ/γR +k1/γR ) . Lastly, the integral over v gives a factor of 2−j |β | because of its finite support. Altogether, this gives an operator norm less than some uniform constant times 2−j Re(z)−kn +2k(Re(sL )1/γL +Re(sr )1/γR ) (recalling the condition on z). Recall, however, that this estimate is derived under the assumption that |xi − yi | 2−k−j αi for i = r + 1, . . . , n . To achieve this condition, one must consider truncations of the form χl JγsLL Tj Pj k JγsRR , where χl is a multiplication operator restricting xi to a suitably small interval. Now

sL sR χ J T P J l γL j j k γR

2→2

l

1

χl J sL Tj Pj k J sR 2 γL

2

γR 2→2

l

by orthogonality of the truncated operators (because the truncation can, of course, be performed in a locally finite way). The sum over l has at most C2k(n −r) terms, yielding the estimates (44) and (45). 4.3. Application of the Cotlar–Stein lemma At this point, the inequalities (40)–(43) can be combined with (44) and (45) to show that, | when Re(sL ), Re(sR ) 0 and |α |+|β − Re(sL ) γα˜L − Re(sR ) γβR = 0, 2 ∞

sL Tj Qj JγsRR JγL j =0

1,

(47)

2→2

∞

sL Tj Pj k JγsRR JγL j =0

2

−k 2r +k(Re(sL ) γ1 +Re(sR ) γ1 ) L

R

,

(48)

2→2

uniformly in k and Im(sL ) and Im(sR ). The proof is simply an application of the Cotlar–Stein almost-orthogonality lemma. Let Rj k := JγsLL Tj Pj k JγsRR . The Littlewood–Paley-type projections Pj k ensure that Rj1 k Rj∗2 k = 0 when |j1 − j2 | is greater than some fixed constant (because the frequency supports are disjoint). On the other hand, Rj∗1 k Rj2 k = Rj∗1 k Q0 +

k k3 =0

P0k Rj2 k +

∞ j3 =0

Rj∗1 k Pj3 k Rj2 k .


1421

By (42), the first term has operator norm at most equal to some constant times 2−M(j1 +j2 +k) provided that j1 and j2 are sufficiently large (independent of k, Im(sL ), and Im(sR )). Likewise, when |j1 − j3 | is sufficiently large, each term in the sum has norm at most 2−M(j1 +j3 +k) by (40). On the other hand, when |j2 − j3 | is sufficiently large, the terms have norm at most 2−M(j2 −j3 +k) . In fact, when |j1 − j2 | is sufficiently large, both of the previous two cases must occur if either one occurs separately. Thus, for any value of j2 , it must be the case that ∞ ∗ ∗ 2k(Re(sL ) γ1 +Re(sR ) γ1 ) R Rj k L R j1 k 2 2→2 + Rj1 k Rj2 k 2→2 2

j1 =0

uniformly in j2 , k, and the imaginary parts of sL and sR . This gives (48) by the Cotlar–Stein lemma. The proof of (47) proceeds in essentially the same manner after a (crucial) summation by parts. In particular, ∞

Tj Qj =

j =0

∞

(Uj − Uj +1 )Qj = U0 Q0 +

j =0

∞

Uj (Qj − Qj −1 ).

j =1

Now the operator JγsLL U0 Q0 JγsRR is clearly bounded on L2 uniformly in the imaginary parts of sL and sR (the argument does not differ from that of (28)). Now let Rj := JγsLL Uj (Qj − Qj −1 )JγsRR . As before, Rj1 Rj∗2 = 0 when |j1 − j2 | is sufficiently large. But the identity Rj∗1 Rj2 = Rj∗1 Q0 Rj2 +

∞ j3 =1

Rj∗1 (Qj3 − Qj3 −1 )Rj2

and the inequalities (41) and (43) guarantee that each term has operator norm rapidly decaying in both |j1 − j3 | and |j2 − j3 | when |j1 − j2 | is sufficiently large. 5. Interpolation and summation 5.1. Lp –Lq inequalities In this section, the inequalities (20)–(25) and (45) are combined to obtain the promised Lp improving estimates for the averaging operator (1). The key is to establish the restricted weaktype estimates at the vertices of the appropriate polygon in the Riesz diagram, then interpolate with the Marcinkiewicz interpolation theorem. To begin, consider the operator j Tj Qj . Riesz–Thorin interpolation of (20) and (22) gives that Tj Qj is bounded on Lp with an operator norm at most some fixed constant times 2−j |α |/p−j |β |/p where p1 + p1 = 1. Choose any such p, and for simplicity, let θ := p1 . Now for any two measurable sets E and F ,

1422


∞ Tj Qj χE (x) dx χF (x) j =0

C

∞

min 2−j (θ|α |+(1−θ)|β |) |E|θ |F |1−θ , 2j |β | |E||F |

j =0

by Lp -boundedness of Tj Qj as well as L1 –L∞ boundedness coming from (24). Now there is a single value of j (call it j0 and note that j0 possibly negative and almost assuredly not an integer) for which the two terms appearing in the minimum on the right-hand side are equal. Away from this special value j0 , the minimum must decay geometrically with a ratio that is independent of |E| and |F |. Therefore the sum of all terms with j > j0 is dominated by some constant times the size of the term with j = j0 , and likewise for the terms with j j0 . Solving the equation |E|1−θ |F |θ 2j0 (|β |+θ|α |+(1−θ)|β |) = 1 and substituting gives that ∞ | θ|α|+(1−θ)|β ˜ 1− ˜ θ|β | ˜ Tj Qj χE (x) dx C |E| θ|α|+(1−θ)|β| |F | θ|α|+(1−θ)|β| . χF (x) j =0

From here, varying θ ∈ [0, 1], using the Marcinkiewicz interpolation theorem, and doing some arithmetic give that j Tj Qj maps Lp to Lq provided that ˜ |β| |α| − = |β | p q and 1 < p < q < ∞. As for the operator j k Tj Pj k , the procedure is in principle the same. First of all, the inequalities (21), (25), and (45) (with sL = sR = 0) give that ∞ ∞ Tj Pj k χE (x) dx χF (x) k=0 j =0

C

∞ ∞

|α |+|β | r 1 1 min 2j |β |+kn |E||F |, 2−j |α | |E|, 2−j 2 −k 2 |E| 2 |F | 2 .

k=0 j =0

|+|β | Now provided that nr > |α |β , there is a unique pair of real numbers j0 and k0 at which | the expression being summed attains a maximum. See Fig. 2 for a schematic illustration of the regions on which the first, second, and third term of the minimum, respectively, is the minimum. Note that it is the condition on nr which guarantees that the level lines of the operator norms

| (i.e., the lines where j |β | + kn is constant in region I, j |α | in region II, and j |α |+|β − k 2r in 2 region III) form closed triangles. Now in each region, the operator norms decay geometrically as one moves away from (j0 , k0 ). Furthermore, the number of terms of any fixed magnitude grows linearly with the distance from (j0 , k0 ). Therefore, it is also true that the sum over all j and k is dominated by some constant times the value of the single term j = j0 , k = k0 . At this particular point,

˜ 0 n |F | = 1 = 2j0 2j0 |α|+k

|α |−|β | −k0 2r 2

1

1

|E|− 2 |F | 2 ;


1423

Fig. 2. The heavy lines indicate the regions in which one term is smaller than the other two; the finer lines indicate where the appropriate operator norm is constant.

solving gives j0 =

n log2 |E|−(n +r) log2 |F | |−|β |)n |α|r+(|α ˜

and k0 =

−|α| ˜ log2 |E|+|β| log2 |F | . |−|β |)n |α|r+(|α ˜

Substituting gives

∞ ∞ |α |n |α |(n +r) 1− ˜ |−|β |)n |−|β |)n ˜ Tj Pj k χE (x) dx C |E| |α|r+(|α |F | |α|r+(|α , χF (x) k=0 j =0

which gives precisely the vertex of the Riesz diagram circled in Fig. 1 and lying above the line 1 1 p + q = 1. Performing the same procedure using (23) for the second term instead of (21) gives the second nontrivial vertex in Fig. 1. 5.2. Sobolev inequalities To begin, observe that it suffices to replace the constraint (6) by the a priori stronger constraint that |α | 1 + |β | 1 − . s max α1 , . . . , αn , β1 , . . . , βn p p

Suppose that αj > βk . Fix η0 ∈ Rn to have kth coordinate equal to 1 and all other coordinates equal to zero; it follows from (2) and (9) that the matrix H P (x , x , y , η0 ) does not depend on xj . Now fix x0 to have j th coordinate equal to 1 and all others zero. The matrix H P (x0 , 0, 0, η0 ) = H P (0, 0, 0, η0 ) must have rank r, so there must be distinct indices l1 , . . . , lr and m1 , . . . , mr = β and so on through α + β = β . From this, it follows, (again distinct) such that αl1 + βm mr k lr k 1 however, that βk < r n

|α |+|β | |β |

|α |+|β | . r

Thus, if there were an αi greater than all entries of β , the condition

> could not hold. Modulo this small change, Theorem 2 follows somewhat more directly than do the Lp –Lq inequalities. Theorem 4 in Chapter IV, Section 5.2 of Stein [23] is easily adapted to yield an

1424


analytic interpolation theorem for an analytic family of operators Rz where Riτ maps L2 –L2 α (with operator norm bounded for all τ ∈ R) and R1+iτ maps L∞ to L∞ x BMOx . The key is to consider a partial sharp function of the form f (x , x ) := sup f (x , x ) − f B,x dx B

where B ranges over all boxes in Rn centered at x with appropriately nonisotropic side lengths. The usual techniques (for example, a distributional inequality relating the sharp function to the associated maximal operator) demonstrate that p g(x , x ) dx Cp g (x , x )p dx (for a.e. x ) for some finite constant Cp provided p < ∞; and Fubini’s theorem guarantees that the coercivity inequality f p Cp f p must hold for p < ∞ as well. The linearization technique found in Stein [23] now applies without further modification. The result is that, for any fixed 2 p < ∞ and any real sL , sR , γL , γR for which α β |α | 1 + |β | 1 − = sL + sR p p γL γR (taking sL and sR real) and any > 0, ∞

sL Tj Qj JγsRR < ∞, JγL j =0 p→p ∞

k(− r +s 1 +s 1 +) sL Tj Qj JγsRR 2 p L γL R γR JγL j =0

p→p

uniformly in k. Fixing sR = 0, for example, it follows that the fractional differentiation JγsLL applied to the sum (19) (summed over j first, then k) converges in the strong operator topology provided that pr > sL γ1L and where |αp | + |β |(1 − p1 ) = sL γαL . Taking γL = 1 gives boundedness p of T from Lp to Ls for p 2 as stated in Theorem 2. The inequalities for p 2 are proved by duality: when sL = 0 instead, fixing γR = 1 and |αp | + |β |(1 − p1 ) = sR γαR , and pr > sR γ1R give

p

that T is bounded from L−sR to Lp , so T ∗ must map Lp to LsR . Since T ∗ satisfies all the same homogeneity and rank conditions (with the roles of α and β suitably interchanged), the portion of Theorem 2 for p 2 follows from this estimate just derived for dual operators T ∗ . p

5.3. Necessity Necessity is shown by means of a Knapp-type example. Consider the condition (4) first. Let Es be a box in Rn with side lengths 2βi t for i = 1, . . . , n, and let Fs be a box in Rn with side lengths 2α˜ i t (here t is, of course, real). Consider the integral χFt (x)T χEt (x) dx.


1425

For all s sufficiently negative, the homogeneity conditions guarantee that the quantity χEs (y , x + S(x, y )) is identically one provided that (x , x ) ∈ Et and (y , x ) ∈ Ft for some fixed constant > 0 (here Et is the set Et scaled linearly and isotropically down by a factor of ). It follows that, when ψ is greater than 12 near the origin, one has

1 χFt (x)T χEt (x) dx 2n +n 2t (|α |+|β |+|β |) , 2

and taking the limit as t → −∞, it follows that 1 1 χF (x)T χE (x) dx C|Et | p |Ft |1− q t t |α| ˜ can hold uniformly for all s only if |β| p − q |β | (i.e., (4) must be satisfied for any appropriate choice of S). p As for condition (6), standard arguments give that, when T maps Lp to Ls for 1 0, one has

λ P T i

p→p

Cp λ−s

uniformly in λ, where λ ∧ Pi f (ξ ) = ψ λ−1 ξi fˆ(ξ ) for any smooth ψ supported in [−2, −1] ∪ [1, 2]; choose ψ to be nonnegative as well. Now consider the integral

λ Pi χFt (x)

χEt y , x + S(x, y ) ψ(x, y ) dy dx.

Choosing λ = 2−tβi for some fixed, small , the function (Piλ χFt )(x) will be larger than some small constant times the characteristic function χFt (x) provided that |xi | 2tβi +1 (which is true of the support of T χEt when t is sufficiently negative). Thus, Sobolev boundedness implies that

2t (|α |+|β |+|β

|)

˜ C 2tsβi 2|β|/p+|α|(1−1/p)

for all t < 0; letting t → −∞ and taking a supremum over i gives (6). 6. Genericity considerations The proof of Theorem 3 is an application of the implicit function theorem. To begin, it is necessary to recall certain basic facts about degenerate matrices (in particular, the point of view adopted here is standard in the literature of singularity theory; see, for example [5]). Suppose M

1426


is an n × n matrix of rank r. Transposing the order of rows and columns as necessary, it may be assumed that M has the following block form: ⎤

⎡ ⎢ ⎢ ⎣

A

B ⎥ ⎥, ⎦

C

D

where A is an r × r invertible submatrix, and B, C, and D are r × (n − r), (n − r) × r, and (n − r) × (n − r) submatrices, respectively. The usual row-reduction arguments guarantee that D − CA−1 B = 0 for the matrix M. Furthermore, this equation continues to be satisfied for all small perturbations of M which are also rank r matrices (where A, B, C, D are, of course, replaced by their perturbations). Suppose that M is some smooth mapping from a neighborhood of the origin in Rk into the space of n × n matrices such that M0 (that is, the matrix to which the origin maps) is equal to M. The implicit function theorem, then, guarantees that the codimension (in Rk ) of the set of points near the origin mapping to a matrix of rank r is at least equal to the rank of the differential of M at the origin minus n 2 − (n − r)2 . Now let Pαl ,α ,β be the space of polynomials p in x , x and y (as always, x , y ∈ Rn and

x ∈ Rn ) for which p(2α x , 2α x , 2β y ) = 2l p(x , x , y ). For convenience, the subscripts α , α , and β will be suppressed as these multiindices are considered “fixed.” Now given any multiindex β , consider the following mapping from P β1 × · · · × P βn × R2n into the space of n × n matrices given by

(p1 , . . . , pn , x, y , η ) →

n ∂ 2 η k pk ∂xi yj x,y k=1

.

(49)

i,j =1,...,n

The goal is to compute the codimension in the space of “pairings” of polynomials and points, i.e., (p1 , . . . , pn , x, y , η ), of those whose mixed Hessian has rank r. In particular, if the codimension is large enough, then for a generic choice of polynomials (p1 , . . . , pn ) there will be no point (aside from the origin) at which the mixed Hessian has low rank. To compute the rank of the differential of this map, it suffices by rescaling to assume that the coordinates of x , y , x and η are either 0 or 1; and of course one may assume that η = 0 and that at least one of x , x , or y is also nonzero. Let K1 be the least common multiple of all the entries of α , α and β . Let Λ be the set of positive integers m which can be expressed as a sum m = αi + βj for some indices i and j . Now for any nonnegative integer k,

# (i, j ) K1 divides l − αi − βj = (n )2 .

l∈Λ+kK1

Fixing K2 to be the cardinality of Λ, it follows that for at least one value of l ∈ Λ + kK1 , there are , β at least K2−1 (n )2 pairs of indices (i, j ) for which K1 divides l − αi − βj (and therefore, αm m divide this difference as well for all appropriate values of m). It will now be shown that and βm the rank of the differential of (49) is at least equal to K2−1 (n )2 provided that all the entries of β are congruent to some element of Λ modulo K1 .


1427

Suppose that β is as described, i.e., the entries of β are all congruent to some element of Λ modulo K1 . Suppose that ηk0 = 0. For any pair of indices (i, j ) such that βk0 −αi −βj is divisible by k, it must be the case that there is a monomial in P p

βk

0

p

p

of the form xl xi yj , xl xi yj and

yl xi yj for any indices l and appropriate values of p in each case. If xi happens to be nonzero, p+1

then differentiating the k0 th polynomial of (49) in the direction of the monomial xi yj gives a matrix whose only nonzero entry is its (i, j )-entry. Likewise, if yj is nonzero, differentiation p+1

in the direction of xi yj

gives a matrix with only the (i, j )-entry nonzero. Finally, if both xi p

and yj are zero, then differentiating in the direction of one of the remaining monomials xl xi yj , p

p

xl xi yj or yl xi yj for which xl , xl or yl is nonzero also gives a matrix with only the (i, j )-entry nonzero. It follows that the codimension of points in P β1 × · · · × P βn × R2n which have mixed Hes−1 sians of rank r is at least (n − r)2 − (1 − K2 )(n )2 provided that the entries of β satisfy the congruence condition. If this codimension is greater than 2n, then it follows from projecting onto the space P β1 × · · · × P βn that such n -tuples of polynomials generically have mixed Hessians with rank everywhere (except the origin) greater than r. Thus, whenever r < n −

1 − K2−1 (n )2 + 2n,

the mixed Hessians (3) have rank everywhere equal to r or greater (except at the origin). References [1] J.-G. Bak, D.M. Oberlin, A. Seeger, Two endpoint bounds for generalized Radon transforms in the plane, Rev. Mat. Iberoamericana 18 (1) (2002) 231–247. [2] M. Christ, Restriction of the Fourier transform to submanifolds of low codimension, PhD thesis, University of Chicago, 1982. [3] M. Christ, A. Nagel, E.M. Stein, S. Wainger, Singular and maximal Radon transforms: Analysis and geometry, Ann. of Math. (2) 150 (2) (1999) 489–577. [4] S. Cuccagna, Sobolev estimates for fractional and singular Radon transforms, J. Funct. Anal. 139 (1) (1996) 94–118. [5] M. Golubitsky, V. Guillemin, Stable Mappings and Their Singularities, Grad. Texts in Math., vol. 14, SpringerVerlag, New York, 1973. [6] M. Greenblatt, Sharp L2 estimates for one-dimensional oscillatory integral operators with C ∞ phase, Amer. J. Math. 127 (3) (2005) 659–695. [7] A. Greenleaf, M. Pramanik, W. Tang, Oscillatory integral operators with homogeneous polynomial phases in several variables, J. Funct. Anal. 244 (2) (2007) 444–487. [8] A. Greenleaf, A. Seeger, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994) 35–56. [9] A. Greenleaf, A. Seeger, Fourier integral operators with cusp singularities, Amer. J. Math. 120 (5) (1998) 1077– 1119. [10] S. Lee, Endpoint Lp –Lq estimates for degenerate Radon transforms in R2 associated with real-analytic functions, Math. Z. 243 (4) (2003) 817–841; corrected reprint of Math. Z. 243 (2) (2003) 217–241 [MR1961865]. [11] S. Lee, Endpoint Lp –Lq estimates for some classes of degenerate Radon transforms in R2 , Math. Res. Lett. 11 (1) (2004) 85–101. [12] D.M. Oberlin, A restriction theorem for a k-surface in R1 , Canad. Math. Bull. 48 (2) (2005) 260–266. [13] D.H. Phong, E.M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math. 157 (1–2) (1986) 99–157. [14] D.H. Phong, E.M. Stein, Hilbert integrals, singular integrals, and Radon transforms. II, Invent. Math. 86 (1) (1986) 75–113. [15] D.H. Phong, E.M. Stein, Radon transforms and torsion, Int. Math. Res. Not. 4 (1991) 49–60.

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[16] D.H. Phong, E.M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. (2) 140 (3) (1994) 703–722. [17] D.H. Phong, E.M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1) (1997) 105–152. [18] D.H. Phong, E.M. Stein, J. Sturm, Multilinear level set operators, oscillatory integral operators, and Newton polyhedra, Math. Ann. 319 (3) (2001) 573–596. [19] M. Pramanik, C.W. Yang, Decay estimates for weighted oscillatory integrals in R2 , Indiana Univ. Math. J. 53 (2) (2004) 613–645. [20] V.S. Rychkov, Sharp L2 bounds for oscillatory integral operators with C ∞ phases, Math. Z. 236 (3) (2001) 461– 489. [21] A. Seeger, Radon transforms and finite type conditions, J. Amer. Math. Soc. 11 (4) (1998) 869–897. [22] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970. [23] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [24] T. Tao, J. Wright, Lp improving bounds for averages along curves, J. Amer. Math. Soc. 16 (3) (2003) 605–638. [25] A.N. Varˇcenko, Newton polyhedra and estimates of oscillatory integrals, Funktsional. Anal. i Prilozhen. 10 (3) (1976) 13–38.


Approximation of the semigroup generated by the Robin Laplacian in terms of the Gaussian semigroup Robin Nittka 1 Institute of Applied Analysis, University of Ulm, Germany Received 15 August 2008; accepted 8 May 2009 Available online 20 May 2009 Communicated by C. Kenig

Abstract We consider the Laplacian R subject to Robin boundary conditions ∂u ∂ν + βu = 0 on the space C(Ω), where Ω is a smooth, bounded, open subset of RN . It is known that R generates an analytic contraction semigroup. We show how this semigroup can be obtained from the Gaussian semigroup on C0 (RN ) via a Trotter formula. As the main ingredient, we construct a positive, contractive, linear extension operator Eβ from C(Ω) to C0 (RN ) which maps an operator core for R into the domain of the generator of the Gaussian semigroup. © 2009 Elsevier Inc. All rights reserved. Keywords: Trotter’s formula; Robin boundary conditions; Extension operator; Degenerate semigroup

1. Introduction Let Ω ⊂ RN be a bounded, open set of class C∞ and let β ∈ C∞ (∂Ω) be a non-negative function. We consider the Laplacian R on Ω subject to Robin boundary conditions ∂u + βu = 0 ∂ν

on ∂Ω.

(1.1)

E-mail address: [email protected]. URL: http://www.uni-ulm.de/mawi/iaa/members/assistants/nittka.html. 1 Supported by the graduate school M ATHEMATICAL A NALYSIS OF E VOLUTION , I NFORMATION AND C OMPLEXITY during the work on this article. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.009

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R. Nittka / Journal of Functional Analysis 257 (2009) 1429–1444

This operator, defined in a suitable way, generates an analytic contractive C0 -semigroup (TR (t))t0 on the space C(Ω). The aim of this article is to show how TR (t) can be obtained from the Gaussian semigroup (G0 (t))t0 on C0 (RN ) via a Trotter product formula. More precisely, we construct a positive, contractive, linear extension operator Eβ : C(Ω) → C0 (RN ) which maps an operator core for R into D(0 ), where 0 denotes the generator of (G0 (t))t0 . Let R : C0 (RN ) → C(Ω) denote the restriction operator Ru = u|Ω . The mentioned properties of Eβ turn out to be sufficient for n t RG0 Eβ n→∞ n

TR (t) = lim

(1.2)

to hold in the strong operator topology of C(Ω) uniformly for t ∈ [0, t0 ] for every t0 > 0. Defining the degenerate C0 -semigroup SR (t) := Eβ TR (t)R on C0 (RN ) and the projection β := Eβ R, identity (1.2) can equivalently be stated as E SR (t) = lim

n→∞

n n t β G0 t = lim G0 E Eβ , n→∞ n n

(1.3)

β is a trivial example see Proposition 4.3. This slightly changes the point of view. The projection E of a degenerate semigroup, so (1.3) takes the form of Trotter’s formula, compare [23]. Kato proved a very general Trotter formula in Hilbert spaces for sectorial forms [15,16] which also covers the case of degenerate semigroups. Since then, a vast number of research articles on Trotter formulae in Hilbert spaces have appeared. On the other hand, Trotter formulae in Banach spaces (as in our article) have seldom been studied, though recently convergence in operator norm has been examined by Cachia and Zagrebnov [5] and Blunck [4]. A particular special case of Trotter’s formula which has attracted some interest occurs when, as in (1.3), one of the semigroups is a projection, see for example the articles of Arendt and Batty [1] or Matolcsi and Shvidkoy [17,18]. The investigation of this situation has also been strongly motivated by its physical interpretation, compare [10–12,14,19]. However, in these articles typically the space L2 (Ω) and Dirichlet boundary conditions are considered. The important difference to our setting is that in the case of Dirichlet boundary conditions in L2 (Ω) one can use the natural projection of L2 (RN ) onto L2 (Ω). For Robin boundary conditions, however, it β . In particular, we want E β to be contractive in is more delicate to find a suitable projection E order to keep the Trotter products bounded. The article is organized as follows. In Section 2 we recall some geometric facts and some results about the Laplacian. In Section 3 we construct the extension operator Eβ related to the Robin Laplacian and prove that it has the aforementioned properties. The main result is contained in Section 4 where we prove formulae (1.2) and (1.3). Section 5 deals with the somewhat different case of Dirichlet boundary conditions. Finally, Section 6 suggests an application of our result. 2. Notation and preliminary results Let Ω be a smooth, bounded, open subset of RN . Here and in what follows, smooth always means of class C∞ . It is known that a neighborhood of ∂Ω can be decomposed into tangential and normal coordinates. But for the sake of completeness and because we use certain features of this decomposition later on, the following proposition recalls the precise statement of this fact.


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For this, we first need to explain how we make ∂Ω into a smooth manifold. Let x0 ∈ ∂Ω be arbitrary. After rotation Ω can locally at x0 be represented as the subgraph of a smooth function ϕ : U → R (U ⊂ RN −1 ). We will apply such a rotation frequently without further notice whenever we do calculations in local coordinates. Then z (2.1) ∂Ω ∩ V = z∈U ϕ(z) z for an open set V ⊂ RN . Thus z → ϕ(z) is a bijection of an open subset of RN −1 onto a neighborhood of x0 in ∂Ω. Using these mappings as charts we have made ∂Ω into a smooth manifold whose topology is the one induced by RN . Let f be a map from ∂Ω to Rk . As usual, we say that f is smooth if the representation f ∗ of f in local coordinates is smooth, i.e., if f ∗ : U → Rk z ∗ given by f (z) = f ϕ(z) is smooth for every chart. Similarly, f is called a diffeomorphism if f ∗ is a diffeomorphism for every chart and if in addition f is injective. Proposition 2.1. Let δ > 0 and consider T : ∂Ω × (−δ, δ) → RN ,

(p, t) → p + tν(p).

If δ > 0 is small enough, then T is a smooth diffeomorphism onto a neighborhood of ∂Ω. Here ν denotes the outwards pointing unit normal of Ω. For the proof we need to write ν is local coordinates. It is easily checked that in a given chart ν

z ϕ(z)

−∇ϕ(z) −1 −∇ϕ(z) = . 1 1

(2.2)

always pick a neighborhood of x0 in ∂Ω such that after Note, moreover, that for x0 ∈ ∂Ω we can z a suitable rotation we have x0 = ϕ(z00 ) and ∇ϕ(z0 ) = 0. Here ϕ is as in (2.1). This means that we can assume ν(x0 ) = eN , the N th unit vector in RN , without loss of generality. Proof. We first check that T is a local diffeomorphism z in local coordinates. For this, let x0 ∈ ∂Ω be arbitrary and fix a chart at x0 such that x0 = ϕ(z00 ) and ∇ϕ(z0 ) = 0. The representation of T in local coordinates is z z + tν . (2.3) T ∗ : U × (−δ, δ) → RN , (z, t) → ϕ(z) ϕ(z) Using (2.2), we see that T ∗ (z0 , 0) = I . Now the inverse function theorem asserts that T ∗ and hence T is a local diffeomorphism. It remains to show that T is injective for small δ. We have already shown that for every x ∈ ∂Ω there exist δx > 0 and an open neighborhood Ox of x in ∂Ω such that T is a smooth diffeomorphism from Ox × (−δx , δx ) onto a neighborhood of x in RN . Due to the compactness of ∂Ω we can choose finitely many Oxi , i = 1, . . . , m, which already cover ∂Ω. It is easily proved by contradiction that we can find δ > 0 such that for every x ∈ ∂Ω there exists an index k(x) ∈ {1, . . . , m} with the property that B4δ (x) ∩ ∂Ω ⊂ Oxk(x) , where Br (a) denotes the open

1432


ball in RN with center a and radius r. Without loss of generality we can assume that δ < δxi for all i = 1, . . . , m. For this choice of δ, T is injective. To see this, assume T (y1 , t1 ) = T (y2 , t2 ) where y1 , y2 ∈ ∂Ω and t1 , t2 ∈ (−δ, δ). Then 0 = T (y1 , t1 ) − T (y2 , t2 ) |y1 − y2 | − t1 ν(y1 ) + t2 ν(y2 ) |y1 − y2 | − 2δ. This shows |y1 −y2 | 2δ and thus y2 ∈ B4δ (y1 ), hence y1 , y2 ∈ Oxk , k = k(y2 ). By construction, T is injective on Ok × (−δ, δ), hence y1 = y2 and t1 = t2 , proving the claim. 2 Next we collect the definitions and some properties of the involved semigroups and their generators for reference. Let β : ∂Ω → [0, ∞) be a smooth function. The operator D(R ) := u ∈ H 1 (Ω) ∩ C(Ω) u ∈ C(Ω),

∇u∇ϕ +

Ω

uϕ +

Ω

uϕβ dσ = 0 for every ϕ ∈ H (Ω) , 1

∂Ω

R u := u on C(Ω) is called the Laplacian with Robin boundary conditions (1.1). A standard elliptic regularity result implies that D(R ) ⊂ C1 (Ω), see [13, Theorem 2.4.2.6]. However, D(R ) is not contained in C2 (Ω). In fact, a classical example of Sobolev [20, §22.2] shows that there exist functions u ∈ H 2 (R3 ) with compact support such that u ∈ C(RN ) and u∈ / C2 (RN ). But if a function u is in C2 (Ω), then it is in D(R ) if and only if (1.1) holds. It is known that under the above assumptions R generates a compact, analytic, contractive C0 -semigroup TR (t) on C(Ω), see the proof of [24, Theorem 3.3]. Lemma 2.2. The set D := D(R ) ∩ C∞ (Ω) is an operator core for R , i.e., D is dense in D(R ) with respect to the graph norm. Proof. Since R is the generator of a semigroup, the space n∈N D(nR ) is a core for R [8, Proposition II.1.8]. Moreover, R(1, R ) H m (Ω) ∩ C(Ω) ⊂ H m+2 (Ω) for every m ∈ N0 by the regularization properties of elliptic operators [13, Remark 2.5.1.2]. By a standard Sobolev embedding theorem [9, Section 5.6], N D nR = R(1, R )n C(Ω) ⊂ H 2n (Ω) ⊂ C2n−[ 2 ]−1 (Ω) for all n >

N 4.

Letting n → ∞ we obtain the assertion.

2

Let G2 (t) denote the C0 -semigroup on L2 (RN ) with generator D(2 ) := u ∈ L2 RN u ∈ L2 RN , 2 u := u.


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The semigroup G2 (t) leaves the space C0 (RN ) ∩ L2 (RN ) invariant, and its restriction extends continuously to a positive, contractive C0 -semigroup on C0 (RN ), denoted by G0 (t). The generator of this semigroup is D(0 ) := u ∈ C0 RN u ∈ C0 RN , 0 u := u. We will refer to both semigroups as the Gaussian semigroup. For more details about the Gaussian semigroup see for example [2, Chapter 3.7]. 3. Extension operator Let Ω be a smooth, bounded, open subset of RN and let β : ∂Ω → [0, ∞) be a smooth function. In this section we construct a positive, contractive, linear extension operator Eβ such that Eβ D ⊂ D(0 ) where D is an operator core for R . For β = 0, the operator is similar to the extension operator in [9, Section II.5.4]. For the whole section, let δ and T be as in Proposition 2.1. We start by fixing a “kinking function” . First choose a function 1 having the following properties. (a) (b) (c) (d) (e)

1 ∈ C∞ ([0, ∞) × [0, ∞)). 0 1 (γ , t) 1 for all γ , t 0.

1 (γ , t) = 0 for all t 2δ and γ 0.

1 (γ , 0) = 1 for all γ 0. ∂ ∂t 1 (γ , 0) = −2γ for all γ 0.

(f)

∂2

(γ , 0) = 4γ 2 ∂t 2 1

for all γ 0.

Here ∂t∂ 1 denotes the partial derivative of 1 with respect to the second argument. For example, we may choose 1 (γ , t) := exp(−2γ t)χ(t) where χ is a smooth cut-off function such that χ = 1 in a neighborhood of 0. Now define : Ω C → R to be

1 (β(z), t), if x = T (z, t), 0 t < δ,

(x) := 0, otherwise. Note that is well defined since T is injective, and it is smooth by construction. Definition 3.1 (Reflection at the boundary). Let x ∈ T (∂Ω × (−δ, δ)), x = T (z, t). We call x˜ := T (z, −t) the (orthogonal) reflection of x at the boundary ∂Ω. For a function u : Ω → R we define the reflected function u˜ : T ∂Ω × (0, δ) → R, u(x) ˜ := u(x). ˜ (3.1) We define the extension operator Eβ corresponding to β as N u, Eβ : C(Ω) → C0 R , Eβ u :=

u, ˜

on Ω, on Ω C .

Here u˜ is defined to be 0 outside T (∂Ω × (0, δ)); note that equals 0 in that region.

(3.2)

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Lemma 3.2. The operator Eβ is well defined, linear, positive, contractive and an extension operator, i.e., REβ = I , where R : C0 (RN ) → C(Ω), Ru := u|Ω . Proof. Let u ∈ C(Ω). By property (d), the function Eβ u is continuous on RN . Since it has compact support, Eβ u ∈ C0 (RN ). Positivity and contractivity follow from property (b). That Eβ is a linear extension operator is obvious from (3.2). 2 We now turn towards a more interesting property of Eβ : we prove that it maps the operator core D := D(R ) ∩ C∞ (Ω) into D(0 ). For this, we need several lemmata. ∂g denotes In the following we will adhere to the usual notation for normal derivatives, i.e., ∂ν the directional derivative of g along the outwards pointing unit normal with respect to the domain ∂g = −∇g · ν, where as of definition of g. Note that for functions defined on Ω C this means that ∂ν usual ν denotes the outwards pointing unit normal of Ω. Lemma 3.3. Let u ∈ D. Then Eβ u ∈ D(2 ) and (2 (Eβ u))|Ω = R u. Proof. The continuous function Eβ u has compact support, hence Eβ u ∈ L2 (RN ). Moreover, Eβ u is smooth on (∂Ω)C being the composition of smooth functions. Thus the function f :=

u, ( u), ˜

on Ω, on Ω C

is measurable and defined on all of RN aside from a set of measure zero. As u and u˜ are smooth up to ∂Ω, f is bounded. Note that f vanishes outside a bounded set, hence f ∈ L2 (RN ). It suffices to show that f = (Eβ u) in the sense of distributions. For this we calculate the (classical) normal derivative of u˜ using that u satisfies (1.1). For x ∈ ∂Ω we have u(x ˜ + hν(x)) − u(x) ˜ u(x − hν(x)) − u(x) ∂ u˜ (x) = − lim = − lim = −β(x)u(x) h→0 h→0 ∂ν h h and ∂

(x + hν(x)) − (x)

1 (β(x), h) − 1 (β(x), 0) (x) = − lim = − lim = 2β(x). h→0 h→0 ∂ν h h This implies ∂( u) ˜ ∂ u˜ ∂

∂u ∂

(x) = (x) (x) + (x)u(x) ˜ = (x) + (x)u(x) = β(x)u(x). ∂ν ∂ν ∂ν ∂ν ∂ν Now let ϕ ∈ D(RN ) be an arbitrary test function. From the above calculations, the classical Green formula [6, Section II.1.3] and u| ˜ ∂Ω = u|∂Ω , we obtain

(Eβ u)ϕ = uϕ +

uϕ ˜ RN

Ω

ΩC

uϕ +

= Ω

(∂Ω)+

∂ϕ ∂u u − ϕ dσ ∂ν ∂ν


+

=

( u)ϕ ˜ +

ΩC

uϕ + Ω

ΩC

(∂Ω)−

1435

∂ϕ ∂( u) ˜

u˜ − ϕ dσ ∂ν ∂ν

( u)ϕ ˜ =

f ϕ,

RN

where (∂Ω)+ is understood as the (oriented) boundary of Ω, whereas (∂Ω)− denotes the (oriented) boundary of Ω C . This shows (Eβ u) = f in the sense of distributions. 2 The remaining calculations will be carried out in local coordinates, i.e., locally at a point x0 = T ∗ (z0 , 0) ∈ ∂Ω. To be precise, we work with the following functions which are defined in a neighborhood of (z0 , 0): u∗ (z, t) := u T ∗ (z, t) , u˜ ∗ (z, t) := u˜ T ∗ (z, t) , β ∗ (z) := β T ∗ (z, 0) ,

∗ (z, t) := T ∗ (z, t) = 1 β ∗ (z), t . Here T ∗ is as in (2.3). It is a very useful feature that the Laplacian is invariant under rotation. Thus without loss of generality we can assume that ν(x0 ) = eN . Here and in the following, en denotes the nth unit vector in RN . Lemma 3.4. For n ∈ {1, . . . , N},

∂ ∗ −1 T (x0 ) = en , ∂xn ⎧ 0 2 ⎨ ∂ ∗ −1 ∂2 if n = N, − 2 ϕ(z0 ) T (x0 ) = ∂zn ⎩ ∂xn2 0 if n = N. Proof. Due to the assumption ν(z0 ) = eN we have T ∗ (z0 , 0) = I as in the proof of Proposition 2.1. By the inverse function theorem, −1 ∗ −1

(x) = T ∗ T ∗ −1 (x) . T For the partial derivatives at x0 this means ∂ ∗ −1 T (x0 ) = T ∗ (z0 , 0)−1 en = I en = en . ∂xn To calculate the second derivatives, we employ a differentiation rule for matrices, −A−1 (t)A (t)A−1 (t):

d −1 dt (A(t) ) =

∂ 2 ∗ −1 ∂ ∗ ∗ −1 −1 en T (x) = T T (x) ∂xn ∂xn2 −1 ∂ ∗ ∗ −1 ∗ ∗ −1 −1 T T = − T ∗ T ∗ −1 (x) (x) T T (x) en . ∂xn

1436


Denoting the entries of T ∗ by tij (i, j = 1, . . . , N ), we can proceed as follows: ∗ −1 ∂ ∗ −1 ∂ ∗ −1 tij T T (x). (x) = ∇tij T (x) · ∂xn ∂xn For x = x0 this yields ∂ ∂ tij T ∗ −1 (x0 ) = ∇tij (z0 , 0)en = tij (z0 , 0), ∂xn ∂zn where for notational simplicity we use zN as an alias for the variable t. Inserting this expression into the above identity, we arrive at

∂ 2 ∗ −1 ∂ T (x0 ) = − tij (z0 , 0) en ∂zn ∂xn2 i,j =1,...,N ∂ ∂2 =− tin (z0 , 0) = − 2 T ∗ (z0 , 0). ∂zn ∂zn i=1,...,N

In combination with formula (2.3) this finishes the proof.

2

Lemma 3.5. Let f ∗ be the representation of a function f in local coordinates, i.e., f ∗ (z, t) = f (T ∗ (z, t)). Then ∇f (x0 ) = ∇f ∗ (z0 , 0), f (x0 ) =

N −1 n=1

N −1

∂2 ∂2 ∗ ∂2 ∂ f (z0 , 0) + 2 f ∗ (z0 , 0) − f ∗ (z0 , 0) ϕ(z0 ). 2 ∂t ∂zn ∂t ∂zn2 n=1

In particular, ∇ (x0 ) = ( 0

−2β(x0 ) ) ,

(x0 ) = 4β(x0 )2 + 2β(x0 )

N −1 n=1

∂2 ϕ(z0 ). ∂zn2

Proof. Differentiating f (x) = f ∗ (T ∗ −1 (x)) we obtain ∂ ∗ −1 ∂ T (x), f (x) = (∇f ∗ ) T ∗ −1 (x) ∂xn ∂xn ∗ −1 ∂ ∗ −1 ∂2 ∂ ∗ −1 T ∗ T T T (x) f (x) = (x)H (x) f ∂xn ∂xn ∂xn2 ∂ 2 ∗ −1 T (x), + (∇f ∗ ) T ∗ −1 (x) ∂xn2 2

where Hf ∗ = ( ∂z∂i ∂zj f ∗ )i,j =1,...,N denotes the Hessian matrix of f ∗ . By using Lemma 3.4 and summing up, we arrive at the desired formulae for x = x0 .


Concerning we remark that ∗ (z, 0) = (β(z), 0) = 1 implies 1, . . . , N − 1). On the other hand, the derivatives with respect to t equal

1437

∂ ∗ ∂zn (z0 , 0)

= 0 (n =

∂ ∂ ∗

(z0 , 0) = 1 β ∗ (z0 ), 0 = −2β ∗ (z0 ) = −2β(x0 ), ∂t ∂t ∂ ∂ ∗

(z0 , 0) = 2 1 β ∗ (z0 ), 0 = 4β ∗ (z0 )2 = 4β(x0 )2 . ∂t 2 ∂t With this information, the formulae for follow from the general formulae.

2

Finally, it is easy to establish the relation between the derivatives of the function and its reflection at the boundary in local coordinates. It suffices to observe that u˜ ∗ (z, t) = u˜ T (z, t) = u T (z, −t) = u∗ (z, −t). From this we deduce the following formulae: u˜ ∗ (z, t) = u∗ (z, −t), ∂ ∗ ∂ ∗ ∂ ∗ ∂ u˜ (z, t) = − u∗ (z, −t), u˜ (z, t) = u (z, −t), ∂zn ∂zn ∂t ∂t ∂2 ∗ ∂2 ∗ u ˜ (z, t) = u (z, −t), ∂zn2 ∂zn2

∂2 ∗ ∂2 ∗ u ˜ (z, t) = u (z, −t). ∂t 2 ∂t 2

Now we are ready to prove the continuity of the function (Eβ u) at x0 . Note that u as well as ( u) ˜ have a continuous extension to ∂Ω. Lemma 3.6. For every u ∈ D, u(x0 ) = ( u)(x ˜ 0 ). Proof. Note that ∂ ∗ ∂ ∂u u˜ (z0 , 0) = − u∗ (z0 , 0) = − (x0 ) = β(x0 )u(x0 ) = β(x0 )u(x ˜ 0 ). ∂t ∂t ∂ν We use the formulae of this section to obtain ( u)(x ˜ 0 ) = (x0 )u(x ˜ 0 ) + 2∇ (x0 ) · ∇ u(x ˜ 0 ) + (x0 )u(x ˜ 0) ˜ 0 ) + 2β(x0 )u(x ˜ 0) = 4β(x0 ) u(x 2

N −1 n=1

+

N −1 n=1

=

N −1 n=1

N −1

∂2 ∂2 ∗ ∂2 ∂ u˜ (z0 , 0) + 2 u˜ ∗ (z0 , 0) − u˜ ∗ (z0 , 0) ϕ(z0 ) 2 ∂t ∂zn ∂t ∂zn2 n=1

N −1

∂2 ∂2 ∗ ∂2 ∗ ∂ ∗ u u (z , 0) + u (z , t) − (z , 0) ϕ(z0 ) 0 0 0 ∂t ∂zn2 ∂t 2 ∂zn2 n=1

= u(x0 ), i.e., the desired identity.

∂2 ∂ ϕ(z0 ) − 4β(x0 ) u˜ ∗ (z0 , 0) ∂t ∂zn2

2

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Combining the above observations, we can prove the main result of this section. Theorem 3.7. Let Ω ⊂ RN be a smooth, open, bounded subset of RN and let β be a non-negative, smooth function on ∂Ω. Then there exists a positive, contractive, linear extension operator Eβ : C(Ω) → C0 (RN ) such that Eβ maps D into D(0 ), where D := D(R ) ∩ C∞ (Ω) is the operator core for R as in Lemma 2.2. Proof. Define Eβ by (3.2). Then Eβ is a positive, contractive, linear extension operator as seen in Lemma 3.2. Let u ∈ D be arbitrary. It suffices to check that the distributional Laplacian (Eβ u) of Eβ u is in C0 (RN ). It follows from Lemma 3.3 that (Eβ u) is a function in L2 (RN ). The proof of Lemma 3.3 provides as with an explicit formula for (Eβ u). It follows from this formula and Lemma 3.6 that (Eβ u) is a continuous function on RN with compact support, hence a function in C0 (RN ). This finishes the proof. 2 We mention that the usual extension operator for Lipschitz domains as constructed for example in [21, VI.§3, Theorem 5] fails one of the above properties, namely contractivity, and is thus not suitable for the application in Section 4. In fact, no extension operator from C(Ω) to C0 (RN ) having the property that C∞ (Ω) is mapped into C∞ (RN ) can be contractive. 4. Semigroup approximation In this section we prove that if Eβ : C(Ω) → C0 (RN ) is a contractive, linear extension operator mapping an operator core D for R into D(0 ), then identity (1.2) holds true. Such an operator exists, see Theorem 3.7. The tool we use for the proof is the following approximation result for semigroups due to Chernoff. Theorem 4.1. (See [8, Theorem III.5.2].) Let X be a Banach space. Consider a function V : [0, ∞) → L (X) satisfying V (0) = I and V (t) 1 for all t 0. Assume that Ax := lim

h→0

V (h)x − x h

(4.1)

exists for all x ∈ D ⊂ X, where D and (I − A)D are dense subspaces in X. Then (A, D) is closable and A generates a contractive C0 -semigroup T (t), which is given by T (t)x = limn→∞ (V ( nt ))n x for all x ∈ X and uniformly for t ∈ [0, t0 ], t0 > 0. Theorem 4.2. Let Eβ : C(Ω) → C0 (RN ) be a contractive, linear extension operator which maps an operator core D for R into D(0 ). Then n t RG0 Eβ n→∞ n

TR (t) = lim

(4.2)

strongly on C(Ω) and uniformly on t ∈ [0, t0 ]. Proof. We apply Theorem 4.1 to X := C(Ω), V (t) := RG0 (t)Eβ and A := R |D , the restriction of R to D. It is clear that V (0) = I . Since R, G0 (t) and Eβ are contractions, V (t) 1 for every t 0. As R is the generator of a contraction semigroup on C(Ω) and D is an operator


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core for R , the density conditions are fulfilled. It only remains to check (4.1). For this let u ∈ D be fixed. By assumption we have Eβ u ∈ D(0 ), hence G(h)(Eβ u) − Eβ u V (h)u − u =R → R0 Eβ u (h → 0) h h by the definition of the infinitesimal generator. The function Eβ u agrees with u on Ω, hence R0 Eβ u equals u in the sense of distributions D(Ω) . By definition this means Au = R u = R0 Eβ u. Because A = R generates TR (t), the claim follows from Theorem 4.1. 2 β := Eβ R Now let Eβ : C(Ω) → C0 (RN ) be any continuous, linear extension operator, let E and define SR (t) := Eβ TR (t)R for t 0. Then (SR (t))t0 is a degenerate C0 -semigroup on C0 (RN ), i.e., (SR (t))t0 is strongly continuous on [0, ∞) and SR (s + t) = SR (s)SR (t) for all β , and (SR (t))t0 is a C0 -semigroup on the range of Eβ . We show s, t 0. In fact, SR (0) = E that (1.2) and (1.3) are equivalent. Proposition 4.3. The following assertions are equivalent, where every limit is in the strong operator topology and uniformly for t ∈ [0, t0 ], t0 > 0: (a) TR (t) = limn→∞ (RG0 ( nt )Eβ )n . β G0 ( t ))n . (b) SR (t) = limn→∞ (E n β )n . (c) SR (t) = limn→∞ (G0 ( nt )E Proof. Assume that (a) is true. The uniform convergence allows us to replace t on the right-hand nt → t, hence side by the sequence tn := n−1 TR (t) = lim RG0 n→∞

n−1 n−1 t t t Eβ Eβ = lim RG0 Eβ RG0 , n→∞ n−1 n−1 n−1

where the latter identity follows from the fact that the product of strongly convergent sequences converges strongly. As a consequence,

n n−1 t t β G0 t Eβ → Eβ TR (t)RI = SR (t). = Eβ RG0 RG0 E n n n

The other implications can be proved in an analogous manner.

2

5. Dirichlet boundary conditions Next we treat the Laplacian with Dirichlet boundary conditions D on a smooth, bounded, open subset Ω of RN . Surprisingly, for proving a Trotter formula there arise new problems compared to the previously treated case of Robin and Neumann boundary conditions, compare Remark 5.1. This is the reason why we consider it worthwhile to treat this operator separately. The Laplacian with Dirichlet boundary conditions defined by

D(D ) := u ∈ C0 (Ω) u ∈ C0 (Ω) ,

D u := u

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generates a positive, contractive C0 -semigroup TD (t) on C0 (Ω) [2, Theorem 6.1.8]. We want to find an analogue of (1.2) or (1.3) for TD (t) and thus an appropriate extension operator E∞ . Because the Dirichlet boundary condition u = 0 on ∂Ω can be regarded to be the limit of (1.1) as β → ∞, it is natural to pass to the limit β → ∞ in (3.2), hence arriving at E∞ : C0 (Ω) → C0 RN ,

E∞ u :=

u 0

on Ω, on Ω C .

Note that we had to replace the space C(Ω) by C0 (Ω) as we require E∞ u to be continuous. Unfortunately, replacing Eβ by E∞ in formula (1.2) is not suitable because the iteration scheme does not remain in C0 (Ω). However, formula (1.3) is well defined (and true) for L2 (Ω). In fact, L2 (Ω) is a closed subspace of L2 (RN ) if we consider its functions to be extended by zero, and the orthogonal projection onto this space is 1Ω . Thus, the analogue of formula (1.3) for Dirichlet boundary conditions reads n t 1Ω G2 u n→∞ n

TD,2 (t)u = lim

in L2 (Ω) for every u ∈ L2 (Ω),

(5.1)

where TD,2 denotes the C0 -semigroup on L2 (Ω) generated by the Dirichlet Laplacian. In fact, (5.1) is true under very mild regularity assumptions on the boundary of Ω. A description of the limit semigroup for arbitrary open domains can be found in [1, Theorem 5.3 and Proposition 7.1], whereas [3] contains characterizations of domains for which (5.1) holds. Remark 5.1. It is surprising that even for a smooth domain (5.1) cannot be proved using Chernoff’s product formula as in Section 4. In fact, to use the same approach we need that an operator core D for the Dirichlet Laplacian D,2 on L2 (Ω) is contained in D(2 ). Assume to the contrary that this were the case. Then E : L2 (Ω) → L2 (RN ) maps D into D(2 ), where the extension operator E extends functions by zero. Since Eu 2 = u 2 and 2 (Eu) 2 = D,2 u 2 : D → D(2 ) is bounded, where E is the restriction of E to D and for all u ∈ D, we get that E where both spaces carry the graph norm. Hence E extends to a bounded operator from D(D,2 ) = Eu for every u ∈ D(D,2 ). But a function u ∈ D(D,2 ) into D(2 ), and it follows that Eu satisfies Eu ∈ D(2 ) if and only if ∂u ∂ν = 0 on ∂Ω. However, there are functions in D(D,2 ) and even in C0 (Ω) ∩ C∞ (Ω) with non-vanishing normal derivative. This is a contradiction. Despite the aforementioned problems, it is possible to obtain a result for Dirichlet boundary conditions in the spirit of Section 4. For this, we need to replace 1Ω by a sequence of smooth cut-off functions. But we have to assure that they exhaust Ω sufficiently fast. To give the precise condition, we need the following lemma. Lemma 5.2. Let Ω be a smooth, bounded, open subset of RN . Then every natural number m > has the following property. Given t > 0, we can find a neighborhood Ut of ∂Ω such that u(x) t 2 (I − D )m u ∞ for every x ∈ Ω ∩ Ut and every u ∈ D(m D ).

N 2


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Proof. It is well known that TD (t) has a kernel representation with a non-negative, symmetric kernel kt ∈ C0 (Ω × Ω) which is dominated by the Gaussian kernel [7, Sections 3.2, 4.6, and 5.2]. Let m > N2 be fixed. Then also (I − D )−m is a positive kernel operator with the non-negative, symmetric kernel k ∈ C0 (Ω × Ω)

∞ k(x, y) = 0

s m−1 −s e ks (x, y) ds, (m − 1)!

see [8, Corollary 2.1.11]. Using compactness of Ω and ∂Ω we deduce that for any ε > 0 there exists a neighborhood Sε of ∂Ω such that x ∈ Ω ∩ Sε implies k(x, y) ε for all y ∈ Ω. Define t2 Ut := Sε , where ε := |Ω| . ) and define v := (I − D )m u ∈ C0 (Ω). For x ∈ Ω ∩ Ut , i.e., x ∈ Ω ∩ Sε , Now fix u ∈ D(m D we obtain

u(x) k(x, y)v(y) dy ε|Ω| v ∞ = t 2 (I − D )m u . ∞ Ω

This concludes the proof.

2

We have already explained why we cannot use E∞ as an extension operator. Instead, we choose u, on Ω, ED : C0 (Ω) → C0 RN , ED u := − u, ˜ on Ω C , which is similar to (3.2). Here denotes a cut-off function that equals 1 near ∂Ω. Using the same ideas as in Section 3 it can be shown that ED is a contractive extension operator that maps D(D ) ∩ C∞ (Ω) into D(0 ). In fact, the main difference to Section 3 is that we know u ∈ C0 (Ω) for u ∈ D(D ) which makes it easy to check the continuity of (ED u). Now let m > N2 be fixed. Choose a family (Ut )t>0 as in Lemma 5.2. We can assume that Ut is decreasing and t>0 Ut ∩ Ω = ∂Ω. For every t > 0 we fix a cut-off function χt ∈ C0 (Ω) satisfying 0 χt 1 and χt (x) = 1 for x ∈ Ω \ Ut . For convenience, define χ0 := 1Ω . To simplify notation, we use the multiplication operator χt also as an operator from C0 (RN ) to C0 (Ω) and χ0 also as the restriction from C0 (RN ) to C(Ω). We remark that in view of the kernel of TD (t) (t > 0) being strictly positive in the interior of Ω due to the strong maximum principle, it can be seen that for every compact set K ⊂ Ω there exists t0 > 0 such that Ut and K are disjoint whenever t < t0 , implying that χt → 1Ω pointwise as t → 0. In this sense, the next result is another flavor of formula (5.1). Theorem 5.3. Let m ∈ N and (χt )t0 be as above. Then n t ED u TD (t)u = lim χ t G0 n n→∞ n for every u ∈ C0 (Ω) uniformly on [0, t0 ] for every t0 > 0.

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Proof. We apply Theorem 4.1 to the operators V (t) : C0 (Ω) → C0 (Ω),

u → χt G0 (t)ED u.

∞ The properties V (0) = I and V (t) 1 for t 0 are obvious. Define D := D(m D ) ∩ C (Ω), which is a core for D . This choice makes the density conditions automatic once we have shown that the limit operator is D . It only remains to check (4.1). For this, let u ∈ D. Then χt G0 (t)ED u − u V (t)u − u − D u = − D u t t ∞ ∞ G0 (t)ED u − ED u − χ u D t t ∞ χt ED u − u + χt D u − D u ∞ . + t ∞

We estimate the three summands separately as t → 0. Since χt 1, ED u ∈ D(0 ) and 0 (ED u) = D u on Ω, the first expression tends to zero. The third term can be estimated by supx∈Ut |D u(x)| using that χt = 1 on Ω \ Ut . This expression tends to zero because D u ∈ C0 (Ω). The second summand can be estimated according to the choice of Ut . We obtain χt ED u − u = 1 χt u − u ∞ 1 sup u(x) t (I − D )m u → 0 ∞ t t t x∈Ut ∞ as t → 0. Together, these three estimates show that the remaining assumption of Theorem 4.1 is fulfilled. This completes the proof. 2 6. Applications Let Ω be a smooth, bounded, open subset of RN and let β : ∂Ω → [0, ∞) be a smooth function. We consider the (autonomous, homogeneous) diffusion equation ⎧ u = u, on (0, ∞) × Ω, ⎪ ⎨ t ∂u (6.1) (t, z) = −β(z)u(t, z), for t > 0 and z ∈ ∂Ω, ⎪ ⎩ ∂ν u(0, x) = u0 (x), for x ∈ Ω, subject to Robin boundary conditions. Here u0 ∈ C(Ω) is an arbitrary initial function. Typically one considers the unique mild solution u(t) := TR (t)u0 to be the right notion for a solution of (6.1), see for example [8, Section II.6]. If we want to calculate this solution numerically, a typical question is how to handle the boundary conditions. For simplicity, assume that Ω = (0, 1) and that we want to apply an explicit finite difference method. Then one replaces the derivatives ut and uxx by appropriate difference quotients and successively calculates approximations u(tn , xj ) of the exact solution u(t, x) by the relation u(tn+1 , xj ) − u(tn , xj ) u(tn , xj +1 ) − 2u(tn , xj ) + u(tn , xj −1 ) = , k h2


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where tn = n · k and xj = j · h for given small numbers k, h > 0. However, at the moment this does not work for the calculation of u(tn+1 , 0) and u(tn+1 , 1) because u(tn , −h) and u(tn , 1 + h) are not defined. For Dirichlet boundary conditions, one usually assigns u(tn , −h) := u(tn , 1 + h) := 0. On the other hand, for Neumann boundary conditions the solution is less obvious. One common technique is to use the auxiliary values u(tn , −h) := u(tn , h)

and u(tn , 1 + h) := u(tn , 1 − h),

(6.2)

which comes from a second order accurate approximation of the derivative at the boundary, compare [22, Section 8.3]. The operator E0 , i.e., Eβ for β = 0, as defined in (3.2) is the natural space-continuous analogue to (6.2). Thus the right-hand side of formula (1.2) reflects the iteration scheme we have described for discrete time, but in continuous space. In this sense, Theorem 4.2 shows that the numerical approach converges to the exact solution TR (t)u0 as k → 0. This justifies (6.2) for Neumann boundary conditions from a semigroup perspective. Moreover, our result immediately provides an extension of this numerical scheme to general Robin boundary conditions with the same convergence result. References [1] W. Arendt, C. Batty, Absorption semigroups and Dirichlet boundary conditions, Math. Ann. 295 (3) (1993) 427– 448. [2] W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monogr. Math., vol. 96, Birkhäuser Verlag, Basel, 2001. [3] W. Arendt, D. Daners, Varying domains: Stability of the Dirichlet and the Poisson problem, Discrete Contin. Dyn. Syst. 21 (1) (2008) 21–39. [4] S. Blunck, Some remarks on operator-norm convergence of the Trotter product formula on Banach spaces, J. Funct. Anal. 195 (2) (2002) 350–370. [5] V. Cachia, V. Zagrebnov, Operator-norm convergence of the Trotter product formula for holomorphic semigroups, J. Operator Theory 46 (1) (2001) 199–213. [6] R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1. Physical Origins and Classical Methods, Springer-Verlag, Berlin, 1990. [7] E. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92, Cambridge University Press, Cambridge, 1990. [8] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., vol. 194, Springer-Verlag, New York, 2000. [9] L. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, American Mathematical Society, Providence, RI, 1998. [10] P. Exner, T. Ichinose, A product formula related to quantum Zeno dynamics, Ann. H. Poincaré 6 (2) (2005) 195–215. [11] P. Exner, T. Ichinose, H. Neidhardt, V. Zagrebnov, Zeno product formula revisited, Integral Equations Operator Theory 57 (1) (2007) 67–81. [12] P. Facchi, V. Gorini, G. Marmo, S. Pascazio, E. Sudarshan, Quantum Zeno dynamics, Phys. Lett. A 275 (1–2) (2000) 12–19. [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math., vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. [14] K. Gustafson, A Zeno story, arXiv:quant-ph/0203032, 2002. [15] T. Kato, On the Trotter–Lie product formula, Proc. Japan Acad. 50 (1974) 694–698. [16] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in: Topics in Functional Analysis, in: Adv. Math. Suppl. Stud., vol. 3, Academic Press, New York, 1978, pp. 185–195, essays dedicated to M.G. Kre˘ın on the occasion of his 70th birthday. [17] M. Matolcsi, On the relation of closed forms and Trotter’s product formula, J. Funct. Anal. 205 (2) (2003) 401–413. [18] M. Matolcsi, R. Shvidkoy, Trotter’s product formula for projections, Arch. Math. (Basel) 81 (3) (2003) 309–317. [19] A. Schmidt, Zeno dynamics in quantum statistical mechanics, J. Phys. A 36 (4) (2003) 1135–1148.

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[20] S. Sobolev, Partial Differential Equations of Mathematical Physics, Pergamon Press, Oxford, 1964, translated from the third Russian edition by E.R. Dawson; English translation edited by T.A.A. Broadbent. [21] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton University Press, Princeton, NJ, 1970. [22] J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, second ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004. [23] H. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc. 10 (1959) 545–551. [24] M. Warma, The Robin and Wentzell–Robin Laplacians on Lipschitz domains, Semigroup Forum 73 (1) (2006) 10–30.


The Dixmier trace of Hankel operators on the Bergman space Miroslav Engliš a,b,1 , Richard Rochberg c,∗,2 a Mathematics Institute AS CR, Zitna 25, 11567 Prague 1, Czech Republic b Mathematics Institute, Silesian University, Na Rybncku 1, 74601 Opava, Czech Republic c Department of Mathematics, Washington University, St. Louis, MO 63130, USA

Received 4 September 2008; accepted 1 May 2009 Available online 6 June 2009 Communicated by N. Kalton

Abstract We give a formula for the Dixmier trace of (big) Hankel operators on the Bergman space of the disk or of finitely connected domains. For harmonic symbols we find the regularity required of the symbol for the formula to hold. © 2009 Elsevier Inc. All rights reserved. Keywords: Hankel operator; Bergman space; Dixmier trace

1. Introduction and summary Suppose f is a smooth function on the closed unit disk D. Let Hf be the (big) Hankel operator with symbol function f acting on the Bergman space of the unit disk, L2hol (D). We will show


E-mail addresses: [email protected] (M. Engliš), [email protected] (R. Rochberg). 1 The author’s work was supported by GA AV CR ˇ Grant No. IAA100190802 and Ministry of Education research plan

No. MSM4781305904. 2 The author’s work was supported by the National Science Foundation under Grant No. 0070642. Also he would like to thank the Mathematics Institute of the Czech Republic for their hospitality and support during the time this work began. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.005

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M. Engliš, R. Rochberg / Journal of Functional Analysis 257 (2009) 1445–1479

that |Hf | has a finite Dixmier trace and that, with T = ∂D, 1 Trω |Hf | = 2π

|∂f | dθ.

(1.1)

T

More generally let f1 , f2 , . . . , fk be additional smooth functions on the disk and let Tfi be the Toeplitz operators with symbol functions fi . Then T = Tf1 . . . Tfk |Hf | has a finite Dixmier trace given by 1 Trω (T ) = 2π

f1 . . . fk |∂f | dθ.

(1.2)

T

In the next section we present background information. In the section after that we prove that (1.1) and (1.2) hold. The basic idea is to recast the issue as one about pseudodifferential operators on the circle and then use the relationship between the Dixmier trace of a pseudodifferential operator and the integral of the principal symbol of the operator. We also show that, under appropriate restrictions on f, the function Tr(|Hf |z ) extends to a meromorphic function on the entire complex plane whose only singularities are simple poles at z = 1, 0, −1, −2, . . . . In Section 4 we consider the regularity that is necessary for (1.1); if f is harmonic and if either side of (1.1) is finite then so is the other and the two are equal. In the section after that we extend our results to Hankel operators on the Bergman space of finitely connected plane domains. In Section 6 we describe operators closely related to the Bergman space Hankel operator. The final section presents instances in which the right side of (1.1), or its analog for multiply connected domains, can be evaluated using considerations from function theory. In some cases that produces quantities which determine the conformal type of the domain. 2. Background 2.1. Spaces and operators Let L2 (D) be the Lebesgue space of the disk with respect to the normalized measure π −1 r dr dθ . Let P be the orthogonal projection onto the Bergman space, L2hol (D), the subspace of holomorphic functions and let P 0 be the projection onto conjugate holomorphic functions of mean zero. For a symbol function f on D one commonly defines the Toeplitz, big Hankel, and small Hankel operators on the Bergman space by Tf φ = P (f φ),

Hf φ = (I − P )(f φ),

hf φ = P 0 (f φ),

φ ∈ L2hol (D).

However it is convenient for us to extend these operators to all of L2 (D) by setting them to zero on L2hol (D)⊥ . Thus we adopt the definitions Tf := Pf P ,

Hf := (I − P )f P ,

hf := P 0 f P .

(2.1)

(The use of P 0 rather than P is to enhance the analogy with the situation in the Hardy space.)


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Let L2 (T) be the Lebesgue space of the circle with respect to the normalized measure (2π)−1 dθ . Recall that the Fourier coefficients of f ∈ L2 (T) are given by 1 f(n) = 2π

2π

f eiθ e−niθ dθ.

0

The Hardy space, H 2 , is the subspace of L2 (T) consisting of those f ∈ L2 (T) for which f(−n) = 0 for n = 1, 2, . . . . We write S for the orthogonal (Szegö) projection of L2 (T) to H 2 ; thus u(n), Su(n) = χ+ (n)

(2.2)

where χ+ is the characteristic function of Z0 . The Toeplitz and Hankel operators for the Hardy space, now for a symbol function f defined on the circle, are given by TfH := Sf S,

HfH = hH f := (I − S)f S.

To prevent confusion, we will reserve the undecorated symbols Tf , Hf for the Bergman space operators given in (2.1). 2.2. The Dixmier trace Recall that if A is a compact operator acting on a Hilbert space then its sequence of singular ∗ 1/2 arranged in nonincreasing values {sn (A)}∞ n=1 is the sequence of eigenvalues of |A| = (A A) order. In particular if A 0 this will also be the sequence of eigenvalues of A in decreasing p order. For 0 0 ). If A 0 is in S1 , the trace class, then A has a finite trace and, in fact, Tr(A) = sn (A). If however we only know that sn (A) = O n−1 or that σn (A) :=

n

sn (A) = O log(1 + n)

k=1

then A may have infinite trace. However in this case we may still try to compute its Dixmier trace, Trω (A). Informally Trω (A) = limN log1N N 1 sn (A) and this will actually be true in the cases of interest to us. We begin with the definition. Select a continuous positive linear functional ω on l ∞ (Z>0 ) and denote its value on a = (a1 , a2 , . . .), by Limω (an ). We require of this choice that Limω (an ) = lim an if the latter exists. We require further that ω be scale invariant; a technical requirement that is fundamental for the theory but will not be of further concern to us. For a positive operator A with

σn (A) log(1 + n)

∈ l∞

(2.3)

σn (A) we define the Dixmier trace of A, Trω (A), by Trω (A) = Limω ( log(1+n) ). Trω (·) is then extended by linearity to the full class of operators which satisfy (2.3). Although this definition does depend

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on ω the operators A we consider are measurable, that is, the value of Trω (A) is independent of the particular instance of Trω considered. We refer to [9] and [8] for details and for discussion of the role of these functionals. For 0 < p < ∞ a Hankel operator with conjugate holomorphic symbol function acting on the Hardy space is in Sp if and only if its symbol function is in the diagonal Besov space B p (D) and the same is true for the small Hankel operator on the Bergman space. The result for the Hardy space is in [22]. Also, Theorem 8.9 there, together with the natural unitary map from the Hardy space to the Bergman space, gives the Bergman space case. A similar result holds for the big Hankel operator on the Bergman space for p > 1. However at p = 1 the story changes, if Hf is in the trace class then Hf is the zero operator [2]. On the other hand, if f is smooth then it is always true that sn (Hf ) = O(n−1 ) [21]. Thus it is natural to consider Trω (|Hf |) and that is what we do here. 2.3. Related results A direct predecessor of this paper is the paper of Engliš, Guo, and Zhang [11]. A particular result there is that if Hf is the big Hankel operator acting on the Bergman space of the unit ball in Cd , d > 1, and f is holomorphic then we have Theorem 1. Trω |Hf¯ |2d =

d |∇f |2 − |Rf |2 dσ.

S

Here S is the boundary of the ball, dσ is its normalized surface measure and R is radial differentiation. In one dimension there is a rich relationship between the theory of Hankel operators and the geometric function theory. For instance we have the following: Theorem 2. Suppose φ is a holomorphic univalent map of the unit disk to a domain of finite area; Area( ) < ∞. The Hankel operator Hφ¯ is in the Hilbert–Schmidt class S2 and 1/2 Tr |Hφ¯ |2 = Area( ). Our theorem leads to statements of a similar spirit. Theorem 3. Suppose φ is a holomorphic univalent map of the unit disk to a domain which has a boundary of finite length, Length(∂ ) < ∞. Then Trω |Hφ¯ | = Length(∂ ). Furthermore, if f is holomorphic on then Trω Tf ◦φ |Hφ¯ | =

f (ζ ) |dζ |.

∂

The second equality recalls the following result of Connes and Sullivan [9, Ch. IV.3, Theorems 17, 26]. Suppose now that is a bounded domain and that ∂ is the limit set of a


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quasi-Fuchsian group. Suppose that the Hausdorff dimension of ∂ is p > 1 and that dp is the p-dimensional Hausdorff measure on ∂ . For the moment we consider operators on the Hardy space. Theorem 4. Suppose φ is a holomorphic univalent map of the unit disk to a domain such as described above. There is a nonzero number c so that if f is holomorphic on then H H p Trω Tf ◦φ Hφ¯ = c f (ζ ) dp (ζ ). ∂

3. Boutet de Monvel–Guillemin theory for the disc Let K denote the Poisson extension operator, acting from functions on the unit circle into harmonic functions on the unit disc and let γ be its inverse, i.e. the operator of taking the (suitably interpreted) boundary values. The operator K has a well known description in terms of the Fourier coefficients: namely, (3.1) Kf reiθ ≡ (Kf )r eiθ = f(m)r |m| emiθ . m∈Z

Here and below we will denote by Fr eiθ := F reiθ ,

0 r < 1, θ ∈ R,

the restriction of F to the circle rT. Viewing K as a (bounded linear) operator from L2 (T) into L2 (D) its adjoint K∗ is given by ∗ F (n) = K

1

r |n| Fr (n)2r dr.

(3.2)

0

In particular, K∗ K = , where is the Fourier multiplier (n) = f

1 f(n). |n| + 1

(3.3)

Setting U := K−1/2 it therefore follows that U is a unitary isomorphism of L2 (T) onto the subspace L2harm (D) of all harmonic functions in L2 (D). From the relations −1 K∗ K = I = γ K we see that γ = −1 K∗

on Ran K.

1450


Also, K−1 K∗ = Pharm

(3.4)

is the projection in L2 (D) onto L2harm (D). Indeed, K−1 K∗ is obviously selfadjoint, is an idempotent, vanishes on (Ran K)⊥ (since K∗ does), and acts as the identity on Ran K. Recall that a pseudodifferential operator ( DO for short) on R is an operator of the form 1 eixξ a(x, ξ )f(ξ ) dξ, (3.5) Af (x) = √ 2π R

where (abusing the notation slightly, but there is no danger of confusion) 1 f(ξ ) := √ f (x)e−ixξ dξ 2π

(3.6)

R

denotes the Fourier transform of a function f on R. Here a, called the “symbol” σA of A, is a function in C ∞ (R × R); it is usually required to satisfy the estimates n l ∂ ∂ a(x, ξ ) cn,l 1 + |ξ | m−l , x ξ

∀n, l = 0, 1, 2, . . . ,

for some m ∈ R — that is, to belong to Hörmander’s class S m (R × R). The DO (3.5) is called classical if a admits the asymptotic expansion a(x, ξ ) ∼

∞

(3.7)

am−j (x, ξ ),

j =0

where am−j ∈ S m−j (R × R) is positive homogeneous of degree m − j in ξ for |ξ | > 1, and “∼” means that a(x, ξ ) −

N −1

am−j (x, ξ ) ∈ S m−N (R × R),

∀N = 0, 1, 2, . . . .

j =0

We denote the vector space of all classical DOs of order m (that is, with a ∈ S m (R × R)) by m (R). One calls am the “leading”, or “principal”, symbol of A. The “total symbol” a(x, ξ ) can be recovered from A by a(x, ξ ) = e−ixξ A ei·ξ ·=x . If A ∈ m (R) and B ∈ n (R), then AB ∈ m+n (R) and σAB (x, ξ ) ∼

∞ j (−i)j j ∂ξ σA (x, ξ ) ∂x σB (x, ξ ) . j! j =0

See e.g. Folland [12,13] or Treves [29] for these properties of DO.

(3.8)


1451

It further turns out that DOs also behave well under coordinate changes, making it possible to define a DO on a compact manifold by declaring that it be of the form (3.5) when restricted to each coordinate chart; see again [12] or [29] for the details. In particular, we can define in this way classical DOs of order m on the unit circle T; we denote the vector space of all such operators by m (T). The above material on DOs is, of course, very standard; it turns out that for the particular case of the circle T there is a much more convenient variant using the Fourier coefficients instead of the Fourier transform. Recall that the sth order Sobolev space, W s (T) ≡ W s , on the unit circle consists of all distributions u on T for which 2 2s u(n) < ∞.

u 2s := 1 + |n| n∈Z

The intersection of all W s is C ∞ (T), and the usual Frechet topology on C ∞ (T) coincides with the one induced by the seminorms · s , s ∈ R (or s ∈ Z). Any continuous operator A : C ∞ (T) → W s1 into W s2 for some s1 and s2 ; and since the Fourier C ∞ (T) thus extends to annitoperator from u(n)e of u ∈ W s1 converges in W s1 , it follows that series u(eit ) = n u(n)Aenit Au eit = n∈Z

=

σA eit , n u(n)enit

(3.9)

n∈Z

(convergence in W s2 ), where σA eit , n := e−nit Aenit .

(3.10)

One calls operators of the form (3.9) periodic (or discrete) DOs (p DOs for short), with “periodic symbol” a = σ (A) ∈ C ∞ (T × Z). The periodic analogue of the Hörmander class is the space S m (T × Z) ≡ S m of all periodic symbols a satisfying j k it ∂t a e , n cj,k 1 + |n| m−k , n

∀j, k = 0, 1, 2, . . . ,

where n stands for the difference operator n a eit , n := a eit , n + 1 − a eit , n . One has also the obvious analogue of the asymptotic expansion (3.7) of a symbol, and of the notion of a classical DO; we denote the vector space of all classical p DOs of order m m by m per (T) ≡ per . Periodic DOs on T were studied by many authors, see e.g. Turunen and Vainikko [31] and the references therein. In particular, it was proved by McLean [17] that m m per (T) = (T),

i.e. that the “ordinary” and “periodic” DOs on T coincide. See also Saranen and Wendland [26], Melo [20], and Turunen [30]. The symbol calculus of p DOs was worked out by Turunen and

1452


Vainikko [31], who proved also the “periodic” analogue of the product formula (3.8): namely, n m+n for A ∈ m per and B ∈ per , AB belongs to per and ∞ 1 j it (j ) it n σA e , n ∂t σB e , n . σAB eit , n ∼ j!

(3.11)

j =0

(j )

Here ∂t

stands for the “shifted derivative” (j )

∂t

:=

j −1 k=0

1 ∂ −k . i ∂t

Note that, in particular, the operator from (3.3) and the Szegö projection S in (2.2) are p DOs of order −1 and 0, respectively, with symbols σ eit , n =

1 , |n| + 1

1 σS eit , n = χ+ (n) ≡ 0

for n 0, for n < 0.

After all these preparations, we can return to the Toeplitz and Hankel operators. Our strategy will be to transfer the operators (2.1) on L2 (D), via the isomorphism U , to operators on L2 (T), which turn out to be of the form (3.9), i.e. periodic DOs. We claim, first of all, that P K = KS.

(3.12)

Indeed, let u ∈ L2 (T) and set v = γ P Ku. Then for all w ∈ H 2 Ku, Kw = P Ku, Kw = Kγ P Ku, Kw = Kv, Kw, so

(u − v), w = 0.

It follows that S(u − v) = 0. As, by (3.3) and (2.2), S = S, and is invertible, we get Su = Sv = v. Thus KSu = Kv = P Ku, proving the claim. Combining (3.12) with (3.4) we see that γ P = γ P Pharm = γ P K−1 K∗ = γ KS−1 K∗ ,

(3.13)


1453

i.e. γ P = S−1 K∗ . Finally, let f ∈ L∞ (D). Then for any u ∈ L2 (T) U ∗ Tf U u = −1/2 K∗ Pf P K−1/2 u = −1/2 SK∗ f KS−1/2

by (3.12)

= S−1/2 K∗ f K−1/2 S

by (3.13).

That is, U ∗ Tf U = SBf S,

(3.14)

where Bf = −1/2 K∗ f K−1/2 . In other words, upon transferring the Toeplitz operator Tf on L2 (D), by means of the isomorphism U , to an operator on L2 (T), it becomes basically a Hardy-space Toeplitz operator but with the multiplication by f replaced by the operator Bf above. One of the starting points of the theory of Boutet de Monvel and Guillemin [7,6,15] is the following. Theorem 5. For f ∈ C ∞ (D), Bf is a classical p DO of order 0. More precisely, if f vanishes −k . The beginning of the expansion of the symbol of B is at ∂D to order k 0, then Bf ∈ per f ∂r f (eit ) σBf eit , n ∼ f eit + 2(|n| + 1) +

∂t2 f (eit ) + 2∂r f (eit ) + 2∂r2 f (eit ) − 2i∂t ∂r f (eit ) + ··· 8(|n| + 1)2

(3.15)

for n > 0; for n < 0, one replaces i by −i. Note that the theorem implies that SBf S also belongs to −k per , and σSBf S eit , n = σBf eit , n χ+ (n). In fact, for any p DO A we have by (3.11) σAS ∼ σSA ∼ σS σA = χ+ σA .

(3.16)

(For σAS , the right-hand side of (3.11) in fact coincides with σA σS ; for σSA , it differs from σS σA only at finitely many values of n.) Before giving the proof of the theorem, let us list some corollaries for Hankel and little Hankel operators.

1454


Corollary 1. For f ∈ C ∞ (D), the operator |Hf |2 = Hf∗ Hf = Tf f − Tf Tf , when transferred to L2 (T) via the isomorphism U , becomes the p DO U ∗ |Hf |2 U = SRf S

(3.17)

where Rf = Bf f − Bf SBf is a p DO of order −2, with leading symbol |∂r f (eit ) − i∂t f (eit )|2 |∂f (eit )|2 = 4(|n| + 1)2 (|n| + 1)2 for n > 0, and |∂f |2 /(|n| + 1)2 for n < 0. Proof. By (3.14), we immediately get (3.17). The assertion about the symbol follows from the formula (3.15) and the product rule (3.11) by a routine computation. 2 Corollary 2. For f ∈ C ∞ (D), Hf belongs to the Dixmier class, and 1 Trω |Hf | = 2π

|∂f | dθ. T

Proof. By the classical result of Wodzicki [32], if T is a DO of order −n on a compact manifold of real dimension n, then T is in the Dixmier class and Trω (T ) equals the integral of the principal symbol of T over the unit cosphere bundle |ξ | = 1. Now by Seeley’s work [28] on powers of elliptic DOs, McLean’s result about the coincidence of the ordinary and periodic DOs, and the preceding corollary, it follows that U ∗ |Hf |U = (SRf S)1/2 is a DO on T of order −1 with leading symbol |∂f (eit )|χ+ (ξ )/|ξ |. Taking T = U ∗ |Hf |U , the assertion follows. 2 Corollary 3. For any f1 , . . . , fk , f ∈ C ∞ (D), T = Tf1 . . . Tfk |Hf | belongs to the Dixmier class and 1 Trω (T ) = f1 . . . fk |∂f | dθ. 2π T

Proof. The first part is immediate from the preceding corollary since the Dixmier class is an ideal. Concerning the second part, observe that by (3.16), U ∗ T U = SBf1 SBf2 S . . . (SRf S)1/2 ∼ SBf1 Bf2 . . . Bfk S(SRf S)1/2 is a p DO of order −1 with leading symbol f1 f2 . . . fk |∂f |χ+ (n)/(n + 1). Appealing to Wodzicki’s result as in the preceding corollary completes the proof. 2


1455

Corollary 4. For f ∈ C ∞ (D), U ∗ |hf |U is a smoothing operator. Consequently, |hf | is in the Dixmier class and Trω (|hf |) = 0. Also, for any f1 , . . . , fk ∈ C ∞ (D), Trω (Tf1 . . . Tfk |hf |) = 0. Proof. Note that hf = (Pharm − P )f P . Thus h∗f hf = P f Pharm f P − P f Pf P . Using (3.4), (3.12), (3.14) and (3.13), this becomes U ∗ |hf |2 U = −1/2 K∗ P f K−1 K∗ f P K−1/2 − U ∗ Tf U U ∗ Tf U = −1/2 SK∗ f K−1 K∗ f KS−1/2 − SBf SBf S = S−1/2 K∗ f K−1 K∗ f K−1/2 S − SBf SBf S = SBf Bf S − SBf SBf S = SBf [Bf , S]S. But by (3.16), this is ∼ 0, proving the first part of the corollary. For the second part, note that U ∗ |hf |2 U =: T ∼ 0 implies that −2 T −2 ∈ −∞ per is a bounded operator on L2 (T); thus so is its square root (−2 T −2 )1/2 and, using polar decomposition, also T 1/2 −2 . Since 2 is trace class, it follows that T 1/2 = U ∗ |hf |U is trace class, and hence has vanishing Dixmier trace. The last part of the corollary also follows immediately, since trace class operators form an ideal. 2 Let us now turn to the proof of Theorem 5. We begin with a preparatory lemma. Lemma 1. For G ∈ C ∞ (T) and M = 0, 1, 2, . . . , the sum 1

M emiθ r 2n+m 1 − r 2 G(m)2r dr

|m|n 0

has the asymptotic expansion ∞ l=0

l i ∂ l−j iθ 1 j l+M cM (M + j ) G e (−1) l−j 2 ∂θ (n + 1)l+M+1

(3.18)

j =0

as n → +∞. Here cM (m) are the numbers defined recursively by c0 (m) = δm,0 , m−1 m

cM (l). cM+1 (m) = l l=M

(3.19)

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One can show that c1 (m) = 1,

c2 (m) = 2m − 2,

and, generally,

M M−j M cM (m) = (−1) j m. j j =0

Proof of Lemma 1. Let us first consider the case of M = 0, i.e.

e

1

miθ

r 2n+m 2r dr =

G(m)

|m|n

emiθ G(m) ≡ Qn . m n+ 2 +1

|m|n

0

Using the summation formula for geometric progression,

k m/2 N N −1 ) (−1)N ( n+1 n+1 k m/2 (−1) + = m m/2 n+ 2 +1 n+1 1 + n+1 k=0 (where N = 1, 2, 3, . . .), we get Qn =

N −1

m k (−1)k emiθ G(m) 2 (n + 1)k+1 |m|n

k=0

+

emiθ G(m)

|m|n

≡

N −1

(−m/2)N (n + 1)N +1 (n + m2 + 1)

Qn,k + Qn,N .

(3.20)

k=0

By integrating by parts, for m = 0, G(m)e

2π miθ

=

dt = G eit emi(θ−t) 2π

0

2π

emi(θ−t) N it dt , ∂ G e (mi)N t 2π

(3.21)

0

which yields the estimate |Qn,N |

−N 2n + 1 ∂ N G · 2 n ∞ (n + 1)N +1 t 2 +1

22−N ∂tN G ∞ , (n + 1)N +1

(3.22)


1457

i.e. the last summand in (3.20) is O((n + 1)−N −1 ), uniformly in θ . On the other hand, by (3.21) again but with k + j in the place of N (j = 2, 3, . . .), ∂ k+j G ∞ G(m) t k+j , m

(3.23)

so

k ∞ k+j

∂t G ∞ miθ −m 2 G(m)e 2 mj m=n |m|>n

k+j ≈ ∂t G∞ n1−j = O n−∞

since j can be taken arbitrary. Hence Qn,k + O n−∞ =

(−1)k m k miθ e G(m) 2 (n + 1)k+1 m∈Z

=

1 ∂ k 1 − emiθ G(m) 2i ∂θ (n + 1)k+1 m∈Z

=

1 ∂ 1 − 2i ∂θ (n + 1)k+1

k

G eiθ .

Combining (3.22) and (3.24) gives Qn =

N −1 k=0

k

iθ i 1 1 . ∂θ G e + O (n + 1)k+1 2 (n + 1)N +1

Since N was arbitrary, this proves the lemma for M = 0. For general M, note that n r 2n+m = − 1 − r 2 r 2n+m . Thus

−n

1 |m|n 0

e

miθ 2n+m

r

G(m)2r dr =

1

emiθ 1 − r 2 r 2n+m G(m)2r dr

|m|n 0

−

1

emiθ r 2n+2+m G(m)2r dr.

|m|=n+1 0

By (3.23) again, the last term is O(n−∞ ). Repeating the same argument M times, we get

(3.24)

1458

M. Engliš, R. Rochberg / Journal of Functional Analysis 257 (2009) 1445–1479 1

M emiθ r 2n+m 1 − r 2 G(m)2r dr + O n−∞

|m|n 0

1 = (−1)

M

M n

e

miθ 2n+m

r

G(m)2r dr

0

= (−1)

∞ i ∂ k iθ 1 ≈ G e (−1)M M . n 2 ∂θ (n + 1)k+1 M

M n Qn

(3.25)

k=0

By Taylor’s formula, we have for any ν ∈ C and |z| < 1, −ν

(1 − z)

=

∞ (ν)j j =0

j!

zj .

(3.26)

Here (ν)j := ν(ν + 1) . . . (ν + j − 1) is the Pochhammer symbol (raising factorial). Taking z = 1 − n+1 , we get −n

1 1 1 = − ν ν (n + 1) (n + 1) (n + 2)ν

−ν 1 1 1− 1+ = (n + 1)ν n+1 =−

∞ (ν)j (−1)j , j !(n + 1)ν+j j =1

with the series converging for n > 0, and also as an asymptotic expansion as n → +∞. Iterating the last formula yields M n

∞ 1 (−1)m (ν)m = cM (m), ν (n + 1) m!(n + 1)ν+m

(3.27)

m=M

with the numbers cM (m) given by (3.19). Take ν = k + 1; since (k + 1)m = (k + m)!/k!, we can also write the formula as M n

∞ k+m 1 cM (m) (−1)m = . k+1 m (n + 1) (n + 1)k+m+1 m=M

Substituting this into (3.25) yields (3.18), completing the proof of the lemma.

2

Proof of Theorem 5. That K∗ f K and, hence, Bf = −1/2 K∗ f K−1/2 , is a p DO of order 0 with leading symbol f |T is, of course, a fact from the theory of “Poisson” (like K) and “trace” (like γ ) operators initiated by Boutet de Monvel [5], combined with McLean’s result


1459

[17] that m = m per for any m ∈ R. For the disc, however, one can also proceed by a much simpler argument: namely, if u ∈ C ∞ (T), so that u(m) = O(m−∞ ) by (3.23), it follows from ∞ ∞ (3.1) that Ku ∈ C (D), and K maps C (T) into C ∞ (D) continuously. Similarly if F ∈ C ∞ (D), ∗ F (m) = O(m−∞ ), then by (3.2) and integration by parts in θ (as in (3.21)) it follows that K ∗ ∞ ∗ ∞ ∞ or K F ∈ C (T), and K maps C (D) into C (T) continuously. Hence for f ∈ C ∞ (D), K∗ f K maps C ∞ (T) into itself continuously. By the remarks preceding (3.9), it is therefore a p DO with symbol σK∗ f K ∈ C ∞ (T × Z), establishing the claim. It thus remains to show that σK∗ f K is classical, belongs to S −k (T × Z) if f vanishes at the boundary to order k, and has the asymptotic expansion as asserted. By (3.10), σK∗ f K eit , n = e−nit K∗ f Kenit .

(3.28)

Since both K and K∗ commute with complex conjugation, it follows that σK∗ f K eit , −n = σK∗ f K eit , n . Thus it is enough to consider n → +∞, i.e. to prove (3.15). So we will assume n > 0 from now on. From (3.28), (3.1) and (3.2), σK∗ f K eit , n = e−itn emit

1 2π

m

=

0 0

1 2π e(m−n)it

m

=

r dr dθ r |m| e−miθ f reiθ r |n| eniθ π

r dr dθ r |m|+|n| e(n−m)iθ f reiθ π

0 0

1 2π e

mit

m

r dr dθ . r |m+n|+|n| f reiθ e−miθ π

(3.29)

0 0

We claim that the contribution from |m| > n to the last sum is O(n−∞ ). Indeed, by (3.21), we have for any k 0 2π iθ −miθ dθ ∂θk f ∞ , f re e 2π mk 0

so 1 2π iθ −miθ dθ |m+n|+|n| mit r e f re e 2r dr 2π |m|>n

0 0

(3.30)

1460

M. Engliš, R. Rochberg / Journal of Functional Analysis 257 (2009) 1445–1479 1 ∂ k f ∞ θ r |m+n|+|n| 2r dr mk |m|>n

0

∂ k f ∞ 2 θ · mk n |m|>n 2∂θk f ∞ · n−k = O n−∞ , since k was arbitrary. (Here, as well as everywhere else in this paper, the various O(n... ) terms are always understood to hold uniformly in t or θ .) Thus

σ

K∗ f K

1 2π it −∞ r dr dθ mit . e ,n = O n + e r 2n+m f reiθ e−miθ π |m|n

(3.31)

0 0

Since f ∈ C ∞ (D), we can write, for any N = 1, 2, 3, . . . , −1 N j N f reiθ = 1 − r 2 Fj eiθ + 1 − r 2 GN reiθ ,

(3.32)

j =0

with Fj ∈ C ∞ (T) given by (−1)j ∂ j √ iθ Fj eiθ := f r e r=1 j ! ∂r j

(3.33)

and GN ∈ L∞ (D). Substituting this into (3.31), the contribution from the term containing GN can be estimated by 1 2π iθ −miθ r dr dθ mit 2n+m 2 N 1−r e r GN re e π |m|n

0 0

1

|m|n

=

N r 2n+m 1 − r 2 2r dr

GN ∞

0

GN ∞

|m|n

(n +

(2n + 1) GN ∞

( n2

m 2

N! + 1) . . . (n +

m 2

+ N + 1)

N! = O n−N . N +1 + 1)

On the other hand, the contributions from the Fj , |m|n

1 e

mit 0

j j (m)2r dr r 2n+m 1 − r 2 F


1461

are precisely of the form handled by the last lemma. Combining everything together, we thus obtain σK∗ f K

l i ∂ l−j 1 e ,n = FM eit l+M+1 2 ∂θ (n + 1) M=0 l=0 j =0

1 j l+M · (−1) cM (M + j ) + O . l−j (n + 1)N it

N −1 N −1−M

Since N was arbitrary, we see that σK∗ f K has the asymptotic expansion (−1)j cM (M + j ) l + M i ∂ l−j FM eit , σK∗ f K eit , n = l+M+1 l−j 2 ∂θ (n + 1)

(3.34)

M0, 0j l

uniformly in t, as n → +∞. Coming now back momentarily to (3.29), note that owing to the estimate (3.30) it is legitimate to differentiate (3.29) with respect to t term by term. Using again integration by parts, we thus get, for any j = 0, 1, 2, . . . , j ∂t σK∗ f K eit , n =

1 2π j mit

(mi) e

m

=

0 0

1 2π e

mit

m

=

r dr dθ r |m+n|+|n| f reiθ e−miθ π

r dr dθ r |m+n|+|n| f reiθ (−∂θ )j e−miθ π

0 0

1 2π e

mit

m

r dr dθ j r |m+n|+|n| ∂θ f reiθ e−miθ π

0 0

= σK∗ (∂ j f )K eit , n . θ

j

Consequently, by (3.34), ∂t σK∗ f K has also an asymptotic expansion as n → +∞, and in fact it j is the one obtained by applying ∂t to (3.34) term by term. Finally, as n r 2n+m = − 1 − r 2 r 2n+m for n 0, it follows from (3.31) that n σK∗ f K (eit , n) is again given by (3.31) except that f (reiθ ) is replaced by −(1 − r 2 )f (reiθ ). On the level of (3.32) and, hence, (3.34), this amounts to a sign change combined with the shift Fj → Fj −1 , which by (3.27) amounts in turn to applying n to the right-hand side of (3.34) term by term. j Combining the observations from the last two paragraphs, we thus see that we can apply kn ∂t to the right-hand side of (3.34) term by term, so that (3.34) is not only an asymptotic expansion

1462


(uniform in t) as n → +∞ in the sense of values, but even the asymptotic expansion (3.7) in the sense of symbols. Thus K∗ f K ∈ −1 per , and

σ

K∗ f K

it e ,n ∼

1

j,k,M0

(n + 1)j +k+M+1

i ∂θ 2

k

FM eit

j +k+M cM (M + j ), · (−1) k j

(3.35)

with FM given by (3.33). The first three terms (i.e. j + k + M 2) in the expansion are f (eit ) i∂t f (eit ) − ∂r f (eit ) σK∗ f K eit , n ∼ + n+1 2(n + 1)2 +

−∂t2 f (eit ) + ∂r f (eit ) − 2i∂t ∂r f (eit ) + ∂r2 f (eit ) + ···. 4(n + 1)3

(3.36)

If f vanishes at ∂D to order l, then F0 = · · · = Fl−1 = 0, so the summation in (3.35) is only −l−1 . over M l. This means that K∗ f K ∈ per Finally, using the fact that √ σ−1/2 eit , n = n + 1 and the product formula (3.11), the facts just established for K∗ f K are easily transferred into j the ones about −1/2 K∗ f K−1/2 = Bf . (Note that the differences n σ−1/2 can be handled 1 0 using (3.27) with ν = − 2 .) Thus, in particular, Bf always belongs to per , belongs even to −k per if f vanishes at ∂D to order k, and using (3.36) to evaluate the first terms in the expansion of σBf yields (3.15). This completes the proof of Theorem 5. 2 These ideas can be extended to show that Tr(|Hf |z ) is meromorphic with only simple poles. We will prove that but will not develop explicit formulas for the general residues. Theorem 6. Suppose f ∈ C ∞ (D) is such that ∂f does not vanish on T. Then the function ζ (|Hf |, z) := Tr(|Hf |z ) which is holomorphic in {z: Re z > 1} extends to a meromorphic function on the entire complex plane C, whose only singularities are simple poles at z = 1, 0, −1, −2, . . . , and Resz=1 ζ |Hf |, z = Trω |Hf | =

|∂f | dθ. T

Proof. We will use standard facts on complex powers Az and zeta functions ζ (A, z) = Tr(Az ) of positive elliptic DOs A, cf. e.g. Shubin [27]. We have seen that U ∗ |Hf |U = SQf S


1463

for a DO Qf on T of order −1 with asymptotic expansion Qf = |∂f | +

∞

gj j ,

j =2

with some gj ∈ C ∞ (T). If ∂f does not vanish on T, then Qf is elliptic, and thus by the standard theory of Seeley has complex powers Qzf = |∂f |z z +

∞

gj,z z+j ,

z ∈ C,

j =1

with some gj,z ∈ C ∞ (T) depending holomorphically on z ∈ C. For uniformity of notation we also set g0,z := |∂f |z . Since z j j g(ζ ) g ζ , ζ = dζ = ζ (z) g dθ Tr Sgz S = (j + 1)z j 0

j 0 T

T

(implying, in particular, that Tr(Az ) is finite and holomorphic in {z: Re z > 1} for any DO A on T of order 0), we obtain Tr |Hf |z = Tr SQzf S =

N −1 j =0

ζ (z + j )

gj,z + (a function holomorphic on Re z > 1 − N ), T

for any N = 1, 2, 3, . . . . Since ζ (z) extends to be holomorphic on C \ {1} and has a simple pole at z = 1 with residue 1, the theorem follows. 2 In principle, the use of periodic DOs can be circumvented by passing from the disc to the upper half-plane U = {x + yi ∈ C: y > 0}. The Cayley transform C(z) =

z−i z+i

is a biholomorphism of U onto D, and the weighted composition operator UC : f → (f ◦ C) · C is a unitary isomorphism of L2 (D) onto L2 (U), as well as of the corresponding Bergman subspaces L2hol (D) onto L2hol (U). A Toeplitz operator Tf , f ∈ L∞ (D), on D corresponds under this isomorphism to the Toeplitz operator Tf ◦C on L2 (U), and similarly for Hankel operators. The role of the Fourier coefficients is taken over by the Fourier transform (3.6), and the formula for the Poisson operator becomes 1 eixξ −y|ξ | f(ξ ) dξ, Kf (x + yi) ≡ (Kf )y (x) = √ 2π R

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where we are now denoting by Fy (x) := F (x + yi), x ∈ R, y > 0, the restriction of a function F on U to the line R + yi. The adjoint of K is given by ∗ F (ξ ) = K

∞

y (ξ ) dy e−y|ξ | F

0

and K∗ K = is again a Fourier multiplier (ξ ) = 1 f(ξ ). f 2|ξ | Also, for any f ∈ C ∞ (D), the function g = f ◦ C satisfies the estimates m n ∂ ∂ g(x + yi) x

y

cm,n , (|x| + y + 1)m+n+2

which serve as a substitute for the estimates (3.30). Using all this, the proof of Theorem 5 carries over with minor modifications also to the half-plane setting. However, the serious trouble that arises is that now the operator is no longer bounded on L2 (R) (and, in particular, K is also only densely defined and unbounded as an operator from L2 (R) into L2 (U)). This has the effect that the various DOs like K∗ f K, Bf , etc., have symbols with singularities of the form |ξ |−m at the origin. Although this technical difficulty can probably be circumvented, it seems much simpler to use the periodic DOs instead. Another difficulty with the half-plane approach is that the little Hankel operators on D are not mapped into the ones on U by the Cayley isomorphism UC (the reason being that UC maps holomorphic functions into holomorphic functions, but not conjugate-holomorphic into conjugate-holomorphic); thus for hf we need to work in D directly. (For the same reason, however, our Corollary 4 cannot be transferred to little Hankel operators on U.) We also remark that in higher dimensions (i.e. for the disc replaced by a bounded strictly pseudoconvex domain in Cn with smooth boundary), the Boutet de Monvel–Guillemin theory gets much more complicated. The main differences against the one-dimensional case are that S itself is no longer a DO on ∂ ; the operators S and need no longer commute; likewise, P K = KS in general; and [Bf , S] need not be smoothing. (In fact, one of the cornerstones of the theory is the result that there exists a DO Q such that SQS = 0 and [Bf + Q, S] ∼ 0.) For the case of the unit ball Bn of Cn , an analysis similar to ours has recently been done by Zhang, Guo and one of the authors [11], using instead of the Boutet de Monvel–Guillemin theory a related technique due to Howe [16]. It turns out that for Bn with n 2, Bf f − Bf Bf is of order −1 not −2 (its leading symbol being ∇f 2 − |Rf |2 — where R stands for the radial derivative — which vanishes if n = 1); hence, it is enough to evaluate one less term in the asymptotic expansions like (3.35) and (3.15), thus paradoxically making the case n 2 easier than the case n = 1 of the unit disc. In principle, it should not be difficult to obtain also our results by Howe’s method (“pseudo-Toeplitz operators” on the Fock space), and it would be no less interesting to have explicit formulas like (3.15) also for some higher-dimensional situations, e.g. for the unit ball Bn .


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4. Optimal regularity for harmonic symbols Theorem 5 was proved under the a priori assumption that f is smooth. In general we do not know how much that requirement can be relaxed; however if f is harmonic we can give a precise statement. First note that such f is uniquely decomposable as f = f1 + f¯2 with both fi ¯ = f and Hf = H ¯ . Hence we can restrict attention to H ¯ holomorphic and f1 (0) = 0. Also ∂f f2 f 2 for holomorphic f. A holomorphic function g is said to be in the Hardy space H 1 if

g H 1

1 = sup 00 is in weak l 1 , {sn } ∈ l 1,∞ , if sn∗ = −1 1,∞ quasinorm of such a sequence is sup ns ∗ . We say that the sequence is in l 1+ if O(n n n ).∗ The l n 1+ norm of such a sequence is sup(log(1 + n))−1 ∗ k=1 sn = O(log(1 + n)). The l k=1 sn . We 1 1,∞ 1+ then have the proper inclusions l ⊂ l ⊂ l . (A WARNING ABOUT NOTATION : The notation l 1,∞ just introduced is traditional for Lorentz sequence spaces. However in the noncommutative geometry literature, the set of operators on a Hilbert space with singular values in l 1+ , often called the Dixmier ideal, is sometimes denoted L1,∞ .) The local oscillation of the symbol function is closely related to the singular values of Hankel operators. When the symbol function is smooth the needed oscillation information is captured by the normalized derivative and it is sufficient to consider those quantities on an appropriately thick discrete set. Pick and fix r > 0 and M, ε > 0 with M very large and ε very small. Select a set of points Z = {zi } in the disk so that the hyperbolic balls centered at zi and of radius εr, {B(zi , εr)}, are disjoint and that the expanded balls {B(zi , Mr)} cover the disk with bounded overlap; i.e. χB(zi ,Mr) is bounded. For given holomorphic g we define the oscillation numbers, Osc(g(zi )) by Osc g(zi ) = sup 1 − |zi |2 g (z): z ∈ B(zi , Mr) . We will prove the following regularity result. Earlier work in this direction was done by Li and Russo in [18]. Theorem 7. Suppose f is a holomorphic function on the disk and select a choice of Trω . The following are equivalent: f is in I H 1 . The numbers {Osc(f (zi ))} are in the sequence space l 1,∞ . The numbers {Osc(f (zi ))} are in the sequence space l 1+ . Trω (|Hf¯ |) < ∞. 1 (5) Trω |Hf¯ | = |f | dθ < ∞. 2π

(1) (2) (3) (4)

T

(4.1)

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Furthermore the inclusions in (2) and (3) do not depend on the particular choice of Z and when (4) or (5) hold for one choice for Trω they hold for every choice. In particular |Hf¯ | is measurable. Finally, the quantities in (4.1) are comparable to both the l 1,∞ quasinorm and the l 1+ norm of the sequence {Osc(f (zi ))}. Proof. The sequence space inclusion shows that (2) implies (3) and it is automatic that (5) implies (4). We will show that (1) implies (2), (3) implies (4), (4) implies (1), and finally that (1) is equivalent to (5). The equivalence of the norms and quasinorm are implicit in the proof. (1) implies (2): It is proved as Theorem C of [25] that if f is in the Besov space B 1 then (2) holds. However that proof starts by noting that B 1 ⊂ I H 1 and then gives a direct argument that condition (1) implies condition (2). (3) implies (4): In Theorem 4 of [19] Luecking gives conditions on general functions f which ensure that |Hf¯ | is in the Schatten ideal Sp . He also notes on page 262 that his proof actually gives more. In particular it shows that if the parameters used in constructing {zi } are chosen appropriately then there is a c > 0 so that for all n, sn (Hf¯ ) c(b(zi ))∗ (n). Hence |Hf¯ | is in the domain of Trω for any ω. (4) implies (1): We again use the ideas and some of the computations in [19]. It will be convenient to be more specific about the choice of Z. We do that in two steps. Pick and fix a, 1 < a < 2 and K large. On the circle {z: |z| = 1 − a −n } distribute Ka n points, uniformly spaced. Enumerate Z˜ so that points closer to the origin have lower indices. This Z˜ satisfies the covering and separation conditions described earlier. Hence, the linear map T˜ of an abstract Hilbert space H with orthonormal basis {ei } into the Bergman space which takes ei to the normalized reproducing kernel at zi ; T˜ (ei ) = (1 − |zi |2 )(1 − z¯ i z)−2 , is bounded [33]. Furthermore the operator norm ˜ Pick and fix the is bounded by a number that depends only on the separation constants of Z. symbol function f¯. We now adjust Z˜ to a new set Z. The point zi ∈ Z˜ is on a circle centered at the origin. On that circle it sits in an arc connecting its two nearest (on that circle) neighbors. Let zi∗ be the point on that arc where |f | is largest. Set Z = {zi∗ }. This new set will have essentially ˜ that is, large balls centered at the z∗ will cover the disk and there the same covering data as Z, i will be an upper bound on the depth of the covering. We define T analogously to T˜ but now ˜ We now study H ¯ T . using the set Z instead of Z. f Luecking also constructs an additional auxiliary operator S from H to the Bergman space. With his construction on page 264 of [19] Luecking obtains the estimate that, for some R inf

1 |B(zi , R)|

|f − h|2 : h ∈ Hol B(zi , 2R)

1/2

C S ∗ Hf¯ T ei , ei

B(zi ,R)

Straightforward estimates shows that this gives Osc f (zi ) = 1 − |zi |2 f (zi ) C S ∗ Hf¯ T ei , ei .

(4.2)

We now sum this over all the Ka n indices which give points on the same circle as zi . Setting K = Ka/(a − 1) we find

M. Engliš, R. Rochberg / Journal of Functional Analysis 257 (2009) 1445–1479 n Ka

1467

n

Ka f (zi ) 1 − |zi |2 C S ∗ H ¯ T ei , ei f

Ka n

si S ∗ Hf¯ T CσK a n S ∗ Hf¯ T

C

CσK a n (Hf¯ ). On the other hand the sum on the left is, up to a constant factor, an upper Riemann sum for on that circle; hence I |zi | :=

|f |

f (z)|dz| CσK a n (H ¯ ). f

|z|=zi |

Now pick and fix a large number M and we repeat this analysis on the circles of radius 1 − a −(n+1) , . . . , 1 − a −(n+M) . The number of points involved is now ≈ a n+M . Recall that because f is holomorphic I (r) is an increasing function of r. Combining these facts we have MI |zi | CσJ (Hf¯ ) with J ≈ a n+M . Dividing by log J we have M 1 I |zi | C σJ (Hf¯ ). M +n log J Letting M → ∞ we obtain I |zi | C lim

1 σJ (Hf¯ ). J →∞ log J

We know f ∈ I H 1 if and only if the left-hand side is bounded and thus if the right-hand side is bounded. This completes the proof. (1) is equivalent to (5): We already have the equivalence of the first four conditions and (5) certainly implies (4). To finish we show (1) through (4) imply (5). For 0 < r < 1 define fr by fr (z) = f (rz). By Theorem 5 we know that Trω |Hf¯r | = (fr ) H 1 . We know that as r → 1 the right-hand side converges to f H 1 . Set gr = f¯r − f¯. We have Hf¯r − Hf¯ = Hgr . Thus limTrω |Hf¯r | − Trω |Hf¯ | C lim Trω |Hgr | r

r

C lim lim r

N

1 σN (Hgr ) log N

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M. Engliš, R. Rochberg / Journal of Functional Analysis 257 (2009) 1445–1479 N 1 Osc gr (zi ) r N log N C lim lim Osc g (zi ) 1

C lim lim r

r

N

C lim g r

lw

r H1

= 0. Here the passage from the second line to the third uses the fact that (3) implies (1) and the norm and quasinorm equivalences. The passage from the fourth to the fifth uses the fact that condition (1) implies condition (2). The fact that the first two conditions do not depend on the choice of Z is based on standard estimates such as can be found in [33]. The proof did not use any particulars related to the choice of Trω and hence it holds for any choice. The right-hand side of (4.1) does not involve the choice of Trω and hence all choices give the same value. 2 There are two places in the proof where aspects of holomorphy play a role. First, the equivalence of (1) and (2) is the statement that a certain potential space (defined by integrability of a derivative) coincides with a Besov type space (defined by global control of local oscillation). Such occurrences are unusual when the spaces are not Hilbert spaces. This is discussed in the appendix of [10] where it is shown that the space of functions in d dimensions with one derivative in Ld coincides with a weak type Besov space with index d. It is noted there that the result fails for d = 1, their proof only yielding the conclusion that the boundary values of f have bounded variation and hence that f is a finite measure. However in our context we have the additional hypothesis of holomorphy and hence can appeal to the F. and M. Riesz theorem to see that the measure is absolutely continuous giving a direct proof that (2) implies (1). Second, the passage from (2) to (3) follows from the obvious sequence space inclusion. However we eventually obtain that (3) implies (2). At its heart that result is based on the fact which we used in proving that (4) implies (1): the integral means |f (reiθ )| dθ are an increasing function of r, a fact proved using considerations of subharmonicity. 5. Other Bergman spaces Once we have Theorem 5 we can obtain similar results for Hankel operators on Bergman spaces of multiply connected domains. Let be a bounded domain in the plane with boundary consisting of finitely many smooth disjoint curves {i }ni=1 . Let dA be Lebesgue area measure. (An easy calculation shows that normalizing dA to, say, total mass one does not affect the singular values of Hankel operators.) Let dγ be arclength rescaled on each boundary component to give the component unit mass; dγ = (length(i ))−1 χi |dz|. The Bergman space of such a domain, L2hol ( ), is the closed subspace of L2 ( ) = L2 ( , dA) consisting of holomorphic functions. For convenience in this section we will write B( ) for L2hol ( ). We write P for the orthogonal projection of L2 ( , dx dy) to B( ). ¯ we define the Hankel operator with symbol f, H , as a linear For a function f ∈ C ∞ ( ) f 2 2 operator from L ( ) to L ( ) given by Hf g = (I − P )f P g.


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Theorem 8. For any choice of Dixmier trace for operators on L2 ( ) we have Trω Hf =

¯ | dγ . |∂f

(5.1)

∂

Proof. Suppose first that n = 1, i.e. that is simply connected. Let φ : D → be a univalent holomorphic map of D onto . Define U = Uφ : L2 ( , dA) → L2 (D) by U (g(z)) = g(φ(z))φ (z). The following facts are straightforward: U is unitary, U maps the subspace B( ) into and onto the subspace B(D), and, with Mg denoting the operator of multiplication by g, U Mf = Mf ◦φ U . Using these facts it is immediate that U Hf = HfD◦φ U . Recall also that ¯ ◦ φ)(z) = ∂f ¯ (φ(z))φ¯ (z). Hence the case n = 1 of the theorem follows from (1.1). ∂(f We just noted that the hypotheses and conclusion transform well under a biholomorphic change of variable. Hence, without loss of generality, we can, and do, suppose that all j are circles, with 2 the unit circle and all other j , j = 2, contained in the unit disc D. Let j , j = 1, . . . , n, be the component of C \ j which contains , and let Bj be the subspace of B consisting of all the functions which extend to be holomorphic in j , and which vanish at ∞ if j is unbounded (i.e. for j = 2). It is not hard to see that B = B1 + B2 + · · · + Bn , a non-orthogonal direct sum decomposition. We denote the associated (oblique) projections from B onto Bj by Qj . We now consider the case of n = 2. This is for notational convenience; the details for n > 2 are straightforward extensions and we will omit them. For compact operators A and B we will write A ≈ B if the singular values satisfy sn (A − B) = O(cn ) for some c, 0 < c < 1. This is enough to ensure that Trω (|A|) = Trω (|B|). (In fact, sn (A − B) = O(n−2 ) would do; cf. [14, Lemma II.4.2], and [9, Ch. IV.3, Lemma 9 on p. 320] (with α = 12 ).) To prove the theorem we will replace Hf by a sequence of simpler operators all related through ≈. For i = 1, 2 let fi be smooth functions on that are supported in small disjoint neighborhoods of i and which agree with f in those neighborhoods. We will verify first that there is no loss replacing f by f1 + f2 : Claim 1. Hf ≈ Hf 1 +f2 = Hf 1 + Hf 2 . We then show that for each summand there is no loss in restricting the operator to functions which are large near that boundary component: Claim 2. Hf 1 ≈ Hf 1 Q1 ; Hf 2 ≈ Hf 2 Q2 . To analyze |Hf 1 Q1 + Hf 2 Q2 | we verify Claim 3. ∗ ∗ ∗ Hf1 Q1 + Hf 2 Q2 Hf 1 Q1 + Hf 2 Q2 ≈ Hf 1 Q1 Hf 1 Q1 + Hf 2 Q2 Hf 2 Q2 ∗ = Hf 1 Q1 ⊕ Hf 2 Q2 Hf 1 Q1 ⊕ Hf 2 Q2 .

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This will ensure us that |Hf 1 Q1 +Hf 2 Q2 | ≈ |Hf 1 Q1 |⊕|Hf 2 Q2 | and hence that Trω (|Hf 1 Q1 + = Trω (|Hf 1 Q1 |) + Trω (|Hf 2 Q2 |). The two summands on the right are handled similarly; we just look at the second. The operator Hf 2 Q2 maps L2 ( ) into itself. We extend it to L2 ( 2 ) by writing L2 ( 2 ) = L2 ( ) ⊕ L2 ( 2 \ ), identifying L2 ( ) with L2 ( ) ⊕ {0}. We then extend Hf 2 Q2 as Hf 2 Q2 ⊕ 0. D = (I − P )f Let Hf D 2 PD be the Hankel operator on the Bergman space of the unit disc with Hf 2 Q2 |)

2

symbol f2 , the extension of f2 to D by zero; also a map of L2 ( 2 ) into itself. We will show D. Claim 4. Hf 2 Q2 ⊕ 0 ≈ Hf 2

The earlier claims reduced the issue to considering Hf 2 Q2 . With this final claim we complete the proof because Trω Hf 2 Q2 = Trω Hf 2 Q2 ⊕ 0 D = Trω Hf 2 = |∂f2 | dγ 2

=

|∂f | dγ . 2

The evaluation of Trω used the case n = 1 of the theorem. We now proceed to the claims. (Claim 1) Just note that Hf − Hf 1 +f2 = Hg with g supported on a compact subset of . For such g, the next proposition implies that Hg ≈ 0. Proposition 1. Let be a domain in C, g a bounded function supported on a compact subset of , and let Mg = gP be the restriction to B( ) of the operator f → gf on L2 ( ) of multiplication by g. Then Mg ≈ 0. Proof. For x ∈ , let Dx and dx denote the discs with center x and radii dist(x, ∂ ) and 1 2 dist(x, ∂ ), respectively. There exist finitely many dxj ≡ dj , j = 1, . . . , m, that cover the support of g; and we can decompose g as a sum g = m j =1 gj with gj supported in dj . (For instance, j −1 take for gj the restriction of g to dj \ k=1 dk .) Then Mg = j Mg j = j ιj Mgjj rj , where ιj : L2 (Dj ) → L2 ( ) and rj : B( ) → B(Dj ) are the inclusion and the restriction maps from to Dj ≡ Dxj , respectively. Since ij and rj are bounded (in fact — even contractive), it is enough

to prove that Mgjj ≈ 0. We have thus reduced to the situation when = D and g is a bounded function supported in {z: |z| < 12 } ≡ 12 D. Clearly, we may also assume that g ∞ 1. Let χ denote the indicator function of 12 D. Since (we will drop the superscripts D in the rest of this proof) Mg∗ Mg = T|g|2 Tχ = Mχ∗ Mχ , whence sn (Mg ) sn (Mχ ) for all n, it is in fact enough to deal


1471

with g = χ . However, an easy calculation using the fact that the monomials zn are an orthogonal basis of eigenvectors for Mχ∗ Mχ = Tχ shows that sn (Mχ ) = 2−n−1 . This completes the proof. 2 The argument works, without modifications, for an arbitrary domain in Cn . (Claim 2) We need to show that Hf 1 Q2 ≈ 0, or (I − P )f1 Q2 P ≈ 0. We claim that even f1 Q2 P ≈ 0.

(5.2)

To see this, let ρ : B(D) → B2 be the restriction map, and ι = ρ −1 : B2 → B(D) the inclusion of B2 into B(D). Then f1 Q2 P = f1 ριQ2 P so it is enough to show that f1 ρ ≈ 0. However, for any φ, ψ ∈ B(D),

|f1 |2 φψ = T|D φ, ψ D , f |2

f1 ρφ, f1 ρψ =

1

(5.3)

where f1 , the extension by zero of f1 to D, is compactly supported in D. Consequently, (f1 ρ)∗ (f1 ρ) = T|D ≈ 0 by the preceding proposition. Thus f1 ρ ≈ 0. f |2 1

(Claim 3) We need to show that (Hf 1 Q1 )∗ (Hf 2 Q2 ) ≈ 0 and also that a similar result holds with the indices interchanged. The two are similar and we will just look at the first. This is slightly more delicate than the previous claims, and will require some particulars of the Bergman kernels. We have ∗ Hf1 Q1 Hf2 Q2 = Q∗1 P f 1 (I − P )f2 Q2 P = −Q∗1 P f 1 Pf2 Q2 P . From (5.2) we have f 1 P ≈ f 1 Q1 P , and similarly, taking adjoints, Pf2 ≈ Q∗2 Pf2 .

(5.4)

Thus we can continue with ∗ Hf1 Q1 Hf2 Q2 ≈ −Q∗1 P f 1 Q1 Q∗2 Pf2 Q2 P . We claim that we even have Q1 Q∗2 ≈ 0. Indeed, for any orthonormal basis {ej }j 0 of B( ), Q1 Q∗2 = operator on with integral kernel k(x, y) := Q1,x Q2,y

j 0

(5.5)

j 0 · , Q2 ej Q1 ej

ej (x)ej (y) = Q1,x Q2,y K (x, y),

is an integral

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K being the Bergman kernel of , where the subscripts x, y indicate the variable to which the operator Qj applies, and Q2 f := Q2 f . Since ∂ consists of circles and, hence, is realanalytic, it is a result of Bell [3] that K extends to a holomorphic function of x, y for (x, y) in a neighborhood of the closure of × minus the boundary diagonal; thus, in particular, to x ∈ ∪ (a neighborhood of 1 ) and y ∈ ∪ (a neighborhood of 2 ). By the next lemma, it follows that k(x, y) is actually holomorphic in x, y for x in a neighborhood of 1 and y in a neighborhood of 2 (= D); and the next proposition then implies that Q1 Q∗2 ≈ 0. Lemma 2. If f ∈ B( ) extends to be holomorphic in a neighborhood of j , then so does Qj f ; that is, Qj f extends to be holomorphic in a neighborhood of j . Proof. We give the proof for j = 2. By Cauchy’s formula, Q2 f is given by f (ξ ) 1 Q2 f (z) = dξ, 2πi ξ −z rT

for any r, |z| < r < 1, the value of the integral being independent of the choice of such r. If f is even holomorphic on |z| < 1 + δ, then we can even take any r with |z| < r < 1 + δ, showing that Q2 f likewise extends to be holomorphic in |z| < 1 + δ. 2 Proposition 2. Let T be an integral operator on a bounded domain , Tf (x) = k(x, y)f (y) dy,

whose integral kernel k(x, y) belongs to the complex conjugate of B( ) for each fixed x, and is holomorphic on 0 ⊃ for each fixed y. Then T ≈ 0. Proof. Let 1/2 be a domain containing but such that its closure is contained in 0 . Morera’s and Fubini’s theorems imply that the integral k(x, y)k(z, y) dy

is holomorphic (hence — continuous) in x, z on 0 × 0 ; taking x = z it follows, in particular, that k(x, ·) 2 C ∀x ∈ 1/2 L ( ) for some finite C. Straightforward estimates then show that the operator Tf (x) := k(x, y)f (y) dy

is bounded from B( ) into B( 1/2 ) (with norm not exceeding C| |1/2 ). Now T = τ T where τ is the restriction map τ : B( 1/2 ) → B( ); by the same argument as in (5.3), it follows from

Proposition 1 that τ ∗ τ = Tχ 1/2 ≈ 0. Hence τ ≈ 0 and T ≈ 0.

2


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(Claim 4) To cope with the various identifications, we denote by R : L2 (D) → L2 ( ) the restriction map, and by E = R ∗ : L2 ( ) → L2 (D) the map of prolonging by zero. We are thus D , where f 2 = Ef2 . claiming that EHf 2 Q2 P R ≈ Hf 2 We have D EHf 2 Q2 P R − Hf = E(I − P )f2 Q2 P R − (I − PD )f2 PD 2

= E(I − P )f2 Q2 P R − (I − PD )Ef2 RPD

= E(I − P )f2 (Q2 P R − RPD ) + E(I − P ) − (I − PD )E f2 RPD = E(I − P )f2 (Q2 P R − RPD ) − (EP − PD E)f2 RPD = E(I − P )f2 (Q2 P R − RPD ) − (EQ2 P − PD E)f2 RPD − EQ1 Pf2 RPD . Thus it is enough to show that Q2 P R − RPD ≈ 0,

(5.6)

EQ2 P − PD E ≈ 0,

(5.7)

Q1 Pf2 ≈ 0.

(5.8)

From (5.4) we have Q1 Pf2 ≈ Q1 Q∗2 Pf2 and thus (5.8) is immediate from (5.5). For (5.6), observe that for F ∈ L2 (D) and x ∈ , (Q2 P R − RPD )F (x) =

Q2,x K (x, y)F (y) dy −

=

KD (x, y)F (y) dy D

Q2,x K (x, y) − KD (x, y) F (y) dy −

KD (x, y)F (y) dy.

D\

The second summand is just RTχDD\ F (x), and TχDD\ ≈ 0 by Proposition 1. The first summand vanishes if RF ⊥ B( ), while on B( ) it acts as integral operator with kernel Q2,x K (x, y) − KD (x, y) = Q2,x [K (x, y) − KD (x, y)]. From Theorem 23.4 of [4] we know that the difference K (x, y) − KD (x, y) extends to be holomorphic in x, y in a neighborhood of 2 = ∂D; by Lemma 2 we thus conclude that Q2,x [K (x, y) − KD (x, y)] is in fact holomorphic for x in a neighborhood of D and y in ∪ (a neighborhood of 2 ). By Proposition 2, the corresponding integral operator is ≈ 0, thus proving (5.6). With (5.6) in hand, it follows that 0 ≈ E(Q2 P R − RPD )E = EQ2 P − ERPD E = EQ2 P − PD E + (I − ER)PD E. As (I − ER)PD = χD\ PD ≈ 0 by Proposition 1, (5.7) follows.

2

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6. Related operators In an effort to understand better the difference between the singular value behavior of the small and big Hankel operators the authors of [23] studied operators built from multiplication followed by projection onto subspaces of L2 (D) which sit between L2hol (D) and L2hol (D)⊥ . Set D¯ = z¯ ∂¯ and note that the Bergman space, which we will denote by A0 in this section, is the ¯ Let A1 = ker(D¯ 2 ), set A1 = A1 A0 and let closure of the smooth functions in the kernel of D. 1 PA1 be the orthogonal projection onto A . ¯ . Pick and fix a smooth holomorphic symbol function b = bn zn and recall that Hb¯ = P ⊥ bP ¯ We will compare this with the intermediate Hankel operator Kb¯ = PA1 bP ; that is Kb¯ (f ) = ¯ ). The difference K ¯ − H ¯ is, in the notation of [23], the operator −H 1 . Theorem 5 of PA1 (bf b b b [23] states that for b in the Besov space B 1 the operator Hb1 will be in the trace class. A trace class perturbation of an operator A will not change Trω (|A|). Thus we have the following corollary. Theorem 9. For any choice of Trω , Trω (|Kb¯ |) = Trω (|Hb¯ |). Theorem 1 in [23] gives orthonormal bases for both A0 and A1 . An orthornomal basis for A0 is given by the functions e0,n =

√ n + 1zn ,

n = 0, 1, 2, . . . ,

and for A1 by the functions e1,n =

√ n + 1 2(n + 1) log r + 1 zn ,

n = 0, 1, 2, . . . .

Using these bases we can compute the matrix of Kb¯ ; the matrix Mb¯ = (βij ) with β¯i,j = Kb¯ (e0,i ), e1,j . Proposition 3. The entries of the matrix Mb¯ are given by βi,j =

(j + 1)1/2 (i − j )b¯i−j (i + 1)3/2

if i j

= 0 otherwise. Proof. The βi,j depend conjugate linearly on b so it suffices to do the computation for the monomial b = zN . We have ¯ 0,i , ei,j βi,j = Kb¯ (e0,i ), ei,j = PA1 be ¯ 0,i , ei,j = e0,i , bei,j = be = e0,i , zN ei,j . Computing the inner product by first doing the θ integration shows that this quantity is zero unless i = j + N. In that case the remaining integral is


βi,j

1 √ = 2π i + 1 j + 1 π √ =2 i +1 j +1

1

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r 2i+1 2(j + 1) log r + 1 dr

0

2(j + 1) 1 − 2i + 2 (2i + 2)2

= (j + 1)1/2 (i + 1)−3/2 (i − j ) which gives the required result.

2

We can regard Mb¯ as the matrix of an operator on the Hardy spaces with respect to the monomial basis and give that operator a function theoretic description. Recall the operator introduced in (3.3) and for any real α let α be the generalized differentiation or integration operator on L2 (T) defined through −α inθ α einθ = 1 + |n| e . Recall that TfH denotes the Toeplitz operator on the Hardy space with symbol f. It is a straightforward computation that Mb¯ is the matrix of the operator Sb¯ = −1/2 T H 3/2 and thus we have zb T rω (|Sb¯ |) = |b |. In fact we could have applied the ideas of Section 3 directly to Sb¯ and then used the results of [23] to pass the results of that analysis back to Bergman space Hankel operators and obtained (1.1). That would have the advantage of staying in the Hardy space where the computations are a bit simpler but would use the results of [23] which are more computational than conceptual. Also, it is not clear that approach could also yield (1.2). 7. Computations In some cases evaluating the integrals in (1.1) or (5.1) is straightforward. For instance if f (z) = z¯ then Trω Hz¯ = number of components of ∂ . Also, on the disk if f = g¯ and g is a finite Blaschke product, or any inner function, then Trω (|Hf |) = 1. For some f it is possible to use the Cauchy–Riemann equations to give a geometric or function theoretic interpretations to the values of the integrals in (1.1) or (5.1). Suppose is a real analytic simple closed curve bounding the bounded domain . Suppose is another simple closed curve which is inside (and which we think of as being near and roughly parallel ). Denote by the subdomain of bounded by and . If dh is a harmonic differential on a domain in the plane we denote by ∗dh the harmonic differential which is conjugate to dh (see Ch. II of [1]).

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Proposition 4. ¯ and holomorphic on . Suppose further that |f (z)| ≡ 1 (1) Suppose f (z) is continuous on on and that for some c > 0, c |f (z)| 1. It follows that |∂¯ f¯||dz| = d arg(f ) = ∗d log |f |. (7.1)

¯ , harmonic and negative on , and h ≡ 0 on then (2) Suppose h is continuous on h ∂e ¯ |dz| = 1 ∗dh. 2

(7.2)

(3) If, instead, were the inner boundary then analogous statements hold with a negative sign inserted on the right-hand side. The variation in which |f | or h has a minimum on also introduces a negative sign. Proof. We know f has constant modulus on and that is real analytic hence we can use the reflection principle to extend f to be holomorphic in a small neighborhood of . Pick and fix x ∈ and U a small simply connected neighborhood of x. Let arg f (z) be a choice of argument which is harmonic in U and thus log f (z) = log |f (x)| + i arg f (z) is holomorphic there. In U the integrand is |∂¯ f¯| = |∂f | = |f ∂ log f | = |∂ log f | = ∂ log |f | + ∂ arg f . The integration is along and log |f | is constant on thus the integrand simplifies to |∂ arg f |. ¯ at Now note that the directional derivative of log |f | in the direction of the outward normal to x is positive. Hence, by the Cauchy–Riemann equations the directional derivative of arg f in the direction of the positively oriented tangent to at x is positive. Thus we can drop the absolute value and find that ¯ ¯ |∂ f ||dz| = d arg(f ). ∩U

∩U

Piecing together these local results gives (7.1). We could obtain the second statement by working directly with the fact that h is harmonic. Alternatively note that, locally, eh = f f¯ for a holomorphic function f with log |f | = 12 h. Hence ¯ f¯| = |f ∂f ¯ | = |∂f ¯ | and the desired conclusion follows from the first statement. ¯ h | = |∂f |∂e The third statement is straightforward. 2 On the disk it is immediate from (1.1) that Trω (|Hz¯ n |) = n. Using the proposition we see that the same conclusion holds if zn is replaced by any Blaschke product with n factors. Suppose now that is bounded by n smooth curves. Pick ζ ∈ and consider the holomorphic function g(z) which solves the following extremal problem: maximize Re g (ζ ) subject to g(ζ ) = 0 and supg(z) 1.


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This function, often called the Ahlfors function, represents as an n-sheeted cover of the disk with the boundary going to the boundary [3]. In particular we can use (5.1) and (7.1) to conclude that Trω (|Hg |) = n. (Recall that the boundary measure dγ in (5.1) is built from normalized arc length measures.) We now consider symbol functions of the form g(z) = φ(z)eh(z) where h is real valued and harmonic and φ is a localizing function. We suppose φ is smooth, is identically one on a neighborhood of one boundary component, say 1 , and identically zero on neighborhoods of the other boundary components. In that case, by (5.1) we have h ¯ dγ . ¯ dγ = |∂g| ¯ dγ = ∂e (7.3) Trω Hg = |∂g| 1

∂

1

In some situations we can use the previous proposition to continue the computation. First we consider double connected domains. Select r, 0 < r < 1, and let = (r) be the ring domain with outer boundary 1 the circle centered at the origin with radius 1 and with inner boundary r the concentric circle with radius r. Let h(z) be the harmonic function on with boundary values 0 on 1 and −1 on r . Pick the smooth function φ which is one near 1 and 0 near r . Again setting g = φeh and combining (7.2) with the previous equality we find 1 1 Trω Hg = ∗ dh. (7.4) 2 2π |z|=1

We know that h must be of the form A + B log |z| for some real A, B and we find

2 2 log |z| = Re 1 + log z . h(z) = 1 + log r log r Hence ∗dh(z) = d Im 1 +

2 2 2 log z = d arg z = dθ. log r log r log r

We combine this with the earlier computation and conclude that Trω Hg =

1 . log r

˜ is any other doubly connected domain we can do the same analysis. That is, we can let If h˜ be the harmonic function which is 0 on the outer boundary and −1 on the inner boundary. Then construct g˜ by localizing exp h˜ to a neighborhood of the outer boundary and consider ˜ ˜ is conformally equivalent to (¯r ) and we can choose Tω (|Hg˜ |). There is a unique r¯ so that the conformal map to take one outer boundary to the other outer boundary. We noted earlier ˜ that Trω (|H· |) behaves well under conformal maps. Also, the conformal map takes harmonic functions to harmonic functions. Combining these facts we find. ˜ Tω Hg˜ =

1 . log r˜

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˜ Conversely the conformal type In particular the trace is determined by the conformal type of . of a doubly connected domain is completely determined by the parameter r. We conclude, with ˘ and g, the natural interpretation of ˘ that ˜ and ˘ are conformally equivalent iff Trω H ˜ = Trω H ˘ . g˜ g˘ This analysis extends to multiply connected domains. Suppose is a domain bounded by n real analytic curves 1 , . . . , n . Select a i and consider the associated harmonic measure, that is, the harmonic function hi with boundary values 1 on i and 0 on the other components. For each index j, where i = j is allowed, set hij = φj hi where φj is smooth, one near j and zero in neighborhoods of the other boundary components. The straightforward extension of the previous argument gives αij := Trω Hexp hij = ± ∗ dhi . j

As before, the numbers {αi,j } are conformal invariants of . Also, they again determine the conformal structure of the domain, but now only up to reflection. That last fact is Proposition 4.10 in [24]. References [1] L. Ahlfors, L. Sario, Riemann Surfaces, Princeton University Press, Princeton, NJ, 1960. [2] J. Arazy, S. Fisher, J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (6) (1988) 989– 1053. [3] S. Bell, Extendibility of the Bergman kernel function, in: Complex Analysis II, College Park, 1985–1986, in: Lecture Notes in Math., vol. 1276, Springer, Berlin–New York, 1987, pp. 33–41. [4] S. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992. [5] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971) 11–51. [6] L. Boutet de Monvel, On the index of Toeplitz operators in several complex variables, Invent. Math. 50 (1979) 249–272. [7] L. Boutet de Monvel, V. Guillemin, The Spectral Theory of Toeplitz Operators, Ann. of Math. Stud., vol. 99, Princeton University Press, Princeton, NJ, 1981. [8] A. Carey, F. Sukochev, Dixmier traces and some applications to noncommutative geometry, Uspekhi Mat. Nauk 61 (6(372)) (2006) 45–110 (in Russian). English version: arXiv:math/0608375. [9] A. Connes, Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994. [10] A. Connes, D. Sullivan, N. Teleman, Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes, Topology 33 (4) (1994) 663–681. [11] M. Engliš, K. Guo, G. Zhang, Toeplitz and Hankel operators and Dixmier traces on the unit ball of Cn , Proc. Amer. Math. Soc., in press. [12] G.B. Folland, Introduction to Partial Differential Equations, second ed., Princeton University Press, Princeton, NJ, 1995. [13] G.B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud., vol. 122, Princeton University Press, Princeton, NJ, 1989. [14] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18, 1969. [15] V. Guillemin, Toeplitz operators in n dimensions, Integral Equations Operator Theory 7 (1984) 145–205. [16] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980) 188–254. [17] W. McLean, Local and global descriptions of periodic pseudodifferential operators, Math. Nachr. 150 (1991) 151– 161.


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[18] S.-Y. Li, B. Russo, Hankel operators in the Dixmier class, C. R. Acad. Sci. Paris Sér. I Math. 325 (1) (1997) 21–26. [19] D. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk, J. Funct. Anal. 110 (2) (1992) 247–271. [20] S.T. Melo, Characterization of pseudodifferential operators on the circle, Proc. Amer. Math. Soc. 125 (1997) 1407– 1412. [21] K. Nowak, Weak type estimate for singular values of commutators on weighted Bergman spaces, Indiana Univ. Math. J. 40 (4) (1991) 1315–1331. [22] V. Peller, Hankel Operators and Their Applications, Springer-Verlag, New York, 2003. [23] L. Peng, R. Rochberg, Z. Wu, Orthogonal polynomials and middle Hankel operators on Bergman spaces, Studia Math. 102 (1) (1992) 57–75. [24] R. Rochberg, Function algebra invariants and the conformal structure of planar domains, J. Funct. Anal. 13 (1973) 154–172. [25] R. Rochberg, S. Semmes, End point results for estimates of singular values of singular integral operators, in: Contributions to Operator Theory and Its Applications, Mesa, AZ, 1987, in: Oper. Theory Adv. Appl., vol. 35, Birkhäuser, Basel, 1988, pp. 217–231. [26] J. Saranen, W.L. Wendland, The Fourier series representation of pseudodifferential operators on closed curves, Complex Var. 8 (1987) 55–64. [27] M. Shubin, Pseudodifferential Operators and Spectral Theory, Springer, Berlin, 1987. [28] R.T. Seeley, Complex powers of an elliptic operator, in: Singular Integrals, in: Proc. Sympos. Pure Math., vol. X, American Mathematical Society, Providence, RI, 1967, pp. 288–307. [29] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Plenum, New York, 1980. [30] V. Turunen, Commutator characterization of periodic pseudodifferential operators, Z. Anal. Anwend. 19 (2000) 95–108. [31] V. Turunen, G. Vainikko, On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anwend. 17 (1998) 9–22; also in: Differential and Integral Equations: Theory and Numerical Analysis, Est. Math. Soc., Tartu, 1999, pp. 107–114. [32] M. Wodzicki, Non-commutative Residue I, in: Lecture Notes in Math., vol. 1289, Springer, New York, 1987, pp. 320–399. [33] K. Zhu, Operator theory in function spaces, Marcel Dekker, Inc., New York, 1990.


Simple and prime crossed products of C ∗ -algebras by compact quantum group coactions Raluca Dumitru a,b,∗ a Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville,

FL 32224, United States b Institute of Mathematics of the Romanian Academy, Bucharest, Romania

Received 16 September 2008; accepted 30 May 2009 Available online 11 June 2009 Communicated by S. Vaes

Abstract Let G = (A, ) be a compact quantum group and δ a coaction of G on a C ∗ -algebra B. We give necessary and sufficient conditions for the simplicity and primeness of the crossed product B ×δ G in terms of certain fixed point algebras. © 2009 Elsevier Inc. All rights reserved. Keywords: Crossed product; Compact quantum group

1. Introduction A coaction of a compact quantum group G on a C*-algebra B gives rise to the reduced crossed product B ×δ G. The aim of this paper is to study the ideal structure of this crossed product. We give necessary and sufficient conditions for the simplicity and primeness of the crossed product obtained by compact quantum group coactions. The problem of the ideal structure of the crossed product has been extensively studied in the last decades. In his fundamental paper [3], Connes establishes a connection between the ideal structure of the fixed points of compact, abelian group actions on von Neumann algebras and that of the crossed product. With α an action of an abelian group G on a von Neumann algebra A, he * Address for correspondence: Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville, FL 32224, United States. E-mail address: [email protected].


R. Dumitru / Journal of Functional Analysis 257 (2009) 1480–1492

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obtained that the crossed product A ×α G is a factor if and only if the fixed point algebra Aα is a factor and the Arveson spectrum, Sp(α) is equal to the dual group G. Subsequently, these topics have been considered in the framework of C ∗ -algebras by Olesen, Pedersen, Elliott and others for abelian groups (see for instance [10]), Rieffel for finite, nonabelian groups in [13], and Landstad in [7], Peligrad in [11] for compact, non-abelian groups. With α an action of a compact, not necessary abelian group G on a C ∗ -algebra A, in [11] Peligrad finds necessary and sufficient conditions for the simplicity or primeness of the crossed product A ×α G in terms of the Arveson Spectrum and the simplicity or primeness of all the fixed point algebras associated to the action, (A ⊗ B(Hπ ))α⊗Ad π , for all irreducible representations π of G. He also shows that in the case of a compact, non-abelian group, it is not enough anymore to consider only the simplicity or primeness of the fixed point algebra Aα , as in the abelian case. These results have been further extended to the case of actions of finite dimensional Hopf ∗-algebras on C ∗ -algebras by Szymanski and Peligrad in [14]. In the case of compact quantum groups, the simplicity of the crossed product was studied in the ergodic case by Landstad in [8] (see also [2]). In this paper we find conditions for the simplicity or primeness of the crossed product (obtained by compact quantum group coactions) in α terms of the simplicity or primeness of all the fixed point algebras (B ⊗ Md )δ⊗Ad(u ) , associated to the coaction δ. The structure of this paper is as follows: in Section 2 we recall the necessary preliminaries and we set some notations. In Section 3 we construct certain subspaces of the crossed product associated to the irreducible representations of G, which we denote by Iα , and discuss their properties. These subspaces of the crossed product constructed here are the quantum version of the subspaces of spherical functions introduced by Landstad in [7]. In Section 4 we state and prove the main results on the simplicity and primeness of the crossed product B ×δ G. 2. Preliminaries 2.1. Compact quantum groups We will first recall the definition of a compact quantum group [17,18]. Definition 2.1. Let A be a unital C ∗ -algebra and : A → A ⊗ A a ∗-homomorphism such that (a) ( ⊗ ι) = (ι ⊗ ), where ι is the identity map, and (b) (A)(1 ⊗ A) = A ⊗ A and (A)(A ⊗ 1) = A ⊗ A. Then the pair (A, ) is called a compact quantum group. We recall [18, p. 12] that u is a unitary representation of a compact quantum group G on a Hilbert space H if u is a unitary element in the multiplier algebra M(K(H ) ⊗ A) such that (ι ⊗ )u = u12 u13 . Furthermore, a unitary representation is called irreducible if the operators intertwining u are scalar multiples of the identity. Note that if u is an irreducible unitary representation then the Hilbert space on which it acts is finite dimensional. denote the dual of G, i.e. the set of all unitary equivalence classes of irreducible repLet G denote by uα a representative of each class and by resentations of G [17,18]. For each α ∈ G, α α α (uij )1i,j dα ∈ Mdα (A) the matrix form of uα . Set χα = di=1 uii . Then χα is called the character of α. By [17,18] there is a unique invertible operator Fα ∈ B(Hα ), where Hα is the finite

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dimensional Hilbert space of the representation uα , that intertwines uα with its double contragradient representation (uα )cc such that tr(Fα ) = tr(Fα−1 ). Set Mα = tr(Fα ). Using an orthonormal basis in Hα , the Hilbert space of the representation uα , the operator Fα can be represented as a (dα × dα ) matrix, Fα = (f1 (uαij ))1i,j dα . With h the Haar state on G (see [17,18]), let (Hh , πh ) be the GNS representation of A associated to h, and denote by ξh its cyclic vector. We will denote by vr the right regular representation of G = (A, ) (see [9,18]), which is defined as follows: vr is a unitary operator in the multiplier algebra M(K(Hh ) ⊗ A) defined by vr πh (a)ξh ⊗ η = (πh ⊗ πh ) (a) (ξh ⊗ η) for all a ∈ A and η ∈ H . We denote by vl the left regular representation of the compact quantum group G = (A, ), i.e. vl ∈ M(K(Hh ) ⊗ A) is a unitary such that vl∗ πh (a)ξh ⊗ η = (πh ⊗ πh ) op (a) (ξh ⊗ η), for all a ∈ A, η ∈ Hh . Here op = σ ◦ denotes the co-multiplication of Gop = (A, op ), the opposite group of G, and σ : A ⊗ A → A ⊗ A, σ (a1 ⊗ a2 ) = a2 ⊗ a1 , ∀a1 , a2 ∈ A denotes the flip operation (see for instance [9]). We will use the following notations (see [17,18]): a ∗ ξ = (ξ ⊗ ι) (a) ;

ξ ∗ a = (ι ⊗ ξ ) (a) ,

(h · a)(b) = h(ba); (a · h)(b) = h(ab), α i, j = 1, . . . , dα A = lin uij α ∈ G, for all a, b ∈ A and for all linear functionals ξ on A. denote: For each α ∈ G Aα = lin uαij 1 i, j dα , hα = Mα h · (χα ∗ f1 )∗ ;

aα = Mα (χα ∗ f1 )∗

(1)

Remark 2.2. Recall [18, Theorem 3.1] that the Fourier transform is defined by: a = Fvr (a) = (id ⊗ ha) vr∗ ,

∀a ∈ A.

the norm closure of the set of all operators of the form Fvr (a), where a ∈ A. Denote by A As noticed by Tomatsu in [15], one can choose a representative of the class α such that Fα is a diagonal matrix. Choose such a representative, let i, j ∈ {1, . . . , dα } and define Eijα =

∗ 1 Mα Fvr uαij ∗ f −1 (uii )

Then Eijα , i, j = 1, . . . , dα are operators on πh (Aα )ξh .


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Remark 2.3. Using standard computations one can show that the operators Eijα satisfy the following properties: (1) Eijα uαkl ∗ ξh = δj l uαki ∗ ξh , for all i, j, k, l ∈ {1, . . . , dα }; α = δ E α , for all i, j, k, l ∈ {1, . . . , d }. (2) Eijα Ekl j k il α Using the remark above, Eijα can be represented as a (dα2 × dα2 ) block diagonal matrix with equal diagonal entries given by mij (where mij is the (dα × dα ) matrix with 1, as the only nonzero entry, in position (i, j )). consider For any α ∈ G pα = (id ⊗ hα ) vr∗ .

(2)

FurtherNote that pα = Fvr (aα ) and that pα are pairwise orthogonal central projections in A. more, pα is the orthogonal projection onto the finite dimensional subspace Aα ξh ⊂ Hh . 2.2. Coactions of compact quantum groups on C ∗ -algebras Definition 2.4. (See [1].) Let G = (A, ) be a compact quantum group, let B be a C ∗ -algebra and let δ : B → B ⊗ A be a one-to-one ∗-homomorphism of B into the minimal tensor product B ⊗ A. If: (a) (δ ⊗ ι)δ = (ι ⊗ )δ and (b) δ(B)(1 ⊗ A) = B ⊗ A then δ is called a coaction of G on B. consider Pα (x) = (id ⊗ hα )(δ(x)). Note that Pα : B → B is a linear map which For any α ∈ G becomes a conditional expectation from B to B δ = {x ∈ B | δ(x) = x ⊗ 1)}, in the case when in the quantum α is the trivial representation. The spectral subspace corresponding to α ∈ G case has been defined in [2] to be the closed subspace Pα (B) of B, and it was denoted by Bα . Recall that the algebraic direct sum of the spectral subspaces is a dense ∗-subalgebra of B (see for instance [2] or [5, Lemma 2.3]). Let δ be a coaction of G on a C ∗ -algebra B and let u be a representation of G on a Hilbert space H . It is straightforward to check that the following is a coaction of G on B ⊗ K(H ): δu (a ⊗ k) = u23 δ(a)13 (1 ⊗ k ⊗ 1) u∗23 .

(3)

Denote by B ×δ G the crossed product between B and G via a coaction δ, as defined in [1]. Recall that B ×δ G is the C ∗ -algebra generated by elements of the form (πu ⊗ πh ) δ(b) 1 ⊗ Fvr (a) with a ∈ A and b ∈ B. where πu : B → B(Hu ) is the universal representation of the C ∗ -algebra B, and πh : A → B(Hh ) is the GNS representation of A associated to the Haar state h.

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Remark that the crossed product is a fixed point subalgebra of B ⊗ K(Hh ) (see [2, Remark 20]), more precisely: δ B ×δ G = B ⊗ K(Hh ) vl

(4)

3. Subspaces of the crossed product be two classes of irreducible representations of the compact quantum group G. Let α1 , α2 ∈ G We make the following notations: Using Eq. (2), note that 1 ⊗ pα ∈ M(B ×δ G), for each α ∈ G. Sα1 ,α2 = (1 ⊗ pα1 )(B ×δ G)(1 ⊗ pα2 ). When α = α1 = α2 , we define Sα = Sα,α . Then Sα1 ,α2 are closed subspaces of the crossed product. We collect some of their properties in the following lemma: Lemma 3.1. (1) (2) (3) (4)

Sα is a hereditary C ∗ -subalgebra of B ×δ G; For all α ∈ G, and α2 = α3 then Sα1 ,α2 Sα3 ,α4 = 0; If α1 , α2 , α3 , α4 ∈ G then (Sα1 ,α2 )∗ = Sα2 ,α1 ; If α1 , α2 ∈ G then Sα1 ,α2 Sα2 ,α1 is a two-sided ideal of Sα1 . If α1 , α2 ∈ G

Proof. (1) follows using the following property: If B is a C ∗ -subalgebra of A then B is hereditary if and only if bab ∈ B for all b, b ∈ B and a ∈ A. Properties (2), (3), and (4) follow from the definitions and the relation pα1 pα2 = δα1 α2 pα1 . This relation holds since pα1 and pα2 are projections onto orthogonal spaces if α1 = α2 . 2 Let ι be the trivial representation of the compact quantum group G. The next proposition establishes a connection between the spectral subspaces Bα and the subspaces of the crossed product, Sα1 ,α2 introduced above. Sα,ι = 0 if and only if Bα = 0. Proposition 3.2. For all α ∈ G, Proof. Using straightforward computations and the fact that pι is a central projection in A, ∗ ∗ one can check that pι Fvr (a) pι cξh = h(a )pι cξh , for all a, c ∈ A. Therefore Apι = Cpι . This implies that Sα,ι = 0 if and only if there exists b ∈ B such that (1 ⊗ pα )δ(b)(1 ⊗ pι ) = 0. On the other hand, using the definition of Bα , it is clear that Bα = 0 if and only if there exists a non-zero b ∈ B such that δ(b) ∈ B ⊗ Aα . Using that pα is an orthogonal projection onto Aα ξh ∈ Hh the proof is completed. 2 Although we are not going to use it in this paper, it is easy to see that the map b → (1 ⊗ pα )δ(b)(1 ⊗ pι ) is a Banach space isomorphism between Bα and Sα,ι . We now make the following notation: ad(vr ) : B ×δ G → (B ×δ G) ⊗ A, ad(vr )(z) = (1 ⊗ vr )(z ⊗ 1) 1 ⊗ vr∗ , ∀z ∈ B ×δ G.


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We are going to show next that ad(vr ) is a coaction of G on the crossed product. In fact, this coaction corresponds at the level of compact groups to the following action considered by Landstad in [7] and Peligrad in [11]: β : G → Aut(B ×δ G) defined by βk (f )(g) = αk (f (k −1 gk)), for all k ∈ K, g ∈ G, where f : G → B is a continuous function. The fixed points of the action β in the crossed product are called K-central elements in [7,11], where K is the compact group under consideration. Lemma 3.3. The application ad(vr ) defined above is a coaction of the compact quantum group G on the crossed product B ×δ G. Proof. Indeed, it is straightforward to check that ad(vr ) is a unital ∗-homomorphism and that

ad(vr ) ⊗ id ad(vr )(z) = (id ⊗ ) ad(vr )(z)

for all z ∈ B ×δ G. The second condition in Definition 2.4, ad(vr )(B ×δ G)(1 ⊗ A) = (B ×δ G) ⊗ A follows from [16], Section 2, remarks and using that vr is the sum of all irreducible representations of G. 2 Let now (B ×δ G)ad(vr ) be the fixed point subalgebra of the crossed product corresponding to the coaction ad(vr ), i.e. (B ×δ G)ad(vr ) = z ∈ B ×δ G (1 ⊗ vr )(z ⊗ 1) 1 ⊗ vr∗ = z ⊗ 1 . in the crossed Lemma 3.4. The fixed point algebra above is the relative commutant of 1 ⊗ A product. That is, ∩ (B ×δ G). (B ×δ G)ad(vr ) = (1 ⊗ A) Proof. We have z ∈ (B ×δ G)ad(vr ) if and only if (z ⊗ 1)(1 ⊗ vr ) = (1 ⊗ vr )(z ⊗ 1), and the 2 conclusion follows since (id ⊗ B(H )∗ )(vr ) is dense in A. We make the following notations Let α ∈ G. Iα = (B ×δ G)ad(vr ) ∩ Sα

α. and I (α) = Ap

there is a ∗-algebra isomorphism Remark 3.5. For every α ∈ G, Sα I (α) ⊗ Iα . = I (α) , where the commutant is taken in Sα . Since Indeed, using Lemma 3.4, Iα (pα A) I (α) is a matrix algebra, using a result of Jacobson (see [6]), we get Sα I (α) ⊗ I (α) I (α) ⊗ Iα .

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Using Remark 3.5, we obtain that Iα and Sα are strongly Morita equivalent. Furthermore, using [12, Theorem 3.1], we obtain: Lemma 3.6. Iα is strongly Morita equivalent with Sα . Therefore (1) Iα is simple ⇔ Sα is simple; (2) Iα is prime ⇔ Sα is prime. Proposition 3.7. Let δ be a coaction of a compact quantum group G on a C ∗ -algebra B. Then Sι is full in B ×δ G (i.e. the two-sided ideal generated by Sι is equal to B ×δ G) if and only if where ι is the trivial representation of G. Sα,ι Sι,α = Sα , for all α ∈ G, Then Proof. Assume Sι is full in B ×δ G and let α ∈ G. (B ×δ G)(1 ⊗ pι )(B ×δ G) = B ×δ G, and, by multiplying both sides with (1 ⊗ pα ), (1 ⊗ pα )(B ×δ G)(1 ⊗ pι )(B ×δ G)(1 ⊗ pα ) = (1 ⊗ pα )(B ×δ G)(1 ⊗ pα ). Hence Sα,ι Sι,α = Sα . Let α ∈ G and z ∈ B ×δ G such that Conversely, assume that Sα,ι Sι,α = Sα , for all α ∈ G. 0 = z ∈ (1 ⊗ pα )(B ×δ G). Note that (1 ⊗ pα )(B ×δ G) = (0) since it contains at least the element 1 ⊗ pα . Since Sα,ι Sι,α is a dense ideal in Sα , there exists an approximate identity (eλ ) of Sα contained in Sα,ι Sι,α . Then, since zz∗ ∈ Sα , we get lim eλ z − z2 = lim (eλ z − z) z∗ eλ∗ − z∗ λ

λ

= lim eλ zz∗ eλ − eλ zz∗ − zz∗ eλ + zz∗ = 0. λ

Since eλ ∈ Sα,ι Sι,α , then eλ z ∈ (B ×δ G)(1 ⊗ pι )(B ×δ G). Therefore z ∈ (B ×δ G)(1 ⊗ pι )(B ×δ G). On the other hand, since

= 1, then

pα α∈G

(1 ⊗ pα )(B ×δ G) = B ×δ G,

α∈G

and the conclusion follows.

2

Remark 3.8. The elements z of Iα have the form Λ ⊗ Idα , where Λ ∈ Mdα (B). Note that Sα ⊂ B ⊗ B(pα Hh ) and since B(pα Hh ) Mdα2 (C), then any element z ∈ Sα can be represented as a (dα2 × dα2 ) matrix over B, z = [zkl ]. Using moreover the appropriate


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identification Mdα2 Mdα ⊗ Mdα , then the elements z of Iα ⊂ Sα have the form Λ ⊗ Idα , where Λ ∈ Mdα (B). Indeed, let z ∈ Iα = (B ×δ G)ad(vr ) ∩ Sα . In particular, (1 ⊗ vr )(z ⊗ 1) = (z ⊗ 1)(1 ⊗ vr ) and hence, by applying (id ⊗ uαij ∗ · h) to both sides, we get 1 ⊗ Eijα (z ⊗ 1) = (z ⊗ 1) 1 ⊗ Eijα for all i, j = 1, . . . , dα . Using now Remark 2.3, the identity above implies that z = Λ ⊗ Idα , for some Λ ∈ Mdα (B). 4. Simplicity and primeness of the crossed product We begin this section by giving necessary and sufficient conditions for the simplicity of the crossed product B ×δ G. Theorems 4.1 and 4.4 are extensions of [11, Theorems 3.4 and 3.11], respectively, to the case of compact quantum groups. Theorem 4.1. Let δ be a coaction of a compact quantum group G on a C ∗ -algebra B. The following are equivalent: (1) B ×δ G is simple; and (2) (a) Sα,ι = (0), ∀α ∈ G, (equivalently, Sα is simple, ∀α ∈ G). (b) Iα is simple, ∀α ∈ G In this case, B ×δ G is strongly Morita equivalent with Sα , ∀α ∈ G. such that Sα,ι = (0). Then Proof. Assume (1) holds and suppose there exists α ∈ G (1 ⊗ pα )(B ×δ G)(1 ⊗ pι ) = (0) and hence (1 ⊗ pα )(B ×δ G)(1 ⊗ pι )(B ×δ G) = (0). The ideal I = (B ×δ G)(1 ⊗ pι )(B ×δ G) is a proper ideal of B ×δ G, since 1 ⊗ pι is a non-zero element in the multiplier algebra M(B ×δ G), so I = (0) and I = B ×δ G since (1 ⊗ pα )I = (0). Hence we obtain that B ×δ G is not simple and, by contradiction, (2)(a) is proved. Since Sα is a hereditary subalgebra of B ×δ G, then Sα is simple. Using Lemma 3.6, Iα is simple if and only if Sα is simple and hence (2)(b) follows. Conversely, assume (2) holds. Since Sα is simple and Sα,ι Sι,α is a two-sided ideal of Sα , we get Sα,ι Sι,α = Sα . By Proposition 3.7, B ×δ G is strongly Morita equivalent with Sι , which is simple by hypothesis. Applying [12, Theorem 3.1], it follows that B ×δ G is simple. 2 Using Proposition 3.2 we obtain the following corollary:

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Corollary 4.2. Let δ be a coaction of a compact quantum group G on a C ∗ -algebra B. The following are equivalent: (1) B ×δ G is simple; and (2) (a) Bα = (0), ∀α ∈ G, (b) Sα is simple, ∀α ∈ G. The results we obtain next are related to the primeness of the crossed product B ×δ G. First, we need the following remark: Remark 4.3. (1) Every hereditary C ∗ -subalgebra of a prime C ∗ -algebra is prime. (2) If a C ∗ -algebra A contains a prime essential ideal then A is prime. Theorem 4.4. Let δ be a coaction of a compact quantum group G on a C ∗ -algebra B. The following are equivalent: (1) B ×δ G is prime; and (2) (a) Sα,ι = (0), ∀α ∈ G, (b) Sα is prime, ∀α ∈ G. Proof. Assume (1) holds. Since 0 = 1 ⊗ pα ∈ B ×δ G and 0 = 1 ⊗ pι ∈ B ×δ G, then Sα,ι = (1 ⊗ pα )(B ×δ G)(1 ⊗ pι ) = (0). Also, since Sα are hereditary C ∗ -subalgebras of the prime C ∗ -algebra B ×δ G, using Remark 4.3, it follows that Sα are prime for all α ∈ G. Conversely, assume (2) holds. Let I = (B ×δ G)(1 ⊗ pι )(B ×δ G) be the two-sided ideal of B ×δ G generated by 1 ⊗ pι . Using Remark 4.3 it is enough to prove that I is a prime, essential ideal of B ×δ G. We prove first that I is an essential ideal, by showing that zI = (0), for all 0 = z ∈ B ×δ G. Let 0 = z ∈ B ×δ G. Since pα are orthogonal projections with sum 1, then there exists α ∈ G such that z(1 ⊗ pα ) = 0. Then (1 ⊗ pα )z∗ z(1 ⊗ pα ) = 0.

(5)

Using (2)(b) and since Sα,ι Sι,α is a two-sided ideal of Sα , we obtain (1 ⊗ pα )I (1 ⊗ pα ) = (1 ⊗ pα )(B ×δ G)(1 ⊗ pι )(B ×δ G)(1 ⊗ pα ) = Sα,ι Sι,α = (0). Therefore (1 ⊗ pα )I (1 ⊗ pα ) = (0).

(6)


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Using relations (5) and (6) above we get (1 ⊗ pα )z∗ z(1 ⊗ pα )I (1 ⊗ pα ) = (0) and hence z(1 ⊗ pα )I = (0), so zI = (0). Therefore I is an essential ideal of B ×δ G. Furthermore, I is also a prime ideal of B ×δ G, since I is strongly Morita equivalent with (1 ⊗ pι )(B ×δ G)(1 ⊗ pι ) which is assumed prime. 2 The next corollary follows from the theorem above using Proposition 3.2. Corollary 4.5. Let δ be a coaction of a compact quantum group G on a C ∗ -algebra B. The following are equivalent: (1) B ×δ G is prime; and (2) (a) Bα = (0), ∀α ∈ G, (2) Sα is prime, ∀α ∈ G. We dedicate the rest of this section to a study of the form of the elements of Iα . We obtain that these elements are matrices over B, fixed under a certain coaction. We start by making a few remarks. In [4] we give an explicit decomposition of the right regular representation, vr , as a direct sum of irreducible representations. One can show the similar decomposition for vl as well. Using the arguments of the proof of [4, Proposition 9], with vr replaced by vl one can show that (pα ⊗ 1)vl (pα ⊗ 1) is a multiple of uα , in the sense that it can be seen as a dα2 -dimensional matrix over A: (pα ⊗ 1)vl (pα ⊗ 1) =

dα

(mij ⊗ Idα ) ⊗ uαij .

(7)

i,j =1

Let now z ∈ Iα = (B ×δ G)ad(vr ) ∩Sα . Then z has the form given in Remark 3.8, z = di,jα =1 λij ⊗ (mij ⊗ Idα ). In what follows we will use the notation δα = δuα , defined as in Eq. (3). We are going to show that there exists a ∗-isomorphism between Iα and the fixed point algebra (B ⊗ Mdα (C))δα . Define first the application Φ : Iα → B ⊗ Mdα (C), Φ(z) = Λ := [λij ] ∈ B ⊗ Mdα (C), where by Mdα (C) we denote the dα -dimensional matrices over C. Note that Φ is a ∗-homomorphism. Furthermore, we obtain the following result: Then Proposition 4.6. Let G be a compact quantum group and let α ∈ G. δ Φ(Iα ) ⊆ B ⊗ Mdα (C) α . Proof. Let z ∈ Iα . Then z has the form in Remark 3.8 and using Eq. (4) it follows that:

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z ⊗ 1 = Λ ⊗ Idα ⊗ 1 =

dα

(1 ⊗ vl )(id ⊗ σ )(δ ⊗ id) λij ⊗ (mij ⊗ Idα ) 1 ⊗ vl∗

i,j =1

=

dα

(1 ⊗ vl )(id ⊗ σ ) δ(λij ) ⊗ (mij ⊗ Idα ) 1 ⊗ vl∗

i,j =1

=

dα

(1 ⊗ vl ) δ(λij )13 1 ⊗ (mij ⊗ Idα ) ⊗ 1 1 ⊗ vl∗

i,j =1

=

dα

1 ⊗ (mkl ⊗ Idα ) ⊗ uαkl δ(λij )13 1 ⊗ (mij ⊗ Idα ) ⊗ 1

i,j,k,l,r,s=1

× 1 ⊗ (mrs ⊗ Idα ) ⊗ uαsr ∗ .

Therefore, Λ⊗1=

dα

1 ⊗ mkl ⊗ uαkl δ(λij )13 (1 ⊗ mij ⊗ 1) 1 ⊗ mrs ⊗ uαsr ∗

i,j,k,l,r,s=1

= 1 ⊗ uα δι (Λ) 1 ⊗ uα ∗ where δι is defined as in Eq. (3) with u = ι, the trivial representation. Hence Φ(z) ⊗ 1 = (1 ⊗ uα )δι (Φ(z))(1 ⊗ uα ∗ ). Therefore δ Φ(Iα ) ⊆ B ⊗ Mdα (C) α .

2

Conversely, let Λ = [λ]ij ∈ B ⊗ Mdα (C) be a matrix with the property that Λ ⊗ 1 = 1 ⊗ uα δι (Λ) 1 ⊗ uα ∗ , and define Ψ (Λ) = Λ ⊗ Idα . We are going to show next that, under these assumptions, Ψ (Λ) ∈ Iα . It is then clear that Ψ is a ∗-homomorphism and that Ψ and Φ are inverse to each other, creating thus an isomorphism between Iα and (B ⊗ Mdα (C))δα . Proposition 4.7. Let Λ ∈ (B ⊗ Mdα (C))δα . Then Ψ (Λ) ∈ Iα . Proof. Let Λ = [λij ] ∈ (B ⊗ Mdα (C)) with the property δα (Λ) = Λ ⊗ 1A . We will first show that Ψ (Λ) ∈ B ×δ G. Using Eq. (4), it enough to check that Ψ (Λ) ∈ (B ⊗ K(Hh ))δvl . We have δvl Ψ (Λ) = (1 ⊗ vl )δ(λij )13 1 ⊗ (mij ⊗ Idα ) ⊗ 1 1 ⊗ vl∗ .

(8)


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Since Λ ⊗ 1 = (1 ⊗ uα )δι (Λ)(1 ⊗ uα ∗ ), using the proof of Proposition 4.6 we obtain Ψ (Λ) ⊗ 1 = (1 ⊗ vl )(id ⊗ σ )(δ ⊗ id) Ψ (Λ) 1 ⊗ vl∗ , and hence Ψ (Λ) ⊗ 1 = (1 ⊗ vl )(id ⊗ σ ) δ(λij ) ⊗ (mij ⊗ Idα ) 1 ⊗ vl∗ = (1 ⊗ vl )δ(λij )13 1 ⊗ (mij ⊗ Idα ) ⊗ 1 1 ⊗ vl∗ . Using the relation above together with Eq. (8) we obtain δvl (Ψ (Λ)) = Ψ (Λ) ⊗ 1, and hence Ψ (Λ) ∈ B ×δ G. We show next that Ψ (Λ) ∈ Iα . It is easy to check that Ψ (Λ) commutes with Eij in the following sense: (1 ⊗ Eij )Ψ (Λ) = Ψ (Λ)(1 ⊗ Eij ),

(9)

for all i, j = 1, . . . , dα . In particular, this means that (1 ⊗ pα )Ψ (Λ) = Ψ (Λ)(1 ⊗ pα ) and hence Ψ (Λ) ∈ Sα . Fur . Using Lemma 3.4 it follows that Ψ (Λ) ∈ thermore, relation (9) implies that Ψ (Λ) ∈ (1 ⊗ A) ad(v ) (B ×δ G) r and since Ψ (Λ) ∈ Sα we obtain Ψ (Λ) ∈ Iα . 2 Using Propositions 4.6 and 4.7 and the discussion between them we obtain the following result: there is an isomorphism Proposition 4.8. Let G be a compact quantum group. For every α ∈ G, δ α between Iα and (B ⊗ Mdα (C)) . Finally, using the proposition above we obtain the following corollary of Theorems 4.1 and 4.4, which gives a description of the simplicity and primeness of the crossed product in terms of the simplicity and primeness of fixed point subalgebras of matrices over B. Corollary 4.9. Let δ be a coaction of a compact quantum group G on a C ∗ -algebra B. The following are equivalent: (1) B ×δ G is simple ( prime); and (2) (a) Bα = (0), ∀α ∈ G, (b) (B ⊗ Mdα (C))δα is simple ( prime), ∀α ∈ G. Proof. The result follows from Theorems 4.1 and 4.4 and Proposition 4.8.

2

Acknowledgments We are indebted to the referee for very useful suggestions that contributed to the simplification of some proofs and the clarification of notations. We also thank Costel Peligrad for useful discussions related to this work.

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References [1] S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de C ∗ -algèbres, Ann. Sci. École Norm. Sup. (4) 26 (4) (1993) 425–488. [2] F. Boca, Ergodic actions of compact matrix pseudogroups on C ∗ -algebras, in: Recent Advances in Operator Algebras, Orléans, 1992, Astérisque 232 (1995) 93–109. [3] A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. 6 (1973) 133–252. [4] R. Dumitru, Unitary representations of compact quantum groups, in: Perspectives in Operator Algebras and Math. Physics, Bucharest, 2005, pp. 85–90, arXiv:math/0609676. [5] R. Dumitru, C. Peligrad, Compact quantum group actions on C ∗ -algebras and invariant derivations, Proc. Amer. Math. Soc. 135 (12) (2007) 3977–3984. [6] N. Jacobson, The Theory of Rings, Math. Surveys, vol. I, Amer. Math. Soc., New York, 1943. [7] M. Landstad, Algebras of spherical functions associated with covariant systems over a compact group, Math. Scand. 47 (1980) 137–149. [8] M. Landstad, Simplicity of crossed products from ergodic actions of compact matrix pseudogroups, in: Recent Advances in Operator Algebras, Orléans, 1992, Astérisque 232 (1995) 111–114. [9] A. Maes, A. Van Daele, Notes on compact quantum groups, Neuw Archief Voor Wiskunde, Vierde Serie, Deel 16 (1–2) (1998) 73–112, arXiv:math/9803122v1. [10] G.K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, Academic Press, New York, 1979. [11] C. Peligrad, Locally compact group actions on C ∗ -algebras and compact subgroups, J. Funct. Anal. 76 (1) (January 1988) 126–139. [12] M.A. Rieffel, Unitary representations of group extensions: An algebraic approach to the theory of Mackey and Blattner, Adv. Math. Suppl. Ser. 4 (1979) 364–368. [13] M.A. Rieffel, Actions of finite groups on C ∗ -algebras, Math. Scand. 47 (1980) 157–176. [14] W. Szymanski, C. Peligrad, Saturated actions of finite dimensional Hopf ∗-algebras on C ∗ -algebras, Math. Scand. 75 (1994) 217–239. [15] R. Tomatsu, Compact quantum ergodic systems, J. Funct. Anal. 254 (2008) 1–83. [16] S. Wang, Ergodic actions of universal quantum groups on operator algebras, Comm. Math. Phys. 203 (2) (1999) 481–498. [17] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (4) (1987) 613–665. [18] S.L. Woronowicz, Compact quantum groups, in: Quantum Symmetries, Proceedings of the Les Houches Summer School, 1995, North-Holland, Amsterdam, 1998, pp. 845–884.


Two versions of the Nikodym maximal function on the Heisenberg group Joonil Kim Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea Received 23 September 2008; accepted 22 May 2009 Available online 9 June 2009 Communicated by J. Bourgain

Abstract The classical Nikodym maximal function on the Euclidean plane R2 is defined as the supremum over averages over rectangles of eccentricity N ; its operator norm in L2 (R2 ) is known to be O(log N ). We consider two variants, one on the standard Heisenberg group H1 and the other on the polarized Heisenberg group H1p . The latter has logarithmic L2 operator norm, while the former has the L2 operator norm which grows essentially of order O(N 1/4 ). We shall imbed these two maximal operators in the family of operators associated to the hypersurfaces {(x1 , x2 , αx1 x2 )} in the Heisenberg group H1 where the exceptional blow up in N occurs when α = 0. © 2009 Elsevier Inc. All rights reserved. Keywords: Nikodym maximal function; Heisenberg group; Oscillatory integral operator

1. Introduction For each integer N 2, let RN be the family of all rectangles centered at the origin whose eccentricity (the length of long side divided by the length of the short side) is N . Then the classical Nikodym maximal function on the Euclidean plane is defined by 1 |R| R ∈ RN

MRN f (x) = sup

f x + y dy,

R

E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.020

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J. Kim / Journal of Functional Analysis 257 (2009) 1493–1518

where x ∈ R2 andf is a locally integrable function on R2 . A. Córdoba [2] proved that MRN L2 (R2 )→L2 (R2 ) C(1 + log N )a

(1.1)

with a = 2 to obtain results on the Bochner–Riesz means on R2 . The sharp bound a = 1 in (1.1) was obtained by Strömberg [10]. In this article we study the classical Nikodym maximal function on the Euclidean plane in the setting of the three-dimensional Heisenberg group. We consider two realizations of the Heisenberg group. First let H1 be the usual Heisenberg group identified with R3 endowed with the group multiplication 1 x · y = x1 + y1 , x2 + y2 , x3 + y3 + (x1 y2 − x2 y1 ) 2 where we use coordinates x = (x1 , x2 , x3 ) and y = (y1 , y2 , y3 ). Next let H1p be the polarized Heisenberg group endowed with the group law x ·p y = (x1 + y1 , x2 + y2 , x3 + y3 + x1 y2 ). Associated with the two cases of Heisenberg groups, we consider two maximal averages over all rectangles of eccentricity N supported on the hyperplane Π = (x1 , x2 , 0): x1 , x2 ∈ R . We define the Nikodym maximal operator MN associated with the standard Heisenberg group H1 by 1 |R| R ∈ RN

f x · (y1 , y2 , 0) dy1 dy2 .

MN f (x) = sup

(1.2)

R p

On the polarized Heisenberg group H1p , the Nikodym maximal operator MN is defined by p

1 R ∈ RN |R|

MN f (x) = sup

f x ·p (y1 , y2 , 0) dy1 dy2 .

R

The main purpose of this article is to prove the following results. Theorem 1. There are positive constants c and C independent of N such that

p

c log N MN L2 (H1 )→L2 (H1 ) C(log N )3/2 . p

p

Theorem 2. There are positive constants c and C independent of N such that cN 1/4 MN L2 (H1 )→L2 (H1 ) CN 1/4 (log N )2 .

(1.3)


1495

p

Remark 1. We regard MN f (x) and MN f (x) as the maximal averages of f over all tubes of eccentricity N that are essentially embedded on the variable planes Π(x) containing x whose normal vectors are a(x) = (x2 /2, −x1 /2, 1) and a(x) = (0, −x1 , 1) respectively. We recently have found an interesting relation between the normal vector fields a and the norms of the corresponding operator on L2 (R3 ). We intend to take up these matters on the forthcoming paper [6]. Remark 2. In the Euclidean space R2 , if the collection RN of all rectangles of eccentricity N is replaced by that of all 1 × 1/N rectangles, then the norm of the corresponding Nikodym maximal operator on L2 (R2 ) is also known to be O((log N )1/2 ) in [2] with the lower bound c0 (log N)1/2 /(log(log N ))c . Fairly recently, the bound (log N )1/2 is shown to be sharp (with respect to N ) by U. Keich [4], where the sharp Lp bounds with p > 2 are also obtained. p

Remark 3. It would be interesting to relate the behavior of MN and MN to the results of curved Nikodym and Kakeya maximal operators considered by Bourgain [1], Minicozzi and Sogge [7] and Wisewell [12]. Remark 4. Consider the spherical maximal operator S2n on the 2n-dimensional hyperplane of the 2n + 1-dimensional Heisenberg group Hn . A. Seeger and D. Müller [8] proved that S2n is 2n when n 2. The unresolved case n = 1 leads us to consider bounded on Lp (Hn ) when p > 2n−1 the Nikodym maximal function on the plane of the Heisenberg group H1 . The proofs are based on the induction argument on the scale N of eccentricity introduced in the Euclidean setting by S. Wainger [11], and the application of group Fourier transform on the Heisenberg group in combination with the Cotlar–Stein lemma used in [5]. We employ the induction argument [11] to reduce the L2 (H1 ) estimation of the maximal operator to that of certain square sum operators. The application of the group Fourier transform for these square sums leads to the uniform L2 (R1 ) estimations of oscillatory integral operators. It turns out that the phase functions of these integral operators are degenerate in the sense that their mixed second derivatives vanish in the case of standard Heisenberg group H1 , but do not vanish in the case of the polarized group H1p . This non-vanishing curvature enables us to obtain the desired uniform estimation of the oscillatory integrals. Organization. In Section 2, we employ the induction argument of [11] via the group Fourier transform to reduce to the uniform L2 (R) estimations of a certain family of one-dimensional oscillatory integral operators. In Section 3, we prove Theorem 1. For this purpose, we combine T T ∗ methods and the Cotlar–Stein lemma to show the uniform L2 (R) estimation of the oscillatory integrals. In Section 4, we obtain the lower bound in Theorem 2. In Section 5, we obtain the upper bound by using a similar argument to that of Section 3. Notation. As usual, the notation A B for two scalar expressions A, B will mean A CB for some positive constant C independent of A, B and A ≈ B will mean A B and B A.

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2. Induction and group Fourier transform 2.1. Hypersurfaces {(x1 , x2 , αx1 x2 )} on H1 We interpret our two maximal operators as the members of the class of Nikodym type maximal operators associated with the hypersurfaces Πα = (x1 , x2 , αx1 x2 ): x1 , x2 ∈ R

where α ∈ R

on the Heisenberg group H1 . In proving Theorems 1 and 2, it suffices to assume that the angle between the long side of the rectangle R ∈ RN and the x1 -axis is restricted to [0, π/4]. For such a rectangle R, we can find a parallelogram R(k, r) with some k ∈ DN = {1, . . . , N } and r > 0 given by, R(k, r) =

y1 , y 2 +

k y1 : −r < y1 < r, −r/N < y2 < r/N N

(2.1)

satisfying the following engulfing property R(k, r/10) ⊂ R ⊂ R(k, 10r).

(2.2)

Thus we now define PN as the family of all parallelograms of the form (2.1), PN = R(k, r): k ∈ DN , r > 0 .

(2.3)

Using the group isomorphism I : H1 → H1p defined by I (x1 , x2 , x3 ) = (x1 , x2 , x3 + x1 x2 /2) where I (x) ·p I (y) = I (x · y), p

1 R∈PN |R|

MN f (x) ≈ sup

1 |R| R ∈ PN

−1 f I I (x1 , x2 , x3 ) ·p I I −1 (y1 , y2 , 0) dy1 dy2

R

= sup

−1 fI I (x1 , x2 , x3 ) · (y1 , y2 , −y1 y2 /2) dy1 dy2

R

where fI (x) = f (I (x)). Thus, in order to prove Theorem 1, we work with the Nikodym maximal

p associated with the hypersurface Π operator M −1/2 : N 1 R ∈ PN |R|

p f (x) = sup M N

f x · (y1 , y2 , −y1 y2 /2) dy1 dy2 .

R

From (2.1), the support of the above integral {(y1 , y2 , −y1 y2 /2): (y1 , y2 ) ∈ R} with R = R(k, r) is written as k 1 k y1 , y2 + y1 , − y2 + y1 y1 : −r < y1 < r, −r/N < y2 < r/N . N 2 N


1497

Due to the group multiplication, k 1 k k 1 k 2 y1 . y1 , y2 + y1 , − y2 + y1 y1 = (0, y2 , 0) · y1 , y1 , − N 2 N N 2N Therefore we have,

p f (x) M M −1/2 f (x) M 2 N N

(2.4)

where MNα f (x) =

1 sup k∈DN , r>0 2r

r f (x1 , x2 , x3 ) · t, k t, α k t 2 dt, N N

−r

1 M2 f (x) = sup r>0 2r

r

f (x1 , x2 , x3 ) · (0, t, 0) dt.

−r

The operator M2 is bounded on Lp (H1 ) for all 1 0

r

fI (x1 , x2 + t, x3 − x1 x2 /2) dt,

−r

which is the directional maximal function along the second axis. Hence, in proving Theorem 1, we have only to prove that for α = −1/2,

α

M

N L2 (H1 )→L2 (H1 )

C(log N )3/2 .

(2.5)

By the change of variable t = −t, we rewrite MNα f (x) =

1 sup k∈DN ,r>0 2r

−1 r dt f x · t, k t, −α k t 2 N N

(2.6)

−r

where we note that (y1 , y2 , y3 )−1 = (−y1 , −y2 , −y3 ), which is the group inverse of (y1 , y2 , y3 ) in H1 . In proving (2.5), it suffices to consider the case that the support of integral is restricted to [0, r] in (2.6) because of similarity. Let us choose a positive smooth function ϕ supported [1/2, 4] and ϕ(t) ≡ 1 on [1, 2]. Put ϕj (t) = ϕ t/2j /2j . For each j ∈ Z, k ∈ DN and α ∈ R, we define a measure μαj,k,N by μαj,k,N (f ) =

k k f t, t, α t 2 ϕj (t) dt. N N

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Fix k ∈ D and r > 0 where 2m−1 < r 2m for some m ∈ Z. Then for f 0, 1 r

r 2j m k 2 −1 k 2 −1 2j k k dt dt f x · t, t, −α t f x · t, t, −α t N N 2m 2j N N j =−∞

0

=

2j −1

sup

f ∗ μ−α j,k,N (x1 , x2 , x3 )

sup

μαj,k,N ∗ [f ]3 (x1 , x2 , −x3 ),

j ∈Z, k∈DN j ∈Z, k∈DN

where [f ]3 (x1 , x2 , x3 ) = f (x1 , x2 , −x3 ) and ∗ is convolution on the Heisenberg group H1 . Here the last line follows from the identity [f ]3 ∗ [g]3 (x1 , x2 , x3 ) = [g ∗ f ]3 (x1 , x2 , x3 ).

(2.7)

Thus, we let MαN f (x1 , x2 , x3 ) =

sup

j ∈Z,k∈DN

μαj,k,N ∗ |f |(x1 , x2 , x3 ),

(2.8)

and prove the following general result in Section 3. Proposition 1. For any α = 0 where α = −1/2 corresponds to Theorem 1,

α

M f

N

L2 (H1 )

C(log N )3/2 f L2 (H1 ) .

2.2. Group Fourier transform Let B(L2 (R)) be the space of all bounded linear operators on L2 (R). The group Fourier transform of f ∈ L1 (H1 ) ∩ L2 (H1 ) is defined as an operator-valued function from R \ {0} to B(L2 (R)) such that λ ∈ R \ {0} → f(λ) ∈ B(L2 (R)), given by (2.9) f(λ)h (x) = F2,3 f x − y, λ(x + y)/2, λ h(y) dy R

where F2,3 is the Fourier transform with respect to the second and third variables. Note that f(λ) ∈ B(L2 (R)) is well defined since the integral kernel is square integrable with respect to x, y variables. We introduce the criteria of the L2 (H1 ) boundedness of convolution type operators on the Heisenberg group. Proposition 2. Let G be a convolution operator defined by Gf = K ∗ f for f ∈ S(H1 ) and where the convolution kernel K is a tempered distribution in S (H1 ). Then GL2 (H1 ) →L2 (H1 ) supλ∈R\{0} K(λ) L2 (R1 ) →L2 (R1 ) . Proof. For the proof of Proposition 2, we refer the reader to Chapter 1 of [3] and Chapter 11 of [9]. 2


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2.3. Non vanishing versus vanishing mixed Hessian We write k k μαj,k,N (y1 , y2 , y3 ) = ϕj (y1 )δ y2 − y1 δ y3 − α y12 , N N where δ is the Dirac mass. By applying Proposition 2 and the formula (2.9), one is led to estimate the one-dimensional oscillatory integral operator, α (λ)h (x) = μ

j,k,N

=

2,3 α F μj,k,N x − y, λ(x + y)/2, λ h(y) dy k exp −iλ Pα (x, y) ϕj (x − y)h(y) dy, N

(2.10)

where the phase function Pα (x, y) is given by 1 Pα (x, y) = (x − y)(x + y) + α(x − y)2 . 2

(2.11)

Note that the mixed second derivative of the oscillatory function Pα for α = 0 does not vanish as [Pα ] xy (x, y) = −2α = 0.

(2.12)

This non-vanishing mixed second derivative condition enables us to apply integration by parts for the kernel of the operator α ∗ α μ j,k,N (λ) μj,k,N (λ) to prove Proposition 1. But the vanishing mixed second derivative condition α = 0 in (2.12) yields the lower bound of Theorem 2, which shall be proved in Section 4. 2.4. Induction argument on the scale of N The proof of Proposition 1 is based on the following induction argument. Assume that

α

M

N L2 (H1 ) →L2 (H1 )

C(log N )3/2 ,

(2.13)

where C does not depend on N . Under this assumption we show that

α

M

2N L2 (H1 ) →L2 (H1 )

C(log 2N )3/2 ,

(2.14)

where C in (2.13) and (2.14) is to be the same one. This combined with the obvious case N = 21 implies that (2.13) holds for every positive integer N of the form 2 by induction on , which yields the desired result for every positive integer N . Now we prove (2.14) under the assumption of (2.13). For any j ∈ Z,

1500


sup μαj,k,2N ∗ f (x1 , x2 , x3 ) sup μαj,2k−1,2N − μαj,2k,2N ∗ f (x1 , x2 , x3 )

k∈D2N

k∈DN

+ sup μαj,2k,2N ∗ f (x1 , x2 , x3 ).

(2.15)

k∈DN

Note that on the last term of (2.15), μαj,2k,2N = μαj,k,N

for all j ∈ Z, k ∈ DN .

Hence we take the suprema over j ∈ Z on both sides of (2.15) to obtain that

α

M f 2 1 Gα f 2 1 + Mα f 2 1 , N 2N L (H ) L (H ) L (H )

(2.16)

where G f (x1 , x2 , x3 ) = α

α μ

α j,2k−1,2N − μj,2k,2N

2 ∗ f (x1 , x2 , x3 )

1/2 .

j ∈Z, k∈DN

In proving (2.14) it suffices to show that for α = 0,

α

G f

L2 (H1 )

C ∗ (log N )1/2 f L2 (H1 ) ,

(2.17)

since (2.17), (2.16) and (2.13) with C > 10C ∗ lead us to obtain that

α

3/2 ∗ 1/2

M 2 1 + C(log N )3/2 C log(2N ) . 2N L (H )→L2 (H1 ) C (log N ) For the estimate in (2.17), we now use the group Fourier transform. For each k ∈ DN , j ∈ Z and fixed λ, α = 0, put α,λ α (λ) − μ Tj,k,N = μαj,2k−1,2N j,2k,2N (λ) α where μ j,k,N (λ) is the group Fourier transform which is expressed in (2.10). In proving (2.17), by Proposition 2, it suffices to prove that there exists a constant C > 0 independent of λ such that for α = 0,

α

G (λ)h

L2 (R1 )

C(log N )1/2 hL2 (R1 ) ,

(2.18)

where α G (λ)h (x) =

2 α,λ T j,k,N h(x)

1/2 .

j ∈Z, k∈DN

3. Proof of Theorem 1 In this section we shall prove (2.18) in order to prove Proposition 1.


1501

3.1. Decomposition into local and global parts Take a function ψ ∈ Cc∞ ([−2, 2]) such that 0 ψ 1 and ψ(u) = 1 for |u| 1/2. Put η(u) = ψ(u) − ψ(2u). We choose the integer a satisfying 2a−1 < 210 |α| + 1 2a .

(3.1)

Note that a > 10. For fixed λ and α, let k y exp iλ Pα (x, y) ϕj (x − y)ψ j +a h(y) dy, N 2 k y m Sj,k,N h(x) = exp iλ Pα (x, y) ϕj (x − y)η m h(y) dy, N 2

loc Sj,k,N h(x) =

(3.2)

where m > j + a. Then, ∞

loc α μ j,k,N (λ) = Sj,k,N +

m Sj,k,N .

m=j +a+1 m loc and Sj,k,N for simplicity. We set We omit λ and α from the notation Sj,k,N loc loc loc Tj,k,N = Sj,2k−1,2N − Sj,2k,2N , glo

Tj,k,N =

∞

m m Sj,2k−1,2N − Sj,2k,2N .

m=j +a+1

In proving (2.18), we show that there is a constant C > 0 independent of λ such that

loc 2 T j,k,N h

1/2

glo 2 T j,k,N h

1/2

j ∈Z, k∈DN

L2 (R1 )

C(log N )1/2 hL2 (R1 ) ,

(3.3)

C(log N )1/2 hL2 (R1 ) .

(3.4)

and

j ∈Z, k∈DN

L2 (R1 )

3.2. Local part estimate (3.3) It suffices to show that for each fixed nonzero α and λ, k 2j −1/2 1 2j 2 λ min 1, 2 λ , op N N

loc

T j,k,N

since from (3.5)

(3.5)

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loc 2 T j,k,N h

j ∈Z, k∈DN

1/2 2

2

L (R1 )

j ∈Z, k∈DN

2 k 2j −1 2 λ min 1, 1 22j λ h2 2 1 N N L (R )

(log N )h2L2 (R1 ) . Proof of (3.5). Our proof is based on the T T ∗ method. In order to estimate the L2 norm of loc loc loc Tj,k,N = Sj,2k−1,2N − Sj,2k,2N , loc [S loc ]∗ : we compute the integral kernel S(x, z) of the operator Sj,k,N j,k,N

S(x, z) =

2 y k exp iλ Pα (x, y) − Pα (z, y) ϕj (x − y)ϕj (z − y)ψ j +a dy. N 2

where we recall 1 Pα (x, y) = (x − y)(x + y) + α(x − y)2 . 2 The derivative of the phase function with respect to the y variable is [Pα ] y (x, y) − [Pα ] y (z, y) = −2α(x − z),

(3.6)

which enables us to apply integration by parts to obtain that S(x, z)

Cα 2−j | Nk 2j λ(x − z)|2 + 1

.

By Schur’s test,

loc

S

j,k,N op

S loc S loc ∗

j,k,N j,k,N op −1/2 k sup S(x, z) dx 22j λ . N z

=

(3.7)

Hence

loc

T

j,k,N op

S loc

j,2k−1,2N op

+ S loc

j,2k,2N

k 2j −1/2 2 λ . op N

(3.8)

In order to gain | N1 22j λ| in (3.5), we need to use the mean value theorem as well as integration loc = S loc loc by parts. Since Tj,k,N j,2k−1,2N − Sj,2k,2N , we write loc f (x) = Tj,k,N

k

eiλ N Pα (x,y) Uj,k,N (x, y)f (y) dy,


1503

where λ Uj,k,N (x, y) = e−i 2N Pα (x,y) − 1 ϕj (x − y)ψ y/2j +a . loc [T loc ]∗ is Thus the integral kernel T (x, z) of Tj,k,N j,k,N

T (x, z) =

k exp iλ Pα (x, y) − Pα (z, y) Uj,k,N (x, y)Uj,k,N (z, y) dy. N

By using the mean value theorem and the support condition of the above integral such that |y| 2j , |x| 2j and |x − y| ≈ 2j , Uj,k,N (x, y) min 1, 1 22j λ 2−j , N 1 j −j ∂Uj,k,N ∂y (x, y) N 2 λ2 , 2 λ 1 j 2 −j ∂ Uj,k,N ∂y 2 (x, y) max N , N 2 λ 2 .

(3.9)

The above inequalities also hold when Uj,k,N (x, y) is replaced by Uj,k,N (z, y). Using the first inequality of (3.9), 2 T (x, z) λ 22j 2−j . N

(3.10)

Applying integration by parts twice, we get T (x, z) =

2 Uj,k,N (x, y)Uj,k,N (z, y) k ∂ dy. exp iλ Pα (x, y) − Pα (z, y) N ∂y (λ Nk 2α(x − z))2

By using (3.9) and the support condition, we obtain that T (x, z)

| Nλ 22j |2 2−j | Nk 2j λ(x − z)|2

(3.11)

.

From (3.10) and (3.12), T (x, z)

| Nλ 22j |2 2−j | Nk 2j λ(x − z)|2 + 1

.

Thus by Schur’s test,

loc loc ∗

| λ 22j |2 2−j

T

N . T j,k,N j,k,N op | Nk 2j λ|

1504


Hence, k 2j −1/2 1 2j 2 λ, 2 λ N N op

loc

T

j,k,N

(3.12)

2

which combined with (3.8) yields (3.5). 3.3. Global part estimate (3.4)

m m m In view of Tj,k,N = Sj,2k−1,2N − Sj,2k,2N ,

2k −1 Pα (x, y) exp iλ Pα (x, y) −1 exp iλ 2N 2N y × ϕj (x − y)η m h(y) dy. 2

m Tj,k,N h(x) =

glo

Thus the integral kernel for Tj,k,N =

∞

m m=j +a+1 Tj,k,N

(3.13)

is supported in the set given by

(x, y): |x − y| ≈ 2j and |y| > 2j +a ⊂ (x, y): |x| ≈ |y| > 2j +a , where a > 10. By using this support condition and (3.13), glo 2 T j,k,N h(x) dx = j ∈Z, k∈DN

j ∈Z, k∈DN m∈Z

2m−1 2j +10 in (5.11), it suffices to replace Uj,k,N in (5.20) m , by only one piece Uj,k,N glo,1

j ∈Z, k∈DN

m ∗ m

Uj,k,N Uj,k,N

1.

op

For this, it suffices to show that

m

U

j,k1 ,N

∗

m Uj,k 1 ,N

|k1 − k2 | j +m −5/2 1 j +m 3

m ∗ m

(5.22) 2 Uj,k2 ,N Uj,k2 ,N op λ min 1, 2 λ N N

since the Cotlar–Stein lemma with (5.22) yields that

1516


m m ∗

Uj,k,N Uj,k,N

op

j,k

m ∗

m

Uj,k,N Uj,k,N

j ∈Z

op

k

∞ 1 j +m −5/4 1 j +m 1/4 1 λ , 2 λ min 2 N N k 5/4 j ∈Z

k=1

1. By using the support condition |y| ≈ 2m > 2j +10 in (5.11), j +m exp iλ 2k − 1 x 2 − y 2 − exp iλ 2k x 2 − y 2 C min λ2 ,1 . 2N 2N N So, we get

m

U

j,k,N

j +m λ2 C min ,1 . op N

(5.23)

Next we show that −5 m ∗

|k1 − k2 | 2j +m λ . U j,k1 ,N j,k2 ,N N op

m

U

(5.24)

If (5.24) is true, then by interpolating (5.24) and (5.23), −5/2 j +m m ∗

λ2

|k1 − k2 | 2j +m λ ,1 . U min j,k1 ,N j,k2 ,N N op N

m

U

(5.25)

We see that (5.22) follows from (5.23) and (5.25). In proving (5.24), from (5.2) it suffices to show that |k1 − k2 | j +m −5 m ∗

2 λ . j,k1 ,N Vj,k2 ,N op N

m

V

(5.26)

m m [Vj,k ]∗ is in (5.14). From the observation The integral kernel S(x, z) of the operator Vj,k 1 ,N 2 ,N that |x + y| ≈ |z + y| ≈ 2m > 2j +10 in (5.16) and (5.17), we obtain that

∂ Θ(x, y, z) C . 23j ∂y

(5.27)

By using (5.15) and (5.27) with the support condition |y| ≈ 2m > 2j +10 , we apply integration by parts on the integral (5.14) to obtain that 5 −j CN V (x, z) λ|k − k |2j +m 2 1 2 which yields (5.26). The proof of (5.20) is finished.

2


1517

Proof of (5.21). Note that our difficulty here comes from the fact that for the case j − 10 < m < j + 10, |x + y| can vanish in (5.16), which prevent us from having (5.27). Instead, we have some bigger bound, j ∂ Θ(x, y, z) C max 1 , λ 2 . ∂y 22j 2j N

(5.28)

However, we can overcome this obstacle by the following observation. The integral kernel K(x, y) of the operator glo,2 ∗ glo,2 Uj,k,N Uj,k,N h (y) = K(x, y)h(y) dy is supported on the set

(x, y): 2j −15 < |x| < 2j +15 , 2j −15 < |y| < 2j +15 .

Thus it suffices to prove for fixed j and m where j − 10 < m < j + 10,

m ∗ m

U U j,k,N j,k,N 1.

k∈DN

op

For proving this we show that

m

U

m ∗ m m

Uj,k2 ,N Uj,k Uj,k 1 ,N 2 ,N op 1 2j 3 |k1 − k2 | 2j −5/2 −5/2 min 1, 2 λ 2 λ max , |k1 − k2 | N N ∗

j,k1 ,N

|k1 − k2 |−5/2

(5.29)

since the Cotlar–Stein lemma with (5.29) yields that

1

m m ∗

U U 1. j,k,N j,k,N

k 5/4 op k

k

For the proof of (5.29), we replace (5.27) by (5.28) and apply the same estimation of (5.22) with the support condition |y| ≈ 2j instead of 2m . 2 Acknowledgments I would like to express my deep gratitude to the referee for many careful suggestions. His suitable and crucial comments were able to make a great improvement of the original manuscript. I am also indebted to Professor Andreas Seeger who gave me many valuable suggestions that have significantly improved the exposition of the manuscript. I have furthermore to thank Professor Seungsu Hwang for providing me with some illustration which confirmed me of a bad behavior of our Nikodym maximal function. This work was supported by Korea Research Foundation (KRF) grant funded by the Korea government (MEST) (No. 0074705).

1518


References [1] J. Bourgain, Lp -estimates for oscillatory integrals in several variables, Geom. Funct. Anal. 1 (1991) 321–374. [2] A. Córdoba, The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977) 1–22. [3] G.B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud., vol. 122, Princeton Univ. Press, 1989. [4] U. Keich, On Lp bounds for Kakeya maximal functions and the Minkowski dimension in R2 , Bull. London. Math. Soc. 4 (1997) 221–237. [5] J. Kim, Hilbert transforms along curves in the Heisenberg group, Proc. London Math. Soc. 80 (2000) 611–642. [6] J. Kim, Nikodym maximal functions associated with variable planes, preprint. [7] W. Minicozzi, C. Sogge, Negative results for Nikodym maximal functions and related oscillatory integrals in curved space, Math. Res. Lett. 4 (1997) 221–237. [8] D. Müller, A. Seeger, Singular spherical maximal operators on a class of two step nilpotent Lie groups, Israel J. Math. 141 (2004) 315–340. [9] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Press, 1993. [10] J.-O. Stromberg, Maximal functions associated to rectangles with uniformly distributed directions, Ann. of Math. 107 (1978) 399–402. [11] S. Wainger, Applications of Fourier transform to averages over lower dimensional sets, in: Pro. Sympos. Pure Math., vol. 35 (1), Amer. Math. Soc., Providence, RI, 1979, pp. 85–94. [12] L. Wisewell, Kakeya sets of curves, Geom. Funct. Anal. 15 (2005) 1319–1362.


Large deviation principles for 2-D stochastic Navier–Stokes equations driven by Lévy processes Tiange Xu, Tusheng Zhang ∗ Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, UK Received 26 September 2008; accepted 4 May 2009 Available online 20 May 2009 Communicated by Paul Malliavin

Abstract In this paper, we establish a large deviation principle for the two-dimensional stochastic Navier–Stokes equations driven by Lévy processes, which involves the study of the Lévy noise and the investigation of the effect of the highly nonlinear, unbounded drifts. © 2009 Published by Elsevier Inc. Keywords: Stochastic Navier–Stokes equation; Lévy process; Large deviation principle

1. Introduction It is well known that the two-dimensional Navier–Stokes equation with Dirichlet boundary condition describes the time evolution of an incompressible fluid and is given by ⎧ du − νu dt + (u · ∇)u dt + ∇p dt = g dt, ⎪ ⎨ (∇ · u)(t, x) = 0, for x ∈ D, t > 0, u(t, x) = 0, for x ∈ ∂D, t > 0, ⎪ ⎩ u(0, x) = u0 (x), for x ∈ D, where D is a bounded domain in R2 with smooth boundary ∂D, u(t, x) ∈ R2 denotes the velocity field at time t and position x, p(t, x) denotes the pressure field, ν > 0 is the viscosity and g is a deterministic force. * Corresponding author.

E-mail address: [email protected] (T. Zhang). 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.05.007

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T. Xu, T. Zhang / Journal of Functional Analysis 257 (2009) 1519–1545

To formulate the Navier–Stokes equations, we introduce the following standard spaces: V = v ∈ H01 D; R2 : ∇ · v = 0, a.e. in D , with the norm

vV :=

1 2

|∇v| dx 2

= v,

D

and denote by (, . ,) the inner product of V . H is the closure of V in the L2 -norm

|v|H :=

1 |v| dx 2

2

= |v|.

D

The inner product on H will be denoted by (· , ·). Define the operator A (Stokes operator) in H by the formula Au := −νPH u,

∀u ∈ H 2 D; R2 ∩ V ,

where the linear operator PH (Helmhotz–Hodge projection) is the projection operator from L2 (D; R2 ) to H , and the nonlinear operator B B(u, v) := PH (u · ∇)v , with the notation B(u) = B(u, u). Obviously the domain of B requires that (u · ∇)v belongs to the space L2 (D; R2 ). By applying the operator PH to each term of the above Navier–Stokes equation (NSE), we can rewrite the NSE in the following abstract form: du(t) + Au(t) dt + B u(t) dt = f (t) dt

in L2 [0, T ]; V ,

(1.1)

with the initial condition u(0) = x

in H.

(1.2)

The purpose of this paper is to establish a large deviation principle for Eq. (1.1) driven by the additive Lévy noise, that is ⎧ ⎪ ⎨ dun (t) = −Aun (t) dt − B un (t) dt + b dt + √1 dW (t) + 1 f (x)N˜ n (dt, dx), n n X ⎪ ⎩ n u (0) = x ∈ H,

(1.3)

where W (·) is an H -valued Brownian motion, b is a constant vector in H , f is a measurable mapping from some measurable space X to H , and N˜ n (dt, dx) is a compensated Poisson measure on [0, ∞) × X with intensity measure nν, where ν is a σ -finite measure on B(X).


1521

There exists a great amount of literature on the stochastic Navier–Stokes equation. A good reference for stochastic Navier–Stokes equations driven by additive noise is the book [5] and the references therein. The existence and uniqueness of solutions for the 2-D stochastic Navier– Stokes equations with multiplicative Gaussian noise were obtained in [11,15,17]. The ergodic properties and invariant measures of the 2-D stochastic Navier–Stokes equations were studied in [10] and [13]. The small Gaussian noise large deviation of the 2-D stochastic Navier–Stokes equations was established in [17] and the large deviation of occupation measures was considered in [12]. Large deviations for stochastic equations and stochastic partial differential equations have been investigated in many papers, see [1–4,19]. There is not much work on large deviations for stochastic evolution equations driven by Lévy noises in infinite dimensions. To the best of our knowledge, [16] is the first paper on this topic, where the Lipschitz coefficients are considered. For this paper, in addition to the difficulties caused by the Lévy noise, much of the problem is to deal with the highly nonlinear term B(u, u). For this purpose, we need to prove a number of exponential estimates for the energy of the solutions as well as the exponential convergence of the approximating solutions. We mention that the large deviation principle for the solution of the stochastic equation driven by jump processes in finite dimensions has been established in [8]. The organization of this paper is as follows. In Section 2, we collect some preliminary facts which are frequently used in the sequel. In Section 3, we prove a number of exponential estimates for the solutions, which will play an important role in the rest of the paper. Section 4 is devoted to establish a large deviation principle. 2. Preliminaries Identifying H with its dual H , we consider Eq. (1.3) in the framework of Gelfrand triple: V ⊂H ∼ = H ⊂ V . In this way, we may consider A as a bounded operator from V into V . Moreover, we also denote by · , ·, the duality between V and V . Hence, for u = (ui ) ∈ V , w = (wi ) ∈ V , we have

Au, w = ν

∂i uj ∂i wj dx = ν((u, w)).

(2.1)

i,j D

Introduce a trilinear form on H × H × H by setting b(u, v, w) =

2

ui ∂i vj wj dx,

i,j D

whenever the integral in (2.2) makes sense. In particular, if u, v, w ∈ V , then 2 B(u, v), w = (u · ∇)v, w =

i,j D

ui ∂i vj wj dx = b(u, v, w).

(2.2)

1522


By integration by parts, b(u, v, w) = −b(u, w, v),

(2.3)

therefore b(u, v, v) = 0,

∀u, v ∈ V .

(2.4)

There are some well-known estimates for b (see [18] for example), which will be required in the rest of this paper and we list them here. Throughout the paper, we denote various generic positive constants by the same letter c. We have b(u, v, w) cu · v · w b(u, v, w) c|u| · v · |Aw| b(u, v, w) cu · |v| · |Aw| b(u, v, w) 2u 21 · |u| 12 · w 21 · |w| 12 · v

(2.5) (2.6) (2.7) (2.8)

for suitable u, v, w. Moreover, combining (2.3) and (2.8), we obtain a useful estimate as follows: B(u)

V

= sup b(u, u, v) = sup b(u, v, u) 2u · |u|. v1

v1

(2.9)

Before ending this section, let us set up the stochastic basis. Let (Ω, F , P ) be a probability space equipped with a filtration {Ft , t 0} satisfying the usual conditions. Let W (·) be a H -valued Brownian motion on (Ω, F , P ) with the covariance operator Q, which is a positive, symmetric, trace class operator on H . Let (X, B(X)) be a measurable space and ν(dx) a σ finite measure on it. Let p = (p(t), t ∈ Dp ) be a stationary Ft -Poisson point process on X with characteristic measure ν(dx), where Dp is a countable subset of [0, ∞) depending on random parameter ω (see[14]). Denote by N (dt, dx) the Poisson counting measure associated with p, i.e., N(t, A) = s∈Dp , st IA (p(s)). Let N˜ (dt, dx) := N (dt, dx) − dtν(dx) be the compensated Poisson measure. Denoted by N˜ n (dt, dx) the compensated Poisson measure with the characteristic measure nν. Let b be a constant vector in H and f be a measurable mapping from X to H . Throughout this paper, we assume that

f (x)2 exp a f (x) ν(dx) < +∞,

for all a > 0.

(2.10)

X

Using approaches similar to that in [17], we can easily show in this additive case that Eq. (1.3) has a unique solution in L2 ([0, T ]; V ) ∩ D([0, T ]; H ), where D([0, T ]; H ) denotes the space of all the càdàg pathes from [0, T ] to H endowed with the uniform convergence topology.


1523

3. Exponential estimates To establish the large deviation principle, we first prove some exponential estimates. Let un· be the solution of the following stochastic Navier–Stokes equation t unt

=x−

t Auns ds

0

−

B

uns

1 1 ds + bt + √ Wt + n n

t

f (x)N˜ n (ds, dx).

(3.1)

0 X

0

Let X·n = nun· , then X·n is the solution of the following equation t Xtn

= nx −

AXsn ds

1 − n

0

t

√ B Xsn ds + nbt + nWt +

t


(3.2)

0 X

0

Denote by {ek }∞ k=1 an orthonormal basis of H that consists of eigenvectors of Q in V with {λk }∞ k=1 being the corresponding eigenvalues. g

Lemma 3.1. For g ∈ Cb2 (H ), Mt = exp(g(Xtn ) − g(nx) − where

t 0

h(Xsn ) ds) is a Ft -local martingale,

∞ n 1 h(y) = − Ay + B(y), g (y) + n b, g (y) + λk g (y) ⊗ g (y) + g (y) ek , ek n 2 k=1 exp g y + f (x) − g(y) − 1 − g (y), f (x) ν(dx). (3.3) +n X

Proof. Applying Itô’s formula to exp(g(Xtn )), we get exp g Xtn − g(nx) −

t

exp g Xsn − g(nx) h Xsn ds

0

is a local martingale. The lemma follows by another integration by parts.

2

1

In the rest of this section, we always set g(y) := (1 + λ|y|2 ) 2 (λ > 0). It is easy to see that supg (y) λ, y

1 supg (y) λ 2 . y

Denote by Tr Q the trace of the operator Q, i.e., Tr Q := We have the following results.

∞

i=1 (Qei , ei ) =

Lemma 3.2. limr→∞ lim supn→∞ n1 log P (sup0t1 |unt | > r) = −∞.

∞

i=1 λi .

1524


Proof. By the proof of Proposition 4.2 in [16], we know that exp g y + f (x) − g(y) − 1 − g (y), f (x) ν(dx) X

2 1 λf (x) exp λ 2 f (x) ν(dx) := Mλ .

X

Since

1 2λ B(y), g (y) = B(y), y = 0, 1 2 (1 + λ|y|2 ) 2

we have, 1 h Xtn n|b|λ 2 + nλ Tr Q + nMλ ,

(3.4)

where h(·) is defined in Lemma 3.1. Observe that P sup un > r = P sup X n > nr 0t1

t

=P

t

0t1

1 sup g Xtn > 1 + λn2 r 2 2

0t1

=P

sup 0t1

g Xtn − g(nx) −

1 > 1 + λn2 r 2 2 P

sup 0t1

P

sup 0t1

h Xsn ds + g(nx) +

0

g Xtn − g(nx) −

1 > 1 + λn2 r 2 2

t

t

g Xtn − g(nx) −

t

t n h Xs ds + g(nx) + sup h Xsn ds 0t1

h Xsn ds

0

0

1 > 1 + λn2 r 2 2 − g(nx) − sup 0t1

Due to (3.4) and Doob’s inequality,

h Xsn ds

0

0

t

t 0

n h Xs ds .

(3.5)


sup

P

0t1

g Xtn − g(nx) −

t 1 n h Xs ds > 1 + λn2 r 2 2 − g(nx) − sup h Xsn ds 0t1

0

P

t

sup 0t1

g Xtn − g(nx) −

t

h Xsn ds

0

sup E exp g Xtn − g(nx) −

0t1

t

0

1 1 > 1 + λn2 r 2 2 − g(nx) − n|b|λ 2 − nλ Tr Q − nMλ

1525

h Xsn ds

0

1 1 × exp − 1 + λn2 r 2 2 + g(nx) + n|b|λ 2 + nλ Tr Q + nMλ 1 1 exp − 1 + λn2 r 2 2 + g(nx) + n|b|λ 2 + nλ Tr Q + nMλ ,

(3.6)

where in the last step, we used the fact that

sup E exp g Xtn − g(nx) −

0t1

t

h Xsn ds

1,

0

t because exp(g(Xtn ) − g(nx) − 0 h(Xsn ) ds) is a nonnegative local martingale with the initial value 1. Putting (3.5) and (3.6) together, we have 1 (1 + λn2 r 2 ) 2 (1 + λn2 |x|2 ) 2 1 log P sup unt > r − + + |b|λ 2 + λ Tr Q + Mλ . (3.7) n n n 0t1 1

1

Hence, lim sup n→∞

1 1 1 1 log P sup unt > r − λr 2 2 + λ|x|2 2 + |b|λ 2 + λ Tr Q + Mλ . n 0t1

Taking r → ∞ in the above inequality, one obtains the result. Lemma 3.3. limr→∞ lim supn→∞ n1 log P ((

1 0

2

1

unt 2 dt) 2 > r) = −∞.

Proof. As 1 P

t

0

and

n 2 u dt

1 2

1

>r =P

n 2 nu dt t

0

1 2

1

> nr = P

n 2 X dt t

0

1 2

> nr ,

1526


1

n 2 X dt

1

1

2

=

t

0

2 − 1 n 2 1 2 · X · 1 + λX n 2 2 dt 1 + λXtn t t

1 2

0

2 1 12 sup 1 + λXtn 2 ·

1

0t1

2 − 1 n 2 2 · X dt 1 + λXtn t

1 2

0

2 1 1 1 sup 1 + λXtn 2 + 2 0t1 2

1

2 − 1 n 2 2 · X dt, 1 + λXtn t

0

it is sufficient to show lim lim sup

r→∞ n→∞

2 1 1 log P sup 1 + λXtn 2 > nr = −∞, n 0t1

(3.8)

and

1 lim lim sup log P r→∞ n→∞ n

1

n 2 − 1 n 2 2 1 + λ Xt · Xt dt > nr = −∞.

(3.9)

0

(3.8) follows from the previous lemma. We prove (3.9). 1 1 Set Y1n := 0 (1 + λ|Xtn |2 )− 2 · Xtn 2 dt. Then 2 − 1 n 2 1 n n 2 · X = AX n + λν 1 + λXtn B X , g X t t t t , n and 1

1 h Xsn ds + λνY1n n|b|λ 2 + nλ Tr Q + nMλ .

0

Therefore, P Y1n > nr = P λνY1n > λνnr P g X1n + λνY1n > λνnr =P

g X1n − g(nx) −

1 0

h Xsn ds + g(nx) +

1 0

n n h Xs ds + λνY1 > λνnr


g X1n − g(nx) −

P

1

1527

1 n h Xs ds > λνnr − 1 + λn2 |x|2 2

0

1 − n |b|λ 2 + λ Tr Q + Mλ

1 1 exp −λνnr + 1 + λn2 |x|2 2 + n |b|λ 2 + λ Tr Q + Mλ ,

(3.10)

which yields lim lim sup

r→∞ n→∞

completing the proof.

1 log P Y1n > nr = −∞, n

2

Define the projection operator Pm by Pm x :=

m

(x, ei )ei ,

x ∈ H.

i=1

Let Ztn,m , Ztn be the solutions of the following linear equations respectively, t Ztn,m = −

AZsn,m ds +

1 n

t

Pm f (x)N˜ n (ds, dx),

(3.11)


(3.12)

0 X

0

and t Ztn

=−

1 + n

AZsn ds

t 0 X

0

Put Z˜ tn,m := n(Ztn,m − Ztn ), then Z˜ tn,m is the solution of the equation Z˜ tn,m = −

t

AZ˜ sn,m ds +

t

Pm f (x) − f (x) N˜ n (ds, dx).

(3.13)

0 X

0

Similar to the proof of Lemma 3.1, one has

exp g Z˜ tn,m − g(0) −

t 0

where

h˜ Z˜ sn,m ds

is a Ft -local martingale,

(3.14)

1528


˜ h(y) = − Ay, g (y) + n

exp g y + Pm f (x) − f (x) − g(y) − 1

X

− g (y), Pm f (x) − f (x) ν(dx). Lemma 3.4. For any δ > 0,

1 lim lim sup log P m→∞ n→∞ n

1

n,m n 2 Z − Zs ds > δ = −∞. s

0

Proof. By Lemma 5.6 in [16], we know that, for any δ > 0, 1 log P sup Ztn,m − Ztn > δ = −∞. n 0t1

lim lim sup

m→∞ n→∞

(3.15)

Define a stopping time by τ1n,m := inf t 0, Ztn,m − Ztn > 1 , then 1 P

n,m 2 Z − Zsn ds > δ, sup Ztn,m − Ztn 1 s

0t1

0

1 P

n,m 2 Z − Z n ds > δ, 1 τ n,m s

s

1

0 n,m 1∧τ 1 n,m 2 n Z P − Z ds > δ

s

P

s

0

sup 0t1∧τ1n,m 1∧τ1n,m

×

g

−1

n,m n g nZt− − nZt−

n,m n,m n n 2 2 nZs − nZs · nZs − nZs ds > n δ ,

0 n,m n = nZ n,m − nZ n } is countable. where in the last step, we have used the fact that {s: nZs− − nZs− s s n,m By the definition of g and the stopping time τ1 , we get


sup

P

0t1∧τ1n,m

1∧τ1n,m

n,m n g nZt− − nZt− ·

2 g −1 nZsn,m − nZsn · nZsn,m − nZsn ds > n2 δ

0

t 0

0

n,m 1∧τ 1 2 P g −1 nZsn,m − nZsn · nZsn,m − nZsn ds >

Set Zˆ tn,m :=

1529

n2 δ 1

(1 + λn2 ) 2

.

g −1 (Z˜ sn,m ) · Z˜ sn,m 2 ds, then 1∧τ1n,m

n,m h˜ Z˜ tn,m + λν Zˆ 1∧τ n,m nMλ,m ,

(3.16)

1

0

where Mλ,m = λ

2 1 exp λ 2 Pm f (x) − f (x) · Pm f (x) − f (x) ν(dx).

X

As in the proof of (3.10), we have

n,m P Zˆ 1∧τ n,m > 1

n2 δ 1

(1 + λn2 ) 2

n,m = P λν Zˆ 1∧τ n,m > 1

λνn2 δ 1

(1 + λn2 ) 2

n,m ˆ n,m n,m > P g Z˜ 1∧τ n,m + λν Z 1∧τ 1

1

n,m g Z˜ 1∧τ n,m − g(0) −

P

1

(1 + λn2 ) 2

1∧τ1n,m

h˜ Z˜ sn,m ds + g(0) +

1

>

˜h Z˜ sn,m ds + λν Zˆ n,m n,m 1∧τ

1∧τ1n,m

0

λνn2 δ

λνn2 δ

1

0

1

(1 + λn2 ) 2

n,m P g Z˜ 1∧τ n,m − g(0) −

1

exp −

˜h Z˜ sn,m ds >

1∧τ1n,m

0

λνn2 δ 1

(1 + λn2 ) 2

+ 1 + nMλ,m ,

where we used the fact in (3.14). Therefore,

λνn2 δ 1

(1 + λn2 ) 2

− g(0) − nMλ,m )

1530


1 lim sup log P n→∞ n

1

n,m n,m 1 n 2 n Z − Zs ds > δ, sup Zt − Zt 1 −νδλ 2 + Mλ,m . (3.17) s 0t1

0

Since limm→∞ Mλ,m = 0, let m → ∞ in (3.17) to obtain 1 lim lim sup log P m→∞ n→∞ n

1

n,m n,m 1 n 2 n Z − Zs ds > δ, sup Zt − Zt 1 −νδλ 2 . s 0t1

0

Since λ is arbitrary, taking λ → ∞ in the above inequality, we obtain 1 lim lim sup log P m→∞ n→∞ n

1

n,m n,m n 2 n Z − Zs ds > δ, sup Zt − Zt 1 = −∞. s 0t1

0

Combining with (3.15) proves the lemma.

2

4. Large deviation principle First, we state the main result of this paper. For l ∈ H , define F (l) := exp f (x), l − 1 − f (x), l ν(dx) + (Ql, l) + (b, l). X

Set, for z ∈ H, F ∗ (z) = sup (z, l) − F (l) . l∈H

Let Lnt

1 1 := bt + √ Wt + n n

t

f (x)N˜ n (ds, dx),

0 X

then by [7], we know that the laws of {Ln· , n 1} satisfy a large deviation principle on D([0, 1]; H ) with the rate function I0 , which is defined by I0 (g) : =

1

F ∗ (g (s)) ds, ∞, otherwise. 0

if g ∈ D([0, 1]; H ), g ∈ L1 ([0, 1]; H ),

For g ∈ D([0, 1]; H ) with g ∈ L1 ([0, 1]; H ), define φ(g) ∈ D([0, 1]; H ) ∩ L2 ([0, 1]; V ) to be the solution of the following equation t φt (g) = x −

t Aφs (g) ds −

0

0

B φs (g) ds + g(t).

(4.1)


1531

Set, for h ∈ D([0, 1]; H ), I (h) := inf I0 (g): h = φ(g), g ∈ D [0, 1]; H with the convention inf{∅} = ∞. Theorem 4.1. Let μn be the law of the solution un of Eq. (1.3), then {μn , n 1} satisfies a large deviation principle on D([0, 1]; H ) endowed with the uniform topology with the rate function I (·), i.e., (i) For any closed subset F ⊂ D([0, 1]; H ), lim sup n→∞

1 log μn (F ) − inf I (h). h∈F n

(ii) For any open set G ⊂ D([0, 1]; H ), lim inf n→∞

1 log μn (G) − inf I (h). h∈G n

According to the generalized contraction principle in the theory of large deviations (see Theorem 4.1 in [9]), Theorem 4.1 will follow from the following Lemmas 4.1–4.3. Lemma 4.1. The mapping φ defined in (4.1) is continuous from D([0, 1]; V ) into D([0, 1]; H ) ∩ L2 ([0, 1]; V ) in the topology of uniform convergence. Proof. Let vt (g) = φt (g) − g(t), then vt (g) satisfies the following equation t vt (g) = x −

t Avs (g) ds −

0

t Ag(s) ds −

0

B vs (g) + g(s) ds.

0

It is sufficient to show that v(·) : D [0, 1]; V → D [0, 1]; H ∩ L2 [0, 1]; V is continuous, that is, take {gn }∞ n=1 , g ∈ D([0, 1]; V ) such that limn→∞ sup0t1 gn (t) − g(t) = 0, then lim

n→∞

2 ν sup vt (g) − vt (gn ) + 2 0t1

1

2 vs (g) − vs (gn ) ds = 0.

0

To this end, we need some energy estimates for vt (g). In view of (2.3), (2.4), (2.5), and (2.9), we have

1532


vt (g)2 + 2ν

t

vs (g)2 ds

0

t

Ag(s), vs (g) ds − 2

= |x| − 2 2

0

vs (g)2 ds + 1 ν

|x| + ν 2

0

t

g(s)2 ds + 2

0

t

vs (g)2 ds + 1 ν

|x| + ν 2

0

+2

B vs (g) + g(s) , vs (g) ds

0

t

t

t

t

B vs (g) + g(s) , vs (g) ds

0

t

g(s)2 ds + 2

0

t

b vs (g), g(s), vs (g) ds

0

b g(s), g(s), vs (g) ds

0

t

vs (g)2 ds + 1 ν

|x| + ν 2

0

t + 4c

t

g(s)2 ds + 4

0

t

vs (g) · g(s) · vs (g) ds

0

g(s)2 vs (g) ds

0

3ν |x| + 2

t

2

vs (g)2 ds + 1 ν

0

16c2 + ν

t

t

g(s)2 ds + 16 ν

0

t

vs (g)2 g(s)2 ds

0

g(s)4 ds.

(4.2)

0

Applying Gronwall’s inequality, we have t t t 2 2 4 2 2 16 1 16c 2 g(s) ds + g(s) ds exp g(s) ds sup vs (g) |x| + ν ν ν 0st |x|2 + t

0

0

2 16c2 4 1 sup g(s) + sup g(s) ν 0st ν 0st

0

2 16 exp t sup g(s) . ν 0st (4.3)

Furthermore,


ν 2

t

1533

vs (g)2 ds

0

2 2 4 1 |x|2 + t sup g(s) + 16 sup g(s) Mt (g) + 16c2 sup g(s) , ν 0st 0st 0st

(4.4)

where

2 16c2 4 2 16 1 exp Mg (t) = |x|2 + t sup g(s) + sup g(s) t sup g(s) . ν 0st ν 0st ν 0st It is easy to see that, if g is replaced by gn , (4.3), (4.4) still hold. Since 2 lim sup gn (t) − g(t) = 0,

n→∞ 0t1

there exists a constant Cg (x) depending on sup0t1 g(t) and |x|2 (which may change line to line) such that 1

vs (gn )2 ds Cg (x)

(n 1).

(4.5)

0

Now by the chain rule, one has vt (gn ) − vt (g)2 + 2ν

t

vs (gn ) − vs (g)2 ds

0

t = −2

Agn (s) − Ag(s), vs (gn ) − vs (g) ds

0

t −2

B vs (gn ) + gn (s) − B vs (g) + g(s) , vs (gn ) − vs (g) ds

0

t ν

vs (gn ) − vs (g)2 ds + 1 ν

0

t

gn (s) − g(s)2 ds

0

t +2 0

B vs (gn ) + gn (s) − B vs (g) + g(s) , vs (gn ) − vs (g) ds.

(4.6)

1534


Now, due to (2.3), (2.4), (2.5) and (2.9), B vs (gn ) + gn (s) − B vs (g) + g(s) , vs (gn ) − vs (g) b vs (gn ) + gn (s), vs (gn ) + gn (s), vs (gn ) − vs (g) − b vs (g) + g(s), vs (gn ) + gn (s), vs (gn ) − vs (g) + b vs (g) + g(s), vs (g) + g(s), vs (gn ) − vs (g) − b vs (g) + g(s), vs (gn ) + gn (s), vs (gn ) − vs (g) b vs (gn ) − vs (g), vs (gn ) + gn (s), vs (gn ) − vs (g) + b gn (s) − g(s), vs (gn ) + gn (s), vs (gn ) − vs (g) + b vs (g) + g(s), gn (s) − g(s), vs (gn ) − vs (g) 2vs (gn ) − vs (g) · vs (gn ) − vs (g) · vs (gn ) + gn (s) + 2cgn (s) − g(s) · vs (gn ) + gn (s) · vs (gn ) − vs (g) + 2cvs (g) + g(s | · gn (s) − g(s) · vs (gn ) − vs (g)

ν vs (gn ) − vs (g)2 + 12 vs (g) + gn (s) 2 · vs (gn ) − vs (g)2 12 ν ν 12c2 2 vs (gn ) + gn (s) 2 · gn (s) − g(s)2 + vs (gn ) − vs (g) + 12 ν 2 12c2 ν vs (g) + g(s) 2 · gn (s) − g(s)2 . + vs (gn ) − vs (g) + 12 ν

Combining (4.6) and (4.7), one obtains vt (gn ) − vt (g)2 + ν 2

t


0

1 ν

t

gn (s) − g(s)2 ds + 24 ν

0

+

24c2 ν

t

vs (g) + gn (s) 2 · vs (gn ) − vs (g)2 ds

0

t

vs (gn ) + gn (s) 2 · gn (s) − g(s)2 ds

0

24c2 + ν

t

vs (g) + g(s) 2 · gn (s) − g(s)2 ds.

0

Applying Gronwall’s inequality and (4.5), we arrive at

(4.7)


2 ν sup vt (gn ) − vt (g) + 2 0t1

48c2 ν

1

1

1535


0

vs (gn ) + gn (s) 2 + vs (g) + g(s) 2 · gn (s) − g(s)2 ds

0

2 + ν

1 0

1 2 2 48 vs (gn ) + gn (s) ds gn (s) − g(s) ds × exp ν

48c2

2 + ν ν

0

2 Cg (x) sup gn (t) − g(t) . 0t1

Let n → ∞ to prove the lemma.

2

Now, let un,m be the solution of the equation · t un,m t

=x−

t Aun,m ds s

−

0

1 1 ds + bm t + √ Wtm + B un,m s n n

t

f m (x)N˜ n (ds, dx), (4.8)

0 X

0

where bm = Pm b, Wtm = Pm Wt , f m (x) = Pm f (x). Recall that Z n,m , Z n are defined as in (3.11) := un,m − Ztn,m , u¯ nt := unt − Ztn . Then u¯ n,m and u¯ nt satisfy and (3.12). Set u¯ n,m t t t t u¯ n,m t

=x−

t Au¯ n,m ds s

−

0

1 B u¯ n,m + Zsn,m ds + bm t + √ Wtm , s n

0

and t u¯ nt

=x−

t Au¯ ns ds

−

0

1 B u¯ ns + Zsn ds + bt + √ Wt . n

0

Lemma 4.2. For any δ > 0, lim lim sup

m→∞ n→∞

1 n > δ = −∞. log P sup un,m − u t t n 0t1

Proof. Note that P sup un,m − unt > δ t 0t1

P

δ n,m δ n n > Z > + P sup . sup u¯ n,m − u ¯ − Z t t t t 2 2 0t1 0t1

(4.9)

1536


By Lemma 5.6 in [16], we know that lim lim sup

m→∞ n→∞

1 log P n

δ sup Ztn,m − Ztn > 2 0t1

= −∞.

(4.10)

= −∞.

(4.11)

It suffices to prove, for any δ > 0, lim lim sup

m→∞ n→∞

1 log P n

δ sup u¯ n,m − u¯ nt > t 2 0t1

For δ0 > 0, define a stopping time by !

τδn,m 0

:= inf t 0, Ztn,m − Ztn > δ0 or

t

" n,m 2 n Z − Zs ds > δ0 . s

0 n,m and Z n , when m is fixed. Define Note that Lemma 3.2 and Lemma 3.3 still hold for un,m · , Z· · n n stopping times τu,1,M , τu,2,M by

n := inf t 0, un (t) > M , τu,1,M and ! n τu,2,M

t

:= inf t 0,

" n 2 u ds > M . s

0 n,m and Z n . We denote these stopping times We can also define similar stopping times for un,m · , Z· · n,m n,m n,m n,m n n by τu,1,M , τu,2,M , τZ,1,M , τZ,2,M , τZ,1,M and τZ,2,M , respectively. Let n,m n,m n,m n,m n,m n n n n := τu,1,M ∧ τu,2,M ∧ τu,1,M ∧ τu,2,M ∧ τZ,1,M ∧ τZ,2,M ∧ τZ,1,M ∧ τZ,2,M , τM

and set An,m (ω) :=

#

$ # $ # $ M ∩ sup un M ∩ sup Ztn,m M sup un,m t t

0t1

∩

#

$ sup Ztn M ,

0t1

0t1

0t1

! 1 B

n,m

(ω) : =

" ! 1 " ! 1 " n,m 2 n 2 n,m 2 u ds M ∩ u ds M ∩ Z ds M s s s

0

0

! 1 ∩ 0

"

n 2 Z ds M , s

0


! C

n,m

1

sup Ztn,m − Ztn δ0 ,

(ω) :=

0t1

1537

" n,m Zt − Z n 2 dt δ0 . t

0

Then, # P

$ sup u¯ n,m − u¯ nt > δ ∩ An,m ∩ B n,m ∩ C n,m t

0t1

P P =P

n,m sup u¯ n,m − u¯ nt > δ, 1 τM ∧ τδn,m t 0

0t1

sup n,m 0t1∧τM ∧τδn,m 0

sup n,m 0t1∧τM ∧τδn,m

n,m u¯ t − u¯ n > δ t

n,m u¯ t − u¯ n 2 > δ 2 .

(4.12)

t

0

Applying Itô’s formula to |u¯ n,m − u¯ nt∧τ n,m ∧τ n,m |2 , we have t∧τ n,m ∧τ n,m δ0

M

M

n,m u¯ n,m n,m − u¯ n n,m n,m 2 + 2ν t∧τ ∧τ t∧τ ∧τ

n,m t∧τM ∧τδn,m

0

n,m 2 u¯ − u¯ n ds s

δ0

M

δ0

M

δ0

s

0


0

n,m B u¯ s + Zsn,m − B u¯ ns + Zsn , u¯ n,m − u¯ ns ds s

= −2 0


0

n,m 1 u¯ s − u¯ ns , bm − b ds + n

+2


0

2 +√ n

0

∞

λi ds

i=m+1

0

n,m t∧τM ∧τδn,m 0

n,m u¯ s − u¯ ns , dWsm − dWs ,

(4.13)

0

thus,

sup n,m 0st∧τM ∧τδn,m 0

n,m 2 u¯ − u¯ n + 2ν s

t∧τmn,m ∧τδn,m

0

s


s

0


4

0

n,m B u¯ + Z n,m − B u¯ n + Z n , u¯ n,m − u¯ n ds s

0

s

s

s

s

s

1538

T. Xu, T. Zhang / Journal of Functional Analysis 257 (2009) 1519–1545 n,m t∧τM ∧τδn,m

0

+4

n,m 2 u¯ − u¯ ns , bm − b ds + s n

0

t ∞

λi ds

0 i=m+1

s 4 n,m n m u¯ r − u¯ r , dWr − dWr . +√ sup n 0st∧τ n,m ∧τ n,m δ0

M

(4.14)

0

Write, t

n,m B u¯ s + Zsn,m − B u¯ ns + Zsn , u¯ n,m − u¯ ns ds s

0

t

b u¯ n,m + Zsn,m , u¯ n,m + Zsn,m , u¯ n,m − u¯ ns − b u¯ ns + Zsn , u¯ ns + Zsn , u¯ n,m − u¯ ns ds s s s s

= 0

t

b u¯ n,m ¯ n,m ¯ n,m − u¯ ns − b u¯ ns , u¯ ns , u¯ n,m − u¯ ns ds s ,u s ,u s s

= 0

t +

n,m n,m b u¯ n,m ¯ s − u¯ ns − b u¯ ns , Zsn , u¯ n,m − u¯ ns ds s , Zs , u s

0

t +

b Zsn,m , u¯ n,m ¯ n,m − u¯ ns − b Zsn , u¯ ns , u¯ n,m − u¯ ns ds s ,u s s

0

t +

b Zsn,m , Zsn,m , u¯ n,m − u¯ ns − b Zsn , Zsn , u¯ n,m − u¯ ns ds s s

0

=: II 1 + II 2 + II 3 + II 4 .

(4.15)

By virtue of the properties of b(· , · , ·), we have, t |II 1 | 2

n,m u¯ − u¯ n · u¯ n · u¯ n,m − u¯ n ds s

s

s

s

s

0

ν 24

t

n,m 2 24 u¯ − u¯ ns ds + s ν

0

t |II 2 | 0

s

s

n,m 2 2 u¯ − u¯ n · u¯ n ds, s

s

(4.16)

s

0

n,m b u¯ − u¯ n , Z n,m , u¯ n,m − u¯ n ds + s

t

s

t

s

n n,m b u¯ , Z − Z n , u¯ n,m − u¯ n ds s

0

s

s

s

s


t

1539

n,m u¯ − u¯ n · Z n,m · u¯ n,m − u¯ n ds

2

s

s

s

s

s

0

t

n 1 n 1 n,m 1 1 u¯ 2 · u¯ 2 · Z − Zsn 2 · Zsn,m − Zsn 2 · u¯ n,m − u¯ ns ds s s s s

+2 0

t

ν 24

n,m 2 24 u¯ − u¯ ns ds + s ν

0

t

n,m 2 2 ν u¯ − u¯ ns · Zsn,m ds + s 24

0

t


s

0

24 n · sup u¯ ns− · sup Z n,m − Zs− + ν 0st s− 0st

t

n,m 2 Z − Z n ds s

s

1 t 1 2 2 n 2 u¯ ds , · s

0

0

(4.17) t

n,m 1 1 1 1 Z − Zsn 2 · Zsn,m − Zsn 2 · u¯ ns 2 · u¯ ns 2 · u¯ n,m − u¯ ns ds s s

|II 3 | 2 0

ν 24

t 0

n,m 2 24 n u¯ · sup u¯ ns− sup Z n,m − Zs− − u¯ ns ds + s ν 0st s− 0st

t

n 2 u¯ ds

×

t


1 2

s

0

1 2

(4.18)

,

s

0

and similarly t |II 4 |

n,m b Z − Z n , Z n,m , u¯ n,m − u¯ n ds + s

s

s

s

s

0

ν 24

t

n n,m b Z , Z − Z n , u¯ n,m − u¯ n ds s

s

s

s

s

0

t 0

n,m 2 n,m n,m 24 n u¯ · sup Zs− sup Zs− − u¯ ns ds + − Zs− s ν 0st 0st

t ×

n,m 2 Z ds

t


1 2

s

0

1 2

s

0

ν + 24

t 0

n,m 2 n,m 24 n u¯ − u¯ ns ds + − Zs− sup Zs− s ν 0st

n × sup Zs− 0st

t


0

s

1 t 1 2 2 n 2 Z ds . · s 0

(4.19)

1540


n } is countable P-a.s. This fact is also true for Note that we have used the fact that {s: Zsn = Zs− Z·n,m and Z·n,m −Z·n . t − u¯ ns , dWsm − dWs ), and putting (4.16)–(4.19) and (4.14) together, one Set Mt := √4n 0 (u¯ n,m s obtains

1 sup 2 0st∧τ n,m ∧τ n,m

s


0

s


δ0

M

n,m 2 u¯ − u¯ n + ν

s

0

∞ 3 c 2t λi + sup |Mt | + 8|bm − b|2 t 2 + (δ0 M) 2 n,m n,m n ν 0st∧τ ∧τ i=m+1

δ0

M


c + ν

0

n,m 2 2 2 u¯ − u¯ n · u¯ n + Z n,m ds. s

s

s

(4.20)

s

0

Note that c is a constant independent of n, m. Applying Gronwall’s inequality, we get sup n,m 0st∧τM ∧τδn,m 0

n,m 2 u¯ − u¯ n s

s

∞ 3 4t c λi + 2 sup |Mt | + 16|bm − b|2 t 2 + (δ0 M) 2 n ν 0st∧τ n,m ∧τ n,m i=m+1

δ0

M

n,m n,m t∧τM ∧τδ0 c (u¯ ns 2 +Zsn,m 2 ) ds ν 0

× exp ∞ 3 4t c 2 2 2 λi + 2 sup |Mt | + 16|bm − b| t + (δ0 M) n ν 0st∧τ n,m ∧τ n,m i=m+1

× exp

δ0

M

2Mc . ν

(4.21)

Set CM := exp ( 2Mc ν ). Applying the martingale inequality in [6] to M· , it follows that % E

sup n,m 0st∧τM ∧τδn,m

n,m 2p & p2 u¯ − u¯ n s

s

0

∞ 3 4t c 2 λi + 16|bm − b|2 t 2 + (δ0 M) 2 n ν i=m+1

+8 E

sup n,m 0st∧τM ∧τδn,m 0

|Mt |p

2

p

2 CM

2 2 CM


∞ 3 4t c 2 λi + 16|bm − b|2 t 2 + (δ0 M) 2 n ν

1541

2 2 CM

i=m+1

2 cCM

+

n


0

p E

λ2i u¯ n,m s

p 2

∞ 3 4t c 2 λi + 16|bm − b|2 t 2 + (δ0 M) 2 n ν

2 2 CM

i=m+1

cC 2 + Mp n

t

2

2 − u¯ ns ds

p

i=m+1

0

∞

cC 2 + Mp n

2

∞

λ2i

t

i=m+1

2p 2 n,m p ds. E u¯ s∧τ n,m ∧τ n,m − u¯ ns∧τ n,m ∧τ n,m

0

(4.22)

δ0

M

δ0

M

Applying Gronwall’s inequality to (4.22), one obtains % E

sup n,m 0s1∧τM ∧τδn,m

n,m 2p & p2 u¯ − u¯ n s

s

0

∞ 3 4 c 2 λi + 16|bm − b|2 + (δ0 M) 2 n ν

2 2 CM

i=m+1

cC 2 + Mp n

2 cCM p . × exp n

∞

2 λ2i

i=m+1

(4.23)

Take p = 2n to obtain lim lim sup

m→∞ n→∞

n,m 2 1 u¯ log P sup − u¯ ns > δ 2 s n 0s1∧τ n,m ∧τ n,m M

% lim lim sup log E m→∞ n→∞

δ0

sup n,m 0s1∧τM ∧τδn,m

n,m & p2 n 2p u¯ − u ¯ − log δ 4 s s

0

√ 3 2c 2 2 (δ0 M) CM + 2cCM 2 log − 4 log δ. ν

(4.24)

Because of Lemma 3.2 and Lemma 3.3, for any R > 0, there exists a M > 0 such that lim lim sup

m→+∞ n→+∞

c 1 log P An,m −R, n

lim lim sup

m→+∞ n→+∞

c 1 log P B n,m −R. n

By Lemma 5.6 in [16] and Lemma 3.4.5, for any δ0 > 0, we have lim lim sup

m→+∞ n→+∞

c 1 log P C n,m = −∞. n

(4.25)

1542


Thus for the above choice of M, we have 2 1 log P sup u¯ n,m − u¯ nt > δ 2 t m→+∞ n→+∞ n 0t1 n,m 1 u¯ t − u¯ n 2 > δ 2 sup lim lim sup log P t m→+∞ n→+∞ n 0t1∧τ n,m ∧τ n,m lim lim sup

δ0

M

∨ lim lim sup m→+∞ n→+∞

c c 1 1 log P An,m ∨ lim lim sup log P B n,m m→+∞ n→+∞ n n

c 1 log P C n,m m→+∞ n→+∞ n

√ 3 2c 2 (δ0 M) 2 CM + 2cCM − 4 log δ ∨ −R. 2 log ν ∨ lim lim sup

(4.26)

Letting δ0 go to 0, one obtains lim lim sup

m→+∞ n→+∞

1 n 2 2 log P sup u¯ n,m −R. − u ¯ > δ t t n 0t1

(4.27)

2

Since R is arbitrary, (4.11) follows, hence the lemma.

For g ∈ D([0, 1]; H ), define φtm (g) as the solution of the following equation: t φtm (g) = x

−

t Aφsm (g) ds

−

0

B φsm (g) ds + Pm g(t).

0

Lemma 4.3. For any r > 0, lim

sup φtm (g) − φt (g) = 0.

sup

m→∞ {g: I (g)r} 0t1 0

Proof. Let g ∈ {g: I0 (g) r}. Let Ztm (g) and Zt (g) be the solutions of the following equations, t Ztm (g) = −

t AZsm (g) ds

0

+

Pm g (s) ds,

0

and t Zt (g) = −

t AZs (g) ds +

0

g (s) ds.

0

Set vtm (g) := φtm (g) − Ztm (g), vt (g) := φt (g) − Zt (g). Then vtm (g), vt (g) satisfy


t vtm (g) = x

−

t Avsm (g) ds

−

0

1543

B vsm (g) + Zsm (g) ds,

(4.28)

B vs (g) + Zs (g) ds.

(4.29)

0

and t vt (g) = x −

t Avs (g) ds −

0

0

As, 2 2 1 d Zt (g) = −ν Zt (g) + g (t), Zt (g) , 2 dt we have, 2 sup Zt (g) + 2ν 0t1

1 0

Zs (g)2 ds 1 sup Zt (g)2 + 8 2 0t1

1

g (s) ds

2 ,

0

so that 2 sup Zt (g) + 4ν 0t1

1

Zs (g)2 ds 16

1

0

g (s) ds

2 .

(4.30)

0

Note that (4.30) also holds for Ztm (g). By Lemma 5.3 in [16], it follows that there exists a M such that 1 sup

{g: I0 (g)r}

g (s) ds M.

0

Hence,

sup

2 1 sup Zt (g) CM ,

{g: I0 (g)r} 0t1

1 sup

{g: I0 (g)r}

Zt (g)2 dt C 2 , M

(4.31)

0

1 , C 2 are the constants depending on M. By the chain rule, where CM M

2 2 1 d vt (g) = −ν vt (g) − b vt (g) + Zt (g), vt (g) + Zt (g), vt (g) 2 dt 2 −ν vt (g) + b vt (g), Zt (g), vt (g) + b Zt (g), Zt (g), vt (g) 2 −ν vt (g) + 2vt (g) · vt (g) · Zt (g) + 2Zt (g) · vt (g) · Zt (g).

1544


It follows that vt (g)2 + ν

t

t

vs (g)2 ds 8 ν

0

t

vs (g)2 · Zs (g)2 ds + 8 ν

0

Zs (g)2 · Zs (g)2 ds.

0

In view of (4.31), applying Gronwall’s inequality, we obtain 1

2 3 sup vt (g) CM ,

sup

sup

{g: I0 (g)r} 0t1

{g: I0 (g)r}

vt (g)2 dt C 4 , M

0

3 , C 4 are the constants depending on M. where CM M It follows from (4.28), (4.29) that,

m v (g) − vt (g)2 + 2ν

t

t

m v (g) − vs (g)2 ds s

0

t 2

m B v (g) + Z m (g) − B vs (g) + Zs (g) , v m (g) − vs (g) ds. s

s

s

0

By a similar estimate to (4.15), it turns out that m v (g) − vt (g)2 + 2ν

t

t

m v (g) − vs (g)2 ds s

0

t ν

m v (g) − vs (g)2 ds + s

0

t +ν

sup

sup Ztm (g) − Zt (g)CM

{g: I0 (g)r} 0t1

m v (g) − vs (g)2 vs (g)2 + Z m (g)2 ds, s

s

0

where CM is the constant depending on M. By Gronwall’s inequality, one obtains sup

2 sup vtm (g) − vt (g)

{g: I0 (g)r} 0t1

sup

sup Ztm (g) − Zt (g)CM · CM .

{g: I0 (g)r} 0t1

By Lemma 5.7 in [16], we know that lim

sup

2 sup Ztm (g) − Zt (g) = 0.

m→∞ {g: I (g)r} 0t1 0

Letting m → ∞ on the both sides of (4.32), we prove the lemma.

2

(4.32)


1545

References [1] C. Cardon-Weber, Large deviations for Burgers’-type SPDE, Stochastic Process. Appl. 84 (1990) 53–70. [2] S. Cerrai, M. Röckner, Large deviations for stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab. 32 (1) (2004) 1100–1139. [3] F. Chenal, A. Millet, Uniform large deviations for parabolic SPDEs and applications, Stochastic Process. Appl. 72 (1997) 161–186. [4] P. Chow, Large deviation problem for some parabolic Itô equations, Comm. Pure Appl. Math. XLV (1992) 97–120. [5] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992. [6] B. Davis, On the Lp -norms of stochastic integrals and other martingales, Duke Math. J. 43 (1976) 697–704. [7] A. De Acosta, Large deviations for vector valued Lévy processes, Stochastic Process. Appl. 51 (1994) 75–115. [8] A. De Acosta, A general non-convex large deviation result with applications to stochastic equations, Probab. Theory Related Fields (2000) 483–521. [9] A. Dembo, O. Zeitouni, Large Deviations Techniques, Jones and Bartlett, Boston, 1993. [10] F. Flandoli, Dissipativity and invariant measures for stochastic Navier–Stokes equations, Nonlinear Differential Equations Appl. 1 (1994) 403–426. [11] F. Flandoli, D. Gatarek, Martingale and stationary solution for stochastic Navier–Stokes equations, Probab. Theory Related Fields 102 (1995) 367–391. [12] M. Gourcy, A large deviation principle for 2D stochastic Navier–Stokes equation, Stochastic Process. Appl. 117 (7) (2007) 904–927. [13] M. Hairer, J.C. Mattingly, Ergodicity of the 2-D Navier–Stokes equation with degenerate stochastic forcing, Ann. of Math. (2) 164 (3) (2006) 993–1032. [14] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1989. [15] R. Mikulevicius, B.L. Rozovskii, Global L2 -solutions of stochastic Navier–Stokes equations, Ann. Probab. 33 (1) (2005) 137–176. [16] M. Rockner, T.S. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles, Potential Anal. 26 (2007) 255–279. [17] S.S. Sritharan, P. Sundar, Large deviation for the two dimensional Navier–Stokes equations with multiplicative noise, Stochastic Process. Appl. 116 (2006) 1636–1659. [18] R. Teman, Navier–Stokes Equations and Nonlinear Functional Analysis, Society for Industrial and Applied Mathematics, 1983. [19] T.S. Zhang, Large deviations for stochastic nonlinear beam equations, J. Funct. Anal. 248 (1) (2007) 175–201.


Weyl–Titchmarsh functions of vector-valued Sturm–Liouville operators on the unit interval Dmitry Chelkak a,∗,1 , Evgeny Korotyaev b,2 a Dept. of Math. Analysis, St. Petersburg State University, Universitetskij pr. 28, Staryj Petergof,

198504 St. Petersburg, Russia b School of Math., Cardiff University, Senghennydd Road, CF24 4AG Cardiff, Wales, UK

Received 16 October 2008; accepted 6 May 2009 Available online 2 June 2009 Communicated by J. Bourgain

Abstract The matrix-valued Weyl–Titchmarsh functions M(λ) of vector-valued Sturm–Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M(λ)) and the residues of M(λ) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N × N Weyl–Titchmarsh functions) corresponding to N × N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct. © 2009 Elsevier Inc. All rights reserved. Keywords: Inverse problem; Matrix potentials; M-function; Sturm–Liouville operators

1. Introduction We start with a short description of known results in the inverse spectral theory for scalar Strum–Liouville operators on a finite interval. We recall only some important steps mostly focusing on the characterization problem, i.e., the complete description of spectral data that * Corresponding author.

E-mail addresses: [email protected] (D. Chelkak), [email protected] (E. Korotyaev). 1 Partly funded by the RF President grants MK-4306.2008.1, NSh-2409.2008.1, and P. Deligne’s 2004 Balzan prize in

Mathematics. 2 Partly supported by EPSRC grant EP/D054621. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.010

D. Chelkak, E. Korotyaev / Journal of Functional Analysis 257 (2009) 1546–1588

1547

correspond to some fixed class of potentials. More information about different approaches to inverse spectral problems can be found in the monographs [15,33,39,43], survey [17] and references therein. The inverse spectral theory goes back to the seminal paper [2] (see also [31]). Borg showed that spectra of two Sturm–Liouville problems −y + q(x)y = λy, x ∈ [0, 1], with the same boundary conditions at 1 but different boundary conditions at 0, determine the potential q(x) and the boundary conditions uniquely. Later on, Marchenko [38] proved that the so-called spectral function ρ(λ) (or, equivalently, the Weyl–Titchmarsh function m(λ)) determines the potential uniquely. Note that the spectral function is piecewise-linear outside the spectrum {λn }+∞ n=1 and its jump at λn is equal to the so-called normalizing constant [αn (q)]−1 given by (1.3). At the same time, a different approach to this problem was developed by Krein [27–29]. An important result was obtained by Gel’fand and Levitan [16]. They gave an effective method to reconstruct the potential q from its spectral function. More precisely, they derived an integral equation and expressed q(x) explicitly in terms of the solution of this equation. At that time, there was some gap between necessary and sufficient conditions for the spectral functions corresponding to fixed classes of q(x). Some characterization of spectral data for q such that q (m) ∈ L1 (0, 1) was derived by Levitan and Gasymov [34] for all m = 0, 1, 2, . . . . Also, they gave the solution of the characterization problem in the case q ∈ L2 (0, 1). Marchenko and Ostrovski [40] obtained a sharpening of this result. Namely, for all m = 0, 1, 2, . . . they gave the complete solution of the inverse problem in terms of two spectra, if q (m) ∈ L2 (0, 1). Trubowitz and co-authors (Isaacson [23], McKean [22], Dahlberg [12], Pöschel [43]) suggested another approach. It is based on the analytic properties of the mapping {potentials} → {spectral data} and the explicit transforms corresponding to the change of only a finite number of spectral parameters (λn (q), νn (q))+∞ n=1 . Their norming constants νn (q) differ slightly from the normalizing constants (1.3), but the characterizations are equivalent (see Appendix B). Also, this approach was applied to other scalar inverse problems with purely discrete spectrum (singular Sturm–Liouville operator on [0, 1] [19]; perturbed harmonic oscillator [7,10,41]). Thus, nowadays the inverse spectral theory for the scalar Sturm–Liouville operators is well understood. By contrast, until recently only some particular results were known for vector-valued operators. In our paper we consider the inverse problem for the self-adjoint operators Lψ = −ψ + V (x)ψ,

ψ(0) = ψ(1) = 0,

ψ ∈ L2 [0, 1]; CN ,

(1.1)

where V = V ∗ ∈ L2 ([0, 1]; CN ×N ) is a self-adjoint N × N matrix-valued potential. Denote by ϕ(x) = ϕ(x, λ, V ) and χ(x) = χ(x, λ, V ) the matrix-valued solutions of the equation −ψ + V (x)ψ = λψ such that ϕ(0) = χ(1) = 0,

ϕ (0) = −χ (1) = IN ,

here and below IN denotes the identity N × N matrix. Note that χ(x, λ, V ) = ϕ 1 − x, λ, V ,

where V (x) ≡ V (1 − x), x ∈ [0, 1].

1548


The matrix-valued Weyl–Titchmarsh function for this problem is given by ∗ M(λ) = M(λ, V ) = χ χ −1 (0, λ, V ) = M(λ) ,

λ ∈ C.

(1.2)

In the scalar case, the Weyl–Titchmarsh function m(λ, q) is a meromorphic function having simple poles at Dirichlet eigenvalues λn (q) and −1 =− res m(λ, q) = − αn (q)

1

λ=λn (q)

ϕ(x, λn , q)2 dx

−1 .

(1.3)

0

So, the sharp characterization of all scalar Weyl–Titchmarsh functions (or, equivalently, all spec2 tral data (λn (q), αn (q))+∞ n=1 ) that correspond to potentials q ∈ L (0, 1) is available due to [40] or [43] (see also Appendix B). Namely, the necessary and sufficient conditions are +∞ λn − π 2 n2 − q0 n=1 ∈ 2 λ1 < λ 2 < λ 3 < · · · , +∞ and πn · 2π 2 n2 αn (q) − 1 n=1 ∈ 2 .

for some q0 ∈ R, (1.4)

In the vector-valued case, it is known that the Weyl–Titchmarsh function determines V uniquely (see [37] or [46]). Some other miscellaneous results concerning vector-valued Schrödinger operators were obtained in [5,6,8,11,24,25,44,45]. Nevertheless, to the best of our knowledge, no solutions of the characterization problems have been available until recently. Following [9], we denote by λ1 < λ2 < · · · < λα < · · · the eigenvalues of L and by kα = dim Eα ∈ [1, N] their multiplicities, where Eα ⊂ L2 ([0, 1]; CN ) is the eigenspace corresponding to the eigenvalue λα . Then (see details in [9]), the Weyl–Titchmarsh function M(λ) is meromorphic outside the Dirichlet spectrum σ (V ) = {λα (V )}α1 and res M(λ) = −Bα = −pα∗ gα−1 pα ,

λ=λα

where

pα : CN → Eα = Ker ϕ(1, λα , V ) = h ∈ CN : ψα;h = ϕ(·, λα , V )h ∈ Eα is the orthogonal projector acting to kα -dimensional spaces Eα (below we also use the notation Pα = pα∗ pα for the same operators considered as acting to CN ) and 1 g α = pα

∗ ϕ ϕ (x, λα , V ) dx pα∗ = gα∗ > 0

0

is the self-adjoint operator (or the normalizing matrix) acting in Eα . Note that 1

ψα;h1 , ψα;h2 L2 ([0,1];CN ) = 0

h∗2 ϕ ∗ ϕ (x, λα , V )h1 dx = h1 , gα h2 Eα


1549

for all h1 , h2 ∈ Eα . We call (λα , Pα , gα )+∞ α=1 the spectral data of L. If kα = 1, then gα acts in the one-dimensional space Eα , so we consider gα as a positive real number (and call it, as in the scalar case, the normalizing constant). The spectral data determine (e.g., see Proposition 2.6) the function M(λ), and so the potential V (x), uniquely. The main result of our paper is the following solution of the characterization problem. 0 be the standard coordinate basis and P 0 = ·, e0 e0 be the standard coordiLet e10 , e20 , . . . , eN j j j nate projectors in CN . We denote the Euclidean norm of vectors h ∈ CN and the operator norm of matrices A ∈ CN ×N by |h| and |A|, respectively. Nevertheless, below we denote by V the L2 -norm of a matrix-valued potential on [0, 1], i.e., 1 V = 2

N 1 ∗ vij (x)2 dx. Tr V V (x) dx = i,j =1 0

0

Such notations are chosen in order to emphasize that · is related to “functions” depending on x ∈ [0, 1] while | · | operates with “fixed” values. Theorem 1.1 (Characterization of spectral data). For all v10 < v20 < · · · < vn0 the mapping V → (λα , Pα , gα )+∞ α=1 is a bijection between the space of potentials V = V ∈ L [0, 1]; CN ×N ∗

2

1 such that

0 V (x) dx = diag v10 , v20 , . . . , vN

(1.5)

0

and the class of spectral data satisfying the following conditions (A)–(C): (A) The spectrum is asymptotically simple, i.e., there exist α 0, n 1 such that k1 + k2 + · · · + kα

= N n − 1 and kα = 1 for all α α + 1. It allows us to define the double-indexing (n, j ), n n , j = 1, 2, . . . , N , instead of α > α . Namely, we set λn,j = λα +N (n−n )+j , Pn,j = Pα +N (n−n )+j and so on for n n . (B) The following hold true for all j = 1, 2, . . . , N : +∞ λn,j − π 2 n2 − vj0 n=n ∈ 2 , Pn,j − P 0 +∞ ∈ 2 j n=n

+∞ πn · 2π 2 n2 gn,j − 1 n=n ∈ 2 , +∞

N and πn · IN − Pn,j ∈ 2 .

j =1

(1.6)

n=n

(C) The collection (λα ; Pα )+∞ α=1 satisfies the following property: N Let ξ : C √ → C be an entire vector-valued function. If Pα ξ(λα ) = 0 for all α 1, ξ(λ) = |Im λ| O(e ) as |λ| → ∞ and ξ ∈ L2 (R+ ), then ξ(λ) ≡ 0.

1550


Remark 1.2. Let V = V ∗ ∈ L2 ([0, 1]; CN ×N ). Applying some unitary transform in CN , one 1 0 }, v 0 v 0 · · · v 0 . Our assumpmay always assume that 0 V (x) dx = diag{v10 , v20 , . . . , vN N 1 2 0 tion (1.5) states that all the vj are distinct. It simplifies the analysis, since otherwise infinitely many eigenvalues λα can be multiple. In particular, in the general case, one has to introduce some other parameters instead of (Pn,j , gn,j ). We give also a simple reformulation of the algebraic restriction (C) (note that it does not depend on the shift of the spectrum). Proposition 1.3 (Reformulation of (C)). Let λα > 0 for all α 1 and Pα = hα h∗α , where (j ) (1) (k ) hα = (hα ; . . . ; hα α ) consists of kα orthonormal vectors hα ∈ CN . Then the condition (C) is equivalent to the following: √

Vector-valued functions e±i λα t hα , j = 1, . . . , kα , α 1, together with the constant vectors 0 span L2 ([−1, 1]; CN ). e10 , . . . , eN (j )

Remark 1.4. In the scalar case, (C) always holds true due to the well-known result of Paley and Wiener (e.g., see [30, p. 47]). In the vector-valued case, this condition is not trivial. Some discussion of (C) is given in Appendix A (see Propositions A.3, A.4). Note that, if Pn,j = Pj0 for all n m + 1 and j = 1, 2, . . . , N , then one can reformulate (C) as the condition det T = 0 for some Nm × Nm matrix T (see Proposition A.5). As usual, Theorem 1.1 consists of several different parts: (i) Uniqueness theorem (spectral data determine the potential uniquely). (ii) Direct problem (spectral data constructed by a given potential satisfy (A)–(C)). (iii) Surjection (any data satisfying (A)–(C) are spectral data of some potential). We do not discuss the uniqueness theorem (i) in our paper and refer to [37,46] (or [9]) for this fact. The direct problem (ii) is considered in Section 2. Note that the spectrum is asymptotically 0 (see also Remark 1.2). As in the scalar case, simple due to our assumption v10 < v20 < · · · < vN the Fourier coefficients of V appear as leading terms in the asymptotics of the spectral data (Propositions 2.1 and 2.5). We also give the explicit expression for M(λ) in terms of the spectral data in Section 2.4. The main part of our paper (Section 3) is devoted to the surjection (iii). The general strategy of the proof is described in detail in Section 3.1. Here we give only a short sketch of our arguments. We start with some admissible data (λ α , Pα† , gα† )α1 satisfying (A)–(C). Using the well-known characterization (1.4) for the scalar case, we construct some special diagonal potential V such that σ (V ) = {λ α }α1 . In Sections 3.2–3.4 we introduce some essential modification of the spectral data in order (a) to control the splitting of multiple eigenvalues and (b) to join together all asymptotics in (1.6). We prove that the mapping Φ : {potentials} → {modified spectral data} is real-analytic3 near V . 3 The mapping F : U → H (2) between real Hilbert spaces U ⊂ H (1) and H (2) is real-analytic iff it has continuation (2) (1) FC : UC → HC into some complex neighborhood U ⊂ UC ⊂ HC that is differentiable as the mapping between the (1) (2) (1) (2) complexifications HC , HC of the real spaces H , H .


1551

The main purpose of involving analyticity arguments here is the well-known equivalence of the analyticity and the weak-analyticity4 for mappings between complex Hilbert spaces. Thus, we immediately derive the smoothness of the whole mapping Φ from the smoothness of its components. In Sections 3.5, 3.6 we use the Fredholm Alternative in order to show that Φ is a local isomorphism near V (i.e., dV Φ is invertible). Thus, all additional spectral data sufficiently close to (Pα (V ), gα (V ))α1 can be obtained from potentials having the same spectrum {λ α }α1 as V . In particular, if α • is large enough, then there exists V • such that σ (V • ) = {λ α }α1 and (Pα (V • ), gα (V • )) = (Pα† , gα† ) for all α > α • . We complete the proof in Section 3.7 using the explicit isospectral transforms constructed in our recent paper [9]. As usual in Trubowitz’s approach, we need to change only some finite number α • of additional spectral data (Pα , gα ). Note that the condition (C) and the restrictions introduced in [9] in terms of “forbidden” subspaces are equivalent (see Proposition A.4). Thus, one can change any finite number of projectors Pα in an arbitrary way that does not violate (C) (see details in Section 3.7). Note that we do not present any explicit reconstruction procedure for the potential, if there are infinitely many perturbed spectral data. The natural idea is to use some passage to the limit changing the residues Bα (V ) → Bα† , α = 1, 2, . . . , of the Weyl–Titchmarsh function step by step. Each step is doable due to isospectral transforms constructed in [9] but we do not prove the convergence of this procedure. We finish the introduction with several remarks concerning some possible further developments of our approach to this inverse problem. Remark 1.5. The isospectral transforms constructed in [9] generalize the scalar isospectral flows (see [43]) and some specific class of isospectral transforms given in [24]. Nevertheless, to the best of our knowledge, no analogues of the explicit flows changing the eigenvalues (see [43]) are known in the vector-valued case. We think that such a construction would simplify the inverse theory a lot. Remark 1.6. One may be interested in the characterization for other parameters, e.g. the spectra of several boundary problems (similarly to the original paper [2]). It is known (even in the nonselfadjoint case, see [37,46]) that N 2 + 1 spectra determine the potential uniquely. Moreover, the number of spectra can be reduced to 12 N (N + 1) + 1 in the self-adjoint case (see Corollary 4.4 in [37]). On the other hand, the naive count says that such an inverse problem is overdetermined. Note that, in the spirit of Appendix B, this question can be considered as a parametrization problem for some class of matrix-valued functions. In connection with Borg type results we mention also the paper [36]. Here a generalization of Borg’s result for first order (Dirac-type) systems as well as results on unique recovery of the potential matrix by a part of the monodromy matrix was obtained. The method used in [37] is just an adaptation of that applied in [36] to the first order (Dirac-type) systems. Remark 1.7. Consider the Schrödinger operator Hy = −y + V y on R with a N × N potential V = V ∗ such that R (1 + |x|)|V (x)| dx < +∞ (e.g., see [42]). It has a finite number of 4 In Hilbert spaces, the weak-analyticity is equivalent to the analyticity of particular coordinates and the local boundedness, see nice Appendix A in [43] or the monograph [13] for details.

1552


eigenvalues λ1 < · · · < λm < 0 with the multiplicities kα = dim Eα , where Eα is the eigenspace corresponding to λα . In order to solve the inverse scattering problem completely, one needs to characterize the residues of the transmission coefficient at λα . Unfortunately, we do not know any results in this direction. For the scattering problem on the half-line a characterization was given in [1] but it involves implicit conditions for spectral data (much more complicated than our condition (C)). Remark 1.8. In the scalar case, the Dirichlet eigenvalues and the norming constants are canonically conjugate variables for the Korteweg–de Vries equation with periodic initial conditions (see [14]). Similarly, the (negative) eigenvalues and the corresponding normalizing constants of the (scalar) Schrödinger operator −y + q(x)y on R with a decreasing potential q(x) are canonically conjugate variables for the Korteweg–de Vries equation (see [47]). The vector-valued case is more complicated (see [3,4,42]). We hope that our results could be useful from this point of view. 2. Direct problem 2.1. Asymptotics of the eigenvalues and the individual projectors Denote by (0) N (0) = V vij i,j =1 =

1

(cn) N (cn) = V vij i,j =1 =

V (t) dt, 0

1 V (t) cos 2πnt dt 0

and

(sn)

V

(sn) N = vij i,j =1 =

1 V (t) sin 2πnt dt,

n 1,

0

the (matrix) Fourier coefficients of V . We start with some elementary asymptotics of the fundamental solutions ϕ(x, λ, V ) and χ(x, λ, V ) = ϕ(1 − x, λ, V ) for λ close to π 2 n2 . It is well known that sin zx 1 IN + 2 ϕ x, z2 , V = z z

x sin z(x − t) · V (t) sin zt dt + O 0

e|Im z|x . |z|3

(2.1)

Here and below constants in O-type estimates depend on the potential. In this section we do not pay the attention to the nature of this dependence. Let z2 = π 2 n2 + μ,

μ = O(1),

so z = πn +

1 μ +O 3 . 2πn n


1553

Then, sin zx 1 IN + 2 2 ϕ x, z2 , V = πn π n

x

1 sin πn(x − t) · V (t) sin πnt dt + O 3 . n

0

In particular, 2 (−1)n 1 (0) (cn) ϕ 1, z , V = μIN − V + V . +O 2 2 n 2π n

(2.2)

(0) = diag{v 0 , v 0 , . . . , v 0 } with v 0 < Proposition 2.1. Let V = V ∗ ∈ L2 ([0, 1]; CN ×N ) satisfy V N 1 2 1 0 0 v2 < · · · < vN . Then: (i) there exists n = n (V ) V such that: (a) there are exactly N (n − 1) eigenvalues counting with multiplicities in the interval (−π 2 (n − 1)2 − 3 V ; π 2 (n − 1)2 + 3 V ), (b) for each n n there are exactly N simple eigenvalues λn,1 , λn,2 , . . . , λn,N in the interval (π 2 n2 − 3 V ; π 2 n2 + 3 V ), (c) there are no other eigenvalues; (ii) for each j = 1, 2, . . . , N the following asymptotics hold true as n → ∞: (cn)

vjj λn,j = π 2 n2 + vj0 −

+ O δn (V ) ,

(cn) 2 1 + ; where δn (V ) = V n

(iii) if Pn,j = ·, hn,j hn,j , where hn,j ∈ CN is such that |hn,j | = 1, hn,j , ej0 > 0, then the asymptotics hn,j =

(cn)

v1,j

v10 −vj0

(cn)

···

vj −1,j vj0−1 −vj0

(cn)

1

vj +1,j vj0+1 −vj0

(cn)

···

vN,j

0 −v 0 vN j

+ O δn (V )

hold true for each j = 1, 2, . . . , N as n → ∞. Note that the condition n (V ) V guarantees that the mentioned intervals do not intersect each other. We need the following simple matrix version of Rouche’s theorem: Lemma 2.2. Let F, G : B(w, r) → C be analytic matrix-valued functions such that |G(λ)| · |F −1 (λ)| < 1 for all λ on the boundary of some disc B(w, r) ⊂ C. Then, the scalar functions det F and det(F + G) have the same number of zeros in B(w, r) counting with multiplicities. Proof. We check that C arg(det F ) = C arg(det(F + G)), where C arg f denotes the increment of arg f along the circumference C = {λ: |λ − w| = r}. Note that, if λ ∈ C, then all eigenvalues of I + G(λ)F −1 (λ) have strictly positive real parts since |G(λ)F −1 (λ)| < 1. Thus, the result follows from C arg det(F + G) − C arg(det F ) = C arg det I + GF −1 = 0 and the classical argument principle.

2

1554


Proof of Proposition 2.1. (i) Firstly, we apply Lemma 2.2 to the function χ(0, λ, V ) = ϕ 1, λ, V = F (λ) + G(λ) in the discs √

sin λ 2 2 λ: |λ| < π n + 3 V with F (λ) = √ IN λ (see asymptotics (2.1)) and

λ: λ = π 2 n2 + μ, |μ| < 3 V

with F (λ) =

(−1)n 2 2 (0) λ − π I − V n 2π 2 n2

(see asymptotics (2.2)). Thus, if n is sufficiently large, then there are exactly N n and N eigenvalues (zeros of det χ(0, · ,V )), respectively, inside these discs counting with multiplicities. Secondly, let d=

0 1 min vj +1 − vj0 . 2 j =1,...,N −1

(cn) | is small and one can apply Lemma 2.2 (with the same If n is sufficiently large, then |V functions F as above) in the discs

λ: λ = π 2 n2 + vj0 + μ, |μ| < d ,

j = 1, 2, . . . , N.

So, if n n , then there are exactly one simple eigenvalue λn,j = π 2 n2 + μn,j inside each small disc B(π 2 n2 + vj0 , d) and there are no other eigenvalues. (ii) Recall that det ϕ(1, λn,j , V ) = 0. Therefore, due to (2.2) and the standard perturbation (0) + V (cn) has at least one eigenvalue τ such that theory, the self-adjoint matrix μn,j IN − V −1 (0) − V (cn) are τs = vs0 − |τ | = O(n ). On the other hand, the eigenvalues of the matrix V (cn) (cn) 2 | ), s = 1, 2, . . . , N . Hence, for some s, vss + O(|V (cn) 2 + O n−1 . μn,j − vs0 + vss(cn) = O V Due to (i), s = j . (iii) Let j = 1 for the simplicity and dk0 = v10 − vk0 , k = 2, . . . , N . In view of (2.2) and (ii), ⎛ (−1)n ϕ(1, λn,1 , V ) = 2π 2 n2

0

⎜ (cn) ⎜ ⎜ v21 ⎜ . ⎜ . ⎝ . (cn)

vN 1 δn (V ) +O . n2

(cn)

v12

(cn)

(cn)

d20 − v11 + v22 .. . (cn)

v2N

⎞

(cn)

···

v1N

··· .. .

v2N .. .

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(cn)

(cn)

(cn)

· · · dN0 − v11 + vN N


1555

Recall that ϕ(1, λn,1 , V )hn,1 = 0. Thus, (cn) | + δn (V )) for all k = 2, . . . , N ,

ϕ(1, λn,1 , V )hn,1 , ek0 = 0 gives hn,1 , ek0 = O(|V 0 |hn,1 | = 1 gives hn,1 , e1 = 1 + O(δn (V )) and, using ϕ(1, λn,1 , V )hn,1 , ek0 = 0 again, one obtains (cn) + dk0 · hn,1 , ek0 + O δn (V ) = 0, vk1

k = 2, . . . , N.

Note that (ii), (iii) are standard results for the perturbation of a simple eigenvalue.

2

2.2. Asymptotics of the norming constants and the averaged projectors Due to Proposition 2.1, all sufficiently large eigenvalues are simple. Therefore, for all sufficiently large n n and j = 1, 2, . . . , N we may introduce the factorization Pn,j = hn,j h∗n,j ,

−1 ∗ −1 Bn,j = − res M(λ) = hn,j gn,j hn,j = gn,j Pn,j , λ=λn,j

where gn,j > 0, hn,j ∈ CN , |hn,j | = 1 and hn,j , ej0 > 0. Denote Bn = Bn (V ) =

N

n n .

Bn,j ,

j =1

We begin with some simple reformulations of the needed asymptotics. Note that Proposition 2.1 gives hn,j = ej0 + 2

for all j = 1, 2, . . . , N.

Here and below we write an = bn + 2k iff +∞

k +∞ 2 |an − bn | n=n ∈ 2k = (cn )+∞ n=n : n cn n=n ∈ for k = 0, 1, 2, . . . . Note that 2 = 20 ⊂ 21 ⊂ 22 ⊂ · · · . Lemma 2.3. The following asymptotics are equivalent: N 2 (i) j =1 Pn,j = IN + 1 ; (ii) hn,j , hn,k = 21 for all j = k, j, k = 1, 2, . . . , N . Proof. Introduce N × N matrices hn = ( hn,1 ; hn,2 ; . . . ; hn,N ). Then hn h∗n =

N j =1

hn,j h∗n,j =

N j =1

Pn,j

(2.3)

1556


and N N h∗n hn = h∗n,j , hn,k j,k=1 = hn,k , hn,j j,k=1 . The matrices hn h∗n and h∗n hn are unitary equivalent (since hn h∗n = un (h∗n hn )u∗n , where hn = un sn is the polar decomposition of hn ). Thus, the asymptotics hn h∗n = IN + 21 are equivalent to the asymptotics h∗n hn = IN + 21 (note that hn,j , hn,j = |hn,j |2 = 1). 2 Lemma 2.4. The collection of asymptotics −1 = 2π 2 n2 1 + 21 gn,j

for all j = 1, 2, . . . , N,

and

N

Pn,j = IN + 21

j =1

is equivalent to Bn = 2π 2 n2 IN + 21 . 1 1 1 Proof. As in Lemma 2.3, we set Hn = g − 2 hn,1 ; g − 2 hn,2 ; . . . ; g − 2 hn,N . Note that Bn = n,N n,1 n,2 Hn Hn∗ while −1 −1 N Hn∗ Hn = gn,j2 gn,k2 · hn,k , hn,j j,k=1 . Thus, as above, asymptotics Bn = 2π 2 n2 (IN + 21 ) and Hn∗ Hn = 2π 2 n2 (IN + 21 ) are equiv−1 −1 , so gn,j = 2π 2 n2 (1 + 21 ). Asymptotics of the alent. The diagonal entries of Hn∗ Hn are gn,j 1/2 1/2

non-diagonal entries give hn,k , hn,j = 2π 2 n2 gn,j gn,k · 21 = 21 , j = k, which is equivalent to N 2 j =1 Pn,j = IN + 1 due to Lemma 2.3. 2 Note that, for sufficiently large n, Bn (V ) = −

N j =1

res M(λ) = −

λ=λn,j

1 2πi

M(λ) dλ. |λ−π 2 n2 |=3 V

This formula allows us to determine sharp asymptotics of Bn (V ). Moreover, it defines the analytic continuation of Bn (V ) for non-selfadjoint potentials. Proposition 2.5. The following asymptotics hold true

1 (sn) 1 (1 − t)V +O 2 Bn (V ) = 2π n IN − πn n 2 2

uniformly on bounded subsets of potentials V ∈ L2 ([0, 1]; CN ×N ).


1557

Proof. It is well known that sin z 1 χ 0, z2 , V = ϕ 1, z2 , V = IN + 2 z z

1 sin z(1 − t) · V (t) sin zt dt 0

1 + 3 z

1

x dx sin z(1 − x) · V (x)

0

e|Im z| sin z(x − t) · V (t) sin zt dt + O |z|4

0

uniformly on bounded subsets of V . Substituting z2 = π 2 n2 + μ, |μ| = 3 V = O(1), one obtains χ 0, π 2 n2 + μ, V 1 (−1)n μ IN + 2 2 = 2π 2 n2 π n

1 sin πn(1 − t) sin πnt · V (t) dt 0

μ + 2πn

1

(1 − t) cos πn(1 − t) sin πnt + t sin πn(1 − t) cos πnt · V (t) dt

0

1 + 3 3 π n =

(−1)n 2π 2 n2

1

x dx

0

1 sin πn(1 − x) sin πn(x − t) sin πnt · V (x)V (t) dt + O 4 n

0

μKn + Ln + O

1 n2

,

where the matrices 1 (sn) 1 (sn) (1 − 2t)V (1 − 2t)V = IN + , 2πn 2πn (0) + V (cn) + O 1 = −V (0) + V (cn) + O 1 Ln = − V n n

Kn = IN +

do not depend on μ. Hence, if μ = 3 V and n is sufficiently large, then −1 1 (−1)n 2 2 −1 χ 0, π n + μ, V = [μK + L ] + O . n n 2 2 2π n n2 Also, note that 1 χ 0, z2 , V = −ϕ 1, z2 , V = −cos zIN − z

1 cos z(1 − t) · V (t) sin zt dt + O 0

e|Im z| . |z|2

1558


Therefore, 1 (sn) 1 2 2 n−1 IN − +O 2 χ 0, π n + μ, V = (−1) V 2πn n

and 1 (sn) −1 1 1 −1 2 2 −1 −1 Kn μIN + Ln Kn 0, π n + μ, V = IN − +O 2 . − 2 2 χχ V 2πn 2π n n Since Ln Kn−1 does not depend on μ and 3 V = |μ| > |Ln Kn−1 | for sufficiently large n, we have 1 2πi

−1 μIN + Ln Kn−1 dμ = IN ,

|μ|=3 V

and so 1 1 (sn) −1 1 1 (sn) 1 Kn + O 2 = IN − (1 − t)V Bn = IN − +O 2 . V 2 2 2πn πn 2π n n n

2

2.3. Proof of the direct part in Theorem 1.1 Proof. In fact, all needed asymptotics have been obtained in Sections 2.1, 2.2. First, asymptotics of the eigenvalues and the individual projectors have been derived in Proposition 2.1. Second, asymptotics of the norming constants and the averaged projectors follow from Proposition 2.5 and Lemma 2.4. In order to prove (C) suppose that ξ : C →√CN is some entire vector-valued function such that Pα ξ(λα ) = 0 for all α 1, ξ(λ) = O(e|Im λ| ) as |λ| → ∞ and ξ ∈ L2 (R+ ). Due to Lemma 2.2 of [9], −1 −1 −1 ∗ = ϕ (1, λ, V ) = Zα + O(λ − λα ) (λ − λα )−1 Pα + Pα⊥ as λ → λα , χ(0, λ, V ) for some Zα such that det Zα = 0. Hence, the (vector-valued) function −1 ω(λ) = χ(0, λ, V ) ξ(λ) is entire. It follows from (2.1) that 1/2 1 2 2 as |λ| = π n + ω(λ) = O |λ| → ∞. 2 Thus, the Liouville theorem gives ω(λ) ≡ ω(0) = ω0 ∈ CN and ξ(λ) ≡ χ(0, λ, V )ω0 . If ω0 = 0, then this contradicts to ξ ∈ L2 (R+ ) in view of asymptotics (2.1). 2


1559

2.4. Explicit formula for the Weyl–Titchmarsh function In this section we prove that the Weyl–Titchmarsh function M(λ, V ) can be written as the regularized sum over all its poles. In other words, we give the explicit formula for M(λ, V ) involving only the spectral data λα (V ) and Bα (V ) = − resλ=λα M(λ, V ). The proof is quite standard. Proposition 2.6. Let V = V ∗ ∈ L2 ([0, 1]; CN ×N ) satisfy (1.5). Then M(λ) +

N

λ − vj0 cot

λ − vj0

· Pj0

=

j =1

α α=1

+

n −1 N 2π 2 n2 Pj0 Bα − λα − λ π 2 n2 + vj0 − λ

n=1 j =1

+∞ N 2π 2 n2 Pj0 Bn,j . − λn,j − λ π 2 n2 + vj0 − λ n=n j =1

(2.4)

The series converge uniformly on compact subsets of C that do not contain poles. Proof. Note that Dn,j (λ) = =

2π 2 n2 Pj0 Bn,j − λn,j − λ π 2 n2 + vj0 − λ Bn,j − 2π 2 n2 Pj0 π 2 n2 − λ

−

vj0 (Bn,j − 2π 2 n2 Pj0 ) (π 2 n2 − λ)(π 2 n2 + vj0 − λ)

−

(λn,j − π 2 n2 − vj0 )Bn,j (λn,j − λ)(π 2 n2 + vj0 − λ)

.

Due to Proposition 2.5, for the first terms one has Dn(1) (λ) =

N B 2 2 0 n,j − 2π n Pj j =1

π 2 n2 − λ

N =

2 where (xn )+∞ n=n ∈ . In particular, the series larities. Moreover,

j =1 Bn,j − 2π π 2 n2 − λ

+∞

n=n

2 n2 I

N

=

n · xn , π 2 n2 − λ

(1)

Dn (λ) uniformly converges outside singu-

+∞ +∞ 1 1 2 |xn | (1) 2 Dn (λ) 2 → ∞. → 0 as |λ| = π m + π 2 |n − (m + 12 )| n=n n=n Since Bn,j = 2π 2 n2 (Pj0 + 2 ) and λn,j = π 2 n2 + vj0 + 2 , the similar results hold true for the sums of second and third terms of Dn,j (λ). Thus, the right-hand side of (2.4) converges outside singularities and tends to zero as |λ| = π 2 n2 (m + 12 )2 → ∞. It follows from the standard asymptotics of fundamental solutions that the left-hand side of (2.4) also tends to zero as |λ| = π 2 n2 (m + 12 )2 → ∞. Since the residues of both sides at singularities coincide, (2.4) holds true for all λ. 2

1560


3. Inverse problem 3.1. Proof of the surjection part in Theorem 1.1. General strategy Step 1. Let some data (λ α , Pα† , gα† )α1 satisfy conditions (A)–(C) in Theorem 1.1 and Bα† = Pα† (gα† )−1 Pα† (we use different superscript for eigenvalues in order to make the further presentation more clear). Consider eigenvalues λ α (possibly multiple for several first α). One can split them into N simple series {λ n,j }∞ n=1 , j = 1, 2, . . . , N such that +∞

+∞ +∞ λn,1 n=1 ∪ λn,2 n=1 ∪ · · · ∪ λ n,N n=1 = λ α α1 (counting with multiplicities) and λ n,j = π 2 n2 + vj0 + 2 for all j = 1, 2, . . . , N . Using the well-known scalar inverse theory (see (1.4)) we construct some scalar potentials

∈ L2 ([0, 1]) such that vjj 1

vjj (t) dt = vj0

+∞ = λn,j n=1 . and σ vjj

0

Note that the corresponding isospectral sets are infinite-dimensional manifolds, so there are in

. For technical reasons, we choose vjj such that finitely many choices for each vjj

gn−1 vjj = − res m λ, vjj = 2π 2 n2 λ=λn,j

for all sufficiently large n,

) is the Weyl–Titchmarsh function of the scalar potential vjj , and where m(λ, vjj

χ 0, λ α , vjj = 0,

i.e.,

m λ α , vjj = 0 for all α 1

(one can always choose such vjj in two steps: taking the scalar m-function with all residues

equal to −2π 2 n2 and changing the first residue slightly in order to guarantee m(λ α , vjj ) = 0 for all α 1). Let

V = diag v11 , v22 , . . . , vN N . Thus, σ (V ) = {λ α }α1 counting with multiplicities. Denote −1 ∗ pα = Bα V . Bα = pα gα Since V is a diagonal potential, each subspace Eα is spanned by some (one, if α is large enough) standard coordinate vectors ej0 and all Pα are coordinate projectors. Step 2. Let Aα (V ) = M −1 λ α = χ(χ )−1 0, λ α , V


1561

∗ ∗ ⊥

A11 A12 : Eα → Eα , α = pα Aα pα : Eα → Eα , α = pα Aα qα ∗ ∗ ⊥ ⊥ ⊥

A21 , A22 : Eα → Eα , α = qα Aα pα : Eα → Eα α = qα Aα qα

(3.1)

and

where pα : CN → Eα , qα : CN → (Eα )⊥ are the coordinate projectors. Note that A11 = 0, α V

A12 = 0, α V

A21 = 0 and α V

det A22 = 0 α V

for all α 1

due to pα χ(0, λ α , V ) = [ϕ(1, λ α , V )(pα )∗ ]∗ = 0 and det χ (0, λ α , V ) = 0. In order to describe some neighborhood of the isospectral set Iso(V ) near V , we introduce kα × kα matrices (more accurate, operators in the coordinate subspaces Eα ) 22 −1 21 12 !α (V ) = A11 Aα (V ), A α − Aα Aα

α 1.

(3.2)

Then (see Proposition 3.2 and Lemma 3.3): !α (V ) are well defined in some complex neighborhood B(V , r ) of V ; (i) all A !α (V ) = [A !α (V )]∗ and the following holds: (ii) for V = V ∗ ∈ B(V , r ) one has A !α (V ) = 0 iff A

λ α is an eigenvalue of V of multiplicity kα .

Furthermore, for potentials V sufficiently close to V , we set !α (V ) = − 1 B 2πi

M(λ, V ) dλ,

where d =

|λ−λ α |=d

1 min λ α+1 − λ α > 0. 2 α1

(3.3)

!α (V ) = Bα (V ). If If kα = 1, then M(λ) has exactly one simple pole inside this contour, so B !α (V ) denotes the kα > 1, we do not know precisely how the multiple eigenvalue λ α is split, so B sum of all corresponding residues. Then (see Proposition 3.2, Lemma 3.3): !α (V ) are well defined in some complex neighborhood B(V , r ) of V ; (i) all B !α = B !α∗ , rank B !α = kα and the following holds: (ii) for V = V ∗ ∈ B(V , r ) one has B !α (V ) = 0 A

⇒

!α (V ) = Bα (V ). B

!α (V ) is the analytic continuation of Bα (V ) from the isospectral set Iso(V ) In other words, B into some complex neighborhood of V (emphasize that, due to the possible splitting of the eigenvalue λ α in case kα > 1, the original function Bα (V ) is discontinuous even for self-adjoint potentials close to V ). Step 3. We introduce the mapping !α (V ); B !α (V ) ! : V → A Φ α1

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which is defined in some complex neighborhood B(V , r ) of V (see Section 3.2). We prove ! maps B(V , r ) into some “proper” 2 -type space. In order to have the “nice” descripthat Φ tion of the image space, we consider some modification Φ, see details in Sections 3.3, 3.4. The modified mapping Φ is analytic in B(V , r ), so its restriction onto self-adjoint potentials close !α and N × N matrix B !α , to V is real-analytic. Note that, if V = V ∗ , then both kα × kα matrix A !α (V ), B !α (V )) is !α = kα , are self-adjoint. So, the total number of (real) parameters in (A rank B (kα )2 + kα (2N − kα ) = 2N kα . Step 4. We check that the Fréchet derivative dV Φ of the modified mapping Φ at the point V is invertible (see details in Sections 3.5, 3.6) . Therefore, due to the Implicit Function Theorem, for each sequence (Bα• )α1 sufficiently close to (Bα )α1 there exists some potential V • (close !α (V • ) = A !α (V ) = 0 and B !α (V • ) = Bα• for all α 1. If α • is large enough, to V ) such that A then the sequence Bα• := Bα ,

if α α • ,

and Bα• := Bα† ,

if α > α • ,

is close to (Bα )α1 . Thus, we obtain some potential V • such that !α V • = 0 for all α 1, A

i.e.,

σ V • = λ α α1

(counting with multiplicities) and !α V • = Bα† Bα V • = B

for α > α • .

Finally, using the isospectral transforms constructed in [9], we change the finite number of residues Bα , α = 1, 2, . . . , α • (see details in Section 3.7), and obtain the potential having the given spectral data (λ α , Bα† )α1 or, equivalently, (λ α , Pα† , gα† )α1 . 2 !α (V ) !α (V ) and B 3.2. Rough asymptotics of A This section contains some preliminary calculations. Loosely speaking, we consider the diagonal potential V as the unperturbed case and derive some rough asymptotics of spectral data for V close to V . The main results are formulated in Proposition 3.2 and Lemma 3.3. Let ϕ , ϑ , χ , η be the standard diagonal matrix-valued solutions (recall that V is diagonal) of the equation −ψ (x) + V (x)ψ(x) = λψ(x) satisfying the following boundary conditions: ϑ (0) = ϕ (0) = IN , ϑ (0) = ϕ (0) = 0,

η (1) = − χ (1) = IN , η (1) = χ (1) = 0.

We denote ϕα (x) = ϕ (x, λ α ), ϑα (x) = ϑ(x, λ α ) and so on. Let J (x, t) = ϕ (x)ϑ (t) − ϑ (x)ϕ (t) = −χ (x)η (t) + η (x)χ (t) be the (diagonal) solution of the same equation such that J (t, t) = 0, (J )x (t, t) = IN .


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Let V = V + W be some complex potential close to V . Then χ(x, λ, V ) can be easily constructed by iterations with the kernel J (x, t) (note that |J (x, t; z2 )| = O(|z|−1 e|Im z|·|x−t| )) starting with χ (x, λ). Thus, χ 0, z , V = χ 0, z2 +

2

1 0

χ 0, z2 , V = χ 0, z2 −

2 W 2 e|Im z|

, ϕ t, z W (t)χ t, z dt + O |z|3

(3.4)

2 2 W 2 e|Im z|

ϑ t, z W (t)χ t, z dt + O |z|2

(3.5)

1 0

2

uniformly on bounded subsets of W . In particular (see (2.2)), if μ = O(1), then

(−1)n 0 χ 0, λ n,j + μ, V = diag μ − vj0 + v10 , . . . , μ − vj0 + vN + o(1) + O W , 2π 2 n2 χ 0, λ n,j + μ, V = (−1)n−1 IN + O n−1 as n → ∞, (3.6) uniformly on bounded subsets of W . Recall that Aα (V ) = [χ(χ )−1 ](0, λ α , V ) and its block 1

∗

A22 α = qα Aα (qα ) are given by (3.1) and d = 2 minα1 (λα+1 − λα ) > 0. Lemma 3.1. There exists r > 0 such that for all (possibly non-selfadjoint) potentials "

" V ∈ B V , r = V ∈ L2 [0, 1]; CN ×N : "V − V " < r the following is fulfilled for all α 1: det χ 0, λ α , V = 0,

det A22 α (V ) = 0 and

det χ 0, λ α + μ, V = 0,

if |μ| = d .

Moreover, for all j = 1, 2, . . . , N and |μ| = d , 22 −1 An,j (V ) = O n2

and

−1 χ 0, λ n,j + μ, V = O n2

(3.7)

uniformly on B(V , r ).

Proof. It follows from (3.6) that all matrices χ (0, λ n,j , V ), A22 n,j (V ), χ(0, λn,j + μ, V ) are

non-degenerate and (3.7) holds, if n n∗ is sufficiently large and r is sufficiently small. So, one needs to consider only some finite number of first indices α = 1, 2, . . . , α∗ .

Note that det χ (0, λ α , V ) = 0, det A22 α (V ) = 0, det χ(0, λα + μ, V ) = 0 for all α and all

these matrices (as functions of V ) are continuous at V . Therefore, if W r and r > 0

is small enough, then all χ (0, λ α , V ), A22 α (V ), χ(0, λα + μ, V ), α = 1, 2, . . . , α∗ , are nondegenerate too. 2

Proposition 3.2. !α (V ), B !α (V ), α 1, are well defined by (3.2), (3.3) and (i) There exists r > 0 such that all A

analytic in B(V , r ).

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(ii) For all j = 1, 2, . . . , N the asymptotics εn (W ) ! , An,j (V ) = O n2

!n,j (V ) − B = O n2 εn (W ) , B n,j

(cn) W + , εn (W ) = W n hold true uniformly for potentials 0

V ∈ B V , r

#

= V = V + W ∈ B V , r :

$

1

W (t) dt = 0 . 0

!α (V ) are well defined in some complex neighbor!α (V ), B Proof. (i) Due to Lemma 3.1, all A hood B(V , r ) of V . These functions are analytic in this neighborhood since χ(0, λ, V ) and χ (0, λ, V ) are analytic for each λ as functions of V . (ii) Let λ = π 2 n2 + μ and |μ| = O(1), thus ϕ (t, λ) = (πn)−1 sin πnt + O n−2 and (−1)n−1 χ (t, λ) = (πn)−1 sin πnt + O n−2 . Using (3.4), (3.5) and

1 0

W (t) dt = 0, we get

χ(0, λ, V ) = χ (0, λ) + O

εn (W ) , n2

W χ (0, λ, V ) = χ (0, λ) + O n

(note that n−1 W εn (W ) by definition). Due to (3.6), it gives εn (W ) . An,j (V ) = χ(χ )−1 0, λ n,j , V = An,j V + O n2 12 21 22

−1 = O(n2 ), we have Since A11 n,j (V ) = 0, An,j (V ) = 0, An,j (V ) = 0 and (An,j (V ))

22 −1 21 11 εn (W ) 12 ! . An,j (V ) = O An,j (V ) = An,j − An,j An,j n2 Due to the similar arguments, if λ = λ n,j + μ, |μ| = d , then −1 −1 χχ (0, λ, V ) = χ χ (0, λ) + O n2 εn (W ) . !n,j (V ) = B + O(n2 εn (W )). Integrating over the contour |μ| = d , we obtain B n,j Lemma 3.3. For some r > 0 and all V = V ∗ ∈ B(V , r ) the following hold: (i) (ii) (iii)

!α (V )]∗ , B !α (V ) = [B !α (V )]∗ and rank B !α (V ) = kα ; !α (V ) = [A A

! Aα (V ) = 0 if and only if λα is an eigenvalue of V of multiplicity kα ; !α (V ) = Bα (V ). !α (V ) = 0, then B if A

2


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!α (V ) = [B !α (V )]∗ , Aα (V ) = Proof. (i) If V = V ∗ , then M(λ) ≡ [M(λ)]∗ , λ ∈ C. In particular, B !α (V ) = [A !α (V )]∗ . Due to Lemma 3.1, det χ(0, λ, V ) has no zeros on the circle [Aα (V )]∗ and A |λ − λα | = d for all V ∈ B(V , r ). Since the spectrum depends on the potentials continuously, for each self-adjoint potential V = V ∗ ∈ B(V , r ) there are exactly kα eigenvalues in the interval (λ α − d , λ α + d ) counting with multiplicities. !α (V ) = rank Bα (V ) = 1. If α α , then rank B !α (V ) kα . If α > α , then kα = 1 and rank B !α (V ) = kα and B !α is a continuous function of V . Thus, if r is small enough, Note that rank B !α (V ) kα for all α α and V ∈ B(V , r ). then rank B (ii) Recall that λ α is an eigenvalue of V of multiplicity kα iff dim Ker χ(0, λ α , V ) = kα . Since det χ (0, λ α , V ) = 0 (see Lemma 3.1), this is equivalent to say that dim Ker χ(χ )−1 0, λ α , V = kα ,

i.e.,

rank Aα (V ) = N − kα .

Due to Lemma 3.1, det A22 α (V ) = 0 for all V ∈ B(V , r ). Then, the last statement is equivalent 11 12 22 −1 21 ! to Aα (V ) = [Aα − Aα (Aα ) Aα ](V ) = 0. !α (V ) = 0, then λ α is an eigenvalue of multiplicity kα and there are no other eigenval(iii) If A ues in the disc |λ − λ α | < d . Thus,

!α (V ) = − res M(λ, V ) = Bα (V ). B

λ=λα

2

3.3. Analyticity. Expanded mapping Ψ !α (V ), α 1, are well defined in some !α (V ), B Proposition 3.2(i) guarantees that all matrices A neighborhood B(V , r ) of V . Let α 0 and n 1 be such that k1 + k2 + · · · + kα

= N n − 1 and kα = 1

for all α α + 1,

so the double-indexing (n, j ), j = 1, 2, . . . , N , is well-defined starting with n . Also, let n be −1 sufficiently large such that gn,j (V ) = 2π 2 n2 for all n n (see Step 1, Section 3.1). Recall that N !n,j (V ) for n n . Bn (V ) = j =1 B Definition 3.4. Introduce the (formal) mapping α +∞ Ψ : V → Ψ (1) (V ); Ψ (2) (V ) = Ψα(1) (V ) α=1 ; Ψn(2) (V ) n=n , !α ; B !α ), Ψα(1) = (A N ! Bn,j Bn 0 !n,j N ; . Ψn(2) = 2π 2 n2 · A − P ; πn − I N j j =1 2π 2 n2 2π 2 n2 j =1 (1)

(2)

Note that Ψα and Ψn

map B(V , r ) into some finite-dimensional spaces. Namely,

Ψα(1) : B V , r → Ckα ×kα ⊕ CN ×N

N and Ψn(2) : B V , r → CN ⊕ CN ×N ⊕ CN ×N .

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Since Ψ (1) has the finite number of components, it also acts into finite-dimensional Hilbert % !(1) = α [Ckα ×kα ⊕ CN ×N ]. It has been shown in Section 3.2 that the (Euclidean) space H α=1 components of Ψ (2) have “nice” asymptotics for potentials # $ 1 V ∈ B0 V , r = V = V + W ∈ B V , r : W (t) dt = 0 . 0

= {A = A∗ ∈ Cm×m } be the real component of the comLet Nn = {n ∈ N: n n } and Cm×m R m×m plex Hilbert space C , i.e., the real space of all self-adjoint m × m matrices. Lemma 3.5. !(2) = 2 (Nn ; CN ⊕[CN ×N ]N ⊕CN ×N ). Moreover, the image (i) Ψ (2) maps B 0 (V , r ) into H C !(2) . Ψ (2) [B 0 (V , r )] is bounded in H !=H !(1) ⊕ H !(2) is an analytic mapping between complex Hilbert (ii) Ψ : B 0 (V , r ) → H spaces. Moreover, the Fréchet derivative dV Ψ of Ψ at V is given by the Fréchet deriva (1) (2) tives of its components: (dV Ψ )W = (((dV Ψα )W )αα=1 ; ((dV Ψn )W )+∞ n=n ). (1) (2) N ×N 0

0

2 ! ! ! (iii) Ψ : BR (V , r ) = B (V , r ) ∩ L ([0, 1]; CR ) → HR = HR × HR is a real-analytic mapping between real Hilbert spaces and the Fréchet derivative dV Ψ is given by the Fréchet derivatives of its components, where

!(1) = H R

α & kα ×kα ×N CR , ⊕ CN R

!(2) = 2 Nn ; RN ⊕ CN ×N N ⊕ CN ×N . H R R R R

α=1

Proof. (i) Due to Proposition 3.2, for all j = 1, 2, . . . , N , !n,j (V ) = O n−2 εn (W ) A

!n,j (V ) − B = O n2 εn (W ) and B n,j

uniformly on B(V , r ), where (cn) W + , εn (W ) = W n

so

+∞ εn (W )2 = O W 2 . n=n

−1 0 Since Bn,j = (gn,j ) Pj = 2π 2 n2 Pj0 , n n , we obtain

2 2 !n,j (V ) +∞ ∈ 2 2π n A n=n

+∞ ! Bn,j (V ) 0 and − Pj ∈ 2 , 2π 2 n2 n=n

j = 1, 2, . . . , N,

uniformly on B 0 (V , r ). Also, due to Proposition 2.5,

Bn (V ) − IN πn 2π 2 n2

+∞ n=n

∈ 2

uniformly on B 0 V , r .


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(ii) Due to Proposition 3.2, all coordinates Ψα , α = 1, 2, . . . , α , are analytic in B(V , r ). (2) Hence, Ψ (1) is analytic too. Similarly, all coordinates Ψn , n n , are analytic in B(V , r ). (2) is also locally bounded in B 0 (V , r ). Therefore (e.g., see [43] It follows from (i), that Ψ (Appendix A, Theorem 3) or [13] (Chapter 3, Proposition 3.7)), Ψ (2) is analytic as the mapping between Hilbert spaces and its Fréchet derivative (or, equivalently, gradient) is given by the Fréchet derivatives (gradients) of its components. !R . Ψ is real-analytic due to (ii). 2 (iii) By Lemma 3.3, Ψ maps BR0 (V , r ) into H (1)

3.4. Analyticity. Modified mapping Φ The expanded mapping Ψ introduced in Definition 3.4 is real-analytic but overdetermined. In other words, its coordinates, obviously, are not independent from each other. In particular, there are no chances that the Fréchet derivative dV Ψ is invertible. On the other hand, the co!α (V ), α 1, of the original mapping Φ !α (V ), B ! are independent, but we have no ordinates A “nice” description of the image space. The next goal is to construct some modified mapping Φ = (Φ (1) , Φ (2) ) (see Definitions 3.6, 3.8, 3.9) such that !α (V ), α 1; !α (V ), B (i) it keeps the full information about A (ii) it is real-analytic as the mapping between Hilbert spaces; (iii) its coordinates are “independent” from each other (more precisely, in Sections 3.5, 3.6 we will show that dV Φ is an invertible linear operator). !α (V ), α = 1, 2, . . . , α . Recall We start with a slight modification of the first coordinates B 0

∗ ! ! ! that, if V ∈ BR (V , r ), then Bα (V ) = [Bα (V )] , rank Bα (V ) = kα and ∗ Bα = pα Bα pα ,

∗ −1 −1 ∗ pα Bα pα = gα = gα >0

(moreover, gα is diagonal, since V is diagonal). Therefore, if r > 0 is sufficiently small, then for each α = 1, 2, . . . , α we have the (unique) factorization !α = B

∗ ∗ !α11 : Eα → Eα , Cα = Cα∗ = B pα + qα Eα Cα pα + Eα∗ qα , 11 −1 ⊥ !α !α21 B : Eα → Eα , Eα = B

(3.8)

!α21 = qα B !α = kα . !α (pα )∗ , B !α (pα )∗ , etc. Note that Cα > 0, since rank B !α11 = pα B where B Definition 3.6. We introduce the first component of the mapping Φ by

α & kα ×kα

k ×k (1) 0 Φ (1) : BR V , r → HR = CR ⊕ CRα α ⊕ C(N −kα )×kα ,

Φ

(1)

α (V ) = Φα(1) (V ) α=1 ,

α=1

!α (V ); Cα (V ); Eα (V ) . Φα(1) (V ) = A

(3.9)

0 (V , r ), if Remark 3.7. Due to Lemma 3.5(ii), Φ (1) is well defined and real-analytic in BR

(1) (1) ! r > 0 is small enough. Note that Φ can be reconstructed from Φ and the total number of real parameters containing in Φ (1) is 2N (k1 + k2 + · · · + kα

) = 2N 2 (n − 1).

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We pass to the design of the second component Φ (2) . The main purpose of (rather technical) Definition 3.8 is to combine heterogeneous objects from (1.6) into one object having “nice” asymptotics as n → ∞ (see Proposition 3.10). Due to Proposition 3.2, if r > 0 is sufficiently small, then A !n,j (V ) = O n−2 εn (W )

and ! Bn,j (V ) − 2π 2 n2 Pj0 = O n2 εn (W ) .

(3.10)

0 (V , r ), then factorization (3.8) is well defined for all n n . Recall In particular, if V ∈ BR

! that kn,j = 1, so An,j (V ) and Cn,j (V ) > 0 are real numbers. 0 (V , r ) and r > 0 be sufficiently small. Introduce two numbers Definition 3.8. Let V ∈ BR an,j (V ), cn,j (V ) ∈ R and one vector en,j (V ) ∈ CN such that en,j , ej0 = 1 as

!n,j (V ), an,j (V ) = 2π 2 n2 A

cn,j (V ) =

2 2 −1 1 2π n Cn,j (V ) 2 ,

en,j (V ) = ej0 + En,j (V )ej0 .

Furthermore, define N × N matrix Yn = Yn (V ) ∈ CN ×N by Yn = exp[ian,1 ] · cn,1 · en,1

;

exp[ian,2 ] · cn,2 · en,2

; · · · ; exp[ian,N ] · cn,N · en,N

and let Yn (V ) = Un (V )Sn (V ),

Un∗ = Un−1 ,

Sn∗ = Sn > 0,

be its polar decomposition. !n,j , j = 1, 2, . . . , N , can be easily reconstructed from Un , Sn . Factoriza!n,j , B Note that all A tion (3.8) reads now as 2 2 −1 2 ∗ !n,j = cn,j 2π n · en,j en,j , B so (3.10) gives an,j (V ), cn,j (V ) − 1, en,j (V ) − e0 = O εn (W ) j

uniformly for n n . Hence, Yn (V ) − IN , Un (V ) − IN , Sn (V ) − IN = O εn (W ) 0 (V , r ), if r is small enough. uniformly for n n and det Yn (V ) = 0 for all V ∈ BR

Definition 3.9. Formally introduce the second component of the mapping Φ by +∞ Φ (2) : V → Φ (2) (V ) = Φn(2) (V ) n=n , 0 ×N ×N V , r → CN ⊕ CN , Φn(2) = −i log Un ; 2πn · (Sn − IN ) : BR R R where log Un = (Un − IN ) − 12 (Un − IN )2 + 13 (Un − IN )3 − · · · .

(3.11)


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!n,j (V ) = 0 and B !n,j (V ) = 2π 2 n2 P 0 for all n n . Thus, Recall that A j Yn V = Un V = Sn V = IN

and Φn(2) V = (0; 0) for all n n .

Proposition 3.10. There exists r > 0 such that the mapping ×N ×N 0 Φ (2) : BR V , r → 2R Nn ; CN × CN R R 0 (V , r ). Moreover, the Fréchet derivative d Φ (2) of is well defined and real-analytic in BR V (2)

Φ at V is given by the Fréchet derivatives of its components.

Proof. Due to (3.11) and

+∞

2 n=1 |εn (W )|

= O( W 2 ), for sufficiently small r > 0 the mapping

+∞ Y : V → Yn (V ) − IN n=n ,

0 V , r → 2R Nn ; CN ×N BR

!n,j and B !n,j , j = 1, 2, . . . , N (see is well defined. Recall that Yn is some simple function of A Definition 3.8). Using real-analyticity of the first two components of the expanded mapping Ψ (2) (see Definition 3.4 and Lemma 3.5), we conclude that Y is real-analytic as a composition of real-analytic mappings. Since Sn = (Yn∗ Yn )1/2 and Un = Yn Sn−1 , both mappings +∞ S : V → Sn (V ) − IN n=n ,

×N 0 V , r → 2R Nn ; CN , BR R

and +∞ U : V → −i log Un (V ) n=n ,

×N 0 V , r → 2R Nn ; CN , BR R

are real-analytic too as compositions of Y with some simple coordinate-wise transforms. In order to complete the proof it is sufficient to show that S actually acts into “better” space 21 . Note that Yn Yn∗ =

N

2 ∗ cn,j · en,j en,j =

j =1

N 1 ! Bn . Bn,j = 2 2 2π n 2π 2 n2 j =1

Due to Lemma 3.5, the mapping +∞ Z : V → 2πn · Yn Yn∗ − IN n=n ,

×N 0 V , r → 2R Nn , CN BR R

(which is the third component of Ψ (2) ) is real-analytic. Using Sn = [Un−1 (Yn Yn∗ )Un ]1/2 , we obtain that the mapping +∞ S! : V → 2πn · Sn (V ) − IN n=n ,

×N 0 V , r → 2R Nn ; CN , BR R

is real-analytic as a result of some coordinate-wise transforms with Z and U . Note that Φ (2) = ! Since the Fréchet derivative dV Ψ is given by the Fréchet derivatives of its components, (U; S). ! 2 the same holds true for all mappings Y, S, U , Z and S.

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Remark 3.11. The mapping Φ = (Φ (1) ; Φ (2) ) is real-analytic too, since both Φ (1) , Φ (2) are real(1) (2) analytic, and its Fréchet derivative is given by the Fréchet derivatives of Φα , Φn . Note that (2)

2 each Φn , n n , contains 2N real parameters, i.e., exactly “the same amount of information” (n) . as, say, the nth Fourier coefficient V 3.5. Explicit form of the Fréchet derivative dV Φ We denote by (0) P 0 : W (x) → W (x) − W ×N ×N (0) = 0}. the orthogonal projector in L2 ([0, 1]; CN ) onto {W ∈ L2 ([0, 1]; CN ): W R R Recall that the mapping Φ was introduced in Definitions 3.6 and 3.9. Due to Remark 3.11, ×N ) is given by (dV Φ)W for W ∈ P 0 L2 ([0, 1]; CN R

!α )W, (dV A

(dV Cα )W,

and (dV Un )W,

(dV Eα )W

(dV Sn )W

for α = 1, 2, . . . , α ,

for n n .

We need some preliminary calculations. Let χα = χ ·, λ α , V ,

ϕα = ϕ ·, λ α , V

and so on.

Since V is a diagonal potential, all these matrix-valued functions are diagonal. For short, we will use (a bit careless) notations like −1 −1 χα (t) := χα (t) χα (0) = χα (0) χα (t).

(χα ) (0) Recall that pα : CN → Eα and qα : CN → (Eα )⊥ are some coordinate projectors. Note that Ker[χα (0)(qα )∗ ] = {0}, Ker[(χα ) (0)(pα )∗ ] = {0} and Ker[χ˙ α (0)(pα )∗ ] = {0}. Thus, expressions −1 ∗ χα (0) qα ,

−1 ∗ pα χα (0)

and

−1 ∗ pα χ˙ α (0)

(and their conjugates) are well defined. ×N ) the following hold: Proposition 3.12. For all α 1 and W ∈ P 0 L2 ([0, 1]; CN R

!α )W = pα (dV A

1 0

(dV Eα )W

= −qα

∗ χα (t) χα (t) W (t) dt pα ,

(χα ) (0) (χα ) (0)

1 0

∗ χα (t) χα (t) W (t) dt pα

χα (0) (χα ) (0)

(3.12)

(3.13)


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and

(dV Cα )W

= pα

1 0

∗ ξα (t) χα (t) χα (t) ξα (t) W (t) + W (t) dt pα ,

χ˙ α (0) χ˙ α (0) χ˙ α (0) χ˙ α (0)

(3.14)

where ξα (t) ≡ χ˙ α (t) −

χ¨α (0) χ (t). 2χ˙ α (0) α

(3.15)

Proof. It follows from (3.4) and (3.5) that dV χ 0, λ α W =

1

ϕα (t)W (t)χα (t) dt,

dV χ 0, λ α W = −

0

1

ϑα (t)W (t)χα (t) dt

0

and dV χ˙ 0, λ α W =

1

ϕ˙ α (t)W (t)χα (t) + ϕα (t)W (t)χ˙ α (t) dt.

0 12 22 −1 21 !α = A11 Recall that A α − Aα (Aα ) Aα , where

Aα (V ) = χ(χ )−1 0, λ α , V ,

∗

A11 α = pα Aα pα ,

∗

A12 α = pα Aα qα

and so on.

21

Due to A12 α (V ) = 0, Aα (V ) = 0 and pα χα (0) = 0, one obtains

−1 ∗

!α )W = dV A11 (dV A pα α W = pα dV χ 0, λα W χα (0) 1 ∗ χα (t)

dt pα . = pα ϕα (t)W (t) (χα ) (0) 0

This gives (3.12), since pα ϕα (t) ≡ pα χα (t)[(χα ) (0)]−1 . Next, !α )W = − 1 (dV B 2πi 1 = 2πi

dV χ χ −1 (0, λ) W dλ

|λ−λ α |=d

|λ−λ α |=d

(χ ) (0, λ) dλ − dV χ (0, λ) W + dV χ(0, λ) W .

χ (0, λ) χ (0, λ)

Note that the diagonal matrix-valued function [χ (0, λ)]−1 has the unique pole (at λ α ) inside of the contour of integration and

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IN χ¨α (0) IN IN

Pα + Q α Q α + O λ − λ α = P − α

2 χ (0, λ) χ˙ α (0)(λ − λα ) 2[χ˙ α (0)] χα (0)

as λ → λα ,

!α21 [B !α11 ]−1 and B !α21 (V ) = 0. Thus, where Q α = (qα )∗ qα = IN − Pα . Recall that Eα = B

11 −1 !α V !α21 W · B !α21 W · pα χ˙ α (0) pα ∗ (dV Eα )W = dV B = − dV B (χα ) (0)

and !α21 W = qα dV B

1 0

ϑα (t) +

∗ (χα ) (0) IN

ϕα (t) W (t)χα (t) dt pα .

χα (0) χ˙ α (0)

Using χα (0)ϑα (t) + (χα ) (0)ϕα (t) ≡ χα (t), one obtains (3.13). !α11 (V ) = pα B !α (V )(pα )∗ . In contrast to (dV B !α21 )W , we do not have Furthermore, Cα (V ) = B cancellations of the singularities by the projectors, so one should find the residue at the second order pole λ α . Straightforward calculations give IN ∗ (χ ) (0, λ)

(dV Cα )W = res pα − dV χ (0, λ) W + d p χ(0, λ) W V

λ=λα χ (0, λ) χ (0, λ) α 1 (χ˙ α ) (0) (χα ) (0)χ¨α (0)

= pα ϑα (t)W (t)χα (t) + ϕα (t)W (t)χα (t) − χ˙ α (0) 2[χ˙ α (0)]2 0

IN (χ ) (0) ϕ˙ α (t)W (t)χα (t) + ϕα (t)W (t)χ˙ α (t) + α χ˙ α (0) χ˙ α (0) ∗ IN χ (t)χ¨ (0) χα (0)ϕα (t)W (t) α α 2 dt pα . − χ˙ α (0) 2[χ˙ α (0)] Using the identities ϑα (t) +

(χ˙ α ) (0) (χα ) (0) χ˙ α (t) ϕ ϕ ˙ (t) + (t) ≡ χ˙ α (0) α χ˙ α (0) α χ˙ α (0)

and pα (χα ) (0)ϕα (t) ≡ pα χα (t), one obtains (3.14).

2

Introduce the functions

χα ,j (t) ≡ χα (t) jj ≡ χ t, λ α , vjj

and ξα ,j (t) ≡ ξα (t) jj ≡ ξ t, λ α , vjj ,

where ξα is given by (3.15).

)} (by definition, the set I (α) consists of Corollary 3.13. Let α 1 and I (α) = {s: λ α ∈ σ (vss N ×N

0 2 kα indices). Then, for all W ∈ P L ([0, 1]; CR ),


k) !α )W (dV A , = Wj k , u(j α jk

(dV Cα )W j k = Wj k ,! uα(j k) ,

where for all λ α ∈ σ (vjj ) ∩ σ (vkk ) the functions uα

(j k)

(j k)

and ! uα

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j, k ∈ I (α),

are given by

,j ,k −1 ,j χα (0) χα (0) · χα (t)χα ,k (t), ,j −1 ,j k)

,k · ξα (t)χα ,k (t) + χα ,j (t)ξα ,k (t) . ! u(j α (t) ≡ χ˙ α (0)χ˙ α (0) k) u(j α (t) ≡

(3.16)

Furthermore, k) , (dV Eα )W j k = Wj k , u(j α

where for all λ α ∈ σ (vkk ) \ σ (vjj ) the function uα

(j k)

j∈ / I (α), k ∈ I (α), is given by

,j ,k −1 ,j k) (0) · χα (t)χα ,k (t). u(j α (t) ≡ − χα (0) χα

(3.17)

Proof. Since χα , ξα are diagonal matrices, this is exactly the result of Proposition 3.12 rewritten in the coordinate form. 2 Proposition 3.14. Let n n and j, k = 1, 2, . . . , N be such that j = k. Then for all W ∈ ×N ) the following identities hold: P 0 L2 ([0, 1]; CN R −1 (jj ) (jj ) Wjj ,! un,j + i · 2π 2 n2 Wjj , un,j , (dV Yn )W jj = 4π 2 n2 (jj )

(3.18)

(jj )

where the functions un,j and ! un,j are given by (3.16), and (j k) (dV Yn )W j k = Wj k , un,k ,

(3.19)

(j k)

where the functions un,k are given by (3.17). Furthermore, ∗ 1 (dV Yn )W + (dV Yn )W , 2 ∗ 1 (dV Un )W = (dV Yn )W − (dV Yn )W . 2 (dV Sn )W =

(3.20)

Proof. By definition of Yn , (dV Yn )W j k = dV exp(ian,k ) · cn,k · en,k W, ej0 . Recall that an,k (V ) = 0, cn,k (V ) = 1, en,k (V ) = ek0 + En,k (V )ek0 and En,k (V ) = 0. Thus, (dV Cn,j )W !n,j )W + i · 2π 2 n2 (dV A (dV Yn )W jj = (dV cn,j )W + i · (dV an,j )W = 4π 2 n2

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and (dV Yn )W j k = (dV En,k )W j . Due to Corollary 3.13, one obtains (3.18) and (3.19). Recall that Sn = (Yn∗ Yn )1/2 , Un = Yn Sn−1 and Yn (V ) = Un (V ) = Sn (V ) = IN . This immediately gives (3.20). 2 3.6. Invertibility of the Fréchet derivative dV Φ Due to Remark 3.11, !α )W ; (dV Cα )W ; (dV Eα )W , α = 1, 2, . . . , α , dV Φα(1) W = (dV A dV Φn(2) W = −i(dV Un )W ; 2πn(dV Sn )W , n = n , n + 1, . . . . Recall that Wkj = Wj k for all 1 k j N . It immediately follows from Corollary 3.13 and Proposition 3.14 that the entries of the components of (dV Φ)W are !α , Cα and (b) Un , Sn ): (1) for all j = 1, 2, . . . , N (diagonal entries of (a) A (jj ) (jj )

(a) Wjj , uα , Wjj ,! uα , where α α are such that λ α ∈ σ (vjj ); un,j , for all n n ; (b) 2π 2 n2 · Wjj , un,j , (2πn)−1 · Wjj ,! !α , Cα ; (b) Eα ; (c) Un , Sn ): (2) for all 1 k < j N (non-diagonal entries of (a) A (j k) (j k) (kj ) (kj ) (a) Wj k , uα , Wj k ,! uα and their complex-conjugates Wj k , uα , Wj k ,! uα ,

where α α : λα ∈ σ (vjj ) ∩ σ (vkk ); (jj )

(jj )

) \ σ (vjj ); Wj k , uα , where α (b) Wj k , uα , where α α are such that λ α ∈ σ (vkk

α are such that λα ∈ σ (vjj ) \ σ (vkk ); (j k)

(c)

1 2i

(kj )

(j k)

(kj )

(j k)

(kj )

· Wj k , [un,k − un,j ], πn · Wj k , [un,k + un,j ] and their conjugates

(kj ) [un,j

(j k) − un,k ],

(j k)

Note that uα

(kj )

= uα

(kj ) πn · Wj k , [un,j (j k)

and ! uα

(kj )

=! uα

(j k) + un,k ],

for all

1 2i

· Wj k ,

n n .

, if λα ∈ σ (vjj ) ∩ σ (vkk ).

Definition 3.15. For each 1 k j N we introduce the collection of real scalar functions

2 2 (jj ) (jj ) ) ∪ 2π n un,j , (2πn)−1! uα(jj ) , α α : λ α ∈ σ vjj un,j , n n , U (jj ) = u(jj α ,!

k) (j k) U (j k) = u(j ∩ σ vkk uα , α α : λ α ∈ σ vjj α ,!

k) (kj )

∪ u(j ∪ uα , α α : λ α ∈ σ vjj \ σ vkk α , α α : λα ∈ σ vkk \ σ vjj ( ' (j k) 1 (j k) (kj ) (kj ) un,k − un,j , πn un,k + un,j , n n , ∪ 2 (j k)

(j k)

where the functions uα and ! uα are given by (3.16) and (3.17). Note that each collection U (j k)

contains exactly 2(n − 1) functions with “small” indices α α .


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Remark 3.16. Due to the arguments given above, in order to prove that [dV Φ]−1 is bounded, it is sufficient to prove that each P 0 U (j k) is a Riesz basis of P 0 L2 (0, 1). Lemma 3.17. For each 1 k j N there exists some collection of functions V (j k) ⊂ P 0 L2 (0, 1) which is biorthogonal to U (j k) (and, therefore, to P 0 U (j k) ). Proof. Taking into account definitions (3.16), (3.17) and (3.15), it is sufficient to construct some !(j k) , where !(j k) ⊂ P 0 L2 (0, 1) which is biorthogonal to P 0 U collection V

!(j k) = χα ,j χα ,k , for all λ α ∈ σ v ∪ σ v U jj kk

,j ,k

,j ,k

∩ σ vkk , ∪ χ˙ α χα + χα χ˙ α , for all λα ∈ σ vjj !(j k) and U (j k) are related by some simple linear transformations (namely, multiplications since U by fixed constants, (χ, ξ = χ˙ + cχ) ↔ (χ, χ˙ ) and (u1 , u2 ) ↔ (u1 + u2 , u1 − u2 )). Note that we consider both cases k = j and k < j simultaneously. Let !(j k) = V

,j ,k ∪ σ vkk ϕβ ϕβ , for all λ β ∈ σ vjj

,j

,j ∩ σ vkk , ∪ ϕ˙ β ϕβ ,k + ϕβ ϕ˙ β ,k , for all λ β ∈ σ vjj

!(j k) ⊂ P 0 L2 (0, 1). Let λα = λβ and {χ, ϕ} = χϕ − χ ϕ. The standard trick (e.g., by definition, V see [43, pp. 44–45] for the similar calculation in the scalar case) shows

,j χα ,j χα ,k , ϕβ ϕβ ,k 1 = 2

1

,j ,k ,j ,k ,j ,k ,j ,k χα χα ϕβ ϕβ ϕβ ϕβ (t) dt − χα χα

0

1 = 2

1

,j ,j ,k ,k ,j ,j ,k ,k χα , ϕβ χα ϕβ + χα ϕβ χα , ϕβ (t) dt

0

,j

=

,j

{χα , ϕβ }{χα ,k , ϕβ ,k }|10 2(λα − λβ )

,j

=

,j

[ϕβ ϕβ ,k ](1) − [χα χα ,k ](0) 2(λα − λβ )

,j

.

(3.21)

,j

If both λα , λβ ∈ σ (vjj ) ∪ σ (vkk ), then ϕβ (1)ϕβ ,k (1) = χα (0)χα ,k (0) = 0. Hence,

,j χα ,j χα ,k , ϕβ ϕβ ,k = 0.

) ∩ σ (vkk ) (the case λβ ∈ σ (vjj ) ∩ σ (vkk ) is similar), then the right-hand Moreover, if λα ∈ σ (vjj side in (3.21), as a function of λα , has a double zero, so we can differentiate this identity (with respect to λα ) and obtain

,j χ˙ α ,j χα ,k + χα ,j χ˙ α ,k , ϕβ ϕβ ,k = 0.

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Also, if both λα , λβ ∈ σ (vjj ) ∩ σ (vkk ), then

,j ,k ,j

,j χ˙ α χα + χα ,j χ˙ α ,k , ϕ˙ β ϕβ ,k + ϕβ ϕ˙ β ,k = 0.

,j

,j

) \ σ (vkk ) (or λα = λβ ∈ σ (vkk ) \ σ (vjj )). Then {χα , ϕα } = 0, Let λα = λβ ∈ σ (vjj

,k

,k {χα , ϕα } = 0 and

,j ,k ,j ,k {χα ,k , ϕα ,k } χα χα , ϕα ϕα = 2

1

χα ,j (t)ϕα ,j (t) dt = 0.

0

,j

,j

) ∩ σ (vkk ). Then {χα , ϕα } = {χα ,k , ϕα ,k } = 0 and Let λα = λβ ∈ σ (vjj

χα ,j χα ,k , ϕα ,j ϕα ,k = 0.

Using (3.21) for λβ → λα , one gets

,j

ϕβ (1)ϕβ ,k (1) [ϕ˙α ,j ϕ˙α ,k ](1) ,j ,k ,j ,k

,j ,k χ˙ α χα + χα χ˙ α , ϕα ϕα = 0. = lim = λβ →λα 2(λα − λβ )2 2 Similarly,

,j [χ˙ α χ˙ α ,k ](0) χα ,j χα ,k , ϕ˙α ,j ϕα ,k + ϕα ,j ϕ˙α ,k = − = 0. 2

!(j k) slightly, replacing the functions [ϕ˙α ,j ϕα ,k + ϕα ,j ϕ˙α ,k ] for all Finally, one needs to correct V

) ∩ σ (vkk ) by λα ∈ σ (vjj ,j ,k ϕ˙α ϕα + ϕα ,j ϕ˙ α ,k + cα ϕα ,j ϕα ,k with appropriate constants cα , in order to guarantee χ˙ α ,j χα ,k + χα ,j χ˙ α ,k , ϕ˙α ,j ϕα ,k + ϕα ,j ϕ˙α ,k + cα ϕα ,j ϕα ,k = 0.

!(j k) becomes biorthogonal to U !(j k) . After these corrections, V

2

Proposition 3.18. P 0 U (j k) is a Riesz basis of P 0 L2 (0, 1) for all 1 k j N . Proof. Since P 0 U (j k) admits the biorthogonal system, it is sufficient to check that elements of P 0 U (j k) are asymptotically close (say, in 2 -sense) to some unperturbed Riesz basis (note

and vkk , and we that these functions are in one-to-one correspondence with eigenvalues of vjj (j k) (j k) for common eigenvalues). Those u ∈ U that correspond to first have two functions in U eigenvalues λ n,j , λ n,k , n < n , do not affect the asymptotical behavior, so it is sufficient to consider n n0 .


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We need some simple asymptotics. Let λ = π 2 n2 + μ, μ = O(1), and v ∈ L2 (0, 1) be some (scalar) potential. Then (1 − t) cos πn(1 − t) 1 sin πn(1 − t) 1 , + O 2 , χ˙ (t, λ, v) = + O 2 2 πn n 2π n n3 1 1 1 (−1)n n−1 χ (0, λ, v) = (−1) +O + O 3 , χ¨ (0, λ, v) = O 4 , χ(0, ˙ λ, v) = n 2π 2 n2 n n χ(t, λ, v) =

as n → ∞. In particular, ξ(t, λ, v) = χ(t, ˙ λ, v) −

χ¨ (0, λ, v) (1 − t) cos πn(1 − t) 1 . χ(t, λ, v) = + O 2χ(0, ˙ λ, v) 2π 2 n2 n3

If k = j , one obtains 2 2 ,j 2π n [χn,j (t)]2 1 (jj ) P 0 2π 2 n2 · un,j = P 0 = − cos 2πnt + O

,j 2 n [(χn,j ) (0)] and

,j ,j ξ (t)χn,j (t) 1 (jj ) −1 0 n,j 0 P (2πn) · ! un,j = P = −P (1 − t) sin 2πnt + O .

,j n πn[χ˙ n,j (0)]2 0

It is easy to see that the collection

R = cos 2πnt, P 0 (1 − t) sin 2πnt , n 1

(3.22)

is a Riesz basis of P 0 L2 (0, 1). Indeed, all functions ( 12 − t) sin 2πnt, n 1, are linear combinations of cos 2πmt, m 1, since they are symmetric with respect to 12 . Hence,

f, cos 2πnt+∞ n=1

f, P 0 [(1 − t) sin 2πnt]+∞ n=1

=

I A

0 1 2I

f, cos 2πnt+∞ n=1

f, sin 2πnt+∞ n=1

+∞ 0 1 2 and the linear operator f, cos 2πnt+∞ n=1 → f, P [( 2 − t) sin 2πnt]n=1 , f ∈ L (0, 1), is 1 2 2 bounded in , since the operator f → ( 2 − t)f is bounded in L (0, 1). Thus, R is a Riesz basis of P 0 L2 (0, 1) and P 0 U (jj ) is 2 -close to R (note that in both P 0 U (jj ) and R there are exactly 2(n − 1) functions with n < n ). Due to Lemma 3.17, the elements of P 0 U (jj ) are linearly independent. Therefore, P 0 U (jj ) is a Riesz basis of P 0 L2 (0, 1) by the Fredholm Alternative (see, e.g., [43, p. 163]).

,j

,j

−1 Let k < j and n n . Due to [(χn,k ) (χ˙ n,k )−1 ](0) = −(gn,k ) = −2π 2 n2 , one has

,j

(j k)

un,k (t) = −

,k χn,k (t)χn,k (t)

,j

,k χn,k (0)(χn,k ) (0)

,j

=

,k χn,k (t)χn,k (t)

,j

,k 2π 2 n2 χn,k (0)χ˙ n,k (0)

.

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Note that ,j

,j

,j χn,k (t) = χn,j (t) + λ n,k − λ n,j χ˙ n,j (t) + O n−3 and ,j

,j χn,k (0) = λ n,k − λ n,j · χ˙ n,j (0) + O n−4 ,

,j

,j

since χn,j (0) = 0 and χ¨n,j (0) = O(n−4 ). Therefore, (j k)

un,k (t) =

,j

,j

,k

,k (t) (t) χn,j (t)χn,k χ˙ n,j (t)χn,k 1 1 . · + + O λ n,k − λ n,j 2π 2 n2 χ˙ ,j (0)χ˙ ,k (0) 2π 2 n2 χ˙ ,j (0)χ˙ ,k (0) n2 n,k n,k n,j n,j

Thus, P

0

1 1 (j k) cos 2πnt (kj ) un,k − un,j = −

+O 2 λn,k − λn,j n (j k)

and, since the first term of un,k (t) is antisymmetric with respect to j and k, (j k) 1 (kj ) P 0 πn · un,k + un,j = −P 0 (1 − t) sin 2πnt + O . n As above, we see that P 0 U (j k) (up to some uniformly bounded multiplicative constants) is 2 close to the Riesz basis R given by (3.22). So, P 0 U (j k) is a Riesz basis due to the Fredholm alternative and Lemma 3.17. 2 Corollary 3.19. The Fréchet derivative (1) (2) ×N dV Φ = dV Φ (1) ; dV Φ (2) : P 0 L2 [0, 1]; CN → HR ⊕ HR , R

(1) HR

α & kα ×kα k ×k (N −kα )×kα CR , = ⊕ CRα α ⊕ C

(2) ×N ×N HR = 2R Nn ; CN ⊕ CN R R

α=1

is a linear isomorphism (in other words, dV Φ is invertible). Proof. See Remark 3.16 and Proposition 3.18.

2

3.7. Completion of the proof. Changing of the finite number of first residues Let {(λ α , Pα† , gα† )}α1 be some data which satisfy conditions (A)–(C) in Theorem 1.1 and † † −1 Bα† = Pα† (gα† )−1 Pα† . Recall that Pn,j = Pj0 + 2 and (gn,j ) = 2π 2 n2 (1 + 21 ). Similarly to Definition 3.8, if n is sufficiently large, then we may introduce the (unique) factorization 2 2 −1 † † 2 † † ∗ 2π n Bn,j = cn,j · en,j en,j ,

† † † cn,j ∈ R+ , en,j ∈ CN , en,j , ej0 = 1.


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† † Note that en,j = en0 + 2 and cn,j = 1 + 2 . Define

) † † Yn† = cn,1 · en,1

† † ; cn,2 · en,2

; ...

† † ; cn,N · en,N

*

∈ CN ×N .

Since Yn† = IN + 2 , the matrix Yn† is non-degenerate for all sufficiently large n, and we may introduce its (unique) polar decomposition Yn† = Un† Sn† ,

† ∗ † −1 † ∗ Un = Un , Sn = Sn† > 0.

Note that Un† = IN + 2 and Sn† = IN + 2 . By our assumptions, † −1 ) (gn,j

N

† j =1 Pn,j

= IN + 21 and

= 2π 2 n2 (1 + 21 ), so Lemma 2.4 gives N 2 ∗ ∗ −1 † Un† Sn† Un† = Yn† Yn† = 2π 2 n2 Bn,j = IN + 21 . j =1

Therefore, Sn† = IN + 21 . Recall that Un = Un (V ) = IN and Sn = Sn (V ) = IN for all n n , so Φ (2) (V ) = 0. Since the Fréchet derivative dV Φ is invertible, the mapping Φ = (Φ (1) ; Φ (2) ) is a local bijection near 0 (V , r ) such that V . Therefore, if α • is large enough, then there exists some potential V • ∈ BR Φ (1) V • = Φ (1) V ,

Φn(2) V • = Φn(2) V = 0 for all n n n• ,

and Φn(2) V • = −i log Un† ; 2πn · Sn† − IN

for all n n• ,

where α • − α = N (n• − n ) (i.e., α • + 1 corresponds to the double-index (n• , 1)). Since the ! can be reconstructed from Φ, one has original mapping Φ !α V = 0 for all α α • , !α V • = A A • !n,j V • = B † !n,j V • = 0 and B A n,j for all n n . Due to Lemma 3.3, it gives σ V • = λ α α1

† and Bn,j V • = Bn,j

for all n n• . •

•

At last, we need to change the finite number of first residues (Bα (V • ))αα=1 to (Bα† )αα=1 . Recall that the isospectral transforms constructed in [9] allow to modify each particular residue Bα in !α ) an almost arbitrary way. The only one restriction (concerning the change of projector Pα to P is !α = {0}, Fα ∩ Ran P

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where Fα , dim Fα = N − kα is some “forbidden” subspace that is uniquely determined by the spectrum and all other subspaces (Eβ )β=α . It is not hard to conclude (see Proposition A.4) that this restriction is equivalent to the following: One can modify Bα in an arbitrary way such that (C) holds true. In general situation one can change all Bα (V • ) to Bα† by α • steps. Nevertheless, it may happen that at some intermediate step the desired residue Bα† violates (C). In order to overcome this !α† which is arbitrary close to Bα† in the difficulty note that one can always change Bα to some B !• such that natural topology. Then, in any case, after α • steps one can obtain some potential V !• ) = B !α† for all α = 1, . . . , α • (and, of course, Bα (V !• ) = Bα (V • ) = Bα† for all α > α • ). By Bα (V • Corollary A.2, the set of all admitted by (C) sequences (Bα )αα=1 is open in the natural topology. • • !α† )α and (Bα† )α are close enough, then all changes B !α† → Bα† are permitted. Therefore, if (B α=1 α=1 • So, after another at most α steps one obtains the potential V such that Bα (V ) = Bα† for all α = 1, . . . , α • (and still Bα (V ) = Bα† for all α > α • ). The proof is finished. 2 Appendix A. Property (C) Let λα > 0 for all α 1. Note that (C) does not depend on shifts of the spectrum, so we do not lose the generality. We begin with the following simple: Remark A.1. If√ an entire function ξ is bounded on the real positive half-line, then the condition ξ(λ) = O(e|Im λ| ) is equivalent to say that ξ(z2 ) is an entire function of exponential type no greater than 1 (see [26, p. 28]). Recall that the Paley–Wiener space P W[−1,1] consists of all entire functions f (z) of exponential type no greater than 1 such that f ∈ L2 (R). The Paley–Wiener theorem (see [26, p. 30]) claims

f ∈ P W[−1,1]

iff

1 f (z) = 2π

1

φ(t)e−izt dt,

where φ ∈ L2 (−1, 1).

(A.1)

−1

Proof of Proposition 1.3. If φ ∈ L2 ([−1, 1]; CN ) is some vector-valued function such that 1 φ(t) dt = 0 and

h∗α

−1

1

φ(t)e±i

√

λα t

dt = 0 for all α 1,

−1

then 1 2π

1 −1

φ(t)e−izt dt = zf (z)

+ and Pα f (± λα ) = 0 for all α 1,

(A.2)


where zf (z) ∈ P W[−1,1] . Denote ξ(z2 ) = 12 [f (z) + f (−z)] or ξ(z2 ) = √

1581

1 2z [f (z) − f (−z)]. Then,

λ| ) and ξ ∈ L2 (R ). This contradicts to (C). Pα ξ(λα ) = 0, α 1, ξ(λ) = O(e|Im + √ Conversely, let ξ(λ) = O(e|Im λ| ) and ξ ∈ L2 (R+ ). Then f (z) = zξ(z2 ) ∈ P W[−1,1] , so it admits representation (A.1) with some φ ∈ L2 (−1, 1). It is easy to check that Pα ξ(λα ) = 0 and f (0) = 0 imply (A.2). Hence, φ ≡ 0. 2

We have the immediate: Corollary A.2. If one fixes the spectrum {λα }α1 and all projectors Pα , α α • + 1, for some • α • 0, then the set of all finite sequences (Pα )αα=1 satisfying the condition (C) is open in the natural topology. Introduce the function

ξβ (λ) ≡

χ(0, λ, V )Pβ λ − λβ

(A.3)

,

where Pβ : CN → Eβ is the orthogonal projector onto the subspace Eβ = Ker χ(0, λβ , V ). ×N ). Then: Proposition A.3. Let β 1 and V = V ∗ ∈ L2 ([0, 1]; CN R √

(i) ξβ : C → CN ×N is an entire matrix-valued function, ξβ (λ) = O(e|Im λ| ) as |λ| → ∞, ξβ ∈ L2 (R+ ) and Pα ξβ (λα ) = 0 for all α = β. √ (ii) If ξ : C → CN is an entire vector-valued function such that ξ(λ) = O(e|Im λ| ) as |λ| → ∞, ξ ∈ L2 (R+ ) and Pα ξ(λα ) = 0 for all α = β, then ξ = ξβ h for some h ∈ CN .

Proof. (i) The function ξβ is entire due to χ(0, λβ , V )Pβ = 0. Furthermore, √ 3 ξβ (λ) = O |λ|− 2 e|Im λ|

as |λ| → ∞ and Pα ξβ (λα ) = 0 for all α = β,

since Pα χ(0, λα ) = Pα [ϕ(1, λα )]∗ = [ϕ(1, λα )Pα ]∗ = 0. (ii) Lemma 2.2 [9] claims −1 −1 −1 ∗ = ϕ (1, λ, V ) = Zα + O(λ − λα ) (λ − λα )−1 Pα + Pα⊥ χ(0, λ, V )

as λ → λα

for some Zα , α = β, such that det Zα = 0 and −1 −1 χ(0, λ, V ) = ϕ 1, λ, V ⊥ −1 Zβ + O(λ − λβ ) = (λ − λβ )−1 Pβ + Pβ

as λ → λβ ,

for some Zβ , det Zβ = 0. Due to Pα ξ(λα ) = 0, α = β, the (vector-valued) function −1 ω(λ) = χ(0, λ, V ) ξ(λ)

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is analytic except λβ and ω(λ) = (λ − λβ )−1 Pβ h + O(1) as λ → λβ for some h ∈ CN . Since ω(λ) = O(|λ|1/2 ) as |λ| = π 2 (n + 12 )2 → ∞, the Liouville theorem gives

ξ(λ) ≡ χ(0, λ, V ) (λ − λβ )−1 Pβ h + ω0 ≡ ξβ (λ)h + χ(0, λ, V )ω0 Finally, ξ ∈ L2 (R+ ) implies ω0 = 0. V∗

for some ω0 ∈ CN .

2

Recall the construction of the “forbidden” subspaces Fα ⊂ CN , α 1, given in [9]. Let V = ∈ L2 ([0, 1]; CN ×N ). For each α 1 denote ⊥ Fα = Sα (Eα ) ,

1 where Sα = Sα (V ) =

∗ ϕ ϕ (t, λα , V ) dt = Sα∗ > 0

0

and Eα = Ran Pα . Note that dim Fα = N − dim Eα = N − kα . The main result of [9] is that one can modify each particular projector Pα (keeping the spectrum and all other projectors fixed) in an arbitrary way such that Fα ∩ Ran Pα = {0}. It is quite natural that this restriction is equivalent to property (C) as shows: Proposition A.4 (Connection between subspaces Fα and property (C)). Let β 1 and +∞ N ×N ∗ 2 ). Then, the collection (λα ; Pα )+∞ α=1 = (λα (V ); Pα (V ))α=1 for some V = V ∈ L ([0, 1]; CR +∞ ! ! (λα ; Pα )α=1 , where Pα = Pα for all α = β, satisfies (C) iff !β = {0}, Fβ ∩ Ran P

⊥ where Fβ = Sβ (Eβ ) .

(A.4)

Moreover, Fβ = [Ran ξβ (λβ )]⊥ , where ξβ is given by (A.3). !α )+∞ Proof. It follows from Proposition A.3(ii) that (C) holds true for the new collection (λα ; P α=1 !β ξβ (λβ )h = 0 for all h ∈ E , h = 0. In other words, (C) is equivalent to if and only if P β !β = {0}. Ran ξβ (λβ ) ∩ Ker P

(A.5)

One has (see Lemmas 2.4 and 2.1 of [9] for details) ξβ (λβ ) = χ˙ (0, λβ )Pβ = ϕ˙ ∗ (1, λβ )Pβ = −ϕ˙ ∗ (1, λβ )ϕ (1, λβ )χ (0, λβ )Pβ .

Moreover, Ran χ (0, λβ )Pβ = Eβ and

Ran ξβ (λβ ) = Ran ϕ˙ ∗ (1, λβ )ϕ (1, λβ )Pβ = Ran Sβ Pβ = Sβ (Eβ ). !β = N − kβ = N − dim Sβ (Eβ ), (A.5) is equivalent to (A.4). Since dim Ker P

2

We finish our discussion by the consideration of the special case when only finite number of Pα differ from the standard unperturbed coordinate projectors.


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Let A = {α1 , α2 , . . . , αm } be some finite set of exceptional indices. Assume that Pα = Pα0 coincides with some coordinate projector Pα0 for all α ∈ / A (we admit multiple eigenvalues). Introduce the sets

A0j = α ∈ / A: Pα ej0 = 0 (possible multiple eigenvalues belong to several A0j ). Assume that there exists C > 0 such that the set {λα , α ∈ A0j } ∩ (−∞, π 2 n2 + C] consists of exactly n − m points for all j = 1, 2, . . . , N , if n is large enough. Let kα1 + kα2 + · · · + kαm = N m. We give the simple description of all finite sequences (Pαs )m s=1 , rank Pα = kα , such that the whole collection {(λα ; Pα )}+∞ satisfies (C): α=1 Proposition A.5. Let (λα ; Pα )+∞ α=1 be as described above. Then (C) holds true iff ⎛ ⎜ ⎜ T =⎜ ⎝

T0 T1 .. .

T1 T2 .. .

··· ··· .. .

Tm−1

Tm

· · · T2m−2

Tm−1 Tm .. .

⎞ ⎟ ⎟ ⎟ = T ∗ > 0, ⎠

where Tk =

λkα F (λα )Pα F (λα ) = Tk∗ ,

k = 0, 1, . . . , 2m − 2,

α∈A

F (λ) ≡ diag f1 (λ), f2 (λ), . . . , fN (λ)

and fj (λ) ≡

λ 1 − . α∈A0j λα

,

Remark A.6. Since T 0 in any case, the condition T > 0 is equivalent to det T = 0. √

Proof. Indeed, let ξ(λ) = (ξ1 (λ), ξ2 (λ), . . . , ξN (λ)) be such that ξ(λ) = O(e|Im λ| ), ξ ∈ L2 (R+ ) and Pα ξ(λα ) = 0 for all α 1. In particular, Pj0 ξ(λα ) = 0 for all α ∈ A0j . In order words, zξj (z2 ) ∈ P W[−1,1] and ξj (λα ) = 0 for all α ∈ A0j . Therefore, ξj (λ) ≡ Qj (λ)fj (λ),

deg Qj m − 1,

for some polynomials Qj . Let m−1 λp yp , Q(λ) = Q1 (λ), Q2 (λ), . . . , QN (λ) = p=0

Nm yp ∈ CN , and y = (yp )m−1 . p=0 ∈ C

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Then, y∗T y =

m−1

yp∗ Tp+q yq =

p,q=0

m−1 p,q=0

yp∗

λp+q F (λ )P F (λ ) yq α α α α

α∈A

∗ ∗ Q(λα ) F (λα )Pα F (λα )Q(λα ) = ξ(λα ) Pα ξ(λα ). = a∈A

α∈A

Hence, the Nm × N m matrix T is degenerate iff there exists ξ such that Pα ξ(λα ) = 0 for all / A by the construction). 2 α ∈ A (recall that Pα0 ξ(λα ) = 0 holds true for all α ∈ Appendix B. Three classical choices of additional spectral data in the scalar case In the scalar case, it is well known that the Dirichlet spectrum σ (q) = {λn (q)}+∞ n=1 determines only “one half” of the potential q. Thus, in order to determine q uniquely, one needs either to assume that some partial information about q is known or to consider some additional spectral data besides σ (q). Note that there are two classical assumptions about the potential that make the knowledge of the spectrum sufficient: symmetry q(x) ≡ q(1 − x) (see, e.g., [43]) or the knowledge of q(x) as x ∈ [0, 12 ] (the Hochstadt–Lieberman theorem [20], see also [18,21,35] for generalizations available in the scalar case and [36] for the vector-valued case). Also, there are several classical choices of additional spectral data: (1) The second spectrum. This setup goes back to the original paper of Borg [2]. The most natural choice is the spectrum {μn (q)}n=1 of the mixed problem −y + qy = λy,

y(0) = y (1) = 0.

+∞ Note that {μn (q)}+∞ n=1 ∪ {λn (q)}n=1 is the Dirichlet spectrum of the symmetric potential q(2 − x) ≡ q(x), x ∈ [0, 1], defined on the doubled interval [0, 2]. (2) The normalizing constants (firstly appeared in Marchenko’s paper [38])

−1 = αn (q)

1

−1 2

ϕ (x, λn ) dx 0

= [ϕϕ ˙ ]−1 (1, λn ) = −

χ (0, λn ) = − res m(λ). λ=λn χ˙ (0, λn )

(3) The norming constants introduced by Trubowitz and co-authors (see [43]) n n−1 ϕ(·, λn ) . νn (q) = log (−1) ϕ (1, λn ) = log (−1) χ(·, λn ) It is quite well known in the folklore that the characterization problems in the setups (1)–(3) are equivalent. Unfortunately, we do not know the good reference for this fact. So, the main purpose of this appendix is to give the short proof of these equivalences (note that our arguments 1 are quite similar to [32]). For the simplicity, we assume that q ∈ L2 (0, 1), 0 q(x) dx = 0, i.e., 2 {λn (q) − π 2 n2 }+∞ n=1 ∈ (the similar arguments work well for other classes of potentials and corresponding classes of spectral data).


1585

Note that 1 2 and μn = π 2 n − + O(1) 2

μ 1 < λ1 < μ2 < λ2 < μ2 < · · ·

as n → ∞.

(B.1)

Also, the Hadamard factorization implies f (λ) = ϕ(1, λ) =

+∞ , m=1

λm − λ π 2 m2

and g(λ) = ϕ (1, λ) =

+∞ ,

μm − λ . 2 π (m + 12 )2 m=1

(B.2)

2 Recall that we write an = bn + 2k iff {nk |an − bn |}+∞ n=1 ∈ .

Proposition B.1. Let λn = π 2 n2 + 2 , (B.1) hold and f (λ), g(λ) be given by (B.2). Then, the following conditions are equivalent: (1) The asymptotics μn = π 2 (n − 12 )2 + 2 hold true. (2) The asymptotics αn = g(λn )f˙(λn ) = (2π 2 n2 )−1 (1 + 21 ) hold true. (3) The asymptotics νn = log[(−1)n g(λn )] = 21 hold true. Proof. We start with the equivalence (2) ⇔ (3). Denote ! λn = π −2 λn − n2 = O(1) as n → ∞. Then ! λm − ! (−1)n , λn 1 , λm − λn (−1)n , λm − λn ˙ 1+ 2 . = = f (λn ) = − 2 2 π n π 2 m2 2π 2 n2 π 2 (m2 − n2 ) 2π 2 n2 m − n2 m=n

m=n

m=n

Note that log

, m=n

! ! λm − ! λm − ! λn λn 1 1+ 2 = +O m − n2 m2 − n2 (m2 − n2 )2 m=n

=

! λm − ! λn

1 +O 2 2 2 m −n n

m=n

=

m=n

! 1 λm +O 2 . 2 2 m −n n

2 Then, it immediately follows from (! λn )+∞ n=1 ∈ and simple properties of the discrete Hilbert transform (see Lemma B.2(ii) below) that f˙(λn ) = (−1)n (2π 2 n2 )−1 (1 + 21 ). Thus, (2) ⇔ (3). The proof of the equivalence (1) ⇔ (3) is similar. Indeed,

g(λn ) =

+∞ ,

μ m − λn

π 2 (m − 12 )2 m=1

= (−1)n

+∞ ,

1+

m=1

= (−1)n

+∞ ,

μm − λn 2 π ((m − 12 )2 − n2 ) m=1

! μm − ! λn (m − 12 )2 − n2

,

where ! μm = π −2 μm − (m + 12 )2 = O(1) as m → ∞. As above,

1586

D. Chelkak, E. Korotyaev / Journal of Functional Analysis 257 (2009) 1546–1588 +∞ log (−1)n g(λn ) =

! μm − ! λn

1 +O 2 1 2 2 n (m − 2 ) − n m=1

+∞

! μm

1 = +O 2 1 2 2 n (m − 2 ) − n m=1

and the equivalence (1) ⇔ (3) follows by Lemma B.2(i).

2

Lemma B.2. +∞ (i) The linear operator (am )+∞ m=1 → (bn )n=1 , where

bn =

+∞ +∞ am am 1 1 = 2 − (m − 1 )2 2πn π n n − m+ 2 m=1 m=1

1 2

+

am n − (1 − m) +

, 1 2

is an isometry in 2 . +∞ (ii) The linear operator (am )+∞ m=1 → (bn )n=1 , where +∞ +∞ am am 1 am + , = bn = 2n n − m n − (−m) n2 − m2 m=1

m=1

is bounded in 2 . Proof. Both results easily follows by the Fourier transform and the identities (in L2 (T)) +∞ k=−∞

ζk k+

1 2

iφ πi = √ = πie− 2 ζ

where ζ = eiφ = 1, φ ∈ (0, 2π).

and

ζk k=0

k

= −i(φ − π),

2

Remark B.3. The similar technique can be applied for other inverse problems in order to derive the characterization of some additional spectral parameters (e.g., similar to αn (q)) from the characterization of other parameters (e.g., similar to νn (q)). In general, these characterizations may differ from each other substantially, see [10]. Acknowledgments Some parts of this paper were written at Mathematisches Forschungsinstitut Oberwolfach, Institut fu¨r Mathematik Humboldt-Universität zu Berlin and Section de Mathématiques Université de Genève. The authors are grateful to the Institutes for the hospitality. The stay of the authors at the MFO was provided by the Oberwolfach-Leibniz Fellowship of the first author. The authors are grateful to the MFO for its stimulating atmosphere. References [1] Z.S. Agranovich, V.A. Marchenko, The Inverse Problem of Scattering Theory, Translated from the Russian by B.D. Seckler, Gordon and Breach Science Publishers, New York, 1963, xiii+291 pp.


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[2] G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946) 1–96 (in German). [3] F. Calogero, A. Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform, I, Nuovo Cimento B (11) 32 (2) (1976) 201–242. [4] F. Calogero, A. Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform, II, Nuovo Cimento B (11) 39 (1) (1977) 1–54. [5] R. Carlson, An inverse problem for the matrix Schrödinger equation, J. Math. Anal. Appl. 267 (2) (2002) 564–575. [6] S. Clark, F. Gesztesy, H. Holden, B.M. Levitan, Borg-type theorems for matrix-valued Schrödinger operators, J. Differential Equations 167 (1) (2000) 181–210. [7] D. Chelkak, P. Kargaev, E. Korotyaev, Inverse problem for harmonic oscillator perturbed by potential, characterization, Comm. Math. Phys. 249 (1) (2004) 133–196. [8] D. Chelkak, E. Korotyaev, Spectral estimates for Schrödinger operators with periodic matrix potentials on the real line, Int. Math. Res. Not. (2006), Art. ID 60314, 41 pp. [9] D. Chelkak, E. Korotyaev, Parametrization of the isospectral set for the vector-valued Sturm–Liouville problem, J. Funct. Anal. 241 (1) (2006) 359–373. [10] D. Chelkak, E. Korotyaev, The inverse problem for perturbed harmonic oscillator on the half-line with a Dirichlet boundary condition, Ann. Henri Poincaré 8 (6) (2007) 1115–1150. [11] Hua-Huai Chern, Chao-Liang Shen, On the n-dimensional Ambarzumyan’s theorem, Inverse Problems 13 (1) (1997) 15–18. [12] B.E.J. Dahlberg, E. Trubowitz, The inverse Sturm–Liouville problem, III, Comm. Pure Appl. Math. 37 (2) (1984) 255–267. [13] S. Dineen, Complex Analysis on Infinite-Dimensional Spaces, Springer Monogr. Math., Springer-Verlag London, Ltd., London, 1999, xvi+543 pp. [14] H. Flaschka, D.W. McLaughlin, Canonically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys. 55 (2) (1976) 438–456. [15] G. Freiling, V. Yurko, Inverse Sturm–Liouville Problems and Their Applications, Nova Science Publishers, Inc., Huntington, NY, 2001, x+356 pp. [16] I.M. Gel’fand, B.M. Levitan, On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951) 309–360 (in Russian); English translation: Amer. Math. Soc. Transl. (2) 1 (1955) 253–304. [17] F. Gesztesy, Inverse spectral theory as influenced by Barry Simon, in: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Part 2, in: Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 741–820. [18] F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, Trans. Amer. Math. Soc. 352 (6) (2000) 2765–2787. [19] J.G. Guillot, J.V. Ralston, Inverse spectral theory for a singular Sturm–Liouville operator on [0, 1], J. Differential Equations 76 (1988) 353–373. [20] H. Hochstadt, B. Lieberman, An inverse Sturm–Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (4) (1978) 676–680. [21] M. Horvath, Inverse spectral problems and closed exponential systems, Ann. of Math. (2) 162 (2) (2005) 885–918. [22] E. Isaacson, H. McKean, E. Trubowitz, The inverse Sturm–Liouville problem, II, Comm. Pure Appl. Math. 37 (1) (1984) 1–11. [23] E. Isaacson, E. Trubowitz, The inverse Sturm–Liouville problem, I, Comm. Pure Appl. Math. 36 (6) (1983) 767– 783. [24] M. Jodeit Jr., B.M. Levitan, Isospectral vector-valued Sturm–Liouville problems, Lett. Math. Phys. 43 (2) (1998) 117–122. [25] Max Jodeit Jr., B.M. Levitan, A characterization of some even vector-valued Sturm–Liouville problems, Zh. Mat. Fiz. Anal. Geom. 5 (3–4) (1998) 166–181. [26] P. Koosis, The Logarithmic Integral, I, Cambridge Stud. Adv. Math., vol. 12, Cambridge University Press, Cambridge, 1988, xvi+606 pp. [27] M.G. Kre˘ın, Solution of the inverse Sturm–Liouville problem, Doklady Akad. Nauk SSSR (N.S.) 76 (1951) 21–24 (in Russian). [28] M.G. Kre˘ın, On the transfer function of a one-dimensional boundary problem of the second order, Doklady Akad. Nauk SSSR (N.S.) 88 (1953) 405–408 (in Russian). [29] M.G. Kre˘ın, On a method of effective solution of an inverse boundary problem, Doklady Akad. Nauk SSSR (N.S.) 94 (1954) 987–990 (in Russian).

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Realizations of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras Takeshi Katsura a , Aidan Sims b , Mark Tomforde c,∗ a Department of Mathematics, Keio University, Yokohama, 223-8522, Japan b School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia c Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA

Received 22 October 2008; accepted 1 May 2009 Available online 2 June 2009 Communicated by Alain Connes

Abstract We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C ∗ -algebra, an Exel–Laca algebra, and an ultragraph C ∗ -algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C ∗ -algebras and Exel–Laca algebras, and that all simple AF-algebras are either graph C ∗ -algebras or Exel–Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C ∗ -algebra of a row-finite graph with no sinks. © 2009 Elsevier Inc. All rights reserved. Keywords: Graph C ∗ -algebras; Exel–Laca algebras; Ultragraph C ∗ -algebras; AF-algebras; Bratteli diagrams

Contents 1. 2.

Introduction . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . 2.1. Graph C ∗ -algebras . . . 2.2. Exel–Laca algebras . . . 2.3. Ultragraph C ∗ -algebras

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E-mail addresses: [email protected] (T. Katsura), [email protected] (A. Sims), [email protected] (M. Tomforde). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.002

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2.4. AF-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Realizations of AF-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. A construction of an ultragraph from a certain type of Bratteli diagram . 4.2. Sufficient conditions for realizations . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Obstructions to realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. A summary of known containments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Simple AF-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. More general AF-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. 4.

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1. Introduction In 1980 Cuntz and Krieger introduced a class of C ∗ -algebras constructed from finite matrices with entries in {0, 1} [4]. These C ∗ -algebras, now called Cuntz–Krieger algebras, are intimately related to the dynamics of topological Markov chains, and appear frequently in many diverse areas of C ∗ -algebra theory. Cuntz–Krieger algebras have been generalized in a number of ways, and two very natural generalizations are the graph C ∗ -algebras and the Exel–Laca algebras. For graph C ∗ -algebras one views a {0, 1}-matrix as an edge adjacency matrix of a graph, and considers the Cuntz–Krieger algebras as C ∗ -algebras of certain finite directed graphs. For a (not necessarily finite) directed graph E, one then defines the graph C ∗ -algebra C ∗ (E) as the C ∗ algebra generated by projections pv associated to the vertices v of E and partial isometries se associated to the edges e of E that satisfy relations determined by the graph. Graph C ∗ -algebras were first studied using groupoid methods [17,18]. Due to technical constraints, the original theory was restricted to graphs that are row-finite and have no sinks; that is, the set of edges emitted by each vertex is finite and nonempty. In fact much of the early theory restricted to this case [2,17,18], and it was not until later [1,8,12] that the theory was extended to infinite graphs that are not row-finite. Interestingly, the non-row-finite setting is significantly more complicated than the row-finite case, with both new isomorphism classes of C ∗ -algebras and new kinds of C ∗ -algebraic phenomena exhibited. Another approach to generalizing the Cuntz–Krieger algebras was taken by Exel and Laca, who defined what are now called the Exel–Laca algebras [10]. In this definition one allows a possibly infinite matrix with entries in {0, 1} and considers the C ∗ -algebra generated by a set of partial isometries indexed by the rows of the matrix and satisfying certain relations determined by the matrix. The construction of the Exel–Laca algebras contains the Cuntz–Krieger construction as a special case. Furthermore, for row-finite matrices (i.e., matrices in which each row contains a finite number of nonzero entries) with nonzero rows, the construction produces exactly the class of C ∗ -algebras of row-finite graphs with no sinks. Despite the fact that the classes of graph C ∗ -algebras and Exel–Laca algebras agree in the row-finite case, they are quite different in the non-row-finite setting. In particular, there are C ∗ algebras of non-row-finite graphs that are not isomorphic to any Exel–Laca algebra, and there are Exel–Laca algebras of non-row-finite matrices that are not isomorphic to the C ∗ -algebra of any graph [23]. In order to bring graph C ∗ -algebras and Exel–Laca algebras together under one theory, Tomforde introduced the notion of an ultragraph and described how to associate a C ∗ algebra to such an object [22,23]. These ultragraph C ∗ -algebras contain all graph C ∗ -algebras


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Fig. 1. The relationship among graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ -algebras.

and all Exel–Laca algebras, as well as examples of C ∗ -algebras that are in neither of these two classes. The relationship among these classes is summarized in Fig. 1. Given the relationship among these classes of C ∗ -algebras, it is natural to ask the following question. Question. “How different are the C ∗ -algebras in the three classes of graph C ∗ -algebras, Exel– Laca algebras, and ultragraph C ∗ -algebras?” There are various ways to approach this question, and one such approach was taken in [16], where it was shown that the classes of graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ -algebras agree up to Morita equivalence. More specifically, given a C ∗ -algebra A in any of these three classes, one can always find a row-finite graph E with no sinks such that C ∗ (E) is Morita equivalent to A. Thus the three classes cannot be distinguished by Morita equivalence classes of C ∗ -algebras. The natural next question is to what extent they can be distinguished by isomorphism classes of C ∗ -algebras. A starting point for these investigations is to ask about AF-algebras. While no Cuntz–Krieger algebra is an AF-algebra, the classes of graph C ∗ -algebras and Exel– Laca algebras each include many AF-algebras. In fact, one of the early results in the theory of graph C ∗ -algebras shows that if A is any AF-algebra, then there is a row-finite graph E with no sinks such that C ∗ (E) is Morita equivalent to A [7]. From this fact and the result in [16] mentioned above, our three classes (graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ algebras) each contain all AF-algebras up to Morita equivalence. The purpose of this paper is to examine the three classes of graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ -algebras and determine which AF-algebras are contained, up to isomorphism, in each class. This turns out to be a difficult task, and we are unable to give a complete solution to the problem. Nonetheless, we are able to give a number of sufficient conditions and

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a number of necessary conditions for a given AF-algebra to belong to each of these three classes (see Sections 4.2 and 4.3). As special cases of our sufficient conditions, we obtain the following. • If A is a stable AF-algebra, then A is isomorphic to the C ∗ -algebra of a row-finite graph with no sinks. • If A is a simple AF-algebra, then A is isomorphic to either an Exel–Laca algebra or a graph C ∗ -algebra. In particular, if A is finite dimensional, then A is isomorphic to a graph C ∗ algebra; and if A infinite dimensional, then A is isomorphic to an Exel–Laca algebra. • If A is an AF-algebra with no nonzero finite-dimensional quotients, then A is isomorphic to an Exel–Laca algebra. From our necessary conditions, we obtain the following. • If an ultragraph C ∗ -algebra is a commutative AF-algebra then it is isomorphic to c0 (X) for an at most countable discrete set X. • No finite-dimensional C ∗ -algebra is isomorphic to an Exel–Laca algebra. • No infinite-dimensional UHF algebra is isomorphic to a graph C ∗ -algebra. Moreover, we are able to give a characterization of AF-algebras that are isomorphic to C ∗ algebras of row-finite graphs with no sinks in Theorem 4.7. Theorem. Let A be an AF-algebra. Then the following are equivalent: (1) A has no unital quotients. (2) A is isomorphic to the C ∗ -algebra of a row-finite graph with no sinks. Our results allow us to make a fairly detailed analysis of the AF-algebras in each of our three classes, and in Fig. 2 at the end of this paper we draw a Venn diagram relating various classes of AF-algebras among the graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ -algebras. Our results are powerful enough that we are able to give examples in each region of the Venn diagram, and also state definitively whether or not there are unital and nonunital examples in each region. Finally, we remark that a particularly useful aspect of our sufficiency results is their constructive nature. When one first approaches the problem of identifying which AF-algebra are in our three classes, one may be tempted to use the K-theory classification of AF-algebras. There are, however, two problems with this approach: (1) Since any AF-algebra is Morita equivalent to the C ∗ -algebra of a row-finite graph with no sinks, we know that all ordered K0 -groups are attained by the AF-algebras in each of our three classes. Thus we need to identify which scaled ordered K0 -groups are attained by the AF-algebras in each class. Unfortunately, however, little is currently known about the scale for the K0 -groups of C ∗ -algebras in these three classes. (2) More importantly, even if we could decide exactly which scaled ordered K0 -groups are attained by, for example, graph AF-algebras, we would obtain at best an abstract characterization of which AF-algebras are graph C ∗ -algebras. Unless our understanding of the scaled ordered K0 -groups achieved by AF graph C ∗ -algebras extended to an algorithm for producing a graph whose C ∗ -algebra achieved a given scaled ordered K0 -group, we would be unable to take a given AF-algebra A and view it as a graph C ∗ -algebra. Most notably, we could not expect to “see” the canonical generators of C ∗ (E) in A.


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With an awareness of the limitations of an abstract characterization, we instead present constructive methods for realizing AF-algebras as C ∗ -algebras in our three classes. Given a certain type of AF-algebra A we show how to build an ultragraph G from a certain type of Bratteli diagram for A so that C ∗ (G) is isomorphic to A (see Section 4.1). This ultragraph C ∗ -algebra is always an Exel–Laca algebra, and in special situations (see Section 4.2) it is also a graph C ∗ -algebra. Furthermore, one can extract from G a {0, 1}-matrix for the Exel–Laca algebra or a directed graph for the graph C ∗ -algebra as appropriate. This paper is organized as follows. In Section 2 we establish definitions and notation for graph C ∗ -algebras, Exel–Laca algebras, ultragraph C ∗ -algebras, and AF-algebras. In Section 3 we establish some technical lemmas regarding Bratteli diagrams and inclusions of finite-dimensional C ∗ -algebras. In Section 4 we state the main results of this paper. Specifically, in Section 4.1 we describe how to take a Bratteli diagram for an AF-algebra A with no nonzero finite-dimensional quotients and build an ultragraph G. In Section 4.2 we prove that the associated ultragraph C ∗ -algebra C ∗ (G) is isomorphic to A. We also show that C ∗ (G) is always isomorphic to an Exel–Laca algebra, and describe conditions which imply C ∗ (G) is also a graph C ∗ -algebra. These results give us a number of sufficient conditions for AF-algebras to be contained in our three classes of graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ -algebras. We also present examples showing that none of our sufficient conditions are necessary. In Section 4.3 we give several necessary conditions for AF-algebras to be in each of our three classes. These conditions allow us to identify a number of obstructions to realizations of various AF-algebras in each class. We conclude in Section 5 by summarizing our containments. First, we characterize precisely which simple AF-algebras fall into each of our classes. Second, we summarize many of the relationships we have derived, including containments for the finite-dimensional and stable AF-algebras, and draw a Venn diagram to represent these containments. We are able to use our results from Section 4 to exhibit examples in each region of the Venn diagram, thereby showing these regions are nonempty. We are also able to describe precisely when unital and nonunital examples occur in these regions. 2. Preliminaries In the following four subsections we establish definitions and notation for graph C ∗ -algebras, Exel–Laca algebras, ultragraph C ∗ -algebras, and AF-algebras. Since the literature for each of these classes of C ∗ -algebras is large and well developed, we present only the definitions and notation required in this paper. However, for each class we provide introductory references where more detailed information may be found. 2.1. Graph C ∗ -algebras Introductory references include [2,20,25]. Definition 2.1. A graph E = (E 0 , E 1 , r, s) consists of a countable set E 0 of vertices, a countable set E 1 of edges, and maps r : E 1 → E 0 and s : E 1 → E 0 identifying the range and source of each edge. A path in a graph E = (E 0 , E 1 , r, s) is a sequence of edges α := e1 . . . en with s(ei+1 ) = r(ei ) for 1 i n − 1. We say that α has length n. We regard vertices as paths of length 0 and edges as paths of length 1, and we then extend our notation for the vertex set and the edge set by writing

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n E n for the set of paths of length n for all n 0. We write E ∗ for the set ∞ n=0 E of paths of ∗ finite length, and extend the maps r and s to E by setting r(v) = s(v) = v for v ∈ E 0 , and r(α1 . . . αn ) = r(αn ) and s(α1 . . . αn ) = s(α1 ). If α and β are elements of E ∗ such that r(α) = s(β), then αβ is the path of length |α| + |β| obtained by concatenating the two. Given α, β ∈ E ∗ , and a subset X of E ∗ , we let αXβ := {γ ∈ E ∗ : γ = αγ β for some γ ∈ X}. So when v and w are vertices, we have vX = γ ∈ X: s(γ ) = v , Xw = γ ∈ X: r(γ ) = w , and vXw = γ ∈ X: s(γ ) = v and r(γ ) = w . In particular, vE 1 w denotes the set of edges from v to w and |vE 1 w| denotes the number of edges from v to w. We say a vertex v is a sink if vE 1 = ∅ and an infinite emitter if vE 1 is infinite. A graph is called row-finite if it has no infinite emitters. Definition 2.2 (Graph C ∗ -algebras). If E = (E 0 , E 1 , r, s) is a graph, then the graph C ∗ -algebra C ∗ (E) is the universal C ∗ -algebra generated by mutually orthogonal projections {pv : v ∈ E 0 } and partial isometries {se : e ∈ E 1 } with mutually orthogonal ranges satisfying (1) se∗ se =pr(e) for all e ∈ E 1 ; (2) pv = e∈vE 1 se se∗ for all v ∈ E 0 such that 0 < |vE 1 | < ∞; (3) se se∗ ps(e) for all e ∈ E 1 . We write v w to mean that there is a path α ∈ E ∗ such that s(α) = v and r(α) = w. A cycle in a graph E is a path α ∈ E ∗ of nonzero length with r(α) = s(α). [17, Theorem 2.4] says that C ∗ (E) is an AF-algebra if and only if E has no cycles. 2.2. Exel–Laca algebras Introductory references include [10–12,21]. Definition 2.3 (Exel–Laca algebras). Let I be a finite or countably infinite set, and let A = {A(i, j )}i,j ∈I be a {0, 1}-matrix over I with no identically zero rows. The Exel–Laca algebra OA is the universal C ∗ -algebra generated by partial isometries {si : i ∈ I } with commuting initial projections and mutually orthogonal range projections satisfying si∗ si sj sj∗ = A(i, j )sj sj∗ and 1 − sy∗ sy = sx∗ sx A(X, Y, j )sj sj∗ (2.1) x∈X

j ∈I

y∈Y

whenever X and Y are finite subsets of I such that X = ∅ and the function j ∈ I → A(X, Y, j ) := 1 − A(y, j ) A(x, j ) x∈X

y∈Y

is finitely supported. (We interpret the unit in (2.1) as the unit in the multiplier algebra of OA .)


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We will see in Remark 2.10 that for a {0, 1}-matrix A with no identically zero rows, the canonical ultragraph GA of A satisfies C ∗ (GA ) ∼ = OA . With this notation, [23, Theorem 4.1] implies that the Exel–Laca algebra OA is an AF-algebra if and only if GA has no cycle. The latter condition can be restated as: there does not exist a finite set {i1 , . . . , in } ⊆ I with A(ik , ik+1 ) = 1 for all 1 k n − 1 and A(in , i1 ) = 1. It is well known that the class of graph C ∗ -algebras of row-finite graphs with no sinks and the class of Exel–Laca algebras of row-finite matrices coincide. However, we have been unable to find a reference, so we give a proof here. Lemma 2.4. The class of graph C ∗ -algebras of row-finite graphs with no sinks and the class of Exel–Laca algebras of row-finite matrices coincide. In particular, (1) If E = (E 0 , E 1 , r, s) is a row-finite graph with no sinks, and if we define a {0, 1}-matrix AE over E 1 by

AE (e, f ) :=

1 0

if r(e) = s(f ), otherwise,

then AE is a row-finite matrix with no identically zero rows and C ∗ (E) ∼ = OAE . (2) If A is a row-finite {0, 1}-matrix over I with no identically zero rows, and if we define a graph 0 := I and drawing an edge from v ∈ I to w ∈ I if and only if A(v, w) = 1, EA by setting EA then EA is a row-finite graph with no sinks and OA ∼ = C ∗ (EA ). Proof. For (1) let E = (E 0 , E 1 , r, s) be a row-finite graph with no sinks, and define the matrix 1 AE as above. Since E is row-finite, AE is also row-finite. Let {Se : e∗ ∈ E } be a generating Exel– 0 Laca AE -family in OAE . For v ∈ E we define Pv := s(e)=v Se Se in OAE . (Note that this sum is always finite since AE is row-finite.) We now show that {Se , Pv : e ∈ E 1 , v ∈ E 0 } is a Cuntz– Krieger E-family in OAE . The Se ’s have mutually orthogonal range projections by the Exel–Laca relations, and hence the Pv ’s are also mutually orthogonal projections. In addition, conditions (2) and (3) in the definition of graph C ∗ -algebras obviously hold from our definition of Pv . It remains to show condition (1) holds. If e ∈ E 1 , let X := {e} and Y := ∅. Then for j ∈ E 1 , we have AE (X, Y, j ) := 1 if and only if s(j ) = r(e). Since j → AE (X, Y, j ) E is row-finite, the function is finitely supported, and (2.1) gives Se∗ Se = j ∈E 1 A(X, Y, j )Sj Sj∗ = s(j )=r(e) Sj Sj∗ = Pr(e) , so condition (1) holds. Thus {Se , Pv : e ∈ E 1 , v ∈ E 0 } is a Cuntz–Krieger E-family, and by the universal property of C ∗ (E) we obtain a ∗-homomorphism φ : C ∗ (E) → OAE with φ(se ) = Se and φ(pv ) = Pv where {se , pv } is a generating Cuntz–Krieger E-family for C ∗ (E). By checking on generators, one can see that φ is equivariant with respect to the gauge actions on C ∗ (E) and OAE , and thus the Gauge-Invariant Uniqueness Theorem [2, Theorem 2.1] implies that φ is injective. Since the image of φ contains the generators {Se : e ∈ E 1 } of OAE , φ is also surjective. Thus C ∗ (E) ∼ = OAE . For (2) let A be a row-finite {0, 1}-matrix with no identically zero rows. Let GA be the canonical ultragraph of A (see Remark 2.10). Then the source map of GA is bijective and C ∗ (GA ) ∼ = OA . Since A is a row-finite matrix, the range of each edge in GA is a finite set. Thus C ∗ (GA ) is isomorphic to the C ∗ -algebra of the graph formed by replacing each edge in GA with a set of edges from s(e) to w for all w ∈ r(e) [16, Remark 2.5]. But this is precisely the graph EA described in the statement above. 2

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2.3. Ultragraph C ∗ -algebras Introductory references include [15,16,22,23]. For a set X, let P(X) denote the collection of all subsets of X. Definition 2.5. (See [22, Definition 2.1].) An ultragraph G = (G0 , G 1 , r, s) consists of a countable set of vertices G0 , a countable set of ultraedges G 1 , and functions s : G 1 → G0 and r : G 1 → P(G0 ) \ {∅}. Note that in the literature, ultraedges are typically just referred to as edges. However, since we will frequently be passing back and forth between graphs and ultragraphs in this paper, we feel that using the term ultraedge will serve as a helpful reminder that edges in ultragraphs behave differently than in graphs. Definition 2.6. For a set X, a subset C of P(X) is called an algebra if (i) ∅ ∈ C, (ii) A ∩ B ∈ C and A ∪ B ∈ C for all A, B ∈ C, and (iii) A \ B ∈ C for all A, B ∈ C. Definition 2.7. For an ultragraph G = (G0 , G 1 , r, s), we let G 0 denote the smallest algebra in P(G0 ) containing the singleton sets and the sets {r(e): e ∈ G 1 }. Definition 2.8. A representation of an algebra C is a collection of projections {pA }A∈C in a C ∗ -algebra satisfying p∅ = 0, pA pB = pA∩B , and pA∪B = pA + pB − pA∩B for all A, B ∈ C. Observe that a representation of an algebra automatically satisfies pA\B = pA − pA pB . Definition 2.9. For an ultragraph G = (G0 , G 1 , r, s), the ultragraph C ∗ -algebra C ∗ (G) is the universal C ∗ -algebra generated by a representation {pA }A∈G 0 of G 0 and a collection of partial isometries {se }e∈G 1 with mutually orthogonal ranges that satisfy (1) se∗ se = pr(e) for all e ∈ G 1 , (2) se se∗ ps(e) for all e ∈ G 1 , (3) pv = e∈v G 1 se se∗ whenever 0 < |vG 1 | < ∞, where we write pv in place of p{v} for v ∈ G0 . As with graphs, we call a vertex v ∈ G0 a sink if vG 1 = ∅ and an infinite emitter if vG 1 is infinite. A path in an ultragraph G is a sequence of ultraedges α = e1 e2 . . . en with s(ei+1 ) ∈ r(ei ) for 1 i n − 1. A cycle is a path α = e1 . . . en with s(e1 ) ∈ r(en ). [23, Theorem 4.1] implies that C ∗ (G) is an AF-algebra if and only if G has no cycles. Remark 2.10. A graph may be regarded as an ultragraph in which the range of each ultraedge is a singleton set. The constructions of the two C ∗ -algebras then coincide: the graph C ∗ -algebra of a graph is the same as the ultragraph C ∗ -algebra of that graph when regarded as an ultragraph (see [22, §3] for more details).


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For a {0, 1}-matrix A over I with nonzero rows, the canonical associated ultragraph GA = (G0A , GA1 , r, s) is defined by G0A = GA1 = I , r(i) = {j ∈ I : A(i, j ) = 1} and s(i) = i for i ∈ GA1 (see [22, Definition 2.5]). It follows from [22, Theorem 4.5] that C ∗ (GA ) ∼ = OA . The ultragraph GA has the property that s is bijective. Conversely an ultragraph G = (G0 , G 1 , r, s) with bijective s is isomorphic to GA where A is the edge matrix of G. Thus one can say that an Exel–Laca algebra is a C ∗ -algebra of a ultragraph with bijective source map. From these observations, one can see that the class of ultragraph C ∗ -algebras contains both the class of graph C ∗ -algebras and the class of Exel–Laca algebras. 2.4. AF-algebras Introductory references include [3,9,13] as well as [5, Chapter 6] and [19, §6.1, §6.2, and §7.2]. Definition 2.11. An AF-algebra is a C ∗ -algebra that is the direct limit of a sequence of finitedimensional C ∗ -algebras. Equivalently, a C ∗ -algebra A is an AF-algebra if and only if A = ∞ ∗ n=1 An for a sequence of finite-dimensional C -subalgebras A1 ⊆ A2 ⊆ · · · ⊆ A. To discuss AF-algebras, we need first to briefly discuss C∗ m inclusions of finite-dimensional n ∗ algebras. Fix finite-dimensional C -algebras A = i=1 Mai (C) and B = j =1 Mbj (C). Let M = (mi,j )i,j be an m × n nonnegative integer matrix with no zero rows such that m

mi,j aj bj

for all j .

(2.2)

i=1

There exists an inclusion φM : A → B with the following property. For an element x = n (xi )m i=1 ∈ A, the image φM (x) of x has the form (yj )j =1 ∈ B where for each j n, the matrix yj is block-diagonal with mi,j copies of each xi along the diagonal and 0’s elsewhere. (Eq. (2.2) ensures that this is possible.) The map φM is not uniquely determined by this property, but its unitary equivalence class is. Every inclusion φ of A into B is unitarily equivalent to φM for some matrix M. Specifically, M = (mi,j )i.j is the matrix such that mi,j is equal to the rank of 1Bj φ(pi ) where 1Bj is the unit for the j th summand of B, and where pi is any rank-1 projection in the ith summand of A. We refer to M as the multiplicity matrix of the inclusion φ. Definition 2.12. A Bratteli diagram (E, d) consists of a directed graph E = (E 0 , E 1 , r, s) together with a collection d = {dv : v ∈ E 0 } of positive integers satisfying the following conditions. (1) (2) (3) (4)

E has no sinks; E 0 is partitioned as a disjoint union E 0 = ∞ n=1 Vn where each Vn is a finite set; for each e ∈ E 1 there exists n ∈ N such that s(e) ∈ Vn and r(e) ∈ Vn+1 ; and for each vertex v ∈ E 0 we have dv e∈E 1 v ds(e) for all v ∈ E 0 .

If (E, d) is a Bratteli diagram, then E is a row-finite graph with no sinks. We regard d as a labeling of the vertices by positive integers, so to draw a Bratteli diagram we sometimes just draw the directed graph, replacing each vertex v by its label dv .

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Remark 2.13. Those experienced with Bratteli diagrams will notice that our definition of a Bratteli diagram is slightly nonstandard. Specifically, a Bratteli diagram is traditionally specified as undirected graph in which each edge connects vertices in consecutive levels. Of course, an orientation of the edges is then implicitly chosen by the decomposition E 0 = Vn , so it makes no difference if we instead draw a directed edge pointing from the vertex in level n to the vertex in level n + 1. Example 2.14. The following is an example of a Bratteli diagram.

0 Given a Bratteli diagram (E, d), we construct an AF-algebra A as follows. For each v ∈ E , let Av be an isomorphic copy of Mdv (C), and for each n ∈ N, let An := v∈Vn Av . For each n let φn : An → An+1 be the homomorphism whose multiplicity matrix is (|vE 1 w|)v∈Vn ,w∈Vn+1 . We then define A to be the direct limit lim −→(An , φn ). Since the φn are determined up to unitary equivalence by (E, d), the isomorphism class of A is also uniquely determined by (E, d).

Example 2.15. In Example 2.14, we see that ⎛

A1 = C

A2 = M4 (C) ⊕ C ⊕ C A3 = M8 (C) ⊕ M2 (C) ⊕ C .. . An = M2n (C) ⊕ Mn−1 (C) ⊕ C .. .

x ⎜0 φ1 (x) = ⎝ 0 0 x φ2 (x, y, z) = 0 x φ3 (x, y, z) = 0 .. . φn (x, y, z) =

x 0

⎞ 0 0⎟ ⎠⊕x ⊕0 0 0 0 y 0 ⊕ ⊕z x 0 z 0 y 0 ⊕ ⊕z x 0 z 0 0 0 0

0 x

0 0 0 0

⊕

y 0

0 z

⊕z

.. .

The following telescoping operation on a Bratteli diagram preserves the associated AFalgebra. Given (E, d), we choose an increasing subsequence {nm }∞ m=1 of N. The set of the ∞ vertices of the new Bratteli diagram is m=1 Vnm , the set of the edges of the new Bratteli ∞ dia∗V gram is ∞ (V E ), and the new function d is the restriction of the old d to n n m m+1 m=1 m=1 Vnm . For example, if we have the portion of a Bratteli diagram shown below on the left and remove


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the middle column of vertices, we obtain the portion of the Bratteli diagram shown below on the right.

We say that two Bratteli diagrams (E, d) and (E , d ) are equivalent if there is a finite sequence (E1 , d1 ), . . . , (En , dn ) such that (E1 , d1 ) = (E, d), (En , dn ) = (E , d ) and for each 1 i n − 1, one of (Ei , di ) and (Ei+1 , di+1 ) is a telescope of the other. Bratteli proved in [3] that two Bratteli diagrams give rise to isomorphic AF-algebras if and only if the diagrams are equivalent (see [3, §1.8 and Theorem 2.7] for details). The class of AF-algebras is closed under forming ideals and quotients. On the other hand, the three classes of graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ -algebras are not closed under forming ideals nor quotients. However we can show the following. Lemma 2.16. The class of graph AF-algebras is closed under forming ideals and quotients. Proof. If E is a graph and the graph C ∗ -algebra C ∗ (E) is an AF-algebra, then E has no cycles by [17, Theorem 2.4]. Thus E vacuously satisfies Condition (K), and it follows that every ideal of C ∗ (E) is gauge-invariant by [1, Corollary 3.8]. Thus every ideal of C ∗ (E) as well as its quotient is a graph C ∗ -algebra by [6, Lemma 1.6] and [1, Theorem 3.6]. 2 Remark 2.17. A quotient of an Exel–Laca AF-algebra need not be an Exel–Laca algebra. For example, if K+ is the minimal unitization of the compact operators K on a separable infinitedimensional Hilbert space, then M2 (K+ ) is an Exel–Laca AF-algebra that has a quotient, M2 (C), that is not an Exel–Laca algebra — for details see Example 4.11 and Corollary 4.19. Whether ideals of Exel–Laca AF-algebras are necessarily Exel–Laca algebras is an open question. We also do not know whether ideals and quotients of ultragraph AF-algebras are necessarily ultragraph C ∗ -algebras. As we shall see later, this uncertainty causes problems in the analyses of Exel–Laca AF-algebras and ultragraph AF-algebras. Lemma 2.18. The three classes of graph AF-algebras, Exel–Laca AF-algebras, and ultragraph AF-algebras are closed under taking direct sums. Proof. Each of the four classes of AF-algebras, graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ -algebras is closed under forming direct sums. The result follows. 2 3. Some technical lemmas In this section we establish some technical results for Bratteli diagrams and inclusions of finite-dimensional C ∗ -algebras. We will use these technical results to prove many of our realization results in Section 4.

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Lemma 3.1. Suppose A is an AF-algebra that has no quotients isomorphic to C, and suppose that (E, d) is a Bratteli diagram for A. Let H = {v ∈ E 0 : dv = 1}, and let F be the subgraph of E such that F 0 := E 0 \ H and F 1 := {e ∈ E 1 : s(e) ∈ / H } with r, s : F 1 → F 0 inherited from E. Let d : F 0 → N be the restriction of d : E 0 → N. Then (F, d) is a Bratteli diagram for A. Proof. First note that if e ∈ E 1 with r(e) ∈ H , then dr(e) = 1 and hence ds(e) = 1 and s(e) ∈ H . Hence F is in fact a subgraph of E. We claim that for any n ∈ N and v ∈ Vn , there exists m ∈ N such that whenever w ∈ Vn+m and v w, we have dw 2. We fix n ∈ N and v ∈ Vn , suppose that there is no such m, and seek a contradiction. Let v0 := v. Inductively choose ei ∈ E 1 such that s(ei ) = vi−1 and such that for each m ∈ N there exists w ∈ Vn+i+m with r(ei ) w and dw = 1, setting vi := r(ei ). Then the infinite path e1 e2 . . . satisfies ds(en ) = 1 for all n. Hence {x ∈ E 0 : x s(en ) for any n} is a saturated hereditary subset and the quotient of A by the corresponding ideal is an AF-algebra with Bratteli diagram

Hence this quotient is isomorphic to C, which contradicts our hypothesis on A. This establishes the claim. Let B be the AF-algebra associated to the Bratteli diagram F , and let ιn : Bn → An denote obvious inclusion of the nth approximating subalgebra of B determined by F into the nth apE : A → A be the connecting maps in proximating subalgebra of A determined by E. Let φn,m n m E the directed system associated to E, and let φn,∞ : An → A be the inclusion of An into the direct F : B → B be the connecting maps in the directed system limit algebra A. Likewise, let φn,m n m F : B → B be the inclusion of B into the direct limit algebra B. associated to F , and let φn,∞ n n E F ◦ιn = ιn+1 ◦φn,n+1 for all n, and thus by the universal property of the direct We see that φn,n+1 F E F limit B = lim −→(Bn , φn ), there is a ∗-homomorphism ι∞ : B → A with φn,∞ ◦ ιn = ι∞ ◦ φn,∞ . Since each ιn is injective, it follows that ι∞ is injective. We shall also show that ι∞ is also surjective and hence an isomorphism. It suffices to show that for any v ∈ Vn and for any a in E (a) ∈ im ι . By the previous the direct summand Av of An corresponding to v, we have φn,∞ ∞ paragraph we may choose m so that whenever w ∈ Vn+m and v w, then dw 2. It follows that E φn,n+m (a) ∈

Mdw (C) ⊆ ιn+m (Bn+m ),

w∈Vn+m dw 2 E so that φn,n+m (a) = ιn+m (b) for some b ∈ Bn+m . Thus E E E E F φn,∞ (a) = φn+m,∞ ◦ φn,n+m (a) = φn+m,∞ ◦ ιn+m (b) = ι∞ ◦ φn+m,∞ (b) ∈ im ι∞

and ι∞ is surjective. Hence ι∞ is an isomorphism as required.

2

Lemma 3.2. Suppose A is an AF-algebra with no nonzero finite-dimensional quotients. Then any Bratteli diagram for A can be telescoped to obtain a second Bratteli diagram (E, d) for A such that for all n ∈ N and for each v ∈ Vn+1 either dv > e∈E 1 v ds(e) or there exists w ∈ Vn with |wE 1 v| 2.


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Proof. Let (F, d) be a Bratteli diagram for A with F 0 partitioned into levels as F 0 = ∞ n=1 Wn . satisfying It suffices to show that for every m there exists n m such that for every v ∈ W n dv = α∈Wm F ∗ v ds(α) , there exists w ∈ Wm with |wF ∗ v| 2. We suppose not, and seek a contradiction. That is, we suppose that there exists m such that for every n m the set

Xn := x ∈ Wn : dx =

∗

ds(α) and |wF x| 1 for all w ∈ Wm

α∈Wm F ∗ x

is nonempty. By telescoping (F, d) to ∞ n=m Wn we may assume m = 1. We claim that if n p, x ∈ Xp , and v ∈ Wn with v x, then v ∈ Xn . Indeed,

dx =

ds(α)

α∈W1 F ∗ x

=

β∈Wn F ∗ x

γ ∈W1

ds(γ )

(3.1)

F ∗ s(β)

(3.2)

ds(β)

β∈Wn F ∗ x

dx . Thus we have equality throughout, and the equality of (3.1) and (3.2) implies ds(β) = ∗ γ ∈W1 F ∗ s(β) ds(γ ) for each β ∈ Wn F x. In particular, since v x, we have that dv = γ ∈W1 F ∗ v ds(γ ) . Moreover for each w ∈ W1 , 1 |wF ∗ x| |wF ∗ v||vF ∗ x|, so v x implies that |wF ∗ v| 1, and v ∈ Xn as required. We shall now construct an infinite pathλ = λ1 λ2 . . . in F such that s(λn ) ∈ Xn for all n. If x ∈ Xn , then since dx is nonzero and dx = α∈W1 F ∗ x ds(α) , there exists w ∈ W1 such that w x. Since W1 is finite, there exists w1 ∈ W1 such that for infinitely many n there exists x ∈ Xn with w1 x. Since w1 F 1 is finite, there exists λ1 ∈ w1 F 1 such that for infinitely many n, we have r(λ1 ) x for some x ∈ Xn . We set w2 := r(λ1 ) which is in X2 by the claim above. Proceeding in this way, we produce an infinite path λ = λ1 λ2 . . . in F such that s(λn ) ∈ Xn for all n. For each w ∈ W1 such that w s(λn ) for some n, we define nw := min{n: w s(λn )}. Let N := max{nw : w ∈ W1 and w s(λn ) for some n}. We claim that F 1 r(λn ) = {λn } for all n N . Fix n N , and e ∈ F 1 r(λn ). Since r(λn ) = s(λn+1 ) ∈ Xn+1 , we have s(e) ∈ Xn . Hence W1 F ∗ s(e) is nonempty, so we may fix β ∈ W1 F ∗ s(e). Now βe is the unique path in s(β)F ∗ r(λn ) by definition of Xn+1 . Let α be the unique path from s(β) to s(λns(β) ). Since ns(β) N n, we have αλns(β) λns(β) +1 . . . λn in s(β)F ∗ r(λn ), and the uniqueness of this path then forces βe = αλns(β) λns(β) +1 . . . λn , and in particular e = λn . Thus F 1 r(λn ) = {λn } as required. Since F 1 r(λn ) = {λn }, we have W1 F ∗ r(λn ) = W1 F ∗ λn = {βλn : β ∈ W1 F ∗ s(λn )}. Hence that r(λn ) ∈ Xn+1 and that s(λn ) ∈ Xn imply that dr(λn ) =

α∈W1 F ∗ r(λn )

ds(α) =

β∈W1 F ∗ s(λn )

ds(β) = ds(λn )

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for all n N . This implies ds(λn ) = ds(λN ) for all n N . Moreover, {y ∈ F 0 : y s(λn ) for all n} is a saturated hereditary subset, and the quotient of A by the ideal corresponding to this set is an AF-algebra with a Bratteli diagram of the form

Hence this quotient is isomorphic to Mds(λN ) (C), which contradicts the hypothesis that A has no finite-dimensional quotients. 2 Lemma 3.3. Let A be an AF-algebra. Then A has no nonzero finite-dimensional quotients if and only if there exists a Bratteli diagram (E, d) for A satisfying the following two properties: (1) dv 2 for all v ∈ E 0 ; and (2) for all n ∈ N and for each v ∈ Vn+1 either dv > e∈E 1 v ds(e) or there exists w ∈ Vn with |wE 1 v| 2. Proof. If A has no nonzero finite-dimensional quotients, then by Lemma 3.1 there exists a Bratteli diagram for A satisfying (1). Lemma 3.2 shows that this Bratteli diagram may be telescoped to obtain a Bratteli diagram for A satisfying (2). The vertices of the telescoped diagram are a subset of those of the original diagram, and the values of dv are the same for those vertices v common to both. In particular, telescoping preserves property (1), so the telescoped Bratteli diagram will then satisfy both (1) and (2). Conversely, suppose that there exists a Bratteli diagram (E, d) for A satisfying (1) and (2). If I is a proper ideal of A, then I corresponds to a saturated hereditary subset H , and the complement (E \ H, d) of H in (E, d) is a Bratteli diagram for A/I . Fix a vertex v in this complement. Since H is saturated hereditary, there exists an edge e1 ∈ E 1 with s(e1 ) = v and r(e1 ) in the complement also. Inductively, we may produce an infinite path e1 e2 . . . in the complement. It follows from property (2) that ds(ei ) < ds(ei+1 ) for all i, which implies that the function d : (E \ H )0 → N is unbounded. Hence A/I is infinite dimensional. 2 Lemma 3.4. Suppose A is an AF-algebra with no unital quotients. Then any Bratteli diagram for A can be telescoped to obtain a second Bratteli diagram (E, d) for A such that for all v ∈ E 0 we have dv > e∈E 1 v ds(e) . Proof. Let (F, d) be a Bratteli diagram for A with F 0 partitioned into levels as F 0 = ∞ n=1 Wn . It suffices to show that for every m there exists n m such that for every v ∈ W we have n dv > α∈Wm F ∗ v ds(α) . Suppose not, and seek a contradiction. That is, we suppose that there exists m such that for every n m the set

Yn := x ∈ Wn : dx =

ds(α)

α∈Wm F ∗ x

is nonempty. By telescoping (F, d) to

∞

n=m Wn

we may assume m = 1. If we let

T := w ∈ F 0 : for infinitely many n there exists x ∈ Yn with w x ,


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then the complement of T is a saturated hereditary subset, and the quotient of A by the ideal corresponding to this complement has a Bratteli diagram obtained by restricting to the vertices in T . Along similar lines to Lemma 3.2, one can show that if n p, x ∈ Yp , and v ∈ Wn with v x, thenv ∈ Yn . Hence each v ∈ T ∩ Wn is in Yn . This implies that each v ∈ T has the property that dv = e∈F 1 v ds(e) , and hence all the inclusions in the corresponding directed system are unital. Thus the quotient of A considered above is unital. This contradicts the hypothesis that A has no unital quotients. 2 Lemma 3.5. Let A be an AF-algebra. Then A has no unital quotients if and only if A has a Bratteli diagram (E, d) such that for all v ∈ E 0 we have both dv 2 and dv > e∈E 1 v ds(e) . Proof. If A has no unital quotients, then the existence of such a Bratteli diagram follows from Lemmas 3.3 and 3.4. Conversely, suppose that A has such a Bratteli diagram (E, d), and fix a nonzero quotient A/I of A.There is a subdiagram (F, d) of (E, d) which is a Bratteli diagram for A/I . In particular dv > e∈F 1 v ds(e) for all v ∈ F 0 . It follows that the inclusions in the direct limit decomposition of A corresponding to (F, d) are all nonunital. Hence A/I is nonunital. 2 ∗ Lemma 3.6. Let A be a C which is generated by finite-dimensional B -algebra subalgebras v v w ∼ and C. Suppose that B = v∈V B where each B = Mbv (C) and that C = w∈W C where each C w ∼ = Mcw (C). For each v ∈ V suppose that q v is a minimal projection in B v such that v q ∈ C and (1B v − q v )C = {0}. For each v, w, let mv,w denote the rank of q v 1C w in C w , and let

aw := cw +

(bv − 1)mv,w .

v∈V

Then A = w∈W Aw where each Aw ∼ = Maw (C). Moreover, the inclusion C w → Aw has multiplicity 1 for w ∈ W , and the inclusion B → A has multiplicity matrix (mv,w )v∈V ,w∈W . Finally, the unit 1A of A is equal to (1B − v∈V q v ) + 1C . Proof. The assumptions on the q v imply that (1B − v∈V q v ) + 1C is the unit of A. To obtain the desired decomposition of A, we construct a family of matrix units for A. We begin by fixing convenient systems of matrix units for the B v and the C w . v : 0 r, s b − 1} be a family of matrix units for B v such that β v = q v . For v ∈ V , let {βr,s v 0,0 w : 0 k, l c − 1} be a family of matrix units for C w such that for Similarly, for w ∈ W let {γk,l w w . Note each v ∈ V there exists a subset κv,w ⊂ {0, 1, . . . , cw − 1} satisfying q v 1C w = k∈κv,w γk,k that the subsets {κv,w }v∈V of {0, 1, . . . , cw − 1} are mutually disjoint and satisfy |κv,w | = mv,w . We are now ready to define the desired matrix units for A; these matrix units will be indexed by the set Iw := {0, 1, . . . , cw − 1} × {0} κv,w × {1, 2, . . . , bv − 1} v∈V

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w for w ∈ W . We have |Iw | = cw + v∈V |κv,w |(bv − 1) = aw . Define elements {α(k,r),(l,s) : w ∈ W, (k, r), (l, s) ∈ Iw } by ⎧ w γk,l if r = s = 0, ⎪ ⎪ ⎪ w βv ⎪ ⎨ γk,l if r = 0, l ∈ κv,w and s 1, 0,s w := α(k,r),(l,s) v γw if k ∈ κv ,w , r 1 and s = 0, βr,0 ⎪ k,l ⎪ ⎪ ⎪ ⎩ β v γ w β v r,0 k,l 0,s if k ∈ κv ,w , r 1, l ∈ κv,w and s 1. We first claim that for each w, w ∈ W , each (k, r), (l, s) ∈ Iw and each (k , r ), (l , s ) ∈ Iw ,

w α(k,r),(l ,s ) if w = w and (l, s) = (k , r ), w w (3.3) α(k α(k,r),(l,s) ,r ),(l ,s ) = 0 otherwise. To verify (3.3), we consider four cases. w are matrix units and since the C w are orthogonal, we have Case 1. s = r = 0. Since γk,l

w w w γk,l γk ,l = γk,l 0

if w = w and l = k , otherwise.

This implies (3.3) in the case s = r = 0.

v γ w = β v β v γ w = 0 because β v Case 2. s 1 and r = 0. Then β0,s s,s 0,s s,s k ,l k ,l

bv −1

w (k ,r ),(l ,s )

w q v which is orthogonal to C by assumption. This shows α(k,r),(l,s) α

s=1

v =1 v − βs,s B

= 0.

Case 3. s = 0 and r 1. This case follows from Case 2 by taking adjoints. Case 4. s 1 and r 1. Then

w v w w v γk,l β0,s βrv ,0 γkw ,l = γk,l β0,0 γk ,l 0

if v = v and s = r , otherwise.

w β v = γ w we have Since γk,l k,l 0,0

w w v w w β0,0 γkw ,l = γk,l γk ,l = γk,l γk,l 0

if w = w and l = k , otherwise.

These show (3.3) in Case 4, completing the proof of the claim. w : (k, r), (l, s) ∈ Iw } ⊂ A. From (3.3), we see that For each w ∈ W , let Aw := span{α(k,r),(l,s) w is isomorphic to Ma w (C) for each w ∈ W , and that {A }w∈W are orthogonal to each other. w v A . To see this, it suffices to show that all the matrix units βr,s We next show that A = w∈W w w and γk,l for B and C belong to w∈W A . If l ∈ κv,w , then

Aw

w w v w γk,l β0,0 = γk,l 1C w q v = γk,l

l ∈κv,w

w γlw ,l = γk,l .


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v γ w = γ w if k ∈ κ . We may deduce from these two equalities that Similarly, we get β0,0 v ,w k,l k,l

w v γ w β v for all k ∈ κ , all r 0, all l ∈ κ α(k,r),(l,s) = βr,0 v,w and all s 0. For each v ∈ V , we v ,w k,l 0,s have v w = qv = q v 1C w = γk,k . β0,0 w∈W

w∈W k∈κv,w

It follows that v v v v = βr,0 β0,0 β0,s = βr,s

v w v βr,0 γk,k β0,s =

w∈W k∈κv,w

w α(k,r),(k,s) ∈

w∈W k∈κv,w

Aw

w∈W

w = αw for all v ∈ V and all 0 r, s bv − 1. We also have γk,l (k,0),(l,0) for w ∈ W and 0 k, l cw − 1. Thus we get A = w∈W Aw . It is clear that the inclusion C w → Aw has multiplicity 1 for w ∈ W . To see that the inclusion B → A has multiplicity matrix (mv,w )v∈V ,w∈W , it suffices to see that for each v ∈ V and w ∈ W , the product of the minimal projection q v ∈ B v and the unit 1Aw of Aw has rank mv,w in Aw ∼ = Maw (C). Since q v ∈ C, we have

q v 1Aw = q v 1C w =

w γk,k =

k∈κv,w

w α(k,0),(k,0) .

k∈κv,w

This shows that the rank of q v 1Aw ∈ Aw is |κv,w | = mv,w .

2

4. Realizations of AF-algebras 4.1. A construction of an ultragraph from a certain type of Bratteli diagram In this section we show how to construct ultragraphs from certain Bratteli diagrams and use these ultragraphs to realize particular classes of AF-algebras as ultragraph C ∗ -algebras, Exel– Laca algebras, and graph C ∗ -algebras. Definition 4.1. Let A be an AF-algebra with no nonzero finite-dimensional quotients. By Lemma 3.3 there exists a Bratteli diagram (E, d) for A satisfying the following two properties: (1) dv 2 for all v ∈ E 0 ; and (2) for all n ∈ N and for each v ∈ Vn+1 either dv > e∈E 1 v ds(e) or there exists w ∈ Vn with |wE 1 v| 2. We define

v := dv −

(ds(e) − 1).

e∈E 1 v

The symbol has been chosen to connote “difference.” Note that from the property (1), v = dv if and only if v is a source. In addition, it follows from the properties of our Bratteli diagram that

v 2 for all v ∈ E 0 .

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We claim that for each v ∈ E 0 there exists an injection kv : E 1 v → {0, 1, . . . , v − 1} such that there exists e ∈ E 1 v with kv (e) = 0 if and only if dv = e∈E 1 v ds(e) , and in this case e is not the only element of s(e)E 1 v. To justify this claim, first observe that

v = dv − ds(e) − 1 = dv − ds(e) + 1 = dv − ds(e) + E 1 v . e∈E 1 v

e∈E 1 v

e∈E 1 v

e∈E 1 v

Hence if dv > e∈E 1 v ds(e) we may always choose an injection kv : E 1 v → {0, 1, . . . , v − 1} so that its image does not contain 0. On the other hand if dv = e∈E 1 v ds(e) , then by hypothesis on the Bratteli diagram there exists w ∈ E 0 with |wE 1 v| 2 so we may choose a bijection kv : E 1 v → {0, 1, . . . , v − 1} such that e ∈ E 1 v with kv (e) = 0 satisfies s(e) = w. This establishes the claim. We now define an ultragraph G = (G0 , G 1 , rG , sG ) by G0 := vi : v ∈ E 0 and 1 i v − 1

and G 1 := evi : vi ∈ G0

with sG (evi ) := vi

for all vi ∈ G0 ,

rG (evi ) := {vi−1 }

for 2 i v − 1

and rG (ev1 ) := wk : there exists a path λ = λ1 λ2 . . . λn such that s(λ) = v, r(λ) = w, kr(λi ) (λi ) = 0 for i = 1, 2, . . . , n − 1, and kw (λn ) = k 1 . To check that G is an ultragraph, we only need to see that rG (ev1 ) = ∅. Lemma 4.2. For all n and v ∈ Vn , the set rG (ev1 ) is nonempty and satisfies rG (ev1 ) = wkw (e) : w ∈ Vn+1 , e ∈ vE 1 w, kw (e) 1 ∪

rG (ew1 ).

w∈Vn+1 , e∈vE 1 w, kw (e)=0

Proof. The latter equality follows from the definition of rG (ev1 ). For each v ∈ Vn , there exists w ∈ Vn+1 such that vE 1 w = ∅. By the assumption on kw , there exists e ∈ vE 1 w such that kw (e) 1. Thus wkw (e) ∈ rG (ev1 ). This shows that rG (ev1 ) is nonempty. 2 Remark 4.3. By definition, rG (ev1 ) ⊂ ∞ k=n+1 Vk for v ∈ Vn . One can show that this property together with the equality in Lemma 4.2 uniquely determines {rG (ev1 )}v∈E 0 . Example 4.4 (An example of the ultragraph construction). Consider a Bratteli diagram (E, d) satisfying conditions (1) and (2) of Lemma 3.3 and whose first three levels are as illustrated


1607

below. In the diagram, each vertex is labeled with its name, and above the label a appears the integer da .

The values of for the vertices visible in the diagram are

s = 2

v = 5

x = 2

t = 2

w = 3

y = 3

u = 3

z = 3

So the corresponding section of the resulting ultragraph G will have vertices G0 = {s1 , t1 , u1 , u2 , v1 , v2 , v3 , v4 , w1 , w2 , x1 , y1 , y2 , z1 , z2 , . . .}, and each of these vertices ai will emit exactly one ultraedge eai . For i = 1, we have rG (eai ) = {ai−1 }. To determine the ranges of the ea1 , we must choose injections ka : E 1 a → {0, 1, . . . ,

a − 1} for a ∈ E 0 with the properties described above; in particular, this necessitates that 0 is in the image of ka only when a = w or a = y, and also that kw (f ) = 0 and ky (h) = 0. One possible set of choices of injections ka is kv (e) = 1, kv (e ) = 3, kv (e ) = 4, kw (f ) = 0, kw (f ) = 2, kw (f ) = 1,

kx (g) = 1, ky (h) = 1, kz (k) = 2,

ky (h ) = 0, kz (k ) = 1.

ky (h ) = 2,

We can calculate rG (es1 ) = {v1 },

rG (et1 ) = {v3 , v4 , w2 , y2 , z1 } ∪ rG (ey1 ),

r(ev1 ) = {x1 , y1 , z2 },

rG (eu1 ) = {w1 },

and r(ew1 ) = {z1 , y2 } ∪ rG (ey1 ).

We may now draw the fragment of the ultragraph G corresponding to the given fragment of the Bratteli diagram (E, d).

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Note that by definition of the ultragraph G, each vertex emits exactly one ultraedge, so in the picture any multiple arrows leaving the same vertex actually have the same label and constitute a single ultraedge of G. 4.2. Sufficient conditions for realizations Theorem 4.5. Let A be an AF-algebra with a Bratteli diagram satisfying the conditions of Lemma 3.3. If G is an ultragraph constructed from this Bratteli diagram as in Definition 4.1, then A ∼ = C ∗ (G). In addition, C ∗ (G) is an Exel–Laca algebra. Proof. Let (E, d) be a Bratteli diagram for A with the vertices partitioned into levels as E 0 = ∞ n=1 Vn and satisfying the conditions of Lemma 3.3, and let G be an ultragraph constructed from (E, d) as in Definition 4.1. Our strategy is to find a direct limit decomposition of C ∗ (G) so that at each level we may apply Lemma 3.6 to see that the inclusion of finite-dimensional algebras is the same as the corresponding inclusion in the direct limit decomposition of A determined by (E, d). For each v ∈ E 0 let C v := C ∗ {sevi : 1 i v − 1} . We have sevi se∗v = pvi for 1 i v − 1 and se∗v sevi = pvi−1 for 2 i v − 1. We define i i a projection q v := prG (ev1 ) = se∗v sev1 ∈ C v , which is orthogonal to pvi for 1 i v − 1. 1 v : 0 k, l − 1} in C v such that These computations show that there exist matrix units {γk,l v v v v v ∈ C v is given by v γ0,0 = q , γi,i = pvi and γi,i−1 = sevi for 1 i v − 1. Explicitly, γk,l v γk,l := sevk sevk−1 · · · sev1 q v se∗v se∗v · · · se∗v 1

2

l

for 0 k, l v − 1. This shows that C v is isomorphic to M v (C) with minimal projection q v v −1 and the unit i=1 pvi + q v . For each n ∈ N Cn := C ∗ {sevi : v ∈ Vn and 1 i v − 1}


is equal to

v∈Vn

1609

C v . Moreover, for n ∈ N, define

Bn := C

∗

n

! =C

Cj

∗

sevi : v ∈

j =1

n

" Vj and 1 i v − 1

.

j =1

v −1 Claim. For each n ∈ N, the unit 1Bn of Bn is given by v∈n Vj i=1 pvi + v∈Vn q v , j =1 and there exists a decomposition Bn = v∈Vn B v such that each B v ∼ = Mdv (C) with minimal projection q v ; and for each n ∈ N, the inclusion Bn → Bn+1 has multiplicity matrix (|vE 1 w|)v∈Vn ,w∈Vn+1 . We proceed by induction on n. When n = 1, let B v := C v for v ∈ V1 . Then B1 = C1 has the decomposition B1 = v∈V1 B v . For each v ∈ V1 , we have v = dv because v is a source. Hence v −1 pv i + q v . B v = C v is isomorphic to Mdv (C) with minimal projection q v and the unit i=1 This shows the claim in the case n = 1. For the inductive step, assume that Bn has the desired decomposition. To apply Lemma 3.6 to the C ∗ -algebra Bn+1 which is generated by Bn and Cn+1 , we check that for each v ∈ Vn the minimal projection q v ∈ B v is in Cn+1 and satisfies (1B v − q v )Cn+1 = {0}. We see that 1B v − q v = 1Bn − qv = v∈Vn

v∈Vn

v∈

v −1

n

pv i

i=1 j =1 Vj

which is orthogonal to Cn+1 . This proves (1B v − q v )Cn+1 = {0} for all v ∈ Vn . For each v ∈ Vn , Lemma 4.2 implies q = prG (ev1 ) = v

w∈Vn+1

=

pwkw (e) +

e∈vE 1 w

kw (e)1

prG (ew1 )

e∈vE 1 w

kw (e)=0

γkww (e),kw (e) .

(4.1)

w∈Vn+1 e∈vE 1 w

Hence q v ∈ Cn+1 . Thus we can apply Lemma 3.6 to obtain the decomposition Bn+1 = w w w w w∈Vn+1 B . Since the inclusion C → B has multiplicity 1 for w ∈ W , the projection q is w v 1 w minimal in B . From (4.1), q 1C w has rank |vE w| in C for w ∈ Vn+1 . The definition of w implies that d w = w +

(dv − 1)vE 1 w .

w∈Vn+1

Hence B w is isomorphic to Mdw (C) for w ∈ Vn+1 . The conclusion of Lemma 3.6 also shows that the inclusion Bn → Bn+1 has multiplicity matrix (|vE 1 w|)v∈Vn ,w∈Vn+1 , and that the unit of v −1 Bn+1 is equal to v∈n+1 Vj i=1 pvi + w∈Vn+1 q w . This proves the claim. j =1 n {se : e ∈ G 1 }. Since We see that ∞ n=1 B contains ∞eachnvertex v in G emits exactly one ultra∗ n . Thus ∗ B edge e, pv = se se is contained in ∞ n=1 n=1 B contains all the generators of C (G).

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n Hence C ∗ (G) = ∞ n=1 B is an AF-algebra, and the preceding paragraphs show that (E, d) is a Bratteli diagram for C ∗ (G), giving A ∼ = C ∗ (G). Since every vertex of G emits exactly one ∗ ultraedge, C (G) is an Exel–Laca algebra (see Remark 2.10). 2 Corollary 4.6. If A is an AF-algebra with no nonzero finite-dimensional quotients, then A is isomorphic to an Exel–Laca algebra. Proof. Since A has no nonzero finite-dimensional quotients, Lemma 3.3 implies that A has a Bratteli diagram satisfying the conditions stated. It follows from Theorem 4.5 that A is isomorphic to an Exel–Laca algebra. 2 The following result is important in that it is one of the few instances where we can give a complete characterization of AF-algebras in a certain graph C ∗ -algebra class. In particular, we give necessary and sufficient conditions for an AF-algebra to be the C ∗ -algebra of a row-finite graph with no sinks. Theorem 4.7. Let A be an AF-algebra. Then the following are equivalent: (1) A has no (nonzero) unital quotients. (2) A is isomorphic to the C ∗ -algebra of a row-finite graph with no sinks. Proof. We shall first prove that (1) implies (2). Suppose that A has no unital quotients. By 0 Corollary 3.5 there is a Bratteli diagram (E, d) for A such that for all v ∈ E we have both dv 2 and dv > e∈E 1 v ds(e) . Let G be an ultragraph constructed from (E, d) as in Definition 4.1. ∗ (G). Furthermore, since d > Theorem 4.5 implies that A ∼ C = v e∈E 1 v ds(e) , we have kv (e) 1 0 1 0 for all v ∈ E and e ∈ E v. For v ∈ E , Lemma 4.2 implies rG (ev1 ) = {wkw (e) : w ∈ Vn+1 , e ∈ vE 1 w, kw (e) 1}. Thus, rG (e) is finite for every e ∈ G 1 . Hence C ∗ (G) is isomorphic to a graph C ∗ -algebra of a row-finite graph with no sinks (see [16, Remark 5.25]). We next prove that (2) implies (1). Suppose that A ∼ = C ∗ (E), where E is a row-finite graph ∗ with no sinks. Since C (E) is an AF-algebra, it follows from [17, Theorem 2.4] that E has no cycles. Thus E satisfies Condition (K), and [2, Theorem 4.4] implies that every ideal of C ∗ (E) is gauge invariant. Suppose I is a proper ideal of C ∗ (E). Then I = IH for some saturated hereditary proper subset H ⊂ E 0 , and C ∗ (E)/IH ∼ = C ∗ (EH ), where EH is the nonempty subgraph of E 0 1 0 1 / H } (see [2, Theorem 4.1]). Since H is saturated with EH := E \ H and EH := {e ∈ E : r(e) ∈ hereditary, that E has no sinks implies that EH has no sinks. Since E has no cycles, EH also has 0 is infinite. Thus no cycles. Because EH is a nonempty graph with no cycles and no sinks, EH ∗ C (EH ) is nonunital [17, Proposition 1.4]. 2 Corollary 4.8. Let A be a stable AF-algebra. Then there is a row-finite graph E with no sinks such that A ∼ = C ∗ (E). In particular, A is isomorphic to a graph C ∗ -algebra, to an Exel–Laca algebra, and to an ultragraph C ∗ -algebra. Proof. Since any nonzero quotient of a stable C ∗ -algebra is stable, every quotient of A is stable, and in particular nonunital. The result then follows from Theorem 4.7. 2


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# = (G #1 , r˜ , s˜ ) be the ultragraph #0 , G Lemma 4.9. Let G = (G0 , G 1 , r, s) be an ultragraph. Let G #1 := G 1 {e0 } with #0 := G0 {v0 } and G defined by G s˜ |G 1 = s,

s˜ (e0 ) = v0 ,

r˜ |G 1 = r,

and r˜ (e0 ) = G0 .

# ∼ Then C ∗ (G) = M2 (C ∗ (G)+ ), where C ∗ (G)+ is the minimal unitization of C ∗ (G). #0 is generated by the algebra G 0 ⊆ P(G #0 ) and the two Proof. We first notice that the algebra G 0 ∗ # # elements G0 , {v0 } ∈ P(G ). The universal property of C (G) implies that there is a ∗-homomor# → M2 (C ∗ (G)+ ) satisfying phism φ : C ∗ (G) pA 0 s 0 φ(pA ) = for all A ∈ G 0 and φ(se ) = e for all e ∈ G 1 0 0 0 0 and φ(pG0 ) =

1 0 , 0 0

φ(pv0 ) =

0 0 , 0 1

and φ(se0 ) =

0 0 . 1 0

The Gauge-Invariant Uniqueness Theorem [22, Theorem 6.8] shows that φ is injective. Standard # # calculations show that the image under φ of the generating Cuntz–Krieger G-family in C ∗ (G) + ∗ generates M2 (C (G) ). Hence φ is an isomorphism. 2 Corollary 4.10. Let A be a C ∗ -algebra, and let A+ denote the minimal unitization of A. If A is isomorphic to an Exel–Laca algebra, then M2 (A+ ) is isomorphic to an Exel–Laca algebra. Proof. If A is isomorphic to an Exel–Laca algebra, then by Remark 2.10 A ∼ = C ∗ (G) where G + ), and since G # ∼ # is an M (A is an ultragraph with bijective source map. By Lemma 4.9 C ∗ (G) = 2 # is an Exel–Laca algebra. 2 ultragraph with bijective source map, C ∗ (G) The following example shows that the converse of Corollary 4.6 does not hold. Example 4.11. Let A be a nonunital, simple AF-algebra (such as K). By Corollary 4.22 A is isomorphic to an Exel–Laca algebra, and by Corollary 4.10 M2 (A+ ) is an Exel–Laca algebra. However, M2 (A+ ) has a quotient isomorphic to the finite-dimensional C ∗ -algebra M2 (C). Thus the converse of Corollary 4.6 does not hold. (It is also worth mentioning that M2 (C) is a quotient of an Exel–Laca algebra, but M2 (C) is not itself an Exel–Laca algebra; cf. Corollary 4.19.) The following elementary example shows that the C ∗ -algebra of a row-finite graph with sinks may admit unital quotients (cf. Theorem 4.7). Example 4.12. The AF-algebra M2 (C) ⊕ M2 (C) is isomorphic to the C ∗ -algebra of the graph • ←− • −→ • by [17, Corollary 2.3]. However, this C ∗ -algebra has M2 (C) as a unital quotient. Thus graphs with sinks can have associated C ∗ -algebras that are AF-algebras with proper unital quotients. The next example is more intriguing. Before considering this example, one is tempted to believe that if E is a row-finite graph, then C ∗ (E) is isomorphic to a direct sum of a countable

1612


collection of algebras of compact operators on (finite or countably infinite dimensional) Hilbert spaces and the C ∗ -algebra of a row-finite graph with no sinks (see Proposition 4.14). This would give a characterization of AF-algebras associated to row-finite graphs along similar lines to Theorem 4.7. However, the example shows that this is not the case in general. Example 4.13. Let E be the graph

Then for each n ∈ N the set Hn := {vn , vn+1 , . . .} ∪ {wn , wn+1 , . . .} is a saturated hereditary subset of E, and C ∗ (E)/IHn is a finite-dimensional C ∗ -algebra. Thus C ∗ (E) is an AF-algebra with infinitely many finite-dimensional quotients. This shows that, unlike what occurs for rowfinite graphs with no sinks (cf. Theorem 4.7), the situation with sinks is much more complicated. It also shows that C ∗ (E) does not have a Bratteli diagram of the types described in Lemma 3.4 or Lemma 3.5. Hence our construction of the ultragraph described in Section 4.1 cannot be applied. By eliminating the bad behavior arising in the preceding example, we obtain a limited extension of Theorem 4.7 to graphs containing sinks. Proposition 4.14. Let A be an AF algebra. Then the following are equivalent: (1) A is isomorphic to the C ∗ -algebra of a row-finite graph in which each vertex connects to at most finitely many sinks; and (2) A has the form ( x∈X Mnx (C)) ⊕ A where X is an at most countably-infinite index set, each nx is a positive integer, and A is an AF algebra with no unital quotients. Proof. To see that (1) implies (2), we let E be a row-finite graph in which each vertex connects to at most finitely many sinks and such that A ∼ = C ∗ (E). Since A is an AF-algebra, E has no cycles. Let sinks(E) denote the collection {v ∈ E 0 : vE 1 = ∅} of sinks in E. Let H be the smallest saturated hereditary subset of E 0 containing sinks(E). Since each vertex connects to at most finitely many sinks, H is equal to the set of v ∈ E 0 such that vE n = ∅ for some n. Let F be the graph with vertices F 0 := E 0 \ H , edges F 1 = {e ∈ E 1 : r(e) ∈ / H } and range and source maps inherited from E. Note that the description of H above implies that F has no sinks; moreover F is row-finite because E is. We claim that C ∗ (E) ∼ =

K 2 (E ∗ v) ⊕ C ∗ (F ).

v∈sinks(E) 0 1 To prove this, we first define a Cuntz–Krieger E-family {qv : v ∈ E }, {te : e ∈ E } in ( v∈sinks(E) K(2 (E ∗ v))) ⊕ C ∗ (F ). We will denote the universal Cuntz–Krieger F -family by {pvF : v ∈ F 0 }, {seF : e ∈ F 1 }, and we will denote the matrix units in each K(2 (E ∗ v)) by


1613

v : α, β ∈ E ∗ v}. As a notational convenience, for v ∈ E 0 \ F 0 , we write p F = 0, and simi{Θα,β v larly for e ∈ E 1 \ F 1 , we write seF = 0. For v ∈ E 0 , let w ⊕ pvF qv := Θα,α w∈sinks(E) α∈vE ∗ w

and for e ∈ E 1 , let

w∈sinks(E)

α∈r(e)E ∗ w

te :=

v Θeα,α

⊕ seF .

Routine calculations show that {qv : v ∈ E 0 }, {te : e ∈ E 1 } is a Cuntz–Krieger E-family. This family clearly generates ( v∈sinks(E) K(2 (E ∗ v))) ⊕ C ∗ (F ), and each qv is nonzero because if w . An application of pvF = 0 then v must connect to a sink w in which case qv dominates some Θα,α the Gauge-Invariant Uniqueness Theorem [2, Theorem 2.1] implies that there is an isomorphism πq,t : C ∗ (E) →

K 2 (E ∗ v) ⊕ C ∗ (F )

v∈sinks(E)

such that πq,t (pv ) = qv and πq,t (se ) = te . To complete the proof of (1) implies (2), let X ⊂ sinks(E) denote the subset {v ∈ sinks(E): |E ∗ v| < ∞}, and for each v ∈ X let nv := |E ∗ v|. We have K(2 (E ∗ v)) = Mnv (C) for each v ∈ X. Recall that F is row-finite and has no sinks, so Theorem 4.7 implies that C ∗ (F ) has no unital quotient. For each v ∈ sinks(E) \ X, the C ∗ -algebra K(2 (E ∗ v)) is simple and nonunital. Thus 2 ∗ A := K (E v) ⊕ C ∗ (F ) v∈sinks(E)\X

has no finite-dimensional quotients. We get A∼ K 2 (E ∗ v) ⊕ C ∗ (F ) ∼ Mnv (C) ⊕ A = C ∗ (E) ∼ = = v∈X

v∈sinks(E)

as required. To see that (2) implies (1), let A = ( x∈X Mnx (C)) ⊕ A as in (2). By Theorem 4.7, there is a row-finite graph E with no sinks such that C ∗ (E ) ∼ = A . For each x ∈ X, let Ex be a copy of the graph A standard argument shows that C ∗ (Ex ) ∼ = Mnx (C). Moreover E := ( x∈X Ex ) E satisfies C ∗ (E) ∼ C ∗ (Ex ) ⊕ C ∗ (E ) ∼ = =A x∈X

as required.

2

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For completeness, we conclude the section with the following well-known result. Lemma 4.15. A C ∗ -algebra A is finite dimensional if and only if it is isomorphic to the C ∗ algebra of a finite directed graph with no cycles. Proof. If E is a finite directed graph with no cycles, then E ∗ is finite, and hence C ∗ (E) = span{sμ sν∗ : μ, ν ∈ E ∗ } is finite dimensional. On the other hand, if A is finite-dimensional, then there exist an integer n 1 and nonnegative n integers d1 , . . . , dn such that A ∼ = i=1 Mdi (C), and [17, Corollary 2.3] then implies that A is isomorphic to the C ∗ -algebra of a finite directed graph with no cycles. (Moreover, we remark that the last part of the proof of Proposition 4.14 actually shows that every finite-dimensional C ∗ -algebra is the C ∗ -algebra of a finite graph with no cycles.) 2 4.3. Obstructions to realizations Here we present a number of necessary conditions for an AF algebra to be an ultragraph C ∗ -algebra, an Exel–Laca algebra, or a graph C ∗ -algebra. Recall that an ultragraph C ∗ -algebra C ∗ (G) is an AF-algebra if and only if G has no cycles by [23, Theorem 4.1]. Proposition 4.16. Let G be an ultragraph and suppose that C ∗ (G) is an AF-algebra. If C ∗ (G) is commutative, then the ultragraph G has no ultraedges, and C ∗ (G) ∼ = c0 (G0 ). Proof. It suffices to show that G has no ultraedges. Suppose that e is an ultraedge in G, and let v = s(e). Since C ∗ (G) is commutative, we have pr(e) = se∗ se = se se∗ ps(e) , and hence r(e) = {s(e)}. Thus e is a cycle. This contradicts the hypothesis that C ∗ (G) is an AF-algebra. 2 Proposition 4.17. Let A be an AF-algebra that is also an Exel–Laca algebra. Then A does not have a quotient isomorphic to C, and for each n ∈ N there is a C ∗ -subalgebra of A isomorphic to Mn (C). Proof. There exists an ultragraph G = (G0 , G 1 , r, s) with bijective s such that C ∗ (G) ∼ = A (see Remark 2.10). The ultragraph G has no cycles. Let {pv }v∈G0 and {se }e∈G 1 be the generator of C ∗ (G) as in Definition 2.9. Suppose, for the sake of contradiction, that there exists a nonzero ∗-homomorphism χ : C ∗ (G) → C. Since χ is nonzero, there exists v ∈ G0 with χ(pv ) = 0. Let e ∈ G 1 be the unique ultraedge with s(e) = v. Since G has no cycles, we have v ∈ / r(e). Hence pv is orthogonal to se∗ se . Thus χ(se )2 χ(pv ) = χ(se )χ(se )χ(pv ) = χ s ∗ se pv = 0, e

and since χ(pv ) = 0, it follows that |χ(se )|2 = 0 and χ(se ) = 0. But then χ(pv ) = χ(se se∗ ) = χ(se )χ(se∗ ) = 0, which is a contradiction. Hence C ∗ (G) has no quotients isomorphic to C. Let n ∈ N. We will construct a C ∗ -subalgebra of C ∗ (G) isomorphic to Mn (C). Choose v1 ∈ 0 G and let e1 ∈ G 1 be the unique ultraedge with s(e1 ) = v1 . Then choose a vertex v2 ∈ r(e1 ). Since G has no cycles, we have v2 = v1 . Continuing in this manner, we can find distinct vertices


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v1 , v2 , . . . , vn ∈ G0 such that vk+1 ∈ r(ek ) for k = 1, 2, . . . , n − 1, where ek ∈ G 1 is the unique ultraedge with s(ek ) = vk . For 1 i, j n, we define Θi,j := sei sei+1 . . . sen−1 pvn se∗n−1 se∗n−2 . . . se∗j . One can check that {Θi,j : 1 i, j n} is a family of matrix units, and thus the C ∗ -subalgebra of C ∗ (G) generated by {Θi,j : 1 i, j n} is isomorphic to Mn (C). 2 Corollary 4.18. If A is an AF-algebra that is also an Exel–Laca algebra, then A has a Bratteli diagram (E, d) such that dv 2 for all v ∈ E 0 . Proof. Since A has no quotient isomorphic to C, the result follows from Lemma 3.1.

2

Corollary 4.19. No finite-dimensional C ∗ -algebra is isomorphic to an Exel–Laca algebra. Definition 4.20. We recall that a C ∗ -algebra A is said to be Type I if whenever π : A → B(H) is a nonzero irreducible representation, then K(H) ⊆ π(A). In the literature, the terms postliminary, GCR, and smooth are all synonymous with Type I. Proposition 4.21. Let C ∗ (E) be a graph C ∗ -algebra that is also an AF-algebra. Then every unital quotient of C ∗ (E) is Type I and has finitely many ideals. Proof. By Lemma 2.16, it suffices to show that if a graph C ∗ -algebra C ∗ (E) is a unital AFalgebra then C ∗ (E) is Type I and has finitely many ideals. Note that C ∗ (E) is a unital AF-algebra if and only if E has a finite number of vertices and no cycles. We first show that C ∗ (E) has finitely many ideals. Since E has no cycles, it satisfies Condition (K). Hence any ideal of C ∗ (E) is of the form I(H,S) for a saturated hereditary subset H of E 0 and a subset S ⊆ E 0 of the set of breaking vertices for H [8, Theorem 3.5]. Since the set E 0 of vertices of E is finite, there are only a finite number of such pairs (H, S). Thus C ∗ (E) has finitely many ideals. To prove that C ∗ (E) is of Type I, first observe that any graph with finitely many vertices and no cycles contains a sink v, and the ideal Iv generated by pv is then a nontrivial gaugeinvariant ideal which is Morita equivalent to C and hence of Type I (see [14, Proposition 2] and the subsequent remark in [14]). We shall show by induction on the number of nonzero ideals of C ∗ (E) that C ∗ (E) is Type I. Our basis case is when has just one nontrivial ideal I . That is, C ∗ (E) is simple, and then the Type I ideal Iv of the preceding paragraph is C ∗ (E) itself, proving the result. Now suppose as an inductive hypothesis that the result holds whenever C ∗ (E) has at most n distinct nonzero ideals, and suppose that C ∗ (E) has n + 1 such. Let v be a sink in E and let Iv be the corresponding nonzero Type I ideal as in the preceding paragraph. If C ∗ (E)/Iv is trivial, then C ∗ (E) = Iv is of Type I, so we may assume that C ∗ (E)/Iv is nonzero. Then Lemma 2.16 implies that C ∗ (E)/Iv is a unital AF-algebra that is a graph C ∗ -algebra. Moreover, C ∗ (E)/Iv has strictly fewer ideals than C ∗ (E), so the inductive hypothesis implies that C ∗ (E)/Iv is of Type I. Since an extension of a Type I C ∗ -algebra by a Type I C ∗ -algebra is Type I (see [19, Theorem 5.6.2]), it follows that C ∗ (E) is of Type I. 2

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Theorem 4.22. For a simple AF-algebra A we have the following. (1) If A is finite dimensional then A is isomorphic to a graph C ∗ -algebra but not isomorphic to an Exel–Laca algebra. (2) If A is infinite dimensional and unital then A is isomorphic to an Exel–Laca algebra but not isomorphic to a graph C ∗ -algebra. (3) If A is infinite dimensional and nonunital then A is isomorphic to a C ∗ -algebra of a rowfinite graph with no sinks (which is also isomorphic to the Exel–Laca algebra of a row-finite matrix by Lemma 2.4). In particular, each simple AF-algebra A is isomorphic to either an Exel–Laca algebra or a graph C ∗ -algebra. Proof. The statement in (1) follows from Lemma 4.15 and Corollary 4.19. For (2) we observe that if A is simple, infinite dimensional, and unital, then it follows from Corollary 4.6 that A is isomorphic to an Exel–Laca algebra. Since A is in particular unital, to see that A is not a graph C ∗ -algebra, it suffices by Proposition 4.21 to show that it is not of Type I. If we suppose for contradiction that A is of Type I, then as it is simple, we must have A ∼ = K(H) for some Hilbert space H. Since A is unital, H and hence K(H) must be finite-dimensional, contradicting that A is infinite dimensional. The statement in (3) follows from Theorem 4.7. The final assertion follows from (1), (2), and (3). 2 Corollary 4.23. If A is an infinite-dimensional UHF algebra, then A is not isomorphic to a graph C ∗ -algebra. 5. A summary of known containments In this section we use our results to describe how various classes of AF-algebras are contained in the classes of graph C ∗ -algebras, Exel–Laca algebras, and ultragraph algebras. We first examine the simple AF-algebras, where we have a complete description. Moreover, we see that the simple AF-algebras allow us to distinguish among the four classes of C ∗ -algebras of row-finite graphs with no sinks, graph C ∗ -algebras, Exel–Laca algebras, and ultragraph algebras. Second, we consider general AF-algebras, and while our description in this case is not complete, we are able to describe how the finite-dimensional and stable AF-algebras are contained in the classes of graph C ∗ -algebras, Exel–Laca algebras, and ultragraph algebras. Furthermore, we use our results to show that there are numerous other AF-algebras in the various intersections of these classes. 5.1. Simple AF-algebras Consider the following partition of the simple AF-algebras. simple

AFfinite := finite-dimensional simple AF-algebras, simple

AF∞,unital := infinite-dimensional simple AF-algebras that are unital, simple

AF∞,nonunital := infinite-dimensional simple AF-algebras that are nonunital.


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Fig. 2. A Venn diagram summarizing AF-algebra containments.

Theorems 4.22 and 4.7 imply that AF∞,nonunital = simple AF-algebras that are C ∗ -algebras of simple

row-finite graphs with no sinks, simple AFfinite

simple ∪ AF∞,nonunital

simple

simple

= simple AF-algebras that are graph C ∗ -algebras,

AF∞,unital ∪ AF∞,nonunital = simple AF-algebras that are Exel–Laca algebras and simple

simple

simple

AFfinite ∪ AF∞,unital ∪ AF∞,nonunital = simple AF-algebras that are ultragraph algebras. Hence these three classes of simple AF-algebras allow us to distinguish among the four classes of C ∗ -algebras of row-finite graphs with no sinks, graph C ∗ -algebras, Exel–Laca algebras, and ultragraph algebras. However, they do not allow us to distinguish between the classes of C ∗ algebras of row-finite graphs with no sinks and the intersection of graph C ∗ -algebras and Exel– Laca algebras. Nor do they allow us to distinguish between the classes of ultragraph C ∗ -algebras and the union of graph C ∗ -algebras and Exel–Laca algebras. To distinguish these classes we will need nonsimple examples. 5.2. More general AF-algebras For nonsimple AF-algebras, we cannot give such an explicit description. Nevertheless, in Fig. 2 we present a Venn diagram summarizing the relationships we have established for finite-

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Table 1 Examples of C ∗ -algebras lying in each region of Fig. 2. Region

Unital C ∗ -algebra

Nonunital C ∗ -algebra

(a) (b) (c) (d) (e) (f)

cc K+ M2∞ ⊕ C M2 (K+ ) — M2∞

c0 ⊕ cc c0 M2∞ ⊕ C ⊕ K M2 (K+ ) ⊕ K C ∗ (F2 ) M2∞ ⊕ K

dimensional and stable AF-algebras, and also give various examples in the intersections of our classes of graph C ∗ -algebras, Exel–Laca algebras, and ultragraph C ∗ -algebras. Table 1 presents, for each region of the Venn diagram of Fig. 2, both a unital and a nonunital example belonging to that region, with three exceptions: we give no examples of finitedimensional or stable AF algebras, nor any example of a unital AF algebra which is the C ∗ algebra of a row-finite graph with no sinks. Our reasons for these omissions are as follows: examples of finite-dimensional and stable AF algebras are obvious, and necessarily unital and nonunital respectively; and no unital example exists in region (e) by Theorem 4.7. In Table 1, we use the following notation: • • • • •

M2∞ denotes the UHF algebra of type 2∞ . K denotes the compact operators on a separable infinite-dimensional Hilbert space. K+ denotes the minimal unitization of the C ∗ -algebra K. c0 denotes the space {f : N → C | limn→∞ f (n) = 0}. cc denotes the space {f : N → C | limn→∞ f (n) ∈ C}.

• F2 denotes the graph

We now justify that the examples listed have the desired properties. (a) • The unital AF-algebra cc is not an ultragraph C ∗ -algebra since it is commutative and its spectrum is not discrete (see Proposition 4.16). • The nonunital AF-algebra c0 ⊕ cc is not an ultragraph algebra for precisely the same reason that cc is not. (b) • The minimal unitization K+ of the compact operators is isomorphic to the C ∗ -algebra of the graph

with two vertices v, w and infinitely many edges from v to w. Since, K+ has a quotient isomorphic to C, it is not an Exel–Laca algebra by Proposition 4.17. • The nonunital AF-algebra c0 is the C ∗ -algebra of the graph with infinitely many vertices and no edges. It is not an Exel–Laca algebra by Proposition 4.17. (c) • Since M2∞ is an infinite-dimensional simple AF-algebra, Theorem 4.22 implies that M2∞ is an Exel–Laca algebra and hence also an ultragraph algebra. In addition, C is a graph


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C ∗ -algebra so also an ultragraph C ∗ -algebra. Since the class of ultragraph C ∗ -algebras is closed under direct sums, M2∞ ⊕ C is a unital ultragraph C ∗ -algebra. It is not an Exel– Laca algebra because it has a quotient isomorphic to C (see Proposition 4.17), and it is not a graph C ∗ -algebra because it has a unital quotient M2∞ that is not Type I (see Proposition 4.21). • Since K and M2∞ ⊕ C are both ultragraph C ∗ -algebras, the direct sum M2∞ ⊕ C ⊕ K is a nonunital ultragraph C ∗ -algebra. It is neither a graph C ∗ -algebra nor an Exel–Laca algebra as above. (d) • The unital AF-algebra M2 (K+ ) is isomorphic to the C ∗ -algebra of the following graph

and it is also isomorphic to the Exel–Laca algebra of the matrix ⎛

0 ⎜0 ⎜ ⎜0 ⎜ ⎝0 .. .

1 0 0 0

1 1 0 0

1 0 1 0

1 0 0 1

···

..

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

.

It is not isomorphic to the C ∗ -algebra of a row-finite graph with no sinks by Theorem 4.7. • The nonunital AF-algebra M2 (K+ ) ⊕ K is isomorphic to both a graph C ∗ -algebra and an Exel–Laca algebra because its two direct summands have this property. It is not the C ∗ algebra of a row-finite graph with no sinks by Theorem 4.7 because it admits the unital quotient M2 (K+ ). (e) • There is no unital example in this region by Theorem 4.7. • Let F2 denote the graph

Then C ∗ (F2 ) is a graph C ∗ -algebra, and since F2 is cofinal with no cycles and no sinks, C ∗ (F2 ) is simple by [17, Corollary 3.10]. In addition, C ∗ (F2 ) is nonunital because F2 has infinitely many vertices. Since C ∗ (F2 ) is the C ∗ -algebra of a row-finite graph with no sinks, it is both a graph C ∗ -algebra and an Exel–Laca algebra (see Lemma 2.4). The function g : F20 → R+ defined by g(vi ) = 2−i is a graph trace with norm 1 (see [24, Definition 2.2]), and the existence of such a function implies that C ∗ (F2 ) is not stable by (a) ⇒ (c) of [24, Theorem 3.2]. (f) • As in example (c), the unital AF-algebra M2∞ is an Exel–Laca algebra but not a graph C ∗ -algebra. • As in example (c), the nonunital AF-algebra M2∞ ⊕ K is an Exel–Laca algebra but not a graph C ∗ -algebra.

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References [1] T. Bates, J.H. Hong, I. Raeburn, W. Szymański, The ideal structure of the C ∗ -algebras of infinite graphs, Illinois J. Math. 46 (2002) 1159–1176. [2] T. Bates, D. Pask, I. Raeburn, W. Szymański, The C ∗ -algebras of row-finite graphs, New York J. Math. 6 (2000) 307–324. [3] O. Bratteli, Inductive limits of finite dimensional C ∗ -algebras, Trans. Amer. Math. Soc. 171 (1972) 195–234. [4] J. Cuntz, W. Krieger, A class of C ∗ -algebras and topological Markov chains, Invent. Math. 56 (1980) 251–268. [5] K. Davidson, C ∗ -Algebras by Example, Fields Inst. Monogr., vol. 6, Amer. Math. Soc., Providence, RI, 1996. [6] K. Deicke, J.H. Hong, W. Szymański, Stable rank of graph algebras. Type I graph algebras and their limits, Indiana Univ. Math. J. 52 (4) (2003) 963–979. [7] D. Drinen, Viewing AF-algebras as graph algebras, Proc. Amer. Math. Soc. 128 (2000) 1991–2000. [8] D. Drinen, M. Tomforde, The C ∗ -algebras of arbitrary graphs, Rocky Mountain J. Math. 35 (2005) 105–135. [9] E.G. Effros, Dimensions and C ∗ -Algebras, Conference Board of the Mathematical Sciences, Washington, DC, 1981, v+74 pp. [10] R. Exel, M. Laca, Cuntz–Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999) 119–172. [11] R. Exel, M. Laca, The K-theory of Cuntz–Krieger algebras for infinite matrices, K-Theory 19 (2000) 251–268. [12] N. Fowler, M. Laca, I. Raeburn, The C ∗ -algebras of infinite graphs, Proc. Amer. Math. Soc. 8 (2000) 2319–2327. [13] K.R. Goodearl, D.E. Handelman, Classification of ring and C ∗ -algebra direct limits of finite-dimensional semisimple real algebras, Mem. Amer. Math. Soc. 69 (1987). [14] A. an Huef, I. Raeburn, D.P. Williams, Properties preserved under Morita equivalence of C ∗ -algebras, Proc. Amer. Math. Soc. 135 (2007) 1495–1503. [15] T. Katsura, P. Muhly, A. Sims, M. Tomforde, Ultragraph C ∗ -algebras via topological quivers, Studia Math. 187 (2008) 137–155. [16] T. Katsura, P. Muhly, A. Sims, M. Tomforde, Graph algebras, Exel–Laca algebras, and ultragraph algebras coincide up to Morita equivalence, J. Reine Angew. Math., in press. [17] A. Kumjian, D. Pask, I. Raeburn, Cuntz–Krieger algebras of directed graphs, Pacific J. Math. 184 (1998) 161–174. [18] A. Kumjian, D. Pask, I. Raeburn, J. Renault, Graphs, groupoids, and Cuntz–Krieger algebras, J. Funct. Anal. 144 (1997) 505–541. [19] G.J. Murphy, C ∗ -algebras and Operator Theory, Academic Press, San Diego, 1990. [20] I. Raeburn, Graph Algebras, CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., Providence, RI, 2005, vi+113 pp. Published for the Conference Board of the Mathematical Sciences, Washington, DC. [21] W. Szymański, Simplicity of Cuntz–Krieger algebras of infinite matrices, Pacific J. Math. 199 (2001) 249–256. [22] M. Tomforde, A unified approach to Exel–Laca algebras and C ∗ -algebras associated to graphs, J. Operator Theory 50 (2003) 345–368. [23] M. Tomforde, Simplicity of ultragraph algebras, Indiana Univ. Math. J. 52 (2003) 901–926. [24] M. Tomforde, Stability of C ∗ -algebras associated to graphs, Proc. Amer. Math. Soc. 132 (6) (2004) 1787–1795 (electronic). [25] M. Tomforde, Structure of graph C ∗ -algebras and their generalizations, in: Gonzalo Aranda Pino, Francesc Perera Domènech, Mercedes Siles Molina (Eds.), Graph Algebras: Bridging the Gap between Analysis and Algebra, Servicio de Publicaciones de la Universidad de Málaga, Málaga, Spain, 2006.


Optimal cubature formulas on compact homogeneous manifolds Alexander Kushpel Department of Mathematics, IMECC-UNICAMP, CXP 6065, 13083-859, Campinas, SP, Brazil Received 19 January 2009; accepted 28 April 2009 Available online 21 May 2009 Communicated by J. Bourgain

Abstract We find lower bounds for the rate of convergence of optimal cubature formulas on sets of differentiable functions on compact homogeneous manifolds of rank I or two-point homogeneous spaces. It is shown that these lower bounds are sharp in the power scale in the case of S2 , the unit sphere in R3 . © 2009 Elsevier Inc. All rights reserved. Keywords: Reconstruction; Data points; Polynomial; Volume

1. Introduction Let (Ω, Σ, η) be a measure space, where Ω is a compact domain in Rd , and K ∈ C(Ω) be a given set of real continuous functions, f : Ω → R. Let {x1 , . . . , xn } ⊂ Ω be a fixed set of points. It is natural to approximate the integral f dη Ω

by a cubature formula n

αk f (xk ),

k=1

E-mail addresses: [email protected], [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.018

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A. Kushpel / Journal of Functional Analysis 257 (2009) 1621–1629

where α = (α1 , . . . , αn ) ∈ Rn and to minimize the error of approximation n κn (K) := inf inf sup f dη − αk f (xk ). {x1 ,...,xn }⊂Ω {α1 ,...,αn }⊂R f ∈K Ω

(1)

k=1

The theory of cubature formulas has a long history and the first references are coming from the unmemorable times. In the modern epoch simple cubature (quadrature) formulas have been constructed by Kepler and Torricelli (1664), Simpson (1743), Newton and Cotes (1722). Different important methods of computing of integrals have been developed by Lagrange, Chebyshev, Bernstein, Krylov, Nikol’skij, Sobolev and many others. In the one-dimensional case, on S1 , the unit circle, it is known that the formula of rectangles S1

n 2πk 1 f (x) dx ≈ f n n k=1

r (S1 ) = {f | f (r) ∈ U (S1 )}, where U (S1 ) = {φ | is an optimal on Sobolev classes W∞ ∞ ∞ φ∞ 1} and r ∈ N. Remark that an analogous result is unknown for fractional values of r > 0. Observe that the extremal problem (1) and their discrete analogs are in spirit of the classical Kolmogorov n-widths of finite-dimensional sets. This range of problems has been extensively studied by Tikhomorov, Makovoz, Kashin, Gluskin and others (see [17] for more details and references). The problem of numerical integration over the surface of the unit sphere Sd in Rd+1 , d 2, is one of the most important in Numerical Analysis and Applications. The theory of functions on S2 has been initiated in the eighteenth century in works of Laplace and Legendre when the first cubature formulas appeared. Consequently, the problem on an optimal cubature formula on S2 (in general, Sd , d 2) remains open since that time. Therefore, the problem on the best cubature formula on S2 or (in general) on compact Riemannian manifolds Md it is natural to call the Laplace–Legendre problem. A fundamental problem in this area is connected with an optimal distribution of data points x1 , . . . , xn and finding an optimal coefficients α1 , . . . , αn to approximate “well” the integral. Even in the case of S2 , the two-dimensional sphere in R3 , it is not possible to construct, in general, an equidistributed set of data points since there are finitely many polyhedral groups. Different attempts to find sets of points on the sphere which imitate the role of the roots of unity on the unit circle usually led to deep problems of the Geometry of Numbers, Theory of Potential, etc., and usually these approaches give just a measure of a uniform distribution like cup discrepancy or minimum energy configurations. r (Md ) ⊂ C(Md ) on a We consider here optimal cubature formulas for the Sobolev classes W∞ d compact two-point homogeneous manifold M defined in Section 2. r (Md ) (see The respective extremal problem can be formulated as following. Let f ∈ W∞ d Section 2 for the definitions) and {x1 , . . . , xn } ⊂ M . Consider an information operator Tn ∈ L(C(Md ), Rn ),

Tn : C M d → R n , f (·) → f (x1 ), . . . , f (xn ) .


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Let f (x1 ), . . . , f (xn ) = ◦ Tn f be a given function, : Rn → R (a recovery operator). Consider the extremal problem r d M , R := κn W∞

inf

inf

sup

{x1 ,...,xn }⊂Md ∈R f ∈W r (Md ) ∞

f dν − ◦ Tn f ,

(2)

Md

where R is a given class of functions : Rn → R and dν is the invariant normalized measure r (Md )) instead of κ (W r (Md ), R) if R = RRn (the set of all on Md . We shall write κn (W∞ n ∞ functions : Rn → R). In particular, if R is the set of linear functionals over Rn then for some α1 , . . . , αn , ◦ Tn f =

n

αk f (xk ),

k=1

and (2) reduces to the Laplace–Legendre problem κn

r d W∞ M :=

n inf inf sup f dν − αk f (xk ). {x1 ,...,xn }⊂Md {α1 ,...,αn }⊂R f ∈W r (Md ) ∞

Md

(3)

k=1

The problem of construction of optimal cubature formulas splits into two parts, finding of a lower bond in (2) or (3) and obtainment of respective upper bounds. The lower bounds are of independent interest because they allow us to compare and classify a wide range of cubature forr (Md )), mulas. In this article we develop a new method of obtainment of lower bounds for κn (W∞ d r > 0 on general compact two-point homogeneous manifolds M which are sharp in the power scale in the case of S2 . On the first step we apply the result of Smolyak [1] to reduce the problem (2) to the linear case (3). Then, to find respective lower bounds for the rate of convergence of a cubature formula on r (Md ) we consider the set W r (Md ) ∩ ker T ∩ T , where T is the set of Sobolev classes W∞ n M M ∞ polynomials of order M on Md and dim TM = m n. Then, applying Bernstein’s inequality we reduce the problem to the consideration of the set m−r/d U∞ (Md ) ∩ TM , where U∞ (Md ) is the unit ball in L∞ (Md ). Finally, we need to find a polynomial γ , deg γ CM, where C > 0 is an absolute constant, such that Tn γ = 0, γ ∞ = 1 and the value Md γ dν is sufficiently big. Even in the case of the circle, S1 , it is very difficult to construct a polynomial with such properties. We show the existence of such objects using methods of geometry of Banach spaces. Remark that in applications we have a little information concerning special convex bodies in Rn which are connected with the structure of a fixed system of spherical harmonics on Md . This is a source of fundamental difficulties which occur if we try to apply the results of geometry of Banach spaces to various open problems in different spaces of functions. A useful tool in this range of problems are Levy means defined in Section 3. We employ estimates of Levy means in combination with the Bieberbach

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inequality and the Brunn–Minkowski theorem. Note that estimates of Levy means connected with different orthonormal systems have been obtained in [10–15]. The results we derive are apparently new even in the one-dimensional case and the method’s possibilities are not confined to the statements proved but can be applied in studying more general problems. We use several universal constants which enter into the estimates. These positive constants are mostly denoted by the letter C. We did not carefully distinguish between the different constants, neither did we try to get good estimates for them. The same letter will be used to denote different universal constants in different parts of the article. For ease of notation we will write an bn for two sequences, if an Cbn for n ∈ N and an bn , if C1 bn an C2 bn for all n ∈ N and some constants C, C1 and C2 . 2. Harmonic analysis A Riemannian manifold is two-point homogeneous if for any set of four points x1 , y1 , x2 , y2 with d(x1 , y1 ) = d(x2 , y2 ), d being the Riemannian metric on Md , there exists φ ∈ G such that φ(x1 ) = x2 and φ(y1 ) = y2 . A complete classification of the two-point homogeneous spaces was given in [18]. For information on this classification see, e.g., [4,6–9] They are: the spheres Sd , d = 1, 2, 3, . . . ; the real projective spaces P d (R), d = 2, 3, 4, . . . ; the complex projective spaces P d (C), l = d/2, d = 4, 6, 8, . . . ; the quaternionic projective spaces P d (H), d = 8, 12, . . . ; the Cayley elliptic plane P 16 (Cay). The superscripts here denote the dimension over the reals of the underlying manifolds Md . Each Md can be considered as the orbit space of some compact subgroup H of the orthogonal group G, that is Md = G/H. Let π : G → G/H be the natural mapping and e be the identity of G. On any such manifold there is an invariant Riemannian metric d(·,·), and an invariant Haar measure dν. The point o = π(e) which is invariant under all motions of H is called the pole of Md . A function Z = Z o : Md → R is called zonal with the pole o ∈ Md if Z o (h−1 ·) = Z o (·) for any h ∈ H. Consider a two-point homogeneous space Md . Let g be its metric tensor, ν its normalized volume element and its Laplace–Beltrami operator. In local coordinates xl , 1 l d, −1/2

= −(g)

∂ jk 1/2 ∂ , g (g) ∂xk ∂xj k

j

where gj k := g(∂/xj , ∂/xk ), g := |det(gj k )|, and (g j k ) := (gj k )−1 . It is well known that is an elliptic, self-adjoint, invariant under isometries, second order operator. The eigenvalues θk , k 0 of are discrete, nonnegative and form an increasing sequence 0 θ0 θ1 · · · θn · · · with +∞ the only accumulation point and θk k 2 . Corresponding eigenspaces Hk ,

k 0 are finite-dimensional, dk := dim Hk , orthogonal and L2 (Md , ν) = ∞ H . It is possible

M k=0 k d−1 d to show that dk := dim Hk k , m := dim TM M , where TM := k=0 Hk . Let us fix a real k of Hk . orthonormal basis {Ylk }dl=1 For an arbitrary φ ∈ Lp (Md ), 1 p ∞ with the formal Fourier series φ ∼ c0 +

dk k∈N l=1

ck,l (φ)Ylk ,

ck,l (φ) = Md

φYlk dν,


1625

the rth fractional integral φr , r > 0, is defined as φr ∼ c0 +

−r/2 θk

dk

ck,l (φ)Ylk ,

c0 ∈ R.

(4)

l=1

k∈N

The last equation defines the operator of fractional integration φr = Ir φ. The function φ (r) ∈ Lp (Md ), 1 p ∞, is called the rth fractional derivative of φ if φ (r) ∼

r/2

θk

dk

k∈N

ck,l (φ)Ylk .

l=1

Sobolev classes Wpr (Md ) are defined as sets of functions with formal Fourier expansions (4) where φp 1 and Md φ dν = 0 (see [2] for details). 3. The results The main result of this article is the following Theorem 1. For any r > 0 and ε > 0 we have r d r d M = κn W∞ M Cε n−r/d (log n)−(1+ε) , κn W∞ where C depends just on ε > 0. Proof. It was discovered by S.A. Smolyak and then published in [1] that for the recovery of linear functionals using linear information it is sufficient to use linear methods. More precisely, let K be a convex centrally symmetric subset of a Banach space X, Tn : X → Rn , Tn x = x1 , x , . . . , xn , x , xk ∈ X , 1 k n, and ξ a linear functional (ξ(x) = x0 , x, x0 ∈ X ). Then there is an optimal linear method of recovery, that is n inf n sup x0 , x − a k x k , x , inf sup x0 , x − ◦ Tn x = (a1 ,...,an )∈R x∈K ∈R x∈K k=1

r (Md ) where R = RR is the set of all functions : Rn → R. Remark that the Sobolev classes W∞ are convex and centrally symmetric. Hence, from (2) and (3) we get r d r d f dν , M = κn W∞ M inf sup κn W∞ n

Tn f ∈W r (Md )∩ker T n ∞

Md

where Tn f = (f (x1 ), . . . , f (xn )). Clearly, codim ker Tn n. In what follows we show that for any {x1 , . . . , xn } ⊂ Md there is a polynomial ∗ γ(x ∈ W∞ Md ∩ T2M ∩ ker Tn , dim TM = m n 1 ,...,xn )

1626


such that for any ε > 0

∗ γ(x dν C n−r/d (log n)−(1+ε) , 1 ,...,xn )

(5)

Md

where C > 0 depends just on . Let α = (α1 , . . . , αm ) ∈ Rm , β = (β1 , . . . , βm ) ∈ Rm and 1/2 be the Euclidean norm on Rm , Sm−1 = {α ∈ Rm | [α, β] = m k=1 αk βk . Let α(2) = [α, α] m m α(2) = 1} be the unit sphere in R , B(2) = {α ∈ Rn | α(2) 1} be the unit ball in Rm and Volm be the standard m-dimensional volume of subsets in Rm . Let us fix a norm · on Rm and denote by E the Banach space E = (Rm , · ) with the unit ball BE = V . The Levy mean MV is defined by m MV = M R , · = α dμ(α), Sm−1

where dμ is the invariant normalized measure on Sm−1 . Consider the coordinate isomorphism J : Rm → TM ,

dim TM = m,

that assigns to α = (α1 , . . . , αm ) = (αk,l , 0 k M, 1 l dk ) ∈ Rm the function J α = ξ α =

M dk k α m k=0 l=1 αk,l Yl ∈ TM . The definition α(∞) = ξ ∞ induces a norm on R . Put m B(∞) := α α ∈ Rm , α(∞) 1 , m := J B m . The following estimate plays an important role in our method [15] then B∞ (∞)

M Rm , · (∞) C(log m)1/2 .

(6)

Let V be a convex centrally symmetric body in Rm which is the unit ball in E(Rm , · ). A direct calculation (see, e.g., [16]) shows that Volm (V ) = α−m dμ(α) m ) Volm (B(2) Sm−1

and by convexity we get α Sm−1

−m

1/m −1 dμ(α) α dμ(α) = MV−1 . Sm−1

Hence, by (6) it follows that m m C m (log m)−m/2 Volm B(2) . Volm B(∞)

(7)

Let Ls ⊂ Rm be any s-dimensional subspace, (Ls )⊥ be the orthogonal complement of Ls and m ) be the orthogonal projection of B m onto (L )⊥ . It is easy to check that B m ⊂ P(Ls )⊥ (B(∞) s (∞) (∞) m−s m and therefore P m )⊂P m ) and Vol m )) Vol (B (B (P (B (B B(2) ⊥ ⊥ ⊥ m−s m−s (Ls ) (Ls ) (Ls ) (2) ). (∞) (2) (∞)


1627

Hence, m = Volm B(∞)

m Vols B(∞) ∩ (y + Ls ) dy.

dx =

m B(∞)

P(L

s )⊥

m ) (B(∞)

Thus, involving standard arguments connected with the Brunn–Minkowski theorem we get m ∩ (y + L )) Vol (B m ∩ (L )) for any y ∈ P m Vols (B(∞) s s s (Ls )⊥ (B(∞) ). Consequently, (∞) m m m Volm B(∞) Vols B(∞) ∩ Ls · Volm−s P(Ls )⊥ B(∞) m m−s . ∩ Ls · Volm−s B(2) Vols B(∞)

(8)

Hence, by (7) and (8), m ) Volm (B(2) m Vols B(∞) ∩ Ls C m (log m)−m/2 m−s . Volm−s (B(2) )

(9)

Comparing (9) with the Bieberbach inequality [3, p. 93], diam(V ) 2

Vols (V ) s ) Vols (B(2)

1/s ,

which is valid, in particular, for any convex centrally symmetric body V ⊂ Rs we get the lower m ∩L , bound for the diameter of the set B(∞) s m diam B(∞) ∩ Ls C m/s (log n)−m/(2s)

m ) Volm (B(2) m−s s ) Volm−s (B(2) )Vols (B(2)

Recall that m m/2 −1 m Volm B(2) = π Γ +1 2 and Γ (z) = zz−1/2 e−z (2π)1/2 1 + O z−1 ,

z → ∞.

Hence,

1/s

m ) Volm (B(2) m−s s ) Volm−s (B(2) )Vols (B(2)

m → ∞,

s → ∞,

m s

−1/2

m−1/(2s) ,

s m.

It means that for any 0 < λ < 1 and s = [λm] we have m diam B(∞) ∩ Ls Cλ (log m)−1/(2λ)

1/s .

1628


or for any fixed {x1 , . . . , xn } ⊂ Md , n = m − s, there is a polynomial tm ∈ TM , dim TM = m, such that tm (xk ) = 0, 1 k n, tm ∞ = 1 and tm 2 Cλ (log m)−1/(2λ) . Consider the polyno∗ 2 . Clearly, γ ∗ ∈ T ∗ ∗ −r/d , = (dim T2M )−r/d tm mial γ ∗ = γ(x 2M , γ 0, γ ∞ = (dim T2M ) 1 ,...,xn ) ∗ γ (xk ) = 0, 1 k n and γ ∗ dν = (dim T2M )−r/d tm 22 Cλ (dim T2M )−r/d (log m)−1/λ . Md r (Md ) or Applying Bernstein’s inequality [5] we get Cr,d (dim T2M )−r/d U∞ (Md ) ∩ T2M ⊂ W∞ r ∩T , from which (5) follows. 2 γ ∗ ∈ CW∞ 2M

Remark 1. Let x = (θ, φ) ∈ [0, π] × [0, 2π), b = 2l, l ∈ N, θl = πl/(2b), 0 l 2b − 1, φk = πk/b, 0 k 2b − 1 and alb

b/2−1 (2s + 1)lπ 1 1 lπ sin . = 2 sin 2b 2s + 1 2b 2b s=0

It has been shown in [14] that for any r > 0 r 2 r 2 κn W∞ S = κn W∞ S

2b−1 2b−1 sup f (x) dx − alb f (θl , φk ) n−r/2 , f ∈W r (S2 ) ∞

S2

l=0

k=0

where n = Card{(θl , φk ), 0 l 2b − 1, 0 k 2b − 1} b2 . From Theorem 1 it follows that r 2 κn W∞ S n−r/2 (log n)−(1+ε) for any ε > 0. Hence, the lower bound given by Theorem 1 is sharp in the power scale in the case of Sobolev’s classes on S2 . Acknowledgments I would like to thank the referees and the Communicating Editor for the useful suggestions and comments. This research has been partially supported by the research grant FAPESP 2007/56162-8 (Brazil). References [1] N.S. Bakhvalov, On optimal linear methods of approximation of operators on convex classes of functions, Zh. Vychisl. Mat. Mat. Fiz. 11 (1971) 1014–1018. [2] B. Bordin, A. Kushpel, J. Levesley, S. Tozoni, Estimates of n-widths of Sobolev’s classes on compact globally symmetric spaces of rank 1, J. Funct. Anal. 202 (2003) 307–326. [3] Yu.D. Burago, V.A. Zalgaller, Geometric Inequalities, Springer-Verlag, Berlin, Aidelberg, New York, London, Paris, Tokyo, 1980. [4] E. Cartan, Sur la determination d’un systeme orthogonal complet dans un espace de Riemann symetrique clos, Circ. Mat. Palermo 53 (1929) 217–252. [5] Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81 (4) (1998) 323–348.


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[6] R. Gangolli, Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters, Ann. Inst. H. Poincaré 3 (1967) 121–226. [7] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. [8] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965) 153–180. [9] T. Koornwinder, The addition formula for Jacobi polynomials and spherical harmonics, SIAM J. Appl. Math. 25 (2) (1973) 236–246. [10] A.K. Kushpel, Estimates of Lévy means and medians of some distributions on a sphere, in: Fourier Series and Their Applications, Inst. Math., Kiev, 1992, pp. 49–53. [11] A.K. Kushpel, Estimates of the Bernstein widths and their analogs, Ukrainian Math. J. 45 (1) (1993) 54–69. [12] A.K. Kushpel, Levy means associated with two-point homogeneous spaces and applications, in: 49 Seminário Brasileiro de Análise, Campinas, SP, 1999, pp. 807–823. [13] A.K. Kushpel, Estimates of n-widths and -entropy of Sobolev’s sets on compact globally symmetric spaces of rank 1, in: 50 Seminário Brasileiro de Análise, São Paulo, SP, 1999, pp. 53–66. [14] A.K. Kushpel, Optimal distribution of data points on S2 and approximation of Sobolev’s classes in L∞ (S2 ), J. Concr. Appl. Math. 2 (1) (2004) 77–89. [15] A.K. Kushpel, S.A. Tozoni, On the problem of optimal reconstruction, J. Fourier Anal. Appl. 13 (4) (2007) 459–475. [16] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press, Cambridge, 1989. [17] V.M. Tikhomirov, Approximation theory, in: R. Gamkrelidze (Ed.), Modern Problems in Mathematics, VINITI, Moscow, 1987. [18] H.C. Wang, Two-point homogeneous spaces, Ann. of Math. 55 (1952) 177–191.


Dyadic-like maximal operators on L log L functions Antonios D. Melas Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece Received 15 April 2008; accepted 5 June 2009 Available online 27 June 2009 Communicated by J. Bourgain

Abstract We study the following well-known property of the dyadic maximal operator Md on Rn (see [E.M. Stein, Note on the class L log L, Studia Math. 32 (1969) 305–310] for the case of the Hardy–Littlewood maximal function): If φ is integrable and supported in a dyadic cube Q then Md φ is integrable over sets of finite measure if and only if |φ| log(1 + |φ|) is integrable and the integral of Md φ can be estimated both from above and from below in terms of the integral of |φ| log(1 + |φ|) over Q. Here we define and explicitly evaluate Bellman functions related to this property and the corresponding estimates (both upper and lower) for the integrals thus producing sharp improved versions of the behavior of Md on the local L log L spaces. © 2009 Elsevier Inc. All rights reserved. Keywords: Bellman; Dyadic; Maximal

1. Introduction It is well known [13] that the Hardy–Littlewood maximal operator M on Rn has the folover B if and only if lowing property: If φ is supported in a ball B then Mφ is integrable , C > 0, depending only on B, |φ| log(1 + |φ|) < +∞ and that there are constants C , C , C 1 2 1 2 B such that C1 |φ| log 1 + |φ| − C1 Mφ C2 |φ| log 1 + |φ| + C2 (1.1) B

B

B

holds for all such φ. An easy scaling argument shows that C1 , C2 cannot be removed. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.005

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A.D. Melas / Journal of Functional Analysis 257 (2009) 1631–1654

Clearly the same holds for the case of the dyadic maximal operator on Rn

1 Md φ(x) = sup |Q|

φ(u) du: x ∈ Q, Q ⊆ Rn is a dyadic cube .

(1.2)

Q

In this paper we will produce sharp versions of the above property for the dyadic maximal operator. First we study the upper bound in (1.1) and we introduce the Bellman type function:

1 Blog (F, f, k) = sup |Q|

Md φ: φ nonnegative, measurable, E

AvQ (φ + 1) log(φ + 1) F, AvQ (φ) = f, E ⊆ Q measurable with |E| = k .

(1.3)

The exact determination of this will give further information on the deeper analytic properties of the dyadic maximal operator on functions φ supported in a dyadic cube and related to the integral of Md φ on sets of finite measure (note that Md φ outside the cube Q where φ is supported is trivially determined depending only on f ). Bellman functions relating different norms of φ and Md φ have been studied extensively in [5]. However the one defined in (1.3) cannot be studied by the methods there. Here we will use a combination of some of the methods from [5] with those in Section 7 of [3] in order to determine it. For more on Bellman functions and their relation to harmonic analysis we refer to [7–9,17]. For the exact evaluation of Bellman functions in certain cases we refer to [2,1,3,5,6,11,12,14–16,10]. We also note the recent approach initiated in [10], and also used in [16], to certain Bellman functions via PDE methods which has given alternative proofs of the results in [3] plus certain more general ones. Actually as in [3] we will take the more general approach of defining Bellman functions with respect to the maximal operator on a nonatomic probability space (X, μ) equipped with a tree T (see Section 2) thus defining

T Blog (F, f, k) = sup

MT φ dμ: φ 0 is measurable with E

(φ + 1) log(φ + 1) dμ F, X

φ dμ = f, X

E ⊆ X is measurable with μ(E) = k .

(1.4)

T (F, f ) the function B T (F, f, 1) corresponding to E = X in (1.4) and we will We denote by Dlog log evaluate this first. To describe the result consider the function V : [1, +∞) → [1, +∞) given by

V (z) =

ez−1 . z

(1.5)


1633

Clearly this is strictly increasing and we let U : [1, +∞) → [1, +∞) denote the inverse V −1 (z)−1) of V . Moreover differentiating exp(U = z we get U (z) U (z) =

U (z) z(U (z) − 1)

(1.6)

on z > 1. Then our first main theorem is the following. Theorem 1. For any nonatomic probability space (X, μ), any tree T on (X, μ) and any F, f with (f + 1) log(f + 1) < F the corresponding Bellman function is given by ⎧ ⎨ (f + 1)U ( eF /(f +1) ) − 1 if F < f (f + 1), f +1 T Dlog (F, f ) = ⎩ F + f + f log F −f if f (f + 1) F, f2

(1.7)

where U : [1, +∞) → [1, +∞) is the inverse V −1 of V . After this theorem and since the right-hand side in (1.7) is strictly increasing in F (for each T (F, f ) = fixed f ) it follows easily that we may replace the F in the definition (1.4) of Dlog T (F, f, 1) by = F . But we have initially used F instead of the usual = F in (1.4) for Blog

T (F, f ) is a phenomenon technical reasons. Also the double formula in (1.7) for the function Dlog not appearing in the Bellman functions studied in [3] but in those studied [5] (mixed norms) where it is also explained. F /(f +1) It is easy to see that U ( e f +1 ) < 1 + f F+1 whenever F < f (f + 1). Thus we get

T (F, f ) < F + f in this case. Of course due to scaling reasons no estimate of the form Dlog

T (F, f ) < C(F + f ) can hold for all F, f . However using the rough estimate log x < x for Dlog x = F /f 2 > 1 in the formula of the case f (f + 1) F we have the following estimate T (F, f ) < F + f + Dlog

√

F

(1.8)

T (F, f ) is holding for all F, f . This shows that as F → +∞ and f is fixed the main term in Dlog actually F . T is Moreover one can verify directly (although the computation is messy) that the function Dlog concave. However it is much more instructive to show the concavity using the Bellman function dynamics of the problem (see [7]). This has to do with the way the main variables F and f as T (F, f ) are behaved when the space (X, μ) is “split” into probability spaces on the well as Dlog children of X in T . We will describe the situation only in the case where X = [0, 1), μ = |.| is Lebesgue measure and T consists of all (left closed, right open) dyadic subintervals of X. This T (F, f ) is actually independent of T . is enough since we infer from Theorem 1 that Dlog We let X− = [0, 1/2) and X+ = [1/2, 1) be the two children of X in T and given F, f > 0 with (f + 1) log(f + 1) < F and φ 0 measurable with X (φ + 1) log(φ + 1) = F and X φ = f we denote by φ± the restrictions of φ to the probability spaces (X± , 2|.|) (which are equivalent

1634


to (X, |.|)) and we let F± = 2 have

X± (φ±

+ 1) log(φ± + 1) and f± = 2

1 F = (F− + F+ ) 2

X±

φ± . Then we clearly

1 and f = (f− + f+ ) 2

(1.9)

these equations constituting the Bellman function dynamics of the problem. On the other hand given F± , f± > 0 with (f± + 1) log(f± + 1) < F± let F, f be defined from (1.9) and for any ε > 0 we choose φ± 0 measurable on X± (respectively) satisfying T F± = 2 X± (φ± + 1) log(φ± + 1), f± = 2 X± φ± and 2 X± MT± φ± > Dlog± (F± , f± ) − ε, where T± are the subtrees of dyadic subintervals of X± . But (X± , 2|.|) are equivalent to (X, |.|) so T T and so considering the function φ which is equal to φ on X and to φ on X we Dlog± = Dlog − − + + easily get T Dlog (F, f )

MT φ =

X

MT− φ− +

X−

MT+ φ+

X+

1 T 1 T > Dlog− (F− , f− ) + Dlog+ (F+ , f+ ) − ε 2 2 1 T T (F+ , f+ ) − ε = Dlog (F− , f− ) + Dlog 2 T (F, f ) (for this and hence for any tree T ) which as ε → 0+ and in view of (1.9) implies that Dlog is concave. T (F, f, k). Given 0 < k 1 we define Next, using Theorem 1, we evaluate the function Blog

τk (x) = log

x(x + k) 1 + x + 1 k(x + 1)

(1.10)

on x 0. Since τk (x) = x(x+1−k) > 0 on x > 0 and τk (0) = 0 we let Tk : [0, +∞) → [0, +∞) k(x+1)2 denote the inverse function of τk . If F, f > 0 are such that F > (f + 1) log(f + 1) it is clear that Tk ( f F+1 − log(f + 1)) < f if and only if τk (f ) > f F+1 − log(f + 1) which is equivalent to f ( fk + 1) > F . If this happens we write ξk (F, f ) = Tk ( f F+1 − log(f + 1)) ∈ (0, f ). Then we can state the following. Theorem 2. Given F, f, k > 0 with k 1 and (f + 1) log(f + 1) < F we have: ⎧ ⎨ (f +1)(ξk (F,f )+k) U ( keξk (F,f )/k ) − k

T Blog (F, f, k) = ⎩

ξk (F,f )+1

F + f + f log

ξk (F,f )+k k(F −f ) f2

if F < f ( fk + 1), if f ( fk + 1) F.

(1.11)

It is easy to see, noting that the equation satisfied by ξ = ξ1 (F, f ) is ξ − log(ξ + 1) = f F+1 − log(f + 1) that by taking k = 1 in (1.11) one obtains (1.7). However we have stated Theorem 1 first since it constitutes an essential step in proving the more general Theorem 2. Now we turn to sharp forms of the lower estimate in (1.1). We will see that no nontrivial lower estimate exists for all trees T . Because of that we will restrict our attention to N -homogeneous


1635

trees that resemble the dyadic tree in Rn which is 2n -homogeneous (the definition is given in the next section). Then assuming that T is N -homogeneous on the probability space (X, μ) we define the following function MT φ dμ: φ 0 is measurable, L (F, f ) = inf T

X

φ dμ = f and X

φ log+

φ dμ = F . f

(1.12)

X

Then we have the following. Theorem 3. If the tree T is N -homogeneous and F, f > 0 then LT (F, f ) =

N −1 F + f. N log N

(1.13)

Thus in particular for the dyadic maximal operator in Rn we get for any φ 0 measurable and supported in the cube Q0 = [0, 1]n that the following sharp estimate holds

2n − 1 Md φ n 2 n log 2

Q0

Q0

φ φ log φ1 +

+ φ1 .

(1.14)

Also by taking N → ∞ we conclude that there is no uniform lower estimate, other than the trivial one X MT φ dμ X φ dμ, holding for all trees T . Theorem 3 provides an example where an infimum Bellman type function is evaluated. For another interesting example related to the dyadic Carleson embedding theorem we refer to [16]. Note however that LT (F, f ) does not satisfy the usual Bellman type dynamics (that is (1.9) in our case) since the definition of F involves the variable f (see (1.12)). We have chosen the growth functions in the above theorems in order to get results that are readable (this would not be the case if we had chosen x log(1 + x) instead). Of course these combined with the L1 norm are equivalent size conditions on φ. In Section 2 we give a general procedure that can be used to evaluate Bellman functions involving the integral of MT φ. We are not aiming at a general theory as in [5] but rather on a more direct computation scheme that can be used for specific growth functions, and especially for the ones like x log x where the theory in [5] does not apply. As applications other than the proof of Theorem 1, which is given in Section 3, we will compute here the corresponding to (1.12) supremum Bellman function as well as one related to the L∞ norm of φ (the last one has been also found in [5]). Other applications of this will be given elsewhere. In Section 3 we will also prove Theorem 2 by a detailed study of the function in (1.7) combined with certain methods from [3]. Then in Section 4 we will prove Theorem 3. 2. Trees and maximal operators As in [3] we let (X, μ) be a nonatomic probability space (i.e. μ(X) = 1). Then we give the following.

1636


Definition 1. (a) A set T of measurable subsets of X will be called a tree if the following conditions are satisfied: (i) X ∈ T and for every I ∈ T we have μ(I ) > 0. (ii) For every I ∈ T there corresponds a finite subset C(I ) ⊆ T containing at least two elements such that the elements of C(I ) are pairwise disjoint subsets of I and I = C(I ). (iii) T = m0 T(m) where T(0) = {X} and T(m+1) = I ∈T(m) C(I ). (iv) limm→∞ supI ∈T(m) μ(I ) = 0. (b) A tree T on (X, μ) will be called N -homogeneous (where N > 1 is an integer) if it satisfies the following additional conditions: (i) For every I ∈ T the set C(I ) consists of exactly N elements of T each having measure equal to N −1 μ(I ). (ii) The family T differentiates L1 (X, μ). We remark that the above definition can be given under the assumption that the elements of each C(I ) are only pairwise almost disjoint, that is ifA, B ∈ C(I ) and A = B then μ(A ∩ B) = 0. However by considering X \ E(T ), where E(T ) = I ∈T J1 ,J2 ∈C (I ), J1 =J2 (J1 ∩ J2 ) clearly has measure 0, the above makes no difference. Examples. 1. If Q0 is the unit cube Rn we let E be the union of all the boundaries of all dyadic cubes in Q0 then let X = Q0 \ E and T be the set of all open dyadic cubes Q ⊆ Q0 . Here N = 2n and each C(Q) is the set of the 2n subcubes of Q obtained by bisecting its sides. More generally for any integer m > 1 we may consider all m-adic cubes Q ⊆ Q0 with C(Q) being the set of the mn open subcubes of Q obtained by dividing each side of it into m equal parts. 2. Given the integers d1 , . . . , dn 1 and m > 1 we can define T on X equal to Q0 minus a certain set of measure 0 by setting for each open parallelepiped R the family C(R) to consist of the open parallelepipeds formed by dividing the dimensions of R into md1 , . . . , mdn equal parts respectively. For example if n = 2, m = 2, d1 = 1 and d2 = 2 we get the set of dyadic parabolic rectangles contained in [0, 1]2 . 3. The family of rectangles {[0, 1) × I : I is a dyadic subinterval of [0, 1)} on the probability space [0, 1)2 equiped with the Lebesgue measure is a tree that satisfies condition (i) of Definition 1(b) with N = 2 but is not 2-homogeneous since it does not satisfy condition (ii) of the same definition. An easy induction shows that each family T(m) consists of pairwise disjoint sets whose union is X. Moreover if x ∈ X \ E(T ) then for each m there exists exactly one Im (x) in T(m) containing x. For every m > 0 there is a J ∈ T(m−1) such that Im (x) ∈ C(J ). Since then x ∈ J we must have J = Im−1 (x). Hence the set A(x) = {I ∈ T : x ∈ I } forms a chain I0 (x) = X I1 (x) · · · with Im (x) ∈ C(Im−1 (x)) for every m > 0. From this remark it easily follows that if I, J ∈ T and I ∩ J is nonempty then I ⊆ J or J ⊆ I . In particular for any I, J ∈ T we have either I ∩ J is empty or one of them is contained in the other. The condition (ii) in Definition 1(b) can now be described as follows: “For any ψ ∈ L1 (X, μ) we have limm→∞ in X.”

1 μ(Im (x)) Im (x) ψ

dμ = ψ(x) for μ-almost every x


1637

This condition will be needed only in Theorem 3 (see Section 4) and all other results in this paper hold without this assumption. In Example 3 above it is easy to see that one can construct 1 functions φ with 0 φ(x, y) dx = 1 for all y but with [0,1)2 φ log+ φ arbitrary large. Thus Theorem 3 cannot hold in this case. However Theorems 1 and 2 hold. The following lemma gives another property of T that will be useful later. For a proof see [3]. Lemma 1. For every I ∈ T and every α such that 0 < α < 1 there F (I ) ⊆ T exists a subfamily consisting of pairwise disjoint subsets of I such that μ( J ∈F (I ) J ) = J ∈F (I ) μ(J ) = (1 − α)μ(I ). Next let S be a finite subset of T such that X ∈ S. For any I ∈ S with I = X we let I ∗ denote the unique minimal ancestor of I in S (i.e. the minimal element of {J ∈ S: I J }) and setting

AI = I \

aI = μ(AI ),

J,

(2.1)

J ∈S : J ∗ =I

we easily get

I=

and so μ(I ) =

AJ

S J ⊆I

aJ

(2.2)

S J ⊆I

for any I ∈ S (if I is a minimal element of S then clearly AI = I ). Then we will need the following (this is a special case of Theorem 1 in [3] but we include it here since its proof is much simpler). Lemma 2. For any finite tree S and any increasing and convex function Ψ : [0, +∞) → R we have

aI Ψ

I ∈S

I ⊆J

aJ μ(J )

∞

Ψ (u)e−u du.

(2.3)

0

Proof. Since Ψ (x) = Ψ (0) + Ψr (0)x

∞ + (x − λ)+ dΨr (λ)

(2.4)

0

where x + = max(x, 0) and Ψr denotes the right derivative of Ψ , it suffices to prove (2.3) when Ψ (x) = Ψλ (x) = (x − λ)+ where λ 0 is fixed. In this case (2.3) reads I ∈S

aI

I ⊆J

aJ −λ μ(J )

+

e−λ .

(2.5)

We will now prove (2.5) for all λ by induction on the size of S. If S = {X} then aX = 1 and so (2.5) becomes (1 − λ)+ e−λ which holds since e−x > 1 − x whenever 0 < x < 1. Now assuming (2.5) for any λ 0 and any tree (on any (X, μ, T ))

1638


having less elements than S we let {J1 , . . . , Jk } be all the elements J of S with J ∗ = X. Then when λ aX the induction hypothesis applied to the subtrees of S with tops J1 , . . . , Jk on the 1 probability spaces (Ji , μ(J μ) and with λ − aX 0 instead of λ gives i)

Bi =

aI

I ∈S I ⊆Ji

I ⊆J ⊆Ji

+ aJ − (λ − aX ) μ(J )

μ(Ji )e−λ+aX

(2.6)

for any i and so I ∈S

aI

I ⊆J

aJ −λ μ(J )

+

= aX (aX − λ)+ +

k

Bi

i=1

=

k

Bi

i=1

k

μ(Ji )e−λ+aX = (1 − aX )e−λ+aX < e−λ .

(2.7)

i=1

On the other hand if 0 λ < aX < 1 the left-hand side in (2.5) becomes I ∈S

aI

I ⊆J

aJ aJ −λ = aI − λ aI μ(J ) μ(J ) J ∈S

=

I ⊆J

I ∈S

aJ − λ = 1 − λ < e−λ

(2.8)

J ∈S

and this completes the induction.

2

Now given any tree T we define the maximal operator associated to it as follows MT ψ(x) = sup AvI |ψ| : x ∈ I ∈ T

(2.9)

1 for every ψ ∈ L1 (X, μ) where for any nonnegative φ ∈ L (X, μ) and for any I ∈ T we have 1 written AvI (φ) = μ(I ) I φ dμ. Given an integer m > 0 and λP 0 for each P ∈ T(m) we consider the function φ given by

φ=

λP χP

(2.10)

P ∈T(m)

(where χP denotes the characteristic function of P ). For every x ∈ X we let Iφ (x) denote the unique largest element of the set {I ∈ T : x ∈ I and MT φ(x) = AvI (φ)} (which is nonempty since AvJ (φ) = AvP (φ) whenever P ∈ T(m) and J ⊆ P ). Next for any I ∈ T we define the set AI = A(φ, I ) = x ∈ X: Iφ (x) = I

(2.11)


1639

and we let S = Sφ denote the set of all I ∈ T such that AI is nonempty. It is clear that each such AI is a union of certain P ’s from T(m) and moreover

MT φ =

AvI (φ)χAI .

(2.12)

I ∈S

We also define the correspondence I → I ∗ with respect to S as before. This is defined for every I in S that is not maximal with respect to ⊆. We also write yI = AvI (φ) for every I ∈ S. The main properties of the above are given in the following (see also [3] and [4]). Lemma 3. (i) For every I ∈ S we have I = S J ⊆I AJ . (ii) For every I ∈ S we have AI = I \ J ∈S : J ∗ =I J and so μ(AI ) = μ(I ) − J ∈S : J ∗ =I μ(J ) 1 and AvI (φ) = μ(I J ∈S : J ⊆I AJ φ dμ. ) (iii) For an I ∈ T we have I ∈ S if and only if AvQ (φ) < AvI (φ) whenever I ⊆ Q ∈ T , I = Q. In particular X ∈ S and so I → I ∗ is defined for all I ∈ S such that I = X. (iv) If I, J ∈ S are such that J ∗ = I then yI < yJ

μ(F ) yI μ(J )

(2.13)

where F is the unique element of the whole tree T such that J ∈ C(F ). In particular if T is N -homogeneous then yI < yJ NyI . Proof. (i) Clearly X = J ∈S AJ . Fix I ∈ S. Supposing that x ∈ A(φ, J ) ∩ I for some J we have x ∈ I ∩ J = ∅ and so either I ⊆ J or J ⊆ I . Suppose now that I J . Then also AvJ (φ) = MT φ(x) AvI (φ) and so I cannot be an Iφ (z) for any z ∈ I . Therefore A(φ, I ) = ∅ contradicting the assumption I ∈ S. Hence we must have J ⊆ I and this easily implies that I is the union of all AJ ’s for J ⊆ I . (ii) Follows easily from (i). (iii) One direction follows from the definition of the Iφ ’s. For the other assume that I ∈ T(s) satisfies the assumption. Since AvJ (φ) =

F ∈C (J ) μ(F ) AvF (φ)

F ∈C (J ) μ(F )

(2.14)

we conclude that for each J ∈ T there exists J ∈ C(J ) such that AvJ (φ) AvJ (φ). Starting from I and applying the above m − s times we get a chain I = I0 ⊇ I1 ⊇ · · · ⊇ Im−s such that I(r) ∈ T(s+r) for each s and moreover AvIm−s (φ) AvIm−s+1 (φ) · · · AvI1 (φ) AvI0 (φ) = AvI (φ). Now from this and the assumption on I it is clear that Iφ (x) = I for every x ∈ Im−s and therefore I ∈ S. (iv) The inequality yI < yJ follows from (iii). For the other first note that clearly F ⊆ I . We claim that AvF (φ) yI . Indeed I ∈ S implies that AvQ (φ) < yI whenever I ⊆ Q, I = Q and so if AvF (φ) > yI there would exist F ∈ T such that F ⊆ F ⊆ I , F = I and AvF (φ) > AvQ (φ)

1640


whenever F ⊆ Q, F = Q. But this combined with (iii) implies that F must be in S contradicting our assumption J ∗ = I . Thus we get since J ⊆ F yJ =

1 μ(J )

φ dμ

1 μ(J )

J

which completes the proof.

φ dμ =

μ(F ) μ(F ) AvF (φ) yI μ(J ) μ(J )

(2.15)

F

2

The above lemma shows that this linearization MT φ may be viewed as a multiscale version of the classical Calderon–Zygmund decomposition. Now writing aI = μ(AI ) and xI = aI−1 AI φ dμ for every I ∈ S the above lemma and (2.12) imply that: MT φ =

1 μ(I )

I ∈S

aJ xJ χAI

φ dμ =

and

J ∈S , J ⊆I

aI xI .

(2.16)

I ∈S

X

Next let G : [0, +∞) → [0, +∞] and Ψ : [0, +∞) → [0, +∞) be two convex and increasing functions such that G(0) = Ψ (0) = 0 and G is the Legendre transform of Ψ , that is for every x > 0 we have G(x) = sup xy − Ψ (y) .

(2.17)

y>0

Note that G is allowed to be extended-valued (but Ψ is not). We thus have xy G(x) + Ψ (y)

(2.18)

for all x, y 0. Moreover we define Ψ+ : R → [0, +∞) to be the function which is equal to the right derivative Ψr of Ψ on [0, +∞) and to 0 on (−∞, 0). Noting that for any x, z 0 we have xΨr (z) − Ψ (x) zΨr (z) − Ψ (z) we conclude that u+ Ψ+ (u) = Ψ u+ + G Ψ+ (u)

(2.19)

for all u ∈ R. We define also the following Bellman function related to G T DG (F, f ) = sup

MT φ dμ: φ 0 is measurable, X

X

when 0 < f and G(f ) < F .

G ◦ φ dμ F,

φ dμ = f X

(2.20)


1641

Using the decomposition of MT φ (and φ) given in (2.16) we can now prove the following: Lemma 4. Given a nonnegative function φ of the form (2.10) with G ◦ φ dμ F and given any c > 0, λ ∈ R we have X

1 F MT φ dμ + c c

X

∞

X φ dμ

= f and

Ψ c(u + λ)+ e−u du − λf.

(2.21)

0

Proof. Using the above notation we note that by Jensen’s inequality

aI G(xI )

I ∈S

G(φ) dμ F.

(2.22)

I ∈S A

I

Now if λ 0 we use (2.18) and Lemma 2 to write

MT φ dμ + λf

c

=c

aJ aI xI μ(J )

J ∈S

X

=

J ∈S , I ⊆J

aI G(xI ) +

I ∈S

aI xI + cλf

J ∈S , I ⊆J

caI xI

I ∈S

aJ +λ μ(J )

aI Ψ c

I ∈S

∞ F +

J ∈S , I ⊆J

aJ + cλ μ(J )

Ψ c(u + λ) e−u du.

(2.23)

0

If λ < 0 then we write S ∗ = {I ∈ S: in (2.23) c

aI xI

I ∈S ∗

aI J ∈S , I ⊆J μ(I )

J ∈S , I ⊆J

aI G(xI ) +

I ∈S ∗

aJ μ(J )

I ∈S ∗

aI Ψ c F + I ∈S

∞ F + 0

J ∈S , I ⊆J

I ∈ S ∗ aI xI

to get as

+ λf ∗

aI Ψ c

> −λ} and f ∗ =

J ∈S , I ⊆J

aJ +λ μ(J )

Ψ c(u + λ)+ e−u du

aJ + cλ μ(J )

+

(2.24)

1642


using Lemma 2 for the convex increasing function x → Ψ (c(x + λ)+ ), whereas

c

I ∈S \S ∗

aI xI

J ∈S , I ⊆J

aJ μ(J )

−cλ

aI xI = −cλ f − f ∗ .

(2.25)

I ∈S \S ∗

Adding now (2.24) and (2.25) and using the first equality in (2.23) we get (2.21).

2

Using this we have the following. Proposition 1. Let f, F > 0 be given such that G(f ) < F . Assume that c > 0 and λ ∈ R satisfy ∞

Ψ+

∞

c(u + λ) e−u du = f

and

0

G ◦ Ψ+ c(u + λ) e−u du = F.

(2.26)

0

Then we have T DG (F, f ) =

∞

uΨ+ c(u + λ) e−u du.

(2.27)

0

Proof. Taking any θ > 0 we define α = α(θ ) = 1 − e−θ ∈ (0, 1) and as in [3], using Lemma 1, we choose for every I ∈ T a family F (I ) ⊆ T of pairwise disjoint subsets of I such that J ∈F (I ) μ(J ) = (1 − α)μ(I ). Then we define S = Sα to be the smallest subset of T such that X ∈ S and for every I ∈ S, F (I ) ⊆ S. It is clear that defining the correspondence ∗ I → I ∗ with respect to this S we have J = I ∈ S if and only if J ∈ F (I ) and so writing AI = I \ J ∈S : J ∗ =I J we have aI = μ(AI ) = μ(I ) − J ∈S : J ∗ =I μ(J ) = αμ(I ) for every I ∈ S. We define rank(I ) = r(I ) of any I ∈ S to be the unique integer m such that I ∈ S(m) and we define the xI ’s by setting 1 x I = γm = α(1 − α)m

(m+1)θ

Ψ+ c(u + λ) e−u du

(2.28)

mθ

for every I ∈ S where m = rank(I ) and let φθ =

xI χAI .

(2.29)

I ∈S

For every I ∈ S and every m 0 we have bm (I ) =

S J ⊆I r(J )=r(I )+m

hence

μ(J ) = (1 − α)m μ(I )

(2.30)


φθ dμ =

aI xI =

I ∈S

X

=α

γm αμ(J ) = α

m0 I ∈S(m)

γm (1 − α) = m

m0

γm bm (X)

m0

∞

1643

Ψ+ c(u + λ) e−u du = f

(2.31)

0

and by Jensen’s inequality G(φθ ) dμ =

aI G(xI ) = α

I ∈S

X

∞

G(γm )(1 − α)m

m0

G Ψ+ c(u + λ) e−u du = F.

(2.32)

0

On the other hand if I ∈ S and m = rank(I ) AvI (φθ ) = =

1 μ(I )

aJ xJ

J ∈S : J ⊆I

α γ +rank(I ) μ(I )

0

=α

0

μ(J )

S J ⊆I rank(J )=rank(I )+

1 γ +m (1 − α) = (1 − α)m

∞

Ψ+ c(u + λ)+ e−u du

(2.33)

mθ

and so MT φθ dμ

aI AvI (φθ )

I ∈S

X

=

m0

=

1 α(1 − α) (1 − α)m

1 − e−θ m0

=

1 − e−θ θ

∞

m

∞


mθ


mθ

∞ θ

u + 1 Ψ+ c(u + λ)+ e−u du θ

(2.34)

0

where [.] denotes the integer part. Therefore taking θ = θs = 2−s → 0+ (s integer), and using the monotone convergence theorem we get

1644


∞ MT φθ dμ

lim sup θ→0+

X

uΨ+ c(u + λ)+ e−u du

(2.35)

0

which proves the lower bound in (2.27). Next given a nonnegative φ ∈ L1 (X, μ) satisfying X φ dμ = f and X G ◦ φ dμ F we consider the sequence (φm ) where φm = I ∈T(m) AvI (φ)χI and set Φm =

max AvJ (φ): I ⊆ J ∈ T χI = MT φm

(2.36)

I ∈T(m)

since AvJ (φ) = AvJ (φm ) whenever I ⊆ J ∈ T when I ∈ T(m) and note that

φm dμ = X

φ dμ = f,

X

Fm =

G(φm ) dμ

X

G(φ) dμ F

(2.37)

X

for all m and that Φm converges monotonically almost everywhere to MT φ. Also since each φm is of the form (2.10) we can apply (2.21), using the values of c and λ satisfying (2.26), and then combining this with the monotone convergence theorem we get

MT φ dμ = lim

m→∞

X

F 1 Φm dμ + c c

X

∞

Ψ c(u + λ)+ e−u du − λf.

(2.38)

0

But now using (2.26) and (2.19) we have

1 F + c c

∞

Ψ c(u + λ)+ e−u du − λf

0

∞

(u + λ)+ − λ Ψ+ c(u + λ) e−u du

= 0

∞ =

uΨ+ c(u + λ) e−u du

(2.39)

0

the last equality holding since Ψ+ (c(u + λ)) = 0 whenever u + λ < 0. This combined with (2.38) and (2.35) completes the proof of the proposition. 2 To illustrate the applicability of the above proposition we will give two examples before turning to the case in Theorem 1.


1645

First let us consider a σ > 0 and define Ψ (x) = σ x

which implies that G(x) =

if 0 x σ, if x > σ.

0 +∞

(2.40)

Then we can easily compute the corresponding functions from (2.26) when λ < 0 to be σ eλ and 0 respectively. Thus with f such that 0 < f < σ and F = 0 the system has always a solution with λ < 0 given by λ = log( fσ ). Hence we need not examine it for λ 0 and we infer from (2.27) T (0, f ) = σ (−λ + 1)eλ = f + f log( σ ). Examining what D T (0, f ) means we have an that DG G f alternative proof of the following formula proven also in [5]

σ . sup MT φL1 (X) : φL1 (X) = f, φL∞ (X) = σ = f + f log f

(2.41)

Next we take Ψ (x) =

x ex−1

if 0 x 1, if x > 1,

which implies that G(x) = x log+ x,

(2.42)

one computes that again the corresponding functions from (2.26) when λ < 0 are eλ [1 + c exp(−1/c) ] and ceλ exp(−1/c) valid only for 0 < c < 1. Then with f = 1 and F > 0 the cor1−c (1−c)2 responding system of equations is equivalent to λ = − log[1 + negative) and to

c exp(−1/c) ] 1−c

(which is always

1 1 (z − 1)ez + 1 = q(z) = 1 − z F

(2.43)

where z = 1c > 1. Observing that q is strictly decreasing, q(1) = 0 and limz→+∞ q(z) = +∞ we conclude that the system has always a solution c, λ with λ < 0, 0 < c < 1 and then computing the integral in (2.27) to be equal to Fc + 1 − λ we have found the value of U T (F, f ) when f = 1 where T MT φ dμ: φ 0 is measurable, U (F, f ) = sup X

φ dμ = f and

X

φ log+

φ dμ F f

(2.44)

X

is the corresponding to (1.12) supremum Bellman function. But since it is easy to see that U T (F, f ) = f U T ( Ff , 1), denoting by W the inverse function of q, straightforward manipulations with Eq. (2.43) give the following. Corollary 1. For any tree T and any F, f > 0 we have

f U (F, f ) = f + F W F T

exp(−1/W (f/F )) + f log 1 + . W (f/F ) − 1

(2.45)

1646


The above provide examples where the corresponding Bellman type function for any F and for a fixed f is given by a single formula coming from solutions (c, λ) with λ < 0 always. This is the exact opposite of what happens in the Bellman function for the (p, p) inequality in [3] (where the single formula comes from solutions (c, λ) with λ 0 always, see [5]). 3. The upper estimate To prove Theorem 1 we take in Proposition 1 ¯ Ψ¯ (x) = ex − x − 1 which implies that G(x) = (x + 1) log(x + 1) − x

(3.1)

(this makes certain computations easier) and note that Ψ¯ + (x) = (ex − 1)+ . Also if φ satisfies the ¯ conditions in (1.4) then X G◦φ dμ F −f . Moreover one easily computes that when 0 < c < 1 ∞ ∞ ¯◦ and λ ∈ R the corresponding functions a1 (c, λ) = 0 Ψ¯ + (c(u + λ)+ )e−u du, a2 (c, λ) = 0 G ∞ + −u + −u Ψ¯ + (c(u + λ) )e du and b(c, λ) = 0 uΨ¯ + (c(u + λ) )e du are given by a1 (c, λ) =

a2 (c, λ) =

ecλ 1−c ceλ 1−c

− 1 if λ > 0,

⎧ ⎨ cecλ ( ⎩

(3.2)

if λ 0,

1 1−c 1−c + λ) − a1 (c, λ) λ ce − a1 (c, λ) (1−c)2

if λ > 0,

(3.3)

if λ 0,

and

b(c, λ) =

⎧ ⎨ ⎩

ecλ −1 (1−c)2 ceλ 1 1−c ( 1−c + 1 − λ)

if λ > 0,

(3.4)

if λ 0.

These functions are infinity when c 1. The corresponding system of Eqs. (2.26) with F − f replacing F , which thus is equivalent to a1 (c, λ) = f and a2 (c, λ) + a1 (c, λ) = F , can be solved as follows. F −f ceλ ceλ If this system has a solution with λ 0 then 1−c = f , (1−c) 2 = F thus c = F which is in 2

(0, 1) and so eλ = Ff−f . However to have λ 0 we must assume that F f 2 + f . On the other hand when F f 2 + f the above values furnish a solution to the system for which we have

ceλ 1 F −f b(c, λ) = + 1 − λ = F + f + f log . 1−c 1−c f2 cλ

(3.5)

cλ

e 1 = f + 1, ce Next if the system has a solution with λ > 0 then 1−c 1−c ( 1−c + λ) = F thus c F 1 F 1−c + cλ = f +1 . Thus setting z = 1−c > 1 we get z exp( f +1 − z + 1) = f + 1 thus V (z) =

exp(F /(f +1)) . This has always a unique solution z > 1 since F > (f +1) log(f +1) thus the rightf +1 +1)) +1)) hand side is greater than 1. Thus z = U ( exp(Ff/(f ) and so cλ = f F+1 + 1 − U ( exp(Ff/(f ). +1 +1


1647

+1)) But we must have λ > 0 for this to work which is equivalent to V ( f F+1 + 1) > exp(Ff/(f which +1 2 in turn is equivalent to F < f + f . If this happens the solution is unique and we have

F /(f +1) e ecλ − 1. − 1 = (f + 1)z − 1 = (f + 1)U b(c, λ) = f +1 (1 − c)2

(3.6)

The above complete the proof of Theorem 1. Note also that the corresponding system has always a unique solution. This holds in much more general situations as derived in [5]. However we will not need this here. Next to prove Theorem 2 we consider the convex function G(x) = (x +1) log(x +1) and argue in a similar as in Section 7 of [3] manner. The basic ingredient here is the fact that, as explained in T (x, y) given in Theorem 1 is concave (and independent of the the Introduction, the function Dlog

T (F, f, k) where 0 < k < 1 tree T ). To proceed further we let φ, E be as in the definition of Dlog and choose u > 0 such that

μ {MT φ > u} k μ {MT φ u}

(3.7)

and then choose a measurable D such that V1 = {MT φ > u} ⊆ D ⊆ {MT φ u} = V2 and μ(D) = k. Since MT φ u on E \ V1 it is easy to see that E MT φ dμ D MT φ dμ and defining s ∈ [0, 1] by μ(D) = sμ(V1 ) + (1 − s)μ(V2 ) we also have (since MT φ = u on V2 \ V1 )

MT φ dμ = s

D

MT φ dμ + (1 − s)

V1

MT φ dμ.

(3.8)

V2 (1)

(2)

Now each of the V1 , V2 is a union of families {Ij }, {Ir } consisting of pairwise disjoint elements maximal under AvI (φ) > u (resp. u) and we clearly have MT φ = MT (I ) φ (where T (I ) 1 is the subtree of T with top I on the probability space (I, μ(I ) μ)) for each of those I ’s. Hence, using Theorem 1 for all these trees, arguing as in [3] and using the concavity of the function in Theorem 1 we get

T MT φ dμ kDlog

A B , k k

(3.9)

E

where A=s

G ◦ φ dμ + (1 − s)

V1

G ◦ φ dμ F

(3.10)

φ dμ f.

(3.11)

V2

and

φ dμ + (1 − s)

B =s V1

V2

1648


Letting η = sχV1 + (1 − s)χV2 Jensen’s inequality implies that

(G ◦ φ)η dμ A B X φη dμ G = =G X k k η dμ X X η dμ

(3.12)

φ(1 − η) dμ (G ◦ φ)(1 − η) dμ F − A f −B = G X . G X = 1−k 1−k (1 − η) dμ X X (1 − η) dμ

(3.13)

B f −B + kG F (1 − k)G 1−k k

(3.14)

and

Thus

T (x, y) is strictly increasing in x, (3.9) and (3.13) imply that and since Dlog

T MT φ dμ kDlog

E

1 f −B B F − (1 − k)G , . k 1−k k

(3.15)

Conversely supposing B is in (0, f ) and satisfies (3.14) we fix δ < 1, choose, using Lemma 1, a family {I1 , I2 , . . .} of pairwise disjoint elements of T such that j μ(Ij ) = k, we write −B ) kG( Bk ) and using Theorem 1 for each j we choose E = j Ij , A = F − (1 − k)G( f1−k a nonnegative measurable φj on Ij such that AvIj (G ◦ φj ) = Ak , AvIj (φj ) = Bk and

T MT (Ij ) (φj ) dμ δμ(Ij )Dlog

A B , . k k

(3.16)

Ij

Next we choose a nonnegative measurable ψ on X \ E such that X\E G ◦ ψ dμ = F − A > 0 and X\E ψ dμ = f − B > 0 which is possible by (3.14) and defining φ = ψχX\E + j φj χIj we have X G ◦ φ dμ = F , X φ dμ = f and

T MT φ dμ δkDlog

A B , . k k

(3.17)

E

Letting now δ → 1− we have proved the following: T (F, f, k) is equal to the supremum of the function Proposition 2. Blog T Rk (B) = kDlog

1 f −B B F − (1 − k)G , k 1−k k

on the set of B in [0, f ] that satisfy the estimate (3.14).

(3.18)


1649

To proceed further we fix F, f, k and define the following functions on [0, f ]

B f −B + kG , h(B) = (1 − k)G 1−k k

f −B A(B) = F − (1 − k)G 1−k

(3.19)

−B −B and y(B) = f1−k . Since h (B) = −G ( f1−k ) + G ( Bk ) the convexity of G implies that h (B) has the same sign as B − kf and since h(kf ) = G(f ) < F we conclude that the set of all B in [0, f ] satisfying (3.14) is a closed interval of the form [B1 , B2 ] where 0 B1 < kf < B2 f . Moreover B2 = f if h(f ) F otherwise B2 < f and h(B2 ) = F and similarly B1 = 0 if h(0) F otherwise B1 > 0 and h(B1 ) = F . T (x, y) is given by a double formula one must also compare A(B) with B( B + 1). Since Dlog k Hence we also consider the function

B f −B +B +1 . (3.20) σ (B) = (1 − k)G 1−k k

Now using Theorem 1 it is easy to see that on y 2 + y < x we have T ∂Dlog

x (x, y) = >0 ∂x x−y

and

T ∂Dlog

∂y

(x, y) = −

x x −y x + log 2 > − x −y x −y y

(3.21)

−B so since A (B) = G ( f1−k ) = 1 + log(y(B) + 1) > 1 we get Rk (B) > 0 for every B ∈ [B1 , B2 ] such that σ (B) < F . Next on the set where (y + 1) log(y + 1) < x < y 2 + y we compute using Theorem 1 and (1.6)

T ∂Dlog

∂x

(x, y) =

U (z) >0 U (z) − 1

and

T ∂Dlog

∂y

(x, y) =

U (z) x U (z) − 2 − U (z) − 1 y+1

(3.22)

T where z = exp(x/(y + 1)). Comparing (3.21) and (3.22) at x = y 2 + y we conclude that Dlog and hence Rk is actually C 1 . Also it easily follows from (3.22) that if B ∈ [B1 , B2 ] is such that σ (B) > F then Rk (B) has the same sign as the expression

U

A(B) k exp(A(B)/(B + k)) − − 1 + log y(B) + 1 B +k B +k

(3.23)

which since V is strictly decreasing has the same sign as k exp(A(B)/(B + k)) exp(A(B)/(B + k) − log(y(B) + 1)) − A(B) B +k B+k + 1 − log(y(B) + 1)

(3.24)

if A(B) > (B + k) log(y(B) + 1) and is positive otherwise. But A(B) > (B + k) log(y(B) + 1) holds if and only if F > (f + 1) log y(B) + 1

(3.25)

1650


and if this also holds we conclude now that Rk (B) has the same sign as (B + k)(B − kf ) F − b(B) = F − (f + 1) log y(B) + 1 − k(f + 1 − B − k)

(3.26)

where b(B) is defined by the above equality. But now comparing (3.25) and (3.26) and also since Rk (B) is positive if σ (B) < F we conclude that Rk (B) > 0 on the whole interval (B1 , kf ). )(2(f +1)−B−k) > 0 and so since Next if B ∈ (kf, f ) then it is easy to see that b (B) = (B−kf k(f +1−B−k)2 b(kf ) = (f + 1) log(f + 1) < F we conclude that if b(B2 ) F then Rk (B) 0 for every T (F, f, k) = R (B ). B ∈ [B1 , B2 ] hence Blog k 2 Assume now that b(B2 ) > F . Then there exists a unique B0 ∈ (kf, B2 ) such that b(B0 ) = F . By (3.26) this B0 clearly satisfies (3.25). We will show that B0 satisfies also the following F σ (B0 ) and therefore Rk (B0 ) = 0. Indeed it suffices to prove that b(B) σ (B) for every B ∈ (kf, f ). But writing B = kf + (1 − k)x where 0 < x < f straightforward calculations show that b(B) σ (B) is equivalent to g(x) =

f +1 + k log(f + 1 − x) (1 − k)x + kf + 1 f +1−x

(3.27)

which holds since g is convex and trivially (3.27) holds at the endpoints x = 0 and x = f . Using the same substitution we also have b(B) > h(B) on (kf, f ) since this can be easily computed to x be equivalent to the inequality log(f + 1 − x) + k(f +1−x) > log(f + 1 − x + xk ) which clearly holds. Hence if B2 < f then σ (B2 ) b(B2 ) > h(B2 ) = F and so B0 exists and since Rk is C 1 we T (F, f, k) = R (B ). Considering also get that B0 is its absolute maximum on [B1 , B2 ] thus Blog k 0 the case B2 = f (that is when (f + k) log( fk + 1) F ) and since b(f ) = f ( fk + 1) we get using Proposition 2 the following (noting that if B2 < f then f ( fk + 1) > (f + k) log( fk + 1) > F ) T Blog (F, f, k) =

Rk (B0 )

if F < f ( fk + 1),

Rk (f )

if f ( fk + 1) F.

(3.28)

) and on the other hand if B0 exists then as we have Obviously Rk (f ) = F + f + f log k(Ff−f 2

T ( A(B0 ) , B0 ) is given by the first part of the formula in (1.7). But now writing seen Dlog k k

B0 − kf = ξ0 f + 1 − B0 − k

(3.29)

we observe that ξ0 satisfies the inequalities 0 < ξ0 < f (since B0 ∈ (kf, f )) and since B0 + k = (f +1)(ξ0 +k) F , y(B0 ) + 1 = ξf0+1 ξ0 +1 +1 the equation b(B0 ) = F becomes τk (ξ0 ) = f +1 − log(f + 1) thus

T ( 1 (F − (1 − k)G( f −B0 )), B0 ) ξ0 = ξk (F, f ). Then substituting F with b(B0 ) in Rk (B0 ) = kDlog k 1−k k and using ξ0 it is straightforward to get that Rk (B0 ) is equal to the second expression in (1.10). This completes the proof of Theorem 2.


1651

4. The lower estimate Here we will prove Theorem 3. Assuming that T is N -homogeneous we let φ be a nonnegative function of the form (2.10) such that

φ dμ = F φ log f

φ dμ = f

+

and

X

(4.1)

X

and let S = Sφ be the corresponding subtree of T . Using the notation from Section 2 we make the following two simple observations. First, by Lemma 3(iv), we have yI ∗ < yI NyI ∗ for all I ∈ S \ {X} and second, by condition (ii) in Definition 1(b), φ(x) yI whenever I ∈ S and x ∈ AI . ˜ We consider the function G(x) = x log+ ( fx ) which is convex on x 0. The second remark combined with the convexity of G gives

1 aI

˜ φ(x) dμ(x) 1 G aI

AI

φ(x) ˜ xI ˜ G(yI ) dμ(x) = G(y I) yI yI

(4.2)

AI

for all I ∈ S. Now Lemma 3 implies that MT φ dμ =

μ(I ) − aI yI =

I ∈S

X

=f +

I ∈S

μ(J ) yI

J ∈S : J ∗ =I

μ(I )(yI − yI ∗ )

(4.3)

I ∈S , I =X

and by using (4.2) we get F=

˜ I) φ G(y dμ φ log aI xI f yI +

X

μ(I )yI − = I ∈S

=

I ∈S ,I =X

I ∈S

yI μ(J )yJ log+ f J ∈S : J ∗ =I

yI yI ∗ − log+ = μ(I )yI log+ f f

μ(I )yI log

I ∈S , I =X

yI yI ∗

(4.4)

since by Lemma 3 aI xI = μ(I )yI − J ∈S : J ∗ =I μ(J )yJ and yI yX = f for all I . Next for any I ∈ S, I = X we have 1 < yyI∗ N and so using the easy to verify fact that 1 t

I

1 log 1−t is increasing for t ∈ (0, 1) we obtain by taking t = 1 −

yI ∗ yI

yI N log N (yI − yI ∗ ). yI log yI ∗ N −1

∈ (0, 1 −

1 N)

the following

(4.5)

1652


Using (4.5) in (4.4) and in view of (4.3) we get MT φ dμ

N −1 F +f N log N

(4.6)

X

for functions φ of the form (2.10). Now for the general case, given φ 0 measurable satisfying (4.1) we define φm , Φm as in the proof of Proposition 1 and for each φm we can apply (4.6) to get

MT φ dμ

X

N −1 Φm dμ N log N

X

+

φm log

φm dμ + f. f

(4.7)

X

We will now show that the sequence ψm = φm log+ ( φfm ) is uniformly integrable. Since φm → φ almost everywhere, by the second condition in the definition of the homogeneous trees, we get the estimate (4.6) for the general measurable φ. To show the uniform integrability of ψm we note that for any λ > e and any m the set where fλ fλ ψm > λf is contained in the set where φm > log λ and therefore in Eλ = {MT φ > log λ }. On the other hand given any I ∈ T(m) Jensen’s inequality implies that ψm = φm log+

φm f

1 μ(I )

φ log+

φ dμ f

(4.8)

I

on I thus integrating and summing we get for any m the following ψm dμ {ψm >λf }

I ∈T(m) , I ⊆Eλ I

φ + φ dμ φ log dμ φ log f f +

(4.9)

Eλ

which tends to 0 as λ → +∞ since φ log+ ( fφ ) is integrable and μ(Eλ )

log λ λ .

−1 These prove that LT (F, f ) NNlog NF +f. To prove the reverse inequality we let X = I0 ⊇ I1 ⊇ · · · Is ⊇ Is+1 ⊇ · · · be a chain such that Is ∈ T(s) for all s 0 (and so μ(Is ) = N −s ). We write ∞

k F = m0 + f log N Nk

(4.10)

k=1

F for the expansion of f log N in base N , thus m0 , 1 , . . . , k , . . . are nonnegative integers such that

k < N for all k 1, and we define the strictly increasing sequence of integers m0 < m1 < · · · < mk < · · · by the rule mk − mk−1 = k + 1 > 0 for all k 1. Then we define

φ=f

∞ k=0

N mk −k χImk \Imk +1 .

(4.11)


1653

We have φ dμ = f

k=0

X

∞

φ log+

∞ 1 =f N mk −k N −mk − N −mk −1 = f N −k 1 − N

(4.12)

k=0

∞ φ N mk −k N −mk − N −mk −1 (mk − k) log N dμ = f f k=0

X

= f log N

∞ mk − k

Nk

k=0

−

∞ mk−1 − k + 1

Nk

k=1

∞

k = f log N m0 + Nk

=F

(4.13)

k=1

and if mk−1 < s mk then AvIs (φ) = N s f

∞

N mj −j N −mj − N −mj −1 = f N s−k

(4.14)

j =k

and this increases as s increases (if s = mk then AvIs (φ) = AvIs+1 (φ)). We next claim that MT φ(x) = AvIs (φ) whenever x ∈ Is \ Is+1 and s 0. Indeed suppose that x ∈ Is \ Is+1 and let J be the unique element of T(s+1) such that x ∈ J (clearly J ∈ C(Is ) and J = Is ). Then the set of all I ’s in T containing x consists of I0 , . . . , Is and J and certain subintervals of it. But AvIs (φ) AvIr (φ) for all 0 r < s and since φ is either 0 on J (if s is not an mk ) or if s = mk it is equal to AvIs (φ) on J we get that MT φ(x) = AvIs (φ). Hence MT φ = f

∞

N s−k(s) χIs \Is+1

(4.15)

s=0

where k(s) is the smallest integer k with mk s and this implies that

∞

1 MT φ dμ = f 1 − N

X

1 =f 1− N =

N −k(s)

s=0

m0 + 1 +

∞ mk − mk−1 k=1

Nk

N −1 F +f N log N

by (4.10). This completes the proof of Theorem 3.

∞

k + 1 f (N − 1) = m0 + 1 + N Nk k=1

(4.16)

1654


References [1] D.L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984) 647–702. [2] D.L. Burkholder, Martingales and Fourier analysis in Banach spaces, in: C.I.M.E. Lectures, Varenna (Como), Italy, 1985, in: Lecture Notes in Math., vol. 1206, 1986, pp. 61–108. [3] A.D. Melas, The Bellman functions of dyadic-like maximal operators and related inequalities, Adv. in Math. 192 (2005) 310–340. [4] A.D. Melas, A sharp Lp inequality for dyadic A1 weights in Rn , Bull. Lond. Math. Soc. 37 (2005) 919–926. [5] A.D. Melas, Sharp general local estimates for Dyadic-like maximal operators and related Bellman functions, Adv. in Math. 220 (2009) 367–426. [6] A.D. Melas, E. Nikolidakis, Dyadic-like maximal operators on integrable functions and Bellman functions related to Kolmogorov’s inequality, Trans. Amer. Math. Soc., in press. [7] F. Nazarov, S. Treil, The hunt for a Bellman function: Applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8 (5) (1996) 32–162. [8] F. Nazarov, S. Treil, A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (4) (1999) 909–928. [9] F. Nazarov, S. Treil, A. Volberg, Bellman function in stochastic optimal control and harmonic analysis (how our Bellman function got its name), Oper. Theory Adv. Appl. 129 (2001) 393–424, Birkhäuser Verlag. [10] L. Slavin, A. Stokolos, V. Vasyunin, Monge–Ampère equations and Bellman functions: The dyadic maximal operator, C. R. Acad. Paris Sér. I Math., I 346 (2008). [11] L. Slavin, V. Vasyunin, Sharp results in the integral-form John–Nirenberg inequality, submitted for publication. [12] L. Slavin, A. Volberg, The explicit BF for a dyadic Chang–Wilson–Wolff theorem. The s-function and the exponential integral, Contemp. Math. 444 (2007). [13] E.M. Stein, Note on the class L log L, Studia Math. 32 (1969) 305–310. [14] V. Vasyunin, The sharp constant in the reverse Holder inequality for Muckenhoupt weights, Algebra i Analiz 15 (1) (2003) 73–117. [15] V. Vasyunin, A. Volberg, The Bellman functions for a certain two weight inequality: The case study, Algebra i Analiz 18 (2) (2006). [16] V. Vasyunin, A. Volberg, Monge–Ampère equation and Bellman optimization of Carleson embedding theorem, preprint. [17] A. Volberg, Bellman approach to some problems in harmonic analysis, in: Seminaire des Equations aux derivées partielles, Ecole Polytéchnique, 2002, eposé. XX, pp. 1–14.

Further reading [18] L. Grafakos, S. Montgomery-Smith, Best constants for uncentered maximal functions, Bull. London Math. Soc. 29 (1) (1997) 60–64. [19] G. Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. 112 (1991) 579–586.


Multicyclicity of unbounded normal operators and polynomial approximation in C Béla Nagy ∗,1 Inst. of Mathematics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary Received 11 June 2008; accepted 25 June 2009

Communicated by N. Kalton

Abstract A remarkable and much cited result of Bram [J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75–94] shows that a star-cyclic bounded normal operator in a separable Hilbert space has a cyclic vector. If, in addition, the operator is multiplication by the variable in a space L2 (m) (not only unitarily equivalent to it), then it has a cyclic vector in L∞ (m). We extend Bram’s result to the case of a general unbounded normal operator, implying by this that the (classical) multiplicity and the multicyclicity of the operator (cf. [N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, 2002]) coincide. It follows that if m is a sigma-finite Borel measure on C (possibly with noncompact support), then there is a nonnegative finite Borel measure τ equivalent to m and such that L2 (C, τ ) is the norm-closure of the polynomials in z. © 2009 Elsevier Inc. All rights reserved. Keywords: Unbounded normal operator; Star-cyclic vector; Cyclic vector; Multiplicity; Multicyclicity; Polynomial approximation in L2 (C, m); Equivalent measures

1. Introduction In 1955 J. Bram [2] studied bounded subnormal and cyclic normal operators, and obtained a remarkable result which showed also an interesting connection to the theory of polynomial * Fax: +361 4633172.

E-mail address: [email protected]. 1 The work was supported by the Hungarian National Foundation OTKA No. T047276 and K77748.


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B. Nagy / Journal of Functional Analysis 257 (2009) 1655–1665

approximation in the space L2 . His results have been widely applied on these areas (cf., e.g., [4]) and also in the theory of spectral multiplicities of bounded normal operators (see, e.g., [15,16]). The aim of this paper is to show that Bram’s basic result can be extended to the case of unbounded operators with the “side effect” of extending the mentioned connection (see, e.g., [18]) to polynomial approximation in the space L2 (C) in noncompact domains. Though the proof is an extension of Bram’s method, we believe that the extension is far from trivial. A serious technical problem is how a key application of a classical result of Lavrentev (a special case of a famous result of Mergelyan, see, e.g., [8, Theorems II.8.7 and 9.1]) on uniform polynomial approximation on compact domains can be used under the more general conditions, a second one is how we can dispense with the continuity of linear operators in the unbounded case. It may be interesting to recall that classical results on polynomial approximation in Bergman spaces L2a (G) (with the [sigma-finite] area measure, cf. e.g., Mergelyan [11] and Dzhrbashyan [6], or see some recent results in [1]) often restricted the basic domain in C or introduced and used weight functions. Both approaches (under clearly different circumstances) may be regarded as possibly pointing toward a general type of result as our Corollary 1 to Theorem 3.1. 2. Terminology and notation Let X denote a separable Hilbert space over C, and let N denote a (possibly unbounded) normal linear operator in X, i.e. N be closed, densely defined and satisfy N ∗ N = N N ∗ (together with its adjoint N ∗ ). Let E denote its resolution of the identity (the unique orthogonal projection valued measure defined on the Borel sets of C, for which N = C z E(dz)). A scalar-valued spectral measure m for N is a nonnegative [finite] measure on the spectrum σ (N ) of N such that m(b) = 0 if and only if E(b) = 0, i.e. m and E are mutually absolutely continuous, in other words: they are equivalent, in sign: m ≡ E. A vector v ∈ X is a star-cyclic vector for N if for all nonnegative integers j , k the domains of definition D(N ∗j N k ) contain v, and the closed linear span of the vectors (N ∗j N k v; j, k 0) is all of X. A vector v is (simply) cyclic if these conditions hold when we fix j := 0. It is known (see, e.g., [5, p. 333]) that if m is a finite nonnegative Borel measure on C such that every polynomial in z and z (its conjugate) belongs to X = L2 (C, m), and Nm f := zf for f in the maximal possible domain in X, then Nm is a normal operator and the identically 1 function is a star-cyclic vector for it. In the converse direction: if N is any normal operator with a star-cyclic vector v, then there are a finite nonnegative Borel measure m on C such that every polynomial in z and z belongs to L2 (C, m) and a unitary operator mapping v onto 1 and satisfying U N U −1 = Nm . Hence, with clear notation, Nm is unitarily equivalent to Nτ if and only if m ≡ τ . The (classical) multiplicity of an unbounded normal operator can be defined in a number of equivalent ways. It originates in the papers of Friedrichs [7], Wecken [19] and Nakano [12,13] and modern recapitulations and versions can be found in the general (nonseparable) case in Halmos [9], A. Brown [3] and Plessner [17]. We shall essentially use the von Neumann model of a normal operator N (cf. e.g., [16, p. 243]), which makes use of the fact that N is unitarily equivalent to the (unbounded) multiplication operator M : f → zf (z) on a vector valued L2 space Y defined by the direct integral Y :=

⊕X(z) ν(dz) ≡ g ∈ L2 σ (N), ν, X : g(z) ∈ X(z) ν-a.e. ,


1657

where ν is a finite nonnegative Borel measure on the (possibly unbounded) σ (N ) and {X(z)} is a measurable family of subspaces in X. The function d(N, z) := dim X(z) is called the local spectral multiplicity function of N . The (classical, global) multiplicity d(N ) of N is defined as d(N) := ν- ess.sup. d(N, z): z ∈ C ∈ N ∪ {∞}. The multicyclicity of a bounded operator has been defined and used by a number of writers (see, e.g., [10,14,16]). Its natural extension for an unbounded normal operator T is as follows: define the family of T -cyclic subspaces by ∞ n

k D T : span T C: n = 0, 1, 2, . . . = X Cyc(T ) := C ⊂ k=1

(here span denotes closed linear hull) and its multicyclicity by μ(T ) := inf dim C: C ∈ Cyc(T ) ∈ N ∪ {∞}. It is clear that for an unbounded normal operator N the global multiplicity d(N ) is 1 if and only if N is unitarily equivalent to Nm , where m is a scalar-valued spectral measure for N or, equivalently, N has a star-cyclic vector s such that the unique (orthogonally) N -reducing subspace containing s is X. On the other hand, its multicyclicity μ(N ) is 1 if and only if N has a cyclic vector c such that the unique N -invariant subspace containing c is X. Hence μ(N ) = 1 implies d(N) = 1. Our main result (Theorem 3.1) proves that the converse is valid also for an unbounded normal N : a star-cyclic N is cyclic. 3. A star-cyclic N is cyclic Theorem 3.1. Let N be a (possibly unbounded) normal operator in the complex Hilbert space X with (classical) multiplicity d(N) = 1 and with a scalar spectral measure m. Then there is a nonnegative Borel measure σ on C equivalent to m and such that P2 (σ ) = L2 (σ ), i.e. the closure of the complex polynomials in L2 (σ ) is the whole space. It follows that the multicyclicity μ(N ) of the operator N is 1. If N = Nm , there is a cyclic vector γ for N in L∞ (m). Proof. It is a modification and development of the proof of Bram [2, Lemma 6 and Theorem 6]. Since N is unitarily equivalent to Nm , we can and shall assume that N = Nm . Let C(1) be the closed unit disk in C, and let C(j ) := z ∈ C: j − 1 < |z| j

(j = 2, 3, . . .).

Consider the restriction N (j ) := N|E[C(j )]X of the normal operator N to the indicated subspace (here E is its resolution of the identity). N (j ) is the bounded normal operator of multiplication by the variable z in the Hilbert space L2 [C(j ) ∩ σ (N ), m(j )], where m(j ) is the restriction of m to C(j ) ∩ σ (N). At first we modify a construction of Bram [2] with respect to the annulus C(j ) (with interior I (j )) so that it be connected in a certain way to a corresponding construction in the annulus C(j + 1) (with interior I (j + 1), and with respect to the restriction N(j + 1)). We warn the reader that in the next paragraphs (almost) everything will depend on the positive integer j , though we shall denote it explicitly so only in the necessary cases.

1658


Assume that z1 ∈ I (j ), z2 ∈ I (j ) ∪ I (j + 1);

zm = ρm eiφm

(0 ρm , 0 φm < 2π) (m = 1, 2)

are their exponential representations. The path z(t) := ρ1 + t (ρ2 − ρ1 ) ei[φ1 +t (φ2 −φ1 )]

(0 t 1)

connects z1 to z2 (within I (j ) if z2 ∈ I (j )). Fixing z1 , z2 , define now oriented (curvilinear) quadrangles (z1 , z2 , z3 , z4 ) as follows: assume first that ρ1 = ρ2 =: ρ, φ1 < φ2 . Then let z3 := (ρ + h)eiφ2 , z4 := (ρ + h)eiφ1 for h > 0 sufficiently small, and define similarly if ρ1 = ρ2 =: ρ, φ1 > φ2 . The (oriented) paths connecting z2 to z3 , then z3 to z4 , then z4 to z1 shall be all of the type z(t) above. Assume now that ρ1 < ρ2 , φ1 φ2 , and let z3 := ρ2 ei(φ2 +h) , z4 := ρ1 ei(φ1 +h) for h > 0 sufficiently small, and define similarly if ρ1 < ρ2 , φ1 > φ2 . The (oriented) boundary paths shall be again as before. Finally, if ρ1 > ρ2 , then proceed exactly as in the preceding paragraph. For short, we shall call all types of the defined quadrangles simply (oriented) quadrangles. Let R(j ) := {r(j )n ≡ rn : n ∈ N} be a countable dense subset of I (j ) that contains no atoms of the measure m(j ); hence m[R(j )] = 0. For each k ∈ N and rn ∈ R(j ) let D(j ; k, n) be an open disk in I (j ) with center rn such that m[D(j ; k, n)] < 1/(k2n ), and let D(j ; k) :=

∞ n=1 D(j ; k, n). Then D(j ; k) ⊂ I (j ) is open, contains R(j ), and m[D(j ; k)] < 1/k. Let T (j ; k, n) be a nonvoid open oriented quadrangle (as above) with r(j )n , r(j )n+1 as two fixed endpoints of one of its (oriented) sides, and such that m[T (j ; k, n)] < 1/(k2n ). Note that this is possible, because the intersection

of all open oriented quadrangles with one oriented side [rn , rn+1 ] is void. Let T (j ; k) := ∞ n=1 T (j ; k, n). Finally, connect the point r(j )1 ∈ I (j ) to r(j + 1)1 ∈ I (j + 1), and form an open oriented quadrangle S(j ; k) := (r(j )1 , r(j + 1)1 , a, b) (depending on the situation of the first two points as above) with sufficiently small positive ρ = ρ(k) such that the m-measure of the open quadrangle S(j ; k) be < 1/k. Define now U (j ; k) := D(j ; k) ∪ T (j ; k) ∪ S(j ; k). Then U (j ; k) ⊂ I (j ) ∪ S(j ; k), and m[U (j ; k)] < 3/k. By induction on k we see that the construction above can be made so that for each k we have U (j ; k + 1) ⊂ U (j ; k), and these sets are open and connected. Define now V (n; k) :=

n

U (j ; k).

j =1

Then V (n; k) is an open connected subset of the disk d(n + 1) := d(n) ∪ S(n; k). Let F (n; k) := d(n) \ V (n; k).

n+1

j =1 C(j ),

and V (n; k) ⊂


1659

Then F (n; k) is a compact set with connected complement F (n; k)c ≡ C \ F (n; k) = [C \ d(n)] ∪ V (n; k). Further, n

R(j ) ⊂ V (n; k) ∩ d(n) ⊂ F (n; k)c ∩ d(n),

j =1

and the left-hand side union is dense in d(n). Hence the interior of F (n; k) is empty. By Lavrentev’s theorem [8, Theorem II.8.7], each continuous complex function on F (n; k) can be uniformly approximated by polynomials. By its definition, we also have m d(n) m F (n; k) > m d(n) − 3n/k.

(1)

Since the sequence of the open sets {U (j ; k): k ∈ N} is decreasing, so is the sequence {V (n; k): k ∈ N}. Hence the sequence {F (n; k): k ∈ N} is increasing, and lim m F (n, k) = m d(n) .

k→∞

The sequence {F (n; k): n ∈ N} is also increasing. Indeed, F (n + 1; k) = d(n) ∪ C(n + 1) ∩

n

U (j ; k)c ∩ U (n + 1; k)c

j =1

n+1 c c = F (n; k) ∩ U (n + 1; k) ∪ C(n + 1) ∩ U (j ; k) . j =1

Here we have U (n + 1; k)c ⊃ I (n + 1)c ∩ S(n + 1; k)c ⊃ d(n), which implies F (n + 1; k) ⊃ F (n; k) ∩ d(n) = F (n; k). Consider the inequalities (1), the increasing sequence {k(n) := 3n2 : n ∈ N}, and for each n ∈ N define F (n) := F (n; k(n)). Then the sequence {F (n): n ∈ N} is increasing, and for each n we have m d(n) m F (n) > m d(n) − 1/n. Let F :=

∞

n=1 F (n),

and define D := C \ F . Then we obtain m(C) m(F ) m(C),

hence m(D) = 0.

Recall that each F (n) is compact, has void interior, and does not separate the plane.

1660


Consider the continuous function f (z) := z. By Lavrentev’s cited theorem, for every n ∈ N we can find a polynomial pn such that pn (z) − f (z) < 1/n

z ∈ F (n) .

Let Mn := maxpn (z)e−|z|

M := max|z|e−|z| , z∈C

z∈C

(n ∈ N).

Further, let {Bn : n ∈ N} be an increasing sequence of reals such that for every n we have Bn max[Mn , 1]. Define the function h0 (z) := 1

z ∈ F (1) ,

:= 1/Bn

z ∈ F (n + 1) \ F (n), n 1 ,

:= 0 [z ∈ D],

and let h(z) := h0 (z)e−|z| . Then h is defined on all of C, is bounded and Borel measurable, and is positive m-almost everywhere on C. For every Borel set b ⊂ C let ν(b) :=

h2 (z) m(dz). b

Then ν is a finite positive Borel measure on C with Radon–Nikodym derivative dν/dm = h2 . Since it is positive m-a.e., ν is equivalent to m (see, e.g., [9, Theorem 47.2, p. 78]). Let P denote the set of all polynomials, and note that for every p ∈ P we have

p 2L2 (C,ν)

≡

p(z)2 e−2|z| h2 (z) m(dz) < ∞, 0

C

i.e., P ⊂ L2 (C, ν). We see similarly that f ∈ L2 (C, ν). We show that 2

pn − f 2L2 (C,ν) ≡ pn (z) − z h2 (z) m(dz) → 0 (n → ∞).

(2)

C

For every n ∈ N the left-hand side is equal to

pn (z) − z2 h2 (z) m(dz) +

pn (z) − z2 h2 (z) m(dz) < ν F (n) / n2 + pn − f 2 ,

C\F (n)

F (n)

where the last norm denotes (from now on till the end of this paragraph) the norm in the space L2 (C \ F (n), ν). It follows that C

pn (z) − z2 h2 (z) m(dz) < ν(C)/ n2 + pn + f 2 .


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We have here

pn 2 =

pn (z)2 h(z)2 m(dz) M 2

n

C\F (n)

h0 (z)2 m(dz).

C\F (n)

For every z in the last domain of integration either z ∈ D, hence h0 (z) = 0, or z ∈ F (p +1)\F (p) for some p n, hence h0 (z) = 1/Bp 1/Bn . We obtain that

pn 2 m C \ F (n) → 0 (n → ∞). Similarly,

f 2 =

h0 (z)2 m(dz) M 2 ν0 C \ F (n) → 0 (n → ∞),

|z|2 h(z)2 m(dz) M 2

C\F (n)

C\F (n)

where ν0 is the finite positive Borel measure on C with Radon–Nikodym derivative dν0 /dm= h20 . This implies (2). Let σ denote the finite positive Borel measure on C with Radon–Nikodym derivative dσ/dν(z) := e−|z| . Since σ ν, we have P ⊂ L2 (C, σ ). Let H 2 (σ ) denote the closure of any set H in L2 (σ ) ≡ L2 (C, σ ). We show that zP ⊂ P2 (σ ). Indeed, there is a sequence {pn } ⊂ P converging in L2 (ν) to f (z) = z. For every p ∈ P we obtain

zp(z) − pn p(z)2 dσ =

C

2 |z − pn |2 p(z) e−|z| dν → 0 (n → ∞),

C

since the function |p(z)|2 e−|z| is bounded on C. Next we show that z P2 (σ ) ⊂ [zP]2 (σ ). Let x ∈ L2 (σ ) [zP]2 (σ ), the orthogonal complement of [zP]2 (σ ), and let ( , ) denote the usual inner product in L2 (σ ). For every polynomial p ∈ P, then 0 = (x, zp) = (zx, p). It follows that zx(z) is orthogonal to P2 (σ ). Let p2 ∈ P2 (σ ). Then we obtain (x, zp2 ) = (zx, p2 ) = 0, hence x ∈ L2 (σ ) z[P2 (σ )] as stated. Thus we have proved that z P2 (σ ) ⊂ [zP]2 (σ ) ⊂ P2 (σ ).

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We show now that the above relations hold also when we replace the function z by its conjugate, i.e. z P2 (σ ) ⊂ [zP]2 (σ ) ⊂ P2 (σ ). Indeed, let p2 ∈ P2 (σ ) and p ∈ P. Then z p2 (z) − p(z) 2 dσ = |z|2 p2 (z) − p(z)2 dσ = zp2 − zp 2 . σ C

C

Hence we obtain the first containment relation. The second one is a consequence of the proved fact that zP ⊂ P2 (σ ). It is well known that x ∈ D(Nσ ) if and only if 2

1 + |z|2 x(z) dσ < ∞. C

Let Q denote the orthogonal projection of L2 (σ ) onto the subspace P2 (σ ). Then Qx ∈ D(Nσ ) if and only if 2

1 + |z|2 [Qx](z) dσ < ∞. C

Since Qx σ x σ , the finiteness of x σ implies that of Qx σ . Further, the function z[Qx](z) belongs to z[P2 (σ )] ⊂ P2 (σ ) ⊂ L2 (σ ). Hence C |z|2 |[Qx](z)|2 dσ = zQx 2σ < ∞. We have proved that the orthogonal projection Q maps the set D(Nσ ) into itself, and its range P2 (σ ) is invariant with respect to the closed normal operator Nσ . We can similarly obtain the same statement for the dual operator Nσ∗ , which is the operator of multiplication by z. Hence the subspace P2 (σ ) is not only invariant, but also orthogonally reducing for Nσ . By assumption, Nσ is a star-cyclic normal operator for which the identically 1 function is a star-cyclic vector. Thus the only orthogonally Nσ -reducing subspace containing 1 is L2 (σ ) = P2 (σ ). So we have also proved that for any nonnegative finite Borel measure m in C with possibly unbounded support there is an equivalent measure σ such that the above equality holds. Let γ denote the nonnegative square root of the Radon–Nikodym derivative dσ/dm. Then k 2 γ (z) = exp( −3|z| 2 )h0 (z) 1 m-a.e. and σ -a.e. Hence the function z γ (z) is in L (σ ) for evk 2 2 ery k ∈ 2N0 . In other words, Nσ γ ∈ L (σ ). The statement g ∈ L (m) is clearly equivalent to C |g/γ | dσ < ∞, i.e. to g/γ ∈ L2 (σ ) = P2 (σ ). This implies that for every ε > 0 there is a polynomial p ∈ P such that |g − pγ |2 dm = |g/γ − p|2 dσ < ε. C

C


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Since N = Nm is the multiplication operator by the variable z in L2 (C, m), this means that g − p(N)γ 2 < ε. m This shows that the multicyclicity μ(N ) of the operator N is equal to 1, and the function γ ∈ L∞ (m) is a cyclic vector for N . 2 Corollary 1. If m is a sigma-finite signed Borel measure on C (possibly with noncompact support), then there is a nonnegative finite Borel measure τ equivalent to m and such that L2 (C, τ ) = P2 (τ ). Proof. Since m is sigma-finite, we can find a nonnegative finite Borel measure m1 equivalent to m. Consider then the normal operator Nm1 of multiplication by the variable z in L2 (C, m1 ), and apply Theorem 3.1. 2 Corollary 2. If m is a sigma-finite signed Borel measure on C (possibly with noncompact support) and f is any complex-valued Borel function on C, then there is a sequence {pn } of polynomials in z such that pn converges to f m-almost everywhere. Proof. As in the preceding proof, we can consider again the normal operator N ≡ Nm1 . Define the function g to be 1/|f | where |f | 1, and to be 1 where |f | < 1. Then 0 < g 1, 0 |f g| 1, hence f g ∈ L2 (m1 ). By Theorem 3.1, there is a cyclic vector γ ∈ L∞ (m1 ) for N . It follows that for any h ∈ L2 (m1 ) the equalities 0 = (N k γ g, h) = (zk γ , gh) for every k ∈ N0 imply gh = 0, hence h = 0 ∈ L2 (m1 ). This shows that γ g ∈ L∞ (m1 ) is a cyclic vector for N , hence is nonzero m1 -a.e. On the other hand, f gγ ∈ L2 (m1 ). Hence there is a sequence {pn } of polynomials such that pn (N )γ g − f γ g → 0 in L2 (m1 ) (n → ∞). It follows that a subsequence {pnk (z)γ (z)g(z): k ∈ N} converges to f (z)γ (z)g(z) m1 -a.e. Since γ g is nonzero m1 -a.e., we obtain that {pnk (z): k ∈ N} converges to f (z) m-a.e. 2 As a consequence of Theorem 3.1, the proof of the next result can essentially be a repetition of that in [16, Theorem 2.3.3, p. 244] for the bounded case. We give it here in view of its importance and for the reader’s convenience. Theorem 3.2. If N is any (unbounded) normal operator, then its global multiplicity and multicyclicity coincide, i.e. d(N) = μ(N ). Proof. Let ν denote, as in Section 2, the measure in the direct integral Y . Let k := d(N ) ≡ ν-ess.sup.{d(N, z): z ∈ C} < ∞, and let σk := {z ∈ C: d(N, z) = k}. Then ν(σk ) > 0 and, since {X(z): z ∈ C} is a measurable family of subspaces of X, there is an orthonormal basis {b1 (z), . . . , bk (z)} of each X(z) for z ∈ σk such that all functions z → bj (z) (j = 1, . . . , k) are measurable, and can be continued (as identically 0) to all of C\σk . Assume that C ∈ Cyc(N ) ⊂ X has dimension m < ∞ with a basis {c1 , . . . , cm } (if there is no such C, then we would have

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μ(N) = ∞ > d(N)). Then there are polynomials in z p(i, j, n; z) such that for every 1 j k we have B(j, n; z) :=

m

p(i, j, n; N )ci → bj (z)

in L2 (C, ν)

as n → ∞.

i=1

For each polynomial p and each ci ∈ C ⊂ X we have here [p(N )ci ](z) ≡ p(z)ci ∈ C ⊂ X(z) ν-a.e. on σk . Hence for every j = 1, . . . , k and for z ∈ σk ν-a.e. the vectors B(j, n; z) and bj (z) can be considered as column vectors in the bases {b1 (z), . . . , bk (z)}. Forming k × k matrices B(n; z) and b(z) of both these families of columns, we obtain that in the norm of L2 (C, ν), hence for some subsequence ν-a.e. on σk lim det B(n; z) = det b(z) ≡ 1.

n→∞

Hence det B(n; z) = 0 if n is sufficiently large. It follows that m k, i.e. μ(N ) d(N ). dTo prove the converse inequality, we may2 assume that d := d(N ) < ∞. Then N = j =1 ⊕Nj , where [Nj fj ](z) = zfj (z), fj ∈ L (τj , ν), where τj is a Borel subset of σ (N ) for j = 1, . . . , d. These normal operators Nj are multiplications by the variable on L2 spaces of scalar-valued functions, hence Theorem 3.1 yields μ(Nj ) = 1. It is easy to see that then μ(N )

d

μ(Nj ) = d = d(N ).

j =1

Hence μ(N) = d(N), and the proof is complete.

2

References [1] A. Boivin, C. Zhu, On the completeness of the system {zτn } in L2 , J. Approx. Theory 118 (2002) 1–19. [2] J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75–94. [3] A. Brown, A version of multiplicity theory, in: Topics in Operator Theory, in: Math. Surveys, vol. 13, Amer. Math. Soc., 1974, pp. 129–160. [4] J.B. Conway, Subnormal Operators, Pitman, Boston, 1981. [5] J.B. Conway, A Course in Functional Analysis, Springer, New York, 1985. [6] M.M. Dzhrbashyan, Metric theorems on the completeness and on the representation of analytic functions, Uspekhi Mat. Nauk 5 (37) (1950) 194–198 (in Russian). [7] K. Friedrichs, Die unitären Invarianten selbstadjungierter Operatoren im Hilbertschen Raum, Jahresber. Deutsch. Math.-Verein. 45 (1935) 79–82. [8] T.W. Gamelin, Uniform Algebras, Prentice Hall, Englewood Cliffs, NJ, 1969. [9] P.R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea, New York, 1951. [10] D.A. Herrero, On multicyclic operators, Integral Equations Operator Theory 1 (1978) 57–102. [11] S.N. Mergelyan, On the completeness of systems of analytic functions, Uspekhi Mat. Nauk VIII, 4 (56) (1953) 3–63 (in Russian); Amer. Math. Soc. Transl. Ser. 2 19 (1962) 109–166. [12] H. Nakano, Unitärinvarianten hypermaximaler normaler Operatoren im Hilbertschen Raum, Ann. of Math. 42 (1941) 657–664. [13] H. Nakano, Unitärinvarianten im allgemeinen euklidischen Raum, Math. Ann. 118 (1941) 112–133. [14] N.K. Nikolski, Multicyclicity phenomenon. I. An introduction and maxi-formulas, in: Toeplitz Operators and Spectral Function Theory, in: Oper. Theory Adv. Appl., vol. 42, Birkhäuser, Basel, 1989, pp. 9–57. [15] N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 1, Math. Surveys Monogr., vol. 92, Amer. Math. Soc., Providence, 2002.


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[16] N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, 2002. [17] A.I. Plessner, Spectral Theory of Linear Operators, Nauka, Moscow, 1965 (in Russian). [18] J.E. Thomson, Approximation in the mean by polynomials, Ann. of Math. 133 (1991) 477–507. [19] F.J. Wecken, Unitärinvarianten selbstadjungierter Operatoren, Math. Ann. 116 (1939) 422–455.


Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators Daniel Alpay a,1 , Jussi Behrndt b,∗ a Department of Mathematics, Ben-Gurion University of the Negev, PO Box 653, 84105 Beer-Sheva, Israel b Technische Universität Berlin, Institut für Mathematik, MA 6-4,

Straße des 17. Juni 136, 10623 Berlin, Germany Received 1 July 2008; accepted 17 June 2009 Available online 1 July 2009 Communicated by C. Kenig

Abstract The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an H 2 -framework are obtained. © 2009 Elsevier Inc. All rights reserved. Keywords: Q-function; Weyl function; Nevanlinna function; Elliptic operator; Dirichlet-to-Neumann map; Krein’s formula; Trace formula

1. Introduction The notion of a Q-function associated to a pair {S, A} consisting of a symmetric operator S and a selfadjoint extension A of S in a Hilbert or Pontryagin space was introduced by M.G. Krein and H. Langer in [37,38]. A Q-function contains the spectral information of the selfadjoint extensions of the underlying symmetric operator and therefore these functions play a very important role in the spectral and perturbation theory of selfadjoint operators. Q-functions appear also nat* Corresponding author.

E-mail addresses: [email protected] (D. Alpay), [email protected] (J. Behrndt). 1 Earl Katz Family Chair in algebraic system theory.


D. Alpay, J. Behrndt / Journal of Functional Analysis 257 (2009) 1666–1694

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urally in the description of the resolvents of the selfadjoint extensions of a symmetric operator with the help of Krein’s formula and they can be used to construct functional models for selfadjoint operators. In the theory of boundary triplets associated to symmetric operators Q-functions can be interpreted as so-called Weyl functions; cf. [16–19,29]. A prominent example for a Qfunction is the classical Titchmarsh–Weyl coefficient in the theory of singular Sturm–Liouville operators. The main objective of this paper is to extend the concept of Q-functions in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be identified as a generalized Q-function. In the abstract part of the paper we introduce the notion of generalized Q-functions and we show that these functions have similar properties as classical Q-functions. Besides a symmetric operator S and a selfadjoint extension A also an operator T whose closure coincides with S ∗ is used. Some of the ideas here parallel [9], where a more abstract approach with isometric and unitary relations in Krein spaces was used. The main result in the abstract part is Theorem 2.6 which states that an operator function is a generalized Q-function if and only if it coincides up to a possibly unbounded constant on a dense subspace with the restriction of a Nevanlinna function with an invertible imaginary part and a certain asymptotic behaviour. Sections 3 and 4 deal with second order elliptic operators on bounded and unbounded domains, and with the coupling of such operators. Suppose first that the domain Ω ⊂ Rn , n > 1, is bounded with a smooth boundary ∂Ω. Let AD and AN be the selfadjoint realizations of a formally symmetric uniformly elliptic differential expression L=−

n j,k=1

∂ ∂ aj k +a ∂xj ∂xk

(1.1)

in L2 (Ω) defined on H 2 (Ω) and subject to Dirichlet and Neumann boundary conditions, respectively. If T denotes the realization of L on H 2 (Ω), then the closure of T in L2 (Ω) coincides with the maximal operator associated to L in L2 (Ω), and AD and AN are both selfadjoint restrictions of T . For a function f ∈ H 2 (Ω) denote the trace and the trace of the conormal derivative by f |∂Ω and ∂f ∂ν |∂Ω , respectively. Then for each λ ∈ ρ(AD ) the Dirichlet-to-Neumann map ∂fλ , Q(λ)(fλ |∂Ω ) := − ∂ν ∂Ω

where Tfλ = λfλ ,

(1.2)

is well defined and will be regarded as an operator in L2 (∂Ω) defined on H 3/2 (∂Ω) with values in H 1/2 (∂Ω). The minus sign in (1.2) is used for technical reasons. It turns out that the operator function λ → Q(λ) is a generalized Q-function in the sense of Definition 2.2 and an explicit variant of Krein’s formula for the resolvents of AD and AN is obtained in Theorem 3.4, see also [9,13,25,26,47–50] for more general problems. In particular, in the case n = 2 it follows from results due to M.S. Birman that the difference of these resolvents is a trace class operator. As a consequence we obtain the trace formula d −1 −1 −1 = tr Q(λ) tr (AD − λ) − (AN − λ) Q(λ) dλ

(1.3)

is a Nevanfor λ ∈ ρ(AD ) ∩ ρ(AN ). Here Q(λ)−1 is the closure of Q(λ)−1 in L2 (∂Ω) and Q linna function which differs from the Dirichlet-to-Neumann map by a symmetric constant. Trace

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formulas for canonical differential expressions and in more abstract situations for finite dimensional resolvent differences can be found in, e.g., [2,3,10]. In Section 4 we consider a so-called coupling of elliptic operators. Such couplings are of great interest in problems of mathematical physics, e.g., in the description of quantum networks; for more details and further references we refer the reader to the recent works [20,21,44–46]. Suppose that Rn , n > 1, is decomposed in a bounded domain Ω with smooth boundary C and the unbounded domain Ω = Rn \Ω. The orthogonal sum of the selfadjoint Dirichlet operators AD and AD associated to L in L2 (Ω) and L2 (Ω ), respectively, is regarded as a selfadjoint diagonal block operator matrix in L2 (Rn ). The resolvent of AD ⊕ AD is then compared with the of L in L2 (Rn ) defined on H 2 (Rn ). In order to resolvent of the usual selfadjoint realization A express this difference in the Krein type formula −1 − λ)−1 = Γ (λ)Q(λ)−1 Γ (λ) ¯ ∗ AD ⊕ AD − λ − (A

(1.4)

with a generalized Q-function an analogon of the Dirichlet-to-Neumann map is constructed which measures the jump of the conormal derivative of L2 (Ω) and L2 (Ω )-solutions of Lu = λu on the boundary C, see (4.21). The operator Γ (λ) : L2 (C) → L2 (Rn ) in (1.4) is closely connected with the generalized Q-function and is identified with a Poisson-type operator solving a certain Dirichlet problem. As a consequence of the representation (1.4) we also obtain a trace formula of the type (1.3) in the coupled case. 2. Generalized Q-functions In this section we introduce the notion of generalized Q-functions associated to symmetric operators in Hilbert spaces. The class of generalized Q-functions is characterized in Theorem 2.6, where it turns out that generalized Q-functions are closely connected with operatorvalued Nevanlinna or Riesz–Herglotz functions. We also note in advance that for the case of finite deficiency indices of the underlying symmetric operator the concept of generalized Qfunctions coincides with the classical notion of (ordinary) Q-functions studied by M.G. Krein and H. Langer in [37,38], see also [35,36]. Let H be a separable Hilbert space and let S be a densely defined closed symmetric operator with equal (in general infinite) deficiency indices n± (S) = dim ker S ∗ ∓ i ∞ in H. It is well known that under this assumption S admits selfadjoint extensions in H. In the following let A be a fixed selfadjoint extension of S in H, so that, S ⊂ A = A∗ ⊂ S ∗ . Furthermore, let T be a linear operator in H such that A ⊂ T ⊂ S ∗ and T = S ∗ hold, i.e., the domain dom T of T is a core of dom S ∗ (see [34]), dom T contains dom A and Af = Tf holds for all f ∈ dom A. For λ ∈ C belonging to the resolvent set ρ(A) of the selfadjoint operator A define the defect spaces Nλ (T ) = ker(T − λ) and Nλ (S ∗ ) = ker(S ∗ − λ). Then the decompositions ˙ Nλ S ∗ and dom S ∗ = dom A +

˙ Nλ (T ) dom T = dom A +

(2.1)

hold for all λ ∈ ρ(A) and the closure Nλ (T ) of Nλ (T ) in H coincides with Nλ (S ∗ ). Recall that the symmetric operator S is said to be simple if there exists no nontrivial subspace D in dom S


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such that S restricted to D is a selfadjoint operator in the Hilbert space D. It is important to note that S is simple if and only if

H = span Nλ S ∗ : λ ∈ C\R

(2.2)

holds; cf. [36]. Here span denotes the closed linear span. As Nλ (T ) = Nλ (S ∗ ) it is clear that the right-hand side in (2.2) coincides with

span Nλ (T ): λ ∈ C\R . Fix some λ0 ∈ ρ(A), let G be a Hilbert space with the same dimension as Nλ0 (T ) and let Γλ0 be a densely defined bounded operator from G into H such that ran Γλ0 = Nλ0 (T ) and ker Γλ0 = {0} hold. The domain dom Γλ0 of Γλ0 will be denoted by G0 . Observe that the closure Γ λ0 of the operator Γλ0 is the bounded extension of Γλ0 which is defined on G 0 = G. We write Γ λ0 ∈ L(G, H), where L(G, H) is the space of bounded linear operators defined on G with values in H. Lemma 2.1. The operator function λ → Γ (λ) := (I + (λ − λ0 )(A − λ)−1 )Γλ0 defined on ρ(A) satisfies Γ (λ0 ) = Γλ0 , Γ (λ) = I + (λ − μ)(A − λ)−1 Γ (μ),

λ, μ ∈ ρ(A),

and Γ (λ) is a bounded operator from G into H which maps dom Γ (λ) = G0 bijectively onto Nλ (T ) for all λ ∈ ρ(A). Moreover, λ → Γ (λ)g is holomorphic on ρ(A) for every g ∈ G0 . Proof. Let us show that ran Γ (λ) = Nλ (T ) is true. The other assertions in the lemma are obvious or follow from a straightforward calculation. Since T is an extension of the selfadjoint operator A we have (T − λ)(A − λ)−1 = I for λ ∈ ρ(A) and therefore (T − λ)Γ (λ)h = (T − λ) I + (λ − λ0 )(A − λ)−1 Γλ0 h = (T − λ0 )Γλ0 h = 0 shows that ran Γ (λ) ⊂ Nλ (T ) holds. Now let fλ ∈ Nλ (T ). Then it follows as above that fλ0 := I + (λ0 − λ)(A − λ0 )−1 fλ is an element in Nλ0 (T ) and hence there exists h ∈ G0 such that fλ0 = Γλ0 h. Now a simple calculation shows fλ = Γ (λ)h, thus ran Γ (λ) = Nλ (T ). 2 In the following definition the concept of generalized Q-functions is introduced. Definition 2.2. Let S, A, T , and Γ (·) be as above. An operator function Q defined on ρ(A) whose values Q(λ) are linear operators in G with dom Q(λ) = G0 for all λ ∈ ρ(A) is said to be a generalized Q-function of the triple {S, A, T } if Q(λ) − Q(μ)∗ = (λ − μ)Γ ¯ (μ)∗ Γ (λ),

λ, μ ∈ ρ(A),

(2.3)

holds on G0 . If, in addition, G0 = G and T = S ∗ , then Q is called an ordinary Q-function of {S, A}.

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We note that the values Q(λ), λ ∈ ρ(A), of a generalized Q-function can be unbounded nonclosed operators. The adjoint Q(μ)∗ in (2.3) is well defined since dom Q(μ) is dense in G and by (2.3) also Q(μ) ⊂ Q(μ) ¯ ∗ holds for all μ ∈ ρ(A). In particular, the operators Q(λ) are closable in G and symmetric for λ ∈ ρ(A) ∩ R. The real and imaginary parts of the operators Q(λ) are defined as usual: Re Q(λ) =

1 Q(λ) + Q(λ)∗ 2

and

Im Q(λ) =

1 Q(λ) − Q(λ)∗ . 2i

Since (Re Q(λ)h, h) and (Im Q(λ)h, h) are real for all h ∈ G0 the operators Re Q(λ) and Im Q(λ) are symmetric. Remark 2.3. We note that the concept of generalized Q-functions is closely connected with the theory of boundary triplets and associated Weyl functions. The Weyl function of an ordinary or generalized boundary triplet (see [16,18,19,29]) is also a generalized Q-function, but the converse is not true. The class of generalized Q-functions studied here coincides with the class of Weyl functions of so-called quasi boundary triplets introduced in [9]. Furthermore, we note that generalized Q-functions are no subclass of the Weyl families associated to boundary relations, see [17] and Theorem 2.6. The concept of generalized Q-functions differs from the classical notion of ordinary Qfunctions only in the case n± (S) = ∞. Proposition 2.4. Let Q be a generalized Q-function of the triple {S, A, T } and assume, in addition, that the deficiency indices n± (S) are finite. Then T = S ∗ and Q is an ordinary Q-function of the pair {S, A}. Proof. If the deficiency indices of the closed operator S are finite, then T is a finite dimensional extension of S and hence also T is closed. Therefore T = T = S ∗ . Moreover, in this case also dim G = dim Nλ0 (T ) is finite and hence G0 = dom Γ (λ) = dom Q(λ) = G, λ ∈ ρ(A). 2 The representation of a generalized Q-function with the help of the resolvent of A in the next proposition is formally the same as for ordinary Q-functions, see [37–39]. Proposition 2.5. Let Q be a generalized Q-function of the triple {S, A, T } and let λ0 ∈ ρ(A). Then Q can be written as the sum of the possibly unbounded operator Re Q(λ0 ) and a bounded holomorphic operator function, Q(λ) = Re Q(λ0 ) + Γλ∗0 (λ − Re λ0 ) + (λ − λ0 )(λ − λ¯ 0 )(A − λ)−1 Γλ0 ,

(2.4)

and, in particular, any two generalized Q-functions of {S, A, T } differ by a constant. Proof. Let h ∈ G0 and set μ = λ0 in (2.3). Making use of the definition of Γ (λ) in Lemma 2.1 we obtain Q(λ)h = Q(λ0 )∗ h + (λ − λ¯ 0 )Γλ∗0 I + (λ − λ0 )(A − λ)−1 Γλ0 h.


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As Q(λ0 )h − Q(λ0 )∗ h = (λ0 − λ¯ 0 )Γλ∗0 Γλ0 h we see that the above formula can be rewritten as Q(λ)h = Q(λ0 )h + (λ − λ0 )Γλ∗0 Γλ0 h + Γλ∗0 (λ − λ0 )(λ − λ¯ 0 )(A − λ)−1 Γλ0 h. The representation (2.4) follows by inserting Q(λ0 )h = Re Q(λ0 )h + i Im Q(λ0 )h and Im Q(λ0 )h = Im λ0 Γλ∗0 Γλ0 h into this expression. 2 Generalized Q-functions are closely connected with the class of Nevanlinna functions; cf. Theorem 2.6 below. Let L(G) be the space of everywhere defined bounded linear operators in G. which is holomorphic on C\R and satisfies Recall that an L(G)-valued operator function Q Im Q(λ) λ) ∗ ¯ = Q(λ) 0 and Q( Im λ

(2.5)

is an L(G)-valued for λ ∈ C\R is said to be an L(G)-valued Nevanlinna function. We note that Q admits an integral representation of the form Nevanlinna function if and only if Q = α + λβ + Q(λ)

R

1 t dΣ(t), − t − λ 1 + t2

λ ∈ C\R,

(2.6)

where α = α ∗ ∈ L(G), 0 β = β ∗ ∈ L(G) and t → Σ(t) ∈ L(G) is a selfadjoint nondecreasing L(G)-valued function on R such that R

1 dΣ(t) ∈ L(G). 1 + t2

It is well known that Nevanlinna functions can be represented with the help of selfadjoint operators or relations in Hilbert spaces in a very similar form as in (2.4). Such operator and functional models for Nevanlinna functions can be found in, e.g., [1,7,12,15,19,27,33,39,41]. In the next theorem we characterize the class of generalized Q-functions. Roughly speaking, it turns out that up to a symmetric constant a generalized Q-function is a restriction of an L(G) with invertible imaginary part on dom Q(λ) and Q satisfies certain valued Nevanlinna function Q limit properties at ∞. Theorem 2.6. Let G0 be a dense subspace of G, λ0 ∈ C\R, and let Q be a function defined on C\R whose values Q(λ) are linear operators in G with dom Q(λ) = G0 , λ ∈ C\R. Then the following is equivalent: (i) Q is a generalized Q-function of a triple {S, A, T }, where S is a simple symmetric operator in some separable Hilbert space H, A is a selfadjoint extension of S in H and A ⊂ T ⊂ S ∗ with T = S ∗ ; with the properties (α), (β) (ii) there exists a unique L(G)-valued Nevanlinna function Q and (γ ): (α) The relations Q(λ)h − Re Q(λ0 )h = Q(λ)h

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and ∗h Q(λ)∗ h − Re Q(λ0 )h = Q(λ) hold for all h ∈ G0 and λ ∈ C\R; (β) Im Q(λ)h = 0 for some h ∈ G0 and λ ∈ C\R implies h = 0; (γ ) the conditions lim

η→+∞

1 Q(iη)k, k = 0 η

and

lim η Im Q(iη)k, k =∞

η→+∞

are valid for all k ∈ G, k = 0. Proof. We start by showing that (i) implies (ii). For this, let Q be a generalized Q-function of the triple {S, A, T } and suppose that S is simple. Let Γλ0 be a bounded operator defined on dom Q(λ) = G0 such that ran Γλ0 = Nλ0 (T ) and ker Γλ0 = {0}. According to Proposition 2.5 for each λ ∈ C\R Q(λ) − Re Q(λ0 ) = Γλ∗0 (λ − Re λ0 ) + (λ − λ0 )(λ − λ¯ 0 )(A − λ)−1 Γλ0 is a bounded operator in G defined on the dense subspace G0 and hence admits a unique bounded extension onto G which is given by := Γλ∗ (λ − Re λ0 ) + (λ − λ0 )(λ − λ¯ 0 )(A − λ)−1 Γ λ0 , Q(λ) 0

(2.7)

where Γ λ0 ∈ L(G, H) is the closure of Γλ0 . Obviously we have Q(λ)h − Re Q(λ0 )h = Q(λ)h for all h ∈ G0 and λ ∈ C\R, which is the first relation in (α). Recall that for a generalized Qfunction Q(λ¯ )∗ is an extension of Q(λ). This implies Re Q(λ0 ) ⊂ (Re Q(λ0 ))∗ , ∗ ∗ Q(λ)∗ − Re Q(λ0 ) ⊂ Q(λ) − Re Q(λ0 ) = Q(λ) ∗ h is true for all h ∈ G0 and λ ∈ C\R. Hence we and therefore also Q(λ)∗ h − Re Q(λ0 )h = Q(λ) have shown (α). in (2.7) is a holomorphic L(G)-valued function on C\R. Denote by Γ (λ) the cloClearly Q sure of Γ (λ) = (I + (λ − λ0 )(A − λ)−1 )Γλ0 . Then Γ (λ) = I + (λ − λ0 )(A − λ)−1 Γ λ0 ,

λ ∈ C\R,

and it is not difficult to see that (2.3) extends to − Q(μ) ∗ = (λ − μ)Γ ¯ (μ)∗ Γ (λ). Q(λ) Hence 2 Im Q(λ)k, k = (Im λ) Γ (λ)∗ Γ (λ)k, k = (Im λ) Γ (λ)k


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is a Nevanlinna function; cf. (2.5). Furthermore, for holds for all k ∈ G and this implies that Q h ∈ G0 we have Im Q(λ)h = (Im λ)Γ (λ)∗ Γ (λ)h and from the property ker Γ (λ) = {0} (see Lemma 2.1) we conclude that Im Q(λ)h = 0 for h ∈ G0 implies h = 0, i.e., condition (β) holds. The same arguments as in [39, Theorem 2.4, Corollaries 2.5 and 2.6] together with the assumption that S is a densely defined closed simple satisfies the conditions in (γ ). symmetric operator show that Q is an L(G)-valued Nevanlinna function, λ0 ∈ Let us now verify the converse direction. If Q C\R and the first condition in (γ ) holds, then it is well known that there exist a Hilbert space H, a selfadjoint operator A in H and a mapping Γ ∈ L(G, H) such that the representation = Re Q(λ 0 ) + Γ∗ (λ − Re λ0 ) + (λ − λ0 )(λ − λ¯ 0 )(A − λ)−1 Γ Q(λ)

(2.8)

is valid for all λ ∈ C\R, see, e.g., [33,39]. Furthermore, the space H can be chosen minimal, i.e.,

H = span I + (λ − λ0 )(A − λ)−1 Γk: k ∈ G, λ ∈ C\R .

(2.9)

We define the mapping Γλ0 to be the restriction of Γ onto G0 . As Γ is bounded the closure Γ λ0 of Γλ0 coincides with Γ. We claim that Γλ0 is injective. In fact, if Γλ0 h = 0 for some 0 )h. Therefore Im Q(λ)h h ∈ G0 then Γh = 0 and by (2.8) we have Q(λ)h = Re Q(λ = 0 and by assumption (β) this implies h = 0. Define the operator S by Sf = Af,

dom S = f ∈ dom A: (A − λ¯ 0 )f, Γλ0 h = 0 for all h ∈ G0 .

Then S is a closed symmetric operator in the Hilbert space H and the identities ran(S − λ¯ 0 ) = (ran Γλ0 )⊥ and ker(S ∗ − λ0 ) = ran Γλ0 hold. Let Γ (λ) = I + (λ − λ0 )(A − λ)−1 Γλ0 ,

λ ∈ C\R.

(2.10)

It is not difficult to check that ran(S − λ¯ ) = (ran Γ (λ))⊥ is true for all λ ∈ C\R and the conditions in (γ ) together with (2.9) now yield in the same way as in [39, Theorem 2.4, Corollaries 2.5 and 2.6] that S is densely defined and simple. Note that dom A ∩ ran Γλ0 = {0} since λ0 ∈ ρ(A) and ran Γλ0 ⊂ Nλ0 (S ∗ ). Let us define a ˙ ran Γλ0 by linear operator T in H on dom T := dom A + T (f + fλ0 ) := Af + λ0 fλ0 ,

f ∈ dom A, fλ0 ∈ ran Γλ0 .

Obviously T is an extension of A and since Nλ0 (T ) = ran Γλ0 and ran Γλ0 is dense in Nλ0 (S ∗ ) ˙ Nλ0 (S ∗ ) (see (2.1)) that T ⊂ S ∗ and T = S ∗ hold. we obtain from dom S ∗ = dom A + and the function Q are related by According to condition (α) the Nevanlinna function Q Q(λ)h = Q(λ)h + Re Q(λ0 )h

∗ h + Re Q(λ0 )h and Q(λ)∗ h = Q(λ)

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for all h ∈ G0 and λ ∈ C\R. It remains to show that Q satisfies (2.3). Observe first that for λ, μ ∈ C\R we have ∗ h. − Q(μ) Q(λ)h − Q(μ)∗ h = Q(λ)h

(2.11)

Denote the closures of the operators Γ (λ), λ ∈ C\R, in (2.10) by Γ(λ). Then Γ(λ) = Γ (λ) = I + (λ − λ0 )(A − λ)−1 Γ λ0 = I + (λ − λ0 )(A − λ)−1 Γ and it follows from (2.8) with a straightforward calculation that − Q(μ) ∗ = (λ − μ) ¯ Γ(μ)∗ Γ(λ), Q(λ)

λ, μ ∈ C\R,

(2.12)

holds. As Γ(μ)∗ = (Γ (μ))∗ = Γ (μ)∗ we conclude ¯ (μ)∗ Γ (λ)h, Q(λ)h − Q(μ)∗ h = (λ − μ)Γ

h ∈ G0 ,

from (2.11). Therefore Q is a generalized Q-function of the triple {S, A, T }.

2

Remark 2.7. The definition of a generalized Q-function can be extended to the case that A is a selfadjoint relation, S is a non-densely defined symmetric operator or relation and T is a linear relation which is dense in the relation S ∗ . We refer to [39] for ordinary Q-functions in this more general situation. In this case the condition (γ ) in Theorem 2.6 can be dropped. For ordinary Q-functions Theorem 2.6 reads as follows; cf. [39, Theorems 2.2 and 2.4]. is an ordinary Q-function of some pair Theorem 2.8. An L(G)-valued Nevanlinna function Q {S, A}, where S is a densely defined closed simple symmetric operator in some Hilbert space H and A is a selfadjoint extension of S in H, if and only if condition (γ ) in Theorem 2.6 and 0 ∈ ρ(Im Q(λ)) holds for some, and hence for all, λ ∈ C\R. be the L(G)-valued Corollary 2.9. Let Q be a generalized Q-function of {S, A, T } and let Q Nevanlinna function in Theorem 2.6. Then for all λ ∈ C\R and h ∈ G0 we have d d ¯ ∗ Γ (λ)h. Q(λ)h = Q(λ)h = Γ (λ) dλ dλ Proof. It follows from (2.12) that − Q(μ) ∗ Q(λ) d ¯ ∗ Γ(λ) = Γ(λ) Q(λ) = lim μ→λ ¯ dλ λ − μ¯ holds. Hence condition (α) in Theorem 2.6 and Γ(λ) = Γ (λ) imply ∗h d Q(λ)h − Q(μ)∗ h Q(λ)h − Q(μ) ¯ ∗ Γ (λ)h Q(λ)h = lim = lim = Γ (λ) μ→λ ¯ μ→λ ¯ dλ λ − μ¯ λ − μ¯ for h ∈ G0 .

2


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3. Elliptic operators and the Dirichlet-to-Neumann map Let Ω ⊂ Rn be a bounded or unbounded domain with compact C ∞ -boundary ∂Ω. Let L be the “formally selfadjoint” uniformly elliptic second order differential expression (Lf )(x) := −

n ∂ ∂f (x) + a(x)f (x), aj k ∂xj ∂xk

(3.1)

j,k=1

x ∈ Ω, with bounded infinitely differentiable real valued coefficients aj k ∈ C ∞ (Ω) satisfying aj k (x) = akj (x) for all x ∈ Ω and j, k = 1, . . . , n; the function a ∈ L∞ (Ω) is real valued and n

aj k (x)ξj ξk C

j,k=1

n

ξk2

(3.2)

k=1

holds for some C > 0, all ξ = (ξ1 , . . . , ξn ) ∈ Rn and x ∈ Ω. We note that the assumptions on the domain Ω and the coefficients of L can be relaxed but it is not our aim to treat the most general setting here. We refer the reader to e.g. [30,40,43,52] for possible generalizations. In the following we consider the selfadjoint realizations of L in L2 (Ω) subject to Dirichlet and Neumann (or oblique Neumann) boundary conditions. For a function f in the Sobolev space H 2 (Ω) we denote the trace by f |∂Ω and the trace of the conormal derivative is defined by n ∂f ∂f := a n ; jk j ∂ν ∂Ω ∂xk ∂Ω j,k=1

here n(x) = (n1 (x), . . . , nn (x)) is the unit vector at the point x ∈ ∂Ω pointing out of Ω. Recall that the mapping C ∞ (Ω) f → {f |∂Ω , ∂f ∂ν |∂Ω } extends by continuity to a continuous surjective mapping

∂f ∈ H 3/2 (∂Ω) × H 1/2 (∂Ω). H (Ω) f → f |∂Ω , ∂ν ∂Ω 2

(3.3)

The kernel of this map is H02 (Ω) =

∂f 2 f ∈ H (Ω): f |∂Ω = = 0 ∂ν ∂Ω

which coincides with the closure of C0∞ (Ω) in H 2 (Ω). We refer the reader to the monographs [40,43,52] for more details. In the following the scalar products in L2 (Ω) and L2 (∂Ω) are denoted by (·,·)Ω and (·,·)∂Ω , respectively. Then Green’s identity ∂g ∂f (Lf, g)Ω − (f, Lg)Ω = f |∂Ω , − , g| ∂Ω ∂ν ∂Ω ∂Ω ∂ν ∂Ω ∂Ω

(3.4)

holds for all functions f, g ∈ H 2 (Ω). We note that (3.4) is even true for f ∈ H 2 (Ω) and g belonging to the domain of the maximal operator associated to L in L2 (Ω) if the (·,·)∂Ω scalar

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product in L2 (∂Ω) is extended by continuity to H 3/2 (∂Ω) × H −3/2 (∂Ω) and H 1/2 (∂Ω) × H −1/2 (∂Ω), respectively, see [40,52]. However, we shall make use of (3.4) only for the case f, g ∈ H 2 (Ω). It is known that the realizations AD and AN of L subject to Dirichlet and Neumann boundary conditions defined by AD f = Lf, AN f = Lf,

dom AD = f ∈ H 2 (Ω): f |∂Ω = 0 ,

∂f 2 dom AN = f ∈ H (Ω): =0 , ∂ν ∂Ω

(3.5)

are selfadjoint operators in L2 (Ω). The following statement is known and can be found in, e.g., [40]. It can be proved with similar methods as Theorem 4.1 in the next section. Proposition 3.1. Let L be the elliptic differential expression in (3.1). Then the operator Sf = Lf,

dom S = H02 (Ω),

(3.6)

is a densely defined closed symmetric operator in L2 (Ω) with infinite deficiency indices n± (S) and the adjoint S ∗ of S coincides with the maximal operator associated to L, S ∗ f = Lf,

dom S ∗ = f ∈ L2 (Ω): Lf ∈ L2 (Ω) .

The operator Tf = Lf,

dom T = H 2 (Ω),

is not closed as an operator in L2 (Ω) and T satisfies T = S ∗ and T ∗ = S. Furthermore, the selfadjoint operators AD and AN in (3.5) are extensions of S and restrictions of T . In order to define a mapping Γλ0 for the definition of a generalized Q-function associated to the triple {S, AD , T } we make use of the decomposition (2.1) in the present situation. More precisely, for all points λ in the resolvent set ρ(AD ) of the selfadjoint Dirichlet operator AD we have the direct sum decomposition of dom T = H 2 (Ω):

˙ Nλ (T ), ˙ Nλ (T ) = f ∈ H 2 (Ω): f |∂Ω = 0 + H 2 (Ω) = dom AD +

(3.7)

where

Nλ (T ) = ker(T − λ) = fλ ∈ H 2 (Ω): Lfλ = λfλ . Let now ϕ be a function in H 3/2 (∂Ω) and let λ0 ∈ ρ(AD ). Then it follows from (3.3) and (3.7) that there exists a unique function fλ0 ∈ H 2 (Ω) which solves the equation Lfλ0 = λ0 fλ0 , i.e., fλ0 ∈ Nλ0 (T ), and satisfies fλ0 |∂Ω = ϕ. We shall denote the mapping that assigns fλ0 to ϕ by Γλ0 , H 3/2 (∂Ω) ϕ → Γλ0 ϕ := fλ0 ∈ Nλ0 (T ),

(3.8)


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and we regard Γλ0 as an operator from L2 (∂Ω) into L2 (Ω) with dom Γλ0 = H 3/2 (∂Ω) and ran Γλ0 = Nλ0 (T ). Proposition 3.2. Let λ0 ∈ ρ(AD ), let Γλ0 be as in (3.8) and let λ ∈ ρ(AD ). Then the following hold: (i) Γλ0 is a bounded operator from L2 (∂Ω) in L2 (Ω) with dense domain H 3/2 (∂Ω); (ii) the operator Γ (λ) = (I + (λ − λ0 )(AD − λ)−1 )Γλ0 is given by Γ (λ)ϕ = fλ ,

where fλ ∈ Nλ (T ) and fλ |∂Ω = ϕ;

(iii) the mapping Γ (λ¯ )∗ : L2 (Ω) → L2 (∂Ω) satisfies ¯ ∗ (AD − λ)f = − Γ (λ)

∂f , ∂ν ∂Ω

f ∈ dom AD .

Proof. Statement (i) will be a consequence of (iii). We prove assertion (ii). Recall that by Lemma 2.1 the range of the operator Γ (λ), λ ∈ ρ(AD ), is Nλ (T ). Let ϕ ∈ dom Γ (λ) = H 3/2 (∂Ω) and choose elements fλ ∈ Nλ (T ) and fλ0 ∈ Nλ0 (T ) such that fλ |∂Ω = ϕ = fλ0 |∂Ω holds. According to (3.7) the functions fλ and fλ0 are unique. Then Γλ0 ϕ = fλ0 and hence we obtain Γ (λ)ϕ = Γλ0 ϕ + (λ − λ0 )(AD − λ)−1 Γλ0 ϕ = fλ0 + (λ − λ0 )(AD − λ)−1 Γλ0 ϕ. Since (λ − λ0 )(AD − λ)−1 Γλ0 ϕ belongs to dom AD it is clear that the trace of this element vanishes. Therefore, the traces of the functions Γ (λ)ϕ ∈ Nλ (T ) and fλ0 coincide, Γ (λ)ϕ ∂Ω = fλ0 |∂Ω = ϕ = fλ |∂Ω . Thus we have that the traces of Γ (λ)ϕ ∈ Nλ (T ) and fλ ∈ Nλ (T ) coincide and from (3.7) we conclude Γ (λ)ϕ = fλ . (iii) Let ϕ ∈ H 3/2 (∂Ω) and choose the unique function gλ¯ ∈ Nλ¯ (T ) with the property ¯ = gλ¯ and for f ∈ dom AD it follows gλ¯ |∂Ω = ϕ. Hence we have Γ (λ)ϕ ¯ λ¯ , f )Ω = (gλ¯ , AD f )Ω − (T gλ¯ , f )Ω . ¯ Γ (λ)ϕ, (AD − λ)f Ω = (gλ¯ , AD f )Ω − (λg Making use of Green’s identity (3.4) we find (gλ¯ , AD f )Ω − (T gλ¯ , f )Ω =

∂gλ¯ ∂f , f |∂Ω − gλ¯ |∂Ω , ∂ν ∂Ω ∂ν ∂Ω ∂Ω ∂Ω

and since the trace of f ∈ dom AD vanishes the first summand on the right-hand side is zero. Therefore

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∂f ∂f Γ (λ¯ )ϕ, (AD − λ)f Ω = − gλ¯ |∂Ω , = ϕ, − ∂ν ∂Ω ∂Ω ∂ν ∂Ω ∂Ω holds for all ϕ ∈ dom Γ (λ¯ ) = H 3/2 (∂Ω). This gives (AD − λ)f ∈ dom Γ (λ¯ )∗ and ¯ ∗ (AD − λ)f = − Γ (λ)

∂f . ∂ν ∂Ω

Moreover, as λ ∈ ρ(AD ) and f ∈ dom AD was arbitrary we see that Γ (λ¯ )∗ is defined on the ¯ ∗ is closed implies whole space L2 (Ω). This together with the fact that Γ (λ) ¯ ∗ ∈ L L2 (Ω), L2 (∂Ω) Γ (λ) ¯ ⊂ Γ (λ) ¯ = Γ (λ) ¯ ∗∗ is bounded. Inserting λ0 = λ¯ this yields for λ ∈ ρ(AD ) and, in particular, Γ (λ) assertion (i). 2 In the study of elliptic differential operators the so-called Dirichlet-to-Neumann map plays an important role, we mention only [4,14,22–26,31,42,44–49,51]. Roughly speaking this operator maps the Dirichlet boundary value fλ |∂Ω of an H 2 (Ω)-solution of the equation Lu = λu onto λ the Neumann boundary value ∂f ∂ν |∂Ω of this solution. In the following definition also a minus sign arises, which is needed to obtain a generalized Q-function in Theorem 3.4. Otherwise −Q would turn out to be a generalized Q-function. Definition 3.3. Let λ ∈ ρ(AD ) and assign to ϕ ∈ H 3/2 (∂Ω) the unique function fλ ∈ Nλ (T ) such that fλ |∂Ω = ϕ, see (3.3) and (3.7). The operator Q(λ) in L2 (∂Ω) defined by ∂fλ Q(λ)ϕ = Q(λ)(fλ |∂Ω ) := − , ∂ν ∂Ω

ϕ ∈ dom Q(λ) = H 3/2 (∂Ω),

(3.9)

is called the Dirichlet-to-Neumann map associated to L. Note that by (3.3) the range of the Dirichlet-to-Neumann map Q(λ), λ ∈ ρ(AD ), lies in H 1/2 (∂Ω). We remark that the Dirichlet-to-Neumann map can be extended, e.g., to an operator from H 1 (∂Ω) in L2 (∂Ω) if instead of H 2 (Ω) the operator T is defined on a suitable subspace of H 3/2 (Ω); cf. [4–6,9,32,40]. However, for our purposes this is not necessary since AD and AN are defined on subspaces of H 2 (Ω). In the next theorem we show that the Dirichlet-to-Neumann map is a generalized Q-function and we illustrate the usefulness of this object in the representation of the difference of the resolvents of the Dirichlet and Neumann operators AD and AN in (3.5). Similar Krein type resolvent formulas can also be found in [9,13,25,26,47–50]. The fact that the difference of the resolvents belongs to some von Neumann–Schatten class depending on the dimension of the space is well known and goes back to M.S. Birman; cf. [11]. Theorem 3.4. Let L be the elliptic differential expression in (3.1) and let AD and AN be the selfadjoint realizations of L in (3.5). Denote by S the minimal operator associated to L and let T = L H 2 (Ω) be as in Proposition 3.1. Define Γ (λ) as in Proposition 3.2 and let Q(λ), λ ∈ ρ(AD ), be the Dirichlet-to-Neumann map. Then the following hold:


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(i) Q is a generalized Q-function of the triple {S, AD , T }; (ii) the operator Q(λ) is injective for all λ ∈ ρ(AD ) ∩ ρ(AN ) and the resolvent formula (AD − λ)−1 − (AN − λ)−1 = Γ (λ)Q(λ)−1 Γ (λ¯ )∗

(3.10)

holds; (iii) for p > n−1 2 the difference of the resolvents in (3.10) belongs to the von Neumann–Schatten class Sp (L2 (Ω)). Proof. In order to prove assertion (i) we have to check the relation ¯ (μ)∗ Γ (λ), Q(λ) − Q(μ)∗ = (λ − μ)Γ

λ, μ ∈ ρ(AD ),

(3.11)

on dom Q(λ) = H 3/2 (∂Ω). For this it will be first shown that H 3/2 (∂Ω) is a subset of dom Q(μ)∗ and that Q(μ)∗ is an extension of Q(μ). ¯ Let ψ ∈ H 3/2 (∂Ω) and choose the unique function fμ¯ ∈ Nμ¯ (T ) such that fμ¯ |∂Ω = ψ . For an arbitrary ϕ ∈ dom Q(μ) = H 3/2 (∂Ω) let fμ ∈ Nμ (T ) be the unique function that satisfies fμ |∂Ω = ϕ. By the definition of the Dirichletto-Neumann map we have ∂fμ Q(μ)ϕ = − ∂ν ∂Ω

∂fμ¯ and Q(μ)ψ ¯ =− ∂ν ∂Ω

and hence Green’s identity (3.4) shows ∂fμ Q(μ)ϕ, ψ ∂Ω = − , fμ¯ |∂Ω ∂ν ∂Ω ∂Ω ∂fμ¯ ∂fμ ∂fμ¯ = fμ |∂Ω , − , fμ¯ |∂Ω + ϕ, − ∂ν ∂Ω ∂Ω ∂ν ∂Ω ∂ν ∂Ω ∂Ω ∂Ω ∂fμ¯ = (Tfμ , fμ¯ )Ω − (fμ , Tfμ¯ )Ω + ϕ, − . ∂ν ∂Ω ∂Ω Since fμ ∈ Nμ (T ) and fμ¯ ∈ Nμ¯ (T ) it is clear that (Tfμ , fμ¯ )Ω = (fμ , Tfμ¯ )Ω holds and therefore we obtain ∂fμ¯ Q(μ)ϕ, ψ ∂Ω = ϕ, − ∂ν ∂Ω ∂Ω for all ϕ ∈ dom Q(μ). Thus ψ ∈ dom Q(μ)∗ and ∂fμ¯ Q(μ) ψ = − = Q(μ)ψ. ¯ ∂ν ∂Ω ∗

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Next we prove the relation (3.11). Let ϕ, ψ ∈ H 3/2 (∂Ω) and choose the functions fλ ∈ Nλ (T ) and gμ ∈ Nμ (T ) such that fλ |∂Ω = ϕ and gμ |∂Ω = ψ . Hence we have Q(λ)ϕ = −

∂fλ , ∂ν ∂Ω

Q(μ)ψ = −

∂gμ , ∂ν ∂Ω

Γ (λ)ϕ = fλ

and Γ (μ)ψ = gμ .

Note that ϕ ∈ H 3/2 (Ω) belongs to dom Q(μ)∗ by the above considerations. With the help of Green’s identity (3.4) we find Q(λ) − Q(μ)∗ ϕ, ψ ∂Ω ∂gμ ∂fλ =− , g | + f | , μ ∂Ω λ ∂Ω ∂ν ∂Ω ∂ν ∂Ω ∂Ω ∂Ω = (Tfλ , gμ )Ω − (fλ , T gμ )Ω = (λ − μ)(f ¯ λ , gμ )Ω = (λ − μ) ¯ Γ (λ)ϕ, Γ (μ)ψ Ω = (λ − μ)Γ ¯ (μ)∗ Γ (λ)ϕ, ψ ∂Ω . This holds for all ψ in the dense subset H 3/2 (∂Ω) of L2 (∂Ω) and therefore (3.11) is valid on dom Q(λ) = dom Γ (λ) = H 3/2 (∂Ω), i.e., the Dirichlet-to-Neumann map is a generalized Qfunction of the triple {S, AD , T }. (ii) Let λ ∈ ρ(AD ) ∩ ρ(AN ) and suppose that we have Q(λ)ϕ = 0 for some ϕ ∈ H 3/2 (∂Ω). There exists a unique fλ ∈ Nλ (T ) such that fλ |∂Ω = ϕ and for this fλ by assumption we have ∂fλ ∂ν |∂Ω = 0. Hence fλ ∈ dom AN ∩ Nλ (T ) and from λ ∈ ρ(AN ) we conclude fλ = 0, that is, ϕ = fλ |∂Ω = 0. Therefore Q(λ)−1 , λ ∈ ρ(AD ) ∩ ρ(AN ) exists and, roughly speaking, Q(λ)−1 maps the negative Neumann boundary values of H 2 (Ω)-solutions of Lu = λu onto their Dirichlet boundary values. Let us prove the formula (3.10) for the difference of the resolvents of AD and AN . Observe first, that the right-hand side in (3.10) is well defined. In fact, by Proposition 3.2(iii) and (3.3) the range of Γ (λ¯ )∗ lies in H 1/2 (∂Ω) and it follows from the surjectivity of the mapping in (3.3) that Q(λ)−1 is defined on the whole space H 1/2 (∂Ω) and maps H 1/2 (∂Ω) onto H 3/2 (∂Ω), the domain of Γ (λ). Let now f ∈ L2 (Ω). We claim that the function ¯ ∗f g = (AD − λ)−1 f − Γ (λ)Q(λ)−1 Γ (λ)

(3.12)

belongs to dom AN . It is clear that g is in H 2 (Ω) since (AD − λ)−1 f ∈ dom AD and the second term on the right-hand side belongs to Nλ (T ), the range of Γ (λ). In order to verify ∂g ∂ν |∂Ω = 0 we choose fD ∈ dom AD such that f = (AD − λ)fD , so that (3.12) becomes ¯ ∗ (AD − λ)fD = fD + Γ (λ)Q(λ)−1 g = fD − Γ (λ)Q(λ)−1 Γ (λ)

∂fD , ∂ν ∂Ω

(3.13)

where we have used Proposition 3.2(iii). Let fλ := Γ (λ)Q(λ)−1 ∂f∂νD |∂Ω . Then fλ ∈ Nλ (T ) and the trace of fλ is given by . ∂ν ∂Ω

−1 ∂fD

fλ |∂Ω = Q(λ)


1681

Hence Q(λ)fλ |∂Ω = ∂f∂νD |∂Ω , but on the other hand, by the definition of the Dirichlet-toλ Neumann map Q(λ)fλ |∂Ω = − ∂f ∂ν |∂Ω . Therefore, the sum of the Neumann boundary value of the function fλ and the Neumann boundary value of fD is zero and we conclude from (3.13) ∂g ∂fD ∂ −1 ∂fD = ∂fD + ∂fλ = 0. Γ (λ)Q(λ) = + ∂ν ∂Ω ∂ν ∂Ω ∂ν ∂ν ∂Ω ∂Ω ∂ν ∂Ω ∂ν ∂Ω We have shown that g in (3.12) belongs to dom AN . As T is an extension of AN and AD , and ran Γ (λ) = ker(T − λ) we obtain ¯ ∗ f = f. (AN − λ)g = (T − λ)(AD − λ)−1 f − (T − λ)Γ (λ)Q(λ)−1 Γ (λ) Together with (3.12) we find ¯ ∗f (AN − λ)−1 f = (AD − λ)−1 f − Γ (λ)Q(λ)−1 Γ (λ) for all λ ∈ ρ(AD ) ∩ ρ(AN ) and f ∈ L2 (Ω), and therefore the resolvent formula (3.10) is valid. Up to some small modifications assertion (iii) was proved in [11]. 2 We mention that for λ, λ0 ∈ ρ(AD ) the Dirichlet-to-Neumann map is connected with the resolvent of AD via Q(λ) = Re Q(λ0 ) + Γλ0 (λ − Re λ0 ) + (λ − λ0 )(λ − λ¯ 0 )(AD − λ)−1 Γλ0 . This follows from the fact that Q is a generalized Q-function and Proposition 2.5. The following two corollaries collect some properties of the Dirichlet-to-Neumann map and its inverse. Corollary 3.5. For λ, λ0 ∈ ρ(AD ) the Dirichlet-to-Neumann map Q(λ) has the following properties: (i) Q(λ) is a non-closed unbounded operator in L2 (∂Ω) defined on the dense subspace H 3/2 (∂Ω) with ran Q(λ) ⊂ H 1/2 (∂Ω); (ii) Q(λ) − Re Q(λ0 ) is a non-closed bounded operator in L2 (∂Ω) defined on H 3/2 (∂Ω); (iii) the closure Q(λ) of the operator Q(λ) − Re Q(λ0 ) in L2 (∂Ω) satisfies d ¯ ∗ Γ (λ) Q(λ) = Γ (λ) dλ is an L(L2 (∂Ω))-valued Nevanlinna function. and Q Proof. Besides the statement that Q(λ) is a non-closed unbounded operator the assertions follow from the fact that Q is a generalized Q-function and the results in Section 2. In Corollary 3.6 it will turn out that Q(λ)−1 is a compact operator and that Q(λ)−1 is not closed. This implies that Q(λ) and Q(λ) are unbounded and that Q(λ) is not closed. 2 Corollary 3.6. For λ ∈ ρ(AD ) ∩ ρ(AN ) the inverse Q(λ)−1 of the Dirichlet-to-Neumann map Q(λ) has the following properties:

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(i) Q(λ)−1 is a non-closed bounded operator in L2 (∂Ω) defined on the dense subspace H 1/2 (∂Ω) with ran Q(λ)−1 = H 3/2 (∂Ω); (ii) the closure Q(λ)−1 is a compact operator in L2 (∂Ω); (iii) the function λ → −Q(λ)−1 is an L(L2 (∂Ω))-valued Nevanlinna function. Proof. It is clear that (i) is an immediate consequence of (ii). Statement (iii) follows from Theorem 2.6 and general properties of the Nevanlinna class. Assertion (ii) is essentially a consequence of the classical results in [40], see also [32, Theorem 2.1]. Namely, for λ ∈ ρ(AD ) ∩ ρ(AN ) the operator Q(λ) : H 3/2 (∂Ω) → H 1/2 (∂Ω) is an isomorphism and can be extended to an isomor −1 is a phism Q(λ) : H 1 (∂Ω) → L2 (∂Ω) which acts as in (3.9). Therefore Q(λ)−1 ⊂ Q(λ) 2 1 densely defined operator in L (∂Ω) which is bounded as an operator in H (∂Ω) and hence also bounded when considered as an operator in L2 (∂Ω). Its closure Q(λ)−1 in L2 (∂Ω) is a bounded everywhere defined operator in L2 (∂Ω) with values in H 1 (∂Ω) and coincides with −1 . As H 1 (∂Ω) is compactly embedded in L2 (∂Ω) it follows that Q(λ)−1 is a compact Q(λ) operator in L2 (∂Ω). 2 The next corollary is a simple consequence of Theorem 3.4 for the case that the difference of the resolvents is a trace class operator. be the Nevanlinna function from Corollary 3.7. Let the assumptions be as in Theorem 3.4, let Q Corollary 3.5 and suppose, in addition, n = 2. Then d tr (AD − λ)−1 − (AN − λ)−1 = tr Q(λ)−1 Q(λ) dλ

(3.14)

holds for all λ ∈ ρ(AD ) ∩ ρ(AN ). Proof. The resolvent formula (3.10) can be written in the form ¯ ∗, (AD − λ)−1 − (AN − λ)−1 = Γ (λ)Q(λ)−1 Γ (λ)

(3.15)

where the closures Γ (λ) and Q(λ)−1 are everywhere defined bounded operators; cf. Corollary 3.6(ii). In the case n = 2 it follows from Theorem 3.4(iii) that (3.15) is a trace class operator and from Corollaries 2.9, 3.5(iii) and well-known properties of the trace of bounded operators (see [28]) we conclude (3.14). 2 4. Coupling of elliptic differential operators In this section we study the uniformly elliptic second order differential expression L from (3.1) on two different domains and a coupling of the associated Dirichlet operators. More precisely, let Ω ⊂ Rn be a simply connected bounded domain with C ∞ -boundary C := ∂Ω and let Ω = Rn \Ω be the complement of the closure of Ω in Rn . Clearly, Ω is an unbounded domain with the compact C ∞ -boundary ∂Ω = C. Let again L be given by Lh = −

n j,k=1

∂ ∂h aj k + ah ∂xj ∂xk

(4.1)


1683

with bounded real valued coefficients aj k ∈ C ∞ (Rn ) satisfying aj k (x) = akj (x) for all x ∈ Rn and j, k = 1, . . . , n; the function a ∈ L∞ (Rn ) is real valued and suppose that L is uniformly elliptic; cf. (3.2). The restriction of L on functions f defined on Ω or functions f defined on Ω will be denoted by LΩ and LΩ , respectively. Then it is clear that the differential expressions LΩ and LΩ are of the type as in Section 3. In the following we will usually denote functions defined on Rn by h or k, and we denote functions defined on Ω or Ω by f, g or f , g , respectively. The scalar products of L2 (Ω) and L2 (Ω ) are indexed with Ω and Ω , respectively, whereas the scalar product of L2 (Rn ) is just denoted by (·,·). For the trace of a function f ∈ H 2 (Ω) and f ∈ H 2 (Ω ) we write f |C and f |C , and the trace of the conormal derivatives are n ∂f ∂f = aj k n j ∂ν C ∂xk C j,k=1

and

n ∂f ∂f = aj k n j ; ∂ν C ∂xk C

(4.2)

j,k=1

here n(x) = (n1 (x), . . . , nn (x)) and n (x) = −n(x) are the unit vectors at the point x ∈ C = ∂Ω = ∂Ω pointing out of Ω and Ω , respectively. Note also that the coefficients aj k in (4.2) are the restrictions of the coefficients in (4.1) onto Ω and Ω , respectively. The Dirichlet operators

dom AΩ = f ∈ H 2 (Ω): f |C = 0 ,

AΩ f = LΩ f , dom AΩ = f ∈ H 2 (Ω ): f |C = 0 , AΩ f = LΩ f,

are selfadjoint operators in L2 (Ω) and L2 (Ω ), respectively. Hence the orthogonal sum A=

AΩ 0

0 AΩ

,

dom A = dom AΩ ⊕ dom AΩ ,

(4.3)

is a selfadjoint operator in L2 (Rn ) = L2 (Ω) ⊕ L2 (Ω ). Observe that A(f ⊕ f ) = L(f ⊕ f ) = LΩ f ⊕ LΩ f ,

dom A = f ⊕ f ∈ H 2 (Ω) ⊕ H 2 (Ω ): f |C = 0 = f |C ,

(4.4)

and that A is not a usual second order elliptic differential operator on Rn since for a function ∂f f ⊕ f ∈ dom A the traces of the conormal derivatives ∂f ∂ν |C and − ∂ν |C at the boundary C of the domains Ω and Ω in general do not coincide. Besides the operator A we consider the usual selfadjoint operator associated to L in L2 (Rn ) defined by = Lh, Ah

= H 2 Rn , h ∈ dom A

(4.5)

and A with the help of and our aim is to prove a formula for the difference of the resolvents of A a generalized Q-function in a similar form as in the previous section. The following theorem indicates how S and T in the triple {S, A, T } for the definition of a generalized Q-function can be chosen.

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Theorem 4.1. The operator Sh = Lh,

dom S = h = f ⊕ f ∈ H 2 Rn : f |C = 0 = f |C ,

(4.6)

is a densely defined closed symmetric operator in L2 (Rn ) with infinite deficiency indices n± (S). The operator T (f ⊕ f ) = L(f ⊕ f ),

dom T = f ⊕ f ∈ H 2 (Ω) ⊕ H 2 (Ω ): f |C = f |C ,

(4.7)

is not closed as an operator in L2 (Rn ) and T satisfies T = S ∗ and T ∗ = S. Furthermore, the in (4.3)–(4.5) are extensions of S and restrictions of T . selfadjoint operators A and A Proof. The operator S is a restriction of the selfadjoint operator A and hence S is symmetric. The fact that dom S is dense follows, e.g., from the fact that H02 (Ω) and H02 (Ω ) are dense subspaces of L2 (Ω) and L2 (Ω ), respectively, and H02 (Ω) ⊕ H02 (Ω ) ⊂ dom S. Since for any function h ∈ H 2 (Rn ) decomposed as h = f ⊕ f , where f ∈ H 2 (Ω), f ∈ is an extension of S and a restricH 2 (Ω ), we have f |C = f |C ∈ H 3/2 (C) it follows that A tion of the operator T . Moreover, S ⊂ A ⊂ T is obvious. Let us verify that S = T ∗ holds. In particular this implies that S is closed and that T = S ∗ is true. We start with the inclusion S ⊂ T ∗ . Let h = f ⊕ f ∈ dom S and k = g ⊕ g ∈ dom T , where f, g ∈ H 2 (Ω) and f , g ∈ H 2 (Ω ). First of all we have (T k, h) − (k, Sh) = (LΩ g, f )Ω − (g, LΩ f )Ω + (LΩ g , f )Ω − (g , LΩ f )Ω and Green’s identity (3.4) shows that this is equal to ∂f ∂g ∂f ∂g g|C , − , f |C + g |C , − , f |C . ∂ν C C ∂ν C ∂ν C C ∂ν C C C Since h = f ⊕ f ∈ dom S we have f |C = f |C = 0

and

∂f ∂f = − , ∂ν C ∂ν C

and for k = g ⊕ g ∈ dom T by definition g|C = g |C holds. Hence we conclude (T k, h) − (k, Sh) = 0 and therefore every h ∈ dom S belongs to dom T ∗ and T ∗ h = Sh, i.e., S ⊂ T ∗ . Let us now prove the converse inclusion T ∗ ⊂ S. For this it is sufficient to check that every function h ∈ dom T ∗ we belongs to dom S. From the fact that T is an extension of the selfadjoint operators A and A conclude


T ∗ ⊂ A∗ = A ⊂ T

1685

∗ = A ⊂ T , and T ∗ ⊂ A

Hence every function h in dom T ∗ belongs also to so that T ∗ is a restriction of A and A. dom A and dom A. Thus h = f ⊕ f ∈ H 2 (Rn ) and f ∈ H 2 (Ω) and f ∈ H 2 (Ω ) satisfy f |C = f |C = 0. Therefore dom T ∗ ⊂ dom S and we have shown T ∗ = S. Next it will be verified that T is not closed. The arguments are similar as in [8, Proof of Proposition 4.5] and could also be formulated in terms of unitary relations between Krein spaces; cf. [17]. Assume that T is closed, i.e., T = T , and consider the subspace ⎧⎡ ⎪ f ⊕f ⎪ ⎪ ⎪ ⎨⎢ ⎢ T (f ⊕ f ) M= ⎢ ⎢ f |C ⎪ ⎪⎣ ⎪ ⎪ ⎩ ∂f | + ∂f | ∂ν C

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎤

∂ν C

⎥ ⎥ ⎥ : f ⊕ f ∈ dom T ⊂ L2 Rn ⊕ L2 Rn ⊕ L2 (C) ⊕ L2 (C). ⎥ ⎪ ⎪ ⎦ ⎪ ⎪ ⎭

Observe that by (3.3) and the definition of T the mapping

∂f ∂f dom T f ⊕ f → f |C , + ∈ H 3/2 (C) × H 1/2 (C) ∂ν C ∂ν C

(4.8)

is onto. Setting N = L2 (Rn ) ⊕ L2 (Rn ) ⊕ {0} ⊕ {0} it is clear that the sum of the subspaces M and N is M + N = L2 Rn ⊕ L2 Rn ⊕ H 3/2 (C) × H 1/2 (C) .

(4.9)

We will calculate the orthogonal complements of M and N in L2 (Rn ) ⊕ L2 (Rn ) ⊕ L2 (C) ⊕ L2 (C) and show that M⊥ + N ⊥ is closed. First of all we have N ⊥ = {0} ⊕ {0} ⊕ L2 (C) ⊕ L2 (C)

(4.10)

and in order to determine M⊥ suppose that ⎤ l ⊕ l ⎢ g ⊕ g ⎥ ⊥ ⎦∈M , ⎣ ϕ ψ ⎡

g, l ∈ L2 (Ω), g , l ∈ L2 (Ω ), ϕ, ψ ∈ L2 (C),

(4.11)

is an element in L2 (Rn ) ⊕ L2 (Rn ) ⊕ L2 (C) ⊕ L2 (C) which is orthogonal to M. Then we have ∂f ∂f T (f ⊕ f ), g ⊕ g + (f ⊕ f , l ⊕ l ) = −(f |C , ϕ)C − + , ψ ∂ν C ∂ν C C for all f ⊕ f ∈ dom T . In particular, for f ⊕ f ∈ dom S we have ∂f ∂f =− ∂ν C ∂ν C

and f |C = f |C = 0,

(4.12)

1686


so that (4.12) becomes T (f ⊕ f ), g ⊕ g = S(f ⊕ f ), g ⊕ g = −(f ⊕ f , l ⊕ l ) and hence g ⊕ g ∈ dom S ∗ and S ∗ (g ⊕ g ) = −l ⊕ l . But we have assumed that T is closed and hence from S = T ∗ we conclude S ∗ = T ∗∗ = T = T , so that g ⊕ g ∈ dom T

and T (g ⊕ g ) = −l ⊕ l .

(4.13)

From Green’s identity we then obtain T (f ⊕ f ), g ⊕ g − f ⊕ f , T g ⊕ g = (LΩ f, g)Ω − (f, LΩ g)Ω + (LΩ f , g )Ω − (f , LΩ g )Ω ∂g ∂f ∂g ∂f = f |C , − , g| + f | , − , g | C C C ∂ν C C ∂ν C ∂ν C C ∂ν C C C ∂g ∂g ∂f ∂f = f |C , + − + , g|C , ∂ν C ∂ν C C ∂ν C ∂ν C C where we have used that f ⊕ f , g ⊕ g ∈ dom T satisfy f |C = f |C and g|C = g |C . Inserting (4.13) in (4.12) and comparing this with the above relation shows that the identity

∂g ∂g ∂f ∂f f |C , + + ϕ = + , g| − ψ C ∂ν C ∂ν C ∂ν C ∂ν C C C

(4.14)

holds for all f ⊕ f ∈ dom T . As the mapping (4.8) is surjective and H 3/2 (C) × H 1/2 (C) is dense in L2 (C) ⊕ L2 (C) we conclude from (4.14) that ϕ=−

∂g ∂g + ∂ν C ∂ν C

and ψ = g|C

hold. Hence we have seen that the element (4.11) in M⊥ is of the form ⎡ −T (g ⊕ g ) ⎤ g ⊕ g ⎥ ⎢ ⎥ ⎢ ∂g ⎣ − |C − ∂g |C ⎦ ∂ν

(4.15)

∂ν

g|C for some g ⊕ g ∈ dom T . It is not difficult to check that conversely an element as in (4.15) belongs to M⊥ . Therefore the orthogonal complement of M is given by ⎧⎡ −T (g ⊕ g ) ⎤ ⎫ ⎪ ⎪ ⎪ ⎪ ⎨⎢ ⎬ g⊕g ⎥ ⊥ ⎢ ⎥ : g ⊕ g ∈ dom T ⊂ L2 Rn ⊕ L2 Rn ⊕ L2 (C) ⊕ L2 (C) M = ⎣ ∂g ∂g ⎦ ⎪ ⎪ − | − | ⎪ ⎪ ∂n C ∂ν C ⎩ ⎭ g|C


1687

and together with (4.10) we find that the sum of M⊥ and N ⊥ is ⊥

⊥

M +N =

−T (g ⊕ g ) : g ⊕ g ∈ dom T ⊕ L2 (C) ⊕ L2 (C). g ⊕ g

The assumption that T is closed implies that M⊥ + N ⊥ is a closed subspace of L2 (Rn ) ⊕ L2 (Rn ) ⊕ L2 (C) ⊕ L2 (C). But then according to [34, IV Theorem 4.8] also M + N is a closed subspace of L2 (Rn ) ⊕ L2 (Rn ) ⊕ L2 (C) ⊕ L2 (C) which is a contradiction to (4.9). Thus T cannot be closed. 2 The following lemma will be useful later in this section. be the selfadjoint realization of L Lemma 4.2. Let S and T be as in Theorem 4.1 and let A 2 n 2 n in L (R ) defined on H (R ). For a function f ⊕ f ∈ dom T , where f ∈ H 2 (Ω) and f ∈ H 2 (Ω ), we have if and only if f ⊕ f ∈ dom A

∂f ∂f = − . ∂ν C ∂ν C

= H 2 (Rn ) it is clear that Proof. For a function f ⊕ f ∈ dom A versely, let f ⊕ f ∈ dom T and assume

∂f ∂ν |C

= − ∂f ∂ν |C holds. Con-

∂f ∂f = − . ∂ν C ∂ν C

(4.16)

satisfies Then also f |C = f |C and since every g ⊕ g ∈ dom A

g|C = g |C

and

∂g ∂g =− ∂ν C ∂ν C

Green’s identity implies ⊕ g ), f ⊕ f − g ⊕ g , T (f ⊕ f ) A(g ∂f ∂g ∂f ∂g = g|C , − , f | + g | , − , f | = 0. C C C ∂ν C C ∂ν C ∂ν C C ∂ν C C C ∗ = dom A. Therefore f ⊕ f ∈ dom A

2

Next we define a mapping Γλ0 which satisfies the assumptions in the definition of a generalized Q-function. For this let A be the selfadjoint operator in L2 (Rn ) in (4.3) and (4.4) which is the orthogonal sum of the Dirichlet operators AΩ and AΩ in L2 (Ω) and L2 (Ω ), respectively. For λ ∈ ρ(A) the domain of the operator T in Theorem 4.1 can be decomposed in ˙ Nλ (T ) dom T = dom A +

˙ Nλ (T ); = f ⊕ f ∈ H 2 (Ω) ⊕ H 2 Ω : f |C = f |C = 0 +

(4.17)

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cf. (2.1). Let us fix some λ0 ∈ ρ(A). The decomposition (4.17) and the surjectivity of the map

∂f ∂f dom T f ⊕ f → f |C , + ∈ H 3/2 (C) × H 1/2 (C) ∂ν C ∂ν C

(4.18)

(see (3.3) and (4.8)) imply that for a given function ϕ ∈ H 3/2 (C) there exists a unique function fλ0 ⊕ fλ 0 ∈ Nλ0 (T ) such that fλ0 |C = fλ 0 |C = ϕ. Let Γλ0 be the mapping that assigns fλ0 ⊕ fλ 0 to ϕ, H 3/2 (C) ϕ → Γλ0 ϕ := fλ0 ⊕ fλ 0 .

(4.19)

Similarly as in the previous section Γλ0 will be regarded as an operator from L2 (C) to L2 (Rn ) with dom Γλ0 = H 3/2 (C) and ran Γλ0 = Nλ0 (T ). Observe that the function Γλ0 ϕ = fλ0 ⊕ fλ 0 consists of an H 2 (Ω)-solution fλ0 of LΩ u = λ0 u and an H 2 (Ω )-solution fλ 0 of LΩ u = λ0 u satisfying the boundary conditions ϕ = fλ0 |C = fλ 0 |C . The following proposition parallels Proposition 3.2. Proposition 4.3. Let λ0 ∈ ρ(A), let Γλ0 be as in (4.19) and let λ ∈ ρ(A). Then the following hold: (i) Γλ0 is a bounded operator from L2 (C) in L2 (Rn ) with dense domain H 3/2 (C); (ii) the operator Γ (λ) = (I + (λ − λ0 )(A − λ)−1 )Γλ0 is given by Γ (λ)ϕ = fλ ⊕ fλ ,

where fλ ⊕ fλ ∈ Nλ (T ) and fλ |C = ϕ = fλ |C ;

¯ ∗ : L2 (Rn ) → L2 (C) satisfies (iii) the mapping Γ (λ) Γ (λ¯ )∗ (A − λ)h = −

∂f ∂f − , ∂ν C ∂ν C

h = f ⊕ f ∈ dom A.

Proof. We start with the proof (ii). Let ϕ ∈ H 3/2 (C) and choose the unique elements fλ ⊕ fλ ∈ Nλ (T ) and fλ0 ⊕ fλ 0 ∈ Nλ0 (T ) such that fλ |C = fλ C = ϕ = fλ0 |C = fλ 0 C holds. By definition Γλ0 ϕ = fλ0 ⊕ fλ 0 and therefore Γ (λ)ϕ = Γλ0 ϕ + (λ − λ0 )(A − λ)−1 Γλ0 ϕ = fλ0 ⊕ fλ 0 + (λ − λ0 )(A − λ)−1 Γλ0 ϕ. Since (λ − λ0 )(A − λ)−1 Γλ0 ϕ is a function belonging to dom A we have (λ − λ0 )(A − λ)−1 Γλ0 ϕ C = 0;


1689

cf. (4.4). This implies Γ (λ)ϕ C = (Γλ0 ϕ)|C = fλ0 ⊕ fλ 0 C = fλ0 |C = fλ 0 C = ϕ and since ran Γ (λ) = Nλ (T ) (see Lemma 2.1) and fλ ⊕ fλ is the unique function in Nλ (T ) with fλ |C = fλ |C = ϕ we conclude Γ (λ)ϕ = fλ ⊕ fλ . ¯ ∗ , λ ∈ ρ(A), is a closed operator which is defined Next we verify (iii). Observe that then Γ (λ) ∗ on the whole space, i.e., Γ (λ¯ ) is bounded and hence assertion (i) follows by setting λ0 = λ¯ . Let ϕ ∈ H 3/2 (C) and choose the unique function fλ¯ ⊕ fλ¯ ∈ Nλ¯ (T ) such that fλ¯ |C = fλ¯ C = ϕ

(4.20)

¯ = fλ¯ ⊕ f and for each h = f ⊕ f ∈ dom A, where f ∈ H 2 (Ω), f ∈ holds. Then Γ (λ)ϕ λ¯ H 2 (Ω ), we have Γ (λ¯ )ϕ, (A − λ)h = fλ¯ ⊕ fλ¯ , A(f ⊕ f ) − T fλ¯ ⊕ fλ¯ , f ⊕ f = (fλ¯ , LΩ f )Ω − (LΩ fλ¯ , f )Ω + fλ¯ , LΩ f Ω − LΩ fλ¯ , f Ω . With the help of Green’s identity this can be rewritten as

∂fλ¯ ∂fλ¯ ∂f ∂f , f | − f | , + , f | − f , . C C λ¯ C λ¯ C ∂ν ∂ν C ∂ν C C ∂ν C C C C C

Since for h = f ⊕ f ∈ dom A we have f |C = f |C = 0 we conclude from the above calculation and (4.20) that ∂f ∂f ¯ Γ (λ)ϕ, (A − λ)h = − ϕ, + ∂ν C ∂ν C C holds for every ϕ ∈ H 3/2 (C) = dom Γ (λ¯ ). Hence (A − λ)h ∈ dom Γ (λ¯ )∗ and Γ (λ¯ )∗ (A − λ)h = −

∂f ∂f − , ∂ν C ∂ν C

h = f ⊕ f ∈ dom A.

Furthermore, for λ ∈ ρ(A) we have ran(A − λ) = L2 (Rn ), so that Γ (λ¯ )∗ is a bounded operator defined on L2 (Rn ). 2 Next we define a function Q in a similar way as the Dirichlet-to-Neumann map in Definition 3.3. For this we make use of the decomposition (4.17). Namely, for λ ∈ ρ(A) and ϕ ∈ H 3/2 (C) there exists a unique function fλ ⊕ fλ ∈ Nλ (T ) such that fλ |C = fλ |C = ϕ. The operator Q(λ) in L2 (C) is now defined by ∂fλ ∂fλ Q(λ)ϕ := − − , ∂ν C ∂ν C

ϕ ∈ dom Q(λ) = H 3/2 (C).

(4.21)

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Observe that ran Q(λ) ⊂ H 1/2 (C) holds. Roughly speaking, up to a minus sign Q(λ) maps the Dirichlet boundary value of the H 2 -solutions of LΩ u = λu and LΩ u = λu , u|C = u |C , onto the sum of the Neumann boundary values of these solutions. We mention that in the analysis of so-called intermediate Hamiltonians a modified form of such a Dirichlet-to-Neumann map has been used in [44]. In the following theorem it turns out that Q can be interpreted as a generalized Q-function is expressed with the help of Q. and the difference of the resolvents of A and A be the selfTheorem 4.4. Let L be the elliptic differential expression in (4.1) and let A and A adjoint realizations of L in (4.3)–(4.4) and (4.5), respectively. Let S and T be the operators in Theorem 4.1, define Γ (λ) as in Proposition 4.3 and let Q(λ), λ ∈ ρ(A), be as in (4.21). Then the following hold: (i) Q is a generalized Q-function of the triple {S, A, T }; and the resolvent formula (ii) the operator Q(λ) is injective for all λ ∈ ρ(A) ∩ ρ(A) − λ)−1 = Γ (λ)Q(λ)−1 Γ (λ¯ )∗ (A − λ)−1 − (A

(4.22)

holds; (iii) for p > n−1 2 the difference of the resolvents in (4.22) belongs to the von Neumann–Schatten class Sp (L2 (Rn )). Proof. Let us prove assertion (i). Before the defining relation (2.3) for a generalized Q-function ¯ μ ∈ ρ(A). For this will be verified we show that the operator Q(μ)∗ is an extension of Q(μ), let ψ ∈ H 3/2 (C) and choose the unique element fμ¯ ⊕ fμ¯ ∈ Nμ¯ (T ) with the property fμ¯ |C = fμ¯ |C = ψ . For ϕ ∈ H 3/2 (C) let fμ ⊕ fμ ∈ Nμ (T ) be such that fμ |C = fμ |C = ϕ holds. By the definition of Q in (4.21) we have ∂fμ ∂fμ Q(μ)ϕ = − − ∂ν C ∂ν C From (fμ |C ,

∂fμ¯ ∂ν |C )C

∂fμ¯ ∂fμ¯ . and Q(μ)ψ ¯ =− − ∂ν C ∂ν C

= ( ∂νμ |C , fμ¯ |C )C and (fμ |C , ∂f

∂fμ¯ ∂ν |C )C

∂f

= ( ∂νμ |C , fμ¯ |C )C we then conclude

∂fμ¯ ∂fμ ∂fμ ∂fμ¯ Q(μ)ϕ, ψ = − , fμ¯ |C − ,f = − ϕ, + ∂ν C ∂ν C μ¯ C C ∂ν C ∂ν C C C and therefore ψ ∈ dom Q(μ)∗ and Q(μ)∗ ψ = Q(μ)ψ ¯ . Let Γ (·) be as in Proposition 4.3. We prove now that ¯ (μ)∗ Γ (λ), Q(λ) − Q(μ)∗ = (λ − μ)Γ

λ, μ ∈ ρ(A),

(4.23)

holds on dom Γ (λ) = H 3/2 (C). For this let ϕ, ψ ∈ H 3/2 (C) and choose the unique elements fλ ⊕ fλ ∈ Nλ (T ), fμ ⊕ fμ ∈ Nμ (T ) with the properties fλ |C = fλ C = ϕ

and fμ |C = fμ C = ψ.

(4.24)


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Then according to Proposition 4.3(ii) Γ (λ)ϕ = fλ ⊕ fλ and Γ (μ)ψ = fμ ⊕ fμ and the definition of Q(·) in (4.21) shows ∂fμ ∂fλ ∂fμ ∂fλ ∗ + ,ψ + ϕ, + . Q(λ) − Q(μ) ϕ, ψ C = − ∂ν C ∂ν C ∂ν C ∂ν C C C By inserting (4.24) and making use of Green’s identity we obtain Q(λ) − Q(μ)∗ ϕ, ψ C

= (LΩ fλ , fμ )Ω − (fλ , LΩ fμ )Ω + LΩ fλ , fμ Ω − fλ , LΩ fμ Ω ¯ fλ ⊕ fλ , fμ ⊕ fμ = (λ − μ) ¯ (fλ , fμ )Ω + fλ , fμ Ω = (λ − μ) = (λ − μ) ¯ Γ (λ)ϕ, Γ (μ)ψ = (λ − μ)Γ ¯ (μ)∗ Γ (λ)ϕ, ψ C ,

i.e., (4.23) holds and Q is a generalized Q-function for the triple {S, A, T }. Assume that Q(λ)ϕ = 0 (ii) We check first that ker Q(λ) = {0} holds for λ ∈ ρ(A) ∩ ρ(A). 3/2 for some ϕ ∈ H (C) and let fλ ⊕ fλ ∈ Nλ (T ) be the unique element with the property fλ |C = fλ |C = ϕ. Then the definition of Q and the assumption Q(λ)ϕ = 0 imply ∂fλ ∂fλ = − . ∂ν C ∂ν C ∩ Nλ (T ). But as λ ∈ ρ(A) we conclude According to Lemma 4.2 this yields fλ ⊕ fλ ∈ dom A fλ = 0 and fλ = 0, and hence ϕ = 0. By the above Now we prove the formula (4.22) for the difference of the resolvents of A and A. −1 argument Q(λ) exists for λ ∈ ρ(A) ∩ ρ(A). Furthermore, (4.18) implies ran Q(λ) = H 1/2 (C) and it follows from Proposition 4.3 that the right-hand side in (4.22) is well defined. Let h ∈ L2 (Rn ) and define the function k as k = (A − λ)−1 h − Γ (λ)Q(λ)−1 Γ (λ¯ )∗ h.

(4.25)

First of all it is clear that k ∈ dom T since (A − λ)−1 h ∈ dom A ⊂ dom T We show k ∈ dom A. and Γ (λ) maps into Nλ (T ). Therefore k = g ⊕ g , where g ∈ H 2 (Ω), g ∈ H 2 (Ω ), and g|C = it is sufficient to check g |C . According to Lemma 4.2 for k ∈ dom A ∂g ∂g + = 0. (4.26) ∂ν C ∂ν C We proceed in a similar way as in the proof of Theorem 3.4. Let hA = fA ⊕ fA ∈ dom A be such that h = (A − λ)hA . Making use of Proposition 4.3(iii) we obtain k = hA + fλ ⊕ fλ ,

fλ ⊕ fλ := Γ (λ)Q(λ)−1

∂fA ∂fA + ∈ Nλ (T ), ∂ν C ∂ν C

(4.27)

from (4.25). Then Proposition 4.3(ii) together with the definition of Q(λ) in (4.21) implies ∂fA ∂fλ ∂fA ∂fλ + = Q(λ)(fλ |C ) = Q(λ) fλ C = − − . ∂ν C ∂ν C ∂ν C ∂ν C

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From Hence we conclude that the function k = g ⊕ g in (4.27) fulfills (4.26), i.e., k ∈ dom A. ⊂ T we obtain (4.25) and A, A − λ)k = (T − λ)(A − λ)−1 h − (T − λ)Γ (λ)Q(λ)−1 Γ (λ¯ )∗ h = h (A − λ)−1 h and (4.25) imply (4.22). and now k = (A Assertion (iii) is a direct consequence of [11, Theorem 1.3].

2

The following corollaries can be proved in the same way as Corollaries 3.5 and 3.6. Corollary 4.5. For λ, λ0 ∈ ρ(A) the following hold: (i) Q(λ) is a non-closed unbounded operator in L2 (C) defined on the dense subspace H 3/2 (C) with ran Q(λ) ⊂ H 1/2 (C); (ii) Q(λ) − Re Q(λ0 ) is a non-closed bounded operator in L2 (C) defined on H 3/2 (C); (iii) the closure Q(λ) of the operator Q(λ) − Re Q(λ0 ) in L2 (C) satisfies d Q(λ) = Γ (λ¯ )∗ Γ (λ) dλ is an L(L2 (C))-valued Nevanlinna function. and Q the following hold: Corollary 4.6. For λ ∈ ρ(A) ∩ ρ(A) (i) Q(λ)−1 is a non-closed bounded operator in L2 (C) defined on the dense subspace H 1/2 (C) with ran Q(λ)−1 = H 3/2 (C); (ii) the closure Q(λ)−1 is a compact operator in L2 (C); (iii) the function λ → −Q(λ)−1 is an L(L2 (C))-valued Nevanlinna function. As a corollary of Theorem 4.4 we obtain a trace formula for the difference of the resolvents of A and A. be the Nevanlinna function from Corollary 4.7. Let the assumptions be as in Theorem 4.4, let Q Corollary 4.5 and suppose, in addition, n = 2. Then − λ)−1 = tr Q(λ)−1 d Q(λ) tr (A − λ)−1 − (A dλ holds for all λ ∈ ρ(A) ∩ ρ(A). Acknowledgments The authors are grateful to Gerd Grubb for helpful comments and valuable suggestions. Jussi Behrndt thanks the Institute for Advanced Studies in Mathematics at Ben-Gurion University of the Negev for support and hospitality during a research stay in May 2008.


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Second-order elliptic equations with variably partially VMO coefficients ✩ N.V. Krylov 127 Vincent Hall, University of Minnesota, Minneapolis, MN 55455, United States Received 14 July 2008; accepted 11 June 2009 Available online 26 June 2009 Communicated by Paul Malliavin

Abstract The solvability in Wp2 (Rd ) spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This generalizes to a very large extent the case of equations with continuous or VMO coefficients. © 2009 Elsevier Inc. All rights reserved. Keywords: Second-order elliptic equations; Vanishing mean oscillation; Partially VMO coefficients; Sobolev spaces

1. Introduction and main result In this article we are concerned with the solvability in Wp2 = Wp2 (Rd ) of the equation Lu(x) − λu(x) = f (x),

(1.1)

where L is a uniformly nondegenerate elliptic differential operator with bounded coefficients of the form Lu(x) = a ij (x)ux i x j (x) + bi (x)ux i (x) + c(x)u(x) in ✩

The work was partially supported by NSF Grant DMS-0653121. E-mail address: [email protected].


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N.V. Krylov / Journal of Functional Analysis 257 (2009) 1695–1712

Rd = x = x 1 , . . . , x d : x 1 , . . . , x d ∈ R . We generalize the main result of [7] where the solvability is established in the case that, roughly speaking, the coefficients a ij are measurable with respect to x 1 and are in VMO with respect to (x 2 , . . . , x d ). Owing to a standard localization procedure, this result admits an obvious extension to the case in which for each ball B ⊂ Rd of a fixed radius there exists a sufficiently regular diffeomorphism that transforms equation (1.1) in B into a similar equation with coefficients satisfying the conditions of [7] in B. In particular, one obtains the solvability if the matrix a = (a ij ) depends on |x| in a measurable way, is in VMO with respect to the angular coordinates, and, say, is continuous at the origin. The main goal of the present article is to show that in the above described generalization the radius of balls need not be fixed. In the end of this section we give an example in which our result is applicable contrarily to the result of [7]. We develop a new technique which seems to be applicable in many situations for elliptic and parabolic equations with partially VMO coefficients as, for instance, in [6,5]. We only concentrate on elliptic equations in order to make simpler the presentation of the method. Generally, the theory of elliptic equations with partially VMO coefficients is quite new and originated in [7] in contrast with the case of completely VMO coefficients, which appeared in [4], or the classical case of equations with continuous coefficients treated in [1]. The reader can find further references to articles and books related to equations with VMO and partially VMO coefficients in the above cited articles and the references therein. In [1] the main technical tool was the theory of singular integrals, in particular, the Calderón– Zygmund theorem. With development of Real Analysis later on in many sources the theory of singular integrals in applications to PDEs was replaced with using the John–Nirenberg theorem or Stampacchia interpolation theorem applied to sharp functions. However, the theory of singular integrals was used again in the paper [4], the results of which came as a real breakthrough in the theory of PDEs. The methods of [4] are based on singular-integral representations of second-order derivatives of solutions, Calderón–Zygmund theorem, and the Coifman– Rochberg–Weiss commutator theorem. Again later it turned out that using singular integrals can be replaced with appropriate other tools from Real Analysis such as the Fefferman–Stein theorem. To the author it seems highly unlikely that the theory of singular integrals can be used to obtain even the main auxiliary result of [7] (see our Lemma 3.1), which is the basis of the present paper along with a new inequality of the Fefferman–Stein type proved in Theorem 2.7. Roughly speaking we use this theorem along with pointwise estimates of the sharp functions of some second-order derivatives of solutions in terms of the maximal function of the right-hand side. In connection with this new development it is instructive to recall that L. Bers and M. Schechter said in 1964 (see [2]) that the linear theory of second order elliptic PDEs “is at present probably nearing completion”. This paper deals with elliptic equations in nondivergence form. A different technique is developed in several articles by the authors of [3] for treating divergence type equations. It would be interesting to know if their methods could be applied to divergence or nondivergence type equations with coefficients satisfying our conditions. This could lead to extending our results to equations in domains. So far we can only deal with equations in the whole space or, for that matter, with interior estimates. Another restriction is that p > 2. Now we state our assumptions rigorously.


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Assumption 1.1. The coefficients a ij , bi , and c are measurable functions defined on Rd , a j i = a ij for all i, j = 1, . . . , d. There exist positive constants δ ∈ (0, 1) and K such that i b (x) K,

i = 1, . . . , d,

c(x) K,

δ|ξ |2 a ij (x)ξ i ξ j δ −1 |ξ |2 for any x ∈ Rd and ξ ∈ Rd . To state the second assumption denote by A the set of d × d symmetric matrix-valued measurable functions a¯ = (a¯ ij (t)) of one variable t ∈ R such that δ|ξ |2 a¯ ij (t)ξ i ξ j δ −1 |ξ |2 for any t ∈ R and ξ ∈ Rd . Introduce Ψ as the set of mappings ψ : Rd → Rd such that (i) the mapping ψ has an inverse ψ −1 : Rd → Rd ; (ii) the mappings ψ and φ = ψ −1 are twice continuously differentiable and |ψx | + |ψxx | δ −1 ,

|φy | + |φyy | δ −1 .

The following assumption contains a parameter γ > 0, which will be specified later. We denote by |B| the volume of a Borel set B ⊂ Rd . Assumption 1.2. (γ ) There exists a constant R0 > 0 such that for any ball B ⊂ Rd of radius less than R0 one can find an a¯ ∈ A and a ψ = (ψ 1 , . . . , ψ d ) ∈ Ψ such that

a(x) − a¯ ψ 1 (x) dx γ |B|.

(1.2)

B

Remark 1.3. Assumption 1.2(γ ) is obviously satisfied with any γ > 0 if a is uniformly continuous as, for instance, in [1]. If Assumption 1.2(γ ) is satisfied with any γ > 0 and constant a¯ (perhaps, changing with B), then one says that a belongs to VMO. This case was first treated in [4]. In [7] the solvability in Wp2 was proved under Assumption 1.2(γ ) with a fixed function ψ , which is not allowed to change with B. (Actually, ψ = x in [7], but changing coordinates shows that the result holds for any ψ ∈ Ψ .) By using partitions of unity the latter restriction on ψ can be easily somewhat relaxed to allow mappings such that in each ball B of radius exactly R0 there is a mapping ψ which would suit all subballs inside B. As usual, by Wp2 = Wp2 (Rd ) we mean the Sobolev space on Rd . Set Lp = Lp (Rd ). Here is our main result. Theorem 1.4. Take a p ∈ (2, ∞). Then there exists a constant γ = γ (d, δ, p) > 0 such that if Assumptions 1.1 and 1.2(γ ) are satisfied then for any λ λ0 (d, δ, K, p, R0 ) 1 and any f ∈ Lp , there exists a unique u ∈ Wp2 satisfying (1.1) in Rd .

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Furthermore, there is a constant N , depending only on d, δ, K, p, and R0 , such that, for any λ λ0 and u ∈ Wp2 , λuLp +

√ λux Lp + uxx Lp N Lu − λuLp .

(1.3)

The proof of this theorem is given in Section 4 after we prepare the necessary auxiliary results in Section 3, which in turn require some general facts proved in Section 2. We finish the section by giving the example we were talking about above. Let f be a measurable function on R with support in the interval (1/2, 1) and such that |f | 1. Introduce ξ(x) = ln(|x| ∧ 1), x ∈ R. It is well known that ξ ∈ BMO. Then for ε > 0 the function εξ is also in BMO and its BMO-norm can be made as small as we like on the account of choosing ε small enough. The same is true for η = sin(εξ ) and ζ (x) = η(4x − 3) with the latter function having support in the interval (1/2, 1). Next take a large κ 4 and for real x, y and z = (x, y) introduce a(z) =

∞ ∞ f κ 2n x ζ κ 2n y + f κ 2n+1 y ζ κ 2n+1 x . n=0

n=0

Notice that the support of f (κ r ·)ζ (κ r ·) belongs to Qr := (κ −r /2, κ −r )2 . Now, for a square Q = I × J ⊂ R2 we are going to estimate the left-hand side of (1.2) with Q and z in place of B and x, respectively, and with ψ equal to either x or y. For brevity we denote the modified left-hand side of (1.2) by M. Define τ as the least integer k 0 such that Q ∩ Qk = ∅. If there are no such k’s, then M = 0. If τ is an even number we set ψ = x and a(x) ¯ = f (κ τ x)ζ¯ , where ζ¯ is the integral average of τ ζ (κ y) over J . Then M I

τ f κ x dx

∞ τ ζ κ y − ζ¯ dy + |Q ∩ Qi |.

J

(1.4)

i=τ +1

On the right, the first term is less than |Q| |ζ |BMO . Also observe that if i τ + 1 and Q ∩ Qi = ∅, then the lengths of I and J are at least κ −τ /2 − κ −i , which is larger than κ −τ /4 since κ 4. Hence, in that case |Q ∩ Qi | |Qi | = 4−1 κ −2i 4κ 2τ −2i |Q| implying that the infinite sum in (1.4) is less than 4(κ 2 − 1)−1 |Q|. We see that in the case that τ is an even number M γ |Q| with any fixed γ > 0 provided that we choose sufficiently small ε and sufficiently large κ. In case τ is odd interchanging x and y leads to the same conclusion and this easily shows that (1.2) holds indeed in its original form. Obviously, functions like the above a cannot be treated by methods of [7] even modified in the way outlined in Remark 1.3. The author wishes to thank Hongjie Dong for pointing out several flaws in the first draft of the article. The critical comments made by the referee are also greatly appreciated. 2. A partial version of the Fefferman–Stein theorem First we recall a few standard notions and facts related to partitions and stopping times. All of them can be found in many books; we follow the exposition in [8].


1699

Let (Ω, F , μ) be a complete measure space with a σ -finite measure μ, such that μ(Ω) = ∞. Let F 0 be the subset of F consisting of all sets A such that μ(A) < ∞. For p ∈ [1, ∞) set Lp (Ω) = Lp (Ω, F , μ). By L0 we denote a fixed dense subset of L1 (Ω). For any A ∈ F we set |A| = μ(A). For A ∈ F 0 and functions f summable on A we use the notation 1 fA = − f μ(dx) := f (x) μ(dx) |A| A

0 := 0 0

A

for the average value of f over A. Definition 2.1. Let Z = {n: n = 0, ±1, ±2, . . .} and let (Cn , n ∈ Z) be a sequence of partitions of Ω each consisting of countably many disjoint sets C ∈ Cn and such that Cn ⊂ F 0 for each n. For each x ∈ Ω and n ∈ Z there exists (a unique) C ∈ Cn such that x ∈ C. We denote this C by Cn (x). The sequence (Cn , n ∈ Z) is called a filtration of partitions if the following conditions are satisfied. (i) The elements of partitions are “large” for big negative n’s and “small” for big positive n’s: inf |C| → ∞ as n → −∞,

C∈Cn

lim fCn (x) = f (x)

n→∞

(a.e.), ∀f ∈ L0 .

(ii) The partitions are nested: for each n and C ∈ Cn there is a (unique) C ∈ Cn−1 such that C ⊂ C . (iii) The following regularity property holds: for any n, C, and C as in (ii) we have |C | N0 |C|, where N0 is a constant independent of n, C, C . Observe that since the elements of partition Cn become large as n → −∞, we have N0 > 1. The only example of a filtration of partitions important for this article in the case that Ω = Rd with Lebesgue measure μ is given by dyadic cubes, that is, by Cn = Cn (i1 , . . . , id ), i1 , . . . , id ∈ Z , where Cn (i1 , . . . , id ) = i1 2−n , (i1 + 1)2−n × · · · × id 2−n , (id + 1)2−n .

1700


In this case, to satisfy requirement (i) in Definition 2.1, one can take L0 as the set of continuous functions with compact support. However, the methods of the present article can also be used in the case of parabolic equations where one should take the filtration of parabolic “cubes”. One can also consider other types of equations with mixed homogeneity and/or have in mind working in Sobolev spaces with weights. Definition 2.2. Let Cn , n ∈ Z, be a filtration of partitions of Ω. (i) Let τ = τ (x) be a function on Ω with values in {∞, 0, ±1, ±2, . . .}. The function τ is called a stopping time (relative to the filtration) if, for each n = 0, ±1, ±2, . . . , the set x: τ (x) = n is either empty or else is the union of some elements of Cn . (ii) For a function f ∈ L1 (Ω) and n ∈ Z, we denote f|n (x) = − f (y) μ(dy). Cn (x)

We read f|n as “f given Cn ”, continuing to borrow the terminology from probability theory. If we are also given a stopping time τ , we let f|τ (x) = f|τ (x) (x) for those x for which τ (x) < ∞ and f|τ (x) = f (x) otherwise. The simplest example of a stopping time is given by τ (x) ≡ 0. It is also known that if g ∈ L1 (Ω) and a constant λ > 0, then τ (x) = inf n ∈ Z: g|n (x) > λ

(inf ∅ := ∞)

is a stopping time and if, in addition, g 0, then g|τ N0 λ (a.e.). For f ∈ L1 (Ω) we denote Mf = sup |f ||n . n∈Z

It is known that for any f ∈ L1 (Ω) and p ∈ (1, ∞) Mf Lp (Ω) qf Lp (Ω) ,

(2.1)

where q = p/(p − 1). In the remaining part of the section we consider two functions u, v ∈ L1 (Ω) and a nonnegative measurable function g on Ω. Below by IMv(x)>αλ we mean the indicator function of the set {x: Mv(x) > αλ}.


1701

Lemma 2.3. Assume that 0 u v and for any n ∈ Z and C ∈ Cn we have (u − vC )+ μ(dx) g(x) μ(dx).

(2.2)

C

C

Then for any λ > 0 x: u(x) λ 2λ−1

(2.3)

g(x)IMv(x)>αλ μ(dx), Ω

where α = (2N0 )−1 . Proof. Fix a λ > 0 and define τ (x) = inf n ∈ Z: v|n (x) > αλ . We know that τ is a stopping time and if τ (x) < ∞, then v|n (x) λ/2,

∀n τ (x).

We also know that v|n → v (a.e.) as n → ∞. It follows that (a.e.) x: u(x) λ = x: u(x) λ, τ (x) < ∞ = x: u(x) λ, v|τ λ/2 = An (C), n∈Z C∈Cτn

where An (C) := x ∈ C: u(x) λ, v|n λ/2 , and Cτn is the family of disjoint elements of Cn such that x: τ (x) = n = C. C∈Cτn

Next, for each n ∈ Z and C ∈ Cn on the set An (C), if it is not empty, we have v|n = vC and u − vC λ/2, so that by Chebyshev’s inequality and assumption (2.2) An (C) 2λ−1 g μ(dx), C

x: u(x) λ 2λ−1 g μ(dx) = 2λ−1 gIτ αλ}. The lemma is proved.

2

1702


Remark 2.4. Obviously, the conditions of Lemma 2.3 are satisfied with g = (1/2)v if u = v. One of the nice features of the lemma is that under its conditions, for any measurable function a such that 1 a 2, the functions au, 2v, and 2g also satisfy its conditions. To give conditions to verify assumption (2.2) which are convenient in this article, we need the following. Assumption 2.5. We have |u| v and for any n ∈ Z and C ∈ Cn there exists a measurable function uC given on C such that |u| uC v on C and

C C u − uC μ(dx) g(x) μ(dx). |u − uC | μ(dx) ∧

C

C

(2.4)

C

Lemma 2.6. Under Assumption 2.5 for any λ > 0 we have x: u(x) λ 2λ−1

g(x)IMv(x)>αλ μ(dx),

(2.5)

Ω

where α = (2N0 )−1 . Moreover if u 0, then one can replace 2λ−1 in (2.5) with λ−1 . Proof. First assume that u 0. Take an n ∈ Z and a C ∈ Cn . If

|u − uC | μ(dx) C

g(x) μ(dx) C

then, since u v, we have uC vC and (u − uC ) + |u − uC | = 2(u − uC )+ 2(u − vC )+ , implying that (2.2) is satisfied with g/2 in place of g. In case that

C u − uC μ(dx)

g(x) μ(dx)

C

C

C

we observe that uC u, uC C vC , so that C C C C C u − uC C + u − uC = 2 u − uC + 2(u − vC )+ , which again implies that (2.2) is satisfied with g/2 in place of g. In the general case we need only show that condition (2.4) is almost preserved if we take |u| in place of u. However, for any measurable set C we have


1703

− u(x) − |u|C μ(dx) = − − u(x) − u(y) μ(dy) μ(dx) C

C

C

− − u(x) − u(y) μ(dy) μ(dx) C C

2 − u(x) − c μ(dx),

(2.6)

C

where c is any constant. If we take c = uC , then we see that |u| satisfy (2.4) with 2g in place of g. The lemma is proved. 2 Now we are ready to prove a partial version of the Fefferman–Stein theorem about sharp functions. Theorem 2.7. Under Assumption 2.5 for any p ∈ (1, ∞) we have p

p−1

uLp (Ω) N (p, N0 )gLp (Ω) vLp (Ω) .

(2.7)

The same conclusion holds under the assumptions of Lemma 2.3. Proof. We have p uLp (Ω)

∞ =

x: u(x) λ1/p dλ

0

∞

2

g(x) Ω

= 2qα 1−p

λ

−1/p

IMv(x)>αλ1/p dλ μ(dx)

0

g(Mv)p−1 μ(dx), Ω

where q = p/(p − 1). By using Hölder’s inequality and (2.1), we come to (2.7). The theorem is proved. 2 Remark 2.8. In the dyadic version of the original Fefferman–Stein theorem uC = u, v = |u|, and g is the sharp function u of u. In that case, assuming that u ∈ Lp (Ω), we get from (2.7) the Fefferman–Stein inequality uLp (Ω) N u Lp (Ω) . 3. Auxiliary results We denote by Br (x) the open ball in Rd of radius r centered at x. Set Br = Br (0) and introduce B as the family of balls in Rd . For a Borel set B ⊂ Rd of nonzero Lebesgue measure and a measurable function f we define

1704


1 fB := − f (x) dx := f (x) dx, |B| B

B

whenever the last integral is finite. The following is Lemma 4.8 of [7]. Lemma 3.1. Take an a¯ ∈ A and set ¯ Lu(x) = a¯ ij x 1 ux i x j (x).

(3.1)

There exists a constant N = N (d, δ) such that, for any κ 4, r > 0, u ∈ C0∞ (Rd ), and i, j ∈ {1, . . . , d} satisfying ij > 1 we have 2 ¯ 2 − ux i x j − (ux i x j )Br dx N κ d |Lu| + N κ −2 |uxx |2 B . B κr

κr

Br

We need a version of this lemma for operators of a more general form. Lemma 3.2. Take an a¯ ∈ A and a ψ ∈ Ψ and set j ¯ Lu(x) = a¯ kn y 1 φyi n (y)φy k (y)ux i x j (x),

(3.2)

where y = ψ(x) and φ = ψ −1 . Then there exist constants N = N (d, δ) and β = β(d, δ) 1 such that, for any κ 4, r > 0, u ∈ C0∞ (Rd ), and i, j ∈ {1, . . . , d} satisfying ij > 1 we have 2 ¯ 2 − uij − (uij )Br dx N κ d |Lu| B

βκr

Br

+ N κ d |ux |2 B

βκr

+ N κ −2 |uxx |2 B

βκr

(3.3)

,

where uij (x) are defined by uij φ(y) = vy i y j (y),

v(y) = u φ(y) ,

φ = ψ −1 .

(3.4)

¯ and observe that Proof. Without loss of generality we assume that ψ(0) = 0. Also set f = Lu a¯ kn y 1 vy k y n (y) + b˜ k (y)vy k (y) = f φ(y) , where j b˜ k (y) = a¯ mn y 1 φyi n (y)φy m (y)ψxki x j (x), Next we apply Lemma 3.1 to the operator L¯ y v(y) = a¯ kn y 1 vy k y n (y)

x = φ(y).

(3.5)


1705

and for any ρ > 0 find 2 − vy i y j − (vy i y j )Bρ dy N κ d |L¯ y v|2 B + N κ −2 |vyy |2 B . κρ

κρ

(3.6)

Bρ

To transform this inequality we use the simple observation that there exist constants N, β < ∞ depending only on d and δ such that for any nonnegative measurable function g we have − f (x) dx N − f φ(y) dy,

− f φ(y) dy N − f (x) dx.

Bρ √β

Bρ

Bρ

Bρ √β

Using this and closely following (2.6) we find 2 2 − uij − (uij )Br dx − − uij (x1 ) − uij (x2 ) dx1 dx2 Br

Br Br

N −

2 − vy i y j (y1 ) − vy i y j (y2 ) dy1 dy2

Br √β Br √β

2 N − vy i y j − (vy i y j )Br √β dy. Br √β

Furthermore, for y = ψ(x) obviously |vyy (y)| N (|uxx (x)| + |ux (x)|) and by (3.5) also ¯ |L¯ y v(y)| |Lu(x)| + N |ux (x)|. By combining the above observations we immediately obtain (3.3) from (3.6). The lemma is proved. 2 Set L0 u(x) = a ij (x)ux i x j (x). In the following lemma we prepare to check Assumption 2.5 for some functions to be introduced later and closely related to uij . However, we still have Br in place of C. Lemma 3.3. (i) Suppose that Assumptions 1.1 and 1.2(γ ) are satisfied. (ii) Let μ, ν ∈ (1, ∞), κ 4, and r > 0 be some numbers such that 1/μ + 1/ν = 1. Then there exist a mapping ψ ∈ Ψ and constants N = N (d, δ, μ) and β = β(d, δ) 1 such that, for any C0∞ function u, vanishing outside a ball of radius R R0 , and i, j ∈ {1, . . . , d} satisfying ij > 1 we have

1706


2 − uij − (uij )Br dx N κ d |L0 u|2 B

βκr

Br

+ N κ d |ux |2 B

βκr

+ N κ d R 2 + κ −2 |uxx |2 B

βκr

1/μ + N κ d γ 1/ν |uxx |2μ B ,

(3.7)

βκr

where uij (x) are defined by (3.4). Proof. We take β from Lemma 3.2 and split the proof into two parts. Case βκr < R. Take a ψ ∈ Ψ and an aˆ ∈ A such that − a(x) − aˆ ψ 1 (x) dx γ .

(3.8)

Bβκr

Reducing δ if necessary we may assume that, for an a¯ ∈ A, we have j

aˆ ij (t) = a¯ kn (t)φyi n (y0 )φy k (y0 ),

(3.9)

where y0 = ψ(0). Then introduce L¯ by (3.2) and set ˆ Lu(x) = aˆ ij ψ 1 (x) ux i x j (x). Observe that for y = ψ(x) and |x| βκr we have |y − y0 | N (d, δ)βκr and kn 1 i = a¯ y φ n (y)φ j k (y) − φ i n (y0 )φ j k (y0 ) u i j (x) (L¯ − L)u(x) ˆ x x y y y y N R uxx (x).

(3.10)

This and (3.3) yield 2 ˆ 2 − uij − (uij )Br dx N κ d |Lu| B

βκr

Br

+ N κ d |ux |2 B

+ N κ d R 2 |uxx |2 B

βκr

βκr

+ N κ −2 |uxx |2 B

βκr

.

(3.11)

After that it only remains to notice that ˆ 2 |Lu| B

βκr

2 |L0 u|2 B

βκr

2 + 2 (Lˆ − L0 )u B

βκr

and by Hölder’s inequality and (3.8) (Lˆ − L0 )u2

Bβκr

which yields (3.7).

1/μ N |uxx |2μ B γ 1/ν , βκr

(3.12)


1707

Case βκr R. Let u = 0 outside BR (x0 ). Take a ψ ∈ Ψ and an aˆ ∈ A such that −

a(x) − aˆ ψ 1 (x) dx γ ,

BR (x0 )

define a¯ by (3.9) with y0 = ψ(x0 ), and define Lˆ and L¯ as above. Then on the support of u we still have (3.10) and hence (3.11) holds again. Finally, (Lˆ − L0 )u2

Bβκr

2 = IBR (x0 ) (Lˆ − L0 )u B βκr 1/μ N |uxx |2μ B J, βκr

where J :=

1

ν

|Bβκr |

a(x) − aˆ ψ 1 (x) dx

Bβκr ∩BR (x0 )

1 |BR (x0 )|

a(x) − aˆ ψ 1 (x) dx γ .

BR (x0 )

It is seen that (3.12) is true again and the lemma is proved.

2

In the next lemma by Cn , n ∈ Z, we mean the filtration of dyadic cubes in Rd and by Mf the classical maximal function of f defined by Mf (x) =

sup B∈B:Bx

− f (y) dy. B

Lemma 3.4. (i) Suppose that Assumptions 1.1 and 1.2(γ ) are satisfied. (ii) Let μ, ν ∈ (1, ∞), and κ 4 be some numbers such that 1/μ + 1/ν = 1. Then for any n ∈ Z and C ∈ Cn there exist a mapping ψ ∈ Ψ and a constant N = N (d, δ, μ) such that, for any C0∞ function u, vanishing outside a ball of radius R R0 , and i, j ∈ {1, . . . , d} satisfying ij > 1 we have C

uij − (uij )C dx N

g dx, C

where uij (x) are defined by (3.4) and g is a nonnegative function satisfying g 2 = κ d M |L0 u|2 + M |ux |2 + κ d R 2 + κ −2 M |uxx |2 1/μ . + κ d γ 1/ν M |uxx |2μ

(3.13)

1708


Furthermore, |uxx | N

|uij | + N |ux | + N |L0 u|.

(3.14)

ij >1

Proof. Let B be the smallest ball containing C and let B be the concentric ball of radius βκr, where r is the radius of B and β is taken from Lemma 3.3. One can certainly shift the origin in the situation of Lemma 3.3 and hence for ij > 1 and an appropriate ψ ∈ Ψ 2 − uij − (uij )B dx N1 κ d |L0 u|2 B + N1 κ d |ux |2 B B

+ N1 κ d R 2 + κ −2 |uxx |2 B 1/μ + N1 κ d γ 1/ν |uxx |2μ B ,

(3.15)

where N1 = N(d, δ, μ). Obviously, the right-hand side of (3.15) is less than N1 g 2 (x) for any x ∈ C (and for that matter, for any x ∈ B ). In particular, the square root of the right-hand side of (3.15) is less than 1/2

N1

− g dx. C

After that, to finish proving the first assertion of the lemma, it only remains to use Hölder’s inequality showing that

1/2 2 J := − uij − (uij )B dx − uij − (uij )B dx B

B

and observe that − uij − (uij )C dx − − uij (x) − uij (y) dx dy C

C C

N (d) − − uij (x) − uij (y) dx dy N J. B B

To prove the second assertion, define f = L0 u, v(ψ(x)) = u(x), and by changing variables ˆ introduce an operator Lˆ such that Lv(y) = f (φ(y)). Then |vyy | N

ˆ + N |vy |. |vy i y j | + N |Lv|

ij >1

By adding to this that |uxx (x)| N |vyy (y)| + N|ux (x)| for y = ψ(x), we come to (3.14). The lemma is proved. 2


1709

Lemma 3.5. Let p ∈ (2, ∞). We assert that there exist constants γ = γ (d, δ, p) > 0 and R = R(d, δ, p, R0 ) ∈ (0, R0 ] such that if Assumptions 1.1 and 1.2(γ ) are satisfied, then for any C0∞ function u vanishing outside a ball of radius R we have uxx Lp N L0 uLp + ux Lp ,

(3.16)

where N = N(d, δ, p). Proof. For the moment we suppose that Assumptions 1.1 and 1.2(γ ) are satisfied with a constant γ > 0 and will choose it appropriately near the end of the proof. Take a number κ 4 and set μ = (2 + p)/4 (μ > 1, 2μ < p). Also take an n ∈ Z and a C ∈ Cn and take a ψ ∈ Ψ from Lemma 3.4. Finally, take a C0∞ function u vanishing outside a ball of radius R, introduce uij by (3.4), and set L0 u = f,

U = |uxx |,

UC =

|uij | + |ux | + |f |,

V = |uxx | + |ux | + |f |.

ij >1

We want to apply Theorem 2.7. Estimate (3.14) says that U N U C . Furthermore, obviously U C NV . Also, similarly to (2.6) C C − uij − (uij )C dx + 2 − ux − (ux )C dx + 2 − |f − fC | dx. − U − UC dx 2 ij >1 C

C

C

C

We estimate the sum over ij > 1 by using Lemma 3.4 and observe that − |f − fC | dx 2|f |C 2Mf (x), ∀x ∈ C, C

− |f − fC | dx 2 − Mf dx,

− ux − (ux )C dx 2 − M|ux | dx.

C

C

C

C

Hence C C − U − UC dx N − g + M|ux | + Mf dx, C

C

where g is defined in Lemma 3.4. Since this holds for any n ∈ Z and any C ∈ Cn , by Theorem 2.7 we conclude 1/p (p−1)/p . uxx Lp = U Lp N g + M|ux | + Mf L V Lp p

By observing that V Lp uxx Lp + ux Lp + f Lp and by Young’s inequality

1710


a 1/p b(p−1)/p N (ε, p)a + εb,

∀a, b, ε > 0,

we easily get that uxx Lp N g + M|ux | + Mf L + ux Lp + f Lp . p

Next, by applying the Hardy–Littlewood maximal function theorem and using the fact that p/(2μ) > 1 and p > 2 we find uxx Lp N1 κ d/2 f Lp + N1 κ d/2 ux Lp + N1 κ d/2 R + κ −1 + κ d/2 γ 1/(2ν) uxx Lp , where ν = μ/(μ − 1), N1 = N (d, δ, p), and κ 4 is an arbitrary number. After choosing R = R(d, δ, p) ∈ (0, R0 ] and κ = κ(d, δ, p) 4 so that N1 κ −1 1/4,

N1 κ d/2 R 1/4,

and finally choosing γ = γ (d, δ, p) > 0 so that N1 κ d/2 γ 1/(2ν) 1/4, we come to (3.16). The lemma is proved.

2

4. Proof of Theorem 1.4 We take a p ∈ (2, ∞) and take γ from Lemma 3.5 and suppose that Assumptions 1.1 and 1.2(γ ) are satisfied. As usual, bearing in mind the method of continuity, one sees that it suffices to prove the a priori estimate (1.3). Notice that L0 u − λuLp Lu − λuLp + N ux Lp + KuLp , where N = N(d, K). Since we only consider large λ, this shows that it suffices to prove (1.3) with L0 in place of L. Therefore, below we assume that b = c = 0. In that case by using partitions of unity one easily derives from Lemma 3.5 that for any u ∈ Wp2 uxx Lp N LuLp + ux Lp + uLp , where N = N(d, δ, p, R0 ). Using the interpolation inequality ux Lp εuxx Lp + N (d, p)ε −1 uLp ,

ε > 0,

shows that uxx Lp N LuLp + uLp . It follows that for any λ 0

(4.1)


λuLp +

1711

√ λux Lp + uxx Lp N Lu − λuLp + (λ + 1)uLp ,

which implies that we only need to find λ0 (d, δ, p, R0 ) 1 such that for λ λ0 we have λuLp N Lu − λuLp

(4.2)

with N = N(d, δ, p, R0 ). As is usual in such situations, we will follow an idea suggested by S. Agmon. Consider the space Rd+1 = z = (x, y): x ∈ Rd , y ∈ R and the function u(z) ˜ = u(t, x)ζ (y) cos(μy), where μ =

√ λ and ζ is a C0∞ (R) function, ζ ≡ 0. Also introduce the operator ˜ Lv(t, z) = a ij (x)vx i x j (z) + vyy (z).

As is easy to see, the operator L˜ satisfies Assumption 1.2 (γ ) (relative to Rd+1 ) with γ = N(d)γ . Therefore, by reducing the γ taken from Lemma 3.5 if necessary, we may apply the above results to the operator L˜ and in light of (4.1) applied to u˜ and L˜ we get ˜ Lp (Rd+1 ) + u ˜ Lp (Rd+1 ) . u˜ zz Lp (Rd+1 ) N L˜ u It is not hard to see that

ζ (y) cos(μy)p dy

R

is bounded away from zero for μ ∈ R. Therefore, p

uL

p

= μ−2p (Rd )

×

ζ (y) cos(μy)p dy

−1

R

u˜ yy (z) − u(x) ζ

(y) cos(μy) − 2μζ (y) sin(μy) p dz

Rd+1

p N μ−2p u˜ zz L

p (R

d+1 )

p + μp + 1 uL

p (R

d)

.

This and (4.3) yield ˜ Lp (Rd+1 ) + N (μ + 1)uLp . μ2 uLp N L˜ u Since

(4.3)

1712


L˜ u˜ = ζ cos(μy)[Lu − λu] + u ζ

cos(μy) − 2μζ sin(μy) , we have L˜ u ˜ Lp (Rd+1 ) N Lu − λuLp + N (μ + 1)uLp , so that √ λuLp N1 Lu − λuLp + N2 ( λ + 1)uLp . For λ λ0 = 16N22 + 4N2 we have √ N2 λ (1/4)λ,

N2 (1/4)λ,

√ N2 ( λ + 1) (1/2)λ

and we arrive at (4.2) with N = 2N1 . The theorem is proved. References [1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959) 623–727; II, Comm. Pure Appl. Math. 17 (1964) 35–92. [2] L. Bers, F. John, M. Schechter, Partial differential equations, With Special Lectures by Lars Gårding and A.N. Milgram, Proceedings of the Summer Seminar, Boulder, Colorado, 1957, in: Lect. Appl. Math., vol. III, Interscience Publishers John Wiley & Sons, Inc., New York/London/Sydney, 1964. [3] S.-S. Byun, L. Wang, Lp -estimates for general nonlinear elliptic equations, Indiana Univ. Math. J. 56 (6) (2007) 3193–3221. [4] F. Chiarenza, M. Frasca, P. Longo, Interior W 2,p estimates for nondivergence elliptic equations with discontinuous coefficients, Ric. Mat. 40 (1) (1991) 149–168. [5] Hongjie Dong, Doyoon Kim, Parabolic and elliptic systems with VMO coefficients, preprint, 2008. [6] Doyoon Kim, Parabolic equations with measurable coefficients in Lp -spaces with mixed norms, http://aps.arxiv.org/ abs/0705.3808. [7] Doyoon Kim, N.V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal. 39 (2) (2007) 489–506. [8] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, RI, 2008.


Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps Jinsung Park School of Mathematics, Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul 130-722, Republic of Korea Received 20 August 2008; accepted 16 June 2009 Available online 1 July 2009 Communicated by Daniel W. Stroock

Abstract In this paper we derive a relationship of the leading coefficient of the Laurent expansion of the Ruelle zeta function at s = 0 and the analytic torsion for hyperbolic manifolds with cusps. Here, the analytic torsion is defined by a certain regularized trace following Melrose [R.B. Melrose, The Atiyah–Patodi–Singer Index Theorem, Res. Notes Math., vol. 4, A.K. Peters, Ltd., Wellesley, MA, 1993]. This extends the result of Fried, which was proved for the compact case in [D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (3) (1986) 523–540], to a noncompact case. © 2009 Elsevier Inc. All rights reserved. Keywords: Analytic torsion; Ruelle zeta function; Selberg trace formula

1. Introduction In this paper we derive an equality between the leading coefficient of the Laurent expansion of the Ruelle zeta function at s = 0 and the analytic torsion for hyperbolic manifolds with cusps. This extends the result of Fried, which was proved for the compact case in [6], to a noncompact case. Here the analytic torsion for manifolds with cusps is defined by a certain regularized trace following the idea of the b-trace of Melrose [16]. This paper can be considered as a continuation of our previous study [21] of the relationship between a special value of the odd type Selberg zeta function and the eta invariant which extends the result of Millson [18] to hyperbolic manifolds with cusps. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.012

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J. Park / Journal of Functional Analysis 257 (2009) 1713–1758

Let XΓ denote a d-dimensional hyperbolic manifold with cusps, which is given by XΓ = Γ \ SO0 (d, 1)/SO(d) where Γ is a co-finite discrete subgroup in SO0 (d, 1). We assume that XΓ is equipped with the constant negative curvature −1. We also assume that Γ is neat (hence torsion free), that is, the group generated by the eigenvalues of Γ contains no root of unity. A consequence of this is ΓP := Γ ∩ P = Γ ∩ N (P )

for P ∈ PΓ

(1.1)

where N(P ) is the nilpotent part of P and PΓ = {P1 , . . . , Pκ } denotes a complete set of Γ conjugacy classes of Γ -cuspidal subgroups of G. The Ruelle zeta function Rχ (s) over XΓ is now defined by Rχ (s) :=

−1 det Id − χ(γ )e−s l(Cγ )

γ ∈PΓhyp

for Re(s) > (d − 1). Here PΓhyp denotes the set of Γ -conjugacy classes of the primitive hyperbolic elements in Γ , the determinant denoted by “det” is taken over the representation space Vχ of a unitary representation χ of Γ , and l(Cγ ) denotes the length of the closed geodesic determined by a hyperbolic element γ . Let us recall some results about Rχ (s) in [10] when d = 2n + 1. First, by Theorem 1.1 of [10], the Ruelle zeta function Rχ (s) has a meromorphic extension to C. Second, let N0 be the order of the singularity of Rχ (s) at s = 0 such that Rχ∗ (0) := lim s N0 Rχ (s) ∈ C − {0, ∞}. s→0

By Theorem 1.2 of [10], we have that if d = 2n + 1, n n−1 2n k k+1 n 2n − 2 + dc (χ)(−1) N0 = 2 (−1) (n + 1 − k)βk + (−1) bk k n−1 k=0

(1.2)

k=0

where βk := dim kerL2 (k ) with the Hodge Laplacian k acting on the space of differential k-forms twisted by χ over XΓ , bk denotes the order of the singularity of the determinant of a certain scattering operator Cχk (σk , s) at s = d−1 2 − k, and dc (χ) is the sum of the dimensions of the maximal subspaces of Vχ ’s where χ|Γ ∩P acts trivially for P ∈ PΓ (see (3.2)). From (1.2), we can see that if d = 2n + 1 the behavior of the Ruelle zeta function Rχ (s) at s = 0 is related to the spectral data of the Hodge Laplacians k ’s, and it is a natural question whether the leading coefficient Rχ∗ (0) may have a relationship with another spectral data. In [6], it was proved that this is equal to the analytic torsion (up to a constant) for odd-dimensional compact hyperbolic manifold XΓ . Since we do not have an analytic torsion for our noncompact case, we need to introduce an analytic torsion T (XΓ , χ) which is linked to the leading coefficient Rχ∗ (0). To do this, first of all we define the spectral zeta function of the Hodge Laplacian k using a certain regularized trace of the heat operator of k . In Theorem 6.1 we show that this spectral zeta function of k is regular at s = 0. Then we can define the regularized determinants of k ’s and the analytic torsion T (XΓ , χ) in the usual way as in the compact case. Actually this


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approach was suggested by Melrose in [16] and the regularized trace in this paper is essentially the same as the b-trace of Melrose. Following this idea, Hassell defined b-analytic torsion for certain noncompact manifolds in [12]. We refer to Section 6 for the precise definitions of the spectral zeta function of k and T (XΓ , χ). The following theorem states a relationship of Rχ∗ (0) with the analytic torsion T (XΓ , χ) where some defect terms are given from the cusps (geometrically) and the scattering data of k ’s (analytically). Theorem 1.1. For a (2n + 1)-dimensional hyperbolic manifold XΓ with cusps, the following equality holds up to sign, Rχ∗ (0)−1 = C(XΓ , χ) · C(d)dc (χ) · S(XΓ , χ) · T (XΓ , χ).

(1.3)

Here C(XΓ , χ) :=

n−1

(−1)k αk −4(n − k)2 ,

C(d) :=

k=0

n−1

Sχ (k)(−1)

2(−1)

k+1 e

1

· (n − k)(−1)

ke

2

k=0

where αk := βk − βk−1 + βk−2 − · · · ± β0 , e1 = S(XΓ , χ) :=

n−1

k+1 2n k

( )

k=0

2n k

−

with Sχ (k) =

2n−1 k

lim

2n−1 , e2 = (2n − 2k + 1) 2n k − k ,

s→−(n−k)

(s + n − k)−bk det Cχk (σk , s).

This result was announced in [22]. Remark 1.2. Let us observe that C(d) depends only on the dimension d, not on Γ although C(XΓ , χ), S(XΓ , χ) depend on Γ sensitively. When XΓ is compact, the equality (1.3) is reduced to the formula of Fried in [6]. Actually we can see that the same formula holds under a more general condition that dc (χ) = 0 even if XΓ may have cusps. In fact, if dc (χ) = 0, then N0 is given only by the βk ’s in (1.2) and C(d)dc (χ) = S(XΓ , χ) = 1. Moreover the sign ambiguity in Theorem 1.1 disappears since this comes from the scattering operators Cχk (σk , s). Remark 1.3. In [26,27], Sugiyama studied the geometric analogues of the Iwasawa conjecture for 3-dimensional hyperbolic manifolds. He proved that the Laurent expansion of the Ruelle zeta function Rχ (s) at s = 0 satisfies several analogues of the Iwasawa conjecture in the algebraic number theory under the condition dc (χ) = 0. In particular, in [27] it is proved that Rχ∗ (0) is essentially given by the Reidemeister torsion for (XΓ , χ) if dc (χ) = 0. Our Theorem 1.1 is crucially used in its proof. It seems to be interesting to understand the equality (1.3) in Theorem 1.1 for general cases in the view point of the geometric analogues of the Iwasawa conjecture. Comparing the formulae of the order of the singularity N0 for even- and odd-dimensional cases (see (1.2) and Theorem 1.2 of [10]), one can expect that there is less relationship of Rχ∗ (0) with the spectral data in the even-dimensional case. Actually because of a certain symmetry (see (8.6)) we can not link Rχ∗ (0) with T (XΓ , χ) in the even-dimensional case. It is also known that the analytic torsion T (XΓ , χ) is trivial for even-dimensional compact manifold. However, it

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turned out that this is not true anymore for noncompact hyperbolic manifold XΓ with cusps, and that T (XΓ , χ) has an following explicit expression. Theorem 1.4. For a 2n-dimensional hyperbolic manifold XΓ with cusps, the following equality holds T (XΓ , χ) =

n−1

dc (χ) (−1)k n( 2n−1 k+1

(n − 1/2 − k)

(

)−(

2n−2 k

))

.

(1.4)

k=0

Remark 1.5. The right-hand side of (1.4) originates from the non-invariant part of the weighted unipotent orbital integral on the geometric side of the Selberg trace formula. Geometrically this is the defect for the Hodge theorem of the de Rham complex for hyperbolic manifolds with cusps. For the odd-dimensional case, the corresponding term is also contained in the factor C(d) in Theorem 1.1. Remark 1.6. For the case of d = 2n, we can also obtain an expression of Rχ∗ (0) in terms of similar factors on the right-hand side of (1.3) except T (XΓ , χ). This easily follows from the functional equation of Rχ (s) presented in Theorem 1.1 of [10]. (The simplest case of d = 2 was also mentioned on p. 162 in [5].) Now let us explain the structure of this paper. In Section 2, we review the basics of harmonic analysis over hyperbolic spaces to fix notations and normalizations used in this paper. In Section 3, we study the spectral side of the Selberg trace formula. This will explain the motivation of the regularized trace that is used to define the analytic torsion for hyperbolic manifolds with cusps. In Section 4, we explain the Selberg trace formula for the nontrivial homogeneous vector bundles over hyperbolic manifolds with cusps. In Section 5, we completely compute the contribution of the weighted orbital integrals for our case applying the result in [13]. In Section 6, we define the spectral zeta functions of the Hodge Laplacians using the regularized trace following Melrose [16] and show that they have meromorphic extensions over C. This enables us to define the regularized determinant and analytic torsion. In Sections 7 and 8, we prove Theorems 1.1 and 1.4 combining all the results proved in the previous sections. In Appendix A, we perform an algebraic computation which gives the proof of Theorem 5.3. 2. Harmonic analysis over real hyperbolic space 2.1. Algebraic structures The d-dimensional real hyperbolic space is the manifold

2 = −1, xd+1 > 0 Hd (R) = x ∈ Rd+1 x12 + x22 + · · · + xd2 − xd+1 equipped with the metric of curvature −1. The orientation preserving isometries of Hd (R) form the group G = SO0 (d, 1) which is the identity component of SO(d, 1). The isotropy subgroup K of the base point (0, . . . , 0, 1) is isomorphic to SO(d). Hence the real hyperbolic space Hd (R) can be identified with the symmetric space G/K. We denote the Lie algebras of G, K by g = so(d, 1), k ∼ = so(d) respectively. The Cartan involution θ on g gives us the decomposition


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g = k ⊕ p where k, p are the 1, −1 eigenspaces of θ , respectively. The subspace p can be identified with the tangent space To (G/K) ∼ = g/k at o = eK ∈ G/K. The invariant metric of curvature −1 over Hd (R) corresponds to the normalized Cartan–Killing form X, Y := −

1 C(X, θ Y ) 2(d − 1)

(2.1)

where the Killing form is defined by C(X, Y ) = Tr(ad X ◦ ad Y ) for X, Y ∈ g. Let a be a fixed maximal abelian subspace of p. Then the dimension of a is one. Let M ∼ = SO(d − 1) be the centralizer of A = exp(a) in K with Lie algebra m. (When d = 2, M ∼ = Z2 .) Let TM be a Cartan subgroup in M so that T = TM · A is a Cartan subgroup of G. Let ΣM be the system of the positive roots for (mC , tmC ). We choose the system ΣA of positive roots of (gC , tC ) which do not vanish on aC so that ΣA is compatible with ΣM . Then the union of ΣM with ΣA gives the system of positive roots for (gC , tC ), which is denoted by ΣG . With respect to the inner product on t∗C induced from ·,· in (2.1), we choose an orthonormal basis {ei } of t∗C such that e1 ∈ a∗C . Then we have: (1) When d = 2n + 1,

ΣG = ei + ej (1 i < j n + 1), ei − ej (1 i < j n + 1) ,

ΣA = e1 + ej (1 < j n + 1), e1 − ej (1 < j n + 1) . (2) When d = 2n,

ΣG = ei (1 i n), ei − ej (1 i < j n), ei + ej (1 i < j n) ,

ΣA = e1 , e1 − ej (1 < j n), e1 + ej (1 < j n) . We put β = e1 , which is the positive restricted root of (g, a). Let ρ denote the half sum of the positive roots of (g, a), that is, ρ = (d−1) 2 β. Later on, we shall use the identification a∗C ∼ =C

by λβ −→ λ.

(2.2)

Let n be the positive root space of β and N = exp(n) ⊂ G. The Iwasawa decomposition is given by G = KAN. From now on we fix the following Haar measure on G, dg = a 2ρ dk da dn = a −2ρ dn da dk

(2.3)

where g = kan is the Iwasawa decomposition and a 2ρ = exp(2ρ(log a)). Here dk is the Haar

measure over K with K dk = 1, da is the Euclidean Lebesgue measure on A given by the identification A ∼ = R via at = exp(tH ) with H ∈ a, β(H ) = 1, and dn is the Euclidean Lebesgue measure on N induced by the normalized Cartan–Killing form ·,· given in (2.1).

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2.2. Homogeneous vector bundle Let us recall the homogeneous vector bundle over the symmetric space Hd (R) ∼ = G/K. If τ is a unitary finite-dimensional representation of K, then the sections of the associated homogeneous vector bundle G ×τ Vτ over G/K consist of the map f : G → Vτ with the condition f (gk) = τ (k)−1 f (g)

for g ∈ G, k ∈ K.

(2.4)

Equivalently, the sections of G ×τ Vτ are equivalence classes of the pairs (g, v) under (gk, v) ∼ (g, τ (k)v). For such a section f of G ×τ Vτ , there is a G-action defined by g0 · f (g) = g0 f (g0−1 g). For instance, the space of k-forms over G/K is given by this construction: Choose an orthonormal basis for p∗ . This basis determines left invariant 1-forms ω1 , . . . , ωd on G. A complex-valued k-form w on G/K pulls back to a k-form ω on G given by ω =

fi1 ,...,ik ωi1 ∧ . . . ωik .

The component functions (fi1 ,...,ik ) give a map f : G → k Cd satisfying the condition (2.4) k d with τ = τk acting on Vτk ∼ C . All the representations τk are irreducible representations = of K except when d = 2n and k = n. In this case, τn decomposes into two irreducible represen 2 tations τn+ , τn− acting on n+ C2n , n− C2n , respectively. Here n± C2n denotes the ± exp( n2 πi)n 2n C . Let us recall that the highest weight μk of the eigenspace of the Hodge operator ∗ on representation τk is given by μk = e2 + e3 + · · · + ek+1

for 1 k n,

d = 2n + 1,

μk = e2 + e3 + · · · + ek+1

for 1 k n − 1,

d = 2n,

μ± n

for

d = 2n.

= e2 + e3 + · · · + en ± en+1

Let us denote the irreducible fundamental representations of M = SO(d − 1) by σk ’s if d = 2n, and σk ’s with k = n, σn± if d = 2n + 1. These satisfy the following branching laws: (1) For k = n with d = 2n or d = 2n + 1, [τk |M : σ ] = 1

if and only if

σ = σk or σ = σk−1 .

(2) For k = n and d = 2n,

τn± |M : σ = 1 if and only if σ = σn = σn−1 .

(3) For k = n and d = 2n + 1, [τn |M : σ ] = 1

if and only if

σ = σn−1 or σ = σn± .


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2.3. Heat kernel of the Hodge Laplacian The Hodge Laplacian k on the space of k-forms for the curvature −1 metric is an invariant differential operator constructed as follows: We choose a basis Ei of k and a basis Ej of p that are orthonormal with respect to the normalized Cartan–Killing form ·,· . Then Ω := −

Ei2 +

Ej2

(2.5)

is the normalized Casimir element in the center of the universal enveloping algebra of g. For a representation τ of K, let Qτ = R(k) ⊗ τ (k) dk K

∼ (L2 (G) ⊗ Vτ )K where R denotes the right be the projection from L2 (G) ⊗ Vτ to L2 (G, τ ) = 2 regular representation of G on L (G). Then the Hodge Laplacian k is given by k = Qτk R(−Ω) ⊗ IdVτk Qτk .

(2.6)

That is, the Hodge Laplacian k is the restriction to the τk -invariant part of the corresponding invariant differential operator to −Ω. and The subgroup P0 := NAM is a minimal parabolic subgroup of G. Given (σ, Hσ ) ∈ M ∗ λ ∈ aC , the following action

1 ⊗ eiλ ⊗ σ (nam) = a iλ σ (m)

defines a representation of P0 on Hσ where a iλ denotes exp(iλ(log a)). Then the principal series iλ representation πσ,λ := IndG P0 (1 ⊗ e ⊗ σ ) of G acts on the space

Hσ,λ := f : G → Hσ f (nam x) = a (iλ+ρ) σ (m)f (x), f |K ∈ L2 (K) by the right translation πσ,λ (g)f (x) = f (xg). The following proposition whose proof is similar to Lemma 1 of [6] gives the action of k over Hσ ,λ if [τk |M : σ ] = 0. 2 Proposition 2.1. If [τk |M : σ ] = 0, the Hodge Laplacian k acts on Hσ ,λ by λ2 + ( d−1 2 − ) ± where σn means σn if d = 2n + 1. k To deal with the heat operator e−t ,2we follow the discussion in Section 2 of [2] or Section 3 of [19]. Let us denote by ΩK = − Ei the normalized Casimir operator of K. Recalling (2.5), let G denote the corresponding left invariant Laplace operator over G; that is,

G = −Ω + 2ΩK = −

Ei2 −

Ej2 .

Using the following well-known formula (for instance, see (A.10) of [1]) τk (ΩK ) = μk + ρK , μk + ρK − ρK , ρK

(2.7)

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where ρK denotes the half sum of the positive roots of K = SO(d), one can show that τk (ΩK ) = λk IdVτk

with λk = (d − k)k.

(This formula is even true for τn with d = 2n.) By (2.6) and (2.7), k = Qτk (G ⊗ IdVτk )Qτk − 2λk IdVτk . Now let e−tG denote the heat semi-group operator given by a smooth kernel Pt , e−tG f (g1 ) =

Pt g2−1 g1 f (g2 ) dg2 ,

for f ∈ L2 (G), g1 ∈ G.

G

Hence, the heat semi-group operator of k satisfies e−tk = e2λk t · Qτk e−tG ⊗ IdVτk Qτk ,

(2.8)

which implies e−tk g2−1 g1 = e2λk t ·

τk (k2 )Pt k1 g2−1 g1 k2 τk (k1 ) dk1 dk2 .

K×K

Therefore, the kernel e−tk (g1 , g2 ) := e−tk (g2−1 g1 ) satisfies the following covariance relation e−tk (g1 k1 , g2 k2 ) = τk (k1 )−1 e−tk (g1 , g2 )τk (k2 )

for k1 , k2 ∈ K.

(2.9)

For a fixed t > 0, by Lemma 2.3 in [2], e−tG belongs to the Harish–Chandra Lp -Schwartz space C p (G) for any p > 0. Here C p (G) is the space of all functions f ∈ C ∞ (G) such that m −2 sup 1 + σ (g) Ψ (g) p D1 D2 f (g) < ∞

for any m 0, D1 , D2

g∈G

where σ (g) is the geodesic distance between the cosets eK and gK in G/K, Ψ (g) =

e−ρ(H (gk)) dk

K

for the Iwasawa decomposition gk = K(gk) exp(H (gk))N (gk), and D1 , D2 denote the right,

left invariant differential operators, respectively. Let us remark that C p (G) ⊆ C p (G) if p p (see [8, p. 4]). Now we can conclude Proposition 2.2. For any t > 0, the heat kernel e−tk belongs to (C p (G) ⊗ End(Vτk ))K×K for any p > 0 and e−tk satisfies the covariance relation in (2.9).


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Now let hkt := tr(e−tk ) where tr is given over Vτk . Then hkt belongs to C p (G) for p > 0 by Proposition 2.2. Hence one can define Θσ,λ hkt = Tr πσ,λ hkt = Tr

hkt (g)πσ,λ (g) dg. G

For a given unitary representation π of G, we define its matrix block of π corresponding to Pτ ∈ HomK (π|K , τ ) by Φτπ (g) := Pτ π g −1 Pτ∗

for g ∈ G.

We note that Φτπ (g) ∈ End(Vτ ) for a fixed g ∈ G and Φτπ (k1 gk2 ) = τ (k2 )−1 Φτπ (g)τ (k1 )−1

for k1 , k2 ∈ K.

This is an τ -spherical function on G on which the normalized Casimir operator Ω in (2.5) acts by its infinitesimal character. We refer to Section VIII 4–6 in [14] for more details of these facts. Hence, by Proposition 2.1, for τ = τk , π = πσ ,λ we have

2 (d − 1) Φτπ (g1 ) − e−tk g2−1 g1 Φτπ (g2 ) dg2 = exp −t λ2 + 2

(2.10)

G

if [τk |M : σ ] = 0 and both sides of (2.10) vanish if [τk |M : σ ] = 0. Taking the trace over Vτ and putting g1 = e, we get dim Vτ

2 π (d − 1) τπ (e) τ (g2 ) dg2 = exp −t λ2 + Φ − hkt g2−1 Φ 2

(2.11)

G

τπ := tr Φτπ . On the other by the orthogonality relations for the matrix elements of τ where Φ hand, Tr π hkt =

π τ (g) dg hkt g −1 Φ

for π = πσ,λ .

G

τπ (e) = dim Vτ , we obtain Comparing (2.11) and (2.12) and noting Φ Proposition 2.3. Θσ ,λ hkt =

2 exp −t λ2 + (d−1) if [τk |M : σ ] = 0, 2 − 0 if [τk |M : σ ] = 0.

(2.12)

1722


3. Spectral decomposition for hyperbolic manifolds with cusps 3.1. Hodge Laplacian for hyperbolic manifolds with cusps Let us choose a unitary representation χ of Γ on a finite-dimensional hermitian vector space Vχ . We now consider the right quasi-regular representation Rχ on

Hχ := φ : G → Vχ φ(γ x) = χ(γ )φ(x) for γ ∈ Γ, x ∈ G, |φ| ∈ L2 (Γ \ G) given by (Rχ (x)φ)(y) = φ(yx). As in [29], this representation Rχ of G on Hχ decomposes into a discrete part and a continuous part. That is, Rχ = Rχd ⊕ Rχc

acts on Hχ = Hdχ ⊕ Hcχ .

The action Rχd on Hdχ is a Hillbert sum of irreducible representations, each of them occurring with finite multiplicity and the action of Rχc on Hcχ is a direct integral, with no irreducible subrepresentations, of principle series. For a test function h ∈ C p (G) with 0 < p < 1, which is of right K-finite, the induced representation Rχd (h) is of trace class and Tr Rχd (h) = Tr

h(g)Rχd (g) dg =

mχ (π) Tr π(h)

(3.1)

π∈G

G

in Hdχ . where mχ (π) denotes the multiplicity of π ∈ G Let us recall that a d-dimensional noncompact hyperbolic manifold with cusps is given by XΓ = Γ \ G/K = Γ \ SO0 (d, 1)/SO(d) where Γ is a cofinite discrete subgroup of G = SO0 (d, 1) satisfying the conditions imposed in the introduction. The vector bundle Eχk over XΓ of k-forms twisted by χ is given by Eχk = Vχ ×χ G ×τk Vτk . The Hodge Laplacian k acting on the space of sections of Vτk over G/K can be naturally pushed down to a differential operator acting on C0∞ (XΓ , Eχk ). By abuse of notation, we use the same notation k to denote its self-adjoint extension on

L2 XΓ , Eχk = |f | ∈ L2 (XΓ , Vτk ) f (γ x) = χ(γ )f (x) for γ ∈ Γ , which consists of the τk -isotypic component of Hχ . In general, the operator k on L2 (XΓ , Eχk ) has discrete spectrum σp (k ) as well as continuous spectrum σc (k ). The continuous spectrum of k is mainly controlled by the scattering operators Cχk (σk , s) and Cχk (σk−1 , s), which will be explained in the next subsection, for purely imaginary numbers s = iλ ∈ C. These scattering operators have the matrix forms of size dc (χ) where dc (χ) =

κ j =1

dj (χ).

(3.2)


1723

Here dj (χ) denotes the dimension of the maximal subspace of Vχ over which χ|Γ ∩Pj acts trivially for Pj ∈ PΓ . When d = 2n + 1, the scattering operator Cχn (σn , s) has the size 2dc (χ) since σn± is un-ramified. 3.2. Scattering operators and Maass–Selberg relation Let

L2 (M) =

⊕ d(σ )Hσ ,

RM =

σ ∈M

⊕d(σ )σ

σ ∈M

be the decomposition of the right regular representation RM of M on L2 (M) where d(σ ) = dim Hσ . A similar induction procedure to the principal series representation starting with RM gives rise to a unitary representation of G, instead of σ ∈ M

⊕π(σ, λ) acts on

σ ∈M

⊕H π(σ, λ)

σ ∈M

where π(σ, λ) =

d(σ )πσ,λ d(σ )πσ,λ ⊕ d(wσ )πwσ,λ

if wσ = σ, if wσ = σ.

Here w is the nontrivial element in W (G, A). Now for Pj ∈ PΓ with the corresponding decomposition Pj = Nj Aj Mj where Pj = xj P0 xj−1 , Nj = xj N xj−1 , Aj = xj Axj−1 , Mj = xj Mxj−1 for certain xj ∈ K, the above definitions carry over to each Mj with obvious changes of notation with [τ |M : σj ] = 0, let us observe such as π(σj , λj ) for 1 j κ. For τ ∈ K H(σj , τ ) := (Hσj ,λj ⊗ Vτ )K ∼ = (Hσj ⊗ Vτ )M , and that the τ -isotypic component of H(π(σj , λj )) can be identified with the d(σj ) copies of (Hσj ⊗ Vτ )M if wσj = σj , or d(σj ) copies of ((Hσj ⊕ Hwσj ) ⊗ Vτ )M if wσj = σj . The second case happens if and only if τ = τn , σ = σn± with d = 2n + 1. For P = Pj ∈ PΓ and Φ ∈ VP ⊗ H(σ, τ ) where VP denotes the maximal invariant subspace of Vχ under χ|Γ ∩P , the Eisenstein series attached to Φ is defined as E(P , Φ, s, x) :=

χ(γ )e(s+ρ)(H (γ

−1 x))

γ ∈Γ /Γ ∩P

d −1 Φ γ −1 x for Re(s) > 2

where H (x) = Hj (x) is given by the decomposition x = Nj (x) exp(Hj (x))K(x). This is absolutely and uniformly convergent on compact sets in the half plane Re(s) > d−1 2 , and extends meromorphically to C. These facts can be proved as in [11,29]. For Pi , Pj ∈ PΓ , the constant term of E(Pi , Φ, s, x) along Pj is defined by 1 EPj (Pi , Φ, s, x) = vol(Γ ∩ Nj \ Nj )

E(Pi , Φ, s, nx) dn Γ ∩Nj \Nj

1724


and has the following expression along Pj , EPj (Pi , Φ, s, x) =

e(ws+ρ)(Hj (x)) Cjτi (w, s)Φ (x)

w∈W (Ai ,Aj )

where W (Ai , Aj ) denotes the set of all bijections w : Ai → Aj defined by wai = xai x −1 for x ∈ K and Cjτi (w, s) : VPi ⊗ H(σi , τ ) −→ VPj ⊗ H(σj , τ ),

w ∈ W (Ai , Aj ).

Now combining the operators Cjτi (xi wxj−1 , xj · s) with the nontrivial element w ∈ W (A, A) defines the scattering operator Cχτ (σ, s) on Hχ (σ, τ ) :=

κ

⊕VPj ⊗ H(σj , τ ).

j =1

When τ = τk , we denote Cχτk (σ, s) by Cχk (σ, s) for simplicity. In a natural way, we see that Hχ (σ, τ ) :=

κ

⊕VPj ⊗ H(σj , τ ) ∼ = Vc ⊗ H(σ, τ )

where Vc :=

j =1

κ

⊕VPj .

j =1

The scattering operator has a meromorphic extension over C and it satisfies the well-known functional equations Cχτ (σ, s)Cχτ (σ, −s) = Id,

Cχτ (σ, s)∗ = Cχτ (σ, s¯ ).

(3.3)

Now we analyze Rχc (h) for h ∈ C p (G) (0 < p < 1) assuming that h is of fixed τ -type. We also assume that Θσ,λ (h) = Θwσ,wλ (h) if wσ = σ . (The function hkt defined in the previous section satisfies these conditions.) Let us choose an orthonormal basis {Φmn = vm ⊗ ξn } of Hχ (σ, τ ). We put E(s, x) :=

E(Φmn , s, x),

m,n

where E(Φmn , s, x) is defined as the usual Eisenstein series E(P , Φ, s, x). Then the kernel K c (h : x, y) of Rχc (h) on Hcχ is given by K (h : x, y) = c

σ ∈M

[τ |M

d(σ ) : σ] 4π

∞

πχ (σ, λ)(h)E(iλ, x) ⊗ E(iλ, y)∗ dλ

−∞

where πχ (σ, λ) is the representation of G on Hχ (σ, τ ) defined by the π(σj , λj )’s. For Pj ∈ PΓ , the subset Cj (u) = Nj Aj (u)K ⊂ G is called a cylindrical domain where

Aj (u) = at ∈ Aj at = exp(tHj ), t u .


1725

Then there is a u0 0 such that the sets Cj (u) := p(Cj (u)) ⊂ XΓ = Γ \ G/K are disjoint to each other for u u0 and 1 j κ where p : G → XΓ denotes the natural projection. The measure in (2.3) induces the metric dt 2 + e−2t dn2 over Cj (u) where dn2 is the flat metric over (Γ ∩ Nj ) \ Nj . We put XΓ (u) := XΓ − κj =1 Cj (u).

Now we have an expansion formula of XΓ (u) tr(K c (h : x, x)) dx as u → ∞ in the following theorem which can be proved as in Section 6 of [29]. Theorem 3.1 (Maass–Selberg relation). For u u0 0, we have

tr K c (h : x, x) dx

XΓ (u)

=

σ ∈M

[τ |M

∞ ∞ d(σ ) : σ] Θσ,λ (h) dλ − Θσ,λ (h) tr Cχτ (σ, −iλ)∂iλ Cχτ (σ, iλ) dλ 2u 4π −∞

τ + π Θσ,0 (h) tr Cχ (σ, 0) + O u−1 .

−∞

4. Selberg trace formula 4.1. Trace formula For 0 < p < 1 the Selberg trace applied to h ∈ C p (G) has the following form, Tr Rχd (h) = Iχ (h) + Hχ (h) + Uχ (h) + Wχ (h) + Sχ (h) + Jχ (h)

(4.1)

where the left-hand side has the form in (3.1). The terms on the right-hand side are explained as follows. Here Iχ , Hχ , Uχ are given by the identity, hyperbolic, unipotent orbital integrals respectively. These are invariant tempered distributions on G, which were fully analyzed in [25]. First, for Iχ (h) we have Iχ (h) = dim Vχ · vol(Γ \ G) · h(e). By the Plancherel theorem, h(e) =

d ω∈G

∞ 1 d(ω)Θω (h) + Θσ,λ (h)p(σ, λ) dλ 4π σ ∈M

−∞

d and p(σ, λ) denotes the Plancherel measure. where d(ω) denotes the formal degree of ω ∈ G Let us recall that for G = SO0 (2n + 1, 1) there is no discrete series so that there are no terms d in the above formula. For G = SO0 (2n, 1), the discrete series may give a nontrivial from G contribution in general. For h = hkt , we can see that this contribution is nontrivial only when

1726


k = n, d = 2n and it is the harmonic part of n in L2 (G, Vτn ) by Theorem 3.2 in [23]. Repeating d as in Section 2.3, we see that the argument for the τ -spherical function defined for ω ∈ G Θω (hkt ) is a nonzero constant only when k = n, d = 2n and Θω (hkt ) = 0 otherwise. Now we have k Iχ ht = dim Vχ · vol(Γ \ G) · δñ (k)c(τn )

+

[τk |M :σ ]=0

1 4π

∞ e

2 −t (λ2 +( (d−1) 2 −) )

p(σ , λ) dλ

(4.2)

−∞

where δñ (k) = 1 if k = n, d = 2n, δñ (k) = 0 otherwise, and c(τn ) is a constant only depending on τn . By Theorem 3.1 in [17] and taking care of the normalization, the Plancherel measure corresponding to (πσk ,λ , Hσk ,λ ) is given by p(σk , λ) = π2

−4(n− 12 )

1 Γ n+ 2

−2 d(σk )

k n 2 2 λ + (n − j + 1)2 λ + (n − j )2

×

j =1

p(σk , λ) = π2

if d = 2n + 1,

j =k+1

−4(n−1)

−2

Γ (n) d(σk ) tanh(πλ) n−1 k 1 2 1 2 λ2 + n − j + λ2 + n − j − if d = 2n, ×λ 2 2 j =1

(4.3)

j =k+1

where σn means σn± when d = 2n + 1. The term Hχ (h) is given by

Hχ (h) =

h g −1 γ g d(Gγ g)

tr χ(γ ) · vol(Γγ \ Gγ ) ·

γ ∈Γhyp

(4.4)

Gγ \G

where Γhyp denotes the set of the Γ -conjugacy classes of the hyperbolic elements in Γ , and Γγ , Gγ denote the centralizers of γ in Γ , G respectively. We may assume that a hyperbolic element γ ∈ Γ has the form aγ mγ ∈ A+ M where A+ = {etH , t > 0}. By Section 6 in [28], we have vol(Γγ \ Gγ ) ·

h g −1 γ g d(Gγ g)

Gγ \G

=

σ ∈M

−1

l(Cγ )j (γ )

−1

D(aγ mγ )

1 tr σ (mγ ) 2π

∞ −∞

Θσ,λ (h)e−il(Cγ )λ dλ

(4.5)


1727

where l(Cγ ) denotes the length of the closed geodesic determined by γ , j (γ ) denotes the positive j (γ ) integer such that γ = γ0 with a primitive γ0 , and D(aγ mγ ) = aγρ det Ad(aγ mγ )−1 − Id|n . For h = hkt , we have 1 Hχ hkt = √ tr χ(γ ) l(Cγ0 ) 4πt γ ∈Γ [τ | :σ ]=0 h

k M

l(Cγ )2 (d−1)l(Cγ ) −1 (d−1) 2 2 × det Ad(aγ mγ )−1 − Id|n tr σ (mγ )e− 4t e−t ( 2 −) e−

ρ

(4.6)

(d−1)l(Cγ )

2 by Proposition 2.3, the Fourier integral of the Gaussian and aγ = e . The terms Uχ (h), Wχ (h) will be discussed later. The scattering term Sχ (h) and the residual term Jχ (h) have the following form,

∞ 1 Sχ (h) = [τ |M : σ ] d(σ ) Tr πχ (σ, λ)(h)Cχτ (σ, −iλ)∂iλ Cχτ (σ, iλ) dλ, 4π σ ∈M τ ∈K

−∞

1 [τ |M : σ ] d(σ ) Tr πχ (σ, 0)(h)Cχτ (σ, 0) . Jχ (h) = − 4 σ ∈M τ ∈K

For h = hkt , these term are 1 Sχ (h) = 4π

[τk |M :σ ]=0

∞ d(σ ) −∞

1 Jχ (h) = − 4

2 +( (d−1) −)2 ) 2

e−t (λ

tr Cχk (σ , −iλ)∂iλ Cχk (σ , iλ) dλ,

d(σ ) e−t (

(d−1) 2 2 −)

tr Cχk (σ, 0) ,

(4.7)

(4.8)

[τk |M :σ ]=0

which are the finite terms as |u| → ∞ on the right-hand side of the Maass–Selberg relation in Theorem 3.1 when h = hkt . 4.2. Unipotent terms By the computation in [20], the terms Uχ (h) and Wχ (h) are given by the sum over P = NAM ∈ PΓ of the following term d vol ΓP \ N (P ) lim sζP (s, χ)TP (h, s) s→0 ds under our normalization. Here the Epstein type zeta function ζP (s, χ) is defined by tr χ(η)|Xη |−(d−1)(s+1) for Re(s) > 0, ζP (s, χ) = η∈ΓP , η=e

(4.9)

1728


where η = exp(Xη ) and |Xη |2 = Xη , Xη . The other term TP (h, s) is given by 1 TP (h, s) = h knk −1 | log n|(d−1)s dk dn A(n) N K

where A(n) is the volume of the unit sphere in n. By Section 1 of [20] (and Section 7 of [29]), we know that s → TP (h, s) is holomorphic over a certain strip containing the imaginary axis. Let us remark that condition (1.1) is used in the derivation of (4.9) and will be used in the forthcoming analysis of ζP (s, χ). Now let us observe that χ|ΓP decomposes into one-dimensional representations χθ ’s of ΓP (since ΓP is abelian by (1.1)) such that χθ (η) = exp 2πi(n1 θ1 + · · · + nd−1 θd−1 )

for η =

d−1

n

ηj j

j =1

where {ηi } denotes a fixed basis of ΓP . For P ∈ PΓ , we decompose V = VP ⊕ VP ⊥ where VP ⊂ V is the maximal subspace over which χ|ΓP acts trivially, so that χ decomposes into a direct sum of IdVP and χθ ’s with nontrivial θ = (θ1 , . . . , θd−1 ), that is, one of θi is not an integer. Proposition 4.1. The Epstein type zeta function χθ (η)|Xη |−(d−1)(s+1) ζP (s, χθ ) := η∈ΓP , η=e

has a meromorphic extension over C. This meromorphic function is entire if θ is nontrivial and has a simple pole at s = 0 if θ is trivial. Proof. Since ζP (s, χθ ) is absolutely convergent for Re(s) > 0, it is enough to consider a meromorphic extension of 1 ζP (s, χθ ) = Γ (z)

∞ t

z−1

0

χθ (η)e

−t|Xη |2

dt

with z =

η∈ΓP , η=e

(d − 1) (s + 1), 2

over the left half plane Re(s) 0. By a standard argument, one can obtain such a meromorphic extension over C if we have the asymptotic expansion of 2 χθ (η)e−t|Xη | as t → 0. (4.10) η∈ΓP ,η=e

To this end, we recall the Jacobi type identity η∈ΓP

χθ (η)e

−t|Xη |2

(d−1) 2 λk −1 π = vol ΓP \ N (P ) e− 4t t λk

(4.11)


1729

where λk denotes the eigenvalues of the Laplacian θ acting on the space of the sections of the flat vector bundle defined by χθ over ΓP \ N (P ). The equality (4.11) follows by putting t = (4s)−1 at the following equality

Tr e−sθ =

(4πs)−

d−1 2

χθ (η)e−

d(n,ηn)2 4s

dn

η∈ΓP

ΓP \N (P )

where d(n, ηn) denotes the Euclidean distance given by the normalized Cartan–Killing form, which equals to |Xη |. Then θ has the zero eigenvalue if and only if θ is trivial. Hence

χθ (η)e−t|Xη | = −1 + R(t) 2

η∈ΓP , η=e (d−1)

c

c

where R(t) = O(e− t ) for a certain c > 0 as t → 0 if θ is nontrivial and t − 2 + O(e− t ) as t → 0 if θ is trivial. It follows that the meromorphic extension of ζP (s, χθ ) is entire if θ is nontrivial and has a simple pole at s = 0 if θ is trivial. 2 Now we have ζP (s, χ) = dP (χ) ·

|Xη |−(d−1)(s+1) +

η∈ΓP ,η=e

χθ (η)|Xη |−(d−1)(s+1) (4.12)

θ η∈ΓP , η=e

where dP (χ) = dim VP and the second sum runs over the nontrivial θ . The first and second sums on the right-hand side of (4.12) have a simple pole, and is regular at s = 0, respectively, by Proposition 4.1. Therefore we conclude d P TP (h) sζP (s, χ)TP (h, s) = dP (χ) CP TP (h) + RP TP (h) + C s→0 ds lim

(4.13)

where CP , RP denote, respectively, the constant term and the residue of the ordinary Epstein P denotes the sum of the constant terms of ζP (s, χθ ) with nontrivial θ zeta function at s = 0, C at s = 0, and 1 TP (h) = A(n)

h knk −1 dk dn,

N K

TP (h) =

(d − 1) A(n)

h knk −1 log | log n| dk dn.

N K

The term Uχ (h) is the sum over P ∈ PΓ of the invariant part of the right-hand side of (4.13), that is, Uχ (h) =

Pj ∈PΓ

Pj TPj (h) vol ΓPj \ N (Pj ) dPj (χ)CPj + C

(4.14)

1730


with ∞ 1 1 TPj (h) = Θσ,λ (h) dλ A(n) 2π σ ∈M

(4.15)

−∞

by Section 6 in [28]. The remaining part is

Wχ (h) =

vol ΓPj \ N (Pj ) dPj (χ)RPj TP j (h).

Pj ∈PΓ

By the computation in [3], (d − 1) vol ΓP \ N (P ) RP =1 A(n)

for P ∈ PΓ

under our normalization. Hence Wχ (h) = dc (χ)

h knk −1 log | log n| dk dn.

(4.16)

N K

5. Computation of the weighted orbital integral 5.1. Weighted orbital integral The weighted orbital integral given in (4.16) Wχ (h) = dc (χ)

h knk −1 log | log n| dk dn

(5.1)

N K

is a non-invariant tempered distribution. To explain this, let us recall that the intertwining operator JP |P (σ, λ)φ :=

φ(x n) ¯ d n¯ : Hσ,λ (P ) → Hσ,λ (P ) N

where the notation Hσ,λ (P ) denotes the principal series representation with its dependence on P . The restriction to K defines an isomorphism from Hσ,λ (P ) to

L2 (K, Hσ ) := f : K → Hσ f (mk) = σ (m)f (k), |f | ∈ L2 (K) . By this isomorphism, JP |P (σ, λ) can be regarded as a family of operators acting on L2 (K, Hσ ). Let JP (σ, λ : h) = − Tr πσ,λ (h)JP |P (σ, λ)−1 ∂iλ JP |P (σ, λ)


1731

where ∂iλ denotes the derivative under the identification (2.2) for a family of operators acting on L2 (K, Hσ ). Now we can get the invariant part of Wχ (h) by subtracting the non-invariant part as follows, h knk −1 log | log n| dk dn IP (h) = N K

∞ 1 1 n(σ ) − p.v. Θσ,0 (h) d(σ ) JP (σ, λ : h) dλ − d(σ ) 2 2π 2 σ ∈M

(5.2)

σ ∈M

−∞

where 2n(σ ) is the order of the zero of p(σ, λ) at λ = 0. Now one can consider the Fourier transform of invariant tempered distribution IP for h ∈ C 2 (G), which is expressed in terms of the discrete series and the principal series. Let Hα ∈ tC be the coroot corresponding to α ∈ ±ΣG , that is, α(Hα ) = 2, α (Hα ) ∈ Z for all α, α ∈ ±ΣG , and let Hα , (5.3) Π= α∈ΣM

which is an element of the symmetric algebra S(tmC ). We denote the simple reflection corresponding to α by sα for α ∈ ΣG . By Corollary on p. 96 of [13] (taking λP = β2 with β(Hβ ) = 2), we have Proposition 5.1. For h ∈ C 2 (G) − C02 (G) where C02 (G) is the subspace of the cusp form in C 2 (G), ∞ 1 1 IP (h) = · Ω(σ, −λ)Θσ,λ (h) dλ 2 2π

(5.4)

−∞ σ ∈M

where Ω(σ, λ) = 2d(σ )ψ(1) −

Π(sα λσ ) 1 ψ 1 + λσ (Hα ) + ψ 1 − λσ (Hα ) . (5.5) β(Hα ) 2 Π(ρM ) α∈ΣA

× ia. Here ψ is the digamma function and λσ − ρM is the highest weight of (σ, iλ) ∈ M Remark 5.2. By Lemma 5 in [3], if G = SO0 (d, 1) for d 3, the equality (5.4) still holds without any contribution from the discrete series for any h ∈ C 2 (G). 5.2. Computation for σk we use (5.2) and (5.5). To express Wχ (h) in terms of the elements in G, First let us investigate the last term on the right-hand side of (5.2). From (4.3), we have n(σk ) = 1 (0 k n − 1), n σn± = 0 if d = 2n + 1, n(σk ) = 1 (0 k n − 1)

if d = 2n.

1732


Next we consider the term given by JP (σ, λ : h) in (5.2). For a fixed irreducible representation τ , it is known that the Harish–Chandra C-function Cτ (σ, iλ) satisfies Tτ JP |P (σ, λ)−1 ∂iλ JP |P (σ, λ) = Cτ (σ, iλ)−1 ∂iλ Cτ (σ, iλ)Tτ where Tτ is the projection to τ -isotypic component of Hσ,λ . Hence, if h is of τ -type, we have JP (σ, λ : h) = −Θσ,λ (h)Cτ (σ, iλ)−1 ∂iλ Cτ (σ, iλ)

(5.6)

when [τ |M : σ ] = 0. By Theorem 8.2 in [4], we can derive the following equalities: (1) When d = 2n + 1, 1 1 1 ∂iλ log Cτk (σk , iλ) = − + ··· + , iλ + n − k iλ iλ + n 1 1 1 ∂iλ log Cτk (σk−1 , iλ) = − + ··· + iλ − n + k − 1 iλ iλ + n

(5.7)

where σn means σn± . (2) When d = 2n, ∂iλ log Cτk (σk , iλ) = ∂iλ log Cτk (σk−1 , iλ) =

1 iλ + n − k −

1 2

1 iλ − n + k −

1 2

1 + 2 log 2, + ψ(iλ) − ψ iλ + n + 2 1 + ψ(iλ) − ψ iλ + n + + 2 log 2 (5.8) 2

where τn means τn± . Now the remaining task to compute Wχ (h) is to obtain an explicit form of Ω(σk , λ) which express IP (h) in terms of the principal series. Theorem 5.3. For the representations σk of SO(d − 1) for 0 k [ d−1 2 ], we have d −1 d −1 d(σk ) ψ iλ − + ψ −iλ − + ψ(iλ + 1) + ψ(−iλ + 1) 2 2 2 k−1 d−1 (−1)k ( d−1 j j +1 2 − k) + (−1) d(σj ) + (−1) d(σj ) − Pkd (λ) 2 λ2 + ( d−1 − k) 2 j =0 j =k+1

Ω(σk , λ) = −

where σn denotes σn± if d = 2n + 1 and Pkd (λ) is an even polynomial of 2n − 4 degree for d = 2n + 1 5 or d = 2n 4 and a constant for d = 3, 2. The proof of this theorem will be given in Appendix A.


1733

For τk , σ with [τk |M : σ ] = 0, we put Φ(τk , σ , λ) := −d(σ )

1 ∂iλ log Cτk (σ , iλ) − ∂iλ log Cτk (σ , −iλ) . 2

Then by equalities (5.7), (5.8) and Theorem 5.3, denoting a possibly different polynomial by the same notation Pkd (λ) (the change happens in the constant term of Pkd (λ) only when d = 2n) we have Corollary 5.4. The following equalities hold, (−1)k ( d−1 2 − k) Ω(σk , λ) + Φ(τk , σk , λ) = −d(σk ) ψ(iλ + 1) + ψ(−iλ + 1) + d−1 2 λ + ( 2 − k)2 k−1 d−1 j j +1 × (−1) d(σj ) + (−1) d(σj ) − Pkd (λ), j =k

j =0

(−1)k ( d−1 2 − k) Ω(σk , λ) + Φ(τk+1 , σk , λ) = −d(σk ) ψ(iλ + 1) + ψ(−iλ + 1) + d−1 2 λ + ( 2 − k)2 k d−1 × (−1)j d(σj ) + (−1)j +1 d(σj ) − Pkd (λ). j =0

j =k+1

6. Zeta regularized determinant for hyperbolic manifolds with cusps Now let us recall that the heat operator e−tk over XΓ is not of trace class, so that we can not take its usual trace. To overcome this, we follow the idea of Melrose in [16] as follows. If the heat operator e−tk would be of trace class, then its trace is the same as XΓ tr(e−tk (x, x)) dx, although this integral diverges in our case. However, we could remove the diverging part of the expansion of

tr e−tk (x, x) dx

as u → ∞,

XΓ (u)

by Theorem 3.1 and define the regularized trace Trr (·) of e−tk to be the remaining finite part of it. Then we have Trr e−tk =

[τk |M :σ ]=0

λj ∈σp (k )

d(σ ) − 4π

e−tλj + ∞ −∞

e−t (λ

2 +d 2 )

d(σ ) −td 2 k e tr Cχ (σ , 0) 4

tr Cχk (σ , −iλ)∂iλ Cχk (σ , iλ) dλ

(6.1)

1734


where σp (k ) denotes the point spectrum of k , d = ( d−1 2 − ), d(σ ) = dim(Vσ ). Let us observe that the right-hand side of (6.1) is the same as the geometric side of the Selberg trace formula applied to the test function hkt over G, so that Trr e−tk = Iχ hkt + Hχ hkt + Uχ hkt + Wχ hkt

(6.2)

by (4.1). Using this regularized trace, let us define the spectral zeta function of k by 1 ζk (s) := Γ (s)

∞

1 + 0

t s−1 Trr e−tk − Pk dt

(6.3)

1

where Pk denotes the orthogonal projection onto kerL2 (k ). Here the small and large time inte 1 ∞ grals 0 , 1 are defined for Re(s) 0 and Re(s) 0 respectively. This decomposition of the integral over the small and large times is needed when d = 2n + 1, k = n since the continuous spectrum of k reaches zero, that is, the heat operator e−tk does not decay exponentially as t → ∞. To state the theorem on the meromorphic extension of ζk (s), we introduce a notation d(d, k) := (−1)

k−1

k−1 d −1 d −2 j (−1) d(σj ) = − . k k j =0

Then we have −2 d(d, k) = −2 (−1)

k−1

= (−1)k

k−1

k−1 j (−1) d(σj ) j =0

d−1 (−1)j d(σj ) + (−1)j +1 d(σj ) , j =k

j =0

which appeared in Corollary 5.4. Now we have Theorem 6.1. The spectral zeta function ζk (s) = ζd−k (s) has a meromorphic extension over C, which has the following form if d = 2n Γ (s)ζk (s) =

∞ j =−n

aj + s+j

∞

aj

j =−(n−2)

s+j −

1 2

+

∞ j =0

bj (s + j −

1 2 2)

−

βk + δn (k)ηe + He (s), s

and if d = 2n + 1 Γ (s)ζk (s) =

∞ j =−n

aj s+j −

+ δn (k)

∞ j =0

1 2

+

∞

bj

j =0

(s + j − 12 )2

cj s−j −

1 2

−

βk + δn (k)ηo + Ho (s) s


1735

for some constants aj , aj , bj , cj where βk = dim kerL2 (k ), δn (k) equals 1 if k = n or k = (d − n) and vanishes otherwise, ηe := dc (χ)d(d, n),

ηo := −

1 dc (χ)d(d, n − 1) 2

and He (s), Ho (s) are entire functions. In particular, ζk (s) is regular at s = 0.

∞ Proof. Let us first deal with the large time contribution 1 dt in (6.3). The continuous spectrum 2 of k is given by the union of the half intervals [( d−1 2 − ) , ∞) for = k, k − 1, hence the bottom of the continuous spectrum of k does not reach zero unless d = 2n + 1 and = k = n. Equivalently, Trr (e−tk ) decays exponentially as t → ∞ for other cases, which we can see easily from the right-hand side of (6.1). Therefore, the large time contribution to the meromorphic extension is trivial unless d = 2n + 1 and = k = n. Now for this case, we observe the following expansion at λ = 0, ∞ a2j λ2j , tr Cχn (σn , −iλ)∂iλ Cχn (σn , iλ) = j =0

which follows from (3.3). From this, we see that 1

∞ 1 2 e−tλ tr Cχn (σn , −iλ)∂iλ Cχn (σn , iλ) dλ ∼ bj t −(j + 2 )

as t → ∞.

(6.4)

j =0

−1

The corresponding integrals over (−∞, 1]λ ∪ [1, ∞)λ decay exponentially as t → ∞. For d = 2n + 1, σn is un-ramified so that tr(Cχn (σn , 0)) = 0. Hence the residual term vanishes for this case.

∞ Now the expansion (6.4) and these facts imply that the large time integral 1 is well defined for Re(s) < 12 and extends meromorphically to the whole complex plane with the following form ∞

∞ t s−1 Trr e−tn − Pn dt =

cj

s−j − j =0

1

1 2

+ H1 (s)

(6.5)

for some constants cj and a holomorphic function H1 (s).

1 Next to deal with the small time integral 0 · dt, we use the right-hand side of equality (6.2). For Iχ (hkt ), we separate the cases d = 2n and d = 2n + 1. First, if d = 2n recall that the Plancherel measure P (σ , λ) is a sum of λ2k+1 tanh(πλ) with 0 k n − 1 from (4.3). Now we observe ∞ e −∞

−tλ2 2k+1

λ

∞ tanh(πλ) dλ = (−1)k ∂tk

e−tλ λ tanh(πλ) dλ 2

−∞

∞ = (−1)k ∂tk 0

√ e−tx tanh(π x ) dx

1736


= (−1)k ∂tk

t

−1 π

∞ e

2

−tx

√ cosh−2 (π x) dx √ x

0

= (−1)k ∂tk

t

−1 π

2

∞ ∞ 0 j =0

(−tx)j j!

√ cosh−2 (π x) dx . √ x

(6.6)

Hence if d = 2n, we conclude ∞ 2 Iχ hkt = e−td · aj t −n+j + δñ (k)c(Γ, ˜ χ, τn )

(6.7)

j =0

=k,k−1

where aj are some constants and δñ (k)c(Γ, ˜ χ, τn ) is the contribution from the first term on the right-hand side of (4.2). Second, if d = 2n + 1 recall that the Plancherel measure p(σ , λ) is a polynomial of order 2n from (4.3). Hence we can easily see n 1 2 e−td · aj t −n+j − 2 Iχ hkt =

(6.8)

j =0

=k,k−1

for some constants aj . For HΓ (hkt ), by (4.6), we have c2 Hχ hkt ∼ ae− 4t

as t → 0

(6.9)

for a constant a and c := min{γ :hyperbolic} l(Cγ ) is a positive real number. For Uχ (hkt ), by (4.14) and (4.15) we can see that this can be dealt as Iχ (hkt ) for d = 2n + 1 and it consists of the terms with j = n in (6.8). For Wχ (hkt ), by Corollary 5.4, we have dc (χ) Wχ hkt = 4π

∞

e−t (λ

2 +d 2 )

P (λ) + Q (λ) + Rk (λ) dλ.

(6.10)

=k,k−1−∞

(There may be an additional constant from the discrete series on the right-hand side of (6.10) when d = 2, k = 1.) Here P (λ) is an even polynomial of degree at most (2n − 4) for d = 2n + 1 or d = 2n, Q (λ) = −d(σ ) ψ(iλ + 1) + ψ(−iλ + 1) ,

Rk (λ) = −(−1)k− 2 d(d, k)

λ2

d . + d2

It is easy to see that the contribution of P (λ) is just the same as (6.8) replacing n by (n − 2). For Q (λ), we use the following asymptotic expansion ∞

ψ(z + 1) ∼ log z +

B2k 1 − 2z (2k)z2k k=1

as z → ∞,


1737

where the B2k ’s are Bernoulli numbers, to obtain ∞

e−tλ Q (λ) dλ ∼ 2

∞

1

1

aj t j − 2 + b0 t − 2 log t

as t → 0

(6.11)

j =0

−∞

for some constants aj and b0 . By an elementary computation, d dt

∞ e

√ d π 2 dλ = −d √ e−td , 2 2 t λ + d

−t (λ2 +d2 )

−∞

(6.12)

which implies ∞

e−t (λ

2 +d 2 )

−∞

∞ 1 d dλ = π + aj t j + 2 2 2 λ + d j =0

(6.13)

for some constants aj . By (6.8), (6.9), (6.11), (6.13), and the Taylor expansion of e−td at t = 0, if d = 2n 2

Iχ hkt + Hχ hkt + Uχ hkt + Wχ hkt ∼ −δn (k)ηe +

∞

∞

aj t j +

j =−n

1

aj t j − 2 +

∞

1

bj t j − 2 log t

as t → 0

j =0

j =−(n−2)

for some constants aj , aj , bj where ηe = dc (χ)d(d, n), and if d = 2n + 1 Iχ hkt + Hχ hkt + Uχ hkt + Wχ hkt ∞

∼ −δn (k)ηo +

1

aj t j − 2 +

j =−n

∞

1

bj t j − 2 log t

as t → 0

j =0

1 for constants aj , bj where ηo = − 12 (dc (χ)d(d, n − 1)). Therefore the small time integral 0 is well defined for Re(s) > d2 and extends meromorphically on C with the following form if d = 2n Γ (s)ζk (s) =

∞ j =−n

aj + s +j

∞ j =−(n−2)

aj s +j −

1 2

+

∞ j =0

bj (s + j −

1 2 2)

−

βk + δn (k)ηe + H2 (s), s

and if d = 2n + 1 Γ (s)ζk (s) =

∞ j =−n

aj s +j −

1 2

+

∞

bj

j =0

(s + j − 12 )2

−

βk + δn (k)ηo + H2 (s) s

1738


for some (new) constants aj , aj , bj and a holomorphic function H2 (z). For d = 2n + 1 and k = n, combining this and (6.5) completes the proof. 2 By Theorem 6.1, we can define the regularized determinant of k by d detζ k := exp − ζk (s) ds s=0 and the analytic torsion T (XΓ , ρ) by T (XΓ , χ) :=

detζ 1 (detζ 3 )3 d d+1 · . . . (detζ d−1 )(−1) (d−1) · (detζ d )(−1) d . 2 (detζ 2 ) (detζ 4 )4

Note that our definition of analytic torsion is a generalization of the original one given in [24], which reduces to (the square of) the original one in [24] when XΓ is compact. We also remark that a similar definition of the analytic torsion was introduced by Hassell in [12] using the b-trace. For a hyperbolic manifold XΓ with cusps, T (XΓ , χ) is nontrivial even if d = 2n as we will see in Section 8. In the following section, we will relate T (XΓ , χ) with the leading coefficient Rχ∗ (0) of the Ruelle zeta function Rχ (s) at s = 0. To do this, we will need following expression of Rχ (s) (for instance, see [6, p. 532]), Rχ (s) =

d−1

Zχ (σk , s + k)(−1)

k+1

(6.14)

k=0

in terms of the Selberg zeta function Zχ (σ , s) defined by d−1 Zχ (σk , s) := exp − tr χ(γ )j (γ )−1 D(γ )−1 tr σk (mγ )e−(s− 2 )l(Cγ )

(6.15)

γ ∈Γhyp

for Re(s) > d − 1. Here we may assume that γ is conjugate to aγ mγ ∈ A+ M and D(γ ) = D(aγ mγ ). We also put Zχ (σn , s) = Zχ (σn+ , s) · Zχ (σn− , s) when d = 2n + 1. By Theorem 4.6 in [10], the Selberg zeta function Zχ (σ , s) has a meromorphic extension over C. 7. Proof of Theorem 1.1 Throughout this section, we assume d = 2n + 1. First, taking the Mellin transform M(·) of equality (6.2), we have M Trr e−tk − βk = MIχ hkt + MHχ hkt + MUχ hkt + MWχ hkt − M(βk ). (7.1) For the left-hand side of (7.1), we have Lemma 7.1. The following equality holds, lim M Trr e−tk − βk (s) + Γ (s) βk + δn (k)ηo = ζ k (0). s→0


1739

Proof. From the definition, we have ζ k (0) = lim

s→0

1 ζk (s) − ζk (0) = lim M Trr e−tk − βk − Γ (s)ζk (0) . s→0 s

Moreover, by Theorem 6.1, ζk (0) = −βk − δn (k)ηo . Our lemma now follows easily.

2

Now we want to obtain the explicit form of each term at s = 0 on the right-hand side of (7.1). First, the last term M(βk ) is defined by 1 M(βk )(s) =

∞ t

s−1

βk dt +

0

t s−1 βk dt 1

where the first (second) term on the right-hand side has a meromorphic extension from the half plane with Re(s) 0 (Re(s) 0). These terms equal βsk and − βsk , respectively so that M(βk )(s) ≡ 0.

(7.2)

The term MIχ (hkt ) is computed in Lemma 3 on p. 533 in [6]. Let us present some details of these computations for completeness. By Proposition 2.3, (4.2) and (4.3), ∞ k 2 −t (n−)2 1 Iχ ht = dimVχ · vol(Γ \ G) e e−tλ p(σ , λ) dλ 4π =k,k−1

−∞

where p(σ , λ) is an even polynomial. Now for the integral part, each monomial can be treated as ∞ e

∞ 1 d a 2 λ dλ = − e−tλ dλ = ba t −a− 2 dt

−tλ2 2a

−∞

where ba =

(7.3)

−∞

√ 1 3 k π 2 · 2 . . . 2a−1 2 so that MIχ (ht )(s) consists of ∞

Ea (s) := ba

t

s−a− 32 −t (n−)2

e

1 1 (n − )−2(s−a− 2 ) dt = ba Γ s − a − 2

0

for Re(s) > a + 12 . When < n, MIχ (hkt ) has a meromorphic extension over C and

(7.4)

1740


1 (n − )2a+1 Ea (0) = ba Γ −a − 2 = (−1)

a+1

n− (iλ)2a dλ.

2π (n − )2a+1 = −2π 2a + 1

0

When = n, we split the integral defining Ea (s) as we did for M(βk ), 1 ∞ 3 s−a− 32 dt + ba t s−a− 2 dt Ea (s) = ba t 0

1

where the first (second) term on the right-hand side has a meromorphic extension from the half plane with Re(s) 0 (Re(s) 0). These terms equal ba 1 and − ba 1 , respectively, so that s−a− 2

s−a− 2

Ea (s) ≡ 0 if = n. In conclusion, 1 MIχ hkt (0) = − dim Vχ · vol(Γ \ G) 2

n− p(σ , iλ) dλ.

=k,k−1 0

The term Uχ (hkt ) and the part with P (λ) in (6.10) denoted by Wχ1 (hkt ) can be dealt in the same way as we did for Iχ (hkt ) and we have 1 M Iχ hkt + Uχ hkt + Wχ1 hkt (0) = − 2

n− (iλ) dλ. P

(7.5)

=k,k−1 0

Here (s) = dimVχ · vol(Γ \ G)p(σ , s) − dc (χ)P d (s) + C(χ, k) P

(7.6)

where C(χ, k) is a constant from Uχ (hkt ), which is determined by (4.14), (4.15). For MHχ (hkt ), first we recall ∞ −s 1 s−1 x(x + 2c) e−x(x+2c)t (2x + 2c) dx for Re(s) < 0, = t Γ (1 − s) 0

where c > 0 following [6,18]. Now using ∞ l2 1 2 e−x(x+2c)t (2x + 2c) √ e− 4t e−tc dt = e−l(x+c) 4πt 0

and putting c = (n − ), l = l(Cγ ) in (4.6), we have MHχ (hkt )(s) =

=k,k−1

1 Γ (1 − s)

∞ −s d log Zχ (σ , 2n − + x) dx (7.7) x x + 2(n − ) dx 0


1741

for Re(s) < 0. By Theorem 4.6 in [10], the Selberg zeta function Zχ (σ , s) has a meromorphic extension over C. In particular, it follows that Zχ (σ , 2n − + x) has the following form near x = 0, (7.8) Zχ (σ , 2n − + x) = Z2n− x −r2n− 1 + O(x) where r2n− denotes the order of singularity of Zχ (σ , s) at s = 2n − . By Theorem 2.1 of [9] or Theorem 4.6 of [10], if = n, −α r2n− = (7.9) −2αn if = n, where αk := βk − βk−1 + βk−2 − · · · ± β0 . Using this, the integral part on the right-hand side of (7.7) can be analyzed as ∞ −s d log Zχ (σ , 2n − + x) dx x x + 2(n − ) dx 0

∞ =

d log Zχ (σ , 2n − + x) dx dx

+ O(s) + O() − r2n−

−s x −s−1 x + 2(n − ) dx

(7.10)

0

where x

−s−1

−s x + 2(n − ) dx =

0

log(2(n − )) − log −

1 2s

1 s

+ O(s)

+ O(s) + O( 1−|s| )

if < n, if = n.

(7.11)

From (7.7), (7.9), (7.10) and (7.11), for small s < 0, 1 k α MHχ ht (s) = − log Z2n− − r2n− log 2(n − ) − + O(s) + O 2 . s =k,k−1

Hence we obtain lim MHχ hkt (s) − βk Γ (s) = − log Z2n− + α log 2(n − ) s→0

(7.12)

=k,k−1

where the term log(2(n − )) disappears when = n. To analyze the remaining terms of Wχ (hkt ) in (6.10), let us denote the corresponding parts of Wχ (hkt ) with Q (λ), R (λ) for = k, k − 1 by Wχ2 (hkt ), Wχ3 (hkt ) respectively, that is,

1742


Wχ2

k dc (χ) ht = − 4π

Wχ3

∞

d(σ )

=k,k−1

e−t (λ

2 +(n−)2 )

ψ(iλ + 1) + ψ(−iλ + 1) dλ,

−∞

k dc (χ)d(d, k) ht = − 2π

∞

k−

(−1)

=k,k−1

−∞

e−t (λ

2 +(n−)2 )

(n − ) dλ. λ2 + (n − )2

Now, we deal with Wχ2 (hkt )(0). Lemma 7.2. The following equality holds, MWχ2 hkt (0) = dc (χ) d(σ ) log Γ (n − + 1) + C =k,k−1

where C is a constant which does not depend on . Proof. For c ∈ R and Re(s) 0, let ∞ fc (s) =

∞ t

s−1

e−t (λ

2 +c2 )

ψ(iλ + 1) + ψ(−iλ + 1) dλ dt.

−∞

0

It can be shown that the c-family of functions fc (s) extends meromorphically over C and that fc (s) is regular over C − { 12 , − 12 , − 32 , . . .} in the same way as the proof of Theorem 6.1. Denoting by fc (s) the derivative of fc (s) with respect to c, for Re(s) 0, fc (s) = −2c

∞

∞ t

0

s

e−t (λ

2 +c2 )

ψ(iλ + 1) + ψ(−iλ + 1) dλ dt = −2cf (s + 1),

−∞

which also holds over C − { 12 , − 12 , − 32 , . . .} by the meromorphic extension. In particular, fc (0) is smooth for c ∈ R, and fc (0) = −2c

∞ ∞

e−t (λ

2 +c2 )

ψ(iλ + 1) + ψ(−iλ + 1) dλ dt

0 −∞

∞ = −2c −∞

λ2

∞ = −2i −∞

1 ψ(iλ + 1) + ψ(−iλ + 1) dλ 2 +c

1 1 ψ(iλ + 1) dλ = −4πψ(1 + c). − λ + ic λ − ic

From this formula, we see that fc (0) = −4π log Γ (1 + c) + a for a constant a for c ∈ [0, ∞). Applying this to the formula of Wχ2 (hkt ), we obtain the expected equality. 2


1743

Next, for Wχ3 (hkt )(0) we have Lemma 7.3. The following equality holds lim

s→0

k ηo ht (s) + δn (k) s

MWχ3

= dc (χ)d(d, k) log(n − k) − log(n − k + 1)

where the right-hand side is trivial if k = n. Proof. For c ∈ (0, ∞), we put ∞ F (t) =

e−t (λ

2 +c2 )

−∞

λ2

c dλ. + c2

By (6.12), we have √ d π 2 F (t) = −c √ e−tc . dt t Hence, for Re(s) 0, ∞ t

s−1

1 F (t) dt = − s

0

∞ √ π 2 t s −c √ e−tc dt t 0

√ ∞ √ c π π −2s 1 s− 12 −tc2 = c . t e dt = Γ s+ s s 2 0

This implies √ π dc (χ)d(d, k) MWχ3 hkt (s) = − 1 − 2 log(n − )s + O s 2 (−1)k− 2π s =k,k−1 √ × π + Γ (1/2)s + O(s) π dc (χ)d(d, k) − 2π log(n − ) + πψ(1/2) + O(s) , (−1)k− =− 2π s =k,k−1

which completes the proof.

2

Combining (7.5), (7.12) and Lemmas 7.1, 7.2, and 7.3, detζ k =

=k,k−1

n− −α 1 (iλ) dλ Z2n− 2(n − ) exp P 2 0 (−1)k−+1 dc (χ)d(d,k)

× (n − )

Γ (n − + 1)−dc (χ)d(σ ) e−dc (χ)d(σ )C

(7.13)

1744


where the terms (2(n − ))−α disappear if = n. Let us remark that the order (−1)k−+1 of (n − ) depends on both k and and this is due to the non-invariant property of the weighted orbital invariant. From Theorem 2.2 of [9] or Theorem 4.14 of [10], we have Proposition 7.4. For s ∈ C, the following equalities hold Zχ (σk , s + k)Γ (s − n + k + 1)−dc (χ) d(σk ) s −dc (χ) d(d,k) −d (χ) d(d,k) = Zχ (σk , 2n − k − s)Γ (n − k − s + 1)−dc (χ) d(σk ) 2(n − k) − s c s+k−n k d(σk ) k −d(σk ) k (iz) dz , × det Cχ (σk , n − k − s) det Cχ (σk , 0) exp − P 0 −2dc (χ) d(σn )

Zχ (σn , s + n)Γ (s + 1)

= Zχ (σn , n − s) · Γ (−s + 1)−2dc (χ) d(σn )

s − 2Pn (iz) dz .

× det Cχn (σn , −s)d(σn ) det Cχn (σn , 0)−d(σn ) exp

0

By Proposition 7.4 and recalling Sχ () :=

lim

s→−(n−)

= (−1)b

(s + n − )−b det Cχ (σ , s) lim

s→(n−)

−1 (s − n + )b det Cχ (σ , s) ,

we obtain n− (iλ) dλ Z2n− exp P 0

d (χ)d(d,) d(σ ) = Z det Cχ (σ , 0)Sχ () 2(n − ) c dc (χ)d(σ ) d (χ)d(σ ) (n − − 1)! × (n − )! c (−1)α +(n−−1)dc (χ)d(σ ) where we also used the fact resz=−n Γ (z) = (detζ k )2 =

(−1)n n! .

(7.14)

Combining (7.13) and (7.14),

−2α +dc (χ)d(d,) d(σ ) 2(n − ) Z2n− Z det Cχ (σ , 0)Sχ ()

=k,k−1

× (n − )−dc (χ)(d(σ )+(−1)

k− 2d(d,k))

e−2dc (χ)d(σ )C (−1)α +(n−−1)dc (χ)d(σ ) . (7.15)

Using detζ k = detζ 2n+1−k , we have T (XΓ , χ) = (detζ 0 )2n+1 · (detζ 1 )−(2n−1) · (detζ 2 )(2n−3) . . . (detζ n )(−1) . (7.16) n


1745

From (6.14), we also have −1 Z0 Z2 Z2n−2 n (−1)n+1 (−1)n−1 = . . . Zn−1 Zn(−1) Zn+1 ... Z2n . lim s N0 Rχ (s) s→0 Z1 Z3 Z2n−1 Now combining this and (7.15), (7.16) and recalling det Cχ (σ , 0) = ±1, finally we conclude that the following equality holds up to sign, −1 lim s N0 Rχ (s) = C(XΓ , χ) · C(d)dc (χ) · S(XΓ , χ) · T (XΓ , χ)

(7.17)

s→0

Here C(XΓ , χ) :=

n−1

(−1)k αk −4(n − k)2 ,

S(XΓ , χ) :=

k=0

n−1

Sχ (k)(−1)

k+1 d(σ

k)

k=0

and C(d) :=

n−1

2(−1)

k+1 d(d,k)

· (n − k)(−1)

k (2d(σ

k )(n−k)+d(d,k))

.

k=0

Note that the terms e−2dc (χ)d(σ )C ’s are combined to be 1 by the equality n (−1) d(σ ) = 0. =0

8. Proof of Theorem 1.4 Throughout this section, we assume d = 2n. As in the odd-dimensional case, we start with M Trr e−tk − βk = MIχ hkt + MHχ hkt + MUχ hkt + MWχ hkt − M(βk ). The Mellin transform of each term except Iχ (hkt ) of this equality can be treated as in the odddimensional case. For Iχ (hkt ), we have Lemma 8.1. When d = 2n, the following equality holds, dimVχ · vol(Γ \ G) MIχ hkt (s) = 22(2n−1) Γ (n)2 ×

=k,k−1

d(σ )

n−1 j =0

Γ (s) bj s−j −1

∞ −2 √ 2 −(s−j −1) π cosh (π x) x + d dx √ 2 x 0

where the bj ’s are given by p(σ , λ) = π2−4(n−1) Γ (n)−2 d(σk )λ tanh(πλ)

n−1

j =0 bj (λ

2

+ d2 )j .

1746


Proof. As in the derivation of (6.6), for Re(s) 0 we obtain ∞

∞ t

s−1

e−t (λ

2 +d 2 )

j λ tanh(πλ) λ2 + d2 dλ dt

−∞

0

∞ = (s − 1) · . . . · (s − j )

t 0

=

Γ (s) s −j −1

s−j −2

∞

e−t (x+d ) 2

√ π cosh−2 (π x) dx dt √ 2 x

0

√ −(s−j −1) π cosh−2 (π x) dx, x + d2 √ 2 x

∞ 0

which, by analytic continuation, also holds for s ∈ C. Since the term c(Γ, ˜ χ, τn ) in (6.7) is a constant with respect to t, its Mellin transform vanishes as in (7.2). Then this completes the proof. 2 By Lemma 8.1, as also expected from Theorem 6.1, the limit of MIχ (hkt ) as s → 0 does not exist by itself and we need to remove the simple pole of MIχ (hkt ) at s = 0. Lemma 8.1 immediately implies Proposition 8.2. lim MIχ hkt (s) − Γ (s)ak = dimVχ · vol(Γ \ G)a(n, k)

s→0

where ak is the residue of MIχ (hkt )(s) at s = 0 and a(n, k) is a constant that is independent of Γ , but depends only on G. Following [7], we write a∼b

if a = exp c · dimVχ · vol(Γ \ G) b

(8.1)

for a constant c that is independent of Γ . We can proceed as in the odd-dimensional case and obtain

detζ k ∼

Z2d0 −

=k,k−1

d0 − −α 1 (iλ) dλ 2(d0 − ) exp P 2 0

(−1)k−+1 dc (χ)d(d,k)

× (d0 − )

Γ (d0 − + 1)−dc (χ)d(σ ) e−dc (χ)d(σ )C

(8.2)

where Z2d0 − denotes the leading coefficient of the Laurent expansion of Zχ (σ , 2d0 − + x) d at x = 0 as in (7.8), d0 = d−1 2 , P (s) = −dc (χ)P (s) + C(χ, k) with the constant C(χ, k) from k Uχ (ht ), which is determined by (4.14), (4.15). Now by Theorem 4.14 of [10],


1747

Proposition 8.3. For s ∈ C, the following equality holds Zχ (σk , s + k)Γ (s − d0 + k + 1)−dc (χ)d(σk ) s −dc (χ) d(d,k) Γd (σk , s + k) = Zχ (σk , 2d0 − k − s)Γ (d0 − k − s + 1)−dc (χ)d(σk ) −d (χ) d(d,k) × 2(d0 − k) − s c Γd (σk , 2d0 − k − s) × det Cχk (σk , d0

− k − s)

d(σk )

det Cχk (σk , 0)−d(σk ) exp

s+k−d 0

k (iz) dz P

− 0

where Γd (σk , s) =

k

(−1) ( d ) k− Γd (s − )Γd (s + + 1)

− dim Vχ E(XΓ ) .

=0

Here Γd (s) is the multiple gamma function of order d introduced in [15] and E(XΓ ) denotes the Euler characteristic of XΓ . From Proposition 8.3, d0 − (iλ) dλ P Z2d0 − exp 0

d(σ ) d (χ)d(d,) ∼ Z det Cχ (σ , 0)Sχ () 2(d0 − ) c × Γ (d0 − + 1)dc (χ)d(σ ) Γ (−d0 + + 1)−dc (χ)d(σ ) (−1)α

(8.3)

where Sχ () :=

lim

s→−(d0 −)

(s + d0 − )−b det Cχ (σ , s).

The ambiguity ‘∼’ in (8.3) comes from the constant term of the Laurent expansion of Γd (σ , 2d0 − s)Γd (σ , s)−1 at s = and the following equality given in Proposition 4.4 of [9], vol(Γ \ G)

2n − 1 = (−1)n E(XΓ ). n 24n−3 n

Combining (8.2) and (8.3), (detζ k )2 ∼

d(σ ) Z2d0 − Z det Cχ (σ , 0)Sχ ()

=k,k−1

−2α +dc (χ)d(d,) −d (χ)d(σ ) Γ (d0 − + 1)Γ (−d0 + + 1) c × (−1)α 2(d0 − ) × (d0 − )(−1)

k−+1 2d (χ) d(d,k) c

e−2dc (χ)d(σ )C .

(8.4)

1748


Using detζ k = detζ 2n−k , we have T (XΓ , χ) = (detζ 0 )−2n · (detζ 1 )2n · (detζ 2 )−2n . . . (detζ n−1 )(−1) · (detζ n )(−1)

n+1 n

n 2n

(8.5)

. k−+1

2dc (χ)d(d,k) By (8.4) and the symmetry in (8.5), all terms cancel except the term (d0 − )(−1) if we plug (8.4) into (8.5). For instance, the leading terms Z of Zχ (σ , ) combined to

(Z0 Z2n−1 )−n · (Z1 Z2n−2 )n · (Z0 Z2n−1 )n . . . (Zn−1 Zn )(−1) · (Zn−1 Zn )(−1)

n+1 n

nn

· (Zn−2 Zn+1 )(−1)

= 1.

(8.6)

The combination of terms (d0 − )(−1)

T (XΓ , χ) ∼

nn

k−+1 2d (χ)d(d,k) c

n−1

results in dc (χ)

2n−2 (−1)k n((2n−1 k+1 )−( k ))

(d0 − k)

.

k=0

Recalling the definition in (8.1), dc (χ) n−1 2n−2 (−1)k n((2n−1 − ) ) ( ) k+1 k (d0 − k) (8.7) T (XΓ , χ) = exp a(G)dimVχ · vol(Γ \ G) k=0

where a(G) is a constant depending only on G, not on Γ . Now let us observe that the equality (8.7) still holds with dc (χ) = 0 if XΓ is compact, that is, 1 = T (XΓ , χ) = exp a(G)dimVχ · vol(Γ \ G) for any co-compact discrete group Γ ⊂ G. Hence it follows that the constant a(G) = 0. Finally we conclude

T (XΓ , χ) =

n−1

dc (χ) 2n−2 (−1)k n((2n−1 k+1 )−( k ))

(d0 − k)

.

(8.8)

k=0

Acknowledgments The author wishes to thank P. Loya for some comments which improved the exposition of this paper. He also wishes to thank the referee for pointing out some minor errors in the first version and helpful suggestions concerning general overview of this paper.


1749

Appendix A. Proof of Theorem 5.3 A.1. Odd-dimensional case: d = 2n + 1 The case of n = 1 can be computed as in the cases n 2. Hence we assume that n 2 in the ± following proof. The highest weights μk , μ± n of the representations σk , σn of M = SO(2n) ⊂ K = SO(2n + 1) are given by μk = e2 + e3 + · · · + ek+1

(0 k n − 1),

μ± n = e2 + e3 + · · · + en ± en+1 .

Recalling ρM = (n − 1)e2 + (n − 2)e3 + · · · + en , we have λσk = iλe1 + μk + ρM = iλe1 + ne2 + (n − 1)e3 + · · · + (n − k + 1)ek+1 + (n − k − 1)ek+2 · · · + en , λσn± = iλe1 + μ± n + ρM = iλe1 + ne2 + (n − 1)e3 + · · · + 2en ± en+1 . First we consider Π(sα λσ ) for α ∈ ΣA , which are given by e1 − e , e1 + e for 2 n + 1. Then we have s(e1 −e ) (iλe1 + μk + ρM ) ⎧ iλe + ne2 + · · · + (n − + 2)e1 + · · · + (n − k + 1)ek+1 ⎪ ⎪ ⎪ ⎪ ⎨ + (n − k − 1)ek+2 + · · · + en = iλe + ne2 + · · · + (n − k + 1)ek+1 ⎪ ⎪ + (n − k − 1)ek+2 + · · · + (n − + 1)e1 + · · · + en ⎪ ⎪ ⎩ iλen+1 + ne2 + · · · + (n − k + 1)ek+1 + (n − k − 1)ek+2 + · · · + en s(e1 −e ) iλe1 + μ± n + ρM iλe + ne2 + · · · + (n − + 2)e1 + · · · + 2en ± en+1 if 2 n, = iλen+1 + ne2 + · · · + 2en ± e1 if = n + 1, s(e1 +e ) (iλe1 + μk + ρM ) ⎧ −iλe + ne2 + · · · − (n − + 2)e1 + · · · + (n − k + 1)ek+1 ⎪ ⎪ ⎪ ⎪ + (n − k − 1)ek+2 + · · · + en ⎪ ⎪ ⎨ −iλej + ne2 + · · · + (n − k + 1)ek+1 = + (n − k − 1)ek+2 + · · · − (n − + 1)e1 + · · · + en ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −iλen+1 + ne2 + · · · + (n − k + 1)ek+1 + (n − k − 1)ek+2 + · · · + en

if 2 k + 1, if k + 2 n, if = n + 1,

if 2 k + 1, if k + 2 n, if = n + 1,

1750


s(e1 +e ) iλe1 + μ± n + ρM −iλe + ne2 + · · · − (n − + 2)e1 + · · · + 2en ± en+1 = −iλen+1 + ne2 + · · · + 2en ∓ e1

if 2 n, if = n + 1.

The above computations give us the following equalities, Π s(e1 ±e ) (iλe1 + μk + ρM ) 2 k = C−1 λ + n2 · λ2 + (n − 1)2 · · · λ2 + (n − + 3)2 · −λ2 − (n − + 1)2 · · · × −λ2 − (n − k + 1)2 · −λ2 − (n − k − 1)2 · · · −λ2 if 2 k + 1, Π s(e1 ±e ) (iλe1 + μk + ρM ) = Ck λ2 + n2 · λ2 + (n − 1)2 · · · λ2 + (n − k + 1)2 · λ2 + (n − k − 1)2 · · · × λ2 + (n − + 2)2 · −λ2 − (n − )2 · · · −λ2 if k + 2 n + 1, Π s(e1 ±e ) iλe1 + μ± n + ρM 2 n = C−1 λ + n2 · λ2 + (n − 1)2 · · · λ2 + (n − + 3)2 · −λ2 − (n − + 1)2 · · · × −λ2 − 22 −λ2 − 1 where Ck =

2 b − a2

0a −1, for any l we have ∞ eiλt t α ln(t)m γ (t) dt =

∂ m Γ (α + 1) + O λ−l m α+1 ∂α (−iλ)

(1.10b)

0

Inserting (1.4a) and (1.4b) into (1.10a) and using Theorem 1.1 and (1.10b) gives the theorem.

2

Comment 1. One does need that d > 1 for Theorem 1.2 to hold. When S(x, y) has a nondegenerate saddle critical point, there are coordinates where S(x, y) = xy. This falls under Case 2, and one has that c0 (φ) = C0 (φ). This means that the two terms of (1.9b) will cancel. And in fact when φ(0, 0) = 0, |IS,φ ()|, |I−S,φ ()| ∼ | ln()|, while |JS,φ (λ)| ∼ λ−1 . Comment 2. In Cases 1 and 2, the expressions (1.9a) and (1.9b) for d0 (φ) and d0 (φ) will always be nonzero when d > 1 and φ(0, 0) = 0. This is because the expressions given by Theorem 1.1 π for c0 (φ), C0 (φ), c0 (φ), and C0 (φ) are real multiples of φ(0, 0), while the ratio of the ei 2d and π e−i 2d factors is never real when d > 1. Comment 3. In any dimension, when the phase satisfies an appropriate nondegeneracy condition there are reasonably explicit formulas for the leading coefficient of the leading term of the asymptotic expansion of oscillatory integrals such as (1.1). Such formulas are proven in [6] and [5]. ∞ Next, we give some less precise

C analogues for the IS,φ (). The lower bounds involve I|S|,φ () = IS,φ () + I−S,φ () = {(x,y): |S(x,y)| 1. If φ is supported in a sufficiently small neighborhood of the origin, then there is a positive BS,φ such that: In Cases 1 and 3 one has IS,φ () < BS,φ d1

(1.11)

IS,φ () < BS,φ | ln()| d1

(1.12)

In Case 2 one has

One has some analogous lower bounds for I|S|,φ (). They are sharp in Cases 1 and 2, and almost sharp in Case 3. (Sharp lower bounds do not hold in general in Case 3, as explicit examples show.) Theorem 1.3b. Suppose we are in the setting of Theorem 1.3a. Suppose also that φ(0, 0) = 0. In Case 1 there exists an AS,φ > 0 such that for sufficiently small we have I|S|,φ () > AS,φ d1

(1.13a)

1766

M. Greenblatt / Journal of Functional Analysis 257 (2009) 1759–1798

In Case 2, one similarly has I|S|,φ () > AS,φ ln() d1

(1.13b)

In Case 3, one has analogous almost-sharp lower bounds, at least if φ(x, y) is nonnegative. Namely, for any δ > 0 one has I|S|,φ () > AS,φ,δ d1 +δ

(1.13c)

The next lemma will be used to show that the three cases of superadapted coordinates are mutually exclusive. Lemma 1.4. Assume S(x, y) is smooth and is in Case 3 of superadapted coordinates with d > 1. Then for any M one can find a smooth function SM (x, y) such that SM − S has a zero of order at least M at the origin, but such that in a small enough neighborhood U of the origin one has

1 SM (x, y)− d dx dy < ∞

(1.14)

U

Proof. Suppose for example that the bisectrix intersects N (S) in the interior of the horizontal ray. Then SM (x, y) = S(x, y) + x M agrees with S(x, y) to order M at the origin and has Newton distance less than that of S(x, y). The Newton distance is also greater than 1 for M large and the relevant polynomials (SM )e (1, y) and (SM )e (−1, y) have zeroes of order at most 1. Hence SM (x, y) is in Case 1 superadapted coordinates and one can apply Theorem 1.1(a) to conclude that (1.14) is finite for a small enough neighborhood U of the origin. 2 Lemma 1.5. Any smooth degenerate phase S(x, y) can be put in superadapted coordinates in exactly one of Cases 1, 2, or 3. Proof. In Section 7 we will show that one can always put S(x, y) into some superadapted coordinate system. Eqs. (1.11) (for S(x, y) and −S(x, y)) and (1.13b) cannot simultaneously hold, so a Case 2 coordinate system cannot be put in a Case 1 or 3 coordinate system. Suppose S(x, y) has a Case 3 coordinate system as well as a Case 1 coordinate system; we will derive a contradiction. Since it has a Case 3 coordinate system, we may adjust S(x, y) to arbitrarily high order and cause (1.14) to hold. In its Case 1 coordinates, an adjustment of high enough order will not affect the fact that it is in Case 1 and thus Eq. (1.11) will still hold. Using the well-known relationship between Lp norms and distribution functions (on the function (I|S|,φ ())−1 ), one then gets that the integral (1.14) in the new coordinates is infinite, a contradiction. Thus the three cases are mutually exclusive. 2 For oscillatory integrals with smooth phase, one has some analogues of Theorem 1.3. It should be noted that Theorem 1.6a can be proved using the results of [10]. Theorem 1.6a. Suppose S(x, y) is smooth and is in superadapted coordinates with d > 1 In Cases 1 and 3 as λ → ∞ one has


1767

JS,φ (λ) < Cλ− d1

(1.15a)

JS,φ (λ) < Cλ− d1 ln(λ)

(1.15b)

In Case 2 one has

Theorem 1.6b. Suppose φ(x, y) is nonnegative with φ(0, 0) > 0. In Case 1, one has JS,φ (λ) >0 lim sup 1 λ→∞ λ− d

(1.16a)

In Case 2, one has JS,φ (λ) >0 lim sup 1 λ→∞ λ− d ln(λ)

(1.16b)

JS,φ (λ) =∞ lim sup 1 λ→∞ λ− d −δ

(1.16c)

In Case 3, for any δ > 0 one has

Although we will not prove it here, it can be shown with some additional argument that the conditions on φ(x, y) in (1.16a)–(1.16b) can be weakened to just that φ(0, 0) = 0. In all of Cases 1–3 we will divide the domain of integration of the expressions (1.1) and (1.3) for JS,φ and IS,φ into 4 parts, depending on whether or not x and y are positive or negative. Adding the resulting formulas and estimates will give the theorems. Without loss of generality we will always focus on the x, y > 0 as the other quadrants are always dealt with the same way. + + and JS,φ , where Hence our goal is to understand IS,φ

+ IS,φ () =

φ(x, y) dx dy

(1.17a)

{(x,y): x>0,y>0,0<S(x,y)0,y>0}

In turn, the domains of (1.17a)–(1.17b) will be written as the union of various “curved triangles” (such as those of Lemma 2.0 below). On a given curved triangle, one typically Taylor expands S(x, y) or one of its derivatives and then uses Van der Corput-type lemmas in the x or y direction to get a desired estimate. For the oscillatory integrals, the traditional van der Corput (see Chapter 8 of [16]) is used, while for sublevel integrals the version of [4] is used. Van der Corput-type lemmas have been considered in some detail, such as in [1] and [3], as well as the early work of Vinogradov [18]. We refer to [3] for further results and references. Throughout this paper, we will often have a constant C appearing on the right-hand side of an inequality. This always denotes a constant depending on S and φ. Occasionally we will need further constants C , C , etc which also depend on S and φ.

1768


2. Some useful lemmas for Cases 1 and 2 Suppose G is an open subset of R2 . Then throughout the course of this paper we will make G () and J G (λ) defined by frequent use of IS,φ S,φ G IS,φ () =

φ(x, y) dx dy

(2.1a)

{(x,y)∈G: 0<S(x,y) 0, 0 < δ < 1, we let G = {(x, y) ∈ [0, δ] × [0, δ]: 0 < y < Ax m }, and suppose there are nonnegative integers a and b with a > b and a 2 such that for some constant C0 the following holds on G, ∂yb S(x, y) > C0 x a

(2.2a)

∂y2 S(x, y) < C0 x a−m

(2.2b)

If b = 1, assume also that

If b = 0, instead of (2.2b) assume also that a > m + 1 and that for some constant C1 we have ∂x S(x, y) > C1 x a−1 ,

∂x2 S(x, y) < C0 x a−2

(2.2c)

Then for some ζab > 0 and C depending on S, φ, and C0 (and C1 if b = 0), if the support of φ is contained in [−δ, δ] × [−δ, δ] one has G m+1 I () < C a+mb Aζab S,φ G m+1 J (λ) < C|λ|− a+mb Aζab S,φ

(2.3a) (2.3b)

Proof. We first consider the case where b > 0; we will do the b = 0 argument afterwards. By (2.2a) and the Van der Corput’s lemma in the y direction (see [4] for example) one has that for a given x we have y: S(x, y) < < C b1 x − ab

(2.4)

As a result, if Gx denotes the vertical cross section of G at x, of length Ax m , then one has y ∈ Gx : S(x, y) < < C min Ax m , b1 x − ab

(2.5)


1769

Consequently, we have δ

G I < C S,φ

1 a min Ax m , b x − b dx

(2.6)

0 1

a

It is natural to break the integral (2.6) into two parts, depending on whether or not Ax m > b x − b . b 1 The two quantities are equal at x0 = A− a+mb a+mb . The left integral becomes x0 Ax m dx = 0

a−b m+1 A 1 x m+1 = A a+mb a+mb m+1 0 m+1

(2.7)

The right integral is computed to be

1 b

δ x

− ab

dx <

1 b

x0

∞

a

x − b dx

x0 a−b m+1 1 a−b b b = b x0 b = A a+mb a+mb a−b a−b a−b

(2.8)

m+1

G | < CA a+mb a+mb as needed. Adding together, we obtain that |IS,φ G (λ) for b 2 are done in a similar fashion. First suppose b 2. Then The estimates for JS,φ one can use the usual Van der Corput’s lemma (see [16, Chapter 8]) in the y direction to obtain eiλS(x,y) φ(x, y) dy < C|λ|− b1 x − ab (2.9) Gx

This is the analogue to (2.4) with replaced by |λ|−1 . As a result, similar to (2.6) we get G J < C S,φ

δ

1 a min Ax m , |λ|− b x − b dx

(2.10)

0

The result is G a−b m+1 J CA a+mb |λ|− a+mb S,φ

(2.11)

This gives (2.3b). We next prove (2.3b) when b = 1. If one integrates by parts in the y variable one gets several terms each of which can be bounded using (2.2a) and (2.2b). If one works it out, one gets that these terms are bounded by C|λ|−1 x −(a+m) . It is thus natural to split the integral into 1 two parts at the x0 satisfying |λ|−1 x0 a+m = 1, in other words, at x0 = |λ|− a+m . We accordingly 1 write G = G1 ∪ G2 , with G1 the portion where x0 < |λ|− a+m . We then have m+1

G1 (λ) < C|G1 | < C A|λ|− m+a JS,φ

(2.12)

1770


G2 As for JS,φ (λ), we integrate by parts in y, obtaining the C|λ|−1 x −(a+m) factor, and one gets that

G2 J (λ) < C

S,φ

1 Ax m |λ|−1 x −(m+a) C |λ|−1 x −(m+a) dy dx

G2

0

1

|λ|− a

1

m+1

|λ|−1 x −a dx < CA|λ|− m+a

= CA

(2.13)

1

|λ|− a+m

Adding (2.12) to (2.13) give the oscillatory integral estimates for b = 1. We now consider the oscillatory integral when b = 0. Here we do the integrations by parts in the x direction. This time, by (2.2a) and (2.2c) an integration by parts incurs a factor of 1 |λ|−1 x −a . Hence we subdivide G = G1 ∪ G2 , where G1 = {(x, y) ∈ G: 0 < x < |λ|− a }. Note m+1 that the measure of G1 is A|λ|− a , so that G1 J (λ) < CA|λ|− m+1 a S,φ

(2.14)

For the G2 piece one obtains G2 J (λ) < C

−1 −a

|λ|

S,φ

x

G

1 Ax m = |λ|−1 x −a dy dx 1

|λ|− a −1

0

1

= C|λ|

Ax m−a dx 1

|λ|− a

= CA|λ|−

m+1 a

(2.15)

(For the last equality we use the hypothesis that a > m + 1.) Adding (2.14) to (2.15) gives the G () when b = 0. In this case, since estimate we seek (2.3b). Lastly, we prove the bounds for IS,φ a |S(x, y)| > C0 x , we have G I < C (x, y) ∈ G: C0 x a < < C (x, y) ∈ G: x < a1 = C A m+1 a S,φ This concludes the proof of Lemma 2.0.

(2.16)

2

In Lemmas 2.1 and 2.2 below, S(x, y) is a smooth phase function in Case 1 or 2 of superadapted coordinates with d > 1. If there is a compact edge E of N (S) such that the bisectrix contains either the upper vertex of E or an interior point of E, then we denote the equation of this edge by t1 + mt2 = α, and for some large but fixed number N we let A1 = {(x, y) ∈ [0, 1] × [0, 1]: y < N1 x m }. Similarly, if there is some edge E with equation t1 + m t2 = α such that the bisectrix contains either the lower vertex of E or an interior point of E , we let


1771

1

A2 = {(x, y) ∈ [0, 1] × [0, 1]: x < N1 y m }. Note that in Case 1 both A1 and A2 exist. We foAi Ai cus our attention on IS,φ () and JS,φ (λ). The relevant information about them (if they exist) is provided by the following lemma. Lemma 2.1. There exists an η > 0 such that if the support of φ(x, y) is sufficiently small, then for i = 1, 2 we have Ai I () < C d1 N −η S,φ Ai J (λ) < C|λ|− d1 N −η S,φ

(2.17) (2.18)

Proof. By symmetry, it suffices to prove the bounds for A1 . Let (a, b) denote the lowest vertex of E. Thus b < a. Since (a, b) and (d, d) are both on E, we have a + mb = (1 + m)d or 1 m+1 d = a+mb . We will show that the hypotheses of Lemma 2.0 hold for these values of a, b, and m+1 , Lemma 2.1 will follow. For a large but fixed M, we m, setting A = N −1 . Since d1 = a+mb p write S(x, y) = p<M, q<M spq x y q + EM (x, y). By standard estimates, for 0 α, β M we have α β ∂ ∂ EM (x, y) < C |x|M−α + |y|M−β x y

(2.19)

We can write

∂yb S(x, y) =

spq x p y q + ∂yb EM (x, y)

(2.20)

p<M, q<M−b = 0. We next show that the sum in (2.20) is dominated by the term s x a in a suffiHere sa0 a0 ciently small neighborhood of the origin. To this end, note that (a, 0) is a vertex of the Newton polygon of ∂yb S(x, y), and that a horizontal ray and an edge of this Newton polygon with equa x p y q in the sum of (2.20) tion t1 + mt2 = a intersect at (a, 0). As a result, for any term spq a other than sa0 x , either p a, or p < a, q > 0, and p + mq a. Correspondingly, we let T1 = {(p, q): p < a, 0 < q < M − b, p + mq a} and T2 = {(p, q): a p M, 0 q b − a, (p, q) = (a, 0)} and we rewrite (2.20) as ∂yb S(x, y) = sa0 xa +

T1

spq xpyq +

spq x p y q + ∂yb EM (x, y)

(2.21)

T2

x p y q in the T sum. Since (x, y) is in the domain A , one has We examine a given term spq 1 1 m x p q −q p+mq −q a y < N . As a result, |spq x y | < |spq N x | |spq N x |. Thus so long as N is chosen sufficiently large (depending on M), we may assume that the absolute value of the T1 |x a . We next examine a term s x p y q in the T sum. Here we have sum is at most 14 |sa0 2 pq p q a |spq x y | < |spq x |(|x| + |y|). Hence if we are in a sufficiently small neighborhood of the ori |x a . Lastly, by (2.19), gin, we can assume the absolute value of the T2 sum is also at most 14 |sa0 |x a . in a sufficiently small neighborhood of the origin |∂yb EM (x, y)| is also bounded by 14 |sa0 Putting these together, we conclude that on the domain A1 we have

b ∂ S(x, y) > 1 s x a y 4 a0

(2.22)

1772


Note that when b > 1, (2.22) ensures hypotheses of Lemma 2.0 hold. Thus one may apply Lemma 2.0, giving Lemma 2.1. When b = 0, in order to apply Lemma 2.0 one needs also that a > m + 1 and that (2.2c) holds. But taking an x derivative of S(x, y) just shifts the Newton polygon to the left by 1, so exactly as in (2.22) the first and second derivative conditions of (2.2c) will hold. As for the requirement that a > m + 1, note that (d, d) and (a, 0) are on the line t1 + mt2 = α and that d > 1. This means a = a + m0 = d + md > 1 + m as needed. Lastly, we show that the supplemental hypothesis (2.2b) holds when b = 1. It helps to view things in (x, y ) coordinates where (x, y) = (x, x m y ). The condition (2.2b) becomes that |∂y2 S(x, x m y )| Cx a+m . Also, since the terms spq x p y q of S(x, y)’s Taylor expansion with mq (= α) are exactly the terms of SE (x, y), the expansion S(x, y) = minimal p + p q p<M, q<M spq x y + EM (x, y) becomes of the following form, where TM is a polynomial in y and a fractional power of x, S x, x m y = x α SE (1, y ) + x α+ TM (x, y ) + EM x, x m y

(2.23)

Using (2.23) and the error estimates (2.19) we have |∂y2 S(x, x m y )| Cx α . But (a, b) = (a, 1) is on the line t1 + mt2 = α and therefore a + m = α. This gives the second derivative bounds of (2.2b), and thus the hypotheses of Lemma 2.0 hold. This concludes the proof of Lemma 2.1. 2 Lemma 2.2. Let S(x, y) be a smooth phase function in Case 1 or 2 superadapted coordinates with d > 1. Suppose e is a compact edge of N (S) intersecting the bisectrix and has equation m given by t1 + mt2 = α. Define B = {(x, y) ∈ [0, 1] × [0, 1]: xN < y < N x m }, where N is some large but fixed constant. Then if the support of φ(x, y) is sufficiently small, depending on N , one B ()| < C d1 and |J B (λ)| < C |λ|− d1 . has the estimates |IS,φ N N S,φ Proof. In the above (x, y ) coordinates one has

x m φ x, x m y dx dy

B () = IS,φ {(x,y )∈[0,1]×[N −1 ,N ]: 0<S(x,x m y ) 0 are in [N −1 , N], and denote these zeroes by z1 , . . . , zk (if there are any). Let v1 , . . . , vk denote the orders of these zeroes, and let Ii denote the interval [zi − N1 , zi + N1 ]. By (2.23) and the error term derivative estimates (2.19) we can assume that on the (sufficiently small, depending on N ) support of φ(x, x m y ), if (x, y ) ∈ [0, 1] × Ii then vi ∂ S x, x m y Cx α y

(2.24a)


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In the case that vi = 1, we similarly have 2 ∂ S x, x m y Cx α y

(2.24b)

We now translate this into the original (x, y) coordinates. Each [0, 1] × Ii becomes a set Di of the form {(x, y) ∈ [0, 1] × [0, 1]: (zi − N1 )x m < y < (zi + N1 )x m }, and on Di (2.24a) becomes v ∂ i S(x, y) Cx α−mvi y

(2.25a)

In the case of vi = 1, (2.24b) becomes 2 ∂ S(x, y) Cx α−2m y

(2.25b)

Since we are superadapted coordinates, 0 < vi < d. Thus (m + 1)vi < (m + 1)d. Since (d, d) is on the edge e, we have α = (m + 1)d. Thus (m + 1)vi < α or α − mvi > vi . As a result, the sets Di have vertical cross sections of length N2 x m and satisfying (2.25a)–(2.25b) with α − mvi > vi . Hence after doing a coordinate change of the form (x, y) → (x, y − f (x)), we are in the set-up of Lemma 2.0 and for some η > 0 we get Di I C m+1 α N −η S,φ Di J C|λ|− m+1 α N −η S,φ Since

m+1 α

=

1 d

(2.26a) (2.26b)

the above becomes Di I C d1 N −η S,φ Di J C|λ|− d1 N −η S,φ

(2.27a) (2.27b)

Next, write [N −1 , N] − ∪i Ii as the union of intervals Ji . Then since Se (1, y ) has no zeroes on any Ji , by the expansion (2.23) and the error derivative bounds (2.19), if δN is sufficiently small then on [0, δN ] × Ji we have ∂x S x, x m y > CN x α−1 ,

2 ∂ S x, x m y < C x α−2 x N

(2.28)

1

Separating at x = |λ|− α and integrating the right portion by parts in x using (2.28) as in the proof of Lemma 2.0 gives iλS(x,x m y ) m m − m+1 e x φ x, x y (2.29a) dx dy < CN |λ| α [0,δN ]×Ji

Converting back into (x, y) coordinates and using that d1 = m+1 α , (2.29a) becomes the following, where Ei denotes the set [0, δN ] × Ji in the (x, y) coordinates. eiλS(x,y) φ(x, y) dx dy C |λ|− d1 (2.29b) N Ei

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Moving now to IS,φ (), by (2.23) one has |S(x, x m y )| > CN x α on [0, δN ] × Ji for sufficiently small δN > 0 and some CN (not necessarily the same constant as above). As a result,

m x φ x, x y dx dy

m

{(x,y )∈[0,δN ]×Ji : 0<S(x,x m y ) α Cx on each Ei . Hence since φ(x, y) is zero on a neighborhood of the origin, |S(x, y)| is E bounded below on the support of the integrand of IS,φi () and (3.2) again holds trivially. This completes the proof of Lemma 3.2. 2

We now move to the Case 1 formulas of Theorems 1.1. We write φ = φ1 + φ2 , where φ1 is supported in a smaller neighborhood of the origin and φ2 is zero on a neighborhood of the IS,φ2 () IS,φ1 () I () = 0 and therefore lim→0 S,φ1 = lim→0 . Hence origin. By Lemma 2.3, lim→0 1 1 d

d

d

when proving the limit (1.6) one can always replace φ by φ1 at will, regardless of how small the support of φ1 is. E Our strategy will involve fixing N and analyzing the IS,φi 1 , where φ1 has small support deDi Ai pending on N . Lemma 2.1 and (2.27a) will ensure that the contributions of the IS,φ and IS,φ 1 1

will be O(N −η ) smaller than that of the IS,φi 1 as goes to zero. Letting N go to infinity will Ei + IS,φ IS,φ () () 1 1 = lim . Adding up the latter limits along with their analogues give lim→0 →0 1 1 i E

d

d

in the other three quadrants will give (1.6). Proof of (1.6). It suffices to assume φ(0, 0) > 0 as the general case can be obtained by writing φ = φ − φ where φ (0, 0) and φ (0, 0) are positive. We work in the (x, y ) coordinates like above. Namely, we write

E IS,φi () = {(x,y )∈[0,1]×J

i:

0<S(x,x m y ) 0 such that on [0, δ] × Ji either S(x, x m y ) is negative, or we have 1 1 S x, x m y x α S0 (1, y ) 1 + S x, x m y 0< 1− N N

(3.4)

Shrinking δ further if necessary, we assume δ is small enough so that on [0, δ] × Ji we have 1 φ(0, 0) φ x, x m y < 1 + N

(3.5)

As described above, one can multiply φ(x, y) by a cutoff function supported on |x| < δ without E

affecting lim→0

i () IS,φ 1 1

d

. Hence we assume φ(x, y) is supported on |x| < δ, and that the multi-

plying cutoff was chosen so that for (x, y ) ∈ [0, δ] × Ji , Eq. (3.5) holds. We also assume the multiplying cutoff was chosen so that for (x, y ) ∈ [0, 2δ ] × Ji one has

1 1− φ(0, 0) < φ x, x m y N

(3.6)

We now proceed to our main estimates. If S(x, x m y ) is negative on [0, δ] × Ji , IS,φi () becomes zero. If on the other hand (3.4) holds, then by (3.4) and (3.5) one has E

Ei IS,φ ()

1 φ(0, 0) 1+ N

{(x,y )∈[0,δ]×J

x m dx dy

(3.7a)

x m dx dy

(3.7b)

1 α i : 0<x S0 (1,y ) 1. Hence the bisectrix intersects N (S) at (d, d). As in the Case 1 proofs, we consider + + () and JS,φ (λ) and we will do some subdivisions of the domains of these integrals to prove IS,φ the estimates and formulas. We proceed as follows. If (d, d) is the lower vertex of a compact edge e2 , denote its equation by t1 + m2 t2 = α2 . Where N is a large natural number, fixed for now, we let A2 = {(x, y) ∈ [0, 1] × [0, 1]: x < 1 m2

1

1 m2 } Ny

and B2 = {(x, y) ∈ [0, 1] × [0, 1]:

1

1 m2 Ny

m2 . The estimates for D1 and D2 are proven the same way, so we restrict our attention to proving the


estimates for D1 . Taylor expand S(x, y) around the origin as S(x, y) = EM (x, y), where like in (2.19) for 0 α, β M we have

a<M, b<M sab x

α β ∂ ∂ EM (x, y) < C |x|M−α + |y|M−β x y

1779 a yb

+

(4.3)

Correspondingly, we have ∂yd S(x, y) =

sab x a y b + ∂yd EM (x, y)

(4.4)

a<M, b<M−d = 0. If (d, d) is the lower The Newton polygon of ∂yd S(x, y) has a vertex at (d, 0) and thus sd0 vertex of some compact edge e2 , then there is a compact edge e of the Newton polygon of x a y b appear∂yd S(x, y) containing (d, 0) with equation t1 + m2 t2 = d. Hence every nonzero sab ing in (4.4) satisfies a + m2 b d and we can rewrite (4.4) as xd + ∂yd S(x, y) = sd0

sab xa yb

a 0 such that on D, φ(x, y) = 0 for x < δ. Doing the y integration first we have D1 I () < C

1

S,φ

y ∈ (D1 )x : 0 < S(x, y) < dx

(4.13)

δ

By (4.6), |∂yd S(x, y)| is bounded below on x > δ. Hence by the Van der Corput’s lemma in the 1

y direction, we have |{y ∈ (D1 )x : 0 < S(x, y) < }| < C d uniformly in x > δ. Inserting this back into (4.13) gives the desired bounds. For the oscillatory integral, one similarly uses the Van

1 der Corput’s lemma in the y direction to get | (D1 )x eiλS(x,y) φ(x, y) dy| < C|λ|− d uniformly in x > δ. Thus 1 D1 J (λ) = S,φ δ

1 φ(x, y) dy dx < C|λ|− d

e

iλS(x,y)

(4.14)

(D1 )x

D (λ) and we are done. These are the sought-after bounds for JS,φ We now proceed to the proof of the explicit formula (1.7). The general methodology is similar to that of the Case 1 arguments of Section 3. If one writes φ = φ1 + φ2 , where φ1 is supported in a smaller neighborhood of the origin and φ2 is zero on a neighborhood of the

origin then by Lemma 4.3, lim sup→0 lim inf→0

+ IS,φ () 1

1

+ () IS,φ

1 ln() d

= lim sup→0

+ IS,φ () 1

1 ln() d

and lim inf→0

+ IS,φ () 1

ln() d

=

. So when proving (1.7) one can always replace φ by φ1 at will, regardless

ln() d

of how small the support of φ1 is. We will show if one first chooses the parameter L of (4.2b) sufficiently large, and then chooses φ1 to be supported in a sufficiently small neighborhood of the origin, then the above limsup and liminf, added to their analogues from the other three quadrants, can both be made as arbitrarily close to the limit given in (1.7). To do this, in view of (4.2b), it suffices to show that if the support of φ1 is sufficiently small, the quantities lim sup→0

D IS,φ

1

1 ln() d

and lim inf→0

D IS,φ

1

1

, can

ln() d

be made arbitrarily close to the appropriate expression. For a fixed L we will find lower bounds for the limsup and upper bounds for the liminf. In doing so, we will choose the parameter M of the Taylor expansions in terms of L, and then the parameter N of (4.1a)–(4.1b) in terms of L and M. Analogous to in Section 3 taking limits as L goes to infinity, both expressions will converge to the same limit. Adding this limit to its analogues in the other 3 quadrants will give (1.7).

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We start with the following lemma. As before, sdd x d y d denotes the (d, d) term of the Taylor expansion S(x, y) = a,b sab x a y b of S at the origin. Lemma 4.4. There are constants β and C depending on S(x, y), and a neighborhood U of the origin depending on S(x, y) and L, such that |S(x, y) − sdd x d y d | < |CL−β x d y d | on D ∩ U . Proof. Analogous to (4.5), we may Taylor expand

S(x, y) − sdd x d y d =

sab xa yb

M>ad, M>bd, (a,b)=(d,d)

+

sab x a y b

ab>d, a+m2 bα2

+

sab x a y b

M>a>d, b d in each term in the second sum, the entire sum can be written as y (x α2 (y )d )f (x, y ) for some f (x, y ) which is a polynomial in y and a fractional power of x. Thus the sum is of absolute value at most CM N −m2 x α2 (y )d in a small enough neighborhood of the origin. Note that sdd x d y d = sdd x d+dm2 (y )d , and this is equal to sdd x α2 (y )d since (d, d) is on the edge with equation t1 + m2 t2 = α2 . As a result, in the original (x, y) coordinates, the sum is of absolute value at most CM N −m2 x d y d . Thus if one chooses N sufficiently large for fixed L and M, one has the desired bounds. The third sum is dealt with in exactly the same way, reversing the roles of the x and y axes. 1 Since D necessarily lies in the range x L > y > x L , the error term EM (x, y) can be made less than 1 d d L x y by making the radius of U sufficiently small. This completes the proof of Lemma 4.4. 2 Proof of (1.7). As before it suffices to assume φ(0, 0) > 0 as the general case can be obtained by writing φ = φ − φ where φ (0, 0) and φ (0, 0) > 0 are positive. Let δL > 0 be such that on the ball B(0, δL ) one has 1 − L−β φ(0, 0) < φ(x, y) < 1 + L−β φ(0, 0)

(4.16)

Further assume that δL is small enough that B(0, δL ) ⊂ U , where U is as in the previous lemma. Let ψ(x, y) be a nonnegative cutoff function such that 0 ψ 1, ψ(x, y) is supported on B(0, δL ), and ψ(x, y) = 1 on [0, δ2L ] × [0, δ2L ]. Then by the discussion following (4.14) we may replace φ(x, y) by φ(x, y)ψ(x, y) without affecting the liminf or limsup. We have D () < 1 + L−β φ(0, 0) (x, y) ∈ D: 0 < S(x, y) < IS,φ


1783

By Lemma 4.4 this is bounded by 1 + L−β φ(0, 0) (x, y) ∈ D: 0 < sdd − CL−β x d y d <

(4.17a)

In addition, δL δL D IS,φ × 0, : 0 < S(x, y) < () > 1 − L−β φ(0, 0) (x, y) ∈ D ∩ 0, 2 2 > 1 − L−β φ(0, 0)

δL δL × 0, : 0 < sdd + CL−β x d y d < (4.17b) × (x, y) ∈ D ∩ 0, 2 2 D () = 0 for large enough L. For s If sdd < 0, then by (4.17a) IS,φ dd > 0, we need the following lemma, whose proof is routine.

Lemma 4.5. Suppose 0 < m2 < m1 and δ0 > 0. Then as t → 0, (x, y) ∈ (0, δ0 ] × (0, δ0 ]: x m1 < y < x m2 , y < t x 1 1 = − t ln(t) + O(t) m1 + 1 m2 + 1 (x, y) ∈ (0, δ0 ] × (0, δ0 ]: 0 < y < x m2 , y < t = − 1 t ln(t) + O(t) x m2 + 1 We now apply Lemma 4.5 to (4.17a)–(4.17b). We get that D IS,φ ()
1 − L−β sdd

(When (d, d) is on the horizontal ray one substitutes m1 = L, and when it is on the vertical ray one substitutes m2 = L1 .) Hence we have

lim sup →0

lim inf →0

D () IS,φ 1

d ln() D () IS,φ 1

d ln()

− 1 1 + L−β sdd − CL−β d φ(0, 0) − 1 1 − L−β sdd + CL−β d φ(0, 0)

1 1 − m1 + 1 m2 + 1 1 1 − m1 + 1 m2 + 1

(4.19a) (4.19b)

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We take limits as L → ∞. Both expressions converge to φ(0, 0)( m11+1 − m21+1 ), where now m1 is taken as ∞ when (d, d) is on the horizontal ray and m2 is taken as 0 when it is on the vertical ray. Hence by the discussion following (4.14) we conclude that lim

→0

+ IS,φ () 1

d ln()

−1 = sddd φ(0, 0)

1 1 − m1 + 1 m2 + 1

Letting si be the slope − m1i of the edge t1 + mi t2 = αi , this becomes lim

→0

+ IS,φ () 1

d ln()

−1 = sddd φ(0, 0)

1 1 − s1 − 1 s2 − 1

(4.20)

In summary, if sdd > 0 then (4.20) gives the contribution to (1.7) from the upper right-hand quadrant. If sdd < 0 then the contribution is zero as mentioned above Lemma 4.5. Adding this to its analogues over the other three quadrants gives exactly the formula of (1.7) and we are done. 2 Our final task is to prove (1.13b): Proof of (1.13b). Since the result is symmetric in S and −S, we may replace S by −S if necessary and assume that sdd > 0. As in the proof of (1.13a), we write φ = φ1 + φ2 , where φ1 is I|S|,φ2 () nonnegative and φ2 is zero on a neighborhood of the origin. By Lemma 4.3, lim→0 1 = 0. ln() d

1

So to prove (1.13b) it suffices to show |I|S|,φ1 ()| > AS,φ1 | ln()| d for some AS,φ1 > 0. 1

Since |I|S|,φ1 ()| |IS,φ1 ()|, it further suffices to show that |IS,φ1 ()| > AS,φ | ln()| d for some AS,φ1 . For this we use Lemma 4.4, which implies that there is a δ > 0 such that on ([0, δ] × [0, δ]) ∩ D we have 3 S(x, y) < sdd x d y d 2

(4.21)

As a result, shrinking δ to ensure that |φ(x, y)| > 12 |φ(0, 0)| on [0, δ] × [0, δ] if necessary, we have IS,φ () > 1 φ(0, 0) (x, y) ∈ D ∩ [0, δ] × [0, δ] : 3 sdd x d y d < (4.22) 1 2 2 1

Using Lemma 4.5, we conclude that there is some AS,φ1 with |IS,φ1 ()| > AS,φ1 | ln()| d as needed. This gives (1.13b) and we are done. 2 5. Case 3 proofs In this section, S(x, y) is a smooth phase function in Case 3 of superadapted coordinates with d > 1. We restrict ourselves to the situation where the bisectrix intersects the horizontal ray in its interior, as the case of a vertical ray is entirely analogous. Thus the lowest vertex of N (S) is of the form (c, d), where c < d and d is also the Newton distance of S. As in Sections 3 and 4


1785

+ + we will focus our attention on the analysis of IS,φ () and JS,φ (λ). We divide [0, 1] × [0, 1] into two parts. For a sufficiently large positive integer k (to be determined by our arguments), we let D1 = {(x, y) ∈ [0, 1] × [0, 1]: y < x k } and D2 = {(x, y) ∈ [0, 1] × [0, 1]: y > x k }. Our first lemma is the following. 1

1

D1 D1 Lemma 5.1. |IS,φ ()| < C d and |JS,φ (λ)| < C|λ|− d .

Proof. As in Cases 1 and 2, we write the Taylor expansion of S at the origin as S(x, y) = a b a<M, b<M sab x y + EM (x, y), where for 0 α, β M the function EM (x, y) satisfies the error estimates α β ∂ ∂ EM (x, y) < C |x|M−α + |y|M−β x y

(5.1)

The dth y-derivative can be written as ∂yd S(x, y) =

sab x a y b + ∂yd EM (x, y)

(5.2)

a<M, b<M−d

Furthermore the Newton polygon of ∂yd S(x, y) has a vertex at (c, 0), contained either in the vertical ray of N (∂yd S) or an edge of N (∂yd S) with equation t1 + mt2 = c with m > 0. Hence each (a, b) in the sum of (5.2) satisfies a + mb c. Analogous to (4.5) we rewrite (5.2) as c x + ∂yd S(x, y) = sc0

sab xa yb

a 1 |r0d |y d x 2 Thus if c 2, one can apply Lemma 2.0 and immediately get this lemma. If c = 1, to apply 1 Lemma 2.0 one also needs that |∂x2 S(x, y)| < Cy d− k . But in fact since the Newton polygon of ∂x2 S(x, y) is a subset of {(x, y): y d}, by expanding as in (5.9) one even has the stronger estimate |∂x2 S(x, y)| < Cy d . Thus Lemma 2.0 applies here. If c = 0, to apply Lemma 2.0 one needs (2.2c) to hold (with the x and y variables reversed) which here means one needs |∂y S(x, y)| > Cy d−1 and |∂y2 S(x, y)| < C y d−2 . Since the Newton polygon of ∂yi S(x, y) has a single vertex at (0, d − i) this holds as in the c = 1 case. Lastly, to apply Lemma 2.0 for c = 0


1787

one also needs that d > k1 + 1. We can make this true simply choosing k large enough since d is at least 2. This completes the proof of Lemma 5.2. 2 Corollary 5.3. (1.11) and (1.15a) hold in Case 3. Proof. Add the estimates from Lemmas 5.1 and 5.2 and their analogues from the other three quadrants. Next, we prove the lower bounds of (1.13c). Since we are not trying to prove sharp estimates, the arguments are not that intricate. Assume φ(x, y) 0 with φ(0, 0) > 0, and let M be some N large positive integer. We examine the behavior of S(x, y) on the set ZN = {(x, y): x > 0, x < y < 2x N } for N sufficiently large. We Taylor expand S(x, y) = a<M, b<M sab x a y b + EM (x, y) as above. If N is large enough, the term scd x c y d dominates this Taylor expansion much the way x c dominates (5.3). Hence in a small enough neighborhood U of r0d y d dominates (5.9) or sc0 the origin, on ZN we have S(x, y) < 2|scd |x c y d

(5.10)

Shrinking U , we may assume that |φ(x, y)| > 12 |φ(0, 0)|. Hence for cφ = 12 |φ(0, 0)| we have I|S|,φ () > cφ (x, y) ∈ U ∩ ZN : 2|scd |x c y d <

(5.11) 1

It is easy to compute that the curve y = 2x N intersects the curve 2|scd |x c y d = at x = a Nd+c for some a depending on S(x, y). Hence for sufficiently small the measure of the set in the 1 right-hand side of (5.11) is at least the measure of the portion of ZN between x = a2 Nd+c and 1

N+1

x = a Nd+c , given by a Nd+c where now a also depends on N . Thus we can write N+1

I|S|,φ () > cS,φ,N Nd+c

(5.12)

Since d > c, the exponent in (5.12) is larger than d1 but as N → ∞ it tends to d1 . This gives us (1.13c). We now move to the case of real-analytic phase. Our goal here is to prove Theorem 1.1(c). I () So assume S(x, y) is real-analytic. It suffices to show that lim→0 S,φ1 exists and is given by (1.8). As in Cases 1 and 2, we will give an expression for lim→0

d + IS,φ () 1

and the full limit

d

will follow by adding this and the analogues from the other quadrants. Also, by Lemma 5.2, D

lim→0

IS,φ2 () 1 d

D

= 0 so it suffices to show lim→0

IS,φ1 () 1

exists and has the desired value. Because

d

the bisectrix intersects N (S) in the interior of its horizontal ray and (c, d) is the lowest vertex of N(S), the real-analytic S(x, y) can be written as S(x, y) = scd x c y d + x c+1 y d g(x) +

bd+1

sab x a y b

(5.13a)

1788


Here g(x) is real-analytic. Changing coordinates from (x, y) to (x, y ) where y = x k y , we have S x, x k y = scd x c+kd (y )d + x c+kd+1 (y )d g(x) + sab x a+kb (y )b

(5.13b)

bd+1

Since the line t1 + mt2 = α is an edge of N (S) containing (c, d), each (a, b) in the sum (5.13a) satisfies a + mb c + md. Furthermore, b > d, and therefore a + kb = a + mb + (k − m)b c + md + (k − m)b > c + md + (k − m)d = c + kd Hence we can rewrite bd+1 sab x a+kb (y )b as x c+kd+1 (y )d+1 f (x, y) where f (x, y) is realanalytic. Thus we have S x, x k y = scd x c+kd (y )d + x c+kd+1 (y )d g(x) + x c+kd+1 (y )d+1 f (x, y )

(5.14)

D1 () becomes In the (x, y ) coordinates IS,φ

D1 IS,φ () =

{(x,y )∈[0,1]×[0,1]:

x k φ x, x k y dx dy

(5.15)

0<S(x,x k y ) 0. Then by (5.14) and (5.17) we have

1 c+kd Ex ⊂ y ∈ [0, 1]: 0 < scd x c+kd y d < 2 ⊂ 0, C d x − d

(5.18)


1789

1

Since x d (k+1) , we have 1

d x−

c+kd d

1

d

kd +d

− d(dc+kd )(k+1)

= d(d )(k+1)

− d(dc+kd )(k+1)

=

k(d −d)+(d −c) dd (k+1) 1

Hence if d were chosen close enough to d, there is some η > 0 such that for x d (k+1) one 1 c+kd has d x − d < η and thus Ex ⊂ 0, C η

(5.19)

() = I () + I (), where Next, we write IS,φ S,φ S,φ

IS,φ () =

1 {(x,y )∈[ d (k+1) ,1]×[0,1]:

() = IS,φ

1 {(x,y )∈[ d (k+1) ,1]×[0,1]:

x k φ(x, 0) dx dy

(5.20a)

x k φ x, x k y − φ(x, 0) dx dy

(5.20b)

0<S(x,x k y ) 12 x c+kd and c < d the integral (5.29) is finite as needed. Going back to the definition of g(x), S(x, y) = (scd x c + x c+1 g(x))y d + O(y d+1 ). As a result, (5.29) translates into the part of Eq. (1.8) coming from the upper right-hand quadrant. (Recall that (5.29) is for scd > 0 and that the limit is zero when scd < 0.) Adding the analogous expressions from the remaining three quadrants gives (1.8) and we are done. 2 6. Proof of Theorem 1.6b Suppose S(x, y) is a smooth phase function in Case 1 superadapted coordinates, and φ(x, y) is nonnegative with φ(0, 0) > 0. Let ψ(t) be a nonnegative function in Cc (R) such that ψ(t) > 1 on [−1, 1]. Then by (1.13a), if j is sufficiently large we have ψ 2j S(x, y) φ(x, y) dx dy >

j φ(x, y) dx dy > AS,φ 2− d

{(x,y):

(6.1)

|S(x,y)| 0, we may let Mδ be such that for |λ| > Mδ we λ

d

1

have |JS,φ (λ)| < δ|λ|− d . We then have −j −j 2 ˆ ψ 2 λ JS,φ (λ) dλ −j = 2 ψˆ 2−j λ JS,φ (λ) dλ + 2−j |λ|<Mδ

< C2−j Mδ + 2−j δ

−j ˆ ψ 2 λ JS,φ (λ) dλ

|λ|>Mδ

−j − 1 ψˆ 2 λ |λ| d dλ

(6.3)

|λ|>Mδ

In turn, Eq. (6.3) is bounded by C2−j Mδ + 2−j δ

R

−j − 1 j ψˆ 2 λ |λ| d dλ = C2−j Mδ + 2− d δ

1 ψ(λ) ˆ |λ|− d dλ

R j

< C2−j Mδ + C δ2− d

(6.4)

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M. Greenblatt / Journal of Functional Analysis 257 (2009) 1759–1798 j

When j is sufficiently large, (6.4) is at most 2C δ2− d . On the other hand, by (6.1), it must also j AS,φ be at least AS,φ 2− d . This gives a contradiction if δ were chosen less than 2C . This contradiction implies that lim supλ→∞ |

JS,φ (λ) − d1

λ

| is in fact positive, giving (1.16a).

That the lim sup of (1.16c) is positive is proven from (1.13c) exactly as (1.16a) is proven from (1.13a), so we do not include a proof here. Since it holds for all δ > 0 the lim sup is automatically infinite. Eq. (1.16b) is proved similarly to (1.16a), using (1.13b) in place of (1.13a). Namely, suppose S(x, y) is a smooth phase function in Case 2 superadapted coordinates, and φ(x, y) is a nonnegative function with φ(0, 0) > 0. Using (1.13b) we have ψ 2j S(x, y) φ(x, y) dx dy >

φ(x, y) dx dy

{(x,y): |S(x,y)| AS,φ 2− d d

(6.5)

Exactly as above we also have

ψ 2j S(x, y) φ(x, y) dx dy = 2−j

ψˆ 2−j λ JS,φ (λ) dλ

(6.6)

Proceeding by contradiction again, suppose (1.16b) does not hold. Therefore for every δ > 0 1 there is some Lδ such that for |λ| > Lδ we have |JS,φ (λ)| < δ|λ|− d ln |λ|. Analogous to (6.3) we have 2−j ψˆ 2−j λ JS,φ (λ) dλ < C2−j Lδ + 2−j δ

−j − 1 ψˆ 2 λ |λ| d ln |λ| dλ

|λ|>Lδ

C2

−j

Lδ + 2

−j

δ

−j − 1 ψˆ 2 λ |λ| d ln |λ| dλ

(6.7)

R

Changing variables, this in turn is equal to C2

−j

Lδ + 2

− dj

δ

1 ψ(λ) ˆ |λ|− d ln 2j |λ| dλ

R j

= C2−j Lδ + 2− d δ

j 1 ψ(λ) ˆ |λ|− d ln |λ| dλ + j 2− d ln(2)δ

R

< C2−j Lδ + C δj 2

1 ψ(λ) ˆ |λ|− d dλ

R

− dj

(6.8) j

j

If j is sufficiently large, (6.4) is at most 2C δj 2− d , while by (6.5) it is at least AS,φ dj 2− d . This is a contradiction if δ
0 we have y Se (x, y) = x Q m x α

(7.2 )

Plugging in x = 1, we see that Q(y) = Se (1, y). We now show that m must in fact be an integer. To see this, note that if m were not an integer, then the degrees of the powers of y appearing in Se (1, y) would have to be separated by at least 2. Hence Se (1, y) would have to be of the form y β R(y c ) for some β 0, c 2, where R is a polynomial. Next, since (d(S), d(S)) is on N (S), we have α = (1 + m)d(S). Since m > 1 when m 1 is not an integer, the maximum possible value of y on the line t1 + mt2 = (1 + m)d(S) β c for t1 , t2 0 is m+1 m d(S) < 2d(S). Thus the degree of y R(y ) is less than 2d(S), and hence the 2d(S) degree of R(y) is less than c d(S). Hence the zeroes of R(y) are of order less than d(S), implying the zeroes of Se (1, y) = y β R(y c ) other than y = 0 are of order less than d(S). This contradicts our assumption that Se (1, y) has a zero r = 0 of order k d(S) and we conclude that m is an integer. Note that since m is an integer so is α. Consequently by (7.2 ) if m is even, then Se (1, y) = ±Se (−1, y), while if m is odd one has Se (1, y) = ±Se (−1, −y). Hence in either case both Se (1, y) and Se (−1, y) have a zero of order k not at the origin. This completes the proof of Lemma 7.2. 2 The next lemma is the crux of this section. To set it up, suppose S(x, y) is not in superadapted coordinates and the bisectrix intersects the interior of an edge e. Then since Se (1, y) or Se (−1, y) has a zero r = 0 of order k d(S), Se (x, y) has zeroes of order k at any point on a curve of the form y = rx m . Hence Se (x, 1) or Se (x, −1) has a zero of order k away from the origin. Thus we may switch the roles of the x and y axes if we want and assume e has equation t1 + mt2 = α for m 1; by Lemma 7.2 m is an integer and Se (1, y) has a zero r = 0 of order k. Lemma 7.3. Suppose S(x, y) is not in superadapted coordinates and the bisectrix intersects the interior of an edge e. As described above, switching the x and y axes if necessary, write the equation of e as t1 + mt2 = α for an integer m 1 and assume Se (1, y) has a zero r = 0 of order k d(S). Then there is a coordinate change of the form (x, y) → (x, y + a(x)) such that a(x)


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is smooth with a(0) = 0, after which one is either Case 1 or 3 of superadapted coordinates, or the following more general version of Case 2. Case 2 . The bisectrix intersects N (S) at a vertex (d, d). Proof. Let Q(y) = Se (1, y), and let (p, q) denote the upper vertex of the edge e; necessarily q > d(S). We will find a smooth function a(x) such that S (x, y) = S(x, y + a(x)) is in one of the following two mutually exclusive categories. Category 1. S (x, y) is either in Cases 1, 2 , or Case 3. Category 2. The bisectrix intersects the interior of an edge e of N (S ) with equation t1 + m t2 = α , m > m 1, such that the upper vertex (p , q ) of e satisfies q < q and such that S (x, y) is not in Case 1. (In particular by Lemma 7.2 Se (1, y) has a zero of order d(S ).) Lemma 7.3 will then follow; there can be at most q iterations of Category 2. We first consider the case where k < q. The function Q(y + r) has a root at y = 0 of order k. We choose a(x) = rx m and define S (x, y) = S(x, y + a(x)) = S(x, y + rx m ). Note that t1 + mt2 = α is a supporting line of N (S ) as it was for N (S), and that there is an edge E of N (S ) on this line whose upper vertex is (p, q). Observe that SE (x, y) = Se (x, y + rx m ) = x α Q( xym + r). Since Q has a zero of order k at r, the lowest power of y appearing in SE (x, y) is y k and therefore E’s lower vertex is at a point (j, k) for some j . Since both vertices of E have ycoordinates at least d(S), they are both in the portion of the line t1 + mt2 = α on or above (d(S), d(S)). Thus the edge E lies wholly on or above the bisectrix. If the bisectrix intersects N(S ) at a vertex or inside the horizontal or vertical rays, one is in Category 1. Otherwise, it must intersect N (S ) in the interior of an edge e whose upper vertex is either (j, k) or a lower vertex. And because t1 + mt2 = α is a supporting line for N (S ) and e lies below E, e will have equation t1 + m t2 = α for some m > m 1. Thus we are either in Case 1 superadapted coordinates (which is in Category 1) or we are in Category 2. Hence when k < q, S (x, y) is in either Category 1 or 2 and we are done. It remains to consider the situation where r is a zero of Q(y) of order q. In this case we have Q(y) = c(y − r)q for some c. For a large integer n we expand S(x, y) as S(x, y) = cx

α

y −r xm

q + Tn (x, y) + En (x, y)

(7.3)

Here the polynomial Tn (x, y) are the terms of S’s Taylor expansion with exponents less than n. For all 0 β, γ < n one has β+γ ∂ n−β En + |y|n−γ ∂x β ∂y γ (x, y) < C |x|

(7.4)

S x, x m y = cx α (y − r)q + x α+1 Tn (x, y) + En x, x m y

(7.5)

Note that

1796


Here Tn (x, y) is also a polynomial. We define s(x, y) =

S(x,x m y) , xα

so that

s(x, y) = c(y − r)q + xTn (x, y) + x −α En x, x m y

(7.6)

We claim that the function s(x, y) is smooth on a neighborhood of (0, r). Off the y-axis smoothness holds because S(x, y) is smooth. One can show that a given derivative of s(x, y) exists when x = 0 and equals that of c(y − r)q + xTn (x, y) for large enough n by examining the difference quotient of a one-lower order derivative of (7.6), inductively assuming this lower-order derivative exists and has the right value when x = 0. Eq. (7.4) ensures that the difference quotient of the lower derivative of x −α En (x, x m y) tends to zero as x goes to zero. We conclude that s(x, y) is smooth on a neighborhood of (0, r). ∂ q−1 s We next use the smooth implicit function theorem on ∂y q−1 and find a smooth function k(x) defined in a neighborhood of x = 0 such that k(0) = r and back to S(x, y) we have

∂ q−1 s (x, k(x)) = 0. ∂y q−1

Transferring this

∂ q−1 S x, x m k(x) = 0 q−1 ∂y

(7.7)

Thus if we let a(x) = x m k(x) and S (x, y) = S(x, y + x m k(x)), for all x we consequently have ∂ q−1 S (x, 0) = 0 ∂y q−1

(7.8)

Thus for every a the Taylor series coefficient Sa q−1 is zero. Next, since t1 + mt2 = α is a supporting line for N (S), this line is also a supporting line for N(S ) and intersects N (S ) at the single vertex (p, q). If S (x, y) is in Category 1 we have nothing to prove, so we may assume we are not in Category 1. Let e denote the edge of N (S ) intersecting the bisectrix and denote its equation by t1 + m t2 = α . Since e lies within the set t1 + mt2 α and is no higher than the vertex (p, q) of N (S ) that is on the supporting line t1 + mt2 = α, we have m > m 1. If the upper vertex (p , q ) of e satisfies q < q, one is in Category 2 and we are done. So we assume this upper vertex is (p, q) itself. If Se (1, y) has a real zero r = 0 of order k < q, one is in the situation above (7.3); there is a smooth b(x) such that S (x, y + b(x)) = S(x, y + a(x) + b(x)) is in Category 1 or 2 as needed. The only other possibility is that Se (1, y) has a single zero r = 0 of order q. But this cannot happen. For this would imply Se (x, y) = c x α ( ym − r )q has a nonvanishing y q−1 term. Conx sequently, for some a the Taylor series coefficient Sa q−1 would be nonzero, contradicting (7.8). Thus the case where Se (1, y) has a single zero of order q does not occur, and we are done with the proof of Lemma 7.3. 2 The final step of the proof of the existence of superadapted coordinates is the following. Lemma 7.4. Suppose one is in Case 2 ; that is, the bisectrix intersects N (S) at a vertex (d, d). Then there exists a smooth coordinate change fixing the origin after which one is in Case 2 of superadapted coordinates.


1797

Proof. Suppose the bisectrix intersects N (S) at a vertex (d, d) but S(x, y) is not in superadapted coordinates. Then (d, d) is on an edge e of N (S) such that Se (1, y) or Se (−1, y) has a zero r = 0 of order d or greater. By Lemma 7.1, Se (x, y) is of the form cx α ( xym − r)d or cy α ( yxm − r)d for positive integers α, m and some nonzero c, r, the first corresponding to the case where e lies below (d, d) and the second corresponding to where it lies above (d, d). Switching axes if necessary, assume that Se (x, y) = cx α ( xym − r)d . One can argue as in (7.3)–(7.8) to obtain a function of the form T (x, y) = S(x, y + a(x)) such that the bisectrix intersects N (T ) at the vertex (d, d), but such that as in the last paragraph of the proof of Lemma 7.3 if (d, d) is the upper vertex of an edge e of N (T ) then Te (1, y) does not have a zero of order d (or greater). We also must have that Te (−1, y) has no zero of order d or greater; for if it did by Lemma 7.1 we could write Te (x, y) in the form cx α ( xym − r)d , which would imply Te (1, y) also has such a zero, a contradiction. There still remains the possibility that (d, d) is the lower vertex of a compact edge f of N (T ) such that Tf (1, y) has a zero of order d or greater. By Lemma 7.1, if this happens Tf (x, y) is of the form cy a ( yxg − r)d where g is an integer. Again using the argument from (7.3) onwards, this time reversing the roles of the x and y variables, there is a smooth coordinate change of the form β : (x, y) → (x − y g h(x, y), y) such that if one denotes T ◦ β by S , then if (d, d) is the lower vertex of an edge f of N (S ) then Sf (x, 1) and Sf (x, −1) do not have any zeroes of order d or greater other than x = 0. This means Sf (1, y) and Sf (−1, y) also have no such zero. For if one of the functions did, Sf (x, y) would have zeroes of order d along some curve y = sx n , which would imply either Sf (x, 1) or Sf (x, −1) had zeroes of order d or greater away from the origin, a contradiction. Furthermore, the slope of e is of the form − m1 for m a positive integer and g > m1 . As a result, e is an edge of N (S ) containing (d, d) and the coordinate change β did not change any of the terms of Te (x, y). Thus Se (x, y) = Te (x, y) and all zeroes of Se (1, y) and Se (−1, y) other than y = 0 have order less than d. Hence we are in superadapted coordinates and the proof of Theorem 7.4 is complete. 2 References [1] G.I. Arkhipov, A.A. Karacuba, V.N. Cubarikov, Trigonometric integrals, Izv. Akad. Nauk SSSR Ser. Mat. 43 (5) (1979) 971–1003. [2] V. Arnold, S. Gusein-Zade, A. Varchenko, Singularities of Differentiable Maps, vol. II, Birkhäuser, Basel, 1988. [3] A. Carbery, M. Christ, J. Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12 (4) (1999) 981–1015. [4] M. Christ, Hilbert transforms along curves, I, Nilpotent groups, Ann. of Math. (2) 122 (3) (1985) 575–596. [5] J. Denef, J. Nicaise, P. Sargos, Oscillating integrals and Newton polyhedra, J. Anal. Math. 95 (2005) 147–172. [6] J. Denef, P. Sargos, Polyedre de Newton et distribution f+s , II, Math. Ann. 293 (2) (1992) 193–211. [7] M.V. Fedoryuk, The Saddle-Point Method, Nauka, Moscow, 1977. [8] M. Greenblatt, Resolution of singularities, asymptotic expansions of integrals over sublevel sets, and applications, submitted for publication. [9] M. Greenblatt, Sharp L2 estimates for one-dimensional oscillatory integral operators with C ∞ phase, Amer. J. Math. 127 (3) (2005) 659–695. [10] I. Ikromov, M. Kempe, D. Müller, Sharp Lp estimates for maximal operators associated to hypersurfaces in R3 for p > 2, preprint. [11] I. Ikromov, D. Müller, On adapted coordinate systems, Trans. Amer. Math. Soc., in press. [12] D.H. Phong, E.M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997) 107– 152. [13] D.H. Phong, E.M. Stein, J. Sturm, On the growth and stability of real-analytic functions, Amer. J. Math. 121 (3) (1999) 519–554.

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[14] V. Rychkov, Sharp L2 bounds for oscillatory integral operators with C ∞ phases, Math. Z. 236 (2001) 461–489. [15] H. Schulz, Convex hypersurfaces of finite type and their Fourier transforms, Indiana Univ. Math. J. 40 (4) (1991) 1267–1275. [16] E. Stein, Harmonic Analysis; Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., vol. 43, Princeton University Press, Princeton, NJ, 1993. [17] A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175– 196. [18] I.M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Tr. Mat. Inst. Steklova 23 (1947), 109 pp.


Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations Pedro J. Mendez-Hernandez a , Minoru Murata b,∗ a Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica b Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan

Received 5 September 2008; accepted 31 May 2009 Available online 27 June 2009 Communicated by L. Gross

Abstract We consider nonnegative solutions of a parabolic equation in a cylinder D × I , where D is a noncompact domain of a Riemannian manifold and I = (0, T ) with 0 < T ∞ or I = (−∞, 0). Under the assumption [SSP] (i.e., the constant function 1 is a semismall perturbation of the associated elliptic operator on D), we establish an integral representation theorem of nonnegative solutions: In the case I = (0, T ), any nonnegative solution is represented uniquely by an integral on (D × {0}) ∪ (∂M D × [0, T )), where ∂M D is the Martin boundary of D for the elliptic operator; and in the case I = (−∞, 0), any nonnegative solution is represented uniquely by the sum of an integral on ∂M D × (−∞, 0) and a constant multiple of a particular solution. We also show that [SSP] implies the condition [SIU] (i.e., the associated heat kernel is semi-intrinsically ultracontractive). © 2009 Elsevier Inc. All rights reserved. Keywords: Semismall perturbation; Semi-intrinsic ultracontractivity; Parabolic equation; Nonnegative solution; Integral representation; Martin boundary; Elliptic equation

* Corresponding author. Fax: +81 3 5734 2738.

E-mail addresses: [email protected] (P.J. Mendez-Hernandez), [email protected] (M. Murata). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.028

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P.J. Mendez-Hernandez, M. Murata / Journal of Functional Analysis 257 (2009) 1799–1827

1. Introduction This paper is a continuation of [34]. It is concerned with integral representations of nonnegative solutions to parabolic equations and perturbation theory for elliptic operators. We consider nonnegative solutions of a parabolic equation (∂t + L)u = 0 in D × I,

(1.1)

where ∂t = ∂/∂t, L is a second order elliptic operator on a noncompact domain D of a Riemannian manifold M, and I is a time interval: I = (0, T ) with 0 < T ∞ or I = (−∞, 0). During the last few decades, much attention has been paid to the structure of all nonnegative solutions to a parabolic equation, perturbation theory for elliptic operators, and their relations. (See [1,2,4–6,11,14,17,19,20,22,25–34,36–38,40–42].) Among others, Murata [34] has established integral representation theorems of nonnegative solutions to Eq. (1.1) under the condition [IU] (i.e., intrinsic ultracontractivity) on the minimal fundamental solution p(x, y, t) for (1.1). Furthermore, he has shown that [IU] implies [SP] (i.e., the constant function 1 is a small perturbation of L on D). It is known [30] that [SP] implies [SSP] (i.e., 1 is a semismall perturbation of L on D). In this paper, we show that [SSP] implies [SIU] (i.e., semi-intrinsic ultracontractivity) and give integral representation theorems of nonnegative solutions to (1.1) under the condition [SSP]. We consider that [SSP] is one of the weakest possible condition for getting “explicit” integral representation theorems. Now, in order to state our main results, we fix notations and recall several notions and facts. Let M be a connected separable n-dimensional smooth manifold with Riemannian metric of class C 0 . Denote by ν the Riemannian measure on M. Tx M and T M denote the tangent space to M at x ∈ M and the tangent bundle, respectively. We denote by End(Tx M) and End(T M) the set of endomorphisms in Tx M and the corresponding bundle, respectively. The inner product on T M is denoted by X, Y , where X, Y ∈ T M; and |X| = X, X1/2 . The divergence and gradient with respect to the metric on M are denoted by div and ∇, respectively. Let D be a noncompact domain of M. Let L be an elliptic differential operator on D of the form Lu = −m−1 div(mA∇u) + V u,

(1.2)

where m is a positive measurable function on D such that m and m−1 are bounded on any compact subset of D, A is a symmetric measurable section on D of End(T M), and V is a realvalued measurable function on D such that V

p ∈ Lloc (D, m dν)

n for some p > max , 1 . 2

p

Here Lloc (D, m dν) is the set of real-valued functions on D locally pth integrable with respect to m dν. We assume that L is locally uniformly elliptic on D, i.e., for any compact set K in D there exists a positive constant λ such that λ|ξ |2 Ax ξ, ξ λ−1 |ξ |2 ,

x ∈ K, (x, ξ ) ∈ T M.


1801

We assume that the quadratic form Q on C0∞ (D) defined by Q[u] =

A∇u, ∇u + V u2 m dν

D

is bounded from below, and put ∞ 2 u m dν = 1 . λ0 = inf Q[u]; u ∈ C0 (D), D

Then, for any a < λ0 , (L − a, D) is subcritical, i.e., there exists the (minimal positive) Green function of L − a on D. We denote by LD the selfadjoint operator in L2 (D; m dν) associated with the closure of Q. The minimal fundamental solution for (1.1) is denoted by p(x, y, t), which is equal to the integral kernel of the semigroup e−tLD on L2 (D, m dν). Let us recall several notions related to [SSP]. [IU] λ0 is an eigenvalue of LD ; and there exists, for any t > 0, a constant Ct > 0 such that p(x, y, t) Ct φ0 (x)φ0 (y),

x, y ∈ D,

where φ0 is the normalized positive eigenfunction for λ0 . This notion was introduced by Davies and Simon [13], and investigated extensively because of its important consequences (see [7–10,12,23,24,31,34,42], and references therein). It looks, on the surface, not related to perturbation theory. But it has turned out [34] that [IU] implies the following condition [SP] for any a < λ0 . [SP] The constant function 1 is a small perturbation of L − a on D, i.e., for any ε > 0 there exists a compact subset K of D such that G(x, z)G(z, y)m(z) dν(z) εG(x, y),

x, y ∈ D \ K,

D\K

where G is the Green function of L − a on D. This condition is a special case of the notion introduced by Pinchover [37]. Recall that [SP] implies the following condition [SSP] (see [30]). [SSP] The constant function 1 is a semismall perturbation of L − a on D, i.e., with x 0 being a fixed reference point in D, for any ε > 0 there exists a compact subset K of D such that D\K

G x 0 , z G(z, y)m(z) dν(z) εG x 0 , y ,

y ∈ D \ K.

1802


This condition [SSP] implies that LD admits a complete orthonormal base of eigenfunctions {φj }∞ j =0 with eigenvalues λ0 < λ1 λ2 · · · repeated according to multiplicity; furthermore, for any j = 1, 2, . . . , the function φj /φ0 has a continuous extension [φj /φ0 ] up to the Martin boundary ∂M D of D for L − a (see Theorem 6.3 of [38]). We show in this paper that [SSP] also implies the following condition [SIU]. [SIU] λ0 is an eigenvalue of LD ; and there exist, for any t > 0 and compact subset K of D, positive constants A and B such that Aφ0 (x)φ0 (y) p(x, y, t) Bφ0 (x)φ0 (y),

x ∈ K, y ∈ D.

This notion was introduced by Bañuelos and Davis [9], where they called it one half IU. Here we should recall that [IU] implies that for any t > 0 there exists a constant ct > 0 such that ct φ0 (x)φ0 (y) p(x, y, t),

x, y ∈ D.

We see that the same argument as in the proof of Theorem 3.1 in [25] (or the argument in the proof of Theorem 1.2 below) shows that [SIU] implies the following condition [NUP] (i.e., nonuniqueness for the positive Cauchy problem). [NUP] The Cauchy problem (∂t + L)u = 0 in D × (0, T ),

u(x, 0) = 0 on D

(1.3)

admits a nonnegative solution u which is not identically zero. We say that [UP] holds for (1.3) when any nonnegative solution of (1.3) is identically zero. We note that [UP] implies that the constant function 1 is a “big” perturbation of L − a on D in some sense (see Theorem 2.1 of [32]). Fix a < λ0 , and suppose that [SSP] holds. Let D ∗ = D ∪ ∂M D be the Martin compactification of D for L − a, which is a compact metric space. Denote by ∂m D the minimal Martin boundary of D for L − a, which is a Borel subset of the Martin boundary ∂M D of D for L − a. Here, we note that ∂M D and ∂m D are independent of a in the following sense: if [SSP] holds, then for any b < λ0 there is a homeomorphism Φ from the Martin compactification of D for L − a onto that for L − b such that Φ|D = identity, and Φ maps the Martin boundary and minimal Martin boundary of D for L − a onto those for L − b, respectively (see Theorem 1.4 of [30]). Now, we are ready to state our main results. In the following theorems we assume that [SSP] holds for some fixed a < λ0 . Theorem 1.1. The condition [SSP] implies [SIU]. Theorem 1.2. Assume [SSP]. Then, for any ξ ∈ ∂M D there exists the limit lim

D y→ξ

p(x, y, t) ≡ q(x, ξ, t), φ0 (y)

x ∈ D, t ∈ R.

(1.4)


1803

Here, as functions of (x, t), {p(x, y, t)/φ0 (y)}y converges to q(x, ξ, t) as y → ξ uniformly on any compact subset of D × R. Furthermore, q(x, ξ, t) is a continuous function on D × ∂M D × R such that q > 0 on D × ∂M D × (0, ∞), q =0

(1.5)

on D × ∂M D × (−∞, 0],

(1.6)

(∂t + L)q(·, ξ, ·) = 0 on D × R.

(1.7)

Theorem 1.3. Assume [SSP]. Consider Eq. (1.1) for I = (0, T ) with 0 < T ∞. Then, for any nonnegative solution u of (1.1) there exists a unique pair of Borel measures μ on D and λ on ∂M D × [0, T ) such that λ is supported by the set ∂m D × [0, T ), and u(x, t) = p(x, y, t) dμ(y) + q(x, ξ, t − s) dλ(ξ, s) (1.8) D

∂M D×[0,t)

for any (x, t) ∈ D × I . Conversely, for any Borel measures μ on D and λ on ∂M D × [0, T ) such that λ is supported by ∂m D × [0, T ) and p x 0 , y, t dμ(y) < ∞, 0 < t < T , (1.9)

D

q x 0 , ξ, t − s dλ(ξ, s) < ∞,

0 < t < T,

(1.10)

∂M D×[0,t)

where x 0 is a fixed point in D, the right-hand side of (1.8) is a nonnegative solution of (1.1) for I = (0, T ) with 0 < T ∞. The proof of this theorem will be given in Sections 4 and 5. It is based upon the abstract integral representation theorem established in [34], without assuming [IU], via a parabolic Martin representation theorem and Choquet’s theorem (see [18,21,35]). Its key step is to identify the parabolic Martin boundary. This theorem is an improvement of Theorem 1.2 of [34]; where the condition [IU], which is more stringent than [SSP], is assumed. It is also an answer to a problem raised in Remark 4.13 of [34]. Note that (1.8) gives explicit integral representations of nonnegative solutions to (1.1) provided that the Martin boundary ∂M D of D for L − a is determined explicitly. We consider that [SSP] is one of the weakest possible condition for getting such explicit integral representations. Let us recall that when [UP] holds for (1.3), the structure of all nonnegative solutions to (1.1) for I = (0, T ) is extremely simple. Namely, the following theorem holds (see [5]). Fact AT. Assume [UP]. Then, for any nonnegative solution u of (1.1) with I = (0, T ), there exists a unique Borel measure μ on D such that (1.11) u(x, t) = p(x, y, t) dμ(y), (x, t) ∈ D × I. D

1804


Conversely, for any Borel measure μ on D satisfying (1.9), the right-hand side of (1.11) is a nonnegative solution of (1.1) with I = (0, T ). It is quite interesting that when [UP] holds, the elliptic Martin boundary disappears in the parabolic representation theorem; while it enters in many cases of [NUP]. Finally, we state an integral representation theorem for the case I = (−∞, 0). Theorem 1.4. Assume [SSP]. Consider Eq. (1.1) for I = (−∞, 0). Then, for any nonnegative solution u of (1.1) there exists a unique pair of a nonnegative constant α and a Borel measure λ on ∂M D × (−∞, 0) supported by the set ∂m D × (−∞, 0) such that u(x, t) = αe−λ0 t φ0 (x) +

q(x, ξ, t − s) dλ(ξ, s)

(1.12)

∂M D×(−∞,t)

for any (x, t) ∈ D × (−∞, 0). Conversely, for any nonnegative constant α and a Borel measure λ on ∂M D × (−∞, 0) such that it is supported by ∂m D × (−∞, 0) and

q x 0 , ξ, t − s dλ(ξ, s) < ∞,

−∞ < t < 0,

(1.13)

∂M D×(−∞,t)

the right-hand side of (1.12) is a nonnegative solution of (1.1). This theorem is an improvement of Theorem 6.1 of [34], where [IU] is assumed instead of [SSP]. Here, in order to illustrate a scope of Theorems 1.3 and 1.4, we give a simple example. Further examples will be given in Section 7. Example 1.5. Let D be a domain in R2 with finite area. Then, by Theorem 6.1 of [33], the constant function 1 is a small perturbation of L = − on D. Thus Theorems 1.3 and 1.4 hold true for the heat equation (∂t − )u = 0 in D × I. Note that there exist many bounded planar domains for which the heat semigroup is not intrinsically ultracontractive (see Example 1 of [13] and Section 4 of [9]). Thus, the last assertion of this example is new for such domains. The remainder of this paper is organized as follows. In Section 2 we prove Theorem 1.1, and Theorem 1.2 is proved in Section 3. Sections 4 and 5 are devoted to the proof of Theorem 1.3. In Section 4 we show it in the case of I = (0, ∞). In Section 5 we show it in the case of I = (0, T ) with 0 < T < ∞ by making use of results to be given in Section 4. Theorem 1.4 is proved in Section 6. Finally we shall give two more concrete examples in Section 7 with emphasis on sharpness of concrete sufficient conditions of [SSP].


1805

2. [SSP] implies [SIU] In this section we prove Theorem 1.1. Proof of Theorem 1.1. We may and shall assume that a = 0 < λ0 . Let G be the Green function of L on D. For any t > 0, put ∞ Gt (x, y) =

p(x, y, s) ds, t

t G (x, y) = t

p(x, y, s) ds. 0

Then G = Gt + Gt . Let us show that for any t > 0 and any compact subset K of D there exists a constant A > 0 such that Aφ0 (x)φ0 (y) p(x, y, t),

x ∈ K, y ∈ D.

(2.1)

Fix a compact subset K. We may assume that x 0 ∈ K. Let K1 ⊂ D be a compact neighborhood of K. Then the same argument as in the proof of Theorem 1.5 of [30] shows that C −1 G x 0 , z φ0 (z) CG x 0 , z ,

z ∈ D \ K1 ,

(2.2)

By [SSP] and (2.2), there exists a compact subset K2 ⊃ K1 such that φ0 (z) G(z, y) dμ(z) t φ0 (y), y ∈ D \ K2 ,

(2.3)

for some constant C > 0. Fix t > 0, and put t =

1 1 − e−tλ0 . 2λ0

D\K2

where dμ(z) = m(z) dν(z). Since φ0 (y) = λ0

G(y, z)φ0 (z) dμ(z), D

and G(y, z) = G(z, y), (2.3) yields φ0 (y) λ0

Gt (z, y) φ0 (z) dμ(z) +

K2

for any y ∈ D \ K2 . By Fubini’s theorem,

K2

Gt (z, y)φ0 (z) dμ(z) + t φ0 (y)

(2.4)

1806


∞ Gt (z, y)φ0 (z) dμ(z) =

ds t

D

p(z, y, s)φ0 (z) dμ(z) D

∞ =

e−λ0 s φ0 (y) ds

t

1 −λ0 t e φ0 (y). λ0

= Thus

Gt (z, y)φ0 (z) dμ(z) K2

1 −λ0 t e φ0 (y). λ0

This together with (2.4) implies t φ0 (y)

Gt (z, y)φ0 (z) dμ(z).

(2.5)

K2

Choose a compact subset K3 whose interior includes K2 . By the parabolic Harnack inequality, there exists a constant C1 depending on t, K2 , K3 such that p(z, y, s) C1 p(x, y, 2t), for any x, z ∈ K2 , y ∈ D \ K3 , and 0 < s t. We have t G (z, y) = t

p(z, y, s) ds C1 tp x 0 , y, 2t ,

z ∈ K2 , y ∈ D \ K3 .

(2.6)

0

Thus

G (z, y)φ0 (z) dμ(z) C1 t t

K2

φ0 (z) dz p x 0 , y, 2t .

K2

This together with (2.5) implies φ0 (y) C2 p x 0 , y, 2t ,

y ∈ D \ K3 ,

where C2 =

1 C1 t t

φ0 (z) dμ(z). K2

(2.7)


1807

By the parabolic Harnack inequality, p x 0 , y, 2t Cp(x, y, 3t),

x ∈ K, y ∈ D,

for some constant C > 0. This together with (2.7) yields the desired inequality (2.1). It remains to show that for any t > 0 and a compact subset K of D there exists a constant B such that p(x, y, t) B φ0 (x)φ0 (y),

x ∈ K, y ∈ D.

(2.8)

Fix a compact subset K. We may assume that x 0 ∈ K. Let K1 ⊂ D be a compact neighborhood of K. By the parabolic Harnack inequality there exists a constant c > 0 such that cp x 0 , y, t p(z, y, 2t),

z ∈ K1 , y ∈ D.

Thus, for any y ∈ D, e−2tλ0 φ0 (y) =

φ0 (z) p(z, y, 2t) dμ(z) D

φ0 (z)p(z, y, 2t) dμ(z)

K1

c

φ0 (z) dμ(z) p x 0 , y, t .

K1

This implies (2.8), since Cp x 0 , y, t p(x, y, t/2),

x ∈ K, y ∈ D,

for some constant C > 0. (We should note that in proving (2.8) we have only used the consequence of [SSP] that φ0 is a positive eigenfunction.) 2 Remark 2.1. It is an open problem whether [SIU] implies [SSP] or not. Furthermore, the problem whether [SSP] implies [SP] or not in the case n > 1 is still open. 3. Parabolic Martin kernels In this section we prove Theorem 1.2. Throughout the present section we assume [SSP]. We may and shall assume that a = 0 < λ0 . Let G be the Green function of L on D. For any 0 < δ < t, put t Gtδ (x, y) =

p(x, y, s) ds.

(3.1)

δ

We denote by ∂M D the Martin boundary of D for L. In order to prove Theorem 1.2, we need two lemmas.

1808


Lemma 3.1. Let ξ ∈ ∂M D. Suppose that a sequence {yn }∞ n=1 ⊂ D converges to ξ , and there exists the limit Gtδ (z, yn ) = w(z, t), n→∞ φ0 (yn )

z ∈ D.

lim

(3.2)

Then lim

n→∞ D

Gt (z, yn ) dμ(z) = G(x, z) δ φ0 (yn )

G(x, z)w(z, t) dμ(z)

(3.3)

D

for any x ∈ D, where dμ(z) = m(z) dν(z). Proof. Fix x ∈ D. Let K1 ⊂ D be a compact neighborhood of x. By [SSP], there exists a constant C > 0 such that C −1 φ0 (y) G(x, y) Cφ0 (y),

y ∈ D \ K1 .

(3.4)

Let > 0. Then there exists a compact subset K ⊃ K1 such that G(x, z)

G(z, y) dμ(z) < , G(x, y) 3C

y ∈ D \ K.

D\K

Thus, for n sufficiently large, D\K

t Gδ (z, yn ) CG(z, yn ) dμ(z) dμ(z) < . G(x, z) G(x, z) φ0 (yn ) G(x, yn ) 3 D\K

By Fatou’s lemma,

G(x, z)w(z, t) dμ(z) . 3

D\K

By Theorem 1.1, there exist constants A1 and A2 such that A1 φ0 (x)φ0 (y) p(x, y, δ) A2 φ0 (x)φ0 (y),

x ∈ K, y ∈ D.

Then, for any t > δ, the semigroup property yields A1 e−λ0 (t−δ) φ0 (x)φ0 (y) p(x, y, t) A2 e−λ0 (t−δ) φ0 (x)φ0 (y) for any x ∈ K, y ∈ D. Thus there exists a constant B > 0 such that for any n Gtδ (z, yn ) Bφ0 (z), φ0 (yn )

z ∈ K.

(3.5)


1809

Then Lebesgue’s dominated convergence theorem yields lim

n→∞ K

t G (z, yn ) dμ(z) = G(x, z)w(z, t) dμ(z). G(x, z) δ φ0 (yn ) K

Therefore, for n sufficiently large,

t G(x, z) Gδ (z, yn ) dμ(z) − G(x, z) w(z, t) dμ(z) < . φ (y ) 0

D

n

D

2

This shows (3.3).

By Lemma 6.1 of [38], it follows from [SSP] that there exists the limit lim

D y→ξ

G(y, z) = h(ξ, z), φ0 (y)

(ξ, z) ∈ ∂M D × D,

(3.6)

and h is a positive continuous function on ∂M D × D. From this we show the following lemma. Lemma 3.2. Under the same assumptions as in Lemma 3.1, one has

h(ξ, z) Gtδ (z, x) dμ(z) = lim

n→∞

D

=

D

G(yn , z) t G (z, x) dμ(z) φ0 (yn ) δ

G(x, z) w(z, t) dμ(z)

(3.7)

D

for any x ∈ D. Proof. Fix x ∈ D. Let K1 ⊂ D be a compact neighborhood of x. By Theorem 1.1, (3.4) and (3.5), there exists a constant C1 > 0 such that C1 G(z, x) Gtδ (z, x) G(z, x),

z ∈ D \ K1 .

Let > 0. By [SSP], there exists a compact subset K ⊃ K1 such that D\K

G(yn , z) t Gδ (z, x) dμ(z) < , φ0 (yn ) 3

(3.8)

for n sufficiently large. By Fatou’s lemma, D\K

h(ξ, z)Gtδ (z, x) dμ(z) . 3

(3.9)

1810


On the other hand, for any sufficiently large n

G(yn , z) t Gδ (z, x) C2 , φ0 (yn )

z ∈ K,

where C2 is a positive constant. By Lebesgue’s dominated convergence theorem, lim

n→∞ K

G(yn , z) t G (z, x) dμ(z) = φ0 (yn ) δ

h(ξ, z)Gtδ (z, x) dμ(z).

(3.10)

K

Combining (3.8), (3.9) and (3.10), we get the first equality. It remains to show the second equality of (3.7). By Fubini’s theorem and the symmetry p(x, y, t) = p(y, x, t), we have

∞ G(yn , z)Gtδ (z, x) dμ(z) =

D

t ds p(yn , x, r + s) =

dr δ

0

G(x, z)Gtδ (z, yn ) dμ(z). D

2

This together with Lemma 3.1 implies the second equality.

Proof of Theorem 1.2. Let {yj }∞ j =1 ⊂ D be any sequence converging to ξ ∈ ∂M D. Put uj (x, t) =

p(x, yj , t) φ0 (yj )

uj (x, t) = 0 for t 0.

for t > 0,

(3.11)

Since [SIU] holds, it follows from the parabolic Harnack inequality and local a priori estimates for nonnegative solutions to parabolic equations (see [6] and [16]) that there exists a subsequence {ujk }∞ k=1 such that ujk converges, as k → ∞, uniformly on any compact subset of D × R to a solution u of the equation (∂t + L)u = 0 in D × R satisfying u > 0 on D × (0, ∞) and u = 0 on D × (−∞, 0]. Thus, in order to prove Theorem 1.2, it suffices to show that the limit function u is independent of {yjk }∞ k=1 and uniquely determined }∞ be two sequences in D converging to ξ . Define u by (3.11), and by ξ . Let {yj }∞ and {y j j n=1 n=1 }∞ converge to u and u , and {u uj by (3.11) with yj replaced by yj . Suppose that {uj }∞ j j =1 j =1 respectively. For any t > δ > 0, put t w(z, t) =

u(z, s) ds, δ

t

w (z, t) = δ

u (z, s) ds.


1811

Then we have Gtδ (z, yn ) = w(z, t), n→∞ φ0 (yn ) lim

Gtδ (z, yn ) = w (z, t). n→∞ φ0 (yn ) lim

By Lemma 3.2, G(x, z)w(z, t) dμ(z) = h(ξ, z)Gtδ (z, x) dμ(z) = G(x, z)w (z, t) dμ(z). D

D

D

Thus w(x, t) = w (x, t), which implies u(x, t) = u (x, t). This completes the proof of Theorem 1.2. 2 4. Integral representations; the case I = (0, ∞) In this section we prove Theorem 1.3 in the case T = ∞. We first state an abstract integral representation theorem which holds without [SSP]. For x ∈ D and r > 0, we denote by B(x, r) the geodesic ball in the Riemannian manifold M with center x and radius r. Let x 0 be a reference point in D. Choose a nonnegative continuous function a on D such that a(x) = 1 on B(x 0 , r 0 ) and a(x) = 0 outside B(x 0 , 2r 0 ) for some r 0 > 0 with B(x 0 , 3r 0 ) D. Choose a nonnegative continuous function b on R such that 0 < b(t) < eγ t on (1, ∞) for some γ < λ0 , and b(t) = 0 on (−∞, 1]. Denote by β the measure defined by dβ(x, t) = a(x)b(t)m(x) dν(x) dt. For any nonnegative measurable function u on Q = D × (0, ∞), we write u(x, t) dβ(x, t). β(u) = Q

Denote by P (Q) the set of all nonnegative solutions of (1.1) with I = (0, ∞), and put

Pβ (Q) = u ∈ P (Q); β(u) < ∞ . Note that for any u ∈ P (Q) there exists a function b as above such that β(u) < ∞; thus P (Q) = β Pβ (Q). Furthermore, the parabolic Harnack inequality shows that if β(u) = 0, then u = 0. β

Now, let us define the β-Martin boundary ∂M Q of Q with respect to ∂t + L along the line given in [21] and [18]. Put p(x, y, t − s), t > s, x, y ∈ D, p(x, t; y, s) = 0, t s, x, y ∈ D. Define the β-Martin kernel Kβ by Kβ (x, t; y, s) =

p(x, t; y, s) , β(p(·; y, s))

(x, t), (y, s) ∈ Q,

where β(p(·; y, s)) = Q p(z, r; y, s) dβ(z, r). Note that β(p(·; y, s)) < ∞ for any (y, s) ∈ Q, since 0 < b(t) < eγ t on (1, ∞) for some γ < λ0 . Let {Dj }∞ j =1 be an exhaustion of D such

1812


that each Dj is a domain with smooth boundary, Dj Dj +1 D, ∞ j =1 Dj = D, and 0 0 B(x , 3r ) D1 . Put Qj = Dj × (1/j, j ). For Y = (y, s), Z = (z, r) ∈ Q, let δβ (Y, Z) =

∞

2−j sup

j =1

X∈Qj

|Kβ (X; Y ) − Kβ (X; Z)| . 1 + |Kβ (X; Y ) − Kβ (X; Z)|

Then we see that δβ is a metric on Q, and the topology on Q induced by δβ is equivalent to the original topology of Q. Denote by Qβ∗ the completion of Q with respect to the metric δβ . Put β k ∞ ∂M Q = Qβ∗ \ Q. A sequence {Y k }∞ k=1 in Q is called a fundamental sequence if {Y }k=1 has no ∞ k point of accumulation in Q and {Kβ (·; Y )}k=1 converges uniformly on any compact subset of Q to a nonnegative solution of (1.1) with I = (0, ∞). By the local a priori estimates for solutions β of (1.1), for any Ξ ∈ ∂M Q there exist a unique nonnegative solution Kβ (·; Ξ ) of (1.1) and a fundamental sequence {Y k }∞ k=1 in Q such that lim

k→∞

∞ j =1

2−j sup X∈Qj

|Kβ (X; Y k ) − Kβ (X; Ξ )| = 0. 1 + |Kβ (X; Y k ) − Kβ (X; Ξ )|

Thus the metric δβ is canonically extended to Qβ∗ . Furthermore, Qβ∗ becomes a compact metric space, since by the parabolic Harnack inequality, any sequence {Y k }∞ k=1 with no point of β accumulation in Q has a fundamental subsequence. We call Kβ (·; Ξ ), ∂M Q and Qβ∗ the βMartin kernel, β-Martin boundary and β-Martin compactification for (Q, ∂t + L), respectively. Note that β(Kβ (·; Ξ )) 1 by Fatou’s lemma; and so Kβ (·; Ξ ) ∈ Pβ (Q). A nonnegative solution u ∈ Pβ (Q) is said to be minimal if for any nonnegative solution v u there exists a nonnegative constant C such that v = Cu. Put

β β Q = Ξ ∈ ∂M Q; Kβ (·; Ξ ) is minimal and β Kβ (·; Ξ ) = 1 , ∂m which we call the minimal β-Martin boundary for (Q, ∂t + L). β Observe that D × [0, ∞) is embedded into Qβ∗ , and D × {0} ⊂ ∂M Q. Indeed, with y ∈ D k k fixed, for any sequence {Y k }∞ k=1 in Q with limk→∞ Y = (y, 0) we have limk→∞ Kβ (x, t; Y ) = p(x, t; y, 0)/β(p(·; y, 0)); furthermore, Kβ (·; y, 0) = Kβ (·; z, 0) if y = z. We also note that k any sequence {Y k = (y k , s k )}∞ k=1 in Q with limk→∞ s = ∞ is a fundamental sequence, since β k limk→∞ Kβ (·; Y ) = 0. We denote by the point in ∂M Q corresponding to the Martin kernel which is identically zero: Kβ (·; ) = 0. Put β Lβm Q = ∂m Q \ D × {0} ∪ { } . We obtain the following abstract integral representation theorem in the same way as in the proof of Theorem 2.1 and Lemma 2.2 of [34]. Theorem 4.1. For any u ∈ Pβ (Q), there exists a unique pair of finite Borel measures κ on D and β β λ on ∂M Q \ (D × {0}) such that λ is supported by the set Lm Q, p(x, t; y, 0) u(x, t) = dκ(y) + Kβ (x, t; Ξ ) dλ(Ξ ) (4.1) β(p(·; y, 0)) D

Lβm Q


1813

for any (x, t) ∈ Q, and β(u) = κ(D) + λ Lβm Q .

(4.2)

Furthermore, the function

p(x, t; y, 0) dκ(y) β(p(·; y, 0))

v(x, t) = u(x, t) − D

is a nonnegative solution of the equation (∂t + L)v = 0

in D × R

such that v = 0 on D × (−∞, 0]. β Conversely, for any finite Borel measures κ on D and λ on ∂M Q \ (D × {0}) such that λ is β supported by the set Lm Q, the right-hand side of (4.1) belongs to Pβ (Q). We put Pβ0 (Q) = v ∈ Pβ (Q); lim v(x, t) = 0 on D . t↓0

We show Theorem 1.3 on the basis of Theorem 4.1. To this end it suffices to show (1.8) for β u ∈ Pβ0 (Q). The key step in the proof is to identify Lm Q. Under the condition [SSP], we shall β

show that Lm Q = ∂m D × [0, ∞). In the remainder of this section we assume [SSP]. We may and shall assume that a = 0 < λ0 . Lemma 4.2. For any domains U and W with U W D, there exist positive constants C and α such that p(x, y, t) Cf (t)φ0 (x)φ0 (y),

x ∈ U, y ∈ D \ W, t > 0,

(4.3)

where f (t) = e−α/t for 0 < t < 1, and f (t) = e−λ0 t for t 1. Furthermore, q(x, ξ, t) Cf (t)φ0 (x), G(x, y) Cφ0 (x)φ0 (y),

x ∈ U, ξ ∈ ∂M D, t > 0, x ∈ U, y ∈ D \ W,

(4.4) (4.5)

where G is the Green function of L on D. This lemma is shown in the same way as Lemmas 4.2 and 4.4 of [34]. Let K(x, ξ ) be the Martin kernel for L on D with reference point x 0 ∈ D, i.e., K(x 0 , ξ ) = 1, ξ ∈ ∂M D. The following lemma gives a relation between K and q.

1814


Lemma 4.3. For any ξ ∈ ∂M D, ∞ G(x, y) = q(x, ξ, t) dt, x ∈ D, lim D y→ξ φ0 (y) ∞ 0 0 q(x, ξ, t) dt K(x, ξ ) = ∞ , x ∈ D. 0 0 q(x , ξ, t) dt

(4.6)

(4.7)

This lemma is shown in the same way as Lemma 4.5 of [34]. Lemma 4.4. Let ξ, η ∈ ∂M D, 0 s, r < ∞ and C > 0. If q(x, ξ, t − s) = Cq(x, η, t − r),

(x, t) ∈ Q,

then ξ = η, s = r and C = 1. Proof. Since q(x, ξ, τ ) > 0 for τ > 0 and q(x, ξ, τ ) = 0 for τ 0, we obtain that s = r. Thus q(x, ξ, τ ) = q(x, η, τ ). This together with (4.7) implies that K(·, ξ ) = K(·, η) on D. Hence ξ = η, and so C = 1. 2 Now, let β be a measure on Q = D × (0, ∞) as described in the beginning of this section: dβ(x, t) = a(x)b(t)m(x) dν(x) dt. The following proposition determines the β-Martin boundary β ∂M Q, β-Martin compactification Qβ∗ , and β-Martin kernel Kβ for (∂t + L, Q). Recall that p(x, t; y, s) = p(x, y, t − s) and Kβ (·; y, s) = p(·; y, s)/β(p(·; y, s)). We write q(x, t; ξ, s) = q(x, ξ, t − s) for ξ ∈ ∂M D and 0 s < ∞. Proposition 4.5. β

(i) The β-Martin boundary ∂M Q of Q for ∂t + L is equal to the disjoint union of D × {0}, ∂M D × [0, ∞) and the one point set { }: β

∂M Q = D × {0} ∪ ∂M D × [0, ∞) ∪ { }.

(4.8)

β

In particular, ∂M Q does not depend on β. (ii) The β-Martin compactification Qβ∗ of Q for ∂t + L is homeomorphic to the disjoint union of the topological product D ∗ × [0, ∞) and the one point set { }, where a fundamental neighborhood system of is given by the family { } ∪ D ∗ × (N, ∞), N > 1. In particular, Qβ∗ does not depend on β. (iii) The β-Martin kernel Kβ is given as follows. For (x, t) ∈ Q, p(x, t; y, 0) , (y, 0) ∈ D × {0}, β(p(·; y, 0)) q(x, t; ξ, s) , (ξ, s) ∈ ∂M D × [0, ∞), Kβ (x, t; ξ, s) = β(q(·; ξ, s)) Kβ (x, t; y, 0) =

and Kβ (x, t; ) = 0.

(4.9) (4.10)


1815

This proposition is shown in the same way as Proposition 4.8 of [34]. Lemma 4.6. Let (ξ, s) ∈ (∂M D \ ∂m D) × [0, ∞). Then there exists a finite Borel measure γ on ∂M D supported by ∂m D such that q(·; ξ, s) =

(4.11)

q(·; η, s) dγ (η).

∂m D

Proof. For reader’s convenience, we give a sketch of the proof for the case s = 0. (For details, see the proof of Lemma 4.10 of [34].) By the elliptic Martin representation theorem, there exists a unique finite Borel measure μ on ∂M D supported by ∂m D such that K(x, ξ ) =

K(x, η) dμ(η).

∂m D

This together with (4.7) implies

∞

∞ q(x, ξ, t) dt =

∂m D

0

(4.12)

q(x, η, t) dt dγ (η), 0

where dγ (η) = [H (x 0 , ξ )/H (x 0 , η)] dμ(η) with ∞ H (x, η) =

q(x, η, t) dt. 0

For α > 0, denote by Gα the Green function of L + α on D. By the resolvent equation and [SSP], we then have ∞ e

−αt

q(x, η, t) dt =

0

∞

∞ q(x, η, t) dt − α

Gα (x, z)

q(z, η, t) dt m(z) dν(z),

D

0

(4.13)

0

for any η ∈ ∂M D. By combining (4.12) and (4.13), we get ∞ e 0

−αt

∞

q(x, η, t) dγ (η) dt = ∂m D

Thus the Laplace transforms of q(x, ξ, t) and holds. 2

e−αt q(x, ξ, t) dt.

0

∂m D q(x, η, t) dγ (η)

coincide; and so (4.11)

Lemma 4.7. Let (ξ, s) ∈ (∂M D \ ∂m D) × [0, ∞). Then q(·; ξ, s) is not minimal.

1816


Proof. For reader’s convenience, we give a proof. We have (4.11). Suppose that q(·; ξ, s) is minimal. Then, along the line given in the proof of Lemma 12.12 of [15], we obtain from (4.11) that the support of γ consists of a single point. Thus, for some η ∈ ∂m D and constant C q(·; ξ, s) = Cq(·; η, s). Hence, by Lemma 4.4, ξ = η; which is a contradiction.

2

Lemma 4.8. Let (ξ, s) ∈ ∂m D × (0, ∞). Then q(·; ξ, s) is minimal if and only if q(·; ξ, 0) is minimal. Proof. Assume that q(·; ξ, 0) is minimal. Suppose that a nonnegative solution u of (1.1) satisfies u(·) q(·; ξ, s) on Q. Put v(x, t) = u(x, t + s). Then v(·) q(·; ξ, 0). Thus v(·) = Cq(·; ξ, 0) for some constant C. Hence u(x, t) = Cq(x, t; ξ, s) for t > s, and u(x, t) = 0 = Cq(x, t; ξ, s) for t s. This shows that q(·; ξ, s) is minimal. Next, assume that q(·; ξ, s) is minimal. Suppose that a nonnegative solution u of (1.1) satisfies u(·) q(·; ξ, 0) on Q. Put v(x, t) = u(x, t − s) for t > s, and v(x, t) = 0 for 0 < t s. Then v(·) q(·; ξ, s). Thus v(·) = Cq(·; ξ, s) for some constant C. Hence u(x, t) = Cq(x, t; ξ, 0). This shows that q(·; ξ, 0) is minimal. 2 By Theorem 4.1 and Lemmas 4.7 and 4.8, we have the following proposition. Proposition 4.9. There exists a Borel subset R of ∂M D such that R ⊂ ∂m D,

Lβm Q = R × [0, ∞),

for any u ∈ Pβ0 (Q) there exists a unique Borel measure λ on ∂M D × [0, ∞) which is supported by R × [0, ∞) and satisfies u(x, t) =

q(x, ξ, t − s) dλ(ξ, s),

(x, t) ∈ Q.

(4.14)

R×[0,∞)

Lemma 4.10. Let (ξ, s) ∈ ∂m D × [0, ∞). Then q(·; ξ, s) is minimal. Proof. Suppose that q(·; ξ, 0) is not minimal. Then ξ ∈ / R and q(x, ξ, t) =

q(x, η, t − s) dλ(η, s)

R×[0,∞)

for some Borel measure λ. We have ∞ K(x, ξ ) 0

q x 0 , ξ, t dt =

∞ q(x, ξ, t) dt = 0

R×[0,∞)

∞ dλ(η, s)K(x, η) 0

q x 0 , η, t dt.


1817

Thus K(x, ξ ) =

K(x, η) dΛ(η) R

for some Borel measure Λ. But ξ ∈ ∂m D \ R and R ⊂ ∂m D. This contradicts the uniqueness of a representing measure in the elliptic Martin representation theorem. Hence q(·; ξ, 0) is minimal; which together with Lemma 4.8 shows Lemma 4.10. 2 Completion of the proof of Theorem 1.3 in the case I = (0, ∞). By Lemma 4.10, R = ∂m D and Lβm Q = ∂m D × [0, ∞). 2

Thus Proposition 4.9 shows Theorem 1.3.

5. Proof of Theorem 1.3; the case 0 < T < ∞ In this section we prove Theorem 1.3 in the case 0 < T < ∞ by making use of the results in Section 4. To this end, the following proposition plays a crucial role. Proposition 5.1. Let ξ ∈ ∂M D and 0 s < r < ∞. Then p(x, y, t − r)q(y, r; ξ, s) dμ(y) = q(x, t; ξ, s),

x ∈ D, t > r,

(5.1)

D

where dμ(y) = m(y) dν(y). Proof. We first show (5.1) for ξ ∈ ∂m D. Define u(x, t) by u(x, t) =

u(x, t) = q(x, t; ξ, s),

0 < t r,

p(x, y, t − r)q(y, r; ξ, s) dμ(y),

r < t < ∞.

(5.2)

D

(We call u the minimal extension of q from t = r.) Then we see that u is a nonnegative solution of (∂t + L)u = 0 in D × (0, ∞) such that u(·) q(·; ξ, s) on D × (0, ∞). By Lemma 4.10, u(·) = Cq(·; ξ, s) for some constant C. But u(x, t) = q(x, t; ξ, s) for 0 < t r. Thus C = 1, and so u(·) = q(·; ξ, s). Next, let ξ ∈ / ∂m D. By Lemma 4.6, there exists a finite Borel measure γ on ∂M D supported by ∂m D such that q(·; ξ, s) = ∂m D

Thus

q(·; η, s) dγ (η).

(5.3)

1818


p(x, y, t − r)q(y, r; ξ, s) dμ(y) = D

p(x, y, t − r)q(y, r; η, s) dμ(y)

dγ (η)

∂m D

D

=

q(x, t; η, s) dγ (η)

∂m D

= q(x, t; ξ, s). This proves (5.1).

2

Lemma 5.2. Let ξ, η ∈ ∂M D, 0 s, r < T and C > 0. If q(x, ξ, t − s) = Cq(x, η, t − r),

x ∈ D, 0 < t < T ,

(5.4)

then ξ = η, s = r and C = 1. Proof. Choose u such that max(r, s) < u < T , and construct minimal extensions of both sides of (5.4) from t = u. Then, by (5.1) we have q(x, ξ, t − s) = Cq(x, η, t − r),

x ∈ D, 0 < t < ∞.

By Lemma 4.4, this implies that ξ = η, s = r and C = 1.

2

Now, let β be a measure on Q = D × (0, T ) defined by dβ(x, t) = a(x)b(t)m(x) dν(x) dt. Here a(x) is a nonnegative continuous function on D as described in the beginning of Section 4, and b(t) is a nonnegative continuous function on R such that b(t) > 0 on (T /2, T ) β β and b(t) = 0 on R \ (T /2, T ). Let Kβ (·; Ξ ), ∂M Q, ∂m Q, and Qβ∗ be the β-Martin kernel, β-Martin boundary, minimal β-Martin boundary, and β-Martin compactification for (Q, ∂t + L) with Q = D × (0, T ), respectively. The following proposition is an analogue of Proposition 4.5, and is shown in the same way. Proposition 5.3. β

(i) The β-Martin boundary ∂M Q of Q for ∂t + L is equal to the disjoint union of D × {0}, ∂M D × [0, T ) and the one point set { }: β

∂M Q = D × {0} ∪ ∂M D × [0, T ) ∪ { }. β

(5.5)

In particular, ∂M Q does not depend on β. (ii) The β-Martin compactification Qβ∗ of Q for ∂t + L is homeomorphic to the disjoint union of the topological product D ∗ × [0, T ) and the one point set { }, where a fundamental neighborhood system of is given by the family { } ∪ D ∗ × (T − ε, T ), 0 < ε < T /2. In particular, Qβ∗ does not depend on β.


1819

(iii) The β-Martin kernel Kβ is given as follows: for (x, t) ∈ Q, p(x, t; y, 0) , (y, 0) ∈ D × {0}, β(p(·; y, 0)) q(x, t; ξ, s) , (ξ, s) ∈ ∂M D × [0, T ), Kβ (x, t; ξ, s) = β(q(·; ξ, s)) Kβ (x, t; y, 0) =

(5.6) (5.7)

and Kβ (x, t; ) = 0. Lemma 5.4. Let (ξ, s) ∈ (∂M D \ ∂m D) × [0, T ). Then q(·; ξ, s) is not minimal. Proof. Suppose that q(·; ξ, s) is minimal. Then we obtain from (5.3) that q(x, ξ, t − s) = Cq(x, η, t − s),

x ∈ D, 0 < t < T ,

for some η ∈ ∂m D and C > 0. By Lemma 5.2, this is a contradiction.

2

Lemma 5.5. Let (ξ, s) ∈ ∂m D × [0, T ). Then q(·; ξ, s) is minimal. Proof. Let u be a nonnegative solution of (∂t + L)u = 0 in Q such that u(·) q(·; ξ, s) in Q. For r ∈ (s, T ), let ur be the minimal extension of u from t = r. By Proposition 5.1, ur (x, t) q(x, t; ξ, s),

x ∈ D, t > 0.

By Lemma 4.10, there exists a constant Cr such that ur (x, t) = Cr q(x, t; ξ, s) for t > 0. But ur (x, t) = u(x, t) for 0 < t < r. Thus Cr is independent of r; and so u(·) = Cq(·; ξ, s) in Q for some constant C. 2 Completion of the proof of Theorem 1.3 in the case 0 < T < ∞. Put β Lβm Q = ∂m Q \ D × {0} ∪ { } . By Proposition 5.3, Lemmas 5.4 and 5.5, we get Lβm Q = ∂m D × [0, T ). Thus, Theorem 2.1 of [34] which is an analogue of Theorem 4.1 completes the proof.

2

6. Integral representations; the case I = (−∞, 0) In this section we prove Theorem 1.4. We begin with the following proposition, which can be shown in the same way as in the proof of Theorem 1 of [9] (see also [39]). Proposition 6.1. Assume [SIU]. Then eλ0 t p(x, y, t) = 1 uniformly in (x, y) ∈ K × D t→∞ φ0 (x)φ0 (y) lim

for any compact subset K of D.

(6.1)

1820


In the rest of this section we assume [SSP]. We may and shall assume that a = 0 < λ0 . By Theorem 1.1, we have the following corollary of Proposition 6.1. Corollary 6.2. Assume [SSP]. Then, for any compact subset K of D and N > 1, lim

s→−∞

p(x, y, t − s) = e−λ0 t φ0 (x) eλ0 s φ0 (y)

uniformly in (x, y, t) ∈ K × D × (−N, 0).

Lemma 6.3. The solution e−λ0 t φ0 (x) is minimal. Proof. Suppose that e−λ0 t φ0 (x) is not minimal. Then, in view of Corollary 6.2, the same argument as in the proof of Theorem 1.3 shows that for any nonnegative solution u of the equation (∂t + L)u = 0 in Q = D × (−∞, 0) there exists a unique Borel measure λ on ∂M D × (−∞, 0) supported by the set ∂m D × (−∞, 0) such that u(x, t) = q(x, ξ, t − s) dλ(ξ, s), (x, t) ∈ Q. ∂M D×(−∞,t)

Thus

e−λ0 t φ0 (x) =

q(x, ξ, t − s) dλ(ξ, s),

(x, t) ∈ Q,

(6.2)

∂M D×(−∞,t)

for such a measure λ. Now, fix x. It follows from Theorems 1.1 and 1.2 that for any δ > 0 there exists a positive constant Cδ such that Cδ −1

q(x, ξ, τ ) Cδ , e−λ0 τ φ0 (x)

τ δ, ξ ∈ ∂M D.

(6.3)

ξ ∈ ∂M D, 0 < τ < 1,

(6.4)

By (4.4), q(x, ξ, τ ) Ce−α/τ φ0 (x),

for some positive constants α and C. By (6.2) and (6.3),

C1−1 e−λ0 (−1−s) dλ(ξ, s).

e φ0 (x) λ0

∂M D×(−∞,−2)

Thus eλ0 s dλ(ξ, s) C1 φ0 (x). ∂M D×(−∞,−2)

(6.5)


For t < −2 and 0 < δ < 1, we have φ0 (x) =

eλ0 (t−s) q(x, ξ, t − s)eλ0 s dλ(ξ, s).

1821

(6.6)

∂M D×{(−∞,t−δ]∪(t−δ,t)}

In view of (6.4) and (6.5), we choose δ so small that the integral on ∂M D × (t − δ, t) of the righthand side of (6.6) is smaller than φ0 (x)/3. Then, in view of (6.3) and (6.5), we choose t < −2 with |t| being so large that the integral on ∂M D × (−∞, t − δ] of the right-hand side of (6.6) is smaller than φ0 (x)/3. This is a contradiction. 2 Completion of the proof of Theorem 1.4. By virtue of Corollary 6.2 and Lemma 6.3, the same argument as in the proof of Theorem 1.3 shows Theorem 1.4. 2 7. Examples In this section we give two examples in order to illustrate a scope of Theorem 1.3. Throughout this section L0 is a uniformly elliptic operator on Rn of the form L0 u = −

n

∂i aij (x) ∂j u ,

i,j =1

where a(x) = [aij (x)]ni,j =1 is a symmetric matrix-valued measurable function on Rn satisfying, for some Λ > 0, Λ−1 |ξ |2

n

aij (x)ξi ξj Λ|ξ |2 ,

x, ξ ∈ Rn .

i,j =1 n n 7.1. Let V (x) be a measurable function in L∞ loc (R ), and L = L0 + V (x) on D = R .

Theorem 7.1. Suppose that there exist a positive constant c < 1 and a positive continuous increasing function ρ on [0, ∞) such that 2 2 c ρ |x| V (x) ρ |x| , x ∈ Rn , c ρ(r), r 0. cρ r + ρ(r)

(7.1) (7.2)

Assume that ∞

dr < ∞. ρ(r)

1

Then 1 is a small perturbation of L on Rn . Thus Theorem 1.3 holds true. Remark. Compare this theorem with a non-uniqueness theorem of [26].

(7.3)

1822


Proof. We first note that (7.2) yields c c c ρ r − , cρ(r) cρ r − + c ρ(r) ρ(r − ρ(r) ) ρ(r)

r

c , ρ(0)

since ρ is increasing. We show the theorem by using the same approach as in the proof of Theorem 5.1 of [31]. Put b = c−2 and

= inf j ∈ Z; ρ(0) < bj . For k , put rk = sup{r 0; ρ(r) bk }. By the continuity of ρ and (7.3), ρ(rk ) = bk and limk→∞ rk = ∞. By (7.2), ρ rk + cb−k c−1 ρ(rk ) = b1/2 bk < bk+1 = ρ(rk+1 ). Thus rk + cb−k < rk+1 for k . Define a positive continuously differentiable increasing function ρ on [0, ∞) as follows. Put ρ (r) = b for r r , ρ (r) = bk+1

for rk + cb−k r rk+1

(k );

and ρ (r) = ρk (r) for rk r rk + cb−k (k ) by choosing a continuously differentiable function ρk on [rk , rk + cb−k ] such that ρk (rk ) = 0, ρk rk + cb−k = bk+1 , ρk rk + cb−k = 0, ρk (rk ) = bk , and 0 ρk (r) Bb2k ,

rk r rk + cb−k ,

for some constant B > 0 independent of k. Then we have C −1

ρ (r) C, ρ(r)

0ρ (r) Cρ(r)2 ,

r 0,

(7.4)

for some positive constant C. Introduce a Riemannian metric g = (gij )ni,j =1 by gij = ρ (|x|)2 δij . n Then M = R with this metric g becomes a complete Riemannian manifold. Furthermore, by (7.2) and (7.4), M has the bounded geometry property (1.1) of [4]. The associated gradient ∇ and divergence div are written as −2 ∇ =ρ |x| ∇ 0 ,

−n n div = ρ |x| ◦ div0 ◦ ρ |x| ,

where ∇ 0 and div0 are the standard gradient and divergence on Rn . Put 2−n , m(x) = ρ |x| Then

−2 L=ρ |x| L, n A(x) = aij (x) i,j =1 ,

−2 γ (x) = ρ |x| V (x).


1823

0 1 1 0 A∇ m, ∇u + γ , Lu = − div(mA∇u) + γ = − div(A∇u) − m m where ·,·0 is the standard inner product on Rn . Since the inner product ·,· associated with the metric g is written as 0 2 X, Y = ρ X, Y , we have

−2 A∇

Lu = − div(A∇u) − ρ

0m

m

, ∇u + γ .

(7.5)

By (7.4), 0 ∇ m(x) C 3 |n − 2| ρ |x| m(x). From this we have 0 0 2 2 −2 A∇ m −2 A∇ m ρ ,ρ ρ −2 Λ2 C 3 |n − 2| ρ Λ C 3 |n − 2| . m m By (7.1) and (7.4), c C −2 γ (x) C 2 . Thus the operator L − c C −2 /2 has the Green function; and L belongs to the class DM (θ, ∞, ) introduced by Ancona [4], where θ = max Λ, Λ C 3 |n − 2| , C 2 , = c C −2 /2. Put −2 −2 L2 = ρ |x| (L + 1) = L + ρ |x| . In order to apply the results of [4], we proceed to estimate ρ (|x|)−2 . Let d(x) be the Riemannian distance dist(0, x) from the origin 0 to x, and put r ψ(r) =

ρ (s) ds. 0

Then we see that d(x) = ψ(|x|). Denote by ψ −1 the inverse function of ψ , and put −1 −2 , Φ(s) = ρ ψ (s)

s 0.

−2 0 0 there exists a compact subset K of D = M such that −2 n H (x, z) ρ |z| H (z, y) ρ |z| dz εH (x, y), x, y ∈ D \ K, D\K

where dz is the Lebesgue measure on Rn , and H (x, z) is the Green function of L on D with respect to the measure ρ (|z|)n dz. Denote by G(x, z) the Green function of L on D with respect to the measure dz. Since L = ρ (|x|)−2 L, we have 2−n H (x, z) = G(x, z) ρ |z| Thus

(2−n)−2 2−n n 2−n G(x, z) ρ |z| G(z, y) ρ |y| ρ |z| dz εG(x, y) ρ |y|

D\K

for any x, y ∈ D \ K. Hence 1 is a small perturbation of L on Rn .

2

Remark. A sufficient condition for (7.2) is the following: ρ is a positive differentiable function on [0, ∞) satisfying 0 ρ (r)ρ(r)−2 C,

r 0,

for some positive constant C. Indeed, from (7.6) we have δ ρ(r)−1 exp CδX(δ) , X(δ) ≡ ρ r + ρ(r)

(7.6)

r 0, δ > 0.

Put δ = (2Ce)−1 , and let γ ∈ (1, e) be the solution of the equation exp[X/2e] = X. Then we get 1 X(δ) γ . Thus (7.2) holds with c = min(δ, 1/γ ). Condition (7.3) is sharp, since Theorem 6.2 of [17] yields the following uniqueness theorem. Theorem 7.2. Suppose that there exists a positive continuous increasing function ρ on [0, ∞) such that V (x) ρ |x| 2 , x ∈ Rn . (7.7)


1825

Assume that ∞

dr = ∞. ρ(r)

(7.8)

1

Then [UP] holds. Thus Fact AT holds true. 7.2. Throughout this subsection we assume that D is a bounded domain of Rn . Let L be an elliptic operator on D of the form L=

1 L0 , w(x)

where w is a positive measurable function on D such that w, w −1 ∈ L∞ loc (D). Theorem 7.3. Let D be a Lipschitz domain. Suppose that there exists a positive function ψ on (0, ∞) such that s 2 ψ(s) is increasing and w(x) ψ δD (x) , x ∈ D, (7.9) where δD (x) = dist(x, ∂D). Assume that 1 sψ(s) ds < ∞.

(7.10)

0

Then 1 is a small perturbation of L on D. Thus Theorem 1.3 holds true. Remark. (i) The first assertion of this theorem is implicitly shown in [17] (see Theorem 7.11 and Remark 7.12(ii) there). (ii) The Lipschitz regularity of the domain D is assumed only for the Hardy inequality to hold for any function in C0∞ (D). Thus, for this theorem to hold, it suffices to assume (for example) that D is uniformly -regular John domain or a simply connected domain of R2 (see [3,4]). Proof of Theorem 7.3. For x ∈ D, put δD (x) . Dx = y ∈ D; |x − y| < 2 Then 1 3 δD (x) δD (y) δD (x), 2 2

y ∈ Dx .

Thus 2 3 3 δD (x)2 w(y) 4δD (y)2 ψ δD (y) 4 δD (x) ψ δD (x) . 2 2

1826


Put Ψ (s) = 9s 2 ψ((3/2)s). Then Ψ (s) is increasing, and satisfies 2

δD (x)

sup w(y) Ψ δD (x) , y∈Dx

1

Ψ (s) ds < ∞. s

0

Hence, by virtue of Proposition 9.2, Theorem 9.1 and Corollary 6.1 of [4], w is a small perturbation of L0 on D. This implies that 1 is a small perturbation of L on D. 2 Condition (7.10) is sharp, since Theorem 7.8 and Lemma 7.6 of [17] yield the following uniqueness theorem. Theorem 7.4. Suppose that there exists a positive continuous increasing function ψ on (0, ∞) such that cψ δD (x) w(x) ψ δD (x) ,

x ∈ D,

(7.11)

for some positive constant c, and ν

ψ(ηs) ν −1 , ψ(s)

s > 0,

1 η 2, 2

(7.12)

for some positive constant ν. Assume 1 1/2 ψ(s) inf r 2 ψ(r) ds = ∞. sr1

(7.13)

0

Then [UP] holds. Thus Fact AT holds true. References [1] H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions, Math. Ann. 312 (1998) 289–318. [2] H. Aikawa, M. Murata, Generalized Cranston–McConnell inequalities and Martin boundaries of unbounded domains, J. Anal. Math. 69 (1996) 137–152. [3] A. Ancona, On strong barriers and an inequality of Hardy for domains in RN , J. London Math. Soc. 34 (1986) 274–290. [4] A. Ancona, First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains, J. Anal. Math. 72 (1997) 45–92. [5] A. Ancona, J.C. Taylor, Some remarks on Widder’s theorem and uniqueness of isolated singularities for parabolic equations, in: Dahlberg, et al. (Eds.), Partial Differential Equations with Minimal Smoothness and Applications, Springer, New York, 1992, pp. 15–23. [6] D.G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sc. Norm. Sup. Pisa 22 (1968) 607–694. [7] R. Bañuelos, Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, J. Funct. Anal. 100 (1991) 181–206. [8] R. Bañuelos, Lifetime of conditioned Brownian motion and related estimates for heat kernels, eigenfunctions and eigenvalues in Euclidean domains, MSRI Lectures, 1998, http://www.msri.org/lectures. [9] R. Bañuelos, B. Davis, Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains, J. Funct. Anal. 84 (1989) 188–200.


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[10] R. Bañuelos, B. Davis, A geometric characterization of intrinsic ultracontractivity for planar domains with boundaries given by the graphs of functions, Indiana Univ. Math. J. 41 (1992) 885–913. [11] R.F. Bass, K. Burdzy, Lifetimes of conditioned diffusions, Probab. Theory Related Fields 91 (1992) 405–443. [12] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. [13] E.B. Davies, B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and the Dirichlet Laplacians, J. Funct. Anal. 59 (1984) 335–395. [14] A. Grigor’yan, W. Hansen, A Liouville property for Schrödinger operators, Math. Ann. 312 (1998) 659–716. [15] L.L. Helms, Introduction to Potential Theory, Wiley, New York, 1969. [16] K. Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J. 155 (1999) 1–26. [17] K. Ishige, M. Murata, Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains, Ann. Sc. Norm. Sup. Pisa 30 (2001) 171–223. [18] K. Janssen, Martin boundary and H p -theory of harmonic spaces, in: Seminar on Potential Theory II, in: Lecture Notes in Math., vol. 226, Springer, Berlin, 1971, pp. 102–151. [19] A. Koranyi, J.C. Taylor, Minimal solutions of the heat equation and uniqueness of the positive Cauchy problem on homogeneous spaces, Proc. Amer. Math. Soc. 94 (1985) 273–278. [20] V. Lin, Y. Pinchover, Manifolds with group actions and elliptic operators, Mem. Amer. Math. Soc. 112 (540) (1994). [21] F.-Y. Maeda, Martin boundary of a harmonic space with adjoint structure and its applications, Hiroshima Math. J. 21 (1991) 163–186. [22] B. Mair, J.C. Taylor, Integral representation of positive solutions of the heat equation, in: Lecture Notes in Math., vol. 1096, Springer, Berlin, 1984, pp. 419–433. [23] P.J. Méndez-Hernández, Toward a geometrical characterization of intrinsic ultracontractivity of the Dirichlet Laplacian, Michigan Math. J. 47 (2000) 79–99. [24] P.J. Méndez-Hernández, K. Wong, Intrinsic ultracontractivity for Schrödinger operators with mixed boundary conditions, Potential Anal. 24 (4) (2006) 333–355. [25] M. Murata, Uniform restricted parabolic Harnack inequality, separation principle, and ultracontractivity for parabolic equations, in: Functional Analysis and Related Topics, 1991, in: Lecture Notes in Math., vol. 1540, Springer, Berlin, 1993, pp. 277–288. [26] M. Murata, Sufficient condition for non-uniqueness of the positive Cauchy problem for parabolic equations, in: Spectral and Scattering Theory and Applications, in: Adv. Stud. Pure Math., vol. 23, Kinokuniya, Tokyo, 1994, pp. 275–282. [27] M. Murata, Uniqueness and non-uniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds, Proc. Amer. Math. Soc. 123 (1995) 1923–1932. [28] M. Murata, Non-uniqueness of the positive Cauchy problem for parabolic equations, J. Differential Equations 123 (1995) 343–387. [29] M. Murata, Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Funct. Anal. 135 (1996) 456–487. [30] M. Murata, Semismall perturbations in the Martin theory for elliptic equations, Israel J. Math. 102 (1997) 29–60. [31] M. Murata, Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations, J. Funct. Anal. 194 (2002) 53–141. [32] M. Murata, Heat escape, Math. Ann. 327 (2003) 203–226. [33] M. Murata, Uniqueness theorems for parabolic equations and Martin boundaries for elliptic equations in skew product form, J. Math. Soc. Japan 57 (2005) 387–413. [34] M. Murata, Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries, J. Funct. Anal. 245 (2007) 177–212. [35] R. Phelps, Lectures on Choquet’s Theorem, Van Nostrand, New York, 1966. [36] Y. Pinchover, Representation theorems for positive solutions of parabolic equations, Proc. Amer. Math. Soc. 104 (1988) 507–515. [37] Y. Pinchover, Criticality and ground states for second-order elliptic equations, J. Differential Equations 80 (1989) 237–250. [38] Y. Pinchover, Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Math. Ann. 314 (1999) 555–590. [39] Y. Pinchover, Large time behavior of the heat kernel, J. Funct. Anal. 206 (2004) 191–209. [40] R.G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Univ. Press, Cambridge, 1995. [41] R. Smits, Necessary and sufficient conditions for finiteness of lifetimes, Potential Anal. 5 (1996) 513–521. [42] M. Tomisaki, Intrinsic ultracontractivity and small perturbation for one-dimensional generalized diffusion operators, J. Funct. Anal. 251 (2007) 289–324.


Quantum scattering at low energies ✩ J. Dereziński a,∗ , E. Skibsted b a Dept. of Math. Methods in Physics, Warsaw University, Ho˙za 74, 00-682, Warszawa, Poland b Institut for Matematiske Fag, Aarhus Universitet, Ny Munkegade 8000 Aarhus C, Denmark

Received 9 October 2008; accepted 29 May 2009 Available online 21 June 2009 Communicated by L. Gross

Abstract For a class of negative slowly decaying potentials, including V (x) := −γ |x|−μ with 0 < μ < 2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki–Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are welldefined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ |x|−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ = 0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities. © 2009 Elsevier Inc. All rights reserved. Keywords: Schrödinger operators; Wave operators; Scattering operator; WKB method

✩

The research of J.D. is supported in part by the grant N N201 270135. Part of the research was done during a visit of both authors to the Erwin Schrödinger Institute. One of us (E.S.) thanks H. Tamura for drawing our attention to the paper [A.A. Kvitsinskii, Scattering by long-range potentials at low energies, Theoret. and Math. Phys. 59 (1984) 629–633, translation of Theoret. Mat. Fiz. 59 (1984) 472–478]. * Corresponding author. E-mail addresses: [email protected] (J. Dereziński), [email protected] (E. Skibsted). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.026

J. Dereziński, E. Skibsted / Journal of Functional Analysis 257 (2009) 1828–1920

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Contents 1.

2. 3.

4.

5.

6.

7.

8.

9.

Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Classical orbits at positive energies . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Wave and scattering matrices at positive energies . . . . . . . . . . . . . . . . 1.3. Short-range wave and scattering operators . . . . . . . . . . . . . . . . . . . . 1.4. Dollard wave and scattering operators . . . . . . . . . . . . . . . . . . . . . . . 1.5. Asymptotic normalized velocity operator . . . . . . . . . . . . . . . . . . . . . 1.6. Low-energy asymptotics of classical orbits . . . . . . . . . . . . . . . . . . . . 1.7. Low-energy asymptotics of wave and scattering matrices . . . . . . . . . . 1.8. Geometric approach to scattering theory . . . . . . . . . . . . . . . . . . . . . . 1.9. Low energy asymptotics of short-range and Dollard operators . . . . . . . 1.10. Location of singularities of the zero energy scattering matrix . . . . . . . . 1.11. Kernel of S(0) as an explicit oscillatory integral . . . . . . . . . . . . . . . . 1.12. Generalized eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13. Propagation of singularities for zero-energy generalized eigenfunctions 1.14. Sommerfeld radiation condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15. Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Scattering orbits at positive energies . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Scattering orbits at low energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Radially symmetric potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary values of the resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Low energy resolvent estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Scattering wave front set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Wave front set bounds of the boundary value of the resolvent . . . . . . . 4.4. Sommerfeld radiation condition . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The WKB-ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The improved WKB-ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Solving transport equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Constructions in incoming region . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Fourier integral operators at fixed energies . . . . . . . . . . . . . . . . . . . . Wave matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Wave operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Wave matrices at positive energies . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Wave matrices at low energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Asymptotics of short-range wave matrices . . . . . . . . . . . . . . . . . . . . 6.5. Asymptotics of Dollard-type wave matrices . . . . . . . . . . . . . . . . . . . Scattering matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Scattering matrices at positive energies . . . . . . . . . . . . . . . . . . . . . . 7.2. Scattering matrices at low energies . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Asymptotics of short-range scattering matrices . . . . . . . . . . . . . . . . . 7.4. Asymptotics of Dollard-type scattering matrices . . . . . . . . . . . . . . . . Generalized eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Representations of generalized eigenfunctions . . . . . . . . . . . . . . . . . . 8.2. Scattering matrices – an alternative construction . . . . . . . . . . . . . . . . 8.3. Geometric scattering matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous potentials – location of singularities of S(0) . . . . . . . . . . . . . . . 9.1. Reduced classical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.2. 9.3. 9.4. Appendix A. A.1. A.2. A.3. A.4. References .


Propagation of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . Location of singularities of the kernel of the scattering matrix . . . Distributional kernel of S(0) as an oscillatory integral . . . . . . . . Elements of abstract scattering theory . . . . . . . . . . . . . . . . . . . Wave operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of rigged Hilbert spaces applied to wave operators . . . . . Method of rigged Hilbert spaces applied to the scattering operator ...............................................

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1902 1906 1909 1914 1914 1916 1917 1918 1919

1. Introduction and results Scattering theory of 2-body systems, both classical and quantum, both short- and long-range, is nowadays a well understood subject [4,13,15,18,19,31,32]. In particular, for large natural classes of potentials we know a lot about the properties of wave and scattering matrices at positive energies. Zero – the only threshold energy – in most works on the subject is avoided, since scattering at zero energy is much more difficult to describe and strongly depends on the choice of the potential. In this paper we consider a class of potentials that have an especially well behaved, nontrivial and interesting low energy scattering theory. Precise conditions used in our paper are described in Section 2. Roughly speaking, the potentials that we consider have a dominant negative radial term V1 (x) similar to −γ |x|−μ with γ > 0 and 0 < μ < 2, plus a faster decaying perturbation. Similar classes of potentials appeared in the literature already in [10]. A systematic study of such 2-body systems at low energies was undertaken in [8], where a complete expansion of the resolvent at the zero-energy threshold was obtained, and in [6], where classical low-energy scattering theory was developed. This paper can be viewed as a continuation of [6,8]. In this paper we show that quantum scattering theory for such potentials is well behaved down to the energy zero. In particular, we study appropriately defined modified wave and scattering matrices for a fixed energy. We show that they have limits at zero energy. Our results were partly announced in [5]. Let us mention also our recent paper [7], where some closely related results about the zeroenergy scattering matrix are proven for a class of radial potentials. [7] and this paper can be viewed as companion papers, even though they can be read independently. For positive energies most (but probably not all) of our results are contained in the literature, scattered in many sources [13,15,18,19,31,32]. Almost all our material about the zero energy case is new. In the introduction we will first review scattering for positive energies for a rather general class of potentials. Then we will describe a simplified version of the main results of our paper, which concerns scattering at low energies for a more restrictive class of potentials. 1.1. Classical orbits at positive energies For the presentation of known results about positive energies we assume that the potentials satisfy the following condition:


1831

Condition 1.1. V = V1 + V3 is a sum of real measurable functions on Rd such that V1 is smooth and for some μ > 0, ∂xα V1 (x) = O |x|−μ−|α| ,

|α| 0,

(1.1)

V3 is compactly supported and V3 (H0 + 1)−1 is a compact operator on the Hilbert space L2 (Rd ). Here H0 := 2−1 p 2 with p := −i ∇x . The Hamiltonian H = H0 + V does not have positive eigenvalues. Let us first consider the classical Hamiltonian h1 (x, ξ ) := 12 ξ 2 + V1 (x) on the phase space Rd × Rd , using h0 (x, ξ ) := 12 ξ 2 as the free Hamiltonian. (The analysis of the classical case is needed in the quantum case.) One can prove that for any ξ ∈ Rd , ξ = 0, and x in an appropriate outgoing/incoming region the following problem admits a solution (strictly speaking, meaning one solution for t → +∞ and one for t → −∞): ⎧ ¨ = −∇V1 (y(t)), ⎨ y(t) y(±1) = x, ⎩ ξ = limt→±∞ y(t). ˙

(1.2)

One obtains a family y ± (t, x, ξ ) of solutions smoothly depending on parameters. All (positive energy) scattering orbits, i.e. orbits satisfying limt→±∞ |y(t)| = ∞, are of this form (the energy is λ = 12 ξ 2 ). Using these solutions, in an appropriate incoming/outgoing region one can construct a solution φ ± (x, ξ ) to the eikonal equation 2 1 1 ∇x φ ± (x, ξ ) + V1 (x) = ξ 2 2 2

(1.3)

satisfying ∇x φ ± (x, ξ ) = y(±1, ˙ x, ξ ). 1.2. Wave and scattering matrices at positive energies Let us turn to the quantum case. Following Isozaki–Kitada, see [18,19,25,31], one can use the functions φ ± (x, ξ ) in the quantum case to construct appropriate modifiers, which can be taken to be ± J ± f (x) := (2π)−d ei φ (x,ξ )−i ξ ·y a ± (x, ξ )f (y) dy dξ. (1.4) Here a ± (x, ξ ) is an appropriate cut-off supported in the domain of the definition of φ ± , equal to one in the incoming/outgoing region. Then one constructs modified wave operators W ± f := lim ei tH J ± e−i tH0 f, t→±∞

fˆ ∈ Cc Rd \ {0} ,

(1.5)

and the modified scattering operator S = W + ∗W −.

(1.6)

1832


We remark that W ± are isometric with range given by the projection onto the continuous spectrum of H 1c (H )L2 Rd = 1]0,∞[ (H )L2 Rd . (Whence S is unitary.) Throughout our paper the modified wave operators W ± and the modified scattering operators S defined using certain well chosen modifiers will be the main object of study. In what follows we will call them simply wave and scattering operators, dropping the word modified. The free Hamiltonian H0 can be diagonalized by the direct integral ∞ H0 =

⊕L2 S d−1 dλ,

0

F0 (λ)f (ω) = (2λ)

(d−2)/4

√ fˆ 2λω ,

(1.7) f ∈ L2 R d .

(1.8)

Here fˆ refers to the d-dimensional Fourier transform. The operator F0 (λ) can be interpreted as a bounded operator from the weighted space L2,s (Rd ) := x−s L2 (Rd ), s > 12 , to L2 (S d−1 ). One can ask whether the wave and scattering operators can be restricted to a fixed energy λ. This question is conceptually simpler in the case of the scattering operator S. Due to the intertwining property, W ± H0 = H W ± it satisfies SH0 = H0 S, so abstract theory guarantees the existence of a decomposition S

⊕S(λ) dλ,

]0,∞[

where S(λ) are unitary operators on L2 (S d−1 ) defined for almost all λ. One can prove that, under Condition 1.1, S(λ) can be chosen to be a strongly continuous function (which fixes uniquely S(λ) for all λ ∈ ]0, ∞[). S(λ) is called the scattering matrix at the energy λ. The case of wave operators is somewhat more complicated. By the intertwining property it is natural to use the direct integral decomposition (1.7) only from the right and the question is whether we can give a rigorous meaning to W ± F0 (λ)∗ . Again, under Condition 1.1 one can show that there exists a unique strongly continuous function ]0, ∞[ λ → W ± (λ) with values in the space of bounded operators from L2 (S d−1 ) to L2,−s (Rd ) with s > 12 such that for f ∈ L2,s (Rd ) W ±f =

W ± (λ)F0 (λ)f dλ.

]0,∞[

The operator W ± (λ) is called the wave matrix at energy λ. One can also extend the domain of W ± (λ) so that it can act on the delta-function at ω ∈ S d−1 , denoted δω . Now w ± (ω, λ) := W ± (λ)δω is an element of L2,−p (Rd ) for p > d2 . It satisfies 1 − + V (x) − λ w ± (ω, λ) = 0. 2

(1.9)


1833

It behaves in the outgoing/incoming region as a plane wave. It will be called the generalized eigenfunction of H at energy λ and at asymptotic normalized velocity ω; this terminology is justified in Section 1.5. 1.3. Short-range wave and scattering operators Let us recall that in the short-range case, that is μ > 1, the standard definitions of wave and scattering operators are Wsr± f := lim ei tH e−i tH0 f,

(1.10)

Ssr := Wsr+ ∗ Wsr− .

(1.11)

t→±∞

We will call Wsr± and S the standard short-range wave and scattering operators. They differ from W ± and S by a momentum-dependent phase factor: ±

W ± = Wsr± ei ψsr (p) , +

(1.12)

−

S = e−i ψsr (p) Ssr ei ψsr (p) .

(1.13)

Note that Wsr± and Ssr are canonically defined given the potential V , whereas W ± , S are not. They depend on the phase functions φ ± , which are non-canonical. Nevertheless, we will see that W ± and S have better properties in the low energy regime than Wsr± and Ssr . 1.4. Dollard wave and scattering operators Similarly, in the case μ >

1 2

one can use the so-called Dollard construction: ± Wdol f := lim ei tH Udol (t)f,

(1.14)

t→±∞

Udol (t) := e−i

t

0 (p

2 /2+V (sp)1 {|sp|R0 } ) ds

+∗ − Wdol . Sdol := Wdol

,

R0 > 0,

(1.15) (1.16)

Analogously, we have ±

± i ψdol (p) W ± = Wdol e , +

−

S = e−i ψdol (p) Ssr ei ψdol (p) .

(1.17) (1.18)

Dollard wave and scattering operators are non-canonical (they depend on R0 ). Again, W ± ± and S have better properties in the low energy regime than Wdol and Sdol .

1834


1.5. Asymptotic normalized velocity operator We mentioned above that the main objects of our study, W ± and S are non-canonical, given the potential V . This does not mean that they have no physical content. The operator W ± is an element of the family of incoming/outgoing wave operators, and S is an element of the family of scattering operators, which are canonically defined. In this subsection we briefly describe a possible definition of these families, following essentially [3,4]. Suppose that V satisfies (1.1) (or even much weaker conditions). Then it can be shown that there exists the following operator: ˆ −i tH 1c (H ), v ± := s − lim ±ei tH xe t→±∞

xˆ =

x . |x|

(1.19)

v ± can be called the asymptotic normalized velocity operator. It is a vector of commuting selfadjoint operators (on the space 1c (H )L2 (Rd )) satisfying ± 2 = 1c (H ), v

± v , H = 0.

(1.20)

We say that W˘ ± is an outgoing/incoming wave operator associated with H if it is isometric and satisfies W˘ ± H0 = H W˘ ± ,

W˘ ± pˆ = v ± W˘ ± ,

(1.21)

p ˘ ˘ + ∗ W˘ − for some wave |p| . We say that S is a scattering operator iff it is of the form W operators W˘ ± . Note that if W˘ 1± and W˘ 2± are two wave operators associated with a given H , then there exists a function ψ ± such that

where pˆ =

± W˘ 1± = W˘ 2± ei ψ (p) .

(1.22)

Therefore, scattering cross sections |S(λ)(ω, ω )|2 , which are usually considered to be the only measurable quantities in scattering theory, are insensitive to the choice of a scattering operator. ± It is easy to show that W ± , Wsr± , Wdol are all wave operators in the sense of the above definition. Likewise, S, Ssr , Sdol are all scattering operators in the sense of the above definition. Clearly, the standard short-range wave and scattering operators Wsr± , Ssr are canonically distinguished. However their definition is possible only if μ > 1. In the long-range case, μ 1, apparently there are no distinguished wave and scattering operators. Therefore in the long-range case the families of wave and scattering operators as defined above seem to be the natural basic objects of scattering theory. Nevertheless, as we will show in our paper, the operators W ± , S that we consider are useful also in the short-range case, even though they are non-canonical. Let us remark in parenthesis that in the case of scattering on [0, ∞[, every unitary operator commuting with H0 is a scattering operator according to our definition. Therefore, our definition of a scattering operator is not very interesting in this case. On Rd , however, the families of wave operators and scattering operators defined above constitute nontrivial and interesting families of operators.


1835

1.6. Low-energy asymptotics of classical orbits In the remaining part of the introduction we consider a more restricted class of potentials. To simplify the presentation, in this introduction let us assume that the potential takes the form (1.23) V (x) = −γ |x|−μ + O |x|−μ− , where μ ∈ ]0, 2[ and γ , > 0. For derivatives, assume that ∂ β (V (x) + γ |x|−μ ) = O(|x|−μ− −|β| ). Compactly supported singularities can be included. Let us note in parenthesis that in all our results, even though we suppose that the dominant part of the potential is radial, we allow for a non-radial perturbation. This lack of radial symmetry requires additional technical complications as compared with the radial case in some of our arguments, especially in [6]. We are convinced, however, that our results are interesting also in the purely radial case. For potentials satisfying (1.23) we would like to extend the results described in Section 1.1 down to the energy λ = 0. To this end√we change variables to “blow up” the discontinuity at λ = 0. This amounts to looking at ξ = 2λω as depending on two independent variables λ 0 and ω ∈ S d−1 . It is proven in [6] that for any ω ∈ S d−1 , λ ∈ [0, ∞[ and x from an appropriate outgoing/incoming region there exists a solution of the problem ⎧ y(t) ¨ = −∇V (y(t)), ⎪ ⎪ ⎨ λ = 1 y(t) 2 + V (y(t)), 2˙ (1.24) ⎪ y(±1) = x, ⎪ ⎩ ω = ± limt→ ±∞ y(t)/|y(t)|. One obtains a family y ± (t, x, ω, λ) of solutions smoothly depending on parameters. All scattering orbits are of this form. Using these solutions one can construct a solution φ ± (x, ω, λ) to the eikonal equation 2 1 ∇x φ ± (x, ω, λ) + V (x) = λ 2

(1.25)

satisfying ∇x φ ± (x, ω, λ) = y(±1, ˙ x, ω, λ). 1.7. Low-energy asymptotics of wave and scattering matrices In the quantum case, we can use the new functions φ ± (x, ω, λ) in the modifiers J ± , which lead to the definitions of the wave operators W ± and the scattering operator S. We can also improve on the choice of the symbols a ± (x, ξ ) by assuming that in the incoming/outgoing region they satisfy the appropriate transport equations. The first main new result of our paper concerns wave operators and their corresponding wave matrices and is expressed in Theorems 6.5, 6.6 and Corollary 6.7. Its simplified version can be stated as follows: Theorem 1.2. There exists the norm limit of wave matrices at zero energy: W ± (0) = lim W ± (λ) λ0

in the sense of operators in B(L2 (S d−1 ), L2,−s (Rd )), where s >

1 2

+ μ4 .

1836


The operator W ± (0) can be called the wave matrix at zero energy. We can introduce w ± (ω, 0) := W ± (0)δω , called the generalized eigenfunction of H at zero energy and fixed asymptotic normalized velocity ω. It belongs to the weighted space L2,−p (Rd ) where p > μ dμ d 2 2 + 2 − 4 . We shall also show weighted L -bounds on its ω-derivatives. It is interesting to note that the behaviour of the generalized eigenfunction w ± (ω, 0) depends strongly on the dimension. In dimension 1 it is unbounded, in dimension 2 it is almost bounded and in dimension greater than 2 it decays at infinity (without being square integrable). The next main result of our paper concerns scattering matrices. It is given in Theorem 7.2. Its simplified version reads: Theorem 1.3. There exists the strong limit of scattering matrices at zero energy S(0) = s- lim S(λ) λ0

in the space B(L2 (S d−1 )). This limit S(0) is unitary on L2 (S d−1 ). We remark that neither W (λ) nor S(λ) are smooth in λ 0 at the threshold 0, which can seem somewhat surprising given the fact that the boundary value of the resolvent R(λ + i 0) = (H − λ − i 0)−1 (interpreted as acting between appropriate weighted spaces) has this property (see [2] for explicit expansions in the purely Coulombic case). 1.8. Geometric approach to scattering theory There exists an alternative approach to scattering theory, based on the study of generalized eigenfunctions. It allows us to characterize scattering matrices by the spatial asymptotics of generalized eigenfunctions. It was used in particular in Vasy [26] or [27, Remark 19.12]. We shall study this approach, including the case of the zero energy, in Section 8.3. 1.9. Low energy asymptotics of short-range and Dollard operators Let us stress again that the existence of the limits of wave and scattering matrices at zero energy is made possible not only by appropriate assumptions on the potentials, but also by the use of appropriate modifiers. Wave matrices Wsr± (λ) defined by the standard short-range procedure, ± as well as the Dollard modified wave operators Wdol (λ), do not have this property. They differ ± from our W (λ) by a momentum dependent phase factor that has an oscillatory behaviour as λ 0. In particular, 1 1 Wsr± (λ) = W ± (λ) exp i O λ 2 − μ , 1 < μ < 2; 1 ± Wdol (λ) = W ± (λ) exp i O λ− 2 ln λ , μ = 1;

(1.26b)

1 1 ± (λ) = W ± (λ) exp i O λ 2 − μ , Wdol

(1.26c)

1 < μ < 1. 2

(1.26a)

By Theorem 1.2, we can replace W ± (λ) with W ± (0) in (1.26a), (1.26b) and (1.26c). Thus ± study of W ± gives asymptotics of more conventional kinds of wave operators: Wsr± and Wdol .


1837

We remark that scattering theory for slowly decaying potentials at low energies in the 1dimensional setting was studied in [28] (for both negative and positive potentials). In particular, an oscillatory behaviour similar to (1.26a) was proved in dimension 1 in [28]. Thus applied to radially symmetric potentials our results concerning the low energy asymptotics of wave matrices have an overlap with [28]. The asymptotics (1.26b) and (1.26c) seem to be new. 1.10. Location of singularities of the zero energy scattering matrix A recurrent idea of scattering theory is the parallel behaviour of classical and quantum systems. One of its manifestations is the relationship between scattering orbits at a given energy and the location of singularities of the scattering matrix. In the case of positive energies the relationship is simple and well-known. To describe it note that scattering orbits of positive energy have the deflection angle that goes to zero when the distance of the orbit to the center goes to infinity. In the quantum case this corresponds to the fact that the integral kernel of scattering matrices S(λ)(ω, ω ) at positive energies λ are smooth for ω = ω and has a singularity at ω = ω . This picture changes at the zero energy. For potentials considered in our paper, the deflection angle of zero-energy orbits does not go to zero for orbits far from the center. The angle of deflection is small for small μ and goes to infinity as μ approaches 2. For the strictly homogeneous potential, V (r) = −γ r −μ , one can solve the equations of motion at zero energy. The (non-collision) zero-energy orbits are given by the implicit equation (in polar coordinates) μ r(t) −1+ 2 μ θ (t) = , sin 1 − 2 rtp

(1.27)

μπ see [6, Example 4.3]. Whence the deflection angle of such trajectories equals − 2−μ . In particular, for attractive Coulomb potentials it equals −π , which corresponds to the well-known fact that in this case zero-energy orbits are parabolas (see [23, p. 126] for example). One of the main results of our paper is a quantum analogue of this fact:

Theorem 1.4. The integral kernel of the zero-energy scattering matrix S(0)(ω, ω ) is smooth μπ . away from ω, ω satisfying ω · ω = cos 2−μ We note that for the attractive Coulomb potential this result can be proven using known formulas (which can be found e.g. in [30]). In fact, in this case one can compute that S(0) = ei c P , where (P τ )(ω) = τ (−ω), as well as the following asymptotics −1/2 {C ln λ+C +o(λ0 )} 1 2

Sdol (λ) = ei λ

P + o λ0 .

(1.28)

Note that Theorem 1.4 implies that the scattering cross section at zero energy |S(0)(ω, ω )|2 μπ can have a singularity only at ω · ω = cos 2−μ . 1.11. Kernel of S(0) as an explicit oscillatory integral μ

In the case V = −γ |x|−μ + O(|x|−1− 2 − ), > 0, it is possible to represent the distributional kernel of the scattering matrix S(0) (modulo a smoothing term) in terms of a fairly explicit

1838


oscillatory integral. This provides an alternative way to prove Theorem 1.4 on the location of singularities of the scattering matrix – given the stronger conditions on the potential (we remark that our proof of Theorem 1.4 is rather abstract, see Section 1.13). Let us remark that in [7], which can be viewed as a companion paper to this one, we present an independent study of the zero-energy scattering matrix for the class of radial potentials satisfying μ

V = −γ r −μ +O(r −1− 2 − ), > 0. Using the 1-dimensional WKB-method, [7] gives an explicit formula for S(0), up to a compact term. 1.12. Generalized eigenfunctions A solution of the equation − + V (x) − λ u = 0

(1.29)

in s L2,−s (Rd ) will be called a generalized eigenfunction with energy λ. One of our results says that each generalized eigenfunction with positive or zero energy is of the form W ± (λ)τ , where τ is a distribution on the sphere S d−1 . Such generalized eigenfunctions are never square-integrable. A rough method to describe their behaviour for large x is to use weighted spaces L2,s (Rd ) with appropriate s. A more precise description is provided by the so-called Besov spaces. One of our results says that the range of (incoming and outgoing) wave matrices can be described precisely by an appropriate Besov space. One can also describe quite precisely their spatial asymptotics. In the case of zero energy, these results are new. 1.13. Propagation of singularities for zero-energy generalized eigenfunctions It is well known that some of the properties of solutions of PDE’s of the form P (x, D)u = 0 can be explained by the behaviour of classical Hamiltonian dynamics given by the principal symbol of P . One of the best known expressions of this idea is Hörmander’s theorem about propagation of singularities. Similar ideas are true in the case of Schrödinger operators. This is well understood for positive energies. In the case of zero energy a similar analysis is possible. It has an especially clean formulation if we assume that the potential is V (x) = −γ |x|−μ . Under this condition, the set of orbits of the classical system given by h(x, ξ ) is invariant with respect to an appropriate scaling. This allows us to reduce the phase space. In the quantum case, we introduce an appropriate concept of a wave front set adapted to the solutions to (1.29), different from Hörmander’s. One of our main results describes a possible location of this special wave front set for solutions to (1.29) for λ = 0 – the statement is very similar to the statement of the original Hörmander’s theorem; it is used in a proof of Theorem 1.4. 1.14. Sommerfeld radiation condition Another of our main results is a version of the Sommerfeld radiation condition for zero energies. It says that given v in a certain weighted space a solution u of the equation (H − λ)u = v satisfying appropriate outgoing/incoming phase space localization is always of the form u = R(λ ± i 0)v.


1839

This somewhat technical result has a number of interesting applications. In particular, we use it in our proof that S(0) can be expressed in terms of an oscillatory integral, and also in the description of the asymptotics of generalized eigenfunctions at large distances. 1.15. Organization of the paper The paper is organized as follows: In Section 2 we impose conditions on the potential. In the case we allow the potential to have a non-spherically symmetric term we shall need certain regularity properties of the leading spherically symmetric term. These properties are stated in Condition 2.2; they are fulfilled for the example (1.23) discussed above. In Section 3 we describe and extend some of results from our previous papers. In particular, we recall the construction of scattering phases in [6] (given there under the same conditions). We describe and to some extend the study of the properties of these objects. In Section 4 we recall various microlocal resolvent estimates from [8] (slightly extended). We also introduce the concept of the scattering wave front set adapted to energy zero. We give its applications, in particular a result about the Sommerfeld radiation condition at zero energy. In Section 5 we describe the modifiers used in our paper. They are given by a WKB-type ansatz, which involves solving transport equations. In Section 6 we introduce wave operators and wave matrices. We describe their low-energy asymptotics. In Section 7 we introduce scattering operators and matrices. We analyse their low-energy asymptotics. In Section 8 we study properties of generalized eigenfunctions for non-negative energies. In Section 9 we restrict our attention to potentials of the form (1.23). We show the classical μ rule, ω · ω = cos 2−μ π , for the location of zero-energy singularities (cf. Theorem 1.4). We also show a “propagation of scattering singularities result”, see Proposition 9.1, on generalized zeroenergy eigenfunctions. Under stronger conditions than (1.23) we represent the kernel of S(0) as an explicit oscillatory integral. In Appendix A we present, in an abstract setting, various elements of stationary scattering theory used in our paper. 2. Conditions We shall consider a classical Hamiltonian h = 12 ξ 2 + V on Rd × Rd where V satisfies Condition 2.1 (in classical mechanics we can take V3 = 0) and possibly Condition 2.2 (both stated below). Throughout the paper we shall use the non-standard notation x for x ∈ Rd to denote a function x = f (r); r = |x|, where here f ∈ C ∞ ([0, ∞[) is taken convex, and obeys f = 12 for r < 14 and f = r for r > 1. We shall often use the notation xˆ = x/r for vectors x ∈ Rd \ {0}. Let L2,s = L2,s (Rdx ) = x−s L2 (Rdx ) for any s ∈ R(the corresponding norm will be denoted by · s ). Introduce also L2,−∞ (= L2,−∞ (Rd )) = s∈R L2,s and L2,∞ = s∈R L2,s . The notation F (s > ) denotes a smooth increasing function = 1 for s > 34 and = 0 for s < 12 ; F (· < ) √ := 1 − F (· > ). The symbol g will be used extensively; it stands for the function g(r) = 2λ − 2V1 (r) (for V1 obeying Condition 2.1 and λ ∈ [0, ∞[). Condition 2.1. The function V can be written as a sum of three real-valued measurable functions, V = V1 + V2 + V3 , such that, for some μ ∈ ]0, 2[, we have:

1840


(1) V1 is a smooth negative function that only depends on the radial variable r in the region r 1 (that is V1 (x) = V1 (r) for r 1). There exists 1 > 0 such that V1 (r) − 1 r −μ ,

r 1.

(2) For all γ ∈ (N ∪ {0})d there exists Cγ > 0 such that

xμ+|γ | ∂ γ V1 (x) Cγ . (3) There exists ˜1 > 0 such that rV1 (r) −(2 − ˜1 )V1 (r),

r 1.

(2.1)

(4) V2 = V2 (x) is smooth and there exists 2 > 0 such that for all γ ∈ (N ∪ {0})d

xμ+ 2 +|γ | ∂ γ V2 (x) Cγ . (5) V3 = V3 (x) is compactly supported. The following condition will be needed only in the case V2 = 0: Condition 2.2. Let V1 be given as in Condition 2.1 and α := max(0, 1 − α(μ + 2 2 )) such that

lim sup r r→∞

−1

r

V1 (r)

− 1 −2V1 (ρ) 2 dρ

1

r

lim sup V1 (r) r→∞

− 1 −2V1 (ρ) 2 dρ

2

2

2 2+μ .

< 4−1 1 − ¯12 ,

< 4−1 1 − ¯12 .

There exists ¯1 >

(2.2)

(2.3)

1

We notice that (2.1) and (2.2) tend to be somewhat strong conditions for μ ≈ 2. On the other hand Conditions 2.1 and 2.2 hold for all 2 > 0 for the particular example V1 (r) = −γ r −μ (with

1 = γ , ˜1 = 2 − μ and some ¯1 < 1 − αμ). In quantum mechanics we consider H = H0 + V , H0 = 12 p 2 , p = −ı∇, on H = L2 (Rd ), and we need the following additional condition. Clearly Condition 2.3(1) assures that H is selfadjoint. For an elaboration of Condition 2.3(2), see [8]; it guarantees that zero is not an eigenvalue of H . Condition 2.3(3) is included here only for convenience of presentation; with the other conditions there are no small positive eigenvalues, cf. [8]. Condition 2.3. In addition to Condition 2.1 (1) V3 (H0 + ı)−1 is a compact operator on L2 (Rd ). (2) H satisfies the unique continuation property at infinity. (3) H does not have positive eigenvalues.


1841

3. Classical orbits In this section we recall and extend the results of [6] about low energy classical orbits that we will need in our paper. 3.1. Scattering orbits at positive energies We introduce for R 1 and σ > 0 + ΓR,σ (ω) = y ∈ Rd y · ω (1 − σ )|y|, |y| R ; ω ∈ S d−1 , + + = (y, ω) ∈ Rd × S d−1 y ∈ ΓR,σ (ω) . ΓR,σ Lemma 3.1. Suppose that V1 satisfies (1.1). Let σ√∈ ]0, 2[. Then there exists a decreasing function ]0, ∞[ λ → R0 (λ) such that for all |ξ | 2λ and x ∈ ΓR+0 (λ),σ (ξˆ ) there exists a unique (ξˆ ) for t > 1. If we set solution y(t) = y + (t, x, ξ ) of the problem (1.2) such that y(t) ∈ Γ + R0 (λ),σ

F + (x, ξ ) := y˙ + (1, x, ξ ), then rotx F + (x, ξ ) = 0. + For any ξ = 0 we let λ = 2−1 ξ 2 , ω = ξˆ and R = R0 (λ). For (x, ω) ∈ ΓR,σ we choose a path + [0, 1] l → γ (l) ∈ ΓR,σ (ω) such that γ (0) = Rω and γ (1) = x. We set

1

+

φ (x, ξ ) :=

√ √ dγ (l) dl + 2λR. F + γ (l), 2λω · dl

0

Note that φ + (x, ξ ) does not depend on the choice of the path γ . For instance, we can take the interval joining these two points and then 1

+

√ √ F + l(x − Rω) + Rω, 2λω dl + 2λR.

φ (x, ξ ) = (x − Rω) ·

(3.1)

0

Another possible choice is the radial interval from Rω to |x|ω and then the arc towards x: |x| √ φ (x, ξ ) = F + lω, 2λω · ω dl +

R arccos ω·xˆ

√ √ dvα dα + 2λR, F + |x|vα , 2λω · |x| dα

+ 0 ˆ ω·xˆ . where vα := cos αω + sin α √x−ω 2 1−(ω·x) ˆ

(3.2)

1842


The phase function constructed above essentially coincides with the Isozaki Kitada (outgoing) phase function, cf. [16], [18, Definition 2.3] or [4, Proposition 2.8.2]. In particular, for any ξ = 0, there are bounds γ ∂ξκ ∂x φ + (x, ξ ) − ξ · x = O |x|δ−|γ |

for |x| → ∞,

δ > max(1 − μ, 0).

(3.3)

These bounds are not uniform in ξ = 0, they are however uniform on compact subsets of Rd \ {0}. 3.2. Scattering orbits at low energies Let us now recall some results about scattering orbits taken from [6]. We assume Conditions 2.1 and 2.2 (only Condition 2.1 if V2 = 0). The fact that our Condition 2.1 includes a possibly singular potential V3 is irrelevant for this subsection since by assumption this term is compactly supported. More precisely we just need to make sure that the R0 1 in Lemma 3.2 stated below is taken so large that V3 (x) = 0 for |x| R0 , then [6] applies. Lemma 3.2. There exist R0 1 and σ0 > 0 such that for all R R0 and for all positive + σ σ0 the problem (1.24) is solved for all data (x, ω) ∈ ΓR,σ and λ 0 by a unique func+ + + tion y (t, x, ω, λ), t 1, such that y (t, x, ω, λ) ∈ ΓR,σ (ω) for all t 1. Define a vector field F + (x, ω, λ) on ΓR+0 ,σ0 (ω) by F + (x, ω, λ) = y˙ + (t = 1; x, ω, λ).

(3.4)

Then rotx F + (x, ω, λ) = 0. Note that under the assumptions of Lemma 3.2, we can suppose that R0 (λ), introduced in + Lemma 3.1, equals R0 for all λ > 0. We can define φ + (x, ω, λ) on (x, ω, λ) ∈ ΓR,σ × [0, ∞[. For further reference let us record the analogues of (3.1) and (3.2): 1

+

φ (x, ω, λ) = (x − R0 ω) ·

√ F + l(x − R0 ω) + R0 ω, ω, λ dl + 2λR0 ,

0

|x| φ (x, ω, λ) = F + (lω, ω, λ) · ω dl + +

R0

arccos ω·xˆ

√ dvα dα + 2λR0 . F + |x|vα , ω, λ · dα

0

We will add the subscript “sph” to all objects where V is replaced by the (spherically symmetric) potential V1 . The following result is proven in [6]: Proposition 3.3. There exists ˘ = ˘ (μ, ¯1 , 2 ) > 0 and uniform bounds + F + (x) − Fsph (x) = O |x|−μ/2−˘ .

(3.5a)


1843

In particular, for constants C, c > 0 independent of x, ω and λ + + (x) Fsph F (x) −˘ |F + (x)| − |F + (x)| C|x| , sph

(3.5b)

F + (x) · xˆ 1 − C(1 − xˆ · ω) − C|x|−˘ , |F + (x)|

(3.5c)

F + (x) · xˆ 1 − c(1 − xˆ · ω) + C|x|−˘ , |F + (x)|

(3.5d)

F + (x) · ω 1 − C(1 − xˆ · ω) − C|x|−˘ . |F + (x)|

(3.5e)

and

More generally (with the same ˘ > 0), for all multiindices δ and γ there are uniform bounds γ ∂ωδ ∂x F + (x) = x−|γ | O g |x| , γ + ∂ωδ ∂x F + (x) − Fsph (x) = x−˘ −|γ | O g |x| .

(3.5f) (3.5g)

The vector field F + (x, ω, λ), as well as all derivatives ∂ωδ ∂x F + , are jointly continuous in the variables (x, ω) ∈ ΓR+0 ,σ0 and λ 0. γ

The problem (1.24) in the case of t → −∞ can also be solved. We introduce for R 1 and σ >0 − ΓR,σ (ω) = y ∈ Rd y · ω (σ − 1)|y|, |y| R , ω ∈ S d−1 ; − − ΓR,σ = (y, ω) ∈ Rd × S d−1 y ∈ ΓR,σ (ω) . Mimicking the previous procedure, starting from the mixed problem (1.24) in the case of t → −∞, we can similarly construct a solution φ − (x, ω, λ) to the eikonal equation in some − ΓR,σ (ω). This amounts to setting φ − (x, ω, λ) = −φ + (x, −ω, λ),

x ∈ ΓR−0 ,σ0 (ω) = ΓR+0 ,σ0 (−ω).

(3.6)

3.3. Radially symmetric potentials In this subsection we assume that V2 = 0, which means that the potential is spherically symmetric. More precisely, we assume that for r R0 n ∂ V (r) cn r −n−μ , r

V (r) −cr −μ ,

c > 0,

rV (r) + 2V (r) < 0.

Note that motion in such a potential is confined to a 2-dimensional plane. In the case of the ˆ It is also convenient to introduce trajectory y + (t, x, ω, λ), it is the plane spanned by ω and x.

1844


θ1 xˆ ˆ θ1 ω the vectors x ⊥ := ω−cos and ω⊥ := x−cos sin θ1 sin θ1 , where ω · xˆ = cos θ1 . Therefore, we can restrict temporarily our attention to a 2-dimensional system. We will use the polar coordinates (r cos θ, r sin θ ). Note that the energy λ and the angular momentum L are preserved quantities. Therefore, the Newton equations (for outgoing orbits) can be reduced to

θ˙ = Lr −2 , r˙ = 2λ − 2V (r) − L2 r −2 .

(3.7)

Lemma 3.4. For some θ0 > 0, for all r1 R0 , |θ1 | θ0 and λ 0 we can find a solution of (3.7) satisfying r˙ (1) > 0,

r(1) = r1 ,

lim θ (t) = 0,

θ (1) = θ1 .

t→∞

There exists a function (r1 , θ1 , λ) → L(r1 , θ1 , λ) ∈ R specifying the total angular momentum of the solution y + (t, x, ω, λ). This function L is an odd function in θ1 . We have the following estimates: ∂rn1 ∂θm2 L2 = O r12−n g(r1 )2 ,

n, m 0;

(3.8a)

L = O r11−n g(r1 ) , θ1

n, m 0.

(3.8b)

1

∂rn1 ∂θm2 1

This allows us to compute the initial velocity of the trajectory: F + (x, ω, λ) =

2λ − 2V (r) − L2 /r 2 xˆ −

L ⊥ x . r

The function φ + equals, with r = |x| and cos θ = xˆ · ω, √ 2λ − 2V (r ) dr + L(r, θ , λ) dθ . φ (x, ω, λ) = 2λR0 + r

θ

+

R0

(3.9)

0

Therefore, using also that ∇ω θ = −ω⊥ , ∇ω φ + = −L(r, θ, λ)ω⊥ .

(3.10)

This gives the following estimates (in any dimension): Lemma 3.5. There exist constants C, c > 0 such that xˆ · F + (x) − g |x| C(1 − xˆ · ω)g |x| , + F (x) − xˆ xˆ · F + (x) C 1 − xˆ · ωg |x| , ∇ω φ + c 1 − xˆ · ωg |x| |x|, γ ∂ωδ ∂x φ + = x1−|γ | O g |x| .

(3.11a) (3.11b) (3.11c) (3.11d)


1845

We calculate for λ > 0: 1

1

∇ξ F + = (2λ)− 2 ∇ω F + + (2λ) 2 ∂λ F + ⊗ ω, − 1 L ∇ω F + = L∂θ L 2λ − 2V (r) + L2 r −2 2 r −2 ω⊥ ⊗ xˆ + ∂θ Lr −1 ω⊥ ⊗ x ⊥ − ∇ω x ⊥ , r 1 + 2 −2 − 2 −2 −1 ⊥ 1 − L∂λ Lr xˆ − ∂λ Lr x . ∂λ F = 2λ − 2V (r) − L r Specifying to x parallel to ω and noting that L(x, x, ˆ λ) = 0, we obtain 1/2 ∇ξ F + = (2λ)1/2 ∂λ 2λ − 2V |x| xˆ ⊗ xˆ − (2λ)−1/2 |x|−1 ∂θ L x ⊥ ⊗ x ⊥ −1/2 = (2λ)1/2 2λ − 2V |x| xˆ ⊗ xˆ −1 ∞ −1/2 −1/2 −1 −2 + (2λ) 2λ − 2V (r) |x| r dr x⊥ ⊗ x⊥,

(3.12)

|x|

cf. [6, (4.5)]. In an arbitrary dimension, the formula is the same except that the second term is repeated d − 1 times on the diagonal. Therefore, √ 1/2 (d−1)/2 det ∇ξ ∇x φ + x, 2λxˆ = (2λ)(2−d)/4 g(r)−1/2 r −1 h(r) ,

(3.13)

where we have introduced the notation ∞ h(r) :=

−1 r

−2

−1

g(r )

dr

.

(3.14)

r

Note the (uniform) bounds crg(r) h(r) Crg(r).

(3.15)

Whence, combining (3.13) and (3.15), √ 1/2 c(2λ)(2−d)/4 g(r)(d−2)/2 det ∇ξ ∇x φ + x, 2λxˆ C(2λ)(2−d)/4 g(r)(d−2)/2 .

(3.16)

4. Boundary values of the resolvent In this section we impose Conditions 2.1 and 2.3. We shall recall (and extend) some resolvent estimates of [8]. They are important tools used throughout our paper. In Section 4.2 we will also introduce the notion of the scattering wave front set, which is well adapted to scattering theory at various energies. We will return to this concept in particular in Section 9, where we will prove a theorem about propagation of singularities for potentials with

1846


a homogeneous principal part. A somewhat cruder version of this theorem is given already in Section 4.2 (valid, however, for a more general class of potentials). In Section 4.4 we prove a version of the Sommerfeld radiation condition for the zero energy. 4.1. Low energy resolvent estimates Let c be a function on the phase space Rd × Rd . The left and right Kohn–Nirenberg quantization of the symbol c are the operators Opl (c) and Opr (c) acting as l Op (c)f (x) = (2π)−d/2 ei x·ξ c(x, ξ )fˆ(ξ ) dξ, r Op (c)f (x) = (2π)−d ei(x−y)·ξ c(y, ξ )f (y) dy dξ, respectively. Notice that Opl (c)∗ = Opr (c). ¯ In Proposition 4.1 stated below we use for convenience both of these quantizations, although they can be used interchangeably. Alternatively one can use Weyl quantization denoted by Opw (c), cf. [8]. We will often use the following (λ-dependent) symbols: a(x, ξ ) =

ξ2 , g(|x|)2

b(x, ξ ) =

x ξ · . g(|x|) x

(4.1)

It is convenient to introduce the following symbol class: Let c ∈ S(m, gμ,λ ), gμ,λ =

x−2 dx 2 + g −2 dξ 2 and m = mλ = mλ (x, ξ ) be a uniform weight function [12]. Here λ ∈ [0, λ0 ] (for an arbitrarily fixed λ0 > 0) is considered as a parameter; the function m obeys bounds uniform in this parameter (see [8, Lemma 4.3(ii)] for details). For a uniform weight function m, the symbol class Sunif (m, gμ,λ ) is defined to be the set of parameter-dependent smooth symbols c = cω,λ satisfying δ γ β ∂ ∂x ∂ cω,λ (x, ξ ) Cδ,γ ,β mλ (x, ξ ) x−|γ | g −|β| . ω

ξ

(4.2)

We notice that the “Planck constant” for this class is x−1 g −1 . The corresponding class of quantizations is denoted by Ψunif (m, gμ,λ ) (it does not depend on whether left or right quantization is used). Finally we remark that the quantizations appearing in Proposition 4.1 stated below belong to Ψunif (1, gμ,λ ), and hence they are bounded uniformly in λ (these symbols are independent of ω). We can obtain the following estimates by mimicking the proof of [8, Theorem 4.1] (first for the smooth case V3 = 0, and then the general case by a resolvent equation, see [8, Subsection 5.1]; here the unique continuation assumption Condition 2.3(2) comes into play). In particular, Proposition 4.1(i) follows from [8, Corollary 3.5] and a resolvent identity (cf. [8, (5.12)]). Similarly Proposition 4.1(ii) follows from [8, Lemma 4.5] and the proof of [8, Lemma 4.6] (notice that it suffices to show the bounds (4.3b) and (4.3c) for t = 0 due to this proof), while Proposition 4.1(iii) follows from [8, Lemma 4.9] and the same minor modification of the proof of [8, Lemma 4.6]. As for the continuity statement at the end of the proposition we refer the reader to the end of this subsection. The notation R(λ + i 0) refers to the limit of the resolvent R(λ + i ) as → 0+ in the sense of a form on the Schwartz space S(Rd ), cf. Remark 4.2(2).


Proposition 4.1. [0, λ0 ]: (i) For all δ >

1 2

1847

Fix any λ0 > 0. The following conclusions, (i)–(v), hold uniformly in λ ∈

there exists C > 0 such that −δ 1 x g 2 R(λ + i 0)g 21 x−δ C.

(4.3a)

(ii) There exists C0 1 such that if χ+ ∈ C ∞ (R), supp(χ+ ) ⊂ ]C0 , ∞[ and χ+ ∈ Cc∞ (R), then for all δ > 12 and all s, t 0 there exists C > 0 such that xg s xt−δ g 21 Opl χ+ (a) R(λ + i 0)g 12 x−t−δ xg −s C, xg −s x−t−δ g 12 R(λ + i 0) Opr χ+ (a) g 21 xt−δ xg s C.

(4.3b) (4.3c)

(iii) Let σ¯ > 0 and χ− ∈ Cc∞ (R). Suppose χ˜ − , χ˜ + ∈ C ∞ (R) satisfy sup supp χ˜ − 1 − σ¯ , Then for all δ >

1 2

inf supp χ˜ + σ¯ − 1.

and all s, t 0 there exists C > 0 such that

xg s xt−δ g 21 Opl χ− (a)χ˜ − (b) R(λ + i 0)g 12 x−t−δ xg −s C, xg −s x−t−δ g 12 R(λ + i 0) Opr χ− (a)χ˜ + (b) g 21 xt−δ xg s C.

(4.3d) (4.3e)

(iv) Suppose χ−1 , χ−2 ∈ Cc∞ (R), χ˜ − and χ˜ + satisfy the assumptions from (3) and in addition sup supp χ˜ − < inf supp χ˜ + . Then for all s 0 there exists C > 0 such that s l 1 x Op χ (a)χ˜ − (b) R(λ + i 0) Opr χ 2 (a)χ˜ + (b) xs C. −

−

(4.3f)

(v) Suppose χ+ is given as in (2), some functions χ˜ + , χ˜ − , χ− are given as in (3) and suppose dist(supp χ− , supp χ+ ) > 0. Then for all s 0 there exists C > 0 such that s l x Op χ+ (a) R(λ + i 0) Opr χ− (a)χ˜ + (b) xs C, s l x Op χ− (a)χ˜ − (b) R(λ + i 0) Opr χ+ (a) xs C.

(4.3g) (4.3h)

All the forms appearing in (i)–(v) are continuous in λ 0. In fact the families of corresponding operators are continuous B(L2 (Rd ))-valued functions.

1848


Remarks 4.2. (1) Although this will not be needed we have in fact (2) with C0 = 1; see Corollary 4.4 for a related result. (2) The paper [8] contains a stronger version of the so-called limiting absorption principle than can be read from Proposition 4.1(i): For all δ > 12 there exists C > 0 such that 1 1 sup x−δ g 2 R(λ + i )g 2 x−δ C;

M := [0, λ0 ] × i ]0, 1],

λ+i ∈M μ

μ

and the B(L2 (Rd )-valued function x−δ− 4 R(ζ ) x−δ− 4 is uniformly Hölder continuous in ζ ∈ M. The (well-known) positive energy analogue of this assertion states that for any positive λ1 < λ0 the B(L2 (Rd )-valued function x−δ R(ζ ) x−δ is uniformly Hölder continuous in ζ ∈ M \ {Re ζ < λ1 }; see (4) for a related remark. (3) The paper [8] also contains an extension of Proposition 4.1 to powers of the resolvent, however this will not be useful in the forthcoming sections; see Example 7.5 for a discussion. This is related to the fact that our classical constructions are not smooth in λ at zero energy, cf. [6, Remarks 4.7(1)]. The collection of all estimates in Proposition 4.1 (more precisely a collection of similar estimates with a complex spectral parameter) yields similar estimates for powers of the resolvent by a completely algebraic reasoning, cf. [8, Appendix A]. (4) Assume that the potential satisfies Condition 1.1. Then all the bounds of Proposition 4.1 remain true uniformly in λ ∈ [λ1 , λ0 ] for any positive λ1 < λ0 provided we replace a → a :=

ξ2 , 2λ

ξ x b → b := √ · 2λ x

and g → 1.

(4.4)

(Under the stronger Conditions 2.1 and 2.3 the validity of this modification is a direct consequence of the bounds of Proposition 4.1.) Also in this case the families of associated operators are norm continuous (now in λ > 0 only). Proof of continuity statements in Proposition 4.1. Due to Remark 4.2(2) and the calculus of pseudodifferential operators all appearing forms in Proposition 4.1 are continuous in λ 0. Norm continuity of the corresponding operator-valued functions also follows from Re1 1 mark 4.2(2). This can be seen as follows for Bδ (λ) := x−δ g 2 R(λ + i 0)g 2 x−δ (appearing in (i)): Pick δ ∈ ] 12 , δ[, insert for (small) κ > 0 the identity I = F (κ|x| < 1) + F (κ|x| > 1) on both sides of Bδ (λ) and expand (into three terms). This yields Bδ (λ) − F κ|x| < 1 Bδ (λ)F κ|x| < 1 Cκ δ−δ Bδ (λ).

Due to Proposition 4.1(i) the right-hand side is O(κ δ−δ ) uniformly λ 0. On the other hand due to Remark 4.2(2) (and the calculus of pseudodifferential operators) for fixed κ > 0 the B(L2 (Rd ))-valued function F (κ|x| < 1)Bδ (·)F (κ|x| < 1) is continuous. Hence Bδ (·) is a uniform limit of continuous functions and therefore indeed continuous. The other operator-valued functions can be dealt with in the same fashion. 2


1849

4.2. Scattering wave front set The remaining subsections of Section 4 are devoted to a number of somewhat technical estimates on solutions to the equation (H − λ)u = v for a fixed λ 0. Although they are proved under Conditions 2.1 and 2.3 we remark that there are similar estimates under Condition 1.1 for a fixed λ > 0. The reader may skip this material on the first reading. Throughout the remaining part of this section we use the notation ξ 1 = (1 + |ξ |2 )1/2 and X = (1 + |x|2 )1/2 for ξ, x ∈ Rd . With reference to the symbol class Sunif (m, gμ,λ ) from Section 4.1 clearly h1 , h2 ∈ Sunif (m, gμ,λ ) with h1 := 12 ξ 2 + V1 , h2 := 12 ξ 2 + V1 + V2 and m = g 2 ξ/g21 . In the remaining part of Section 4 we shall however only need a reminiscence of this symbol class given by disregarding the uniformity in λ 0. Whence we shall consider symbols c ∈ S(m, gμ,λ ) meaning, by definition, that γ β ∂x ∂ c(x, ξ ) Cγ ,β m(x, ξ ) x−|γ | g −|β| .

(4.5)

ξ

The corresponding class of standard Weyl quantizations Opw (c) is denoted by Ψ (m, gμ,λ ). It is convenient to introduce the following constants: μ, for λ = 0, 1 + μ2 /2, 1 − μ2 , s1 = s2 = (4.6) s0 = 0, for λ > 0. 1/2, 1, If > 0, then x−s0 − will be a typical weight that appears in resolvent estimates. (Notice 1 1 that in the uniform estimates of Proposition 4.1 the corresponding weight is g 2 x− 2 − .) The weight x−s1 plays the role of the “Planck constant” for the class Ψ (m, gμ,λ ). Finally, x−s2 will appear in the “elliptic regularity estimate” of Proposition 4.3. Clearly s0 > s2 and s1 > 0. Let us decompose the normalized momentum ξ/g as follows: x x x ξ x ξ ξ =b + c, ¯ b := · and c¯ := I − . (4.7) g

x

x g

x x g Notice that b was already defined in Section 4.1, besides for r = |x| 1, b2 + c¯2 = a with a also defined in Section 4.1. Moreover for r 1 we have the identification b = xˆ · gξ ∈ R

and c¯ = (I − |x

ˆ x|) ˆ gξ ∈ Txˆ∗ (S d−1 ) with xˆ = x/r ∈ S d−1 , which obviously constitute canonical coordinates for “the phase space” T∗ := T ∗ (S d−1 ) × R = S d−1 × Rd . This partly motivates the following definition: The wave front set W Fscs (u) of a distribution u ∈ L2,−∞ is the subset of T∗ given by the condition / W Fscs (u) z1 = (ω1 , c¯1 , b1 ) = (ω1 , b1 ω1 + c¯1 ) = (ω1 , η1 ) ∈

⇔

∃ neighbourhoods Nω1 ω1 , Nη1 η1 ∀χω1 ∈ Cc∞ (Nω1 ), χη1 ∈ Cc∞ (Nη1 ): Opw χz1 F (r > 2) u ∈ L2,s where χz1 (x, ξ ) = χω1 (x)χ ˆ η1 (bxˆ + c). ¯

(4.8)

Notice that this quantization is defined by the substitution bxˆ + c¯ → ξ/g, cf. (4.7). Keep in mind that the whole concept depends on the given energy λ ∈ [0, ∞[ in consideration (through g, which enters in the definition of b and c). ¯

1850


The above notion of wave front set is of course adapted to the problem in hand. The classical definition is taylored to measure decay in momentum space; see for example [14, Chapter VIII]. Our definition concerns decay in position space, and thus it is more related to the wave front set introduced in [21, Section 7] (dubbed there as “the scattering wave front set”). Obviously u ∈ L2,s

⇒

W Fscs (u) = ∅.

Conversely (by a compactness argument), if for some χ ∈ Cc∞ (Rd ) u − Opw χ(ξ/g) u ∈ L2,s ,

(4.9)

then W Fscs (u) = ∅

⇒

u ∈ L2,s .

Proposition 4.3. Let λ 0 and s2 be defined in (4.6). Let u ∈ L2,−∞ , v ∈ L2,s+s2 and (H − λ)u = v. Then the estimates (4.9) and (4.10) W Fscs (u) ⊆ z ∈ T∗ b2 + c¯2 = 1 hold. More generally, suppose u ∈ L2,−∞ , g −1 v ∈ L2,s and (H − λ)u = v. Then the following estimates hold: (4.11a) For all > 0: g Opw F b2 + c¯2 − 1 > u ∈ L2,s , 2 w 2 2 2,s For all > 0, g Op ξ/g1 F b + c¯ − 1 > u ∈ L , (4.11b) For all > 0: g Opw F 1 − b2 − c¯2 > u ∈ L2,s , (4.11c) W Fscs (gu) ⊆ z ∈ T∗ b2 + c¯2 = 1 . (4.11d) Proof. Obviously (4.11b) is stronger than (4.11a). Notice also that (4.11a) in some sense is stronger than Proposition 4.1(ii) (involves weaker weights). It is also obvious that (4.11d) is a consequence of (4.11b) and (4.11c). The proof of (4.11b) given below is somewhat similar to the proof of the analogue of Proposition 4.1(ii) given in [8]. For convenience we have divided the proof into four steps. For the calculus of pseudodifferential operators, used tacitly below, we refer to [14, Theorems 18.5.4, 18.6.3, 18.6.8] (the reader might find it more convenient to consult [8] for an elaboration). The bounds (4.11c) can be proved by mimicking Steps III and IV below. We note that the complication due to high energies, cf. Step II below, is absent. For this reason (4.11c) is somewhat easier to establish than (4.11b) and we shall leave the details of proof to the reader. Step I. At various points in the proof of (4.11b) we need to control the possibly existing local singularities of the potential V3 . This is done in terms of the following elementary bounds: T1 := xt g −1 V3 (H − i)−1 g −1 x−t ∈ B L2 , t, t ∈ R; (4.12a) −1 (4.12b) T1 := xt g −1 V3 1 + p 2 g −1 x−t ∈ B L2 , t, t ∈ R; T2 := xt 1 + p 2 g(H − i)−1 g −1 x−t ∈ B L2 , t ∈ R. (4.12c)


1851

Step II. Suppose gu ∈ L2,t for some fixed t s. We shall prove that then Agu ∈ L2,t for all A ∈ Ψ ( ξ/g21 , gμ,λ ), more precisely, that for all A ∈ Ψ ξ/g21 , gμ,λ :

Agut C gut + g −1 v s .

(4.13)

For any such an operator A and any m ∈ R, we decompose

xt A = Bm xt Opw ξ/g21 + Rm ,

(4.14)

where Bm ∈ Ψ (1, gμ,λ ) and Rm ∈ Ψ ( ξ/g21 x−m , gμ,λ ). Now, cf. [8, proof of Lemma 4.5], Opw ξ/g21 = g −1 p 2 g −1 + Opw (a1 ) = 2g −1 (H − λ)g −1 + Opw (a2 ) − 2g −2 V3 ; 2 a2 = a1 + 1 − 2g −2 V2 ∈ S(1, gμ,λ ). (4.15) a1 = 1 − ∇g −1 + 4−1 g −2 , We substitute (4.15) in (4.14), expand into altogether four terms and apply the resulting sum to the state gu. The contribution from the first term of (4.15) is estimated as Bm xt 2g −1 (H − λ)g −1 (gu) C1 g −1 v C2 g −1 v . t s Similarly, the contribution from the second term of (4.15) is estimated as Bm xt Opw (a2 )gu Cgut . As for the third term of (4.15) we use (4.12a) with t = t to bound 2Bm xt g −2 V3 gu 2Bm T1 xt g(H − i)u C1 gvt + (λ − i)gut C2 gut + g −1 v s . To treat the contribution from the second term of (4.14) we note that Ψ ξ/g21 x−m , gμ,λ ⊆ Ψ ξ 21 x2−m , gμ,λ . Whence, using (4.12c) and choosing m = 2 − t, Rm gu C1 T2 xt g(H − i)u C2 gut + g −1 v s . We conclude (4.13). Step III. Suppose gu ∈ L2,t for some fixed t < s. Fix s ∈ ]t, t + 1 − μ/2] with s s. We shall show that (4.11a) holds with s replaced by s . We set F := F (b2 + c¯2 − 1 > ). − 2−μ 2

We need a regularization in x-space given in terms of ικ = Xκ 1/2 . Xκ := 1 + κ|x|2

, where for κ ∈ ]0, 1] we let (4.16)

1852


Mimicking [8, proof of Lemma 4.5], for R > 1 large enough we clearly have F 2 F (r > R)2

2h2 − 2λ 3 F 2 F (r > R)2 . Re

g2

Let

Pκ = Opw (pκ ),

1−s −1 −s d = ξ/g−1 = h¯ −1 ξ/g−1 ; 1 x 1 g x 6 qκ = xs F ικ F (r > R). Re(h2 − λ) − g 2 , pκ = qκ2

D = Opw (d),

Since 0 pκ ∈ S(h¯ −2 d −2 , gμ,λ ), D ∗ Pκ D −C uniformly in κ. Since 0 < d ∈ S(d, gμ,λ ), we can for any m ∈ R find em ∈ S(d −1 , gμ,λ ) such that DEm − I ∈ Ψ x−2m , gμ,λ ;

Em = Opw (em ).

Consequently, we have the uniform bound ∗ E m + Rm , Pκ −CEm

Rm ∈ Ψ ξ/g21 g 2 x2s −2m , gμ,λ ,

and therefore by choosing m = s − t and by using (4.13) that the expectation 2

Pκ u −C gut + g −1 v s .

(4.17)

On the other hand, for any δ ∈]0, 1[ 2

Pκ u C gut + g −1 v s − (1 − δ) Q∗κ Qκ gu ,

Qκ = Opw (qκ ).

(4.18)

Here we use that Opw qκ2 Re(h2 − λ) = Re (Qκ g)∗ Qκ g −1 (H − V3 − λ) + Rκ , Rκ ∈ Ψ ξ/g21 h¯ 2 x2s g 2 , gμ,λ ⊆ Ψ ξ/g21 x2t g 2 , gμ,λ , and the fact that Rκ is bounded in κ ∈ ]0, 1] in the class Ψ ( ξ/g21 x2t g 2 , gμ,λ ). Notice that 2 6 Re (Qκ g)∗ Qκ g −1 (H − λ) u CQκ gug −1 v s δQκ gu2 + Cδ g −1 v s ,

and that the contributions from V3 and the term Rκ can be treated by (4.12a) and (4.13), respectively.


1853

Now, combining (4.17) and (4.18) we conclude that 2 Qκ gu2 C gut + g −1 v s uniformly in κ ∈ ]0, 1]. Letting κ → 0 completes Step III. Step IV. Note that (4.11b) is equivalent to the following, seemingly stronger statement: For all > 0, A ∈ Ψ ξ/g21 , gμ,λ implies Ag Opw (F )u ∈ L2,s .

(4.19)

We will show (4.19) by induction. By assumption, gu ∈ L2,t for a sufficiently small t s and consequently, due to Step II, it follows that Agu ∈ L2,t for all A ∈ Ψ ( ξ/g21 , gμ,λ ). Consider for all k ∈ N the following claim given in terms of tk := min(s, t + (1 − μ/2)(k − 1)): The bound/localization (4.11b) holds for all > 0 and all A ∈ Ψ ( ξ/g21 , gμ,λ ) provided u → u := Opw (F /2 )u and s is replaced by tk . (Notice that this implies in particular that the state gu2 ∈ L2,tk and, since > 0 is arbitrary, that gu ∈ L2,tk .) We have seen that this claim holds for k = 1. So suppose k > 1 and that the claim is true for k → k − 1. To show the claim for k, we can assume that tk−1 < s. First, we notice that v := (H − λ)u obeys the condition g −1 v ∈ L2,tk due to the induction hypothesis, (4.13), (4.12a) and (4.12b). Notice at this point that H − V3 − λ, Opw (F /2 ) ∈ Ψ g 2 ξ/g21 h¯ , gμ,λ , and that in fact (for any m ∈ R) H − V3 − λ, Opw (F /2 ) = gAg Opw (F /4 ) + Rm , A ∈ Ψ ξ/g21 xμ/2−1 , gμ,λ , Rm ∈ Ψ ξ/g21 x−m , gμ,λ . Now, by Step III, (4.11a) applies to u → u , t → sk−1 and with s replaced by s = tk . Next, by applying Step II to the state u → u˜ := Opw (F )u (note that as above g −1 (H − λ)u˜ ∈ L2,tk ), we conclude that indeed the bound (4.11b) holds with u → u and s replaced by tk . The induction is complete. Finally we obtain, using the above claim, that the bound (4.11b) holds without changing u and with s replaced by tk . Since clearly tk = s for k sufficiently large, (4.11b) follows. 2 The following corollary follows immediately from Proposition 4.3. At a fixed energy, it strengthens Proposition 4.1(ii). Corollary 4.4. Let χ ∈ Cc∞ (R), χ = 1 around 1. Then for any s > s0 we have (with λ 0, and s0 and s2 as given in (4.6)) s−s x 2 Opw a 2 + 1 1 − χ(a) R(λ ± i 0) x−s C. (4.20) The following proposition is similar to Proposition 9.1 stated later, although the flavour is somewhat “global”. These results (as well as their proofs) are modifications of [12, Proposition 3.5.1] (and its proof), see also [21] and [11]. The condition (4.21) is similar to (4.11b); it implies that W Fscs (u) ⊆ {b2 + c¯2 1} and hence that W Fscs (u) is compact.

1854


Proposition 4.5. Let λ 0 and s0 be defined in (4.6). Suppose u, v ∈ L2,−∞ , (H − λ)u = v, s ∈ R, k ∈ ]−1, 1[ and {b = k} ∩ W Fscs (u) = ∅. Suppose the following condition: For all δ > 0,

Opw ξ/g21 F b2 + c¯2 − 1 > δ u ∈ L2,s .

(4.21)

Define ˜ ∩ W Fscs (u) = ∅ , k + = sup k˜ k b ∈ [k, k] ˜ k] ∩ W Fscs (u) = ∅ . k − = inf k˜ k b ∈ [k,

(4.22) (4.23)

Then k+ < 1

⇒

k − > −1

⇒

b = k + ∩ W Fscs+2s0 (v) = ∅, {b = k − } ∩ W Fscs+2s0 (v) = ∅.

(4.24) (4.25)

Proof. We shall only deal with the case of superscript “+”; the case of “−” is similar. For convenience we shall assume that 2 2 − μ and divide the proof into two steps. Step I. We will first show the following weaker statement: Suppose u ∈ L2,s− 2 /2 , v ∈ L2,s+2s0 and (H − λ)u = v (in this case (4.21) follows from Proposition 4.3). Then k + 1.

(4.26)

Suppose on the contrary that k + < 1. By a compactness argument we can then find a point in W Fscs (u) of the form z1 = (ω1 , c¯1 , k + ). For > 0 chosen small enough (less than (k + − k)/2 suffices here)

b ∈ k + − 2 , k + ∩ W Fscs (u) = ∅.

(4.27)

We can assume that J := ]k + − 2 , k + + [ ⊆ ]−1, 1[. Pick a non-positive f ∈ Cc∞ (J ) with f 0 on [k + − , ∞[ and f (k + ) < 0, and consider for K > 0 and κ ∈ ]0, 1] the symbol bκ = X s 0 aκ ,

aκ = X s Xκ− 2 /2 F (r > 2) exp(−Kb)f (b)F b2 + c¯2 < 3 ;

(4.28)

here Xκ is defined by (4.16). We compute the Poisson bracket g 2 V1 (b2 − 1) x · ∇V2 c¯ + − r g g x g 1 − rV1 g −2 c¯2 + rV1 g −2 b2 + c¯2 − 1 + O r − 2 = r g 1 − rV1 g −2 1 − b2 + g −2 2(h2 − λ) + O r − 2 . = r

{h2 , b} =

(4.29) (4.30)


1855

We expand the right-hand side of (4.30) into three terms and notice that due to (2.1) the first term has the following positive lower bound on supp bκ : g ··· c ; r

c=

˜1 1 − sup t 2 t ∈ supp f . 2

First we fix K: A part of the Poisson bracket with bκ2 is g h2 , X 2s+2s0 Xκ− 2 = Yκ bX 2s+2s0 Xκ− 2 , r

(4.31)

where Yκ = Yκ (r) is uniformly bounded in κ. We pick K > 0 such that for all κ r 2Kc |Yκ | + 2 X −2s0 g

on supp bκ .

From (4.30), (4.31) and the properties of K and f , we conclude the following bound at {f (b) 0}:

h2 , bκ2 −2aκ2 + g −2 (h2 − λ)aκ O r s + O r 2s F 2 b2 + c¯2 < 3 + O r 2s− 2 .

To use this bound effectively, we introduce a partition of unity: Let f1 , f2 ∈ Cc∞ (J ) be chosen such that supp f1 ⊆ ]k + − 2 , k + [, supp f2 ⊆ ]k + − , k + + [ and f12 + f22 = 1 on supp f . We multiply both sides by f22 (= 1 − f12 ) and obtain after a rearrangement h2 , bκ2 −2aκ2 + g −2 (h2 − λ)aκ dκ + K1 f12 F b2 + c¯2 < 3 x2s − K2 F 2 b2 + c¯2 < 3 x2s + K3 x2s− 2 , (4.32) dκ ∈ S xs , gμ,λ ; here K1 , K2 , K3 > 0 are independent of κ, and the symbols dκ are bounded in κ in the indicated class. We introduce Aκ = Opw (aκ ), Bκ = Opw (bκ ) and the regularization uR = F (|x|/R < 1)u in terms of a parameter R > 1. First we compute i H, Bκ2 u = lim i H, Bκ2 u = −2 Im v, Bκ2 u . R→∞

R

(4.33)

Using (4.33) and the calculus, cf. [14, Theorems 18.5.4, 18.6.3, 18.6.8], we estimate i H, B 2 C1 vs+2s Aκ u + us− /2 1 Aκ u2 + C2 . κ u 0 2 2 On the other hand, using (4.21), (4.27) and (4.32), we infer that i H − V3 , Bκ2 u = lim i H − V3 , Bκ2 u R→∞

R

−2Aκ u2 + C3 (H − V3 − λ)us+μ Aκ u + C4 ,

(4.34)

1856


and whence, using (4.12a) to bound (H − V3 − λ)us+μ C(vs+μ + us− 2 /2 , that 3 i H − V3 , Bκ2 u − Aκ u2 + C5 . 2

(4.35)

Clearly another application of (4.12a) yields i V3 , Bκ2 u C6 .

(4.36)

Combining (4.34)–(4.36) yields Aκ u2 C7 = C2 + C5 + C6 , which in combination with the property that f (k + ) < 0 in turn gives a uniform bound − /2 w X 2 Op χz F (r > 2) u2 C8 ; κ 1 s

(4.37)

here χz1 signifies any phase-space localization factor of the form entering in (4.8) supported in a sufficiently small neighbourhood of the point z1 = (ω1 , c¯1 , k + ). / W Fsc (u), which is a contradiction; whence (4.26) We let κ → 0 in (4.37) and infer that z1 ∈ is proven. Step II. We need to remove the conditions of Step I, u ∈ L2,s− 2 /2 and v ∈ L2,s+2s0 . This will be accomplished by an iteration and modification of the procedure of Step I. Pick t1 ∈ R such that v ∈ L2,t1 . Pick t < s such that u ∈ L2,t and define sm = + be given by (4.22) with s → s . Clearly min(s, t + m 2 /2) for m ∈ N. Let correspondingly km m + + km−1 ; km

m = 2, 3, . . .

(4.38)

+ and s → s follows from If u ∈ L2,sm − 2 /2 and v ∈ L2,sm +2s0 then (4.24) with k + → km m Step I. Although we shall not verify these conditions we remark that a suitable micro-local modification will come into play in an inductive procedure, see (4.41) and (4.43) below. We shall + and s → s , i.e. that indeed (inductively) show (4.24) with k + → km m + 0, J and f be chosen as in Step I with k + → km c ˜ with f = 1 on J . It follows from (4.40), possibly by taking > 0 smaller than needed in Step I, that

I v ∈ L2,sm +2s0 ;

I = Opw f˜(b)F b2 + c¯2 < 6 .

(4.41)


1857

Next, we introduce the symbol bκ by (4.28) (with s → sm ) and proceed as in Step I. As for the bounds (4.34), we can replace v by I v up to addition of a term of the form C(v2t1 + u2sm − 2 /2 ). Similarly we can verify (4.35) and (4.36) (using conveniently (4.12b)). So again we obtain (4.37) (with s → sm ), and therefore a contradiction as in Step I. We have shown (4.39) for m = 1. Now suppose m 2 and that (4.39) is verified for m − 1. We need to show the statement for the given m. Due to (4.38) and the induction hypothesis, we can assume that + + < km−1 . km

(4.42)

Again we argue by contradiction assuming (4.40). We proceed as above noticing that it follows from (4.42) that in addition to (4.41) we have I u ∈ L2,sm−1 ;

(4.43)

+ + )/2). By replacat this point we possibly need choosing > 0 even smaller (viz. < (km−1 − km ing v by I v and u by I u at various points in the procedure of Step I (using (4.41) and (4.43), respectively) we obtain again a contradiction. Whence (4.39) follows. 2

Corollary 4.6. Let s ∈ R, u ∈ L2,−∞ , v ∈ L2,s+2s0 , (H − λ)u = v, k ∈ ]−1, 1[ and {b = k} ∩ W Fscs (u) = ∅. Then W Fscs (u) ⊆ {b = 1} ∪ {b = −1}.

(4.44)

Proof. The condition (4.21) is guaranteed by Proposition 4.3. Then we apply Proposition 4.5. 2 4.3. Wave front set bounds of the boundary value of the resolvent Proposition 4.1 implies that the symbol R(λ ± i 0) in many cases can be treated as an operator, although initially it is defined in terms of a quadratic form. Notice that Remark 4.2(2) in one situation gives a slightly different and direct interpretation of R(λ ± i 0) (as a limit of operators and hence avoiding quadratic forms). It will however be convenient to investigate possible other interpretations of states R(λ ± i 0)v (for which in particular Remark 4.2(2) does not apply) and study associated wave front set bounds. The case of R(λ − i 0) is similar to that of R(λ + i 0) and will not be elaborated regarding proofs. For sufficiently decaying states v we have (using in (ii) the slightly abused notation a := ¯ b) = (ω, bω + c) ¯ ∈ T∗ ): b2 + c¯2 for generic points z = (ω, c, Proposition 4.7. Let s > s0 and v ∈ L2,s . Then the following is true: (i) For any t > s0 , R(λ ± i 0)v = lim R(λ ± i )v

0

exists in L2,−t .

(ii) W Fscs−s2 R(λ ± i 0)v ⊆ {a = 1}. (iii) For any > 0, W Fscs−2s0 − R(λ ± i 0)v ⊆ {b = ±1}.

(4.45)

1858


Proof. Re (i). This statement follows from Remark 4.2(2); notice that the notation for the limit conforms with Proposition 4.1(i). Re (ii). We have (H − λ)u = v. Therefore (ii) follows from Proposition 4.3 (alternatively by using Corollary 4.4). Re (iii). Let χ− ∈ Cc∞ (R) such that χ− is zero around 1. Let χ ∈ Cc∞ (R). Then by Proposition 4.1(iii), for any > 0 Opw χ(a)χ− (b) R(λ + i 0)v ∈ L2,s−2s0 − .

2

Based completely on Proposition 4.1 one can give a meaning to R(λ ± i 0)v also for some states v with a slower decay provided they have an appropriate phase space localization. (In the statement below C0 1 is given in agreement with Proposition 4.1(ii).) Proposition 4.8. Let s s0 and v ∈ L2,s . Suppose that for some t > s0 and k ∈ ]−1, 1] (or k ∈ [−1, 1[) W Fsct (v) ∩ {b < k, a < 2C0 } = ∅

or W Fsct (v) ∩ {b > k, a < 2C0 } = ∅ .

(4.46)

(i) For any > 0 there exists R(λ + i 0)v = lim R(λ + i 0)vκ κ0

R(λ − i 0)v := lim R(λ − i 0)vκ in L2,s−2s0 − , κ0

where vκ (x) := F (κ|x| < 1)v(x). (ii) W Fscs−s2 R(λ + i 0)v ⊆ {a = 1} W Fscs−s2 R(λ − i 0)v ⊆ {a = 1} . (iii) For any > 0, W Fsct−2s0 − R(λ + i 0)v ∩ {b < k, a C0 } = ∅ W Fsct−2s0 − R(λ − i 0)v ∩ {b > k, a C0 } = ∅ .

(4.47)

Proof. Re (i). Let χ ∈ Cc∞ (]−∞, 2C0 [), χ = 1 around [0, C0 ]. Let χ− ∈ C ∞ (R) be chosen such that χ− = 1 around ]−∞, −1] and χ− = 0 in [(k − 1)/2, ∞[. Then by the condition (4.46) and the calculus of pseudodifferential operators Opw χ(a)χ− (b) vκ −→ Opw χ(a)χ− (b) v

in L2,t as κ 0.

Whence by Proposition 4.1(i), for any > 0, u1 := lim R(λ + i 0) Opw χ(a)χ− (b) vκ κ0

exists in L2,−s0 − .

By Proposition 4.1(ii) we have u2 := lim R(λ + i 0) Opw 1 − χ(a) vκ κ0

exists in L2,s−2s0 − .


1859

By Proposition 4.1(iii) we have u3 := lim R(λ + i 0) Opw χ(a) 1 − χ− (b) vκ κ0

exists in L2,s−2s0 − .

But s − 2s0 −s0 . Hence R(λ + i 0)v := lim R(λ + i 0)vκ = u1 + u2 + u3 ∈ L2,s−2s0 − . κ0

Re (ii). This statement is proven as (ii) of the previous proposition. Re (iii). Let χ 1 , χ 2 ∈ Cc∞ (] − ∞, 2C0 [), χ 2 = 1 around [0, max(sup supp χ 1 , C0 )]. Let 1 χ− ∈ Cc∞ (]−∞, k[) and χ−2 ∈ C ∞ (R) such that χ−2 = 1 around ]−∞, sup supp χ−1 ] and supp χ−2 ⊆ ]−∞, k[. Then by the condition (4.46) Opw χ 2 (a)χ−2 (b) v ∈ L2,t . Whence, by Proposition 4.1(i), noting that t > s0 , we obtain R(λ + i 0) Opw χ 2 (a)χ−2 (b) v ∈ L2,−s0 − and W Fsct−2s0 − R(λ + i 0) Opw χ 2 (a)χ−2 (b) v ⊆ {b = 1}.

(4.48)

By Proposition 4.1(iv), Opw χ 1 (a)χ−1 (b) R(λ + i 0) Opw χ 2 (a) 1 − χ−2 (b) v ∈ L2,∞ ,

(4.49)

and by Proposition 4.1(v), Opw χ 1 (a)χ−1 (b) R(λ + i 0) Opw 1 − χ 2 (a) v ∈ L2,∞ .

(4.50)

Now (4.48)–(4.50) yields Opw χ 1 (a)χ−1 (b) R(λ + i 0)v ∈ L2,t−2s0 − , which implies (4.47).

2

We have yet another interpretation very similar to Proposition 4.7(i): Proposition 4.9. Fix real-valued χ ∈ Cc∞ (R) and χ˜ ∈ C ∞ (R) such that inf supp χ˜ > −1 (or sup supp χ˜ < 1). Let A := Opw (χ(a)χ˜ (b)). Suppose v ∈ L2,s for some s s0 . For any > 0 there exists R(λ + i 0)Av = lim R(λ + ıκ)Av κ0

in L2,s−2s0 −

or R(λ − i 0)Av = lim R(λ − ıκ)Av in L2,s−2s0 − . κ0

Moreover this limit agrees with the interpretation of Proposition 4.8(i).

1860


Proof. We need to invoke an extended version of the bound (4.3e), see [8, Lemma 4.10]. First notice that the symbols g, and hence also a and b, obviously depend on λ. Let ζ = λ + ıκ and define gζ , aζ and bζ by replacing λ by |ζ | in the definition of g in Section 2.1 and of a and b in (4.1), respectively. Now we have the following extension of the bound (4.3e): For all δ > 12 and all s, t 0, there exists C > 0 such that for all κ ∈ ]0, 1] 1 1 xgζ −s x−t−δ g 2 R(ζ ) Opw χ− (aζ )χ˜ + (bζ ) g 2 xt−δ xgζ s C.

ζ

ζ

(4.51)

Although this will not be needed, the bound (4.51) is in fact locally uniform in λ 0. We pick in (4.51) the functions χ− and χ˜ + in agreement with Proposition 4.1(iii) such that in addition χ− = 1 around [0, sup supp χ] and χ˜ + = 1 around [min(0, inf supp χ˜ ), ∞[. Using the bounds g gζ , aζ a and |bζ | |b| we then obtain that for any m ∈ R w (4.52) Op χ− (aζ )χ˜ + (bζ ) − 1 A ∈ Ψ xm , gμ,λ . By combining Remark 4.2(2), (4.51) (with s = 0, t = s0 − s + 2 and δ = obtain the uniform bound: For all κ ∈ ]0, 1] −t−δ 1 1 x g 2 R(ζ )Ag 2 xt−δ C.

1 2

+ 2 ) and (4.52) we (4.53)

Obviously we obtain from (4.53) and a density argument that indeed there exists the limit in L2,s−2s0 − .

u := lim R(λ + ıκ)Av κ0

Since u = R(λ + i 0)Av for v ∈ L2,∞ we are done (by using density and interchanging limits). 2 4.4. Sommerfeld radiation condition In this subsection we describe a version of the Sommerfeld radiation condition close in spirit to [13, Theorem 30.2.7], [17] and [21]. We introduce for s > 0 Besov spaces Bs and corresponding duals Bs∗ as in [1] (see [13, Section 14.1] for details about these spaces). They consist of local L2 functions with a certain (norm) expression being finite. Throughout this subsection we shall actually only use the duals Bs∗ , for which we can take the norm squared to be 2 −2s |u|2 dx. uBs∗ := sup R R>1

|x| −1}, (ii) W F (Bs∗0 ,0 , u) ⊆ {b > 0}. −s

This solution is given by u = R(λ + i 0)v ∈ L2,−s for all s > s0 and W Fsc 0 (u) ⊆ {b = 1}. Similarly, under the same condition on v, the equation (H − λ)u = v has a unique solution u ∈ L2,−∞ obeying one of the following conditions: −s

(i) W Fsc 0 (u) ⊆ {b < 1}, (ii) W F (Bs∗0 ,0 , u) ⊆ {b < 0}; −s

and this solution is given by u = R(λ − i 0)v ∈ L2,−s for all s > s0 and W Fsc 0 (u) ⊆ {b = −1}. Proof. We shall only consider the first mentioned cases (i) or (ii) (they will be treated in parallel); the other cases can be treated similarly. By Proposition 4.7, the function u = u˜ := R(λ + i 0)v is a solution to (H − λ)u = v enjoying the stated properties (including (i) and (ii)). Suppose −s in the sequel that u ∈ L2,−t for some t > s0 , (H − λ)u = v and W Fsc 0 (u) ⊆ {b > −1} or ∗ ˜ W F (Bs0 ,0 , u) ⊆ {b > 0}. It remains to be shown that u = u.

1862


Step I. We shall show that u ∈ L2,−s for all s > s0 . By Proposition 4.3, W Fsc−s0 (u) ⊆ b2 + c¯2 = 1 , A Opw F b2 + c¯2 > 3 u ∈ L2,−s0 for all A ∈ Ψ ξ/g21 , gμ,λ .

(4.56) (4.57)

It follows from (4.55), Propositions 4.3 and 4.5 and a compactness argument that W Fsc−s (u) ⊆ {b = 1} for all s > s0 .

(4.58)

Pick a real-valued decreasing ψ ∈ Cc∞ ([0, ∞)) such that ψ(r) = 1 in a small neighbourhood of 0 and ψ (r) = −1 if 1/2 r 1. Let ψR (x) = ψ(|x|/R); R > 1. We also introduce δ = max(t − s0 , 2t − 2s0 + μ − 2), and check that δ + s0 t,

s0 + δ/2 + 1 − μ/2 t

and s0 + δ/2 < t.

By undoing the commutator we have on one hand that i H, X −δ ψR u = −2 Im v, X −δ ψR u ,

(4.59)

i H, X −δ ψR C1 vs u−δ−s C2 vs u−t = O R 0 . u 0 0 0

(4.60)

yielding the estimate

On the other hand i H, X −δ ψR = Re g xhδ,R Opw (b) ; −1 hδ,R (x) = −δX −2−δ ψR (x) + X −δ |x|R ψ |x|/R , yielding by using (4.57), (4.58) and the calculus (cf. [14, Theorems 18.5.4, 18.6.3, 18.6.8]) i H, X −δ ψR u = Re g xhδ,R Opw bF (b > 1/2)F b2 + c¯2 < 6 u + O R 0 , which in turn (by the same arguments) implies that i H, X −δ ψR u −δ4−1 g xX −2−δ ψR u + O R 0 .

(4.61)

By combining (4.60) and (4.61) we obtain

g xX −2−δ ψR

u

C,

(4.62)

for some constant C which is independent of R > 1. Whence, letting R → ∞ we see that u ∈ L2,−t1 ; t1 := s0 + δ/2.


1863

More generally, we define for k ∈ N tk = s0 + 2−1 max(tk−1 − s0 , 2tk−1 − 2s0 + μ − 2),

t0 := t,

and iterate the above procedure. We conclude that u ∈ L2,−tk , and hence that indeed u ∈ L2,−s for all s > s0 . Step II. Due to Step I, it suffices to show that u = 0 is the only solution to the equation −s (H − λ)u = 0 subject to the conditions u ∈ L2,−s for all s > s0 and either W Fsc 0 (u) ⊆ {b > −1} or W F (Bs∗0 ,0 , u) ⊆ {b > 0}. In the following Steps III and IV we consider this problem. Step III. We shall show that u ∈ Bs∗0 ,0 . Under condition (i) the bound (4.58) holds for s = s0 (by Proposition 4.5) which implies that There exists > 0 such that W F (Bs∗0 ,0 , u) ⊆ {b > }. Under condition (ii), we have the same conclusion due to (4.56) and a compactness argument. Next, we apply the same scheme as in Step I, now with δ = 0 and using a factor of F (b > ) instead of a factor of F (b > 1/2). This leads to R −1 g x|x|−1 ψ | · |/R u = o R 0 , and hence u ∈ Bs∗0 ,0 . Step IV. We shall show that u = 0. For convenience we assume that 2 2 − μ. First, letting s ∈ ]s0 − 2 /2, s0 [ be given arbitrarily, our goal is to show that u ∈ L2,−s . For that consider for κ ∈ ]0, 1/2] bκ = X a κ ;

aκ =

s0

X Xκ

−s

Xκ−s0 F (−b > 1/2)F b2 + c¯2 < 3 .

Here we use the regularization factor of (4.16). We calculate the Poisson bracket 2s0 −2s−1 2s0 −2s X −1 −3 X = (1 − κ)(2s0 − 2s) xX Xκ h2 , gb. Xκ Xκ Obviously this is negative on the support of bκ , with the (uniform) upper bounds · · · −8

−1

(2s0 − 2s) xX 2s0 −2 g Xκ−2

−cXκ−2

X Xκ

−s

Xκ−s0

2 ,

X Xκ

−s

Xκ−s0

2

c > 0.

Similarly, by (4.29),

h2 , F 2 (−b > 1/2) g = − F 2 (−b > 1/2) 1 − rV1 g −2 c¯2 + rV1 g −2 g −2 2(h2 − λ) + O r − 2 , r

(4.63)

1864


where we expand the right-hand side into a sum of three terms and note that the first term is non-positive. We introduce the quantizations Aκ = Opw (aκ ) and Bκ = Opw (bκ ), and the states uR (x) = ψR (x)u(x), R > 1. By Step III, lim i H, Bκ2 u = 0.

(4.64)

R

R→∞

On the other hand, due to the above considerations the expectation of i[H, Bκ2 ] in uR tends to be negative. Keeping the precise upper bounds in mind, we can let R → ∞ (using the calculus, (4.12a), to deal with a contribution from V3 and (4.64)) obtaining 2 2 ! cXκ−1 Aκ u = lim cXκ−1 Aκ uR C, R→∞

where the constants c (the one given above) and C are positive and independent of κ. Whence, letting κ → 0, we conclude that Opw F (−b > 1/2)F b2 + c¯2 < 3 u ∈ L2,−s .

(4.65)

Upon replacing the factor F (−b > 1/2) in (4.63) by F (b > 1/2), we can argue similarly and obtain Opw F (b > 1/2)F b2 + c¯2 < 3 u ∈ L2,−s .

(4.66)

In combination with Proposition 4.5, the bounds (4.65) and (4.66) and the fact that (4.56) holds with s0 replaced by s (note this is trivial since, by assumption, now v = 0) yield that u ∈ L2,−s . Next, the above procedure can be iterated: Assuming that u ∈ L2,−s for all s > tk := s0 − k 2 /2 (for some k ∈ N), the procedure leads to u ∈ L2,−s for all s > tk+1 . Consequently, u ∈ L2,s for all s ∈ R. In particular u ∈ L2 , and therefore u = 0. 2 5. Fourier integral operators In this section we construct and study certain modifiers in the form of Fourier integral operators; they will enter in the construction of wave operators in Section 6. 5.1. The WKB-ansatz Assume first that Condition 1.1 holds. Fix σ0 ∈ ]0, 2[. Recall from Lemma 3.1 that there exists a decreasing function ]0, ∞[ λ → R0 (λ) such that on the set (x, ξ ) ∈ Rd × Rd \ {0} x ∈ ΓR+ (|ξ |2 /2),σ (ξˆ ) 0

0

we can construct a solution φ + of the eikonal equation satisfying the (non-uniform in energy) bounds (3.3).


1865

We fix 0 < σ < σ < σ0 . Next we introduce smoothed out characteristic functions χ1 (r) =

for r 2, for r 1,

1, 0,

(5.1)

and χ2 (l) =

1, for l 1 − σ, 0, for l 1 − σ .

(5.2)

Define a0+ (x, ξ ) := χ2 (xˆ · ξˆ )χ1 |x|/R0 |ξ |2 /2 . The basic idea of Isozaki–Kitada is to use the modifier given by a Fourier integral operator J0+ on L2 (Rd ) of the form + J0 f (x) = (2π)−d/2

ei φ

+ (x,ξ )

a0+ (x, ξ )fˆ(ξ ) dξ,

(5.3)

where fˆ(ξ ) := (2π)−d/2

e−i x·ξ f (x) dx

denotes the (unitary) Fourier transform of f . If we assume that the potentials satisfy Conditions 2.1 and 2.2, then we can assume that the function R0 (λ) is the constant R0 given by Lemma 3.2. Thus in this case the solution φ + (x, ω, λ) of the eikonal equation is defined in ΓR+0 ,σ0 × [0, ∞[ (here σ0 is also given by Lemma 3.2; possibly it is much smaller than 2), and the amplitude a0 is simply given by a0+ (x, ξ ) := χ2 (xˆ · ξˆ )χ1 |x|/R0 . 5.2. The improved WKB-ansatz The modifier J0+ (and its incoming counterpart, say J0− ) is sufficient only for the most basic purposes, such as the existence of the outgoing (incoming) wave operator. To study finer properties of wave operators it is useful to use a more refined construction suggested by the WKB method. This more refined construction is possible and useful already under Condition 1.1. However, for simplicity of presentation, in the remaining part of the section we will assume that the potentials satisfy the more restrictive Conditions 2.1 and 2.2. These conditions allow us to extend this and related constructions (see Section 5.5) down to (and including) λ = 0. Therefore, it will be convenient to switch between the two notations φ + (x, ξ ) and φ + (x, ω, λ). This will be done tacitly in the following, and in fact, we shall often slightly abuse notation by writing (x, ξ ) ∈ ΓR+0 ,σ0 instead of (x, ω, λ) ∈ ΓR+0 ,σ0 × [0, ∞[.

1866


The WKB method suggests to approximate the wave operator by a Fourier integral operator J + on L2 (Rd ) of the form + J f (x) = (2π)−d/2

ei φ

+ (x,ξ )

a + (x, ξ )fˆ(ξ ) dξ,

(5.4)

where the symbol a + (x, ξ ) is supported in ΓR+0 ,σ0 and constructed by an iterative procedure to + for some R > R0 , make the difference T + := i(H J + − J + H0 ) small in an outgoing region ΓR,σ σ < σ0 . We have + T f (x) = (2π)−d/2

ei φ

+ (x,ξ )

t + (x, ξ )fˆ(ξ ) dξ,

(5.5)

where t + (x, ξ ) =

1 i ∇x φ + (x, ξ ) · ∇x + #x φ + (x, ξ ) a + (x, ξ ) − #x a + (x, ξ ). 2 2

(5.6)

As it is well known from the WKB method, it is possible to improve on the ansatz by putting (here we need ξ = 0) 1/2 + a + (x, ξ ) := det ∇ξ ∇x φ + (x, ξ ) b (x, ξ ), 1/2 + r (x, ξ ). t + (x, ξ ) := det ∇ξ ∇x φ + (x, ξ )

(5.7) (5.8)

We have

1/2 1 + + ∇x φ (x, ξ ) · ∇x + #x φ (x, ξ ) det ∇ξ ∇x φ + (x, ξ ) = 0, 2

and therefore r + (x, ξ ) = ∇x φ + (x, ξ ) · ∇x b+ (x, ξ ) −1/2 i 1/2 + #x det ∇ξ ∇x φ + (x, ξ ) − det ∇ξ ∇x φ + (x, ξ ) b (x, ξ ). 2 It is useful to introduce 1/2 ; ζ + (x, ξ ) = ln det ∇ξ ∇x φ(x, ξ )

ξ = 0.

(5.9)

1 ∇x φ(x, ξ ) · ∇x ζ + (z, ξ ) + #x φ(x, ξ ) = 0. 2

(5.10)

Note that it satisfies the equation


1867

+ Proposition 5.1. For (x, ξ ) ∈ ΓR,σ , ξ = 0,

1 ζ (x, ξ ) = 2 +

∞

#y φ + y + (t; x, ξ ), ξ dt.

(5.11)

1

Proof. Both ζ + (x, ξ ) and the right-hand side of (5.11) satisfy the first order Eq. (5.10). Both go to zero as |x| → ∞. In particular, they go to zero along the characteristics t → y + (t, x, ξ ). Therefore, they coincide. 2 Lemma 5.2. There exist the uniform limits γ + lim ∂ωδ ∂x ζ + (x, ξ ) − ζsph (x, ξ ) .

λ0

Besides, we have uniform estimates with ˘ given as in Proposition 3.3 γ + ∂ωδ ∂x ζ + (x, ξ ) − ζsph (x, ξ ) = O |x|−|γ |−˘ ,

|δ| + |γ | 0.

Proof. Below div and ∇ will always involve the derivatives with respect to the first argument. We compute:

ζ

+

+ (x, ξ ) − ζsph (x, ξ ) =

∞

1 + + div F + y + (t), ξ − div Fsph ysph (t), ξ dt 2

1

∞ =

1 dt 2

1

1

+ ∇ div F + yl+ (t), ξ · y + (t) − ysph (t) dl

0

∞ +

+ 1 + + div F + ysph ysph (t), ξ dt (t), ξ − div Fsph 2

1

= I + II, + (t). where yl+ (t) = ly + (t) + (1 − l)ysph Now I can be estimated (cf. (3.5f) and [6, (6.43)]) by

∞ ∞ + −2 + α− −2 (α− )/α + y C1 g y t dt C2 y + y + dy 1

|x|

= O |x|− /α = O |x|−˘ .

Here α = 2/(2 + μ) and > 0 is specified in [6, Subsection 6.1]. We used that d|y + | cg y + , dt

+ y ct α , c > 0.

(5.12)

1868


T

∞ Splitting the time-integral as 1 0 dt + T0 dt, the argument above yields (uniform) smallness of the second term (provided T0 is chosen big). As for the contribution from the first term, we can apply the dominated convergence theorem; whence we obtain the existence of limλ0 I . γ Next ∂ωδ ∂x I is a sum integrals of terms of the following form: γ γ + ∂ωδ1 ∂x 1 yl+ · · · ∂ωδn ∂x n yl+ ∂yn+ ∂ων ∇ div F yl+ , ξ · ∂ωα ∂xβ y + (t) − ysph (t) , l

where δ1 + · · · + δn + ν + α = δ and γ1 + · · · + γn + β = γ . This can be estimated (cf. (3.5f) and [6, (4.41) and (6.43)]) by −2 C|x|−|γ | y + g y + t α− . γ

γ

We argue as above to obtain uniform bounds on ∂ωδ ∂x I , as well as the existence of limλ0 ∂ωδ ∂x I . Now II is bounded (cf. (3.5g)) by ∞ ∞ + −1−˘ + −1−˘ + C1 y g y dt C2 y + dy = O |x|−˘ .

(5.13)

|x|

1

Then we apply the dominated convergence theorem as above, and we obtain the existence of limλ0 II. γ ∂ωδ ∂x II is a sum of integrals of terms of the form γ γ + + y ,ξ , ∂ωδ1 ∂x 1 y + · · · ∂ωδn ∂x n y + ∂yn+ ∂ων div F + y + , ξ − div Fsph where δ1 + · · · + δn + ν = δ and γ1 + · · · + γn = γ . This can be estimated (cf. (3.5g) and [6, (4.41) and (6.43)]) by −1−˘ + C|x|−|γ | y + g y . Then we can argue as above.

2

Define √ ζ˜ + (x, ω, λ) := ζ + x, 2λω − ln(2λ)(2−d)/4 . Proposition 5.3. (i) There exist (uniform) estimates + ζ˜ (x, ω, λ) − ln g |x| (d−2)/2 C, γ ∂ωδ ∂x ζ˜ + (x, ω, λ) = O |x|−|γ | , for |δ| + |γ | 1.

(5.14a) (5.14b)


1869

(ii) There exist (uniform) estimates 1/2 (d−2)/2 −|γ | γ (2λ)(d−2)/4 ∂ωδ ∂x det ∇ξ ∇x φ + (x, ξ ) , = g |x| O |x| for |δ| + |γ | 0.

(5.14c)

(iii) There exist the locally uniform limits ∂ωδ ∂x ζ˜ + (x, ω, 0) := lim ∂ωδ ∂x ζ˜ + (x, ω, λ). γ

γ

λ0

Proof. Let us first prove the estimates (5.14b) for |δ| = 0, |γ | 1 in the spherically symmetric γ + (x, ξ ) is an integral of terms of the form case. ∂x ζsph γ γ + + ∂x 1 y · · · ∂x n y∂yn div Fsph y (t), ξ , where γ1 + · · · + γn = γ . Using ∂x y + = O(|x|1−|γ | g(|x|)g(|y + |)−1 ), cf. [6, Proposition 4.9], these integrals are bounded by γ

∞ C1

n −n+1 + −n−1 y |x|−|γ |+n g |x| g y + dt

1

∞ C2

n −n −n−1 + |x|−|γ |+n g |x| g y + y + dy = O |x|−|γ | .

|x|

Thus γ + ∂x ζsph (x, ξ ) = O |x|−|γ | ,

for |γ | 1.

Clearly we can argue as above for |δ| > 0 as well. If |γ | = 0, we can use the formula (valid due to spherical symmetry) + + −1 ζsph Rη x, ξ , (x, Rη ξ ) = ζsph for any d-dimensional rotation Rη . Clearly this converts ω-derivatives to x-derivatives, and consequently we have shown (5.14b) in the general case. Taking into account Lemma 5.2 we obtain the estimates (5.14b) in the general case (when V is not necessarily radial). We have √ + + ζ˜sph x, 2λxˆ + (x, ξ ) = ζ˜sph

θ

√ + x, 2λω(l) · ω⊥ (l) dl, ∇ω ζ˜sph

0

where [0, θ ] l → ω(l) is the arc joining xˆ and ω and ω⊥ (l) is the tangent vector. Using (3.16) and (5.14b) with |δ| = 1, |γ | = 0 and Lemma 5.2 we obtain (5.14a).

1870


The above arguments in conjunction with the proof of Lemma 5.2 can be used to prove that there exist the limits γ lim ∂ωδ ∂x ζ˜ + (x, ω, λ),

λ0

|δ| + |γ | 1.

+ (x, x, ˆ λ) exists locally uniformly We know from the explicit formula (3.13) that limλ0 ζ˜sph + in x. Hence so does limλ0 ζ˜ (x, ω, λ) locally uniformly in (x, ω) ∈ Γ + . As for the bounds (5.14c), we use (5.14a) and (5.14b). 2

5.3. Solving transport equations Introduce the operator −1/2 i 1/2 M = det ∇ξ ∇x φ + (x, ξ ) #x det ∇ξ ∇x φ + (x, ξ ) 2 i + + ˜ ˜ = e−ζ (x,ξ ) #x eζ (x,ξ ) 2 i = #x + 2∇x ζ + (x, ξ ) · ∇x + #x ζ + (x, ξ ) + ∇x ζ + (x, ξ )2 . 2 Notice that due to Proposition 5.3 this operator is well defined at λ = 0 (more precisely, for (x, ω, λ) ∈ ΓR+0 ,σ0 × {0}). + : We define inductively for (x, ξ ) ∈ ΓR,σ 0 b0+ (x, ξ ) := 1; + bm+1 (x, ξ ) :=

∞

+ y(t, x, ξ, t), ξ dt. Mbm

1

Proposition 5.4. There exist the following (uniform) estimates: γ + ∂ωδ ∂x bm (x, ξ ) = O |x|−m(1−μ/2)−|γ | , γ + ∂ωδ ∂x Mbm (x, ξ ) = O |x|−2−m(1−μ/2)−|γ | .

(5.15a) (5.15b)

Proof. For a given m, (5.15a) easily implies (5.15b). γ γ Integrating ∂ωδ ∂x Mbm (x, ξ ) we can bound ∂ωδ ∂x bm+1 (x, ξ ) by ∞ ∞ + −2−m(1−μ/2)−|γ | −2−m(1−μ/2)−|γ | + −1 + y dt C1 y + g y d y |x|

1

∞ −2−m(1−μ/2)−|γ |+μ/2 + dy = O |x|−(m+1)(1−μ/2)−|γ | . C2 y + |x|

This shows the induction step.

2


1871

We set b+ (x, ξ ) := χ2 (xˆ · ω)b˘ + (x, ξ ),

b˘ + (x, ξ ) =

∞ "

+ bm (x, ξ )χ1 |x|/Rm

m=0

for an appropriately chosen sequence Rm → ∞ (this is an example of the so-called Borel construction, cf. [13, Proposition 18.1.3]). There are (uniform) bounds γ ∂ωδ ∂x b+ (x, ξ ) = O |x|−|γ | . We introduce r + (x, ξ ) = ∇x φ + (x, ξ ) · ∇x + M b+ (x, ξ ), + (x, ξ ) = χ2 (xˆ · ω) ∇x φ + (x, ξ ) · ∇x + M b˘ + (x, ξ ), rpr + + (x, ξ ) = r + (x, ξ ) − rpr (x, ξ ). rbd

(5.16)

(The subscript pr stands for the propagation and bd stands for the boundary.) Proposition 5.5. There exist (uniform) bounds γ + ∂ωδ ∂x rpr (x, ξ ) = O |x|−∞ , + and rbd (x, ξ ) is supported away from ΓR+0 ,σ and

γ + ∂ωδ ∂x rbd (x, ξ ) = O g |x| |x|−1−|γ | . 5.4. Constructions in incoming region Using the phase function φ − = φ − (x, ω, λ) given in (3.6) we can construct a symbol − − − + r − ), r − = O(|x|−∞ ) and the symbol r − = O(g(|x|)|x|−1 ) = eζ b− with t − = eζ (rpr pr bd bd − ⊆ ΓR−0 ,σ0 and obeying appropriate analogues of the conditions of the vanishing on a given ΓR,σ previous subsection. Similar to (5.4) we consider the Fourier integral operator J − on L2 (Rd ) given by − − −d/2 (J f )(x) = (2π) (5.17) ei φ (x,ξ ) a − (x, ξ )fˆ(ξ ) dξ.

a−

5.5. Fourier integral operators at fixed energies For all τ ∈ L2 (S d−1 ) we introduce ± J (λ)τ (x) := (2π)−d/2

± T (λ)τ (x) := (2π)−d/2

ei φ

± (x,ω,λ)

ei φ

a˜ ± (x, ω, λ)τ (ω) dω,

± (x,ω,λ)

t˜± (x, ω, λ)τ (ω) dω,

(5.18) (5.19)

1872


where √ a˜ ± (x, ω, λ) := (2λ)(d−2)/4 a ± x, 2λω , √ t˜± (x, ω, λ) := (2λ)(d−2)/4 t ± x, 2λω . The functions a˜ ± and t˜± are continuous in (x, ω, λ) ∈ Rd × S d−1 × [0, ∞). This fact will be very important in the forthcoming sections. Due to these properties we can define J ± (λ) and T ± (λ) at λ = 0 by the expressions (5.18) and (5.19), respectively. We can split T ± (λ) = ± Tbd (λ) + Tpr± (λ) in agreement with the decomposition (5.16) (cf. (5.8)). Throughout this subsection ˘ signifies the ˘ > 0 appearing in Proposition 3.3 (it is tacitly assumed that ˘ < 1 − μ/2)). For the problems at hand we can use coordinates for ω ∈ S d−1 sufficiently close to the dth standard vector ed ∈ Rd specified as follows (using a partition of unity in the x-variable ˆ and a rotation of coordinates this is without loss of generality): ω = ω ⊥ + ω d ed ;

ωd =

2, 1 − ω⊥

ω⊥ ∈ Rd−1 , |ω⊥ | is small.

(5.20)

Proposition 5.6. There exist a (large) R R0 and a (small ) σ˜ ∈ ]0, σ0 ] such that for all + |x| R there exists a unique ω ∈ S d−1 satisfying ω · xˆ 1 − σ˜ (alternatively: x ∈ ΓR, σ˜ (ω)) + + + and ∂ω φ (x, ω, λ) = 0. We introduce the notation ωcrt = ωcrt (x, λ) for this vector. It is smooth in x and we have γ + ∂x ωcrt − xˆ = O |x|−˘ −|γ | . Let + φ(x, λ) = φ + x, ωcrt (x, λ), λ .

(5.21)

This function solves the eikonal equation 2 ∂x φ(x, λ) /2 + V (x) = λ. + = xˆ and In the spherically symmetric case we have ωcrt |x|

√ φsph (x, λ) = 2λR0 + 2λ − 2V (r) dr.

(5.22)

R0

The proposition is obvious in the case V2 = 0, cf. (3.9). The general case follows by an application of the fixed point theorem, cf. the proof of the similar statement [15, Lemma 4.1]. At this point one needs some control of the Hessian; we refer the reader to the proof of Theorem 5.7. Of course, there is an analogue of Proposition 5.6 in the − case; we then need to replace φ + − + with φ − , and xˆ with −x. ˆ We obtain ωcrt (x, λ) = −ωcrt (x, λ). Note the identity − φ(x, λ) = −φ − x, ωcrt (x, λ), λ .


1873

Theorem 5.7. Let τ ∈ C ∞ (S d−1 ). Then ± 1 d−1 1 d−1 J (λ)τ (x) = (2π)− 2 e∓i π 4 g − 2 (r, λ)r − 2 e±ıφ(x,λ) τ (±x) ˆ + O r −˘ .

(5.23)

Moreover (5.23) is uniform in (x, ˆ λ) ∈ S d−1 × [0, ∞[. The same asymptotics holds for ±g −1 xˆ · pJ ± (λ)τ (x). Proof. We invoke the method of stationary phase (with a parameter given by the expression h = h(r) of (3.14)), cf. [14, Theorem 7.7.6] or [15, Theorem 4.3]. For simplicity we consider + . This method yields (up to a minor point that is only the + case and we abbreviate ωcrt = ωcrt resolved below) that + − 1 d d−1 J (λ)τ (x) = (2π)− 2 e−i π 4 det ∂ω2 φ + (x, ωcrt , λ)/2π 2 d−2 + × ei φ (x,ωcrt ,λ) a˜ + (x, ωcrt , λ)τ (ωcrt ) + g 2 O r −˘ .

(5.24)

Let us consider the Hessian. We first compute it in the case V2 = 0 choosing coordinates such that xˆ = ed and using (5.20): + + (ω = x) ˆ = −∂ω⊥ ∂xˆ φsph (ω = x), ˆ ∂ω2 ⊥ φsph

and using the fact that + (ω = x) ˆ = hI, ∂ω⊥ ∂xˆ φsph

(5.25)

d−1 × T Sωd−1 given by the Euclidean cf. the computation (3.12) (here I refers to the form on T Sx=ω ˆ metric), we obtain that + (ω = x) ˆ = −hI. ∂ω2 ⊥ φsph

(5.26)

In particular the critical point is non-degenerate in this case. Since ωcrt is a critical point, the second derivative has an invariant geometric meaning. Therefore, we can drop the reference to the special coordinates ω⊥ and we can write simply ∂ω2 for ∂ω2 ⊥ in the left-hand side of (5.26). The formula (5.26) is then valid for all xˆ ∈ S d−1 . The general case is similar. In particular, after applying Proposition 3.3 and (5.26), we obtain 2 + det ∂ φ (x, ωcrt , λ) = hd−1 1 + O r −˘ . (5.27) ω We conclude by combining (3.13), Lemma 5.2, Proposition 5.3, (5.27) and the construction of the symbol a˜ + that (5.24) and indeed also (5.23) hold. The second part of the theorem follows similarly. 2 6. Wave matrices In this section we study (modified) wave matrices. We prove that they have a limit at zero energy, in the sense of maps into an appropriate weighted space. This implies asymptotic oscillatory formulas for the standard short-range and Dollard scattering matrices.

1874


6.1. Wave operators The following theorem is essentially well known (follows from (3.3)). It describes a construction of modified wave operators similar to that of Isozaki–Kitada [18,19]. Notice, however, that the original construction involved energies strictly bounded away from zero. Notice also that the construction of J ± in Section 5, although given under Conditions 2.1 and 2.2, in fact can be done under Condition 1.1 as well. Theorem 6.1. Suppose that V satisfies Condition 1.1. Then W ± f = lim ei tH J0± e−i tH0 f = lim ei tH J ± e−i tH0 f ; t→±∞

t→±∞

fˆ ∈ Cc Rd \ {0} .

(6.1)

The “wave operator” W ± extends to an isometric operator on L2 (Rd ) satisfying H W ± = W ± H0 , and its range is the absolutely continuous spectral subspaces of H . Moreover, 0 = lim ei tH J0± e−i tH0 f = lim ei tH J ± e−i tH0 f ; t→∓∞

t→∓∞

fˆ ∈ Cc Rd \ {0} .

(6.2)

Remarks. We know that J0± 1] ,∞[ (H0 ) and J ± 1] ,∞[ (H0 ) are bounded for any > 0, but we do not know if J0± and J ± are bounded (not even under Conditions 2.1 and 2.2). This is the reason for restricting the choice of vectors in (6.1) and (6.2). An alternative, and equivalent, definition of W ± as a bounded operator on L2 (Rd ) is the following: W ± = s- lim s- lim ei tH J ± 1] ,∞[ (H0 )e−i tH0 . t→±∞

0

The following general fact serves as the basic formula in stationary scattering theory, see Appendix A for a derivation. Lemma 6.2. Suppose there are densely defined operators J˘± and T˘ ± on L2 (Rd ) such that J˘± 1] ,∞[ (H0 ) and T˘ ± 1] ,∞[ (H0 ) are bounded for any > 0 and that T˘ ± f = i(H J˘± − J˘± H0 )f for any f ∈ L2 (Rd ) with fˆ ∈ Cc (Rd \ {0}). Suppose there exists W˘ ± f := lim ei tH J˘± e−i tH0 f, t→±∞

fˆ ∈ Cc Rd \ {0} .

Then we have the following formula W˘ ± f = lim

0

where δ (λ) =

R0 (λ+i )−R0 (λ−i ) 2πı

± J˘ + i R(λ ∓ i )T˘ ± δ (λ)f dλ,

(6.3)

= π ((H0 − λ)2 + 2 )−1 .

6.2. Wave matrices at positive energies For any s ∈ R we recall the definition of weighted spaces L2,s (Rd ) := (1 + x 2 )−s/2 L2 (Rd ). Let ω denote the Laplace–Beltrami operator on the sphere S d−1 . For n ∈ R we define the Sobolev spaces on the sphere L2,n (S d−1 ) := (1 − ω )−n/2 L2 (S d−1 ).


1875

For λ > 0 we introduce F0 (λ) by √ F0 (λ)f (ω) = (2λ)(d−2)/4 fˆ 2λω . Let s > 12 and n 0. Note that F0 (λ) is a bounded operator in the space B(L2,s+n (Rd ), L2,n (S d−1 )) and depends continuously on λ > 0. Likewise, F0 (λ)∗ ∈ B(L2,−n (S d−1 ), L2,−s−n (Rd )) and it also depends continuously on λ > 0. Note also that the operator

⊕F0 (λ) dλ : L Rd →

∞

2

⊕L2 S d−1 dλ

(6.4)

0

is unitary; consequently the operators F0 (λ) diagonalize the operator H0 . Finally, s-lim δ (λ) = F0 (λ)∗ F0 (λ)

0

in B L2,s Rd , L2,−s Rd .

(6.5)

Due to the limiting absorption principle we have the following partial analogue of (6.5) for the full Hamiltonian, defined under Condition 1.1: Let s > 12 and δ V (λ) :=

R(λ + i ) − R(λ − i ) . 2πı

(6.6)

Then there exists δ V (λ) := s-lim δ V (λ)

0

in B L2,s Rd , L2,−s Rd .

(6.7)

The operator-valued function δ V (·) is a strongly continuous function of λ > 0. If Conditions 2.1–2.3 are true then we can extend the definition of δ V (λ) to include λ = 0 if we demand that s > 12 + μ4 , and the corresponding operator-valued function will be a strongly continuous (in fact, norm continuous) function of λ 0, cf. Remark 4.2(2). In the remaining part of this section we shall assume that the positive parameter σ in (5.2) is sufficiently small (this requirement can be fulfilled uniformly in λ 0). Notice that the condition conforms well with Lemma 3.2; we need it at various points, see for example the proof of Lemma 6.9. Formally, we have J ± (λ) = J ± F0 (λ)∗ and T ± (λ) = T ± F0 (λ)∗ . This suggests that (6.3) can be used to define wave operators at a fixed energy. This idea is used in the following theorem (which is essentially well known). Theorem 6.3. Suppose that the potential satisfies Condition 1.1. Let > 0, n 0 and λ > 0. Then W ± (λ) := J ± (λ) + i R(λ∓i 0)T ± (λ)

(6.8)

1876

J. Dereziński, E. Skibsted / Journal of Functional Analysis 257 (2009) 1828–1920 1

defines a bounded operator in B(L2,−n (S d−1 ), L2,− 2 − −n (Rd )), which depends continuously on λ > 0. It depends only on the splitting of the potential V into V1 and V3 (but does not depend 1 on the details of the construction of J ± ). For all f ∈ L2, 2 + (Rd ) and g ∈ Cc (]0, ∞[), we have ±

∞

W g(H0 )f =

g(λ)W ± (λ)F0 (λ)f dλ.

(6.9)

0

Moreover, W ± (λ)W ± (λ)∗ = δ V (λ).

(6.10)

We set w ± (ω, λ) = W ± (λ)δω , where δω denotes the delta-function at ω ∈ S d−1 . Then for all multiindices δ the function S d−1 × ]0, ∞[ (ω, λ) → ∂ωδ w ± (ω, λ) ∈ L2,−p Rd ;

p > |δ| + d/2,

is continuous. Remark. The operator W ± (λ) : D (S d−1 ) → L2,−∞ is called the wave matrix at the energy λ. Its range consists of generalized eigenfunction at the energy λ. The function w ± (ω, λ) (which 1 belongs to W ± (λ)L2, 2 −p (S d−1 ) for p > d2 ) is called the generalized eigenfunction at the energy λ and outgoing (or incoming) asymptotic normalized velocity ω. Let us explain the steps of a proof of Theorem 6.3 (in the case of “+”-superscript only); our (main) results contained in Theorems 6.5 and 6.6 will be proved by a parallel procedure. First one introduces a partition of unity of the form I = Opr χ+ (a) + Opr χ− (a)χ˜ − (b) + Opr χ− (a)χ˜ + (b) =: Opr (χ1 ) + Opr (χ2 ) + Opr (χ3 ).

(6.11)

Here a and b are the symbols introduced in (4.4) (rather than in (4.1) since we do not here impose Conditions 2.1–2.3) and χ+ is a real-valued function as in Proposition 4.1(ii) such that χ+ (t) = 1 for t 2C0 , and χ− = 1 − χ+ . Moreover, χ˜ − , χ˜ + ∈ C ∞ (R) are real-valued functions obeying χ˜ − + χ˜ + = 1 and supp χ˜ − ⊆ (−∞, 1 − σ¯ ],

(6.12)

supp χ˜ + ⊆ [1 − 2σ¯ , ∞[.

(6.13)

The number σ¯ needs to be taken (small) positive depending on the parameter σ of Section 5.1. (For the proof of Theorems 6.5 and 6.6 to be elaborated on later we refer at this point to (6.38) for the precise requirement.) The proof of Theorem 6.3 is based on the following lemma:


1877

Lemma 6.4. Suppose that the potential satisfies Condition 1.1. (i) For all n 0 and > 0, J + (λ) is a continuous function in λ > 0 with values in 1 B(L2,−n (S d−1 ), L2,− 2 − −n (Rd )). + (λ) is a continuous function in λ > 0 with values in (ii) For all n ∈ R and > 0, Tbd 1

B(L2,−n (S d−1 ), L2, 2 − −n (Rd )). + (λ) is a continuous function in λ > 0 with values in (iii) For all m, n ∈ R, Opr (χ3 )Tbd 2,−n d−1 2,m d B(L (S ), L (R )). (iv) For all m, n ∈ R, Tpr+ (λ) is a continuous function in λ > 0 with values in B(L2,−n (S d−1 ), L2,m (Rd )).

More general statements than Lemma 6.4 (i)–(iv) will be given and proven in the context of treating small energies (see Lemma 6.8); these statements are under Conditions 2.1 and 2.2. Let us here use (i)–(iv) in an Outline of the proof of Theorem 6.3. The expression (6.8) is a well-defined element of 1 B(L2,−n (S d−1 ), L2,− 2 − −n (Rd )) due to the positive energy version of Proposition 4.8 and Lemma 6.4; this is for any > 0 and n 0. (Notice that (4.46) holds for any t ∈ R by Lemma 6.4.) Effectively, this argument is based on the following scheme (to be used below): We insert the right-hand side of (6.11) to the right of the resolvent in (6.8) and expand into three terms. Whence, by using Remark 4.2(4) and Lemma 6.4, we see that W + (λ) is a sum of four 1 well-defined operators in B(L2,−n (S d−1 ), L2,− 2 − −n (Rd )), hence well-defined. Next note that λ → W + (λ) is norm continuous, due to the norm continuity of each of the above mentioned four operators, which in turn may be seen by combining the continuity statements of Remark 4.2(4) and Lemma 6.4. The statement on the independence of details of construction of J ± is based on the positive energy version of Proposition 4.10; the interested reader will realize this by using arguments from the proof of Lemma 6.10 stated later. The formula (6.9) can be verified by combining (6.3) with arguments used above, see Appendix A for an abstract approach. The identity (6.10) is a consequence of (6.9). 1 Finally, due to the fact that ∂ωδ δω ∈ L2, 2 −p (S d−1 ) for p > |δ| + d2 (with continuous dependence of ω ∈ S d−1 ), we conclude that indeed ∂ωδ w + (ω, λ) ∈ L2,−p (Rd ) with a continuous dependence of ω and λ. 6.3. Wave matrices at low energies Until the end of this section we assume that Conditions 2.1–2.3 are true. The main new result of this section is expressed in the following two theorems which concern the low-energy behaviour of the wave matrices of Theorem 6.3: Theorem 6.5. For s >

1 2

+

μ 4

and n 0, W ± (0) := J ± (0) + i R(∓i 0)T ± (0)

(6.14)

1878


defines a bounded operator in B(L2,−n (S d−1 ), L2,−s−n(1−μ/2) (Rd )). It depends only on the splitting of the potential V into V1 + V2 and V3 (but does not depend on the details of the construction of J ± ). We have W ± (0)W ± (0)∗ = δ V (0).

(6.15)

If we set w ± (ω, 0) = W ± (0)δω , then we obtain an element of L2,−p (Rd ) with p > d2 + μ2 − dμ 4 depending continuously on ω. In δ ± 2,−p d fact, more generally, ∂ω w (ω, 0) ∈ L (R ) with p > (|δ| + d2 )(1 − μ2 ) + μ2 with continuous dependence on ω. Theorem 6.6. For all > 0 and n 0, −n 1 1

xg x− 2 − g 2 W ± (λ)

(6.16)

is a continuous B(L2,−n (S d−1 ), L2 (Rd ))-valued function in λ ∈ [0, ∞[. For all > 0 and all multiindices δ, the function −|δ|+ 1 − d 1 1 2 2 x− 2 − g 2 ∂ δ w ± (ω, λ) ∈ L2 Rd S d−1 × [0, ∞[ (ω, λ) → xg ω is continuous. The following corollary interprets Theorem 6.6 in terms of the usual weighted spaces: Corollary 6.7. Let n 0. We have W ± (0) = lim W ± (λ) λ0

in the sense of operators in B(L2,−n (S d−1 ), L2,−˜sn (Rd )), where sñ > For all multiindices δ, the function

1 2

+ n + max(0, μ4 − n μ2 ).

S d−1 × [0, ∞[ (ω, λ) → ∂ωδ w ± (ω, λ) ∈ L2,−p˜ Rd is continuous, with p˜ >

d 2

+ |δ| for d 2 and p˜ >

1 2

+ |δ| + max(0, (1 − 2|δ|) μ4 ) for d = 1.

The proof of Theorems 6.5 and 6.6 is based on the following analogue of Lemma 6.4 (for convenience we focus as before on the case of “+”-superscript only). The symbol χ3 appearing in the statement (iii) below is specified as before, i.e. by (6.11) and the subsequent discussion.


1879

Lemma 6.8. (i) For all n 0 and > 0, −n − 1 − 1 +

xg

x 2 g 2 J (λ)

(6.17a)

is a continuous B(L2,−n (S d−1 ), L2 (Rd ))-valued function in λ ∈ [0, ∞[. (ii) For all n ∈ R and > 0, −n 1 − − 1 +

xg

x 2 g 2 Tbd (λ)

(6.17b)

is a continuous B(L2,−n (S d−1 ), L2 (Rd ))-valued function in λ ∈ [0, ∞[. (iii) For all m, n ∈ R, +

xm Opr (χ3 )Tbd (λ)

(6.17c)

is a continuous B(L2,−n (S d−1 ), L2 (Rd ))-valued function in λ ∈ [0, ∞[. (iv) For all m, n ∈ R,

xm Tpr+ (λ)

(6.17d)

is a continuous B(L2,−n (S d−1 ), L2 (Rd ))-valued function in λ ∈ [0, ∞[. Later on we will actually need a slightly stronger bound than the one of Lemma 6.8(i) with n = 0, which we state below (referring to notation of (4.6) and (4.54)): Lemma 6.9. For all τ ∈ L2 (S d−1 ), J + (λ)τ ∈ Bs∗0 . In fact, with a bounding constant independent of λ 0, 1 g 2 J + (λ) ∈ B L2 S d−1 , B ∗1 . 2

Proof. We need to bound the operator PR := R −1 J + (λ)∗ g1{|x| 1 and λ 0. Writing PR = R −1 0 dr Sr Qr dx with Sr = {|x| = r}, it thus suffices to

bound the operator Sr Qr dx independently of r > 0 and λ 0.

Step I. Analysis of Sr Qr dx. The kernel of Qr is given by Qr (ω, ω ) = ei (φ

+ (x,ω ,λ)−φ + (x,ω,λ))

a(x, ω, ω , λ),

where a(x, ω, ω , λ) = (2π)−d g |x|, λ a˜ + (x, ω, λ)a˜ + (x, ω , λ). For simplicity, we shall henceforth omit the superscript +, r > 0 and λ 0 in the notation.

1880


Our goal is to show that Sr Qr dx is a PsDO on L2 (S d−1 ) with symbol b(ω, ω , z) obeying uniform bounds (uniform in r > 0 and λ 0) β β2 α ∂ 1 ∂ ∂ b Cβ ,β ,α z−|α| . (6.18) ω ω z 1 2 Clearly this would prove the lemma. We can use a partition of unity on S d−1 , and therefore we can assume that the vectors ω, ω and xˆ are close to the dth standard vector ed ∈ Rd . Consequently, we can use coordinates ω = ω ⊥ + ω d ed , x = x⊥ + x d ed ,

2, ωd = 1 − ω⊥ 2. x d = r 2 − x⊥

(6.19) (6.20)

Next we write

φ(x, ω

) − φ(x, ω) = (ω⊥ − ω⊥ ) · z,

1 z=−

∂ω⊥ φ x, s(ω − ω) + ω ds.

0

Step II. We shall show that the map Sr ⊃ U x → T x = z ∈ Rd−1 is a diffeomorphism onto its range.

(6.21)

Here U is an open neighbourhood of ed containing the supports of a(·, ω, ω ). To this end we investigate the bilinear form ∂x ∂ω φ(x, ω) on T Sxd−1 × T Sωd−1 . Note that + ∂x ∂ω φsph (xˆ = ω) = r −1 hI,

(6.22)

cf. (5.25). In the coordinates (6.19) and (6.20), the identity (6.22) reads for zsph = (T x)sph (here we consider the case where V2 = 0) ∂xj zsph, i (ω = ω = x) ˆ = −r −1 h δij + ωd−2 ωi ωj ,

i, j d − 1.

(6.23)

Due to (3.15), Proposition 3.3 and (6.23) we obtain the more general result ∂xj zi = −r −1 h δij + ωd−2 ωi ωj + O(σ ) + O r −˘ ,

i, j d − 1.

(6.24)

Here O(σ ) refers to a term obeying |O(σ )| Cσ , where σ > 0 is given in (5.2) (assumed to be small). In particular, T is a local diffeomorphism with inverse determinant 1−d 2 ωd 1 + O(σ ) + O r −˘ . |∂xj zi |−1 = −r −1 h

(6.25)

For a later application we note the uniform bounds β ∂ωβ1 ∂ω2 ∂xα |∂xj zi |−1 = g 1−d r −|α| O r 0 .

(6.26)


1881

Also, T is injective: Suppose T x 1 = T x 2 , then 1 0=

∂xj zi s x 1 − x 2 + x 2 xj1 − xj2 ds

0

= −r −1 h δij + ωd−2 ωi ωj + O(σ ) + O r −˘ xj1 − xj2 . Using the invertibility of the matrix δij + ωd−2 ωi ωj , it follows that x 1 = x 2 . II, we can change coordinates and obtain that

Step III. Analysis of symbol b. Due to Step −1 Q dx is a PsDO with a symbol b = |∂ z | a. It remains to show (6.18). For zero indices r x i j Sr β1 = β2 = α = 0, we obtain the bound by combining Proposition 5.3 and (6.25). For derivatives, we note the bounds β β2 α ∂ 1 ∂ ∂ z Cβ ,β ,α gr 1−|α| , (6.27) ω ω x 1 2 which by a little bookkeeping yields β β2 γ ∂ 1 ∂ ∂z x Cβ ,β ,γ r z−|γ | . ω ω 1 2 Another bookkeeping using Proposition 5.3, (6.25) and (6.28) yields (6.18).

(6.28) 2

Proof of Lemma 6.8. We drop the superscript “+” and the parameter λ in the notation. We first prove uniform boundedness on any compact interval [0, λ1 ]. Re (i). We replace J = J (·) by J χ(ω), where χ ∈ C ∞ (S d−1 ) with a sufficiently small support. We can assume that n is a non-negative integer. Instead of studying J (1 − ω )n/2 , it then suffices to study J ∂ων for |ν| n. Integrating by parts, we observe that the corresponding integral kernel equals ˜ ω) = ei φ(x,ω) a˜ ν (x, ω), C∂ων ei φ(x,ω) a(x, where a˜ ν is a linear combinations of terms of the form ˜ ω), ∂ων1 φ(x, ω) · · · ∂ωνk φ(x, ω)∂ων0 a(x, with ν0 + ν1 + · · · + νk = ν. Thus, using that |∂ωδ φ| C xg (cf. (3.11d)) and Proposition 5.3, we obtain d−2 γ ∂ωδ ∂x a˜ ν (x, ω) = O xn−|γ | g n+ 2 .

(6.29)

Then we follow the proof of Lemma 6.9. Re (ii). Assume first that n 0. Then we follow the same scheme as above. The bound on the relevant kernel needs to be replaced by d γ ∂ωδ ∂x t˜ν (x, ω) = O xn−1−|γ | g n+ 2 , cf. Proposition 5.5. Using (6.30) we can proceed as before.

(6.30)

1882


Assume next that n < 0. We can assume that n is a negative integer. For fixed x, we decompose 2 x, ω = ω⊥ + 1 − ω⊥ ˆ where ω⊥ · x = 0. By (3.11c), we have the uniform lower bound ∇ω φ(x, ω) c|x|g ⊥

for xˆ · ω 1 − σ,

(6.31)

and by (3.11d) the uniform upper bounds δ ∂ φ(x, ω) C|x|g. ω⊥

(6.32)

We apply the non-stationary method based on the identity

∇ω⊥ φ i · ∇ω⊥ |∇ω⊥ φ|2

−n

ei φ

+ (x,ω)

= ei φ(x,ω) .

After performing −n integrations by parts, the bounds (6.31) and (6.32) yield T χτ =

" |ν|−n

t˜ν (x, ω)∂ων ⊥ τ (ω) dω,

where the functions t˜ν also satisfy the bounds (6.30). Then we proceed as before. Re (iii). The kernel of Opr (χ3 )Tbd (·) is given by the integral

dξ ei x·ξ

ei (φ(y,ω)−y·ξ ) k(ω, y, ξ ) dy,

k(ω, y, ξ ) = (2π)−3d/2 χ3 (y, ξ )t˜bd (y, ω).

It suffices to show that β δ i (φ(y,ω)−y·ξ ) Cβ,δ ∂ ∂ k(ω, y, ξ ) dy e ξ ω

uniformly in ξ, ω and λ.

(6.33)

Notice that the symbol k is compactly supported in ξ . First we observe that (using notation of Section 4.1) d k = kω,λ ∈ Sunif g 2 x−1 , gμ,λ . ¯ We can substitute k → k = F (|y| > 2R)k(ω, y, ξ ). Next we integrate by parts, writing first ξ − ∇y φ i · ∇ ei (φ(y,ω)−y·ξ ) = ei (φ(y,ω)−y·ξ ) . y |ξ − ∇y φ|2 We need to argue that ξ − ∂y φ = 0 on the support of the involved symbol. For that we recall the following elementary inequality valid for all z1 , z2 ∈ Rd and κ1 , κ2 > 0: |z1 − z2 |2 min κ12 /2, κ2 − κ22 /2 |z1 |2 + |z2 |2 ,

(6.34)


1883

provided one of the following three conditions holds: |z2 | (1 − κ1 )|z1 |,

|z1 | (1 − κ1 )|z2 |

or z1 · z2 (1 − κ2 )|z1 ||z2 |.

Now, on the support of the symbol k we have (1 − σ )|y| y · ω (1 − σ )|y|, cf. (5.2). We use these inequalities in (3.5c) and (3.5d), yielding 1 − Cσ − C|y|−˘

∇y φ(y, ω) · yˆ 1 − cσ + C|y|−˘ , |∇y φ(y, ω)|

which in turn (if R¯ is taken large enough) implies that 1 − 2Cσ

∇y φ(y, ω) y c · 1 − σ. g(|y|)

y 2

(6.35)

We claim that there exists a small c = c (σ, σ ) > 0 such that ξ − ∇y φ(y, ω) c |ξ | + ∇y φ(y, ω)

(6.36)

on the support of k (showing in particular that ξ − ∂y φ = 0). Obviously, (6.36) follows from (6.34) with z1 =

ξ g(|y|)

and z2 =

∇y φ(y, ω) , g(|y|)

provided one of the above three conditions hold. If all of those conditions fail, so that intuitively z1 ≈ z2 , we can replace z2 in (6.35) by z1 yielding c 1 − 3Cσ b(x, ξ ) 1 − σ. 3

(6.37)

Here we applied (6.34) for some κ1 and κ2 , depending on σ and σ . Now, the second inequality of (6.37) is violated on the support of χ˜ + (b(y, ξ )), provided that σ¯ > 0 of (6.13) is chosen such that c 2σ¯ < σ. 3

(6.38)

We have shown the bound (6.36) on the support of the symbol k, and therefore in particular on the support of the relevant symbol, after performing the y-integrations by parts. The estimate (6.33) follows. Re (iv). First we assume that n 0. Integrating by parts in ω, as in the proof of (i), and using Proposition 5.5, which says that tpr with all its derivatives is O( x−∞ ), we obtain that

xm Tpr (λ) is in B(L2,−n (S d−1 ), L2 (Rd )) for any m. The case n < 0 then follows trivially. Let us now prove the continuity. Consider for instance (i). Let τ ∈ C ∞ (S d−1 ) and set −n 1 1 Jn, (λ) := xg x− 2 − g 2 J (λ).

1884


Clearly, for (small) κ > 0, Jn, (λ)τ = F κ|x| < 1 Jn, (λ)τ + F κ|x| > 1 x− /2 Jn, /2 (λ)τ.

(6.39)

We know that Jn, /2 (λ) is bounded uniformly in λ. Hence the second term on the right of (6.39) is O(κ /2 ). We know that a(x, ω, λ), φ(x, ω, λ) and g(x, λ)±1 are continuous down to λ = 0. The first term on the right of (6.39) involves only variables in a compact set. Therefore it is continuous in λ. Hence Jn, (λ)τ is continuous as the uniform limit of continuous functions. By the uniform bound, which we proved before, we conclude that Jn, (λ) is strongly continuous in B(L2,−n (S d−1 ), L2 (Rd )). Now Jn, (λ) = g /2 Jn+ /2, /2 (λ)(1 − ω )− /4 (1 − ω ) /4 , where g /2 is strongly continuous, Jn+ /2, /2 (λ) is strongly continuous in B(L2,−n− /2 (S d−1 ), L2 (Rd )), (1 − ω )− /4 is a compact operator on L2,−n− /2 (S d−1 ) and (1 − ω ) /4 is a unitary element of B(L2,−n (S d−1 ), L2,−n− /2 (S d−1 )). We invoke the general fact that the product of a strongly continuous operator-valued function and a compact operator is norm continuous. Whence we obtain the norm continuity of Jn, (λ) in B(L2,−n (S d−1 ), L2 (Rd )). The proof of the norm continuity of the operators in the remaining parts of the lemma is similar. 2 Outline of the proof of Theorems 6.5 and 6.6. The proof goes along the lines of the proof of Theorem 6.3. In particular this amounts to inserting the right-hand side of (6.11) to the right of the resolvent in (6.8) and expanding into three terms. Next, using Proposition 4.1 and Lemma 6.8, we conclude that W + (λ) is well defined as a sum of four operators, say Tj (λ). In fact, all of the four maps −n 1 1 [0, ∞[ λ → xg x− 2 − g 2 Tj (λ) ∈ B L2,−n S d−1 , L2 Rd are continuous. For the independence of W + (λ) of cutoffs, we use Propositions 4.8 and 4.10 in the same way as in the arguments for deducing (6.40) stated below. The formula (6.15) follows by combining (6.10), Remark 4.2(2) and the shown continuity properties of W + (λ) and W + (λ)∗ . 2 Lemma 6.10. For any λ 0, R(λ±i 0)T ± (λ) is well defined as a map from D (S d−1 ) to L2,−∞ and 0 = J ± (λ) + i R(λ±i 0)T ± (λ).

(6.40)

Proof. Note that we can extend Lemma 6.8 as follows: Let χ− ∈ Cc∞ (R) and χ˜ − ∈ Cc∞ (R) with supp χ˜ − ⊆ ]−∞, 2σ¯ − 1[ for some small σ¯ > 0. Then, for all m, n ∈ R, + Opr χ− (a)χ˜ − (b) Tbd (λ), Opr χ− (a)χ˜ − (b) J + (λ) ∈ B L2,−n S d−1 , L2,m Rd ,


1885

cf. (6.37) (recall the standing hypothesis of this subsection that the positive parameter σ in (5.2) is sufficiently small). Therefore, for all τ ∈ D (S d−1 ) and s ∈ R, W Fscs T + (λ)τ ∪ W Fscs J + (λ)τ ∩ {b < σ¯ − 1} = ∅.

(6.41)

By the definition of T + (λ), (H − λ)J + (λ)τ = −i T + (λ)τ = −i(H − λ)R(λ + i 0)T + (λ)τ.

(6.42)

Notice that due to (6.41) and Proposition 4.8(iii), the vector u = R(λ+i 0)T + (λ)τ is in fact well-defined and W Fscs (u) ∩ {b < σ¯ − 1} = ∅.

(6.43)

Using (6.41)–(6.43) and Proposition 4.10, we conclude that the generalized eigenfunction satisfies J + (λ)τ + i R(λ + i 0)T + (λ)τ = 0.

2

(6.44)

Remark. There exists an alternative time-dependent proof of Lemma 6.10 that avoids the use of Proposition 4.10: Due to (6.2) 0 = lim

0

± J + i R(λ ± i )T ± δ (λ)f dλ,

fˆ ∈ Cc Rd \ {0} ,

cf. Lemma 6.2 or Appendix A. The right-hand is given by

± J + i R(λ ± i 0)T ± δ0 (λ)f dλ,

cf. Appendix A. Whence, by a density argument, (6.40) follows. We complete this subsection by discussing a certain refined mapping property of W ± (λ). Besides its own interest its application (see Corollary 6.12 stated below) will be needed in Section 8. The result is related to the fact that the continuity in λ of the operators in (6.16) and (6.17a) is proven only for n 0 while the continuity in λ of the operator in (6.17b) is valid for all n ∈ R. Theorem 6.11. Fix real-valued χ, χ˜ − ∈ Cc∞ (R) and χ+ ∈ C ∞ (R) such that supp χ˜ − ⊂ ]−1, 1[, χ+ ∈ Cc∞ (R) and supp χ+ ⊂ ]C0 , ∞[. Let A˜ := Opw (χ(a)χ˜ − (b)) and A+ := Opw (χ+ (a)) for λ 0. For all n ∈ R, > 0 and with A = A˜ or A = A+ , −n − 1 − 1 ± (λ) := xg

x 2 g 2 AW ± (λ) Wn, is a continuous B(L2,−n (S d−1 ), L2 (Rd ))-valued function in λ ∈ [0, ∞[.

(6.45)

1886


Proof. With reference (4.2) (this class of symbols is used extensively in [8]) −n n 1 1 1 1

B(λ) := xg x− 2 − g 2 Ag − 2 x 2 + 2 xg ∈ Ψunif x− 2 , gμ,λ . Whence, by the calculus, B(λ) ∈ B(L2 (Rd )) with a bound locally independent of λ 0, and in fact B(·) is norm continuous. By using this continuity and Theorem 6.6, we conclude that it suffices to consider the case n < 0. ˜ Since the construction of W + (λ) is independent of the (small) parameters σ and Re A = A. σ in (5.2), we can take them smaller (if needed) to assure that sup supp χ− < 1 − 3Cσ .

(6.46)

Here we refer to the left-hand side of (6.37). + (λ) is an element of B(L2,−n (S d−1 ), L2 (Rd )), we consider for λ > 0 Now, to show that Wn, the two terms of (6.8) separately (if λ = 0 we use instead (6.14)): The contribution from the first term (i.e. from J + (λ)) has better mapping properties than specified, cf. Lemma 6.8(iii). In fact, using (6.46) we can mimic the proof of Lemma 6.8(iii) to handle this contribution. As for the contribution from the second term (i.e. from i R(λ − i 0)T + (λ)), we combine Lemma 6.8 (ii) and (iv) and Proposition 4.1(iii). By the same arguments, continuity in λ 0 is valid for the contribution from each of the + (λ). mentioned two terms, hence for Wn, Re A = A+ . Again we consider for λ > 0 the two terms of (6.8) separately (if λ = 0 we use instead (6.14)). The contribution from the first term J + (λ) has again better mapping properties than needed. More precisely, we have the following analogue of Lemma 6.8(iii): For all m ∈ R the family of operators xm A+ J + (λ) constitutes a continuous B(L2,−n (S d−1 ), 2 L (Rd ))-valued function of λ ∈ [0, ∞[. To show this, we can again follow the proof of Lemma 6.8(iii). It suffices to show locally uniform boundedness in the indicated topology and we may replace A+ → Opr (χ+ (a)). The kernel of Opr (χ+ (a))J + (λ) is given by the integral i x·ξ ei (φ(y,ω)−y·ξ ) kω,λ (y, ξ ) dy, dξ e 2 kω,λ (y, ξ ) = (2π)−3d/2 χ+ ξ 2 /g |y|, λ a˜ + (y, ω, λ). It suffices to show that β

ξ d+1 ∂ξ ∂ωδ ei (φ(y,ω)−y·ξ ) kω,λ (y, ξ ) dy Cβ,δ

uniformly in ξ, ω and λ.

For that we notice that d k = kω,λ ∈ Sunif g 2 −1 , gμ,λ . ¯ ω,λ (y, ξ ). It suffices to show (6.47) with k → k = F (|y| > 2R)k Next we integrate by parts, writing first

ξ − ∇y φ i · ∇y |ξ − ∇y φ|2

ei (φ(y,ω)−y·ξ ) = ei (φ(y,ω)−y·ξ ) ,

(6.47)


1887

and then we invoking the uniform bounds C|ξ | ξ − ∇y φ(y, ω) c |ξ | + ∇y φ(y, ω) ,

(6.48)

which are valid on the support of k (provided R¯ is chosen sufficiently large). Clearly, we obtain (6.47) by this procedure if (i.e. the number of integrations by parts) is chosen sufficiently large. As for the contribution from the second term i R(λ − i 0)T + (λ), we combine Lemma 6.8 (ii) and (iv) and Proposition 4.1(ii). 2 We can extend the identities (6.10) and (6.15) (which below corresponds to s = 0) as follows: Corollary 6.12. Let χ, χ˜ − ∈ Cc∞ (R) be given as in Theorem 6.11. Fix λ 0. Let again A˜ := Opw (χ(a)χ˜ − (b)). For all δ > 12 and s 0, there exists the strong limit ˜ 2 = g 2 δ V (λ)Ag ˜ 2 = g 2 W ± (λ)W ± (λ)∗ Ag ˜ 2 s-lim g 2 δ V (λ)Ag 1

1

1

1

1

1

0

in B(L2,s+δ (Rd ), L2,s−δ (Rd )). Proof. It follows from Proposition 4.9 that indeed there exists the limit ˜ 2 B := s-lim g 2 δ V (λ)Ag 1

1

0

in B L2,s+δ Rd , L2,s−δ Rd .

Let n = s/s1 , where s1 is given as in (4.6). Due to Theorem 6.11 1 1 ˜ 2 = g 2 AW ˜ ± (λ) ∗ ∈ B L2,s+δ Rd , L2,n S d−1 , W ± (λ)∗ Ag and due to Theorem 6.6 1 g 2 W ± (λ) ∈ B L2,n S d−1 , L2,s−δ Rd . We have shown that 1 1 ˜ 2 ∈ B L2,s+δ Rd , L2,s−δ Rd . g 2 W ± (λ)W ± (λ)∗ Ag Since ˜ 2v Bv = g 2 W ± (λ)W ± (λ)∗ Ag 1

1

cf. (6.10) and (6.15), we are done by a density argument.

for v ∈ L2,∞ , 2

(6.49)

1888


6.4. Asymptotics of short-range wave matrices Clearly, if μ > 1, there exists Wsr± f = lim ei tH e−i tH0 f, t→±∞

(6.50)

which is the usual definition of wave operators in the short-range case. In the case μ ∈ ]1, 2[, we can compare our wave matrices with the wave matrices defined by (6.50). Recall pˆ := p/|p|. Theorem 6.13. For μ ∈ ]1, 2[, the operators ψsr+ (p) := i

∞ |p| − F + l p, ˆ p, ˆ p 2 /2 · pˆ dl, R0

ψsr− (p) := −i

∞ |p| + F + −l p, ˆ −p, ˆ p 2 /2 · pˆ dl R0

are well-defined. If V2 = 0, then ψsr± (p) = ψsr± (|p|) with ψsr±

|p| = ±i

∞ |p| − p 2 − 2V1 (r) dr. R0

We have +

Wsr+ = W + ei ψsr (p) , Wsr− = W − e

i ψsr− (p)

(6.51a) (6.51b)

.

Whence in particular, for all λ > 0, +

Wsr+ (λ) = W + (λ)ei ψsr ( Wsr− (λ) = W − (λ)e

√ 2λ·)

i ψsr− (

,

√ 2λ·)

.

(6.52a) (6.52b)

Proof. One can readily show the theorem from well-known properties of the free evolution and the fact that ∞ √ √ φ (x, ω, λ) + 2λ − F + (lω, ω, λ) · ω dl = 2λω · x + o |x|0 , +

(6.53)

R0

which in turn follows from [6, (4.50)] and a change a contour of integration. The asymptotics is locally uniform in (ω, λ) ∈ S d−1 × ]0, ∞[. 2


1889

Remark 6.14. ψsr± is indeed oscillatory. Notice that for V1 (r) = −γ r −μ , as λ → 0+ , we have √ √ 2λ = 2λ − 2 λ + γ r −μ dr ∞

ψsr+

R0

= (2λ)

∞

1 1 2−μ

1 − 1 + 2γ s −μ ds 1

R0 (2λ) μ

= (2λ)

1 1 2−μ

∞ 1 − 1 + 2γ s −μ ds + O λ0 , 0

cf. [28, (7.11)]. See Remark 6.16 for a similar result. 6.5. Asymptotics of Dollard-type wave matrices For μ >

1 2

and μ + 2 > 1, the Dollard-type wave operators are given by ± Wdol f = lim ei tH Udol (t)f, t→±∞

where Udol (t) = e−i

t

0 (p

2 /2+V

1 (sp)1{|sp|R0 } ) ds

.

We have the following analogue of Theorem 6.13. Theorem 6.15. For

1 2

< μ < 2, 2 < 1 and μ + 2 > 1, the operators

+ (p) = i ψdol

∞ |p| − F + l p, ˆ p, ˆ p 2 /2 · pˆ − |p|−1 V1 (l) dl, R0

− ψdol (p) = −i

∞ |p| + F + −l p, ˆ −p, ˆ p 2 /2 · pˆ − |p|−1 V1 (l) dl R0

± ± (p) = ψdol (|p|) and are well-defined. If V2 = 0, then ψdol

± ψdol |p| = ±i

∞ |p| − p 2 − 2V1 (r) − |p|−1 V1 (r) dr. R0

We have

1890

J. Dereziński, E. Skibsted / Journal of Functional Analysis 257 (2009) 1828–1920 +

+ Wdol = W + ei ψdol (p) ,

(6.54a)

−

− Wdol = W − ei ψdol (p) .

(6.54b)

Whence in particular, for all λ > 0 +

+ (λ) = W + (λ)ei ψdol ( Wdol − Wdol (λ) = W − (λ)e

√ 2λ·)

(6.55a)

,

√ − i ψdol ( 2λ·)

(6.55b)

.

± are well-defined due to the fact that Proof. First we notice that ψdol

+ F + − Fsph (lω, ω, λ) = O l −δ

+ for any δ < min(μ + 2 , 2μ), and hence integrable. Here Fsph refers to the F + for the case + V2 = 0, whence Fsph (lω, ω, λ) = g(l, λ)ω. For this estimate, we refer to [6, Remarks 6.2 2)] and the proof of [6, Lemma 6.4]. It appears stronger at the price of not being uniform in (small) λ. There is an extension of this estimate that allows us to integrate along the line segment joining x and Rω and taking the limit:

∞ω Rω + + + + F − Fsph (x, ¯ ω, λ) · d x¯ = lim F − Fsph (x, ¯ ω, λ) · dx¯ R→∞

x

x

= o |x|0 .

(6.56)

Introduce the auxillary phases ± (x, ω, λ) = φdol

±x·ω √ − 12 2λx · ω ∓ (2λ) V1 (l) dl, R0

± ± φaux (x, ω, λ) = φaux

± = φdol

±∞ω

+ ± Fsph − ∇x φdol · dx, ¯

− x

and corresponding modifiers ± J f (x) = (2π)−d/2

±

ei φ (x,ξ ) χ(x, ±ξˆ )fˆ(ξ ) dξ ;

ξ=

√ 2λω.

Here we can take the function χ of the form χ(x, ω) = χ1 (|x|/R)χ2 (xˆ · ω) with χ1 and χ2 given as in (5.1) and (5.2), respectively. By the stationary phase method, [14, Theorem 7.7.6], one derives the following asymptotics in L2 (Rd ) for any state f with fˆ ∈ Cc∞ (Rd \ {0}): ± i tH0 ± i tH0 Udol (t)f & Jdol e f & Jaux e f

as t → ±∞.


1891

Next we notice the following analogue of (6.53), cf. (6.56): ∞ + φ (x, ω, λ) + ∇φdol − F + (±lω, ±ω, λ) · ω dl ±

R0

± = φaux (x, ω, λ) + o |x|0 . Again this asymptotics is locally uniform in (ω, λ) ∈ S d−1 × ]0, ∞[.

2

Remark 6.16. The first factor on the right-hand side of (7.11) is oscillatory. Let us state the following asymptotics for the special case where V1 (r) = −γ r −μ for r R0 : √ √ 1 2λ = 2λ − 2 λ + γ r −μ + (2λ)− 2 γ r −μ dr ∞

+ ψdol

R0

= (2λ)

1 1 2−μ

∞

1 − 1 + 2γ s −μ + γ s −μ ds. 1

R0 (2λ) μ

For λ 0, this behaves as 1 1 1 (2λ) 2 − μ Cμ + O λ− 2 , 1

1 < μ < 1; 2 1

−γ (2λ)− 2 ln 2λ + (2λ)− 2 C1 + O(1), 1

(2λ)− 2

μ = 1;

1−μ

1 1 R0 γ + O λ2−μ , μ−1

1 < μ < 2.

Here ∞ 1 − 1 + 2γ s −μ + γ s −μ ds, Cμ := 0

∞

1

1

0

1 − 1 + 2γ s −1 + γ s −1 ds +

C1 :=

1 − 1 + 2γ s −1 ds − γ ln R0 .

7. Scattering matrices In this section we study (modified) scattering matrices. We prove that they have a limit at zero energy. This implies low energy oscillatory asymptotics for the standard short-range and Dollard scattering matrices.

1892


7.1. Scattering matrices at positive energies The scattering operator commutes with H0 , which is diagonalized by the direct integral decomposition (6.4). Because of that, the general theory of decomposable operators says that there exists a measurable family ]0, ∞[ λ → S(λ), with S(λ) unitary operators on L2 (S d−1 ) defined for almost all λ, such that ∞ ⊕S(λ) dλ,

S

(7.1)

0

using the decomposition (6.4). The following theorem is (essentially) well known: Theorem 7.1. Assume Condition 1.1. Then S(λ) = −2πJ + (λ)∗ T − (λ) + 2πi T + (λ)∗ R(λ + i 0)T − (λ) = −2πW + (λ)∗ T − (λ)

(7.2a) (7.2b)

defines a unitary operator on L2 (S d−1 ) depending strongly continuously on λ > 0. Moreover, (7.1) is true. Furthermore, for all n ∈ R and > 0, S(λ) ∈ B L2,n S d−1 , L2,n− S d−1 , depending norm continuously on λ > 0. (Hence in particular S(λ) maps C ∞ (S d−1 ) into itself.) For a derivation of the formula (7.2a) we refer the reader to Appendix A. For the remaining part of the theorem we refer the reader to the proof of Theorem 7.2 stated below (one can use Theorem 6.3 and Lemma 6.4 as substitutes for Theorem 6.6 and Lemma 6.8, respectively). 7.2. Scattering matrices at low energies Until the end of this section we assume that Conditions 2.1–2.3 are true. The main new result of this section is the following theorem: Theorem 7.2. The result of Theorem 7.1 is true for all λ ∈ [0, ∞[. Specifically, if we define S(0) = −2πJ + (0)∗ T − (0) + 2πi T + (0)∗ R(+i 0)T − (0) +

∗

−

= −2πW (0) T (0),

(7.3a) (7.3b)

then S(0) is unitary, s-limλ0 S(λ) = S(0) in the sense of B(L2 (S d−1 )) and limλ0 S(λ) = S(0) in the sense of B(L2,n (S d−1 ), L2,n− (S d−1 )) for any n ∈ R and > 0. Proof. First we notice that the expression S(λ) = −2πW + (λ)∗ T − (λ) ∈ B L2,n S d−1 , L2,n− S d−1 for n > 0,


1893

has a norm continuous dependence of λ 0. Indeed, fix n > 0 and ∈ ]0, n], and pick 1 , 2 ∈ R such that μ2 < 1 < and 2 = 12 ( − 1 ). We write W + (λ)∗ T − (λ) −n+ − − n 1 1 1 1 = W + (λ)∗ g 2 x− 2 − 2 xg g x 1 xg x 2 − 2 g − 2 T − (λ) .

(7.4)

We shall use the analogues of Lemma 6.8 (ii) and (iv) with T + (λ) replaced by T − (λ) (proved in the same way). The third factor on the right of (7.4) is continuous in λ with values in B(L2,n (S d−1 ), L2 (Rd )). The second factor is continuous in λ as an operator on L2 (Rd ). The first factor is continuous in λ as an operator in B(L2 (Rd ), L2,n− (S d−1 )) due to Theorem 6.6. This proves the norm continuity of S(λ) in B(L2,n (S d−1 ), L2,n− (S d−1 )) for n > 0. Let us prove the same property for n 0 using a slight extension of the above scheme: Notice that the positive sign condition above entered only in the condition n − 0 needed for applying Theorem 6.6. Since n 0 we have n − < 0 and therefore we need a substitute for Theorem 6.6. This is provided by Theorem 6.11 and an analogue of Lemma 6.8 for T − (λ). In fact, choose for (small) σ¯ > 0 real-valued χ˜ − ∈ Cc∞ (R) and χ+ ∈ C ∞ (R) such that supp χ˜ − ⊂ ]−1, 1[, χ˜ − = 1 in [σ¯ − 1, 1 − σ¯ ], supp χ+ ⊂ ]C0 , ∞[ and χ+ = 1 in [2C0 , ∞[. Let χ = 1 − χ+ , A˜ = Opw (χ(a)χ˜ − (b)), A+ = Opw (χ+ (a)) and A¯ = Opw (χ(a)(1 − χ˜ − (b))). We insert the identity I = A˜ + A+ + A¯ − ∗ ¯ (λ) . (7.5) W + (λ)∗ T − (λ) = (A˜ + A+ )W + (λ) T − (λ) + W + (λ)∗ AT Due to Theorem 6.11 the above argument can be repeated for the first term on the right-hand side, and if σ¯ > 0 is chosen sufficiently small we have the following analogue of Lemma 6.8 ¯ − (λ) (iii) and (iv) (here stated in combination): For all m ∈ R the family of operators xm AT 2,−n d−1 2 d (S ), L (R ))-valued function of λ ∈ [0, ∞[. By choosing constitutes a continuous B(L m > 12 + μ4 and using Theorem 6.6 we conclude norm continuity of the second term of (7.5). But from the isometricity of S we see that S(λ) is isometric for almost all λ as a map on L2 (S d−1 ). Therefore, it is isometric and strongly continuous as a map on L2 (S d−1 ) for all λ 0. By repeating this argument for S ∗ (not to be elaborated on) we obtain that S(λ)∗ is isometric and strongly continuous in λ 0 as a map on L2 (S d−1 ). Whence S(λ) is unitary as a map on L2 (S d−1 ). 2 Remark. There is an alternative and completely stationary approach to proving the unitarity of the scattering matrices. In fact taking (7.2b) and (7.3b) as definitions the unitarity is a consequence of the formula (8.11), which in turn can be verified directly along the lines of Section 8. 7.3. Asymptotics of short-range scattering matrices In the case μ ∈ ]1, 2[ we can compare S(λ) with the S-matrix Ssr (λ) defined similarly Ssr = Wsr+ ∗ Wsr−

∞

⊕Ssr (λ) dλ. 0

Under the condition of radial symmetry Yafaev considered in [28] the component of Ssr (λ) for each sector of fixed angular momentum. He computed an explicit oscillatory behaviour as λ → 0.

1894


The following result is a consequence of Theorem 6.13. In combination with Theorem 7.2, it yields oscillatory behaviour in a more general situation than considered in [28]. Theorem 7.3. For μ ∈ ]1, 2[, the operators Ssr and S are related by +

−

Ssr = e−i ψsr (p) Sei ψsr (p) .

(7.6)

In particular, for all λ > 0, +

Ssr (λ) = e−i ψsr (

√

2λ·)

−

S(λ)ei ψsr (

√

2λ·)

,

(7.7)

S(λ).

(7.8)

and if V2 = 0 then Ssr (λ) = e

−i 2

∞ √ √ R ( 2λ− 2(λ−V1 (r))) dr 0

7.4. Asymptotics of Dollard-type scattering matrices For μ >

1 2

and μ + 2 > 1, the Dollard-type S-matrix is diagonalized as before: +∗ − Sdol = Wdol Wdol

∞

⊕Sdol (λ) dλ. 0

We have the following analogue of Theorem 7.3, cf. Theorem 6.15: Theorem 7.4. For

1 2

< μ < 2, 2 < 1 and μ + 2 > 1, the operators Sdol and S are related by +

−

Sdol = e−i ψdol (p) Sei ψdol (p) .

(7.9)

In particular, for all λ > 0, +

Sdol (λ) = e−i ψdol (

√

2λ·)

−

S(λ)ei ψdol (

√ 2λ·)

(7.10)

,

and if V2 = 0 then Sdol (λ) = e

−i 2

∞ √ √ −1/2 V (r)) dr 1 R ( 2λ− 2(λ−V1 (r)−(2λ) 0

S(λ).

(7.11)

Example 7.5. For the purely Coulombic case V = −γ r −1 in dimension d 3 one can compute S(0) = ei c P ,

c ∈ R,

(7.12)

where (P τ )(ω) = τ (−ω). This formula can be verified using (7.11) and Remark 6.16, the explicit formula [30, (4.3)] for the Coulombic (Dollard) scattering matrix (slightly different from our definition), asymptotics of the Gamma function (see for example the reference [3] of [30]) and, for example, the stationary phase formula [14, Theorem 7.7.6] (alternatively one can use the formula [30, (3.4)]). √ It also follows (up to a compact term) from [7], where the constant c is computed as c = 4 2γ R0 − π d−2 2 .


1895

It follows from (7.12) that the singularities of the kernel S(0)(ω, ω ) in this particular case are located at {(ω, ω ) ∈ S d−1 × S d−1 | ω = −ω }. We devote Section 9 to an extension of this result. We also note# that for the purely Coulombic case there is in fact a complete asymptotic exj/2 . Here (of course) S is given by (7.12), and one can readily check pansion S(λ) & ∞ 0 j =0 Sj λ that S1 = 0. In particular we see that S(λ) is not smooth at λ = 0, cf. Remark 4.2(3). We refer to [2] (and references cited therein) for explicit expansions √ of the generalized purely Coulombic eigenfunctions at zero energy (for d = 3); those are also in λ. 8. Generalized eigenfunctions Throughout this section we impose Conditions 2.1–2.3. For any λ 0, we define V −∞ (λ) = u ∈ L2,−∞ (H − λ)u = 0 ⊆ S Rd . Elements of V −∞ (λ) will be called generalized eigenfunctions of H at energy λ. In this section we study all generalized eigenfunctions of H . Remark. Note that by Proposition 4.3, for any u ∈ V −∞ (λ) and s ∈ R, W Fscs (u) ⊆ b2 + c¯2 = 1 .

(8.1)

8.1. Representations of generalized eigenfunctions In this subsection we show that all generalized eigenfunctions can be represented by their incoming or outgoing data. Theorem 8.1. For any λ 0 the map W ± (λ) : D S d−1 → V −∞ (λ) ⊆ L2,−∞ is continuous and bijective. Proof. Step I. Clearly W ± (λ) : D (S d−1 ) → V −∞ (λ) is well defined and continuous, cf. Theorem 6.6. Step II. We show that W ± (λ) is onto. Let u ∈ V −∞ (λ) be given. Let 1 χ ± = χ− (a)χ˜ ± (b) + χ+ (a), 2

(8.2)

where χ+ = 1 − χ− is a real-valued function as in Proposition 4.1(ii) such that χ+ (t) = 1 for t 2C0 , and χ˜ − , χ˜ + ∈ C ∞ (R) are real-valued functions obeying χ˜ − + χ˜ + = 1 and

Now

supp χ˜ − ⊆ ]−∞, 1/2[,

(8.3)

supp χ˜ + ⊆ ]−1/2, ∞[.

(8.4)

1896


lim R(λ ± i )(H − λ) Opr χ ± u

↓0

= Opr χ ± u ± lim i R(λ ± i ) Opr χ ± u.

↓0

(8.5)

Note that lim ↓0 R(λ ± i ) Opr (χ ± )u exists, due to Propositions 4.3, 4.7, and 4.9. Therefore the second term on the right of (8.5) is zero. Therefore, we have 0 = Opr χ ± u − R(λ±i 0)(H − λ) Opr χ ± u.

(8.6)

Adding the two equations of (8.6) yields u = 2πıδ V (λ)(H − λ) Opr χ + u, which in turn in conjunction with Proposition 4.3, (6.10), (6.15) and Corollary 6.12 yields u = W ± (λ)τ,

τ = ±2πi W ± (λ)∗ H, Opr χ ± u ∈ D S d−1 .

(8.7)

Step III. We show that W ± (λ) is injective. For convenience we shall only treat the case of superscript +. By (8.7) we need to show that for all τ ∈ D (S d−1 ) τ = 2πi W + (λ)∗ (H − λ) Opr χ + W + (λ)τ.

(8.8)

By continuity it suffices to verify (8.8) for τ ∈ C ∞ (S d−1 ). This can be done as follows. Pick

∞

t/R non-negative f ∈ Cc∞ (R) with 0 f (s) ds = 1, and let FR (t) = 1 − 0 f (s) ds; R > 1. We write the right-hand side of (8.8) as w- lim 2πi W + (λ)∗ FR x (H − λ) Opr χ + W + (λ)τ R→∞

(8.9)

and pull the factor (H − λ) to the left. Thus (8.9) equals w- lim 2πR −1 W + (λ)∗ f x/R g Opr bχ + W + (λ)τ. R→∞

If λ 0, we insert (6.8) for W + (λ) (if λ = 0, we use instead (6.14)). By Proposition 4.1 (ii) and (iii) and Lemma 6.8 (ii) and (iv), we can replace each factor of W + (λ) by a factor of J + (λ), cf. the proof of Theorem 6.11. Moreover, we can replace the factor Opr (bχ + ) by the operator g −1 xˆ · p. Therefore, (8.9) becomes w- lim 2πR −1 J + (λ)∗ f x/R xˆ · pJ + (λ)τ. R→∞

By Theorem 5.7, (8.10) equals τ . The identity (8.8) follows.

(8.10)

2

Remarks. (1) A somewhat similar representation formula has been derived for representing positive solutions to a PDE, see for example [22]. This involves the so-called Martin boundary. In our case, the notion analogous to the “Martin boundary” would be S d−1 .


1897

(2) For V3 = 0, we have V −∞ (λ) = u ∈ S Rd (H − λ)u = 0 , and hence the set V −∞ (λ) is closed in S (Rd ) (with respect to the weak-∗ topology of S (Rd )). Moreover, in this case W ± (λ) maps D (S d−1 ) bicontinuously onto V −∞ (λ). In fact, suppose u ∈ S (Rd ) obeys (H − λ)u = 0. Then for some m ∈ N we have

p−2m u ∈ L2,−∞ . But (H − λ + i)−m p2m is bounded on any L2,s . Whence, showing that indeed u ∈ V −∞ (λ), i−m u = (H − λ + i)−m u = (H − λ + i)−m p2m p−2m u ∈ L2,−∞ . 8.2. Scattering matrices – an alternative construction The construction of scattering matrices given in Sections 7.1 and 7.2 involved a detailed knowledge of appropriate operators, see the proof of Theorem 7.2. However, given the theory of wave matrices developed in Section 8.1 and the basic formulas (6.10) and (6.15) for the spectral resolution, we could have constructed the scattering matrix more easily. Recall from Theorem 8.1 that W ± (λ) : D (S d−1 ) → L2,−∞ is injective. Hence, W ± (λ)∗ : L2,∞ → C ∞ (S d−1 ) has a dense range. For τ ∈ L2 (S d−1 ) of the form τ = W − (λ)∗ v with v ∈ L2,∞ , we define S(λ)τ := W + (λ)∗ v. By (6.10) and (6.15), we know that + ∗ 2 − ∗ 2 V W (λ) v = W (λ) v = v, δ (λ)v . Hence S(λ) is indeed well-defined and isometric. But W ± (λ)∗ L2,∞ is dense in C ∞ (S d−1 ), and therefore also in L2 (S d−1 ). Whence S(λ) extends to an isometric operator on L2 (S d−1 ). Reversing the role of + and −, we obtain that S(λ) is actually unitary. By construction, it satisfies S(λ)W − (λ)∗ = W + (λ)∗ ,

λ 0.

(8.11)

8.3. Geometric scattering matrices The following type of result was proved for a class of constant coefficient Hamiltonians (with no potential) in [1], and generalized to Schrödinger operators with long-range potentials (for a class including the one given by Condition 1.1) at positive energies by [9]. It gives a characterization of the space W ± (λ)L2 (S d−1 ), which in turn yields yet another characterization of the scattering matrix S(λ). Let s0 = s0 (λ) be given as in (4.6), and introduce in terms of a dual Besov space V −s0 (λ) := Bs∗0 ∩ V −∞ (λ) endowed with the topology of Bs∗0 . The statement (iv) below is given in terms of the phase function φ = φ(x, λ) of (5.21).

1898


Theorem 8.2. (i) For all τ ∈ L2 (S d−1 ), W Fsc−s0 W ± (λ)τ ⊆ {b = −1} ∪ {b = 1}. (ii) The operator W ± (λ) maps L2 (S d−1 ) bijectively and bicontinuously onto V −s0 (λ). ⊇ Bs0 ) maps Bs0 onto L2 (S d−1 ). (iii) The operator W ± (λ)∗ (defined a priori on Bs∗∗ 0 (iv) For all τ ∈ L2 (S d−1 ), W − (λ)τ (x) − W + (λ)τ (x) −

ei π

d−1 4

e−i φ(x,λ) τ (−x) ˆ + e−i π 1 2

1 2

d−1 4

(2π) g (r, λ)r e−i π

d−1 4

ei φ(x,λ) τ (x) ˆ + ei π 1 2

1 2

d−1 4

ei φ(x,λ) (S(λ)τ )(x) ˆ

d−1 2

e−i φ(x,λ) (S(λ)∗ τ )(−x) ˆ d−1 2

(2π) g (r, λ)r √ 1 π g 2 (r, λ)W ± (λ)τ 2 dx. τ 2L2 (S d−1 ) = lim R −1

∈ Bs∗0 ,0 ,

(8.12)

∈ Bs∗0 ,0 ,

(8.13)

R→∞

(8.14)

r 0 be given as before, cf. (6.12) and (6.13). As for the partition functions (8.2), we modify (8.3) and (8.4) by replacing here χ˜ ± → χ˜ ±,righ supp χ˜ −,righ ⊆ ]−∞, 1 − σ¯ /4[,

(8.15)

supp χ˜ +,righ ⊆ ]1 − σ¯ /2, ∞[.

(8.16)

Then it follows from Propositions 4.1 and 4.3 and Lemmas 6.8 and 6.9 that + + Opr χrigh W (λ) ∈ B L2 S d−1 , Bs∗0 .

(8.17)

(The fact that this bound holds for W + (λ) → J + (λ) is indeed a consequence of Lemma 6.9 due to interpolation, cf. [13, Theorem 14.1.4], but it can also be proved concretely along the lines of the proofs of Lemma 6.9 and Theorem 6.11.) + )]W + (λ)τ = 0, we conclude from (4.30) and (8.17) that Since W + (λ)τ, i[H, FR Opr (χrigh sup Re W + (λ)τ, Opw FR χ− (a)χ˜ righ (b)gr −1 W + (λ)τ Cτ 2 .

(8.18)

R>1

Here we used the calculus of pseudodifferential operators, cf. [14, Theorem 18.6.8]. In combination with Propositions 4.3 and 4.5, we conclude that {−1 < b < 1} ∩ W Fsc−s0 W + (λ)τ = ∅. Re (ii) (Boundedness). To proceed from here we change (8.15) and (8.16) as follows:

(8.19)


1899

supp χ˜ −,lef ⊆ ]−∞, −1 + σ¯ /2[,

(8.20)

supp χ˜ +,lef ⊆ ]−1 + σ¯ /4, ∞[.

(8.21)

With these cutoffs we can show analogously that − Opr (χlef )W − (λ) ∈ B L2 S d−1 , Bs∗0 .

(8.22)

Using (8.11), this leads to − Opr (χlef )W + (λ) ∈ B L2 S d−1 , Bs∗0 .

(8.23)

+ − Finally, writing (with χmiddle := 1 − χrigh − χlef )

+ + − W (λ) + Opr (χlef )W + (λ) + Opr (χmiddle )W + (λ), W + (λ) = Opr χrigh we conclude from (4.54), (8.17), (8.19) and (8.23) that indeed W + (λ) ∈ B L2 S d−1 , Bs∗0 .

(8.24)

Whence W + (λ) maps L2 (S d−1 ) continuously into V −s0 (λ). Re (ii) (Bijectiveness). We shall show that W + (λ) maps L2 (S d−1 ) onto V −s0 (λ). Using the expression (8.7) for the inverse τ ∈ D (S d−1 ), mimicking the first part of Step III in the proof of Theorem 8.1 and using the Riesz’ representation theorem (see for example [33]) in conjunction with (8.24), we obtain that indeed τ ∈ L2 (S d−1 ). This argument also shows that W + (λ)−1 ∈ B V −s0 (λ), L2 S d−1 . (8.25) Re (iii). The result follows from (ii) by the Banach’s closed range theorem, see [33]. Re (iv). Let 1

u±,τ (x) = (2π)− 2 e∓i π

d−1 4

1

g − 2 (r, λ)r −

d−1 2

e±ıφ(x,λ) τ (±x). ˆ

Clearly u±,τ ∈ Bs∗0 with a continuous dependence on τ . We claim (with reference to (8.2)) that Opr χ ± W ± (λ)τ − u±,τ ∈ Bs∗0 ,0 .

(8.26)

Notice that also the first term is in Bs∗0 with a continuous dependence on τ , cf. (8.17) and (8.19), hence it suffices to show (8.26) for τ ∈ C ∞ (S d−1 ), in which case the asymptotics follows from Theorem 5.7, cf. Step III of the proof of Theorem 8.1. Now, combining (8.26) and the identity (8.11), we obtain Opr (χ − )W + (λ)τ − u−,S(λ)∗ τ ∈ Bs∗0 ,0 . (8.27) Opr χ + W − (λ)τ − u+,S(λ)τ , By (8.26) and (8.27), W − (λ)τ − (u−,τ + u+,S(λ)τ ), showing (8.12) and (8.13).

W + (λ)τ − (u+,τ + u−,S(λ)∗ τ ) ∈ Bs∗0 ,0 ,

1900


As for (8.14) we use (8.12) and (8.13); notice that the cross terms do not contribute to the limit which can be seen by an integration by parts with respect to the variable r = |x|, invoking Proposition 3.3. 2 On the basis of Theorem 8.2, we can characterize the scattering matrix S(λ) geometrically as follows: Corollary 8.3. For all τ − ∈ L2 (S d−1 ), there exist a uniquely determined u ∈ V −s0 (λ) and τ + ∈ L2 (S d−1 ) such that u−

ei π

d−1 4

e−i φ(x,λ) τ − (−x) ˆ + e−i π 1

1

(2π) 2 g 2 (r, λ)r

d−1 4

ei φ(x,λ) τ + (x) ˆ

d−1 2

∈ Bs∗0 ,0 .

(8.28)

We have τ + = S(λ)τ − , u = W − (λ)τ − = W + (λ)τ + . Proof. The existence part (with τ + = S(λ)τ − ) follows from (8.12). To show the uniqueness, suppose that ui , τi+ , i = 1, 2, satisfy the requirements of (8.28) with the same τ− . Then for the difference, u = u1 − u2 , we have (H − λ)u = 0 and W F (Bs∗0 ,0 , u) ⊆ {b = 1}. Hence by Proposition 4.10, u = 0. 2 Corollary 8.4. Let d 2 and λ 0. Suppose

∞(in addition to Conditions 2.1 and 2.3) that V2 and V3 are spherically symmetric and that 0 r|V3 (r)| dr < ∞. (Condition 2.2 is not needed since V2 can be absorbed into V1 ). Then there exists a real-valued continuous function σl (·) such that for all spherical harmonics Y of order l we have S(λ)Y = ei 2σl (λ) Y . Let ul (r) denote the regular solution of the reduced Schrödinger equation on the half-line ]0, ∞[ −u + Vl u = 0,

(l + Vl (r) = 2 V (r) − λ +

d 2

− 1)2 − 4−1 , r2

l 0;

d−1

where “regular” refers to the asymptotics u(r) & r l+ 2 as r → 0. Then σl (·) is uniquely determined mod 2π by the asymptotics r √ √ π + σl (λ) sin R0 2(λ − V (r )) dr + 2λR0 − d−3+2l ul (r) 4 −C ∈ Bs∗0 ,0 , (8.29) d−1 1 d−1 r 2 (λ − V (r)) 4 r 2 where C = C(l, λ) is a (uniquely determined ) positive constant. Proof. Let Y be a spherical harmonic of order l. Note that its parity is (−1)l , i.e. Y (−ω) = d−1 ˆ solves (H − λ)u = 0. We apply Corollary 8.3 with (−1)l Y (ω). Besides, u := r − 2 ul (r)Y (x) this u and with τ − = Y , so that τ + = ei 2σl (λ) Y . Then ei π

d−1 4

e−i φ(x,λ) τ − (−x) ˆ + e−i π

d−1 4

ei φ(x,λ) τ + (x) ˆ

d−1 d−1 = ei π 4 −i φ(x,λ)+i πl + e−i π 4 +i φ(x,λ)+i 2σl (λ) Y (x) ˆ l d − 3 + 2l π + σl (λ) Y (x). ˆ = 2ei π 2 +i σl (λ) sin φ(x, λ) − 4

We finish the proof using (5.22).

2


1901

Let us remark that, because of the spherical symmetry of the potential, the asymptotics (8.29) can be improved: 1 λ − V (r) 4 ul (r) − C sin

r

d − 3 + 2l 2 λ − V (r ) dr + 2λR0 − π + σl (λ) 4

R0

→ 0 for r → ∞.

(8.30)

The equivalence of (8.29) and (8.30) follows from the 1-dimensional WKB-method described for instance in [24] (see also [7]). 9. Homogeneous potentials – location of singularities of S(0) In this section we impose Conditions 2.1–2.3 with d 2 and the condition V1 (r) = −γ r −μ −μ + O(r −μ− 2 ), cf. (1.23). Throughout the section g = for r 1 and √ hence V (r) = −γ r g(λ = 0) = −2V1 . Our goal is to prove a statement about the localization of the singularities of the (Schwartz) kernel S(0)(ω, ω ). The purely Coulombic case for which μ = 1 and d 3 was treated explicitly in Example 7.5. Under an additional condition we can write down a fairly explicit integral that carries the singularities. This section is also closely related to our recent paper [7], which is, however, restricted to radial potentials. 9.1. Reduced classical equations Consider the classical system given by the Hamiltonian h1 (x, ξ ) = 12 ξ 2 − γ |x|−μ for x = 0. The equations of motion for h1 (x, ξ ) are invariant with respect to the transformation (x, ξ ) → λx, λ−μ/2 ξ ,

λ ∈ R+ ,

(9.1)

upon rescaling of time t → tλ1+μ/2 . Let T∗ := Rd \ {0} × Rd /∼, where (x1 , ξ1 ) ∼ (x2 , ξ2 ) iff there exists λ > 0 such that (x1 , ξ1 ) = (λx2 , λ−μ/2 ξ2 ). Note that T∗ can be conveniently identified with T ∗ (S d−1 ) × R. We shall introduce coordinates of T∗ by ˆ x|) ˆ gξ ∈ Txˆ∗ (S d−1 ) with xˆ ∈ S d−1 . (At this point we are setting b = xˆ · gξ ∈ R and c¯ = (I − |x

slightly abusing the notation of Section 4.2, however as noticed there the b and c¯ given by (4.7) agree with the above definition for r 1.) The equations of motion for the Hamiltonian h1 can be reduced to T∗ . Introducing the “new time” τ by dτ dt = g/r we have the following system of reduced equations of motion: ⎧ d ¯ ⎪ ⎨ dτ xˆ = c, μ d 2ˆ dτ c¯ = −(1 − 2 )bc¯ − c¯ x, ⎪ ⎩ d μ 2 μ 2 2 dτ b = (1 − 2 )c¯ + 2 (b + c¯ − 1).

(9.2)

1902


(Notice that the last equation follows from (4.29).) The maximal solution of (9.2) that passes z = (x, ˆ b, c) ¯ ∈ T∗ at τ = 0 is denoted by γ (τ, z). Beside (9.2), we shall consider a related dynamics given by the equations ⎧ d ¯ ⎪ ⎨ dτ xˆ = c, μ d 2ˆ dτ c¯ = −(1 − 2 )bc¯ − c¯ x, ⎪ ⎩ d μ 2 dτ b = (1 − 2 )c¯ .

(9.3)

The (maximal) solution of the system (9.3) that passes z = (x, ˆ b, c) ¯ ∈ T∗ at τ = 0 will be denoted by γ0 (τ, z). Clearly the equation c¯ = 0 defines the fixed points, and the system is complete. Notice that the surface h−1 ˆ b, c) ¯ corresponds to the condition 1 (0) in the coordinates (x, 2 b + c¯2 = 1. This surface is preserved both by the flow γ and γ0 , and on this surface both flows coincide. Note that the flow γ0 is exactly solvable. The variable b is always increasing and k = b2 + c¯2 is a conserved quantity; of course the relevant value is k = 1. For non-fixed points we can compute its dependence on the modified time √ √ μ b(τ ) = k tanh k 1 − (τ − τ0 ). 2

(9.4)

Values k = 1 correspond in this picture to replacing the coupling constant

γ → kγ . More 2 + c¯2 for a solution to (9.3), we can define r(τ ) = r exp( τ b dτ ), introduce precisely, if k = b 0 0

τ r dτ and check that indeed t = 0 g(r)

x(t) = r x, ˆ ξ(t) = g(r)(bxˆ + c), ¯

(9.5)

defines a zero energy solution to Hamilton’s equations with V → kV . The equation b = 0 corresponds to a turning point (at which |x(t)| has the smallest √ value). √ Clearly, it follows from (9.4) that limτ →∞ b = k, limτ →−∞ b = − k. Upon writing x(τ ˆ ) · x(∞) ˆ = cos θ (τ ) for some monotone continuous function θ (·), we obtain from (1.27) that θ (∞) − θ (−∞) =

2 π. 2−μ

(9.6)

9.2. Propagation of singularities We will use the scattering wave front set at zero energy, introduced in Section 4.2. The following proposition is somewhat similar to Hörmander’s theorem about propagation of singularities adapted to scattering at the zero energy. It is a “local” version of Proposition 4.5 which takes into account the fact that in the case of a homogeneous potential we can use the dynamics in the reduced phase space. Again the proof is a modification of that of [12, Proposition 3.5.1], see also [21] and [11].


1903

Proposition 9.1. Suppose u, v ∈ L2,−∞ , H u = v, s ∈ R, z ∈ T∗ and z ∈ / W Fscs (u). Define τ + := sup τ 0 γ0 (τ˜ , z) ∈ / W Fscs (u) for all τ˜ ∈ [0, τ ] , τ − := inf τ 0 γ0 (τ˜ , z) ∈ / W Fscs (u) for all τ˜ ∈ [τ, 0] . If τ + < ∞, then γ0 (τ + , z) ∈ W Fsc

s+2s0

(v). If τ − > −∞, then γ0 (τ − , z) ∈ W Fsc

s+2s0

(v).

Proof. The proof is similar to the one of Proposition 4.5. We shall only deal with the case of forward flow; the case of superscript “−” is similar (actually it follows from the case of “+” by time reversal invariance). For convenience, we shall assume that 2 2 − μ.

2

Step I. We will first show the following weaker statement: Suppose u ∈ L2,s− 2 , v ∈ L2,s+2s0 and H u = v. Then γ0 (τ, z) ∈ / W Fscs (u)

for all τ 0.

(9.7)

Suppose on the contrary that (9.7) is false. Then we obtain from Proposition 4.3 that the flows of (9.2) and (9.3), starting at z, coincide. Letting γ (τ ) = γ (τ, z), it thus needs to be shown that / W Fscs (u) for all τ˜ ∈ [0, τ ] = ∞. τ + := sup τ 0 γ (τ˜ ) ∈

(9.8)

Suppose on the contrary that τ + is finite. Then γ (τ + ) is not a fixed point. Consequently, we can pick a slightly smaller τ˜ + < τ + and a transversal (2d − 2)-dimensional submanifold at γ (τ˜ + ), say M, such that with J = ]− + τ˜ + , τ + + [, for some small > 0, the map J × M (τ, m) → Ψ (τ, m) = γ τ − τ˜ + , m ∈ T∗ is a diffeomorphism onto its range. We pick χ ∈ Cc∞ (M) supported in a small neighbourhood of γ (τ˜ + ) such that χ(γ (τ˜ + )) = 1 and Ψ − + τ˜ + , τ˜ + × supp χ ∩ W F s (u) = ∅.

(9.9)

We pick a non-positive function f ∈ Cc∞ (J ) such that f 0 on a neighbourhood of [τ˜ + , τ + + ) and f (τ + ) < 0. Let fK (τ ) = exp(−Kτ )f (τ ) for K > 0, and Xκ = (1 + κr 2 )1/2 for κ ∈ ]0, 1]. We consider the symbol bκ = g −1/2 X 1/2 aκ ;

aκ = X s Xκ− 2 /2 F (r > 2)(fK ⊗ χ) ◦ Ψ −1 .

(9.10)

First we fix K. A part of the Poisson bracket with bκ2 is

h2 , g −1 X 2s+1 Xκ− 2 = r −1 Yκ bX 2s+1 Xκ− 2 ,

(9.11)

where Yκ = Yκ (r) is uniformly bounded in κ. We fix K such that 2K |Yκ b| + 2 on supp bκ .

1904


We compute g h1 , (fK ⊗ χ) ◦ Ψ −1 = r

$

% d fK ⊗ χ ◦ Ψ −1 . dτ

From (9.11) and (9.12), and by the choice of f and K, we conclude that at P ⊆ T∗ h2 , bκ2 −2aκ2 + O r 2s− 2

(9.12)

(9.13)

given by P =Ψ

τ ∈ J f (τ ) 0 × supp χ .

Introducing Aκ = Opw (aκ ) and Bκ = Opw (bκ ), we have i H, Bκ2 u = −2 Im v, Bκ2 u ,

(9.14)

and we estimate the right-hand side using the calculus of pseudodifferential operators, cf. [14, Theorems 18.5.4, 18.6.3, 18.6.8], to obtain the uniform bound i H, B 2 C1 vs+2s Aκ u + C2 Aκ u2 + C3 . (9.15) κ u 0 On the other hand, using (9.9) and (9.13), we infer that i H − V3 , Bκ2 u −2Aκ u2 + C4 .

(9.16)

An application of (4.12a) yields i V3 , Bκ2 u C5 .

(9.17)

Combining (9.15)–(9.17) yields Aκ u2 C6 = C3 + C4 + C5 , which in turn gives a uniform bound − /2 w X 2 Op χγ (τ + ) F (r > 2) u2 C7 . κ

s

(9.18)

Here χγ (τ + ) signifies a phase-space localization factor of the form entering in (4.8) supported in a sufficiently small neighbourhood of the point γ (τ + ). / W Fscs (u), which is a contradiction. We have proved We let κ → 0 in (9.18) and infer that τ + ∈ (9.8) and hence (9.7). Step II. To relax the assumptions on u and v used in Step I, we modify the above proof (using localization) in an iterative procedure very similar to Step II of the proof of Proposition 4.5. Pick t < s such that u ∈ L2,t and define sm = min(s, t + m 2 /2) for m ∈ N. Let correspondingly τm+ be given as τ + , upon replacing s → sm . Clearly, + τm+ τm−1 ;

m = 2, 3, . . . .

(9.19)


1905

We shall show that τm+ < ∞

⇒

γ0 τm+ , z ∈ W Fscsm +2s0 (v).

(9.20)

We are done by using (9.20) for an m taken so large that sm = s. Let us consider the start of induction given by m = 1, in which case obviously u ∈ L2,sm − 2 /2 . Suppose on the contrary that (9.20) is false. Then we consider the following case: τm+ < ∞

and γ0 τm+ , z ∈ / W Fscsm +2s0 (v).

(9.21)

It follows from (9.21) and an ellipticity argument that b2 + c¯2 = 1 at γ0 (τm+ , z) (using that s +μ / W Fscm (H u)). Consequently we can henceforth use the flow of (9.2), γ (τ ) = γ0 (τm+ , z) ∈ γ (τ, ·), exactly as in Step I. We let > 0, J , f , fK , χ and Ψ be chosen as in Step I with τ + → τm+ and τ˜ + → τ˜m+ . Let f˜ ∈ Cc∞ (]τ˜m+ − 2 , τm+ + 2 [) with f˜ = 1 on J . Similarly, let χ˜ ∈ Cc∞ (M) be supported in ˜ (τ˜m+ )) = 1 in a neighbourhood of supp χ . a small neighbourhood of γ (τ˜m+ ) such that χ(γ It follows from (9.21), possibly by shrinking the supports of f˜ and χ˜ , that I v ∈ L2,sm +2s0 ,

I = Opw F (r > 2)(f˜K ⊗ χ˜ ) ◦ Ψ −1 .

(9.22)

Next, we introduce the symbol bκ by (9.10) (with s → sm ) and proceed as in Step I. As for the bounds (9.15), we can replace v by I v up to addition of a term that is bounded uniformly in κ. Clearly, we can verify (9.16) and (9.17). So again we obtain (9.18) (with s → sm ), and therefore a contradiction as in Step I. We have shown (9.20) for m = 1. Now suppose m 2 and that (9.20) is verified for m − 1. We need to show the statement for the given m. Due to (9.19) and the induction hypothesis, we can assume that + . τm+ < τm−1

(9.23)

Again we argue by contradiction assuming (9.21). We proceed as above noticing that it follows from (9.23) that in addition to (9.22) we have I u ∈ L2,sm−1 ;

(9.24)

at this point we possibly need to shrink the supports of f˜ and χ˜ even more (viz. taking < + − τm+ )/2). By replacing v by I v and u by I u at various points in the procedure of (τm−1 Step I (using (9.22) and (9.24), respectively) we obtain again a contradiction. Whence (9.20) follows. 2 Remark 9.2. Suppose u ∈ L2,t1 , v ∈ L2,t2 and H u = v. Suppose z0 ∈ / W Fscs (u) for some s > t1 . s+2s0 + Fix τ˜ ∈ ]0, ∞[ and suppose that γ0 (τ, z0 ) ∈ / W Fsc (v) for all τ ∈ [0, τ˜ + ]. Write γ0 (τ˜ + , z0 ) = (ω1 , c¯1 , b1 ) = (ω1 , η1 ). Then there exist neighbourhoods Nω1 ω1 and Nη1 η1 such that for all χω1 ∈ Cc∞ (Nω1 ) and χη1 ∈ Cc∞ (Nη1 ) we have Opw χz1 F (r > 2) u ∈ L2,s . Here χz1 (x, ξ ) = ˆ η1 (ξ/g). Notice that this conclusion is already contained in Proposition 9.1; however the χω1 (x)χ above proof yields an additional bound: First, writing z0 = (ω0 , η0 ), we can pick any similarly defined localization factor, say denoted by χz0 , with χω0 = 1 and χη0 = 1 around the points ω0 and η0 , respectively, and such that

1906


Opw (χz0 F (r > 2))u ∈ L2,s (this is by assumption). Next we pick a small neighbourhood U of γ0 ([0, τ˜ + ], z0 ) ⊂ T∗ and χ ∈ Cc∞ (U ) with χ = 1 around this orbit segment. If U is small enough we have (again by assumption) that Opw (χγ0 F (r > 2))v ∈ Ls+2s0 , χγ0 (x, ξ ) := χ(x, ˆ ξ/g). Now, there are neighbourhoods Nω1 ω1 and Nη1 η1 depending only on χz0 and χγ0 such that for all χω1 ∈ Cc∞ (Nω1 ) and χη1 ∈ Cc∞ (Nη1 ) we have w Op χz F (r > 2) u 1 s w C Op χz0 F (r > 2) us + ut1 + Opw χγ0 F (r > 2) v s+2s + vt2 , 0

where the constant C only depends on the various localization factors. 9.3. Location of singularities of the kernel of the scattering matrix In this subsection we describe the location of the singularities of the scattering matrix at zero energy. Theorem 9.3. Suppose that V1 (r) = −γ r −μ for r 1. Then the kernel S(0)(ω, ω ) is smooth μ outside the set {(ω, ω ) | ω · ω = cos 2−μ π}. To analyse S(0)(ω, ω ) we shall use the representation (7.3a), which we write (formally) as S(0)(ω, ω ) = −2π j + (·, ω), v − (·, ω ) + 2πı v + (·, ω), R(+i 0)v − (·, ω ) , where ± j ± (x, ω) = (2π)−d/2 ei φ a˜ ± (x, ω, 0), ± v ± (x, ω) = (2π)−d/2 ei φ t˜± (x, ω, 0). + Let φsph denote the solution of the eikonal equation for the potential V1 at zero energy, cf. (3.9). It is given by

+ φsph (x, ω) =

√ 2γ 1−μ/2 1−μ/2 r , cos(1 − μ/2)θ − R0 1 − μ/2

where cos θ = xˆ · ω. Using x ⊥ =

ω−xˆ cos θ sin θ

⊥

and ∇x θ = − xr , we can also compute

+ + Fsph (x, ω) = ∇x φsph (x, ω) = 2γ r −μ/2 xˆ cos(1 − μ/2)θ + x ⊥ sin(1 − μ/2)θ .

(9.25)


1907

Lemma 9.4. For all s ∈ R, ω ∈ S d−1 and multiindices δ, W Fscs ∂ωδ v ± (·, ω) + Fsph (x, ˆ ±ω) ∗ , ⊆ z = (x, ˆ c, ¯ b) ∈ T 1 − σ ±xˆ · ω 1 − σ, bxˆ + c¯ = ± (2γ )1/2 W Fscs ∂ωδ j ± (·, ω) + Fsph (x, ˆ ±ω) ∗ . ⊆ z = (x, ˆ c, ¯ b) ∈ T 1 − σ ±xˆ · ω, bxˆ + c¯ = ± (2γ )1/2

(9.26)

(9.27)

Suppose in addition that χ+ ∈ C ∞ (R), χ+ ∈ Cc∞ (R) and supp χ+ ⊂ ]1, ∞[. Then Opw χ+ (a) ∂ωδ v ± (·, ω), Opw χ+ (a) ∂ωδ j ± (·, ω) ∈ L2,s .

(9.28)

Proof. Only the “+” case needs to be considered (can be seen by complex conjugation). Upon multiplying by a localization operator supported outside of the right-hand side of (9.26), we need to demonstrate that the result is in L2,s , cf. the definition (4.8). Using the right Kohn–Nirenberg quantization (instead of the Weyl quantization) this can be done by integrating by parts in explicit integrals, exactly as in the proofs of Lemma 6.8(iii) and Theorem 6.11. The arguments for (9.27) and (9.28) are the same, in particular, (9.28) follows from the proof of Theorem 6.11. 2 Proof of Theorem 9.3. Due to Proposition 4.8 and Lemma 9.4 we are allowed to act by R(+i 0) on ∂ωδ v − (·, ω )). In fact, for all τ ∈ C ∞ (S d−1 ) R(+i 0)T − (0)τ =

R(+i 0)v − (·, ω )τ (ω ) dω .

(9.29)

S d−1

Using the representation (7.3a) interpreted as a form on C ∞ (S d−1 ), and (9.29), we have Sκ (0) → S(0) as κ 0, where the kernel of Sκ (0) is the well-defined smooth expression Sκ (0)(ω, ω ) = −2π j + (·, ω), F κ| · | < 1 v − (·, ω ) + 2πı v + (·, ω), F κ| · | < 1 R(+i 0)v − (·, ω ) . μπ It remains to be shown that Sκ (0)(·, ·) has a limit in C ∞ ({ω · ω = cos 2−μ }). By integration by parts, it follows that the first term has a limit, in fact in C ∞ (S d−1 × S d−1 ), cf. the proof of Lemma 9.4. Whence we only look at the second term. By Lemma 9.4 and Proposition 3.3, for all s

& W Fscs ∂ωδ v + (·, ω) ⊆ c¯ = 0, b2 + c¯2 = 1 ∩ z lim x(τ ˆ ) = ω, τ →+∞ ' where γ0 (τ, z) = x(τ ˆ ), b(τ ), c(τ ¯ ) . Here γ0 (τ, z) refers to the flow defined by (9.3).

(9.30)

1908


By Propositions 4.8 and 9.1, for all s, W Fscs R(+i 0)∂ωδ v − (·, ω ) ⊆ γ0 (τ, z) τ 0, z ∈ W Fscs ∂ωδ v − (·, ω ) ∪ {c¯ = 0, b > 0} & ' ⊆ z lim x(τ ˆ ) = −ω ∪ {c¯ = 0, b > 0}. τ →−∞

(9.31)

By invoking (9.6), we see that the sets on the right-hand side of (9.30) and (9.31) are disjoint μπ away from {ω · ω = cos 2−μ }. Hence also W Fscs ∂ωδ v + (·, ω) ∩ W Fscs R(+i 0)∂ωδ v − (·, ω ) = ∅, which implies, upon taking s = 0 and using (9.28) and a suitable partition of unity, that

∂ωδ v + (·, ω, 0), R(+i 0)∂ωδ v − (·, ω , 0)

is well defined. By the same arguments ∂ωδ ∂ωδ v + (·, ω, 0), F κ| · | < 1 R(+i 0)v − (·, ω , 0) → ∂ωδ v + (·, ω, 0), R(+i 0)∂ωδ v − (·, ω , 0) μπ locally uniformly in {ω · ω = cos 2−μ }. Notice that the bound (9.28) is uniform in ω; a similar statement is valid for the bounds underlying (9.26), and we also need at this point to invoke Remark 9.2. 2

Remarks 9.5. (1) The somewhat abstract procedure of the proof of Theorem 9.3 does not provide information μ π . In the study of the sinabout the nature of the singularities at the cone ω · ω = cos 2−μ gularities at the diagonal of the kernel of scattering matrices for positive energies (see [19] and [31]) it is important that the eikonal and transport equations can be solved in sufficiently big sectors. In combination with resolvent estimates this allows one to put the singularities in a rather explicit term similar to the first one on the right-hand side of (7.2a). A very similar procedure can be used (at least for V2 = 0) for S(0)(ω, ω ) provided μ < 1. However, for μ ∈ [1, 2[ there is a “glueing problem” due to the fact that in order to apply resolvent estimates in this case the constructed solutions to the eikonal equations φ ± need to be exπ . Therefore, multivalued φ ± are needed. We devote tended, viz. as to including some θ > 2−μ Section 9.4 to a discussion of this question. (2) Under Condition 1.1, it follows essentially by the same method of proof that, for λ > 0, the kernel S(λ)(ω, ω ) is smooth outside the set {(ω, ω ) | ω = ω }; for that we use (9.3) with μ = 0. See [27, Chapter 19] for a related result and procedure. (3) There is a discrepancy between our results and the main result of [20]. The idea of [20] is to use a partial wave analysis to obtain an asymptotic expression of the scattering amplitude for λ → 0 (with the assumption of radial symmetry and under the short-range condition μ > 1). Unfortunately [20, (17)] is incompatible with Theorems 7.2, 7.3 and 9.3.


1909

9.4. Distributional kernel of S(0) as an oscillatory integral In addition to the previous assumption V1 (r) = −γ r −μ for r 1, we shall here assume that V2 = 0, see though Remark 9.6(1). We shall explain a procedure which in principle allows us to calculate the singularities of the kernel S(0)(ω, ω ); a fairly explicit oscillatory integral will be specified. Using this integral we derive below the location of the singularities of S(0) by the method of non-stationary phase, which gives an alternative proof of Theorem 9.3 (under the condition that V2 = 0). We shall improve on the representation (7.3a) for S(0). Notice that the functions a˜ + and φ + used up to now are supported near the forward region cos θ = xˆ · ω ≈ 1 only. Now we shall take advantage of the fact that the expression (9.25) defines a solution to the eikonal equation for all values of θ . We shall consider a cut-off at larger values of θ , in fact slightly to the left of the critical angle θ = (1 − μ/2)−1 π . The basic idea is similar to the one applied in the study of the kernel of scattering matrices for positive energies, cf. Remark 9.5(1). If we can extend the construction of the phase and amplitude as indicated above, then we can apply a “two-sided” resolvent estimate to deal with the second term on the right-hand side of (7.3a), i.e. to show that it contributes by a smooth kernel; in our case the appropriate “two-sided” estimate is given by (4.3f). Now besides the problem of extending the phase up to θ = (1 − μ/2)−1 π , there is obviously the issue of well-definedness, since θ as a function of x is multi-valued; for the case of positive energies this problem does not occur since the cut-off in this case occurs before the angle θ = π . We have + J τ (x) = (2π)−d/2

i φ+ + e a˜ (x, ω, 0)τ (ω) dω.

(9.32)

S d−1

In fact, in the present spherically symmetric case the dependence of the variables x and ω is through r = |x| and xˆ · ω only. Writing ω = cos θ xˆ + sin θ ω, where ω · xˆ = 0, (9.32) can be written as −d/2

π d ω

(2π)

S d−2

iφ e a˜ (r, θ )τ (cos θ xˆ + sin θ ω) sind−2 θ dθ ;

(9.33)

0

for convenience we dropped the superscript. The phase φ is given by (9.25), and using this expression and the orbit (1.27), we can extend the support of a˜ by solving transport equations as in Section 5.3; the cut-off is now taken slightly to the left of θ = (1 − μ/2)−1 π . More precisely, the cut-off is defined as follows: First pick L ∈ N such that (1 − μ/2)L < 1 while (1 − μ/2) × (L + 1) 1. We shall assume that the analogue of σ for the construction of J − , entering in (5.2) for the construction of J + , is so small that (1 − μ/2) Lπ + cos−1 (1 − σ ) < π.

(9.34)

1910


Next the version of (5.2) that we need is given in terms of the σ of the construction of J − as follows: Choose angles πL < θ0 < θ0 < π(L + 1) such that (1 − μ/2)θ0 < π and (1 − μ/2)(θ0 + cos−1 (1 − σ )) > π. Introduce a smoothed out characteristic function 1, for s θ0 , (9.35) χ2 (s) = 0, for s θ0 ; and with this choice the new cut-off function takes the (essentially same) form χ = χ1 (r)χ2 (θ ). The extended a˜ has similar properties as before due to the cut-off. Whence we are lead to consider the following modification of the expression (9.33): ∞

d ω S d−2

f (r, θ )τ (cos θ xˆ + sin θ ω)sind−2 θ dθ ;

f = (2π)−d/2 ei φ a, ˜

0

where the θ -integration (due to the cut-off) effectively takes place on the interval [0, (1 − μ/2)−1 π]. The next step is to change variable, writing for θ in intervals of the form (2kπ, (2k + 1)π], cos θ xˆ + sin θ ω = cos ψ xˆ + sin ψ ω;

ψ = θ − 2kπ,

while on intervals of the form ((2k + 1)π, (2k + 2)π], cos θ xˆ + sin θ ω = cos ψ xˆ + sin ψ(− ω);

ψ = (2k + 2)π − θ,

respectively; here k ∈ N ∪ {0}. Whence we consider the expression F (r, ψ)τ (ω) dω, S d−1

where F (r, ψ) =

∞ " f (r, ψ + 2kπ) + f r, (2k + 2)π − ψ , k=0

and as above ω = cos ψ xˆ + sin ψ ω with ω · xˆ = 0 and ψ ∈ [0, π], i.e. ψ = cos−1 xˆ · ω. We claim that F (r, ψ) is smooth in x and ω. Notice that this is not an obvious fact, since although the function ψ = cos−1 xˆ · ω is continuous, it has a cusp singularity at xˆ · ω = ±1. However, as can easily verified, ψ 2 is smooth at xˆ · ω = 1 and (π − ψ)2 is smooth at xˆ · ω = −1, respectively. Moreover, f (r, ψ) and f (r, ψ + 2(k + 1)π) + f (r, (2k + 2)π − ψ) are in fact smooth functions of ψ 2 near xˆ · ω = 1, and similarly f (r, ψ + 2kπ) + f (r, (2k + 2)π − ψ) = f (r, (2k + 1)π − (π − ψ)) + f (r, (2k + 1)π + (π − ψ)) is a smooth function of (π − ψ)2 at xˆ · ω = −1.


1911

Recall that we have the representation (7.3b) − S(0)(ω, ω ) = −2π w + (ω, 0), ei φ t˜− (·, ω , 0) ,

(9.36)

where w + (ω, 0) is the generalized eigenfunction of Theorem 6.5. Define w = w(x, ω) = F (r, ψ) − R(−i 0)H F . Due to Proposition 4.10, Proposition 4.1(iii) and Lemma 6.8(iii), this w agrees with the eigenfunction w + (ω, 0), cf. the proof of Lemma 6.10. Therefore, our (extended) version of (7.3a) reads − − S(0)(ω, ω ) = −2π F, ei φ t˜− (·, ω , 0) + 2π R(−i 0)H F, ei φ t˜− (·, ω , 0) .

(9.37)

As indicated above, the contribution to S(0)(ω, ω ) from the second term on the right-hand side of (9.37) is smooth in ω and ω , if we use a cut-off sufficiently close (but to the left of) the critical angle θ = (1 − μ/2)−1 π ; this is indeed accomplished by using (9.35) as cut-off function. We conclude that the singularities of the kernel of S(0) are the same as those of the kernel of the operator S(0) given by i φ− − S(0)τ2 = −2π e t˜ (·, ω , 0)τ2 (ω ) dω . τ1 , F (r, ψ)τ1 (ω) dω, Whence (formally) S(0)(ω, ω ) = −2π

− F (r, ψ) ei φ t˜− (·, ω , 0) dx.

(9.38)

Next we introduce the variable θ = cos−1 xˆ · (−ω ) ∈ [0, π/2); we can represent The integrand on the right-hand side of (9.38) is given as

φ − (x, ω , 0) = −φ(r, θ ), cf. (3.6). # ∞ k=0 fk , where fk has the form

e−i (φ(r,ψ+2kπ)+φ(r,θ )) g(r, ψ + 2kπ, θ ) + e−i (φ(r,(2k+2)π−ψ)+φ(r,θ )) g r, (2k + 2)π − ψ, θ .

(9.39)

μ Let us argue that the integral (9.38) is well-defined in {ω · ω = cos 2−μ π}, in agreement with Theorem 9.3. The argument is based on the method of non-stationary phase. First we notice that the cusp singularities at ψ = 0 and ψ = π correspond to non-stationary points. More precisely, we can write

˜ˆ x = r(cos ψ ω + sin ψ x), and perform the x-integration as

π · · · dx =

sin 0

d−2

ψ dψ S d−2

dx˜ˆ

∞ · · · r d−1 dr.

(9.40)

0

Now on the support of g the factor cos(1 − μ/2)θ cos θ 1 − σ , while the factors cos(1 − μ/2)(ψ + 2kπ) and cos(1 − μ/2)((2k + 2)π − ψ) stay sufficiently away from −1

1912


(given that ψ ≈ 0 or ψ ≈ π ) to ensure that the sum of phases does not vanish; here we use (9.34). Thus the phases of fk are nonzero near the ψ -endpoints of integration, and consequently integration by parts with respect to r regularizes the integral (9.38) (upon first substituting (9.40) and localizing near the ψ -endpoints). By the same reasoning as above, depending on whether L is even or odd (viz. L = 2l or L = 2l + 1), only the integral of one term of (9.39) (and only with k = l) carries singularities. We first look at the case for which only e−i φ(r,ψ+2lπ)+φ(r,θ )) g(r, ψ + 2lπ, θ ) contributes by singularities. Clearly, for a stationary point cos (1 − μ/2)(ψ + 2lπ) + cos (1 − μ/2)θ = 0,

(9.41)

which leads to the condition cos(ψ + θ ) = cos

2 π . 2−μ

(9.42)

There are three cases to consider. Case I. ω = −ω . In this case θ = ψ , so that d φ(r, ψ + 2lπ) + φ(r, θ ) dψ = − 2γ r 1−μ/2 sin(1 − μ/2)(ψ + 2lπ) + sin(1 − μ/2)ψ < 0.

(9.43)

Whence there are no stationary points. Case II. ω = ω . In this case θ = π − ψ so that (9.42) reads ω · ω = 1 = − cos

μ 2 π = cos π . 2−μ 2−μ

This agrees with the “rule” of Theorem 9.3. Case III. ω = Cω . In dimension d 3 the vectors x˜ˆ = ±y/|y| where y = ω − ω · ωω are the only possible critical points of the map ˜ˆ · ω ∈ R. S d−2 x˜ˆ → θ = cos−1 −(cos ψω + sin ψ x) Consequently, for any stationary point, xˆ must belong to the plane spanned by ω and ω (like for d = 2). Let us introduce the angle γ = cos−1 ω · (−ω ). There are three possible relationships to be considered (a) γ = |ψ − θ |, (b) γ = ψ + θ and (c) γ = 2π − (ψ + θ ). For (a), θ = ψ ∓ γ can be substituted into the sum of phases and we compute as in (9.43). Again there will not be any stationary point. For (b) we can use (9.42) to compute ω · ω = − cos γ = − cos

μ 2 π = cos π , 2−μ 2−μ


1913

which agrees with the “rule” of Theorem 9.3. Similarly, for (c) we compute ω · ω = − cos γ = − cos(ψ + θ ) = − cos

2 μ π = cos π . 2−μ 2−μ

Next we look at the case for which only e−i (φ(r,2(l+1)π−ψ)+φ(r,θ )) g(r, 2(l + 1)π − ψ, θ ) contributes to singularities. The stationary point is given by cos (1 − μ/2) 2(l + 1)π − ψ + cos (1 − μ/2)θ = 0,

(9.44)

which leads to the condition cos(ψ − θ ) = cos

2 π . 2−μ

(9.45)

Again there are three cases to consider. Case I. ω = −ω . In this case θ = ψ , so that ω · ω = −1 = − cos

2 μ π = cos π , 2−μ 2−μ

which agrees with Theorem 9.3. Case II. ω = ω . We have θ = π − ψ , so that d φ r, 2(l + 1)π − ψ + φ(r, θ ) dψ = 2γ r 1−μ/2 sin(1 − μ/2) 2(l + 1)π − ψ + sin(1 − μ/2)(π − ψ) > 0;

(9.46)

whence there are no stationary points. Case III. ω = Cω . As in the previous “Case III”, for any stationary point the vector xˆ must belong to the plane spanned by ω and ω . Again we define γ = cos−1 ω · (−ω ), and there are three possible relationships to be considered: (a) γ = |ψ − θ |, (b) γ = ψ + θ and (c) γ = 2π − (ψ + θ ). For (a), ω · ω = − cos γ = − cos(ψ − θ ) = − cos

μ 2 π = cos π , 2−μ 2−μ

which agrees with Theorem 9.3. For (b) and (c) we compute as in (9.46); there are no stationary points. Remarks 9.6. (1) For the above considerations (on the location of singularities), it is not strictly needed that V2 = 0. In fact we can include a V2 as in Condition 2.1 with 2 > 1 − 12 μ and solve transport equations as before using the same phase function (the one determined by V1 only).

1914


(2) Suppose in addition to (1) that V2 is spherically symmetric. Then the operators T = S(0), as ˜ well as T = S(0), obey that RT R −1 = T for all d-dimensional rotations R. This means that the kernel T (ω, ω ) of these operators is a function of ω · ω only. Using the staμ ∈ / Z to write (as a possible continuation of tionary phase method it is feasible for 2−μ ˜ the above analysis) the singular part of the kernel of S(0) as a sum of terms of the form s (ω · ω − ν ± i 0)− 2 a(ω · ω ) (at least for poly-homogeneous V2 ); we shall not elaborate. Our recent paper [7] is devoted to an alternative approach that we find more elementary, and μ ∈Z besides, by that method we can extract the singular part in the exceptional cases 2−μ too. Appendix A. Elements of abstract scattering theory Various versions of stationary scattering theory can be found in the literature. In this appendix we give, in an abstract setting, a self-contained presentation of its elements used in our paper. It is a version of the standard approach contained e.g. in [29], adapted to our paper. In our stationary formulas for the scattering operator we use in addition ideas due to Isozaki–Kitada, see the proof of [19, Theorem 3.3]. A.1. Wave operators Let H0 and H be two self-adjoint operators on a Hilbert space H. We assume that H0 has only continuous spectrum. Throughout this appendix, let Λn , n ∈ N, be a sequence of compact subsets of σ (H0 ) such that Λn is a subset of the interior of Λn+1 , and such that σ (H0 ) \ n Λn ∞ has the Lebesgue measure zero. Pick a sequence hn ∈ Cc (Λn+1 ) with hn = 1 on Λn . Let D := n Ran 1Λn (H0 ); it is dense in H. / σ (H0 ), and We will write R(ζ ) = (H − ζ )−1 and R0 (ζ ) = (H0 − ζ )−1 for ζ ∈ δ (λ) =

= R0 (λ − i )R0 (λ + i ), π((H0 − λ)2 + 2 ) π

> 0.

Note that if I is an interval and f ∈ H, then π R0 (λ − i )R0 (λ + i )f dλ f ,

(A.1)

I

lim

0

R0 (λ − i )R0 (λ + i )f dλ = 1I (H0 )f. π

(A.2)

I

Theorem A.1. Suppose J ± is a densely defined operator whose domain contains D such that Jn± := J ± hn (H0 ) is bounded for any n, and 2 lim J ± ei tH0 f = f 2 ,

t→±∞

f ∈ D.

We also suppose that there exists the wave operator W ± f := lim ei tH J ± e−i tH0 f, t→±∞

f ∈ D.

(A.3)


1915

Then (i) W ± extends to an isometric operator and W ± H0 = H W ± . (ii) For any interval I and f ∈ D,

W ± 1I (H0 )f = lim R(λ ∓ i )J ± R0 (λ ± i )f dλ.

0 π

(A.4)

I

(iii) For any continuous function g : R → C vanishing at infinity, interval I and f ∈ D,

g(λ)R(λ ∓ i )J ± R0 (λ ± i )f dλ. W ± g(H0 )1I (H0 )f = lim

0 π

(A.5)

I

(iv) Suppose in addition that J ± maps D into Dom H . Suppose that T ± is a densely defined operator such that Tn± := T ± hn (H0 ) is bounded for any n and that T ± f = i(H J ± − J ± H0 )f for any f ∈ D. Then we have the following modifications of (A.4) and (A.5): ± J + i R(λ ∓ i )T ± δ (λ)f dλ, (A.6) W ± 1I (H0 )f = lim

0

I

W ± g(H0 )1I (H0 )f = lim

0

g(λ) J ± + i R(λ ∓ i )T ± δ (λ)f dλ.

(A.7)

I

Proof. (i) is well known. Let us prove (ii): By (A.3), ∞

±

W f = lim 2

0

e−2 t e±i tH J ± e∓i tH0 f dt.

0

By the vector-valued Plancherel formula, we obtain

R(λ ∓ i )J ± R0 (λ ± i )f dλ. W ± f = lim

0 π Therefore,

±

W 1I (H0 )f = lim

0 I

R(λ ∓ i )J ± R0 (λ ± i )f dλ π

− lim

0

R(λ ∓ i )J ± R0 (λ ± i )1R\I (H0 )f dλ π

I

+ lim

0 R\I

R(λ ∓ i )J ± R0 (λ ± i )1I (H0 )f dλ. π

(A.8)

1916


We need to show that the last two terms vanish. The proof for both terms is identical. Consider the last one term. Let f1 ∈ H and pick an n so that f = 1Λn (H0 )f . Then (using (A.1) in the last estimation)

± f1 , R(λ ∓ i )J R0 (λ ± i )1I (H0 )f dλ π R\I

Jn±

R\I

R(λ ± i )f1 2 dλ π

C f1 ;

C := Jn±

1 2 R\I

R0 (λ ± i )1I (H0 )f 2 dλ π

R0 (λ ± i )1I (H0 )f 2 dλ π

R\I

1 2

1 2

.

Due to (A.2), C → 0 as → 0. Whence (ii) follows. Let us prove (iii): Let f1 ∈ H and pick an n so that f = 1Λn (H0 )f . Any continuous function g vanishing at infinity can be uniformly approximated by gm , finite linear combinations of characteristic functions of intervals. By (ii) and (A.1),

±

W gm (H0 )1I (H0 )f = lim

0

gm (λ)R(λ ∓ i )J ± R0 (λ ± i )f dλ. π

I

Now

± g (λ) − g(λ) f , R(λ ∓ i

)J R (λ ± i

)f dλ 1 0 π m I

1 1 2 2

R(λ ± i )f1 2 dλ R0 (λ ± i )f 2 dλ sup |gm − g| n π π ± Cm f1 ; Cm := Jn f sup |gm − g|.

J ±

Since Cm → 0 we are done. To prove (iv), we use (iii) and the identity R(λ ∓ i )J ± = J ± + i R(λ ∓ i )T ± R0 (λ ∓ i ).

2

Remark. In the context of our paper, we can take Λn = [ n1 , n]. A.2. Scattering operator Define the scattering operator by S := W + ∗ W − . Clearly, H0 S = SH0 . Theorem A.2. Suppose that the conditions of Theorem A.1 hold. Let the operator J − satisfy lim ei tH J − e−i tH0 f = 0,

t→+∞

f ∈ D.

(A.9)


1917

Then for all f ∈ D Sf = − lim 2π

0

δ (λ)W + ∗ T − δ (λ)f dλ.

(A.10)

Proof. W − f = − lim ei tH J − e−i tH0 − e−i tH J − ei tH0 f t→+∞

t = − lim

t→+∞ −t

∞ = − lim

0

= − lim

0

ei sH T − e−i sH0 f ds

e− t dt

t

ei sH T − e−i sH0 f ds

−t

0

e− |s| ei sH T − e−i sH0 f ds.

Then we use the definition of S and the intertwining property of W + ∗ to obtain Sf = − lim

0

e− |s| ei sH0 W + ∗ T − e−i sH0 f ds.

Finally, we use the vector-valued Plancherel theorem.

2

A.3. Method of rigged Hilbert spaces applied to wave operators Consider a family of separable Hilbert spaces H and Vs , s > 12 , such that Vs is densely and continuously embedded in H, and similarly, Vs is densely and continuously embedded in Vt if s > t. Let Vs∗ be the space dual to Vs , so that we have nested Hilbert spaces Vs ⊆ Vt ⊆ H ⊆ Vt∗ ⊆ Vs∗ ;

s > t.

We remark that H equipped with such a structure is sometimes called a rigged Hilbert space. The following theorem allows us to introduce wave matrices: Theorem A.3. Fix s > t > 12 . Suppose that there exists for almost all λ the limit s- lim δ (λ) =: δ0 (λ) ∈ B Vt , Vt∗ .

→0

Suppose the conditions of Theorem A.1 and that the operators Jn± and R(λ ∓ i )Tn± with λ ∈ Λn and > 0 extend to elements of B(Vt∗ , Vs∗ ). Suppose that for fixed n and almost everywhere in Λn there exists R(λ ∓ i 0)Tn± := s- lim R(λ ∓ i )Tn± ∈ B Vt∗ , Vs∗ .

0

1918


Suppose furthermore that for any n there exists n > 0 such that sup sup δ (λ)V →V ∗ , sup sup R(λ ∓ i )Tn± V ∗ →V ∗ < ∞. t

λ∈Λn < n

t

t

λ∈Λn < n

s

(A.11)

Let I be an interval with I ⊆ Λn for some n, and let f ∈ Vt be given such that f = hn (H0 )f (in particular this means that f ∈ D ∩ Vt ). Then (in terms of an integral of a Vs∗ -valued function), for all g ∈ C(R), W ± g(H0 )1I (H0 )f =

g(λ) Jn± + i R(λ ∓ i 0)Tn± δ0 (λ)f dλ.

(A.12)

I

Proof. We can replace T ± → Tn± , J ± → Jn± in the integrand of (A.7). Then, by the assumptions, it has a pointwise limit as an element of Vs∗ . Due to (A.11), we can apply the dominated convergence theorem. 2 Remark. In the context of our paper, we take Vs := L2,s . A.4. Method of rigged Hilbert spaces applied to the scattering operator The method of rigged Hilbert spaces allows us to introduce scattering matrices: Theorem A.4. Suppose that the conditions of Theorem A.3 hold for some s > t > 12 . Suppose (A.9). Fix r > s. Suppose that for all n ∈ R and > 0 the operators Tn− δ (λ) ∈ B(Vr , Vs ) with a measurable dependence on λ ∈ R. Suppose that for fixed n and almost everywhere in Λn there exists the limit s- lim Tn− δ (λ) =: Tn− δ0 (λ) ∈ B(Vr , Vs ).

→0

Suppose furthermore that for any n there exists n > 0 such that sup sup Tn− δ (λ)V

λ∈R < n

r →Vs

< ∞.

(A.13)

Let I be an interval with I ⊆ Λn for some n, and let f1 ∈ D ∩ Vt and f2 ∈ D ∩ Vr be given such that f1 = 1I (H0 )f1 and f2 = hn (H0 )f2 . Then

f1 , Sf2 = −2π I

f1 , δ0 (λ)Jn+ ∗ Tn− δ0 (λ)f2 dλ

+ 2πi

f1 , δ0 (λ)Tn+ ∗ R(λ + i 0)Tn− δ0 (λ)f2 dλ.

I

Proof. Set r (λ) :=

. π(λ2 + 2 )

We insert (A.12) with g(λ) = r (λ − λ1 ) into (A.10):


f1 , Sf2 = − lim 2π

0

1919

f1 , δ (λ1 )W + ∗ T − δ (λ1 )f2 dλ1

r (λ − λ1 ) f1 , δ0 (λ) Jn+ ∗ − i Tn+ ∗ R(λ + i 0) Tn− δ (λ1 )f2

= − lim 2π

0

I

= − lim 2π

0

f1 , δ0 (λ) Jn+ ∗ − i Tn+ ∗ R(λ + i 0) Tn− δ2 (λ)f2 dλ.

I

In the last step we interchanged integrals using (A.13) and the Fubini theorem, and we used that δ (λ1 )r (λ − λ1 ) dλ1 = δ2 (λ). Then we pass with → 0 using (A.13) and the dominated convergence theorem.

2

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Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces Franz Luef a,b,∗,1 a Department of Mathematics, University of California, Berkeley, CA 94720-3840, United States b Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria

Received 13 October 2008; accepted 1 June 2009 Available online 4 July 2009 Communicated by Alain Connes

Abstract In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita–Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger’s algebra or in Schwartz space. © 2009 Elsevier Inc. All rights reserved. Keywords: Gabor frames; Noncommutative tori; Twisted group C ∗ -algebras; Spectral invariance; Standard Hilbert C ∗ -module frames

* Address for correspondence: Department of Mathematics, University of California, Berkeley, CA 94720-3840, United States. E-mail address: [email protected]. 1 Supported by the Marie Curie Excellence grant MEXT-CT-2004-517154 and the Marie Curie Outgoing fellowship PIOF 220464.


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F. Luef / Journal of Functional Analysis 257 (2009) 1921–1946

1. Introduction We start with a short review of the first theme of our study: projective modules over C ∗ algebras and the relevance of Rieffel’s work on Morita equivalence of operator algebras. Rieffel introduced (strong) Morita equivalence for C ∗ -algebras in [37,38], which we call Rieffel–Morita equivalence. The seminal work of Rieffel was motivated by his formulation of Mackey’s imprimitivity theorem in terms of C ∗ -algebras. Rieffel–Morita equivalence allows a classification of C ∗ -algebras which is weaker than a classification up to isomorphisms. The classification of unital C ∗ -algebras with respect to Rieffel–Morita equivalence requires the construction of projective modules over C ∗ -algebras. During the 1980’s, the research of many operator algebraists concerned projective modules and K-theory for C ∗ -algebras. Another reason for the relevance of projective modules has its origins in Connes’ theory of noncommutative geometry [7]. In noncommutative geometry projective modules over noncommutative C ∗ -algebras appear as noncommutative analogue of vector bundles over manifolds, and projective modules over smooth subalgebras of a C ∗ -algebra are viewed as noncommutative analogue of smooth vector bundles over manifolds [6]. Recall that Connes calls a subalgebra of a C ∗ -algebra smooth if it is stable under the holomorphic function calculus. As a demonstration of the power of noncommutative geometry Connes has constructed projective modules over smooth noncommutative tori in [5]. Rieffel extended Connes’ projective modules over noncommutative tori to higher-dimensional noncommutative tori in [41]. After these groundbreaking results of Connes and Rieffel, projective modules over noncommutative tori found many applications in mathematics and physics, e.g. Bellissard’s interpretation of the integer quantum Hall effect [3], the work of Marcolli and Mathai on the fractional quantum Hall effect, or the relevance of Rieffel–Morita equivalence of operator algebras in mathematical physics [29]. The classification of noncommutative tori up to Morita–Rieffel equivalence relies on the construction of projective modules over noncommutative tori. Rieffel found a general method to construct such in [41]. In [32,33] we have shown that Rieffel’s construction of projective modules over noncommutative tori [41] has a natural formulation in terms of Gabor analysis and we were able to extend his construction to the setting of twisted group algebras. The present work is a continuation of this line of research. We especially want to stress that Connes’ theorem [6] on the correspondence between projective modules over a C ∗ -algebra and projective modules over smooth subalgebras of a C ∗ -algebra for noncommutative tori appears naturally in the research about good window classes in Gabor analysis. In joint work with Manin we have shown the relevance of this interpretation for the understanding of quantum theta functions in [34]. Before we are in the position to describe the main theorems of our investigation we want to give a brief exposition of Gabor analysis, the other theme of our investigation. Gabor analysis arose out of Gabor’s seminal work in [21] on the foundation of information theory. After the groundbreaking work of Daubechies, Grossmann and Meyer, frames for Hilbert spaces have become central objects in signal analysis [9], especially wavelets and Gabor frames. In the last years various other classes of frames have been introduced by workers in signal analysis, e.g. curvelets, ridgelets and shearlets. The relevance of Hilbert C ∗ -modules for signal analysis was pointed out by Packer and Rieffel [35,36] and Wood in [43] for wavelets. A Gabor system G(g, Λ) = {π(λ)g: λ ∈ Λ} consists of a Gabor atom g ∈ L2 (Rd ) and a Rd , where π(λ) denotes the time-frequency shift π(λ)f (t) = e2πiλω ·t f (t − λx ) lattice Λ in Rd × for a point λ = (λx , λω ) in Λ. If there exist finite constants A, B > 0 such that


Af 22

f, π(λ)g 2 Bf 2 2

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(1)

λ∈Λ

holds for all f ∈ L2 (Rd ), then G(g, Λ) is called a Gabor frame for L2 (Rd ). There is a natural operator associated with a Gabor system G(g, Λ), namely the Gabor frame operator Sg,Λ defined as follows Sg,Λ f =

f, π(λ)g π(λ)g,

for f ∈ L2 Rd .

(2)

λ∈Λ

The Gabor frame operator Sg,Λ is a self-adjoint operator on L2 (Rd ). If G(g, Λ) is a Gabor frame for L2 (Rd ), then an element f ∈ L2 (Rd ) has a decomposition with respect to the Gabor system G(g, Λ). More precisely, f=

f, π(λ)(Sg,Λ )−1 g π(λ)g

λ∈Λ

=

λ∈Λ

=

f, π(λ)g π(λ)(Sg,Λ )−1 g

f, π(λ)(Sg,Λ )−1/2 g π(λ)(Sg,Λ )−1/2 g

λ∈Λ

for all f ∈ L2 (Rd ). We call g0 := (Sg,Λ )−1 g the canonical dual Gabor atom and g˜ := (Sg,Λ )−1/2 g the canonical tight Gabor atom of a Gabor frame G(g, Λ). Therefore the invertibility of the Gabor frame operator is essential for the decomposition of a function in terms of Gabor frames. Janssen proved that for Gabor frames G(g, Λ) for L2 (Rd ) with g ∈ S (R d ) their canonical dual and tight Gabor atoms g0 , g˜ are again in S (Rd ). In other words he demonstrated that Gabor frames with good Gabor atoms have dual atoms of the same quality, i.e. all ingredients of the reconstruction formulas are elements of S (Rd ). The key ingredient in the proof of this deep theorem is the so-called Janssen representation of the Gabor frame operator [28], which relies on the fact that a Gabor frame operator Sg,Λ commutes with time-frequency shifts π(λ) for λ in Λ, i.e. π(λ)Sg,Λ = Sg,Λ π(λ), for all λ ∈ Λ. These commutation relations for Gabor frame operators are the very reason for the rich structure of Gabor systems and the differences between Gabor frames and wavelets, see e.g. [23] for further information on this topic. The Janssen representation of a Gabor frame operator allows one to express the Gabor frame operator Sg,Λ in terms of time-frequency shifts of the adjoint lattice Λ◦ . The adjoint lattice Λ◦ consists of all time-frequency shifts of R2d that commute with all time-frequency shifts of Λ, see Section 3 for an extensive discussion. Now, the Janssen representation of the Gabor frame operator Sg,Λ of G(g, Λ) with g ∈ S (Rd ) is the following Sg,Λ f = vol(Λ)−1

g, π λ◦ g π λ◦ f,

(3)

λ◦ ∈Λ◦

where vol(Λ) denotes the volume of a fundamental domain of Λ. The Janssen representation links the original Gabor system G(g, Λ) with a dual system with respect to the adjoint lattice in such a way that the original Gabor frame operator becomes a superposition of time-frequency

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shifts over the adjoint lattice Λ◦ acting on the function f . Therefore Janssen introduced the following Banach algebras [28] for s 0, where the multiplication is given by twisted convolution, the so-called noncommutative Wiener algebras: A1s (Λ, c) =

2 s/2 a(λ) 1 + |λ| a(λ)π(λ): 0. Since a∗∗ 1v Ca∗ 1v C 2 a1v , and a∗∗ 1v = a1v , it follows that C = 1 and v(−λ) = v(λ). This completes our proof. 2 We refer the interested reader to the survey article [26] of Gröchenig for a thorough treatment of weights in time-frequency analysis. Now, we want to represent 1v (Λ, c) as superposition of time-frequency shifts on L2 (Rd ). For (x, ω) ∈ R2d we define the time-frequency shift π(x, ω)f (t) of f by π(x, ω)f (t) = Mω Tx f (t), where Tx f (t) = f (t − x) is the translation by x ∈ Rd and Mω f (t) = e2πit·ω f (t) is the modulation by ω ∈ Rd .


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Observe, that (x, ω) → π(x, ω) is a projective representation of R2d on L2 (Rd ). This is essentially due to the following commutation relation between translation and modulation operators: Mω Tx = e2πixω Tx Mω

for (x, ω) ∈ R2d .

(8)

The commutation relation (8) yields a composition law for time-frequency shifts π(x, ω) and π(y, η): π(x, ω)π(y, η) = c (x, ω), (y, η) c (y, η), (x, ω) π(y, η)π(x, ω) = csymp (x, ω), (y, η) π(y, η)π(x, ω),

(9) (10)

where c denotes the continuous 2-cocycle c on R2d defined by c((x, ω), (y, η)) = e2πiy·ω for (x, ω), (y, η) ∈ R2d and csymp is an anti-symmetric bicharacter or symplectic bicharacter on R2d . More explicitly, csymp is given by csymp (x, ω), (y, η) = e2πi(y·ω−x·η) = e2πiΩ((x,ω),(y,η)) ,

(11)

where Ω((x, ω), (y, η)) = y · ω − x · η is the standard symplectic form on R2d . Rd the mapping of λ → π(λ) is a projective representation of Λ on For a lattice Λ in Rd × 2 d L (R ). Now, a projective representations of a lattice Λ in R2d gives a non-degenerate involutive representation of 1v (Λ, c) by πΛ (a) :=

a(λ)π(λ) for a = a(λ) ∈ 1v (Λ).

λ∈Λ

In other words, πΛ (a b) = πΛ (a)πΛ (b) and πΛ (a∗ ) = πΛ (a)∗ . Moreover, this involutive representation of 1v (Λ, c) is faithful, i.e. πΛ (a) = 0 implies a = 0. We refer the reader to [41] for a proof of the last assertion. We denote the image of the map a → πΛ (a) by A1v (Λ, c). More explicitly,

A1v (Λ, c) = A ∈ B L2 Rd : A = a(λ)π(λ), a1v < ∞ λ

is an involutive Banach algebra with respect to the norm AA1v (Λ) = λ |a(λ)|v(λ). We call A1v (Λ, c) the noncommutative Wiener algebra because it is the noncommutative analogue of Wiener’s algebra of Fourier series with absolutely convergent Fourier coefficients. The involutive Banach algebra 1v (Λ, c) is not a C ∗ -algebra. There exists a canonical construction, which associates to an involutive Banach algebra A a C ∗ -algebra C ∗ (A), the universal enveloping C ∗ -algebra of A. If a ∈ 1v (Λ, c), then one defines a C ∗ -algebra norm aC ∗ (Λ,c) as the supremum over the norms of all involutive representations of 1v (Λ, c) and the twisted group C ∗ -algebra C ∗ (Λ, c) as the completion of 1v (Λ, c) by · C ∗ (Λ,c) . In the literature C ∗ (Λ, c) is also known as noncommutative torus or quantum torus. If we represent C ∗ (Λ, c) as a subalgebra of bounded operators on L2 (Rd ), then A1v (Λ, c) is a dense subalgebra of C ∗ (Λ, c). Now we use the noncommutative Wiener algebras A1v (Λ, c) as building blocks for a class ∗ of subalgebras A∞ v (Λ, c) of C (Λ, c) that are noncommutative analogues of smooth functions on a compact manifold. More concretely, we want to deal with smooth noncommutative tori

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with respect to a general submultiplicative weight. If v is a submultiplicative weight, then we 1 call A∞ (Λ, c) = v s0 Av s (Λ, c) a generalized smooth noncommutative torus. The subalgebra ∗ (Λ, c) is a complete locally convex algebra whose topology is defined by a A∞ (Λ, c) of C v family of submultiplicative seminorms { · A1s | s 0} with v

AA1s = v

a(λ)v s (λ)

for A ∈ A∞ v (Λ, c).

λ∈Λ

In the literature a complete locally convex algebra A equipped with a family of submultiplicative seminorms is called a locally convex m-algebra or m-algebra. It is well known that m-algebras are precisely the projective limits of Banach algebras. An important class of m-algebras are Frechet algebras with submultiplicative seminorms. By construction several special properties of A∞ v (Λ, c) are consequences of the structure of 1 Av s (Λ, c), e.g. the spectral invariance in C ∗ (Λ, c). Recall that a unital Banach algebra A is spectrally invariant in a unital Banach algebra B with common unit, if for A ∈ A with A−1 ∈ B implies A−1 ∈ A. The spectral invariance of A1v (Λ, c) in C ∗ (Λ, c) was investigated by Gröchenig and Leinert in [25]. Their main result shows that this problem only depends on properties of the weight v, see [26] for the following formulation: Theorem 2.2 (Gröchenig–Leinert). Let Λ be a lattice in R2d . Then the noncommutative Wiener algebra A1v (Λ, c) is spectrally invariant in C ∗ (Λ, c) if and only if v is a GRS-weight, i.e. lim v(nλ)1/n = 1 for all λ ∈ Λ. For the proof we refer the reader to [26]. As a consequence we get the spectral invariance of A∞ (Λ, c) in C ∗ (Λ, c). Corollary 2.3. Let Λ be a lattice in R2d . Then the smooth noncommutative torus A∞ v (Λ, c) is spectrally invariant in C ∗ (Λ, c) if and only if v is a GRS-weight. Proof. Note that we have A1v s (Λ, c) ⊂ A1v s−1 (Λ, c) ⊂ A1 (Λ, c) ⊂ C ∗ (Λ, c). Therefore A1v s (Λ, c) is spectrally invariant in C ∗ (Λ, c) for all s. Consequently, A∞ v (Λ, c) is spectrally invariant in C ∗ (Λ, c). 2 Remark. A submultiplicative weight grows at most exponentially and a GRS-weight has at most sub-exponential growth. For an extensive discussion of weights we refer the reader to Chapter 11 in [23] and to [26]. ∗ By the above remark the spectral invariance of A1v (Λ, c) and A∞ v (Λ, c) in C (Λ, c) forces v to be sub-exponential. Therefore in the case that v grows faster than a polynomial, the smooth ∞ noncommutative torus A∞ v (Λ, c) is a subspace of A (Λ, c). An important fact about Gabor frames is that Gabor frames G(g, Λ) with g in the Schwartz space S (Rd ) provide a discrete description of the Schwartz space S (Rd ) in terms of its Gabor coefficients. Namely, f ∈ S (Rd ) if and only if (f, π(λ)g) ∈ S (Λ). The key to such statements is that the Janssen representation of the Gabor frame operator is in A∞ (Λ, c). In an analogous manner the classes A∞ v (Λ, c) for v that grows faster than a polynomial provide a


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description of subspaces Sv (Rd ) of S (Rd ) in terms of Gabor frames, see Section 3 for a further discussion of this aspect. The classes Sv (Rd ) for v that grows faster than a polynomial are natural spaces of test functions for ultra-distributions [8]. The last theorem has various applications in Gabor analysis, see [16,25], most notably that the Gabor frame operator has the same spectrum on all modulation spaces for good Gabor atoms and that the canonical dual and tight Gabor window for good Gabor systems have the same quality as the Gabor atom. These results are based on two observations about spectrally invariant Banach algebras and m-convex algebras A in C ∗ (Λ, c): (1) The spectrum σA (A) = σC ∗ (Λ,c) (A) for A ∈ A, where σA (A) = {z ∈ C: (z − A)−1 does not exist in A} is the spectrum of A ∈ A. (2) If A is spectrally invariant in C ∗ (Λ, c), then A is stable under holomorphic function calculus of C ∗ (Λ, c). Now, we want to explore the consequences of Gröchenig–Leinert’s Theorem 2.2 for an understanding of the deeper properties of A1v (Λ, c) and A∞ v (Λ, c), e.g. their topological stable rank. These results will allow us to draw some important conclusions about the deeper structure of Gabor frames in Section 4. In [40] the topological stable rank of a Banach algebra was introduced as a noncommutative analogue of the notion of covering dimension of a compact space. In the remaining part of this section we derive some upper bounds for the topological stable rank of the noncommutative Wiener algebras and smooth noncommutative tori. The left(right) topological stable rank of a unital topological algebra A, denoted by ltsr(A) (rtsr(A)), is the smallest number n such that the set of n-tuples of elements of A which generate A as a left(right) ideal is dense in An . We denote the set of n-tuples of elements of A which generate An as a left(right) ideal by Lgn (A) (Rgn (A)). If ltsr(A) = rtsr(A), then we call it the topological stable rank of A, and we denote it by tsr(A). Proposition 2.4. Let Λ be a lattice in R2d and let v be a GRS-weight. Then 1 ∗ tsr A∞ v (Λ, c) = tsr Av (Λ, c) = tsr C (Λ, c) . 1 Furthermore, we have that tsr(A∞ v (Λ, c)) = tsr(Av (Λ, c)) 2d + 1.

Proof. Recall that our assumptions on v, i.e. v(−λ) = v(λ) for all λ ∈ Λ, implies that A1v (Λ, c) has a continuous involution. If A is a unital Banach algebra or m-convex algebra with a continuous involution, then Rieffel proved in [40] that ltsr(A) = rtsr(A). Now, we invoke a result of Badea that tsr(A) = tsr(B) if A is spectrally invariant in B [1]. Since A1v (Λ) is spectrally invariant in C ∗ (Λ, c) if v satisfies the GRS-condition [26]. Finally, the upper bound for the topological stable rank of C ∗ (Λ, c) is due to Rieffel, see [40,41]. This completes the proof. 2 A well-known fact about topological stable rank is that for a topological algebra A with topological stable rank-one the invertible elements are dense in A, e.g. [40]. By the preceding 1 theorem tsr(C ∗ (Λ, c)) = 1 implies tsr(A∞ v (Λ, c)) = tsr(Av (Λ, c)). It is quite a challenge to ∗ determine the topological stable rank of a specific C -algebra. In the case of noncommutative tori Putnam has shown that the irrational noncommutative 2-torus has topological stable rankone. Later Blackadar, Kumjian and Roerdam extended this result to simple noncommutative tori in [4]. In our setting this amounts to the following: C ∗ (Λ, c) is simple if and only if Λ is completely irrational in the sense that for every λ ∈ Λ there is a μ ∈ Λ such that Ω(λ, μ) is an irrational number.

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Theorem 2.5. Let Λ be completely irrational and v a GRS-weight. Then tsr(A∞ v (Λ, c)) = tsr(A1v (Λ)) = 1. Proof. The GRS-condition is equivalent to the spectral invariance of A1v (Λ, c) in C ∗ (Λ, c) and 1 ∗ therefore by Badea’s result [1] to tsr(A∞ v (Λ, c)) = tsr(Av (Λ, c)) = tsr(C (Λ, c)). By Theo∗ rem 1.5 in [4] we know that tsr C (Λ, c) = 1 for Λ completely irrational. That completes the argument. 2 In Section 3 our main theorems deal with the construction of smooth projective modules over A1v (Λ, c) and A∞ v (Λ, c) in the sense of Connes [6]. In other words, we have to construct projections in the algebras Mn (A1v (Λ, c)) and Mn (A∞ v (Λ, c)) of n × n matrices with entries in A1v (Λ, c) or A∞ (Λ, c), respectively. v In [30] Leptin has proved that Mn (A) = Mn ⊗ A is spectrally invariant in Mn (B) if A is a spectrally invariant Banach subalgebra of B. Note that it is elementary to extend Leptin’s result to m-convex algebras. Connes obtained this result for Frechet algebras independently, see [6], see [42] for a discussion of this theorem for m-convex algebras. These observations and the char∗ acterization of the spectral invariance of A1v (Λ, c) and A∞ v (Λ, c) in C (Λ, c), see Theorem 2.2 and our Proposition 2.4, yield the following result. Theorem 2.6. Let Λ be a lattice in R2d . Then Mn (A1v (Λ, c)) and Mn (A∞ v (Λ, c)) are spectrally invariant in Mn (C ∗ (Λ, c)) if and only if v is a GRS-weight. Following Connes we deduce from the previous theorem the density result for K-groups, see [7]. Corollary 2.7. Let Λ be a lattice in R2d and v a GRS-weight. Then the inclusion i of ∗ A1v (Λ, c) and A∞ v (Λ, c) into C (Λ, c), respectively, gives an isomorphism of K0 -groups i0 : ∗ K0 (A) → K0 (C (Λ, c)) and of K1 -groups i1 : K0 (A) → K0 (C ∗ (Λ, c)) for A = A1v (Λ, c) and A = A∞ v (Λ, c). If the weight v grows at most polynomially, then the preceding corollary corresponds to the well-known result for the smooth noncommutative torus A∞ (Λ, c) [5]. One of the main goals of this section was to demonstrate that Connes’s results on the spectral invariance of the smooth noncommutative torus A∞ (Λ, c) in C ∗ (Λ, c) is the special case A∞ v (Λ, c) for the radial weight c ) of C ∗ (Λ, c) and that these algebras v(λ) = (1 + |λ|2 ) of a general class of subalgebras A∞ (Λ v are intimately tied with recent developments in time-frequency analysis. 3. Projective modules over noncommutative tori and noncommutative Wiener algebras In [32] we have shown that Feichtinger’s algebra S0 (R) provides a convenient class of functions for the construction of Hilbert C ∗ (Λ, c)-modules. In thepresent section we extend the results in [32] to modulation spaces Mv1 (Rd ) and to Sv (Rd ) = s0 Mv1s (Rd ) for a non-trivial submultiplicative weight v. At this point it will be convenient to formally introduce an important class of function spaces, invented by Feichtinger in [12]. Modulation spaces have found many applications in harmonic analysis and time-frequency analysis, see the interesting survey article [13] for an extensive bib-


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liography. If g is a window function in L2 (Rd ), then the short-time Fourier transform (STFT) of a function or distribution f is defined by Vg f (x, ω) = f, π(x, ω)g =

f (t)g(t − x)e−2πix·ω dt.

(12)

Rd

The STFT of f with respect to the window g measures the time-frequency content of a function f . Modulation spaces are classes of function spaces, where the norms are given in terms of integrability or decay conditions of the STFT. In the present section we restrict our interest to Feichtinger’s algebra M 1 (Rd ) and its weighted versions Mv1 (Rd ) for a submultiplicative weight v. We introduce the full class of modulation spaces in Section 4, where we interpret the projective modules over C ∗ (Λ, c) of the present section as multi-window Gabor frames over modulation spaces. In time-frequency analysis the modulation space Mv1 (Rd ) has turned out to be a good class of 2 windows. If ϕ(t) = e−πt is the Gaussian, then the modulation space Mv1 (Rd ) is the space

Mv1 Rd = f ∈ L2 Rd : f Mv1 := Vϕ f (x, ω)v(x, ω) dx dω < ∞ . Rd

The space M 1 (Rd ) is the well-known Feichtinger algebra S0 (Rd ), which he introduced in [10] as the minimal strongly character invariant Segal algebra. Rd . Then a Let v be a submultiplicative weight on R2d such that v is not constant on Rd and natural generalization of Schwartz’s class of test functions is given by Sv Rd := Mv1s Rd s0

with seminorms f M 1s for s 0. If v is of at most polynomially growth, then Sv (Rd ) is v

the Schwartz class of test functions S (Rd ) [23]. For a submultiplicative weight v that grows faster than a polynomial the Gelfand–Shilov space S 1 , 1 (Rd ) is contained in Sv (Rd ), see [8]. 2 2

In the main results about projective modules over C ∗ (Λ, c) the space Mv1 (Rd ) serves as preequivalence bimodule between A1v (Λ, c) and A1v (Λ◦ , c) and in an analogous manner Sv (Rd ) is ∞ ◦ shown to be a pre-equivalence bimodule between A∞ v (Λ, c) and Av (Λ , c). 1 d d The spaces Mv (R ) and Sv (R ) have many useful properties, see [12,23]. In the following proposition we collect those facts which we need in the construction of the projective modules over C ∗ (Λ, c). Proposition 3.1. Let v be a non-trivial submultiplicative weight. (1) For g ∈ Mv1 (Rd ) we have π(y, η)g ∈ Mv1 (Rd ) for (y, η) ∈ R2d with π(y, η)g

Mv1

v(y, η)gMv1 .

1 (R2d ). (2) If f, g are in Mv1 (Rd ), then Vg f ∈ Mv⊗v

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(3) Let a = (a(λ)) be in 1v (Λ) and g ∈ Mv1 (Rd ). Then a(λ)π(λ)g λ∈Λ

Mv1

λ∈Λ a(λ)π(λ)g

is in Mv1 (Rd ) with

a1v gMv1 .

(4) If f, g are in Mv1 (Rd ), then (Vg f (λ)) ∈ 1v (Λ). Corollary 3.2. Let v be a submultiplicative weight. (1) For g ∈ Sv (Rd ) we have π(y, η)g ∈ Sv (Rd ) for (y, η) ∈ R2d with π(y, η)g

Mv1s

v(y, η)gM 1s

for all s 0.

v

(2) If f, g are in Sv (Rd ), then Vg f∈ Sv⊗v (R2d ). (3) Let a = (a(λ)) be in Sv (Λ) = s0 1v s (Λ) and g ∈ Sv (Rd ). Then λ∈Λ a(λ)π(λ)g is in Sv (Rd ) with a(λ)π(λ)g λ∈Λ

Mv1s

a1s gM 1s , v

v

for all s 0.

(4) If f, g are in Sv (Rd ), then (Vg f (λ)) ∈ Sv (Λ). We refer the reader to [23] for a proof of these statements about g ∈ Mv1 (Rd ) and Sv (Rd ). We continue our presentation with a brief discussion of the Fundamental Identity of Gabor analysis, which is an identity for the product of two STFTs. This identity is the essential tool in Rieffel’s construction of projective modules over noncommutative tori in [41]. Later, Janssen, Tolimieri, Orr observed independently the relevance of this identity in Gabor analysis, therefore Janssen called it the Fundamental Identity of Gabor analysis (FIGA). Feichtinger and Kozek generalized these results in [17] to Gabor frames with lattices in elementary locally compact abelian groups, because they realized that the Poisson summation formula for the symplectic Fourier transform is the main ingredient in the proof of the FIGA. Actually, in the approach of Feichtinger and Kozek to FIGA they have rediscovered the main arguments of Rieffel’s discussion in [41]. In [18] we have extended the results of Feichtinger, Kozek and Rieffel in a discussion of dual pairs of Gabor windows. In the following we present a slightly more general version of the main theorem in [18]. We already mentioned that the FIGA follows from an application of the Poisson summation formula for the symplectic Fourier transform. In the symplectic version of the Poisson summation formula the adjoint lattice of a lattice Λ is the substitute of the dual lattice in the Euclidean Poisson summation formula. More precisely, if Λ is a lattice in R2d , then in [17] Feichtinger and Kozek defined its adjoint lattice by Λ◦ = (x, ω) ∈ R2d : csymp (x, ω), λ = 1 for all λ ∈ Λ . In [41] Rieffel denoted the lattice Λ◦ by Λ⊥ and called it the orthogonal lattice.


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Theorem 3.3 (FIGA). Let Λ be lattice in R2d . Then for f1 , f2 , g1 , g2 ∈ Mv1 (Rd ) or in Sv (Rd ) the following identity holds f1 , π(λ)g1 π(λ)g2 , f2 = vol(Λ)−1 f 1 , π λ ◦ f 2 π λ ◦ g2 , g1 ,

(13)

λ◦ ∈Λ◦

λ∈Λ

where vol(Λ) denotes the volume of a fundamental domain of Λ. The case v(x, ω) = (1 + |x|2 + |ω|2 ) for Sv (Rd ) was proved by Rieffel in Proposition 2.11 in [41]. In [32] we observed that Rieffel’ construction holds for M 1 (Rd ). In the present investigation we want to emphasize that the method of Rieffel also works for Mv1 (Rd ) and Sv (Rd ) and provides new classes of pre-Hilbert C ∗ (Λ, c)-modules. We define a left action of A1v (Λ, c) on Mv1 (Rd ) by πΛ (a) · g =

a(λ)π(λ) g

for a ∈ 1v (Λ), g ∈ Mv1 Rd .

λ∈Λ

If f, g are in Mv1 (Rd ), then (Vg f (λ)) is in 1v (Λ). Consequently, we have that Λ f, g =

f, π(λ)g π(λ)

λ∈Λ

is an element of A1v (Λ, c). The crucial observation is that Λ f, g is a A1v (Λ, c)-valued inner product. In the following theorem we prove that Mv1 (Rd ) becomes a full left Hilbert C ∗ (Λ, c)1/2 module Λ V when completed with respect to the norm Λ f = Λ f, f op for f ∈ Mv1 (Rd ). Theorem 3.4. Let Λ be a lattice in R2d . If v is a submultiplicative weight, then Mv1 (Rd ) is a left pre-inner product A1v (Λ, c)-module for the left action of A1v (Λ, c) on Mv1 (Rd ) πΛ (a) · g =

a(λ)π(λ)g

for a = a(λ) ∈ 1v (Λ), g ∈ Mv1 Rd ,

λ∈Λ

the A1v (Λ, c)-inner product Λ f, g =

f, π(λ)g π(λ)

for f, g ∈ Mv1 Rd

λ∈Λ 1/2

and the norm Λ f = Λ f, f op . Proof. We briefly sketch the main steps of the proof, since the discussion follows similar lines as in [32,41]. (a) If a ∈ 1v (Λ) and g ∈ Mv1 (Rd ), then a → πΛ (a) · g is in Mv1 (Rd ), see Proposition 3.1. Therefore, the left action πΛ (a) · g is a well defined and bounded map on Mv1 (Rd ). (b) The compatibility of the left action with the A1v (Λ, c)-inner product amounts to the following identity

1934


Λ

πΛ (a) · f, g = πΛ (a)Λ f, g

for all f, g ∈ Mv1 (Rd ) and a ∈ 1v (Λ), which follows from the following computation: πΛ (a) · f, g = πΛ (a) · f, π(μ)g π(μ)

Λ

μ∈Λ

=

a(λ) π(λ)f, π(μ)g π(μ)

μ∈Λ λ∈Λ

=

a(λ) f, π(λ)∗ π(μ)g π(μ)

λ,μ

=

a(λ) f, π(μ − λ)g π(μ)c(λ − μ, μ)

λ,μ

=

a Vg f (μ)π(μ) = πΛ (a)Λ f, g.

μ

Therefore, the compatibility condition is actually a statement about the twisted convolution of (Vg f (λ)) and a in 1v (Λ, c). (d) Λ f, g∗ = Λ g, f amounts to

∗ f, π(λ)g π(λ) = f, π(λ)g π(λ)∗

λ

λ

=

f, π(λ)g c(λ, λ)π(−λ)

λ

=

λ

f, π(−λ)g c(λ, λ)π(λ)

=

λ

=

π(λ)f, g π(λ)

g, π(λ)f π(λ) = Λ g, f .

λ

The previous argument is equivalent to the fact that the involution of (Vg f (λ)) is (Vf g(λ)) in 1 (Λ, c). (e) The positivity of Λ f, f for f ∈ Mv1 (Rd ) in C ∗ (Λ, c) is a non-trivial fact. It is a consequence of the Fundamental Identity of Gabor analysis, see [32,41]. Recall, that the representation of A1v (Λ, c) is faithful on L2 (Rd ). Therefore, it suffices to verify the positivity of Λ f, f in B(L2 (Rd )). Consequently, we have to check the positivity just for the dense subspace Mv1 (Rd ):

Λ f, f · g, g

=

f, π(λ)f π(λ)g, g

λ∈Λ

= vol(Λ)−1

λ◦ ∈Λ◦

f, π(λ)g π(λ)g, f 0.


1935

In the previous statement we invoked FIGA as in [41]. The statements (a)–(e) yield that Mv1 (Rd ) 1/2 becomes a Hilbert C ∗ (Λ, c)-module when completed with respect to Λ f = Λ f, f op . 1 d ∗ ∗ Since the ideal span{Λ f, g: f, g ∈ Mv (R )} is dense in C (Λ, c), the Hilbert C (Λ, c)-module is full. 2 Suppose A is a unital C ∗ -algebra. If (A V , A ·,·) and (A W, A ·,·) are left Hilbert Amodules, then a map T : A V → A W is adjointable, if there is a map T ∗ : A W → A V such that A Tf, g = A f, T

∗

g

for all f, g ∈ A V .

We denote the set of all adjointable maps from A V to A W by L(A V , A W ). If we view C ∗ (Λ, c) as a full left Hilbert C ∗ (Λ, c)-module, then the map CgΛ f := Λ f, g is an adjointable operator from Λ V to C ∗ (Λ, c) and its adjoint is the map DgΛ (a) := πΛ (a) · g. More precisely, A1v (Λ, c) is a left inner product A1v (Λ, c)-module with respect to πΛ (a) · πΛ (b) = πΛ (a)πΛ (b) and C ∗ (Λ,c) πΛ (a), πΛ (b) = πΛ (a)πΛ (b)∗ for a, b ∈ 1v (Λ, c) and the modulenorm of πΛ (a) equals the operator norm of πΛ (a). If we complete the inner product A1v (Λ, c)module with respect to this norm, then we obtain a full left Hilbert C ∗ (Λ, c)-module C ∗ (Λ,c) V . Lemma 3.5. The map CgΛ is an element of L(Λ VC ∗ (Λ,c) V ) and DgΛ is in L(C ∗ (Λ,c) V , Λ V ). Furthermore, CgΛ and DgΛ are adjoints of each other. Proof. By the faithfulness of the representation of C ∗ (Λ, c) it suffices to check the statement for the dense subalgebra A1v (Λ, c). Let a ∈ 1v (Λ, c) and f, g ∈ Mv1 (Rd ). Then we have, that C ∗ (Λ,c)

πΛ (a), CgΛ f = πΛ (a)Λ g, f = Λ πΛ (a) · g, f = Λ DgΛ (a), f .

2

The preceding lemma is a Hilbert C ∗ -module analog of the well-known fact that the coefficient mapping Cg,Λ and the synthesis mapping Dg,Λ are adjoint operators for a Gabor system G(g, Λ), π(λ)g)λ is a map from L2 (Rd ) to 2 (Λ) and the synthesis mapping is where Cg,Λ f := (f, defined by Dg,Λ a = λ∈Λ a(λ)π(λ)g for a ∈ 2 (Λ) and maps 2 (Λ) into L2 (Rd ). Therefore, the mappings CgΛ and DgΛ are noncommutative analogs of the coefficient and synthesis mappings of a Gabor system. In the Hilbert space setting a central role is played by the Gabor frame operator Sg,Λ = Dg,Λ ◦ Cg,Λ , i.e. Sg,Λ f =

f, π(λ)g π(λ)g

for f ∈ L2 Rd .

λ∈Λ

Analogously we define the noncommutative frame operator SgΛ as the composition DgΛ ◦ CgΛ , which is by definition a C ∗ (Λ, c)-module map. If f, g ∈ Mv1 (Rd ), then Sg,Λ f = Λ f, g · g = πΛ (Vg f ) · g.

(14)

In other words, the Gabor frame operator on Mv1 (Rd ) may be considered as a Hilbert C ∗ (Λ, c)module map. Furthermore, the Gabor frame operator is a so-called rank-one Hilbert C ∗ (Λ, c)module operator. Recall, that on a left Hilbert C ∗ -module (A V , A ·,·) a rank-one operator

1936


A is defined by Θ A f := f, gh. Consequently, S Λ Θg,h g,Λ f is the rank-one operator Θg,g f . A g,h Λ A general rank-one operator Θg,h is given by Λ f= Θg,h

f, π(λ)g π(λ)h

for f, g, h ∈ Λ V ,

(15)

λ∈Λ

which in Gabor analysis are called Gabor frame type operators and denoted by Sg,h,Λ . In the next section we will have to deal with finite sums of rank-one operators in our description of projective modules over C ∗ (Λ, c). At the moment we want to take a closer look at adjointable operators on Λ V . By definition, a map T on Λ V is adjointable if there exists a map T ∗ on Λ V such that Λ Tf, g = Λ f, T

∗

g,

f, g ∈ Λ V .

More explicitly, the last equation amounts to Tf, π(λ)g = f, π(λ)T ∗ g .

λ∈Λ

λ∈Λ

If we restrict our interest to elements of the inner product A1v (Λ, c)-module, then an adjointable C ∗ (Λ, c)-module map is bounded on 1v (Λ), because every adjointable module map is bounded and the operator norm of the module map can be controlled by the 1v -norm. Rieffel made the following crucial observation in [41], that C ∗ (Λ, c) and the opposite algebra of C ∗ (Λ◦ , c) are closely related, namely they are Morita–Rieffel equivalent. We recall the notion of Morita–Rieffel equivalence for C ∗ -algebras after the discussion of right Hilbert C ∗ -modules over the opposite algebra of C ∗ (Λ◦ , c). Note that opposite time-frequency shifts π(x, ω)op are given by Tx Mω , which satisfy Tx Mω = e−2πix·ω Mω Tx = e−2πix·ω π(x, ω). Lemma 3.6. The opposite algebra of C ∗ (Λ◦ , c) is C ∗ (Λ◦ , c). Theorem 3.4 gives the completion of Mv1 (Rd ) the structure of a left Hilbert C ∗ (Λ◦ , c)-module ∗ ◦ Λ◦ V with respect to the left action and C (Λ , c)-valued inner product ·,·Λ◦ defined above, but we need a right module structure. There is a well-known procedure, which we describe in the following Lemma 3.7. Let A be a C ∗ -algebra and Aop its opposite C ∗ -algebra. Furthermore, we denote by V op the opposite vector space structure on a Banach (Frechet) space V . We have a one–one correspondence between A-left modules V and Aop -right modules V op . Lemma 3.7. Let A be a C ∗ -algebra and (A V , A ·,·) a left Hilbert A-module. Then the opposite module V op is a right Hilbert module for the opposite algebra Aop with the Aop -valued inner product ·,·Aop : V op × V op → Aop given by (f op , g op ) → A g, f op . Proof. Let f op , g op ∈ V op and Aop ∈ Aop . Then ·,·Aop is compatible with the right action of V op :

op f op , g op Aop Aop = A f op , (Ag)op = A Ag, f = AA g, f = A g, f op Aop = f op , g op Aop Aop .


1937

Since the compact A-module operators are defined in terms of rank-one operators, we have to A f = Θ Aop op demonstrate that Θg,h hop ,g op f . By definition we have that op A Θg,h f = A f, hg = g op A f, h = g op hop , f op = ΘhAop ,g op f op .

2

Therefore, Lemma 3.7 gives the following right action of A1v (Λ◦ , c) on Mv1 (Rd ) by f · πΛ◦ (b) = vol(Λ)−1

∗ π λ◦ f b λ◦ ,

(16)

λ◦ ∈Λ◦

and C ∗ (Λ, c)-valued inner product ·,·Λ◦ : f, gΛ◦ = vol(Λ)−1

∗ π λ◦ π λ◦ g, f .

λ◦ ∈Λ◦

We summarize all these observations and statements in the following theorem: Theorem 3.8. Let Λ be a lattice in R2d . If v is a submultiplicative weight, then the completion of Mv1 (Rd ) becomes a full right Hilbert C ∗ (Λ◦ , c)-module VΛ◦ for the right action of A1v (Λ◦ , c) on Mv1 (Rd ) g · πΛ◦ (b) = vol(Λ)−1

∗ π λ◦ g b λ◦

for b = b λ◦ ∈ 1v Λ◦ , g ∈ Mv1 Rd ,

λ◦ ∈Λ◦

with the C ∗ (Λ◦ , c)-inner product f, gΛ◦ = vol(Λ)−1

∗ π λ◦ g, π λ◦ f for f, g ∈ Mv1 Rd

λ◦ ∈Λ◦ 1/2

when completed with respect to the norm f Λ◦ = f, f Λ◦ op . Rieffel introduced in [38] the notion of strong Morita equivalence for C ∗ -algebras, which we state in the following definition. Definition 3.9. Let A and B be C ∗ -algebras. Then an A–B-equivalence bimodule A V B is an A–B-bimodule such that (a) A V B is a full left Hilbert A-module and a full right Hilbert B-module; (b) for all f, g ∈ A V B , A ∈ A and B ∈ B we have that A · f, gB = f, A∗ · gB

and A f · B, g = A f, g · B ∗ ;

(c) for all f, g, h ∈ A V B , A f, g · h = f

· g, hB .

1938


The C ∗ -algebras A and B are called Morita–Rieffel equivalent if there exists an A–B equivalence bimodule. In words, condition (b) in Definition 3.9 says that A acts by adjointable operators on VB and that B acts by adjointable operators on A V , and condition (c) in Definition 3.9 is an associativity condition between the A-inner product and the B-inner product. Theorems 3.4 and 3.8 give an C ∗ (Λ, c)–C ∗ (Λ◦ , c) equivalence bimodule Λ VΛ◦ . The associativity condition between Λ ·,· and ·,·Λ◦ is a statement about rank-one Hilbert C ∗ -module operators for C ∗ (Λ, c) and C ∗ (Λ◦ , c), which in Gabor analysis is known as the Janssen representation of a Gabor frame-type operator. Theorem 3.10. Let Λ be a lattice in R2d . Then for all f, g, h ∈ Mv1 (Rd ) Λ f, g · h = f

· g, hΛ◦ ,

(17)

or in terms of Gabor frame-type operators: Sg,h,Λ f = vol(Λ)−1 Sh,f,Λ◦ g.

(18)

Proof. The identity (17) is equivalent to

Λ f, g · h, k

= f · g, hΛ◦ , k

for all k ∈ Mv1 (Rd ). More explicitly, the associativity condition reads as follows f, π(λ)g π(λ)h, k = vol(Λ)−1 f, π λ◦ k π λ◦ h, g .

λ◦ ∈Λ◦

λ∈Λ

In other words, the associativity condition is the Fundamental Identity of Gabor analysis. Therefore, Theorem 3.3 gives the desired result. 2 The observation, that the associativity condition for Λ ·,· and ·,·Λ◦ is the Fundamental Identity of Gabor analysis, allows one to link projective modules over noncommutative tori and Gabor frames for modulation spaces. The last step in the construction of an equivalence bimodule between C ∗ (Λ, c) and C ∗ (Λ◦ , c) is to establish that C ∗ (Λ, c) acts by adjointable maps on C ∗ (Λ◦ , c), which in the present setting is a non-trivial task. The main difficulty lies in the fact, that we actually have just a pre-equivalence bimodule. Therefore, in addition to the condition (b) in Definition 3.9 one has to check that the actions are bounded:

πΛ (a) · g, πΛ (a) · g

Λ◦

2 πΛ (a)op gΛ◦

and 2 g · πΛ (b), g · πΛ (b) πΛ (b)op Λ g

Λ


1939

for all a ∈ 1v (Λ), b ∈ 1v (Λ◦ ) and g ∈ Mv1 (Rd ). These inequalities are formulated in Proposition 2.14 in [41] and the proof of these inequalities holds also in the present context. In words, the first inequality yields the boundedness of the left action of A1v (Λ, c) on the right Hilbert C ∗ module VΛ◦ and the second inequality amounts to an analogous statement for the right action of A1v (Λ◦ , c) on the left Hilbert C ∗ -module Λ V . 1/2 Therefore, we have that the completion of Mv1 (Rd ) with respect to Λ f = Λ f, f op 1/2 or equivalently by f Λ◦ = f, f Λ◦ op , becomes an equivalence bimodule Λ VΛ◦ between C ∗ (Λ, c) and C ∗ (Λ◦ , c). We summarize the previous discussion in the following theorem, which includes one of the main results in [41]. Theorem 3.11. Let Λ be a lattice in R2d . Then the completion of Mv1 (Rd ) with respect to Λ f = 1/2 Λ f, f op becomes an equivalence bimodule Λ VΛ◦ between C ∗ (Λ, c) and C ∗ (Λ◦ , c). In the present setting we do not just have an equivalence bimodule Λ VΛ◦ between C ∗ (Λ, c) and C ∗ (Λ◦ , c), but we have dense subspaces Mv1 (Rd ) of Λ VΛ◦ that give rise to equivalence bimodules between A1v (Λ, c) and A1v (Λ◦ , c). In Connes’ work on noncommutative geometry this kind of structure is very important, see [5]. Connes discussed the general framework in [6]. For further motivation and results we refer the interested reader to [7]. We briefly recall Connes’s general result, see also Proposition 3.7 and its proof in [41]. Theorem 3.12 (Connes). Let A and B be unital C ∗ -algebras that are Morita equivalent via an equivalence bimodule A VB . Suppose we have dense ∗-Banach (or Frechet) subalgebras A0 and B0 of A and B respectively containing the identity elements. Furthermore we assume that A0 and B0 are spectrally invariant in A and B respectively. Let V0 be a dense subspace of A VB which is closed under the actions of A0 and B0 , and such that the restrictions of the inner products A ·,· and ·,·B have values in A0 and B0 respectively. Then V0 is a finitely generated projective left A0 -module and the mapping from A ⊗A0 V0 to A VB defined by A ⊗ f → Af is an isomorphism of left A-modules. In addition we have that V0 is a finitely generated projective right B0 -module and the mapping from V0 ⊗B0 B to A VB defined by f ⊗ B → f B is an isomorphism of left B-modules. Therefore V0 is an equivalence bimodule between A0 and B0 . The result of Connes suggests the following definition. If we are in the situation of Theorem 3.12, then we call the algebras A0 and B0 Morita–Rieffel equivalent. Now, we are in the position to draw an important consequence concerning the structure of the equivalence bimodule Λ VΛ◦ from the spectral invariance of A1v (Λ◦ , c) in C ∗ (Λ◦ , c) Theorem 3.13. Let Λ be a lattice in R2d . Then the noncommutative Wiener algebras A1v (Λ, c) and A1v (Λ◦ , c) are Morita-equivalent through Mv1 (Rd ) if and only if v is a GRS-weight. Consequently, Mv1 (Rd ) is a finitely generated projective left A1v (Λ, c)-module and a finitely generated right A1v (Λ◦ , c)-module. Proof. We follow closely the discussion of Proposition 3.7 in [41]. Let A = C ∗ (Λ, c) and B = C ∗ (Λ◦ , c). Let Λ V be the projective left C ∗ (Λ, c)-module completion of Mv1 (Rd ) with C ∗ (Λ, c)-valued inner product Λ f, g =

f, π(λ)g π(λ)

λ∈Λ

for f, g ∈ Mv1 Rd .

1940


Furthermore, we have the C ∗ (Λ◦ , c)-valued inner product

f, gΛ = vol(Λ)−1

π ∗ λ◦ π λ◦ g, f

for f, g ∈ Mv1 Rd .

λ◦ ∈Λ◦

Now, we consider the dense involutive unital subalgebras A0 = A1v (Λ, c) and B0 = A1v (Λ◦ , c) of C ∗ (Λ, c) and C ∗ (Λ◦ , c), respectively. Then Mv1 (Rd ) is a dense subspace of Λ V that is closed under the actions of A1v (Λ, c) and A1v (Λ◦ , c) given by πΛ (a) · f =

a(λ)π(λ)f

for a ∈ 1v (Λ), f ∈ Mv1 Rd

λ∈Λ

and πΛ◦ (b) · f = vol(Λ)−1

∗ π λ◦ f

for b λ◦ b ∈ 1v Λ◦ , f ∈ Mv1 Rd .

λ◦ ∈Λ◦

Furthermore, the restrictions of the inner products Λ ·,· and ·,·Λ◦ to M 1 (Rd ) have values in A1v (Λ, c) and A1v (Λ◦ , c), respectively. The final ingredient in our proof is the spectral invariance of A1v (Λ◦ , c) in C ∗ (Λ◦ , c), which is equivalent to v being a GRS-weight by Theorem 2.2. An application of Proposition 3.7 in [41] gives the desired assertion that Mv1 (Rd ) is an equivalence bimodule between A1v (Λ, c) and A1v (Λ◦ , c). 2 All the results in this section hold for A1v s (Λ, c), A1v s (Λ◦ , c) and Mv1s (Rd ). Therefore all ∞ ◦ d d theorems remain true for A∞ v s (Λ, c), Av s (Λ , c) and Sv s (R ). By construction Sv (R ) is a pro1 d 1 d jective limit of Mv (R ); consequently statements for Mv (R ) translate into ones about Sv (Rd ). Consequently, the preceding results allow us to prove the Morita equivalence of A∞ v (Λ, c) and ◦ , c). (Λ A∞ v ∞ ◦ Theorem 3.14. Let Λ be a lattice in R2d . Then A∞ v (Λ, c) and Av (Λ , c) are Morita equivalent d through the equivalence bimodule Sv (R ) if and only if v is a GRS-weight.

The case v(x, ω) = 1 + |x|2 + |ω|2 for (x, ω) ∈ R2 is the famous theorem of Connes in [5] about the Morita equivalence of smooth noncommutative tori A∞ (Λ, c) and on the level of C ∗ algebras the theorem was proved by Rieffel in [39]. 4. Applications to Gabor analysis In the present section we link the results about projective modules over A1v (Λ◦ , c) and with multi-window Gabor frames for modulation spaces. Modulation spaces [12] were introduced by Feichtinger in 1983. Later Feichtinger described modulation spaces in terms of Gabor frames [11]. In collaboration with Gröchenig he developed the coorbit theory [14], which associates to an integrable representation of a locally compact group a class of function spaces. The coorbit spaces for the Schrödinger representation of the Heisenberg group is the class of modulation spaces. In the coorbit theory [14] modulation spaces are introduced as subspaces of the space of conjugate linear functionals (Mv1 )¬ (Rd ). Recall that a weight m is called v-moderate, if there exists a constant C > 0 such that m(x + y, ω + η) Cv(x, ω)m(y, η) for all ◦ A∞ v (Λ , c)


1941

(x, ω), (y, η) ∈ R2d . Let m be a v-moderate weight on R2d and g a non-zero window in Mv1 (Rd ). p,q Then the modulation spaces Mm (Rd ) are defined as p,q d

R q/p 1/q

Vg f (x, ω)p m(x, ω)p dx = f ∈ Sv Rd : f Mmp,q = dω 1 we have |fn (s) − fn (t)| 2C . |s − t| |s − t| Combining these two estimates, we get fn (s) − fn (t) C(s) . 1 + |t| s−t Because (1 + |t|)−1 dμ(t) < ∞, we can apply the Dominated Convergence Theorem.

2

To prove Theorem 2.1 in the general case, let us first remind the reader of a few well-known facts about form bounded perturbations. 2.2. Form bounded perturbations and resolvent formula For an unbounded self-adjoint operator A in a Hilbert space H, one can define the standard scale of spaces · · · ⊂ H2 (A) ⊂ H1 (A) ⊂ H0 (A) = H ⊂ H−1 (A) ⊂ H−2 (A) ⊂ · · · , where Hr (A) := {ψ ∈ H: (1 + |A|)r/2 ψH < ∞} for r 0. Here |A| is the modulus of the operator A, i.e. |A| = (A∗ A)1/2 .

C. Liaw, S. Treil / Journal of Functional Analysis 257 (2009) 1947–1975

1953

If r < 0, it is defined by Hr (A) := [H−r (A)]∗ with the duality inherited from the inner product in H. Or, speaking more carefully, one can say that the space H−r , r > 0, is defined by introducing the norm −r/2 f H−r = I + |A| f H on H and taking the completion of H in this norm. In the case when A is the multiplication operator Mt by the independent variable t in L2 (μ), we simply have 2 r r Hr = L2 1 + |t| dμ = f : f (t) 1 + |t| dμ(t) < ∞ . Note, if A is a bounded operator, then Hr = H for all r. It is well known that it is possible to define the rank one perturbation Aα = A + α(·, ϕ)ϕ of the operator A for unbounded perturbations (·, ϕ)ϕ, i.e. when ϕ ∈ / H, but ϕ belongs to some Hk . Such perturbations are called singular, and the case ϕ ∈ H−1 \ H is probably the simplest case of a singular perturbation. Perturbations with ϕ ∈ H−1 \H are called form bounded, the term form bounded used because the quadratic form of the perturbation (·, ϕ)ϕ is bounded by the quadratic form of the operator I + |A|. When ϕ ∈ / H, but ϕ belongs to some Hk , we can define the quadratic form of the perturbed operator Aα = A + α(·, ϕ)ϕ on some dense subset of H. The question is whether or not this form gives rise to a unique self-adjoint extension. It is well known that the answer is affirmative for form bounded perturbations. Without going into details about how the form bounded perturbation is defined, let us mention the main facts we will be using. The first one is the following resolvent formula (Aα − λI)−1 f = (A − λI)−1 f −

α((A − λI)−1 f, ϕ) (A − λI)−1 ϕ 1 + α((A − λI)−1 ϕ, ϕ)

(2.4)

which initially holds for f ∈ H, λ ∈ C \ R (see, e.g. Eq. (17) of [1] or Proposition 2.1 and Theorem 3.3 of [6]). Note, the inner product ((A − λI)−1 ϕ, ϕ) is well defined for ϕ ∈ H−1 (A) and (A − λI)−1 is an isomorphism between Hr−2 (A) and Hr (A). Probably the easiest way to see that is to invoke the Spectral Theorem. The following three well known lemmata are corollaries of the resolvent formula (2.4). Lemma 2.4. The resolvent formula (2.4) can be extended to f ∈ H−1 (A). Moreover, for any λ ∈ C \ R the operator (Aα − λI)−1 is an isomorphism between H−1 (A) and H1 (A). Proof. Take f ∈ H−1 (A). We have (A − λI)−1 f ∈ H1 (A). So the right-hand side of (2.4) defines a bounded operator from H−1 (A) to H1 (A) (for λ ∈ C \ R). To complete the proof, take a sequence of vectors fn ∈ H, n ∈ N, such that fn → f in the norm of H−1 (A). The boundedness of the right side of (2.4) implies that the sequence gn = (Aα − λI)−1 fn converges in H1 (A). Let g be its limit.

1954


The boundedness of the right-hand side of the identity (2.4) implies the estimate gH1 (A) Cf H−1 (A) . Since Aα − λI ∈ B(H1 (A), H−1 (A)), we can conclude that (Aα − λI)g = f . The second statement follows trivially from the first one. 2 Let us recall that a vector ϕ is called cyclic for a self-adjoint operator A, if the span of the vectors (A − λI)−1 ϕ, λ ∈ C \ R, is dense in H. Note that for this definition, one does not need to assume that ϕ ∈ H, but only that (A − λI)−1 ϕ ∈ H, i.e. that ϕ ∈ H−2 (A). If ϕ ∈ H−1 (A), then for Aα = A + α(·, ϕ)ϕ we trivially have (Aα − λI)−1 ϕ ∈ H. So ϕ ∈ H−2 (Aα ). Lemma 2.5. Let ϕ ∈ H−1 (A) be a cyclic vector for A and let Aα = A + α(·, ϕ)ϕ. Then ϕ is cyclic for Aα . Proof. Recall Lemma 2.4. So since α((A − λI)−1 ϕ, ϕ) = 1 ∀λ ∈ C \ R, 1 + α((A − λI)−1 ϕ, ϕ) the resolvent formula (2.4) implies that (Aα − λI)−1 ϕ = c(λ)(A − λI)−1 ϕ, c(λ) = 0, for all λ ∈ C \ R. 2 Lemma dμα (s) 2.6. If ϕ ∈ H−1 (A), then ϕ ∈ H−1 (Aα ) for all α ∈ R. In particular, we have R 1+|s| < ∞. Remark. If the operator A is semibounded, i.e. if A aI for some a ∈ R, then the proof of the lemma is almost trivial. Indeed, if A is semibounded, then Aα is semibounded, and for semibounded operators f ∈ H−1 (A) if and only if ((A − λI)−1 f, f ) is defined and bounded for some (or equivalently, for all) λ ∈ C \ R. We learned the proof below, which works for the general case, from Pavel Kurasov. Proof of Lemma 2.6. Recall that we assume ϕ ∈ H−1 (A). Define Fα (z) := (Aα − zI)−1 ϕ, ϕ =

R

1 dμα (x) x −z

for all z ∈ C \ R, α ∈ R. It is not hard to see that ∀K > 0 ∃C(K) > 0:

C(K)−1 1 + |x|

∞ Im K

C(K) 1 dy . x − iy y 1 + |x|

(2.5)

Further we have the statement ∃C

∀|y| C:

Im F (iy) ∼ Im Fα (iy),

(2.6)


1955

1 where F := F0 . For the proof of statement (2.6), first notice that for |y| 1 it holds | iy−x | dμ(x) 1 | i−x | by a geometric argument. Since ϕ ∈ H−1 (A), we have |i−x| < ∞. By the Dominated Convergence Theorem, we obtain

1 1 dμ(x) = dμ(x) = 0. lim y→∞ iy − x y→∞ iy − x

lim F (iy) lim

y→∞

F Recall the Aronszajn–Krein formula Fα = 1+αF which follows from the resolvent formula (2.4), see e.g. Eq. (15) of [1]. To see statement (2.6) note that

Im Fα = Im

F Im F . = 1 + αF |1 + αF |2

Let us complete the proof that ϕ ∈ H−1 (Aα ). The inclusion ϕ ∈ H−1 (A) means that ∞ Im F (iy) dμ(x) dy < ∞ for all K > 0. R 1+|x| < ∞. By the right inequality of (2.5) it follows that K y For the interchange of order of integration, note that the latter integrand is positive for all y. Ac∞ cording to (2.6) it follows that K Im Fyα (iy) dy < ∞ for all K > C. By the left inequality of (2.5), α (x) we obtain R dμ 1+|x| < ∞, that is ϕ ∈ H−1 (Aα ). 2

2.3. Proof of Theorem 2.1 for unbounded A Recall that A = Mt in L2 (μ), Aα = A + α(·, ϕ)ϕ, ϕ ≡ 1t and that Vα Aα = Ms Vα , where Ms is the multiplication by the independent variable s in L2 (μα ). Recall also that Vα 1t = 1s . Using the resolvent equality (2.4) for f = ϕ = 1t and A = Mt , we get (Ms − λI)−1 1s = Vα (Aα − λI)−1 Vα−1 1s = Vα (Aα − λI)−1 1t −1 = 1 + α (Mt − λI)−1 1t , 1t L2 (μ) Vα (Mt − λI)−1 1t for λ ∈ C \ R. So multiplying both sides by the term in square brackets and recalling that (Mx − λI)−1 1x = (x − λ)−1 , we have Vα

1 1 dμ(t) = 1+α t −λ t −λ s −λ

∀λ ∈ C \ R.

R

Rewriting

1 s−λ

·

1 t−λ

=−

Vα

1 s−λ

−

1 1 t−λ s−t

we obtain

1 1 = −α t −λ s −λ

R

1 s−λ

−

1 t−λ

s −t

dμ(t)

for λ ∈ C \ R. By linearity we get that formula (2.1) holds for f in the space B := span

1 : λk ∈ C \ R . t − λk

1956


Let us show that formula (2.1) holds on C01 (R). Let f ∈ C01 (R), supp f ⊂ [−L, L], and let ε Pε be the Poisson kernel, Pε (x) = π1 x 2 +ε 2. Assume for a moment that the formula (2.1) holds for functions of the form Pε ∗ f , f ∈ C01 . Convolution Pε ∗ f converges to f uniformly on R. So |Pε ∗ f (x)| C (C does not depend on ε and x) for all sufficiently small ε. Moreover, for |x| > 2L we have |(Pε ∗ f )(x)| Cε/x 2 , so |(Pε ∗ f )(x)| C/(1 + |x|). Since (Pε ∗ f ) = Pε ∗ f , we conclude (Pε ∗ f ) → f uniformly on R, so |(Pε ∗ f ) (x)| C for all sufficiently small ε. If εn 0, then the functions fn = Pεn ∗ f satisfy the assumptions of Lemma 2.3, and (2.1) holds for f . To complete the proof of Theorem 2.1, we need to show that formula (2.1) holds for the functions of the form Pε ∗ f , f ∈ C01 . Let us (for a fixed ε > 0) approximate the convolution g(x) := Pε ∗ f (x) = Pε (x − t)f (t) dt by its Riemann sums. 1 1 1 Since Pε (x − t) = 2πi t−iε−x − t+iε−x , we can choose the Riemann sums gn (x) to be elements of B. So representation formula (2.1) holds for f = gn . Uniform continuity and boundedness of f and Pε imply that gn ⇒ g. It is also easy to see that for |x| > 2L we can estimate |gn (x)| C/x 2 , thus |gn (x)| C/(1 + |x|). Finally, taking the derivative we get the uniform estimate |gn | C. Notice, that C = C(ε) here, we do not need uniform in ε estimate. Functions gn satisfy the assumptions of Lemma 2.3 and we can extend formula (2.1) to functions of form Pε ∗ f , f ∈ C01 . 2 2.4. Proof of the rigidity Theorem 2.2 Assume the hypotheses of the rigidity theorem, Theorem 2.2, are satisfied. Recall that Mt and Ms denote the multiplication operators by the independent variable in L2 (μ) and L2 (ν), respectively. Note that if Ms is unbounded, commuting with Ms means commuting with its spectral measure, or equivalently, with its resolvent. We utilize two lemmata. Lemma 2.7. Under the assumptions of Theorem 2.2 operator V V ∗ commutes with Ms . In particular, we have V V ∗ = Mψ for some ψ ∈ L∞ (ν). Proof. Let us first present an easier proof for the case of bounded and compactly supported measures μ and ν. Let us begin by showing that Ms V = V Mt + α(·, 1t )L2 (μ) 1t .

(2.7)

Notice, that we can extend formula (2.1) from C01 to polynomials by multiplying the polynomials by an appropriate cut-off function h ∈ C01 , h ≡ 1 on supp μ ∪ supp ν. Let us prove (2.7) for monomials t n . For f ≡ 1t , formula (2.1) yields V 1t = 1s . Then application of (2.1) to t n and t n+1 yields for n 1 (Ms V − V Mt )t n (s) = s V t n (s) − V t n+1 (s)


1957

s(s n − t n ) s n+1 − t n+1 dμ(t) − s −t s −t = α t n dμ(t) = α t n , 1t L2 (μ) 1s .

= −α

So (2.7) holds for monomials t n . By linearity and continuity (2.7) holds on polynomials. By assumption the polynomials are dense in L2 (μ), operator V : L2 (μ) → L2 (ν) is bounded and measures ν, μ are bounded and of compact support. Therefore (2.7) holds as an operator from L2 (μ) to L2 (ν). Denoting Aα = Mt + α(·, 1t )1t we rewrite (2.7) as Ms V = V Aα , and take the adjoint to get V ∗ Ms = Aα V ∗ . So V V ∗ commutes with Ms : Ms V V ∗ = V Aα V ∗ = V V ∗ Ms . To prove the theorem in the general case we need an analogue of (2.7) with resolvents instead of the operators Ms and Aα , see (2.8) below. First, taking a test function f ∈ C01 , f 0, f L2 (μ) > 0, and noticing that |(Vf )(s)| C/|s| for large |s|, we can see that the boundedness of the operator V implies that (1 + s 2 )−1 dν(s) < ∞. Next, we want to show that the representation formula (2.1) holds on functions of the form (t − λ)−1 for all λ ∈ C \ R. Take f (t) = (t − λ)−1 . Notice that f ∈ L2 (μ) ∩ L2 (ν). Consider a family of cut-off functions hn , n ∈ N, such that 0 hn 1, hn ≡ 1 on [−n, n] and |h n (t)| 1. Then for each λ ∈ C \ R the family of functions {fn }, fn (t) := hn (t)(t − λ)−1 satisfies the assumptions of Lemma 2.3, so the representation formula (2.1) holds for the functions f , f (t) = (t − λ)−1 . With this extension of formula (2.1) we prove an identity that is the unbounded analog of the intertwining identity (2.7). Namely, for λ ∈ C \ R we have V (Aα − λI)−1 = (Ms − λ)−1 V

(2.8)

on L2 (μ), where Aα = Mt + α(·, 1t )L2 (μ) 1t . 1 1 s−t To show this, fix λ ∈ C \ R. Since s−λ − t−λ = − (t−λ)(s−λ) , the representation formula (2.1) gives us V (t − λ)−1 (s) = 1 + α (t − λ)−1 , 1t L2 (μ) (s − λ)−1 . That is V (Mt − λI)−1 1t = 1 + α (Mt − λI)−1 1t , 1t (Ms − λI)−1 1s . With this identity and resolvent equality (2.4) for A = Mt and f = ϕ = 1t , we get −1 (Mt − λI)−1 1t (Ms − λI)−1 1s = V 1 + α (Mt − λI)−1 1t , 1t α((Mt − λI)−1 1t , 1t ) (Mt − λI)−1 1t =V 1− 1 + α((Mt − λI)−1 1t , 1t ) = V (Aα − λI)−1 1t .

1958


For τ ∈ C \ R, we have the (usual) resolvent identity, (Aα − λI)−1 (Aα − τ I)−1 = (Aα − λI)−1 − (Aα − τ I)−1 (λ − τ )−1 . Combination of the latter two equations and

1 s−λ

−

V (Aα − λI)−1 (Aα − τ I)−1 1t =

1 s−τ

=

λ−τ (s−λ)(s−τ )

yields

1 V (Aα − τ I)−1 1t . (s − λ)

Identity (2.8) now follows from cyclicity of 1t for Aα , see Lemma 2.5. Writing identity (2.8) for λ¯ instead of λ and taking the adjoint, we have (Aα − λI)−1 V ∗ = V ∗ (Ms − λI)−1 .

(2.9)

Combination of (2.8) and (2.9) yields V V ∗ (Ms − λI)−1 = (Ms − λI)−1 V V ∗

∀λ ∈ C \ R,

i.e. V V ∗ commutes with the spectral measures of Ms . The second statement is a standard result in operator theory.

2

Lemma 2.8. Under the assumptions of Theorem 2.2, Ker V ∗ = {0}. Proof. Since Ker V ∗ = Ker V V ∗ and V V ∗ commutes with Ms (so V V ∗ is a multiplication operator Mψ ), the kernel Ker V ∗ is a spectral subspace of Ms . Namely, there exists a Borel subset E ⊂ R such that Ker V ∗ = f ∈ L2 (ν): χR\E f = 0 . Assume Ker V ∗ = {0}. Then ν(E) > 0. We obtain a contradiction by constructing a function / Ker V ∗ . f ∈ {f ∈ L2 (ν): χR\E f = 0} such that f ∈ By assumption supp μ consists of at least two points. Let a ∈ R such that there exist I1 (−∞, a), I2 (a, ∞) with μ(I1 ) > 0, μ(I2 ) > 0. We need to consider two cases. Ifν(E ∩ [a, ∞)) > 0, we can pick b ∈ R such that ν(E ∩ [a, b]) > 0. Let f = χE∩[a,b] . Recall that (1 + s 2 )−1 dν(s) < ∞ (see proof of Lemma 2.7). Hence f ∈ L2 (ν) and χR\E f = 0. Take g ∈ C01 such that g|I1 = 1 and so that g and f have separated compact support. We have

(f, V g)L2 (ν) = E∩[a,b] I1

f (s)g(t) dμ(t) dν(s) > 0, s −t

since I1 g(t) s−t dμ(t) > 0 for all s ∈ E ∩ [a, b]. / Ker V ∗ . Because (V ∗ f, g)L2 (μ) = (f, V g)L2 (ν) , we have f ∈ Consider the case ν(E ∩ [a, ∞)) = 0. Recall ν(E) > 0. So ν(E ∩ (∞, a]) > 0 and an analogous argument yields the desired contradiction. The assumption Ker V ∗ = {0} was wrong. 2


1959

With these two lemmata, we prove the rigidity theorem, Theorem 2.2. Proof of Theorem 2.2. Assume the hypotheses of Theorem 2.2. In particular, Ker V = {0}. With Lemmata 2.7 and 2.8 we have V V ∗ = Mψ , ψ ∈ L∞ (ν), and it holds Ker V ∗ = {0}. Let us conclude the first statement. Since V V ∗ 0, we have ψ 0 and the existence of |V ∗ | for some operator |V ∗ | = (V V ∗ )1/2 = Mψ 1/2 . Writing polar decomposition, we get V ∗ = U −1 1/2 ∞ ∗ = Mh V . It remains to show . Note that h = ψ ∈ L (ν). Taking U := U partial isometry U is a unitary operator. We have Ker U = Ker V ∗ = {0}. Let us show surjectivity. From that U ∗ = {0}. By definition (polar decomposition) we Ker V = {0} and h−1 ∈ L∞ (ν) it follows Ker U is closed. So also Ran U = [Ker U ∗ ]⊥ = L2 (μ) and U is unitary. have that Ran U Let us show the second part of the rigidity theorem, namely U Aα = Ms U , where Ms is the multiplication by the independent variable s in L2 (ν). Consider the case of bounded A. From the proof of the first statement we extract U = Mψ −1/2 V and ψ 1/2 ∈ L∞ (ν). Substitution of V into identity (2.7) yields Ms Mψ 1/2 U = Mψ 1/2 U Aα . Because multiplication operators commute, we get the second part of the rigidity theorem for bounded operators. The unbounded case follows in analogy using (2.8) instead of (2.7). 2 3. Singular integral operators Functions f and g are said to be of separated compact supports, if supp f and supp g are compact sets and dist(supp f, supp g) > 0. Let K(s, t) be a function (kernel) which is bounded on each set {(s, t): |s − t| > ε}, ε > 0. By a singular integral operator (see [8]), henceforth referred to as SIO, T : L2 (μ) → L2 (ν) with kernel K(s, t) we mean a bounded operator T : L2 (μ) → L2 (ν) such that for f ∈ L2 (μ) and g ∈ L2 (ν) with separated compact supports (Tf, g)L2 (ν) =

K(s, t)f (t)g(s) dμ(t) dν(s).

Notice, due to the condition of separated compact supports, the integral is well defined. 3.1. Unitary operator Vα is a singular integral operator Lemma 3.1. Operator Vα : L2 (μ) → L2 (μα ) from Theorem 2.1 is an SIO with kernel K(s, t) = −α(s − t)−1 . In particular, we have (Vα f, g)L2 (μα ) = −α

f (t)g(s) dμ(t) dμα (s) s −t

(3.1)

for all f ∈ L2 (μ) and g ∈ L2 (μα ) with separated compact supports. Proof. Formula (2.1) implies that (3.1) holds for f ∈ C01 and g ∈ L2 (μα ), if f and g have separated compact supports. To show that the same formula holds for arbitrary f ∈ L2 (μ) and g ∈ L2 (μα ) with separated compact supports, let us take a compact set K such that supp f K and dist(K, supp g) > 0 and a sequence {fn } of C01 functions so that supp fn ⊂ K for all n and such that fn → f in L2 (μ).

1960


Trivially limn→∞ (Vα fn , g) = (Vα f, g). Since |s − t|−1 1/ dist{K, supp g} for t ∈ K, s ∈ supp f , one can easily see that lim

n→∞

fn (t)g(s) dμ(t) dμα (s) = s −t

which proves the lemma.

f (t)g(s) dμ(t) dμα (s), s −t

2

3.2. Cauchy transform acting L2 (μ) → L2 (μα ) and its regularizations It is well known in the theory of singular integral operators, that if a singular operator T with a Calderon–Zygmund kernel2 K is bounded on L2 , then the truncated operators Tε , where

Tε f (s) =

K(s, t)f (t) dt,

|t−s|>ε

are uniformly (in ε) bounded. Also, this fact remains true, if instead of truncations, one considers any reasonable regularization of the kernel K. However, the classical theory does not apply in our case, because we integrate with respect to the measure μ which does not satisfy the doubling condition. Moreover, even the recently developed theory, see [8], of singular integral operators on non-homogeneous spaces (i.e. with non-doubling measure) does not work here, because, first this theory works only for one weighted case (the same measure in the target space), and second, the measure μ has to satisfy a growth condition (μ([a − ε, a + ε]) Cε uniformly in a and ε). The measure μ appearing in our situation can be any Radon measure. So no known result about singular integrals can be applied here. Nevertheless, it still can be shown that the following regularized operators are uniformly bounded operators acting from L2 (μ) to L2 (μα ). Let Tε = (Tμ )ε , ε > 0, be the integral operator with kernel (s − t + iε)−1 , Tε f (s) :=

f (t) dμ(t), s − t + iε

(3.2)

and let Tε = (Tμ )ε be the truncated operator, Tε f (s) :=

|t−s|>ε

f (t) dμ(t). s −t

Note, it is trivial that both Tε and Tε are well defined for compactly supported f . It is also not hard to show — using Cauchy–Schwartz inequality — that, if (1 + x 2 )−1 dμ(x) < ∞, then the operators are well defined for all f ∈ L2 (μ). 2 Calderon–Zygmund means that the kernel K satisfies some growth and smoothness estimates. Without giving the definition let us only mention that 1/(s − t) is one of the classical examples of a Calderon–Zygmund kernel.


1961

Theorem 3.2. Let μ and μα be the spectral measures of A and Aα , correspondingly. Then the regularized operators Tε = (Tμ )ε : L2 (μ) → L2 (μα ) defined by (3.2) are uniformly bounded Tε L2 (μ)→L2 (μα ) 2|α|−1 . Moreover, the weak limit T of Tε exists as ε → 0+ , and operator Vα has the alternative representation Vα f (s) = f (s)(1 − αT 1) + αTf

(3.3)

for all f ∈ L2 (μ). Finally, for any f ∈ C01 we have lim (Tε f )(s) = Tf (s)

ε→0+

μα almost everywhere. Remark. If 1 ∈ / L2 (μ), then the function T 1 can be defined, for example, by duality,

T ∗ f dμ

T 1f dμα =

for all compactly supported f ∈ L2 (μα ). Note, since (1 + |x|)−1 dμ(x) < ∞, the integral T ∗ f dμ is well defined. It is easy to see from the proof that T 1 coincides with −F (x + i0+ ), F (x + i0+ ) := limε→0+ F (x + iε), where F (z) =

dμ(t) . t −z

Remark. For purely singular measure μ the μα -a.e. convergence of Tε f for all f ∈ L2 (μ) (not only for f ∈ C01 ) was settled by Poltoratski˘ı’s theorem in [9]. Apparently, as it came out of our communications with A. Poltoratski˘ı and other experts in this area, it is possible to prove μα a.e. convergence for all f ∈ L2 (μ) in the general case, although it is hard to present a formal reference. However, for our purposes a simpler fact of μα -a.e. convergence for all f ∈ C01 , is sufficient. Proof of Theorem 3.2. To prove the first statement, let Vα : L2 (μ) → L2 (μα ) be the spectral representation of Aα from Theorem 2.1. Using formula (2.1) it is easy to see that for all a ∈ R and f ∈ C01 it holds Vα f (s) − e

ias

ia(s−t) )

Note that the kernel (1−es−t functions f ∈ L2 (μ).

Vα e−iat f (s) = α

f (t)(1 − eia(s−t) ) dμ(t). s−t

is bounded, so the integral is well defined for compactly supported

1962


Since Vα is unitary and multiplication by e−iax is a unitary operator on L2 (μ) and L2 (μα ), f (t)(1 − eia(s−t) ) dμ(t) s−t

L2 (μ

2|α|−1 f L2 (μ) .

(3.4)

α)

For ε > 0 we have ∞ ε 0

1 − eia(s−t) −εa 1 da = e t −s s − t + iε

∞ and ε 0 e−εa da = 1. So, by averaging the integral in the left side of (3.4) over all a 0 with weight εe−εa , we get f (t) 2|α|−1 f L2 (μ) s − t + iε dμ(t) 2 L (μα ) for compactly supported f ∈ C 1 and all ε > 0. Let us show the existence of the weak limit of Tε . Take a convergent (in weak operator topology) sequence Tεk → T, εk → 0, as k → ∞. For f ∈ C01 , we have that Tε f → Tf pointwise μα -a.e. for some operator T . Indeed, Tε f (s) =

f (t) dμ(t) = s − t + iε

f (t) − f (s) dμ(t) + f (s)Tε 1 s − t + iε

and note that the integrand on the right-hand side remains bounded as ε → 0 for compactly supported C 1 functions f . For the second term on the right-hand side, recall that we denote by w the density function of operator A’s spectral measure. Aronszajn–Donoghue theory on rank one perturbations says that −F (· + iε) = Tε 1 → −πw a.e. with respect to the Lebesgue measure and −F (· + iε) = Tε 1 → α −1 a.e. with respect to (μα )s as ε → 0, see e.g. [10]. So for f ∈ C01 we have that Tε f → Tf pointwise μα almost everywhere. Lemma 3.3 below shows that Tf = Tf for all f ∈ C01 , so w.o.t.-limk→∞ Tεk = T = T . Since the operators Tε are uniformly bounded, any sequence εk → 0 has a subsequence εkn such that Tεkn converges in weak operator topology. As we discussed above, this limit must be T . And that means w.o.t-limε→0 Tε = T . Let us prove representation formula (3.3). Take f ∈ C01 . By the Dominated Convergence Theorem we have

f (s) − f (t) dμ(t) = lim s −t ε→0+

f (s) dμ(t) − s − t + iε

f (t) dμ(t) s − t + iε

for all real s. The first part of the theorem implies that the second integral converges weakly in L2 (μα ) and μα -a.e. to Tf as ε → 0. It is an easy exercise to show that the first integral converges weakly in L2 (μα ) to f T 1 = −f F (· + i0+ ), and μα -a.e. convergence was shown above in the proof. Representation (3.3) now immediately follows from (2.1). 2


1963

The lemma below is well known. We present the proof only for the sake of completeness. Lemma 3.3. Let η be a measure. If a sequence of functions fn converges to f weakly in L2 (η) and to g pointwise η-a.e., then we have f = g in L2 (η). Proof. Recall that a closed convex subset of a Banach space is weakly closed (it is a simple corollary of the Hahn–Banach theorem). So we have f ∈ w-clos(conv{fn , fn+1 , . . .}) = fn+1 , . . .}) for all n ∈ N. Hence for clos(conv{fn , every n there exists a non-negative sequence {αkn }k∈N with kn αkn = 1 and such that gn = kn αkn fn converge to f in L2 (η). Therefore, one can find a subsequence gnk such that gnk → f η-a.e. On the other hand lim gnk (x) = lim gn (x) = g(x), so f = g η-a.e. 2 3.3. Regularization of the Cauchy transform in the general case The situation we considered in the previous section is very special, because measures μ and μα are rigidly related to each other. The theorem below shows that for very general measures, a rather natural and weak assumption of boundedness implies the uniform boundedness of the regularized operators. Let us recall that two Borel measures μ and ν are called mutually singular (notation μ ⊥ ν) if they are supported on disjoint sets, i.e. if there exist Borel sets E and F such that E ∩ F = ∅ and μ(E c ) = ν(F c ) = 0. Theorem 3.4. Let μ and ν be Radon measures on R such that for their singular parts μs ⊥ νs , and such that f (t)g(s) Cf 2 g 2 dμ(t) dν(s) (3.5) L (μ) L (ν) s −t for all f and g with separated compact supports. Then for all ε > 0 Tε f L2 (ν) 4Cf L2 (μ)

∀f ∈ L2 (μ),

and the truncated operators Tε : L2 (μ) → L2 (ν) are also uniformly bounded. Remark. By a well-known Aronszajn–Donoghue theorem, the singular parts of μα and μβ are mutually singular for all α, β ∈ R with α = β, see e.g. [10]. So the above theorem can be used in the situation when μ is the spectral measure of A and ν = μα is the spectral measure of Aα . On the other hand, it is not hard to show that the uniform boundedness of Tε implies that μs ⊥ νs , so Theorem 3.2 gives a different proof of this Aronszajn–Donoghue theorem. Proof of Theorem 3.4 for Tε . Estimate (3.5) holds, if we replace function f by e−iat f (t) and g by e−ias g(s), a ∈ R. So for all a ∈ R 1 − eia(s−t) f (t)g(s) dμ(t) dν(s) 2Cf L2 (μ) gL2 (ν) s −t (again for f and g with separated compact supports).

1964


In analogy to the proof of Theorem 3.2, we obtain f (t)g(s) dμ(t) dν(s) 2Cf L2 (μ) gL2 (ν) s − t + iε

(3.6)

independent of ε and for f and g with separated compact supports. The lemma below shows that the estimate holds for arbitrary compactly supported functions, not necessarily with separated supports, which proves the theorem for Tε . 2 Lemma 3.5. Let μ and ν be Radon measures such that μs ⊥ νs . Let T : L2 (μ) → L2 (ν) be a compact operator. If |(Tf, g)| Cf L2 (μ) gL2 (ν) for all pairs f ∈ L2 (μ) and g ∈ L2 (ν) with separated compact supports, then T 2C. If one restricts everything to an interval (−R, R), the integral operator with kernel 1/(s − t + iε) is clearly compact. So Lemma 3.5 gives the estimate |(Tε f, g)| 4Cf L2 (μ) gL2 (ν) for compactly supported f and g, which is all we need to prove Theorem 3.4. Proof of Lemma 3.5. Consider first the case when μ and ν are absolutely continuous with respect to Lebesgue measure. Pick small δ > 0, define functions h1 = 1[0,1/2−δ] , h2 = 1[1/2,1−δ] on [0, 1) and extend them to 1-periodic functions on the whole real line. For f ∈ L2 (μ) and g ∈ L2 (ν), define functions fn , gn by fn (t) := f (t)h1 (nt),

gn (s) := g(s)h2 (ns).

For each n, the functions fn , gn have separated support. We claim that fn → (1/2 − δ)f,

gn → (1/2 − δ)g

(3.7)

gn 2L2 (ν) → (1/2 − δ)g2L2 (ν)

(3.8)

weakly in L2 (μ) and L2 (ν), respectively, and that fn 2L2 (μ) → (1/2 − δ)f 2L2 (μ) ,

as n → ∞. Both statements follow immediately from the fact that for arbitrary φ ∈ L1 (with respect to the Lebesgue measure) and for h = h1 or h = h2

φ(t)h(nt) dt = (1/2 − δ)

lim

n→∞ R

φ(t) dt. R

This fact is trivial for characteristic functions of intervals, extends by linearity for their finite linear combinations and from this dense set to all L1 by ε/3 theorem, since the functionals φ → R φ(t)h(nt) dt are uniformly bounded. Since T is compact, the weak convergence of fn and gn implies that (Tfn , gn ) → (1/2 − δ)2 (Tf, g). Therefore


1965

(1/2 − δ)2 (Tf, g) = lim (Tfn , gn ) C lim fn L2 (μ) gn L2 (ν) n→∞

n→∞

= (1/2 − δ)Cf L2 (μ) gL2 (ν) , so T (1/2 − δ)−1 C. Since δ can be arbitrary small, the conclusion of the lemma follows in the case when μ and ν are absolutely continuous. The reasoning in the above paragraph works for general Radon measures. So to prove lemma for the general case it is sufficient for arbitrary f ∈ L2 (μ), g ∈ L2 (ν) to construct functions fn , gn satisfying (3.7), (3.8) and such that for each n the supports of fn and gn are separated. Let E and F be disjoint Borel subsets of Lebesgue measure zero supporting the singular parts of μ and ν, respectively, meaning that μs (E c ) = 0, νs (F c ) = 0. Denote G := (E ∪ F )c . Radon measures on R are inner regular. So there exist compact subsets En ⊂ E, Fn ⊂ F , Gn ⊂ G such that μ(En ) → μ(E), ν(Fn ) → ν(F ), μ(Gn ) + ν(Gn ) → μ(G) + ν(G) as n → ∞. Let fa = f χG , ga = gχG be “absolutely continuous” parts of f and g, and fs = f χE , gs = gχF be the “singular” parts of f and g. Take δ > 0 and define fn (t) := fa (t)h1 (nt)χGn (t) + (1/2 − δ)fs (t)χEn (t), gn (t) := ga (t)h2 (nt)χGn (t) + (1/2 − δ)gs (t)χFn (t). Clearly, for each n supports of fn and gn are separated. Let us show that fn → (1/2 − δ)f weakly in L2 (μ). Clearly, due to absolute continuity of integral fs χEn − fs L2 (μ) → 0 as n → ∞. Take arbitrary k ∈ L2 (μ). Then fa (t)h1 (nt)χGn (t)k(t) dμ(t) → (1/2 − δ)(fa , k)L2 (μ) R

because, as it was discussed above, fa (t)h1 (nt) converges weakly to (1/2 − δ)fa , and trivially kχGn converges strongly to kχG . As for the norms, it is not hard to show that lim fn 2L2 (μ) = (1/2 − δ)fa 2L2 (μ) + (1/2 − δ)2 fs 2L2 (μ) (1/2 − δ)f 2L2 (μ) .

n→∞

Similarly gn → (1/2 − δ)g weakly in L2 (ν) and limn→∞ gn 2L2 (ν) (1/2 − δ)g2L2 (ν) . And the same reasoning as for the absolutely continuous case completes the proof. 2 In order to show Theorem 3.4 for Tε , we prove the necessity of an A2 -type condition for Tε to be uniformly bounded. Lemma 3.6. Assume that the operators Tε : L2 (μ) → L2 (ν) are uniformly bounded. Then there exists a constant C > 0 such that 2 Im a 2 Im a dμ(t) dν(s) C 2 |t − a| |s − a|2 R

R

for all a, Im a > 0. In particular |I |−2 μ(I )ν(I ) C < ∞ for all intervals I = ∅.

1966


Proof. Let ba (x) = x−a x−a¯ , a ∈ C \ R. Consider auxiliary operators Rε := Tε − Mba Tε Mba : 2 2 L (μ) → L (ν), where Mϕ is the multiplication operator, Mϕ f = ϕf . Since Mb and Mb are isometries, the operators Rε are uniformly bounded with respect to ε and a. Since (s − a)(t ¯ − a) 2i Im a(s − t) 1 − = , s − t + iε (s − a)(s − t + iε)(t − a) ¯ (s − t + iε)(s − a)(t − a) ¯ we have for compactly supported f ∈ L2 (μ) Rε f (s) =

2i Im a(s − t)f (t) dμ(t). (s − t + iε)(s − a)(t − a) ¯

It follows from the Dominated Convergence Theorem that for compactly supported f ∈ L2 (μ), g ∈ L2 (ν) lim (Rε f, g)L2 (ν) =

ε→0+

2i(Im a)f (t)g(s) dμ(t)dν(s), (s − a)(t − a)

so the weak limit R0 := w.o.t.-limε→0+ Rε . Its norm can be easily computed (for example, the operator norm of a rank one operator coincides with its Hilbert–Schmidt norm): R

2 Im a dμ(t) |t − a|2

R

2 Im a dν(s) = R0 2 4 lim sup Tε 2 < ∞. |s − a|2 ε→0+

(3.9)

But that is exactly the conclusion of the theorem. To prove the statement about intervals, take a non-empty interval I . Set Im a = |I | and Re a = 1/2(sup I − inf I ). Integrating in (3.9) only over I × I and using that 1/|t − a| 1/(2|I |) for t ∈ I we get that |I |−2 μ(I )ν(I ) C < ∞. 2 Proof of Theorem 3.4 for Tε . To prove Theorem 3.4 for the operators Tε it is sufficient to show that the difference operators Tε − Tε are uniformly bounded. The difference operator is defined for compactly supported f ∈ L2 (μ) by (Tε − Tε )f (s) =

Kε (s − t)f (t) dμ(t)

where Kε (x) = √ (x + iε)−1 − x −1 χ[−ε,ε]c . Note |Kε (x)| is bounded from above by the decaying 2ε 1 in |x| function x 2 +ε 2 with uniformly bounded (with respect to ε) L norm. So it can be majorated by a convex combination of characteristic functions |I |−1 χI ,

Kε (x) ck (ε)|Ik |−1 χIk (x) =: Mε (x), k

with intervals Ik centered at the origin.

ck (ε) 0,

k

ck (ε) C < ∞


1967

Clearly Mε (s − t)f (t) · g(s) dμ(t) dν(s). K (s − t)f (t)g(s) dμ(t) dν(s) ε So to prove uniform boundedness of Tε − Tε , it is sufficient to show that the operators TI : L2 (μ) → L2 (ν) given by TI f (s) = |I |−1

χI (s − t)f (t) dμ(t)

are uniformly bounded. To prove this uniform estimate let k∈Z Jk be a cover of R by non-intersecting half open intervals of length |Jk | = |I |. Let Jk := Jk−1 ∪ Jk ∪ Jk+1 . For all s ∈ Jk , we have TI f (s) 3|Jk |−1

|f | dμ 3|Jk |−1

Jk

1/2

|f |2 dμ

1/2 μ(Jk ) .

Jk

(The last inequality is just Cauchy–Schwartz.) So we obtain

|TI f | dν 9|Jk |−2 μ(Jk )ν(Jk )

|f |2 dμ.

2

Jk

Jk

Summing over all k and taking into account that |Jk |−2 μ(Jk )ν(Jk ) |Jk |−2 μ(Jk )ν(Jk ) C and that each x ∈ R is covered by 3 intervals Jk , we get

|TI f |2 dν 27C

R

|f |2 dμ.

2

R

4. Absence of singular spectrum In this section we are going to investigate the absence of the singular spectrum of the perturbed operator Aα . < ∞, let For a complex-valued Borel measure η on R such that |dη(t)| 1+t 2 Kη(s) := lim

ε→0+

dη(t) . s − t + iε

It is a standard fact that this limit exists almost everywhere with respect to Lebesgue measure. We will need the result below about the boundary values of the Cauchy transform of a measure, cf. [3], where it was proved for the case of the unit circle. The case of the real line can be treated absolutely the same way.

1968


Theorem 4.1. Let I ⊂ R be a bounded open interval. Then tχ({|Kη|>t}∩I ) dx → 2χI d|ηs | + χ∂I d|ηs | in the weak∗ -sense as t → ∞. The following corollary is an immediate consequence of this theorem. Corollary 4.2. If I ⊂ R is a bounded closed interval such that ηs |I = 0, then there exists a C > 0 such that |{|Kη| > t} ∩ I | C/t for large t. Assume the setting of rank one perturbations, see e.g. Section 1.1. Let F (z) = R

dμ(x) , x−z

Fα (z) = R

dμα (x) , x−z

where Im z > 0, μ and μα are the spectral measures of A and Aα , respectively. By the well-known Aronszajn–Krein formula we have Fα =

1 1 F = 1− . 1 + αF α 1 + αF

And it is also well known that Im Kμ = limε→0+ Im F (x + iε) = πw(x) a.e. with respect to Lebesgue measure (w is the density of the absolutely continuous part of μ). Therefore, we get |Kμα |

1 1 + 2 . |α| α πw

Combining this with Corollary 4.2 one immediately gets the following proposition. Proposition 4.3. If for a bounded closed interval I we have x ∈ I : 1/w(x) > t = o(1/t)

as t → +∞,

then the measures μα do not have singular part on I for all α ∈ R, α = 0. The above reasoning is probably well known to specialists. We have learned it from E. Abakumov (personal communication). Using the fact about uniform (in ε not in α) boundedness of the operators Tε = (Tμ )ε : L2 (μ) → 2 L (μα ) we can get a stronger result in this direction. For a bounded interval I and a weight w, define the distribution function Dw = Dw,I (t) := |{x ∈ I : w(x) < t}| of w|I . Consider its inverse function, the increasing rearrangement w ∗ = wI∗ of w|I , i.e. w ∗ = (Dw )−1 . Lemma 4.4. Let μ and ν be Radon measures on R such that the operators Tε = (Tμ )ε : L2 (μ) → L2 (ν),


Tε f (s) = R

1969

f (t) dμ(t), s − t + iε

are uniformly (in ε) bounded, and let dμ = w dt + dμs be the Lebesgue decomposition of the measure μ. Assume that for a bounded closed interval I the increasing rearrangement w ∗ = wI∗ of w|I satisfies ε

x −2 w ∗ (x) dx = ∞

(4.1)

0

for some (all) ε > 0. Then the measure ν is non-singular on I , i.e. νs |I ≡ 0. Lemma 4.5. Condition (4.1) can equivalently be expressed in terms of the distribution function Dw = Dw,I as δ

1 dy = ∞. Dw (y)

(4.2)

0

Proof. If w ∗ (x) cx, then Dw (y) Cy and both (4.1) and (4.2) are satisfied. So it is sufficient to consider the case when limn→∞ w ∗ (εn )/εn = 0 for some sequence εn → 0+ . Denoting y = w ∗ (x), so x = Dw (y), and integrating by parts, we get ε x εn

−2

∗

ε

w (x) dx = −

∗

w (x) d(1/x) = −w

∗

δ (x)/x|εεn

εn

+

x −1 dy,

δn

where δ = w ∗ (ε), δn = w ∗ (εn ). Taking limit as n → ∞ we can see that the conditions (4.1) and (4.2) are equivalent. 2 Remark. Condition (4.1) is satisfied if for small x, w ∗ (x) x, or if w ∗ (x) cx ln−p (1/x), p < 1, or even if w ∗ (x) cx/ (ln 1/x)(ln ln 1/x) · · · ( ln ln · · · ln 1/x)( ln · · · ln 1/x)p ln m times

m+1 times

(p < 1). Similarly, (4.2) holds if for p < 1 and t → ∞ x ∈ I : 1/w(x) > t Ct −1 (ln t)(ln ln t) · · · ( ln ln · · · ln t)( ln ln · · · ln t)p . m times

m+1 times

1970


Proof of Lemma 4.4. Since (Tμ )∗ε = −(Tν )ε , the operators (Tν )ε : L2 (ν) → L2 (μ) are uniformly bounded, and therefore they are uniformly bounded as operators L2 (ν) → L2 (w). Therefore we can pick a subsequence (Tν )εk , εk → 0+ which converges in the weak operator topology of B(L2 (ν), L2 (w)). Since for any f ∈ L2 (ν) the Cauchy integral Kf ν exists a.e. with respect to Lebesgue measure,3 Lemma 3.3 implies that the corresponding weak limit is the operator f → Kf ν. Since this operator is clearly bounded, applying it to f = χI , we get

|K ν| ˜ 2 w(x) dx

|K ν| ˜ 2 w(x) dx CχI 2L2 (ν) = Cν(χI ), R

I

where d ν˜ = χI dν. Using the distribution function we get that

∞ |K ν| ˜ w(x) dx = 2

I

2t 0

w(x) dx dt. {|K ν˜ |>t}∩I

Let us assume that ν has a non-trivial singular part on I , i.e. that ν˜ has a non-trivial singular part. By Corollary 4.2, we have |{|K ν| ˜ > t} ∩ I | C/t > 0 for all sufficiently large t (t A for some A > 0). Let L = |{|K ν| ˜ > t} ∩ I |. Since C/t L for t A, we have C/t L ∗ w (x) dx w ∗ (x) dx 0

w(x) dx.

{|K ν˜ |>t}∩I

0

Multiplying this inequality by 2t and integrating we get ∞ A

C/t ∞ ∗ 2t w (x) dx dt 2t

∞

w(x) dx dt {|K ν˜ |>t}∩I

A

0

2t 0

=

w(x) dx dt {|K ν˜ |>t}∩I

|K ν| ˜ 2 w(x) dx Cν(χclos I ) < ∞.

(4.3)

I

|f (t)| dν(t) < ∞, but one does not need to show that, 1+|t| because in the proof it is sufficient to consider only compactly supported f . 3 It is not difficult to show that under assumptions of the lemma


1971

Using Tonelli’s theorem to change the order of integration, we can write the left side as ∞ A

C/A C/t ∗ (C/x)2 − A2 w ∗ (x) dx. 2t w (x) dx dt = 0

(4.4)

0

C/A Clearly 0 w ∗ (x) dx < ∞. So combining (4.3) and (4.4), we get that if the measure ν˜ has a non-trivial singular part, then ε

x −2 w ∗ (x) dx < ∞,

0

where ε = C/A.

2

Let Aα = A + α(·, ϕ)ϕ be the family of rank one perturbations of the operator A as described in Section 1.1, and let μα be their spectral measures (corresponding to ϕ), μ = μ0 being the spectral measure of A. The following theorem is an immediate corollary of Lemma 4.4. Theorem 4.6. Let dμ = w dt + dμs be the Lebesgue decomposition of the spectral measure μ = μϕ . If for a bounded closed interval I the distribution function Dw = Dw,I satisfies (4.2) (equivalently, its inverse w ∗ satisfies (4.1)), then for all α ∈ R \ {0} operator Aα has empty singular spectrum on I . Let us state a similar result that incorporates the averages of the spectral measures into the hypothesis. Theorem 4.7 (Averaged condition). For a finite Borel measure σ on R, define the average spec tral measure τ = μβ dσ (β), and let dτ = w dt + dτs be its Lebesgue decomposition. Consider dσ (β) the set E := α ∈ R: |α−β| 2 0, bn ∈ R for all n ∈ N. The free Jacobi matrix T0 , is the Jacobi matrix with bn = 0 and an = 1 for all n ∈ N. We assume also that supn |an | + |bn | < ∞, so a Jacobi matrix can be viewed as a bounded operator on 2 = 2 (N) (the Jacobi operator). As it is well known, see e.g. [2], there is a one-to-one correspondence between compactly supported Borel measures on R satisfying the normalization condition μ(R) = 1 and bounded Jacobi operators. Namely, any such measure is the spectral measure (corresponding to the cyclic 2 vector e1 ) of the corresponding Jacobi matrix; here {en }∞ n=1 is the standard basis in . So all that was said above in Section 5.1 about perturbations of multiplication operator can be trivially said about perturbations Tα of a Jacobi matrix T , Tα = T + α(·, e1 )e1 ; note that Tα is obtained from T by replacing the entry b1 in the Jacobi matrix by b1 + α. What is more interesting, the same can be said about Jacobi matrices that are Hilbert–Schmidt perturbations of the free Jacobi matrix, i.e. about Jacobi matrices such that ∞

(an − 1)2 + bn2 < ∞. n=1

In [4], the following complete description of spectral measures of such matrices was obtained. Theorem 5.1. (Killip–Simon [4].) Let J be a Jacobi matrix and μ be the corresponding spectral measure (corresponding to the vector e1 ). Operator T − T0 is Hilbert–Schmidt if and only if all four conditions hold: − ± (1) Blumenthal–Weyl: supp dμ = [−2, 2] ∪ {λ+ j } ∪ {λj }, where {λj } denote the sequences of + + − eigenvalues of J in R \ [−2, 2] and λ1 > λ2 > · · · > 2 and λ1 < λ− 2 < · · · < −2, − + 3/2 3/2 (2) Lieb–Thirring: j (λj − 2) + j (λj + 2) < ∞, 2 (3) Quasi-Szegö: −2 (4 − t 2 )1/2 log(w(t)) dt > −∞, where w is the density function of μ, i.e. dμ = w dt + dμs , (4) Normalization: μ(R) = 1.

It is easy to see, that one can construct a measure μ satisfying all four conditions of Theorem 5.1, and such that condition (4.2) is satisfied for the interval [−2, 2]. Notice, that the conditions of Theorem 5.1 and condition (4.2) pose no restriction (except the trivial one μs (R) < 1) on the singular part of μ on the interval. So, the reasoning of the previous subsection applies to this case and the perturbations Tα of T have no singular spectrum on [−2, 2]. Considering perturbations of Tα0 , α0 = 0, one comes up with the example of a family of rank one perturbations Tα0 + α(·, e1 )e1 such that the singular spectrum on σess (T ) appears only for one value of α. Note, operator Tα0 is a Hilbert–Schmidt perturbation of the free Jacobi matrix. 5.3. Schrödinger operators The same idea as in Section 5.1 can be applied to (half-line) Schrödinger operators H := d2 2 2 2 − dx 2 + V with L potentials (V ∈ L (R+ )) on L (R+ ), R+ := (0, ∞). 2

d 2 Let us recall that for a formal differential operator H = HV = − dx 2 + V on R+ , V ∈ L (R+ ), one can define a family of self-adjoint operators Hϑ on L2 (R+ ) with different boundary conditions at 0; that is, these operators differ by their respective domains,

1974


D(Hϑ ) = u ∈ L2 R+ : u, u are locally absolutely continuous,

u(0) cos(ϑ) + u (0) sin(ϑ) = 0 for 0 ϑ < π, Hϑ u ∈ L2 R+ .

Note that ϑ = 0 corresponds to the Dirichlet boundary condition and ϑ = π/2 corresponds to the Neumann boundary condition. Recall that if V ∈ L2 (R+ ), then H is limit point, see e.g. [7], meaning that Dirichlet boundary conditions (and also the boundary conditions for Hϑ ) define a self-adjoint operator. A recent theorem of Killip and Simon [5] gives a complete description of spectral measures of Schrödinger operators with L2 potentials (with Dirichlet boundary condition). Without stating Killip–Simon theorem here, we will only mention that it is not hard to construct a measure μ satisfying the conditions of this theorem and such that its weight w satisfies condition (4.2) for all intervals I [0, ∞). Moreover, it is not hard to show that the singular part of μ can be essentially arbitrary, i.e. given a singular Radon measure τ on R+ one can find μ satisfying the conditions of the Killip–Simon theorem and such that the singular part μs of μ is mutually absolutely continuous with τ (and the density w of μ satisfies (4.2)). It is well known that the Schrödinger operators with mixed boundary conditions are viewed as self-adjoint rank one perturbations of the Schrödinger H0 operator with Dirichlet boundary conditions. Unfortunately, our results cannot be applied directly, because to get from the H0 to Hϑ the perturbation should formally be written as H0 + α(ϑ)(·, δ0 )δ0 , where δ0 is the derivative of delta function at zero. The spectral measure, which is traditionally defined via the Weyl M-function, is also the spectral measure with respect to δ0 . But vector δ0 is not in H−1 (H0 ), one can only prove that it is in H−2 (H0 ). However, there is a simple workaround: one just needs to consider resolvents. Namely, fix ϑ and consider λ < 0 which is not an eigenvalue of H0 or Hϑ . Then the difference of the resolvents can be formally written as (Hϑ − λI)−1 = (H0 − λI)−1 + α (ϑ) ·, (H0 − λI)−1 δ0 (H0 − λI)−1 δ0 ,

(5.1)

where α (ϑ) = α(ϑ)/[1 + α(ϑ)((H0 − λI)−1 δ0 , δ0 )]. The fact that the difference of resolvents is a rank one operator follows from the standard theory of differential operators, and knowing the resolvent one defines the operator. Thus, in this case, one can avoid the rather complicated construction of rank one perturbations with ϕ ∈ H−2 . This construction is described, for example, in [1]. The spectral measure ν of the resolvent (H0 − λI)−1 can be easily computed from the spectral measure μ of H0 , and it is clear that if the density of μ satisfies the assumption (4.2) on any subinterval I ⊂ [0, ∞), then the density of ν satisfies the same condition (4.2) for any subinterval of (0, −1/λ]. So, one can apply Theorem 4.6 to the resolvents. By doing so, one can obtain all the phenomd2 2 ena discussed in Section 5.1. For example, one can get H = − dx 2 + V , V ∈ L (R+ ), such that H0 (Dirichlet boundary conditions) has dense in R+ singular spectrum, but for all other boundary conditions the operators Hϑ have no singular spectrum on R+ . And the density of the spectral measure of Hϑ will exhibit some weird behavior: In particular, it will not satisfy condition (4.2) on any bounded subinterval of R+ .


1975

References [1] S. Albeverio, P. Kurasov, Rank one perturbations of not semibounded operators, Integral Equations Operator Theory 27 (4) (1997) 379–400. [2] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, Amer. Math. Soc., Providence, RI, 2000. [3] M.G. Goluzina, On the multiplication and division of Cauchy–Stieltjes-type integrals, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1981) 8–15, 124. [4] R. Killip, B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. 158 (1) (2003) 253–321. [5] R. Killip, B. Simon, Sum rules and spectral measures of Schrödinger operators with L2 potentials, Ann. of Math., in press. [6] P. Kurasov, Singular and supersingular perturbations: Hilbert space methods, in: Spectral Theory of Schrödinger Operators, in: Contemp. Math., vol. 340, Amer. Math. Soc., Providence, RI, 2004, pp. 185–216. [7] M.A. Naimark, Linear Differential Operators, Ungar, New York, 1968. [8] F. Nazarov, S. Treil, A. Volberg, The Tb-theorem on non-homogeneous spaces, Acta Math. 190 (2) (2003) 151–239. [9] A.G. Poltoratski˘ı, Boundary behavior of pseudocontinuable functions, Algebra i Analiz 5 (2) (1993) 189–210; English translation in: St. Petersburg Math. J. 5 (2) (1994) 389–406. [10] B. Simon, Spectral analysis of rank one perturbations and applications, in: Mathematical Quantum Theory. II. Schrödinger Operators, Vancouver, BC, 1993, in: CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149; Also available as part of B. Simon, Trace Ideals and Their Applications, second ed., Math. Surveys Monogr., vol. 120, Amer. Math. Soc., Providence, RI, 2005, viii+150 pp.


A Grobman–Hartman theorem for general nonuniform exponential dichotomies ✩ Luis Barreira ∗ , Claudia Valls Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal Received 5 November 2008; accepted 22 June 2009

Communicated by C. Kenig

Abstract For a nonautonomous dynamics with discrete time given by a sequence of linear operators Am , we establish a version of the Grobman–Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(n) , determined by a sequence ρ(n). For all sufficiently small Lipschitz perturbations Am + fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators Am . We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ(n) = n, but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ(n) = log n. © 2009 Elsevier Inc. All rights reserved. Keywords: Conjugacies; Growth rates; Nonuniform hyperbolicity

1. Introduction 1.1. Motivation and uniform hyperbolicity A fundamental problem in the study of the local behavior of a map or a flow is whether the linearization along a given solution approximates well the solution itself. This problem goes ✩

Partially supported by FCT through CAMGSD, Lisbon.


E-mail addresses: [email protected] (L. Barreira), [email protected] (C. Valls). URL: http://www.math.ist.utl.pt/~barreira/ (L. Barreira). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.023

L. Barreira, C. Valls / Journal of Functional Analysis 257 (2009) 1976–1993

1977

back to the pioneering work of Poincaré. It can be interpreted as looking for an analytic change of variables, called a conjugacy, that takes the system to a linear one. Moreover, as a means to distinguish various dynamics further than in the topological category, we would like the change of variables to be as regular as possible. For example, we would like to know whether it is possible to distinguish between different types of nodes. In the case of hyperbolic fixed points, the Grobman–Hartman theorem gives a complete answer in the topological category, by constructing a topological conjugacy between the original dynamics and its linearization. The original references are Grobman [11,12] and Hartman [15,16]. Using the ideas in Moser’s proof in [21] of the structural stability of Anosov diffeomorphisms, the Grobman–Hartman theorem was extended to Banach spaces independently by Palis [23] and Pugh [26]. On the other hand, the work of Sternberg [29,30] showed that there are algebraic obstructions, expressed in terms of resonances between the eigenvalues of the linearization, that prevent the existence of conjugacies with a prescribed high regularity (see also [8,9,20,27] for further related work). In spite of this unavoidable drawback, the linearization problem still stands today as a fundamental step in the study of the local behavior of a dynamical system. Having this is mind, it is crucial to understand what is the most general class of systems with some hyperbolic behavior for which the problem can be solved. Nevertheless, there exist large classes of linear dynamics with uniform hyperbolic behavior, and the corresponding theory and its applications are widely developed. We refer to the books [10,14,17,28] for details and references related to uniform hyperbolic behavior. 1.2. Nonuniform hyperbolicity and its ubiquity On the other hand, the classical notion of uniform hyperbolicity is very stringent for the dynamics and it is important to look for more general types of hyperbolic behavior that can be much more typical. This is precisely what happens with the notion of nonuniform hyperbolicity. Roughly speaking, a nonuniform exponential behavior includes the usual exponential contraction and expansion, but it also allows a “spoiling” of the contraction and expansion along each trajectory as the initial time increases. In other words, instead of having uniform asymptotic stability along the stable direction into the future and along the unstable direction into the past, in general we have a nonuniform asymptotic stability. This causes that at a given time, the “size” of the neighborhood in the stable and unstable directions, where respectively the exponential stability or instability of the trajectory is guaranteed, may decay with exponential rate. We refer to [1,2] for detailed expositions of large parts of the theory of nonuniform hyperbolicity, which goes back to the landmark works of Oseledets [22] and particularly Pesin [24]. As we already mentioned, the notion of nonuniform hyperbolicity (here reformulated in terms of nonuniform exponential dichotomies) is much more typical than uniform hyperbolicity. For example, almost all trajectories with nonzero Lyapunov exponents of a dynamical system preserving a finite invariant measure (such as for example any compact level set of any Hamiltonian system) are nonuniformly hyperbolic. We refer to [2,7] for a precise formulation of the results, and for related detailed discussions. Among the most important properties due to nonuniform hyperbolicity is the existence of stable and unstable manifolds, and their absolute continuity property established by Pesin in [24]. The theory also describes the ergodic properties of dynamical systems with a finite invariant measure absolutely continuous with respect to the volume [25], and it expresses the Kolmogorov–Sinai entropy in terms of the Lyapunov exponents by the Pesin entropy formula [25] (see also [19]). In another direction, combining the nonuniform hyperbolicity with the nontrivial recurrence given by the existence of a finite invariant measure, the fundamental work of Katok [18] revealed a very rich and complicated orbit structure, including an

1978


exponential growth rate for the number of periodic points measured by the topological entropy, and an approximation of the entropy of an invariant measure by uniformly hyperbolic horseshoes. We point out that the smallness of the nonuniformity is a rather common phenomenon from the point of view of ergodic theory: namely, almost all linear variational equations obtained from a measure-preserving flow have a nonuniform exponential dichotomy with arbitrarily small nonuniformity. Nevertheless, even if arbitrarily small, in general the nonuniformity cannot be discarded a priori. In particular, it follows from work of Barreira and Schmeling in [3] that for some classes of measure-preserving transformations, the nonuniformity cannot be made arbitrarily small in a set of full topological entropy and full Hausdorff dimension. In other words, also from the topological and the dimensional points of view it is crucial to study nonuniform hyperbolicity. We also would like to mention that if an autonomous linear dynamics has a nonuniform exponential dichotomy, then in fact the dichotomy must be uniform. This is why in the context of nonuniform exponential behavior we are only interested in perturbations of a nonautonomous linear dynamics. 1.3. Brief description of our results Our main objective is to generalize the Grobman–Hartman theorem to perturbations of a nonautonomous dynamics with discrete time zm+1 = Am zm ,

m ∈ Z,

(1)

given by a sequence of linear operators Am that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates ecρ(n) , determined by a sequence ρ(n) (see the following paragraph for a detailed motivation for considering this general situation). Namely, for a sequence of sufficiently small Lipschitz perturbations Am + fm we construct topological conjugacies between the dynamics zm+1 = Am zm + fm (zm ),

m ∈ Z,

and the linear dynamics in (1). We emphasize that in strong contrast with the usual (exponential) stable and unstable behaviors, we allow asymptotic rates of the form ecρ(n) determined by an arbitrary sequence ρ(n). The usual exponential behavior corresponds to take ρ(n) = n. We point out that it is easy to construct large classes of linear dynamics as in (1) for which all or some Lyapunov exponents are infinite (either +∞ or −∞). In this situation, one is not able to apply the existing stability theory. Nevertheless, we may still be able to distinguish between different growth rates in different directions, specified by an appropriate sequence ρ(n). It is quite reasonable, and we would even say compelling, to take advantage of such a decomposition, which allows us to develop a corresponding stability theory virtually for all linear dynamics and not only for some particular classes. From a more practical perspective, we show in [6] that for a large class of growth rates ρ(n) there exist many linear dynamics exhibiting this asymptotic behavior. We refer to that paper for more details on the notion of nonuniform exponential dichotomy in this general context, and for a discussion of its ubiquity. Moreover, we show that all conjugacies are Hölder continuous, with Hölder exponent determined by the constants c in the asymptotic rates ecρ(n) . We note that in the classical case of uniform exponential dichotomies, the Hölder regularity of the conjugacies seems to have been known by some experts for quite some time, although apparently, to the best of our knowledge, no correct published proof appeared in the


1979

literature before our proof in [5] (see also [4] for the nonuniform setting). Indeed, it was claimed by some authors that the conjugacy is always Hölder continuous. Others have announced proofs of this property, but either they were never published or were not correct. We refer to [5] for detailed references to these works. It is however pointed out in [13] that the statement about the Hölder regularity is contained in a 1994 preprint of Belitski˘ı, although it remains unpublished. We follow the strategy of proof in [4], which involves three steps: 1. to show that there exist unique continuous functions um satisfying Am ◦ um = um+1 ◦ (Am + fm ) such that the sequence um − Id has a certain boundedness property (see Theorem 1); 2. to show that there exist unique continuous functions vm satisfying vm+1 ◦ Am = (Am + fm ) ◦ vm such that the sequence vm − Id has a certain boundedness property (see Theorem 2); 3. to verify that for each m ∈ Z these functions satisfy um ◦ vm = vm ◦ um = Id, and thus that they are the desired conjugacies (see Corollary 1). 2. Preliminaries Let X be a Banach space. We consider invertible linear operators Am , m ∈ Z such that with respect to some decomposition X = E × F (independent of m) we can write Am =

Bm 0

0 Cm

,

m ∈ Z.

Each sequence (zm )m∈Z in X satisfying zm+1 = Am zm for every m ∈ Z can be written in the form zm = B(m, n)xn , C(m, n)yn ,

m, n ∈ Z,

where zn = (xn , yn ) ∈ E × F , and ⎧ ⎪ ⎨ Bm−1 · · · Bn , m > n, m = n, B(m, n) = Id, ⎪ ⎩ B −1 · · · B −1 , m < n, m n−1

⎧ ⎪ ⎨ Cm−1 · · · Cn , C(m, n) = Id, ⎪ ⎩ C −1 · · · C −1 , m n−1

m > n, m = n, m < n.

Now consider an increasing function ρ : Z → Z with ρ(−m) = −ρ(m) for each m ∈ Z. We say that the sequence (Am )m∈Z admits a ρ-nonuniform exponential dichotomy if there exist constants a < 0 < b,

ε 0 and D 1

1980


such that for every m, n ∈ Z with m n we have C(m, n)−1 De−bμ(m,n)+ε|ρ(m)| , B(m, n) Deaμ(m,n)+ε|ρ(n)| ,

(2)

where μ(m, n) = ρ(m) − ρ(n). Now we introduce new norms, with respect to which the nonuniform behavior in (2) becomes uniform. Choose σ > 0 such that σ < min{−a, b}. For each m ∈ Z we set B(k, m)x e−(a+σ )μ(k,m)

xm =

for x ∈ E,

km

C(m, k)−1 y e(b−σ )μ(m,k)

ym =

for y ∈ F,

(3)

km

and for each (x, y) ∈ E × F ,

(x, y) = max x , y . m m m By (2) we have xm D

eε|ρ(m)| e−σ μ(k,m) x

km

= Deε|ρ(m)| eσρ(m)

e−σρ(k) x

km

Deε|ρ(m)| eσρ(m)

wρ(m)

e−σ w x

D eε|ρ(m)| x, 1 − e−σ

with similar estimates for ym . Thus, each series in (3) converges and there exists C > 0 such that 1 z zm Ceε|ρ(m)| z C

(4)

for every z ∈ X. Furthermore, since μ(k, n) − μ(k, m) = μ(m, n) for every m, n, k ∈ Z, we can easily show that B(m, n) = sup B(m, n)xm e(a+σ )μ(m,n) , xn x∈E\{0} −1 C(m, n)−1 = sup C(m, n) yn e(−b+σ )μ(m,n) ym y∈F \{0}

for every m n. This shows that with respect to the norms · m the sequence of operators (Am )m∈Z has a uniform exponential behavior.


1981

3. Construction of topological conjugacies We establish in this section a version of the Grobman–Hartman theorem, by constructing topological conjugacies between the sequences Am and Fm = Am + fm , for a large class of nonlinear perturbations fm . Namely, we consider continuous maps fm : X → X, m ∈ Z and a constant δ > 0 such that for each m ∈ Z and x, y ∈ X: 1. fm (0) = 0 and the map Fm is a homeomorphism;

2. fm ∞ := sup fm (x): x ∈ X δe−ε|ρ(m+1)| ; fm (x) − fm (y) δe−ε|ρ(m+1)| x − y. 3.

(5)

We also consider the space X of sequences u = (um )m∈Z of continuous functions um : X → X such that

u∞ := sup um m : m ∈ Z < ∞,

(6)

where

um m := sup um (x)m : x ∈ X . One can easily verify that X is a complete metric space with the norm · ∞ . We start with a preliminary result. Theorem 1. If the sequence (Am )m∈Z admits a ρ-nonuniform exponential dichotomy, then there is a unique (um )m∈Z ∈ X such that for every m ∈ Z we have Am ◦ um = um+1 ◦ (Am + fm ),

where um = Id +um .

(7)

Proof. Write um = (bm , cm ) and fm = (gm , hm ), with values in E × F . We can easily verify that (7) holds for every m ∈ Z if and only if −1 = bm , b¯m := (Bm−1 ◦ bm−1 − gm−1 ) ◦ Fm−1

(8)

and −1 c¯m := Cm ◦ (cm+1 ◦ Fm + hm ) = cm

for every m ∈ Z. Given u = (um )m∈Z = (bm , cm )m∈Z ∈ X, we write S(u) = (b¯m , c¯m )m∈Z . We show that S(X) ⊂ X, and that S is a contraction in the complete metric space X. Since each map Fm is a homeomorphism, (b¯m , c¯m ) is continuous for every m ∈ Z. Furthermore, for each z ∈ X we have b¯m (z) B(k, m)Bm−1 bm−1 F −1 (z) e−(a+σ )μ(k,m) m−1 m km

+

B(k, m)gm−1 F −1 (z) e−(a+σ )μ(k,m) m−1

km

1982


B(k, m − 1)bm−1 F −1 (z) e−(a+σ )μ(k,m−1) m−1

e(a+σ )μ(m,m−1)

km−1

B(k, m) · gm−1 ∞ e−(a+σ )μ(k,m) + km

e

−1 bm−1 Fm−1 (z) m−1

(a+σ )μ(m,m−1)

+ Dδ

eaμ(k,m)+ε|ρ(m)| e−ε|ρ(m)| e−(a+σ )μ(k,m)

km

−1 e(a+σ )μ(m,m−1) bm−1 Fm−1 (z) m−1 + Dδ e−σ μ(k,m) km

−1 e(a+σ )μ(m,m−1) bm−1 Fm−1 (z) m−1 +

Dδ . 1 − e−σ

(9)

Therefore, for the sequences b = (bm )m∈Z and b¯ = (b¯m )m∈Z we obtain

¯ ∞ = sup b¯m m : m ∈ Z b∞ + b

Dδ < ∞, 1 − e−σ

since a + σ < 0 and ρ is increasing. In an analogous manner, for each z ∈ X we have C(m, k)−1 C −1 cm+1 Fm (z) e(b−σ )μ(m,k) c¯m (z) m m km

+

C(m, k)−1 C −1 hm (z)e(b−σ )μ(m,k) m

km

C(m + 1, k)−1 cm+1 Fm (z) e(b−σ )μ(m+1,k)

e(−b+σ )μ(m+1,m)

km+1

C(m + 1, k)−1 · hm ∞ e(b−σ )μ(m,k) + km

e

cm+1 Fm (z) m+1

(−b+σ )μ(m+1,m)

+ Dδ

e−bμ(m+1,k)+ε|ρ(m+1)| e−ε|ρ(m+1)| e(b−σ )μ(m,k)

km

e(−b+σ )μ(m+1,m) cm+1 Fm (z) m+1 + Dδe−bμ(m+1,m) eσ μ(k,m) km

e(−b+σ )μ(m+1,m) cm+1 Fm (z) m+1 +

D δe−bμ(m+1,m) . 1 − e−σ

Therefore, for the sequences c = (cm )m∈Z and c¯ = (c¯m )m∈Z we obtain c ¯ ∞ c∞ +

Dδ < ∞, 1 − e−σ

(10)


1983

since −b + σ < 0 and ρ is increasing. This shows that S(X) ⊂ X. Now we prove that S is a contraction. Given u1 = (b1,m , c1,m )m∈Z and u2 = (b2,m , c2,m )m∈Z in X, for each z ∈ X we have b¯1,m (z) − b¯2,m (z) e(a+σ )μ(m,m−1) b1,m−1 F −1 (z) − b2,m−1 F −1 (z) m−1 m−1 m m−1 e(a+σ )μ(m,m−1) b1,m−1 − b2,m−1 m−1 , and c¯1,m (z) − c¯2,m (z) e(−b+σ )μ(m+1,m) c1,m+1 Fm (z) − c2,m+1 Fm (z) m m+1 e(−b+σ )μ(m+1,m) c1,m+1 − c2,m+1 m+1 . Thus b¯1 − b¯2 ∞ ea+σ b1 − b2 ∞ ,

(11)

c¯1 − c¯2 ∞ e−b+σ c1 − c2 ∞ .

(12)

and

By (11) and (12) the operator S is a contraction, and there exists a unique sequence u ∈ X such that S(u) = u. This completes the proof of the theorem. 2 Theorem 2. If the sequence (Am )m∈Z admits a ρ-nonuniform exponential dichotomy and δ is sufficiently small, then there is a unique (vm )m∈Z ∈ X such that for every m ∈ Z we have vm , vm+1 ◦ Am = (Am + fm ) ◦

where vm = Id +vm .

(13)

Proof. Write vm = (dm , em ) and fm = (gm , hm ), with values in E × F . We can easily verify that (13) holds for every m ∈ Z if and only if (dm , em ) = (d¯m , e¯m ) for every m ∈ Z, where vm−1 ) ◦ A−1 d¯m := (Bm−1 ◦ dm−1 + gm−1 ◦ m−1 = dm ,

(14)

−1 ◦ (em+1 ◦ Am − hm ◦ v m ) = em e¯m := Cm

(15)

and

for every m ∈ Z. Given v = (vm )m∈Z = (dm , em )m∈Z ∈ X, we write T (v) = (d¯m , e¯m )m∈Z . Clearly, (d¯m , e¯m ) is continuous for every m ∈ Z. For each z ∈ X we have d¯m (z) B(k, m)Bm−1 dm−1 A−1 z e−(a+σ )μ(k,m) m−1 m km

+

−(a+σ )μ(k,m) e B(k, m)gm−1 vm−1 A−1 m−1 z

km

e

(a+σ )μ(m,m−1)

B(k, m − 1)dm−1 A−1 z e−(a+σ )μ(k,m−1) m−1 km−1

B(k, m) · gm−1 ∞ e(a+σ )μ(k,m) . + km

(16)

1984


¯ ∞ < ∞. In an analogous manner, Proceeding as in (9) this implies that d C(m, k)−1 C −1 em+1 (Am z)e(b−σ )μ(m,k) e¯m (z) m m km

+

C(m, k)−1 hm vm (z) e(b−σ )μ(m,k)

km

e(−b+σ )μ(m+1,m)

C(m + 1, k)−1 em+1 (Am z)e(b−σ )μ(m+1,k) km+1

C(m, k)−1 · hm ∞ e(b−σ )μ(m,k) , +

(17)

km

and proceeding as in (10) this implies that e ¯ ∞ < ∞. Thus, T (X) ⊂ X. Now we prove that T is a contraction. Given v1 = (d1,m , e1,m )m∈Z and v2 = (d2,m , e2,m )m∈Z in X, let vi,m = Id +vi,m

and Gi,m = vi,m ◦ A−1 m−1 .

Proceeding as in (16), for each z ∈ X we have d¯1,m (z) − d¯2,m (z) m e(a+σ )μ(m,m−1)

B(k, m − 1)(d1,m−1 − d2,m−1 ) A−1 z e−(a+σ )μ(k,m) m−1

km−1

B(k, m) gm−1 G1,m−1 (z) − gm−1 G2,m−1 (z) e−(a+σ )μ(k,m) + km

−1 d1,m−1 A−1 m−1 z − d2,m−1 Am−1 z m−1 −ε|ρ(m)| B(k, m)δe G1,m−1 (z) − G2,m−1 (z)e−(a+σ )μ(k,m) +

e

(a+σ )μ(m,m−1)

km

−1 d1,m−1 A−1 m−1 z − d2,m−1 Am−1 z m−1 v1,m−1 A−1 + θ v2,m−1 A−1 m−1 z − m−1 z

e

(a+σ )μ(m,m−1)

ea+σ d1,m−1 − d2,m−1 m−1 + θ Cv1,m−1 − v2,m−1 m−1 , where θ = Dδ/(1 − e−σ ), using (4) in the last inequality. Analogously, proceeding as in (17) we have e¯1,m (z) − e¯2,m (z) m e(−b+σ )μ(m+1,m)

C(m + 1, k)−1 e1,m+1 (Am z) − e2,m+1 (Am z) e(b−σ )μ(m,k) km+1

C(m + 1, k)−1 hm−1 v1,m (z) − hm−1 v2,m (z) e(b−σ )μ(m,k) + km

e(−b+σ )μ(m+1,m) e1,m+1 − e2,m+1 m+1 + θ C v1,m − v2,m m .


1985

Thus d¯1,m − d¯2,m m ea+σ d1,m−1 − d2,m−1 m−1 + θ Cv1,m−1 − v2,m−1 m−1 , and e¯1,m − e¯2,m m e−b+σ e1,m+1 − e2,m+1 m+1 + θ Cv1,m − v2,m m . This implies that

T (v1 ) − T (v2 ) max ea+σ , e−b+σ + 2θ C v1 − v2 . ∞ ∞ Since σ < min{−a, b}, for δ sufficiently small the operator T is a contraction, and there exists a unique sequence v ∈ X such that T (v) = v. This completes the proof of the theorem. 2 We finally obtain a version of the Grobman–Hartman theorem. Corollary 1. If the sequence (Am )m∈Z admits a ρ-nonuniform exponential dichotomy and δ is sufficiently small, then the maps um = Id +um and vm = Id +vm in Theorems 1 and 2 are homeomorphisms, and vm = vm ◦ um = Id, um ◦

m ∈ Z.

Proof. By (7) and (13) we obtain vm+1 ◦ Am = um+1 ◦ Fm ◦ vm = Am ◦ um ◦ vm um+1 ◦

(18)

for every m ∈ Z. Moreover, since vm − Id = vm + um ◦ vm , um ◦ we have

vm − Id m : m ∈ Z < ∞, sup um ◦ vm )m∈Z ∈ X. It follows from (18) and the uniqueness in Theorems 1 or 2 (for the maps and ( um ◦ fm = 0) that um ◦ vm = Id for every m ∈ Z. 2 The following is another consequence. Corollary 2. If the sequence (Am )m∈Z admits a ρ-nonuniform exponential dichotomy, δ is sufficiently small, and there exist maps A and f such that Am = A

and fm = f

for every m ∈ Z,

then there is a homeomorphism h : X → X with um = h for m ∈ Z.

(19)

1986


Proof. For each continuous function u : X → X we have

pu := (um )m∈Z : um = u for every m ∈ Z ∈ X. Furthermore, when (19) holds, the contraction maps S and T in the proofs of Theorems 1 and 2 take the set D = {pu : u is continuous} into itself. Moreover, since D is a closed nonempty subset of X, the unique fixed points of the maps S and T are also in D. 2 4. Hölder regularity of the conjugacies We show in this section that the topological conjugacies in Corollary 1 are always Hölder continuous. 4.1. Preliminaries We say that the sequence (Am )m∈Z admits a strong ρ-nonuniform exponential dichotomy if there exist constants −c a < 0 < b −d,

ε 0 and D 1

such that for every m, n ∈ Z with m n we have B(m, n) Deaμ(m,n)+ε|ρ(n)| ,

C(m, n)−1 De−bμ(m,n)+ε|ρ(m)| ,

and for every m, n ∈ Z with m n we have B(m, n) Decμ(n,m)+ε|ρ(n)| ,

C(m, n)−1 De−dμ(n,m)+ε|ρ(m)| .

We also introduce new norms. Choose σ > 0 such that σ < min{−a, b}. For each m ∈ Z we set x∗m =

B(k, m)x e−(a+σ )μ(k,m) + B(k, m)x e−(c+σ )μ(m,k) k<m

km

for x ∈ E, y∗m =

C(m, k)−1 y e(d−σ )μ(k,m) + C(m, k)−1 y e(b−σ )μ(m,k) k<m

km

for y ∈ F , and

(x, y)∗ = max x∗ , y∗ . m m m Again there exists C > 0 such that for every z ∈ X, 1 z z∗m C eε|ρ(m)| z. C

(20)


Lemma 1. For each m ∈ Z we have Am ∗ := sup z∈X\{0}

Am z∗m+1 z∗m

e(−d+σ )μ(m+1,m) ,

−1 ∗ −1 ∗ A := sup Am zm e(c+σ )μ(m,m−1) . m ∗ z∈X\{0} zm+1

Proof. Setting z = (x, y) ∈ E × F we have

Am z∗m+1 = max Bm x∗m+1 , Cm y∗m+1 . Furthermore B(k, m)x e−(a+σ )μ(k,m)

Bm x∗m+1 = e(a+σ )μ(m+1,m)

km+1

+ e−(c+σ )μ(m+1,m)

B(k, m)x e−(c+σ )μ(m,k)

km

e

(a+σ )μ(m+1,m)

x∗m ,

and C(m, k)−1 y e(d−σ )μ(k,m)

Cm y∗m+1 = e(−d+σ )μ(m+1,m)

km+1

+ e(b−σ )μ(m,m+1)


km

e

(−d+σ )μ(m+1,m)

y∗m .

Since −d + σ > a + σ , we obtain Am z∗m+1 e(−d+σ )μ(m+1,m) z∗m . Similarly, −1 ∗ −1 ∗

−1 ∗ A m−1 v m−1 = max Bm−1 x m−1 , Cm−1 y m−1 , with −1 ∗ −(a+σ )μ(m,m−1) B B(k, m)x e−(a+σ )μ(k,m) m−1 x m−1 = e km−1

+e

(c+σ )μ(m,m−1)

B(k, m)x e−(c+σ )μ(m,k)

k<m−1

e

(c+σ )μ(m,m−1)

x∗m ,

1987

1988


and −1 ∗ (d+σ )μ(m,m−1) C(m, k)−1 y e(d+σ )μ(k,m) C m−1 y m−1 = e km−1

+e

(−b+σ )μ(m,m−1)


k<m−1

e(−b+σ )μ(m,m−1) y∗m . Since c + σ > −b + σ , we obtain −1 ∗ (c+σ )μ(m,m−1) A z∗m . m−1 z m−1 e This completes the proof of the lemma.

2

4.2. Hölder regularity of the conjugacies Now we establish the Hölder regularity of the conjugacies in Corollary 1. In this section we replace condition (5) by the stronger condition fm (x) − fm (y) δe−3ε|ρ(m+1)| e(c+σ )μ(m+1,m) x − y for every m ∈ Z and x, y ∈ X. Theorem 3. If the sequence (Am )m∈Z admits a strong ρ-nonuniform exponential dichotomy, then for each positive number −a b , , (21) γ < min c −d provided that δ is sufficiently small (depending on γ ) there exists a constant K = K(γ , δ) > 0 (independent of the maps fm ) such that for every m ∈ Z and x, y ∈ X with x − y∗m < 1 we have vm (x) − vm (y)∗ K x − y∗ γ . (22) m m Proof. Let σ > 0 be so small such that −a − σ b − σ , . γ < min c + σ −d + σ

(23)

Given constants K > 0 and γ ∈ (0, 1), we consider the subset Xγ ⊂ X of the sequences (vm )m∈Z satisfying (22) for every m ∈ Z and x, y ∈ X with x − y∗m < 1. One can easily verify that Xγ is closed with respect to the norm · ∞ in (6). Now let v = (vm )m∈Z = (dm , em )m∈Z , with values in E × F , be a sequence in Xγ . By (14) we have d¯m (x) − d¯m (y)∗ Bm−1 δm−1 (x) − Bm−1 δm−1 (y)∗ m m ∗ + gm−1 v¯m−1 (x) − gm−1 v¯m−1 (y) m ,

(24)


1989

where δm = dm ◦ A−1 m

and v¯m = vm ◦ A−1 m .

Since a +σ > −(c +σ ), the two terms in the right-hand side of (24) can be estimated respectively by B(k, m − 1) δm−1 (x) − δm−1 (y) e−(a+σ )μ(k,m) km

+

B(k, m − 1) δm−1 (x) − δm−1 (y) e−(c+σ )μ(m,k) k<m

=e

(a+σ )μ(m,m−1)

B(k, m − 1) δm−1 (x) − δm−1 (y) e−(a+σ )μ(k,m−1) km

+ e−(c+σ )μ(m,m−1) δm−1 (x) − δm−1 (y) B(k, m − 1) δm−1 (x) − δm−1 (y) e−(c+σ )μ(m−1,k) + e−(c+σ )μ(m,m−1) k<m−1

B(k, m − 1) δm−1 (x) − δm−1 (y) e−(a+σ )μ(k,m−1)

e(a+σ )μ(m,m−1)

km−1

+ e(a+σ )μ(m,m−1)

B(k, m − 1) δm−1 (x) − δm−1 (y) e(c+σ )μ(m−1,k)

k<m−1

∗ δm−1 (x) − δm−1 (y)m−1 ∗ γ e(a+σ )μ(m,m−1) K A−1 m−1 (x − y) m−1 , e

(a+σ )μ(m,m−1)

and by δ

B(k, m)e−3ε|ρ(m)| e−(c+σ )μ(m−1,m−2) v¯m−1 (x) − v¯m−1 (y)e−(a+σ )μ(k,m) km

+δ

B(k, m)e−3ε|ρ(m)| e−(c+σ )μ(m−1,m−2) v¯m−1 (x) − v¯m−1 (y)e−(c+σ )μ(m,k) k<m

∗ C δe−(c+σ )μ(m−1,m−2) v¯m−1 (x) − v¯m−1 (y)m−1 C δe−(c+σ )μ(m−1,m−2) L + KLγ , where ∗ L = A−1 m−1 (x − y) m−1 . By Lemma 1, for x = y with x − y∗m < 1 we obtain

(25)

1990


d¯m (x) − d¯m (y)∗m K e(a+σ )μ(m,m−1) + C δe−(c+σ )μ(m−1,m−2) ∗ γ (x − ym ) × eγ (c+σ )μ(m−1,m−2) + C δ.

(26)

In an analogous manner, by (15) we have ∗ −1 e¯m (x) − e¯m (y)∗ C −1 em+1 (y)m m em+1 (x) − Cm m −1 ∗ −1 + C m (hm ◦ vm )(x) − Cm (hm ◦ vm )(y)m ,

(27)

where em+1 = em+1 ◦ Am . The two terms in the right-hand side of (27) are respectively e(d−σ )μ(m+1,m)

C(m + 1, k)−1 em+1 (x) − em+1 (y) e(d−σ )μ(k,m+1) km

+e

(−b+σ )μ(m+1,m)

C(m + 1, k)−1 em+1 (x) − em+1 (y) e(b−σ )μ(m+1,k) k<m

e

C(m + 1, k)−1 em+1 (x) − em+1 (y) e(d−σ )μ(k,m+1)

(−b+σ )μ(m+1,m)

km+1

+ e(−b+σ )μ(m+1,m)

C(m + 1, k)−1 em+1 (x) − em+1 (y) e(b−σ )(m+1−k)

k<m+1

∗ em+1 (x) − =e em+1 (y)m+1 ∗ γ e(−b+σ )μ(m+1,m) K Am (x − y) , (−b+σ )μ(m+1,m)

m+1

(28)

and δ

C(m + 1, k)−1 e−3ε|ρ(m+1)| e−(c+σ )μ(m,m−1) vm (x) − vm (y)e(d−σ )μ(k,m) k>m

+δ

C(m + 1, k)−1 e−3ε|ρ(m+1)| e−(c+σ )μ(m,m−1) vm (x) − vm (y)e(b−σ )μ(m,k) km

∗ e(d−σ )μ(m+1,m) e(−b+σ )μ(m+1,m) vm (x) − δDe + vm (y)m 1 − e−σ 1 − e−σ γ 2δD −(c+σ )μ(m,m−1) x − y∗m + K x − y∗m . e −σ 1−e −(c+σ )μ(m,m−1)

(29)

By Lemma 1, for x = y with x − y∗m < 1 we obtain 2(1 + K)δD e¯m (x) − e¯m (y)∗m Ke(−b+σ )μ(m+1,m) eγ (−d+σ )μ(m+1,m) + . (x − y∗m )γ 1 − e−σ It follows readily from (23) that ea+σ eγ (c+σ ) < 1 and e−b+σ eγ (−d+σ ) < 1.

(30)


1991

Hence, by (26) and (30), for each sufficiently small δ there exists K > 0 such that ∗ ∗

γ max d¯m (x) − d¯m (y)m , e¯m (x) − e¯m (y)m K x − y∗m for every m ∈ Z and x, y ∈ X with x − y∗m < 1. Therefore, T (Xγ ) ⊂ Xγ . This completes the proof of the theorem. 2 Theorem 4. If the sequence (Am )m∈Z admits a strong ρ-nonuniform exponential dichotomy, then for each positive γ as in (21), provided that δ is sufficiently small (depending on γ ) there exists a constant K = K(γ , δ) > 0 (independent of the maps fm ) such that for every m ∈ Z and x, y ∈ X with x − y∗m < 1 we have um (x) − um (y)∗ K x − y∗ γ . m m Proof. Set α = e(−d+σ )μ(m+1,m) + (C )2 δe−(c+σ )μ(m,m−1)

and β = (1 − C δ)−1 .

Lemma 2. For each x, y ∈ X we have Fm (x) − Fm (y)∗ αx − y∗ , m m+1 −1 F (x) − F −1 (y)∗ βe(c+σ )μ(m,m−1) x − y∗ . m m m+1 m Proof. In view of Lemma 1 and (20), we have Fm (x) − Fm (y)∗

m+1

∗ ∗ Am (x − y)m+1 + fm (x) − fm (y)m+1 e(−d+σ )μ(m+1,m) x − y∗m + C eε|ρ(m+1)| fm (x) − fm (y) e(−d+σ )μ(m+1,m) x − y∗m + C δe−(c+σ )μ(m,m−1) x − y,

and Fm (x) − Fm (y)∗

Am (x − y)∗ − fm (x) − fm (y)∗ m+1 m+1 m+1 −(c+σ )μ(m,m−1) 2 −(c+σ )μ(m,m−1) x − y∗m . e − (C ) δe

This yields the desired inequalities.

2

Now take x, y ∈ X with x − y∗m < 1. By (8), proceeding as in (25) we obtain γ b¯m (x) − b¯m (y)∗ e(a+σ )μ(m,m−1) K F −1 (x) − F −1 (y)∗ m−1 m−1 m m−1 ∗ −1 −1 + C δe−(c+σ )μ(m−1,m−2) Fm−1 (x) − Fm−1 (y)m−1 , and by Lemma 2,

1992


b¯m (x) − b¯m (y)∗ e(a+σ )μ(m,m−1) Kβ γ eγ (c+σ )μ(m−1,m−2) x − y∗ γ + C δβx − y∗ m m m (a+σ )μ(m,m−1) γ (c+σ )μ(m−1,m−2) γ ∗ γ Ke e β + C δβ x − ym (31) (since x − y∗m < 1). Furthermore, proceeding as in (29) we obtain −1 C hm (x) − hm (y) ∗ m m

2δD −(c+σ )μ(m,m−1) e x − y∗m . 1 − e−σ

Thus, proceeding as in (28) and using Lemma 2, c¯m (x) − c¯m (y)∗ m ∗ γ 2δD −(c+σ )μ(m,m−1) e(−b+σ )μ(m+1,m) K Fm (x) − Fm (y)m+1 + e x − y∗m 1 − e−σ γ Ke(−b+σ )μ(m+1,m) e(−d+σ )μ(m+1,m) + C δe−(c+σ )μ(m,m−1) +

γ 2δD −(c+σ )μ(m,m−1) x − y∗m . e −σ 1−e

(32)

By (31) and (32), for each sufficiently small δ there exists K > 0 such that ∗ ∗

γ max b¯m (x) − b¯m (y)m , c¯m (x) − c¯m (y)m K x − y∗m , for each m ∈ Z and x, y ∈ X with x − y∗m < 1. This completes the proof of the theorem.

2

References [1] L. Barreira, Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lecture Ser., vol. 23, Amer. Math. Soc., 2002. [2] L. Barreira, Ya. Pesin, Nonuniform Hyperbolicity, Encyclopedia Math. Appl., vol. 115, Cambridge University Press, 2007. [3] L. Barreira, J. Schmeling, Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000) 29–70. [4] L. Barreira, C. Valls, A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations 228 (2006) 285–310. [5] L. Barreira, C. Valls, Hölder Grobman–Hartman linearization, Discrete Contin. Dyn. Syst. 18 (2007) 187–197. [6] L. Barreira, C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst. 22 (2008) 509–528. [7] L. Barreira, C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Math., vol. 1926, Springer, 2008. [8] G. Belicki˘ı, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, Funct. Anal. Appl. 7 (1973) 268–277. [9] G. Belicki˘ı, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys 33 (1978) 107– 177. [10] C. Chicone, Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr., vol. 70, Amer. Math. Soc., 1999. [11] D. Grobman, Homeomorphism of systems of differential equations, Dokl. Akad. Nauk SSSR 128 (1959) 880–881. [12] D. Grobman, Topological classification of neighborhoods of a singularity in n-space, Mat. Sb. (N.S.) 56 (98) (1962) 77–94. [13] M. Guysinsky, B. Hasselblatt, V. Rayskin, Differentiability of the Hartman–Grobman linearization, Discrete Contin. Dyn. Syst. 9 (2003) 979–984.


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[14] J. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol. 25, Amer. Math. Soc., 1988. [15] P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960) 610–620. [16] P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc. 14 (1963) 568–573. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer, 1981. [18] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980) 137–173. [19] F. Ledrappier, L.-S. Young, The metric entropy of diffeomorphisms I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985) 509–539. [20] P. McSwiggen, A geometric characterization of smooth linearizability, Michigan Math. J. 43 (1996) 321–335. [21] J. Moser, On a theorem of Anosov, J. Differential Equations 5 (1969) 411–440. [22] V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968) 197–221. [23] J. Palis, On the local structure of hyperbolic points in Banach spaces, An. Acad. Brasil. Cienc. 40 (1968) 263–266. [24] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv. 10 (1976) 1261–1305. [25] Ya. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math. Surveys 32 (1977) 55–114. [26] C. Pugh, On a theorem of P. Hartman, Amer. J. Math. 91 (1969) 363–367. [27] G. Sell, Smooth linearization near a fixed point, Amer. J. Math. 107 (1985) 1035–1091. [28] G. Sell, Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci., vol. 143, Springer, 2002. [29] S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957) 809–824. [30] S. Sternberg, On the structure of local homeomorphisms of euclidean n-space. II, Amer. J. Math. 80 (1958) 623– 631.


Prescribing Q-curvature problem on Sn Juncheng Wei a , Xingwang Xu b,∗ a Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, China b Department of Mathematics, National University of Singapore, 2 Science drive 2, Singapore 117543,

Republic of Singapore Received 30 October 2007; accepted 11 June 2009 Available online 16 July 2009 Communicated by I. Rodnianski

Abstract Let Pn be the n-th order Paneitz operator on Sn , n 3. We consider the following prescribing Qcurvature problem on Sn : Pn u + (n − 1)! = Q(x)enu

on Sn ,

where Q is a smooth positive function on Sn satisfying the following non-degeneracy condition: (Q)2 + |∇Q|2 = 0. Let G∗ : Sn → Rn+1 be defined by G∗ (x) = −Q(x), ∇Q(x) . ∗

G , Sn ) = 0, then the above equation has a solution. We show that if Q > 0 is non-degenerate and deg( |G ∗| When n is even, this has been established in our earlier work [J. Wei, X. Xu, On conformal deformation of metrics on Sn , J. Funct. Anal. 157 (1998) 292–325]. When n is odd, Pn becomes a pseudo-differential operator. Here we develop a unified approach to treat both even and odd cases. The key idea is to write it as an integral equation and use Liapunov–Schmidt reduction method. © 2009 Published by Elsevier Inc.

Keywords: Q-curvature; Existence; Pseudo-differential operator


E-mail addresses: [email protected] (J. Wei), [email protected] (X. Xu). 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.06.024

1996

J. Wei, X. Xu / Journal of Functional Analysis 257 (2009) 1995–2023

1. Introduction On a general Riemannian manifold M with metric g, a metrically defined operator Ag is said to be conformally invariant if under the conformal change in metric gu = e2u g, the pair of corresponding operators Agu and Ag are related by Agu (ϕ) = e−bu Ag eau ϕ

(1.1)

for all ϕ ∈ C ∞ (M) and some constants a and b. One such well known second order conformally invariant operator is the conformal Laplacian which is closely related to the Nirenberg problem (or Yamabe problem) and more generally, to the problem of prescribing Gauss (or scalar) curvature: Given a smooth positive function K (or R) defined on a compact Riemannian manifold (M, g0 ) of dimension two (or n 3), does there exist a metric g conformal to g0 for which K (or R) is the Gauss (or scalar) curvature of 4 the new metric g? If g = e2u g0 for n = 2 or g = u n−2 g0 for n 3, our problem is reduced to finding solutions to the following nonlinear elliptic equations: g0 u + Ke2u = K0

(1.2)

for n = 2, or n+2 4(n − 1) g0 u + Ru n−2 = R0 u, n−2

u > 0 on M

(1.3)

for n 3. (Here g0 denotes the Laplace–Beltrami operator of g0 , K0 is the Gauss curvature of g0 for n = 2 and while R0 is the scalar curvature of g0 for n 3.) The problem is to determine for which K, Eq. (1.2) (or (1.3)) admits a solution (in the case of (1.3), one requires u > 0). This problem has been studied extensively. See [7,8,11,14–16,23,24,31,35] and the references therein. In search for a higher order conformally invariant operator, Paneitz [28] discovered an interesting fourth order operator on compact 4-manifolds

2 P4 ϕ = ϕ + δ RI − 2 Ric dϕ 3 2

where δ denotes the divergence, d the differential and Ric the Ricci curvature of the metric g. Under the conformal change gu = e2u g, P4 undergoes the transformation (P4 )u = e−4u P4 (i.e., a = 0, b = 4 in (1.1) ). See [3,6,9,12] and [13] for a discussion of general properties of Paneitz operators. On a general compact manifold of dimension n, the existence of such an operator Pn with (Pn )u = e−nu Pn for even dimension is established in [22]. However Pn ’s explicit form is known n only for Euclidean space Rn with standard metric (Pn = (−) 2 ) and hence only for the sphere n n S with standard metric g0 . The explicit formula for Pn on S which appears in [3] and [2] is Pn =

⎧ n−2 ⎨ 2

k=1 (− + k(n − k − 1)), ⎩ (− + ( n−1 )2 ) 12 n−3 2 (− + k(n − k − 1)), 2

k=0

for n even, for n odd.


1997

In analogy to the second order case there exists some naturally defined curvature invariant Qn of order n which, under the conformal change of metric gu = e2u g0 , is related to Pn u through the following differential equation Pn u + (Qn )0 = (Qn )u enu

on M.

(1.4)

Motivated by the problem of the prescribing Gaussian curvature on S 2 , we pose the following prescribing Qn -curvature problem on Sn : Given a smooth function Q on Sn , find a conformal metric gu = e2u g0 for which (Qn )u = Q. We remark that there is a similar problem for general compact Riemannian manifolds. But since, in this case, the explicit expression for the operator Pn is unknown, we will not address the general prescribing Qn curvature problem. When n = 4, there have been many works recently. See [4,5,9,17,18,27] and the references therein. Clearly the above question is equivalent to finding a solution of the differential equation Pn u + (n − 1)! = Qenu

on Sn .

(1.5)

In our previous paper [32], we have treated the case n = 2m, i.e. n is even. (A different approach, based on curvature flows, was given recently in [26], and also a recent work [1].) In that case, the operator Pn is a point-wise operator and by stereographic projection to Rn , it simply becomes (−)m . In this paper, we shall consider both even and odd cases. Note that when n is n odd, the operator Pn involves (−) 2 which is a pseudo-differential operator. Our basic idea is to transform (1.5) into an integral equation: u(x) =

1 βn

log Rn

|y| Q(y)enu(y) dy + C0 , |x − y|

(1.6)

1 ) = βn δy (x). with βn being given by (−x )n/2 (log |x−y| This approach was first taken in [10] and then later in [34]. As in [7] and [8], there are three main steps in the proofs: first a priori estimates, then a perturbation result, and finally a continuation argument. For a priori estimates, we work directly with the integral equation (1.6). (We note that a similar idea has been used in a recent paper [19].) For perturbation result, we use a direct Liapunov–Schmidt reduction method. The continuation argument is the same as before. The novelty of our approach is that we don’t use any type of Moser–Trudinger inequalities. This also gives a new proof of the results in [7] and [8]. It is interesting to compare our approach here with the original approach of Chang and Yang in [7] and [8] and the flow-approach of Struwe [31]. To state the main results of this paper, we use the function introduced in [24]. For any smooth positive function Q on Sn , Q is called non-degenerate if it satisfies the non-degeneracy condition:

(Q)2 + |∇Q|2 = 0

(1.7)

on Sn . For a non-degenerate function Q on Sn , we can define the mapping G∗ : Sn → Rn+1 by G∗ (x) = −Q(x), ∇Q(x) .

(1.8)

1998

J. Wei, X. Xu / Journal of Functional Analysis 257 (2009) 1995–2023 G∗ |G∗ | C 1 on Sn ,

This mapping is well defined and it is never zero if the condition (1.7) is satisfied. Thus from Sn into Sn will be well defined. If we assume Q is C 3 function, then hence its degree

G∗ n deg( |G ∗| , S )

G∗ |G∗ |

is

is well defined. Now we can state our main result as the following. ∗

G n Theorem 1.1. Suppose that Q > 0 on Sn (n 3) is non-degenerate and deg( |G ∗ | , S ) = 0, then Eq. (1.5) has a solution.

For example, if Q satisfies (1.7) and Q(x)Q(−x) − ∇Q(x) · ∇Q(−x) 0,

∀x ∈ Sn

then Eq. (1.5) has a solution. The proof is similar to that of Corollary 1.1 of [24]. More examples of Q can also be found in [24]. 2. Preliminary results The main aim in this section is to prove the Pohozev identity for integral equations. In order to prove this, we need some regularity estimates and control on the solutions. First, we present the classification of solutions. Then two lemmas following it are on the regularity of the solutions. We would like to point out that those estimates clearly depend on the solution itself. Finally we study the linearized equation at the standard solutions for the integral equation. In particular, we try to classify the kernels of this linear operator. Let us start with the following classification of the solutions of an integral equation: Lemma 2.1. Let Q(x) ≡ (n − 1)! be a constant. Assume u ∈ C 2 (Rn ) is a solution to the integral equation u(x) =

1 βn

Rn

|y| log Q(y)enu(y) dy + C0 , |x − y|

(2.1)

for some constant C0 such that enu ∈ L1 (Rn ). Then u is given by u(x) = log

2λ , λ2 + |x − x0 |2

λ ∈ R + , x0 ∈ R n .

(2.2) 1

Proof. See [10] or [33,34]. We just remark that the constant λ is defined by λ = e− 2 C0 (e−C0 − |x0 |2 eC0 )1/2 + e−C0 > 0. 2 As a starting step for estimates, we show the following lemma. Lemma 2.2. Suppose w is a C 2 function on Rn (n 3) such that (a) Qenw is in L1 (Rn ) with 0 < m Q M for some constants m, M;


1999

(b) in the sense of weak derivative, w satisfies the following equation: (n − 2) w + βn

Rn

Q(y)enw(y) dy = 0. |x − y|2

(2.3)

Then there is a constant C > 0, depending on w, such that |w|(x) C on Rn .

Proof. Set α = Rn Q(y)enw(y) dy. Then assumption (a) implies that 0 < α < ∞. This also im plies that Rn enw(y) dy is finite with upper bound depending only on m and α. Therefore there exists a large constant R > 0 such that

1 (2.4) Q(y)enw(y) dy . 8 Rn \BR (0)

Now notice that (2.3) holds almost everywhere. Since w ∈ C 2 , it holds everywhere. For any x0 ∈ Rn \ BR+8 (0), we consider the solution h of the equation

Q(y)enw(y) [(−)h](x) = (n−2) B4 (x0 ) |x−y|2 dy in B4 (x0 ), βn (2.5) h=0 on ∂B4 (x0 ). Let 1 v1 (x) = βn

B4 (x0 )

log

16 |x − y|

Q(y)enw(y) dy

(2.6)

for all x ∈ B4 (x0 ). Since for all x, y ∈ B4 (x0 ), we have |x − y| |x − x0 | + |y − x0 | 4 + 4 = 8, hence, we conclude that v1 (x) 0 in B4 (x0 ). It is a routine calculation to check that (n − 2) (−)v1 (x) = βn

B4 (x0 )

Q(y)enw(y) dy. |x − y|2

Combining (2.5) and (2.7), we obtain (−)[±h − v1 ] 0 in B4 (x0 ), on ∂B4 (x0 ), ±h − v1 0 in weak sense. The maximum principle [21, Theorem 8.16] allows us to conclude that h(x) v1 (x), x ∈ B4 (x0 ).

(2.7)

(2.8)

(2.9)

2000


Now let us denote the measure Q(y)enw(y) dy/ B4 (x0 ) Q(y)enw(y) dy by dμ. Notice that βn = 1 n 2 Vol(S ) > 1. Thus Jensen’s inequality, together with (2.4), implies that

exp 4nh(x) dx

B4 (x0 )

B4 (x0 )

=

nβn v1 (x) dx exp 2 B4 (x0 ) Q(y)enw(y) dy

n exp 2

B4 (x0 )

B4 (x0 )

B4 (x0 ) B4 (x0 )

log

= B4 (x0 ) B4 (x0 )

16 |x − y| 16 |x − y|

16 |x − y|

n/2

n/2

dμ dx

dμy dx dx dμy

C,

(2.10)

where C is just a dimensional constant. Now we consider the function q(x) = w(x)−h(x) in the smaller ball B3 (x0 ). First we observe that, in weak sense, (q)(x) = (w)(x) − (h)(x)

(n − 2) Q(y)enw(y) =− dy − βn |x − y|2 Rn

=−

(n − 2) βn

Rn \B4 (x0 )

B4 (x0 )

Q(y)enw(y) dy |x − y|2

Q(y)enw(y) dy. |x − y|2

If x ∈ B3 (x0 ) and y ∈ Rn \ B4 (x0 ), then |x − y| |y − x0 | − |x − x0 | 1. Therefore we have 0 (−q)(x)

(n − 2) α, βn

(2.11)

in weak sense. Hence it follows from the weak Harnack principle ([21], Theorem 8.17) that sup q(x) C q + L2 (B

B2 (x0 )

3 (x0 ))

+ q L∞ (B3 (x0 )) .

(2.12)

We have seen in (2.11) the second term on the right is bounded independent of x0 . To estimate the previous one, we note that q + (x) = (w − h)+ (x) w + (x) + |h(x)| and also we have for all t 0, 2et t 2 . Thus we have


B3 (x0 )

+ 2 q (x) dx 2

eq

+ (x)

2001

dx

B3 (x0 )

2

ew

+ (x)

e|h(x)| dx

B3 (x0 )

2

1 + ew(x) e|h(x)| dx

B3 (x0 )

1/n

2

e

nw(x)

dx

B3 (x0 )

(n−1)/n e

n|h(x)|/(n−1)

dx

+2

B3 (x0 )

e|h(x)| dx

B3 (x0 )

C,

(2.13)

where C is independent of x0 by using (2.10), Hölder inequality as well as assumption (a). It follows that w(y) = q(y) + h(y) C + |h(y)| in the even smaller ball B2 (x0 ). This leads us to reach the estimate:

e4nw(y) dy e4nC e4n|h(y)| dy C1 , (2.14) B2 (x0 )

B2 (x0 )

where we have used (2.10). Next by Eq. (2.3), we have, for any |x0 | sufficiently large,

βn |w|(x0 ) = (n − 2) Rn

Q(y)enw(y) dy |x0 − y|2

M(n − 2)

Rn \B2 (x0 )

(n − 2)M 4

enw(y) dy + |x0 − y|2

B2 (x0 )

enw(y) dy |x0 − y|2

enw(y) dy Rn

+ (n − 2)M B2 (x0 )

1 dy |x0 − y|2p

1/p

·

1/q eqnw(y) dy

,

B2 (x0 )

where p and q are such that 1/p + 1/q = 1. Since n 3, choosing p = 4n−1 8 , then p > 1 and < 4. Clearly with those choices of p and q, the first integral of the second term in the q = 4n−1 4n−9 right side of above equation is bounded. The other two integrals are also bounded by (2.14) and the assumption (a). Therefore |w| is bounded almost everywhere on Rn \ BR+8 (0). But if w ∈ C 2 (Rn ), then w is continuous and hence is also bounded on B¯ R+8 (0). This finishes the proof of Lemma 2.2. 2 We also have the following estimate.

2002


Lemma 2.3. Suppose w is a C 2 function on Rn such that 0 (−)w(x) A on Rn for some

constant A and Rn Q(y)enw(y) dy = α < ∞ with 0 < m Q M. Then there exists a constant B, depending only on A, n, m, M and α, such that w(x) B on Rn . Proof. For any point x0 in Rn , let w1 be the solution of the Poisson’s problem

(−)w1 = (−w) := f w1 = 0

in B1 (x0 ), on ∂B1 (x0 ).

(2.15)

It follows from the elliptic estimate of Poisson’s equation (for example, [21], p. 189, Theorem 8.16) that w1 (x) sup w1 (x) C(n) sup |f | CA, B1 (x0 )

(2.16)

B1 (x0 )

since w1 (x) = 0 on ∂B1 (x0 ). Now we set w2 (x) = w(x) − w1 (x) in B1 (x0 ). Then it is obvious that (−)w2 = 0 in the unit ball B1 (x0 ) in weak sense. By mean value property for harmonic functions, we reach at the estimate

+ + w ∞ C (n) w dx , (2.17) 2 2 L (B (x )) 2 1/2

0

B1 (x0 )

where w2+ is the positive part of w2 . However, by the definition of w2 , we have w2+ w + + |w1 |. Notice that we have the obvious inequality

+

nw dx B1 (x0 )

enw dx α.

B1 (x0 )

Thus, combining those estimates, we get

w2+ dx

α A + ωn := C3 (n, A, α), n n

B1 (x0 )

where ωn is the volume of solid unit ball in Rn . Thus it follows from estimate (2.17) that + w 2

L∞ (B1/2 (x0 ))

C2 (n)C3 (n, A, α).

Finally by definition again, we have w = w1 + w2 ,


2003

thus, w + w2+ + |w1 | C2 (n)C3 (n, A, α) + CA, which is independent of x0 . Since w is C 2 , lemma follows.

2

The next lemma is the so-called Pohozev’s identity which implies some necessary conditions for the integral equation (2.1) to have a solution. Lemma 2.4. Suppose a C 2 function u satisfies the integral equation (2.1). Assume there exist constants 0 < m < M such that m Q M. Assume that eu ∈ Ln (Rn ). (1) If |x, C for some constant C > 0 and |x| sufficiently large, then

∇Q(x) | nu(x) 2 x, ∇Q(x) e dx = γ (γ − 2); n nβn R (2) If there exists a constant C > 0 such that |∇Q|(x) C for |x| sufficiently large, then

nu(x) dx = 0, ∇Q(x)e Rn where γ is given by 1 γ= βn

Q(x)enu(x) dx.

(2.18)

Rn

Proof. Part (1) has been shown in [34]. Notice that the fixed sign condition on Q plus the assumption on the integrability of the function eu implies that the constant γ is finite and Q(x)enu(x) is absolutely integrable over Rn . Therefore Theorem 1 in [34] can be applied. We should point out that the proof provided there is not complete. It only provided the formal calculation. Notice that the proof of Lemma 2.1 in [34] depends on this part. But when Q is constant, the conditions in this lemma is clearly fulfilled. Here, for completeness of our argument, we fulfill the detail with those extra assumption as we have stated. The method for both cases is the n same. We only deal with the second case. Since Qenu ∈ L∞ loc (R ) is absolutely integrable and u is of class C 2 , both sides of Eq. (2.1) are C 2 functions and we can take twice weak derivatives. This is to say that we can get: n−2 u(x) = − βn

Rn

Q(y)enu(y) dy, |x − y|2

(2.19)

in the sense of weak derivatives. Therefore, by Lemma 2.2, there exists a constant A > 0 such that 0 (−u) A and by Lemma 2.3, there exists a constant B such that u B which can be applied to conclude that Qenu is in Lp (Rn ) for any p 1. In fact, we have Qp epnu M p enu e(p−1)B since u B. Now through routine argument, we can see that we have the following: ∇u(x) = −

1 βn

Rn

(x − y)Q(y)enu(y) dy, |x − y|2

(2.20)

2004


in the sense of weak derivatives. By property of weak derivatives, we also have Q(x)∇enu(x) = −

nQ(x)enu(x) βn

Rn

(x − y)Q(y)enu(y) dy. |x − y|2

(2.21)

Now choose a smooth compact supported function η(t) such that η(t) = 1 for t 1 and η(t) = 0 for t 2 and also −2 η (t) 0 for all t. Multiplying both sides of (2.21) by η( |x| R) for all real number R > 0 and integrate over Rn , we obtain:

Q(x)η Rn

=−

1 βn

|x| nu(x) e ∇u(x) dx R

η

Rn

Rn

|x| x − y nu(y) Q(y)e dy Q(x)enu(x) dx. R |x − y|2

(2.22)

But the left-hand side of the same equation transforms, by integration by parts,

Q(x)η Rn

=−

1 n

Rn

|x| nu(x) e ∇u(x) dx R

|x| x |x| 1 enu(x) dx + Q(x)enu(x) dx. ∇Q(x) η η R nR R |x|

(2.23)

Rn

Notice that |η ( |x| R )| 2 for R |x| 2R and otherwise it vanishes. Then integrability of |Q|enu implies that the second integral approaches zero as R → ∞. Clearly the first integral goes to the integral of (∇Q)enu with the help of the integrability of (|∇Q|)enu over Rn . Now we consider the right-hand side. For each fixed R, by the Hardy–Littlewood–Sobolev x−y n(u(x)+u(y)) is absolutely integrable over Rn × Rn since inequality, the function |x−y| 2 Q(x)Q(y)e

we have Q(x)enu(x) ∈ L2n/(2n−1) (Rn ). Hence we can take the limit under the integral sign as R → ∞ by dominated convergence theorem. This implies that 1 n

∇Q(x)enu(x) dx = − Rn

1 βn

Rn

Rn

x −y nu(y) Q(y)e dy Q(x)enu(x) dx. |x − y|2

(2.24)

Again the right-hand side is absolutely integrable as a function over Rn × Rn , thus we can conclude that the integral vanishes by interchange variables x and y. Hence the second part of Lemma 2.4 follows. 2 Remark 2.5. In Lemma 2.1, we may multiply the function Q by a suitable constant to make γ = 2. Then the standard Pohozev’s identity holds.


2005

Finally, we discuss the non-degeneracy for the linearized integral equation of (2.1) at standard solutions. For simplicity, we set: 2Λ , UΛ,a (x) = log 2 Λ + |x − a|2

(2.25)

for Λ > 0 and a ∈ Rn . By changing variables, we only need to prove Theorem 2.6. Suppose that the bounded function φ(x) satisfies the integral equation φ(x) =

n! βn

|y| log enU1,0 (y) φ(y) dy + C. |x − y|

(2.26)

Rn

Then there are constants Cj for j = 0, 1, 2, . . . , n such that the function φ(x) is given as φ(x) =

n

(2.27)

Cj ψj (x),

j =0

where ψ0 (x) =

|x|2 − 1 , |x|2 + 1

ψj (x) =

2xj , 1 + |x|2

j = 1, 2, . . . , n.

(2.28)

Proof. First of all, we want to show that if φ(x) satisfies Eq. (2.26), then n! h := βn

enU1,0 (x) φ(x) dx = 0.

(2.29)

Rn

In fact, it follows from Eq. (2.26) and the definition of h that the following is true:

x φ |x|2

n! − h log |x| = βn =

n! βn

n! = βn

log

|y|

Rn

|x|| |x|x 2 − y|

Rn

1 | |y|y 2 − x|

log

log

Rn

1 |x − z|

enU1,0 (y) φ(y) dy + C

enU1,0 (y) φ(y) dy + C

z nU1,0 (z) dz + C, e φ |z|2

(2.30)

where we have used the special form for the function U1,0 . Since φ is bounded, the term h log |x| has to be bounded for x near zero which forces h = 0. Now it follows from (2.30) that, by the dominated convergence theorem, φ has a limit at ∞ since the right-hand side of Eq. (2.30) has limit as |x| → 0. Thus, since h = 0, the function 2 n n n g(x) = φ(y) − h2 log 1+|y| 2 is a well defined function on S where π : S → R is the standard

2006


stereographic projection of the sphere Sn into Rn and x = π −1 (y) for all y ∈ Rn . And also it is well known that Pn is a conformal invariant operator, so

Pn g(x) g(x) dσx =

Sn

(−)n/2 φ(y) φ(y) dy

Rn

= n!

enU1,0 (y) φ 2 (y) dy

Rn

= n!

g(x)2 dσx ,

(2.31)

Sn

where the second equality follows from (2.26). Notice that any function satisfying (2.26) also satisfies the following differential equation

(−)n/2 φ (y) = n!enU1,0 (y) φ(y).

(2.32)

Now just observe that, by definition of the operator Pn , it follows from Eq. (2.32) that Pn g = n!g. Thus g is a first eigenfunction with eigenvalue n!. It is well known [9] that the first eigenspace of Pn on Sn is spanned by {ψ0 , ψ1 , . . . , ψn } under stereographic projection. Theorem 2.6 follows. 2 Remark 2.7. We would like to point out that several facts, specially the properties of the operator Pn , we have used in the proof of above theorem can be traced back to Chang and Yang’s earlier paper [8]. The bound for φ can also be seen by direct calculation from the integral representation (2.26) and potential estimate. 3. Some a priori estimates In this section, we want to prove the a priori estimates for the solutions of Eq. (2.1) with given bounded positive smooth function Q satisfying the following non-degeneracy condition (Q)2 + |∇Q|2 = 0.

(3.1)

Since Pn is a pseudo-differential operator, standard elliptic regularity estimates do not apply. We have to work with the integral equation (2.1). We start with the estimates on the derivatives of the solution under the assumption that the solution has an upper bound and Laplace of the solution is bounded. Lemma 3.1. Let w satisfy (2.1). Suppose there exist positive constants A and B such that w B and 0 (−w) A. For each positive integer 0 < k n − 1, there exists a constant Ck (A, B) such that k ∇ w (x) Ck . And furthermore, w is C ∞ .


2007

Proof. It follows from the integral representation of the function w, we obtain: k ∇ w (x) Bk

Rn

Q(y)enw(y) dy |x − y|k

Bk Me

nB B1 (x)

dy + Bk |x − y|k

Rn \B1 (x)

Q(y)enw(y) dy |x − y|k

Ck .

(3.2)

In this estimate, Bk is just a constant depending only on n and k and B is the constant in the assumption. To see that w ∈ C ∞ , we simply notice that Eq. (2.3) implies βn (w)(x) = − n−2

Rn

Q(y)enw(y) dy |x − y|2

x log = Rn

=

y log

Rn

log = Rn

1 |x − y| 1 |x − y|

Q(y)enw(y) dy Q(y)enw(y) dy

|y| y Q(y)enw(y) dy + c0 |x − y|

(3.3)

through integration by parts, where c0 = − Rn (log |y|)y (Q(y)enw(y) ) dy. Let us explain this last step in more detail: fix a point x, for R sufficiently large, the integral can be decomposed into the integral over the ball BR (x) with center at x and the part on Rn \ BR (x). For the integral over the ball, we use Green’s second identity to convert it into the integral with on Qenw plus the boundary terms. For the integral over the outside of the ball, the integral equals to

1 nw(y) ) dy which goes to zero as R → ∞. Then the integrability of Qenw n R \BR (x) |x−y|2 (Q(y)e will imply that those two boundary integrals in the integral over the finite ball also tend to 0 as R → ∞. Thus the formula holds. Repeat above estimates to conclude that w is in C n−1 . Inductively, we see that w ∈ C ∞ . 2 We also need more accurate estimate. Lemma 3.2. Let w satisfy (2.1) with assumptions as in Lemma 3.1. Then w ∈ C n−1,α for any 0 α < 1. Furthermore, there is a constant C(n, α, A, B) > 0 such that enw C n−1,α C(n, α, A, B). Proof. When n is even, it is clear since (−)n/2 w = Q(x)enw(x) is bounded, standard elliptic estimate implies the result.

2008


When n is odd, we write it as n = 2k + 1. Then from the integral representation of w, we have (−)k w(x) =

Ck βn

Rn

Q(y)enw(y) dy. |x − y|2k

By the inequality, |x − y|−2k − |x1 − y|−2k 2k|x − x1 |α |x − y|−2k−α + |x1 − y|−2k−α , for any real number 0 α < 1 (Eq. (3) on p. 225 [25]), we have (−)k w(x) − (−)k w(x1 )

2kCk Q(y)enw(y) Q(y)enw(y) |x − x1 |α dy + dy βn |x − y|2k+α |x1 − y|2k+α Rn

Rn

C|x − x1 | , α

(3.4)

where the last estimate is just same as the estimate (3.2). The last statement follows from our assumptions as well as above estimates. This completes the proof of Lemma 3.2. 2 Now we are ready to state and prove the main result of this section. Theorem 3.3. Suppose Q is a positive smooth function defined on Sn (n 3), satisfying the non-degeneracy condition (3.1). Then there are constants C1 and C2 such that any solution u of the equation: Pn u + (n − 1)! = Qenu

on Sn ,

(3.5)

satisfies the bounds −C1 u C2 .

(3.6)

Proof. First let us show that there exists a constant C2 such that u C2 . Suppose this is not the case, there would exist a sequence {uk } such that maxSn uk (x) = uk (xk ) → ∞ and xk → x0 as k → ∞. Choose stereographic projection π −1 : Sn → Rn with north pole at −x0 and set wk (y) = 2 uk (π(y)) + log 1+|y| 2 . Then one checks that w satisfies the integral representation wk (x) =

1 βn

log Rn

|y| Q(y)enwk (y) dy + Ck , |x − y|

(3.7)

where Ck = wk (0) and Q(y) = Q(π(y)). Translating wk by a constant, and still denoting it by wk , we get

ρk |y| Q(y)enwk (y) dy, wk (x) = log (3.8) βn |x − y| Rn


where ρk = enCk → ∞ as k → ∞. An important fact is that

ρk Q(y)enwk (y) dy = 2βn .

2009

(3.9)

Rn

(This follows from integration of (3.5) on Sn and a change of variable.) Since wk (π −1 (xk )) = uk (xk ) + log 1+|π −12 (x )|2 , when k → ∞, π −1 (xk ) → 0, we conclude k

that wk (yk ) → ∞ where yk = π −1 (xk ). Now set vk (z) = wk (yk + k z) − wk (yk ) = uk (π(yk + 1+|yk |2 k z)) − uk (π(yk )) + log[ 1+|y 2 ] with k to be determined later. k +k z| Observe that the fact that yk → 0 as k → ∞ implies that we can assume for all k, |yk | 1. Hence there exists an absolute constant C > 0 such that vk (z) C for all z ∈ Rn no matter what k > 0 is. For example, we can take C = log 2. This follows from the assumption that uk (π(yk )) is the maximum for uk on the sphere S n as well as the definition of vk . It follows from the integral representation of wk that ρk enwk (yk ) vk (z) = βn =

=

|yk − y| Q(y)en[wk (y)−wk (yk )] dy log |yk + k z − y|

Rn

kn ρk enwk (yk ) βn

Rn

kn ρk enwk (yk ) βn

Rn

|t| Q(yk + k t)en[wk (yk +k t)−wk (yk )] dt log |z − t| |t| Q(yk + k t)envk (t) dt. log |z − t|

(3.10)

Now we choose k > 0 such that kn ρk enwk (yk ) = 1. The advantage for this choice of k is that, by Eq. (3.9), we have

Q(yk + k t)envk (t) dt = 2βn . (3.11) Rn

Observe that we also have k → 0 as k → ∞, since wk (yk ) + Ck → ∞ as k → ∞. Therefore Eq. (3.10) implies that vk (z) =

1 βn

log Rn

|t| Q(yk + k t)envk (t) dt. |z − t|

(3.12)

Similarly to the argument in Lemma 2.4, it can be shown that vk (z) satisfies Eq. (2.3). Since vk (z) is bounded from above, envk (z) C for some constant C, independent of k. As a consequence of this fact, |vk (z)| will be also uniformly bounded with upper bound independent of k which can be shown as follows: for any point z ∈ Rn , by Eq. (2.3), we have

nvk (y) vk (z) = n − 2 Q(y)e dy 2 βn |z − y| Rn

2010


n−2 MC βn

B1 (z)

1 dz + (n − 2)M |z − y|2

MC (n − 2)M ωn−1 + βn β

Rn \B1 (z)

envk (y) dy |z − y|2

envk (y) dy Rn

:= C1 .

n Clearly the constant C1 is independent of k since, by Eq. (3.11), Rn envk (y) dy 2β m , where m is the lower bound of the function Q. We can apply Lemma 3.2 to conclude that there is a constant C, independent of k but dependn−1,α ing on α such that envk (z) C n−1,α C. Thus for some 0 < α0 < 1, vk → v0 in Cloc 0 (Rn ) as 2λ k → ∞. By Lemma 2.1, v0 = log[ λ2 +|y−y 2 ]. By the definition of vk , we have v0 = U1,0 . 0| We need the following lemma on the decay of vk (z): Lemma 3.4. For all δ ∈ (0, 2), there exist Rδ , Cδ > 0 such that vk (z) (2 − δ) ln

1 + Cδ , |z|

∀|z| Rδ .

(3.13)

Proof. The proof is standard. For the reader’s convenience, we include it here. Let δ ∈ (0, 2) be fixed. Since vk → U1,0 locally in Rn , (log |z|)envk (z) is in L1 (Rn ). Thus by (3.11), we may choose k large and Rδ such that δ βn . dt 2 − 2

Q(yk + k t)e

nvk (t)

R |t| 2δ

We then compute 1 vk (z) = βn

+

R |t| 2δ

log

+

R |t| 2δ

, |t|2|z−t|

|t|

Rδ 2

× Q(yk + k t)envk (t) dt 1 1 δ 1 2βn − βn log +C + βn 2 |z| βn 1 δ log + Cδ 2− 2 |z|

, |t|2|z−t|

log |t|

|t| |z − t|

Rδ 2

1 Qenvk dt |z − t|

, |t|2|z−t|

where C depends on the integral of (log |z|)envk (z) and βn and Cδ combines C and some constants depending on δ. Let us explain a little bit more on first inequality: for the first part we note that |z||t| |t| |t| |z−t| (1 + |z−t| )|t| and |z−t| 1 if |t| (Rδ )/2 and |z| (Rδ )/2. For the second part of the |t| log 2. For the third part, the term with log |t| will integral, if |t| 2|z − t|, we have log |z−t| be absorbed to the first term with log |t| while the another term is left as it is. For the second last


2011

inequality, simply notice that if |z − t| 1, then the integral is negative while the integration over the ball B1 (z) is finite as we can easily see. Thus the lemma follows. 2 Let us continue the proof of Theorem 3.3. So far we have shown that the assumptions for Lemma 2.4 have fulfilled, so it is ready to be for use. Observe that in the definition of γ in Eq. (2.18), we have γ = 2 in current situation as shown by Eq. (3.11). Since Q is a smooth function on Sn , |∇Q| is clearly bounded, by second case of Lemma 2.4, we have

0=

∇Q(yk + k t)envk (t) dt Rn

=

∇Q(yk + k t) − ∇Q(yk ) envk (t) dt +

Rn

∇Q(yk )envk (t) dt.

(3.14)

Rn

The first term, as k → +∞, approaches zero by the Lebesgue’s dominated convergence theorem since ∇Q is bounded and vk has decay (3.13). Thus we obtain: ∇Q(yk ) → 0,

that is,

∇Q(0) = 0.

(3.15)

In fact, more can be concluded from Eq. (3.14). Namely, we have ∇Q(yk ) = O(k ).

(3.16)

Next since Q is a smooth function on S n , we have that |y, ∇Q(y) |, where y = π −1 (x) for x ∈ S n , is bounded. Hence, by Lemma 2.4(1), we have

0=

t, ∇Q(yk + k t) envk (t) dt

Rn

=

nv (t) nvk (t) k t, ∇Q(yk + k t) − ∇Q(yk ) e dt + ∇Q(yk ), te dt

Rn

=

Rn

Qij (yk )k ti tj envk (t) dt + o(k ).

(3.17)

Rn

To check the last equality above, we observe that v0 , the limit of the sequence {vk } is radially symmetric with respect to the point 0. It is because v0 does satisfy the integral equation (2.1) with Q(y) = Q(0). Hence we easily conclude that

zenv0 (z) dz = 0. Rn

Thus we obtain:

(3.18)

2012


∇Q(yk ),

tenvk (t) dt = ∇Q(yk ), t envk (t) − env0 (t) dt

Rn

Rn

= ∇Q(yk ),

t en(vk (t)−v0 (t)) − 1 env0 (t) dt

Rn

= o(k ),

(3.19)

where in the last step, we have used the estimate (3.16) and the decay (3.13). Now, by similar reasons, Eq. (3.17) implies that

0 = k Q(yk ) envk (t) |t|2 dt + o(k ).

(3.20)

Rn

Thus we have Q(0) = 0 which contradicts the non-degeneracy assumption on Q (see Eq. (3.1)). Once the upper bound on the solution u is available, the lower bound is easy. Notice that every 2 solution u can be written as w(x) − log 1+|x| 2 with w satisfying the integral equation (2.1). It 2 will not be hard to see that w(x) − log 1+|x| 2 has a lower bound independent of w. Observe that by Eq. (2.1). Hence we have

x−y 1 ∇w(x) = − Q(y)enw(y) dy. βn |x − y|2 Rn

This implies that 1/2 1 1 + |x|2 |∇w|(x) βn

Rn

(1 + |x|2 )1/2 Q(y)enw(y) dy |x − y|

[2(1 + |x|2 )]1/2 MenC

1/2

βn

Rn

1 2 ( )n dy 2 |x − y| 1 + |y|2

1/2

C0 ,

(3.21)

2 where we have used the Hölder inequality, the fact that w(x) − log 1+|x| 2 = u C and

Rn

n 2 1 2 . dy = C (−) log 1 |x − y|2 1 + |y|2 1 + |x|2

2 The latter identity is due to the fact that log 1+|x| 2 is a solution of Eq. (2.1) with Q = constant, 2 take Laplace both sides to get the equation. Now since (−)(log 1+|x| 2)=

2n 1+|x|2

−

4|x|2 , (1+|x|2 )2

it

follows from a simple argument that (1 + |x|2 )1/2 |∇(w(x) − log for some constant, n n which in turn implies that |∇u| C on S and diameter of S is finite, hence u is bounded from below too. 2 )| C 1+|x|2


This finishes the proof of Theorem 3.3.

2013

2

Remark 3.5. With the help of this theorem, we can conclude that solutions of Eq. (3.5) are uniformly bounded in C n−1,α (S n ) for any constant 0 < α < 1. We shall use this conclusion in the rest of our argument implicitly. 4. Perturbation result In this section, we use Liapunov–Schmidt reduction method to solve Eq. (2.1) with Q − (n − 1)! C 2 (Rn ) < and sufficiently small. (Similar approach was used by Rey and Wei [29,30].) This approach is different from the usual one adopted by Chang and Yang [7,8]. Here we don’t use any type of Moser–Trudinger inequalities. ˆ Of course if Q is non-degenerate in Let us rewrite the function Q as Q = (n − 1)!(1 + Q). ˆ We consider the integral equation (2.1). To be more precise, we write the sense of (1.7), so is Q. the equation through a non-local operator form: S[u] := (−)n/2 u −

2βn (n − 1)!Q(x)enu(x)

. nu(x) dx Rn Q(x)e

(4.1)

In this section, we should construct a function u such that S[u] = 0 and such that it can be lift to Sn so that this is a solution we are looking for. The solution will have the form u(x) = UΛ,a (x) + φ(x),

(4.2)

where (Λ, a) ∈ (0, 1] × Sn will be chosen later, φ(x) is relatively small and UΛ,a (x) is given by UΛ,a (x) = log

2Λ . Λ2 + |x − a|2

(4.3)

Observe that if u(x) takes the form (4.2) with φ uniformly bounded on Rn and lim|x|→∞ φ(x) exists, clearly we can lift it to Sn by stereographic projection. Now we substitute (4.2) into Eq. (4.1) to obtain S[UΛ,a + φ] = S[UΛ,a ] + L[φ] + N [φ],

(4.4)

where 2βn (n − 1)!Q(x)enUΛ,a (x)

, nUΛ,a (x) dx Rn Q(x)e

2βn n!enUΛ,a φ 2βn n!enUΛ,a Rn enUΛ,a (x) φ(x) dx n/2

+ L[φ] = (−) φ − , nUΛ,a dx ( Rn enUΛ,a (x) dx)2 Rn e S[UΛ,a ] = (−)n/2 UΛ,a −

(4.5) (4.6)

and N [φ] = O y 2n |φ| + |φ|2 y 2n ,

(4.7)

2014


where y 2 = (1+|y|2 ). Here the estimate (4.7) follows from the definition and Taylor expansion. In fact, by definition, we have N(φ) = 2βn (n − 1)!

QenUΛ,a Qen(UΛ,a (x)+φ)

− nUΛ,a (x) dx n(UΛ,a (x)+φ(x)) dx Rn Q(x)e Rn Q(x)e

enUΛ,a (x) Rn enUΛ,a (x) φ(x) dx enUΛ,a φ

. − + 2βn n! nUΛ,a (x) dx ( Rn enUΛ,a (x) dx)2 Rn e

Then by double Taylor expansion, one in φ and another in (since Q depends on ), we can easily get the result. Note that

S[Uλ,a ] + N[φ] dy = 0, (4.8) Rn

since Rn L(φ) dy = 0. Now we begin with Lemma 4.1. Suppose the bounded function φ satisfies

nUΛ,a ψ Λ,j φ = 0 for j = 0, 1, 2, . . . , n with ψΛ,j given by Rn e ψΛ,0 (x) =

Rn

enUΛ,a φ dx = 0 and

|x − a|2 − Λ2 , Λ(Λ2 + |x − a|2 )

(4.9)

2(xj − aj ) . Λ2 + |x − a|2

(4.10)

and ψΛ,j (x) =

2 n If φ − log 1+|x| 2 can be lifted to be a smooth function on S , then there is a constant c0 > 0 such that

(−)n/2 φ φ dx − n!(1 + c0 ) enUΛ,a (x) φ 2 (x) dx 0. (4.11) Rn

Rn

Proof. When we consider it as the eigenvalue problem on Sn for generalized Paneitz operator Pn , the inequality we stated above is clear since it is well know that the first non-zero eigenvalue of Pn on Sn is always equal to n!. 2 Now we shall adopt the following notation in the future argument:

φ ∗ = sup φ(y),

(4.12)

y∈Rn

and

f ∗∗ = sup y 2n f (y). y∈Rn

(4.13)


2015

We still use C(Rn ) to denote the functions in C(Rn ) which have limits at infinity. Note that such functions can be lifted to Sn . Lemma 4.2. Let f be a function on Rn such that f ∗∗ is finite and bounded function φ is a solution of the equation L[φ] + f +

n

Rn

f dy = 0. Assume the

Cj enUΛ,a ψΛ,j = 0,

(4.14)

j =0

for some constants Cj such that 0, 1, 2, . . . , n. Then we have

Rn

enUΛ,a φ dx = 0 and

Rn

enUΛ,a ψΛ,j φ dx = 0 for j =

φ ∗ C f ∗∗ ,

(4.15)

for some positive constant C > 0 which depends on the upper bound of n + Λ + |a| only.

Proof. Since Rn enUΛ,a (x) dx = 2βn which is independent of Λ, a, by taking the derivative with respect to Λ and a, we find that

enUΛ,a (x) ψΛ,j (x) dx = 0, (4.16) Rn

for j = 0, 1, 2, . . . , n. Therefore, together with the fact that L[ψΛ,j ] = 0 for every j , we have

L[φ]ψΛ,j dx = 0, (4.17) Rn

for all j by integration by parts and the assumption. Thus if we multiply Eq. (4.14) by ψΛ,j and integrate it over the space Rn , we obtain the estimate |Cj | = O f ∗∗ , j = 0, 1, 2, . . . , n. (4.18) By the integral representation of Eq. (4.14) we have, for some constant β, n! φ(x) = βn

Rn

1 + βn

|y| log enUΛ,a (y) φ(y) dy |x − y|

n Cj |y| log f (y) dy + ψΛ,j + β. |x − y| n!

(4.19)

j =0

Rn

Since φ is bounded, arguments as in (2.30) show that φ(x) has limit as |x| tends to infinity. Therefore it can be lifted to be a smooth function on Sn . Next multiplying Eq. (4.14) by φ and integrating the resulting equation, we obtain

(−)n/2 φ φ dx − n! enUΛ,a (x) φ 2 (x) dx + f φ dx = 0. (4.20) Rn

Rn

Rn

2016


It follows from the estimate of Lemma 4.1 and (4.20) that there is a constant B0 such that

enUΛ,a (x) φ 2 (x) dx B0 |f ||φ| dx. (4.21) Rn

Rn

This implies that

enUΛ,a (x) φ 2 (x) dx B φ ∗ f ∗∗ .

(4.22)

Rn

Now by taking the derivative of Eq. (4.19), we have the estimate:

n Cj nU (y) −nU (y) Λ,a Λ,a ∇φ(x) C φ(y) + e f (y) dy + e n! j =0

Rn

C f ∗∗ ,

(4.23)

by using the Hölder inequality together with (4.21) in the first term and definition for · ∗∗ in the second term. The last term follows from (4.18). Then we have estimate on the function φ at 0 as follows:

2βn φ(0) = enUΛ,a (y) φ(0) − φ(y) dy Rn

∇φ L∞

enUΛ,a (y) |y| dy Rn

C f ∗∗ .

(4.24)

Here we have used the estimate (4.23). Now it also follows from (4.19) that the following estimate holds true: |x||y| nUΛ,a (y) φ(x) − φ(0) = n! log e φ(y) dy β |x − y| n Rn

n 1 Cj |x||y| ψΛ,j (x) − ψΛ,j (0) + log f (y) dy + β |x − y| n! n

Rn

j =0

C f ∗∗ ,

(4.25)

where the first line was achieved by the observation that Rn enUΛ,a (y) φ(y) dy = 0 and

Rn f (y) dy = 0, while the second inequality follows the standard potential estimates and (4.18). Now clearly the required estimate follows from the triangle inequality. 2

Lemma 4.3. For every function f with the property that f ∗∗ < ∞ and Rn f dx = 0 and every point (Λ, a), there exist constants Cj (Λ, a, f ) for j = 0, 1, 2, . . . , n such that Eq. (4.14) has a unique solution.


2017

Proof. The uniqueness of φ has been shown in Lemma 4.2. Notice that for every fixed f , there is only one set of constants Cj for j = 0, 1, 2, . . . , n such that Eq. (4.14) has a solution. This follows from the estimate (4.18). Now for a fixed function f with the required property and for a given point (Λ, a), we choose the constants Cj according to the equation

ψΛ,j f dx + Cj

Rn

2 enUΛ,a (x) ψΛ,j (x) dx = 0.

(4.26)

Rn

Now we define the pre-Hilbert space H by

enUΛ,a (x) ψΛ,j (x)φ(x) dx = 0 (4.27) H := φ ∈ C Rn enUΛ,a (x) φ(x) dx = 0; Rn

Rn

with the inner product defined by

|x||y| φ, ψ = log en(UΛ,a (x)+UΛ,a (y)) φ(x)ψ(y) dx dy, |x − y|

(4.28)

Rn Rn

for any φ, ψ in H . Notice that it is a simple consequence of potential theory that H is a pre-Hilbert space. In fact, with the definition of the inner product as above, bi-linear and symmetric properties are easy to see. For non-negativity, we note that φ, φ =

|x||y| log en(UΛ,a (x)+UΛ,a (y)) φ(x)φ(y) dx dy |x − y|

Rn Rn

|x||y| δ [ |x−y| ] − 1 lim en(UΛ,a (x)+UΛ,a (y)) φ(x)φ(y) dx dy δ δ→0+

= Rn Rn

[ |x||y| ]δ − 1 |x−y| en(UΛ,a (x)+UΛ,a (y)) φ(x)φ(y) dx dy = lim δ δ→0+ Rn Rn

0.

(4.29)

Now Eq. (4.14) can be written as n! φ− βn

|x||y| log enuΛ,a (y) φ(y) dy |x − y|

Rn

1 = βn

n Cj |x||y| log f (y) dy + ψΛ,j (x) + β |x − y| n!

Rn

:= fˆ,

j =0

(4.30)

2018


where the constants Cj are given by Eq. (4.26) and β is chosen so that the integral over Rn of enUλ,a fˆ is equal to 0. If we denote the second term on the left of Eq. (4.30) by T [φ], then the equation can be simplified to be φ − T [φ] = fˆ.

(4.31)

Clearly T is a linear operator from C(Rn ) into itself. Suppose S is a bounded set in C(Rn ). It follows from the definition that T (S) is also a bounded set in C(Rn ). Furthermore for any φ ∈ S with φ ∗ A, we have |x2 − y| nUΛ,a (y) T (φ)(x1 ) − T (φ)(x2 ) = n! log e φ(y) dy βn |x1 − y| Rn

|x1 − x2 |

1/2

C(A) Rn

enUΛ,a (y) dy + |x1 − y|1/2

Rn

enUΛ,a (y) dy |x2 − y|1/2

C1 |x1 − x2 |1/2 .

(4.32)

1/2 (|x − y|−1/2 + |x − y|−1/2 ). 2 −y| Here we have used the inequality: | log |x 1 2 |x1 −y| | |x1 − x2 | On the other hand, similar to (2.30), we have

x |y| n! T (φ) = log enUΛ,a (y) φ(y) dy βn |x|| |x|x 2 − y| |x|2 Rn

=

n! βn

n! = βn

log

Rn

log

Rn

1 y | |y|2 − x| 1 |x − z|

enUΛ,a (y) φ(y) dy

z nUΛ,a (z) dz. e φ |z|2

(4.33)

Similar to (4.32), we obtain x2 T (φ) x1 C1 |x1 − x2 |1/2 . − T (φ) |x |2 |x |2 1

(4.34)

2

Using (4.32)–(4.34) and the Ascoli–Arzela theorem, T (S) is a relatively compact set in C(Rn ). It follows that T : C(Rn ) → C(Rn ) is a compact operator. Thus we conclude that I − T : C(Rn ) → C(Rn ) is a Fredholm operator. By Fredholm’s alternative, (4.31) has a solution provided fˆ is perpendicular to the kernel of I − T in the sense that fˆ is in H . However, this latter condition does satisfy due to the choice of β as well as Eq. (4.26) and our previous estimates imply that fˆ ∗ is bounded with bound in terms of f ∗∗ . Hence the existence follows and we finish the proof of Lemma 4.3. 2 Notice that in previous lemma, the constants Cj as well as β all depend on Λ and a. We need to get rid of those constants. Let us denote the map f → φ in Lemma 4.3 by A(φ).


2019

Lemma 4.4. There exists a unique φ = φΛ,a such that L[φ] + S[UΛ,a ] + N [φ] +

n

Cj enUΛ,a (x) ψΛ,j = 0.

(4.35)

j =0

Moreover, there is a constant C > 0 such that φ ∗ C with = Q − (n − 1)! C 2 (Rn ) . Also, the map (Λ, a) → φΛ,a is continuous. Proof. The tool we are going to use is the contraction mapping principle. In order to do so, first of all, let us rewrite Eq. (4.35) in its equivalent form: φ = A S[UΛ,a ] + N [φ] := B[φ] (4.36) where A(φ) is just defined above. For a positive constant C1 , define a convex set in H by Z := φ | φ ∗ < C1 .

(4.37)

Observe that Eq. (4.8) makes the definition of the operator B meaningful. Next we have B[φ1 ] − B[φ2 ] C + φ2 ∗ + φ1 ∗ φ1 − φ2 ∗ . ∗

(4.38)

Finally there exists a constant C > 0 such that the following inequality holds true: S[UΛ,a ] + N [φ] C 1 + φ ∗ φ 2 . ∗ ∗

(4.39)

Eqs. (4.8), (4.38) and (4.39) together imply that the operator B is a contraction mapping from Z into Z, maybe with a different constant C if we choose sufficiently small. Hence B has a fixed point. The continuity of φΛ,a on the parameters Λ, a follows from the integral representation formula. Hence Lemma 4.4 holds true. 2 Lemma 4.5. The solution given in previous Lemma 4.4 can always be lifted to be a smooth function on Sn . Proof. It is not hard to see that φΛ,a as well as its derivatives has a limit at infinity by the dominated convergence theorem. Keep in mind that φΛ,a is bounded by its nature. 2 Let us now compute the asymptotic expansions of Cj (Λ, a). Multiplying Eq. (4.35) by ψΛ,l , we obtain n

j =0 Rn

−L[φΛ,a ] − S[UΛ,a ] − N [φΛ,a ] ψΛ,l dx

enUΛ,a (x) ψΛ,j (x)ψΛ,l (x) dx = Rn

=− Rn

S[UΛ,a ]ψΛ,l dx + O 2 .

(4.40)

2020


We should point out that in this calculation we have used Eq. (4.17) as well as the 2 fact

that N [φΛ,a ] ∗∗ C . With this calculation, we have left just to compute the term S[U ]ψ dx. To do this, let us recall the definition of S[UΛ,a ] to have Λ,a Λ,l Rn Q(x)enUΛ,a (x) βn

2 Rn Q(x)enUΛ,a (x) dx ˆ 1 + Q(x) βn enUΛ,a (x) . = 1−

nU (x) ˆ Λ,a 2 dx Rn (1 + Q(x))e

S[UΛ,a ] = (−)n/2 UΛ,a −

(4.41)

Thus it follows from this equation that

Rn

ˆ βn 1 + Q(x) 1− enUΛ,a (x) ψΛ,l dx S[UΛ,a ]ψΛ,l dx =

nUΛ,a (x) dx ˆ 2 (1 + Q(x))e n R Rn

βn nUΛ,a (x) ˆ ψΛ,l (x) dx Q(x)e = nUΛ,a (x) dx ˆ 2 Rn (1 + Q(x))e n R

nUΛ,a (x) ˆ (4.42) ψΛ,l (x) dx + O 2 . = − Q(x)e Rn

Notice that the functions ψΛ,j satisfy the relations:

enUΛ,a (x) ψΛ,i (x)ψΛ,j (x) dx = γj−1 δij .

(4.43)

Rn

Now Eqs. (4.40)–(4.43) imply that

Cj (Λ, a) = −γj

nUΛ,a (x) ˆ ψΛ,l (x) dx + O 2 . Q(x)e

(4.44)

Rn

Now we define a mapping G from Bn+1 into Rn+1 by ˆ G(z) = G0 (z), G1 (z), . . . , Gn (z) ∈ Rn+1 where

Gl (z) =

ˆ Q(x)e

nU

1 z (x) 1−|z| ,π( |z| )

ψ

1 1−|z| ,l

(x) dx,

(4.45)

Rn

for l = 0, 1, 2, . . . , n. ˆ Bn+1 , 0) = 0, then there exists a Next by Eq. (4.44), we know that if the degree deg(G, n+1 point z0 ∈ B such that Cj (Λ, a) = 0 for all j = 0, 1, 2, . . . , n where Λ = (1 − |z0 |) and a = π −1 (z0 /|z0 |).


2021

ˆ B n+1 , 0) = 0, we need to do several calculations. First of all, we To see the fact that deg(G, note that for j 1, we have

ˆ j (Λ, a) = G

nUΛ,a (x) ˆ ψΛ,j (x) dx Q(x)e

Rn

1 = 2Λ =c

ˆ Q(Λy + a)yj

Rn

2 1 + |y|2

n+1 dy

∂ Qˆ (a) + o(Λ), ∂xj

(4.46)

as Λ → 0 for some constant c > 0 depending only on the dimension n. Similarly for j = 0, we have ˆ 0 (Λ, a) = G

nUΛ,a (x) ˆ ψΛ,0 (x) dx Q(x)e

Rn

1 = 2Λ

2 ˆ Q(Λy + a) |y| − 1

Rn

2 1 + |y|2

n+1 dy

ˆ + o(Λ), = −c1 ΛQ(a)

(4.47)

as Λ → 0 and some constant c1 > 0 depending only on n which might be different from c in above. Set δ = c1 Λ/c to obtain: ˆ ˆ ˆ G(z) = c δ(−Q)(a), ∇ Q(a) + o(Λ),

(4.48)

where a = π −1 (z/|z|) with |z| 1. Now we define another mapping as follows: ˆ ˆ δ∇ Q(a) . Gδ (z) = −Q(a),

(4.49)

Then it is clear that 2 ˆ 2 + Q(a) ˆ ˆ δ + o(δ) . G(z) · Gδ (z) = c ∇ Q(a) ˆ if δ is sufficiently small, By the non-degeneracy assumption on the function Q, hence on Q, ˆ ˆ G(z) · Gδ (z) > 0 on ∂B1−δ (0). Now we fix δ small, then G(a) · Gδ (a) > 0 on Sn . By simple property of the degree theory (see Proposition 1.27 of [20]), we have ˆ G|, ˆ Sn . ˆ B n+1 (0), 0 = deg G/| deg G, 1−δ 1−δ ˆ ˆ G| ˆ on Then if we set H (t, z) = t G(z) + (1 − t)Gδ (z), we have shown that the degree of G/| n+1 is same as that of Gδ /|Gδ | on S1−δ . However, by natural definition of Gδ , we can see that

Sn1−δ

2022


it is also well defined on the sphere Sn . Finally for any real numbers, λ > 0 and s > 0, we define the map: ˆ ˆ λ∇ Q(a) . Gλ,s (z) = −sQ(a), Since this map never vanishes for all a ∈ Sn , the degree of the maps is well defined and they all have same degree which implies that n ˆ ˆ ˆ G|, ˆ Sn deg G/| 1−δ = deg Gδ /|Gδ |, S1−δ = deg G1−r0 ,1 /|G1−r0 ,1 |, Sn1−δ = deg G1,1 /|G1,1 |, Sn1−δ = deg G∗ /|G∗ |, Sn = 0. Keep in mind that here we have identified the domains Sn and Sn1−δ for our map G∗ which is clearly true since they give the same values for G∗ . Thus we have finished the proof of the main theorem for the case Q − (n − 1)! is small in C 2 topology. 5. Proof of Theorem 1.1 The proof of our main theorem is exactly the same as in our previous paper [32], since all the eigenvalues and eigenfunctions of Pn are known [2,3]. We will not reproduce it here again. For interested readers, we refer them to Section 5 of [32]. Acknowledgments The research of the first author is supported by an Earmarked Grant from RGC of Hong Kong. This research was initiated when the first author visited the Institute for Mathematical Sciences. He thanks the Institute of Mathematical Science at National University of Singapore for its hospitality. The research of the second author is partially supported by NUS research grant R146-000-077-112. The part of this work was done when the second author visited Mathematical Institute at Nanjing University, China. He would like to thank them for hospitality and providing excellent work condition. Both authors wish to thank the anonymous referees for careful reading and useful comments. References [1] P. Baird, A. Fardoun, R. Regbaoui, Prescribed Q-curvature on manifolds of even dimension, J. Geom. Phys. 59 (2) (2009) 221–233. [2] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. of Math. 138 (1993) 213–242. [3] T. Branson, Group representations arising from the Lorentz conformal geometry, J. Funct. Anal. 74 (1987) 199–293. [4] S. Brendle, Global existence and convergence for a higher order flow in conformal geometry, Ann. of Math. 158 (2003) 323–343. [5] S. Brendle, Prescribing a higher order conformal invariant on S n , Comm. Anal. Geom. 11 (5) (2003) 837–858. [6] S.-Y.A. Chang, On a fourth order differential operator—the Paneitz operator—in conformal geometry, in: Harmonic Analysis and Partial Differential Equations, in: Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999, pp. 127–150.


2023

[7] S.-Y.A. Chang, P.C. Yang, Prescribing Gaussian curvature on S 2 , Acta Math. 159 (1987) 215–259. [8] S.-Y. Chang, P.C. Yang, A perturbation result in prescribing scalar curvature on Sn , Duke Math. J. 64 (1991) 27–69. [9] S.-Y.A. Chang, P.C. Yang, Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math. 142 (1995) 171–212. [10] S.-Y.A. Chang, P.C. Yang, On uniqueness of solutions of n-th order differential equations in conformal geometry, Math. Res. Lett. 4 (1997) 91–102. [11] S.-Y.A. Chang, M. Gursky, P.C. Yang, Prescribing scalar curvature on S 2 and S 3 , Calc. Var. Partial Differential Equations 1 (1993) 205–229. [12] S.-Y.A. Chang, J. Qing, The zeta functional determinants on manifolds with boundary I—The formula, J. Funct. Anal. 147 (1997) 327–362. [13] S.-Y.A. Chang, J. Qing, The zeta functional determinants on manifolds with boundary II, J. Funct. Anal. 147 (1997) 363–399. [14] K.C. Chang, J. Liu, On Nirenberg’s problem, Int. J. Math. 4 (1993) 35–58. [15] C.C. Chen, C.S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. LV (2002) 728–771. [16] C.C. Chen, C.S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. LVI (2003) 1667–1727. [17] Z. Djadli, A. Malchiodi, A forth order uniformization theorem on some four manifolds with large total Q-curvature, C. R. Math. Acad. Sci. Paris 340 (2005) 341–346. [18] Z. Djadli, A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Ann. of Math. (2) 168 (3) (2008) 813–858. [19] O. Druet, F. Robert, Bubbling phenomena for fourth-order four dimensional PDEs with exponential growth, Proc. Amer. Math. Soc. 134 (2006) 897–908. [20] I. Fonseca, W. Gangbo, Degree Theory in Analysis and Applications, Oxford Science Publications, 1995. [21] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Springer-Verlag, 1983. [22] C.R. Graham, R. Jenne, L. Mason, G. Sparling, Conformally invariant powers of the Laplacian, I: Existence, J. London Math. Soc. (2) 46 (1992) 557–565. [23] Z.C. Han, Prescribing Gaussian curvature on S 2 , Duke Math. J. 61 (1990) 679–703. [24] M. Ji, On positive scalar curvature on D 2 , Calc. Var. Partial Differential Equations 19 (2) (2004) 165–182. [25] E.H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, AMS, Providence, 2001. [26] A. Malchiodi, M. Struwe, Q-curvature flow on S 4 , J. Differential Geom. 73 (2006) 1–44. [27] C.B. Ndiaye, Constant Q-curvature metric in arbitrary dimension, J. Funct. Anal. 251 (1) (2007) 1–58. [28] S. Paneitz, A quadratic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint, 1983. [29] O. Rey, J. Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. I. N = 3, J. Funct. Anal. 212 (2004) 472–499. [30] O. Rey, J. Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. I. N 4, Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 459–484. [31] M. Struwe, A flow approach to Nirenberg’s problem, Duke Math. J. 128 (2005) 19–64. [32] J. Wei, X. Xu, On conformal deformation of metrics on Sn , J. Funct. Anal. 157 (1998) 292–325. [33] J. Wei, X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999) 207–228. [34] X. Xu, Uniqueness and non-existence theorems for conformally invariant equations, J. Funct. Anal. 222 (2005) 1–28. [35] X. Xu, P.C. Yang, Remarks on prescribing gaussian curvature on S 2 , Trans. Amer. Math. Soc. 336 (1993) 831–840.


The Faraday effect revisited: Thermodynamic limit Horia D. Cornean a,∗ , Gheorghe Nenciu b,c a Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg, Denmark b Dept. Theor. Phys., University of Bucharest, PO Box MG11, RO-76900 Bucharest, Romania c Inst. of Math. “Simion Stoilow” of the Romanian Academy, PO Box 1-764, RO-014700 Bucharest, Romania

Received 1 July 2008; accepted 22 June 2009 Available online 10 July 2009 Communicated by J. Bourgain

Abstract This paper is the second in a series revisiting the (effect of) Faraday rotation. We formulate and prove the thermodynamic limit for the transverse electric conductivity of Bloch electrons, as well as for the Verdet constant. The main mathematical tool is a regularized magnetic and geometric perturbation theory combined with elliptic regularity and Agmon–Combes–Thomas uniform exponential decay estimates. © 2009 Elsevier Inc. All rights reserved. Keywords: Thermodynamic limit; Magnetic perturbation theory; Faraday rotation

Contents 1.

2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Generalities . . . . . . . . . . . . . . . . . . . . 1.2. The main result . . . . . . . . . . . . . . . . . 1.3. A short description of the proof strategy . Proof of (i) . . . . . . . . . . . . . . . . . . . . . . . . . Proof of (ii) . . . . . . . . . . . . . . . . . . . . . . . . . Proof of (iii) . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Partitions and cut-offs . . . . . . . . . . . . . 4.2. Proof of (1.12) . . . . . . . . . . . . . . . . . . 4.3. Proof of (1.13) . . . . . . . . . . . . . . . . . .

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E-mail addresses: [email protected] (H.D. Cornean), [email protected] (G. Nenciu). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.020

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H.D. Cornean, G. Nenciu / Journal of Functional Analysis 257 (2009) 2024–2066

4.3.1. Only UL (B) counts . 4.3.2. The boundary terms . 4.3.3. The bulk contribution Acknowledgments . . . . . . . . . . . . . . . . . . Appendix A. Uniform exponential decay . References . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction 1.1. Generalities The rotation of the polarization of a plane-polarized electromagnetic wave passing through a material immersed in a homogeneous magnetic field oriented parallel to the direction of propagation, is known in physics as the Faraday effect (called sometimes also Faraday rotation). The experiment consists in sending a monochromatic light wave, parallel to the 0z direction and linearly polarized in the plane x0z. When the light enters the sample, the polarization plane starts rotating. A simple argument based on (classical) Maxwell equations shows that there exists a linear relation between the angle θ of rotation of the plane of polarization per unit length and the transverse component of the conductivity tensor of the material (see e.g. formula (1) in [29]). For most materials – and we will restrict ourselves to this case – the transverse component of the conductivity tensor vanishes when the magnetic field is absent and is no longer zero when the magnetic field is turned on. Under the proviso that the dependence of the conductivity tensor upon the strength B of the magnetic field is smooth, for weak fields one expands the conductivity tensor to the first order and neglect the higher terms. The coefficient of the linear term is known as the Verdet constant of the corresponding material. It follows that the basic object is the conductivity tensor and the main goal of the theory (classical or quantum) is to provide a workable formula for it, in particular for the Verdet constant. The problem has a long and distinguished history in solid state physics theory and the spectrum of possible applications ranges from astrophysics to optics and general quantum mechanics (see e.g. [13,16,22,23,28,29,33] and references therein). Using quantum theory in the setting in which the sample is modeled by a system of independent electrons subjected to a periodic electric potential, Laura Roth [29] obtained (albeit only at a formal level) a formula for the Verdet constant in full generality and applied it to metals as well as semiconductors. Roth’s method is based on an effective Hamiltonian approach for Bloch electrons in the presence of a weak constant magnetic field (see [31] and references therein) which in turn is based on a (proto) magnetic pseudodifferential calculus (for recent mathematical developments see [21] and references therein). But Roth’s theory is far from being free of difficulties. Due to her highly formal way of doing computations, it seems almost hopeless – even with present day mathematical tools – to control the errors or push the computations to higher orders in B except maybe the case of simple bands. Even more, the final formula contains terms which are singular at the crossings of Bloch bands. Accordingly, in spite of the fact that it has been considered a landmark of the subject, it came as no surprise that at the practical level this theory only met a moderate success and a multitude of unrelated, simplified models have been tailored for specific cases. Our paper is the second in a series aiming at a complete, unitary and mathematically sound theory of Faraday effect having the same generality as Roth’s theory (i.e. a theory of the con-

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ductivity tensor for electrons subjected to a periodic electric potential and to a constant magnetic field in the linear response approximation), but free of its shortcomings. More precisely in the first paper [13] we started by a rigorous derivation of the transverse component of the conductivity tensor in the linear response regime for a finite sample. It is given as the formula (1.10) below. To proceed further, we employ a method going back at least Sondheimer and Wilson [32] and which has been also used in the rigorous study of the Landau magnetism [1,5–7,10]. The basic idea is that the traces involved in computing various physical quantities can be written as integrals involving Green functions (i.e. integral kernels in the configuration space of either the resolvent or the semigroup of the Hamiltonian of the system), which are more robust and easier to control. As it stands, the conductivity tensor depends upon the shape of the sample and of boundary conditions which define the quantum Hamiltonian. The physical idea of the thermodynamic limit for an intensive physical quantity is that in the limit of large samples it approaches a limit which is independent of the shape of the sample, boundary conditions, etc. The existence of the thermodynamic limit is one of the basic (and far from trivial) problem of statistical mechanics (see e.g. [3,30]). In [13] we took for granted that the thermodynamic limit of the transverse component of the conductivity tensor exists and the limit is smooth as a function of the magnetic field strength B. Moreover, we assumed that the thermodynamic limit commuted with taking the derivative with respect to B. Under these assumptions, we gave – among other things – explicit formulas for the Verdet constant in terms of zero magnetic field Green functions, free of any divergences. The proof of the thermodynamic limit, which from the mathematical point of view is the most delicate part of the theory of the Faraday effect, was left aside in [13], and is the content of the current paper. The mathematical problem behind it is hard due to the singularity induced by the long range magnetic perturbation. Even for a simpler problem involving constant magnetic field – namely the Landau diamagnetism of free electrons – the existence of the thermodynamic limit leading to a correct thermodynamic behavior was a long standing problem. Naive computations led to unphysical and contradictory results (see [1] for historical remarks). Accordingly, the first rigorous results came as late as 1975 [1] and were based on various identities expressing the gauge invariance which was crucial in dealing with the singular terms appearing in the thermodynamic limit. Even though the importance of gauge invariance was already highlighted in [1], an efficient way to implement this idea at a technical level was still lacking. Only recently a regularized magnetic perturbation theory based on factorizing the (singular in the thermodynamic limit) magnetic phase factor has been fully developed in [10–12,27]. This regularized magnetic perturbation theory has been already used in [5–7,10] in order to prove far reaching generalizations of the results in [1]. Coming back to the Faraday effect, we would like to stress that the object at hand is much more singular than the one encountered in the Landau diamagnetism. This adds an order of magnitude to the mathematical difficulty and requires an elaborate and tedious combination of regularized magnetic perturbation theory with techniques like Combes–Thomas exponential decay, trace norm estimates and elliptic regularity. We expect that the method developed here in order to control the thermodynamic limit in the presence of an extended magnetic field to be useful in related problems, e.g. to obtain an elegant and complete study of the diamagnetism and de Haas–van Alphen effect for electrons in metals. The content of the paper is as follows. In the rest of this Introduction we state the mathematical problem, give the main result in Theorem 1.1, and since the proof is quite long and technical we will briefly describe the main points. The other sections are devoted to the proof of our main


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theorem. The core of the proof heavily involving regularized magnetic perturbation theory is contained in Section 4. Some technical estimates about exponential decay with uniform control in the spectral parameter are given as Appendix A. 1.2. The main result Consider a simply connected open and bounded set Λ1 ⊂ R3 , which contains the origin. We assume that the boundary ∂Λ1 is smooth. Consider a family of scaled domains ΛL = x ∈ R3 : x/L ∈ Λ1 ,

L > 1.

(1.1)

Area(∂ΛL ) ∼ L2 .

(1.2)

We have the estimates Vol(ΛL ) ∼ L3 ,

We are interested in the thermodynamic limit, which will mean L → ∞, that is ΛL will fill out the whole space. The one particle Hilbert space is HL := L2 (ΛL ). Note that we include the case L = ∞. The one body Hamiltonian of a non-confined particle, subjected to a constant magnetic field (0, 0, B), in an external potential V , formally looks like this: H∞ (B) = P2 (B) + V ,

(1.3)

P(B) = −i∇ + Ba = P(0) + Ba.

(1.4)

with

Let us explain the various terms. Here a(x) is a smooth magnetic vector potential which generates a magnetic field of intensity B = 1, i.e. ∇ ∧ a(x) = (0, 0, 1). A frequently used magnetic vector potential is the symmetric gauge: 1 A(x) = n3 ∧ x = (−x2 /2, x1 /2, 0), 2

(1.5)

where n3 is the unit vector along z axis. We neglect the spin structure since it only complicates the notation and does not influence the mathematical problem. On components, (1.4) reads as: Pj (B) = Dj + Baj =: Pj (0) + Baj ,

j ∈ {1, 2, 3}.

(1.6)

We will from now on assume that V is a C ∞ (R3 ) function, periodic with respect to the lattice Z3 . Standard arguments then show that H∞ (B) is essentially self-adjoint on C0∞ (R3 ). When L < ∞ we need to specify a boundary condition. We will only consider Dirichlet boundary conditions, that is we start with the same expression as in (1.3), defined on C0∞ (ΛL ), and we define HL (B) to be the Friedrichs extension of it. This is indeed possible, because our operator can be written as −D + W, where D is the Dirichlet Laplacian and W is a first

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order differential operator, relatively bounded to −D (remember that L < ∞). The form domain of HL (B) is the Sobolev space H01 (ΛL ), while the operator domain (use the estimates in Section 10.5, Lemma 10.5.1, in [19]) is H 2 (ΛL ) ∩ H01 (ΛL ). Moreover, HL (B) is essentially ∞ (Λ ), i.e. functions with support in Λ and indefinitely differentiable in Λ self-adjoint on C(0) L L L up to the boundary. Another important operator is (−i∇ +Ba)2D , i.e. the usual free magnetic Schrödinger operator defined with Dirichlet boundary conditions. We know that its spectrum is non-negative for all L > 1. By adding a positive constant, we can always assume that the spectrum of HL (B) is non-negative, uniformly in L > 1. Let us now introduce the physical quantity we want to study. Consider ω ∈ C and (ω) < 0. For some fixed μ ∈ R and β > 0, define the Fermi–Dirac function on its maximal domain of analyticity: fFD (z) = Define

1 . eβ(z−μ) + 1

π | Im ω| , , d := min 2β 2

(1.7)

(1.8)

and introduce a counter-clockwise oriented contour given by Γω = {x ± id: a x < ∞} ∪ {a + iy: −d y d}

(1.9)

where a + 1 lies below the spectrum of HL (B). By adding a positive constant to V , we can take a = −1 uniformly in L 1 and B ∈ [0, 1]. We introduce the transverse component of the conductivity tensor (see [13,29]) as σL (B) = −

1 Tr Vol(ΛL )

−1 −1 fFD (z) P1 (B) HL (B) − z P2 (B) HL (B) − z − ω

Γω

−1 −1 dz. + P1 (B) HL (B) − z + ω P2 (B) HL (B) − z

(1.10)

Here Tr assumes that the integral is a trace-class operator. Now we are prepared to formulate our main result. Theorem 1.1. The above defined transverse component of the conductivity tensor admits the thermodynamic limit; more precisely: (i) The following operator, defined by a B(L2 (ΛL ))-norm convergent Riemann integral, −1 −1 FL := fFD (z) P1 (B) HL (B) − z P2 (B) HL (B) − z − ω Γω

−1 −1 dz, + P1 (B) HL (B) − z + ω P2 (B) HL (B) − z

is in fact trace-class;

(1.11)


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(ii) Consider the operator F∞ defined by the same integral but with H∞ (B) instead of HL (B), and defined on the whole space. Then F∞ is an integral operator, with a kernel F (x, x ) which is jointly continuous on its variables. Moreover, the continuous function defined by R3 x → sB (x) := F (x, x) ∈ R is periodic with respect to Z3 ; (iii) Denote by Ω the unit cube in R3 . The thermodynamic limit exists: σ∞ (B) := lim σL (B) = −

sB (x) dx.

L→∞

(1.12)

Ω

Moreover, the mapping B → sB ∈ L∞ (Ω) is differentiable at B = 0 and: ∂B σ∞ (0) = −

∂B sB |B=0 (x) dx = lim ∂B σL (0). L→∞

(1.13)

Ω

Remark 1. The formula (1.12) is only the starting point in the study of the Faraday rotation. A related problem is the diamagnetism of Bloch electrons, where the main object is the integrated density of states of magnetic Schrödinger operators (see [17,18,20]). For a systematic treatment of magnetic pseudo-differential operators which generalizes our magnetic perturbation theory, see [21,24–26]. Remark 2. The Dirichlet boundary conditions are important for us. Even though we suspect that our main result should also hold for Neumann conditions and for less regular domains (see [2, 14]), we do not see an easy way to prove it. Remark 3. We believe that the method we use in the proof of (1.13) can be used in order to obtain a stronger result: the mapping B → sB ∈ L∞ (Ω) is smooth and for any n 1: ∂Bn σ∞ (B) = −

∂Bn sB (x) dx = lim ∂Bn σL (B). L→∞

(1.14)

Ω

We leave this statement as an open problem. In the rest of the paper we give the proof of Theorem 1.1. 1.3. A short description of the proof strategy Since the proof is rather long, we list here the main steps and ideas. Let us start with some general considerations about the thermodynamic limit. If we are interested in the thermodynamic limit of a quantum physical quantity, the object we need to control is the trace of the operator representing the corresponding quantity. The basic idea consists in writing this trace as an integral of the diagonal value of the operator’s integral kernel (Schwartz kernel) over the confining box. This procedure makes the quantum thermodynamic limit look very similar to what happens in classical statistical mechanics. More precisely, we need to show that the difference between the integral kernel for the finite box and the one for the entire space decays sufficiently rapidly with the distance from the boundary of the box, so that

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the replacement of the integral kernel for the box with the one corresponding to the entire space gives an error term increasing slower than the volume, which then disappears in the limit. It turns out that for the transverse conductivity this kernel is far more complicated than say the heat kernel – whose behavior has been extensively studied in the literature. Notice for example, that the integrand in (1.10) contains two resolvents sandwiched with magnetic momentum operators. Thus we need a good control of their integral kernels, in particular when the distance between their arguments increases to infinity, and all that uniformly in the spectral parameter z. Since a constant magnetic field is present, the biggest difficulty is to deal with the linear growth of the vector potential. Here, the use of gauge covariance is crucial. Now let us list the main ideas of the proof. For the first statement of the theorem one simply uses integration by parts with respect to z in order to transform the integrand into a product of Hilbert–Schmidt operators. The proof of the second statement is based on elliptic regularity. The main technical difficulty is to control the z behavior of all our bounds, especially the exponential localization of the magnetic resolvents sandwiched with momentum operators. We also have to control the linear growth of the magnetic potential. We turn the operator norm bounds which we obtain in Appendix A into pointwise bounds for certain integral kernels. It is a long road using magnetic perturbation theory, but nevertheless, we use nothing more than well-known Combes–Thomas exponential bounds, local gauge covariance, the Cauchy–Schwarz inequality, and integration by parts. The third statement of Theorem 1.1 contains the main result and is proved in Section 4. We start with a bit strange three-layered partition of unity defined in (4.1)–(4.12). This idea goes back at least to [4,5]. The main effort consists in isolating the bulk of ΛL – where only operators defined in the whole space will act – from the region close to the boundary. Note that in the absence of the magnetic field, it would be enough to work with only two cut-off functions: one isolating the bulk from the boundary, and the other one supported in a tubular neighborhood of the boundary. When long-range magnetic fields are present, this is not enough. The tubular neighborhood needs to be chopped up in many small pieces, in order to apply local gauge transformations (see below why we need them). The central idea in proving (1.12) is to show that the contribution to the total trace of the region close to the boundary grows slower than the volume. Technically, this is obtained by approximating the true resolvent (HL (B) − z)−1 with an operator UL (B, z) given in (4.16). UL (B, z) contains the bulk term, plus a boundary contribution which consists from a sum of terms each locally approximating (HL (B) − z)−1 and containing a specially tailored local gauge given in (4.13). These locally defined vector potentials are made globally bounded with the help of our third layer of cut-off functions g˜˜ γ ’s. The switch to the local gauge is performed through the central identity (3.18). An explanation of why UL (B, z) is a convenient approximation for the full resolvent can be found right after (4.18), and there is the place where we first fully use the exponential estimates of Appendix A. In Proposition 4.1 we prove that we can replace (HL (B) − z)−1 with UL (B, z) in the conductivity formula, without changing the value of the limit. In Proposition 4.1 we show that only the bulk term from UL (B, z) will contribute. In Proposition 4.6 we show that removing the cut-offs only gives a surface contribution. The proof of (1.13) is heavily based on magnetic perturbation theory. Although the technical estimates are considerably more involved than at the previous point, the main idea is the same: the boundary terms can be discarded. The full power of the magnetic phases is used in Lemma 4.10; an heuristic explanation of how and why they manage to kill the linear growth of the magnetic potential is given right after (4.73).


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2. Proof of (i) The integrand in (1.11) is a bounded operator, with an L2 norm bounded by a constant times |(z)|2 , uniformly in L, as we can see from (1.6) and (A.2). Because fFD has an exponential decay in |(z)|, the integral converges and defines a bounded operator. Let us note that the integrand is not a trace class operator under our conditions. But the total integral is a different matter. The point is that we can integrate by parts with respect to z by using anti-derivatives of fFD which are still decaying exponentially at infinity. By doing this at least four times, we obtain integrals of the form

−m −n P2 (B) HL (B) − z − ω dz f˜(z)P1 (B) HL (B) − z

(2.1)

Γω

where m + n 5, hence max{m, n} 3. Assume that m 3. Then we can write the above integral as

−m+2 −2 −n HL (B) − z P2 (B) HL (B) − z − ω dz. f˜(z)P1 (B) HL (B) − z

(2.2)

Γω

The main point is that (HL (B) − z)−1 is Hilbert–Schmidt since we can write −1

−1

−1 HL (B) − z (−i∇ + Ba)2D + 1 HL (B) − z , = (−i∇ + Ba)2D + 1

(2.3)

and by using (A.7) with δ = 0, and (A.13), we obtain HL (B) − z −1

B2

const · Vol(ΛL ) (z) ,

(2.4)

where the above constant does not depend on L and z. Thus (HL (B) − z)−2 is trace class, and the trace norm of the integrand in (2.2) is bounded by 4 const · f˜(z) · (z) · Vol(ΛL ) where again the constant is uniform in L and z. This now is integrable on the contour, thus the integral defines a trace class operator. Moreover, its trace grows at most like the volume of ΛL , hence lim supL→∞ |σL | < ∞. Remark. The same type of argument may be used to show that σL (B) is smooth in B, by repeatedly using the formal identity −1 −1 −1 ∂B HL (B) − z ∂B HL (B) HL (B) − z = − HL (B) − z

(2.5)

in the sense of bounded operators. Note the important fact that ∂B HL (B) will generate some linear growing terms coming from the magnetic vector potential a(x), therefore the trace norm of the new integrand will grow like L4 . We therefore cannot conclude here that the derivatives |∂Bn σL (B)| will admit a finite lim sup when L → ∞.

2032


3. Proof of (ii) We are going to prove the regularity statement for the kernel without using the periodicity of V , only the fact that the potential is smooth and bounded on R3 together with all its derivatives. The strategy consists in integrating by parts with respect to z many times, such that we obtain high powers of the resolvent (H∞ (B) − ζ )−N . Then we will prove that Pj (B)(H∞ (B) − ζ )−N Pk (B) has a smooth kernel which does not grow too fast with |ζ |. Let us now be more precise and start with some technical results. Proposition 3.1. Fix 0 < η < 1 and choose z ∈ C with dist{z, [0, ∞)} = η > 0. Let r = |(z)|. Then the operator (H∞ (0) − z)−1 has an integral kernel G1 (x, x ; z) which is smooth away from the diagonal x = x . There exists δ > 0 and some M 1 such that for any multi-index α ∈ N3 with |α| 1 we have the estimate δ

sup |x − x ||α|+1 e r |x−x | Dxα G1 (x, x ; z) = C1 (α, η)rM < ∞.

(3.1)

x=x ∈R3

Proof. The result without the exponential decay is essentially contained in [15]. The symbol 2 (R3 × R3 ), H (0) ∈ L2 (R3 ) (see Example 3.1 of H∞ (0), denoted by h0 (x, ξ ) belongs to S1,0 ∞ 1,0 in [15]), and is uniformly elliptic. Fix λ > 0 large enough. We can apply Theorem 4.1 in [15] and construct a parametrix for −2 (R3 × R3 ). The symbol H∞ (0) + λ starting from the symbol q0 (x, ξ ) := 1/(h(x, ξ ) + λ) ∈ S1,0 giving the parametrix is an asymptotic sum of symbols, starting with q0 , then the next one is in −3 and so on. Each term gives a contribution to the integral kernel of the parametrix. The most S1,0 singular contribution is (in the sense of oscillatory integrals): 1

ei(x−x )·ξ q0 (x, ξ ) dξ. (2π)3 R3 −N for large N generate We only have to consider a few terms besides this one, since symbols in S1,0 more and more regular kernels at the diagonal. By standard “integration by parts” tricks, and using the fact that we work in three dimensions, one can prove the estimate

sup |x − x ||α|+1 Dxα G1 (x, x ; −λ) = const(α, λ) < ∞.

(3.2)

x=x ∈R3

In fact, outside the region |x − x | 1 we can integrate by parts several times with respect to ξ and prove that G1 (x, x ; −λ) decays faster than any power of |x − x |. But the Combes–Thomas method will give a better, exponential localization. The important thing is that the L2 estimates from the Combes–Thomas argument can be transferred into pointwise estimates for the kernel. Let us now prove this. Using (A.13) at L = ∞ and B = 0, together with the triangle and Cauchy–Schwarz inequalities, we get that (− + λ)−1 with exponential weights maps L2 into L∞ . The key estimate is (0 < c < 1) e

√ −c λ|x−x0 |

√

√ e−(1−c) λ|x−x | K∞ (x, x )ec λ|x −x0 | . 4π|x − x |


2033

Since we can write: −1 −1 H∞ (0) + λ = (− + λ)−1 (− + λ) H∞ (0) + λ ,

(3.3)

using (A.12) (at L = ∞ and B = 0), it follows that this resolvent with exponential weights is a bounded map from L2 into L∞ . More precisely, for any x0 ∈ R3 , there exists 0 < c < 1 small enough such that: √ −1 √ supe−c λ·−x0 H∞ (0) + λ ec λ·−x0 B(L2 ,L∞ ) const, x0

λ λ0 .

(3.4)

Now if we look at the map C0∞

3 R Ψ →

G1 (x0 , x; −λ)eδλ x−x0 Ψ (x) dx

R3

(it makes sense to fix x0 since the resolvent maps smooth functions into smooth functions), we see that by using (3.4) we can extend this map to a linear and bounded functional on L2 . Riesz’ representation theorem then gives: √ √ sup ec λ·−x0 G1 (x0 , ·; −λ)L2 = sup ec λ·−x0 G1 (·, x0 ; −λ)L2 const,

x0 ∈R3

(3.5)

x0 ∈R3

uniformly in λ λ0 . Using this, together with the Cauchy–Schwarz and the triangle inequality, we get that the integral kernel G2 (x, x ; −λ) of (H∞ (0) + λ)−2 obeys uniformly in λ λ0 : sup ec

√

λ|x−x |

x,x

sup x,x

G2 (x, x ; −λ)

√

c√λx−x

e G1 (x, x

; −λ)ec λx −x G1 (x

, x ; −λ) dx

const .

(3.6)

R3

Now if |x − x | 1, write

∞

G1 (x, x ; −λ) =

G2 (x, x ; −λ1 ) dλ1 ,

(3.7)

λ c√

which together with (3.6) and the integrability of e− 2 c

sup e 2

|x−x |1

√ λ|x−x |

λ1

imply that

G1 (x, x ; −λ) const .

(3.8)

We can also deal with derivatives with respect to x, by showing that the operator Dj (H∞ (0) + λ)−N (N large enough) has an integral kernel Dj GN (x, x ; −λ) obeying the same type of estimate as in (3.6). This is done by commuting Dj several times with a few resolvents; let us see how it works.

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First, by commuting we have: −1 −1 −2 Dj H∞ (0) + λ = H∞ (0) + λ Dj + H∞ (0) + λ T1 ,

−1 T1 = H∞ (0), Dj − H∞ (0), H∞ (0), Dj H∞ (0) + λ ,

(3.9)

where T1 is bounded due to the fact that [H∞ (0), Dj ] is bounded, while [H∞ (0), [H∞ (0), Dj ]] is relatively bounded with respect to H∞ (0). Second, by commuting twice we have: −2 −2 −3 Dj H∞ (0) + λ = H∞ (0) + λ Dj + H∞ (0) + λ T1 −2 −1 + H∞ (0) + λ T1 H∞ (0) + λ .

(3.10)

This identity allows us to write: −N +2 −2 −N +5 Dj H∞ (0) + λ = H∞ (0) + λ T H∞ (0) + λ , −1 −1 −1 H∞ (0) + λ , T = Dj + H∞ (0) + λ T1 + T1 H∞ (0) + λ

(3.11)

where T with exponential weights is bounded from L2 to L2 . Then we prove that the integral kernel of Dj (H∞ (0) + λ)−N +2 obeys an L2 estimate like in (3.5), then from the identity −N −N +2 −2 Dj H∞ (0) + λ H∞ (0) + λ = Dj H∞ (0) + λ we get the needed L∞ estimate by mimicking (3.6). Then we write (−1)N Dj G1 (x, x ; −λ) = (N − 1)!

∞

∞ dλ1

λ

∞ dλ2 . . .

λ1

dλN Dj GN (x, x ; −λ )

λN−1

and propagate the exponential decay over the integrals in λ. Therefore we can state the first result regarding the exponential localization. For λ large enough we have:

sup |x − x ||α|+1 e|x−x | Dxα G1 (x, x ; −λ) = const(α, λ) < ∞.

(3.12)

x=x ∈R3

Now let us investigate the z dependence. Let us apply the resolvent identity several times and get (N 2): −1 −1 −2 H∞ (0) − z = H∞ (0) + λ + (z + λ) H∞ (0) + λ + ··· −N −1 H∞ (0) − z . + (z + λ)N H∞ (0) + λ

(3.13)

The idea is to keep the z dependence to the right in the last term, and to keep a regular kernel to the left. We start with a norm estimate. From the usual resolvent identity:


2035

−1 −1 −1 −1 H∞ (0) − z H∞ (0) − z , = H∞ (0) + λ + (z + λ) H∞ (0) + λ the use (3.5) and (A.5) (with L = ∞) provides us with some N1 > N such that: δ −1 δ supe− r ·−x0 H∞ (0) − z e r ·−x0 B(L2 ,L∞ ) const(η) · r N1 .

(3.14)

x0

This estimate implies that the map (initially defined on compactly supported functions) L2 R 3 Ψ →

G1 (x0 , x; z)e r x−x0 Ψ (x) dx ∈ C δ

R3

is a bounded linear functional. Riesz’ representation theorem leads us to: δ δ sup e r ·−x0 G1 (x0 , ·; z)L2 = sup e r ·−x0 G1 (·, x0 ; z)L2 const(η) · r N1 .

x0 ∈R3

(3.15)

x0 ∈R3

We are only left with the case in which we have a derivative on the left. Using (3.15) in (3.13) and the other results we have obtained for the kernel where z = λ, it is not hard to obtain the exponential decay claimed in (3.1). 2 Proposition 3.2. Assume that α ∈ {0, 1} and 0 < η < 1. Let dist{z, [0, ∞)} = η and N is large enough. Then the operator P1α (B)(H∞ (B) − z)−N P21−α (B) has a jointly continuous integral kernel KN,B (x, x ; z), and there exists δ > 0 small enough and M large enough such that: δ

sup e r |x−x | KN,B (x, x ; z) const(N, B, η) · rM .

(3.16)

x,x ∈R3

Proof. Although this particular result might be obtained with other methods, we will employ the magnetic perturbation theory as developed in [5,11,27]. For different approaches involving magnetic pseudo-differential calculus, see [17,18,21,24,25]. We can assume that the magnetic vector potential is expressed in the transverse gauge, given in (1.5). Define the antisymmetric magnetic gauge phase 1 1 ϕ0 (x, y) := −A(y) · x = (y2 x1 − y1 x2 ) = e3 · (x ∧ y), 2 2

(3.17)

where e3 denotes the unit vector (0, 0, 1) ∈ R3 . Then we have the following identity (true for instance on Schwartz functions), valid for every vector y kept fixed in R3 : Px (0) + BA(x) eiBϕ0 (x,y) = eiBϕ0 (x,y) Px (0) + BA(x − y) .

(3.18)

For every z ∈ C \ [0, ∞) define

SB (x, x ; z) := eiBϕ0 (x,x ) G1 (x, x ; z).

(3.19)

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This integral kernel generates an L2 bounded operator (use the estimate (3.1) and employ the Schur–Holmgren criterion). We denote this operator by SB (z). There are two important facts related to this kernel. First, G1 (x, x ; z) solves the distributional equation 2 Px (0) + V (x) − z G1 (x, x ; z) = δ(x − x ). Second, one can prove that SB (z) maps Schwartz functions into Schwartz functions. Moreover, employing (3.18) and integrating by parts, we can establish an identity which holds at first in the weak sense on Schwartz functions:

H∞ (B) − z Ψ, SB (z)Ξ = Ψ, 1 + BTB (z) Ξ ,

(3.20)

where TB (z) is the operator generated by the following integral kernel: 2

TB (x, x ; z) := eiBϕ0 (x,x ) −2iA(x − x ) · ∇x G1 (x, x ; z) + B A(x − x ) G1 (x, x ; z) . (3.21) Using (3.1), and the fact that |A(x − x )| |x − x |, we obtain the following pointwise estimate true for all x = x : δ

e− r |x−x | · rM . max SB (x, x ; z), TB (x, x ; z) const |x − x |

(3.22)

Clearly, TB (z) can be extended to a bounded operator. Now let us prove that (3.20) also holds true in the strong sense. Because H∞ (B) is essentially self-adjoint on the set of Schwartz functions, (3.20) can be extended to any Ξ ∈ L2 and any Ψ ∈ Dom(H∞ (B)). It means that the range of SB (z) belongs to the domain of H∞ (B) and H∞ (B) − z SB (z) = 1 + BTB (z).

(3.23)

At this point we can establish the following identity, valid for all z of interest: −1 −1 H∞ (B) − z = SB (z) − B H∞ (B) − z TB (z).

(3.24)

Denote by TˆB (z) := TB∗ (z) the bounded operator generated by the kernel TˆB (x, x ; z) := TB (x , x; z). Now replace z with z in (3.24) and then take the adjoint. We obtain: −1 −1 H∞ (B) − z = SB (z) − B TˆB (z) H∞ (B) − z .

(3.25)

Denote by K1,B (x, x ; z) the integral kernel of (H∞ (B) − z)−1 . With (3.25) as starting point, together with (A.5), we can use the same argument as in the zero magnetic field case, in order to show that the resolvent (with exponential weights) maps L2 into L∞ . Thus its kernel obeys an estimate like in (3.15). Moreover, (3.25) implies via the Cauchy–Schwarz inequality that

H.D. Cornean, G. Nenciu / Journal of Functional Analysis 257 (2009) 2024–2066 δ

2037

− r |x−x | K1,B (x, x ; z) const e · rM . |x − x |

(3.26)

Moreover, we can repeat the arguments from the proof of Proposition 3.1 and get the boundedness and joint continuity for the kernel of Pj (B)(H∞ (B) − z)−N for large N , since all we have to do is to change Dj with Pj (B) and to notice that the formal commutator [Pj (B), H∞ (B)] is only linear in Pk (B)’s, therefore Pj (B) “commutes well” with (H∞ (B) − z)−1 . Now let us show that (H∞ (B) − z)−1 maps L2 into Hölder continuous functions. In fact, one can prove the following estimate: Lemma 3.3. Fix a compact set U ⊂ R3 , and fix β ∈ (0, 1/2). Take ψ with ψL2 = 1. Then there exist two positive constants C and M such that −1 −1 sup r−M H∞ (B) − z ψ (x) − H∞ (B) − z ψ (y) C · |x − y|β ,

(3.27)

z∈Γω

for any x, y ∈ U . The same estimate holds true for SB (z). Proof. The domain of H∞ (B) is locally H 2 , hence for some positive λ, the function (H∞ (B) + λ)−1 ψ is locally H 2 . From the Sobolev embedding lemma, we obtain that (H∞ (B) + λ)−1 ψ is β-Hölder continuous for every β ∈ [0, 1/2). The estimate (3.27) follows from the resolvent identity −1 −1 −1 −1 H∞ (B) − z . H∞ (B) − z = H∞ (B) + λ + (z + λ) H∞ (B) + λ The same result for SB (z) follows from (3.24).

2

We now can start the actual proof of (ii). We integrate by parts N times with respect to z in the expression of F∞ , and N is supposed to be large. The terms we obtain in the integrand will look like this one: −N1 −N2 Pk (B) H∞ (B) − z2 , Pj (B) H∞ (B) − z1

(3.28)

where N1 + N2 = N + 2, and either N1 or N2 is large. Here z1 and z2 are complex numbers like z ∈ Γω or z ± ω. By repeated commutations, we can always write this operator as (H∞ (B)−z1 )−1 W (H∞ (B)− z2 )−1 , where W is a sum of terms like this one: −n −n −n W1 := H∞ (B) − z1 1 vα1 Pα1 (B) H∞ (B) − z1 2 vα2 Pα2 (B) H∞ (B) − z1 3 . Here n1 , n2 , n3 1, vα ’s are smooth and uniformly bounded functions. Note that such a term always starts and ends with a resolvent. Using the fact that Pα1 (B)(H∞ (B) − z1 )−n2 Pα2 (B) is a bounded operator, then we can always write W as the product of the form (H∞ (B) − z1 )−1 W˜ where W˜ is a bounded operator whose norm increases at most polynomially in r. Thus e−·δ/r W is a Hilbert–Schmidt operator, with a H–S norm which increases polynomially in r. Hence it is an integral operator which obeys the estimate:

2038


2 e−2yδ/r W (y, y ; z) dy dy const r M ,

(3.29)

where the constant and M are independent of r. Thus we have: −N1 −N2 Pj (B) H∞ (B) − z1 Pk (B) H∞ (B) − z2 −1 −1 = H∞ (B) − z1 e·δ/r e−·δ/r W H∞ (B) − z2 .

(3.30)

Using an identity like the one in (A.8), we can rewrite the above operator as: −1 −1 e·δ/r H∞ (B) − z1 T e−·δ/r W H∞ (B) − z2 , where T is a bounded operator uniformly in r if δ is small enough. Thus W := T e−·δ/r W is Hilbert–Schmidt, and has an integral kernel whose norm in L2 (R6 ) is polynomially bounded in r. Therefore we are left with the investigation of the joint continuity in x and x of the integral kernel defined by:

f∞ (x, x ) :=

K1,B (x, y; z1 )W (y, y )K1,B (y , x ; z2 ) dy dy ,

where |W (y, y )|2 dy dy const r M . Using this, (3.26), and the Cauchy–Schwarz inequality, U . Since W is Hilbert–Schmidt, we obtain |f (x, x )| const r M1 W B2 , uniformly in x, x ∈

gj (y)hj (y ) where g’s and h’s we can approximate W (y, y ) with a finite sum of the type 2

are L . Thus we can (uniformly in x, x ∈ U ) approximate the function f∞ with functions of the type n −1 −1 H∞ (B) − z1 gj (x) H∞ (B) − z2 hj (x ), j =1

which from Lemma 3.3 we know are continuous on compacts. Hence f∞ is jointly continuous on its variables. As for the integral kernel of F∞ , we see that it can be written as the integral with respect to z of a finite number of kernels of the same type as f∞ . Due to the exponential decay of fFD (see (1.7)), we see that we can approximate F∞ (x, x ) uniformly on compacts with continuous functions, and we are done. Therefore the function sB as defined in Theorem 1.1 is a continuous function. If V is periodic with respect to Z3 , then H∞ (B) commutes with the magnetic translations, defined for every γ ∈ Z3 as (see also (3.18)): [Mγ ψ](x) := eiBϕ0 (x,γ ) ψ(x − γ ),

M−γ Mγ = 1.

Hence we have that as operators, M−γ F∞ Mγ = F∞ , which for kernels gives

e−iBϕ0 (x,γ ) F (x + γ , x + γ )eiBϕ0 (x +γ ,γ ) F (x, x ).

(3.31)

Now since ϕ0 (γ , γ ) = 0, when we put x = x we get sB (x + γ ) = sB (x) and we are done with (ii).


2039

4. Proof of (iii) We start by proving the thermodynamic limit for the conductivity, that is (1.12). We need to introduce a special partition of unity in ΛL . 4.1. Partitions and cut-offs Fix 0 < α < 1 (small enough, to be chosen later) and define for t > 0: ΞL (t) := x ∈ ΛL : dist{x, ∂ΛL } tLα .

(4.1)

This models a “thin” compact subset of ΛL , near the boundary, with a volume of order tL2+α . Because we assumed that the boundary ∂Λ1 was smooth, all points of ΞL (t) have unique projections on ∂ΛL , if L is large enough. Then if t1 < t2 we have ΞL (t1 ) ⊂ ΞL (t2 ) and dist ΞL (t1 ), ΛL \ ΞL (t2 ) (t2 − t1 )Lα .

(4.2)

The subset ΛL \ ΞL (1) models the “bulk region” of ΛL , which is still “far-away” from the boundary. Now consider the inclusion of ΞL (2) in the dilated lattice Lα Z3 . That is we cover ΞL (2) with disjoint closed cubic boxes parallel to the coordinate axis, centered at points in Lα Z3 , of side length Lα . Denote by E ⊂ Lα Z3 the set of centers of those cubes which have common points with ΞL (2). Clearly, due to volume considerations, #E ∼ L2−2α . In order to fix notation, let us denote by K(γ , s) the cube centered at γ ∈ E, with side length equal to s Lα . Moreover, denote by E˜ := E ∪ {(0, 0, 0)} (note that the origin cannot belong to E if L is large enough). Now choose a partition of unity {gγ }γ ∈E˜ of ΛL which has the following properties: supp(g0 ) ⊂ ΛL \ ΞL (1); supp(gγ ) ⊂ K γ , 2Lα , γ ∈ E; gγ (x) = 1, ∀x ∈ ΛL ; 0 gγ 1, β D g γ

∞

(4.3) (4.4) (4.5)

γ ∈E˜

∼ L−α|β| ,

˜ ∀β ∈ N3 , ∀γ ∈ E.

(4.6)

This partition has the property that if we restrict ourselves to E, then uniformly in L, the number of gγ ’s which are not zero at the same time is bounded by a constant. Only g0 has ∼ L2−2α neighbors whose supports have common points with supp(g0 ). Now we choose another set of functions, {g˜ γ }γ ∈E˜ having the following properties: g˜ 0 (x) = 1 if x ∈ ΛL \ ΞL (1/2); supp(g˜ 0 ) ⊂ ΛL \ ΞL (1/4); α supp(g˜ γ ) ⊂ K γ , 10L , γ ∈ E; g˜ γ (x) = 1 if x ∈ K γ , 9Lα ; β D g˜ γ ∼ L−α|β| , ∀β ∈ N3 , ∀γ ∈ E. ˜ ∞

(4.7) (4.8) (4.9)

2040


These functions were chosen “wider” than the gγ ’s, and obey g˜ γ gγ = gγ ,

dist supp(D g˜ γ ), supp(gγ ) ∼ Lα ,

˜ γ ∈ E.

(4.10)

By D g˜ γ we mean that we take at least one derivative of g˜ γ . Now let us define the third type of cut-offs, {g˜˜ γ }γ ∈E : supp g˜˜ γ ⊂ K γ , 12Lα , γ ∈ E; β D g˜˜ γ ∼ L−α|β| , ∞

g˜ γ (x) = 1 if x ∈ K γ , 11Lα ; ∀β ∈ N3 , ∀γ ∈ E.

(4.11) (4.12)

Note that we have g˜˜ γ g˜ γ = g˜ γ (the origin is not considered here). 4.2. Proof of (1.12) Define for every γ ∈ E: Aγ (x) := g˜˜ γ (x)A(x − γ ).

(4.13)

Due to the support properties of our cut-off functions, we have the estimates g˜ γ Aγ = g˜ γ A(x − γ ), Aγ C 1 (R3 ) const · Lα .

(4.14)

Define for every γ ∈ E (see also (1.3) and (1.4)): Pγ (B) := P(0) + BAγ ,

HL (B, γ ) := Pγ (B)2 + V ,

(4.15)

where the Hamiltonian is defined with Dirichlet boundary conditions. Note that HL (B, γ ) − HL (0) is a relatively bounded perturbation of HL (0). Define the operator −1 −1 UL (B, z) := g˜ 0 H∞ (B) − z g0 + eiBφ0 (·,γ ) g˜ γ HL (B, γ ) − z e−iBφ0 (·,γ ) gγ . γ ∈E

(4.16) One can prove that the range of UL (B, z) is in the domain of HL (B) and we have: HL (B) − z UL (B, z) = 1 + VL (B, z), −1 VL (B, z) := −2i(∇ g˜ 0 ) · P(B) − (g˜ 0 ) H∞ (B) − z g0 −1 + eiBφ0 (·,γ ) −2i(∇ g˜ γ ) · Pγ (B) − (g˜ γ ) HL (B, γ ) − z e−iBφ0 (·,γ ) gγ . γ ∈E

(4.17)


2041

In order to obtain this equality we used the locality of our operators and various support properties of our cut-off functions, the identity (3.18), definition (4.13), and (4.5). Then we can write −1 −1 HL (B) − z = UL (B, z) + HL (B) − z VL (B, z).

(4.18)

The good thing about VL (B, z) is that its operator norm is exponentially small in Lα . This is because we have the boundedness from (A.5) and (A.6) (valid also for HL (B, γ ), as can easily be seen from the proofs), and because of the estimate in (4.10). Indeed, for terms involving γ = 0, put x0 = γ in the two exponential estimates, and take +δ on the left and −δ on the right. Then we gain an overall decaying term from the left as (4.10) implies: sup

δ

sup

x∈supp(D g˜ γ ) x ∈supp(gγ )

δ1

e− r (x−γ −x −γ ) e− r L , α

γ = 0,

(4.19)

where δ1 > 0 is small enough and L is larger than some L0 . For γ = 0 the situation is slightly different, because we did not assume convexity for ΛL . But one of the terms whose norm we need to estimate is (see (4.17)) −1 (∇ g˜ 0 ) · P(B) H∞ (B) − z g0 . From (3.1) it follows that the integral kernel of this operator is bounded by δ

(∇ g˜ 0 ) · P(B) H∞ (B) − z −1 g0 (x, x ) const(η)rM e− r |x−x | .

(4.20)

Because x and x are always separated by ∼ Lα (see the support properties for our cut-offs), we can write: δ δ

α (∇ g˜ 0 ) · P(B) H∞ (B) − z −1 g0 (x, x ) const(η)rM e− r1 |x−x | e− r2 L ,

(4.21)

where δ1 and δ2 are smaller than δ. Therefore we can write for all N 1: δ α VL (B, z) const(η) · L2−2α rM e− r3 L const(η, α, N ) · L−N rM1 ,

(4.22)

where we have to remember that we have ∼ L2−2α of gγ ’s. The second estimate says that the norm decays faster than any power of L, with the price of a higher power in r. We now want to show that VL (B, z) does not contribute to the thermodynamic limit of σL (B). We have the following result: Proposition 4.1. σL (B) = −

1 · Tr Vol(ΛL )

fFD (z) P1 (B)UL (B, z)P2 (B)UL (B, z + ω)

Γω

+ P1 (B)UL (B, z − ω)P2 (B)UL (B, z) dz + O L−∞ .

(4.23)

2042


Proof. The main idea is to show that when we replace (HL (B) − z)−1 in FL with the right-hand side of (4.18), all terms containing VL (B) will generate (after integrating by parts with respect to z) some operators which in the trace norm will decay faster than any power of L. If the differentiation does not act on VL (B) but on the other resolvents, then it is enough to know that in the norm of B(L2 ) it goes to zero faster than any power in L, as we saw in (4.22). If the differentiation with respect to z acts on VL (B), then there will be a few terms which must be separately considered, and prove their smallness in the trace norm. To give an example, after differentiating N − 1 times with respect to z (N large), we obtain a term containing a factor like: 3 −N (∂j g˜ 0 )Pj (B) H∞ (B) − z g0 .

(4.24)

j =1

We will prove (in the trace norm) that it decays faster than any power of L, times some polynomially bounded factor in r. Note that all the other factors multiplying the above operator are bounded operators, with a norm which is polynomially bounded in r. Let us start with a technical result: Lemma 4.2. Let Q1 and Q2 be two compact unit cubes such that dist(Q1 , Q2 ) = d > 1, and let χ1 , χ2 denote their characteristic functions. Let α ∈ {0, 1} and j ∈ {1, 2, 3}. Then if N is large enough, there exist three constants δ2 > 0, N1 > 1 and C > 0, all three independent of z ∈ Γω , d, α, j and Q’s such that χ1 P α (B) H∞ (B) − z −N χ2 j

B1

Cr N1 exp{−dδ2 /r}.

(4.25)

Proof. We assume that α = 1, the other case being similar. The strategy is to write our operator as a product of two Hilbert–Schmidt operators. By commuting Pj (B) with one resolvent, we can rewrite our operator as: −1 −N +1 χ1 H∞ (B) − z Tj H∞ (B) − z χ2 , where Tj is a bounded operator which contains factors like Pk (B)(H∞ (B) − z)−1 . Denote by x2 an arbitrary point in the support of χ2 . We insert some exponentials in the following way: δ2 δ2 −1 δ2 χ1 e− r ·−x2 χ1 e r ·−x2 H∞ (B) − z e− r ·−x2 −N +1 − δ2 ·−x2 δ2 ·−x2 δ2 e r χ2 e r χ2 . · e r ·−x2 Tj H∞ (B) − z

(4.26)

If δ2 < δ we can write: δ2

−δ

δ2

e− r x−x2 e r x−x e r x −x2 const e−

δ−δ2

r |x−x |

.

Now the two factors containing resolvents in (4.26) are Hilbert–Schmidt due to the presence of the cut-offs χ and the exponential decay of our kernels (Proposition A.2 and (3.1)). Their


2043

Hilbert–Schmidt norm will grow polynomially with r, but be independent of d. The choice of x2 provides the decaying exponential factor on the right-hand side of (4.25). 2 Let us go back to (4.24) and try to use the previous lemma. We will show that in the trace norm, this operator decays exponentially in Lα . Consider only one j . Cover both supp{∂j g˜ 0 } and supp{g0 } with disjoint cubes centered at points in Z3 and side length equal to 1; we only use cubes which have common points with the respective supports. We then have −N −N (∂j g˜ 0 )Pj (B) H∞ (B) − z g0 = χ˜ s (∂j g˜ 0 )Pj (B) H∞ (B) − z g0 χs ,

(4.27)

s,s

where χ˜ s and respectively χs denote the characteristic function of such unit cubes which cover supp{∂j g˜ 0 } and respectively supp{g0 }. The number of cubes needed to cover the support of g0 is of order L3 , while for the other one is of order L3α ; hence we have about L3+3α terms in the above double sum. But each operator of the form −N χ˜ s (∂j g˜ 0 )Pj (B) H∞ (B) − z g0 χs is exponentially small in the trace norm due to Lemma 4.2, since the distance between any two supports of χ˜ s and χs is of order Lα . Hence the entire sum in (4.27) will be (r-dependent) exponentially small in Lα . But then we can trade off the fading exponential decay with a polynomial decay in L and a polynomial growth in r as we did in (4.22). So this term is under control. Now let us go back to the beginning of the proof of Proposition 4.1. Other “bad” terms from the remainder in (4.23) after differentiation with respect to z will contain powers of (HL (B, γ ) − z)−1 , like for example −N gγ . (∂j g˜ γ )Pj,γ (B) HL (B, γ ) − z

(4.28)

Here we cannot easily commute with P ’s due to various boundary terms. But we do not need δ to do that. Look at (A.8), where we put s = r and x0 = γ . Because Aγ is bounded from above α by L (see (4.14)), it follows: (−D + 1) HL (B, γ ) − z −1 rM Lα . δ

·−γ

−

(4.29) δ

·−γ

It means that for small δ, the resolvent e r (HL (B, γ ) − z)−1 e r sandwiched with exponentials remains Hilbert–Schmidt (with a norm which does not grow faster than the B2 norm of (−D + 1)−1 times Lα and some polynomial in r). Since we have N resolvents, the product of two of them will give a trace class operator. Now we can repeat the insertion of exponentials as we did in (4.26), and use (4.19) for getting the exponential decay in Lα . 2 Now let us show that all terms involving the sum over γ ∈ E in (4.16) will not contribute at the end. The explanation is that these terms are “localized near boundary”. We can formulate the result as follows:

2044


Proposition 4.3. If α > 0 is small enough, then: lim

L→∞

σL (B)

1 + · Tr Vol(ΛL )

−1 −1 fFD (z) P1 (B)g˜ 0 H∞ (B) − z g0 P2 (B)g˜ 0 H∞ (B) − z − ω g0

Γω

−1 −1

+ P1 (B)g˜ 0 H∞ (B) − z + ω g0 P2 (B)g˜ 0 H∞ (B) − z g0 dz = 0.

(4.30)

Proof. If we compare this with (4.16), we see that we need to show that all terms containing factors localized near boundary will converge to zero. Let us look at one such term, and prove the next lemma: Lemma 4.4. Assume that 0 < α < 1/3. Then we have that 1 lim · Tr L→∞ Vol(ΛL ) · P2 (B)

−1 fFD (z)P1 (B)g˜ 0 H∞ (B) − z g0

Γω

e

iBφ0 (·,γ )

−1 g˜ γ HL (B, γ ) − z − ω e−iBφ0 (·,γ ) gγ dz = 0.

(4.31)

γ ∈E

Proof. Using (3.18), and the fact that g˜˜ γ g˜ γ = g˜ γ , we can rewrite the above term as −1 1 · Tr fFD (z)P1 (B)g˜ 0 H∞ (B) − z g0 g˜˜ γ Vol(ΛL ) γ ∈E

Γω

−1 · eiBφ0 (·,γ ) P2,γ (B)g˜ γ HL (B, γ ) − z − ω e−iBφ0 (·,γ ) gγ dz.

(4.32)

After integrating by parts N − 1 times with respect to z, we have to deal with several situations. Let us take one resulting term (just the operator in the integrand): −1 −N −iBφ (·,γ ) 0 P1 (B)g˜ 0 H∞ (B) − z g0 g˜˜ γ eiBφ0 (·,γ ) P2,γ (B)g˜ γ HL (B, γ ) − z e gγ , (4.33) and let us estimate its trace norm. The factor containing H∞ (B) is just bounded, and we cannot use it as a Hilbert–Schmidt factor. What we do is to commute g˜ γ over one resolvent to the right and get the identity: −N gγ P2,γ (B)g˜ γ HL (B, γ ) − z − ω −1 −N +1 = P2,γ (B) HL (B, γ ) − z − ω g˜ γ HL (B, γ ) − z gγ

−1 −N HL (B, γ ), g˜ γ HL (B, γ ) − z − ω + P2,γ (B) HL (B, γ ) − z − ω gγ . (4.34)


2045

Now the second term will again contain at least one derivative of g˜ γ and reasoning as we did for (4.28) we can show it will be exponentially small. Let us look at the first term. The operator P2,γ (B)(HL (B, γ ) − ζ )−1 is only bounded. But g˜ γ (HL (B, γ ) − ζ )−N +1 gγ is trace class and g˜ γ HL (B, γ ) − ζ −N +1 gγ B

1

1 g˜ γ HL (B, γ ) − ζ −1

ηN −2

B2

−1 · HL (B, γ ) − ζ gγ B . 2

(4.35)

Hence using (4.29) we have g˜ γ HL (B, γ ) − ζ −1

B2

constrM g˜ γ (−D + 1)−1 B · Lα . 2

But the Hilbert–Schmidt norm of g˜ γ (−D + 1)−1 is of order of the square root of the support of g˜ γ , that is L3α/2 (use here (A.13) with B = 0). The other factor comes with a similar contribution, hence we can write g˜ γ HL (B, γ ) − z − ω −N +1 gγ

B1

constrM · L5α .

Now using this in the integral with respect to z, this particular term will give a contribution of L5α for each γ . Since we have ∼ L2−2α different γ ’s, the total contribution will be bounded by L2+3α . But if α < 1/3, after we divide with the volume of ΛL it will converge to zero. Now let us go back to (4.32), and see that after integration by parts we can get a term like −N −1 P1 (B)g˜ 0 H∞ (B) − z g0 g˜˜ γ eiBφ0 (·,γ ) P2,γ (B)g˜ γ HL (B, γ ) − z − ω gγ .

(4.36)

Here the operator P2,γ (B)g˜ γ (HL (B, γ ) − z)−1 is not Hilbert–Schmidt, so we have to look at the first factor. We can write −N g0 g˜˜ γ P1 (B)g˜ 0 H∞ (B) − z −N +1 −1 = P1 (B)g˜ 0 H∞ (B) − z g˜˜ γ H∞ (B) − z g0

−N −1 H∞ (B), g˜˜ γ H∞ (B) − z g0 . + P1 (B)g˜ 0 H∞ (B) − z

(4.37)

In the first term, the function g˜˜ γ makes the two resolvents next to it become Hilbert–Schmidt, each having a norm proportional with L3α/2 and some power of r (use the exponential decay of the kernels). So this term is “good”, considering that we have to divide with L3 in the end. The second term contains at least one derivative of g˜˜ γ (here [H∞ (B), g˜˜ γ ] is linear in Pj (B)’s), together with factors like −N −1 P1 (B)g˜ 0 H∞ (B) − z Pj (B) (∂j g˜˜ γ ) H∞ (B) − z g0 . Now here we can use the estimate from Proposition 3.2 and see that we again have a product of two Hilbert–Schmidt operators: the first Hilbert–Schmidt norm will be proportional with L3/2 ,

2046


while the other one will behave like L3α/2 . When we divide by L3 , this contribution will go to zero. All the other terms resulting from integrating by parts with respect to z can be treated in a similar way. 2 Now let us go back to (4.23) and analyze another boundary term: Lemma 4.5. For every 0 < α < 1/3 we have: −1 1

· Tr fFD (z) · P1 (B) eiBφ0 (·,γ ) g˜ γ HL (B, γ ) − z e−iBφ0 (·,γ ) gγ lim L→∞ Vol(ΛL )

· P2 (B)

Γω

γ ∈E

−1 eiBφ0 (·,γ ) g˜ γ HL (B, γ ) − z − ω e−iBφ0 (·,γ ) gγ dz = 0.

(4.38)

γ ∈E

Proof. Let us note that when keeping γ fixed, only a finite number (L-independent) of γ ’s will have an overlapping support. This means that the above double sum will only contain around L2−2α non-zero terms. Now use again (3.17) and integration by parts with respect to z. Each non-zero term in the double sum will be a product of two Hilbert–Schmidt operators, each with a Hilbert–Schmidt norm of the order of L5α/2 . The total trace norm will grow at most like L2+3α , hence if α < 1/3 this term will not contribute. We do not give more details. 2 The last ingredient in proving (1.12) is contained in the following result (see also (4.30)): Proposition 4.6. lim −

L→∞

1 · Tr Vol(ΛL )

−1 −1 fFD (z) P1 (B)g˜0 H∞ (B) − z g0 P2 (B)g˜0 H∞ (B) − z − ω g0

Γω

−1 −1 · P1 (B)g˜0 H∞ (B) − z + ω g0 P2 (B)g˜0 H∞ (B) − z g0 dz = − sB (x) dx.

(4.39)

Ω

Proof. First, due to support properties, we have g0 P2 (B)g˜ 0 = g0 P2 (B) = P2 (B) − (1 − g0 )P2 (B).

(4.40)

Because 1 − g0 is supported outside of a thin region around the boundary of ΛL , all terms generated by (1 − g0 )P2 (B) will converge to zero; let us prove this. After integrating N − 1 by parts with respect to z, we obtain several terms from the integrand which look like this one: (N1 + N2 N 10): −N1 −N2 P1 (B)g˜0 H∞ (B) − z (1 − g0 )P2 (B) H∞ (B) − z − ω g0 .

(4.41)

Due to the symmetry of this term, assume without loss of generality that N1 5. Then by writing 1 − g0 = (1 − g0 )χΛL + (1 − g0 )(1 − χΛL ), we get two types of contributions. The one coming


2047

from (1 − g0 )(1 − χΛL ) = (1 − χΛL ) is localized outside ΛL , and its trace norm will be exponentially small in Lα ; we can apply Lemma 4.2 since the distance between ΛcL and the supports of g˜ 0 and g0 is of order Lα . The number of needed covering unit cubes is polynomially bounded in L. Thus the only contribution from (4.41) can come from: −N1 −N2 P1 (B)g˜0 H∞ (B) − z (1 − g0 )χΛL P2 (B) H∞ (B) − z − ω g0 .

(4.42)

Here we can write: −N1 (1 − g0 )χΛL P1 (B)g˜0 H∞ (B) − z −2 −N1 +2 · H∞ (B) − z · P1 (B)g˜0 H∞ (B) − z (1 − g0 )χΛL ,

(4.43)

where both factors are Hilbert–Schmidt, with kernels exponentially localized near diagonal as in Propositions 3.1 and 3.2. The Hilbert–Schmidt norm of the first factor is bounded by L3/2 , while for the second one we have a bound of L1+α/2 (square roots of certain volumes). The trace norm of the product is thus bounded by L5/2+α/2 and some polynomial in r. After integrating with respect to z, and dividing with L3 (the volume of ΛL ), this term will converge to zero provided α < 1. Now we can go back to (4.40) and analyze the term generated by P2 (B). Because there are no other cut-offs in the middle, and because the commutator [P1 (B), g˜ 0 ] will generate another fast decaying term, we see that we have just proved the following identity (see Theorem 1.1(ii) for the definition of F∞ ): lim

L→∞

1 σL (B) + Tr{g˜ 0 F∞ g0 } . Vol(ΛL )

(4.44)

But the operator g˜ 0 F∞ g0 is trace class, with a jointly continuous kernel, hence (see (4.1), (4.3) and (4.7)) Tr{g˜ 0 F∞ g0 } =

sB (x, x)g0 (x) dx.

supp(g0 )

Using the periodicity of sB with respect to Z3 and the support properties of g0 , we finally get: lim σL (B) = −

sB (x, x) dx

L→∞

(4.45)

Ω

and the proof of (1.12) is over.

2

4.3. Proof of (1.13) We start by investigating ∂B σL (0) and try to put it in a form which is better suited for the thermodynamic limit.

2048


Note that we have already argued that σL was smooth in B near zero (see the remark around (2.5)). The hard part is to show that the polynomial growth in L of the trace norm does not appear in the actual trace. Similar difficulties involving magnetic semigroups were encountered in [1,6,7,10]. 4.3.1. Only UL (B) counts Notation. In order to shorten our formulas, we will sometimes write (z → z − ω), which means that we have to repeat the previously appearing product of operators in which we have to change z with z − ω. First, let us prove that even if we differentiate with respect to B, we still have a result similar to Proposition 4.1. Proposition 4.7. At B = 0 we have: ∂B σL (B) +

1 · Tr Vol(ΛL )

= O L−∞ .

fFD (z) P1 (B)UL (B, z)P2 (B)UL (B, z + ω) + z → z − ω dz

Γω

(4.46)

Proof. As usual, before differentiating with respect to B one has to perform a certain number of integrations by parts with respect to z. There will appear many terms in the remainder containing VL (B, z) (see (4.17)), but all of them will have the distinctive feature of containing pairs of cut-off functions whose supports are at a distance ∼ Lα one from another. We do not want to treat all the situations which can appear here, and instead we will only prove a typical estimate needed for the result. Lemma 4.8. Assume that N is large enough, and choose z such that dist{z, [0, ∞)} = η > 0, r = (z). Then the map −N (−1, 1) B → (g˜ 0 ) H∞ (B) − z g0 ∈ B1 L2 is differentiable in the trace norm, and there exists δ1 small enough and M large enough such that δ α ∂B (g˜ 0 ) H∞ (B) − z −N g0 const(η, N )e− r1 L rM . B=0 1

(4.47)

Moreover, (g˜ 0 ) H∞ (B) − z −N g0 − (g˜ 0 ) H∞ (0) − z −N g0 δ1 α −N − B ∂B (g˜ 0 ) H∞ (B) − z g0 B=0 B const(η, N )e− r L rM . 1

(4.48)


2049

Proof. We only prove differentiability at B = 0. The key ingredient is to give a proper sense to the formal identity: −1 −1 H∞ (B) − z − H∞ (0) − z −1 −1 H∞ (B) − H∞ (0) H∞ (0) − z . = − H∞ (B) − z

(4.49)

As it stands, the right-hand side makes no sense because H∞ (B) − H∞ (0) contains terms like −2iBA(x) · ∇x which are not relatively bounded to the free Laplacian due to the linear growth of our magnetic potential. For 0 < δ2 < δ3 and x0 ∈ R3 , we can write (still formally): δ2 −1 −1 − δ3 ·−x0 e r − H∞ (0) − z e r ·−x0 H∞ (B) − z δ2

= −e r

·−x0

H∞ (B) − z

−1 −1 − δ3 ·−x0 H∞ (B) − H∞ (0) H∞ (0) − z e r .

(4.50)

Note the fact that the growing exponential is weaker that the decaying one. This identity now holds in the Hilbert–Schmidt class; in order to see this, introduce the notation 1 < α1 < α2
1: −1 C sup [Dα + Baα ] (−i∇ + Ba)2D − iλ √ . λ L>1 Since V is bounded we have:

−1 C sup V (−i∇ + Ba)2D − iλ . λ L>1 Choosing a λ0 large enough and using the Neumann series in V for the resolvent we have −1

−1 sup V HL (B) − iλ0 + (−i∇ + Ba)2D − iλ0 HL (B) − iλ0 1/2.

L>1

Using the resolvent identity we obtain:

−1 sup (−i∇ + Ba)2D − iλ0 HL (B) − z C|z|/η.

(A.4)

L>1

Hence writing −1 [Dα + Baα ] HL (B) − z

−1

−1 (−i∇ + Ba)2D − iλ0 HL (B) − z = [Dα + Baα ] (−i∇ + Ba)2D − iλ0 we obtain the result.

2

We will need a certain type of uniform exponential localization, stated in the proposition below. If x0 is some√point in ΛL and α ∈ R, then let eα·−x0 denote the multiplication operator

with the function eα |x−x0 | +1 . Note that multiplication with the exponential weight is a bounded operator if L < ∞ and leaves invariant the domain of HL (B). 2

Proposition A.2. Fix x0 ∈ ΛL and dist{z, [0, ∞)} = η > 0. Denote by r := (z). Then there exists a δ0 > 0 and a constant C such that for every 0 δ δ0 we have


2063

±δ ∓δ −1 sup sup sup e·−x0 r HL (B) − z e·−x0 r C,

(A.5)

±δ ∓δ −1 sup sup sup r−1 [Dα + Baα ]e·−x0 r HL (B) − z e·−x0 r C,

(A.6)

L>1 r∈R x0 ∈Λ

L>1 r∈R x0 ∈Λ

and ±δ ∓δ

−1 sup sup sup r−1 e·−x0 r (−i∇ + Ba)2D + 1 HL (B) − z e·−x0 r C. (A.7)

L>1 r∈R x0 ∈Λ

Proof. An heuristic explanation of (A.5) is the following: if we apply the resolvent on a funcδ

tion which is exponentially localized near x0 and decays like e− r x−x0 , then we do not loose exponential decay. Moreover, the L2 bounds are uniform in z, L and the location of x0 . For s ∈ R, the well-known Combes–Thomas rotation [9] gives: 3 es·−x0 HL (B) − z e−s·−x0 = HL (B) − z + s wj [Dj + Baj ] + sV1 + s 2 V2 , j =1

where wj , V1 , V2 are bounded functions, uniformly in L and x0 . Now put s = δ/r, and use (A.2). If δ is small enough, we get that uniformly in L, x0 and r 3 −1 2 wj [Dj + Baj ] + sV1 + s V2 HL (B) − z 1/2, s j =1

which gives −1 es·−x0 HL (B) − z e−s·−x0 3 −1 −1 −1 2 = HL (B) − z wj (Dj + Baj ) + sV1 + s V2 HL (B) − z . 1+ s j =1

(A.8) This implies (A.5), and together with (A.2) we also get (A.6). Let us now concentrate ourselves on the last estimate (A.7). Up to a commutation, (A.6) gives δ δ −1 sup sup sup r−1 e·−x0 r [Dα + Baα ] HL (B) − z e−·−x0 r C.

(A.9)

L>1 r∈R x0 ∈Λ

Thus again up to a commutation, (A.7) follows if we can prove δ δ

−1 sup sup sup r−1 (−i∇ + Ba)2D + 1 e·−x0 r HL (B) − z e−·−x0 r C.

L>1 r∈R x0 ∈Λ

(A.10) But this estimate follows from (A.8) and (A.4).

2

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Corollary A.3. Let λ λ0 > 0. Then there exists c > 0 such that √ √ −1 const(λ0 ) . sup sup sup e±c·−x0 λ HL (B) + λ e∓c·−x0 λ λ L>1 λλ0 x0 ∈Λ

(A.11)

Proof. We use the key estimate (A.1) for the case when η λ and (z) = −λ < 0. This gives us √

(Dα + Baα ) (−i∇ + Ba)2 + λ −1 const / λ, D hence √

(Dα + Baα ) HL (B) + λ −1 const / λ. Now we proceed as in (A.8) and we get the result. Finally, note that by repeating the argument of Proposition A.2 we can obtain a uniform estimate in λ, L and x0 : √ ±c·−x √λ

−1 0 e (−i∇ + Ba)2D + λ HL (B) + λ e∓c·−x0 λ const .

2

(A.12)

Proposition A.4. The operator [(−i∇ + Ba)2D + λ]−1 has an integral kernel KL (x, x ) which is jointly continuous away from the diagonal x = x , and obeys the estimate √

− λ|x−x | KL (x, x ) e , 4π|x − x |

(A.13)

for every x = x in ΛL . Proof. The argument is based on several well known results. First, one uses the Feynman–Kac– 2 Itô representation for the kernel of the semigroup e−t (−i∇+Ba)D , t > 0 (see [8]) and obtains a diamagnetic inequality in ΛL : −t (−i∇+Ba)2 |x−x |2 e D (x, x ) e tD (x, x ) (4πt)−3/2 e − 4t , Second, we perform a Laplace transform and obtain the result.

x, x ∈ ΛL .

2

Proposition A.5. All the results in this section are also valid if the operators are defined on the whole space R3 (formally L = ∞). Proof. The argument relies on various standard limiting and cut-off arguments, which are necessary because the exponential growing factors do not invariate the operator domain of H∞ (B). The most important ingredient (uniformity in L > 1 of all our previous estimates) has been already proved. 2


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References [1] N. Angelescu, M. Bundaru, G. Nenciu, On the Landau diamagnetism, Comm. Math. Phys. 42 (1975) 9–28. [2] V. Bonnaillie-Noël, S. Fournais, Superconductivity in domains with corners, Rev. Math. Phys. 19 (6) (2007) 607– 637. [3] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 2. Equilibrium States. Models in Quantum Statistical Mechanics, Texts Monogr. Phys., Springer-Verlag, Berlin, 1997. [4] P. Briet, J.M. Combes, P. Duclos, Spectral stability under tunneling, Comm. Math. Phys. 126 (1) (1989) 133–156. [5] P. Briet, H.D. Cornean, Locating the spectrum for magnetic Schrödinger and Dirac operators, Comm. Partial Differential Equations 27 (5–6) (2002) 1079–1101. [6] P. Briet, H.D. Cornean, D. Louis, Generalized susceptibilities for a perfect quantum gas, Markov Process. Related Fields 11 (2) (2005) 177–188. [7] P. Briet, H.D. Cornean, D. Louis, Diamagnetic expansions for perfect quantum gases, J. Math. Phys. 47 (8) (2006) 083511. [8] K. Broderix, D. Hundertmark, H. Leschke, Continuity properties of Schrödinger semigroups with magnetic fields, Rev. Math. Phys. 12 (2) (2000) 181–225. [9] J.M. Combes, L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys. 34 (1973) 251–270. [10] H.D. Cornean, On the magnetization of a charged Bose gas in the canonical ensemble, Comm. Math. Phys. 212 (1) (2000) 1–27. [11] H.D. Cornean, G. Nenciu, On eigenfunction decay for two-dimensional magnetic Schrödinger operators, Comm. Math. Phys. 198 (3) (1998) 671–685. [12] H.D. Cornean, G. Nenciu, Two-dimensional magnetic Schrödinger operators: Width of mini bands in the tight binding approximation, Ann. Henri Poincaré 1 (2) (2000) 203–222. [13] H.D. Cornean, G. Nenciu, T.G. Pedersen, The Faraday effect revisited: General theory, J. Math. Phys. 47 (1) (2006) 013511. [14] S. Fournais, B. Helffer, On the third critical field in Ginzburg–Landau theory, Comm. Math. Phys. 266 (1) (2006) 153–196. [15] A. Grigis, J. Sjöstrand, Microlocal Analysis for Differential Operators. An Introduction, London Math. Soc. Lecture Note Ser., vol. 196, Cambridge University Press, Cambridge, 1994. [16] S. Hacyan, R. Jauregui, Faraday effect and Bessel beams in a magneto-optic medium, J. Phys. B 41 (1) (2008) 015402. [17] B. Helffer, J. Sjöstrand, On diamagnetism and de Haas–van Alphen effect, Ann. Inst. H. Poincaré Phys. Théor. 52 (4) (1990) 303–375. [18] B. Helffer, J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 118–197. [19] L. Hörmander, Linear Partial Differential Operators. I, Grundlehren Math. Wiss., vol. 116, Academic Press/ Springer-Verlag, New York/Berlin–Göttingen–Heidelberg, 1963. [20] V. Iftimie, Uniqueness and existence of the integrated density of states for Schrödinger operators with magnetic field and electric potential with singular negative part, Publ. Res. Inst. Math. Sci. 41 (2) (2005) 307–327. [21] V. Iftimie, M. M˘antoiu, R. Purice, Magnetic pseudodifferential operators, Publ. Res. Inst. Math. Sci. 43 (3) (2007) 585–623. [22] H.A. Kastrup, Quantization of the canonically conjugate pair angle and orbital angular momentum, Phys. Rev. A 73 (2006) 052104. [23] D. Martinez, C. Plechaty, R. Presura, Magnetic fields for the laboratory simulation of astrophysical objects, Astrophys. Space Sci. 307 (1–3) (2007) 109–114. [24] M. M˘antoiu, R. Purice, Strict deformation quantization for a particle in a magnetic field, J. Math. Phys. 46 (5) (2005) 052105. [25] M. M˘antoiu, R. Purice, The magnetic Weyl calculus, J. Math. Phys. 45 (4) (2004) 1394–1417. [26] M. M˘antoiu, R. Purice, S. Richard, Spectral and propagation results for magnetic Schrödinger operators: A C ∗ algebraic framework, J. Funct. Anal. 250 (1) (2007) 42–67. [27] G. Nenciu, On asymptotic perturbation theory for quantum mechanics: Almost invariant subspaces and gauge invariant magnetic perturbation theory, J. Math. Phys. 43 (3) (2002) 1273–1298. [28] T.G. Pedersen, Tight-binding theory of Faraday rotation in graphite, Phys. Rev. B 68 (2003) 245104. [29] L.M. Roth, Theory of the Faraday effect in solids, Phys. Rev. 133 (1964) A542–A553.

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[30] D. Ruelle, Statistical Mechanics: Rigorous Results, World Scientific Publishing Co./Imperial College Press, River Edge, NJ/London, 1999. [31] L.M. Roth, Theory of Bloch electrons in a magnetic field, J. Phys. Chem. Solids 23 (1962) 433. [32] E.H. Sondheimer, A.H. Wilson, The theory of the magneto-resistance effects in metals, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 190 (1023) (1947) 435–455. [33] A. Zarifi, T.G. Pedersen, Theoretical analysis of the Faraday effect in zigzag carbon nanotubes, Phys. Rev. B 77 (2008) 085409.


H p –H q estimates for dispersive equations and related applications ✩ Yong Ding a , Xiaohua Yao b,∗ a School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (Beijing Normal University),

Ministry of Education, Beijing Normal University, Beijing 100875, PR China b Department of Mathematics, Huazhong Normal University, Wuhan 430079, PR China

Received 25 August 2008; accepted 1 July 2009 Available online 18 July 2009 Communicated by I. Rodnianski

Abstract This paper studies the H p –H q estimates of the solutions for a class of dispersive equations under the assumption that principal operators are homogeneous elliptic and the corresponding level hypersurfaces are convex and of finite type. These estimates allow us to show higher order Schrödinger operator generates a fractionally integrated group in Lp (Rn ). © 2009 Elsevier Inc. All rights reserved. Keywords: Dispersive equation; H p –H q estimate; Convex hypersurface; Integrated semigroup

1. Introduction In this paper, we are mainly concerned with the H p –H q estimates of the solution for the following Cauchy problem of dispersive type:

∂t u(t, x) = iP (D)u(t, x), u(0, x) = u0 (x),

(t, x) ∈ R × Rn , x ∈ Rn ,

✩

(1.1)

The first author was supported by NSFC (No. 10571015) and SRFDP of China (No. 20050027025), the second author was supported by NSFC (No. 10801057) and the Key Project of Chinese Ministry of Education (No. 109117). * Corresponding author. E-mail addresses: [email protected] (Y. Ding), [email protected] (X. Yao). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.002

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Y. Ding, X. Yao / Journal of Functional Analysis 257 (2009) 2067–2087

where n 2, D = −i(∂/∂x1 , . . . , ∂/∂xn ), P : Rn → R is a real homogeneous elliptic polynomial of order m 2 (m must be even), and H p (Rn ) (0 < p < ∞) denotes Hardy space, as well as H ∞ (Rn ) denotes BMO(Rn ) (bounded mean oscillation space) for the convenience of notation. As we known, H p (Rn ) = Lp (Rn ) for 1 < p < ∞ and H 1 (Rn ) (resp. L∞ (Rn )) is a proper subspace of L1 (Rn ) (resp. BMO(Rn )) (see [32, Chapters III–IV]). For every initial value u0 ∈ S(Rn ) (the Schwartz space), by Fourier transform the solution of (1.1) can be given by u(t, ·) := eitP (D) u0 = F −1 eitP uˆ 0 (ξ ) , where F (or ˆ ) denotes Fourier transform and F −1 is its inverse. When P (ξ ) = |ξ |2 , it is well known that Eq. (1.1) represents free Schrödinger equation and the solution operator e−it satisfies with the following sharp Lp –Lp -estimates −it e

n

Lp –Lp

C|t| 2

( p1 − p1 )

,

t = 0, 1 p 2

(1.2)

where p is the conjugate index of p (see e.g. [33, p. 71]). In particular, a remarkable fact is that the estimates (1.2) imply the famous Strichartz inequalities which are very useful in the well-posedness of nonlinear Schrödinger equations (see e.g. [8,10,24,33]). In order to treat with the H p –H q (also Lp –Lq )-estimates for Eq. (1.1) with a real homogeneous elliptic polynomial P , we need estimate its fundamental solution F −1 (eitP ) (t = 0). For this, the geometry property of the level set Σ := ξ ∈ Rn : P (ξ ) = 1

(1.3)

plays a crucial role. Note that Σ is always a compact connected smooth hypersurface of Rn due to the homogeneous elliptic condition of P . If the hypersurface Σ has nonvanishing Gaussian curvature everywhere (e.g. spherical surface Sn−1 of Rn corresponding to polynomial P (ξ ) = |ξ |2N ), then the optimal H p –H q estimates for the evolution operator eitP (D) (t = 0) can be deduced from [27, Theorem 4.1]. Actually, Miyachi [27] mainly considered the H p –H q boundedness of a class of singular multipliers a ψ(ξ )|ξ |−b ei|ξ | (a > 0, b ∈ R) where ψ(ξ ) is equal to 1 for large ξ and 0 near origin. But from [27, Rem. 4.2] one knows that his results even hold for a positively homogeneous phase function P of degree m with nonzero Gaussian curvature’s Σ . Moreover, we remark that, based on one or other nondegenerate conditions on P which are all equivalent that Σ has nonzero Gaussian curvature everywhere, the Lp –Lq estimates and some related topics have also been extensively generalized to nonhomogeneous polynomials P (ξ ) + Q(ξ ), where Q(ξ ) is any real polynomial of order less than m (see e.g. [2,5,6,9,15,23,25]). In particular, all one-dimensional cases have been covered in these papers. On the other hand, if dropping the restriction of the nonzero Gaussian curvature at some points of Σ, then it would become more difficult to estimate the oscillatory integral F −1 (eitP ) (t = 0) due to the failure of the principle of stationary phase (see e.g. [32, p. 344]). In fact, there exist many elliptic polynomials such that their level hypersurfaces have zero Gaussian curvature at some points, for instance ξ1m + · · · + ξnm (m = 4, 6, . . .) and ξ14 + 6ξ12 ξ22 + ξ24 . Motivated by these examples, based on a powerful result of [7, Theorem B], Zheng et al. [38] recently made an interesting work about Lp –Lq estimates for the evolution operators eitP (D) for Σ is


1 q

2069

F (1,1)

1

2kn(m−1) τ = m(2n+k−2)−2kn ,

1 2

O

B = (1, τ1 ),

A ( 12 , 12 ) •

1 2

1

D = ( τ , 0) B(1, 1 ) • q0 • B D E( 1 , 0) 0 •• • p • D( 1 , 0) q0

1 p

C (1,0)

Fig. 1. The H p –H q estimates of eitP (D) (t = 0) when k > 2.

a convex hypersurface of finite type k 2 (k ∈ N). More precisely, they proved that (see [38, Theorem 2.4]) itP (D) e

n

Lp –Lq

C|t| m

( q1 − p1 )

,

t = 0,

(1.4)

where ( p1 , q1 ) ∈ 2AB CD \ {B , D }, and 2AB CD is a closed quadrangle by the four vertex points

(see Fig. 1): A= ( 12 , 12 ), B = (1, τ1 ), C= (1, 0), and D = ( τ1 , 0), where τ is the conjugate index of τ and τ=

2kn(m − 1) . m(2n + k − 2) − 2kn

(1.5)

If Σ is convex and k = 2, i.e. equivalently, Σ has nonzero Gaussian curvature everywhere, then the range 2AB CD is sharp for (1.4) on Lp (Rn ). Thus a natural question we ask is: does the sharpness of 2AB CD remain if k > 2? To this problem, although we don’t affirm how large is the optimal range for (1.4) as k > 2, but we certainly negate the sharpness of 2AB CD for k > 2. Indeed, in this paper we firstly establish the H p –H q estimates of eitP (D) under the same conditions (see Theorem 2.3 below), from which it follows that, as k > 2, the estimates (1.4) for eitP (D) can be extended to the larger region 2ABCD (see Fig. 1) if L1 (Rn ) (resp. L∞ (Rn )) is replaced by H 1 (Rn ) (resp. BMO(Rn )), and as k = 2, 2ABCD is again identical with 2AB CD . Therefore Theorem 2.3 improves the estimates (1.4) and extends to H p (Rn ) (0 < p 1). More generally, we investigate the H p –H q -estimates of regularized operators J σ eitP (D) (t = 0) and I σ eitP (D) (t = 0) where, J σ = (1 − )σ/2 and I σ = (−)σ/2 denote the Bessel potential and Riesz potential of order −σ , respectively. In particular when σ > 0, we remark that the kind of estimates give so-called global smoothing effects of Eq. (1.1), as shown by [2,23] on Lp for nondegenerate polynomials. Moreover, as an application of the Lp –Lq -estimates of J σ eitP (D) (t = 0), another aim of the paper is to show that higher order Schödinger operator iP (D) + V (x, D) in Lp (Rn ) generates a

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fractionally integrated group where V (x, D) can be a differential operator of lower order than P (see Theorem 4.3). As we known, the semigroup of operator is an abstract tool to treat Cauchy problems. However, the elliptic operator iP (D) in Eq. (1.1) cannot generate a classical C0 -semigroup on Lp (Rn ) (p = 2) (see [21]). Since then, several generalizations of C0 -semigroup were introduced, such as distribution semigroup, integrated semigroup and regularized semigroup et al., as well as applied to general differential operators (including iP (D) and even nonelliptic class) and associated Cauchy problems (see e.g. [1,4,16,17,19,20,37]). Here, the corresponding Cauchy problem with iP (D) + V (x, D) is the following generalized Schödinger equation:

∂t u(t, x) = iP (D)u(t, x) + V (x, D)u(t, x), u(0, x) = u0 (x),

(t, x) ∈ R × Rn , x ∈ Rn ,

(1.6)

As a consequence of Theorem 4.3, certain Lp –Lp estimates of solution for Eq. (1.6) can be obtained by employing Straub’s fractional powers (see Theorem 4.5). When V (x, D) is a suitable integrable complex function V (x), similar arguments for Eq. (1.6) have also been considered in [5,26,30,36,38] by semigroup methods. Nevertheless, none of them can deal with Eq. (1.6) with a differential perturbed operator V (x, D). Finally, with respect to the classical Schrödinger equation, i.e. Eq. (1.6) where iP (D) + V (x, D) = i(− + V (x)) and V (x) is a real potential, the study of Lp –Lq -estimates for the Schrödinger group e−it (−V ) has received great attentions in recent years motivated by nonlinear problems (see [8, pp. 17–27], also refer to [31] for a detail survey on the subject). However, for higher order Schrödinger group eit (P (D)+V ) , such an analysis similar to (1.4) seems undeveloped as much as we best know. Clearly, it would be very interesting to further study the problem. The paper is organized as follows: Section 2 is to state the main results and Section 3 is to give their proofs. In Section 4 we show that higher order Schödinger operator iP (D) + V (x, D) is a generator of a fractionally integrated group in Lp (Rn ) by a perturbation method. Finally, the proofs of two useful propositions are given as Appendix A. 2. The H p –H q estimates of solution In the sequel, throughout the paper, let P be a real homogeneous elliptic polynomial of order m on Rn , n 2, and Σ is always a hypersurface defined by P in (1.2). Now we recall that Σ is of finite type if there exist k ∈ N and C > 0 such that k η, ∇ j P (ξ ) C > 0,

ξ ∈ Σ and η ∈ Sn−1 ,

(2.1)

j =1

where η, ∇ = Σ is convex if

n

l=1 ηl ∂/∂xl .

The least k in (2.1) is called the type order of Σ. Also, say that

Σ ⊂ η ∈ Rn η − ξ, ∇P (ξ ) 0 ,

ξ ∈ Σ;

Σ ⊂ η ∈ Rn η − ξ, ∇P (ξ ) 0 ,

ξ ∈ Σ.

or


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For a given P , we knew that corresponding Σ is always of finite type and 2 k m by direct calculations (or see [38, Prop. 2.1]). But it is obviously not always convex. In fact, the following proposition gives some criterions to determine whether Σ is a convex hypersurface or not. Proposition 2.1. The following statements are equivalent. (a) The hypersuface Σ is convex. (b) All principal curvatures of Σ are nonnegative everywhere with respect to the inward unit normal orientation. (c) The Hessian matrix [∂i ∂j P (ξ )]n×n of P is semi-definite for each ξ ∈ Rn . Additionally, if the type order of Σ is exactly 2, then we also have further consequences: Proposition 2.2. The following statements are equivalent. (a) The hypersuface Σ is convex and k = 2. (b) Σ has nonzero Gaussian curvature everywhere. (c) P is nondegenerate, i.e. the determinant of Hessian matrix satisfies that det ∂i ∂j P (ξ ) n×n = 0,

ξ = 0.

The proofs of Propositions 2.1 and 2.2 are partially elementary, which are given as Appendix A in view of completion. By exploiting the arguments above one easily checks that all examples below are degenerate and satisfy with the condition of convexity: ξ1m + · · · + ξnm (m = 4, 6, . . .) and ξ14 + 6ξ12 ξ22 + ξ24 . Theorem 2.3. Suppose Σ is a convex hypersurface of finite type k for 2 k m. Then for each pair ( p1 , q1 ) ∈ 0 we have itP (D) e

n

H p –H q

C|t| m

( q1 − p1 )

,

t = 0,

(2.2)

where 0 is a closed triangle by the three vertex points (see Fig. 1): A = ( 12 , 12 ), D = ( q1 , 0) and 0

E = ( p10 , 0), where q0 is the conjugate index of q0 and 1 m(2n + k − 2) , = p0 2kn

1 1 − p0 m(2n + k − 2) − 2kn . = = q0 2 − p0 2m(2n + k − 2) − 2kn

Remark 2.1. (i) Note that the point B(1, q10 ) is the point of intersection between line CF and line AE. Since 1 1 1 1 m−2 q0 > τ if and only if k > 2, and q0 = τ = 2(m−1) if and only if 2AB CD is the proper subset of 2ABCD when k > 2 as described

k = 2, thus it follows that in above Fig. 1, and again

the two sets are identical if k = 2. (ii) The conclusion of Theorem 2.4 in [38] actually showed the Lp –Lq estimates of eitP (D) (t = 0) only for these points belonging to the set Ξ := {(1/p, 1/q): 1 p 2, q ∈ Ip } where

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⎧ ⎨ (q(p), ∞], ) Ip = (q(p), p(2−τ ), p−τ ⎩ {2},

if 1 p < τ ; if τ p < 2; if p = 2.

1 1 Here q(p) = τp + τ 1p and τ is given in Fig. 1. It is easy to check that the set Ξ is exactly equal to (2AB CD \ {AB , AD }) ∪ {A}. However, as claimed for (1.4) in the introduction, the authors of [38] essentially obtained a slight larger set 2AB CD \ {B , D }. Indeed, using

−n/τ , F −1 e±iP (x) = O 1 + |x|

as |x| → ∞,

(see [38, Theorem 2.3] or Proposition 3.2 below), by scaling we can obtain F −1 eitP ∈ L∞ Rn ∩ Lτ,∞ Rn ,

t = 0,

where Lτ,∞ (Rn ) is the weak-Lτ space. Since eitP (D) u0 = F −1 eitP ∗ u0 ,

u0 ∈ S Rn ,

thus by the Young (or weak-Young) inequality (see e.g. [18, p. 22]) it follows the L1 –L∞ and L1 – Lτ,∞ estimates for eitP (D) (t = 0), i.e. corresponding to points C (1,0) and B (1, τ1 ), respectively. Next combining with the trivial case A( 12 , 12 ), the desired claim immediately follows from the Marcinkiewicz interpolation theorem (cf. [18, p. 38]) and a dual argument. As a consequence, Theorem 2.3 can immediately follow from Theorem 2.4 below, which deals with the H p –H q -estimates of regularized operators J σ eitP (D) (t = 0) and I σ eitP (D) (t = 0) where J σ = (1 − )σ/2 and I σ = (−)σ/2 for σ 0. Theorem 2.4. Suppose Σ is a convex hypersurface of type k for 2 k m. If 0 σ m(2n+k−2) − n, then for each pair ( p1 , q1 ) ∈ σ , we have 2k σ itP (D) J e

H p –H q

n 1 1 σ ( − ) C 1 + |t|− m |t| m q p ,

t = 0,

(2.3)

and σ itP (D) I e

H p –H q

n

C|t| m

σ ( q1 − p1 )− m

,

t = 0,

(2.4)

where σ is a closed triangle by the three vertex points (see Fig. 2): A = ( r1σ , r1 ), D = ( q1 , 0) and E = ( p1σ , 0), where qσ (resp. rσ ) is the conjugate index of qσ (resp. rσ ), 1 = rσ and

1

2, 1 kσ 2 + m(2n+k−2)−2kn ,

if m = 2; if m > 2;

σ

σ


1 q

2073

F (1,1)

1

B= (1, q1σ ) 1 2

1 2

O

C= (1, 0) • A ( r1 , 1 ) σ rσ B E( 1 , 0) • pσ • D( 1 , 0) qσ

1 p

C

Fig. 2. The H p –H q estimates of J σ eitP (D) (t = 0) for some σ 0.

1 (1 − rσ )(1 − pσ ) = . qσ pσ − rσ

1 m(2n + k − 2) σ − , = pσ 2kn n

Remark 2.2. (i) If Σ is convex and k = 2, i.e. Σ has nonzero Gaussian curvature everywhere, then 0 σ n(m − 2)/2, and the triangle σ is optimal range for the estimates (2.3) by testing m the special multiplier ψ(ξ )|ξ |σ ei|ξ | where ψ(ξ ) is equal to 1 for large ξ and 0 near origin (see [27, Theorem 4.1] for the details). In particular, if taking σ = n(m − 2)/2, then the set n(m−2)/2 only consists of one point C= (1, 0). Hence we obtain the following only decay estimate: n(m−2) itP (D) I 2 e

n

H 1 –H ∞

C|t|− 2 ,

t = 0.

(2.5)

Notice that n(m − 2)/2 is an integer, so the operator I n(m−2)/2 can be replaced by any partial derivative D α with |α| = n(m − 2)/2. Moreover, H 1 –H ∞ in (2.5) also can be replaced by L1 –L∞ in view of the proof of Theorem 2.4, and if (2.5) is equipped with L1 –L∞ -norm, then its another related estimate is W (t)

n

L1 –L∞

C|t|− 2 ,

t = 0,

where W (t)u0 = F −1 (|H P (ξ )|1/2 eitP (ξ ) uˆ 0 (ξ )) and H P (ξ ) = det([∂i ∂j P (ξ )]n×n ) is a homogeneous elliptic polynomial of order n(m − 2) (see [23, Theorem 3.2]). (ii) If m = 2, then P (ξ ) = ξ, Aξ where A is a positively defined matrix, and corresponding Σ is an ellipsoid of Rn . Therefore we only have that k = 2, σ = 0, and 0 = {( p1 , p1 ); 1 p 2}. Thus from (2.3) and Remarks (i) it follows that itD,AD e

Lp –Lp

n

C|t| 2

( p1 − p1 )

,

t = 0, 1 p 2,

which indicates that the solution u(t, x) of second order dispersive equation (1.1) cannot gain any global smoothing effect on Lp (Rn ) if t = 0 and initial value u0 in Lp (Rn ) (1 p 2).

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Nevertheless, the solution u(t, x) satisfies with the following important local smoothing effect: T

(1 − ) 14 u(t, x)2 dx dt C(T , R)u0 2 . L

−T |x|R

As we know, this type estimate was first proved by T. Kato [22] for solutions of the Korteweg–de Vries (KdV) equation (see e.g. [33, Chapter 4]), and later extended to very general dispersive type equation (cf. [3,12,13,23] and references therein). (iii) In Theorem 2.4, (2.3) also can be extended to J −σ . For this case, a crucial different point is that there exist nontrivial H p –H p -estimates of J −σ eitP (D) for any real elliptic polynomial P . Let σ = nm(1/r − 1/2). If 0 σ < nm/2 (i.e. 1 < r 2), then we have −σ itP (D) J e

Lp –Lp

σ C 1 + |t| m ,

t ∈ R, r p r .

(2.6)

σ C 1 + |t| m ,

t ∈ R, r p ∞.

(2.7)

If σ nm/2 (i.e. r 1), then we get −σ itP (D) J e

H p –H p

When r p 2, (2.6) and (2.7) can be directly concluded by a multiplier theorem (e.g. see [28, Theorem 1]), and for the remainder cases, the claims follow from the cases r p 2 by a dual argument. Here, we also comment that the estimate (2.6) and its some variants with other type regularized operator are known for some time, which are mainly related to the theory on regularized semigroup and fractionally integrated semigroup (e.g. see [16,19,20, 37]). 3. The proofs of theorems Now begin to prove above theorems. For this, let us first recall the definition and some important properties of Fourier multipliers in H p (Rn ) and Lp (Rn ). Let X p be one of the following spaces: H p (0 < p ∞), Lp (1 p ∞). Given a ∈ S (the tempered distribution space), we define aM(Xp ,Xq ) := sup

F −1 (a fˆ)Xq f ∈ S ∩ X p , f Xp = 0 , f Xp

and M(X p , X q ) is the space of all a ∈ S such that aM(Xp ,Xq ) < ∞. It was well known that Fourier multiplier space M(X p , X q ) has very abundant theory. In the following, we only collect some necessary properties for our proof. For more contents, refer to Hömander [21] and Miyachi [27].


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Lemma 3.1. (i) Let a ∈ M(X p , X q ), δ > 0, then aδ ∈ M(X p , X q ) and aδ M(Xp ,Xq ) = δ

n( q1 − p1 )

aM(Xp ,Xq )

where aδ is the dilation of a by aδ (ϕ) = a(δ −n ϕ(δ −1 ·)) for each ϕ ∈ S(Rn ). (ii) (Fractional integral.) Let 0 < p < ∞, 0 < s < n/p and 1/q = 1/p − s/n, then the operator Is : f → F −1 |ξ |−s fˆ(ξ ) is well defined on H p and bounded from H p to H q . (iii) (Mihlin multiplier theorem.) Let 0 < p ∞ and l = [n|1/p − 1/2|] + 1. If a ∈ C k (Rn \ {0}) and satisfies with μ D a(ξ ) Cμ |ξ |−|μ| ,

for |μ| l,

then a ∈ M(H p , H p ). As for the proof of Lemma 3.1, (i) can be easily deduced from the definition and equality f (δ · )Xp = δ −n/p f Xp for δ > 0. (ii) and (iii) are well-known results in harmonic analysis, we refer to [14, pp. 162–171] and [27, p. 282] for their proofs. Next, we give the decay estimates of a class of oscillatory integrals marked with a parameter z. For this, let ψ be a fixed smooth function on R such that 0 ψ(s) 1, ψ(s) = 0 if s 1 and ψ(s) = 1 if s 2. Proposition 3.2. If Σ is a convex hypersurface of finite type k and 2 k m, then (F −1 az± )(x) ∈ C ∞ (Rn ) and there exists a constant C independent of β, x such that −1 ± F a (x) C 1 + |β| N 1 + |x| −h(m,n,k,α) , z

x ∈ Rn ,

(3.1)

where z = α + iβ for (α, β) ∈ R2 , N is a nonnegative integer > n + α, az± (ξ ) = ψ |P |1/m |P |z/m e±iP and h(m, n, k, α) =

α m(2n + k − 2) − 2kn − . 2k(m − 1) m−1

Proof. We may assume that P (ξ ) > 0 for any ξ = 0. Let φ = P 1/m , then φ is a positively smooth homogeneous function of degree 1, and Σ = {ξ ∈ Rn | φ(ξ ) = 1}. Because of analogy, it suffices to treat with (F −1 az+ )(x). Now consider the integral + Kε,z (x) :=

Rn

e−εφ(y)+iφ

m (y)+ix,y

ψ φ(y) φ z (y) dy,

ε > 0.

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+ (x) is a smooth function of Rn in x variable. As ε → 0, one can show Clearly for each ε > 0, Kε,z + that Kε,z (x) converges uniformly to (F −1 az+ )(x) in each compact subsets of Rn . So it follows that F −1 az+ ∈ C(Rn ). Similarly, we can repeat the procedure and conclude (F −1 az+ ) ∈ C ∞ (Rn ) + (x) (∀μ ∈ Nn ). Next, in order to get (3.1) for F −1 (a + ), it suffices to show that by D μ Kε,z z

+ K (x) C 1 + |β| N 1 + |x| −h(m,n,k,α) , ε,z

x ∈ Rn ,

(3.2)

where N > n + α and C is independent of β, x, ε. To prove (3.2), we divide Rn into two regions to discuss: |x| m/2M and |x| < m/2M where M = maxη∈Σ |η|. We begin with the case |x| m/2M. By a polar coordinates transform (y = sξ for s > 0 and ξ ∈ Σ), rewrite + (x) = Kε,z

∞

isrη,ξ

e m dσ (ξ ) ds, e−εs+is s n−1+z ψ(s) |∇φ(ξ )| Σ

0

where r = |x|, x = rη, z = α + iβ, and dσ is the induced surface measure on Σ . For each η ∈ S n−1 , let ξ± be the two points of Σ whose the outward normal directions are ±η, then by Theorem B in [7], we have Σ

eiλη,ξ

dσ (ξ ) = eiλη,ξ+ H+ (λ) + eiλη,ξ− H− (λ) + H∞ (λ), |∇φ(ξ )|

λ > 0.

where H± ∈ C ∞ ((0, ∞)), and there exist constants Cj independent on the hypersurface Σ such that (j ) H (λ) Cj λ−j −(n−1)/k ±

for j ∈ N0 ,

and H∞ (λ) Cj λ−j

for j ∈ N.

Therefore + Kε,z (x)

∞ =

e−εs+is

m +isrη,ξ

+

s n−1+z ψ(s)H+ (sr) ds

0

∞ +

e−εs+is

m +isrη,ξ

−

s n−1+z ψ(s)H− (sr) ds

0

∞ +

e−εs+is s n−1+z ψ(s)H∞ (sr) ds m

0

:= I1ε

+ I2ε + I3ε .


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To get (3.2), it suffices to estimate Ilε (l = 1, 2, 3). For this, if carrying out a similar process as Jlε in [38, pp. 129–132], and keeping track of the variable β at the same time, then we can conclude that + K (x) C 1 + |β| N |x|−h(m,n,k,α) , ε,z

|x| m/2M.

Here omitting these details, refer readers to consult [38]. + (x), we get Now turning to case |x| < m/2M. By commuting the integral order in the Kε,z + (x) = Kε,z

∞ e Σ

−εs+is m +isx,ξ n−1+z

s

ψ(s) ds

1

dσ (ξ ) . |∇φ(ξ )|

Let ϕ(s) = −εs + is m + isx, ξ . Note that when |x| < m/2M and s 1, m−1 ϕ (s) ms + x, ξ ms m−1 − M|x| (m/2)s m−1 m/2. So N -times integrations by parts yield that + K (x) ε,z

∞ dσ N ϕ(s) n−1+z C 1 + |β| , ψ(s) ds e s |∇φ|

Σ

|x| < m/2M.

1

Therefore combining with two cases above, Eq. (3.2) follows.

2

Proof of Theorem 2.4. First we show (2.3). As above, we may assume that P (ξ ) > 0 for ξ = 0. Rewrite 1 −σ 1 σ ±iP (|t| m1 ξ ) · |t| m ξ e , ξ σ eitP (ξ ) = ξ σ |t| m ξ

t = 0,

1

where ξ := (1 + |ξ |2 ) 2 . Note that μ σ 1 −σ Cμ 1 + |t|− mσ |ξ |−μ , D ξ |t| m ξ

μ ∈ Nn0 ,

so it follows from Lemma 3.1(iii) that for any 0 < p < ∞, 1 −σ ∈ M H p, H p , ξ σ |t| m ξ

t = 0,

and σ 1 −σ · |t| m ·

M(H p ,H p )

σ C 1 + |t|− m ,

t = 0.

Thus in view of Lemma 3.1(i) it suffices to consider multipliers ξ σ e±iP (ξ ) .

(3.3)

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Decomposing as follows: 1 1 σ ξ σ e±iP (ξ ) = 1 − ψ P m ξ σ e±iP (ξ ) + ψ˜ P m ξ σ P − m · aσ± (ξ ), ˜ ˜ ˜ where ψ(s) is a smooth function on R such that 0 ψ(s) 1, ψ(s) = 0 if s if s 1, and

1 2

˜ and ψ(s) =1

1 σ aσ± (ξ ) = ψ P m P m e±iP (ξ ) , here ψ is defined in Proposition 3.2. Since 1 1 − ψ P m ξ σ e±iP (ξ ) are smooth function with compact support, it follows that 1 1 − ψ P m · σ e±iP ∈ M H p , H q ,

for any 0 < p q ∞.

(3.4)

Moreover, by using Lemma 3.1(iii) we also have 1 σ ψ˜ P m ξ σ P − m ∈ M H p , H p , for any 0 < p < ∞.

(3.5)

Thus ξ σ e±iP (ξ ) can be reduced to multiplier aσ± (ξ ). For 0 σ (m(2n + k − 2)/2k) − n, if we can prove that aσ± (ξ ) ∈ M H p , H q ,

(3.6)

for any pair ( p1 , q1 ) ∈ σ , then the desired (2.3) can immediately follows from (3.3)–(3.6) above. To obtain (3.6), by interpolation and duality it suffices to show the following two end-points: A( r1σ , r1 ), and E( p1σ , 0) (see Fig. 2). σ

Let σmax = (m(2n + k − 2)/2k) − n. Note that F −1 (aσ±max (·)) ∈ L∞ by Proposition 3.2, hence we obtain that aσ±max (ξ ) ∈ M H 1 , L∞ .

Thus from Lemma 3.1(ii) and (iii) it follows that σ −σ 1 aσ± (ξ ) = aσ±max (ξ ) · P m |ξ |−1 max · |ξ |σ −σmax ∈ M H pσ , L∞ , which proves the point E( p1σ , 0).

(3.7)

± To the point A( r1σ , r1 ), we introduce a family of multipliers aα+iβ (ξ ) where 0 α σmax σ and β ∈ R. When α = σmax , by Proposition 3.2 again we have

± a

σmax +iβ (·) M(H 1 ,L∞ )

N C 1 + |β| ,


2079

where N > n + σmax . On the other hand, the Plancherel’s theorem also yields that ± a (·) iβ

M(L2 ,L2 )

C.

Therefore by a complex interpolation argument (see e.g. [14, pp. 151–152], or [32, p. 185]) it follows that aσ± (ξ ) ∈ M Lrσ , Lrσ . Thus (3.6) follows from (3.7) and (3.8) and the proof of (2.3) is completed.

(3.8) 2

Next turn to show (2.4). Let t = ±1, then for each ( p1 , q1 ) ∈ σ , (2.3) yields that σ ±iP (D) p q C J e H –H from which it immediately follows that σ ±iP (D) p q I σ J −σ q q J σ e±iP (D) p q C I e H –H H –H H –H

(3.9)

where I σ J −σ is bounded on H q (Rn ) (0 < q ∞) by Lemma 3.1(iii). Thus by scaling the desired (2.4) follows from Lemma 3.1(i) and (3.9). Proof of Theorem 2.3. In fact, if taking σ = 0, then the conclusion immediately follows from Theorem 2.4. 2 4. Application to iP (D) + V (x, D) In this section, using the Lp –Lq estimates of J σ eitP (D) (t = 0) we will show that higher order Schrödinger operator iP (D) + V (x, D) generates a fractionally integrated group on Lp (Rn ) (see Theorem 4.3). To this, let us start with the definition (see e.g. [19]). Let A be a linear operator on a Banach space X and α 0. Then a strongly continuous family T : [0, ∞) → L(X) is called an α-times integrated semigroup on X with generator A if there exist constants C, ω 0 such that T (t) Ceωt for t 0, (ω, ∞) ⊂ ρ(A) (the resolvent set of A), and −1

(λ − A)

∞ x=λ

α

e−λt T (t)x dt

for λ > ω and x ∈ X.

(4.1)

0

If A and −A both are generators of α-times integrated semigroups on X, we say A is the generator of an α-times integrated group on X. In view of our applications, we need the following perturbation result of the fractionally integrated semigroup (see [19, Theorem 5.1] and [26, Theorems 3.1, 3.3]). Particularly, the second part of the following lemma is the special case of Theorem 3.3(a) in [26], where the same conclusion can hold on a class of Banach spaces of Fourier type s ∈ [1, 2].

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Lemma 4.1. Let (A, D(A)) be a generator of an α-times integrated semigroup on X and let (B, D(B)) be a linear operator on X such that D(A) ⊆ D(B) and there exist constants M, ω 0 such that B(λ − A)−1 M < 1 for Re λ > ω. Then (A + B, D(A)) generates a β-times integrated semigroup on X, where β > α + 1. Moreover, if X = Lp (Rn ) (1 α + max{1/p, 1/p }. Now turning to study the higher order Schödinger operator of the form:

iP (D) + V (x, D) = iP (D) +

N

hj (x)Qj (D),

(4.2)

j =1

where P (ξ ) is a real homogeneous elliptic polynomial of order m in Rn where n 2, hj (x) is a real (or complex) valued measurable function on Rn and Qj (ξ ) belongs to the pseudo-differential m symbol class S1,0j (mj 0) for each 1 j N (see e.g. [32, p. 271]). Of course, Qj (D) can be a partial differential operator of order mj if mj is a positive integer. When V (x, D) ≡ 0, it was well known that the elliptic operator iP (D) (even any general elliptic operator) generates an α-times integrated semigroup T (t) on Lp (Rn ) where α n| 12 − p1 | and 1 < p < ∞ (see e.g. [20,37]). Hence when V (x, D) = 0, in order to show that iP (D) + V (x, D) in Lp (Rn ) generates a fractionally integrated semigroup, it would use V (x, D) to perturb iP (D) by Lemma 4.1 above. For this, we need establish the Lp –Lq estimates of the operator J σ (λ − iP (D))−1 by the use of Theorem 2.4, where −1 −1 J σ λ − iP (D) u0 := F −1 ξ σ λ − iP (ξ ) uˆ 0 ,

u0 ∈ S Rn , Re λ = 0.

(4.3)

Before giving these main results, for 2 k m and 0 σ (m(2n + k − 2)/2k) − n, we define a set Γ (σ, k) as follows Γ (σ, k) := 2σABCD ∩ (1/p, 1/q); 1/p − 1/q < (m − σ )/n ,

(4.4)

where 2σABCD is a closed quadrangle as described in Fig. 2 of Section 2. Under the help of the Fig. 2, one easily verifies that Γ (σ, k) = ∅ if and only if ⎧ if m = 2, k = 2; ⎪ ⎨ {0}, 2kn [0, m − ), if m > 2, 2 k (2n − 2) ∧ m; σ∈ 2n+k−2 ⎪ ⎩ [0, m(2n+k−2) − n], if m > 2, 2n − 2 < k m; 2k

(4.5)

where a ∧ b denotes the smaller one of a, b. Note that when m > 2 and 2 k m, the following inequalities constantly hold: m−

2kn > 0, 2n + k − 2

m(2n + k − 2) − n > 0. 2k

Hence this means that as m > 2, there are always some σ > 0 such that Γ (σ, k) = ∅ for each 2 k m. In particular when m > 2 and k = 2, we can take any σ ∈ [0, m − 2).


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Proposition 4.2. Suppose Σ is a convex hypersurface of type k for 2 k m. If σ satisfies with (4.5), then for any Re λ = 0, we have σ J λ − iP (D) −1

Lp –Lq

n 1 1 σ ( − )+ σ −1 C 1 + | Re λ|− m | Re λ| m p q m ,

(4.6)

where J σ = (1 − )σ/2 , (1/p, 1/q) ∈ Γ (σ, k) and 1 < p, q < ∞. Proof. When Re λ > 0, by (4.3) one has

J

σ

−1 λ − iP (D) u0 =

∞

for u0 ∈ S Rn .

e−λt J σ eitP (D) u0 dt

0

Therefore for each (1/p, 1/q) ∈ Γ (σ, k) and 1 < p, q < ∞, from (2.3) it follows that ∞

σ J λ − iP (D) −1

Lp –Lq

C

σ n ( 1 − 1 )− σ e−(Re λ)t 1 + t m t m q p m dt

0 n 1 1 σ ( − )+ σ −1 C 1 + | Re λ|− m | Re λ| m p q m .

When Re λ < 0, we notice that −1 J λ − iP (D) u0 =

∞

σ

eλt J σ e−itP (D) u0 dt

for u0 ∈ S Rn ,

0

and thus the desired estimate also similarly holds.

2

Remark 4.1. In the same way, using (2.3) one also can obtain that σ I λ − iP (D) −1

Lp –Lq

n

C| Re λ| m

σ ( p1 − q1 )+ m −1

,

Re λ = 0.

where I σ = (−)σ/2 , (1/p, 1/q) ∈ Γ (σ, k) and 1 < p, q < ∞. Let σ satisfies with (4.5) and 1 < p rσ , we denote by Λ(p, σ, k) the following set: 1 1 1 1 1 , ∈ Γ (σ, k), 1 < q < ∞ , Λ(p, σ, k) := s ∈ R; = − , s p q p q

(4.7)

where Γ (σ, k) is defined in (4.4) and rσ is defined in Theorem 2.4. Clearly, Γ (σ, k) = ∅ implies that Λ(p, σ, k) = ∅ for some p ∈ (1, rσ ]. More precisely, Λ(p, σ, k) =

⎧ ⎨ (p, qσ p rσ ], n (rσ −p)rσ ,∞ ∩ ⎩ [s(σ, p), qσ p rσ ], m−σ (rσ −p)r σ

if 1 < p qσ ; if qσ < p rσ ;

(4.8)

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where rσ − p 2 1 = + − 1. s(σ, p) p(rσ − qσ ) rσ Here qσ and rσ are defined in Theorem 2.4., and qσ and rσ are the conjugate indexes of qσ and rσ , respectively. Notice that the following inequality qσ p rσ n > (rσ − p)rσ m−σ holds as p sufficiently close to rσ , therefore this claims that there always exists some p ∈ (1, rσ ] such that Λ(p, σ, k) = ∅ when 2 k m and σ satisfies with (4.5). Theorem 4.3. Suppose Σ is a convex hypersurface of type k for 2 k m. Let σ satisfies with (4.5) and V (x, D) is defined by (4.2). (a) If hj ∈ Lsj (Rn ) with sj ∈ Λ(p, σ, k) for some 1 n| 12 − p1 | + p1 . (b) If hj ∈ Lsj (Rn ) with sj ∈ Λ(p , σ, k) for some rσ p < ∞, and degree (Qj ) = mj σ (j = 1, 2, . . . , N ), then the dual operator L¯ ∗ (x, D) generates a β-times integrated group on Lp (Rn ), where β > n| 12 − p1 | + p1 . Proof. Since iP (D) + V (x, D) and −(iP (D) + V (x, D)) satisfy with the same assumptions, it suffices to show that iP (D) + V (x, D) generates a β-times integrated semigroup on Lp (Rn ). We consider first the case (a). Let q1j = p1 − s1j for some 1 < p rσ (j = 1, 2, . . . , N ). Then sj ∈ Λ(p, σ, k) implies (p, qj ) ∈ Γ (σ, k) by Eq. (4.7). So we obtain by Proposition 4.2 and Hölder’s inequality that V (x, D) λ − iP (D) −1

Lp –Lp

N j =1

−1 hj Lqj –Lp Qj (D) λ − iP (D) Lp –Lqj

N −1 hj Lsj Qj (D)J −σ Lp –Lp J σ λ − iP (D) Lp –Lqj j =1

C

N n σ + σ −1 1 + | Re λ|− m | Re λ| mrj m j =1

where the boundedness of the operator Qj (D)J −σ in Lp (Rn ) is from Lemma 3.1(iii). In view σ − 1 < 0, there exists ω 1 such that of msn j + m V (x, D) λ − iP (D) −1

Lp –Lp

1/2,

Re λ > ω.


2083

Consequently, by Lemma 4.1 and the fact that iP (D) generates an α-times integrated semigroup T (t) on Lp (Rn ) where α n| 12 − p1 |, it follows that iP (D) + V (x, D) generates a β-times integrated semigroup on Lp (Rn ) where β > n| 12 − p1 | + p1 . ¯ Next, we consider the case (b). From the proof of (a) one sees that L(x, D) = −iP (D) + p n ∗ ¯ ¯ V (x, D) is densely defined on L (R ), and thus its dual operator L (x, D) exists and is also ¯ D) densely defined on Lp (Rn ). Since sj ∈ Λp and n| 12 − p1 | = n| 12 − p1 |, it follows that L(x,

generates a β-times integrated semigroup on Lp (Rn ). Thus by a dual argument we can obtain the desired conclusion for L¯ ∗ (x, D) on Lp (Rn ). Hence we have completed the proof of Theorem 4.3. 2

If k = 2 and m μ4 (even integer), then in Theorem 4.3 we can take σ = m − 3 and V (x, D) = |μ|m−3 hμ (x)D where hμ (x) is a suitable integrable function. Moreover, if σ = 0, then r0 = 2 and from (4.8) we have Λ(p, 0, k) =

⎧ ⎨ (p, q0 p ], n 2−p ,∞ ∩ ⎩ [ p(2−q0 ) , q0 p ], m 2−p

2−p

if 1 < p q0 ; if q0 < p 2.

where q0 is defined in Theorem 2.3 and q0 is the conjugate index of q0 . Thus from Theorem 4.3 we also obtain the following simple consequences: Corollary 4.4. Let Σ is a convex hypersurface of type k for 2 k m. (a) If V (x) ∈ Ls (Rn ) with s ∈ Λ(p, 0, k) for some 1 n| 12 − p1 | + p1 . (b) If V (x) ∈ Ls (Rn ) with s ∈ Λ(p , 0, k) for some 2 p < ∞, then an extension of i(P (D) + V (x)), i.e. L¯ ∗ (x, D) generates a β-times integrated group on Lp (Rn ), where β > n| 12 − p1 | + p1 . Remark 4.2. n (i) Theorem 4.4 in [38] has proved the conclusion (a) for s ∈ ( m , ∞] ∩ Ip and β > n| 12 − p1 | + 1, where

Ip =

⎧ τ p ⎪ ⎨ [p, 2−p ),

if 1 < p < τ ;

( p(2−τ ) , τ p ), ⎪ ⎩ 2−p 2−p {∞},

if τ p 2; if p = 2.

Note that q0 τ (i.e. q0 τ ) (see Fig. 1), thus we can immediately improve the set n , ∞] ∩ Ip ), which is just equal Λ(p, 0, k) in Corollary 4.4 to the larger set Λ(p, 0, k) ∪ (( m to

2084


⎧ q0 p ⎪ [p, ], ⎪ 2−p ⎪ ⎨

n q0 p , ∞ ∩ (p, 2−p ], ⎪ m ⎪ ⎪ ⎩ p(2−q0 ) q0 p [ 2−p , 2−p ],

if 1 < p < τ ; if τ p q0 ;

(4.9)

if q0 < p 2.

Moreover, here the range of β is larger than one in [38, Theorem 4.4] because 1 1 1 1 1 1 , < n − + 1, n − + max 2 p p p 2 p

1 < p < ∞.

p (ii) If P (D) = −, then m = k = 2, σ = 0, τ = q0 = ∞, and s ∈ (( n2 , ∞] ∩ { 2−p }) for 1 n| 12 − p1 | + max{ p1 , p1 }. In particular, we have 1 < p < ∞ when n = 2.

Finally, in order to give Lp –Lp estimates of the solution for Eq. (1.6), we need Straub’s fractional powers (cf. [29]). Let α0 0. If A is the generator of an α-times integrated group for every α > α0 , then the fractional powers (ω ± A)α are well defined for large ω ∈ R and their domains all contain the dense subspace D(A[α]+1 ). The following result is a consequence of Theorem 1.1 in [29] and Theorem 4.3(a) above. Theorem 4.5. Suppose P , V , p and β satisfy the assumptions of Theorem 4.3(a). Then there exist constants C, ω > 0 such that for every data u0 ∈ D((ω + iP (D) + V (x, D))β ) ∩ D((ω − iP (D) − V (x, D))β ), Eq. (1.6) has a unique solution u ∈ C(R, Lp (Rn )) and u(t, ·)

Lp

β Ceω|t| ω ± iP (D) + V (x, D) u0 Lp ,

t ∈ R,

where we choose + (resp. −) if t 0 (resp. < 0). Acknowledgments The first author would like to thank Professor Akihiko Miyachi for giving him the papers [27] and [28]. The second one wishes to thank his tutor Professor Quan Zheng for giving him the kind guidance and interests in this topic when he was a PhD student several years ago. Appendix A In this appendix, we will show Proposition 2.1 and 2.2 (also see [35, p. 20]). Let us start to recall some basic concepts related to Gaussian curvature of Σ (cf. e.g. [34, Chapters 6, 9, 12, 13]). For a given ξ ∈ Σ, let Tξ denote the tangent space of Σ at ξ , then we can define the Weingarten map Lξ : Tξ → Tξ by Lξ (v) = (−∇v N )(ξ ), where N (η) = −∇P (η)/|∇P (η)| is the Gauss map on Σ. It was known that Lξ is a self-adjoint transformation on Tξ and n −1 ∂ 2 P (ξ ) Lξ (v), w = v, Lξ (w) = ∇P (ξ ) v i wj , ∂ξi ∂ξj

i,j =1

v, w ∈ Tξ .

(A.1)


2085

Also, the quadratic function Lξ (v), v on Tξ is called the second fundamental form of Σ at ξ , the eigenvalues k1 (ξ ), . . . , kn−1 (ξ ) of Lξ are called principal curvatures of Σ at ξ , and their products is called Gaussian curvature of Σ at ξ . Now denote by Ψξ the symmetric bilinear form associated with the Hessian matrix Hess P := [∂i ∂j P (ξ )]n×n of P with respect to the standard basis of Rn , i.e. n ∂ 2 P (ξ ) Ψξ (v, w) = v, (Hess P )w = v i wj , ∂ξi ∂ξj

v, w ∈ Rn .

(A.2)

i,j =1

If we choose another basis {α1 , . . . , αn } on Rn where αi is the unit eigenvector of Lξ corresponding to the ki (ξ ), i.e. Lξ (αi ) = ki (ξ )αi for i = 1, . . . , n − 1, and αn = ξ , then exploiting (A.1) and (A.2), we can easily show that the matrix for Ψξ with respect to this basis is Bξ := Ψξ (αi , αj ) 1i,j n =

0 , m(m − 1)

|∇P (ξ )|Aξ 0

(A.3)

where Aξ = diag(k1 (ξ ), . . . , kn−1 (ξ )). Clearly, Hess P is congruent with Bξ . Next, we come to prove Proposition 2.1. Proof of Proposition 2.1. In the sequel, without the loss of generality we may assume that P (ξ ) > 0 for ξ = 0. which means that Hess P (ξ ) 0 everywhere in the statement (c). Here A 0 represents the positive semi-definite matrix A. (a) ⇒ (b). By Theorem 1 of [34, p. 95], we immediately have ki (ξ ) = Lξ (αi ), αi 0 for every ξ ∈ Σ and i ∈ {1, . . . , n − 1}. (b) ⇒ (c). For each ξ ∈ Σ , by the assumption (b) and (A.3) one see that Bξ 0. Since Hess P is congruent to Bξ , thus Hess P 0, which implies (c) by the homogeneity of P . (c) ⇒ (a). Clearly, from the above proof of (b) ⇒ (c), one get that Gaussian curvature k(ξ ) = n−1 i=1 ki (ξ ) 0 for every ξ ∈ Σ. To further get the convexity of Σ , we will proceed by induction on n ( 2). In the following, also denote by Σn (P ) the hypersurface associated with the polynomial P : Rn → R. When n = 2, it is easy to check that Σ2 (P ) is a simple closed curve. Thus Σ2 (P ) is convex is exactly equivalent that Gaussian curvature of Σ2 (P ) is nonnegative everywhere (cf. [11]). So the implication is true for n = 2. Now suppose it is true for n-dimensional case. To prove the convexity of Σn+1 (P ), it suffices to show the intersection between Σn+1 (P ) and any n-dimensional hyperplane through origin is convex on the hyperplane. Firstly, we consider the case {ξ ∈ Rn+1 ; ξn+1 = 0}. Set Q(ξ ) = P (ξ , 0) for ξ ∈ Rn . Regarding ξ as (ξ , 0), one gets Σn (Q) = Σn+1 (P ) ∩ {ξn+1 = 0}. Note that Q : Rn → R is also a homogeneous elliptic polynomial of order m by the assumptions on P , therefore it is suffices to prove that the Hessian matrix [∂i ∂j Q(ξ )]n×n of Q is positive semi-definite everywhere. To the end, write P (ξ ) = Q(ξ ) + ξn+1 P0 (ξ ) for ξ ∈ Rn+1 where P0 is a polynomial. Thus

∂ 2 P (ξ ) ∂ξi ∂ξj

1i,j n

=

∂ 2 Q(ξ ) ∂ξi ∂ξj

1i,j n

+ ξn+1

∂ 2 P0 (ξ ) ∂ξi ∂ξj

. 1i,j n

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When ξn+1 = 0, we see that the matrix [∂i ∂j Q(ξ )]n×n is one of principal sub-matrixes of Hess P (ξ , 0), and naturally positive semi-definite everywhere by the assumption (c). Thus it follows from the induction assumption that Σn (Q) is convex, as desired. Next to deal with the general n-dimensional subspace Ω of Rn+1 . Choose an orthogonal translation A which rotates Ω to the hyperplane ξn+1 = 0, then A Σn+1 (P ) ∩ Ω = Σn+1 (P1 ) ∩ {ξn+1 = 0}, where P1 (ξ ) = P (A ξ ). Notice that Hess P1 (ξ ) = A(Hess P (A ξ ))A , therefore H P1 (ξ ) 0 for any ξ ∈ Rn+1 . Thus the desired assertion again follows from the special case {ξn+1 = 0} and the whole proof is completed. 2 Proof of Proposition 2.2. Since the equivalence of (b) and (c) was known in [36, Prop. 2.1], it suffices to prove that (a) ⇔ (b). Consider (a) ⇒ (b). Since k = 2, it follows from (2.1) and (A.1) that ki (ξ ) = Lξ (αi ), αi = ∇P (ξ )−1 αi , ∇ 2 P (ξ ) C > 0,

ξ ∈ Σ,

for each i ∈ {1, . . . , n − 1}. Obviously, this implies that the statement (b) is true. Conversely, suppose Σ has nonzero curvature everywhere, i.e. (b) is true, then it follows that Σ must be strictly convex (i.e. see [34, p. 104]). So it just remains to prove the type order of Σ is 2. First, it is clear that v, ∇ P (ξ ) = v, ∇P (ξ ) > 0,

v ∈ Tξc ∩ Sn−1 , ξ ∈ Σ.

On the other hand, by Theorem 6 in [34, p. 92], we can obtain v, ∇ 2 P (ξ ) = ∇P (ξ ) Lξ (v), v > 0,

v ∈ Tξ ∩ Sn−1 , ξ ∈ Σ.

Therefore, combining the two equalities above, the desired (2.1) for k = 2 immediately follows from the compactness of Σ and Sn−1 . Thus the implication (b) ⇒ (a) is proved, and the whole proof is also concluded. 2 References [1] W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987) 327–352. [2] M. Balabane, On the regularizing effect on Schrödinger type group, Ann. Inst. H. Poincaré 6 (1989) 1–14. [3] M. Ben-Artzi, A. Devinatz, Local smoothing and convergence properties of Schrödinger type equations, J. Funct. Anal. 101 (1991) 231–254. [4] M. Balabane, H.A. Emami-Rad, Smooth distribution group and Schrödinger equation in Lp , J. Math. Anal. Appl. 70 (1979) 61–71. [5] M. Balabane, H.A. Emami-Rad, Lp estimates for Schrödinger evolution equations, Trans. Amer. Math. Soc. 292 (1985) 357–373. [6] M. Ben-Artzi, H. Koch, J.-C. Saut, Dispersion estimates for fourth order Schriödinger equations, C. R. Math. Acad. Sci. Paris 330 (2000) 87–92. [7] J. Bruna, A. Nagel, S. Wainger, Convex hypersurfaces and Fourier transforms, Ann. of Math. 127 (1988) 333–365. [8] J. Bourgain, Global Solutions for Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ., vol. 46, Amer. Math. Soc., 1999.


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[9] M. Ben-Artzi, J.-C. Saut, Uniform estimates for a class of oscillatory integrals and applications, Differential Integral Equations 12 (1999) 137–145. [10] T. Cazenave, Semilinear Schriödinger Equations, Courant Lect. Notes Math., vol. 10, New York Univ., Courant Inst. Math. Sci., Amer. Math. Soc., 2003. [11] S.S. Chern, Curves and surfaces in Euclidean space, in: S.S. Chern (Ed.), Global Geometry and Analysis, in: MMA Stud. Math., vol. 4, 1967, pp. 16–56. [12] H. Chihara, Smoothing effects of dispersive pseudo-differential equations, Comm. Partial Differential Equations 27 (2002) 1953-2002. [13] P. Constantin, J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988) 413–439. [14] A.P. Calderón, A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Adv. Math. 24 (1977) 101–171. [15] S. Cui, Point-wise estimates for oscillatory integrals and related Lp –Lq estimates, J. Fourier Anal. Appl. 12 (2006) 605–627. [16] E.B. Davies, M.M.H. Pang, The Cauchy problem and a generalization of the Hille–Yosida theorem, Proc. London Math. Soc. 55 (1987) 181–208. [17] R. DeLaubenfels, Existence Families, Functional Calculi and Evolution Equation Equations, Lecture Notes in Math., vol. 1570, Springer-Verlag, Berlin, 1994. [18] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, New Jersey, 2003. [19] M. Hieber, Integrated semigroups and differential operators on Lp , Dissertation, Tübingen, 1989. [20] M. Hieber, Integrated semigroups and differential operators on Lp , Math. Ann. 291 (1991) 1–16. [21] L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960) 93–140. [22] T. Kato, On the Cauchy problem for the (generalized) Korteweg–de Vries equation, Adv. Math. Suppl. Stud., Stud. Appl. Math. 8 (1983) 93–128. [23] C.E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991) 33–69. [24] M. Keel, T. Tao, Endpoint Strichart estimates, Amer. J. Math. 120 (1998) 360–413. [25] J. Kim, X. Yao, Q. Zheng, Global estimates of fundamental solutions for higher-order Schrödinger equations with application, preprint. [26] C. Kaiser, L. Weis, Perturbation theorems for a-times integrated semigroups, Arch. Math. 81 (2003) 215–228. [27] A. Miyachi, On some singular Fourier multipliers, J. Fac. Sci. Univ. Tokyo 28 (1981) 267–315. [28] A. Miyachi, On some Fourier multipliers for H p (R n ), J. Fac. Sci. Univ. Tokyo 27 (1980) 157–179. [29] J. van Neerven, B. Straub, On the existence and growth of mild solutions of the abstract Cauchy problem for operators with polynomially bounded resolvent, Houston J. Math. 24 (1998) 137–171. [30] M.M.H. Pang, Resolvent estimates for Schrödinger operators in Lp (R n ) and the theory of exponentially bounded C-bounded semigroups, Semigroup Forum 41 (1990) 97–114. [31] W. Schlag, Dispersive estimates for Schrödinger operators: A survey, in: J. Bourgain, C.E. Kenig, S. Klainerman (Eds.), Ann. of Math. Stud., vol. 163, 2007, pp. 255–285. [32] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. [33] T. Tao, Nonlinear dispersive equations, local and global analysis, CBMS Reg. Conf. Ser. Math. 106 (2006). [34] J.A. Thorpe, Elementary Topics in Differential Geometry, Undergrad. Texts Math., Springer-Verlag, New York, 1979. [35] X. Yao, Lp estimates for Higher order Schrödinger equations, PhD Dissertation, Lanzhou University, PR China, 2004 (in Chinese). [36] X. Yao, Q. Zheng, Oscillatory integral and Lp estimates for Schrödinger equations, J. Differential Equations 244 (2008) 741–752. [37] Q. Zheng, Y. Li, Abstract parabolic systems and regularized semigroups, Pacific J. Math. 182 (1998) 183–199. [38] Q. Zheng, X. Yao, D. Fan, Convex hypersurfaces and Lp estimates for Schrödinger equations, J. Funct. Anal. 208 (2004) 122–139.


Fourier transform, null variety, and Laplacian’s eigenvalues ✩ Rafael Benguria a , Michael Levitin b,∗ , Leonid Parnovski c a Facultad de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile b Cardiff School of Mathematics, Cardiff University, and WIMCS, Senghennydd Road, Cardiff CF24 4AG,

United Kingdom c Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom

Received 29 September 2008; accepted 22 June 2009 Available online 21 July 2009 Communicated by J. Bourgain

Abstract We consider a quantity κ(Ω)—the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximised, among all convex balanced domains Ω ⊂ Rd of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians. © 2009 Elsevier Inc. All rights reserved. Keywords: Laplacian; Dirichlet eigenvalues; Neumann eigenvalues; Eigenvalue estimates; Fourier transform; Characteristic function; Pompeiu problem; Schiffer’s conjecture; Convex sets

✩

The research has been supported by the Royal Society International collaborative grant with Chile. The work of R.B. has also been supported by CONICYT/PBCT (Chile) Proyecto Anillo de Investigación en Ciencia y Tecnología ACT30/2006. The work of L.P. has been also supported by a Leverhulme Trust grant. * Corresponding author. E-mail addresses: [email protected] (R. Benguria), [email protected] (M. Levitin), [email protected] (L. Parnovski). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.022

R. Benguria et al. / Journal of Functional Analysis 257 (2009) 2088–2123

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1. Introduction Let Ω be a bounded open domain in Rd with boundary ∂Ω, let x = (x1 , . . . , xd ) be a vector of Cartesian coordinates in Rd , and let 1, if x ∈ Ω, χΩ (x) = 0, if x ∈ /Ω denote the characteristic function of Ω. The complex Fourier transform of χΩ (x), χ Ω (ξ ) = F [χΩ ](ξ ) :=

eiξ ·x dx

Ω

or, more importantly, its complex null variety, or null set, NC (Ω) := ξ ∈ Cd : χ Ω (ξ ) = 0 has been studied extensively. Particular attention has been attracted by the role it plays in numerous attempts to prove the famous Pompeiu problem and Schiffer’s conjecture. We can refer for example to [2,4–7,11,13,14]; this list is by no means complete. Although our paper is not directly related to these still open questions, we recall them as part of the motivation for further study of the null variety. Let M(d) be a group of rigid motions of Rd , and Ω be a bounded simply connected domain with piecewise smooth boundary. The Pompeiuproblem is to prove that the existence of a nonzero continuous function f : Rd → R such that m(Ω) f (x) dx = 0 for all m ∈ M(d) implies that Ω is a ball. Schiffer’s conjecture is that the existence of an eigenfunction v (corresponding to a non-zero eigenvalue μ) of a Neumann Laplacian on a (simply connected) domain Ω such that v ≡ const along the boundary ∂Ω (or, in other words, the existence of a non-constant solution v to the over-determined problem −v = μv, ∂v/∂n|∂Ω = 0, v|∂Ω = 1) implies that Ω is a ball. It is known that the positive answer to the Pompeiu problem is equivalent to Schiffer’s conjecture. Moreover, a domain Ω would be a counterexample to both if there exists r > 0 such that NC (Ω) contains the complex sphere {ξ ∈ Cd : dj =1 ξj2 = r 2 }. One of the common tools in attacking the conjectures has been an asymptotic analysis of the null variety far from the origin in an attempt to prove that such counterexample cannot exist. In many cases, the study of the null variety in the papers cited above has been restricted to the case of a convex domain Ω. Additionally, it is convenient to assume that Ω is balanced (i.e., centrally symmetric with respect to the origin), and to deal instead with the real null variety d (ξ ) = 0 = ξ ∈ R : cos(ξ · x) dx = 0 . N (Ω) := NC (Ω) ∩ Rd = ξ ∈ Rd : χ Ω Ω

We assume that Ω is convex and balanced in most parts of this paper. The purpose of this paper is to study the behaviour of the null variety near the origin, and its relation with the classical spectral theory. Namely, we define the numbers

κC (Ω) := dist NC (Ω), 0 = min |ξ |: ξ ∈ NC (Ω)

2090


and

κ(Ω) := dist N (Ω), 0 = min |ξ |: ξ ∈ N (Ω) (if there are no real zeros, we set κ(Ω) = ∞). Throughout most of the paper, we will be dealing with the real zeros of Fourier transform and the quantity κ(Ω), so, unless specified otherwise, we always assume that the argument of the Fourier transform F [χΩ ] is real. On the basis of some partial cases presented below in Section 3, we conjecture that, firstly, κ(Ω) is maximised, among all convex balanced domains of the same volume as Ω, by a ball (see Conjecture 2.2), and, secondly, that for all convex balanced domains κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω (see Conjecture 2.3). Note that it is very easy to see that κ(Ω) is always (i.e. without the convexity and central symmetry conditions) bounded below by the square root of the second Neumann eigenvalue of Ω, see Lemma 3.3. Unfortunately we are unable to prove Conjectures 2.2 and 2.3 as stated. Even in the planar case d = 2, when the geometry of convex domains is easier to deal with, we are only able to establish some weaker versions of these conjectures, see Theorems 2.4 and 2.5. However, even these weaker results shed some extra light on the links between κ(Ω) and Dirichlet and Neumann eigenvalues, and in particular show some surprising links with Friedlander’s inequalities between the eigenvalues of these two problems, see Remark 3.9 and Remark 4.9. Additionally, we can also establish the validity of Conjectures 2.2 and 2.3 for small star-shaped perturbations of a disk, see Theorem 2.7. We also indicate that our results and conjectures cannot be extended to wider classes of domains, in particular when the convexity condition is dropped, see Theorems 2.8, 2.9, and Corollary 2.10. The rest of this paper is organised as follows. Section 2 contains the statements of our Conjectures and main Theorems. Some particular cases making the conjectures plausible are collated in Section 3. Some preliminary estimates (which in particular imply the validity of Conjecture 2.2 for relatively “long and thin” planar convex balanced domains) are presented and proved in Section 4. Extra notation and facts from convex geometry are in Section 5. Section 6 contains the proof of Theorem 2.4; some auxiliary technical lemmas used in the proofs are collected in a separate Section 7. The perturbation-type results are proved in Section 8, and the counterexamples for non-convex domains are proved in Section 9. We finish this section by introducing some additional notation used throughout the paper. We write vold (·) for a d-dimensional Lebesgue measure of a set. Given a unit vector e ∈ S d−1 , we write xe = x · e and x e = x − xe e. We write a real vector ξ ∈ Rd in spherical coordinates as ξ = (ρ, ω), with ρ = |ξ | and ω = ξ /ρ ∈ S d−1 . Bd (R) = {x ∈ Rd : |x| < R} denotes a ball of radius R centred at 0, and a shorthand for a unit ball will be Bd = Bd (1). Ω ∗ stands for a ball in Rd centred at 0 and of the same volume as Ω. Additionally, for a direction e ∈ S d−1 , we define κj (e) = κ j (e; Ω) as the j -th positive real ∞ ρ-zero of χ Ω (ρe) (counting multiplicities); note that N (Ω) = j =1 Nj (Ω), where Nj (Ω) :=

κj (e; Ω)e.

e∈S d−1

Finally, Ja (r) are the usual Bessel functions of order a, and ja,k are their positive zeros numbered in increasing order. The eigenvalues of the Dirichlet Laplacian on Ω are denoted by λk (Ω), k = 1, . . . , and of the Neumann Laplacian by μj (Ω), j = 1, . . . (μ1 = 0).


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2. Conjectures and statements Definition 2.1. Ω is balanced if it is invariant with respect to the mapping x → −x. Conjecture 2.2. If Ω is convex and balanced, then κ(Ω) κ(Ω ∗ ),

(2.1)

with the equality iff Ω is a ball. Conjecture 2.3. If Ω is convex and balanced, then κ(Ω)

(2.2)

λ2 (Ω),

with the equality iff Ω is a ball. In the next section we consider several explicit examples for which we demonstrate the validity of these conjectures. Although we believe these Conjectures to be true, we are unable to prove them without some additional assumptions. We can however establish somewhat weaker forms in the twodimensional case as stated in the next two theorems. Also, we can prove (2.1) subject to some additional conditions on Ω, see Corollaries 4.3 and 4.4, and Remark 4.5. Theorem 2.4. If d = 2, and Ω is convex and balanced, then κ(Ω) Cκ(Ω ∗ ),

(2.3)

with := C=C

2j0,1 ≈ 1.2552. j1,1

(2.4)

Theorem 2.5. If d = 2, and Ω is convex and balanced, then κ(Ω) 2 λ1 (Ω).

(2.5)

Remark 2.6. Note that Theorem 2.5 immediately follows from Theorem 2.4 by the Faber–Krahn inequality, λ1 (Ω) λ1 (Ω ∗ ) =

2 πj0,1

vol2 (Ω)

,

and rescaling properties of Lemma 3.2. Note also that (2.5) is clearly weaker than (2.2) in the two-dimensional case, since, by the Payne–Pólya–Weinberger inequality [18], in two dimensions λ2 (Ω) < 3λ1 (Ω),

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or by the even stronger Ashbaugh–Benguria inequality [3], λ2 (Ω)

j1,1 j0,1

2 λ1 (Ω) ≈ 2.539λ1 (Ω).

Finally, in the one-dimensional case, a convex balanced domain is an interval (−a, a) = B1 (a) for some a > 0, and

π

κ B1 (a) = λ2 B1 (a) = , a so that (2.1) and (2.2) hold with equality. We can also establish the validity of (2.1) and (2.2) for balanced star-shaped (but not necessarily convex) domains which are close to a disk. Namely, let F : S1 → R be a C 2 function on the unit circle; we additionally assume that F is periodic with period π : F (θ + π) = F (θ ). For 0, define a domain in polar coordinates (r, θ ) as Ω F := (r, θ ): 0 r 1 + F (θ ) .

(2.6)

(2.7)

Condition 2.6 implies that Ω F is balanced. Assume additionally that F is area preserving, that is 2π F (θ ) dθ = 0,

(2.8)

0

and so

vol2 (Ω F ) = π + O 2 . As we shall see from the re-scaling properties summarised in Lemma 3.2, condition (2.8) can be assumed without any loss of generality. The unperturbed domain (when = 0), Ω0F , is just a unit planar disk B2 . We have Theorem 2.7. Let us fix a non-zero function F as above satisfying (2.6) and (2.8). Then the one-sided derivatives satisfy dκ(Ω F ) < 0, (2.9) d =0+

and √ d λ2 (Ω F ) dκ(Ω F ) < . d d =0+ =0+

(2.10)


2093

Consequently, for sufficiently small > 0 (depending on F ), Conjectures 2.2 and 2.3 with Ω = Ω F hold. On the other hand, there exist arbitrarily small star-shaped non-convex perturbations of the disk for which at least (2.1) does not hold. Namely, we have Theorem 2.8. For each positive δ , there exists a balanced star-shaped domain Ω with vol2 (Ω) = π and such that B(0, 1 − δ ) ⊂ Ω ⊂ B(0, 1 + δ ), for which κ(Ω) > j1,1 . Continuing formulating negative results, we have the following Theorem 2.9. There is no C such that (2.3) holds uniformly for all (not necessarily connected) balanced one-dimensional domains Ω. From this, we immediately have Corollary 2.10. There is no C such that (2.3) holds uniformly for all balanced connected twodimensional domains Ω. Theorem 2.8 and Corollary 2.10 show that convexity plays a crucial role in Theorem 2.4 and Conjecture 2.2. 3. Motivation and elementary domains We start with two trivial results, which are immediate by the change of variables, and which in particular show that our conjectures are scale invariant. Let Rα denotes a mapping (x1 , x2 , . . . , xd ) → (αx1 , x2 , . . . , xd ), α > 0. Lemma 3.1. For any Ω ⊂ Rd , N (Rα Ω) = R1/α N (Ω). Lemma 3.2. Let Ω be the image of Ω ⊂ Rd under a homothety with coefficient α > 0. Then κ(Ω ) =

1 κ(Ω), α

λj (Ω ) =

1 λj (Ω). α2

The following result illustrates that there exists a relation between the null variety and eigenvalues of the Neumann Laplacian, which makes Conjecture 2.3 even more intriguing. Lemma 3.3. For any Ω ⊂ Rd , κ(Ω) κC (Ω)

μ2 (Ω).

(3.1)

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Proof. Let ξ 0 ∈ NC (Ω), and so Ω eiξ 0 ·x dx = 0. This means that eiξ 0 ·x , 1L2 (Ω) = 0, so that φ := eiξ 0 ·x is a test function for μ2 (Ω) (obviously, φ ∈ H 1 (Ω)). But, by direct computation, ∇φ2L2 (Ω) φ2L2 (Ω)

= |ξ 0 |2 .

Thus, |ξ 0 |2 μ2 (Ω) for any ξ 0 ∈ NC (Ω), whence the result.

2

In fact, as was shown to us by N. Filonov [9], one can improve this result to obtain Lemma 3.4. For any Ω ⊂ Rd , κ(Ω) 2 μ2 (Ω).

(3.2)

Proof. By the variational principle, μ2 (Ω) sup

φ∈L2

∇φ2L2 (Ω) φ2L2 (Ω)

for any linear subspace L2 ⊂ H 1 (Ω) such that dim L2 = 2. Choose ξ 0 ∈ N (Ω), and set L2 = span(eiξ 0 ·x/2 , e−iξ 0 ·x/2 ). The elements of L are linearly independent, and the result immediately follows by direct computation. 2 Example 3.5 (A ball in Rd ). For a unit ball Bd and real ξ , we have: d/2 χ Bd (ξ ) = (2π)

Jd/2 (|ξ |) , |ξ |d/2

(3.3)

and so κ(Bd ) = jd/2,1 .

(3.4)

On the other hand,

2 2 λ2 (Bd ) = λ3 (Bd ) = · · · = λ1+d (Bd ) = jd/2,1 = κ(Bd ) . For illustration, we give a proof of (3.3) in dimension d = 2. We choose the direction of ξ as the x1 -axis, and write, in polar coordinates, x = (r cos θ, r sin θ ). Thus, 12π χ Bd (ξ ) =

ei|ξ |r cos θ r dr dθ.

0 0

Then we use formula [1, formula 9.1.18], i.e., 1 J0 (z) = 2π

2π cos(z cos θ ) dθ, 0


2095

to express the previous integral as 1 χ Bd (ξ ) = 2π 0

2π J0 |ξ |r r dr = 2 |ξ |

|ξ | J0 (s)s ds. 0

Finally, we use the raising and lowering relations for Bessel functions embodied in [1, formula 9.1.27] (third formula with ν = 1), i.e., 1 J1 (r) = J0 (r) − J1 (r), r which can be expressed in the more convenient form,

rJ0 (r) = rJ1 (r) . Thus, we get 2π χ Bd (ξ ) = |ξ |2

|ξ |

2π J1 |ξ | , sJ1 (s) ds = |ξ | 0

which is the desired equality (3.3) in two dimensions. The corresponding formula in any dimension is equally simple to establish. Example 3.6 (A cuboid in Rd ). Consider, for d 2, a cuboid P with edge lengths a1 a2 · · · ad > 0. We have: λ2 (P ) = π

2

4a1−2

d −2 + (aj ) .

(3.5)

j =2

On the other hand, if ξ ∈ N (P ), we have vector (2π/a1 , 0, . . . , 0), giving

d

κ(P ) =

j =1 sin(ξj aj /2) = 0,

and so |ξ | is minimised by the

2π < λ2 (P ). a1

Proving (2.1) for P requires a bit more effort. We have P ∗ = Bd (R)

with R =

(a1 · a2 · · · ad · Γ (1 + d/2))1/d , √ π

and, after some transformations, the required inequality is reduced to 1/d √ d jd/2,1 2 π Γ 1 + . 2

(3.6)

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This, in turn, is proved using a combination of Stirling’s formula, Lorch’s lower bound jν,1 √ (ν + 1)(ν + 5) [16], and numerical checks for low d. Example 3.7 √ (A right-angled triangle in R2 ). Let T = T1,a be a right-angled triangle with sides 1, a > 1, and 1 + a 2 . One can check, after some computations, that κ(T1,a ) = 2π 1 + a −2 . We remark that both inequalities (2.1) and (2.2) with Ω = T hold for values of a sufficiently close to one, but fail for large a or small a. This can be checked either by direct computation (in case of (2.1)) or by domain monotonicity (in case of (2.2)), by comparing λ2 (T ) with either λ2 (Ta,a ) = 10π 2 /a 2 (for small a) or with the second eigenvalue of the rectangle with sides 3/4 and a/4 (for large a). Note that T is not balanced and we do not conjecture that (2.1) and (2.2) √ hold in general for such domains. It may be plausible that κC (Ω) κ(Ω ∗ ) and κC (Ω) λ2 (Ω) for general convex domains, however the study of complex null varieties is outside the scope of this paper. Example 3.8 (Numerics). We have also verified Conjectures 2.2 and 2.3 numerically. We have conducted (jointly with Brian Krushave, an undergraduate student at Heriot–Watt University, whose research was funded by a Nuffield Foundation undergraduate bursary) a large number of calculations for different multiparametric families of balanced convex domains in the twodimensional case. A typical example would be a family of rectangles with different circular or elliptic segments added along their sides, in order to produce some stadium-like domains. The zeros of Fourier transform were found by analytic or numerical integration and minimisation, and the eigenvalues of the Dirichlet Laplacian by the finite element method. Remark 3.9 (Estimates of the spectrum). We would like to show how to use estimates of κ(Ω) in spectral inequalities between the eigenvalues λn = λn (Ω) of the Dirichlet Laplacian on Ω and the eigenvalues μn = μn (Ω) of the Neumann Laplacian on the same domain. It is known that for general domains we have μn+1 < λn

(3.7)

for each n, and, moreover, for convex domains in Rd we have [15] μn+d < λn .

(3.8)

It was conjectured that (3.8) holds for all domains; this conjecture remains open, and we remark that a ‘counterexample’ given in the paper by Levine and Weinberger is erroneous. Estimate (3.7) was proved by Friedlander [10] for domains with smooth boundaries; later, an elegant proof for arbitrary domains was obtained by Filonov [8]. Filonov’s proof goes like this. Let n be fixed. Denote by φj the Dirichlet eigenfunctions of Ω. By the min–max principle, in order to prove μn+1 λn , it is enough to find a subspace L of H 1 (Ω) such that dim L = n + 1 and for each φ ∈ L \ {0} we have |∇φ|2 dx λn φ 2 dx. (3.9) Ω

Ω


2097

Put L = span(φ1 (x), . . . , φn (x), eiξ ·x ), where ξ is any real vector satisfying |ξ |2 = λn . Obviously, dim L = n + 1. Suppose now that φ ∈ L. This means that φ=

n

aj φj + beiξ ·x .

j =1

Then the left-hand side of (3.9) is n

|aj | λj + |b| λn vold (Ω) + 2 2

2

j =1

n

Re aj bλn

j =1

φj e

iξ ·x

dx

Ω

(in the last sum, we have integrated by parts using the fact that φj satisfies Dirichlet boundary conditions on ∂Ω and that |ξ |2 = λn ). The right-hand side of (3.9) is λn

n

|aj |2 + |b|2 vold (Ω) + 2

j =1

n

Re aj b φj eiξ ·x dx .

j =1

Ω

Comparing the last two expressions leads to (3.9). Now suppose we want to improve this result and to show (3.8) that for some class of (not necessarily convex) domains μn+2 λn . The natural approach to try is to add one more exponential to L, namely to put

L = span φ1 (x), . . . , φn (x), eiξ 1 ·x , eiξ 2 ·x , |ξ j |2 = λn ,

j = 1, 2.

(3.10)

Then, in order for (3.9) to hold, we must get rid of the cross-term with two exponentials, i.e. we must assume that

ei(ξ 1 −ξ 2 )·x dx = 0.

Ω

In the notation introduced above, this means that ξ 1 − ξ 2 ∈ N (Ω).

(3.11)

√ Obviously, we can choose vectors ξ 1 , ξ 2 satisfying both (3.10) and (3.11) iff κ(Ω) 2 λn (Ω). Thus, if we could show that for some, not necessarily convex, d-dimensional domain Ω, the estimate (2.5) holds, then the inequality μn+2 (Ω) λn (Ω) will hold for each n. Similarly, for any Ω, if we know a number n0 such that κ(Ω) 2 λn0 (Ω), then the inequality μn+2 (Ω) λn (Ω) is guaranteed to hold for n n0 .

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4. Some estimates of κ(Ω) for convex balanced domains Throughout this section Ω is convex and balanced, and Ω dependence is frequently dropped; also we always work with real zeros of the Fourier transform. Our aim here is to prove the following Theorem 4.1. Suppose that d = 2 and D(Ω) is the diameter of Ω. Then κ(Ω)

4π . D(Ω)

(4.1)

Remark 4.2. After this paper was written, we have discovered that Theorem 4.1 had been previously proved in [20]. Note that Theorem 4.1 immediately implies Corollary 4.3. Conjecture 2.2 holds for convex, balanced domains Ω ⊂ R2 such that the diameter D(Ω) satisfies √ 2π πD(Ω) . (4.2) √ 2 vol2 (Ω) j1,1 The scaling in (4.2) is chosen in such a way that its left-hand side equals one for a disk. Let r− (Ω) be the inradius of a convex balanced domain Ω. Then it is easy to see that there exists a rectangle with sides 2r− (Ω) and D(Ω) which contains Ω. Thus 2r− (Ω)D(Ω) vol2 (Ω), which together with Corollary 4.3 immediately implies Corollary 4.4. Conjecture 2.2 holds for convex, balanced domains Ω ⊂ R2 such that inradius r− (Ω) satisfies √ πr− (Ω) j1,1 . (4.3) √ 8 vol2 (Ω) Remark 4.5. In the same spirit, one can also establish the validity of Conjecture 2.3 subject to additional geometric constraints: if a domain is sufficiently “long” (i.e. the left-hand side of (4.2) is sufficiently large or the left-hand side of (4.3) is sufficiently small), then (2.2) holds. However such an estimate would be non-explicit, as there is no explicit isoperimetric bound on the second Dirichlet eigenvalue for convex domains, see [12]. Before proving Theorem 4.1, we need to introduce some auxiliary notation, and establish some technical facts. Fix e ∈ S d−1 , and define the function νe : R → R by

νe (t) = vold−1 {x: xe = t} ∩ Ω . It is easy to see that νe is an even function and has a compact support supp νe = [−w(e), w(e)], where w is the support function of Ω, i.e. w(e) is a half-breadth of Ω in direction e. If Ω is convex and d = 2, then νe is a concave function on [−w(e), w(e)] (this is not true if d 3,


2099

e.g. when Ω is a cube and e is a diagonal but in general Brunn–Minkowski inequality implies that (νe (ρ))1/(d−1) is concave on [−w(e), w(e)]). Thus, νe (t) νe (0) and the function νe is non-increasing on [0, w(e)]. As we are working with real zeros of the Fourier transform, we can instead work with (ρe) = χ e (ρ) := χ

w(e)

cos(ρe · x) dx = 2 Ω

cos(tρ)νe (t) dt. 0

Lemma 4.6. Let Z : [0, z] → R be non-increasing and concave. Then 2π(k+1)

Z(t) cos(t) dt 0,

(4.4)

2πk 2π(k+3/2)

Z(t) cos(t) dt 0,

(4.5)

2π(k+1/2) 2π(k+1/2)

Z(t) cos(t) dt 0,

(4.6)

Z(t) cos(t) dt 0

(4.7)

2πk 2π(k+1)

2π(k+1/2)

for k ∈ N (assuming that all intervals of integration are inside [0, z]). Proof. Let L(t) be a linear function such that L(2π(k + 1/4)) = Z(2π(k + 1/4)) and L(2π(k + 3/4)) = Z(2π(k + 3/4)). Then, by concavity of Z(t), we have Z(t) L(t) for t ∈ [2π(k + 1/4), 2π(k + 3/4)] (note that cos(t) 0 for these values of t) and also Z(t) L(t) for t ∈ [2πk, 2π(k + 1/4)] ∪ [2π(k + 3/4), 2π(k + 1)] (note that cos(t) 0 for these values of t). Therefore, 2π(k+1)

2π(k+1)

Z(t) cos(t) dt 2πk

L(t) cos(t) dt = 0, 2πk

the last equality easily checked by a direct computation. This proves (4.4), and (4.5) is being dealt with similarly. Further, 2π(k+1/2)

2π(k+1/4)

Z(t) − Z 2π(k + 1/2) − t cos(t) dt 0,

Z(t) cos(t) dt = 2πk

2πk

since the integrand is non-negative. Inequality (4.7) is similar.

2

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Remark 4.7. Note that (4.4) and (4.5) require only concavity of a function Z, whereas (4.6) and (4.7) require only its monotonicity. Recall that κj (e) denotes the j -th ρ-root (counted in increasing order with account of multiplicities) of χ e (ρ). Lemma 4.8. Let d = 2, then κj (e) Proof. We have χ e (0) > 0. Set ρj =

jπ w(e) .

π(j + 1) . w(e)

Let us show that

χ e (ρj ) = χ e

jπ w(e)

is non-negative when j is odd, and is non-positive when j is even. Assume j = 2k. Then w χ e (ρj ) = 2

2πk τw 2πkt w νe (t) dt = dτ cos cos(τ )νe w πk 2πk

0

0

k−1 2π(+1)

w = πk

cos(τ )νe

=0 2π

τw dτ, 2πk

which is non-positive by (4.4). Assume now that j = 2k + 1. Then w χ e (ρj ) = 2

w 2π(k + 1)t νe (t) dt = cos w πk

0

=

w πk

2π(k+1)

cos(τ )νe 0

π cos(τ )νe 0

k−1 w + πk

τw dτ 2π(k + 1)

τw dτ 2π(k + 1)

2π(+3/2)

=0 2π(+1/2)

cos(τ )νe

τw dτ, 2π(k + 1)

which is non-negative by (4.5) and (4.6). The result now follows from the Intermediate Value Theorem (if, for example, χ e (ρ) is positive except at the points ρ2k where it is zero, then each point ρ2k is a zero of multiplicity (at least) +1) two, so we still have κj (e) π(j w(e) ). 2 Lemma 4.8 immediately leads to the main result of this section.


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Proof of Theorem 4.1. By Lemma 4.8, κ(Ω) = inf κ1 (e) inf e∈S 1

e∈S 1

2π 2π 4π = = . w(e) supe∈S 1 w(e) D(Ω)

2

Remark 4.9. It was proved in [20] that the function κ1 (e) is continuous. Using this fact, one can establish further relationship between this function and Neumann eigenvalues, similar to Lemma 3.3. For example, we have: max κ1 (e) e∈S 1

(4.8)

μ3 (Ω).

Indeed, recall that N1 = N1 (Ω) = e∈S 1 κ1 (e)e. Obviously, N1 (Ω) ⊂ N (Ω). Assuming the continuity of κ1 (e), we see that N1 is a continuous closed curve having the origin inside it. Let e0 be arbitrary unit vector so that p0 := κ1 (e0 )e0 ∈ N1 . Then the closed curve p0 + N1 obviously contains both the points inside N1 (the origin) and outside N1 (for example, the point 2p0 ). Therefore, the intersection (p0 + N1 ) ∩ N1 is non-empty, say p1 ∈ (p0 + N1 ) ∩ N1 . Then three points p0 , p1 , and p0 − p1 all belong to N1 . Now we can argue as in the proof of Lemma √ 3.3, with eip0 ·x , eip1 ·x and 1 being three mutually orthogonal test-functions. This shows that μ3 (Ω) max(|p1 |, |p2 |) maxe∈S 1 κ1 (e). √ Remark 4.10. Using the results of this section and the fact that D(Ω) 2 vol2 (Ω)/π , we obtain κ(Ω)

4π 2π 3/2 2π √ = κ(Ω ∗ ), D(Ω) vol2 (Ω) j1,1

thus proving (2.3) with a numerical constant 1 = 2π ≈ 1.6398. C=C j1,1 Remark 4.11. It should be noted that there is no analog of Theorem 4.1 in dimensions higher than two, i.e. one cannot estimate κ(Ω) in terms of the diameter D(Ω). Indeed, let S = {(x1 , x2 ) ∈ R2 : |x1 | + |x2 | < 1}, and let T be the three-dimensional body of revolution obtained by rotating S around the x1 -axis. Also, choose α > 0, and set Tα = {x ∈ R3 : (x1 , αx2 , αx3 ) ∈ T }. Then, as α → ∞, the distances to origin of all zeros of χ Tα , which are not proportional to e1 = (1, 0, 0), tend to ∞ by Lemma 3.1. On the other hand,

χ Tα (ξ e1 ) = α

−2

χ T (ξ e1 ) = 2πα

−2

1 (1 − x)2 cos(xξ ) dx =

2πα −2 (ξ − sin ξ ) > 0 ξ3

0

for all ξ ∈ R. Thus, κ(Tα ) → ∞ as α → ∞, while D(Tα ) = 2. A similar example works in any higher dimension.

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5. Geometric notation for planar balanced star-shaped domains We set, for a balanced star-shaped domain Ω ⊂ R2 , and r 0,

η(r; Ω) := vol1 Ω ∩ |x| = r and 1 α(r; Ω) := vol2 (Ω)

r η(ρ; Ω) dρ = 0

vol2 (Ω ∩ B2 (r)) vol2 (Ω)

(the normalising factor 1/ vol2 (Ω) will simplify the computations later on). Let us also define the numbers r− = r− (Ω) = min w(e), e∈S 1

r+ := max w(e). e∈S 1

Obviously, r− is the inradius of Ω and 2r+ is its diameter. Some properties of the functions η and α and the numbers r± are obvious: • Both η(r) and α(r) are non-negative; additionally, α(r) is non-decreasing; • η(r) ≡ 2πr and α(r) ≡ πr 2 / vol2 (Ω) for r r− ; moreover, r− = sup{r: η(r) = 2πr} = sup{r: α(r) = πr 2 / vol2 (Ω)}; • η(r) ≡ 0 and α(r) ≡ const = 1 for r r+; moreover, r+ = D/2 = inf{r: η(r) = 0} = inf{r: α(r) = 1} and supp η = [0, D/2]. An additional important property is valid for convex domains. Lemma 5.1. Let Ω ⊂ R2 be a balanced convex domain. Then for r ∈ [r− (Ω), r+ (Ω)], the function η(r) is decreasing and the function α(r) is concave. Proof. Let us prove that the function η is decreasing in the given interval. Indeed, suppose r− < r1 < r2 . Since η(r1 ) < 2πr1 , we have:

Ω ∩ |x| = r1 = |x| = r1 . Thus, the set G := Ω ∩ {|x| = r} consists of several (possibly, infinitely many, but at least two) circular arcs, say G1 , . . . , Gn , . . . . Note that G is obviously symmetric with respect to the origin, j is also a part of G. Let Sj be the strip so if Gj is one of the arcs of G, then the symmetric arc, G j , see based on Gj and Gj (i.e. Sj is the smallest centrally symmetric strip containing Gj and G Fig. 1). Then a little thought shows that the convexity of Ω implies

Ω ∩ |x| r1 ⊂

j

Sj .


2103

j and strip Sj . Fig. 1. Arcs Gj , G

Thus, η(r2 ) vol1

Sj

∩ |x| = r2 .

j

However, for each j we have:

vol1 Sj ∩ |x| = r2 < 2 vol1 Gj (see Fig. 1). Summing this over j , we obtain η(r1 ) > η(r2 ). The concavity of α follows immediately from its definition as an integral of η.

2

Remark 5.2. In a similar manner, one can define the analogues of functions η and α in a higherdimensional setting. Unfortunately, in general, the function η is no longer decreasing on the interval [r− , r+ ]; the simplest counterexample is a strip Ω = {x = (x1 , x2 , x3 ) ∈ R3 , |x1 | < 1}. 6. Proof of Theorem 2.4 Set τ := 2j0,1 . Without loss of generality we assume that 2 vol2 (Ω) = πτ 2 = 4πj0,1 ,

(6.1)

and so Ω ∗ = B(τ ). Thus, κ(Ω ∗ ) =

j1,1 j1,1 1 = = τ 2j0,1 C

as in (2.4), and in order to prove Theorem 2.4, we need to prove with C κ(Ω) 1.

(6.2)

We prove (6.2), and therefore Theorem 2.4 by a sequence of lemmas. Some of them are rather technical, and for convenience the proofs of these lemmas are collected in the next section. First, Theorem 4.1 implies that if D(Ω) 4π , then the statement is proved.

2104


2 /2, then, by TheoCorrespondingly, if the half-breadth of Ω in some direction e, w(e) < j0,1 rem 4.1, the statement is proved since 2r− D Vol2 (Ω) as in Corollary 4.4. Thus without loss of generality we can assume that

r+ = D/2 < 2π

(6.3)

and r− >

2 j0,1

2

.

(6.4)

The following averaging result is one of the central points of the proof. Lemma 6.1. Suppose that

J0 |x| dx 0.

(6.5)

Ω

Then (6.2) holds. Proof. To prove (6.2), it is enough to show that there exists e ∈ S 1 such that cos(xe ) dx 0. Ω

Suppose this inequality is wrong for all e ∈ S 1 . Then cos(xe ) dx de > 0.

(6.6)

S1 Ω

Changing the order of integration and acting as in the proof of (3.3), we get

J0 |x| dx > 0. Ω

Now the lemma follows by contradiction.

2

We now show that the condition of Lemma 6.1 in fact follows from some integral inequality being satisfied by a class of functions. Namely, consider a class A of continuous functions α : [0, ∞) → R with the following properties: (a) (b) (c) (d) (e)

α(r) is non-negative and non-decreasing; 2 ) for 0 r r ; α(r) = r 2 /(4j0,1 − α(r) = 1 for r r+ ; α(r) is concave for r− r r+ ; 2 /2 < r 2j j0,1 − 0,1 r+ < 2π .


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Table 1 Decimal values of constants appearing along the horizontal axis. 2π − 4π 2 − τ 2 2.240206980 2 /2 τ 2 /8 = j0,1 2.891592982 j1,1 3.831705970 τ = 2j0,1 4.809651116 2π 6.283185308 7.015586670 j1,2 8.653727913 j0,3 2π + 4π 2 − τ 2 10.326163640

Lemma 6.2. If j0,3 sup α(r)J1 (r) dr 0

α∈A

(6.7)

0

holds, then (6.2) holds for all planar convex balanced domains Ω normalised by (6.1). Proof. We continue the calculations in the proof of Lemma 6.1. Using the geometric notation introduced in the previous section, we have:

J0 |x| dx =

Ω

∞

∞ η(r)J0 (r) dr = vol2 (Ω)

0

α(r)J1 (r) dr 0

(the last identity is proved by integration by parts using J0 = −J1 ). By Lemma 6.1, we need to show that ∞ α(r)J1 (r) dr 0. 0

If for some k ∈ N, α(j0,k ) = 1, then α(r) = 1 for r j0,k , and so, after integration by parts, ∞

∞ α(r)J1 (r) dr =

j0,k

J1 (r) dr = J0 (j0,k ) = 0.

j0,k

We need to choose which k to take. In our case α(r) = 1 whenever r D/2, so by (6.3) we need to choose k such that j0,k > 2π and we can take k = 3, see Table 1. Thus, we need to show that j0,3 I := α(r)J1 (r) dr 0. 0

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Fig. 2. Graph of J1 (r). Table 2 Decimal values of constants appearing along the vertical axis. −0.0852948043 −0.0072444612 0.0386824043

L (see (6.14)) ymin L + M (see (6.16)) M (see (6.15)) 2 /τ 2 ) = c(j1,1

2 τ 2 −j1,1

τ 2 (2π −j1,1 )

(see (6.9))

0.3403496255

ymin (see (6.8)) 2 /τ 2 j1,1

0.5384485717 0.6346834915

The conditions (a)–(d) are just the re-statement of the properties of the function α summarised at the start of the previous section with account of normalisation (6.1); condition (e) re-states 2 vol (Ω) πr 2 . 2 (6.4), (6.3), and also the obvious inequalities πr− 2 + It is useful here to plot the function J1 (r) (see Fig. 2) and other quantities appearing above. For future use, we also give two tables of approximate decimal values of various constants appearing here and below. Table 1 lists the values appearing along the horizontal axis in various graphs, and Table 2 lists the values along the vertical axis. In both tables the values are sorted out in increasing order. The key points of the proof are the estimates of the function α(r) which are collected in the following sequence of lemmas. We start by denoting y1,1 := α(j1,1 ), and we also introduce a new constant ymin := 1 −

(2π − j1,1 )(64 − τ 2 ) . 8(16π − τ 2 )

Lemma 6.3. For the functions α(r) satisfying the conditions (a)–(e) above, y1,1 ymin .

(6.8)


2107

Fig. 3. A typical graph of α(r) (solid line) and αapprox (r) (dashed line). The points have coordinates A− = (r− , α(r− )) = 2 /τ 2 ), A = (j , α(j )) = (j , y ), and A = (2π, α(2π )) = (2π, 1). (r− , r− + 1 1,1 1,1 1,1 1,1

The proof of this lemma is in the next section. Now, given the function α(r) and using the value of y1,1 = α(j1,1 ) as a parameter, we construct two new functions. One of them is a linear function v(r) = c(y1,1 )r + d(y1,1 ), where the coefficients c and d are chosen to be c = c(y1,1 ) =

1 − y1,1 ; 2π − j1,1

d = d(y1,1 ) := 1 − 2πc =

2πy1,1 − j1,1 . 2π − j1,1

(6.9)

The graph of v(r) is a straight line joining the points A1 = (j1,1 , α(j1,1 )) = (j1,1 , y1,1 ) and A+ = (2π, α(2π)) = (2π, 1). The other function is a piecewise-continuous one (cf. Fig. 3) given by ⎧ ⎪ r 2 /τ 2 ⎪ ⎨ y1,1 αapprox (r) := ⎪ v(r) ⎪ ⎩ 1

for r ∈ [0, τ 2 /8]; for r ∈ (τ 2 /8, j1,1 ]; for r ∈ [j1,1 , 2π]; for r ∈ [2π, j0,3 ].

(6.10)

Obviously, α(r) ≡ αapprox (r) ≡ 1 for r 2π . Lemma 6.4. Let α(r) satisfy conditions (a)–(e). Then α(r) αapprox (r)

for r ∈ [0, j1,1 ]

(6.11)

for r ∈ [j1,1 , 2π].

(6.12)

and α(r) αapprox (r) The proof of this lemma is in the next section.

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Lemma 6.4 immediately implies, with account of the fact that J1 (r) changes sign from plus to minus at r = j1,1 , the following Corollary 6.5. j0,3 j0,3 α(r)J1 (r) dr αapprox (r)J1 (r) dr. 0

(6.13)

0

The integral in the right-hand side of (6.13) can be explicitly calculated as a function of the parameter y1,1 , although the expressions are quite complicated. We introduce two constants,

1 π 2 J1 (2π)H0 (2π) − π 2 J0 (2π)H1 (2π) 2π − j1,1 πj1,1 J0 (j1,1 )H1 (j1,1 ) + j1,1 J0 (j1,1 ) + 2πJ0 (2π) + 2

L := J0

τ2 8

−

(6.14)

and 2 1 τ 1 M := J2 π 2 J1 (2π)H0 (2π) − π 2 J0 (2π)H1 (2π) + 8 8 2π − j1,1 πj1,1 J0 (j1,1 )H1 (j1,1 ) − j1,1 J0 (j1,1 ) + 2πJ0 (2π) ; + 2

(6.15)

in the above formulae H denote the Struve functions [1, Chapter 12]. The numerical values of L and M can be found in Table 2 above. Lemma 6.6. j0,3 αapprox (r)J1 (r) dr = Ly1,1 + M. 0

With account of Lemmas 6.2, 6.3, and 6.6, and Corollary 6.5 we immediately have j0,3 α(r)J1 (r) dr Lymin + M ≈ −0.00724446126 < 0,

(6.16)

0

which finishes the proof of the theorem. 7. Proofs of lemmas j2

Proof of Lemma 6.3. There are two possibilities. If r− j1,1 , then α(j1,1 ) = τ1,1 2 by condition (b), and the claim of the lemma is true. We thus need to consider a case when r− j1,1 .


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Let us introduce a linear function y(r) := a(r− )r + b(r− ), where the constants a and b depend upon r− as a parameter and are chosen to be a = a(r− ) =

2 τ 2 − r− ; τ 2 (2π − r− )

b = b(r− ) := 1 − 2πa =

r− (2πr− − τ 2 ) . τ 2 (2π − r− )

(7.1)

2 /τ 2 ) and A = The graph of y(r) is a straight line joining the points A− = (r− , α(r− )) = (r− , r− + (2π, α(2π)) = (2π, 1). The function α(r) is concave on the interval [r− , 2π] by conditions (c) and (d), and its graph passes through the points A− and A+ . Thus, this graph lies above the straight line joining A− and A+ , and therefore

α(r) y(r)

for r ∈ [r− , 2π].

(7.2)

As j1,1 ∈ [r− , 2π], (7.2) implies α(j1,1 ) a(r− )j1,1 + b(r− ) = 1 − (2π − j1,1 )a(r− ), and as in our case r− can take values only in the interval [τ 2 /8, j1,1 ], we have α(j1,1 ) 1 − (2π − j1,1 )

max

r− ∈[τ 2 /8,j1,1 ]

a(r− ).

We have 2 − 4πr + τ 2 da(r− ) r− − = 2 . dr− τ (2π − r− )2 √ The roots of the numerator in the right-hand side are 2π ± 4π 2 − τ 2 , and as seen from the table above the derivative is negative for r− ∈ [τ 2 /8, j1,1 ]. Thus

y1,1 1 − (2π − j1,1 )a τ 2 /8 . (7.3)

It is an easy manipulation to check that the right-hand side of (7.3) equals ymin .

2

Proof of Lemma 6.4. As αapprox (r) = α(r) for r in the interval [0, τ 2 /8], and αapprox (r) = y1,1 = α(j1,1 ) for r in the interval [τ 2 /8, j1,1 ], inequality (6.11) follows immediately from the monotonicity condition (a). In order to prove (6.12), we again need to consider two cases. First, if r− < j1,1 , then α(r) is concave for r ∈ [j1,1 , 2π], and its graph between the points A1 and A+ lies above the straight 2 /τ 2 ). line joining this points. Thus, it remains to consider the case r− j1,1 (and so y1,1 = j1,1 We now show that in this case

2

2 r2 v(r) = c j1,1 /τ 2 r + d j1,1 /τ 2 2 τ

for r j1,1 .

Indeed, consider u(r) :=

2

2 r2 − c j1,1 /τ 2 r − d j1,1 /τ 2 . τ2

(7.4)

2110


We have u(j1,1 ) = 0, and also for r j1,1 ,

2 du r j1,1 τ 2 − j1,1 (r) = 2 2 − c j1,1 >0 /τ 2 2 2 − 2 dr τ τ τ (2π − j1,1 ) (see Table 2 for numerical values), which proves (7.4). Thus, α(r− ) αapprox (r− ). As, by concavity, the graph of α(r) between the points A− and A+ lies above the straight line joining these points, and the graph of αapprox (r) is a straight line joining the point (r− , αapprox (r− )) (which is located below A− ) with A+ , inequality (6.12) follows. 2 Proof of Lemma 6.6. The result follows from straightforward integration of (6.10) using the standard relations J1 (x) dx = −J0 (x) [1, formula 11.1.6];

xJ1 (x) dx = − =

xJ0 (x) dx

= −xJ0 (x) +

J0 (x) dx

πx

J1 (x)H0 (x) − J0 (x)H1 (x) 2

x 2 J1 (x) dx = x 2 J2 (x)

[1, formula 11.1.7];

[1, formula 11.3.20].

2

8. Perturbation-type results In this section, √ we prove Theorem 2.7. In order to do this, we need to compute the one-sided derivatives of λ2 (Ω F ) and κ(Ω F ) with respect to the parameter describing the deformations of the disk. The first derivative is easily computable from the following classical result (see e.g., [12,19]) which we state without proof: Theorem 8.1 (Derivative of a multiple Dirichlet eigenvalue). Let Ω0 ⊂ Rd be a bounded domain with C 2 boundary. Assume that λk (Ω0 ) = · · · = λk+p−1 (Ω0 ) is a multiple Dirichlet eigenvalue of order p 2. Let us denote by uk1 , uk2 , . . . , ukp an orthonormal family of eigenfunctions associated to λk . Let S(t) : Rd → Rd , t ∈ [0, t0 ), be a continuously differentiable with respect to t family of mappings such that S(0) is an identity, and let Ωt = S(t)(Ω0 ). Then, the function t → λk (Ωt ) has a (directional) derivative at t = +0 which is one of the eigenvalues of the p × p matrix M = [mi,j ] defined by mi,j = −

∂uki ∂ukj S (0)(σ ) · n(σ ) dσ, ∂n ∂n

∂Ω

where n(σ ) is an exterior normal to ∂Ω at the point σ ∈ ∂Ω.

i, j = 1, . . . p,

(8.1)


2111

Theorem 8.1 implies Lemma 8.2. Let Ω F be as in Theorem 2.7. Then 2π √ d λ2 (Ω F ) j1,1 2iθ = − F (θ )e dθ . d 2π =+0

(8.2)

0

Proof. In our case, Ω0 = Ω0F is a disk of radius 1, λ2 (Ω0 ) is doubly degenerate, so the matrix M is of dimension 2, and we can choose the orthonormal eigenfunctions u2 (r, θ ) = N J1 (j1,1 r) cos θ , u3 (r, θ ) = N J1 (j1,1 r) sin θ . The constant N is introduced in order for the eigenfunctions u2 and u3 to have L2 -norm one in the unit disk. Using standard properties of Bessel functions (in particular [1, formulas 11.45 and 9.1.30]) one gets

1 2 . π |J0 (j1,1 )|

N=

Using the lowering property of Bessel functions [1, formula 9.1.30], 1 d

zJ1 (z) = J0 , z dz the value of N just obtained, and the fact that J1 (j1,1 ) = 0 and J0 (j1,1 ) < 0, in the expression for u2 and u3 we obtain ∂u2 2 = −j1,1 cos θ, ∂n π and ∂u3 2 = −j1,1 sin θ, ∂n π at the boundary of the disk (i.e., at r = 1). Taking these remarks into account, the elements of the matrix M in our case are given by m1,1 =

2 2j1,1

π F (θ ) cos2 θ dθ,

π 0

m1,2 = m2,1 =

2 2j1,1

π F (θ ) cos θ sin θ dθ,

π 0

and m2,2 =

2 2j1,1

π F (θ ) sin2 θ dθ.

π 0

2112


For area preserving deformations of the ball, i.e. for functions F satisfying (2.8), we can write the matrix M in the simple form, M=

Re a Im a

Im a , − Re a

where a=

2 j1,1

2π e2iθ F (θ ) dθ.

π 0

It is simple to compute the two eigenvalues of M in this case, and they are given by ±|a|. Hence, using Theorem 8.1 we obtain the (directional) derivative of the second eigenvalue of the perturbed domain by using the smallest of these two eigenvalues: 2 2π j1,1 dλ2 (Ω F ) 2iθ = − F (θ )e dθ . dt π t=0

(8.3)

0

2 , we finally get (8.2). From (8.3), taking into account that λ2 = j1,1

2

Now, we will compute the derivative of κ(Ω F ) at = 0, for area preserving deformations of the disk. For brevity, we shall use the notation f (ξ ) := χ Ω F (ξ ) for the Fourier transform of the characteristic function of Ω F and N = N (Ω F ) = {ξ ∈ Rd : f (ξ ) = 0} for its null variety. We know that N0 contains a circle of radius j1,1 , and we seek to characterise the elements of N . Pick an element ξ 0 ∈ N0 . For definiteness, we choose coordinates in such a way that ξ 0 = (1, 0)j1,1 ,

(8.4)

and we write an element of N as ξ = ξ 0 + ξ 1 . It is precisely ξ 1 which we would like to determine by requiring f (ξ ) = 0 to hold up to first order in . Using polar coordinates, we write ξ 1 = (cos ω, sin ω)ρ1 . With the above notation, we have 2π 1+ F (θ) f (ξ ) = eij1,1 r cos θ ei ρ1 r cos(θ−ω) r dr dθ. 0

0

(8.5)


2113

In the sequel, we use the fact that f0 (j1,1 ) = 0, i.e., 2π1 eij1,1 r cos θ r dr dθ = 0,

(8.6)

0 0

and split the integral in the variable r in (8.5) as an integral from r = 0 to r = 1 plus an integral from r = 1 to r = 1 + F (θ ). After some algebraic computations we obtain

f (ξ ) = N (ρ1 , ω) + O 2 ,

(8.7)

where

2π 1 N(ρ1 , ω) = iρ1

e 0

2π

r cos(θ − ω) dr +

ij1,1 r cos θ 2

0

F (θ )eij1,1 cos θ dθ.

(8.8)

0

Using the fact that the perturbed domain is balanced, i.e., that F (θ ) = F (θ + π), we get 2π

2π F (θ )eij1,1 cos θ dθ =

0

Since we also have

2π 0

F (θ ) cos(j1,1 cos θ ) dθ. 0

eij1,1 r cos θ sin θ dθ = 0, we arrive at

2π 1 N (ρ1 , ω) = iρ1

e 0

ij1,1 r cos θ

2π

cos θ cos ω r dr + 2

0

F (θ ) cos(j1,1 cos θ ) dθ.

(8.9)

0

Integrating the first integral in (8.9) by parts in r gives ρ1 cos ω N(ρ1 , ω) = j1,1

2π

1 2π eij1,1 cos θ dθ −

0

2reij1,1 r cos θ dr dθ 0 0

2π +

F (θ ) cos(j1,1 cos θ ) dθ. 0

Now, we use the integral representation 1 J0 (x) = 2π

2π eix cos θ dθ 0

(8.10)

2114


to simplify the first two terms in (8.10). We finally get ρ1 N (ρ1 , ω) = 2π cos ω J0 (j1,1 ) + j1,1

2π F (θ ) cos(j1,1 cos θ ) dθ.

(8.11)

0

1 2 = 0, since j Here we have used the fact that 0 rJ0 (j1,1 r) dr = J1 (j1,1 )/j1,1 1,1 is a zero of J1 . The vector ξ 1 = ρ1 (cos ω, sin ω) is determined by the condition N (ρ1 , ω) = 0. Therefore, (8.11) implies j1,1 ρ1 cos ω = − 2πJ0 (j1,1 )

2π F (θ ) cos(j1,1 cos θ ) dθ.

(8.12)

0

In the case when ξ 0 is not given by (8.4), but by ξ0 = (cos α, sin α)j1,1 , we have j1,1 ρ1 cos ω = − 2πJ0 (j1,1 )

2π F (θ + α) cos(j1,1 cos θ ) dθ.

(8.13)

0

In order to compute κ(Ω F ) to first order in all we have to compute is |ξ 0 + ξ 1 | to first order in , which in turn is given by

j1,1 + ρ1 cos ω + O 2 . Using (8.13), we obtain κ(Ω F ) = min j1,1 α

1− 2πJ0 (j1,1 )

2π

F (θ + α) cos(j1,1 cos θ ) dθ + O 2 . (8.14)

0

From this result, we immediately have Lemma 8.3. Let F be a C 2 function on a unit circle satisfying periodicity condition (2.6). Then 2π dκ(Ω F ) j1,1 = min − F (θ + α) cos(j1,1 cos θ ) dθ . α d 2πJ0 (j1,1 ) =0 0

Remark 8.4. Note that Lemma 8.3 does not assume the area preservation condition (2.8).

(8.15)


2115

In order to finish the proof of Theorem 2.7 for perturbations around the circle, we need to prove that the right-hand side of (8.15) is always less or equal than the right-hand side of (8.2), i.e., π min α

π F (θ + α) cos(j1,1 cos θ ) dθ −A F (θ )e2iθ dθ ,

0

(8.16)

0

where A = −J0 (j1,1 ) ≈ 0.408 . . . , assuming additionally that F satisfies the area preservation condition (2.8). For future reference we denote the left-hand side and the right-hand side of (8.16) by LF and RF , respectively. Remark π 8.5. Note also that if the average of F is 0 and, additionally, F has zero two-modes, i.e., 0 F (θ )e2iθ dθ = 0, then (8.16) is valid. In fact, in this case, the right-hand side RF vanishes whereas the left-hand side is given by π LF = min

F (θ + α) cos(j1,1 cos θ ) dθ,

α

0

so we have π LF

F (θ + α) cos(j1,1 cos θ ) dθ,

(8.17)

0

for every α, and averaging over α we get 1 LF π

π π 0

1 F (θ + α) cos(j1,1 cos θ ) dθ dα = π

0

π

π F (θ + α) dα = 0,

dθ cos(j1,1 cos θ ) 0

0

and we are done. Before we conclude, we need to analyse the case of equality in (2.10), i.e., we need to show that equality is only attained in (2.10) if the domain is a ball, or, in other words, if F (θ ) ≡ 0. In order to have equality in (2.10), we need equality in (8.17), which in turn implies π F (θ + α) cos(j1,1 cos θ ) dθ = 0,

(8.18)

0

for all α. Since F (θ ) has zero average, and moreover F (θ + π) = F (θ ) (which is required so that the perturbed domain is balanced), the Fourier series of F (θ ) can be written as F (θ ) =

∞ k=−∞

ck eikθ =

m=0

c2m e2imθ ,

(8.19)

2116


with c0 = 0 (because F has zero average) and c2k+1 = 0, for all k (because the domain is balanced). Replacing (8.19) in (8.18) we get

π e2im(θ+α) cos(j1,1 cos θ ) dθ = 0.

c2m

m=0

(8.20)

0

Using the integral representation for Jn (z), i.e., 1 Jn (z) = π

π cos(z sin θ − nθ ) dθ, 0

after some computation we can write (8.20) as c2m (−1)m J2m (j1,1 )e2imα = 0,

(8.21)

m=0

for all 0 α 2π . Since the exp(2imα) form an orthogonal set of functions, we finally get c2m (−1)m J2m (j1,1 ) = 0, for all m. Since jm,1 > j1,1 for all m > 1, J2m (j1,1 ) = 0, thus, c2m = 0 for all m, hence, F (θ ) ≡ 0 as it was to be shown. A similar argument can be used to show that equality is attained only for the ball in the general case. Before we go into the proof of (8.16) for a general F satisfying both the periodicity condition (giving a balanced domain) and the zero average condition (area preserving domain perturbation), we need the following result. Lemma 8.6. Assuming F averages up to zero, it is always possible to rotate F in such a way that the following two conditions are fulfilled simultaneously: π F (θ + φ) sin(2θ ) dθ = 0,

(8.22)

F (θ + φ) cos(2θ ) dθ 0.

(8.23)

0

and π 0

Here F (θ + φ) is F rotated by an angle φ. Proof. Consider the function π T (φ) :=

F (θ + φ) sin(2θ ) dθ. 0


Since F averages to zero, 0, and (8.22) holds. Now, consider

π 0

2117

T (φ) dφ = 0, so there exists a point φ1 ∈ [0, π], such that T (φ1 ) =

π Q(φ) :=

F (θ + φ) cos(2θ ) dθ. 0

Clearly, T (φ1 ) = T (φ1 + π/2) = 0. On the other hand, Q(φ1 ) = −Q(φ1 + π/2). So, either Q(φ1 ) 0, or Q(φ1 + π/2) 0, and we have obtained (8.23) by choosing φ = φ1 or φ = φ1 + π/2. 2 After proving this lemma we are ready π π to prove (8.16). Consider F with zero average and such that 0 F (θ ) cos(2θ ) dθ 0 and 0 F (θ ) sin(2θ ) dθ = 0. In this case, the right-hand side of (8.16) is given by π RF = −A

F (θ ) cos(2θ ) dθ.

(8.24)

0

On the other hand, the left-hand side LF satisfies (8.17) for each α. Now, multiply (8.17) by 2

cos α

!

π cos2 α dα ≡ (2/π) cos2 α 0

and integrate in α from 0 to π (notice that (2/π) cos2 α 0). We thus have 2 LF π

π π

F (θ + α) cos(j1,1 cos θ ) dθ cos2 α dα. 0

(8.25)

0

Now, split cos2 α = (e2iα + e−2iα + 2)/4 in (8.25). If we do the integral over α first, using the fact that the average of F vanishes, we get π

1 F (θ + α) cos α dα = 4

π

2

0

e

2iα

1 F (θ + α) dα + 4

0

π

e−2iα F (θ + α) dα

0

1 = e−2iθ 4

π e

2iβ

1 F (β) dβ + e2iθ 4

0

π

e−2iβ F (β) dβ.

0

By Lemma 8.6, and the choice of orientation of F , we have π

π e

0

2iβ

F (β) dβ =

e 0

−2iβ

π F (β) dβ =

cos(2β)F (β) dβ =: P , 0

2118


and so π F (θ + α) cos2 α dα =

1 cos(2θ )P . 2

0

Then, 21 LF π2

π cos(2θ ) cos(j1,1 cos θ ) dθ P = P J0 (j1,1 ).

(8.26)

0

π Here we have used the fact that 0 cos(2θ ) cos(j1,1 cos θ ) dθ = πJ0 (j1,1 ) (this follows by taking real part in [1, formula 9.1.21], with n = 2, and the fact that J2 (j1,1 ) = −J0 (j1,1 ) [1, 9.1.27]). Hence, L J0 (j1,1 )P = −AP = RF . This proves (8.16) and therefore Theorem 2.7. 9. Non-convex domains: counterexamples We start by proving Theorem 2.8. First, we introduce some notation. For a domain Ω we put ζ (r) = ζΩ (r) :=

η(r) . 2πr

Then the function ζ satisfies the following properties: ζ (r) 1; if Ω is star-shaped, ζ is non D/2 increasing, supp ζ = supp η = [0, D(Ω)/2], and vol2 (Ω) = 2π 0 rζ (r) dr. The strategy of the proof is the following: first, we construct a function ζ which is non-increasing, ζ (r) = 1 for 1+δ 0 r 1 − δ , supp ζ ⊂ [0, 1 + δ ], 0 r ζ (r) dr = π and, finally, ∞

r ζ (r)J0 (γ r) dr > 0

(9.1)

0

for all γ j1,1 . Then we construct a domain Ω such that ζ = ζΩ and Ω cos(γ xe ) dx is close to the left-hand side of (9.1) for all e, |e| = 1 and all γ j1,1 . This Ω will be a required domain. Let ζ0 (r) =

1, 0,

0 < r 1, r >1


2119

be the ζ -function for the ball of radius one. Suppose that δ is fixed. Let δ be a small positive parameter, and put " ξδ (r) =

−1/2, 1 − δ < r < 1, a, 1 < r < 1 + δ, 0, otherwise.

Here, we choose a = a(δ) from the condition ∞ rξδ (r) dr = 0,

(9.2)

0

which is equivalent to 1

r dr . a = 1−δ 1+δ 2 1 r dr

(9.3)

Obviously, a → 0 as δ → 0. Note also that for small δ we have ∞ rξδ (r)J0 (j1,1 r) dr > 0.

(9.4)

0

Indeed, we obviously have d

1 0

rξδ (r)J0 (j1,1 r) dr 1d =− dδ 2

1

1−δ rJ0 (j1,1 r) dr

dδ

=−

J0 (j1,1 ) , 2

so 1 rξδ (r)J0 (j1,1 r) dr ∼

−J0 (j1,1 )δ . 2

(9.5)

0

Similarly, using (9.3) we obtain ∞

1+δ d(a 1 rJ0 (j1,1 r) dr) rξδ (r)J0 (j1,1 r) dr b := = dδ dδ 1 1+δ 1+δ rJ0 (j1,1 r) dr d 1−δ r dr rJ0 (j1,1 r) dr = 1 = 1 , 1+δ 1+δ dδ 2 r dr 2 r dr d

1

1

(9.6)

1

so ∞ rξδ (r)J0 (j1,1 r) dr ∼ bδ 1

(9.7)

2120

R. Benguria et al. / Journal of Functional Analysis 257 (2009) 2088–2123 J (j

)

as δ → 0. Since j1,1 is a local minimum of J0 , we have b > 0 21,1 . Now formulas (9.5) and (9.7) imply (9.4). We now fix a small δ < δ for which (9.4) holds and put ζ (r) = ζ0 (r) + ξδ (r). Then, since ∞ rζ0 (r)J0 (j1,1 r) dr = 0,

(9.8)

0

we have ∞

r ζ (r)J0 (j1,1 r) dr > 0.

(9.9)

0

It is easy to show that in fact for all positive γ j1,1 we have ∞

r ζ (r)J0 (γ r) dr > 0.

(9.10)

0

Indeed, the function ∞ l(γ ) :=

r ζ (r)J0 (γ r) dr

0

decreases for γ < j1,1 , since its derivative ∞

l (γ ) = −

ζ (r)J1 (γ r) dr r 2

0

is negative as J1 (r) is positive for r ∈ [0, j1,1 ]. Note also that (9.2) implies ∞

r ζ (r) dr = 1.

(9.11)

0

Now let us construct a sequence of domains Ωn which satisfy the following properties: (i) the domain Ωn is invariant under the rotation on 2π n around the origin; (ii) in the sector − πn φ πn in polar coordinates (r, φ) the domain Ωn is given by {(r, φ), |φ|
0

(9.13)

Ωn

for all positive γ j1,1 and all unit vectors e. Thus, for this domain Ωn we have κ(Ωn ) > j1,1 , finishing the proof of Theorem 2.8. It remains to prove Theorem 2.9 and Corollary 2.10. Suppose that we have proved Theorem 2.9, and thus constructed a sequence In of one-dimensional balanced domains for which κ(In ) vol1 (In ) → ∞ as n → ∞. Consider An := (In )2 = x = (x1 , x2 ), x1 , x2 ∈ In .

(9.14)

√ Then vol2 (An ) = (vol1 (In ))2 , and κ(An ) = κ(In ), and so κ(An ) vol2 (An ) → ∞ as n → ∞. Now it remains to connect the disjoint rectangles in An by narrow corridors to construct conn with κ(A n ) vol2 (A n ) → ∞ as n → ∞, proving Corollary 2.10. nected domains A Let us prove Theorem 2.9. We formulate the following Lemma 9.1. For each positive C there exist a natural number nand real numbers w1 , . . . , wn such that w1 1, wj +1 wj + 1, and the function f (ξ ) := nj=1 cos(wj ξ ) is positive for ξ ∈ [−C/n, C/n].

2122


Proof. The proof is due to F. Nazarov [17]. Put, for real t, g(t) := (1 − |t|)2+ . Then the Fourier transform g (s) of g is positive for real s. Denote, for real x, G(x) := 1 + 2

n

1−

k=1

k n

2 cos(kx) =

k eikx . g n

(9.15)

k∈Z

Then the Poisson summation formula implies that G(x) = n

g n(x + 2πm) n g (nx),

(9.16)

m∈Z

and so G(x) cn whenever |x| C/n. Now put F (x) :=

n

akn cos(kx),

(9.17)

k=1

where akn is a collection of independent random variables such that akn = 1 with probability (1 − nk )2 ; otherwise akn = 0. Then the standard probabilistic arguments based on the large deviation principle imply that for each fixed point x the probability of the event

F (x) − G(x) − 1 /2 n3/4

(9.18)

is O(e−n ). In particular, putting x = 0 in (9.18), we see that the number of coefficients akn 1/4 which are equal to one, is at least n/10 with probability 1 − O(n3 e−n ). Put xj := nj3 , j = 0, . . . , n3 . Then the probability of the event that for all j = 0, . . . , n3 we have 1/4

F (xj ) − G(xj ) − 1 /2 n3/4

(9.19)

is at least 1 − O(n3 e−n ) and thus is positive for sufficiently large n. Since the derivative of both F and G is O(n2 ), this means that the probability that (9.18) is satisfied for all x ∈ [0, 1] is positive when n is large. Therefore, for each large n there is at least one F such that (9.18) is satisfied for all x ∈ [0, 1]. Thus chosen F satisfies F (x) cn/2 for |x| C/n. 2 1/4

To finish the proof of Theorem 2.9, we now take, for a given C > 0, the numbers n and wj , j = 1, . . . , n, from Lemma 9.1, and define In := {x ∈ R, ||x| − wj | 1/2 for some j }. Then vol1 (In ) 2n, and 4 sin(ξ/2) cos(wj ξ ). ξ n

χ In (ξ ) =

j =1

Therefore by Lemma 9.1 for any constant C we have κ(In ) C/n for sufficiently large n.


2123

Acknowledgments We would like to express our gratitude to I. Polterovich for stimulating discussions, to F. Nazarov and N. Sidorova for helping with the proof of Theorem 2.9, and to N. Filonov for letting us use his Lemma 3.4. We are also grateful to the referee for helpful suggestions. References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1964. [2] M.L. Agranovsky, On the stability of the spectrum in the Pompeiu problem, J. Math. Anal. Appl. 178 (1) (1993) 269–279. [3] M.S. Ashbaugh, R.D. Benguria, Proof of the Payne–Pólya–Weinberger conjecture, Bull. Amer. Math. Soc. (N.S.) 25 (1) (1991) 19–29. [4] P. Aviles, Symmetry theorems related to Pompeiu’s problem, Amer. J. Math. 108 (5) (1986) 1023–1036. [5] C.A. Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem, J. Anal. Math. 37 (1980) 128–144. [6] L. Brown, J.-P. Kahane, A note on the Pompeiu problem for convex domains, Math. Ann. 259 (1) (1982) 107–110. [7] L. Brown, B.M. Schreiber, B.A. Taylor, Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier (Grenoble) 23 (3) (1973) 125–154. [8] N. Filonov, On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator, Algebra i Analiz 16 (2) (2004) 172–176 (in Russian); translation in St. Petersburg Math. J. 16 (2) (2005) 413–416. [9] N. Filonov, Private communication, 2008. [10] L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Ration. Mech. Anal. 116 (2) (1991) 153–160. [11] N. Garofalo, F. Segàla, Asymptotic expansions for a class of Fourier integrals and applications to the Pompeiu problem, J. Anal. Math. 56 (1991) 1–28. [12] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006. [13] T. Kobayashi, Bounded domains and the zero sets of Fourier transforms, in: 75 Years of Radon Transform, Vienna, 1992, in: Conf. Proc. Lecture Notes Math. Phys., vol. IV, Int. Press, Cambridge, MA, 1994, pp. 223–239. [14] T. Kobayashi, Perturbation of domains in the Pompeiu problem, Comm. Anal. Geom. 1 (3–4) (1993) 515–541. [15] H.A. Levine, H.F. Weinberger, Inequalities between Dirichlet and Neumann eigenvalues, Arch. Ration. Mech. Anal. 94 (1986) 193–208. [16] L. Lorch, Some inequalities for the first positive zeros of Bessel functions, SIAM J. Math. Anal. 24 (1993) 814–823. [17] F. Nazarov, Private communication, 2008. [18] L.E. Payne, G. Pólya, H.F. Weinberger, Sur le quotient de deux fréquences propres consécutives, C. R. Acad. Sci. Paris 241 (1955) 917–919; L.E. Payne, G. Pólya, H.F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. Phys. 35 (1956) 289–298. [19] F. Rellich, Perturbation Theory of Eigenvalue Problems, Gordon and Breach, New York, 1969. [20] V.P. Zastavnyi, Zero set of the Fourier transform of measures and the summation of double Fourier series of methods of Bernshtein–Ragozinskii type, Ukr. Math. J. 36 (1984) 459–464.


Elliptic operators, conormal derivatives and positive parts of functions (with an appendix by Haïm Brezis) Alano Ancona Département de Mathématiques, Université Paris-Sud 11, Orsay 91405, France Received 29 September 2008; accepted 15 December 2008 Available online 21 March 2009 Communicated by H. Brezis

Abstract Haïm Brezis and Augusto Ponce introduced and studied several extensions of Kato’s inequality, in particular Kato’s inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a W 1,1 function in a smooth domain [H. Brezis, A.C. Ponce, Kato’s inequality when u is a measure, C. R. Acad. Sci. Paris Sér. I 338 (2004) 599–604; H. Brezis, A.C. Ponce, Kato’s inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241]. Using potential theoretic methods we answer here some questions raised in [H. Brezis, A.C. Ponce, Kato’s inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241] about the relations between the normal derivative of a function u and the normal derivative of its positive part u+ . The results apply to a large class of domains and elliptic operators in divergence form and finally an expression of the normal derivative of a function of u is given. In the final appendix, H. Brezis solves an old question of J. Serrin about pathological solutions of certain elliptic equations [J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa (3) 18 (1964) 385–387]. This is used in the paper to extend the first version of our main result. © 2008 Elsevier Inc. All rights reserved. Keywords: Second order elliptic equations; Potential theory; Boundary values problems; Weak solutions


A. Ancona / Journal of Functional Analysis 257 (2009) 2124–2158

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1. Introduction Let U be an open subset of Rd and let u be a locally integrable function in U whose Laplacian u as a distribution is a locally integrable function in U . Kato’s inequality [26] says that (u+ ) is a measure and that (u+ ) 1{u0} u. Brezis and Ponce [9] have extended this result to the case where u is an arbitrary Radon measure in U . On another side, using methods of the fine potential theory, B. Fuglede has improved in [17] results of De la Vallée Poussin, M. Brelot and M.A. Grishin about the positivity of the trace on the set {u = 0} of the measure u associated to a function u which is in U the difference of two subharmonic function—an important point being to define precisely the set {u = 0}. Fuglede’s result also improves and extends Kato’s inequality (see Section 5) but seems to have remained unnoticed by the followers of Kato’s work. In [10] H. Brezis and A. Ponce have introduced and studied forms of Kato’s inequality up to the boundary. If U is a smooth bounded domain in Rd , if u ∈ W 1,1 (U ) is such that u is a finite Radon measure in U whose normal derivative ∂n u—in some appropriate weak sense—is a measure on ∂U , it is shown in [10] that u+ , ∂n (u+ ) are finite measures (in U and ∂U respectively) such that u+ + ∂n u+ u + ∂n u (where . denotes the total mass). If moreover ∂n u ∈ L1 (∂Ω), then ∂n (u+ ) 1{u>0} ∂n u − 1{u=0} (∂n u)− ,—and even ∂n (u+ ) = 1{u>0} ∂n u − 1{u=0} (∂n u)− if u ∈ W 2,1 (U )—see [10] where other results on the normal derivative ∂n u+ are established. The aim of this paper is to solve the questions in [10] about possible improvements of the above results (see [10, Section 1]). It will be also shown that these natural improvements hold in a quite general framework. Relying in particular on Fuglede’s result mentioned above (see Theorem 5.1) and on a systematic use of the fine potential theory, we establish in particular (Theorem 6.1 in Section 6) the formula ∂n (u+ ) = 1{u>0} ∂n u − 1{u=0} (∂n u)− assuming only that U is C 1,1 (or even C 1,α , α > 0), that u ∈ W 1,1 (U ) and that u and ∂n u are finite measures in U and ∂U respectively. The assumption on U can be further relaxed if u ∈ W 1,2 (U ). We will also show that the Laplacian can be replaced by a quite general second order uniformly elliptic operator in divergence form and that some other results of [10] can be extended to our framework. In Section 7, we prove as an application a formula giving the normal derivative ∂n f (u) for a class of functions f in R. This is also a generalization of Theorem 6.1. I am pleased to thank Haïm Brezis for attracting my attention and interest to the questions introduced by him and Augusto Ponce in [10] and for supplementing this paper by his recent solution of Serrin’s conjecture in [32]. See Theorems A1.1 and A1.2 in Appendix A. The conjecture is about the non-existence of pathological solutions for certain elliptic equations (see Section 3 and Appendix A). It will be seen that Brezis results (announced in [8]) allow us to relax to a certain extent the required regularity assumptions in the paper’s main results (see Section 6 and [4]). 2. The setting We will consider a differentiable manifold M of class C 1 , separable and of dimension d 2, equipped with a second order elliptic operator of a type described below. It would in fact be more natural—but perhaps somewhat heavier—to consider Lipschitz manifolds. We first state some natural definitions and simple facts needed in the sequel. The reader may just as well glance through this section and return to it when necessary.

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2.1. Required Sobolev spaces and distributions A density m in M is a positive Borel measure m on M such that for any chart x : U → Rd of M, one has dm = f (x) dx1 . . . dxd with f > 0 and continuous on x(U ); equivalently, for one (or for any) Riemannian C 0 -metric g in M, dm(z) = h(z)dσg (z) in M with h > 0 and continuous. Here and in the sequel σg denotes the Riemannian volume induced by g. A set A ⊂ M is negligible if negligible w.r. to any density. For U open in M, the Frechet space L1loc (U ; m) is independent of the chosen density m and p will be denoted L1loc (U ). For 1 p ∞ one defines similarly the Lebesgue spaces Lloc (U ) and p p p Lloc (U ). The subspace of functions f ∈ Lloc (U ) with compact support in U is noted Lc (U ). If p moreover, U is relatively compact in M, L (U ; m) does not depend on m and we set Lp (U ) = Lp (U ; m). One defines similarly, using local charts—or an arbitrary C 0 -metric in M—the space of vector fields whose pth power is locally integrable in U (1 p ∞) and, for U relatively compact, the integrability of a vector field in U . 1,p The Sobolev space Wloc (U ), for U open in M and 1 p ∞, is the space of functions p 1,p f ∈ Lloc (U ) such that for any chart ϕ : U → V ⊂ Rd , U ⊂ U , one has f ◦ ϕ −1 ∈ Wloc (V ). We 1,p 1,p define Wc (U ) := {f ∈ Wloc (U ): supp(f ) compact in U }. The space Wc1,∞ (U ) = Lipc (U ) is the space of (locally) Lipschitz functions with compact support in U . This space is a natural space of test functions in U in our setting. 1,p If g is a C 0 -metric in M and if f ∈ Wloc (U ) the gradient ∇g f is well defined as a class (modulo negligible subsets of U ) of locally integrable fields in U by the following property: for V open in Rd and φ : V → W ⊂ U a C 1 -diffeomorphism, one has g(∇g f [φ(x)], v) = ∇(f ◦ φ)(x).(Dx φ)−1 v, v ∈ Tφ(x) (M), for a.a. x ∈ V . The integrability of ∇g f on a relatively compact subset of M is independent of the choice of g. If Σ is an open subset of ∂U consisting of points having a neighborhood in which U is a 1,p 1,p C 1 -smooth domain, Wloc (U ∪ Σ) denotes the set of functions f ∈ Wloc (U ) such that each point in Σ admits an open neighborhood V in M such that f and ∇g f are of class Lp in V ∩ U (for any C 0 -metric g in M). If U is relatively compact and C 1 -smooth in M, we set W 1,p (U ) := W 1,p (U ). Distributions. The spaces of distributions that will be needed are the duals D1 (U ) = 1 [Cc (U ; R)] for U open in M. The space D1 (U ) is the set of linear forms : Cc1 (U ; R) → R such that (fn ) → 0 for any sequence {fn } in Cc1 (U ; R) such that ∪n1 supp(fn ) is relatively compact in U and lim[fn ∞ + ∇g fn g,∞ ] = 0 for some (or any)C 0 -metric g in M. In fact we will mostly consider distributions T ∈ D1 (U ) in the form: T (f ) = U f dμ + U g(V , ∇f ) dσg , ∀f ∈ Cc1 (U ), where V is a locally integrable vector field in U and μ is a Radon measure in U . 2.2. Standard elliptic operators in M We consider divergence form second order elliptic operators in M defined by a symmetric bilinear form β : Wc1,2 (M) × Wc1,2 (M) → R of the following type: for a certain C 0 -metric g in M there exists a measurable section A = Ag of End(T (M)) such that (i) A(x) is g-symmetric for all x ∈ M, (ii) A is locally uniformly bounded and accretive, i.e., for each compact subset K


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of M there is a constant C 1 for which C −1 gx (u, u) gx (A(x)u, u) Cgx (u, u) when x ∈ K and u ∈ Tx (M), (iii) the form β is given by β(u, v) =

A(∇g u), ∇g v g dσg

(2.1)

1,1 (U ), v ∈ Wc1,∞ (U ) and when u, v ∈ Wc1,2 (M) (β(u, v) is then also meaningful for say u ∈ Wloc U open in M). If g1 is another C 0 -Riemannian metric in M, β admits a similar representation with respect to g1 . For if B is the continuous section of End(T (M)) such that g1 (ξ, η) = g(Bξ, η) for ξ, η ∈ T (M), then ∇g1 ϕ = B −1 (∇g ϕ) for ϕ ∈ Wc1,2 (M) and

β(u, v) =

A(B∇g1 u), ∇g v g dσg =

A(B∇g1 u), ∇g1 v

g1

dσg .

(2.2)

Hence β(u, v) = A1 (∇g1 u), ∇g1 vg1 dσg1 with A1 = J1g A ◦ B where the jacobian Jg1 = 1 √ detB is the density of σg1 with respect to σg . Let us also notice that for a given metric g, the section A is unique (up to almost everywhere ˜ equality). If A is another section representing β withrespect to g and if A = A − A then for 1 1 ˜ ˜ ˜ u, v ∈ Cc (M) and ϕ ∈ C (M), A∇u, ∇vϕ dσ = − A∇ϕ, ∇vu dσ = A∇ϕ, ∇uv dσ = ˜ ˜ − A∇v, ∇uϕ dσ (using uϕ, uv and vϕ). Hence A∇u, ∇vϕ dσ = 0. It follows that ˜

A∇u, ∇v = 0 a.e. Thus A˜ = 0 a.e. in M. 1,1 The Dirichlet form β induces, for each open subset U of M, a map L : Wloc (U ) → D1 (U ) determined by the relations L(u)(v) = −

A(∇g u), ∇g v g dσg

(2.3)

U

for all v ∈ Cc1 (U ). These maps are independent of g and are local with the obvious meaning. They will be viewed as an elliptic operator L and in this paper such an operator will be called a standard elliptic operator in M. We will say that L is associated to β, or equivalently that β is the Dirichlet form associated to L and denote β = βL . Remark 2.1. To give a meaning to L(u) as a function (for u sufficiently regular) the choice of a density m in M is required. This density determines canonical embeddings L1loc (U ) → D1 (U ) for each open subset U in M (by f (ϕ) = f ϕ dm for ϕ ∈ Cc1 (U )) and L can be seen as the elliptic operator in divergence form which can be written L(u) :=

1 divg (A∇g u) θ

(2.4)

with respect to any given C 0 -metric g, where A = Ag is as in (2.1) and where θ is the density 1,1 of m with respect to σg . By definition, for f ∈ L1loc (U ) and u ∈ Wloc (U ), one has Lu = f (in the 1 weak sense) if and only if β(u, v) = − f v dm for all v ∈ Cc (U ).

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Remark 2.2. If M is moreover a C 2 -differentiable manifold and if A = Ag is locally Lipschitz 2,1 in M, then for u ∈ Wloc (U ), h := L(u) ∈ L1loc (M) can be directly expressed through formula (2.4) which gives a meaning to h almost everywhere in U (for any fixed C 1 -metric g). Direct image by a diffeomorphism. Let Φ : M → N be a C 1 -diffeomorphism (or just a locally bilipschitz homeomorphism) between two (separable) C 1 -manifolds and let L be a standard elliptic operator in M. The direct image Φ ∗ (L) is the standard elliptic operator L in N associated to the Dirichlet form βL such that βL (f, g) = βL (f ◦ Φ, g ◦ Φ) for f, g ∈ Cc1 (N ). Equivalently for f ∈ Cc1 (N ), L (f ) = Φ ∗ [L(f ◦ Φ)] in D1 (N ), where for S ∈ D1 (M) the direct image distribution Φ ∗ (S) ∈ D1 (N ) is defined by the relations Φ ∗ (S)(f ) = S(f ◦ Φ) for all f ∈ Cc1 (N ). 2.3. The conormal derivative on the boundary Let U be an open subset of M. Using the procedure in [10] one may define the conormal 1,1 (U ) as a distribution supported by ∂U — derivative, with respect to L, of a function u ∈ Wloc 0 provided that u is sufficiently regular. Let g be a C -metric in M. 1,1 (U ) be such that ∇g u is integrable in a neighborhood of every point Definition 2.3. Let u ∈ Wloc of ∂U and λ := L(u) is a signed Radon measure in U satisfying |λ|(U ∩ K) < ∞ for every compact K ⊂ M. The conormal derivative ∂n u of u along ∂U is the distribution ∂n u ∈ D1 (M) defined by (∂n u)(v) := v dλ + Ag (∇g u), ∇v dσg (2.5) U

U

for v ∈ Cc1 (M). This distribution is independent of the chosen metric g. Obviously, (2.5) is also meaningful for v ∈ Lipc (M) and ∂n (u) extends in a natural way to Lipc (M). If U is C 1 -smooth, it is clear that ∂n u depends solely on the C 1 -structure of the manifold with boundary U , the function u and the restriction L|U (or (βL )|U ). Remark 2.4. It is easily checked that ∂n u is supported by ∂U and that the map u → ∂n u is local (if u = 0 in a neighborhood in U of P ∈ ∂U , then ∂n u vanishes in a neighborhood of P in M). Another important property is that if U is C 1 -smooth then ∂n u is in fact a distribution on the submanifold ∂U (this was already noticed in the first version of [10]): that is ∂n (u)(v) = 0 for v ∈ Cc1 (M) vanishing in ∂U . For, in that case, ∂n (u)(v) = ∂n u(vN ) if vN = min(max{v, − N1 }; N1 ) and A(∇u), (∇vN ) dσg = A(∇u), ∇v dσ → 0 U

U ∩{0 0}. Assume moreover that L is symmetric with respect to xd (i.e., aij (x , xd ) = aij (x , −xd ) if 1 i, j < d or if i = j = d, and aid (x , xd ) = −aid (x , −xd ) for 1 i < d). Then L is regular.

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The assumptions imply that—after perhaps modifying the coefficients on the hyperplane {xd = 0}—the aij with 1 i, j d − 1 or i = j = d are C ε in BR (these aij being even with respect to xd ). The point is that the aid , i < d, may well be non-extendable by continuity on this hyperplane. Proof. (a) To establish Theorem 3.3 it will be shown that there exists a small R ∈ (0, R) and a bilipschitz homeomorphism Ψ : BR → U , U ⊂ Rd , that commutes with the orthogonal symmetry σ = σd with respect to the hyperplane {xd = 0}, that is C 1,ε on the manifold with boundary ∗ ε B+ R = BR ∩ {xd 0} and which is such that Ψ (L) has C coefficients in Ψ (BR )—and not only + ∗ in Ψ (B R ). This will show, since Ψ (L) is regular by Theorem A1.2, that L is regular in a neighborhood of 0. Similarly, L is regular in a neighborhood of each point in BR ∩ {xd = 0}. Since L is clearly regular in BR \ {xd = 0}—using again Theorem A1.2—it follows that L is regular in BR . The map Ψ will be constructed in B + R = BR ∩ {xd 0} by using the following procedure. (b) Construction of a class of diffeomorphism. Let V = (V1 , . . . , Vd ) be a C ε vector function in ΣR = B R ∩ {xd = 0} with Vd = 1. Let V˜ be a Whitney extension of V to Rd (Ref. [36]). Of course V˜d = 1 and as is well known V˜ is C ∞ in Rd \ ΣR , C ε in Rd and for each multi-index α ∈ Nd one has an estimate in the form |D α V˜ (x)| Cα |xd |ε−|α| for some positive constant Cα . In particular V˜ is C ε in BR . Consider then the map m → F (m) = m + md V˜ (1) (m) in BR , where V˜ (1) := (V˜1 , . . . , V˜d−1 , 0). Standard verifications show that F is C 1,ε -smooth in BR . It suffices to show that fj k (x) := ∂Vj xd ∂xk (x) is C ε in BR+ which can be seen as follows. Consider x, x ∈ BR+ , xd xd , then

|fj k (x) − fj k (x )| |fj k (x)| + |fj k (x )| C(xdε + xd ε ) 21+ε C|x − x |ε if |x − x | 12 xd . And if |x − x | 12 xd the mean value theorem and the above estimates of |D α V˜ | yield |fj k (x) − fj k (x )| C xdε−1 |x − x | C 2ε−1 |x − x |ε . Computing D0 F it is seen that the map F is even a local C 1,ε -diffeomorphism at 0 that fixes each point of ΣR . Its differential maps the normal field N := (0, 0, . . . , 0, 1) on ΣR to the field V . So taking Φ := F −1 one gets a C 1,ε -diffeomorphism Φ on a ball BR , that fixes every point in ΣR ∩ V and maps V to N in BR ∩ ΣR . (c) The construction of Ψ . Let now Φ be a C 1,ε diffeomorphism from BR onto an open set Ω ⊂ BR , associated as above to a C ε vector field V in BR which is transverse to ΣR (and to be chosen below). Let us denote Ψ the bilipschitz homeomorphism which coincides with Φ in − + B+ R = BR ∩ {m; md 0} and is equal to σ ◦ Φ ◦ σ in B R = −B R . The operator L = Ψ ∗ (L) in Ω = Ψ (BR ) is—as is well known—easily computed. One has L (u) = f , u ∈ W 1,1 (Ω), f ∈ D1 (Ω), if and only if L(u ◦ Ψ ) = Ψ ∗ −1 (f ), which means that for all v ∈ Cc1 (Ω) i,j B R

aij (x)(u ◦ Ψ )i (x)(v ◦ Ψ )j (x) dx = −f (v)


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or α,β B

R

i,j

aij (x)

∂Ψβ ∂Ψα (x) (x) uα (Ψ )vβ (Ψ ) dx = −f (v) ∂xi ∂xj

(3.1)

and α,β Ω

i,j

−1 ∂Ψα −1 ∂Ψβ −1 1 Ψ x Ψ x uα (x )vβ (x ) aij Ψ x dx = −f (v) ∂xi ∂xj J (x )

(3.2)

where J (x ) = | det{DΦ −1 (x ) Ψ }|. Recall that by assumption the aij are ε-Hölder continuous in + BR+ = BR ∩ {xd > 0}; let aij+ denote the continuous extension of aij |B + to B + R and let aL (x) R denote the bilinear form associated to {aij (x)}1i,j d . The above computation shows that the elliptic operator L has the required type if along + Σd = {xd = 0} ∩ BR the coefficients bij := aL (∇Φi , ∇Φj ) vanish when 1 i < d = j . But, (1) ˜ since Φ is in the form Φ(x) = x + xd V (x), where V is a C ε vector field (with Vd = 1) in ΣR , one has on Σd , ∇Φi (x) = (0, . . . , 0, 1, 0, . . . , 0, Vi (x)) (where 1 is the ith coordinate) for i < d and ∇Φd (x) = (0, . . . , 0, 1) . Thus the condition to satisfy in order that bi,d , with 1 i < d, a + (x)

+ + (x)Vi (x) + ai,d (x) = 0 for x ∈ Σd , or Vi (x) = − a +id (x) in Σd . Now vanishes in Σd is that add dd

these relations for 1 i < d together with Vd = 1 define a C ε vector field V in BR for which, by the previous calculations, the corresponding map Ψ has the desired property. 2 Remark 3.4. There is a version of Theorem 3.3 for the case ε = 0 (i.e., Hölder continuity is replaced by continuity). The conclusion being now that L is “weakly regular,” that is every weak 1,p 1,2 (U ). The proof is similar to the proof L-solution u ∈ Wloc (U ), U ⊂ BR , with p > 1 is in Wloc of Theorem 3.3 above, using Theorem A1.1 instead of Theorem A1.2. 4. Potential theory In this section we collect some known basic facts from potential theory. Let L be a standard elliptic operator defined on the C 1 manifold M. It is well known [34,23,24] that L defines a Brelot type potential theory (Refs. [7,22]) in M. The corresponding harmonic functions 1,2 (U ) (or L-harmonic functions) are the continuous representatives of weak solutions u ∈ Wloc of L(u) = 0, U being an open subset of M (local charts reduce us to the more usual case 1,2 (U ) and such where M = Rd ). More generally, an L-local supersolution s in U (i.e., s ∈ Wloc 1,∞ that A∇g s, ∇g ϕ dσg 0 for all ϕ ∈ Wc (U ) with the notations of 2.2) admits a unique L-superharmonic representative in U . B.1. Green’s function, potentials. Cf. [34,23,24]. If U is an open subset in M where there exists an L-superharmonic function which is non-constant in each component of U —we then say that U is admissible—, there exists an L-Green’s function G = GL U : U × U → R+ which is continuous, symmetric, finite off the diagonal and for every positive measure μ compactly supported in U , the function Gμ := G(., y) dμ(y) is an L-potential (i.e., it is L-superharmonic and its 1,r d greater L-harmonic minorant vanishes in U ). Moreover Gμ ∈ Wloc (U ) for r < d−1 , LGμ = −μ

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1 (U \ supp(μ)) [34,23]. One has also L[G ] = −μ in the in the weak sense of [34] and Gμ ∈ Hloc μ weak sense (2.3). This easily follows from the approximation result [34, Théorème 9.2]. Every open subset of an admissible domain is admissible and if M is connected and not admissible, it is well known (Myrberg’s theorem) that an open subset U is admissible if and only if M \ U is not polar (see next paragraph). We will denote P(U ) (resp. S(U )) the set of all L-potentials (resp. L-superharmonic functions) in U .

B.2. Local behavior of G, polar sets. For every compact K ⊂ U and every fixed C 0 -metric g in M, there is an estimate: c−1 [dg (x, y)]2−d G(x; y) c dg (x, y)2−d for x, y ∈ K and a conC stant c > 0 (when dim(M) = 2, [dg (x, y)]2−d is to be replaced by log dg (x,y) , C > 0 sufficiently large). A polar set is a subset A ⊂ M such that in a neighborhood of each of its points, A is contained in a set in the form {p = +∞} with p superharmonic in this neighborhood. Equivalently, A is polar in every local chart in the sense of classical potential theory [6,15]. B.3. Thinness, fine topology. Cf. [7,15,16]. The set A ⊂ M is thin at a ∈ A \ A if there exists U a open and p ∈ P(U ) such that p(a) < lim infx∈A, x→a p(x). By definition, A is thin at every a ∈ / A and for a ∈ A, thinness at a is the same as thinness at a of A \ {a}. Using the estimates in B.2 one may show that thinness does not depend on the given standard operator L [23]. So thinness at a is the same as classical thinness in one (or all) local chart at a and can be characterized by the classical Wiener criterion [15]. One says that V ⊂ M is a fine neighborhood of a if a ∈ V and if V c is thin at a. To this notion of neighborhood corresponds a topology called the fine topology and for which all L-superharmonic functions are continuous. If p = Gμ and q = Gν are two L-potentials in M (assuming that M is admissible), then μ and ν coincide on the fine interior of the set {p = q} (see Lemma 8.4). Also if p = q almost everywhere (with respect to a density) in a finely open subset U , then p = q everywhere in U since every finely open subset is non-negligible (cf. e.g. [6]). B.4. Balayage. Let p = Gμ be a potential in the admissible open subset U of M (μ is the positive measure in U associated to p) and let A ⊂ U . Recall that the réduite RpA (with respect to U ) is the infimum of all nonnegative L-superharmonic functions in U that are larger than p in A; its lower semicontinuous regularization Rˆ pA is an L-potential and is equal quasi-everywhere to RpA in U (cf. [22,24]). The measure μA = −L(Rˆ pA ) associated to this potential is the swept-out of μ on A—with respect to U . It is known that μA = εxA dμ(x) where εx denotes the Dirac measure at x (in particular, μ → μA is linear). Also the swept-out measure εxA is distinct from εx if and only if A is thin at x, and in this case εxA does not charge polar sets. In fact for an arbitrary set A ⊂ M and for x ∈ / A, the swept-out εxA is concentrated on ∂f (A) the fine boundary of A (more precisely, on an ordinary Kσ subset of ∂f (A)). Cf. [16], [15, pp. 183–186] or Theorem 8.3 in Section 8. 5. A precise form of Kato’s inequality This section is devoted to a precise form of Kato’s inequality based on fine potential theory considerations and given by Bent Fuglede in [17]. Let us note that Brezis and Ponce [9] have independently obtained an extension of Kato’s inequality for functions whose Laplacian is an arbitrary Radon measure. The reader should consult [17] and [9] for older related results. Again M denotes a C 1 manifold and L is a standard elliptic operator in M. In all this section, except in—and after—the final Remark 5.8, we assume that L is regular.


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1,1 Let u ∈ Wloc (M) be such that −L(u) (in the distribution sense (2.3) in M) is a Radon measure 1,1 such that μ in M. In each relatively compact open subset ω of M, v = u − GL ω (μ|ω ) is W L(v) = 0 in ω so that v is L-harmonic. Hence, u is (locally) equal almost everywhere to a difference of two L-superharmonic functions and u admits a representative u which is finite and finely continuous outside a Gδ polar subset of M. This function u is unique up to modification on a polar subset. As in [17] let us precisely define (not only up to a polar subset) the set {u > 0} ⊂ M as follows: a point a ∈ M belongs to {u > 0} if and only if the fine liminf at a of u is strictly positive—which is meaningful, a polar set being thin at every point. Since a nonempty finely open subset of M is non-negligible [6], we may more simply set

{u > 0} = {a ∈ M: ∃ε > 0, ∃A ⊂ M thin at a and s.t. u ε a.e. in M \ A}

(5.1)

where on the right-hand side u is seen as an element of L1loc (M). Clearly {u > 0} is a finely open set which is disjoint from the finely open set {u < 0} := {−u > 0}. Moreover this set is Borel-measurable (a Fσ set, see [16] or Section 8). Its fine boundary ∂f {u > 0} is also Borelmeasurable—more precisely a Gδ set (see Section 8). We may now state the following result which is essentially contained in [17]. Theorem 5.1 (A precise form of Kato’s inequality). The distribution L(u+ ) is a measure and L(u+ ) = 1{u>0} . L(u) + λ+ in M where λ+ is a positive measure concentrated on the finely closed set ∂f {u > 0} (recall that ∂f means the fine boundary). Moreover, if M is L-admissible and if u = p − q with p, q ∈ P(M), the measure λ+ is smaller than the swept-out on {u > 0}c of some positive measure in M supported by {u > 0}. We will give here a proof of this theorem which relies on the next lemma and is somewhat different from Fuglede’s proof [17]. The following classical Fatou–Doob type property will be needed: if p = Gμ and q = Gν are potentials in M (generated by the measures μ and ν) and if A ⊂ M is a Borel polar set such that ν(A) = 0, then p/q admits the fine limit +∞ at μ-almost all points a of A ([15, p. 172], or see Theorem 8.1 in Section 8 below). A fact which contains the even more classical property that 1{p 0} and μj = −L(pj ), j = 1, 2. Then L(u) = 1V .(μ2 − μ1 ) + ([1V .μ1 ]V − [1V .μ2 ]V ). Proof. We may assume that μ1 ∧ μ2 = 0 (after subtracting μ1 ∧ μ2 to each of μ1 and μ2 ). Since the pj are finely continuous, we have p1 = p2 in V := {u > 0}c by the very definition of V . By the general property that have just been recalled, neither μ1 nor μ2 may charge a polar subset of V c . Thus V c is unthin at μj -almost all a ∈ V c (recall that the finely isolated points of c c V c form a polar set) and, by the balayage properties, we have L(Rˆ pVj ) = −1V c .μj − (1V .μj )V for j = 1, 2. Whence the equality in the statement on applying L to the equality p1 − p2 = c c (p1 − Rˆ pV1 ) − (p2 − Rˆ pV2 ). 2 Note the following particular case: if under the conditions of Lemma 5.2, one has 1V .μ2 1V .μ1 then L(u) = 1{u>0} .L(u) + λ+ where λ+ is a positive measure concentrated on ∂f {u > 0} (in fact, the swept-out of ν = 1V .(μ1 − μ2 ) on V c , a measure which does not charge polar subsets).

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Proposition 5.3. Let p1 , p2 be two L-potentials in M such that p2 p1 in M, let u = p1 − p2 and set V = {u > 0}. Then L(u) = 1V .L(u) + λ+ where λ+ is a positive measure supported by ∂f V and smaller than the balayée on V c of the positive measure μ1 = −1V L(p1 ) (which is supported by V ). We want to study the trace of L(u) on the finely closed set V c , the last assertion being in fact ensured by Lemma 5.2. Set again μj = −L(pj ) and assume—as possible—that μ1 ∧ μ2 = 0. We have already remarked in the beginning of the proof of Lemma 5.2 that the μj charge no polar subset of V c . Similarly by the property reminded just before the statement of Lemma 5.2 we have ν(V c ∩ {p1 = +∞}) = 0 for every positive measure ν in M such that Gν p1 . Let ε > 0 be an arbitrary positive number and write u = u ∧ ε + (u − ε)+ . Observe that u ∧ ε = p1 ∧ (p2 + ε) − p2 and (u − ε)+ = u − u ∧ ε = p1 − p1 ∧ (p2 + ε) are also (almost everywhere equal to) differences of potentials. The measure L[(u − ε)+ ] does not charge the finely open set W = {p1 < p2 + ε} since p1 = p1 ∧ (p2 + ε) in W . Since moreover, V c \ W ⊂ {p1 = +∞}, the measure L[(u − ε)+ ] vanishes in V c by the remarks in the above paragraph. Consider then w = u ∧ ε. We have L(w) = L(p1 ∧ (p2 + ε)) − L(p2 ) and thus L(w) μ2 . Moreover in the finely open set Wε = {p1 > p2 + ε} ⊂ V we have w = ε and hence L(w) = 0 in Wε . It is seen in that way that L(w) 1V \Wε .μ2 in V . c c Lemma 5.2 says now that 1V c L(u) = 1V c L(w) −[1V (L(w))+ ]V −[1V \Wε .μ2 ]V (observe that V = {w > 0}). Since for ε ↓ 0 the measure 1V \Wε .μ2 decreases to zero, its swept-out also decreases to zero and hence 1V c L(u) 0. 1,1 (U ), u 0, be such that L(u) is a Radon Corollary 5.4. Let U be open in M and let u ∈ Wloc measure in U . If V := {u > 0} (in the sense of (5.1) in U ) we have Lu = 1V .L(u) + λ+ in U , where λ+ is a positive measure in U supported by U ∩ ∂f V (and hence singular w.r. to 1V .L(u)).

Proof. Repeating the argument of [17], we observe that the required properties are local so that we may assume U to be L-admissible and that u = s1 − s2 with sj ∈ S+ (U ); taking the réduites of sj over large compact subsets of U , it is seen that without altering u in the neighborhood of a given point, we may also assume sj to be a potential in U , j = 1, 2. We are then reduced to Proposition 5.3. 2 The next observation also follows from Proposition 5.3. Remark 5.5. Locally, the measure λ+ in Corollary 5.4 is smaller than the swept-out on V c of a positive measure supported by V . More precisely, if U1 , U2 are relatively compact open subsets c of U such that U 1 ⊂ U2 , U 2 ⊂ U , then 1U1 .λ+ = τ V ∩U2 in U1 where τ is a finite positive Borel measure supported by V ∩ U2 , and the sweeping is made with respect to the ambient space U2 . The following corollary will be used to extend to our setting inequalities due to Brezis and Ponce [10]. Recall that if μ is a Radon measure in an open subset U of M, we denote μ the total mass of |μ|. Corollary 5.6. If u ∈ Wc1,1 (U ) is such that L(u) is a Radon measure in U , then L(u+ ) L(u).


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Proof. Since u and u+ are compactly supported we have dL(u) = 0 and dL(u+ ) = 0 (write βL (u, ϕ) = βL (u+ , ϕ) = 0 for ϕ ∈ Cc1 (M) with compact support in U and equal to 1 in a neighborhood of the support of u). Hence using the notations of Theorem 5.1 we have λ+ = − {u>0} dL(u) and | {u>0}c dL(u)| = λ+ . So L(u+ ) = {u>0} | dL(u)| + λ+ {u>0} | dL(u)| + | {u>0}c dL(u)| L(u). 2 Let us close this section by two final observations. The first is independent of Proposition 5.3 and complements a remark made in the beginning of Section 5. It will be used in Section 6 (Remark 6.8). 1,1 Remark 5.7. Let u ∈ Wloc (M) be such that Lu is a measure and let λ = |L(u)|. Observing that u is locally the difference of two L-potentials and using the quotient limit theorem (Theorem 8.1) reminded before the statement of Lemma 5.2, it is seen that u admits a (non-necessarily finite) representative (or version) uˆ that is finely continuous outside a polar and λ-negligible subset of M.

Remark 5.8. If the standard operator L is not assumed to be regular the results above (in particular Theorem 5.1) apply to every u which is locally a difference of two L-superharmonic 1,2 functions. This is in particular the case when u ∈ Wloc (M) and L(u) is a measure in M. Indeed in every open and relatively compact subset ω of M, we have u = w + h with h ∈ W 1,2 (ω) satisfying L(h) = 0 (so h is L-harmonic) and w ∈ W01,2 (ω) is such that Lw = −μ in ω. Since w ∈ W01,2 (ω) we have also L(w) = −μ in the weak sense of [34] and w = GL ω (μ) = L L Gω (μ+ ) − Gω (μ− ). Note also that thanks to Theorem A1.1 the first sentence in Remark 5.8 applies also if u ∈ W 1,p (M) for some p > 1, L(u) is a measure and L has continuous coefficients in any local C 1 chart. 6. Kato’s inequality up to the boundary In this section we will first assume that M is a C 1,α -manifold with α ∈ (0, 1] and that the standard elliptic operator L has C α -smooth coefficients. This means that in the representations (2.1), (2.3) of L with respect to a C α -smooth metric g the section A = Ag is locally Hölder continuous of exponent α. Equivalently at each point of M there is a chart in which the standard elliptic operator L is in the form L = i,j ∂i (aij ∂j ) with C α coefficients aij . 1,1 Let U be a C 1,α relatively compact open subset of M and let u ∈ Wloc (U ) be such that L(u) is a Radon measure in U . We precisely define the set {u > 0} ⊂ U as in Section 5: if u˜ is a representative of u in U which is finely continuous outside some polar subset of U , {u > 0} is the finely open subset of U of all point a ∈ U where u˜ admits a > 0 fine lower limit. In other words, considering u as an element of L1loc (U ), {u > 0} is the set of all points a ∈ U for which there exist ε > 0 and A ⊂ U thin at a such that u ε a.e. in U \ A. We may now state our main result. Two variants are given at the end of the section.

Theorem 6.1. Under the above assumptions on M, U and L, if u ∈ W 1,1 (U ) is such that λ := L(u) and ∂n u are finite measures—in U and ∂U respectively—, then L(u+ ) and ∂n (u+ ) are also

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finite measures and ∂n (u+ ) = 1{u>0} .(∂n u) − λ+ where λ+ is a positive measure concentrated on ∂U ∩ ∂fr {u > 0} where ∂fr means the fine boundary in U . More precisely we have ∂n (u+ ) = 1{u>0} .(∂n u) − 1{|u|>0}c (∂n u)−

(6.1)

Remark 6.2. In particular if ∂n u ∈ L1 (∂U ) then ∂n (u+ ) ∈ L1 (∂U ) and we have the following equality in L1 (∂U ): ∂n (u+ ) = (1{u>0} .∂n u) − 1{|u|>0}c (∂n u)− . Thus Theorem 6.1 solves open problems 1 and 2 of [10, Section 1].1 Remark 6.3. The proof will also show that L(u+ ) + ∂n u+ L(u) + ∂n u

(6.2)

which extends inequalities obtained by Brezis and Ponce in [10]. The proof of the first claim in Theorem 6.1 will be reduced to an application of Theorem 5.1. We start with the following elementary lemma (which as well as the next lemma is valid in the general context of Section 2, that is when M and U are C 1 and L is an arbitrary standard operator in M). Let ω, ω be two disjoint open subsets of M such that in the open region B ⊂ M, the set Σ := B \ (ω ∪ ω ) is a C 1 -hypersurface separating ω and ω . 1,1 Lemma 6.4. If v ∈ Wloc (B) is such that L(v) is a finite measure μ in B \ Σ and if one denotes ∂n (v), ∂n (v) the conormal derivatives (with respect to L) of v|ω∩B (resp. v|ω ∩B ) along Σ , then L(v) = μ − (∂n (v) + ∂n (v)) in the sense of distributions in B. In particular the distribution L(v) is a measure in B if and only if ∂n (v) + ∂n (v) is a measure in B (supported a priori by Σ ).

Proof. If ϕ ∈ Cc1 (B) is a test function in B, we have using the notations of Section 2: L(v)(ϕ) = −

A∇v, ∇ϕ dσ B

=−

A∇v, ∇ϕ dσ −

ω∩B

=

ω∩B

=−

ϕ∂n (v) +

Σ

ϕ ∂n (v) + ∂n (v) +

Σ

which is the desired result.

ϕ dμ −

A∇v, ∇ϕ dσ

ω ∩B

ω ∩B

ϕ dμ,

ϕ dμ −

ϕ∂n (v)

Σ

(6.3)

B

2

1 If uˆ is the L1 trace of u on ∂U , {u > 0} ∩ ∂U = {uˆ > 0} ∩ ∂U (mod negligible sets in ∂U ). Observing—see after Proposition 6.6—that we may assume u := GV θ in U , with θ a finite measure in V ⊃ U , the assertion easily follows.


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We will use the following consequence of Lemma 6.4. In the sequel we say that a C 1 diffeomorphism Φ : V → V between two open subsets of M leaves L invariant if L|V = Φ ∗ (L|V ) (see Section 2). In the next lemma we maintain the assumptions and notations of Lemma 6.4. It is also assumed that ω ⊂ B and ω ⊂ B. Lemma 6.5. Let Φ : B → B be an involutive C 1 -diffeomorphism (so Φ ◦ Φ = IdB ) applying ω 1,1 (ω ∪ Σ) is such onto ω , fixing every point of Σ = ∂ω ∩ B and leaving L invariant. If v ∈ Wloc 1,1 that ν := L(v) is a finite Radon measure in ω and if v˜ is the function in Wloc (B) obtained by extending v by symmetry (that is v(x) ˜ = v(Φ(x)) when x ∈ B \ ω) we have L(v) ˜ = ν + Φ ∗ (ν) − 2∂n (v).

(6.4)

Here ν is considered as a finite measure in B supported by ω, Φ ∗ (ν) is its direct image under Φ, ˜ is a measure in B if, and ∂n v is seen as a distribution in B (supported by Σ ). In particular L(u) and only if, ∂n (v) is a measure in Σ . 1,1 To check that v˜ ∈ Wloc (B), one can, using local charts, reduce itself to the case where B = d M = R , ω = {xd < 0}, ω = {xd > 0}, Σ = Rd−1 × {0} and where v˜ is compactly supported. It suffices then to observe that if v = lim vj in W 1,1 (ω), vj ∈ C ∞ (ω), supp vj ⊂ B(0, R) then v˜j ∈ W 1,1 (Rd ) and v˜j − v˜k W 1,1 = 2vj − vk W 1,1 (ω) . Thus v˜ is the limit of the sequence v˜j in W 1,1 (Rd ). Set v = v˜|U = v ◦ Φ. We have seen that L(v ) coincides in ω with the direct image measure ν = Φ(ν) of ν under Φ. Moreover by Definition 2.3 of the conormal derivative, we have for ψ ∈ Cc1 (B) and ϕ = ψ ◦ Φ,

(∂n v )(ψ) =

Ag (∇g v ), ∇g ψ dσg

ψ dν + ω

ω

=

ϕ dν +

ω

Ag (∇g v), ∇g ϕ dσg

ω

= (∂n v)(ϕ) = (∂n v)(ψ)

(6.5)

where we have used in the last line the fact that ∂n v|∂B vanishes on test functions which are null on ∂ω—see Remark 2.4. Whence ∂n v = ∂n v in B and the statement follows from the previous Lemma 6.4. Proof of Theorem 6.1. In most of what follows we will retain only the C 1 structures, and so use only the standard character of L (locally the “coefficients” of L are bounded measurable). We will return to the extra regularity assumptions (C 1,α regularity of M and U , and C α regularity of the “coefficients” of L) to establish Proposition 6.6 below; it is only there that they intervene and for a while it will be convenient to ignore them. Let us now proceed with the first step in the proof of Theorem 6.1. First part. Let us introduce a compact C 1 -manifold M˜ which is a double of the C 1 -manifold with boundary U : topologically it is obtained by gluing U with a copy U = U × {1} of U by

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identification of corresponding points of ∂U and ∂U . It is provided with a natural bicontinuous symmetry Φ : M˜ → M˜ such that Φ(x) = (x, 1), Φ(x, 1) = x, ∀x ∈ U . We may then fix a C 1 -differentiable structure on M˜ using the following known fact. There exists an open neighborhood V of ∂U in M and a C 1 -diffeomorphism (of C 1 -manifolds with boundaries) θ : V = V ∩ U → ∂U × [0, 1); Whitney’s theorem asserting the existence of a C ∞ structure on M compatible with its C 1 -structure [25] reduces us to a classical property (I owe this argument to J.-B. Bost). If W denotes the open collar V ∪ Φ −1 (V ) in M˜ and if s is the natural symmetry (x, t) → (x, −t) of the C 1 -manifold N = ∂U × (−1, 1), there exists a unique C 1 -structure on M˜ satisfying the following: (i) the map θ˜ : W → N equal to θ on V and such that θ˜ ◦ Φ = s ◦ θ˜ is a C 1 diffeomorphism, (ii) this structure coincides with the initially given structure in U and Φ : U → U is a C 1 -diffeomorphism. ˜ the initial C 1,α -manifold with boundary U is a C 1 submanifold For this C 1 -structure in M, with boundary of M˜ and Φ is an involutive C 1 -diffeomorphism of M˜ such that Φ ◦ Φ = IdM˜ and Φ(U ) = U . There is not uniqueness in general of the C 1 -structure that has been so obtained, but the induced Lipschitz structure is unique and much easier to define. We may then fix a Φ invariant C 0 -metric in M˜ (take any C 0 -metric g in M˜ and set g0 = g + Φ ∗ (g) for example). We may also extend L|U to a Φ-invariant standard second order elliptic operator in M˜ (cf. Section 2.2): if L is associated to the section A = Ag with respect to g0 in U , it suffices to extend A to a Φ-invariant measurable section of End(T (M)) with A(x) = Id for ˜ x ∈ ∂U (the values of A on ∂U are unimportant since ∂U is negligible in M). We will now exploit the regularity assumptions of Theorem 6.1 to establish the following proposition. ˜ Proposition 6.6. The operator L˜ is regular in M. ˜ Proof. It is plain that L˜ is regular in U since L is regular and L˜ = L in U . And since L˜ = Φ ∗ (L) ˜ it is clear that L˜ is regular in M˜ \ Σ where Σ denotes the boundary of U in M. It remains to show that L˜ is regular in a neighborhood of each point m0 ∈ Σ . By assumption, since U is a C 1,α -submanifold with boundary of M, there is a chart ϕ : V ∩ U → BR+ = BR ∩ Rd+ which is C 1,α for the initial structure in U and transforms L˜ |U ∩V into a standard elliptic operator with α-Hölder continuous coefficients in BR+ . Extending ϕ by symmetry, one gets a bilipschitz homeomorphism ϕ˜ : V → BR transforming L˜ into a standard elliptic operator L in BR to which Proposition 3.3 applies. This operator L is thus regular in BR and by regularity invariance under bilipschitz homeomorphism we see that L˜ is regular in a neighborhood of m0 . 2 ˜ To establish Theorem 6.1, we thus may (and will) from now on assume that (i) M = M, ˜ (ii) U being seen as a C 1 -open subset of the C 1 -manifold M˜ and ∂U as its boundary in M, L = L˜ is regular Φ-invariant. Proof of Theorem 6.1. Continuation. Denote by u˜ the extension by symmetry of u: u(x) ˜ = 1,1 (M) u(Φ(x)) for x ∈ U . Since Φ is a C 1 -automorphism, Lemma 6.5 says that u˜ ∈ Wloc (= W 1,1 (M), M being now compact) and that L(u) ˜ is a measure in M. Combining now the precise form of Kato’s inequality (Theorem 5.1) and Lemma 6.5, we 1,1 (M) is will obtain the first assertion of Theorem 6.1. Indeed, L(u) ˜ is a measure so u˜ + ∈ Wloc


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such that L(u˜ + ) is a measure in M and one has L(u˜ + ) = 1{u>0} .L(u) ˜ + λ˜ + where λ˜ + is a ˜ positive Radon measure in M supported by ∂f (u˜ > 0). Thus by Lemma 6.5 the distribution ∂n u+ is a Radon measure supported by ∂U and one has L(u) ˜ = 1U ∪Φ(U ) L(u) ˜ − 2∂n (u), L(u˜ + ) = 1U ∪Φ(U ) L(u˜ + ) − 2∂n (u+ ). Passing to traces on ∂U we get ∂n (u+ ) = 1{u>0} .∂n (u) − 12 1∂U λ˜ + . This establishes the first claim of Theorem 6.1 with λ+ = 12 1∂U λ˜ + . To prove the second claim in Theorem 6.1 let us first notice that by considering −u we also have ∂n u− = −1{−u>0} ∂n u − λ− where λ− is a finite positive measure supported by ∂U ∩ ∂fr {−u > 0}. Moreover we know from Theorem 5.1 that λ+ is “locally" dominated by the swept-out on {u˜ > 0}c of a finite positive measure supported by {u˜ > 0}. More precisely (see Remark 5.5) for each x0 ∈ ∂U every admissible open neighborhood V of x0 in M, λ+ is near x0 smaller than the swept-out (w.r. to V ) on {u˜ > 0}c ∩ V of a positive measure concentrated in {u˜ > 0} ∩ V and with compact support in V . Similarly λ− is smaller in the vicinity of x0 than the swept-out on {−u˜ > 0}c ∩ V of a positive measure concentrated in {−u˜ > 0} ∩ V and compactly supported in V . Proposition 6.9 stated and established below will show that λ+ ∧ λ− = 0. The second claim of Theorem 6.1 follows by observing that since ∂n u = ∂n (u+ ) − ∂n (u− ) = −λ+ + λ− on the set ∂U ∩ {|u| > 0}c one has λ+ = 1{|u|>0}c .[∂n u]− . The proof of Theorem 6.1 is then complete. 2 Remark 6.7. Let us observe that at this stage Remark 6.3 easily follows from Remark 5.6—i.e., the case where u is compactly supported in U : indeed using the above and applying Lemma 6.5

L(u˜ + ) = 2 1U ∩{u>0} L(u) + ∂n u+

(6.6)

˜ by and similarly L(u) ˜ = 2{1U L(u) + ∂n u}. Whence the result since L(u˜ + ) L(u) Remark 5.6. Remark 6.8. In view of the next section, let us also notice that an application of Remark 5.7 to u˜ shows that the function u admits in U a (non-necessarily finite) finely continuous representative outside a Borel polar subset which is also negligible with respect to |∂n u| + 1U |L(u)|. In order to work now with an admissible (with respect to L) connected manifold we assume as we may that U is connected and consider from now on M = M \ (T1 ∪ T2 ) where T1 is a compact subset with nonempty interior in U and T2 = Φ(T1 ) is its symmetric image. The next proposition relies on the C 1 -regularity of the hypersurface Σ = ∂U . Proposition 6.9. Let V , W be two finely open disjoint and Φ-invariant subsets of M and let μ, ν be two finite positive measures supported by V and W respectively (i.e., μ∗ (V c ) = ν ∗ (W c ) = 0). c c Let μ (resp. ν ) be the trace on Σ = ∂U of the swept-out measure μV (resp. ν W ) in M . Then μ ∧ν =0 Proof. Adding to V the set of all points of M where M \ V is thin and modifying similarly W we may assume that V and W are Borel sets (and even Kσ sets, cf. [6,15]). Arguing by contradiction and assuming that μ ∧ ν = 0 there exists a compact set K ⊂ ∂U \ V ∪ W which does not separate M and is such that the traces of μ and ν on K are non-vanishing mutually absolutely continuous positive measures.

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Since 1K .μ is smaller than the swept-out measure μK in M and since εxK is the L-harmonic measure of x in M \ K, we see that the harmonic measure class with respect to M \ K does not vanish and dominates the class of μ (or ν ) on K. To pursue, we will consider the Martin boundary of Ω := M \ K (w.r. to L) and use several properties known in the case at hand (K contained in a Lipschitz hypersurface of M ). For the Martin boundary theory, the reader is referred to [29,31,15] and the exposition [3]. Recall that this whose main part consists theory associates to each admissible region Ω of M a boundary ∂Ω of the “minimal” boundary points. Having fixed a reference point x0 ∈ Ω, to each minimal point corresponds on the one hand a unique positive L-harmonic function Kζ in Ω which is ζ ∈ ∂Ω minimal and normalized at x0 , and on the other hand a notion of “minimal thinness” at ζ : A ⊂ Ω A ≡ K (the réduite is performed with respect to the domain Ω). is minimally thin at ζ if RK ζ ζ A point a ∈ Ω is a pole of ζ if, for all r > 0, the set Ω ∩ B(a, r) is not minimally thin at ζ , Ref. [31]. We will use here a variant of the following well-known property. Let F be a closed subset of Ω, let ν be the harmonic measure in Ω \ F of some point a ∈ Ω \ F . Then ν-almost every such that F is minimally point z ∈ ∂Ω is the unique pole of at least one minimal point ζ ∈ ∂Ω thin at ζ [31, p. 247 and Chapter V]. The simple variant we need is stated in the next lemma. Lemma 6.10. Let V be a finely open subset of M , let K be a compact subset of ∂f V not separating M , let μ be a finite positive measure supported by V and let μ denote the trace on K of the swept-out measure (in M ) of μ on V c . Then with respect to Ω := M \ K, μ -almost all x ∈ K is the unique pole of at least one minimal point ζ in the Martin boundary of Ω such that Ω \ V is minimally thin at ζ . Let us sketch for the reader’s convenience a proof of Lemma 6.10. It is easily seen that we may assume that V is relatively compact and by adding to V a polar subset that V c is thin at no point of V c ∩ Kc . Then V is an ordinary Fσ set (cf. [6] or [16]). c Let L be a compact subset of K such that μ (L) > 0. The function x → u(x) := εxV (L)— sweeping with respect to M —is subharmonic in M \ L (see Proposition 8.5 below). It vanishes quasi everywhere in V c \ L and 0 u 1. Moreover u ≡ 0 since V u(x) dμ(x) = Vc Vc V εx (L) dμ(x) = [ V εx dμ(x)](L) = μ (L) > 0). It follows that in Ω = M \ K the function h(x) = ω(x; L; Ω) (the harmonic measure of x w.r. to Ω and L) is not stable by reduction—with respect to the domain Ω—on V c ∩ Ω. Indeed by the maximum principle, we have u h in Ω (note that h = lim ↓ sn where the sn are Lpositive superharmonic in Ω and lim inf sn 1 at every point of L) so h − u is nonnegative Ω\V L-superharmonic, h − u = h q.e. on V c ∩ Ω and [Rˆ h ]Ω h − u. Now, if ω denotes the harmonic measure on the minimal Martin boundary of the fixed nor malization point x0 ∈ Ω, we have h = π −1 (L) Kζ dω(ζ ) (here π(ζ ) is the unique pole of ζ when c c c it exists2 ). Thus Rˆ V = −1 Rˆ V dω(ζ ). As Rˆ V ≡ h, the set AL of all points ζ in π −1 (L) h

π

(L)

Kζ

h

where V c is minimally thin has positive harmonic measure. In other words, the set of all points for which Ω \ V is x ∈ L such that x is the unique pole of at least one minimal point ζ ∈ ∂Ω minimally thin at ζ has > 0 harmonic measure. 2 2 We need only the case where K is contained in a C 1 hypersurface. Then every Martin point over K has a well-defined pole [2].


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Continuation of the proof of Proposition 6.9. In our case, the compact subset K is contained in a C 1 -hypersurface and one has a rather precise description of the part X of the minimal Martin boundary of M \ K lying above K ([2, Sections 7 and 8]—see generalizations in [1,5] and references therein). In particular (being of local nature the results in [2] extend to the setting of C 1 manifolds) there is a continuous projection π of X onto K, which associates to each point ζ ∈ X its unique pole x ∈ K, each point x ∈ K being a pole of one or two minimal points (compare also with the striking general result in [37] about triple points)—in the first case the point x is said to be simple and in the other case it is a double point. Moreover when x is a double point, Φ exchanges the two minimal points above x. Indeed the arguments in [12] show that from the Harnack boundary principle of [2] it follows that: (a) each sequence {xn } in M \ K converging non-tangentially to some point z ∈ K, admits only minimal points as cluster values (b) every minimal point ζ associated to z ∈ K is the limit of such on the Martin boundary ∂Ω, a sequence. In particular for a connected subset C ⊂ M \ ∂U which is non-tangential for ∂U at is reduced to one minimal boundary point. z ∈ K ∩ C, the cluster set C ∩ ∂Ω We note here that the symmetry of the elliptic operator L is used again since the proof of the main result in [2, Section 7] (and final remark in Section 8) relies in an essential way on the symmetry of the elliptic operators under consideration. We now deduce the following consequence using the invariance of L and V under Φ. Consequence. If under the assumptions of Lemma 6.10 it is assumed moreover that V is Φsymmetric and K ⊂ ∂U , then for μ -a.a. x ∈ K, M \ V is minimally thin at each minimal point with pole x. For if x is simple, the claim is already contained in Lemma 6.10, and if x ∈ K admits two corresponding minimal points in the Martin boundary of Ω and is such that Ω \ V is minimally thin w.r. to one of these points, M \ Φ(V ) = M \ V is also minimally thin with respect to the other minimal point. Conclusion. Proposition 6.9 is now established since for μ ∧ ν almost all points x ∈ K the two subsets M \ V and M \ W are both minimally thin at each point in π −1 (x) which is impossible. We have thus reached a contradiction. 2 We now state a variant of Theorem 6.1, where using a stronger assumption on u, the problems related to the non-regularity of L vanish so that the smoothness assumptions on M and U can be notably relaxed. Theorem 6.11. Let M be a C 1 -manifold, let L be a standard second order elliptic operator in M, and let U be a Lipschitz relatively compact open subset of M. If u ∈ W 1,2 (U ) is such that L(u) and ∂n (u) are finite measures, L(u+ ) and ∂n (u+ ) are also finite measures and the following formula holds ∂n (u+ ) = 1{u>0} .(∂n u) − 1{|u|>0}c (∂n u)− .

(6.7)

Observe first that we may assume U to be C 1 -smooth (assumptions and results are of local nature and invariant under bilipschitz homeomorphism). The point is then (see Remark 5.8) that 1,2 (Ω) such that μ = −L(v) is a Radon measure in Ω, can for Ω open in M, an element v ∈ Wloc be written locally as the difference of two L-superharmonic functions—even without assuming

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that L is regular. This remark applies to u and u+ and an inspection of the arguments used above shows that the proof of Theorem 6.1 extends to the case at hand (and can also be made simpler—Proposition 6.6 being now superfluous). Remark 6.12. Let us also notice another variant of Theorem 6.1 which can be proved along the same lines (using now Remark 3.4 instead of Theorem 3.3) and which is based on Brezis improvement (Theorem A1.1 in Appendix A) of Hager and Ross result [20]. Let M, U and L be as above in Theorem 6.11, U being C 1 -smooth and L having continuous coefficients. If u ∈ W 1,p (U ) with p > 1 is such that L(u) and ∂n (u) are finite measures, then the conclusions in Theorem 6.11 still hold. Added in proofs. One may extend [10, Theorem 1.2] to our framework as follows. Assumptions and notations are as in the beginning of Section 6. Proposition 6.13. If u ∈ W01,1 (U ) is such that L(u) is a measure of finite total mass in U , then ∂n u is an absolutely continuous Radon measure on ∂U (i.e. ∂n u ∈ L1 (∂U )). Moreover (i) L(u+ ) L(u), (ii) ∂n u L(u) and (iii) if u 0, then ∂n u 0. Proof. Set μ := −L(u) and denote GU the L-Green’s function in U . We know [34] that GU μ ∈ W01,1 (U ) and therefore by the uniqueness principle Theorem A5.1 in Appendix A and [30, Chapter 5] we have that u = GU μ (Lipschitz regularity for U suffices here). To prove the first claim we may assume that μ is positive. Then, if V is an open neighborhood of U , writing GU μ = GV μ − GV μ where μ is the swept-out of μ on V \ U in V , it is easily checked using the definitions that ∂n u = −μ . Now μ is the limit of an increasing sequence {μp } of positive measures with compact supports in U and since μp − μ → 0, up = GU μp → u in W01,1 (U ) and ∂n up decreases to ∂n u as p → ∞. Since up is C 1,α in a neighborhood of ∂U in U , ∂n up is absolutely continuous (and coincides with the standard conormal derivative if one fixes a C α -metric in M). Hence ∂n u ∈ L1 (∂U ). To prove (ii), write u = GU μ+ − GU μ− . Since ∂n u = −μ+ + μ− , μ± μ± we obtain (ii). Taking U as the ambient manifold, setting W := {u > 0}, W = {−u > 0}, and using Theoc c rem 5.1 and Lemma 5.2, we have Lu+ = −1W (μ+ − μ− ) + λ, with λ = [1W μ+ ]W − [1W μ− ]W and λ is positive and supported by A = U \ (W ∪ W ). Using the similar formulas for u− and the c c relation u = u+ − u− , we see that λ = −1A (μ+ − μ− ) + (−[1W μ+ ]W + [1W μ− ]W ). Since sweeping-out decreases total masses, (i) easily follows. Finally, if u 0 in U , Theorem 5.1 and Lemma 5.2 yield μ+ − μ− 0 (using the same notations as above). Whence (iii) (which can also be deduced from Theorem 6.1). 2 7. An application and extension of Theorem 6.1 We return here to the assumptions of the beginning of Section 6. In particular U is a relatively compact C 1,α -smooth open subset of M and u ∈ W 1,1 (U ) satisfies the following conditions: (i) L(u) is a finite measure in U and (ii) ∂n (u) is a finite measure in ∂U . Recall (see Remark 6.8) that in U , u admits a representative which is finely continuous outside a Borel polar subset N of U , N being moreover negligible with respect to the measure λ := 1U |L(u)| + |∂n u| (a Radon measure in U ). We fix such a representative which will still be denoted u and observe that up to


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λ-negligible sets the usual sets {u > θ }, θ ∈ R, coincide with the precise sets {(u − θ ) > 0} as defined in Section 6. Similarly for the sets {u = θ } = {|u − θ | > 0}c . Let now f : R → R be a continuous function in R whose second derivative in the sense of distributions is a Radon measure in R with finite total mass. Thus the right and left derivatives fd and fg exist everywhere, have finite total variations and {t ∈ R; fg (t) = fd (t)} is at most enumerable. Moreover by taking limits fg (±∞) and fd (±∞) will be considered as welldefined reals: fd (+∞) = fg (+∞) = limtθ λ − 1u=θ λ− } dν(θ ) +

{−1u0

+ 1u0 {fg (u)λ+ − fd (u)λ− } + 1u 0, the finely closed set FC = {p Cq} is thin at μ-almost all a ∈ A. We know that the set of points where FC is not thin (this set is called the basis of FC ) is an ordinary Gδ . If FC is unthin at each point of the compact set K ⊂ A, if {Kn } is a decreasing sequence of compact neighborhoods of K shrinking to K in M, if μ = 1K μ and Kn ∩FC dν(y) q. Since for each if p = G(μ ), we have p = RpKn ∩FC C Rˆ qKn ∩FC = C Rˆ G y K n y∈ / K, Rˆ decreases to 0 outside K when n → ∞, we see that p (x) = 0 for all x ∈ M \ K Gy

such that q(x) < ∞. This means that p ≡ 0 and so μ(K) = 0.


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A.2. The finely open set V := {u > 0} (Sections 5–7) is an Fσ set in M: for if Bn is the set of points where the set (defined up to a polar set) {u n1 } is unthin, the complement V c is the intersection n1 Bn . But Bn is a basis (Bn is equal to the set of points where it is unthin) and so is a Gδ set [6,15]. The fine boundary ∂f V = ∂f {u > 0} is also a Gδ since ∂f V = {u > 0}f \ {u > 0} and {u > 0}f is the basis set {u > 0}. A.3. Next we consider balayage and start with the following simple lemma. Lemma 8.2. Let p, q be two nonnegative L-superharmonic functions in the region U of M. If c c p q on the compact subset K of U , then RpK RqK in K. Proof. It is well known that there exists a strictly increasing sequence {qn }n0 of continuous functions in S+ (U ) such that q = supn1 qn . Then {p > qn } is an open set Un containing K and c

taking U open and such that K ⊂ U ⊂ U Un the minimum principle gives that RpK RqUn c c c in U . Letting U decrease to K one gets RpK RqKn = Rˆ qKn in K. Letting then n go to infinity the desired result follows. 2 c

Theorem 8.3. (See [15, pp. 183–186], [13].) If A ⊂ M is thin at x the swept-out measure εxA is concentrated on the fine boundary ∂f (A) of A. Replacing A by its basis, we may assume that A is a basis (in particular A and ∂f (A) = A ∩ b(Ac ) are Gδ sets). If p ∈P(M) and μ = −L(p) (so p = Gμ ), we have (for arbitrary A (x) = R A dμ = G dμ = G dε A and since R A is stable by reduction on x ∈ M) RG μ εxA x Gx Gμ μ A A . Taking for p a strict potential A we obtain by replacing p by RpA that Gμ dεxA = RG dε x μ ([7], [15, p. 180], [16]) it is known that M \ A = {RpA < p} and so we get εxA (M \ A) = 0. Thus it have been shown that for any set B ⊂ M thin at x, εxB is supported by the fine closure of B. It remains to see that for x ∈ / A, εxA does not charge the fine interior V of a basis A. Set c A p = RGx . By Lemma 8.2 applied to p and Gx we have p = RpK in M for every compact K ⊂ A. This means that εxA is equal to its swept-out in K c . So by the above εxA does not charge the fine interior of K. This gives the desired result (since V = {q > RqA = Rˆ qA } if q ∈ P(M) is continuous and strict [15]). Lemma 8.4. Let p, q be two L-superharmonic functions in the open subset U of M. If p = q on a finely open subset V of U then the measures L(p) and L(q) coincide in V . Proof. By the assumptions p = q on the finely open set V = {x; V c is thin at x} which contains V . Thus one may assume that V c is a basis (in particular an ordinary Gδ set). Since the properties in the statement are of local nature we may also assume that p and q are L-potentials in U . By definition of the réduites Rˆ pV = RpV = RqV = Rˆ qV , so L(Rˆ pV ) = L(Rˆ qV ). Since (εx )V = εx when x ∈ V we obtain that (1V c .L(p))V + 1V .L(p) = (1V c .L(q))V + 1V .L(q). But for x ∈ / V, the balayée (εx )V is supported by the fine boundary ∂f V of V and thus vanishes in V . Restricting to V in the previous relation we get 1V .L(p) = 1V .L(q). 2

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A.4. Finally we turn to a property used in the proof of Lemma 6.10 and for which a reference seems difficult to locate. Proposition 8.5. Let A ⊂ M and let L be a compact subset of A. Then u(x) := εxA (L) is an L-subharmonic function in M \ L. Replacing A by its fine closure we may and will assume that A is finely closed. Let then π := Rˆ 1L be the equilibrium potential of L in M, and let R˜ denote the réduite operator with respect to U := M \ L (i.e. R˜ sB = inf{w ∈ S+ (U ); w s in B}). Then, u(x) = π(x) − R˜ πA∩U (x),

x ∈ U,

(8.1)

which implies the result since R˜ πA∩U is L-superharmonic in U and π is L-harmonic in U . To prove (8.1) we remark another formula about the reduites: if s ∈ Pc (M), RsA = Rˆ sL + R˜ A∩Uˆ L , s−Rs

x ∈ U,

(8.2)

which follows at once from Lemma 8.6 below. Now, (8.2) means that

s dεxA =

s dεxL +

s d ε˜ xA∩U −

sdεyL d ε˜ xA∩U (y),

x ∈ U,

(8.3)

U

which then also holds for s ∈ Pc (M) − Pc (M). Since 1L is the limit of a decreasing sequence of such sj (with supports shrinking to L), we get letting j → ∞ εxA (L) = π(x) −

π d ε˜ xA∩U ,

x ∈ U,

which is exactly (8.1). Lemma 8.6. Let p ∈ P(M) then p − Rˆ pL ∈ P(U ). If moreover L(p) is concentrated in the Borel set B and p is locally bounded, then p − Rˆ pL = R˜ B∩Uˆ L . p−Rp

Let h be nonnegative L-harmonic in U and such that h p − Rˆ pL . Note again h its extension by zero outside U . Choosing w ∈ S+ (M) such that w(x) = +∞ for all x ∈ L where L is thin, clearly that (h − εw)+ is subharmonic in M and less than p (for every ε > 0). So (h − εw)+ = 0 and letting ε → 0, h = 0 in U . This proves that p − Rˆ pL ∈ P(U ). The second claim follows then by the domination principle [6,15,13].


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Appendix A. Solution of a conjecture by J. Serrin by Haïm Brezisa,b,3 a Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA b Department of Mathematics, Technion, 32.000 Haifa, Israel

A1. Introduction 1,1 Let Ω ⊂ RN , N 2, be a bounded domain and let u ∈ Wloc (Ω) be a weak solution of the equation

∂ ∂u aij = 0 in Ω, ∂xj ∂xi

(A1.1)

i,j

where the coefficients aij (x) are bounded measurable and elliptic, i.e., λ|ξ |2

aij (x)ξi ξj Λ|ξ |2 ,

x ∈ Ω, ξ ∈ RN ,

i,j 1,1 with 0 < λ Λ < ∞. A weak solution u ∈ Wloc (Ω) satisfies, by definition,

aij

i,j

∂u ∂ϕ = 0 ∀ϕ ∈ Cc1 (Ω), ∂xi ∂xj

(A1.2)

where the subscript c indicates compact support. A celebrated result of E. DeGiorgi [14] asserts that if u is a weak solution of (A1.1) and more1 (Ω), then u is locally Hölder continuous, and in particular u ∈ L∞ (Ω) (see also over u ∈ Hloc loc [35]). Subsequently J. Serrin produced in [32] a striking example showing that the assumption 1 (Ω) is essential; more precisely, for every p, 1 < p < 2, and all N 2, he constructed u ∈ Hloc an equation of the form (A1.1) which has a solution u ∈ W 1,p (Ω) and u ∈ / L∞ loc (Ω). J. Serrin conjectured in [32] that if the coefficients aij are locally Hölder continuous, then any weak solu1,1 1 (Ω). Serrin’s conjecture tion u ∈ Wloc (Ω) of (A1.1) must be a “classical” solution, i.e., u ∈ Hloc was established by R.A. Hager and J. Ross [20] provided u is a weak solution of class W 1,p (Ω) for some p with 1 < p < 2. We present here the solution of Serrin’s conjecture in full generality, starting with u ∈ 1,1 Wloc (Ω), or even with u ∈ BVloc (Ω), i.e., u ∈ L1loc (Ω) and its derivatives (in the sense of distributions) are measures. ¯ The first result is an improvement of the theorem of Hager and Ross: instead of aij ∈ C 0,α (Ω) 0 ¯ for some α ∈ (0, 1), we assume only aij ∈ C (Ω). 3 E-mail address: [email protected].

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¯ and u ∈ W 1,p (Ω) for some p > 1. If u is a weak solution Theorem A1.1. Assume aij ∈ C 0 (Ω) 1,q of (A1.1), then u ∈ Wloc (Ω) for every q < ∞. Moreover uW 1,q (ω) CuW 1,p (Ω) , for every ω Ω, where C depends only on N , λ, Λ, p, q, ω, Ω, and the modulus of continuity ¯ of aij on Ω. Open problem 0. We do not know whether the conclusion of Theorem A1.1 holds in the two limiting cases: p = 1 and/or q = ∞ . (The answer to both questions is positive if the coefficients aij are Dini continuous; see Theorem A1.2 below). We now turn to Serrin’s conjecture. Here we assume that the coefficients aij are Dini contin¯ i.e., aij ∈ C 0 (Ω), ¯ and uous in Ω, A(r) =

aij (x) − aij (y),

sup

r > 0,

(A1.3)

i,j x,y∈Ω,|x−y| 1 (since 1,r (Ω), ∀r < ∞), but the value t = 1 is not admissible since we do not know whether v ∈ Wloc 1,∞ v ∈ Wloc (Ω). Fix ζ ∈ Cc∞ (Ω) with ζ = 1 on ω. We may choose ϕ = ζ u in (A2.8) (here we use the assumption u ∈ W 1,p (Ω) and p > 1). This yields ∂u ∂u ∂ζ ∂v ∂ζ . (A2.9) aij ζ +u = fj ζ +u ∂xi ∂xi ∂xj ∂xj ∂xj i,j Ω

j Ω

On the other hand, by (A1.2), and a density argument we have ∂u ∂w 1,p aij = 0 ∀w ∈ Wc (Ω). ∂xi ∂xj

(A2.10)

i,j Ω

1,r Next we choose w = ζ v in (A2.10) (this w is admissible since v ∈ Wloc (Ω) ∀r < ∞ and p < ∞; here we use once more the assumption p > 1). We obtain ∂v ∂u ∂ζ ζ = 0. (A2.11) aij +v ∂xi ∂xj ∂xj i,j Ω

Comparing (A2.9) and (A2.11) we find ∂u ∂u ∂ζ ∂ζ ∂v ∂ζ ζ fj = − aij v + aij u − fj u ∂xj ∂xi ∂xj ∂xi ∂xj ∂xj j Ω

i,j Ω

i,j Ω

= I + II + III.

j Ω

(A2.12)

Recall that p < 2 N and, by the Sobolev embedding, uLp (Ω) CuW 1,p (Ω) ,

(A2.13)

1 1 1 = − . p p N

(A2.14)

where

Finally we choose s ∈ ( NN−1 , 2] according to the following dichotomy: (a) When p 2 we choose s = p . Note that p > NN−1 because p > 1. Then we have, since s = p , 1 1 1 1 1 1 1 =1− = + − = − , p p p s p s N

(A2.15)


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and by the Sobolev embedding we have vLp (ω) CvW 1,s (ω) , which is valid since s =

p p −1

(A2.16)

< N (because p > 1). Therefore, from (A2.7), vLp (ω) C

(A2.17)

|I | CuW 1,p (Ω) ,

(A2.18)

and thus (with ω ⊃ Supp ζ ),

where I is defined in (A2.12). Next we have, by (A2.13) (with s = p ), |II| CuW 1,p (Ω) vW 1,s (ω) CuW 1,p (Ω)

by (A2.7).

(A2.19)

Finally |III| C

fj Ls (Ω) uLs (Ω)

j

CuW 1,p (Ω)

(A2.20)

by (A2.13) and the choice s = p . Combining (A2.12), (A2.18), (A2.19) and (A2.20) yields ∂u ζ fj CuW 1,p (Ω) , ∂xj j Ω

for every (fj ) in Cc∞ (Ω) satisfying (A2.3) (where the constant C depends on ζ ). ∂u Therefore ζ ∂x ∈ Ls (Ω) and j ∂u ζ ∂x

j

Ls (Ω)

CuW 1,p

∀j.

In particular, u ∈ W 1,p (ω) with uW 1,p (ω) CuW 1,p (Ω) .

(A2.21)

(b) When p > 2 we choose s = 2. Then vH 1 (Ω) C

by (A2.6)

(A2.22)

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and thus vLp (Ω) C

(A2.23)

since p < 2 when N 3 (this is equivalent to p > 2) and p < ∞ when N = 2 (here we use once more the assumption p > 1). From (A2.22) and (A2.23) we deduce that |I | + |II| + |III| CuW 1,p (Ω) , because uL2 (Ω) CuLp (Ω) CuW 1,p (Ω) . We now conclude as above that u ∈ H 1 (ω) with uH 1 (ω) CuW 1,p (Ω) .

(A2.24) 1,p

Iterating the preceding argument of case (a) in the dichotomy yields u ∈ Wloc (Ω), u ∈

1,p

Wloc (Ω), etc. until we reach the first value bigger than 2. At that point we use part (b) of the dichotomy. Thus we have proved that any u ∈ W 1,p (Ω) with 1 < p < 2 satisfying (A1.1), must belong to 1 Hloc (Ω) and uH 1 (ω) CuW 1,p (Ω) . 1,q

Applying once more Lemma A2.1 with fj = 0 ∀j , gives u ∈ Wloc (Ω) ∀q < ∞ and uW 1,q (ω) CuW 1,p (Ω) , and the proof of the theorem is complete.

2

A3. Proof of Theorem A1.2 For the proof of Theorem A1.2 we will need the following extension of the Schauder regularity theory for elliptic equations in divergence form with Dini continuous coefficients: ¯ satisfy (A1.3)–(A1.4) and let u ∈ H 1 (Ω) be a weak solution Lemma A3.1. Assume aij ∈ C 0 (Ω) ∞ of (A2.1) with fj ∈ Cc (Ω) ∀j , then u ∈ C 1 (Ω). The conclusion of Lemma A3.1 comes with an estimate of the Dini modulus of continuity of Du involving the Dini modulus of continuity of aij . However we do not need such an estimate—we use only the qualitative form of Lemma A3.1; this explains Remark 1. It is not easy to find an early reference for Lemma A3.1. According to the experts (I am quoting M. Giaquinta), it was common knowledge in Pisa in the late 60s—the proof being based on Campanato’s approach to Schauder estimates (as presented in [19], or [11]), combined with a result of S. Spanne (Corollary 1 in [33]). A complete proof may be found e.g. in [28, Theorem 5.1]. Y. Li [27] has


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obtained a similar conclusion (also valid for systems) under weaker assumptions on the coefficients aij . Proof of Theorem A1.2. We follow the same duality strategy as in the proof of Theorem A1.1. We start with a weak solution u ∈ BV (Ω) of (A1.1). We fix some 1 < s < NN−1 , so that 2 N < s < ∞. Let (fj ) in Cc∞ (Ω) with

fj Ls (Ω) 1.

(A3.1)

j

Let v ∈ H01 (Ω) be the solution of (A2.5). By Lemma A3.1 we know that v ∈ C 1 (Ω). Clearly vH 1 (Ω) C

fj L2 (Ω) C

since s > 2,

(A3.2)

j

and by Lemma A2.1 (which uses only continuous coefficients aij ) we have vW 1,s (ω) C.

(A3.3)

On the other hand we know from the DeGiorgi–Stampacchia theory (which uses only bounded measurable coefficients aij ) that v ∈ L∞ (Ω) since s > N ; see e.g. [35] and [21]. Moreover vL∞ (Ω) C

fj Ls (Ω) C.

(A3.4)

j

By density we have i,j Ω

aij

∂ϕ ∂v = ∂xi ∂xj

fj

j Ω

∂ϕ ∂xj

∀ϕ ∈ BVc (Ω),

(A3.5)

and we choose ϕ = ζ u in (A3.5) with ζ as in the proof of Theorem A1.1. This gives (A2.9). On the other hand we may choose ϕ = ζ v in (A1.2). This gives (A2.11), which yields (A2.12). Next we have |I | CuBV (Ω) vL∞ (Ω) CuBV (Ω)

by (A3.3),

|II| CuLN/N−1 (Ω) vW 1,N (ω) CuL(N/N−1) (Ω) CuBV (Ω) by (A3.3) since s > N, uLN/N−1 (Ω) fj LN (Ω) CuBV (Ω) by (A3.1) since s > N. |III| C j

We conclude that ∂u fj CuBV (Ω) ζ ∂xj j

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∂u for every f = (fj ) in Cc∞ (Ω) satisfying (A3.1). Therefore ζ ∂x ∈ Ls (Ω) and, in particular, j

u ∈ W 1,s (ω) with uW 1,s (ω) CuBV (Ω) . 1 (Ω) with We may now apply Theorem A1.1 and conclude that u ∈ Hloc

uH 1 (ω) CuBV (Ω) ,

(A3.6)

where C depends only N, λ, Λ, ω, Ω and the modulus of continuity of aij on Ω¯ (and not the Dini modulus of continuity of aij ). A4. More on the Open problem 0 As we already mentioned briefly in the Introduction there are several open problems related to Serrin’s conjecture. ¯ and u ∈ W 1,1 (Ω) is a weak solution of (A1.1). Is it true Open problem 1. Assume aij ∈ C 0 (Ω) 1 that u ∈ Hloc (Ω)? Does one have an estimate of the form uH 1 (ω) CuW 1,1 (Ω) ,

(A4.1)

for every ω Ω where C depends only on N, λ, Λ, ω, Ω and the modulus of continuity of aij ¯ Same questions when u ∈ BV (Ω) instead of W 1,1 . on Ω? ¯ and u ∈ H 1 (Ω) is a weak solution of (A1.1). Is it true Open problem 2. Assume aij ∈ C 0 (Ω) 1,∞ 1 that u ∈ Wloc (Ω) (resp. u ∈ C (Ω))? Does one have an estimate of the form uW 1,∞ (ω) CuH 1 (Ω) ,

(A4.2)

¯ for every ω Ω where C depends on N, λ, Λ, ω, Ω and the modulus of continuity of aij on Ω? Note that estimate (A1.5) in Theorem A1.2 gives some evidence in favor of a positive answer to Open problem 1. (We do not have any evidence in favor of a positive answer to Open problem 2.) A natural strategy in trying to solve Open problem 1 is to smooth the coefficients aij by aijε preserving the ellipticity and the modulus of continuity. Following the proof of Theorem A1.1 we have ∂u ∂w aij = 0 ∀w ∈ Cc1 (Ω). (A4.3) ∂xi ∂xj i,j Ω

Fix some s such that 1 < s
1. If u is a weak solution Theorem A5.1. Assume aij ∈ C 0 (Ω) 0 of (A5.1), then u ≡ 0. The same conclusion holds if u ∈ W01,1 (Ω) provided the coefficients aij ¯ are Dini continuous on Ω. 1,p


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Here is a question reminiscent of Open problem 1: ¯ and let u ∈ W 1,1 (Ω) satisfy (A5.1). Is it true that u ≡ 0? Open problem 3. Assume aij ∈ C 0 (Ω) 0 Theorem A5.1 is an immediate consequence of the following: ¯ Lemma A5.2. (See e.g. [30, Theorem 5.5.5’], [28, Theorem 5.1], or [27].) Assume aij ∈ C 0 (Ω) 1 and let v ∈ H0 (Ω) be the weak solution of ∂ ∂v aij =F ∂xj ∂xi

in Ω,

(A5.4)

i,j

with F ∈ Cc∞ (Ω), then v ∈ W 1,r (Ω), for every r < ∞. If the coefficients aij are Dini continuous ¯ ¯ then v ∈ C 1 (Ω). on Ω, Added in proofs. The answers to Open problems 1, 2 and 3 are negative. Interesting examples have been constructed by T. Jin, V. Maz’ya and J. Van Schaftingen (paper in preparation). Acknowledgments I am grateful to Alano Ancona and Yanyan Li for very fruitful discussions. I thank L. Boccardo, M. Giaquinta, G. Mingione, L. Orsina and J. Serrin for useful informations regarding references. The author is partially supported by NSF Grant DMS-0802958. References [1] H. Aikawa, K. Hirata, T. Lundh, Martin boundary points of a John domain and unions of convex sets, J. Math. Soc. Japan 58 (1) (2006) 247–274. [2] A. Ancona, Régularité d’accès des bouts et frontière de Martin d’un domaine euclidien, J. Math. Pures Appl. 63 (1984) 215–260. [3] A. Ancona, Théorie du potentiel sur les graphes et les variétés, in: École d’été de Probabilités de Saint-Flour XVIII—1988, in: Lecture Notes in Math., vol. 1427, Springer, Berlin, 1990, pp. 1–112. [4] A. Ancona, Sur l’inégalité de Kato jusqu’au bord, C. R. Acad. Sci. Paris Sér. I 346 (2008) 939–944. [5] A. Ancona, Sur la théorie du potentiel dans les domaines de John, Publ. Mat. 51 (2) (2007) 345–396. [6] M. Brelot, Eléments de la théorie classique du potentiel, 3e édition, Les cours de Sorbonne, 3e cycle, Centre de Documentation Universitaire, Paris, 1965. [7] M. Brelot, Axiomatique des fonctions harmoniques, Les Presses de l’Université de Montréal, 1969. [8] H. Brezis, On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338. [9] H. Brezis, A.C. Ponce, Kato’s inequality when u is a measure, C. R. Acad. Sci. Paris Sér. I 338 (2004) 599–604. [10] H. Brezis, A.C. Ponce, Kato’s inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241. [11] Y.-Z. Chen, L.-C. Wu, Second Order Elliptic Equations and Elliptic Systems, Transl. Math. Monogr., vol. 174, Amer. Math. Soc., 1998. [12] N. Chevallier, Frontière de Martin d’un domaine de R n dont le bord est inclus dans une hypersurface Lipschitzienne, Ark. Mat. 27 (1) (1989) 29–48. [13] C. Constantinescu, A. Cornea, Potential Theory on Harmonic Spaces, Grundlehren Math. Wiss., vol. 158, SpringerVerlag, 1972. [14] E. DeGiorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino 3 (1957) 25–43. [15] J.L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Classics Math., Springer-Verlag, Berlin, 2001, Reprint of the 1984 edition.

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[16] B. Fuglede, Finely Harmonic Functions, Springer Lecture Notes, vol. 289, 1972. [17] B. Fuglede, Some properties of the Riesz charges associated with a δ-subharmonic function, Potential Anal. 1 (4) (1992) 355–371. [18] M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser, 1978. [19] E. Giusti, Equazioni Ellitiche del Secondo Ordine, Pitagora Editice, Bologna, 1978. [20] R.A. Hager, J. Ross, A regularity theorem for linear second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa (3) 26 (1972) 283–290. [21] P. Hartman, G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966) 271–310. [22] R.-M. Hervé, Recherche sur la théorie axiomatique des fonctions surharmoniques et du Potentiel, Ann. Inst. Fourier (Grenoble) XII (1962) 415–471. [23] R.-M. Hervé, Quelques propriétés des fonctions surharmoniques associées à une équation uniformément elliptique ∂u ) = 0, Ann. Inst. Fourier (Grenoble) 15 (2) (1965) 215–223. de la forme Lu = − i ∂x∂ ( j aij ∂x i

j

[24] M. Hervé, R.-M. Hervé, Les fonctions surharmoniques associées à un opérateur elliptique du second ordre à coefficients discontinus, Ann. Inst. Fourier XIX (1969) 305–359. [25] M.W. Hirsch, Differential Topology, Grad. Texts in Math., vol. 33, Springer-Verlag, New York, 1994. [26] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972) 135–148. [27] Y. Li, in preparation. [28] G. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. 148 (1987) 77–99. [29] R.S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941) 137–172. [30] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Grundlehren Math. Wiss., vol. 130, Springer-Verlag, New York, 1966. [31] L. Naïm, Sur le rôle de la frontière de R.S. Martin dans la théorie du Potentiel, Ann. Inst. Fourier 7 (1957) 183–281. [32] J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa (3) 18 (1964) 385–387. [33] S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Sc. Norm. Super. Pisa 19 (1965) 593–608. [34] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier 15 (1) (1965) 189–258. [35] G. Stampacchia, Contributi alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellittiche, Ann. Sc. Norm. Super. Pisa 12 (1958) 223–245. [36] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton Univ. Press, Princeton, NJ, 1970. [37] B. Tsirelson, Triple points: From non-Brownian filtrations to harmonic measures, Geom. Funct. Anal. 7 (6) (1997) 1096–1142.


On the construction of frames for spaces of distributions George Kyriazis a,∗ , Pencho Petrushev b,c,1 a Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus b Department of Mathematics, University of South Carolina, Columbia, SC 29208, United States c Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria

Received 13 October 2008; accepted 25 June 2009 Available online 10 July 2009 Communicated by N. Kalton

Abstract We introduce a new method for constructing frames for general distribution spaces and employ it to the construction of frames for Triebel–Lizorkin and Besov spaces on the sphere. Conceptually, our scheme allows the freedom to prescribe the nature, form or some properties of the constructed frame elements. For instance, our frame elements on the sphere consist of smooth functions supported on small shrinking caps. © 2009 Elsevier Inc. All rights reserved. Keywords: Frames; Spaces of distributions; Perturbations; Triebel–Lizorkin spaces on the sphere; Besov spaces on the sphere

1. Introduction Bases and frames are a workhorse in Harmonic analysis in making various spaces of functions and distributions more accessible for study and utilization. Wavelets [18] are one of the most striking example of bases playing a pivotal role in Theoretical and Computational Harmonic analysis. The ϕ-transform of Frazier and Jawerth [6–8] is an example of frames which have had a significant impact in Harmonic analysis. Orthogonal expansions were recently used for the development of frames of a similar nature in non-standard settings such as on the sphere [19,20], interval [15,23] and ball [16,24] with weights, and in the context of Hermite [25] and Laguerre [12] expansions. * Corresponding author. Fax: +357 22 892601.

E-mail addresses: [email protected] (G. Kyriazis), [email protected] (P. Petrushev). 1 Supported by NSF grant DMS-0709046.


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G. Kyriazis, P. Petrushev / Journal of Functional Analysis 257 (2009) 2159–2187

Our aim is to construct bases and frames with prescribed nature or form for different spaces of distributions by using a particular “small perturbation argument” method. Here we only present our scheme in the case of frames. The somewhat simpler version of our method for construction of bases with some meaningful applications will be reported elsewhere. To describe the idea of our construction of bases and frames, assume that H is a separable Hilbert space of functions (e.g. some L2 -space) and S ⊂ H ⊂ S , where S is a linear space of test functions and S is the associated space of distributions. Suppose L ⊂ S is a quasi-Banach space of distributions with associated sequence space (X ) which is a quasiBanach space as well. Targeted spaces L are the Triebel–Lizorkin and Besov spaces on the unit sphere Sn in Rn+1 , on the unit ball or cube in Rn with weights as well as Triebel–Lizorkin and Besov spaces in the context of Hermite and Laguerre expansions. We assume that there is a basis or frame Ψ = {ψξ }ξ ∈X in H which allows to characterize L in terms of (X ). The central idea of our method is to construct a new system Θ = {θξ }ξ ∈X ⊂ H which approximates Ψ sufficiently well in a specific sense, while at the same time the nature, form or some specific properties of the elements {θξ } can be prescribed in advance. To make this scheme work we rely on two basic principles: Localization and Approximation. The measure of localization is in terms of the size of the various inner products of the form ψξ , ψη , θη , ψξ , ψξ , θη , more precisely, in terms of boundedness of the respective operators on 2 (X ) and (X ). The measure of approximation is in terms of the size of the inner products of the form ψη , ψξ − θξ , ψη − θη , ψξ . In fact, the critical step is to construct {θξ } so that the operators with matrices

ψη , ψξ − θξ ξ,η∈X

and

ψη − θη , ψξ ξ,η∈X

have sufficiently small norms on 2 (X ) and (X ). The good localization and approximation properties of the new system Θ will guarantee that it is a basis or frame for the distribution spaces of interest. The goal of this paper is two-fold: First, to develop our “small perturbation argument” method for construction of frames in a general setup of distribution spaces, and second, to apply these results for developing new frames for specific spaces of distributions. Choosing from various possible applications, we consider one key example that best demonstrates the versatility of our general scheme. Building upon the recently developed needlet frame on the sphere [20] we shall construct a new frame for Triebel–Lizorkin and Besov spaces on the sphere with elements supported on small shrinking caps. These frames are reminiscent of compactly supported wavelets on Rn . The situation on the sphere, however, is much more complicated than on Rn since there are no dilation or translation operators on the sphere. Other meaningful applications of our scheme would be to the construction of frames on the cube and ball with weights, and in the context of Hermite and Laguerre expansions, which we shall not pursue here. Our “small perturbation argument” method for construction of frames is related to the method of Christensen and Heil [1] for construction of atomic decompositions. We shall explain the similarities and differences of the two approaches in Section 2.5.


2161

A relevant theme is the study of the localization and self-localization of frames, initiated by Gröchenig in [9,10] and further generalized and extended by Fornasier and Gröchenig in [5], using Banach algebra techniques, and in [4]. Our understanding of localization is different but related to the one in [4,5,9,10]. Our idea of using the basic principles of localization and approximation mentioned above for constructing bases and frames for spaces of distributions has its roots in our previous developments, where bases and frames were constructed for Triebel–Lizorkin and Besov spaces on Rn . Most of our previous results on bases and frames from [13,14,22] can now be derived as applications of our general theory. The rest of the paper is organized as follows: In Section 2 we develop our general method for construction of frames for distribution spaces. In Section 3 we make an application of our general results from Section 2 to the construction of frames for the Triebel–Lizorkin and Besov spaces on the sphere. Some useful notation: We shall denote |x| := ( i |xi |2 )1/2 for x ∈ Rn . Positive constants will be denoted by c, c1 , c2 , . . . and they will be allowed to vary at every occurrence; a ∼ b will stand for c1 a b c2 a. 2. General scheme for construction of frames 2.1. The setting We assume that H is a separable complex Hilbert space (of functions) and S ⊂ H is a linear subspace (of test functions) furnished with a locally convex topology induced by a sequence of norms or semi-norms. Let S be the dual of S consisting of all continuous linear functionals on S. We also assume that H ⊂ S . The pairing of f ∈ S and φ ∈ S will be denoted by f, φ := f (φ) and we assume that it is consistent with the inner product f, g in H . Typical examples are: (a) H := L2 (Rn ), S = S∞ (Rn ) is the set of all functions φ in the Schwartz class S(Rn ) such that φ(x)x α = 0 for α ∈ Zn+ , and S is its dual; (b) H := L2 (Sn ), S := C ∞ (Sn ) with Sn being the unit sphere in Rn+1 , and S is its dual; (c) H := L2 (B n , μ), where B is the unit ball in Rn and dμ := (1 − |x|)γ −1/2 dx, S := C ∞ (B n ), and S is its dual; (d) H := L2 (I, μ), where I := I1 × · · · × In is a box in Rn and μ is a product Jacobi measure on I , S := C ∞ (I ), and S is its dual. Our next assumption is that L ⊂ S with norm · L is a quasi-Banach space of distributions, which is continuously embedded in S . Further, we assume that S ⊂ H ∩ L and S is dense in H and L with respect to their respective norms. We also assume that (X ) with norm · (X ) is an associated to L quasi-Banach space of complex-valued sequences with domain a countable index set X . Coupled with a frame Ψ the sequence space (X ) will be utilized for characterization of the space L. In addition to being a quasi-norm we assume that · (X ) obeys the conditions: (i) For any ξ ∈ X the projections Pξ : (X ) → C defined by Pξ (h) = hξ for h = (hη ) ∈ (X ) are uniformly bounded on (X ), i.e. |hξ | c h (X ) for ξ ∈ X . (ii) For any sequence (hξ )ξ ∈X ∈ (X ) one has (hξ ) (X ) = (|hξ |) (X ) . (iii) If the sequences (hξ )ξ ∈X , (gξ )ξ ∈X ∈ (X ) and |hξ | |gξ | for ξ ∈ X , then (hξ ) (X ) c (gξ ) (X ) . (iv) Compactly supported sequences are dense in (X ).

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2.2. Frames in Hilbert spaces: Background Here we collect some basic facts from the theory of frames (cf. [2,11]). Let H with inner product ·,· be a separable Hilbert space. A family Ψ := {ψξ : ξ ∈ X } ⊂ H , where X is a countable index set, is called a frame for H if there exist constants A, B > 0 such that A f 2H

f, ψξ 2 B f 2

H

for f ∈ H.

(2.1)

ξ ∈X

It is not hard to see that the frame operator S : H → H defined by Sf =

f, ψξ ψξ

(2.2)

ξ ∈X

is a bounded linear operator and AI S BI . Therefore, S is self-adjoint, S is invertible, and B −1 I S −1 A−1 I . Also, S −1 f =

f, S −1 ψξ S −1 ψξ

in H.

(2.3)

ξ ∈X

The family S −1 Ψ := {S −1 ψξ }ξ ∈X is a frame for H as well. Furthermore, for every f ∈ H

f, S −1 ψξ ψξ

f = SS −1 f =

in H

(2.4)

ξ ∈X

and f=

f, ψξ S −1 ψξ

in H.

(2.5)

ξ ∈X

Thus Ψ and S −1 Ψ provide (like Riesz bases) stable representations of all f ∈ H . However, unlike a basis, Ψ may be redundant and (2.4) is not necessarily a unique representation of f in terms of {ψξ }. A similar observation holds for S −1 Ψ . The frame Ψ is termed a tight frame if A = B in (2.1). 2.3. The old frame We adhere to the setting describe in Section 2.1. We also assume that for any f ∈ H f=

ξ ∈X

f, ψξ ψξ

in H

and f H ∼ f, ψξ 2 (X ) .

Thus Ψ := {ψξ : ξ ∈ X } ⊂ S is a frame for H . More importantly, we assume also that Ψ is a frame for L in the following sense:

(2.6)


2163

A1. For any f ∈ L

f=

f, ψξ ψξ

in L.

(2.7)

ξ ∈X

A2. For any f ∈ L, (f, ψξ )ξ ∈ (X ), and

c1 f L f, ψξ (X ) c2 f L .

(2.8)

Our aim is by using the idea of “small perturbation argument” to construct a new system Θ := {θξ : ξ ∈ X } ⊂ S with some prescribed features, which is a frame for L in the following sense: Definition 2.1. We say that Θ := {θξ : ξ ∈ X } ⊂ H is a frame for the space L with associated sequence space (X ) if the following conditions are obeyed: B1. There exist constants c1 , c2 > 0 such that

c1 f L f, θξ (X ) c2 f L where f, θξ is defined by f, θξ := B2. The operator S : L → L defined by

for f ∈ L,

(2.9)

η∈X f, ψη ψη , θξ .

Sf =

f, θξ θξ

ξ ∈X

is bounded and invertible on L; S −1 is also bounded on L and f, S −1 θξ S −1 θξ in L. S −1 f = ξ ∈X

B3. There exist constants c3 , c4 > 0 such that

c3 f L f, S −1 θξ

(X )

where as above by definition f, S −1 θξ := B4. For any f ∈ L f=

c4 f L

for f ∈ L,

η∈X f, ψη ψη , S

f, S −1 θξ θξ = f, θξ S −1 θξ

ξ ∈X

−1 θ

(2.10)

ξ .

in L.

(2.11)

ξ ∈X

Remark 2.2. Above and throughout the rest of this section when we write “in H ” or “in L” it means that the convergence of the respective series is unconditional in H or in L. For unconditional convergence we refer the reader to [17]. Observe that if L is a Hilbert space then properties B2–B4 are byproducts of B1 (see Section 2.2). However, this is no longer true for more general spaces.

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2.4. Construction of a new frame The key of our method for constructing a new frame Θ := {θξ : ξ ∈ X } for L (as described above) is to build {θξ } with appropriate localization and approximation properties with respect to the given tight frame Ψ . The localization of Θ will be measured in terms of the size of the inner products ψξ , ψη , θη , ψξ , ψξ , θη . More precisely, we construct {θξ } so that the operators with matrices A := (aξ,η )ξ,η∈X ,

aξ,η := ψη , ψξ ,

B := (bξ,η )ξ,η∈X ,

bξ,η := θη , ψξ ,

C := (cξ,η )ξ,η∈X ,

cξ,η := ψη , θξ ,

(2.12)

are bounded on 2 (X ) and (X ). Notice that C = B∗ the adjoint of B. The approximation property of Θ will be measured in terms of the size of the inner products ψη , ψξ − θξ , ψη − θη , ψξ . Namely, we construct {θξ } so that the operators with matrices D := (dξ,η )ξ,η∈X ,

dξ,η := ψη , ψξ − θξ ,

E := (eξ,η )ξ,η∈X ,

eξ,η := ψη − θη , ψξ ,

(2.13)

are bounded on 2 (X ) and (X ) and, more importantly, for sufficiently small ε > 0

D 2 (X ) →2 (X ) ε,

E 2 (X ) →2 (X ) ε,

(2.14)

D (X ) →(X ) ε,

E (X ) →(X ) ε.

(2.15)

Notice that E = D∗ . Before we treat the case of general distribution spaces, we shall give sufficient conditions which guarantee that the new system Θ is a frame for the Hilbert space H itself. Proposition 2.3. As above, let Ψ = {ψξ }ξ ∈X be a frame for the Hilbert space H such that (2.6) holds. Suppose Θ = {θξ }ξ ∈X ⊂ H is constructed so that the operators with matrices C and D defined in (2.12)–(2.13) are bounded on 2 (X ) and for a sufficiently small ε > 0

D 2 (X ) →2 (X ) ε.

(2.16)

Then Θ is a frame for H , that is, there exist constants c1 , c2 > 0 such that

c1 f H f, θξ ξ 2 (X ) c2 f H , f ∈ H. Proof. Note that f =

η∈X f, ψη ψη

(2.17)

for f ∈ H and hence

=

f, ψη ψη , θξ 2 (X )

f, θξ

η∈X

C 2 (X ) →2 (X )

ξ 2 (X )

f, ψξ

Thus the right-hand side estimate in (2.17) is established.

2 (X )

c f H .

(2.18)


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For the proof of the left-hand side of (2.17), we have using (2.1)

f H c f, ψξ 2 (X )

c f, ψξ − θξ 2 (X ) + f, θξ 2 (X ) .

(2.19)

Observe that

f, ψξ − θξ 2

= f, ψ ψ , ψ − θ η η ξ ξ

(X )

ξ 2 (X )

η∈X

D 2 (X ) →2 (X ) f, ψξ 2 (X ) ε f H .

(2.20)

From (2.19)–(2.20) we obtain for sufficiently small ε > 0 (ε < 1/c will do)

f H

c

f, θξ 2 c f, θξ 2 (X ) , ( X ) 1 − cε

which confirms the left-hand side estimate in (2.17).

2

We now come to the main result of this section. Theorem 2.4. Let Ψ := {ψξ : ξ ∈ X } ⊂ S be the old frame for H and L as described in Section 2.3. Suppose the system Θ := {θξ : ξ ∈ X } ⊂ H is constructed so that the operators with matrices A, B, C, D, E from (2.12)–(2.13) are bounded on (X ) and C, D are bounded on 2 (X ) as well. Then if for sufficiently small ε > 0 the matrices D, E obey (2.14)–(2.15), the sequence Θ is a frame for L in the sense of Definition 2.1. Most importantly, if f ∈ S , then f ∈ L if and only if (f, S −1 θξ ) ∈ (X ), and for f ∈ L f=

f, S −1 θξ θξ

ξ ∈X

in L and f L ∼ f, S −1 θξ (X ) .

(2.21)

Proof. We first note that by Proposition 2.3 Θ is a frame for H . We next prove that Θ obeys condition B1. From the definition of f, θξ (see Definition 2.1), the boundedness of C, and (2.8) we infer

=

f, ψη ψη , θξ

(X )

f, θξ

η∈X

(X )

C (X ) →(X ) f, ψξ

(X )

c f L ,

which confirms the right-hand side estimate in (2.9). For the proof of the left-hand side of (2.9), we have by (2.8)

f L c f, ψξ (X ) c f, ψξ − θξ (X ) + c f, θξ (X ) and we next estimate the first term above using (2.15) and (2.8):

(2.22)

2166


f, ψξ − θξ

= f, ψ ψ , ψ − θ η η ξ ξ

(X )

(X )

η∈X

D (X ) →(X ) f, ψξ

(X )

c ε f L .

Substituting this above, we get

f L

c

f, θξ

, (X ) 1 − cc ε

yielding the left-hand side estimate in (2.9) if ε > 0 is sufficiently small, namely, if ε < 1/cc . The following lemma will play a key role in the sequel. Lemma 2.5. The operators T h := ξ ∈X hξ θξ and V h := ξ ∈X hξ ψξ are well defined and bounded as operators from (X ) to L. Proof. We shall only prove the boundedness of T ; the proof of the boundedness of V is easier and will be omitted. Let h = (hξ )ξ ∈X be a compactly supported sequence of complex numbers. Then using (2.8) and the boundedness of B, we get

h θ , ψ

T h L c

ξ ξ η

ξ ∈X

η (X )

= c

h θ , ψ ξ ξ η

ξ ∈X

η (X )

c B (X ) →(X ) h (X ) c h (X ) . By condition (iv) on (X ) compactly supported sequences are dense in (X ) and, therefore, the operator T can be uniquely extended to a bounded operator from (X ) to L. Furthermore, it is easy to show that the series ξ ∈X hξ ψξ converges unconditionally in L. 2 We now prove that Θ satisfies B2. By definition Sf = ξ ∈X f, θξ θξ , but by (2.22) we have (f, θξ )ξ ∈X ∈ (X ). Therefore, by Lemma 2.5 the operator S : L → L is bounded. The space L is a quasi-Banach space, but nevertheless it is easily seen that if I − S L →L < 1, then S −1 exists on L. In fact, S −1 can be constructed by the Neumann series, ∞ and is bounded −1 k i.e. S = k=0 (I − S) . To prove that I − S L →L < 1 for sufficiently small ε, let us denote G = (gξ,η )ξ,η∈X , where gξ,η := (I − S)ψη , ψξ . Then, assuming that G is bounded on (X ), we get

(I − S)f c (I − S)f, ψξ

= c f, ψ (I − S)ψ , ψ η η ξ

L (X )

c G (X ) →(X ) f, ψξ

(X )

η∈X

(X )

c G (X ) →(X ) f L .

(2.23)

Here for the equality we used that the operator I − S is bounded on L. We next estimate

G (X ) →(X ) . Evidently, we have Sψη , ψξ =

ω∈X

ψη , θω θω , ψξ and ψη , ψξ =

ω∈X

ψη , ψω ψω , ψξ


2167

and hence gξ,η = ψη , ψξ − Sψη , ψξ = ψη , ψω − θω ψω , ψξ + ψη , θω ψω − θω , ψξ ω∈X

ω∈X

= (AD)ξ,η + (EC)ξ,η . Thus G = AD + EC and by the boundedness of the respective operators and (2.15)

G (X ) →(X ) c A (X ) →(X ) D (X ) →(X ) + E (X ) →(X ) C (X ) →(X ) cε. Substituting this in (2.23) we get (I − S)f L c ε f L and hence for sufficiently small ε we have I − S L →L c ε < 1 (ε < 1/c will do). Then the operator S −1 exists and is bounded on L. For the rest of the proof of the theorem we need the following lemma: Lemma 2.6. The operators with matrices H := ψη , S −1 θξ ξ,η∈X , J := ψη , Sψξ ξ,η∈X ,

H∗ := S −1 θξ , ψη ξ,η∈X , J1 := ψη , S −1 ψξ ξ,η∈X

are bounded on (X ). Proof. We shall only prove the boundedness of H and H∗ ; the proof of the boundedness of J and J1 is simpler and will be omitted. Let d = (dξ ) be a compactly supported sequence and set f := ξ ∈X dξ ψξ . Then (Hd)ξ =

−1 dη ψη , S −1 θξ = dη S ψ η , θ ξ = dη S −1 ψη , θξ

η∈X

η∈X

η∈X

−1 S f, ψω ψω , θξ . dη ψη , θξ = S −1 f, θξ = = S −1 η∈X

ω∈X

Here for the second equality we used that S −1 is self-adjoint on H . Now, similarly as before we get

Hd (X ) C (X ) →(X ) S −1 f, ψω (X ) c S −1 f L c f L c d (X ) . Here for the last inequality we used Lemma 2.5. Since compactly supported sequences are dense in (X ) then the operator H can be uniquely extended to a bounded operator on (X ). ∗ The proof of the boundedness of H goes along similar lines. Given a compactly supported sequence d = (dξ ), we set g := η∈X d η ψη and then

2168


(H∗ d)ξ =

dη S −1 θξ , ψη = dη θξ , S −1 ψη = θξ , S −1 d η ψη .

η∈X

η∈X

η∈X

As above, using the boundedness of S −1 on L and B1, we obtain

H∗ d (X ) = S −1 g, θξ (X ) c S −1 g L c g L c d (X ) = c d (X ) . Here for the first and last equalities we used condition (ii) on (X ). Now the boundedness of H∗ follows as above. 2 Just as in (2.22) the boundedness on (X ) of the operator with matrix H from Lemma 2.6 implies

f, S −1 θξ

c f L for f ∈ L. (X ) Furthermore, the boundedness on (X ) of the operator with matrix H∗ defined in Lemma 2.6 yields that the operator hξ S −1 θξ U h := ξ ∈X

is bounded as an operator from (X ) to L (see the proof of Lemma 2.5). Combining these two facts shows that the operator S−1 defined by S−1 f :=

f, S −1 θξ S −1 θξ

(2.24)

ξ ∈X

is well defined and bounded on L. On the other hand, by a well known property of frames (see (2.3)) for any f ∈ H S −1 f = f, S −1 θξ S −1 θξ . (2.25) ξ ∈X

Since by assumption S ⊂ H is dense in L, this leads to S −1 = S−1 on L. Therefore, representation (2.25) of S −1 holds on L as well. This completes the proof of B2. We need one more lemma. Lemma 2.7. For any f ∈ L Sf, ψξ = f, Sψξ and

−1 S f, ψξ = f, S −1 ψξ for ξ ∈ X .

(2.26)

Proof. The proof relies on the fact that S and S −1 are self-adjoint operators on H and S ⊂ H ∩ L is dense in L. We shall only prove the left-hand side identity in (2.26); the proof of the right-hand side identity is the same. Let f ∈ L and choose a sequence fn ∈ S so that f − fn L → 0. Using that Sfn , ψξ = fn , Sψξ as fn ∈ S, we get Sf, ψξ − f, Sψξ S(f − fn ), ψξ + f − fn , Sψξ . (2.27)


2169

By condition (i) on (X ), (2.8), and the boundedness of S on L, it follows that

S(f − fn ), ψξ c S(f − fn ), ψξ

c S(f − fn ) L c f − fn L . (X ) By definition f − fn , Sψξ = (X ) and Lemma 2.6, we get

η∈X f

(2.28)

− fn , ψη ψη , Sψξ and using again condition (i) on

f − fn , Sψξ

f − f , ψ ψ , Sψ n η η ξ

(X )

η∈X

J (X ) →(X ) f − fn , ψη (X ) c f − fn L . We use this and (2.28) in (2.27) to obtain Sf, ψξ − f, Sψξ c f − fn L → 0, which implies the left-hand side identity in (2.26).

2

that Θ obeys B3-4. Given f ∈ L, by definition Sf = We are now prepared to prove −1 f we arrive at f, θ θ and from f = SS ξ ξ ξ ∈X f=

S −1 f, θξ θξ = f, S −1 θξ θξ

ξ ∈X

in L,

(2.29)

ξ ∈X

where we used Lemma 2.7. Thus the left-hand side identity in (2.11) holds. Similarly f = S −1 Sf and using (2.25) in L and Lemma 2.7, we get Sf, S −1 θξ S −1 θξ = f, SS −1 θξ S −1 θξ = f, θξ S −1 θξ , f= ξ ∈X

ξ ∈X

ξ ∈X

which gives the right-hand side identity in (2.11). Therefore, B3 holds. Going further, we have by definition f, S −1 θξ := η∈X f, ψη ψη , S −1 θξ and using the boundedness of H (Lemma 2.6), we get

f, S −1 θξ

H (X ) →(X ) f, ψη (X ) c f L , (X ) which confirms the validity of the right-hand side estimate in (2.10). In the other direction, by (2.29) f, ψη = ξ ∈X f, S −1 θξ θξ , ψη and hence

−1

f, S θξ θξ , ψη = c

(X )

f L c f, ψη

ξ ∈X

c B (X ) →(X ) f, S −1 θξ

(X )

η (X )

c f, S −1 θξ

(X )

.

Thus B3 is established. and (f, S −1 θξ ) ∈ (X ), then by Lemma 2.5 F := observe that if f ∈ S Finally, −1 −1 f, S θ θ ∈ L. Since g = ξ ξ ξ ∈X ξ ∈X g, S θξ θξ for g ∈ H and S ⊂ H ∩ L is dense in L, then F = f . The proof of Theorem 2.4 is complete. 2

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2.5. Comparison of our method with the method of Christensen and Heil Our approach to constructing frames is related to the work of Christensen and Heil [1], where they use perturbations of atomic decompositions to construct new atomic decompositions. To be more specific, using our notation from above, a pair {ψξ }ξ ∈X , {ψ˜ ξ }ξ ∈X is said to be an atomic decomposition of the Banach space L with respect to the sequence space (X ) if each f ∈ L has the representation f = ξ ∈X f, ψ˜ ξ ψξ and f L ∼ f, ψ˜ ξ (X ) . The most relevant Theorem 2.3 in [1] says that if {θξ }ξ ∈X is in a sense a “small” perturbation of {ψξ }ξ ∈X , then the operator Tf = ξ f, ψ˜ ξ θξ is invertible in L and the pair {θξ }, {(T −1 )∗ ψ˜ ξ } is a new atomic decomposition of L. In contrast, we have shown in Section 2.4 that the usual frame operator Sf = ξ f, θξ θξ is bounded and invertible in L. This enabled us to establish, as in the Hilbert space case, that both Theorem 2.4), more precisely, for all f ∈ L we have f = {θξ } and {S −1 θξ } are frames in L (see−1 −1 θ θ = f, S f, θ S θξ and f L ∼ (f, θξ ) (X ) ∼ (f, S −1 θξ ) (X ) . ξ ξ ξ ξ ∈X ξ ∈X Thus, although the two approaches bear some similarities, our goal is not only to construct atomic decompositions but rather to extend the basic elements of the frame theory in Hilbert spaces to the case of a general quasi-Banach space L. 3. Frames with elements supported on shrinking caps on the sphere In this section we utilize the scheme from Section 2.4 to the construction of frames for Triebel–Lizorkin (F) and Besov (B) spaces on the unit sphere Sn in Rn+1 (n > 1) of the form n {θξ }ξ ∈X , where X = ∞ j =0 Xj is a multilevel index set of points on S and for ξ ∈ Xj the frame element θξ is supported on a spherical cap of radius ∼ 2−j centered at ξ . The F- and B-spaces on the sphere are introduced and explored in [20] as a natural progression of the Littlewood–Paley theory on Sn . These spaces are also characterized in [20] via frames with elements of nearly exponential localization, called “needlets”. We next give a short account of the development in [20], which we shall build upon. 3.1. Spaces of distribution on the sphere: Background Denote by Hν the space of all spherical harmonics of order ν on Sn . As is well known the kernel of the orthogonal projector onto Hν is given by Pν (ξ · η) =

ν + λ (λ) P (ξ · η), λωn ν

λ = λn :=

n−1 , 2

(λ)

(3.1)

where ωn is the hypersurface area of Sn and Pν is the Gegenbauer polynomial of degree ν normalized with Pν(λ) (1) = ν+2λ−1 ; ξ · η is the inner product of ξ, η ∈ Sn . ν ∞ n Let S := C (S ) be the space of all test functions on Sn and let S := S (Sn ) be its dual, the space of all distributions on Sn . The action of f ∈ S on φ ∈ S is denoted by f, φ := f (φ). For functions Φ ∈ L∞ [−1, 1] and f ∈ L1 (Sn ) the nonstandard convolution Φ ∗ f is defined by Φ ∗ f (ξ ) := Φ(ξ · σ )f (σ ) dσ, Sn

where the integration is over Sn , and it extends by duality from S to S .


2171

To define the Triebel–Lizorkin and Besov spaces on the sphere, one first introduces a sequence of functions {Φj } of the form Φ0 := P0

and Φj :=

∞ ν aˆ j −1 Pν , 2

j 1,

(3.2)

ν=0

with aˆ obeying the conditions: aˆ ∈ C ∞ [0, ∞), supp aˆ ⊂ [1/2, 2], a(t) ˆ > c > 0 if t ∈ [3/5, 5/3].

(3.3) (3.4)

Hence, Φj , j = 0, 1, . . . , are band limited. sq

Definition 3.1. Let s ∈ R, 0 < p < ∞, and 0 < q ∞. The Triebel–Lizorkin space Fp := sq Fp (Sn ) is defined as the set of all f ∈ S such that

f Fpsq

∞

q 1/q

sj 2 Φj ∗ f (·) :=

j =0

< ∞,

(3.5)

Lp (Sn )

where the q -norm is replaced by the sup-norm if q = ∞. We note that as in the classical case on Rn by varying the indexes s, p, q one can recover most of the classical spaces on Sn , e.g. Fp02 = Lp (Sn ) if 1 0,

(iii)

aˆ 2 (t) + aˆ 2 (2t) = 1,

aˆ 0,

supp aˆ ⊂ [1/2, 2],

if t ∈ [3/5, 5/3], if t ∈ [1/2, 1]

(3.7)

2172


and hence, ∞

aˆ 2 2−j t = 1,

t ∈ [1, ∞).

(3.8)

j =0

We select j0 −2 so that 2j0 +1 λ < 2j0 +2 (λ := and

n−1 2 ) and define the kernels {Ψj } by Ψj0

∞ ν+λ Ψj := aˆ Pν , 2j

j > j0 .

:= P0

(3.9)

ν=0

A Calderón type reproducing formula follows from (3.8)-(3.9): For any f ∈ S f=

∞

Ψj ∗ Ψj ∗ f

in S .

(3.10)

j =j0

As in [20] (see also [19]) there exist a set Xj ⊂ Sn (j j0 ) and weights {cξ }ξ ∈Xj such that the cubature formula f (σ ) dσ ∼ cξ f (ξ ) (3.11) ξ ∈ Xj

Sn

is exact for all spherical polynomials of degree 2j +1 . Here, in addition, cξ ∼ 2−j n and the points in Xj are almost uniformly distributed, i.e. there exist constants c2 > c1 > 0 such that Bξ (c1 2−j ) ∩ Bη (c1 2−j ) = ∅ whenever ξ = η, ξ, η ∈ Xj , and Sn = ξ ∈Xj Bξ (c2 2−j ), where Bξ (r) := {η ∈ Sn : d(η, ξ ) < r} with d(η, ξ ) being the geodesic distance between η, ξ on Sn . The j th level needlets are defined by 1/2

ψξ (x) := cξ Ψj (ξ · x),

ξ ∈ Xj ,

(3.12)

and the whole needlet system by Ψ := {ψξ }ξ ∈X ,

where X :=

∞

Xj .

(3.13)

j =j0

Here equal points from different levels Xj are regarded as distinct points of the index set X . By discretization of (3.10) using cubature formula (3.11) one arrives at the representation formula: For any f ∈ S f=

f, ψξ ψξ

in S .

ξ ∈X

The same representation holds in Lp for functions f ∈ Lp (Sn ) as well.

(3.14)


2173

The key feature of the functions ψξ , ξ ∈ X , is their superb localization: For any M > 0 there exists a constant cM > 0 such that ψξ (x) cM

2j n/2 , (1 + 2j d(ξ, x))M

x ∈ Sn ,

(3.15)

where as mentioned above d(ξ, η) := arccos(ξ · η). sq sq We next define the sequence spaces fp and bp associated to X , where for ξ ∈ Xj , Gξ −j denotes the spherical cap Bξ (c2 2 ), introduced above. sq

sq

Definition 3.3. Let s ∈ R, 0 < p < ∞, and 0 < q ∞. Then fp := fp (X ) is defined as the space of all complex-valued sequences h := (hξ )ξ ∈X such that

h fpsq

q 1/q

−s/n−1/2

|Gξ | :=

|hξ |1Gξ (·)

0, we construct an even function b ∈ C ∞ (R) obeying the following conditions: (i) supp b ⊂ [−R, R] for some R > 0, −M (ii) a (r) (t) − b(r) (t) ε 1 + |t| for 0 r N + n − 1, t r b(t) dt = 0 for 0 r N + n − 2. (iii)

(3.20)

R

Note that the Fourier transform bˆ of b is even and belongs to S. A scheme for constructing this sort of functions b will be given below. Just as in the construction of needlets we shall use X = ∪∞ j =j0 Xj (see (3.13)) as an index set as well as a set of localization points for the new elements. For each ξ ∈ Xj (j j0 ) we define the function θξ on the sphere by 1/2

θξ (x) := cξ

∞ ν+λ Pν (ξ · x), bˆ 2j

λ := (n − 1)/2,

(3.21)

ν=0

and then Θ := {θξ }ξ ∈X is our new system on Sn . With the next theorems we show that for appropriately selected parameters M, N , and ε, Θ is a frame for the F- and B-spaces with the claimed support property. Let J := n/ min{1, p, q} in the case of F-spaces and J := n/ min{1, p} for B-spaces. Theorem 3.7. Suppose s ∈ R, 0 < p, q < ∞ and let Θ := {θξ }ξ ∈X be constructed as above with b satisfying (3.20), where M > J and N > max{s, J − n − s, 1}. Then for sufficiently small sq sq ε > 0 the system Θ is a frame for the spaces L2 (Sn ), Fp , and Bp in the sense of Definition 2.1. In particular, we have:


2175

(a) The operator Sf :=

f, θξ θξ ,

(3.22)

ξ ∈X

sq sq where f, θξ := η∈X f, ψη ψη , θξ , is bounded and invertible on L2 (Sn ), Fp , Bp , and sq sq S −1 is also bounded on L2 (Sn ), Fp , Bp , and S −1 f =

f, S −1 θξ S −1 θξ .

(3.23)

ξ ∈X

(b) If f ∈ S , then f ∈ Fp if and only if (f, S −1 θξ ) ∈ fp , and for f ∈ Fp sq

f=

sq

f, S −1 θξ θξ

ξ ∈X

sq

and f Fpsq ∼ f, S −1 θξ f sq .

(3.24)

p

(c) If f ∈ S , then f ∈ Bp if and only if (f, S −1 θξ ) ∈ bp , and for f ∈ Bp sq

f=

sq

f, S −1 θξ θξ

ξ ∈X

sq

and f Bpsq ∼ f, S −1 θξ bsq .

(3.25)

p

sq

sq

The convergence in (3.22)–(3.25) is unconditional in the respective space L2 , Fp , or Bp . Above, (b) and (c) also hold with the roles of θξ and S −1 θξ interchanged. Moreover, for any ξ ∈ Xj , j j0 , the element θξ is supported on the spherical cap Bξ (R2−j ), where R > 0 is the constant from (3.20). Several remarks are in order: (a) Atomic decompositions are available for various spaces and in particular for Triebel– Lizorkin and Besov spaces on Rn (see [7]). Theorem 3.7 provides atomic decompositions for Triebel–Lizorkin and Besov spaces on Sn . These atomic decompositions have the advantage that they involve atoms from a fixed sequence Θ, while in general the atoms in the atomic decompositions may vary with the distributions. (b) Note that the function b ∈ C ∞ from our construction is not necessarily compactly supported. As long as b satisfies conditions (ii)–(iii) in (3.20) it will induce a frame for the F- and Bspaces on Sn . In addition to this the nature of b or bˆ can be prescribed, e.g. b or bˆ can be a low degree rational function or a linear combination of a small number of dilations and shifts 2 of the Gaussian e−t . (c) We would like to point out that the elements of Θ are essentially rotations and spectral dilations of a single function supported on a cap on the sphere and hence bear some resemblance with compactly supported wavelets. We start with the construction of a function b obeying (3.20). Then we shall carry out the proof of Theorem 3.7 in several steps. The gist of the proof will be the interplay between the spherical harmonics and the classical Fourier transform related by the Dirichlet–Mehler representation of Gegenbauer polynomials.

2176


3.4. Construction of b A first step in constructing the frame {θξ } is the construction of a function b satisfying conditions (3.20), which we give in the next theorem. As will be seen this construction allows to ˆ prescribe the nature of b or b. Theorem 3.8. For given M > 0, N 1, and ε > 0, here exists an even real-valued function b ∈ C ∞ which satisfies conditions (3.20). Proof. The construction of a function b with the claimed properties follows the same lines as in the proof of Theorem 4.1 in [13]. Therefore, we shall only outline the main steps in this construction. We pick an even function φ ∈ C ∞ such that supp φ ⊂ [−1, 1] and R φ = 1. Write φk (t) := kφ(kt) and denote by Φk the set of all finite linear combinations of shifts of φk , i.e. functions g of the form g(t) = j aj φk (t + bj ), where the sum is finite. We first show that for every ε > 0 and an even (or odd) function h ∈ C ∞ there exist k > 0 (sufficiently large) and an even (or odd) function g ∈ Φk such that (r) h (t) − g (r) (t) ε 1 + |t| M ,

t ∈ R, r = 0, 1, . . . , N0 ,

where N0 := N + n − 1. Indeed, define gk := h ∗ φk . Since h(r) (t) − gk(r) (t) =

R φk

(3.26)

= 1, then

(r) h (t) − h(r) (t − y) φk (y) dy

R

and taking k sufficiently large one easily shows that (r) h (t) − g (r) (t) (ε/2) 1 + |t| −M , k

t ∈ R, r = 0, 1, . . . , N0 .

(3.27)

Notice that gk is even (odd) if h is even (odd). To discretize the approximant gk we first observe that since h ∈ S, there exists R > 0 such that (r) h (t) ε 1 + |t| −M ,

|t| R, r = 0, 1, . . . , N0 .

(3.28)

Now, we choose sufficiently large S > 0 so that J := SR is an integer and consider the points j +1/2 tj := j −1/2 S , j = 1, . . . , J , and tj := S , j = −1, . . . , −J . We define g(t) := S −1

h(tj )φk (t − tj ),

−J j J, j =0

R which can be viewed as a Riemann sum for the integral R h(y)φk (t − y) dy. Notice that gk (t) = R h(y)φk (t − y) dy. As in the proof of Theorem 4.1 in [13], one easily shows, using (3.27)–(3.28), that for sufficiently large S this function satisfies (3.26). In addition to this, evidently g is even (odd) if h is even (odd) and g ∈ Φk .


2177

Our second step is to utilize the result of the first step to construct the desired function b. Consider the shift operator Tδ f (t) := f (t + δ). Then the sth centered difference is defined by sδ f := (Tδ − T−δ )s f and it is easy to see that its Fourier transform satisfies (sδ f )∧ (ξ ) = (2i sin δξ )s fˆ(ξ ). ˆ ) := We choose s := N0 and 0 < δ 1/s, and define the function h from the identity h(ξ a(ξ ˆ ) ˆ ) = 0 for ξ ∈ [−1/2, 1/2], then hˆ ∈ S and hence h ∈ S. Further, since aˆ is (2i sin δξ )s . Since a(ξ even, then hˆ and h are even (odd) if s is even (odd). Moreover, by the construction a = sδ h. We now use the result of the first step to construct a function g ∈ Φk such that g satisfies (3.26) with h from above. After this preparation, we define b := sδ g and claim that b has the desired properties. Indeed, note that a (r) − b(r) = sδ (h(r) − g (r) ) and by (3.26) we infer (r) a (t) − b(r) (t) ε2s+M 1 + |t| −M , r = 0, 1, . . . , N0 . (3.29) On the other hand r r s s t b(t) dt = t δ g(t) dt = (−1) g(t)sδ t r dt = 0, R

R

r = 0, 1, . . . , s − 1.

R

Also, note that b := sδ g is even if g and s are both odd or even and evidently b ∈ Φk and hence b is compactly supported. We finally observe that since ε is independent of M and s the factor ε2s+M in (3.29) can be replaced by ε. 2 3.5. Almost diagonal matrices To show that the new system Θ := {θξ : ξ ∈ X } is a frame for Triebel–Lizorkin and Besov sq sq sq sq spaces we shall use Theorem 2.4 with L := Fp (Sn ) or Bp (Sn ) and (X ) := fp (X ) or bp (X ), respectively. Then L2 (Sn ) is the natural selection of an associated Hilbert space. By Theorem 2.4 sq sq it readily follows that Θ is a frame for Fp (or Bp ) if the operators with matrices

sq

A := (aξ,η )ξ,η∈X ,

aξ,η := ψη , ψξ ,

B := (bξ,η )ξ,η∈X ,

bξ,η := θη , ψξ ,

C := (cξ,η )ξ,η∈X ,

cξ,η := ψη , θξ ,

D := (dξ,η )ξ,η∈X ,

dξ,η := ψη , ψξ − θξ ,

E := (eξ,η )ξ,η∈X ,

eξ,η := ψη − θη , ψξ ,

(3.30)

sq

are bounded on fp (or bp ), and D fpsq →fpsq ε, E fpsq →fpsq ε (respectively,

D bpsq →bpsq ε, E bpsq →bpsq ε) for sufficiently small ε. In analogy with the classical case on Rn (see [7]), we shall show the boundedness of the above operators by using the machinery of the almost diagonal operators. It will be convenient to us to denote (ξ ) := 2−j

for ξ ∈ Xj , j j0 .

Evidently, (ξ ) is a constant multiple of the radius of the cap Gξ .

(3.31)

2178

G. Kyriazis, P. Petrushev / Journal of Functional Analysis 257 (2009) 2159–2187 sq

sq

Definition 3.9. Let A be a linear operator acting on fp (X ) or bp (X ) with associated matrix (aξ η )ξ,η∈X . We say that A is almost diagonal if there exists δ > 0 such that |aξ η | < ∞, ξ,η∈X ωδ (ξ, η) sup

where s

−J −δ d(ξ, η) 1+ ωδ (ξ, η) := max{(ξ ), (η)} (n+δ)/2 (ξ ) (η) (n+δ)/2+(J −n) , × min , (η) (ξ )

(ξ ) (η)

sq

sq

with J := n/ min{1, p, q} for fp and J := n/ min{1, p} for bp . sq

sq

The almost diagonal operators are bounded on fp and bp . More precisely, with the notation |aξ η | ξ,η∈X ωδ (ξ, η)

A δ := sup

(3.32)

the following result holds: Theorem 3.10. Suppose s ∈ R, 0 < q ∞, and 0 0. Then there exists a constant sq c > 0 such that for any sequence h := {hξ }ξ ∈X ∈ fp

Ah fpsq c A δ h fpsq ,

(3.33)

sq

and for any sequence h := {hξ }ξ ∈X ∈ bp

Ah bpsq c A δ h bpsq .

(3.34)

The proof of this theorem is quite similar to the proof of Theorem 3.3 in [7]. For completeness we give it in Appendix A. sq sq The above theorem indicates that to prove that Θ is a frame for Fp (or Bp ) it suffices to show that the operators with matrices A, B, C, D, and E, defined in (3.30) , are almost diagonal and

D δ ε, for a fixed δ > 0 and sufficiently small ε > 0.

E δ ε

(3.35)


2179

3.6. Representation and localization of kernels. Estimation of supp θξ Kernels of the form ΛN (ξ · η) :=

ν + λ gˆ Pν (ξ · η), N

ξ, η ∈ Sn , N 1,

(3.36)

ν0

will play an important role in the proof of Theorem 3.7. Here as everywhere else Pν and λ are from (3.1). Lemma 3.11. For an even function gˆ ∈ S the kernel ΛN from above has the representation cn ΛN (cos α) = (sin α)n−2

π (cos α − cos ϕ)λ−1 KN (ϕ) dϕ,

0 α π,

(3.37)

α

where d ν(n−1) KN (α) = (π/2)N g N (α + 2πν) (−1) Rn dα

(3.38)

−z sin λπ, n even, 2 2 Rn (z) := −z − (λ − r) × cos λπ, n odd,

(3.39)

ν∈Z

with n−1 2 r=1

and cn > 0 depends only on n. This lemma is in essence contained in [19, Proposition 3.2]. For completeness we give its proof in Appendix A. We next give an estimate of the localization of the kernels ΛN from (3.36) provided g and its derivatives are well localized. Lemma 3.12. If g ∈ C n−1 (R) is even and (m) g (t)

A , (1 + |t|)M

t ∈ R, 0 m n − 1,

(3.40)

for some constants M > 1 and A > 0, then ΛN (cos α) where c > 0 depends only on M and n.

cAN n , (1 + N α)M

0 α π,

(3.41)

2180


Proof. We use (3.40) and that Rn (z) from (3.39) is a polynomial of degree n − 1 to obtain KN (α) cAN ν∈Z

N n−1 cAN n . (1 + N |α + 2πν|)M (1 + N α)M

Now, precisely as in [19, §3.4] one shows that the above estimate used in (3.37) yields (3.41). We skip the details. 2 Lemma 3.13. For every ξ ∈ Xj , j j0 , θξ is supported on the spherical cap of radius R2−j centered at ξ , where R is from (3.20)(i). Proof. Let ξ ∈ Xj , j j0 . Then by the definition of θξ in (3.21) along with Lemma 3.11, we have cn θξ (x) = (sin φ)n−2

π (cos φ − cos ϕ)λ−1 Kj (ϕ) dϕ,

ξ · x =: cos φ,

(3.42)

φ

where 1/2

Kj (ϕ) := (π/2)cξ 2j

(−1)ν(n−1) Rn (

ν∈Z

d j )b 2 (ϕ + 2πν) . dϕ

By construction supp b ⊂ [−R, R] and, hence, supp Kj ⊂ [−R2−j , R2−j ] whenever R2−j < π . This and (3.42) apparently lead to supp θξ ⊂ Bξ (R2−j ). The case when R2−j π is trivial. 2 3.7. Estimation of inner products We shall need an estimate on the localization of the convolution of two well localized functions. In the following, for a given function g on R we denote gj (t) := 2j g(2j t). Lemma 3.14. Suppose the functions g ∈ C N (R) and h ∈ C(R) satisfy the conditions: (r) g (t)

A1 , (1 + |t|)M1

0 r N,

h(t)

A2 , (1 + |t|)M2

and t r h(t) dt = 0

for 0 r N − 1,

R

where N 1, M2 M1 , M2 > N + 1, and A1 , A2 > 0. Then for k j gj ∗ hk (t) cA1 A2 2−(k−j )N where c > 0 depends only on M1 , M2 , and N .

2j , (1 + 2j |t|)M1


2181

The proof of this lemma is almost identical to the proof of Lemma B.1 in [7] and will be omitted. The only difference is in the normalization of the functions. We now come to the main lemma which will enable us to estimate the inner products involved in (3.30). For simplicity, in the following we assume that g, h ∈ S. Then their Fourier transforms g, ˆ hˆ ∈ S as well, with S being the Schwartz class. For ξ ∈ Xj , j j0 , and η ∈ Xk , k j0 , we define 1/2 Gξ (x) := cξ

∞ ν+λ gˆ Pν (ξ · x), 2j

Hη (x) := cη1/2

ν=0

∞ ˆh ν + λ Pν (η · x), 2k

(3.43)

ν=0

where cξ , cη are from (3.11). Lemma 3.15. Suppose g, h ∈ S are both even and real-valued, (m) g (t)

A1 (1 + |t|)M

and h(m) (t)

A2 , (1 + |t|)M

0 m N + n − 1,

(3.44)

and

t r g(t) dt = R

t r h(t) dt = 0,

0 m N + n − 2,

(3.45)

R

where N > 1 and M > N + 1. Then for ξ ∈ Xj and η ∈ Xk Gξ , Hη cA1 A2 2−|k−j |(N +n/2) 1 + 2min{j,k} d(ξ, η) −M where c > 0 depends only on N , M, and n. Proof. Assume that k j and let ξ · η =: cos α, 0 α π . Then using that Pν (ξ · x)P (ξ · x) dx = δν, Pν (ξ · η), Sn

cξ ∼ 2−j n if ξ ∈ Xj , and cη ∼ 2−kn if η ∈ Xk , we have Gξ , Hη ∼ 2

−(k+j ) n2

∞ ν +λ ˆ ν +λ gˆ Pν (ξ · η). h 2j 2k ν=0

It is easy to see that ν +λ ˆ ν +λ ∧ ∧ ν+λ gˆ = (gj ∗ hk ) (ν + λ) = (g ∗ hk−j ) . h 2j 2k 2j On the other hand, (g ∗ hk−j )(m) (t) = g (m) ∗ hk−j (t)

(3.46)

2182


and therefore, by Lemma 3.14, −(k−j )N (g ∗ hk−j )(m) (t) cA1 A2 2 , (1 + |t|)M

0 m n − 1.

We now invoke Lemma 3.12 to obtain Gξ , Hη cA1 A2 2−(k+j ) n2 2−(k−j )N

n

2j n 2−(k−j )(N + 2 ) cA A . 1 2 (1 + 2j α)M (1 + 2j α)M

2

Proof of Theorem 3.7. Evidently, Theorem 3.7 will follow by Theorem 2.4, applied with H := sq sq sq sq L2 (Sn ), L := Fp and (X ) := fp (or L := Bp and (X ) := bp ), and Ψ the frame from Theorem 3.5, if we prove that the matrices defined in (3.30) are almost diagonal and D δ < ε,

E δ < ε for some δ > 0 and sufficiently small ε (see (2.15)). Here, we only give the argument regarding the estimate D δ < ε; the proof of the estimate

E δ < ε is the same. By the definition of the needlet ψξ for ξ ∈ Xj (j j0 ) we have 1/2

ψξ (x) := cξ

∞ ν +λ aˆ Pν (ξ · x). 2j ν=0

ˆ = 0 for t ∈ [−1/2, 1/2], there exists a constant Since aˆ ∈ C ∞ is compactly supported and a(t) A1 > 0 such that (r) a (t) A1 1 + |t| −M , 0 r N + n − 1, and t r a(t) dt = 0, r 0. R

On the other hand, from the definition of θξ in (3.21) it follows that ψη (x) − θη (x) = cη1/2

∞ ν +λ (a − b)∧ Pν (η · x), 2k

η ∈ Xk ,

ν=0

and from the construction of b we have (a − b)(r) (t) ε 1 + |t| −M , 0 r N + n − 1, t r (a − b)(t) dt = 0, 0 r N + n − 2.

and

R

We now apply Lemma 3.15 with g = a and h := a − b to obtain N + n −M 2 d(ξ, η) ψξ , ψη − θη cA1 ε min (ξ ) , (η) 1+ (η) (ξ ) max{(ξ ), (η)} and since M > J and N > max{s, J − n − s}, we get D δ < cA1 ε. However, ε is independent of c, A1 , M, and N , therefore, cA1 ε above can be replaced by ε. 2


2183

Appendix A Proof of Theorem 3.10. We need the maximal operator on Sn . Denote by G the set of all spherical caps on Sn , i.e. G ∈ G if G is of the form: G := {x ∈ Sn : d(x, η) < ρ} with η ∈ Sn and ρ > 0. The maximal operator Mt (t > 0) is defined by Mt f (x) :=

sup

G∈G : x∈G

1 |G|

f (ω)t dω

1/t x ∈ Sn .

,

G

We shall use the Fefferman–Stein vector-valued maximal inequality (see [26]): If 0 < p < ∞, 0 < q ∞, and 0 < t < min{p, q}, then for any sequence of functions f1 , f2 , . . . on Sn

∞

∞ 1/q p 1/q p

q q

fj (·) Mt fj (·)

c

j =1

j =1

L

(A.1)

L

where c = c(p, q, t, n). The next lemma will also be needed. Lemma A.1. Let 0 < t 1 and M > d/t. For any sequence of complex numbers {hη }η∈Xm , m 0, we have for x ∈ Gξ , ξ ∈ X ,

|hη | 1 +

η∈Xm

d(ξ, η) max{(ξ ), (η)}

−M

d (ξ ) t c max , 1 Mt |hη |1Gη (x). (η) η∈Xm

When (ξ ) (η), this lemma is Lemma 4.8 in [20]. The proof in the case (ξ ) > (η) is similar and will be omited (see also Remark A.3 in [7]). We shall only prove estimate (3.33). The proof of (3.34) is similar and we omit it. Let A be sq sq an almostdiagonal operator on fp with associated matrix (aξ η )ξ,η∈X and let h ∈ fp . Then (Ah)ξ = η∈X aξ η hη , where the series converges absolutely (see proof below). Then

Ah fpsq

q 1/q

−s/n−1/2

(Ah)ξ 1Gξ |Gξ | :=

Lp

ξ ∈X

q 1/q

−s−n/2

(ξ ) c

|a ||h |1 ξη η Gξ

ξ ∈X

c(Σ1 + Σ2 ),

Lp

η∈X

where

(ξ )−s−n/2 Σ1 :=

ξ ∈X

(η)≤(ξ )

(ξ )−s−n/2 Σ2 :=

ξ ∈X

q 1/q

|aξ η ||hη |1Gξ

(η)>(ξ )

and

Lp

q 1/q

|aξ η ||hη |1Gξ

Lp

.

2184


Since A δ < ∞ , we have whenever (η) (ξ ) |aξ η | c A δ

(η) J −s−n/2+δ/2 d(ξ, η) −J −δ . 1+ (ξ ) (ξ )

Choose 0 < t < min{1, p, q} so that J − d/t + δ/2 > 0. Let λξ := (ξ )−s−n/2 1Gξ . Then we have

Σ1 c

A δ ξ ∈X

n δ q 1

(η) J −s− 2 + 2 q

d(ξ, η) −J −δ

1+ |hη |λξ

p (ξ ) (ξ ) L

(η)(ξ )

q 1

q

(j −m)(J −s− n + δ ) −J −δ j

2 2 1 + 2 d(ξ, η) = c

2 |hη |λξ

j 0 ξ ∈Xj

.

Lp

η∈Xm

mj

We now apply Lemma A.1 and the maximal inequality (A.1) to obtain

q 1

q

(j −m)(J −s− n + δ − n ) Σ1

2 2 t c

2 Mt |hη |1Gη λξ

p

A δ L j 0 ξ ∈Xj

η∈Xm

mj

q 1

q

(j −m)(J − nt + 2δ )

c

2 M |h |λ t η η

j 0

q 1

q

Mt c

|hξ |λξ

Lp

ξ ∈ Xj

j 0

Lp

η∈Xm

mj

c h fpsq .

If (η) > (ξ ), then |aξ η | c A δ

(ξ ) (η)

s+d/2+δ/2 1+

d(ξ, η) (η)

−J −δ

and hence

Σ2

c

A δ ξ ∈X

n δ q 1

(ξ ) s+ 2 + 2 q

d(ξ, η) −J −δ

1+ |hη |λξ

p (η) (η) L

(η)>(ξ )

q 1

q

(m−j )(s+ n + δ ) −J −δ m

2 2 = c

2 d(ξ, η) |h |λ 1 + 2 η ξ

j 0 ξ ∈Xn

m<j

η∈Xm

Employing again Lemma A.1 and the maximal inequality (A.1) we obtain

Lp

.


2185

q 1

q

Σ2 (m−j )(s+ n2 + 2δ )

c

2 Mt |hη |1Gη λξ

p

A δ L j 0 ξ ∈Xj

η∈Xm

m<j

q 1

q

(m−j )(δ/2)

c

2 M |h |λ t η η

j 0

Lp

η∈Xm

m<j

q 1

q

M c

|h |λ t ξ ξ (x)

Lp

ξ ∈ Xj

j 0

The above estimates for Σ1 and Σ2 yield (3.33).

c h fpsq .

2

Proof of Lemma 3.11. Recall first the Dirichlet–Mehler integral representation of Gegenbauer polynomials [3, p. 177]: Pν(λ) (cos α) =

2λ Γ (λ + 12 )Γ (ν + 2λ)(sin α)1−2λ √ π ν!Γ (λ)Γ (2λ)

π α

cos (ν + λ)ϕ − λπ dϕ. (cos α − cos ϕ)1−λ

On account of (3.1), then (3.37) holds with KN (α) =

∞ ν + λ (ν + λ)(ν + n − 2)! sin λπ sin(ν + λ)α, × gˆ cos λπ cos(ν + λ)α, N ν! ν=0

Evidently,

(ν+λ)(ν+n−2)! ν!

n even, n odd.

= (ν + λ)(ν + n − 2) . . . (ν + 1) and a little algebra shows that

2 (ν + λ)(ν + n − 2)! ν + λ, 2 2 = (ν + λ) − (λ − r) × ν! 1, n−1

r=1

n even, n odd.

Let now Qn (z) be the degree n − 1 polynomial z sin λπ, z2 − (λ − r)2 × cos λπ,

n−1 2

Qn (z) :=

r=1

n even, n odd.

Then ∞ ν+λ sin(ν + λ)α, n even, KN (α) = Qn (ν + λ) × gˆ cos(ν + λ)α, n odd. N ν=0

Note that Qn (−z) = (−1)n−1 Qn (z) and Qn has zeros ±(λ − r), r = 1, . . . , n−1 2 . The critical step now is that since gˆ is even and because of the symmetry and zeros of Qn ν + λ sin(ν + λ)α, Qn (ν + λ) × KN (α) = (1/2) gˆ cos(ν + λ)α, N ν∈Z

n even, n odd.

(A.2)

2186


Let n−1 2

Rn (z) :=

r=1

−z sin λπ, n even, −z2 − (λ − r)2 × cos λπ, n odd,

which is a polynomial of degree n − 1 (related to Qn ). Then (A.2) can be rewritten in the form

d ν+λ cos(ν + λ)α KN (α) = (1/2)Rn gˆ dα N ν∈Z d ν + λ i(ν+λ)α = (1/4)Rn e gˆ . dα N

(A.3)

ν∈Z

Here we again used that the part of the sum in (A.2) with indices −(n − 1) < ν < 0 is void. Recall the Poisson summation formula: f (2πν) = (2π)−1 fˆ(ν), where fˆ(ξ ) := f (y)e−iξy dy, ν∈Z

ν∈Z

R

+λ i(ξ +λ)t and set fˆ(ξ ) := g( ˆ ξN )e . Then f (y) = N e−iλy a(N (y + t)) and (A.3) along with the summation formula give d KN (α) = (π/2)N Rn e−2πiνλ g N (α + 2πν) , dα ν∈Z

which implies (3.38).

2

References [1] O. Christensen, C. Heil, Perturbation of Banach frames and atomic decompositions, Math. Nachr. 185 (1997) 33–47. [2] R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366. [3] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. II, McGraw–Hill, New York, 1953. [4] M. Fornasier, On some stability results of localized atomic decompositions, Rend. Mat. Appl. 26 (2006) 315–325. [5] M. Fornasier, K. Gröchenig, Intrinsic localization of frames, Constr. Approx. 22 (2005) 395–415. [6] M. Frazier, B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985) 777–799. [7] M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution, J. Funct. Anal. 93 (1990) 34–170. [8] M. Frazier, B. Jawerth, G. Weiss, Littlewood–Paley Theory and the Study of Function Spaces, CBMS Reg. Conf. Ser. Math., vol. 79, Amer. Math. Soc., 1991. [9] K. Gröchenig, Localization of frames, Adv. Comput. Math. 18 (2003) 149–157. [10] K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl. 10 (2004) 105–132. [11] C.E. Heil, D.F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989) 628–666. [12] G. Kerkyacharian, P. Petrushev, D. Picard, Yuan Xu, Decomposition of Triebel–Lizorkin and Besov spaces in the context of Laguerre expansions, J. Funct. Anal. 256 (2009) 1137–1188. [13] G. Kyriazis, P. Petrushev, New Bases for Treibel–Lizorkin and Besov spaces, Trans. Amer. Math. Soc. 354 (2002) 749–776. [14] G. Kyriazis, P. Petrushev, On the construction of frames for Treibel–Lizorkin and Besov spaces, Proc. Amer. Math. Soc. 134 (2006) 1759–1770.


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[15] G. Kyriazis, P. Petrushev, Yuan Xu, Jacobi decomposition of weighted Triebel–Lizorkin and Besov spaces, Studia Math. 186 (2008) 161–202. [16] G. Kyriazis, P. Petrushev, Yuan Xu, Decomposition of weighted Triebel–Lizorkin and Besov spaces on the ball, Proc. London Math. Soc. 97 (2008) 477–513. [17] J. Lindenstrauss, L. Tsafriri, Classical Banach Spaces I, Springer-Verlag, 1977. [18] Y. Meyer, Ondelettes et Opérateurs I, Hermann, Paris, 1990. [19] F.J. Narcowich, P. Petrushev, J.D. Ward, Localized tight frames on spheres, SIAM J. Math. Anal. 38 (2006) 574– 594. [20] F.J. Narcowich, P. Petrushev, J.D. Ward, Decomposition of Besov and Triebel–Lizorkin spaces on the sphere, J. Funct. Anal. 238 (2006) 530–564. [21] J. Peetre, New Thought on Besov Spaces, Duke Univ. Math. Ser., Durham, N.C., 1993. [22] P. Petrushev, Bases consisting of rational functions of uniformly bounded degrees or more general functions, J. Funct. Anal. 174 (2000) 18–75. [23] P. Petrushev, Yuan Xu, Localized polynomial frames on the interval with Jacobi weights, J. Fourier Anal. Appl. 11 (2005) 557–575. [24] P. Petrushev, Yuan Xu, Localized polynomial frames on the ball, Constr. Approx. 27 (2008) 121–148. [25] P. Petrushev, Yuan Xu, Decomposition of spaces of distributions induced by Hermite expansion, J. Fourier Anal. Appl. 14 (2008) 372–414. [26] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. [27] H. Triebel, Theory of Function Spaces, Birkhäuser, 1993.


Optimal bounds on the Kuramoto–Sivashinsky equation Felix Otto Institute of Applied Mathematics, University of Bonn, Germany Received 13 October 2008; accepted 26 January 2009 Available online 23 July 2009 Communicated by C. Villani

Abstract In this paper, we consider solutions u(t, x) of the one-dimensional Kuramoto–Sivashinsky equation, i.e. ∂t u + ∂x

1 2 u + ∂x2 u + ∂x4 u = 0, 2

which are L-periodic in x and have vanishing spatial average. Numerical simulations show that for L 1, solutions display complex spatio-temporal dynamics. The statistics of the pattern, in particular its scaled power spectrum, is reported to be extensive, i.e. not to depend on L for L 1. More specifically, after an initial layer, it is observed that the spatial quadratic average (|∂x |α u)2 of all fractional derivatives |∂x |α u of u is bounded independently of L. In particular, the time-space average (|∂x |α u)2 is observed to be bounded independently of L. The best available result states that (|∂x |α u)2 1/2 = o(L) for all 0 α 2. In this paper, we prove that

2 1/2 = O ln5/3 L |∂x |α u

for 1/3 < α 2. To our knowledge, this is the first result in favor of an extensive behavior—albeit only up to a logarithm and for a restricted range of fractional derivatives. As a corollary, we obtain u2 1/2 O(L1/3+ ), which improves the known bounds. © 2009 Elsevier Inc. All rights reserved. Keywords: Kuramoto–Sivashinsky; Burgers; Oleinik’s estimate


F. Otto / Journal of Functional Analysis 257 (2009) 2188–2245

2189

1. Introduction 1.1. Motivation In this paper, we consider solutions u(t, x) of the one-dimensional Kuramoto–Sivashinsky equation, that is, of 1 2 u + ∂x2 u + ∂x4 u = 0, ∂t u + ∂x (1) 2 which are L-periodic in x, i.e. ∀(t, x) ∈ [0, ∞) × R,

u(t, x + L) = u(t, x).

(2)

d Clearly, because of (2), (1) preserves the spatial average of u, i.e. dt u = 0, where for any L −1 L-periodic function v(x), we use the abbreviation v := L 0 v dx. Because (1) is invariant under the Galilean transformation

t = tˆ,

x = xˆ + U t,

u = uˆ + U,

(3)

one may restrict oneself to the study of solutions with vanishing spatial average, that is ∀t ∈ [0, ∞),

u = 0.

(4)

Another important quantity is the average energy 12 u2 . Because of (2), we have 2 d 1 2 u = (∂x u)2 − ∂x2 u . dt 2

(5)

Hence it is the second-order term in (1) which pumps in energy, while the fourth-order term dissipates energy. The total average energy is not affected by the quadratic term in (1). Consider the linearization of (1) around the trivial solution u ≡ 0: ∂t u + ∂x2 u + ∂x4 u = 0.

(6)

L The spatial Fourier transform (F u)(t, q) = L−1 0 exp(iqx)u(t, x) dx, q ∈ 2πL−1 Z, evolves according to (7) ∂t (F u)(t, q) = q 2 − q 4 (F u)(t, q). Hence we see that (6) amplifies waves of length > 2π while it damps those of length < 2π . For L 1, the most unstable wave-length is O(1) and grows at a rate of O(1); the number of unstable modes is of the order O(L). However, the rate at which a wave of length 2π/|q| grows is of the order O(q 2 ) for |q| 1. Clearly, the quadratic term in (1), which does not affect the total energy density, provides an energy transfer between the modes q ∈ 2πL−1 Z. This interaction is highlighted by taking the spatial Fourier transform of (1): i ∂t (F u)(t, q) − q 2 − q 4 (F u)(t, q) = − q 2

(F u)(t, q )(F u)(t, q − q ).

q ∈2πL−1 Z

2190


To get a better feeling of how this transfer is realized, we consider the effect of the inviscid Burgers equation, i.e. ∂t u + ∂x

1 2 u = 0, 2

on initial data corresponding to a mode of length scale 2π/|q|, i.e. u(t = 0, x) = U cos |q|x . By the method of characteristics, we see that after a time of the order O(|U q|), the solution develops length scales of the order 2π/|q|. Hence the nonlinear term provides an energy transfer from long wave-lengths to short wave-lengths (and vice versa), see Fig. 2. Numerical simulations display a spatio-temporal chaotic behavior, see Fig. 1. More careful numerical experiments, for instance by Wittenberg and Holmes [16], reveal that the time-averaged power spectrum

L lim T

−1

T

T ↑∞

(F u)(t, q)2 dt

(8)

0

is independent of L for L 1, see [16, Fig. 2]. Moreover, they find

L lim T

−1

T

T ↑∞

(F u)(t, q)2 dt = O(1)

for |q| 1,

0

L lim T −1

T

T ↑∞

(F u)(t, q)2 dt = decays exponentially as |q| → ∞.

0

Incidentally, the exponential decay of the power spectrum also follows from the analyticity of solutions (established in [4]), but the exponential decay rate has not yet been proven to be Lindependent, while the above numerics imply that it should be. The numerical experiments show a similar, but less smooth, pointwise-in-time behavior of the spectrum. This suggests that for all α 0, 2 1/2 |∂x |α u =

2α

|q|

2 (F u)(t, q)

1/2 = O(1),

(9)

q∈2πL−1 Z

or at least 2 1/2 = |∂x |α u

q∈2πL−1 Z

1/2 |q|2α

= O(1),

(10)


2191

Fig. 1. A typical solution.

Fig. 2. Fourier multiplier.

T where v 2 := lim supT ↑∞ T −1 0 v 2 dt the time-space average. This conjecture is supported by a universal bound on all stationary periodic solutions of (1) with mean zero, due to Michelson [10]. Surprisingly, (10) is difficult to prove and so far, only grossly suboptimal bounds have been obtained. In the two following subsections, we sketch the methods behind the existing bounds. The method in this paper is quite different. We note that (10) is just a first step in a rigorous analysis of the Kuramoto–Sivashinsky equation: A more subtle analysis, introduced by Foias, Nicolaenko, Sell and Teman [7] to the Kuramoto–Sivashinsky equation, consists in proving the existence of a finite-dimensional global attractor or even a finite-dimensional inertial manifold, and in estimating its dimension in terms of the system size L. An extensive behavior of the Kuramoto–Sivashinsky equation, as reflected by the estimate (10), would suggest that the dimension is O(L). The work [7], starting from an estimate of the l.h.s. of (9) for α = 0 (or more precisely, of the radius of an absorbing ball in L2 ) by O(L2 ), constructs an inertial manifold of dimension O(L7/2 ). 1.2. Bounds by the background flow method The historically first bound, which is of the form 1/2 lim sup u2 O Lp , t↑∞

(11)

2192


was established by Nicolaenko, Scheurer and Temam [12]. An easy argument based on (5) shows that (11) implies 2 1/2 |∂x |α u O Lp , for all 0 α 2. Nicolaenko, Scheurer and Temam obtained (11) with p = 2, provided u is an odd function in x. They used the “background flow method” we will sketch below. The oddness assumption was removed by Goodman [9]. By the same method, Collet, Eckmann, Epstein and Stubbe [3] improved the result to p = 11/10. Recently, still by the same method, Bronski and Gambill [2] improved the result to p = 1, i.e. 1/2 O(L). (12) lim sup u2 t↑∞

Following Wittenberg [15], Bronski and Gambill [2] also argue that (12) is optimal within the background flow method. The background flow method is based on the construction of a “background flow” ζ (x), an L-periodic function. The function ζ is used to “unfold” the energy estimate (5) as follows: d 1 1 (u − ζ )2 = − ∂x ζ u2 + (∂x u − ∂x ζ )∂x u − ∂x2 u − ∂x2 ζ ∂x2 u dt 2 2 1 2 2 ∂x u − (∂x u)2 + ∂x ζ u2 − 2 1 2 2 + ∂x ζ + (∂x ζ )2 . 2

(13) (14)

In order to derive (11), one constructs ζ such that the quadratic form in u defined through line (13) is coercive, i.e. 2 2 (15) ∂x u − (∂x u)2 + ∂x ζ u2 u2 . In view of the term in line (14), ζ has then to be controlled as follows: 2 1/2 1/2 2 2 1/2 ζ + (∂x ζ )2 + ∂x ζ O Lp .

(16)

Using (∂x u)2 = −u∂x2 u 12 u2 1/2 + 12 (∂x2 u)2 1/2 , one sees that (15) is a consequence of 3 1 2 2 ∂x u + ∂x ζ u2 u2 . 2 2

(17)

Based on (4), the arguments in [9] and [3] allow to restrict the coercivity requirement (17) to L-periodic functions which in addition satisfy u(0) = 0. In view of (18), the first impulse is to take a saw-tooth profile, i.e. ∂x ζ (x) = 3/2 1 − Lδ(x) , x ∈ [−L/2, L/2),

(18)

(19)


2193

where δ denotes the Dirac distribution. However, while (17) is satisfied, the l.h.s. of (16) is infinite. Thus one has to mollify the saw-tooth profile by replacing −3/2Lδ(x) by a smooth φ(x) which is compactly supported in (−L/2, L/2) and satisfies

L/2

3 φ dx = − L while 2

−L/2

1 2

L/2

L/2

2 2 ∂x u dx +

−L/2

φu2 dx 0

(20)

−L/2

for all u(x) with (18). The ingenious idea of Bronski and Gambill was to rewrite (20) in terms of v = x −1 u and ψ = x 2 φ. Because of 1 2

L/2

L/2

(18) 2 2 ∂x u dx

−L/2

(∂x v)2 dx,

−L/2

(20) follows from

L/2 x

−2

−L/2

3 ψ dx = − L 2

L/2

L/2 (∂x v) dx +

ψv 2 dx 0

2

while −L/2

−L/2

for all v(x). L can be scaled out by x = L−1/3 x, ˆ

ψ = L2/3 ψˆ

ˆ and thus φ = L4/3 φ,

(21)

ˆ x), so that it suffices to construct a smooth ψ( ˆ supported in xˆ ∈ (−1/2, 0) ∪ (0, 1/2) s.t.

1/2 xˆ −1/2

−2

3 ψˆ d xˆ = − 2

1/2

1/2 (∂xˆ v) d xˆ + 2

while −1/2

ˆ 2 d xˆ 0 ψv

−1/2

for all v(x). ˆ This is indeed possible [2, Lemma 5], with an even ψˆ which changes sign from negative near zero to positive away from zero. In view of (21), one has as desired 1/2 2 2 1/2 1/2 2 1/2 + (∂x ζ )2 + ∂x ζ ≈ (∂x φ)2 ζ =

L−1

1/2

L/2 (∂x φ)2 dx

−L/2 (21)

1/2

= L −1/2

= O(L). This yields (12).

1/2 ˆ d xˆ (∂xˆ φ) 2

2194


1.3. Bounds by the entropy method A rather different method was introduced by Giacomelli and Otto [8] and yields a slight improvement of (12), namely 1/2 lim sup u2 = o(L).

(22)

t↑∞

The method is not oblivious to the fact that the linear part’s dispersion relation q 2 − q 4 , cf. (7), vanishes for q → 0. We now sketch the argument. It exploits the fact that (1) can be written in conservation form, i.e. 1 2 3 u + ∂x u + ∂x u = 0. (23) ∂t u + ∂x 2 Consider the local version of (5), i.e. ∂t

2 1 2 1 3 u + ∂x u + u∂x u + u∂x3 u − ∂x u∂x2 u = (∂x u)2 − ∂x2 u , 2 3

which one rewrites as ∂t

2 1 2 1 3 3 2 u + ∂x u + u∂x u − ∂x u∂x u = −u∂x2 u − ∂x2 u 2 3 2 1 1 u + ∂x2 u = u2 − 4 2 1 u2 . 4

(24)

Upon the rescaling x = Lxˆ

and u = Lu, ˆ

(25)

(23) and (24) turn into

1 2 −2 −4 3 ∂t uˆ + ∂xˆ uˆ + L ∂xˆ uˆ + L ∂xˆ uˆ = 0, 2 1 2 1 3 1 ∂t uˆ + ∂xˆ uˆ + L−4 u∂ ˆ x3ˆ uˆ − L−4 ∂xˆ u∂x2ˆ uˆ uˆ 2 . 2 3 4

(26) (27)

Hence it is natural to expect that for L 1, uˆ behaves like an entropy solution of the inviscid Burgers equation. However, entropy solutions uˆ of the inviscid Burgers equation that are 1-periodic and that have mean zero decay in time. By an indirect argument, this establishes (22).


2195

The two main ingredients for carrying out this strategy are the following: The first ingredient is that, according to [5, Theorem 2.4], uˆ ∈ L4loc ((0, ∞) × R) is an entropy solution to the inviscid Burgers equation if

1 2 uˆ = 0, 2 1 2 1 3 ∂t uˆ + ∂xˆ uˆ μ, 2 3 ∂t uˆ + ∂xˆ

in the distributional sense, provided the “entropy production measure” μ has vanishing upper H1 -density in time-space, i.e. lim r↓0

μ(Br (t, x)) = 0 for every (t, x) ∈ (0, ∞) × R. r

In particular, this condition is satisfied for

1 2 ∂t uˆ + ∂xˆ uˆ = 0, 2 1 2 1 3 1 ∂t uˆ + ∂xˆ uˆ uˆ 2 . 2 3 4 In order to apply this characterization, we need an a priori bound for uˆ in L4loc ((0, ∞) × R) that is uniform in L 1. It follows from the estimate

1

4 3/2 uˆ dt C u(0, ˆ ·)2

0

which is established in [8, Lemma 5.1]. This estimate is the second main ingredient of [8]. Its proof is motivated by a well-known qualitative argument in the theory of conservation laws, which goes back to Tartar [14] and DiPerna [6]. The argument is based on the observation that (26) and (27) imply that ∂t uˆ 1 2 uˆ , · 1 2 = ∂t uˆ + ∂xˆ ∂xˆ ˆ 2 2u 1 3 ˆ 1 2 1 3 ∂t 3u × = −∂t uˆ − ∂xˆ uˆ ∂xˆ 2 3 − 12 uˆ 2 are controlled. By a quantification of Murat’s div-curl lemma [11], this yields some control of

1 3 ˆ uˆ 1 3u = uˆ 4 . 1 2 · 1 2 ˆ 12 − 2 uˆ 2u

2196


1.4. Main result of this paper We start by introducing a couple of notations. Definition 1. (i) For any suitable L-periodic function u(x) we define the spatial average via u := L

−1

L u(x) dx. 0

(ii) For any function u(t, x) which is L-periodic in x and defined for all t 0, and for any p ∈ [1, ∞) we define the time-space average of the pth power via p |u| := lim sup T −1 T ↑∞

T

u(t, ·)p dt.

0

We note that by Jensen’s inequality we have for 1 p0 p1 < ∞: p 1/p0 p 1/p1 |u| 0 |u| 1 . We will express our result in terms of fractional, homogeneous Sobolev spaces. Definition 2. (i) For any suitable L-periodic function u(x) we define the Fourier series (F u)(q) via (F u)(q) = L−1

L exp(iqx) u(x) dx

for q ∈ 2πL−1 Z.

0

(ii) For α ∈ R, the α-fractional derivative of u, that is, the L-periodic function (|∂x |α u)(x), is defined via F |∂x |α u (q) = |q|α (F u)(q). (28) We will consider expressions of the form |∂x |α up 1/p . This is a homogeneous fractional Sobolev norm. Without the L−1 in the spatial average, it is usually denoted uH˙ α , see for instance [1, Chapter 6.2]. Notice that p

2 2 |∂x |α u = ∂xα u for α ∈ N and |∂x |α u = (−1)α/2 ∂xα u

for α ∈ 2N.

However, the proof relies on Besov spaces rather than Sobolev spaces.


2197

Definition 3. (i) For a Schwartz function φ(x), x ∈ R, we define its Fourier transform (F φ)(q), q ∈ R via

(F φ)(q) =

eiqx φ(x) dx.

(29)

R

(ii) We select a family {φk (x)}k∈Z of Schwartz functions such that their Fourier transforms {(F φk )(q)}k∈Z satisfy only for q with |q| ∈ 2−1 , 2 , (F φk )(q) = (F φ0 ) 2−k q for all k and q, (F φk )(q) = 1 for all q = 0.

(F φ0 )(q) = 0

(30) (31) (32)

k∈Z

See for instance [1, 6.1.7 Lemma] for a construction. (iii) For a suitable function u(x) which is L-periodic in x, we define the Littlewood–Paley decomposition {uk }k∈Z via uk = φk ∗ u,

(33)

where ∗ denotes convolution in the x-variable. (iv) For α ∈ R and 1 < p < ∞ and a suitable function u(x) which is L-periodic in x, we will consider expressions of the form

2

αpk

1/p

|uk |

p

.

k∈Z

This is a homogeneous Besov norm. Without the L−1 in the spatial average, it is usually denoted uB˙ α , see for instance [1, Chapter 6.2]. Notice that for p = 2, the Besov norm uB˙ α is not p,p p,p equivalent to the Sobolev norm uH˙ α . p

Theorem 1. For any γ > 5/3 there exists a constant C < ∞ with the following property: Consider any L 2 and any function u(t, x) that satisfies (1), (2) and (4). (i) Then we have |∂x |α up 1/p C lnγ L

(34)

for all (α, p) with 1/3 < α < 2

and 1 < p
5/2 there exists a constant C < ∞ with the following property: Consider any L 2 and any smooth functions u(t, x) and f (t, x), which are L-periodic in x with mean zero, and satisfy

1 2 ∂t u + ∂x u + ∂x4 u = |∂x |f , 2 1/2 lim sup u2 L. t↑∞

(38) (39)


2199

4k −γ k 2 |uk |3 + ln−γ /3 L 2 |uk |2 C 2k |fk |3/2 . ln L

(40)

Then we have

k∈Z

k∈Z

k∈Z

The assumption (39) can be replaced by any polynomial bound of the form lim supt↑∞ u2 1/2 C0 Lp . The constant C in (40) will then depend on C0 and p. Let us comment a bit on (40). This estimate suggests that on the level of the Littlewood–Paley decomposition, Eq. (38) acts, up to a logarithm, as ∂t uk + 2k uk |uk | + 24k uk = 2k fk .

(41)

Indeed, multiplication of (41) with uk , averaging in time-space and applying Hölder’s and Young’s inequalities on the l.h.s. yields (40). The insight of Theorem 2 thus is that the nonlinear term ∂x ( 12 u2 ) acts as the coercive term 2k uk |uk |. Theorem 2 relies on the Propositions 1 and 2. Proposition 1. Let 0 < α, β < 1 and 1 < p, q < ∞ be given. Let 1 < p ∗ , q ∗ < ∞ denote the dual exponents, i.e. 1/p + 1/p ∗ = 1/q + 1/q ∗ = 1. Then there exists a constant C < ∞ with the following property: Consider any L 2 and any smooth functions u(t, x), f (t, x), g(t, x) which are L-periodic in x with mean zero and satisfy

1 2 u = |∂x |(f + g), ∂t u + ∂x 2 1/2 L. lim sup u2

(42) (43)

t↑∞

Then we have 1/p∗ −1 k 1/p (1−α)p∗ k ∗ ln L |uk |p 2 |uk |3 C 2αpk |fk |p 2 k∈Z

k∈Z

+C

k∈Z

2

(1−β)q ∗ k

q∗

|gk |

1/q ∗

k∈Z

4k + C ln−1 L 2 |uk |2 .

2

βqk

1/q

|uk |

q

k∈Z

(44)

k∈Z

Let us comment a bit on (44). This estimate suggests that on the level of the Littlewood–Paley decomposition, Eq. (42) acts, up to a logarithm, as ∂t uk + 2k uk |uk | = 2k fk + 2k gk .

(45)

Indeed, multiplication of (45) with uk , averaging in time-space and applying Hölder’s and Young’s inequalities on the l.h.s. yields (44). As for Theorem 2, the main insight of Proposition 1

2200


is that the nonlinear term ∂x ( 12 u2 ) acts as the coercive term 2k uk |uk |. However, as opposed to what (45) suggests, the range of the (fractional) order of derivatives is restricted to 0 < α, β < 1. In order to apply Proposition 1 to (38), we need to set g = −|∂x |−1 ∂x4 u in (42). Hence the 4−β r.h.s. of (44) involves a norm of u in the Besov space Bq ∗ ,q ∗ , where β < 1 and thus 4 − β > 3, i.e. with more than 3 derivatives. This requires the following proposition. Proposition 2. Let 1 < p, q, p ∗ < ∞ and α ∈ R be given and related by 1/p + 1/p ∗ = 1,

p + 1 q 2p

and 0 < (6 + α)p/q − 3 < 1.

(46)

Then there exists a constant C < ∞ with the following property: If u(t, x), f (t, x) are smooth, L-periodic in x, and satisfy ∂t u + ∂x4 u = −∂x

1 2 u + |∂x |f, 2

(47)

(i) then we have

2

(3+α)pk

|uk |

p

C

k∈Z

2

((6+α)p/q−3)qk

|uk |

q

k∈Z

+

2

αpk

|fk |

p

,

(48)

k∈Z

(ii) and k∈Z

2

4k

|uk |

2

C

k∈Z

2

αpk

|fk |

p

1/p

2

(1−α)p ∗ k

p∗

|uk |

1/p∗

.

(49)

k∈Z

Let us comment a bit on (48). This estimate suggests that on the level of the Littlewood–Paley decomposition, (47) acts as ∂t uk + 24k uk 2k |uk |2 + 2k fk .

(50)

Indeed, multiplication of (50) with uk |uk |p−2 , averaging in time-space and applying Hölder’s and Young’s inequalities on the r.h.s. yields (48). The insight of Proposition 2(i) therefore is twofold: • The effect of the nonlinear term −∂x ( 12 u2 ) on the r.h.s. of (47) can be estimated by the effect of the diagonal term 2k |uk |2 . • The fourth-order term ∂x4 u on the l.h.s. of (47) acts as the coercive term 24k uk under the nonlinear test function u|u|p−2 . In this sense, it behaves as the second-order analogue −∂x2 u, which is surprising since (∂x4 u) (u|u|p−2 ) typically is non-positive as opposed to (−∂x2 u)(u|u|p−2 )—reflecting the absence of a maximum principle for the operator ∂t + ∂x4 . The crucial ingredient for Proposition 1 is a generalization of Oleinik’s E condition for the homogeneous inviscid Burgers equation, i.e. (42) with f, g ≡ 0. Oleinik’s E condition [13] states that


2201

(entropy) solutions u of the homogeneous inviscid Burgers equation satisfy a one-sided Lipschitz bound which improves over time. More precisely, for arbitrary t, τ 0 it holds τ ∂x u(t, ·) 1

⇒

∀s ∈ (0, ∞): (τ + s)∂x u(t + s, ·) 1 ,

more on this in Remark 3. We found a form in which this feature survives for the inhomogeneous inviscid Burgers equation: We consider the L2 -distance to the set of all functions with a given one-sided Lipschitz bound: Definition 4. For u(x) L-periodic and τ > 0 we define D + (u, τ ) := inf (u − ζ )2 ζ (x) smooth, L-periodic, τ ∂x ζ 1 , D − (u, τ ) := inf (u − ζ )2 ζ (x) smooth, L-periodic, τ ∂x ζ −1 . If u(t, x) is L-periodic in x we use the abbreviation D ± (t, τ ) := D ± u(t, ·), τ . The idea of monitoring the square of the L2 -distance of u to a translation-invariant set of functions has been introduced in [9, p. 296] in the context of the Kuramoto–Sivashinsky equation. Proposition 3. Let 0 < α, β < 1 and 1 < p, q < ∞ be given. Let 1 < p ∗ , q ∗ < ∞ denote the dual exponents, i.e. 1/p + 1/p ∗ = 1/q + 1/q ∗ = 1. Let the functions u(t, x), f (t, x), g(t, x) be smooth, L-periodic in x, and satisfy

1 2 u = |∂x |(f + g), ∂t u + ∂x 2 lim sup u2 < ∞.

(51) (52)

t↑∞

Let u denote the spatial shift of u by , i.e. u (t, x) = u(t, x + ). Then we have: (i) The function D + (t, τ ) satisfies the differential inequality ∂t D + + ∂τ D + + τ −1 D + ∞

1/p ∞

1/p∗ ∗ u − u p d f − f p d 4 α 1−α π 0

0

∞

1/q ∞

1/q ∗ ∗ u − u q d g − g q d 4 + β 1−β π 0

in the distributional sense.

0

(53)

2202


(ii) We have sup τ −1 D + + sup τ −1 D − τ >0

τ >0

∞

1/p ∞

1/p∗ ∗ u − u p d f − f p d 8 α 1−α π 0

0

∞

1/q ∞

1/q ∗ ∗ u − u q d g − g q d 8 + , β 1−β π 0

(54)

0

where D ± := lim supT ↑∞ T −1

T 0

D ± dt denotes the time average.

The merit of (54) is that the l.h.s. is cubic in u while the r.h.s. is only quadratic in (u, f ) and (u, g). The scale of Besov norms is appropriate to characterize both the l.h.s. and the r.h.s. of (54): Proposition 4. For any 0 < α < 1 and 1 < p < ∞ there exists a constant C < ∞ with the following property: Let u(t, x) be smooth and L-periodic in x with mean zero. (i) Then we have

2 |uk | k

3

∞ C

k∈Z

dτ , τ −1 D + + D − τ

(55)

0

(ii) and

∞ 0

dτ τ −1 D + + D − τ (ln M) sup τ −1 D + + D − τ >0

1/2 2 2 ∂x u + CM −1/2 L3 lim sup u2

for all M 1,

(56)

t↑∞

(iii) and

∞ u − u p d C 2αpk |uk |p . α 0

k∈Z

(57)


2203

2. Proofs 2.1. Proof of Theorem 1 In this subsection, we derive Theorem 1 from Theorem 2. Fix a γ > 5/3 and an 1/3 < α < 2 and let C < ∞ denote a generic constant which only depends on γ and α. In applying Theorem 2, we first note that (39) is satisfied because of (22). We then note that (38) turns into (1) for f = −|∂x |−1 ∂x2 u.

(58)

Step 1. Rewriting the r.h.s. of (40) in terms of u. It holds

2k |fk |3/2 C 2(5/2)k |uk |3/2 .

k∈Z

k∈Z

(59)

Indeed, we notice that because φk in the support of (30)–(32) we have 1 = k =k−1,k,k+1 F

of F φk , so that F φk = F φk k =k−1,k,k+1 F φk , which turns into φk = φk ∗ k =k−1,k,k+1 φk . Hence we have (33)

fk = φk ∗ f = φk ∗

φk ∗ f

k =k−1,k,k+1

(58)

= −|∂z |−1 ∂z2 φk ∗

φk ∗ u

k =k−1,k,k+1

(33)

= −|∂z |−1 ∂z2 φk ∗

uk .

k =k−1,k,k+1

Using the fundamental estimate |φ ∗ f |p 1/p

3/2 2/3

|fk |

−1 2 |∂z | ∂ φk dz z

p 1/p , R |φ| dz |f |

R

R (31) k

= 2

3/2 2/3 u k

k =k−1,k,k+1

−1 2 |∂z | ∂ φk dz z

we obtain

|uk |3/2

2/3

k =k−1,k,k+1

−1 2 |∂zˆ | ∂ φ0 d zˆ zˆ

|uk |3/2

2/3

k =k−1,k,k+1

R

C 2(3/2)k

2/3

|uk |

3/2

.

k =k−1,k,k+1

Raising to the 3/2th power, multiplying with 2k , and summing over k ∈ Z yields (59).

2204


Step 2. Estimate for the pivotal Besov norms. In this step, we argue that (36) holds for (α, p) = (1/3, 3), (2, 2). To this purpose, we derive the estimate

2(5/2)k |uk |3/2 C

k∈Z

1/2

2k |uk |3

+

k∈Z

3/4 .

24k |uk |2

(60)

k∈Z

We first argue that (60) is sufficient. In view of (59) and (60), the estimate of Theorem 2 ((40) with γ replaced by (3/2)γ > 5/2) now turns into −(3/2)γ k 4k ln L 2 |uk |3 + ln−(1/2)γ L 2 |uk |2 k∈Z

C

k∈Z

1/2

2 |uk | k

3

+

k∈Z

=C

2

4k

3/4

|uk |

2

k∈Z

1/2 (3/2)γ 1/2 −(3/2)γ k ln L L 2 |uk |3 ln k∈Z

3/4

1/4 −(1/2)γ 4k ln + ln(3/2)γ L L 2 |uk |2

.

k∈Z

By Young’s inequality, this yields 4k −(3/2)γ k L 2 |uk |3 + ln−(1/2)γ L 2 |uk |2 C ln(3/2)γ L , ln k∈Z

k∈Z

or

1/3

2k |uk |3

k∈Z

+

1/2

24k |uk |2

C lnγ L ,

k∈Z

which is (36) for (α, p) = (1/3, 3), (2, 2). It remains to argue in favor of (60), which we split into

1/2 2(5/2)k |uk |3/2 C 2k |uk |3 ,

k0

k∈Z

k0

2(5/2)k |uk |3/2 C

3/4

24k |uk |2

k∈Z

Both estimates follow from Jensen’s and Hölder’s inequalities:

.


2k k 1/2 2(5/2)k |uk |3/2 2 2 |uk |3

k0

k0

2

4k

1/2

k0

C

1/2

2 |uk | k

2205

3

k0

1/2

2k |uk |3

k∈Z

and

−(1/2)k 4k 3/4 2 |uk |2 2(5/2)k |uk |3/2 2

k0

k0

2

1/4

−2k

k0

C

2

4k

3/4

|uk |

2

k0

3/4

24k |uk |2

.

k∈Z

Step 3. Estimate of all Besov norms. In this step, we argue that (36) holds for (α, p) with (37). As we shall see, this follows from Step 2 and a straightforward interpolation inequality, which we address first. We set for abbreviation r(α) :=

10 3+α

and observe that for any 0 θ 1, α = (1 − θ )α0 + θ α1

(61)

1 1 1 = (1 − θ ) +θ . r(α) r(α0 ) r(α1 )

(62)

yields

The required interpolation estimate is

2

α r(α) k

|uk |

r(α)

1 r(α)

k∈Z

2

α0 r(α0 )k

|uk |

r(α0 )

1 r(α0 ) (1−θ)

k∈Z

2

α1 r(α1 )k

|uk |

r(α1 )

k∈Z

Indeed, it follows via Hölder’s inequality in (t, x):

|uk |r(α)

1 r(α)

(62)

|uk |r(α0 )

r(α1

0)

(1−θ)

|uk |r(α1 )

1 r(α1 ) θ

,

1 r(α1 ) θ

.

(63)

2206


which can be rewritten as 1 (61) 1 1−θ α k 1 θ 2 1 |uk |r(α1 ) r(α1 ) . 2αk |uk |r(α) r(α) 2α0 k |uk |r(α0 ) r(α0 ) Thus (63) follows from Hölder’s inequality in k. From Step 2 and (63) for α0 = 1/3 and α1 = 2 we learn that (36) holds for all (α, r(α)) with 1/3 α 2. By Jensen’s inequality we obtain the full range of (37). Step 4. Estimate of Sobolev norms. In this step, we argue that (34) holds. As we shall see, this follows from Step 3 and the following interpolation estimate (in additive form):

||∂x |α u|p

1/p

C

1/p

2α0 pk |uk |p

+

k∈Z

1/p

2α1 pk |uk |p

,

(64)

k∈Z

which holds provided (65)

α0 < α < α1 .

Indeed, since the set S of all (α, p) satisfying (35) is open, we have that for any (α, p) ∈ S, there exist α0 , α1 with (65) such that (α0 , p), (α1 , p) ∈ S. Hence (36) yields (34) via (64). The interpolation estimate (64) is standard, we reproduce its elementary proof for the convenience of the reader. Since (33) and (32) imply that u = k∈Z uk , we have by the triangle inequality |∂x |α up 1/p |∂x |α uk p 1/p . k∈Z

As in Step 1 we argue that

|∂x |α uk p 1/p C

1/p 2αk |uk |p ,

k =k−1,k,k+1

so that 1/p |∂x |α up 1/p C 2αk |uk |p . k∈Z

In view of this, (64) follows from k0

k0

2

αk

p 1/p

|uk |

C

2

α1 pk

1/p

|uk |

p

,

k∈Z

1/p 1/p 2αk |uk |p C 2α0 pk |uk |p . k∈Z

These estimates in turn follow from Hölder’s inequality:


2207

1/p 1/p 2αk |uk |p = 2(α−α1 )k 2α1 pk |uk |p

k0

k0

2

(α−α1 )kp ∗

1/p∗

k0 (65)

C

2

α1 pk

1/p

|uk |

p

k0

1/p

2α1 pk |uk |p

,

k∈Z

and similarly for the second estimate. 2.2. Proof of Theorem 2 In this subsection, we derive Theorem 2 from Proposition 1 and Proposition 2. For notational convenience, we introduce the abbreviations J :=

2k |fk |3/2 ,

k∈Z

I (α) :=

2αr(α)k |uk |r(α) ,

(66)

k∈Z 10 . We note that where r(α) is defined as in Step 3 of the proof of Theorem 1, i.e. r(α) := 3+α 1/3 1/2 I (1/3) and I (2) are just the pivotal norms. Let C < ∞ denote a generic constant which only depends on β, where we think of β as very close to 1 (in a first reading, we recommend to think of β = 1).

Step 1. Reformulation of the results of Propositions 1 and 2. In this step, we argue that for any β ∈ (1/3, 1),

7−β 3+β ln−1 L I (1/3) C J 2/3 I (1/3)1/3 + I (4 − β) 10 I (β) 10 + I (2) , I (11/3) C I (β) + J ,

(67)

I (2) CJ 2/3 I (1/3)1/3 .

(69)

(68)

We start with (67) which we derive from Proposition 1 with g = −|∂x |−1 ∂x4 u,

(70)

so that (38) turns into (42). As in Step 1 of the proof of Theorem 1, we have

2(1−β)q

∗k

|gk |q

k∈Z

∗

C

2(4−β)q

∗k

∗ |uk |q .

(71)

k∈Z

Furthermore, we apply Proposition 1 with the exponents α = 2/3,

p = 3/2,

and q = r(β)

β∈(1/3,1)

∈

(5/2, 3),

(72)

2208


so that (72)

(72)

(72)

p ∗ = 3 = r(1/3) = r(1 − α),

q ∗ = r(4 − β).

(73)

Therefore, (44) turns as desired into (67) (where we use ln L ln 2 > 0 to neglect ln−1 L in front of I (2)). We use Proposition 2(i) for the same choice of exponents, cf. (72). The last statement in (72) turns into the middle admissibility condition (46). Also the last admissibility condition in (46) is automatically satisfied since by (72): (6 + α)p/q − 3 = 10/q − 3 = β. Notice furthermore that (72)

(72)

p = 3/2 = r(11/3) = r(3 + α), so that (48) turns as desired into (68). Finally, we use Proposition 2(ii) for the same choice of exponents. In view of (73), (49) turns into (69). Step 2. Conclusion. In this step, we show that for any γ > 5/2, there exists a C < ∞ such that −γ ln L I (1/3) + ln−γ /3 L I (2) CJ, (74) which is a reformulation of (40). We will use the results from Step 1 and the interpolation inequalities I (4 − β) I (11/3) I (β) I (2)

3β−1 3+β

4(2−β) 7−β

3β−1

(75)

I (2) 7−β ,

I (1/3)

2(2−β) 3+β

(76)

.

Inequality (75) (and analogously (76)) are special cases of the interpolation inequality established in Step 3 of the proof of Theorem 1. This can best be seen by rewriting (75) as 1 1 3(2−β) 1 3β−1 I (4 − β) r(4−β) I (11/3) r(11/3) 5 I (2) r(2) 5 ,

and noticing that 1=

3(2 − β) 3β − 1 + , 5 5

4−β =

3(2 − β) 3β − 1 11/3 + 2. 5 5

The inequalities (67), (68), (69), (75) and (76) are all we need to conclude. We first eliminate I (2) with help of (69) in (67), (75) and (76): −1 7−β 3+β ln L I (1/3) C J 2/3 I (1/3)1/3 + I (4 − β) 10 I (β) 10 , I (4 − β) CJ

2(β−1/3) 7−β

I (β) CJ

I (11/3)

2(β−1/3) 3+β

4(2−β) 7−β

I (1/3)

I (1/3)

11/3−β 3+β

.

β−1/3 7−β

,

(77) (78) (79)


2209

We now substitute I (11/3) in (78) according to (68): β−1/3 22/3−2β β−1/3 4(2−β) 2(β−1/3) I (4 − β) C J 7−β I (β) 7−β I (1/3) 7−β + J 7−β I (1/3) 7−β .

(80)

We thus obtain 7−β

3+β

I (4 − β) 10 I (β) 10 (80) 2(β−1/3) β−1/3 22/3−2β β−1/3 11−3β 3+β C J 10 I (β) 10 I (1/3) 10 + J 10 I (β) 10 I (1/3) 10 6β 2 −29β+59 −6β 2 +44β−14 C J 15(3+β) I (1/3) 15(3+β) + J 2/3 I (1/3)1/3 .

(79)

Inserting this into (77) yields 6β 2 −29β+59 −6β 2 +44β−14 −1 ln L I (1/3) C J 15(3+β) I (1/3) 15(3+β) + J 2/3 I (1/3)1/3 .

An application of Young’s inequality yields −1 6β 2 −29β+59 ln L I (1/3) C ln −6β 2 +44β−14 L J + ln1/2 L J .

(81)

We notice that 6β 2 − 29β + 59 = 3/2. β↑1 −6β 2 + 44β − 14

lim

Since ln L ln 2 > 0, this implies that the first term on the r.h.s. of (81) is dominant. Hence for any γ > 5/2 there exists a constant C < ∞ such that −γ ln L I (1/3) CJ. Together with (69) this implies (74). 2.3. Proof of Proposition 1 In this subsection, we derive Proposition 1 from Propositions 3 and 4. Step 1. Sobolev vs. Besov norm. In this step, we argue that 2 2 C 24k u2k . ∂x u

(82)

k∈Z

This estimate is standard, we reproduce the argument for the convenience of the reader. By Plancherel and (33), (82) translates into the following inequality on the level of Fourier multipliers: 2 24k (F φk )(q) . |q|4 C k∈Z

2210


In order to establish this inequality, we fix an arbitrary q = 0 and let k ∈ Z be such that |q| ∈ 2k−1 , 2k .

(83)

(F φk−1 )(q) + (F φk )(q) = 1,

(84)

Then we have by (30)–(32)

so that as desired (83)

|q|4 24k 2 24k (F φk−1 )(q) + (F φk )(q) 2 2 25 24(k−1) (F φk−1 )(q) + 24k (F φk )(q) .

(84)

Step 2. Conclusion. Since (43) implies (52), we may apply Proposition 3(ii). We start from estimate (54), we replace the r.h.s. according to Proposition 4(iii). For the l.h.s. we combine Proposition 4 (i) and (ii), where we choose M = L8 , with (43) and Step 1 to

(55) 2 |uk | C k

∞

3

k∈Z

dτ τ −1 D + + D − τ

0

8(ln L) sup τ −1 D + + D −

(56)

τ >0

+ CL

−1

1/2 2 2 ∂x u lim sup u2 t↑∞

2 8(ln L) sup τ −1 D + + D − + C ∂x2 u

(43)

τ >0

8(ln L) sup τ −1 D + + D − + C 24k u2k .

(82)

τ >0

k∈Z

We use this estimate to replace the l.h.s. of (54), yielding (44). 2.4. Proof of Proposition 2 Proof of Proposition 2(i). Let C < ∞ denote a generic constant that only depends on α, p, and q. Step 1. Narrow-bandedness in Fourier space. Before embarking on the proof of Proposition 2(i) proper, we establish the following: Assume that 0 < δ 2/3 and that u(x) is a smooth L-periodic function which is narrow-banded in Fourier space in the sense that F (u)(q) = 0 for all q with |q| − 1 δ.

(85)


2211

We claim that in this case we have 4 ∂ u − up 1/p Cδ |u|p 1/p x

(86)

for some universal constant C. To showthis, we select a Schwartz function φ(z), z ∈ R, such that its Fourier transform (F φ)(q) = R exp(iqz)φ(z) dz, q ∈ R, satisfies (F φ)(q) =

1 for |q| 1 0 for |q| 2

.

(87)

Consider the rescaled and modulated version φδ of φ: φδ (z) := 2(cos z)δφ(δ z).

(88)

Since 2(cos z) = exp(iz) + exp(−iz) we have (F φδ )(q) = (F φ)((q + 1)/δ) + (F φ)((q − 1)/δ) so that by (87), we have due to δ 2/3 (F φδ )(q) = 1 for |q| − 1 δ.

(89)

Since (F (φδ ∗ u))(q) = (F φδ )(q)(F u)(q), q ∈ 2πL−1 Z, (85) and (89) imply (F (φδ ∗ u))(q) = (F u)(q), which means that φδ leaves u invariant under convolution, i.e. u = φδ ∗ u.

(90)

Now (86) follows easily because (90) implies the representation ∂x4 u − u = ∂z4 φδ − φδ ∗ u. Indeed, we obtain on the one hand 4 ∂ u − up 1/p

x

4 ∂ φδ − φδ dz |u|p 1/p . z

(91)

R

On the other hand, because of ∂z4 cos z = cos z we obtain from the representation (88) that 4 ∂z φδ − φδ (z) = 8(sin z)δ 2 ∂z φ(δz) − 12(cos z)δ 3 ∂z2 φ(δz) − 8(sin z)δ 4 ∂z3 φ(δz) + 2(cos z)δ 5 ∂z4 φ(δz), so that, since φ is a Schwartz function and δ

Inserting (92) into (91) yields (86).

1, we have

4 ∂ φδ − φδ dz Cδ. z

R

2 3

(92)

2212


Step 2. Energy estimate. In this step we argue that there exists a universal δ > 0 and C with the following property: Consider smooth functions u(t, x), v(t, x), g(t, x) which are L-periodic in x. Suppose that u satisfies the “capillary” advection equation ∂t u + v∂x u + ∂x4 u = f + g

(93)

and is narrow-banded in Fourier space as in Step 1, i.e. F u(t, ·) (q) = 0 for all t and q with |q| − 1 δ.

(94)

We claim that under these assumptions we have p |u| C |u|q + |∂x v|q + |g|q/2 + |f |p .

(95)

Indeed, let A(z) be a smooth approximation of A(z) =

1 p |z| . p

(96)

We obtain from (93): ∂t A(u) = − A (u)v∂x u − A (u)∂x4 u + A (u)g + A (u)f . Because of A (u)v∂x u = ∂x (A(u))v = −A(u)∂x v, the above turns into ∂t A(u) + A (u)∂x4 u = A(u)∂x v + A (u)g + A (u)f .

(97)

At this stage, we may carry out our approximation argument in A so that (97) holds for (96). Unfortunately, as opposed to the second-order term −A (u)∂x2 u, the fourth-order term A (u)∂x4 u is in general not a non-negative term for convex A. However, we will show with help of Step 1 that the narrow-bandedness (94) implies the positivity of this term: 1 A (u)∂x4 u = sign u|u|p−1 ∂x4 u |u|p , 2

(98)

for δ small enough. Indeed, by Step 1 we have for δ small enough: 4 ∂ u − up 1/p 1 |u|p 1/p . x 2

(99)

Inequality (98) follows from (99) via Hölder’s inequality with (p/(p − 1), p):

sign u|u|p−1 ∂x4 u = sign u|u|p−1 u + sign u|u|p−1 ∂x4 u − u 1−1/p u − ∂ 4 up 1/p . |u|p − |u|p x

We now return to (97) with A given by (96), in which we insert (98), yielding 1 p 1 p 1 p ∂t |u| + |u| |u| |∂x v| + |u|p−1 |g| + |u|p−1 |f | . p 2 p

(100)


2213

We obtain from (100) by taking the time average 1 p 1 p |u| |u| |∂x v| + |u|p−1 |g| + |u|p−1 |f | . 2 p

(101)

For the third term of the r.h.s. of (101) we use Hölder’s and Young’s inequality with (p/(p − 1), p) yielding (p−1)/p p 1/p 1 p |u| + C |f |p . |f | |u|p−1 |f | |u|p 12

(102)

Since p−1= 1−

1 1 2 p+ − q, q −p q −p q 1 2 2 1 + − + , 1= 1− q −p q −p q q 1−

1 (46) 0, q −p

2 (46) 1 − 0, q −p q

we obtain by Hölder’s and Young’s inequalities for the middle term on the r.h.s. of (101)

|u|

p−1

|g| =

1 1 2 1− q−p p q−p − q q |u| |u| |g|

1− 1 q 1 − 2 q/2 2/q q−p |u| q−p q |g| |u|p

1 p |u| + C |u|q + |g|q/2 . 12

(103)

Likewise, since

1 1 1 p= 1− p+ − q, q −p q −p q 1 1 1 1 + − + , 1= 1− q −p q −p q q we obtain for the first term on the r.h.s. of (101)

|u| |∂x v| = p

|u|

1 1− q−p p

|u|

1 1 q−p − q

q

|∂x v|

1− 1 q 1 − 1 q−p |u| q−p q |∂ v|q 1/q |u|p x

1 p |u| + C |u|q + |∂x v|q . 12

Inserting (102), (103) and (104) into (101) yields (95).

(104)

2214


Step 3. Commutator estimates. Let φ be a (universal) Schwartz function to be fixed in Step 4. Given a smooth L-periodic function v(x), we are interested in the commutator [v, φ∗] of the operation “multiplication with v” and the operation “convolution with φ”, that is, [v, φ∗]w := v(φ ∗ w) − φ ∗ (vw)

(105)

for any smooth L-periodic function w. In this step, we shall establish the two estimates [v, φ∗]∂x w q/2 C |∂x v|q 1/2 |w|q 1/2 , [v, φ∗]w q/2 C |∂x v|q 1/2 |w|q 1/2 .

(106) (107)

Both estimates rely on the elementary inequality

φ(z)v(· − tz)w(· − z) dz

q/2 2/q

R

φ(z) dz |v|q 1/q |w|q 1/q ,

(108)

R

where t ∈ [0, 1], which we shall establish for the convenience of the reader. Using Hölder’s inequality with (q/(q − 2), q, q), we obtain for the inner integral of (108):

φ(z)v(x − tz)w(x − z) dz

R

=

φ(z)1−2/q φ(z)1/q v(x − tz) φ(z)1/q w(x − z) dz

R

q/2

φ(z) dz

q/2−1

R

φ(z)v(x − tz)q dz

1/2

R

q/2

φ(z)w(x − z)q dz

1/2 .

R

Using Cauchy–Schwarz in x we conclude

R

φ(z)v(· − tz)w(· − z) dz

φ(z) dz

R

=

= R

φ(z)v(· − tz)q dz

1/2

R

φ(z) dz

R

q/2−1

q/2

φ(z) dz

q/2−1

q/2

|v|q

1/2

R

φ(z) v(· − tz)q dz

R

φ(z)w(· − z)q dz

1/2

1/2

R

|w|q

1/2

,

φ(z) w(· − z)q dz

1/2


2215

where we have used the L-periodicity in form of |v(· − tz)|q = |v|q , |w(· − z)|q = |w|q in the last identity. This establishes (108). We now turn to (107). For this purpose, we write [v, φ∗]w (x) = v(x)

=

φ(z)w(x − z) dz −

R

φ(z)v(x − z)w(x − z) dz R

φ(z) v(x) − v(x − z) w(x − z) dz

R

1

=

φ(z)z∂x v(x − tz)w(x − z) dz dt, 0 R

so that

[v, φ∗]w (x) sup t∈[0,1]

φ(z)z∂x v(x − tz)w(x − z) dz.

R

Applying (108) with φ(z) replaced by φ(z)z and v replaced by ∂x v, we obtain as desired [v, φ∗]w q/2 2/q

φ(z)z dz |∂x v|q 1/q |w|q 1/q

R

1/q q 1/q |w| C |∂x v|q . For (106) we write [v, φ∗]∂x w (x) =

φ(z) v(x) − v(x − z) ∂x w(x − z) dz

R

= R

∂z φ(z) v(x) − v(x − z) w(x − z) dz

+

φ(z)∂x v(x − z)w(x − z) dz R

1

=

∂z φ(z)z∂x v(x − tz)w(x − z) dz dt 0 R

+

φ(z)∂x v(x − z)w(x − z) dz, R

so that

2216


[v, φ∗]∂x w (x) sup t∈[0,1]

+

∂z φ(z)z∂x v(x − tz)w(x − z) dz

R

φ(z)∂x v(x − z)w(x − z) dz.

R

Thus the triangle inequality and (108) yield as desired [v, φ∗]∂x w q/2 2/q

∂z φ(z)z dz |∂x v|q 1/q |w|q 1/q

R

+

φ(z) dz |∂x v|q 1/q |w|q 1/q

R

C |∂x v|q

1/q

|w|q

1/q

.

Step 4. Non-dyadic Littlewood–Paley decomposition. Set θ˜ = 1 + δ where δ > 0 is as in Step 2. Let {φ˜ k (x)}k∈Z be a family of Schwartz functions such that their Fourier transforms {(F φ˜ k )(q)}k∈Z satisfy (F φ˜ 0 )(q) = 0 only for q with |q| ∈ θ˜ −1 , θ˜ , (F φ˜ k )(q) = (F φ˜ 0 ) θ˜ −k q for all k and q, (F φ˜ k )(q) = 1 for all q.

(109) (110) (111)

k∈Z

Usually, this construction is carried out for δ = 1, see for instance [1, 6.1.7 Lemma], but it easily adapts to the general case. We consider the related Littlewood–Paley decomposition u˜ k (t, ·) := φ˜ k ∗ u(t, ·)

and f˜k (t, ·) := φ˜ k ∗ f (t, ·),

(112)

where u and f are as in the statement of Proposition 2. In this step, we investigate u˜ 0 , v˜0 :=

k−1

u˜ k ,

and w˜ 0 :=

u˜ k .

(113)

k0

We claim that p |u˜ 0 | C |u˜ 0 |q + |∂x v˜0 |q + |w˜ 0 |q +

|f˜k |p

k=−1,0,1

.

(114)


2217

We will derive (114) from Step 2 with (u˜ 0 , v˜0 ) playing the role of (u, v). Notice that because of (109) and (112), u˜ 0 satisfies (94) (using (θ˜ −1 , θ˜ ) = ((1 + δ)−1 , 1 + δ) ⊂ (1 − δ, 1 + δ)). It thus remains to show that we have ∂t u˜ 0 + v˜0 ∂x u˜ 0 + ∂x4 u˜ 0 = |∂x |f˜0 + g,

(115)

with ||∂x |f˜0 |p C

|f˜k |p ,

(116)

k=−1,0,1

|g|q/2 C |∂x v˜0 |q + |w˜ 0 |q .

(117)

Estimate (116) follows as in Step 1 of the proof of Theorem 1. For (117), we start by rewriting Eq. (47). Since by (111) and (112) and (113) we have u = v˜0 + w˜ 0 , we obtain ∂x

1 2 u = u∂x u 2 = v˜0 ∂x u + w˜ 0 ∂x u = v˜0 ∂x u + w˜ 0 ∂x v˜0 + w˜ 0 ∂x w˜ 0 1 2 w˜ 0 . = v˜0 ∂x u + w˜ 0 ∂x v˜0 + ∂x 2

Hence we obtain from (47) ∂t u + v˜0 ∂x u + ∂x4 u = −w˜ 0 ∂x v˜0 − ∂x

1 2 w˜ 0 + |∂x |f. 2

We now apply the operator φ˜ 0 ∗ to this identity and recall the definitions (112). Because of φ˜ 0 ∗ (v˜0 ∂x u) = v˜0 (φ˜ 0 ∗ ∂x u) − [v˜0 , φ˜ 0 ∗]∂x u = v˜0 ∂x u˜ 0 − [v˜0 , φ˜ 0 ∗]∂x u, we obtain (115) with r.h.s. g defined by g = [v˜0 , φ˜ 0 ∗]∂x u − φ˜ 0 ∗ (w˜ 0 ∂x v˜0 ) − φ˜ 0 ∗ ∂x = [v˜0 , φ˜ 0 ∗]∂x v˜0

1 2 w˜ 2 0

(118)

+ [v˜0 , φ˜ 0 ∗]∂x w˜ 0

(119)

− φ˜ 0 ∗ (w˜ 0 ∂x v˜0 ) 1 + ∂z φ˜ 0 ∗ w˜ 02 . 2

(120) (121)

It remains to check the estimates (116) and (117). For (118) we obtain by (107) applied to ˜ v, w) = (φ˜ 0 , v˜0 , ∂x v˜0 ) (φ, [v˜0 , φ˜ 0 ∗]∂x v˜0 q/2 C |∂x v˜0 |q .

2218


˜ v, w) = (φ˜ 0 , v˜0 , w˜ 0 ) For (119) we appeal to (106) applied to (φ, [v˜0 , φ˜ 0 ∗]∂x w˜ 0 q/2 C |∂x v˜0 |q 1/2 |w˜ 0 |q 1/2 C |∂x v˜0 |q + |w˜ 0 |q . For (120) we argue as follows: φ˜ 0 ∗ (w˜ 0 ∂x v˜0 )q/2

|φ˜ 0 | dz

q/2

|w˜ 0 ∂x v˜0 |q/2

R

1/2 1/2 C |∂x v˜0 |q |w˜ 0 |q C |∂x v˜0 |q + |w˜ 0 |q . For (121) finally we notice ∂z φ˜ 0 ∗ w˜ 2 q/2 C 0

|∂z φ˜ 0 | dz

q/2

|w˜ 02 |q/2 C |w˜ 0 |q .

R

This concludes the proof of (117). Step 5. Scaling. In this step, we argue that for any ∈ Z ˜θ (3+α)p |u˜ |p C θ˜ ((6+α)p/q−4)q |∂x v˜ |q + θ˜ ((6+α)p/q−3)q |w˜ |q θ˜ αp |f˜k |p ,

+ θ˜ ((6+α)p/q−3)q |u˜ |q +

(122)

k=−1,,+1

where v˜ and w˜ are defined analogously to v˜0 and w˜ 0 in (113), that is, v˜ := u˜ k , w˜ := u˜ k . k−1

(123)

k

Indeed, we notice that the change of variables x = θ˜ − x, ˆ

t = θ˜ −4 tˆ,

ˆ u = θ˜ 3 u,

leaves (47) invariant. Notice that (110) translates into φ˜ k (z) = θ˜ k φ˜ 0 θ˜ k z ,

f = θ˜ 6 fˆ

(124)

(125)

so that θ˜ − φ˜ k θ˜ − zˆ = φ˜ k− (ˆz). Hence we deduce from (124) the following relation between the Littlewood–Paley decompositions u˜ k = θ˜ 3 u˜ k− ,

˜k− . f˜k = θ˜ 6 f


2219

In particular, we have u˜ = θ˜ 3 u ˜ 0,

∂x v˜ = θ˜ 4 ∂xˆ v ˜0 ,

w˜ = θ˜ 3 w ˜ 0.

Hence (114), applied to (tˆ, x, ˆ u, ˆ fˆ) yields in terms of (t, x, u, f ): ˜θ −3p |u˜ |p C θ˜ −4q |∂x v˜ |q + θ˜ −3q |w˜ |q

+ θ˜ −3q |u˜ |q + θ˜ −6p

|f˜k |p

.

k=−1,,+1

Multiplication with θ˜ (6+α)p yields (122). Step 6. Estimate in Besov norm based on a non-dyadic Littlewood–Paley decomposition. In this step we establish the analogue of the statement of Proposition 2(i) for the Besov norm based on the non-dyadic Littlewood–Paley decomposition from Step 4. More precisely, we will make use of (46)

α := (6 + α)p/q − 3 ∈ (0, 1) to show that

αp p ˜ ˜θ (3+α)p |u˜ |p C ˜θ α q |u˜ |q + ˜θ . |f |

∈Z

∈Z

(126)

∈Z

Starting point is (122), which we sum over ∈ Z to obtain

α q

θ˜ (3+α)p |u˜ |p C θ˜ (α −1)q |∂x v˜ |q + θ˜ |w˜ |q

∈Z

∈Z

+

θ˜

∈Z

α q

αp |u˜ |q + |f˜ |p . θ˜

∈Z

∈Z

Hence it remains to show

α q

θ˜ (α −1)q |∂x v˜ |q θ˜ |u˜ |q ,

∈Z

(127)

∈Z

α q

θ˜ α q |w˜ |q θ˜ |u˜ |q .

∈Z

(128)

∈Z

Estimate (127) relies on α < 1, whereas (128) requires α > 0. Since the argument for (128) is similar, we restrict ourselves to (127). We start by noticing that with help of the triangle inequality we obtain from (123)

|∂x v˜ |q

1/q

|∂x u˜ k |q

k−1

1/q

.

(129)

2220


We further remark that as in Step 1 of the proof of Theorem 1

|∂x u˜ k |q

1/q

C θ˜ k

|u˜ k |q

1/q

.

(130)

k =k−1,k,k+1

We combine (129) and (130) to

1/q 1/q

θ˜ (α −1)+k |u˜ k |q C . θ˜ (α −1) |∂x v˜ |q

(131)

k

Introducing the abbreviations 1/q

A := θ˜ (α −1) |∂x v˜ |q ,

1/q

Bk := θ˜ α k |u˜ k |q ,

(131) and (127) translate into A C

θ˜ −(1−α )(−k) Bk ,

k

q

A C

∈Z

q

(132) (133)

Bk .

k∈Z

Notice that (132) states that {A }∈Z can be estimated by the discrete convolution of {Bk }k∈Z with the sequence

0

for k > 0

θ˜ −(1−α )|k |

for k 0

.

Since this sequence is summable (because of α < 1), this observation entails (133). Step 7. Conclusion of Proposition 2(i). In view of Step 6, we just have to show that the Besov norms based on the non-dyadic Littlewood–Paley decomposition introduced in Step 4 and those based on the dyadic Littlewood–Paley decomposition are comparable. Hence let {g }∈Z be the dyadic Littlewood–Paley decomposition of some given L-periodic function g, see Definition 3. In view of (126), it remains to show for arbitrary 1 < p < ∞ and α ∈ R

2αpk |gk |p , θ˜ αp |g˜ |p C

∈Z

2

∈Z

k∈Z

θ˜ αpk |g˜ k |p . |g |p C

αp

k∈Z

To symmetrize the argument, we generalize {g }∈R to be a Littlewood–Paley decomposition w.r.t. 2 replaced by some θ > 1. It then remains to show ∈Z

θ˜ αpk |g˜ k |p . θ αp |g |p C k∈Z

(134)


2221

We set for abbreviation λ :=

ln θ ∈ (0, ∞) ln θ˜

so that θ = θ˜ λ .

(135)

Notice that −1 +1 [(−1)λ] [(+1)λ]+1 θ ,θ ⊂ θ˜ , , θ˜

(136)

where [x] denotes the largest integer x. We retain for later purpose that for all ∈ Z ( + 1)λ + 1 − ( − 1)λ ( + 1)λ + 1 − ( − 1)λ − 1 2λ + 2 C

(137)

and that for all k ∈ Z # ∈ Z k ∈ ( − 1)λ , . . . , ( + 1)λ + 1 # ∈ Z k > ( − 1)λ − 1 and k ( + 1)λ + 1 1 1 = # ∈ Z < (k + 1) + 1 and (k − 1) − 1 λ λ 1 1 (k + 1) + 1 − (k − 1) − 1 λ λ =

2 + 2 C. λ

(138)

By the properties (109)–(111) we have supp F φ ⊂ θ −1 , θ +1 ∪ −θ −1 , −θ +1 , k+

F φ˜ k = 1 on θ˜ k− , θ˜ k+ ∪ −θ˜ k− , −θ˜ k+ ,

k=k−

so that (136) implies F φ = F φ

[(+1)λ]+1

F φ˜ k .

k=[(−1)λ]

As in Step 6, this translates into g = φ ∗ g = φ ∗

[(+1)λ]+1 k=[(−1)λ]

From this representation we obtain the estimate

φ˜ k ∗ g = φ ∗

[(+1)λ]+1 k=[(−1)λ]

g˜ k .

2222


p 1/p

|g |

R

[(+1)λ]+1 p 1/p |φ | dz g˜ k k=[(−1)λ]

|φ0 | d zˆ

|g˜ k |p

1/p

k=[(−1)λ]

R

C

[(+1)λ]+1

[(+1)λ]+1

|g˜ k |p

1/p

.

k=[(−1)λ]

We thus obtain by Hölder’s inequality in k

|g |

p

p−1 [(+1)λ]+1 C ( + 1)λ + 1 − ( − 1)λ |g˜ k |p k=[(−1)λ] (137)

C

[(+1)λ]+1

|g˜ k |p .

k=[(−1)λ]

Summation over ∈ Z yields

[(+1)λ]+1 θ αp |g |p C θ αp |g˜ k |p

∈Z

∈Z k=[(−1)λ] (135)

C

[(+1)λ]+1

θ˜ αpk |g˜ k |p

∈Z k=[(−1)λ]

= C # ∈ Z k ∈ ( − 1)λ , . . . , ( + 1)λ + 1 θ˜ αpk |g˜ k |p k∈Z (138)

C

θ˜ αpk |g˜ k |p .

k∈Z

This establishes (134). Proof of Proposition 2(ii). Let C < ∞ denote a generic universal constant. Step 1. Energy estimate. In this step, we argue 2 2 ∂x u |∂x |f u . Indeed, multiplication of (47) with u, averaging over x and integration by parts yields d 1 2 2 2 u + ∂x u = |∂x |f u |∂x |f u . dt 2 Averaging over t yields (139).

(139)


2223

Step 2. Sobolev vs. Besov norms. In this step, we argue

2 24k u2k C ∂x2 u .

(140)

k∈Z

Estimate (140) is standard; we reproduce the easy argument for the convenience of the reader. By Plancherel and (33), (140) translates into the following inequality on the level of Fourier multipliers:

2 24k (F φk )(q) C|q|4 .

k∈Z

In order to establish this inequality, we fix an arbitrary q = 0 and let ∈ Z be such that |q| ∈ 2−1 , 2 .

(141)

Then we have by (30) and (31)

2 (141) 4k 2 24k (F φk )(q) = 2 (F φk )(q) k=−1,

k∈Z

2 = sup(F φ0 )(q) ˆ 24(−1) + 24 qˆ

(141)

C|q|4 .

Step 3. In this step, we argue ∗ 1/p (1−α)p∗ k 1/p αpk p p∗ |∂x |f u C |fk | |uk | 2 2 . k∈Z

(142)

k∈Z

Also estimate (142) is classical, we reproduce the easy argument for the convenience of the reader. By (33) and (32) we have u=

f=

uk ,

k∈Z

fk .

k∈Z

We thus obtain by Hölder’s inequality |∂x |f u (30),(31) =

k∈Z k =k−1,k,k+1

|∂x |fk uk

|∂x |α fk |∂x |1−α uk

k∈Z k =k−1,k,k+1

k∈Z k =k−1,k,k+1

1/p

|∂x |α fk p

2224


×

1/p∗

∗ |∂x |1−α uk p

k∈Z k =k−1,k,k+1

C

∗ 1/p ∗ 1/p |∂x |α fk p |∂x |1−α uk p . k∈Z

(143)

k∈Z

As in Step 1 of the proof of Theorem 1, we have |∂x |α fk p C 2αpk |fk |p , k∈Z

k∈Z

∗ ∗ ∗ |∂x |1−α uk p C 2(1−α)p k |uk |p . k∈Z

k∈Z

Inserting this into (143), we obtain ∗ 1/p (1−α)p∗ k 1/p αpk p p∗ |∂x |f u = C |fk | |uk | 2 2 . k∈Z

k∈Z

Estimate (142) follows from this by averaging in t and applying Hölder’s inequality. 2.5. Proof of Proposition 4 Remark 2. After time average, Proposition 4(i) follows from

∞

2 |uk | C k

3

k∈Z

dτ . τ −1 D + (u, τ ) + D − (u, τ ) τ

(144)

0

We remark that (144) would reduce to a classical statement in real interpolation theory if the sum D + (u, τ ) + D − (u, τ ) of the one-sided controls is replaced by the two-sided control D(u, τ ) = inf (u − ζ )2 τ |∂x ζ | 1 . Indeed, with this substitution, (144) would turn into u; B˙ 1/3 3 C

∞

3,3

dτ 1 τ −1 τ −1 inf u − ζ ; L2 2 ζ ; H˙ ∞ τ

0

∞ =C

ds 1 s s inf u − ζ ; L2 2 ζ ; H˙ ∞ s

0

∞ 2 ds 1/2 1 s . =C s inf u − ζ ; L2 ζ ; H˙ ∞ s 0

(145)


2225

The validity of (145) can be seen as follows. On the one hand, we have by the relation of approximation and real interpolation spaces 1 3 u; H˙ , L2 C ∞ 2/3,3

∞ 2 ds 1/2 1 s , s inf u − ζ ; L2 ζ ; H˙ ∞ s 0

1 , L ), α = 1/2, r = 2 and thus θ = 2/3, q = 3. cf. [1, 7.1.7 Theorem] with A¯ = (H˙ ∞ 2 On the other hand, we have by the standard results on real interpolation of Besov spaces

u; B˙ 1/3 C u; B˙ 1 , B˙ 0 ∞,∞ 2,2 2/3,3 3,3 1 C u; H˙ ∞ , L2 2/3,3 .

(146) (147)

Inequality (146) is an application of [1, 6.4.5 Theorem (3)] with s0 = 1, p0 = ∞, q0 = ∞, s1 = 0, p1 = 2, q1 = 2, θ = 2/3 and thus p ∗ = q ∗ = 3, s ∗ = 1/3. Inequality (147) follows from 0 ([1, 6.4.4 Theorem]). 1 ⊂B 1 ˙ ∞,∞ ([1, 6.3.1 Theorem]) and L2 = B˙ 2,2 H˙ ∞ Proof of Proposition 4(i). Let C < ∞ denote a generic universal constant. Step 1. Adapted Littlewood–Paley decomposition. Due to the only one-sided control of ∂x ζ in the definition D ± (τ ), we cannot right away work with the Littlewood–Paley decomposition from Definition 3. We need to replace {φk }k∈Z by {φ˜ k }k∈Z where φ˜ k are derivatives of non-negative functions. More precisely, we select a Schwartz function ψ0 with the properties

ψ0 (z) 0 for z ∈ R, (F ψ0 )(q) > 0 for q ∈ R, ψ0 dz = 1 (148) R

(take for instance a Gaussian) and define {φ˜ k }k∈Z via φ˜ k (z) = 2k φ˜ 0 2k z .

φ˜ 0 = ∂z ψ0 ,

(149)

For a given L-periodic function v(x) we introduce {v˜k }k∈Z via v˜k = φ˜ k ∗ v.

(150)

In this step, we argue that {vk }k∈Z and {v˜k }k∈Z are comparable in the sense of v˜k2 C v 2 , k∈Z

2k |vk |3 C 2k |v˜k |3 .

(152)

k∈Z

We start with (151). By definition (150) and Plancherel, (151) follows from (F φ˜ k )(q)2 C for all q ∈ R. k∈Z

(151)

k∈Z

(153)

2226


This estimate can be seen as follows. We note that by definition, (149) (149) (F φ˜ k )(q) = (F φ˜ 0 ) 2−k q = 2−k q(F ψ0 ) 2−k q .

(154)

Since ψ0 is a Schwartz function, we have in particular (F ψ0 )(q) ˆ −2 , ˆ C min 1, |q| so that (154) yields (F φ˜ k )(q) C min 2−k |q|, 2k |q|−1 . Hence we obtain as desired (F φ˜ k )(q)2 C min 4−k |q|2 , 4k |q|−2 k∈Z

k∈Z

4−k |q|2 +

=C

k,4−k |q|2 1

4k |q|−2

k,4−k |q|2 1

C. We now address (152). We actually will show that for every k ∈ Z, |vk |3 C |v˜k |3 .

(155)

In view of (149) and (150) and by rescaling (cf. (31) and (149)), it is sufficient to show |v0 |3 C |v˜0 |3 .

(156)

For that purpose, we note that (33)

(150)

F v0 = (F φ0 )(F v) =

F φ0 F v˜0 = mF v˜0 , F φ˜ 0

where the Fourier multiplier m(q) :=

(F φ0 )(q) (149) −1 (F φ0 )(q) = |q| (F ψ0 )(q) (F φ˜ 0 )(q)

is a Schwartz function in q ∈ R by the second property in (148) and since (F φ0 )(q) = 0 for |q| ∈ / (1/2, 2), cf. (30). Hence there exists a Schwartz function η(x) with F η = m and thus v0 = η ∗ v˜0 , which yields (156). Step 2. In this step, we argue that

2 max |u˜ k | − 2−k τ −1 , 0 C D + (τ ) + D − (τ ) .

k∈Z

(157)


2227

We split (157) into

2 max u˜ k − 2−k τ −1 , 0 CD + (τ ),

k∈Z

2 max −u˜ k − 2−k τ −1 , 0 CD − (τ ).

(158)

k∈Z

˜ k , it suffices to show (158). By definition Because of D − (u˜ k , τ ) = D + (−u˜ k , τ ) and −u˜ k = −u of D + (τ ), there exists an L-periodic ζ (x) such that (159) τ ∂x ζ 1 and (u − ζ )2 2D + (τ ). It thus suffices to show

2 max u˜ k − 2−k τ −1 , 0 C (u − ζ )2 .

(160)

k∈Z

Notice that by definition of {φ˜ k }k∈Z (149) (149) φ˜ k (z) = 2k φ˜ 0 2k z = 2k ∂z ψ0 2k z = ∂z ψ0 2k z , and thus by definition of {ζ˜k }k∈Z (150) τ 2k ζ˜k = τ 2k φ˜ k ∗ ζ = 2k ψ0 2k · ∗ (τ ∂x ζ ). Since by the choice of ψ0 , (148) 2k ψ0 2k z 0,

2k ψ0 2k z dz =

R

(148)

ψ0 (ˆz)d zˆ = 1, R

we obtain from (161) and τ ∂x ζ 1, cf. (159), τ 2k ζ˜k 1,

ζ˜k 2−k τ −1 .

i.e.

From this we deduce max u˜ k − 2−k τ −1 , 0 max{u˜ k − ζ˜k , 0}, so that

2 max u˜ k − 2−k τ −1 , 0 max{u˜ k − ζ˜k , 0}2

k∈Z

k∈Z

k∈Z

Now (160) follows from this and Step 1, cf. (151).

(u˜ k − ζ˜k )2 .

(161)

2228


Step 3. Conclusion. We have according to Steps 1 and 2:

∞ τ

−1

∞ + dτ (157) −1 2 dτ − −k −1 D (τ ) + D (τ ) C max |u˜ k | − 2 τ , 0 τ τ2

0

= C

−1

k∈Z

0

k

2

k∈Z

= C −1

max{|u˜ k | − s, 0} ds 2

2k

k∈Z (152)

C −1

∞ 0

1 |u˜ k |3 3

2k |uk |3 .

k∈Z

Proof of Proposition 4(ii). Step 1. In this step, we argue that for all t, τ 0 D ± (t, τ ) u(t, ·)2 , 2 1/2 1. D ± (t, τ ) = 0 provided τ L ∂x2 u(t, ·)

(162) (163)

The variable t is just a parameter in (162) and (163), which we suppress. Inequality (162) is obvious, since ζ ≡ 0 is admissible in the definition of D ± (t, τ ). We now address inequality (163). We observe that since ∂x u is L-periodic with mean zero, (∂x u)2 vanishes at some point in (0, L). Since any point has at most distance L/2 to that point, we have

sup(∂x u)2 x

1 2

L

∂x (∂x u)2 dx

0

L =

|∂x u|∂x2 u dx

0

L

L (∂x u)2 dx

0

2 2 ∂x u dx

0

L L

1/2

1/2

sup |∂x u| x

2 2 ∂x u dx

1/2

0

so that L sup |∂x u| L

1/2

x

0

2 2 ∂x u dx

1/2 ,

,


2229

which can be rewritten as 2 1/2 sup |∂x u| L ∂x2 u .

(164)

x

Hence for τ with τ L(∂x2 u)2 1/2 1, ζ = u is admissible in the definition of D ± (·, τ ). This yields (163). Step 2. In this step, we argue that for all 0 < τ− τ+ < ∞

∞

dτ 2 τ+−1 L4 ∂x2 u , τ −1 D ± τ

(165)

dτ 2 τ− L2 lim sup u2 ∂x2 u . τ −1 D ± τ t↑∞

(166)

τ+

τ− 0

Inequalities (165) and (166) follow from

∞

2 dτ τ+−1 L4 ∂x2 u , τ

(167)

2 dτ τ− L2 u2 ∂x2 u τ

(168)

τ −1 D ±

τ+

τ−

τ −1 D ±

0

by averaging in time t. The variable t is just a parameter in (167) and (168), which we suppress. We first address (167). Since u is L-periodic with mean zero, we have analogously to (164) 1/2 sup |u| L (∂x u)2 . x

The combination of (164) and (169) yields 2 1/2 2 1/2 u L2 ∂x2 u , so that (162) implies 2 D ± (·, τ ) L4 ∂x2 u . Inequality (167) now follows by integration in τ ∈ (τ+ , ∞). For (168) we notice that (162) and (163) combine to 2 1/2 2 2 D ± (·, τ ) u2 τ L ∂x2 u = τ 2 L2 u2 ∂x2 u . Integration in τ ∈ (0, τ− ) yields (168).

(169)

2230


Step 3. Conclusion. In this step, we establish (56). Splitting the τ -integral into (0, τ− ), [τ− , τ+ ], and (τ+ , ∞), for 0 < τ− τ+ < ∞ to be optimized later, we obtain from (165) and (166):

∞ 0

dτ 2 τ −1 D + + D − τ− L2 lim sup u2 ∂x2 u τ t↑∞ τ+ sup τ −1 D + + D − + ln τ− τ >0 2 + τ+−1 L4 ∂x2 u .

We reformulate this inequality in terms of M =

∞ 0

τ+ τ−

1 and s = τ+ τ− ∈ (0, ∞):

dτ 2 M −1/2 s 1/2 L2 lim sup u2 ∂x2 u τ −1 D + + D − τ t↑∞ + (ln M) sup τ −1 D + + D − τ >0

+M

−1/2 −1/2 4

s

L

2 2 ∂x u .

Optimization in s ∈ (0, ∞) at fixed M yields (56). Proof of Proposition 4(iii). This is a classical statement, see for instance [1, 6.2.5 Theorem]. For the convenience of the reader, we reproduce the elementary proof. Let C < ∞ denote a generic constant which only depends on α and p. Estimate (57) follows from the estimate

∞ u − u p d 2αpk |uk |p α C

(170)

k∈Z

0

by integration in time. Since time is just a parameter in (170), we restrict our attention to functions u(x). Step 1. In this step, we argue that u − u p 1/p 1/p 1−α k −α αk 2 |uk |p min 2k , 2 . α

(171)

k∈Z

By (32) and (33) we have u =

k∈Z uk ,

so that we obtain by the triangle inequality

p 1/p u − u p 1/p u k − uk . α α k∈Z

(172)


2231

As in Step 1 of the proof of Theorem 1 we have uk = φk ∗ k =k−1,k,k+1 uk so that uk . u k − uk = φk − φk ∗ k =k−1,k,k+1

This entails by the triangle inequality u − uk p 1/p k

φ − φk dz

k

|uk |p

1/p

.

k =k−1,k,k+1

R

On the other hand, we have

2k φ − φk dz (31) φ = − φ0 d zˆ C min 2k , 1 . k 0 R

R

The combination of the two last estimates yields uk − uk p 1/p 1/p |uk |p C min 2k 1−α , −α α k =k−1,k,k+1 1/p k 1−α k −α αk 2 |uk |p C min 2 , 2 . k =k−1,k,k+1

We sum this estimate over k, plug the result into (172); this yields (171). Step 2. Conclusion. The idea is to divide the -axis into dyadic intervals. By Step 1 we have 2−

1/p u − u p d B := α 2−−1

u − u p 1/p α ∈(2−−1 ,2− )

(171)

C

max

k∈Z

=: C

1−α k− −α αk 1/p 2 |uk |p min 2k− , 2 1−α k− −α Ak . min 2k− , 2

k∈Z

Hence {B }∈Z is estimated by the discrete convolution of {Ak }k∈Z with the sequence 1−α k −α . , 2 min 2k Because of 0 < α < 1, the latter sequence is summable. This implies p p B C Ak , ∈Z

k∈Z

which in view of the definition of {Ak }k∈Z and {B }∈Z turns into (170).

2232


2.6. Proof of Proposition 3 The most transparent way of proving Proposition 3(i) is by operator splitting. More precisely, we decompose the evolution defined through (51) into the three components ∂t u + ∂x

1 2 u = 0, 2

(173)

∂t u = |∂x |f,

(174)

∂t u = |∂x |g.

(175)

We monitor the evolution of D + (t, ·) individually in Lemma 1 and Lemma 2 for (173) and (174) and (175), respectively. Moreover, it is more revealing to generalize the definition of D + (t, ·) from the exponent r = 2 to a general 1 r < ∞: Definition 5. Let 1 r < ∞ be given. For any smooth function u(x) which is L-periodic in x and any τ > 0 we consider Dr (u, τ ) defined through (176) Dr (u, τ ) := inf |u − ζ |r ζ (x) smooth, L-periodic, τ ∂x ζ 1 . For any smooth function u(t, x) which is L-periodic in x we write Dr (t, τ ) = Dr (u(t, ·), τ ). We note that Dr (t, τ ) is locally Lipschitz continuous in (t, τ ). Indeed, we easily obtain from (176) and the triangle inequality that for t1 t0 and for τ1 τ0 r 1/r 1/r 1/r Dr (t1 , τ ) − Dr (t0 , τ ) u(t1 , ·) − u(t0 , ·) , r 1/r τ0 1/r 1/r u(t, ·) . Dr (t, τ1 ) − Dr (t, τ0 ) 1 − τ1 Lemma 1. Let u(t, x) be a smooth function which is L-periodic in x and satisfies the homogeneous inviscid Burgers equation (173). Then for any 1 r < ∞, Dr (t, τ ) satisfies the differential inequality ∂t Dr + ∂τ Dr + (r − 1)τ −1 Dr 0

(177)

in a distributional sense. Remark 3. (i) Notice that the linear differential inequality (177) can be rewritten as d (t + τ )r−1 Dr (t, τ + t) 0. dt Its integrated version reads Dr (t, t + τ )

τ t +τ

r−1 Dr (0, τ ).

(178)


2233

Hence we have in particular for any fixed τ > 0: Dr (0, τ ) 0

⇒

∀t 0 Dr (t, τ + t) 0,

which in view of the Definition 5 translates into τ ∂x u(0, ·) 1

⇒

∀t 0 (τ + t)∂x u(t, ·) 1.

(179)

This is Oleinik’s celebrated E-condition. (ii) Notice that for r ↓ 1, (178) turns into D1 (t, τ + t) D1 (0, τ ).

(180)

This inequality also follows easily from well-known principles, as we shall point out now: Let ζ0 (x) be smooth and near-optimal in the definition of D1 (0, τ ), i.e. u(0, ·) − ζ0 ≈ D1 (0, τ ). (181) Solve the inviscid homogeneous Burgers equation ∂t ζ + ∂x ( 12 ζ 2 ) = 0 with initial data ζ0 ; ζ (t, x) will be smooth for sufficiently small times. • On the one hand, by Oleinik’s E-condition (179), τ ∂x ζ0 1 implies (τ + t)∂x ζ (t, ·) 1. Hence ζ (t, ·) is admissible in D1 (t, τ + t) so that (182) D1 (t, τ + t) u(t, ·) − ζ (t, ·) . • On the other hand, we have by the L1 -contraction principle for the inviscid Burgers equation (173) that u(t, ·) − ζ (t, ·) u(0, ·) − ζ0 . (183) The combination of (181)–(183) yields (180). Whereas r = 2 does not play a special role for (173), it does so for (174). Lemma 2. Let u(t, x) and f (t, x) be a smooth functions which are L-periodic in x and satisfy (174). Then for any 0 < α < 1 and 1 < p < ∞, D2 (t, τ ) satisfies the differential inequality 4 ∂t D2 π

∞

1/p ∞

1/p∗ ∗ u − u p d f − f p d α 1−α 0

0

in a distributional sense. Proof of Lemma 1. Since Dr is Lipschitz continuous in (t, τ ), cf. Definition 5, and by translation invariance in time, it is enough to show (177) in the following integrated version:

t Dr (t, τ + t) + (r − 1) 0

(τ + t )−1 Dr (t , τ + t ) dt Dr (0, τ )

(184)

2234


for all t 0 and τ > 0. Indeed, a Lipschitz function is classically differentiable almost everywhere and its classical derivative agrees with its weak derivative. Since u is smooth, it can be uniformly approximated (locally in time) by solutions u (t, x) of the homogeneous viscous Burgers equation, i.e. ∂t u + ∂x

1 2 u − ∂x2 u = 0. 2

(185)

This can be seen by interpreting u as an approximate solution of (185) with a right-hand side which uniformly tends to zero as ↓ 0, and using the comparison principle for the viscous Burgers equation to compare u to u . Therefore, w.l.o.g., we may assume that u solves (185) for some > 0 instead of (173). To establish (184), we employ a strategy like in Remark 3(ii). Let τ > 0 be arbitrary and ζ0 be admissible in the definition of Dr (0, τ ), i.e. τ ∂x ζ0 1.

(186)

Since u is smooth and since > 0, there exists a smooth function ζ (t, x), L-periodic in x, that solves the inhomogeneous viscous Burgers equation with initial data ζ0 : ∂t ζ + ∂x

1 2 1 ζ − ∂x2 ζ = 1 − (u − ζ ) (τ + t)−1 − ∂x ζ , 2 r

ζ (0, ·) = ζ0 .

(187)

The role of the r.h.s. of (187) will become apparent in Step 2 below. Step 1. Maximum principle. In this step, we argue that (τ + t)∂x ζ (t, ·) 1 for all t 0.

(188)

ρ(t, x) := (τ + t)−1 − ∂x ζ (t, x)

(189)

To see (188), we introduce

and will argue that (187) can be rewritten in terms of ρ as 1 1 u+ ζ − ∂x2 ρ = 0. ∂t ρ + (τ + t)−1 ρ + ∂x ρ 1 − r r

(190)

Notice that (190) is an advection–diffusion equation for the “density” ρ and thus satisfies a comparison principle. In particular, since ρ ≡ 0 is a solution of (190), (190) preserves nonnegativity of ρ. Hence ρ(0, ·) 0, which follows from (186) and (187) and (189), yields ρ(t, ·) 0 for all t 0, which is a reformulation of (188).


2235

We now argue that (187) turns into (190). Indeed, the r.h.s. of (187) can be rewritten as (189) 1 1− (u − ζ ) (τ + t)−1 − ∂x ζ = ρ 1 − r = ρ 1−

1 u+ r 1 u+ r

1 ζ r 1 ζ r

− (τ + t)−1 − ∂x ζ ζ

−1

− (τ + t)

ζ + ∂x

1 2 ζ , 2

so that (187) turns into ∂t ζ + (τ + t)−1 ζ − ρ

1−

1 1 u + ζ − ∂x2 ζ = 0. r r

We apply −∂x to this equation: −1

∂t (−∂x ζ ) + (τ + t)

1 1 u+ ζ + ∂x3 ζ = 0. (−∂x ζ ) + ∂x ρ 1 − r r

This equation turns into (190) since (189)

∂t ρ + (τ + t)−1 ρ − ∂x2 ρ = =

−(τ + t)−2 + ∂t (−∂x ζ ) + (τ + t)−2 − (τ + t)−1 ∂x ζ + ∂x3 ζ

∂t (−∂x ζ ) + (τ + t)−1 (−∂x ζ ) + ∂x3 ζ.

Step 2. Contraction in Lr . In this step, we argue that d |u − ζ |r + (r − 1)(τ + t)−1 |u − ζ |r 0. dt

(191)

We start by noting that (187) can be rewritten as 1 1 (τ + t)−1 (u − ζ ) + (u − ζ )∂x ζ. = 1− r r

∂t ζ + u∂x ζ

− ∂x2 ζ

(192)

Eq. (185), which we rewrite as ∂t u + u∂x u − ∂x2 u = 0 and (192) combine to 1 1 (τ + t)−1 (u − ζ ) = 0. ∂t (u − ζ ) + u∂x (u − ζ ) − ∂x2 (u − ζ ) + (u − ζ )∂x ζ + 1 − r r For any smooth and convex A(z) this yields by multiplication with A (u − ζ ) ∂t A(u − ζ ) + u∂x A(u − ζ ) − ∂x2 A(u − ζ ) 1 1 + (u − ζ )A (u − ζ )∂x ζ + 1 − (τ + t)−1 (u − ζ )A (u − ζ ) 0, r r

2236


where the convexity is used as follows: 2 −A (u − ζ )∂x2 (u − ζ ) = −∂x A (u − ζ )∂x (u − ζ ) + A

(u − ζ ) ∂x (u − ζ ) −∂x2 A(u − ζ ). Averaging in x and integration by parts gives 1 d A(u − ζ ) − A(u − ζ )∂x u + (u − ζ )A (u − ζ )∂x ζ dt r 1 (τ + t)−1 (u − ζ )A (u − ζ ) 0. + 1− r Letting A(z) approximate A(z) = |z|r we obtain because of zA (z) = r|z|r : d |u − ζ |r − |u − ζ |r ∂x u + |u − ζ |r ∂x ζ dt + (r − 1)(τ + t)−1 |u − ζ |r 0. The two middle terms cancel:

1 r r r − |u − ζ | ∂x u + |u − ζ | ∂x ζ = ∂x = 0, (u − ζ )|u − ζ | r +1

which yields (191). Step 3. Conclusion. In this step, we argue that (184) holds. Integration in time of the result of Step 2, i.e. (191), yields u(t, ·) − ζ (t, ·)r

t + (r − 1)

r r (τ + t )−1 u(t , ·) − ζ (t , ·) dt u(0, ·) − ζ0 .

0

According to Step 1, ζ (t , ·) is admissible in D(t , τ + t ) so that the above yields

t Dr (t, τ + t) + (r − 1)

r (τ + t )−1 Dr (t , τ + t ) dt u(0, ·) − ζ0 .

0

Finally, since ζ0 was an arbitrary admissible function in Dr (0, τ ), the last inequality turns into (184). 2 Proof of Lemma 2. By redefining f and an approximation argument, we may assume that u satisfies ∂t u − ∂x2 u = π|∂x |f for some > 0.

(193)


2237

It is enough to establish the additive version ∞

∗

∞ u − u p d λ−p∗ f − f p d 1 λp ∂t D2 2 α + p ∗ 1−α 2 p 0

0

for an arbitrary λ ∈ (0, ∞). By approximation, it is enough to show 1 ∂t D2 2 2

∞

∞ d u − u d ∗ f −f A A + , α 1−α 0

0

where A(z) denotes a smooth, even and uniformly convex function and A∗ (z) its Legendre transform. Again, we use the same strategy as in Remark 3(ii). Fix τ > 0 and let ζ0 be arbitrary in the definition of D2 (0, τ ). We introduce the nonlocal, nonlinear elliptic operator A via

∞

A(ζ ) (x) =

d 1

ζ (x) − ζ (x + )

ζ (x − ) − ζ (x) A − A , α α α

(194)

0

where ζ (x) is a smooth L-periodic function. We remark that the integral (194) converges absolutely: For ↑ ∞ we note that the boundedness of ζ implies

A

ζ (x) − ζ (x + ) , α

A

ζ (x − ) − ζ (x) α

= O(1),

so that the integral converges because of α > 0. For ↓ 0 we note that because of the smoothness of A and of ζ , A

ζ (x) − ζ (x + )

ζ (x − ) − ζ (x) − A α α x+

=

A

ζ (x − ) − ζ (x ) ∂x ζ (x − ) − ∂x ζ (x )

dx α α

x

1 =− α

x+

x

A x

x −

ζ (x − ) − ζ (x ) 2 ∂x ζ (x

) dx

dx

α

= O 2−α ,

(195)

so that the integral converges because of α < 1. From the representation (195) we also learn that A(ζ ) is a compact perturbation w.r.t. to −∂x2 ζ so that there exists a smooth ζ (t, x), L-periodic in x, that solves the initial value problem ∂t ζ − ∂x2 ζ + A(ζ ) = 0,

ζ (0, ·) = ζ0 .

(196)

2238


The crucial structural properties of the operator A are • A satisfies the maximum principle on the level of ∂x ζ , cf. Step 1, ∞ d • A is the subdifferential of the convex potential 0 A( ζ −ζ α ) , cf. Step 2. Step 1. Maximum principle. In this step, we argue that τ ∂x ζ (t, ·) 1 for all t 0.

(197)

In order to obtain (197), we rewrite (196) in terms of ρ := −∂x ζ : ∂t ρ − ∂x2 ρ + ∂Aρ = 0,

(198)

where the linear inhomogeneous operator ∂A is defined via

∞ (∂Aρ)(t, x) =

1

ζ (t, x) − ζ (t, x + ) A ρ(t, x) − ρ(t, x + ) α 2α

0

− A

d ζ (t, x − ) − ζ (t, x) ρ(t, x − ) − ρ(t, x) . α

(199)

We note that because of A

(z) 0, (199) has the property: If (t ∗ , x ∗ ) is a spatial minimum point of ρ, i.e. ρ t ∗ , x ρ t ∗ , x ∗ for all x, then we have (∂Aρ) t ∗ , x ∗ 0. This implies that (198) satisfies a comparison principle. Since ρ ≡ 0 is a solution of (198), (198) preserves non-negativity. This yields (197). Step 2. Contraction in L2 . In this step, we argue that d 1 (u − ζ )2 2 dt 2

∞

∞ d u − u d ∗ f −f A A + . α 1−α 0

0

We combine (193) and (196) to ∂t (u − ζ ) − ∂x2 (u − ζ ) = π|∂x |f + A(ζ ), which yields after multiplication with u − ζ , average in x and integration by parts d 1 (u − ζ )2 + (∂x u − ∂x ζ )2 = (u − ζ )(π|∂x |f + A(ζ ) . dt 2


2239

It thus remains to show (u − ζ ) π|∂x |f + A(ζ ) ∞

∞ d u − u d f − f A A∗ + 2 . α 1−α 0

(200)

0

The ingredients for the inequality (200) are the two identities for an arbitrary L-periodic v(x):

∞

∞ v − v f − f d d v−v f −f = vπ|∂x |f = , α 2 1−α

0

(201)

0

vA(ζ ) =

∞

v − v ζ − ζ A α α

d .

(202)

0

Indeed, we immediately obtain from (201) and (202) for v = u − ζ (u − ζ ) π|∂x |f + A(ζ )

∞ =

u − u ζ − ζ − α α

f −f

ζ −ζ . + A α 1−α

(203)

0

By the characterizing property of the Legendre transform A∗ (and the evenness of A) we have u − u f − f u − u ∗ f −f + A , A α α 1−α 1−α ζ − ζ f − f ζ − ζ ∗ f −f − + A . A α α 1−α 1−α By the convexity of A we have

u − u ζ − ζ u − u ζ − ζ

ζ −ζ A A − A . − α α α α α

The combination of these inequalities yields

u − u ζ − ζ f −f

ζ −ζ − +A α α α 1−α u − u ∗ f −f 2 A + A . α 1−α

Inserting (204) into (203) yields (200).

(204)

2240


We now give the argument for (201). Since F |∂x |f (q) = |q|(F f )(q)

and

F v − v (q) = 1 − ei q (F v)(q),

(201) reduces to the Fourier multiplier identity

∞ 2 d π|q| = 1 − ei q 2 , 0

which is classical:

∞

∞ 2 d d i q 2 1 − e q = 4 sin 2 2 2 0

0

∞ = 2|q|

ˆ sin2 ( )

0

ˆ d ˆ2

= π|q|. We finally address (202). Because of − −ζ

ζ ζ −ζ vA = v A , α α

(205)

we have by definition (194) of A:

vA(ζ ) =

∞

1 α

− d −ζ

ζ −ζ

ζ vA − vA α α

1 α

d

ζ −ζ ζ −ζ vA − v A α α

0 (205)

∞

=

0

∞ =

v − v ζ − ζ d . A α α

2

0

Proof of Proposition 3. Part (i) follows formally from Lemmas 1 and 2 by operator splitting. Instead of an attempt to make the operator splitting argument rigorous, we combine the proofs of Lemma 1 and of Lemma 2, just sketching the main steps: As in Lemma 2 we assume that u satisfies 1 2 ∂t u + ∂x u − ∂x2 u = π|∂x |f + π|∂x |g (206) 2


2241

for some > 0. Indeed, (206) turns into (51) if f is replaced by πf − |∂x |−1 ∂x2 u and g is replaced by πg. Hence by an approximation argument in , it is enough to establish the following time-integrated version of (53):

t

+

(τ + t )−1 D + (t , τ + t ) dt

D (t, τ + t) + 0 +

D (0, τ )

1/p t ∞

1/p∗ t ∞ ∗ u − u p d

f − f p d

dt +2 α dt 1−α 0 0

0 0

t ∞

1/q t ∞

1/q ∗ ∗ u − u q d

g − g q d

dt +2 . β dt 1−β 0 0

0 0

By an optimization argument in λ, μ > 0, it is enough to establish the following additive version:

t

+

D (t, τ + t) +

(τ + t )−1 D + (t , τ + t ) dt

0 +

D (0, τ )

t ∞ ∗

t ∞ u − u p d λ−p∗ f − f p d

λp dt +2 α dt + p ∗ 1−α p 0 0

μq +2 q

0 0

∗

t ∞ u − u q d μ−q ∗ g − g q d

dt . β dt + q ∗ 1−β

t ∞ 0 0

0 0

As in Lemma 2, we approximate the homogeneous expression on the r.h.s. by smooth, even and uniformly convex functions A(z) and B(z):

t

+

D (t, τ + t) +

(τ + t )−1 D + (t , τ + t ) dt

0 +

D (0, τ )

t ∞

t ∞ d u − u d

∗ f −f

A A dt + dt +2 α 1−α 0 0

0 0

t ∞

t ∞ d u − u d

∗ g−g

+2 B B dt + dt , β 1−β 0 0

0 0

where A∗ and B ∗ denote the Legendre transforms of A and B, respectively.

(207)

2242


We now fix τ > 0 and let ζ0 be arbitrary in the definition of D + (0, τ ). Let the nonlocal, nonlinear elliptic operator A be defined as in (194), and B be defined analoguously on the basis of B. Combining (187) and (196), we define ζ as the solution of ∂t ζ + ∂x

1 2 1 ζ − ∂x2 ζ = (u − ζ ) (τ + t)−1 − ∂x ζ − A(ζ ) − B(ζ ) 2 2

with initial data ζ0 . Since > 0 and the nonlinear differential operators A, B are of order strictly less than 2, a unique smooth solution of this nonlinear initial value problem exists. As in Step 1 (maximum principle) of Lemma 1 and Lemma 2, one argues that (τ + t)∂x ζ (t, ·) 1 for all t 0.

(208)

Following Step 1 of Lemma 1, this is done by showing that ρ(t, x) := (τ + t)−1 − ∂x ζ satisfies the differential equation −1

∂t ρ + (τ + t)

1 1 − ∂x2 ρ + ∂Aρ + ∂Bρ = 0, ρ + ∂x ρ u + ζ 2 2

where the linear inhomogeneous operator ∂A is defined in (199) and ∂B is defined analoguously on the basis of B. As in Step 1 of Lemma 1 and Lemma 2, one argues that this differential equation for ρ satisfies a comparison principle and thus preserves the nonnegativity of ρ. This yields (208). With the same manipulations as in Step 2 (L2 -contraction) of Lemma 1 and Lemma 2, one shows that d 1 1 (u − ζ )2 + (τ + t)−1 (u − ζ )2 + (∂x u − ∂x ζ )2 dt 2 2 = (u − ζ ) π|∂x |f + A(ζ ) + (u − ζ ) π|∂x |f + B(ζ ) . As in Step 2 of Lemma 2, one sees that this identity yields the differential inequality d 1 1 (u − ζ )2 + (τ + t)−1 (u − ζ )2 dt 2 2 ∞

∞ d u − u d f − f A A∗ + 2 α 1−α 0

0

∞

∞ d u − u d g − g B B∗ +2 + . β 1−β 0

0

By integration in time from t = 0 to t = t, this yields

(209)


1 1 (u − ζ )2 (t = t, ·) + 2 2

t

2243

(τ + t )−1 (u − ζ )2 dt

0

2 1

u(t = 0, ·) − ζ0 2 t ∞

t ∞ d u − u d

∗ f −f

A A dt + dt +2 α 1−α 0 0

0 0

t ∞

t ∞ d u − u d

∗ g−g

+2 B B dt + dt . β 1−β 0 0

(210)

0 0

As in the proof of Lemma 1 and Lemma 2, one combines (210) with (208) to obtain (207) by definition of D + . We now turn to part (ii). It follows from Definition 4 that D + (t, τ ) is monotone increasing in τ , which implies that ∂τ D + 0 distributionally, so that (53) implies

+

∂t D + τ

−1

4 D π +

∞

1/p ∞

1/p∗ ∗ u − u p d f − f p d α 1−α 0

0

∞

1/q ∞

1/q ∗ ∗ u − u q d g − g q d 4 + β 1−β π 0

0

in the distributional sense. Integration over t ∈ (0, T ), for given T < ∞, and Hölder’s inequality yields

D u(T , ·), τ + τ −1 +

T

D + (u, τ ) dt

0

∞ T

1/p ∞ T

1/p∗ ∗ u − u p f − f p 4 d d α dt 1−α dt π 0 0

0 0

∞ T

1/q ∞ T

1/q ∗ ∗ u − u q g − g q d d 4 + β dt 1−β dt π 0 0

+ D + u(0, ·), τ . We now divide by T and take the limit T ↑ ∞:

0 0

(211)

2244


τ

−1

4 D (u, τ ) π +

∞

1/p ∞

1/p∗ ∗ u − u p d f − f p d α 1−α 0

0

∞

1/q ∞

1/q ∗ ∗ u − u q d g − g q d 4 + . β 1−β π 0

(212)

0

For D − (u, τ ), we fix T < ∞ and note that the change of variables tˆ = T − t,

uˆ = −u

(213)

leaves (51) invariant. We apply (211) to (t, u) replaced by (tˆ, u). ˆ Since D + u( ˆ tˆ, ·), τ = D + −u(T − tˆ, ·), τ = D − u(T − tˆ, ·), τ , and since (0, T ) is invariant under (213), (211) turns into D − u(0, ·), τ + τ −1

T

D − (u, τ ) dt

0

4 π

∞ T

1/p ∞ T

1/p∗ ∗ u − u p f − f p dt d dt d α 1−α 0 0

0 0

∞ T

1/q ∞ T

1/q ∗ ∗ u − u q g − g q d d 4 + β dt 1−β dt π 0 0

0 0

+ D u(T , ·), τ . −

By Definition 4, D − (u(T , ·), τ ) u(T , ·)2 , so that because of (52), division by T and the limit T ↑ ∞ yields τ −1 D − (u, τ ) ∞

1/p ∞

1/p∗ ∗ u − u p d f − f p d 4 α 1−α π 0

0

∞

1/q ∞

1/q ∗ ∗ u − u q d g − g q d 4 + . β 1−β π 0

Now (54) follows by summing (212) and (214).

0

2

(214)


2245

Acknowledgments The author acknowledges discussions with Ralf Wittenberg and thanks PIMS and Simon Fraser University for its hospitality. He also thanks Lorenzo Giacomelli for discussions and Helmut Abels and Michael Westdickenberg for pointing out references. This research has been partially supported by the German Science Foundation through the Hausdorff Center for Mathematics and Project B3 of the SFB 611. References [1] J. Bergh, J. Löfström, Interpolation Spaces, an Introduction, Springer, 1976. [2] J.C. Bronski, T.N. Gambill, Uncertainty estimates and L2 bounds for the Kuramoto–Sivashinsky equation, Nonlinearity 19 (2006) 2023–2039. [3] P. Collet, J.-P. Eckmann, H. Epstein, J. Stubbe, A global attracting set for the Kuramoto–Sivashinsky equation, Comm. Math. Phys. 152 (1993) 203–214. [4] P. Collet, J.-P. Eckmann, H. Epstein, J. Stubbe, Analyticity for the Kuramoto–Sivashinsky equation, Phys. D 67 (1993) 321–326. [5] C. DeLellis, F. Otto, M. Westdickenberg, Minimal entropy conditions for Burgers equation, Quart. Appl. Math. 62 (2004) 687–700. [6] R.J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Ration. Mech. Anal. 82 (1983) 27–70. [7] C. Foias, B. Nicolaenko, G.R. Sell, R. Teman, Inertial manifolds for the Kuramoto–Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl. 67 (1988) 197–226. [8] L. Giacomelli, F. Otto, New bounds for the Kuramoto–Sivashinsky equation, Comm. Pure Appl. Math. 58 (2005) 297–318. [9] J. Goodman, Stability of the Kuramoto–Sivashinsky and related systems, Comm. Pure Appl. Math. 47 (1994) 293– 306. [10] D. Michelson, Steady solutions of the Kuramoto–Sivashinsky equation, Phys. D 19 (1986) 89–111. [11] F. Murat, Compacité par compensation: Condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 8 (1981) 69–102. [12] B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of the Kuramoto–Sivashinsky equations: Nonlinear stability and attractors, Phys. D 16 (1985) 155–183. [13] O.A. Oleinik, Discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk 12 (1957) 3–73, English translation in: Amer. Math. Soc. Transl. 26 (1963) 1155–1163. [14] L. Tartar, The compensated compactness method applied to systems of conservation laws, in: J.M. Ball (Ed.), Systems of Nonlinear Partial Differential Equations, Oxford, 1982, in: NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, 1983, pp. 263–285. [15] R.W. Wittenberg, Dissipativity, analyticity and shocks in the (de)stabilized Kuramoto–Sivashinsky equation, Phys. Lett. A 300 (2002) 407–416. [16] R.W. Wittenberg, P. Holmes, Scale and space localization in the Kuramoto–Sivashinsky equation, Chaos 9 (2) (1999) 452–464.


A general Trotter–Kato formula for a class of evolution operators Pierre-A. Vuillermot a,∗ , Walter F. Wreszinski b , Valentin A. Zagrebnov c a UMR-CNRS 7502, Institut Élie Cartan, Nancy, France b Departamento de Física Matemática, Universidade de São Paulo, Brazil c UMR-CNRS 6207, Université d’Aix-Marseille II, France

Received 31 October 2008; accepted 23 June 2009

Communicated by L. Gross

Abstract In this article we prove new results concerning the existence and various properties of an evolution system UA+B (t, s)0stT generated by the sum −(A(t) + B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B (t, s)0stT as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter–Kato type under very general conditions which allow the domain D(A(t) + B(t)) to evolve with time provided there exists a fixed set D ⊂ t∈[0,T ] D(A(t) + B(t)) everywhere dense in B. We obtain a special case of our formula when B(t) = 0, which, in effect, allows us to reconstruct UA (t, s)0stT very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial–boundary value problems, including one related to the theory of timedependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics. © 2009 Elsevier Inc. All rights reserved. Keywords: Evolution operators; Trotter–Kato formula


E-mail address: [email protected] (P.-A. Vuillermot). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.026

P.-A. Vuillermot et al. / Journal of Functional Analysis 257 (2009) 2246–2290

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1. Introduction and outline It is well known that the Hille–Yosida theory of semigroups and its numerous extensions regarding the construction of evolution operators on Banach spaces has had and still has far reaching applications to the analysis of certain linear or nonlinear, deterministic or stochastic, partial differential equations with time-independent or time-dependent coefficients. In many instances that may encompass parabolic equations, hyperbolic equations or Schrödinger equations, to name only a few, it is indeed possible to reformulate a given initial and boundary-value problem as one related to evolution equations on suitably chosen functional spaces. The mathematical investigation of such a problem concerning for example the existence and the uniqueness of various types of solutions, the relations among them, their various representations and their asymptotic behavior for large times, then becomes intimately related to the properties of the corresponding linear propagator (see, for instance, [22,28,34,37] for general references regarding the deterministic case as well as [9] for the stochastic case). Among those properties, perturbation formulae of the Trotter–Kato type such as those stated in [6,7,14,25,26,35] for holomorphic or more general semigroups are of particular importance for the understanding of certain basic questions in applied mathematics or mathematical physics. For example, a strongly convergent product formula of the form n t t exp − A exp − B n→+∞ n n

exp −t (A + B) = lim

(1)

with t ∈ R+ and A, B time-independent linear operators on a Banach space satisfying certain conditions, allows one to relate the solutions of certain evolution problems to the theory of Wiener integrals through the celebrated Feynman–Kac formula (see, for instance, [30]). On the other hand, in the realm of quantum mechanics a slightly modified version of (1) also allows a rigorous construction of the so-called Feynman path integral representation of the solutions to Schrödinger equations with time-independent potentials (see, for instance, [4,19,26]). Consequently, a question that arises naturally is whether formulae of the form (1) can be generalized to the case where the linear operators A(t) and B(t) depend explicitly on the time variable. It turns out that such a generalization was indeed carried out in [13] when both A(t) and B(t) are the infinitesimal generators of C0 -contraction semigroups on a Banach space for every t, under the additional restriction that the domain D(A(t) + B(t)) of the operator sum A(t) + B(t) be time-independent. This was nonetheless sufficient to enable the author of [13] to give a precise mathematical meaning to the Feynman path integral representation in the case of Schrödinger equations with certain time-dependent potentials. With further hypotheses regarding the continuity properties of A(t) and B(t) as functions of t, a generalization of (1) was also obtained in [18] where the authors were able to prove the convergence of their approximations in the operator norm-topology rather than just in the strong topology. There are, however, a host of important situations where D(A(t) + B(t)) does depend explicitly on time, thereby making some of the arguments of [13] and [18] inapplicable; as a concrete class of examples which will motivate some of the hypotheses of the theory we develop below, let D ⊂ Rd be an open bounded domain with a smooth boundary ∂D (see, for instance, [2] for + a definition of this and related concepts); let T ∈ R+ ∗ := R {0} and let us consider parabolic initial–boundary value problems of the form

2248


∂u(x, t) = div k(x, t)∇u(x, t) − u(x, t), ∂t u(x, 0) = u0 (x), x ∈ D, ∂u(x, t) = 0, (x, t) ∈ ∂D × (0, T ], ∂n(k)

(x, t) ∈ D × (0, T ],

(2)

with ∈ R+ a parameter and where the last relation in (2) stands for the conormal derivative of u relative to the matrix-valued function k. We assume that the following hypotheses hold (here and below we use the standard notations for the usual spaces of Lebesgue integrable functions and for the corresponding Sobolev spaces on regions of Euclidean space; we also write c for all the irrelevant constants that occur in the various estimates unless we specify these constants otherwise): 2

(K) The function k : D × [0, T ] → Rd is matrix-valued and for every i, j ∈ {1, . . . , d} we have ki,j = kj,i ∈ L∞ (D × (0, T ), R); moreover, there exists a constant k ∈ R+ ∗ such that the inequality

k(x, t)q, q Rd k|q|2 (3) holds uniformly in (x, t) ∈ D × [0, T ] for all q ∈ Rd , where (.,.)Rd and |.| denote the Euclidean inner product and the induced norm in Rd , respectively; finally, there exist constants c∗ ∈ R+ ∗, σ ∈ ( 12 , 1], such that the Hölder continuity estimate max

i,j ∈{1,...,d}

ki,j (x, t) − ki,j (x, s) c∗ |t − s|σ

is valid for every x ∈ D and every s, t ∈ [0, T ]. (I) The initial datum satisfies u0 ∈ L2 (D, R). As is well known, Hypothesis (K) allows one to construct a self-adjoint, positive realization of the elliptic partial differential operator with conormal boundary conditions in (2). In fact, let us write (.,.)2 and . 2 for the inner product and the induced norm in L2 (D, C), respectively, together with (.,.)1,2 and . 1,2 for the inner product and the induced norm in H 1 (D, C), respectively; let (.,.)Cd be the standard inner product in Cd . Then, for the Hermitian sesquilinear form a : [0, T ] × H 1 (D, C) × H 1 (D, C) → C defined by

a(t, v, w) := dx k(x, t)∇v(x), ∇w(x) Cd (4) D

we have the estimates a(t, v, w) c v 1,2 w 1,2 ,

a(t, v, v) k v 21,2 − v 22 0 uniformly in t ∈ [0, T ] for every v, w ∈ H 1 (D, C), as well as a(t, v, w) − a(s, v, w) c|t − s|σ v 1,2 w 1,2

(5)

(6)


2249

for every s, t ∈ [0, T ]; consequently, the operator

A(t) := − div k(., t)∇ +

(7)

is indeed self-adjoint and positive in L2 (D, C) on the time-dependent domain given by

D A(t) = v ∈ H 1 (D, C): A(t)v ∈ L2 (D, C), A(t) − v, w 2 = a(t, v, w)

(8)

where the last relation in (8) holds for every w ∈ H 1 (D, C). Then for any t ∈ [0, T ], −A(t) is the infinitesimal generator of a holomorphic semigroup of contractions exp[−sA(t)]s0 in L2 (D, C), and also generates there an evolution system UA (t, s)0stT given by

v UA (t, s)v =

D dy GA (., t; y, s)v(y)

if t = s, if t > s,

(9)

whose range satisfies

Ran UA (t, s) ⊆ D A(t) for every s, t with 0 s < t T . Here we denote by GA the parabolic Green’s function associated with (2) (see, for instance, [23,24,28,34] for other typical constructions of this kind). This means that it becomes possible to investigate the existence and the various properties of solutions to (2) through the integral relation u(., t) = dy GA (., t; y, 0)u0 (y) D

in L2 (D, R). Let us now perturb the partial differential operator in (2) by considering initial–boundary value problems of the form

∂u(x, t) = div k(x, t)∇u(x, t) − l(x, t), ∇u(x, t) Rd − + εm(x, t) u(x, t), ∂t (x, t) ∈ D × (0, T ], u(x, 0) = u0 (x), x ∈ D, ∂u(x, t) = 0, (x, t) ∈ ∂D × (0, T ], ∂n(k)

(10)

with ε ∈ R+ a parameter and with the following additional hypotheses regarding the lower-order differential operators, where we assume without restricting the generality that the constants c∗ and σ are the same as in Hypothesis (K): (L) Each component of the vector-field l : D × [0, T ] → Rd satisfies li ∈ L∞ (D × (0, T ), R) and the Hölder continuity estimate max li (x, t) − li (x, s) c∗ |t − s|σ i∈{1,...,d}

holds for every x ∈ D and every s, t ∈ [0, T ].

2250


(M) We have m ∈ L∞ (D × (0, T ), R+ ) along with m(x, t) − m(x, s) c∗ |t − s|σ for every x ∈ D and every s, t ∈ [0, T ]. As is the case for (7), it is also possible to construct a realization of the partial differential operator

Cε (t) := − div k(., t)∇ + + l(., t), ∇ Rd + εm(., t) := A(t) + Bε (t)

(11)

in (10) by considering the sesquilinear form cε : [0, T ] × H 1 (D, C) × H 1 (D, C) → C defined by cε (t, v, w) := a(t, v, w) + (v, w)2 + bε (t, v, w)

(12)

with a(t, v, w) given by (4) and bε (t, v, w) :=

dx l(x, t), ∇v(x) Cd w(x) + ε

D

dx m(x, t)v(x)w(x).

(13)

D

In fact, thanks to Hypotheses (L), (M) and by elementary arguments we get the estimates bε (t, v, w) c v 1,2 w 2 , k Re bε (t, v, v) − v 21,2 − c v 22 2

(14)

uniformly in t ∈ [0, T ] for every v, w ∈ H 1 (D, C), as well as bε (t, v, w) − bε (s, v, w) c|t − s|σ v 1,2 w 2

(15)

for every s, t ∈ [0, T ], with the norm w 2 rather than w 1,2 in (14) and (15). Consequently, this leads to the realization of the lower-order operator Bε (t) in L2 (D, C) on the time-independent domain

D Bε (t) = H 1 (D, C) with

Bε (t)v, w 2 = bε (t, v, w) and Bε (t)v c v 1,2 2

(16)


2251

for every t ∈ [0, T ], any v ∈ H 1 (D, C) and each w ∈ L2 (D, C), and thereby to the realization of (11) as an operator in L2 (D, C) on the time-dependent domain

D Cε (t) = D A(t) ∩ H 1 (D, C) = D A(t) for every t ∈ [0, T ]. Moreover, as is the case for −A(t) the operator −Cε (t) also generates a holomorphic semigroup and an evolution system UA+Bε (t, s)0stT given by

v UA+Bε (t, s)v =

D dy GA+Bε (., t; y, s)v(y)

if t = s, if t > s

(17)

in L2 (D, C), whose range satisfies

Ran UA+Bε (t, s) ⊆ D A(t) for every s, t with 0 s < t T and where GA+Bε is the parabolic Green’s function associated with the differential operator in (10). These two assertions follow from the general theory developed in [34] since we can infer successively from (5), (6), (14) and (15) that the estimates cε (t, v, w) c v 1,2 w 1,2 , k Re cε (t, v, v) v 21,2 − (k + c) v 22 2 hold uniformly in t ∈ [0, T ] for every v, w ∈ H 1 (D, C), and that cε (t, v, w) − cε (s, v, w) c|t − s|σ v 1,2 w 1,2

(18) (19)

(20)

holds for every s, t ∈ [0, T ]. In the realm of this class of examples the natural questions we want to ask are whether we can reconstruct the evolution system UA (t, s)0stT in terms of the contraction semigroup exp[−sA(t)]s0 in a simple manner, and more generally whether we can express (17) in terms of the unperturbed system (9) through some kind of generalization of (1). Even the first question is not trivial, as the various relations known thus far between UA (t, s)0stT and exp[−sA(t)]s0 are notoriously complicated ones (see, for instance, [28] and [34]). In order to motivate further the theory we develop below, it is worth noting here that under the above hypotheses the operator Bε (t) is always a relatively bounded perturbation of the operator A(t) in the sense of [22]. In fact, aside from the inclusion

D A(t) ⊆ D Bε (t) we also have

v 21,2 k −1 A(t) − v, v 2 + v 22 k −1 A(t) − v 2 v 2 + v 22 as a consequence of (5), (8) and Schwarz inequality, which implies

v 1,2 c A(t)v 2 + v 2

2252


for every v ∈ D(A(t)) since v −1 1,2 v 2 1 when v = 0. Consequently, from the last relation and (16) we obtain

Bε (t)v c A(t)v + v 2 2 2 for every t ∈ [0, T ] and any v ∈ D(A(t)), which is the desired assertion. As we shall see in Section 4, similar questions can be raised for other classes of concrete examples, a case in point being the class of time-dependent singular perturbations of self-adjoint differential operators which are supported on a finite or discrete set of points in Euclidean space (see, for instance, [3,8,11,12,15] for general references concerning such problems). Although (10) is inherently variational, it is equally plain that it is formally a particular example of an abstract evolution problem of the form

du(t) = − A(t) + B(t) u(t), dt u(s) = us

t ∈ (s, T ], (21)

defined in a complex and separable Banach space B. In the sequel we shall investigate (21) from the point of view we just outlined under appropriate hypotheses concerning A(t) and B(t) when D(A(t) + B(t)) may be time-dependent, but without reference to any kind of variational structure in the abstract setting. Accordingly, we shall organize the remaining part of this article in the following way: in Section 2 we state and discuss our main theorem regarding the existence of an evolution system UA+B (t, s)0stT concerning (21) and a related extension of (1), for a suitable class of A(t)’s and of time-dependent perturbations B(t)0tT . There we also put our result into a broader perspective by comparing our way of constructing the UA+B (t, s)’s with other known methods such as those put forward in [1,29] or in the review article [32]. We prove our main result in Section 3; our general framework in that section is the theory of evolution operators as developed in [34], which indeed motivated our choice of the A(t)’s and the B(t)’s in the first place. We illustrate our main statements by means of several examples in Section 4, aside from also considering there examples showing that some of our hypotheses, albeit natural, sufficient and indeed verifiable in a host of important situations, are not necessary for our product formula to hold. In this context it is worth pointing out that there are two well-known analytical tools which play an important rôle in our analysis of some of those examples, namely, Euler’s summation formula and Krein’s formula for resolvents (see, for instance, [17] and [3], respectively). Finally, we refer the reader to [36] for a short announcement of our result and a very brief sketch of its proof. 2. Statement and discussion of the main result In the sequel we write . for the norm in B and . ∞ for the usual operator-norm in L(B), the Banach algebra of all bounded linear operators on B. According to what we outlined in the preceding section, we wish to construct an evolution system UA+B (t, s)0stT for Problem (21) which we can express in terms of the semigroups generated by A(t) and B(t) through a suitable generalization of (1), without ever requiring that the domains D(C δ (t)) of the fractional powers of C(t) for δ ∈ (0, 1] be time-independent, where C(t) := A(t) + B(t). To this end we assume that the following hypotheses are valid (see, for instance, [34] for the basic definitions and properties):


2253

(A1) The linear operator −A(t) is the infinitesimal generator of a holomorphic semigroup exp[−sA(t)]s0 on B for every t ∈ [0, T ] and we have 0 ∈ ρ(A(t)) for any such t, where ρ(A(t)) denotes the resolvent set of A(t). (A2) The function t → A−1 (t) is continuously differentiable with respect to the normtopology of L(B) and there exist constants a2 ∈ R+ ˜ 2 ∈ (0, 1] such that the Hölder continuity ∗,a estimate −1 dA (t) dA−1 (s) a2 |t − s|a˜ 2 − dt ds ∞

(22)

is valid for every s, t ∈ [0, T ]. As is well known, Hypothesis (A1) implies the existence of constants θ ∈ (0, π2 ), c∗ ∈ R+ ∗ such that the inclusion Sθ ⊆ ρ(A(t)) and the inequality

R A(t), λ

∞

−1 c∗ 1 + |λ|

(23)

hold for every t ∈ [0, T ] and any λ ∈ Sθ , where

−1 R A(t), λ := A(t) − λ and

Sθ := λ ∈ C: | arg λ| θ ∪ {0}.

(24)

Furthermore, Hypotheses (A1) and (A2) also imply the differentiability of the function t → R(A(t), λ) on [0, T ] with respect to the norm-topology of L(B), whose derivative we require to satisfy the following hypothesis: (A3) There exist constants a3 ∈ R+ ˜ 3 ∈ (0, 1] such that the inequality ∗,a ∂

R A(t), λ ∂t

∞

a3 |λ|−a˜ 3

(25)

holds for every t ∈ [0, T ] and every λ ∈ Sθ {0}. Hypotheses (A1)–(A3) are the building blocks of the existence theory of solutions to nonautonomous linear parabolic equations developed in [34] when D(A(t)) varies with time, thereby providing an evolution system UA (t, s)0stT for Problem (21) when B(t) = 0; however, they are by far not the only sufficient conditions that allow the construction of the UA (t, s)’s, and we shall indeed dwell a bit on this point and on related questions immediately after the statement of our theorem. Since we have in mind a generalization of (1) to the time-dependent case, it is then natural to ask whether those conditions remain stable under a suitable class of perturbations B(t)0tT of the A(t)’s. We shall see that this is indeed the case provided we impose the following hypotheses:

2254


(B1) The linear operator B(t) is closed in B for every t ∈ [0, T ] and we have D(B(t)) ⊇ D(A(t)) for any such t; moreover, there exist constants a ∈ [0, 1), b ∈ [0, (1 − a)c∗−1 ) where c∗ is the constant in (23), such that the inequality

B(t)v a A(t) − λ v + b v

(26)

holds for every v ∈ D(A(t)), any t ∈ [0, T ] and each λ ∈ Sθ . (B2) The function t → B(t)A−1 (t) is continuously differentiable on [0, T ] with respect to ˜ the norm-topology of L(B) and there exist constants b2 ∈ R+ ∗ , b2 ∈ (0, 1] such that the Hölder continuity estimate d(B(t)A−1 (t)) d(B(s)A−1 (s)) b2 |t − s|b˜2 − dt ds ∞ is valid for every s, t ∈ [0, T ]. (B3) The function t → B(t)R(A(t), λ) is continuously differentiable on [0, T ] with respect to the norm-topology of L(B) and there exists a constant c∗∗ ∈ R+ ∗ such that the inequality ∂

B(t)R A(t), λ ∂t

∞

c∗∗

holds for every t ∈ [0, T ] and each λ ∈ Sθ . While Hypothesis (B1) is evidently some kind of relative boundedness condition, we remark that it also imposes a smallness condition on the constant b in (26). This will allow us to prove a crucial ingredient for our upcoming arguments to work, namely, the bounded invertibility of A(t) + B(t) for every t ∈ [0, T ], which means that even in the case of bounded B(t)’s the admissible perturbations will be limited to those of small norm. Furthermore, whereas the preceding hypotheses indeed guarantee the existence of the evolution system we alluded to above (see Proposition 1 of Section 3), we note that they are not quite sufficient to allow the generalization of (1) that we want. For this we still impose the following three conditions. (A4) The semigroup exp[−sA(t)]s0 in Hypothesis (A1) is contractive on B for every t ∈ [0, T ]. (B4) The operator −B(t) is the infinitesimal generator of a holomorphic semigroup of contractions exp[−sB(t)]s0 on B for every t ∈ [0, T ]; moreover, the function t → R(B(t), λ) is continuous on [0, T ] uniformly in λ ∈ Sθ ∗ in the strong topology of L(B), where Sθ ∗ is given by (24) but with

Sθ ∗ ⊆ ρ B(t) for some θ ∗ ∈ (0, π2 ). (D) There exists a dense set D ⊂ B satisfying D⊂

t∈[0,T ]

D A(t) + B(t)

(27)


2255

such that for every v ∈ D we have sup A(t)v < +∞

(28)

sup B(t)v < +∞.

(29)

t∈(0,T ]

and

t∈(0,T ]

As is the case for the operator A(t), Hypothesis (B4) also implies the existence of a constant c∗ ∈ R + ∗ such that the resolvent estimate

R B(t), λ

∞

−1 c∗ 1 + |λ|

(30)

holds for every t ∈ [0, T ] and every λ ∈ Sθ ∗ ; moreover, our arguments below will show that in the particular case of time-independent B’s, we can weaken Hypothesis (B4) by only requiring that exp[−sB]s0 be a C0 -contraction semigroup. Under these conditions we can formulate our main result as follows. Theorem. Assume that Hypotheses (A1)–(A3) and (B1)–(B3) hold. Then there exists an evolution system UA+B (t, s)0stT solving Problem (21) such that the following properties are valid for all s, t with 0 s < t T : (1) The range of UA+B (t, s) satisfies

Ran UA+B (t, s) ⊆ D A(t) + B(t) = D A(t) .

(31)

Moreover, the operator-valued function t → UA+B (t, s) is continuously differentiable with respect to the norm-topology of L(B) and we have

∂UA+B (t, s) = − A(t) + B(t) UA+B (t, s) ∈ L(B) ∂t with the estimate

∂UA+B (t, s) ∂t

∞

c(t − s)−1

for some c ∈ R+ ∗ independent of s, t. Finally, the operator-valued function s → UA+B (t, s) is also differentiable with respect to the norm-topology of L(B) and we have ∂UA+B (t, s) ∈ L(B) ∂s with the same estimate as above, namely, ∂UA+B (t, s) c(t − s)−1 ∂s ∞ where

∂UA+B (t,s) ∂s

stands for the bounded linear extension of UA+B (t, s)(A(s) + B(s)) on B.

2256


(2) In addition to the above hypotheses, if (A4), (B4) and (D) hold then for all s, t with 0 s t < T we have the Trotter–Kato product formula UA+B (t, s) = lim

n→+∞

0 γ =n−1

γ t −s γ t −s A s + (t − s) exp − B s + (t − s) exp − n n n n

(32)

in the strong topology of L(B). Remarks. (1) Aside from Hypotheses (A1)–(A3), there exist several other sufficient conditions that would have allowed the construction of the UA (t, s)’s when B(t) = 0; we refer the reader for instance to [1] for a general and thorough investigation of such conditions and of the relations among them. In particular, we could have used Hypotheses I and II of that paper in the somewhat stronger form introduced in [29] and [32] to get such a result. There also exist various sufficient conditions which could have led to the existence of perturbed evolution systems UA+B (t, s) for suitable classes of B(t)’s, for example those put forward in [29] and [32]. However, a basic difficulty emerges there when one tries to prove a product formula such as (32) for them; thus, while the UA+B (t, s)’s of Theorem 9.19 in [32] are lacking good differentiability properties, those of Theorem 4.2 in [29] are only weakly locally differentiable relative to the time variable and satisfy an equation such as (21) only almost everywhere. A direct consequence of this is that such evolution systems are not directly amenable to the way we prove (32) in the next section, which requires the UA+B (t, s)’s to be once continuously differentiable in t relative to the strong topology of L(B); furthermore, such a strong smoothness property for the UA+B (t, s)’s of [29] and [32] does not readily follow from our hypotheses regarding the B(t)’s unless we assume more regularity properties on the perturbations. In short, it is by far the general framework of [34] that has allowed us to prove the above theorem and to deal with all the examples we have in mind in a relatively simple and direct way. Of course, whether one can prove a Trotter–Kato formula such as (32) under the sole conditions of [29,32], or under even more general conditions, remains an interesting open problem at this time. (2) It is clear that the condition 0 ∈ ρ(A(t)) ∩ ρ(B(t)) stemming from Hypotheses (A1), (B4) is imposed only for convenience, as the conclusions of our theorem still hold without this restriction; in particular, (32) remains unaltered by the addition of constants to A(t) or B(t). It is also clear that if both A(t) and B(t) are independent of t, formula (32) reduces to the form (1). However, in the time-dependent case we ought to point out that the first factor on the right-hand side of (32) only involves the contraction semigroup exp[−sA(t)]s0 and not the full evolution system UA (t, s)0stT . Then, by choosing B = 0 in (32) we obtain UA (t, s) = lim

n→+∞

0 γ =n−1

γ t −s A s + (t − s) , exp − n n

(33)

which provides the new and simple way of reconstructing the UA (t, s)’s from the exp[−sA(t)]’s we alluded to above. In Section 4 we shall also consider two examples for which we can prove (33) more directly by means of Euler’s summation formula. (3) While (32) and (33) hold in the strong topology of L(B), an issue of independent interest is whether there might exist simple and natural conditions which would imply the convergence


2257

of these approximations in the norm-topology of that space. We refer the reader to [5,18,25,38] for some results and discussions in this direction in a different context. (4) Whereas the above conditions are sufficient to ensure the validity of the theorem, they are certainly not optimal since we did not strive for maximal generality. In particular, they are not all necessary when applied to parabolic problems that exhibit a variational structure; this is easy to understand in light of the theory developed in [34] since, in that case, proofs can as a rule be obtained under a weaker set of hypotheses. Typical hypotheses of this kind are, for instance, (4)–(6) and (18)–(20) in the case of (9) and (17), respectively. In particular, it would be highly desirable to get a proof of (32) under hypotheses of that kind, which, in effect, raises the more general question of proving product formulae by means of the theory of time-dependent quadratic forms. To the best of our knowledge this is an open problem, whereas the time-independent case was settled in [21], of which a special case can be found in [10]. We shall come back to this point in Section 4. (5) Our theorem offers an alternative approach to Kato’s theory of nonautonomous parabolic evolution equations which was developed many years ago in [20]. Since that time this theory has been successfully applied to numerous specific problems particularly when the domains of the operators involved are time-independent (see, for instance, [19,20,28,34] and the references therein). However, when those domains become time-dependent Kato’s theory imposes rather strong invariance conditions which are as a rule very difficult to check in practice, particularly in concrete examples of partial differential equations with time-varying boundary conditions such as (10). This remark applies, for instance, to the verification of the first product formula in [27], which, incidentally, does bear some formal resemblance with (33). By contrast, our result does not require any such invariance conditions and thereby allows us to treat a wide class of such models as we shall see below. Finally, we also would like to mention [16] and its numerous references for a systematic account of certain recent probabilistic developments of Kato’s theory in the nonautonomous case, including the analysis of the related Feynman–Kac propagators. We devote the next section of this article to the proof of the above theorem. 3. Proof of the main result Our preliminary remark is the following lemma, whose proof is immediate by induction and therefore omitted. Lemma 1. For every n ∈ N+ ∩ [3, +∞) let (Uγ )γ ∈{1,...,n} and (Vγ )γ ∈{1,...,n} be two families of operators in L(B); then the identity 1 γ =n

Uγ −

1 γ =n

Vγ =

2

Vα × (U1 − V1 )

α=n

+

+1 n−1 γ

Vα × (Uγ − Vγ ) ×

γ =2 α=n

1

Uβ + (Un − Vn ) ×

β=γ −1

1

Uβ

(34)

β=n−1

holds. Furthermore, for every n ∈ N+ and any U, V ∈ L(B) we have Un − V n =

n γ =1

U n−γ (U − V )V γ −1 .

(35)

2258


In what follows we write I for the identity operator on B and recall that C(t) = A(t) + B(t). The stability of the basic properties of the A(t)’s relative to the perturbation by the B(t)’s is stated in the following result. Lemma 2. (a) Assume that Hypotheses (A1) and (B1) hold. Then for any t ∈ [0, T ] the operator −C(t) is the infinitesimal generator of a holomorphic semigroup on B. Moreover, for every such t the operator C(t) is invertible and we have C −1 (t) ∈ L(B). (b) Assume that Hypotheses (A1), (A2), (B1) and (B2) hold. Then the function t → C −1 (t) is continuously differentiable with respect to the norm-topology of L(B) and there exist constants c2 ∈ R + ∗ , c˜2 ∈ (0, 1] such that the Hölder continuity estimate −1 dC (t) dC −1 (s) c2 |t − s|c˜2 − dt ds ∞ is valid for every s, t ∈ [0, T ]. (c) Assume that Hypotheses (A1), (A3), (B1) and (B3) hold. Then there exist constants c3 ∈ R+ ∗ , c˜3 ∈ (0, 1] such that the inequality ∂

R C(t), λ ∂t

∞

c3 |λ|−c˜3

(36)

holds for every t ∈ [0, T ] and each λ ∈ Sθ {0}. Proof. In order to prove (a), it is sufficient to show that Sθ ⊆ ρ(C(t)) and that

R C(t), λ

∞

−1 c˜∗ 1 + |λ|

(37)

for every t ∈ [0, T ] and each λ ∈ Sθ for some c˜∗ ∈ R+ ∗ independent of t and λ, where Sθ is given by (24). Let λ ∈ Sθ and let us choose v = R(A(t), λ)w in (26) where w ∈ B{0} is arbitrary; then, by virtue of (23) and the choice of a, b in Hypothesis (B1) we have

B(t)R A(t), λ w (a + bc∗ ) w < w ,

(38)

so that (I + B(t)R(A(t), λ))−1 ∈ L(B). Therefore we get

−1 R C(t), λ = R A(t), λ I + B(t)R A(t), λ +∞

m

= R A(t), λ (−1)m B(t)R A(t), λ ∈ L(B),

(39)

m=0

which implies (37) as a consequence of (23) and (38). In order to prove (b), we first remark that (39) implies C −1 (t) = A−1 (t)D(t)

(40)


2259

when λ = 0, where we have defined

−1

D(t) := I + B(t)A−1 (t)

=

+∞

m (−1)m B(t)A−1 (t)

m=0

for every t ∈ [0, T ]. According to Hypothesis (A2) and (40), it is then sufficient to prove that the function t → D(t) is continuously differentiable with respect to the norm-topology of L(B) on [0, T ] and that its derivative t → d Ddt(t) is Hölder continuous there; but this follows from Hypothesis (B2) and standard arguments based on the decomposition formulae (34) and (35). The starting point for the proof of (c) is (39), which we rewrite as

R C(t), λ = R A(t), λ D(t, λ) by analogy with (40), where we have defined

D(t, λ) := I + B(t)R A(t), λ

−1

=

+∞

m (−1)m B(t)R A(t), λ .

m=0

On the one hand, it follows from the preceding expression and (38) that the function t → D(t, λ) is bounded in the norm-topology of L(B) on [0, T ] uniformly in λ ∈ Sθ . On the other hand, it also follows from standard arguments that the function t → D(t, λ) is continuously differentiable with respect to the norm-topology of L(B) on [0, T ], and that the representation +∞

m−1

m=1

k=0

m−k−1

k ∂(B(t)R(A(t), λ)) ∂ D(t, λ) = B(t)R A(t), λ B(t)R A(t), λ (−1)m ∂t ∂t holds as a convergent series in the Banach space of all continuous mappings from [0, T ] into L(B) endowed with the uniform topology. From this, (38) and Hypothesis (B3) we then infer that the estimate +∞ ∂(B(t)R(A(t), λ)) ∂ D(t, λ) m−1 c < +∞ sup mρ ∂t ∂t t∈[0,T ] ∞ ∞ m=1

is valid uniformly in λ ∈ Sθ , where ρ := a + bc∗ . Consequently, since we have

∂ D(t, λ)

∂ ∂ R C(t), λ = R A(t), λ + R A(t), λ × D(t, λ) ∂t ∂t ∂t we evidently get ∂

R C(t), λ ∂t

∂

R A(t), λ c R A(t), λ ∞ + ∂t ∞ ∞

−1

−a˜ 3 −a˜ 3 c3 |λ| + a3 |λ| c c∗ 1 + |λ|

2260


for some c3 ∈ R+ ˜ 3 ∈ (0, 1]; we ∗ for every λ ∈ Sθ {0}, by virtue of (23), (25) and the fact that a may thus choose c˜3 = a˜ 3 . 2 Lemma 2 along with Tanabe’s theory developed in [34] then imply the following result. Proposition 1. Assume that Hypotheses (A1)–(A3) and (B1)–(B3) hold. Then, there exists an evolution system UA+B (t, s)0stT for Problem (21) such that Statement (1) of the theorem holds for every s, t with 0 s < t T . The remaining results are, therefore, preparatory statements which will lead to our proof of the product formula. From now on we may assume that t ∈ (s, T ) since Statement (2) of the theorem trivially holds for t = s, and begin with the following uniformity result which is the consequence of an elementary compactness argument. Lemma 3. Let (R(h, r))(h,r)∈(0,T −t]×(s,t] ⊂ L(B) be a family of operators satisfying R(h, r) < +∞. sup ∞

(41)

(h,r)∈(0,T −t]×(s,t]

Furthermore let K ⊂ B be compact, let I be any subinterval of (s, t] and assume that the limit lim R(h, r)v = 0

(42)

h→0

exists for every v ∈ K in the strong topology of B uniformly in r ∈ I . Then, (42) holds uniformly in v ∈ K. Proof. On the one hand, if the preceding conclusion does not hold there exist ∈ R+ ∗ along with a sequence (hn ) ⊂ (0, T − t] satisfying hn < n1 , together with sequences (rn ) ⊂ I , (vn ) ⊂ K, such that the inequality R(hn , rn )vn > (43) holds true for every n ∈ N+ . On the other hand, because of the compactness of K we may assume that vn → v ∗ ∈ K in the strong topology of B when n → +∞, so that by virtue of (41) and (42) the estimate R(hn , rn )vn R(hn , rn )(vn − v ∗ ) + R(hn , rn )v ∗ c vn − v ∗ + R(hn , rn )v ∗ is valid for every n N (, v ∗ ) for some N (, v ∗ ) ∈ N+ , thereby contradicting (43).

2

We now introduce three families of linear operators on B whose properties will be crucial in our proof of Statement (2) below; indeed we define E(h, r), F (h, r), G(h, r) by

E(h, r) = h−1 I − exp −hA(r) − A(r),

F (h, r) = h−1 exp −hA(r) I − exp −hB(r) − B(r),

G(h, r) = h−1 I − UA+B (r + h, r) − C(r)

(44)


2261

for every (h, r) ∈ (0, T − t] × (s, t], respectively. From these relations and the definition of the generator C(r) we then obtain E(h, r) + F (h, r) − G(h, r)

= h−1 UA+B (r + h, r) − exp −hA(r) exp −hB(r) .

(45)

The following result unveils the behavior of these operators when h → 0, and part of its proof is a consequence of a repeated application of Lemma 3. Lemma 4. Assume that the same hypotheses as in Proposition 1 hold; moreover, assume that Hypotheses (A4) and (D) hold, along with the C0 -continuity and the contractive property of Hypothesis (B4). Then we have lim sup E(h, r)UA+B (r, s)w = lim sup F (h, r)UA+B (r, s)w

h→0 r∈(s,t]

h→0 r∈(s,t]

= lim sup G(h, r)UA+B (r, s)w = 0 h→0 r∈(s,t]

(46)

for every t ∈ (0, T ) and each w ∈ D. Proof. In order to prove the first two relations in (46), it is sufficient to show that the two limits lim

sup E(h, r)UA+B (r, s)w

h→0 r∈[s+μ,t]

= lim

sup F (h, r)UA+B (r, s)w = 0

h→0 r∈[s+μ,t]

(47)

hold uniformly in μ ∈ (0, t − s), respectively. From the definition of E(h, r) and a general property of C0 -semigroups we may write −1

h

E(h, r)UA+B (r, s)w = h

dk exp −kA(r) − I A(r)UA+B (r, s)w

0

since UA+B (r, s)w ∈ D(A(r)) for every r ∈ [s + μ, t] and each w ∈ D as a consequence of (31). Therefore we obtain the inequality

E(h, r)UA+B (r, s)w sup exp −kA(r) − I A(r)UA+B (r, s)w , k∈[0,h]

so that in order to get the first relation in (47) it is sufficient to have

lim sup exp −hA(r) − I A(r)UA+B (r, s)w = 0

(48)

for every w ∈ D uniformly in μ. To this end let us define R(h, r) and Kμ by R(h, r) = exp −hA(r) − I

(49)

h→0 r∈[s+μ,t]

2262


and

Kμ = v ∈ B: v = A(r)UA+B (r, s)w, r ∈ [s + μ, t] respectively; since the semigroup in (49) is contractive, it is clear that R(h, r) satisfies (41). Furthermore we also have lim

sup

h→0 r∈[s+μ,t]

R(h, r)v = 0

(50)

for every v ∈ B uniformly in μ; in fact, from (28), (49) and for every v ∈ D we obtain the estimate R(h, r)v

h

dk exp −kA(r) A(r)v h sup A(r)v → 0 r∈(0,T ]

0

as h → 0 uniformly in r and μ, which then leads to (50) since D ⊂ B is dense. Finally, in order to prove the compactness of Kμ it is sufficient to prove the continuity of the mapping r → A(r)UA+B (r, s)w in the strong topology of B for r ∈ [s + μ, t]; but this is an immediate consequence of Hypothesis (B2), (40) and Statement (1) of the theorem since we have

A(r)UA+B (r, s)w = I − B(r)A−1 (r)D(r) A(r) + B(r) UA+B (r, s)w. Therefore, (48) indeed emerges as a consequence of Lemma 3. We now proceed in much the same way to prove the second relation in (47). We start with the integral representation

−1

h

F (h, r)UA+B (r, s)w = h

dk exp −hA(r) exp −kB(r) − I B(r)UA+B (r, s)w

0

valid for every w ∈ D, which leads to the inequality F (h, r)UA+B (r, s)w

sup exp −hA(r) exp −kB(r) − I B(r)UA+B (r, s)w .

(51)

k∈[0,h]

This time we define R(h, r) and Kμ by R(h, r) = exp −hA(r) exp −kB(r) − I and

Kμ = v ∈ B: v = B(r)UA+B (r, s)w, r ∈ [s + μ, t]

(52)


2263

respectively; from the contractive properties of the semigroups involved it follows again that (52) satisfies (41), while the relation lim

sup

h→0 r∈[s+μ,t]

R(h, r)v = 0

(53)

is in this case a consequence of the identity

R(h, r) = exp −hA(r) exp −kB(r) − I + exp −hA(r) − I. In fact, by using the same kind of integral representation as above along with (29) we obtain

exp −hA(r) exp −kB(r) − I v k

dl exp −lB(r) B(r)v h sup B(r)v → 0 r∈(0,T ]

0

(54)

as h → 0 uniformly in r and μ for every v ∈ D and hence for every v ∈ B, while from the first part of the proof we get

(55) sup exp −hA(r) − I v → 0 r∈[s+μ,t]

as h → 0 for every v ∈ B, both (54) and (55) implying (53) for every such v. Finally, for the same reasons as above Kμ is compact, so that from Lemma 3 we infer that the relation lim

sup exp −hA(r) exp −kB(r) − I B(r)UA+B (r, s)w = 0

h→0 r∈[s+μ,t]

holds for every w ∈ D uniformly in μ; together with (51) this implies the desired result. The proof of the third relation in (46) follows a slightly different line, the starting point being the identity −1

G(h, r)UA+B (r, s)w = h

r+h

dk C(k)UA+B (k, s)w − C(r)UA+B (r, s)w

(56)

r

which is a simple consequence of Statement (1) of the theorem and of the third relation in (44). Since that statement also implies the uniform continuity of the function k → C(k)UA+B (k, s)w − C(r)UA+B (r, s)w with respect to the strong topology of B on the compact interval [r, r + h], we + may conclude that for every ∈ R+ ∗ there exists an h ∈ R∗ such that the inequalities 0 k − r h h together with (56) lead to the estimate G(h, r)UA+B (r, s)w uniformly in r ∈ (s, t], which is the desired result.

2

Finally, we will still need the following continuity result.

2264


Lemma 5. Assume that Hypotheses (A1), (A2), (A4), (B4) and (D) hold. Then, the functions r → exp −(r − s)A(r) (57) and r → exp −(r − s)B(r)

(58)

are continuous on the interval [s, t] in the strong topology of L(B). Proof. The proof of (57) follows from the standard arguments of [34]. As for (58) we first prove the right-continuity at r = s; for this we start by noticing that the analyticity part of Hypothesis (B4) allows us to write

1 dλ e−(r−s)λ R B(r), λ v (59) exp −(r − s)B(r) v = 2πi Γr,s

for every v ∈ B and each r ∈ (s, t], where Γr,s := Γ1,r,s ∪ Γ2,r,s ∪ Γ3,r,s is the union of the three paths 1 μ < +∞ , r −s 1 iμ ∗ ∗ e : θ μ 2π − θ = r −s

∗ Γ1,r,s = μe−iθ : Γ2,r,s and ∗ Γ3,r,s = μeiθ :

1 μ < +∞ . r −s

The orientation we choose for Γr,s is that of increasing values of Im λ. From the residue theorem and the chosen orientation of Γr,s we then easily obtain 1 2πi

dλ e−(r−s)λ λ−1 = −1,

Γr,s

so that we may write 1 exp −(r − s)B(r) v − v = 2πi

dλ e−(r−s)λ R B(r), λ + λ−1 v

Γr,s

=

1 2πi

Γr,s

=

1 2πi

Γ

dλ e−(r−s)λ λ−1 R B(r), λ B(r)v dλ e−λ λ−1 R B(r),

λ B(r)v r −s

(60)


2265

for every v ∈ D, the dense set of Hypothesis (D), where the new integration path Γ := (r −s)Γr,s in (60) is independent of r and s. Therefore, owing to (29) and (30) we obtain exp −(r − s)B(r) v − v c sup B(r)v (r − s) |dλ|e−λ |λ|−2 → 0 r∈(0,T ]

Γ

as r s; since D is dense and the semigroup exp[−sB(t)]s0 contractive, the desired rightcontinuity at r = s follows. As for the proof of the continuity away from the left endpoint, we find it more convenient to choose the integration path Γ := Γ1 ∪ Γ2 in (59) rather than the path Γr,s , where

∗ Γ1 = μe−iθ : 0 μ < +∞ and

∗ Γ2 = μeiθ : 0 μ < +∞ , the orientation being the same as before. Let us fix r ∈ (s, t), let (rn )n∈N+ be any sequence such that rn > r with rn → r as n → +∞ and write exp −(r − s)B(r) v − exp −(rn − s)B(rn ) v

1 dλ e−(r−s)λ R B(r), λ − R B(rn ), λ v = 2πi Γ

1 + dλ e−(r−s)λ 1 − e−(rn −r)λ R B(rn ), λ v. (61) 2πi Γ

From the continuity part of Hypothesis (B4) we have

lim sup R B(r), λ − R B(rn ), λ v = 0, n→+∞ λ∈S

θ∗

which indeed implies that

lim

n→+∞

dλ e−(r−s)λ R B(r), λ − R B(rn ), λ v = 0

(62)

Γ

strongly in B for every v since

|dλ|e−(r−s)λ < +∞.

Γ

Furthermore, the norm of the integrand in the second term on the right-hand side of (61) goes to zero as n → +∞ for every λ ∈ Γ . Moreover, owing to (30) and to our choice of the rn ’s we can estimate that norm as −(r−s)λ

|e−(r−s)λ | e v 1 − e−(rn −r)λ R B(rn ), λ v c 1 + |λ|

2266


uniformly in n, so that we eventually get

dλ e−(r−s)λ 1 − e−(rn −r)λ R B(rn ), λ v = 0 lim n→+∞

(63)

Γ

strongly in B for every v by dominated convergence since |dλ| Γ

|e−(r−s)λ | < +∞. 1 + |λ|

A similar argument holds for r ∈ (s, t] if (rn )n∈N+ is any sequence such that rn < r with rn → r as n → +∞. This, together with (61)–(63) proves the desired continuity away from r = s. Evidently, if the B’s are independent of r the C0 -continuity of exp[−sB]s0 alone gives the result. 2 We are now ready for the following. Proof of the theorem. By virtue of Proposition 1 it remains to prove Statement (2). For every n ∈ N+ sufficiently large we set h = t−s n and define the sequence of products (Pn (t, s)) ⊂ L(B) by Pn (t, s) = UA+B (t, s) −

1

exp −hA s + (γ − 1)h exp −hB s + (γ − 1)h .

γ =n

Since both exp[−sA(t)]s0 and exp[−sB(t)]s0 are semigroups of contractions for every t ∈ [0, T ] the sequence (Pn (t, s)) is bounded in L(B), so that in order to prove the product formula it is sufficient to show that Pn (t, s)v → 0 as n → +∞ in the strong topology of B for every v ∈ D, the dense set of Hypothesis (D). To this end we choose the two families (Uγ )γ ∈{1,...,n} , (Vγ )γ ∈{1,...,n} of Lemma 1 as

Uγ = UA+B s + γ h, s + (γ − 1)h ,

Vγ = exp −hA s + (γ − 1)h exp −hB s + (γ − 1)h , respectively; owing to the basic composition law of the UA+B (t, s)’s and by virtue of Lemma 1 we then have after some rearrangements Pn (t, s) =

+1 n−1 γ

Vα × (Uγ − Vγ )UA+B s + (γ − 1)h, s

γ =1 α=n

+ (Un − Vn )UA+B s + (n − 1)h, s .

Therefore, for every v ∈ D and by using again the estimate γ +1 Vα α=n

∞

1


2267

we obtain the inequalities n

Pn (t, s)v (Uγ − Vγ )UA+B s + (γ − 1)h, s v γ =1

n

sup

UA+B (r + h, s)v − exp −hA(r) exp −hB(r) UA+B (r, s)v

r∈[s,s+(n−1)h]

n sup UA+B (r + h, s)v − exp −hA(r) exp −hB(r) UA+B (r, s)v

(64)

r∈[s,t]

after a simple change of summation variable, where we may now write r = s + κnh for κ ∈ [0, 1]. Furthermore, expressing h in (64) as a function of r by means of this last relation and by using Lemma 5, we see that the function r −s r −s r −s r → UA+B r + , s − exp − A(r) exp − B(r) UA+B (r, s) κn κn κn is continuous on the interval [s, t] in the strong topology of L(B) since the evolution system UA+B (t, s)0stT also enjoys this property. Consequently, this allows us to replace the interval [s, t] by (s, t] in (64), so that we finally obtain Pn (t, s)v

r −s r −s r −s c sup E ,r + F ,r − G , r UA+B (r, s)v →0 κn κn κn r∈(s,t]

for every v ∈ D when n → +∞, as a consequence of (45) and Lemma 4.

2

We devote the next section to the discussion of some examples illustrating the statements of our main theorem. 4. Some simple examples While it is clear that our theorem has a wide range of potential applications, we shall restrict ourselves here to the simplest situations. We first consider a particular case of (10), namely, the class of parabolic initial-value problems given by

∂u(x, t) = div k(x, t)∇u(x, t) − + εm(x, t) u(x, t), ∂t u(x, 0) = u0 (x), x ∈ D, ∂u(x, t) = 0, (x, t) ∈ ∂D × (0, T ]. ∂n(k) In this case (13) reduces to bε (t, v, w) = ε

dx m(x, t)v(x)w(x) D

(x, t) ∈ D × (0, T ],

(65)

2268


and thus extends to a Hermitian sesquilinear form on B =L2 (D, C), so that the associated multiplication operator Bε (t) := εB(t) is bounded and self-adjoint there. In order for (65) to fit the theory of the preceding section, however, we need to impose stronger conditions on the coefficients than (K) and (M) do. The following smoothness requirements are sufficient for this purpose. (K ) We have k : D × [0, T ] → Rd and for every i, j ∈ {1, . . . , d} the functions (x, t) → ki,j (x, t) = kj,i (x, t) are continuously differentiable on D × [0, T ]; moreover, the ellipticity condition (3) holds and there exist constants c∗ ∈ R+ ∗ , σ ∈ (0, 1] such that the Hölder continuity estimate 2

∂ki,j (x, t) ∂ki,j (x, s) c |t − s|σ − max ∗ i,j ∈{1,...,d} ∂t ∂s

(66)

is valid for every x ∈ D and every s, t ∈ [0, T ]. (M ) We have m ∈ L∞ (D × [0, T ], R+ ) and t → m(x, t) is continuously differentiable on ∞ [0, T ] uniformly in x ∈ D with ∂m ∂t ∈ L (D × [0, T ], R); moreover, the Hölder continuity estimate ∂m(x, t) ∂m(x, s) c |t − s|σ − ∗ ∂t ∂s holds for every x ∈ D and every s, t ∈ [0, T ]. We then have the following result. Proposition 2. Assume that Hypotheses (K ) and (M ) hold; then, all the conclusions of the theorem are valid for the evolution system given by (17). In particular, for all s, t with 0 s t < T and every ε ∈ R+ sufficiently small we have UA+Bε (t, s) = lim

n→+∞

0 γ =n−1

γ γ t −s t −s A s + (t − s) exp − Bε s + (t − s) (67) exp − n n n n

in the strong topology of L(L2 (D, C)), where exp[−sA(t)]s0 and exp[−sBε (t)]s0 are the semigroups generated by (7) and −Bε (t), respectively. Thus, the reconstruction formula (33) also holds in this case. The proof of Proposition 2 rests on several lemmas and remarks. Without restricting the generality, we first choose k in (65). Then, the operator given by (7) satisfies Hypotheses (A1), (A4) as an immediate consequence of standard Lax–Milgram arguments and elliptic regularity theory; moreover, Hypothesis (B1) trivially holds for ε sufficiently small since Bε (t) is bounded. The verification of the remaining hypotheses requires more work; we settle the question regarding (A2) with the following result.


2269

Lemma 6. Assume that Hypothesis (K ) holds; then, the function t → A−1 (t)f is strongly difd −1 ferentiable in L2 (D, C) for every f ; moreover, we have dt A (t)f ∈ H 1 (D, C) and there exists a constant a2 ∈ R+ such that the Hölder continuity estimate ∗ d −1 A (t)f − d A−1 (s)f a2 |t − s|σ f 2 dt ds 1,2

(68)

holds for all s, t ∈ [0, T ] and every f ∈ L2 (D, C), with σ the Hölder exponent in (66). Proof. Let us write u(s) := A−1 (s)f with u(s) ∈ D(A(s)); from (8) we then have

a s, u(s), w + u(s), w 2 = (f, w)2

(69)

and a t, u(t), w + u(t), w 2 = (f, w)2

(70)

for all s, t ∈ [0, T ] and every w ∈ H 1 (D, C), so that by subtracting (69) from (70) we obtain

a t, u(t) − u(s), w + u(t) − u(s), w 2 = a s, u(s), w − a t, u(s), w .

(71)

Next, for every t ∈ [0, T ] we introduce the shorthand notation (v, w)1,2,t := a(t, v, w) + (v, w)2

(72)

and observe that these new sesquilinear forms defined on H 1 (D, C) × H 1 (D, C) induce norms . 1,2,t equivalent to . 1,2 on H 1 (D, C) by virtue of the boundedness of the ki,j ’s and (3), the equivalence constants being independent of t. Then, fix t and set h = s − t in (71); owing to (4) and (72) this gives

u(t + h) − u(t) ,w h =−

d

i,j =1 D

dx

1,2,t

ki,j (x, t + h) − ki,j (x, t) uxi (x, t + h)w xj (x), h

(73)

from which equality we now want to prove that lim

h→0

u(t + h) − u(t) ,w h

=− 1,2,t

d

dx

i,j =1 D

Subtracting the right-hand side of (74) from (73) we have

∂ki,j (x, t) uxi (x, t)w xj (x). ∂t

(74)

2270


d u(t + h) − u(t) ∂ki,j (x, t) ,w uxi (x, t)w xj (x) + dx h ∂t 1,2,t i,j =1 D

ki,j (x, t + h) − ki,j (x, t) ∂ki,j (x, t) ux (x, t)wx (x) − dx j i h ∂t

d i,j =1 D

+

d i,j =1 D

ki,j (x, t + h) − ki,j (x, t) ux (x, t + h) − ux (x, t)wx (x) dx i i j h

and proceed by showing that these two terms each go to zero when h → 0. For the first one this follows from Hypothesis (K ) and dominated convergence since the estimate ki,j (x, t + h) − ki,j (x, t) ∂ki,j (x, t) ux (x, t)wx (x) cux (x, t)wx (x) − i j i j h ∂t holds uniformly in h as a consequence of the boundedness of continuity of t → ki,j (x, t). For the second one we have d i,j =1 D

∂ki,j (x,t) ∂t

and the uniform Lipschitz

ki,j (x, t + h) − ki,j (x, t) ux (x, t + h) − ux (x, t)wx (x) dx i i j h

cu(t + h) − u(t)1,2,t w 1,2,t ,

so that it remains to prove the relation lim u(t + h) − u(t)1,2,t = 0.

h→0

(75)

But going back to (73) we have

u(t + h) − u(t), w

1,2,t

c|h|A−1 (t + h)f

1,2,t

w 1,2,t c|h| f 2 w 1,2,t

from which (75) follows immediately, thereby completing the proof of (74). Therefore, there 1 exists du(t) dt ∈ H (D, C) such that

du(t) ,w dt

=− 1,2,t

d i,j =1 D

dx

∂ki,j (x, t) uxi (x, t)w xj (x) ∂t

for every t ∈ [0, T ] and every w ∈ H 1 (D, C), which implies that lim

h→0

u(t + h) − u(t) du(t) = h dt


2271

strongly in L2 (D, C) by virtue of the compact embedding H 1 (D, C) → L2 (D, C). Hence, d −1 the function t → A−1 (t)f is indeed strongly differentiable in L2 (D, C) with dt A (t)f ∈ 1 H (D, C) for every f ; furthermore, we have

du(t) du(s) − ,w dt ds

d

=− 1,2,t

dx

i,j =1 D

−

d

dx

i,j =1 D

∂ki,j (x, t) ∂ki,j (x, s) uxi (x, t)w xj (x) − ∂t ∂s

∂ki,j (x, s) uxi (x, t) − uxi (x, s) wxj (x), ∂s

which, together with (66) and arguments similar to those we just used, leads to the estimate du(t) du(s) c|t − s|σ f 2 w 1,2,t − , w dt ds 1,2,t and thereby to (68).

2

It is plain that the preceding construction implies the existence of a linear bounded operator dA−1 (t) : L2 (D, C) →H 1 (D, C) satisfying du(t) f , so that the validity of Hypothesis dt = dt (A2) indeed emerges as a direct consequence of Lemma 6. We now turn to Hypothesis (A3), whose verification rests on the following result. dA−1 (t) dt

Lemma 7. Assume that Hypothesis (K ) holds; then, the function t → R(A(t), λ)f is strongly differentiable in L2 (D, C) for every f ; moreover, we have ∂t∂ R(A(t), λ)f ∈ H 1 (D, C) and there exists a constant a3 ∈ R+ ∗ such that the estimate ∂

R A(t), λ f ∂t

1

a3 |λ|− 2 f 2

(76)

1,2

holds for every t ∈ [0, T ], every f ∈ L2 (D, C) and every λ ∈ Sθ {0}. Proof. Let us fix λ ∈ ρ(A(t)); it is easy to prove the strong differentiability of t → R(A(t), λ)f by relating R(A(t), λ)f to A−1 (t)f by means of the resolvent identity; we obtain

dA−1 (t)

∂ R A(t), λ f = I + λR A(t), λ I + λR A(t), λ f ∂t dt

(77) −1

for every t ∈ [0, T ] and every f ∈ L2 (D, C). Furthermore, (77) and the definition of dAdt (t) give ∂ π 1 ∂t R(A(t), λ)f ∈ H (D, C). Let us now fix θ ∈ (0, 2 ) and λ ∈ Sθ {0} in order to prove (76). For this we rely again on the variational structure of the problem; writing u(t, λ) := R(A(t), λ)f with u(t, λ) ∈ D(A(t)) and arguing exactly as in the proof of Lemma 6 we eventually get the relation

2272


∂u(t, λ) ∂u(t, λ) ∂u(t, λ) ,w + ,w − λ ,w a t, ∂t ∂t ∂t 2 2 d ∂ki,j (x, t) =− dx uxi (x, t, λ)w xj (x) ∂t i,j =1 D

valid for every t ∈ [0, T ] and every w ∈ H 1 (D, C), which reduces to ∂u(t, λ) 2 ∂u(t, λ) 2 ∂u(t, λ) ∂u(t, λ) , + a t, − λ ∂t ∂t ∂t 2 ∂t 2 d ∂ki,j (x, t) ∂ u(x, ¯ t, λ) uxi (x, t, λ) =− dx ∂t ∂t xj

(78)

i,j =1 D

by choosing w = ∂u(t,λ) ∂t . We first prove (76) for arg λ θ with Re λ > 0, Im λ > 0; for this we take the real and imaginary parts of (78) to obtain ∂u(t, λ) 2 ∂u(t, λ) 2 ∂u(t, λ) ∂u(t, λ) , + = a t, Re λ ∂t 2 ∂t ∂t ∂t 2 d ∂ki,j (x, t) ∂ u(x, ¯ t, λ) uxi (x, t, λ) + Re dx ∂t ∂t xj

(79)

i,j =1 D

and d ∂u(t, λ) 2 ∂ki,j (x, t) ∂ u(x, ¯ t, λ) = Im u dx (x, t, λ) , Im λ xi ∂t ∂t ∂t 2 xj

(80)

i,j =1 D

respectively. From (79), (80), the fact that the form a + is coercive on H 1 (D, C) and from the ∂k boundedness of the ∂ti,j ’s we then get d ∂u(t, λ) 2 ∂ki,j (x, t) ∂ u(x, ¯ t, λ) 1 k Im uxi (x, t, λ) dx ∂t 1,2 tan θ ∂t ∂t xj i,j =1 D

− Re

d

dx

i,j =1 D

∂ki,j (x, t) ∂ u(x, ¯ t, λ) uxi (x, t, λ) ∂t ∂t xj

∂u(t, λ) cu(t, λ)1,2 ∂t , 1,2 that is, ∂

R A(t), λ f ∂t

1,2

cR A(t), λ f 1,2

(81)


2273

for every t ∈ [0, T ] and every f ∈ L2 (D, C). But from standard estimates for the resolvent of time-dependent sectorial operators (see, for instance, [34]) we have in this case

R A(t), λ f

1

1,2

c|λ|− 2 f 2

(82)

so that (76) indeed follows from (81) and (82). The proof of (76) when arg λ −θ with Re λ > 0, Im λ < 0, or when Re λ 0 with λ = 0, follows from similar arguments and is thereby omitted. 2 It remains to verify Hypotheses (B2)–(B4) and (D). As far as (B2) and (B3) are concerned, it is sufficient to prove that the function t → B(t) is continuously differentiable with respect to the norm-topology of L(L2 (D, C)) and that its derivative dB(t) dt is Hölder continuous there, for then the result follows from Lemma 6 and (77), respectively; but these required properties of B(t) are immediate consequences of Hypothesis (M ). As for Hypothesis (B4), the semigroup generated by −B(t) is the multiplication operator exp −sB(t) f = exp −sm(., t) f on L2 (D, C) and is clearly holomorphic and contractive since B(t) is self-adjoint and m(., t) 0 for every t ∈ [0, T ]; consequently, the only point that requires attention is the continuity of t → R(B(t), λ), although we can easily establish the continuity of (58) directly in this case since the B(t)’s are bounded. However, we wish to present an independent argument which easily carries over to the case of certain unbounded B(t)’s. For this we assume without restricting the generality that m := inf(x,t)∈D×[0,T ] m(x, t) > 0. Lemma 8. The mapping t → R(B(t), λ) is Lipschitz continuous on [0, T ] in the norm-topology of L(B) uniformly in λ ∈ Sθ ∗ for every θ ∗ ∈ ( π4 , π2 ). Proof. Let us write Ran m for the range of m; if λ ∈ CRan m then from the relation

R B(t), λ f (x) =

f (x) m(x, t) − λ

and the fact that t → m(x, t) is Lipschitz continuous uniformly in x as a consequence of Hypothesis (M ) we readily obtain

R B(t), λ f − R B(s), λ f c |t − s| f 2 2 2 dm,λ

(83)

for every f ∈ L2 (D, C) and every s, t ∈ [0, T ], where dm,λ :=

inf

m(x, t) − λ > 0

(x,t)∈D×[0,T ]

is the distance between λ and Ran m. In order to get the desired uniformity in (83), it is thus sufficient to prove that dm := inf dm,λ > 0. λ∈Sθ ∗

(84)

2274


Let us fix θ ∗ ∈ ( π4 , π2 ); we first prove (84) for arg λ θ ∗ with Re λ > 0, Im λ > 0. On the one hand, if Re λ ∈ (0, m) we have 2 dm,λ (Im λ)2 + (m − Re λ)2

(Re λ)2 tan2 θ ∗ + m2 − 2m Re λ

(Re λ)2 tan2 θ ∗ − + m2 1 − −1 2 ∗ for every ∈ R+ ∗ by using Cauchy’s interpolated inequality, so that by choosing = tan θ we obtain

2 dm,λ m2 1 − tan−2 θ ∗ > 0 thanks to our choice of θ ∗ . On the other hand, if Re λ ∈ [m, +∞) we get dm,λ Im λ Re λ tan θ ∗ m tan θ ∗ > 0. The remaining cases when arg λ −θ ∗ with Re λ > 0, Im λ < 0, or when Re λ 0 can be dealt with in a similar way, thereby proving (84). 2 Finally, Hypothesis (D) is a straightforward consequence of (K ), (M ) and Gauss’ divergence theorem if we choose, for instance, D = C 20 (D, C), the space of all complex-valued, twice continuously differentiable functions with compact support in D. Remarks. (1) The statement of Proposition 2 is, therefore, a direct consequence of the above considerations and our main theorem since (K ) obviously implies (K) while (M ) implies (M). Indeed, by uniqueness, the evolution systems UA (t, s)0stT and UA+Bε (t, s)0stT of Proposition 2 are then exactly the same as those defined by (9) and (17), respectively. But the natural question that is now emerging is whether the product formula (67) might hold under (K) and (M) alone; this is not immediate for Hypothesis (D) is not necessarily verified under these two conditions and, furthermore, some aspects of our proof of (32) are not completely independent of the existence proof for UA+B (t, s)0stT . In fact, a rigorous proof of (67) under the sole set of conditions (K) and (M) is lacking at the moment, though we conjecture that this result is true. In any case, this brings us back to the third remark following the statement of the theorem. (2) The fact that (67) holds with UA+Bε (t, s) given by (17) where GA+Bε is now the parabolic Green’s function associated with the differential operator in (65) allows one to express the solution to this problem in the form of a Feynman–Kac formula. This is of course invaluable information for what regards the analysis of solutions to related semilinear initial–boundary value problems. However, we will not dwell on this any further in this paper, as we want to defer such detailed applications to a separate publication. The above conjecture is all the more reinforced by the fact that some of the hypotheses of the preceding section are not necessary for our formulae to be valid in some simpler models. A case in point is the following example, which is a particular case of (65), namely, the class of parabolic initial-value problems of the form


∂u(x, t) = k(t)u(x, t) − u(x, t), ∂t u(x, 0) = u0 (x), x ∈ D, ∂u(x, t) = 0, ∂n(x)

2275

(x, t) ∈ D × (0, T ],

(x, t) ∈ ∂D × (0, T ],

(85)

where k ∈ C 1 ([0, T ], R+ ∗ ). Hypothesis (K) is here trivially satisfied and the self-adjoint, positive operator A(t) := −k(t) + in L2 (D, C) is defined on the time-independent domain

D A(t) = v ∈ H 2 (D, C): ∇v(x), n(x) Cd = 0, x ∈ ∂D since k is a scalar function. Furthermore, Hypothesis (A1) holds if is sufficiently large but Hypothesis (A2) does not since we cannot expect (22) to be satisfied without requiring the derivative k to be Hölder continuous. Nevertheless, there exists an evolution system UA (t, s)0stT for (85), namely, UA (t, s) = e

−(t−s)

t exp

dy k(y) ,

(86)

s

and our point with this example is to show that we can also reconstruct (86) by means of (33). In fact, on the one hand we have 0 γ =n−1

=e

γ t −s A s + (t − s) exp − n n n−1 t −s γ exp k s + (t − s) , n n

−(t−s)

(87)

γ =0

and on the other hand we may write n−1 n−1 1 n−1 γ y k(s) + k s + (t − s) k s + (t − s) = dy k s + (t − s) + n n 2 n

γ =0

0

t −s + n

n−1 y 1 k s + (t − s) dy y − [y] − 2 n

(88)

0

by Euler’s summation formula, with [y] the integral part of y (see, for instance, [17]). Regarding the first term on the right-hand side of (88) we have t −s lim n→+∞ n

n−1 t y dy k s + (t − s) = dy k(y) n 0

s

(89)

2276


since k is smooth, while for the remaining two terms we get n−1 t −s k(s) + k s + (t − s) n→+∞ 2n n lim

= lim

n→+∞

t −s n

2 n−1 y 1 k s + (t − s) = 0, dy y − [y] − 2 n

(90)

0

the last equality in the preceding expression being a consequence of the boundedness of y → y − [y] − 12 and k . Consequently, owing to (87)–(90) and to the C0 -continuity of the underlying diffusion semigroup generated by the Laplacian we get t exp

n−1 t −s γ dy k(y) = lim exp k s + (t − s) n→+∞ n n

(91)

γ =0

s

in the strong topology of L(L2 (D, C), as desired. Along with (91), we remark that in the preceding example Hypothesis (D) also holds if we choose once more D= C 20 (D, C). Since that hypothesis plays an important rôle in the proof of (32) and (33) within our abstract setting, we may then tend to believe that it is also necessary for those product formulae to hold. We now show that even this is not the case by considering a third example related to one very briefly mentioned at the end of [20]. Let us consider the initial value problem u(x, t) du(x, t) =− , dt (t − x)2 u(x, 0) = u0 ,

(x, t) ∈ (0, 1) × (0, 1],

x ∈ (0, 1)

(92)

in L2 ((0, 1), C), that is, (21) with T = 1, B(t) = 0 and the A(t)’s self-adjoint, multiplication operators defined by A(t)v(x) :=

v(x) (t − x)2

(93)

on the maximal, time-dependent domains 1

|v(x)|2 2 D A(t) = v ∈ L (0, 1), C : dx < +∞ (t − x)4

(94)

0

where t ∈ [0, 1]. In this case Hypothesis (D) does not hold since we have the rather extreme opposite situation where t∈[0,1]

D A(t) = {0}.

(95)


2277

In fact, let v ∈ t∈[0,1] D(A(t)) and ∈ R+ ∗ sufficiently small; on the one hand, by using Schwarz inequality we have −1

(2)

t+ 1 t+ 2 −1 4 A(t)v c 32 A(t)v → 0 dx v(x) (2) dx (t − x) 2 2

t−

(96)

t−

for every t ∈ (0, 1) as → 0+ . On the other hand, we infer from standard one-dimensional Lebesgue integration theory that −1

lim (2)

→0+

t+ dx v(x) = v(t)

t−

for almost every t ∈ (0, 1), which, together with (96), indeed implies v = 0 in L2 ((0, 1), C). In spite of this fact and by means of yet another application of Euler’s summation formula, we now prove that (33) holds for all s, t with 0 s t 1 in the strong topology of L2 ((0, 1), C), thereby showing that the reconstruction of the full evolution system from the semigroups generated by the A(t)’s is possible in this case as well. On the one hand, the holomorphic semigroup generated by −A(t) is the contraction semigroup given by exp −sA(t) v(x) = exp −

s v(x) (t − x)2

(97)

for every s ∈ R+ and any v ∈ L2 ((0, 1), C). On the other hand, an explicit calculation from (92) shows that the corresponding evolution system UA (t, s)0st1 in B = L2 ((0, 1), C) also exists in the form of the multiplication operators UA (t, s)v(x) =

exp[(t − x)−1 − (s − x)−1 ]v(x) 0

if x ∈ (0, s) ∪ (t, 1), if x ∈ (s, t).

(98)

We begin our analysis of the reconstruction with the following auxiliary result. Proposition 3. For every v ∈ L2 ((0, 1), C) and all s, t with 0 s < t 1 we have t

dx exp −2n(t − s)

lim

n→+∞

n−1 γ =0

s

1 v(x)2 = 0. (γ (t − s) − n(x − s))2

Proof. It is sufficient to prove that t


s

n−1 γ =0

In order to achieve this we write

1 v(x)2 exp − 4n v 2 . 2 t −s (γ (t − s) − n(x − s))2

(99)

2278


t


n−1 γ =0

s

=

s+(δ+1) (t−s) n

n−1

1 v(x)2 (γ (t − s) − n(x − s))2


δ=0

n−1 γ =0

s+δ (t−s) n

1 v(x)2 2 (γ (t − s) − n(x − s))

(100)

and observe that the inequalities (γ − δ − 1)(t − s) < γ (t − s) − n(x − s) < (γ − δ)(t − s) hold for every x and every γ , δ ∈ {0, . . . , n − 1} in each of the integrals on the right-hand side of (100). Consequently, if γ δ we get the lower bound 1 1 > (γ (t − s) − n(x − s))2 (γ − δ − 1)2 (t − s)2 while if γ δ + 1 we have 1 1 > . 2 2 (γ (t − s) − n(x − s)) (γ − δ) (t − s)2 Therefore we obtain the estimate n−1 γ =0

δ n−1 1 1 1 = + (γ (t − s) − n(x − s))2 (γ (t − s) − n(x − s))2 (γ (t − s) − n(x − s))2 γ =0

1 > (t − s)2

γ =δ+1

δ γ =0

n−1 1 1 + 2 (γ − δ − 1) (γ − δ)2

γ =δ+1

>

2 (t − s)2

uniformly in x, δ and n, so that the substitution of the preceding inequality into the right-hand side of (100) indeed leads to

t


γ =0

s

n−1 4n exp − t −s δ=0

which is (99).

n−1

2

1 v(x)2 2 (γ (t − s) − n(x − s))

s+(δ+1) (t−s) n

s+δ (t−s) n

2 4n v 22 , dx v(x) exp − t −s


2279

It is more complicated to get the relevant estimates when x ∈ (0, s) ∪ (t, 1). To this end let us introduce the functions fn,t,s (., x) : [0, n − 1] → R+ ∗ defined by fn,t,s (y, x) :=

1 (y(t − s) − n(x − s))2

(101)

for every n ∈ N+ ∩ [2, +∞), along with the function ft,s : (0, s) ∪ (t, 1) → R− ∗ given by ft,s (x) := (t − x)−1 − (s − x)−1 .

(102)

Our second auxiliary result is the following. Proposition 4. For every v ∈ L2 ((0, 1), C) and all s, t with 0 s < t 1 we have s lim

n→+∞

2 n−1 2 dx exp −n(t − s) fn,t,s (γ , x) − exp ft,s (x) v(x) = 0 γ =0

0

and 1 lim

n→+∞ t

2 n−1 2 dx exp −n(t − s) fn,t,s (γ , x) − exp ft,s (x) v(x) = 0. γ =0

The proof of this proposition rests on one crucial lemma. We first remark that the fn,t,s (., x)’s are well defined and continuously differentiable on [0, n − 1] for every x ∈ (0, s) ∪ (t, 1) since their denominators do not vanish there. Consequently, we may write n−1 γ =0

n−1

1 fn,t,s (γ , x) = dy fn,t,s (y, x) + fn,t,s (0, x) + fn,t,s (n − 1, x) 2 0

n−1 dy ψ(y)fn,t,s (y, x) −

(103)

0

as a variant of Euler’s summation formula, where we have defined y ψ(y) :=

1 . dz z − [z] − 2

0

We remark that ψ satisfies the inequalities 1 − ψ(y) 0 8 for every y ∈ R; regarding (101) we then have the following.

(104)

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Lemma 9. For every x ∈ (0, s) ∪ (t, 1) and all s, t with 0 s < t 1 we have n−1 t − s −1 −1 dy fn,t,s (y, x) = exp t − x − − (s − x) exp −n(t − s) n

(105)

0

along with

n(t − s) fn,t,s (0, x) + fn,t,s (n − 1, x) exp − 2 1 n t −s . + = exp − 2 n(x − s)2 (n(t − x) − (t − s))2

(106)

Moreover, we have the estimate n (t − s)2 1 exp + 4 (n(t − x) − (t − s))3 n2 (x − s)3 n−1 dy ψ(y)fn,t,s (y, x) 1.

exp n(t − s)

(107)

0

Proof. Relations (105) and (106) follow from an explicit evaluation of the first two terms on the right-hand side of (103). Furthermore, from (104) we have

n(t − s) n(t − s) − fn,t,s (n − 1, x) − fn,t,s (0, x) = − 8 8

n−1 dy fn,t,s (y, x) 0

n−1

dy ψ(y)fn,t,s (y, x) 0

n(t − s) 0

since fn,t,s (., x) is convex, from which (107) follows immediately.

2

For the sake of simplicity we now introduce a shorthand notation for all three exponential arguments above, namely, n−1 Θn,t,s (x) := n(t − s) dy fn,t,s (y, x),

(108)

0

Φn,t,s (x) :=

n(t − s) fn,t,s (0, x) + fn,t,s (n − 1, x) 2

(109)


2281

and n−1 dy ψ(y)fn,t,s (y, x). Ψn,t,s (x) := n(t − s)

(110)

0

We then have the following. Proof of Proposition 4. For every x ∈ (0, s) ∪ (t, 1) and all s, t with 0 s < t 1 we may write exp −n(t − s)

n−1

fn,t,s (γ , x) − exp ft,s (x)

γ =0

= exp −Φn,t,s (x) exp Ψn,t,s (x) exp −Θn,t,s (x) − exp ft,s (x)

+ exp −Φn,t,s (x) exp Ψn,t,s (x) − 1 exp ft,s (x) as a consequence of (103) and (108)–(110); moreover, from (102), (106) and (107) we have exp ft,s (x) 1, exp −Φn,t,s (x) exp Ψn,t,s (x) 1 and lim exp −Φn,t,s (x) exp Ψn,t,s (x) = 1,

(111)

n→+∞

respectively. For every v ∈ L2 ((0, 1), C), almost every x ∈ (0, s) ∪ (t, 1) and all s, t with 0 s < t 1 we then get 2 n−1 2 fn,t,s (γ , x) − exp ft,s (x) v(x) exp −n(t − s) γ =0

2 2 2 exp −Θn,t,s (x) − exp ft,s (x) v(x) 2

2 + exp −Φn,t,s (x) exp Ψn,t,s (x) − 1 v(x) , so that by taking (105) and (111) into account we obtain 2 n−1 2 lim exp −n(t − s) fn,t,s (γ , x) − exp ft,s (x) v(x) = 0. n→+∞ γ =0

The result then follows from a simple dominated convergence argument.

2

It is now plain that (33) emerges as a consequence of Propositions 3 and 4; in fact, thanks to (97) and (101) we have

2282

P.-A. Vuillermot et al. / Journal of Functional Analysis 257 (2009) 2246–2290 0 γ =n−1

n−1 γ t −s A s + (t − s) v(x) = exp −n(t − s) exp − fn,t,s (γ , x) v(x) n n γ =0

for every v ∈ L2 ((0, 1), C), almost every x ∈ (0, 1) and all s, t with 0 s < t 1. Therefore, from (98), (102) along with Propositions 3 and 4 we indeed get 0 2 γ t −s A s + (t − s) v exp − UA (t, s)v − n n γ =n−1

s =

2

2 n−1 2 dx exp −n(t − s) fn,t,s (γ , x) − exp ft,s (x) v(x) γ =0

0

t +


n−1

2 fn,t,s (γ , x) v(x)

γ =0

s

2 1 n−1 2 + dx exp −n(t − s) fn,t,s (γ , x) − exp ft,s (x) v(x) → 0 γ =0

t

as n → +∞. Our last example is motivated by some questions related to the theme of singular perturbations of self-adjoint operators. It also illustrates the fact that the theory we developed in the preceding section can be applied to evolution problems defined on unbounded domains of Euclidean space and, of course, to the case of unbounded B(t)’s. Thus, let us consider the parabolic initial-value problem ∂u(x, t) = ∂t

d2 − s(t)δx ∗ − − εm(x, t) u(x, t), dx 2

u(x, 0) = u0 (x),

(x, t) ∈ R × (0, T ],

x ∈ R,

(112)

corresponding to a time-dependent, zero-range perturbation at x ∗ ∈ R involving Dirac’s distribution δx ∗ , with u0 ∈ L2 (R, R) and , ε ∈ R+ parameters as before. Problems such as (112) with one or several perturbations supported on a discrete set of points in one or several space dimensions may play an important rôle in the mathematical analysis of the dynamics of one particle diffusing through a set of very small obstacles varying with time (see, for instance, [11] for further information on the subject). Regarding the strength of the zero-range perturbation we introduce the following condition: (S) The function s : [0, T ] → R+ is differentiable, and its derivative s is Hölder continuous with Hölder exponent σ ∈ (0, 1]. As for the lower-order term we impose the following hypothesis:


(M ) The function m : R × [0, T ] → R+ is measurable with

x → M(x) := sup m(x, t) ∈ L2 R, R+ . t∈[0,T ]

2283

(113)

Furthermore, the function t → m(x, t) is differentiable on [0, T ] for every x ∈ R and there exist 2 + a constant c∗ ∈ R+ ∗ , a function H ∈ L (R, R ) such that the Hölder continuity estimate ∂m(x, t) ∂m(x, s) c H(x)|t − s|σ − (114) ∗ ∂t ∂s holds for every x ∈ R and every s, t ∈ [0, T ], where σ ∈ (0, 1] may be chosen to be the same as in Hypothesis (S). Finally, we have ∂m(x, t)

∈ L2 R, R+ . (115) x → N(x) := sup ∂t t∈[0,T ] As above, it is here also possible to construct a self-adjoint, positive realization of the operator A(t) := −

d2 + s(t)δx ∗ + dx 2

(116)

in L2 (R, C), this time as a form sum by considering the Hermitian sesquilinear form a : [0, T ] × H 1 (R, C) × H 1 (R, C) → C defined by a(t, v, w) := dx v (x)w (x) + s(t)v(x ∗ )w(x ∗ ) + dx v(x)w(x). (117) R

R

In this case, the corresponding time-dependent domain for (116) is given by

∗

∗

D A(t) = v ∈ H 1 (R, C) ∩ H 2 R{x ∗ }, C : v x+ − v x− = s(t)v(x ∗ ) for every t ∈ [0, T ], where ∗

v x+ := lim v (x ∗ + y) y0

and ∗

v x− := lim v (x ∗ − y) y0

(see, for instance, [3]). From Hypothesis (S) and standard one-dimensional Sobolev theory, it follows that (117) satisfies estimates similar to (5) and (6). Consequently, since s(t) 0 for every t ∈ [0, T ] we infer from the general theory of [34] that −A(t) generates a holomorphic semigroup of contractions exp[−sA(t)]s0 in L2 (R, C), which means that Hypotheses (A1) and (A4) hold provided we choose again sufficiently large, for instance 1. Furthermore, −A(t) also generates an evolution system UA (t, s)0stT in L2 (R, C).

2284


Therefore, in order to prove (32) for (112) we can begin by verifying (A2), (A3) and for this we wish to exploit the fact that the resolvent operator for (116) is known quite explicitly, rather than rely on the variational structure of the problem. More precisely, for every λ ∈ CR+ we write k2 := − λ with Re k > 0 and then have by Krein’s formula for resolvents (see, for instance, [3])

R A(t), λ f (x) −1 d2 f (x) = − 2 + s(t)δx ∗ + k2 dx s(t) 1 ∗ −k|x−y| −k|x ∗ −y| dy e f (y) − dy e f (y) e−k|x−x | = 2k 2k(s(t) + 2k) R

(118)

R

for every f ∈ L2 (R, C) and almost every x ∈ R, from which we obtain

s (t) ∂ ∗ −k|x ∗ −y| R A(t), λ f (x) = − dy e f (y) e−k|x−x | 2 ∂t (s(t) + 2k)

(119)

R

thanks to the differentiability of s. From this we first get the following result. Lemma 10. Assume that Hypothesis (S) holds; then, there exists a constant a2 ∈ R+ ∗ such that the Hölder continuity estimate d −1 A (t)f − d A−1 (s)f a2 |t − s|σ f 2 (120) dt ds 2 holds for all s, t ∈ [0, T ] and every f ∈ L2 (R, C), with σ the Hölder exponent in (S). Proof. Relation (119) with λ = 0 reduces to d −1 s (t) A (t)f (x) = − √ dt (s(t) + 2 )2

dy e

√ − |x ∗ −y|

√ ∗ f (y) e− |x−x | ,

(121)

R

and furthermore we infer from Hypothesis (S) that the function t →

s (t) √ (s(t) + 2 )2

is Hölder continuous on [0, T ] with Hölder exponent σ . Moreover, we can estimate the integral in (121) by means of Schwarz inequality, so that we eventually get 1 √ 2 d −1 −2 |x| A (t)f − d A−1 (s)f c|t − s|σ dx e f 2 dt ds 2 R

σ

c|t − s| f 2 for every s, t ∈ [0, T ] and every f ∈ L2 (R, C).

2


2285

While it is plain that (120) leads to Hypothesis (A2), we now turn to the verification of (A3). For this we have the following result. Lemma 11. Assume that Hypothesis (S) holds; then, there exists a constant a3 ∈ R+ ∗ such that the estimate ∂

R A(t), λ f a3 |λ|−1 f 2 (122) ∂t 2 holds for every t ∈ [0, T ], any f ∈ L2 (R, C) and each λ ∈ Sθ {0}. Proof. From (119) we easily obtain ∂

c −2 Re k|y| R A(t), λ f f 2 dy e ∂t |s(t) + 2k|2 2 R

c 2 f 2 |k| Re k

(123)

for every f ∈ L2 (R, C) and every λ ∈ CR+ , where the last inequality follows from an explicit evaluation of the integral and the fact that s 0, Re k > 0. Without restricting the generality we now take θ ∈ ( π4 , π2 ) and first prove the existence of a constant cθ ∈ R+ ∗ such that the inequality

|k|2 cθ 1 + |λ|

(124)

holds for every λ ∈ Sθ {0}; this is obvious with a constant independent of θ if Re λ 0 (with λ = 0 when Re λ = 0) since 2 2 k = 2 − 2 Re λ + |λ|2 . Furthermore, if arg λ θ with Re λ > 0, Im λ > 0, or if arg λ −θ with Re λ > 0, Im λ < 0 we get from the preceding relation and Cauchy’s interpolated inequality the estimate 2 2

k 1 − −1 2 + 1 − tan−2 θ |λ|2

(125)

1 2 for every ∈ R+ ∗ , so that by choosing for instance = 2 (1 + tan θ ) we can make the two terms in (125) positive, which indeed leads to (124). We proceed by proving that

inf

λ∈Sθ {0}

Re k > 0.

(126)

We have Re k =

Re k2 + |k|2 2

1 2

(127)

2286


so that if Re λ we get Re k2 0 and hence Re k cθ

(128)

from (124) and (127). In order to get a similar bound for the case Re λ > , it is sufficient to prove that Re k is bounded from below by a function of Re λ having a positive minimum at Re λ = ; to this end, let us define Fθ : [, +∞) → R+ ∗ by 1

Fθ (ξ ) :=

− ξ + ( 2 − 2ξ + ξ 2 (1 + tan2 θ )) 2 . 2

(129)

A direct calculation shows that Fθ is monotone increasing, so that the comparison of (127) and (129) gives (Re k)2 Fθ (Re λ) Fθ ()

tan θ >0 2

and thereby indeed a bound identical to (128). Therefore (126) holds, which, together with (123) and (124), gives (122). 2 We proceed with the verification of (B1)–(B4). The multiplication operators Bε (t) := εB(t) defined by B(t)v := m(., t)v

(130)

are in this case self-adjoint and positive on the maximal, time-dependent domains 2

D B(t) = v ∈ L2 (R, C): dx m(x, t)v(x) < +∞ R

for every t ∈ [0, T ] and, although the B(t)’s are in general unbounded, the crucial fact that entails the validity of (B1) is the boundedness of the operators B(t)R(A(t), λ) on L2 (R, C). More precisely we have the following result. Lemma 12. Assume that Hypothesis (S) and (113) hold; then, there exists a constant cθ ∈ R+ ∗ such that the inequality

B(t)R A(t), λ f cθ M 2 f 2 (131) 2 is valid for every t ∈ [0, T ], any f ∈ L2 (R, C) and each λ ∈ Sθ . Thus, the Bε (t)’s satisfy (B1) for every ε ∈ R+ sufficiently small. Proof. From (118), (130) and estimates similar to those carried out in the proofs of the last two lemmas we easily obtain

B(t)R A(t), λ f 2 2

c 2 2 1 + | k | dx m (x, t) f 22 , |k|4 Re k R


2287

of which (131) is a consequence because of (113), (124) and (128). The remaining statement of the lemma is then immediate by setting v = R(A(t), λ)f in (131) for every v ∈ D(A(t)). 2 Next, we have the following result whose proof is relatively long but similar to the last three and therefore omitted. Lemma 13. Assume that Hypothesis (S) and (113)–(115) hold; then, there exists a constant c∗ ∈ R + ∗ depending on θ , M 2 , H 2 and N 2 such that the Hölder continuity estimate d B(t)A−1 (t)f − d B(s)A−1 (s)f c∗ |t − s|σ f 2 dt ds 2 is valid for all s, t ∈ [0, T ] and every f ∈ L2 (R, C). Moreover, the function t → ∂t∂ (B(t)R(A(t), λ)) ∞ is continuously differentiable on [0, T ] with respect to the normtopology of L(L2 (R, C)) and there exists a constant cθ ∈ R+ ∗ such that the inequality ∂

B(t)R A(t), λ f cθ M 2 + N 2 f 2 ∂t 2

holds for every t ∈ [0, T ], any f ∈ L2 (R, C) and each λ ∈ Sθ . Thus, the Bε (t)’s satisfy (B2) and (B3) for every ε ∈ R+ . As for the verification of (B4), we can either proceed as in Lemma 8 or prove (58) directly by observing that exp −(r − s)B(r) f = exp −(r − s)m(., r) f in L2 (R, C). Then, for any r ∈ [s, t] and any sequence (rn )n∈N+ ⊂ [s, t] such that rn → r we have exp −(rn − s)m(x, rn ) f (x) → exp −(r − s)m(x, r) f (x) for almost every x ∈ R when n → +∞, as well as exp −(rn − s)m(x, rn ) f (x) − exp −(r − s)m(x, r) f (x)2 cf (x)2 uniformly in n. Therefore, we get lim exp −(rn − s)B(rn ) f = exp −(r − s)B(r) f

n→+∞

strongly in L2 (R, C) by dominated convergence, which is the desired property. Finally, Hypothesis (D) can be verified with

D = v ∈ C02 R{x ∗ }, C : v(x ∗ ) = 0

(132)

2288


which is dense in L2 (R, C); indeed (27) and (29) trivially hold, as does (28) since the restriction d2 of (116) to the domain (132) coincides with the time-independent operator − dx 2 + (see, for instance, [3] or [11]). The preceding considerations thus lead to the following result. Proposition 5. Assume that Hypotheses (S) and (M ) hold; then, all the conclusions of the theorem are valid for (112) for every ε ∈ R+ sufficiently small. In particular, the Trotter–Kato formula (32) and the reconstruction formula (33) hold in the strong topology of L(L2 (R, C)). Remark. The preceding example shows that in the particular case where the B(t)’s are selfadjoint multiplication operators on a Hilbert space, there is a much more direct way of proving the strong continuity of r → exp[−(r − s)B(r)] than that stemming from Hypothesis (B4), as it is sufficient to invoke the spectral theorem. However, in the general case the full force of (B4) is indeed deemed appropriate according to the proof of Lemma 5. We conclude this article by establishing a connection between the above theory and the corresponding evolution problems for Schrödinger-type equations of quantum mechanics, namely, i

du(t) = A(t) + B(t) u(t), dt u(s) = us

t ∈ (s, T ], (133)

defined in a complex and separable Hilbert space H, with A(t) + B(t) self-adjoint there. In this case, only partial results regarding the existence of dynamics are known, for example when the domain of A(t) + B(t) is independent of time (see, for instance, [30,33] and the references therein); but to the best of our knowledge a Trotter–Kato product formula for this is not available. For instance, in the case of (112) the corresponding quantum mechanical equation reads d2 ∂u(x, t) ∗ = − 2 + s(t)δx + + εm(x, t) u(x, t), i ∂t dx u(x, 0) = u0 (x),

(x, t) ∈ R × (0, T ],

x ∈ R,

(134)

and under Hypotheses (S) and (M ) there exist the unitary groups exp[−isA(t)]s∈R and exp[−isBε (t)]s∈R for every t ∈ [0, T ], where the A(t)’s, Bε (t)’s are given by (116), (130), respectively. However, whether the strong limit

lim

n→+∞

0 γ =n−1

γ γ t −s t −s A s + (t − s) exp −i Bε s + (t − s) exp −i n n n n

exists in L(L2 (R, C)) and describes the true dynamics generated by (134) seems to be an open problem at this time. The same remark applies to other unitary evolution systems generated by Schrödinger equations in the presence of time-dependent singular perturbations of zero-range, such as those constructed in [31] and more recently in [8] and [12]. Away from the onedimensional case, these constructions rest essentially on von Neumann’s theory of self-adjoint extensions for symmetric operators.


2289

Acknowledgments The authors would like to thank the referee for having pointed out to them references [1,29,32], which allowed them to put this work into a broader perspective and to formulate several important open problems along the way. The research of P.A.V. regarding this paper was supported in part by the Brazilian FAPESP and by the Forschungsinstitut für Mathematik der ETH in Zürich, the financial support and hospitality of which are gratefully acknowledged. The research of W.F.W. was supported in part by the Brazilian CNPq, while that of V.A.Z. was also supported in part by FAPESP. Last but not least, P.A.V. and V.A.Z. would like to take this opportunity to thank the Departamento de Física Matemática of the University of São Paulo where this work was begun for its very kind hospitality. References [1] P. Acquistapace, B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova 78 (1987) 47–107. [2] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Pure Appl. Math., vol. 140, Academic Press, New York, 2003. [3] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Texts Monogr. Phys., Springer-Verlag, New York, 1988. [4] D. Babbitt, The Wiener integral and perturbation theory of the Schrödinger operator, Bull. Amer. Math. Soc. 70 (1964) 254–259. [5] V. Cachia, V.A. Zagrebnov, Operator-norm approximation of semigroups by quasi-sectorial contractions, J. Funct. Anal. 180 (2001) 176–194. [6] P.R. Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal. 2 (1968) 238–242. [7] P.R. Chernoff, Semigroup product formulas and addition of unbounded operators, Bull. Amer. Math. Soc. 76 (1970) 395–398. [8] M. Correggi, G.F. Dell’Antonio, R. Figari, A. Mantile, Ionization for three dimensional time-dependent point interactions, Comm. Math. Phys. 257 (2005) 169–192. [9] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge, 1992. [10] E.B. Davies, One-Parameter Semigroups, London Math. Soc. Monogr., vol. 15, Academic Press, New York, 1980. [11] G.F. Dell’Antonio, R. Figari, A. Teta, A limit evolution problem for time-dependent point interactions, J. Funct. Anal. 142 (1996) 249–275. [12] G.F. Dell’Antonio, R. Figari, A. Teta, The Schrödinger equation with moving point interactions in three dimensions, in: Stochastic Processes, Physics and Geometry: New Interplays I, in: CMS Conference Proceedings Series, vol. 28, American Mathematical Society, Providence, 2000, pp. 99–113. [13] W.G. Faris, Product formulas for perturbations of linear propagators, J. Funct. Anal. 1 (1967) 93–108. [14] W.G. Faris, The product formula for semigroups defined by Friedrichs extensions, Pacific J. Math. 22 (1967) 47–70. [15] W.G. Faris, Self-Adjoint Operators, Lecture Notes in Math., vol. 433, Springer-Verlag, New York, 1975. [16] A. Gulisashvili, J. van Casteren, Non-Autonomous Kato Classes and Feynman–Kac Propagators, World Scientific, Singapore, 2006. [17] G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949. [18] T. Ichinose, H. Tamura, Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations, Osaka J. Math. 35 (1998) 751–770. [19] G.W. Johnson, M.L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus, Oxford Math. Monogr., Oxford University Press, Oxford, 2000. [20] T. Kato, Abstract evolution equations of parabolic type in Banach and Hilbert spaces, Nagoya Math. J. 5 (1961) 93–125. [21] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in: I. Gohberg, M. Kac (Eds.), Topics in Functional Analysis, Academic Press, New York, 1978, pp. 185–195. [22] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, New York, 1984. [23] J.L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Grundlehren Math. Wiss. in Einzeldarstellungen, vol. 111, Springer-Verlag, New York, 1961.

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Strong convergence towards self-similarity for one-dimensional dissipative Maxwell models G. Furioli a,∗ , A. Pulvirenti b , E. Terraneo c , G. Toscani b,∗∗ a University of Bergamo, viale Marconi 5, 24044 Dalmine, Italy b Department of Mathematics, University of Pavia, via Ferrata 1, 27100 Pavia, Italy c Department of Mathematics, University of Milano, via Saldini 50, 20133 Milano, Italy

Received 1 November 2008; accepted 15 June 2009 Available online 10 July 2009 Communicated by C. Villani

Abstract We prove the propagation of regularity, uniformly in time, for the scaled solutions of the one-dimensional dissipative Maxwell models introduced in [D. Ben-Avraham, E. Ben-Naim, K. Lindenberg, A. Rosas, Selfsimilarity in random collision processes, Phys. Rev. E 68 (2003) R050103]. This result together with the weak convergence towards the stationary state proven in [L. Pareschi, G. Toscani, Self-similarity and powerlike tails in nonconservative kinetic models, J. Stat. Phys. 124 (2–4) (2006) 747–779] implies the strong convergence in Sobolev norms and in the L1 norm towards it depending on the regularity of the initial data. As a consequence, the original nonscaled solutions are also proved to be convergent in L1 towards the corresponding self-similar homogeneous cooling state. The proof is based on the (uniform in time) control of the tails of the Fourier transform of the solution, and it holds for a large range of values of the mixing parameters. In particular, in the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity. © 2009 Elsevier Inc. All rights reserved. Keywords: Dissipative Boltzmann equation; Granular gases; Asymptotic behavior

* Corresponding author. Fax: +39 035 2052 077. ** Principal corresponding author.

E-mail addresses: [email protected] (G. Furioli), [email protected] (A. Pulvirenti), [email protected] (E. Terraneo), [email protected] (G. Toscani). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.016

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G. Furioli et al. / Journal of Functional Analysis 257 (2009) 2291–2324

1. Introduction and preliminary results In 2003 Ben-Avraham and coworkers [3] introduced a one-dimensional model of the Boltzmann equation, in which binary collision processes are given by arbitrary linear collision rules v ∗ = pv + qw,

w ∗ = qv + pw,

p q > 0.

(1)

The positive constants p and q represent the mixing parameters, namely the portion of the pre-collisional velocities (v, w) which generate the post-collisional ones (v ∗ , w ∗ ). Under the hypothesis of constant collision frequency, this mechanism of collision leads to the following integro-differential equation of Boltzmann type for the distribution of velocities f (v, t) ∂t f (v, t) = R

1 f (v∗ , t)f (w∗ , t) − f (v, t)f (w, t) dw J

(2)

where now (v∗ , w∗ ) are the pre-collisional velocities v = pv∗ + qw∗ ,

w = qv∗ + pw∗

that generate the couple (v, w) after the interaction and J = p 2 − q 2 is the Jacobian of the transformation of (v, w) into (v ∗ , w ∗ ). Special cases include the elastic model (p 2 + q 2 = 1), which is the analogous of the well-known Kac model [16,21], the inelastic collisions (p + q = 1) [1], the granules model (p + q < 1) [24], the inelastic Lorenz gas (q = 0, p < 1) [19], and in addition energy producing models (p > 1) [28]. The existence and uniqueness of a solution f ∈ C 1 ([0, +∞), L1 (R)) of a Cauchy problem coupled with Eq. (2) for any normalized initial data f0 satisfying f0 0,

f0 (v) dv = 1,

vf0 (v) dv = 0,

R

R

v 2 f0 (v) dv = 1,

(3)

R

can be established in the same way as for the elastic Kac equation [22,23]. Writing Eq. (2) in weak form as d dt

(v)f (v, t) dv = R

f (v, t)f (w, t) (v ∗ ) − (v) dw dv,

R2

and

(v)f (v, t) dv =

lim

t→0

R

(v)f0 (v) dv R

for any bounded and continuous on R, it can be shown that the integrals R v n f (v, t) dv, n 0, obey a closed hierarchy of equations [4] and so moments can be evaluated recursively, starting from mass conservation. In particular, choosing as initial density a normalized probability density


2293

f0 satisfying (3), it follows that both mass and momentum are preserved in time, while the second moment varies according to the law E(t) =

v 2 f (v, t) dv = exp p 2 + q 2 − 1 t .

(4)

R

It is worth noticing that in the case p + q = 1, one has additionally that the moment is always preserved (even if not initially vanishing). By (4) it follows that the second moment of the solution is not conserved, unless the collision parameters satisfy p 2 + q 2 = 1. If this is not the case, the energy can grow to infinity or decrease to zero, depending on the sign of p 2 +q 2 −1. In both cases, however, stationary solutions of finite energy do not exist, and the large-time behavior of the system can at best be described by selfsimilarity properties. As observed in [3], while in the long-time limit velocity distributions are generically self-similar, there is a wide spectrum of possible behaviors. The velocity distributions are characterized by algebraic or stretched exponential tails and the corresponding exponents depend sensitively on the collision parameters. Interestingly, when there is energy or momentum conservation, the behavior is universal. The standard way to look for self-similarity is to scale the solution according to the rule g(v, t) =

E(t)f v E(t), t .

This scaling implies that R v 2 g(v, t) dv = 1 for all t 0. The large-time behavior of the densities f (v, t) and g(v, t) have been studied in [22], by resorting to the Fourier transform version of the Boltzmann-type equation (2), which reads ∂t fˆ(ξ, t) = fˆ(pξ, t)fˆ(qξ, t) − fˆ(ξ, t)

(5)

and which convertes for the scaled solution g(t) into the following ∂t g(ξ, ˆ t) +

1 2 p + q 2 − 1 ξ ∂ξ g(ξ, ˆ t) = g(pξ, ˆ t)g(pξ, ˆ t) − g(ξ, ˆ t). 2

(6)

It is worth recognizing from Eq. (5) an equivalent formulation of Eq. (2) making use of the convolution operator ∂t f (v, t) = fp ∗ fq (v, t) − f (v, t) where we used the shorthand fp (v) =

v 1 f . p p

The key argument for studying Eqs. (5) and (6) is the use of a metric for probability densities with finite and equal moments. More precisely, for 0 δ < 1 we introduce on the space

2294


M2+δ = f 0:

f (v) dv = 1,

R

vf (v) dv = 0,

R

v 2 f (v) dv = 1, R

|v|2+δ f (v) dv < ∞ R

the distance |fˆ(ξ ) − g(ξ ˆ )| . 2+δ |ξ | ξ ∈R

d2+δ (f, g) = sup

(7)

This metric has been introduced in [14] to investigate the trend to equilibrium of the solutions to the Boltzmann equation for Maxwell molecules. Further applications of d2+δ can be found in [10,15,23,27]. The study of the time evolution of the d2+δ -metric for some suitable 0 < δ < 1, enlightened the range of the mixing parameters for which one can expect that the scaled function g(t) converges (weakly) towards a steady profile g∞ at an exponential rate. More precisely, let us define, for fixed p and q the function Sp,q (δ) = p 2+δ + q 2+δ − 1 −

2+δ 2 p + q2 − 1 . 2

(8)

Then, the sign of Sp,q determines the asymptotic behavior of the distance d2+δ (g1 (t), g2 (t)) between two scaled solutions g1 (t), g2 (t) issued from two initial data f1,0 , f2,0 . In particular, ˜ it has been proved in [22] that if there exists δ˜ ∈ (0, 1) such that Sp,q (δ) < 0 for 0 < δ < δ, we can conclude that d2+δ (g1 (t), g2 (t)) converges exponentially to zero if initially finite. Note that, by construction, Sp,q (0) = 0, and thus minδ∈(0,1) {Sp,q } 0. A numerical evaluation of the region where the minimum of the function Sp,q is negative for p, q ∈ [0, 2] is reported in [22]. This region includes the relevant cases of both inelastic (p + q = 1) and elastic (p 2 + q 2 = 1) collisions, as well as the case, among others, in which p = q = 1. It has been also proved in [22] that in this case a unique steady state g∞ exists for Eq. (6). We recall this statement here. Theorem 1. (See Pareschi and Toscani [22].) Assume 0 < q p and such that there exists δ˜ ∈ ˜ Let g(t) be the weak solution of Eq. (6), correspond(0, 1) for which Sp,q (δ) < 0 for 0 < δ < δ. ˜ ing to the initial density g0 satisfying the normalization conditions (3) and R |v|2+δ g0 (v) dv < +∞. Then g(t) satisfies for 0 < δ δ˜ and for cδ > 0: |v|2+δ g(v, t) dv cδ ,

t 0.

R

˜ Moreover, there exists a unique stationary state g∞ to Eq. (6) which satisfies for 0 < δ < δ: |v|2+δ g∞ (v) dv < +∞, R


2295

g(t) converges exponentially fast in Fourier metric towards g∞ and the following bound holds ˜ for 0 < δ < δ: d2+δ g(t), g∞ e−|Sp,q (δ)|t d2+δ (g0 , g∞ ),

t 0.

(9)

In the relevant case p + q = 1, Eq. (6) admits an explicit stationary state [1] gˆ ∞ (ξ ) = 1 + |ξ | e−|ξ | . In all the cases p + q = 1 (excluding p 2 + q 2 = 1) the stationary state is not explicit, even if some properties can be extracted from the analysis of the evolution equation. On the original nonscaled solution f (t), the limit behavior corresponds to a self-similar state 1 v f∞ (v, t) = √E(t) g∞ ( √E(t) ). Despite the fact that the large-time behavior of the solution to the Boltzmann-like equation (2) can be described in terms of the d2+δ -metric, which is equivalent to the weak∗ convergence of measures [12], the strong convergence of the scaled density g(t) towards g∞ has never been proved before. A similar problem occurs in dissipative kinetic theory, where the weak convergence of the (scaled) solution to the inelastic Boltzmann equation for Maxwell molecules towards the homogeneous cooling state with polynomial tails is known to hold [5] in the d2+δ -metric framework, but the strong convergence is still unknown in full generality. A recent paper by Carlen, Carrillo and Carvalho [11] shows that in some cases one can prove that the strong convergence holds. Their result, however, requires a small inelasticity regime, which in our setting of the mixing parameters means p + q = 1, and at the same time 1 − p 2 − q 2 1. In this case, in fact, one can resort to methods close to the elastic situation, in which the (controlled) growth of the Fisher information, coupled with the exponential decay of the d2+δ -metric allows to prove the uniform propagation of Sobolev regularity, and from this, by interpolation, the strong convergence. Propagation of Sobolev regularity for both Kac equation and the elastic Boltzmann equation for Maxwellian molecules, together with the precise exponential rate of the strong convergence to the Maxwellian equilibrium M has been proved in [10]. The advantage of working with the classical elastic Boltzmann equation relies on the fact that one can resort to the H -theorem. A careful reading of [10], however, allows to conclude that the proof of uniform propagation of regularity makes use of the following condition for the distance between the solution f (t) and the Maxwellian M:

) → 0, sup fˆ(ξ, t) − M(ξ

ξ ∈R

t → +∞.

(10)

While the convergence of H (f )(t) to H (M) implies the L1 -convergence and therefore (10), the same condition continues to hold provided the Fourier transform of the solution to the (scaled) Boltzmann equation g(t) and of the stationary state g∞ satisfy (9) together with a suitable (uniform) decay at infinity in the ξ variable, of the type

μ ˆ t) K, 1 + κ|ξ | g(ξ,

ξ ∈ R, t 0

for some positive constants κ, K and μ. In fact, in this case, for any given R > 0,

(11)

2296


2K

g(ξ, ˆ t) − gˆ ∞ (ξ ) d2+δ g(t), g∞ R 2+δ + , (κR)μ which implies, optimizing over R,

μ/(2+δ+μ)

g(ξ, ˆ t) − gˆ ∞ (ξ ) C(δ, μ, κ, K)d2+δ g(t), g∞ ,

ξ ∈ R, t 0.

Therefore, in presence of condition (11), the decay to zero of the d2+δ -metric implies the decay to zero of supξ ∈R |g(ξ, ˆ t) − gˆ ∞ (ξ )| for t → +∞. We will remark in Section 5 that condition (11) on the initial density f0 holds provided the square root of f0 has some regularity. For example, the Fisher information of f0 , which controls the H 1 -norm of the square root, controls |ξ ||fˆ0 (ξ )| (cf. the proof in [18]). Condition (11) is difficult to prove directly from the equation satisfied by the scaled density g(t), due to the presence of the drift term in Eq. (6). To simplify the proof of the various bounds we will introduce a semi-implicit discretization of Eq. (6), that is, for a small time interval t, we will consider the solution to g(ξ, ˆ t + t) − g(ξ, ˆ t) 1 2 + p + q 2 − 1 ξ ∂ξ g(ξ, ˆ t + t) = g(pξ, ˆ t)g(qξ, ˆ t) − g(ξ, ˆ t). t 2

(12)

ˆ t + t) Inspired by the integral formulation of the stationary state g∞ in [7], the solution g(ξ, to (12) at time t + t is shown to be a convex combination of the probability densities g(ξ, ˆ t) and g(pξ, ˆ t)g(qξ, ˆ t). Precisely, for the dissipative case p 2 + q 2 < 1 we have r g(ξ, ˆ t + t) = t

+∞ dτ t g(τpξ, ˆ t)g(τ ˆ qξ, t) + (1 − t)g(τ ˆ ξ, t) r +1 , τ t

(13)

1

where 1 1 − p2 − q 2 = r 2 and for the energy producing case p 2 + q 2 > 1 we have r g(ξ, ˆ t + t) = − t

1 0

dτ t g(τpξ, ˆ t)g(τ ˆ qξ, t) + (1 − t)g(τ ˆ ξ, t) r +1 . τ t

(14)

Eventhough the solution of the semi-implicit discretization is in both dissipative and nondissipative case a uniform approximation of the scaled solution g(t) (see Proposition 9 in Section 2), we have been able to exploit this approximation only in the dissipative case. The integral formulation is also useful to recover regularity properties of the steady solution g∞ . In particular, we shall prove in the dissipative case that the steady state has the Gevrey regularity of class λ, namely

λ eμ|ξ | gˆ ∞ (ξ ) 1,

|ξ | > ρ,


2297

where the constants λ, μ and ρ are related both to p, q and to the number of moments which are bounded initially. Our main results are summarized into the following statements. In what follows we will denote g0 the initial data of a scaled solution g(t), even if of course g0 = f0 , the initial data of the original, nonscaled solution f (t). Theorem 2. Assume 0 < q p satisfying p + q 1, p 2 + q 2 < 1 and such that there is δ˜ ∈ (0, 1) ˜ Let g(t) be the weak solution of Eq. (6), corresponding to for which Sp,q (δ) < 0, for 0 < δ < δ. ˜ the initial density g0 satisfying the normalization conditions (3), and R |v|2+δ g0 (v) dv < +∞. If in addition

gˆ 0 (ξ )

1 , (1 + β|ξ |)ν

|ξ | > R,

(15)

for some R > 0, ν > 0 and β > 0, then there exist ρ > 0, k > 0, ν > 0 such that g(t) satisfies

g(ξ, ˆ t)

1 , 1+kξ 2 1 , (1+β|ξ |)ν

|ξ | ρ, t 0, |ξ | > ρ, t 0.

(16)

Theorem 2 is proven in Section 3. The second result is concerned with the regularity of the steady state g∞ and is proven in Section 4. Theorem 3. Assume 0 < q p satisfying p 2 + q 2 < 1 and such that there exists δ˜ ∈ (0, 1) for which Sp,q (δ) < 0 for 0 < δ < δ˜ so that a non-trivial steady state g∞ to the Boltzmann equation (6) exists. Let us denote λ ∈ (0, 2) the exponent such that p λ + q λ = 1. Then g∞ is a smooth function and belongs to the λ-th Gevrey class Gλ (R), i.e.

gˆ ∞ (ξ ) exp −μ|ξ |λ ,

|ξ | > ρ

with suitable positive numbers ρ and μ. As a byproduct of both proofs of Theorem 2 and Theorem 3, we also get the uniform propagation of the Gevrey regularity for solutions g(t) issued from Gevrey initial data. Theorem 4. Assume 0 < q p satisfying p 2 + q 2 < 1 and such that there exists δ˜ ∈ (0, 1) for which Sp,q (δ) < 0 for 0 < δ < δ˜ and let us denote λ ∈ (0, 2) the exponent such that p λ + q λ = 1. Let g(t) be the weak solution of Eq. (6), corresponding to the initial density g0 satisfying the ˜ normalization conditions (3), and R |v|2+δ g0 (v) dv < +∞. If in addition

gˆ 0 (ξ ) e−β|ξ |ν ,

|ξ | > R,

for some R > 0, ν > 0 and β > 0, then there exist ρ > 0 and κ > 0 such that g(t) satisfies −κξ 2

g(ξ, ˆ t) e−κ|ξ |,min(ν,λ) e ,

|ξ | ρ, t 0, |ξ | > ρ, t 0.

2298


The previous results are enough to prove the convergence in strong norms towards the steady state g∞ . In consequence of both Theorems 2 and 3, we can show in fact the uniform in time propagation of regularity in Sobolev spaces of high degree

2

ˆ ) dξ

g 2H˙ η (R) = |ξ |2η g(ξ R

with η > 0. It is enough to apply the technique developed in [10] for the Boltzmann equation for Maxwell molecules for showing that whenever the equation propagates a tiny degree of regularity, as in Theorem 2, this implies that the equation propagates regularity of any degree. Then, using the regularity in high Sobolev spaces, we can pass from the weak convergence in d2+δ metric obtained in [22] into convergence in all Sobolev norms, and strong L1 convergence at an explicit exponential rate for a certain class of initial data. This is the objective of Section 5 and the main result is summarized as follows. Theorem 5. Assume 0 < q p satisfying p + q 1, p 2 + q 2 < 1 and such that there exists δ˜ ∈ (0, 1) for which Sp,q (δ) < 0 for 0 < δ < δ˜ and let g∞ be the unique stationary solution of (6). ˜ Let the initial density g0 satisfy the normalization conditions (3), and R |v|2+δ g0 (v) dv < +∞. √ If in addition g0 ∈ H η (R) for some η > 0 large enough, g0 ∈ H˙ ν (R) for some ν > 0, then the solution g(t) of (6) converges strongly in L1 with an exponential rate towards the stationary solution g∞ , i.e., there exist positive constants C and γ explicitly computable such that g(t) − g∞

L1 (R)

Ce−γ t ,

t 0.

Thanks to the scaling invariance of the L1 norm, Theorem 5 allows to deduce also the strong convergence of the original nonscaled solution f (t) to the self-similar state f∞ (v, t) = √ 1 g∞ ( √ v ): E(t) E(t) f (t) − f∞ (t)

L1 (R)

Ce−γ t ,

t 0.

Finally, in Section 6 we will discuss in details the relevant case in which p + q = 1, which corresponds to the one-dimensional inelastic Boltzmann equation for Maxwell molecules [1]. In this case, in fact, it is known that an explicit stationary solution to Eq. (6) exists, gˆ ∞ |ξ | = 1 + |ξ | e−|ξ | , or, in the physical space g∞ (v) =

2

π 1 + v2

2 .

This stationary solution is independent of the values of p and q, and within the set of functions f which satisfy the normalization conditions (3), is the minimum of the convex functional

f (v) dv. H (f ) = − R


2299

This suggests the idea that H is an entropy functional for the scaled equation (6), but the proof of this conjecture would require to satisfy an inequality which is the analogous of the Shannon entropy power inequality [6,25], we are not able to prove. It is interesting to remark that the ideas of the present paper, which are valid for the p + q = 1 case, may be fruitfully used for the three-dimensional dissipative Boltzmann equation for Maxwell molecules, extending the validity of the recent analysis of [11] to a general coefficient of restitution. However, the proof of the analogous of Theorem 2 to the three-dimensional case requires heavy computations, we will publish separately elsewhere. The semi-implicit discretization for Eq. (6) can be profitably replaced by the Duhamel formula t g(ξ, ˆ t) = e gˆ 0 e r ξ + −t

t

t−s t−s e−(t−s) gˆ p e r ξ, s gˆ q e r ξ, s ds.

0

The crucial estimates in the proofs of Theorems 2 and 4 are indeed essentially the same. We thank the anonymous referee for pointing out this interesting remark. 2. An iteration process The goal of this section is to build up a sequence of functions {g N (ξ, t)} which approximates uniformely the solution g(ξ, ˆ t). In order to do this, for any fixed T > 0 we consider firstly a semiimplicit discretization in time of Eq. (6) by partitioning the interval [0, T ] into N subintervals T and we define thus the approximate solution at any time t = j N for j = 0, . . . , N . Secondly, we N define g (ξ, t) on the whole interval [0, T ] by interpolation and lastly we show the convergence of the approximation to the solution. In order to lighten the reading, we will postpone almost all the proofs of this technical section at the end of the paper in Appendix A. The approximate equation T In this paragraph we define the approximate solution at any time t = j N for j = 0, . . . , N by an iteration process and study some of its properties. T Let T > 0 and t = N for N ∈ N, N > T . Let ϕˆ jN (ξ ), j = 0, . . . , N be the sequence:

⎧ N ⎪ ⎨ ϕˆ 0 (ξ ) = gˆ 0 (ξ ), ϕˆ jN+1 (ξ ) − ϕˆ jN (ξ ) 1 d N ⎪ ⎩ = ξ ϕˆ (ξ ) + ϕˆ jN (pξ )ϕˆjN (qξ ) − ϕˆ jN (ξ ), t r dξ j +1 where

1 r

=

j = 0, . . . , N − 1,

(17)

1−p 2 −q 2 . 2

Proposition 6. Assume 0 < q p. If g0 verifies the normalization conditions (3), then there exists a unique sequence of bounded function ϕˆjN for j = 1, . . . , N satisfying (17).

2300


Proof. Let us begin by proving that ϕˆ 1N is well defined. In a similar way as in [7] we multiply r r ) sgn ξ |ξ |− t −1 and obtain Eq. (17) by (− t r r d N r ϕˆ 1 (ξ )|ξ |− t = − sgn ξ |ξ |− t −1 t ϕˆ0N (pξ )ϕˆ 0N (qξ ) + (1 − t)ϕˆ0N (ξ ) . dξ t We assume now p 2 + q 2 < 1. For any positive ξ we integrate on [ξ, +∞) and since ϕˆ 0N (ξ ) is bounded, we get: r ϕˆ1N (ξ )|ξ |− t

r = t

+∞ r t ϕˆ 0N (ps)ϕˆ0N (qs) + (1 − t)ϕˆ0N (s) s − t −1 ds. ξ

Finally by the change of variables τ = s/ξ we are led to:

ϕˆ1N (ξ ) =

r t

+∞ dτ t ϕˆ0N (pτ ξ )ϕˆ 0N (qτ ξ ) + (1 − t)ϕˆ0N (τ ξ ) r +1 . τ t

(18)

1

For any negative ξ we integrate on (−∞, ξ ] and in a similar way we obtain that equality (18) holds for any ξ = 0. Moreover since g0 satisfies the conditions (3), then ϕˆ 0N = gˆ 0 belongs to C 1 (R) and there exists C > 0 such that gˆ 0 (0) = 1,

d gˆ 0 (ξ )

d ξ C,

d gˆ 0 (0) = 0. dξ

(19)

Therefore the function ϕˆ1N can be defined by continuity in ξ = 0 and it is the unique, bounded and C 1 (R) solution of (17). By an iteration argument the same conclusion holds for any ϕˆjN obtaining for j = 0, . . . , N − 1,

ϕˆ jN+1 (ξ ) =

r t

+∞ dτ t ϕˆjN (pτ ξ )ϕˆ jN (qτ ξ ) + (1 − t)ϕˆjN (τ ξ ) r +1 . τ t

(20)

1

For p 2 + q 2 > 1, we repeat the same argument by integrating on [0, ξ ] and [ξ, 0] and we get in the end

ϕˆ jN+1 (ξ ) = −

r t

1 0

dτ t ϕˆ jN (pτ ξ )ϕˆjN (qτ ξ ) + (1 − t)ϕˆjN (τ ξ ) r +1 . τ t

2

Applying Fubini’s theorem we can remark that for p 2 + q 2 < 1 any ϕˆjN+1 (ξ ) is the Fourier transform of ϕjN+1 (v) where for j = 0, . . . , N − 1,


⎧ N ⎪ ϕ0 (v) = g0 (v), ⎪ ⎪ ⎨ +∞ v 1 N 1 N v dτ r N N ⎪ t ϕj,p ∗ ϕj,q + (1 − t) ϕj ϕj +1 (v) = , ⎪ r ⎪ t τ τ τ τ ⎩ τ t +1

2301

(21)

1

N (v) = 1 ϕ N ( v ) and similarly for ϕ N . Analogously, for p 2 + q 2 > 1, ϕˆ N (ξ ) is the with ϕj,p j,q j +1 p j p Fourier transform of ϕjN+1 (v) where for j = 0, . . . , N − 1,

⎧ N ϕ0 (v) = g0 (v), ⎪ ⎪ ⎪ ⎨ r ⎪ ϕjN+1 (v) = − ⎪ ⎪ t ⎩

1 0

v 1 N 1 N v dτ N t ϕj,p ∗ ϕj,q + (1 − t) ϕj . r τ τ τ τ τ t +1

(22)

In what follows, the function Sp,q (δ) is defined as in (8). Proposition 7. Assume 0 < q p such that there exists δ˜ ∈ (0, 1) for which Sp,q (δ) < 0 for 0 < ˜ Let ϕ N , for j = 0, . . . , N defined as in (21) or (22), with g0 satisfying the normalization δ < δ. j ˜ conditions (3) and R |v|2+δ g0 (v) dv < +∞. Then, for 0 < δ < δ˜ there exists Cδ > 0 such that for N large enough (depending on δ, T , p and q), for j = 0, . . . , N , we get ϕjN (v) 0,

ϕjN (v) dv

R

= 1, R

v 2 ϕjN (v) dv = 1, R

vϕjN (v) dv = 0,

|v|2+δ ϕjN (v) dv Cδ .

(23)

R

Remark 8. The equalities in (23) imply that there exists C > 0 such that

N

ϕˆ (ξ ) 1, j

N

dϕˆ j (ξ )

dξ C

2 N

d ϕˆj (ξ )

1 and dξ 2

(24)

for any ξ ∈ R, for N large enough and for j = 0, . . . , N . The first two inequalities have already been proved in the proof of Proposition 6. Definition of the sequence {g N (ξ, t)}N T We are now in position to define the sequence {g N (ξ, t)}N . We define g N (ξ, j N ) = ϕˆjN (ξ ) for j = 0, . . . , N . We extend the definition on the whole interval by interpolation. More precisely, let us define

g (ξ, t) = N

gˆ 0 (ξ ), N N (ξ ), α(t)ϕˆK (ξ ) + (1 − α(t))ϕˆK N N −1

t = 0, 0 < t T,

2302


T T where for 0 < t T we have (KN − 1) N < t KN N for KN ∈ {1, . . . , N } and more precisely T T + (1 − α(t))KN N . Any g N (ξ, t) there is a function 0 α(t) < 1 such that t = α(t)(KN − 1) N 2 is continuous on R × [0, T ] and for any t ∈ [0, T ] it belongs to C (R). The result of convergence is therefore as follows.

Proposition 9. There is a subsequence {g Nl (ξ, t)}l of {g N (ξ, t)}N which converges uniformely on any compact set of R × [0, T ] to the solution g(ξ, ˆ t). 3. Propagation of regularity In this section we prove Theorem 2. Thanks to the uniform convergence of a subsequence of the approximate solutions g N (ξ, t) to the solution g(ξ, ˆ t) and to the definition of g N (ξ, t), it is N enough to prove the bounds (16) for any ϕˆj (ξ ) uniformly for N ∈ N and j = 0, . . . , N . We will exploit an idea introduced in [13] to prove the propagation of Gevrey regularity for the solution of the homogeneous Boltzmann equation for Maxwellian molecules. The main difference here is that the uniform profile for low and high frequencies for the solution g(t) and for the approximations ϕjN for all N and j cannot be both obtained by a recursive procedure on the integral equation (20) as in [13]. In fact the low frequencies bound is a direct consequence of the properties (3) of the initial data and of the convergence to zero of the distance d2+δ (g(t), g∞ ). On the contrary, the uniform bound on high frequencies is obtained through a recursive procedure on Eq. (20) exploiting in a crucial way the low frequencies bound already proved. In the following lemma we prove the low frequencies bound. Lemma 10. Assume 0 < q p satisfying p 2 + q 2 < 1 and such that there is δ˜ ∈ (0, 1) for which ˜ Let g(t) be the weak solution of Eq. (6), corresponding to the initial Sp,q (δ) < 0 for 0 < δ < δ. ˜ density g0 satisfying the normalization conditions (3), and R |v|2+δ g0 (v) dv < +∞. Let ϕˆjN the approximation defined in (17). For any 0 < k < 12 there exists ρ > 0 such that for any fixed T > 0 and any N ∈ N large enough we get

g(ξ, ˆ t)

N

ϕˆ (ξ ) j

1 , 1 + kξ 2

1 , 1 + kξ 2

|ξ | ρ, t 0,

(25)

|ξ | ρ, j = 0, . . . , N.

(26)

Proof. Let us begin by proving (25). We remark first that since for all t 0, g(v, t) 0,

g(v, t) dv = 1,

R

vg(v, t) dv = 0,

R

R

we can deduce at once g(ξ, ˆ t) = 1 − and so for t 0 and 0 < k
0 such that


g(ξ, ˆ t)

1 , 1 + kξ 2

2303

|ξ | ρ(t)

but this is not enough because we need an estimate independent of t. We have therefore to proceed differently as already observed in [11] and [12]. By Theorem 1, for any fixed 0 < δ < δ˜ we have |g(ξ, ˆ t) − gˆ ∞ (ξ )| e−|Sp,q (δ)|t d2+δ (g0 , g∞ ), d2+δ g(t), g∞ = sup 2+δ |ξ | ξ ∈R Moreover, since g∞ 0,

R g∞ (v) dv

= 1,

gˆ ∞ (ξ ) = 1 −

R vg∞ (v) dv

ξ2 + o ξ2 , 2

= 0,

Rv

2g

∞ (v) dv

t 0. = 1, we have

ξ → 0,

so we get uniformly for t 0 and ξ = 0:

|g(ξ,

ˆ t) − gˆ ∞ (ξ )| 2+δ

g(ξ, |ξ | ˆ t) gˆ ∞ (ξ ) + |ξ |2+δ 1− This shows that for any 0 < k
0, such that for all t 0,

g(ξ, ˆ t) 1 − kξ 2 , or, which is equivalent, for any 0 < k
0 such that for all t 0,

g(ξ, ˆ t)

1 , 1 + kξ 2

|ξ | ρ.

In order to prove (26), we would like to exploit again the d2+δ distance. For N ∈ N and j = 1, . . . , N − 1 let us estimate first d2+δ (ϕjN+1 , gˆ ∞ ). We recall that 1 d 0 = ξ gˆ ∞ (ξ ) + gˆ ∞ (pξ )gˆ ∞ (qξ ) − gˆ ∞ (ξ ) r dξ with

1 r

=

1−p 2 −q 2 . 2

By considering the two Eqs. (17) and (27) we have

ϕˆ jN+1 (ξ ) − gˆ ∞ (ξ ) − (ϕˆjN (ξ ) − gˆ ∞ (ξ )) t

d 1 d N = ξ ϕˆj +1 (ξ ) − gˆ ∞ (ξ ) r dξ dξ + ϕˆjN (pξ )ϕˆ jN (qξ ) − gˆ ∞ (pξ )gˆ ∞ (qξ ) − ϕˆjN (ξ ) − gˆ ∞ (ξ ) .

(27)

2304


In the same way as in Proposition 6 we obtain the integral form:

ϕˆjN+1 (ξ ) − gˆ ∞ (ξ ) =

r t

+∞ t ϕˆ jN (pτ ξ )ϕˆjN (qτ ξ ) − gˆ ∞ (pτ ξ )gˆ ∞ (qτ ξ ) 1

dτ + (1 − t) ϕˆjN (τ ξ ) − gˆ ∞ (τ ξ ) r +1 . τ t Therefore for ξ = 0 we get |ϕˆjN+1 (ξ ) − gˆ ∞ (ξ )| |ξ |2+δ r t

+∞ |ϕˆ N (pτ ξ )ϕˆ N (qτ ξ ) − gˆ (pτ ξ )gˆ (qτ ξ )| ∞ ∞ j j t 2+δ |ξ | 1

+ (1 − t)

|ϕˆjN (τ ξ ) − gˆ ∞ (τ ξ )|

|ξ |2+δ

dτ τ

r t +1

.

Now for τ = 0, |ϕˆjN (pτ ξ )ϕˆjN (qτ ξ ) − gˆ ∞ (pτ ξ )gˆ ∞ (qτ ξ )| |ξ |2+δ

d2+δ ϕjN , g∞ p 2+δ + q 2+δ τ 2+δ

and so |ϕˆjN+1 (ξ ) − gˆ ∞ (ξ )| |ξ |2+δ

+∞ N 2+δ dτ r d2+δ ϕj , g∞ t p + q 2+δ + 1 − t τ 2+δ r +1 t τ t 1

r 2+δ t p + q 2+δ − 1 + 1 d2+δ ϕjN , g∞ t

+∞ 1

For N >

T (2+δ) r

τ

we get

d2+δ ϕjN+1 , g∞ d2+δ ϕjN , g∞

r t r t

− (2 + δ)

2+δ + q 2+δ − 1 + 1 t p

and remembering that 2+δ Sp,q (δ) = p 2+δ + q 2+δ − 1 + 1 − p2 − q 2 , 2 we get

dτ r t −1−δ

.

G. Furioli et al. / Journal of Functional Analysis 257 (2009) 2291–2324 r t r t

− (2 + δ)

= =

2305

2+δ t p + q 2+δ − 1 + 1

t (p 2+δ + q 2+δ − 1) + 1 1 − t

2+δ 2

(1 − p 2 − q 2 )

2 2 2+δ + q 2+δ − 1) 1 + Sp,q (δ) t + (t)2 2+δ 2 (1 − p − q )(p 2 2 2 1 − (t)2 ( 2+δ 2 (1 − p − q ))

.

Since Sp,q (δ) < 0 and 1 2 1 − (t)2 ( δ+2 2 (1 − p

− q 2 ))2

= 1 + o(t),

t → 0,

we get, for N ∈ N large enough and j = 1, . . . , N − 1: |Sp,q (δ)| t d2+δ ϕjN , g∞ d2+δ ϕjN , g∞ . d2+δ ϕjN+1 , g∞ 1 − 2 Recursively, we get d2+δ ϕjN+1 , g∞ d2+δ (g0 , g∞ ) where d2+δ (g0 , g∞ ) depends only on the quantities in (23). Therefore, for j = 1, . . . , N and ξ = 0, we get

N

ϕˆ (ξ ) gˆ ∞ (ξ ) + |ξ |2+δ d2+δ ϕ N , g∞ gˆ ∞ (ξ ) + |ξ |2+δ C j

j

for C = d2+δ (g0 , g∞ ) independent of N and j . Since gˆ ∞ (ξ ) = 1 − we get that for any 0 < k
0 such that for j = 0, . . . , N ,

N

ϕˆ (ξ ) j

which achieves the proof.

ξ2 + o ξ2 , 2

1 , 1 + kξ 2

|ξ | ρ

2

We are now in position to prove Theorem 2. Theorem 2. Assume 0 < q p satisfying p + q 1, p 2 + q 2 < 1 and such that there is δ˜ ∈ (0, 1) ˜ Let g(t) be the weak solution of Eq. (6), corresponding to for which Sp,q (δ) < 0, for 0 < δ < δ. ˜ the initial density g0 satisfying the normalization conditions (3), and R |v|2+δ g0 (v) dv < +∞. If in addition

gˆ 0 (ξ )

1 , (1 + β|ξ |)ν

|ξ | > R,

(15)

2306


for some R > 0, ν > 0 and β > 0, then there exist ρ > 0, k > 0, ν > 0 such that g(t) satisfies ⎧ 1

⎨ 1+kξ 2 ,

g(ξ, ˆ t) 1 ⎩ , ν (1+β|ξ |)

|ξ | ρ, t 0,

(16)

|ξ | > ρ, t 0.

Proof. The bound on the low frequencies |ξ | ρ has been established in Lemma 10. Moreover, as a consequence of Proposition 2.4 in [13], we can suppose that condition (15) holds for any |ξ | > ρ with a possibly smaller exponent ν . We will prove that for any N ∈ N and j = 0, . . . , N , we get

N

ϕˆ (ξ ) j

1 , (1 + β|ξ |)ν

|ξ | > ρ

for positive ν small enough. By induction we have only to check the bound on

ϕˆ1N (ξ ) =

r t

+∞ t gˆ 0 (pτ ξ )gˆ 0 (qτ ξ ) + (1 − t)gˆ 0 (τ ξ ) 1

1 τ

r t +1

Let |ξ | > ρ and τ > 1. We are faced with three different situations. Case I: If |qτ ξ | > ρ, then

gˆ 0 (pτ ξ )gˆ 0 (qτ ξ )

1 1 (1 + βpτ |ξ |)ν (1 + βqτ |ξ |)ν

1 (1 + β(p + q)|ξ |)ν

1 . (1 + β|ξ |)ν

Case II: If |pτ ξ | > ρ and |qτ ξ | ρ, then


1 1 ν (1 + βpτ |ξ |) (1 + kq 2 τ 2 |ξ |2 ) 1 1 . 2 |ξ |2 ) ν (1 + kq (1 + βp|ξ |)

By choosing ν > 0 small enough we can show that 1 1 1 . (1 + βp|ξ |)ν (1 + kq 2 |ξ |2 ) (1 + β|ξ |)ν Indeed, since

1+βx 1+βpx

1 p

for any x 0, we obtain

dτ.


1 + β|ξ | 1 + βp|ξ |

ν

2307

ν 1 1 p (1 + kq 2 |ξ |2 ) ν 1 1 . p (1 + kq 2 ρ 2 )

1 (1 + kq 2 |ξ |2 )

Finally the last term is smaller than 1 for ν log 1 (1 + kq 2 ρ 2 ). p

Case III: If |pτ ξ | ρ, then


1 1 2 2 2 (1 + kp τ |ξ | ) (1 + kq 2 τ 2 |ξ |2 ) 1 1 2 2 (1 + kp |ξ | ) (1 + kq 2 |ξ |2 )

and we have to show that 1 1 . (1 + kp 2 |ξ |2 )(1 + kq 2 |ξ |2 ) (1 + β|ξ |)ν Since 1 (1 + kp 2 |ξ |2 )(1 + kq 2 |ξ |2 )

(1 + k(p 2

1 + q 2 )|ξ |2 )

and p 2 + q 2 = C > 0, in order to establish the desired estimate we have only to prove that for ν small enough we have 1 1 1 + kC|ξ |2 (1 + β|ξ |)ν or ν log 1 + β|ξ | log 1 + kC|ξ |2 for any |ξ | > ρ. This is true since the function F (ξ ) =

log(1 + kC|ξ |2 ) log(1 + β|ξ |)

is bounded below by a positive constant on |ξ | > ρ and so by choosing ν > 0 smaller than this constant we get the desired estimate. 2 Remark 11. The bounds in (16) are also equivalent to

g(ξ, ˆ t)

C , (1 + κ|ξ |)μ

ξ ∈ R, t 0

for some positive constants κ, μ and C suitably chosen.

2308


Remark 12. We would like to stress again that the proof by induction of Theorem 2 we have just given doesn’t allow to obtain the low frequencies bound on the solution and this is why we had to exploit the d2+δ convergence and the behavior of the steady state as in Lemma 10. 4. Smoothness of the steady state In this section we prove the following Theorem 3. Theorem 3. Assume 0 < q p satisfying p 2 + q 2 < 1 and such that there is δ˜ ∈ (0, 1) for which ˜ so that a non-trivial steady state g∞ to the Boltzmann equation (6) Sp,q (δ) < 0 for 0 < δ < δ, exists. Let us denote λ ∈ (0, 2) the exponent such that p λ + q λ = 1. Then g∞ is a smooth function and belongs to the λ-th Gevrey class Gλ (R), i.e.

gˆ ∞ (ξ ) exp −μ|ξ |λ ,

|ξ | > ρ

with suitable positive numbers ρ and μ. Proof. We are going to prove that g∞ ∈ Gλ (R), following the scheme already used in [20]. As in the proof of Lemma 10, we know the behavior of gˆ ∞ close to zero. In particular, by the same reasoning, there exist k ∈ (0, 12 ) and ρ > 0 such that

gˆ ∞ (ξ ) e−kξ 2 ,

|ξ | ρ.

As a consequence of Proposition 2.4 in [13] and without any loss in generality, we can also suppose ρ > 1. Let us remember that gˆ ∞ satisfies the equation d gˆ ∞ (ξ ) 1 2 p + q2 − 1 ξ = gˆ ∞ (pξ )gˆ ∞ (qξ ) − gˆ ∞ (ξ ) 2 dξ and so, denoting again r =

2 1−p 2 −q 2

and following [7], we get

∞ gˆ ∞ (ξ ) = r

gˆ ∞ (pτ ξ )gˆ ∞ (qτ ξ ) d τ, τ r+1

ξ ∈ R.

1

Let us consider the space

Xρ = ψ ∈ L∞ (R), ψ(ξ ) 1, ψ(ξ ) = gˆ ∞ (ξ ) for |ξ | ρ endowed with the metric d2+δ defined in (7), where δ satisfies the assumptions of Theorem 3. The space Xρ is a Fréchet space. Let us consider then the function R defined by R(ψ)(ξ ) =

gˆ ∞ (ξ ) |ξ | ρ, ∞ ψ(pτ ξ )ψ(qτ ξ ) r 1 dτ, |ξ | > ρ. τ r+1

(28)


2309

We are going to prove that R is a contraction on Xρ and so gˆ ∞ is its unique fixed point. It is straightforward that R : Xρ −→ Xρ . As for the contractiveness, for ψ and ϕ ∈ Xρ and |ξ | > ρ, we have ∞ τ 2+δ (p 2+δ + q 2+δ ) |R(ψ)(ξ ) − R(ϕ)(ξ )| r dτ d2+δ (ψ, ϕ) |ξ |2+δ τ r+1

1

r p 2+δ + q 2+δ

∞

dτ τ r−1−δ

d2+δ (ψ, ϕ).

1

We remark that the assumption Sp,q (δ) = p 2+δ + q 2+δ − 1 +

2+δ 1 − p2 − q 2 < 0 2

implies 2+δ 1 − p 2 − q 2 < 1 − p 2+δ + q 2+δ < 1 2 and so, δ 1 − p2 − q 2 < p2 + q 2 2 which is precisely r − 1 − δ > 1. So we get r(p 2+δ + q 2+δ ) d2+δ R(ψ), R(ϕ) d2+δ (ψ, ϕ). r −2−δ Remembering that r =

2 , 1−p 2 −q 2

we get

2+δ r(p 2+δ + q 2+δ ) 2 = p + q 2+δ 2 2 r −2−δ 1−p −q

1 2 1−p 2 −q 2

−2−δ

and this allows to conclude. Choosing ψ0 ∈ Xρ for example as ψ0 (ξ ) = and defining by induction for n 0,

gˆ ∞ (ξ ), 0,

|ξ | ρ, |ξ | > ρ,

ρ,

we get authomatically d2+δ (ψn , gˆ ∞ ) → 0 for n → +∞, which implies convergence pointwise. We show now that there exists μ > 0 such that for all n ∈ N,

ψn (ξ ) e−μ|ξ |λ ,

|ξ | > ρ

and this uniform estimate passes therefore to the limit and allows to conclude. The only thing to control is that for τ > 1 and |ξ | > ρ we have

ψ0 (pτ ξ )ψ0 (qτ ξ ) e−μ|ξ |λ . We distinguish three cases: Case I: If qτ |ξ | > ρ, since p λ + q λ = 1 and ρ > 1, we get

λ λ λ λ λ λ λ eμ|ξ | · ψ0 (pτ ξ ) · ψ0 (qτ ξ ) eμ|ξ | (1−τ (p +q )) eμ|ξ | (1−τ ) 1. Case II: If pτ |ξ | ρ, then |ξ | > 1 implies ξ 2 |ξ |λ . Denoting p 2 + q 2 = C, we conclude

λ λ 2 2 2 2 λ λ 2 2 λ eμ|ξ | · ψ0 (pτ ξ ) · ψ0 (qτ ξ ) eμ|ξ | −kτ ξ (p +q ) eμ|ξ | −k|ξ | (p +q ) e|ξ | (μ−k C) 1, provided that μ kC. Case III: Now assume that qτ |ξ | ρ while pτ |ξ | > ρ. Using the condition p λ + q λ = 1 once again, one finds

λ λ 2 2 2 λ λ λ λ λ 2 eμ|ξ | · ψ0 (qτ ξ ) · ψ0 (pτ ξ ) eμ|ξ | −k τ q ξ −μτ p |ξ | e|ξ | μ(1−p )−kq e|ξ |

λ (μq λ −kq 2 )

provided that μ

kq 2 . qλ

1

2

Remark 13. By the proof of Theorem 3, the condition g∞ ∈ Gλ (R) seems to be sharp, since in the estimate of Case I, it wouldn’t have been possible to replace λ by σ > λ. Moreover, in the case p + q = 1 the explicit stationary state is gˆ ∞ (ξ ) = (1 + |ξ |)e−|ξ | and this proves sharpness at least in this special case. It is worth noticing that Bobylev and Cercignani in [7] (Theorem 5.3) proved that for p + q 1 and p 2 + q 2 1 the stationary state g∞ satisfies the bounds e−

ξ2 2

gˆ ∞ (ξ ) 1 + |ξ | e−|ξ | .

For the values of p and q for which in addition there is δ˜ ∈ (0, 1) such that Sp,q (δ) < 0 for ˜ our result improves the upper bound and gives also a new result for some p and q in 0 < δ < δ, the range p + q < 1.


2311

By a careful reading of both proofs of Theorem 2 and Theorem 3, we can deduce that not only polynomial tails of the Fourier transform of the initial data g0 are uniformly propagated by the solution g(t), but also exponential ones, as long as the exponent does not exceed the exponent λ characterizing the mixing parameters p and q. More precisely, the result is as follows. Theorem 4. Assume 0 < q p satisfying p 2 + q 2 < 1 and such that there is δ˜ ∈ (0, 1) for ˜ Let us denote λ ∈ (0, 2) the exponent such that p λ + q λ = 1. which Sp,q (δ) < 0 for 0 < δ < δ. Let g(t) be the weak solution of Eq. (6), corresponding to the initial density g0 satisfying the ˜ normalization conditions (3), and R |v|2+δ g0 (v) dv < +∞. If in addition

gˆ 0 (ξ ) e−β|ξ |ν ,

|ξ | > R,

for some R > 0, ν > 0 and β > 0, then there exist ρ > 0 and κ > 0, such that g(t) satisfies −κξ 2

g(ξ, ˆ t) e−κ|ξ |,min(ν,λ) e ,

|ξ | ρ, t 0, |ξ | > ρ, t 0.

Proof. The proof follows the same lines as the proof of Theorem 2, replacing the polynomial decreasing by the exponential one. The key argument for the low frequencies is that the normalization assumptions (3) on the steady state imply gˆ ∞ (ξ ) = 1 −

ξ2 + o ξ2 , 2

ξ → 0,

which means

gˆ ∞ (ξ )

1 , 1 + kξ 2

|ξ | ρ

or equivalently

gˆ ∞ (ξ ) e−kξ 2 ,

|ξ | ρ

for any k ∈ (0, 12 ) and ρ depending on k. The whole proof follows then without any particular difficulty, exploiting for the high frequencies the estimates performed in the proof of the Gevrey regularity of the steady state. 2 5. Strong convergence In this section, we are going to prove Theorem 5 on the strong L1 convergence of the scaled solution g(t) to the stationary state g∞ . Theorem 5. Assume 0 < q p satisfying p + q 1, p 2 + q 2 < 1 and such that there exists δ˜ ∈ (0, 1) for which Sp,q (δ) < 0 for 0 < δ < δ˜ and let g∞ be the unique stationary solution of (6). ˜ Let the initial density g0 satisfy the normalization conditions (3), and R |v|2+δ g0 (v) dv < +∞. √ If in addition g0 ∈ H η (R) for some η > 0 large enough, g0 ∈ H˙ ν (R) for some ν > 0, then the

2312


solution g(t) of (6) converges strongly in L1 with an exponential rate towards the stationary solution g∞ , i.e., there exist positive constants C and γ explicitly computable such that g(t) − g∞

L1 (R)

Ce−γ t ,

t 0.

Let us begin by the following lemma. Lemma 14. Let the initial density g0 satisfy the normalization conditions (3), and ˜ |v|2+δ g0 (v) dv < +∞. R

√ If in addition g0 ∈ H˙ ν (R) for some ν > 0, then g0 satisfies

gˆ 0 (ξ )

C , (1 + β|ξ |)ν

ξ ∈R

for positive constants C and β and the solution g(t) of (6) satisfies

sup g(ξ, ˆ t) − gˆ ∞ (ξ ) C1 e−C2 t ,

ξ ∈R

t 0

(29)

for positive constants C1 and C2 . Proof. Since g0 =

√ √ √ √ g0 g0 , then gˆ 0 = g0 ∗ g0 . So, for ξ ∈ R, we get

|ξ |ν gˆ 0 (ξ )

R

√ √ |ξ |ν g0 (ξ − τ ) g0 (τ ) dτ

K

√ √ |ξ − τ |ν + |τ |ν g0 (ξ − τ ) g0 (τ ) dτ

R

√ 2K g0 H˙ ν . Since moreover |gˆ 0 (ξ )| 1, we can find positive C and β such that

gˆ 0 (ξ )

C , (1 + β|ξ |)ν

ξ ∈ R.

Thanks to Theorem 2 and to Remark 11, we get

g(ξ, ˆ t)

C , (1 + κ|ξ |)μ

ξ ∈ R, t 0

κ and μ. The steady state g∞ belongs to a Gevrey class, so it satisfies an analogous for suitable C, estimate, with suitable constants which we can suppose to be the same. Let now R > 0 to be chosen in a moment. We get, for ξ ∈ R:


g(ξ, ˆ t) − gˆ ∞ (ξ ) d2+δ g(t), g∞ R 2+δ +

2313

2C , (κR)μ

which implies, optimizing over R,

μ/(2+δ+μ)

g(ξ, ˆ t) − gˆ ∞ (ξ ) C1 d2+δ g(t), g∞ = C1 e−C2 t , for C1 and C2 positive constants.

ξ ∈ R, t 0,

2

Proof of Theorem 5. The result consists in converting the weak convergence in the Fourier distance of the solution g(t) to the stationary state g∞ (Theorem 1) into a L1 convergence by interpolating this weak distance d2+δ with the uniformly boundedness in time of suitable moment and Sobolev norm of the solution itself. The only missing ingredient at this point is the boundedness of the Sobolev norm and we will go through the proof of it in a moment. Let us recall first how we can interpolate these results, following the scheme introduced in [10] and fruitfully applied afterward in several papers ([5,9] for instance). First of all, it is easy to prove the two following interpolation bounds (see Theorems 4.1 ˜ there exists a positive constant C such that and 4.2 in [10]): for δ ∈ (0, δ), 2(2+δ) 1

h L1 C |v|2+δ hL1+2(2+δ)

h L1+2(2+δ) 1 2

and for any s 0 there exist positive constants M, N , β and γ such that

h H s

ˆ )| |h(ξ C sup 2+δ R |ξ |

β

γ

h H M + h H N .

(30)

So, letting h = g(t) − g∞ , and s = 0, we get g(t) − g∞ 1 C |v|2+δ g(t) 1 + |v|2+δ g∞ 1 α˜ d2+δ g(t), g∞ β˜ L L L γ˜ × g(t)H M + g(t)H N + g∞ H M + g∞ H N ˜ γ˜ . Concerning the stationary state g∞ , it has been proved in Thefor suitable exponents α, ˜ β, 2+δ ˜ and thanks to the Gevrey regularity, we have orem 1 that |v| g∞ L1 < ∞ for δ ∈ (0, δ) s g∞ ∈ H (R) for all s 0. As for the scaled solution g(t), the uniform boundedness of the (2 + δ)-th moment have been also proved in Theorem 1, so we will get the L1 exponential convergence as soon as we prove the uniform boundedness in time of g(t) in a suitable Sobolev space H max(M,N ) (R). Of course, we need to assume g0 ∈ H max(M,N ) (R). As a byproduct of the uniform boundedness of Sobolev norms, we will also get from (30) the convergence of g(t) to g∞ in Sobolev spaces. Let us recall how to prove the uniform boundedness of g(t) in a generic homogeneous Sobolev space H˙ η for η 0 under the assumption g0 ∈ H˙ η . First of all, let us remark that g(t) ∈ H˙ η for all t > 0, without any uniformity in time. Indeed, coming back to the original nonscaled solution f (t), we get

2314


d f (t)2˙ η = d H dt dt

|ξ |2η fˆ(ξ, t)fˆ(ξ, t) dξ

R

|ξ |2η Re fˆ(ξ, t) fˆ(pξ, t)fˆ(qξ, t) − fˆ(ξ, t) dξ

=2 R

−

2

|ξ |2η fˆ(ξ, t) dξ +

R

− R

Since g(e ˆ

p 2 +q 2 −1 t 2

2

2

|ξ |2η fˆ(pξ, t) fˆ(qξ, t) dξ

R

2 1 1 1 |ξ |2η fˆ(ξ, t) dξ + + 2η+1 2η+1 2 q p

2 |ξ |2η fˆ(ξ, t) dξ.

R

ξ, t) = fˆ(ξ, t), we get

2 1−p 2 −q 2 d 1−p2 −q 2 (2η+1)t 1 1 (2η+1)t ˙η g(t)2˙ η −1 + 1 2 g(t) e e 2 + H H dt 2 q 2η+1 p 2η+1 and so 2 2 1 1 d g(t)2˙ η g(t)2˙ η −1 − 1 − p − q (2η + 1) + 1 + H H 2η+1 2η+1 dt 2 2 q p 2 = Cp,q,η g(t)H˙ η ,

(31)

which leads to g(t)2˙ η g0 2 η eCp,q,η t . H H˙ Since p 2 + q 2 < 1, it is not difficult to be convinced that Cp,q,η > 0 as soon as for example q is small enough. Let us make estimate (31) more accurate. The goal is to get for example the following differential inequality: for two positive constants H and K and t0 > 0: d g(t)2˙ η −H g(t)2˙ η + K, H H dt

t t0

g(t)2˙ η C max g(t0 )2˙ η , 1 , H H

t t0 .

so that

Let us come back to inequality d f (t)2˙ η − H dt

2 |ξ |2η fˆ(ξ, t) dξ +

R

which reads, on the scaled solution g(t),

R

2

2 |ξ |2η fˆ(pξ, t) fˆ(qξ, t) dξ

(32)


2315

d 1−p2 −q 2 (2η+1)t g(t)2˙ η e 2 H dt 2 2

2

2

2

1−p −q 1−p 2 −q 2 (2η+1)t 2η (2η+1)t

2 2 ˆ t) dξ + e ˆ t) g(qξ, ˆ t) dξ −e |ξ | g(ξ, |ξ |2η g(pξ, R

R

and so 2 2

2

2 d

g(qξ,

dξ. g(t)2˙ η −1 − 1 − p − q (2η + 1) g(t)2˙ η + |ξ |2η g(pξ, ˆ t) ˆ t) H H dt 2 R

Since p 2 + q 2 < 1, it would be enough to obtain for example the following inequality

2

2 2

1 ˆ t) g(qξ, ˆ t) dξ g(t)H˙ η + K, |ξ |2η g(pξ, 2

R

t t0

(33)

where K > 0 is independent of t. Let us prove inequality (33) by following the scheme introduced in [10] and whose application is here much simpler. We split the integral in (33) into two parts

2

2

ˆ t) g(qξ, ˆ t) dξ = |ξ |2η g(pξ,

R

+

=A+B

|ξ |>R

|ξ |R

where R will be chosen later. Let us estimate first the term in A (we will denote ε a constant which is allowed to vary from one line to another, depending at most on p and q). Since |g(ξ, ˆ t)| 1 for ξ ∈ R and t 0 we simply get

2

2

ˆ t) g(qξ, ˆ t) dξ |ξ |2η g(pξ,

|ξ |R

|ξ |2η dξ =

|ξ |R

2 R 2η+1 , 2η + 1

t 0.

Let us come to the term in B, where we are going to exploit Lemma 14. We remark that gˆ ∞ (ξ ) → 0 for ξ → +∞ and so, by Lemma 14, for any ε > 0 there exist R > 0 and t0 depending on ε and p such that

g(pξ, ˆ t) − gˆ ∞ (pξ ) + gˆ ∞ (pξ ) 2ε, ˆ t) g(pξ,

|ξ | > R, t t0 .

We can deduce for t t0 :

2

2

ˆ t) g(qξ, ˆ t) dξ (2ε)2 |ξ |2η g(pξ,

|ξ |>R

2

ˆ t) dξ |ξ |2η g(qξ,

|ξ |>R

ε q 2η+1

R

2

ˆ t) dξ |ξ |2η g(ξ,

2 = ε g(t) ˙ η . H

2316


We have obtained 2

2

2

ˆ t) g(qξ, ˆ t) dξ ε g(t)H˙ η + |ξ |2η g(pξ, R

2 R 2η+1 , 2η + 1

Letting ε be fixed such that ε 12 , we get the desired estimate.

t t0 .

2

6. Lyapunov functionals and open questions The case p + q = 1 separates in a natural way from the others. It corresponds to a onedimensional dissipative Boltzmann equation in which the momentum is preserved in a microscopic collision of type (1). Eq. (2) with p + q = 1 as been intensively studied in a series of papers [1,4,7,22]. Among other properties, this model possesses an explicit self similar solution, which has been first discovered in [1]. In fact, condition p + q = 1 implies 1 2 p + q 2 − 1 = −pq. 2 Hence the Fourier transformed version of the scaled equation (6) can be written as ˆ t) − pqξ ∂ξ g(ξ, ˆ t). g(pξ, ˆ t)g(qξ, ˆ t) − g(ξ, ˆ t) = ∂t g(ξ,

(34)

The choice gˆ ∞ (ξ ) = 1 + |ξ | e−|ξ | leads to gˆ ∞ (pξ )gˆ ∞ (qξ ) − gˆ ∞ (ξ ) = pq|ξ |2 e−|ξ | = −pqξ ∂ξ gˆ ∞ (ξ ) and so gˆ ∞ solves (34) as a stationary solution for any choice of the parameters p and q such that p + q = 1. It can be easily verified that in physical variables the steady solution reads g∞ (v) =

2

π 1 + v2

2 .

(35)

Note that this function satisfies the normalization conditions (3). Under these constraints, however, it can be shown [26] that g∞ is the (unique) minimizer of the convex functional

f (v) dv. H (f ) = −

(36)

R

It is a natural question to investigate whether the functional H is a Lyapunov functional for the scaled equation for g(v, t), which can be formally written as ∂t g(v, t) = gp ∗ gq (v, t) − g(v, t) +

1 2 p + q 2 − 1 ∂v vg(v, t) . 2

(37)


2317

The results of both Section 3 and Section 5 lead to conclude that, under suitable regularity assumptions on the initial value, one can study the time derivative of the functional H (g)(t) along solutions to Eq. (37), obtaining d 1 H (g)(t) = − dt 2

gp ∗ gq (v, t) 1 + p2 + q 2 dv − √ 2 g(v, t)

R

g(v, t) dv . R

Hence, the functional H is a Lyapunov functional for Eq. (37) provided the inequality 1 + p2 + q 2 2

g(v) dv

R

R

gp ∗ gq (v) dv, √ g(v)

p + q = 1,

(38)

is verified for all functions satisfying constraints (3). We remark that inequality (38) is saturated by choosing g = g∞ , with g∞ defined as in (35). To our knowledge, this inequality has never been investigated before, but it can be conjectured that it holds true, even if we are not able to prove it. A different way to attach the problem is to resort to the Fourier version of Eq. (37). This idea has been fruitfully employed in [8] to recover Lyapunov functionals for the Boltzmann equation for Maxwell molecules. Let us consider the approximate solution (13) which is a convex combination of the probability densities g(ξ, ˆ t) and g(pξ, ˆ t)g(qξ, ˆ t). For any convex functional acting on gˆ we obtain H r g(ξ, H ˆ t + t) t

+∞ dτ g(τpξ, g(τ t H ˆ t)g(τ ˆ qξ, t) + (1 − t)H ˆ ξ, t) . r τ t +1

(39)

1

(g) If H ˆ = H (g) is defined by (36), we have √ g(τ g (ξ ) , H ˆ ξ) = τH and this implies that inequality (39) becomes g(ξ, H ˆ t + t)

r t r t

−

1 2

g(pξ, g(ξ, t H ˆ t)g(qξ, ˆ t) + (1 − t)H ˆ t) .

Let us suppose that there exists A(p, q) such that for all functions g satisfying conditions (3) we get g(pξ g(ξ H ˆ )g(qξ ˆ ) A(p, q)H ˆ ) . Then, since 1 1 − p2 − q 2 = , r 2

(40)

2318


the condition A(p, q) =

3 + p2 + q 2 4

(41)

would imply r t r t

−

1 2

t A(p, q) + (1 − t) = 1.

In this case, inequality (39) would imply g(ξ, g(ξ, H ˆ t + t) H ˆ t) , and H would be a Lyapunov functional. Note that inequality (40) corresponds to a reverse Young inequality first derived by Leindler [17]: for 0 < α, β, ρ 1 and f , g non-negative [2]

f ∗ g ρ f α g β ,

1/α + 1/β = 1 + 1/ρ.

(42)

In our case, ρ = α = 1/2, β = 1 together with the second condition in (3) implies

fp ∗ fq 1/2 p f 1/2 , namely inequality (40) with A(p, q) = p. Unlikely, the direct application of inequality (42) is not enough to obtain (40). It remains an open question to prove that, under constraints (3) it holds the Young-type reverse inequality 1/2

1/2

fp ∗ fq 1/2 A(p, q) f 1/2 , where p + q = 1 and A(p, q) is given by (41). Appendix A Proof of Proposition 7. We will consider only the dissipative case p 2 + q 2 < 1, since the other case adapts straightforwardly. It is easy to get for all N ∈ N and j = 0, . . . , N − 1: ϕjN+1 (v) 0,

ϕjN+1 (v) dv = 1,

R

vϕjN+1 (v) dv = 0.

(43)

R

Let us consider therefore the evolution of the two other moments of the sequence ϕjN . First of all, it is worth noticing that for any function h ∈ L1 (R), α > 0 and τ = 0, we have R

v 1 dv = |τ |α |v|α hL1 . |v|α h τ τ

(44)


2319

Let us compute now the second moment of ϕjN+1 : v 2 ϕjN+1 (v) dv R

+∞

r = t

v N N v ϕ ∗ ϕj,q dv τ j,p τ

21

t R

1

v dτ 1 dv + (1 − t) v 2 ϕjN r τ τ t τ +1 R

2 N r N N 1 t v 2 ϕj,p = ∗ ϕj,q 1 + (1 − t) v ϕj L L t

+∞

r

1

For N >

2T r

τ 2 dτ τ t +1

.

, we get

r t

+∞

dτ r

1

τ t −1

=

1 1−

2t r

1 1 + t (p 2 + q 2 − 1)

=

N ∗ ϕ N . We have and so we are left with v 2 ϕj,p j,q L1

v2 R2

1 N v−w 1 N w ϕj ϕj dw dv p p q q

=

1 v−w 1 N w ϕj dw dv. (v − w)2 + w 2 + 2(v − w)w ϕjN p p q q

R2

Thanks to (43), we get 2 N v ϕ ∗ ϕ N 1 = p 2 + q 2 v 2 ϕ N 1 j,p j,q L j L and we end up with 2 N v ϕ

j +1 L1

=

(t (p 2 + q 2 − 1) + 1) v 2 ϕ N 1 = v 2 ϕ N 1 j j L L 1 + t (p 2 + q 2 − 1)

and so by a recursive procedure v 2 ϕjN+1 (v) dv = 1, R

j = 0, . . . , N − 1.

(45)

2320


As for

2+δ ϕ N (v) dv, j +1 R |v|

|v|2+δ ϕjN+1 (v) dv R

we proceed in the same way:

r = t

+∞

|v|

t R

1

N v N ϕ ∗ ϕj,q dv + τ j,p τ

2+δ 1

v dτ 1 + (1 − t) dv |v|2+δ ϕjN r τ τ τ t +1 R

r N N t |v|2+δ ϕj,p = ∗ ϕj,q

L1 + (1 − t)|v|2+δ ϕjN L1 t

+∞

(2+δ)T r

r t

τ t +1 r

1

Now, for N >

τ 2+δ dτ

, we get

+∞ 1

dτ τ

r t −1−δ

=

1 1−

t r (δ

+ 2)

=

1 2 1 − t δ+2 2 (1 − p

− q 2)

.

N ∗ ϕ N . We will denote Let us estimate |v|2+δ ϕj,p j,q L1

djN =

|v|2+δ ϕjN (v) dv. R

Since 0 < δ < 1, we can write |v|2+δ = v 2 |v|δ = (v − w + w)2 |v − w + w|δ (v − w)2 + w 2 + 2(v − w)w |v − w|δ + |w|δ = |v − w|2+δ + w 2 |v − w|δ + 2(v − w)|v − w|δ w + (v − w)2 |w|δ + |w|2+δ + 2(v − w)|w|δ w so, thanks to (43) and (45), we get djN+1

1 2 2 1 − t δ+2 2 (1 − p − q ) N 2+δ × dj + t p + q 2+δ − 1 djN + t p δ q 2 + q δ p 2 |v|δ ϕjN L1 .

By Hölder inequality and the conservation of the mass, we obtain δ N |v| ϕ j

L1

δ 2−δ |v|2 ϕjN L2 1 ϕjN L21 = 1

.


2321

and so we get the recursive estimate djN+1

1 2 1 − t δ+2 2 (1 − p

Remembering that t =

T N,

− q 2)

N dj + t p 2+δ + q 2+δ − 1 djN + t p δ q 2 + q δ p 2 .

we would like to neglect the low order terms. We have

1 djN + t p 2+δ + q 2+δ − 1 djN + t p δ q 2 + q δ p 2 δ+2 1 − t 2 (1 − p 2 − q 2 ) 2 2 N 1 + t δ+2 2 (1 − p − q ) = dj + t p 2+δ + q 2+δ − 1 djN + t p δ q 2 δ+2 2 2 2 2 1 − (t) ( 2 (1 − p − q )) N N 2 2 δ 2 δ 2 2 2 djN +t (p 2+δ +q 2+δ −1+ δ+2 2 (1−p −q ))dj +t (p q +q p )+(t) K dj +(t) H

=

2 2 2 1−(t)2 ( δ+2 2 (1−p −q ))

+ q δ p2

,

where H , K are positive constants, depending only on p, q and δ. Denoting Sp,q (δ) = p 2+δ + q 2+δ − 1 +

2+δ 1 − p2 − q 2 , 2

Bp,q (δ) = p δ q 2 + q δ p 2 ,

we got djN+1

djN + tSp,q (δ)djN + tBp,q (δ) + (t)2 H djN + (t)2 K 2 2 2 1 − (t)2 ( δ+2 2 (1 − p − q ))

where, by assumption, we have Sp,q (δ) < 0. Moreover, 1 2 1 − (t)2 ( δ+2 2 (1 − p

− q 2 ))2

= 1 + o(t),

t → 0,

so we get, for N ∈ N large enough, j = 0, . . . , N − 1:

1 djN+1 djN − t Sp,q (δ) djN + 2tBp,q (δ). 2 This recursive relation implies |v|2+δ ϕjN+1 (v) dv Cδ ,

j = 0, . . . , N − 1,

R

for any N ∈ N large enough.

2

Proof of Proposition 9. We divide it into several steps. I Step: Existence of the limit g ∗ (ξ , t) of a subsequence. Thanks to inequalities (24) and to the definition of g N we have therefore

2322


sup g N (ξ, t) 1,

sup ∂ξ g N (ξ, t) C,

[0,T ]×R

[0,T ]×R

sup ∂ξ2 g N (ξ, t) 1,

[0,T ]×R

(46)

where C is the same constant as in (24) and N is large enough. Moreover since g N (ξ, t) satisfies 1 d N N N N ∂t− g N (ξ, t) = ξ ϕˆK (ξ ) + ϕˆK (pξ )ϕˆ K (qξ ) − ϕˆK (ξ ) N −1 N −1 N −1 r dξ N

(47)

then for any compact set K ⊂ R there exists a constant C > 0 such that

sup ∂t− g N (ξ, t) C,

[0,T ]×K

N

2T . r

(48)

For any compact K ⊂ R the function g N (ξ, t) belongs to C([0, T ] × K) and thanks to properties (46) and (48) the sequence is equibounded and equicontinuous. Therefore by Ascoli–Arzelà theorem and by taking the diagonal, there exists a subsequence {g Nl (ξ, t)} which converges uniformly on [0, T ] × K for any compact K ⊂ R. Let us call g ∗ (ξ, t) the limit function. Since g Nl (ξ, 0) = gˆ 0 (ξ ) then g ∗ (ξ, 0) = gˆ 0 (ξ ). Moreover for any t ∈ [0, T ] the function g ∗ (ξ, t) ∈ C 1 (R): indeed by Ascoli–Arzelà theorem and the diagonal argument applied now to the sequence {∂ξ g Nl (ξ, t)}, for any t ∈ [0, T ] we get a subsequence that converges uniformly to ∂ξ g ∗ (ξ, t) on any compact set K ⊂ R. Since the limit function is ∂ξ g ∗ (ξ, t), the convergence holds for the original sequence {∂ξ g Nl (ξ, t)} and it is not necessary to pass to a subsequence. In order to get a uniform convergence in both frequency and time, we remark that by (46) we have that sup[0,T ]×R |∂ξ2 g N (ξ, t)| 1 and thanks to (47) and (24) we control ∂t ∂ξ g N (ξ, t) therefore ∂ξ g N (ξ, t) is Lipschitz continuous on [0, T ] × K and uniformely bounded. Again by Ascoli–Arzelà we prove that {∂ξ g Nl (ξ, t)} converges uniformly to ∂ξ g ∗ (ξ, t) on [0, T ] × K for any compact set K ⊂ R. From the uniform convergence of g Nl (ξ, t) to g ∗ (ξ, t) and of ∂ξ g Nl (ξ, t) to ∂ξ g ∗ (ξ, t) we get that both g ∗ (ξ, t) and ∂ξ g ∗ (ξ, t) belong to C([0, T ] × K). II Step: g ∗ (ξ , t) is a solution of Eq. (17). By a direct computation we obtain: 1 ∂t− g Nl (ξ, t) = ξ ∂ξ g Nl (ξ, t) + g Nl (pξ, t)g Nl (qξ, t) − g Nl (ξ, t) + RNl (ξ, t), r

(49)

where d Nl 1 d Nl ϕˆ KN −1 (ξ ) − ϕˆ K RNl (ξ, t) = − ξ (ξ ) l r dξ dξ Nl N N N N + (1 − α) α ϕˆ KNl (qξ ) − ϕˆ KNl −1 (qξ ) ϕˆ KNl (pξ ) − ϕˆKNl −1 (pξ ) l l l l N N N N N N + ϕˆ KNl −1 (pξ ) ϕˆ KNl −1 (qξ ) − ϕˆKNl (qξ ) + ϕˆ KNl (qξ ) ϕˆKNl −1 (pξ ) − ϕˆ KNl (pξ ) l l l l l l N N + (1 − α) ϕˆ KNl (ξ ) − ϕˆ KNl −1 (ξ ) . (50) l

l

Since {RNl (ξ, t)} converges to zero for any t ∈ [0, T ] uniformely on any compact set K ⊂ R, the whole right-hand side of Eq. (49) converges; let us call L(ξ, t) the limit function, so that


2323

1 L(ξ, t) = ξ ∂ξ g ∗ (ξ, t) + g ∗ (pξ, t)g ∗ (qξ, t) − g ∗ (ξ, t) r

(51)

and {∂t− g Nl (ξ, t)} converges to L(ξ, t) for any t ∈ [0, T ] uniformely on any compact set K ⊂ R. Thanks to property (48) and to Lebesgue’s dominated convergence theorem we remark that {∂t− g Nl (ξ, t)} converges to L(ξ, t) in D (]0, T [ × K ◦ ). Since {g Nl (ξ, t)} converges to g ∗ (ξ, t) uniformly on [0, T ] × K and so in D (]0, T [ × K ◦ ) we have that {∂t g Nl (ξ, t)} converges in distributions to ∂t g ∗ (ξ, t). For the uniqueness of the limit and the fact that in distributions ∂t− g Nl (ξ, t) = ∂t g Nl (ξ, t), we obtain ∂t g ∗ (ξ, t) = L(ξ, t) in the sense of distributions. Finally since g ∗ (ξ, t) and ∂ξ g ∗ (ξ, t) belongs to C([0, T ] × K), for any compact K the right-hand side of (51) is continuous in both variables therefore the same is true for ∂t g ∗ (ξ, t). This implies that g ∗ (ξ, t) belongs to C ([0, T ] × R). √ III Step: We show that g ∗ (ξ , t) = g(ξ ˆ , t). Defining F ∗ (ξ, t) = g ∗ (ξ E(t), t) and fˆ(ξ, t) = √ g(ξ ˆ E(t), t) we obtain two solutions of Eq. (5). Both F ∗ (ξ, t) and fˆ(ξ, t) are C ([0, T ] × R) and bounded by 1. Denoting h(ξ, t) = fˆ(ξ, t) − F ∗ (ξ, t) by an easy computation we are led to |∂t h + h| 2h(t)∞ . By Gronwall Lemma we have h(t)

∞

et h(0)∞ ,

ˆ t). and since fˆ(ξ, 0) = F ∗ (ξ, 0) this implies the desired equality g ∗ (ξ, t) = g(ξ,

2

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Metric aspects of noncommutative homogeneous spaces Hanfeng Li 1 Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, USA Received 7 December 2008; accepted 26 May 2009 Available online 6 June 2009 Communicated by Alain Connes Dedicated to Marc A. Rieffel in honor of his seventieth birthday

Abstract For a closed cocompact subgroup Γ of a locally compact group G, given a compact abelian subgroup K of G and a homomorphism ρ : Kˆ → G satisfying certain conditions, Landstad and Raeburn constructed ˆ ρ) of the homogeneous space G/Γ , generalizing Riequivariant noncommutative deformations C ∗ (G/Γ, effel’s construction of quantum Heisenberg manifolds. We show that when G is a Lie group and G/Γ is ˆ ρ) induced by the derivaconnected, given any norm on the Lie algebra of G, the seminorm on C ∗ (G/Γ, tion map of the canonical G-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on ρ continuously, with respect to quantum Gromov–Hausdorff distances. © 2009 Elsevier Inc. All rights reserved. Keywords: Noncommutative homogeneous space; quantum Heisenberg manifold; Compact quantum metric space; Gromov–Hausdorff distance

1. Introduction In recent years, the quantum Heisenberg manifolds have received quite some attention. These interesting C ∗ -algebras were constructed by Rieffel [28] as deformation quantizations of the Heisenberg manifolds, and carry natural actions of the Heisenberg group. The classification of these C ∗ -algebras up to isomorphism (in most cases) and Morita equivalence (in all cases) has E-mail address: [email protected]. 1 Partially supported by NSF Grant DMS-0701414.


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H. Li / Journal of Functional Analysis 257 (2009) 2325–2350

been achieved by Abadie and her collaborators [1–4]. These C ∗ -algebras also appear in the work of Connes and Dubois-Violette on noncommutative 3-spheres [10,11]. Aiming partly at giving a mathematical foundation for various approximations in the string theory, such as the fuzzy spheres, namely the matrix algebras Mn (C), converging to the 2sphere S 2 , Rieffel developed a theory of compact quantum metric spaces and quantum Gromov– Hausdorff distance between them [31–33]. As the information of the metric on a compact metric space X is encoded in the Lipschitz seminorm on the algebra of continuous functions on X, a quantum metric on (the compact quantum space represented by) a unital C ∗ -algebra A is a (possibly +∞-valued) seminorm on A satisfying suitable conditions (see Section 5 below for detail). One important class of examples of compact quantum metric spaces comes from ergodic actions of a compact group G on a unital C ∗ -algebra A, which should be thought of as the translation action of G on a noncommutative homogeneous space of G. Given any length function on G, such an ergodic action induces a quantum metric on A [30] (see [25] for a generalization to ergodic actions of co-amenable compact quantum groups). This class of examples includes the (fuzzy) spheres above and the noncommutative tori. When G is a compact connected Lie group and the length function comes from the geodesic distance associated to some bi-invariant Riemannian metric on G, this seminorm can also be defined in terms of the derivation map on the space of once differentiable elements of A with respect to the G-action [31, Proposition 8.6]. Explicitly, denote by σX (b) the derivation of a once differentiable element b of A with respect to an element X of the Lie algebra g of G (see Section 3 below for detail). Then the seminorm L(b) is defined as the norm of the linear map g → A sending X to σX (b) when b is once differentiable, or ∞ otherwise. It is natural to ask what conditions are needed to guarantee that L defined above gives rise to a quantum metric when G is not compact. Rieffel raised the question about the quantum Heisenberg manifolds in [33]. In [38] Weaver studied some sub-Riemannian metric on the quantum Heisenberg manifolds, which does not quite fit into the above framework. In [9] Chakraborty showed that certain seminorm associated to some 1 -norm does define a quantum metric on the quantum Heisenberg manifolds. Since the 1 -norm is bigger than the C ∗ -norm, this seminorm is bigger than the seminorm L defined above. Thus the result in [9] is weaker than what Rieffel’s question asks for. Our first main result in this article is an affirmative answer to Rieffel’s question. In fact, we shall deal more generally with Landstad and Raeburn’s noncommutative homogeneous spaces. In [22] Landstad and Raeburn generalized Rieffel’s construction to obtain equivariant deformations of compact homogeneous spaces G/Γ , starting from a locally compact group G, a closed cocompact subgroup Γ of G, a compact abelian subgroup K of G, and a homomorphism ˆ ρ) and ρ : Kˆ → G satisfying certain conditions. These C ∗ -algebras were denoted by Cr∗ (G/Γ, were further studied in [20]. We shall see in Proposition 2.7 below that these algebras coincide ˆ ρ). For our result to be valid with certain universal C ∗ -algebras, which we denote by C ∗ (G/Γ, for these algebras, we shall assume conditions (S1)–(S5) (see Sections 2, 3, and 4 below). Among these conditions, (S1)–(S3) are essentially the same but slightly weaker than the conditions of Landstad and Raeburn. The conditions (S4) and (S5) are just that G is a Lie group and G/Γ is connected. Theorem 1.1. Let G, Γ, K and ρ satisfy the conditions (S1)–(S5). Fix a norm on the Lie algeˆ ρ) defined above for the canonical action α bra g of G. Denote by Lρ the seminorm on C ∗ (G/Γ, ∗ ∗ ˆ ˆ of G on C (G/Γ, ρ). Then (C (G/Γ, ρ), Lρ ) is a C ∗ -algebraic compact quantum metric space.


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Since Rieffel introduced his quantum Gromov–Hausdorff distance in [31], several variations have appeared [18,19,24–26,35,39]. Among these quantum distances, probably the most suitable one in our current situation is the distance distnu discussed in [19, Section 5], which is the unital version of the quantum distance introduced in [26, Remark 5.5]. As pointed out in [19, Section 5], this distance is no less than the distances introduced in [18,31]. It is also no less than the distances in [35] (see Appendix A below). Our second main result says that the compact quantum metric ˆ ρ), Lρ ) depend on ρ continuously. Let us mention that among the conditions spaces (C ∗ (G/Γ, (S1)–(S5), only the conditions (S1) and (S2) involve ρ. Theorem 1.2. Fix G, Γ , and K so that there exists ρ satisfying the conditions (S1)–(S5). Denote by Ω the set of all ρ satisfying the conditions (S1) and (S2), equipped with the weakest topology ˆ Then Ω is a making the maps Ω → G sending ρ to ρ(s) to be continuous for each s ∈ K. locally compact metrizable space. Fix a norm on the Lie algebra g of G. Then for any ρ ∈ Ω, ˆ ˆ ρ), C ∗ (G/Γ, ρ )) → 0 as ρ → ρ . distnu (C ∗ (G/Γ, This paper is organized as follows. In Section 2 we recall Landstad and Raeburn’s construction of noncommutative homogeneous spaces, and establish some general properties of these noncommutative spaces. The relation between the derivations coming from two canonical group ˆ ρ) is established in Section 3. In Section 4 we show that in the nondeformed actions on C ∗ (G/Γ, case Lρ is essentially the Lipschitz seminorm corresponding to some metric on G/Γ . A general result of establishing certain seminorm being a quantum metric by the help of a compact group action is proved in Section 5. Theorems 1.1 and 1.2 are proved in Sections 6 and 7 respectively. In Appendix A we compare the distance distnu and the proximity Rieffel introduced in [35]. 2. Noncommutative homogeneous spaces In this section we recall Landstad and Raeburn’s construction of noncommutative deformations of homogeneous spaces, discuss some examples, and establish some general properties of these noncommutative homogeneous spaces. These properties are of independent interest themselves. Let G be a locally compact group. Throughout this paper, we make the following standard assumptions: (S1) K is a compact abelian subgroup of G, and ρ : Kˆ → G is a group homomorphism from its ˆ commutes with K. Pontryagin dual Kˆ into G such that ρ(K) (S2) Γ is a closed subgroup of G commuting with K and satisfies ˆ γ ∈ Γ, Ωγ (s) := γρ(s)γ −1 ρ(−s) is in K for all s ∈ K, ˆ γ ∈ Γ, Ωγ (s), t = Ωγ (t), s for all s, t ∈ K,

and

ˆ where ·,· denotes the canonical pairing between K and K. Denote by Cb (G) the Banach algebra of bounded continuous C-valued functions on G, equipped with the pointwise multiplication and the supremum norm. Endow K with its normalized Haar measure. Consider the action of K on Cb (G) induced by the right multiplication of K on G. For f ∈ Cb (G), let fs ∈ Cb (G) for s ∈ Kˆ be the partial Fourier transform defined

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by fs (x) := K k, sf (xk) dk for x ∈ G (this is denoted by fˆ(x, s) in (1.3) of [22]). Note that although the action of K on Cb (G) may not be strongly continuous, we do have fs ∈ Cb (G). Then Cb,1 (G) := f ∈ Cb (G) f ∞,1 := fs < ∞ s∈Kˆ

is a Banach ∗-algebra [21, Proposition 5.2] with norm · ∞,1 and operations f ∗ g(x) =

fs xρ(t) gt xρ(−s) ,

(1)

s,t

f ∗ (x) = f (x).

(2)

Fix a left invariant Haar measure on G. For each s ∈ Kˆ denote by Ps the projection on L2 (G) corresponding to the restriction of the left regular representation L|K of K in L2 (G), i.e., Ps =

k, sLk dk, K

where Ly ξ(x) = ξ(y −1 x) for ξ ∈ L2 (G), x, y ∈ G. Then Cb,1 (G) has a faithful ∗-representation V on L2 (G) [22, Proposition 1.3] given by V (f ) =

Pt Lρ(s) M(f )Lρ(−t) Ps ,

(3)

s,t

where M is the representation of Cb (G) on L2 (G) given by M(f )ξ(x) = f (x −1 )ξ(x). Denote by C0 (G/Γ ) the C ∗ -algebra of continuous C-valued functions on G/Γ vanishing at ∞, and think of it as a C ∗ -subalgebra of Cb (G) via the quotient map G → G/Γ . The space C0,1 (G/Γ, ρ) := C0 (G/Γ ) ∩ Cb,1 (G, ρ) is a closed ∗-subalgebra of Cb,1 (G, ρ), and the nonˆ commutative homogeneous space Cr∗ (G/Γ, ρ) of Landstad and Raeburn is defined as the closure of V (C0,1 (G/Γ, ρ)) [22, Theorem 4.3]. Clearly the left translations αy defined by αy (f )(x) = f (y −1 x) for y ∈ G extend to isometric ˆ ρ) [22, ∗-automorphisms of C0,1 (G/Γ, ρ). They also extend to ∗-automorphisms of Cr∗ (G/Γ, ˆ ρ) is strongly continuous. Theorem 4.3]. We shall see later that this action of G on Cr∗ (G/Γ, Before discussing properties of these noncommutative homogeneous spaces, let us look at some examples. Example 2.1. Let H1 be the 3-dimensional Heisenberg group consisting of matrices of the form ⎛

1 y

⎝0 0

1 0

z

⎞

x⎠ 1


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as a subgroup of GL(3, R). Denote by Z the subgroup consisting of elements with x = y = 0 and z ∈ Z. Then we can write the elements of G := H1 /Z as (x, y, e2πiz ) for x, y, z ∈ R. Fix a positive integer c. Take Γ =

x, y, e2πiz ∈ G x, y, cz ∈ Z ,

K=

0, 0, e2πiz ∈ G z ∈ R .

Take μ, ν ∈ R and define ρ : Z = Kˆ → G by

2 ρ(s) = sμ, sν, eπis μ·ν . ˆ The C ∗ -algebra Cr∗ (G/Γ, ρ) is isomorphic to Rieffel’s quantum Heisenberg manifold D1 in [28, Theorem 5.5] (see [22, page 493]). Example 2.2. (Cf. [22, Example 4.17].) Let Hn be the (2n + 1)-dimensional Heisenberg group consisting of matrices of the form ⎛

1 y1

y2

···

yn

1 0 .. . 0

0 1 .. . 0

··· ··· .. .

0 0 .. . 0

⎜0 ⎜ ⎜ ⎜0 ⎜ ⎜ .. ⎝. 0

···

z

⎞

x1 ⎟ ⎟ ⎟ x2 ⎟ ⎟ .. ⎟ . ⎠ 1

as a subgroup of GL(n + 2, R). Denote by Z the subgroup consisting of elements with x1 = · · · = xn = y1 = · · · = yn = 0 and z ∈ Z. Then we can write the elements of G := Hn /Z as (x, y, e2πiz ) for x, y ∈ Rn and z ∈ R. Fix positive integers b1 , . . . , bn , d1 , . . . , dn and c such that bj dj |c for all j . Set b = (b1 , . . . , bn ) and d = (d1 , . . . , dn ) ∈ Zn . Take Γ =

x, y, e2πiz ∈ G b · x, d · y, cz ∈ Z ,

K=

0, 0, e2πiz ∈ G z ∈ R .

Take μ, ν ∈ Rn and define ρ : Z = Kˆ → G by

2 ρ(s) = sμ, sν, eπis μ·ν . ˆ The C ∗ -algebra Cr∗ (G/Γ, ρ) is a higher-dimensional generalization of Example 2.1. Example 2.3. Let n 3. Let W be the subgroup of GL(n, Z) consisting of upper triangular matrices (aj,l ) with diagonal entries all being 1. Denote by Z the subgroup consisting of matrices whose entries are all 0 except diagonal ones being 1 and a1,n being an integer. Then we can write the elements of G := W/Z as (aj,l ) with a1,n ∈ T. Fix a positive integer c. Take c Γ = (aj,l ) ∈ G a1,n = 1 and aj,l ∈ Z if (j, l) = (1, n) , K = (aj,l ) ∈ G aj,l = 0 if j < l and (j, l) = (1, n) . Take μ, ν ∈ R and define ρ : Z = Kˆ → G by

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⎧ sμ ⎪ ⎪ ⎪ ⎨ sν

ρ(s) j,l = ⎪ eπis 2 μ·ν ⎪ ⎪ ⎩ 0

if (j, l) = (2, n), if (j, l) = (1, n − 1), if (j, l) = (1, n), for other j < l.

For n = 3 we get the quantum Heisenberg manifold in Example 2.1 again. ˆ ˆ In the rest of this section we establish some properties of Cr∗ (G/Γ, ρ). Denote by C ∗ (G/Γ, ρ) ∗ the enveloping C -algebra of the Banach ∗-algebra C0,1 (G/Γ, ρ) [36, page 42]. By the uniˆ ˆ versality of C ∗ (G/Γ, ρ) there is a canonical surjective ∗-homomorphism C ∗ (G/Γ, ρ) → ∗ ˆ ρ) such that the diagram Cr (G/Γ, C0,1 (G/Γ, ρ)

ˆ C ∗ (G/Γ, ρ)

ˆ Cr∗ (G/Γ, ρ) commutes. Clearly the right translations βk (f )(x) = f (xk) for k ∈ K extend to isometric ∗-automorphisms of C0,1 (G/Γ, ρ). Recall the action α of G on C0,1 (G/Γ, ρ) defined before Example 2.1. ˆ Then α and β induce actions of G and K on C ∗ (G/Γ, ρ) respectively, which we still denote by ˆ set α and β respectively. For each s ∈ K, Bs := f ∈ C0 (G/Γ ) f = fs .

(4)

ˆ ρ) resp.) commute Lemma 2.4. The actions α and β of G and K on C0,1 (G/Γ, ρ) (C ∗ (G/Γ, with each other and are strongly continuous. The spectral spaces {f ∈ C0,1 (G/Γ, ρ) | βk (f ) = ˆ k, sf for all k ∈ K} and {a ∈ C ∗ (G/Γ, ρ) | βk (a) = k, sa for all k ∈ K} of β corresponding ˆ ˆ ρ) is exactly the to s ∈ K are exactly Bs , and the norm of Bs in C0,1 (G/Γ, ρ) and C ∗ (G/Γ, supremum norm. Proof. Clearly α and β commute with each other. It is also clear that Bs = {f ∈ C0,1 (G/Γ, ρ) | βk (f ) = k, sf for all k ∈ K} and that the norm of Bs in C0,1 (G/Γ, ρ) is exactly the supremum norm. It follows that the restrictions of the actions α and β on Bs ⊆ C0,1 (G/Γ, ρ) are strongly ˆ For any f ∈ C0,1 (G/Γ, ρ), one has fs ∈ Bs for each s ∈ K. ˆ For continuous for each s ∈ K. any ε > 0 take a finite subset F ⊆ Kˆ such that s∈K\F fs < ε. Then f − s∈F fs ∞,1 = ˆ fs < ε. Therefore ˆ s∈K\F s∈Kˆ Bs is dense in C0,1 (G/Γ, ρ). It follows that the actions α and β are strongly continuous on C0,1 (G/Γ, ρ). Note that the canonical homomorphism ˆ C0,1 (G/Γ, ρ) → C ∗ (G/Γ, ρ) is contractive [36, Proposition 5.2]. Consequently, the induced ˆ actions of α and β on C ∗ (G/Γ, ρ) are also strongly continuous. Note that the subalgebra B0 of C0,1 (G/Γ, ρ) is a C ∗ -algebra, which can be identiˆ fied with C0 (G/KΓ ). Since the natural homomorphism C0,1 (G/Γ, ρ) → Cr∗ (G/Γ, ρ) is ∗ ˆ injective, so is the canonical homomorphism C0,1 (G/Γ, ρ) → C (G/Γ, ρ). As injective ∗homomorphisms between C ∗ -algebras are isometric, we conclude that the homomorphism of B0


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ˆ into C ∗ (G/Γ, ρ) is isometric. For any f ∈ Bs one has f ∗ ∗ f ∈ B0 and the supremum norm of f ∗ ∗ f is equal to the square of the supremum norm of f . It follows that the homomorphism ˆ ˆ C0,1 (G/Γ, ρ) → C ∗ (G/Γ, ρ) is isometric on Bs . In particular, the image of Bs in C ∗ (G/Γ, ρ) is closed. ˆ ρ) is strongly continuous, the spectral space Since the action β of K on C ∗ (G/Γ, ˆ {a ∈ C ∗ (G/Γ, ρ) | βk (a) = k, sa for all k ∈ K} is the image of the continuous linear opˆ ˆ erator C ∗ (G/Γ, ρ) → C ∗ (G/Γ, ρ) sending a to K k, sβk (a) dk. It follows that the imˆ age of Bs = {f ∈ C0,1 (G/Γ, ρ) | βk (f ) = k, sf for all k ∈ K} in C ∗ (G/Γ, ρ) is dense in ∗ ˆ ˆ {a ∈ C (G/Γ, ρ) | βk (a) = k, sa for all k ∈ K}. Therefore the image of Bs in C ∗ (G/Γ, ρ) is ˆ exactly {a ∈ C ∗ (G/Γ, ρ) | βk (a) = k, sa for all k ∈ K}. 2 We refer the reader to [8, Chapter 2] for the basics of nuclear C ∗ -algebras. ˆ Proposition 2.5. The C ∗ -algebra C ∗ (G/Γ, ρ) is nuclear. ˆ Proof. By Lemma 2.4 the action β of K on C ∗ (G/Γ, ρ) is strongly continuous, and its fixed∗ point subalgebra is B0 , a commutative C -algebra, and hence is nuclear [8, Proposition 2.4.2]. For any C ∗ -algebra carrying a strongly continuous action of a compact group, the algebra is nuclear if and only if the fixed-point subalgebra is nuclear [14, Proposition 3.1]. Consequently, ˆ C ∗ (G/Γ, ρ) is nuclear. 2 We shall need the following well-known fact a few times (see for example [8, Proposition 4.5.1]). Lemma 2.6. Let H be a compact group, and let σj be a strongly continuous action of H on a C ∗ -algebra Aj for j = 1, 2. Let ϕ : A1 → A2 be an H -equivariant ∗-homomorphism. Then ϕ is injective if and only if the restriction of ϕ on the fixed-point subalgebra AH 1 is injective. In particular, if ϕ is surjective and ϕ|AH is injective, then ϕ is an isomorphism. 1

ˆ ˆ ρ) → Cr∗ (G/Γ, ρ) is an isomorProposition 2.7. The canonical ∗-homomorphism C ∗ (G/Γ, phism. Proof. We shall apply Lemma 2.6 to show that the canonical ∗-homomorphism ˆ ˆ ϕ : C ∗ (G/Γ, ρ) → Cr∗ (G/Γ, ρ) is an isomorphism. By [22, Lemma 4.4] the action β on ˆ ρ), which we denote by β . Clearly ϕ is C0,1 (G/Γ, ρ) extends to an action of K on Cr∗ (G/Γ, ˆ ρ). Since ϕ is contractive, K-equivariant. By Lemma 2.4 β is strongly continuous on C ∗ (G/Γ, ˆ it follows that β is strongly continuous on Cr∗ (G/Γ, ρ). By Lemma 2.4 the fixed-point subalgeˆ ˆ ρ))K is B0 . Since the homomorphism C0,1 (G/Γ, ρ) → Cr∗ (G/Γ, ρ) is injective, bra (C ∗ (G/Γ, ∗ K ˆ ρ)) is injective. Therefore the conditions of we see that the restriction of ϕ on (C (G/Γ, Lemma 2.6 are satisfied and we conclude that ϕ is an isomorphism. 2 We refer the reader to [15] for a comprehensive treatment of C ∗ -algebraic bundles, which are usually called Fell bundles now. Notice that for fs ∈ Bs and gt ∈ Bt the product fs ∗ gt is in Bs+t and fs∗ is in B−s . Also fs∗ ∗ fs = fs 2 . Therefore we have a Fell bundle B ρ = {Bs }s∈Kˆ over Kˆ with operations given by (1) and (2). It is easy to see that C0,1 (G/Γ, ρ) is exactly the

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ˆ L1 -algebra of B ρ (cf. the proof of [21, Proposition 5.2]). Thus the C ∗ -algebra C ∗ (G/Γ, ρ) is also the enveloping C ∗ -algebra C ∗ (B ρ ) of the Fell bundle B ρ . Next we discuss what happens if we let ρ vary continuously. We refer the reader to [13, Chapter 10] for the basics of continuous fields of Banach spaces and C ∗ -algebras. On page 505 of [22] Landstad and Raeburn pointed out that it seems reasonable that we shall get a continuous field of C ∗ -algebras, but no proof was given there. This is indeed true, and we give a proof here. To be precise, fix G, Γ and K, let W be a locally compact Hausdorff space and for each w ∈ W we assign a ρw satisfying (S1) and (S2) such that the map w → ρw (s) is continuous for each ˆ Notice that B ρ as a Banach space bundle over Kˆ do not depend on ρ. For clarity we s ∈ K. denote the product and ∗-operation in (1) and (2) by fs ∗w gt and fs∗w . For any fs ∈ Bs and gt ∈ Bt , clearly the maps w → fs ∗w gt and w → fs∗w are both continuous. This leads to the next lemma, which is a slight generalization of [5, Proposition 3.3, Theorem 3.5]. The proof of [5, Proposition 3.3, Theorem 3.5], which in turn follows the lines of [29], is easily seen to hold also in our case. Lemma 2.8. Let H be a discrete group and Ah be a vector space for each h ∈ H . Let W be a locally compact Hausdorff space and for each w ∈ W assign norms and algebra operations making Aw = {Ah }h∈H into a Fell bundle in such a way that for any fs ∈ As and gt ∈ At the map w → fs w ∈ R is continuous (then we have a continuous field of Banach spaces (As , · w )w∈W over W for each s ∈ H ) and the sections w → fs ∗w gt ∈ Bst and w → fs∗w ∈ Bs −1 are continuous in the above continuous fields of Banach spaces (Bst , · w )w∈W and (B s −1 , · w )w∈W respectively. Then the map w → f w is upper semi-continuous for each f ∈ s∈H As , where · w is the norm on the enveloping C ∗ -algebra C ∗ (Aw ) and extends the norm of As as part of Aw for each s ∈ H . Moreover, if H is amenable, then {C ∗ (Aw )}w∈W is a continuous field ∗ -algebras with the field structure determined by the continuous sections w → f for all of C f ∈ s∈H As . Since every discrete abelian group is amenable [27, page 14], from Proposition 2.7 we get Proposition 2.9. Fix G, Γ and K. Let W be a locally compact Hausdorff space and for each ˆ w ∈ W let ρw satisfy (S1) and (S2) such that the map w → ρw (s) is continuous for each s ∈ K. ˆ ρw )}w∈W is a continuous field of C ∗ -algebras with the field structure determined Then {C ∗ (G/Γ, by the continuous sections w → f for all f ∈ s∈Kˆ Bs . 3. Derivations In this section we prove Proposition 3.3, to establish the relation between derivations coming from α and β. Throughout the rest of this paper, we assume: (S3) G/Γ is compact. (S4) G is a Lie group. The examples in Section 2 all satisfy these conditions. We refer the reader to [17, Section 1.3] for the discussion about differentiable maps into Fréchet spaces. We just recall that a continuous map ψ from a smooth manifold M into a Fréchet space A is continuously differentiable if for any chart (U, φ) of G, where U is an open subset


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of some Euclidean space Rn and φ is a diffeomorphism from U onto an open set of M, the derivative D(ψ ◦ φ)(x, h) = lim

Rν→0

ψ ◦ φ(x + νh) − ψ ◦ φ(x) ν

exists for all (x, h) ∈ (U, Rn ) and is a jointly continuous map from (U, Rn ) into A. In such case, D(ψ ◦ φ)(x, h) is linear on h, and depends only on ψ and the tangent vector u := φ∗ (vx,h ) of M at φ(x), where vx,h denotes the tangent vector h at x. Thus we may denote D(ψ ◦ φ)(x, h) by ∂u ψ . Then ∂u ψ is linear on u. Denote by g and k the Lie algebras of G and K respectively. For a strongly continuous action σ of G on a Banach space A as isometric automorphisms, we say that an element a ∈ A is once differentiable with respect to σ if the orbit map ψa from G into A sending x to σx (a) is continuously differentiable. Then the set A1 of once differentiable elements is a linear subspace of A. For any a ∈ A and any compactly supported smooth C-valued function ϕ on G, it is easily checked that G ϕ(x)σx (a) dx is in A1 . As a can be approximated by such elements, we see that A1 is dense in A. Thinking of g as the tangent space of G at the identity element, for each X ∈ g we have the linear map σX : A1 → A sending a to ∂X ψa . Fix a norm on g. We define a seminorm L on A1 by setting L(a) to be the norm of the linear map g → A sending X to σX a. Lemma 3.1. Let σ be a strongly continuous action of G on a Banach space A as isometric automorphisms. For any a ∈ A1 , one has σeX (a) − a . X 0=X∈g

L(a) = sup

Proof. The proof is similar to that of [31, Proposition 8.6]. Let X ∈ g with X = 1. One has σeνX (a) − a σeνX (a) − a lim = σX (a). + ν ν ν→0 ν>0

sup

For any ν > 0, one also has σ

ν ν

σezX σX (a) dz σezX σX (a) dz eνX (a) − a = 0

ν =

0

σX (a) dz = ν σX (a).

0

Therefore σeνX (a) − a = σX (a). ν ν>0

sup

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Thus σeX (a) − a σ νX (a) − a = sup sup e X ν 0=X∈g X∈g, X =1 ν>0 = sup σX (a) = L(a). sup

X∈g, X =1

2

Lemma 3.2. Let σ be a strongly continuous action of G on a Banach space A as isometric automorphisms. Then A1 is a Banach space with the norm p(a) := L(a) + a . Suppose that σ is a strongly continuous isometric action of a topological group H on A, commuting with σ . Then H preserves A1 , and the restriction of σ on A1 preserves the norm p and is strongly continuous with respect to p. Proof. Let {an }n∈N be a Cauchy sequence in A1 under the norm p. Then as n goes to infinity, an converges to some a ∈ A, and σX (an ) converge to some bX in A uniformly on X in bounded subsets of g. Let : [0, 1] → G be a continuously differentiable curve in G. Then (a )−σ (a ) σ limz→0 ν+z n z ν n = σν (σν (an )) for all ν ∈ [0, 1]. Thus ν σν (an ) − σ0 (an ) =

σz σz (an ) dz.

0

Letting n → ∞ we get ν σν (a) − σ0 (a) =

σz (bz ) dz. 0

σ (a)−σ (a)

Therefore limz→0 z z 0 = σ0 (b0 ). It follows easily that a ∈ A1 and σX (a) = bX for all X ∈ g. Consequently, an converges to a in A1 under the norm p, and hence A1 is a Banach space under the norm p. Clearly σ preserves A1 and the norm p. For any a ∈ A1 , the set of σX (a) for X in the unit ball of g is compact. Then for any h ∈ H and ε > 0, when h ∈ H is close enough to h, one has σh (a) − σh (a) < ε and σX (σh (a)) − σX (σh (a)) = σh (σX (a)) − σh (σX (a)) < ε for all X in the unit ball of g. Consequently, p(σh (a) − σh (a)) = L(σh (a) − σh (a)) + σh (a) − σh (a) < 2ε. Therefore the restriction of σ on A1 is strongly continuous with respect to p. 2 ˆ By Lemma 2.4 the actions α and β on C ∗ (G/Γ, ρ) commute with each other and are strongly 1 ˆ ˆ ρ) continuous. Denote by C (G/Γ, ρ) the space of once differentiable elements of C ∗ (G/Γ, with respect to the action α. Recall the Bs defined in (4). Proposition 3.3. Let X1 , . . . , Xn be a basis of g. For Y ∈ k say Adx (Y ) =

j

Fj,Y (x)Xj ,


2335

ˆ where Ad denotes the adjoint action of G on g. Then Fj,Y ∈ B0 . Any f ∈ C 1 (G/Γ, ρ) is once differentiable with respect to the action β and βY (f ) = −

Fj,Y ∗ αXj (f ).

(5)

j

ˆ commute Proof. Clearly Fj,Y is a smooth function on G. Since the subgroups Γ , K and ρ(K) with K, if y is in any of these subgroups, then Ady (Y ) = Y , and hence

Fj,Y (x)Xj = Adx (Y ) = Adx Ady (Y ) = Adxy (Y ) = Fj,Y (xy)Xj ,

j

j

which means that Fj,Y is invariant under the right translation of y. Thus Fj,Y ∈ C(G/KΓ ) = C0 (G/KΓ ) = B0 . For each X ∈ g denote by X # (X# resp.) the corresponding right (left resp.) translation invariant vector field on G. Then Y# = j Fj,Y Xj# . ˆ ˆ By Lemma 2.4 the norm on Bs ⊆ C ∗ (G/Γ, ˆ Let f ∈ C 1 (G/Γ, ρ) ∩ Bs for some s ∈ K. ρ) 1 is exactly the supremum norm. Thus f belongs to the space C (G) of continuously differentiable functions on G. For any continuous vector field Z on G denote by ∂Z the corresponding derivation map C 1 (G) → C(G). Then ∂Y# (f ) =

Fj,Y ∂X# (f ) = −

j

j

Fj,Y αXj (f ).

j

Since Fj,Y is invariant under the right translation of Γ and ρ(K), we have Fj,Y (x)gt (x) = ˆ Fj,Y ∗ gt (x) for any gt ∈ Bt and x ∈ G. By Lemma 2.4 the actions α and β on C ∗ (G/Γ, ρ) commute with each other. Thus α preserves B , and hence α (f ) ∈ B for every X ∈ g. Therefore s X s ∂Y# (f ) = − j Fj,Y ∗ αXj (f ). Let : [0, 1] → K be a continuously differentiable curve in K. Then

f (xν+z ) − f (xν ) lim = ∂(ν )# (f ) (xν ) = − Fj,ν ∗ αXj (f ) (xν ) z→0 z j

for all ν ∈ [0, 1] and x ∈ G, and hence we have the integral form f (xν ) − f (x0 ) =

ν − Fj,z ∗ αXj (f ) (xz ) dz 0

(6)

j

for all ν ∈ [0, 1] and x ∈ G. The left-hand νside of (6) is the value of βν (f ) − β0 (f ) at x, while the right-hand side of (6) is the value of 0 βz (− j Fj,z ∗ αXj (f )) dz at x, where the integral ˆ ρ). Therefore is taken in Bs ⊆ C ∗ (G/Γ, ν βν (f ) − β0 (f ) = 0

for all ν ∈ [0, 1].

βz − Fj,z ∗ αXj (f ) dz j

(7)

2336


ˆ ˆ Clearly (7) also holds for f ∈ s∈Kˆ (C 1 (G/Γ, ρ) ∩ Bs ). By Lemma 3.2 C 1 (G/Γ, ρ) is a 1 ˆ Banach space with norm p(·) = L(·) + · , β preserves C (G/Γ, ρ) and p, and the restriction ˆ ˆ of β on C 1 (G/Γ, ρ) is strongly continuous on C 1 (G/Γ, ρ) with respect to p. By Lemma 2.4 ∗ ˆ the spectral subspace of C (G/Γ, ρ) corresponding to s ∈ Kˆ for the action β is equal to Bs . ˆ It follows that the spectral subspace of C 1 (G/Γ, ρ) corresponding to s ∈ Kˆ for the restric1 (G/Γ, 1 (G/Γ, ˆ ˆ tion of β on C ρ) is exactly C ρ) ∩ Bs . Then standard techniques tell us that 1 (G/Γ, 1 (G/Γ, ˆ ˆ (C ρ) ∩ B ) is dense in C ρ) with respect to p. Notice that both sides s s∈Kˆ 1 ˆ ˆ ρ). Therefore (7) holds for all of (7) define continuous maps from C (G/Γ, ρ) to C ∗ (G/Γ, 1 ˆ f ∈ C (G/Γ, ρ). Consequently, βz (f ) − β0 (f ) lim = β0 − Fj, ∗ αXj (f ) 0 z→0 z j

ˆ ρ). It follows easily that f is once differentiable with respect to β and for all f ∈ C 1 (G/Γ, ˆ βY (f ) = − j Fj,Y ∗ αXj (f ) for all f ∈ C 1 (G/Γ, ρ) and Y ∈ k. 2 We shall need the following lemma (compare [34, Proposition 2.5]). Lemma 3.4. Let σ be a strongly continuous action of G on a Banach space A as isometric automorphisms. Let a ∈ A. Then for any ε > 0, there is some b ∈ A such that b is smooth with σ X (a)−a X (b) respect to σ , b a , b − a ε, and sup0=X∈g σ X sup0=X∈g e X . If A has an isometric involution being invariant under σ , then when a is self-adjoint, we can choose b also to be self-adjoint. Proof. Endow G with a left-invariant Haar measure. Let U be a small open neighborhood of the identity element in G with compact closure, which we shall determine later. Let ϕ be a nonnegative smooth function on G with support contained in U such that G ϕ(x) dx = 1. Set b = G ϕ(x)σx (a) dx. Then b is smooth with respect to σ , and b a . When U is small enough, we have a − b ε/2. For any X ∈ g, setting ψ(x) = Adx −1 (X), we have

σ X (b) − b = ϕ(x) σ X (a) − σx (a) dx e e x G

= ϕ(x)σx σeψ(x) (a) − a dx

G

ϕ(x)σx σeψ(x) (a) − a dx

G

sup σeψ(x) (a) − a x∈U

σeY (a) − a · sup ψ(x). Y x∈U 0=Y ∈g

sup


2337

Set δ = ε/(2 + 2 a ). When U is small enough, we have Adx −1 (X) (1 + δ) X for all X ∈ g and x ∈ U . Then σeX (b) − b (1 + δ) X sup0=Y ∈g we get

σeY (a)−a Y

for all X ∈ g. By Lemma 3.1

σX (b) σeX (b) − b σeX (a) − a = sup (1 + δ) sup . X X 0=X∈g X 0=X∈g 0=X∈g sup

Now it is clear that b = b/(1 + δ) satisfies the requirement. Note that b is self-adjoint if a is so. 2 4. Nondeformed case In this section we consider the nondeformed case, i.e., the case ρ is the trivial homomorphism ρ0 sending the whole Kˆ to the identity element of G. In Proposition 4.2 we identify Lρ0 ˆ on C 1 (G/Γ, ρ0 ) with the Lipschitz seminorm for certain metric on G/Γ . Note that C0,1 (G/Γ, ρ0 ) is sub-∗-algebra of C0 (G/Γ ) = C(G/Γ ). By the universality ˆ ˆ of C ∗ (G/Γ, ρ0 ) we have a natural ∗-homomorphism ψ of C ∗ (G/Γ, ρ0 ) into C(G/Γ ), extending the inclusion C0,1 (G/Γ, ρ0 ) → C(G/Γ ). The right translation of K on G induces a strongly continuous action β of K on C(G/Γ ), and clearly ψ intertwines β and β . An application of Lemmas 2.6 and 2.4 tells us that ψ is injective. By definition Bs is the spectral subspace ˆ Thus of C(G/Γ ) corresponding to s ∈ K. B is dense in C(G/Γ ). As s ˆ s∈K s∈Kˆ Bs is in the image of ψ, we see that ψ is surjective and hence is an isomorphism. We shall identify ˆ C ∗ (G/Γ, ρ0 ) and C(G/Γ ) via ψ . ˆ ρ0 ). If it corresponds to The seminorm Lρ0 describes the size of derivatives of f ∈ C 1 (G/Γ, some metric on G/Γ , this metric should be kind of geodesic distance. In order for the geodesic distance to be defined, throughout the rest of this paper we assume: (S5) G/Γ is connected. The examples in Section 2 all satisfy this condition. Fix an inner product on g. Then we obtain a right translation invariant Riemannian metric on G in the usual way. Denote by dG the geodesic distance on connected components of G. We extend dG to a semi-distance on G via setting dG (x, y) = ∞ if x and y lie in different connected components of G. Lemma 4.1. The function d on G/Γ × G/Γ defined by d(xΓ, yΓ ) := infx ∈xΓ, y ∈yΓ dG (x , y ) is equal to infy ∈yΓ dG (x, y ). It is a metric on G/Γ and induces the quotient topology on G/Γ . Proof. Let V be a connected component of G. Then V Γ is clopen in G, and hence V Γ /Γ is clopen in G/Γ for the quotient topology. As G/Γ is connected, we conclude that V Γ /Γ = G/Γ . Therefore d is finite valued. Since dG is right translation invariant, we have infx ∈xΓ, y ∈yΓ dG (x , y ) = infy ∈yΓ dG (x, y ). It follows easily that d is a metric on G/Γ . Let x ∈ G. Let W be a neighborhood of xΓ in G/Γ for the quotient topology. Then there exists ε > 0 such that if dG (x, y) < ε , then yΓ ∈ W . It follows that if d(xΓ, yΓ ) < ε, then yΓ ∈ W . Therefore the topology induced by d on G/Γ is finer than the quotient topology. For

2338


any ε > 0, set U = {y ∈ G | dG (x, y) < ε }. Then U is an open neighborhood of x. Thus U Γ /Γ is an open neighborhood of xΓ for the quotient topology. For any zΓ ∈ U Γ /Γ , we can find z ∈ zΓ ∩ U and hence d(xΓ, zΓ ) dG (x, z ) < ε . Therefore the quotient topology on G/Γ is finer than the topology induced by d. We conclude that d induces the quotient topology. 2 ˆ ˆ Proposition 4.2. For any f ∈ C 1 (G/Γ, ρ0 ) ⊆ C ∗ (G/Γ, ρ0 ) = C(G/Γ ), we have Lρ0 (f ) = sup

xΓ =yΓ

|f (xΓ ) − f (yΓ )| . d(xΓ, yΓ )

Proof. The right-hand side of the above equation is equal to supx=y show Lρ0 (f ) = sup x=y

|f (x)−f (y)| dG (x,y) .

So it suffices to

|f (x) − f (y)| . dG (x, y)

(8)

The proof is similar to that of [31, Proposition 8.6]. Let : [0, 1] → G be a continuously differentiable curve. Denote by () the length of . Then (f ◦ ) (ν) = (α− Adν (ν ) f )(ν ) for all ν ∈ [0, 1], and hence 1 1 f (1 ) − f (0 ) = (f ◦ ) (ν) dν (f ◦ ) (ν) dν 0

1 =

0

ν (ν ) f (ν ) dν

α− Ad

0

1 αAdν (ν ) f dν 0

1 Lρ0 (f )

Ad dν = Lρ (f )(), ν ν 0

0

where in the last equality we use the fact that the Riemannian metric on G is right translation invariant. It follows easily that |f (1 ) − f (0 )| Lρ0 (f )() holds if is only piecewise continuously differentiable. Considering all piecewise continuously differentiable curves connecting x and y we obtain |f (x) − f (y)| Lρ0 (f )dG (x, y) for all x, y ∈ G. Denote by eG the identity element of G. For any 0 = X ∈ g, we have sup

x=y

|f (x) − f (y)| |f (x) − f (eνX x)| sup sup dG (x, y) dG (x, eνX x) x ν=0 = sup sup

|f (x) − f (eνX x)| dG (eG , eνX )

sup sup

|f (x) − f (eνX x)| |ν| X

ν=0 x

ν=0 x

f − αe−νX f |ν| X ν=0

= sup


2339

αeνX f − f αX (f ) . |ν| X X ν=0

= sup Therefore supx=y

|f (x)−f (y)| dG (x,y)

Lρ0 (f ). This proves (8).

2

5. Lip-norms and compact group actions In this section we recall the definition of compact quantum metric spaces and prove Theorem 5.2, which enables one to show that certain seminorm defines a quantum metric, via the help of a compact group action. Rieffel has set up the theory of compact quantum metric spaces in the general framework of order-unit spaces [31, Definition 2.1]. We shall need it only for C ∗ -algebras. By a C ∗ -algebraic compact quantum metric space we mean a pair (A, L) consisting of a unital C ∗ -algebra A and a (possibly +∞-valued) seminorm L on A satisfying the reality condition

L(a) = L a ∗

(9)

for all a ∈ A, such that L vanishes exactly on C and the metric dL on the state space S(A) defined by dL (ψ, φ) = sup ψ(a) − φ(a)

(10)

L(a)1

induces the weak∗ -topology. The radius of (A, L), denote by rA , is defined to be the radius of (S(A), dL ). We say that L is a Lip-norm. Let A be a unital C ∗ -algebra and let L be a (possibly +∞-valued) seminorm on A vanishing on C. Then L and · induce (semi)norms L˜ and · ∼ respectively on the quotient space A˜ = A/C. Recall that a character of a compact group is the trace function of a finite-dimensional complex representation of the group [7, Section II.4]. Lemma 5.1. Let H be a compact group and H0 be a closed normal subgroup of H of finite index. Then for any linear combination of finitely many characters of H , its multiplication with the characteristic function of H0 is also a linear combination of finitely many characters of H . Proof. The products and sums of characters of H are still characters [7, Proposition II.4.10]. Thus it suffices to show that the characteristic function of H0 on H is a linear combination of finitely many characters of H . Since H /H0 is finite, every C-valued class function on H /H0 , i.e., functions being constant on conjugate classes, is a linear combination of characters of H /H0 [16, Proposition 2.30]. Thus the characteristic function of {eH /H0 } on H /H0 , where eH /H0 denotes the identity element of H /H0 , is a linear combination of characters of H /H0 . Then the characteristic function H0 on H is a linear combination of characters of H . 2 Recall that a length function on a topological group H is a continuous R0 -valued function, , on H such that (h) = 0 if and only if h is equal to the identity element eH of H , that (h1 h2 ) (h1 ) + (h2 ) for all h1 , h2 ∈ H , and that (h−1 ) = (h) for all h ∈ H .

2340


Suppose that a compact group H has a strongly continuous action σ on a Banach space A as isometric automorphisms. Endow H with its normalized Haar measure. For any continuous C-valued function ϕ on H , define a linear map σϕ : A → A by σϕ (a) =

ϕ(h)σh (a) dh H

for a ∈ A. Denote by Hˆ the set of isomorphism classes of irreducible representations of H . For each s ∈ Hˆ , denote by As the spectral subspace of A corresponding to s. For a finite subset J of Hˆ , set AJ = s∈J As . The main tool we use for the proof of Theorem 1.1 will be the following slight generalization of [23, Theorem 4.1]. Theorem 5.2. Let A be a unital C ∗ -algebra, let L be a (possibly +∞-valued) seminorm on A satisfying the reality condition (9), and let σ be a strongly continuous action of a compact group H on A by automorphisms. Assume that L takes finite values on a dense subspace of A, and that L vanishes on C. Suppose that the following conditions are satisfied: (1) there are some length function on a closed normal subgroup H0 of H of finite index and some constant C > 0 such that L C · L on A, where L is the (possibly +∞-valued) seminorm on A defined by σh (a) − a L (a) = sup h ∈ H0 , h = eH ; (h)

(11)

(2) for any linear combination ϕ of finitely many characters on H we have L ◦ σϕ ϕ 1 · L on A, where ϕ 1 denotes the L1 norm of ϕ; (3) for each s ∈ Hˆ not being the trivial representation s0 of H , the set {a ∈ As | L(a) 1, a r} is totally bounded for some r > 0, and the only element in As vanishing under L is 0; (4) there is a unital C ∗ -algebra A containing the fixed-point subalgebra Aσ , with a Lipnorm LA , such that LA extends the restriction of L to Aσ ; (5) for each s ∈ H /H0 ⊆ Hˆ not equal to s0 , there exists some constant Cs > 0 such that · Cs L on As . Then (A, L) is a C ∗ -algebraic compact quantum metric space with rA C H0 (h) dh + Cs (dim(s))2 + rA , where H0 is endowed with its normalized Haar measure. s =s∈H /H 0

0

We need some preparation for the proof of Theorem 5.2. The following lemma generalizes [23, Lemma 3.4]. Lemma 5.3. Let H be a compact group, and let H0 be a closed normal subgroup of H of finite index. Let f be a continuous C-valued function on H with f (eH ) = 0. Then for any ε > 0 there is a nonnegative function ϕ on H with support contained in H0 such that ϕ is a linear combination of finitely many characters of H , ϕ 1 = 1, and ϕ · f 1 < .


2341

Proof. Denote by χ the characteristic function of H0 on H . Set g = f χ + ε(1 − χ). Then g ∈ C(H ) and g(eH ) = 0. By [23, Lemma 3.4] we can find a nonnegative function φ on H such that φ is a linear combination of finitely many characters, φ 1 = 1, and φ · g 1 < ε/2. Then ε H \H0 φ(h) dh φ · g 1 < ε/2, and hence χφ 1 = φ 1 −

φ(h) dh > 1 − 1/2 = 1/2.

H \H0

Set ϕ = χφ/ χφ 1 . By Lemma 5.1 ϕ is a linear combination of finitely many characters of H . One has ϕ · f 1 = χφf 1 / χφ 1 = χφg 1 / χφ 1 < (ε/2)/(1/2) = ε.

2

For a compact group H and a finite subset J of Hˆ , set J¯ = {¯s | s ∈ J }, where s¯ denotes the contragradient representation. Replacing [23, Lemma 3.4] by Lemma 5.3 in the proof of [23, Lemma 4.4], we get Lemma 5.4. Let H be a compact group. For any ε > 0 there is a finite subset J = J¯ in Hˆ , containing the trivial representation s0 , depending only on and ε/C, such that for any strongly continuous isometric action σ of H on a complex Banach space A with a (possibly +∞-valued) seminorm L on A satisfying conditions (1) and (2) in Theorem 5.2, and any a ∈ A, there is some a ∈ AJ with a a ,

L(a ) L(a),

and a − a εL(a).

If A has an isometric involution being invariant under σ , then when a is self-adjoint we can choose a also to be self-adjoint. We are ready to prove Theorem 5.2. Proof of Theorem 5.2. Most part of the proof of [23, Theorem 4.1] carries over here. In fact, conditions (2)–(4) here are the same as the conditions (2)–(4) in [23, Theorem 4.1]. Since the proof of Lemma 4.5 in [23] does not involve condition (1) there, this lemma still holds in our current situation. Replacing [23, Lemma 4.4] by Lemma 5.4 in the proof of Lemma 4.6 of [23], we see that the latter also holds in our current situation. To finish the proof of Theorem 5.2, we only need to prove the following analogue of Lemma 4.7 of [23]: Lemma 5.5. We have · ∼ C (h) dh +

H0

s0 =s∈H /H0

2 Cs dim(s) + rA L∼

˜ sa , where H0 is endowed with its normalized Haar measure. on (A)

2342


Proof. By Lemma 5.1 the characteristic function ϕ of H0 on H is a linear combination of characters of H . Set n = |H /H0 |. Let a ∈ Asa . Then σnϕ (a) belongs to Asa and is fixed by σ |H0 . We have a − σnϕ (a) = a dh − σh (a) dh a − σh (a) dh H0

L (a)

H0

H0

(h) dh C · L(a)

H0

(h) dh,

H0

where the last inequality comes from the condition (1). By the condition (2) we have

L σnϕ (a) nϕ 1 · L(a) = L(a). A . Say, σnϕ (a) = s∈H a with as ∈ As . For each s ∈ H /H0 , Note that Aσ |H0 = s∈H /H0 s /H0 s denote by χs the corresponding character of H /H0 , thought of as a character of H . Then as = σdim(s)χs (σnϕ (a)) [23, Lemma 3.2]. Thus

2 L(as ) = L σdim(s)χs σnϕ (a) dim(s)χs 1 L σnϕ (a) dim(s) L(a), where the first inequality comes from the condition (2). Note that as0 ∈ Asa . By the condition (5) we have

2 as Cs L(as ) Cs dim(s) L(a) for each s ∈ H /H0 not equal to s0 . By the condition (4), we have b ∼ rA L∼ (b) for all b ∈ (As0 )sa = (Aσ )sa [30, Proposition 1.6, Theorem 1.9], [23, Proposition 2.11]. Thus as0 ∼ rA L∼ (as0 ) = rA L(as0 ) rA L(a). Therefore we have a ∼ a − σnϕ (a) + as0 ∼ +

(h) dh + rA L(a) +

2

This finishes the proof of Theorem 5.2.

s0 =s∈H /H0

H0

as desired.

as

s0 =s∈H /H0

C · L(a)

2

2 Cs dim(s) L(a)


2343

6. Proof of Theorem 1.1 In this section we prove Theorem 1.1. Denote by K0 the connected component of K containing the identity element eK . Take an inner product on k and use it to get a translation invariant Riemannian metric on K in the usual way. For each x ∈ K0 set (x) to be the geodesic distance from eK to x. Then is a length function on K0 . In order to prove Theorem 1.1, we just need to verify the conditions in Theorem 5.2 for ˆ (A, L, H, H0 , σ ) = (C ∗ (G/Γ, ρ), Lρ , K, K0 , β). Recall that we are given a norm on g, and Lρ (a) =

sup0=X∈g ∞

αX (a) X

ˆ if a ∈ C 1 (G/Γ, ρ); otherwise,

(12)

ˆ ρ). for a ∈ C ∗ (G/Γ, ˆ By Lemma 2.4 the actions α and β on C ∗ (G/Γ, ρ) commute with each other. Thus β preˆ ρ) and Lρ . serves C 1 (G/Γ, Choose the basis X1 , . . . , Xdim(G) of g in Proposition 3.3 to be of norm 1. Denote by C1 the supremum of Fj,Y for all 1 j dim(G) and Y in the unit sphere of k (with respect to the inner product on k above) in Proposition 3.3. ˆ ρ). Lemma 6.1. We have L (dim(G)C1 ) · Lρ on C ∗ (G/Γ, ˆ Proof. It suffices to show L (dim(G)C1 ) · Lρ on C 1 (G/Γ, ρ). By Proposition 3.3 every 1 ˆ ρ) is once differentiable with respect to the action β. By [31, Proposition 8.6] a ∈ C (G/Γ, we have L (a) = supY ∈k, Y =1 βY (a) . Then from (5) in Proposition 3.3 we get L (a) (dim(G)C1 )Lρ (a). 2 Lemma 6.2. For any linear combination ϕ of finitely many characters of K we have Lρ ◦ βϕ ˆ ϕ 1 · Lρ on C ∗ (G/Γ, ρ). Proof. We have remarked above that β preserves Lρ . By Lemma 3.1 one has αeX (a) − a X 0=X∈g

Lρ (a) = sup

(13)

ˆ ˆ for every a ∈ C 1 (G/Γ, ρ). It follows that Lρ is lower semi-continuous on C 1 (G/Γ, ρ) equipped ˆ ˆ ρ) ⊆ C ∗ (G/Γ, ρ). By Lemma 3.2 the action β is also with the relative topology from C 1 (G/Γ, ˆ strongly continuous on C 1 (G/Γ, ρ) with respect to the norm defined in Lemma 3.2. Then βψ is 1 ˆ also well defined on C (G/Γ, ρ) for any continuous C-valued function ψ on K. By [23, Remark 4.2.(3)] we get Lemma 6.2. 2 ˆ ρ), Lρ , K, The conditions (1) and (2) in Theorem 5.2 for (A, L, H, H0 , σ ) = (C ∗ (G/Γ, K0 , β) follow from Lemmas 6.1 and 6.2 respectively. ˆ ρ) defined by (12) Fix an inner product on g, and denote by Lρ the seminorm on C ∗ (G/Γ, but using this inner product norm instead. Since g is finite dimensional, any two norms on g

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are equivalent. Therefore there exists some constant C2 > 0 not depending on ρ such that Lρ C2 Lρ . ˆ By Lemma 4.1 and Proposition 4.2 the restriction of Lρ0 on C 1 (G/Γ, ρ0 ) ⊆ C(G/Γ ) is the Lipschitz seminorm associated to some metric d on G/Γ . The Arzela–Ascoli theorem [12, ˆ ρ0 ) | Lρ0 (a) r1 , a r2 } is totally Theorem VI.3.8] tells us that the set {a ∈ C ∗ (G/Γ, bounded for any r1 , r2 > 0. Since for each s ∈ Kˆ neither the seminorm Lρ nor the C ∗ -norm ˆ on Bs ⊆ C ∗ (G/Γ, ρ) depends on ρ, the condition (3) in Theorem 5.2 for (A, L, H, H0 , σ ) = ∗ ˆ (C (G/Γ, ρ), Lρ , K, K0 , β) follows. From the criterion of Lip-norms in [30, Proposition 1.6, Theorem 1.9] (see also [23, Proposition 2.11]) one sees that the Lipschitz seminorm associated to the metric on any compact metric space is a Lip-norm on the C ∗ -algebra of continuous functions on this space. Since ˆ ρ0 ) = C(G/Γ ) is no less than the Lipschitz seminorm associated to the Lρ0 on C ∗ (G/Γ, metric d on G/Γ , from [30, Proposition 1.6, Theorem 1.9] one concludes that Lρ0 is also a Lip-norm on C(G/Γ ). Therefore we may take (A, LA ) in condition (4) of Theorem 5.2 to be ˆ ρ), Lρ , K, K0 , β). (C(G/Γ ), Lρ0 ) for (A, L, H, H0 , σ ) = (C ∗ (G/Γ, Let s ∈ Kˆ not being the trivial representation of K, and let a ∈ Bs . Then Lρ0 (a) C2 Lρ0 (a) = C2 Lρ (a). Thus for any λ in the range of a on G/Γ one has a −λ1C(G/Γ ) C(G/Γ ) C2 C3 Lρ (a), where C3 denotes the diameter of G/Γ under the metric d. We have a C ∗ (G/Γ,ρ) = a C(G/Γ ) = k, sβk (a − λ1C(G/Γ ) ) dk ˆ K

C(G/Γ )

a − λ1C(G/Γ ) C(G/Γ ) C2 C3 Lρ (a). ˆ ρ), Lρ , This establishes the condition (5) of Theorem 5.2 for (A, L, H, H0 , σ ) = (C ∗ (G/Γ, K, K0 , β). ˆ ρ), We have shown that the conditions in Theorem 5.2 hold for (A, L, H, H0 , σ ) = (C ∗ (G/Γ, Lρ , K, K0 , β). Thus Theorem 1.1 follows from Theorem 5.2. 7. Quantum Gromov–Hausdorff distance In this section we prove Theorem 1.2. We recall first the definition of the distance distnu from [19, Section 5]. To simplify the notation, for fixed unital C ∗ -algebras A1 and A2 , when we take infimum over unital C ∗ -algebras B containing both A1 and A2 , we mean to take infimum over all unital isometric ∗-homomorphisms of A1 and A2 into some unital C ∗ -algebra B. Denote by distB H the Hausdorff distance between subsets of B. For a C ∗ -algebraic compact quantum metric spaces (A, LA ), set E(A) = a ∈ Asa LA (a) 1 . For any C ∗ -algebraic compact quantum metric spaces (A1 , LA1 ) and (A2 , LA2 ), the distance distnu (A1 , A2 ) is defined as

distnu (A1 , A2 ) = inf distB H E(A1 ), E(A2 ) , where the infimum is taken over all unital C ∗ -algebras B containing A1 and A2 .


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Throughout the rest of this section, we fix G, Γ , K such that there exists ρ satisfying the conditions (S1)–(S5). We also fix a norm on g. Denote by Ω the set of all ρ satisfying the conditions (S1) and (S2), equipped with the weakest topology making the maps Ω → G sending ρ to ρ(s) ˆ to be continuous for each s ∈ K. Every closed subgroup of a Lie group is also a Lie group [37, Theorem 3.42]. Thus K is a compact abelian Lie group. Then K is the product of a torus and a finite abelian group [7, Corolˆ lary 3.7]. Therefore Kˆ is finitely generated. Let s1 , . . . , sn be a finite subset of Kˆ generating K. Then the map ϕ : Ω → nj=1 G sending ρ to (ρ(s1 ), . . . , ρ(sn )) is injective, and its image is closed. Furthermore, it is easily checked that the topology on Ω is exactly the pullback of the relative topology of ϕ(Ω) in nj=1 G. Since G is a Lie group, it is locally compact metrizable. Thus nj=1 G and Ω are also locally compact metrizable. ˆ For clarity and convenience, we shall denote the actions α and β on C ∗ (G/Γ, ρ) by αρ and βρ ∗ ∗ ˆ respectively, and denote the C -norm on C (G/Γ, ρ) by · ρ . Consider the (possibly +∞ˆ valued) auxiliary seminorm Lρ on C ∗ (G/Γ, ρ) defined by Lρ (a) = sup

αρ,eX (a) − a ρ X

0=X∈g

.

Lemma 7.1. Let W be a locally compact Hausdorff space with a continuous map W → Ω sending w to ρw . Let f be a continuous section of the continuous field of C ∗ -algebras over W in Proposition 2.9. Then the function w → Lρw (fw ) is lower semi-continuous on W . Proof. Let w ∈ W . To show that the above function is lower semi-continuous at w , we consider the case Lρw (fw ) < ∞. The case Lρw (fw ) = ∞ can be dealt with similarly. Let ε > 0. Take 0 = X ∈ g such that Lρw (fw ) X < αρw ,eX (fw ) − fw ρ + ε X . w

It is easily checked that w → αρw ,eX (fw ) is also a continuous section of the continuous field. Then when w is close enough to w , we have α

ρw ,eX (fw ) − fw ρw

< αρw ,eX (fw ) − fw ρ + ε X w

and hence

Lρw (fw ) X < αρw ,eX (fw ) − fw ρ + 2ε X Lρw (fw ) + 2ε X . w

Therefore Lρw (fw ) Lρw (fw ) + 2ε.

2

Note that although the ∗-algebra structure of C0,1 (G/Γ, ρ) (Cb,1 (G, ρ) resp.) depends on ρ, the Banach space structure, the left translation action of G and the right translation action of K on C0,1 (G/Γ, ρ) (Cb,1 (G, ρ) resp.) do not depend on ρ. Thus we may denote by C0,1 (G/Γ ), 1 (G/Γ ) the set of α and β this Banach space and these actions respectively. Also denote by C0,1 once differentiable elements of C0,1 (G/Γ ) with respect to α. Lemma 7.2. For any a in

s∈Kˆ (Bs

1 (G/Γ )), the function ρ → L (a) is continuous on Ω. ∩ C0,1 ρ

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1 ˆ Proof. Say, a = s∈F as for some finite subset F of K and as ∈ Bs ∩C0,1 (G/Γ ) for each s ∈ F . ∈ Ω. Since α commutes with β, we Then Lρ (a) = supX∈g, X =1 s∈F αX (as ) ρ for each ρ have αX (as ) ∈ Bs . By Proposition 2.9 the function ρ → s∈F αX (as ) ρ is continuous on Ω for each X ∈ g. Since g is a finite-dimensional vector space and αX (as ) depends on X linearly, it follows easily that the function (X, ρ) → s∈F αX (as ) ρ is continuous on g× Ω. As the unit sphere of g is compact, one concludes that the function ρ → supX∈g, X =1 s∈F αX (as ) ρ is continuous on Ω. 2 Fix ρ ∈ Ω. Let Z be a compact neighborhood of ρ in Ω. ˆ ρ) for a in some Bs and f ∈ C(Z) is Note that the linear span of ρ → f (ρ)a ∈ C ∗ (G/Γ, dense in the C ∗ -algebra of continuous sections of the continuous field over Z in Proposition 2.9. Since Z is a compact metrizable space, C(Z) is separable. As G is a Lie group, it is separable. Then G/Γ is separable, and hence is a compact metrizable space. Thus C(G/Γ ) is separable, ˆ On the other hand, since Kˆ is finitely generated, Kˆ is and hence Bs is separable for each s ∈ K. countable. Therefore the C ∗ -algebra of continuous sections of the continuous field over Z in Proposition 2.9 is separable. ˆ ρ) is nuclear. Every separable continuous field of unital By Proposition 2.5 each C ∗ (G/Γ, nuclear C ∗ -algebras over a compact metric space can be subtrivialized [6, Theorem 3.2]. Thus we ˆ ρ) → B for all ρ ∈ Z such that, can find a unital C ∗ -algebra B and unital embeddings C ∗ (G/Γ, ˆ ρ) with its image in B, the continuous sections of the continuous via identifying each C ∗ (G/Γ, field over Z in Proposition 2.9 are exactly the continuous maps Z → B whose images at each ρ ˆ ρ). are in C ∗ (G/Γ, For any C ∗ -algebraic compact quantum metric space (A, LA ) and any constant R no less than the radius of (A, LA ), the set DR (A) := {a ∈ Asa | LA (a) 1, a R} is totally bounded and every a ∈ E(A) can be written as x + λ for some x ∈ DR (A) and λ ∈ R [30, Proposition 1.6, Theorem 1.9]. In Section 6 we have seen that the conditions in Theorem 5.2 hold for ˆ ρ), Lρ , K, K0 , β) with some C, Cs and (A, LA ) not depending (A, L, H, H0 , σ ) = (C ∗ (G/Γ, ˆ ρρ ), Lρ ) on ρ. Thus, by Theorem 5.2 there is some constant R such that the radius of (C ∗ (G/Γ, is no bigger than R for all ρ ∈ Ω. For any ε > 0, by Lemmas 5.4 and 2.4 there is a fiˆ ρ)) there is some nite subset F ⊆ Kˆ satisfying that for any ρ ∈ Ω and any x ∈ E(C ∗ (G/Γ, ∗ ˆ y ∈ E(C (G/Γ, ρ)) ∩ s∈F Bs with y ρ x ρ and x − y ρ < ε. Lemma 7.3. Let ε > 0. Then there is a neighborhood U of ρ in Z such that for any ρ ∈ U and ˆ ˆ ρ )) there is some b ∈ E(C ∗ (G/Γ, ρ)) with a − b B < ε. any a ∈ E(C ∗ (G/Γ, ˆ Proof. According to the discussion above we can find a finite subset Y of E(C ∗ (G/Γ, ρ )) ∩ ∗ ˆ s∈F Bs such that for every a ∈ E(C (G/Γ, ρ )) there are some z ∈ Y and λ ∈ R with a − (z + λ) ρ < ε. For each y ∈ Y , write y as s∈F ys with ys ∈ Bs . Since Lρ (y) < ∞, y is once differentiable with respect to αρ . It is easy to see that each ys is once differentiable with respect to αρ . Thus, by Lemma 7.2 the function ρ → Lρ (y) is continuous on Ω. Then we can find a constant δ > 0 and a neighborhood U of ρ in Z such that δ yρ ρ < ε, yρ − yρ B < ε, and ˆ Lρ (yρ ) < 1 + δ for all y ∈ Y and ρ ∈ U , where yρ denotes y as an element in C ∗ (G/Γ, ρ). Fix ρ ∈ U . Set b = zρ /(1 + δ). Then Lρ (b + λ) = Lρ (b) < 1, and a − (b + λ) a − (zρ + λ) + zρ − zρ B + zρ − b ρ B ρ < ε + ε + ε = 3ε.

2


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Lemma 7.4. Let ε > 0. Then there is a neighborhood U of ρ in Z such that for any ρ ∈ U and ˆ ˆ ρ)) there is some b ∈ E(C ∗ (G/Γ, ρ )) with a − b B < ε. any a ∈ E(C ∗ (G/Γ, Proof. According to the discussion before Lemma 7.3, it suffices to show that there is a neighˆ ρ)) ∩ s∈F Bs satisfying borhood U of ρ in Z such that for any ρ ∈ U and any a ∈ E(C ∗ (G/Γ, ˆ ρ )) with a − b B < ε. Suppose that this fails. Then we a ρ R there is some b ∈ E(C ∗ (G/Γ, ˆ ρn ))∩ s∈F Bs satiscan find a sequence {ρn }n∈N in Z converging to ρ and an an ∈ E(C ∗ (G/Γ, ∗ ˆ fying an ρn R for each n ∈ N such that an − b B ε for all n ∈ N and b ∈ E(C (G/Γ, ρ )). Write an as s∈F an,s with an,s ∈ Bs . Then an,s = K k, sβρn ,k (an ) dk. Thus an,s ρn an ρn R and Lρn (an,s ) Lρn (an ) 1 by Lemma 6.2. Since the restriction of Lρ on Bs does not depend on ρ, and the set {a ∈ Bs | Lρ (a) 1, a R} is totally bounded, passing to a subsequence if necessary, we may assume that an,s converges to some as in Bs when n → ∞ for each s ∈ F . Set a = s∈F as . Then (an )ρn converges to aρ in B as n → ∞, where (an )ρn and ˆ ˆ aρ denote an and a as elements in C ∗ (G/Γ, ρn ) and C ∗ (G/Γ, ρ ) respectively. In particular, a is self-adjoint and a ρ limn→∞ an ρn R. By Lemma 3.1 we have Lρn (an ) = Lρn (an ) 1 for all n ∈ N. On the one-point compactification W = N ∪ {∞} of N, consider the continuous map W → Ω sending n ∈ N to ρn and ∞ to ρ . ˆ ˆ ρn ) for n ∈ N and f∞ = a ∈ C ∗ (G/Γ, ρ) Then the section f defined as fn = an ∈ C ∗ (G/Γ, is a continuous section of the continuous field on W in Proposition 2.9. Thus, by Lemma 7.1 we have Lρ (a) lim infn→∞ Lρn (an ) 1. By Lemma 3.4 we can find some self-adjoint ˆ b ∈ C 1 (G/Γ, ρ ) with b ρ a ρ R, b − a ρ ε/2, and Lρ (b) L (a) 1. Then ρ

ˆ b ∈ E(C ∗ (G/Γ, ρ )), and b − an B → b − a ρ ε/2

as n → ∞. Therefore, when n is large enough, we have b − an B < ε, contradicting our assumption. This finishes the proof of the lemma. 2 From Lemmas 7.3 and 7.4 we conclude that Theorem 1.2 holds. Acknowledgment I am grateful to Wei Wu for comments. Appendix A. Comparison of distnu and prox In this appendix we compare the distance distnu and the proximity Rieffel introduced in [35]. A (possibly +∞-valued) seminorm L on a unital (possibly incomplete) C ∗ -norm algebra A is called a C ∗ -metric [35, Definition 4.1] if (1) L is lower semi-continuous, satisfies the reality condition (9), and is strongly-Leibniz in the sense that L(ab) L(a) b + a L(b) for all a, b ∈ A, L(1A ) = 0, and L(a −1 ) a −1 2 L(a) for all a being invertible in A, ¯ (2) L extended to the completion A¯ of A by L(a) = ∞ for a ∈ A¯ \ A is a Lip-norm on A, ¯ (3) the algebra {a ∈ A | L(a) < ∞} is spectrally stable in A. In such case, the pair (A, L) is called a compact C ∗ -metric space.

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The seminorm Lρ in Theorem 1.1 may fail to be a C ∗ -metric since it may fail to be lower ˆ ρ) by Lemma 3.1. Thus its semi-continuous. However, it is lower semi-continuous on C 1 (G/Γ, ∗ ˆ restriction on the algebra of smooth elements in C (G/Γ, ρ) with respect to α is a C ∗ -metric. ˆ By [35, Proposition 3.2] its closure L¯ ρ is a C ∗ -metric on C ∗ (G/Γ, ρ). Lemma 3.4 tells us that αeX (a) − a L¯ ρ (a) = sup X 0=X∈g ˆ for all a ∈ C ∗ (G/Γ, ρ). In [35, Definition 5.6, Section 14] Rieffel introduced the notions of proximity prox(A, B) and complete proximity proxs (A, B) between two compact C ∗ -metric spaces (A, LA ) and (B, LB ). In general, one has proxs (A, B) prox(A, B). For each q ∈ N, denote by UCPq (A) the set of unital completely positive linear maps from the completion A¯ of A to Mq (C). Define proxq (A, B) as the infimum of the Hausdorff distance of UCPq (A) and UCPq (B) in q UCPq (A ⊕ B) under the metric dL , for L running through C ∗ -metrics L on A ⊕ B whose quoq tients on A and B agree with LA and LB on Asa and Bsa respectively. Here the metric dL is defined as q

dL (ϕ, ψ) =

sup ϕ(a, b) − ψ(a, b). L(a,b)1

Then proxs (A, B) is defined as supq proxq (A, B). Note that the definition of distnu extends to compact C ∗ -metric spaces (A, LA ) and (B, LB ) directly. Theorem A.1. For any compact C ∗ -metric spaces (A, LA ) and (B, LB ), one has distnu (A, B) proxs (A, B). Proof. The proof is similar to those of [24, Proposition 4.7] and [19, Theorem 3.7]. Let A be ¯ Set c = distA (E(A), E(B)). Let ε > 0. Define a semia unital C ∗ -algebra containing A¯ and B. H norm L on A ⊕ B by a − b . L(a, b) = max LA (a), LB (b), c+ε It was pointed in the proof of [24, Proposition 4.7] that L extended to A ⊕ B = A¯ ⊕ B¯ as in the condition (2) of the definition of C ∗ -metrics above is a Lip-norm, and that the quotients of L on A and B agree with LA and LB on Asa and Bsa respectively. It is readily checked that L satisfies the conditions (1) and (3) in the definition of C ∗ -metrics. Thus L is a C ∗ -metric on A ⊕ B. For any q ∈ N and ϕ ∈ UCPq (A), by Arveson’s extension theorem [8, Theorem 1.6.1] extend ϕ to ¯ For any (a, b) ∈ E(A ⊕ B) one has a φ in UCPq (A). Set ψ to be the restriction of φ on B. ϕ(a, b) − ψ(a, b) = ϕ(a) − ψ(b) = φ(a − b) a − b c + ε.


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Thus dL (ϕ, ψ) c + ε. Similarly, for any ψ ∈ UCPq (B), we can find some ϕ ∈ UCPq (A) with q dL (ϕ , ψ ) c + ε. Therefore proxq (A, B) c + ε. It follows that proxq (A, B) distnu (A, B), and hence proxs (A, B) distnu (A, B) as desired. 2 q

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Reduced free products of finite dimensional C ∗ -algebras Nikolay A. Ivanov Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada, K7L 3N6 Received 7 November 2006; accepted 17 July 2009 Available online 30 July 2009 Communicated by D. Voiculescu

Abstract We find a necessary and sufficient conditions for the simplicity and uniqueness of trace for reduced free products of finite families of finite dimensional C ∗ -algebras with specified traces on them. Published by Elsevier Inc. Keywords: C ∗ -algebra; Reduced free product; Trace; Simplicity

1. Introduction and definitions The notion of reduced free product of a family of C ∗ -algebras with specified states on them was introduced independently by Avitzour [3] and Voiculescu [26]. We will recall this notion and some of its properties here. Definition 1.1. The couple (A, φ), where A is a unital C ∗ -algebra and φ a state is called a C ∗ noncommutative probability space or C ∗ -NCPS. Definition 1.2. Let (A, φ) be a C ∗ -NCPS and {Ai | i ∈ I } be a family of C ∗ -subalgebras of A, s.t. 1A ∈ Ai , ∀i ∈ I , where I is an index set. We say that the family {Ai | i ∈ I } is free if φ(a1 . . . an ) = 0, whenever aj ∈ Aij with i1 = i2 = · · · = in and φ(aj ) = 0, ∀j ∈ {1, . . . , n}. A family of subsets {Si | i ∈ I } ⊂ A is ∗-free if {C ∗ (Si ∪ {1A }) | i ∈ I } is free. E-mail address: [email protected]. 0022-1236/$ – see front matter Published by Elsevier Inc. doi:10.1016/j.jfa.2009.07.012

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N.A. Ivanov / Journal of Functional Analysis 257 (2009) 2351–2377

Let {(Ai , φi ) | i ∈ I } be a family of C ∗ -NCPS such that the GNS representations of Ai assodef

ciated to φi are all faithful. Then there is a unique C ∗ -NCPS (A, φ) = ∗i∈I (Ai , φi ) with unital embeddings Ai → A, s.t. (1) (2) (3) (4)

φ|Ai = φi ; the family {Ai | i ∈ I } is free in (A, φ); A is the C ∗ -algebra generated by i∈I Ai ; the GNS representation of A associated to φ is faithful.

And also: (5) if φi are all traces then φ is a trace too [26]; (6) if φi are all faithful then φ is faithful too [9]. In the above situation A is called the reduced free product algebra and φ is called the free product state. Also the construction of the reduced free product is based on defining a free product Hilbert space, which turns out to be HA – the GNS Hilbert space for A, associated to φ. One can define von Neumann algebra free products, similarly to reduced free products of C ∗ -algebras. Example 1.3. If {Gi | i ∈ I } is a family of discrete groups and Cr∗ (Gi ) are the reduced group C ∗ algebras, corresponding to the left regular representations of Gi on l 2 (Gi ) respectively, and if τi are the canonical traces on Cr∗ (Gi ), i ∈ I , then we have ∗i∈I (Cr∗ (Gi ), τi ) = (Cr∗ (∗i∈I Gi ), τ ), where τ is the canonical trace on the group C ∗ -algebra Cr∗ (∗i∈I Gi ). Reduced free products satisfy the following property: Lemma 1.4. (See [12].) Let I be an index set and let (Ai , φi ) be a C ∗ -NCPS (i ∈ I ), where each φi is faithful. Let (B, ψ) be a C ∗ -NCPS with ψ faithful. Let (A, φ) = ∗ (Ai , φi ). i∈I

Given unital ∗-homomorphisms, πi : Ai → B, such that ψ ◦ πi = φi and {πi (Ai )}i∈I is free in (B, ψ), there is a ∗-homomorphism, π : A → B such that π|Ai = π and ψ ◦ π = φ. From now on we will be concerned only with C ∗ -algebras equipped with tracial states. The study of simplicity and uniqueness of trace for reduced free products of C ∗ -algebras, one can say, started with the paper of Powers [21]. In this paper Powers proved that the reduced C ∗ algebra of the free group on two generators F2 is simple and has a unique trace – the canonical one. In [6] Choi showed the same for the “Choi algebra” Cr∗ (Z2 ∗ Z3 ) and then Paschke and Salinas in [20] generalized the result to the case of Cr∗ (G1 ∗ G2 ), where G1 , G2 are discrete groups, such that G1 has at least two and G2 at least three elements. After that Avitzour in [3] gave a sufficient condition for simplicity and uniqueness of trace for reduced free products of C ∗ -algebras, generalizing the previous results. He proved:


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Theorem 1.5. (See [3].) Let (A, τ ) = (A, τA ) ∗ (B, τB ), where τA and τB are traces and (A, τA ) and (B, τB ) have faithful GNS representations. Suppose that there are unitaries u, v ∈ A and w ∈ B, such that τA (u) = τA (v) = τA (u∗ v) = 0 and τB (w) = 0. Then A is simple and has a unique trace τ . Note. It is clear that uw satisfies τ ((uw)n ) = 0, ∀n ∈ Z\{0}. Unitaries with this property we define below. 2. Statement of the main result and preliminaries

p0

We adopt the following notation: If A0 , . . . , An are unital C ∗ -algebras equipped with traces τ0 , . . . , τn respectively, then A = p1

pn

α1

αn

A0 ⊕ A1 ⊕ · · · ⊕ An will mean that the C ∗ -algebra A is isomorphic to the direct sum of A0 , . . . , α0

An , and is such that Ai are supported on the projections pi . Also A comes with a trace (let’s call it τ ) given by the formula τ = α0 τ0 + α1 τ1 + · · · + αn τn . Here of course α0 , α1 , . . . , αn > 0 and α0 + α1 + · · · + αn = 1. Definition 2.1. If (A, τ ) is a C ∗ -NCPS and u ∈ A is a unitary with τ (un ) = 0, ∀n ∈ Z\{0}, then we call u a Haar unitary. If 1A ∈ B ⊂ A is a unital abelian C ∗ -subalgebra of A we call B a diffuse abelian C ∗ subalgebra of A if τ |B is given by an atomless measure on the spectrum of B. We also call B a unital diffuse abelian C ∗ -algebra. From [13, Proposition 4.1(i), Proposition 4.3] we can conclude the following: Proposition 2.2. If (B, τ ) is a C ∗ -NCPS with B-abelian, then B is diffuse abelian if and only if B contains a Haar unitary. p

1−p

q

1−q

α

1−α

β

1−β

C ∗ -algebras of the form (C ⊕ C ) ∗ (C ⊕ C ) have been described explicitly in [2] (see also [11]): Theorem 2.3. Let 1 > α β

1 2

and let

p 1−p q 1−q (A, τ ) = C ⊕ C ∗ C ⊕ C . α

1−α

β

1−β

If α > β then A=

p∧(1−q)

C

α−β

⊕ C [a, b], M2 (C) ⊕

for some 0 < a < b < 1. Furthermore, in the above picture

p∧q

C ,

α+β−1

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0 ⊕ 1, 0 √ t t (1 − t) √ q =0⊕ ⊕ 1, t (1 − t) 1−t p=1⊕

1 0

and the faithful trace τ is given by the indicated weights on the projections p ∧ (1 − q) and p ∧ q, together with an atomless measure, whose support is [a, b]. If α = β > 12 then

A = f : [0, b] → M2 (C) f is continuous and f (0) is diagonal ⊕

p∧q

C ,

α+β−1

for some 0 < b < 1. Furthermore, in the above picture

1 0 ⊕ 1, 0 0 √ t t (1 − t) √ q= ⊕ 1, t (1 − t) 1−t p=

and the faithful trace τ is given by the indicated weight on the projection p ∧ q, together with an atomless measure on [0, b]. If α = β = 12 then

A = f : [0, 1] → M2 (C) f is continuous and f (0) and f (1) are diagonal . Furthermore in the above picture

1 0 , 0 0 √ t t (1 − t) √ q= , t (1 − t) 1−t p=

and the faithful trace τ is given by an atomless measure, whose support is [0, 1]. The question of describing the reduced free product of a finite family of finite dimensional abelian C ∗ -algebras was studied by Dykema in [10]. He proved the following theorem: Theorem 2.4. (See [10].) Let p0 p1 pn q0 q1 qm (A, φ) = A0 ⊕ C ⊕ · · · ⊕ C ∗ B0 ⊕ C ⊕ · · · ⊕ C , α0

α1

αn

β0

β1

βm

where α0 0 and β0 0 and A0 and B0 are equipped with traces φ(p0 )−1 φ|A0 , φ(q0 )−1 φ|B0 and A0 and B0 have diffuse abelian C ∗ -subalgebras, and where n 1, m 1 (if α0 = 0 or β0 = 0, or both, then, of course, we don’t impose any conditions on A0 or B0 , or both respectively). Suppose also that dim(A) 2, dim(B) 2, and dim(A) + dim(B) 5.


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Then r0

A = A0 ⊕

pi ∧qj

C

(i,j )∈L+

,

αi +βi −1

where L+ = {(i, j ) | 1 i n, 1 j m and αi + βj > 1}, and where A0 has a unital, diffuse abelian sublagebra supported on r0 pi and another one supported on r0 qj for each i = 1, . . . , n and each j = 1, . . . , m. Let L0 = {(i, j ) | 1 i n, 1 j m and αi + βj = 1}. If L0 is empty then A0 is simple and φ(r0 )−1 φ|A0 is the unique trace on A0 . If L0 is not empty, then for each (i, j ) ∈ L0 there is a ∗-homomorphism π(i,j ) : A0 → C such that π(i,j ) (r0 pi ) = 1 = π(i,j ) (r0 qj ). Then: def (1) A00 = (i,j )∈L0 ker(π(i,j ) ) is simple and nonunital, and φ(r0 )−1 φ|A00 is the unique trace on A00 .

(2) For each i ∈ {1, . . . , n}, r0 pi is full in A0 ∩ (i ,j )∈L0 ker(π(i ,j ) ).

i =i (3) For each j ∈ {1, . . . , m}, r0 qj is full in A0 ∩ (i,j )∈L0 ker(π(i,j ) ). j =j

We will denote by Mn the C ∗ -algebra (von Neumann algebra) of n × n matrices with complex coefficients. Dykema studied the case of von Neumann algebra free products of finite dimensional (von Neumann) algebras: Theorem 2.5. (See [7].) Let p0

p1

pk

α0

α1

αk

q0

q1

ql

β0

β1

βl

A = L(Fs ) ⊕ Mn1 ⊕ · · · ⊕ Mnk and B = L(Fr ) ⊕ Mm1 ⊕ · · · ⊕ Mml , where L(Fs ), L(Fr ) are interpolated free group factors, α0 , β0 0, and where dim(A) 2, dim(B) 2 and dim(A) + dim(B) 5. Then for the von Neumann algebra free product we have: A ∗ B = L(Ft ) ⊕

fij

MN (i,j ) ,

(i,j )∈L+

where L+ = {(i, j ) | 1 i k, 1 j l, ( α2i ) + ( ni

N(i, j )2 · ( α2i + ni

βj m2j

− 1), and fij pi ∧ qj .

βj ) m2j

γij

> 1}, N (i, j ) = max(ni , mj ), γij =

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Note. t can be determined from the other data, which makes sense only if the interpolated free group factors are all different. We will use only the fact that L(Ft ) is a factor. For definitions and properties of interpolated free group factors see [22] and [8]. In this paper we will extend the result of Theorem 2.4 to the case of reduced free products of finite dimensional C ∗ -algebras with specified traces on them. This resolves [10, Conjecture 7.3] in the tracial case. We will prove: Theorem 2.6. Let p0 p1 pk q0 q1 ql (A, φ) = A0 ⊕ Mn1 ⊕ · · · ⊕ Mnk ∗ B0 ⊕ Mm1 ⊕ · · · ⊕ Mml , α0

α1

β0

αk

β1

βl

where α0 , β0 0, αi > 0, for i = 1, . . . , k and βj > 0, for j = 1, . . . , l, and where φ(p0 )−1 φ|A0 and φ(q0 )−1 φ|B0 are traces on A0 and B0 respectively,. Suppose that dim(A) 2, dim(B) 2, dim(A) + dim(B) 5, and that both A0 and B0 contain unital, diffuse abelian C ∗ -subalgebras (if α0 > 0, respectively β0 > 0). Then f

A = A0 ⊕ γ

αi n2i

where L+ = {(i, j ) |

+

βj m2j

fij

MN (i,j ) ,

(i,j )∈L+

γij

> 1}, N (i, j ) = max(ni , mj ), γij = N (i, j )2 ( α2i + C ∗ -subalgebra

ni

βj m2j

− 1), fij

of A0 , supported on fp1 and another pi ∧ qj . There is a unital, diffuse abelian one, supported on f q1 . β If L0 = {(i, j ) | α2i + j2 = 1}, is empty, then A0 is simple with a unique trace. If L0 ni

mj

is not empty, then ∀(i, j ) ∈ L0 , ∃π(i,j ) : A0 → MN (i,j ) a unital ∗-homomorphism, such that π(i,j ) (fpi ) = π(i,j ) (f qj ) = 1. Then: def (i,j )∈L0

ker(π(i,j ) ) is simple and nonunital, and has a unique trace φ(f )−1 φ|A00 .

(2) For each i ∈ {1, . . . , k}, fpi is full in A0 ∩ (i ,j )∈L0 ker(π(i ,j ) ).

i =i (3) For each j ∈ {1, . . . , l}, f qj is full in A0 ∩ (i,j )∈L0 ker(π(i,j ) ). (1) A00 =

j =j

The way to prove the theorem is as follows: First we study the special case p1 pm C ⊕ · · · ⊕ C ∗ (Mn , trn ). α1

αm

For this we use the fact that Mn ∼ = (C ⊕ · · · ⊕ C) Zn and Lemma 3.1 below, which is a wellknown result. Then the question is reduced to that of deciding the simplicity and the uniqueness of trace of a certain cross product C ∗ -algebra, which is done by using few results on cross product C ∗ -algebras and the results from [10]. Then we proceed by a kind of induction and study C ∗ -algebras of the form


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p0 pk p1 p1 pl (M, τ ) = A0 ⊕ Mm1 ⊕ · · · ⊕ Mmk ⊕ C ⊕ · · · ⊕ C ∗ (Mn , trn ), α0

α1

αk

α1

αl

using the knowledge about the C ∗ -algebra p0 p1 pk p1 pl (N , τ |N ) = C ⊕ C ⊕ · · · ⊕ C ⊕ C ⊕ · · · ⊕ C ∗ (Mn , trn ) α0

α1

αk

α1

αl

and the techniques developed in [10]. Then we move to the general case, again using induction and the techniques from [10]. I want to thank the reviewer for informing me that Lemma 3.1 is a well-known result and for suggesting to reorganize the paper. 3. Beginning of the proof – a special case In order to prove this theorem we will start with a simpler case. We will study first the C ∗ def p1

pm

α1

αm

algebras of the form (A, τ ) = ( C ⊕ · · · ⊕ C ) ∗ (Mn , trn ) with 0 < α1 · · · αm . We chose a set of matrix units for Mn and denote them by {eij | i, j ∈ {1, . . . , n}} as usual. Let’s take the (trace zero) permutation unitary ⎛

0 1 ... . . def ⎜ . u=⎝ 0 0 ... 1 0 ...

⎞ 0 .⎟ ⎠ ∈ Mn . 1 0

Then we have the well-known representation Mn ∼ = (Ce11 ⊕ · · · ⊕ Cenn ) Ad(u) Zn . Clearly A = C ∗ {p1 , . . . , pm }, {eii }ni=1 , u . We will need the following C ∗ -subalgebras of A: n−1 def B = C ∗ uk p1 u−k , . . . , uk pm u−k k=0 , {e11 , . . . , enn } and if l be an integer and l|n, 1 < l < n (if such l exists) then define E = C∗ def

k l−1 u p1 u−k , . . . , uk pm u−k k=0 , {e11 , . . . , enn }, ul , u2l , . . . , un−l .

Note that C ∗ {e11 , . . . , enn }, ul , u2l , . . . , un−l = M nl ⊕ · · · ⊕ M nl ⊂ Mn . l -times

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The k-th summand (1 k l) correspond to Span{eij | i, j = k (mod l)}. If w ∈ Mn is the unitary which is such that kl + t → kl + t + 1 for 0 t l − 2 and (k + 1)l − 1 → kl. Then clearly w l = 1 and also C ∗ {e11 , . . . , enn }, ul , u2l , . . . , un−l Ad(w) ∼ = Mn . We will adopt the following notation from [11]: Let (D, ϕ) be a C ∗ -NCPS and 1D ∈ D1 , . . . , Dk ⊂ D be a family of unital C ∗ -subalgebras def

of D, having a common unit 1D . We denote by D ◦ = {d ∈ D | ϕ(d) = 0}. We denote by Λ◦ (D1◦ , D2◦ , . . . , Dk◦ ) the set of all words of the form d1 d2 · · · dj and of nonzero length, where dt ∈ Di◦t , for some 1 it k and it = it+1 for any 1 t j − 1. The following lemma is well known. In [5] it was proved and used to show simplicity for certain free product C ∗ -algebras. Lemma 3.1. Consider a C ∗ -NCPS (W ∗ -NCPS) (D, ϕ) and consider a C ∗ -NCPS (W ∗ -NCPS) (F rAd(.) G, ψ), where (F, ψ ), with ψ = ψ|F , is a C ∗ -NCPS (W ∗ -NCPS) and where G is a countable discrete group of unitaries on the GNS Hilbert space of (F rAd(.) G, ψ). Assume that ϕ and ψ are faithful states. Assume that ψ is invariant under the action of G, i.e. ψ(Ad(g)(f )) = ψ(f ) for all g ∈ G and f ∈ F . Finally assume that ψ(gf ) = 0 for all g ∈ G\{id} and all f ∈ F . Then (D, ϕ) ∗ F rAd(.) G, ψ ∼ ∗ (Dg , ϕg ) ∗ (F, ψ ) rλ G, ω , = g∈G

where we have a fixed set of ∗-isomorphisms (normal ∗-isomorphisms) Ig : (Dg , ϕg ) → (D, ϕ) for each g ∈ G, λ(g )(dg ) = Ig−1 ◦ Ig (dg ) for all g, g ∈ G and all dg ∈ Dg , and λg |F = Ad(g) for g ∈ G. ω coinsides with the free product state on (∗g∈G (Dg , ϕg )) ∗ (F, ψ ) and ω(rg) = 0 for all g ∈ G\{id} and all r ∈ (∗g∈G (Dg , ϕg )) ∗ (F, ψ ). Finally assume that ω is invariant under λ. Since eii uk is an offdiagonal element (1 k n − 1) we apply this lemma to B and Ad(u). It follows that n−1 B = (C · e11 ⊕ · · · ⊕ C · enn ) ∗ ∗ C · uk p1 u−k ⊕ · · · ⊕ C · uk pm u−k k=0

n−1 ∼ = (C ⊕ · · · ⊕ C) ∗ ∗ ( C ⊕ · · · ⊕ C ) 1 n

1 n

k=0 α1

αm

and that A∼ = B Ad(u) Zn . Since also eij w k is an offdiagonal element for 1 k l and i = j (mod l) we can apply the lemma to E and to Ad(w). It follows that l−1 E = C ∗ e11 , . . . , enn , ul , u2l , . . . , un−l ∗ ∗ C · uk p1 u−k ⊕ · · · ⊕ C · uk pm u−k k=0

l−1 ∼ = (M nl ⊕ · · · ⊕ M nl ) ∗ ∗ ( C ⊕ · · · ⊕ C ) l n

l n

k=0 α1

αm


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and that A∼ = E Ad(w) Zl . To study simplicity in this situation, we can invoke [18, Theorem 4.2] and [19, Theorem 6.5], or with the same success, use the following result from [16]: Theorem 3.2. (See [16].) Let Γ be a discrete group of automorphisms of a C ∗ -algebra B. If B is simple and if each γ is outer for the multiplier algebra M(B) of B, ∀γ ∈ Γ \{1}, then the reduced crossed product of B by Γ , B Γ , is simple. An automorphism ω of a C ∗ -algebra B, contained in a C ∗ -algebra A is outer for A, if there doesn’t exist a unitary w ∈ A with the property ω = Ad(w). A representation π of a C ∗ -algebra A on a Hilbert space H is called nondegenerate if there doesn’t exist a vector ξ ∈ H, ξ = 0, such that π(A)ξ = 0. The idealizer of a C ∗ -algebra A in a C ∗ -algebra B (A ⊂ B) is the largest C ∗ -subalgebra of B in which A is an ideal. We will not give a definition of multiplier algebra of a C ∗ -algebra. Instead we will give the following property from [1], which we will use (see [1] for more details on multiplier algebras): Proposition 3.3. (See [1].) Each nondegenerate faithful representation π of a C ∗ -algebra A extends uniquely to a faithful representation of M(A), and π(M(A)) is the idealizer of π(A) in its weak closure. Suppose that we have a faithful representation π of a C ∗ algebra A on a Hilbert space H. If ¯ (in H) the weak closure of π(A) in B(H). confusion is impossible we will denote by A To study uniqueness of trace we invoke a theorem of Bédos from [4]. Let A be a simple, unital C ∗ -algebra with a unique trace ϕ and let (πA , HA , 1 A ) denote the GNS-triple associated to ϕ. The trace ϕ is faithful by the simplicity of A and A is isomorphic to πA (A). Let α ∈ Aut(A). The trace ϕ is α-invariant by the uniqueness of ϕ. Then α is implea ) = α(a) · 1 mented on HA by the unitary operator Uα given by Uα ( A , a ∈ A. Then we denote def ¯ the extension of α to the weak closure A (in HA ) of πA (A) on B(HA ) by α˜ = Ad(Uα ). We will ¯ say that α is ϕ-outer if α˜ is outer for A. Theorem 3.4. (See [4].) Suppose A is a simple unital C ∗ -algebra with a unique trace ϕ and that Γ is a discrete group with a representation α : Γ → Aut(A), such that αγ is ϕ-outer ∀γ ∈ Γ \{1}. Then the reduced crossed product A Γ is simple with a unique trace τ given by τ = ϕ ◦ E, where E is the canonical conditional expectation from A Γ onto A. p1

pm

α1

αm

Let’s now return to the C ∗ -algebra (A, τ ) = ( C ⊕ · · · ⊕ C ) ∗ (Mn , trn ), with α1 α2 · · · αm . If B ⊂ E ⊂ A are as in the beginning of this section, then the representations of B, E and A on HA are all nondegenerate. Since the trace on B is just τ |B and the trace on E is just τ |E it is easy to see that we have the following Lemma 3.5. The weak closure of B in B(HB ) and the one in B(HA ) are the same (which we will denote by B¯ (in HB ) ∼ = B¯ (in HA )). Analoguously, E¯ (in HE ) ∼ = E¯ (in HA ).

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We will state the following theorem from [10], which we will frequently use: Theorem 3.6. (See [10].) Let A and B be unital C ∗ -algebras with traces τA and τB respectively, whose GNS representations are faithful. Let (C, τ ) = (A, τA ) ∗ (B, τB ). Suppose that B = C and that A has a unital, diffuse abelian C ∗ -subalgebra D (1A ∈ D ⊆ A). Then C is simple with a unique trace τ . Using repeatedly Theorem 2.4 we see that B = (C · e11 ⊕ · · · ⊕ C · enn ) ∗

∼ = U⊕

p˜

C

max{nαm −n+1,0}

n−1 ∗ C · uk p1 u−k ⊕ · · · ⊕ C · uk pm u−k k=0

e11 enn ∗ C ⊕··· ⊕ C , 1 n

1 n

where U has a unital, diffuse abelian C ∗ -subalgebra, and where p˜ = We will consider the following 3 cases, for α1 α2 · · · αm : (I) αm < 1 − (II) αm = 1 − (III) αm > 1 −

n−1 i=0

ui pm u−i .

1 . n2 1 . n2 1 . n2

We will organize those cases in few lemmas: (I) Lemma 3.7. If A is as above, then for αm < 1 − n12 we have that A is simple with a unique trace. Proof. We consider: (1) αm 1 − n1 . e11 enn Then B ∼ = U ∗ ( C ⊕ · · · ⊕ C ) with U containing a unital, diffuse abelian C ∗ -subalgebra (from 1 n

1 n

Theorem 2.4). From Theorem 3.6 we see that B is simple with a unique trace. (2) 1 − n1 < αm < 1 − n12 . Then B ∼ = (U ⊕

p˜

C

nαm −n+1

e11

enn

1 n

1 n

) ∗ ( C ⊕ · · · ⊕ C ) with U having a unital, diffuse abelian C ∗ -

subalgebra. Using Theorem 2.4 one more time we see that B is simple with a unique trace in this case also. We know that A = B G, where G = Ad(u) ∼ = Zn . Since B is unital then the multiplier algebra M(B) coinsides with B. We note also that since B¯ (in HB ) is isomorphic to B¯ (in HA ) to prove that some element of Aut(B) is τB -outer it’s enough to prove that this automorphism is outer for B¯ (in HA ) (and it will be outer for M(B) = B also). Making these observations


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and using Theorems 3.2 and 3.4 we see that if we prove that Ad(ui ) is outer for B¯ (in HA ), ∀0 < i n − 1, then it will follow that A is simple with a unique trace. We will show that Ad(ui ) is outer for B¯ (in HA ) (we will write just ∗¯ for ∗¯ (in HA ) and omit writing HA – all the closures will be in B(HA )) for the case αm 1 − n12 . ¯ such / B¯ (in HA ). Suppose ∃w ∈ B, Fix 0 < k n − 1. Since uk HB ⊥ HB it follows that uk ∈ ¯ Then uk wu−k = www ∗ = w and uk w ∗ u−k = ww ∗ w ∗ = w ∗ and that Ad(uk ) = Ad(w) on B. this implies that uk , u−k , w and w ∗ commute, so it follows uk w ∗ commutes with C ∗ (B, uk ), so it belongs to its center. If k n then C ∗ (B, uk ) = A¯ and by Theorem 2.5 A¯ (in HA ) is a factor, ¯ If k = l | n, then C ∗ (B, uk ) = E¯ so uk w ∗ is a multiple of 1A , which contradicts the fact uk ∈ / B. ∼ and E¯ (in HA ) = E¯ (in HE ) is a factor too (by Theorem 2.5), so this implies again that uk w ∗ is a ¯ multiple of 1A = 1E , so this is a contradiction again and this proves that Ad(uk ) are outer for B, ∀0 < k n − 1. This concludes the proof. 2 (III) Lemma 3.8. If A is as above, then for αm > 1 − n12 we have A = A0 ⊕ a unique trace. Proof. In this case B ∼ = (U ⊕

p˜

C

Mn , n2 αm −n2 +1

e11

enn

1 n

1 n

where A0 is simple with

) ∗ ( C ⊕ · · · ⊕ C ), where U has a unital, diffuse

nαm −n+1

p˜ 0

abelian C ∗ -subalgebra. Form Theorem 2.4 we see that B ∼ =B0 ⊕

e11 ∧p˜

C

nαm −n+ n1

⊕··· ⊕

enn ∧p˜

C

with

nαm −n+ n1

p˜ 0 = 1 − e11 ∧ p˜ − · · · − enn ∧ p, ˜ and B0 being a unital, simple and having a unique trace. It’s easy to see that Ad(u) permutes {eii | 1 i n} and that Ad(u) permutes {ui pj u−i | 0 i n − 1} i −i we see that Ad(u)(p) for each 1 j m. But since p˜ = n−1 ˜ = p. ˜ This shows that i=0 u pm u Ad(u) permutes {eii ∧ p˜ | 1 i n}. This shows that Ad(p˜ 0 u) is an automorphism of B0 and e11 ∧p˜

enn ∧p˜

that Ad((1 − p˜ 0 )u) is an automorphism of C ⊕ · · · ⊕ C . If we denote G1 = Ad(p˜ 0 u) e11 ∧p˜

enn ∧p˜

and G2 = Ad((1 − p˜ 0 )u), then we have A = B0 G1 ⊕ ( C ⊕ · · · ⊕ C ) G2 . Now e11 ∧p˜

enn ∧p˜

it’s easy to see that ( C ⊕ · · · ⊕ C ) G2 = C ∗ ({e11 ∧ p, ˜ . . . , enn ∧ p}, ˜ (1 − p˜ 0 )u) = def ∗ ∼ (1− p˜ 0 ).C ({e11 , . . . , enn }, u) = Mn (because p˜ 0 is a central projection). To study A0 = B0 G1 we have to consider the automorphisms Ad(p˜ 0 u). From Lemma 3.5 we see that e ∧p˜

e ∧p˜

e ∧p˜

e ∧p˜

nn nn 11 11 B0 ⊕ C ⊕ · · · ⊕ C (in HB ) ∼ = B0 ⊕ C ⊕ · · · ⊕ C (in HA ).

This implies B¯ 0 (in HB0 ) ∼ = B¯ 0 (in HA0 ). This is because HA0 = p˜ 0 HA and HB0 = p˜ 0 HB (which is clear, since HA0 and HB0 are direct summands in HA and HB respectively). For some def

l|n if we denote E0 = p˜ 0 E then by the same reasoning as above E = E0 ⊕ (1 − p˜ 0 ).C ∗ {e11 , . . . , enn }, ul ∼ = E0 ⊕ (M nl ⊕ · · · ⊕ M nl ). l -times

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So we similarly have E¯0 (in HE0 ) ∼ = E¯0 (in HA0 ). We use Theorem 2.5 and see that A¯ ∼ = L(Ft ) ⊕ Mn and that E¯ ∼ = L(Ft ) ⊕ (M nl ⊕ · · · ⊕ M nl ), l -times

for some 1 < t, t < ∞. This shows that A¯0 and E¯0 are both factors. Now for Ad(p˜ 0 uk ), 1 k n − 1 we can make the same reasoning as in the case (I) to show that Ad(p˜ 0 uk ) are all outer for B0 , ∀1 k n − 1. Now we use Theorems 3.2 and 3.4 to finish the proof. Notice that the trace of the support projection of Mn , e11 ∧ p˜ + · · · + enn ∧ p, ˜ is n2 αm − n2 + 1. 2 (II) ¯ ∀1 k n − 1. Using Theorem 2.4 we We already proved that Ad(uk ) are outer for B, p˜

e

e

nn 11 see B ∼ = (U ⊕ C ) ∗ ( C ⊕ · · · ⊕ C ) with U having a unital, diffuse abelian C ∗ -subalgebra.

1− n1

1 n

1 n

There are ∗-homomorphisms πi : B → C, 1 i n with πi (p) ˜ = πi (eii ) = 1, and such that def n−1 B0 = i=0 ker(πi ) is simple with a unique trace. Now if 1 k n − 1, then B0 ∩ Ad(uk )(B0 ) = either 0 or B0 , because B0 and Ad(uk )(B0 ) are simple ideals in B. The first possibility is actually impossible, because of dimension reasons, so this shows that B0 is invariant for Ad(uk ), 1 k n − 1. In other words Ad(uk ) ∈ Aut(B0 ). Similarly as above it can be shown that

def A0 = C ∗ B0 ⊕ B0 u ⊕ · · · ⊕ B0 un−1 ∼ = B0 Ad uk 0 k n − 1 ⊂ A. Lemma 3.9. We have a short split-exact sequence:

0 → A0 → A → Mn → 0. Proof. It’s clear that we have the short exact sequence π

· · ⊕ C → 0, 0 → B0 → B −→ C ⊕ · n-times def

where π = (π1 , . . . , πn ). We think π to be a map from B to Diag(Mn ), defined by ⎛

π1 (b) ⎜ 0 π(b) = ⎝ . 0

0 π2 (b) . 0

⎞ ... 0 ... 0 ⎟ ⎠. . . . . . πn (b)

Now since πi (p) ˜ = πi (eii ) = 1 and Ad(u)(e11 ) = ue11 u∗ = enn and for 2 i n, Ad(u)(eii ) = ueii u∗ = e(i−1)(i−1) , then πi ◦ Ad(u)(e(i+1)(i+1) ) = πi ◦ Ad(u)(p) ˜ = 1 for 1 i n − 1 and πn ◦ Ad(u)(e11 ) = πn ◦ Ad(u)(p) ˜ = 1. So since two ∗-homomorphisms of a C ∗ -algebra, which coinside on a set of generators of the C ∗ -algebra, are identical, we have πi ◦ Ad(u) = πi+1 for 1 i n − 1 and πn ◦ Ad(u) = π1 . Define π˜ : A → Mn by


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n−1

n−1 k → k=0 π(bk )W (with bk ∈ B), where W ∈ Mn is represented by the matrix, which represent u ∈ Mn ⊂ A, namely k k=0 bk u

⎛

0 1 ... . . def ⎜ . W =⎝ 0 0 ... 1 0 ...

⎞ 0 .⎟ ⎠. 1 0

We will show that if b ∈ B and 0 k n − 1, then π(uk bu−k ) = W k π(b)W −k . For this it’s enough to show that π(ubu−1 ) = W π(b)W −1 . For the matrix units {Eij | 1 i, j n} we have as above W Eii W ∗ = E(i−1)(i−1) for 2 i n − 1 and W E11 W ∗ = Enn . So ⎞ π1 (b) 0 ... 0 0 ⎟ ∗ π2 (b) . . . ⎜ 0 W⎝ ⎠W . . . . 0 0 . . . πn (b) ⎞ ⎛ 0 ... 0 π2 (b) 0 ⎟ π3 (b) . . . ⎜ 0 =⎝ ⎠ . . . . 0 0 . . . π1 (b) ⎛ π1 (Ad(u)(b)) 0 0 π2 (Ad(u)(b)) ⎜ =⎝ . . 0 0 = π Ad(u)(b) , ⎛

... ... . ...

⎞ 0 0 ⎟ ⎠ . πn (Ad(u)(b))

just what we wanted. Now for b ∈ B and 0 k n − 1 we have ∗ π˜ buk = π˜ u−k b∗ = π˜ u−k b∗ uk u−k = π u−k b∗ uk W −k = W −k π(b∗ )W k W −k , ∗ ∗ = W −k π(b)∗ = π(b)W k = π˜ buk . Also if b, b ∈ B and 0 k, k n − 1, then π˜ b uk . buk = π˜ b uk bu−k uk+k = π b uk bu−k W k+k = π(b )π uk bu−k W k+k = π(b )W k π(b)W −k W k+k = π˜ b uk π˜ buk . This proves that that π˜ is a ∗-homomorphism. Continuity follows from continuity of π and i the representation A = n−1 i=0 Bu as a sum of closed subspaces of the Banach space A. From the construction of the map π˜ we see that π˜ (eii ) = Eii , since π(eii ) = Eii and also π˜ (uk ) = W k . Since {Eii | 1 i n} ∪ {W k | 0 k n − 1} generate Mn , then we have π˜ (eij ) = Eij , so the inclusion map s : Mn → A given by Eij → eij is a right inverse for π˜ .

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Clearly A0 is a closed subspace of A. By the definition of π˜ we have A0 ⊂ ker(π˜ ). Therefore A = A0 ⊕ Span{eij | i, j ∈ {1, . . . , n}} as a sum of closed subspaces of the Banach space A. Since π˜ is continuous and π(e ˜ ij ) = Eij = 0, then ker(π) ˜ = A0 . 2 From this lemma follows that A = A0 ⊕ Mn as a Banach space. ˜ defined by η(a ˜ 0 ⊕ M) = Lemma 3.10. If η is a trace on A0 , then the linear functional on A η, η(a0 ) + trn (M), where a0 ∈ A0 and M ∈ Mn is a trace and η˜ is the unique extension of η to a trace on A (of norm 1). Proof. The functional η can be extended in at most one way to a tracial state on A, because of the requirement η(1 ˜ A ) = 1, the fact that Mn sits as a subalgebra in A, and the uniqueness on trace on Mn . Since η(1 ˜ A ) = 1, to show that η˜ is a trace we need to show that η˜ is positive and satisfies the trace property. For the trace property: If x, y ∈ A then we need to show η(xy) ˜ = η(yx). ˜ It is easy to see, that to prove this it’s enough to prove that if a0 ∈ A0 and M ∈ Mn , then η(a0 M) = η(Ma0 ). Since η is linear and a0 is a linear combination of 4 1/2 1/2 positive elements we can think, without loss of generality, that a0 0. Then a0 = a0 a0 1/2 1/2 1/2 1/2 and Ma0 , a0 M ∈ A0 , so since η is a trace on A0 , we have η(Ma0 ) = η((Ma0 )a0 ) = 1/2 1/2 1/2 1/2 1/2 1/2 η(a0 (Ma0 )) = η((a0 M)a0 ) = η(a0 (a0 M)) = η(a0 M). This shows that η˜ satisfies the trace property. to show positivity. It remains a0 ⊕ M 0. We must show η(a0 ⊕ M) 0. Suppose Write M = ni=0 nj=0 mij eij and a0 = ni=0 nj=0 eii a0 ejj . Since η˜ is a trace if i = j , then η(e ˜ ii a0 ejj ) = η(e ˜ jj eii a0 ) = 0, so this shows that η(a ˜ 0 ⊕ M) = ni=0 ( mnii + η(eii a0 eii )). Clearly a0 ⊕ M 0 implies ∀1 i n, eii (a0 ⊕ M)eii 0. So to show positivity we only need to show ∀1 i n, η(e ˜ ii (a0 + M)eii ) 0, given ∀1 i n, mii eii + eii a0 eii 0. Suppose that for some i, mii < 0. Then it follows that eii a0 eii −mii eii , so eii a0 eii ∈ eii A0 eii is invertible, which implies eii ∈ A0 , that is not true. So this shows that mii 0, and mii eii −eii a0 eii . If {γ } is an approximate unit for A0 , then positivity of η implies 1 = η = limγ η(γ ). Since η is 1/2 1/2 1/2 1/2 a trace we have limγ η(γ eii ) = n1 . Since ∀γ , mii γ eii γ −γ eii a0 eii γ , then trn (mii eii ) =

mii = lim η(mii eii γ ) = lim η mii γ1/2 eii γ1/2 lim η γ1/2 eii a0 eii γ1/2 γ γ γ n

= lim η(eii a0 eii γ ) = η(eii a0 eii ). γ

This finishes the proof of positivity and the proof of the lemma.

2

Remark 3.11. We will show below that τ |A0 is the unique trace on A0 . Since we have A = A0 ⊕ Mn as a Banach space, then clearly the free product trace τ on A is given by τ (a0 ⊕ M) = τ |A0 (a0 ) + trn (M), where a0 ⊕ M ∈ A0 ⊕ Mn = A. All tracial positive linear functionals of norm 1 on A0 are of the form tτ |A0 , where 0 t 1. Then there will be no other traces on A then def

the family λt = tτ |A0 ⊕ trn . To show that these are traces indeed, we can use the above lemma (it is still true, no mater that the norm of tτA0 can be less than one), or we can represent them as a convex linear combination λt = tτ + (1 − t)μ of the free product trace τ and the trace μ, defined by μ(a0 ⊕ M) = trn (M) = trn (π˜ (a0 ⊕ M)).


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Lemma 3.12. B¯0 (in HA ) = B¯ (in HA ). def 1−p˜

p˜

e11

e22 +···+enn

def

) ⊂ B. Denote D0 = D ∩ B0 . From TheoProof. Let’s take D = ( C ⊕ C) ∗ ( C ⊕ C p∧(1−e ˜ 11 ) rem 2.3 follows that D ∼ = {f : [0, b] → M2 | f is continuous and f (0)-diagonal} ⊕ C , where 0 < b < 1 and τ |D is given by an atomless measure μ on {f : [0, b] → M2 | f is continuous and f (0)-diagonal}, p˜ is represented by 10 00 ⊕ 1, and e11 is represented by 1−t √t (1−t) √ ⊕ 0. A ∗-homomorphism, defined on the generators of a C ∗ -algebra can t (1−t) t be extended in at most one way to the whole C ∗ -algebra. This observation, together with π1 (e11 ) = π1 (p) ˜ = 1 and πi (e22 + · · · + enn ) = π(p) ˜ = 1 implies that π1 |D (f ⊕ c) = f11 (0) and πi |D (f ⊕ c) = c for 2 i n − 1. This means that D0 = {f : [0, b] → M2 | f is continuous and f11 (0) = f12 (0) = f21 (0) = 0} ⊕ 0. Now we see D¯0 (in HD ) ∼ = M2 ⊗ L∞ ([0, b], μ) ⊕ 0, so ¯ then e11 ∈ D0 (in HD ). So we can find sequence {εn } of self-adjoined elements (functions) of D0 , supported on e11 , weakly converging to e11 on HD and such that {εn2 } also converges weakly to e11 on HD . Then take a1 , a2 ∈ A. In HA we have a1 , (εn2 − e11 ) a2 = τ ((εn2 − e11 )a2 a1∗ ) = ∗ ∗ 2 τ ((εn − e11 )a2 a1 (εn − e11 )) 4a2 a1 τ (εn − e11 ). (The last inequality is obtained by representing a2 a1∗ as a linear combination of 4 positive elements and using Cauchy–Bounjakovsky– Schwartz inequality.) This shows that e11 ∈ D¯0 (in HA ) ⊂ B¯0 (in HA ). Analoguously eii ∈ B¯0 (in HA ), so this shows B¯0 = B¯ (in HA ). 2 It easily follows now that Corollary 3.13. A¯0 (in HA ) = A¯ (in HA ). The representation of B0 on HA is faithful and nondegenerate, and we can use Proposition 3.3, together with Theorem 3.2 and the fact that Ad(uk ) are outer for B¯ = B¯0 to get: Lemma 3.14. A0 = B0 G is simple. For the uniqueness of trace we need to modify a little the proof Theorem 3.4 (which is a special case of [4, Theorem 1]). Lemma 3.15. A0 = B0 G has a unique trace, τ |A0 . Proof. Above we already proved that {Ad(uk ) | 1 k n − 1} are τ |B0 -outer for B0 . Suppose that η is a trace on A0 . We will show that τ |A0 = η. We consider the GNS representation of B, associated to τ |B . By repeating the proof of Lemma 3.10 we see that B¯0 (in HB ) = B¯ (in HB ). The simplicity of B0 allows us to identify B0 with πτ |B (B0 ). We will also identify B0 with it’s canonical copy in A0 . A0 is generated by {b0 ∈ B0 } ∪ {uk | 0 k n − 1} and {Ad(uk ) | 0 k n − 1} extend to B¯0 (in HA ), so also to B¯0 (in HB ) (∼ = B¯ (in HA )). Now def ¯ k ˜ we can form the von Neumann algebra crossed product A = B0 {Ad(u ) | 0 k n − 1} ∼ = B¯ {Ad(uk ) | 0 k n − 1}, where the weak closures are in HB . Clearly A˜ ∼ = A¯ (in HA ). Denote by τ! B0 the extension of τ |B0 to B¯0 (in HA ), given by τ! B0 (x) = x(1 A ), 1 A HA . By [25, ¯ Chapter V, Proposition 3.19], τ! B0 is a faithful normal trace on B0 (in HA ). Now from the fact that B¯0 (in HA ) is a factor and using [17, Lemma 1] we get that τ! B0 is unique on B¯0 (in HA ). By

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∼ ¯ (in HA ) is unique, the same argument we have that the extension τ! A0 of τ |A0 to A¯0 (in HA ) = A ¯ since A¯0 (in HA ) ∼ A (in H ) is a factor. = A We take the unique extension of η to A. We will call it again η for convenience. We denote by HC the GNS Hilbert space for C, corresponding to η|C (for C = A, B, B0 , A0 ). Since η|B0 = τ |B0 it follows that B¯0 (in HB0 ) ∼ = B¯ (in HB ) and of course HB0 = HB . Then similarly as in Lemma 3.9 we get that A¯0 (in HA0 ) ∼ = A¯ (in HA ), so HA0 = HA (this can be def

˜ = done, since τ |B0 = η|B0 ). Now again by [25, Chapter V, Proposition 3.19] we have that η(x) x(1 A ), 1 A HA (1 A is abuse of notation – in this case it’s the element, corresponding to 1A in HA )

˜ π (B) is a faithful normal trace defines a faithful normal trace on πA (A) (in HA ). In particular η| A

on πA (B) (in HA ). By uniqueness of τ |B0 we have τ |B0 = η|B0 , so for b0 ∈ B0 we have τ˜ (b0 ) = ˜ A (b0 )). τ (b0 ) = η(b0 ) = πA (b0 )(1 A ), 1 A HA = η(π Since B0 is simple, it follows that πA |B0 is a ∗-isomorphism from B0 onto πA (B0 ) and from [14, Exercise 7.6.7] it follows that πA |B0 extends to a ∗-isomorphism from B¯0 (in HA ) ∼ = πA (B) (in HA ). We will denote this ∗-isomorphism = B¯ (in HA ) onto πA (B0 ) (in HA ) ∼ def

def

by θ . We set w = πA (u), β = θ Ad(u)θ −1 ∈ Aut(πA (B)) (in HA ). For b0 ∈ B0 we have wπA (b0 )w ∗ = πA (ub0 u∗ ) = πA ((Ad(u))(b0 )) = β(πA (b0 )). So by weak continuity follows β = Ad(w) on πA (B) (in HA ). Since B¯ (in HA ) is a factor and {Ad(uk ) | 1 k n − 1} are all outer, Kallman’s Theorem [15, Corollary 1.2] gives us that {Ad(uk ) | 1 k n − 1} act ¯ then freely on B¯ (in HA ). Namely if b¯ ∈ B¯ (in HA ), and if ∀b¯ ∈ B¯ (in HA ), b¯ b¯ = Ad(uk )(b¯ )b, b¯ = 0. Then by the above settings it is clear that {Ad(w k ) | 1 k n − 1} also act freely on πA (B) (in HA ). Since η˜ is a faithful normal trace on πA (A) (in HA ), then by [25, Chapter V, Proposition 2.36] there exists a faithful conditional expectation P : πA (A) → πA (B) (both weak closures are in HA ). ∀x ∈ πA (B) (in HA ), and ∀1 k n − 1, Ad(w k )(x)w k = w k x. Applying P we get Ad(w k )(x)(P (w k )) = P (w k )x, so by the free action of Ad(w k ) we get that P (w k ) = 0, ∀1 k n − 1. It’s clear that {πA (B)} ∪ {w k | 1 k n − 1} generates πA (A) (in HA ) as a von Neumann algebra. Now we use [24, Proposition 22.2]. It gives us a ∗-isomorphism Φ : πA (A) (in HA ) → B¯ {Ad(uk ) | 1 k n − 1} ∼ = A¯ (last two weak closures are in HA ) with Φ(θ (x)) = x, x ∈ B¯ (in HA ), Φ(w) = u. So since A¯ (in HA ) is a finite factor, so is πA (A) (in HA ), and so it’s trace η˜ is unique. Hence, η˜ = τ˜ ◦ Φ, and so ∀b ∈ B, and ∀1 k n − 1 we have η(buk ) = η(π ˜ A (b)πA (uk )) = τ˜ (Φ(πA (b))Φ(πA (uk ))) = τ˜ (Φ(θ (b))Φ(w k )) = τ˜ (buk ) = k τ (bu ). By continuity and linearity of both traces we get η = τ , just what we want. 2 We conclude this section by proving the following Proposition 3.16. Let def

(A, τ ) =

p1 pm C ⊕ · · · ⊕ C ∗ (Mn , trn ), α1

αm

where α1 α2 · · · αm . Then: (I) If αm < 1 −

1 , n2

then A is unital, simple with a unique trace τ .


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(II) If αm = 1 − n12 , then we have a short exact sequence 0 → A0 → A → Mn → 0, where A has no central projections, and A0 is nonunital, simple with a unique trace τ |A0 . 1−f Mn , where n2 −n2 αm n2 αm −n2 +1 trace (n2 − n2 αm )−1 τ |A0 .

(III) If αm > 1 − n12 , then A = simple and has a unique

f

A0

⊕

1 − f pm , and where A0 is unital,

Let f means the identity projection for cases (I) and (II). Then in all cases for each of the projections fp1 , . . . , fpm we have a unital, diffuse abelian C ∗ -subalgebra of A, supported on it. In all the cases pm is a full projection in A. Proof. We have to prove the second part of the proposition, since the first part follows from Lemmas 3.7–3.9, 3.14 and 3.15. From the discussion above we see that in all cases we have f A = f B {Ad(f uk f ) | 0 k n − 1}, where B and {Ad(f uk ) | 0 k n − 1} are as above. So the existence of the unital, diffuse abelian C ∗ -subalgebras follows from Theorem 2.4, applied to B. In the case (I) pm is clearly full, since A is simple. In the case (III) it’s easy to see that pm ∧ f = 0 and pm (1 − f ), so since A0 and Mn are simple in this case, then pm is full in A. In case (II) it follows from Theorem 2.4 that pm is full in B, and consequently in A. 2 4. The general case In this section we prove the general case of Theorem 2.6, using the result from the previous section (Proposition 3.16). The prove of the general case involves techniques from [10]. So we will need two technical results from there. The first one is [10, Proposition 2.8] (see also [7]): Proposition 4.1. Let A = A1 ⊕ A2 be a direct sum of unital C ∗ -algebras and let p = 1 ⊕ 0 ∈ A. def

Suppose φA is a state on A with 0 < α = φA (p) < 1. Let B be a unital C ∗ -algebra with a state φB and let (A, φ) = (A, φA ) ∗ (B, φB ). Let A1 be the C ∗ -subalgebra of A generated by (0 ⊕ A2 ) + Cp ⊆ A, together with B. In other words p 1−p (A1 , φ|A1 ) = C ⊕ A2 ∗ (B, φB ). α

1−α

Then pAp is generated by pA1 p and A1 , which are free in (pAp, α1 φ|pAp ). In other words 1

1

1

∼ ∗ A . φ φ pAp, φ

p, , pA

= 1 1 A

A1 α pAp α pA1 p α Remark 4.2. This proposition was proved for the case of von Neumann algebras in [7]. It is true also in the case of C ∗ -algebras. The second result is [10, Proposition 2.5(ii)], which is easy and we give its proof also: Proposition 4.3. Let A be a C ∗ -algebra. Take h ∈ A, h 0, and let B be the hereditary subalgebra hAh of A (∗ means norm closure). Suppose that B is full in A. Then if B has a unique trace, then A has at most one tracial state.

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Proof. It’s easy to see that Span{xhahy | a, x, y ∈ A} is norm dense in A. If τ is a tracial state on A then τ (xhahy) = τ (h1/2 ahyxh1/2 ). Since h1/2 ahyxh1/2 ∈ B, τ is uniquely determined by τB . 2 It is clear that Proposition 3.16 agrees with Theorem 2.6, so it is a special case. As a next step we look at a C ∗ -algebra of the form p0 pk p1 p1 pl (M, τ ) = A0 ⊕ Mm1 ⊕ · · · ⊕ Mmk ⊕ C ⊕ · · · ⊕ C ∗ (Mn , trn ), α0

α1

αk

α1

αl

where A0 comes with a specified trace and has a unital, diffuse abelian C ∗ -subalgebra with unit p0 . Also we suppose that α0 0, 0 < α1 · · · αk , 0 < α1 · · · αl , m1 , . . . , mk 2, and def

def

p1

pk

α1

αk

either α0 > 0 or k 1, or both. Let’s denote p0 = p0 + p1 + · · · + pk , B0 = Mm1 ⊕ · · · ⊕ Mmk , def

and α0 = α0 + α1 + · · · + αk = τ (p0 ). Let’s have a look at the C ∗ -subalgebras N and N of M given by p0 p1 pl (N, τ |N ) = C ⊕ C ⊕ · · · ⊕ C ∗ (Mn , trn ) α0

α1

αl

and p0 p1 pk p1 pl (N , τ |N ) = C ⊕ C ⊕ · · · ⊕ C ⊕ C ⊕ · · · ⊕ C ∗ (Mn , trn ).

α0

α1

αk

α1

αl

We studied the C ∗ -algebras, having the form of N and N in the previous section. A brief description is as follows: If α0 , αl < 1 − n12 , then N is simple with a unique trace and N is also simple with a unique trace. For each of the projections p0 , p1 , . . . , pk , p1 , . . . , pl we have a unital, diffuse abelian C ∗ -subalgebra of N , supported on it. If α0 , or αl = 1 − n12 , then N has no central projections, and we have a short exact sequence 0 → N0 → N → Mn → 0, with N0 being simple with a unique trace. Moreover p0 or pl respectively is full in N . For each of the projections p0 , p1 , . . . , pk , p1 , . . . , pl we have a unital, diffuse abelian C ∗ -subalgebra of N , supported on it. q

If α0 or αl > 1 − n12 , then N =N0 ⊕ Mn , with N0 being simple and having a unique trace. We consider 2 cases: (I) case: αl α0 . (1) αl < 1 − n12 . In this case N and N are simple and has unique traces, and p0 is full in N and consequently 1M = 1N is contained in p0 N – the ideal of N , generated by p0 . Since p0 N ⊂ p0 M it follows that p0 is full also in M. From Proposition 4.1 we get p0 Mp0 ∼ = (A0 ⊕ B0 ) ∗ p0 Np0 . Then from Theorem 3.6 follows that p0 Mp0 is simple and has a unique trace. Since p0 is a full projection, Proposition 4.3 tells us that M is simple and τ is its unique trace. For each of


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the projections p0 , p1 , . . . , pk , p1 , . . . , pl we have a unital, diffuse abelian C ∗ -subalgebra of M, supported on it, and coming from N . (2) αl = 1 − n12 . In this case it is also true that for each of the projections p0 , p1 , . . . , pk , p1 , . . . , pl we have a unital, diffuse abelian C ∗ -subalgebra of M, supported on it, and coming from N . It is easy to see that M is the linear span of p0 Mp0 , p0 M(1 − p0 )N (1 − p0 ), (1 − p0 )Np0 Mp0 , (1 − p0 )Np0 Mp0 N (1 − p0 ) and (1 − p0 )N (1 − p0 ). We know that we have a ∗-homomorphism π : N → Mn , such that π(pl ) = 1. Then it is clear that π(p0 ) = 0, so we can extend π to a linear map π˜ on M, defining it to equal 0 on p0 Mp0 , p0 M(1 − p0 )N (1 − p0 ), (1 − p0 )Np0 Mp0 and (1 − p0 )Np0 Mp0 N (1 − p0 ). It is also clear then that π˜ will actually be a ∗-homomorphism. Since ker(π) is simple in N and p0 ∈ ker(π), then p0 is full in ker(π) ⊂ N , so by the above representation of M as a linear span we see that p0 is full in ker(π) ˜ also. From Proposition 4.1 follows that p0 Mp0 ∼ = (A0 ⊕ B0 ) ∗ (p0 Np0 ). Since p0 Np0 has a unital, diffuse abelian C ∗ subalgebra with unit p0 , it follows from Theorem 3.6 that p0 Mp0 is simple and has a unique trace (to make this conclusion we could use Theorem 1.5 instead). Now since p0 Mp0 is full and hereditary in ker(π˜ ), from Proposition 4.3 follows that ker(π˜ ) is simple and has a unique trace. (3) αl > 1 − n12 . In this case N =

1−q

q

N0 ⊕

n2 −n2 αl

Mn

n2 αl −n2 +1

and also N =

q

N0 ⊕

1−q

Mn

n2 αl −n2 +1 qp0 , qp1 , . . . , qpk ,

n2 −n2 αl

with N0 and N0 being

qp1 , . . . , qpl we have simple with unique traces. For each of the projections a unital, diffuse abelian C ∗ -subalgebra of M, supported on it, and coming from N0 . Since p0 q we can write M as a linear span of p0 Mp0 , p0 Mp0 N0 (1 − p0 ), (1 − p0 )N0 p0 Mp0 , (1 − p0 )N0 p0 Mp0 N0 (1 − p0 ), (1 − p0 )N0 (1 − p0 ) and Mn . So we can q

write M = M0 ⊕ n2 −n2 αl

1−q Mn , n2 αl −n2 +1

def

where M0 = qMq ⊃ N0 . We know that p0 is full in N0 , so as

before we can write 1M0 = 1N0 ∈ p0 N0 ⊂ p0 M0 , so p0 M0 = M0 . Because of Proposition 4.1, we can write p0 M0 p0 ∼ = (A0 ⊕ B0 ) ∗ (p0 N0 p0 ). Since p0 N0 p0 has a unital, diffuse abelian C ∗ subalgebra with unit p0 , then from Theorem 3.9 (or from Theorem 1.5) it follows that p0 M0 p0 is simple with a unique trace. Since p0 M0 p0 is full and hereditary in M0 , Proposition 4.3 yields that M0 is simple with a unique trace. (II) α0 > αl . (1) α0 1 − n12 . In this case p0 is full in N and also in N , so 1M = 1N ∈ p0 N , which means p0 is full in M also. p0 Mp0 is a full hereditary C ∗ -subalgebra of M and p0 Mp0 ∼ = (A0 ⊕ B0 ) ∗ p0 Np0 by Proposition 4.1. Since p0 Np0 has a diffuse abelian C ∗ -subalgebra, Theorem 3.6 (or Theorem 1.5) shows that p0 Mp0 is simple with a unique trace and then by Proposition 4.3 follows that the same is true for M. For each of the projections p0 , p1 , . . . , pk , p1 , . . . , pl we have a unital, diffuse abelian C ∗ -subalgebra of M, supported on it, coming from N . (2) α0 > 1 − n12 . We have 3 cases: (2 ) α0 > 1 − n12 . q q In this case N ∼ = N0 ⊕ Mn and N ∼ = N0 ⊕ Mn , where q q , with N0 and N0 being simple and having unique traces. It is easy to see that p1 , . . . , pk , p1 , . . . , pl q , so for each of the projections p1 , . . . , pk , p1 , . . . , pl we have a unital, diffuse abelian C ∗ -subalgebra of N ,

2370


supported on it. So those C ∗ -subalgebras live in M also. We have a unital, diffuse abelian C ∗ subalgebra of A0 , supported on 1A0 , which yields a unital, diffuse abelian C ∗ -subalgebra on M, supported on p0 . It is clear that p0 is full in N , so as before, 1M = 1N ∈ p0 N , so p0 is full in M also, so p0 Mp0 is a full hereditary C ∗ -subalgebra of M. From Proposition 4.1 we have p0 Mp0 ∼ = (A0 ⊕ B0 ) ∗ (p0 N0 p0 ⊕ Mn ). It is easy to see that Mn , for n 2 contains two trn -orthogonal zero-trace unitaries. Since also p0 N0 p0 has a unital, diffuse abelian C ∗ -subalgebra, supported on 1N0 , it is easy to see (using Proposition 2.2) that it also contains two τ |N0 -orthogonal, zero-trace unitaries. Then the conditions of Theorem 1.5 are satisfied. This means that p0 Mp0 is simple with a unique trace and Proposition 4.3 implies that M is simple with a unique trace also. (2 ) αk > 1 − n12 . Let’s denote pk−1 p0 p1 pk p1 pl N = A0 ⊕ Mm1 ⊕ · · · ⊕ Mmk−1 ⊕ C ⊕ C ⊕ · · · ⊕ C ∗ (Mn , trn ).

α0

α1

αk

αk−1

α1

αl

q

∼ N ⊕ Mn . Clearly p , p , Then N satisfies the conditions of case (I)(3) and so N = 0 0 1 . . . , pk−1 , p1 , . . . , pl q, so for each of the projections p0 , p1 , . . . , pk−1 , p1 , . . . , pl we have a unital, diffuse abelian C ∗ -subalgebra of N0 , supported on it. Those C ∗ -algebras live in M also. From case (I)(3) we have that pk is full in N and as before 1M = 1N ∈ pk N implies that pk is full in M also. From Proposition 4.1 follows that pk Mpk ∼ = (pk N0 pk ⊕ Mn ) ∗ Mmk . Since N0 has a unital, diffuse abelian C ∗ -subalgebra, supported on qpk , then an argument, similar to the one we made in case (II)(2 ), allows to apply Theorem 1.5 to get that pk Mpk is simple with a unique trace. By Proposition 4.3 follows that the same is true for M. The unital, diffuse abelian C ∗ -subalgebra of M, supported on pk , we can get by applying the note after Theorem 1.5 to pk Mpk ∼ = (pk N0 pk ⊕ Mn ) ∗ Mmk . (2 ) α0 and αk 1 − q

1 . n2

In this case N ∼ = N0 ⊕ Mn , with N0 being simple and having a unique trace. Moreover N has no central projections and for each of the projections p0 , p1 , . . . , pk , p1 , . . . , pl we have a unital, diffuse abelian C ∗ -subalgebra of N , supported on it. So those C ∗ -subalgebras live in M also. It is clear that p0 is full in N , so as before 1M = 1N ∈ p0 N , so p0 is full in M also, so p0 Mp0 is a full hereditary C ∗ -subalgebra of M. From Proposition 4.1 we have p0 Mp0 ∼ = (A0 ⊕ B0 ) ∗ (p0 N0 p0 ⊕ Mn ). Since A0 and p0 N0 p0 both have unital, diffuse abelian C ∗ -subalgebras, supported on their units, it is easy to see (using Proposition 2.2), that the conditions of Theorem 1.5 are satisfied. This means that p0 Mp0 is simple with a unique trace and Proposition 4.3 yields that M is simple with a unique trace also. We summarize the discussion above in the following Proposition 4.4. Let p0 pk p1 p1 pl (M, τ ) = A0 ⊕ Mm1 ⊕ · · · ⊕ Mmk ⊕ C ⊕ · · · ⊕ C ∗ (Mn , trn ), def

α0

α1

αk

α1

αl


2371 p0

where n 2, α0 0, α1 α2 · · · αk , α1 · · · αl , m1 , . . . , mk 2, and A0 ⊕ 0 has a unital, diffuse abelian C ∗ -subalgebra, having p0 as a unit. Then: (I) If αl < 1 − n12 , then M is unital, simple with a unique trace τ . (II) If αl = 1 − n12 , then we have a short exact sequence 0 → M0 → M → Mn → 0, where M has no central projections and M0 is nonunital, simple with a unique trace τ |M0 . (III) If αl > 1 − n12 , then f

M = M0 ⊕ n2 −n2 αl

1−f Mn , n2 αl −n2 +1

where 1−f pl , and where M0 is unital, simple and has a unique trace (n2 −n2 αl )−1 τ |M0 . Let f means the identity projection for cases (I) and (II). Then in all cases for each of the projections fp0 , fp1 , . . . , fpk , fp1 , . . . , fpl we have a unital, diffuse abelian C ∗ -subalgebra of M, supported on it. In all the cases pl is a full projection in M. To prove Theorem 2.6 we will use Proposition 4.4. First let’s check that Proposition 4.4 agrees with the conclusion of Theorem 2.6. We can write p0 pk p1 q1 p1 pl def (M, τ ) = A0 ⊕ Mm1 ⊕ · · · ⊕ Mmk ⊕ C ⊕ · · · ⊕ C ∗ Mn , α0

α1

α1

αk

αl

β1

where q1 = 1M and β1 = 1. It is easy to see that L0 = {(l, 1) | 1α2l + n12 = 1} = {(l, 1) | αl = 1 − n12 }, which is not empty if and only if αl = 1 − n12 . Also L+ = {(l, 1) | 1α2l + n12 > 1} = {(l, 1) | αl > 1 − n12 }, and here L+ is not empty if and only if αl > 1 − n12 . If both L+ and L0 are empty, then M is simple with a unique trace. If L0 is not empty, then clearly L+ is empty, so we have no central projections and a short exact sequence 0 → M0 → M → Mn → 0, with M0 being simple with a unique trace. In this case all nontrivial projections are full in M. If L+ is not empty, then clearly L0 is empty and so 1−q

q

M = M0 ⊕ n2 −n2 αl

Mn

n2 (

αl + 1 −1) 12 n2

,

where M0 is simple with a unique trace. pl is full in M. Proof of Theorem 2.6. Now to prove Theorem 2.6 we start with p0 p1 pk q0 q1 ql (A, φ) = A0 ⊕ Mn1 ⊕ · · · ⊕ Mnk ∗ B0 ⊕ Mm1 ⊕ · · · ⊕ Mml , α0

α1

αk

β0

β1

βl

where A0 and B0 have unital, diffuse abelian C ∗ -subalgebras, supported on their units (we allow α0 = 0 or/and β0 = 0). The case where n1 = · · · = nk = m1 = · · · = ml = 1 is treated in

2372


Theorem 2.5. The case where α0 = 0, k = 1, and nk > 1 was treated in Proposition 4.4. So we can suppose without loss of generality that nk 2 and either k > 1 or α0 > 0 or both. To prove that the conclusions of Theorem 2.6 takes place in this case we will use induction on card{i | ni 2}+card{j | mj 2}, having Theorem 2.5 (card{i | ni 2}+card{j | mj 2} = 0) as first step of the induction. We look at p0 pk−1 p1 q1 ql pk q0 (B, φ|B ) = A0 ⊕ Mn1 ⊕ · · · ⊕ Mnk−1 ⊕ C ∗ B0 ⊕ Mm1 ⊕ · · · ⊕ Mml ⊂ (A, φ). α0

α1

αk−1

αk

β0

β1

βl

We suppose that Theorem 2.6 is true for (B, φ|B ) and we will prove it for (A, φ). This will be the induction step and will prove Theorem 2.6. Denote " #

α βj def

i = (i, j ) + = 1 , LA

0 n2i m2j " # " #

α βj βj αi def

k i k − 1 and LB = (i, j ) + = 1 ∪ (k, j ) + = 1

2 0 1 n2i m2j m2j and similarly " #

α βj def

i LA = (i, j ) + > 1 ,

+ n2i m2j and def LB + =

" # " #

α βj βj αi

k (i, j ) i k − 1, and 2 + 2 > 1 ∪ (k, j ) 2 + 2 > 1 . 1 ni mj mj

Clearly B LA 0 ∩ {1 i k − 1} = L0 ∩ {1 i k − 1}

and similarly B LA + ∩ {1 i k − 1} = L+ ∩ {1 i k − 1}.

Let NA (i, j ) = max(ni , mj ), let NB (i, j ) = NA (i, j ), 1 i k − 1, and let NB (k, j ) = mj . By assumption g

B = B0 ⊕ δ

(i,j )∈LB +

gij

MNB(i,j ) . δij

We want to show that f

A = A0 ⊕ γ

(i,j )∈LA +

fij

MNA(i,j ) . γij

(1)


2373

We can represent A as the span of pk Apk , pk Apk B(1 − pk ), (1 − pk )Bpk Apk , (1 − pk )Bpk Apk B(1 − pk ), and (1 − pk )B(1 − pk ). From the fact that gkj pk and gij 1 − pk , ∀1 i k − 1 we see that pk B(1 − pk ) = pk B0 (1 − pk ), (1 − pk )Bpk = (1 − pk )B0 pk , and (1 − pk )B(1 − pk ) = (1 − pk )B0 (1 − pk ) ⊕ (i,j )∈LB MN (i,j ) . All this +

tells us that we can represent A as the span of

i=k pk Apk , pk Apk B0 (1 − pk ), (1 − pk )B0 pk Apk , gij

(1 − pk )B0 pk Apk B0 (1 − pk ), (1 − pk )B0 (1 − pk ), and

In order to show that A has the form (1), we need to look have g pk Apk ∼ = (pk Bpk ) ∗ Mnk ∼ = pk B0 pk ⊕ δ αk

MN (i,j ) . (i,j )∈LB + δij i=k at pk Apk . From Proposition

4.1 we

gkj MN (k,j ) ∗ Mnk .

(k,j )∈LB +

δkj αk

Since by assumption pk B0 pk has a unital, diffuse abelian C ∗ -subalgebra, supported on 1pk B0 pk , we can use Proposition 4.4 to determine the form of pk Apk . Thus pk Apk : (i) Is simple with a unique trace if whenever for all 1 r l with N (k, r) = 1 we have 1−

1 . n2k

(ii) Is an extension 0 → I → pk Apk → Mnk → 0 if ∃1 r l, with N (k, r) = 1, and 1−

1 . n2k

1−

1 . n2k

− 1), so when r satisfies the conditions of case (iii) above,

n2 αk βr αk + βr − 1 1 + 2 −1 = k + − 1 , αk αk n2k 12 nk

just what we needed to show. Defining fij A 1 − fij , A0 = 1 − def

(i,j )∈LA +

(i,j )∈LA +

we see that A has the form (1). We need to study A0 now. Since clearly g f , we see that Apk B0 = Apk gB0 = Agpk B0 = A0 pk B0 and similarly Apk B0 = A0 pk B0 . From this and from what we proved above follows that:

2374


(1 − pk )B0 pk A0 pk ,

A0 is the span of pk A0 pk , pk A0 pk B0 (1 − pk ),

(1 − pk )B0 pk A0 pk B0 (1 − pk ),

and (1 − pk )B0 (1 − pk ).

(2)

We need to show that for each of the projections fps , 0 s k and f qt , 1 t l, we have a unital, diffuse abelian C ∗ -subalgebra of A0 , supported on it. The ones, supported on fps , 1 s k − 1 come from (1 − pk )B0 (1 − pk ) by the induction hypothesis. The one with unit fpk comes from the representation pk Apk ∼ = (pk Bpk ) ∗ Mnk and Proposition 4.4. For 1 s l we have f qs

qs Aqs ∼ = q s A0 q s ⊕ γ βs

fis

fks

γis βs

γks βs

gis

gks

δis βs

δks βs

MNA(i,s) ⊕ MNA(k,s)

(i,s)∈LA +

(3)

1ik−1

and gqs

qs Bqs ∼ = qs B0 qs ⊕ δ βs

MNB(i,s) ⊕ MNB(k,s) .

(i,s)∈LB + 1ik−1

(4)

From what we showed above follows that for 1 i k − 1 we have γis = δis and fis = gis . βs / LA If (k, s) ∈ / LB + (or αk < 1 − m2 ), then (k, s) ∈ + and by (3) and (4) we see that gqs = f qs s and so in A0 we have a unital, diffuse abelian C ∗ -subalgebra with unit gqs = f qs , which comes from B0 . If (k, s) ∈ LB + , then gqs f qs and since we have a unital, diffuse abelian C ∗ -subalgebra of A0 , supported on gqs , coming from B0 , we need only to find a unital, diffuse abelian C ∗ -subalgebra of A0 , supported on f qs − gqs and its direct sum with the one supported on gqs will be a unital, diffuse abelian C ∗ -subalgebra of A0 , supported on f qs . But from the form (3) and (4) it is clear that f qs − gqs gks , since from (3) and (4) (f1s +· · ·+f(k−1)s )qs Aqs (f1s +· · ·+f(k−1)s ) = (g1s +· · ·+g(k−1)s )qs Bqs (g1s +· · ·+g(k−1)s ). It is also clear then that f qs − gqs = f gks pk , since gqs ⊥ gks . We look for this C ∗ -subalgebra in fpk

γ αk

(k,j )∈LA +

pk Apk = pk A0 pk ⊕ g ∼ = pk B0 pk ⊕ δ αk

fkj

MNA (k,j ) ∼ = (pk Bpk ) ∗ Mnk

(k,j )∈LB +

γkj αk

gkj MNB(k,j ) ∗ Mnk . δkj αk

Proposition 4.4 gives us a unital, diffuse abelian C ∗ -subalgebra of pk A0 pk , supported on (fpk )gks = f gks = f qs − gqs . This proves that we have a unital, diffuse abelian C ∗ -subalgebra of A0 , supported on f qs . Now we have to study the ideal structure of A0 , knowing by the induction hypothesis, the form of B. We will use the “span representation” of A0 (2). B0 For each (i, j ) ∈ LB 0 we know the existence of ∗-homomorphisms π(i,j ) : B0 → MNB (i,j ) . B

B

For i = k we can write those as π(i,j0) : B0 → MNA (i,j ) and since the support of π(i,j0) is contained

N.A. Ivanov / Journal of Functional Analysis 257 (2009) 2351–2377 B

2375

A

in (1 − pk ), using (2), we can extend linearly π(i,j0) to π(i,j0 ) : A0 → MNA (i,j ) , by defining it to be zero on pk A0 pk , (1 − pk )B0 pk A0 pk , pk A0 pk B0 (1 − pk ), and (1 − pk )B0 pk A0 pk B0 (1 − pk ). A Clearly π(i,j0 ) is a ∗-homomorphism also.

B By the induction hypothesis we know that gpk is full in (i,j )∈LB ker(π(i,j0) ) ⊂ B0 and i=k

by (2), and the way we extended

B π(i,j0) ,

Then pk A0 pk is full and hereditary in from [23], we have that pk A0 pk and Above we saw that

we see that fpk is full in

ker(π(i,j0 ) ) ⊂ A0 .

ker(π(i,j0 ) ), so by the Rieffel correspondence A

(i,j )∈LA 0 i=k

fpk

γ αk

(k,j )∈LA +

pk Apk = pk A0 pk ⊕ gp k ∼ = pk B0 pk ⊕ δ αk

A

(i,j )∈LA 0 i=k

A

(i,j )∈LA 0 i=k

0

ker(π(i,j0 ) ) have the same ideal structure.

fkj

MNA (k,j ) ∼ = (pk Bpk ) ∗ Mnk γkj αk

gkj MNB(k,j ) ∗ Mnk .

(k,j )∈LB +

(5)

δkj αk

From Proposition 4.4 follows that pk A0 pk is not simple if and only if ∃1 s m, such δks 1 that (k, s) ∈ LB + , ms = 1 with αk = 1 − 2 , where δks = αk + βs − 1. This means that αk +βs −1 αk

=1−

1 , n2k

which is equivalent to

nk βs + α2k 12 nk

= 1, so this implies (k, s) ∈ LA 0 . If this is

: pk A0 pk → Mnk , the case (5), together with Proposition 4.4 gives us a ∗-homomorphism π(k,s) such that ker(π(k,s) ) ⊂ pk A0 pk is simple with a unique trace. Using (2) we extend π(k,s) linA

A

0 0 : A0 → Mnk , by defining π(k,s) to be zero on (1 − pk )B0 pk A0 pk , early to a linear map π(k,s) pk A0 pk B0 (1 − pk ), (1 − pk )B0 pk A0 pk B0 (1 − pk ), and (1 − pk )B0 (1 − pk ). Similarly as A0 before, π(k,s) turns out to be a ∗-homomorphism. By the Rieffel correspondence of the ideals of

A ) ⊂ p k A 0 pk pk A0 pk and (i,j )∈LA ker(π(i,j0 ) ), it is easy to see that the simple ideal ker(π(k,s)

i=k

0

corresponds to the ideal

A

(i,j )∈LA 0

ker(π(i,j0 ) ) ⊂

A

(i,j )∈LA 0 i=k

ker(π(i,j0 ) ), so

(i,j )∈LA 0

A

ker(π(i,j0 ) )

A

ker(π(i,j0 ) ) has a unique trace we notice that from the

A A0 A ) = pk ker(π(k,s) )pk = pk (i,j )∈LA ker(π(i,j0 ) )pk construction of π(i,j0 ) we have ker(π(k,s) 0

A (the last equality is true because pk A0 pk ⊂ (i,j )∈LA ker(π(i,j0 ) )). Now we argue simiis simple. To see that

(i,j )∈LA 0

i=k

0

) has a unique trace: larly as in the proof of Proposition 4.3, using the fact that ker(π(k,s)

A0 Suppose that ρ is a trace on (i,j )∈LA ker(π(i,j ) ). It is easy to see that Span{xpk apk y | 0

A A ) is x, y, a ∈ (i,j )∈LA ker(π(i,j0 ) ), a 0} is dense in (i,j )∈LA ker(π(i,j0 ) ), since ker(π(k,s) 0 0

A0 full in (i,j )∈LA ker(π(i,j ) ). Then since pk apk 0 we have ρ(xpk apk y) = ρ((pk apk )yx) = 0

ρ((pk apk )1/2 yx(pk apk )1/2 ) and since (pk apk )1/2 yx(pk apk )1/2 is supported on pk , it follows

2376


A that (pk apk )1/2 yx(pk apk )1/2 ∈ pk (i,j )∈LA ker(π(i,j0 ) )pk = ker(π(k,s) ), so ρ is uniquely deter0

A0 mined by ρ|ker(π ) and hence (i,j )∈LA ker(π(i,j ) ) has a unique trace. (k,s)

0

If 1 s m with (k, s) ∈ LA 0 it follows from what we said above, that pk A0 pk is sim A ple with a unique trace. But since pk A0 pk is full and hereditary in (i,j )∈LA ker(π(i,j0 ) ) =

A ker(π(i,j0 ) ) (i,j )∈LA 0

i=k

0

A ker(π(i,j0 ) ) (i,j )∈LA 0

it follows that is simple with a unique trace in this case too.

A We showed already that fpk is full in (i,j )∈LA ker(π(i,j0 ) ). Now let 1 r k − 1. We need 0

to show that fpr is full in

(i,j )∈LA 0 i=r

i=k A ker(π(i,j0 ) ).

From (3) and (4) follows that f − g pk . A

So fpr = gpr for all 1 r k − 1. From the way we constructed π(i,j0 ) is clear that fpr ∈

A0 A / ker(π(r,j0 ) ) for any 1 j l. So the smallest (i,j )∈LA ker(π(i,j ) ). It is also true that fpr ∈ i=r

0

ideal of A0 , that contains fpr , is

A0 (i,j )∈LA ker(π(i,j ) ). i=r

A

(i,j )∈LA 0 i=r

ker(π(i,j0 ) ), meaning that we must have fpr A0 =

0

Finally, we need to show that for all 1 s l we have that f qs is full in

A ker(π(i,j0 ) ). (i,j )∈LA 0 j =s g)qs pk , the way

B Let (i, j ) ∈ LA 0 with i = k, j = s. Since gqs ∈ ker(π(i,j ) ) and since (f − B to π A B A we extended π(i,j ) (i,j ) shows that f qs ∈ ker(π(i,j ) ). Let (i, s) ∈ L0 and i = k. Then we B A know that gqs ∈ / ker(π(i,j ) ), which implies f qs ∈ / ker(π(i,j ) ). Suppose (k, s) ∈ LA 0 . Then ms = 1 0 to π(k,s) show, that f gks = f qs − gqs is and (5), Proposition 4.4, and the way we extended π(k,s)

A

A

0 ). full in pk A0 pk , meaning that f qs − gqs , and consequently f qs , is not contained in ker(π(k,s) A B Finally let j = s, and suppose (k, j ) ∈ L0 . This means that (k, j ) ∈ L+ and also that the trace / LB / LB of qj is so big, that (i, s) ∈ + and (i, s) ∈ 0 for any 1 i k. Then (4) shows that qs g. A0 A0 The way we defined π(k,j ) using (5) and Proposition 4.4 shows us that B0 ⊂ ker(π(k,j ) ) in this

A

0 case. This shows qs = gqs = f qs ∈ ker(π(k,j ) ). All this tells us that the smallest ideal of A0 ,

A0 A containing f qs , is (i,j )∈LA ker(π(i,j ) ), and therefore f qs A0 = (i,j )∈LA ker(π(i,j0 ) ).

j =s

0

This concludes the proof of Theorem 2.6.

2

j =s

0

Acknowledgments I would like to thank Ken Dykema, my adviser, for the many helpful conversations I had with him, for the moral support and for reading the first version of this paper. I would also like to thank Ron Douglas and Roger Smith for some discussions. Finally I want to thank the reviewer for informing me that Lemma 3.1 is a well-known fact, for suggesting me to reorganize the paper and for pointing out some minor mistakes. References [1] C. Akeman, G. Pedersen, J. Tomiyama, Multipliers of C ∗ -algebras, J. Funct. Anal. 13 (1973) 277–301.


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[2] J. Anderson, B. Blackadar, U. Haagerup, Minimal projections in the reduced group C ∗ -algebra of Zn ∗ Zm , J. Operator Theory 26 (1991) 3–23. [3] D. Avitzour, Free products of C ∗ -algebras, Trans. Amer. Math. Soc. 271 (1982) 423–435. [4] E. Bédos, On the uniqueness of the trace on some simple C ∗ -algebras, J. Operator Theory 30 (1993) 149–160. [5] M. Choda, K. Dykema, Purely infinite, simple C ∗ -algebras arizing from free product constructions. III, Proc. Amer. Math. Soc. 128 (11) (2000) 3269–3273. [6] M.D. Choi, A simple C ∗ -algebra generated by two finite-order unitaries, Canad. J. Math. 31 (4) (1979) 867–880. [7] K. Dykema, Free products of hyperfinite von Neumann algebras and free dimension, Duke Math. J. 69 (1) (1993) 97–119. [8] K. Dykema, Interpolated free group factors, Pacific J. Math. 163 (1994) 123–134. [9] K. Dykema, Faithfulness of free product states, J. Funct. Anal. 154 (2) (1998) 323–329. [10] K. Dykema, Simplicity and stable rank of some free product C ∗ -algebras, Trans. Amer. Math. Soc. 351 (1) (1999) 1–40. [11] K. Dykema, Free Probability Theory and Operator Algebras, Seoul National University GARC lecture notes, in preparation. [12] K. Dykema, M. Rørdam, Projections in free product C ∗ -algebras, Geom. Funct. Anal. 8 (1998) 1–16. [13] K. Dykema, U. Haagerup, M. Rørdam, The stable rank of some free product C ∗ -algebras, Duke Math. J. 90 (1) (1997) 95–121. [14] R. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, Boston, New York, 1986. [15] R. Kallman, A generalization of free action, Duke Math. J. 36 (1969) 781–789. [16] A. Kishimoto, Outer automorphisms and reduced crossed products of simple C ∗ -algebras, Comm. Math. Phys. 81 (1981) 429–435. [17] R. Longo, A remark on crossed product of C ∗ -algebras, J. London Math. Soc. 23 (1981) 531–533. [18] D. Olesen, Inner ∗-automorphisms of simple C ∗ -algebras, Comm. Math. Phys. 44 (1975) 175–190. [19] D. Olesen, G. Pedersen, Application of the connes spectrum to C ∗ -dynamical systems, J. Funct. Anal. 30 (1978) 179–197. [20] W. Paschke, N. Salinas, C ∗ -algebras associated with free products of groups, Pacific J. Math. 82 (1) (1979) 211–221. [21] R. Powers, Simplicity of the C ∗ -algebra, associated with the free group on two generators, Duke Math. J. 42 (1975) 151–156. [22] F. R˘adulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994) 347–389. [23] M. Rieffel, Morita equivalence for operator algebras, Proc. Sympos. Pure Math. 38 (1982) 285–298. [24] S. Str˘atil˘a, Modular Theory in Operator Algebras, Editura Academiei, Abacus Press, 1981. [25] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York, Heidelberg, Berlin, 1979. [26] D. Voiculescu, Symmetries of some reduced free product C ∗ -algebras, in: Operator Algebras and Their Connection with Topology and Ergodic Theory, in: Lecture Notes in Math., vol. 1132, Springer-Verlag, 1985, pp. 556–588.


Spectral theory for algebraic combinations of Toeplitz and composition operators Thomas L. Kriete a , Barbara D. MacCluer a,∗ , Jennifer L. Moorhouse b a Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville, VA 22904-4137, United States b Department of Mathematics, Colgate University, Hamilton, NY 13356, United States

Received 7 November 2008; accepted 28 May 2009 Available online 24 June 2009 Communicated by N. Kalton

Abstract We determine the essential spectra of algebraic combinations of Toeplitz operators with continuous symbol and composition operators induced by a class of linear-fractional non-automorphisms of the unit disk. The operators in question act on the Hardy space H 2 on the unit disk. Our method is to realize the C ∗ -algebra that they generate as an extension of the compact operators by a concrete C ∗ -algebra whose invertible elements are easily characterized. © 2009 Elsevier Inc. All rights reserved. Keywords: Composition operator; Toeplitz operator; Essential spectrum; C ∗ -algebra

1. Introduction Given any analytic self-map ϕ of the open unit disk D in the complex plane, one can define the composition operator Cϕ : f → f ◦ ϕ, which acts as a bounded linear operator on the Hardy space H 2 , as well as a number of other Hilbert spaces of analytic functions on D. The study of these objects over four decades has produced a substantial literature centered until relatively recently on their properties as single operators. The monographs [36] and [15] give a good picture of the field before 1995. Subsequent to the characterization of compact composition operators * Corresponding author.

E-mail addresses: [email protected] (T.L. Kriete), [email protected] (B.D. MacCluer), [email protected] (J.L. Moorhouse). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.025

T.L. Kriete et al. / Journal of Functional Analysis 257 (2009) 2378–2409

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[25,35,34,39], several authors have turned to a study of the algebra of composition operators modulo the ideal K of compact operators, beginning with the problem of determining for which pairs ϕ and ψ the difference Cϕ − Cψ lies in K and a related issue of identifying the connected components in the topological space of composition operators [24,38,37]. Considerable recent progress has been made on these questions [30,2,31,29,23,16], as well as the problems of characterizing compact linear combinations [29,23], compact products with adjoints [8], and compact commutators with adjoints [4,26,27,5,7]. Composition operators associated with linearfractional self-maps of D play a role in some of this work, and have also provided fertile ground for the investigation of other operator-theoretic phenomena, including adjoints [13], cyclicity [6], subnormality [14,33], spectra [12], invariant subspaces [17,32,28], and exact norm calculations [18,3,1]. Recently, M. Jury [19,20] and the present authors [21,22] have begun the study of C ∗ -algebras generated by linear-fractional composition operators. Jury’s work considers automorphisms while we are concerned with linear-fractional non-automorphisms. The present paper continues the latter investigation. Throughout, the underlying Hilbert space is H 2 . One motivation for our work here is provided by a result from [23]. Suppose that ϕ has the following properties: (i) There are a finite number of distinct points ζ1 , . . . , ζn on the unit circle such that ϕ extends analytically across ∂D in a neighborhood of each ζj , and ϕ(ζ1 ), . . . , ϕ(ζn ) all lie on ∂D. (ii) For each j , the curvature κj of the curve ϕ(eiθ ) when eiθ = ζj satisfies κj > 1. (iii) For any open set U in the plane containing ζ1 , . . . , ζn , |ϕ(z)| is bounded away from 1 on D \ U. Under these conditions, there exist a compact operator K and linear-fractional non-automorphisms ϕ1 , . . . , ϕn of D such that ϕj (∂D) is the osculating circle of the curve ϕ(eiθ ) at eiθ = ζj and Cϕ = Cϕ1 + Cϕ2 + · · · + Cϕn + K. Each ϕj is determined by the requirement that it has the same second order data as ϕ at ζj : ϕj (ζj ) = ϕ(ζj ),

ϕj (ζj ) = ϕ (ζj ),

ϕj (ζj ) = ϕ (ζj ).

The spectral theory of such Cϕ , and operators that can be algebraically derived from it, is then encoded in the C ∗ -algebra generated by the ideal K and the building blocks Cϕ1 , . . . , Cϕn . A second motivation comes from the theory of Toeplitz operators. By identifying an H 2 function with its (almost everywhere existing) non-tangential limit function on the circle, we can think of H 2 as the subspace of L2 (the Lebesgue space associated with normalized arc-length measure on ∂D) spanned by the orthonormal set {einθ : n 0}. From this point of view, any bounded, measurable, complex-valued function w on ∂D induces a Toeplitz operator Tw on H 2 , given by Tw f = P (wf ), where P is the orthogonal projection of L2 onto H 2 . Taking w to be the independent variable z yields the shift operator Tz , which (together with the identity operator) generates the unital C ∗ -algebra C ∗ (Tz ) known as the continuous Toeplitz algebra. Coburn’s theorem [9,10] asserts that C ∗ (Tz ) contains the ideal K, as well as all Toeplitz operators with continuous symbol. Moreover, the map sending w to the coset of Tw is a ∗-isomorphism of C(∂D), the algebra of continuous functions on ∂D, onto the quotient algebra C ∗ (Tz )/K. In [21] the authors observed that Cϕ and Tw can interact nicely, and established an analogue of Coburn’s

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theorem for the unital C ∗ -algebra C ∗ (Tz , Cϕ ) generated by Tz and a single composition operator Cϕ with ϕ satisfying

ϕ is a linear-fractional self-map of D, not an automorphism, having ϕ(ζ ) = η for distinct points ζ, η in ∂D.

(1)

In this article we investigate the unital C ∗ -algebra C ∗ (Tz , Cϕ1 , . . . , Cϕn ) generated by Tz and a finite collection {Cϕ1 , Cϕ2 , . . . , Cϕn } of composition operators with each ϕj of type (1). For each such ϕj , let ζj and ηj be related to ϕj as ζ and η are to ϕ in (1). The configuration of points ζ1 , η1 , . . . , ζn , ηn on the circle determines a (possibly disconnected) graph with n edges if we think of each ϕj as representing an edge in this graph, and the input point ζj and target point ηj as the corresponding vertices. We require each component of this graph to be a tree; i.e., to contain no cycles. If there are q components, and the ith component has ni edges, we construct a concrete C ∗ -subalgebra D of C(∂D) ⊕ Mn1 +1 C [0, 1] ⊕ · · · ⊕ Mnq +1 C [0, 1] and a ∗-homomorphism Ψ : C ∗ (Tz , Cϕ1 , . . . , Cϕn ) → D that yields a short exact sequence i

Ψ

0 → K → C ∗ (Tz , Cϕ1 , . . . , Cϕn ) → D → 0 of C ∗ -algebras. Here i is inclusion, C([0, 1]) is the algebra of complex-valued continuous functions on [0, 1], and Mk (C([0, 1])) is the C ∗ -algebra of k × k matrices over C([0, 1]), which we can also think of as the algebra of Mk -valued continuous functions on [0, 1], equipped with the supremum operator norm. Since invertibility in D is transparent, we can in principle read off the essential spectrum of any operator in C ∗ (Tz , Cϕ1 , . . . , Cϕn ). 2. Some linear-fractional maps and their composition operators For A in B(H 2 ), the algebra of bounded operators on H 2 , we write [A] for the coset of A in the Calkin algebra B(H 2 )/K. If B is also in B(H 2 ) we sometimes indicate the relation A − B ∈ K (that is, [A] = [B]) by A ≡ B (mod K). We write σ (A) for the spectrum of A, and σe (A) for the essential spectrum, that is, the spectrum of [A] in B(H)/K, or equivalently, the set of complex λ for which A − λI is not Fredholm. 2.1. The adjoint of Cϕ modulo K Consider a linear-fractional self-map ϕ of D given by ϕ(z) =

az + b . cz + d


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The Krein adjoint of ϕ, defined by σ (z) =

az − c −bz + d

,

also carries D into D. If ϕ is not an automorphism of D and if ζ and η = ϕ(ζ ) are both in ∂D, Cϕ∗ ≡

1 |ϕ (ζ )|

Cσ

(mod K),

(2)

see [21]. Moreover, σ is also a non-automorphism, σ (η) = ζ , and σ (η)ϕ (ζ ) = 1. The Krein adjoint of σ is ϕ, and if σ1 and σ2 are the respective Krein adjoints of ϕ1 and ϕ2 , then σ2 ◦ σ1 is the Krein adjoint of ϕ1 ◦ ϕ2 . If we write x and tw for the respective cosets [Cϕ ] and [Tw ], where w is continuous on ∂D, then we have the relations tw x = w(ζ )x,

xtw = w(η)x,

tw x ∗ = w(η)x ∗

and x ∗ tw = w(ζ )x ∗

established in [21]. Also, according to Coburn’s theorem, tuw = tu tw for continuous u and w on ∂D. 2.2. Parabolic non-automorphisms of D Recall that a linear-fractional self-map ρ of D is parabolic if it fixes some γ in ∂D and has the property that β ◦ ρ ◦ β −1 acts as “translation by a”, where β(z) = (γ + z)/(γ − z), a conformal map of D onto the half plane {w: Re w > 0} carrying γ to infinity. The translation number a necessarily satisfies Re a 0 and, when ρ is a non-automorphism, Re a > 0. We sometimes denote the map ρ by ργ ,a . Among all linear-fractional self-maps of D fixing γ , the parabolics are also characterized by the condition ρ (γ ) = 1. The collection {ργ ,a : Re a 0} is a commutative semigroup under composition satisfying ργ ,a ◦ ργ ,b = ργ ,a+b . Moreover, the Krein adjoint of ργ ,a is ργ ,a . It follows from Eq. (2) that when Re a > 0 Cρ∗γ ,a ≡ Cργ ,a

(mod K).

(3)

Recall that a bounded operator T on H 2 is normal if T ∗ T = T T ∗ , and is essentially normal provided T and T ∗ commute modulo the ideal K. The only normal composition operators Cψ are those with ψ(z) = az for some complex a with |a| 1; see Theorem 8.2 in [15]. Bourdon, Levi, Narayan and Shapiro [4] showed that if ψ is linear-fractional with ψ ∞ = 1 (so that Cψ is not compact) and Cψ is not normal, then Cψ is essentially normal if and only if ψ is a parabolic non-automorphism. Indeed, the “if” implication follows from Eq. (3) and the semigroup property of the parabolic maps fixing γ . A recent theorem of Montes-Rodríguez, Ponce-Escudero and Shkarin [28] shows that Cργ ,a is irreducible for Re a > 0. The unital C ∗ -algebra C ∗ (Pγ ) generated by Pγ ≡ {Cργ ,a : Re a > 0} is thus irreducible, and it contains non-zero compact operators (the commutator of any Cργ ,a and Cρ∗γ ,a , for example). Thus it must contain the ideal K; see [11, p. 74]. Moreover, it is clear

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from the above remarks that C ∗ (Pγ )/K is commutative. It was shown in [22] that there is a ∗-isomorphism Γ : C [0, 1] → C ∗ (Pγ )/K satisfying Γ (t a ) = [Cργ ,a ], Re a > 0, where t ∈ [0, 1] is the independent variable. It follows that C ∗ (Pγ ) is generated as a C ∗ -algebra by any single Cργ ,a . If ϕ : D → D is linear-fractional, not an automorphism, and satisfies ϕ(ζ ) = η for ζ, η in ∂D, its Krein adjoint σ has the property that ϕ ◦ σ and σ ◦ ϕ are positive parabolic maps (meaning their translation numbers are positive) fixing η and ζ respectively. Moreover σ (Cϕ◦σ ) = σ (Cσ ◦ϕ ) = σe (Cϕ◦σ ) = σe (Cσ ◦ϕ ) = [0, 1], see [12] and [21]. 2.3. Linear-fractional maps and composition operators related to ϕ Let ϕ be a linear-fractional non-automorphism of D with ϕ(ζ ) = η for some ζ and η in ∂D, and suppose σ is its Krein adjoint. Here we collect some results about related maps and their composition operators. For any function h analytic in a neighborhood of z0 , we call

h(z0 ), h (z0 )

and

h(z0 ), h (z0 ), h (z0 )

the first and second order data, respectively, of h at z0 . We will need the fact that a linearfractional map is uniquely determined by its second order data at any point of analyticity. The translation numbers of the parabolic maps σ ◦ ϕ and ϕ ◦ σ play a role in the sequel, and we begin with some relations between them and the geometry of ϕ. Proposition 1. Suppose ϕ is a linear-fractional non-automorphism of D with ϕ(ζ ) = η for some ζ, η ∈ ∂D, and let σ be its Krein adjoint. (a) If b and c are the unique positive numbers with ρη,b (D) = ϕ(D) and ρζ,c (D) = σ (D), then b = κϕ − 1 and c = κσ − 1, where κϕ and κσ are the curvatures of the circles ϕ(∂D) and σ (∂D), respectively. Moreover, c = |ϕ (ζ )|b. (b) The positive parabolic maps ϕ ◦ σ and σ ◦ ϕ have the specific forms ϕ ◦ σ = ρη,2b and σ ◦ ϕ = ρζ,2c where b and c are as in part (a). (c) For any complex number d1 , ρη,d1 ◦ ϕ = ϕ ◦ ρζ,d2 where d2 = |ϕ (ζ )|d1 . Proof. With b and c as defined in part (a), the statements c = ϕ (ζ )b

(4)


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Table I Linear-fractional ψ with Cψ in C ∗ (Cϕ , K) Condition on ψ in Proposition 2 (d) (b)

ψ

Distinguished representative of [Cψ ]

ρη,a , Re a > 0 ρζ,a , Re a > 0

s − 2b (Cϕ∗ Cϕ ) 2b

(c)

a

a

a

s − 2c (Cϕ Cϕ∗ ) 2c

ρη,a ◦ ϕ, Re a > −b ρζ,a ◦ σ , Re a > −c

(a)

a

a

1

a

s − 2b U (Cϕ∗ Cϕ ) 2 + 2b a

1

a

s − 2c −1 U ∗ (Cϕ Cϕ∗ ) 2 + 2c

and ϕ ◦ σ = ρη,2b ,

σ ◦ ϕ = ρζ,2c

(5)

appear in Theorem 6 of [22]. To relate b to the curvature of ϕ(∂D), we use the fact that ρη,b is conjugate via the map β(z) = (η + z)/(η − z) to translation by b in the right-half plane. Thus ϕ(∂D) = ρη,b (∂D) is the image of the line Re w = b under β −1 (w) = η(w − 1)/(w + 1). We check that this image is a circle of radius (1 + b)−1 , and so κϕ = 1 + b. Similarly, if ρζ,c (D) = σ (D), then κσ = 1 + c. This completes the proof of the statements in (a) and (b). The proof of (c) appears in Lemma 5 of [22]. 2 Assume that ϕ satisfies (1) and σ is its Krein adjoint. In [22] the authors considered the question of which composition operators lie in C ∗ (Cϕ , K) and in particular showed the following: Proposition 2. (See [22].) If ψ, not the identity, is a linear-fractional self-map of D with supremum norm ψ ∞ = 1, then Cψ ∈ C ∗ (Cϕ , K) if and only if ψ is not an automorphism and one of the following holds: (a) (b) (c) (d)

ψ(ζ ) = η and ψ (ζ ) = ϕ (ζ ). ψ(ζ ) = ζ and ψ (ζ ) = 1. ψ(η) = ζ and ψ (η) = 1/ϕ (ζ ). ψ(η) = η and ψ (η) = 1.

More concretely, the maps satisfying (b) and (d) are exactly ρζ,a and ρη,a with Re a > 0, respectively. If b is the unique positive number with ρη,b (D) = ϕ(D), then the maps satisfying (a) are exactly ρη,a ◦ ϕ, Re a > −b (although ρη,a (D) does not contain D when Re a < 0, ρη,a ◦ ϕ is a non-automorphism self-map of D as long as Re a > −b). Similarly, if ρζ,c (D) = σ (D), the maps satisfying (c) are exactly ρζ,a ◦ σ, Re a > −c. The first two columns in Table I summarize this situation. An operator-theoretic description from [22] of the maps satisfying (a), (b), (c) and (d) in Proposition 2 will also be useful. Assume ϕ satisfies (1) and consider the polar decomposition U Cϕ∗ Cϕ = Cϕ Cϕ∗ U

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Fig. 1.

of Cϕ . As shown in [21], U is unitary, and every operator B in C ∗ (Cϕ , K) has a representation B = Y + K where K is compact and Y = cI + f Cϕ∗ Cϕ + g Cϕ Cϕ∗ + U h Cϕ∗ Cϕ + U ∗ k Cϕ Cϕ∗ , where f, g, h and k are continuous on σ (Cϕ∗ Cϕ ) = σ (Cϕ Cϕ∗ ) and vanish at 0 (both spectra contain 0). If s = 1/|ϕ (ζ )|, the essential spectra σe (Cϕ∗ Cϕ ) and σe (Cϕ Cϕ∗ ) coincide with [0, s], and the restrictions of f , g, h and k to [0, s] are uniquely determined by the coset [B] in B(H 2 )/K. We call Y a distinguished representative of [B]. The third column in Table I presents distinguished representatives of [Cψ ] with ψ as in the first two columns. 3. Outgoing star graphs We begin with a special case to which the general case will be reduced. We have n linearfractional maps ϕj : D → D, j = 1, . . . , n, such that {ζ ∈ ∂D: |ϕj (ζ )| = 1} is a singleton set for each j . In this section we consider the case where these singleton sets are the same for each j , that is, there exists η0 in ∂D with ϕj (η0 ) ∈ ∂D for j = 1, . . . , n. We write ϕj (η0 ) = ηj and assume further that the points η0 , η1 , η2 , . . . , ηn are all distinct. We picture this situation by the directed graph in Fig. 1, an “outgoing star” centered at η0 . Let σi denote the Krein adjoint of ϕi , and put si =

1 . |ϕi (η0 )|

(6)

By Eq. (2) Cϕ∗i ≡ si Cσi

(mod K), i = 1, 2, . . . , n.

Since C ∗ (Tz ) contains the ideal of compact operators, so does C ∗ (Tz , Cϕ1 , . . . , Cϕn ). For j = 1, 2, . . . , n we write xj = [Cϕj ]. Again, with tw = [Tw ], we find from Section 2 that • • • •

tw xj = w(η0 )xj , xj tw = w(ηj )xj , tw xj∗ = w(ηj )xj∗ , xj∗ tw = w(η0 )xj∗ .


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If i = j , then σj ◦ ϕi maps D into a compact subset of itself, so Cσj ◦ϕi is a compact operator on H 2 , that is, xi xj∗ = [Cϕi ][sj Cσj ] = sj [Cσj ◦ϕi ] = 0. On the other hand, xi xi∗ = si [Cσi ◦ϕi ] and xi∗ xi = si [Cϕi ◦σi ] are non-zero, and since σi ◦ ϕi and ϕi ◦ σi are positive parabolic maps fixing η0 and ηi respectively, σ (xi xi∗ ) = σ (xi∗ xi ) = [0, si ]. In a similar vein, xi∗ xj = [si Cσi ][Cϕj ] = si [Cϕj ◦σi ]. Since (ϕj ◦ σi )(ηi ) = ηj ∈ ∂D, the map ϕj ◦ σi has a finite angular derivative at ηi and Cϕj ◦σi is not compact [40]. Hence xi∗ xj is non-zero for all i and j . Similar arguments show that xi xj = 0 and xi∗ xj∗ = 0 for all i and j . We want to express the above identities more generally using the polar decomposition Cϕi = Ui Cϕ∗i Cϕi = Cϕi Cϕ∗i Ui where Ui is unitary (see [21]). Putting ui = [Ui ] (a unitary element of B(H 2 )/K), we find xi = ui xi∗ xi = xi xi∗ ui

and xi∗ = u∗i xi xi∗ = xi∗ xi u∗i .

(7)

For c > 0, we write C0 ([0, c]) for the algebra of continuous functions f on [0, c] with f (0) = 0, equipped with the supremum norm. The map f → f (xi xi∗ ) is an isometry from C0 ([0, si ]) onto the closed subspace Mi ≡ {f (xi xi∗ ): f ∈ C0 ([0, si ])} of B(H 2 )/K. Moreover, the linear man ifold {p(xi xi∗ ) xi xi∗ : p a polynomial} is norm-dense in Mi ; see Lemma 1 in [21]. It follows that the relations xi xj = 0, xi∗ xj∗ = 0 for all i, j , and xi xj∗ = 0 for i = j , imply their more general respective counterparts f xi xi∗ ui g xj xj∗ uj = 0,

u∗i f xi xi∗ u∗j g xj xj∗ = 0

for all i, j , and f xi xi∗ ui u∗j g xj xj∗ = 0 for i = j , where f ∈ C0 ([0, si ]) and g ∈ C0 ([0, sj ]). Indeed, using the well-known relations ui f (xi∗ xi ) = f (xi xi∗ )ui and their adjoints, valid for all f in C([0, si ]), we can reduce the above equations to more basic relations: f xi∗ xi g xj xj∗ = 0 = g xj xj∗ f xi∗ xi for all i, j and f (xi∗ xi )g(xj∗ xj ) = 0 if i = j provided f and g are in C0 ([0, si ]) and C0 ([0, sj ]), respectively. On the other hand, u∗i f (xi xi∗ )g(xj xj∗ )uj is non-zero for all i, j and all non-trivial continuous functions f, g defined on [0, si ] and [0, sj ], respectively. Since tw xi∗ = w(ηi )xi∗ , we

2386


also have tw u∗i f (xi xi∗ ) = w(ηi )u∗i f (xi xi∗ ) for w continuous and f in C0 ([0, si ]), see [21, Section 4.2]. Similarly, f (xi xi∗ )ui tw = w(ηi )f (xi xi∗ )ui with the same restrictions on w and f . Since the maps σi ◦ ϕi are all positive parabolic non-automorphisms fixing η0 , the corresponding operators Cσi ◦ϕi (i = 1, . . . , n) commute, and the relations xi xi∗ = si [Cσi ◦ϕi ] imply that x1 x1∗ , . . . , xn xn∗ are commuting positive elements in the Calkin algebra. If ai > 0 is the translation number of σi ◦ ϕi , then C ∗ x1 x1∗ , . . . , xn xn∗ = C ∗ (Pη0 )/K ∼ = C [0, 1]

(8)

and the map Γ described in Section 2 acts by Γ : si t ai → xi xi∗ ,

i = 1, . . . , n.

By Proposition 1, ai = 2(κi − 1)/si where κi is the curvature of the circle ϕi (∂D). The algebra in Eq. (8) is generated by any single xi xi∗ alone. Let P denote the set of linear combinations of {tw : w ∈ C(∂D)} and all words in x1 , . . . , xn , x1∗ , . . . , xn∗ . Then P is a unital ∗-algebra which is dense in C ∗ (Tz , Cϕ1 , . . . , Cϕn )/K. Here, the identities tw xi = w(η0 )xi and their companions are needed to conclude that P contains products like tw xi . The only non-identity words in x1 , . . . , xn , x1∗ , . . . , xn∗ are those of the forms ∗ r1 r r ∗ r1 x1 x1 · · · xn xn∗ n , x1 x1 · · · xn xn∗ n xj , r r r r xi∗ x1 x1∗ 1 · · · xn xn∗ n and xi∗ x1 x1∗ 1 · · · xn xn∗ n xj for 1 i, j n, where r1 , . . . , rn are non-negative integers, and, for words of the first type, at least one rj is positive. The closed linear span of r ∗ r1 x1 x1 · · · xn xn∗ n : each ri is a non-negative integer is the commutative C ∗ -algebra in Eq. (8). By Gelfand theory, the elements of this algebra have the form f (x1 x1∗ , . . . , xn xn∗ ) where f is continuous on the joint spectrum σ (x1 x1∗ , . . . , xn xn∗ ) of the elements x1 x1∗ , . . . , xn xn∗ . The map Γ discussed above allows us to realize this joint spectrum as σ x1 x1∗ , . . . , xn xn∗ = s1 t a1 , . . . , sn t an : 0 t 1 , and further, to represent C ∗ (x1 x1∗ , . . . , xn xn∗ ) as {g(xi xi∗ ): g ∈ C([0, si ])} for any given i = 1, . . . , n. Writing xi = xi xi∗ ui , xi∗ = u∗i xi xi∗ , and defining M to be ∗ f x1 x1 , . . . , xn xn∗ : f is continuous on σ x1 x1∗ , . . . , xn xn∗ and f (0, . . . , 0) = 0 , we see that M, Muj , u∗i M and u∗i Muj are, respectively, the closed linear spans of our four types of words; here i and j range from 1 to n. We simplify notation: Let u0 = [I ], the identity in the Calkin algebra, and let z denote the commuting n-tuple (x1 x1∗ , . . . , xn xn∗ ). We write A for the collection of all elements of C ∗ (Tz , Cϕ1 , . . . , Cϕn )/K of the form

T.L. Kriete et al. / Journal of Functional Analysis 257 (2009) 2378–2409 n

b = tw +

u∗i fij (z)uj

2387

(9)

i,j =0

where w ∈ C(∂D) and fij lies in C(σ (z)) with fij (0, . . . , 0) = 0. Lemma 1. The representation in Eq. (9) is unique. Proof. Suppose that tw +

n

u∗i fij (z)uj = 0.

(10)

i,j =0

We will show that each of the terms in the left-hand side of Eq. (10) is zero. For k, l = 1, 2, . . . , n, multiply (10) on the left by xk and on the right by xl∗ . For i 1 and i = k we claim that xk u∗i fij (z)uj xl∗ = 0, and that for j 1 and j = l we again have xk u∗i fij (z)uj xl∗ = 0. To verify these claims, recall that we may write any fij (z) as a function of xk xk∗ alone, for any desired choice of k, 1 k n. We adopt some temporary notation for this, and let hkij (xk xk∗ ) = fij (z) for 1 k n; here hkij is in C0 ([0, sk ]). Thus xk u∗i fij (z)uj xl∗ = xk u∗i hiij xi xi∗ uj xl∗ = uk xk∗ xk hiij xi∗ xi u∗i uj xl∗ = 0 for i = k and j j xk u∗i fij (z)uj xl∗ = xk u∗i hij xj xj∗ uj xl∗ = xk u∗i uj hij xj∗ xj xl∗ xl u∗l = 0 for j = l. With these observations the result of our left and right multiplication is xk tw xl∗ + xk

n

j =0

f0j (z)uj xl∗ + xk

n

u∗i fi0 (z) xl∗ + xk u∗k fkl (z)ul xl∗ = 0.

i=1

Next check that xk f0j (z)uj xl∗ = uk xk∗ xk hk0j xk xk∗ uj xl∗ = 0 and xk u∗i fi0 (z)xl∗ = xk u∗i hli0 xl xl∗ xl∗ xl u∗l = 0

(11)

2388


for j = 0, 1, . . . , n and i = 1, . . . , n. Thus Eq. (11) simplifies further as xk tw xl∗ + xk u∗k fkl (z)ul xl∗ = 0

(12)

for 1 k, l n. At this point we distinguish two cases: k = l and k = l. In the first case, (12) becomes w(ηk )xk xk∗ + xk u∗k fkk (z)uk xk∗ = 0. Writing xk =

xk xk∗ uk and xk∗ = u∗k xk xk∗ we have w(ηk )xk xk∗ +

xk xk∗ fkk (z) xk xk∗ = 0.

(13)

We claim that this forces w(ηk ) = 0 and fkk (z) = 0. Since fkk (z) = hkkk (xk xk∗ ) we may rewrite Eq. (13) as w(ηk )xk xk∗ + xk xk∗ hkkk xk xk∗ = 0. This requires hkkk (t) to have constant value −w(ηk ). Since hkkk (0) = 0, this implies that w(ηk ) = 0 and hkkk ≡ 0, and so fkk (z) = 0 as desired. When k = l we proceed similarly to obtain w(ηk )xk xl∗ +

xk xk∗ fkl (z) xl xl∗ = 0.

Since k = l, xk xl∗ = 0, and we find

xk xk∗ fkl (z) xl xl∗ = 0.

With notation as above, we write fkl (z) = hkkl (xk xk∗ ). Moreover, for some constant clk , a /a xl xl∗ = clk xk xk∗ l k where ai denotes the translation number of σi ◦ ϕi (see the sentence above containing Eq. (8)). Thus a /(2a ) xk xk∗ hkkl xk xk∗ xk xk∗ l k = 0. If follows that fkl (z) = hkkl (xk xk∗ ) = 0. Thus for each k = 1, . . . , n and l = 1, . . . , n, we have fkl (z) = 0. At this point we have shown that if Eq. (10) holds, then tw +

n

i=0

u∗i fi0 (z) +

n

j =1

f0j (z)uj = 0.

(14)


2389

Multiply (14) on the left by x1∗ and on the right by x1 , noting that x1∗ u∗i fi0 (z)x1 = x1∗ u∗i hii0 xi xi∗ x1 = u∗1 x1 x1∗ hii0 xi∗ xi u∗i x1 = 0 for i 1 and j j x1∗ f0j (z)uj x1 = x1∗ h0j xj xj∗ uj x1 = x1∗ uj h0j xj∗ xj x1 x1∗ u1 = 0 for j 1. Thus w(η0 )x1∗ x1 + x1∗ f00 (z)x1 = 0, which we may rewrite using the equations in (7) as w(η0 )x1∗ x1 +

x1∗ x1 u∗1 f00 (z)u1 x1∗ x1 = 0.

Upon setting f00 (z) = h100 (x1 x1∗ ) and writing h100 (x1 x1∗ )u1 = u1 h100 (x1∗ x1 ) we find w(η0 )x1∗ x1 + x1∗ x1 h100 x1∗ x1 = 0. This forces h100 (t) to have constant value −w(η0 ), and since h100 (0) = 0, we conclude that w(η0 ) = 0 and h100 ≡ 0. In particular, f00 (z) = 0, and Eq. (14) becomes tw +

n

u∗i fi0 (z) +

n

f0j (z)uj = 0.

(15)

j =1

i=1

∗ , m = 1, . . . , n, we see that Multiplying in (15) on the left and right by xm ∗ ∗ xm f0m (z)um xm = 0,

since the left multiplication kills all of the terms u∗i fi0 (z), 1 i n, and the right multiplication ∗ = u∗ x x ∗ we obtain kills all of the terms f0j (z)uj except for the j = m term. Writing xm m m m ∗ f (z) x x ∗ = 0. u∗m xm xm 0m m m ∗ Since um is unitary, we conclude that f0m (z) = hm 0m (xm xm ) = 0 for m = 1, . . . , n. Finally we return to (15), now in the form

tw +

n

u∗i fi0 (z) = 0,

i=1

and multiply on the left and right by xj , j = 1, . . . , n, to conclude in a similar fashion that fj 0 (z) = 0 for all j 1, and thus also that w = 0. 2

2390


Now given a continuous function f on the joint spectrum σ (z), define f˜ in C([0, 1]) by f˜(t) = f s1 t a1 , . . . , sn t an ,

0 t 1.

Let Φ denote the map Φ : A → C(∂D) ⊕ Mn+1 C [0, 1] given by Φ tw +

n

= w, diag w(ηj ) + [f ij ]

u∗i fij (z)uj

i,j =0

where ⎡ f00 ..

⎣ [fij ] = . f n0

⎤ f 0n .. ⎦ . ··· f nn ···

(16)

and ⎡ w(η0 ) ⎢ diag w(ηj ) = ⎣

0

0 ..

.

⎤ ⎥ ⎦

w(ηn )

By Lemma 1, Φ is well defined, and it is clearly linear and ∗-preserving. The direct sum C(∂D)⊕ Mn+1 (C([0, 1])) is a C ∗ -algebra with operations defined coordinatewise and norm (w, M) = max w ∞ , sup M(t) . 0t1

Lemma 2. The set A is an algebra and the map Φ preserves products. Proof. Suppose that b 1 = tw +

n

u∗i fij (z)uj

i,j =0

and b 2 = tv +

n

i,j =0

are in A. Their product b1 b2 is of the form

u∗i gij (z)uj


twv + tw

n

u∗i gij (z)uj

+

i,j =0

+

n

n

u∗i fij (z)uj

2391

tv

i,j =0

u∗i fij (z)uj u∗k gkl (z)ul .

(17)

i,j =0 k,l=0

Since u∗i fij (z)uj u∗k gkl (z)u∗k = 0 if j = k and u∗i fij (z)uj u∗j gj l (z)ul = u∗i fij (z)gj l (z)ul , we see that n n

u∗i fij (z)uj u∗k gkl (z)ul =

i,j =0 k,l=0

n

u∗i

n

fij (z)gj l (z) ul ∈ A.

j =0

i,l=0

Furthermore, tw

n

u∗i gij (z)uj =

i,j =0

n

w(ηi )u∗i gij (z)uj

(18)

i,j =0

and

n

i,j =0

u∗i fij (z)uj

tv =

n

v(ηj )u∗i fij (z)uj

(19)

i,j =0

are both in A. Since A is clearly a linear space, this gives the first statement in the theorem. To see that Φ preserves products, notice that Φ(b1 ) = (w, diag(w(ηj )) + [f ij ]) and Φ(b2 ) = (v, diag(v(ηj )) + [ gij ]). Using Eqs. (17), (18), and (19) we have Φ(b1 b2 ) = wv, diag w(ηj )v(ηj ) + M where M is equal to ⎡

w(η0 )g 00 .. ⎣ . w(ηn )g n0

⎤ ⎡ v(η0 )f w(η0 )g 0n 00 .. .. ⎦+⎣ . .

· · · w(ηn )g v(η nn 0 )f n0 ···

⎤ v(ηn )f 0n .. ⎦ + [f gij ] ij ][ . · · · v(ηn )f nn ···

gij ] similarly. It is straightforward to check that this is equal to for [f ij ] as in Eq. (16) and [ Φ(b1 )Φ(b2 ). 2 Now let D denote the collection of all (w, [hij ]) in C(∂D) ⊕ Mn+1 (C([0, 1])) with hii (0) = w(ηi ) for i = 0, 1, . . . , n and hij (0) = 0 whenever i = j . Clearly D is a closed unital ∗-subalgebra of C(∂D) ⊕ Mn+1 (C([0, 1])), hence a unital C ∗ -algebra in its own right. Also, clearly D coincides with the range of Φ. Lemma 3. We have A = C ∗ (Tz , Cϕ1 , . . . , Cϕn )/K.

2392


Proof. We know that the map Φ is a ∗-homomorphism of A onto D. Suppose b is an element in A described as in Eq. (9) with w, diag w(ηj ) + [f ij ] = Φ(b) = 0. Clearly then we must have w ≡ 0 on ∂D, and moreover each f ij vanishes on [0, 1], and so fij vanishes on the joint spectrum σ (z). Thus fij (z) = 0 for each i and j . It follows that b = 0, so that Φ is one-to-one. Thus Φ −1 is a one-to-one ∗-homomorphism of D into C ∗ (Tz , Cϕ1 , . . . , Cϕn )/K. It follows that Φ −1 is isometric, and hence its range A is closed. Since A is dense in C ∗ (Tz , Cϕ1 , . . . , Cϕn )/K, we are done. 2 Lemmas 2 and 3 allow us to define a ∗-homomorphism Ψ : C ∗ (Tz , Cϕ1 , . . . , Cϕn ) → D by Ψ (B) = Φ([B]). We can summarize our conclusions so far in the following theorem. Theorem 1. We have a short exact sequence i

Ψ

0 → K → C ∗ (Tz , Cϕ1 , . . . , Cϕn ) → D → 0 of C ∗ -algebras, where i is inclusion. 4. Disconnected graphs with outgoing star components As before, we consider linear-fractional non-automorphisms ϕ : D → D with ϕ(ζ ) = η for distinct points ζ and η in ∂D. For a finite collection of such maps ϕ1 , ϕ2 , . . . , ϕn with ϕi (ζi ) = ηi , we allow the possibility that for some i = j , ζi = ζj or ηi = ζj , or ηi = ηj , but we do not simultaneously allow ζi = ζj and ηi = ηj , or ζi = ηj and ηi = ζj . Such a collection of maps determines a directed graph, with the directed edge determined by ϕi connecting the input vertex ζi to the target vertex ηi . In Section 3 our directed graph was an outgoing star. We remove this restriction in two stages, and ultimately replace our star by an undirected graph, each component of which is required to be a tree (meaning it contains no cycles). For stage 1, suppose our graph has q connected components, all of which are outgoing stars. We re-label our maps as q

q

ϕ11 , . . . , ϕn11 , ϕ12 , . . . , ϕn22 , . . . , ϕ1 , . . . , ϕnq ,

(20)

where ϕ1k , . . . , ϕnkk determine the kth outgoing star; see Fig. 2 for a picture. Here, and throughout this section, a superscript appears as part of the label of the object, and does not denote a power. Note that n1 + · · · + nq = n, the total number of maps in our collection. For each k, the set of vertices for the kth star, which we re-label as {η0k , η1k , . . . , ηnk k }, is disjoint from the sets of vertices for any of the remaining stars. As before, we assume η0k , η1k , . . . , ηnk k are all distinct and that ϕik (η0k ) = ηik for i = 1, 2, . . . , nk , so that η0k is the center of the kth star.


2393

Fig. 2.

For 1 k q and 1 i nk , let 1

sik =

|(ϕik ) (η0k )|

.

If σik is the Krein adjoint of ϕik , we have Cϕ∗k ≡ sik Cσ k

(mod K).

i

i

The polar decompositions are Cϕ k = Uik C ∗k Cϕ k = Cϕ k C ∗k Uik , i

ϕi

i

i

ϕi

and if xik = [Cϕ k ], i

xik =

∗ xik xik uki

and k ∗ k ∗ k k ∗ xi = ui xi xi . If y (j ) denotes a word in j ∗ j j j ∗ x1 , . . . , xnj , x1 , . . . , xnj , then for k = r we have y (k) y (r) = 0.

(21)

This allows us to apply the analysis from Section 3, appropriately modified, to the current set-up. Let A denote the linear manifold in C ∗ (Tz , Cϕ1 , . . . , Cϕn )/K of all elements having the form

2394


b = tw +

q

nk

k ∗ k k k ui fij z uj

(22)

k=1 i,j =0

where ∗ ∗ zk = x1k x1k , . . . , xnkk xnkk , fijk is continuous on the joint spectrum σ (zk ) and vanishes at the origin, and uk0 = [I ]. We have the following analogue of Lemma 1. Lemma 4. The representation in Eq. (22) is unique. Proof. Suppose that q

nk

k ∗ k k k ui fij z uj = 0. tw + k=1 i,j =0 r and on the right by (x r )∗ for 1 m, l n . By Eq. (21), We multiply both sides on the left by xm r l k k k r ∗ k xm (ui ) fij (z )uj = 0 unless k = r, so that we obtain the analogue of Eq. (11) with superscripts r throughout (note that here m is playing the role of k in Eq. (11)). The proof now follows that of Lemma 1 to the conclusion that for any 1 r q and all 1 i, j nr , fijr (zr ) = 0. Hence also tw = 0, and we are done. 2

Now let aik > 0 be the translation number of σik ◦ ϕik for k = 1, 2, . . . , q and i = 1, 2, . . . , nk . If f is continuous on the joint spectrum σ (zk ), define f˜ in C([0, 1]) by k ak f˜(t) = f s1k t a1 , . . . , snkk t nk ,

0 t 1.

We define a map Φ : A → C(∂D) ⊕ Mn1 +1 C[0, 1] ⊕ · · · ⊕ Mnq +1 C[0, 1] given by 1 q q Φ(b) = w, diag w ηj1 + f + fij ij , . . . , diag w ηj where b is given by Eq. (22). This map is well defined by Lemma 4 and the fact that fijk (zk ) is determined by the restriction of fijk to the joint spectrum σ (zk ). By Eq. (21) and approximation as discussed in Section 3, r ∗ r r r k ∗ k k k ui fij z uj um fms z us = 0 unless k = r. We can follow the proof of Lemma 2 to show that A is a ∗-algebra and Φ is a one-to-one ∗-homomorphism. The range D of Φ consists of all elements q 1 2 w, hij , hij , . . . , hij


2395

where w is in C(∂D) and [hkij ] is in Mnk +1 (C[0, 1]) for k = 1, 2, . . . , q and satisfies hkjj (0) = w(ηjk ) for j = 0, 1, . . . , nk , and hkij (0) = 0 if i = j . Again D is clearly a closed unital ∗-subalgebra of C(∂D) ⊕ Mn1 +1 C[0, 1] ⊕ · · · ⊕ Mnq +1 C[0, 1] and thus a C ∗ -algebra itself. Hence Φ −1 is a one-to-one ∗-homomorphism from D into B(H 2 )/K with range A. Thus Φ −1 is isometric, A is closed, and so A = C ∗ (Tz , Cϕ1 , . . . , Cϕn )/K. If we again define Ψ (B) = Φ([B]) for any operator B in C ∗ (Tz , Cϕ1 , . . . , Cϕn ), we have the following. Theorem 2. Suppose that the directed graph associated to ϕ1 , . . . , ϕn has q components, each of which is an outgoing star as described above. With D and Ψ as defined above, there is a short exact sequence i

Ψ

0 → K → C ∗ (Tz , Cϕ1 , . . . , Cϕn ) → D → 0 of C ∗ -algebras, where i is inclusion. 5. Converting undirected trees to outgoing stars We turn to stage 2. Now suppose that we start with n maps with the property that the associated undirected graph G is a tree, i.e. G is connected and contains no cycles. We have the option of undirectedness at the outset for this reason: If some map in our collection takes ζ in ∂D to η in ∂D, then its Krein adjoint takes η to ζ . In view of Eq. (2), if we replace the original map by its Krein adjoint, we reverse the direction of the associated edge without changing the generated C ∗ -algebra. We first describe how we will label the maps and vertices in G in order to realize G as a directed tree outgoing from an arbitrarily chosen vertex. Then we will show how to convert the outgoing directed tree G into an outgoing star, bringing us into the purview of Theorems 1 and 2. Fix any vertex of G and denote it by ζ0 . Consider all of those maps from our original collection whose associated edges are incident to ζ0 . If ζ0 is an input point of one of these maps, leave it alone, but if ζ0 is a target point for a map, replace the map by its Krein adjoint. The resulting maps incident to ζ0 are relabeled ϕ1 , ϕ2 , . . . , ϕk0 ; we call this collection of maps the (unique) first generation outgoing star with central point ζ0 . The target vertices ϕj (ζ0 ), 1 j k0 , are labeled ζj . In the second generation, we will follow this procedure using appropriate remaining maps to form at most k0 outgoing stars, with respective centers chosen from ζ1 , . . . , ζk0 . We continue labeling our maps in increasing order as we proceed. For example, if we start at ζ1 , we consider all maps, excluding ϕ1 from the first generation, having edges incident to ζ1 . If there are none, no outgoing star centered at ζ1 is created. If there are some, any such maps that have ζ1 as target point are replaced by their Krein adjoint, the rest are unchanged, and the total collection is labeled ϕk0 +1 , . . . , ϕk1 . Their target vertices are labeled so that ϕj (ζ1 ) = ζj , k0 + 1 j k1 . Note that ϕk0 +1 , . . . , ϕk1 determines an outgoing star with central point ζ1 . Next turn to ζ2 . If ϕ2 is the only map incident to ζ2 , no outgoing star is created. Otherwise, excluding ϕ2 , and replacing as necessary a map by its Krein adjoint, we arrive at maps ϕk1 +1 , . . . , ϕk2 which determine an outgoing star centered at ζ2 . Repeat this procedure at ζ3 , . . . , ζk0 , completing the second generation.

2396


Fig. 3.

Each target point ζ of a second generation star will now serve as the center of a third generation star, unless there is only one edge incident to ζ , in which case no third generation star at ζ is created. Continuing, we see that eventually the process terminates, leaving us with a directed tree outgoing from ζ0 ; see Definition 5.1 below. A key point in the above construction is that maps ϕi and ϕj occurring, respectively, in an earlier and later generation always have i < j . The vertices of our directed outgoing tree are always labeled so that ζi is the target of ϕi . In particular, ϕ1 , . . . , ϕn and ζ0 , ζ1 , . . . , ζn satisfy the following: Definition 5.1. Suppose that ϕ1 , ϕ2 , . . . , ϕn satisfy Eq. (1). Assume that ζ0 , ζ1 , . . . , ζn in ∂D are distinct, ζ0 is the input point for ϕ1 and ζk is the target point of ϕk for k = 1, . . . , n. If for every k = 1, . . . , n there exists j < k with ϕk (ζj ) = ζk we say that ϕ1 , . . . , ϕn determine an outgoing tree centered at ζ0 . For example, in Fig. 3 there is a first generation star {ϕ1 , ϕ2 , ϕ3 } centered at the (arbitrarily chosen) point ζ0 . The second generation stars produced are {ϕ4 , ϕ5 } centered at ζ2 and {ϕ6 , ϕ7 , ϕ8 } centered at ζ3 . There is one third generation star, {ϕ9 , ϕ10 } centered at ζ5 , and one fourth generation star, {ϕ11 } centered at ζ10 . To summarize, we have shown that if the undirected graph associated to a collection of n maps is a tree having ζ0 as a vertex, upon replacing maps by their Krein adjoints as necessary, we may label the collection of maps as ϕ1 , . . . , ϕn so that they determine an outgoing tree centered at ζ0 . For the final step, we will need the following consequence of Proposition 2. Lemma 5. Let ζ0 , ζ1 and ζ2 be distinct points in ∂D and suppose that ϕ1 and ϕ2 are linearfractional non-automorphisms of D with ϕ1 (ζ0 ) = ζ1 and ϕ2 (ζ1 ) = ζ2 . If ψ = ϕ2 ◦ ϕ1 , then C ∗ (Cϕ1 , Cϕ2 , K) = C ∗ (Cϕ1 , Cψ , K). Proof. Clearly Cψ = Cϕ1 Cϕ2 lies in C ∗ (Cϕ1 , Cϕ2 , K), giving one containment. For the other, note that the Krein adjoint σ1 for ϕ1 satisfies σ1 (ζ1 ) = 1/ϕ1 (ζ0 ). Let β = ψ ◦ σ1 , and note that β(ζ1 ) = ϕ2 (ζ1 ) and β (ζ1 ) = ψ (ζ0 )σ1 (ζ1 ) = ϕ2 (ζ1 )ϕ1 (ζ0 )/ϕ1 (ζ0 ) = ϕ2 (ζ1 ).


2397

By Proposition 2, Cϕ2 ∈ C ∗ (Cβ , K) = C ∗ (Cσ1 Cψ , K) ⊆ C ∗ (Cσ1 , Cψ , K) = C ∗ (Cϕ1 , Cψ , K).

2

The next theorem completes stage 2. Theorem 3. Let ϕ1 , . . . , ϕn determine an outgoing tree centered at ζ0 and suppose that for each j = 1, . . . , n, ζj is the target point of ϕj . Then there exist maps ψ1 , . . . , ψn such that for each k = 1, . . . , n the following hold: (a) ψk lies in the semigroup under composition generated by ϕ1 , . . . , ϕk . In particular, either ψk = ϕk or ψk = ϕk ◦ ψj where j < k. (b) ψk (ζ0 ) = ζk for k = 1, . . . , n. (c) C ∗ (Cψ1 , . . . , Cψk , K) = C ∗ (Cϕ1 , . . . , Cϕk , K) for k = 1, . . . , n. Note that condition (b) states that ψ1 , . . . , ψn determine an outgoing star with center ζ0 . Proof. We proceed by induction. Clearly (a), (b), and (c) hold for k = 1, if we put ψ1 = ϕ1 . Suppose that ψ1 , . . . , ψk satisfy the conclusions of the theorem for all k m − 1, where m n. By Definition 5.1, either ϕm (ζ0 ) = ζm or there exists positive j < m with ϕm (ζj ) = ζm . In the former case, put ψm = ϕm . The induction hypothesis guarantees that (a), (b), and (c) hold for this m. In the latter case, set ψm = ϕm ◦ ψj . Lemma 5 now implies that C ∗ (Cψj , Cϕm , K) = C ∗ (Cψj , Cψm , K). This equation, together with (c) for index m − 1, implies (c) for index m. Conditions (a) and (b) for index m are immediate. 2 Suppose ϕ1 , ϕ2 , . . . , ϕn form an outgoing tree centered at ζ0 , as defined by Definition 5.1. Let ψ1 , ψ2 , . . . , ψn be as in Theorem 3, so that ψ1 , . . . , ψn form an outgoing star centered at ζ0 . Let Ψ : C ∗ (Tz , Cψ1 , . . . , Cψn ) → D = ran Ψ ⊆ C(∂D) ⊕ Mn+1 C [0, 1] be associated to this outgoing star as in Section 3. There remains the problem of explicitly calculating the value of Ψ at each of the original operators Cϕ1 , . . . , Cϕn . We turn to this next. In the following lemma and theorem, we sometimes write τ (ρ) for the translation number of a parabolic map ρ. Lemma 6. Suppose that ψk = ϕk ◦ ψj , where j < k, ψj (ζ0 ) = ζj and ϕk (ζj ) = ζk . Let ai denote the translation number of σi ◦ ψi , where σi is the Krein adjoint of ψi , and write ϕk for the Krein adjoint of ϕk . Then we have ak = aj + ψk (ζ0 )τ (ϕk ◦ ϕk ). Proof. Taking Krein adjoints in the equation ψk = ϕk ◦ ψj yields σk = σj ◦ ϕk . Thus we have ψk ◦ σk = ϕk ◦ ψj ◦ σj ◦ ϕk .

(23)

2398


By Proposition 1(a) and (b), τ (ψi ◦ σi ) =

ai τ (σi ◦ ψi ) = |ψi (ζ0 )| |ψi (ζ0 )|

for i = j, k, and in particular, ψj ◦ σj = ρζj ,aj /|ψj (ζ0 )| . If we insert this formula into Eq. (23) and use Proposition 1(c), we find ψk ◦ σk = ρζk ,γ ◦ ϕk ◦ ϕk where γ=

aj aj = . |ϕk (ζj )||ψj (ζ0 )| |ψk (ζ0 )|

(24)

By the semigroup property of the parabolic maps, ak = τ (ψk ◦ σk ) = γ + τ (ϕk ◦ ϕk ). |ψk (ζ0 )| The conclusion follows upon multiplying by |ψk (ζ0 )|.

2

In the next theorem we write Ej,k for the (n + 1) × (n + 1) matrix unit with a 1 in the (j, k) position and 0’s elsewhere, for 0 j, k n. Recall that by our labeling convention (e.g. Eq. (16)), the upper left corner is the (0, 0) entry. We retain the notation of Theorem 3 and Lemma 6 in the next result. Theorem 4. Suppose that ϕ1 , . . . , ϕn form an outgoing tree centered at ζ0 , and that ψ1 , . . . , ψn is the related outgoing star centered at ζ0 as constructed in Theorem 3. Define Ψ : C ∗ (Tz , Cψ1 , . . . , Cψn ) → C(∂D) ⊕ Mn+1 C[0, 1] (B) for the coordinate as in Section 3, and for any operator B in C ∗ (Tz , Cψ1 , . . . , Cψn ) write Ψ of Ψ (B) that lies in Mn+1 (C([0, 1])). If ψk = ϕk , then 1 (Cϕk )(t) = sk t ak 2 E0,k = Ψ

1 t pk E0,k |ϕk (ζ0 )|1/2

(25)

for 0 t 1, where pk = |ϕk (ζ0 )|(κϕk − 1). If ψk = ϕk ◦ ψj for some 1 j < k, then (Cϕk )(t) = Ψ

sk sj

1 2

t (ak −aj )/2 Ej,k =

1 |ϕk (ζj )|1/2

t pk Ej,k

(26)


2399

for 0 t 1, where pk = |ψk (ζ0 )|(κϕk − 1). In both cases, si = 1/|ψi (ζ0 )|, ai is the translation number of σi ◦ ψi where σi is the Krein adjoint of ψi , and κϕk is the curvature of the circle ϕk (∂D). Proof. By our construction of an outgoing star from an outgoing tree as described in Theorem 3, we have either ϕk = ψk or ψk = ϕk ◦ ψj for some positive j < k. In the first case, by the polar decomposition of Cψk and the definition of Ψ , (Cϕk )(t) = Ψ (Cψk )(t) = sk t ak 1/2 E0,k Ψ where sk = 1/|ψk (ζ0 )| = 1/|ϕk (ζ0 )|, and ak is the translation number of σk ◦ ψk . By Proposition 1, ak = 2|ϕk (ζ0 )|(κϕk − 1), and the alternate expression in (25) follows. On the other hand, if ψk = ϕk ◦ ψj for some 1 j < k, Proposition 1 implies that ψk ◦ σj = ϕk ◦ ψj ◦ σj = ϕk ◦ ρζj ,aj /|ψj (ζ0 )| = ρζk ,γ ◦ ϕk with γ as in Eq. (24). Thus ϕk = ρζk ,−γ ◦ ψk ◦ σj .

(27)

By the proof of Lemma 6, the translation numbers τ (ψk ◦ σk ) and τ (ϕk ◦ ϕk ) are related by τ (ψk ◦ σk ) = γ + τ (ϕk ◦ ϕk ),

(28)

where ϕk is the Krein adjoint of ϕk . Thus we can choose with 0 < < γ so that τ (ψk ◦ σk ) > γ + . If we set γ1 = (γ + )/2 and γ2 = (γ − )/2, note that γ1 + γ2 = γ , and use Proposition 1(c), we can rewrite Eq. (27) as ϕk = ρζk ,−γ1 ◦ ρζk ,−γ2 ◦ ψk ◦ σj = (ρζk ,−γ1 ◦ ψk ) ◦ (ρζ0 ,−γ2 |ψk (ζ0 )| ◦ σj ). We want to show that each of the composition factors in parentheses here is a non-automorphism self-map of D. For the factor ρζk ,−γ1 ◦ ψk it is enough to check (by Proposition 1 and the paragraph following Proposition 2) that 1 −γ1 > − τ (ψk ◦ σk ). 2 This inequality follows from our choice of and γ1 . Similarly, for the factor ρζ0 ,−γ2 |ψk (ζ0 )| ◦ σj , we need to check that aj 1 −γ2 ψk (ζ0 ) > − τ (σj ◦ ψj ) = − 2 2

2400


or equivalently, (γ − )|ψk (ζ0 )| < aj ; this follows from Eq. (24). As a consequence, the operator factorization Cϕk = Cρζ

◦σj 0 ,−γ2 |ψk (ζ0 )|

Cρζk ,−γ1 ◦ψk

makes sense. Now, for i = 1, 2, let μi = γi |ψk (ζ0 )|. By Table I, μ1 1 − μ1 a Cρζk ,−γ1 ◦ψk ≡ sk k Uk Cψ∗ k Cψk 2 ak

(mod K)

μ1 1 − μ1 a = sk k Cψk Cψ∗ k 2 ak Uk

and Cρζ

◦σj 0 ,−γ2 |ψk (ζ0 )|

μ2 aj −1

≡ sj

1 − μ2 Uj∗ Cψj Cψ∗ j 2 aj

(mod K).

(B) = Φ([B]) Since Ψ when B is in C ∗ (Tz , Cψ1 , . . . , Cψn ), we have μ1 1 μ1 u∗0 xk xk∗ 2 − ak uk (t) (Cρζ ,−γ ◦ψk )(t) = s ak Φ Ψ k k 1 μ1 1 − μ1 a = sk k sk t ak 2 ak E0,k 1

= sk2 t

ak 2

−μ1

E0,k ,

for 0 t 1

and (Cρ Ψ ζ

◦σj 0 ,−γ2 |ψk (ζ0 )|

μ2 aj −1

1 μ2 u∗j xj xj∗ 2 − aj u0 (t) Φ

μ2 a −1

a 12 − μa 2 j E sj t j j,0

)(t) = sj

= sj j

−1

= sj 2 t

aj 2

−μ2

for 0 t 1.

Ej,0 ,

preserves products and μ1 + μ2 = aj (see Eq. (24)), Since Ψ 1 a 1 aj (Cϕk )(t) = s − 2 t 2 −μ2 Ej,0 s 2 t 2k −μ1 E0,k Ψ j k =

sk sj

1 2

t (ak −aj )/2 Ej,k .

Since ψk = ϕk ◦ ψj , we have sk /sj = 1/|ϕk (ζj )|. By Lemma 6 and Proposition 1, ak − aj = ψk (ζ0 )τ (ϕk ◦ ϕk ) = 2ψk (ζ0 )(κϕk − 1). (Cϕk ) in Eq. (26) then follows. The alternate formula for Ψ

2


2401

As a special case of this theorem, suppose that our starting outgoing tree is a chain; that is, that our initial maps ϕ1 , . . . , ϕn satisfy ϕ1 (ζ0 ) = ζ1 , ϕ2 (ζ1 ) = ζ2 , . . . , ϕn (ζn−1 ) = ζn for distinct points ζ0 , . . . , ζn in ∂D. In converting to an outgoing star we have ψ1 = ϕ1 , and (Cϕk ) as ψj = ϕj ◦ ψj −1 for j 2. For k 2 we recover Ψ (Cϕk )(t) = Ψ

sk sk−1

1 2

t (ak −ak−1 )/2 Ek−1,k .

6. Essential spectra for operators in C ∗ (Tz , Cϕ1 , . . . , Cϕn ) We consider the general set-up in Section 4. The graph G associated to the maps ϕ1 , . . . , ϕn splits into q components G1 , . . . , Gq , 1 q n. We re-label our maps as in the display (20) so that ϕ1k , . . . , ϕnkk correspond to the nk edges of Gk . Upon converting each Gk to an outgoing star as in Section 5, we can define our ∗-homomorphism Ψ : C ∗ (Tz , Cϕ1 , . . . , Cϕn ) → C(∂D) ⊕ Mn1 +1 C[0, 1] ⊕ · · · ⊕ Mnq +1 C[0, 1] . Any operator B in C ∗ (Tz , Cϕ1 , . . . , Cϕn ) has a representation B = Tw + Y1 + · · · + Yq

(29)

where w ∈ C(∂D), and for k = 1, . . . , q, Yk lies in C0∗ (Cϕ k , . . . , Cϕnk , K), the nonunital C ∗ 1 k algebra generated by Cϕ k , . . . , Cϕnk and the ideal K. The Toeplitz summand is uniquely deter1 k mined by B, and each Yk is unique up to compact perturbation. In the representation (22) for b = [B], nk

k ∗ k k k ui fij z uj = [Yk ] i,j =0

k (B) denote the coordinate of Ψ (B) lying and Eq. (21) implies that Yj Yk ∈ K if j = k. Let Ψ 0 (B) = w so that ψ(B) = (Ψ 0 (B), Ψ 1 (B), . . . , in Mnk +1 (C[0, 1]) for k = 1, . . . , q. Also put Ψ q (B)). It is easy to verify that Ψ C0∗ (Cϕ k , . . . , Cϕnk , K) = 1

k

j Ker Ψ

0j q, j =k

for k = 1, . . . , q. We denote the j × j identity matrix by Ij . Theorem 5. Suppose that B ∈ C ∗ (Tz , Cϕ1 , . . . , Cϕn ) and that for 1 k q, k (B)(t) − λInk +1 = 0 for some t in [0, 1] . Xk = λ ∈ C: det Ψ

2402


Then σe (B) = w(∂D) ∪ X1 ∪ · · · ∪ Xq . Proof. By Theorem 2 and spectral permanence, the spectrum of [B] in B(H)/K coincides with the spectrum of Ψ (B) in C(∂D) ⊕ Mn1 +1 C[0, 1] ⊕ · · · ⊕ Mnq +1 C[0, 1] . Moreover, an element of this direct sum is invertible exactly when each of its coordinates is pointwise invertible. The result follows. 2 If for each k = 1, . . . , q the Toeplitz symbol w is constant on the set of vertices of the subgraph Gk , then σe (B) can be expressed directly in terms of σe (Tw ) and σe (Yk ), k = 1, . . . , q. For any complex number c and subset E ⊆ C, write c + E for the translate {c + λ: λ ∈ E}. Corollary 1. Let B in C ∗ (Tz , Cϕ1 , . . . , Cϕn ) be given by Eq. (29). Suppose that for each k = 1, . . . , q, the Toeplitz symbol w takes the same value ck at each vertex of the subgraph Gk . Then σe (B) = σe (Tw ) ∪ c1 + σe (Y1 ) ∪ · · · ∪ cq + σe (Yq ) . k (Yj ) = 0 if j = k. By the definition of Ψ , Proof. Note that Ψ k (B) = ck Ink +1 + Ψ k Ψ

q

Yj

k (Yk ). = ck Ink +1 + Ψ

j =0

k (Yk ) in Mnk +1 (C[0, 1]) and σe (B) By Theorem 4, σe (Yk ) coincides with the spectrum of Ψ coincides with the spectrum of (w, Ψ1 (B), . . . , Ψq (B)). 2 Example 1. Our first example uses the map ϕ(z) = into itself, maps each point ζj = e2πi j/n ,

zn +n−1 n

where n 2, which takes the disk D

j = 0, 1, . . . , n − 1,

to the point 1, and maps no other points of ∂D into ∂D. For each j = 0, . . . , n − 1 we find the unique linear-fractional self-map σj of D with σj (ζj ) = 1,

σj (ζj ) = ϕ (ζj ) = ζj

and σj (ζj ) = ϕ (ζj ) = (n − 1)ζj 2


2403

so that ϕ and σj have the same second order data at ζj . By direct computation we determine that σj (z) =

(n − 3)ζj z − n + 1 (n − 1)ζj z − n − 1

,

which has Krein adjoint ψj (z) = ζj

(n − 3)z − n + 1 . (n − 1)z − n − 1

In particular, σ0 (z) = ψ0 (z) = ((n − 3)z − n + 1)/((n − 1)z − n − 1), a positive parabolic map fixing 1 and having translation number n − 1. For each j = 0, 1, . . . , n − 1, σj ◦ ψj has the form (2 − n)z + n − 1 , (1 − n)z + n

(σj ◦ ψj )(z) =

a parabolic non-automorphism fixing 1 and having translation number aj = 2n − 2. The maps ψ1 , ψ2 , . . . , ψn−1 determine an outgoing star graph centered at ζ0 = 1 with ψj (ζ0 ) = ζj for j = 1, . . . , n − 1, and we can invoke the setup in Section 3. Note that sj = 1/|ψj (ζ0 )| = 1 for each j . For j 1, let xj = [Cψj ], so that xj∗ = [Cσj ] and xj xj∗ = [Cσj ◦ψj ]. By Proposition 1, the circles ψj (∂D) and σj (∂D) have common curvature κj = n, j = 0, . . . , n − 1. By the construction of σ0 , . . . , σn−1 and Corollary 5.16 of [23] (discussed above in the introduction), Cϕ ≡ Cσ0 + Cσ1 + · · · + Cσn−1

(mod K).

By Proposition 2, the operator Cσ0 lies in C ∗ (Cψ1 , K). It follows from Proposition 2 and Table I that 1/2 Cσ0 ≡ Cψ1 Cψ∗ 1 Passing to cosets modulo K we have [Cσ0 ] =

(mod K).

x1 x1∗ and

∗ x1 x1∗ + x1∗ + · · · + xn−1 ∗ u . = u∗0 x1 x1∗ u0 + u∗1 x1 x1∗ u0 + · · · + u∗n−1 xn−1 xn−1 0

[Cϕ ] =

Thus, in the notation of Eq. (9), f00 (z) = x1 x1∗ , fj 0 (z) = xj xj∗ for j = 1, . . . , n − 1 and 2n−2 , . . . , t 2n−2 ) = t n−1 for k = 0, j 0 fj k (z) = 0 for k 1 and all j . Note that f j k (t) = fj k (t and fj k (z) = 0 for k 1, j 0. Therefore ⎡

t n−1 ⎢ t n−1 (Cϕ )(t) = ⎢ . Ψ ⎣ .. t n−1

⎤ 0 ··· 0 0 ··· 0⎥ ⎥ ∈ Mn . ⎦ 0 ··· 0

We compute the essential spectra of several operators associated with Cϕ .

2404


(Cϕ )(t) above and Theorem 5, it is immediate (a) First consider Cϕ itself. From the formula for Ψ that σe (Cϕ ) = [0, 1]. (b) Consider next the problem of finding the essential spectrum of the operator A = Cϕ + Cϕ∗ . We have ⎡ n−1 n−1 ⎤ 2t t · · · t n−1 ⎢ t n−1 0 ··· 0 ⎥ ⎥. (A)(t) = ⎢ . Ψ ⎣ .. ⎦ t n−1

···

0

0

(A)(t) − The essential spectrum σe (A) consists of those points λ for which the determinant of Ψ λIn is zero for some t ∈ [0, 1]. Note that (A)(t) − λIn = 2t n−1 − λ (−λ)n−1 − (n − 1)t 2n−2 (−λ)n−2 det Ψ √ and this is zero for√ λ = 0 and √ λ = t n−1 ± nt n−1 . As t ranges over [0, 1], λ sweeps out the line segment from 1 − n to 1 + n. In particular we see that the essential spectrum of the real part of Cϕ is the interval σe (Re Cϕ ) =

√ √ 1− n 1+ n , . 2 2

(c) For an operator of the form B = Tw + Cϕ − Cϕ∗ we have ⎡ (B)(t) = ⎢ Ψ ⎣

0

w(ζ0 ) ..

0

.

⎡

⎤

0

t n−1 ⎥ ⎢ ⎦+⎢ ⎣ ... w(ζn−1 ) t n−1

−t n−1 0 0

⎤ · · · −t n−1 ··· 0 ⎥ ⎥. ⎦ ···

0

(B)(t) are seen to be the roots of The eigenvalues of Ψ n−1

n−1

2n−2 w(ζj ) − λ + t

j =0

n−1

w(ζi ) − λ .

(30)

j =1 i=1,i=j

It follows from Theorem 5 and a brief calculation that if

1 −1 , w(ζ0 ) − λ w(ζj ) − λ n−1

f (λ) =

j =1

then σe (B) = w(∂D) ∪ f −1 [1, ∞) . When w ≡ 0, the above formula shows that σe (Cϕ − Cϕ∗ ) is the imaginary line segment √ √ [− n − 1 i, n − 1 i]. On choosing w(z) = zn , we see that the essential spectrum of


2405

σe (Tzk + Cϕ − Cϕ∗ ), ϕ(z) = (zn + n − 1)/n. Fig. 4.

√ Tzn + Cϕ − Cϕ∗ consists of the unit circle and the vertical line segment from 1 − n − 1 i to √ 1 + n − 1 i (this also follows from Corollary 1). For B = Tzk + Cϕ − Cϕ∗ and various choices of k and n, numerically sampled subsets of σe (B) are shown in Fig. 4; for more precise pictures, “connect the dots”. (d) Next we consider the self-commutator [Cϕ∗ , Cϕ ] = Cϕ∗ Cϕ − Cϕ Cϕ∗ , where ϕ is as above. Since is a ∗-homomorphism, Ψ ⎡

(1 − n)t 2n−2 ⎢ t 2n−2 ⎢ ∗ .. ⎢ Cϕ , Cϕ (t) = Ψ . ⎢ ⎣ t 2n−2 t 2n−2

t 2n−2 t 2n−2 ··· t 2n−2

⎤ · · · t 2n−2 · · · t 2n−2 ⎥ ⎥ ⎥. ⎥ t 2n−2 ⎦

(31)

· · · t 2n−2

To determine σe ([Cϕ∗ , Cϕ ]), we must compute the determinant of the n × n matrix M(t) − λIn , where M(t) is the matrix on the right-hand side in Eq. (31). To facilitate this, we perform n − 1 row operations on M(t) − λIn , subtracting the nth row from the j th row, for j = 1, 2, . . . , n − 1. The resulting matrix, which has the same determinant as M(t) − In , is ⎡

(−nt 2n−2 − λ) ⎢ 0 ⎢ .. ⎢ . ⎢ ⎣ 0 t 2n−2

0 −λ ··· t 2n−2

··· ···

0 0

0 −λ · · · t 2n−2

λ λ λ (t 2n−2 − λ)

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

where for j = 2, . . . , n − 1, the j th row has two non-zero entries. The determinant of this matrix is readily calculated; setting it equal to 0 gives λn−2 (−1)n λ2 + (−1)n−1 n(n − 1)t 4n−4 = 0

2406


σe (Tz + [Cϕ∗ , Cϕ ]), ϕ(z) = (zn + n − 1)/n. Fig. 5.

√ whose solutions are√λ = 0 and λ√= ± n(n − 1) t 2n−2 . Letting t range over the unit interval gives σe ([Cϕ∗ , Cϕ ]) = [− n(n − 1), n(n − 1)]. (e) Finally consider the operator D = Tz + [Cϕ∗ , Cϕ ]. We have ⎡ (D)(t) = ⎢ Ψ ⎣

0

ζ0 ..

0

.

⎤ ⎥ ⎦ + M(t)

ζn−1

with M(t) as above. In the case n = 3, a modification of the above calculations show that the essential spectrum of D is the union of ∂D with the set of solutions to ζ0 − 3t 4 − z (ζ1 − z) ζ2 + t 4 − z − t 4 (ζ1 − z)(z − ζ2 ) − t 4 (z − ζ2 ) ζ0 − 3t 4 − z = 0 as t ranges over [0, 1]. Equivalently, the essential spectrum is equal to the union of ∂D with the collection of roots of the depressed cubic p(z) = z3 + 3r(1 − 2r)z + 3r(1 − r) − 1 as r ≡ t 4 ranges over [0, 1]. For values of r in the interval [0, r0 ), where r0 is approximately 0.7397 (the root of the discriminant of p(z) in the range [0, 1]), there is one real root and two complex conjugate roots. For values of r in (r0 , 1] there are three distinct real roots. For r = r0 there is a negative root of multiplicity two, and one positive root. Numerically sampled pictures of σe (Tz + [Cϕ∗ , Cϕ ]) for n = 3, 4, and 5 appear in Fig. 5. The darkening in pictures (b) and (c) is an artifact of magnifying Mathematica output. Example 2. This example makes use of the special case discussed following Theorem 4. Begin with a chain of linear-fractional non-automorphisms ϕj of D, 1 j n, with ϕj (ζj −1 ) = ζj for distinct points ζ0 , . . . , ζn on ∂D. Complete the chain to a loop by including an (n + 1)st linear-fractional non-automorphism ϕn+1 of D mapping ζn to ζ0 that is required to satisfy (ζn )ϕn (ζn−1 ) · · · ϕ2 (ζ1 )ϕ1 (ζ0 ) = 1. (ϕn+1 ◦ ϕn ◦ · · · ◦ ϕ1 ) (ζ0 ) = ϕn+1

(32)

Define the maps ψ1 , . . . , ψn by ψ1 = ϕ1 and ψk = ϕk ◦ ϕk−1 ◦ · · · ◦ ϕ1 = ϕk ◦ ψk−1 for 2 k n. The associated graph is an outgoing star centered at ζ0 . Let σk and ϕk denote the Krein adjoints


2407

of ψk and ϕk , respectively. Eq. (32) states that the composition ϕn+1 ◦ ϕn ◦ · · · ◦ ϕ1 is parabolic, so that ϕn+1 ◦ ϕn ◦ · · · ◦ ϕ1 = ϕn+1 ◦ ψn = ρζ0 ,α for some α with positive real part. Eq. (32) also forces ϕn+1 and σn to have the same first order (ζn ) = 1/ψn (ζ0 ) = σn (ζn ). By the discussion following Proposition 2, we can data at ζn : ϕn+1 write ϕn+1 = ρ ◦ σn , where ρ = ρζ0 ,β is a parabolic linear-fractional map fixing ζ0 (but not necessarily mapping D into D), or (in the case β = 0) the identity map. Also by Proposition 2, Cϕn+1 ∈ C ∗ (Cψn , K) ⊆ C ∗ (Cϕ1 , . . . , Cϕn , K). Let w ∈ C(∂D) and suppose that c1 , . . . , cn+1 are non-zero complex numbers. We will find the essential spectrum of B = Tw + c1 Cϕ1 + · · · + cn+1 Cϕn+1 .

(33)

Since ϕn+1 = ρζ0 ,β ◦ σn , Proposition 1(b) and the last line in Table I imply that − aβn −1

Cϕn+1 ≡ sn

1+ β Un∗ Cψn Cψ∗ n 2 an

(mod K)

where ak is the translation number of σk ◦ ψk for k = 1, . . . , n. The above relations between ψn , σn and ϕn+1 yield ρζ0 ,α = ϕn+1 ◦ ψn = ρζ0 ,β ◦ σn ◦ ψn = ρζ0 ,β ◦ ρζ0 ,an so that α = β + an . Consequently, − aαn

Cϕn+1 ≡ sn

α −1 Un∗ Cψn Cψ∗ n an 2

(mod K);

necessarily, Re α > an /2. Thus α 1 u∗n xn xn∗ an − 2 u0 (t) Φ α −1 −α = sn an sn t an an 2 En,0 − aαn

(Cϕn+1 )(t) = sn Ψ

−1

an

= sn 2 t α− 2 En,0 . (B)(t) is equal to It follows that Ψ √ ⎡ w(ζ0 ) c1 s1 t a1 ⎢ 0 w(ζ1 ) ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣ −1

cn+1 sn 2 t α−

an 2

0

0 c2 ss21 t (a2 −a1 )

··· ···

··· w(ζn−1 ) ···

0

⎤

⎥ ⎥ ⎥ ⎥ 0 sn (an −an−1 ) ⎥ ⎥ cn sn−1 t ⎦ 0

w(ζn )

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so that (B)(t) − λIn+1 = (−1)n+1 p(λ) − (c1 c2 · · · cn+1 )t α det Ψ where p(λ) =

!n

k=0 (λ − w(ζk )).

It follows from Theorem 5 that σe (B) = w(∂D) ∪ p −1 (Λ),

where Λ = {(c1 c2 · · · cn+1 )t α : 0 t 1}. The curve Λ is the complex line segment from c1 c2 · · · cn+1 to 0 if α is real, and is otherwise the union of {0} and a spiral starting at c1 c2 · · · cn+1 and converging to the origin. Acknowledgments We are grateful to William T. Ross of the University of Richmond for writing the Mathematica program which produced Figs. 4 and 5. We also thank the referee for suggestions which simplified the proofs of Proposition 1 and Lemma 1. References [1] E. Basor, D. Retsek, Extremal non-compactness of composition operators with linear fractional symbol, J. Math. Anal. Appl. 322 (2006) 749–763. [2] P. Bourdon, Components of linear-fractional composition operators, J. Math. Anal. Appl. 279 (2003) 228–245. [3] P. Bourdon, E. Fry, C. Hammond, C. Spofford, Norms of linear-fractional composition operators, Trans. Amer. Math. Soc. 356 (2004) 2459–2480. [4] P. Bourdon, D. Levi, S. Narayan, J. Shapiro, Which linear fractional composition operators are essentially normal? J. Math. Anal. Appl. 280 (2003) 30–53. [5] P. Bourdon, B. MacCluer, Selfcommutators of automorphic composition operators, Complex Var. Elliptic Equ. 52 (2007) 85–104. [6] P. Bourdon, J. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (596) (1997). [7] J. Clifford, D. Levi, S. Narayan, Commutator of composition operators, preprint. [8] J. Clifford, D. Zheng, Composition operators on the Hardy space, Indiana Univ. Math. J. 48 (1999) 1585–1616. [9] L. Coburn, The C ∗ -algebra generated by an isometry I, Bull. Amer. Math. Soc. 73 (1967) 722–726. [10] L. Coburn, The C ∗ -algebra generated by an isometry II, Trans. Amer. Math. Soc. 137 (1969) 211–217. [11] J. Conway, A Course in Operator Theory, Amer. Math. Soc., Providence, 1999. [12] C. Cowen, Composition operators on H 2 , J. Operator Theory 9 (1983) 77–106. [13] C. Cowen, Linear fractional composition operators on H 2 , Integral Equations Operator Theory 11 (1988) 151–160. [14] C. Cowen, T. Kriete, Subnormality and composition operators on H 2 , J. Funct. Anal. 81 (1988) 298–319. [15] C. Cowen, B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [16] E. Gallardo-Guttiérez, M. González, P. Nieminen, E. Saksman, On the connected component of compact composition operators on the Hardy space, Adv. Math. 219 (2008) 986–1001. [17] J. Guyker, On reducing subspaces of composition operators, Acta Sci. Math. (Szeged) 53 (1989) 369–376. [18] C. Hammond, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged) 69 (2003) 813–829. [19] M. Jury, C ∗ -algebras generated by groups of composition operators, Indiana Univ. Math. J. 56 (2007) 3171–3192. [20] M. Jury, The Fredholm index for elements of Toeplitz-composition C ∗ -algebras, Integral Equations Operator Theory 58 (2007) 341–362. [21] T. Kriete, B. MacCluer, J. Moorhouse, Toeplitz-composition C ∗ -algebras, J. Operator Theory 58 (2007) 135–156. [22] T. Kriete, B. MacCluer, J. Moorhouse, Composition operators within singly generated composition C ∗ -algebras, Israel J. Math., in press, arXiv:math/0610077. [23] T. Kriete, J. Moorhouse, Linear relations in the Calkin algebra for composition operators, Trans. Amer. Math. Soc. 359 (2007) 2915–2944.


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[24] B. MacCluer, Components in the space of composition operators, Integral Equations Operator Theory 12 (1989) 725–738. [25] B. MacCluer, J. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986) 878–906. [26] B. MacCluer, R. Weir, Essentially normal composition operators on Bergman spaces, Acta Sci. Math. (Szeged) 70 (2004) 799–817. [27] B. MacCluer, R. Weir, Linear-fractional composition operators in several variables, Integral Equations Operator Theory 53 (2005) 373–402. [28] A. Montes-Rodríguez, M. Ponce-Escudero, S. Shkarin, Invariant subspaces of parabolic self-maps in the Hardy space, preprint. [29] J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005) 70–92. [30] J. Moorhouse, C. Toews, Differences of composition operators, in: Trends in Banach Spaces and Operator Theory, Memphis, TN, 2001, in: Contemp. Math., vol. 321, Amer. Math. Soc., Providence, RI, 2003, pp. 207–213. [31] P.J. Nieminen, E. Saksman, On the compactness of the difference of composition operators, J. Math. Anal. Appl. 298 (2004) 501–522. [32] E. Nordgren, P. Rosenthal, F. Wintrobe, Invertible composition operators on H p , J. Funct. Anal. 73 (1987) 324–344. [33] A. Richman, Subnormality and composition operators on the Bergman space, Integral Equations Operator Theory 45 (2003) 105–124. [34] D. Sarason, Composition operators as integral operators, in: C. Sadosky (Ed.), Analysis and Partial Differential Equations, Marcel Dekker, New York, 1990, a volume dedicated to M. Cotlar. [35] J.H. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987) 375–404. [36] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. [37] J.E. Shapiro, Aleksandrov measures used in essential norm inequalities for composition operators, J. Operator Theory 40 (1998) 133–146. [38] J.H. Shapiro, C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990) 117–152. [39] J.H. Shapiro, C. Sundberg, Compact composition operators on L1 , Proc. Amer. Math. Soc. 108 (1990) 443–449. [40] J.H. Shapiro, P. Taylor, Compact, nuclear, and Hilbert–Schmidt composition operators on H 2 , Indiana Univ. Math. J. 23 (1973) 471–496.


Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces ✩ Jan Maas, Jan van Neerven ∗ Delft Institute of Applied Mathematics, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands Received 11 November 2008; accepted 1 July 2009 Available online 24 July 2009 Communicated by I. Rodnianski Dedicated to Professor Alan Mc Intosh on the occasion of his 65th birthday

Abstract Let (E, H, μ) be an abstract Wiener space and let DV := V D, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space H . Given a bounded operator B on H , coercive on the range R(V ), we consider the operators A := V ∗ BV in H and A := V V ∗ B in H , as well as the realisations of the operators L := DV∗ BDV and L := DV DV∗ B in Lp (E, μ) and Lp (E, μ; H ) respectively, where 1 < p < ∞. Our main result asserts that the following four assertions are equivalent: (1) (2) (3) (4)

√

√

√

D( L) = D(DV ) with Lf p DV f p for f ∈ D( L); ∞ -functional calculus on R(D ); L admits a bounded H √ √ √ V D( A) = D(V ) with Ah V h for h ∈ D( A); A admits a bounded H ∞ -functional calculus on R(V ).

Moreover, if these conditions are satisfied, then D(L) = D(DV2 ) ∩ D(DA ). The equivalence (1)–(4) is a nonsymmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where H = √H , V = I , B = 12 I ). A one-sided version of (1)–(4), giving Lp -boundedness of the Riesz transform DV / L in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C0 -contraction

✩

The authors are supported by VIDI subsidy 639.032.201 (JM and JvN) and VICI subsidy 639.033.604 (JvN) of the Netherlands Organisation for Scientific Research (NWO). The first named author acknowledges partial support by the ARC Discovery Grant DP0558539. The second named author was partially supported by the ARC Discovery Grant DP0559465. * Corresponding author. E-mail addresses: [email protected] (J. Maas), [email protected] (J. van Neerven). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.001

J. Maas, J. van Neerven / Journal of Functional Analysis 257 (2009) 2410–2475

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semigroup on a Hilbert space H and let −L be the Lp -realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an Lp -domain characterisation for the operator L. © 2009 Elsevier Inc. All rights reserved. Keywords: Divergence form elliptic operators; Abstract Wiener spaces; Riesz transforms; Domain characterisation in Lp ; Kato square root problem; Ornstein–Uhlenbeck operator; Meyer inequalities; Second quantised operators; Square function estimates; H ∞ -functional calculus; R-boundedness; Hodge–Dirac operators; Hodge decomposition

1. Introduction Let (E, H, μ) be an abstract Wiener space, i.e., E is a real Banach space and μ is a centred Gaussian Radon measure on E with reproducing kernel Hilbert space H . In this paper we prove square function estimates and boundedness of Riesz transforms for abstract second order elliptic operators L in divergence form acting on Lp (E, μ), 1 < p < ∞. Our main result (Theorem 2.1) gives necessary and sufficient conditions for the domain equality √ D( L) = D(DV ) (1.1) in Lp (E, μ) with equivalence of norms √ Lf p DV f p

(1.2)

for a class of divergence form elliptic operators of the form L = DV∗ BDV . Here DV := V D, where D is the Malliavin derivative in the direction of H , V : D(V ) ⊆ H → H is a closed and densely defined operator, and B is a bounded operator on H which is coercive on R(V ). Our main result asserts that (1.1) and (1.2) hold if and only if the sectorial operator A := V V ∗ B on H admits a bounded H ∞ -functional calculus. In particular, if (1.1) and (1.2) hold for one 1 < p < ∞, then they hold for all 1 < p < ∞. By well-known examples, cf. [38, Theorem 4 and its proof], sectorial operators on H of the form T B with T : D(T ) ⊆ H → H positive and self-adjoint and B coercive on H need not always have a bounded H ∞ -calculus. In our setting, such examples can √ be translated into examples of operators L for which (1.2) fails (e.g., take H = H and V = T ). Returning to (1.2), we shall prove the more precise result that the inclusions √ D( L) → D(DV ), respectively √

D( L) ← D(DV ),

hold in Lp (E, μ) if and only if the operator A satisfies a lower, respectively upper square function estimate in R(V ). The simplest example to which our results apply is the classical Ornstein–Uhlenbeck operator of Malliavin calculus. This example is obtained by taking H = H , V = I , and B = 12 I . With

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these choices, DV reduces to the Malliavin derivative D in the direction of H and L = 12 D ∗ D is the classical Ornstein–Uhlenbeck operator. The equivalences (1.1) and (1.2) then reduce to the classical Meyer inequalities [40]. Various proofs of these inequalities have been given; see, e.g., [21,47]. For further references on this subject we refer to Nualart [45]. A second and non-trivial application concerns the computation of the Lp -domains of second quantised operators. Let (E, H, μ) be an abstract Wiener space and suppose that S = (S(z))z∈Σ is an analytic C0 -contraction semigroup defined on the closed sector Σ . By this we mean that S is a C0 -semigroup of contractions on Σ which is analytic on the interior of Σ . Let −A denote the generator of S. For 1 < p < ∞, by second quantisation (see Section 3 for the details) we obtain an analytic C0 -contraction semigroup (Γ (S(t)))z∈Σ , with generator −L, on Lp (E, μ). As we will show, the operators A and L are always of the form A = V ∗ BV and L = DV∗ BDV for suitable choices of V and B, and Theorem 2.1 implies that √

D( L) = D(DV )

√ with Lf p DV f p ,

1 < p < ∞,

(1.3)

if and only if A admits a bounded H ∞ -calculus on the homogeneous form domain associated with A. As before, one-sided versions of this result can be formulated in terms of square function estimates. By restricting (1.3) to the first Wiener–Itô chaos of Lp (E, μ) (see Section 3) and using that the Lp -norms are pairwise equivalent on every chaos, we see that a necessary condition for (1.3) is given by √

D( A) = D(V )

√ with Ah V h.

Since D(V ) equals the domain of the form associated with A, this is nothing but Kato’s square root property for A. Thus our main result asserts that this necessary condition is also sufficient. Second quantised operators arise naturally as generators of transition semigroups associated with solutions of linear stochastic evolution equations with additive noise; see for instance [11,12]. In a forthcoming paper we shall apply our results to obtain Meyer inequalities for nonsymmetric analytic Ornstein–Uhlenbeck operators in infinite dimensions. These extend previous results of Shigekawa [49] and Chojnowska–Michalik and Goldys [12] for the symmetric case, and of Metafune, Prüss, Rhandi, and Schnaubelt [39] for the finite dimensional case, and they improve results of [36] where a slightly more general class of non-symmetric Ornstein–Uhlenbeck operators was considered. Preliminary versions of this paper have been presented during the Semester on Stochastic Partial Differential Equations at the Mittag–Leffler institute (Fall 2007) and the 8th International Meeting on Stochastic Partial Differential Equations and Applications in Levico Terme (January 2008). 2. Statement of the main results The domain, kernel, and range of a (possibly unbounded) linear operator T are denoted by D(T ), N(T ), and R(T ), respectively. When considering an operator T acting consistently on a scale of (vector-valued) Lp -spaces, Dp (T ), Np (T ), and Rp (T ) denote the domain, kernel, and

range of the Lp -realisation of T . We introduce the setting studied in this paper in the form of a list of assumptions which will be in force throughout the paper, with the exception of the intermediate Sections 3, 6, and 7.


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Assumption (A1). (E, H, μ) is an abstract Wiener space. More precisely, we assume that E is real Banach space, H is a real Hilbert space with inner product [·,·], and μ is a centred Gaussian Radon measure on E with reproducing kernel Hilbert space H . Recall that this implies that H is continuously embedded in E; we shall write i : H → E for the inclusion mapping. The covariance operator of μ equals i ◦ i ∗ (here and in what follows, we identify H ∗ and H via the Riesz representation theorem). For h ∈ H we may define a linear function φh : iH → R by φh (ig) := [h, g]. Although μ(iH ) = 0 if H is infinite dimensional [7, Theorem 2.4.7], there exists a μ-measurable linear extension φh : E → R which is uniquely defined μ-almost everywhere [7, Theorem 2.10.11]. Note that for x ∗ ∈ E ∗ we have φi ∗ x ∗ (x) = x, x ∗ μ-almost everywhere. The identity

x, x ∗ 2 dμ(x) = i ∗ x ∗ 2 ,

x∗ ∈ E∗,

E

shows that h → φh , as a mapping from H into L2 (E, μ), is an isometric embedding. Assumption (A2). V is a closed and densely defined linear operator from H into another real Hilbert space H . When H0 is a linear subspace of H and k 0 is an integer, we let F Cbk (E; H0 ) denote the vector space of all (μ-almost everywhere defined) functions f : E → R of the form f (x) := ϕ φh1 (x), . . . , φhn (x) with n 1, ϕ ∈ Cbk (Rn ), and h1 , . . . , hn ∈ H0 . Here Cbk (Rn ) is the space consisting of all bounded continuous functions having bounded continuous derivatives up to order k. In case H0 = H we simply write F Cbk (E). For f ∈ F Cb1 (E; D(V )) as above the gradient DV f ‘in the direction of V ’ is defined by n DV f (x) := V Df (x) = ∂j ϕ φh1 (x), . . . , φhn (x) ⊗ V hj , j =1

where D denotes the Malliavin derivative and ∂j ϕ denotes the j -th partial derivative of ϕ. We shall write Lp := Lp (E, μ),

Lp := Lp (E, μ; H )

for brevity. As in [19, Theorem 3.5], the proof of which can be repeated almost verbatim, the operator DV is closable as an operator from Lp into Lp for all 1 p < ∞. From now on, DV denotes its closure; domain and range of this closure will be denoted by Dp (DV ) and Rp (DV ) respectively. Assumption (A3). B is a bounded operator on H which is coercive on R(V ).

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More precisely, B is a bounded operator on H which satisfies the coercivity condition [BV h, V h] kV h2 ,

h ∈ D(V ),

where k > 0 is a constant independent of h ∈ D(V ). Clearly B satisfies (A3) if and only if B ∗ satisfies (A3). At this point we pause to observe that Assumptions (A1), (A2), (A3) continue to hold after complexifying. In what follows we shall be mostly dealing with the complexified operators, which we do not distinguish notationally from their real counterparts as this would only overburden the notations. The complexified version of (A3) reads Re[BV h, V h] kV h2 ,

h ∈ D(V ).

If (A1), (A2), (A3) hold, the operator L := DV∗ BDV is well defined, and −L generates an analytic C0 -contraction semigroup (P (t))t0 on Lp for all 1 < p < ∞, which coincides with the second quantisation of the analytic C0 -contraction semigroup on H generated by −A, where A := V ∗ BV (Theorem 4.4). In the converse direction we show that every second quantised analytic C0 contraction semigroup arises in this way (Theorem 3.2). It is not hard to see (Proposition 5.1) that the operator A := V V ∗ B is sectorial on H and that −A generates a bounded analytic C0 -semigroup on this space. Associated with this operator is the operator L = DV DV∗ B on Rp (DV ). This operator is well defined and sectorial on Rp (DV ), and −L generates a bounded analytic C0 -semigroup on this space (Theorem 5.6 and Definition 5.8). The main results of this paper read as follows. Theorem 2.1 (Domain of assertions are equivalent: (1) (2) (3) (4)

√

√ L). Assume (A1), (A2), (A3), and let 1 < p < ∞. The following √

√

Dp ( L) = Dp (DV ) with Lf p DV f p for f ∈ Dp ( L); ∞ -functional calculus on R (D ); L admits a bounded H√ √ √p V D( A) = D(V ) with Ah V h for h ∈ D( A); A admits a bounded H ∞ -functional calculus on R(V ).

For the precise definition of operators admitting a bounded H ∞ -functional calculus we refer to Section 7. Moreover we shall see (Lemma 10.2) that if the equivalent conditions of Theorem 2.1 hold, then their analogues where B is replaced by B ∗ hold as well. Some further equivalent conditions are given at the end of Section 10.


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Theorem 2.2 (Domain of L). Let 1 < p < ∞ and let the equivalent conditions of Theorem 2.1 be satisfied. Then Dp (L) = Dp DV2 ∩ Dp (DA ) with equivalence of norms f p + Lf p f p + DV f p + DV2 f p + DA f p . Here DA = AD, where D is the Malliavin derivative in the direction of H . For the precise definitions of DV2 and DA we refer to Section 11 where Theorem 2.2 is proved. The conditions of Theorem 2.1 are automatically satisfied in each of the following two cases: (i) B is self-adjoint. In this case A is self-adjoint and therefore (3) holds by the theory of symmetric forms (since A is associated with a closed symmetric form with domain D(V ); see Section 4). (ii) V has finite dimensional range. In this case (4) is satisfied; since A is injective on the (closed) range of V (see Lemma 5.4), the H ∞ -functional calculus of A is given by the Dunford calculus. In fact we shall prove the stronger result that one-sided inclusions in (1) and (3) of Theorem 2.1 hold if and only if A and/or L satisfies a corresponding square function estimate. In particular, Lp -boundedness of the Riesz transform √ (DV / L)f f p p is characterised by the square function estimate ∞ 1/2 2 dt tA S(t)u , u t

u ∈ R(V ).

0

Here S is the bounded analytic semigroup on H generated by −A. In Section 3 we present two applications of Theorem 2.1. The first gives an extension of the classical Meyer inequalities for the Ornstein–Uhlenbeck operator. The second concerns Lp estimates for the square root of the Lp -realisation of generators of the second quantisation of analytic C0 -contraction semigroups on Hilbert spaces. We show that for such semigroups the square root property of Theorem 2.1(3) is preserved under second quantisation. The proof of Theorem 2.1 depends crucially on the following gradient bounds for the semigroup P generated by −L and the first part of the Littlewood–Paley–Stein inequalities below. Theorem 2.3 (Gradient bounds). Assume (A1), (A2), (A3), and let 1 0 we have, for μ-almost all x ∈ E, √ 1/2 t DV P (t)f (x) P (t)|f |2 (x) . √ (2) The set { tDV P (t): t 0} is R-bounded in L (Lp , Lp ).

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The notion of R-boundedness is a strengthening of the notion of uniform boundedness and is discussed in Section 6. Theorem 2.4 (Littlewood–Paley–Stein inequalities). Assume (A1), (A2), (A3), and let 1 < p < ∞. For all f ∈ Lp we have the square function estimate ∞ 1/2 2 dt √ tDV P (t)f f − PNp (L) f p f p , t 0

p

where PNp (L) is the projection onto Np (L) along the direct sum decomposition Lp = Np (L) ⊕ Rp (L). Theorem 2.1 is proved in Sections 10 and 12, and Theorems 2.3 and 2.4 are proved in Section 8. At this point we emphasise that in the present non-symmetric setting, it is not possible to derive the cases p > 2 from the cases 1 < p 2 by means of duality arguments (as is done, for example, in [12,49]). New ideas are required; see Section 8.2. It follows from [6, Proposition 2.2] that the following Hodge decompositions hold: H = R(V ∗ B) ⊕ N(V ),

H = R(V ) ⊕ N(V ∗ B).

(2.1)

Here V ∗ B is interpreted as a closed densely defined operator from H to H . The second decomposition, however, shows that the closures of the ranges of V ∗ B and its restriction to R(V ) are the same. Therefore, in the first decomposition we may just as well interpret V ∗ B as an unbounded operator from R(V ) to H . This observation is relevant for the formulation of the following Gaussian Lp -analogues of the above decompositions, which are proved in Section 12. Theorem 2.5 (Hodge decompositions). Assume (A1), (A2), (A3), and let 1 < p < ∞. One has the direct sum decomposition Lp = Rp DV∗ B ⊕ Np (DV ), where DV∗ B is interpreted a closed densely defined operator from Rp (DV ) to Lp . If the equivalent conditions of Theorem 2.1 hold, then the above decomposition remains true when DV∗ B is interpreted as a closed densely defined operator from Lp to Lp . In that case one has the direct sum decomposition Lp = Rp (DV ) ⊕ Np DV∗ B , where DV∗ B is interpreted as a closed densely defined operator from Lp to Lp . In the proofs of these theorems we use the Hodge–Dirac formalism introduced recently by Axelsson, Keith, and Mc Intosh [6] in the context of the Kato square root problem. In the spirit of this formalism, let us define the Hodge–Dirac operator Π associated with DV and DV∗ B by the operator matrix

0 DV∗ B . Π := DV 0 Using Theorems 2.3, 2.4, and 2.5 we shall prove:


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Theorem 2.6 (R-bisectoriality). Assume (A1), (A2), (A3), and let 1 < p < ∞. The operator Π is R-bisectorial on Lp ⊕ Rp (DV ). If the equivalent conditions of Theorem 2.1 hold, then Π is R-bisectorial on Lp ⊕ Lp . For the definition of R-(bi)sectorial operators we refer to Section 7. The analogue of Theorem 2.6 for the more general framework considered in [6] generally fails for p = 2. It this therefore a non-trivial fact that the theorem does hold in the special case considered here. Its proof depends on Theorems 2.3, 2.4, and a delicate Lp -analysis of the operators DV and DV∗ B, which is carried out in Section 9. From the fact that on Lp ⊕ Rp (DV ) one has

L 0 Π2 = 0 L we deduce that Π admits a bounded H ∞ -functional calculus on Lp ⊕ Rp (DV ) if and only if L admits a bounded H ∞ -functional calculus on Rp (DV ), i.e., if and only if condition (2) in Theorem 2.1 is satisfied. An alternative proof of the implication (2) ⇒ (1) of Theorem 2.1√can now be derived from the H ∞ -functional calculus of Π applied to the function sgn(z) = z/ z2 ; this is done in the final Section 12. 3. Consequences Before we start with the proofs of our main results we discuss a number of situations where operators of the form studied in this paper arise naturally. 3.1. The Ornstein–Uhlenbeck operator Let (A1) be satisfied. Taking H = H and V = I , the derivative DV reduces to the Malliavin derivative in the direction of H . Assumption (A2) is then obviously satisfied. Let B be an arbitrary operator satisfying (A3). Since A = B is bounded and sectorial, condition (4) of Theorem 2.1 is satisfied. For the special choice B = 12 I , the resulting operator L = 12 DV∗ DV is the classical Ornstein– √ Uhlenbeck operator of Malliavin calculus, and the two-sided Lp -estimate for L of Theorem 2.1 reduces to the celebrated Meyer inequalities. 3.2. Linear stochastic evolution equations In this subsection we shall describe an application of our results to stochastic evolution equations. This application will be worked out in more detail in a forthcoming paper. For unexplained terminology and background material we refer to [16,42]. Consider the following linear stochastic evolution equation in a Banach space E: dU (t) = A U (t) dt + σ dW (t), t 0, U (0) = x. Here, A is assumed to generate a C0 -semigroup on E, σ is a bounded linear operator from a Hilbert space H to E, and W is an H -cylindrical Brownian motion. For later reference we put H := H N(σ ).

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Let us assume now that for each initial value x ∈ E the above problem admits a unique weak solution U x = (U x (t))t0 and that these solutions admit an invariant measure; necessary and sufficient conditions for this to happen can be found in [16,42,43]. Under this assumption one has weak convergence limt→∞ μt = μ, where μt is the distribution of the E-valued centred Gaussian random variable U 0 (t) corresponding to the initial value x = 0. The limit measure μ is invariant as well; in a sense that can be made precise it is the minimal invariant measure associated with the above problem. The reproducing kernel Hilbert space associated with μ is denoted by H and the corresponding inclusion operator H → E by i. Define, for bounded continuous functions f : E → R, P (t)f (x) := Ef U x (t) ,

t 0, x ∈ E.

The operators P (t) extend in a unique way to a C0 -contraction semigroup on Lp (E, μ) for all 1 p < ∞. It has been shown in [37] that if P is analytic for some (equivalently, for all) 1 < p < ∞, then its infinitesimal generator −L is of the form considered in Section 2. More precisely, there exists a unique coercive operator B on H such that L = DV∗ BDV , where V : D(V ) ⊆ H → H is the closed linear operator defined by V (i ∗ x ∗ ) := σ ∗ x ∗ ,

x∗ ∈ E∗.

It is easy to see that DV is nothing but the Fréchet derivative on E in the direction of H (where we think of H as a Hilbert subspace of E under the identification u → σ u). As a consequence of our main results we obtain the following result. Theorem 3.1. In the above situation, suppose that the transition semigroup P is analytic on Lp (E, μ) for some (all ) 1 < p < ∞. (1) The C0 -semigroup generated by A leaves H invariant and restricts to a bounded analytic C0 -semigroup√on H ; √ (2) We have Dp ( L) = Dp (DV ) with equivalence of norms Lf p DV f p if and only if the negative generator of the restricted semigroup admits a bounded H ∞ -functional calculus on H . 3.3. Second quantised operators Theorem 2.1 can be applied to the second quantisation of an arbitrary generator −A of an analytic C0 -contraction semigroup on a Hilbert space H . The idea is to prove that such operators A can be represented as V ∗ BV for certain canonical choices of operators V and B satisfying (A2) and (A3). If E and μ are given such that (A1) holds, the second observation is that the generator of the second quantised semigroup on Lp = Lp (E, μ) equals the operator −DV∗ BDV and therefore Theorem 2.1 can be applied. We begin with recalling the definition and elementary properties of second quantised operators. For more systematic discussions we refer to [24,50]. We work over the real scalar field and complexify afterwards.


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Let H (0) := R1 and define H (n) inductively as the closed linear span of H ((n−1)) together with all products of the form φh1 · . . . · φhn with h1 , . . . , hm ∈ H . Then we let H (0) := R1 and define H (n) as the orthogonal complement of H ((n−1)) in H (n) . The space H (n) is usually referred to as the n-th Wiener–Itô chaos. We have the orthogonal Wiener–Itô decomposition L = 2

∞

H (n) .

n=0

It is well known that for all 1 p q < ∞ there exist constants Cn,p,q > 0 such that F p F q Cn,p,q F p ,

F ∈ H (n) .

Denoting by In the orthogonal projection in L2 onto H (n) , we have the identity

1 In (φh1 · . . . · φhn ), In (φh1 · . . . · φhn ) = [h1 , hσ (1) ] · . . . · [hn , hσ (n) ], n! σ ∈Sn

where Sn is the permutation group on n elements. This shows that H (n) is canonically isometric to the n-fold symmetric tensor product H s n , the isometry being given explicitly by 1 In (φh1 · . . . · φhn ) → √ hσ (1) ⊗ · · · ⊗ hσ (n) . n! σ ∈S n

Thus the Wiener–Itô decomposition induces a canonical isometry of L2 and the (symmetric) Fock space Γ (H ) :=

∞

H s n .

n=0

Let T ∈ L (H ) be a contraction. We denote by Γ (T ) ∈ L (Γ (H )) the (symmetric) second quantisation of T , which is defined on H s n by Γ (T )

σ ∈Sn

hσ (1) ⊗ · · · ⊗ hσ (n) =

T hσ (1) ⊗ · · · ⊗ T hσ (n) .

σ ∈Sn

By the Wiener–Itô isometry, T induces a contraction on L2 and we have Γ (T )In (φh1 · . . . · φhn ) = In (φT h1 · . . . · φT hn ).

(3.1)

Moreover, Γ (T ) is a positive operator on L2 . We have the identities Γ (I ) = I,

Γ (T1 T2 ) = Γ (T1 )Γ (T2 ),

∗ Γ (T ) = Γ (T ∗ ).

(3.2)

For all 1 p ∞, Γ (T ) extends to a positive contraction on Lp and (3.2) continues to hold.

2420


For later reference we collect some further properties of second quantised operators which will not be used in the present section. The following formula is known as Mehler’s formula [45]: if f = ϕ(φh1 , . . . , φhn ) with ϕ ∈ Cb (Rn ) and h1 , . . . , hn ∈ H , then for μ-almost all x ∈ E we have Γ (T )f (x) =

ϕ φT h1 (x) + φ√I −T ∗ T h1 (y), . . . , φT hn (x) + φ√I −T ∗ T hn (y) dμ(y). (3.3)

E

For h ∈ H define ∞ 1 n 1 2 In φh = exp φh − h . Eh = n! 2

(3.4)

n=0

This sum converges absolutely in Lp for all 1 p < ∞, and the linear span of the functions Eh is dense in Lp [45, Chapter 1]. From a routine approximation argument using the closedness of DV we obtain that h ∈ D(V ) implies Eh ∈ Dp (DV ) and DV Eh = Eh ⊗ V h.

(3.5)

From (3.1) and (3.4) one has the identity Γ (T )Eh = ET h .

(3.6)

Let us now turn to the situation where −A be the generator of an analytic C0 -contraction semigroup on a Hilbert space H . It is well known [46, Theorem 1.57, Theorem 1.58 and the remarks following these results] that A is associated with a sesquilinear form a on (the complexification of ) H which is densely defined, closed and sectorial, i.e., there exists a constant C 0 such that Im a(h, h) C Re a(h, h),

h ∈ D(a).

The next result may be known to experts, but as we could not find an explicit reference we include a proof for the convenience of the reader. Theorem 3.2. There exists a Hilbert space H , a closed operator V : D(V ) ⊆ H → H with dense domain D(V ) = D(a) and dense range, and a bounded coercive operator B ∈ L (H ) such that A = V ∗ BV . More precisely, this identity means that we have a(g, h) = [BV g, V h] for all g, h ∈ D(V ); cf. Section 4. Proof. Writing a(h) := a(h, h) by [46, Proposition 1.8] we have a(g, h) Re a(g) 1/2 Re a(h) 1/2 ,

g, h ∈ D(a).

We claim that N := {h ∈ D(a): Re a(h) = 0} is a closed subspace of D(a). Indeed, if hn → h in D(a) and Re a(hn ) = 0, then


2421

Re a(h) a(h) − a(hn )

1/2 1/2 1/2 1/2 Re a(h) Re a(h − hn ) Re a(hn ) + Re a(h − hn ) ,

which becomes arbitrary small as n → ∞. On the quotient D(a)/N we define a sesquilinear form [V g, V h] :=

1 a(g, h) + a(h, g) , 2

g, h ∈ D(a),

where V denotes the canonical mapping from D(a) onto D(a)/N. This form is well defined, since for n, n ∈ N we have a(g + n, h + n ) − a(g, h) Re a(n) 1/2 Re a(h) 1/2 + Re a(g) 1/2 Re a(n ) 1/2 1/2 1/2 Re a(n ) + Re a(n) = 0. Since Re a(h) = 0 implies [h] = [0], the form [·,·] is an inner product on D(a)/N. We put H := D(a)/N , where the completion is taken with respect to the norm induced by [·,·]. We interpret V as a linear operator from H into H with dense domain D(V ) = D(a) and dense range. To show that V is closed, we take a sequence (hn )n1 in D(a) such that hn → h in H and V hn → u in H . Since Re a(hn − hm ) = V (hn − hm )2 → 0 as m, n → ∞, the sequence (hn )n1 is Cauchy in D(a). Thus the closedness of a implies that (hn )n1 has a limit in D(a), which is h since hn → h in H . Consequently, V hn − V h2 = Re a(hn − h) → 0. We conclude that V is closed. Now we define a sesquilinear form b on R(V ) by b(V g, V h) := a(g, h). This is well defined, since V g = V g˜ and V h = V h˜ imply that a(g, h) − a(g, ˜ ˜ a(g − g, ˜ h − h) ˜ h) + a(g, ˜ h)

1/2 ˜ 1/2 Re a(g − g) ˜ Re a(h) + Re a(g) ˜ Re a(h − h) ˜ = 0. ˜ V (h − h) ˜ V h + V g = V (g − g)

Moreover, the associated operator B extends to a bounded operator on H , since b(V g, V h) = a(g, h) Re a(g) 1/2 Re a(h) 1/2 = V gV h. We conclude that a(g, h) = [BV g, V h]. By the identity V h2 = Re a(h) = Re[BV h, V h]

2422


and the boundedness of B we infer that u2 = Re[Bu, u] for all u ∈ H , and the coercivity of B follows. 2 Although the triple (H , V , B) is not unique, the next result implies that the statements in Theorem 2.1 do not depend on the choice of (H , V , B). Proposition 3.3. Let −A be the generator of an analytic C0 -contraction semigroup on H . Let , V , B) be triples with the properties as stated in Theorem 3.2. Then: (H , V , B) and (H coincide; (i) The coercivity constants k and k˜ of B and B ) with V h V h. (ii) D(V ) = D(V If in addition to the above assumptions (E, H, μ) is an abstract Wiener space, then for 1 p < ∞ we have (iii) Dp (DV ) = Dp (DV ) with DV f p DVf p . V h, V h] for h ∈ D(a) and Proof. (i): This follows from the identity [BV h, V h] = a(h, h) = [B the fact that V and V have dense range. (ii): For h ∈ D(A) we have V h, V h] B V h2 . kV h2 Re[BV h, V h] = Re[Ah, h] = Re[B ) the result follows. Since D(A) is a core for both D(V ) and D(V (iii): Let D denote the Malliavin derivative, which is well defined as a densely defined closed operator from Lp (E, μ) to Lp (E, μ; H ), 1 p < ∞. For f ∈ F Cb1 (E; D(V )) we have, by (ii), p p Df p dμ = DV f pp . DV f p = V Df dμ V E

E

The claim follows from this since F Cb1 (E; D(V )) is a core for Dp (DV ) and Dp (DV ).

2

Let N := {h ∈ D(a): Re a(h) = 0} and let D˙ (a) := H be defined as in the proof of Theorem 3.2, i.e., D˙ (a) is the completion of D(a)/N with respect to the norm V hD˙ (a) := Re a(h) , where V denotes the canonical operator from D(a) onto D(a)/N . In the proof of Theorem 3.2 we showed that V is a closed operator from H into D˙ (a) with dense domain and dense range. We also constructed a coercive operator B ∈ L (D˙ (a)) such that a(g, h) = [BV g, V h] for g, h ∈ D(a). In Lemma 5.3 below we show that the semigroup S generated by −A induces a bounded analytic C0 -semigroup S on D˙ (a) in the sense that S(t)V h = V S(t)h for all h ∈ D(V ). Its generator will be denoted by −A. Now let (E, H, μ) be an abstract Wiener space and let DV := V D as before. For cylindrical functions f = f (φh1 , . . . , φhn ) with hj ∈ D(V ) = D(a) we have DV f =

n j =1

∂j f (φh1 , . . . , φhn ) ⊗ V hj .


2423

As will be shown in Section 4, the realisation on L2 (E, μ) of the operator L := DV∗ BDV extends to a sectorial operator L on Lp (E, μ) for 1 < p < ∞, and −L equals the generator of the second quantisation on Lp (E, μ) of the semigroup S generated by −A on H , i.e., P (t) = Γ S(t) ,

t 0.

As a consequence, Theorem 2.1 can be translated into the following result. Theorem 3.4. Assume (A1) and let −A be the generator of an analytic C0 -contraction semigroup S on H . Let 1 < p < ∞ and let −L denote the realisation on Lp (E, μ) of the generator of the second quantisation of S. The following assertions are equivalent: √ √ √ (1 ) Dp√ ( L) = Dp (DV ) with Lf p DV f p for f ∈ Dp ( L); √ √ (3 ) D( A) = D(a) with Ah a(h) for h ∈ D(a); (4 ) the realisation of A in D˙ (a) admits a bounded H ∞ -functional calculus. The main equivalence here is (1 ) ⇔ (3 ). It asserts that the square root property with homogeneous norms is preserved when passing from H to Lp (E, μ) by means of second quantisation. The equivalence (3 ) ⇔ (4 ) is probably known, although we could not find a reference for it. The related equivalence (3 ) ⇔ (4 ), with √ (3 ) D( A) = D(a); (4 ) the realisation of A in D(a) admits a bounded H ∞ -functional calculus, is stated explicitly in [2, Theorem 5.5.2]. We have already mentioned that Theorem 2.1 is obtained by combining two one-sided versions of it involving Riesz transforms. The same is true for Theorem 3.4. 4. The operator L In this section we give a rigorous definition of the operator L as a closed and densely defined operator acting in Lp := Lp (E, μ), where 1 < p < ∞. We begin with an analysis in the space L2 := L2 (E, μ). Associated with the (complexified) operators B : H → H , V : D(V ) ⊆ H → H , and DV : D(DV ) ⊆ L2 → L2 are the sesquilinear forms a on H and l on L2 defined by D(a) := D(V ) and a(h1 , h2 ) := [BV h1 , V h2 ], and D(l) := D(DV ) and l(f1 , f2 ) := [BDV f1 , DV f2 ], where in the second line we identify B with the operator I ⊗ B. Here, and in what follows, we write D(DV ) := D2 (DV ),

R(DV ) := R2 (DV ).

2424


The forms a and l are easily seen to be closed, densely defined and sectorial in H and L2 , respectively. The operators associated with these forms are denoted by A and L, respectively; their domains will be denoted by D(A) and D(L). We may write A = V ∗ BV ,

L = DV∗ BDV ;

this notation is justified by the observation that h ∈ D(A) f ∈ D(L)

⇐⇒ ⇐⇒

h ∈ D(V ) and BV h ∈ D(V ∗ ) ,

f ∈ D(DV ) and BDV f ∈ D DV∗ ,

in which case we have Ah = V ∗ (BV h),

Lf = DV∗ (BDV )f.

Let us also note (cf. [46, Lemma 1.25]) that D(A) and D(L) are cores for D(V ) and D(DV ), respectively. For later use we observe that if B satisfies (A3), then also B ∗ satisfies (A3) and we have A∗ = V ∗ B ∗ V ,

L∗ = DV∗ B ∗ DV

with similar justifications. The proof of the next lemma can be found in [19]. Recall that φ : H → L2 is the isometric embedding defined in Section 2. Lemma 4.1. Let 1 < p < ∞. For all f ∈ F Cb1 (E; D(V )) and u ∈ D(V ∗ ) we have and DV∗ (f ⊗ u) = f φV ∗ u − [DV f, u]. f ⊗ u ∈ Dp DV∗ Lemma 4.2. Identifying H with its image φ(H ) in L2 , A is the part of L in H . Proof. Suppose first that h ∈ D(A). Then, by the form definition of A, we have h ∈ D(V ) and BV h ∈ D(V ∗ ). Hence Lemma 4.1 gives us 1 ⊗ BV h ∈ D(DV∗ ) and DV∗ (1 ⊗ BV h) = φV ∗ BV h = φAh . From this, combined with the identity DV φh = 1 ⊗ V h which follows from the definition of DV , we deduce that [BDV φh , DV f ] = [1 ⊗ BV h, DV f ] = [φAh , f ] for all f ∈ D(DV ). Therefore, φh ∈ D(L) and Lφh = φAh . Denoting the part of L in H by LH for the moment, this argument shows that A ⊆ LH .


2425

On the other hand, if φh ∈ D(LH ), then φh ∈ D(L) and Lφh = φh for some h ∈ H . Hence for all g ∈ D(V ) we obtain [BV h, V g] = [BDV φh , DV φg ] = l(φh , φg ) = [Lφh , φg ] = [φh , φg ] = [h , g]. It follows that h ∈ D(A) and [Ah, g] = [h , g]. This shows that Ah = h , and we have proved the opposite inclusion A ⊇ LH . 2 It follows from the theory of forms (cf. [46, Proposition 1.51]) that −A and −L generate analytic C0 -contraction semigroups S = (S(t))t0 and P = (P (t))t0 on H and L2 , respectively. In fact we have the following more precise result. Recall that the constant k > 0 has been introduced in Assumption (A3). Proposition 4.3. The operators −A and −L generate analytic C0 -contraction semigroups on H 1 and L2 of angle arctan γ , where γ = 2k B − B ∗ . Proof. We prove this for L; the proof of A is similar (alternatively, the result for A follows from the result for L via Lemma 4.2). By the Lumer–Phillips theorem it suffices to show that L has numerical range in the sector of angle arctan γ . Using that B is a (complexified) real operator, for f ∈ D(L) we have, with F = Re DV f and G = Im DV f , Im[Lf, f ] = (B − B ∗ )F, G 1 B − B ∗ F 2 + G2 2 1 B − B ∗ [BF, F ] + [BG, G] 2k 1 2 = B − B ∗ Re[Lf, f ]. 2k

(4.1)

The first main result of this section identifies P as the second quantisation of S. Theorem 4.4. For all t 0 we have P (t) = Γ (S(t)). Proof. We recall from Lemma 4.2 that P (t)φh = S(t)h for all h ∈ H . First we check that for all h ∈ D(A), the functions Eh ∈ L2 are in the domains of L and L, where −L is the generator of Γ (S), and that both generators agree on those functions. Using (3.5) and Lemma 4.1 we obtain LEh = DV∗ BDV Eh = DV∗ (Eh ⊗ BV h) = Eh φV ∗ BV h − [BV h, V h]Eh = φAh − [Ah, h] Eh ,

2426


while on the other hand, using (3.4) and (3.6) combined with a simple approximation argument, we have h = lim 1 (ES(t)h − Eh ) LE t↓0 t 2 d 1 = Eh φS(t)h − S(t)h dt t=0 2 = φAh − [Ah, h] Eh . The set lin{Eh : h ∈ D(A)} is dense in L2 and invariant under the semigroup Γ (S). As a con It follows that D(L) ⊆ D(L). Since both −L and −L are sequence, this set is a core for D(L). generators this implies D(L) = D(L) and therefore L = L. 2 So far we have considered P as a C0 -semigroup in L2 . Having identified P as a second quantised semigroup on L2 , we are in a position to prove that P extends to the spaces Lp . Theorem 4.5. For 1 p < ∞, the semigroup P extends to a C0 -semigroup of positive contractions on Lp satisfying P (t)f ∞ f ∞ for f ∈ L∞ . The measure μ is an invariant measure for P , i.e., P (t)f dμ = f dμ, f ∈ Lp , t 0. E

E

For 1 < p < ∞, P is an analytic C0 -contraction semigroup on Lp . Proof. The extendability to a C0 -contraction semigroup on Lp , as well as the L∞ -contractivity and positivity follow from general results on second quantisation. The invariance of μ follows from P (t)f dμ = f P ∗ (t)1 dμ = f dμ, f ∈ Lp . E

E

E

Here we use that P ∗ is a second quantised semigroup as well, and therefore satisfies P ∗ (t)1 = 1 for all t 0. It remains to prove the last statement. We have seen in Proposition 4.3 that P extends to an analytic C0 -contraction semigroup on L2 . The extension to an analytic C0 -contraction semigroup on Lp , 1 < p < ∞, follows from a standard argument involving the Stein interpolation theorem and duality. 2 Remark 4.6. We mention that a different argument to establish analyticity in Lp has been given in the case of Ornstein–Uhlenbeck semigroups in [10,37]. This argument also works in the more general setting considered here and yields an angle of analyticity which is better than the one obtained by Stein interpolation. For Ornstein–Uhlenbeck semigroups (for which we have B + B ∗ = I , see [37]), this angle is optimal. Definition 4.7. On Lp we define the operator L as the negative generator of the semigroup P .


2427

Lemma 4.8. For all 1 < p < ∞, F Cb∞ (E; D(A)) is a P -invariant core for Dp (L). Moreover, for f, g ∈ F Cb∞ (E; D(A)) and ψ ∈ Cb∞ (R) we have (1) (Product rule) f g ∈ F Cb∞ (E; D(A)) and

L(fg) = f Lg + gLf − (B + B ∗ )DV f, DV g ; (2) (Chain rule) ψ ◦ f ∈ F Cb∞ (E; D(A)) and L(ψ ◦ f ) = (ψ ◦ f )Lf − (ψ ◦ f )[BDV f, DV f ]. Proof. First we show that F Cb∞ (E; D(A)) is contained in Dp (L); we thank Vladimir Bogachev for pointing out an argument which simplifies our original proof. Pick a function f ∈ F Cb∞ (E; D(A)) and notice that f ∈ D(L) ∩ Lp . The space Lp being reflexive, by a standard result from semigroup theory (cf. [9]) it suffices to show that 1 lim sup P (t)f − f p < ∞. t t↓0 Using that L = DV∗ BDV in L2 , an explicit calculation using Lemma 4.1 shows that Lf ∈ L2 ∩ Lp . Moreover, in L2 we have the identity 1 1 P (t)f − f = t t

t P (s)Lf ds. 0

Since Lf ∈ Lp , the right-hand side can be interpreted as a Bochner integral in Lp , which for 0 < t 1 can be estimated in Lp by t 1 P (s)Lf ds Lf p . t 0

p

This gives the desired bound for the limes superior. To show that F Cb∞ (E; D(A)) is invariant under P , we take f of the form f = ϕ(φh1 , . . . , φhn ), with ϕ ∈ Cb∞ (Rn ) and h1 , . . . , hn ∈ D(A). Let R(t) := for μ-almost all x ∈ E we have P (t)f (x) =

√ I − S ∗ (t)S(t). By Mehler’s formula,

ϕ φS(t)h1 (x) + φR(t)h1 (y), . . . , φS(t)hn (x) + φR(t)hn (y) dμ(y)

E

= ψt φS(t)h1 (x), . . . , φS(t)hn (x) ,

(4.2)

2428


where

ϕ ξ1 + φR(t)h1 (y), . . . , ξn + φR(t)hn (y) dμ(y).

ψt (ξ1 , . . . , ξn ) = E

Since ψt ∈ Cb∞ (Rn ) and S(t)hj ∈ D(A) for j = 1, . . . , n, it follows that the subspace F Cb∞ (E; D(A)) is invariant under P . Since it is dense in Lp and contained in Dp (L), it is a core for Dp (L). The identities (1) and (2) follow by direct computation, using the identity L = DV∗ BDV and Lemma 4.1. 2 Remark 4.9. The same proof shows that F Cb∞ (E; D(Ak )) is a P -invariant core for Dp (L) for every k 1. 5. The operator L Our next aim is to give a rigorous description of the operator L on the spaces Rp (DV ), 1 < p < ∞, where the closure is taken in Lp := Lp (E, μ; H ). On H and L2 we consider the sesquilinear forms a : D(V ∗ ) × D(V ∗ ) → C, a(u1 , u2 ) := [V ∗ u1 , V ∗ u2 ] and l : D(DV∗ ) × D(DV∗ ) → C,

l(F1 , F2 ) := DV∗ F1 , DV∗ F2 . Here, DV∗ : D(DV∗ ) ⊆ L2 → L2 is the adjoint of the operator DV : D(DV ) ⊆ L2 → L2 . The forms a and l are closed, densely defined and sectorial. The operators associated with these forms are denoted by AI and LI respectively, with domains D(AI ) and D(LI ). We may write AI = V V ∗ ,

LI = DV DV∗

with similar justifications as before. These operators are self-adjoint; see e.g. [46, Proposition 1.31]. We introduce next the operators

D(A) := h ∈ H : Bh ∈ D(AI ) ,

D(L) := F ∈ L2 : BF ∈ D(LI ) ,

A := AI B; L := LI B.

Note that A = V V ∗ B,

L = DV DV∗ B.

It follows from standard operator theory [6, Lemma 4.1] that A and L are closed and densely defined and satisfy A = (B ∗ AI )∗ ,

L = (B ∗ LI )∗ .


2429

Proposition 5.1. The operators A and L are sectorial on H and L2 of angle arctan γ , where 1 γ := 2k B − B ∗ . For all u ∈ D(A) we have 1 ⊗ u ∈ D(L) and L(1 ⊗ u) = 1 ⊗ Au. Proof. Writing v := Re u and w := Im u, by estimating as in (4.1) we obtain Im[Bu, u] 1 B − B ∗ Re[Bu, u]. 2k This shows that the numerical range of B is contained in the closed sector around R+ of angle arctan γ . The same is true for the operator B as an operator acting on L2 . Hence it follows from [5, Proposition 7.1] (in which ‘positive’ may be weakened to ‘non-negative’) that the operators A = AI B and L = LI B are sectorial of angle arctan γ . The final identity follows from L(1 ⊗ u) = DV DV∗ (1 ⊗ Bu) = DV (φV ∗ Bu ) = 1 ⊗ V V ∗ Bu = 1 ⊗ Au.

2

As a consequence, −A and −L generate bounded analytic C0 -semigroups of angle arccot γ on H and L2 . In what follows we denote these semigroups by S and P . Lemma 5.2. If h ∈ D(A) and Ah ∈ D(V ), then V h ∈ D(A) and AV h = V Ah. Proof. Since h ∈ D(A), the definition of A as the operator associated with the form (h, g) → [BV h, V g] implies that h ∈ D(V ), BV h ∈ D(V ∗ ), and Ah = V ∗ (BV h). To check that we have V h ∈ D(A), in view of the identity A = (B ∗ AI )∗ we must find h ∈ H such that [B ∗ AI g, V h] = [g, h ] for all g ∈ D(AI ). But h := V Ah does the job, since [g, V Ah] = [g, V V ∗ BV h] = [B ∗ AI g, V h]; this implies that V h ∈ D(A) and AV h = V Ah. 2 Lemma 5.3. For all h ∈ D(V ) and t 0 we have S(t)h ∈ D(V ) and V S(t)h = S(t)V h. Proof. We may assume that t > 0. First let g ∈ D(A2 ). Then Ag ∈ D(A) ⊆ D(V ), and therefore Lemma 5.2 implies that V g ∈ D(A) and AV g = V Ag. For λ > 0 it follows that (I + λA)V g = V (I + λA)g. Applying this to g = (I + λA)−1 h with h ∈ D(A) we obtain V (I + λA)−1 h = (I + λA)−1 V h. Taking λ =

t n

and repeating this argument n times we obtain, for all h ∈ D(A), −n −n t t V I+ A h= I + A V h. n n

2430


Taking limits n → ∞ and using the closedness of V , we obtain S(t)h ∈ D(V ) and V S(t)h = S(t)V h. We are still assuming that h ∈ D(A). However, this assumption may now be removed by recalling the fact that D(A) is a core for D(V ). 2 Lemma 5.4. For all t 0 we have S(t)R(V ) ⊆ R(V ). Moreover, the part of A in R(V ) is injective. Proof. The first assertion follows from Lemma 5.3. Suppose that Au = V V ∗ Bu = 0 for some u belonging to the domain of the part of A in R(V ). Then V ∗ Bu2 = 0, so Bu ∈ N(V ∗ ). Thus [Bu, V h] = 0 for all h ∈ D(V ). Since u ∈ R(V ) it follows that [Bu, u] = 0, and therefore u = 0 by the coercivity of B on R(V ). 2 Next we show that the semigroups P and P ⊗ S agree on R(DV ). We need two lemmas which are formulated, for later reference, for the Lp -setting. Lemma 5.5. For 1 < p < ∞, F Cb∞ (E; D(A)) is a core for Dp (DV ). Proof. First let f = ϕ(φh1 , . . . , φhn ) with ϕ ∈ Cb1 (Rn ) and h1 , . . . , hn ∈ D(V ). Choose sequences (hj k )k1 in D(A) with hj k → hj in D(V ) as k → ∞. Then fk → f in Lp and DV fk → DV f in Lp , where fk = ϕ(φh1k , . . . .φhnk ). Since F Cb1 (E; D(V )) is a core for Dp (DV ), this proves that F Cb1 (E; D(A)) is a core for Dp (DV ). Now a standard mollifier argument, convolving ϕ with a smooth function of compact support, shows that F Cb∞ (E; D(A)) is a core for Dp (DV ). 2 The next result is well known in the context of Ornstein–Uhlenbeck semigroups; see, e.g., [12, Lemma 2.7], [36, Proposition 3.5]. Theorem 5.6. For all 1 < p < ∞, the semigroup P ⊗ S restricts to a bounded analytic C0 semigroup on Rp (DV ). For f ∈ Dp (DV ) and t 0 we have P (t)f ∈ Dp (DV ) and DV P (t)f = P (t) ⊗ S(t) DV f. Proof. First we show that for all f ∈ Dp (DV ) we have P (t)f ∈ Dp (DV ) and DV P (t)f = (P (t) ⊗ S(t))DV f . Since DV is closed and F Cb∞ (E; D(A)) is a core for Dp (DV ) by Lemma 5.5, it suffices to check this for functions f ∈ F Cb∞ (E; D(A)). We use the notations of Lemma 4.8. By (4.2) and Lemma 5.3, for functions f = ϕ(φh1 , . . . , φhn ) we have, for μ-almost all x ∈ E, DV P (t)f (x) =

n

∂j ψt φS(t)h1 (x), . . . , φS(t)hn (x) ⊗ V S(t)hj

j =1

=

n j =1 E

∂j ϕ φS(t)h1 (x) + φR(t)h1 (y), . . . , φS(t)hn (x)


2431

+ φR(t)hn (y) dμ(y) ⊗ S(t)V hj = P (t) ⊗ S(t) DV f (x). This identity shows that P (t) ⊗ S(t) maps Rp (DV ) into itself, and therefore P ⊗ S restricts to a bounded C0 -semigroup on Rp (DV ). The invariance of Rp (DV ) under the operators P (z) ⊗ S(z), where z ∈ C is in the sector of bounded analyticity of P , follows by uniqueness of analytic continuation (consider the quotient mapping from Lp to Lp /Rp (DV )). 2 In the next result we return to the L2 -setting and show that the semigroups P ⊗ S and P on L2 agree on R(DV ). Theorem 5.7. Both P and P ⊗ S restrict to bounded analytic C0 -semigroups on R(DV ), and their restrictions coincide: for t 0, P (t)F = P (t) ⊗ S(t)F,

F ∈ R(DV ).

Proof. The invariance of R(DV ) under P ⊗ S follows from the previous theorem. Let us write −N for the generator of P ⊗ S on R(DV ). From V (D(A2 )) ⊆ D(A) (cf. the proof of Lemma 5.3) and F Cb∞ (E; D(A2 )) ⊗ D(A) ⊆ D(L) ⊗ D(A) we see that the subspace U := {DV f : f ∈ F Cb∞ (E; D(A2 ))} is contained in D(N ). This subspace is dense in R(DV ) since F Cb∞ (E; D(A2 )) is a core for D(L) (by Lemma 4.8 and the remark following it) and D(L) is a core for D(DV ). Since (P ⊗ S)U ⊆ U by Theorem 5.6, it follows that U is a core for D(N ). For functions f ∈ F Cb∞ (E; D(A2 )) we obtain N DV f = DV Lf = LDV f. The first identity follows from Theorem 5.6 and the second from a direct computation. Alternatively, the second identity can be deduced from the analogue of Lemma 5.2 for DV and L. Thus N = L on the core U of D(N ). It follows that D(N ) ⊆ D(L) and N = L on D(N ). Let λ > 0. Multiplying the identity λ + N = λ + L from the right with (λ + N )−1 and from the left with (λ + L)−1 , we obtain (λ + N )−1 = (λ + L)−1 on R(DV ). In particular, (λ + L)−1 maps R(DV ) into itself. As in Lemma 5.3 it follows that P leaves R(DV ) invariant and that the restriction of P to R(DV ) equals the semigroup generated by −N , which is P ⊗ S|R(DV ) . 2 Definition 5.8. Let 1 < p < ∞. On Rp (DV ) we define P := P ⊗ S|Rp (DV ) . The negative generator of P is denoted by L. By Theorem 5.7, for p = 2 this definition is consistent with the one given at the beginning of this section. 6. Intermezzo I: R-boundedness and radonifying operators Before proceeding with the proofs of the main results we insert a section containing a concise discussion of the notions of R-boundedness, radonifying operators, and square functions. For more information and further results we refer to the excellent sources [17,31] as well as the papers [41,42] and the references given therein. The notations in this section will be independent of those in the previous ones.

2432


6.1. R-boundedness Throughout this section, unless otherwise stated (M, μ) is an arbitrary σ -finite measure space and H is an arbitrary Hilbert space. In analogy to previous notations we write Lp := Lp (M, μ) and Lp := Lp (M, μ; H ). Let X and Y be Banach spaces and let (rj )j 1 be a sequence of independent Rademacher variables, i.e., P(rj = 1) = P(rj = −1) = 12 for each j . A collection of bounded linear operators T ⊆ L (X, Y ) is said to be R-bounded if there exists C 0 such that for all k = 1, 2, . . . , and all choices of x1 , . . . , xk ∈ X and T1 , . . . , Tk ∈ T we have 2 2 k k E rj Tj xj C 2 E rj xj . j =1

j =1

The smallest constant C for which this inequality holds is denoted by R(T ). By the Kahane– Khintchine inequalities one may replace the exponents 2 by arbitrary p ∈ [1, ∞); this only changes the value of the constant C. Every bounded subset of operators on a Hilbert space is R-bounded. If T is R-bounded, then the closure with respect to the strong operator topology of the absolutely convex hull of T is R-bounded as well, with constant at most C (in the real case) or 2C (in the complex case). A useful consequence of this is the following result [31, Corollary 2.14] which we formulate for real spaces X and Y (in the complex case an extra constant 2 appears). Proposition 6.1. Let T ⊆ L (X, Y ) be R-bounded, and let f : M → L (X, Y ) be a function with values in T such that ξ → f (ξ )x is strongly μ-measurable for all x ∈ E. For φ ∈ L1 define Tφ,f x :=

φ(t)f (t)x dμ(t),

x ∈ X.

M

Then the collection {Tφ,f : φL1 1} is R-bounded in L (X, Y ). The next result may be known to specialists, but since we couldn’t find a reference for it we include a proof. Proposition 6.2. Let 1 p < ∞. If T ⊆ L (Lp ) is R-bounded and S ⊆ L (H ) is bounded, then T ⊗ S ⊆ L (Lp ) is R-bounded. Proof. Since T ⊗ S = (T ⊗ I )(I ⊗ S) and I ⊗ S is R-bounded by the Fubini theorem, it suffices to show that T ⊗ I is R-bounded. Let (hi )ni=1 be an orthonormal system in H and let F1 , . . . , Fk be functions in Lp of the form Fj := ni=1 fij ⊗ hi . Let (ri )i1 and (˜ri )i1 be independent Rademacher sequences. Then, putting gi := kj =1 rj Tj fij ,


2433

p p k k n E rj (Tj ⊗ I )Fj = E rj Tj fij ⊗ hi j =1

p

j =1

i=1

p

p n k =E rj Tj fij ⊗ hi dμ M

j =1

M

i=1

i=1

p n gi ⊗ hi dμ =E p n E r˜i gi dμ E i=1

M

k n p rj Tj r˜i fij = EE j =1

i=1

k n p EE rj r˜i fij j =1

i=1

p k E rj Fj . j =1

p

p

p

The last step follows by performing the computation in reverse order. The result follows by an application of the Kahane–Khintchine inequalities. 2 We need the following duality result for R-bounded families [26, Proposition 3.5]. Let I1 ∈ L (L2 (E, μ)) be the orthogonal projection defined in Section 3.3 and let IX be the identity operator on a Banach space X. Then X is said to be K-convex if the operator I1 ⊗ IX on L2 (E, μ) ⊗ X extends to a bounded operator on the Lebesgue–Bochner space L2 (E, μ; X) (see, e.g., [18,48]). Proposition 6.3. If X and Y are K-convex Banach spaces, then a family T ⊆ L (X, Y ) is Rbounded if and only if the adjoint family T ∗ ⊆ L (Y ∗ , X ∗ ) is R-bounded. We shall apply this proposition to the K-convex spaces X = Lp and Y = Lp for 1 < p < ∞. 6.2. Radonifying operators It will be convenient to exploit the connection between square functions and radonifying norms. Let (γn )n1 be a Gaussian sequence, i.e., a sequence of independent standard normal random variables. If H is a Hilbert space and X is a Banach space, we denote by γ (H , X) the completion of the finite rank operators from H to X with respect to the norm k 2 hj ⊗ xj j =1

γ (H ,X)

2 k = E γj x j , j =1

2434


where it is assumed that the vectors h1 , . . . , hk are orthonormal in H . We have a continuous inclusion γ (H , X) → L (H , X). Operators in L (H , X) belonging to γ (H , X) are called radonifying; this terminology is explained by the fact that an operator T ∈ L (H , X) is radonifying if and only if there exists a centred Gaussian Radon measure on X whose covariance operator equals T T ∗ . We continue with an observation about repeated radonifying norms which follows from the Kahane–Khintchine inequalities and Fubini’s theorem. For a proof see, e.g., [44]. Proposition 6.4. Let (S1 , σ1 ), (S2 , σ2 ) be σ -finite measure spaces, and let 1 p < ∞. The mapping f1 ⊗ (f2 ⊗ g) → (f1 ⊗ f2 ) ⊗ g,

fi ∈ L2 (Si , σi ), g ∈ Lp ,

extends uniquely to an isomorphism of Banach spaces γ L2 (S1 , σ1 ), γ L2 (S2 , σ2 ), Lp γ L2 (S1 × S2 , σ1 ⊗ σ2 ), Lp . The next multiplier result is a slight extension of a result due to Kalton and Weis [27] and can be proved in the same way. Proposition 6.5. Let X and Y be Banach spaces, and let K : M → L (X, Y ) be a function such that K(·)x is strongly μ-measurable for all x ∈ X. If the set TK = {K(ξ ): ξ ∈ M} is R-bounded, then the mapping TK : f (·) ⊗ x → f (·) ⊗ K(·)x,

f ∈ L2 , x ∈ X,

extends uniquely to a bounded operator TK from γ (L2 , X) to γ (L2 , Y ) of norm TK R(TK ). 6.3. Square functions In this subsection we recall how R-bounded families in Lp -spaces and radonifying operators into Lp -spaces can be characterised by square functions. The first result follows from a standard application of the Kahane–Khintchine inequalities; see [31]. Proposition 6.6. A family T ⊆ L (Lp , Lp ) is R-bounded if and only if there exists a constant C such that for all T1 , . . . , TN ∈ T and f1 , . . . , fN ∈ Lp , N N 1/2 1/2 2 2 Tn fn |fn | C . n=1

p

n=1

p

The next result is another consequence of the Kahane–Khintchine inequalities; see [8,41]. Proposition 6.7. Let (S, σ ) be a σ -finite measure space and let 1 p < ∞ and p1 + q1 = 1. Let k : S → Lp be a strongly σ -measurable function such that for all g ∈ Lq the function s → k(s), g is square integrable. The following assertions are equivalent:


(1) The operator Ik ∈ L (L2 (S, σ ), Lp ) defined by Ik f, g = f (s) k(s), g dσ (s),

2435

f ∈ L2 (S, σ ), g ∈ Lq ,

S

is radonifying; (2) The square function ( S k(s)2 dσ (s))1/2 defines an element of Lp . In this situation we have an equivalence of norms 1/2 . k(s)2 dσ (s) Ik γ (L2 (S,σ ),Lp ) p

S

In what follows we always identify k with the operator Ik . 7. Intermezzo II: H ∞ -functional calculi In this section we recall some basic facts concerning H ∞ -functional calculi. For more information we refer to the monographs [17,23], the lecture notes [1,31], and the references given therein. For ω ∈ (0, π) we consider the open sector Σω+ := z ∈ C: z = 0, | arg z| < ω . A closed operator A acting on a Banach space X is said to be sectorial of angle ω ∈ (0, π) if / Σθ+ } is bounded for all θ ∈ (ω, π). The least angle σ (A) ⊆ Σω+ and the set {λ(λ − A)−1 : λ ∈

of sectoriality is denoted by ω+ (A). If A is sectorial and the set {λ(λ − A)−1 : λ ∈ / Σθ+ } is Rbounded for all θ ∈ (ω, π), then A is said to be R-sectorial of angle ω ∈ (0, π). The least angle + of R-sectoriality is denoted by ωR (A). We will frequently use the fact [23, Proposition 2.1.1(h)] that a sectorial operator A on a reflexive Banach space X induces a direct sum decomposition X = N(A) ⊕ R(A).

(7.1)

Let H ∞ (Σθ+ ) be the space of all bounded holomorphic functions on Σθ+ , and let H0∞ (Σθ+ ) denote the linear subspace of all ψ ∈ H ∞ (Σθ+ ) which satisfy an estimate ψ(z) C

|z| 1 + |z|2

α ,

z ∈ Σθ+ ,

for some α > 0 and C 0. If A is a sectorial operator and ψ is a function in H0∞ (Σθ+ ) with 0 < ω+ (A) < θ < θ < π , we may define the bounded operator ψ(A) on X by the Dunford integral 1 ψ(z)(z − A)−1 x dz, x ∈ X, ψ(A)x := 2πi ∂Σθ+

2436


where ∂Σ + is parametrised counter-clockwise. By Cauchy’s theorem this definition does not θ

depend on the choice of θ . A sectorial operator A on X is said to admit a bounded H ∞ (Σθ+ )-functional calculus, or a bounded H ∞ -functional calculus of angle θ , if there exists a constant Cθ 0 such that for all ψ ∈ H0∞ (Σθ+ ) and all x ∈ X we have ψ(A)x Cθ ψ∞ x,

+ where ψ∞ = supz∈Σ + |ψ(z)|. The infimum over all possible angles θ is denoted ωH ∞ (A). We θ ∞ say that a sectorial operator A admits a bounded H -functional calculus if it admits a bounded H ∞ (Σθ+ )-functional calculus for some 0 < θ < π . The following result is well known; see, e.g., [31, Theorem 2.20]. + (A) < 12 π on X, and let S be the bounded analytic Lemma 7.1. Let A be R-sectorial of angle ωR C0 -semigroup generated by −A. The family {S(t): t 0} is R-bounded in L (X).

In the remainder of this section we work in an Lp -setting and use the notations of the previous section. As before we write Lp = Lp (M, μ) and Lp = Lp (M, μ; H ), where (M, μ) is a σ -finite measure space and H is a Hilbert space. The following result is taken from [15,33] where the result is proved for scalar-valued Lp spaces. An extension to more a general class of Banach spaces can be found in [26]. + Proposition 7.2. Let 1 < p < ∞ and let A be an R-sectorial operator on Lp . Let ωR (A) < θ < π . + ∞ For all non-zero ϕ, ψ ∈ H0 (Σθ ) we have

∞ ∞ 1/2 1/2 2 dt 2 dt ϕ(tA)F ψ(tA)F , t t p

0

p

0

with implied constants independent of F . Moreover, the following assertions are equivalent: (1) A admits a bounded H ∞ -calculus; (2) For some (equivalently, for all ) non-zero ψ ∈ H0∞ (Σθ+ ) we have ∞ 1/2 2 dt F − PN(A) F p ψ(tA)F F p , t 0

F ∈ Lp .

p

In (2), PN(A) is the projection onto N(A) with kernel R(A) along the decomposition (7.1). If these + + (A) = ωH equivalent conditions are fulfilled, then ωR ∞ (A). In the next result we let 1 < p < ∞ and consider two R-sectorial operators L and A. We assume that −L and −A generate R-bounded analytic C0 -semigroups P and S on Lp and H . We denote by −L the generator of the tensor product C0 -semigroup P = P ⊗ S on Lp . This + (L), ω+ (A)} < 12 π on Lp . operator is R-sectorial of angle max{ωR We consider the following three square function norms:


∞ 1/2 2 dt tA S(t)u uA := , t

2437

u ∈ H;

0

∞ 1/2 2 dt tLP (t)f f p,L := , t

f ∈ Lp ;

p

0

∞ 1/2 2 dt tL P (t)F F p,L := , t

F ∈ Lp .

p

0

Proposition 7.3. Under the above assumptions we have: (1) If uA u for all u ∈ H and f p,L f p for all f ∈ Lp , then F p,L F p for all F ∈ Lp . (2) If uA (I − PN(A) )u for all u ∈ H and f p,L (I − PN(L) )f p for all f ∈ Lp , then F p,L (I − PN(L) )F p for all F ∈ Lp . As a consequence, if A and L have bounded H ∞ -functional calculi of angles less than 12 π , then L has a bounded H ∞ -functional calculus of angle less than 12 π . Proof. Let us first show that (1) implies (2). It is well known that the assumptions of (2) imply the dual estimates uA∗ u and f q,L∗ f q , where p1 + q1 = 1. By (1) we find that F q,L∗ F q and by duality we obtain the conclusion of (2). The final assertion follows by combining (1) and (2) with Proposition 7.2. It remains to prove (1). We proceed in three steps. Step 1. We prove that t (I ⊗ A) I ⊗ S(t) F 2 γ (L (R

F p .

dt p + , t ),L )

For F ∈ Lp we have, for μ-almost all x ∈ M, 1/2 ∞ t (I ⊗ A) I ⊗ S(t) F (x)2 dt F (x). t 0

Integrating this estimate over M yields ∞ 1/2 2 dt t (I ⊗ A) I ⊗ S(t) F F p . t 0

p

Step 2. We prove that t (L ⊗ I ) P (t) ⊗ I F

γ (L2 (R+ , dtt ),Lp )

F p .

2438


Let (hj )kj =1 be a finite orthonormal system in H and pick F := kj =1 fj ⊗ hj ∈ Lp . For f ∈ Lp let (Uf )(t) := tLP (t)f , and notice that U is a bounded operator from Lp into γ (L2 (R+ , dtt ), Lp ) by the assumption in (1) and Proposition 6.7. Let (rj )j 1 and (γj )j 1 be a Rademacher and a Gaussian sequence respectively on a probability space (Ω , P ). Noting the pointwise equality k t (L ⊗ I ) P (t) ⊗ I F 2 = Ufj (t)2 j =1

we have ∞ 1/2 2 dt t (L ⊗ I ) P (t) ⊗ I F t 0

p

∞ k 1/2 2 dt Ufj (t) = t 0 j =1

p

∞ k 2 1/2 dt = E rj Ufj (t) t j =1

0

p

k rj Ufj

γ (L2 (R+ ×Ω , dtt ⊗P ),Lp )

j =1

k (∗) U rj fj j =1

k (∗∗) rj fj

γ (L2 (Ω ,P ),γ (L2 (R+ , dtt ),Lp ))

γ (L2 (Ω ,P ),Lp )

j =1

2 1/2 k = E γj f j p

(∗∗∗)

j =1

L

j =1

L

p 1/p k E γj f j p p 1/p k = E γj f j j =1

k 1/2 |fj |2 j =1

Lp

Lp

= F p . In (∗) we used Proposition 6.4, in (∗∗) we used the boundedness of U from Lp into γ (L2 (R+ , dtt ), Lp ), and in (∗ ∗ ∗) the definition of the radonifying norm.


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Step 3. We combine the previous estimates. By Lemma 7.1 the family {P (t): t 0} is Rbounded on Lp . Hence by Proposition 6.2 the family {P (t) ⊗ I : t 0} is R-bounded on Lp . Also, by a simple application of Fubini’s theorem, {I ⊗ S(t): t 0} is R-bounded. Combining these facts with Proposition 6.5, for F ∈ Lp we obtain ∞ 1/2 dt tL P F 2 tL P F γ (L2 (R+ , dt ),Lp ) t t p

0

I ⊗ S(t) t (L ⊗ I ) P (t) ⊗ I F γ (L2 (R , dt ),Lp ) + t + P (t) ⊗ I t (I ⊗ A) I ⊗ S(t) F γ (L2 (R , dt ),Lp ) + t t (L ⊗ I ) P (t) ⊗ I F 2 dt p γ (L (R+ ,

+ t (I ⊗ A) I ⊗ S(t) F γ (L2 (R F p .

t

),L )

dt p + , t ),L )

2

Remark 7.4. The final assertion in Proposition 7.3 is due to Lancien, Lancien, and Le Merdy [32, Theorem 1.4] who proved it using operator-valued H ∞ -functional calculi. The next proposition has been proved in [33, Theorem 3.5, Remark 3.6] (for H = C) and can be extended to a more general class of Banach spaces including the spaces Lp [22,44] (in [44] a generalisation of the crucial ingredient [33, Proposition 3.3] is obtained). + (L) < 12 π , and let P be Proposition 7.5. Let 1 < p < ∞, let L be R-sectorial on Lp of angle ωR p the bounded analytic C0 -semigroup P on L generated by −L. Let U : Dp (L) → Lp be a linear operator, bounded with respect to the graph norm of Dp (L). Consider the following statements.

(1)

∞ 1/2 2 dt √ tU P (t)f f p , t

f ∈ Dp (L);

p

0

√ (2) The family { tU P (t): t > 0} is R-bounded in L (Lp , Lp ). Then (1) implies (2). If L satisfies the square function estimate ∞ 1/2 2 dt tLP (t)f f p , t 0

f ∈ Lp ,

p

then (2) implies (1). Remark 7.6. In [33] and other works in the mathematical systems theory literature, condition (2) is replaced by the following equivalent condition: (2 ) The family {tU (I + t 2 L)−1 : t > 0} is R-bounded.

2440


That (2) implies (2 ) follows by taking Laplace transforms and the opposite direction is observed in [33, (3.12)]. Since our computations involve semigroups rather than resolvents we find it more natural to use (2) here. Below we shall also need the notion of an (R-)bisectorial operator, which is analogous to that of an (R-)sectorial operator, the only difference being that the sector Σθ+ is replaced by the bisector Σθ = Σθ+ ∪ Σθ− , where Σθ− = −Σθ+ . Many results in the literature on (R-)sectorial operators carry over to (R-)bisectorial operators, with only minor changes in the proofs. We refer to the lecture notes [4] for more details. 8. Proof of Theorems 2.3 and 2.4 We return to the main setting of the paper and take up our study of the operators L and L introduced in Sections 4 and 5. Notations are again as in these sections. For functions f ∈ F Cb∞ (E; D(A)) we consider the Littlewood–Paley–Stein square functions ∞ 1/2 2 dt √ tDV P (t)f (x) , H f (x) := t

x ∈ E,

0

∞ 1/2 2 dt tDV Q(t)f (x) G f (x) := , t

x ∈ E,

0

√ where Q denotes the analytic C0 -semigroup generated by − L. The functions t → DV P (t)f are analytic in a sector containing R+ , and therefore a wellknown result of Stein [51] allows us to select a pointwise version (t, x) → DV P (t)f (x) which is analytic in t for every fixed x. Using such a version, we see that H f is well defined almost everywhere (but possibly infinite). The square function G f is well defined by similar reasoning. In Section 10 we shall need the following inequality. The argument is taken from [13]. Lemma 8.1. For all f ∈ F Cb∞ (E; D(A)) we have G f H f μ-almost everywhere. Proof. Using the representation 1 Q(t)f = √ π

∞

2 e−u t f du √ P 4u u

0

and the closedness of DV , G 2 f (x) =

∞ tDV Q(t)f (x)2 dt t 0

1 π

−u 2 2 ∞ ∞ e t dt tDV P . f (x) √ du 4u t u 0

0


Since

∞ 0

e√−u u

du =

2441

√ π we may apply Jensen’s inequality to obtain

1 G f (x) √ π 2

2 2 ∞ ∞ e−u dt tDV P t f (x) √ du u 4u t 0

1 =√ π

2 −u 2 tDV P t f (x) dt e√ du t 4u u

0

2 =√ π

0

∞ ∞ 0

∞ ∞

√ √ −u sDV P (s)f (x)2 ds ue du s

0

0

= H f (x).

2

2

The main results of this section are the following two theorems, which together imply part (2) of Theorem 2.3 as well as Theorem 2.4. Part (1) of Theorem 2.3 is contained in Theorem 8.10. Theorem 8.2 (R-Gradient bounds). Assume (A1), (A2), (A3), and let 1 0

and

−1 tDV I + t 2 L : t > 0

are R-bounded in L (Lp , Lp ). Theorem 8.3 (Littlewood–Paley–Stein inequalities). Assume (A1), (A2), (A3), and let 1 < p < ∞. For all f ∈ F Cb∞ (E; D(A)) we have the square function estimate f − PNp (L) f p H f p f p , where PNp (L) is the projection onto Np (L) along the direct sum decomposition Lp = Np (L) ⊕ Rp (L). By Theorem 8.3 the square function H f is actually well defined for arbitrary f ∈ Lp , and by approximation Theorem 8.3 extends to all of Lp . Since we do not need these observations we leave the details to the reader. For the proofs of both theorems we distinguish between the cases 1 < p 2 and 2 < p < ∞. For 1 < p 2 we show by a direct argument that H is Lp -bounded and deduce from this that Dp (L) is a core for Dp (DV ). Theorem 8.2 is then a consequence of Proposition 7.5. For 2 < p < ∞ we first derive Theorem 8.2 from a pointwise gradient bound and a duality argument involving maximal functions. Since L has a bounded H ∞ -calculus of angle < 12 π by Lemma 8.4, the right-hand side estimate of Theorem 8.3 then follows by an application of Proposition 7.5. Finally, the left-hand side inequality of Theorem 8.3 is proved, for 1 < p < ∞, by a duality argument. We begin with an easy observation.

2442


Lemma 8.4. Let 1 < p < ∞. The operator L is R-sectorial and admits a bounded H ∞ -calculus + + 1 on Lp of angle ωH ∞ (L) = ωR (L) < 2 π . Moreover, (1) The family {P (t): t 0} is R-bounded in L (Lp ); (2) The family {P (t): t 0} is R-bounded in L (Rp (DV )). Proof. Since −L generates an analytic C0 -semigroup of positive contractions on Lp for all 1 < p < ∞, the first part follows from [28, Corollary 5.2 and Theorem 5.3]. Assertion (1) follows from Lemma 7.1, and assertion (2) follows by combining (1) with the identity P = P ⊗ S and Proposition 6.2. 2 We continue with a simple extension of a well-known result of Cowling [14, Theorem 7] (see also [52]). For the convenience of the reader we give a sketch of the proof. Proposition 8.5. Let (M, μ) be a σ -finite measure space and let T be an analytic C0 -semigroup of positive operators on L2 := L2 (M, μ) satisfying T (t)f p f p for all f ∈ L2 ∩ Lp , t 0 and 1 p ∞. Let T f (x) := supT (t)f (x). t>0

Then for 1 < p < ∞ we have T f p f p ,

f ∈ Lp .

Proof. Let −L denote the generator of T in Lp . By [28, Corollary 5.2], L has a bounded H ∞ calculus of angle ω < 12 π . The key idea of the proof is to write 1 T (t)f = t

t

1 T (s)f ds + T (t)f − t

0

t

1 T (s)f ds = t

0

t T (s)f ds + m(tL)f, 0

1 where m(z) := e−z − 0 e−sz ds. By the Hopf–Dunford–Schwartz ergodic theorem [30, Theorem 6.12] we have t 1 T (s)f ds f p , sup t>0 t 0

p

so that it remains to prove that supt>0 |m(tL)f |p f p . Let n := m ◦ exp and let nˆ be its Fourier transform. Using the identities 1 iu n(u)z ˆ du, n(u) ˆ = 1 − (1 + iu)−1 Γ (iu), m(z) = 2π R

1

and the estimate |n(u)| ˆ Ce− 2 π|u| (see [14]) we obtain 1 1 1 n(u) ˆ (tL)iu F du supm(tL)F sup e− 2 π|u| Liu F du. 2π t>0 t>0 2π R

R


2443

From the H ∞ -calculus of L we obtain Liu f p eω|u| f p . Taking Lp -norms we obtain 1 1 1 1 e− 2 π|u| Liu F p du e(ω− 2 π)|u| F p du F p . supm(tL)F p 2π 2π t>0 R

2

R

8.1. The case 1 0. Using Lemma 5.3 and the fact that S(t)h ∈ D(A) by analyticity, we obtain

S(t)V h2 = V S(t)h2 k −1 BV S(t)h, V S(t)h

= k −1 AS(t)h, S(t)h = −(2k)−1

d S(t)h2 . dt

Hence ∞ T S(t)V h2 dt (2k)−1 lim sup − d S(t)h2 dt dt T →∞ 0

0

2 = (2k)−1 h2 − lim infS(T )h T →∞

(2k)−1 h2 .

2

Lemma 8.7. Let f ∈ F Cb∞ (E; D(A)) and F ∈ F Cb∞ (E; D(A)) ⊗ D(A) be such that DV f = (I ⊗ V )F . Then for all 1 < p < ∞ we have H f ∈ Lp and H f p F p . Proof. By Proposition 6.2 and Lemma 7.1, the set {P (t) ⊗ I : t 0} is R-bounded in L (Lp ). Hence, by Propositions 6.5, 6.7, and Lemma 8.6, ∞ 1/2 2 P (t)DV f dt H f p = 0

p

∞ 1/2 2 P (t) ⊗ I I ⊗ S(t) (I ⊗ V )F dt = 0

p

2444


∞ 1/2 2 I ⊗ S(t) (I ⊗ V )F dt

p

0

(2k)−1/2 F p .

2

The following proof is based on a classical argument which goes back to Stein [51]. The same idea has been applied in the related works [12,13,36,49]. For the convenience of the reader we include a proof. Proof of the first part of Theorem 8.3, 1 0. Fix f = ϕ(φh1 , . . . , φhk ) ∈ F Cb∞ (E; D(A)) of the usual form. Pick functions mn ∈ Cb∞ (Rk ) satisfying mn 0, supp(mn ) ⊆ [− n1 , n1 ]k , and mn 1 = 1, and put 1 ∗ mn , ψn,± := ϕ ± + n gn,± := ψn,± (φh1 , . . . , φhk ), gn,±,j := ∂j ψn,± (φh1 , . . . , φhk ). Clearly gn,± ∈ F Cb∞ (E; D(A)) satisfy

1 n

gn,± ϕ∞ + 1, and

± 1 f + − gn,± →0 n p by dominated convergence. From Lemma 8.7 it follows that H f − H (gn,+ − gn,− ) H f − (gn,+ − gn,− ) p p k fj − (gn,+,j − gj,n,−,j ) ⊗ hj , j =1

p

where fj = ∂j ϕ(φh1 , . . . , φhk ). Since the functions gn,±,j = ∂j ϕ(φh1 , . . . , φhk )1{±ϕ(φh1 ,...,φhk )>0} ∗ mn , belong to L∞ uniformly in n, we conclude by dominated convergence that fj − (gn,+,j − gn,−,j )p → 0. Therefore H (gn,+ − gn,− ) → H f in Lp as n → ∞. Hence if H gn,± p gn,± p with constants not depending on n, then H f p = lim H (gn,+ − gn,− )p n→∞ lim sup H gn,+ p + H gn,− p n→∞

lim sup gn,+ p + gn,− p n→∞


2445

= f + p + f − p 2f p . Thus it suffices to prove the result for f ∈ F Cb∞ (E; D(A)) satisfying f ε for some ε > 0. Set u(t, x) := P (t)f (x),

x ∈ E, t > 0,

and notice that by Mehler’s formula (3.3) we have u(t, x) ε for all x ∈ E and t 0. By Lemma 4.8 we have u(t, ·) ∈ F Cb∞ (E; D(A)) ⊆ Dp (L) for all t 0. Arguing as in [12,13,49], for 1 < p 2 we use Lemma 4.8 and a truncation argument to obtain that u(t, ·)p ∈ Dp (L) and (∂t + L)u(t, x)p = pu(t, x)p−1 (∂t + L)u(t, x)

− p(p − 1)u(t, x)p−2 BDV u(t, x), DV u(t, x)

= −p(p − 1)u(t, x)p−2 BDV u(t, x), DV u(t, x) . Hence, using the coercivity Assumption (A3),

DV u(t, x)2 k −1 BDV u(t, x), DV u(t, x) =−

1 u(t, x)2−p (∂t + L)u(t, x)p . kp(p − 1)

Now we set ∞ K(x) := − (∂t + L)u(t, x)p dt 0

and u (x) := sup u(t, x) t>0

to obtain H f (x)2 =

∞ DV u(t, x)2 dt 0

∞ −Cp,k

u(t, x)2−p (∂t + L)u(t, x)p dt 0

Cp,k u (x)2−p K(x). Hölder’s inequality with exponents

2 2−p

and

2 p

implies

2446


p 2

H f (x) dμ(x) Cp,k p

u (x)

E

(2−p)p 2

p

K(x) 2 dμ(x)

E p 2

2−p p 2 2 u (x) dμ(x) K(x) dμ(x) .

Cp,k

p

E

(8.1)

E

Using the invariance of μ and the Lp -contractivity of P we obtain

∞ K(x) dμ(x) = −

(∂t + L)u(t, x)p dμ(x) dt

E

0 E

∞ =−

∂t u(t, x)p dμ(x) dt 0 E

∞ = − ∂t u(t, x)p dμ(x) dt E

0

p p lim sup f p − u(t, ·)p t→∞ p f p ,

(8.2)

where the use of Fubini’s theorem is justified by the non-negativity of the integrand K, and the interchange of differentiation and integration by the fact that f ∈ F Cb∞ (E; D(A)). Combining (8.1), (8.2) and Proposition 8.5 we conclude that (2−p)p 2

p

H f p u p

p2

p

f p2 f p .

2

Proof of Theorem 8.2, 1 < p 2. First we show that Dp (L) is contained in Dp (DV ). Once we know this, Lemmas 4.8 and 5.5 imply that Dp (L) is even a core for Dp (DV ). Fix a function f ∈ F Cb∞ (E; D(A)). From Theorem 5.6 it follows that s → e−s DV P (s)f = −s e P (s)DV f is Bochner integrable in Lp and ∞

e−s DV P (s)f ds = (I + L)−1 DV f.

0

Since s → e−s P (s)f is Bochner integrable in Lp , the closedness of DV implies that ∞ (I + L)−1 f = 0 e−s P (s)f ds ∈ Dp (DV ) and −1

DV (I + L)

∞ f = DV

e

−s

∞ P (s)f ds =

0

Moreover, by the Cauchy–Schwarz inequality,

0

e−s DV P (s)f ds.


∞ −1 −s DV (I + L) f e DV P (s)f ds p 0

2447

p

∞ 1/2 2 1 DV P (s)f ds √ 2

p

0

1 = √ H f p f p . 2 It follows that DV (I + L)−1 extends to a bounded operator from Lp to Lp . In view of the closedness of DV and Lemma 4.8, the desired inclusion follows from this. This concludes the proof that Dp (L) is a core for Dp (DV ). The R-boundedness assertions follow from Proposition 7.5 and Remark 7.6. 2 8.2. The case 2 < p < ∞ In case that P is symmetric it is possible to use a variant of a duality argument of Stein [51] to prove the boundedness of H . This approach has been taken in [12], but the proof breaks down if L is non-symmetric and we have to proceed in a different way. First we derive an explicit formula for the semigroup P which allows us to prove suitable gradient bounds. Having obtained those gradient bounds we give a general argument √ involving a maximal inequality for P ∗ to prove the R-boundedness of the collection { tDV P (t): t > 0}. Since L has a bounded H ∞ -calculus, we obtain the boundedness of H by an appeal to Proposition 7.5. We begin with some preliminary observations. For 0 < t < ∞ we define the operators Qt ∈ L (E ∗ , E) by Qt x ∗ := ii ∗ x ∗ − iS ∗ (t)S(t)i ∗ x ∗ , where i : H → E is the inclusion operator. The operators Qt are positive and symmetric, i.e., for all x ∗ , y ∗ ∈ E ∗ we have Qt x ∗ , x ∗ 0 and Qt x ∗ , y ∗ = Qt y ∗ , x ∗ . Let Ht be the reproducing kernel Hilbert space associated with Qt and let it : Ht → E be the inclusion mapping. Then, it it∗ = Qt . Since Qt x ∗ , x ∗ ii ∗ x ∗ , x ∗ for all x ∗ ∈ E ∗ , the operators Qt are covariances of centred Gaussian measures μt on E; see, e.g., [20]. This estimate also implies that we have a continuous inclusion Ht → H and that the mapping Vt : i ∗ x ∗ → it∗ x ∗ ,

x∗ ∈ E∗,

is well defined and extends to a contraction from H into Ht . It is easy to check that the adjoint operator Vt∗ is the inclusion from Ht into H . Let us also note that for s t and x ∗ ∈ E ∗ we have 2 2 Qs x ∗ , x ∗ = i ∗ x2 − S(s)i ∗ x ∗ i ∗ x2 − S(t)i ∗ x ∗ = Qt x ∗ , x ∗ by the contractivity of S.

2448

J. Maas, J. van Neerven / Journal of Functional Analysis 257 (2009) 2410–2475 μ

In the next proposition we fix t > 0 and h ∈ Ht and denote by φh t : E → R the (μt -essentially μ unique; see Section 2) μt -measurable linear extension of the function φh t (it g) := [g, h]Ht . Proposition 8.8. For all f = ϕ(φh1 , . . . , φhn ) ⊗ h ∈ F Cb (E) ⊗ H , where H is some Hilbert space, the following identity holds for μ-almost all x ∈ E: P (t) ⊗ I f (x) =

μ μ ϕ φS(t)h1 (x) + φVtth1 (y), . . . , φS(t)hn (x) + φVtthn (y) h dμt (y).

E

Proof. Defining ψ : E × Rn → H by ψ(x, ξ ) := ϕ φS(t)h1 (x) + ξ1 , . . . , φS(t)hn (x) + ξn h, we have

μ μ ϕ φS(t)h1 (x) + φVtth1 (y), . . . , φS(t)hn (x) + φVtthn (y) h dμt (y)

E

=

μ μ ψ x, φVtth1 (y), . . . , φVtthn (y) dμt (y)

E

=

ψ(x, ξ ) dγt (ξ ), Rn

where γt is the centred Gaussian measure on Rn whose covariance matrix equals ([Vt hi , Vt hj ])ni,j =1 . √ On the other hand, writing R(t) = I − S ∗ (t)S(t), by Mehler’s formula (3.3) we have P (t) ⊗ I f (x) =

ϕ φS(t)h1 (x) + φR(t)h1 (y), . . . , φS(t)hn (x) + φR(t)hn (y) h dμ(y)

E

=

ψ x, φR(t)h1 (y), . . . , φR(t)hn (y) dμ(y)

E

=

ψ(x, ξ ) d γ˜t (ξ ), Rn

where γ˜t is the centred Gaussian measure on Rn whose covariance matrix equals ([R(t)hi , R(t)hj ])ni,j =1 . The result follows from the observation that

[Vt hi , Vt hj ] = [hi , hj ] − S(t)hi , S(t)hj = R(t)hi , R(t)hj .

2


2449

Lemma 8.9. For all u ∈ H and t > 0 we have S ∗ (t)u ∈ D(V ∗ ), V ∗ S ∗ (t)u ∈ Ht , and ∗ ∗ V S (t)u

Ht

1 √ u. t

Proof. First we observe that S(s) maps H into D(A) ⊆ D(V ) for s > 0. For t > 0 we claim that Jt : Vt h → V S(·)h extends to a bounded operator from Ht into L2 (0, t; H ) of norm √1 . 2k Indeed, by the coercivity of B and the definition of Ht , we obtain for h ∈ H , t

V S(s)h2 ds 1 k

0

t

BV S(s)h, V S(s)h ds

0

1 =− 2k

t

d S(s)h2 ds ds

0

2 1 h2 − S(t)h = 2k 1 = Vt h2Ht . 2k Recall that Vt∗ is the inclusion mapping Ht → H . Noting that S ∗ (t) maps H into D(A∗ ) ⊆ D(V ∗ ) and using Lemma 5.3, the adjoint mapping Jt∗ : L2 (0, t; H ) → Ht is given by Vt∗ Jt∗ f

t =

V ∗ S ∗ (s)f (s) ds,

f ∈ L2 (0, t; H ).

0

The resulting identity V ∗ S ∗ (t)u = 1t Vt∗ Jt∗ (S ∗ (t − ·)u) shows that V ∗ S ∗ (t)u can be identified with the element 1t Jt∗ (S ∗ (t − ·)u) of Ht and we obtain ∗ ∗ V S (t)u

Ht

1 = Jt∗ S ∗ (t − ·)u H t t 1 √ S ∗ (t − ·)uL2 (0,t;H ) t 2k 1 supS ∗ (s)L (H ) u. √ 2kt s0

2

The following pointwise gradient bound is included for reasons of completeness. We shall only need the special case corresponding to r = 2, for which a simpler proof can be given; see Remark 8.11.

2450


Theorem 8.10 (Pointwise gradient bounds). Let 1 < r < ∞. For f ∈ F Cb (E) and t > 0 we have, for μ-almost all x ∈ E, √ 1/r t DV P (t)f (x) P (t)|f |r (x) . Proof. For notational simplicity we take f of the form f = ϕ(φh ) with ϕ ∈ Cb (R) and h ∈ H . It is immediate to check that the argument carries over to general cylindrical functions in F Cb (E). By Lemma 8.9 we have S ∗ (t)V ∗ u ∈ Ht for u ∈ D(V ∗ ) and therefore, for all h ∈ H ,

μ φS(t)h (iV ∗ u) = S(t)h, V ∗ u = h, S ∗ (t)V ∗ u = φVtth iS ∗ (t)V ∗ u . By Proposition 8.8 (with H = R) we find that for all g ∈ H ,

∗

P (t)f (x + iV u) =

μ ϕ φS(t)h (x + iV ∗ u) + φVtth (y) dμt (y)

E

=

μ ϕ φS(t)h (x) + φVtth y + iS ∗ (t)V ∗ u dμt (y).

E

Recalling that D denotes the Malliavin derivative we have, for all u ∈ D(V ∗ ),

DV P (t)f (x), u = DP (t)f (x), V ∗ u 1 P (t)f (x + εiV ∗ u) − P (t)f (x) ε 1 μ ϕ φS(t)h (x) + φVtth y + εiS ∗ (t)V ∗ u = lim ε↓0 ε = lim ε↓0

E

μ − ϕ φS(t)h (x) + φVtth (y) dμt (y). Using Lemma 8.9 and the Cameron–Martin formula [7, Corollary 2.4.3] we obtain

1 DV P (t)f (x), u = lim ε↓0 ε

μt μ EεS ∗ (t)V ∗ u (y) − 1 ϕ φS(t)h (x) + φVtth (y) dμt (y),

E μ

μ

where Eh t (y) = exp(φh t (y) − 12 h2Ht ). It is easy to see that for each h ∈ Ht the family μ ( 1ε (Eεht − 1))00 |T ∗ (t)f | is measurable and Lq -bounded, where p1 + q1 = 1. Then, for all f1 , . . . , fn ∈ Lp and all t1 , . . . , tn > 0, n n T (tk )|fk | |fk | . k=1

k=1

p

p

Proof. Taking the supremum over all g = (gk )nk=1 ∈ Lq (∞ n ) of norm one we obtain n T (tk )|fk | = T (t(·) )|f(·) | Lp (1 ) n k=1

p

= sup g

T (tk )|fk | · gk dμ

E k=1

= sup g

n n

|fk | · T ∗ (tk )gk dμ

E k=1

|f(·) |

Lp (1n )

sup T ∗ (t(·) )g(·) Lq (∞ ) . n

g

Using the positivity of T ∗ on Lq to obtain sup1kn T∗ |gk | T∗ (sup1kn |gk |) we estimate ∗ T (t(·) )g(·)

Lq (∞ n )

= sup T ∗ (tk )gk 1kn

Lq

sup T∗ |gk |

Lq

1kn

T∗ sup |gk | 1kn

sup |gk |

Lq

1kn

= (gk )Lq (∞ ) . n

This completes the proof.

2

The previous two results are now combined to prove:

Lq


Proof of Theorem 8.2 (for 2 0

is bounded on Lq . Using Theorem 8.10 (for r = 2) and Proposition 8.12 we obtain, for all f1 , . . . , fn ∈ F Cb (E), n n 1/2 1/2 √ 2 tk DV P (tk )fk P (tk )|fk |2 k=1

p

k=1

1/2 n 2 = P (tk )|fk | k=1

1/2 n 2 |fk | k=1

p

p/2

p/2

n 1/2 2 = |fk | . k=1

p

p By an approximation argument this estimate √ extends to arbitrary f1 , . . . , fn ∈ L . Now Proposition 6.6 implies the R-boundedness of { tDV P (t): t > 0}. Taking Laplace transforms and using Proposition 6.1, it follows that Dp (L) ⊆ Dp (DV ) and that the collection {tDV (I + t 2 L)−1 : t > 0} is R-bounded from Lp into Lp . As in the case 1 < p 2, Lemmas 4.8 and 5.5 imply that Dp (L) is even a core for Dp (DV ). 2

Proof of the first part of Theorem 8.3 (for 2 < p < ∞). By Lemma 8.4, L has a bounded H ∞ -calculus of angle < 12 π , and the result follows from Proposition 7.5. 2 8.3. Completion of the proof of Theorem 8.3 It remains to prove, for 1 < p < ∞, the left-hand side inequality of Theorem 8.3. We adapt a standard duality argument (see, e.g., [3, Section 7, Step 8]). It is enough to prove the estimate for f ∈ Rp (L); for such f we have f − PNp (L) f = f . First let f = Lg with g ∈ Dp (L2 ). Then by [31, Lemma 9.13], t lim P (t)Lg − Lg = − lim

t→∞

t P (s)L g ds = − lim 2

t→∞

ψ(sL)Lg

t→∞

0

ds = −Lg, s

0

where ψ(z) = ze−z . Hence, limt→∞ P (t)Lg = 0. By a density argument, this implies lim P (t)f = 0,

t→∞

f ∈ R(L).

2454


Fix f ∈ L2 ∩ Rp (L) and g ∈ L2 ∩ Lq with −∂t

P (t)f g dμ =

E

E

=

1 p

+

1 q

= 1. For any t > 0,

L P (t)f g dμ 1 1 t f P ∗ t g dμ L P 2 2

E

=

BDV P

1 1 t f, DV P ∗ t g dμ. 2 2

E

Since E f dμ = 0 we obtain, using Theorem 8.3 applied to the adjoint semigroup P ∗ (which is generated by −L∗ = −DV∗ B ∗ DV ) in Lq ,

f g dμ = E

f g dμ −

E

= lim

f dμ

E

g dμ E

P (ε)f g dμ − lim

t→∞

ε↓0

E

P (t)f g dμ

E

∞ = − ∂t P (t)f g dμ dt E

0

∞ =

1 ∗ 1 BDV P t f, DV P t g dμ dt 2 2

0 E

∞ 2 1/2 √ tDV P 1 t f dt B 2 t

p

0

∞ 1/2 2 dt √ tDV P (t)f gq . t

∞ 2 1/2 √ tDV P ∗ 1 t g dt 2 t

q

0

p

0

This implies that ∞ 1/2 2 dt √ tDV P (t)f f p . t 0

p

So far we have assumed that f ∈ L2 ∩ Rp (L). The extension to general f ∈ Rp (L) follows by a density argument (using the first part of the theorem to see that the right-hand side can be approximated as well).


2455

9. The operators DV and DV∗ B In this section we study some Lp -properties of the operators DV and DV∗ B and provide a rigorous interpretation of the identities L = DV∗ BDV and L = DV DV∗ B in Lp and Rp (DV ). From these operators we build operator matrices which will play an important role in the proofs of Theorems 2.1, 2.5, and 2.6. Throughout this section we fix 1 < p < ∞. The operator DV∗ B is closed and densely defined as an operator from Lp to Lp with domain

Dp DV∗ B = F ∈ Lp : BF ∈ Dp DV∗

.

Moreover, since BF p F p for F ∈ Rp (DV ), DV∗ B = (B ∗ DV )∗ , where B ∗ DV is interpreted as a operator from Lq to Lq , p1 + q1 = 1. For the next result we recall that C := F Cb∞ (E; D(A)) is a P -invariant core for Dp (L). We set C ∗ := F Cb∞ (E; D(A∗ )); this is a P ∗ -invariant core for Dp (L∗ ). Proposition 9.1. In Lp we have L = (DV∗ B)DV . More precisely, f ∈ Dp (L) if and only if f ∈ Dp (DV ) and DV f ∈ Dp (DV∗ B), in which case we have Lf = (DV∗ B)DV f . Proof. First note that for all f, g ∈ C we have Lf, g = DV f, B ∗ DV g . Since C is a core for Dp (L), and Dp (L) is core for Dp (DV ) by the first part of Theorem 8.2, this identity extends to all f ∈ Dp (L) and g ∈ Dp (DV ). This implies that DV f ∈ Dp ((B ∗ DV )∗ ) and (B ∗ DV )∗ DV f = Lf . Since (B ∗ DV )∗ = DV∗ B, we find that L ⊆ (DV∗ B)DV . To prove the other inclusion we take f ∈ Dp (DV ) such that DV f ∈ Dp (DV∗ B). We have f, L∗ g = DV f, B ∗ DV g = (DV∗ B)DV f, g for all g ∈ C ∗ , where the second identity follows from DV f ∈ Dp (DV∗ B) = Dp ((B ∗ DV )∗ ). Since C ∗ is a core for Dq (L∗ ) this implies that f ∈ Dp (L) and Lf = (DV∗ B)DV f . 2 We shall be interested in the restriction DV∗ B|Rp (DV ) of DV∗ B to Rp (DV ). As its domain we take

Dp DV∗ B|Rp (D ) := F ∈ Rp (DV ): BF ∈ Dp DV∗ V

= Dp DV∗ B ∩ Rp (DV ).

In the middle expression, as before we consider DV∗ as a densely defined operator from Lp to Lp . Corollary 9.2. The restriction DV∗ B|Rp (DV ) is closed and densely defined. Proof. Let f ∈ Dp (DV ). By the first part of Theorem 8.2 there exist functions fn ∈ Dp (L) such that fn → f in Dp (DV ). Proposition 9.1 implies that DV fn ∈ Dp (DV∗ B|Rp (DV ) ). This shows that DV∗ B|Rp (DV ) is densely defined on Rp (DV ). Closedness is clear.

2

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Proposition 9.3. The domain Dp (L) is a core for Dp (DV∗ B|Rp (DV ) ). Moreover, for all t > 0 the operators (I + t 2 L)−1 DV∗ B|Rp (DV ) and P (t)DV∗ B|Rp (DV ) (initially defined on

Dp (DV∗ B|Rp (D ) )) extend uniquely to bounded operators from Rp (DV ) to Lp , and for all V

F ∈ Rp (DV ) we have −1 −1 I + t 2 L DV∗ BF = DV∗ B I + t 2 L F and P (t)DV∗ BF = DV∗ BP (t)F. Proof. We split the proof into four steps. Step 1. By Proposition 9.1, for all f ∈ Dp (L) we have f ∈ Dp (DV ) and DV f ∈ Dp (DV∗ B|Rp (D ) ), and for all t > 0 we have V P (t) DV∗ B DV f = P (t)Lf = LP (t)f = DV∗ B DV P (t)f = DV∗ BP (t)DV f. By taking Laplace transforms and using the closedness of DV∗ B, this gives (I + t 2 L)−1 DV f ∈ Dp (DV∗ B|R(D ) ) and V −1 −1 ∗ DV B DV f = DV∗ B I + t 2 L DV f. I + t 2L

(9.1)

Step 2. By Theorem 8.2, for all t > 0 the operator T (t) := B ∗ DV (I + t 2 L∗ )−1 is bounded from Lq into Lq , p1 + q1 = 1. For all F ∈ Dp (DV∗ B|Rp (DV ) ) and g ∈ Lq we have −1 −1 F, T (t)g = F, B ∗ DV I + t 2 L∗ g = I + t 2 L DV∗ BF, g .

(9.2)

Now let F ∈ Rp (DV ) be arbitrary and take a sequence (Fn )n1 ⊆ Dp (DV∗ B|Rp (DV ) ) converging to F in Rp (DV ). By Proposition 9.1 and the fact that Dp (L) is a core for Dp (DV ) we may take the Fn of the form DV fn with fn ∈ Dp (L). Then (I + t 2 L)−1 Fn → (I + t 2 L)−1 F , and from (9.1) we obtain −1 −1 DV∗ B I + t 2 L Fn = I + t 2 L DV∗ BFn = T ∗ (t)Fn → T ∗ (t)F. The closedness of DV∗ B implies that (I + t 2 L)−1 F ∈ Dp (DV∗ B|Rp (DV ) ). This proves the domain inclusion Dp (L) ⊆ Dp (DV∗ B|Rp (DV ) ), along with the identity −1 DV∗ B I + t 2 L F = T ∗ (t)F,

F ∈ Rp (DV ).

Note that for F ∈ Dp (DV∗ B), from (9.2) we also obtain −1 −1 DV∗ B I + t 2 L F = T ∗ (t)F = I + t 2 L DV∗ BF.

(9.3)


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Step 3. By Step 2 the operator DV∗ B(I + t 2 L)−1 is bounded from Rp (DV ) to Lp . Therefore, by (9.1), the operator (I + t 2 L)−1 DV∗ B (initially defined on the dense domain Dp (DV∗ B|Rp (DV ) )) uniquely extends to a bounded operator from Rp (DV ) to Lp , and for this extension we obtain the identity −1 −1 I + t 2 L DV∗ B = DV∗ B I + t 2 L . On Dp (DV∗ B|Rp (DV ) ), the identity DV∗ BP (t) = P (t)DV∗ B follows from (9.3) by real Laplace inversion (cf. the proof of Lemma 5.3). The existence of a unique bounded extension of P (t)DV∗ B is proved in the same way as before. Step 4. It remains to prove that Dp (L) is a core for Dp (DV∗ B|Rp (DV ) ). Take F ∈ Dp (DV∗ B|Rp (DV ) ). Then limt→0 (I + t 2 L)−1 F = F

limt→0 DV∗ B(I sult.

2

+

t 2 L)−1 F

= limt→0 (I +

t 2 L)−1 DV∗ BF

=

in Rp (DV ) and, by (9.3)

DV∗ BF

in Lp . This gives the re-

Proposition 9.4. For all F ∈ Dp (L) we have F ∈ Dp (DV∗ B), DV∗ BF ∈ Dp (DV ), and DV (DV∗ B)F = LF . Proof. Since Dp (L) is a core for Dp (DV ), the set P := {DV (I + L)−1 g: g ∈ Dp (DV )} is a P -invariant dense subspace of Rp (DV ). To see that P is contained in Dp (L), note that if g ∈ Dp (DV ), then f := (I + L)−1 g ∈ Dp (L) and DV f = DV (1 + L)−1 g = (1 + L)−1 DV g ∈ Dp (L) as claimed. It follows that P is a core for Dp (L), and hence a core for Dp (DV∗ B|Rp (DV ) ) by Proposition 9.3. Moreover, (1 + L)DV f = DV g = DV (I + L)f , and therefore LDV f = DV Lf . For F ∈ P, say F = DV f with f = (I + L)−1 g for some g ∈ Dp (DV ), we then have LF = LDV f = DV Lf = DV DV∗ B DV f = DV DV∗ B DV f = DV DV∗ B F. To see that this above identity extends to arbitrary F ∈ Dp (L), let Fn → F in Dp (L) with all Fn in P. It follows from Proposition 9.3 that Fn → F in Dp (DV∗ B). In particular, DV∗ BFn → DV∗ BF in Lp . Since DV (DV∗ B)Fn = LFn → LF in Rp (DV ), the closedness of DV then implies that DV∗ BF ∈ Dp (DV ) and DV (DV∗ B)F = LF . 2 In the remainder of this section we consider DV∗ B as a closed and densely defined operator from Rp (DV ) to Lp and write DV∗ B instead of using the more precise notation DV∗ B|Rp (DV ) . For the proof of Proposition 9.5 we need the first part of Theorem 2.6. Its proof uses the Hodge–Dirac formalism, introduced by Axelsson, Keith, and Mc Intosh [6] in their study of the Kato square root problem. It was by using this formalism that the main results of this paper suggested themselves naturally. On the Hilbertian direct sum H ⊕ R(V ) we consider the closed and densely defined operator

0 T := V

V ∗B . 0

By [5, Theorem 8.3] T is bisectorial on H ⊕ R(V ).

(9.4)

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On the direct sum Lp ⊕ Rp (DV ) we introduce the closed and densely defined operator Π by Π :=

0 DV

DV∗ B . 0

Proof of Theorem 2.6, first part. By Theorems 4.5 and 5.6, L and L are sectorial on Lp and Rp (DV ), respectively. From this it is easy to see that on Lp ⊕ Rp (DV ) we have that iR \ {0} is contained in the resolvent set of Π and

it (I + t 2 L)−1 DV∗ B (1 + t 2 L)−1 −1 , t ∈ R \ {0}; (I − itΠ) = (I + t 2 L)−1 itDV (I + t 2 L)−1 the rigorous interpretation of this identity is provided by the above propositions. Note that the off-diagonal entries are well defined and bounded by Theorem 8.2 and Proposition 9.3; the proof of the latter result also shows that (I + t 2 L)−1 DV∗ B is the adjoint of B ∗ DV (I + t 2 L∗ )−1 . We check the R-boundedness of the entries of the right-hand side matrix for t ∈ R \ {0}. For the upper left and the lower right entry this follows from the R-sectoriality of L and L on Lp and Rp (DV ) respectively. Theorem 8.2 ensures the R-boundedness of the lower left entry, and the R-boundedness of the upper right entry follows from Proposition 6.3 (applied with B and L replaced by B ∗ and L∗ ). 2 As a consequence of the bisectoriality of Π , the operator Π 2 is sectorial. Moreover,

(DV∗ B)DV Π = 0 2

0 L = DV (DV∗ B) 0

0 . L

To justify the latter identity, we appeal to Propositions 9.1 and 9.4 to obtain the inclusion

L 0 1 2 0 L ⊆ Π . Since both operators are sectorial of angle < 2 π , they are in fact equal. Proposition 9.5. On Lp and Rp (DV ) the following identities hold:

Rp (L) = Rp DV∗ B ,

Np (L) = Np (DV ),

Rp (L) = Rp (DV ),

Np (L) = Np DV∗ B = {0}.

Moreover, Lp = Rp (DV∗ B) ⊕ Np (DV ). We recall that DV∗ B is interpreted as a densely defined closed operator from Rp (DV ) to Lp . In the final section we will show that under the assumptions of Theorem 2.1 we have Rp (DV∗ B) = Rp (DV∗ ) and that in this situation the space Rp (DV∗ B) does not change if we consider DV∗ B as an unbounded operator from Lp to Lp . Proof. The bisectoriality of Π on Lp ⊕ Rp (DV ) implies that

Rp Π 2 = Rp (Π)

and Np Π 2 = Np (Π).


2459

The result follows from this by considering both coordinates separately. The fact that Np (DV∗ B) = {0} follows from the bisectorial decomposition Lp ⊕ Rp (DV ) = Rp (Π) ⊕ Np (Π) and considering the second coordinate. The final identity follows by inspecting the first coordinate of the same decomposition. 2 10. Proof of Theorem 2.1 The main effort in this section is directed towards proving the following comprehensive version of Theorem 2.1. Theorem 10.1. Assume (A1), (A2), (A3), and let 1 < p < ∞. (a) The following √ assertions are equivalent: √ (a1) Dp ( L) ⊆ Dp (DV ) with DV f p Lf p ; (a2) L satisfies a square function estimate on Rp (DV ): ∞ 1/2 2 dt tL P (t)F F p ; t 0

p

√ √ (a3) D( A) ⊆ D(V ) with V h Ah; (a4) A satisfies a square function estimate on R(V ): ∞ 1/2 2 dt tA S(t)u . u t 0

(b) The same result holds with ‘’ and ‘⊆’ replaced by ‘’ and ‘⊇’. (c) The following √ assertions are equivalent: √ (c1) Dp ( L) = Dp (DV ) with DV f p Lf p ; (c2) L admits a bounded H ∞ -functional √ √ calculus on Rp (DV ); (c3) D( A) = D(V ) with V h Ah; (c4) A admits a bounded H ∞ -functional calculus on R(V ). The plan of the proof is as follows. First we consider (a). The equivalence of (a3) and (a4) will be proved in Lemma 10.2, while the implications (a1) ⇒ (a3) and (a2) ⇒ (a4) follow by considering functions of the form f = φh and F = 1 ⊗ u respectively, and using the equivalence of Lp -norms on the first Wiener–Itô chaos. In Proposition 9.5 we have shown that L is injective on Rp (D), and then Proposition 7.3 asserts that (a4) implies (a2), so that it remains to show that (a2) implies (a1). Next we turn to part (b). The equivalence of (b3) and (b4) follows from Lemma 10.2, and the implications (b1) ⇒ (b3) and (b2) ⇒ (b4) follow as in part (a). Proposition 7.3 asserts that (b4) implies (b2), so that it suffices to show that (b4) implies (b1). Finally, part (c) follows by putting together the estimates obtained in (a) and (b) and appealing to Proposition 7.2.

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The next lemma is a variation of Theorem 10.1 in [5]. We are grateful to Alan Mc Intosh for showing us the argument below. Keeping in mind that B satisfies (A3) if and only if B ∗ satisfies (A3), we write A∗ := V V ∗ B ∗ and we denote the semigroup generated by −A∗ by S ∗ . Lemma 10.2. Assume (A2) and (A3). For h ∈ D(A) we have ∞ 1/2 √ 2 dt tA S(t)V h Ah. t

(10.1)

0

As a first consequence, the following assertions are equivalent: √ √ √ (1) D( A) ⊆ D(V ) with Ah V h, h ∈ D( A); (2) A satisfies a square function estimate on R(V ): 1/2 ∞ 2 dt tA S(t)u u; t 0

√ √ (3) D( A∗ ) ⊇ D(V ) with A∗ h V h, h ∈ D(V ); (4) A∗ satisfies a square function estimate on R(V ): 1/2 ∞ 2 dt tA∗ S ∗ (t)u u. t 0

As a second consequence, the following assertions are equivalent: √ √ (1 ) D( A) = D(V ) with equivalence of norms Ah V h; a bounded H ∞ (2 ) A admits √ √-functional calculus on R(V ); ∗ (3 ) D( A ) = D(V ) with A∗ h V h; (4 ) A∗ admits a bounded H ∞ -functional calculus on R(V ). Proof. To prove (10.1), let ω ∈ (ω(T ), 12 π), where T is defined by (9.4). By [5, Proposition 8.1] + we have for all ψ ∈ H0∞ (Σω ) and ψ˜ ∈ H0∞ (Σ2ω ), ∞ ∞ ψ(tT )u2 dt ψ˜ tT 2 u2 dt , t t 0

u ∈ R(T ).

0

Using this, the fact that A has a bounded H ∞ -calculus, and the fact that ϕ := sgn ·ψ ∈ H0∞ (Σω ), for h ∈ D(A) we obtain √ 2 dt √ ˜ Ah Ah2 ψ(tA) t ∞

0

2 ∞ dt 2 2 h ˜ tT ψ T = 0 t 0


2461

∞ h 2 dt ψ(tT ) T 2 0 t 0

2 ∞ h dt ϕ(tT )T = 0 t 0

2 ∞ 0 dt = ϕ(tT ) Vh t 0

∞ 2 0 2 dt ˜ ψ tT Vh t 0

∞ 2 dt ˜ . ψ(tA)V h t 0

˜ The equivalence of (1) and (2) follows immediately by taking ψ(z) = ze−z . Replacing B by B ∗ we obtain the equivalence of (3) and (4). Finally, the equivalence of (1) and (3) is a well-known consequence of the duality theory of forms [29,35] (see also [5, Theorem 10.1]). The equivalence of the primed statements follows in the same way (or can alternatively be deduced from the equivalence of the un-primed statements). 2 Proof of Theorem 10.1. Fix 1 < p < ∞ and let p1 + q1 = 1. Part (a): It remains to prove that (a2) implies (a1). Perhaps the shortest proof of this implication is based on a lower bound for the square function associated with the semigroup Q generated 10.2 to the Lp -setting. by − L. Alternatively, one could adapt the argument in Lemma √ −√z −z and ψ(z) = ze . These functions belong to Consider the functions ϕ(z) = ze H0∞ (Σθ+ ) for θ < 12 π . Substituting t = s 2 we obtain, from Proposition 7.2, ∞ ∞ 1/2 1/2 √ 2 dt 2 ds ψ(tL)F s LQ(s)F = 2 . t s p

0

p

0

Using (a2) and the first part of Proposition 7.2, the identity of Theorem 5.6 (which extends to √ the semigroup Q generated by − L), and Lemma 8.1 and Theorem 8.3, for all f ∈ Dp (L) we obtain ∞ 1/2 2 dt tL P (t)DV f DV f p t p

0

∞ 1/2 2 ds s LQ(s)DV f s

p

0

√ = G ( Lf )

p

2462


√ H ( Lf )p √ Lf p . √ Since Dp (L) is a core for both Dp ( L)√and Dp (DV ), the desired domain inclusion follows and the norm estimate holds for all f ∈ Dp ( L). Part (b): It remains to show that (b4) implies (b1). By Lemma 10.2 (applied with the roles of B and B ∗ reversed), (b4) implies the estimate ∞ 1/2 2 dt tAB ∗ S B ∗ (t)u u, t

u ∈ R(V ).

0

√ It follows from Part (a) (with B replaced by B ∗ ) that Dq ( L∗ ) ⊆ Dq (DV ) and, for f ∈ √ Dq ( L∗ ), √ DV f q L∗ f q , 1 p

1 < q < ∞.

√ We will use next a standard duality argument to prove the estimate Lgp DV gp where + q1 = 1. For g ∈ F Cb∞ (E; D(A)) we have √ √ Lgp = sup Lg, f˜ . f˜q 1

√ √ √ The sectoriality of L∗ allows us to use the decomposition f˜ = f˜0 + f˜1 ∈ N( L∗ ) ⊕ R( L∗ ) = √ Lq , and since F Cb∞ (E; D(A∗ )) is a core for Dq ( L∗ ), it suffices to consider f˜ of the form f˜ = √ √ √ f˜0 + L∗ f , with f ∈ F Cb∞ (E; D(A∗ )) and f˜q 1. Since f˜0 ∈ N( L∗ ) and L∗ f p P √ ∗ p f˜p , we obtain R( L )

√ Lgp = = =

√ √ Lg, f˜0 + L∗ f

sup

√ f˜0 + L∗ f q 1

sup

√ √ Lg, L∗ f

sup

Lg, f

√ L∗ f q 1 √ L∗ f q 1

sup

Lg, f

sup

BDV g, DV f

sup

B DV gp DV f q

DV f q 1 DV f q 1 DV f q 1

= B DV gp . Since F Cb∞ (E; D(A)) is a core for Dp (L), the result of Step 2 follows.


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Part (c): The equivalences follow immediately from (a) and (b) combined with Proposition 7.2. 2 We finish this section by pointing out two further equivalences to the ones of Theorem 2.1 and their one-sided extensions in Theorem 10.1. The conditions (1)–(4) of Theorem 2.1 are equivalent to (5) Dp ( L) = Dp (DV∗ B) with LF p DV∗ BF p for F ∈ Dp ( L); (6) D( A) = D(V ∗ B) with Aup V ∗ Bup for u ∈ D( A). Here, in the spirit of Theorem 2.1, we interpret A as an operator in R(V ). This is immaterial, however, in view of the definition A = V V ∗ B and the (not necessarily orthogonal) Hodge decomposition H = R(V ) ⊕ N(V ∗ B) (see (2.1)) by virtue of which (6) also holds on the full space H . To see that (1) implies (5), note that for f ∈ Dp (L) we have √ ∗ D B DV f = Lf p DV Lf p = LDV f p . V p Since DV (Dp (L)) is a core for both Dp (DV∗ B) and Dp ( L), (5) follows. The converse implication that (5) implies (1) is proved similarly. The equivalence (3)⇔(6) is proved in the same way. It is clear from the proofs that the one-sided versions of these implications hold as well. 11. Proof of Theorem 2.2 We continue with the proof of Theorem 2.2. It will be a standing assumption that the equivalent conditions of Theorem 2.1 are satisfied. As we have already observed (in Lemma 10.2, see also the discussion below Theorem 2.1), the corresponding equivalences obtained by replacing B with B ∗ then also hold. Below, for k = 1, 2 we will use the bounded analytic C0 -semigroups P (k) (t) := P (t) ⊗ S ⊗k (t), which are defined on the spaces p L(k) := Lp E, μ; H ⊗k . p

Note that L(1) = Lp and P (1) coincides with P on the closed subspace Rp (DV ). The generators of P (k) will be denoted by −L(k) . The semigroups generated by − I + L(k) will be denoted by Q(k) . We also consider the operator DV ⊗ I , initially defined on the algebraic tensor product p Dp (DV ) ⊗ H , which is viewed as a dense subspace of L(1) . Using that DV is a closed operator from Lp into Lp , it is straightforward to check that DV ⊗ I extends to a closed operator p

p

D V : Dp (D V ) ⊆ L(1) → L(2) . On the algebraic tensor product Lp ⊗ H , for t > 0 we define the operators

2464


D V P (1) (t) := DV P (t) ⊗ S(t), D V P ∗(1) (t) = DV P ∗ (t) ⊗ S ∗ (t). p

p

By Theorem 8.2 these operators extend uniquely to bounded operators from L(1) to L(2) . Proposition 11.1. Let 1 0} and { tD V P ∗(1) (t): t > 0} are R-bounded in p p L (L(1) , L(2) ). p (ii) The following square function estimates hold for F ∈ L(1) : ∞ 1/2 2 dt √ tD V P (1) (t)F F p , t p

0

∞ 1/2 2 dt √ ∗ tD V P (t)F F p . (1) t p

0

(iii) The domain inclusions Dp ( L(1) ) ⊆ Dp (D V ) and Dp ( L∗(1) ) ⊆ Dp (D V ) hold with norm estimates D V F p F p + L(1) F p and D V F p F p + L∗(1) F p . Proof. (i): The R-boundedness is a consequence from (an easy Hilbert space-valued extension of) Proposition 6.2 combined with (11.1) and Theorem 8.2. (ii): Since A has a bounded H ∞ -calculus on H of angle < 12 π , the same holds for A∗ . Propop sition 7.3 implies that L(1) and L∗(1) have bounded H ∞ -functional calculi on L(1) of angle < 12 π . The domain inclusions Dp (L(1) ) ⊆ Dp (D V ) and Dp (L∗(1) ) ⊆ Dp (D V ) follow from (i) by taking Laplace transforms. By combining(i) and Proposition 7.5 we obtain the desired result. (iii): Combining the fact that I + L(2) has a bounded H ∞ -calculus of angle < 12 π with Proposition 7.2, the commutation relation D V P (1) (t) = P (2) (t)D V , the H -valued analogue of Lemma 8.1, and the first estimate of (ii), for all F ∈ Dp (L(1) ) we obtain ∞ 1/2 2 dt t I + L(2) Q(2) (t)D V F D V F p t 0

p

∞ 1/2 2 dt tD V Q(1) (t) I + L(1) F = t 0

p


2465

∞ 1/2 2 dt √ −t tD V e P (1) (t) I + L(1) F t

p

0

∞ 1/2 2 dt √ tD V P (1) (t) I + L(1) F t

p

0

I + L(1) F p F p + L(1) F p . This gives the first estimate. Since Dp (L(1) ) is a core for Dp ( L(1) ), the domain inclusion follows as well. To prove the second estimate we put T := P ∗ ⊗ S ∗ ⊗ S ∗ , where S ∗ is the bounded analytic semigroup generated by −V V ∗ B ∗ ; this notation is as in Section 10. Note that the negative generator C of T has a bounded H ∞ -calculus of angle < 12 π ; this follows from the fact that if Theorem 2.1 holds for B, then it also holds for B ∗ (see Lemma 10.2) and therefore the negative generators of S ∗ and S ∗ both have bounded H ∞ -calculi of angle < 12 π . Let R be the semigroup √ generated by − I + C. Using the identity D V P ∗(1) (t)F = T (t)D V F, and arguing as above, for all F ∈ Dp (L∗(1) ) we obtain ∞ 1/2 2 dt √ t I + CR(t)D V F D V F p t 0

p

∞ 1/2 2 dt ∗ ∗ tD V Q(1) (t) I + L(1) F = t 0

p

∞ 1/2 √ dt 2 tD V e−t P ∗ (t) I + L∗ F (1) (1) t

p

0

∞ 1/2 √ dt 2 tD V P ∗ (t) I + L∗ F (1) (1) t 0

I + L∗(1) F p F p + L∗(1) F p .

p

The second domain inclusion now follows from the fact that Dp (L∗(1) ) is a core for Dp ( L∗(1) ). 2 In the following theorem we give a characterisation of Dp ( L(1) ). Since L = L(1) on Rp (DV ), this gives a further equivalence of norms for L on Rp (DV ), different from the one

2466


in Theorem 2.1. In the proof of Theorem 2.2 we use both equivalences to determine the domain of L. First we need a simple lemma. Lemma 11.2. Let 1 < p < ∞. The semigroup Q(1) restricts to C0 -semigroups on the space Dp (D V ) ∩ Dp ( I ⊗ A). Proof. It suffices to prove the result with Q(1) replaced by P (1) ; the latter is readily seen to restrict to a C0 -semigroup on Dp (D V ) ∩ Dp ( I ⊗ A) by the identities D V P(1) (t) = P (2) (t)D V and I ⊗ A P (1) (t) = P (1) (t) I ⊗ A. 2 Theorem 11.3. Let 1 < p < ∞. We have equality of domains

Dp ( L(1) ) = Dp (D V ) ∩ Dp ( I ⊗ A),

with equivalence of norms F p + L(1) F p F p + D V F p + I ⊗ AF p , Proof. By a result of Kalton and Weis [28, Theorem 6.3], applied to the sums L(1) = L ⊗ I + I ⊗ A and L∗(1) = L∗ ⊗ I + I ⊗ A∗ , we have the estimates (I ⊗ A)F F p + L(1) F p , p (I ⊗ A∗ )F F p + L∗ F , (1) p p

F ∈ Dp (L(1) ) F ∈ Dp L∗(1) .

Since the square root domains equal the complex interpolation spaces at exponent 12 for sectorial operators with bounded imaginary powers [23, Theorem 6.6.9], by interpolating the inclusions Dp (L(1) ) → Dp (I ⊗ A),

Dp (L∗(1) ) → Dp (I ⊗ A∗ ),

with the identity operator, we obtain the estimates I ⊗ AF p F p + L(1) F p , F ∈ Dp (L(1) ), I ⊗ A∗ F p F p + L∗(1) F p , F ∈ Dp L∗(1) .

(11.1)

Combining these estimates with Proposition 11.1 we obtain F p + D V F p + I ⊗ AF p F p + L(1) F p , F ∈ Dp (L(1) ), F p + D V F p + I ⊗ A∗ F p F p + L∗(1) F p , F ∈ Dp L∗(1) .

( p1

Next we prove the reverse estimates. For F ∈ Dp (L) ⊗ Dp (A) and G ∈ Dq (L∗ ) ⊗ Dp (A∗ ) + q1 = 1) we have F ∈ Dp (L(1) ), G ∈ Dq (L∗(1) ), and


2467

I + L(1) F, G = (I + L(1) )F, 1/ L∗(1) + I G = F, 1/ I + L∗(1) G + (L ⊗ I )F, 1/ I + L∗(1) G + (I ⊗ A)F, 1/ I + L∗(1) G = F, 1/ I + L∗(1) G + BD V F, D V / I + L∗(1) G + I ⊗ AF, I ⊗ A∗ / I + L∗(1) G .

Using the boundedness of the three operators 1/ I + L∗(1) , D V / I + L∗(1) (by Proposi tion 11.1(iii)), and I ⊗ A∗ / I + L∗(1) (by the second estimate in (11.1)), we find I + L(1) F p = sup I + L(1) F, G Gq 1

sup F p 1/ I + L∗(1) Gq Gq 1

+ B D V F p D V / I + L∗(1) Gq + I ⊗ AF p I ⊗ A∗ / I + L∗(1) Gq F p + D V F p + I ⊗ AF p . The estimate I + L∗ F F p + D V F p + I ⊗ A∗ F p (1) p is proved similarly and will not be needed. It remains to prove theequality of domains. Since Dp (L) ⊗ D(A) is a core for Dp (L(1) ), it is also a core for Dp ( L(1) ). Using this, the domain inclusion Dp ( L(1) ) ⊆ Dp (D V ) ∩ Dp ( I ⊗ A) follows, and the equivalence of norms extends to all F ∈ Dp ( L(1) ). Again by the equivalence of norms, Dp ( L(1) ) is closed in Dp (D V ) ∩ Dp ( I ⊗ A). It remains to provethat the inclusion is dense. This follows from Lemma 11.2, since for F ∈ Dp (D V ) ∩ Dp ( I ⊗ A) and t > 0 we have Q(1) (t)F ∈ Dp ( L(1) ) and Q(1) (t)F → F in the norm of Dp (D V ) ∩ Dp ( I ⊗ A) as t ↓ 0. 2 Recall that D denotes the Malliavin derivative. Since A is a closed operator, it follows from the results in [19] that the operator AD, initially defined on F Cb1 (E; D(A)), is closable as an operator from Lp into Lp (E, μ; H ) for 1 < p < ∞. We denote its closure by DA . We also consider the operator DV2 defined by

Dp DV2 := f ∈ Dp (DV ): DV f ∈ Dp (D V ) ,

DV2 := D V DV .

It is easy to check that this operator is closed from Dp (DV ) into Lp (E, μ; H ⊗2 ).

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Lemma 11.4. Let 1 < p < ∞. The semigroup P restricts to a C0 -semigroup on the space Dp (DV2 ) ∩ Dp (DA ). Proof. An easy argument based on Theorem 5.6 shows that P (t)Dp (DV2 ) ⊆ Dp (DV2 ) and DV2 P (t)f := P (2) (t)DV2 f,

f ∈ Dp DV2 .

Similarly, we have P (t)Dp (DA ) ⊆ Dp (DA ) and DA P (t)f = P (t) ⊗ S(t) DA f, These identities easily imply the result.

f ∈ Dp (DA ).

2

Proof ofTheorem 2.2. Using the fact that Dp (L) ⊆ Dp (DV ), Proposition 9.1, the domain equality Dp ( L) = Dp (DV∗ B) (see (5) at the end of Section 10), Theorem 11.3, the domain equality D( A) = D(V ∗ B) on R(V ) (see (6) at the end of Section 10), and the definition of DA , for f ∈ Dp (L) we obtain f p + Lf p f p + DV f p + Lf p = f p + DV f p + DV∗ B DV f p f p + DV f p + LDV f p f p + DV f p + DV2 f p + ADV f p f p + DV f p + D 2 f + (V ∗ B)DV f V

p

f p + DV f p + DV2 f p + DA f p .

p

This proves the equivalence of norms and the domain inclusion Dp (L) ⊆ Dp (DV2 ) ∩ Dp (DA ). To obtain equality of domains it remains to show that this inclusion is both closed and dense. Closedness follows easily from the norm estimate and density follows from Lemma 11.4 in the same way as in Theorem 11.3. 2 Note that Theorem 2.2 is natural in view of the expression Lf (x) = DV∗ BDV f (x) =−

n

[BV hj , V hk ]∂j ∂k ϕ(φh1 , . . . , φhn ) +

j,k=1

n

∂j ϕ(φh1 , . . . , φhn ) · φAhj ,

j =1

which holds for all f ∈ F Cb∞ (E; D(A)) of the form f = ϕ(φh1 , . . . , φhn ).


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12. Proofs of Theorems 2.5 and 2.6 The first part of Theorem 2.5 has already been proved in Proposition 9.5. We begin with some preparations for the proof of the second part. Let us denote by F P (E; D(V )) the vector space of all functions of the form p = ϕ(φh1 , . . . , φhn ) with hj ∈ D(V ) for j = 1, . . . , n and ϕ : Rn → R a polynomial in n variables. In the proof of the next proposition we need the following auxiliary result. Lemma 12.1. For 1 p < ∞, F P (E; D(V )) is a core for Dp (DV ). Proof. A simple approximation argument shows that F P (E; D(V )) ⊆ Dp (DV ). Thus it suffices to approximate elements of F Cb1 (E; D(V )) in the graph norm of Dp (DV ) with elements of F P (E; D(V )). Let f ∈ F Cb1 (E; D(V )) be of the form f = ϕ(φh1 , . . . , φhn ) with hj ∈ D(V ) for j = 1, . . . , n and ϕ ∈ Cb1 (Rn ). By a Gram-Schmidt argument we may assume that the elements h1 , . . . , hn are orthonormal in H . The image measure of μ under the transformation x → (φh1 (x), . . . .φhn (x)) is the standard Gaussian measure γn on Rn . This reduces the problem to finding polynomials pk in n variables such that pk → ϕ in Lp (Rn , γn ) and ∇pk → ∇ϕ in Lp (Rn , γn ; Rn ). It is a classical fact that such polynomials exist. 2 Lp

In the remainder of this section we interpret DV∗ B as a closed densely defined operator from to Lp .

Proof of Theorem 2.5, second part. We shall prove separately that

Rp (DV ) + Np DV∗ B = Lp ,

(12.1)

Rp (DV ) ∩ Np DV∗ B = {0}.

(12.2)

The proof of (12.1) is more or less standard. The idea behind the proof of (12.2) is to note that for p = 2 the Hodge decomposition is obtained as a special case of the Hodge decomposition theorem of Axelsson, Keith, and Mc Intosh [6], and to use this fact together with the fact that the Lp -norm and L2 -norm are equivalent on each summand in the Wiener–Itô decomposition. √ We begin with the proof of (12.1). By Theorem 2.1(1) the operator R := DV / L is well √ √ √ defined on Rp ( L) and bounded. In view of the decomposition Lp = Rp ( L) ⊕ Np ( L) we may extend R to Lp by putting R|Np (√L) := 0. A similar remark applies to the operator R∗ := √ DV / L∗ . p ∗ ∗ ∗ there exists f ∈ For ∗ F ∈ Rp (DV ), where √ F ∈ L we claim that RR√ √ R∗ := (R∗ ∗ ) . Indeed, Np ( L) and a sequence fn ∈ Dp ( L) such that f + Lfn → R∗ F in Lp . Therefore RR∗∗ F = limn→∞ DV fn ∈ Rp (DV ). √ √ Now, for functions ψ ∈ Dp ( L) and φ ∈ Dq ( L∗ ), √ √ DV ψ, B ∗ DV φ = Lψ, φ = Lψ, L∗ φ . Furthermore, approximating a function f ∈ Lp by a sequence (f0 + √ √ Np ( L) and fn ∈ Dp ( L) we obtain

√

Lfn )n1 with f0 ∈

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Rf, B ∗ DV φ = lim DV fn , B ∗ DV φ n→∞ √ √ = lim Lfn , L∗ φ n→∞ √ = f − f0 , L∗ φ √ = f, L∗ φ . Hence for the duality between Lp and Lq we obtain √ F − RR∗∗ BF, B ∗ DV φ = F, B ∗ DV φ − F, B ∗ R∗ L∗ φ = 0.

This shows that F − RR∗∗ BF ∈ Np (DV∗ B). This completes the proof of (12.1). We continue with the proof of (12.2). Assume that G ∈ Dp (DV∗ B) satisfies DV∗ BG = 0. Then for all f ∈ Dq (DV ) we have B ∗ DV f, G = 0, where the duality is between Lq and Lp . Let Ip,m and Iq,m denote the projections in Lp and Lq onto the m-th Wiener–Itô chaoses. The ranges of Ip,m and Iq,m are isomorphic by the equivalence of norms on the Wiener–Itô chaoses. ∗ =I p q Note that Ip,m q,m . Then Ip,m ⊗ I and Iq,m ⊗ I are bounded projections in L and L . Let jp,m denote the induced isomorphism of the range of Ip,m ⊗ I onto the range of I2,m ⊗ I . For cylindrical polynomials f ∈ F P (E; D(V )) ∩ H (m) (where H (m) is as in Section 3) we have the identity B ∗ DV f = (Iq,m−1 ⊗ I )B ∗ DV f and

jp,m−1 (Ip,m−1 ⊗ I )G, B ∗ DV f = (Ip,m−1 ⊗ I )G, B ∗ DV f = G, (Iq,m−1 ⊗ I )B ∗ DV f = G, B ∗ DV f = 0.

(12.3)

In the first term, the duality is the inner product of L2 . ∗ = jq,m−1 On the other hand, if f ∈ F P (E; D(V )) ∩ H (n) for some n = m, then jp,m−1 implies

jp,m−1 (Ip,m−1 ⊗ I )G, B ∗ DV f = (Ip,m−1 ⊗ I )G, B ∗ DV f = (Ip,m−1 ⊗ I )G, (Iq,n−1 ⊗ I )B ∗ DV f

= jp,n−1 (Ip,n−1 ⊗ I )(Ip,m−1 ⊗ I )G, B ∗ DV f = 0,

(12.4)

since DV f is in the (n − 1)-th chaos; in the last step we used the L2 -orthogonality of the chaoses. Since the cylindrical polynomials form a core for D(DV ) by Lemma 12.1 and B is bounded on H , we conclude from (12.3) and (12.4) that jp,m−1 (Im−1 ⊗ I )G annihilates R(B ∗ DV ) and therefore it belongs to N(DV∗ B). Next we claim that if G ∈ Rp (DV ), then jp,m−1 (Ip,m−1 ⊗ I )G ∈ R(DV ). Indeed, from G = limk→∞ DV gk in Lp it follows that jp,m−1 (Ip,m−1 ⊗ I )G = lim DV jp,m (Ip,m ⊗ I )gk ∈ R(DV ). k→∞


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Combining what we have proved, we see that if G ∈ Rp (DV ) ∩ Np (DV∗ B), then jp,m−1 (Ip,m−1 ⊗ I )G ∈ R(DV ) ∩ N(DV∗ B). Hence, jp,m−1 (Ip,m−1 ⊗ I )G = 0 by the Hodge decomposition of L2 [6]. It follows that (Ip,m−1 ⊗ I )G = 0 for all m 1, and therefore G = 0. This concludes the proof of (12.2). 2 The next application is included for reasons of completeness. Corollary 12.2. If the equivalent conditions of Theorem 2.1 hold, then

Rp DV∗ B = Rp DV∗ .

Note that by the second part of Theorem 2.5 it is immaterial whether we view DV∗ B as an unbounded operator from Lp to Lp or from Rp (DV∗ B) to Lp . Proof. By the first part of Theorem 2.5 (first applied to B and then to I ) we have the decompositions Lp = Np (DV ) ⊕ Rp DV∗ B = Np (DV ) ⊕ Rp DV∗ , where both DV∗ B and DV∗ are viewed as closed densely defined operators from Rp (DV ) to Lp . The corollary will follow if we check that Rp (DV∗ B) ⊆ Rp (DV∗ ). This inclusion is trivial if we may interpret DV∗ B and DV∗ as unbounded operators from Lp to Lp . By the preceding remark, we may indeed do so for DV∗ B. The proof will be finished by checking that the conditions of Theorem 2.1 also hold with B replaced by I , since then we may do the same for DV∗ . But this follows from the fact that V V ∗ , being self-adjoint on R(V ), admits a bounded H ∞ -calculus on R(V ). 2 We proceed with the proof of the second part of Theorem 2.6. The first part has been proved in Section 10. Proof of Theorem 2.6, second part. We use the notation X1 := Lp ⊕ Rp (DV )

and X2 := Np DV∗ B .

Fix t ∈ R \ {0}. First we show that it − Π is injective on Lp ⊕ Lp . Theorem 2.5 implies the decomposition Lp ⊕ Lp = X 1 ⊕ X 2 .

(12.5)

Take x = x (1) + x (2) ∈ X1 ⊕ X2 , and suppose that (it − Π)x = 0. Then (it − Π)x (1) = 0 and itx (2) = 0. Thus x (1) = x (2) = 0, since Π|X1 in X1 is bisectorial. Next we show that it − Π is surjective on Lp ⊕ Lp . Let y (1) ∈ X1 and y (2) ∈ X2 . The equation (it − Π)(x (1) + x (2) ) = y (1) + y (2) is solved by x (1) = (it − Π|X1 )−1 y (1) This implies that it − Π is surjective.

and x (2) = (it)−1 y (2) .

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Using (12.5) and the sectoriality of Π on X1 it follows that (1) x + x (2) (it − Π|X )−1 y (1) + |t|−1 y (2) 1 t −1 y (1) + y (2) t −1 y (1) + y (2) , which is the desired resolvent estimate that shows that Π is bisectorial on X1 ⊕ X2 . (1) (2) To show R-bisectoriality of Π on Lp ⊕ Lp we take yj = yj + yj ∈ X1 ⊕ X2 . Let (rj )j 1 be a Rademacher sequence. Using the R-bisectoriality of Π|X1 we obtain k k −1 −1 (1) E rj tj (itj − Π) yj E rj tj (itj − Π|X1 ) yj j =1

p

j =1

p

k −1 (2) + E rj tj (itj − Π|X2 ) yj j =1

p

N k −1 (2) (1) E rj yj + E rj tj tj yj j =1

p

k E rj yj . j =1

j =1

p

p

By an application of the Kahane–Khintchine inequalities we conclude that {t (it − Π)−1 : t ∈ R \ {0}} is R-bounded on Lp ⊕ Lp . This completes the proof. 2 We finish by showing how the first part of Theorem 2.6 can be used to prove the implication (2) ⇒ (1) of Theorem 2.1. We need the following lemma, which is an extension of the corresponding Hilbert space result, cf. [4, Section (H)]. Proposition 12.3. Let 1 < p < ∞ and suppose A is an R-bisectorial operator on a closed subspace U of Lp . Then A 2 is R-sectorial and for each ω ∈ (0, 12 π) the following assertions are equivalent: (1) A admits a bounded H ∞ (Σω )-functional calculus; + )-functional calculus. (2) A 2 admits a bounded H ∞ (Σ2ω Proof. We prove the implication (2) ⇒ (1), the other assertions being well known. Let ψ˜ ∈ + ˜ 2 ). ) and define ψ ∈ H0∞ (Σω ) by ψ(z) := ψ(z H0∞ (Σ2ω + Since A 2 has a bounded H ∞ (Σ2ω )-functional calculus, by Proposition 7.2 A 2 satisfies a square function estimate ∞ 1/2 2 2 dt ψ˜ tA f cf − PN(A 2 ) f p Cf p , t 0

p

where PN(A 2 ) denotes the projection on N(A ) = N(A 2 ) with range R(A ) = R(A 2 ).


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√ √ 2 ) = ψ( tA ), and therefore by the substitution ˜ But, ψ(tA t = s the above estimate is equivalent to ∞ 1/2 2 ds c C ψ(sA )f √ f − PN(A ) f p √ f p . s 2 2 p 0

By the bisectorial version of Proposition 7.2, this estimate implies that A has a bounded H ∞ (Σω )-functional calculus. 2 Alternative proof of Theorem 2.1 (2) ⇒ (1). We adapt an argument of [6], where more details can be found. √ Consider the function sgn ∈ H ∞ (Σω ) given by sgn(z) = 1Σω+ (z) − 1Σω− (z) = z/ z2 . Since Π has a bounded H ∞ -calculus on Lp ⊕ Rp (DV ) by Proposition 12.3,√the operator sgn(Π) is bounded. By the results already proved, this implies that Dp (Π) = Dp ( Π 2 ) with Πxp Π 2 x p ,

x ∈ Dp (Π) = Dp

Π2 .

Clearly, on Lp ⊕ Rp (DV ) we have

Π2

√ L = 0

0 , L

and by restricting to elements of the form x = (f, 0) with f ∈ Dp (DV ) we obtain the desired result. 2 Acknowledgments Part of this work was done while the authors visited the University of New South Wales (JM) and the Australian National University (JvN). They thank Ben Goldys at UNSW and Alan Mc Intosh at ANU for their kind hospitality. The inspiration for this paper came from many discussions with Alan on his recent work on functional calculi for Hodge–Dirac operators. References [1] D. Albrecht, X.T. Duong, A. Mc Intosh, Operator theory and harmonic analysis, in: Instructional Workshop on Analysis and Geometry, Part III, Canberra, 1995, in: Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 34, Austral. Nat. Univ., Canberra, 1996, pp. 77–136. [2] W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, in: Evolutionary Equations, in: Handb. Differ. Equ., vol. I, North-Holland, Amsterdam, 2004, pp. 1–85. [3] P. Auscher, On necessary and sufficient conditions for Lp -estimates of Riesz transforms associated to elliptic operators on Rn and related estimates, Mem. Amer. Math. Soc. 186 (871) (2007). [4] P. Auscher, X.T. Duong, A. Mc Intosh, Boundedness of Banach space-valued singular integral operators and Hardy spaces, preprint, 2004. [5] P. Auscher, A. Mc Intosh, A. Nahmod, Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (2) (1997) 375–403. [6] A. Axelsson, S. Keith, A. Mc Intosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (3) (2006) 455–497. [7] V.I. Bogachev, Gaussian Measures, Math. Surveys Monogr., vol. 62, Amer. Math. Soc., Providence, RI, 1998.

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Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls M.I. Ostrovskii a,∗ , V.S. Shulman b , L. Turowska c a Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway,

Queens, NY 11439, USA b Department of Mathematics, Vologda State Technical University, 15 Lenina street, Vologda 160000, Russia c Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg,

SE-41296, Gothenburg, Sweden Received 11 November 2008; accepted 22 June 2009 Available online 8 July 2009 Communicated by D. Voiculescu

Abstract We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball. © 2009 Elsevier Inc. All rights reserved. Keywords: Hilbert space; Bounded representation; Unitary representation; Hyperbolic space; Fixed point; Normal structure; Biholomorphic transformation; Indefinite quadratic form

* Corresponding author. Fax: +1 718 990 1650.

E-mail addresses: [email protected] (M.I. Ostrovskii), [email protected] (V.S. Shulman), [email protected] (L. Turowska). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.021

M.I. Ostrovskii et al. / Journal of Functional Analysis 257 (2009) 2476–2496

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1. Introduction Let K, H be Hilbert spaces; by L(K, H ) we denote the Banach space of all linear bounded operators from K to H . We will denote the open unit ball of L(K, H ) by B and call it operator ball. We say that a subset M of B is separated from the boundary if it is contained in a ball rB, for some r ∈ [0, 1). A group G of transformations of B will be called elliptic if all its orbits are separated from the boundary (this terminology goes back to [9]). We call G equicontinuous if, for each ε > 0 there is δ > 0 such that if A, B ∈ B and A − B < δ, then g(A) − g(B) < ε for all g ∈ G. This condition can be also called global equicontinuity because it is possible also to consider equicontinuity in a point. Since B is a bounded open set of a Banach space, one may consider holomorphic maps from B to Banach spaces. We will deal with invertible holomorphic maps from B onto B; such maps are called biholomorphic automorphisms of B. Our aim is to prove that if one of the spaces K, H is finite-dimensional and the other is separable, then any elliptic group of biholomorphic automorphisms of B has a common fixed point. More precisely we will prove the following result. Theorem 1.1. Let dim K < ∞ and H be separable. For a group G of biholomorphic automorphisms of B, the following statements are equivalent: (i) (ii) (iii) (iv)

G is elliptic on B; at least one orbit of G is separated from the boundary; G is equicontinuous; G has a common fixed point in B.

Remark 1.2. The assumption dim K < ∞ is essential, some of the results of this paper are known to fail without it, see, for example, the last paragraph of Section 8. As for separability of H , it is just a technical convenience, our approach works for non-separable H also, with a bit more complicated proofs. The result will be applied to the orthogonalization (or similarity) problem for bounded group representations on Hilbert spaces. This problem can be formulated as follows. Let π be a representation of a group G on a Hilbert space H. Under which conditions there is an invertible operator V such that the representation σ of G, defined by the formula σ (g) = V π(g)V −1 , is unitary? Clearly a necessary condition is the boundedness of π : supg∈G π(g) < ∞. In general it is not sufficient. Some sufficient conditions (on G or π ) are known, see the book [12]. We will show that a bounded representation π of a group G on a Hilbert space H is similar to a unitary representation if it preserves a quadratic form η with finite number of negative squares. The last condition means that η(x) = P x2 − Qx2 and P , Q are orthogonal projections in H with P + Q = 1 and dim(QH) < ∞. As a consequence we obtain that each bounded group of J -unitary operators on a Pontryagin space Πk has an invariant dual pair of subspaces. In other words the space can be decomposed into J -orthogonal direct sum H+ + H− of positive and negative subspaces which are invariant for all operators in the group.

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The proof of Theorem 1.1 is based on the analysis of the structure of the operator ball as a metric space with respect to the Carathéodory distance (see Chapters 4 and 5 of [6]). It was proved by Shafrir [15] that B is a hyperbolic space with respect to this distance. Since [15] is not easily accessible, we present a proof of this result in Appendix A, with the kind permission of the author. We will show that B has a restricted form of a normal structure if dim(K) < ∞. In the case where K is one-dimensional Theorem 1.1 was obtained in [17]; a transparent proof can be found in [10, Section 23]. 2. Hyperbolic spaces In our definition of hyperbolic spaces we follow fixed point theory literature (see e.g. [13,14]). In geometric literature (see e.g. [3]) hyperbolic spaces are defined differently. By a line in a metric space (X , ρ) we mean a subset of X which is isometric to the real line R with its usual metric (in the literature lines are also called metric lines or geodesic lines). Let (X , ρ) be a metric space with a distinguished set M of lines. We say that X is a hyperbolic space if the following conditions are satisfied: (1) (Uniqueness of a distinguished line through a given pair of points) For each x, y ∈ X , there is exactly one line ∈ M containing both x and y. (2) (Convexity of the metric) To state the condition (see (2.3)) we need to introduce some more definitions and notation. The segment [x, y] is defined as the part of the line ∈ M containing both x and y, which consists of all z ∈ satisfying ρ(x, y) = ρ(x, z) + ρ(z, y).

(2.1)

z = (1 − t)x ⊕ ty

(2.2)

We write

if z ∈ [x, y], ρ(z, x) = tρ(x, y), and ρ(z, y) = (1 − t)ρ(x, y) (where t ∈ [0, 1]). The convexity condition is: 1 1 1 1 1 ρ x ⊕ y, x ⊕ z ρ(y, z). 2 2 2 2 2

(2.3)

Hyperbolic spaces satisfy also the following stronger form of the condition (2.3): ρ (1 − t)x ⊕ ty, (1 − t)w ⊕ tz (1 − t)ρ(x, w) + tρ(y, z).

(2.4)

(To get (2.4) from (2.3) we observe that, if for some value of t we have the inequalities ρ((1 − t)x ⊕ ty, (1 − t)x ⊕ tz) tρ(y, z) and ρ((1 − t)x ⊕ tz, (1 − t)w ⊕ tz) (1 − t)ρ(x, w), then, by the triangle inequality, we have (2.4) for that value of t. Using this observation repeatedly we prove the inequalities for t of the form 2kn (k ∈ N, 1 k 2n ). Then we use continuity.) A subset C ⊂ X is called convex if x, y ∈ C implies [x, y] ⊂ C. Sometimes we say ρ-convex instead of convex, to avoid confusion with other natural notions of convexity for the same set.


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We use the notation Ea,r for {x ∈ X : ρ(a, x) r} and call such sets closed balls. The condition (2.4) implies that in a hyperbolic space all closed balls are convex. 3. Normal structure Let M be a subset in a metric space (X , ρ). The diameter of M is defined by diam M = sup ρ(x, y): x, y ∈ M .

(3.1)

A point a ∈ M is called diametral if sup ρ(a, x): x ∈ M = diam M. A hyperbolic space X is said to have normal structure if every convex bounded subset of X with more than one element has a non-diametral point. This notion goes back to Brodskii and Milman [4] who proved that uniformly convex Banach spaces (they are hyperbolic spaces) have normal structure. Takahashi [18] introduced and studied normal structure in somewhat more general context. See [2, Chapter 3] for a nice account on those aspects of fixed point theory which are related to the geometry of Banach spaces. Lemma 3.1. Let M be a separable bounded convex subset of a hyperbolic space X and α be the diameter of M. If all points of M are diametral, then M contains a sequence {an } with the property: limn→∞ ρ(an , x) = α for each x ∈ M. Proof. Let {cn } be a dense sequence in M. We define a sequence {bn } of “centers of mass” by n 1 the following rule: b1 = c1 , bn+1 = n+1 bn ⊕ n+1 cn+1 . By convexity of ρ we have 1 ρ(x, ck ) n n

ρ(x, bn )

(3.2)

k=1

for all n ∈ N. Indeed for n = 1 this is obvious. If it is true for some n, then ρ(x, bn+1 ) n 1 n 1 n 1 1 n+1 ρ(x, c ) + ρ(x, b ) ρ(x, c ) + ρ(x, c ) = n+1 n n+1 k k=1 k=1 ρ(x, ck ). n+1 n+1 n+1 n+1 n n+1 By convexity of M we have bn ∈ M for each n ∈ N. Our assumption implies that bn is diametral, hence there is a point an ∈ M with ρ(bn , an ) (1 − n12 )α. It follows that (1 − n12 )α 1 n 1 1 n 1 k=1 ρ(an , ck ). If ρ(an , cj ) < (1 − n )α, for some j n, then n k=1 ρ(an , ck ) < n (1 − n 1 n−1 1 1 n )α + n α = (1 − n2 )α, a contradiction. Hence ρ(an , cj ) (1 − n )α for j n. This shows that limn→∞ ρ(an , cj ) = α for each fixed j . Since the sequence {cj } is dense in M, the lemma is proved. 2 4. The invariant distance in the operator ball Recall that K, H denote Hilbert spaces and B is the open unit ball of L(K, H ). For A, X ∈ B set MA (X) = (1 − AA∗ )−1/2 (A + X)(1 + A∗ X)−1 (1 − A∗ A)1/2 .

(4.1)

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Clearly all MA are holomorphic on B. They are called the Möbius transformations. It can be proved that MA−1 = M−A (see [8, Theorem 2]). Hence each Möbius transformation is a biholomorphic automorphism of B. Since MA (0) = A the group of all biholomorphic automorphisms is transitive on B. We set

(4.2) ρ(A, B) = tanh−1 M−A (B) . It is easy to see that ρ coincides with the Carathéodory distance cB in B. Indeed, by [6, Theorem 4.1.8], cB (0, B) = tanh−1 (B) (this holds for the unit ball of every Banach space). Since cB is invariant and M−A sends A to 0 we get:

(4.3) cB (A, B) = tanh−1 M−A (B) = ρ(A, B). Hence ρ is invariant with respect to biholomorphic automorphisms. I. Shafrir [15] proved that the space (B, ρ) is hyperbolic. We present a proof of this result in Appendix A. A set in B is called bounded if it is contained in some ρ-ball, or equivalently in a multiple rB of the operator ball with r < 1. So a set is bounded if and only if it is separated from the boundary of B in the sense of Section 1. The following lemma is a special case of a more general result proved in [6, Theorem IV.2.2]. Lemma 4.1. On any bounded set the hyperbolic metrics is equivalent to the operator norm. 5. WOT-topology As before, let B be the unit ball of the space of operators from K to H . We suppose that K is finite-dimensional, dim K = n, and that H is separable. We consider biholomorphic maps on B. By WOT we denote the weak operator topology (see [5, p. 476]). Because of the separability, the restriction of this topology to B is metrizable, so in our arguments we may consider only sequences, not nets. Lemma 5.1. If K is finite-dimensional and H is separable, then all biholomorphic maps of B are WOT-continuous. Proof. Let us firstly show that all Möbius transforms MB are WOT-continuous (this was noticed and used already in the paper of Krein [11]). Indeed let B ∈ B be fixed, then the map ϕ : X → 1 + B ∗ X from (B, WOT) to (L(K, K), WOT) is continuous. Moreover, since K is finite-dimensional, ϕ remains continuous if instead of WOT we endow L(K, K) with its norm topology. The map T → T −1 is norm continuous on the group of invertible operators on K. Hence the map ψ : X → (1 + B ∗ X)−1 is continuous from (B, WOT) to L(K, K) with its norm topology. It follows that the map ω : X → (X + B)(1 + B ∗ X)−1 is continuous from (B, WOT) to (B, WOT). Indeed, if Xn → X, then ω(Xn ) − ω(X) = (Xn + B)(ψ(Xn ) − ψ(X)) + (Xn − X)ψ(X), where ψ was defined above. The first summand tends to zero in norm while the second one tends to zero in WOT. By a result of Harris [7], if a biholomorphic map of B preserves the point 0, then it coincides with the restriction to B of an isometric linear map h : L(K, H ) → L(K, H ). Since K is finitedimensional, the WOT-topology on L(K, H ) coincides with the weak topology (indeed L(K, H )


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is linearly homeomorphic to the direct sum of n copies of H ); since any bounded linear map is weakly continuous, h is WOT-continuous. On the other hand, if ϕ is a biholomorphic map of B and A = ϕ(0), then ψ = M−A ◦ ϕ is a biholomorphic map preserving 0. Hence ψ is an isometric −1 ◦ ψ = MA ◦ ψ is a composition of two WOT-continuous maps. Thus linear map and ϕ = M−A ϕ is WOT-continuous. 2 Corollary 5.2. If dim K < ∞ and H is separable, then each ball EA,r is WOT-compact. Proof. Since there is a Möbius transform that maps EA,r onto E0,r , and since all Möbius transforms are WOT-continuous, it suffices to consider the case A = 0. But E0,r is a usual closed operator ball; its WOT-compactness follows from the Banach–Alaoglu theorem. 2 6. Restricted normal structure of B The purpose of this section is to show that in the case when dim K < ∞ and H is separable, the (open) operator ball B with the metric (4.2) has a restricted form of normal structure in the sense that WOT-compact ρ-convex subsets in it have non-diametral points. As we already mentioned B with the metric (4.2) is a hyperbolic space (see Appendix A). Our assumptions on K and H imply that B is separable in the norm-topology and hence, by Lemma 4.1, with respect to ρ. Theorem 6.1. Let K be finite-dimensional and H be separable. Let M be a weakly compact, ρ-convex subset of B endowed with its hyperbolic metric. If M is not a singleton, then M contains a non-diametral point. Proof. Let α = diam M > 0. Assume the contrary, that is, all points in M are diametral. By Lemma 3.1, there is a sequence {An } in M such that limn→∞ ρ(An , X) = α for each X ∈ M. Since M is weakly compact, the sequence {An }∞ n=1 contains a weakly convergent subsequence. Let W be its limit, we have W ∈ M (since M is weakly compact). Throughout this proof we will not change our notation after passing to a subsequence. Since W ∈ M we get lim ρ(W, An ) = α.

n→∞

(6.1)

We will get a contradiction by proving sup ρ(An , Am ) > α.

(6.2)

n,m

We may assume without loss of generality that W = 0 (since a Möbius transformation which maps W to 0 is a ρ-isometry and weak homeomorphism). Let β = tanh α. Then (6.1) leads to limn→∞ An = β and it suffices to show that

sup MAm (−An ) > β. n,m

Since K is finite-dimensional and An ∈ L(K, H ), we can select a strongly convergent subsequence in the sequence {A∗n An }. Assume that A∗n An → P , where P ∈ L(K, K). It is clear that P 0 and P = β 2 .

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Choose ε > 0 and fix a number m with A∗m Am − P < ε. For brevity, denote A∗m Am by Q. We prove that limn→∞ MAm (−An ) > β if ε > 0 is small enough. By the definition, −1/2 −1 1/2 1 − A∗m Am (Am − An ) 1 − A∗m An . MAm (−An ) = 1 − Am A∗m

(6.3)

Since A∗m is of finite rank, A∗m An → 0 in the norm topology. Hence limn→∞ MAm (−An ) = limn→∞ Tn where −1/2 1/2 Tn = 1 − Am A∗m (Am − An ) 1 − A∗m Am −1/2 1/2 = Am − 1 − Am A∗m An 1 − A∗m Am . It follows from the identity (1 − t)−1/2 − 1 =

t (1 − t)(1 + (1 − t)−1/2 )

that the operator (1 − Am A∗m )−1/2 is a finite rank perturbation of the identity operator. Since An → 0 in WOT, we obtain that Tn − Sn → 0, where Sn = Am − An (1 − A∗m Am )1/2 . Denote An (1 − A∗m Am )1/2 by Bn . Since Bn → 0 in WOT, the sequence (Am − Bn )∗ (Am − Bn ) − A∗m Am − Bn∗ Bn = −A∗m Bn − Bn∗ Am tends to zero in norm topology. Furthermore, Bn∗ Bn = (1 − Q)1/2 A∗n An (1 − Q)1/2 tends in norm topology to (1 − Q)1/2 P (1 − Q)1/2 . Therefore (Am − Bn )∗ (Am − Bn ) → Q + (1 − Q)1/2 P (1 − Q)1/2 . Since P − Q < ε, we have that

Q + (1 − Q)1/2 P (1 − Q)1/2 − Q + (1 − Q)Q < ε. The inequalities β 2 − ε Q β 2 imply

Q + (1 − Q)Q 2β 2 − β 4 − 2ε, whence

lim Sn∗ Sn = lim (Am − Bn )∗ (Am − Bn ) 2β 2 − β 4 − 3ε > β 2 ,

n→∞

if ε is sufficiently small.

n→∞

2


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7. Fixed points The main purpose of this section is to establish the existence of a common fixed point for an elliptic group G of biholomorphic transformations of the operator ball B. As it is shown in Appendix A, a biholomorphic transformation of B is a bijective isometric transformation of the metric space (B, ρ) which maps the set M onto itself (and hence segments onto segments). Lemma 7.1. If G is an elliptic group of biholomorphic transformations of B, then there is a non-empty WOT-compact ρ-convex G-invariant subset of B. Proof. Let A ∈ B be such that the orbit G(A) := {g(A): g ∈ G} is bounded. Therefore G(A) is contained in some closed ball Ea,r . Let M be the intersection of all closed balls containing G(A). It is clear that this intersection is non-empty (it contains G(A)), WOT-compact and ρ-convex (as an intersection of WOT-compact ρ-convex sets). It remains to check that it is G-invariant. To see this it suffices to observe that each element g ∈ G maps the set of balls containing G(A) bijectively onto itself. 2 Lemma 7.2. Let G be an elliptic group of biholomorphic transformations of B. Let M be a minimal WOT-compact ρ-convex G-invariant subset in (B, ρ). Then M is a singleton. Proof. We use the approach suggested in [4]. Assume the contrary, let diam M = α > 0. By Theorem 6.1 M contains a non-diametral point N , so that M ⊂ {A: ρ(A, N ) δ} for some δ < α. Consider the set O= EB,δ . B∈M

The set O is non-empty because N ∈ O. The set O is weakly compact and ρ-convex since each of the balls EB,δ is weakly compact and ρ-convex. The set O is a proper subset of M since M has diameter α > δ. Since G is a group of isometric transformations and M is invariant under each element of G, the action of G on M is by isometric bijections. Therefore O is G-invariant. We get a contradiction with the minimality of M. 2 Proof of Theorem 1.1. The implication (i) ⇒ (ii) is obvious. On the other hand if G(X0 ) is separated from the boundary, for some X0 ∈ B, then supg∈G ρ(0, g(X0 )) < ∞ whence, for each X ∈ B, supg∈G ρ(0, g(X)) supg∈G (ρ(0, g(X0 )) + ρ(g(X0 ), g(X))) = supg∈G (ρ(0, g(X0 )) + ρ(X0 , X)) < ∞. This means that the orbit G(X) is separated from the boundary. We proved that (i) ⇔ (ii). The implication (i) ⇒ (iv) can be derived from Lemmas 7.1 and 7.2 as follows. It is clear that families of WOT-compact ρ-convex G-invariant sets with the finite intersection property have non-empty intersections which are also WOT-compact ρ-convex and G-invariant. Therefore, by the Zorn Lemma, there is a minimal non-empty WOT-compact ρ-convex G-invariant set M0 . By Lemma 7.2, M0 is a singleton and (iv) is proved. If (iv) is true and A is a fixed point of G, then G1 = M−A GMA is a group of biholomorphic maps of B preserving 0. Hence it consists of restrictions to B of isometric linear maps (see the beginning of Section 4 in this connection). Thus G1 is equicontinuous.

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Note that each Möbius transform is a Lipschitz map: MA (X) − MA (Y ) CX − Y for each X, Y ∈ B, where the constant C > 0 depends on A. Indeed setting F (X) = (A + X) × (1 + A∗ X)−1 and D = (1 − A)−1 we have

F (X) − F (Y ) = (A + X) (1 + A∗ X)−1 − (1 + A∗ Y )−1 + (X − Y )(1 + A∗ Y )−1

= (A + X)(1 + A∗ X)−1 A∗ (Y − X)(1 + A∗ Y )−1 + (X − Y )(1 + A∗ Y )−1

2D 2 X − Y + DX − Y 3D 2 X − Y . Hence

MA (X) − MA (Y ) = (1 − AA∗ )−1/2 F (X) − F (Y ) (1 − A∗ A)1/2

1

5 D 2 F (X) − F (Y ) 3D 2 X − Y . Since G = MA G1 M−A and the maps MA , M−A are Lipschitz, G is also equicontinuous. We proved that (iv) ⇒ (iii). Let now (iii) hold, we have to prove (ii). We will show that the orbit of 0 is separated from the boundary. Assuming the contrary we get that for any δ > 0 there is g ∈ G with g(0) > 1 − δ. Let A = g(0); we may assume that δ < 1/2 so A > 1/2. By the already mentioned result of [7], g = MA ◦ h where h is a linear isometry. Let P be the spectral projection of T = A∗ A corresponding to the eigenvalue T = A2 (recall that T is an operator in a finite-dimensional space). Then

(1 − T )P = 1 − T 2 1 − A < 2δ. Set X1 = 0, X2 = h−1 ( 12 AP ). Then X2 − X1 = 12 AP = A/2 > 1/4. On the other hand

g(X2 ) − g(X1 ) = MA 1 AP − MA (0)

2

−1

1 ∗ ∗ −1/2 1 ∗ 1/2

= (1 − AA ) AP + A 1 + A AP (1 − A A) − A

2 2

−1

1 −1/2 1 1/2

A(1 − T ) =

P + 1 1 + T P (1 − T ) − A

2 2

−1 −1

1 1 1 1

= A P + 1 1 + T P − A = A(1 − T )P 1 + T P

2 2 2 2

1

1 A (1 − T )P < 2δ = δ. 2 2 This contradicts to the assumption of equicontinuity. Indeed for each δ we get points Yi = g(Xi ) with Y1 − Y2 < δ and g −1 (Y1 ) − g −1 (Y2 ) > 1/4. Thus (ii) holds. 2


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8. Orthogonalization Theorem 8.1. If a bounded representation π of a group G on a Hilbert space H preserves a quadratic form η with finite number of negative squares, then it is similar to a unitary representation. Proof. By our assumptions, H = H1 ⊕ H2 , dim(H2 ) < ∞, and η(x) = P x2 − Qx2 where P , Q are the projections onto H1 and H2 respectively. We write H1 = H and H2 = K, for brevity. We will relate to each invertible operator T on H preserving the form η a biholomorphic map wT of B in such a way that wT1 T2 = wT1 ◦ wT2 .

(8.1)

Let us call a subspace L of H positive (negative) if η(y) > 0 (respectively η(y) < 0) for all non-zero y ∈ L. Since each negative subspace L is finite-dimensional, there is ε > 0 such that η(y) −εy2

for all non-zero y ∈ L.

The supremum of all such ε is called the degree of negativeness of L and is denoted by ε(L). For each operator A ∈ B, the set L(A) = {Ax ⊕ x: x ∈ K} is a negative subspace of H. Furthermore the condition η(y) −εy2

for all y ∈ L(A)

means that −x2 + Ax2 −ε x2 + Ax2 for all x ∈ K. That is ε

1 − A2 . 1 + A2

It follows that the degree of negativeness of L(A) is related to A by the equality 1 − A2 ε L(A) = . 1 + A2

(8.2)

Since dim(L(A)) = dim(K), L(A) is a maximal negative subspace in H. Indeed if some subspace M of H strictly contains L(A), then its dimension is greater than codimension of H , whence M ∩ H = {0}. But all non-zero vectors in H are positive. Conversely, each maximal negative subspace Q of H coincides with L(A), for some A ∈ B. Indeed, since Q ∩ H = {0}, there is an operator A : K → H such that each vector of Q is of the form Ax ⊕ x. Since Q is negative, we have η(Ax ⊕ x) = Ax2 − x2 < 0, and therefore A < 1, so A ∈ B. Thus Q ⊂ L(A); and, by maximality, Q = L(A).

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It is easy to see that the map A → L(A) from B to the set E of all maximal negative subspaces is injective and therefore bijective. Now we can define wT . Indeed, if a subspace L of H is maximal negative, then its image T L under T is also maximal negative (because T is invertible and preserves η). Hence, for each A ∈ B, there is B ∈ B such that L(B) = T L(A). We let wT (A) = B. The equality (8.1) follows easily because L(wT1 (wT2 (A))) = T1 L(wT2 (A)) = T1 T2 L(A) = L(wT1 T2 (A)) and the map A → L(A) is injective. Our next goal is to check that wT is biholomorphic. Since wT−1 = wT −1 it suffices to show that wT is holomorphic. Let T = (Tij )2i,j =1 be the matrix of T with respect to the decomposition H = H1 ⊕ H2 . Then T (Ax ⊕ x) = (T11 Ax + T12 x) ⊕ (T21 Ax + T22 x). Since T (Ax ⊕ x) ∈ L(wT (A)), we conclude that wT (A)(T21 Ax + T22 x) = T11 Ax + T12 x. Thus wT (A) = (T11 A + T12 )(T21 A + T22 )−1 .

(8.3)

This shows that wT is a holomorphic map on B. Suppose now that π is a bounded representation of a group G on H preserving η. Then W = {wπ(g) : g ∈ G} is a group of biholomorphic maps of B. Moreover since π is bounded, the group W is elliptic. To see this, note that for each negative subspace L, one has η(y) −ε(L)y2

for all y ∈ L.

If T is an invertible operator preserving η, then T −1 x ∈ L, for each x ∈ T L, whence

2 η(x) = η T −1 x −ε(L) T −1 x −ε(L)T −2 x2 . Thus ε(T L) ε(L)T −2 . For L = L(A), T L = L(wT (A)). This gives 1 − A2 1 − wT (A)2 T −2 2 1 + wT (A) 1 + A2 if one takes into account (8.2). Thus, if π(g) C for all g ∈ G, then 2 1 − wπ(g) (A)2 −2 1 − A C . 1 + wπ(g) (A)2 1 + A2

Therefore

2 1 − A2 1 − wπ(g) (A) C −2 1 + A2


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and

sup wπ(g) (A) < 1 g∈G

for each A ∈ B. By Theorem 1.1, there is D ∈ B with wπ(g) (D) = D for all g ∈ G. Hence π(g)L(D) = L(D) for all g ∈ G. Let U be an operator on H with the matrix (Uij ) where U11 = (1H − DD ∗ )−1/2 , U12 = −D(1K − D ∗ D)−1/2 , U21 = −D ∗ (1H − DD ∗ )−1/2 , U22 = (1K − D ∗ D)−1/2 . It can be checked that U preserves η and maps L(D) onto K. Then all operators τ (g) = U π(g)U −1 preserve η, and the subspace K is invariant for them. It follows that H is also invariant for operators τ (g). Hence these operators preserve the scalar product on H. Thus g → τ (g) is a unitary representation similar to π . 2 It should be noted that Theorem 8.1 does not extend to the case when η has infinite number of negative (and positive) squares, that is, to the case that both H1 and H2 are infinitedimensional [16]. 9. J -unitary operators on Pontryagin spaces The Pontryagin space is a linear space E supplied with an indefinite scalar product x, y → [x, y] which has a finite number of negative squares. More precisely this means that one can choose a usual scalar product x, y → (x, y) with respect to which E is a Hilbert space and [x, y] = (J x, y), where J is a selfadjoint involutive operator on this Hilbert space with rank(1 − J ) < ∞. An invertible operator T on E is called J -unitary if [T x, T y] = [x, y] for all x, y ∈ E. It should be noted that the terminology does not seem to be successful because the choice of the operator J and the corresponding scalar product is not unique while the set of J -unitary operator is completely determined by the original indefinite scalar product [·,·]. However, this terminology is widely used (see, for example, [1,10] and references therein). It is important that all scalar products defining [·,·] via J -operators are equivalent, so one can speak, for example, about boundedness of a set of operators, without indicating which scalar product is chosen. A subspace X ⊂ E is called positive (negative) if [x, x] > 0 (respectively [x, x] < 0) for all x ∈ X. A dual pair of subspaces in E is a pair Y , Z, where Y is a positive subspace, Z is a negative subspace and Y + Z = E. The study of dual pairs invariant for a given set of J -unitary operators was started by Sobolev and intensively developed by Pontryagin, Krein, Phillips, Naimark and other prominent mathematicians. The previous theorem on the orthogonalization of representations implies the following result. Corollary 9.1. A group of J -unitary operators on a Pontryagin space has an invariant dual pair if and only if it is bounded. Proof. Choose a scalar product (·,·) and the corresponding operator J . Denote by H the Hilbert space (E, (·,·)). Since J is an Hermitian involutive operator, there are orthogonal subspaces H , K of H such that J = PH − PK . By our assumption on J , the subspace K is finite-dimensional. Let G be a group of J -unitary operators. If it is bounded, then the identity map can be regarded as a bounded representation of G on H. Moreover it preserves the form η(x) = [x, x]. Since it

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has a finite number of negative squares, Theorem 8.1 implies that there is an invertible operator V such that the representation τ (g) = T −1 gT is unitary. It follows from [10, Theorem 5.8] that G has an invariant dual pair of subspaces. For completeness we include the proof of this fact. Passing to adjoints in the equality T τ (g) = gT and taking into account that g ∗ = J g −1 J , τ (g)∗ = τ (g −1 ) we obtain that τ (g −1 )T ∗ = T ∗ J g −1 J . Using this identity for g instead of g −1 and multiplying both sides by J T we get: τ (g)T ∗ J T = T ∗ J gJ J T = T ∗ J gT = T ∗ J T τ (g). Thus the invertible selfadjoint operator R = T ∗ J T commutes with the group τ (G) of unitary operators. It follows that its spectral subspaces H1 and K1 corresponding to positive and negative parts of spectrum are invariant for τ (G). Note that (Rx, x) > 0 for x ∈ T −1 H \{0} and (Rx, x) < 0 for x ∈ T −1 K\{0}. It follows that dim K1 = dim K. Now the subspaces H2 = T H1 and K2 = T K1 form an invariant dual pair for G. The converse implication is simple. If G has an invariant dual pair H, K, then the scalar product (h1 + k1 , h2 + k2 ) = [h1 , h2 ] − [k1 , k2 ] is invariant for G. Thus G is a group of unitary operators on H = (E, (·,·)), hence it is bounded. 2 As a consequence we obtain the following result proved in [16]: Corollary 9.2. A J -symmetric representation of a unital C ∗ -algebra on a Pontryagin space is similar to a ∗ -representation. For a proof it suffices to notice that restricting the representation to the unitary group of the C ∗ -algebra we obtain a bounded group of J -unitary operators. Acknowledgments We wish to express our gratitude to Professor Itai Shafrir for informing us about results of his dissertation [15] and to Ekaterina Shulman for providing us with a copy of [15] and for helping us with its translation. The second author also would like to thank Alexei Loginov and Natal’ya Yaskevich for helpful discussions on the subject of this paper many years ago. Appendix A. Hyperbolicity of B (after Itai Shafrir) For any bounded domains D1 , D2 of complex Banach spaces we denote by Hol(D1 , D2 ) the set of all holomorphic maps from D1 to D2 . If D1 = D2 = D, then Hol(D1 , D2 ) is a semigroup with respect to the composition, and by Aut(D) we denote the set of all its invertible elements (biholomorphic automorphisms of D). The group Aut(B) acts transitively on B. Indeed, for each A ∈ B the Möbius transform MA is biholomorphic and sends 0 to A. As usually the Carathéodory metric on B is defined by the equality: cB (A, B) = sup ω f (A), f (B) : f ∈ Hol(B, ) where is the unit disk and ω is the Poincaré distance: −1 z1 − z2 . ω(z1 , z2 ) = tanh 1 − z1 z2


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As it was mentioned in Section 4, cB coincides with the metric ρ defined by the formula (4.2). Clearly cB is invariant under biholomorphic maps of B. We shall prove that B is a hyperbolic space with respect to this metric. Furthermore the differential Carathéodory metrics on B is defined by α(A, V ) =

|Df (A)V | 2 1 f ∈Hol(B ,) − |f (A)|

(A.1)

sup

for all A ∈ B, V ∈ L(K, H ), where Df (A) is the differential of f in A (see [6], where α is denoted by γB ). Lemma A.1. For each A ∈ B, V ∈ L(K, H ) DMB (A)V = (1 − BB ∗ )1/2 (1 + AB ∗ )−1 V (1 + B ∗ A)−1 (1 − B ∗ B)1/2 .

(A.2)

In particular, DMB (0)V = (1 − BB ∗ )1/2 V (1 − B ∗ B)1/2 . Proof. By definition, MB (X) = (1 − BB ∗ )−1/2 (B + X)(1 + B ∗ X)−1 (1 − B ∗ B)1/2 . We have to calculate the coefficient c of t in the Taylor decomposition of the function t → MB (A + tV ). For this, note that if P is an invertible operator then (P + tQ)−1 = P −1 − tP −1 QP −1 + o(t). It follows immediately that c = (1 − BB ∗ )−1/2 V (1 + B ∗ A)−1 − (B + A)(1 + B ∗ A)−1 B ∗ V (1 + B ∗ A)−1 (1 − B ∗ B)1/2 = (1 − BB ∗ )−1/2 1 − (B + A)(1 + B ∗ A)−1 B ∗ V (1 + B ∗ A)−1 (1 − B ∗ B)1/2 = (1 − BB ∗ )−1/2 1 − (B + A)B ∗ (1 + AB ∗ )−1 V (1 + B ∗ A)−1 (1 − B ∗ B)1/2 = (1 − BB ∗ )−1/2 1 + AB ∗ − (B + A)B ∗ (1 + AB ∗ )−1 V (1 + B ∗ A)−1 (1 − B ∗ B)1/2 = (1 − BB ∗ )1/2 (1 + AB ∗ )−1 V (1 + B ∗ A)−1 (1 − B ∗ B)1/2 .

2

Lemma A.2. α(A, V ) = (1 − AA∗ )−1/2 V (1 − A∗ A)−1/2 for all A ∈ B and V ∈ L(K, H ). Proof. By [6, Lemma V.1.5] α(0, V ) = V . Let now A be arbitrary. Then by [6, Proposition V.1.2] α MA (0), DMA (0)X = α(0, X) = X. On the other hand, by Lemma A.1, α MA (0), DMA (0)X = α A, (1 − AA∗ )1/2 X(1 − A∗ A)1/2 . Setting now V = (1 − AA∗ )1/2 X(1 − A∗ A)1/2 , we obtain X = (1 − AA∗ )−1/2 V (1 − A∗ A)−1/2 and hence α(A, V ) = (1 − AA∗ )−1/2 V (1 − A∗ A)−1/2 ). 2

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For any bounded operator D, set D (1) = D, D (3) = DD ∗ D, D (5) = DD ∗ DD ∗ D, . . . , D (2k+1) = (DD ∗ )k D. Let Th D =

∞

a2n+1 D (2n+1)

(A.3)

n=0

2n+1 . where aj are the Taylor coefficients of tanh t, i.e., tanh t = ∞ n=0 a2n+1 t It follows from the definition that Th D = tanh(D) if D is selfadjoint. If D = J |D| is the polar decomposition of D (that is, |D| = (D ∗ D)1/2 and J is a partial isometry such that (J ∗ J )|D| = |D|(J ∗ J ) = |D|), then D (2n+1) = J |D|2n+1 , and hence Th D = J tanh |D|. On the other hand, we can write D = |D ∗ |J , where |D ∗ | = J |D|J ∗ = (DD ∗ )1/2 , therefore Th D = tanh |D ∗ | J. For the space (B, ρ) we define the set M of lines as follows: for A ∈ B, D ∈ ∂B (i.e., D = 1) we let γA,D = γA,D (t) := MA Th(tD) : t ∈ R (A.4) and set M = {γA,D : A ∈ B, D ∈ ∂B}. Proposition A.3. γA,D is a metric line. Proof. It suffices to show that ρ(γA,D (s), γA,D (t)) = |s − t|. Since ρ is invariant with respect to MA we can assume that A = 0. We have ρ(γ0,D (s), γ0,D (t)) = tanh−1 MB (Th(tD)), where B = − Th(sD). Using polar decomposition D = J |D| we have that Th(tD) = J tanh(t|D|), Th(tD)∗ Th(sD) = tanh(t|D|) tanh(s|D|), whence −1/2 MB Th(tD) = 1 − Th(sD) Th(sD)∗ Th(tD) − Th(sD) −1 1/2 1 − Th(sD)∗ Th(sD) × 1 − Th(sD)∗ Th(sD) −1/2 ∗ = J 1 − tanh2 s|D| J J tanh t|D| − tanh s|D| 1/2 −1 1 − tanh2 s|D| × 1 − tanh s|D| tanh t|D| = J tanh (t − s)|D| = Th (t − s)D giving the statement.

2


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We have to prove that γA,D (t) is a metric curve, in the sense that the metric of its derivation equals 1, i.e., α(γ (t), γ (t)) = 1. Lemma A.4. Let γ (t) = Th(tD), D ∈ ∂B. Then γ (t) = D − γ (t)D ∗ γ (t).

(A.5)

Proof. We have γ (t) = J tanh(t|D|) and −2 −2 γ (t) = J |D| cosh t|D| = D cosh t|D| . On the other hand D − γ (t)D ∗ γ (t) = D − J tanh t|D| |D|J ∗ J tanh t|D| = D − D tanh2 t|D| −2 = D cosh t|D| giving (A.5).

2

Lemma A.5. Let γ (t) = Th(tD), D ∈ ∂B. Then (1 − γ γ ∗ )−1/2 (D − γ D ∗ γ )(1 − γ ∗ γ )−1/2 = D. Proof. Setting D = J |D|, we have γ ∗ γ = Th(tD)∗ Th(tD) = tanh t|D| J ∗ J tanh t|D| = tanh2 t|D| . Furthermore γ γ ∗ = tanh t|D ∗ | J J ∗ tanh t|D ∗ | = tanh2 t|D ∗ | . Since D|D| = |D ∗ |D, we have that Df |D| = f |D ∗ | D for any bounded Borel function f . Taking f (x) = 1 − tanh2 (tx), we get D(1 − γ ∗ γ ) = (1 − γ γ ∗ )D. Next D∗γ = γ ∗D because D ∗ Th(tD) = D ∗ J tanh(t|D|) = |D| tanh(t|D|) is selfadjoint.

(A.6)

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Hence (1 − γ γ ∗ )−1/2 (D − γ D ∗ γ )(1 − γ ∗ γ )−1/2 = (1 − γ γ ∗ )−1/2 (1 − γ γ ∗ )D(1 − γ ∗ γ )−1/2 = (1 − γ ∗ γ )1/2 (1 − γ ∗ γ )−1/2 D = D.

2

Proposition A.6. Let γ (t) = γA,D (t), A ∈ B, D ∈ ∂B. Then α γ (t), γ (t) = 1. Proof. It suffices to prove this for A = 0, since α(F (X), DF (X)V ) = α(X, V ) for any F ∈ Aut(B), X ∈ B, V ∈ L(H, K) (see [6, Proposition V.1.2]), and hence α MA Th(tD) , MA Th(tD) = α MA Th(tD) , DMA Th(tD) Th(tD) = α Th(tD), Th(tD) . Assume therefore that γ (t) = Th(tD). By Lemmas A.2 and A.5 we have

α γ (t), γ (t) = (1 − γ γ ∗ )−1/2 (D − γ D ∗ γ )(1 − γ ∗ γ )−1/2 = D = 1.

2

The next step is to prove that the family M of all lines is invariant with respect to the biholomorphic maps of B. Lemma A.7. Let η(t) = MA (γ (t)) where γ (t) = Th(tD). Then, for each biholomorphic map h : B → B, the curve h(η(t)) belongs to the family M. Proof. By [8, Theorems 3 and 4], there is a linear isometry L of the space L(K, H ) to itself satisfying the condition L(AB ∗ A) = L(A)L(B)∗ L(A)

for all A, B ∈ L(K, H )

(A.7)

and such that h = Mh(0) ◦ L = L ◦ M−h(0) . It follows from (A.7) (see a remark after [8, Corollary 5]) that L ◦ MA = ML(A) ◦ L for all A ∈ B. So it suffices to consider the cases h = L and h = MB . Let us firstly prove that L(η(t)) ∈ M. Indeed, L η(t) = L MA γ (t) = ML(A) L γ (t) = ML(A) L Th(tD) = ML(A) Th tL(D) ∈ M.


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Now we have to prove that MB (η(t)) ∈ M. Applying [8, Theorems 3 and 4] to h(x) = MB (MA (x)) we get a linear isometry L satisfying (A.7) and such that MB ◦ MA = MC ◦ L where C = h(0) = MB (A). Thus MB η(t) = MB MA γ (t) = MC L γ (t) = MC Th tL(D) ∈ M.

2

Our next goal is to show that for each A, B ∈ B there is a unique line in M which passes through A, B. Lemma A.8. The set of all lines in M that go through A is {γA,D : D ∈ ∂(B)}. Proof. It suffices to assume that A = 0. Suppose that a line γ (t) = MB (Th(tD)) goes through 0, i.e., γ (s) = 0 for some s ∈ R. Then clearly B = − Th(sD). Using the arguments from the proof of Proposition A.3 we obtain γ (t) = Th((t − s)D). Thus γ = γ0,D . 2 Corollary A.9. For each A, B ∈ B, there is a unique line in M that passes through them. Proof. We may assume that A = 0. Let B = J |B| be the polar decomposition of B and let C = tanh−1 |B|/t0 for t0 > 0 be such that C = 1. Then for D = J C the line γ0,D passes through 0 and B. If there are two lines, γ0,D1 and γ0,D2 , going through B then by the above lemma, B = Th(tD1 ) = Th(sD2 ) for some t, s ∈ R. We may suppose that t, s > 0. Taking polar decompositions of D1 = J1 |D1 | and D2 = J2 |D2 | we see that J1 = J2 and tanh(t|D1 |) = tanh(s|D2 |), which imply that t|D1 | = s|D2 |. But this clearly shows that the lines coincide. 2 Lemma A.10.

A (1 − BB ∗ )−1/2 (A − BA∗ B)(1 − B ∗ B)−1/2

(A.8)

for each A, B ∈ B. Proof. Consider the polar decomposition B = J |B|. Then |B ∗ | = (BB ∗ )1/2 = J |B|J ∗ . Let P = tanh−1 (|B ∗ |), and Q = tanh−1 (|B|). Then (1 − BB ∗ )−1/2 (A − BA∗ B)(1 − B ∗ B)−1/2 = (cosh P )A(cosh Q) − (sinh P )J AJ ∗ (sinh Q). For any ε > 0, there are unit vectors x, y such that

(cosh P )A(cosh Q)x, y (cosh P )y A (cosh Q)x − ε. Since (cosh P )y2 − (sinh P )y2 = y2 , and (cosh Q)x2 − (sinh Q)x2 = x2 one can find numbers a, b such that

(sinh P )y = sinh b,

(sinh Q)x = sinh a,

(cosh P )y = cosh b,

(cosh Q)x = cosh a.

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Hence

(cosh P )A(cosh Q) − (sinh P )J AJ ∗ (sinh Q)

(cosh P )A(cosh Q) − (sinh P )J AJ ∗ (sinh Q) x, y (cosh b)(cosh a)A − ε − (sinh b)(sinh a)A cosh(b − a)A − ε A − ε, giving the statement.

2

Lemma A.11. Let us consider two lines: γ (t) = MA (Th(tC)), η(t) = MA (Th(tD)). Then 2ρ γ (s), η(s) ρ γ (2s), η(2s)

(A.9)

for each s > 0. Proof. Since ρ is invariant with respect to the transformations MA we may assume A = 0. Let C(t) be a curve γB,E (t) which joins γ (2s) with η(2s), we assume that C(0) = γ (2s), C(t0 ) = η(2s) for some t0 > 0 (such curve exists by Corollary A.9). Define now a new curve C1 by C1 = Th

1 −1 Th C . 2

Then C1 (0) = γ (s), C1 (t0 ) = η(s) and −1 C(t) = 2C1 (t) 1 + C1 (t)∗ C1 (t) . As usually we denote by L(C1 ) the length of the curve C1 : L(C1 ) = If we could show that

(A.10)

t0 0

α(C1 (t), C1 (t)) dt.

L(C) 2L(C1 )

(A.11)

for all curves C, C1 satisfying (A.10) then we would obtain that ρ γ (2s), η(2s) = L(C) 2L(C1 ) 2ρ γ (s), η(s) (the first equality follows from Propositions A.3 and A.6, the last inequality holds because the length of any curve is not smaller then the distance between its ends). Thus our goal is the inequality (A.11). It suffices to show that 2α C1 (t), C1 (t) α C(t), C (t) . Since 2C1 = C 1 + C1∗ C1 ,

(A.12)


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we have C 1 + C1∗ C1 + C C1 ∗ C1 + C1∗ C1 = 2C1 , whence C =

−1 2 − CC1∗ C1 − CC1 ∗ C1 1 + C1∗ C1 .

(A.13)

Since −1 −1 −1 = 2 1 + C1 C1∗ , 2 − CC1∗ = 2 − 2C1 1 + C1∗ C1 C1∗ = 2 1 − C1 C1∗ 1 + C1 C1∗ substituting this into (A.13) we obtain −1 −1 −1 C = 2 1 + C1 C1∗ C1 − C1 1 + C1∗ C1 C1∗ C1 1 + C1∗ C1 −1 −1 C1 − C1 C1∗ C1 1 + C1∗ C1 . = 2 1 + C1 C1∗ Now it follows from Lemma A.2 that the inequality (A.11) is equivalent to the following

1 − C1 C ∗ −1/2 C 1 − C ∗ C1 −1/2

1 1 1

−1 −1 (1 − CC ∗ )−1/2 1 + C1 C1∗ C1 − C1 C1 ∗ C1 1 + C1∗ C1 (1 − C ∗ C)−1/2 . (A.14) But −2 −2 1 − CC ∗ = 1 − 4C1 1 + C1∗ C1 C1∗ = 1 − 4C1 C1∗ 1 + C1 C1∗ 2 −2 2 −2 = 1 + C1 C1∗ − 4C1 C1∗ 1 + C1 C1∗ = 1 − C1 C1∗ 1 + C1 C1∗ . Similarly −1 (1 − C ∗ C)−1/2 = 1 + C1∗ C1 1 − C1∗ C1 . It follows now that (A.14) is equivalent to the inequality

1 − C1 C ∗ −1/2 C 1 − C ∗ C1 −1/2

1 1 1

−1 −1

1 − C1 C1∗ C1 − C1 C1 ∗ C1 1 − C1∗ C1 .

(A.15)

But (A.15) follows from Lemma A.10 by substituting B = C1 and A = (1 − C1 C1∗ )−1/2 C1 × (1 − C1∗ C1 )−1/2 into inequality (A.8). 2 The above results establish Theorem A.12. B is a hyperbolic space.

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References [1] T.Ya. Azizov, I.S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, 1989. [2] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, AMS, Providence, RI, 2000. [3] M.R. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss., vol. 319, Springer, Berlin, New York, 1999. [4] M.S. Brodskii, D.P. Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR (N.S.) 59 (1948) 837–840 (in Russian). [5] N. Dunford, J.T. Schwartz, Linear Operators. Part I: General Theory, Interscience Publishers, New York, 1958. [6] T. Franzoni, E. Vesentini, Holomorphic Maps and Invariant Distances, Notas Mat., vol. 69, North-Holland Publishing Company, Amsterdam, New York, 1980. [7] L.A. Harris, Schwarz’s lemma in normed linear spaces, Proc. Natl. Acad. Sci. USA 62 (1969) 1014–1017. [8] L.A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, in: T.L. Hayden, T.J. Suffridge (Eds.), Proceedings on Infinite Dimensional Holomorphy, University of Kentucky, 1973, in: Lecture Notes in Math., vol. 364, Springer-Verlag, 1974, pp. 13–40. [9] J.W. Helton, Operators unitary in an indefinite metric and linear fractional transformations, Acta Sci. Math. (Szeged) 32 (1971) 261–266. [10] E. Kissin, V. Shulman, Representations on Krein Spaces and Derivations of C ∗ -Algebras, Pitman Monographs and Surveys in Pure and Applied Math., vol. 89, Addison–Wesley/Longman, 1997. [11] M.G. Krein, On an application of the fixed-point principle in the theory of linear transformations of indefinite metric spaces, Uspekhi Mat. Nauk 5 (1950) 180–190 (in Russian); English translation: Amer. Math. Soc. Transl. Ser. 2 1 (1955) 27–35. [12] G. Pisier, Similarity Problems and Completely Bounded Maps, second, expanded edition, Lecture Notes in Math., vol. 1618, Springer-Verlag, Berlin, 2001. [13] S. Reich, I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990) 538–558. [14] S. Reich, A.J. Zaslavski, Generic aspects of metric fixed point theory, in: W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer/Springer, 2001, pp. 557–575. [15] I. Shafrir, Operators in hyperbolic spaces, Ph.D. thesis, Technion—Israel Institute of Technology, 1990 (in Hebrew). [16] V.S. Shulman, On representation of C ∗ -algebras on indefinite metric spaces, Mat. Zametki 22 (1977) 583–592 (in Russian); English translation: Math. Notes 22 (3–4) (1977) 816–820 (1978). [17] V.S. Shulman, On fixed points of fractionally linear transformations, Funktsional. Anal. i Prilozhen. 14 (1980) 93–94 (in Russian); English translation: Funct. Anal. Appl. 14 (2) (1980) 162–163. [18] W. Takahashi, A convexity in metric spaces and non-expansive mappings, I, Kodai Math. Sem. Rep. 22 (1970) 142–149.


Strong sums of projections in von Neumann factors Victor Kaftal a,∗ , Ping Wong Ng b , Shuang Zhang a a University of Cincinnati, Department of Mathematics, Cincinnati, OH 45221-0025, USA b University of Louisiana, Department of Mathematics, Lafayette, LA 70504, USA

Received 18 November 2008; accepted 28 May 2009 Available online 21 June 2009 Communicated by N. Kalton

Abstract This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum converging in the strong operator topology if the collection is infinite. A similar necessary condition is given when the operator and the projections are taken in a type II von Neumann factor, and the condition is proven to be also sufficient if the operator is “diagonalizable”. A simpler necessary and sufficient condition is given in the type III factor case. © 2009 Elsevier Inc. All rights reserved. Keywords: Sums of projections

1. Introduction Which positive bounded operators on a separable Hilbert space can be written as sums of projections? For finite sums, Fillmore asked this question and obtained the characterizations of the finite rank operators that are sums of projections [6, Theorem 1] (see Corollary 2.5 below) and of the bounded operators that are sums of two projections [6, Theorem 2] (see Proposition 2.10 below). For infinite sums with convergence in the strong operator topology, this question arose naturally from work on ellipsoidal tight frames by Dykema, Freeman, Kornelson, Larson, Ordower, * Corresponding author.

E-mail addresses: [email protected] (V. Kaftal), [email protected] (P.W. Ng), [email protected] (S. Zhang). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.027

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and Weber in [5]. They proved that a sufficient condition for a positive bounded operator A ∈ B(H )+ to be the sum of projections is that its essential norm Ae is larger than one [5, Theorem 2]. This result served as a basis for further work by Kornelson and Larson [16] and then by Antezana, Massey, Ruiz, and Stojanoff [1] on the decomposition of positive operators into (strongly converging) sums of rank-one positive operators of preset norms. The same question can be asked relative to a von Neumann algebra M. We say that an operator A ∈ M + is a strong sum of projections if there exists a collection of (not necessarily mutually orthogonal or commuting) projections Pj ∈ M with cardinality N ∞, for which A = N j =1 Pj and the series converges in the strong operator topology if N = ∞. The main goal of this article is to answer the question of which operators are strong sums of projections. To simplify the treatment, we consider only von Neumann factors, and we further assume that they are σ -finite (i.e., countably decomposable) so that all infinite projections are equivalent. Thus let H be a complex infinite dimensional Hilbert space and M ⊂ B(H ) be a σ -finite von Neumann factor. If M is of type I, we will identify it with B(H ) (hence we will assume that H is separable), and denote by Tr the usual normalized trace such that Tr P = 1 for any rank-one projection P . If M is of type II, τ will denote the faithful positive semifinite normal trace, unique up to scalar multiples in the type II∞ case and normalized by τ (I ) = 1 in the type II1 case. If M is only assumed to be semifinite, i.e., it is of type I or type II unless specified, we will generically denote its trace by τ . The conditions for A to be a strong sum of projections are expressed in terms of the excess and the defect parts of A. Given A ∈ M + , we denote by χA

RA = χA 0, A

the spectral measure of A,

the range projection of A, A+ := (A − I )χA 1, A the excess part of A,

A− := (I − A)χA (0, 1)

the defect part of A.

Thus we have the decomposition A = A+ − A− + RA .

(1)

A positive operator A is said to be diagonalizable if A = γj Ej for some γj > 0 and mutually orthogonal projections {Ej } in M. Diagonalizable operators are also called discrete and are the most accessible operators in a type II factor (e.g., see [3]). The main results of this article are collected in the following theorem. Theorem 1.1. Assume that M is a σ -finite von Neumann factor and A ∈ M + . (i) Let M be of type I. Then A is a strong sum of projections if and only if either Tr(A+ ) = ∞ or Tr(A− ) Tr(A+ ) < ∞ and Tr(A+ ) − Tr(A− ) ∈ N ∪ {0}. (Theorems 6.6, 4.3, and 3.3.) (ii) Let M be of type II and A be diagonalizable. Then A is a strong sum of projections if and only if τ (A+ ) τ (A− ). The condition is necessary even when A is not diagonalizable. (Theorems 6.6, 5.2, and 3.3.) (iii) Let M be of type III. Then A is a strong sum of projections if and only if either A > 1 or A is a projection. (Corollary 6.4 and Theorem 3.3.)


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Remark 1.2. The statement (i) above extends the sufficient condition obtained in [5, Theorem 2]. In fact, it is elementary to show that Ae > 1 implies that Tr(A+ ) = ∞; however, the reverse implication is false. The necessary conditions in Theorem 1.1 are obtained via the frame theory type construction of Proposition 3.1 that links decomposability of an operator A into a strong sum of projections to the condition that the identity is the “diagonal” of W ∗ AW for some partial isometry W with W ∗ W = RA . For instance, the integrality condition in the B(H ) case (Theorem 1.1(i)) when Tr(A+ ) < ∞ emerges naturally from the fact that Tr(A+ ) − Tr(A− ) coincides with the trace of the projection I − W W ∗ . A modification of these arguments provides an alternative proof of the necessity of the “integrality condition” for diagonals of projections in Kadison’s [9, Theorem 15] that identifies explicitly the integer as the difference of traces of two projections (Corollary 3.6). The basic tool for all the sufficient conditions is provided by a 2 × 2 matrix construction that decomposes certain diagonal matrices into the sum of a projection and a rank-one “remainder” (Lemma 2.1). This lemma serves also several other purposes: when applied to finite matrices it provides in Corollary 2.5 another proof of Fillmore’s characterization of finite sums of projections [6, Theorem 1]. It can be applied to (finite) sums of scalar multiples of mutually orthogonal equivalent projections in a C ∗ -algebra (Lemma 2.6). It also provides in the von Neumann algebra setting a short constructive proof (Proposition 2.10) of Fillmore’s characterization of sums of two projections [6, Theorem 2, Corollary]. As the results of [5] suggest, the most tractable case is the “infinite” one. The key special case (Lemma 6.1) is when A is an infinite sum of scalar multiples of mutually orthogonal equivalent projections in M and the sum of the coefficients in the corresponding expansion of A+ diverges. Based on this lemma we obtain the sufficiency in Theorem 1.1 for part (iii), for part (i) when Tr(A+ ) = ∞, and for part (ii) when τ (A+ ) = ∞. For the more delicate “finite trace” case in B(H ), i.e., when Tr(A+ ) < ∞, we diagonalize A+ and A− and then apply iteratively Lemma 2.1, which provides canonically a sequence of projections. The strong convergence of the series of these projections is proven by reducing the problem to a finite-dimensional construction and to three infinite dimensional special cases (Lemmas 2.3, 4.1, 4.2, and Theorem 4.3). As in [2], the most tractable operators in a type II case are the diagonalizable ones. Applying Lemma 2.1 to a diagonalizable operator, the strong convergence of the “remainders” is obtained by showing that they converge in the trace-norm (Lemma 5.1). Example 5.3 exhibits a nondiagonalizable operator that is the sum of two projections. It remains open whether the condition τ (A+ ) τ (A− ) is always sufficient for A to be the strong sum of projections. Von Neumann algebras are by no means the only setting in which positive operators may be decomposed into sums of projections. In a separate paper [12], we will investigate the same problem for positive operators in the multiplier algebra M(A ⊗ K) where A is a σ -unital purely infinite simple C ∗ -algebra. The first and second named authors were participants in the NSF supported Workshop in Linear Analysis and Probability, Texas A&M University, 2006, where they first heard from David Larson about the results in [5] and [16] that stimulated this project. 2. The matrix construction We start with a simple lemma which will be used in our key constructions.

2500


Lemma 2.1. Let e and f be two orthogonal unit vectors in H . For every μ 0 and 0 λ 1, let (1−λ)μ (1−λ)λ for μ = 0, for μ = 0, (1+μ−λ)(μ+λ) μ+λ ν := and ρ := (2) 1 for μ = 0, 0 for μ = 0, and let w :=

√ ρf − 1 − ρe

and

v :=

√ √ νf + 1 − νe.

(3)

Then w ⊗ w and v ⊗ v are rank-one projections and (1 + μ)(e ⊗ e) + (1 − λ)(f ⊗ f ) = w ⊗ w + (1 + μ − λ)(v ⊗ v).

(4)

Proof. It is immediate to verify that 0 ν, ρ 1, w and v are unit vectors, and hence w ⊗ w, v ⊗ v are rank-one projections with range contained in span{e, f }. Their matrix representations with respect to the basis {e, f } are, respectively, √ √ 1−ρ − ρ(1 − ρ) ν(1 − ν) 1−ν √ and . √ − ρ(1 − ρ) ρ ν(1 − ν) ν An elementary computation shows that

1+μ 0 0 1−λ

and hence (4) holds.

√ 1−ρ − ρ(1 − ρ) = √ − ρ(1 − ρ) ρ √ ν(1 − ν) 1−ν + (1 + μ − λ) √ ν(1 − ν) ν

2

If we do not require the orthogonality of the vectors e and f , we still obtain the decomposition in (4), but the vectors w and v are no longer obtained as simply as in (3). With a slight generalization and a reformulation in terms of rank-one projections, we have Lemma 2.2. Let P , Q be rank-one projections in B(H ) and let a c b. Then there are projections P ∼ Q ∼ P for which aP + bQ = cP + (a + b − c)Q . Proof. The cases when P = Q or when a = 0 or c = a or c = b being trivial, we assume that P = Q and that 0 < a < c < b. Diagonalize the positive rank-two operator A. Then A = a E + b F where E and F are two mutually orthogonal rank-one projections, 0 < a a < c < b b , and a + b = a + b. Without loss of generality we can assume that c = 1, and now the conclusion follows from Lemma 2.1. 2 A generalization of Lemma 2.2 provides the algorithm for constructing frame perturbations in [14]. The following lemma is obtained by iterative applications of Lemma 2.1 and serves several complementary purposes: it illustrates in the simpler finite-dimensional case a construction that


2501

is applicable also in the cases of infinite dimensions, it is a key ingredient in the proof of Theorem 4.3, and it provides another proof of Fillmore’s characterization of finite sums of rank-one projections [6, Theorem 1]. Lemma 2.3. Let A ∈ B(H )+ be a finite rank operator and set A=

n m (1 + μj )(ej ⊗ ej ) + (1 − λi )(fi ⊗ fi ) j =1

i=1

where {e1 , . . . , en , f1 , . . . , fm } is a collection of mutually orthogonal unit vectors, μj > 0 and 0 λi < 1 for all 1 i m and 1 j n. If all the eigenvalues of A are greater than 1, set m = 0, i.e., drop the sum involving the λi . Similarly, if all the eigenvalues of A are less than or equal to 1, set n = 0. (i) Assume that k := Tr(A) − Tr(RA ) ∈ N ∪ {0}. Then A is the sum of n + m + k rank-one projections. (ii) Assume that 0 nj=1 μj − m i=1 λi max{μj }. Then there are n+m rank-one projections P1 , P2 , . . . , Pm+n for which A=

m+n−1

Ph + 1 +

h=1

Proof. First notice that A+ =

n

j =1 μj (ej

n

μj −

j =1

m

λi Pm+n .

(5)

i=1

⊗ ej ) and A− =

Tr(A) − Tr(RA ) = Tr(A+ ) − Tr(A− ) =

m

n j =1

i=1 λi (fi

μj −

⊗ fi ), hence by (1)

m

λi .

i=1

(i) To avoid triviality, assume that A = 0, and in particular that n = 0. If k > 0, let A1 := μ1 (e1 ⊗ e1 ) +

n m (1 + μj )(ej ⊗ ej ) + (1 − λi )(fi ⊗ fi ). j =2

i=1

Then A = e1 ⊗ e1 + A1 , A1 0, RA1 = RA , and Tr(A1 ) = Tr(A) − 1, whence we obtain Tr(A1 ) − Tr(RA1 ) = k − 1. Iterating, we decompose A into the sum of k rank-one projections and a positive operator m Ak with Tr(Ak ) = Tr(RAk ). Thus, we can simply assume that k = 0. Hence n μ = j j =1 i=1 λi and also m = 0. Now we start by the decomposition (1 − λ1 )(f1 ⊗ f1 ) + (1 + μ1 )(e1 ⊗ e1 ) = P1 + (1 + δ1 )(v1 ⊗ v1 ) where δ1 := μ1 − λ1 and P1 and v1 ⊗ v1 are the rank-one projections prescribed by Lemma 2.1. Then either δ1 = 0 and n = m = 1, in which case A is the sum of two rank-one projections, or one of the three conditions hold: δ1 > 0, in which case m > 1; δ1 < 0, in which case n > 1; or

2502


δ1 = 0 and (n, m) = (1, 1), in which case n > 1 and m > 1. Notice that v1 is orthogonal to each ej and fi for j > 1 and i > 1 if any, so Lemma 2.1 yields again the decomposition ⎧ ⎨ (1 + δ1 )(v1 ⊗ v1 ) + (1 − λ2 )(f2 ⊗ f2 ) (1 + δ1 )(v1 ⊗ v1 ) + (1 + μ2 )(e2 ⊗ e2 ) ⎩ (1 − λ2 )(f2 ⊗ f2 ) + (1 + μ2 )(e2 ⊗ e2 )

if δ1 > 0, if δ1 < 0, = P2 + (1 + δ2 )(v2 ⊗ v2 ) if δ1 = 0

where P2 and v2 ⊗ v2 are rank-one projections and ⎧ ⎨ μ 1 − λ1 − λ2 δ2 = μ1 + μ2 − λ1 ⎩ μ2 − λ2

if δ1 > 0, if δ1 < 0, if δ1 = 0.

In general after q steps, we have

q n m (1 + μj )(ej ⊗ ej ) + (1 − λi )(fi ⊗ fi ) = Pj + (1 + δq )(vq ⊗ vq ) Aq := j =1

(6)

j =1

i=1

where δq = nj =1 μj − m i=1 λi and n , m ∈ N, n n, m m. We continue the process until we “run out” of summands to which apply Lemma 2.1. This occurs only when n = n and m m = m. Indeed, if n = n but m < m, then δq = nj=1 μj − m i=m +1 λi > 0 and i=1 λi = thus we can further decompose (1 + δq )vq ⊗ vq + (1 − λm +1 )(fm +1 ⊗ fm +1 ) into the sum of a rank-one projection and a positive remainder. The case when m = m but n = n is similar. But when n = n and m = m, then δq = 0 and hence A = Aq is the sum of Tr(A) = n + m + k rank-one projections. (ii) Assume without loss of generality that max{μj } occurs for j = n, i.e., that n−1

μj

j =1

m

λi

n

μj .

j =1

i=1

We can carry on the same construction process as in (i). If after the q steps that lead to the decomposition (6) we have n = n and m = m, then δq m i=m +1 λi 0 and we can continue the process. If we have m = m but n = n then

δq =

n

μj −

j =1

m i=1

λi

n−1 j =1

μj −

m

λ1 0,

i=1

and in this case too we can continue the process. Thus the process terminates only when n = n and m = m and thus (5) holds. 2 Remark 2.4. The condition in (ii) is necessary, because if (5) holds, then 1+

n j =1

μj −

m i=1

λi A = 1 + max{μj }.


2503

This lemma provides a constructive proof of Fillmore’s characterization of finite sums of finite projections [6, Theorem 1] that does not depend on the mean value theorem (see also [5, Proposition 6]). Corollary 2.5. (See [6, Theorem 1].) Let A ∈ B(H )+ be a finite rank operator. Then A is the sum of projections if and only if Tr A ∈ N and Tr A Tr(RA ). Proof. The sufficiency is given by Lemma 2.3(i). For the necessity, assume that A = kj =1 Pi is a sum of projections and by further decomposing them if necessary, assume that they all have rank one. Then Tr A = k ∈ N and, clearly, rank A k. 2 The matrix construction in Lemma 2.1 extends to C ∗ -algebras and hence in particular to von Neumann algebras. It is well known that given a collection {Ej }nj=1 of mutually orthogonal equivalent projections in a C ∗ -algebra A, we can chose a corresponding set of matrix units and hence an embedding of Mn (C) into A. Thus by Lemma 2.1 and Corollary 2.5 we obtain: Lemma 2.6. Let A be a C ∗ -algebra. (i) If E and F are two mutually orthogonal equivalent projections in A, 0 λ 1, and μ 0, are scalars, then there are two projections P− and P+ in A, with P− ∼ P+ ∼ E, for which (1 + μ)E + (1 − λ)F = P− + (1 + μ − λ)P+ . (ii) If A = nj=1 γj Ej for some mutually orthogonal equivalent projections Ej ∈ A and some scalars γj > 0 with nj=1 γj = k ∈ N and k n, then A is the sum of k equivalent projections in A. Remark 2.7. (i) The embedding of Mn (C) into A depends not only on the projections Ej but also on the matrix units. However, once these matrix units are chosen, the construction in Lemma 2.1 assigns the decomposition in a canonical way. Explicitly for the n = 2 case, let V ∈ A be a partial isometry with E = V ∗ V and F = V V ∗ , then the projections P− and P+ obtained from this embedding and the formulas in Lemma 2.1 are ρ(1 − ρ) V + V ∗ + ρF, P+ := (1 − ν)E + ν(1 − ν) V + V ∗ + νF.

P− := (1 − ρ)E −

(7)

√ √ It can also be verified directly that setting W := 1 − ρE − ρV , we get W ∗ W = E and W W ∗ = P− . Thus W is a partial isometry and hence P− is a projection and P− ∼ E. Similarly, P+ ∼ E. (ii) More generally, if 0 a c b, set ρ :=

a(b−c) c(b−a)

0

if b = c, if b = c,

and ν :=

a(c−a) (a+b−c)(b−a)

1

Then with P− and P+ as in (7), bE + aF = cP− + (a + b − c)P+ .

if b = c, if b = c.

2504


Lemma 2.8. Let P , Q be finite equivalent commuting projections in a von Neumann algebra M and let 0 a < b and a c b. Then there are projections P ∼ Q ∼ P in M for which aP + bQ = cP + (a + b − c)Q . Proof. By the assumption of finiteness, we have the cancellation P − P Q ∼ Q − P Q. By Lemma 2.6 (see also Remark 2.7(ii)), there are projections P ∼ Q ∼ P − P Q in M, with P ∨ Q P − P Q + Q − P Q, for which a(P − P Q) + b(Q − P Q) = cP + (a + b − c)Q . But then, aP + bQ = a(P − P Q) + b(Q − P Q) + (a + b)P Q = c(P + P Q) + (a + b − c)(Q + P Q). Since P ⊥ P Q and Q ⊥ P Q, P + P Q and Q + P Q are projections and both are equivalent to P . 2 Remark 2.9. (i) If the projections P and Q are not finite, cancellation might fail and indeed the property itself might fail. For instance if P is infinite but P = I , then there are no projections P and Q for which 15 P + I = 25 P + 45 Q . Indeed, otherwise I − Q = 25 P − 15 P − 15 Q , whence I − Q 45 and hence Q = I . But then, 15 P + 15 I = 25 P , whence P = P = I , against the assumption. (ii) Lemma 2.8 holds also for every C ∗ -algebra A with the cancellation property (e.g., AFalgebras). The proof of Lemma 2.1 can be used also to obtain a simple constructive proof of Fillmore’s [6, Theorem 2, Corollary] characterization of the operators in B(H ) that are sums of two projections. The same characterization holds for von Neumann algebras. Proposition 2.10. Let M be a von Neumann algebra and A ∈ M with 0 A 2I . Then A is the sum of two projections in M if and only A = E ⊕ B where E is a (possibly zero) projection in M and there is a unitary U ∈ M that commutes with E and for which U BU ∗ = 2RB − B. Proof. To prove the sufficiency, it is obviously enough to consider the case when E = 0 and RA = I , i.e., U AU ∗ = 2I − A. Let A be the (abelian) von Neumann algebra generated by A. Since U AU ∗ = 2I − A ∈ A, it follows that U AU ∗ ⊂ A and hence U 2 A(U 2 )∗ ⊂ U AU ∗ . Since ∗ A = 2I − U AU ∗ = 2I − U 2I − U AU ∗ U ∗ = U 2 A U 2 , it follows that A = U 2 A(U 2 )∗ . Thus A = U AU ∗ , i.e., U · U ∗ is a conjugation of A. In particular, for every Borel set Ω ⊂ [0, 2] there is a Borel set ΩU ⊂ [0, 2] for which U χA (Ω)U ∗ = χA (ΩU ). Let Et := χA [0, t) ∈ A for t ∈ [0, 2] be the spectral resolution of A. Then A= t dEt + χA {1} + t dEt = U (2I − A)U ∗ [0,1)

=U (1,2]

(1,2]

∗ ∗ (2 − t) dEt U + U χA {1}U + U (2 − t) dEt U ∗ . [0,1)


2505

It is now clear that U χA 0, 1)U ∗ = χA (1, 2 , and U χA 1, 2]U ∗ = χA [0, 1 . U χA {1}U ∗ = χA {1}, In particular, (1,2] t dEt = U ( [0,1) (2 − t) dEt )U ∗ . Thus

t dEt + χA {1} + U

A= [0,1)

(2 − t) dEt U ∗ .

[0,1)

Let 2P− :=

t dEt − U

[0,1)

2P+ := [0,1)

t (2 − t) dEt − t (2 − t) dEt U ∗ + U (2 − t) dEt U ∗ , [0,1)

[0,1)

[0,1)

[0,1)

[0,1)

[0,1)

t dEt + U t (2 − t) dEt + t (2 − t) dEt U ∗ + U (2 − t) dEt U ∗ .

Then both P− and P+ belong to M and are selfadjoint. Since χA [0, 1) ⊥ U χA [0, 1)U ∗ , it is simple to verify that P− , P+ are idempotents and hence are projections. Furthermore we have P− + P+ = A − χA {1}, hence P− ⊥ χA {1} and thus P− + χA {1} is also a projection, which completes the proof of the sufficiency. The necessity follows as in Fillmore’s proof in [6, Theorem 2, Corollary] from the analysis of the relative position of two projections which holds for general von Neumann algebras (e.g., see [19, pp. 306–308]), and hence, applies without changes to our setting. 2 Remark 2.11. With the notations of the above proof, if A is a masa, then it cannot be singular, since U belongs to the normalizer N(A) of A but does not belong to A, as otherwise A = I , against the assumption that A is a masa. 3. The necessary condition Proposition 3.1. Let A ∈ M + and let N ∈ N∪{∞}. Then the following conditions are equivalent. identity into N mu(i) There is a partial isometry V with V ∗ V = RA and a decomposition of the N ∗ E , for which tually orthogonal nonzero projections Ej , I = N j j =1 j =1 Ej V AV Ej = I , the convergence of the series being in the strong operator topology if N = ∞. (ii) A is the sum of N nonzero projections, the convergence of the series being in the strong operator topology if N = ∞, and if M is semifinite, then τ (A) = τ (I ). 1

Proof. (i) ⇒ (ii) For every j , let Wj := Ej V A 2 and let Pj := Wj∗ Wj . Since Wj Wj∗ = Ej V AV ∗ Ej = Ej

N

Ei V AV ∗ Ei Ej = Ej ,

i=1

we see that Wj is a partial isometry, and hence, Pj is a projection and Pj ∼ Ej for every j . Then

2506

V. Kaftal et al. / Journal of Functional Analysis 257 (2009) 2497–2529 N j =1

Pj =

N

1 2

1 2

∗

1 2

A V Ej V A = A V

∗

j =1

N

1

1

1

1

1

E j V A 2 = A 2 V ∗ V A 2 = A 2 RA A 2 = A

j =1

N and if N = ∞ the series N j =1 Ej and hence the series j =1 Pj converge in the strong operator topology. Furthermore, if M is semifinite, by the normality of the trace τ we have τ (A) =

N

τ (Pj ) =

j =1

N

τ (Ej ) = τ (I ).

j =1

(ii) ⇒ (i) Let A = N j =1 Pj where {Pj } are nonzero projections. First, we decompose the N identity I = j =1 Ej into N mutually orthogonal projections Ej ∼ Pj . This is immediate if all the projections Pj are infinite, and hence so is I , because then we can decompose I into N mutually orthogonal infinite projections and all infinite projections are equivalent by the assumption that M is σ -finite. Assume henceforth that M is semifinite and that Λ := {j | τ (Pj ) < ∞} = ∅ and let Λ be its (possibly empty) complement. Then τ (I ) = τ (A) =

N

τ (Pj )

j =1

τ (Pj ).

j ∈Λ

Whether M is of type II or it is of type I and then j ∈Λ τ (Pj ) ∈ N∪{∞}, there exists a projection F with τ (F ) = j ∈Λ τ (Pj ). Then it is routine to find mutually orthogonal projections Ej F with τ (Ej ) = τ (Pj ) for every j ∈ Λ. Let E := j ∈Λ Ej . Then E F and τ (E) = τ (F ) = τ (I ). We now consider three cases. In the first case, assume that τ (I ) < ∞. Then τ (I − E) = 0, hence E = I , and we are done. In the second case, assume that τ (I ) = ∞ and Λ = ∅. Then E is an infinite projection, hence there is an isometry W for which W W ∗ = E. Set Ej = W ∗ Ej W . Then Ej ∼ Ej ∼ Pj for every j and I = N j =1 Ej provides the required decomposition. In the third case, assume that τ (I ) = ∞ and Λ = ∅. Modify if necessary F so that I − F is infinite and hence so is I − E I − F . Then decompose I − E into card Λ mutually orthogonal infinite projections Ej , I − E = j ∈Λ Ej . Since Pj ∼ Ej for all j ∈ Λ , the identity I = j ∈Λ Ej + j ∈Λ Ej = N j =1 Ej provides in this case too the required decomposition. Now choose partial isometries Wj with Pj = Wj∗ Wj and Ej = Wj Wj∗ . If N < ∞, define B := N j =1 Wj . If N = ∞ and m > n, then

n j =m

∗

Wj

n j =m

Wj

=

n i,j =m

Vi∗ Wj =

n j =m

Wj∗ Wj =

n

Pj .

(8)

j =m

Thus, by the strong (and hence the weak) convergence of the series ∞ j =1 Pj , we see that the ∞ series j =1 Wj is strongly Cauchy and hence converges in the strong operator topology. Again, 1 2 call its sum B. By the same computation as in (8), we have B ∗ B = N j =1 Pj = A. Let B = V A be the polar decomposition of B. Then V ∗ V = RA and BB ∗ = V AV ∗ . Moreover, Ej B = Wj


2507

for every j , thus N

Ej V AV ∗ Ej =

j =1

N

Ej BB ∗ Ej =

j =1

N

Wj Wj∗ =

j =1

N

Ej = I.

2

j =1

Lemma 3.2. Let A ∈ M + be a strong sum of projections. (i) Either A > 1 (equivalently, A+ = 0) or A is a projection. (ii) If M is semifinite, then τ (RA ) τ (A). Proof. (i) Obvious, since if P , Q are projections, then P + Q = 1 if and only if P Q = 0 if and only if P + Q is a projection. (ii) Let A = N j =1 Pj with Pj nonzero projections and N ∈ N ∪ {∞} and assume without loss of generality that τ (A) < ∞. By Kaplansky’s parallelogram law, [8, Theorem, 6.1.7], for every integer n N we have

τ

n

j =1

Pj

n

τ (Pj ) τ (A).

j =1

n If N < ∞, then RA = N j =1 Pj and we are done. If N = ∞, then j =1 Pj ↑ RA and by the n normality of τ , τ ( j =1 Pj ) ↑ τ (RA ). Thus also τ (RA ) τ (A). 2 Theorem 3.3. Assume that A ∈ M is a strong sum of projections. Then (i) If M is of type I, then Tr(A+ ) Tr(A− ) and either Tr(A+ ) = ∞ or Tr(A+ ) < ∞ and Tr(A+ ) − Tr(A− ) ∈ N ∪ {0}. (ii) If M is of type II, then τ (A+ ) τ (A− ). (iii) If M is of type III, then either A > 1 (equivalently, A+ = 0) or A is a projection. Proof. (iii) is given by Lemma 3.2(i), so assume henceforth that M is semifinite. Let A = N j =1 Pj with Pj nonzero projections and N ∈ N ∪ {∞}. Assume first that τ (RA ) < ∞ and hence also τ (A) < ∞ and τ (A− ) < ∞. Then by (1) and by Lemma 3.2 we have τ (A+ ) − τ (A− ) = τ (A) − τ (RA ) 0. Moreover, if M is of type I, then N < ∞ and both τ (A) and τ (RA ) are positive integers, which proves the integrality condition in (i) for the case when τ (RA ) < ∞ (see also Corollary 2.5). Now assume that τ (RA ) = ∞ and assume furthermore that τ (A+ ) < ∞. Obviously, τ (I ) = ∞ and by Lemma 3.2, τ (A) = ∞, hence τ (A) = τ (I ). Thus by Proposition 3.1 there is a partial isometry V with V ∗ V = RA and a decomposition of the identity I = N j =1 Ej into N ∗ N mutually orthogonal projections Ej for which j =1 Ej V AV Ej = I . Recall that the map

M X → Φ(X) :=

N j =1

Ej XEj ∈ M

2508


is linear, positive, unital, faithful, and in case M is semifinite, it is also trace preserving. Then we have by (1) that I = Φ V AV ∗ = Φ V A+ V ∗ − Φ V A− V ∗ + Φ V RA V ∗ . It follows from V ∗ V = RA that Φ V A+ V ∗ = Φ V A− V ∗ + Φ I − V V ∗ ,

(9)

and hence τ V A− V ∗ = τ Φ V A− V ∗ τ Φ V A+ V ∗ = τ V A+ V ∗ = τ (RA A+ ) = τ (A+ ) < ∞. But then, τ (A− ) = τ (RA A− RA ) = τ V ∗ V A− V ∗ V = τ V V ∗ V A− V ∗ = τ V A− V ∗ . This concludes the proof of the case when M is of type II. If M is of type I and Tr(A+ ) < ∞, it follows from (9) and the above computations that Tr(A+ ) = Tr(A− ) + Tr Φ I − V V ∗ = Tr(A− ) + Tr I − V V ∗ . This shows that Tr(I − V V ∗ ) < ∞, i.e., I − V V ∗ is a finite projection, and therefore it follows that Tr(I − V V ∗ ) ∈ N ∪ {0}. 2 Remark 3.4. Notice that if M is semifinite and A ∈ M + , then τ (A+ ) τ (A− )

⇒ τ (A) τ (RA )

⇒ τ (A+ ) τ (A− ).

However, if τ (RA ) < ∞, then τ (A+ ) τ (A− ) ⇐⇒ τ (A) τ (RA ). Remark 3.5. In the case of M = B(H ), let V(A) := {V AV ∗ | V ∗ V = RA } be the partial isometry orbit of A and let E denote the (unique) normal conditional expectation on the diagonal masa of B(H ) (according to a fixed orthonormal basis). Then Proposition 3.1 states that a positive operator A ∈ B(H ) with infinite trace is a strong sum of rank-one projections if and only if I ∈ E(V(A)). When A is also invertible, this is a special case of [1, Proposition 4.5]. In the case of compact operators, the diagonals of the partial isometry orbit are characterized in terms of majorization of sequences by the infinite dimensional Schur–Horn theorem obtained in [13]. The set E(V(A)) is further studied in [15] for the case of positive not necessarily compact operators. In the case of M = B(H ), an application of Proposition 3.1 together with a modification of the proof of Theorem 3.3(i) provides an alternative proof of the necessity of Kadison’s integrality condition in [9, Theorem 15] that characterizes the diagonals of infinite co-infinite projections and identifies explicitly the integer as the difference of the traces of two projections.


2509

Corollary 3.6. (See [9, Theorem 15].) Let P ∈ B(H ) be an infinite,co-infinite projection, let en be an orthonormal basis, let cn := (P en , en ), and assume that {cn | cn 12 } < ∞ and {1 − cn | cn > 12 } < ∞. Then 1 1 cn cn 1 − cn cn > − ∈ Z. 2 2 Proof. Let W be an isometry with P = W W ∗ . Define wn =

√1 W ∗ en cn

if cn = 0,

e1

if cn = 0,

and Pn := wn ⊗ wn .

Then w n = 1 for every n and hence Pn are rank-one projections. A simple computation shows that I = n cn Pn , with the series converging in the strong operator topology. Define T+ :=

1 , (1 − cn )Pn | cn > 2

T− :=

1 , cn Pn | cn 2

and

T := T+ − T− .

Then T+ , T− , and hence T are trace-class operators and 1 1 cn cn 1 − cn cn > − . Tr T = 2 2 Since {cn Pn | cn > 12 } and {(1 − cn )Pn | cn > 12 } both converge in the strong operator topology, it follows that also {Pn | cn > 12 } converges in the strong operator topology. Set A := {Pn | cn > 12 }. By Proposition 3.1 there is a partial isometry V with V ∗ V = RA for which E(V AV ∗ ) = I , where E is the conditional expectation on the atomic masa (the operation of taking the main diagonal). Since I = A − T , we have V V ∗ = V AV ∗ − V T V ∗ and thus E V V ∗ = E V AV ∗ − E V T V ∗ = I − E V T V ∗ = E(I ) − E V T V ∗ . Thus E(V T V ∗ ) = E(I − V V ∗ ) and hence Tr V ∗ V T = Tr V T V ∗ = Tr I − V V ∗ ∈ Z since T is trace-class and V V ∗ and hence I − V V ∗ are projections. On the other hand, ⊥ ⊥ ∗ ⊥ V V T = V ∗ V (A − I ) = − V ∗ V , hence Tr((V ∗ V )⊥ T ) ∈ Z, and thus ⊥ Tr(T ) = Tr V ∗ V T + Tr V ∗ V T ∈ Z.

2

2510


4. B(H ): the finite trace case In this section we will prove that in the case that A ∈ B(H )+ and Tr(A− ) Tr(A+ ) < ∞ and Tr(A+ ) − Tr(A− ) ∈ N ∪ {0}, then A is a strong sum of projections. Since the trace-class operators A+ and A− are diagonalizable and have orthogonal supports, then by (1), A too is diagonalizable. As in Lemma 2.3, let us denote the eigenvalues of A which are larger than 1, if any, by 1 + μj and those are less or equal than 1, if any, by 1 − λi , i.e., set A :=

N

(1 + μj )(ej ⊗ ej ) +

j =1

K (1 − λi )(fi ⊗ fi ) i=1

where N, K ∈ N ∪ {0} ∪ {∞}, the unit vectors ej , fi are mutually orthogonal, μj > 0, and 0 λi 1 for all i and j . Notice that the series, if infinite, converge in the strong operator topology. Of course, it would be equivalent to assume that μj 0 and 0 < λi < 1 for all i and j . Thus K A+ = N j =1 μj (ej ⊗ ej ) and A− = i=1 λi (fi ⊗ fi ) and hence N

μj −

j =1

K

λi ∈ N ∪ {0}.

i=1

Here too we adopt the convention to set a series 0i=1 as zero, e.g., by K = 0 we mean that A has no non-negative eigenvalues less or equal than 1, and hence, A− = 0; similarly for N = 0. Our proof will depend on iterative applications of Lemma 2.1. Since we will focus on infinite rank operators, i.e., on the case when N +K = ∞, the process will not terminate as in Lemma 2.3 after a finite number of steps and the crux of the proofs will be to establish strong convergence. This will be illustrated by the following lemma which handles two key special cases. Lemma 4.1. Let {go , g1 , . . .} be mutually orthogonal unit vectors. (i) Let A = (1 − λ)(go ⊗ go ) + ∞ j =1 (1 + μj )(gj ⊗ gj ) where μj > 0, 0 λ 1, for all j and ∞ λ = j =1 μj . Then A is a strong sum of projections. (ii) Let A = (1 + μ)(go ⊗ go ) + ∞ j =1 (1 − λj )(gj ⊗ gj ) where μ > 0, 0 λj 1 for all j and λ . Then A is a strong sum of projections. μ= ∞ j =1 j Proof. (i) If λ = 0, then μj = 0 for all j and hence A is already a projection. Thus assume that λ = 0. Define

−λ, δj := j −1 i=1

j = 1, μi − λ, j > 1.

Then δj increases strictly to 0, so we can also define σj :=

0,

j = 1,

(1+δj −1 )δj −1 (1+δj )(2δj −1 −δj ) ,

j > 1.

(10)


2511

Then for every j > 1, σj > 0 and also (1 + δj − δj −1 )(δj −1 − δj ) > 0. (1 + δj )(2δj −1 − δj )

(11)

j = 1, go , vj := √ σj vj −1 + 1 − σj gj −1 , j > 1.

(12)

1 − σj = Define also

Solving this recurrence relation, we get

j −2 j √ 1 − σk+1 σi gk + 1 − σj gj −1 . vj = k=0

(13)

i=k+2

We claim that there is a sequence of rank-one projections Pj for which (1 − λ)(go ⊗ go ) +

n n (1 + μj )(gj ⊗ gj ) = Pj + (1 + δn+1 )(vn+1 ⊗ vn+1 ) j =1

(14)

j =1

for every n. By Lemma 2.1, (1 − λ)(go ⊗ go ) + (1 + μ1 )(g1 ⊗ g1 ) = P1 + (1 + μ1 − λ)(v ⊗ v) = P1 + (1 + δ2 )(v ⊗ v) where P1 is a rank-one projection and by (3) and (2), v = ν=

√

νgo +

√ 1 − νg1 and

(1 − λ)λ (1 + δ1 )(−δ1 ) = = σ2 . (1 + μ1 − λ)(μ1 + λ) (1 + δ2 )(δ2 − 2δ1 )

Thus v = v2 and hence (14) is satisfied for n = 1. Assume that (14) is satisfied for n − 1. Then (1 − λ)(go ⊗ go ) +

n (1 + μj )(gj ⊗ gj ) j =1

=

n−1

Pj + (1 + δn )(vn ⊗ vn ) + (1 + μn )(gn ⊗ gn )

(by the induction hypothesis)

j =1

=

n−1

Pj + Pn + (1 + μn + δn )(v ⊗ v) (by Lemma 2.1)

j =1

=

n

Pj + (1 + δn+1 )(v ⊗ v) (by the definition of δ)

j =1

where Pn is a rank-one projection, and by (3) and (2), v =

√ √ νvn + 1 − νgn and

2512


ν=

(1 + δn )δn (1 + δn )(−δn ) = = σn+1 . (1 + μn + δn )(μn − δn ) (1 + δn+1 )(2δn − δn+1 )

Hence v = vn+1 and thus (14) is satisfied for n. Thus for every n, A=

n

Pi + (1 + δn+1 )(vn+1 ⊗ vn+1 ) +

∞

(1 + μj )(gj ⊗ gj ).

j =n+1

i=1

Since ∞ 0, to prove that A = ∞ j =n+1 (1 + μj )(gj ⊗ gj ) → i=1 Pi where the convergence is in s the strong topology (and hence, to establish the thesis), we need to show that vn+1 ⊗ vn+1 → 0, s

or, equivalently, that vj → 0 weakly. Since vj ∈ span{gi }, it is enough to show that (vj , gq ) → 0 for every q ∈ N ∪ {0}. Indeed, for every j > q + 1, we have from (13) that (vj , gq ) =

j √

1 − σq+1

σi .

i=q+2

j Thus it is enough to show that i=2 σi → 0, or, equivalently, that ∞ i=2 (1 − σi ) = ∞. By (11) and since δj −1 < δj < 0 we have 1 − σj =

δj −1 − δj (1 − δj −1 + δj )(δj −1 − δj ) 1 δj −1 − δj > > > 0. (1 + δj )(2δj −1 − δj ) 2δj −1 − δj 2 δj −1

(15)

Since δj ↑ 0, for every n > m, n n δi−1 − δi δi−1 − δi δm − δn δn = =1− , δi−1 δm δm δm

i=m+1

i=m+1

∞ δi−1 −δi 1 whence ∞ i=m+1 δi−1 > 2 for every m. As a consequence, j =2 ∞ (1 − σ ) = ∞, which completes the proof for this case. j j =2 (ii) Let k := card{j | λj = 1}. By passing to

δi−1 −δi δi−1

= ∞, and thus,

{gj ⊗ gj | λj = 0} (1 − λj )(gj ⊗ gj ) 0 < λj < 1 , = (1 + μ − k)(go ⊗ go ) +

A := A − k(go ⊗ go ) −

we can assume without loss of generality that 0 < λj < 1 for all j . Define δj :=

μ, μ−

j −1 i=1

j = 1, λi ,

j > 1.

Then δj ↓ 0. Let σj and vj be defined by (10) and (12), respectively. We claim that there is a sequence of rank-one projections Pj for which


(1 + μ)(go ⊗ go ) +

2513

n n (1 − λj )(gj ⊗ gj ) = Pj + (1 + δn+1 )(vn+1 ⊗ vn+1 ) j =1

(16)

j =1

for every n. Apply Lemma 4.2 to obtain (1 − λ1 )(g1 ⊗ g1 ) + (1 + μ)(go ⊗ go ) = P1 + (1 + δ2 )(v ⊗ v) where P1 is a rank-one projection and by (3), (2), and (11), v = ν=

√ √ νg1 + 1 − νgo and

(1 + δ2 − δ1 )(δ1 − δ2 ) (1 − λ1 )λ1 = = 1 − σ2 . (1 + μ − λ1 )(μ + λ1 ) (1 + δ2 )(2δ1 − δ2 )

Thus v = v2 and (16) holds for n = 1. The inductive proof of the claim then proceeds as in part (i). Thus A=

n

∞

Pj + (1 + δn+1 )(vn+1 ⊗ vn+1 ) +

j =1

(1 − λj )(gj ⊗ gj ),

j =n+1

and hence, to prove that A = ∞ j =1 Pj we need to show that vj → 0 weakly. Again, by (13) it ∞ suffices to show that j =2 (1 − σj ) = ∞. The only difference from the proof of part (i) is that the inequality used in (15) does no longer hold since δj > 0. However, since δj → 0, we have, for j large enough, 1 − σj =

(1 − δj −1 + δj )(δj −1 − δj ) 1 δj −1 − δj 1 δj −1 − δj > > > 0. (1 + δj )(2δj −1 − δj ) 2 2δj −1 − δj 4 δj −1

Then the same argument as in part (i) proves the claim.

2

The next special case is also based on iterated applications of Lemma 2.1 and shares part of the construction with the previous lemma, but with a different proof of the weak convergence of the vector sequence. ∞ − λi )(fi ⊗ fi ) where {ei , fi } are Lemma 4.2. Let A = ∞ i=1 (1 + μi )(ei ⊗ ei ) + i=1 (1 mutually ∞ ∞ m > 0, 0 < λ < 1, for all i, λ = μ < ∞, and orthogonal unit vectors, μ i i i=1 i i=1 i i=1 λi = n μ for every n, m ∈ N. Then A is a strong sum of projections. i=1 i Proof. Since by hypothesis λ1 = μ1 , we assume that λ1 > μ1 and leave to the reader the similar integer n1 for which proof for the case when λ1 < μ1 . Since λ1 < ∞ j =1 μj , there is a smallest n1 m1 1 μj . From λ1 < j =1 μj . Similarly, there is a smallest integer m1 for which j =1 λj > nj =1 here we obtain recursively the strictly increasing integer sequences {mk }, {nk }, starting with no = 0, mo = 1, for which

nk−1 j =1

μj

n k −1 j =1

mk−1

μj
0,

mk + nk < j mk + nk+1 .

(19)

Thus by (19) by (19) = δnk +mk −1 − λmk by (18) > −λmk

min{δj | mk−1 + nk j mk + nk+1 } = δnk +mk

> −1 (by hypothesis).

(20)

Moreover, 2δj −1 − δj = δj −1 + λj −nk > δj −1 > 0,

mk−1 + nk < j mk + nk ,

2δj −1 − δj = δj −1 − μj −mk < δj −1 < 0,

mk + nk < j mk + nk+1 .

(21)

Define the sequence σj as in (10). From (18), (19), and (21), we see that for every j , δj −1 , δj −1 − δj , and 2δj −1 − δj have the same sign. Since furthermore 1 + δj > 0 by (20) and also 1 + δj − δj −1 > 0 by (19) and (20), it follows that 0 < σj < 1.


2515

Now let Jk := mk + nk+1 . Then we have by (18) that δJk −1 < 0 < δJk , by (20) that 1 + δJk > 0, and hence

1+δJk −1 1+δJk

< 1. As 2δJk −1 − δJk < 0 by (21), we also have 0
mh + nh ,

√ σi (w, fq )

j

(vj , fq ) =

j

and (vj , eq ) =

i=mh +nh +1

√ σi (w, eq ).

i=mh +nh +1

Since 0 < σi < 1 for all i by (10) and (11) and σi < 12 infinitely often by (22), we see that j √ i=mh +nh +1 σi → 0 and hence vj → 0 weakly, which concludes the proof. 2 Theorem 4.3. Let A ∈ B(H )+ and assume that Tr(A− ) Tr(A+ ) < ∞ and also Tr(A+ ) − Tr(A− ) ∈ N ∪ {0}. Then A is a strong sum of projections. Proof. Since A+ and A− are of trace-class and supported in orthogonal subspaces, they are N simultaneously diagonalizable, so we can set A− = M i=1 λi (fi ⊗ fi ), A+ = j =1 μj (ej ⊗ ej ), where M, N ∈ N ∪ {0} ∪ {∞}, {fi , ej } are mutually orthogonal unit vectors, and 0 < λi < 1, μj > 0 or all i and j . Let k := Tr(A+ ) − Tr(A− ) =

N

μi −

j =1

M

λi .

i=1

Since χA {1} is the sum of rank-one projections, we can by (1) assume without loss of generality that A=

N M (1 + μj )(ej ⊗ ej ) + (1 − λi )(fi ⊗ fi ). j =1

(26)

i=1

By the same proof as in Lemma 2.3 we can decompose A as the sum of k rank-one projections and a positive operator A with Tr(A+ ) = Tr(A− ). Thus we assume henceforth that k = 0. We need to consider four cases: (a) (b) (c) (d)

when both A− and A+ have finite rank (i.e., N, M < ∞), when A+ has finite rank and A− does not (i.e., N < ∞, M = ∞), when A− has finite rank and A+ does not (i.e., N = ∞, M < ∞), and when both have infinite rank (i.e., N = M = ∞).

The case (a) is given by Lemma 2.3(i). Consider the case (b). If N > 1, choose an m ∈ N for which N −1 j =1

μj
n. Now, again by Corolj =1 λj = j =1 μj but lary 2.5, n

(1 + μj )(ej ⊗ ej ) +

j =1

m (1 − λj )(fj ⊗ fj ) j =1

is the sum of rank-one projections and its remainder A := A −

n m (1 + μj )(ej ⊗ ej ) + (1 − λj )(fj ⊗ fj ) j =1

j =1

satisfies the same conditions as A, but in addition has ΦA = ∅. Finally, the crucial case when ΦA = ∅ is given by Lemma 4.2. 2 In view of the necessary condition established in Theorem 3.3(i), to conclude our study in B(H ) it remains to consider the case when Tr(A+ ) = ∞. This will be done in Section 6. 5. Type II factors: the finite diagonalizable case In this section, we assume that M is a type II factor with trace τ . The following key lemma is also a consequence of Lemma 2.1, or, more precisely, of Lemma 2.6. Lemma 5.1. Let A = (1 + μ)E + (1 − λ)F where E and F are finite projections, EF = 0, μ 0, 0 λ 1, and τ (A) τ (RA ). Then A is a strong sum of projections. Proof. To avoid triviality, assume that A = 0 and hence E = 0. The case when λ = 1 (resp. λ = 0) is equivalent (resp., implied by) the case when F = 0, so assume that 0 < λ < 1, and hence, RA = E + F . If μ = 0, then −λτ (F ) = τ (A) − τ (RA ) 0, whence λF = 0, and then A = E + F is already a projection. Thus assume henceforth also that μ > 0. Now consider first

2520


the key case when τ (A) = τ (RA ), i.e., μτ (E) = λτ (F ). In summary, assume that μ > 0, 0 < λ < 1,

and μτ (E) = λτ (F ) > 0.

(27)

If μ = λ, then τ (E) = τ (F ) and then A is the sum of two (equivalent) projections by Lemma 2.6. If μ < λ, then τ (E) > τ (F ), and hence, there is some projection E E with τ (E ) = τ (F ). Then E ∼ F and by Lemma 2.6 there are projections R1 , F1 ∈ M, R1 , F1 E + F with R1 ∼ F1 ∼ F for which (1 + μ)E + (1 − λ)F = R1 + (1 + μ − λ)F1 . Set A1 := A − R1 , E1 := E − E , μ1 := μ, and λ1 := λ − μ. Then E1 F1 = 0, E1 + F1 E + F , and ⎧ μ1 = μ > 0, ⎪ ⎪ ⎪ ⎨ 0 < λ1 = λ − μ < 1, τ (E1 ) = τ (E) − τ (F ), ⎪ ⎪ ⎪ ⎩ τ (F1 ) = τ (F ). Moreover, A1 = (1 + μ1 )E1 + (1 − λ1 )F1 and μ1 τ (E1 ) = μ τ (E) − τ (E ) = (λ − μ)τ (F ) = λ1 τ (F1 ). Thus A1 satisfies the same conditions (27) as A does. Similarly, if μ > λ, and hence, τ (E) < τ (F ), choose a projection F F with τ (F ) = τ (E). By the same argument as above, there are projections R1 , E1 ∈ M, R1 E + F and E1 E + F , with R1 ∼ E1 ∼ E for which (1 + μ)E + (1 − λ)F = R1 + (1 + μ − λ)E1 . Set F1 := F − F , μ1 := μ − λ, λ1 := λ, and A1 := A − R1 . Then, again, we have E1 F1 = 0, E1 + F1 E + F , and ⎧ μ1 = μ − λ > 0, ⎪ ⎪ ⎪ ⎨ 0 < λ = λ < 1, 1 ⎪ τ (E1 ) = τ (E), ⎪ ⎪ ⎩ τ (F1 ) = τ (F ) − τ (E). Moreover, A1 = (1 + μ1 )E1 + (1 − λ1 )F1 and μ1 τ (E1 ) = λ1 τ (F1 ), i.e., here too A1 satisfies the conditions (27). We can thus iterate the construction and find nonzero projections Ek , Fk , Rk ∈ M with Ek Fk = 0, Ek + Fk Ek−1 + Fk−1 E + F , positive operators Ak = Ak−1 − Rk , and scalars μk > 0 and 0 < λk < 1 for which μk τ (Ek ) = λk τ (Fk ), Ak = (1 + μk )Ek + (1 − λk )Fk , and


⎧ μk+1 = μk > 0, ⎪ ⎪ ⎪ ⎨0 < λ k+1 = λk − μk < 1, ⎪ τ (Ek+1 ) = τ (Ek ) − τ (Fk ), ⎪ ⎪ ⎩ τ (Fk+1 ) = τ (Fk ) ⎧ μk+1 = μk − λk > 0, ⎪ ⎪ ⎪ ⎨0 < λ k+1 = λk < 1, ⎪ τ (Ek+1 ) = τ (Ek ), ⎪ ⎪ ⎩ τ (Fk+1 ) = τ (Fk ) − τ (Ek )x

2521

if μk < λk ,

(28)

if μk > λk .

(29)

Thus for every k, A = kj =1 Rj + Ak . This construction terminates if for some k we have μk = λk , in which case Ak is the sum of two projections, and hence, A is the sum of k + 2 projections. Thus assume henceforth that μk = λk for every k. By construction, both sequences τ (Ek ) and τ (Fk ) are monotone non-increasing, and hence, both converge. Let α := lim τ (Ek ) and β := lim τ (Fk ). The sequences μk and λk are also monotone non-increasing. If μk+n = μk for some k and n ∈ N, then by (28), λk+n = λk − nμk . Thus there must be a largest such n, i.e., the sequence μk cannot be eventually constant and, similarly, neither can be the sequence λk . Thus, both inequalities μk < λk and μk > λk must occur for infinitely many indices. Thus it follows from (28) that α = α − β, and it follows from (29) that β = α − β, whence α = β = 0. As a consequence, Ek 1 → 0 and Fk 1 → 0, and hence, Ek → 0 and Fk → 0; this implication is well known, the reader is referred to [8, Exercise 8.7.39]. s s Thus Ak → 0, and hence, A = ∞ j =1 Rj where the convergence is also in the strong operator s

topology. We now consider the remaining case when μ > 0, 0 < λ < 1, and μτ (E) > λτ (F ) 0. Since M is of type II, we can decompose E = E1 + E2 + E3 into the sum of three mutually orthogonal projections with the following traces (μ denotes the integer part of μ): τ (E1 ) =

λ τ (F ) 0, μ

τ (E2 ) =

μτ (E) − λτ (F ) > 0, 1 + μ

τ (E3 ) =

(1 − μ + μ)(μτ (E) − λτ (F )) > 0. μ(1 + μ)

Let A1 := (1 + μ)E1 + (1 − λ)F, A2 := μ − μ E2 + (1 + μ)E3 = ((1 − 1 + μ − μ E2 + (1 + μ)E3 , A3 := 1 + μ E2 . Thus A = A1 + A2 + A3 . If τ (E1 ) = 0, then λτ (F ) = 0, and hence, A1 = F is already a projection. If τ (E1 ) = 0, then A1 satisfies the conditions of (27), and hence, it is a strong sum of projections. If μ ∈ N, then A2 = (1 + μ)E3 is the sum of 1 + μ projections. If μ = μ, then it is easy to verify that also A2 satisfies the conditions of (27), and hence, is a strong sum of

2522


projections. Finally, A3 is always trivially the sum of 1 + μ projections, which concludes the proof. 2 Now we consider positive diagonalizable operators in M, namely, those operators of the form A = k γk Gk where Gk ∈ M are mutually orthogonal projections and γk > 0, and the series, if infinite, converges in the strong operator topology. Theorem 5.2. Let M be a type II factor with trace τ and let A ∈ M be a positive diagonalizable operator. If τ (A+ ) τ (A− ), then A is a strong sum of projections. Proof. To avoid triviality, assume that A = 0. By renaming appropriately the coefficients and using the semifiniteness of M to split the projections into sums of projections with finite trace, we rewrite A as N K A= (1 + μj )Ej + (1 − λi )Fi j =1

i=1

with N, K ∈ N ∪ {0} ∪ {∞], Ej , Fi mutually orthogonal finite projections, μj > 0, and 0 λi < 1 for all j and i, and with the series converging strongly if N or K are infinite. Again, we use the convention that if N or K are zero then A is the sum of only one series. Since {(1 − λi )Fi | λi = 0} is already a projection, we can further assume without loss of generality that λi > 0 for all i. Then A+ =

N

μj E j ,

A− =

j =1

K

λi Fi ,

and hence,

N

μj τ (Ej )

j =1

i=1

In particular, N > 0. Assume K > 0. Then λ1 τ (F1 )

N

j =1 N

K

λj τ (Fj ).

j =1

μj τ (Ej ). Since M is of type

II, we can find projections Ej 1 Ej such that λ1 τ (F1 ) = j =1 μj τ (Ej 1 ). Then decompose F1 = N j =1 Fj 1 into mutually orthogonal projections so that λ1 τ (Fj 1 ) = μj τ (Ej 1 ) for every j . If K > 1 we have K i=2

λi τ (Fi )

N

μj τ (Ej − Ej 1 ),

j =1

and hence, we can iterate the process. Thus for every i and j we decompose Fi = N j =1 Fj i into mutually orthogonal projections and further find mutually orthogonal projections Ej i Ej so that λi τ (Fj i ) = μj τ (Ej i ). Set Ej o := Ej − K i=1 Ej i . Then A=

N N K (1 + μj )Ej i + (1 − λi )Fj i + (1 + μj )Ej o . j =1 i=1

j =1

By Lemma 5.1, each summand (1 + μj )Ej i + (1 − λi )Fj i and (1 + μj )Ej o is a strong sum of projections, and hence, so is A. In the case that K = 0, A = N j =1 (1 + μj )Ej , and hence, it is also the strong sum of projections by the same reasoning. 2


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As the following examples show, the condition that A is diagonalizable is not necessary for A to be a strong sum of projections. Example 5.3. Let M be a type II 1 factor, let P ∈ M be a projections with P ∼ P ⊥ , let A (resp., B) be a masa in MP (resp., in MP ⊥ ). By properly scaling the spectral resolution of a generator of A we can find a monotone increasing strongly continuous net of projections {Et }t∈[0, 1 ] 2 in A with τ (Et ) = t. (i) Assume that A and B are conjugate in M, and hence there is a selfadjoint unitary U ∈ M for which U AU ∗ = B. Define

Et := I − U E1−t U

∗

for t ∈

1 ,1 2

1

2

1 (1 + t) dEt +

and A = 0

t dEt . 1 2

Then {Et }t∈[0,1] is flag, namely a monotone increasing strongly continuous net of projections with τ (Et ) = t for all t ∈ [0, 1] and A is not diagonalizable (in fact, it has no eigenvalues). Furthermore, 0 A 32 I 2I , RA = I , and it is easy to verify that 1

2

∗

U AU =

(1 + t) d U Et U ∗ +

1

0

1 2 1

2

1 (1 + t) d(I − E1−t ) +

= 0

t d(I − E1−t ) 1 2

1

1

2 (2 − t) dEt +

=

t d U Et U ∗

2 − (1 + t) dEt

0

1 2

= 2I − A. Thus by Proposition 2.1, A is the sum of two projections. (ii) Assume that A and B are not conjugate in M. Such a case can be easily obtained by choosing P so that M ∼ MP ∼ MP ⊥ , choosing two non-conjugate masas Ao and Bo in M (e.g., a Cartan masa and a singular one) and defining A and B to be the compressions of Ao and Bo to MP and MP ⊥ respectively. Complete {Et }t∈[0, 1 ] to be a flag in M by defining Et := P + Ft 2

for t ∈ ( 12 , 1] where {Ft }t∈[ 1 ,1] is an arbitrary monotone increasing strongly continuous net of 2

projections in B with τ (Ft ) = t − 12 . Define as in (i) 1

2 A=

1 (1 + t) dEt +

0

t dEt . 1 2

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Again, A is not diagonalizable, in fact it has no, 0 A 32 I 2I , and RA = I . By Proposition 2.10, A is the sum of two projections if and only if U AU ∗ = 2I − A for some unitary U ∈ M. Reasoning as in the proof of Proposition 2.10, it is simple to see that if such a unitary existed, we would have B = U AU ∗ and hence A and B would be conjugate, against the assumption. Thus A cannot be the sum of two projections in M. However we do not know whether A is a strong sum of projections in M or not. Question 5.4. Can the condition that A is diagonalizable be removed from Theorem 5.2? 6. The infinite case In this section we assume that M is an infinite factor, i.e., of type I∞ , type II∞ , or type III. The following lemma is the key to the proof of Theorem 1.1 in this case. are mutually orthogonal Lemma 6.1. Let A = ∞ j =1 (1 + μj )Ej + (1 − λ)F where {Ej , F } equivalent projections in M, μj > 0, 0 < λj 1, and sup μj < ∞. If ∞ j =1 μj = ∞, then A is a strong sum of projections in M. Proof. Let n1 1 be the smallest integer for which ∞ j =1 μj = ∞. Set α1 :=

λ 1 −1 λ − jn=1 μj

if n1 = 1, if n1 > 1,

n1

j =1 μj

λ. Such an integer exists because

and 1 − β1 := μn1 − α1 − μn1 − α1

where x denotes the integer part of x. Then μn1 α1 , 0 < α1 1, and 0 < β1 1. The positive operator D1 :=

n 1 −1

(1 + μj )Ej + 1 + α1 + μn1 − α1 En1 + (1 − λ)F

j =1

is a linear combination of n :=

n1 + 1 if λ = 1, n1 if λ = 1,

mutually orthogonal equivalent projections in M and the sum of their coefficients is k1 := n1 + 1 + μn1 − α1 . Since k1 ∈ N and k1 n, by Lemma 2.6(ii), D1 is the sum of k1 (equivalent) projections. Next, we apply the same construction to the “remainder” A − D1 =

∞

(1 + μj )Ej + (1 − β1 )En1

j =n1 +1

where now β1 plays the role of λ and En1 the role of F . Iterating we find an increasing sequence of indices nk and two sequences of positive numbers 0 < αk , βk 1 with μnk αk and 1 − βk = μnk − αk − μnk − αk . Then the positive operator


Dk :=

n k −1

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(1 + μj )Ej + 1 + αk + μnk − αk Enk + (1 − βk−1 )Enk−1

j =nk−1 +1

is by Lemma 2.6 the sum of finitely many (equivalent) projections. But then A−

k j =1

Dj =

∞ j =nk +1

(1 + μj )Ej + (1 − βk )Enk → 0 s

(30)

because the projections {Ej } are mutually orthogonal. Thus A = ∞ j =1 Dj , where the series converges in the strong operator topology. Since each Dk is the sum of projections, so is A. 2 If M is of type I and all projections Ej and F have rank-one, then we can relax the condition that they are mutually orthogonal. Indeed, orthogonality is not necessary to conclude that each positive finite rank operator Dk is the sum of projections (see Corollary 2.6 and also Lemma 2.2), and assuming strong convergence of the series ∞ (1 + μj )Ej is sufficient to guarantee that j =1 ∞ j =nk +1 (1 + μj )Ej + (1 − βk )Enk → 0 in (30). Thus we have: s

Lemma 6.2. Let A = ∞ μj )Ej + (1 − λ)F where Ej , F ∈ B(H ) are rank-one projecj =1 (1 + tions, μj > 0, 0 < λj 1, and ∞ j =1 (1 + μj )Ej converges in the strong operator topology. If ∞ μ = ∞, then A is a strong sum of projections. j =1 j Proposition 6.3. Let A ∈ M + and assume that there is some μ > 0 for which the spectral projection χA [1 + μ, ∞) is infinite. Then A is a strong sum of projections. Proof. Let E := χA [1 + μ, ∞), B := A − (1 + μ)E, and let A be a masa containing A. Then B ∈ A. By [18, Corollary 2.23], B can be decomposed into a norm converging series B= ∞ i=1 (1 − λi )Qi with 0 λi < 1 and with the projections Qi ∈ A. (In fact we can choose 1 − λi = 2−i , but we do not need this fact here.) Some or all of the projections Qi can be zero. Since M is infinite and E ∈ A, by [7, Theorem 3.18] (see also [11, Corollary 31]), we can decompose E = ∞ i=1 Ei into a sum of infinite projections Ei ∈ A. Let Ai := (1 + μ)Ei + (1 − λi )Qi . Then A = ∞ i=1 Ai . Thus it suffices to prove that Ai is a strong sum of projections for each i. Using the fact that Ei , Qi ∈ A, and hence, they commute, it follows that Ai is diagonalizable as Ai = (1 + μ)(Ei − Ei Qi ) + (2 + μ − λi )Ei Qi + (1 − λi )(Qi − Ei Qi ). Since Ei is infinite, at least one of the two orthogonal projections Ei − Ei Qi and Ei Qi must be infinite. Assume that Ei − Ei Qi is infinite. If Qi = 0, then Ai = (1 + μ)Ei and the conclusion follows from Lemma 6.1 by further decomposing Ei into a sum of infinitely many mutually orthogonal equivalent projections. Thus assume that Qi = 0 and decompose 2 + μ − λi = m n=1 (1 − γn ) into the sum of finitely (n) many numbers 0 < 1 − γn < 1. Next, decompose Ei − Ei Qi = m+1 into the sum of n=1 Ei (n) m + 1 mutually orthogonal equivalent (infinite) projections Ei . Then further decompose each (n) (n) projection Ei into a sum of infinitely many mutually orthogonal projections Eij with

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(n)

Eij ∼ Thus Ei − Ei Qi =

m+1 ∞ n=1

Ei Qi

for 1 n m,

Qi − Ei Qi

for n = m + 1.

(n) j =1 Eij .

Define

⎧ (n) ⎨ ∞ j =1 (1 + μ)Eij + (1 − γn )Ei Qi (n) Bi = ⎩ ∞ (1 + μ)E (m+1) + (1 − λ )(Q − E Q ) i i i i j =1 ij

for 1 n m, for n = m + 1.

(n) (n) are strong sums of By construction, Ai = m+1 n=1 Bi . By Lemma 6.1, all the operators Bi projections and hence so is Ai . Finally, the case when Ei Qi is infinite is similar and is left to the reader. 2 An immediate consequence of this proposition is the sufficient condition in Theorem 1.1(iii) for the type III case. Corollary 6.4. Let M be a type III factor, A ∈ M + , and either A be a projection or A satisfy A > 1. Then A is a strong sum of projections. Proof. If A > 1, then there is some μ > 0 for which the spectral projection χA [1 + μ, ∞) is nonzero and hence infinite. Then A is a strong sum of projections by Proposition 6.3. 2 Remark 6.5. (i) The condition that χA [1 + μ, ∞) is infinite for some μ > 0 is equivalent to the condition Aess > 1 where Aess is the essential norm, i.e., the norm in the quotient M/K, where K is the norm closed ideal generated by the finite projections of M. If M = B(H ), then K is the ideal of compact operators K(H ) on H and Proposition 6.3 provides another proof of [5, Theorem 2 ] stating that if Aess > 1, then A is a strong sum of projections. If M is of type II∞ , K is the ideal of compact operators relative to M introduced by Sonis [17] and Breuer [4] (see also [10]). If M is of type III, then K = {0} and Aess = A. (ii) If M is semifinite and A ∈ K + is a strong sum of projections then τ (RA ) < ∞. Proof. (ii) It is well known that τ (χA (γ , ∞)) < ∞ for every γ > 0 (e.g., see [10, Theorem 1.3]). In particular, τ (χA (1, ∞)) < ∞, whence τ (A+ ) < ∞. Thus it follows from Theorem 3.3 that τ (A− ) < ∞. But A− 12 χA (0, 12 ], hence τ (χA (0, 12 )) < ∞. From this follows that τ (χA (0, ∞)) = τ (χA (0, 12 ]) + τ (χA ( 12 , ∞)) < ∞. 2 An alternative proof of (ii) for the case when M = B(H ) is that a strongly converging series of rank-one projections that converges to a compact operator must converge uniformly and hence be finite. We can now prove the last part of the sufficiency in Theorem 1.1. Theorem 6.6. Let M be type I∞ or type II∞ . If τ (A+ ) = ∞. Then A is a strong sum of projections.


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Proof. By Proposition 6.3, we just need to consider the case when χA [1 + μ, ∞) is finite for every μ > 0. Let A be a masa of M containing A. Let E1 := χA [2, ∞) = χA [2, A] and set 1 Ej := χA [1 + j1 , 1 + j −1 ) for j > 1. Then τ (Ej ) < ∞ for all j . Since ∞ 1 j =1

j

∞ Ej A+ A − 1 E1 + j =2

1 E , j −1 j

∞ 1 1 we see that (A − 1)τ (E1 ) + ∞ j =2 j −1 τ (Ej ) = ∞. Then also j =1 j τ (Ej ) = ∞. Further∞ 1 + + more, AχA (1, A] − j =1 (1 + j )Ej ∈ A , AχA [0, 1] ∈ A , and

∞ ∞ 1 1 1+ 1+ Ej + AχA [0, 1] + AχA 1, A − Ej . A= j j j =1

j =1

Now we consider separately the case when M = B(H ) and when M is of type II. If M = B(H ), first decompose by [18, Corollary 2.23] the positive operator

∞ 1 1+ B := AχA [0, 1] + AχA 1, A − Ej j j =1

into a norm converging series B = ∞ i=1 (1 − λi )Qi with 0 < λi 1 and with the projections Qi ∈ A. Some or all of the projections Qi can be zero. Then, further decompose the projections Ej and Qi into rank-one projections. Relabel the ensuing sequence of coefficients 1 + j1 (resp., 1 − λi ) repeated according to the multiplicity of the projections as 1 + μj (resp. 1 − λi ). To take into account the case when there are only finitely many nonzero projections Qi and they all have finite rank, allow λi = 1. Thus A=

∞ ∞ (1 + μj )Ej + (1 − λi )Qi j =1

i=1

where all the projections Ej and Qi have rank-one, μ j > 0, 0 < λi 1 for all i and j , both series converge in the strong operator topology, and ∞ = ∞. Now further decompose j =1 μj ∞ N = i=1 Λi into infinite disjoint subsets Λi so that for each i, j ∈Λi μj = ∞. Then A=

∞ i=1

(1 + μj )Ej + (1 − λi )Qi

j ∈Λi

and each summand being the strong sum of projection by Lemma 6.2, so is A. Now assume that M is of type using [18, Corollary 2.23] decompose separately II and again 1 AχA [0, 1] and AχA (1, A] − ∞ (1 + )E j into two norm converging series of scalar mulj =1 j tiples 1 − λi of projections Qi ∈ A. If a projection Qi in the series decomposing AχA [0, 1] is infinite, by the semifiniteness of M we can further decompose it into a strongly converging sum

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of mutually orthogonal finite projections in M. These projections are not necessarily in A, however, being majorized by χA [0, 1], they are all orthogonal to and hence commute with all the projections Ej . 1 Every projection Qi in the series decomposing AχA (1, A] − ∞ j =1 (1 + j )Ej is in A, is ma∞ jorized by χA (1, A] = j =1 Ej , and hence is the sum Qi = ∞ j =1 Qi Ej of finite projections Qi Ej which belong to A and hence commute with all the projections Ej . Therefore, A=

∞ ∞ 1 1+ Ej + (1 − λi )Qi j j =1

i=1

where for all i, 0 λi < 1 and Qi are finite projections that commute with each Ej . ∞ 1 Since τ (E ) = ∞, we can choose an increasing sequence of indices ni for which ni+1 j1=1 j j ni+1 ∞ 1 i=1 Ai j =ni +1 j τ (Ej ) λi τ (Qi ) and let Ai := j =ni +1 (1+ j )Ej +(1−λi )Qi . Then A = and

ni+1

τ (Ai ) =

j =ni +1

τ

ni+1 1 τ Ej − λi τ (Qi ) τ Ej + τ (Qi ) + j

ni+1

Ej

j =ni +1

∨ Qi

j =ni +1

= τ (RAi ). Since Ai is diagonalizable because Qi commutes with all Ej , Ai is a strong sum of projections by Theorem 5.2 and hence so is A. 2 Note added in proof In subsequent work, the authors have proved that if A is a positive element in a type III factor such that A > 1 or is a positive element in a type I∞ , or type II ∞ factor, or in the multiplier algebra of a σ -unital, nonunital purely infinite simple C ∗ -algebra satisfying Ae > 1, then A is the sum of finitely many projections. References [1] J. Antezana, P. Massey, M. Ruiz, D. Stojanoff, The Schur–Horn Theorem for operators and frames with prescribed norms and frame operator, Illinois J. Math. 51 (2) (2007) 537–560. [2] M. Argerami, P. Massey, A Schur–Horn Theorem in type II 1 factors, Indiana Univ. Math. J. 56 (5) (2007) 2051– 2060. [3] M. Argerami, P. Massey, Towards the Carpenter’s theorem, Proc. Amer. Math. Soc., in press. [4] M. Breuer, Fredholm theories in von Neumann algebras, I and II, Math. Ann. 178 (1968) 243–254, Math. Ann. 180 (1969) 313–325. [5] K. Dykema, D. Freeman, K. Kornelson, D. Larson, M. Ordower, E. Weber, Ellipsoidal tight frames and projection decompositions of operators, Illinois J. Math. 48 (2004) 477–489. [6] P. Fillmore, On sums of projections, J. Funct. Anal. 4 (1969) 146–152. [7] R.V. Kadison, Diagonalizing matrices, Amer. J. Math. 106 (1984) 1451–1468.


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[8] R.V. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras, II, Pure Appl. Math. (N. Y.), Academic Press, London, 1985. [9] R. Kadison, The Pythagorean Theorem II: The infinite discrete case, Proc. Natl. Acad. Sci. USA 99 (8) (2002) 5217–5222. [10] V. Kaftal, On the theory of compact operators in von Neumann algebras, I, Indiana Univ. Math. J. 26 (1977) 447– 457. [11] V. Kaftal, Type decomposition for von Neumann algebra embeddings, J. Funct. Anal. 98 (1) (1991) 169–193. [12] V. Kaftal, P.W. Ng, S. Zhang, Projection decomposition in multiplier algebras, preprint. [13] V. Kaftal, G. Weiss, An infinite dimensional Schur–Horn theorem and majorization theory with applications to operator ideals, preprint. [14] V. Kaftal, D. Larson, Perturbations of finite frames and projections, preprint. [15] V. Kaftal, D. Larson, Admissible sequences for positive operators, preprint. [16] K. Kornelson, D. Larson, Rank-one decompositions of operators and construction of frames, in: Wavelets, Frames and Operator Theory, in: Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 203–214. [17] M.G. Sonis, On a class of operators in von Neumann algebras with Segal measures, Math. USSR Sbornik 13 (1971) 3, 344–359. [18] S. Stratila, L. Zsido, Lectures on Von Neumann Algebras, Abacus Press, Bucharest, 1979. [19] M. Takesaki, Theory of Operator Algebras, I, Springer-Verlag, New York, 1979.


Quantum group of orientation-preserving Riemannian isometries Jyotishman Bhowmick 1 , Debashish Goswami ∗,2 Stat.-Math. Unit, Kolkata Centre, Indian Statistical Institute, 203, B. T. Road, Kolkata 700 108, India Received 21 November 2008; accepted 8 July 2009 Available online 21 July 2009 Communicated by S. Vaes

Abstract We formulate a quantum group analogue of the group of orientation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly R-twisted and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any ‘good’ Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffel-type deformation as well as the 2 are discussed. equivariant spectral triples on SU μ (2) and Sμ,c © 2009 Elsevier Inc. All rights reserved. Keywords: Compact quantum group; Quantum isometry groups; Spectral triples

1. Introduction Motivated by the formulation of quantum automorphism groups by Wang (following Alain Connes’ suggestion) [29,31], and the study of their properties by a number of mathematicians (see, e.g. [1,2,5,34] and references therein), we have introduced in an earlier article [17] a quantum group analogue of the group of Riemannian isometries of a classical or noncommutative manifold. In a follow-up article [3] we have also computed these quantum groups for a number * Corresponding author.

E-mail address: [email protected] (D. Goswami). 1 Support from the National Board of Higher Mathematics, India is gratefully acknowledged. 2 The author gratefully acknowledges support obtained from the Indian National Academy of Sciences through the

grants for a project on ‘Noncommutative Geometry and Quantum Groups’. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.006

J. Bhowmick, D. Goswami / Journal of Functional Analysis 257 (2009) 2530–2572

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of examples. However, our formulation of quantum isometry groups in [17] had a major drawback from the viewpoint of noncommutative geometry since it needed a ‘good’ Laplacian to exist. In noncommutative geometry it is not always easy to verify such an assumption about the Laplacian, and thus it would be more appropriate to have a formulation in terms of the Dirac operator directly. This is what we aim to achieve in the present article. The group of Riemannian isometries of a compact Riemannian manifold M can be viewed as the universal object in the category of all compact metrizable groups acting on M, with smooth and isometric action. Moreover, assume that the manifold has a spin structure (hence in particular orientable, so we can fix a choice of orientation) and D denotes the conventional Dirac operator acting as an unbounded self-adjoint operator on the Hilbert space H of square integrable spinors. Then, it can be proved that a group action on the manifold lifts as a unitary representation on the Hilbert space H which commutes with D if and only if the action on the manifold is an orientation preserving isometric action. Therefore, to define the quantum analogue of the group of orientation-preserving Riemannian isometry group of a possibly noncommutative manifold given by a spectral triple (A∞ , H, D), it is reasonable to consider a category Q of compact quantum groups having unitary (co-)representation, say U , on H, which commutes with D, and the action on B(H) obtained by conjugation maps A∞ into its weak closure. A universal object in this category, if it exists, should define the ‘quantum group of orientation preserving Riemannian isometries’ of the underlying spectral triple. Unfortunately, even in the finite-dimensional (but with noncommutative A) situation this category may often fail to have a universal object, as will be discussed later. It turns out, however, that if we fix any suitable faithful functional on B(H) (to be interpreted as the choice of a ‘volume form’) then there exists a universal object in the subcategory of Q obtained by restricting the object-class to the quantum group actions which also preserve the given functional. The subtle point to note here is that unlike the classical group actions on B(H) which always preserve the usual trace, a quantum group action may not do so. In fact, it was proved by one of the authors in [16] that given an object (Q, U ) of Q (where Q is the compact quantum group and U denotes its unitary co-representation on H), we can find a positive invertible operator R in H so that the given spectral triple is R-twisted in the sense of [16] and the corresponding functional τR (which typically differs from the usual trace of B(H) and can have a nontrivial modularity) is preserved by the action of Q. This makes it quite natural to work in the setting of twisted spectral data (as defined in [16]). Motivated by the ideas of Woronowicz and Soltan [26,35], we actually consider a bigger category. The group of orientation-preserving Riemannian isometries of a classical manifold, viewed as a compact metrizable space (forgetting the group structure), can be seen to be the universal object of a category whose object-class consists of subsets (not necessarily subgroups) of the set of such isometries of the manifold. Then it can be proved that this universal compact set has a canonical group structure. A natural quantum analogue of this has been formulated by us, called the category of ‘quantum families of smooth orientation preserving Riemannian isometries’. The underlying C ∗ -algebra of the quantum group of orientation preserving isometries (whenever exists) has been identified with the universal object in this bigger category and moreover, it is shown to be equipped with a canonical coproduct making it into a compact quantum group. We discuss a number of examples, covering both the examples coming from Rieffel-type de2 . formation as well as the equivariant spectral triples constructed recently on SU μ (2) and Sμ,c It may be relevant to point out here that it was not clear whether one could accommodate the 2 in the framework of our previous work on quantum isomespectral triples on SU μ (2) and Sμ,c try groups, since it is very difficult to give a nice description of the space of ‘noncommutative’ forms and the Laplacian for these examples. However, the present formulation in terms of the

2532


Dirac operator makes it easy to accommodate them, and we have been able to identify Uμ (2) and SOμ (3) as the universal quantum group of orientation preserving isometries for the spec2 2 respectively (the computations for Sμ,c have been presented tral triples on SU μ (2) and Sμ,c in [4]). For readers’ convenience, let us briefly sketch the plan of the paper. Section 2 is devoted to the definition and existence of quantum group of orientation preserving isometries, which begins with a characterization of the group of such isometries in the classical case (Section 2.1), motivating the quantum formulation elaborated in Section 2.2. The other two subsections of Section 2 discuss sufficient conditions for ensuring a C ∗ -action of the quantum group of (orientation preserving) isometries (2.3) and for the existence of universal object without fixing a volume form (2.4). Then in Section 3 we study the connections of our approach with that of [17]. Section 4 is devoted to the explicit examples and computations, and in the last section we sketch a general principle of computing quantum group of orientation preserving isometries of spectral triples obtained by Rieffel deformation. We conclude this section with an important remark about the use of the phrase ‘orientationpreserving’ in our terminology. Let us make it clear that by a ‘classical spectral triple’ we always mean the spectral triple obtained by the Dirac operator on the spinors (so, in particular, manifolds are assumed to be compact Riemannian spin manifolds), and not just any spectral triple on the commutative algebra C ∞ (M). This is absolutely crucial in view of the fact that the Hodge Dirac operator d + d ∗ on the L2 -space of differential forms also gives a spectral triple of compact type on any compact Riemannian (not necessarily with a spin structure) manifold M, but the action of the full isometry group ISO(M) (and not just the subgroup of orientation preserving isometries ISO+ (M), even when M is orientable) lifts to a canonical unitary representation on this space commuting with d +d ∗ . In fact, the category of groups acting on M such that the action comes from a unitary representation commuting with d + d ∗ , has ISO(M), and not ISO+ (M), as its universal object. So, one must stick to the Dirac operator on spinors to obtain the group of orientation preserving isometries in the usual geometric sense. This also has a natural quantum generalization, as we shall see in Section 3. 2. Definition and existence of the quantum group of orientation-preserving isometries 2.1. The classical case We first discuss the classical situation clearly, which will serve as a motivation for our quantum formulation. We begin with a few basic facts about topologizing the space C ∞ (M, N ) where M, N are smooth manifolds. Let Ω be an open set of Rn . We endow C ∞ (Ω) with the usual Fréchet topology coming from uniform convergence (over compact subsets) of partial derivatives of all orders. this topology, so is a Polish space in particular. The space C ∞ (Ω) is complete with respect to Moreover, by the Sobolev imbedding theorem, k0 Hk (Ω) = C ∞ (Ω) as a set, where Hk (Ω) denotes the k-th Sobolev space. Thus, C ∞ (Ω) has also the Hilbertian seminorms coming from the Sobolev spaces, hence the corresponding Fréchet topology. We claim that these two topologies on C ∞ (Ω) coincide. Indeed, the inclusion map from C ∞ (Ω) into k Hk (Ω), is continuous and surjective, so by the open mapping theorem for Fréchet space, the inverse is also continuous, proving our claim.


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Given two second countable smooth manifolds M, N , we shall equip C ∞ (M, N ) with the weakest locally convex topology making C ∞ (M, N ) φ → f ◦ φ ∈ C ∞ (M) Fréchet continuous for every f in C ∞ (N ). For topological or smooth fibre or principal bundles E, F over a second countable smooth manifold M, we shall denote by Hom(E, F ) the set of bundle morphisms from E to F . We remark that the total space of a locally trivial topological bundle such that the base and the fibre spaces are locally compact Hausdorff second countable must itself be so, hence in particular Polish. In particular, if E, F are locally trivial principal G-bundles over a common base, such that the (common) base as well as structure group G are locally compact Hausdorff and second countable, then Hom(E, F ) is a Polish space. We need a standard fact, stated below as a lemma, about the measurable lift of Polish space valued functions. Polish spaces (complete separable Lemma 2.1. Let M be a compact metrizable space, B, B → B. Then any continuous map metric spaces) such that there is an n-covering map Λ : B which is Borel measurable and Λ ◦ ξ˜ = ξ . In particular, ξ : M → B admits a lifting ξ˜ : M → B and B are topological bundles over M, with Λ being a bundle map, any continuous section if B of B admits a lifting which is a measurable section of B. The proof is a trivial consequence of the selection theorem due to Kuratowski and RyllNardzewski (see [27, Theorem 5.2.1]). We shall now give an operator-theoretic characterization of the classical group of orientationpreserving Riemannian isometries, which will be the motivation of our definition of its quantum counterpart. Let M be a compact Riemannian spin manifold, with a fixed choice of orientation. We note that (see, for example, [14]) the spinor bundle S is the associated bundle of a principal Spin(n)-bundle, say P , on M (n = dimension of M), which has a canonical 2-covering bundle-map Λ from P to the frame-bundle F (which is an SO(n)-principal bundle), such that locally Λ is of the form (idM ⊗ λ), where λ : Spin(n) → SO(n) is the canonical 2-covering group homomorphism. Let f be a smooth orientation preserving Riemannian isometry of M, and con = Hom(P , f ∗ (P )) (where Hom denotes the set sider the bundles E = Hom(F, f ∗ (F )) and E of bundle maps). We view df as a section of the bundle E in the natural way. By Lemma 2.1 : M → E, which is a measurable section of E. Using this, we we obtain a measurable lift df define a map on the space of measurable section of S = P ×Spin(n) n (where n is as in [14]) as follows: given a (measurable) section ξ of S, say of the form ξ(m) = [p(m), v], with p(m) (f −1 (m))(p(f −1 (m))), v]. Note that sections in Pm , v in n , we define U ξ by (U ξ )(m) = [df of the above form constitute a total subset in L2 (S), and the map ξ → U ξ is clearly a densely defined linear map on L2 (S), whose fibre-wise action is unitary since the Spin(n) action is so on n . Thus it extends to a unitary U on H = L2 (S). Any such U , induced by the map f , will be denoted by Uf (it is not unique since the choice of the lifting used in its construction is not unique). Theorem 2.2. Let M be a compact Riemannian spin manifold (hence orientable, and fix a choice of orientation) with the usual Dirac operator D acting as an unbounded self-adjoint operator on the Hilbert space H of the square integrable spinors, and let S denote the spinor bundle, with Γ (S) being the C ∞ (M) module of smooth sections of S. Let f : M → M be a smooth oneto-one map which is a Riemannian orientation-preserving isometry. Then the unitary Uf on H

2534


commutes with D and Uf Mφ Uf∗ = Mφ◦f , for any φ in C(M), where Mφ denotes the operator of multiplication by φ on L2 (S). Moreover, when the dimension of M is even, Uf commutes with the canonical grading γ on L2 (S). Conversely, suppose that U is a unitary on H such that U D = DU and the map αU (X) = U XU −1 for X in B(H) maps A = C(M) into L∞ (M) = A , then there is a smooth one-to-one orientation-preserving Riemannian isometry f on M such that U = Uf . We have the same result in the even case, if we assume furthermore that U γ = γ U. Proof. From the construction of Uf , it is clear that Uf Mφ Uf−1 = Mφ◦f . Moreover, since the Dirac operator D commutes with the Spin(n)-action on S, we have Uf D = DUf on each fibre, hence on L2 (S). In the even-dimensional case, it is easy to see that the Spin(n) action commutes with γ , hence Uf does so. For the converse, first note that αU is a unital ∗-homomorphism on L∞ (M, dvol) and thus must be of the form ψ → ψ ◦ f for some measurable f . We claim that f must be smooth. Fix any smooth g on M and consider φ = g ◦ f . We have to argue that φ is smooth. Let δD denote the generator of the strongly continuous one-parameter group of automorphism βt (X) = eitD Xe−itD on B(H) (with respect to the weak operator From the assumption that D and topology, say). n ) into itself and since C ∞ (M) ⊂ D, U commute it is clear that αU maps D := n1 Dom(δD we conclude that αU (Mφ ) = Mφ◦g belongs to D. We claim that this implies the smoothness of φ. Let m in M and choose a local chart (V , ψ) at m, with the coordinates (x1 , . . . , xn ), such that Ω= ψ(V ) ⊆ Rn has compact closure, S|V is trivial and D has the local expression D = i nj=1 μ(ej )∇j , where ∇j = ∇ ∂ denotes the covariant derivative (with respect to ∂xj

the canonical Levi-Civita connection) operator along the vector field

∂ 2 ∂xj on L (Ω) and μ(v) L∞ (Ω) ⊆ L2 (Ω) and it

denotes the Clifford multiplication by a vector v. Now, φ ◦ ψ −1 ∈ is structure of D that [D, Mφ ] has the local expression easy to observe from the above local n −1 is in Dom(d . . . d ) iM ⊗ μ(e ). Thus, the fact M ∈ ∂ j φ j1 jk j n1 Dom(δD ) implies φ ◦ ψ φ ∂xj

for every integer tuples (j1 , . . . , jk ), ji ∈ {1, . . . , n}, where dj :=

∂ ∂xj

. In other words, φ ◦ ψ −1

is in H k (Ω) for all k 1, where H k (Ω) denotes the k-th Sobolev space on Ω (see [24]). By Sobolev’s Theorem (see, for example [24, Corollary 1.21, p. 24]) it follows that φ ◦ ψ −1 is in C ∞ (Ω). We note that f is one-to-one as φ → φ ◦ f is an automorphism of L∞ . Now, we shall show that f is an isometry of the metric space (M, d), where d is the metric coming from the Riemannian structure, and we have the explicit formula (see [8]) d(p, q) =

sup

φ∈C ∞ (M),[D,Mφ ]1

φ(p) − φ(q).

Since U commutes with D, we have [D, Mφ◦f ] = [D, U Mφ U ∗ ] = U [D, Mφ ]U ∗ = [D, Mφ ] for every φ, from which it follows that d(f (p), f (q)) = d(p, q). Finally, f is orientation preserving if and only if the volume form (say ω), which defines the choice of orientation, is preserved by the natural action of df , that is, (df ∧ · · · ∧ df )(ω) = ω. This will follow from the explicit description of ω in terms of D, given by (see [28, p. 26], also see [10]) ω(φ0 dφ1 . . . dφn ) = τ Mφ0 [D, Mφ1 ] . . . [D, Mφn ] ,


2535

where φ0 , . . . , φn belong to C ∞ (M), = 1 in the odd case and = γ (the grading operator) in the even case and τ denotes the volume integral. In fact, τ (X) = Limt→0+ Tr(e

−tD 2 X) 2

Tr(e−tD )

, where

Lim denotes a suitable Banach limit, which implies τ (U XU ∗ ) = τ (X) for all X in B(H) (using the fact that D and U commute). Thus, ω φ0 ◦ f d(φ1 ◦ f ) . . . d(φn ◦ f ) = τ U Mφ0 U ∗ U [D, Mφ1 ]U ∗ . . . U [D, Mφn ]U ∗ = τ U Mφ0 [D, Mφ1 ] . . . [D, Mφn ]U ∗ = τ Mφ0 [D, Mφ1 ] . . . [D, Mφn ] = ω(φ0 dφ1 . . . dφn ).

2

Now we turn to the case of a family of maps. We now prove a useful general fact. Let us introduce a few notations at this point. For a possibly non-unital C ∗ -algebra B we will denote by M(B), its multiplier algebra. For a Hilbert B module E, the set of adjointable B-linear maps on E and the set of all compact B-linear maps will be denoted by L(E) and K(E) respectively. We also note the ∗-isomorphism M(K(H) ⊗ B) ∼ = L(H ⊗ B). Using this, we often identify an element V of M(K(H) ⊗ B) with the map from H to H ⊗ B which sends a vector ξ of H to V (ξ ⊗ 1B ) ∈ H ⊗ B. Lemma 2.3. Let A be a C ∗ -algebra and ω, ωj (j = 1, 2, . . .) be states on A such that ωj → ω in the weak-∗ topology of A∗ . Then for any separable Hilbert space H and for all Y in M(K(H) ⊗ A), we have (id ⊗ ωj )(Y ) → (id ⊗ ω)(Y ) in the strong operator topology. Proof. Clearly, (id ⊗ ωj )(Y ) → (id ⊗ ω)(Y ) (in the strong operator topology) for all Y in Fin(H) ⊗alg A, where Fin(H) denotes the set of finite rank operators on H. Using the strict density of Fin(H) ⊗alg A in M(K(H) ⊗ A), we choose, for a given Y ∈ M(K(H) ⊗ A), ξ ∈ H with ξ = 1, and δ > 0, an element Y0 ∈ Fin(H) ⊗alg A such that (Y − Y0 )(|ξ ξ | ⊗ 1) < δ. Thus, (id ⊗ ωj )(Y )ξ − (id ⊗ ω)(Y )ξ = (id ⊗ ωj ) Y |ξ ξ | ⊗ 1 ξ − (id ⊗ ω) Y |ξ ξ | ⊗ 1 ξ (id ⊗ ωj ) Y0 |ξ ξ | ⊗ 1 ξ − (id ⊗ ω) Y0 |ξ ξ | ⊗ 1 ξ + 2 (Y − Y0 ) |ξ ξ | ⊗ 1 (id ⊗ ωj ) Y0 |ξ ξ | ⊗ 1 ξ − (id ⊗ ω) Y0 |ξ ξ | ⊗ 1 ξ + 2δ, from which it follows that (id ⊗ ωj )(Y ) → (id ⊗ ω)(Y ) in the strong operator topology.

2

We are now ready to state and prove the operator-theoretic characterization of ‘set of orientation preserving isometries’. Theorem 2.4. Let X be a compact metrizable space and ψ : X × M → M is a map such that ψx defined by ψx (m) = ψ(x, m) is a smooth orientation preserving Riemannian isometry and x → ψx ∈ C ∞ (M, M) is continuous with respect to the locally convex topology of C ∞ (M, M) mentioned before.

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Then there exists a (C(X)-linear) unitary Uψ on the Hilbert C(X)-module H ⊗ C(X) (where H = L2 (S) as in Theorem 2.2) such that for all x belonging to X, Ux := (id ⊗ evx )Uψ is a unitary of the form Uψx on the Hilbert space H commuting with D and Ux Mφ Ux−1 = Mφ◦ψ −1 . x If in addition, the manifold is even-dimensional, then Uψx commutes with the grading operator γ . Conversely, if there exists a C(X)-linear unitary U on H ⊗ C(X) such that Ux := (id ⊗ evx )(U ) is a unitary commuting with D for all x, (and Ux commutes with the grading operator γ if the manifold is even-dimensional) and (id ⊗ evx )αU (L∞ (M)) ⊆ L∞ (M) for all x in X, then there exists a map ψ : X × M → M satisfying the conditions mentioned above such that U = Uψ .

= X × F and P

= X × P over X × M, with fibres at (x, m) Proof. Consider the bundles F isomorphic with Fm and Pm respectively, and where F and P are respectively the bundles of orthonormal frames and the Spin(n) bundle discussed before. Moreover, denote by Ψ the map from

, Ψ ∗ (F

)) → X be the obvious X × M to itself given by (x, m) → (x, ψ(x, m)). Let πX : Hom(F map obtained by composing the projection map of the X × M bundle with the projection from

, Ψ ∗ (F

))) X × M to X, and let us denote by B the closed subset of the Polish space C(X, Hom(F in a similar way replacing consisting of those f such that for all x, πX (f (x)) = x. Define B

by P

. The covering map from P to F induces a covering map from B to B as well. Let F dψ : M → B be the map given by dψ (x, m) ≡ dψ (m)(x) = dψx |m . Then by Lemma 2.1 there exists a measurable lift of dψ , say d ψ from M into B. Since dψ (x, m) ∈ Hom(Fm , Fψ(x,m) ), it is (x, m) will be an element of Hom(P , P clear that the lift d m ψ(x,m) ). ψ

We can identify H ⊗ C(X) with C(X → H), and since H has a total set F (say) consisting of sections of the form [p(·), v], where p : M → P is a measurable section of P and v belongs of H ⊗ C(X) consisting of F valued continuous functions from X. to n , we have a total set F Any such function can be written as [Ξ, v] with Ξ : X × M → P , v ∈ n , and Ξ (x, m) ∈ Pm , by U [Ξ, v] = [Θ, v], where and we define U on F −1 −1 Θ(x, m) = d ψ x, ψx (m) Ξ x, ψx (m) . It is clear from the construction of the lift that U is indeed a C(X)-linear isometry which maps onto itself, so extends to a unitary on the whole of H ⊗ C(X) with the desired the total set F properties. Conversely, given U as in the statement of the converse part of the theorem, we observe that for each x in X, by Theorem 2.2, (id ⊗ evx )U = Uψx for some ψx such that ψx is a smooth orientation preserving Riemannian isometry. This defines the map ψ by setting ψ(x, m) = ψx (m). The proof will be complete if we can show that x → ψx ∈ C ∞ (M, M) is continuous, which is equivalent to showing that whenever xn → x in the topology of X, we must have φ ◦ ψxn → φ ◦ ψx in the Fréchet topology of C ∞ (M), for any φ ∈ C ∞ (M). However, by Lemma 2.3, we have (id ⊗ evxn )αU ([D, Mφ ]) → (id ⊗ evx )αU ([D, Mφ ]) in the strong operator topology where αU (X) = U XU −1 . Since U commutes with D, this implies (id ⊗ evxn ) D ⊗ id, αU (Mφ ) → (id ⊗ evx ) D ⊗ id, αU (Mφ ) , that is, for all ξ in L2 (S),


2537

2

L [D, Mφ◦ψxn ]ξ − → [D, Mφ◦ψx ]ξ.

By choosing φ with support in a local trivializing coordinate neighborhood for S, and then using the local expression of D used in the proof of Theorem 2.2, we conclude that L2 dk (φ ◦ ψxn ) − → dk (φ ◦ ψx ) (where dk is as in the proof of Theorem 2.2). Similarly, by taking repeated commutators with D, we can show the L2 convergence with dk replaced by dk1 . . . dkm for any finite tuple (k1 , . . . , km ). In other words, φ ◦ ψxn → φ ◦ ψx in the topology of C ∞ (M) described before. 2 2.2. Quantum group of orientation-preserving isometries of an R-twisted spectral triple We begin by recalling the definition of compact quantum groups and their actions from [36,37]. A compact quantum group (to be abbreviated as CQG from now on) is given by a pair (S, ), where S is a unital separable C ∗ -algebra equipped with a unital C ∗ -homomorphism : S → S ⊗ S (where ⊗ denotes the injective tensor product) satisfying (ai) ( ⊗ id) ◦ = (id ⊗ ) ◦ (co-associativity), and (aii) each of the linear spans Span((S)(S ⊗ 1)) and Span((S)(1 ⊗ S)) is norm-dense in S ⊗ S. It is well known (see [36,37]) that there is a canonical dense ∗-subalgebra S0 of S, consisting of the matrix coefficients of the finite-dimensional unitary (co-)representations (to be defined shortly) of S, and maps : S0 → C (co-unit) and κ : S0 → S0 (antipode) defined on S0 which make S0 a Hopf ∗-algebra. We say that the compact quantum group (S, ) (co-)acts on a unital C ∗ -algebra B, if there is a unital C ∗ -homomorphism (called an action) α : B → B ⊗ S satisfying the following (bi) (α ⊗ id) ◦ α = (id ⊗ ) ◦ α, and (bii) the linear span of α(B)(1 ⊗ S) is norm-dense in B ⊗ S. It is known (see, for example, [22,31]) that (bii) is equivalent to the existence of a norm-dense, unital ∗-subalgebra B0 of B such that α(B0 ) ⊆ B0 ⊗alg S0 and on B0 , (id ⊗ ) ◦ α = id. We shall sometimes say that α is a ‘topological’ or C ∗ -action to distinguish it from a normal action of von Neumann algebraic quantum group. Definition 2.5. A unitary (co-)representation of a compact quantum group (S, ) on a Hilbert space H is a unitary element U of M(K(H) ⊗ S) satisfying (id ⊗ )U = U(12) U(13) , where for an operator X in B(H1 ⊗ H2 ) we have denoted by X(12) and X(13) the operators X ⊗ IH2 in B(H1 ⊗ H2 ⊗ H2 ), and Σ23 X(12) Σ23 respectively and Σ23 is the unitary on H1 ⊗ H2 ⊗ H2 which flips the two copies of H2 . An alternative description of a unitary representation U can be given by identifying it with the isometric map, again denoted by U by an abuse of notation, from H to the Hilbert S module H ⊗ S which sends a vector ξ of H to the element U ξ = U (ξ ⊗ 1) of H ⊗ S. Note that we have used here the isomorphism M(K(H) ⊗ S) ∼ = L(H ⊗ S).

2538


Given a unitary representation U we shall denote by αU the ∗-homomorphism αU (X) = U (X ⊗ 1)U ∗ for X belonging to B(H). For a not necessarily bounded, densely defined (in the weak operator topology) linear functional τ on B(H), we say that αU preserves τ if αU maps a suitable (weakly) dense ∗-subalgebra (say D) in the domain of τ into D ⊗alg S and (τ ⊗ id)(αU (a)) = τ (a)1S for all a in D. When τ is bounded and normal, this is equivalent to (τ ⊗ id)(αU (a)) = τ (a)1S for all a belonging to B(H). We say that a (possibly unbounded) operator T on H commutes with U if T ⊗ I (with the natural domain) commutes with U . Sometimes such an operator will be called U -equivariant. Let us now recall the concept of universal quantum groups as in [29,34] and references therein. We shall use most of the terminologies of [29], for example Woronowicz C ∗ -subalgebra, Woronowicz C ∗ -ideal etc., however with the exception that we shall call the Woronowicz C ∗ algebras just compact quantum groups, and not use the term compact quantum groups for the dual objects as done in [29]. For an n × n positive invertible matrix Q = (Qij ), let Au (Q) be the compact quantum group defined and studied in [31,34], which is the universal C ∗ -algebra Q Q generated by {ukj , k, j = 1, . . . , di } such that u := ((ukj )) satisfies uu∗ = In = u∗ u,

u QuQ−1 = In = QuQ−1 u .

(1)

˜ is given by, Here u = ((uj i )) and u = ((u∗ij )). The coproduct, say , ˜ ij ) = (u

uik ⊗ ukj .

k

It may be noted that Au (Q) is the universal object in the category of compact quantum groups which admit an action on the finite-dimensional C ∗ -algebra Mn (C) which preserves the functional Mn x → Tr(QT x) (see [32]) where we refer the reader to [34] for a detailed discussion on the structure and classification of such quantum groups. In view of the characterization of orientation-preserving isometric action on a classical manifold (Theorem 2.4), we give the following definitions. Definition 2.6. A quantum family of orientation preserving isometries for the (odd, compact type) spectral triple (A∞ , H, D) is given by a pair (S, U ) where S is a separable unital C ∗ algebra and U is a unitary element of M(K(H) ⊗ S) satisfying the following (i) for every state φ on S we have Uφ D = DUφ , where Uφ := (id ⊗ φ)(U ); (ii) (id ⊗ φ) ◦ αU (a) ∈ (A∞ ) for all a in A∞ and state φ on S, where αU (x) := U (x ⊗ 1)U ∗ for x belonging to B(H). In case the C ∗ -algebra S has a coproduct such that (S, ) is a compact quantum group and U is a unitary representation of (S, ) on H, we say that (S, ) acts by orientation-preserving isometries on the spectral triple. In case the spectral triple is even with the grading operator γ , a quantum family of orientation preserving isometries (A∞ , H, D, γ ) will be defined exactly as above, with the only extra condition being that U commutes with γ .


2539

From now on, we will mostly consider odd spectral triples. However let us remark that in the even case, all the definitions and results obtained by us will go through with some obvious modifications. We also remark that all our spectral triples are of compact type. Consider the category Q ≡ Q(A∞ , H, D) ≡ Q(D) with the object-class consisting of all quantum families of orientation preserving isometries (S, U ) of the given spectral triple, and the set of morphisms Mor((S, U ), (S , U )) being the set of unital C ∗ -homomorphisms Φ : S → S satisfying (id ⊗ Φ)(U ) = U . We also consider another category Q ≡ Q (A∞ , H, D) ≡ Q (D) whose objects are triplets (S, , U ), where (S, ) is a compact quantum group acting by orientation preserving isometries on the given spectral triple, with U being the corresponding unitary representation. The morphisms are the homomorphisms of compact quantum groups which are also morphisms of the underlying quantum families of orientation preserving isometries. The forgetful functor F : Q → Q is clearly faithful, and we can view F (Q ) as a subcategory of Q. It is easy to see that any object (S, U ) of the category Q gives an equivariant spectral triple ∞ (A , H, D) with respect to the action of S implemented by U . It may be noted that recently there has been a lot of interest and work (see, for example, [6,9,13]) towards construction of quantum group equivariant spectral triples. In all these works, given a C ∗ -subalgebra A of B(H) and a CQG Q having a unitary representation U on H such that adU gives an action of Q on A, the authors investigate the possibility of constructing a (nontrivial) spectral triple (A∞ , H, D) on a suitable dense subalgebra A∞ of A such that U commutes with D ⊗ 1, i.e. D is equivariant. Our interest here is in the (sort of) converse direction: given a spectral triple, we want to consider all possible CQG representations with respect to which the spectral triple is equivariant; and if there exists a universal object in the corresponding category, i.e. Q , we should call it the quantum group of orientation preserving isometries. Unfortunately, in general Q or Q will not have a universal object. It is easily seen by taking the standard example A∞ = Mn (C), H = Cn , D = I . However, the fact that comes to our rescue is that a universal object exists in each of the subcategories which correspond to the CQG actions preserving a given faithful functional on Mn . On the other hand, given any equivariant spectral triple, it has been shown in [16] that there is a (not necessarily unique) canonical faithful functional which is preserved by the CQG action. For readers’ convenience, we state this result (in a form suitable to us) briefly here. Before that, let us recall the definition of an R-twisted spectral data from [16]. Definition 2.7. An R-twisted spectral data (of compact type) is given by a quadruplet (A∞ , H, D, R) where 1. (A∞ , H, D) is a spectral triple of compact type. 2. R a positive (possibly unbounded) invertible operator such that R commutes with D. 3. For all s ∈ R, the map a → σs (a) := R −s aR s gives an automorphism of A∞ (not necessarily ∗-preserving) satisfying sups∈[−n,n] σs (a) < ∞ for all positive integer n. We shall also sometimes refer to (A∞ , H, D) as an R-twisted spectral triple. Proposition 2.8. Given a spectral triple (A∞ , H, D) (of compact type) which is Q-equivariant with respect to a representation of a CQG Q on H, we can construct a positive (possibly unbounded) invertible operator R on H such that (A∞ , H, D, R) is a twisted spectral data, and moreover, we have αU preserves the functional τR defined at least on a weakly dense

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∗-subalgebra ED of B(H) generated by the rank-one operators of the form |ξ η| where ξ , η are eigenvectors of D, given by τR (x) = Tr(Rx),

x ∈ ED .

Remark 2.9. If Vλ denotes the eigenspace of D corresponding to the eigenvalue, say λ, it is clear 2 2 that τR (X) = etλ Tr(Re−tD X) for all X = |ξ η| with ξ, η belonging to Vλ and for any t > 0. Thus, the αU -invariance of the functional τR on ED is equivalent to the αU -invariance of the 2 functional X → Tr(XRe−tD ) on ED for each t > 0. If, furthermore, the R-twisted spectral triple 2 is Θ-summable in the sense that Re−tD is trace class for every t > 0, the above is also equivalent 2 to the αU -invariance of the bounded normal functional X → Tr(XRe−tD ) on the whole of B(H). In particular, this implies that αU preserves the functional B(H) x → Limt→0+

2

Tr(xRe−tD ) 2 , Tr(Re−tD )

where Lim is a suitable Banach limit discussed in [15]. This motivates the following definition: Definition 2.10. Given an R-twisted spectral data (A∞ , H, D, R) of compact type, a quantum family of orientation preserving isometries (S, U ) of (A∞ , H, D) is said to preserve the R-twisted volume (simply said to be volume-preserving if R is understood) if one has (τR ⊗ id)(αU (x)) = τR (x)1S for all x in ED , where ED and τR are as in Proposition 2.8. We shall also call (S, U ) a quantum family of orientation-preserving isometries of the R-twisted spectral triple. If, furthermore, the C ∗ -algebra S has a coproduct such that (S, ) is a CQG and U is a unitary representation of (S, ) on H, we say that (S, ) acts by volume and orientationpreserving isometries on the R-twisted spectral triple. We shall consider the categories QR ≡ QR (D) and QR ≡ QR (D) which are the full subcategories of Q and Q respectively, obtained by restricting the object-classes to the volumepreserving quantum families. Remark 2.11. We shall not need the full strength of the definition of twisted spectral data here; in particular the condition 3 in Definition 2.7. However, we shall continue to work with the usual definition of R-twisted spectral data, keeping in mind that all our results are valid without assuming condition 3. Let us now fix a spectral triple (A∞ , H, D) which is of compact type. The C ∗ -algebra generated by A∞ in B(H) will be denoted by A. Let λ0 = 0, λ1 , λ2 , . . . be the eigenvalues of D with Vi denoting the (di -dimensional, 0 di < ∞) eigenspace for λi . Let {eij , j = 1, . . . , di } be an orthonormal basis of Vi . We also assume that there is a positive invertible R on H such that (A∞ , H, D, R) is an R-twisted spectral data. The operator R must have the form R|Vi = Ri , say, with Ri positive invertible di × di matrix. Let us denote the CQG Au (RiT ) by Ui , with its T ∼ Cdi , given by βi (eij ) = eik ⊗ uRi . Let U be the canonical unitary representation βi on Vi = k kj free product of Ui , i = 1, 2, . . . , and β = ∗i βi be the corresponding free product representation of U on H. Lemma 2.12. Consider the R-twisted spectral triple (A∞ , H, D) and let (S, U ) be a quantum family of volume and orientation preserving isometries of the given spectral triple. Moreover,


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assume that the map U is faithful in the sense that there is no proper C ∗ -subalgebra S1 of S such that U belongs to M(K(H) ⊗ S1 ). Then we can find a C ∗ -isomorphism φ : U/I → S between S and a quotient of U by a C ∗ -ideal I of U , such that U = (id ⊗ φ) ◦ (id ⊗ ΠI ) ◦ β, where ΠI denotes the quotient map from U to U/I. If, furthermore, there is a compact quantum group structure on S given by a coproduct such that (S, , U ) is an object in QR , the ideal I is a Woronowicz C ∗ -ideal and the C ∗ -isomorphism φ : U/I → S is a morphism of compact quantum groups. (i)

Proof. It is clear that U maps Vi into Vi ⊗ S for each i. Let vkj (j, k = 1, . . . , di ) be the ele (i) (i) ments of S such that U (eij ) = k eik ⊗ vkj . Note that vi := ((vkj )) is a unitary in Mdi (C) ⊗ S. (i) , i 0, j, k 1} must be dense in S by the Moreover, the ∗-subalgebra generated by all {vkj assumption of faithfulness. Consider the ∗-homomorphism αi from the finite-dimensional C ∗ -algebra Ai ∼ = Mdi (C) generated by the rank one operators {|eij eik |, j, k = 1, . . . , di } to Ai ⊗ S given by αi (y) = U (y ⊗ 1)U ∗ |Vi . Clearly, the restriction of the functional τR on Ai is nothing but the functional given by Tr(Ri ·), where Tr denotes the usual trace of matrices. Since αi preserves this functional by assumption, we get, by the universality of Ui , a C ∗ -homomorphism from Ui to S sending RT

ukj ≡ ukji to vkj , and by definition of the free product, this induces a C ∗ -homomorphism, say Π , from U onto S, so that U/I ∼ = S, where I := Ker(Π). In case S has a coproduct making it into a compact quantum group and U is a quantum (i) , i 0, j, k = group representation, it is easy to see that the subalgebra of S generated by {vkj (i) (i) (i) 1, . . . , di } is a Hopf algebra, with (vkj ) = l vkl ⊗ vlj . From this, it follows that Π is Hopfalgebra morphism, hence I is a Woronowicz C ∗ -ideal. 2 (i)

(i)

Before we state and prove the main theorem, let us note the following elementary fact about C ∗ -algebras. Lemma 2.13. Let C be a unital C ∗ -algebra and F be a nonempty collection of C ∗ -ideals (closed two-sided ideals) of C. Let I0 denote the intersection of all I in F , and let ρI denote the map C/I0 x + I0 → x + I ∈ C/I for I in F . Denote by Ω the set {ω ◦ ρI , I ∈ F , ω state on C/I}, and let K be the weak-∗ closure of the convex hull of Ω ∪ (−Ω). Then K coincides with the set of bounded linear functionals ω on C/I0 satisfying ω = 1 and ω(x ∗ + I0 ) = ω(x + I0 ). Proof. We have, by Lemma 4.6 of [17] that for any x belonging to C, sup x + I = x + I0 ,

I ∈F

where x + I = inf {x − y: y ∈ I} denotes the norm in C/I. Clearly, K is a weak-∗ compact, convex subset of the unit ball (C/I0 )∗1 of the dual of C/I0 , satisfying −K = K. If K is strictly smaller than the self-adjoint part of unit ball of the dual of C/I0 , we can find a state ω on C/I0 which is not in K. Considering the real Banach space X = (C/I0 )∗s.a. and using standard separation theorems for real Banach spaces (for example, Theorem 3.4 of [25, p. 58]), we can find a self-adjoint element x of C such that x + I0 = 1, and sup ω (x + I0 ) < ω(x + I0 ).

ω ∈K

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Let γ belonging to R be such that supω ∈K ω (x + I0 ) < γ < ω(x + I0 ). Fix 0 < < ω(x + I0 ) − γ , and let I be an element of F be such that x + I0 − 2 x + I x + I0 . Let φ be a state on C/I such that x + I = |φ(x + I)|. Since x is self-adjoint, either φ(x + I) or −φ(x + I) equals x + I, and φ := ±φ ◦ ρI , where the sign is chosen so that φ (x + I0 ) = x + I. Thus, φ is in K, so x + I0 = φ (x + I) γ < ω(x + I0 ) − . But this implies x + I0 x + I + 2 < ω(x + I0 ) − 2 x + I0 − (since ω is a state), which is a contradiction completing the proof. 2 Theorem 2.14. For any R-twisted spectral triple of compact type (A∞ , H, D), the category QR of quantum families of volume and orientation preserving isometries has a universal (initial) U0 ). Moreover, G has a coproduct 0 such that (G, 0 ) is a compact quantum object, say (G, group and (G, 0 , U0 ) is a universal object in the category QR . The representation U0 is faithful. Proof. Recall the C ∗ -algebra U considered before, along with the representation β in M(K(H) ⊗ U). For any C ∗ -ideal I of U , we shall denote by ΠI the canonical quotient map from U onto U/I, and let ΓI = (id ⊗ ΠI ) ◦ β. Clearly, ΓI is a unitary element of M(K(H) ⊗ U/I). Let F be the collection of all those C ∗ -ideals I of U such that (id ⊗ ω) ◦ αΓI ≡ (id ⊗ ω) ◦ adΓI maps A∞ into A for every state ω (equivalently, every bounded linear functional) on U/I. This collection is nonempty, since the trivial one-dimensional C ∗ -algebra C gives an object in QR and by Lemma 2.12 we do get a member of F . Now, let I0 be the intersection of all ideals in F . We claim that I0 is again a member of F . Indeed, in the notation of Lemma 2.13, it is clear that for a in A∞ , (id ⊗ φ) ◦ ΓI0 (a) belongs to A for all φ in the convex hull of Ω ∪ (−Ω). Now, for any state ω on U/I0 , we can find, by Lemma 2.13, a net ωj in the above convex hull (so in particular ωj 1 for all j ) such that ω(x + I0 ) = limj ωj (x + I0 ) for all x in U/I0 . It follows from Lemma 2.3 that (id ⊗ ωj )(X) → (id ⊗ ω)(X) (in the strong operator topology) for all X belonging to M(K(H) ⊗ U/I0 ). Thus, for a in A∞ , (id ⊗ ω) ◦ αΓI0 (a) is the limit of (id ⊗ ωi ) ◦ αΓI0 (a) in the strong operator topology, hence belongs to A . := U/I0 , ΓI ) is a universal object in QR . To see this, consider any We now show that (G 0 object (S, U ) of QR . Without loss of generality we can assume U to be faithful, since otherwise (i) we can replace S by the C ∗ -subalgebra generated by the elements {vkj } appearing in the proof of Lemma 2.12. But by Lemma 2.12 we can further assume that S is isomorphic with U/I for some I belonging to F . Since I0 ⊆ I, we have a C ∗ -homomorphism from U/I0 onto U/I, sending x + I0 to x + I, which is clearly a morphism in the category QR . This is indeed the unique such morphism, since it is uniquely determined on the dense subalgebra generated by (i) {ukj + I0 , i 0, j, k 1} of G. ⊗ G) given = U/I0 , we first consider U (2) ∈ M(K(H) ⊗ G To construct the coproduct on G by U (2) = (ΓI0 )(12) (ΓI0 )(13) , ⊗ G, U (2) ) is an object in where Uij is the usual ‘leg-numbering notation’. It is easy to see that (G the category QR , so by the universality of (G, ΓI0 ), we have a unique unital C ∗ -homomorphism → G ⊗ G satisfying 0 : G (id ⊗ 0 )(ΓI0 ) = U (2) .


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Letting both sides act on eij , we get

(i) (i) (i) eil ⊗ (πI0 ⊗ πI0 ) ulk ⊗ ukj = eil ⊗ 0 πI0 ulj .

l

k

l

˜ (i) ) = Comparing coefficients of eil , and recalling that (u lj coproduct on U ), we have

˜ = 0 ◦ πI (πI0 ⊗ πI0 ) ◦ 0

(i) k ulk

(i) ˜ denotes the ⊗ ukj (where

(2)

(i) ˜ on the linear span of {uj k , i 0, j, k 1}, and hence on the whole of U . This implies that maps I0 = Ker(πI0 ) into Ker(πI0 ⊗ πI0 ) = (I0 ⊗ 1 + 1 ⊗ I0 ) ⊂ U ⊗ U . In other words, I0 is = U/I0 has the canonical compact quantum group structure as a Hopf C ∗ -ideal, and hence G a quantum subgroup of U . It is clear from the relation (2) that 0 coincides with the canonical coproduct of the quantum subgroup U/I0 inherited from that of U . It is also easy to see that 0 , ΓI ) is universal in the category Q , using the fact that (by Lemma 2.12) any the object (G, 0 R compact quantum group (S, U ) acting by volume and orientation preserving isometries on the given spectral triple is isomorphic with a quantum subgroup U/I, for some Hopf C ∗ -ideal I of U . Finally, the faithfulness of U0 follows from the universality by standard arguments which we is a ∗-subalgebra of G such that U0 is an element of M(K(H) ⊗ G1 ), it is briefly sketch. If G1 ⊂ G it follows easy to see that (G1 , U0 ) is also an object in QR , and by definition of universality of G to G1 . But the map i ◦ j is a morphism from G that there is a unique morphism, say j , from G to itself, where i : G1 → G is the inclusion. Again by universality, we have that i ◦ j = idG, so in 2 particular, i is onto, i.e. G1 = G.

U0 ) is the universal object obtained in Consider the ∗-homomorphism α0 := αU0 , where (G, the previous theorem. For every state φ on G, (id ⊗ φ) ◦ α0 maps A into A . However, in general generated by the α0 may not be faithful even if U0 is so, and let G denote the C ∗ -subalgebra of G ∗ elements {(f ⊗ id) ◦ α0 (a): f ∈ A , a ∈ A}. Remark 2.15. If the spectral triple is even, then all the proofs above go through with obvious modifications. Definition 2.16. We shall call G the quantum group of orientation-preserving isometries of the ∞ R-twisted spectral data (A∞ , H, D, R) and denote it by QISO+ R (A , H, D, R) or even simply + (D). is denoted by QISO as QISO+ (D). The quantum group G R

R

+ (D, γ ) by QISO+ (D, γ ) and QISO If the spectral triple is even, then we will denote G and G R R respectively. 2.3. Stability and topological action G, U0 , In this subsection, we are going to use the notations as in Section 2.2, in particular G, + ∗ α0 . It is not clear from the definition and construction of QISOR (D) whether the C -algebra A

2544


generated by A∞ is stable under α0 in the sense that (id ⊗ φ) ◦ α0 maps A into A for every φ. Moreover, even if A is stable, the question remains whether α0 is a C ∗ -action of the CQG QISO+ R (D). In Section 4.2, we have given an example (which is described in details in [4]) of a spectral triple for which the ∗-homomorphism α0 is not a C ∗ -action. However, one can prove that α0 is a C ∗ -action for a rather large class of spectral triples, including the cases mentioned below. (i) For any spectral triple for which there is a ‘reasonable’ Laplacian in the sense of [17]. This includes all classical spectral triples as well as their Rieffel deformation (with R = I ). (ii) Under the assumption that there is an eigenvalue of D with a one-dimensional eigenspace spanned by a cyclic separating vector ξ such that any eigenvector of D belongs to the span of A∞ ξ and {a ∈ A∞ : aξ is an eigenvector of D} is norm-dense in A (to be proved in Section 2.4). (iii) Under some analogue of the classical Sobolev conditions with respect to a suitable group action on A (see [18]). Now we prove the sufficiency of the condition (i). We begin with a sufficient condition for stability of A∞ under α0 . Let (A∞ , H, D) be a (compact type) spectral triple such that (1) A∞ and {[D, a], a ∈ A∞ } are contained in the domains of all powers of the derivations [D, ·] and [|D|, ·]. We will denote by Tt the one parameter group of ∗-automorphisms on B(H) given by Tt (S) = for all S in B(H). We will denote the generator of this group by δ. For X such that [D, X] is bounded, we have δ(X) = i[D, X] and hence

eitD Se−itD

t Tt (X) − X = Ts [D, X] ds t [D, X] . 0

Let us say that the spectral triple satisfies the Sobolev condition if A∞ = A

Dom δ n .

n1

Then we have the following result, which is a natural generalization of the classical situation, where a measurable isometric action automatically becomes topological (in fact smooth). Theorem 2.17. (i) For every state φ on G, (id ⊗ φ) ◦ α0 (A∞ ) belongs to A (ii) If the spectral triple satisfies the Sobolev condition then der α0 .

n n1 Dom(δ ). A∞ (and hence

A) is stable un-


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Proof. Since U0 commutes with D ⊗ I , it is clear that the automorphism group Tt commutes φ with α0 ≡ (id ⊗ φ) ◦ α0 , and thus by the continuity of α0 in the strong operator topology it is easy to see that, for a in Dom(δ), φ φ Tt (α0 (a)) − α0 (a) φ Tt (a) − a = lim α0 t→0+ t→0+ t t φ = α0 δ(a) . lim

φ

Thus, α0 leaves Dom(δ) invariant and commutes with δ. Proceeding similarly, we prove (i). The assertion (ii) is a trivial consequence of (i) and the Sobolev condition. 2 Let us now assume (2) The spectral triple is Θ-summable, i.e. for every t > 0, e−tD is trace-class and the functional 2

τ (X) = Limt→0

−tD 2

) Tr(Xe 2 , Tr(e−tD )

where Lim is a suitable Banach limit as in [15], is a positive

faithful trace on the ∗-algebra, say S ∞ , generated by eitD ’s, A∞ and {[D, a]: a ∈ A∞ }. The functional τ is to be interpreted as the volume form (we refer to [15,17] for the details). The completion of S ∞ in the norm of B(H) is denoted by S, and we shall denote by a2 and 1 · ∞ the L2 -norm τ (a ∗ a) 2 and the operator norm of B(H) respectively. From the definition of τ, it is also clear that Tt preserves τ, so extends to a group of unitaries on N := L2 (S ∞ , τ ). Moreover, for X such that [D, X] is in B(H), in particular for X in S ∞ , we have 2 ∗ Ts (X) − X 2 = τ Ts (X) Ts (X) − X + τ X ∗ X − Ts (X) 2 X − Ts (X) ∞ X2 2s [D, X] ∞ X2 , which clearly shows that s → Ts (X) is L2 -continuous for X belonging to S ∞ , hence (by unitarity of Ts ) on the whole of N , i.e. it is a strongly continuous one-parameter group of unitaries. Let us ˜ which is a skew adjoint map, i.e. i δ˜ is self adjoint, and Tt = exp(t δ). ˜ denote its generator by δ, ∞ ˜ Clearly, δ = δ = [D, ·] on S . 0 and the restriction of δ˜ to H0 (which is a closable We will denote L2 (A∞ , τ ) ⊂ N by HD D 0 map from HD to N ) by dD . Thus, dD is closable too. We now recall the assumptions made in [17], for defining the ‘Laplacian’ and the corresponding quantum isometry group of a spectral triple (A∞ , H, D), without going into all the technical details, for which the reader is referred to [17]. The following conditions will also be assumed throughout the rest of this subsection: ∗d . (3) A∞ ⊆ Dom(L) where L ≡ LD := −dD D (4) L has compact resolvent. (5) Each eigenvector of L (which has a discrete spectrum, hence a complete set of eigenvectors) belongs to A∞ .

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∞ (6) The complex linear span of the eigenvectors of L, say A∞ 0 (which is a subspace of A by assumption (5)), is norm dense in A∞ . ∞ ∞ It is clear that L maps A∞ 0 into itself. The ∗-subalgebra of A generated by A0 is denoted 2 ˜ by A0 . We also note that L = P0 LP0 , where L := (i δ) (which is a self adjoint operator on N ) 0 . and P0 denotes the orthogonal projection in N whose range is the subspace HD

Theorem 2.18. Let (A∞ , H, D) be a spectral triple satisfying the assumptions (1)–(6) made above. In addition, assume that at least one of conditions (a) and (b) mentioned below is satisfied: 0. (a) A ⊆ HD φ ∞ (b) α0 (A ) ⊆ A∞ for every state φ on G = QISO+ I (D).

Then α0 is a C ∗ -action of QISO+ I (D) on A. Proof. Under either of the conditions (a) and (b), for any fixed φ, the map α0 maps A∞ into 0 of N . Since α φ also commutes with [D, ·] on A∞ , it is clear that α φ maps S ∞ the subset HD 0 0 φ into N . In fact, using the complete positivity of the map α0 and the α0 -invariance of τ , we see that φ

φ φ φ τ α0 (a)∗ α0 (a) τ α0 (a ∗ a) = (id ⊗ φ) (τ ⊗ id)α0 (a ∗ a) = τ (a ∗ a).1, φ

which implies that α0 extends to a bounded operator from N to itself. Since U0 commutes φ with D, it is clear that α0 (viewed as a bounded operator on N ) will commute with the group of = (i δ) ˜ 2. unitaries Tt , hence with its generator δ˜ and also with the self adjoint operator L + On the other hand, it follows from the definition of G = QISOI (D) that (τ ⊗ id)(α0 (X)) = τ (X)1G for all X in B(H), in particular for X belonging to S ∞ , and thus the map S ∞ ⊗alg G (a ⊗ q) → α0 (a)(1 ⊗ q) extends to a G-linear unitary, denoted by W (say), on the Hilbert Gmodule N ⊗G. Note that here we have used the fact that for any φ, (id⊗φ)(W )(S ∞ ⊗alg G) ⊆ N , φ φ since α0 (S ∞ ) ⊆ N . The commutativity of α0 with Tt for every φ clearly implies that W and φ 0 into itself, so W maps H0 ⊗ G into Tt ⊗ idG commute on N ⊗ G. Moreover, α0 maps HD D φ itself, and hence (by unitarity of W ) it commutes with the projection P0 ⊗ 1. It follows that α0 0 as well. commutes with L = P0 LP commutes with P0 , and (since it also commutes with L) φ Thus, α0 preserves each of the (finite-dimensional) eigenspaces of the Laplacian L, and so is a Hopf algebraic action on the subalgebra A0 spanned algebraically by these eigenvectors. Moreover, the G-linear unitary W clearly restricts to a unitary representation on each of the above eigenspaces. If we denote by ((qij ))(i,j ) the G-valued unitary matrix corresponding to one such particular eigenspace, then by the general theory of CQG representations, qij must belong to G0 and we must have (qij ) = δij (Kronecker delta). This implies (id ⊗ ) ◦ α0 = id on each of the eigenspaces, hence on the norm-dense subalgebra A0 of A, completing the proof of the fact that α0 extends to a C ∗ -action on A. 2 Combining the above theorem with Theorem 2.17, we get the following immediate corollary.


2547

Corollary 2.19. If the spectral triple satisfies the Sobolev condition mentioned before, in addi∗ tion to the assumptions (1)–(6), then QISO+ I (D) has a C -action. In particular, for a classical ∗ -action. spectral triple, QISO+ (D) has C I Remark 2.20. Let us remark here that in case the restriction of τ on A∞ is normal, i.e. continuous 0 will contain A , which with respect to the weak operator topology inherited from B(H), then HD ∞ is the closure of A in the weak operator topology of B(H), so the condition (a) of Theorem 2.18 (and hence its conclusion) holds. Remark 2.21. The results obtained in this subsection can be formulated and proved in an R∗ d twisted set-up as well, if the corresponding Laplacian (which is an extension of dD,R D,R , where ∗ dD,R denotes the adjoint of dD ≡ dD,R with respect to the R-twisted volume form) ‘exists’ and satisfies the analogues of the assumptions made in this subsection about LD . In [4], we have made some computations with such an R-twisted Laplacian arising naturally in that context. Remark 2.22. In a private communication to us, Shuzhou Wang has kindly pointed out that a possible alternative approach to the formulation of quantum group of isometries may involve the category of CQG which has a C ∗ -action on the underlying C ∗ -algebra and a unitary representation with respect to which the Dirac operator is equivariant. However, we see from (3), Theorem 4.15 of Section 4.2 that the category proposed by Wang does not admit a universal object in general. 2.4. Universal object in the categories Q or Q We shall now investigate further conditions on the spectral triple which will ensure the existence of a universal object in the category Q or Q . Whenever such a universal object exists we + (D), and denote by QISO+ (D) its largest Woronowicz subalgebra for shall denote it by QISO + (D) on H) is faithful. which αU on A∞ (where U is the unitary representation of QISO + (D) is + (D) exists, by [16], there will exist some R such that QISO Remark 2.23. If QISO + (D), is an object in the category Q (D). Since the universal object in this category, i.e. QISO R

R

+ (D) for this choice of R. + (D) ∼ + (D), we have QISO clearly a sub-object of QISO = QISO R Let us state and prove a result below, which gives some sufficient conditions for the existence + (D). of QISO Theorem 2.24. Let (A∞ , H, D) be a spectral triple of compact type as before and assume that D has an one-dimensional eigenspace spanned by a unit vector ξ , which is cyclic and separating for the algebra A∞ . Moreover, assume that each eigenvector of D belongs to the dense subspace U0 ). Moreover, G has a coproduct 0 such that A∞ ξ of H. Then there is a universal object, (G, 0 ) is a compact quantum group and (G, 0 , U0 ) is a universal object in the category Q . (G, generated by elements of the form If we denote by G the Woronowicz C ∗ subalgebra of G valued inner αU0 (a)(η ⊗ 1), η ⊗ 1G where η, η are in H, a is in A∞ and ·,·G denotes the G ∼ product of H ⊗ G, we have G = G ∗ C(T).

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Proof. Let Vi , {eij } be as before, and by assumption we have eij = xij ξ for a unique xij in A∞ . Clearly, since ξ is separating, the vectors {eij = xij∗ ξ, j = 1, . . . , di } are linearly independent, i so the matrix Qi = ((eij , eik ))dj,k=1 is positive and invertible. Now, given a quantum family of orientation-preserving isometries (S, U ), we must have U (ξ ⊗ 1) = ξ ⊗ q, say, for some q in S, and from the unitarity of U it follows that q is a unitary element. Moreover, U leaves Vi (i) invariant, so let U (eij ⊗ 1) = k eik ⊗ vkj . But this can be rewritten as

αU (xij )(ξ ⊗ q) =

(i)

xik ξ ⊗ vkj .

k

Since ξ is separating and q is unitary, this implies αU (xij ) = U (eij ⊗ 1) = αU (xij )∗ (ξ ⊗ q) =

k xik

⊗ vkj q ∗ , and thus we have (i)

(i) ∗ (i) ∗ ∗ xik ξ ⊗ q vkj q = eik ⊗ q vkj q.

k

k

Taking the S-valued inner product ·,·S on both sides of the above expression, and using the (i) fact that U preserves this S-valued inner product, we obtain Qi = vi Qi vi (where vi = ((vkj ))).

Thus, Q−1 i vi Qi must be the (both-sided) inverse of vi . Thus, we get a canonical surjective (i) morphism from Au (Qi ) to the C ∗ algebra generated by {vkj , j, k = 1, 2, . . . , di }. This induces a surjective morphism from the free product of Au (Qi ), i = 1, 2, . . . , onto S. The rest will be quite similar to the arguments used of the arguments for showing the existence of G in the proof of Theorem 2.14, hence omitted. It is also quite obvious from the proof that = G ∗ C ∗ (q) ∼ G = G ∗ C(T). 2

Remark 2.25. Some of the examples considered in Section 4 will show that the conditions of the + (D) may exist without the existence of a single above theorem are not actually necessary; QISO cyclic separating eigenvector as above. Let (A∞ , H, D) be a spectral triple of compact type satisfying the conditions of the above theorem. Let the faithful vector state corresponding to the cyclic separating vector ξ be denoted by τ. Let A00 = span{a ∈ A∞ : aξ is an eigenvector of D}. Moreover, assume that A00 is norm dense in A∞ .

: A∞ → A00 be defined by Let D 0

D(a)ξ = D(aξ ). This is well defined since ξ is cyclic and separating. Definition 2.26. Let A be a C ∗ algebra and A∞ be a dense ∗-subalgebra. Let (A∞ , H, D) be a spectral triple of compact type as above. Let

C be the category with objects (Q, α) such that Q is a compact quantum group with a ∗ C -action α on A such that 1. α is τ preserving (where τ is as above), i.e., (τ ⊗ id)(α(a)) = τ (a).1. 2. α maps A00 inside A00 ⊗alg Q.

= (D

⊗ I )α. 3. α D


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of the category C and it is isomorphic to the Corollary 2.27. There exists a universal object Q obtained in Theorem 2.24. Woronowicz C ∗ subalgebra G = QISO+ (D) of G Proof. The proof of the existence of the universal object follows verbatim from the proof of

and noting that D has compact resolvent. We denote by Theorem 4.7 in [17] replacing L by D

αˆ the action of Q on A.

is isomorphic to G. Now, we prove that Q Each eigenvector of D is in A∞ by assumption. It is easily observed from the proof of Theorem 2.24 that αU0 maps the norm-dense ∗-subalgebra A0 ∞ into A0 ∞ ⊗alg G0 , and (id ⊗ ) ◦ αU0 = id, so that αU0 is indeed a C ∗ -action of the CQG G. Moreover, it can be easily

Therefore, (G, αU0 ) is an element of seen that τ preserves αU0 and that αU0 commutes with D.

by the universality of Q.

Obj(

C), and hence G is a quantum subgroup of Q

∗ C(T) For the converse, we start by showing that αˆ induces a unitary representation W of Q ˆ on H which commutes with D, and the corresponding conjugated action αW coincides with α.

∗ C(T) where q is a (unitary) Define W (aξ ⊗ b) = α(ξ ˆ ⊗ bq ∗ ) for a in A00 and b in Q

∗ C(T). generator of C(T) viewed as a subalgebra of Q

∗ C(T)-linear) isometry on Since we have (τ ⊗ id)(α(a)) = τ (a).1, it follows that W is a (Q

∗ C(T)) and thus extends to H ⊗ (Q

∗ C(T)) as an isometry. the dense subspace A00 ξ ⊗alg (Q

is norm dense in A ⊗ Q

(by the definition of a CQG action) it is Moreover, since

α (A)(1 ⊗ Q) clear that the range of W is dense, so W is indeed a unitary. It is quite obvious that it is a unitary

∗ C(T). representation of Q We also have,

W (D ⊗ 1)(aξ ⊗ b) = W D(a)ξ ⊗ b = αˆ D(a) (ξ ⊗ bq ∗ ) = (D ⊗ I ) α(a)(ξ ˆ ⊗ bq ∗ ) = (D ⊗ I )W (aξ ⊗ b), i.e. W commutes with D. Moreover, it is easy to observe that αW = α. ˆ This gives a surjective CQG morphism from = G ∗ C(T) to Q

∗ C(T), sending G onto Q,

which completes the proof. 2 G 3. Comparison with the approach of [17] based on Laplacian Throughout this section, we shall assume the set-up of Section 2.3 for the existence of a ‘Laplacian’, including assumptions (1)–(6). Let us also use the notation of that subsection. As in [17], a CQG (S, ) which has an action α on A is said to act smoothly and isometrically ∞ on the noncommutative manifold (A∞ , H, D) if (id ⊗ φ) ◦ α(A∞ 0 ) ⊆ A0 for every state φ on S, ∞ and also (id ⊗ φ) ◦ α commutes with the Laplacian L ≡ LD on A0 . One can consider the category QLD of all compact quantum groups acting smoothly and isometrically on A, where the morphisms are quantum group morphisms which intertwin the actions on A. We make the following additional assumption throughout the present section: (7) There exists a universal object in QLD (the quantum isometry group for the Laplacian L ≡ LD in the sense of [17]), and it is denoted by QL ≡ QLD . Remark 3.1. It is proved in [17] that a sufficient condition for the assumption (7) to hold is the so-called ‘connectedness hypothesis’, that is, the kernel of L is one-dimensional, spanned by

2550


0 . The category QL has a universal object, the identity 1 of A∞ viewed as a unit vector in HD say QL , called the quantum isometry group in [17]. In [17] it was also shown (Lemma 2.5, b ⇒ a) that for an isometric group action on a not necessarily connected classical manifold, the volume functional is automatically preserved. It can be easily seen that the proof goes verbatim for a quantum group action, and consequently we get the existence of QLD for a general (but not necessarily connected) classical manifold.

The following result now follows immediately from Theorem 2.18 of Section 2.3. Corollary 3.2. If (A∞ , H, D) is a spectral triple (of compact type) satisfying any of the two conLD in the category QLD . ditions (a) or (b) of Theorem 2.18, then QISO+ I (D) is a sub-object of Q ∗ Proof. The proof is a consequence of the fact that QISO+ I (D) has the C -action α0 on A, and the observation already made in the proof of Theorem 2.18 that this action commutes with the Laplacian LD . 2

Given a Θ-summable spectral triple (A∞ , H, D), we recall the Hilbert space of forms from pp. 124–127 of [15]. Let Ω k (A∞ ) be the space of universal k-forms on the algebra A∞ which is spanned by a0 δ(a1 ) · · · δ(ak ), ai ∈ A∞ , where δ denotes the universal derivation from A∞ to the bimodule of universal one-forms Ω 1 (A∞ ). There is a natural graded algebra structure on Ω ≡ k0 Ω k (A∞ ), which also has a natural involution given by (δ(a))∗ = −δ(a ∗ ), and using the spectral triple, we get a ∗-representation Π : Ω → B(H) which sends a0 δ(a1 ) · · · δ(ak ) to a0 dD (a1 ) · · · dD (ak ), where dD (a) = [D, a]. Consider the state τ on B(H) given by, τ (X) = 2

Limt→0+

Tr(Xe−tD ) 2 , Tr(e−tD )

where Lim denotes a suitably chosen Banach limit. Using τ , we define a

positive semi definite sesquilinear form on Ω k (A∞ ) by setting w, η = τ (Π(w)∗ Π(η)). Let k be the Hilbert space K k = {w ∈ Ω k (A∞ ): w, w = 0}, for k 0, and K −1 := (0). Let ΩD k ∞ k obtained by completing the quotient Ω (A )/K with respect to the inner product mentioned k := P ⊥ Ω k , where P denotes the projection onto the closed subspace above, and we define HD k D k k ∗ on H := k−1 generated by δ(K ). The map D := d + d ∗ ≡ dD + dD D k0 HD has a selfk has a total set consisting of adjoint extension (which is again denoted by d + d ∗ ). Clearly, HD ∞ elements of the form [a0 δ(a1 ) · · · δ(ak )], with ai ∈ A and where [ω] denotes the equivalence class Pk⊥ (w + K k ) for ω ∈ Ω k (A∞ ). There is a ∗-representation πD : A → B(HD ), given by πD (a)([a0 δ(a1 ) · · · δ(ak )]) = [aa0 δ(a1 ) · · · δ(ak )]. Then it is easy to see that (A∞ , HD , d + d ∗ ) is another spectral triple. We assume that this is of compact type, i.e. d + d ∗ has compact resolvents. We will denote the inner product on the space of k forms coming from the spectral triples (A∞ , H, D) and (A∞ , HD , d + d ∗ ) by , Hk and , Hk ∗ respectively, k = 0, 1. D

d+d

∗ Let Ud+d ∗ be the canonical unitary representation of QISO+ I (d + d ) on HD . The Hilbert space HD breaks up into finite-dimensional orthogonal subspaces corresponding to the distinct eigenvalues of := (d + d ∗ )2 = d ∗ d + dd ∗ . It is easy to see that leaves each of the subspaces i invariant, and we will denote by V i HD λ,i the subspace of HD spanned by eigenvectors of corresponding to the eigenvalue λ. Let {ej,λ,i }j be an orthonormal basis of Vλ,i . Note that LD is 0 . the restriction of to HD Now we recall the result of Section 2.4 of [17]. It was shown there that QLD has a unitary representation U ≡ UL on HD such that U commutes with d + d ∗ . On the Hilbert space of k , U is defined by k-forms, i.e. HD


2551

(1) (2) (2) (1) (1) (2) ⊗ a0 a1 · · · ak q, U a0 δ(a1 ) · · · δ(ak ) ⊗ q = a0 δ a1 · · · δ ak where q ∈ QLD , ai ∈ A∞ 0 (notation as in Section 2.3), and for x in A0 , we write in Sweedler notation α(x) = x (1) ⊗ x (2) ∈ A0 ⊗ (QLD )0 (α denotes the action of QLD ). Thus, (A∞ , HD , d + d ∗ ) is a QLD equivariant spectral triple. It follows from the construction in [17] that QLD is a quotient (by a Woronowicz C ∗ ideal) of the free product of countably many Wang algebras of the type Au (I ), and hence is a compact quantum Kac algebra. It follows that it has tracial Haar state, which implies, by Theorem 3.2 of [16], that αU keeps the functional τI invariant. Summarizing, we have the following result: ∗ Proposition 3.3. The quantum isometry group (QLD , UL ) is a sub-object of (QISO+ I (d + d ), ∗ L Ud+d ∗ ) in the category QI (d + d ), so in particular, Q D is isomorphic to a quotient of ∗ ∗ QISO+ I (d + d ) by a Woronowicz C -ideal.

We shall give (under mild conditions) a concrete description of the above Woronowicz ideal. ∗ Let I be the C ∗ -ideal of QISO+ I (d + d ) generated by P0⊥ ⊗ id Ud+d ∗ (ej λ0 ), ej λi ⊗ 1 : j, i 1 , λ∈σ () 0 and ·,· denotes the QISO+ (d + d ∗ ) valued inner product. where P0 is the projection onto HD I Since Ud+d ∗ keeps the eigenspaces of = (d + d ∗ )2 invariant, we can write

Ud+d ∗ (ej λ0 ) =

ekλ0 ⊗ qkj λ0 +

k

ek λi ⊗ qk j λi ,

i =0,k

∗ for some qkj λ0 , qk j λi in QISO+ I (d + d ). We note that qk j λi is in I if i = 0. ∗ Lemma 3.4. I is a co-ideal of QISO+ I (d + d ). + ∗ ∗ Proof. It is enough to prove the relation (X) ∈ I ⊗ QISO+ I (d + d ) + QISOI (d + d ) ⊗ I for the elements X in I of the form (P0⊥ ⊗ id)Ud+d ∗ (ej λ0 ), ej λi0 ⊗ 1. We have

P0⊥ ⊗ id Ud+d ∗ (emλ0 ), ej λi0 ⊗ 1 = P0⊥ ⊗ id (id ⊗ )Ud+d ∗ (emλ0 ), ej λi0 ⊗ 1 ⊗ 1 = P0⊥ ⊗ id U(12) U(13) (emλ0 ), ej λi0 ⊗ 1 ⊗ 1 ekλ0 ⊗ 1 ⊗ qkmλ0 , ej λi0 ⊗ 1 ⊗ 1 = P0⊥ ⊗ id U(12) k

P0⊥ ⊗ id U(12) (elλi ⊗ 1 ⊗ qlmλi ), ej λi0 ⊗ 1 ⊗ 1 + i =0,l

P0⊥ ⊗ id (ek λ0 ⊗ qk kλ0 ⊗ qkmλ0 ), ej λi0 ⊗ 1 ⊗ 1 = k,k

2552


+

P0⊥ ⊗ id (ek kλi ⊗ qk ,k,λ,i ⊗ qkmλ0 ), ej λi0 ⊗ 1 ⊗ 1 i =0,k,k

+

P0⊥ ⊗ id (el λi ⊗ ql lλi ⊗ qlmλi ), ej λi0 ⊗ 1 ⊗ 1

i =0,l,l

+

⊥ P0 ⊗ id (el λi ⊗ ql lλi ⊗ qlmλi ), ej λi0 ⊗ 1 ⊗ 1

i =0,i =i ,l,l

=

ek λi ⊗ qk kλi ⊗ qkmλ0 , ej λi0 ⊗ 1 ⊗ 1

i =0,k ,k

+

i =0,l,l

+

el λi ⊗ ql lλi ⊗ qlmλi , ej λi0 ⊗ 1 ⊗ 1

el λi ⊗ ql lλi ⊗ qlmλi , ej λi0 ⊗ 1 ⊗ 1,

i =0,i =i ,i =0,l,l + ∗ ∗ which is clearly in I ⊗ QISO+ I (d + d ) + QISOI (d + d ) ⊗ I, as qkj λi is an element of I for i = 0. 2 ∗ Theorem 3.5. If αUd+d ∗ is a C ∗ -action on A, then we have QLD ∼ = QISO+ I (d + d )/I.

Proof. By Proposition 3.3, we conclude that there exists a surjective CQG morphism ∗ LD . By construction (Section 2.4 in [17]), the unitary representation π : QISO+ I (d + d ) → Q i , in particular H0 . It is then clear from the definition of L UL of Q D preserves each of the HD D I that π induces a surjective CQG morphism (in fact, a morphism in the category QI (d + d ∗ )) ∗ LD . π : QISO+ I (d + d )/I → Q ∗ Conversely, if V = (id ⊗ ρI ) ◦ Ud+d ∗ is the representation of QISO+ I (d + d )/I on HD ∗ ) → QISO+ (d + d ∗ )/I denotes the quotient map), induced by Ud+d ∗ (where ρI : QISO+ (d + d I I 0 (by definition of I), so commutes with P . Since V also commutes with then V preserves HD 0 (d + d ∗ )2 , it follows that V must commute with (d + d ∗ )2 P0 = L, that is, V (d ∗ dP0 ⊗ 1) = (d ∗ dP0 ⊗ 1)V . It is easy to show from the above that αV (which is a C ∗ -action on A since αUd+d ∗ is so by ∗ assumption) is a smooth isometric action of QISO+ I (d + d )/I in the sense of [17], with respect ∗ L L to the Laplacian L. This implies that QISO+ I (d + d )/I is a sub-object of Q in the category Q , and completes the proof. 2 Now we prove that under some further assumptions which are valid for classical manifolds as ∗ well as their Rieffel deformation, one even has the isomorphism QLD ∼ = QISO+ I (d + d ). We assume the following: (A) Both the spectral triples (A∞ , H, D) and (A∞ , HD , d + d ∗ ) satisfy the assumptions (1)–(7), so in particular both QLD and QLD exist (here D = d + d ∗ ). (B) For all a, b in A∞ , we have


a, bH0 = a, bH0 ,

δ(a), δ(b) H1 = δ(a), δ(b) H1 .

D

D

2553

D

D

Remark 3.6. For classical compact spin manifolds these assumptions can be verified by comparing the local expressions of D 2 and the ‘Hodge Laplacian’ (D )2 in suitable coordinate charts. In fact, in this case, both these operators turn out to be essentially same, upto a ‘first order term’, which is relatively compact with respect to D 2 or (D )2 . By assumption (B), we observe that the identity map on A∞ extends to a unitary, say Σ , from 0 . Moreover, we have to HD

0 HD

LD = Σ ∗ LD Σ, from which we conclude the following: Proposition 3.7. Under the above assumptions, QLD ∼ = QL D . We conclude this section with the following result, which identifies the quantum isometry group QLD of [17] as the QISO+ I of a spectral triple, and thus, in some sense, accommodates the construction of [17] in the framework of the present article. Theorem 3.8. If in addition to the assumptions already made, the spectral triple (of compact type) (A∞ , HD , D ) also satisfies the conditions of Theorem 2.18, so that QISO+ I (D ) has a ∗ C -action, then we have the following isomorphism of CQG’s: ∼ LD . QL D ∼ = QISO+ I (D ) = Q Proof. By Proposition 3.3 we have that QLD is a sub-object of QISO+ I (D ) in the category + QI (D ). On the other hand, by Theorem 2.18 we have QISOI (D ) as a sub-object of QLD in the category QLD . Combining these facts with the conclusion of Proposition 3.7, we get the required isomorphism. 2

Remark 3.9. The assumptions, and hence the conclusions, of this section are valid also for spectral triples obtained by Rieffel deformation of a classical spectral triple, to be discussed in details in Section 5. 4. Examples and computations 4.1. Equivariant spectral triple on SU μ (2) Let μ belongs to [−1, 1]. The C ∗ -algebra SU μ (2) is defined as the universal unital C ∗ -algebra generated by α, γ satisfying: α ∗ α + γ ∗ γ = 1, ∗

∗

(3)

αα + μ γ γ = 1,

(4)

γ γ ∗ = γ ∗γ ,

(5)

2

2554


μγ α = αγ ,

(6)

μγ ∗ α = αγ ∗ .

(7)

There is a coproduct of SU μ (2) given by (α) = α ⊗ α − μγ ∗ ⊗ γ ,

(γ ) = γ ⊗ α + α ∗ ⊗ γ

which makes it into a CQG. Let h denote the Haar state and H = L2 (SU μ (2)) be the corresponding G.N.S. space. For each n in {0, 1/2, 1, . . .}, there is a unique irreducible representation T n of dimension n the ij -th entry of T n . They form an orthogonal basis of H. Denote by en 2n + 1. Denote by ti,j i,j n s so that {en : n = 0, 1/2, 1, . . . , i, j = −n, −n + 1, . . . , n} is an orthonormal the normalized ti,j i,j basis. We recall from [19] that 1/2

t−1/2,−1/2 = α,

t−1/2,1/2 = −μγ ∗ ,

t1/2,−1/2 = γ , t1/2,1/2 = α ∗ .

1/2

1/2

1/2

(8)

Moreover, if fn,i = an,i α n−i γ ∗n+i (where an,i s are some constants as in [19]) then {fn,i : n = −n i n} is an orthonormal basis of SU μ (2). 0, Now, fn+ 1 ,i = c(n, i)αfn,i+ 1 for some constants cn,i . Applying the coproduct on both sides 2 2 and then comparing coefficients, we have the following recursive relations. 1 3 2 , 1, 2 , . . . ,

l l ti,l+1/2 = c11 (i, l)γ ∗ ti+1/2,l + c12 (i, l)α ∗ ti−1/2,l l+1/2

l = c21 (i, l)γ ∗ ti+1/2,l

i = −l − 1/2

l = c31 (i, l)α ∗ ti−1/2,l

i = l + 1/2,

−l + 1/2 i l − 1/2

(9)

and for j l, l+1/2

ti,j

l l = c(l, i, j )αti+1/2,j +1/2 + c (l, i, j )γ ti−1/2,j +1/2 l ∗ l = d(l, j )αt−l,j +1/2 + d (l, j )γ t−l,j − 1

2

l = d (l, j )αti+1/2,j +1/2

i = −l − 1/2, −l +

i = −l − 1/2, j = −l −

l ∗ l = e(l, j )γ ti−1/2,j +1/2 + e (l, j )α ti− 1 ,j − 1 2

−l + 1/2 i l − 1/2

2

1 1 j l− 2 2

1 2

i = l + 1/2;

(10)

, d (l, j ), e(l, j ), e (l, j ) are all complex numbers. where Cpq (il), c(l, i, j ), d(l, j ), dl,j We recall the following multiplication rule from [19] which we are going to need: 1/2

l ti,j ti ,j =

k=|l−1/2|,...,l+1/2

(ck (l, i, j, i , j ) are scalars).

k ck (l, i, j, i , j )ti+i ,j +j

(11)


2555

Consider the spectral triple on SU μ (2) constructed by Chakraborty and Pal [6] and also disn s, cussed thoroughly in [9] which is defined by (A∞ , H, D) where A∞ is the linear span of ti,j and D is defined by n n = (2n + 1)ei,j D ei,j n = −(2n + 1)ei,j

n = i n = i.

Here, we have a cyclic separating vector 1SU μ (2) , and the corresponding faithful state is the Haar state h. Thus, we are in the set up of Section 2.4, and as ξ = 1, A∞ = A00 in this case.

must keep V l := Span{t l : j = Therefore, an operator commuting with D (equivalently with D) i i,j

is the operator as in Section 2.4. −l, . . . , l} invariant for all fixed l and i where D l : l = 0, 1/2, . . .} = A∞ in this In the notation of Corollary 2.27, we have A00 = Span{ti,j case. All the conditions of Theorem 2.24 and Corollary 2.27 are satisfied. Thus, the universal

object of the category

C exists (notation as in Corollary 2.27) and we denote it by Q. Lemma 4.1. Given a CQG Q with an action Φ on A, the following are equivalent: (1) (Q, Φ) is an element of Obj(

C). 1/2 1/2 (2) The action is linear, in the sense that V−1/2 (equivalently, V1/2 ) is invariant under Φ and 1/2

the representation obtained by restricting Φ to V1/2 is a unitary representation. (3) Φ is linear and Haar state preserving. (4) Φ keeps Vil invariant for each fixed l and i.

Φ keeps each of the eigenspaces of D

invariant Proof. (1) ⇒ (2): Since Φ commutes with D, 1/2 and so in particular preserves V−1/2 , i.e. Φ is linear. The condition (h ⊗ id)Φ = h(·).1 implies the unitarity of the corresponding representation. (2) ⇒ (3): By linearity, write Φ(α) = α ⊗ X + γ ∗ ⊗ Y and Φ(γ ∗ ) = α ⊗ Z + γ ∗ ⊗ W. k } for k = 0 and k = 1 follow from the linearity and the fact Firstly, Φ-invariance of Span{ti,j 2 that Φ(1) = 1. 1 : i, j = −1, 0, 1} invariant. Next, we show that Φ keeps Span{ti,j 1 )): We recall the explicit form of the matrix ((ti,j

α∗2 γ ∗α∗ −μγ ∗ 2

−(μ2 + 1)α ∗ γ 1 − (μ2 + 1)γ ∗ γ −(μ2 + 1)γ ∗ α

−μγ 2 αγ α2

.

By inspection, we see that Φ(Vi1 ) ⊆ Vi1 ⊗ Q for i = −1, 1. Hence, it is enough to check the Φ-invariance for αγ and 1 − (μ2 + 1)γ ∗ γ . Comparing coefficients in Φ(αγ ), we can see that it belongs to V01 if and only if XZ ∗ + Y W ∗ = 0. Similarly, in the case of 1 − (μ2 + 1)γ ∗ γ , we have the condition ∗ Z∗ , ZZ ∗ + W W ∗ = 1. But these conditions follow from the unitarity of the matrix X Y∗ W∗ 1/2

which is nothing but the matrix corresponding to the restriction of Φ to V1/2 . Thus, Φ keeps 1 : i, j = −1, 0, 1} invariant. Span{ti,j Moreover, by using the recursive relations (9), (10) and the multiplication rule (11), we can l+1/2 l−1/2 l+1/2 ) ⊆ Vi ⊕ Vi . easily observe that for all l 3/2, Φ(Vi

2556


l : l 1/2} into itself. Using these observations, we conclude that Φ maps Span{ti,j So, in particular, Ker(h) = Span{Vil , i = −l, . . . , l, l 1/2} is invariant under Φ which (along with Φ(1) = 1) implies that Φ preserves h. (3) ⇒ (4): We proceed by induction. The induction hypothesis holds for l = 12 since linearity means that span {α, γ ∗ } is invariant under Φ and hence Span{α ∗ , γ } is also invariant. The case for l = 1 can be checked by inspection as in the proof of (2) ⇒ (3). Consider the induction hypothesis that Φ keeps Vik invariant for all k, i with k l. From the proof of (2) ⇒ (3) we l+1/2 l−1/2 l+1/2 also have for all l 32 , Φ(Vi ) ⊆ Vi ⊕ Vi , by using linearity only. Thus, Φ leaves l− 12

invariant the Hilbert Q module (Vi

preserving. Since Φ leaves invariant l+ 1 Vi 2

l+ 12

⊕ Vi

l− 1 Vi 2

) ⊗ Q, and is a unitary there since Φ is Haar-state

⊗ Q by the induction hypothesis, it must keep its

invariant as well. orthocomplement, (4) ⇒ (1): That Φ preserves the Haar state follows from arguments used in the proof of the l : l 0, i, j = −l, . . . , l}, and Φ keeps each V l implication (2) ⇒ (3). Since A00 = Span{ti,j i

= (D

⊗ id)Φ. 2 invariant, it is obvious that Φ(A00 ) ⊆ A00 ⊗alg Q0 and Φ D We now introduce the compact quantum group Uμ (2). We refer to [19] for more details. Definition 4.2. As a unital C ∗ -algebra, Uμ (2) is generated by 4 elements u11 , u12 , u21 , u22 satisfying: u11 u12 = μu12 u11 , u11 u21 = μu21 u11 , u12 u22 = μu22 u12 , u21 u22 = μu22 u21 u, uu1212u21 = is a u21 u12 , u11 u22 − u22 u11 = (μ − μ−1 )u12 u21 and the condition that the matrix u11 21 u22 unitary. The CQG structure is given by (uij ) = k=1,2 uik ⊗ ukj , κ(uij ) = uj i ∗ , (uij ) = δij . Remark 4.3. Let the quantum determinant Dμ be defined by Dμ = u11 u22 − μu12 u21 = u22 u11 − μ−1 u12 u21 . Then, Dμ ∗ Dμ = Dμ Dμ ∗ = 1. Moreover, Dμ belongs to the centre of Uμ (2). By Lemma 4.1, we have identified the category

C with the category of CQG having C ∗ actions on SU μ (2) satisfying condition (3) of Lemma 4.1. Let the universal object of this category

Γ ). be denoted by (Q, Then by linearity we can write Γ (α) = α ⊗ A + γ ∗ ⊗ B, Γ (γ ∗ ) = α ⊗ C + γ ∗ ⊗ D. Now we shall exploit the fact that Γ is a ∗-homomorphism to get relations satisfied by A, B,

is generated as a C ∗ -algebra by the elements A, B, C, D. C, D where Q Lemma 4.4. A∗ A + CC ∗ = 1,

(12)

A∗ A + μ2 CC ∗ = B ∗ B + DD ∗ ,

(13)


2557

A∗ B = −μDC ∗ ,

(14)

B ∗ A = −μCD ∗ .

(15)

Proof. The proof follows from the relation (3) by comparing coefficients of 1, γ ∗ γ , α ∗ γ ∗ and αγ respectively. 2 Lemma 4.5. AA∗ + μ2 CC ∗ = 1, ∗

∗

(16)

2

BB + μ DD = μ .1,

(17)

BA∗ = −μ2 DC ∗ .

(18)

2

Proof. From Eq. (4) by equating coefficients of 1 and α ∗ γ ∗ , we get respectively (16) and (18) whereas (17) is obtained by equating coefficients of γ ∗ γ and using (16). 2 Lemma 4.6. C ∗ C = CC ∗ , 1 − μ2 C ∗ C = D ∗ D − DD ∗ ,

(19)

C ∗ D = μDC ∗ .

(21)

(20)

Proof. The proof follows from Eq. (5) by comparing the coefficients of 1, γ ∗ γ , α ∗ γ ∗ , respectively. 2 Lemma 4.7. −μ2 AC ∗ + BD ∗ − μD ∗ B + μC ∗ A = 0,

(22)

AC ∗ = μC ∗ A,

(23)

BC ∗ = C ∗ B,

(24)

AD ∗ = D ∗ A.

(25)

Proof. The proof follows from Eq. (6) comparing the coefficients of γ ∗ γ , 1, α ∗ γ ∗ and αγ respectively. 2 Lemma 4.8. AC = μCA,

(26)

BD = μDB, AD − μCB = DA − μ

(27) −1

BC.

(28)

2558


Proof. The proof follows from (7) from the coefficients of α 2 , γ ∗2 , γ ∗ α respectively.

2

Now we consider the antipode, say κ. From the condition (h ⊗ id)Γ (a) = h(a).1, we see that Γ induces a unitary representation of the compact quantum group, say UΓ , given by UΓ (a ⊗ q) = Γ (a)(1 ⊗ q) for a in SU μ (2), q

in Q. 2 α, 1 + μ2 γ ∗ } Now, the restriction of this unitary representation to the orthonormal set { 1+μ 2 μ A μC is given by the matrix: μ−1 B D . Similarly, with respect the orthonormal set { 1 + μ2 α ∗ , 1 + μ2 γ }, this representation is A∗ Cto ∗ given by the matrix: B ∗ D ∗ . Thus, we have κ(A) = A∗ , κ(A∗ ) = A,

κ(D) = D ∗ , κ(C ∗ ) = B,

κ(C) = μ−2 B ∗ , κ(B ∗ ) = C,

κ(B) = μ2 C ∗ , κ(D ∗ ) = D.

Lemma 4.9. AB = μBA,

(29)

CD = μDC,

(30)

BC ∗ = C ∗ B.

(31)

Proof. The relations (29)–(31) follow by applying κ to Eqs. (26), (27) and (24) respectively.

2

defined by φ(u11 ) = A, φ(u12 ) = Lemma 4.10. There exists a ∗-homomorphism φ : Uμ (2) → Q −1 μC, φ(u21 ) = μ B, φ(u22 ) = D. Proof. It is enough to check that the defining relations of Uμ (2) are satisfied. 1. φ(u11 u12 ) = φ(μu12 u11 ) ⇔ φ(u11 )φ(u12 ) = μφ(u12 )φ(u11 ) ⇔ A(μC) = μ(μC)A ⇔ AC = μCA which is the same as (26). 2. φ(u11 u21 ) = φ(μu21 u11 ) ⇔ A(μ−1 B) = μ(μ−1 B)A ⇔ AB = μBA which is the same as Eq. (29). 3. φ(u12 u22 ) = φ(μu22 u12 ) ⇔ μCD = μD(μC) ⇔ CD = μDC which is the same as Eq. (30). 4. φ(u21 u22 ) = φ(μu22 u21 ) ⇔ μ−1 BD = μDμ−1 B ⇔ BD = μDB which is the same as Eq. (27). 5. φ(u12 u21 ) = φ(u21 u12 ) ⇔ μCμ−1 B = μ−1 BμC ⇔ CB = BC. Now, BC ∗ = C ∗ B follows from Eq. (31). But by (19), C is normal, which implies BC = CB. 6. φ(u11 u22 − u22 u11 ) = (μ − μ−1 )φ(u12 u21 ) ⇔ AD − DA = (μ − μ−1 )μCμ−1 B. From (28), we have AD − DA = μCB − μ−1 BC = (μ − μ−1 )CB, using BC = CB. 2 Lemma 4.11. There is a C ∗ -action Ψ of Uμ (2) on SU μ (2) such that (Uμ (2), Ψ ) is an object of Obj(

C) and Ψ is given by


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Ψ (α) = α ⊗ u11 + γ ∗ ⊗ μu21 , Ψ (γ ∗ ) = α ⊗ μ−1 u12 + γ ∗ ⊗ u22 . Proof. The homomorphism conditions are exactly the conditions (12)–(28) with A, B, C, D replaced by u11 , μu21 , μ−1 u12 and u22 respectively. We check one of the relations and remark ∗ that the proof of the others are similar. We prove (12), i.e., u11 ∗ u11 + (μ−1 u12 )(μ−1 u12 ) = 1. Using the fact that Dμ is a central element of Uμ (2), we have u∗11 u11 + μ−2 u12 u∗12 = u22 Dμ−1 u11 + μ−2 u12 −μu21 Dμ−1 = u22 u11 − μ−1 u12 u21 Dμ−1 = Dμ Dμ−1 = 1. 1/2

Clearly, Ψ keeps V−1/2 invariant and the corresponding representation is a unitary. It follows from Lemma 4.1 that (Uμ (2), Ψ ) is an object of

C. 2

to Uμ (2) sending A, μC, Corollary 4.12. There exists a surjective CQG morphism from Q −1 μ B, and D to u11 , u12 , u21 and u22 respectively. + (D) ∼

∼ Theorem 4.13. We have Q = Uμ (2) and hence QISO = Uμ (2) ∗ C(T). Proof. The first part follows from Lemma 4.10 and Corollary 4.12 and the second part follows from Theorem 2.24. 2 4.2. The Podles spheres 2 is defined to be Let μ ∈ (0, 1), t ∈ (0, 1], c = t −1 − t > 0. The Podles sphere (as in [21]) Sμ,c ∗ the universal unital C -algebra generated by A, B satisfying

A∗ = A, B ∗ B = A − A2 + cI,

AB = μ−2 BA, BB ∗ = μ2 A − μ4 A2 + cI.

We shall consider two different classes of spectral triples on this C ∗ -algebra and describe the + corresponding QISO+ R or QISO , which are computed in [4]. (I) The spectral triple of Dabrowski et al.: 2 is isomorphic with the unital C ∗ -subalgebra of It is known that (see [12]) the C ∗ -algebra Sμ,c and B where SU μ (2) generated by A −1 ∗ −1 2 ∗ −1 ∗ = 1 + t μγ α − t ρ(1 − (1 + μ )γ γ ) + t γ α , A 1 + μ2 −1 2 2 2 2 = t [μα + ρ(1 + μ )αγ − μ γ ] , B 2 1+μ

where ρ 2 =

μ2 t 2 . 2 (μ2 +1) (1−t)

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Identifying elements of SU μ (2) with the operators by left multiplication on the G.N.S. space B leaves the subspace L2 (SU μ (2), h), it can be verified that A, K = Span el

1 3 : l = , , . . . , m = −l, −l + 1, . . . , l 1 ± 2 ,m 2 2

l 2 → invariant (where ek,j are as in the previous subsection) and the ∗-homomorphism π0 : Sμ,c K and B| K respectively is faithful, so we can identify B(K) obtained by sending A and B to A| 2 with the image of π . Sμ,c 0 Let D and R be defined by

l l D e± = (c1 l + c2 )e∓ , 1 1 ,m ,m 2

2

where c1 , c2 ∈ R, c1 = 0, n n R e± = μ−2i e± 1 1 . ,i ,i 2

2

Let A∞ be the unital ∗-subalgebra (without norm closure) generated by A, B. Then is an R-twisted spectral data and we have the following (see [4] for the proof):

(A∞ , K, D, R)

Theorem 4.14. The CQG QISO+ R (D) is isomorphic with SOμ (3), where SOμ (3) is the Woronow1 : i, j = −1, 0, 1}. icz C ∗ -subalgebra of SU μ (2) generated by the elements {ti,j (II) The spectral triple constructed by Chakraborty and Pal in [7] for c > 0: Let H+ = H− = l 2 (N ∪ {0}), H = H+ ⊕ H− . Let en be an orthonormal basis of H+ = H− and N be the operator defined on it by N(en ) = nen . 2 → B(H ) as in [7], given by Consider the irreducible representations π± : Sμ,c ± π± (A)en = λ± μ2n en , 1

π± (B)en = c± (n) 2 en−1 , 1

2

where e−1 := 0, and λ± = 12 ± (c + 14 ) 2 , c± (n) = λ± μ2n − (λ± μ2n ) + c. Let π = π+ ⊕ π− . 2 with π(S 2 ). The representation π is faithful and we identify Sμ,c μ,c 0 N Let D = N 0 . Then (A∞ , π, H, D) is a spectral triple of compact type.


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We showed in [4] that Theorem 4.15. (1) For the above spectral triple, QISO+ (D) exists and is isomorphic with C(Z2 ) ∗ C(T)∗∞ , where C(T)∗∞ denotes the free product of countably infinitely many copies of C(T). (2) If U denotes the unitary representation of QISO+ (D) on H, the ∗-homomorphism αU is not 2 . a C ∗ -action on Sμ,c (3) The subcategory of Q (D) consisting of objects (Q, V ) for which αV is a C ∗ -action does not admit a universal object. Remark 4.16. The above example shows that unlike the classical case, where isometry groups are Lie groups and hence have faithful imbedding into a matrix group, QISO+ in general may fail to be a compact matrix quantum group. In fact, it will be quite interesting to find conditions under which QISO+ will be so. 4.3. A commutative example: Spectral triple on T2 We consider the spectral triple (A∞ , H, D) on T2 given by A∞ = C ∞ (T2 ), H = L2 (T2 ) ⊕ 0 d1 +id2 L2 (T2 ) and D = d1 −id , where we view C(T2 ) as the universal C ∗ -algebra generated 0 2 by two commuting unitaries U and V , and d1 and d2 are derivations on A∞ defined by d1 (U ) = U,

d1 (V ) = 0,

d2 (U ) = 0,

d2 (V ) = V .

(32)

The vectors e1 = (1, 0) and e2 = (0, 1) form an orthonormal basis of the eigenspace corresponding to the eigenvalue zero. The Laplacian in the sense of [17] exists in this case, and is given by L(U m V n ) = −(m2 + n2 )U m V n . We recall that we denote the quantum isometry group from the Laplacian L in the sense of [17] by QLD . W ) be an object of Q (D). Then the ∗-homomorphism α = αW must be of Lemma 4.17. Let (Q, the following form: α(U ) = U ⊗ z1 ,

(33)

α(V ) = V ⊗ z2 ,

(34)

where z1 , z2 are two commuting unitaries. which acts on C(T2 ) faithfully Proof. We denote the maximal Woronowicz C ∗ -subalgebra of Q by Q. We observe that D 2 (aei ) = L(a)ei for i = 1, 2. The fact that U commutes with D implies that U commutes with D 2 as well, and hence α commutes with the Laplacian L. Therefore, Q is a quantum subgroup of QLD . From [3], we conclude that QLD = C(T2 >Z2 3 ). Thus Q must be of the form C(G) for a classical subgroup G of the orientation preserving isometry group of T2 , which is T2 itself and whose (co-)action is given by U → U ⊗ U and V → V ⊗ V . 2

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+ (C ∞ (T2 ), H, D) exists and is isomorphic with Theorem 4.18. The universal CQG QISO 2 ∗ 2 ∼ C(T ) ∗ C(T) = Cr (Z ∗ Z) (as a CQG). Moreover, QISO+ of this spectral triple is isomorphic with C(T2 ). W ) be an object in Q (D) as in Lemma 4.17. Since {e1 , e2 } is an orthonormal Proof. Let (Q, basis for an eigenspace of D, we must have W (e1 ) = e1 ⊗ q11 + e2 ⊗ q12 ,

(35)

W (e2 ) = e1 ⊗ q21 + e2 ⊗ q22 ,

(36)

By comparing coefficients of U e1 and U e2 in the both sides of the equality for some qij in Q. (D ⊗ I )W (U e1 ) = W DU e1 , we have, z1 q12 = z1 q21

(37)

z1 q11 = z1 q22 ,

(38)

and

where z1 , z2 are as in Lemma 4.17. Since z1 is a unitary, we have q11 = q22 and q12 = q21 . Similarly, from the relation (D ⊗ I )W (V e1 ) = W DV e1 , we have q12 = −q21 , q22 = q11 . By the above two sets of relations, we obtain q12 = q21 = 0,

q11 = q22 = q

(say).

q q12 so q is a unitary. But the matrix q11 is a unitary in M2 (Q), 21 q22 Moreover, we note that W (aei ) = α(a)W (ei ) for all a in C ∞ (T2 ). Using Lemma 4.17 and the above observations, we deduce that any CQG which has a unitary representation commuting with the Dirac operator is a quantum subgroup of C(T2 ) ∗ C(T). On the other hand, C(T2 ) ∗ C(T) has a unitary representation commuting with D, given by the formulae (33)–(36) taking q12 = q21 = 0, q11 = q22 = q where q is the generator of C(T) and z1 , z2 to be the generators of C(T2 ). This completes the proof. 2 2 2 2 2 Remark 4.19. The canonical 0 1 grading on C(T ) is given by the operator (id ⊗ γ ) on L (T ⊗ C ) where γ is the matrix −1 0 . The representation of C(T2 ) ∗ C(T) clearly commutes with the grading operator and hence is 2 ), L2 (T2 ⊗ C2 ), D, γ ). isomorphic with QISO(C(T

Remark 4.20. This example shows that the conditions of Theorem 2.24 are not necessary for the + . existence of QISO 4.4. Another class of commutative examples: The spheres We consider the usual Dirac operator on the classical n-sphere S n . In fact, we shall first consider a slightly more general set-up as in Section 3.5, pp. 82–89 of [14], which we very briefly recall here. Let G be a compact Lie group, K a closed subgroup, and let M be the homogeneous


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space G/K with a G-invariant metric. The algebra C ∞ (M) is identified with the algebra (say A∞ ) of K-invariant functions in C ∞ (G), i.e. functions f satisfying f (gk) = f (g) for all g in G, k in K. The Lie algebra g of G splits as a vector space direct sum g = k + m where k is the Lie algebra of K and m is a suitable Ad(K)-invariant subspace of g (see [14] for more details). Thus we have the representation of K given by Ad : K → SO(m), and the corresponding : K → Spin(m). The space of smooth spinors can then be identified with the space of lift Ad −1 ))ψ(g) for all g in G, k in K, where smooth functions ψ : G → satisfying ψ(gk) = κ(Ad(k κ : Spin(m) → GL() denotes the spin representation. The action of C ∞ (M), identified with the K-invariant smooth functions on G, is given by multiplication, and the Dirac operator D is given by Dψ =

m

Xi · Xi (ψ),

i=1

where m = dim(m) and {X1 , . . . , Xm } is an orthonormal basis of m with respect to the suitable invariant inner product described in [14] and · denotesthe Clifford multiplication. From this expression of D, we get [D, f ]ψ = ωf .ψ where ωf = i (Xi f )Xi , by using the fact that Xi ’s 1 , which is are acting as derivations on the algebra of smooth functions. In fact, the space ΩD isomorphic with the (complexified) space of smooth 1-forms on M, can now be identified with the space of (smooth) AdK invariant functions from G to m ∼ = Cm (which is also isomorphic with ∞ m this identification, is nothing but the map which sends C (M)⊗C ), and the map dD is, through an element f of A∞ ∼ = C ∞ (M) to i Xi (f ) ⊗ Xi which is in A∞ ⊗ Cm . The Hilbert space of 1-forms is isomorphic with L2 (M) ⊗ Cm , where L2 (M) is the Hilbert space completion with respect to the G-invariant volume measure, and from the G-invariance of the volume measure it is clear that the adjoint Xi∗ of the (left invariant) vector field Xi , viewed as a closable unbounded m ∗d =− ∗ map on L2 (M), is −Xi . It follows that the Laplacian is given by, dD D i=1 Xi Xi on ∞ ∞ ∼ A = C (M). It is in fact nothing but the Casimir ΩG in the notation of [14], since Yf = 0 for any Y in k. Now, we want to apply the above observations to the special case of n-spheres. The Laplacian on such spheres considered in [28, p. 17] is indeed the Casimir operator and so by Theorem 2.2 and Remark 3.3 of [3] the corresponding quantum isometry group QISOL is commutative as a C ∗ -algebra. However, by Corollary 3.2 of the present paper, any object of the category QI must be a quantum subgroup of QLD , so is in particular commutative as a C ∗ -algebra, and must be of the form C(G) for a subgroup G of the universal group of orientation preserving (classical) Riemannian isometries of S n , i.e. SO(n + 1). To summarize, we have the following: n Theorem 4.21. The quantum isometry group QISO+ I (S ) is isomorphic with C(SO(n + 1)).

5. QISO+ of deformed spectral triples In this section, we give a general scheme for computing orientation-preserving quantum isom+ of a deformed noncommutative manifold coincides with etry groups by proving that QISO R

+ of the original manifold. (under reasonable assumptions) a similar deformation of the QISO R The technique is very similar to the analogous result for the quantum isometry groups in terms of Laplacian discussed in [3], so we often merely sketch the arguments and refer to a similar theorem or lemma in [3].

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We recall the generalities on compact quantum groups from Section 2.2. In particular, given a compact quantum group (S, ), the dense unital ∗-subalgebra S0 of S generated by the matrix coefficients of the irreducible unitary representations has a canonical Hopf ∗-algebra structure. Moreover, given an action γ : B → B ⊗ S of the compact quantum group (S, ) on a unital C ∗ -algebra B, it is known that one can find a dense, unital ∗-subalgebra B0 of B on which the action becomes an action by the Hopf ∗-algebra S0 . We shall use the Sweedler convention of abbreviating γ (b) ∈ B0 ⊗alg S0 by b(1) ⊗ b(2) , for b in B0 . This applies in particular to the canonical action of the quantum group S on itself, by taking γ = . Moreover, for a linear functional f on S and an element c ∈ S we shall define the ‘convolution’ maps f c := (f ⊗ id)(c) and c f := (id ⊗ f )(c). We also define convolution of two functionals f and g by (f g)(c) = (f ⊗ g)((c)). We also need the following: Definition 5.1. Let (S, S ) be a compact quantum group. A vector space M is said to be an algebraic S co-module (or simply S co-module) if there exists a linear map α : M → M ⊗alg S0 such that 1. ( α ⊗ id) α = (id ⊗ S ) α; 2. (id ⊗ ) α (m) = (m)1S for all m in M. Let (A, Tn , β) be a C ∗ -dynamical system and π0 : A → B(H) be a faithful representation, where H is a separable Hilbert space. We shall often identify A with π0 (A). Let A∞ be the algebra of smooth (C ∞ ) elements with respect to the Tn action β corresponding to the Tn action β. Then for each skew-symmetric n × n real matrix J , we refer to [23] for the construction of the ‘deformed’ C ∗ -algebra AJ and their properties. Assume now that we are given a spectral triple (A∞ , π0 , H, D) of compact type. Suppose that D has eigenvalues {λ0 , λ1 , . . .} and Vi denotes the (finite-dimensional) eigenspace of λi and let S00 denote the linear span of {Vi : i = 0, 1, 2, . . .}. n , with a covering map Suppose, furthermore, that there exists a compact abelian Lie group T n n n are isomorphic with Rn and we denote by e n → T . The Lie algebra of both T and T γ :T and e˜ respectively the corresponding exponential maps, so that e(u) = e(2πiu) where u is in Rn and γ (e(u)) ˜ = e(u). By a slight abuse of notation we shall denote the Rn -action βe(u) by βu . We also make the following assumption. n , of T n on H such that There exists a strongly continuous unitary representation Vg˜ , g˜ ∈ T (a) Vg˜ D = DVg˜ for all g, ˜ −1 n , and g = γ (g). ˜ (b) Vg˜ π0 (a)Vg˜ = π0 (βg (a)), where a is in A, g˜ is in T We shall now show that we can ‘deform’ the given spectral triple along the lines of [11]. For each J , the map πJ : A∞ → Lin(H∞ ) (where H∞ is the smooth subspace corresponding to the representation V and Lin(V) denotes the space of linear maps on a vector space V) defined by πJ (a)s ≡ a ×J s :=

v (s)e(u.v) du dv βJ u (a)β

extends to a faithful ∗-representation of the C ∗ -algebra A∞ in B(H) where βv = Ve(v) (which ˜ clearly maps H∞ into H∞ ).


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We can extend the action of Tn on the C ∗ -subalgebra A1 of B(H) generated by A, {eitD : t ∈ R} and elements of the form {[D, a]: a ∈ A∞ } by βg (X) = Vg˜ XVg˜ −1 for all X in A1 where by an abuse of notation, we denote the action by the same symbol β. Let A1 ∞ denote the smooth vectors of A1 with respect to this action. We note that for all a in A∞ 1 , [D, a] is in A∞ . 1 Lemma 5.2. β is a strongly continuous action (in the C ∗ -sense) of Tn on A1 and hence for all v (s)e(u.v) du dv is a bounded operator. X in A1 ∞ , πJ (X) defined by πJ (X)s = βJ u (X)β Proof. We note that β is already strongly continuous on the C ∗ -algebra generated by A, {eitD : t ∈ R}. Thus it suffices to check the statement for elements of the form [D, a] where a is in A∞ . To this end, fix any one parameter subgroup gt of G such that gt goes to the identity of G as t → 0. Let Tt , Tt denote the group of normal ∗-automorphisms on B(H) defined by Tt (X) = Vgt XVg −1 and Tt (X) = eitD Xe−itD . As Vgt and D commute, so do their generators. In particular, t each of these generators leaves the domain of the other invariant. Note also that A∞ is in the domain of the both the generators, and the generator of Tt is given by i[D, ·] there. Thus, for a in A∞ , we have that a, [D, a] must be in the domain of the generator of Tt , say Ξ, and Ξ ([D, a]) = [D, Ξ (a)] belongs to B(H). t Using this, we obtain Tt ([D, a]) − [D, a] = 0 Ts (Ξ ([D, a])) ds tΞ ([D, a]). The required strong continuity follows from this. Then applying Theorem 4.6 of [23] to the C ∗ algebra A1 and the action β, we deduce that πJ (X) is a bounded operator. 2 Lemma 5.3. For each J , (A∞ J , πJ , H, D) is a spectral triple of compact type. Proof. It suffices to prove that [D, πJ (a)] is a bounded operator. Now, [D, πJ (a)](s) = D βJ u (a)βv (s)e(u.v) du dv − βJ u (a)βv (Ds)e(u.v) du dv. Using the expression f (u, v)e(u.v) = limL p∈L (f φp )(u, v)e(u.v) du dv (where notations are as in [23, pp. 4–5]) and closability of D, we have D

βJ u (a)βv (s)e(u.v) du dv =

D βJ u (a)βv (s) e(u.v) du dv.

Since D commutes with V , we get D, πJ (a) (s) =

= =

D βJ u (a)βv (s) e(u.v) du dv −

βJ u (a)D βv (s)e(u.v) du dv

D, βJ u (a) βv (s)e(u.v) du dv VJu [D, a]VJu −1 βv (s)e(u.v) du dv

= πJ [D, a] (s). Thus, [D, πJ (a)] = πJ ([D, a]) which is a bounded operator by Lemma 5.2.

2

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U ) is an element of Obj(Q(A, H, D)), and there exists a unital Lemma 5.4. Suppose that (Q, ∗-subalgebra A0 ⊆ A which is norm dense in every AJ such that αU (π0 (A0 )) ⊆ π0 (A0 ) ⊗alg Q0 , is the smallest Woronowicz C ∗ -subalgebra for which αU (A0 ) ⊆ π0 (A0 ) ⊗ Q, where Q ⊆ Q and Q0 is the Hopf ∗-algebra obtained by matrix coefficients of irreducible unitary (co)representations of Q. Also, let S0 = Span{as: a ∈ A0 , s ∈ S00 }. Then we have the following 0 . (a) U (S0 ) ⊆ S0 ⊗alg Q 0 makes S0 an algebraic Q 0 co-module, satisfying for all a in (b) α˜ := U |S0 : S0 → S0 ⊗alg Q A0 , s in S0 , ˜ α˜ π0 (a)s = αU (a)α(s). n ) is a sub-object of Q in Q(A, H, D), then C(Tn ) is a quantum subgroup of Q. (c) If C(T Proof. Since U commutes with D and hence preserves the eigenspaces of D, U must preserve S00 . Thus, U (as) = α(a)U (s) ⊆ (A0 ⊗ Q0 )(S00 ⊗ Q0 ) ⊆ S0 ⊗ Q0 , which proves (a). The assertion (b) follows from the definition of α and αu . Tn ) is We now prove (c). Let us denote by γ ∗ the dual map of γ , so that γ ∗ : C(Tn ) → C( ∗ ∗ an injective C -homomorphism. It is quite clear that (id ⊗ πQ ) ◦ α(A0 ) ⊆ Im(id ⊗ γ ), hence ∗ ∗ −1 ◦ π is a surjective CQG morphism from Q to we have πQ (Q0 ) ⊆ Im(γ ). Thus, πQ := (γ ) Q C(Tn ), which identifies C(Tn ) as a quantum subgroup of Q. 2 Remark 5.5. From the definitions of A0 and S0 , it follows that (i) π0 (A0 )S0 ⊆ S0 ; (ii) for all g, βg (A0 ) ⊆ A0 . U ) as in the statement of Lemma 5.4. From now on, we will Let us now fix the object (Q, ∞ ) and often write π (a) simply as a. with π (A identify A∞ J 0 J n := eve(u) ◦πQ We define Ω(u) := eve(u) ◦πQ , Ω(u) , for u in R , where evx (respectively evx˜ ) ˜ denotes the state on C(Tn ) (respectively, on C( Tn )) obtained by evaluation of a function at the point x (respectively x). ˜ For a fixed J , we shall work with several multiplications on the vector space A0 ⊗alg Q0 . We shall denote the counit and antipode of Q0 by and κ respectively. Let us define the following xy =

e(−u.v)e(w.s) Ω(−J u) x Ω(J w) Ω(−v) y Ω(s) du dv dw ds,

R4n

where x, y are in Q0 . This is clearly a bilinear map, and will be seen to be an associative multiplication later on. Moreover, we define two bilinear maps • and •J by setting (a ⊗ x) • (b ⊗ y) := ab ⊗ x y and (a ⊗ x) •J (b ⊗ y) := (a ×J b) ⊗ (x y), for a, b in A0 , x, y in Q0 . We have Ω(u) (Ω(v) c) = (Ω(u) Ω(v)) c. Lemma 5.6. 1. The map satisfies


Ω(J u) x Ω(v) y e(u.v) du dv =

R2n

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x Ω(J u) y Ω(v) e(u.v) du dv,

R2n

for x, y belonging to Q0 . 2. ˜ (2) ) , u (s) = s(1) ⊗ id ⊗ Ω(u) (s α β ˜ 2 ). α βu (a) = a1 ⊗ id ⊗ Ω(u) (a 0 , we have 3. For s in S, a in Q α (a ×J s) = a(1) s(1) ⊗

(a(2) J u)(s(2) v)e(u.v) du dv .

4. For s belonging to S0 and a in A0 , α (s) = a(1) s(1) ⊗ α(a) •J

Ω(J u) a(2) Ω(v) s(2) e(u.v) du dv .

5. For a in A0 , s in S we have α(a) •J α(s) = α (a ×J s). Proof. The proofs follow verbatim those in Lemmas 3.2–3.6 respectively in [3].

2

Let us recall at this point the Rieffel-type deformation of compact quantum groups as in [30]. We shall now identify with the multiplication of a Rieffel-type deformation of Q. Since Q has a quantum subgroup isomorphic with Tn , we can consider the following canonical action λ of R2n on Q given by λ(s,u) = Ω(−s) ⊗ id id ⊗ Ω(u) . Now, let J:= −J ⊕ J , which is a skew-symmetric 2n × 2n real matrix, so one can deform Q by defining the product of x and y (x, y belonging to Q0 , say) to be the following

λJ(u,w) (x)λv,s (y)e (u, w).(v, s) d(u, w) d(v, s).

We claim that this is nothing but introduced before. Lemma 5.7. One has x y = x ×J y for all x, y in Q0 . Proof. The proof is the same as Lemma 3.7 in [3].

2

Let us denote by QJ the C ∗ -algebra obtained from Q by the Rieffel deformation with respect to the matrix J described above. It has been shown in [30] that the coproduct on Q0 extends to a coproduct for the deformed algebra as well and (QJ, ) is a compact quantum group.

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We recall Lemma 3.8 of [3], which is stated below for reader’s convenience: Lemma 5.8. The Haar state (say h) of Q coincides with the Haar state on QJ (say hJ ) on the common subspace Q∞ , and moreover, h(a ×J b) = h(ab) for a, b in Q∞ . We note a useful implication of the above lemma. Let us make use of the identification of Q0 as a common vector-subspace of all QJ. To be precise, we shall sometimes denote this identification map from Q0 to QJ by ρJ . Corollary 5.9. Let W be a finite-dimensional (say, n-dimensional) unitary representation of Q, with W ∈ Mn (C) ⊗ Q0 be the corresponding unitary. Then, for any J, we have that WJ := (id ⊗ ρJ )(W ) is unitary in QJ, giving a unitary n-dimensional representation of QJ. In other words, any finite-dimensional unitary representation of Q is also a unitary representation of QJ. Proof. Since the coalgebra structures of Q and QJ are identical, and WJ is identical with W as a linear map, it is obvious that WJ gives a nondegenerate representation of QJ. Let y = (id ⊗ h)(WJ∗ WJ ). It follows from the proof of Proposition 6.4 of [20] that y is invertible positive 1

1

element of Mn and (y 2 ⊗ 1)WJ (y − 2 ⊗ 1) gives a unitary representation of QJ. We claim that y = 1, which will complete the proof of the corollary. For convenience, let us write W in the Sweedler notation: W = w(1) ⊗ w(2) . We note that by Lemma 5.8, we have ∗ ∗ (id ⊗ h) WJ∗ WJ = w(1) w(1) h w(2) ×J w(2) ∗ ∗ w(1) h w(2) w(2) = w(1) = (id ⊗ h)(W ∗ W ) = (id ⊗ h)(1 ⊗ 1) = 1.

2

Let us consider the finite-dimensional unitary representations U (i) := U |Vi , where Vi is the eigenspace of D corresponding to the eigenvalue λi . By the above Corollary 5.9, we can view (i) U (i) as a unitary representation of QJ as well, and let us denote it by UJ . In this way, we obtain a unitary representation UJ on the Hilbert space H, which is the closed linear span of all the Vi ’s. It is obvious from the construction (and the fact that the linear span of Vi ’s, i.e. S0 , is a core for D) that UJ (D ⊗ I ) = (D ⊗ I )UJ . Let αJ := αUJ . With this, we have the following Lemma 5.10. For a belonging to A0 , we have αJ (a) = (α(a))J ≡ (πJ ⊗ ρJ )(α(a)), and hence in particular, for every state φ on QJ, (id ⊗ φ) ◦ αJ (AJ ) ⊆ AJ . Proof. Using Lemma 5.6, we have, for all s in S0 , a in A0 , UJ πJ (a)s = α (a ×J s) = α(a) •J α (s) = α(a) J UJ (s), from which we conclude by the density of S0 in H that αJ (a) = (α(a))J ∈ πJ (A0 ) ⊗ QJ. The lemma now follows using the norm-density of A0 in AJ . 2


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,UJ ) is an orientation preserving isometric action on the spectral triple (A∞ ,H, D). Thus, (Q J J We shall now show that if we fix a ‘volume-form’ in terms of an R-twisted structure, then the ‘deformed’ action αJ preserves it. Lemma 5.11. Suppose, in addition to the set-up already assumed, that there is an invertible positive operator R on H such that (A∞ , H, D, R) is an R-twisted Θ-summable spectral triple of compact type, and let τR be the corresponding ‘volume form’. Assume that αU preserves the functional τR . Then the action αUJ preserves τR too. Proof. Let the (finite-dimensional) eigenspace corresponding to the eigenvalue λn of D be Vn . As U commutes with D, there exist subspaces Vn,k of Vn and an orthonormal basis {ejn,k } for j n . Vn,k such that the restriction of U to Vn,k is irreducible. Write U (ejn,k ⊗ 1) = i ein,k ⊗ ti,j n∗ . Then, U ∗ (ejn,k ) = n,i ein,k ⊗ tj,i Then H will be decomposed as H = n1,k Vn,k . Let R(ejn,i ) = s,t Fn (i, j, s, t)etn,s . By hypothesis, U (· ⊗ id)U ∗ preserves the functional τR (·) = Tr(R ·) on ED where ED is as in Proposition 2.8, i.e., the weakly dense ∗-subalgebra of B(H) generated by the rank one operators |ξ η| where ξ, η are eigenvectors of D. Thus, (τR ⊗ id)(U (X ⊗ id)U ∗ ) = τR (X).1Q for all X in ED . Then, for a in ED , we have: n,i ej ⊗ 1, UJ (a ⊗ 1)UJ∗ Rejn,i ⊗ 1 (τR ⊗ h) UJ (a ⊗ 1)UJ∗ = n,i,j

=

UJ∗ ejn,i ⊗ 1 , (a ⊗ 1)UJ∗ Fn (i, j, s, t)etn,s ⊗ 1

n,i,j,s,t

=

n ∗ n ∗ Fn (i, j, s, t) ekn,i ⊗ tj,k , (a ⊗ 1) eln,s ⊗ tt,l

n,i,j,s,t,k,l

=

n n ∗ ×J tt,l Fn (i, j, s, t) ekn,i , aeln,s hJ tj,k

n,i,j,s,t,k,l

=

n n ∗ Fn (i, j, s, t) ekn,i , aeln,s h0 tj,k tt,l

n,i,j,s,t,k,l

= (τR ⊗ h) U (a ⊗ 1)U ∗ = τR (a).1, n ) × (t n )∗ ) = h (t n t n ∗ ) as deduced by using Lemma 5.8. where hJ ((tj,k J t,l 0 j,k t,l Thus (τR ⊗ h)(UJ (a ⊗ id)UJ ∗ ) = τR (a).1. Let (τR ⊗ h)(UJ (X ⊗ id)UJ∗ ) = (τR ∗ h)(X). As UJ (· ⊗ id)UJ∗ keeps ED invariant, we can , to have use Sweedler notation: UJ (a ⊗ 1)UJ∗ = a(1) ⊗ a(2) , with a, a(1) in ED , a(2) in Q J

(τR ⊗ id) UJ (a ⊗ 1)UJ∗ = (τR ∗ h ⊗ id) UJ (a ⊗ 1)UJ∗ = (τR ∗ h)(a(1) )a(2)

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= (τR ⊗ h ⊗ id)(a(1)(1) ⊗ a(1)(2) ⊗ a(2) ) = (τR ⊗ h ⊗ id)(id ⊗ J) UJ (a ⊗ 1)UJ∗ = τR (a(1) )(h ⊗ id) ◦ J(a(2) ) = τR (a(1) )h(a(2) ).1QJ = (τR ⊗ h)(a(1) ⊗ a(2) ) = (τR ∗ h)(a).1QJ = τR (a).1QJ.

2

+ ∞ ∞ ∗ Remark 5.12. If QISO+ R (A , H, D) (QISO (A , H, D), if it exists) has a C -action, ∗ then from the definition of a C -action, we get a subalgebra A0 as in Lemma 5.4. Thus, ∞ the conclusions of Lemma 5.4 and the subsequent lemmas hold for QISO+ R (A , H, D) + ∞ (QISO (A , H, D)).

, UJ ) is an object in Q(AJ , H, D). Now, proceeding as in the We have already seen that (Q J proof of Theorem 3.13 of [3] we obtain the following result (using Lemma 5.11 for 1). Theorem 5.13. + ∞ ∞ ∗ 1. If QISO+ R (AJ , H, D) and (QISOR (A , H, D))J have C -actions on A and AJ respectively, we have

+ ∞ + A∞ , H, D ∼ QISO = QISO J R R A , H, D J, ∞ + ∞ ∼ QISO+ R AJ , H, D = QISOR A , H, D J. + + (A∞ , H, D) both exist and have C ∗ -actions on (A∞ , H, D) and QISO 2. If moreover, QISO J A and AJ respectively, then

+ ∞ + A∞ , H, D ∼ A , H, D J, QISO = QISO J ∞ + ∼ QISO+ A∞ J , H, D = QISO A , H, D J. As an example, we consider the noncommutative torus Aθ , which is a Rieffel deformation of 0 θ C(T2 ) with respect to the matrix J = −θ and we deform the spectral triple as in Section 4. 0 This is the standard spectral triple on Aθ . + (A∞ , H, D) ∼ + (C ∞ (T2 )) ∼ Theorem 5.14. We have QISO = QISO = C(T2 ) ∗ C(T), and θ + + ∞ ∞ 2 2 QISO (Aθ ) ∼ = QISO (C (T )) = C(T ). Proof. We use Theorem 5.13 and recall that QISO+ (C ∞ (T2 )) ∼ = C(T2 ) which is generated by z1 and z2 , say. Then, from the formula of the deformed product, it can easily be seen after a change of variable that z1 ×J z2 = z2 ×J z1 which proves the theorem. 2 Remark 5.15. In a private communication S. Wang has kindly pointed out that one can possibly formulate and prove an analogue of Theorem 5.13 in the setting of discrete deformation as in [33], and this may give a solution to a problem posed by Connes (see [8, p. 612]). We believe that more work is needed in this direction.


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Acknowledgments We would like to thank S. Wang and T. Banica for many valuable comments and feedback which have led to substantial improvement of an earlier version of this article. References [1] T. Banica, Quantum automorphism groups of small metric spaces, Pacific J. Math. 219 (1) (2005) 27–51. [2] T. Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2) (2005) 243–280. [3] J. Bhowmick, D. Goswami, Quantum isometry groups: Examples and computations, Comm. Math. Phys. 285 (2) (2009) 421–444. [4] J. Bhowmick, D. Goswami, Quantum isometry groups of the Podles spheres, arXiv:0810.0658. [5] J. Bichon, Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (3) (2003) 665–673. [6] P.S. Chakraborty, A. Pal, Equivariant spectral triples on the quantum SU(2) group, K-Theory 28 (2003) 107–126. [7] P.S. Chakraborty, A. Pal, Spectral triples and associated Connes–de Rham complex for the quantum SU(2) and the quantum sphere, Comm. Math. Phys. 240 (3) (2003) 447–456. [8] A. Connes, Noncommutative Geometry, Academic Press, London/New York, 1994. [9] A. Connes, Cyclic cohomology, quantum group symmetries and the local index formula for SUq (2), J. Inst. Math. Jussieu 3 (1) (2004) 17–68. [10] A. Connes, On the spectral characterization of manifolds, preprint, arXiv:0810.2088v1. [11] Alain Connes, Michel Dubois-Violette, Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, Comm. Math. Phys. 230 (3) (2002) 539–579. [12] L. Dabrowski, F. D’Andrea, G. Landi, E. Wagner, Dirac operators on all Podles quantum spheres, J. Noncommut. Geom. 1 (2007) 213–239. [13] L. Dabrowski, G. Landi, A. Sitarz, W. van Suijlekom, Joseph C. Varilly, The Dirac operator on SUq (2), Comm. Math. Phys. 259 (3) (2005) 729–759. [14] T. Friedrich, Dirac Operators in Riemannian Geometry, Grad. Stud. Math., vol. 25, American Math. Soc., Providence, RI, 2000. [15] J. Fröhlich, O. Grandjean, A. Recknagel, Supersymmetric quantum theory and non-commutative geometry, Comm. Math. Phys. 203 (1) (1999) 119–184. [16] D. Goswami, Twisted entire cyclic cohomology, JLO cocycles and equivariant spectral triples, Rev. Math. Phys. 16 (5) (2004) 583–602. [17] D. Goswami, Quantum group of isometries in classical and noncommutative geometry, Comm. Math. Phys. 285 (1) (2009) 141–160. [18] D. Goswami, Some remarks on the action of quantum isometry groups, preprint, arXiv:0811.3063. [19] A. Klimyk, K. Schmudgen, Quantum Groups and Their Representations, Springer, New York, 1998. [20] A. Maes, A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Wiskd. (4) 16 (1–2) (1998) 73–112. [21] P. Podles, Quantum spheres, Lett. Math. Phys. 14 (1987) 193–202. [22] P. Podles, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys. 170 (1995) 1–20. [23] Mark A. Rieffel, Deformation quantization for actions of Rd , Mem. Amer. Math. Soc. 106 (206) (1993). [24] S. Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge University Press, Cambridge, 1997. [25] W. Rudin, Functional Analysis, Tata/McGraw–Hill, New Delhi, 1974. [26] P.M. Soltan, Quantum families of maps and quantum semigroups on finite quantum spaces, J. Geom. Phys. 59 (3) (2009) 354–368. [27] S.M. Srivastava, A Course on Borel Sets, Grad. Texts in Math., vol. 180, Springer-Verlag, New York, 1998. [28] J.C. Varilly, An Introduction to Noncommutative Geometry, EMS Ser. Lect. Math., 2006. [29] S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (3) (1995) 671–692. [30] S. Wang, Deformation of compact quantum groups via Rieffel’s quantization, Comm. Math. Phys. 178 (1996) 747– 764. [31] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998) 195–211. [32] S. Wang, Ergodic actions of universal quantum groups on operator algebras, Comm. Math. Phys. 203 (2) (1999) 481–498. [33] S. Wang, Rieffel type discrete deformation of finite quantum groups, Comm. Math. Phys. 202 (1999) 291–307.

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[34] S. Wang, Structure and isomorphism classification of compact quantum groups Au (Q) and Bu (Q), J. Operator Theory 48 (2002) 573–583. [35] S.L. Woronowicz, Pseudogroups, pseudospaces and Pontryagin duality, in: Proceedings of the International Conference on Mathematical Physics, Lausane, 1979, in: Lecture Notes in Phys., vol. 116, 1980, pp. 407–412. [36] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (4) (1987) 613–665. [37] S.L. Woronowicz, Compact quantum groups, in: A. Connes, et al. (Eds.), Symétries Quantiques (Quantum Symmetries), Les Houches, 1995, Elsevier, Amsterdam, 1998, pp. 845–884.


Two canonical passive state/signal shift realizations of passive discrete time behaviors Damir Z. Arov a,1 , Olof J. Staffans b,∗ a Division of Mathematical Analysis, Institute of Physics and Mathematics, South-Ukrainian Pedagogical University,

65020 Odessa, Ukraine b Åbo Akademi University, Department of Mathematics, FIN-20500 Åbo, Finland

Received 21 November 2008; accepted 21 May 2009 Available online 9 July 2009 Communicated by J. Coron

Abstract A discrete time invariant linear state/signal system Σ with a Hilbert state space X and a Kre˘ın signal x(n) space W has trajectories (x(·), w(·)) that are solutions of the equation x(n + 1) = F u(n) , where F is X a bounded linear operator from into X with a closed domain whose projection onto X is all of X . This W system is passive if the graph of F is a maximal nonnegative subspace of the Kre˘ın space −X [] X [] W. The future behavior Wfut of a passive system Σ is the set of all signal components w(·) of trajectories (x(·), w(·)) of Σ on Z+ = {0, 1, 2, . . .} with x(0) = 0 and w(·) ∈ 2 (Z+ ; W). This is always a maximal nonnegative shift-invariant subspace of the Kre˘ın space k 2 (Z+ ; W), i.e., the space 2 (Z+ ; W) endowed with the indefinite inner product inherited from W. Subspaces of k 2 (Z+ ; W) with this property are called passive future behaviors. In this work we study passive state/signal systems and passive behaviors (future, full, and past). In particular, we define and study the input and output maps of a passive state/signal system, and the past/future map of a passive behavior. We then turn to the inverse problem, and construct two passive state/signal realizations of a given passive future behavior W+ , one of which is observable and backward conservative, and the other controllable and forward conservative. Both of these are canonical in the sense that they are uniquely determined by the given data W+ , in contrast earlier realizations that depend not only on W+ , but also on some arbitrarily chosen fundamental decomposition of the signal space W. From


E-mail address: [email protected] (O.J. Staffans). URL: http://web.abo.fi/~staffans/ (O.J. Staffans). 1 Damir Z. Arov thanks Åbo Akademi for its hospitality and the Academy of Finland and the Magnus Ehrnrooth Foundation for their financial support during his visits to Åbo in 2003–2008. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.029

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D.Z. Arov, O.J. Staffans / Journal of Functional Analysis 257 (2009) 2573–2634

our canonical realizations we are able to recover the two standard de Branges–Rovnyak input/state/output shift realizations of a given operator-valued Schur function in the unit disk. © 2009 Elsevier Inc. All rights reserved. Keywords: Passive; Conservative; Behavior; State/signal; De Branges–Rovnyak model; Input map; Output map; Past/future map; Kre˘ın space; Quotient space

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2. Passive future, full, and past behaviors . . . . . . . . 3. Anti-passive reflected systems and behaviors . . . 4. The Hilbert spaces H(W+ ) and H(W[⊥] − )..... 5. The output and input maps . . . . . . . . . . . . . . . . 6. The past/future map of a passive full behavior . . 7. The observable backward conservative realization 8. The controllable forward conservative realization 9. Frequency domain versions of passive behaviors . 10. Input/output representations of passive behaviors Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2574 2581 2591 2596 2600 2607 2612 2617 2622 2628 2634 2634

1. Introduction In this work we continue our study of passive linear discrete time invariant s/s (state/signal) system begun in [2–5]. However, the approach taken here is somewhat different from the approach in [2–5], and the present article is essentially self-contained. The s/s systems theory differs from the standard i/s/o (input/state/output) systems theory in the sense that no distinction is made between input and output signals, only between an “internal” state x ∈ X and an “external” interaction signal w ∈ W. In [2] it was assumed that both the state space X and the signal space W are Hilbert spaces, but in the subsequent articles [3–5] dealing with passive systems the signal space W was replaced by a Kre˘ın space (the state space X still remains a Hilbert space). A trajectory (x(·), w(·)) of a linear discrete time-invariant s/s system Σ on a discrete time interval I ⊂ Z consists of an X -valued state sequence x(·) and a W-valued signal sequence w(·) satisfying the equations x(n + 1) = F

x(n) , w(n)

n ∈ I,

(1.1)

X where F is a bounded linear operator with closed domain D(F ) ⊂ W and values in X with the property that the projection of D(F ) onto X is all of X . The last property is equivalent to the following property of the set of trajectories of Σ : for every discrete time interval I with finite left end-point m and for every xm ∈ X there exists at least one trajectory (x(·), w(·)) of Σ on I with initial state x(m) = xm . Earlier in [2–5] we primarily restricted our attention to the


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interval I = Z+ := {k ∈ Z | k 0}, but below we shall, in addition, consider the cases I = Z and I = Z− := {k ∈ Z | k < 0}, and occasionally some other intervals. A s/s system is called forward passive if, for every discrete time interval I and every trajectory (x(·), w(·)) of Σ in I , it is true that 2 2 −x(n + 1)X + x(n)X + w(n), w(n) W 0,

n ∈ I,

(1.2)

where · X is the norm in the Hilbert space X and [·,·]W is the inner product in the Kre˘ın space W. In view of the time-invariance of (1.1), it is enough that property (1.2) holds on the interval I = {0}. This property can be dressed in a geometric form in terms of the Kre˘ın (node) space K := −X [] X [] W as follows: condition (1.2) holds if and only if the graph V of the operator F in (1.1) is a nonnegative subspace of K. By replacing F in (1.1) by its graph V we can rewrite (1.1) in the equivalent form

x(n + 1) ∈ V, x(n) w(n)

n ∈ I.

(1.3)

The subspace V above is called the generating subspace of Σ , since condition (1.3) defines the set of all trajectories (x(·), w(·)) of Σ on any interval I . The above discussion can be summarized as follows. By a linear discrete time-invariant s/s system we mean a colligation Σ = (V ; X , W), where X is a Hilbert (state) space, W is a Kre˘ın (signal) space, and V is a generating subspace of the Kre˘ın (node) space K = −X [] X [] W, i.e., a subspace which is the graph of an operator F with the properties described in the connection with (1.1). Given a s/s system Σ = (V ; X , W), there is another s/s system Σ∗ = (V∗ ; X , W∗ ), called the adjoint of Σ , where W∗ = −W (this is the same space as W but with the inner product [·,·]−W = −[·,·]W ), and

0 V∗ = 1X 0

1X 0 0

0 0

V [⊥] ,

(1.4)

1[W∗ ,W ]

where V [⊥] is the orthogonal companion to V in K, and 1[W∗ ,W ] is the identity map from W to W∗ . The system Σ is called backward passive if Σ∗ is forward passive, and Σ is called passive if it is both forward and backward passive. This implies that if a s/s system Σ is passive, then its generating subspace V is a maximal nonnegative subspace of the node space K. Conversely, suppose that V is an arbitrary maximal nonnegative subspace of K. Let W = −Y [] U be a fundamental decomposition of W (i.e., Y and U are Hilbert spaces, and the sum is orthogonal). Then, by standard Kre˘ın space theory, V has a graph representation of the type ⎫ ⎧⎡ ⎤ Ax + Bu ⎪ ⎪ ⎬ ⎨ x ⎢ ⎥ V= ⎣ ⎦ x ∈ X and u ∈ U , ⎪ ⎪ ⎭ ⎩ Cx + Du u

(1.5)

A B where C is a linear contraction X ⊕ U → X ⊕ Y. This means that V is the graph of the D operator F defined by

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F

x0 y0 u0

= Ax0 + Bu0 ,

D(F ) =

x0 y0 u0

X ∈ Y U

y0 = Cx0 + Du0 .

Trivially, this operator F satisfies the conditions listed below (1.1), and hence Σ = (V ; X , W) is a s/s system. This system is passive since V is maximal nonnegative. Thus, we conclude that V is the generating subspace of a passive s/s system if and only if V is maximal nonnegative in the node space K. In this article we discuss only passives s/s systems. In the terminology of [2,3], the existence of the graph representation (1.5) means that every fundamental decomposition A B of W is admissible for the passive s/s system Σ . The corresponding i/s/o system Σi/s/o = ( C ; X , U, Y) is called a (scattering) i/s/o representation of Σ . If we B decompose the signal w(·) in (1.3) into w(·) = u(·) + y(·), where the values of u(·) and y(·) lie in U and Y, respectively, then (1.3) takes the form x(n + 1) = Ax(n) + Bu(n), y(n) = Cx(n) + Du(n),

n ∈ I.

(1.6)

See [2,3] for more details. Since every Kre˘ın space W that is neither a Hilbert space nor an anti-Hilbert spaces has infinitely many fundamental decompositions, this means that a passive s/s system Σ = (V ; X , W) AB with a Kre˘ın signal space W usually has an infinite family Σi/s/o = ( C ; X , U, Y) of scatB tering i/s/o representations (in the exceptional cases Σi/s/o is unique, but it has no input or no output). Each such system Σi/s/o has a scattering matrix D(z) = zC(1 − zA)−1 B + D which is a Schur class function, i.e., a B(U; Y)-valued analytic contractive function in the unit disk. = ∞ D(k)zk with contractive coefficients This function has a power series expansion D(z) k=0 D(k) ∈ B(U; Y). Different choices of the fundamental decomposition gives different systems Σi/s/o and different scattering matrices. Using the coefficients D(k) of each scattering matrix D(z) we can define a block-Toeplitz operator D : 2 (U) → 2 (Y) by (Du)(n) =

n

D(n − k)u(k),

n ∈ Z, u(·) ∈ 2 (Z; U),

k=−∞

and we can also define two additional block Toeplitz operators D+ : 2 (Z+ ; U) → 2 (Z+ ; Y) and D− : 2 (Z− ; U) → 2 (Z− ; Y) by D+ := D|2 (Z+ ;U ) and D− := P2 (Z− ;Y ) D|2 (Z− ;U ) . A crucial fact is that although D, D+ , and D− do depend on the fundamental decomposition W = −Y [] U , the graphs of these three operators do not. We call these three graphs the full, future, and past behaviors, respectively, of Σ. Above we defined the full, future and past behaviors of a passive s/s system Σ in terms of an i/s/o representation of Σ, but they can also be defined directly by means of trajectories of Σ . To do this we first need to introduce the notion of an externally generated stable trajectory of a passive s/s system. A trajectory (x(·), w(·)) of Σ on a discrete time interval I is called stable if x(·) ∈ ∞ (I ; X )

and w(·) ∈ 2 (I ; W)

(see Section 2 for details). If (x(·), w(·)) is a trajectory of Σ on I , then by (1.2),

(1.7)

D.Z. Arov, O.J. Staffans / Journal of Functional Analysis 257 (2009) 2573–2634 n x(n + 1)2 x(m)2 + w(k), w(k) W , X X

m, n ∈ I, m n.

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(1.8)

k=m

Thus, if I is an interval with finite left end-point m, then the first condition x(·) ∈ ∞ (I ; X ) in (1.7) is implied by the second condition w(·) ∈ 2 (I ; W), so to guarantee the stability of the trajectory it suffices to require that w(·) ∈ 2 (I ; W). If x(m) = 0, then we call this trajectory externally generated. If the left end-point of the interval I is −∞, then we call a trajectory externally generated if x(m) → 0 in X as m → −∞. Also such a trajectory is stable if and only if w(·) ∈ 2 (I ; W); this follows from (1.8) by letting m → −∞. The sum in (1.8) (where we allow m = −∞ or n = ∞ or both) can be interpreted as an indefinite inner product in 2 ([m, n]; W), where [m, n] := {k ∈ Z | m k n} (and we replace “” by “ 1} ∪ {∞}, T = {z ∈ C | |z| = 1}, Z = {0, ±1, ±2, . . .}, Z+ = {0, 1, 2, . . .}, and Z− = {−1, −2, −3, . . .}. For any set Ω, we denote the closure of Ω by Ω, andwe denote the closed linear span of a collection {Ωα }α∈A of sets in a Hilbert or Kre˘ın space by α∈A Ωα . The space of bounded linear operators from one Kre˘ın space U to another Kre˘ın space Y is denoted by B(U; Y). The domain, range, and kernel of a linear operator A are denoted by D(A), R(A), and N (A), respectively. The restriction of A to some subspace Z ⊂ D(A) is denoted by A|Z . The identity operator on U is denoted by 1U , or by 1 if the space is clear from the context. The orthogonal projection onto a closed subspace Y of a Kre˘ın space K is denoted by PY . The inner product in a Hilbert space X is denoted by (·,·)X , and the inner product in a Kre˘ın space K is denoted by [·,·]K . The orthogonal sum of U and Y is denoted by U ⊕ Y in the case of Hilbert spaces, and by U [] Y in the case of Kre˘ın spaces. The anti-space −K of a Kre˘ın space is algebraically the same space as K, but it has a different inner product [·,·]−K := −[·,·]K . We denote the product of two Kre˘ın or Hilbert spaces Y and U by [ UY ]. If L is a set of vectors in a Kre˘ın space, then L[⊥] is the orthogonal companion to L, i.e., L[⊥] := x ∈ K [x, y]K = 0 for all y ∈ L . If w(·) is a sequence with values in a Kre˘ın or Hilbert space W defined on some discrete time interval I , then S ±1 w is the sequence w(·) shifted one step to the right or left, respectively


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(this includes a right or left shift of I if I = Z). For sequences w(·) defined on Z+ we define (S+ w)(0) = 0, (S+ w)(n) = w(n − 1), n 1, and for sequences w(·) defined on Z− we define (S− w)(n) = w(n − 1), n ∈ Z− . If we want to emphasize that the values of w lie in W we write S W instead of S. 2. Passive future, full, and past behaviors Passive state/signal systems A passive linear discrete time invariant s/s system Σ = (V ; X , W) has a Hilbert (state) space X , a Kre˘ın (signal) space W, and a (generating) maximal nonnegative subspace V of the Kre˘ın space K = −X [] X [] W. A trajectory of Σ on a discrete time interval I is a pair of sequences (x(·), w(·)) satisfying (1.3). Observe that w(·) is always defined on I , but that x(·) is defined at one extra point at the right end if I is bounded to the right, i.e., if w(·) is defined on I = (m, n) := {k ∈ Z | m < k < n}, then x(·) is defined on (m, n] := {k ∈ Z | m < k n} (here we allow m = −∞; if n = +∞, then these two sets coincide. Earlier, in [2–5], we most of the time took the interval I to be I = Z+ = {0, 1, 2, . . .}, but below we shall also consider other intervals, finite or infinite. In particular, in addition to Z+ we shall frequently take I = Z or I = Z− = {−1, −2, −3, . . .} (in which case x(k) is also defined for k = 0). By a past trajectory we mean a trajectory on Z− , by a full trajectory we mean a trajectory on Z, and by a future trajectory we mean a trajectory on Z+ . In the case where the interval I is bounded to the left we call a trajectory (x(·), w(·)) on I externally generated if x vanishes at the left end-point of I , i.e., x(m) = 0 if I = [m, n) := {z ∈ Z | m z < n} (where we allow n = ∞), and if I is unbounded to the left we call the trajectory externally generated if x(m) → 0 in X as m → −∞. Stable trajectories of passive state/signal systems All the s/s systems in this article will be passive. A trajectory (x(·), w(·)) of the passive s/s system Σ = (V ; X , W) on an interval I is called stable if w(·) ∈ k 2 (I ; W) and x(·) ∈ ∞ (I ; X )

(2.1)

(strictly speaking, the restriction of x(·) to the interval I should belong to ∞ (I ; X )). Here ∞ (I ; X ) is the Banach space of bounded X -valued sequences on the interval I . The space k 2 (I ; W) is a Kre˘ın space whose inner product is defined in (2.3) below. A sequence w(·) with values in W belongs to k 2 (I ; W) if and only if w(k)2 < ∞, W

(2.2)

k∈I

where · W is some admissible Hilbert space norm in the Kre˘ın space W, given by w2W = −[PW− w, PW− w]W + [PW+ w, PW+ w]W for some fundamental decomposition W = −W− [] W+ where W− and W+ are Hilbert spaces with the norms inherited from −X and X , respectively. Different fundamental decompositions give different norms · W , but they are all equivalent, so (2.2) is independent of the chosen

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admissible norm in the sense that if (2.1) holds for one admissible norm · W , then it holds for all admissible norms · W . The space k 2 (I ; W) does not have a unique positive inner product (only a family of equivalent inner Hilbert space inner products), but it does have a natural indefinite inner product, namely w1 (·), w2 (·) k 2 (I ;W ) := w1 (k), w2 (k) W .

(2.3)

k∈I

Because of (2.2), the sum above converges absolutely for all w ∈ k 2 (I ; W). With this inner product k 2 (I ; W) becomes a Kre˘ın space, and each fundamental decomposition W = −Y [] U induces a fundamental decomposition k 2 (I ; W) = −2 (I ; Y) [] 2 (I ; U),

(2.4)

where the norms in Y and U are the norms inherited from −W and W, respectively, and 2 (I ; Y) and 2 (I ; U) stand for the standard Hilbert 2 -spaces on the interval I : if X is a Hilbert space and I an discrete interval then 2 (I ; X ) consists of all X -valued sequences x(·) on I satisfying x(·)22

(I ;X )

:=

x(k)2 < ∞. X

(2.5)

k∈I

In the sequel we abbreviate the cases where I is one of the intervals Z− , Z, or Z+ as follows: 2 k− (W) := k 2 Z− ; W , 2− (X ) := 2 Z− ; X ,

k 2 (W) := k 2 (Z; W), 2 (X ) := 2 (Z; X ),

2 (W) := k 2 Z+ ; W , k+ 2+ (X ) := 2 Z+ ; X .

If I and I are two intervals with I ⊂ I , then we frequently identify k 2 (I ; W) with the subspace

/I w ∈ k 2 (I ; W) w(k) = 0 for k ∈

of k 2 (I ; W), and in the same way we identify 2 (I ; X ) with a subspace of 2 (I ; X ). As the following lemma shows, the condition x ∈ ∞ (I ; X ) in (2.1) is often redundant or almost redundant. Lemma 2.1. Let Σ = (V ; X , W) be a passive s/s system, and let I be an discrete time interval, and let (x(·), w(·)) be a trajectory of Σ on I . (1) If I = [m, ∞) for some finite m, then (x(·), w(·)) is stable if and only if w(·) ∈ k 2 (I ; W). (2) If I is unbounded to the left, then (x(·), w(·)) is stable if and only if w(·) ∈ k 2 (I ; W) and lim supm→−∞ x(m)X < ∞. Proof. It follows from the nonnegativity of V that (1.8) holds. This implies both (1) and (2) since the sum in (1.8) stays bounded as n → ∞ or m → −∞. 2 In the case of externally generated trajectories the preceding result simplifies as follows.


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Lemma 2.2. Let Σ = (V ; X , W) be a passive s/s system, and let I be an discrete time interval, and let (x(·), w(·)) be an externally generated trajectory of Σ on I . Then (x(·), w(·)) is stable if and only if w(·) ∈ k 2 (I ; W). Moreover, if I = [m, ∞) for some finite m, then x(n + 1)2 w(·), w(·) 2 , X k ([m,n];W )

n ∈ I,

(2.6)

and if I = (−∞, k) (where we allow k = ∞), then x(n + 1)2 w(·), w(·) 2 , X k ((−∞,n];W )

n ∈ I.

(2.7)

In particular, if I = Z− , then x(0)2 w(·), w(·) 2 . X k (W ) −

(2.8)

Proof. This follows from Lemma 2.1 and the definition of an externally generated trajectory.

2

Formulas (1.8) and (2.6)–(2.8) explain why the Kre˘ın spaces k 2 (I ; W) appear naturally in connection with passive s/s systems. In the sequel we shall need the following basic facts about stable trajectories of Σ . Lemma 2.3. The set of stable trajectories of a passive s/s system Σ = (V ; X , W) have the following properties. (1) Both the set of all stable trajectories and the set of all externally generated stable trajectories of Σ on some interval I (finite or infinite) are closed subspaces of ∞ (I ; X ) × 2 (I ; W). (2) If (x(·), w(·)) is a stable trajectory of Σ on some interval I and n ∈ Z, then (S n x, S n w) is a stable trajectory of Σ on S n I = {k ∈ Z | k − n ∈ I }, and (x(·), w(·)) is externally generated on I if and only if (S n x, S n w) is externally generated on S n I . (3) The restriction of a stable trajectory on some interval I to a subinterval I ⊂ I is a stable trajectory of Σ on I , and if I and I have the same left end-point, then the restricted trajectory is externally generated if and only if the original trajectory is externally generated. (4) If (x(·), w(·)) is an externally generated stable trajectory of Σ on an interval I = [m, n) (where we allow n = ∞), and if we define x(k) = 0 and w(k) = 0 for k < m, then this extended pair of sequences is an externally generated stable trajectory of Σ on (−∞, n). (5) Let W = −Y [] U be a fundamental decomposition of W. Then, for each x0 ∈ X and each u ∈ 2+ (U) there exists a unique stable future trajectory (x(·), w(·)) of Σ satisfying x(0) = x0 and P2 (U ) = u. + (6) Every stable trajectory on some interval I = (m, n] (where we allow m = −∞) can be extended to a stable trajectory of Σ on (m, ∞). ! x1 " (7) To each x0 ∈ V there exists at least one stable future trajectory (x(·), w(·)) of Σ satisfying w0

x(0) = x0 , x(1) = x1 , and w(0) = w0 . Proof. (1)–(4) Claim (1) follows from (1.3) and the fact that V is maximal nonnegative, and hence closed in the node space K. Properties (2)–(4) follow immediately from the definition of a stable trajectory.

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(5) Let W = −Y [] U be a fundamental decomposition. Then, by Theorem II.5.7, this de+ composition is admissible for Σ, which means that for each x0 ∈ X and u ∈ U Z the system Σ has a unique trajectory (x(·), w(·)) on Z+ satisfying x(0) = x0 and PU Z+ w(·) = u(·). For example, we may take u ∈ 2+ (U). It then follows from (1.8) that the corresponding trajectory is stable, since we have for all n ∈ Z+ , n x(n + 1)2 − w(k), w(k) W X k=0 n n 2 PY w(k)2 − PU w(k)2 = x(n + 1)X + Y U k=0

k=0

x0 X . 2

(2.9)

(6) By property (2), we may without loss of generality suppose that n = −1. Let (x (·), w (·)) be the stable future trajectory of Σ given by (5) that satisfies x (0) = x(0) and P2 (U ) w(·) = 0. + By defining x(k) = x (k) and w(k) = w (k) for k > 0 we get an trajectory on I = (m, ∞) whose restriction to I = (m, −1] is the given trajectory of Σ . (7) This is a special case of (6) with I = {0}.

2

Lemma 2.4. Let Σ = (V ; X , W) be a passive s/s system, and let I = (−∞, n) (where we allow n = ∞). Then the set of all compactly supported externally generated stable trajectories (i.e., trajectories (x(·), w(·)) that satisfy x(k) = 0 and w(k) = 0 for all k in some interval (−∞, m]) is dense in the set of all externally generated stable trajectories of Σ on I in the topology inherited from ∞ (I ; X ) × k 2 (I ; W). Proof. Let (x(·), w(·)) be an externally generated stable trajectory of Σ on I , and let W = −Y [] U be a fundamental decomposition of W. By claims (2)–(5) of Lemma 2.3, for each m ∈ I there is a unique externally generated stable trajectory (xm (·), wm (·)) of Σ on I satisfying (·) = x(·) − x (·) x(k) = 0 and w(k) = 0 for k m and P2 (I ;U ) wm = P2 ([m,n);U ) w. Define xm m (·) = w(·) − w (·). Then (x (·), w (·)) is an externally generated trajectory of Σ on I , and wm m m and by (2.7), for all k ∈ I , 2 2 x (k + 1)2 + P 2 P2 ((−∞,k];U ) wm ((−∞,k];Y ) wm m X P2 ((−∞,m);U ) w2 . This implies that 2 x ∞

m ((−∞,n];X )

2 + P2 ((−∞,n);Y ) wm 2P2 ((−∞,m);U ) w2 ,

where the right-hand side tends to zero as m → −∞. Thus, xm → x in ∞ ((−∞, n]; X ) and wm → w in k 2 (I ; W) as m → −∞. 2


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Behaviors of passive state/signal systems By the (stable) behavior induced by the passive s/s system Σ on the interval I we mean the set w(·) x(·), w(·) is an externally generated stable trajectory of Σ on I , and we denote it by WΣ (I ). Here we sometimes omit the upper index Σ if it is clear from the context which system this behavior is induced by. The cases where I is one of the intervals Z− , Z, and Z+ are especially important, and we refer to these behaviors as the past behavior WΣ past , Σ Σ the full behavior Wfull , and the future behavior Wfut induced by the passive system Σ. Thus, − Σ WΣ past = W Z ,

+ Σ WΣ fut = W Z .

Σ WΣ full = W (Z),

The following result is immediate. ∞ Lemma 2.5. To each w ∈ WΣ fut there exists a unique x ∈ + (X ) such that (x(·), w(·)) is an + externally generated stable trajectory of Σ on Z . The same statement remains true if we replace Σ Σ + − WΣ fut by Wfull or by Wpast and at the same time replace Z by Z or Z , respectively. Σ Σ Proof. This follows from the definitions of WΣ fut , Wfull , and Wpast and Lemma 2.2.

2

2 (W), k 2 (W), and k 2 (W), are denoted by S , S, and S , The right-shift operators on k− − + + 2 (W), the operator S is unitary on k 2 (W), respectively. The operator S− is a co-isometry on k− 2 (W). The operators S and S can be expressed in and the operator S+ is an isometry on k+ − + terms of the operator S by

S− = π− S|k 2 (W ) ,

S+ = S|k 2 (W ) ,

−

+

2 (W). where π− is the orthogonal projection of k 2 (W) onto k− It will be shown in Theorem 2.8 below that the full behavior WΣ full of a passive s/s system Σ = (V ; X , W) is a maximal S-reducing subspace of k 2 (W) (i.e., it is invariant under both S and S −1 ). However, the converse is not true: WΣ fut has one extra property, called causality, Σ which is not a consequence of the fact that Wfull is maximal nonnegative and S-reducing. Let W be a maximal nonnegative subspace of k 2 (W), and let W = −Y [] U be a fundamental decomposition of W. Then k 2 (W) = −2 (Y) [] 2 (U) is a fundamental decomposition of W. It follows from (2.8) that

x(0)2 −P 2 w22 + P2 (U ) w22 (U ) . (Y ) X (Y ) −

In particular, if P2 (Y ) w22 −

− (U )

−

−

−

= 0, then π− w = 0.

Definition 2.6. A maximal nonnegative S-reducing subspace W of k 2 (W) is causal if it is true for some fundamental decomposition W = −Y [] U of W that

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w(·) ∈ W

and P2 (U ) w = 0 −

⇒

π− w(·) = 0.

(2.10)

We shall see later that the choice of the fundamental decomposition W = −Y [] U in Definition 2.6 is not important: if (2.10) holds for one fundamental decomposition, then it holds for every fundamental decomposition of W. Not every maximal nonnegative S-reducing subspace of k 2 (W) is causal, as the following counter-example shows. Example 2.7. Let U be a Hilbert space, and let X be the Kre˘ın space X = −Y [] U where Y = U . Then k 2 (W) = −2 (Y) [] 2 (U). Let # −1 $ SU u 2 W= (2.11) u ∈ (U) , u where SU is the right-shift in 2 (U). It is easy to see that W[⊥] = W, i.e., W is Lagrangian, hence maximal nonnegative (and also maximal nonpositive). It is also S-reducing. However, it is not −1 , so condition (2.10) does not hold. causal: if u ∈ 2+ (U) and u(0) = 0, then (SU u)(−1) = u(0) 0 u(−1)

Theorem 2.8. Let Σ = (V ; X , W) be a passive s/s system. Then the behaviors induced by Σ have the following properties. (1) (2) (3) (4) (5) (6) (7)

2 WΣ fut is a maximal nonnegative S+ -invariant subspace of k+ (W). Σ Wfull is a maximal nonnegative S-reducing causal subspace of k 2 (W). 2 WΣ past is a maximal nonnegative S− -invariant subspace of k− (W). Σ ∩ k (W). WΣ + fut = W full Σ Wfull = n∈Z+ S −n WΣ fut . Σ . WΣ = π W − full past % 2 −n Σ = WΣ n∈Z+ {w(·) ∈ k (W) | π− S w ∈ Wpast }. full

Proof. Step 1: Proofs of (4), (6), and (7). These identities follow from Lemma 2.3. Step 2: Proof of (1). The nonnegativity of WΣ fut follows from (2.6), and the S+ -invariance of WΣ follows from Lemma 2.3. It remains to prove that WΣ fut fut is maximal nonnegative 2 in k+ (W). By definition, w(·) ∈ WΣ fut if and only if there exists (a unique) bounded sequence x(·) such that (x(·), w(·)) is an externally generated stable trajectory of Σ on Z+ . Let W = −Y [] U be a fundamental decomposition of W. Then (2.4) with I = Z+ is a fundamental decomposition 2 (W), and by (2.9) with n = 0 and x(0) = 0, of k+ PY w(·)

2+ (Y )

PU w(·)2 (U ) . +

By part (5) of Lemma 2.3, the function PU w(·) can be an arbitrary function in 2+ (U). This implies that there exists a bounded linear operator D+ such that # WΣ fut =

$ D+ u u ∈ 2+ (U) . u

(2.12)


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2 2 Thus, WΣ fut is the graph of a contraction D+ : + (U) → + (Y) and hence maximal nonnegative. Σ 2 Step 3: WΣ full is closed in k (W). Let wj (·) be a sequence in Wfull converging to some 2 ∞ w ∈ k (W). Then, to each wj there corresponds a sequence xj (·) ∈ (X ) satisfying xj (n) → 0 as n → −∞ such that (xj (·), wj (·)) is an externally generated full stable trajectory of Σ. The sequence wj (·) is a Cauchy sequence in k 2 (W), and it follows from (2.7) that xj (·) is a Cauchy sequence in ∞ (X ). Thus, xj (·) tends to a limit x(·) in ∞ (X ) satisfying x(n) → 0 as n → −∞. The generating subspace V is closed, and it follows from (1.3) that (x(·), w(·)) is an exterΣ nally generated stable trajectory of Σ on Z. Thus, w ∈ WΣ full , and this proves that Wfull is closed. Σ Step 4: Proofs of (2) and (5). The nonnegativity of WΣ full follows from (2.7), and that Wfull is S-reducing follows from Lemma 2.3. 2 Recall that WΣ fut has the graph representation (2.12) for some contraction D+ : + (U) → 2 + (Y), where W = −Y [] U is a fundamental decomposition of W. The S+ -invariance of WΣ fut implies that D+ is shift-invariant in the sense that D+ S+ = S+ D+ . Let 20 (U) be the subset of 2 (U) consisting of those sequences in 2 (U) whose support is bounded to the left. It is possible to define a contraction D : 20 (U) → 2 (Y) in the following way: If u ∈ 20 (U) vanishes on (−∞, n], then we define Du = S −m D+ S m u, where m is chosen to be so large that S m u vanishes on Z− . The result is independent of the particular value of m because D+ S+ = S+ D+ . Since 20 (U) is dense in 2 (U) we can extend D to a contraction 2 (U) → 2 (Y). This contraction is causal in the sense that D2+ (U) ⊂ 2+ (Y), and it is shift-invariant in the sense that DSu = SDu for all u ∈ 2 (U). Moreover, D+ = D|2 (W ) . It follows from (2.12) with D+ = D|2 (W ) that

S

−n

# WΣ fut

=

$ # $ −n −n D+ u+ u+ SY DSY 2 2 u+ ∈ + (U) = u+ ∈ + (U) , SU−n u+ SU−n u+

Du 2 where n∈Z+ 2+ (U) = 2 (U). Thus, z∈Z+ S −n WΣ fut = { u | u ∈ (U)}. This graph repre Σ −n 2 sentation implies that z∈Z+ S Wfut is maximal nonnegative in k (W). & Σ Σ It follows from Lemma 2.3 that z∈Z+ S −n WΣ fut ⊂ Wfull , and since Wfull is closed, we have Σ Σ Σ −n −n Wfut ⊂ Wfull . Here z∈Z+ S Wfut is maximal nonnegative, and WΣ z∈Z+ S full is nonnegaΣ , and hence WΣ is maximal nonnegative and (5) holds. In tive. Thus, z∈Z+ S −n WΣ = W fut full full particular, # WΣ full

=

$ Du 2 u ∈ (U) . u

(2.13)

2 2 That WΣ full is causal follows from this graph representation and the fact that D+ (U) ⊂ + (Y).

Step 5: Proofs of (3). That WΣ past is S− -invariant follows from Lemma 2.3. The graph representation (2.13) together with (6) and the fact that D2+ (U) ⊂ 2+ (Y) implies that WΣ past has the graph representation # WΣ past =

$ # $ π− D− u D− u u ∈ 2 (U) = u ∈ 2− (U) , π− u u

(2.14)

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where D− := π− D|2 (U ) is a contraction 2− (U) → 2− (Y). This graph representation implies −

2 that WΣ past is maximal nonnegative in k− (W).

2

Corollary 2.9. Let Σ = (V ; X , W) be a passive s/s system. Then each one of the past, full, and Σ Σ future stable behaviors WΣ past , Wfull , and Wfut of Σ determines the other two uniquely. Proof. This follows from claims (4)–(7) in Theorem 2.8.

2

Passive future, full, and past behaviors Let Σ = (V ; X , W) be a passive s/s system. According to Theorem 2.8, the future behavior Σ 2 WΣ fut of Σ is a maximal nonnegative S+ -invariant subspace of k+ (W), the full behavior Wfut of 2 Σ is a maximal nonnegative S-reducing causal subspace of k (W), and the past behavior WΣ past 2 (W). It will be shown in Section 7 of Σ is a maximal nonnegative S− -invariant subspace of k− 2 (W) is the future behavior of a that every maximal nonnegative S+ -invariant subspace of k+ passive s/s system, and analogously, it will be shown in Section 8 that every maximal nonnegative 2 (W) is the past behavior of a passive s/s system. We shall also see S− -invariant subspace of k− that every maximal nonnegative S-reducing causal subspace of k 2 (W) is the full behavior of a passive s/s system. In view of these three facts the following definitions are natural. Definition 2.10. Let W be a Kre˘ın space. 2 (W) is called a passive future behavior (1) A maximal nonnegative S+ -invariant subspace of k+ on the Kre˘ın (signal) space W. (2) A maximal nonnegative S-reducing causal subspace of k 2 (W) is called a passive full behavior on the (signal) space W. 2 (W) is called a passive past behavior (3) A maximal nonnegative S− -invariant subspace of k− on the (signal) space W.

The basic connections between passive future, full, and past behaviors are described in the following theorem. Theorem 2.11. Let W be a Kre˘ın space. (1) If W is a passive full behavior on W, and if we define W+ and W− by 2 W+ := W ∩ k+ (W),

W− := π− W,

(2.15)

then W+ and W− are passive future and past behaviors on W, respectively, and W can be recovered from W+ and from W− by the formulas W=

'

S −n W+ ,

(2.16)

n∈Z+

W=

( w(·) ∈ k 2 (W) π− S −n w ∈ W− .

n∈Z+

(2.17)


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(2) If W+ is a passive future behavior on W, and if we define W by (2.16), then W is a passive 2 (W). full behavior on W and W+ = W ∩ k+ (3) If W− is a passive past behavior on W, and if we define W by (2.17), then W is a passive full behavior on W and W− = π− W. Proof. Most of the proof of this theorem is very similar to the proof of Theorem 2.8, but some of the details are different. (1) Let W = −Y [] U be a fundamental decomposition of W. Then k 2 (W) = −2 (Y) [] 2 (U) is a fundamental decomposition of k 2 (W), and the maximal nonnegativity of W implies that it has a graph representation # W=

$ Du 2 u ∈ (U) u

(2.18)

for some contraction D : 2 (U) → 2 (Y). Since W is S-reducing, we have SY D = DSU , and since W is causal, D2+ (U) ⊂ 2+ (Y). This, together with (2.15) implies that W± have the graph representations #

$ D+ u 2 u ∈ (U) , + u # $ D− u W− = u ∈ 2− (U) , u

W+ =

(2.19) (2.20)

where D+ = D|2 (U ) and D− = π− D|2 (U ) are contractions 2± (U) → 2± (Y). These two graph +

−

2 (W) = −2 (Y) [] 2 (U) representations with respect to the fundamental decompositions k± ± ± 2 imply that W± are maximal nonnegative in k± (W). That W+ is S+ -invariant follows from its 2 (W) and the fact that SW = S. The S -invariance of W is proved by definition W+ = W ∩ k+ − − the following computation:

S− W− = π− Sπ− W = π− π(−∞,0] SW = π− W = W− .

(2.21)

Thus, W+ and W− are passive future and past behaviors, respectively. A proof of the fact that z∈Z+ S −n W+ is maximal nonnegative in k 2 (W) is contained in Σ step 4 of the proof of−nTheorem 2.8 (with Wfut replaced & by W+ ), and essentially the same proof shows that z∈Z+ S W+ = W (this time we have z∈Z+ S −n W+ ⊂ W since W is S-reducing and W+ ⊂ W). % Let Wn := {w(·) ∈ k 2 (W) | π− S −n w ∈ W− }, and let W := n∈Z+ Wn . The fact that W is S-reducing and that π− W = W− implies that W ⊂ W . Each Wn is nonnegative in 2 (W). For each w ∈ W we have π k 2 ((−∞, n]; W) since W− is nonnegative in k− (−∞,n] w(·) ∈ Wn , and hence w(·), w(·) k 2 (W ) = lim π(−∞,n] w(·), π(−∞,n] w(·) k 2 ((−∞,n];W ) 0, n→+∞

w ∈ W .

Thus, W ⊂ W where W is maximal nonnegative and W is nonnegative, and hence W = W .

2590


(2) Since W+ is maximal nonnegative, it has a graph representation of the type (2.19) for some of W. contraction D+ : 2+ (U) → 2+ (Y), where W = −Y [] U is a fundamental decomposition The same argument that we used in step 4 in the proof of Theorem 2.8 shows that z∈Z+ S −n W+ is passive full behavior on W, and that z∈Z+ S −n W+ = W whenever W is a S-reducing closed nonnegative subspace of k 2 (W) satisfying W+ ⊂ W. (3) Since W− is maximal nonnegative, it has a graph representation of the type (2.20) for some contraction D− : 2− (U) → 2− (Y), where W = −Y [] U is a fundamental decomposition of W. With the help of D− we can define a contraction D : 2 (U) → 2 (Y) in the following way. We first define the sequence of contractions Dn : 2 (U) → 2 (Y) by Dn u = S n D− π− S −n u, n 0. The right-shift invariance of D− implies that, for all m n, π(−∞,n] Dm = S n π− S m−n D− π− S −m = S n D− π− S m−n π− S −m = Dn . Thus, for each u ∈ 2 (U) and all m ∈ Z+ , Dm u2 (Y ) u2 (U ) , and π(−∞,n] Dm u is independent of m for m n. This implies that Dm u tends weakly to a limit y ∈ 2 (Y). Thus, for each u ∈ 2 (U) and m n, (Dm − Dn )u

2 (Y )

π(n,∞) (Dm − Dn )u2 (Y ) 2π(n,∞) u2 (U ) ,

which tends to zero as n → +∞. Thus, Dn tends strongly to a limit contraction D : 2 (U) → 2 (Y). This contraction is causal in the sense that D2+ (U) ⊂ 2+ (Y), and it is shift-invariant in the sense that DSu = SDu for all u ∈ 2 (U). Define W by (2.18). Then, by construction, DS = SD, D2+ (U) ⊂ 2+ (Y), and D− = π− D|2 (U ) . This implies that W is a passive full behavior on W satisfying W− = π− W. That − formula (2.17) holds follows from claim (1). 2 Lemma 2.12. Let W− be a passive past behavior on a Kre˘ın space W. Then the set of all w(·) ∈ W− with finite support (i.e., w(k) = 0 for all k in some interval (−∞, n]) is a dense subspace of W− . Proof. By (2.15) and (2.16), W− = π− W = π−

' n∈Z+

' 2 2 S −n W ∩ k+ (W) = π− S −n W ∩ k+ (W) , n∈Z+

2 (W)) has finite support. where each sequence in π− S −n (W ∩ k+

2

2 (W) is a bijection from the set of Remark 2.13. By Theorem 2.11, the map W → W ∩ k+ all passive fullbehaviors on W onto the set of all passive future behaviors on W, with inverse W+ → n∈Z+ S −n W+ . Likewise, the map W → π− W is a bijection from the set of all passive % full behaviors on W onto the set of all passive past behaviors on W, with inverse W− → n∈Z+ {w(·) ∈ k 2 (W) | π− S −n w ∈ W− }. Thus, formulas (2.15)–(2.17) define one-toone correspondences between a passive future behavior W+ , a passive full behavior W, and a passive past behavior W− : any one of these can be used to define the two others.

Let us go back to Example 2.7.


2591

Example 2.14. Let W be the Lagrangian subspace of k 2 (−Y [] U) defined in (2.11). As we saw in Example 2.7, W is not causal. Define W± by (2.15). Then $ S −1 u 2 W+ = u(·) ∈ + (U) with u(k) = 0 for all k 0 , u # $ y W− = y(·) ∈ 2 (Y) . S− y #

(2.22) (2.23)

The subspace W+ is not maximal nonnegative since the projection onto the positive component 2 (W) = −2 (Y) [] 2 (U) is not all of 2 (U), and the in the fundamental decomposition k+ + + −1 + subspace W− is not even nonnegative: if u ∈ 2+ (U) with u(0) = 0, then π− S u ∈ W− and u

π−

−1 2 S −1 u S u , π− = −u(0)U < 0. u u k 2 (W ) −

Remark 2.15. Our proof of claim (2) in Theorem 2.11 shows that a stronger statement is true than the one recorded in the theorem: If W is a closed nonnegative S-reducing subspace of k 2 (W) 2 (W), then W is given which contains a maximal nonnegative S+ -invariant subspace W+ of k+ by (2.16). Thus, W is uniquely determined by W+ within the class of all closed nonnegative S-reducing subspaces of k 2 (W), and not only within the class of all maximal nonnegative causal S-reducing subspaces of k 2 (W). A similar extension of claim (3) is also valid, as explained in Remark 3.10 below. 3. Anti-passive reflected systems and behaviors Since the generating subspace V of a passive s/s system Σ = (V ; X , W) is maximal nonnegative, its orthogonal companion V [⊥] is maximal nonpositive, and it generates an anti-passive reflected state/signal system Σ † = (V [⊥] ; X , W). The trajectories (x † (·), w † (·)) of Σ † satisfy

x † (n + 1) † ∈ V [⊥] , x (n) † w (n)

n ∈ I.

(3.1)

It differs from a standard passive s/s system in the sense that trajectories always can be continued backward in time instead of forward in time, and it is not a special case of a state/signal system in the sense of [2–5]. If we define V∗ by (1.4), then V∗ is maximal nonnegative in the Kre˘ın space −X [] X [] −W, and it generates a standard passive s/s system Σ∗ = (V∗ ; X , −W), which we in [3–5] called the adjoint of the s/s system Σ . Here we shall instead refer to Σ∗ as the passive dual of Σ , and call Σ † the anti-passive dual of Σ . The trajectories of Σ∗ and Σ † differ from each other by a time reflection, and, in addition, their signal spaces also differ from each other (the signal space of Σ † is W and the signal space of Σ∗ is −W). Because of the indexing conventions used in (1.3) and (3.1), the reflection in the state component x(·) differs slightly from the reflection in the signal component w(·): (x(·), w(·)) is a trajectory of Σ∗ on an interval I if and only if the function (x † (·), w † (·)) defined by x † (n) = x(−n) and w † (n) = w(−n − 1) is a trajectory of Σ † on I † = {z ∈ Z | −z − 1 ∈ I }.

2592


Stable trajectories of an anti-passive reflected s/s system are defined in the same way as for a passive s/s system, and we still refer to trajectories defined on Z− , Z, and Z+ as past, full, and future trajectories. Past, full, and future trajectories are also defined in the same way as for passive s/s systems, i.e., “past” always refers to the time interval Z− , “full” to the time interval Z, and “future” to the time interval Z+ . However, since the natural direction of evolution of an antipassive reflected s/s system is opposite to the natural direction of evolution of a passive system, a trajectory (x † (·), w † (·)) of an anti-passive reflected s/s system is externally generated if the state vanishes at the right end-point of the interval of definition, i.e., x † (n) = 0 when I = (m, n) and limn→+∞ x(n) = 0 when I = (m, ∞). Lemma 3.1. Let Σ = (V ; X , W) be a passive s/s system, and let Σ † = (V [⊥] ; X , W) be its anti-passive dual. (1) Σ is forward conservative if and only if every trajectory of Σ on every interval I is also a trajectory of Σ † on I . (2) Σ is backward conservative if and only if every trajectory of Σ † on every interval I is also a trajectory of Σ on I . (3) Σ is conservative if and only if Σ and Σ † have the same set of trajectories on every interval I . Proof. This is true, because, by definition, Σ is forward conservative if and only if V ⊂ V [⊥] , Σ is backward conservative if and only if V [⊥] ⊂ V , and Σ is conservative if and only if V = V [⊥] . 2 The trajectories of the original passive s/s system Σ are “orthogonal” to trajectories of the anti-passive dual system Σ † in the following sense: Lemma 3.2. Let Σ = (V ; X , W) be a passive s/s system, and let Σ † = (V [⊥] ; X , W) be the anti-passive dual of Σ. Let I be a subinterval of Z, let (x(·), w(·)) be a stable trajectory of Σ on I , and let (x † (·), w † (·)) be a stable trajectory of Σ † on I . (1) If I = [m, n) for some finite n > m, then x(n), x † (n) X = x(m), x † (m) X + w(·), w † (·) k 2 (I ;W ) .

(3.2)

(2) If I = (−∞, n) for some finite n, then limm→−∞ (x(m), x † (m))X exists, and x(n), x † (n) X = lim x(m), x † (m) X + w(·), w † (·) k 2 (I ;W ) . m→−∞

(3.3)

(3) If I = [m, ∞) for some finite m, then limn→+∞ (x(n), x † (n))X exists, and lim

n→+∞

x(n), x † (n) X = x(m), x † (m) X + w(·), w † (·) k 2 (I ;W ) .

Proof. This follows immediately from (1.3) and (3.1).

2

(3.4)


2593

By the (stable) behavior induced by the anti-passive s/s system Σ † on the interval I we mean the set † † w (·) x (·), w † (·) is an externally generated stable trajectory of Σ † on I , and we denote it by WΣ (I ). We refer to the behaviors on the intervals Z− , Z, and Z+ as the past † Σ† Σ† behavior WΣ past , the full behavior Wfull , and the future behavior Wfut induced by the anti-passive system Σ † . In the next theorem we need the notion of an anti-causal maximal nonpositive S-reducing subspace of k 2 (W). †

Definition 3.3. A maximal nonpositive S-reducing subspace W† of k 2 (W) is anti-causal if it is true for some fundamental decomposition W = −Y [] U of W that w † (·) ∈ W†

and P2 (Y ) w = 0 +

⇒

π+ w(·) = 0.

(3.5)

Note, in particular, that the projection here is onto the negative component in the fundamen2 (W) = −2 (Y) [] 2 (U), and that π in Definition 2.6 now has been tal decomposition k+ − + + replaced by π+ . Σ Σ Theorem 3.4. Let WΣ past , Wfull , and Wfut be the past, full, and future behaviors of a passive s/s †

†

†

Σ Σ system Σ = (V ; X , W), and let WΣ past , Wpast , Wpast be the past, full, and future behaviors of the † [⊥] anti-passive dual Σ = (V ; X , W). Then ∗ 2 (1) WΣ† past is a maximal nonpositive S− -invariant subspace of k− (W). †

2 (2) WΣ full is a maximal nonpositive anti-causal S-reducing subspace of k (W). † ∗ 2 (3) WΣ fut is a maximal nonpositive S+ -invariant subspace of k+ (W). † † Σ Σ 2 (4) Wpast = Wfull ∩ k− (W). † n Σ† (5) WΣ n∈Z+ S Wpast . full = †

†

Σ (6) WΣ fut = π+ Wfull . % † Σ† 2 n (7) WΣ n∈Z+ {w(·) ∈ k (W) | π+ S w ∈ Wfut }. full = † Σ [⊥] , WΣ † = (WΣ )[⊥] , and WΣ † = (WΣ )[⊥] . (8) WΣ past = (Wpast ) full fut full fut

Proof. Claims (1)–(7) are proved in the same way as in Theorem 2.8, either by repeating essentially the same argument with Σ replaced by Σ † , or by applying Theorem 2.8 to the passive dual Σ∗ of Σ and then doing a time reflection and replacing −W by W to get the anti-passive dual Σ † . If one chooses the second alternative one needs to know the connections between W[⊥] , [⊥] W[⊥] + , and W− explained in Lemma 3.5 below. The three identities in claim (8) are in principle proved in the same way, so we only prove one of these. If (x(·), w(·)) and let (x † (·), w † (·)) are stable externally generated trajectories of Σ and Σ † , respectively, then by Lemma 3.2, [w(·), w † (·)]k 2 (W ) = 0. This implies that Σ [⊥] . Since WΣ is maximal nonpositive and (WΣ )[⊥] is nonpositive, this imWΣ full full ⊂ (Wfull ) full † Σ )[⊥] . 2 = (W plies that WΣ full full †

†

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Lemma 3.5. Let W be a closed subspace of k 2 (W), and define W± by (2.15). Then [⊥] 2 W[⊥] ∩ k− (W), − =W

[⊥] . W[⊥] + = π+ W

(3.6)

2 (W) and W = π W. Here W[⊥] is the orthogoConversely, if (3.6) hold, then W+ = W ∩ k+ − − ± 2 [⊥] nal companion of W± in k± (W) and W is the orthogonal companion of W in k 2 (W). 2 (W) and w ∈ k 2 (W) we have [w , w] Proof. For each w− ∈ k− 2 (W ) . This − k 2 (W ) = [w− , π− w]k− gives

2 (π− W)[⊥] 2 (W ) = 0 for all wp ∈ π− W − = w− ∈ k− (W) [w− , wp ]k− 2 (W) [w− , π− w]k 2 (W ) = 0 for all w ∈ W = w− ∈ k − 2 (W) [w− , w]k 2 (W ) = 0 for all w ∈ W = w− ∈ k− 2 (W). = W[⊥] ∩ k− [⊥] ∩ k 2 (W). Conversely, if W[⊥] = W[⊥] ∩ k 2 (W), then Thus, if W− = π− W, then W[⊥] − =W − − − [⊥] , and hence by the above computation, W[⊥] = (π W) − −

[⊥] [⊥] W− = W[⊥] = (π− W)[⊥] = π− W[⊥] . − For the second half of (3.6) we use essentially the same computation to get (recall that (W[⊥] )[⊥] = W since W is closed) [⊥] 2 π+ W[⊥] = w+ ∈ k+ (W) [w+ , wf ]k 2 (W ) = 0 for all wf ∈ π+ W[⊥] − 2 = w+ ∈ k+ (W) [w+ , π+ w]k 2 (W ) = 0 for all w ∈ W[⊥] 2 (W) [w+ , w]k 2 (W ) = 0 for all w ∈ W[⊥] = w+ ∈ k + [⊥] 2 2 = W[⊥] ∩ k+ (W) = W ∩ k+ (W). 2 (W), then W[⊥] = ((π W[⊥] )⊥ )[⊥] = π W[⊥] . Conversely, if W[⊥] = Thus, if W+ = W ∩ k+ + + + + π+ W[⊥] , then the above computation together with the fact that W+ is closed gives

[⊥] [⊥] [⊥] 2 = π+ W[⊥] = π+ W[⊥] = W ∩ k+ (W). W+ = (W+ )[⊥]

2

Definition 3.6. Let W be a Kre˘ın space. ∗ -invariant subspace of k 2 (W) is called a anti-passive past behav(1) A maximal nonpositive S− − ior on the Kre˘ın (signal) space W. (2) A maximal nonpositive S-reducing anti-causal subspace of k 2 (W) is called a anti-passive full behavior on the (signal) space W. ∗ -invariant subspace of k 2 (W) is called a anti-passive future be(3) A maximal nonpositive S+ + havior on the (signal) space W.


2595

Theorem 3.7. Let W be a Kre˘ın space. (1) If W† is an anti-passive full behavior on W, and if we define W†+ and W†− by 2 W†− := W† ∩ k− (W),

W†+ := π+ W† ,

(3.7)

then W†− and W†+ are anti-passive past and future behaviors on W, respectively, and W† can be recovered from W†− and from W†+ by the formulas W† =

' n∈Z+

W† =

S n W†− ,

(3.8)

( w(·) ∈ k 2 (W) π− S n w ∈ W†+ .

(3.9)

n∈Z+

(2) If W†− is an anti-passive past behavior on W, and if we define W† by (3.8), then W† is an 2 (W). anti-passive full behavior on W † and W†+ = W † ∩ k− † (3) If W+ is an anti-passive future behavior on W, and if we define W† by (3.9), then W† is an anti-passive full behavior on W and W†+ = π+ W† . Proof. This is the anti-passive version of Theorem 2.11.

2

Lemma 3.8. Let W+ be a passive future behavior on a Kre˘ın space W. Then the set of all † w † (·) ∈ W[⊥] + with finite support (i.e., w (k) = 0 for all k in some interval [m, ∞)) is a dense subspace of W[⊥] + . Proof. The set W†+ := W[⊥] + is an anti-passive future behavior on W. By (3.7) and (3.8), W†+ = π+ W† = π+

' n∈Z+

' 2 2 S n W ∩ k− (W) = π+ S n W ∩ k− (W) , n∈Z+

2 (W)) has finite support. where each sequence in π+ S −n (W† ∩ k−

2

In some cases the following simple lemma is also useful. Lemma 3.9. Let W be a closed S-reducing subspace of k 2 (W), and define W± by (2.15). Then S + W+ ⊂ W+ ,

[⊥] ∗ S+ W[⊥] + = W+ ,

(3.10)

S − W− = W− ,

[⊥] ∗ S− W[⊥] − ⊂ W− .

(3.11)

Proof. The two inclusions in (3.10) and (3.11) are obvious. That the equality in (3.11) holds follows from (2.21). To prove the equality in (3.10) we use Lemma 3.5 and the fact that W[⊥] is S-reducing to compute ∗ −1 [⊥] = π π −1 [⊥] = π W[⊥] = W[⊥] . W[⊥] S+ + [−1,∞) S W + + = π+ S π+ W +

2

2596


Remark 3.10. The following analogue of Remark 2.15 is true: If W is a S-reducing subspace of k 2 (W) with the property that W[⊥] is nonpositive and that π− W contains some maximal 2 (W), then W is given by (2.17). This can be proved nonnegative S− -invariant subspace W− of k− by applying the extended version of claim (2) to the orthogonal companion W[⊥] of W, using Lemma 3.5. 4. The Hilbert spaces H(W+ ) and H(W[⊥] − ) In this section we shall present two special Hilbert spaces that play a central role throughout the rest of this article. Among others, they will be used as the state spaces of two of our canonical realizations of a passive behavior. These two spaces are special cases of the Hilbert space H(Z) constructed in [6], where Z is a maximal nonnegative subspace of a Kre˘ın space K. We begin with a short review of those results in [6] which are relevant here. The Hilbert space H(Z) Let Z be a maximal nonnegative subspace of the Kre˘ın space K, and let K/Z be the quotient of K modulo Z. We define H(Z) by H(Z) = h ∈ K/Z sup −[x, x]K x ∈ h < ∞ .

(4.1)

It turns out that sup{−[x, x]K | x ∈ h} 0 for all h ∈ H(Z), that H(Z) is a subspace of K, that H(Z) is a Hilbert space with the norm 1/2 , hH(Z ) = sup −[x, x]K x ∈ h

h ∈ H(Z),

(4.2)

and that H(Z) is continuously contained in X /Z. We denote the equivalence class h ∈ K/Z that contains a particular vector x ∈ K by h = x + Z. Thus, with this notation, (4.1) and (4.2) can be rewritten in the form H(Z) = x + Z ∈ K/Z x + Z2H(Z ) < ∞ , x + Z2H(Z ) = sup −[x + z, x + z]K z ∈ Z , x ∈ H(Z).

(4.3) (4.4)

A very important (and easily proved) fact is that if we define H0 (Z) := z† + Z z† ∈ Z [⊥] ,

(4.5)

then H0 (Z) is a subspace of H(Z). However, even more is true: H0 (Z) is a dense subspace of H(Z), and for every z† ∈ Z [⊥] it is true that † z + Z 2

H (Z )

= − z† , z† K ,

z† ∈ Z [⊥] .

(4.6)

Furthermore, it is easy to compute the inner product in H(Z) of a vector in H0 (Z) with any vector in H(Z). To explain how this is done we introduce the notation K(Z) = x ∈ K x + Z ∈ H(Z) .

(4.7)


2597

Thus, H(Z) = {x + Z | x ∈ K(Z)}, and K(Z) is the domain of the restriction of the quotient map πZ := x → x + Z to those x ∈ X for which πZ x ∈ H(Z). Let us denote this restriction by R and interpret it as a map K → H(Z) with domain K(Z). Then R is a closed and surjective linear operator; this follows from the definition of K(Z) and the fact that H(Z) is continuously contained in X /Z (for the closedness it is important that we use the H(Z)-norm in the range space). In particular, R has a bounded right-inverse H(Z) → K. Moreover, if xn ∈ K(Z) and xn + Z → x + Z for some x ∈ K(Z), then there exists a sequence zn ∈ Z such that xn + zn → x in K; this is true because H(Z) is continuously contained in X /Z and πZ has a bounded rightinverse. The rule for computing the inner product of a vector z† + Z ∈ H0 (Z) and a vector x + Z ∈ H(Z) is the following: † z + Z, x + Z H(Z ) = − z† , x K ,

z† ∈ Z [⊥] , x ∈ K(Z).

(4.8)

See [6] for more details. 2 (W) for some Kre˘ın In this article we shall need the results cited above with either K = k+ 2 (W) and space W and Z = W+ for some passive future behavior W+ on W, or K = −k− [⊥] Z = W− for some passive past behavior W− on W, interpreted as a maximal nonnegative 2 (W). subspace of −k− The Hilbert space H(W+ ) Let W+ be a given passive future behavior on a Kre˘ın signal space W, i.e., W+ is a max2 (W). We take K = k 2 (W) and Z = W in the imal nonnegative S+ -invariant subspace of k+ + + discussion above, and adapting our earlier formulas to this case we get the following result. 2 (W). Define Theorem 4.1. Let W+ be a passive future behavior on the Kre˘ın space k+

2 H(W+ ) = h+ ∈ k+ (W)/W+ sup −[w+ , w+ ]k 2 (W ) w+ ∈ h+ < ∞ , +

(4.9)

and define · H(W+ ) by 1/2 h+ H(W+ ) = sup −[w+ , w+ ]k 2 (W ) w+ ∈ h+ , +

h+ ∈ H(W+ ).

(4.10)

Then H(W+ ) is a Hilbert space with the norm · H(W+ ) that is continuously contained in 2 (W)/W . The set k+ + † † + W+ w + ∈ W[⊥] H0 (W+ ) := w+ +

(4.11)

is a dense subspace of H(W+ ), and † w + W+ 2 + H(W

+)

† † = − w+ (·), w+ (·) k 2 (W ) , +

† w+ ∈ W[⊥] + .

(4.12)

The set 2 K(W+ ) = w+ (·) ∈ k+ (W) w+ (·) + W+ ∈ H(W+ )

(4.13)

2598


2 (W), and is a subspace of k+

† w+ (·) + W+ , w(·) + W+ H(W

+)

if

† (·) ∈ W[⊥] w+ +

† = − w+ (·), w+ (·) k 2 (W ) , +

(4.14)

and w+ (·) ∈ K(W+ ).

2 (W) for The restriction R+ of the quotient map πW+ : w+ (·) → w+ (·)+W+ to those w+ (·) ∈ k+ 2 which πW+ w+ ∈ H(W+ ), regarded as an operator k+ (W) → H(W+ ), is closed and surjective k (·) ∈ K(W ) and with domain K(W+ ), and it has a bounded right-inverse. Moreover, if w+ + k w+ (·) + W+ → w+ (·) + W+ in H(W+ ) for some w+ (·) ∈ K(W+ ), then there exists a sequence k (·) ∈ W such that w k (·) + zk (·) → w (·) in k 2 (W). z+ + + + + +

Lemma 4.2. Let W+ be a passive future behavior on the Kre˘ın space W. Then the set † † † H00 (W+ ) := w+ + W+ w + ∈ W[⊥] + and w+ has finite support (which is contained in H0 (W+ )) is a dense subspace of H(W+ ). [⊥] † k Proof. Let w+ ∈ W[⊥] + . Then by Lemma 3.8, there exists a sequence w+ (·) ∈ W+ , where † k has finite support, such that w k → w in k 2 (W) as k → ∞. This implies that each w+ + + + † k k − w† ] [w+ − w+ , w+ → 0 as n → ∞, and according to (4.12), this means that 2 + k (W ) +

k + W → w † + W in H(W ) as n → ∞. Since H0 (W ) is dense in H(W ), this proves w+ + + + + + + the lemma. 2

Lemma 4.3. If w+ (·) ∈ K(W+ ), where W+ is a passive future behavior on the Kre˘ın space W, ∗ w ∈ K(W ) and then S+ + + ∗ S w+ + W+ 2 +

H(W+ )

w+ + W+ 2H(W+ ) + w+ (0), w+ (0) W .

(4.15)

If w+ (·) ∈ W[⊥] + , then w+ (·) ∈ K(W+ ) and (4.15) holds as an equality. Proof. We have for all w+ (·) ∈ K(W+ ) and all z ∈ W+ , ∗ ∗ ∗ ∗ w+ + z, S+ w + + z k 2 (W ) = − S + (w+ + S+ z), S+ (w+ + S+ z) k 2 (W ) − S+ + + = −[w+ + S+ z, w+ + S+ z]k 2 (W ) + w+ (0), w+ (0) W + w+ + W+ 2H(W+ ) + w+ (0), w+ (0) W . From here we get (4.15) by taking the supremum over all z ∈ W+ . If w+ ∈ W[⊥] + , then w+ + W+ ∈ H0 (W+ ) ⊂ H(W+ ), and by (4.10), ∗ 2 S w+ + W+ 2 + H(W+ ) − w+ + W+ H(W+ ) ∗ ∗ = − S+ w+ , S + w+ k 2 (W ) + [w+ , w+ ]k 2 (W ) = w+ (0), w+ (0) W . +

+

2


2599

The Hilbert space H(W⊥ −) Let W− be a given passive past behavior on a Kre˘ın signal space W, i.e., W− is a maximal 2 (W). Then W[⊥] is a maximal nonpositive S ∗ -invariant nonnegative S− -invariant subspace of k− − − 2 (W), and hence it can be interpreted as a maximal nonnegative S ∗ -invariant subspace of k− − 2 (W). This time we take K = −k 2 (W) and Z = W[⊥] in the subspace of the anti-space −k− − − definition of H(Z). Adapting our earlier formulas to this case we get the following result. 2 (W), and interpret W[⊥] Theorem 4.4. Let W− be a passive past behavior on the Kre˘ın space k− − ∗ -invariant subspace of the anti-space −k 2 (W). Define as a maximal nonnegative S− −

2 sup w− (·), w− (·) 2 w− (·) ∈ h− < ∞ , = h− ∈ −k− (W)/W[⊥] H W[⊥] − − k (W ) −

(4.16) and define · H(W[⊥] ) by −

h− 2

H(W[⊥] − )

= sup w− (·), w− (·) k 2 (W ) w− (·) ∈ h− . −

(4.17)

Then H(W[⊥] − ) is a Hilbert space with the norm · H(W[⊥] ) that is continuously contained in −

2 (W)/W[⊥] . The set −k− −

w− (·) ∈ W− = w− (·) + W[⊥] H0 W[⊥] − −

(4.18)

is a dense subspace of H(W[⊥] − ), and w− + W[⊥] 2 −

H(W[⊥] − )

= w− (·), w− (·) k 2 (W ) , −

w− (·) ∈ W− .

(4.19)

The set [⊥] 2 K W[⊥] = w− (·) ∈ k− (W) w− (·) + W[⊥] − − ∈ H W−

(4.20)

2 (W), and is a subspace of k−

[⊥] w− (·) + W[⊥] = w− (·), v− (·) k 2 (W ) , − , v− (·) + W− H(W[⊥] ) − − [⊥] if w− (·) ∈ W− and v− (·) ∈ K W− .

(4.21)

2 The restriction R− of the quotient map πW[⊥] : w− (·) → w− (·) + W[⊥] − to those w− (·) ∈ k− (W) −

[⊥] 2 for which πW[⊥] w− ∈ H(W[⊥] − ), regarded as an operator k− (W) → H(W− ), is closed and sur−

[⊥] k jective with domain K(W[⊥] − ), and it has a bounded right-inverse. Moreover, if w− (·) ∈ K(W− ) [⊥] [⊥] [⊥] [⊥] k (·) + W and w− − → w− (·) + W− in H(W− ) for some w− (·) ∈ K(W− ), then there exists k (·) ∈ W[⊥] such that w k (·) + zk (·) → w (·) in k 2 (W). a sequence z− − − − − −

2600


Lemma 4.5. Let W− be a passive past behavior on the Kre˘ın space W. Then the set w− ∈ W− and w− has finite support H00 := w− + W[⊥] − [⊥] (which is contained in H0 (W[⊥] − )) is a dense subspace of H(W− ).

Proof. The proof of this lemma is analogous to the proof of Lemma 4.2.

2

[⊥] Lemma 4.6. If w− (·) ∈ K(W[⊥] − ), then S− w− ∈ K(W− ) and

S− w− + W[⊥] 2 −

H(W[⊥] − )

2 w− + W[⊥] − H(W[⊥] ) − w− (−1), w− (−1) W . −

(4.22)

If w− (·) ∈ W− , then w− (·) ∈ K(W[⊥] − ) and (4.22) holds as an equality. Proof. The proof of this lemma is analogous to the proof of Lemma 4.3.

2

5. The output and input maps The output map CΣ We begin by presenting the output map of a passive s/s system. Lemma 5.1. Let Σ = (V ; X ; W) be a passive s/s system with future behavior Wfut . If (x(·), w(·)) is a stable future trajectory of Σ, then w(·) ∈ K(Wfut )

and w(·) + Wfut H(W

fut )

x(0)X .

(5.1)

Proof. Let (x(·), w(·)) be a stable future trajectory of Σ , let z(·) ∈ Wfut , and let (x1 (·), z(·)) be the corresponding externally generated stable future trajectory of Σ . Then (x(·) + x1 (·), w(·) + z(·)) is a stable future trajectory of Σ, and by (1.8), 2 2 − w(·) + z(·), w(·) + z(·) k 2 (W ) x(0) + x1 (0)X = x(0)X . +

Taking the supremum over all z ∈ Wfut we find that (5.1) holds.

2

Lemma 5.2. Let Σ = (V ; X ; W) be a passive s/s system with future behavior Wfut . Then the formula # CΣ x0 = w+ + Wfut

$ w+ (·) is the signal part of some stable future trajectory x(·), w+ (·) of Σ with x(0) = x0

defines a linear contraction CΣ : X → H(Wfut ).

(5.2)


2601

Proof. Let (x(·), w(·)) be a stable future trajectory of Σ. If (x1 (·), w1 (·)) is another stable future trajectory of Σ with the same initial state x1 (0) = x(0), then w1 (·)−w(·) ∈ Wfut , and conversely, if w1 (·) − w(·) ∈ Wfut , then there exists a stable future trajectory (x1 (·), w1 (·)) with x1 (0) = x(0). Thus, the set of all signal parts w(·) of the stable future trajectories (x(·), w(·)) of Σ with 2 (W)/W . By (5.1), the map C from fixed initial state x(0) = x0 is an equivalence class in k− fut Σ x0 to this equivalence class is a contraction X → H(Wfut ). It is easy to see that this map is linear, and by part (5) of Lemma 2.3, the domain of CΣ is all of X . 2 Definition 5.3. The contraction CΣ in Lemma 5.2 is called the output map of Σ . 2 In our next lemma we need the subspace SΣ fut of k+ (W) which is defined as follows:

2 SΣ fut = w(·) ∈ k+ (W) w + Wfut ∈ R(CΣ ) .

(5.3)

We remark that, by Lemma 5.1, it is always true that SΣ fut ⊂ K(Wfut ), where K(Wfut ) is the space defined in (4.13). Lemma 5.4. Let Σ = (V ; X ; W) be a passive s/s system with future behavior Wfut and output map CΣ , and define SΣ fut by (5.3). Then every stable future trajectory (x(·), w(·)) of Σ satisfies w(·) ∈ SΣ fut

∗ n and CΣ x(n) = S+ w + Wfut ,

n ∈ Z+ .

(5.4)

Proof. That w(·) ∈ SΣ fut follows immediately from (5.3). To get (5.4) we simply shift the trajectory (x(·), w(·)) to the left n steps and apply (5.2) with x0 replaced by x(n). 2 Definition 5.5. By an unobservable future trajectory of a passive s/s system Σ we mean a (stable) future trajectory of Σ of the type (x(·), 0) (i.e., the signal part is identically zero). The unobservable subspace UΣ of Σ consists of all the initial states x(0) of all unobservable trajectories of Σ . The system Σ is observable if UΣ = {0}. Lemma 5.6. The unobservable subspace UΣ of a passive s/s system Σ = (V ; X , W) is equal to the null space of its output map CΣ . Proof. It follows directly from Definition 5.5 and Lemma 5.4 that if x0 ∈ UΣ , then 0 ⊂ CΣ x0 , and hence CΣ x0 is the zero element in H(Wfut ). Conversely, suppose that x0 ∈ N (CΣ ), i.e., CΣ x0 = Wfut . By part (5) of Lemma 2.3, there exists a stable future trajectory (x1 (·), w1 (·)) of Σ with x1 (0) = x0 , and by Lemma 5.4, w1 (·) ∈ CΣ x0 = Wfut . Let (x2 (·), w1 (·)) be the externally generated future trajectory of Σ whose signal part is w1 (·) (cf. Lemma 2.5), and define x(·) = x1 (·) − x2 (·). Then (x(·), 0) is a stable future trajectory of Σ with x(0) = x0 , and hence x 0 ∈ UΣ . 2 Lemma 5.7. Let Σ = (V ; X , W) be a passive s/s system with output map CΣ , and define SΣ fut by (5.3). Σ Σ ∗ ∗ (1) SΣ fut is invariant under S+ , i.e., S+ w ∈ Sfut whenever w ∈ Sfut .

2602


! x1 " (2) To each

x0 w0

∈ V , there exists some w ∈ SΣ fut such that ∗ CΣ x1 = S+ w + Wfut ,

CΣ x0 = w + Wfut ,

(5.5)

w0 = w(0). (3) A vector

x0 w0

! x1 " X ∈ W satisfies the condition x0 ∈ V for some x1 ∈ X if and only if w0

w0 = w(0) for some w ∈ CΣ x0 .

(5.6)

∗ -invariance of S(Σ) follows from the fact that every left-shifted stable future Proof. (1) The S+ trajectory of Σ is still a stable future trajectory of Σ . ! x1 " (2) Let x0 ∈ V . According to assertion (7) of Lemma 2.3, there exists a stable future trajecw0

tory (x(·), w(·)) with x(0) = x0 , x(1) = x1 , and w(0) = w0 . In particular, w ∈ SΣ fut . By applying (5.4) with n = 0 to this trajectory we see that (5.5) holds. ! x1 " (3) That x0 ∈ V implies (5.6) follows from (5.5). Conversely, if (5.6) holds, then there w0

2 (W) with w(0) = w such that w + W exists some w(·) ∈ k+ 0 fut = CΣ x0 . By definition, this means that there exists some (x1 (·), w(·)) with w(0) = w0 which is a stable future trajectory of Σ . By Lemma 5.4, CΣ x1 (0) = w + Wfut . Thus, CΣ (x0 − x1 (0)) = Wfut , and by Lemma 5.6, x0 − x(0) belongs to the unobservable subspace of X . This means that there exists a stable future trajectory (x2 (·), 0) of Σ (whose signal part is identically zero) with x2 (0) = x0 − x1 (0). Define x(·) = x1 (·) + x2 (·). Then (x(·), w(·)) is a stable future trajectory of Σ with x(0) = x0 and ! x(1)" w(0) = w0 , and hence x0 ∈ V . 2 w0

Lemma 5.8. If the passive s/s system Σ = (V ; X ; W) is observable, then (x(·), w(·)) is a stable future trajectory of Σ if and only if (5.4) holds. Proof. The necessity of (5.4) follows from Lemma 5.4 and (5.3). Conversely, suppose that (5.4) holds. According to (5.3) there exists at least one stable future trajectory (x1 (·), w(·)) of Σ , and by Lemma 5.4, (5.4) holds with x(·) replaced by x1 (·). By Lemma 5.6 and the observability assumption on Σ , CΣ is injective, and hence (5.4) implies that x(n) = x1 (n) for all n ∈ Z+ . This implies that (x(·), w(·)) is a stable future trajectory of Σ . 2 Lemma 5.9. Let Σ = (V ; X , W) be a passive s/s system with output map CΣ . Then (x(·), w(·)) is a stable future trajectory of Σ if and only if x(·) = x1 (·) + x2 (·), where (x1 (·), 0) is an unobservable future trajectory of Σ and (x2 (·), w(·)) is a stable future trajectory of Σ with x2 (0) ∈ (N (CΣ ))⊥ . This decomposition is unique, and (5.4) also holds with x(·) replaced by x2 (·). Proof. Trivially, if x(·) has a decomposition of the type described in the lemma, then (x(·), w(·)) is a stable future trajectory of Σ.


2603

Conversely, let (x(·), w(·)) be a stable future trajectory of Σ . Define x1 (0) = PUΣ x(0) and x2 (0) = PU⊥ x(0). Then x(0) = x1 (0) + x2 (0) and x1 (0) ∈ UΣ . The latter condition implies Σ that x1 (0) is the initial state of some unobservable trajectory (x1 (·), 0) of Σ . Define x2 (·) = x(·) − x1 (·). Then (x2 (·), w(·)) is a stable future trajectory of Σ and x(·) = x1 (·) + x2 (·). That (5.4) also holds x(·) replaced by x2 (·) follows from the fact that (x2 (·), w(·)) is a stable future trajectory of Σ . 2 The input map BΣ We now proceed to the construction of the input map BΣ of a passive s/s system Σ . Lemma 5.10. Let Σ = (V ; X ; W) be a passive s/s system with past behavior Wpast . Then there [⊥] 0 exists a unique linear contraction BΣ : H(W[⊥] past ) → X whose restriction to H (Wpast ) is given by BΣ w− + W[⊥] past = x(0),

w− (·) ∈ Wpast ,

(5.7)

where (x(·), w− (·)) is the unique stable externally generated past trajectory of Σ whose signal part is w− (·) (cf. Lemma 2.5). Proof. Let w(·) ∈ Wpast , and let (x(·), w(·)) be the externally generated stable past trajectory of Σ with signal part w(·). Then by (2.8) and (4.19) 2 x(0)2 w(·), w(·) 2 = w + W[⊥] past H(W[⊥] ) . X k (W ) −

past

[⊥] 0 This implies that the mapping w + W[⊥] past → x(0) is a linear contraction H (Wpast ) → X . Since

[⊥] H0 (W[⊥] past ) is dense in H(Wpast ), this mapping has a unique extension to a linear contraction

BΣ : H(W[⊥] past ) → X .

2

Definition 5.11. The contraction BΣ in Lemma 5.10 is called the input map of Σ . Lemma 5.12. Let Σ = (V ; X ; W) be a passive s/s system with past behavior Wpast , future behavior Wfut , input map BΣ , and output map CΣ . Then (x(·), w(·)) is an externally generated stable past trajectory of Σ if and only if w ∈ Wpast

−n and x(n) = BΣ S− w + W[⊥] past ,

n 0,

(5.8)

and (x(·), w(·)) is an externally generated stable full trajectory of Σ if and only if w ∈ Wfull

and x(n) = BΣ π− S −n w + W[⊥] past ,

n ∈ Z.

(5.9)

In the latter case we have, in addition, CΣ x(n) = π+ S −n w + Wfut ,

n ∈ Z.

(5.10)

2604


Proof. The proof of the claim about past trajectories is an easy modification of the proof of the first claim about full trajectories, so let us only prove the two claims about the full trajectories. Let (x(·), w(·)) be an externally generated stable full trajectory of Σ . Then w(·) ∈ Wfull , and (5.7) implies that (5.9) holds with n = 0. By shifting the trajectory to the left or right |n| steps and applying (5.7) to the shifted trajectory we get (5.8) for all values of n ∈ Z. Conversely, let w(·) ∈ Wfull . Then there exists a sequence x(·) such that (x(·), w(·)) is an externally generated stable full trajectory of Σ , and by the first part of the proof, the sequence x(·) is given by (5.9). That also (5.10) holds follows from Lemma 5.4 and the fact that the restriction to Z+ of any left- or right-shifted externally generated stable full trajectory of Σ is a stable future trajectory of Σ . 2 Definition 5.13. By the finite time exactly reachable subspace of a passive s/s system Σ = (V ; X , W) we mean the set # x0 ∈ X

$ x0 = x(0) for some (stable) past trajectory of Σ with finite support ,

by the infinite time exactly reachable subspace of Σ we mean the set # x0 ∈ X

$ x0 = x(0) for some stable externally , generated past trajectory of Σ

and by the H(W[⊥] past )-exactly reachable subspace of Σ we mean the range of the input map BΣ of Σ. The system Σ is exactly reachable in one of the above senses if the corresponding exactly reachable subspace is all of X . The closure of the first of these three subspaces is called the (approximately) reachable subspace. Finally, Σ is approximately reachable or controllable if the approximately reachable subspace is all of X . Lemma 5.14. All the different types of exactly reachable subspaces in Definition 5.13 have the same closure, equal to the approximately reachable subspace. Proof. The three different types of exactly reachable subspaces defined in Definition 5.13 are (in the order that they appear) the range of the restriction of BΣ to the space H00 (W[⊥] past ) defined in

Lemma 4.2, the range of the restriction of BΣ to the space H0 (W[⊥] past ), and the full range of BΣ . That these three subspaces have the same closure follows from the fact that when one restricts the bounded linear operator BΣ to a dense subset of its domain, then the closure of its range remains the same. 2 Lemma 5.15. If Σ is a passive forward conservative s/s system, then the input map BΣ of Σ is an isometry. If, in addition, Σ is controllable, then BΣ is unitary. Proof. That BΣ is an isometry follows from the fact that we have equality in (2.8) whenever Σ is forward conservative. In particular, R(BΣ ) is closed. If, in addition, Σ is controllable, then R(BΣ ) is dense in X , and hence equal to X . 2


2605

Lemma 5.16. In the setting of Lemma 5.12, the subspace V˚ :=

x(0) X x(·), w(·) is a stable externally x(−1) ∈ X generated past trajectory of Σ w(−1) W

(5.11)

of V is dense in V if and only if the system Σ is controllable, and it is equal to V if and only if Σ is infinite time exactly reachable. Proof. Suppose that V˚ is dense in V . This implies that the infinite time exactly reachable subspace is dense in X , and by Lemma 5.14, this implies that Σ is controllable. Conversely, suppose that Σ is controllable. By Lemma 2.3, every stable externally generated past trajectory of Σ can be extended to a stable externally generated full trajectory of Σ , and Eq. (5.11) can be rewritten in the equivalent form (where we have shifted the extended trajectory one step to the left) V˚ :=

x(1) X x(·), w(·) is a stable externally . x(0) ∈ X generated full trajectory of Σ w(0) W

(5.12)

Let W = −Y [] U be a fundamental decomposition of W. This induces a fundamental decomposition

−X K := X W

−X = 0 −Y

0 [] X U

of the node space K. We claim that the orthogonal projection of V˚ onto the uniformly positive ! 0" ! 0" subspace X in this decomposition is dense in X . This projection is equal to U

U

0 X x(·), w(·) is a stable externally . ∈ X x(0) generated full trajectory of Σ PU w(0) W

The above set does not change if we replace the trajectory (x(·), w(·)) in the parametrization above by (x(·), w(·)) = (x1 (·) + x2 (·), w1 (·) + w2 (·)), where (x1 (·), w1 (·)) is a stable externally generated full trajectory of Σ and (x2 (·), w2 (·)) is a stable externally generated future trajectory of Σ (since the result is still a stable full externally generated trajectory of Σ). By part (4) of Lemma 2.3, if one first fixes (x1 (·), w1 (·)), and hence fixes x(0), then it is still possible to choose (x2 (·), w2 (·)) in such a way that PU w(0) = PU (w1 (0) + w2 (0)) is an arbitrary vector in U . This ! 0" ! 0 " implies that the orthogonal projection of V˚ onto X is X0 , where X0 is the infinite-time U U! " 0 exactly reachable subspace of Σ. This is a dense subspace of X , as claimed. U

Since V is maximal nonnegative, it has a graph representation of the form V=

Ax + Bu x x ∈ X and u ∈ U , Cx + Du

(5.13)

2606


for some contraction

X A B X : U → Y . The subspace V˚ is equal to CD V˚ =

Ax + Bu x x ∈ X0 and u ∈ U . Cx + Du

(5.14)

, this implies that V˚ is dense in V . It is equal to V if and only if Since XU0 is dense in X U X0 = X , i.e., if Σ is infinite time exactly reachable. 2 The adjoints of CΣ and BΣ The rest of this section is devoted to the study of the adjoints of the input and output maps of a passive s/s system. Lemma 5.17. Let Σ = (V ; X , W) be a passive s/s system with past and future behaviors Wpast and Wfut , respectively, and let Σ † = (V [⊥] ; X , W) be the anti-passive dual of Σ with past and [⊥] future behaviors W[⊥] past and Wfut , respectively. (1) There exists a unique contraction BΣ † : H(Wfut ) → X such that (x † (·), w † (·)) is an externally generated stable future trajectory of Σ † if and only if w † ∈ W†fut and ∗ n † x † (n) = BΣ † S+ w ,

n ∈ Z+ .

(5.15)

(2) There exists a unique contraction CΣ † : X → H(W[⊥] past ) satisfying CΣ † x(−n) = (S− )n w † + W[⊥] past

(5.16)

for every stable past trajectory (x † (·), w † (·)) of Σ † . Proof. Claim (1) is the anti-passive version of Lemma 5.12, and claim (2) is the anti-passive version of Lemma 5.4. They can be proved by either repeating the proofs of these two lemmas, or by applying Lemmas 5.12 and 5.4 to the passive dual Σ∗ of Σ . 2 Definition 5.18. The contractions BΣ † and CΣ † are called the input and output maps of Σ † , respectively. Lemma 5.19. Let Σ = (V ; X , W) be a passive s/s system with input map BΣ and output map CΣ , and let Σ † be the anti-passive dual of Σ , with the input map BΣ † and output map CΣ † . Then BΣ † = C∗Σ and CΣ † = B∗Σ . Proof. Let Wpast and Wfut be the past and future behaviors of Σ , respectively. Let (x(·), w(·)) be an externally generated past trajectory of Σ , and let (x † (·), w † (·)) be a stable past trajectory of Σ † . Then, by (3.3) and (5.8),


2607

† † BΣ w + W[⊥] past , x (0) X = x(0), x (0) X = w(·), w † (·) k 2 (W ) − [⊥] = w(·) + Wpast , w † (·) + W[⊥] past H(W[⊥] ) † = w(·) + W[⊥] past , CΣ † x (0) H(W[⊥] ) .

past

past

† This implies that (BΣ h, x † )X = (x, CΣ † x † )H(W[⊥] ) for every h ∈ H0 (W[⊥] past ) and every x ∈ X . past

[⊥] ∗ Since H0 (W[⊥] past ) is dense in H(Wpast ), this implies that BΣ = CΣ † . ∗ The proof of the fact that CΣ † = BΣ is similar to the one above, and it is left to the reader (start by taking a stable future trajectory (x(·), w(·)) of Σ and a stable externally generated future trajectory (x † (·), w † (·)) of Σ † ). 2

Lemma 5.20. If Σ is a backward conservative passive s/s system, then the output map CΣ of Σ is a co-isometry. If, in addition, Σ is observable, then CΣ is unitary. Proof. The first claim follows from the fact that if Σ is backward conservative, then the antipassive dual Σ † is forward conservative, and hence its input map BΣ † = C∗Σ is an isometry. The second claim follows from the first claim since CΣ is injective iff Σ is observable. 2 6. The past/future map of a passive full behavior We begin by constructing the past/future map of a given passive full behavior W, and then investigate what can be said about this map in the case where W is the full behavior of a passive s/s system Σ. Lemma 6.1. Let W be a passive full behavior on W with the corresponding passive past be2 (W). Then there exists a unique havior W− = π− W and passive future behavior W+ = W ∩ k+ [⊥] contraction ΓW : H(W− ) → H(W+ ) satisfying ΓW π− w + W[⊥] = π+ w + W+ , −

w ∈ W.

(6.1)

2 (W), we have for all w ∈ W and Proof. Since W is nonnegative in k 2 (W) and W+ = W ∩ k+ all z ∈ W+ ,

0 [w + z, w + z]k 2 (W ) = [π− w, π− w]k 2 (W ) + [π+ w + z, π+ w + z]k 2 (W ) . −

+

Consequently, 2 −[π+ w + z, π+ w + z]k 2 (W ) [π− w, π− w]k 2 (W ) = π− w + W[⊥] − H(W[⊥] ) +

−

−

for every w ∈ W and every z ∈ W+ . This implies that π+ w + W+ ∈ H(W+ ), and that π+ w + W+ H(W+ ) π− w + W[⊥] − H(W[⊥] ) . −

(6.2)

2608


If both w1 ∈ W and w2 ∈ W and π− (w1 − w2 ) ∈ W[⊥] − , then by the above argument, π+ (w1 − w2 ) ∈ W+ and π+ w1 − π+ w2 + W+ H(W+ ) π− (w1 − w2 ) + W[⊥] − H(W[⊥] ) = 0. −

Consequently, π+ w1 − π+ w2 ∈ W+ . Thus, formula (6.1) defines a (unique) linear contraction [⊥] [⊥] 0 H0 (W[⊥] − ) → H(W+ ), and since H (W− ) is dense in H(W− ), it has a unique extension to a [⊥] linear contraction ΓW : H(W− ) → H(W+ ). 2 Definition 6.2. The contraction ΓW : H(W[⊥] − ) → H(W+ ) in Lemma 6.1 is called the past/future map of the full behavior W. If W is the full behavior of a passive s/s system Σ , then we also call ΓW the past/future map of Σ and denote it by ΓΣ . Lemma 6.3. The past/future map ΓΣ of a passive s/s system Σ = (V ; X , W) factors into the product ΓΣ = CΣ BΣ

(6.3)

of the input map BΣ and the output map CΣ of Σ . In particular, if Σi , i = 1, 2, are two externally equivalent passive s/s systems, with input maps BΣi and output maps CΣi , then CΣ1 BΣ1 = CΣ2 BΣ2 . Proof. Let (x(·), w(·)) be an externally generated stable full trajectory of Σ . Then the restriction of (x(·), w(·)) to Z− is an externally generated stable past trajectory and the restriction of (x(·), w(·)) to Z+ is a stable future trajectory of Σ. Thus, by (5.8), x(0) = BΣ π− w and by (5.4), CΣ x(0) = π+ w + Wfut . Thus, the two contractions ΓW and CΣ BΣ coincide on the dense sub[⊥] [⊥] space H0 (W[⊥] − ) of H(W− ), and hence on all of H(W− ). If the systems Σi , i = 1, 2, are externally equivalent, then they have the same full behavior W and hence the same past/future map ΓW . Thus CΣ1 BΣ1 = ΓW = CΣ2 BΣ2 . 2 Lemma 6.4. Let W be a full behavior with the corresponding past behavior W− and future behavior W+ . Then there is a unique contraction ΓW[⊥] : H(W+ ) → H(W[⊥] − ) satisfying ΓW[⊥] π+ w † + W+ = π− w † + W[⊥] − ,

w † ∈ W[⊥] .

(6.4)

Proof. The proof is the same as the proof of Lemma 6.1 with the following replacements: We [⊥] ∩ k 2 (W) and W ↔ −W[⊥] ∩ interchange π− ↔ π+ , W ↔ −W[⊥] , W+ ↔ −W[⊥] − − = −W − 2 k+ (W). 2 Definition 6.5. The contraction ΓW[⊥] : H(W[⊥] − ) → H(W+ ) in Lemma 6.1 is called the future/past map of the anti-passive full behavior W[⊥] . If W[⊥] is the full behavior of a passive anti-causal s/s system Σ † , then we also call ΓW[⊥] the future/past map of Σ † and denote it by ΓΣ † .


2609

Lemma 6.6. The future/past map ΓΣ † of the anti-passive full behavior W[⊥] full induced by a anti† passive reflected s/s system Σ factors into the product ΓΣ † = CΣ † BΣ †

(6.5)

of the input map BΣ † of Σ † and the output map CΣ † of Σ † . Proof. The proof is analogous to the proof of Lemma 6.3.

2

Lemma 6.7. The adjoint of the past/future map ΓW of the full behavior W is the future/past map ΓW[⊥] of the dual behavior W[⊥] . Proof. This follows from Lemmas 6.3, 5.19, and 6.6.

2

Lemma 6.8. Let W be a passive full behavior with the corresponding passive past be2 (W). Let w ∈ K(W[⊥] ), havior W− = π− W and passive future behavior W+ = W ∩ k+ − − w+ ∈ K(W+ ), and suppose that w+ + W+ = ΓW w− + W[⊥] − .

(6.6)

−n Denote w := w− + w+ Then, for all n ∈ Z+ , π− S −n w ∈ K(W[⊥] − ), π+ S w ∈ K(W+ ),

+ π+ S −n w + W+ = ΓW π− S −n w + W[⊥] − , n∈Z , 2 π− S −n−1 w + W[⊥] 2 = π− S −n w + W[⊥] − H(W[⊥] − ) H(W[⊥] − − ) + w+ (n), w+ (n) W , n ∈ Z+ .

(6.7)

(6.8)

Moreover, there exists a sequence w k ∈ W such that π+ S −n w k + W+ → π+ S −n w+ + W+ [⊥] −n π− S −n w k + W[⊥] − → π− S w + W−

π+ w k → w+

in H(W+ ), n ∈ Z+ , + in H W[⊥] − , n∈Z ,

2 in k+ (W),

(6.9) (6.10) (6.11)

as n → ∞, where the convergence in (6.9) and (6.10) is uniform in n. [⊥] Proof. Step 1: Proofs of (6.9)–(6.11) with n = 0. Since H0 (W[⊥] − ) is dense in H(W− ), there [⊥] [⊥] [⊥] k k exists a sequence w− ∈ W− such that w− + W− → w− + W− in H(W− ) as k → ∞. As k to a function w k ∈ W, i.e., w k = π w k . Then W− = π− W, it is possible to extend each w− − − k (6.10) holds with n = 0 for this sequence w . By the definition of ΓΣ ,

π+ w k (·) + W+ = ΓW π− w k (·) + W[⊥] − , Since ΓW ∈ B(H(W[⊥] − ); H(W+ ), this implies that

k ∈ Z+ .

(6.12)

2610


in H(W+ ). π+ w k (·) + W+ → ΓW w− + W[⊥] − This together with (6.6) gives (6.9) with n = 0. Then, by Theorem 4.1, there exists a sequence k ∈ W such that π w k + zk → w in k 2 (W). If we replace w k by w k , then (6.9) ˜ k = w k + z+ z+ + + + + + and (6.10) remain valid, and also (6.11) holds. Step 2: Proof of (6.8) with n = 0. Let w k be a sequence satisfying (6.9)–(6.11) with n = 0. Then S −1 w k ∈ W, and consequently π− S −1 w k ∈ W− . By Lemma 4.6, S− π− S −1 w k ∈ K(W[⊥] − ) and 2 S− π− S −1 w k + W[⊥] 2 = π− S −1 w k + W[⊥] − H(W[⊥] − ) H(W[⊥] − − ) − π− S −1 w k (−1), π− S −1 w k (−1) W . Here S− π− S −1 w k = π− w k and (π− S −1 w k )(−1) = w k (0) = w+ (0) where w+ = π+ w. Consequently, π− S −1 w k + W[⊥] 2 −

H(W[⊥] − )

2 k k = π− w k + W[⊥] − H(W[⊥] ) + w (0), w (0) W . −

(6.13)

By (6.10) with n = 0 and by (6.11), the right-hand side of this identity tends to the righthand side of (6.8) with n = 0, so to prove (6.8) with n = 0 it suffices to show that [⊥] [⊥] −1 π− S −1 w k + W[⊥] − → π− S w + W− in H(W− ) as k → ∞. We begin by showing that [⊥] [⊥] limk→∞ π− S −1 w k + W− exists in H(W− ). The identity (6.13) also holds with w k replaced by w k − w for all k, ∈ Z+ . From this and conditions (6.9) and (6.10) follows that [⊥] [⊥] −1 k π− S −1 w k + W[⊥] − is a Cauchy sequence in H(W− ), and hence π− S w + W− → h1 in [⊥] [⊥] [⊥] −1 H(W− ) for some h1 ∈ H(W− ). We still have to show that h1 = π− S w + W− . By Thek ∈ W[⊥] such that π w k + zk → w in k 2 (W). Then, orem 4.4, there exists a sequence z− − − − − − by (6.11), k k w k + z− + π+ w k → w− + π+ w = w = π− w k + z− and k → π± S −1 w π± S −1 w k + z−

2 in k± (W)

(6.14)

as k → ∞. Moreover, [⊥] k −1 k π− S −1 w k + z− + W[⊥] − = π− S w + W− → h1

in H W[⊥] −

(6.15)

[⊥] as k → ∞. By Theorem 4.4, the restriction of the quotient map w(·) → w(·) + W[⊥] − to K(W− ) [⊥] [⊥] 2 −1 is a closed operator k− (W) → H(W− ), and thus π− S w + W− = h1 , as claimed.

Step 3: Proof of (6.7) with n = 0. Formula (6.12) also holds with w k replaced by S −1 w k , and by applying ΓW to π− S −1 w k + W[⊥] − we get → ΓW π− S −1 w + W[⊥] in H(W+ ) π+ S −1 w k + W+ = ΓW π− S −1 w k + W[⊥] − −


2611

as k → ∞. By Theorem 4.1, the restriction of the quotient map w(·) → w(·) + W+ to K(W+ ) 2 (W) → H(W ), and, recalling also (6.14), we get (6.7) with n = 0. is a closed operator k+ + Step 4: Proof of (6.7) and (6.8) by induction. Suppose that (6.7) and (6.8) hold with n replaced by m 0. Then (6.6) holds with w− replaced by w˜ − := π− S −m w and w+ replaced by w˜ + := π+ S −m w. We can then repeat steps 2 and 3 above with w− replaced by w˜ − and w+ replaced by w˜ + to get (6.7) and (6.8) with n replaced by m + 1. Step 5: Proof (6.9) and (6.10). The assumption of Lemma 6.8 is still satisfied if we replace w by w k − w (see, in particular, (6.12)), and hence (6.8) holds if we replace w by w k − w. If we furthermore replace n by = 0, 1, . . . , n and add the resulting identities, then we get π− S −n−1 w k − w + W[⊥] 2 − H(W[⊥] − ) k 2 [⊥] = π− w − w + W− H(W[⊥] ) −

+

n

w (n) − w+ (n), w (n) − w+ (n) W .

(6.16)

=0

Here the right-hand side tends to zero as k → ∞, uniformly in n ∈ Z+ , and consequently [⊥] [⊥] −n + π− S −n w k + W[⊥] − → π− S w + W− in H(W− ) as k → ∞, uniformly in n ∈ Z . The uni−n k −n form convergence of π+ S w + W+ to π+ S w+ + W+ in H(W+ ) then follows from (6.7) with w replaced by w k − w. 2 Lemma 6.9. Let Σ = (V ; X , W) be a passive s/s system with input map BΣ , past behavior Wpast , future behavior Wfut , and past/future map ΓΣ . Then the following two conditions are equivalent: (1) (x(·), w+ (·)) is a stable future trajectory of Σ satisfying x(0) ∈ R(BΣ ); (2) There exists some w− (·) ∈ K(W[⊥] past ) such that w+ ∈ K(Wfut ), w+ + Wfut = ΓW w− + W[⊥] past , x(n) = BΣ π− S −n (w− + w+ ) + W[⊥] past ,

(6.17) n ∈ Z+

(in particular, x(0) = BΣ (w− + W[⊥] past )). When these equivalent conditions hold, then (6.17) remains true for every w− (·) ∈ K(W[⊥] past ) satisfying x(0) = BΣ (w− + W[⊥] past ).

Proof. We first suppose that (x(·), w(·)) is a stable future trajectory of Σ satisfying x(0) ∈ R(BΣ ) and show that (6.17) holds for every w− (·) ∈ K(W[⊥] past ) satisfying x(0) = BΣ (w− + W[⊥] past ).

That w+ ∈ K(Wfut ) follows from Lemma 5.1. By assumption,

!

x(1) " x(0) w+ (0)

∈ V and x(0) =

[⊥] BΣ (w− + W[⊥] past ) for some w− ∈ K(Wpast ). By Lemma 5.2, CΣ x(0) = w+ + Wfut , and hence

2612


[⊥] w+ +Wfut = CΣ BΣ (w− +W[⊥] past ) = ΓΣ (w− +Wpast ). This proves the first two claims in (6.17), and it remains to prove the formula for x(n) given in (6.17) for n 1. Denote w = w− + w+ . By Lemma 6.8, there exists a sequence w k ∈ Wfull such that 2 (W) as k → ∞ and π S −n w k + W[⊥] → π S −n w + W[⊥] in H(W[⊥] ) π+ w k → w+ in k+ − − past past past as k → ∞, uniform in n ∈ Z+ . Let (x k (·), w k (·)) be the externally generated stable full trajectory of Σ whose signal part is w k (·) (cf. Lemma 2.5). By Lemma 5.12, x k (n) = [⊥] −n BΣ (π− S −n w k + W[⊥] past ), which tends to x1 (n) := BΣ (π− S w + Wpast ) as k → ∞, uniformly

in n ∈ Z+ . In particular, x k (0) → BΣ (w− + W[⊥] past ) = x1 (0) = x(0). Since the restriction of k k + (x (·), w (·)) to Z is a future trajectory of Σ for each k, it follows from part (1) of Lemma 2.3 that the limit (x1 (·), w+ (·)) is a stable future trajectory of Σ . This trajectory has both the same initial state x(0) and the same signal part w+ (·) as the given trajectory (x(·), w+ (·)), and hence x1 (n) = x(n) for all n ∈ Z+ . This proves that the last claim in (6.17) holds. The proof of the converse direction is based on induction over the length of the interval where (x(·), w(·)) is a solution of Σ. We begin by showing that if (6.17) holds, then (x(·), w(·)) is a trajectory of Σ on the one-point interval [0, 0] = {0}. Suppose that (6.17) holds for n = 0, 1. Thus, in particular, x(0) = BΣ (w− + W[⊥] past ) and w+ + Wfut = ΓΣ (w− + W[⊥] past ). By Lemma 6.3, w+ + Wfut = CΣ x0 . By part (3) of Lemma 5.7, ! x(1) " x(0) ∈ V for some x(1) ∈ X . By part (7) of Lemma 2.3, there exists a stable future trajectory w(0)

(x1 (·), w1 (·)) of Σ satisfying x1 (0) = x(0) and w1 (0) = w+ (0). By the first part of the proof, −1 −1 −1 (w− + w1 ) + W[⊥] x1 (1) = BΣ (π− S− past ). Here π− S− (w− + w1 ) = π− S− (w− + w+ ) since

−1 w1 (0) = w+ (0), and hence x1 (1) = BΣ (π− S− (w− + w+ ) + W[⊥] past ). Since we assume that ! x(1) " (6.17) holds (for n = 1), we get x(1) = x1 (1), and consequently x(0) ∈ V . This proves that w+ (0)

(x(·), w+ (·)) is a trajectory of Σ on the one-point interval {0}. One can use essentially the same argument to show that if we know that (x(·), w+ (·)) is a trajectory of Σ on an interval [0, k], then it is also a trajectory on [0, k + 1], i.e., one shifts the trajectory k + 1 steps to the left, and then apply the above argument. The invariance of the first two conditions in (6.17) under this left-shift follows from Lemma 6.8. Thus, by induction, (x(·), w(·)) is a future trajectory of Σ. By Lemma 2.1, this trajectory is stable. 2 7. The observable backward conservative realization W

W

W

In this section we shall construct a canonical model Σobc+ = (Vobc+ ; Xobc+ , W) of a passive observable backward conservative s/s system with a given passive future behavior W+ . W

Theorem 7.1. Let W+ be a passive future behavior on the Kre˘ın space W. Let Xobc+ = H(W+ ), where H(W+ ) is the space defined in Theorem 4.4, and let ∗

S + w + W+ H(W+ ) W+ Vobc = (7.1) ∈ H(W+ ) w ∈ K(W+ ) , w + W+ w(0) W W

W

where K(W+ ) is the space defined in (4.20). Then Σobc+ = (Vobc+ ; H(W+ ), W) is a passive observable backward conservative s/s system whose future behavior is equal to W+ . Moreover, W (x(·), w(·)) is a stable future trajectory of Σobc+ if and only if


w ∈ K(W+ )

∗ n and x(n) = S+ w + W+ ,

2613

n ∈ Z+ .

(7.2)

W

Proof. In this proof we denote the node space of Σobc+ by K+ := −H(W+ ) [] H(W+ ) [] W. W

W

Step 1: Vobc+ is a nonnegative subspace of K+ . It follows from Lemma 4.3 that Vobc+ ⊂ K+ , W and that Vobc+ is nonnegative in K+ . It is a subspace of K+ since it is a linear image of the 2 (W). subspace K(W+ ) of k+ W W W Step 2: Vobc+ is closed and (Vobc+ )[⊥] ⊂ Vobc+ . Define V˚obc by

⎧⎡ ∗ † ⎫ ⎤ ⎨ S + z + W+ ⎬ V˚obc = ⎣ z† + W+ ⎦ z† ∈ W[⊥] . + ⎩ ⎭ z† (0)

(7.3)

W W Then V˚obc ⊂ Vobc+ since H0 (W+ ) ⊂ H(W+ ). We claim that (V˚obc )[⊥] = Vobc+ . Clearly, this W+ W+ [⊥] W+ W+ [⊥] implies that Vobc is closed, and that (Vobc ) ⊂ Vobc since (Vobc ) = ((V˚obc )[⊥] )[⊥] is the closure of V˚obc . ! " x1 A vector k = x0 belongs to (V˚obc )[⊥] if and only if x1 , x0 ∈ K(W+ ), w0 ∈ W, and w0

∗ † z + W+ H(W − x1 , S+

+)

+ x0 , z† + W+ H(W

+)

+ w0 , z† (0) W = 0,

z† ∈ W[⊥] + . (7.4)

[⊥] ∗ ∗ † Since W+ is S+ -invariant, its orthogonal companion W[⊥] + is S+ -invariant, i.e., S+ z ∈ W+ whenever z† ∈ W[⊥] + . By (4.14), for every v1 ∈ x1 and v0 ∈ x0 , (7.4) can therefore be rewritten in the form

∗ † v 1 , S+ z k 2 (W ) − v0 , z† k 2 (W ) + w0 , z† (0) W = 0, +

+

z† ∈ W[⊥] + .

(7.5)

2 (W) by w(0) = w and w(n) = 0 for n > 0, and let P be the Define the sequence w ∈ k+ 0 0 2 (W) onto the subspace of vectors k(·) satisfying k(n) = 0 for n > 0. orthogonal projection in k+ Then (7.5) can be rewritten as

S+ v1 − v0 + P0 w, z† k 2 (W ) = 0, +

[⊥] = W , this is equivalent to Since (W[⊥] + + )

S+ v1 − v0 + P0 w = z for some z ∈ W+ . Define v = v0 + z. Then v ∈ x0 , and S+ v1 − v + P0 w = 0. This is equivalent to the pair of equations

z† ∈ W[⊥] + .

2614


v(0) = w0 ! x1 " Thus,

x0 w0

∗ and v1 = S+ v.

∗ v + W , and w = v(0) for some ∈ (V˚obc )[⊥] if and only if x0 = v + W+ , x1 = S+ + 0 W

v ∈ K(W+ ), or equivalently, if and only if k ∈ Vobc+ . W

Step 3: Vobc+ is the generating subspace of a passive and backward conservative s/s system W W W = (Vobc+ ; H(W+ ), W). By steps 1 and 2, Vobc+ is closed and nonnegative, and (Vobc+ )[⊥] W is neutral, hence nonpositive. By, e.g., [3, Proposition 2.2(5)], Vobc+ is a maximal nonnegative subspace of K+ , and hence, by [3, Corollary 5.13], it generates a passive backward conservative s/s system. W Σobc+

W

Step 4: Σobc+ is observable. Let (x(·), w(·)) be an unobservable future trajectory of Σ , i.e., † ∗ n † + w(n) = 0 for all n ∈ Z+ . Let z† ∈ W[⊥] + , and define x (n) = (S+ ) z + W+ , n ∈ Z . Then ! x † (n+1)" [⊥] it follows from (7.3) that x † (n) ∈ V˚obc for all n ∈ Z+ . Since V˚obc ⊂ Vobc , this means that z† (n)

W

is a future trajectory of the anti-passive dual of Σobc+ (and also a future trajectory Moreover, x † (n) → 0 in H(W+ ) as n → ∞, because by Theorem 4.1,

(x † (·), z† (·)) of

W Σobc+ ).

† 2 x (n)

H(W+ )

∗ n † ∗ n † = − S+ z , S+ z k 2 (W ) = − z† , z† k 2 ([n,∞);W ) +

which tends to zero as k → ∞. By part (3) of Lemma 3.2 and Theorem 4.1 (recall that w(·) = 0), x(0), z† + W+ H(W

+)

= x(0), x † (0) H(W

+)

= − w(·), z† (·) k 2 (W ) = 0. +

Thus, x(0) is orthogonal to H0 (W+ ), and since H0 (W+ ) is dense in H(W+ ), this implies that W x(0) = 0. Thus, Σobc+ is observable. W

Step 5: If (7.2) holds, then (x(·), w(·)) is a stable future trajectory of Σobc+ . Let w ∈ K(W+ ) ∗ )n w + W , n 0. Then it is easy to see that (x(·), w(·)) is a trajectory of and define x(n) = (S+ + W 2 (W). It is stable since Σ W+ is passive and w(·) ∈ 2 (W) (see Lemma 2.1). Σobc+ with w ∈ k+ + obc W

Step 6: The future behavior of Σobc+ is equal to W+ . It follows from step 5 that the future W+ Σ Σ behavior WΣ + of Σobc contains W+ , and hence W+ = W+ since, by Theorem 2.8, W+ is 2 nonnegative, and by assumption, W+ is maximal nonnegative in k+ (W). W

Step 7: If (x(·), w(·)) is a stable future trajectory of Σobc+ , then (7.2) holds. Let (x(·), w(·)) W be a stable future trajectory of Σobc+ . By Lemma 5.1, w(·) ∈ K(W+ ). As we saw above, if we ∗ n define x1 (n) = (S+ ) w + W+ , n ∈ Z+ , then (x1 (·), w(·)) is another stable future trajectory of W W Σobc+ with the same signal part (·). Since Σobc+ is observable, this implies that x(n) = x1 (n) for + all n ∈ Z , i.e., (7.2) holds. 2 W

Definition 7.2. We call the system Σobc+ the canonical model of an observable passive backward conservative s/s system with future behavior W+ .


2615

W

Corollary 7.3. The system Σobc+ is approximately null-controllable, i.e., the set of all the initial W states x(0) of all those future trajectories of Σobc+ with have finite support is dense in Xobc = H(W+ ). Proof. This follows from Theorem 7.1 and Lemma 4.2.

2

In Theorem 2.11 we established the connections (2.15)–(2.17) between passive past, future, and full behaviors W− = W− , W+ = W+ , and W = W (see Remark 2.13). In particular, they permit us to define unique full behavior W in terms of a given future behavior W+ . Once we have the full behavior W, we can also define the past/future map ΓW by (6.1). W

Lemma 7.4. The input map of Σobc+ is the past/future map ΓW of W, and the output map of W Σobc+ is the identity on H(W+ ). Proof. It follows from Lemma 5.4 and Theorem 7.1 that we for every stable future trajectory W (x(·), w(·)) of Σobc+ have C

Σobc+ W

x(0) = w + W+ = x(0).

W

W

Thus, the output map of Σobc+ is the identity. This implies that the input map of Σobc+ is ΓW , since the product of the input and output maps must be equal to ΓW . 2 Lemma 7.5. A sequence (x(·), w(·)) is an externally generated stable past trajectory of Σobc if n w + W[⊥] ), n 0. and only if w ∈ W− and x(−n) = ΓW (S− − Proof. This follows from Lemmas 5.4 and 7.4.

2

Definition 7.6. A bounded linear operator E : X1 → X2 intertwines the two passive s/s systems Σ1 = (V1 ; X1 ; W) and Σ2 = (V2 ; X2 ; W) (with the same signal space W) if

E 0 0

0 E 0

0 0 1W

X2 V1 = V2 ∩ R(E) . W

(7.6)

In this case we say that Σ1 and Σ2 are boundedly intertwined by E, or contractively intertwined by E if E is a contraction. If E has a bounded inverse, then we say that Σ1 and Σ2 are similar with similarity operator E, and if E is unitary, then we say that Σ1 and Σ2 are unitarily similar. It is also possible to define a more general intertwinement relation where E is allowed to be a closed relation instead of a bounded operator, but Definition 7.6 covers our present needs. Lemma 7.7. The two passive s/s systems Σ1 = (V1 ; X1 ; W) and Σ2 = (V2 ; X2 ; W) are intertwined by the operator E ∈ B(X1 ; X2 ) if and only if the formula x1 (·), w(·) → Ex1 (·), w(·)

(7.7)

2616


defines a map from the set of all stable future trajectories (x1 (·), w(·)) of Σ1 onto the set of all stable future trajectories (x2 (·), w(·)) of Σ2 satisfying x2 (0) ∈ R(E). In particular, if Σ1 and Σ2 are boundedly intertwined by E, then they have the same future behavior. Proof. Let us first comment on the last claim: For externally generated trajectories of Σ2 the condition x2 (0) ∈ R(E) is trivially true, and so there is a one-to-one correspondence between the externally generated future trajectories of Σ1 and Σ2 (an externally generated trajectory is uniquely determined by its signal part w(·)). This implies that the two systems have the same future behavior. Suppose next that (7.6) holds, i.e., that E intertwines Σ1 and Σ2 . Then trivially, if (x1 (·), w(·)) is a stable future trajectory of Σ1 , then (Ex1 (·), w(·)) is a stable future trajectory of Σ2 . Conversely, suppose that (x2 (·), w(·)) is a stable future trajectory of Σ2 . Then

x2 (n + 1) ∈ V2 , x2 (n) w(n)

n ∈ Z+ .

(7.8)

! x1 (1)" Taking n = 1 above we can use (7.6) to conclude that there exists a vector x1 (0) ∈ V1 such w(0) ! x2 (1)" ! Ex1 (1)" that x2 (0) = Ex1 (0) . In particular, x2 (1) ∈ R(E). We can therefore repeat the same argument w(0) w(0) ! x1 (2)" with n = 1 to conclude that there exists (a unique) x1 (2) ∈ X1 such that x1 (1) ∈ V1 and x2 (2) = w(1)

Ex1 (2). By repeating this argument indefinitely (or by using induction) we get a sequence x1 (·) such that (x1 (·), w(·)) is a future trajectory of Σ1 , and such that x2 (·) = Ex1 (·). By Lemma 2.1, the trajectory (x1 (·), w(·)) is stable. Thus, the mapping defined in (7.7) is surjective. We then turn to the converse statement, and suppose that the stable future trajectories of Σ1 and Σ2 are related as described in the lemma. Let (x1 (·), w(·)) be a stable future trajectory of Σ1 . Then, by the assumption, (Ex1 (·), w(·)) be a stable future trajectory of Σ1 . In particular, ! x(1) " ! Ex1 (1)" Ex1 (0) ∈ V2 . By part (7) of Lemma 2.3, the vector x(0) can be an arbitrary vector in V . This w(0)

w(0)

shows that the that the left-hand side of (7.6) is a subset of the right-hand side. On the other hand, if (x2 (·), w(·)) is an arbitrary stable future trajectory of Σ2 satisfying x2 (0) ∈ R(E), then by assumption, there exists a future trajectory (x1 (·), w1 (·)) of Σ1 such that x2 (·) = Ex1 (·). Here ! x2 (1)" x2 (0) represents an arbitrary vector in the right-hand side of (7.6), and we have shown that it w

belongs to the left-hand side of (7.6). Thus, we have equality in (7.6).

2

Theorem 7.8. Let Σ = (V ; X , W) be a passive s/s system with output map CΣ and future beW W W havior W+ . Then Σ and Σobc+ = (Vobc+ ; Xobc+ , W) are contractively intertwined by CΣ , i.e.,

CΣ 0 0

0 CΣ 0

0 0 1W

⎡

W

Xobc+

⎤

⎥ ⎢ W . V = Vobc+ ∩ ⎣ SΣ fut ⎦ W

(7.9)

Proof. Let (x(·), w(·)) be a stable future trajectory of Σ . By Lemmas 5.1 and 5.4, w(·)∈K(Wfut ) ∗ )n w + W , n ∈ Z+ . Define x (·) = C x(·). Then x (n) = (S ∗ )n w + W , and CΣ x(n) = (S+ fut 0 Σ 0 fut +


2617

n ∈ Z+ (where Wfut is the future behavior of Σ ), and by Theorem 7.1, (x0 (·), w(·)) is stable ! x(1) " W future trajectory of Σobc+ . By part (7) of Lemma 2.3, the vector x(0) can be an arbitrary vector w(0) ! x0 (1)" ! CΣ x(1)" W+ in V , and since x0 (0) = CΣ x(0) ∈ Vobc , this implies that the left-hand side of (7.9) is a subset w(0)

w(0)

of the right-hand side. W To prove the converse inclusion we let (x0 (·), w(·)) be a stable future trajectory of Σobc+ , ! x0 (1)" x0 (0) represents and suppose that x0 (0) ∈ SΣ fut . Then, by part (7) of Lemma 2.3, the vector w(0)

an arbitrary vector in the right-hand side of (7.9). Choose some arbitrary x(0) ∈ X such that W CΣ x(0) = x0 (0). Recall that the output map of Σobc+ is the identity. By part (3) of Lemma 5.7 W applied to Σobc+ , w0 ∈ (CΣ x0 (0))(0), and by the same lemma applied to the system Σ, there ! x(1) " exists some x(1) ∈ X such that x(0) ∈ V . By the first inclusion that we already proved, this w(0) ! CΣ x(1)" W+ W implies that CΣ x(0) ∈ Vobc . But here the last two components of Vobc+ determine the first w(0) ! x0 (1)" ! CΣ x(1)" component uniquely, and hence we must have x0 (0) = CΣ x(0). Thus, x0 (0) = CΣ x(0) , where w(0) w(0) ! x(1) " x(0) ∈ V . This proves that the right-hand side of (7.8) is contained in the left-hand side. 2 w(0)

Corollary 7.9. Let Σ = (V ; X , W) be a passive s/s system with output map CΣ and full behavW ior W, and let Σobc+ be the canonical model of an observable backward conservative s/s system with full behavior W. Then the formula x(·), w(·) → CΣ x(·), w(·)

(7.10)

defines a map from the set of all stable future trajectories of Σ onto the set of all stable future W trajectories (x0 (·), w(·)) of Σobc+ satisfying x0 (0) ∈ R(CΣ ). Proof. This follows from Lemma 7.7 and Theorem 7.8.

2

Corollary 7.10. Any two observable and backward conservative realizations of a given full behavior W are unitarily similar to each other. Proof. This is true, because, by Lemma 5.20, the output maps of these two systems are unitary, W and hence, by Corollary 7.9, both systems are unitarily similar to Σobc+ . 2 8. The controllable forward conservative realization W

W

W

In this section we shall construct a canonical model Σcfc − = (Vcfc − ; Xcfc − , W) of a passive controllable forward conservative s/s system with a given passive past behavior W− . The results W for this model are analogous to the results on the model Σobc+ obtained in the preceding section. W The state space of Σcfc − is the Hilbert space H(W[⊥] − ) presented in Theorem 4.4 (whereas the W+ state space of Σobc is the Hilbert space H(W+ ) presented in Theorem 4.1). The full description

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D.Z. Arov, O.J. Staffans / Journal of Functional Analysis 257 (2009) 2573–2634 W

W

of the generating subspace Vcfc − is more complicated than the description of Vobc+ , and in our W next theorem we first give a preliminary definition of Vcfc − as the closure of the set ⎧⎡ ⎫ ⎤ [⊥] ⎤ ⎡ H(W[⊥] ⎨ w − + W− ⎬ − ) ⎦ ∈ ⎣ H(W[⊥] ) ⎦ w− ∈ W− . = ⎣ S− w− + W[⊥] − − ⎩ ⎭ w− (−1) W

W V˚cfc −

(8.1)

Since every w− ∈ W− can be extended to a function w ∈ W, and since π− w ∈ W− whenever w ∈ W, Eq. (8.1) can alternatively be written in the form (where we have shifted the extended function one step to the left)

W V˚cfc −

⎫ ⎧⎡ ⎤ [⊥] ⎤ ⎡ −1 H(W[⊥] ⎬ ⎨ π− S w + W− − ) ⎦ ∈ ⎣ H(W[⊥] ) ⎦ w ∈ W . = ⎣ π− w + W[⊥] − − ⎭ ⎩ w(0) W

(8.2)

W

A full description of Vcfc − will be given later in Theorem 8.6. Theorem 8.1. Let W be a Kre˘ın space, and let W− be a passive past behavior on W. Let W W− ˚ W− Xcfc − := H(W[⊥] − ) and let Vcfc be the closure of the set Vcfc defined in (8.1) in the Kre˘ın W− W− [⊥] [⊥] space K− := −H(W[⊥] − ) [] H(W− ) [] W. Then Σcfc = (Vcfc ; H(W− ), W) is a passive controllable forward conservative s/s system whose past behavior is equal to W− . Moreover, the following claims are true: W

(1) The sequence (x(·), w(·)) is an externally generated stable past trajectory of Σcfc − if and only if w ∈ W−

|n|

and x(n) = S− w + W[⊥] − ,

n 0.

(8.3)

W

(2) If (x(·), w(·)) is a stable past trajectory of Σcfc − , then |n| and x(n) = S− w + W[⊥] w ∈ K W[⊥] − − ,

n 0.

(8.4)

[⊥] [⊥] 0 Proof. Step 1: Vcfc − is a neutral subspace of K. Recall that w + W[⊥] − ∈ H (W− ) ⊂ H(W− ) [⊥] for every w ∈ W− . Since W− is S− -invariant, it is also true that S− w + W[⊥] − ∈ H(W− ) for W W every w ∈ W− . This implies that V˚cfc − is a subspace of K− . To show that Vcfc − is neutral it W W W suffices to show that V˚cfc − is neutral, since V˚cfc − is dense in Vcfc − . However, this follows from Lemma 4.6. W

W

Step 2: Vcfc − is maximal nonnegative in K− . Let W = −Y [] U be a fundamental decomposition of W. This induces a fundamental decomposition of the node space ⎡

⎤

−H(W[⊥] −H(W[⊥] 0 − ) − ) [⊥] ⎦= K− := ⎣ H(W[⊥] [] H(W− ) . 0 − ) U −Y W


2619

Arguing in the same way as we did in the proof of Lemma 5.16 with x(0) replaced by [⊥] ˚ W− π− w + W[⊥] − and x(−1) replaced by π− Sw + W− we find that the projection of Vcfc onto the H0 (W[⊥] ) − positive component of this fundamental decomposition is equal to , which is dense U H(W[⊥] ) W − − . We know that Vcfc is neutral, and hence it is the graph of an isometric operin AU0 B H0 (W[⊥] ) [⊥] − ator C0 D : → H(W− ) (i.e., A0 and C0 are defined on H0 (W[⊥] − ), and B and Y U A B D are defined on U ). This implies that C0 D0 has a unique extension to an isometric operator A B H(W[⊥] ) A B [⊥] W W − : → H(W− ) . Since Vcfc − is the closure of V˚cfc − , it is the graph of C , CD D Y U and hence maximal nonnegative. W

Step 3: Vcfc − is the generating subspace of a passive and forward conservative s/s system W− W Σcfc = (Vcfc − ; H(W− ), W). This follows from steps 1 and 2. Step 4: If (8.3) holds, then (x(·), w(·)) is a stable externally generated past trajectory of ! x(n+1)" W W |n| x(n) ∈ V˚cfc − ⊂ Vcfc − for all When w ∈ W− and x(n) = (S− w) + W[⊥] − , n 0, then

W Σcfc − .

w(n) W

2 (W). To n ∈ Z− . Thus, by definition, (x(·), w(·)) is a past trajectory of Σcfc − . Clearly w ∈ k− W see that x(n) → 0 as n → −∞ we argue as follows. The subspace V˚cfc − is neutral in K− , and hence, for all n ∈ Z− ,

x(n)2

H(W[⊥] − )

−1 2 w(k), w(k) W . = x(0)H(W[⊥] ) − −

k=n

As n → −∞, the last sum tends to [w(·), w(·)]k 2 (W ) . However, by (4.19), −

x(0)2

H(W[⊥] − )

2 = w + Z [⊥] H(W[⊥] ) = w(·), w(·) k 2 (W ) . −

−

This implies that x(n) → 0 in H(W[⊥] − ) as n → −∞. W

Step 5: The past behavior of Σcfc − is equal to W− . It follows from step 4 that the past behavior W− Σ Σ WΣ − of Σcfc contains W− , and hence W− = W− since, by Theorem 2.8, W− is nonnegative, 2 and by assumption, W− is maximal nonnegative in k− (W). Step 6: Σcfc is controllable. It follows from step 4 that if w ∈ W− has compact support, then W− w + W[⊥] − belongs to the reachable subspace of Σcfc . According to Lemma 4.5, this set is dense [⊥] in H(W− ). Thus, the set of states that can be reached in a finite time is dense in the state space W− W− H(W[⊥] − ) of Σcfc , and so Σcfc is controllable. W

Step 7: If (x(·), w(·)) is a stable past trajectory of Σcfc − , then (8.4) holds. By Lemma 3.1, W every stable past trajectory (x(·), w(·)) of Σcfc − is also a stable past trajectory of the anti-passive W dual Σ † of Σcfc − . By applying the reflected version of Theorem 7.1 to the system Σ † we find [⊥] n that w ∈ H(W[⊥] − ) and x(−n) = (S− w) + W− , n 0.

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D.Z. Arov, O.J. Staffans / Journal of Functional Analysis 257 (2009) 2573–2634 W

Step 8: If (x(·), w(·)) is a stable externally generated past trajectory of Σcfc − , then (8.3) holds. This follows from steps 5 and 7. 2 W

Definition 8.2. We call the system Σcfc − the canonical model of a passive controllable forward conservative s/s system with full behavior W. In Theorem 2.11 we established the connections (2.15)–(2.17) between passive past, future, and full behaviors W− = W− , W+ = W+ , and W = W (see Remark 2.13). In particular, they permit us to define unique full behavior W in terms of a given past behavior W− . Once we have the full behavior W, we can also define the past/future map ΓW by (6.1). − Lemma 8.3. The input map of Σcfc − is the identity on H(W[⊥] − ), and the output map of Σcfc is the past/future map ΓW of W.

W

W

Proof. It follows from Lemma 5.12 and Theorem 8.1 that the B H0 (W[⊥] − ),

and since

H0 (W[⊥] − )

is dense in

This implies that the output map of must be equal to ΓW . 2

W Σcfc −

H(W[⊥] − ),

Σcfc − W

acts as the identity on

this means that B

Σcfc − W

is the identity.

is ΓW , since the product of the input and output maps

Corollary 8.4. The system Σcfc − is both H(W[⊥] − )-exactly controllable and constructable (observable in backward time), i.e., if the signal part w(·) of a past stable trajectory (x(·), w(·)) of W Σcfc − is zero, then also the state part x(·) is zero. W

Proof. The first claim follows from Lemma 8.3 and the second claim follows from (8.4).

2

W

Lemma 8.5. The pair of sequences (x(·), w+ (·)) is a stable future trajectory of Σcfc − if and only if w+ ∈ K(Wfut ), w+ + Wfut = ΓW w− + W[⊥] − , x(n) = π− S −n (w− + w+ ) + W[⊥] − ,

(8.5) n ∈ Z+ ,

[⊥] for some sequence w− ∈ K(W[⊥] − ) (in particular, x(0) = w− + W− ). W

Proof. This follows from Lemma 6.9, taking into account that the input map of Σcfc − is the identity on H(W[⊥] − ). 2 W

W

Lemma 8.5 gives us the following description of the generating subspace Vcfc − of Σcfc − : Theorem 8.6. Let W− be a passive past behavior on the Kre˘ın space W. Then the generating W W W W subspace Vcfc − of the canonical model Σcfc − = (Vcfc − ; Xcfc − , W) of a passive controllable forward conservative realization of W− is given by


W

Vcfc −

2621

⎧⎡ ⎫ [⊥] ⎤ −1 [⊥] ⎨ π− S w + W− ⎬ w+ ∈ K(W+ ), ⎦ w = w− + w+ , w− ∈ K W− ,[⊥] = ⎣ π− w + W[⊥] . − and w+ + W+ = ΓW w− + W− ⎩ ⎭ w(0) (8.6) ! x1 "

Proof. This follows from Lemma 8.5 and the fact that

x0 w0

W

∈ Vcfc − if and only if there exists a

W

stable future trajectory (x(·), w(·)) of Σcfc − with x(0) = x0 , x(1) = x1 , and w(0) = w0 .

2

Theorem 8.7. Let Σ = (V ; X , W) be a passive s/s system with input map BΣ and full behavW W W ior W. Then BΣ intertwines Σcfc − = (Vcfc − ; Xcfc − , W) with Σ in the sense that

BΣ 0 0

0 BΣ 0

0 0 1W

W Vcfc −

X = V ∩ R(BΣ ) . W

Proof. This follows from Lemmas 6.9, 7.7, and 8.5.

(8.7)

2

Corollary 8.8. Any two controllable and forward conservative realizations of a given past behavior W− are unitarily similar to each other. Proof. This is true, because by Lemma 5.20, the input maps of these two systems are unitary, W and hence, by Theorem 8.7, both systems are unitarily similar to Σcfc − . 2 W

W

Theorem 8.9. The operator ΓW intertwines the two s/s systems Σcfc − and Σobc+ , i.e.,

ΓW 0 0

0 ΓW 0

0 0 1W

⎡

W Vcfc −

W = Vobc+

W ⎤ Xobc+ ∩ ⎣ R(ΓW ) ⎦ . W

(8.8)

Proof. This follows from Theorem 7.8 and Lemma 7.4, and also from Theorem 8.7 and Lemma 8.3. 2 W

The orthogonal companion of Vcfc − can be characterized as follows: W

Lemma 8.10. The orthogonal companion of Vcfc − is given by ⎫ ⎧⎡ [⊥] ⎤ [⊥] ⎬ W− [⊥] W− [⊥] ⎨ w− + W−[⊥] . = Vcfc = ⎣ S − w − + W− ⎦ w − ∈ K W − V˚cfc ⎭ ⎩ w− (−1)

(8.9)

Proof. The proof of this lemma is analogous to step 2 in the proof of Theorem 7.1 which shows W that (V˚obc )[⊥] = Vobc+ , where V˚obc is the subspace of K+ defined in (7.3). We leave this proof to the reader (interchange the first two components in V˚obc with each other, replace W+ by −W[⊥] − , ∗ ). 2 replace Z+ by Z− , and replace S+ by S−

2622


9. Frequency domain versions of passive behaviors The Fourier transform Up to now we have throughout worked in the time domain, and formulated all our results in terms of sequences in k 2 (I ; W), where I is a discrete time interval. It is also possible to work in the frequency domain instead, replacing all the signal sequences w(·) by their Fourier transforms. In this section we assume, for simplicity, that the signal space W is separable. As is well known, for defined by each Hilbert nspace X , the Fourier transform F , formally 2 (X ) onto the w(n)z , w(·) → w(·) ˆ is a unitary map from (F w(·))(z) := w(z) ˆ = ∞ n=−∞ Lebesgue space L2 (X ) := L2 (T; X ). The restrictions F± = F |2 (X ) of F to 2± (X ) are uni±

tary maps of from 2± (X ) onto the Hardy spaces H 2 (D± ; X ), where D+ := z ∈ Z |z| < 1 , D− := z ∈ Z |z| > 1 ∪ {∞}, T := z ∈ Z |z| = 1 . Functions in H±2 (X ) are analytic in D± , they have boundary values a.e. on T, L2 (X ) = H−2 (X )⊕ H−2 (X ), and the norm in these three spaces are given by the same formula 2 w(·) ˆ 2 L

1 = (X ) 2π

)

2 w(ζ ˆ )X |dζ |.

(9.1)

ζ ∈T

Constant X -valued functions belong to H+2 (X ), and every wˆ ∈ H−2 (X ) satisfies w(∞) ˆ = 0. The inverse Fourier transform is given by 1 w(n) = 2πi

)

ζ −n−1 w(ζ ˆ ) dζ,

n ∈ Z.

(9.2)

ζ ∈T

If w ∈ 2+ (X ) so that wˆ ∈ H+2 (X ), and if n ∈ Z+ , then w(n) =

wˆ (n) (0) , n!

n ∈ Z+ .

(9.3)

A similar formula is valid when w ∈ 2− (X ) and n ∈ Z− , involving derivatives of the function 2 (X ) ⊕ H2 (X ). We w(1/z) ˆ at the origin. Since 2 (X ) = 2− (X ) ⊕ 2+ (X ) also L2 (X ) = H− + 2 2 denote the orthogonal projections of L (X ) onto H± (X ) by πˆ ± . They are explicitly given by (πˆ + w)(z) ˆ =

1 2πi

)

(ζ − z)−1 w(ζ ˆ ) dζ,

wˆ ∈ L2 (W), z ∈ D+ ,

ζ ∈T

1 ˆ =− (πˆ − w)(z) 2πi

)

(9.4) −1

(ζ − z) ζ ∈T

w(ζ ˆ ) dζ,

wˆ ∈ L (W), z ∈ D− . 2


2623

Above we discussed the situation where X is a Hilbert space. A corresponding theory applies 2 (W), k 2 (W), to the case where X is replaced by a Kre˘ın space W. We denote the images of k+ 2 2 2 2 and k− (W) under the Fourier transform by K+ (W) := K (D+ ; W), K (W) := K 2 (T; W), and 2 (W) := K 2 (D ; W), respectively, and define the indefinite inner products in these spaces K− − so that the Fourier transform is a unitary operator in each case. This means that, if we fix some 2 (W), K 2 (W), and K 2 (W) admissible Hilbert space inner product in W, then the functions in K+ − 2 2 2 belong to H+ (W), L (W), and H− (W), respectively, and that these three spaces share the same Kre˘ın space inner product 1 wˆ 1 (·), wˆ 2 (·) K 2 (W ) = 2π

)

wˆ 1 (ζ ), wˆ 2 (ζ ) W |dζ |.

(9.5)

ζ ∈T

Every fundamental decomposition W = −Y [] U of the signal space gives rise to the three fundamental decompositions H+2 (W) = −H+2 (Y) [] H+2 (U), L2 (W) = −L2 (Y) [] L2 (U), H−2 (W) = −H−2 (Y) [] H−2 (U). Under the Fourier transform the three shift operators S+ , S, and S− and their adjoints are mapped into the frequency domain shift operators ˆ := zw(z), ˆ S+ w(z)

2 w(·) ˆ ∈ K+ (W),

1 ∗ 2 w(z) ˆ − w(0) ˆ , w(·) ˆ ∈ K+ w(z) ˆ := (W), S+ z S w(z) ˆ := zw(z), ˆ w(·) ˆ ∈ K 2 (W), 1 ˆ w(·) ˆ ∈ K 2 (W), ˆ := w(z), S −1 w(z) z 2 ˆ := zw(z) ˆ − lim ζ w(ζ ˆ ), w(·) ˆ ∈ K− (W), S− w(z)

(9.6)

ζ →∞

1 ∗ ˆ w(z) ˆ := w(z), S− z

2 w(·) ˆ ∈ K− (W).

Frequency domain behaviors Under the Fourier transform the class of all passive future behaviors W+ on W is mapped 2 (W), the class of + of K+ onto the class of all maximal nonnegative S+ -invariant subspaces W S− all passive past behaviors W− on W is mapped onto the class of all maximal nonnegative 2 (W), and the class of all passive full behaviors W is mapped onto − of K− invariant subspaces W of K 2 (W). The definition the class of all maximal nonnegative S-reducing causal subspaces W of causality in the frequency domain is analogous to the definition of causality in time domain, is causal if it is true for some fundamental i.e., a S-reducing maximal nonnegative subspace W decomposition W = −Y [] U of W that

2624


w(·) ˆ ∈W

and PH 2 (U ) wˆ = 0 −

⇒

πˆ − w(·) ˆ = 0.

(9.7)

+ ), where The frequency domain analogue of the space H(W+ ) is the Hilbert space H(W 2 W+ is a maximal nonnegative S+ -invariant subspace of K+ (W), and the frequency domain ana [⊥] logue of the space H(W[⊥] − ) is the Hilbert space H(W− ), where W− is a maximal nonnegative 2 S− -invariant subspace of K− (W). These spaces are defined in the same way as in Section 4, with 2 (W) replaced by K 2 (W) and with M replaced by W ± . Since the F± are unitary maps of k± ± ± 2 2 k± (W) onto H± (W), and since the frequency domain constructions are identical to the time do ± ) which main constructions, the Fourier transform induces two unitary maps H(W± ) → H(W 0 0 map H (W± ) isometrically onto H (W± ). We shall use the same notation F± for these two unitary maps. Given a passive full behavior W we define the frequency domain version of the past/future −1 maps of W by ΓW * = F+ ΓW F− . Thus, if W is a passive full behavior on W with the corresponding passive future and past behaviors W+ and W− , then ΓW * is the unique linear contrac [⊥] tion H(W ) → H( W ), which is defined by the relation + − + = Γ * πˆ − wˆ + W [⊥] πˆ + wˆ + W − , W

wˆ ∈ W,

[⊥] − [⊥] | wˆ − ∈ W − } of H(W [⊥] ˆ− +W on the dense subspace H0 (W − ) := {w − ) and then extended [⊥] to H(W− ) by continuity. Graph representations of frequency domain behaviors W + , and W − by using the graph representations We next develop graph representations of W, (2.12)–(2.14) of W+ , W, and W− . As is well known and easy to prove, the operators D+ , D, and D− in appearing in (2.12)–(2.14) have the expansions (D+ w+ )(n) =

n

2 w+ ∈ k + (U), n ∈ Z+ ,

(9.8)

D(n − k)w(k),

w ∈ k 2 (U), n ∈ Z,

(9.9)

D(n − k)w(k),

2 w− ∈ k− (U), n ∈ Z− ,

D(n − k)w(k),

k=0

(Dw)(n) =

n k=−∞

(D− w− )(n) =

n

(9.10)

k=−∞

with the same coefficients D(k), k ∈ Z+ , in all the three formulas. If we define Φ(z) by Φ(z) =

∞

D(n)zn ,

(9.11)

n=0

then Φ is a Schur class function in the unit disk D+ , i.e., a B(U, Y)-valued analytic contractive function in D+ . The radial limits


Φ(ζ ) = lim Φ(rζ ), r↑1

ζ ∈ T,

2625

(9.12)

exist in the strong sense a.e. on T. The frequency domain analogues of the three operators D, D+ , = F DF −1 , D + = D| 2 , and D − = πˆ + D| is a Laurent operator 2 . Here D and D− are D H + (U ) H − (U ) − are the appropriate compressions + , and D (multiplication operator) with symbol Φ, and D of D, i.e., w)(ζ (D ˆ ) = Φ(ζ )w(ζ ˆ ),

wˆ ∈ L2 (W), ζ ∈ T,

+ wˆ + )(z) = Φ(z)wˆ + (z), wˆ + ∈ H+2 (W), z ∈ D+ , (D ) Φ(ζ )wˆ − (ζ ) 1 dζ, wˆ − ∈ H−2 (W), z ∈ D− . (D− wˆ − )(z) = − 2πi ζ −z

(9.13)

ζ ∈T

∗ of D is the Laurent operator whose symbol is Φ ∗ (ζ ), ζ ∈ T, and D ∗+ and D ∗+ The adjoint D ∗ ∗ are the appropriate compressions of D . The symbol Φ (ζ ) is the radial boundary value of the function Φ ∗ (1/z), z ∈ D− , which is a Schur class function in D− . In terms of the three operators and D ± the Fourier images W := F W and W ± := F± W± of W and W± have the graph D representations # $ uˆ D 2 wˆ = uˆ ∈ L (U) , uˆ # $ ± uˆ ± D = wˆ ± = uˆ ± ∈ H±2 (U) . uˆ ±

= W ± W

(9.14)

+ ) and H(D ∗− ) The de Branges complementary spaces H(D + ) and H(W [⊥] We next describe how the spaces H(W − ) can be mapped unitarily onto the de ∗ Branges complementary spaces H(D+ ) and H(D− ). + ) and H(D ∗− ) is that both of the operators The most important fact in the construction of H(D ∗− are contractions, and below we describe how one constructs the de Branges com + and D D +→ Y + where U + and Y + are Hilbert spaces. plementary space H(A) for a given contraction A : U This space is defined by the formulas + y ˜ H(A) < ∞ , H(A) = y˜ ∈ Y

(9.15)

+. y ˜ H(A) = sup y˜ − Au ˜ 2Y+ − u ˜ 2U+ u˜ ∈ U

(9.16)

where

+ It was introduced and used in [7,8] with A This is a Hilbert space continuously contained in Y. replaced by D+ as the state space in the canonical de Branges–Rovnyak model of a scattering i/s/o observable backward conservative system with a given Schur class scattering matrix Φ. We shall derive this model from our s/s model in the next section.

2626


Later it was observed that H(A) has another alternative characterization: H(A) = R (1 − AA∗ )1/2 , [−1] y ˜ H(A) = (1 − AA∗ )1/2 y˜ Y+,

y˜ ∈ H(A),

(9.17)

where the upper index [−1] represents a pseudo-inverse, i.e., B [−1] : R(B) → (N (B))⊥ is the inverse of the injective operator B|(N (B))⊥ → R(B). The operator (1 − AA∗ )1/2 is usually called the defect operator of the contraction A∗ . See [1,11] for more details. In [6] it was explained how the space H(Z) described in Section 4 is related to the space H(A), where A is the contraction appearing in the graph representation # Z=

$ Au˜ + u˜ ∈ U u˜

of the maximal nonnegative subspace Z of K with respect to some fundamental decomposition + [] U +. The connection is the following. There exists a unitary map T : H(Z) → H(A) K = −Y with the property that the image of x + Z ∈ H(Z) under T is the unique vector y˜ in this equiva+ is zero. Explicitly this means that lence class whose projection onto U , y˜ y˜ T + Z = y˜ − Au, ˜ ∈ K(Z), u˜ u˜ y˜ T −1 y˜ = + Z, y˜ ∈ H(A). 0

(9.18)

The operator T maps H0 (Z) one-to-one onto the dense subspace R(1 − AA∗ ) of H(A). In the sequel we denote H0 (A) := R(1 − AA∗ ). We now apply the theory described above with the following alternative replacements: +, A = D + , U + = H+2 (U), Y + = H+2 (Y), and T = T+ , (1) Z = W 2 2 [⊥] ∗ + + (2) Z = W − , A = D− , U = H− (Y), Y = H− (U), and T = T− . We leave it to the reader to carry out these substitutions in (9.15)–(9.17). When we do the same substitution in (9.18) we get yˆ+ yˆ+ + = yˆ+ − D + ), + uˆ + , +W ∈ K(W uˆ + uˆ + , [⊥] yˆ− yˆ− [⊥] ∗ − , + W− = uˆ − − D− yˆ− , ∈K W T− uˆ − uˆ − yˆ+ + , yˆ+ ∈ H(W+ ), +W T+−1 yˆ+ = 0 0 −1 [⊥] +W T− uˆ − = ˆ − ∈ H W[⊥] − , u − . uˆ − T+

,

(9.19)


2627

∗− ) to H(D +) The past/future map from H(D [⊥] ∗ By using the unitary maps T− : H(W − ) → H(D− ) and T+ : H(W+ ) → H(D+ ) we can define a version of the past/future map of a passive full behavior which is a contraction from ∗− ) to H(D + ), namely H(D + Γ * T−−1 = T+ F+ ΓW F−−1 T−−1 . Γ(D ∗ ,D + ) := T W −

This map is related to but not identical with the Hankel operator πˆ − : H−2 (U) → H+2 (Y) ˆ +D ΓD := π Before we explaining the exact connection we first prove the following lemma. induced by D. and write wˆ in the form wˆ = Lemma 9.1. Let wˆ ∈ W, (cf. (9.14)). Then

D uˆ uˆ

where uˆ = PL2 (U ) uˆ ∈ L2 (U)

∗− D [⊥] − uˆ − , = 1−D T− πˆ − wˆ + W − + ) = Γ uˆ − , T+ (πˆ + wˆ + W D

(9.20)

where uˆ − = πˆ − u ∈ H−2 (U). Proof. Since πˆ − w =

πˆ − D uˆ πˆ − uˆ

=

D − uˆ − uˆ −

, we get from (9.19),

[⊥] ∗− (D − uˆ − ) = 1 − D ∗− D − uˆ − , = uˆ − + D T− πˆ − wˆ + W − which is the first claim in (9.20). Analogously, πˆ + wˆ =

uˆ + uˆ + ΓD ˆ− πˆ + D D u + = , uˆ + 0 πˆ + uˆ

(9.21)

+ , and hence where uˆ + = πˆ + uˆ ∈ H+2 (Y). The first component in the above sum belongs to W by (9.19), T+ (πˆ + wˆ + W+ ) = ΓD ˆ −. 2 u ∗− ) → H(D + ), which Lemma 9.2. The operator Γ(D ∗ ,D + ) is the unique linear contraction H(D − is defined by the relation ∗− D − [−1] , Γ(D ∗ ,D + ) = ΓD 1−D −

(9.22)

∗− ) = R(1 − D ∗− D ∗− ) and then extended to H(D ∗− ) by − ) of H(D on the dense subspace H0 (D continuity.

2628


Proof. By the Fourier transformed version of Lemma 6.1, [⊥] +, ΓW ˆ − wˆ + W = πˆ + wˆ + W * π −

wˆ ∈ W.

(9.23)

This together with (9.20) gives ∗− D + ) = T+ Γ * πˆ − wˆ + W [⊥] − uˆ − . −−1 1 − D ΓD = T+ ΓW ˆ − = T+ (πˆ + wˆ + W u *T − W Here uˆ − can be an arbitrary function in H−2 (U), and consequently ∗− D − . ΓD ∗ ,D +) 1 − D = Γ(D

(9.24)

−

∗− D − )[−1] to both sides of (9.24) we get the conclusion By applying the pseudo-inverse (1 − D of Lemma 9.2 2 ∗− D − appearing in Lemma 9.2 has a natural interpretation: The operator 1 − D ∗− ) → H−2 (U) is the operator 1 − D ∗− D − : Lemma 9.3. The adjoint of the inclusion map H(D 2 ∗ H− (U) → H(D− ). ∗− ) can be written in the form uˆ − = (1 − D ∗− D − )1/2 uˆ 0 for Proof. By (9.17), every uˆ − ∈ H(D 2 2 some uˆ 0 ∈ H− (U). Therefore, for every uˆ 1 ∈ H− (U) (to get the third equality sign below we polarize the second identity in (9.17)) (uˆ − , uˆ 1 )H 2 (U ) = −

∗− D − 1/2 uˆ 0 , uˆ 1 2 1−D H (U ) −

∗− D − 1/2 uˆ 1 2 = uˆ 0 , 1 − D H (U ) −

∗− D ∗− D − 1/2 uˆ 0 , 1 − D − uˆ 1 = 1−D ∗ ) H (D − ∗ −D − uˆ 1 = uˆ − , 1 − D 2 ∗ . H (D − )

10. Input/output representations of passive behaviors Frequency domain versions of the canonical s/s models W

W

By using the Fourier transform we can map the two canonical models Σobc+ and Σcfc − into * W

* W

the frequency domain, to get two canonical frequency domain models Σobc+ and Σcfc − whose The generating subspace of the frequency domain passive frequency domain full behavior is W. * W+ is given by observable and backward conservative model Σ obc

* W

Vobc+

⎧⎡ ∗ ⎫ ⎤ + +) S+ wˆ + W H(W ⎨ ⎬ +) , + ) wˆ ∈ K(W + ⎦ ∈ H(W = ⎣ wˆ + W ⎩ ⎭ W w(0) ˆ

(10.1)


2629

and the generating subspace of the frequency domain controllable and forward conservative model is * W

Vcfc −

⎧⎡ ⎫ ⎤ [⊥] S −1 wˆ + W [⊥] wˆ = wˆ + wˆ , wˆ ∈ KW ⎨ πˆ − − ˆ + ∈ K(W+ ), ⎬ − + − − ,w ⎦ [⊥] = ⎣ πˆ − wˆ + W . − + = Γ* wˆ − + W [⊥] and wˆ + + W ⎩ ⎭ − W wˆ + (0) (10.2)

The first canonical de Branges–Rovnyak model * W

We continue by developing a description of the i/s/o representation of Σobc+ corresponding to a fundamental decomposition W = −Y [] U of the signal space W. We begin by applying the * *+ W + ) of Σ W unitary similarity transform T+ to Σ + in order to replace the state space H(W by obc

obc

D

+ ) of the new system Σ + with generating subspace the state space H(D obc D Vobc+

T+ := 0 0

0 T+ 0

0 0 1W

*

W . Vobc

+ ) in (10.1) in the form wˆ + = We decompose the parameter w+ ∈ K(W yˆ+ (0) S ∗ yˆ+ ∗w , and ( S+ ˆ + )(z) = ∗+ . Thus, by (9.19), for all z ∈ D+ , uˆ (0)

yˆ+ uˆ +

. Then wˆ + (0) =

S+ uˆ +

+

∗ ∗ ∗ + (z) = + T+ S+ wˆ + + W S+ yˆ+ − D uˆ + (z) S+ =

1 yˆ+ (z) − yˆ+ (0) − Φ(z) uˆ + (z) − uˆ + (0) . z

Denoting + ) = yˆ+ − D + uˆ + , xˆ0 = T+ (wˆ + + W u0 = uˆ + (0), + ) and u0 can be an arbitrary vector in and observing that xˆ0 can be an arbitrary vector in H(D U we get

D

Vobc+

where

⎧⎡ ⎫ ⎤ ⎡ ⎤ Aobc xˆ0 + Bobc u0 H(D+ ) ⎪ ⎪ ⎨ ⎬ +) ⎥ xˆ0 ⎢ ⎥ ⎢ H(D + ) and u0 ∈ U , = ⎣ ⎦ xˆ0 ∈ H(D ⎦∈⎣ Y ⎪ ⎪ ⎩ Cobc xˆ0 + Cobc u0 ⎭ u0 U

(10.3)

2630


1 + ), z ∈ D+ , xˆ0 (z) − xˆ0 (0) , xˆ0 ∈ H(D z 1 Φ(z) − Φ(0) u0 , u0 ∈ U, z ∈ D+ , (Bobc u0 )(z) = z + ), Cobc xˆ0 = xˆ0 (0), xˆ0 ∈ H(D (Aobc xˆ0 )(z) =

(10.4)

Dobc = Φ(0). Here

Aobc

Bobc Cobc Eobc

: H(D+ ) → H(D+ ) is a linear co-isometric operator, and (10.3) is a graph D

Y

U

representation of Vobc+ of the type (1.5). Thus, the i/s/o representation

D+ = Σ i/s/o

,

Bobc + ), U, Y ; H(D Dobc

Aobc Cobc

D

of Σobc+ that we obtain in this way is the canonical de Branges–Rovnyak model of an observable backward conservative scattering system with the scattering matrix Φ mentioned above. This % D system is observable since Σobc+ is observable, i.e., n0 N (Cobc Anobc ) = {0}, and the scattering matrix zCobc (1 − zAobc )−1 Bobc + Dobc of this system is equal to Φ(z). The second canonical de Branges–Rovnyak model * W By applying the unitary similarity transformation T− to the system Σcfc − whose generating D

subspace is given in (10.2) we get another system Σcfc− whose generating subspace is D Vcfc−

T− := 0 0

0 T− 0

0 0 1W

0 T− 0

0 0 1W

* W

Vcfc − .

(10.5)

* W V˚cfc − ,

(10.6)

This subspace contains the dense subspace D V˚cfc−

T− := 0 0

* W W where V˚cfc − is the frequency domain version of the subspaceV˚cfc − defined in (8.2), i.e., * W V˚cfc −

⎫ ⎧⎡ ⎤ ⎡ ⎤ [⊥] [⊥] H(W S −1 wˆ + W ⎬ ⎨ πˆ − − − ) . ⎦ ∈ ⎣ H(W [⊥] [⊥] = ⎣ πˆ − wˆ + W ) ⎦ wˆ ∈ W − − ⎭ ⎩ w(0) ˆ W

We parametrize wˆ in (10.7) by wˆ = ˆ By (9.20), denote uˆ ± = πˆ ± u.

D uˆ uˆ

(10.7)

where uˆ = PL2 (U ) is a free parameter in L2 (U), and

∗− D [⊥] − uˆ − . = 1−D T− πˆ − wˆ + W −


2631

=D − uˆ − + D + uˆ + + Γ uˆ − and using (9.19) we get Recalling that Du D [⊥] (z) T− πˆ − S −1 wˆ + W − ∗− πˆ − − uˆ − + D + uˆ + + Γ uˆ − ) (z) = πˆ − S −1 uˆ (z) − D S −1 (D D =

1 ∗− D − uˆ − (z) − 1 Φ(1/z) Φ(0)uˆ + (0) + (Γ uˆ − )(0) . 1−D D z z

Denoting ∗− D − uˆ − , xˆ0 = 1 − D u0 = uˆ + (0), ∗− ) and u0 can be an and using Lemma 9.2 and the fact that xˆ0 can be an arbitrary vector in H0 (D arbitrary vector in U we get ∗ D V˚cfc−

⎧⎡ ⎫ ⎤ ⎡ H(D ∗− ) ⎤ Acfc xˆ0 + Bcfc u0 ⎪ ⎪ ⎨ ⎬ ∗ −) ⎥ xˆ0 ⎢ ⎥ ⎢ H(D 0 ∗ , x ˆ ∈ D and u = ⎣ ∈ H ∈ U ⎦ ⎣ ⎦ 0 0 − ⎪ ⎪ Y ⎩ Ccfc xˆ0 + Dcfc u0 ⎭ u0 U

(10.8)

where ∗ 1 − , z ∈ D− , xˆ0 (z) − Φ ∗ (1/z)(Γ(D ∗ ,D + ) xˆ 0 )(0) , xˆ 0 ∈ H D − z 1 (Bcfc u0 )(z) = 1U − Φ ∗ (1/z)Φ(0) u0 , u0 ∈ U, z ∈ D− , z ∗ − , Ccfc xˆ0 = (Γ(D xˆ0 ∈ H D ∗ ,D + ) xˆ 0 )(0), (Acfc xˆ0 )(z) =

−

Dcfc = Φ(0).

(10.9)

∗− ) is dense in H(D ∗− ) we find that Since H0 (D D

Vcfc−

Here

⎧⎡ ⎫ ⎤ ⎡ H(D ∗− ) ⎤ Acfc xˆ0 + Bcfc u0 ⎪ ⎪ ⎨ ⎬ ∗ ∗− ) ⎥ xˆ0 ⎢ ⎥ ⎢ H(D − and u0 ∈ U . = ⎣ ⎦∈⎣ ⎦ xˆ0 ∈ H D ⎪ ⎪ Y ⎩ Ccfc xˆ0 + Ccfc u0 ⎭ u0 U

Acfc

tation

∗ ∗ ) Bcfc H(D − → H(D− ) is an isometric operator, Ccfc Ecfc : Y U − D of Vcfc of the type (1.5). Thus, the i/s/o representation

D− = Σ i/s/o D

,

Acfc Ccfc

(10.10)

and (10.10) is a graph represen-

∗ Bcfc ; H D− , U, Y Dcfc

of Σcfc+ that we obtain in this way is the canonical de Branges–Rovnyak model of a controllable forward conservative scattering system with the scattering matrix Φ. This system is

2632


D controllable since Σcfc− is controllable, i.e., n0 R(Ancfc Bcfc ) = X , and the scattering matrix zCcfc (1 − zAcfc )−1 Bcfc + Dcfc of this system is equal to Φ(z). The formulas for the adjoints of the operators Acfc , Bcfc , Ccfc , and Dcfc in (10.9) are simpler than the formulas for these operators themselves, and they are also easier to compute. This can be done without any knowledge of the past/future map Γ(D ∗ ,D + ) . Explicitly, these adjoints are − given by ∗ Acfc xˆ0 (z) = zxˆ0 (z) − lim ζ xˆ0 (ζ ), ζ →∞

∗ xˆ0 Bcfc

= lim ζ xˆ0 (ζ ), ζ →∞

∗ − , xˆ0 ∈ H D

∗ Ccfc y0 (z) = Φ ∗ (1/z) − Φ ∗ (0) y0 , ∗ Dcfc

∗ − , z ∈ D− , xˆ0 ∈ H D (10.11)

y0 ∈ Y, z ∈ D− ,

∗

= Φ (0).

The most straightforward way to compute these adjoints is to repeat the computation leading to ∗ ∗ (10.4) with (7.1) replaced by (8.9), W+ replaced by W[⊥] − , D+ replaced by D− , and S+ replaced by S− . However, they can, of course, also be computed directly from (10.9). We leave the proof of (10.11) to the reader. Input and output maps of i/s/o representations Let Σ = (V ; X , W) be a passive s/s system, and let BΣ : H(W[⊥] − ) → X and CΣ : X → H(W+ ) be the input and output maps of Σ , where W− = WΣ and W+ = WΣ past fut are the past [⊥] [⊥] and future behaviors of Σ. We again map H(W− ) unitarily onto H(W− ) by means of F− and + ) by means of F+ . Under these transformations BΣ and CΣ are H(W+ ) unitarily onto H(W mapped onto the frequency domain input and output maps BΣ = BΣ F−−1 ,

CΣ = F + CΣ .

(10.12)

[⊥] It follows from Lemma 5.10 that BΣ is the unique contraction H(W − ) → X whose restriction [⊥] 0 − ) is given by to H (W [⊥] BΣ wˆ − + W past = x(0),

past , wˆ − (·) ∈ W

(10.13)

where (x(·), F−−1 wˆ − (·)) is the unique stable externally generated past trajectory of Σ whose signal part is F−−1 wˆ − (·). By Lemma 5.2, CΣ is the contraction defined by # CΣ x0 = wˆ + + Wfut

$ w+ := F −1 wˆ + is the signal part of some stable + future trajectory x(·), w+ (·) of Σ with x(0) = x0 .

(10.14)

[⊥] Let W = −Y [] U be a fixed fundamental decomposition of W, and let T− : H(W − )→ + ) → H(D + ) be the two unitary operators in (9.19). Under these trans ∗− ) and T+ : H(W H(D formations BΣ and CΣ are mapped into the two contractions


2633

∗ D − → X , BΣ − = BΣ F−−1 T−−1 : H D

(10.15)

D + ). CΣ + = T+ F + CΣ : X → H(D

A B These two maps can be characterized more explicitly in terms of the coefficient matrix C of D A B the corresponding scattering i/s/o representation Σi/s/o = ( C D ; X , U, Y)) of the s/s system Σ . This coefficient matrix is the contraction appearing in the graph representation ⎧⎡ ⎤ ⎡ ⎤ x X ⎪ ⎨ 1 x1 A ⎢ x0 ⎥ ⎢ X ⎥ V = ⎣ ⎦ ∈ ⎣ ⎦ xˆ0 ∈ X , u0 ∈ U, and = C y y Y ⎪ 0 0 ⎩ u0 U

B D

⎫ ⎪ ⎬

x0 u0 ⎪ ⎭

(10.16)

of the generating subspace V of Σ corresponding to the fundamental decomposition W = −Y [] U . This means that (x(·), w(·)) is a trajectory of Σ on some interval I if and only if y(·) and (1.6) holds. (x(·), u(·), y(·)) is a trajectory of Σi/s/o on I , where w(·) = u(·) D

D

The maps BΣ − and CΣ − are related to but not identical with the standard input and output maps BΣi/s/o and CΣi/s/o of the i/s/o system Σ . These two maps are defined as follows: If (x− (·), u− (·), y− (·)) is a stable externally generated past trajectory of Σi/s/o then BΣi/s/o u− (·) = x− (0), and if (x+ (·), u+ (·), y+ (·)) is a (stable) future trajectory of Σi/s/o with u+ (·) = 0, then CΣi/s/o x+ (0) = y+ (·). By using (1.6) one get the following explicit formulas for these two operators:

BΣi/s/o u− =

A−k Bu− (k),

k∈Z−

CΣi/s/o x0 = CAk x0

u− ∈ 2− (U), (10.17)

k∈Z+

,

x0 ∈ X ;

see, e.g., [12, p. 697]. It follows from (2.8) that BΣi/s/o is a contraction 2− (U) → X , and it follows from (1.8) with m = 0 that CΣi/s/o is a contraction X → 2+ (Y). We denote the frequency domain version of BΣi/s/o and CΣi/s/o by BΣi/s/o := BΣi/s/o F−−1 ,

CΣi/s/o := F+ CΣi/s/o .

(10.18)

D

The operators CΣ + and CΣi/s/o are related in the following way. D + ) → Lemma 10.1. The operator CΣi/s/o is the composition of CΣ − and the inclusion map H(D H+2 (Y).

Proof. This follows from (9.19).

2

2634

D.Z. Arov, O.J. Staffans / Journal of Functional Analysis 257 (2009) 2573–2634 D

∗− ) → X , which is defined Lemma 10.2. The operator BΣ − is the unique linear contraction H(D by the relation D ∗− D − [−1] , BΣ − = BΣi/s/o 1 − D

(10.19)

∗− ) = R(1 − D ∗− D ∗− ) and then extended to H(D ∗− ) by − ) of H(D on the dense subspace H0 (D continuity. Proof. The proof of this is a simplified version of the proof of Lemma 9.1, and it is left to the reader. 2 Acknowledgment The authors thank Prof. James Rovnyak for sharing with us his expertise on the two canonical de Branges–Rovnyak i/s/o scattering models discussed in Sections 7 and 8. References [1] Daniel Alpay, Aad Dijksma, James Rovnyak, Henrik de Snoo, Schur Functions, Operator Colligations, and Reproducing Kernel Hilbert Spaces, Oper. Theory Adv. Appl., vol. 96, Birkhäuser-Verlag, Basel/Boston/Berlin, 1997. [2] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part I: Discrete time systems, in: The State Space Method, Generalizations and Applications, in: Oper. Theory Adv. Appl., vol. 161, BirkhäuserVerlag, Basel/Boston/Berlin, 2005, pp. 115–177. [3] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part II: Passive discrete time systems, Internat. J. Robust Nonlinear Control 17 (2007) 497–548. [4] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part III: Transmission and impedance representations of discrete time systems, in: Operator Theory, Structured Matrices, and Dilations, Tiberiu Constantinescu Memorial Volume, Theta Foundation, Bucharest, Romania, 2007, pp. 101–140, available from American Mathematical Society. [5] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part IV: Affine representations of discrete time systems, Complex Anal. Oper. Theory 1 (2007) 457–521. [6] Damir Z. Arov, Olof J. Staffans, A Kre˘ın space coordinate free version of the de Branges complementary space, J. Funct. Anal. 256 (2009) 3892–3915. [7] Louis de Branges, James Rovnyak, Canonical models in quantum scattering theory, in: Perturbation Theory and Its Applications in Quantum Mechanics, in: Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965, Wiley, New York, 1966, pp. 295–392. [8] Louis de Branges, James Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966. [9] Nikola˘ı K. Nikolski˘ı, Vasily I. Vasyunin, A unified approach to function models, and the transcription problem, in: The Gohberg Anniversary Collection, vol. II, Calgary, AB, 1988, in: Oper. Theory Adv. Appl., vol. 41, Birkhäuser, Basel, 1989, pp. 405–434. [10] Nikola˘ı K. Nikolski˘ı, Vasily I. Vasyunin, Elements of spectral theory in terms of the free function model. I. Basic constructions, in: Holomorphic Spaces, Berkeley, CA, 1995, in: Math. Sci. Res. Inst. Publ., vol. 33, Cambridge Univ. Press, Cambridge, 1998, pp. 211–302. [11] Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, in: University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994, a Wiley–Interscience Publication. [12] Olof J. Staffans, Well-Posed Linear Systems, Cambridge University Press, Cambridge/New York, 2005.


Sharp bounds for the first non-zero Stekloff eigenvalues ✩ Qiaoling Wang a , Changyu Xia a,b,∗,1 a Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília-DF, Brazil b MPI for Mathematics in the Sciences, Inselstr. 22 D-04103 Leipzig, Germany

Received 24 November 2008; accepted 15 June 2009 Available online 24 June 2009 Communicated by L. Gross

Abstract Let (M, ,) be an n( 2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problems u = 0 in M, 2 u = 0 in M,

∂u = pu on ∂M; ∂ν ∂u u = u − q = 0 on ∂M; ∂ν

where is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first nonzero eigenvalues of the above problems will be denoted by p1 and q1 , respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by √ √ a positive constant c, then p1 λ1 ( λ1 + λ1 − (n − 1)c2 )/{(n − 1)c} with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius 1c , here λ1 denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q1 nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius 1c . Finally, we show that q1 A/V and that if the equality holds and if there is a point x0 ∈ ∂M such that the mean curvature of ∂M at x0 is no less than A/{nV }, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively. ✩

Partially supported by CNPq.

* Corresponding author at: Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília-DF, Brazil.

E-mail addresses: [email protected] (Q. Wang), [email protected] (C. Xia). 1 Part of this work was done while Changyu Xia was visiting MPI for Mathematics in the Sciences. The second author

is very grateful to MPI for Mathematics in Leipzig for its hospitality and CAPES in Brazil. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.008

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Q. Wang, C. Xia / Journal of Functional Analysis 257 (2009) 2635–2644

© 2009 Elsevier Inc. All rights reserved. Keywords: Stekloff eigenvalue; Sharp bounds; Non-negative Ricci curvature; Compact manifolds with boundary; Euclidean ball

1. Introduction Let (M, ,) be an n-dimensional compact Riemannian manifold with boundary. The Stekloff problem is to find a solution of the equation u = 0 in M,

∂u = pu ∂ν

on ∂M;

(1.1)

where p is a real number. This problem was first introduced by Stekloff for bounded domains in the plane in [25]. His motivation came from physics. The function u represents the steady state temperature on M such that the flux on the boundary is proportional to the temperature. Problem (1.1) is also important in conductivity and harmonic analysis as it was initially studied by Calderón (cf. [3]). This connection arises because the set of eigenvalues for the Stekloff problem is the same as the set of eigenvalues of the well-known Dirichlet–Neumann map. This map associates to each function u defined on the boundary ∂M, the normal derivative of the harmonic function on M with boundary data u. Let p1 be the first non-zero eigenvalue of the problem (1.1). Many interesting results for p1 have been obtained during the past years, especially when M is a compact domain in an Euclidean space (cf. [1,2,5–8,12–16,18–21]). Here we list a sharp estimate obtained by Payne [20] which says that if M is a convex plane domain, then Kmax p1 Kmin ,

(1.2)

where Kmax and Kmin are the maximum and minimum of the geodesic curvature of ∂M, respectively. The second inequality in (1.2) has been generalized by Escobar to non-negative curvature manifolds of dimension 2 (cf. [5]). The first result in this paper is a sharp upper bound for the first non-zero eigenvalue of the problem (1.1) for compact manifolds with non-negative Ricci curvature. Namely, we have Theorem 1.1. Let (M, ,) be an n-dimensional compact connected Riemannian manifold with non-negative Ricci curvature and boundary ∂M and let ν the outward unit normal vector field of ∂M. Assume that the principal curvatures of ∂M are bounded from below by a positive constant c. Denote by λ1 the first non-zero eigenvalue of the Laplacian acting on functions on ∂M. Then the first non-zero eigenvalue p1 of the following Stekloff eigenvalue problem: u = 0 in M, ∂u − pu = 0 on ∂M ∂ν

(1.3)

√ λ1 p1 λ1 + λ1 − (n − 1)c2 (n − 1)c

(1.4)

satisfies

with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius 1c .


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Remark 1.1. It has been shown by Xia (cf. [27]) that under the same assumptions as in Theorem 1.1, the first non-zero eigenvalue λ1 of the Laplacian acting on functions on ∂M satisfies λ1 (n − 1)c2 with equality holding if and only if M is isometric to an Euclidean n-ball of radius 1c . This result has been used to solve partially a conjecture by Schroeder and Strake (cf. [24,27,28]). In view of this Xia’s estimate, the number on the right-hand side of the inequality (1.4) is a well-defined positive number. We consider now a fourth order Stekloff eigenvalue problem on an n-dimensional compact connected Riemannian manifold (M, ,) given by 2 u = 0 in M, ∂u = 0 on ∂M, u = u − q ∂ν

(1.5)

where q is a real number. Let q1 be the first non-zero eigenvalue of the problem (1.5). As pointed by Kuttler [14], q1 is the sharp constant for a priori estimates for the Laplace equation v = 0 in M,

v=g

on ∂M,

(1.6)

where g ∈ L2 (∂M). Indeed, using Fichera’s principle of duality (cf. [9]), for the solution v of (1.6) one has q1 v2L2 (M) g2L2 (∂M)

(1.7)

and q1 is the largest possible constant for this inequality. The boundary condition in (1.5) has an interesting interpretation in theory of elasticity. Consider the model problem 2 u = 0 in Ω, ∂u = 0 on ∂Ω u = u − (1 − σ )κ ∂ν

(1.8)

where Ω ⊂ R2 is an open bounded domain with smooth boundary, σ ∈ (−1, 1/2) is the Poisson ratio and κ is the geodesic curvature of ∂Ω. Problem (1.8) describes the deformation u of the linear elastic supported plate Ω under the action of the transversal exterior force f = f (x), x ∈ Ω. The Poisson ratio σ of an elastic material is the negative transverse strain divided by the axial strain in the direction of the stretching force. In other words, this parameter measures the transverse expansion (resp. contraction) if σ > 0 (resp. σ < 0) when the material is compressed by an external force (cf. [11,17,25,26]). The restriction on the Poisson ratio is due to thermodynamic considerations of strain energy in the theory of elasticity. As shown in [17], there exist materials for which the Poisson ratio is negative and the limit case σ = −1 corresponds to materials with an infinite flexural rigidity, see [25, p. 456]. This limit value for σ is strictly related to the eigenvalue problem (1.5). In view of the important applications, one is interested in finding both lower and upper bounds for q1 . It has been proven by Payne that if Ω ⊂ R2 is a bounded convex domain with smooth boundary then q1 (Ω) 2ρ0 with equality holding if and only if Ω is a disk, where ρ0 is the minimum geodesic curvature of ∂Ω. This Payne’s theorem has been extended

2638


to higher dimensional Euclidean domains by Ferrero, Gazzola and Weth [8]. We will prove the following sharp estimate without convexity condition on the boundary for compact manifolds with boundary. Theorem 1.2. Let (M, ,) be an n ( 2)-dimensional compact connected Riemannian manifold with boundary ∂M and non-negative Ricci curvature. Denote by ν the outward unit normal vector field of ∂M and assume that the mean curvature of M is bounded below by a positive constant c. Let q1 be the first eigenvalue of the following Stekloff eigenvalue problem: 2 u = 0 in M, ∂u = 0 on ∂M. u = u − q ∂ν Then q1 nc with equality holding if and only if M is isometric to a ball of radius

(1.9) 1 c

in Rn .

Finally, we prove a sharp upper bound for q1 . Theorem 1.3. Let M be an n-dimensional compact connected Riemannian manifold with boundary ∂M and let ν be the outward unit normal vector field of ∂M. Denote by A, V and H the area of ∂M, the volume of M and the mean curvature of ∂M, respectively. Then the first eigenvalue q1 of the following Stekloff eigenvalue problem: 2 u = 0 in M, ∂u = 0 on ∂M u = u − q ∂ν

(1.10)

satisfies q1 VA . Moreover, if in addition that the Ricci curvature of M is non-negative and that A , then q1 = VA implies that M is isometric to an there is a point x0 ∈ ∂M such that H (x0 ) nV n-dimensional Euclidean ball. Remark 1.2. When M is an Euclidean domain, the equality case in Theorem 1.3 has been proved in [8] without the assumption on the mean curvature thanks to a theorem of Serrin [23]. It is therefore natural to know if the condition on the mean curvature in the last part of Theorem 1.3 could be removed. 2. Proofs of the results Before proving our results, let us fix some notations. Let M be n-dimensional compact manifold M with boundary ∂M. We will often write , the Riemannian metric on M as well as that induced on ∂M. Let ∇ and be the connection and the Laplacian on M, respectively. Let ν be the unit outward normal vector of ∂M. The shape operator of ∂M is given by S(X) = ∇X ν and the second fundamental form of ∂M is defined as II(X, Y ) = S(X), Y , here X, Y ∈ T ∂M. The eigenvalues of S are called the principal curvatures of ∂M and the mean curvature H of ∂M is 1 given by H = n−1 tr S, here tr S denotes the trace of S. We can now state the Reilly’s formula (see [22, p. 46]). For a smooth function f defined on an n-dimensional compact manifold M


2639

with boundary ∂M, the following identity holds if h = ∂f ∂ν ∂M , z = f |∂M and Ric denotes the Ricci tensor of M: 2 (f )2 − ∇ 2 f − Ric(∇f, ∇f ) = (n − 1)H h + 2z h + II(∇z, ∇z) . (2.1) M

∂M

Here ∇ 2 f is the Hessian of f ; and ∇ represent the Laplacian and the gradient on ∂M with respect to the induced metric on ∂M, respectively. Proof of Theorem 1.1. Let f be the solution of the following Laplace equation f = 0 in M, f |∂M = z, where z is a first eigenfunction of ∂M corresponding to λ1 , that is z + λ1 z = 0. Set h = then we have from the Rayleigh–Ritz inequality that (cf. [15]) h2 p1 ∂M 2 M |∇f | and

|∇f |2 , 2 ∂M z

p1 M

∂f ∂ν ∂M ;

(2.2)

(2.3)

which gives p12

h2 ∂M 2 . ∂M z

(2.4)

Since the principal curvatures of ∂M are bounded below by c, we have II(∇z, ∇z) c|∇z|2 ,

H c.

(2.5)

It then follows by substituting f into the Reilly’s formula and noticing the non-negativity of the Ricci curvature of M that 2 0 (f )2 − ∇ 2 f − Ric(∇f, ∇f ) M

(n − 1)c ∂M

= (n − 1)c

hz + c

∂M

(n − 1)c ∂M

hz + cλ1

∂M

h2 − 2λ1

1 h2

∂M

z2

∂M

|∇z|2

∂M

h2 − 2λ1

∂M

h2 − 2λ1

2

1 z2

∂M

2

+ cλ1 ∂M

z2 .

(2.6)

2640


Hence, we have

1

2

2

h ∂M

√

1 2 λ1 2 2 λ1 + λ1 − (n − 1)c z . (n − 1)c

(2.7)

∂M

Substituting (2.7) into (2.4), one obtains √ λ1 λ1 + λ1 − (n − 1)c2 . q1 (n − 1)c This finishes the proof of the first part of Theorem 1.1. Assume now that √ λ1 λ1 + λ1 − (n − 1)c2 . q1 = (n − 1)c Then we have

1

h2 ∂M

2

=

√

1 2 λ1 λ1 + λ1 − nc2 z2 (n − 1)c

(2.8)

∂M

and so the inequalities in (2.6) should take equality sign. We infer therefore ∇ 2 f = 0,

(H − c)h = 0

(2.9)

and √ λ1 h= λ1 + λ1 − (n − 1)c2 z. (n − 1)c

(2.10)

Take a local orthonormal fields {ei }n−1 i=1 tangent to ∂M. We infer from (2.9) and (2.10) that 0=

n−1

˜ + (n − 1)H h ∇ 2 f (ei , ei ) = z

i=1

√ λ1 = −λ1 z + (n − 1)c · λ1 + λ1 − (n − 1)c2 z, (n − 1)c

(2.11)

which gives λ1 = (n − 1)c2 . It then follows from Xia’s result as mentioned in Remark 1.1 that M is isometric to an n-dimensional Euclidean ball of radius 1c . It is easy yo see that for the ndimensional Euclidean ball of radius 1c , the equality holds in (1.4). This completes the proof of Theorem 1.1. 2 Proof of Theorem 1.2. Let w be an eigenfunction corresponding to the first eigenvalue q1 of problem (1.8), that is


2 w = 0 in M, ∂w = 0 on ∂M. w = w − q1 ∂ν Set η =

∂w ∂ν ∂M ;

2641

(2.12)

then

(w)2 . 2 ∂M η

q1 = M

(2.13)

Substituting w into Reilly’s formula, we have

2 (w)2 − ∇ 2 w =

M

Ric(∇w, ∇w) +

M

∂M

(n − 1)c

(n − 1)H η2

η2 .

(2.14)

∂M

The Schwarz inequality implies that 2 2 1 ∇ w (w)2 n

(2.15)

with equality holding if and only if ∇ 2 w = w n ,. Combining (2.13)–(2.15), we have q1 nc. This completes the proof of the first part of Theorem 1.2. Assume now that q1 = nc. In this case, the inequalities (2.14) and (2.15) must take equality sign. In particular, we have ∇ 2w =

w ,. n

(2.16)

Take an orthonormal frame {e1 , . . . , en−1 , en } on M such that when restricted to ∂M, en = ν. From 0 = ∇ 2 w(ei , en ), i = 1, . . . , n − 1, and w|∂M = 0, we conclude that η = ρ = const and so w|∂M = q1 η = ncρ is also a constant. Since (2.14) takes equality sign and η is constant, we infer that H ≡ c. Also, we conclude from the fact that w is a harmonic function on M and the maximum principle that w is constant on M. Suppose without loss of generality that w = 1 and so we have 1 ∇ 2 w = ,. n

(2.17)

It then follows by deriving (2.17) covariantly that ∇ 3 f = 0 and from the Ricci identity, R(X, Y )∇w = 0,

(2.18)

for any X, Y tangent vector to M, where R is the curvature tensor of M. From the maximum principle w attains its minimum at some point x0 in the interior of M. From (2.17) it follows that 1 ∂ ∇w = r , n ∂r

(2.19)

2642


where r is the distance function to x0 . Using (2.18), (2.19), Cartan’s theorem (cf. [4]) and w|∂M = 0, we conclude that M is an Euclidean ball whose center is x0 , and f is given by 1 |x − x0 |2 − b2 n

w(x) =

in M, b being the radius of the ball. Since the mean curvature of ∂M is c, the radius of the ball is 1c . The rest of Theorem 1.2 is obvious and this completes the proof of Theorem 1.2. 2 Proof of Theorem 1.3. Let f be the solution of the following Laplace equation f = 1 in M, f |∂M = 0. It follows from the Rayleigh–Ritz characterization of q1 (cf. [15]) that

(f )2 V = , 2 2 g ∂M ∂M g

q1 M where g =

∂f ∂ν ∂M .

(2.20)

Integrating f = 1 on M and using the divergence theorem, it gives V=

g.

∂M

Hence we infer from Schwarz inequality that V A 2

g2.

(2.21)

∂M

Consequently, we have from (2.20) that q1 VA . Assume now that ∂M has non-negative Ricci A for some x0 ∈ ∂M and q1 = VA . In this case, the equality must hold in curvature, H (x0 ) nV (2.21) and so g = VA is a constant. Consider the function φ on M given by 1 f φ = |∇f |2 − . 2 n Using the Bochner formula, f = 1, the Schwarz inequality and the assumption Ric 0, we have 2 1 φ = ∇ 2 f + ∇f, ∇(f ) + Ric(∇f, ∇f ) − n 1 1 (f )2 − = 0. n n

(2.22)

Thus φ is subharmonic. Observe that φ = 12 ( VA )2 on the boundary. We conclude from the strong maximum principle and Hopf Lemma (see [10, pp. 34–35]) that either


1 V 2 φ= 2 A

2643

in M

(2.23)

∀y ∈ ∂M.

(2.24)

or ∂φ (y) > 0, ∂ν From f |∂M = 0, we have 1 = f |∂M = (n − 1)Hg + ∇ 2 f (ν, ν).

(2.25)

Hence it holds on ∂M that ∂φ g = g∇ 2 f (ν, ν) − ∂ν n g = g 1 − (n − 1)Hg − n

V V 1 −H , = (n − 1) A n A

(2.26)

A which shows that (2.24) is not true since H (x0 ) nV . Therefore φ is constant on M. Since its Laplacian then vanishes, we conclude that equality must hold in (2.22). This and the fact f = 1 imply that

1 ∇ 2 f = ,. n

(2.27)

By using the same arguments as in the final part of the proof of Theorem 1.2, we conclude that M is isometric to an Euclidean n-ball. This completes the proof of Theorem 1.3. 2 References [1] G. Alessandrini, R. Magnanini, Symmetry and non-symmetry for the overdetermined Stekloff eigenvalue problem, Z. Angew. Math. Phys. 45 (1994) 44–52. [2] F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, ZAMM Z. Angew. Math. Mech. 81 (2001) 69–71. [3] A.P. Calderón, On a inverse boundary value problem, in: Seminar in Numerical Analysis and Its Applications to Continuum Physics, Soc. Brasileira de Matemática, Rio de Janeiro, 1980, pp. 65–73. [4] M.P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1993. [5] J.F. Escobar, The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150 (1997) 544–556. [6] J.F. Escobar, An isoperimetric inequality and the first Steklov eigenvalue, J. Funct. Anal. 165 (1999) 101–116. [7] J.F. Escobar, A comparison theorem for the first non-zero Steklov eigenvalue, J. Funct. Anal. 178 (2000) 143–155. [8] A. Ferreero, F. Gazzola, T. Weth, On a fourth order Stekloff eigenvalue problem, Analysis 25 (2005) 315–332, (2005). [9] G. Fichera, Su un principio di dualitá per talune formole di maggiorazione relative alle equazioni differenziali, Atti Accad. Naz. Lincei 19 (1955) 411–418. [10] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics Math., Springer-Verlag, Berlin, 2001. [11] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Res. Notes Math., Pitman Advanced Publishing Program, 1985.

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[12] J. Hersch, L.E Payne, M.M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Ration. Mech. Anal. 57 (1974) 99–114. [13] J.R. Kuttler, Remarks on a Stekloff eigenvalue problem, SIAM J. Numer. Anal. 9 (1972) 1–5. [14] J.R. Kuttler, Dirichlet eigenvalues, SIAM J. Numer. Anal. 16 (1979) 332–338. [15] J.R. Kuttler, V.G. Sigillito, Inequalities for membrane and Stekloff eigenvalues, J. Math. Anal. Appl. 23 (1968) 148–160. [16] J.R. Kuttler, V.G. Sigillito, Estimating Eigenvalues with a Posteriori à Priori inequalities, Res. Notes Math., Pitman Advanced Publishing Program, 1985. [17] R.S. Lakes, Foam structures with a negative Poisson’s ratio, Science 235 (1987) 1038–1040. [18] L.E. Payne, New isoperimetric inequalities for eigenvalues and other physical quantities, Comm. Pure Appl. Math. 9 (1956) 531–542. [19] L.E. Payne, Isoperimetric inequalities and their applications, SIAM Rev. 9 (1967) 453–488. [20] L.E. Payne, Some isoperimetric inequalities for harmonic functions, SIAM J. Math. Anal. 1 (1970) 354–359. [21] L.E. Payne, Some overdetermined boundary value problem for harmonic functions, Z. Angew. Math. Phys. 42 (1991) 864–873. [22] R. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977) 459– 472. [23] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 (1971) 304–318. [24] V. Schroeder, M. Strake, Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature, Comment. Math. Helv. 64 (1989) 173–186. [25] M.W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique, Ann. Sci. École Norm. Sup. 19 (1902) 455–490. [26] P. Villaggio, Mathematical Models for Elastic Structures, Cambridge Univ. Press, 1997. [27] C. Xia, Rigidity for compact manifolds with boundary and non-negative Ricci curvature, Proc. Amer. Math. Soc. 125 (1997) 1801–1806. [28] C. Xia, On a conjecture by Schroeder and Strake, Quart. J. Math. 53 (2002) 119–124.


Index theory for boundary value problems via continuous fields of C ∗ -algebras Johannes Aastrup a , Ryszard Nest b , Elmar Schrohe c,∗ a SFB 478 “Geometrische Strukturen”, Hittorfstrasse 27, 48149 Münster, Germany b Department of Mathematics, Copenhagen University, Universitetsparken 5, 2100 Copenhagen, Denmark c Institut für Analysis, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany

Received 2 December 2008; accepted 21 April 2009 Available online 6 June 2009 Communicated by Alain Connes

Abstract We prove an index theorem for boundary value problems in Boutet de Monvel’s calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid T − X generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field Cr∗ (T − X) of C ∗ algebras over [0, 1]. Its fiber in h¯ = 0, Cr∗ (T − X), can be identified with the symbol algebra for Boutet de Monvel’s calculus; for h¯ = 0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K0 (Cr∗ (T − X)) = K0 (C0 (T ∗ X)) → K0 (K) = Z. Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map. © 2009 Elsevier Inc. All rights reserved. Keywords: Index theory; Boundary value problems; Continuous fields of C ∗ -algebras; Groupoids

0. Introduction Let X be a smooth compact manifold with boundary ∂X. An operator in Boutet de Monvel’s calculus on X is a matrix * Corresponding author.

E-mail addresses: [email protected] (J. Aastrup), [email protected] (R. Nest), [email protected] (E. Schrohe). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.019

2646

J. Aastrup et al. / Journal of Functional Analysis 257 (2009) 2645–2692

A=

P+ +G T

K S

C ∞ (X, E2 ) C ∞ (X, E1 ) ⊕ → ⊕ : C ∞ (∂X, F2 ) C ∞ (∂X, F1 )

(0.1)

of operators acting on smooth sections of (hermitian) vector bundles E1 and E2 over X and F1 and F2 over ∂X. of X, and P + is the so-called Here P is a pseudodifferential operator on the double X + + + + ∞ E1 ) denotes extruncated operator given by P = r P e , where e : C (X, E1 ) → L2 (X, + tension by zero and r is the operator of restriction of distributions on X to the open interior X ◦ of X. For a general pseudodifferential operator P , the truncation P + will map C ∞ (X, E1 ) to ∞ C (X ◦ , E2 ), but the result may not be smooth up to the boundary. The operator P is supposed to satisfy the transmission condition to ensure mapping property (0.1). The entry G is a so-called singular Green operator. Roughly speaking it acts like an operatorvalued pseudodifferential operator along the boundary with values in smoothing operators in the normal direction. In the interior, G is regularizing. Singular Green operators come up naturally: If P and Q are pseudodifferential operators on X, then the composition of the associated truncated operators differs from the truncation of the composition by the so-called leftover term L(P , Q) = (P Q)+ − P + Q+ ,

(0.2)

which is a singular Green operator. The operators T and K are trace and potential (or Poisson) operators, respectively, and S is a pseudodifferential operator on ∂X. We skip details here, since for the purpose of index theory it is sufficient to consider the case where there are no bundles over the boundary and the operator A is of the form A = P+ + G with both P and G classical. Moreover we can confine ourselves to the case where A is of order and class zero. In analogy to the classical Lopatinskij–Shapiro condition, the Fredholm property is governed by the invertibility of two symbols. The first, the interior symbol of A, simply is the principal symbol of P . The second, the boundary symbol, is an operator-valued symbol on S ∗ ∂X which we will explain, below. The above calculus was introduced by L. Boutet de Monvel in 1971 [2]. He showed that the operators of the form (0.1) indeed form an algebra under composition, assuming the vector bundles match. Moreover, this calculus contains the parametrices to elliptic elements and even their inverses whenever they exist. He also proved an index theorem: To every elliptic operator A one can associate a class [A] in Kc (T ∗ X ◦ ), the compactly supported K-theory of the cotangent bundle over the interior X ◦ of X. The index of A then is given by the topological index of [A]. It is far from obvious how to assign [A] to A. Boutet de Monvel gave a brilliant construction, combining pseudodifferential analysis and classical topological K-theory in an ingenious way. Still, the proof is hard for readers not familiar with the details of the calculus, and efforts have been made to make it accessible to a wider audience.


2647

Following work by Melo, Nest and Schrohe [8], it has been shown in [9] by Melo, Schick and Schrohe that the mapping A → [A] can be obtained more easily with the help of C ∗ -algebra K-theory, which had not yet been developed in 1971. This proof relies on only a rudimentary knowledge of the calculus and the ideal structure of the algebra of operators of order and class zero, but it lacks the geometric intuition of Boutet de Monvel’s initial idea. In this paper we will make the link to geometry. We will show how the index of a Fredholm operator of order and class 0 can be determined from its two symbols with the help of deformation theory and a continuous field of C ∗ -algebras over [0, 1]. This field, Cr∗ (T − X), is the reduced C ∗ -algebra of the tangent semi-groupoid T − X associated to X; it was introduced in [1]. The construction of T − X is similar to Connes’ construction of the tangent groupoid for a closed manifold, cf. [3, Section II.5]. In the case at hand, the ‘halftangent space’ T − X is glued to X × X × ]0, 1]. Here, fixing a connection, T ± X consists of all tangent vectors (x, v) such that expx (±tv) ∈ X for small t 0, and X × X × ]0, 1] is endowed with the pair groupoid structure; the gluing is performed with the help of the exponential map: (x, v, h) ¯ ¯ → (x, expx (−h¯ v), h). We showed in [1] that – just as in the boundaryless case – the fibers of Cr∗ (T − X) are isomorphic to the algebra K of compact operators for h¯ > 0 so that their K-theory is given by Z. For h¯ = 0, the situation is different. The ‘symbol algebra’ Cr∗ (T − X)(0) = Cr∗ (T − X) is generated by two representations of Cc∞ (T − X). The first is the representation on L2 (T X ◦ ) via convolution in the fibers. The second takes into account the boundary: We associate to f ∈ Cc∞ (T − X) the operator π0∂ (f ) on L2 (T + X|∂X ) given by half-convolution: π0∂ (f )ξ(x, v) =

f (x, v − w)ξ(x, w) dw.

Tx+ X

The K-theory of Cr∗ (T − X) turned out to be given by K0 (C0 (T ∗ X)) = Kc (T ∗ X ◦ ). In order to make use of this, we first compose the operator with an order reducing operator of positive order m > dim X. This gives us an operator A of order m and class 0 in Boutet de Monvel’s calculus (in fact, we might have started with A of this order and class). We denote by p m its principal pseudodifferential symbol, a homogeneous function on the nonzero vectors in the cotangent bundle, and by cm its homogeneous principal boundary symbol, a homogeneous operator-valued function defined outside the zero section in the cotangent bundle over the boundary. We then set out to compute the index of A. To this end we first smooth out both symbols near the zero section in the corresponding cotangent bundle, obtaining a smooth function p on T ∗ X and a smooth boundary symbol operator c on T ∗ ∂X. Following an idea of Elliott, Natsume and Nest [4] we then consider a semiclassical deformation Ah¯ , 0 < h¯ 1 of A with A = A1 and study the associated graph projection G(Ah¯ ) =

(1 + A∗h¯ Ah¯ )−1 Ah¯ (1 + A∗h¯ Ah¯ )−1

(1 + A∗h¯ Ah¯ )−1 A∗h¯ Ah¯ (1 + A∗h¯ Ah¯ )−1 A∗h¯

.

(0.3)

The positivity of the order of A is crucial here; it ensures that all entries of this matrix are compact operators, except for the one in the lower right corner which differs from a compact operator by the identity. The graph projection of Ah¯ therefore is a projection in K∼ , the unitization of the compact operators. It is closely related to the index of A. In fact, denoting by e the

2648


projection 00 10 (we will use the same notation for this projection in various algebras), we will show in Theorem 4.15 that G(A) − [e] = [πker A ] − [πcoker A ]

(0.4)

is the difference of the classes associated to the projections onto the kernel and the cokernel, respectively, thus the index of A. Using formula (0.3) we can also define the graph projections for p and c. The crucial step is to show that

h¯ →

0 < h¯ 1,

G(Ah¯ ), G(p) ⊕ G(c),

h¯ = 0

defines a continuous section of the unitization of Cr∗ (T − X), cf. Proposition 6.24, Theorem 6.25. A technical problem arises from the fact that Boutet de Monvel’s calculus does not in general contain the adjoints of operators of positive order. The analysis of the graph projection therefore takes us out of the calculus. We overcome this difficulty by working with operator-valued symbol classes. Boutet de Monvel’s calculus fits well into this concept, cf. Schrohe and Schulze [14], Schrohe [13]; moreover, it also allows to treat the adjoints. The continuity of the section gives us a natural map associating to an elliptic operator A a class in K0 (Cr∗ (T − X)) = K0 (C0 (T ∗ X ◦ )) = Kc (T ∗ X ◦ ) by evaluating the section in h¯ = 0, more precisely by taking G(p) ⊕ G(c) − [e ⊕ e]. In addition, evaluation in h¯ = 0 and in h¯ = 1 defines a map in K-theory inda : K0 C0 (T ∗ X ◦ ) → K0 (K); it associates to [G(p) ⊕ G(c)] − [e ⊕ e] the class [G(A)] − [e], thus the index of A. In this way we obtain the analytic index map. In a second step we then construct the topological index in order to obtain an index formula in cohomological terms. A cohomological index formula had been established before by Fedosov [5, Chapter 2, Theorem 2.4]: He showed that ind A = T ∗X

ch [p] Td(X) +

ch [c] Td(X),

T ∗ ∂X

with the Chern character of the K-theory class of p and a variant ch of the Chern character of the K-class of c. As usual, Td denotes the Todd class. In this article, we conclude the discussion in the spirit of noncommutative geometry. We extend the fundamental class : Hc∗ (T ∗ X ◦ ) = H P ∗ Cc∞ (T ∗ X ◦ ) → C T ∗ X◦


2649

to a fundamental class ∞ − F : H P ∗ Ctc (T X) → C. Here ∞ Ctc (T − X) = Cc∞ (T X) ⊕ Cc∞ (T ∂X × R+ × R+ )

is the ‘smooth’ algebra associated to the symbol algebra Cr∗ (T − X) which was introduced in [1, Definition 2.16]. The construction leaves us a certain degree of freedom. In fact, we obtain extensions Fω for every choice ω of a closed form on T ∗ X, which is the pull-back of a closed form on X. Using the equality of the analytical and the topological index in the boundaryless case, established by Connes, we then obtain the index formula ind A = FTd(X) ch G(p) ⊕ G(c) − [e ⊕ e] with the Chern–Connes character ch. 1. Operators in Boutet de Monvel’s calculus 1.1. Manifolds with boundary. In the sequel let X be a compact manifold with boundary and X and the associated spaces its double. We use the standard Sobolev space H s (X) , H s (X) = u|X◦ : u ∈ H s (X)

supp u ⊆ X . H0s (X) = u ∈ H s (X):

The space C ∞ (X) is dense in H s (X) for all s, while Cc∞ (X ◦ ) is dense in H0s (X) for all s. The L2 -inner product allows us to identify H s (X) with the dual of H0−s (X). In general, the spaces H s (X) and H0s (X) are quite different. For −1/2 < s < 1/2, however, they can be identified. In particular, we have for s = 0 the natural identification of L2 (X) with which vanish on X \ X. the subset of all functions in L2 (X) 1.2. Operators, symbols, ellipticity. Detailed descriptions of Boutet de Monvel’s calculus were given by Grubb [6] and Rempel and Schulze [10]. In order to overcome technical difficulties we will also rely on the representation of these operators by operator-valued symbols as presented in [13]. To keep the exposition short, we will focus on the elements in the upper left corner, i.e. the operators A : C ∞ (X, E1 ) → C ∞ (X, E2 ) of the form A = P + + G, where P is a pseudodifferential operator satisfying the transmission condition and G a singular Green operator (sGo). We assume all operators to be classical. The operator A is said to have order μ and class d ∈ N0 , if P is of order μ and G is of order μ and class d. We speak of smoothing or regularizing operators, if the order is −∞.

2650


A sGo G of order m and class d can be written G=

d

Gj ∂νj ,

j =0

where Gj , j = 0, . . . , d, are sGo’s of order μ − j and class 0 and ∂ν is a differential operator which coincides with the normal derivative in a neighborhood of the boundary and vanishes farther away from the boundary. This representation differs from the standard one in that it avoids the trace operators. The equivalence becomes clear from 2.4, below. Let ϕ ∈ Cc∞ (X ◦ ), and denote by Mϕ multiplication by ϕ. Then GMϕ is regularizing of class 0 and Mϕ G is regularizing of class d. μ We associate two symbols to A. The first is the pseudodifferential principal symbol, σψ (A). ∗ It is defined as the principal symbol of P , restricted to T X \ 0: μ

σψ (A) = σ μ (P )|T ∗ X\0 . This makes sense as G is smoothing in X ◦ and therefore cannot contribute to the symbol. μ The second is the principal boundary symbol, σ∂ (A), defined on T ∗ ∂X \ 0. In local coordinates near ∂X, it is given by μ 1 ) → S(R+ , E 2 ). σ∂ (A)(x , ξ ) = p μ (x , 0, ξ , Dn )+ + g μ (x , ξ , Dn ) : S(R+ , E

Here p μ and g μ are the homogeneous principal symbols of P and G, respectively. For fixed x , ξ , the operator p μ (x , 0, ξ , Dn ) is the Fourier multiplier with symbol p μ (x , 0, ξ , ξn ), while j we have denoted the g μ (x , ξ , Dn ) is an integral operator with smooth integral kernel. By E fiber in (x , ξ ) of the pullback of Ej to T ∗ X. The principal boundary symbol is homogeneous on T ∗ ∂X \ 0 in a sense we shall explain later, see (2.7), so that it can be viewed as a function on S ∗ ∂X. An operator A of order μ and class d max{μ, 0} is said to be elliptic, if (i) σψ (A)(x, ξ ) : π ∗ E1 → π ∗ E2 is invertible for all (x, ξ ) ∈ T ∗ X \ 0, and μ 1 ) → L2 (R+ , E 2 ) is invertible for all (x , ξ ) ∈ T ∗ ∂X \ 0. (ii) σ∂ (A)(x , ξ ) : H μ (R+ , E μ

Here π : T ∗ X → X is the base point projection. Apart from these symbols we have, of course, in any coordinate neighborhood, the full symbols of P and G in the corresponding classes. 1.3. Theorem. Let A be an operator of order μ and class d in Boutet de Monvel’s calculus. Then A induces a bounded linear map A : H s (X, E1 ) → H s−μ (X, E2 ) for each s > d − 1/2. In general we cannot extend A to H s (X, E1 ) for s d − 1/2. The reason is that neither extension by zero makes sense on these spaces nor do integral operators with smooth (up to the boundary) integral kernels act continuously on them.


2651

1.4. Theorem. Let A1 : C ∞ (X, E1 ) → C ∞ (X, E2 ) and A2 : C ∞ (X, E2 ) → C ∞ (X, E3 ) be operators of orders μ1 and μ2 and classes d1 and d2 , respectively, in Boutet de Monvel’s calculus on X. The composition A2 A1 is an operator of order μ1 + μ2 and class max{d1 , μ1 + d2 }. Its principal symbols are given by μ +μ2

σψ 1

μ1 +μ2

σ∂

μ

μ

μ

μ

(A2 A1 ) = σψ 2 (A2 )σψ 1 (A1 ); (A2 A1 ) = σ∂ 2 (A2 )σ∂ 1 (A1 ).

1.5. Theorem. Let A be an operator of order μ and class d max{μ, 0} in Boutet de Monvel’s calculus and s > d − 1/2. Then A : H s (X, E1 ) → H s−μ (X, E2 ) is Fredholm if and only if A is elliptic. In this case we find an operator B of order −μ and class d max{−μ, 0} such that AB = I + R1

and BA = I + R2

with regularizing operators R1 and R2 of class d and d, respectively. For the symbols we have −μ

σψ (B) = σψ (A)−1 μ

−μ

and σ∂ (B) = σ∂ (A)−1 . μ

1.6. Adjoints. Boutet de Monvel’s calculus is not closed under taking adjoints. For the sake of completeness let us introduce a few basic concepts. To an operator A of order μ and class d max{0, μ} we can associate a minimal adjoint A∗min defined on Cc∞ (X ◦ ) (we omit the bundles from the notation) and taking values in C ∞ (X) by the relation

Au, v = u, A∗min v ,

u ∈ C ∞ (X), v ∈ Cc∞ (X ◦ ).

For s > d − 1/2, the adjoint A∗ of the bounded operator A : H s (X) → H s−μ (X) is then given by extending A∗min by continuity to an operator A∗ : H0 (X) → H0−s (X). If the class is zero, we can explicitly determine the adjoint: We write A = P + + G with a and a singular Green operator G of order μ and class 0. Next pseudodifferential operator P on X we recall from [6, Lemma 1.3.1] that a pseudodifferential operator P of order μ ∈ Z satisfying the transmission condition can be written μ−s

P = S + Q,

(1.1)

where S is a differential operator of order μ and Q is a pseudodifferential operator of order μ to L2 (X) and which maps e+ C ∞ (X), the extensions (by zero) of smooth functions on X to X, ∗ satisfies – with the formal adjoint Qf – + Q u, v = u, Qf∗,+ v ,

u, v ∈ Cc∞ (X).

(1.2)

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Note that for μ 0, equality (1.2) will hold for P . Still it useful to know that we can choose S in such a way that the local symbol q of Q is of order (at most) −1 with respect to ξn ; we say that q is of normal order −1. For u, v ∈ C ∞ (X) it is known [6, Section 1.6] that + S u, v L2 (X) = u, Sf∗,+ v L2 (X) + Aρ + u, ρ + v L2 (∂X) with the formal adjoint Sf∗ of S, the Green matrix A, and the vectors ρ + u and ρ + v of boundary values for u and v. When v lies in Cc∞ (X ◦ ), the boundary terms on the right-hand side vanish, so that

S + u, v

L2 (X)

= u, Sf∗,+ v L2 (X) ,

u ∈ C ∞ (X), v ∈ Cc∞ (X ◦ ).

On Cc∞ (X ◦ ), the operation e+ of extending by zero is trivial; moreover Sf∗ is a differential operator, so that Sf∗ v ∈ Cc∞ (X ◦ ). There are no singular terms arising at the boundary, and Sf∗,+ v = Sf∗ v as a functional on C ∞ (X). For Q the corresponding identity (1.2) is valid by construction. Again, Qf∗,+ v = Q∗f v as a functional on C ∞ (X). Hence

P + u, v

L2 (X)

= u, Pf∗ v L2 (X) ,

u ∈ C ∞ (X), v ∈ Cc∞ (X ◦ ).

(1.3)

As the singular Green part G is assumed to be of class zero, it also has a formal adjoint G∗f , cf. [6, (1.2.47)], and

Gu, vL2 (X) = u, G∗f v L2 (X) .

(1.4)

Hence, as a functional on C ∞ (X), A∗min v = Pf∗ v + G∗f v,

v ∈ Cc∞ (X ◦ ).

If A = P + + G is of order μ 0 and class 0, then A : L2 (X) → L2 (X) is continuous, and is dense in L2 (X). We conclude from (1.3) and (1.4) that the L2 -adjoint of A is

Cc∞ (X ◦ )

A∗ = Pf∗,+ + G∗f with the formal adjoints Pf∗ and G∗f of P and G, extended to L2 . Thus A∗ again is an operator in Boutet de Monvel’s calculus. Its principal symbols are given by σψ 1 (A∗ ) = σψ 1 (A)∗ ; μ

μ

σ∂ 1 (A∗ ) = σ∂ 1 (A)∗ . μ

μ

1.7. Corollary. Suppose A : H μ (X, E1 ) → L2 (X, E2 ) is invertible of order μ 0 and class d μ. Then A−1 is an operator of order −μ and class 0, cf. [12, Theorem 4.5]. It has an adjoint −μ (A−1 )∗ in Boutet de Monvel’s calculus, which extends to a bounded operator from H0 (X, E2 ) −μ to L2 (X, E1 ), considering e+ as a trivial operation on H0 . On the other hand, the minimal


2653

−μ

adjoint A∗min of A extends to an invertible operator A∗ : L2 (X, E2 ) → H0 (X, E1 ). It is easily checked that (A−1 )∗ = (A∗ )−1 . 1.8. Order reducing operators. There exists a family Λm − , m ∈ Z, of classical scalar pseudodifferential operators on X satisfying the transmission condition with the following properties: + s s−m (X) is an isomorphism for all s > −1/2. (i) (Λm − ) : H (X) → H s (ii) In fact, the result of the application of r + Λm − to u in H (X) only depends on the restriction ◦ s s−m (X) for of u to X . The map in (i) extends to an isomorphism r + Λm − es : H (X) → H s s an arbitrary choice of an extension operator es : H (X) → H (X). μ + m+μ + + ) , m, μ ∈ Z. (iii) (Λm − ) (Λ− ) = (Λ− (iv) The (extension of the) formal pseudodifferential adjoint defines an isomorphism Λm + = + )∗ : H s (X) → H s−m (X). ) ((Λm − 0 0 m (v) The operators Λm − , Λ+ can be extended to operators with the same properties but acting in a vector bundle E.

2. Operator-valued symbols It will be helpful to consider the operators in Boutet de Monvel’s calculus as operator-valued pseudodifferential operators. We recall the basic concepts from [13]. We first fix a function Rn ξ → [ξ ] ∈ R0 which is positive for ξ = 0 and coincides with |ξ | for |ξ | 1. 2.1. Group actions. A strongly continuous group action on a Banach space E is a family κ = {κλ : λ ∈ R+ } of isomorphisms in L(E) such that κλ κμ = κλμ and the mapping λ → κλ e is continuous for every e ∈ E. Note that there is an M > 0 such that M κλ max λ, λ−1 .

(2.1)

For the usual Sobolev spaces on R and R+ we shall use the group action defined on functions u by (κλ u)(x) = λ1/2 u(λx).

(2.2)

It will be useful to consider also weighted Sobolev spaces: For s = (s1 , s2 ) ∈ R2 we define H s (R) = H (s1 ,s2 ) (R) = [x]−s2 u: u ∈ H s1 (R) with the usual unweighted space on the right-hand side. Similarly we define H s (R+ ). We then have S(R+ ) = projlims1 ,s2 →∞ H (s1 ,s2 ) (R+ ) and S (R+ ) = indlims1 ,s2 →∞ H (−s1 ,−s2 ) (R+ ). On E = Cl , l ∈ N, we use the trivial group action κλ ≡ id. Sums of spaces of the above kind will be endowed with the sum of the group actions. 2.2. Operator-valued symbols and amplitudes. Let E, F be Banach spaces with strongly continuous group actions κ and κ, ˜ respectively. Let a ∈ C ∞ (Rq × Rq × Rq , L(E, F )) and μ ∈ R. μ q We shall write a ∈ S (R × Rq × Rq ; E, F ) and call a an amplitude of order μ provided that, for all multi-indices α, β, γ , there is a constant C = C(α, β, γ ) with

2654


κ˜

α β γ ˜ η)κ[η] L(E,F ) [η]−1 Dη Dy Dy˜ a(y, y,

C[η]μ−|α| .

If a is independent of y or y˜ we shall write a ∈ S μ (Rq × Rq ; E, F ). For E = F = C we recover the usual pseudodifferential symbol classes. The concept extends to the cases where E is an inductive or F a projective limit. In order to avoid lengthy formulas we shall abbreviate this by saying that a is a symbol of order μ with values in L(E, F ). 2.3. Example: Potential, trace and singular Green boundary symbol operators. The boundary symbol operators associated to potential, trace or singular Green symbols in the usual presentation of Boutet de Monvel’s calculus have simple descriptions in the framework of operatorvalued symbols. (a) The elements in S μ (Rn−1 × Rn−1 ; C, S(R+ )) are precisely the boundary symbol operators associated with potential symbols of order1 μ on Rn+ . (b) The elements of S μ (Rn−1 × Rn−1 ; S (R+ ), C) are precisely the boundary symbol operators associated with trace symbols of order μ and class 0. Those of the form d

tj (x , ξ )Dn j

j =0

with tj in S μ−j (Rn−1 × Rn−1 ; S (R+ ), C) and the derivative Dn on R are the boundary symbol operators associated with trace symbols of order μ and class d on Rn+ . (c) The elements of S μ (Rn−1 × Rn−1 ; S (R+ ), S(R+ )) are precisely the boundary symbol operators associated with singular Green symbols of order μ and class 0 on Rn+ . Those of the form d

gj (x , ξ )Dn j

j =0

with gj in S μ−j (Rn−1 × Rn−1 ; S (R+ ), S(R+ )) are the singular Green boundary symbol operators of order μ and class d. We therefore speak of these operator-valued symbols as potential, trace and singular Green boundary symbol operators or, for short, symbols, of the corresponding orders and classes. j

2.4. Example: Trace operators. Let γj be defined on S(R+ ) by γj u = limt→0+ Dn u(t). It extends to an element of L(H σ (R+ ), C) for σ = (σ1 , σ2 ), σ1 > j + 1/2, by the trace theorem for Sobolev spaces. Viewed as an operator-valued symbol independent of the variables y and η, γj then is a symbol of order j + 1/2 with values in L(H σ (R+ ), C): Recalling that the group action on the Sobolev space is given by (2.2) while on C it is given by the identity, we only have to check that 1 In fact, the notion of order differs slightly in [6,13,14]; this will, however, not play a role in the sequel.


2655

γj κ[η] L(H σ (R+ ),C) = O [η]j +1/2 . This is immediate, since j j ∂t [η]1/2 u [η]t = [η]j +1/2 ∂t u [η]t . Let us next show that γj is a trace symbol of order j + 1/2 and class j + 1 in the sense of 2.3. It clearly suffices to do this for j = 0. Choose ϕ ∈ S(R+ ) with ϕ(0) = 1. The identity ∞ ∞ u(0) = − [ξ ]ϕ [ξ ]s u(s) ds − ϕ [ξ ]s ∂s u(s) ds, 0

u ∈ S(R+ ),

(2.3)

0

shows that γ0 = t0 + it1 Dn , where t0 and t1 are the operator-valued symbols of order 1/2 and −1/2, respectively, with values in L(S (R+ ), C), given by ∞ t0 u = − [ξ ]ϕ [ξ ]s u(s) ds, 0

∞ t1 u = −

ϕ [ξ ]s u(s) ds.

0

Hence γ0 is of class 1. 2.5. Definition. For a ∈ S μ (Rq × Rq × Rq ; E, F ), the pseudodifferential operator op a : S Rq , E → S Rq , F is defined by (op a)u(y) =

˜ ei(y−y)η a(y, y, ˜ η)u(y) ˜ d y˜ d−η;

y ∈ Rq .

Here d−η = (2π)−q dη. If a is independent of y, ˜ this reduces to (op a)u(y) =

eiyη a(y, η)u(η) ˆ d−η;

(2.4)

in this case we call a a left symbol for op a. If a is independent of y, then (op a)u(y) = and a is called a right symbol.

˜ ei(y−y)η a(y, ˜ η)u(y) ˜ d y˜ d−η,

(2.5)

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2.6. Example: Action in the normal direction. Let p ∈ S μ (Rn × Rn ). For fixed (x , ξ ), the function p(x , ·, ξ , ·) is an element of S μ (R × R). For σ ∈ R2 , p(x , ·, ξ , ·) induces a bounded linear operator p(x , xn , ξ , Dn ) = (opxn p)(x , ξ ) : H σ (R) → H (σ1 −μ,σ2 ) (R); by

(opxn p)(x , ξ )u(xn ) =

eixn ξn p(x , xn , ξ , ξn )u(ξ ˆ n ) d−ξn ,

see [11, Theorem 1.7] for the boundedness on weighted spaces. We then have κ[ξ ]−1 (opxn p)κ[ξ ] = opxn p x , xn /[ξ ], ξ , [ξ ]ξn .

(2.6)

In fact, for u ∈ S(R), κ[ξ ]−1 (opxn p)(κ

[ξ ]

u)(xn ) =

eixn ξn /[ξ ] [ξ ]−1 p x , xn /[ξ ], ξ , ξn uˆ ξn /[ξ ] d−ξn ;

and the substitution ηn = ξn /[ξ ] yields the assertion. The theorem, below, shows that opxn p is an operator-valued symbol in the sense of 2.2: 2.7. Proposition. In the above situation we have opxn p ∈ S μ Rn−1 × Rn−1 ; H σ (R), H (σ1 −μ,σ2 ) (R) . Proof. Given multi-indices α, β, we have to estimate β sup [ξ ]|α| κ[ξ ]−1 opxn Dξα Dx p κ[ξ ] L(H (σ1 ,σ2 ) (R),H (σ1 −μ,σ2 ) (R))

x ,ξ

β = sup [ξ ]|α| opxn Dξα Dx p x , xn /[ξ ], ξ , ξn [ξ ] L(H (σ1 ,σ2 ) (R),H (σ1 −μ,σ2 ) (R)) . x ,ξ β

Since Dξα Dx p is of order μ − |α| we may assume that α = β = 0. Now opxn p x , xn /[ξ ], ξ , ξn [ξ ] : H (σ1 ,σ2 ) (R) → H (σ1 −μ,σ2 ) (R) is continuous, and a bound for its norm is given by the suprema sup Dξαn Dxβn p x , xn /[ξ ], ξ , ξn [ξ ] [ξn ]−μ : xn , ξn ∈ R for a finite number of derivatives. Since each of them is O([ξ ]μ ) the proof is complete.

2


2657

2.8. Example: Multiplication operators. (a) Multiplication Mxn by xn is an element of S −1 (Rn−1 × Rn−1 , H s (R+ ), H s+(0,1) (R+ )), s ∈ R2 . We may replace the pair of Sobolev spaces by (H0s (R+ ), H0s+(0,1) (R+ )). (b) Let ϕ ∈ Cb∞ (R+ ) vanish to all orders at 0. Then multiplication Mϕ by ϕ(xn ) is an element of S −∞ (Rn−1 × Rn−1 , H s (R+ ), H s (R+ )). Proof. (a) follows from the fact that Mxn has the symbol a(x , ξ ) = 1 ⊗ Mxn and that κ[ξ−1 ] a(x , ξ )κ[ξ ] u = [ξ ]−1 Mxn u, (b) follows from (a).

u ∈ Cc∞ (R).

2

2.9. Asymptotic summation and classical symbols. A sequence (aj ) of operator-valued symbols of orders μ − j with values in L(E, F ) can be summed asymptotically to a symbol a of order μ, and a is unique modulo symbols of order −∞. A symbol a of order μ is said to be classical, if it has an asymptotic expansion a ∼ ∞ j =0 aj with aj of order μ − j satisfying the homogeneity relation ˜ λη) = λμ−j κ˜ λ aj (y, y, ˜ η)κλ−1 aj (y, y,

(2.7) μ

for all λ 1, |η| R with a suitable constant R. We write a ∈ Scl (Rq × Rq × Rq ; E, F ). For E = Ck , F = Cl we recover the standard notion. The key to many results on compositions is the lemma, below, which is adapted from Kumanogo [7, Chapter 2, Lemma 2.4]. 2.10. Lemma. Let a ∈ S μ (Rq × Rq × Rq ; E, F ). For |θ | 1 define aθ by the oscillatory integral aθ (y, η) =

e−izζ a(y, y + z, η + θ ζ ) dz d−ζ.

(2.8)

Then the family {aθ : |θ | 1} is uniformly bounded in S μ ; its seminorms can be estimated by those for a. 2.11. Theorem. (a) Let a ∈ S μ (Rq × Rq × Rq ; E, F ). Then there is a (unique) left symbol aL = aL (y, η) for ˜ η) acting as in (2.5). They op a acting as in (2.4) and a (unique) right symbol aR = aR (y, are given by the oscillatory integrals aL (y, η) =

e−iyη a(y, y + z, η + ζ ) dz d−ζ

(2.9)

˜ ei yη a(y˜ + z, y, ˜ η + ζ ) dz d−ζ.

(2.10)

and aR (y, ˜ η) =

2658


Moreover, we have

aL (y, η) =

1

1

(1 − θ )N −1 ˜ η) +N ∂ηα Dyα˜ a(y, y, α! γ! y=y ˜

|α| 0, g can be approximated by elements in S −∞ Rq × Rq , S (R+ ), S(R+ ) , which vanish for large |y|, in the topology of S m+ε = S m+ε (Rq × Rq , L2 (R+ ), L2 (R+ )). For the proof we need the following well-known result. 2.19. Lemma. Let K ∈ K(L2 (X)) and ε > 0. Then there exist ϕ1 , . . . , ϕN , ψ1 , . . . , ψN ∈ Cc∞ (X ◦ ) such that N

ϕj ⊗ ψj K − j =1

< ε.

L2 (X)

Proof of Proposition 2.18. By composing with the operator [η]−m−ε ⊗ id, we may assume that m = −ε. Choose ϕ ∈ Cc∞ ([0, ∞)) with ϕ(t) ≡ 1 for t 1, and let gN (y, η) = g(y, η)ϕ |η|/N ,

N ∈ N.

Then gN is a regularizing symbol with values in L(H0(−s1 ,−t1 ) (R+ ), H (s2 ,t2 ) (R+ )). Moreover, gN tends to g in the topology of S 0 . It is therefore sufficient to approximate gN in S 0 . As gN vanishes for (y, η) outside a compact set, we have in fact ˆ π L H0(−s1 ,−t1 ) (R+ ), H (s2 ,t2 ) (R+ ) → S Rq × Rq ⊗ ˆ π K L2 (R+ ) . gN ∈ S R q × R q ⊗ According to Lemma 2.19, each element of K(L2 (R+ )) can be approximated in L(L2 (R+ )) by an integral operator with a rapidly decreasing kernel. Each of these defines a continuous operator S (R+ ) → S(R+ ). Hence gN can be approximated, in the topology of ˆ π L L2 (R+ ) → S −∞ Rq × Rq ; L2 (R+ ), L2 (R+ ) , S Rq × Rq ⊗ ˆ π S(R+ × R+ ), hence also by elements of by elements in the tensor product S(Rq × Rq ) ⊗ S −∞ (Rq × Rq ; S (R+ ), S(R+ )) which vanish for y outside a compact set. 2 2.20. Remark. A symbol g in S −∞ (Rq × Rq ; S (R+ ), S(R+ )) which vanishes for y outside q+1 a compact set induces an operator op a on L2 (R+ ) with an integral kernel which is rapidly decreasing and thus can be approximated by a smooth compactly supported integral kernel. This slightly improves the statement of Proposition 2.18.


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3. Semiclassical operators in Boutet de Monvel’s calculus 3.1. Pseudodifferential operators. Let p ∈ S μ (Rn × Rn ) and u ∈ S(R). Then ∞ opxn (ph¯ )u(xn ) =

− ei(xn −yn )ξn p(x , xn , hξ ¯ , h¯ ξn )u(yn ) dyn d ξn

0

∞ =

− ei(xn /h¯ −yn )ξn p(x , xn , hξ ¯ , ξn )u(h¯ yn ) dyn d ξn

0

∞ = κh¯ −1

ei(xn −yn )ξn p(x , h¯ xn , h¯ ξ , ξn )κh¯ u(yn ) dyn d−ξn .

0

In case p is independent of xn , the last term equals (opxn p)h¯ u(xn ) where the subscript indicates that we use the h¯ -scaled symbol associated to the operator-valued symbol opxn p. 3.2. Potential, trace, and singular Green boundary symbol operators. We define the h¯ -scaled operators as in 2.14, noting that the group action on C is the identity. Specifically, kh¯ (x , ξ ) = κh¯−1 k(x , h¯ ξ ) th¯ (x , ξ ) = t (x , hξ ¯ )κh¯ −1 gh¯ (x , ξ ) = κh¯ g(x , h¯ ξ )κh¯

(potential symbols), (trace symbols), (singular Green symbols).

3.3. Lemma. Let g ∈ S μ (Rn−1 × Rn−1 , S (R+ ), S(R+ )), μ ∈ R, and ϕ ∈ Cb∞ (Rn ), supported in Rn+ . Then, for every N ∈ N, we have ϕ op(gh¯ ) = h¯ N op rN (h¯ ) with a family rN of singular Green symbols such that rN,1/h¯ is bounded in S μ−N . Proof. Write ϕN (x) = xn−N ϕ ∈ Cb∞ (Rn ). In view of the fact that MxnN κh¯−1 = h¯ N κh¯−1 MxnN we have Mϕ op gh¯ = h¯ N MϕN op κh¯−1 xnN g(x , hξ ¯ )κh¯ . This shows the assertion, since xnN g is a singular Green symbol of order μ − N .

2

3.4. Lemma. Let g1 and g2 be singular Green symbols of orders μ1 and μ2 , respectively. Then g1,h¯ g2,h¯ − (g1 g2 )h¯ = h¯ c(h¯ ) for a family c(h¯ ), 0 < h¯ 1, of singular Green symbols with c1/h¯ bounded of order μ1 + μ2 − 1. Proof. This is immediate from 2.14(c) in connection with Definition 3.2.

2

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3.5. Remark. In the following proposition we will show an analog of 2.14(c). The statement becomes more involved, since we have to take care of the leftover term. We thus fix the notation beforehand. μ Let pj ∈ Str j (Rn × Rn ), μj ∈ Z, j = 1, 2. For fixed (x , ξ ), we consider the operators (op+ xn pj )(x , ξ ). Their composition gives rise to a (x , ξ )-dependent leftover term. We denote by l(p1 , p2 ) its operator-valued singular Green symbol. In order to study it we follow Grubb [6, (2.6.18)ff] and decompose as in (1.1): pj = sj + qj

(3.1)

with differential symbols sj and symbols qj of normal order −1. We write s2 (x, ξ ) = μ2 j m j =0 aj (x, ξ )ξn and γm = γ0 ◦ Dn . Then l(p1 , p2 ) =

μ 2 −1

km γm + g + (q1 )g − (q2 )

(3.2)

m=0

with the potential symbol km given by

km (x , ξ )v = ir

+

p1 (x, ξ , Dn )

μ2

al (x, ξ

)Dnl−1−m

(v ⊗ δ);

(3.3)

l=m+1

and the singular Green symbols g ± defined by g + (q1 ) = r + opxn (q1 )e− J

and g − (q2 ) = J r − opxn (q2 )e+

(3.4)

with the reflection operator J : f (x , xn ) → f (x , −xn ). We will need a semiclassical version of the following well-known statement: 3.6. Theorem. Given a pseudodifferential symbol p of order μ with the transmission property and l ∈ N0 , the prescription v → r + op p v ⊗ Dnl δ ,

v ∈ S Rn−1 ,

defines a potential symbol k of order μ + l + 1/2 whose symbol seminorms can be estimated in terms of those of p. Writing p = p 0 + xn p 1 with p 0 (x , ξ ) = p(x , 0, ξ ), we have k = k 0 + k 1 , where k 0 (x , ξ ) = opxn p 0 (x , ξ )ξnl (δxn =0 ),

(3.5)

considered as a multiplication operator on S(R+ ), and k 1 is of order μ + l − 1/2. For a proof in the spirit of operator-valued symbols see [13, Lemma 2.11]. In the semiclassical situation we obtain


2665

3.7. Lemma. Under the above assumptions, v → r + opxn ph¯ v ⊗ h¯ l+1/2 Dnl δ ,

v ∈ S Rn−1

defines a potential boundary symbol operator k(h¯ ). We have k(h¯ ) = kh0¯ + h¯ k 1 (h¯ ) 1 bounded of order μ + l − 1/2. with k 0 defined in (3.5) and k1/ h¯

Proof. Replacing p by pξnl we can assume l = 0. This yields the potential symbol

1/2 +

k(h¯ ; x , ξ ) = h¯

− eixn ξn p(x, hξ ¯ ) d ξn

r

= h¯ −1/2 r + = κh¯−1 r +

− eixn ξn /h¯ p 0 (x , h¯ ξ , ξn ) + xn p 1 (x, hξ ¯ , ξn ) d ξn

opxn p 0 (x , hξ ¯ )δ +

e

ixn ξn

1

−

h¯ xn p (x , h¯ xn , h¯ ξ , ξn ) d ξn

= kh0¯ (x , ξ ) + h¯ k 1 (h¯ ; x , ξ ). Rescaling the potential symbols k 1 (h¯ ) we obtain 1 k1/ ¯;x ,ξ ) = h¯ (h

=

− eixn ξn xn p 1 (x , hx ¯ n , ξ , ξn ) d ξn

eixn ξn (−Dξn )p 1 (x , h¯ xn , ξ , ξn ) d−ξn ,

which is uniformly bounded in S μ−1/2 (Rn−1 × Rn−1 ; C, S(R+ )) by Theorem 3.6.

2

In the following proposition, we shall analyze the relation between the interior symbols and the leftover term. 3.8. Proposition. We use the notation of Remark 3.5. Then + + + op+ ¯ opxn c(h¯ ) + d(h¯ ) xn p1,h¯ ◦ opxn p2,h¯ − opxn (p1 p2 )h¯ − l(p1 , p2 )h¯ = h μ +μ2 −1

for two families c and d with c1/h¯ bounded in Str 1 S μ1 +μ2 −1 (Rn−1 × Rn−1 ; S (R+ ), S(R+ )).

(Rn × Rn ) and d1/h¯ bounded in

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Proof. We already know from 2.14 (for the standard C-valued case) that opxn p1,h¯ opxn p2,h¯ − opxn (p1 p2 )h¯ = h¯ op c(h¯ ) , with c1/h¯ bounded in S μ1 +μ2 −1 . In fact, we even obtain boundedness in the corresponding class with the transmission property, since we have an asymptotic expansion for c with terms bounded in the topology of symbols with the transmission property. Let us next consider the symbol l(p1,h¯ , p2,h¯ ) of the leftover term L(opxn p1,h¯ , opxn p2,h¯ ), which we will compute according to (3.2), (3.3). Replacing ξ by h¯ ξ , we obtain l(p1,h¯ , p2,h¯ ) =

μ 2 −1

km (h¯ )γm + g + (q1,h¯ )g − (q2,h¯ ).

m=0

Using the notation introduced above, the potential symbol km is given by

km (h¯ ; x , ξ )v = ir

+

opxn p1 (x, hξ ¯ )

μ2

l

al (x, hξ ¯ )h¯

Dnl−1−m

(v ⊗ δ).

l=m+1

By Lemma 3.7, h¯ −m−1/2 km (h¯ ) is a family of potential symbols of order μ1 + μ2 − m − 1/2, 0 equal to km, ¯ m+1/2 γm , this shows h¯ modulo lower order terms. In view of the fact that γm,h¯ = h 0 γ that the composition km (h¯ )γm is a singular Green symbol which equals km, h¯ m,h¯ modulo lower order terms of the desired form. The singular Green symbols g ± (p) = g ± (p)(x , ξ ) associated to a pseudodifferential symbol p of negative normal order are the integral operators with the kernels g˜ ± (p)(x , ξ , xn , yn ) =

eizξn p(x , xn , ξ , ξn ) d−ξn

(3.6)

z=±(xn +yn )

so that ±

g˜ (ph¯ )(x , ξ , xn , yn ) = h¯

−1

eizξn p(x , xn , h¯ ξ , ξn ) d−ξn

z=±(xn +yn )/h¯

.

This implies that g ± (ph¯ )(x , ξ ) = κh¯−1 g ± p(x , h¯ xn , h¯ ξ , ξn ) κh¯ . Writing p = p 0 + xn p 1 as before, we have −1 ± 1 g ± (ph¯ )(x , ξ ) = g ± p 0 h¯ + hκ ¯ h¯ xn g p (x , h¯ xn , h¯ ξ , ξn ) κh¯ . As p 1 (x , hx ¯ n , hξ ¯ , ξn ), 0 < h¯ 1, is a bounded family of pseudodifferential operators of order μ with the transmission property and since multiplication by xn lowers the order by 1 according to 2.8(a), we obtain the assertion of the proposition by first applying this consideration to g + (q1,h¯ ) and g − (q2,h¯ ) and then using Lemma 3.4. 2


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4. The graph projection 4.1. Definition. The graph projection of a bounded operator a on a Hilbert space H is the operator G(a) on H ⊕ H given by G(a) =

(1 + a ∗ a)−1 a(1 + a ∗ a)−1

(1 + a ∗ a)−1 a ∗ a(1 + a ∗ a)−1 a ∗

=

(1 + a ∗ a)−1 a(1 + a ∗ a)−1

(1 + a ∗ a)−1 a ∗ 1 − (1 + aa ∗ )−1

.

For unbounded operators, it is not a priori clear that the above definition makes sense, nor is the second identity obvious. We will now have a closer look. 4.2. The framework. Let V → H → V be Hilbert spaces, with V dense in H , and assume that V is the dual space of V with respect to an extension ·,· of the inner product in H . Moreover let a0 : V → H be a bounded operator with adjoint a0∗ : H → V . We assume that a0 is closable in H and denote by a the closure. Explicitly: The domain D of a consists of all x in H for which there exists a sequence (xn ) in V with xn → x in H and a0 xn → y in H for some y ∈ H . In that case, we define ax = y. 4.3. Lemma. D naturally is a Hilbert space with the inner product

x, yD = x, y + ax, ay.

(4.1)

V is dense in D. Proof. Clearly, (4.1) defines an inner product on D. The associated norm xD = (x2 + ax2 )1/2 is the graph norm with respect to which D is complete. Hence D is a Hilbert space. V is dense in D by construction. 2 4.4. Lemma. Let x ∈ H . Then the element a0∗ x ∈ V extends to a continuous linear functional on D, and

a0∗ x, y = x, ay,

x ∈ H, y ∈ D.

(4.2)

Proof. For v in the dense subspace V we have ∗ a x, v = x, av xav xvD 0

so that a0∗ x extends continuously to D. The stated identity follows.

2

We now denote by E the range of the operator 1 + a0∗ a : D → V . 4.5. Lemma. The elements of E define continuous linear functionals on D. The norm of (1 + a0∗ a)x on D is xD , and the operator 1 + a0∗ a is injective. Proof. As D → H , the first statement follows from Lemma 4.4. Note that

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1 + a∗a x 0

L(D ,C)

= sup 1 + a0∗ a x, y = sup x, y + ax, ay yD 1

= sup

x, yD = xD .

yD 1

yD 1

This implies injectivity.

2

4.6. Lemma. E inherits a Hilbert space structure from D. The associated norm is the norm in D = L(D, C). This allows us to identify E with the dual of D with respect to the sesquilinear pairing between V and V . E contains the range of a0∗ . Proof. Let e = x + a0∗ ax and f = y + a0∗ ay be two elements of E. Then we let

e, f E =

x, yD ,

(4.3)

so that the Hilbert space structure of D carries over to E . The associated norm of e = (1 + a0∗ a)x then is equal to xD , which is the norm of (1 + a0∗ a)x in D by Lemma 4.5. Moreover, the identity x + a0∗ ax, y =

x, yD

shows that the action of E on D via the sesquilinear pairing coincides with the pairing with elements of D, which gives the whole dual space. By Lemma 4.4, a0∗ (H ) ⊆ D = E. 2 4.7. Corollary. The operator 1 + a0∗ a : D → E is invertible with a bounded inverse. Moreover, the operators (1 + a0∗ a)−1 , (1 + a0∗ a)−1 a0∗ , a(1 + a0∗ a)−1 , and a(1 + a0∗ a)−1 a0∗ define elements of L(H ). 4.8. Lemma. The restriction of (1 + a0∗ a)−1 to H is self-adjoint, and the operators a(1 + a0∗ a)−1 and (1 + a0∗ a)−1 a0∗ are adjoints of each other in L(H ). Proof. Let x ∈ H ⊆ E and z = (1 + a0∗ a)−1 x ∈ D. For y ∈ H , Lemma 4.4 implies that −1 −1 −1 1 + a0∗ a y, x = 1 + a0∗ a y, 1 + a0∗ a z = y, z = y, 1 + a0∗ a x . This shows that (1 + a0∗ a)−1 is self-adjoint. For x, y ∈ H the element a0∗ x ∈ E has a preimage z in D under 1 + a0∗ a. We infer from (4.2) that −1 −1 −1 a 1 + a0∗ a y, x = 1 + a0∗ a y, a0∗ x = 1 + a0∗ a y, 1 + a0∗ a z −1 = y, z = y, 1 + a0∗ a a0∗ x . This shows the second statement.

2

4.9. Notation. In the above we wrote a0∗ in order to stress the fact that this operator is not the H -valued Hilbert space adjoint of a but an operator with values in V . Now that this has been made clear we shall go back to the simpler notation and write a ∗ .


2669

4.10. The set-up. In the sequel, T : C ∞ (X, E1 ) → C ∞ (X, E2 ) will be a Fredholm operator of order and class zero in Boutet de Monvel’s calculus. Following an idea of Elliott, Natsume and Nest [4] we will associate to T the operator A = Λm,+ − T for some m > n and then study the graph projection. The operator T induces a Fredholm operator Tm : H m (X, E1 ) → H m (X, E2 ). Its adjoint ∗ Tm : H0−m (X, E2 ) → H0−m (X, E1 ) extends the L2 -adjoint T ∗ . We let m 2 A = Λm,+ − Tm : H (X, E1 ) → L (X, E2 ).

This is an operator of order m and class zero; there is no leftover term in the composition. The −m ∗ ∗ m 2 adjoint A∗ is the operator (Λm,+ − Tm ) = Tm Λ+ : L (X, E2 ) → H0 (X, E1 ). ∗ We consider the composition A A as the bounded operator m,+ −m m Tm∗ Λm + Λ− Tm : H (X, E1 ) → H0 (X, E1 ). 2 On the other hand, A = Λm,+ − T extends to a bounded operator from L (X, E1 ) to m m −m ∗ ∗ 2 ∗ ∗ m H (X, E2 ) with adjoint A = T Λ+ : H0 (X, E2 ) → L (X, E1 ), and AA = Λm,+ − T T Λ+ m −m maps H0 (X, E2 ) to H (X, E2 ).

4.11. Lemma. We have natural embeddings H m (X, Ej ) → L2 (X, Ej ) → H0−m (X, Ej ) and H0m (X, Ej ) → L2 (X, Ej ) → H −m (X, Ej ), j = 1, 2, and topological isomorphisms 1 + A∗ A : H m (X, E1 ) → H0−m (X, E1 )

and 1 + AA∗ : H0m (X, E2 ) → H −m (X, E2 ).

Proof. The embeddings are well known. The second statement follows from Corollary 4.7, applied to the operator a=

0 A∗ A 0

H m (X, E1 ) L2 (X, E1 ) ⊕ → ⊕ : , H0m (X, E2 ) L2 (X, E2 )

together with the fact that – due to elliptic regularity – the domain of the closure of A is H m (X, E1 ), that of A∗ is H0m (X, E2 ). 2 As a consequence, the operators A(1 + A∗ A)−1 A∗ and (1 + AA∗ )−1 are bounded operators on L2 (X, E1 ) and L2 (X, E2 ), respectively.

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4.12. Lemma. The restriction of AA∗ to H02m (X, E2 ) maps to L2 (X, E2 ). ∼ =

2m m m m Proof. We know that Λm + : H0 −→ H0 . Now we observe that H0 naturally embeds into H , ∼ =

m 2 and thus T T ∗ defines a bounded map from H0m to H m . Finally, Λm − : H −→ L .

2

4.13. Lemma. A(1 + A∗ A)−1 A∗ = 1 − (1 + AA∗ )−1 on L2 (X, E2 ). Proof. According to Lemma 4.12, the composition A(1 + A∗ A)−1 A∗ (1 + AA∗ ) is defined as a bounded operator from H02m to L2 , and we have A(1 + A∗ A)−1 A∗ (1 + AA∗ ) = A(1 + A∗ A)−1 (1 + A∗ A)A∗ = AA∗ .

(4.4)

We next denote by R the range of the restriction of (1+AA∗ )−1 to L2 , so that 1+AA∗ : R → L2 is an isomorphism. According to 4.11 and 4.12, we have H02m ⊆ R ⊆ H0m . As (4.4) extends to R, the compositions, below, are defined on L2 , and A(1 + A∗ A)−1 A∗ = A(1 + A∗ A)−1 A∗ (1 + AA∗ )(1 + AA∗ )−1 = AA∗ (1 + AA∗ )−1 = 1 − (1 + AA∗ )−1 .

2

4.14. Lemma. G(A) and e = 00 10 define idempotents in K(L2 (X, E1 ⊕ E2 ))∼ , where as usual the tilde indicates the unitization. In particular, the difference [G(A)] − [e] defines a class in K0 (K(L2 (X, E1 ⊕ E2 ))). Proof. Apart from the one in the lower right corner, the entries in G(A) are compact on L2 as a consequence of the compact embeddings H0m → L2 and H m → L2 . By Lemma 4.13, the last entry differs from the identity by the compact operator (1 + AA∗ )−1 . 2 4.15. Theorem. The class [G(A)] − [e] in K0 (K(L2 (X, E1 ⊕ E2 ))) equals [πker A ] − [πker A∗ ] = [πker T ] − [πker T ∗ ]. Here, πV denotes the orthogonal projection onto V with respect to the L2 -inner product. Proof. Replacing A by tA for t 1, we consider G(tA), which is a norm continuous family of idempotents. We claim that G(tA) converges to

πker T 0

0 1 − πker T ∗

as t → ∞. We let H1 = (1 + A∗ A)−1 (L2 ) and H2 = (1 + AA∗ )−1 (L2 ). This allows us to consider A∗ A and AA∗ as unbounded operators on L2 with domains H1 and H2 , respectively. The graph projection is not affected by this change. For the unbounded operator, however, the statements are well known. They are a consequence of the fact that 0 is an isolated point in the spectrum. Alternatively, the statement can be checked by a direct computation. 2


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4.16. Corollary. For m > n, ind T = Tr(1 + A∗ A)−1 − Tr(1 + AA∗ )−1 . Proof. The difference G(A) − e is a trace class operator in L(L2 ) since its four entries are trace class operators. This in turn follows from the fact that the embeddings H m → L2 and L2 → H0−m are trace class. According to Theorem 4.15 we then have ind A = Tr [πker A ] − [πcoker A ] = Tr G(A) − e = Tr(1 + A∗ A)−1 − Tr(1 + AA∗ )−1 .

2

5. The graph projections of the symbols 5.1. Notation. We denote by p m = σψm (A)

and cm = σ∂m (A)

the homogeneous principal pseudodifferential symbol and the homogeneous principal boundary symbol of the operator A in 4.10. Both are invertible. Locally cm (x , ξ ) = p m,+ (x , 0, ξ , Dn ) + g m (x , ξ )

(5.1)

with a suitable strictly homogeneous singular Green part g m . We then choose a smooth function p on T ∗ X which coincides with p m for |ξ | 1 and a smooth singular Green symbol g which coincides with g m for |ξ | 1 and let c(x , ξ ) = p + (x , 0, ξ , Dn ) + g(x , ξ ). We next apply Corollary 4.7 to the operator family 1 ) → H −m (R+ , E 1 ), 1 + c∗ (x , ξ )c(x , ξ ) : H m (R+ , E 0

(x , ξ ) ∈ T ∗ ∂X.

For each (x , ξ ), we denote by D(x ,ξ ) the domain of the closure and by E(x ,ξ ) the range. 5.2. Proposition. For each choice of (x , ξ ) we have 1 ) D(x ,ξ ) = H m (R+ , E

1 ). and E(x ,ξ ) = H0−m (R+ , E

(5.2)

The operator family 1 ) → H −m (R+ , E 1 ) 1 + c∗ (x , ξ )c(x , ξ ) : H m (R+ , E 0 is pointwise invertible. The inverse (1 + c∗ c)−1 is an element of 1 ), H m (R+ , E 1 ) . S −2m Rn−1 × Rn−1 ; H0−m (R+ , E

(5.3)

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We shall improve this result in 5.3, below. 1 ) is the domain of the closure of the operator Proof. By definition, D(x ,ξ ) ⊆ H m (R+ , E 1 ) → L2 (R+ , E 2 ) c(x , ξ ) : H m (R+ , E 1 ). It consists of all v ∈ L2 for which there is a sequence vk in H m with vk → v in in L2 (R+ , E 2 L and c(x , ξ )vk → w in L2 for some w which then is defined to be c(x , ξ )v. 1 ) → S(R+ , E 2 ) is continuous, g(x , ξ )vk will converge for any conAs g(x , ξ ) : S (R+ , E 2 vergent L -sequence (vk ); hence the domain is independent of g. We therefore have D(x ,ξ ) ⊆ v ∈ L2 : p + (x , 0, ξ , Dn )v ∈ L2 . 1 ), and we get the first In view of the fact that p is elliptic, the last set is a subset of H m (R+ , E part of (5.2). According to Lemma 4.6, the space E(x ,ξ ) is the dual space of Dx ,ξ ) with respect to the pairing induced by the L2 inner product. This gives the second statement in (5.2). We conclude that (x , ξ ) → 1 + c∗ (x , ξ )c(x , ξ ) is a smooth family of operators in 1 ), H −m (R+ , E 1 ) . L H m (R+ , E 0 As inversion is continuous, (1 + c∗ (x , ξ )c(x , ξ ))−1 also is a smooth family. Moreover, for |ξ | 1, we have c = cm , and both c(x , ξ ) and c∗ (x , ξ ) are invertible. We can write −1 (1 + c∗ c)−1 = c−1 1 + (c∗ )−1 c−1 (c∗ )−1 .

(5.4)

If ϕ = ϕ(ξ ) is an excision function on R which vanishes for |ξ | 1 and is equal to one for large |ξ |, then the homogeneity of c implies that, in local coordinates, ϕc−1 ∈ S −m Rn−1 × Rn−1 ; L2 (R+ ), H m (R+ ) .

(5.5)

Similarly as in 1.6, c∗ (x , ξ )−1 = (c−1 (x , ξ ))∗ . Consequently, ϕc∗−1 ∈ S −m Rn−1 × Rn−1 ; H0−m (R+ ), L2 (R+ ) .

(5.6)

Next we note that the positivity of m implies that, as |ξ | → ∞, κ[ξ−1 ] (c∗ )−1 (x , ξ ) c−1 (x , ξ )κ[ξ ] → 0

in L H0−m (R+ ), H m (R+ ) .

In particular, (1 + (c∗ )−1 c−1 )−1 is uniformly bounded in L(L2 (R+ )). For d = 1 + (c∗ )−1 c−1 we then deduce that


2673

κ[ξ−1 ] ∂ξj d −1 κ[ξ ] = κ[ξ−1 ] d −1 ∂ξj d d −1 κ[ξ ] = O [ξ ]−1 in L(L2 (R+ )). Iteration shows that −1 ∈ S 0 Rn−1 × Rn−1 ; L2 (R+ ), L2 (R+ ) . ϕ 1 + (c∗ )−1 c−1

(5.7)

The smoothness in (x , ξ ) and Eqs. (5.4)–(5.7) then imply that (1 + c∗ c)−1 ∈ S −2m Rn−1 × Rn−1 ; H0−m (R+ ), H m (R+ ) .

2

5.3. Theorem. (1 + c∗ c)−1 is a boundary symbol operator in Boutet de Monvel’s calculus whose pseudodifferential part is r + (1 + p ∗ p)|−1 xn =0 . Proof. We start with a few preliminaries. We consider the composition c∗ (x , ξ )c(x , ξ ) with c defined in 5.1. For fixed (x , ξ ), the adjoint c∗ (x , ξ ) : H00 (R+ ) ∼ = L2 (R+ ) → H0−m (R+ ) is given by c∗ (x , ξ ) = p ∗ (x , 0, ξ , Dn ) + g ∗ (x , ξ ) with the formal adjoints of p and g. This, however, needs some explanation. In order to keep the notation light, we will simply write c∗ = p ∗ (Dn ) + g ∗ . The adjoint g ∗ is of order m and class 0. It naturally maps L2 (R+ ) to S(R+ ), which can be viewed as a subspace of H0−m (R+ ) via extension by zero. In order to apply p ∗ (Dn ) to u ∈ L2 (R+ ), we first extend u by zero to an element of L2 (R). Applying p ∗ (Dn ) furnishes a distribution in H −m (R), which will in general not vanish on R− : It only induces a functional on H m (R+ ) coming from an element in H0−m (R+ ). Indeed, according to (1.1), we can write p ∗ = s ∗ + q ∗ with a polynomial s ∗ and q ∗ of normal order −1. Then q ∗ (Dn ) maps e+ L2 (R+ ) to L2 (R) so that its restrictions to the positive and the negative halfline are defined. In view of the fact that s ∗ (Dn ) preserves the support, the distribution p ∗ (Dn )u as an element in H0−m (R+ ) is given by p ∗ (Dn )u − r − q ∗ (Dn )u = p ∗ (Dn )u − e− J g − (q ∗ )u with the symbol g − (q ∗ ) introduced in (3.4). It is of order m and class 0. We therefore have c∗ c = p ∗ (Dn )e+ p + (Dn ) + g + e+ g ∗ p + (Dn ) + g − e− J g − (q ∗ ) p + (Dn ) + g = p ∗ (Dn )e+ p + (Dn ) + g + e+ g1 + e− J g2 (5.8) with suitable singular Green symbols g1 and g2 of orders 2m and class m. Hence (with p = p(x , 0, ξ ))

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r + (1 + p ∗ p)−1 (Dn )c∗ c = r + (1 + p ∗ p)−1 p ∗ (Dn )e+ p + (Dn ) + r + (1 + p ∗ p)−1 p ∗ (Dn )e+ g + r + (1 + p ∗ p)−1 (Dn )e+ g1 + r + (1 + p ∗ p)−1 (Dn )e− J g2 . Surprisingly, all the terms on the right-hand side can be treated within Boutet de Monvel’s calculus. The first is the composition of two truncated pseudodifferential operators. It equals + (1 + p ∗ p)−1 p ∗ p (Dn ) − l (1 + p ∗ p)−1 p ∗ , p , where the leftover term of the composition is of order 0 and class m. The second and the third are compositions of a truncated pseudodifferential operator with a singular Green symbol, thus singular Green symbols. Both have order zero and class m. The final term is the composition of the g + -term of the pseudodifferential part with g2 and therefore also a singular Green symbol of order 0 and class m. Putting all this together, we find that r + (1 + p ∗ p)−1 (Dn )(1 + c∗ c) = 1 + g3 with a singular Green symbol g3 of order 0 and class m. Hence (1 + c∗ c)−1 = r + (1 + p ∗ p)−1 (Dn ) − g3 (1 + c∗ c)−1 .

(5.9)

Let us now have a look at c∗ cr + ((1 + p ∗ p)−1 )(Dn ). We first note that we may consider e+ a trivial action on H0−m (R+ ) and that + p + (Dn )r + (1 + p ∗ p)−1 (Dn )e+ = p(1 + p ∗ p)−1 (Dn ) + l p, (1 + p ∗ p)−1 ,

(5.10)

where the leftover term on the right-hand side is of order −m and class 0. As the first term on the right-hand side maps to L2 (R) we can rewrite it as p(1 + p ∗ p)−1 (Dn )e+ − e− r − p(1 + p ∗ p)−1 (Dn )e+ = p(1 + p ∗ p)−1 (Dn )e+ − e− J g − p(1 + p ∗ p)−1 ,

(5.11)

where the g − -term is of order −m and class 0. Taking into account (5.8), (5.10) and (5.11) (1 + c∗ c)r + (1 + p ∗ p)−1 (Dn ) = 1 + p ∗ (Dn ) e+ g4 + e− J g5 + e+ g6 + e− J g7 with singular Green symbols g4 and g5 of order −m and class 0 and g6 and g7 of order and class 0. Note that the image of the sum of the second and the fourth summand necessarily lies in H0−m (R+ ), since this is the case for the others.


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Combining this with (5.9) we conclude that (1 + c∗ c)−1 = r + (1 + p ∗ p)−1 (Dn ) − (1 + c∗ c)−1 p ∗ (Dn ) e+ g4 + e− J g5 − (1 + c∗ c)−1 e+ g6 + e− J g7 = r + (1 + p ∗ p)−1 (Dn ) − r + (1 + p ∗ p)−1 (Dn )p ∗ (Dn )e+ g4

− r + (1 + p ∗ p)−1 (Dn )p ∗ (Dn )e− J g5 − r + (1 + p ∗ p)−1 (Dn ) e+ g6 + e− J g7 + g3 (1 + c∗ c)−1 p ∗ (Dn ) e+ g4 + e− J g5 + g3 (1 + c∗ c)−1 e+ g6 + e− J g7 .

The first term on the right-hand side is the one we want as the pseudodifferential part. The second is the composition of a truncated pseudodifferential operator of order −m with a singular Green symbol of order −m and class 0, thus a singular Green symbol of order −2m and class 0. The third is the composition of a g + -type symbol of order −m and class 0 with a singular Green symbol of order −m and class 0, thus of the same type as the second. The summands of the fourth term are of the same type as the second and the third. As for the sum of the fifth and sixth, we note that e+ g4 + e− J g5 ∈ S −m Rn−1 × Rn−1 ; S (R+ ), S(R+ ) ⊕ S(R− ) and p ∗ (Dn ) ∈ S m Rn−1 × Rn−1 ; S(R+ ) ⊕ S(R− ), H −m (R) . The composition of both therefore is an element of S 0 (Rn−1 × Rn−1 ; S (R+ ), H −m (R)). As S(R+ ) ⊕ S(R− ) → H −m (R), we have e+ g6 + e− J g7 ∈ S 0 Rn−1 × Rn−1 ; S (R+ ), H −m (R) . Moreover, we know that the range of the sum of all these terms is in H0−m (R+ ) so that we can replace H −m (R) in the symbol space by H0−m (R+ ). We saw in Proposition 5.2 that (1 + c∗ c)−1 ∈ S −2m Rn−1 × Rn−1 ; H0−m (R+ ), H m (R+ ) . As g3 is of order 0 and class m we have g3 ∈ S 0 Rn−1 × Rn−1 ; H m (R+ ), S(R+ ) . Hence the total composition is an element of S −2m (Rn−1 × Rn−1 ; S (R+ ), S(R+ )) thus a singular Green symbol of order −2m and class 0. This shows that (1 + c∗ c)−1 is a boundary symbol operator in Boutet de Monvel’s calculus which differs from r + (1 + p ∗ p)−1 (Dn ) by a singular Green symbol of order −2m and class 0. 2

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5.4. The inverse of 1 + cc∗ . For fixed (x , ξ ), the operator c(x , ξ )c∗ (x , ξ ) : H0m (R+ ) → H −m (R+ ) acts on v ∈ Cc∞ (R+ ) by considering the function c∗ (x , ξ )v as an element of H00 (R+ ) ∼ = L2 (R+ ) to which we then apply c. We know that c∗ = p ∗ (Dn ) + g ∗ with the notation introduced above. Here p ∗ (Dn )v is a function in S(R); interpreting it as a distribution in H00 (R+ ) amounts to restricting it to R+ . As e+ can be considered a trivial action on Cc∞ (R+ ), the action of c∗ coincides with that of p ∗,+ (Dn ) + g ∗ . The composition cc∗ therefore coincides with the composition of two boundary symbol operators in Boutet de Monvel’s calculus. As 1 + cc∗ is invertible, the inverse is also given by a boundary symbol in that calculus, and we obtain the statement below: 5.5. Corollary. The inverse (1 + cc∗ )−1 has the following form (1 + cc∗ )−1 = r + (1 + pp ∗ )−1 (Dn ) + g8

(5.12)

with a singular Green symbol g8 of order −2m and class 0. 5.6. Proposition. By G(p) =

(1 + p ∗ p)−1 p(1 + p ∗ p)−1

(1 + p ∗ p)−1 p ∗ p(1 + p ∗ p)−1 p ∗

∼ ∈ C0 T ∗ X, L(E1 ⊕ E2 )

we denote the graph projection of p. The difference of equivalence classes

G(p) −

0 0 0 1

then defines an element in K0 (C0 (T ∗ X, L(E1 ⊕ E2 ))) which is independent of the way the smoothing near zero is performed. Proof. Let p0 and p1 be two smooth extensions of p m . We let pt = (1 − t)p0 + tp1 , 0 t 1. For each t, pt is a smooth function on T ∗ X which coincides with p m on {|ξ | 1}. The associated family G(pt ) is continuous in t; hence the class [G(pt )] is constant. 2 5.7. Proposition. For the graph projection of c, G(c) =

(1 + c∗ c)−1 c(1 + c∗ c)−1

(1 + c∗ c)−1 c∗ c(1 + c∗ c)−1 c∗

1 ⊕ E 2 ) ∼ ∈ C0 T ∗ ∂X, L L2 (R+ , E

the difference of equivalence classes G(c) −

0 0 0 1

1 ⊕ E 2 )))). It does not depend on the way the defines an element in K0 (C0 (T ∗ ∂X, L(L2 (R+ , E smoothing near zero in 5.1 is performed.


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Proof. Let p0 and p1 be two smooth extensions of p m and g0 and g1 two smooth extensions of g m . We let pt = (1 − t)p0 + tp1 , gt = (1 − t)g0 + tg1 and ct (x , ξ ) = pt+ (x , ξ , Dn ) + gt (x , ξ ),

0 t 1.

The associated graph projections G(ct ) depend continuously on t in the topology of 1 ⊕ E 2 )))∼ . Hence the class is independent of t. 2 C0 (T ∗ ∂X, L(L2 (R+ , E 6. The tangent semi-groupoid We recall a few concepts from Aastrup, Nest and Schrohe [1]. X for which expx (±εv) ∈ 6.1. Definition. By T ± X we denote the set of all vectors (x, v) ∈ T X| X for sufficiently small ε > 0. This is a semi-groupoid with addition of vectors, and T ± X = T X ◦ ∪ T ± X|∂X . We define T − X as the disjoint union T − X ∪ (X × X × ]0, 1]), endowed with the fiberwise semi-groupoid structure induced by the semi-groupoid structure on T − X and the pair groupoid structure on X × X. We glue T − X to X × X × ]0, 1] via the charts T − X × [0, 1] ⊇ U (x, v, h¯ ) →

(x, v) (x, expx (−h¯ v), h¯ )

for h¯ = 0, for h¯ = 0

and let T − X(0) = T − X and T − X(h¯ ) = X × X × {h¯ }. In order to avoid problems with the topology of T − X (which is in general not a manifold T −X . with corners) we let Cc∞ (T − X) = Cc∞ (T X)| C ∗ -algebras associated to the semi-groupoids T − X and T − X. Let Cc∞ (T − X) denote the smooth functions on T − X with compact support in T − X. We introduce π0 : Cc∞ (T − X) → L L2 (T X) and π0∂ : Cc∞ (T − X) → L L2 T + X|∂X acting by π0 (f )ξ(x, v) =

f (x, v − w)ξ(x, w) dw,

(6.1)

f (x, v − w)ξ(x, w) dw.

(6.2)

Tx X

π0∂ (f )ξ(x, v) = Tx+ X

Note. As f has compact support in T − X, it naturally extends (by zero) to T X.

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6.2. Definition. Cr∗ (T − X) is the C ∗ -algebra generated by π0 ⊕ π0∂ , i.e. by the map Cc∞ (T − X) f → π0 (f ), π0∂ (f ) ∈ L L2 (T X) ⊕ L2 T + X|∂X . 6.3. Remark. According to Lemmas 2.14 and 2.15 in [1], Cr∗ (T − X) has the dense ∗-subalgebra ∞ (T − X) = Cc∞ (T X) ⊕ Cc∞ (T ∂X × R+ × R+ ) Ctc

with the representation π0 ⊕ π0∂ of Cc∞ (T X) on L2 (T X) ⊕ L2 (T + X|∂X ), defined as above, and the representation π˜ 0∂ of Cc∞ (T ∂X × R+ × R+ ) on L2 (T + X|∂X ) given by

π˜ 0∂ (K)ξ(x, v , vn ) =

K(x, v − w , vn , wn )ξ(x, w , wn ) dw dwn .

T + X|∂X

6.4. An ideal in Cr∗ (T − X). Denote by F : L2 (T X) → L2 (T ∗ X) and F : L2 (T ∂X) → L2 (T ∗ ∂X) the fiberwise Fourier transforms. It was noted in [1, Lemma 2.15] that F : L2 (T ∂X) → L2 (T ∗ ∂X) provides an isomorphism between the ideal of Cr∗ (T − X) generated by the representation π˜ 0∂ in 6.3 and C0 (T ∗ ∂X, K). The Fourier transform allows us two more important identifications: For f ∈ Cc∞ (T X), the operator F π0 (f )F −1 : L2 (T ∗ X) → L2 (T ∗ X) is the operator of multiplication by f= F f . At the boundary, the choice of a Riemannian metric allows us to identify T ± X|∂X with T ∂X × R± and the operator F π0∂ (f )F

−1

: L2 (T ∗ ∂X × R+ ) → L2 (T ∗ ∂X × R+ )

with the boundary symbol operator F (f )(x , 0, ξ , Dn )+ . 6.5. Remark. So far we have been working with the graph projections of operators and symbols acting in vector bundles. Since we want it to give rise to elements in the K-theory of Cr∗ (T − X) and Cr∗ (T − X) we will describe here how this can be achieved: Choose bundles F1 and F2 such that E1 ⊕ F1 and E2 ⊕ F2 are trivial with fiber Cν . We can consider the graph projection of an operator A acting between sections of E1 and E2 as an element of L(L2 (X, C2ν )). Let πF2 be the projection onto F2 . Note that G(A) + πF2


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is a projection in M2ν (K(L2 (X))∼ ) and that G(A) + πF2 − [πE2 ⊕F2 ] is the index class of A in K0 (K(L2 (X))). The same consideration applies to the graph projection of the symbol, so that we get a class in K0 (C ∗ (T − X)). Since the construction is stable under the h¯ -scaling this gives a way of passing from general vector bundles to trivial vector bundles. We will in the rest of the paper use this to identify various graph projections acting in vector bundles with projections in trivial bundles. 6.6. Proposition. With the identification provided by the Fourier transforms described in 6.4, G(p) ⊕ G(c) can be regarded as an element in MN (Cr∗ (T − X)∼ ) for suitable N . Proof. We abbreviate p(Dn ) = p(x , 0, ξ , Dn ) and consider the difference G(c) − r + G p(Dn ) e+

(6.3)

between the graph projection of c and the truncated operator obtained from the graph projection of the operator p(Dn ), i.e. from G p(Dn ) =

(1 + p ∗ p)−1 (Dn ) p(Dn )(1 + p ∗ p)−1 (Dn )

(1 + p ∗ p)−1 (Dn )p ∗ (Dn ) p(Dn )(1 + p ∗ p)−1 (Dn )p ∗ (Dn )

1 ⊕ E 2 ). According to Theorem 5.3, (1 + c∗ c)−1 − r + (1 − p ∗ p)−1 (Dn ) is acting on L2 (R, E 1 ) to a singular Green symbol of negative order and thus a compact operator from H0−m (R+ , E m ∗ H (R+ , E1 ). Hence the difference (6.3) is an element of MN (C0 (T ∂X, K)). As pointed out in 6.4, conjugation by the boundary Fourier transform maps it to an ideal in MN (Cr∗ (T − X)). We only have to show that conjugation by the Fourier transforms maps G(p) ⊕ r + G p(Dn ) e+ to an element of MN (Cr∗ (T − X)∼ ). 1 ⊕ E 2 )) which In order to do this, it actually suffices to find a sequence (fk ) in Cc∞ (T X, L(E converges to f = F −1 (G(p) − e), where e is the usual projection onto the second component, with respect to the norm g → supx∈X g(x, ·)L1 . Indeed, this will imply that π0 (fk ) → π0 (f ) or, equivalently, that as multiplication operators F fk → G(p) − e. Moreover, as π0∂ (·) is dominated by π0 (·), also the operators r + F (fk )(Dn )e+ will approximate r + G(p(Dn ))e+ − e. It remains to find such a sequence. To this end we note that the entries in G(p) − e are symbols of orders −m < −n. Hence, for any K 0, they satisfy −1 2 K ivξ K − F q (x, v) sup e (1 − ξ ) q(x, ξ ) d ξ < ∞. sup 1 + |v| x,v x,v As Cc∞ (T X) is dense in the space of functions for which the weighted sup-norm on the left-hand side is finite and as this norm is larger than supx g(x, ·)L1 whenever K > N/2, we find the desired sequence. 2

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6.7. Parametrix construction. We fix a collar neighborhood ∂X × [0, 2) of ∂X in X. Moreover, we choose a covering of X by open sets Xj and coordinate maps χj : Uj ⊂ Rn → Xj . We assume that a coordinate neighborhood either is contained in a collar neighborhood of the boundary (‘boundary chart’) or else does not intersect a neighborhood of the boundary (‘interior chart’). It is also no restriction to suppose that the boundary charts all lie in the collar neighborhood and that the variable normal to the boundary, xn , is fixed and that changes of coordinates only involve the tangential variables. We fix a partition of unity ϕj subordinate to the coordinate charts and cut-off functions ψj supported in the same charts with ψj (x) ≡ 1 in a neighborhood of the support of ϕj . 6.8. Lemma. Without changing the K-classes of G(p) and G(c) we may assume that the symbol p of 5.1 is independent of xn in a neighborhood of ∂X. In particular, we shall assume that this is the case on the collar neighborhood ∂X × [0, 1) of the boundary. Proof. Apply a smooth deformation in the variable xn . This will imply continuity of the change of the graph projections and thus keep the associated K-class constant. 2 6.9. Scaling. According to 2.14 we define ph¯ (x, ξ ) = p(x, h¯ ξ ), gh¯ (x , ξ ) = κh¯−1 g(x , h¯ ξ )κh¯ , ch¯ (x , ξ ) = κh¯−1 op+ ¯ ξ )κh¯ + gh¯ (x , ξ ). xn (p|xn =0 )(x , h

We denote by p j the function p in the χj -coordinates and by χj∗ the transport of operators from Uj to Xj , i.e. χj∗ op ph¯ is the operator induced on X from the operator op ph¯ on Rn . Of course, this only makes sense when multiplied with suitable cut-off functions from the left and the right. In the boundary charts we will use the h¯ -scaled boundary symbol operators ch¯ ; we write chk¯ for this symbol in the χk -coordinates and op for the quantization map for boundary symbol operators. Then we define the operator family Ah¯ : C ∞ (X, E1 ) → C ∞ (X, E2 ) by j

Ah¯ =

boundary charts

ϕk χk∗ op chk¯ ψk +

j

j ϕj χj∗ op ph¯ ψj , = Ab,h¯ + Ai,h¯

interior charts

consisting of a boundary part Ab,h¯ and an interior part Ai,h¯ . 6.10. Lemma. Let ω, ω1 be smooth and supported in a single boundary neighborhood. Then ω op+ (ph¯ )ω1 = ωκh¯−1 op op+ xn p x

n =0

(x , h¯ ξ ) κh¯ ω1 .

Proof. This follows from the fact that p is independent of xn on the collar neighborhood and the computation in 3.1. 2


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6.11. Lemma. Ah¯ is an operator family in Boutet de Monvel’s calculus. Its pseudodifferential symbol is of the form ph¯ + h¯ q(h¯ ), where q1/h¯ is bounded in Strm−1 . Over a boundary chart, its singular Green symbol is of the form gh¯ + h¯ r(h¯ ) with r1/h¯ bounded in S m−1 (Rn−1 × Rn−1 ; S (R+ ), S(R+ )). Proof. Apply 2.13, in particular (2.16), in connection with 2.14(b) and Lemma 6.10.

2

We shall say that a family C(h¯ ) of operators given by pseudodifferential operators in the interior and operator-valued symbols close to the boundary is semiclassically bounded of order μ, if, for the family q(h¯ ) of pseudodifferential symbols, q1/h¯ is bounded in S μ , and, for the family d(h¯ ) of operator-valued symbols, d1/h¯ is bounded in S μ (Rn−1 × Rn−1 ; E, F ), where E and F have to be specified. 6.12. Proposition. We have

1 + A∗h¯ Ah¯ =

ϕk χk∗ op 1 + chk∗ c k ψk ¯ h¯

boundary charts

+

j∗ j ϕj χj∗ op 1 + ph¯ ph¯ ψj + h¯ R1 (h¯ )

interior charts

=

boundary charts

+

ϕk χk∗ op 1 + ck∗ ck h¯ ψk

interior charts

ϕj χj∗ op 1 + p j ∗ p j h¯ ψj + h¯ R1 (h¯ ),

where the family R1 is semiclassically bounded of order 2m − 1 with operator-valued symbols acting between E = H m (R+ ) and F = H0−m (R+ ). Here, p j,∗ is the adjoint of the symbol p j in local coordinates, and ck,∗ is the adjoint of the operator-valued symbol ck . Proof. For the first identity apply 2.14(c) using Lemma 6.11. The second is obvious.

2

6.13. Proposition. Let ϕ, ψ ∈ Cc∞ (∂X × (0, 1)) be supported in the intersection of an interior and a boundary neighborhood. Then ∗ −1 ϕ op (1 + c∗ c)−1 h¯ ψ − ϕ op(1 + p p)h¯ ψ

is a semiclassically regularizing pseudodifferential operator; i.e. for each N , we can write it as N bounded in S −2m−N . h¯ N op r N (h¯ ) with r1/ h¯ Proof. We know that (1 + c∗ c)−1 is a boundary symbol in Boutet de Monvel’s calculus, whose pseudodifferential part is given by (1 + p|∗xn =0 p|xn =0 )−1 . As the localization of the h¯ -scaled singular Green part to the interior is semiclassically smoothing by Lemma 3.3, this implies the assertion. 2

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6.14. Proposition. Define

B(h¯ ) =

boundary charts

+

−1 ϕk χk∗ op 1 + ck∗ ck h¯ ψk

interior charts

−1 ϕj χj∗ op 1 + p j ∗ p j h¯ ψj .

(6.4)

This is an operator family in Boutet de Monvel’s calculus which is semiclassically bounded of order −2m, and B(h¯ ) 1 + A∗h¯ Ah¯ = 1 + hR ¯ 2 (h¯ ) and 1 + A∗h¯ Ah¯ B(h¯ ) = 1 + h¯ R3 (h¯ ) with families R2 , R3 which are semiclassically bounded of order −1 with operator-valued symbols acting on H m (R+ ) for R2 and on H0−m (R+ ) for R3 . Proof. This follows from 2.14(c) in connection with Propositions 6.12 and 6.13.

2

6.15. Corollary. We infer from Proposition 6.14 that −1 −1 = B(h¯ ) − h¯ B(h¯ )R3 (h¯ ) + h¯ 2 R2 (h¯ ) 1 + A∗h¯ Ah¯ R3 (h¯ ). 1 + A∗h¯ Ah¯

(6.5)

From this we want to deduce that (1 + A∗h¯ Ah¯ )−1 differs from B(h¯ ) by a term which is O(h¯ ) in L(L2 (X)). So far, this is not obvious: The boundary symbol parts of R2 and R3 act on H m and H0−m , respectively, while we only can guarantee boundedness of the inverse on L2 (X). To this end we make the following observations: 6.16. Lemma. Given N ∈ N we find an operator family C(h¯ ), 0 < h¯ 1, in Boutet de Monvel’s calculus such that C(h¯ )Ah¯ = 1 + SN (h¯ ) with C and SN semiclassically bounded of orders −m and −N , respectively, in Boutet de Monvel’s calculus. Proof. Apply a semiclassical parametrix construction in Boutet de Monvel’s calculus, using Proposition 3.8. 2 6.17. Lemma. The operator families (1 + A∗h¯ Ah¯ )−1 , (1 + Ah¯ A∗h¯ )−1 , Ah¯ (1 + A∗h¯ Ah¯ )−1 A∗h¯ , Ah¯ (1 + A∗h¯ Ah¯ )−1 , and (1 + A∗h¯ Ah¯ )−1 A∗h¯ are (after continuous extension) uniformly bounded on the corresponding L2 spaces.


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Proof. For the first two families the statement is obvious as their operator norm is bounded by 1. For the third we use that, by Lemma 4.13, −1 ∗ −1 Ah¯ 1 + A∗h¯ Ah¯ Ah¯ = 1 − 1 + Ah¯ A∗h¯ . For the fourth we note that on a Hilbert space, the norm of an operator T equals T ∗ T 1/2 . We apply this to T = Ah¯ (1 + A∗h¯ Ah¯ )−1 . By 4.8, −1 ∗ −1 −1 −2 T ∗ T = 1 + A∗h¯ Ah¯ Ah¯ Ah¯ 1 + A∗h¯ Ah¯ = 1 + A∗h¯ Ah¯ − 1 + A∗h¯ Ah¯ , which is bounded. Duality yields the boundedness of the fifth family.

2

6.18. Corollary. (1 + A∗h¯ Ah¯ )−1 − B(h¯ ) = O(h¯ ) in L(L2 (X, E1 )). Proof. In the third term on the right-hand side of (6.5) we can write −1 −1 ∗ −1 ∗ 1 + A∗h¯ Ah¯ = C(h¯ )Ah¯ 1 + A∗h¯ Ah¯ Ah¯ C(h¯ )∗ − SN (h¯ ) 1 + A∗h¯ Ah¯ Ah¯ C(h¯ )∗ −1 − C(h¯ )Ah¯ 1 + A∗h¯ Ah¯ SN (h¯ )∗ (6.6) ∗ R are bounded on L2 , uniformly for some N > m. By Lemma 2.17, R2 C, R2 SN , C ∗ R3 and SN 3 in h¯ . The assertion then follows from (6.5). 2

6.19. Corollary. Ah¯ (1 + A∗h¯ Ah¯ )−1 − Ah¯ B(h¯ ) = O(h¯ ) in L(L2 (X, E1 ), L2 (X, E2 )). Proof. We multiply Eq. (6.5) from the left by Ah¯ . The composition Ah¯ B(h¯ ) furnishes an operator family in Boutet de Monvel’s calculus which is semiclassically bounded of order −m. According to Lemma 2.17, the operator family Ah¯ B(h¯ )R3 (h¯ ) is therefore uniformly bounded on L2 . In the second term on the right-hand side we substitute according to Eq. (6.6). We note that Ah¯ R2 C and Ah¯ R2 SN are semiclassically bounded of order 0, with the operator-valued symbols acting on L2 (R+ ). Hence Ah¯ R2 (h¯ )(1 + A∗h¯ Ah¯ )−1 R3 (h¯ ) is uniformly bounded in L(L2 ). 2 In an analogous way we find 6.20. Corollary. (1 + A∗h¯ Ah¯ )−1 A∗h¯ − B(h¯ )A∗h¯ = O(h¯ ) in L(L2 (X, E2 ), L2 (X, E1 )). 6.21. Remark. We know from Lemma 4.13 that, as operators on L2 (X, E2 ), −1 ∗ −1 Ah¯ = 1 − 1 + Ah¯ A∗h¯ . Ah¯ 1 + A∗h¯ Ah¯ In order to determine the structure of the left-hand side, we construct, similarly as before, an h¯ ) to 1 + Ah¯ A∗ , using the structure of the boundary symbol operator approximate inverse B( h¯ ∗ −1 (1 + cc ) determined in Corollary 5.5. We obtain the following result:

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6.22. Proposition.

h¯ ) = B(

boundary charts

−1 ϕk χk∗ op 1 + ck ck∗ h¯ ψk

+

interior charts

−1 ϕj χj∗ op 1 + p j p j ∗ h¯ ψj

(6.7)

defines a semiclassically bounded family in Boutet de Monvel’s calculus of order −2m, and −1 h¯ ) + h¯ R4 (h¯ ); 1 + Ah¯ A∗h¯ = B( with a family R4 which is uniformly bounded on L2 (X, E1 ). 6.23. Definition. The construction of Ah¯ together with Propositions 6.6 allows us to define a section of the continuous field MN (Cr∗ (T − X)∼ ) by

s(h¯ ) =

G(Ah¯ ), G(p) ⊕ G(c),

0 < h¯ 1, h¯ = 0.

We shall now show that this section is continuous. We will distinguish the cases h¯ > 0 and h¯ = 0. 6.24. Proposition. The section s is continuous on (0, 1]. Proof. It follows from the fact that the symbol topology is stronger than the operator topology that the mappings h¯ → Ah¯ ∈ L H m (X, E1 ), L2 (X, E2 ) and h¯ → Ah¯ ∈ L L2 (X, E1 ), H0−m (X, E2 ) depend continuously on h¯ . As taking adjoints and inversion are continuous, we obtain the assertion. 2 6.25. Theorem. The section s is continuous in h¯ = 0. Proof. Consider the four entries of G(Ah¯ ). By Corollary 6.18, (1 + A∗h¯ Ah¯ )−1 differs from B(h¯ ) by a term which vanishes in h¯ = 0. It is therefore enough to show the continuity of B(h¯ ). By construction, the boundary symbol of B(h¯ ) is given by (1 + c∗ c)−1 h¯ while the interior symbol is . (1 + p ∗ p)−1 h¯ Like in Proposition 2.18 we find smoothing symbols qk converging to (1 + p ∗ p)−1 in the topology of S 0 . As p was assumed constant near ∂X, we may assume the same of the qk . As for the boundary symbol, we know that (1 + c∗ c)−1 is a boundary symbol operator in Boutet de Monvel’s calculus whose pseudodifferential part is r + (1 + p ∗ p)|−1 xn =0 (Dn ).


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Denote by h its singular Green part. According to Proposition 2.18 we find a sequence of symbols hk in S −∞ (Rn−1 × Rn−1 ; S (R+ ), S(R+ )) converging to h in the topology of S 0 (Rn−1 × Rn−1 ; L2 (R+ ), L2 (R+ )). Replacing, for h¯ = 1, in the definition of B(1) the pseudodifferential symbols over interior charts by qk and the boundary symbol operators over boundary charts by r + qk |xn =0 (Dn ) + hk , we obtain a sequence of operators Bk . According to Remark 2.20, a further approximation allows us to assume that Bk are integral operators with smooth compactly supported integral kernels. Similarly we approximate the other entries in G(A) − e. Adding then e again and going over to h-scaled symbols, we infer from Lemma 2.17 that the approximation ¯ is uniform for 0 < h¯ 1. Hence we obtain a sequence of sections sk (h¯ ) of MN (Cr∗ (T X)) which approximates G(Ah¯ ) uniformly. By definition, these sections are continuous for h¯ > 0 and have a continuous extension to h¯ = 0 given by the N × N -matrices of their interior and boundary symbols. As this matrix, on the other hand, tends to G(p) ⊕ G(c), we conclude that s is continuous. 2 7. The fundamental class for manifolds with boundary In this section we will describe how the fundamental class

: Hc∗ (T ∗ X ◦ ) = H P ∗ Cc∞ (T ∗ X ◦ ) → C

T ∗ X◦

extends to a fundamental class ∞ − F : H P ∗ Ctc (T X) → C. Let us first assume T ∗ X = T ∗ ∂X × T ∗ R+ so that we can consider the elements of Cc∞ (T − X) ∞ (T − R ). We write an element of C ∞ (T − R ) as the sum of a ˆ π Ctc as elements of Cc∞ (T ∗ ∂X) ⊗ + + tc pseudodifferential symbol p and a singular Green symbol g on the boundary. We then obtain the operator of multiplication by p on T ∗ R+ and the boundary symbol operator c = p(0, Dn ) + g.

(7.1)

Following Fedosov we define tr (p + g) = tr(g), noting that g is an integral operator on L2 (R+ ) with a rapidly decreasing kernel and thus trace class. The functional tr is not quite a trace, but satisfies the following fundamental property [5, (2.19)] tr [p1 + g1 , p2 + g2 ] = −i

∂p1 (0, ξn ) p2 (0, ξn ) dξn = i ∂ξn

p1 (0, ξn )

∂p2 (0, ξn ) dξn . ∂ξn

∞ (T − X)⊗ Given an element in the cyclic periodic complex, i.e. an element in Ctc introduce the boundary functional

m+1

, we first

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F∂ f0 ⊗ (p0 + g0 ) ⊗ f1 ⊗ (p1 + g1 ) ⊗ · · · ⊗ fm ⊗ (pm + gm ) f0 df1 · · · dfm tr (p0 + g0 ) · · · (pm + gm ) . = T ∗ ∂X ∞ (T − X) into tensor factors. For notational convenience we We have used here the splitting of Ctc will omit the tensor symbols and write fj (pj + gj ) instead of fj ⊗ (pj + gj ). The fundamental class is given by

F f0 (p0 + g0 ) ⊗ f1 (p1 + g1 ) ⊗ · · · ⊗ fm (pm + gm ) f0 p0 d(f1 p1 ) · · · d(fm pm ) = T ∗X

+ iF∂

sgn(σ )fσ (0) (pσ (0) + gσ (0) ) ⊗ · · · ⊗ fσ (m) (pσ (m) + gσ (m) ) .

σ cyclic

7.1. Proposition. The fundamental class F is a cocycle on the periodic cyclic complex. ∞ (T − X)). By Stokes’ theoProof. We need to prove that F ((B + b)a) = 0, where a ∈ CC∗ (Ctc rem the boundary part F∂ of F vanishes on Ba. The remaining ‘nonboundary’ part of F clearly vanishes on ba. Computing the nonboundary part we get per

d(f0 p0 )d(f1 p1 ) · · · d(fm pm ) = T ∗X

f0 p0 d(f1 p1 ) · · · d(fm pm ).

∂(T ∗ X)

We want to compute the boundary part, i.e. F∂ , on the cyclic permuted terms appearing in F . A single cyclic permutation of b(a) without sgn(σ ) is of the form fi+1 (pi+1 + gi+1 ) ⊗ · · · ⊗ f0 (p0 + g0 )f1 (p1 + g1 ) ⊗ · · · ⊗ fi (pi + gi ) − fi+1 (pi+1 + gi+1 ) ⊗ · · · ⊗ f1 (p1 + g1 )f2 (p2 + g2 ) ⊗ · · · ⊗ fi (pi + gi ) .. . + (−1)i−1 fi+1 (pi+1 + gi+1 ) ⊗ · · · ⊗ f0 (p0 + g0 ) ⊗ · · · ⊗ fi−1 (pi−1 + gi−1 )fi (pi + gi ) + (−1)i fi (pi + gi )fi+1 (pi+1 + gi+1 ) ⊗ · · · ⊗ f0 (p0 + g0 ) ⊗ · · · ⊗ fi−1 (pi−1 + gi−1 ) .. . + (−1)m fi (pi + gi ) ⊗ · · · ⊗ fm (pm + gm )f0 (p0 + g0 ) ⊗ · · · ⊗ fi−1 (pi−1 + gi−1 ). We split this expression into the sum of the first i terms and the sum of the subsequent m + 1 − i terms. The action of F∂ on the first i terms is


2687

tr (pi+1 + gi+1 )(pi+2 + gi+2 ) · · · (pm + gm )(p0 + g0 ) · · · (pi + gi ) × fi+1 dfi+2 · · · d(f0 f1 ) · · · dfi − fi+1 dfi+2 · · · d(f1 f2 ) · · · dfi + · · · T ∗ ∂X

+ (−1)i−1 fi+1 dfi+2 · · · d(fi−1 fi ) . The second factor in this expression can be rewritten as

f0 fi+1 dfi+2 · · · dfm df1 · · · dfi + f1 fi+1 dfi+2 · · · df0 df2 · · · dfi

T ∗ ∂X

− (f1 fi+1 dfi+2 · · · df0 df2 · · · dfi + f2 fi+1 dfi+2 · · · df1 df3 · · · dfi )

+ · · · + (−1)i−1 (fi−1 fi+1 dfi+2 · · · dfi−2 dfi + fi fi+1 dfi+2 · · · dfi−1 ) = f0 fi+1 dfi+2 · · · dfm df1 · · · dfi + (−1)i−1 fi fi+1 dfi+2 · · · dfi−1 . T ∗ ∂X

A short computation shows that d(f0 fi fi+1 )dfi+2 · · · dfm df1 · · · dfi−1 = f0 fi dfi+1 · · · dfm df1 · · · dfi−1 + (−1)m f0 fi+1 dfi+2 · · · dfm df1 · · · dfi + (−1)m−(i−1) fi fi+1 dfi+2 · · · dfi−1 . Stokes’ theorem then implies that the term (7.2) equals f0 fi dfi+1 · · · dfm df1 · · · dfi−1

(−1)m+1 T ∗ ∂X

= (−1)

mi+1

f0 fi df1 · · · dfi−1 dfi+1 · · · dfm .

T ∗ ∂X

Putting everything together, the action of F∂ on the first sum is given by (−1)mi+1 tr (pi+1 + gi+1 )(pi+2 + gi+2 ) · · · (pm + gm )(p0 + g0 ) · · · (pi + gi ) × f0 fi df1 · · · dfi−1 dfi+1 · · · dfm . T ∗ ∂X

The action of F∂ on the remaining m + 1 − i terms gives tr (pi + gi )(pi+1 + gi+1 ) · · · (pm + gm )(p0 + g0 ) · · · (pi−1 + gi−1 ) (−1)i fi fi+1 dfi+2 · · · dfm df0 · · · dfi−1 × T ∗ ∂X

(7.2)

2688


+ (−1)i+1 (fi fi+1 dfi+2 · · · dfi−1 + fi fi+2 dfi+1 dfi+3 · · · dfi−1 ) + · · ·

+ (−1)m (fi fm dfi+2 · · · dfm−1 df0 · · · dfi−1 + fi f0 dfi+2 · · · dfm df1 · · · dfi−1 ) = (−1)mi tr (pi + gi )(pi+1 + gi+1 ) · · · (pm + gm )(p0 + g0 ) · · · (pi−1 + gi−1 ) × fi f0 df1 · · · dfi−1 dfi+1 · · · dfm . T ∗ ∂X

All in all we get that the action of F∂ on this symmetrization is (−1)mi tr pi + gi , (pi+1 + gi+1 ) · · · (pm + gm )(p0 + g0 ) · · · (pi−1 + gi−1 ) × fi f0 df1 · · · dfi−1 dfi+1 · · · dfm T ∗ ∂X

= (−1)mi+1 i

p0 (0, ξn ) · · · pi−1 (0, ξn )pi+1 (0, ξn ) · · · pm (0, ξn )

T ∗ ∂X R

∂pi (0, ξn ) dξn ∂ξn

× fi f0 df1 · · · dfi−1 dfi+1 · · · dfm = (−1)(m+1)i i p0 (0, ξn ) · · · pi−1 (0, ξn )pi+1 (0, ξn ) · · · pm (0, ξn )fi f0 df1 · · · dfi−1 ∂(T ∗ X)

∂pi (0, ξn ) dξn dfi+1 · · · dfm ∂ξn = sgn(σ )i p0 (0, ξn ) · · · pi−1 (0, ξn )pi+1 (0, ξn ) · · · pm (0, ξn )fi f0 df1 · · · dfi−1 ×

∂(T ∗ X)

×

∂pi (0, ξn ) dξn dfi+1 · · · dfm , ∂ξn

where σ is the corresponding permutation. Hence m

F b(a) = −

p0 (0, ξn ) · · · pi−1 (0, ξn )pi+1 (0, ξn ) · · · pm (0, ξn )fi f0 df1 · · · dfi−1

i=1 ∂(T ∗ X)

×

∂pi (0, ξn ) dξn dfi+1 · · · dfm , ∂ξn

which is equal to −

f0 p0 d(f1 p1 ) · · · d(fm pm ).

2

∂(T ∗ X)

7.2. The fundamental class in general. For general X the restriction of an element a in ∞ (T − X) to ∂(T ∗ X) can be factorized as a sum of elements of the form f ⊗ (p + g), i.e. Ctc the boundary symbol factorizes. We will adopt this notation, i.e. the boundary part of a will be


2689

denoted f ⊗ (p + g). The symbol part in the interior will be denoted by a, ˜ which is a function on T ∗ X. It is then straightforward to generalize the fundamental class to nonproduct cases: Let ω be a closed differential form on T ∗ X, which is the pull back of a closed differential form on X. We first define F∂,ω by F∂,ω f0 ⊗ (p0 + g0 ) ⊗ f1 ⊗ (p1 + g1 ) ⊗ · · · ⊗ fm ⊗ (pm + gm ) = f0 df1 · · · dfm · ω · tr (p0 + g0 ) · · · (pm + gm ) T ∗ ∂X

and then let F (a0 ⊗ a1 ⊗ · · · ⊗ am ) = a˜ 0 d a˜ 1 · · · d a˜ m · ω + iF∂,ω sgn(σ )a0 ⊗ aσ (1) ⊗ · · · ⊗ aσ (m) . σ cyclic

T ∗X

7.3. Proposition. Fω descends to a map ∞ − Fω : H P Ctc (T X) → C. Proof. The same computation as in Proposition 7.1.

2

8. The index formula Like in [3, Section 2.5] the short exact sequence 0 → C0 (]0, 1]) ⊗ K → Cr∗ (T − X) → Cr∗ (T − X) → 0 induces an analytic index map inda : K0 Cr∗ (T − X) → Z. In this section we will give a formula for this index map. Together with the results in the previous sections this will give an index formula for Fredholm operators in Boutet de Monvel’s calculus of order and class zero. We have another short exact sequence coming from the interior of the manifold, namely 0 → C0 (]0, 1]) ⊗ K → Cr∗ (T X ◦ ) → C0 (T ∗ X ◦ ) → 0 (noting that Cr∗ (T X ◦ ) ∼ = C0 (T ∗ X ◦ )) also inducing an analytic index map inda : K0 C0 (T ∗ X ◦ ) → Z. According to Connes [3, Section 2.5] we have in this case inda = indt ,

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where indt denotes the topological index. On the other hand, [1, Theorem 3.2] shows that we have an isomorphism Φ : K0 (Cr∗ (T − X)) → K0 (C0 (T ∗ X ◦ )), and the diagram K0 (Cr∗ (T − X)) → Z ↓ K0 (C0 (T ∗ X ◦ )) commutes. Let T be the Fredholm operator of order and class zero in Boutet de Monvel’s calculus introduced in 4.10, and let G(a) = G(p) ⊕ G(c) denote the graph projection of a complete symbol of Λm,+ − T . From Theorem 6.25 we obtain 8.1. Theorem. The index of T is given by ind T = inda G(a) − [e] . We define the topological index indt : K0 Cr∗ (T − X) → Z as the composition indt ◦Φ. We thus get an index theorem 8.2. Theorem. inda = indt . With this notation we can now prove 8.3. Theorem. ind T = FT d(X) ch G(a) − [e] , where ch denotes the Chern–Connes character. Proof. Let C n (T − X) denote the subalgebra of Cr∗ (T − X) consisting of symbols of order strictly less than −n, n being the dimension of X. We note that C n (T − X)∼ is closed under holomorphic functional calculus and hence K0 (C n (T − X)) = K0 (Cr∗ (T − X)). Also note that Fω is defined on H P (C n (T − X)). Using the cohomological form of the topological index we get the following commutative diagrams: K0 (C n (T ∗ X ◦ ))

K0 (C n (T − X))

ch

ch

H Pev (C n (T ∗ X ◦ ))

H Pev (C n (T − X))·

Hc∗ (T ∗ X ◦ )

FT d(X) T d(X)·

Z


2691

and K0 (C n (T ∗ X ◦ ))

K0 (C0 (T ∗ X ◦ ))

K0 (C0 (T ∗ X ◦ )) indt

inda

ch

H Pev (C n (T ∗ X ◦ ))

Hc∗ (T ∗ X ◦ )

Z

T d(X)·

and K0 (Cr∗ (T − X))

K0 (C0 (T ∗ X ◦ ))

inda

inda

K0 (C n (T − X))

inda

Z from which follows that FT d(X) ◦ ch = inda on K0 (C n (T − X)).

2

Acknowledgment J. Aastrup and E. Schrohe gratefully acknowledge the support of Deutsche Forschungsgemeinschaft (DFG). References [1] J. Aastrup, R. Nest, E. Schrohe, A continuous field of C ∗ -algebras and the tangent groupoid for manifolds with boundary, J. Funct. Anal. 237 (2006) 482–506. [2] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971) 11–51. [3] A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994. [4] G. Elliott, T. Natsume, R. Nest, The Atiyah–Singer index theorem as passage to the classical limit in quantum mechanics, Comm. Math. Phys. 182 (1996) 505–533. [5] B.V. Fedosov, Index Theorems, in: Partial Differential Equations VIII, in: Encyclopaedia Math. Sci., vol. 65, Springer, Berlin, 1996, pp. 155–251; translation from Itogi Nauki Tekh., Sovrem. Mat. Fundam. Napravl. 65 (1991) 165–268. [6] G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, second ed., Birkhäuser, Boston, 1996. [7] H. Kumano-go, Pseudo-Differential Operators, MIT Press, Cambridge/London, 1981. [8] S. Melo, R. Nest, E. Schrohe, C ∗ -structure and K-theory of Boutet de Monvel’s algebra, J. Reine Angew. Math. 561 (2003) 145–175. [9] S. Melo, T. Schick, E. Schrohe, A K-theoretic proof of Boutet de Monvel’s index theorem, J. Reine Angew. Math. 599 (2006) 217–233. [10] S. Rempel, B.-W. Schulze, Index Theory of Elliptic Boundary Problems, Akademie-Verlag, Berlin, 1982. [11] E. Schrohe, Boundedness and spectral invariance for standard pseudodifferential operators on anisotropically weighted Lp Sobolev spaces, Integral Equations Operator Theory 13 (1990) 271–284. [12] E. Schrohe, Fréchet algebra techniques for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance, Math. Nachr. 199 (1999) 145–185.

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[13] E. Schrohe, A short introduction to Boutet de Monvel’s calculus, in: J. Gil, D. Grieser, M. Lesch (Eds.), Approaches to Singular Analysis, in: Oper. Theory Adv. Appl., vol. 125, 2001, pp. 85–116. [14] E. Schrohe, B.-W. Schulze, Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities I, in: Pseudo-Differential Calculus and Mathematical Physics, in: Math. Top., vol. 5, Akademie-Verlag, Berlin, 1994, pp. 97–209.


Regularity results for stable-like operators ✩ Richard F. Bass Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA Received 4 December 2008; accepted 6 May 2009 Available online 21 May 2009

Abstract For α ∈ [1, 2) we consider operators of the form Lf (x) = Rd

A(x, h) f (x + h) − f (x) − 1(|h|1) ∇f (x) · h dh |h|d+α

and for α ∈ (0, 1) we consider the same operator but where the ∇f term is omitted. We prove, under appropriate conditions on A(x, h), that any solution u to Lu = f will be in C α+β if f ∈ C β . © 2009 Elsevier Inc. All rights reserved. Keywords: Semigroups; Holder; Stable-like; Non-local

1. Introduction Many models in mathematical physics, financial mathematics, and mathematical economics are based on diffusions corresponding to second-order elliptic differential operators. In the last decade or so, though, researchers in these areas have found that frequently real world phenomena are better fitted if one allows jumps. To give a very simple example, an outbreak of war or a new discovery may cause the price of a stock to make a sudden jump. Since the operators corresponding to jump processes are non-local, one would like to consider operators that are the sum of an elliptic operator and a non-local term. Such operators are not yet well understood. In order to study them and the influence of the non-local part, it is quite natural to first look at the extreme case, that is, where the operator has ✩

Research partially supported by NSF grant DMS-0601783. E-mail address: [email protected].


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R.F. Bass / Journal of Functional Analysis 257 (2009) 2693–2722

no differential part, and to begin by understanding the potential theory, existence and uniqueness questions, and stochastic differential equations for non-local operators and the associated pure jump processes. The first such purely non-local operator one would want to study is the fractional Laplacian −(−Δ)α/2 , where Δ is the Laplacian and α ∈ (0, 2). Such operators have been much studied; the stochastic processes associated to these operators are known as symmetric stable processes. See [11,9,10] for a sampling of research on these processes and operators. The next simplest class of operators L is a class introduced in [6], known as stable-like operators. These are operators L defined by A(x, h) f (x + h) − f (x) − 1(|h|1) ∇f (x) · h dh (1.1) Lf (x) = |h|d+α Rd \{0}

for f ∈ C 2 (Rd ) when α ∈ [1, 2) and A(x, h) f (x + h) − f (x) dh |h|d+α

(1.2)

Rd \{0}

when α ∈ (0, 1). We use x · y for the inner product in Rd . These stable-like operators bear the same relationship to the fractional Laplacian as elliptic operators in non-divergence form do to the usual Laplacian. The name stable-like (which was introduced in [2] and also used in [12]) refers to the fact that the jump intensity measure A(x, h)/|h|d+α dh is comparable to that of the jump intensity measure of a symmetric stable process. See [6,8,22,23] for some additional results on these operators. See [1,3–5,7,12,13,16,17,20,21,26] for results on operators that are very closely related to (1.1) and (1.2) and which are also sometimes known as stable-like operators. There are some papers concerning regularity for operators with both local and non-local parts; see, e.g., [14,16,17]. Two of the first questions one might ask about stable-like operators given by (1.1) and (1.2) are the Hölder continuity of harmonic functions and whether a Harnack inequality holds for non-negative functions that are harmonic with respect to L when the function A(x, h) only satisfies some boundedness and measurability conditions. These questions were answered in [6]; see also [20] and [23]. A natural question one might then ask is whether one can assert additional smoothness for the solution u to the equation Lu = f if A(x, h) and f also satisfy some continuity conditions. The answer to this last question is the subject of this paper. Let α ∈ (0, 2). We impose the following conditions on A(x, h). Assumption 1.1. Suppose 1. There exist positive finite constants c1 , c2 such that c1 A(x, h) c2 ,

x, h ∈ Rd .

2. There exist β ∈ (0, 1) and a positive constant c3 such that sup supA(x + k, h) − A(x, h) c3 |k|β , x

h

3. Neither β nor α + β is an integer.

k ∈ Rd .


2695

The assumption that A(x, h) is uniformly bounded above and below is the analog of strict ellipticity for an elliptic operator in non-divergence form. The uniform Hölder continuity of A(x, h) in x is the analog of the usual assumptions of Hölder continuity in the Schauder theory; see [18, Chapter 6]. Note that no continuity in h is required here. Finally, the requirement that neither β nor α + β be an integer is quite reasonable; in the theory of elliptic operators, most estimates break down when the coefficients are not in a Hölder space of non-integer order. Our main result is the following. We let C β and C α+β be the usual Hölder spaces. (We recall the definition in (2.3).) Theorem 1.2. Let L be given by (1.1) or (1.2) and suppose Assumptions 1.1 hold. If u ∈ C α+β (Rd ) satisfies Lu = f , then the following a priori estimate holds: there exists c1 not depending on f such that uC α+β c1 uL∞ + c1 f C β .

(1.3)

This is the exact analog of the corresponding estimate for elliptic operators; see [18, Chapter 6]. Lim [22] has obtained some partial results along the lines of Theorem 1.2. Our result here extends his results by weakening the hypotheses and strengthening the conclusions. We show in Section 7 that our result is sharp in several respects. Two additional motivations for Theorem 1.2 are the following. In [6] harmonic functions for L were discussed. There a probabilistic definition of harmonic functions was given because in general a harmonic function, although Hölder continuous, will not be smooth enough to be in the domain of L. This is not surprising, because for elliptic operators this is also the case. Theorem 1.2 gives a sufficient condition for the harmonic function to be in the domain of L; see Remark 5.3. Secondly, when one considers the process associated with L, an essential tool is, as might be expected, Ito’s formula. However the hypotheses of Ito’s formula require the function to be C 2 . Therefore it would be useful to have conditions under which a class of functions associated with the process are at least C 2 . Our proof follows roughly along the lines of the Schauder theory for elliptic equation as presented in [18, Chapter 6]. There are some major differences, however. The estimates for the case when A(x, h) is constant in x are much more difficult than the corresponding estimates for the Laplacian. In addition, because we are dealing with non-local operators, our localization procedure is necessarily quite different. Our results are in the form of an a priori estimate. For the existence of solutions to the related integro-differential equation, see Proposition 7.4. In Section 2 we define the Hölder spaces and prove a few estimates that we will need. Section 3 investigates the derivatives of the semigroup corresponding to the operator L in the case when A(x, h) does not depend on x, while Section 4 is concerned with the smoothing properties of the corresponding potential operator. In Section 5 we obtain estimates on the integrands in (1.1) and (1.2), and we prove Theorem 1.2 in Section 6. We prove a number of results related to Theorem 1.2 in Section 7. For example we examine what happens when we add to L a zero-order term or a first-order differential term and what happens when A(x, h) has further smoothness in x. We also discuss there a number of directions for further research, including the Dirichlet problem for bounded domains, boundary estimates for bounded domains, the parabolic case, the symmetric jump process case, and the case of variable order operators.

2696


The letter c with subscripts denotes a finite positive constant whose value may vary from place to place. 2. Hölder spaces Let β ∈ (0, 1). We define the seminorm |f (x + h) − f (x)| |h|β x∈Rd |h|>0

[f ]C β = sup sup

(2.1)

and the norm f C β = f L∞ + [f ]C β ,

(2.2)

and say f is Hölder continuous of order β if f C β < ∞. We write Di f for ∂f /∂xi , Dij f for ∂ 2 f/∂xi ∂xj , and so on. Suppose β > 1 is not an integer and let m be the largest integer strictly less than β. We define f C β = f L∞ +

d

[Dj1 ···jm f ]C β

(2.3)

j1 ,...,jm =1

and say f ∈ C β if f C β < ∞. It is well known (see the proof of Proposition 2.2 below, for example) that this norm is equivalent to the norm f L∞ +

d

Dj1 f L∞ +

j1 =1

+

d

d

Dj1 j2 f L∞ + · · · +

j1 ,j2 =1

d

Dj1 ···jm f L∞

j1 ,...,jm =1

[Dj1 ···jm f ]C β .

(2.4)

j1 ,...,jm =1

(When we say two norms · 1 and · 2 are equivalent, we mean that there exist constants c1 , c2 such that c1 f 1 f 2 c2 f 1 for all f .) We also use the fact that the C β norm is equivalent to a second difference norm: by [24, Chapter V, Proposition 8], we have Proposition 2.1. For β ∈ (0, 1) ∪ (1, 2), f ∈ C β if and only if f ∈ L∞ and there exists c1 such that f (x + h) + f (x − h) − 2f (x) c1 |h|β ,

h, x ∈ Rd .


2697

The norm |f (x + h) + f (x − h) − 2f (x)| |h|β |h|>0

f L∞ + sup sup x

(2.5)

is equivalent to the C β norm. We will sometimes use the notation Df L∞ =

d i=1

Di f L∞ ,

D 2 f L∞ =

d

Dij f L∞ .

i,j =1

In order to be able to include the case of integer β in the next two results, we introduce the following notation. If a is not an integer, set N (f, a) = f C a ; if a = 1, set N (f, a) = f L∞ + Df L∞ ; and if a = 2, set N (f, a) = f L∞ + Df L∞ + D 2 f L∞ . The following proposition is similar to known results. Proposition 2.2. If 0 < a 0, there exists c1 depending only on a, b, and ε such that N (f, a) c1 f L∞ + εN (f, b).

(2.6)

Proof. We first do the case when 0 < a < b 1. Let h0 = ε 1/(b−a) . If |h| < h0 , then f (x + h) − f (x) N (f, b)|h|b < N (f, b)|h|a ε. If |h| h0 , then f (x + h) − f (x) 2 f L∞ |h|a . ha0 Combining, we have |f (x + h) − f (x)| εN (f, b) + c2 f L∞ . |h|a |h|>0 sup

Taking the supremum over x, (2.6) follows immediately. Second, we do the case a = 1 and b ∈ (1, 2]. Fix 1 i d and let x0 be a point in Rd . The case when N(f, b) = 0 is trivial, so we suppose not. Let R = (f L∞ /N (f, b))1/b . By the mean value theorem, there exists x on the line segment between x0 and x0 + Rei such that Di f (x ) = |f (x0 + Rei ) − f (x0 )| 2 f L∞ . R R Then Di f (x0 ) Di f (x ) + Di f (x ) − Di f (x0 ) 2f L∞ + N (f, b)R b−1 . R

2698


With our choice of R, Di f (x0 ) c3 f 1−1/b N (f, b)1/b . ∞

(2.7)

L

Taking the supremum over x0 ∈ Rd and then applying the inequality x θ y 1−θ x + y,

x, y > 0, θ ∈ (0, 1),

(2.8)

we obtain Di f L∞

c4 f L∞ + εN (f, b). ε

Third, suppose a = 2 and b ∈ (2, 3). Applying (2.7) with f replaced by Dj f and b replaced by b − 1 and setting γ = 1/(b − 1), we have 1−γ

γ

Dij f L∞ c3 Dj f L∞ Dj f C b−1 . Using the well-known inequality g L∞ c5 gL∞ g L∞ (this is a special case of (2.7)) and summing over i and j , we have 1/2

2 D f

L∞

1/2

(1−γ )/2 γ D 2 f L∞ f C b ,

(1−γ )/2

c6 f L∞

and therefore 2 D f

L∞

(1−γ )/(1+γ )

c7 f L∞

2γ /(1+γ )

f C b

.

Applying (2.8) with θ = (1 − γ )/(1 + γ ), we obtain (2.6). For the case a ∈ (0, 1] and b ∈ (1, 2], using the first and second cases above we have N(f, a) c8 f L∞ + c8 Df L∞ c8 f L∞ + c9 f L∞ + εN (f, b), and the remaining cases are treated similarly.

2

Lemma 2.3. If a ∈ (0, 3), there exists c1 such that N (f g, a) c1 N (f, a)N (g, a). Proof. Clearly f gL∞ f L∞ gL∞ . If a ∈ (0, 1), we write f (x + h)g(x + h) − f (x)g(x) = f (x + h) g(x + h) − g(x) + g(x) f (x + h) − f (x) , and it follows that [f g]C a f L∞ gC a + gL∞ gC b .


2699

If a ∈ (1, 2), we use Di (f g) = f (Di g) + (Di f )g. As in the above paragraph, we bound (Di f )g C a−1 Di f L∞ gC a−1 + gL∞ Di f C a−1 c2 f C a gC a , and we bound [f (Di g)]C a similarly. Doing this for each i takes care of the case a ∈ (1, 2). Similarly, if a ∈ (2, 3), we use Dij (f g) = f (Dij g) + g(Dij f ) + (Di f )(Dj g) + (Dj f )(Di g).

(2.9)

As in the first paragraph, (Di f )(Dj g) C a−2 c3 Di f C a−2 Dj gC a−2 c4 f C a gC a . The other terms in (2.9) are similar. The remaining cases, when a = 1 and a = 2, are easy and are left to the reader.

2

We will need the following lemma. Lemma 2.4. Let β ∈ (0, 1). Let ϕ be a non-negative C ∞ symmetric function with compact support such that ϕ(x) dx = 1, and let ϕε (x) = ε −d ϕ(x/ε). Define fε = f ∗ ϕε . Then there exists c1 such that for each i and j f − fε L∞ c1 f C β ε β ,

(2.10)

Di fε L∞ c1 f C β ε β−1 , Dij fε L∞ c1 f C β ε

β−2

and

.

Proof. The first inequality follows from f (x) − fε (x) = f (x) − f (x − y) ϕε (y) dy f C β |y|β ϕε (y) dy = c2 f C β ε β . Since

Di ϕε (y) dy = 0, Di fε (x) = f (x − y) − f (x) Di ϕε (y) dy f C β |y|β Di ϕε (y) dy = c3 f C β ε β−1 .

(2.11) (2.12)

2700


Similarly, since

Dij ϕε (y) dy = 0, then Dij fε (x) = f (x − y) − f (x) Dij ϕε (y) dy f C β |y|β Dij ϕε (y) dy = c4 f C β ε β−2 .

2

3. Derivatives of semigroups Let Qt be the semigroup of a symmetric stable process of order α and let q(t, x) be the density, that is, the function such that Qt f (x) = f (y)q(t, x − y) dy. It is well known that q can be taken to be C ∞ in x. Proposition 3.1. For each k > 0 and each j1 , . . . , jk = 1, . . . , d, we have

D j

1 ···jk

q(1, x) dx < ∞.

This can be proved by generalizing the ideas of [21, Proposition 2.6], which considers the case of first derivatives. See also [26]. It can also be proved using Fourier transforms and complex analytic techniques; see [25], for example. We give a simple proof based on subordination. Proof. Let Wt be a d-dimensional Brownian motion and let Tt be a one-dimensional one-sided stable process of index α/2 independent of W . Then it is well known, by the principle of subordination [15, Section X.7], that Xt = WTt is a symmetric stable process of index α. Hence ∞ r(t, x) P(T1 ∈ dt),

q(1, x) =

(3.1)

0

where r(t, x) = (2πt)−d/2 e−|x| /2t is the density of Wt . The number of jumps of Tt of size larger than λ is a Poisson process with parameter c1 λ−α/2 . So the probability that Tt has no jumps of size λ or larger by time 1 is bounded by exp(−c1 λ−α/2 ). Because Tt is non-decreasing, this implies 2

P(T1 λ) exp −c1 λ−α/2 . Hence for any N > 0, ∞ 1

−N 1+t P(T1 ∈ dt) 2 + t −N P(T1 ∈ dt) 0

0

2+

∞ n=0

2N (n+1) P T1 ∈ 2−n−1 , 2−n


2+

∞

2701

2N (n+1) P T1 2−n

n=0

2+

∞

2N (n+1) e−c1

−n α/2 2

< ∞.

(3.2)

n=0

It is easy to see that for each a > 0 there exist b and c2 depending on a such that

sup 1 + |x|a r(t, x) c2 1 + t −b ,

t > 0.

x

This and (3.2) allow us to use dominated convergence to differentiate under the integral sign in (3.1), and we obtain ∞ Dj1 ···jk q(1, x) =

Dj1 ···jk r(t, x) P(T1 ∈ dt). 0

Then, using (3.2) again and Fubini,

D j

1 ···jk q(1, x) dx

∞

D j

1 ···jk

r(t, x) dx P(T1 ∈ dt)

0

∞ c3

t −k/2 P(T1 ∈ dt) < ∞.

2

0

If f ∈ L∞ , it follows easily that Q1 f is C ∞ for t > 0 and for each j1 , . . . , jk Dj ···j Q1 f (x) c1 f L∞ . k 1 By scaling we have D j

1 ···jk

Qt f (x) ct −k/α f L∞ .

(3.3)

Now we consider Lévy processes whose Lévy measure is comparable to that of a symmetric stable process of index α. Suppose A0 : Rd \ {0} → [κ1 , κ2 ], where κ1 , κ2 are finite positive constants. Define L0 f (x) = Rd \{0}

A0 (h) f (x + h) − f (x) − 1(|h|1) ∇f (x) · h dh |h|d+α

(3.4)

for C 2 functions f when α 1, and without the ∇f (x) term when α < 1. Let Pt be the semigroup corresponding to the generator L0 .

2702


Theorem 3.2. If f ∈ L∞ , then Pt f is C ∞ for t > 0 and for each j1 , . . . , jk = 1, . . . , d, there exists c1 (depending on k) such that Dj ···j Pt f (x) c1 t −k/α f L∞ . k 1 Proof. Let L1 be defined by (3.4) but with A0 (h) replaced by κ1 and let L2 = L0 − L1 . Let Q1t and Q2t be the semigroups for the Lévy processes with generators L1 , L2 , resp., and let X 1 , X 2 be the corresponding Lévy processes. If we take X 1 independent of X 2 , then X 1 + X 2 has the law of the Lévy process corresponding to the generator L. Therefore Pt = Q2t Q1t . We know that Q1t f satisfies the desired estimate by (3.3) and the fact that the process associated with L1 is a deterministic time change of the process considered in Proposition 3.1. By translation invariance, Q2t commutes with differentiation. Therefore Pt f = Q2t Q1t f also satisfies the desired estimate, since Dj1 ···jk Pt f L∞ = Q2t Dj1 ···jk Q1t f L∞ Dj1 ···jk Q1t f L∞ c1 t −k/α f L∞ .

2

4. Potentials and Hölder continuity Let Pt continue to be the semigroup corresponding to the Lévy process in Rd with infinitesimal generator L0 given by (3.4) and define the potential ∞ Rf (x) =

Pt f (x) dt 0

when the function t → Pt f (x) is integrable. We want to prove that R takes functions in C β into functions in C α+β , provided neither β nor α + β is an integer and that Rf is bounded. Proposition 4.1. Suppose β ∈ (0, 1), f ∈ C β , Rf ∈ L∞ , and α + β < 1. Then Rf ∈ C α+β and there exists c1 not depending on f such that Rf C α+β c1 f C β + c1 Rf L∞ . Proof. We first prove that Ps f (x) − Ps f (y)

c2 |y (1−β)/α s

− x|f C β .

(4.1)

Define fε as in Lemma 2.4. We have, using Theorem 3.2 and (2.10), Ps (f − fε )(y) − Ps (f − fε )(x) ∇Ps (f − fε ) ∞ |y − x| L c3

f − fε L∞ |y − x| s 1/α c3 1/α ε β f C β |y − x|. s

(4.2)


2703

Also, using (2.11), Ps fε (y) − Ps fε (x) c4 ∇Ps fε L∞ |y − x| c4 ∇fε L∞ |y − x| c5 ε β−1 f C β |y − x|.

(4.3)

Setting ε = s 1/α and combining (4.2) and (4.3) yields (4.1). If x, y ∈ Rd and we define g(z) = f (y − x + z), then by the translation invariance of Ps (that is, Ps commutes with translation), Ps g(x) = Ps f (y), and then Ps f (y) − Ps f (x) = Ps (g − f )(x) g − f L∞ f C β |y − x|β . So putting t0 = |y − x|α , we have t0

Ps f (y) − Ps f (x) ds t0 f C β |y − x|β = f C β |y − x|α+β .

(4.4)

0

Using (4.1) and noting (1 − β)/α > 1, ∞ ∞ Ps f (y) − Ps f (x) ds t0

c6 f C β |y (1−β)/α s

− x| ds

t0 1−(1−β)/α

= c 7 t0

f C β |y − x|

= c7 f C β |y − x|α+β . Combining this with (4.4) and the fact that Rf (y) − Rf (x)

t0 0

our result follows.

Ps f (y) − Ps f (x) ds +

∞ Ps f (y) − Ps f (x) ds,

(4.5)

t0

2

Next we consider the case when 0 < β < 1 and 1 < α + β < 2. Proposition 4.2. Suppose β ∈ (0, 1), f ∈ C β , Rf L∞ < ∞, and α + β ∈ (1, 2). Then Rf ∈ C α+β and there exists c1 not depending on f such that Rf C α+β c1 f C β + c1 Rf L∞ . Proof. Define Vhs (f )(x) = Ps f (x + h) + Ps f (x − h) − 2Ps f (x).

2704


First we show Vhs (f )(x) c2 |h|2 f C β s −(2−β)/α .

(4.6)

By Theorem 3.2, (2.10), and Taylor’s theorem, Vhs (f − fε )(x) c3 |h|2 D 2 Ps (f − fε ) ∞ L c4 2 2/α |h| f − fε L∞ s c5 2/α |h|2 ε β f C β . s

(4.7)

If we set g1ε (z) = fε (z + h) and g2ε (z) = fε (z − h), by the translation invariance of Ps , Vhs (fε )(x) = Ps g1ε (x) + Ps g2ε (x) − 2Ps fε (x), and therefore by (2.12) Vhs (fε )(x) = Ps (g1ε + g2ε − 2fε )(x) g1ε + g2ε − 2fε L∞ c6 |h|2 D 2 fε L∞ c7 |h|2 ε β−2 f C β .

(4.8)

Letting ε = s 1/α and combining with (4.7), we obtain (4.6). Using (4.6) and noting (2 − β)/α > 1, ∞

Vhs (f )(x) c8 |h|2 f C β

|h|α

∞

s −(2−β)/α ds = c9 f C β |h|α+β .

(4.9)

|h|α

Let g10 (x) = f (x + h), g20 (x) = f (x − h). By translation invariance and the Hölder continuity of f , Vhs (f )(x) = Ps (g10 + g20 − 2f )(x) g10 + g20 − 2f L∞ 2f C β |h|β , and thus |h|

α

Vhs (f )(x) ds 2f C β |h|α+β .

(4.10)

0

Adding (4.9) and (4.10) we conclude Rf (x + h) + Rf (x − h) − 2Rf (x) cf C β |h|α+β . This with Proposition 2.1 completes the proof.

2

Finally we consider the case when α + β ∈ (2, 3). Proposition 4.3. Suppose β ∈ (0, 1), f ∈ C β , Rf L∞ < ∞, and α + β ∈ (2, 3). Then Rf ∈ C α+β and there exists c1 not depending on f such that Rf C α+β c1 f C β + c1 Rf L∞ .


2705

Proof. Necessarily α > 1. In view of Proposition 2.2 it suffices to show Di Rf C α+β−1 c2 f C β ,

i = 1, . . . , d.

(4.11)

Fix i and let Qt = Di Pt . From Theorem 3.2 we have Dj1 j2 Qt f L∞ c3 t −3/α f L∞ ,

j1 , j2 = 1, . . . , d.

Define Whs (f )(x) = Qs f (x + h) + Qs f (x − h) − 2Qs f (x). Note that Qs is translation invariant. Analogously to (4.7) and (4.8), Whs (f − fε )(x) c4 |h|2 D 2 Qs (f − fε ) ∞ L

c5 |h|2 f − fε L∞ s 3/α

c6 |h|2 β ε f C β s 3/α

and Whs (fε )(x) c7 |h|2 D 2 Qs fε

L∞

= c7 |h|2 Qs D 2 fε L∞

c8 |h|2 D 2 fε ∞ c9 |h|2 ε β−2 f C β . L 1/α s s 1/α

Taking ε = s 1/α we obtain Whs (f )(x) c10 |h|2 s (β−3)/α f C β . Integrating this bound over [|h|α , ∞) yields c11 |h|α+β−1 f C β . On the other hand, if g10 and g20 are defined as in the proof of Proposition 4.2, Whs (f )(x) Qs (g10 + g20 − 2f )

L∞

c12 s −1/α g10 + g20 − 2f L∞

c13 s −1/α |h|β f C β , and integrating this bound over s from 0 to |h|α yields c14 |h|α+β−1 f C β ; we use the fact that 1/α < 1 here. Therefore Whs (Di Rf )(x) c14 |h|α+β−1 f C β , which with Proposition 2.1 yields (4.11).

2

We reformulate and summarize the preceding propositions in the following theorem. Let L0 be defined as in (3.4).

2706


Theorem 4.4. Suppose β ∈ (0, 1) and α + β ∈ (0, 1) ∪ (1, 2) ∪ (2, 3). There exists c1 such that if u is in the domain of L0 and L0 u = f with uL∞ < ∞, then uC α+β c1 f C a + c1 uL∞ .

(4.12)

Proof. If L0 u = f and uL∞ < ∞, then we have u = −Rf , and so the result follows by Propositions 4.1, 4.2, and 4.3. 2 5. First and second differences For f bounded define Eh f (x) = f (x + h) − f (x).

(5.1)

Fh f (x) = f (x + h) − f (x) − ∇f (x) · h.

(5.2)

For f ∈ C 1 define

Observe that if g : R → R is in C γ with γ ∈ (1, 2), then t g(t) − g(0) − g (0)t = g (s) − g (0) ds 0

t gC γ

s γ −1 ds c1 gC γ t γ ,

(5.3)

0

while if γ ∈ (2, 3), then t 1 1 2 g(t) − g(0) − g (0)t − g (0)t 2 = g (s) − g (0) ds − (0)t g 2 2 0

t s = g (r) − g (0) dr ds 0 0

t s gC γ

r γ −2 dr ds = c2 gC γ t γ .

0 0

Let Hf be the Hessian of f , so that h · Hf (x)k =

d i,j =1

if h = (h1 , . . . , hd ) and k = (k1 , . . . , kd ).

hi Dij f (x)kj

(5.4)


2707

Theorem 5.1. Suppose f ∈ C γ for γ ∈ (0, 1) ∪ (1, 2) ∪ (2, 3). There exists c1 not depending on f such that the following estimates hold. (a) For all γ ,

Eh f (x) c1 |h|γ ∧1 ∧ 1 f C γ

(5.5)

Fh f (x) c1 |h|γ ∧2 ∧ 1 f C γ .

(5.6)

Eh f (x + k) − Eh f (x) c1 |h|γ ∧1 ∧ |k|γ ∧1 f C γ .

(5.7)

Eh f (x + k) − Eh f (x) c1 |h|γ −1 |k| ∧ |h||k|γ −1 f C γ .

(5.8)

and if γ > 1,

(b) For all γ ,

(c) If γ ∈ (1, 2), then

(d) If γ ∈ (1, 2), then

Fh f (x + k) − Fh f (x) c1 |h|γ ∧ |h||k|γ −1 f C γ .

(5.9)

(e) If γ ∈ (2, 3), then

Fh f (x + k) − Fh f (x) c1 |k|γ −2 |h|2 ∧ |h|γ −1 |k| f C γ .

(5.10)

Proof. (a) The estimate for Eh f follows by the definition of C γ . The one for Fh f follows from (5.3) or (5.4) applied to g(s) = f (x + sh/|h|) with t = |h|. (b) Write Eh f (x + k) − Eh f (x) = f (x + h + k) − f (x + k) − f (x + h) − f (x) ,

(5.11)

and note that because f ∈ C γ , this is bounded by 2|h|γ ∧1 f C γ . We can also write Eh f (x +k)− Eh f (x) as f (x + h + k) − f (x + h) − f (x + k) − f (x) , so we also get the bound 2|k|γ ∧1 f C γ . (c) Using (5.3) f (x + h + k) − f (x + k) = ∇f (x + k) · h + R1 and f (x + h) − f (x) = ∇f (x) · h + R2 ,

(5.12)

2708


where R1 and R2 are both bounded by c2 f C γ |h|γ . By (5.11) Eh f (x + k) − Eh f (x) = ∇f (x + k) − ∇f (x) · h + R1 − R2 , and the right-hand side is bounded by

c3 f C γ |k|γ −1 |h| + |h|γ .

(5.13)

Starting with (5.12) instead of (5.11) we also get the bound

c3 f C γ |h|γ −1 |k| + |k|γ .

(5.14)

Using (5.13) when |h| |k| and (5.14) when |h| > |k| proves (5.8). (d) By (5.4) Fh f (x) c3 f C γ |h|γ , and the same bound holds for Fh f (x + k), so Fh f (x + k) − Fh f (x) c3 f C γ |h|γ .

(5.15)

On the other hand f (x + k + h) − f (x + h) = ∇f (x + h) · k + R3 and f (x + k) − f (x) = ∇f (x) · k + R4 , where R3 and R4 are both bounded by c4 f C γ |k|γ . Also ∇f (x + k) · h − ∇f (x) · h c5 f C γ |h||k|γ −1 and ∇f (x + h) · k − ∇f (x) · k c5 f C γ |k||h|γ −1 . Combining and using the fact that γ < 2,

Fh f (x + k) − Fh f (x) c6 f C γ |k|γ + |h||k|γ −1 + |k||h|γ −1 , which together with (5.15) proves (5.9). (e) Applying (5.4) Fh f (x) − 1 h · Hf (x)h c7 f C γ |h|γ 2

(5.16)


2709

and we obtain the same bound for |Fh f (x + k) − 12 h · Hf (x + k)h|. Since

h · Hf (x + k) − Hf (x) h c8 f C γ |h|2 |k|γ −2 , then

Fh f (x + k) − Fh f (x) c9 f C γ |h|2 |k|γ −2 + |h|γ .

(5.17)

On the other hand, using (5.3) and (5.4), 1 f (x + k + h) − f (x + k) = ∇f (x + h) · k + k · Hf (x + h)k + R5 , 2 1 f (x + k) − f (x) = ∇f (x) · k + k · Hf (x)k + R6 , 2 ∇f (x + k) · h − ∇f (x) · h = k · Hf (x)h + R7 , and ∇f (x + h) · k − ∇f (x) · k = h · Hf (x)k + R8 , where R5 and R6 are both bounded by c10 f C γ |k|γ , R7 is bounded by c10 f C γ |k|γ −1 |h|, and R8 is bounded by c10 f C γ |h|γ −1 |k|. Therefore

Fh f (x + k) − Fh f (x) − 1 k · Hf (x + h) − Hf (x) k |R5 | + |R6 | + |R7 | + |R8 |, 2 which implies

Fh f (x + k) − Fh f (x) c11 f C γ |k|γ + |k|γ −1 |h| + |h|γ −1 |k| + |k|2 |h|γ −2 . Using (5.17) if |h| |k| and (5.18) if |h| > |k| proves (5.10).

(5.18)

2

We have the following corollary. Corollary 5.2. Suppose f ∈ C α+β for some β ∈ (0, 1) and α + β ∈ (0, 1) ∪ (1, 2) ∪ (2, 3). There exists c1 not depending on f such that: (a) If α < 1, then

Eh f (x + k) − Eh f (x)

(b) If α ∈ [1, 2), then Fh f (x + k) − Fh f (x)

dh c1 |k|β f C α+β . |h|d+α

dh + |h|d+α

|h|>1

|h|1

c1 |k| f C α+β . β

Eh f (x + k) − Eh f (x)

(5.19)

dh |h|d+α (5.20)

2710


Proof. If |k| > 1, the left-hand side of (5.19) is less than or equal to

Eh f (x + k)

dh + |h|d+α

Eh f (x)

dh , |h|d+α

which is bounded using Theorem 5.1(a). We treat (5.20) similarly. If |k| 1, we use the bounds in Theorem 5.1(b)–(e), breaking the integrals into three: where |h| < |k|, where |k| |h| 1, and where |h| > 1. The rest is elementary calculus. 2 Remark 5.3. By Theorem 5.1(a), the integrals defining Lu are thus absolutely convergent if u ∈ C α+β for some β > 0. In particular, the domain of L contains C α+β for each β > 0. The following is immediate from Corollary 5.2. Corollary 5.4. Suppose u ∈ C α+β for some β ∈ (0, 1) and α + β ∈ (0, 1) ∪ (1, 2) ∪ (2, 3). Let L0 be defined by (3.4). Then L0 u ∈ C β and there exists c1 such that L0 uC β c1 uC α+β . 6. Proof of Theorem 1.2 Let B(x, r) denote the ball of radius r centered at x. Let ϕ be a cut-off function that is 1 on B(0, 1), 0 on B(0, 2)c , takes values in [0, 1], and is C ∞ . Let ϕr,x0 (x) = r −d ϕ((x − x0 )/r). When r and x0 are clear, we will write just ϕ for ϕr,x0 . Proposition 6.1. Suppose uC α+β < ∞. Suppose for each δ > 0 there exists r and c1 (depending on δ) such that uϕr,x0 C α+β c1 f C β + c1 uL∞ + δuC α+β .

(6.1)

Then there exists c2 depending on δ such that uC α+β c2 f C β + c2 uL∞ .

(6.2)

Proof. First we do the case where α + β ∈ (0, 1) ∪ (1, 2). Recall from Proposition 2.2 that there exist c3 and c4 such that g(x + h) + g(x − h) − 2g(x) c4 gC α+β |h|α+β |h|>0

c3 gC α+β gL∞ + sup sup x

(6.3)

for all g ∈ C α+β . Choose δ = c3 /2c4 and then choose r and c1 using (6.1). If x0 ∈ Rd , let v = uϕr,x0 , and note that u = v in the ball B(x0 , r). If |h| < r, u(x0 + h) + u(x0 − h) − 2u(x0 ) = v(x0 + h) + v(x0 − h) − 2v(x) c4 vC α+β |h|α+β .

(6.4)


2711

On the other hand, if |h| r, u(x0 + h) + u(x0 − h) − 2u(x0 )

4 uL∞ |h|α+β = c5 uL∞ |h|α+β . r α+β

(6.5)

Combining (6.4) and (6.5) and using (6.1),

u(x0 + h) + u(x0 − h) − 2u(x0 ) c4 vC α+β + c5 uL∞ |h|α+β

c1 c4 f C β + (c1 c4 + c5 )uL∞ + c4 δuC α+β |h|α+β . This and (6.3) yield 1 uC α+β c6 f C β + c6 uL∞ + uC α+β . 2 Subtracting 12 uC α+β from both sides and multiplying by 2 gives (6.1). Now we consider the case when α + β ∈ (2, 3). Since u ∈ C α+β if u ∈ L∞ and each Di u ∈ C α+β−1 , by (2.3), (2.4), and Propositions 2.1 and 2.2 there exists c7 such that |Di u(x + h) + Di u(x − h) − 2Di u(x)| uC α+β c7 uL∞ + sup sup sup . |h|α+β−1 x |h|>0 i Let δ = 1/2c7 (1 + c4 ), choose r using (6.1), and let v = uϕr,x0 . If |h| < r, then for any i, Di u(x0 + h) + Di u(x0 − h) − 2Di u(x0 ) = Di v(x0 + h) + Di v(x0 − h) − 2Di v(x0 ) c4 vC α+β |h|α+β−1

c1 c4 f C β + c1 c4 uL∞ + δc4 uC α+β |h|α+β−1 . On the other hand, if |h| r, then Di u(x0 + h) + Di u(x0 − h) − 2Di u(x0 )

4 r α+β−1

Di uL∞ |h|α+β−1 .

Choose ε = r α+β−1 δ/4 and then use Proposition 2.2 to see there exists c8 such that Di uL∞ c8 uL∞ + εuC α+β . Substituting this in (6.6),

Di u(x0 + h) + Di u(x0 − h) − 2Di u(x0 ) c9 uL∞ + δuC α+β |h|α+β−1 . Therefore Di u(x0 + h) + Di u(x0 − h) − 2Di u(x0 )

c10 f C β + c10 uL∞ + (1 + c4 )δuC α+β |h|α+β−1 ,

(6.6)

2712


and hence 1 uC α+β c11 f C β + c11 uL∞ + uC α+β . 2 Subtracting 12 uC α+β from both sides, and multiplying by 2 yields our result.

2

Proof of Theorem 1.2. Step 1. In this step we define a certain function F . Let us suppose for now that α < 1, leaving the case α 1 until later. Fix δ > 0 and let ε > 0 be chosen later. Let x0 ∈ Rd be fixed and choose r such that sup A(x, h) − A(x0 , h) < ε

|h|>0

if |x − x0 | 4r. Let b(x, h) = A(x, h) − A(x0 , h), L0 u(x) =

A(x0 , h) u(x + h) − u(x) dh, |h|d+α

and B = L − L0 . Let ϕ = ϕr,x0 be as in the paragraph preceding Proposition 6.1 and let v = uϕ. We have v(x + h) − v(x) = u(x) ϕ(x + h) − ϕ(x) + ϕ(x) u(x + h) − u(x) + u(x + h) − u(x) ϕ(x + h) − ϕ(x) , and therefore Lv(x) = u(x)Lϕ(x) + ϕ(x)Lu(x) + H (x) = u(x)Lϕ(x) + ϕ(x)f (x) + H (x), where H (x) =

A(x, h) dh. u(x + h) − u(x) ϕ(x + h) − ϕ(x) |h|d+α

On the other hand, Lv(x) = L0 v(x) + Bv(x), and so we have L0 v(x) = u(x)Lϕ(x) + ϕ(x)f (x) + H (x) − Bv(x) = J1 (x) + J2 (x) + J3 (x) + J4 (x).

(6.7)


2713

Set F (x) =

4

(6.8)

Ji (x).

i=1

By Theorem 4.4 we have

vC α+β c1 F C β + vL∞ c1 F C β + uL∞ . So if, given ε, we can show

F C β c2 f C β + uL∞ + εuC α+β ,

(6.9)

we take ε = δ/c2 , we then have (6.1), we apply Proposition 6.1, and we are done. Step 2. We first look at the L∞ norm of F . Since

ϕ(x + h) − ϕ(x)

1 dh |h|d+α

Eh ϕ(x)

1 dh c3 < ∞, |h|d+α

where Eh is defined in (5.1), then u(x)Lϕ(x) c3 uL∞ . Similarly H (x) = u(x + h) − u(x) ϕ(x + h) − ϕ(x) A(x0 , r) dh d+α |h| 1 c4 uL∞ Eh ϕ(x) d+α dh |h| c5 uL∞ . We also have ϕ(x)f (x) f L∞ f C β . It remains to bound Bv(x). If x ∈ / B(x0 , 3r), then since v(x) = 0 and v(x + h) = 0 unless |h| > r, we see Bv(x) = |h|>r

b(x, h) v(x + h) d+α dh c6 uL∞ |h|

|h|−d−α dh = c7 uL∞ .

|h|>r

We have vC α+β c8 ϕC α+β uC α+β c9 uC α+β ,

2714


since ϕ is smooth. By Theorem 5.1(a),

Eh v(x) c10 |h|(α+β)∧1 ∧ 1 vC α+β , and so Bv(x) = Eh v(x) b(x, h) dh d+α |h| (α+β)∧1

c10 ε |h| ∧1 c11 εvC α+β

1 dh vC α+β |h|d+α c12 εuC α+β .

We used the fact that we chose r small so that |b(x, h)| ε. To summarize, in this step we have shown

F L∞ c13 f C β + uL∞ + εuC α+β .

(6.10)

Step 3. We next estimate [F ]C β . Since we have

F (x + k) − F (x) 2F L∞ 2β /r β F L∞ |k|β when |k| r/2 and we have an upper bound of the correct form for F L∞ in (6.10), to bound [F ]C β it suffices to look at F (x + k) − F (x) when |k| r/2. We look at the differences for Ji for i = 1, . . . , 4. We look at J4 first, since this is the most difficult one. First suppose x ∈ / B(x0 , 3r). Then v(x + h + k), v(x + h), v(x + k), and v(x) are all zero if |h| r/2. So Bv(x + k) − Bv(x) = |h|>r/2

|h|>r/2

dh v(x + h + k)b(x + k, h) − v(x + h)b(x, h) |h|d+α

v(x + h + k) − v(x + h)b(x + k, h) dh |h|d+α

+ |h|>r/2

v(x + h)b(x + k, h) − b(x, h) dh |h|d+α

c14 vC β |k|

β |h|>r/2

dh + c11 vL∞ |k|β |h|d+α

|h|>r/2

Since vL∞ uL∞ and vC β c15 uC β ϕC β c16 uC β c17 uL∞ + εuC α+β by Proposition 2.2, we have our required estimate when x ∈ / B(x0 , 3r).

dh . |h|d+α


2715

Now suppose x ∈ B(x0 , 3r). Since |k| r/2, then x + k ∈ B(x0 , 4r), and so |b(x, h)| ε and |b(x + k, h)| ε for all h. We write Bv(x + k) − Bv(x)

Eh v(x + k) − Eh v(x) |b(x + k, h)| dh |h|d+α Eh v(x) |b(x + k, h) − b(x, h)| dh + |h|d+α |h|ζ

+ |h|>ζ

Eh v(x) |b(x + k, h) − b(x, h)| dh |h|d+α

= I1 + I2 + I3 , where ζ will be chosen in a moment. By Theorem 5.1, I1 ε

dh (α+β)∧1 |h| ∧ |k|(α+β)∧1 c18 εvC α+β |k|β . |h|d+α

Suppose for the moment that α + β < 1. For I2 we have I2 c19 vC α+β |h|ζ

α+β

|k|β |h| ∧1 dh εvC α+β , |h|d+α

provided we take ζ small; note that the choice of ζ can be made to depend only on d, α, β, and ε. For I3 we have I3 c20 vC β |h|>ζ

β

|k|β |h| ∧ 1 dh c21 vC β |k|β . |h|d+α

We now use vC α+β c22 uC α+β ϕC α+β and vC β c23 uC β ϕC β εuC α+β + c24 uL∞ . Summing the estimates for I1 , I2 , and I3 , we have the desired bound for J4 when α + β < 1. The case α + β ∈ (1, 2) is very similar; the details are left to the reader. Next we look at J1 . Similarly to the estimates for J4 , we see that LϕC β c25 . We then have J2 C β c26 uC β LϕC β , and then Proposition 2.2 gives our estimate.

2716


The estimate for J2 is quite easy. By Lemma 2.3 ϕf C β c26 ϕC β f C β c27 f C β . It remains to handle J3 . We have

A(x + k, h) dh Eh u(x + k) − Eh u(x) Eh ϕ(x + k) |h|d+α A(x + k, h) + Eh u(x) Eh ϕ(x + k) − Eh ϕ(x) dh |h|d+α A(x + k, h) − A(x, h) + Eh u(x)Eh ϕ(x) dh |h|d+α = I4 + I5 + I6 .

H (x + k) − H (x) =

By Theorem 5.1 |I4 | c28 |k| uC β β

dh β dh c29 |k|β uC β . |h| ∧ 1 |h|d+α

Also by Theorem 5.1 |I5 | c30 uL∞

dh β dh c31 uL∞ |k|β ; |h| ∧ |k|β ∧ 1 |h|d+α

to get the second inequality we split the integral into |h| |k|, |k| < |h| 1, and |h| > 1. Using Theorem 5.1 a third time |I6 | c32 uL∞

|k|β β dh c33 uL∞ |k|β . |h| ∧ 1 |h|d+α

Combining yields [H ]C β c34 uC β , and we now apply Proposition 2.2. Step 4. Finally we consider the case α 1. This is very similar to the α < 1 case, but where we replace the use of Eh f by Fh f . We leave the details to the reader. 2 7. Further results and remarks 7.1. An extension We remark that the proof of Theorem 1.2 really only required that there exist c1 and h0 such that sup sup A(x + k, h) − A(x, h) c1 |k|β . x |h|h0


2717

The observation needed is that one can bound |h|>h0

A(x, h) u(x + h) − u(x) dh β c2 uC β c3 uL∞ + εuC α+β . |h|d+α C

7.2. Zero-order terms We can add a zero-order term to L and have the result remain valid. Theorem 7.1. Let P be a function such that P C β < ∞. Let L u(x) = Lu(x) + P (x)u(x), where L is defined by (1.1) or (1.2) and satisfies the assumptions of Theorem 1.2. Then there exists c1 (which depends on P C β ) such that if L u(x) = f (x) and uC α+β < ∞, then

uC α+β c1 uL∞ + f C β . Proof. We proceed as in the proof of Theorem 1.2, but now in (6.8) we write F (x) = J1 (x) + · · · + J5 (x), where J5 (x) = P (x)v(x). We have, using Proposition 2.2 and Lemma 2.3,

J5 C β c2 P C β ϕC β uC β c3 uL∞ + εuC α+β . Other than this additional term, the rest of the proof goes through as before.

2

7.3. First-order terms If α > 1, we can add a first-order term to L. (We can also keep the zero-order term as in Theorem 7.1, but we omit this in the following discussion for simplicity.) Theorem 7.2. Suppose α > 1. For i = 1, . . . , d, let Qi be functions such that Qi C β < ∞. Let L u(x) = Lu(x) +

d

Qi (x)Di u(x),

i=1

where L is defined by (1.1) or (1.2) and satisfies the assumptions of Theorem 1.2. Then there exists c1 (which depends on di=1 Qi C β ) such that if L u(x) = f (x) and uC α+β < ∞, then

uC α+β c1 uL∞ + f C β .

2718


Proof. As in the proof of Theorem 7.1 we have an additional term in the definition of F , but this time the term is J5 (x) =

d

Qi (x)Di v(x).

i=1

We have

Qi Di vC β c2 Qi C β ϕDi uC β + uDi ϕC β

c3 ϕC β Di uC β + uC β Di ϕC β c4 uL∞ + εuC α+β , using Lemma 2.3 and Proposition 2.2. With J5 handled in this fashion, we proceed as before.

2

7.4. Higher-order smoothness One would expect that if f and A(·, h) have additional smoothness, then the solution u to Lu = f should have additional smoothness. This is indeed the case. One way to show this is to extend the estimates previously proved to C β and C α+β when β > 1. Here is an alternate way. We do the case β ∈ (1, 2) for concreteness, but the case when β ∈ (m, m + 1) for some m is similar. When we write Di A(x, h), we mean the ith partial derivative in the variable x. Theorem 7.3. Suppose β ∈ (1, 2) and there exists c1 such that for each i = 1, . . . , d, sup supDi A(x + k, h) − Di A(x, h) c1 |k|β−1 . x

h

Then there exists c2 such that if f ∈ C β and u ∈ C α+β with Lu = f , we have

uC α+β c1 uL∞ + f C β . Proof. We sketch the proof, and we restrict our attention to α < 1 for simplicity. Differentiating Lu = f yields L(Di u)(x) +

Di A(x, h) u(x + h) − u(x) dh = Di f. |h|d+α

Writing Gi (x) for the second term on the left, L(Di u) = Di f − Gi , and by Theorem 1.2,

Di uC β−1 c3 Di uL∞ + Di f C β−1 + Gi C β−1 .


2719

Note Di f C β−1 c4 f C β and Di uL∞ c5 uL∞ + εuC α+β . Also uC β

c6 di=1 Di uC β−1 . So the key step is to prove that Gi C β−1 c7 uL∞ + εuC α+β .

(7.1)

By arguments similar to the derivation of the estimates for J4 in the proof of Theorem 1.2 but somewhat simpler, Gi C β−1 c8 uC α+β−1 . By Proposition 2.2, the right-hand side is bounded by the right-hand side of (7.1).

2

7.5. Sharpness Our results are sharp in several respects. For example, one might ask if the solution u to Lu = f can be taken to be in C α+β+δ for some δ > 0 when f ∈ C β . The answer is no in general. Let L = L0 , where L0 is defined by (3.4). Let f be a C β function that is not in C β+δ for any δ. If the solution to Lu = f satisfied

uC α+β+δ c1 uL∞ + f C β , then by Corollary 5.4, f = L0 u would be in C β+δ , a contradiction. Another question is whether one can still obtain our main estimate (1.3) if A(x, h) only satisfies sup supA(x + k, h) − A(x, h) c1 |k|β−δ , x

k ∈ Rd ,

(7.2)

h

for some δ > 0. Again the answer is no in general. Let f be a function that is in C β but not in any C β+ζ for ζ > 0. Let w be a function that is in C β−δ for some δ ∈ (0, β) but not in C β−δ+ζ for any ζ > 0. Suppose also that w is bounded below by a positive constant. Let L0 be defined as in (3.4), and define A(x, h) = w(x)A0 (h). Then Lu(x) = w(x)L0 u(x), and A(x, h) satisfies (7.2). Consider the solution to Lu(x) = f (x). We have L0 u(x) = f (x)/w(x). If u were in C α+β , then f (x)/w(x) = L0 u(x) would be in C β , a contradiction. 7.6. The uL∞ term Our main estimate (1.3) has a uL∞ on the right-hand side. When can one dispense with this term? First we give a condition where one can do so. Suppose one considers L u(x) = f (x), where L is defined in Theorem 7.1 and moreover for some λ > 0, P (x) −λ for all x. If Xt is the strong Markov process associated to L (that is, the infinitesimal generator of X is L, for example), the solution to L u(x) is given in probabilistic terms by ∞ u(x) = −E

x

e 0

s 0

P (Xr ) dr

f (Xs ) ds.

2720


Under the condition that P (x) −λ, then u(x) Ex

∞

1 e−λs f (Xs ) ds f L∞ . λ

0

In this case, we have the bound uL∞ f L∞ /λ f C β /λ. On the other hand, if there is no zero-order term, there is no reason to expect that a bound of the form uL∞ c1 f C β

(7.3)

should hold when Lu = f . This bound trivially fails to hold because u plus a constant is still a solution to the equation. Even when we restrict ourselves to solutions that vanish at infinity, (7.3) cannot hold. To see this, let A(x, h) be identically 1, so that L is the infinitesimal generator of a symmetric stable process, let ϕ be defined as in the beginning of Section 6, and let fr (x) = ϕ(x/r). Then fr L∞ = 1 for all r, while [fr ]C β → 0 as r → ∞ for each β ∈ (0, 1). On the other hand, if ur is the solution to Lu = fr , a scaling argument shows that |ur (0)| = c1 r α → ∞ as r → ∞. Let us return to the equation L u = f,

(7.4)

where L is defined in Theorem 7.1, and where for some λ > 0 we have P (x) −λ for all x. Proposition 7.4. If Assumption 1.1 holds and f ∈ C β , then there exists u ∈ C α+β such that L u = f . Proof. As discussed above, in this situation uL∞ c1 f L∞ , and we therefore have uC α+β c2 f C β if u is a solution to (7.4). We can now obtain existence of a solution to (7.4) by the method of continuity [18, Section 5.2]. We follow the proof of [18, Theorem 6.8] exactly, except that we replace the use of the Laplacian by the fractional Laplacian. 2 7.7. Future research We mention some directions for future research. 1. Interior estimates for the Dirichlet problem. Can one give interior estimates for the regularity of harmonic functions (the Dirichlet problem) and the regularity of potentials (the analog of Poisson’s equation) in bounded domains?


2721

2. Boundary estimates. To obtain a satisfactory theory, one would like estimates on harmonic functions and potentials in bounded domains that are valid up to the boundary. 3. Symmetric processes. Suppose instead of L one works instead with the Dirichlet form E(f, g) = Rd Rd

B(x, y) f (y) − f (x) g(y) − g(x) . |x − y|d+α

The generator associated to E is the analog of an elliptic operator in divergence form. The Harnack inequality and Hölder regularity for harmonic functions are known in this setting under the assumption that B(x, y) is symmetric and bounded above and below by positive constants; see [12]. However if one adds some continuity conditions to B, one would expect the corresponding potentials and harmonic functions to have additional smoothness. 4. The parabolic case. One could look at the fundamental solution or heat kernel p(t, x, y), which is equivalent to looking at the transition densities of the associated process. One would expect that if the A(x, h) (and the B(x, h)) have some smoothness, say, Hölder continuous of order β, and are bounded above and below by positive constants, then the p(t, x, y) are not only Hölder continuous in x and y, but will be C α+β in each coordinate. (In the symmetric case Hölder continuity is known, but of a smaller order.) This question could be asked about the transition densities in the whole space Rd and also in bounded domains. 5. Variable order. Consider operators L of the form (7.5) Lf (x) = f (x + h) − f (x) − 1(|h|1) ∇f (x) · h n(x, h) dh, where we assume c2 c1 n(x, h) d+β , d+α |h| |h|

x ∈ Rd , 1 |h| > 0,

0 < α < β < 2, and some appropriate condition is imposed on n(x, h) for |h| 1. Such an operator is of variable order because if one writes it as a pseudo-differential operator, then the order is not fixed; see [19]. Some progress has already been made on operators of variable order; see [3] and [4] for the operators L in (7.5) and see [1] and [5] for nonlocal Dirichlet forms of variable order. Can one give suitable assumptions on n(x, h) so that harmonic functions and potentials have additional smoothness? 6. Diffusions with jumps. If we consider operators that are the sum of an elliptic differential operator and a non-local operator, the same questions could be asked as for the pure jump case: higher-order derivatives, regularity up to the boundary, transition density estimates. (The Harnack inequality was considered in [16] and [17].) References [1] M.T. Barlow, R.F. Bass, Z.-Q. Chen, M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc. 361 (2009) 1963–1999. [2] R.F. Bass, Uniqueness in law for pure jump Markov processes, Probab. Theory Related Fields 79 (1988) 271–287. [3] R.F. Bass, M. Kassmann, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc. 357 (2005) 837–850. [4] R.F. Bass, M. Kassmann, Hölder continuity of harmonic functions with respect to operators of variable order, Comm. Partial Differential Equations 30 (2005) 1249–1259.

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[5] R.F. Bass, M. Kassmann, T. Kumagai, Symmetric jump processes: Localization, heat kernels, and convergence, Ann. Inst. H. Poincaré, in press. [6] R.F. Bass, D.A. Levin, Harnack inequalities for jump processes, Potential Anal. 17 (2002) 375–388. [7] R.F. Bass, D.A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc. 354 (2002) 2933–2953. [8] R.F. Bass, H. Tang, The martingale problem for a class of stable-like processes, Stochastic Process. Appl. 49 (2009) 1144–1167. [9] K. Bogdan, Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian, Studia Math. 181 (2007) 101–123. [10] K. Bogdan, T. Kulczycki, M. Kwa´snicki, Estimates and structure of α-harmonic functions, Probab. Theory Related Fields 140 (2008) 345–381. [11] K. Bogdan, P. Sztonyk, Harnack’s inequality for stable Lévy processes, Potential Anal. 22 (2005) 133–150. [12] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic Process. Appl. 108 (2003) 27–62. [13] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields 140 (2008) 277–317. [14] Z.-Q. Chen, T. Kumagai, A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps, Rev. Mat. Iberoamericana, in press. [15] W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, 2nd ed., John Wiley, New York, 1971. [16] M. Foondun, Harnack inequalities for a class of integro-differential operators, preprint. [17] M. Foondun, Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local parts, preprint. [18] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. [19] N. Jacob, Pseudo Differential Operators and Markov Processes, vol. II. Generators and Their Potential Theory, Imperial College Press, London, 2002. [20] M. Kassmann, The classical Harnack inequality fails for nonlocal operators, preprint. [21] V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. Lond. Math. Soc. 80 (2000) 725–768. [22] A.P.C. Lim, Regularity of solutions to Poisson’s equation for an operator associated to a pure jump process of non-variable order, preprint. [23] R. Song, Z. Vondraˇcek, Harnack inequality for some classes of Markov processes, Math. Z. 246 (2004) 177–202. [24] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. [25] P. Sztonyk, Regularity of harmonic functions for anisotropic fractional Laplacian, Math. Nachr., in press. [26] H. Tang, Uniqueness for the martingale problem associated with pure jump processes of variable order, preprint.

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