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Journal of Functional Analysis 258 (2010) 2865–2883 www.elsevier.com/locate/jfa On the generalized self-similar singula...

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Journal of Functional Analysis 258 (2010) 2865–2883 www.elsevier.com/locate/jfa

On the generalized self-similar singularities for the Euler and the Navier–Stokes equations Dongho Chae 1 Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 12 August 2008; accepted 3 February 2010 Available online 13 February 2010 Communicated by I. Rodnianski

Abstract In this paper we study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the Euler and the Navier–Stokes equations, where the sense of convergence and self-similarity are considered in various generalized senses. We improve substantially, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. Generalization of the self-similar transforms is also considered, and by appropriate choice of the parameterized transform we obtain new a priori estimates for the Euler and the Navier–Stokes equations depending on a free parameter. © 2010 Elsevier Inc. All rights reserved. Keywords: Euler equations; Navier–Stokes equations; Generalized self-similar singularities; A priori estimates

1. Self-similar singularities We are concerned on the following Euler equations for the homogeneous incompressible fluid flows in R3 : ⎧ ∂v ⎪ ⎪ + (v · ∇)v = −∇p, (x, t) ∈ R3 × (0, ∞), ⎨ ∂t (E) div v = 0, (x, t) ∈ R3 × (0, ∞), ⎪ ⎪ ⎩ v(x, 0) = v0 (x), x ∈ R3 , E-mail address: [email protected]. 1 This research was done, while the author was visiting University of Chicago. The work is supported partially by NRF

Grant. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.006

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D. Chae / Journal of Functional Analysis 258 (2010) 2865–2883

where v = (v1 , v2 , v3 ), vj = vj (x, t), j = 1, 2, 3, is the velocity of the flow, p = p(x, t) is the scalar pressure, and v0 is the given initial velocity, satisfying div v0 = 0. The system (E) is first modeled by Euler in [13]. The local well-posedness of the Euler equations in H m (R3 ), m > 5/2, is established by Kato in [17], which says that given v0 ∈ H m (R3 ), there exists T ∈ (0, ∞] such that there exists a unique solution to (E), v ∈ C([0, T ); H m (R3 )). The finite time blowup problem of the local classical solution is known as one of the most important and difficult problems in partial differential equations (see e.g. [20,6,8,7,5] for graduate level texts and survey articles on the current status of the problem). We say a local in time classical solution v ∈ C([0, T ); H m (R3 )) blows up at T if lim supt→T v(t)H m = ∞ for all m > 5/2. The celebrated Beale–Kato–Majda criterion [1] states that the blow-up happens at T if and only if T

ω(t)

L∞

dt = ∞.

0

Although the original result of [1] is the blow-up criterion in the H m (R3 ) norm with m 3, it is easy to extend it to the case of m > 5/2. There are studies of geometric nature for the blowup criterion [9,7,12]. As another direction of studies of the blow-up problem mathematicians also consider various scenarios of singularities and study carefully their possibility of realization (see e.g. [10,11,2,3] for some of those studies). One of the purposes in this paper, especially in this section, is to study more deeply the notions related to the scenarios of the self-similar singularities in the Euler equations, the preliminary studies of which are done in [2,3]. We recall that system (E) has scaling property that if (v, p) is a solution of the system (E), then for any λ > 0 and α ∈ R the functions v λ,α (x, t) = λα v λx, λα+1 t ,

p λ,α (x, t) = λ2α p λx, λα+1 t

(1.1)

are also solutions of (E) with the initial data v0λ,α (x) = λα v0 (λx). In view of the scaling properties in (1.1), a natural self-similar blowing-up solution v(x, t) of (E) should be of the form, v(x, t) = p(x, t) =

1 α

(T − t) α+1 α+1 2α

(T − t) α+1

V

,

1

P

x (T − t) α+1 x

(1.2)

1

(T − t) α+1

(1.3)

for α = −1 and t sufficiently close to T . Substituting (1.2)–(1.3) into (E), we obtain the following stationary system:

αV + (y · ∇)V + (α + 1)(V · ∇)V = −∇P , div V = 0,

(1.4)

the Navier–Stokes equations version of which has been studied extensively after Leray’s pioneering paper [19,23,24,22,3,16]. Existence of solution of the system (1.4) is equivalent to the existence of solutions to the Euler equations of the form (1.2)–(1.3), which blows up in a self-similar fashion. Given (α, p) ∈ (−1, ∞) × (0, ∞], we say the blow-up is α-asymptotically


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self-similar in the sense of Lp if there exists V = V α ∈ W˙ 1,p (R3 ) such that the following convergence holds true:

1 · =0 ∇v(·, t) − ∇V lim (T − t) 1 t→T T −t (T − t) α+1 L∞ if p = ∞, while

1 · =0 1 ω(·, t) − T − t Ω (T − t) α+1 Lp

3 1− (α+1)p

lim (T − t)

t→T

if 0 < p < ∞, where and hereafter we denote Ω = curl V

and Ω = curl V .

The above limit function V ∈ Lp (R3 ) with Ω = 0 is called the blow-up profile. We observe that the self-similar blow-up given by (1.2)–(1.3) is trivial case of α-asymptotic self-similar blow-up with the blow-up profile given by the representing function V . We say a blow-up at T is of type I, if lim sup (T − t)∇v(t)L∞ < ∞. t→T

If the blow-up is not of type I, we say it is of type II. For the use of terminology, type I and type II blow-ups, we followed the literatures on the studies of the blow-up problem in the semilinear heat equations (see e.g. [21,15,14], and the references therein). The use of ∇v(t)L∞ rather than v(t)L∞ in our definition of types I and II is motivated by Beale–Kato–Majda’s blow-up criterion. Theorem 1.1. Let m > 5/2, and v ∈ C([0, T ); H m (R3 )) be a solution to (E) with v0 ∈ H m (R3 ), div v0 = 0. We set lim sup (T − t)∇v(t)L∞ := M(T ).

(1.5)

t→T

Then, either M(T ) 1, or the solution does not blow up at time T , which implies that M(T ) = 0. Hence, one can only have type I blow-up at time T if M(T ) 1. An immediate implication of the above theorem on the self-similar blow-up is the following. Corollary 1.1. There exists no self-similar blow-up for the solution of the 3D Euler equations with the blow-up profile V satisfying ∇V L∞ < 1. Proof. We just observe that if v(x, t) =

1

α

(T −t) α+1

V(

x 1

(T −t) α+1

(T − t)∇v(t)L∞ = ∇V L∞ ,

), then

∀t ∈ (0, T ).

2

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Proof of Theorem 1.1. It suffices to show that M(T ) < 1 implies non-blow-up at T , which, in turn, leads to M(T ) = 0, since ∇v(t)L∞ ∈ C([0, T ]) in this case. We suppose M(T ) < 1. Then, there exists t0 ∈ (0, T ) such that sup (T − t)∇v(t)L∞ := M0 < 1.

t0 1.

(1.18)

Theorem 1.3. Let v ∈ C([0, T ); H m (R3 )), m > 5/2, be local classical solution of the Euler equations. Suppose there exists γ > 1 and R1 > 0 such that the following convergence holds true: 3 γ lim (T − t)(α− 2 ) α+1 v(·, t) −

t→T

1 γα

(T − t) α+1

V

· (T − t)

γ α+1

L2 (BR1 )

= 0,

(1.19)

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where BR1 = {x ∈ R3 | |x| < R1 }. Then, the blow-up profile V ∈ L2loc (R3 ) is a weak solution of the following stationary Euler equations, (V · ∇)V = −∇P ,

div V = 0.

(1.20)

Remark 1.3. Eq. (1.20) seems to not have any immediate applicability in the Euler setting, but see its counterpart (1.27) and Theorem 1.4 below, where it is used to rule out type II asymptotically self-similar singularities for the Navier–Stokes equations. Proof of Theorem 1.3. We introduce a self-similar transform defined by v(x, t) =

1 (T − t)

αγ α+1

p(x, t) =

V (y, s),

1 2αγ

(T − t) α+1

(1.21)

P (y, s)

with y=

1 γ

(T − t) α+1

x,

T γ −1 1 s= −1 . (γ − 1)T γ −1 (T − t)γ −1

(1.22)

Substituting (v, p) in (1.21)–(1.22) into the (E), we have

(E2 )

⎧ γ α 1 ⎪ ⎪ ⎪ ⎨ − s(γ − 1) + T 1−γ α + 1 V + α + 1 (y · ∇)V = Vs + (V · ∇)V + ∇P , div V = 0, ⎪ ⎪ ⎪ γ αγ ⎩ V (y, 0) = V0 (y) = T α+1 v0 T α+1 y .

(1.23)

The hypothesis (1.19) is written as lim V (·, s) − V (·)L2 (B

s→∞

R(s)

= 0, )

R(s) = (γ − 1)s +

1 T γ −1

γ (α+1)(γ −1)

,

(1.24)

which implies that lim V (·, s) − V L2 (B

s→∞

R)

= 0,

∀R > 0,

(1.25)

where V (y, s) is defined by (1.21). Similarly to [16,3], we consider the scalar test function 1 ξ ∈ C01 (0, 1) with 0 ξ(s) ds = 0, and the vector test function φ = (φ1 , φ2 , φ3 ) ∈ C01 (R3 ) with div φ = 0. We multiply the first equation of (E2 ), in the dot product, by ξ(s − n)φ(y), and integrate it over R3 × [n, n + 1], and then we integrate by parts to obtain


α + α+1

2873

1 g(s + n)ξ(s)V (y, s + n) · φ(y) dy ds 0 R3

1 − α+1

1 g(s + n)ξ(s)V (y, s + n) · (y · ∇)φ(y) dy ds 0 R3

1 =

ξs (s)φ(y) · V (y, s + n) dy ds 0

R3

1

ξ(s) V (y, s + n) · V (y, s + n) · ∇ φ(y) dy ds = 0,

+ 0 R3

where we set g(s) =

γ . s(γ − 1) + T 1−γ

1 1 Passing to the limit n → ∞ in this equation, using the facts 0 ξs (s) ds = 0, 0 ξ(s) ds = 0, V (·, s + n) → V in L2loc (R3 ), and finally g(s + n) → 0, we find that V ∈ L2loc (R3 ) satisfies V · (V · ∇)φ(y) dy = 0 R3

for all vector test function φ ∈ C01 (R3 ) with div φ = 0. On the other hand, we can pass s → ∞ directly in the weak formulation of the second equation of (E2 ) to have V · ∇ψ(y) dy = 0 R3

for all scalar test function ψ ∈ C01 (R3 ).

2

In the following theorem we rule out the possibility of the blow-up rate given by (1.18) in the setting of the Navier–Stokes equations. Theorem 1.4. Let p ∈ [3, ∞) and v ∈ C([0, T ); Lp (R3 )) be a local classical solution of the Navier–Stokes equations constructed by Kato [18]. Suppose there exist γ > 1 and V ∈ Lp (R3 ) such that the following convergence holds true: lim (T − t)

t→T

(p−3)γ 2p

(p−3)γ v(·, t) − (T − t)− 2p V

If the blow-up profile V belongs to H˙ 1 (R3 ), then V = 0.

· γ

(T − t) 2

= 0. Lp

(1.26)

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Proof. Since the main part of the proof is essentially identical to that of Theorem 1.3, we will be brief. Introducing the self-similar variables of the form (1.21)–(1.23) with α = 12 , and substituting (v, p) into the Navier–Stokes equations, ∂v

+ (v · ∇)v = v − ∇p, ∂t div v = 0, v(x, 0) = v0 (x),

(NS)

we find that (V , P ) satisfies ⎧ γ ⎪ ⎪ ⎨ − 2s(γ − 1) + 2T 1−γ V + (y · ∇)V = Vs + (V · ∇)V − V + ∇P , div V = 0, ⎪ ⎪ γ γ ⎩ V (y, 0) = V0 (y) = T 2 v0 T 2 y .

The hypothesis (1.26) is now translated as lim V (·, s) − V (·)Lp = 0.

s→∞

Following exactly same argument as in the proof of Theorem 1.3, we can deduce that V is a stationary solution of the Navier–Stokes equations, namely there exists P such that (V · ∇)V = V − ∇P ,

div V = 0. (1.27) In the case V ∈ H˙ 1 ∩ Lp (R3 ), we easily obtain from (1.27) that R3 |∇V |2 dy = 0, which implies V = 0. 2 2. Generalized self-similar singularities Let us consider a classical solution to (E) v ∈ C([0, T ); H m (R3 )), m > 5/2, where we assume T ∈ (0, ∞] is the maximal time of existence of the classical solution. Let p(x, t) be the associated pressure. Let μ(·) ∈ C 1 ([0, T )) be a scalar function such that μ(t) > 0 for all t ∈ [0, T ). We transform from (v, p) to (V , P ) according to the formula, v(x, t) = μ(t)

α α+1

V μ(t)

1 α+1

t x,

μ(σ ) dσ ,

(2.1)

0

p(x, t) = μ(t)

2α α+1

P μ(t)

1 α+1

t x,

μ(σ ) dσ ,

(2.2)

0

where α ∈ (−1, ∞) as previously. This means that the space–time variables are transformed from (x, t) ∈ R3 × [0, T ) into (y, s) ∈ R3 × [0, ∞) as follows: y = μ(t)

1 α+1

t x,

s=

μ(σ ) dσ. 0

(2.3)


2875

Substituting (2.1)–(2.3) into the Euler equations, we obtain the equivalent equations satisfied by (V , P )

(E∗ )

⎧ μ (t) α 1 ⎪ ⎪ ⎪ ⎨ − μ(t)2 α + 1 V + α + 1 (y · ∇)V = Vs + (V · ∇)V + ∇P , div V = 0, ⎪ ⎪ ⎪ α 1 ⎩ V (y, 0) = V0 (y) = μ(0) α+1 v0 μ(0) α+1 y .

We note that the special cases μ(t) =

1 , T −t

μ(t) =

1 , (T − t)γ

γ > 1,

are considered in the previous section. Let us choose μ(t) = exp[±γ Then,

t

±γ α v(x, t) = exp α+1

t 0

∇v(τ )L∞ dτ ], γ 1.

∇v(τ )

L∞

dτ V (y, s),

(2.4)

0

±2γ α p(x, t) = exp α+1

t

∇v(τ )

L∞

dτ P (y, s)

(2.5)

0

with

±γ y = exp α+1 t s=

t

L∞

dτ x,

0

τ

exp ±γ 0

∇v(τ )

∇v(σ )

L∞

dσ dτ

(2.6)

0

respectively for the signs ±. Substituting (v, p) in (2.4)–(2.6) into the (E∗ ), we find that (E∗ ) becomes

(E± )

⎧ α 1 ⎪ ⎪ V+ (y · ∇)V = Vs + (V · ∇)V + ∇P , ⎨ ∓γ ∇V (s) L∞ α+1 α+1 ⎪ div V = 0, ⎪ ⎩ V (y, 0) = V0 (y) = v0 (y)

respectively for ±. Similar equations to the system (E± ), without the term involving (y · ∇)V , are introduced and studied in [4], where similarity type of transform with respect to only time variables was considered. The argument of the global/local well-posedness of the system (E± )

2876


respectively from the local well-posedness result of the Euler equations is as follows. We define S± =

T

τ

exp ±γ 0

∇v(σ )

L∞

dσ dτ.

0

Then, S ± is the maximal time of existence of classical solution for the system (E± ). Indeed, by the BKM criterion we have T

+

S =

τ exp γ ∇v(σ )L∞ dσ dτ

0

0

T

1 ω0 L∞

ω(τ )

L∞

dτ = ∞.

0

We also note the following integral invariant of the transform, T

∇v(t)

0

S ± ∇V ± (s) ∞ dt =

L

L∞

ds.

0

t Below we fix μ(t) := exp[ 0 ∇v(τ )L∞ dτ ]. We assume our local classical solution in H m (R3 ) blows up at T , and hence μ(T − 0) = T exp[ 0 ∇v(τ )L∞ dτ ] = ∞. Given (α, p) ∈ (−1, ∞) × (0, ∞], as previously, we say the blowup is α-asymptotically self-similar in the sense of Lp if there exists V = V α ∈ W˙ 1,p (R3 ) such that the following convergence holds true: 1 lim μ(t)−1 ∇v(·, t) − μ(t)∇V μ(t) α+1 (·) L∞ = 0

t→T

(2.7)

for p = ∞, and 3 −1+ (α+1)p

lim μ(t)

t→T

3 1− (α+1)p

ω(·, t) − μ(t)

1 Ω μ(t) α+1 (·) Lp = 0

(2.8)

for p ∈ (0, ∞). The above limiting function V with Ω = 0 is called the blow-up profile as previously. Proposition 2.1. Let α = 3/2. Then there exists no α-asymptotically self-similar blow-up in the sense of L∞ with the blow-up profile belonging to L2 (R3 ). Proof. Let us suppose that there exists V ∈ W˙ 1,∞ (R3 ) ∩ L2 (R3 ) such that (2.7) holds, then we will show that V = 0. In terms of the self-similar variables (2.7) is translated into lim ∇V (·, s) − ∇V L∞ = 0,

s→∞

where V is defined in (2.1). If ∇V L∞ = 0, then the condition V ∈ L2 (R3 ) implies that V = 0, and there is nothing to prove. Let us suppose ∇V L∞ > 0. As is done in the proof of Theorem 1.3 the equations satisfied V are easily shown to be


⎧ ⎨ ⎩

−∇V L∞

2877

α 1 (y · ∇)V = (V · ∇)V + ∇P , V+ α+1 α+1

(2.9)

div V = 0

for a scalar function P . Taking L2 (R3 ) inner product of the first equation of (2.9) by V we obtain

3 ∇V L∞ α− V L2 = 0. α+1 2 Since ∇V L∞ = 0 and α = 32 , we have V L2 = 0, and V = 0.

2

Proposition 2.2. There exists no α-asymptotically self-similar blowing-up solution to (E) in the 3 sense of Lp if 0 < p < 2(α+1) . Proof. Suppose there exists α-asymptotically self-similar blow-up at T in the sense of Lp . Then, there exists Ω ∈ Lp (R3 ) such that, in terms of the self-similar variables introduced in (2.1)–(2.2), we have lim Ω(s)Lp = ΩLp < ∞.

(2.10)

s→∞

We represent the Lp norm of ω(t)Lp in terms of similarity variables to obtain ω(t)

Lp

Ω(s)

3 1− (α+1)p

= μ(t)

t Lp

μ(t) = exp

,

∇v(τ )

L∞

dτ .

(2.11)

0

Substituting this into the lower estimate part of (1.12), we have 3 −2+ (α+1)p

μ(t) If −2 +

3 (α+1)p

Ω(s)Lp . Ω0 Lp

(2.12)

> 0, then taking the limit t → T in the above inequality we obtain 3 −2+ (α+1)p

∞ = lim sup μ(t)

Ω0 Lp

t→T

lim sup Ω(s)Lp = ΩLp , s→∞

which is a contradiction to (2.10).

2

3. New a priori estimates One of the important advantages of the formulation of (E) in terms of (E± ) of the previous section is that after representing it by the vorticity formulation, the convection term is dominated by the linear term associated with ∓γ ∇V (s)L∞ (see (3.7) below), which enables us to derive new a priori estimates for ω(t)L∞ as follows.

2878


Theorem 3.1. Given m > 5/2 and v0 ∈ H m (R3 ) with div v0 = 0, let ω be the vorticity of the solution v ∈ C([0, T ); H m (R3 )) to the Euler equations (E). Then we have an upper estimate t ω0 L∞ exp[γ 0 ∇v(τ )L∞ dτ ] , t τ 1 + (γ − 1)ω0 L∞ 0 exp[γ 0 ∇v(σ )L∞ dσ ] dτ

(3.1)

t ω0 L∞ exp[−γ 0 ∇v(τ )L∞ dτ ] t τ 1 − (γ − 1)ω0 L∞ 0 exp[−γ 0 ∇v(σ )L∞ dσ ] dτ

(3.2)

ω(t) ∞ L and lower one ω(t)

L∞

for all γ 1 and t ∈ [0, T ). Moreover, the denominator of the right-hand side of (3.2) can be estimated from below as

t 1 − (γ − 1)ω0 L∞

τ

exp −γ 0

∇v(σ )

L∞

0

dσ dτ

1 , (1 + ω0 L∞ t)γ −1

(3.3)

which shows that the finite time blow-up does not follow from (3.2). Remark 3.1. We observe that for γ = 1, the estimates (3.1)–(3.2) reduce to the well-known ones in (1.12) with p = ∞. In this sense the above estimates seem to be a natural extension from the known ones, but their use is not clear at this point. Moreover, combining (3.1)–(3.2) together, we easily derive another new estimate,

t 0

sinh[γ cosh[γ

t 0t τ

∇v(τ )L∞ dτ ] ∇v(σ )L∞ dσ ] dτ

(γ − 1)ω0 L∞ .

(3.4)

Proof of Theorem 3.1. Below we denote V ± for the solutions of (E± ) respectively, and Ω ± = curl V ± . Note that V0± = v0 := V0 and Ω0± = ω0 := Ω0 . We will first derive the following estimates for the system (E± ): + Ω (s)

L∞

Ω0 L∞ , 1 + (γ − 1)sΩ0 L∞

(3.5)

− Ω (s)

L∞

Ω0 L∞ , 1 − (γ − 1)sΩ0 L∞

(3.6)

as long as V ± (s) ∈ H m (R3 ). Taking curl of the first equation of (E± ), we have ∓γ ∇V L∞ Ω −

1 (y · ∇)Ω = Ωs + (V · ∇)Ω − (Ω · ∇)V . α+1

(3.7)


2879

Multiplying Ξ = Ω/|Ω| on the both sides of (3.7), we deduce ∇V (s)L∞ (y · ∇)|Ω| α+1 = Ξ · ∇V · Ξ ∓ ∇V L∞ |Ω| −(γ − 1)∇V L∞ |Ω| ∓ (γ − 1)∇V L∞ |Ω| (γ − 1)∇V L∞ |Ω|

|Ω|s + (V · ∇)|Ω| ∓

for (E+ ), for (E− ),

(3.8)

since |Ξ · ∇V · Ξ | |∇V | ∇V L∞ . Given smooth solution V (y, s) of (E± ), we introduce the particle trajectories {Y± (a, s)} defined by ∇V (s)L∞ ∂Y (a, s) = V± Y (a, s), s ∓ Y (a, s); ∂s α+1

Y (a, 0) = a.

Recalling the estimate

∇V (s) ∞ Ω(s) ∞ Ω(y, s) , L L

∀y ∈ R3 ,

we can further estimate from (3.8) −(γ − 1)|Ω(Y (a, s), s)|2 ∂

Ω Y (a, s), s ∂s (γ − 1)|Ω(Y (a, s), s)|2

for (E+ ), for (E− ).

(3.9)

Solving these differential inequalities (3.9) along the particle trajectories, we obtain that

Ω Y (a, s), s

|Ω0 (a)| 1+(γ −1)s|Ω0 (a)| |Ω0 (a)| 1−(γ −1)s|Ω0 (a)|

for (E+ ),

(3.10)

for (E− ).

Writing the first inequality of (3.10) as

+

Ω Y (a, s), s

1 1 |Ω0 (a)|

+ (γ − 1)s

1 1 Ω0 L∞

+ (γ − 1)s

,

and then taking supremum over a ∈ R3 , which is equivalent to taking supremum over Y (a, s) ∈ R3 due to the fact that the mapping a → Y (a, s) is a diffeomorphism (although not volume preserving) on R3 as long as V ∈ C([0, S); H m (R3 )), we obtain (3.5). In order to derive (3.6) from the second inequality of (3.10), we first write − Ω (s)

L∞

Ω Y (a, s), s

1 1 |Ω0 (a)|

− (γ − 1)s

,

and then take supremum over a ∈ R3 . Finally, in order to obtain (3.1)–(3.2), we just change variables from (3.5)–(3.6) back to the original physical ones, using the fact

2880


Ω + (y, s) = exp −γ

t

∇v(τ )

L∞

dτ ω(x, t),

0

t s=

τ exp γ ∇v(σ )L∞ dσ dτ

0

0

for (3.1), while in order to deduce (3.2) from (3.6) we substitute t Ω (y, s) = exp γ ∇v(τ ) L∞ dτ ω(x, t), −

0

t

τ

exp −γ

s= 0

∇v(σ )

L∞

dσ dτ.

0

Now we can rewrite (3.2) as ω(t)

L∞

t τ 1 d log 1 − (γ − 1)ω0 L∞ exp −γ ∇v(σ )L∞ dσ dτ . − γ − 1 dt 0

0

Thus, t

∇v(τ )

t

L∞

0

dτ

ω(τ )

L∞

dτ

0

t τ 1 log 1 − (γ − 1)ω0 L∞ exp −γ ∇v(σ )L∞ dσ dτ . − γ −1 0

0

(3.11) Set t y(t) := 1 − (γ − 1)ω0 L∞

τ

exp −γ 0

∇v(σ )

L∞

dσ dτ.

0

We find further integrable structure in (3.11), which is γ

y (t) −(γ − 1)ω0 L∞ y(t) γ −1 . Solving this differential inequality, we obtain (3.3).

2

Similar method can also be applied to derive new a priori estimates for the 3D Navier–Stokes equations.


2881

Theorem 3.2. Given v0 ∈ H 1 (R3 ) with div v0 = 0, let ω be the vorticity of the classical solution v ∈ C([0, T ); H 1 (R3 )) ∩ C((0, T ); C ∞ (R3 )) to the Navier–Stokes equations (NS). Then, there exists an absolute constant C0 > 1 such that for all γ C0 the following enstrophy estimate holds true: ω(t)

L2

t ω0 L2 exp[ γ4 0 ω(τ )4L2 dτ ] . t τ 1 {1 + (γ − C0 )ω0 4L2 0 exp[γ 0 ω(σ )4L2 dσ ] dτ } 4

(3.12)

The denominator of (3.12) is estimated from below by t 1 + (γ

− C0 )ω0 4L2 0

τ 4 ω(σ ) L2 dσ dτ exp γ 0

1 (1 − C0 ω0 4L2 t)

(3.13)

γ −C0 C0

for all γ C0 . Proof. Let (v, p) be a classical solution of the Navier–Stokes equations, and ω be its vorticity. We transform from (v, p) to (V , P ) according to the formula, given by (2.1)–(2.3), where t 4 μ(t) = exp γ ω(τ )L2 dτ . 0

Substituting (2.1)–(2.3) with such μ(t) into (NS), we obtain the equivalent equations satisfied by (V , P )

(NS∗ )

⎧ −γ Ω(s)4L2 ⎪ ⎪ ⎨ V + (y · ∇)V = Vs + (V · ∇)V − V − ∇P , 2 ⎪ div V = 0, ⎪ ⎩ V (y, 0) = V0 (y) = v0 (y).

Operating curl on the evolution equations of (NS∗ ), we obtain −γ Ω(s)4L2 2Ω + (y · ∇)Ω = Ωs + (V · ∇)Ω − (Ω · ∇)V − Ω. 2

(3.14)

Taking L2 (R3 ) inner product of (3.14) by Ω, and integrating by part, we estimate 1 d γ Ω2L2 + ∇Ω2L2 + Ω6L2 = 2 ds 4

(Ω · ∇)V · Ω dy R3 3

3

ΩL3 ∇V L2 ΩL6 CΩL2 2 ∇ΩL2 2 ∇Ω2L2 +

C0 Ω6L2 4

(3.15)

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for an absolute constant C0 > 1, where we used the fact ΩL2 = ∇V L2 , the Sobolev imbedding, H˙ 1 (R3 ) → L6 (R3 ), the Gagliardo–Nirenberg inequality in R3 , 1

1

f L3 Cf L2 2 ∇f L2 2 , and Young’s inequality of the form ab a p /p + bq /q, 1/p + 1/q = 1. Absorbing the term ∇Ω2L2 to the left-hand side, we have from (3.15) d γ − C0 Ω2L2 − Ω6L2 . ds 2

(3.16)

Solving the differential inequality (3.16), we have Ω(s)

L2

Ω0 L2 1

[1 + (γ − C0 )sΩ0 4L2 ] 4

(3.17)

.

Transforming back to the original variables and functions, using the relations t s=

τ 4 exp γ ω(σ )L2 dσ dτ,

0

ω(t)

L2

= Ω(s)

0

γ exp L2 4

t

ω(τ )4 2 dτ , L

0

we obtain (3.12). Next, we observe (3.12) can be written as ω(t)4 2 L

τ t 4 d 1 4 ω(σ ) L2 dσ dτ , log 1 + (γ − C0 )ω0 L2 exp γ (γ − C0 ) dt 0

0

which, after integration over [0, t], leads to t

ω(τ )4 2 dτ L

0

τ t 4 1 4 ω(σ ) L2 dσ dτ log 1 + (γ − C0 )ω0 L2 exp γ (γ − C0 ) 0

0

(3.18) for all γ > C0 . Setting t y(t) := 1 + (γ

− C0 )ω0 4L2 0

τ 4 exp γ ω(σ )L2 dσ dτ, 0


2883

we find that (3.18) can be written in the form of a differential inequality, γ

y (t) (γ − C0 )ω0 4L2 y(t) γ −C0 , which can be integrated to provide us with (3.13).

2

Acknowledgment The author would like to thank deeply to the anonymous referee for many invaluable suggestions and comments. References [1] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984) 61–66. [2] D. Chae, Nonexistence of self-similar singularities for the 3D incompressible Euler equations, Comm. Math. Phys. 273 (1) (2007) 203–215. [3] D. Chae, Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations, Math. Ann. 338 (2) (2007) 435–449. [4] D. Chae, On the deformations of the incompressible Euler equations, Math. Z. 257 (3) (2007) 563–580. [5] D. Chae, Incompressible Euler equations: the blow-up problem and related results, in: Handbook of Differential Equations: Evolutionary Partial Differential Equations, vol. IV, Elsevier Science Ltd., 2008, pp. 1–55. [6] J.Y. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998. [7] P. Constantin, Geometric statistics in turbulence, SIAM Rev. 36 (1994) 73–98. [8] P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. 44 (2007) 603–621. [9] P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potential singularity formulation in the 3-D Euler equations, Comm. Partial Differential Equations 21 (3–4) (1996) 559–571. [10] D. Córdoba, C. Fefferman, On the collapse of tubes carried by 3D incompressible flows, Comm. Math. Phys. 222 (2) (2001) 293–298. [11] D. Córdoba, C. Fefferman, R. De La Llave, On squirt singularities in hydrodynamics, SIAM J. Math. Anal. 36 (1) (2004) 204–213. [12] J. Deng, T.Y. Hou, X. Yu, Geometric and nonblowup of 3D incompressible Euler flow, Comm. Partial Differential Equations 30 (2005) 225–243. [13] L. Euler, Principes généraux du mouvement des fluides, Mémoires de l’Académie des Sciences de Berlin 11 (1755) 274–315. [14] Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985) 297–319. [15] Y. Giga, R.V. Kohn, Characterizing blow-up using similarity variables, Indiana Univ. Math. J. 36 (1) (1987) 1–40. [16] T.Y. Hou, R. Li, Nonexistence of locally self-similar blow-up for the 3D incompressible Navier–Stokes equations, Discrete Contin. Dyn. Syst. Ser. A 18 (4) (2007) 637–642. [17] T. Kato, Nonstationary flows of viscous and ideal fluids in R3 , J. Funct. Anal. 9 (1972) 296–305. [18] T. Kato, Strong Lp solutions of the Navier–Stokes equations in Rm with applications to weak solutions, Math. Z. 187 (1984) 471–480. [19] J. Leray, Essai sur le mouvement d’un fluide visqueux emplissant l’espace, Acta Math. 63 (1934) 193–248. [20] A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Univ. Press, 2002. [21] H. Matano, F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. (2004) 1494–1541. [22] J.R. Miller, M. O’Leary, M. Schonbek, Nonexistence of singular pseudo-self-similar solutions of the Navier–Stokes system, Math. Ann. 319 (4) (2001) 809–815. [23] J. Neˇcas, M. Ružiˇcka, V. Šverák, On Leray’s self-similar solutions of the Navier–Stokes equations, Acta Math. 176 (2) (1996) 283–294. [24] T.-P. Tsai, On Leray’s self-similar solutions of the Navier–Stokes equations satisfying local energy estimates, Arch. Ration. Mech. Anal. 143 (1) (1998) 29–51.


Restriction estimates for some surfaces with vanishing curvatures ✩ Sanghyuk Lee a,∗ , Ana Vargas b a School of Mathematical Sciences, Seoul National University, Seoul 151-742, Republic of Korea b Department of Mathematics, University Autonoma de Madrid, 28049 Madrid, Spain

Received 18 June 2009; accepted 13 January 2010 Available online 11 February 2010 Communicated by J. Bourgain

Abstract We obtain bilinear restriction estimates for surfaces with vanishing curvatures. As application we also prove new linear restriction estimates for some class of conic surfaces. © 2010 Elsevier Inc. All rights reserved. Keywords: Restriction estimates; Conic surfaces

1. Introduction In this note we consider Fourier restriction estimates for some class of conic surfaces. It has been known that the curvature plays an important role in determining the boundedness of the restriction operators. Let S be a smooth compact surface in Rn+1 with the induced Lebesgue measure dσ . It is well known [3,7,11] that if k principal curvatures are nonzero at each point of the surface S, for q 2k+4 k f dσ Lq (Rn+1 ) Cf L2 (dσ ) . ✩

(1.1)

The first author was partially supported by the NRF of Korea grant (2009-0072531). The second author was partially supported by the grant MTM2007-60952 (Spain). * Corresponding author. E-mail addresses: [email protected] (S. Lee), [email protected] (A. Vargas). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.014

S. Lee, A. Vargas / Journal of Functional Analysis 258 (2010) 2884–2909

2885

The range on q is optimal as it can be easily seen using Knapp’s example. The natural conjecture is that the estimate f dσ Lq (Rn+1 ) Cf Lp (dσ ) 1 2k+2 holds if k+2 q k(1 − p ) and p > k . It was Bourgain [1] who first obtained a result beyond the sharp L2 –Lq restriction estimates when k 2. Recent development on the restriction problem has been made by considering a suitable bilinear version of the operator [5,8–10,12,13]. To be specific, let S1 , S2 be subsets of S with measures dσ1 , dσ2 . Let us consider the bilinear adjoint restriction estimate

f dσ1 g dσ2 Lp Cf L2 (dσ1 ) gL2 (dσ2 ) .

(1.2)

In this form of estimate one can impose additional conditions which specify the relative position of the two surfaces. One typical condition is transversality. For positively curved surfaces (e.g. the cone and the paraboloid) transversality makes it possible to get a wider range of boundedness than is allowed for the linear estimates. However, for the surfaces with positive and negative principal curvatures transversality is not enough to obtain such improvement. Actually one needs stronger (separation) conditions [5,12]. Unlike the case of linear estimate (1.1), the role of curvature in bilinear restriction estimates does not seem to be clearly understood. In fact, the sharp bilinear restriction estimates (1.2) for the cone and the paraboloid are valid for the same range of q except the endpoint even though the cone has only n − 1 nonzero principal curvatures. The estimate (1.2) has been studied mainly with surfaces with nonvanishing Gaussian curvature or one vanishing curvature. In this note we want to generalize the known bilinear restriction estimates to the surfaces having two or more vanishing curvatures. Let k n − 1 be an integer. We assume that S has k nonvanishing principal curvatures and n − k vanishing curvatures. In other words the surface S has n − k null directions along which the curvatures vanish. To be more precisely, let ±N(ξ ) ∈ Sn be the unite normal vector of S at ξ . By rotation and decomposition we may assume that |N(ξ ) − en+1 | 1/2. Definition 1.1. Suppose that S is a smooth compact surface with (possibly) boundary in Rn+1 . We say that S is of conic type with k nonvanishing curvatures if the following assumptions are satisfied: • The map dN : Tξ (S) → TN (ξ ) (Sn ) has k nonzero eigenvalues and n − k zero eigenvalues. • The nonzero eigenvalues have magnitude ∼ 1.1 We denote by Nξ (S) the span of the eigenvectors with zero eigenvalues of dN at ξ . We say that any nonzero vector v is in a null direction of S at ξ if v ∈ Nξ (S). Definition 1.2. Let S1 , S2 be subsets of a conic type surface S of k nonvanishing curvatures. We say that S1 and S2 are transversal if 1 For A, B > 0, A ∼ B means C −1 A B CA for some constant C > 0.

2886


N(ξ1 ) − N(ξ2 ) ∼ 1

(1.3)

for ξ1 ∈ S1 and ξ2 ∈ S2 . Let z1 , z2 be points in Rn+1 and let us denote the translated surfaces by Szj = Sj + zj , 2

j = 1, 2.

Then by the condition (1.3) we may assume that the intersection of two surfaces Sz1 and Sz2 is a smooth (n − 1)-dimensional immersed submanifold by dividing the surfaces into small pieces (if necessary), as long as the intersection is not empty. We denote the intersection by Πz1 ,z2 = Sz1 ∩ Sz2 . As it is well known, the dispersion of the normal vectors of a given surface accounts for the decay of Fourier transform of the surface measure, which was crucial in obtaining the linear restriction estimate (1.1). As it turns out, for the bilinear estimates the dispersion along the intersection Πz1 ,z2 is important. Roughly, the number of nonzero curvatures along Πz1 ,z2 takes the role that is played by the total number of nonzero curvatures in the linear estimates. (See Theorem 1.4 below.) This explains why the range of bilinear restriction estimates for the cone and paraboloid is essentially the same. Let us denote by Tξ (Πz1 ,z2 ) the tangent space of Πz1 ,z2 at ξ . Now we make an assumption on the surfaces S1 and S2 : dim Tξ (Πz1 ,z2 ) ⊕ Nξ1 (S1 ) = n, dim Tξ (Πz1 ,z2 ) ⊕ Nξ2 (S2 ) = n

(1.4)

for all ξ ∈ Πz1 ,z2 , ξ1 ∈ S1 and ξ2 ∈ S2 as long as Πz1 ,z2 = ∅. This is one of most important assumption which gives the maximal amount of dispersion of normal vectors along the intersection Πz1 ,z2 . Since the surface S has k nonvanishing principal curvatures, the maps Nzi : Szi → Sn have rank k. Here Nzj (ξ ) is the normal vector to Szi at ξ . Hence from the condition (1.4) one can j

easily see that for j = 1, 2, the map Nz1 ,z2 which is given by ξ ∈ Πz1 ,z2 :→ Nzj (ξ ) ∈ Sn j

is also of rank k. That is to say, the rank of dNz1 ,z2 is k, j = 1, 2. Finally we assume that / dN1z1 ,z2 Tξ (Πz1 ,z2 ) ⊕ span Nz1 (ξ1 ) , Nz2 (ξ2 ) ∈ / dN2z1 ,z2 Tξ (Πz1 ,z2 ) ⊕ span Nz2 (ξ2 ) Nz1 (ξ1 ) ∈

(1.5)

2 Obviously, multiplication of e−2π ixξ0 to f dσj does not have any effect on the estimate and the Fourier transform

of e−2π ixξ0 f dσj is supported in Sj + ξ0 . So, it is natural to consider conditions which are valid uniformly for the translated surfaces.


2887

for ξ ∈ Πz1 ,z2 , ξ1 ∈ Sz1 and ξ2 ∈ Sz2 as long as Πz1 ,z2 = ∅. For the positively curved surfaces (e.g. the cone, sphere, or paraboloid) this type of transversality can be obtained by the normal separation condition (1.3) but it is not the case for the surfaces with principal curvatures of different sings. This is actually the separation condition which was used to obtain the best possible bilinear restriction estimates for the hyperboloid [5,12]. Remark 1.3. The condition (1.5) is concerned with the transversality between the cone generated by the normal vectors from the intersection surface Πz1 ,z2 and the normal vectors to the opposite surface. More precisely, for j = 1, 2, let us set Γj = tNzj (ξ ): ξ ∈ Πz1 ,z2 , 1 |t| 2 . Then the condition (1.5) equivalently means that any normal vector N1 (N2 , resp.) of Sz1 (Sz2 , resp.) is transversal to Γ2 (Γ1 , resp.) because the tangent space of Γj is given by j dNz1 ,z2 (Tξ (Πz1 ,z2 )) ⊕ span{Nzj (ξ )}. The following is our main result: Theorem 1.4. Let 1 k n − 1. Suppose that S is a smooth compact surface of conic type in Rn+1 with k-nonvanishing curvatures. If the surfaces S1 , S2 ⊂ S satisfy the assumptions (1.3), (1.4) and (1.5), then for p > k+4 k+2 f dσ1 g dσ2 Lp Cf 2 g2 . This theorem is sharp in the sense that there are surfaces satisfying (1.3), (1.4) and (1.5) but the estimate fails for p < k+4 k+2 . See Remark 3.3 and the proof of Proposition 3.2. Remark 1.5. If one considers two transversal subsets of a cylinder in R3 satisfying (1.3), then the condition (1.5) is trivially satisfied. But Πz1 ,z2 is contained in a line which is parallel with the null direction. Hence (1.4) fails. As it can be easily seen, the best possible bilinear restrictionL2 estimate is L2 × L2 → L2 estimate. Also, considering the n-dimensional cylinder (ξ , ξ , 1 − |ξ |2 ), (ξ , ξ ) ∈ Rk × Rn−k and suitable transversal subsets S1 and S2 it is easy to see that (1.2) fails for p < k+3 k+1 even though the conditions (1.3) and (1.5) are satisfied. On the other hand, if we only assume the conditions (1.3) and (1.5) without (1.4), then one can show for 3 p > k+3 k+1 f dσ1 g dσ2 Lp Cf 2 g2 . But this is still better than the trivial L2 × L2 → L estimate (1.1).

k+2 k

estimate which follows from the linear

3 This can be shown by following the argument below. The only thing one has to observe is that the combinatorial 1

estimates in Lemma 2.3 lose an additional factor of R 2 without (1.4).

2888


The paper is organized as follows. In Section 2 we give the proof of Theorem 1.4. As application we consider some model surfaces of conic type in Section 3 and obtain new restriction estimates. 2. Proof of Theorem 1.4 This section is devoted to proving Theorem 1.4. The proof here is based on the induction on scale argument which was used to obtain the sharp bilinear restriction estimates [8,13] (also see [5,12]). However adaptations are needed to reflect the geometry of conic surfaces. By rotation, translation and breaking the surfaces S into surfaces of small diameter if necessary, we may assume that the surface S is given by the graph of a function. For a δ0 > 0 let φ be a smooth function such that φ : Q = [−δ0 , δ0 ]n → R and φ satisfies φ(0) = 0,

∇φ(0) = 0.

We may also assume that the surfaces S1 , S2 are given by the graphs of the function −φ over the set Q1 , Q2 ⊂ Q, respectively. That is to say, for j = 1, 2, Sj =

x, −φ(x) : x ∈ Qj

for some cubes Q1 , Q2 ⊂ Q. Then for j = 1, 2, let us define the extension operators Ej f (x, t) =

ei(x·ξ −tφ(ξ )) f (ξ ) dξ.

(2.1)

Qj

Since dσi is comparable to dξ , it is enough to show that for p >

k+4 k+2 ,

2 2

Ej fj C fj 2 . j =1

p

j =1

First, we decompose Ej f into a sum of wave packets. The wave packets have good localization properties in both Fourier transform side and (x, t)-space. Unlike the usual decomposition of an O(R −1 )-neighborhood of a conic surface [2,6,13], which takes into account the null directions, we use a more direct decomposition in spirit of [8] (also see [5]). It does not depend on the presence of null directions. 2.1. Wave packet decomposition at scale R For d > 0 and A ⊂ Rn , let us denote by A + O(d) the set x ∈ Rn : dist(x, A) < Cd , for some big constant C > 0. Let R 1. The wave packet decomposition at scale R makes the support of the functions be expanded by O(R −1/2 ) (see Lemma 2.1). So, we need to consider a little bit larger sets


2889

than Q1 , Q2 . For CR −1/2 < δ0 , let us set Vj = Qj + O( ). We define the spatial grid Y by Y = R 1/2 Zn and the frequency grids V1 , V2 , respectively by setting Vj = R −1/2 Zn ∩ Vj ,

j = 1, 2.

Let us set Wj = (y, v): (y, v) ∈ Y × Vj . For wj = (yj , vj ) ∈ Wj , we define the associated tube Twj by Twj = (x, t) ∈ Rn × R: |t| 2R, x − yj + t∇φ(vj ) R 1/2 . Obviously Tyj ,vj contains (yj , 0) and its major direction is parallel to (∇φ(vj ), 1) ∈ Rn × R, which is parallel to the normal vector of the surface Sj at (vj , φ(vj )). That is, ∇φ(vj ), 1 N(vj ,φ(vj )) .

(2.2)

The following is a modification of Lemma 4.1 in [8]. For a proof see [5]. Lemma 2.1 (Wave packet decomposition). Let φ be a smooth function defined on Q. Suppose that f1 , f2 are supported in Q1 , Q2 , respectively. If |t| 10R, we can write Ej fj as Ej fj (x, t) =

Cwj pwj (x, t),

(x, t) ∈ Rn × R,

wj ∈Wj

such that Cwj , pwj satisfy the following conditions: (P1) For j = 1, 2,

1/2 |Cwj |

2

Cfj 2 .

wj ∈Wj

(P2) For j = 1, 2, pwj = Ej pw j (·, 0) .

2890


(P3) If wj = (yj , vj ), then −1/2 . supp p wj (·, t) ⊂ ξ : ξ = vj + O R (P4) For any N > 0, |t| 10R,

|x − (yj + t∇φ(vj ))| −N −n/4 1+ . j ,vj (x, t) CN R R 1/2

py

In particular, if dist((x, t), Tyj ,vj ) R δ+1/2 , then py

j ,vj

(x, t) CR −100n .

(P5) For any S ⊂ Wj , 2 pwj (·, t) C#S. 2

wj ∈S

2.2. Induction on scale Since k 1, the Fourier transform of the surface measures have some decay at infinity. Hence, by the globalization Lemma 2.4 in [9], it is enough to show that for any α > 0 2

Ej fj i=1

k+4 L k+2 (Q(R))

CR α

2

fj 2 .

(2.3)

i=1

Here Q(R) is the cube which is centered at the origin and of side length R. Let us denote the estimates (2.3) by E ∗ (α). We may assume f1 2 = f2 2 = 1. Since |Ei fi (x, t)| Cf 2 , using the wave packet decomposition and the standard pigeonholing argument together with the property (P1) in Lemma 2.1 the proof of (2.3) reduces (modulo loss of (log R)2 in bounds4 ) to obtaining the estimate

w1 ∈W1 , w2 ∈W2

pw1 pw2

k+4

CR α |W1 |1/2 |W2 |1/2

(2.4)

L k+2 (Q(R))

for any subsets W1 ⊂ W1 and W2 ⊂ W2 whenever pwj is the L2 normalized wave packet satisfying (P2), (P3), (P4) and (P5). By rapid decay of pwj away from the associated tube Twj we may assume that Twj meets with Q(2R). The main part of the induction on scale argument is to establish the following implication: 4 Such loss is harmless due to the nature of the estimate.


2891

If (2.4) is valid for R 1, then for 0 < δ 1,

w1 ∈W1 , w2 ∈W2

pw1 pw2

CR α(1−δ)+cδ |W1 |1/2 |W2 |1/2

k+4

(2.5)

L k+2 (Q(R))

with c independent of R and δ. Hence we have the implication E ∗ (α) → E ∗ α(1 − δ) + cδ 5 for any α > 0. Choosing sufficiently small δ and iterating this estimates finitely many times one can show that R ∗ (α) is valid for any α > 0. Pigeonholing further we can specify some of the quantities involved. Let us partition Q(2R) into disjoint R 1/2 -cubes q and denote the collection of those cubes by Q(R). We classify the q, w1 and w2 using dyadic parameters. For each q ∈ Q let us set Wj (q) = wj ∈ Wj : R δ q ∩ Twj = ∅ . Here R δ q is the set having the same center as q and expanded to R δ times from q. We also set for each dyadic number μ1 , μ2 1, Qμ1 ,μ2 = q: Wj (q) ∼ μj , j = 1, 2 . For each wj , define Qμ1 ,μ2 (wj ) = q ∈ Qμ1 ,μ2 : R δ q ∩ wj = ∅ and for dyadic λ 1, we again set λ,μ1 ,μ2

Wj

= wj ∈ Wj : Qμ1 ,μ2 (wj ) ∼ λ .

Then by (P4) in Lemma 2.1 and pigeonholing, it is easy to see

w1 ∈W1 , w2 ∈W2

pw1 pw2

k+4

L k+2 (Q(R))

C(log R)

4

λj ,μ1 ,μ2

wj ∈Wj

, j =1,2

q∈Qμ1 ,μ2

χq pw1 pw2

k+4 L k+2 (Q(R))

+ O R −M

for some dyadic 1 λ1 , λ2 , μ1 , μ2 R C and large M. Hence the matter is reduced to showing 5 The losses of (log R)C are absorbed in R cδ .

2892


λj ,μ1 ,μ2

wj ∈Wj

χq pw1 pw2

q∈Qμ1 ,μ2

, j =1,2

k+4

L k+2 (Q(R))

C R α(1−δ)+cδ |W1 |1/2 |W2 |1/2 .

(2.6)

Now we fix the numbers λ1 , λ2 , μ1 , μ2 for the rest of this section. We partition Q(2R) into O(R (n+1)δ ) cubes b which are of side length R 1−δ and essentially disjoint. So, we get

q∈Qμ1 ,μ2

λj ,μ1 ,μ2 wj ∈Wj , j =1,2

λj ,μ1 ,μ2 wj ∈Wj , j =1,2

b

χq pw1 pw2

k+4

L k+2 (Q(R))

q∈Qμ1 ,μ2 , q⊂2b

χq pw1 pw2

k+4

.

L k+2 (b)

For each wj , let b(wj ) be the cube b which contains the maximal number of q ∈ Qμ1 ,μ2 (wj ). There may be many of such cubes but we simply choose one of them. We now define a relation λ ,μ ,μ between b and wj ∈ Wj j 1 2 by saying b wj

if b ∩ 10b(wj ) = ∅.

Since the number of b is O(R (n+1)δ ), it is obvious that q ∈ Qμ1 ,μ2 : wj ∩ R δ q = ∅, q ∩ 10b = ∅ R −cδ λj provided wj b. Then

λj ,μ1 ,μ2

b

wj ∈Wj

, j =1,2

q∈Qμ1 ,μ2 , q⊂2b

χq pw1 pw2

k+4

L k+2 (b)

I b + I b , b

where

I

I

b

b

=

=

w1 b, w2 b

w1 b, or w2 b

q∈Qμ1 ,μ2 , q⊂2b

χq pw1 pw2

q∈Qμ1 ,μ2 , q⊂2b

k+4

,

L k+2 (b)

χq pw1 pw2

k+4

L k+2 (b)

.


To simplify the notation, in the summations to be in the set

λ ,μ ,μ W1 1 1 2

λ ,μ ,μ × W2 2 1 2 .

w1 b, w2 b ,

2893

w1 b, or w2 b , (w1 , w2 ) is assumed

Therefore we are reduced to showing

I b + I b C R α(1−δ)+cδ |W1 |1/2 |W2 |1/2 . b

From the induction hypothesis (2.3) and (P5), it follows that

I

b

CR

α(1−δ)

2

wj ∈ W λj ,μ1 ,μ2 : b wj 1/2 j i=1

since the side length of b is ∼ R 1−δ . By Cauchy–Schwarz’s inequality it follows that

I b CR α(1−δ)

b

2

1/2 wj ∈ W λj ,μ1 ,μ2 : b wj j b

i=1

CR

α(1−δ)

2

2

{b: b wj }

1/2

λj ,μ1 ,μ2

i=1

CR α(1−δ)

wj ∈Wj

|Wj |1/2 .

i=1

For the second inequality we use the fact that there are only O(1) cubes b with b wj for λ ,μ ,μ

wj ∈ Wj j 1 2 . The induction assumption is used to handle highly concentrating part and under proper relation this gives slightly improved bound because the considered cube is of size R 1−δ . It is in fact the major advantage of the induction scale argument. However, we have to trade off such easiness of improvement with a detailed analysis when we handle less concentrating part. Now we need to show I b CR cδ |W1 |1/2 |W2 |1/2 . It follows from interpolation between the easy L1 -estimate, I b CR|W1 |1/2 |W2 |1/26 and the L2 -estimate

w1 b, or w2 b

q∈Qμ1 ,μ2 , q⊂2b

2 χq pw1 pw2 2

(2.7)

L (b)

λ ,μ1 ,μ2

Here the summation is taken over wj ∈ Wj j 6 This follows from the trace lemma, or (P5).

CR −k/2+cδ |W1 ||W2 |.

, j = 1, 2. Hence it remains to show (2.7).

2894


2.3. L2 estimates over R 1/2 -cubes For given v1 ∈ V1 and v2 ∈ V2 , let us set z1 = v2 , φ v2 ,

z2 = v1 , φ(v1 )

and also set z1 ,z2 = Π

2 Sj + zj + O R −1/2 i=1

which also can be considered as Πz1 ,z2 + O(R −1/2 ). For Uj ⊂ Wj , j = 1, 2, let us set z ,z Π 1 2

Uj

z1 ,z2 . = wj = (yj , vj ) ∈ Uj : vj , φ(vj ) + zj ∈ Π

Lemma 2.2. Let q be an R 1/2 -cube and Uj ⊂ Wj (q), j = 1, 2. Then, 2 cδ −(n−1)/2 p p |U1 ||U2 | min sup U1 Πz1 ,z2 , sup U2 Πz1 ,z2 . w1 w2 R R 2 z ,z z ,z w1 ∈U1 w2 ∈U2

L

1

2

1

2

Proof. Let us write the left-hand side of the above inequality as w1 ∈U1 w2 ∈U2

Iw1 ,w2 ,

where Iw1 ,w2 =

pw1 pw2 ,

w2 ∈U2

w1 ∈U1

pw1 pw2 .

Fixing w1 ∈ U1 and w2 ∈ U2 , it is enough to show that |Iw1 ,w2 | R cδ R −(n−1)/2 min U1 Πz1 ,z2 , U2 Πz1 ,z2 . By symmetry we only need to show |Iw1 ,w | R cδ R −(n−1)/2 U1 Πz1 ,z2 2

(2.8)

with w1 = (y1 , v1 ), w2 = (y2 , v2 ), z1 = (v2 , φ(v2 )) and z2 = (v1 , φ(v1 )). We observe that the Fourier supports of the functions w2 ∈U2 pw1 pw2 , w ∈U1 pw1 pw2 are 1 contained in the sets S2 + z2 + O R −1/2 ,

S1 + z1 + O R −1/2 ,


2895

respectively. So it is obvious that pw1 pw2 , pw1 pw2 = 0 only if w1 = (y1 , v1 ) and w2 = (y2 , v2 ) satisfy z2 + v2 , φ(v2 ) ∈ Πz1 ,z2 + O R −1/2 , z1 + v1 , φ v1 ∈ Πz1 ,z2 + O R −1/2 , and z2 + v2 , φ(v2 ) = z1 + v1 , φ v1 + O R −1/2 .

(2.9)

Therefore, |Iw1 ,w2 | is bounded by

pw pw , pw pw . 1 2

1

{w2 ∈U2 : v1 +v2 =v1 +v2 +O(R −1/2 )}

{w1 ∈U1 Πz1 ,z2 }

2

Hence it is now sufficient to show that for fixed w1 , w2 , w1 ,

pw pw , pw pw R cδ R −(n−1)/2 . 1 2 1 2

{w2 ∈U2 : v2 =v1 +v2 −v1 +O(R −1/2 )}

Since w1 , w2 , w1 are given, there are at most O(1) possible v2 in the inner summation. Note that U2 ⊂ W2 (q). That is, all the tubes Tw2 are passing through R δ q. Hence, there are at most O(R cδ )–w2 = (y2 , v) satisfying v = v2 because y2 ∈ Y are R 1/2 -separated. Using (P4) and the transversality (1.3) between the tubes Tw1 and Tw2 , it is easy to see that |pw1 pw2 , pw1 pw2 | R −(n−1)/2 . Therefore, we get the desired estimate. 2 2.4. Proof of (2.7) λ ,μ1 ,μ2

It should be noticed that wj ∈ Wj j the left-hand side of (2.7) as

w1 b, or w2 b

=

even though it is not explicitly written out. We write

q∈Qμ1 ,μ2 , q⊂2b

q∈Qμ1 ,μ2 , q⊂2b

2 χq pw1 pw2 2

w1 b, or w2 b

L (b)

2 pw1 pw2 2

.

L (q)

Discarding harmless O(R −M ) terms and using Lemma 2.2, we see that the right-hand side of the above is bounded by

2896


CR −(n−1)/2+cδ ×

1

2

q∈Qμ1 ,μ2 , q⊂2b

λ ,μ ,μ ,b Π λ ,μ ,μ ,b Π sup W1 1 1 2 (q) z1 ,z2 + sup W2 2 1 2 (q) z1 ,z2 ,

z1 ,z2

λ ,μ1 ,μ2 ,b

where Wi i

λ1 ,μ1 ,μ2 λ2 ,μ1 ,μ2 W (q)W (q)

z1 ,z2

λ ,μ1 ,μ2

= {w ∈ Wi 1

λ ,μ1 ,μ2

: w b}. Since |Wj i

(q)| μj and

λj ,μ1 ,μ2 W (q) Cλj |Wj |, 7 j

q∈Qμ1 ,μ2

one can easily see that

λ1 ,μ1 ,μ2 λ2 ,μ1 ,μ2 W (q)W (q) C min λ1 μ2 |W1 |, λ2 μ1 |W2 | . 1

2

q∈Qμ1 ,μ2 , q⊂2b

Therefore, to show (2.7) it is enough to show that if q ∈ Qμ1 ,μ2 and q ⊂ 2b, then λ ,μ ,μ ,b Π n−1−k λ1 μ2 W1 1 1 2 (q) z1 ,z2 CR 2 +cδ |W2 |, λ ,μ ,μ ,b Π n−1−k λ2 μ1 W2 2 1 2 (q) z1 ,z2 CR 2 +cδ |W1 |.

(2.10)

By symmetry it is enough to show one of the estimates. In fact, to prove (2.10) we need only to show the following. Lemma 2.3. Let q0 ∈ Q(R) be contained in Q(0, R) and let Q be a subset of Q(R). Additionally, let Uj be subset of Wj for j = 1, 2. Suppose that w ∈ Wj : R δ q ∩ Tw = ∅ ∼ μj

for q ∈ Q,

(2.11)

and for j = 1, 2 and some λj , μj 1, q ∈ Q: R δ q ∩ Tw = ∅, dist(q, q0 ) 2R 1−δ R −cδ λj j

(2.12)

provided wj ∈ Uj . For each q ∈ Q let us set Uj (q) = wj ∈ Uj : R δ q ∩ Twj = ∅ . Then, there are constants C and c such that λj ,μ1 ,μ2 7 It can be shown by interchanging the order of summation. More precisely, use (q)| = q∈Qμ1 ,μ2 |Wj δ λj ,μ1 ,μ2 |{q: R Twj ∩ q = ∅}|. wj ∈Wj


2897

n−1−k z ,z +cδ |W2 | U1 (q0 )Π 2 1 2 CR , λ1 μ2 n−1−k z ,z +cδ |W1 | U2 (q0 )Π 2 1 2 CR λ2 μ1 with c independent of δ.

Proof. By symmetry it is enough to show the estimate for |U1 (q0 )Πz1 ,z2 |. To begin with, let us set (∇φ(v1 ), 1) z1 ,z2 + O R −1/2 . NΠz1 ,z2 = : v1 , φ1 (v1 ) + z1 ∈ Π |(∇φ(v1 ), 1)| We consider the set

Λ1 =

R Tw1 ∩ Q(0, 2R) \ Q q0 , R 1−δ . δ

w1 ∈U1 (q0 )Πz1 ,z2

From the definition of U1 (q0 )Πz1 ,z2 the associated tubes Tw1 have directions which are normal z1 ,z2 . More precisely, the normal directions of the tubes are contained vectors to S1 + z1 along Π z ,z Π 1 2 in the set N , and all the tubes Tw1 meet with R δ q0 . Hence one can see that Λ1 is contained 1/2+cδ -neighborhood of the conic set in an R Γ1 (R) = tN: N ∈ NΠz1 ,z2 , R 1−δ |t| 4R + q0 . 1

This is actually an isotropic dilation of the set Γ1 + (R − 2 ) by factor R. Since the surface S has k nonvanishing curvatures, and by (1.4), the normal map ξ ∈ Πz1 ,z2 :→ Nz1 (ξ ) ∈ Sn has rank k. Since Πz1 ,z2 has dimension n − 1, the tubes Tw1 , w1 ∈ U1 (q0 )Πz1 ,z2 can overlap at most n−1−k O(R cδ+ 2 ) over the set Λ. It can be more clearly seen using implicit function theorem. Hence we have

χR δ Tw CR cδ+

n−1−k 2

(2.13)

.

w∈U1 (q0 )Πz1 ,z2

By (2.12) we have λ1 U1 (q0 )Πz1 ,z2 CR cδ

q ∈ Q: R δ q ∩ Tw = ∅, dist(q, q0 ) R 1−δ .

w∈U1 (q0 )Πz1 ,z2

Hence it follows that n+1 λ1 U1 (q0 )Πz1 ,z2 CR cδ− 2

w∈U1 (q0 )Πz1 ,z2

By (2.13) we get

χR δ Tw Λ

q∈Q

χq dx dt.

2898


n+1 λ1 U1 (q0 )Πz1 ,z2 CR − 2

χR δ Tw

n−1−k 2

χq dx dt

q∈Q

w∈U1 (q0 )Πz1 ,z2

Λ

CR cδ+

{q ∈ Q: q ⊂ Λ1 }.

On the other hand it is not difficult to see that the condition (1.5) implies that the tube Tw2 meets the opposite tube cone Γ1 (R) transversally. Hence we see that Tw2 can intersect only O(R cδ ) number of q ⊂ Λ1 . (See Remark 1.3.) This means that the collections of tubes {Tw2 }w2 ∈W2 (q) are essentially disjoint along q ⊂ Λ1 (overlapping at most R cδ ) so that W2 (q) CR cδ |W2 |. q∈Λ1

Hence by (2.11) it follows that

μ2 {q ∈ Q: q ⊂ Λ1 } C

w ∈ W2 : R δ q ∩ Tw = ∅

q∈Q: q⊂Λ1

CR cδ |W2 |. Therefore we get the required inequality comparing the upper and lower bounds for |{q ∈ Q: q ⊂ Λ}|. 2 3. Applications to linear restriction estimates In this section we apply Theorem 1.4 to obtain linear restriction estimates for some surfaces with vanishing curvatures. It is also possible to obtain results for more general surfaces but instead we do it with some model surfaces to avoid unnecessary complications. Let 1 L n − 1 and let n1 , n2 , . . . , nL 2 be positive integers satisfying n1 + n2 + · · · + nL = n. For each 1 l L, let ηl ∈ Rnl −1 and ρ l ∈ [1, 2]. We set η = η1 , . . . , ηL ∈ Rn−L ,

ρ = ρ 1 , . . . , ρ L ∈ [1, 2]L .

Abusing notation, we sometimes denote (ρ 1 η1 , ρ 2 η2 , . . . , ρ L ηL ) by ρη = ρ 1 η1 , ρ 2 η2 , . . . , ρ L ηL when it is more convenient. Now, the angular variable θ = θ (ξ ) for ξ = (η, ρ) is defined by setting θ = θ 1, θ 2, . . . , θ L =

ηL η1 η2 , , . . . , ρL ρ1 ρ2

=

η ∈ [1, −1]n−L . ρ


2899

Using these notations, we write the variable ξ so that ξ = (η, ρ) = (ρθ, ρ). Let us set Q = (η, ρ): |η| 1, ρ ∈ [1, 2]L . Then we consider the smooth conic surface S=

ξ, −φ(ξ ) : ξ ∈ Q

with φ given by

φ(ξ ) =

L

±

l=1

L 2 |ηl |2 = ±ρ l θ l . l ρ

(3.1)

l=1

Here the sign can be chosen arbitrarily for each summand. Obviously S has L vanishing and n − L nonvanishing curvatures. Instead of dealing with the operator f → f dσ , we consider as before the equivalent operator Eφ given by Eφ f (x, t) =


Q

We define a mixed norm space by

f Lp Lqρ = θ

[1,−1]n−L

q> that

p/q

1

p

dθ

.

[1,2]L

Theorem 3.1. Let φ be given by (3.1). If 2n−2L+8 n−L+2

f (ρθ, ρ)q dρ

n−L+2 q

(n − L)(1 −

1 p)

and p 2, then, for

when n − L 2, and for q > 4 when n − L = 1, there is a constant C such

Eφ f q Cf Lp L2 . θ

ρ

This is obviously stronger than the usual Lp –Lq estimate when p > 2. An interpolation with the trivial estimate Eφ f ∞ Cf L1 L1 further strengthens the restriction estimate slightly. θ

ρ

(n − L)(1 − p1 ) By adapting Knapp’s example one can easily show that the condition n−L+2 q p is necessary. When the surface is the cone, the Lθ L2ρ → Lq estimates were obtained for

2900


the non-endpoint case [13].8 Theorem 3.1 also includes the endpoint case. We prove Theorem 3.1 by obtaining bilinear estimates with suitable separation. The following is what we need. Proposition 3.2. Let Q1 , Q2 be the cubes contained in Q and the extension operators E1 , E2 be defined by (2.1). Suppose that η1 η2 − ∼1 (3.2) ρ ρ2 1 (equivalently |θ1 − θ2 | ∼ 1) for each (η1 , ρ1 ) ∈ Q1 , (η2 , ρ2 ) ∈ Q2 . Then for q >

n−L+4 n−L+2 ,

2 2

Ej fj C fj 2 . j =1

q

j =1

Remark 3.3. Adapting the usual counterexample in [10], one can show that the estimate fails for q < n−L+4 n−L+2 . In fact, for 0 < 1 and i = 1, 2, let fi be the characteristic functions of the sets (η, ρ) ∈ Q: η1 1 + (−1)i 2 , ρ 1 − 1 , η∗ , where η = ((η1 )1 , η∗ ) ∈ R × Rn−L−1 . Let x = (y, z) ∈ Rn−L × RL , y = ((y 1 )1 , y ∗ ) ∈ R × Rn−L−1 and z = (z1 , z∗ ) ∈ R × RL−1 . Then the condition (3.2) is satisfied on the supports of f1 , f2 , and |Ef1 Ef2 (x, t)| ∼ 2(n−L+2) on the set {(x, t) = (y, z, t): |t| c −2 , |(y 1 )1 | c −2 , |y ∗ | c −1 , |z1 − t| c −1 , |z∗ | c} for some small c > 0. The bilinear L2 estimate implies 2(n−L+2) −(n−L+4)/q C (n−L+2) . Hence letting → 0, we see that q n−L+4 n−L+2 . Assuming Proposition 3.2 for the moment, we prove Theorem 3.1. The argument below is an adaptation of the usual argument which was used to derive linear estimates from bilinear estimates [10]. To obtain the endpoint estimates we also use a simple summation argument involving Lorentz spaces and real interpolation (see, for instance [4]). Proof of Theorem 3.1. Since f is supported in a fixed compact set, it is enough to show Theorem 3.1 at the endpoint (p, q) that satisfies

n−L+2 1 = (n − L) 1 − . (3.3) q p We consider separately the cases n − L 2 and n − L = 1 and first prove the case n − L 2. Since the other case can be handled similarly, we only give a brief remark at the end of proof. When n − L 2, due to the known L2 restriction estimate (1.1) [3,7,11] we may assume that p > 2 and q < 4. To exploit the separation condition we make a decomposition of Eφ f Eφ f . Let I = k of side 2−k . Here m is the [−1, 1]n−L . For each k ∈ Z+ we partition I into dyadic cubes Θm 8 That is, n−L+2 < (n − L)(1 − 1 ). q p


2901

k ∼ Θ k if Θ k and Θ k have index running over these cubes of side length 2−k . We write Θm m m m adjacent parents, but are not adjacent. As in [10], we use a Whitney type decomposition of I × I away from its diagonal D, so that

(I × I \ D) =

∞

k k Θm × Θm .

k ∼Θ k k=0 Θm m

We set fmk (η, ρ) = χΘmk

η f (η, ρ) = χΘmk (θ )f (ρθ, ρ). ρ

Then it follows that (Ef )(Ef ) =

E fmk E fmk .

(3.4)

k ∼Θ k k0 Θm m

Here we drop the subscript of Eφ and do so for the rest of the proof. Fixing k, we claim that for q > 2n−2L+8 n−L+2 and p 2 k k E f E f m m k ∼Θ k Θm m

C2

(n−L+2) 2k( (n−L) )−(n−L) p + q

f 2Lp L2 . θ

q/2

(3.5)

ρ

Once one has (3.5), then the desired bound can be shown by using a simple summation argun−L+2 ment. In fact, let us fix p0 , q0 satisfying q0 > 2n−2L+8 = (n − L)(1 − p10 ). n−L+2 , p0 > 2 and q0 Using (3.5), for q >

2n−2L+8 n−L+2

we have

k k E fm E fm k ∼Θ k Θm m

C2

2k(n−L+2)( q1 − q1 ) 0

q/2

f 2 p0

Lθ L2ρ

Let ∞K be an integer to be chosen later. Then using (3.4) and splitting the sum K+1 , we see that (x, t): (Ef Ef )(x, t) > λ (x, t):

(3.6)

. k

to

K

−∞ ,

K k k λ E fm E fm > 2 k −∞ k

+ (x, t):

Θm ∼Θm

∞

k ∼Θ k K+1 Θm

k k λ E fm E fm > . 2

m

Let us choose q1 , q2 such that 2n−2L+8 n−L+2 < q1 < q0 < q2 . Obviously, using triangle inequality and (3.6), by summation of geometric series we get

2902


K

k=−∞ Θ k ∼Θ k

∞

m

m

k=K+1 Θ k ∼Θ k m

k k E fm E fm

C2

1

0

f 2 p0

C2

2K(n−L+2)( q1 − q1 ) 2

,

Lθ L2ρ

q1 /2

k k E fm E fm

m

2K(n−L+2)( q1 − q1 )

0

f 2 p0

Lθ L2ρ

q2 /2

.

Hence by Chebyshev’s inequality and the above we have q1 (x, t): (Ef Ef )(x, t)> λ C2K(n−L+2)(1− q0 ) f q1p 0

Lθ L2ρ

+ C2

q

K(n−L+2)(1− q2 ) 0

f

λ−q1 /2

q2 λ−q2 /2 . p Lθ 0 L2ρ

Then by choosing K which optimizes the right-hand side of the above inequality we get Ef Lq0 ,∞ Cf Lp0 L2 . θ

ρ

Finally, this estimate is valid for p0 , q0 satisfying q0 > 2n−2L+8 n−L+2 , p0 > 2 and (3.3). Real interpolation among these estimates gives the desired because p0 q0 . Hence now it remains to show (3.5). k ∼ Θ k are supported in Observe that for a fixed k the Fourier transforms of E(fmk )E(fmk ), Θm m boundedly overlapping parallelepipeds. Hence, for 2 q 4 we have k k E fm E fm k ∼Θ k Θm m

C

q/2 2/q E f k E f k . m

q/2

m

q/2

(3.7)

k ∼Θ k Θm m

This follows from interpolation between L2 and trivial L1 estimates (see Lemma 6.1 in [10]). By k ∼ Θ k , then this, we are reduced to showing that if Θm m (n−L) (n−L+2) k k E f E f C22k( p + q )−(n−L) f k p 2 f k p 2 . m m L L m m L L q/2 θ

ρ

θ

ρ

In fact, putting this in the right-hand side of the above, we have k k E f E f m m k ∼Θ k Θm m

C2

(n−L+2) 2k( (n−L) )−(n−L) p + q

q/2

×

q/2 q/2 2/q f k p 2 f k p 2 . m L L m L L k ∼Θ k Θm m

θ

ρ

θ

ρ

(3.8)


2903

Therefore the desired inequality (3.5) follows if one can show that

q/2 q/2 2/q q/2 q/2 f k p 2 f k p 2 Cf Lp L2 f Lp L2 . m L L m L L k ∼Θ k Θm m

θ

ρ

θ

ρ

θ

ρ

θ

ρ

q By Cauchy–Schwarz’s inequality, the left-hand side is bounded by ( m fmk Lp L2 )2/q . It is θ

ρ

again bounded by f 2Lp L2 since q p and the angular projections of the supports of fmk are θ

ρ

disjoint. Now we proceed to show (3.8). Let Θ1 , Θ2 ⊂ [−1, 1]n−L be cubes of diam(Θ1 ), diam(Θ2 ) −k 2 , and dist(Θ1 , Θ2 ) 2−k . For j = 1, 2, let us set P Θj , 2−k = (η, ρ) ∈ Q: η = ρθ, θ ∈ Θj . For (3.8) we need show that Ef Egq/2 C2

(n−L+2) 2k( (n−L) )−(n−L) p + q

f Lp L2 gLp L2 θ

ρ

θ

(3.9)

ρ

whenever f , g are supported in P (Θ1 , 2−k ), P (Θ2 , 2−k ), respectively. Let θ0 = (θ01 , θ02 , . . . , θ0L ) ∈ Rn−L be the center of the smallest cube containing both of Θ1 , Θ2 . We write the phase part of the extension operator Ef as x · ξ − tφ(ξ ) = x˜ · η + x¯ · ρ − t

L l=1

|ηl |2 ± l ρ

where x = (x, ˜ x) ¯ ∈ Rn−L × RL . Let us set x˜ = (x˜ 1 , x˜ 2 , . . . , x˜ L ), x˜ l ∈ Rnl −1 and x¯ = 1 2 L (x¯ , x¯ , . . . , x¯ ), x¯ l ∈ R. The phase part is transformed to L 2 l l x˜ · θ0 + x¯ l ∓ t θ0l ρ l − t (x˜ ∓ 2tθ0 ) · η + l=1

L l=1

|ηl |2 ± l ρ

under the change of variables (η, ρ) → (η + ρθ0 , ρ) = η1 + θ01 ρ 1 , η2 + θ02 ρ 2 , . . . , ηL + θ0L ρ L , ρ . Note that this sends the parallelepipeds P (Θ1 , 2−k ), P (Θ2 , 2−k ) to P (Θ1∗ , 2−k ), P (Θ2∗ , 2−k ), respectively. Here Θ1∗ , Θ2∗ are cubes contained in [−2−k+1 , 2−k+1 ]n−L with dist(Θ1∗ , Θ2∗ ) 2−k . Performing the change of variables (η, ρ) → (η + ρθ0 , ρ) for both of the integrals E(f ), E(g), we see (Ef Eg)(x, t) = (E f˜E g)(T ˜ x, t),

2904


where f˜, g˜ are functions satisfying that f˜Lp L2 = f Lp L2 , g ˜ Lp L2 = gLp L2 , and θ

ρ

ρ

θ

θ

ρ

ρ

θ

2 2 T (x) = x˜ ∓ 2tθ, x˜ 1 · θ01 + x¯ 1 ∓ t θ01 , . . . , x˜ L · θ0L + x¯ L ∓ t θ0L . Since |det T | = 1, the matters are reduced to showing that Ef Egq/2 C2

(n−L+2) 2k( (n−L) )−(n−L) p + q

f Lp L2 gLp L2 θ

ρ

θ

ρ

whenever f , g are supported in P (Θ1∗ , 2−k ), P (Θ2∗ , 2−k ). Now we make an additional change of variables (η, ρ) → 2−k η, ρ for both of the integrals Ef , Eg to see we see ˜ x, ¯ 2−2k t (Ef Eg)(x, t) = 2−2k(n−L) (Efk Egk ) 2−k x, where fk , gk are functions satisfying that fk Lp L2 = 2k(n−L)/p f Lp L2 , θ

ρ

θ

ρ

gk Lp L2 = 2k(n−L)/p gLp L2 θ

ρ

θ

ρ

and the supports of fk , gk are contained in P (Θ1o , 12 ), P (Θ2o , 12 ), satisfying Θ1o , Θ2o ⊂ [−1, 1]n−L and dist(Θ1o , Θ2o ) 1. Then, after rescaling, it is sufficient to show that Efk Egk q/2 Cfk Lp L2 gk Lp L2 θ

ρ

θ

ρ

provide that fk , gk are supported in P (Θ1o , 12 ), P (Θ2o , 12 ), respectively. This follows from Proposition 3.2. Hence we get (3.5). This completes the proof of Theorem 3.1 for the case n − L 2. Now we turn to the remaining case n − L = 1. This condition implies that n = 2 and L = 1. We only need to show (3.5) in a neighborhood of the segment determined by 3/q + 1/p = 1, q > 4, because once this is obtained the rest of the argument works without modification. Since 4 < q, we need to replace (3.7) with k k E f E f m m k ∼Θ k Θm m

C q/2

k k (q/2) (2/q) E f E f

m

m

(q/2)

k ∼Θ k Θm m

which is valid for q 4. Then, using (3.9), one can get the desired estimate (3.5) as long as 2(q/2) p. There is a small neighborhood of the segment determined by 3/q + 1/p = 1, q > 4, where this condition is satisfied. 2


2905

Proof of Proposition 3.2. To prove Proposition 3.2, we use Theorem 1.4. It is enough to verify the conditions (1.3), (1.4) and (1.5) when the surface is given by (ξ, φ(ξ )) and (3.2) is satisfied. We only show it for the case φ(ξ ) =

L |ηl |2 l=1

ρl

.

It is not difficult to see that the same is valid for the other cases. The geometry of Πz1 ,z2 can be complicated and the conditions (1.3), (1.4) and (1.5) are not so easy to check directly in practical applications. So we simplify the picture by projecting Πz1 ,z2 to Rn . It generally makes some information be lost but in our example we can still get the sharp results. By finite decomposition we may assume that Q1 and Q2 are as small as we wish. More precisely, let (ρ1 θ1 , ρ1 ), (ρ2 θ2 , ρ2 ) ∈ Q. Then, we may assume that the surfaces are given as graphes of φ over the sets Q1 = (ρθ, ρ) ∈ Q: |θ − θ1 | < 0 , |ρ − ρ1 | < 0 , Q2 = (ρθ, ρ) ∈ Q: |θ − θ2 | < 0 , |ρ − ρ2 | < 0 , respectively, for some small 0 > 0. Hence by continuity, to verify the conditions (1.3), (1.4) and (1.5) it is enough to show them at each point (ρ1 θ1 , ρ1 ), (ρ2 θ2 , ρ2 ) ⊂ Q if 0 is sufficiently small. Let us denote by πz1 ,z2 the projection of Πz1 ,z2 to spatial space. That is, πz1 ,z2 = {v: (v, τ ) ∈ Πz1 ,z2 }. Let us write zi = (vi , τi ) for i = 1, 2. Since Szi = {(u, φ(u)) + zi : u ∈ Qi }, i = 1, 2, one can easily see πz1 ,z2 = v ∈ (Q1 + v1 ) ∩ (Q2 + v2 ): φ(v − v1 ) − φ(v − v2 ) = τ2 − τ1 . Hence πz1 ,z2 is an (n−1)-dimensional immersed surface as long as ∇φ(v −v1 )−∇φ(v −v2 ) = 0 for all v ∈ πz1 ,z2 . Obviously this condition is equivalent to (1.3). Now note that 2 2 ∇φ(ρθ, ρ) = 2θ 1 , 2θ 2 , . . . , 2θ L , −θ 1 , . . . , −θ L . So we have for ξ1 = (ρ1 θ1 , ρ1 ), ξ2 = (ρ2 θ2 , ρ2 ), 2 2 2 2 ∇φ(ξ1 ) − ∇φ(ξ2 ) = 2θ1 − 2θ2 , θ21 − θ11 , . . . , θ2L − θ1L . Hence, the condition (1.3) is satisfied by (3.2). Note that at the point ξ = (ρθ, ρ) the projected null directions9 are given the span of the vectors N 1 (θ ) = θ 1 , 0, . . . , 0, 1, 0, . . . , 0 , N 2 (θ ) = 0, θ 2 , 0, . . . , 0, 0, 1, 0, . . . , 0 , .. .

N L (θ ) = 0, . . . , 0, θ L , 0, . . . , 1 . 9 That is, (N j · ∇)2 φ = 0.

2906


The condition (1.4) is satisfied as long as at least one of the null directions from each surface is transversal to the tangent space of Πz1 ,z2 . Since the surface πz1 ,z2 has dimension n − 1, (1.4) is satisfied if one of the projected null directions N 1 (θ1 ), N 2 (θ1 ), . . . , N L (θ1 ) is transversal to the tangent space of πz1 ,z2 . On the other hand the normal vector to πz1 ,z2 is ∇φ(ρ1 θ1 , ρ1 ) − ∇φ(ρ2 θ2 , ρ2 ) + O( 0 ). So it is enough to show that there are some i, k such that ∇φ(ρ1 θ1 , ρ1 ) − ∇φ(ρ2 θ2 , ρ2 ), N i (θ1 ) ∼ 1, ∇φ(ρ1 θ1 , ρ1 ) − ∇φ(ρ2 θ2 , ρ2 ), N k (θ2 ) ∼ 1. By direct computation one can easily see that the right-hand side of the first expression is equal to |θ1i − θ2i |2 and that of the second is |θ1k − θ2k |2 . Due to the condition (3.2), there must be i, k (in fact, i = k) satisfying the above. Hence (1.4) follows. Now it remains to show (1.5). By Remark 1.3 we need to show that for j = 1, 2, the normal vectors Nj of surfaces Szj are transversal to the opposite tube cones Γ3−j . Instead of considering the sets Γ1 , Γ2 , we consider larger sets which are given by Γj = t ∇φ(v), 1 : v ∈ Qj , 2−1 |t| 2 . Obviously Γj ⊂ Γj . So, if the opposite normal Nj is transversal to Γ3−j , it is also transversal to Γ3−j . By the homogeneity of φ Γj = tΦ(θ ): θj ∈ Θj , 2−1 |t| 2 where Θj = { ρη : (η, ρ) ∈ Qj } and 2 2 Φ(θ ) = t 2θ 1 , 2θ 2 , . . . , 2θ L , −θ 1 , . . . , −θ L , 1 . We only consider the case j = 2 because the other case j = 1 is obvious from symmetry. From the above the tangent space of Γ1 at each point is spanned by the partial derivatives (∂(θ l )k Φ(θ1 ), 0), and (Φ(θ1 ), 1). Here (θ l )k denotes the k-th component of θ l . The direction of opposite tube is also given by (Φ(θ2 ), 1). So the transversality between N1 and Γ2 follows if one can show that these n − L + 1 vectors (∂(θ L )k Φ(θ1 ), 0), and Φ(θ1 ) − Φ(θ2 ) are linearly independent. Equivalently, we need to show that the n × (n − L + 1) matrix M = ∂θ Φ(θ1 ), Φ(θ1 ) − Φ(θ2 ) has a minor of size (n − L + 1) × (n − L + 1) with nonvanishing determinant. Here we write Φ(θ1 ), Φ(θ2 ) as column vectors and ∂θ Φ(θ1 ) is an n × (n − L) matrix. If one removes the last L − 1 rows from M except (n − L + l)-th, then we have the matrix l = 2In−L M v

u α

!


2907

where In−L is the identity matrix of (n − L) × (n − L), u = 2θ1 − 2θ2 , α = |θ2l |2 − |θ1l |2 and v = (0, . . . , 0, −2θ1l , 0, . . . , 0). By a simple computation one see l = 2n−L−1 2α − u, v = 2n−L θ l − θ l 2 . det M 1 2 l with nonzero determinant because Therefore there is an (n − L + 1) × (n − L + 1) minor M det M1 , . . . , det ML cannot be zero simultaneously by the condition (3.2). This completes the proof of Proposition 3.2. 2 Since the estimates are stable under a small smooth perturbation of the surface, it is possible to obtain the same results for other surfaces which are of similar type. For example, let us consider ψ(ξ ) =

L l ξ l=1

where ξ = (ξ 1 , . . . , ξ L ) ∈ Rn1 × · · · × RnL . Then let Eψ be the operator given by


Eψ f (x, t) = {|ξ l |∼1, 1lL}

We now split variable ξ l and write ξ l = (ηl , ρ l ) ∈ Rnl −1 × R. By a finite decomposition of Q and rotation in each ξ l one may assume Q = ξ = ξ 1 , . . . , ξ L : ηl 0 , ρ l ∼ 1, 1 l L with small 0 > 0. Then by the linear change of variables (x, t) → x˜ 1 − t x¯ 1 , x˜ 2 − t x¯ 2 , . . . , x˜ L − t x¯ L , t ˜ )= ψ is replaced by ψ(ξ ˜ )= ψ(ξ

L

l=1 |(η

l , ρ l )| − ρ l .

Then by Taylor’s expansion

L L l 4 l 4 1 l 2 1 |ηl |2 η θ . θ = + O ρ + O l 2 ρl 2 l=1

l=1

This is a small smooth perturbation of ψ . Hence the bilinear estimate can be similarly formulated for Eψ because all the required (1.3), (1.4) and (1.5) are stable under smooth perturbation. By this stability the bilinear → linear argument also works. Let r l , θ l be the spherical coordinates for ξ l such that ξ l = r l θ l and define a mixed norm

f Lp Lqr = θ

Sn1 −1 ×···×SnL −1

f (rθ )q dr

p/q

1

p

dθ

[1,2]L

where θ = (θ 1 , . . . , θ L ) and r = (r 1 , . . . , r L ). Then we get the following.

2908


Corollary 3.4. Let ψ be the function given as in the above. If then for q > that

2n−2L+8 n−L+2

n−L+2 q

(n−L)(1− p1 ) and p 2,

when n − L 2, and for q > 4 when n − L = 1, there is a constant C such

Eψ f q Cf Lp L2 . θ

r

Remark 3.5. Proposition 3.2 can be further generalized. Let A1 , . . . , AL be nonsingular symmetric matrices of (nl − 1) × (nl − 1). Then consider

φ(ξ ) =

L

ρ l Al θ l , θ l ,

l=1

where θ l = ηl /ρ l . Here we use the same notations which are used in Proposition 3.2. Then the same bilinear estimate holds under the conditions L l l A θ − θl , θl − θl ∼ 1 1 2 1 2 l

for (ρ1 θ1 , ρ1 ) ∈ Q1 , (ρ2 θ2 , ρ2 ) ∈ Q2 . This is a generalization of the bilinear restriction estimate for a conic surface which was studied in [5]. However, when Al has eigenvalues of different signs it is still in question whether the sharp linear estimates can be obtained from these bilinear estimates. References [1] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991) 147–187. [2] J. Bourgain, Estimates for cone multipliers, in: Geometric Aspects of Functional Analysis, Israel, 1992–1994, in: Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 41–60. [3] A. Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981) 519–537. [4] S. Lee, Endpoint estimates for the circular maximal function, Proc. Amer. Math. Soc. 131 (5) (2003) 1433– 1442. [5] S. Lee, Bilinear restriction estimates for surfaces with curvatures of different signs, Trans. Amer. Math. Soc. 358 (8) (2006) 3511–3533. [6] G. Mockenhaupt, A note on the cone multiplier, Proc. Amer. Math. Soc. 117 (1) (1993) 145–152. [7] R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (3) (1977) 705–714. [8] T. Tao, A sharp bilinear restriction estimate for paraboloids, Geom. Funct. Anal. 13 (2003) 1359–1384. [9] T. Tao, A. Vargas, A bilinear approach to cone multipliers. I. Restriction estimates, Geom. Funct. Anal. 10 (2000) 185–215. [10] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjecture, J. Amer. Math. Soc. 11 (1998) 967–1000. [11] P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975) 477–478. [12] A. Vargas, Restriction theorems for a surface with negative curvature, Math. Z. 249 (1) (2005) 97–111. [13] T. Wolff, A sharp cone restriction estimate, Ann. of Math. 153 (2001) 661–698.


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Further reading [14] B. Barcelo, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985) 321–333. [15] T. Tao, A. Vargas, A bilinear approach to cone multipliers. II. Application, Geom. Funct. Anal. 10 (2000) 216–258.


Rough paths analysis of general Banach space-valued Wiener processes S. Dereich Philipps-Universität Marburg, Fb. 12 – Mathematik und Informatik, Hans-Meerwein-Straße, D-35032 Marburg, Germany Received 10 July 2009; accepted 14 January 2010 Available online 6 February 2010 Communicated by Paul Malliavin

Abstract In this article, we carry out a rough paths analysis for Banach space-valued Wiener processes. We show that most of the features of the classical Wiener process pertain to its rough path analog. To be more precise, the enhanced process has the same scaling properties and it satisfies a Fernique type theorem, a support theorem and a large deviation principle in the same Hölder topologies as the classical Wiener process does. Moreover, the canonical rough paths of finite dimensional approximating Wiener processes converge to the enhanced Wiener process. Finally, a new criterion for the existence of the enhanced Wiener process is provided which is based on compact embeddings. This criterion is particularly handy when analyzing Kunita flows by means of rough paths analysis which is the topic of a forthcoming article. © 2010 Elsevier Inc. All rights reserved. Keywords: Rough paths; Wiener process; Support theorem; Large deviation principle

1. Introduction The notion rough path was coined by Terry Lyons in 1994 [20]. The corresponding theory provides an extension of Young integrals to less regular driving signals. In the context of probability theory, it allows an alternative representation for solutions to Stratonovich differential equations as solutions to rough path differential equations (RDE). The power of the approach is that once the driving signal has been associated to a rough path, the solution can be written E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.018

S. Dereich / Journal of Functional Analysis 258 (2010) 2910–2936

2911

as continuous function (Itô map) of the rough path signal by Terry Lyons’ universal limit theorem [22]. In general irregular controls admit several extensions to rough paths. Nonetheless, in the context of stochastic analysis there is one canonical choice which is uniquely defined up to a null set. Since the null set does not depend on the choice of the RDE this allows one to pick a random RDE depending on the path itself and, in particular, the concept of filtrations becomes obsolete for the existence of solutions. As eluded by Ledoux, Qian, and Zhang [18] the theory of rough paths leads to natural proofs of support theorems (ST) and large deviation principles (LDP) since both properties behave nicely under an application of the continuous Itô map. Consequently, it suffices to prove a support theorem and a large deviation principle for the canonical rough path of the driving signal and then to infer the corresponding results for the solution of the SDE. This approach has been firstly carried out in [18] for the multi-dimensional Wiener process under the p-variation topology for p > 2. Later on, analogous results were proved under fine Hölder topologies by Friz and Victoir [10] (see also [12]). General Banach space-valued Wiener processes were embedded into the theory of rough paths by Ledoux et al. [17] in 2002. A series of articles by Inahama and Kawabi [14–16] followed which was mainly motivated by its applicability to heat kernel measures on loop spaces. Nowadays the theory of rough paths is well established and we refer the reader to the monographs [21,23], and [11] for a general account on the topic. Our results are manifold. First we establish a representation of the enhanced Wiener process as limit of finite dimensional enhanced Wiener processes. This Itô–Nisio type theorem implies that the enhanced Wiener process has the same scaling properties as the classical Wiener process. For finite dimensional Wiener processes there are various ways to define the canonical rough path (either as solution to a Stratonovich stochastic differential equation or via the limit of certain smooth approximations, see for instance [11]) and it is thus conceived as a universal object. Since we can freely approximate the enhanced infinite dimensional Wiener process by finite dimensional approximations, also the infinite dimensional canonical rough path can be seen as a universal object. We derive a support theorem and a large deviation principle in fine Hölder topologies similar as the one known for finite dimensional processes [10]. By doing so we extend results of [14] who analyzed the problem under p-variation topology. In general, the existence of the canonical rough path is not trivial (at least for projective tensor products), and Ledoux et al. provide a sufficient criterion in [17]. We relate their concept of finite dimensional approximation to entropy numbers of compact embeddings. Since these are known for various embeddings [7,8], we obtain a new sufficient criterion which can be easily verified in many cases. We phrase its implications in the case where the state space of the Wiener process is a Hölder–Zygmund space. Our main interest in the theory developed here is its applicability to stochastic flows generated by Kunita type SDEs. Indeed, we will establish a support theorem and a large deviation principle for Brownian flows in a forthcoming article [5]. We start with summarizing the results of the article. Here we introduce some notation rather in an informal way in order to enhance readability. All notation will be introduced in great detail at the end of this section. Results Let (V , | · |V ) be a separable Banach space, and let X = (Xt )t∈[0,1] denotes a V -valued Wiener process on a probability space (Ω, F , P). More explicitly, X is measurable with respect to the Borel sets of C([0, 1], V ) and satisfies, for 0 s < t 1,

2912


• Xt − Xs is independent of σ (Xw : w s) and • L((t − s)−1/2 (Xt − Xs )) = L(X1 ) is a centered Gaussian distribution. We denote by (H1 , | · |H1 ) and (H, | · |H ) the reproducing kernel Hilbert spaces of X1 and X = (Xt )t∈[0,1] . Note that H can be expressed in terms of H1 as · H=

2

ft dt: f ∈ L [0, 1], H1

,

0

where the integral is to be understood as Bochner integral. For a general account on Gaussian distributions we refer the reader to the books by Lifshits [19] and Bogachev [2]. We let ϕ : (0, 1] → (0, ∞) denote an increasing function with ϕ(δ) =∞ lim √ δ↓0 −δ log δ

(1)

and consider the geometric ϕ-Hölder rough path space GΩϕ (V), which will be rigorously introduced below. Moreover, X is assumed to possess a canonical rough path X in the sense of assumption (E), see Section 2. Theorem 1.1. X is almost surely an element of GΩϕ (V) and its range in that space is the closure of the lift of the reproducing kernel Hilbert space H of X into GΩϕ (V). Theorem 1.2. The family {Xε : ε > 0} with Xε being the canonical rough path of (ε · Xt )t∈[0,1] satisfies a LDP in GΩϕ (V) with good rate function J (h) =

1

2 2 |h|H

∞

if ∃h ∈ H with h = S(h), else,

where S denotes the canonical lift of H into GΩϕ (V). Suppose now that the reproducing kernel Hilbert space H1 of X1 is infinite dimensional and fix a complete orthonormal system (ei )i∈N of H1 . We represent X as the in C([0, 1], V ) almost sure limit Xt = lim

n→∞

n

(i)

ξt ei ,

(2)

i=1

(i)

with {(ξt )t∈[0,1] : i ∈ N} being an appropriate family of independent standard Wiener processes. (i) Theorem 1.3. Each X (n) = ( ni=1 ξt ei )t∈[0,1] possesses a canonical rough path X(n) , and one has almost sure convergence lim X(n) = X in GΩϕ (V).

n→∞


2913

Remark 1.4. The latter theorem can be proved in a more general setting. One can replace the assumption that (ei ) is a complete orthonormal system of the reproducing kernel Hilbert space by the assumption that, for any h ∈ H1 , n lim h, ei H1 ei = h.

n→∞

i=1

We proceed with a sufficient criterion for the existence property (E) for Hölder–Zygmund spaces. For an open and bounded set D, for n ∈ N0 , η ∈ (0, 1], and γ = n + η, we denote by γ C0 (D, Rd ) the set of n-times differentiable functions f : Rd → Rd whose derivatives are ηHölder continuous and satisfy the zero boundary condition f |D c = 0. The space is endowed with a canonical norm · C γ , see (14). Theorem 1.5. Let 0 < γ < γ¯ and D ⊂ D be bounded and open subsets of Rd with D¯ ⊂ D . γ¯ Let μ be a centered Gaussian measure on C0 (D, Rd ). Then there exists a separable and closed γ subset V ⊂ C0 (D , Rd ) and a V -valued Wiener process X = (Xt )t∈[0,1] satisfying the existence property (E) and L(X1 ) = μ. For instance, one may choose V as the closure of H1 . γ¯

Remark 1.6. In the theorem, the two Banach spaces C0 (D, Rd ) and C0 (D , Rd ) can be replaced by arbitrary Banach spaces V1 and V2 for which V1 is compactly embedded into V2 and for which the entropy numbers of the embedding decay at least at a polynomial order, see Section 5 for the details. In particular, one can use the results of Edmunds and Triebel [7,8] on embeddings of Sobolev and Besov spaces. γ

Let us summarize the implications in a language that does not incorporate rough paths. Let W denote a further Banach space, let f : W → L(V , W ) be a Lip(γ )-function for a γ > 2 in the sense of [23, Def. 1.21]. For a fixed absolutely continuous path g : [0, 1] → V with differential in L2 ([0, 1], V ), we consider the Young differential equation dyt = f (yt ) d[x + g]t ,

y0 = ξ.

(3)

By Picard’s theorem, the differential equation possesses a unique solution Ig (x) for any x ∈ BV(V ) (actually unique solutions exist under less restrictive assumptions). For a V -valued Wiener process X and n ∈ N, we denote by X(n) the dyadic interpolation of X with breakpoints Dn = [0, 1] ∩ (2−n Z). We call Y the Wong–Zakai solution of (3) for the control X, if {Ig (X(n)): n ∈ N} is a convergent sequence in Cϕ ([0, 1], W ) with limit Y . As is well known Wong–Zakai solutions are tightly related to stochastic differential equations in the Stratonovich sense: If the state space W is an M-type 2 Banach space and if dgt = v dt for some v ∈ V , then the Wong–Zakai solution solves the corresponding Stratonovich stochastic differential equation, see [3]. Theorem 1.7. Let X be a V -valued Wiener process with reproducing kernel Hilbert space H . If property (E) is valid, then the following is true:

2914


(I) X admits a Wong–Zakai solution Y of (3). (II) For n ∈ N, let X (n) be as in Theorem 1.3 and denote by Y (n) the corresponding Wong–Zakai solution of (3). Then {Y (n) : n ∈ N} converges almost surely to Y in Cϕ ([0, 1], W ). (III) The range of Y in Cϕ ([0, 1], W ) is the closure of Ig (H ). (IV) For ε > 0, we let Y ε denote the Wong–Zakai solution of (3) for the control ε · X. Then {Y ε : ε > 0} satisfies a large deviation principle in Cϕ ([0, 1], W ) with speed (ε 2 )ε>0 and good rate function

1 J (h) = inf x H : x ∈ H, Ig (x) = y . 2 Here and elsewhere, the infimum of the empty set is defined as ∞. Agenda The article is organized as follows. Sections 3 and 4 are concerned with the derivation of Theorems 1.1 and 1.2, respectively. Section 2 has rather preliminary character. Here, we prove a preliminary version of the representation provided by Theorem 1.3 (see Remark 3.3 for the extension to the stronger statement). Moreover, we derive a Fernique type theorem together with Lévy’s modulus of continuity. Section 5 is concerned with the proof of Theorem 1.5. Finally, we explain the implications of the theory of rough paths (Theorem 1.7) in Section 6. Notation We start with introducing the (for us) relevant notation of the theory of rough paths. For a profound background on the topic we refer the reader to the textbooks [23] and [11]. For two Banach spaces V1 and V2 we denote by V1 ⊗a V2 the algebraic tensor product of V1 and V2 . Moreover, we let V1 ⊗ V2 denote the projective tensor product, that is the completion of V1 ⊗a V2 under the projective tensor norm | · |V1 ⊗V2 given by |v|V1 ⊗V2 = inf

n

|fi |V1 |gi |V2 ,

i=1

where the infimum is taken over all representations v=

n

f i ⊗ gi

(n ∈ N, fi ∈ V1 , gi ∈ V2 for i = 1, . . . , n).

i=1

From now on V denotes a separable Banach space. For a (continuous) piecewise linear function x : [0, 1] → V , we may compute its iterated (Young) integrals t x1s,t

=

dxu ∈ V , s

x2s,t = s 0: P X ∈ BΩϕ (f, ε) > 0 if and only if ∀ε > 0:

P T−f X ∈ BΩϕ (0, ε) > 0.

Due to the Cameron Martin Theorem the latter statement is equivalent to ∀ε > 0:

P X ∈ BΩϕ (0, ε) > 0.

¯ On the other hand, we have X = TX(n) (X(n)), and by Corollary 2.9 there exists a constant c = c(ϕ) such that for any ε > 0:

2926


¯ ˙ P( X ϕ 2cε) P X(n)

ϕ ε, X(n)

L2 ([0,1],V ) ε ¯ ˙ P X(n)

ϕ ε P X(n)

L2 ([0,1],V ) ε . →1 (n→∞)

>0

¯ In the last step we have used the independence of X(n) and X(n).

2

4. Large deviations Let again X denote a V -valued Wiener process satisfying assumption (E) with enhanced process X. For ε > 0 let X ε = (Xtε )t∈[0,1] = (εXt )t∈[0,1] and recall that the family {X ε : ε > 0} satisfies a large deviation principle in C([0, 1], V ) with good rate function J (h) =

1

2 2 h H

if h ∈ H, else.

∞

The aim of this section is to prove: Theorem 4.1. The family {Xε : ε > 0} given by Xε = (δε (Xt ))t∈[0,1] = Γ (X ε ) satisfies a LDP in GΩϕ (V) with good rate function J (h) =

1

2 2 h H

∞

if ∃h ∈ H with h = S(h), else.

Similarly as in [17] and [10], the proof uses the concept of exponentially good approximations. It mainly relies on the isoperimetric inequality and the following estimate. Lemma 4.2. Let n ∈ N, and denote by h : [0, 1] → V an absolutely continuous function with h˙ ∈ L2 ([0, 1], V ). Set f = Υn (h) and g = h − f . There exists a universal constant C such that for x, y ∈ Ωϕ (V) and κ = x ϕ + y ϕ , one has Th (x) − Tf (y) 1 + 2 β x ϕ + y ϕ + 4βn h

˙ L2 ([0,1],V ) , ϕ βn where β = supδ∈(0,1]

√ δ ϕ(δ)

and βn = supδ∈(0,1∧21−n/2 ]

√ δ ϕ(δ)

→ 0 as n → ∞.

Proof. We denote x = x1 , y = y1 , f = Υn (h) and g = h − f . By Jensen’s inequality one ˙ L2 ([0,1],V ) and thus the triangle inequality gives that g

˙ L2 ([0,1],V ) has f˙ L2 ([0,1],V ) h

˙ L2 ([0,1],V ) . Moreover, 2 h

t Th (x)s,t − Tf (y)s,t = xs,t + S(f + g)s,t +

t (f + g)s,u ⊗ dxu +

s

t − ys,t − S(f )s,t −

t fs,u ⊗ dyu −

s

xs,u ⊗ d(f + g)u s

ys,t ⊗ dfu s


t = xs,t − ys,t + gs,t +

t hs,u ⊗ dgu +

s

gs,u ⊗ dfu s

t

t

+

fs,u ⊗ d(x − y)u +

gs,u ⊗ dxu

s

s

t

t

+

(x − y)s,u ⊗ dfu + s

xs,u ⊗ dgu .

(11)

s

We need to control the norms of each single term above. We start with | Clearly, t hs,u ⊗ dgu

V ⊗2

s

2927

t s

hs,u ⊗ dgu |V ⊗2 .

˙ 22 hs,· L∞ ([s,t],V ) g

˙ L1 ([s,t],V ) 2(t − s) h

. L ([s,t],V )

For t − s 2−n one can refine this estimate as follows. Let s t0 · · · tN t with {t0 , . . . , tN } = Dn ∩ [s, t], and observe that t hs,u ⊗ dgu s

V ⊗2

t0 hs,u ⊗ dgu s

+ V ⊗2

t + hs,u ⊗ dgu tN

ti+1 hs,u ⊗ dgu

N −1 i=0

ti

V ⊗2

. V ⊗2

˙ 22 Since t0 − s 2−n the first term is bounded by 2−n+1 h

. For v ∈ V one has L ([s,t0 ],V ) dgu = 0 so that ti+1 hs,u ⊗ dgu ti

V ⊗2

ti+1 = hti ,u ⊗ dgu ti

V ⊗2

˙ 22 2−n+1 h

L ([t ,t

i i+1 ],V )

.

Moreover, the remaining term is bounded by t hs,u ⊗ dgu tN

V ⊗2

˙ 22 hs,· L∞ ([s,t],V ) g

˙ L1 ([tN ,t],E) 2−n/2+1 h

. L ([0,1],V )

Combining the above estimates yields t hs,u ⊗ dgu s

V ⊗2

˙ 22 ˙ 22 2 2−n + 2−n/2 h

2−n/2+2 h

L ([0,1],V ) L ([0,1],V )

ti+1 ti

v⊗

2928


so that in general t hs,u ⊗ dgu s

V ⊗2

˙ 22 2 (t − s) ∧ 21−n/2 h

. L ([0,1],V )

Similarly, one finds the same estimate for

t s

gs,u ⊗ dfu and

! ˙ L2 ([0,1],V ) . |gs,t |V 2 (t − s) ∧ 21−n h

We proceed with the next term in (11): t fs,u ⊗ d(x − y)u s

√ ˙ L2 ([0,1],V ) ϕ(t − s) t − s x − y ϕ h

V ⊗2

√ ˙ L2 ([0,1],V ) , 2κϕ(t − s) t − s h

where κ := x ϕ + y ϕ . Analogously one √ finds that also the remaining three terms from (11) ˙ L2 ([0,1],V ) . We now combine the above estihave norm smaller or equal to 2κϕ(t − s) t − s h

mates: Th (x)s,t − Tf (y)s,t ! ˙ L2 ([0,1],V ) κϕ(t − s) + 2 (t − s) ∧ 21−n h

√ ˙ 22 ˙ L2 ([0,1],V ) . + 2 (t − s) ∧ 21−n/2 h

+ 2κϕ(t − s) t − s h

L ([0,1],V ) Next, we use that

!

(t − s) ∧ 21−n/2 βn ϕ(t − s) to conclude that

Th (x)s,t − Tf (y)s,t ˙ L2 ([0,1],V ) + 2 βn2 h

˙ 22 ˙ L2 ([0,1],V ) ϕ(t − s) κ + 2βn h

+ 2κβ h

L ([0,1],V ) β ˙ L2 ([0,1],V ) ϕ(t − s). κ + 4βn h

2 1+2 βn Proof of Theorem 4.1. We will use the concept of exponentially good approximations. Recall that by Lemma A.3, the map S ◦ Υn : C([0, 1], E) → GΩϕ (E) is continuous. Therefore, the processes Xε (n) = S(X ε (n)) (ε > 0) satisfy a large deviation principle with good rate function Jn (h) =

1

2 2 h H

∞

if ∃h ∈ H : h = S(h) and h = Υn (h), else.

It remains to show that the approximation is exponentially good (see [4, Thm. 4.2.23]), in the sense that for every δ > 0,


2929

lim lim sup ε 2 log P dϕ Xε , Xε (n) > δ = −∞

(12)

n→∞

ε↓0

and that for every α > 0,

lim sup dϕ S(x), S Υn (x) : x ∈ H, x H α = 0.

n→∞

(13)

We start with verifying (12). Recall that we can fix κ > 0 such that P(X ∈ A)

1 2

for # " A = x ∈ C [0, 1], V : x0 = 0, Γ (x) exists, Γ (x)ϕ + supΥn (x)ϕ κ . n∈N

With the isoperimetric inequality we conclude that P(X ∈ Ar ) Φ(r) for the sets Ar = A + BH (0, r)

(r 0).

Here Φ denotes again the standard normal distribution function. By Lemma 4.2 it follows that for z = x + h ∈ Ar (with x ∈ A and h ∈ BH (0, r)): z − z(n) = Th (x) − Tf x(n) C βn σ h H + κ(1 + β/βn ) , ϕ ϕ where β and βn → 0 are as in the lemma and σ is the norm of the canonical embedding of H1 into V . Hence, we can find a standard normal random variable N (on a possibly larger probability space) such that for any n ∈ N: X − X(n) C βn σ N+ + κ(1 + β/βn ) , ϕ where N+ = N ∨ 0. Now choose αn = 13 (C 2 σ 2 βn2 )−1 . Then limn→∞ αn = ∞ and 2 2 Cn := Eeαn X−X(n) ϕ E exp αn C 2 βn σ N+ + κ(1 + β/βn )

1 2 = E exp N+ + O(N+ ) < ∞. 3 By Chebychev’s inequality we get for η 0: 2 P X − X(n)ϕ η Cn e−αn η and hence lim sup η→∞

1 log P X − X(n)ϕ η −αn → −∞ as n → ∞. 2 η

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Now assertion (12) is a consequence of ε X − Xε (n) = δε (X) − δε X(n) = δε X − X(n) = ε X − X(n) . ϕ ϕ ϕ ϕ Finally note that (13) is an immediate consequence of Lemma 4.2. Indeed, for h ∈ H and f = Υn (h) one gets S(h) − S(f ) = Th (0) − Tf (0) 3βn h

˙ L2 ([0,1],E) 3σβn h

˙ H ϕ ϕ with limn→∞ βn = 0.

2

5. A sufficient criterion for exactness In this section we introduce a new sufficient criterion for the exactness of a Gaussian random vector attaining values in a Hölder–Zygmund space. For n ∈ N and for an n-times continuously differentiable function f : Rd → Rd , we set

f C n =

sup ∂ α f (x),

|α|n x∈R

d

where the sum is taken over all multiindices α with entries in {1, . . . , d} of length smaller or equal to n. Moreover, for γ = n + η with n ∈ N0 and η ∈ (0, 1], and an n-times continuously differentiable function f : Rd → Rd , we consider the Hölder–Zygmund norm of order γ

f C γ = f C n +

sup

|α|=n x=y

|∂ α f (x) − ∂ α f (y)| |x − y|η

(14)

γ

and we denote by C0 (D, Rd ) (D ⊂ Rd open) the set of n-times continuously differentiable functions satisfying f |D c ≡ 0

and f C γ < ∞.

It is endowed with the norm · C γ . Theorem 5.1. Let 0 γ < γ¯ and let D, D ⊂ Rd denote bounded open sets with D¯ ⊂ D . Every γ¯ centered Gaussian measure μ on C0 (D, Rd ) is exact and Bochner measurable when viewed as γ Gaussian measure on C0 (D , Rd ). The proof is based on the concept of finite dimensional approximation introduced in [17]. For a Banach space V (not necessarily separable) and a V -valued random vector Y , we denote the linear average Kolmogorov widths of Y by

n (Y ) = Vn (Y ) = inf EY − Tn (Y )V : Tn : V → V linear, rk(Tn ) n

(n ∈ N).

Lemma 5.2. Let Y be a V -valued Gaussian random vector and suppose that n (Y ) n−ε for some ε > 0. Then Y is exact and Bochner measurable in V .


2931

Proof. Since n (Y ) decays to zero Y is Bochner measurable. Without loss of generality we assume that V is infinite dimensional. Let G1 be a μ-distributed r.v. For fixed n ∈ N there exists a bounded operator Tn : V → V with n-dimensional range and EG1 − Tn (G1 ) 2n (Y ) =: ε(n). Set F1 := Tn (G1 ) and observe that there are n independent standard normals ξ11 , . . . , ξn1 and n vectors e1 , . . . , en ∈ V such that F1 =

n

ξi1 ei .

i=1

Moreover, set H1 = G1 − F1 . Then for each i = 1, . . . , n one has !

2/π|ei |V = Eξ 1 ei E|F1 | E|G1 | + E|H1 | C,

where C := C(Y ) := 21 (Y ) + E|Y | does only depend on the distribution of Y . ˜ l , F˜l , H˜ l , (ξ˜ l )i=1,...,n )l∈N denote independent Let now (Gl , Fl , Hl , (ξil )i=1,...,n )l2 and (G i 1 copies of (G1 , F1 , H1 , (ξi )i=1,...,n ). Then N

˜l = Gl ⊗ G

l=1

N

Fl ⊗ F˜l +

l=1

N

Fl ⊗ H˜ l +

l=1

N

Hl ⊗ F˜l +

l=1

N

Hl ⊗ H˜ l

(15)

l=1

and N E Fl ⊗ F˜l l=1

Using that E|

N

l ˜l l=1 ξi ξj |

V ⊗2

N n n E ξil ξ˜jl |ei ⊗ ej |V ⊗2 . i=1 j =1

l=1

√ N we arrive at

N E Fl ⊗ F˜l l=1

√ √ π d 2 N max |ei |2V C 2 d 2 N . i=1,...,n 2

(16)

NE|F1 |E|H˜ 1 | N ε(n)E|F1 | CN ε(n).

(17)

V ⊗2

On the other hand, N E Fl ⊗ H˜ l l=1

V ⊗2

The same estimate holds for E|

N

l=1 Hl

N ˜ E Hl ⊗ Hl l=1

⊗ F˜l |V ⊗2 . Finally, the last term in (15) is bounded by 2 N E|H1 | N ε(n)2 . V ⊗2

(18)

2932


When choosing n = n(N ) = N 1/(4+2ε) and letting n tend to infinity one obtains with (16), (17) and (18) that N ˜ l N (4+ε)/(4+2ε) , E Gl ⊗ G

as N → ∞,

l=1

which implies exactness.

2

In the forthcoming proof of Theorem 5.1, we use a result by Pisier [25] that provides an estimate for the average Kolmogorov width against entropy numbers of generating operators. For two Banach spaces E and V , and a compact operator ρ : E → V we define the n-th entropy number as

n−1 2$ BV (bj , ε) for some b1 , . . . , b2n−1 ∈ V . en (ρ) = inf ε > 0: u BE (0, 1) ⊂ j =1

Proof of Theorem 5.1. We denote by ρ the canonical embedding of the reproducing kernel γ¯ Hilbert space of Y into C0 (D, Rd ). By Pajor and Tomczak-Jaegermann [24], one has en (ρ) n−1/2 . The asymptotic behavior of the entropy numbers for general Besov embeddings were studied by γ¯ Edmunds and Triebel [7,8]. In particular, one has for the canonical embedding : C0 (D, Rd ) → γ d C0 (D , R ) that en () n−

γ¯ −γ d

.

Combining the above estimates with the general estimate ek+l−1 ( ◦ ρ) ek ()el (ρ) (k, l ∈ N0 ) gives 1

en ( ◦ ρ) n− 2 −

γ¯ −γ d

.

Now note that ◦ ρ generates the Gaussian random element Y on C0 (D , Rd ). In order to control (Y ) we use a result of Pisier (see [25, Thm. 9.1, p. 141]) combined with the duality of metric entropy found in [1]: γ

n (Y ) C1

k −1/2 (log k)ek ( ◦ ρ)

kC2 n

for two universal constants C1 , C2 > 0. Combining this estimate with the above result on the entropy numbers gives n (Y ) n−

γ¯ −γ d

log n.

2


2933

6. Consequences of Lyons’ universal limit theorem In this section, we use Lyons’ universal limit theorem together with our findings to derive Theorem 1.7. Let V and W denote two Banach spaces and let f : W → L(V , W ) a Lip(γ )function for a γ > 2 in the sense of [23, Def. 1.21]. As in the introduction, Ig denotes the solution operator for controls x ∈ BV(V ) to the differential equation dyt = f (yt ) d[x + g]t ,

y0 = ξ,

where g ∈ C([0, 1], V ) is an absolutely continuous function with g˙ ∈ L2 ([0, 1], V ). Next, fix a real p ∈ (2, 3 ∧ γ ) and an increasing function ϕ¯ : (0, 1] → (0, ∞) that is dominated log δ by ϕ and satisfies limδ↓0 −δϕ(δ) = 0 such that ϕ¯ p is convex. By Lemma A.4, there always exists ¯ an appropriate ϕ¯ and since Cϕ¯ ([0, 1], W ) is continuously embedded into Cϕ ([0, 1], W ), it suffices to prove Theorem 1.7 in Cϕ¯ ([0, 1], W ). By choice of ϕ¯ the space GΩϕ¯ (V) is continuously embedded into GΩp (V), where GΩp (V) denotes the space of all geometric rough paths induced by the p-variation norm. Hence, the universal limit theorem (see for instance [23, Thm. 5.3]) implies that the rough path differential equation dyt = f (yt ) dxt ,

y0 = idD

induces a continuous solution operator I : GΩϕ¯ (V) → GΩϕ¯ (W) (Itô map) and the following diagram commutes for any piecewise linear V -valued path x:

x

Tg

xg

I0

y ξ +πW (·)

S

x

I

y

Assuming assumption (E), the processes {X(n): n ∈ N} converge in GΩϕ¯ (V ) to X. By the continuity of Tg : GΩϕ¯ (V ) → GΩϕ¯ (V ) (Lemma A.2), I : GΩϕ¯ (V ) → GΩϕ¯ (W ) (Lyons’ universal limit theorem), and ξ + πW (·) : GΩϕ¯ (W ) → Cϕ¯ ([0, 1], W ), we conclude that {Y (n): n ∈ N} converges to ξ + πW ◦ I0 ◦ Tg (X) which is statement (I) of the theorem. Assertions (II)–(IV) are now immediate consequences of Theorems 1.3, 1.1, and 1.2, respectively. Appendix A. Preliminary results For u ∈ V we set |||u||| = inf

n

ui ,

(19)

i=1

where the infimum is taken over all representations u = arbitrary.

%n

i=1 ui

with n ∈ N and u1 , . . . , un ∈ V

2934


Lemma A.1. • ||| · ||| satisfies the triangle inequality with respect to ∗. • For t ∈ R and u ∈ V, one has |||δt u||| = |t||||u|||. • Moreover, · and ||| · ||| are equivalent: |||u||| u 2|||x|||. Proof. The proof of the first % two statements is straight forward and we only present the proof of the third statement. Let u = ni=1 ui . Then u1 = ni=1 u1i and |u1 |V ni=1 |u1i |V . Moreover, since u2 = ni=1 u2i + i<j u1i ⊗ u1j we get u 2

V ⊗V

& ' ' n 2 1 1 ( = ui + ui ⊗ uj i=1

& ' n ' ( u 2

i<j

)

i V ⊗V

i=1

+

V ⊗V

n 1 u i V i=1

n 1 u + u2 =

ui

i V i V ⊗V

n i=1

so that u 2|||u|||.

*2

i=1

2

Lemma A.2. Let f ∈ C([0, 1], V ) be an absolutely continuous with f˙ ∈ L2 ([0, 1], V ) and let √ > 0. Then the shift operator ϕ : (0, 1] → (0, ∞) be an increasing function with infδ∈(0,1] ϕ(δ) δ

Tf : Ωϕ (V) → Ωϕ (V) is continuous. Proof. For x, y ∈ Ωϕ (V), one has t Tf (x)s,t − Tf (y)s,t = xs,t − ys,t +

t fs,u ⊗ d(x − y)u +

s

(x − y)s,u ⊗ dfu , s

where x = πV (x) and y = πV (y). The process f is of bounded variation, and one has t fs,u ⊗ d(x − y)u s

f˙ L1 ([0,1],V ) V ⊗V

sup

|xu,v − yu,v |V .

suvt

√ Next, recall that f˙ L1 ([s,t],V ) t − s f˙ L2 ([0,1],V ) and that, by assumption, there exists a √ constant c = c(ϕ) with t − s c ϕ(t − s). Consequently,


t fs,u ⊗ d(x − y)u s

2935

c f˙ L2 ([0,1],V ) x − y ϕ ϕ(t − s)2 , V ⊗V

and ) t * fs,u ⊗ d(x − y)u

c f˙ L2 ([0,1],V ) x − y ϕ .

(s,t)∈ ϕ

s

t Analogously one finds the same estimate for ( s (x − y)s,u ⊗ dfu )(s,t)∈ ϕ so that Tf (x) − Tf (y) x − y ϕ + 2 c f˙ 2 L ([0,1],V ) x − y ϕ . ϕ

2

Lemma A.3. For n ∈ N, the maps S ◦ Υn : C(Dn , V ) → GΩϕ (V)

and

log ◦S ◦ Υn : C(Dn , V ) → Cϕ ( , V)

are continuous. Moreover, for x, y ∈ C([0, 1], V ), x = S ◦ Υn (x), and y = S ◦ Υn (y), one has |log xs,t − log ys,t |V 2n+1 x − y ∞ 1 + 2 x ∞ + 2 y ∞ (t − s),

for (s, t) ∈ .

The proof is straight-forward and therefore omitted. Lemma A.4. For any p > 2 and ϕ : (0, 1] → (0, ∞) increasing with exists a function ϕ¯ : (0, 1] → (0, ∞) such that ϕ¯ p is convex and ! Proof. Set φ(δ) =

√

−δ log δ ϕ(δ) ¯ ϕ(δ),

√ −δ log δ ϕ(δ) there

for δ ∈ (0, 1].

−δ log δ and define φm,u : (0, 1] → [0, ∞) for m ∈ N and u > 0 via

p φm,u (δ) =

mφ p (δ)

if δ u,

mφ p (u) + (mφ p ) (u)(δ − u)

otherwise.

As one easily verifies by taking derivatives, φ p is convex on an appropriate set (0, ε). Hence, p φm,u is convex provided that u ε. Moreover, one can check that, for every m ∈ N, there exists u(m) ∈ (0, ε) such that φm,u(m) ϕ. Consequently, taking ϕ¯ = supm∈N φm,u(m) finishes the proof. 2 References [1] S. Artstein, V. Milman, S.J. Szarek, Duality of metric entropy, Ann. of Math. (2) 159 (3) (2004) 1313–1328. [2] V.I. Bogachev, Gaussian Measures, Math. Surveys Monogr., vol. 62, American Mathematical Society, Providence, RI, 1998. [3] Z. Brze´zniak, A. Carroll, Approximations of the Wong–Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces, in: Séminaire de Probabilités XXXVII, in: Lecture Notes in Math., vol. 1832, Springer, Berlin, 2003, pp. 251–289.

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[4] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, second edition, Appl. Math. (New York), vol. 38, Springer, New York, 1998. [5] S. Dereich, G. Dimitroff, Large deviations and a support theorem for Brownian flows, in preparation. [6] G.A. Edgar, L. Sucheston, Stopping Times and Directed Processes, Encyclopedia Math. Appl., vol. 47, Cambridge University Press, Cambridge, 1992. [7] D.E. Edmunds, H. Triebel, Entropy numbers and approximation numbers in function spaces, Proc. Lond. Math. Soc. (3) 58 (1) (1989) 137–152. [8] D.E. Edmunds, H. Triebel, Entropy numbers and approximation numbers in function spaces, II, Proc. Lond. Math. Soc. (3) 64 (1) (1992) 153–169. [9] P. Friz, H. Oberhauser, Isoperimetry and rough path regularity, preprint. [10] P. Friz, N. Victoir, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist. 41 (4) (2005) 703–724. [11] P. Friz, N. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge University Press, 2010. [12] P.K. Friz, Continuity of the Itô-map for Hölder rough paths with applications to the support theorem in Hölder norm, in: Probability and Partial Differential Equations in Modern Applied Mathematics, in: IMA Vol. Math. Appl., vol. 140, Springer, New York, 2005, pp. 117–135. [13] A.M. Garsia, E. Rodemich, H. Rumsey Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20 (1970/1971) 565–578. [14] Y. Inahama, H. Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. Lond. Math. Soc. (2) 73 (3) (2006) 797–816. [15] Y. Inahama, H. Kawabi, Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths, J. Funct. Anal. 243 (1) (2007) 270–322. [16] Y. Inahama, H. Kawabi, On the Laplace-type asymptotics and the stochastic Taylor expansion for Itô functionals of Brownian rough paths, in: Proceedings of RIMS Workshop on Stochastic Analysis and Applications, in: RIMS Kôkyûroku Bessatsu, vol. B6, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 139–152. [17] M. Ledoux, T. Lyons, Z. Qian, Lévy area of Wiener processes in Banach spaces, Ann. Probab. 30 (2) (2002). [18] M. Ledoux, Z. Qian, T. Zhang, Large deviations and support theorem for diffusion processes via rough paths, Stochastic Process. Appl. 102 (2) (2002) 265–283. [19] M.A. Lifshits, Gaussian Random Functions, Math. Appl. (Dordrecht), vol. 322, Kluwer Academic Publishers, Dordrecht, 1995. [20] T. Lyons, Differential equations driven by rough signals, I: An extension of an inequality of L.C. Young, Math. Res. Lett. 1 (4) (1994) 451–464. [21] T. Lyons, Z. Qian, System Control and Rough Paths, Oxford Math. Monogr., Oxford University Press, Oxford, 2002. [22] T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310. [23] T.L. Lyons, M.J. Caruana, T. Lévy, Differential equations driven by rough paths, in: Ecole d’Eté de Probabilités de Saint-Flour XXXIV – 2004, in: Lecture Notes in Math., vol. 1908, Springer, Berlin, 2007. [24] A. Pajor, N. Tomczak-Jaegermann, Remarques sur les nombres d’entropie d’un opérateur et de son transposé, C. R. Acad. Sci. Paris Sér. I Math. 301 (15) (1985) 743–746. [25] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., vol. 94, Cambridge University Press, Cambridge, 1989.


Quantum isometry groups of the Podles spheres Jyotishman Bhowmick 1 , Debashish Goswami ∗,2 Stat-Math Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700 108, India Received 18 July 2009; accepted 5 February 2010 Available online 13 February 2010 Communicated by Alain Connes

Abstract For μ ∈ (0, 1), c 0, we identify the quantum group SOμ (3) as the universal object in the category of compact quantum groups acting by ‘orientation and volume preserving isometries’ in the sense of 2 constructed Bhowmick and Goswami (2009) [4] on the natural spectral triple on the Podles sphere Sμ,c by Dabrowski, D’Andrea, Landi and Wagner (2007) in [9]. © 2010 Elsevier Inc. All rights reserved. Keywords: Quantum isometry groups; Podles spheres; Spectral triples

1. Introduction In a series of articles initiated by [11] and followed by [3,4], we have formulated and studied a quantum group analogue of the group of Riemannian isometries of a classical or noncommutative manifold. This was motivated by previous work of a number of mathematicians including Wang, Banica, Bichon and others (see, e.g. [20,21,1,2,5,22] and the references therein), who have defined quantum automorphism and quantum isometry groups of finite spaces and finite dimensional algebras. Our theory of quantum isometry groups can be viewed as a natural generalization of such quantum automorphism or isometry groups of ‘finite’ or ‘discrete’ structures to the continuous or smooth set-up. Clearly, such a generalization is crucial to study the quantum * Corresponding author.

E-mail addresses: [email protected] (J. Bhowmick), [email protected] (D. Goswami). 1 The support from National Board of Higher Mathematics, India, is gratefully acknowledged. 2 Partially supported by a project on ‘Noncommutative Geometry and Quantum Groups’ funded by Indian National

Science Academy. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.009

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J. Bhowmick, D. Goswami / Journal of Functional Analysis 258 (2010) 2937–2960

symmetries in noncommutative geometry a la Connes [6], and in particular, for a good understanding of quantum group equivariant spectral triples. 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 the action of a compact group G on the manifold lifts as a unitary ˜ which is topologically a 2-cover of G, see [7] and [8] representation (possibly of some group G for more details) 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 (D) of compact quantum groups having unitary (co-)representation, say U , on H, which commutes with D, and the action on B(H) obtained via conjugation by U 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. Indeed (see [4]), if we consider a classical spectral triple, the subcategory of the category Q (D) consisting of groups has the classical group of orientation preserving isometries as the universal object, which justifies our definition of the quantum analogue. Unfortunately, if we consider quantum group actions, even in the finite dimensional (but with noncommutative A) situation the category Q (D) may often fail to have a universal object. It turns out, however, that if we fix any suitable faithful functional τR on B(H) (to be interpreted as the choice of a ‘volume form’) then there exists a universal object in the subcategory QR (D) of Q (D) 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 [10] that given an object (Q, U ) of Q (D) (where Q is the compact quantum group and U denotes its unitary co-representation on H), we can find a suitable functional τR (which typically differs from the usual trace of B(H) and can have a nontrivial modularity) which is preserved by the action of Q. This makes it quite natural to work in the setting of twisted spectral data (as defined in [10]). It may also be mentioned that in [4] we have actually worked in slightly bigger category QR (D) of the so-called ‘quantum family of orientation and volume preserving isometries’ and deduced that the universal object in QR (D) exists and coincides with that of QR (D). It is very important to explicitly compute the (orientation and volume preserving) quantum group of isometries for as many examples as possible. This programme has been successfully carried out for a number of spectral triples, including classical spheres and tori as well as their Rieffel deformations. The aim of the present article is to identify SOμ (3) as the quantum group of 2 , orientation and volume preserving isometries for the spectral triples on the Podles spheres Sμ,c constructed by Dabrowski et al. in [9]. Let us mention here that although the quantum groups SOμ (3) are ‘deformations’ of the classical SO(3), these are not Rieffel deformations and so the results and techniques of [4] do not apply. Our characterization of SOμ (3) as the quantum isometry group of a noncommutative Riemannian manifold generalizes the classical description of the group SO(3) as the group of orientation preserving isometries of the usual Riemannian structure on the 2-sphere. It may be mentioned here that in a very recent article [19], P.M. Soltan has characterized SOμ (3) as the universal compact quantum group acting on the finite dimensional C ∗ -algebra M2 (C) such that the action


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preserves a functional ωμ defined in [19]. In the classical case, we have three equivalent descriptions of SO(3): (a) as a quotient of SU(2), (b) as the group of (orientation preserving) isometries of S 2 , and (c) as the automorphism group of M2 . In the quantum case the definition of SOμ (3) is an analogue of (a), so the characterization of SOμ (3) obtained in this paper as the quantum isometry group, together with Soltan’s characterization, completes the generalization of all three classical descriptions of SO(3). 2. Notations and preliminaries 2.1. Basics of the theory of compact quantum groups We begin by recalling the definition of compact quantum groups and their actions from [24,23]. 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) the linear spans of (S)(S ⊗ 1) and (S)(1 ⊗ S) are norm-dense in S ⊗ S. It is well known (see [24,23]) 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 below) of S, and maps : S0 → C (co-unit) and κ : S0 → S0 (antipode) defined on S0 which make S0 a Hopf ∗-algebra. A CQG (S, ) is said to (co)-act 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. For a Hilbert B-module E, (where B is a C ∗ algebra) we shall denote the set of adjointable B-linear maps on E by L(E). The norm-closure of the linear span of the finite-rank B-linear maps on E, to be called the set of compact operators on E, will be denoted by K(E). We note that L(E) = M(K(E)), where M(C) denotes the multiplier algebra of a C ∗ -algebra C. We shall also need the ‘leg-numbering’ notation: for an operator X in B(H1 ⊗ H2 ), X(12) and X(13) will denote the operators X ⊗ IH2 in B(H1 ⊗ H2 ⊗ H2 ), and Σ23 X12 Σ23 , respectively, where Σ23 is the unitary on H1 ⊗ H2 ⊗ H2 which flips the two copies of H2 . A unitary (co)-representation of a CQG (S, ) on a Hilbert space H is given by a unitary element U of M(K(H) ⊗ S) ≡ L(H ⊗ S) satisfying (id ⊗ )(U ) = U(12) U(13) . Given a unitary representation U we shall denote by αU the ∗-homomorphism αU (X) = U (X ⊗ 1S )U ∗ for X belonging to B(H). We shall sometimes identify U with the isometric map from the Hilbert space H to the Hilbert module H ⊗ S which sends a vector ξ of H to U (ξ ⊗ 1), and may even denote U (ξ ⊗ 1) by U ξ by a slight abuse of notation. We say that a (possibly unbounded) operator T on H commutes with U if T ⊗ I (with the natural domain) commutes with U . Such an operator will also be called U -equivariant or S-equivariant if U is understood.

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2.2. The quantum group of orientation preserving Riemannian isometries We briefly recall the definition of the quantum group of orientation preserving Riemannian isometries for a spectral triple (of compact type) (A∞ , H, D) as in [4]. We consider the category Q (A∞ , H, D) ≡ Q (D) whose objects (to be called orientation preserving isometries) are the triplets (S, , U ), where (S, ) is a CQG with a unitary representation U in H, satisfying the following: (i) U commutes with D, (ii) for every state ω on S, (id ⊗ ω) ◦ αU (a) belongs to (A∞ ) for all a in A∞ . The category Q (D) may not have a universal object in general, as pointed out in [4]. In case + (D), with the corresponding representathere is a universal object, we shall denote it by QISO + (D) generated tion U , say, and we denote by QISO+ (D) the Woronowicz subalgebra of QISO by the elements of the form ξ ⊗ 1, αU (a)(η ⊗ 1), where ξ, η belong to H, a belongs to A∞ and + (D)-valued inner product of H ⊗ QISO + (D). The quantum group QISO+ (D)

·,· is the QISO will be called the quantum group of orientation-preserving Riemannian isometries of the spectral triple (A∞ , H, D). Although the category Q (D) may fail to have a universal object, we can always get a universal object in suitable subcategories which will be described now. Suppose that we are given an invertible positive (possibly unbounded) operator R on H which commutes with D. Then we consider the full subcategory QR (D) of Q (D) by restricting the object class to those (S, , U ) for which αU satisfies (τR ⊗ id)(αU (X)) = τR (X)1 for all X in the ∗-subalgebra generated by operators of the form |ξ η|, where ξ, η are eigenvectors of the operator D which by assumption has discrete spectrum, and τR (X) = Tr(RX) = η, Rξ for X = |ξ η|. We shall call the objects of QR (D) orientation and (R-twisted) volume preserving isometries. It is clear (see Remark 2.9 in [4]) that 2 when Re−tD is trace-class for some t > 0, the above condition is equivalent to the condition that 2 αU preserves the bounded normal functional Tr(· Re−tD ) on the whole of B(H). It is shown in + (D), and [4] that the category Q (D) always admits a universal object, to be denoted by QISO R

R

the Woronowicz subalgebra generated by { ξ ⊗ 1, αW (a)(η ⊗ 1): ξ, η ∈ H, a ∈ A∞ } (where + (D) in H) will be denoted by QISO+ (D) and called W is the unitary representation of QISO R R the quantum group of orientation and (R-twisted) volume preserving Riemannian isometries of the spectral triple. 2.3. SU μ (2) and the Podles spheres Fix μ in (0, 1). The C ∗ algebra underlying the CQG SU μ (2) is defined as the universal unital C ∗ algebra generated by α, γ such that α ∗ α + γ ∗ γ = 1, αα ∗ + μ2 γ γ ∗ = 1, γ γ ∗ = γ ∗ γ , μγ α = αγ , μγ ∗ α = αγ ∗ . α −μγ ∗ The CQG structure is given by the following fundamental representation: γ α ∗ . The coproduct is defined by: (α) = α ⊗ α − μγ ∗ ⊗ γ , (γ ) = γ ⊗ α + α ∗ ⊗ γ .


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The Haar state of SU μ (2) will be denoted by h and the corresponding G.N.S. Hilbert space will be denoted by L2 (SU μ (2)). We will call the unital ∗-subalgebra of SU μ (2) (without any norm-closure) generated by α, γ the ‘co-ordinate Hopf ∗-algebra’ of SU μ (2) and denote it by O(SU μ (2)) as in [17]. We now recall the definition of the Podles sphere from [9] (see also the original article [15] by Podles). n −μ−n For c 0, let t in (0, 1] be given by c = t −1 − t. Let [n] ≡ [n]μ = μμ−μ −1 , n ∈ N. 2 is defined to be the universal unital C ∗ algebra generated by elements The Podles sphere Sμ,c x−1 , x0 , x1 satisfying the relations:

x−1 (x0 − t) = μ2 (x0 − t)x−1 , x1 (x0 − t) = μ−2 (x0 − t)x1 , −[2]x−1 x1 + μ2 x0 + t (x0 − t) = [2]2 (1 − t), −[2]x1 x−1 + μ−2 x0 + t (x0 − t) = [2]2 (1 − t). 2 is given by The involution on Sμ,c ∗ = −μ−1 x1 , x−1

x0∗ = x0 .

2 as defined above is the same as χ We note that Sμ,c q,α ,β in [13, p. 124] with q = μ, α = t, 2 −2 2 2 β = t + μ (μ + 1) (1 − t). 2 can be Thus, from the expressions of x−1 , x0 , x1 given in [13, p. 125], it follows that Sμ,c realized as a ∗-subalgebra of SU μ (2) by setting:

x−1 =

μα 2 + ρ(1 + μ2 )αγ − μ2 γ 2

, 1 μ(1 + μ2 ) 2 x0 = −μγ ∗ α + ρ 1 − 1 + μ2 γ ∗ γ − γ α ∗ , x1 =

where ρ 2 = Taking

μ2 γ ∗2

− ρμ(1 + μ2 )α ∗ γ ∗

− μα ∗2

1

(1 + μ2 ) 2

,

(1) (2) (3)

μ2 t 2 . (μ2 +1)2 (1−t)

A=

1 − t −1 x0 , 1 + μ2

− 1 B = μ 1 + μ2 2 t −1 x−1 ,

2 coincides with the original description given in [15], one obtains (see [9]) that the C ∗ algebra Sμ,c ∗ i.e., the universal C algebra generated by elements A and B satisfying the relations:

A∗ = A, B ∗ B = A − A2 + cI,

AB = μ−2 BA, BB ∗ = μ2 A − μ4 A2 + cI.

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2 ) the co-ordinate ∗-algebra of S 2 , i.e. the unital ∗-subalgebra generWe will denote by O(Sμ,c μ,c ated by A, B. We recall from [17] the Hopf ∗-algebra Uμ (su(2)) which is generated by elements F , E, K, K −1 with defining relations:

KK −1 = K −1 K = 1,

KE = μEK, F K = μKF, 2 −1 −1 EF − F E = μ − μ K − K −2

with involution E ∗ = F , K ∗ = K and comultiplication: (E) = E ⊗ K + K −1 ⊗ E,

(F ) = F ⊗ K + K −1 ⊗ F,

(K) = K ⊗ K.

The counit is given by (E) = (F ) = (K − 1) = 0 and antipode S(K) = K −1 , S(E) = −μE, S(F ) = −μ−1 F . There is a dual pairing .,. of Uμ (su(2)) and O(SU μ (2)), for which the nonzero values of the pairing among the generators are given below:

1 K ±1 , α ∗ = K ∓1 , α = μ± 2 ,

E, γ = F, −μγ ∗ = 1.

The left action and right action of Uμ (su(2)) on SU μ (2) are given by: f x = f, x(2) x(1) ,

x f = f, x(1) x(2) ,

x ∈ O SU μ (2) , f ∈ Uμ su(2) ,

where we have used the Sweedler notation (x) = x(1) ⊗ x(2) . The actions satisfy the following: (f x)∗ = S(f )∗ x ∗ , f xy = (f(1) x)(f(2) y),

(x f )∗ = x ∗ S(f )∗ , xy f = (x f(1) )(y f(2) ).

The action on generators is given by: E γ = α∗, E γ ∗ = E α ∗ = 0, E α = −μγ ∗ , F −μγ ∗ = α, F α∗ = γ , F α = F γ = 0, 1 1 1 1 K γ ∗ = μ 2 γ ∗, K γ = μ− 2 γ , K α∗ = μ 2 α∗; K α = μ− 2 α, γ E = α,

α ∗ E = −μγ ∗ ,

α E = γ ∗ E = 0,

α F = γ,

−μγ ∗ F = α ∗ ,

γ F = α ∗ F = 0,

1

α K = μ− 2 α,

1

γ ∗ K = μ− 2 γ ∗ ,

1

γ K = μ2 γ,

1

α∗ K = μ 2 α∗.

2 from [18] which we are going to need. We recall an alternative description of Sμ,c


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Let −1 1 1 Xc = μ 2 μ−1 − μ c− 2 1 − K 2 + EK + μF K,

c > 0,

X0 = 1 − K 2 . One has (Xc ) = 1 ⊗ Xc + Xc ⊗ K 2 . Moreover, we have the following [18, p. 9]: Theorem 2.1. We have, 2 O Sμ,c = x ∈ O SU μ (2) : x Xc = 0 . 2 ) is given by {Ak , Ak B l , Ak B ∗ m , k 0, l, m > 0}. A basis of the vector space O(Sμ,c 2 ) can be written as a finite linear combination of elements of the Thus, any element of O(Sμ,c l k k l k ∗ form A , A B , A B . Let ψ be the densely defined linear map on L2 (SU μ (2)) defined by ψ(x) = x Xc . 2 ⊆ Ker(ψ) where ψ is the closed extenLemma 2.2. The map ψ is closable and we have Sμ,c 2 denotes the Hilbert subspace generated by S 2 in L2 (SU (2)). Moreover, sion of ψ and Sμ,c μ μ,c 2 ) = O(SU (2)) ∩ Ker(ψ) = O(SU (2)) ∩ Ker(ψ). O(Sμ,c μ μ

Proof. From the expression of Xc , it is clear that O(SU μ (2)) ⊆ Dom(ψ ∗ ) implying that ψ is 2 )= closable, hence Ker(ψ) is closed. The lemma now follows from the observation that O(Sμ,c Ker(ψ) ⊆ Ker(ψ). 2 We end this subsection with a discussion on the CQG SOμ (3) as described in [16]. It is the universal unital C ∗ algebra generated by elements M, N , G, C, L satisfying: L∗ L = (I − N) I − μ−2 N ,

LL∗ = I − μ2 N I − μ4 N ,

M ∗M = N − N 2, CC ∗ = μ2 N − μ4 N 2 , CN = μ2 NC,

MM ∗ = μ2 N − μ4 N 2 , LN = μ4 N L,

LG = μ4 GL,

LG∗ = μ4 G∗ L,

M 2 = μ−1 LG,

C∗C = N − N 2,

GN = N G,

LM = μ2 ML,

G∗ G = GG∗ = N 2 ,

MN = μ2 N M,

MG = μ2 GM,

M ∗ L = μ−1 (I − N )C,

CM = MC, N ∗ = N.

This CQG can be identified with a Woronowicz subalgebra of SU μ (2) by taking: N = γ ∗γ ,

M = αγ ,

C = αγ ∗ ,

G = γ 2,

L = α2.

2 , i.e. the action obtained by restricting the coproduct of The canonical action of SU μ (2) on Sμ,c 2 SU μ (2) to the subalgebra Sμ,c , is actually a faithful action of SOμ (3). On the subspace spanned

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by {x−1 , x0 , x1 } this action is given by the following SOμ (3)-valued 3 × 3-matrix: ⎛

α2 1 Z := ⎝ (1 + μ−2 ) 2 αγ γ2

1

−μ(1 + μ−2 ) 2 αγ ∗ I − μ(μ + μ−1 )γ ∗ γ 1 (1 + μ−2 ) 2 γ α ∗

⎞ μ2 γ ∗2 1 −μ(1 + μ−2 ) 2 γ ∗ α ∗ ⎠ . α ∗2

3. Spectral triples on the Podles spheres and their quantum isometry groups 3.1. Description of the spectral triples 2 discussed in [9] (see also [17] for the case c = 0). We now recall the spectral triples on Sμ,c 1

Let s = −c− 2 λ− , λ± = For all j in 12 N,

1 2

1

± (c + 14 ) 2 .

uj = α ∗ − sγ ∗ α ∗ − μ−1 sγ ∗ . . . α ∗ − μ−2j +1 sγ ∗ , wj = (α − μsγ ) α − μ2 sγ . . . α − μ2j sγ , u−j = E 2j wj , u0 = w0 = 1, 1 1 1 y1 = 1 + μ−2 2 c 2 μ2 γ ∗2 − μγ ∗ α ∗ − μc 2 α ∗2 , −1 l Nkj = F l−k y1 l−|j | uj . l−|j |

l = N l F l−k (y uj ), l ∈ 12 N0 , j, k = −l, −l + 1, . . . , l. Define vk,j k,j 1 l : l = |N |, Let MN be the Hilbert subspace of L2 (SU μ (2)) with the orthonormal basis {vm,N |N| + 1, . . . , m = −l, . . . , l}. Set

H = M− 1 ⊕ M 1 , 2

2

2 on H by and define a representation π of Sμ,c l−1 l+1 l l = αi− (l, m; N )vm+i,N + αi0 (l, m; N )vm+i,N + αi+ (l, m; N )vm+i,N , π(xi )vm,N

where αi− , αi0 , αi+ are as defined in [9]. 2 ) with S 2 . We will often identify π(Sμ,c μ,c Finally by Proposition 7.2 of [9], the following Dirac operator D gives a spectral triple 2 ), H, D) which we are going to work with: (O(Sμ,c l l D vm,± 1 = (c1 l + c2 )v m,∓ 1 2

where c1 , c2 belong to R, c1 = 0.

2


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It is easy to see that the action of SU μ (2) on itself keeps the subspace H invariant and so induces a unitary representation, say U0 on H. We define a positive, unbounded operator R on H by R(v n 1 ) = μ−2i v n 1 . i,± 2

i,± 2

2 ) Proposition 3.1. αU0 preserves the R-twisted volume. In particular, for x belonging to π(Sμ,c −tD ), and h denotes the restriction and t > 0, we have h(x) = ττRR (x) (1) , where τR (x) := Tr(xRe 2 , which is the unique SU (2)-invariant state of the Haar state of SU μ (2) to the subalgebra Sμ,c μ 2 on Sμ,c . 2

Proof. It is enough to prove that τR is αU0 -invariant. Define R0 (v n

i,± 12

) = μ−2i∓1 v n

i,± 12

, and note

that it has been observed in [12] that Tr(R0 e−tD ) < ∞ (for all t > 0) and one has 2

(τR0 ⊗ id) U0 (x ⊗ 1)U0 ∗ = τR0 (x).1, for all x in B(H), where τR0 (x) = Tr(xR0 e−tD ). Let us denote by P 1 , P− 1 the projections onto the closed subspaces generated by {v l 1 } and 2

2

{v l

i, 2

2

−tD ). 1 } respectively. Moreover, let τ± be the functionals defined by τ± (x) = Tr(xR0 P± 1 e

i,− 2

2

2

We observe that R0 , e−tD and U0 commute with P± 1 so that for x belonging to B(H), 2

2

(τ± ⊗ id) αU0 (x) = (τR0 ⊗ id) αU0 (xP± 1 ) = τR0 (xP± 1 )1 = τ± (x)1, 2

2

i.e. τ± are αU0 -invariant. Moreover, since we have RP± 1 = μ± R0 P± 1 , the functional τR coin-

cides with μ−1 τ+ + μτ− , hence is αU0 -invariant.

2

2

2

Theorem 3.2. (SU μ (2), , U0 ) is an object in QR (D). Proof. The above spectral triple is equivariant with respect to this representation (see [9]) and it preserves τR by Proposition 3.1, which completes the proof. 2 We now note down some useful facts for later use. l and , we observe: Remark 3.3. Using the definition of vi,j

1. The eigenspaces of D corresponding to (c1 l + c2 ) and −(c1 l + c2 ) are span{v l vl 1 : m,− 2

−l m l} and

span{v l 1 m, 2

− vl 1 : m,− 2

−l m l} respectively.

m, 12

2. The eigenspace of |D| corresponding to the eigenvalue (c1 . 12 + c2 ) is span{α, γ , α ∗ , γ ∗ }. Remark 3.4. l−1 l+1 l l 1. π(A)vm,N belongs to Span{vm,N , vm,N , vm,N }, l−1 l+1 l l π(B)vm,N belongs to Span{vm−1,N , vm−1,N , vm−1,N }, l−1 l+1 l l ∗ π(B )vm,N belongs to Span{vm+1,N , vm+1,N , vm+1,N }.

+

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l−k l−k+1 l+k l 2. π(Ak )(vm,N ) belongs to Span{vm,N , vm,N , . . . , vm,N }.

l−(n +m −1)

l−m −n l ) belongs to Span{vm−n 3. π(Am B n )(vm,N ,N , vm−n ,N

l+n +m , . . . , vm−n ,N }.

l−s−r l−s−r+1 l+s+r l 4. π(Ar B ∗s )(vm,N ) belongs to Span{vm+s,N , vm+s,N , . . . , vm+s,N }.

˜ We shall now proceed to show that QISO+ R (D) is isomorphic with SOμ (3). Let (Q, U ) be an object in the category QR (D) of CQG s acting by orientation and R-twisted volume preserving isometries on this spectral triple and Q be the Woronowicz C ∗ subalgebra of Q˜ generated by 2 (where ·,· is the Q-valued ˜

(ξ ⊗ 1), αU (a)(η ⊗ 1)Q˜ , for ξ, η in H, a in Sμ,c inner product Q˜ ˜ We shall denote αU by φ from now on. of H ⊗ Q). The proof has two main steps: first, we prove that φ is ‘linear’, in the sense that it keeps the span of {1, A, B, B ∗ } invariant, and then we shall exploit the facts that φ is a ∗-homomorphism 2 , i.e. the restriction of the Haar state of and preserves the canonical volume form on Sμ,c SU μ (2). Remark 3.5. The first step does not make use of the fact that φ preserves the R-twisted volume, so linearity of the action follows for any object in the bigger category Q (D). 3.2. Linearity of the action For a vector v in H, we shall denote by Tv the map from B(H) to L2 (SU μ (2)) given by Tv (x) = xv ∈ H ⊂ L2 (SU μ (2)). It is clearly a continuous map with respect to the strong operator topology on B(H) and the Hilbert space topology of L2 (SU μ (2)). For an element a in SU μ (2), we consider the right multiplication Ra as a bounded linear map on L2 (SU μ (2)). Clearly the composition Ra Tv is a continuous linear map from B(H) (with the strong operator topology) to the Hilbert space L2 (SU μ (2)). We now define T = Rα ∗ T α + μ 2 R γ T γ ∗ . 2 , we have T (φ (x)) = φ (x) ≡ R (φ (x)) Lemma 3.6. For any state ω on Q˜ and x in Sμ,c ω ω 1 ω 2 ⊆ L2 (SU (2)), where φ (x) = (id ⊗ ω)(φ(x)). belonging to Sμ,c μ ω

Proof. It is clear from the definition of T (using αα ∗ + μ2 γ γ ∗ = 1) that T (x) = x ≡ R1 (x) for x 2 ⊂ B(H), where x in the right-hand side of the above denotes the identification of x ∈ S 2 in Sμ,c μ,c 2 , φ (x) belongs as a vector in L2 (SU μ (2)). Now, the lemma follows by noting that for x in Sμ,c ω 2 ) , which is the closure of S 2 in the strong operator topology, and the continuity of T to (Sμ,c μ,c in this topology discussed before. 2 Let l V l = Span vi,± 1 , −l i l , l l . 2

Since Span{v l

i,± 21

, −l i l} is the eigenspace of |D| corresponding to the eigenvalue c1 l + c2 ,

U and U ∗ must keep V l invariant for all l.


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Lemma 3.7. There is some finite dimensional subspace V of O(SU μ (2)) such that 1

1

Rα ∗ (φω (A)v 2

), Rγ (φω (A)v 2

j,± 12

j,± 12

˜ ) belong to V for all states ω on Q.

The same holds when A is replaced by B or B ∗ . Proof. We prove the result for A only, since a similar argument will work for B and B ∗ . 1

We have φ(A)(v 2 Now, U ∗ (v

j,± 12

1 2

j,± 12

1

⊗ 1) = U (π(A) ⊗ 1)U ∗ (v 2

j,± 12

⊗ 1).

1 ˜ and then using the definition of π as well as Re⊗ 1) belongs to V 2 ⊗ Q, 1

mark 3.4, we observe that (π(A) ⊗ 1)U ∗ (v 2

j,± 12

⊗ 1) belongs to Span{v l

j,± 12

: −l j l , 1

3 ˜ Again, U keeps V 32 ⊗ Q˜ invariant, so Rα ∗ (φω (A)v 2 1 ) belongs to l 32 } ⊗ Q˜ = V 2 ⊗ Q.

±2

1

3

3

Span{vα ∗ : v ∈ V 2 }. Similarly, Rγ (φω (A)(v 2 1 )) belongs to Span{vγ : v ∈ V 2 }. So, the lemma ±2

3

follows for A by taking V = Span{vα ∗ , vγ : v ∈ V 2 } ⊂ O(SU μ (2)). 1

Since α, γ ∗ belong to Span{v 2

j,± 12

2

}, we have the following immediate corollary:

Corollary 3.8. There is a finite dimensional subspace V of O(SU μ (2)) such that for every state ˜ we have T (φω (A)) belongs to V. A similar (hence for every bounded linear functional) ω on Q, ∗ conclusion holds for B and B as well. 2 )⊗ Proposition 3.9. φ(A), φ(B), φ(B ∗ ) belong to O(Sμ,c alg Q.

Proof. We give the proof for φ(A) only, the proof for B, B ∗ being similar. From Corollary 3.8 ˜ T (φω (A)) belongs to and Lemma 3.6 it follows that for every bounded linear functional ω on Q, 2 2 2 V ∩ Sμ,c ⊂ O(SU μ (2)) ∩ Ker(ψ) and hence V ∩ Sμ,c = V ∩ O(Sμ,c ), where V is the finite dimen2 ) is a finite dimensional subspace sional subspace mentioned in Corollary 3.8. Clearly, V ∩ O(Sμ,c 2 of O(Sμ,c ) implying that there must be finite m, say, such that for every ω, T (φω (A)) belongs to Span{Ak , Ak B l , Ak B ∗l : 0 k, l m}. Denote by W the (finite dimensional) subspace of B(H) spanned by {Ak , Ak B l , Ak B ∗l : 0 k, l m}. Since for every state (and hence for every ˜ we have T (φω (A)) = R1 (φω (A)) ≡ φω (A)1, it is clear that bounded linear functional) ω on Q, φω (A) belongs to W for every ω in Q˜ ∗ . Now, let us fix any faithful state ω on the separable unital C ∗ -algebra Q˜ and embed Q˜ in B(L2 (Q, ω)) ≡ B(K). Thus, we get a canonical embedding of ˜ in B(H ⊗K). Let us thus identify φ(A) as an element of B(H ⊗K), and then by choosL(H ⊗ Q) basis in K = L2 (ω), ing a countable family of elements {q1 , q2 , . . .} of Q˜ which is an orthonormal

∞ ij we can write φ(A) as a weakly convergent series of the form i,j =1 φ (A) ⊗ |qi qj |. But φ ij (A) = (id ⊗ ωij )(φ(A)), where ωij (·) = ω(qi∗ · qj ). Thus, φ ij (A) belongs to W for all i, j ,

and hence the sequence ni,j =1 φ ij (A) ⊗ |qi qj | ∈ W ⊗ B(K) converges weakly, and W being finite dimensional (hence weakly closed), the limit, i.e. φ(A), must belong

to W ⊗ B(K). In other words, if A1 , . . . , Ak denotes a basis of W, we can write φ(A) = ki=1 Ai ⊗ Bi for some Bi ∈ B(K). ˜ For any trace-class positive operator ρ in H, say We claim that each Bi must belong to Q. of the form ρ = j λj |ej ej |, where {e1 , e2 , . . .} is an orthonormal basis of H and λj 0,

2948


λj < ∞, let us denote by ψρ the normal functional on B(H) given by x → Tr(ρx), and it is ˜ given by ψ˜ ρ (X) = easy to see that it has a canonical extension ψ˜ ρ := (ψρ ⊗ id) on L(H ⊗ Q)

˜ ˜ λ

e ⊗ 1, X(e ⊗ 1) , where X belongs to L(H ⊗ Q) and

·,· denotes the Q-valued j j j j Q˜ Q˜ ˜ Clearly, ψ˜ ρ is a bounded linear map from L(H ⊗ Q) ˜ to Q. ˜ Now, inner product of H ⊗ Q. since A1 , . . . , Ak in the expression of φ(A) are linearly independent, we can choose trace class operators ρ1 , . . . , ρk such that ψρi (Ai ) = 1 and ψρi (Aj ) = 0 for j = i. Then, by applying ψ˜ ρi ˜ But by definition, Q is the Woronowicz subalgebra on φ(A) we conclude that Bi belongs to Q. 2 ), and hence ˜ of Q generated by ξ ⊗ 1, φ(x)(η ⊗ 1)Q˜ , with η, ξ belonging to H and x in O(Sμ,c it follows that Bi belongs to Q. 2 j

Proposition 3.10. φ keeps the span of 1, A, B, B ∗ invariant. Proof. We prove the result for φ(A) only, the proof for the other cases being quite similar. Using Proposition 3.9, we can write φ(A) as a finite sum of the form:

Ak ⊗ Qk +

Am B n ⊗ Rm ,n +

m ,n ,n =0

k0

Ar B ∗s ⊗ Rr,s .

r,s,s=0

l Let ξ = vm . 0 ,N0 l We have that U (ξ ) belongs to Span{vm,N , m = −l, . . . , l, N = ± 12 }. Let us write

U (ξ ⊗ 1) =

l l vm,N ⊗ q(m,N ),(m0 ,N0 ) ,

m=−l,...,l, N =± 12 l where q(m,N ),(m0 ,N0 ) belong to Q. Since αU preserves the R-twisted volume, we have:

m ,N

l l∗ q(m,N ),(m ,N ) q(m,N ),(m ,N ) = 1.

(4)

l It also follows that U (Aξ ) belongs to Span{vm,N , m = −l , . . . , l , l = l − 1, l, l + 1, N = ± 12 }. Recalling Remark 3.4, we have

φ(A)U (ξ ⊗ 1) =

l l Ak vm,N ⊗ Qk q(m,N ),(m0 ,N0 )

k,m=−l,...,l, N=± 12

+

l l Am B n vm,N ⊗ Rm ,n q(m,N ),(m0 ,N0 )

m ,n , n =0, m=−l,...,l, N=± 12

+

l l Ar B ∗s vm,N ⊗ Rr,s q(m,N ),(m0 ,N0 ) .

r,s, s=0, m=−l,...l, N =± 12

Let m0 denote the largest integer m such that there is a nonzero coefficient of Am B n , n 1 in the expression of φ(A). We claim that the coefficient of v l Rm0 ,n q(m,N ),(m0 ,N0 ) .

l−m0 −n m−n ,N

in φ(A)U (ξ ⊗ 1) is


2949

l−m −n

Indeed, the term vm−n0 ,N can arise in three ways: it can come from a term of the form

l l l or Ak vm,N or Ar B ∗s vm.N for some m , n , k, r, s. Am B n vm,N In the first case, we must have l − m0 − n = l − m − n + t, 0 t 2m and m − n = m − n implying m = m0 + t, and since m0 is the largest integer such that Am0 B n appears in φ(A), we l−m −n

only have the possibility t = 0, i.e. vm−n0 ,N appears only in Am0 B n . In the second case, we have m − n = m implying n = 0 – a contradiction. In the last case, we have m − n = m + s so that −n = s which is only possible when n = s = 0 which is again a contradiction. It now follows from the above claim, using Remark 3.4 and comparing coefficients in the l equality U (Aξ ⊗ 1) = φ(A)U (ξ ⊗ 1), that Rm0 ,n q(m,N ),(m0 ,N0 ) = 0 for all n 1, for all m, N when m0 1. Now varying (m0 , N0 ), we conclude that the above holds for all (m0 , N0 ). Using (4), we conclude that Rm0 ,n

m ,N

l l∗ q(m,N ),(m ,N ) q(m,N ),(m ,N ) = 0 for all n 1,

that is, Rm0 ,n = 0 for all n 1 if m0 1. Proceeding by induction on m0 , we deduce Rm ,n = 0 for all m 1, n 1. = 0 for all r 1, s 1. Similarly, we have Qk = 0 for all k 2 and Rr,s Thus, φ(A) belongs to Span{1, A, B, B ∗ , B 2 , . . . , B n , B ∗2 , . . . , B ∗m }. But the coefficient of l−n vm−n ,N in φ(A)U (ξ ⊗ 1) is R0,n . Arguing as before, we conclude that R0,n = 0 for all n 2. In a similar way, we can prove R0,n 2 = 0 for all n 2. In view of the above, let us write: φ(A) = 1 ⊗ T1 + A ⊗ T2 + B ⊗ T3 + B ∗ ⊗ T4 ,

(5)

φ(B) = 1 ⊗ S1 + A ⊗ S2 + B ⊗ S3 + B ∗ ⊗ S4 ,

(6)

for some Ti , Si in Q. 3.3. Identification of SOμ (3) as the quantum isometry group In this subsection, we shall use the facts that φ is a ∗-homomorphism and it preserves the R-twisted volume to derive relations among Ti , Si in (5), (6). Lemma 3.11. 1 − T2 , 1 + μ2 −S2 S1 = . 1 + μ2

T1 =

Proof. We have the expressions of A and B in terms of the SU μ (2) elements from Eqs. (1), (2) and (3). From these, we note that h(A) = (1 + μ2 )−1 and h(B) = 0. By recalling Proposition 3.1, we use (h ⊗ id)φ(A) = h(A).1 and (h ⊗ id)φ(B) = h(B).1 to have the above two equations. 2

2950


Lemma 3.12. T1∗ = T1 ,

T2∗ = T2 ,

T4∗ = T3 .

Proof. It follows by comparing the coefficients of 1, A and B respectively in the equation φ(A∗ ) = φ(A). 2 Lemma 3.13. 2 2 S2∗ S2 + c 1 + μ2 S3∗ S3 + c 1 + μ2 S4∗ S4 2 2 2 = (1 − T2 ) μ2 + T2 − c 1 + μ2 T3 T3∗ − c 1 + μ2 T3∗ T3 + c 1 + μ2 .1, −2S2∗ S2 + 1 + μ2 S3∗ S3 + μ2 1 + μ2 S4∗ S4 = μ2 + 2T2 − 1 T2 − μ2 1 + μ2 T3 T3∗ − 1 + μ2 T3∗ T3 , S2∗ S2 − S3∗ S3 − μ4 S4∗ S4 = −T22 + μ4 T3 T3∗ + T3∗ T3 , S2∗ S4 + S3∗ S2 = − μ2 + T2 T3∗ + T3∗ (1 − T2 ), S2∗ S3

+ μ2 S4∗ S2 S4∗ S3

(7)

(8) (9) (10)

= −T2 T3 − μ T3 T2 ,

(11)

= −T32 .

(12)

2

Proof. It follows by comparing the coefficients of 1, A, A2 , B ∗ , AB and B 2 in the equation φ(B ∗ B) = φ(A) − φ(A2 ) + cφ(I ) and then using Lemmas 3.11 and 3.12. 2 Lemma 3.14. 2 2 −S2 (1 − T2 ) + c 1 + μ2 S3 T3∗ + c 1 + μ2 S4 T3 2 2 = −μ2 (1 − T2 )S2 + cμ2 1 + μ2 T3 S4 + cμ2 1 + μ2 T3∗ S3 , S2 − 2S2 T2 + 1 + μ2 μ2 S3 T3∗ + S4 T3 = μ2 S2 − 2μ2 T2 S2 + μ4 1 + μ2 T3 S4 + μ2 1 + μ2 T3∗ S3 ,

(13)

(14)

−S2 T3 + S3 (1 − T2 ) = −μ2 T3 S2 + μ2 (1 − T2 )S3 ,

(15)

−S2 T3∗

(16)

T3∗ S2 ,

+ S4 (1 − T2 ) = μ (1 − T2 )S4 − μ S 2 T3 + μ 2 S 3 T2 = μ 2 T2 S 3 + μ 2 T3 S 2 , 2

2

(17)

S 3 T3 = μ 2 T3 S 3 ,

(18)

S4 T3∗ = μ2 T3∗ S4 .

(19)

Proof. It follows by equating the coefficients of 1, A, B, B ∗ , AB, B 2 and B ∗ 2 in the equation φ(BA) = μ2 φ(AB) and then using Lemmas 3.11 and 3.12. 2


2951

−S2 S4∗ − S3 S2∗ = μ2 1 + μ2 T3 − μ4 (1 − T2 )T3 − μ4 T3 (1 − T2 ),

(20)

Lemma 3.15.

S2 S4∗ + μ2 S3 S2∗ = −μ4 T2 T3 − μ6 T3 T2 , S3 S4∗

= −μ

4

(21)

T32 .

(22)

Proof. The lemma is proved by equating the coefficient of B, AB, B 2 in the equation φ(BB ∗ ) = μ2 φ(A) − μ4 φ(A2 ) + cφ(I ) and then using Lemmas 3.11 and 3.12. 2 Now, we compute the antipode, say κ of Q. To begin with, we note that {x−1 , x0 , x1 } is a set of orthogonal vectors. Moreover, they have the same norm. The first assertion being easier, we prove below the second one. Lemma 3.16. ∗ x−1 = h x0∗ x0 = h x1∗ x1 h x−1 −1 2 = t 2 1 − μ2 1 − μ6 μ + t −1 μ4 + 2μ2 + 1 + t −μ4 − 2μ2 − 1 . ∗ x ∗ 2 −2 2 2 2 2 Proof. We have x−1 −1 = t μ (1 + μ )(A − A + cI ), x0 x0 = t (1 − 2(1 + μ )A + (1 + ∗ 2 2 2 2 2 2 4 2 μ ) A ), x1 x1 = t (1 + μ )(μ A − μ A + cI ). We recall from [18] that for all bounded Borel function f on σ (A), ∞ ∞ 2n 2n f λ + μ μ + γ− f λ− μ2n μ2n , h f (A) = γ+ n=0 1

n=0 1

where λ+ = 12 + (c + 14 ) 2 , λ− = 12 − (c + 14 ) 2 , γ+ = (1 − μ2 )λ+ (λ+ − λ− )−1 , γ− = (1 − μ2 )λ− (λ− − λ+ )−1 . ∗ x , x∗x , The lemma follows by applying this relation to the above expressions of x−1 −1 0 0 ∗ x1 x 1 . 2 , x , x is the normalized basis corresponding to {x , x , x }, then from (5) and (6) If x−1 −1 0 1 0 1 along with the fact that each of the vectors x−1 , x0 , x1 has the same norm, it follows that

− 1 φ x−1 = x−1 ⊗ S3 + x0 ⊗ −μ−1 1 + μ2 2 S2 + x1 ⊗ −μ−1 S4 , 1 1 φ x0 = x−1 ⊗ −μ 1 + μ2 2 T3 + x0 ⊗ T2 + x1 ⊗ 1 + μ2 2 T4 , − 1 φ x1 = x−1 ⊗ −μS4∗ + x0 ⊗ 1 + μ2 2 S2∗ + x1 ⊗ S3∗ . Since φ is kept invariant by the Haar state h of SU μ (2) and φ keeps the span of the or , x , x } invariant too, we get a unitary representation of the CQG Q on the thonormal set {x−1 0 1 span of {x−1 , x0 , x1 }. If we denote by Z the M3 (Q)-valued unitary corresponding to this unitary , x , x }, we get by using T = T ∗ from representation with respect to the ordered basis {x−1 4 0 1 3

2952


Lemma 3.12 the following: ⎛

S3 ⎜ √−S2 Z = ⎝ μ 1+μ2 −μ−1 S4

−μ 1 + μ2 T3

T2 1 + μ2 T3∗

⎞ −μS4∗ S2∗ ⎟ √ . 1+μ2 ⎠ S3∗

Recall that (see, for example, [14]), the antipode κ on the matrix elements of a finite dimensional unitary representation U α ≡ (uαpq ) is given by κ(uαpq ) = (uαqp )∗ . Thus, the antipode κ is given by: κ(T2 ) = T2 ,

κ(T3 ) =

κ(S3 ) = S3∗ ,

S2∗ , 2 μ (1 + μ2 )

κ(S4 ) = μ2 S4 ,

κ S2∗ = 1 + μ2 T3 ,

κ(S2 ) = μ2 1 + μ2 T3∗ , κ T3∗ =

κ S3∗ = S3 ,

S2 , 1 + μ2 κ S4∗ = μ−2 S4∗ .

Now we derive some more relations by applying the anti-homomorphism κ on the relations obtained earlier. Lemma 3.17. 3 2 2 −2μ4 1 + μ2 T3∗ T3 + μ2 1 + μ2 S3∗ S3 + μ4 1 + μ2 S4 S4∗ = μ2 1 + μ2 T2 μ2 + 2T2 − 1 − μ2 S2 S2∗ − S2∗ S2 , 4 2 2 μ4 1 + μ2 T3∗ T3 − μ2 1 + μ2 S3∗ S3 − μ6 1 + μ2 S4 S4∗ 2 = −μ2 1 + μ2 T22 + μ4 S2 S2∗ + S2∗ S2 , 2 2 μ2 1 + μ2 S4 T3 + μ2 1 + μ2 T3∗ S3 = −S2 μ2 + T2 + (1 − T2 )S2 , S4 S3 = −

(23)

(24) (25)

−S22 . μ2 (1 + μ2 )2

(26)

Proof. The relations follow by applying κ on (8), (9), (10) and (12) respectively.

2

Lemma 3.18. −μ2 (1 − T2 )T3∗ + cS2 S3∗ + cS2∗ S4 = −μ4 T3∗ (1 − T2 ) + cμ2 S4 S2∗ + cμ2 S3∗ S2 ,

(27)

S3 S2 = μ2 S2 S3 ,

(28)

S2 S4 = μ S4 S2 ,

(29)

2

−S2∗ T3∗ + (1 − T2 )S3∗ = −μ2 T3∗ S2∗ + μ2 S3∗ (1 − T2 ),

(30)

−S2 T3∗

(31)

+ (1 − T2 )S4 = μ S4 (1 − T2 ) − μ 2

2

T3∗ S2 .

Proof. The relations follow by applying κ on (13), (18), (19), (15) and (16) respectively.

2


2953

Lemma 3.19. μ2 S22 S3 S4 = − , (1 + μ2 )2 2 2 −μ2 1 + μ2 S4∗ T3∗ − μ2 1 + μ2 T3 S3∗ = μ2 1 + μ2 S2∗ − μ4 S2∗ (1 − T2 ) − μ4 (1 − T2 )S2∗ , 2 2 1 + μ2 S4∗ T3∗ + μ2 1 + μ2 T3 S3∗ = −μ2 S2∗ T2 − μ4 T2 S2∗ . Proof. The relations follow by applying κ on (22), (20) and (21) respectively.

(32)

(33) (34) 2

Remark 3.20. It follows from (26) and (32) that μ4 S4 S3 = S3 S4 . Lemma 3.21. S2∗ S2 = (1 − T2 ) μ2 + T2 . Proof. Subtracting the equation obtained by multiplying c(1 + μ2 ) with (8) from (7), we have 2 1 + 2c 1 + μ2 S2∗ S2 + c 1 + μ2 1 − μ2 S4∗ S4 = (1 − T2 ) μ2 + T2 − c 1 + μ2 μ2 + 2T2 − 1 T2 2 2 + c 1 + μ2 μ2 − 1 T3 T3∗ + c 1 + μ2 .1.

(35)

Again, by adding (7) with c(1 + μ2 )2 times (9) gives 2 2 1 + c 1 + μ2 S2∗ S2 + c 1 − μ4 1 + μ2 S4∗ S4 2 2 2 = (1 − T2 ) μ2 + T2 − c 1 + μ2 T22 + c 1 + μ2 μ4 − 1 T3 T3∗ + c 1 + μ2 .1.

(36)

Subtracting the equation obtained by multiplying (μ2 + 1) with (35) from (36) we obtain 2 − μ2 + c 1 + μ2 S2∗ S2 2 = (1 − T2 ) μ2 + T2 − c 1 + μ2 T22 2 2 − 1 + μ2 (1 − T2 ) μ2 + T2 − cμ2 1 + μ2 .1 + c 1 + μ2 μ2 + 2T2 − 1 T2 . The right-hand side can be seen to equal −(μ2 + c(1 + μ2 )2 )(1 − T2 )(μ2 + T2 ). Thus, S2∗ S2 = (1 − T2 )(μ2 + T2 ). 2 Lemma 3.22. 2 μ2 1 + μ2 T3∗ T3 = (1 − T2 ) μ2 + T2 , 2 1 + μ2 T3 T3∗ = (1 − T2 ) 1 + μ2 T2 , S2 S2∗ = μ2 (1 − T2 ) 1 + μ2 T2 .

(37) (38) (39)

2954


Proof. Applying κ on Lemma 3.21, we obtain (37). Unitarity of the matrix Z ((2, 2) position of the matrix Z ∗ Z) gives μ2 (1 + μ2 )T3∗ T3 + T22 + (1 + μ2 )T3 T3∗ = 1. Using (37) we deduce −(1 + μ2 )2 T3 T3∗ = (T2 − 1)(1 + μ2 T2 ). Thus we obtain (38). Applying κ on (38), we deduce (39). 2 Lemma 3.23. −2 S4∗ S4 = S4 S4∗ = 1 + μ2 μ2 (1 − T2 )2 . Proof. Adding (23) and (24), we have: 3 2 −μ4 1 + μ2 1 − μ2 T3∗ T3 + μ4 1 + μ2 1 − μ2 S4 S4∗ = −μ2 1 + μ2 1 − μ2 T2 (1 − T2 ) − μ2 1 − μ2 S2 S2∗ . Using μ2 = 1, we obtain, 3 2 −μ4 1 + μ2 T3∗ T3 + μ4 1 + μ2 S4 S4∗ = −μ2 1 + μ2 T2 (1 − T2 ) − μ2 S2 S2∗ . Now using (37) and (39), we reduce the above equation to 2 μ4 1 + μ2 S4 S4∗ = −μ2 (1 − T2 ) T2 + μ2 T2 + μ2 + μ4 T2 + μ2 1 + μ2 (1 − T2 ) μ2 + T2 = μ6 (1 − T2 )2 . Thus, S4 S4∗ = =

μ6 (1 − T2 )2 μ4 (1 + μ2 )2 μ2 (1 − T2 )2 . (1 + μ2 )2

μ2 (1 − T2 )2 . (1+μ2 )2 2 μ (1 − T2 )2 . 2 (1+μ2 )2

Applying κ, we have S4∗ S4 = Thus, S4∗ S4 = S4 S4∗ = Lemma 3.24.

2 μ2 1 + μ2 S3∗ S3 = μ2 + T2 μ2 1 + μ2 − (1 − T2 ) . Proof. Using Lemma 3.21 in (7), we have S3∗ S3 + T3∗ T3 + T3 T3∗ + S4∗ S4 = 1.

(40)

The lemma is derived by substituting the expressions of T3∗ T3 , T3 T3∗ and S4∗ S4 from (37), (38) and Lemma 3.23 in Eq. (40). 2


2955

Lemma 3.25. 2 1 + μ2 S3 S3∗ = 1 + μ2 T2 1 + μ2 − μ4 (1 − T2 ) . Proof. By unitarity of the matrix Z, in particular equating the (1, 1)-th entry of ZZ ∗ to 1 we get S3 S3∗ + μ2 (1 + μ2 )T3 T3∗ + μ2 S4∗ S4 = 1. Then the lemma follows by using (38) and Lemma 3.23 in the above equation. 2 Lemma 3.26. −S2∗ S3 = μ2 + T2 T3 . Proof. By applying the adjoint and then multiplying by μ2 on (10) we have μ2 S2∗ S3 + μ2 S4∗ S2 = −μ2 T3 (μ2 + T2 ) + μ2 (1 − T2 )T3 . Subtracting this from (11) we have (1 − μ2 )S2∗ S3 = −T2 T3 − μ2 T3 T2 + μ2 T3 (μ2 + T2 ) − μ2 (1 − T2 )T3 which implies −S2∗ S3 = (μ2 + T2 )T3 as μ2 = 1. 2 Lemma 3.27. S2 (1 − T2 ) = μ2 (1 − T2 )S2 . Proof. Applying κ to Lemma 3.26 and then taking adjoint, we have 2 μ2 1 + μ2 T3∗ S3 = − μ2 + T2 S2 .

(41)

Adding (33) and (34) and then taking adjoint, we get (by using μ2 = 1) 2 μ2 1 + μ2 T3 S4 = μ4 (1 − T2 )S2 .

(42)

Moreover, (25) gives 2 2 μ2 1 + μ2 S4 T3 = −S2 μ2 + T2 + (1 − T2 )S2 − μ2 1 + μ2 T3∗ S3 . Using (41), the right-hand side of this equation turns out to be S2 (1 − T2 ). Thus, 2 1 + μ2 S4 T3 = μ−2 S2 (1 − T2 ).

(43)

Again, application of adjoint to Eq. (33) gives: 2 2 μ2 1 + μ2 S3 T3∗ = −μ2 1 + μ2 T3 S4 − μ2 1 + μ2 S2 + μ4 (1 − T2 )S2 + μ4 S2 (1 − T2 ). Using (42), we get 2 1 + μ2 S3 T3∗ = −S2 1 + μ2 T2 .

(44)

2956


Using (41)–(44) to Eq. (14), we obtain: −1 −1 S2 − 2S2 T2 − 1 + μ2 μ2 S2 1 + μ2 T2 + μ−2 1 + μ2 S2 (1 − T2 ) −1 −1 2 μ + T2 S 2 . = μ2 S2 − 2μ2 T2 S2 + 1 + μ2 μ6 (1 − T2 )S2 − 1 + μ2 This gives μ2 1 + μ2 (S2 − S2 T2 ) − μ2 S2 − μ2 T2 S2 − μ2 1 + μ2 S2 T2 − μ2 T2 S2 − μ4 S2 − μ6 S2 T2 + S2 (1 − T2 ) − μ8 (S2 − T2 S2 ) + μ4 S2 + μ2 T2 S2 = 0. Thus, μ2 1 + μ2 S2 (1 − T2 ) − μ2 (1 − T2 )S2 + S2 (1 − T2 ) − μ2 (S2 − T2 S2 ) + μ6 S2 (1 − T2 ) − μ2 (1 − T2 )S2 − μ6 (1 − T2 )S2 + μ4 S2 (1 − T2 ) + μ2 S2 (1 − T2 ) − μ2 (1 − T2 )S2 = 0. On simplifying, (μ6 + 2μ4 + 2μ2 + 1)(S2 (1 − T2 ) − μ2 (1 − T2 )S2 ) = 0, which proves the lemma as 0 < μ < 1. 2 Lemma 3.28. T3 (1 − T2 ) = μ2 (1 − T2 )T3 ,

(45)

S3 S4∗ = μ4 S4∗ S3 .

(46)

Proof. Eq. (45) follows by applying κ on Lemma 3.27 and then taking adjoint. We have S4∗ S3 = −T32 from (12). On the other hand we have S3 S4∗ = −μ4 T32 from (22). Combining these two, we get (46). 2 Lemma 3.29. S 4 T2 = T2 S 4 . Proof. Subtracting (31) from (16) we get the required result.

2

Lemma 3.30. T3 S 2 = S 2 T3 . Proof. By applying adjoint on (30) and then subtracting it from (15) we obtain S2 T3 − T3 S2 = 0. 2 Lemma 3.31. S3 (1 − T2 ) = μ4 (1 − T2 )S3 .


2957

Proof. By adding (15) with (17) we obtain S3 (1 − T2 ) + μ2 S3 (T2 − 1) = μ2 μ2 − 1 T3 S2 . Thus, using μ2 = 1, S3 (1 − T2 ) = −μ2 T3 S2 .

(47)

Moreover, by taking adjoint of (30), we obtain μ2 (1 − T2 )S3 = μ2 S2 T3 − T3 S2 + S3 (1 − T2 ). Thus, μ4 (1 − T2 )S3 = μ4 S2 T3 − μ2 T3 S2 + μ2 S3 (1 − T2 ). Hence, to prove the lemma it suffices to prove: S3 (1 − T2 ) = μ4 S2 T3 − μ2 T3 S2 + μ2 S3 (1 − T2 ). After using T3 S2 = S2 T3 obtained from Lemma 3.30 we get this to be the same as (1 − μ2 )S3 (1 − T2 ) = μ2 (μ2 − 1)T3 S2 . This is equivalent to S3 (1 − T2 ) = −μ2 T3 S2 (as μ2 = 1) which follows from (47). 2 Proposition 3.32. The map SOμ (3) → Q sending M, L, G, N , C to −(1 + μ2 )−1 S2 , S3 , −μ−1 S4 , (1 + μ2 )−1 (1 − T2 ), μT3 respectively is a CQG homomorphism. Proof. It is enough to check that the map is ∗-homomorphic, since the coproducts on SOμ (3) and Q are determined in terms of the fundamental unitaries Z and Z respectively, and the map described in the statement of the proposition sends (ij )-th entry of Z to the (ij )-th entry of Z for all (ij ). Now, it can easily be checked that the proof of the homomorphic property of the given map reduces to verification of the relations on Q as derived in Lemmas 3.21–3.31 along with the following equations: S3 S4 = μ4 S4 S3 ,

(48)

S3 S2 = μ2 S2 S3 ,

(49)

S2 S4 = μ S4 S2 ,

(50)

2

S3 S4 = −

μ2

S2, 2 2 2 (1 + μ )

which follow from Remark 3.20, (28), (29), (32) respectively.

(51) 2

Theorem 3.33. We have the isomorphism: 2 ∼ QISO+ R O Sμ,c , H, D = SOμ (3).

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+ (D) as remarked before, and thus one gets a surjective Proof. SU μ (2) is an object in QISO R + (D) to SU (2) which clearly maps QISO+ (D) onto SO (3), identimorphism from QISO μ μ R R fying the latter as a quantum subgroup of QISO+ (D). Let us denote the surjective map from R + QISO+ (D) to SO (3) by Π . On the other hand, Proposition 3.32 implies that QISO μ R R (D) is a quantum subgroup of SOμ (3), and the corresponding surjective CQG morphism from SOμ (3) onto QISO+ R (D) is clearly seen to be the inverse of Π , thereby completing the proof. 2 Remark 3.34. Theorem 3.33 shows that for a fixed μ, the quantum isometry group QISO+ R (D) 2 of Sμ,c does not depend on c. This may appear somewhat surprising, but let us remark that in 2 are the classical situation (that is for μ = 1), c corresponds to the radius of the sphere and S1,c ∗ isomorphic as C algebras for all c 0. We refer the reader to [13, p. 126], for the details regarding this. Although in the noncommutative case, that is, when μ = 1, we do get non-isomorphic 2 for different choices of c, one may still think that the parameter c in some C ∗ -algebras Sμ,c sense determines the ‘radius’ of the noncommutative sphere, and thus one should get the same (quantum) isometry group for different choices of c. In view of the above, it seems impossible to ‘reconstruct’ the quantum homogeneous spaces 2 from the quantum isometry groups SO (3). In this context, it may be mentioned that for Sμ,c μ 2 are quantum homogeneous spaces corresponding to SO (3), only S 2 μ = 1, although all Sμ,c μ μ,0 arises as a quotient of SOμ (3) by a quantum subgroup (see [16] for more details). Thus, it is 2 from the quantum group SO (3). perhaps possible to somehow ‘reconstruct’ Sμ,0 μ + (D) 3.4. Existence of QISO For the above spectral triple, we have been unable to settle the issue of the existence of QISO+ (D) which is the universal object (if it exists) in the category Q (D) mentioned in Section 1. Nevertheless, we shall show that if a universal object in Q (D) exists, then QISO+ (D) must coincide with SOμ (3). + (D) exists, its induced action on S 2 , say α , must preserve the state h Lemma 3.35. If QISO 0 μ,c on the subspace spanned by {1, A, B, B ∗ , AB, AB ∗ , A2 , B 2 , B ∗ }. 2

Proof. Let W0 = C.1, W 1 = Span{1, A, B, B ∗ }, 2

W 3 = Span 1, A, B, B ∗ , AB, AB ∗ , A2 , B 2 , B ∗2 . 2

We note that the proof of Proposition 3.10 and the lemmas preceding it do not use the assumption that the action is R-twisted volume preserving, so the proof of Proposition 3.10 goes through verbatim implying that α0 keeps invariant the subspace spanned by {1, A, B, B ∗ } and hence it preserves W 3 as well. Let W 3 = W 1 ⊕ W be the orthogonal decomposition with respect to the 2

2

2

Haar state (say h0 ) of QISO+ (D). Since SOμ (3) is a sub-object of QISO+ (D), there is a CQG morphism π from QISO+ (D) onto SOμ (3) satisfying (id ⊗ π)α0 = , where is the SOμ (3) 2 . It follows from this that any QISO+ (D)-invariant subspace (in particular W ) is action on Sμ,c also SOμ (3)-invariant. On the other hand, it is easily seen that on W 3 , the SOμ (3)-action de2


2959

composes as W 1 ⊕ W (orthogonality with respect to h, the Haar state of SOμ (3)), where W 2 is a five-dimensional irreducible subspace. We claim that W = W , which will prove that the QISO+ (D)-action α0 has the same horthogonal decomposition as the SOμ (3)-action on W 3 , so preserves C.1 and its h-orthogonal 2 complements. This will prove that α0 preserves the Haar state h on W 3 . 2 We now prove the claim. Observe that V := W ∩ W is invariant under the SOμ (3)-action but due to the irreducibility of on the vector space W or W , it has to be zero or W = W . Now, dim(V) = 0 implies dim(W ) + dim(W ) = 5 + 5 > 9 = dim(W 3 ) which is a contradiction 2 unless W = W . 2 + (D) exists, then we must have that QISO+ (D) ∼ Theorem 3.36. If QISO = SOμ (3). Proof. In the proof of Lemma 3.35, it was noted that Proposition 3.10 follows under the assumption of the present theorem. To complete the proof of the theorem, we just need to observe that the other lemmas used for proving Theorem 3.33 require the conclusion of Lemma 3.35 as the only extra ingredient. 2 Let us conclude the article with brief explanation of the technical difficulties regarding the + (D). Let R be a positive, invertible operator commuting with D issue of existence of QISO

2 such that τR = τR and let φ denote the action of QISO+ R (D) on Sμ,c . The problem of existence + (D) is closely related to the question whether it is possible to identify QISO+ (D) as of QISO R

a quantum subgroup of SOμ (3) for a general R . By Theorem 3.36, a negative answer of this question will prove that Q (D) does not have a universal object. Now, as has been noted in Remark 3.5, φ is linear, that is, it keeps the span of {1, A, B, B ∗ } invariant and hence it is given by an expression similar to Eqs. (5) and (6) with Ti , Si replaced ∗ by some Ti , Si which generate QISO+ R (D) as a C -algebra. We can in principle write down all the relations satisfied by these generators, proceeding as in Section 3.3. These relations will be analogous to Eqs. (7)–(34), and in fact, the relations which make use of the homomorphism property only remain unchanged. However, the ones making use of the fact that φ preserves τR will change, since τR is in general different from τR . In particular, the expression of the antipode will change, which will affect all the relations starting from (23). We need to have a deeper and systematic understanding of the relations satisfied by Ti , Si for a general R , and possibly study their representations in concrete Hilbert spaces, to decide whether QISO+ R (D) is a quantum subgroup of SOμ (3) or not. We are not yet able to do this. + (D) exists, although we can identify QISO+ (D) with the wellMoreover, even if QISO + (D). If U denotes known quantum group SOμ (3), it is not so easy to explicitly compute QISO + (D), the fact that U commutes with D implies the unitary representation corresponding to QISO that U must preserve each of the two-dimensional eigenspaces span{v l and span{v l

m, 12

− vl

m,− 21

m, 21

+ vl

m,− 12

: m = ± 12 }

: m = ± 12 } of D. Suppose that (qij )i,j =1,2 and (rij )i,j =1,2 are the ma-

+ (D)) of U corresponding to these two spaces respectively. Then it trices (with entries in QISO + (D) will be generated by q , r ’s as well as the generators is clear that as a C ∗ algebra QISO ij

ij

Ti , Si of SOμ (3). However, the mutual relations among these generating elements have to be

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determined from the fact that U preserves each of the eigenspaces of D. In principle one gets infinitely many such relations which are quite complicated and it is not clear how to simplify them. 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, arXiv:0707.2648. [4] J. Bhowmick, D. Goswami, Quantum group of orientation preserving Riemannian isometries, J. Funct. Anal. 257 (8) (2009) 2530–2572. [5] J. Bichon, Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (3) (2003) 665–673. [6] A. Connes, Noncommutative Geometry, Academic Press, London, New York, 1994. [7] Alain Connes, Michel Dubois-Violette, Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, Comm. Math. Phys. 230 (3) (2002) 539–579. [8] L. Dabrowski, Spinors and theta deformations, Russ. J. Math. Phys. 16 (3) (2009) 404–408. [9] L. Dabrowski, F. D’Andrea, G. Landi, E. Wagner, Dirac operators on all Podles quantum spheres, J. Noncommut. Geom. 1 (2007) 213–239. [10] D. Goswami, Twisted entire cyclic cohomology, JLO cocycles and equivariant spectral triples, Rev. Math. Phys. 16 (5) (2004) 583–602. [11] D. Goswami, Quantum group of isometries in classical and noncommutative geometry, Comm. Math. Phys. 285 (1) (2009) 141–160, arXiv:0704.0041. [12] D. Goswami, Some noncommutative geometric aspects of SU q (2), arXiv:math-ph/0108003. [13] A. Klimyk, K. Schmudgen, Quantum Groups and Their Representations, Springer, New York, 1998. [14] A. Maes, A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Wiskd. (4) 16 (1–2) (1998) 73–112. [15] P. Podles, Quantum spheres, Lett. Math. Phys. 14 (1987) 193–202. [16] 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. [17] K. Schmudgen, E. Wagner, Dirac operators and a twisted cyclic cocycle on the standard Podles’ quantum sphere, J. Reine Angew. Math. 574 (2004) 219–235. [18] K. Schmudgen, E. Wagner, Representation of cross product algebras of Podles quantum spheres, math.QA/0305309. [19] P.M. Soltan, Quantum SO(3) groups and quantum group actions on M2 , arXiv:0810.0398v1. [20] S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (3) (1995) 671–692. [21] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998) 195–211. [22] S. Wang, Structure and isomorphism classification of compact quantum groups Au (Q) and Bu (Q), J. Operator Theory 48 (2002) 573–583. [23] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (4) (1987) 613–665. [24] 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.


Similarity of analytic Toeplitz operators on the Bergman spaces ✩ Chunlan Jiang a , Dechao Zheng b,∗ a Department of Mathematics, Hebei Normal University, Shijianzhuang 050016, PR China b Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

Received 31 July 2009; accepted 16 September 2009 Available online 3 October 2009 Communicated by N. Kalton

Abstract In this paper we give a function theoretic similarity classification for Toeplitz operators on weighted Bergman spaces with symbol analytic on the closure of the unit disk. © 2009 Elsevier Inc. All rights reserved. Keywords: Bergman spaces; Toeplitz operator; Similarity; Blaschke product; Strongly irreducible

1. Introduction Let dA denote Lebesgue area measure on the unit disk D, normalized so that the measure of D equals 1. For α > −1, the weighted Bergman space A2α is the space of analytic functions on D which are square-integrable with respect to the measure dAα (z) = (α + 1)(1 − |z|2 )α dA(z). For u ∈ L∞ (D, dA), the Toeplitz operator Tu with symbol u is the operator on A2α defined by Tu f = P (uf ); here P is the orthogonal projection from L2 (D, dAα ) onto A2α . Tg is called to be the analytic Toeplitz operator if g ∈ H ∞ (the set of bounded analytic functions on D). In this case, Tg is just the operator of multiplication by g on A2α . In this paper we study the similarity of Toeplitz operators with symbol analytic on the closure of the unit disk on A2α . On the Hardy ✩

The first author was partially supported by NSFC and the second author was partially supported by NSF.

* Corresponding author.

E-mail addresses: [email protected] (C. Jiang), [email protected] (D. Zheng). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.011

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C. Jiang, D. Zheng / Journal of Functional Analysis 258 (2010) 2961–2982

space, Cowen showed that two Toeplitz operators with symbol analytic on the closure of the unit disk are similar if and only if they are unitarily equivalent [3]. However this is not true on the Bergman space. In [17], Sun showed that for two functions f and g analytic on the closure of the unit disk, the analytic Toeplitz operator Tf on the Bergman space A20 is unitarily equivalent to Tg if and only if there is an inner function χ of order one such that g = f ◦ χ . Also in [18], Sun and Yu showed that if the direct sum of two analytic Toeplitz operators is unitarily equivalent to an analytic Toeplitz operator, then they must be constants. So the Bergman space is rigid. But in [12], Jiang and Li showed that if f is a finite Blaschke product, then the analytic Toeplitz operator Tf is similar to the direct sum of finite copies of the Bergman shift Tz on the unweighted Bergman space A20 . In this paper, we will completely determine when two Toeplitz operators with symbol analytic on the closure of the unit disk are similar on the weighted Bergman spaces in terms of symbols, which is analogous to the result on the Hardy space [3]. While the Beurling theorem plays an important role on the Hardy space [3] and the Beurling theorem does not hold on the weighted Bergman spaces [8], we apply the general results [9–11] on similarity classification of the Cowen–Douglas classes to analytic Toeplitz operators. It was shown in [9,11] that two strongly irreducible members of Cowen–Douglas operator class Bn (Ω) [4] are similar if and only if the respective commutant algebras have isomorphic K0 groups and strongly irreducible decomposition operators give the similarity classification for Cowen–Douglas operator classes. As the adjoint of Toeplitz operators with symbol analytic on the closure of the unit disk is in the Cowen–Douglas operator classes, our main ideas are to identify the commutant of analytic Toeplitz operators as the commutant of Toeplitz operators with some finite Blaschke products and to use the strongly irreducible decomposition of analytic Toeplitz operators and K0 -groups of the commutants. The following theorem is our main result. It gives a function theoretic similarity classification of Toeplitz operators with symbol analytic on the closure of the unit disk. Theorem 1.1. Suppose that f and g are analytic on the closure of the unit disk D. Tf is similar to Tg on the weighted Bergman spaces A2α if and only if there are two finite Blaschke products B and B1 with the same order and a function h analytic on the closure of the unit disk such that f =h◦B

and g = h ◦ B1 .

2. Toeplitz operators with symbol as a finite Blashcke product A finite Blashcke product B is given by B=

n z − ak 1 − ak z

k=1

for n numbers {ak }nk=1 in the unit disk and some positive integer n. Here n is said to be the order of the Blaschke product B. In this section we will show that two Toeplitz operators with symbols as finite Blaschke products are similar on the weighted Bergman spaces if and only if their symbols have the same order. This was conjectured in [6] and proved in [12] on the unweighted Bergman space. For another proof, see [7]. Even for a very special Toeplitz operator Tzn on the weighted Bergman space A2α , the following result was established in [14] recently.


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Theorem 2.1. Let B be a Blaschke product with order n. Then TB on the weighted Bergman space A2α is similar to the direct sum nj=1 Tz on nj=1 A2α . We define the composition operator CB on A2α by CB (f )(z) = f B(z) for f ∈ A2α . Since B is a finite Blaschke product, the Nevanlinna counting function of B is equivalent to (1 − |z|2 ) on the unit disk. Thus CB is bounded below and hence has the closed range. For each k, define Mk =

Since T

1 1−ak z

f (B) : f ∈ A2α . 1 − ak z

is invertible, Mk is a closed subspace of A2α .

Assume that the n-th order Blaschke B has n distinct zeros. In [16], Stessin and Zhu showed that the Bergman space A20 is spanned by {M1 , . . . , Mn }. In [12], Jiang and Li showed that the Bergman space A20 is the Banach direct sum M1 M2 · · · Mn . The following theorem extends Jiang and He’s result to the weighted Bergman spaces, which immediately gives Theorem 2.1. Theorem 2.2. Let B be an n-th order Blaschke product with distinct zeros {ak }nk=1 in the unit disk. Then the weighted Bergman space A2α is the Banach direct sum of M1 , . . . , Mn , i.e., Aα = M1 M2 · · · Mn . Before going to the proof of the above theorems we need the following simple lemma. Lemma 2.3. Suppose that {ak }nk=1 are n distinct nonzero numbers in the unit disk. Then the following system ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 1−|a1 |2 1 1−a1 a2 1 1−a1 a3

1 1−a2 a1 1 1−|a2 |2 1 1−a2 a3

.. .

.. .

1 1−a1 an

1 1−a2 an

1 1−a3 a1 1 1−a3 a2 1 1−|a3 |2

.. .

1 1−a3 an

··· ··· ··· .. . ···

1 1−an a1 1 1−an a2 1 1−an a3

.. . 1 1−|an |2

⎞

⎛ ⎞ ⎛ ⎞ c1 0 ⎟ ⎟ ⎜ c2 ⎟ ⎜ 0 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ c3 ⎟ = ⎜ 0 ⎟ ⎟⎜ . ⎟ ⎜ . ⎟ ⎟ ⎝ . ⎠ ⎝ .. ⎠ ⎠ . 0 cn

(2.1)

has only the trivial solution. Proof. Using row reductions and induction we obtain that the determinant of the coefficient matrix of system (2.1) equals

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1 1−|a |2 1 1 1−a1 a2 1 1−a1 a3 .. . 1 1−a1 an

1 1−a2 a1 1 1−|a2 |2 1 1−a2 a3

1 1−a3 a1 1 1−a3 a2 1 1−|a3 |2

.. .

.. .

1 1−a2 an

1 1−a3 an

1 1−|a1 |2 a (a1 −a2 ) (1−a11a2 )(1−|a 2 1| ) a1 (a1 −a3 ) = (1−a1 a3 )(1−|a1 |2 ) .. . a1 (a1 −an )

(1−a1 an )(1−|a1 |2 )

···

1 1−an a1 1 1−an a2 1 1−an a3

··· ··· .. . ···

.. . 1 1−|an |2

1 1−a2 a1 a2 (a1 −a2 ) (1−|a2 |2 )(1−a2 a1 ) a2 (a1 −a3 ) (1−a2 a3 )(1−a2 a1 )

1 1−a3 a1 a3 (a1 −a2 ) (1−a3 a2 )(1−a3 a1 ) a3 (a1 −a3 ) (1−|a3 |2 )(1−a3 a1 )

.. .

.. .

a2 (a1 −an ) (1−a2 an )(1−a2 a1 )

1 a1 (1−a1 a2 ) a1 − aj a1 1 (1−a1 a3 ) = 1 − a1 aj 1 − |a1 |2 .. j >1 . a1

(1−a1 an )

1 0 a1 − aj 1 0 = 1 − a1 aj . 1 − |a1 |2 j >1 .. 0

a3 (a1 −an ) (1−a3 an )(1−a3 a1 )

1 a3 (1−a3 a2 ) a3 (1−|a3 |2 )

.. .

.. .

a2 (1−a2 an )

a3 (1−a3 an )

.. .

an (a1 −an ) (1−|an |2 )(1−an a1 )

··· ···

1 an (1−an a2 ) an (1−an a3 )

··· .. . ···

.. . an (1−|an |2 )

−(a1 −a3 ) (1−a3 a2 )(1−a1 a2 ) −(a1 −a3 ) (1−|a3 |2 )(1−a1 a3 )

.. .

.. .

−(a1 −a2 ) (1−a2 an )(1−a1 an )

··· ···

1

−(a1 −a2 ) (1−|a2 |2 )(1−a1 a2 ) −(a1 −a2 ) (1−a2 a3 )(1−a1 a3 )

1−a2 an

n(n−1) 2

··· .. . ···

1

1

1 1−an a1 an (a1 −a2 ) (1−an a2 )(1−an a1 ) an (a1 −a3 ) (1−an a3 )(1−an a1 )

···

a2 (1−|a2 |2 ) a2 (1−a2 a3 )

1 1−|a2 |2 |a1 − aj |2 1−a12 a3 1 n−1 = (−1) 1 − |a1 |2 (1 − a1 aj )2 .. j >1 . 1

= (−1)

···

··· .. . ···

−(a1 −a3 ) (1−a3 an )(1−a1 an ) 1 1−a3 a2 1 1−|a3 |2

.. . 1 1−a3 an

··· ··· .. . ···

n |aj − ak |2 1 = 0. 2 (1 − aj ak )2 j =1 (1 − |aj | )

n

j =1 j T . Example 21. Consider the case σ = 1, b1 = 0 and b2 (x) is a smooth increasing function such √ −1/2 that b2 (x) = x1 for x1 1. In this case, ∂1 b2 (x) = 12 x1 . As before we consider the path z : [0,√T ] → (1, +∞) to be a piecewise linear function as in the proof of Lemma 16 such that 1 1 z0 = x , zT = y , zT /2 = q > 1 with T zt dt = y 2 − x 2 . 0

V. Bally, A. Kohatsu-Higa / Journal of Functional Analysis 258 (2010) 3134–3164

3161

1 1 Therefore τ = ∞. For √ the sake of the argument we assume that min{x , y } > 1 and q = √ 1+ y1 x 1 2 2 > 1. In this case, we have T /2 (y − x ) − 2

T

−1 −2 220 150 pT (x, y) K1 (μ) 2 zT 1 + (2zt ∂t zt )2 2−1 zt−2 exp −K2 (μ) dt , 0

where √ 3 K1 (μ) = , 4πμ446

−1 2μ296 9 ln 2 × 21π 1/2 μ744 μ¯ 6 T ∧ C ∗ (μ) + 150 + 49μ4 , ∗ T ∧ C (μ) √ √ 1 q− y 1 max{| q−T /2x |, | T /2 |} μ¯ = μ , √ √ 1 q− y 1 min{| q−T /2x |, | T /2 |} √ max{ x 1 , y 1 , q} μ= . √ min{ x 1 , y 1 , q}

K2 (μ) =

√ −1 so we have a degeneracy at infinity. ThereNote that for |x1 | > 1, we have ∂σ b2 (x) = (2 |x|) t 1 2 2 fore xt = (zt ) , φt = 2zt ∂t zt and x2 (t) = x + 0 zs ds. We then have that εφ (t) = (2zt )−1 , Cφ (t) = zt ,

148

, hφ (t) = min h∗ , C ∗ (μ) 2−1 zt−2

where we have that zt , zt−2 ∈ Λμ,h∗ for h∗ = T . Furthermore we also set |φt | = |2zt ∂t zt | = φ¯ t ∈ Λμ,h ¯ ∗ . As before, one can use the lower bound for the density to prove that

ln((y 2 )446 pt (x, y)) −C x 1 , y 1 , x 2 , T . 2 4 2 (y ) y →∞

lim inf

Remark. A variety of similar situations can be treated with the above arguments. For example, in the case that σ = 1, b1 = 0 and b2 (x) is a smooth function such that for |x1 | big enough b2 (x) = e−|x1 | . It is clear from the above argument that if we assume that min{x 1 , y 1 } > 1, a similar argument as the one above can be used to obtain a lower bound for the density in this case. We now give a lower bound in optimal form. For z ∈ CT (x, y) we denote

εz (t) = ∂1 b2 xt (z) ∧ σ xt (z) ∧ 1 > 0,

Cz (t) = 1 + |σ | + |b| + |∂σ σ | + |∂σ b| + |Lb| xt (z) .

3162


We also fix μ > 1 > h and we define the class CT ,μ,h (x, y) = z ∈ CT (x, y): εz , Cz , εz Cz−1 ∈ Λμ,h . Following (24), we define εz (t) 148 hz (t) := min h∗ , C ∗ (μ) , Cz (t) φt (z) =

∂t zt − ∂b b2 (xt (z)) . ∂σ b2 (xt (z))

So, for z ∈ CT ,μ,h (x, y), the hypothesis (H2 (φ)).i) holds with φ = φ(z), (H1 (φ)) holds with εφ = ε(z) and (H3 (φ)) holds with φ¯ = φ, μ and h∗ = 1. Then as an immediate consequence of Theorem 14 we obtain Theorem 22. Suppose that the (H2 (φ)).ii) (see (9)) is satisfied and CT (x, y) = ∅ for some μ > 1 > h. Then Cz (T ) 220 K1 (μ) εz (T ) μ>1>h z∈CT (x,y)

pT (x, y) sup

sup

T ∂t zt − ∂b b2 (xt (z)) 2 Cz (t) 150 1+ × exp −K2 (μ) dt . ∂σ b2 (xt (z)) εz (t) 0

Here, K1 (μ) =

√ 3 , 4πμ448

K2 (μ) =

2μ296 9 ln 2 × 21π 1/2 μ¯ 6 μ744 C ∗ (μ)−1 + 150 + 49μ4 . ∗ C (μ)

In this theorem one may use μ¯ = maxt∈[0,T ] |φt (z)|. Appendix A. Proof of Theorem 2 We recall the reader that the notation appearing in this proof corresponds to the one in [2]. We use that framework with M˜ k =

1 ak Mk ,

H˜ k =

1/2

ak Hk 1/2

ak−1

, a˜ k = ak2 with the set A˜ k redefined as

ρ˜i A˜ k = ω ∈ Ω: |Fi−1 − yi |ı˜ , i = 1, . . . , k . 2 Here the norm |x|ı˜ = M˜ i−1 x, x1/2 and ρ˜i = ai ρi . Then the proofs in [2] apply exactly chang˜ z) by ρ˜ and ing the constant 1 which appears in the definition of the hypothesis (H1 , a, ˜ A, 1/2


3163

(ρ+ ˜ c) ˜2 η ∈ (0, c˜ M˜ ) which gives the same result except that e2 is changed by e 2 . This factor is not considered in the definition of Cd above as it was the case in [2]. Therefore in this set˜ z) is ting the constant e2 in [2] starts to depend on ρ˜ and c. ˜ Similarly, the condition (H2 , a, ˜ A, changed to −1/2 M˜ R t,δ,d+2,p d

1 (ρ+ ˜ c) ˜2 2

a˜ 4(d+1) Cd e 2

.

Proposition 8 becomes 1

pη (z)(ω) 4e

(2π a) ˜ d/2 det M˜

(ρ+ ˜ c) ˜2 2

˜ and some η ∈ (0, c˜ M˜ ). Then Section 2.3 applies for ω ∈ A ⊂ {ω: V (ω) − zM −1 ρ} exactly with the following changes. 1. The estimate in Corollary 9 becomes (where c˜k = ρ˜˜k ) 8Hk

P (A˜ k )

P (A˜ k−1 )ρ˜kd 8d+1 H˜ kd e

(c˜k +ρ˜k−1 /2)2 2

(2dak−1 π)d/2

under the condition that −1/2 M˜ Rk t k

k−1 ,δk ,d+2,pd

1

4(d+1)2

Cd a˜ k

e

(c˜k +ρ˜k−1 /2)2 2

.

2. This estimate naturally leads to the estimate in Theorem 15 with pFN (xN )

e−N dθ 4(2π)d/2 det M˜ N

with ˜ N N N

1 1 1 Hk ρ˜k−1 2 + c˜k + ln(a˜ k ) + ln . θ = ln 82 (2πd)1/2 + 2N N ρ˜k 8dN 2 k=1

k=2

k=2

Finally replacing all the above parameters changes and the inequality c˜k + ρ˜k−1 = a 1/2 ρk−1 + ρk a 1/2 ρk−1 + ρk k−1 k−1 2 2 8Hk 2 one obtains the result.

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References [1] S. Aida, S. Kusuoka, D. Stroock, On the support of Wiener functionals, in: K.D. Elworthy, N. Ikeda (Eds.), Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics, in: Pitman Res. Notes Math. Ser., vol. 284, Longman Scient. Tech., 1993, pp. 3–34. [2] V. Bally, Lower bounds for the density of the law of locally elliptic Ito processes, Ann. Probab. 34 (2006) 2406– 2440. [3] G. Ben-Arous, R. Leandre, Decroissance exponentille du noyau de la chaleur sur la diagonale (II), Probab. Theory Related Fields 90 (1991) 377–402. [4] U. Boscain, S. Polidoro, Gaussian estimates for hypoelliptic operators via optimal control, Rend. Lincei Mat. Appol. 18 (2007) 333–342. [5] F. Delarue, S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, 2009, in preparation. [6] C.L. Fefferman, A. Sanchez-Calle, Fundamental solutions of second order subelliptic operators, Ann. of Math. (2) 124 (1986) 247–272. [7] A. Kohatsu-Higa, Lower bounds for densities of uniform elliptic random variables on Wiener space, Probab. Theory Related Fields 126 (2003) 421–457. [8] S. Kusuoka, D. Stroock, Applications of the Malliavin calculus, part III, J. Fac. Sci. Univ Tokyo Sect. 1A Math. 34 (1987) 391–442. [9] P. Malliavin, E. Nualart, Density minoration of a strongly non-degenerated random variable, J. Funct. Anal. 256 (2009) 4197–4214. [10] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin, 1995. [11] A. Pascucci, S. Polidoro, Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators, Trans. Amer. Math. Soc. 358 (2006) 4873–4893. [12] S. Polidoro, A global lower bound for the fundamental solution of a Kolomgorov–Fokker–Planck equation, Arch. Ration. Math. Anal. 137 (1997) 321–340. [13] A. Sanchez-Calle, Fundamental solutions and geometry of the sum of square of vector fields, Invent. Math. 78 (1) (1984) 143–160. [14] M. Yor, On some exponential functionals of Brownian motion, Adv. in Appl. Probab. 24 (1992) 509–531.


Existence and multiplicity results for a fourth order mean field equation ✩ Mohamed Ben Ayed a , Mohameden Ould Ahmedou b,∗ a Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia b Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Received 1 September 2009; accepted 9 January 2010 Available online 8 February 2010 Communicated by J. Coron

Abstract In this article we consider the following fourth order mean field equation on smooth domain Ω R4 : Keu u Ω Ke

2 u =

in Ω,

u = u = 0 on ∂Ω, where ∈ R and 0 < K ∈ C 2 (Ω). Through a refined blow up analysis, we characterize the critical points at infinity of the associated variational problem and compute their contribution of the difference of topology between the level sets of the associated Euler–Lagrange functional. We then use topological and dynamical methods to prove some existence and multiplicity results. © 2010 Elsevier Inc. All rights reserved. Keywords: Fourth order nonlinear elliptic equation; Critical point at infinity; Morse theory; Topological methods

✩

The research of M. Ould Ahmedou has been partly supported by SFB TR 71.


E-mail addresses: [email protected] (M. Ben Ayed), [email protected] (M. Ould Ahmedou). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.009

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M. Ben Ayed, M. Ould Ahmedou / Journal of Functional Analysis 258 (2010) 3165–3194

1. Introduction and statement of main results In this paper we consider the following fourth order mean field equation:

u

2 u = KeKeu

in Ω,

u = u = 0

on ∂Ω,

Ω

(1.1)

where ∈ R, 0 < K ∈ C 2 (Ω) and Ω is a bounded and smooth domain of R4 . In dimension 2 the analogue problem

u

−u = KeKeu

in Ω,

u=0

on ∂Ω,

Ω

(1.2)

where Ω is a bounded smooth domain in R2 and K ∈ C 2 (Ω) has been extensively studied, see [9,14,13,19,20] and the references therein. Our interest in (1.1) grew up, in particular from its resemblance to the prescribed Q-curvature problem on 4-dimensional riemannian manifold (M 4 , g). Indeed on such a manifold the Paneitz operator is defined as: Pg4 ϕ

= 2g ϕ

− divg

2 Rg g − 2 Ricg dϕ, 3

where Rg and Ricg denote respectively the scalar and Ricci curvature. The Paneitz operator gives rise to a fourth order curvature: the Q-curvature defined as: Q :=

1 −R + R 2 − 3|Ric|2 . 12

Now since under conformal change of metrics g = e2w g, there holds Pg4 w + 2Qg = 2Qg e4w ,

(1.3)

the following natural question arises: Does there exist a metric g˜ conformally equivalent to g such that Qg˜ = K? In view of the above transformation law, this amounts to solve the following nonlinear fourth order equation: Pg4 w + 2Qg = 2Ke4w

in M 4 .

(1.4)

The above equation has been during the last decades extensively studied. See the works [2,7,10,12,11,15–17,25,26] and references therein. Coming back to our fourth order mean field Eq. (1.1), we point out that it has a variational structure, indeed its solutions are in one-to-one correspondence with the critical points of the following functional defined on H := H 2 (Ω) ∩ H01 (Ω) by J (u) := 1/2 Ω

|u|2 dx − Log Keu dx . Ω


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The analytic features of Eq. (1.1) and of its associated Euler–Lagrange functional depend strongly on some critical values of the parameter . Indeed depending on wether is a multiple of 64π 2 or not, the noncompactness of the variational problem and therefore the way of finding critical points of the functional J on H change drastically. Therefore we will first focus on the case = 64mπ 2 , m ∈ N. Theorem 1.1. Assume that ∈ (64mπ 2 , 64(m + 1)π 2 ); m ∈ N∗ and that Ω is not contractible, then the problem (1.1) has at least one solution. Furthermore, for generic K’s, there holds

1 # solutions of (1.1) 1 − χ(Ω) · · · m − χ(Ω) . m! Remark 1.2. Under the stronger condition that χ(Ω) 0, where χ(Ω) denotes the Euler characteristic of Ω, the above theorem has been proved in [22]. Their proof which is drastically different from ours, uses a topological degree argument. We observe that, the embedding of S2 in R4 has a positive Euler characteristic but nontrivial topology. We point out that a main ingredient in the proof of Theorem 1.1 is the fact that for = 64mπ 2 , m ∈ N∗ , although the functional does not satisfy the Palais Smale Condition, the change of topology between any finite level sets is due only to the existence of critical points of the functional J and from another part the topology of the sublevel set J −L := {u ∈ H; J (u) < −L} for very large L is homotopically equivalent to the set of formal barycenter Bm (K) := K m ×σm m−1 , where K Ω is a compact subset of Ω and m−1 denotes the standard simplex. We recall that the role of topology of formal barycenters in noncompact variational problems involving exponential nonlinearities was first discovered by Djadli and Malchiodi (see [16]). Now we address the case = 64π 2 . To state our results in this case, we need to introduce the following notation. Let G4 (a, .) be the Green’s function of 2 under Navier boundary conditions and H4 (a, .) its regular part and set fK : Ω → R defined by fK (y) := Log K(y) − 32π 2 H4 (y, y).

(1.5)

We say that the function K satisfies the condition (C0 ) if fK has only nondegenerate critical points and at each critical point of fK there holds that (Log K)(y) − 64π 2 1 H (y, y) = 0, where 1 H (y, y) denotes the Laplacian of the function H (y, .). We set

K− := y ∈ Ω; ∇fK (y) = 0 and (Log K)(y) − 64π 2 1 H (y, y) < 0 . Now we are ready to state our next result:

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Theorem 1.3. Let = 64π 2 and assume that K satisfies the condition (C0 ). If

(−1)morse(fK ,q) = 1,

q∈K−

then the problem (1.1) has at least one solution. Here morse(fK , q) denotes the Morse index of fK at the critical point q. Furthermore for generic K’s the number of solutions of (1.1) is lower bounded by morse(fK ,q) 1 − (−1) . q∈K−

In view of the above results, one may think about the situation where the total sum equals 1 but a partial one is not equal 1. A natural question arises: Is it possible in this case to use such an information to derive an existence result? In the following theorem we give a partial result to this question. Theorem 1.4. Let = 64π 2 and K be a function satisfying (C0 ). Assume that there exists k ∈ N∗ such that: ∀q ∈ K− ;

(1)

ι(q) = k,

where ι(q) := 4 − morse(fK , q) is the coindex of q. (2) (−1)morse(fK ,q) = 1. q∈K− ; ι(q) 0 and let V (p, ε) be a neighborhood of potential critical points at infinity defined as p C1 V (p, ε) := αi P δai ,λi + w: for each i, |αi − 1| 2 , λi ε −1 , λi i=1 λi dist(ai , ∂Ω) η, and < C1 , |ai − aj | 2η for i = j , λj where C1 is a large positive constant and η is a fixed positive constant. Following the ideas of Bahri and Coron [5], for u ∈ V (p, ε) and ε small, the following minimization problem has, up to permutation, only one solution. p αi P δai ,λi . min u − αi >0; ai ∈Ω; λi >0

(3.1)

i=1

Hence every u ∈ V (p, ε) can be written as u=

p

αi P δai ,λi + w,

(3.2)

i=1

where w satisfies ∂P δai ,λi = 0, w, P δai ,λi = w, ∂λi

∂P δai ,λi w, = 0 and w < ε. ∂ai

Remark that, from the definition of V (p, ε), we have Bη (ai ) := B(ai , η) ⊂ Ω

for each i

and Bη (ai ) ∩ Bη (aj ) = ∅ for i = j.

(3.3)

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In this section, we give an asymptotic expansion of the functional J in V (p, ε) and we start with the following lemma: Lemma 3.1. Let u :=

p

i=1 αi P δai ,λi

∈ V (p, ε). On Bi := Bη (ai ) there holds

λ8i

Keu =

FiA (x) (1 + λ2i |x − ai |2 )4 Log2 λ A 2 2 × 1 + Bi (x) + (αi − 1) 8 Log λi − 4 Log 1 + λi |x − ai | + O λ4 λ8i Log λ , = FiA (x) 1 + O 2 2 4 λ2 (1 + λi |x − ai | )

where, if p 2, A = (a1 , . . . , ap ), FiA (x) := eLog(K(x))−64π

2 (H

4 (ai ,x)−

j =i

G4 (aj ,x))

,

(αj − 1)G4 (aj , x) BiA (x) := −64π 2 (αi − 1)H4 (ai , x) − + 16π 2

j =i

H2 (ai , x) G2 (aj , x) , − 2 λ2i λ i i=j

and if p = 1, then A = a1 , F1A (x) := eLog(K(x))−64π

2H

4 (ai ,x)

,

B1A (x) := −64π 2 (α1 − 1)H4 (a1 , x) + 16π 2

H2 (a1 , x) . λ21

Proof. The lemma follows easily from Lemma 8.1 and the fact that for each j , we have |αj − 1| C1 /λ2j (see the definition of V (p, ε)). 2 Using the above lemma, we can expand the integral part of the functional J . In fact, we have Lemma 3.2. Let u :=

p

Ke dx = u

Ω

i=1 αi P δai ,λi

p 2 π i=1

6

∈ V (p, ε), then we have

λ4i FiA (ai ) +

π2 4 A π2 2 λi Fi (ai )BiA (ai ) + λ FiA (ai ) 6 24 i

5 4π 2 4 A 2 (αi − 1)λi Fi (ai ) Log λi − + O Log λ . + 3 12


3173

Proof. For simplicity, in all the proofs, we will write Fi and Bi instead of FiA and BiA respectively. We notice first that, since the function u in Ω \ ( Bi ) is bounded, we get Ω\(

Keu dx = O(1). Bi )

Now using Lemma 3.1, we derive:

Keu dx = Bi

Bi

λ8i Fi (x) (1 + λ2i |x

− ai |2 )4

+ (αi − 1) Bi

1 + Bi (x) + O Log2 λ

λ8i Fi (x) (1 + λ2i |x − ai |2 )4

8 Log λi − 4 Log 1 + λ2i |x − ai |2 .

Now we remark that, on Bi , Fi is a C ∞ function and we have Fi C ∞ (Bi ) is bounded. Thus, expanding around ai , we obtain

Keu dx = λ4i Fi (ai ) 1 + Bi (ai )

R4

Bi

dy 1 + O (1 + |y|2 )4 λ4

1 + (Fi + Fi Bi )(ai )λ2i 8

R4

|y|2 dy 1 + O Log2 λ + O 2 4 2 (1 + |y| ) λ

4 + (αi − 1) 8λi Log λi Fi (ai ) R4

dy 1 +O 4 2 4 (1 + |y| ) λ

2 Log(1 + |y|2 ) dy Log λ 4 + O λi Log λi . − 4λi Fi (ai ) +O (1 + |y|2 )4 λ4 R4

Finally using easy computations (to obtain the values of the integrals) and the fact that for each j , we have |αj − 1| C1 /λ2j , the lemma follows. 2 In the following, we will show that the w-part in the parametrization of functions in V (p, ε) does not play in the lack of compactness. More precisely we perform a finite dimensional reduction of the functional in such potential neighborhoods of infinity. Indeed we have Proposition 3.3. Let u =

p

i=1 αi P δai ,λi

J (u) = J

p i=1

where

+ w ∈ V (p, ε). Then we have that

αi P δai ,λi

1 − f (w) + Q(w) + o w 2 , 2

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α Pδ

Ke j aj ,λj w

f (v) := Ω αj P δaj ,λj Ω Ke and

α Pδ

Ke j aj ,λj w 2

Q(w) := w 2 − Ω . αj P δaj ,λj Ke Ω Moreover Q is a positive definite quadratic form and f satisfies: for every γ ∈ (0, 1) there exists a constant C(γ ) such that f (w) C(γ ) w

|∇FiA (ai )| Log λ . + λ1−γ λ2−γ

Before giving the proof of Proposition 3.3, we need the following lemma: Lemma 3.4. Let u =

p

i=1 αi P δai ,λi

+ w ∈ V (p, ε).

(i) Let β 1. For every γ ∈ (0, 1), there exists a constant C(γ ) such that

Keu−w w β C(γ )λγ +4 w β ,

Ω

Keu−w ew − 1 − w − w 2 /2 = o λ4 w 2 .

(ii) Ω

Proof. Observe that as u − w is bounded in Ω \ ( Bi ), it follows:

Ω\(

It remains the integral on

Keu−w |w|β C Ω

Bi )

Bi . From Lemma 8.1, it follows that

Keu−w |w|β C

Bi

|w|β C w β .

Bi

λ8i (1 + λ2i |x − ai |2 )4

Cλ8i Bi γ −4

Cλ8i λi

|w|β

1 2 (1 + λi |x − ai |2 )4

4 4−γ

C(γ ) w β ,

4−γ 4

w

β 4

Lγ

(3.4) 4

where we have used the continuity of the embedding H → L γ (Ω).


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Now to prove the second statement we argue as follows: Let t0 be a small positive constant. Using (i) of Lemma 3.4, for every γ ∈ (0, 1), there exists C(γ ) such that

Ke

u−w

Ω∩{|w| 0 such that 2 |w| 2 |w|2 w e − 1 − w − w 2 /2 Ce|w| Ce|w|(1−32π w 2 ) e32π w 2

Ce

t0 (1−32π 2

t0 )

w 2

e

32π 2

|w|2

w 2

.

Now using Moser–Trudinger Inequality [1] we derive that

Ke

u−w

Ω∩{|w|t0 }

w2 e −1−w− dx Cλ8 2

w

Ω∩{|w|t0 }

Cλ8 e

t0 (1−32π 2

2 w e − 1 − w − w dx 2 t0 )

w 2

.

Since w is very small, we get

e

t0 (1−32π 2

t0 )

w 2

ce

−32π 2

t02

w 2

c w 10 .

Finally, using the fact that w Cλ−2 (see the definition of V (p, ε)), the lemma follows. Proof of Proposition 3.3. First, using the orthogonality of w to P δai ,λi , we derive that 1 1 J (u) = u − w 2 + w 2 2 2 u−w 2 u−w w 2 1 + w + w /2 + Ke e − 1 − w − w /2 . − Log Ke Using Lemmas 3.2 and 3.4, we derive that Log Keu−w 1 + w + w 2 /2 + Keu−w ew − 1 − w − w 2 /2

= Log Hence

Ke

u−w

Keu−w w Keu−w w 2 Keu−w w 2 1 + + − + o w 2 . u−w u−w u−w 2 Ke Ke 2 Ke

2

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Keu−w w 1 Keu−w w 2 1 2 J (u) = J (u − w) − + −

w

u−w 2 2 Ke Keu−w Keu−w w 2 1 − + o w 2 . u−w 2 Ke To complete the proof, it remains to estimate |f (w)|. Observe that Keu−w w =

p i=1 B

Ω

i

=

p

Log λ w + O F (x) 1 + O i λ2 (1 + λ2i |x − ai |2 )4

λ4i Fi (ai )

i=1

+O Bi

Bi

λ4i w (1 + λ2i |x

− ai |2 )4

Bi

Log λ + 2 2 2 4 λ (1 + λi |x − ai | )

Bi

Ω\(

Bi )

λ8i (x − ai )w

+ ∇Fi (ai )

λ8i |x − ai |2 |w|

|w|

λ8i

(1 + λ2i |x − ai |2 )4

λ8i |w| (1 + λ2i |x − ai |2 )4

+ w .

Since w is orthogonal to P δi , it holds

P δai ,λi w = c0

λ4i

2

Ω

Ω

(1 + λ2i |x − ai |2 )4

w = 0,

16 3 3 4 λ4i w λi c c w

4 w . (1 + λ2 |x − a |2 )4 2 2 λ 1 + λ |x − a | i i i i Bi

Ω\Bi

As in the proof of (3.4), for every γ ∈ (0, 1), there holds Keu−w w = O

∇Fi (ai )λ3+γ + λ2+γ Log λ w .

Ω

Now using Lemma 3.2, the expansion claimed in Proposition 3.3 follows. Now we can proceed as in [3, pp. 65–68] to prove that the quadratic form Q is positive definite. Proposition 3.3 is thereby proved. 2 Now it follows from the above estimate on |f (w)| and the fact that the quadratic form Q is positive definite that: Corollary 3.5. Let u :=

p

i=1 αi P δai ,λi

∈ V (p, ε). Then there exists a unique w such that

J (u + w) = min J (u + w); u + w ∈ V (p, ε) . Furthermore, for every γ > 0 there exists a constant C(γ ) such that


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|∇FiA (ai )| 1 ,

w C(γ ) + 1−γ 2−γ λi λi where A = (a1 , . . . , ap ) and FiA is defined in Lemma 3.1. Corollary 3.6. Let u := w − w → v such that

p

i=1 αi P δai ,λi

∈ V (p, ε). Then there exists a change of variable

1 J (u + w) = J (u + w) + ∂ 2 J (u + w)(v, v). 2 Using the above estimate (Corollary 3.5), we derive the following expansion. Proposition 3.7. Let u :=

p

i=1 αi P δai ,λi

J (u + w) = 4 × 64π 2

p

∈ V (p, ε). Then we have

αi2 Log λi

i=1

− 64π 2

p i=1

5 2 αi + 32π 2 αi2 H4 (ai , ai ) − αi αj G4 (ai , aj ) 3 j =i

p 1 1 1 1 H (a , a ) − G (a , a ) + 2 i i 2 i j 2 λ2i λ2i λ2j j =i i=1 p π2 4 A − Log λi Fi (ai ) 6 + 16 × 64π 2

i=1

p 1 − p λ4i FiA (ai )BiA (ai ) + λ2i FiA (ai ) 4 A 4 i=1 λi Fi (ai ) i=1 Log λ 5 + w 2 . f (w) + 8(αi − 1)λ4i FiA (ai ) Log λi − +O + 12 λ4

Proof. The proof follows immediately from Lemmas 3.2, 8.3, Proposition 3.3 and Remark 8.4. 2 Corollary 3.8. Let u :=

p

i=1 αi P δai ,λi

∈ V (p, ε). There holds

J (u) = 4 64pπ 2 − Log λ1 + O(1). Proof. Since u ∈ V (p, ε) then there exists a constant C1 such that C1−1

λi C1 λj

∀i, j,

and |αi − 1|

C1 λ2

∀i.

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Therefore Log λi = Log λ1 + O(1)

and

αi2

1 . =p+O λ

It follows then from Proposition 3.7 that J (u) = 4 × 64π 2 p Log λ1 − 4 Log λ1 + O(1). Thus the corollary follows.

2

As an immediate result we have Corollary 3.9. Assume that = 64π 2 m with m 1. (1) There exists L > 0 such that ∀u ∈ V (m, ε),

−L J (u) L.

(2) For each b > L, we can choose ε > 0 small enough such that ∀p > m, ∀u ∈ V (p, ε),

J (u) > b.

(3) Assume that m 2. For each a < −L, we can choose ε > 0 small enough such that ∀p < m, ∀u ∈ V (p, ε),

J (u) < a.

4. Deformation lemma Lemma 4.1. Let a, b ∈ R such that a < b and there is no critical value of J in [a, b]. (1) If = 64mπ 2 , m ∈ N∗ , then J a is a retract by deformation of J b . (2) If = 64mπ 2 , m ∈ N∗ then there are two possibilities J a is a retract by deformation of J b , or J b retracts by deformation onto J a ∪ σ, where σ ⊂ V (m, ε) and for c ∈ R, J c := {u ∈ H: J (u) < c}. Proof. We first point out that it follows from an abstract result of [18,23] that given a < b ∈ R, regarding the difference of topology between the sublevel sets J a and J b there are only two possibilities: or J a is a retract by deformation of J b or there exists a sequence k and a sequence of solutions uk of


⎧ u ⎨ 2 u = Ke k k k uk Ω Ke dx ⎩ uk = uk = 0

in Ω,

3179

(4.1)

on ∂Ω,

such that k →

and a J (uk ) b.

Now observe that if uk were bounded, it would converge to a solution of the problem (1.1) which contradicts our assumption that in [a, b], there is no critical value of J . Therefore the sequence uk should blow up and it follows from the blow up analysis of Lin and Wei [21] that k → 64mπ 2

and uk ∈ V (p, ε)

for some p ∈ N∗ .

It follows then from Corollary 3.9 that p = m and = 64mπ 2 . As a consequence, in case where = 64mπ 2 , we have for every a < b ∈ R, J a is a retract by deformation of J b and if = 64mπ 2 and J a is not homotopically equivalent to J b then J b J a ∪ σ, where σ ⊂ V (m, ε) and means “retracts by deformation”.

2

Corollary 4.2. Let = 64mπ 2 , m ∈ N∗ and assume that (1.1) has a finite number l ∈ N of solutions. Thus there exists a large positive constant L1 such that: H retracts by deformation onto J L1 . Proof. Let w1 , . . . , wl be all the solutions of (1.1) with l ∈ N (if there exist). Thus there exists ˜ where L is defined in L˜ such that J (wi ) L˜ for each i l. Now let b > a > L1 := max(L, L) Corollary 3.9. By Lemma 4.1, we get that J b J a . Hence our corollary follows. 2 5. Expansion of the gradient near its potential end points Proposition 5.1. Let = 64mπ 2 , m 1 and u = P δi := P δai ,λi

m

i=1 αi P δai ,λi

and γi := 1 −

∈ V (m, ε). Setting

mλ4i FiA (ai ) π 2 . u 6 Ω Ke

There holds 1 ∂P δi −2π 2 λ4i ∇FiA (ai ) 64π 2 ∂H4 (ai , ai ) ∇J (u), = γi − u λi ∂ai 3 λi λi ∂a Ω Ke 64π 2 ∂G4 (ai , aj ) 1 + γj (if m 2) + O 2 . λi ∂a λ j =i

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Proof. Using Lemmas 3.1, 8.1 and expanding around ai , we derive that

1 ∂P δi Ke = λi ∂ai

Bi

Bi

λ8i Fi (x)

u

(1 + λ2i |x − ai |2 )4

λ8i

1 1 + (αi − 1) Log +O 2 2 2 4 λ (1 + λi |x − ai | )

λi (x − ai ) 64π 2 ∂H4 (ai , x) 1 × 8 − +O 3 2 2 λ ∂a 1 + λi |x − ai | λi i i

= 8λ3i

π2 π 2 ∂H4 (ai , ai ) ∇Fi (ai ) − 64π 2 λ3i Fi (ai ) + O λ2 . 12 6 ∂a

Now for i = j (if m 2) there holds Keu

1 ∂P δi = λi ∂ai

Bj

Bj

(1 + λ2j |x − aj |2 )4

1+O

Log λ λ2

64π 2 ∂G4 (ai , x) 1 × +O 3 λi ∂a λ

=

π 2 ∂G4 (ai , aj ) 64π 2 4 + O(λ Log λ). λj Fj (aj ) λi 6 ∂a

Finally using Lemma 8.2, the proposition follows. Proposition 5.2. Let = 64mπ 2 , m 1 and u =

λ8j Fj (x)

∂P δi ∇J (u), λi ∂λi

2

p

i=1 αi P δi

∈ V (m, ε). There holds

8π 2 Fi (ai ) 7 + 2 = 4 × 64π γi 1 − 2 H2 (ai , ai ) + Bi (ai ) + 8(αi − 1) Log λi − 24 λi λi Fi (ai ) 2

40 × 64π 2 (αi − 1) − 32 × 64π 2 (αi − 1) Log λi 3 64π 4 Log λ 2 Fi (ai ) . + 32 × 2 − 32π 2 γj G2 (ai , aj ) (if m 2) + O λ4 λi Fi (ai ) λi j =i − 4 × 64π 2 Bi (ai ) +

Proof. From one part we have

∂P δi Ke λi = ∂λi

u

Bi

Bi

λ8i (1 + λ2i |x

− ai

|2 )4

Fi (x) 1 + Bi (x) + (αi − 1)

× 8 Log λi − 4 Log 1 + λ2i |x − ai |2 8 32π 2 Log λ × − 2 H2 (ai , x) + O λ4 1 + λ2i |x − ai |2 λi


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π2 π2 = 8 1 + 8(αi − 1) Log λi λ4i Fi (ai ) + λ2i Fi (ai ) 12 12 2 π π2 + 8λ4i Fi (ai )Bi (ai ) − 32π 2 λ2i Fi (ai )H2 (ai , ai ) 12 6 2 7π + O(Log λ). − 32(αi − 1)λ4i Fi (ai ) 144 On the other hand for j = i

∂P δi Ke λi = ∂λi

u

Bj

Bj

=

λ8j Fj (x) (1 + λ2j |x − aj |2 )4

32π 2 Log λ G2 (ai , x) + O λ4 λ2i

32π 2 π2 4 + O(Log λ). G (a , a )λ F (a ) 2 i j j j j 6 λ2i

The proof follows from Lemma 8.2, the definition of γi and the above estimates.

2

Observe that the above proposition can be written as: Corollary 5.3. Let = 64mπ 2 , m 1 and u =

p

i=1 αi P δi

∈ V (m, ε). There holds

FiA (ai ) ∂P δi 5 2 A = 4 × 64π γi − Bi (ai ) − 8(αi − 1) Log λi − − 2 A ∇J (u), λi ∂λi 12 8λi Fi (ai ) m Log λ Log λ + |γj | . +O λ4 λ2 j =1

From Lemma 3.2, for m = 1, it is easy to get that γ1 = B1 (a1 ) +

Log(λ1 ) 1 F1 (a1 ) 5 , + O + 8(α − 1) Log(λ ) − 1 1 4 λ21 F1 (a1 ) 12 λ41

and therefore Corollary 5.3 can be written as follows: Corollary 5.4. Let = 64π 2 and u = α1 P δa1 ,λ1 ∈ V (1, ε). Then we have A Log2 (λ) ∂P δ1 2 F1 (a1 ) ∇J (u), λ1 = 32π 2 A . +O ∂λ1 λ4 λ1 F1 (a1 ) For m 2, we can obtain a similar result. Indeed: Corollary 5.5. Let u =

m

i=1 αi P δai ,λi

∈ V (m, ε) with m 2 and assume that

|γj | C

Log λ λ2

for each j.

(5.1)

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(i) For each j , we have mλ4j FjA (aj ) Log λ =1+O .

4 A λ2 λi Fi (ai ) F A (ai ) ∂P δi Log2 λ i 2 +O = 32π . ∇J (u), λi ∂λi λ4 λ2i FiA (ai )

(ii)

Proof. Claim (i) follows immediately from the assumption on γj and Lemma 3.2. Now using claim (i), we can improve Lemma 3.2 and it becomes

Ke = u

Ω

π2 4 1 1 Fi (ai ) λi Fi (ai ) 1 + Bi (ai ) + 6 m 4m λ2i Fi (ai )

+

Log2 (λ) 5 8 . (αi − 1) Log λi − +O m 12 λ4

Hence, using the definition of Proposition 5.6. Let u =

γj , claim (ii) follows immediately from Corollary 5.3.

p

i=1 αi P δi

2

∈ V (m, ε) and = 64mπ 2

∂P δi 5 2 ∇J (u), P δi = 2 Log λi − − 16π H4 (ai , ai ) ∇J (u), λi 6 ∂λi 2 F A (ai ) ∂P δj − 64π 2 Log λi 2 iA + 64π 2 G4 (ai , aj ) ∇J (u), λj ∂λj λi Fi (ai ) j =i 1 Log λ |γj | . + 8 × 64π 2 (αi − 1) Log λi + O 2 + 2 λ λ

!

"

Proof.

Ke P δi = u

Bi

Bi

λ8i Fi (x) (1 + λ2i |x

− ai

|2 )4

1 + Bi (x) + (αi − 1)

× 8 Log λi − 4 Log 1 + λ2i |x − ai |2 1 × 8 Log λi − 4 Log 1 + λ2i |x − ai |2 − 64π 2 H4 (ai , x) + O 2 λ

π2 π2 = 8 Log λi 1 + 8(αi − 1) Log λi λ4i Fi (ai ) + λ2i Log λi Fi (ai ) 6 3 2 4 5π − 4 1 + 8(αi − 1) Log λi λi Fi (ai ) 36 π2 − 64π 2 1 + 8(αi − 1) Log λi λ4i Fi (ai )H4 (ai , ai ) 6


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5π 2 π2 + 8 Log λi λ4i Fi (ai )Bi (ai ) + O λ2 36 6 2

5 π 4 2 λ Fi (ai ) 1 + 8(αi − 1) Log λi 2 Log λi − − 16π H4 (ai , ai ) = 6 i 6 − 32(αi − 1) Log λi λ4i Fi (ai )

π2 5π 2 − 32(αi − 1) Log λi λ4i Fi (ai ) 3 36 2 π + O λ2 . + 8 Log λi λ4i Fi (ai )Bi (ai ) 6 + Fi (ai )λ2i Log(λi )

Furthermore for j = i

Keu P δi =

Bj

(1 + 8(αj − 1) Log λj ) (1 + λ2j |x − aj |2 )4

Bj

λ8j Fj (x) 64π 2 G4 (ai , x) + O λ2

π2 + O λ2 . = 1 + 8(αj − 1) Log λj 64π 2 G4 (ai , x) λ4i Fi (ai ) 6 It follows then from the above estimates and Lemma 8.3 that " ! 5 ∇J (u), P δi = 4 × 64π 2 2 Log λi − − 16π 2 H4 (ai , ai ) 6 2 mλ4i Fi (ai ) π6 1 + 8(αi − 1) Log λi + (αi − 1)8 × 64π 2 Log λi × 1− u Ω Ke 2 mλ4j Fj (aj ) π6 2 2 1 + 8(αj − 1) Log λj + 64π G4 (ai , aj ) 1 − u Ω Ke

j =i

π2 5π 2 2 4 λ − 32(α Log λ F (a ) − 1)λ Log λ F (a ) − i i i i i i i i i u 3 36 Ω Ke 2 1 π +O 2 . + 8 Log λi Bi (ai )λ4i Fi (ai ) 6 λ Recall that mλ4i Fi (ai )π 2 /6 = 1 − γi , u Ω Ke it follows 5 2 ∇J (u), P δi = 4 × 64π 2 Log λi − − 16π H4 (ai , ai ) 6 × γi 1 + 8(αi − 1) Log λi − 8(αi − 1) Log λi 2 G4 (ai , aj ) γj 1 + 8(αj − 1) Log λj − 8(αj − 1) Log λj + 64π 2

!

"

2

j =i

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Fi (ai ) λ2i Fi (ai ) 80 1 + 8 × 64π 2 (αi − 1) Log λi + 64π 2 (1 − γi )(αi − 1) Log(λi ) + O 2 . 3 λ

− 8 × 64π 2 (1 − γi ) Log λi Bi (ai ) − 2 × 64π 2 (1 − γi ) Log λi

Finally, using Proposition 5.2 our proposition follows.

2

6. Morse lemma at infinity Using the above estimate, we are now able to construct a pseudogradient near neighborhood of potential critical points at infinity. The analysis of the end points of such a pseudogradient is easier than the genuine gradient flow. In order to construct such a pseudogradient we have to divide V (m, ε) in different regions, to construct an appropriate pseudogradient in each region and then glue up through convex combinations. Proposition 6.1. Let = 64mπ 2 with m 2 (resp. m = 1) and assume that the function K satisfies the condition (C1 ) (resp. (C0 )). Then there exists a pseudogradient W defined in V (m, ε) and satisfying the following properties:

There exists a constant C independent of u = m i=1 αi P δi such that (1)

(2)

m 1 |∇Fi (ai )| −∇J (u), W C + + |αi − 1| . λi λ2i i=1

!

"

m 1 ∂w(W ) |∇Fi (ai )| C −∇J (u + w), W + + + |α − 1| . i ∂(α, λ, a) λi λ2i i=1

(3) |W | is bounded and the only region where the maximum of the λi ’s increases along the flow lines of W is: (a1 , . . . , am ) is near a critical point q := (q1 , . . . , qm ) of F K (resp. fK ) with l(q) :=

q m F (qi ) i C Log λi /λ2i ,

V2 := u ∈ V (m, ε): ∀i, |γi | < 2C Log λi /λ2i and ∃j s.t. ∇Fj (aj ) > c/λj ,

V3 := u ∈ V (m, ε): ∀i, |γi | < 2C Log λi /λ2i ; ∇Fi (ai ) < 2c/λi and l(q) > 0 ,

V4 := u ∈ V (m, ε): ∀i, |γi | < 2C Log λi /λ2i ; ∇Fi (ai ) < 2c/λi and l(q) < 0 , where γi is defined in Proposition 5.1.


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Remark. From Lemma 3.2, it is easy to see that m i=1

Log λ . γi = O λ2

(6.1)

Therefore, if u ∈ / V1 , we derive that γi > −(3/2)C Log λi /λ2i for each i and thus u ∈ V2 ∪ V3 ∪ V4 . For u =

αi P δi ∈ V1 , we define Y1 := −

i∈F1

λi

where F1 := i: γi > C Log λi /λ2i .

∂P δi , ∂λi

From Corollary 5.3, we get m m " Log λi −∇J (u), Y1 c γi c + c |αi − 1|. λ2i i∈F i=1 i=1

!

(6.2)

1

In V1 , we define W1 := Y1 + Wa ,

with Wa :=

m ∇F (ai ) 1 ∂P δi ξ λi ∇F (ai ) , |∇F (ai )| λi ∂ai

(6.3)

i=1

where ξ is a C ∞ positive function satisfying ξ(t) = 0 if t μ and ξ(t) = 1 if t 2μ where μ is a small positive constant. From Proposition 5.1 and (6.2), we derive that !

m m m " Log λi |∇F (ai )| + c |α − 1| + c . − ∇J (u), W1 c i 2 λi λ i i=1 i=1 i=1

(6.4)

For u = αi P δi ∈ V2 , we use the vector field Wa defined in (6.3). Note that, since u ∈ V2 , there exists at least one index i such that ξ(ai ) 1 (since μ is small with respect to c). Now, using Proposition 5.1, we derive " −∇J (u), Wa c

!

i: ξ(ai )1

c

|∇F (ai )| 1 +O 2 λi λ

(6.5)

m m m 1 |∇F (ai )| + c |α − 1| + c . i 2 λi λ i=1 i i=1 i=1

For u = αi P δi ∈ V3 , it is easy to get that the m-tuple (a1 , . . . , am ) is close to a critical point q := (q1 , . . . , qm ) of F K with l(q) > 0. In this case we define W3 := −

m i=1

λi

∂P δi , ∂λi

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and from Corollary 5.5 we derive that m ! " Log(λ) F (ai ) 2 −∇J (u), W3 = 32π . +O λ4 λ2 F (ai ) i=1 i

(6.6)

Now using claim (i) of Corollary 5.5, we get λ4i F (ai ) = λ4j F (aj ) + O λ2 Log(λ)

for each i, j,

which implies that m m Log(λ) 1 F (ai ) F (ai ) . + O = √ √ λ4 λ2 F (ai ) λ21 F (a1 ) i=1 F (ai ) i=1 i Thus, (6.6) becomes !

" −∇J (u), W3 c c

l(A) √ F (a1 )

λ21 m i=1

m m 1 |∇F (ai )| + c |α − 1| + c . i λi λ2i i=1 i=1

Finally, for u = αi P δi ∈ V4 , the m-tuple (a1 , . . . , am ) is close to a critical point q := (q1 , . . . , qm ) of F K with l(q) < 0, in this case we will increase the variables λi ’s and we move the concentration points ai ’s and the variables αi ’s by defining

(αi − 1) 2 Wαi , (αi − 1) W4 := μWa − 4W3 − ξ λ1 | (αi − 1)| where μ is a small positive constant, Wa and ξ are defined in (6.3) and Wαi :=

∂P δj 1 64π 2 P δi − G4 (ai , aj )λj Log(λi ) Log(λi ) ∂λj − 2−

j =i

16π 2 ∂P δi 5 − H4 (ai , ai ) λi . 6 Log(λi ) Log(λi ) ∂λi

Using (6.6) and Propositions 5.1, 5.6, we derive that m m m " 1 |∇F (ai )| −∇J (u), W4 c + c |α − 1| + c . i 2 λi λ i=1 i i=1 i=1

!

Now, we define the pseudogradient W by a convex combination of W1 , . . . , W4 and therefore claim (1) follows. Concerning claim (2), it follows as in [8] since the norm of w 2 is small with respect to the lower bound of claim (1). Finally, claim (3) follows from the definitions of Wi ’s


3187

and the fact that λi ∂P δi /∂λi and λ−1 i ∂P δi /∂ai are bounded. This completes the proof of our proposition in the case where m 2. Now, for m = 1, the proof is similar to the one done for m 2 and it is more easy since for example V1 = ∅ (from (5.1)). 2 As a consequence of Proposition 6.1 we are able to identify the critical points at infinity of J . Indeed we have Corollary 6.2. Let = 64mπ 2 , m 2 (resp. m = 1). The critical points at infinity of J are m

P δqi ,∞ ,

i=1

such that: q := (q1 , . . . , qm ) is a critical point of F K (resp. fK ) with qi = qj if i = j and l(q) :=

q m F (qi ) i < 0 resp. q1 ∈ K− . q Fi (qi ) i=1

Furthermore the energy level of such a critical point at infinity (q1 , . . . , qm )∞ denoted C∞ (q1 , . . . , qm )∞ is given by C∞ (q1 , . . . , qm )∞ = −

m 640mπ 2 π2 − 64mπ 2 Log m − 64π 2 Log K(qi ) 6 6

+

2 1 64π 2 2

m

i=1

H4 (qi , qi ) −

i=1

G4 (qi , qj ) (if m 2) .

j =i

Moreover the Morse index of such a critical point at infinity (q1 , . . . , qm )∞ is given by 5m − 1 − morse F K , (q1 , . . . , qm )

resp. 4 − morse(fK , q1 ) .

7. Proof of the main results First of all we point out that, just like for usual critical points, it is associated to each critical point at infinity x∞ of J stable and unstable manifolds Ws∞ (x∞ ) and Wu∞ (x∞ ), see [4]. These manifolds can be easily described once a finite dimensional reduction like the one we performed in Section 3 is established. The stable manifold is, indeed defined to be the set of points attracted by the asymptote while the unstable one is a shadow object, which is the limit of Wu (xλ ), xλ being critical point of the reduced problem and Wu (xλ ) its associated unstable manifold. Proof of Theorem 1.1. We prove the theorem by contradiction, therefore, we assume that Eq. (1.1) does not have solution. Let K Ω be a compact subset of Ω such that Ω retracts by deformation on K and denote by Bm (K) := K m ×σ m m−1 ,

3188


the set of formal barycenters in K, where m−1 := (α1 , . . . , αm ); αi 0,

m

αi = m .

i=1

It follows from [16,24] that J −L is homotopically equivalent to Bm (K) for some compact subset K Ω which is a retract by deformation of Ω itself. Therefore, it follows from the fact that Ω is not contractible that J −L is not contractible. From another part, according to Lemma 4.1, for L large enough we have that the whole space H retracts by deformation onto J L . Moreover, if (1.1) has no solution, then J L retracts by deformation on J −L and therefore H, which is contractible, retracts by deformation on J −L which is not contractible. A contradiction! To prove the multiplicity part of the statement, we observe that, it follows from Sard–Smale Theorem that for generic K’s, the solutions of (1.1) are all nondegenerate, in the sense that, the associated linearized operator does not admit zero as eigenvalue. Now in case Eq. (1.1) has infinitely many solutions, we are done, otherwise, there exists L1 1 such that all solutions are in the sublevel J L . We choose it to be larger than L in Lemma 4.2. It 1 follows then that J L is contractible. Therefore it follows from the Euler–Poincaré Theorem that

1 = χ J −L +

(−1)m(w) ,

(7.1)

w;∇J (w)=0

where m(w) denotes the Morse index of the solution w. It follows from [16,24] that J −L is homotopically equivalent to Bm (K) for some compact subset K Ω which is a retract by deformation of Ω itself. Therefore 1 −χ(Ω) + 1 (· · ·) −χ(Ω) + m . χ J −L = χ Bm (K) = 1 − m! Hence the lower bound on the number of solutions follows from (7.1) and (7.2).

(7.2) 2

Proof of Theorem 1.3. First, we remark that, for = 64π 2 , the functional J is lower bounded. We prove the existence result by contradiction. Therefore, we assume that Eq. (1.1) does not have solution. We recall, from Lemma 4.2, that there exists a large L > 0 such that H J L. Therefore J L is contractible. Now thanks to Lemma 4.1, we can compute the Euler–Poincaré characteristic of J L using the pseudogradient constructed in Proposition 6.1, whose “zeros”, under the assumption that (1.1) has no solution, are the critical points at infinity of J . It follows then from a theorem of Bahri and Rabinowitz [6] that JL

# {w∞ : critical point at infinity}

Wu (w∞ ).

(7.3)


3189

These critical points at infinity, according to Corollary 6.2, are in one-to-one correspondence to the elements of the set K− . It follows then from the Euler–Poincaré Theorem and the assumption of the theorem that (−1)morse(fK ,q) = 1 χ JL = q∈K−

which contradicts the fact that J L is contractible. Moreover we observe that for generic K’s, Eq. (1.1) admits only nondegenerate solutions. Now using Lemmas 4.1, 4.2 and a theorem of Bahri and Rabinowitz [6], we derive that #

JL

#

Wu (w∞ ) ∪

{w∞ : critical point at infinity}

Wu (w).

{w: critical point}

Now using the Euler–Poincaré Theorem, we derive that 1=

q∈K−

Our result follows.

(−1)morse(fK ,q) +

(−1)morse(J,w) .

w: critical point of J

2

Proof of Theorem 1.4. We set #

X∞ :=

Wu∞ (q∞ ),

{q∈K− ; ι(q) 0, since the functions f1 , . . . , fk depend only on F , we may choose λ so that λ ) < C + . This shows the other inequality. 2 (gi,j Lemma 1.7. Let A be an operator algebra of functions on the set X, then A˜ equipped with the ˜ given in Definition 1.5 is an operator algebra. collection of norms on Mn (A)

M. Mittal, V.I. Paulsen / Journal of Functional Analysis 258 (2010) 3195–3225

3199

Proof. It is clear from the definition of A˜ that it is an algebra. Thus, it is enough to check that the axioms of BRS are satisfied by the algebra A˜ equipped with the matrix norms given in Definition 1.5. ˜ then for > 0 there exists If L and M are scalar matrices of appropriate sizes and G ∈ Mn (A), Gλ ∈ Mn (A) such that limλ Gλ (x) = G(x) for all x ∈ X and supλ Gλ Mn (A) GMn (A˜ ) + . Since A is an operator space, LGλ M ∈ Mn (A) and LGλ MMn (A) LGλ Mn (A) M. Note that it follows that LGMMn (A˜ ) LGMn (A˜ ) M, since LGλ M → LGM pointwise and supλ LGλ MMn (A) L(GMn (A˜ ) + )M for any > 0. ˜ then for every > 0 there exist Gλ , Hλ ∈ Mn (A) such that limλ Gλ (x) = If G, H ∈ Mn (A), G(x) and limλ Hλ (x) = H (x) for x ∈ X. Also, we have that supλ Gλ Mn (A) GMn (A˜ ) + and supλ Hλ Mn (A) H Mn (A˜ ) + . Let L = GH and Lλ = Gλ Hλ . Since A is matrix normed algebra, Lλ ∈ Mn (A) and Lλ Mn (A) Gλ Mn (A) Hλ Mn (A) for every λ. This implies that limλ Lλ (x) = L(x) and that LMn (A˜ ) sup Lλ Mn (A) sup Gλ Mn (A) sup Hλ Mn (A) . λ

λ

λ

This yields LMn (A˜ ) GMn (A˜ ) H Mn (A˜ ) , and so the multiplication is completely contractive. ˜ and H ∈ Mm (A). ˜ Given > 0 Finally, to see that the L∞ conditions are met, let G ∈ Mn (A) there exist Gλ ∈ Mn (A) and Hλ ∈ Mm (A) such that limλ Gλ (x) = G(x), limλ Hλ (x) = H (x) and supλ Gλ Mn (A) GMn (A˜ ) + , supλ Hλ Mn (A) H Mn (A˜ ) + . Note that Gλ ⊕ Hλ ∈ Mn+m (A) and Gλ ⊕ Hλ = max{Gλ Mn (A) , Hλ Mn (A) } for every λ ˜ and which implies that G ⊕ H ∈ Mn+m (A), G ⊕ H Mn+m (A˜ ) sup Gλ ⊕ Hλ = sup max Gλ Mn (A) , Hλ Mm (A) λ

λ

= max sup Gλ Mn (A) , supH λ M (A) m λ λ max GMn (A˜ ) + , H Mm (A˜ ) + . This shows that G ⊕ H Mn+m (A˜ ) max{GMn (A˜ ) , H Mm (A˜ ) }, and so the L∞ condition follows. This completes the proof of the result. 2 Lemma 1.8. If A is an operator algebra of functions on the set X, then A˜ equipped with the norms of Definition 1.5 is a local operator algebra of functions on X. Moreover, for (fi,j ) ∈ Mn (A), (fi,j )Mn (A˜ ) = (fi,j )Mn (AL ) . Proof. It is clear from the definition of the norms on A˜ that the identity map from A to A˜ is completely contractive and thus A ⊆ A˜ as sets. This shows that A˜ separates points of X and contains the constant functions. ˜ and > 0, then there exists a net (f λ ) ∈ Mn (A) such that limλ (f λ (x)) = Let (fij ) ∈ Mn (A) ij ij (fij (x)) for each x ∈ X and supλ (fijλ )Mn (A) (fij )Mn (A˜ ) + . Since A is an operator al-

3200


gebra of functions on the set X, we have that (fijλ )∞ (fijλ )Mn (A) . Thus, supλ (fijλ )∞ (fij )Mn (A˜ ) + . Fix z ∈ X, then fij (z) = lim f λ (z) sup f λ ij

λ

λ

ij

∞

(fij )M (A˜ ) + . n

By letting → 0 and taking the supremum over z ∈ X, we get that (fij )∞ (fij )Mn (A˜ ) . Hence, A˜ is an operator algebra of functions on the set X. ˜ f |F ≡ 0} and let (fij ) ∈ Mn (A). ˜ Then, clearly supF (fij + I˜F ) Set I˜F = {f ∈ A: Mn (A˜ /I˜F ) ˜ (fij ) ˜ . To see the other inequality, assume that supF (fij + IF ) < 1. Then for every fiM n ( A)

˜ such that (hF )|F = (f F )|F and supF hF 1. Fix a nite F ⊆ X there exists (hFij ) ∈ Mn (A) ij ij ij ˜ Then for all finite F ⊆ X there exists (k F ) ∈ Mn (A) such that set F ⊆ X and (hF ) ∈ Mn (A).

ij

ij

(kijF )|F = (hFij )|F and supF kijF 1. In particular, let F = F then (kijF )|F = (hFij )|F = (fij )|F and supF kijF 1. Hence, (fij )Mn (A˜ ) 1, and (fij )Mn (A˜ ) supF (fij + I˜F )Mn (A˜ /I˜ ) . F ˜ Finally, given that (fij ) ∈ Mn (A), (fij ) ˜ = supF (fij + IF ) ˜ ˜ . Note that for M n (A )

M n ( A/ I F )

any F ⊆ X we have (fij + I˜F )Mn (A˜ /I˜ ) (fij + IF )Mn (A/IF ) , since IF ⊆ I˜F . We claim F that for any (fij ) ∈ Mn (A), and for any finite subset F ⊆ X, we have that (fij + IF ) = (fij + I˜F ). To see the other inequality, let (gij ) ∈ Mn (I˜F ). Then for > 0 and G ⊆ X, we may G G choose (hG ij ) ∈ Mn (A) such that (hij )|G = (fij + gij )|G and supG (hij ) (fij + gij ) + . Hence, (fij + IF ) = (hFij + IF ) (hFij ) (fij + gij ) + . Since > 0 was arbitrary, the equality follows. Now it is clear that, (fij ) and so the result follows.

Mn (A˜ )

= sup (fij + IF ) = (fij )M (A ) , n L F

2

Corollary 1.9. If A is a BPW complete operator algebra then AL = A˜ completely isometrically. Proof. Since A is BPW complete, A = A˜ as sets. But by Lemma 1.6, the norm defined on AL ˜ 2 agrees with the norm defined on A. Remark 1.10. In the view of above corollary, we denote the norm on A˜ by · L . Lemma 1.11. If A is an operator algebra of functions on X, then Ball(AL ) is BPW dense in ˜ and A˜ is BPW complete, i.e., A˜˜ = A. ˜ Ball(A) Proof. It can be easily checked that the statement is equivalent to showing that AL is BPW dense ˜ since the other containment follows immediately by ˜ We’ll only prove that AL BPW ⊆ A, in A. ˜ the definition of A. Let {fλ } be a net in AL such that fλ → f pointwise and supλ fλ AL < C. Then for fixed F ⊆ X and > 0, there exists λF such that |fλF (z) − f (z)| < for z ∈ F . Also since supλ fλ < C, there exists gλF ∈ IF such that fλF + gλF < C. Note that the function


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˜ Thus, AL is hF = fλF + gλF ∈ A satisfies hF A < C, and hF → f pointwise. Hence, f ∈ A. ˜ BPW dense in A. Finally, a similar argument yields that A˜ is BPW complete. 2 All the above lemmas can be summarized as the following theorem. Theorem 1.12. If A is an operator algebra of functions on X, then A˜ is a BPW complete local operator algebra of functions on X which contains AL completely isometrically as a BPW dense subalgebra. Definition 1.13. Given an operator algebra of functions A on X, we call A˜ the BPW completion of A. We now present a few examples to illustrate these concepts. We will delay the main family of examples to a later section. Example 1.14. If A is a uniform algebra, then there exists a compact, Hausdorff space X, such that A can be represented as a subalgebra of C(X) that separates points. If we endow A with the matrix-normed structure that it inherits as a subalgebra of C(X), namely, (fi,j ) = (fi,j )∞ ≡ sup{(fi,j )Mn : x ∈ X}, then A is a local operator algebra of functions on X. Indeed, to achieve the norm, it is sufficient to take the supremum over all finite subsets consisting of one point. In this case the BPW completion A˜ is completely isometrically isomorphic to the subalgebra of ∞ (X) consisting of functions that are bounded, pointwise limits of functions in A. Example 1.15. Let A = A(D) ⊆ C(D− ) be the subalgebra of the algebra of continuous functions on the closed disk consisting of the functions that are analytic on the open disk D. Identifying Mn (A(D)) ⊆ Mn (C(D− )) as a subalgebra of the algebra of continuous functions from the closed disk to the matrices, equipped with the supremum norm, gives A(D) it’s usual operator algebra structure. With this structure it can be regarded as a local operator algebra of functions on D or on D− . If we regard it as a local operator algebra of functions on D− , then A(D) A(D). n To see that the containment is strict, note that f (z) = (1 + z)/2 ∈ A(D) and f (z) → χ{1} , the characteristic function of the singleton {1}. However, if we regard A(D) as a local operator algebra of functions on D, then its BPW = H ∞ (D), the bounded analytic functions on the disk, with its usual operator completion A(D) structure. Example 1.16. Let X = D, 0 < < 1 and A = {f |X : f ∈ H ∞ (D)}. If we endow A with the matrix-normed structure on H ∞ (D), then A is an operator algebra of functions on X. Also, it ˜ Indeed, if F = can be verified that A is a local operator algebra of functions and that A = A. (fij ) ∈ Mn (A) with (fij + IY )∞ < 1 for all finite subset Y ⊆ X, then there exists HY ∈ Mn (A) such that HY ∞ 1 and HY → F pointwise on X. Note by Montel’s theorem there will exist a subnet HY and G ∈ Mn (H ∞ (D)) such that G∞ 1 and HY → G uniformly on compact subsets of D. Thus, by the identity theorem F ≡ G on D. Hence, F Mn (A) 1 and so A is a local operator algebra. A similar argument shows that if f is a BPW limit on X, then there exists g ∈ H ∞ (D) such that g|X = f , and so A is BPW complete. By Lemma 1.11, A˜ = A completely isometrically.

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∞ Example 1.17. Let = H (D) but endowed with a new norm. Fix b > 1, and set F = A max{F ∞ , F ( 00 0b )}, F ∈ Mn (A). It can be easily verified that A is a BPW complete operator algebra of functions. However, we also claim that A is local. To prove this we proceed by contradiction. Suppose there exists F = (fij ) ∈ Mn (H ∞ (D)) such that F > 1 > c, where c = supY (fij + IY ). In this case, F = F ( 00 0b ), since (fij + IY ) = F (λ) when Y = {λ}. Let = 1−c 4b and Y = {0, } ⊆ D, then ∃G ∈ Mn (H ∞ (D)) such that G|Y = 0 and F + G < 1+c 2 . Thus, we z− ∞ can write BY (z) = 1−¯ z , so that we can write G(z) = zBY (z)H (z), for some H ∈ Mn (H ). It follows that H ∞ < 2, since G∞ < 2. We now consider

0 b 0 b 0 b + G (F + G) F 1 1. Fix α > 0, such that 1 + 2α < z. For each Y = {z1 , z2 , . . . , zn }, note that z = 10 −1 z−zi we define BY (z) = ni=1 ( 1− z¯i z ) and choose h ∈ A such that h(1) = −BY (1), h(−1) = BY (−1), and h∞ 2. Let g(z) = z + BY (z)h(z)α, then g ∈ A, g(1) = g(−1) and g|Y = z|Y . Hence, πY (z) = πY (g) g = g∞ 1 + 2α < z. Thus, since α was arbitrary, supY ⊆D πY (z) = 1 < z and hence A is not local. Example 1.19. This example shows that one can easily build non-local algebras by adding “values” outside of the set X. Let A be the algebra of polynomials regarded as functions on the set X = D. Then A endowed with the matrix-normed structure as (pij ) = max{(pij )∞ , (pij (2))}, is an operator algebra of functions on the set X. To see that A is not local, let p ∈ A be such that p∞ < |p(2)|. For each finite subset Y = {z1 , . . . , zn } of X, let |p(2)|−p∞ > 0. Note that hY (z) = ni=1 (z − zi ) and gY (z) = p(z) − αhY (z)p(2), where α = 2|p(2)|h Y ∞ gY (1 − α)|p(2)| and gY |Y = p|Y . Hence, πY (p) = πY (gY ) gY (1 − α)|p(2)| < p. It follows that A is not local. Finally, be BPW complete. For example, if we take

observe that in this case A cannot 1 for z ∈ D and pn < f , which implies that pn = 13 ni=0 ( 3z )i ∈ A then pn (z) → f (z) = 3−z ˜ AL A. Example 1.20. It is still an open problem as to whether or not every unital contractive, homomorphism ρ : H ∞ (D) → B(H) is completely contractive. For a recent discussion of this problem see [27]. Let’s assume that ρ is a contractive homomorphism that is not completely contractive.


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Let B = H ∞ (D), but endow it with the family of matrix-norms given by, (fi,j ) = max (fi,j ) , ρ(fi,j ) . ∞ Note that |f | = f ∞ , for f ∈ B. It is easily checked that B is a BPW complete operator algebra of functions on D. However, since every contractive homomorphism of A(D) is completely contractive, we have that for (fi,j ) ∈ Mn (A(D)), |(fi,j )| = (fi,j )∞ . If Y = {x1 , . . . , xk } is a finite subset of D and F = (fi,j ) ∈ Mn (B), then there is G = (gi,j ) ∈ Mn (A(D)), such that F (x) = G(x) for all x ∈ Y , (n) (n) and G∞ = F ∞ . Hence, πY (F ) F ∞ . Thus, supY πY (F ) = F ∞ . It follows that B is not local and that B˜ = BL = H ∞ (D), with its usual supremum norm operator algebra structure. In particular, if there does exist a contractive but not completely contractive representation of H ∞ (D), then we have constructed an example of a non-local BPW complete operator algebra of functions on D. 2. A characterization of local operator algebras of functions The main goal of this section is to prove that every BPW complete local operator algebra of functions is completely isometrically isomorphic to the algebra of multipliers on a reproducing kernel Hilbert space of vector-valued functions. Moreover, we will show that every such algebra is a dual operator algebra in the precise sense of [13]. We will then prove that for such BPW algebras, weak∗ -convergence and BPW convergence coincide on bounded balls. First we need to recall a few basic facts and some terminology from the theory of vectorvalued reproducing kernel Hilbert spaces. Given a set X and a Hilbert space H, then by a reproducing kernel Hilbert space of H-valued functions on X, we mean a vector space L of H-valued functions on X that is equipped with a norm and an inner product that makes it a Hilbert space and which has the property that for every x ∈ X, the evaluation map Ex : L → H, is a bounded, linear map. Recall that given a Hilbert space H, a matrix of operators, T = (Ti,j ) ∈ Mk (B(H)) is regarded as an operator on the Hilbert space H(k) ≡ H ⊗ Ck , which is the direct sum of k copies of H. A function K : X × X → B(H), where H is a Hilbert space, is called a positive definite operator-valued function on X, provided that for every finite set of (distinct) points {x1 , . . . , xk } in X, the operator-valued matrix, (K(xi , xj )) is positive semidefinite. Given a reproducing kernel Hilbert space of H-valued functions, if we set K(x, y) = Ex Ey∗ , then K is positive definite and is called the reproducing kernel of L. There is a converse to this fact, generally called Moore’s theorem, which states that given any positive definite operator-valued function K : X × X → B(H), then there exists a unique reproducing kernel Hilbert space of H-valued functions on X, such that K(x, y) = Ex Ey∗ . We will denote this space by L(K, H). Given v, w ∈ H, we let v ⊗ w ∗ : H → H denote the rank one operator given by (v ⊗ w ∗ )(h) = h, wv. A function g : X → H belongs to L(K, H) if and only if there exists a constant C > 0 such that the function C 2 K(x, y) − g(x) ⊗ g(y)∗ is positive definite. In which case the norm of g is the least such constant. Finally, given any reproducing kernel Hilbert space L of H-valued functions with reproducing kernel K, a function

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f : X → C is called a (scalar) multiplier provided that for every g ∈ L, the function f g ∈ L. In this case it follows by an application of the closed graph theorem that the map Mf : L → L, defined by Mf (g) = f g, is a bounded, linear map. The set of all multipliers is denoted M(K) and is easily seen to be an algebra of functions on X and a subalgebra of B(L). The reader can find proofs of the above facts in [16,4]. Also, we refer to the fundamental work of Pedrick [28] for further treatment of vector-valued reproducing kernel Hilbert spaces. Another good source is [3]. Lemma 2.1. Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H). Then M(K) ⊆ B(L) is a weak∗ -closed subalgebra. Proof. It is enough to show that the unit ball is weak∗ -closed by the Krein–Smulian theorem. So let {Mfλ } be a net of multipliers in the unit ball of B(L) that converges in the weak∗ -topology to an operator T . We must show that T is a multiplier. Let x ∈ X be fixed and assume that there exists g ∈ L, with g(x) = h = 0. Then T g, Ex∗ hL = limλ Mfλ g, Ex∗ hL = limλ Ex (Mfλ g), hH = limλ fλ (x)h2 . This shows that at every such x the net {fλ (x)} converges to some value. Set f (x) equal to this limit and for all other x’s set f (x) = 0. We claim that f is a multiplier and that T = Mf . Note that if g(x) = 0 for every g ∈ L, then Ex = Ex∗ = 0. Thus, we have that for any g ∈ L and any h ∈ H, Ex (T g), hH = limλ Ex (Mfλ g), hH = limλ fλ (x)g(x), hH = f (x)g(x), hH . Since this holds for every h ∈ H, we have that Ex (T g) = f (x)g(x), and so T = Mf and f is a multiplier. 2 Every weak∗ -closed subspace V ⊆ B(H) has a predual and it is the operator space dual of this predual. Also, if an abstract operator algebra is the dual of an operator space, then it can be represented completely isometrically and weak∗ -continuously as a weak∗ -closed subalgebra of the bounded operators on some Hilbert space. For this reason an operator algebra that has a predual as an operator space is called a dual operator algebra. See the book of [13] for the proofs of these facts. Thus, in summary, the above lemma shows that every multiplier algebra is a dual operator algebra in the sense of [13]. Theorem 2.2. Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H) and let M(K) ⊆ B(L) denote the multiplier algebra, endowed with the operator algebra structure that it inherits as a subalgebra. If K(x, x) = 0, for every x ∈ X and M(K) separates points on X, then M(K) is a BPW complete local dual operator algebra of functions on X. Proof. The multiplier norm of a given matrix-valued function F = (fi,j ) ∈ Mn (M(K)) is the least constant C such that 2 C In − F (xi )F (xj )∗ ⊗ K(xi , xj ) 0, for all sets of finitely many points, Y = {x1 , . . . , xk } ⊆ X. Applying this fact to a set consisting of a single point, we have that (C 2 In − F (x)F (x)∗ ) ⊗ K(x, x) 0, and it follows that C 2 In − F (x)F (x)∗ 0. Thus, F (x) C = F and we have that point evaluations are completely contractive on M(K). Since M(K) contains the constants and separates points by hypothesis, it is an operator algebra of functions on X.


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Suppose that M(K) was not local, then there would exist F ∈ Mn (M(K)), and a real num(n) ber C, such that supY πY < C < F . Then for each finite set Y = {x1 , . . . , xk } we could choose G ∈ Mn (M(K)), with G < C, and G(x) = F (x), for every x ∈ Y . But then we would have that ((C 2 In − F (xi )F (xj )∗ ) ⊗ K(xi , xj )) = ((C 2 In − G(xi )G(xj )∗ ) ⊗ K(xi , xj )) 0, and since Y was arbitrary, F C, a contradiction. Thus, M(K) is local. Finally, assume that fλ ∈ M(K), is a net in M(K), with fλ C, and limλ fλ (x) = f (x), pointwise. If g ∈ L with gL = M, then ∗ (MC)2 K(x, y) − fλ (x)g(x) ⊗ fλ (y)g(y) is positive definite. By taking pointwise limits, we obtain that (MC)2 K(x, y) − f (x)g(x) ⊗ (f (y)g(y))∗ is positive definite. From the earlier characterization of functions in L and their norms in a reproducing kernel Hilbert space, this implies that f g ∈ L, with f gL MC. Hence, f ∈ M(K) with Mf C. Thus, M(K) is BPW complete. 2 In general, M(K) need not separate points on X. In fact, it is possible that L does not separate points and if g(x1 ) = g(x2 ), for every g ∈ L, then necessarily f (x1 ) = f (x2 ) for every f ∈ M(K). Following [26], we call C a k-idempotent operator algebra, provided that there are k operators, {E1 , . . . , Ek } on some Hilbert space H, such that Ei Ej = Ej Ei = δi,j Ei , I = E1 + · · · + Ek and C = span{E1 , . . . , Ek }. Proposition 2.3. Let C = span{E1 , . . . , Ek } be a k-idempotent operator algebra on the Hilbert space H, let Y = {x1 , . . . , xk } be a set of k distinct points and define K : Y × Y → B(H) by K(xi , xj ) = Ei Ej∗ . Then K is positive definite and C is completely isometrically isomorphic to M(K) via the map that sends a1 E1 + · · · + ak Ek to the multiplier f (xi ) = ai . Proof. It is easily checked that K is positive definite. We first prove that the map is an isometry. Given B = ki=1 ai ⊗ Ei ∈ C, let f : Y → C be defined by f (xi ) = ai . We have that f ∈ M(K) with f C if and only if P = ((C 2 −f (xi )f (xj )∗ )K(xi , xj )) is positive semidef inite in B(H(k) ). Let v = e1 ⊗ v1 + · · · + ek ⊗ vk ∈ H(k) , let h = kj =1 Ej∗ vj and note that

Ej∗ h = Ej∗ vj . Finally, set h = ki=1 hi . Thus, P v, v =

k 2 C − ai a¯j Ei Ej∗ vj , vi i,j =1

=

k 2 C − ai a¯j Ej∗ h, Ei∗ h = C 2 h2 − B ∗ h, B ∗ h i,j =1

2 = C 2 h2 − B ∗ h . Hence, B C implies that P is positive and so Mf B.

For the converse, given any h let v = kj =1 ej ⊗ Ej∗ h, and note that P v, v 0, implies that ∗ B h C, and so B Mf . The proof of the complete isometry is similar but notationally cumbersome. 2

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Theorem 2.4. Let A be an operator algebra of functions on the set X then there exist a Hilbert space, H and a positive definite function K : X × X → B(H) such that M(K) = A˜ completely isometrically. Proof. Let Y be a finite subset of X. Since A/IY is a |Y |-idempotent operator algebra, by the above lemma, there exists a vector valued kernel KY such that A/IY = M(KY ) completely isometrically. Define KY (x, y) when (x, y) ∈ Y × Y, Y (x, y) = K 0 when (x, y) ∈ / Y ×Y

Y , where the direct sum is over all finite subsets of X. K and set K = Y It is easily checked that K is positive definite. Let f ∈ Mn (M(K)) with Mf 1, which is equivalent to ((In − f (x)f (y)∗ ) ⊗ K(x, y)) being positive definite. This is in turn equivalent to ((In − f (x)f (y)∗ ) ⊗ KY (x, y)) being positive definite for every finite subset Y of X. This last condition is equivalent to the existence for each such Y of some fY ∈ Mn (A) such that πY (fY ) 1 and fY = f on Y . The net of functions {fY } then converges BPW to f . Hence, f ∈ A˜ with f L 1. ˜ isometrically, for every n, and the result follows. 2 This proves that Mn (M(K)) = Mn (A) Corollary 2.5. Every BPW complete local operator algebra of functions is a dual operator algebra. Proof. In this case we have that A = A˜ = M(K) completely isometrically. By Lemma 2.1, this latter algebra is a dual operator algebra. 2 The above theorem gives a weak∗ -topology to a local operator algebra of functions A by using the identification A ⊆ A˜ = M(K) and taking the weak∗ -topology of M(K). The following proposition proves that convergence of bounded nets in this weak∗ -topology on A is same as BPW convergence. Proposition 2.6. Let A be a local operator algebra of functions on the set X. Then the net (fλ )λ ∈ Ball(A) converges in the weak∗ -topology if and only if it converges pointwise on X. Proof. Let L denote the reproducing kernel Hilbert space of H-valued functions on X with kernel K for which A˜ = M(K). Recall that if Ex : L → H, is the linear map given by evaluation at x, then K(x, y) = Ex Ey∗ . Also, if v ∈ H, and h ∈ L, then h, Ex∗ vL = h(x), vH . First assume that the net (fλ )λ ∈ Ball(A) converges to f in the weak∗ -topology. Using the identification of A˜ = M(K), we have that the operators Mfλ of multiplication by fλ , converge in the weak∗ -topology of B(L) to Mf . Then for any x ∈ X, h ∈ L, v ∈ H, we have that fλ (x) h(x), v H = f (x)h(x), v H = Mfλ h, Ex∗ v L → Mf h, Ex∗ v L = f (x) h(x), v H . Thus, if there is a vector in H and a vector in L such that h(x), vH = 0, then we have that fλ (x) → f (x). It is readily seen that such vectors exist if and only if Ex = 0, or equivalently,


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K(x, x) = 0. But this follows from the construction of K as a direct sum of positive definite functions over all finite subsets of X. For fixed x ∈ X and the one element subset Y0 = {x}, we have that the 1-idempotent algebra A/IY0 = 0 and so KY0 (x, x) = 0, which is one term in the direct sum for K(x, x). Conversely, assume that fλ < K, for all λ and fλ → f pointwise on X. We must prove that Mfλ → Mf in the weak∗ -topology on B(L). But since this is a bounded net of operators, it will be enough to show convergence in the weak operator topology and arbitrary vectors can be replaced by vectors from a spanning set. Thus, it will be enough to show that for v1 , v2 ∈ H and x1 , x2 ∈ X, we have that Mfλ Ex∗1 v1 , Ex∗2 v2 L → Mf Ex∗1 v1 , Ex∗2 v2 L . But we have,

Mfλ Ex∗1 v1 , Ex∗2 v2 L = Ex2 Mfλ Ex∗1 v1 , v2 H = fλ (x2 ) K(x2 , x1 )v1 , v2 H → f (x2 ) K(x2 , x1 )v1 , v2 H = Mf Ex∗1 v1 , Ex∗2 v2 L ,

and the result follows.

2

Corollary 2.7. The ball of a local operator algebra of functions is weak∗ -dense in the ball of its BPW completion. 3. Residually finite-dimensional operator algebras A C∗ -algebra B is called residually finite-dimensional (RFD) if it has a separating family of finite-dimensional representations, that is, of ∗-homomorphisms into matrix algebras. Since every ∗-homomorphism of a C∗ -algebra is completely contractive, a C∗ -algebra B is RFD if and only if for all n, and for every (bi,j ) ∈ Mn (B), we have that (bi,j )Mn (B) = sup{(π(bi,j ))} where the suprema is taken over all ∗-homomorphisms, π : B → Mk with k arbitrary. Residually finite-dimensional C∗ -algebras have been studied in [21,12,17,6]. Moreover, for C∗ -algebras, a homomorphism is a ∗ -homomorphism if and only if it is completely contractive. Thus, the following definition gives us a natural way of extending the notion of RFD to operator algebras. Definition 3.1. An operator algebra, B is called RFD if for every n and for every (bi,j ) ∈ Mn (B), (bi,j ) = sup{(π(bi,j ))}, where the suprema is taken over all completely contractive homomorphisms, π : B → Mk with k arbitrary. A dual operator algebra B is called weak∗ -RFD if this last equality holds when the completely contractive homomorphisms are also required to be weak∗ -continuous. The following result is implicitly contained in [26], but the precise statement that we shall need does not appear there. Thus, we refer the reader to [26] to be able to fully understand the proof since we have used some of the definitions and results from [26] without stating them. Lemma 3.2. Let B = span{F1 , . . . , Fk } be a concrete k-idempotent operator algebra. Then S(B ∗ B) is a Schur ideal affiliated with B, i.e., B = A(S(B ∗ B)) completely isometrically.

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Proof. From Corollary 3.3 of [26] we have that the Schur ideal S(B ∗ B) is non-trivial and bounded. Thus, we can define the algebra A(S(B ∗ B)) = span{E1 , . . . , Ek }, where Ei =

n

Q∈Sn−1 (B∗ B )

Q1/2 (In ⊗ Eii )Q−1/2

is the idempotent operator that lives on n Q∈S −1 Mk (Mn ). By using Theorem 3.2 of [26] n we get that S(A(S(B ∗ B))A(S(B ∗ B))∗ ) = S(B ∗ B). This further implies that ∗ A S B∗ B A S B∗ B = B∗ B completely order isomorphically under the map which sends Ei∗ Ej to Fi∗ Fj . Finally, by restricting the same map to A we get a map which sends Ei to Fi completely isometrically. Hence, the result follows. 2 Theorem 3.3. Every k-idempotent operator algebra is weak∗ -RFD. Proof. Let A be an abstract k-idempotent operator algebra. Note that A is a dual operator algebra since it is a finite-dimensional operator algebra. From this it follows that there exist a Hilbert space, H and a weak∗ -continuous completely isometric homomorphism, π : A → B(H). Note that B = π(A) is a concrete k-idempotent algebra generated by the idempotents, B = span{F1 , F2 , . . . , Fk } contained in B(H). Thus, from the above lemma B = A(S(B ∗ B)) completely isometrically. Q For each n ∈ N and Q ∈ Sn−1 , we define πn : B → Mk (Mn ) via πnQ (Fi ) = Q1/2 (In ⊗ Eii )Q−1/2 . Assume for the moment that we have proven that πn is a weak∗ -continuous completely contracQ tive homomorphism. Then for every (bij ) ∈ Mk (B) we must have that supn,Q∈S −1 (πn (bij )) = n (bij ), and hence (bij ) = sup{(ρ(bij ))} where the supremum is taken over all weak∗ continuous completely contractive homomorphisms ρ : B → Mm with m arbitrary. Since π : A → B is a complete isometry and weak∗ -continuous, this would imply the result for A by composition. Q Thus, it remains to show that πn is a weak∗ -continuous completely contractive homomorphism on B. Note that it is easy to check that it is a completely contractive homomorphism and it is completely isometric by the proof of the previous lemma. Q Finally, the fact that πn is weak∗ -continuous follows from the fact that B is finite-dimensional ∗ so that the weak -topology and the norm topology are equal. 2 Q

Theorem 3.4. Every BPW complete local operator algebra of functions is weak∗ -RFD. Proof. Let A be a BPW complete local operator algebra of functions on the set X and let F be a finite subset of X, so that A/IF is an |F |-idempotent operator algebra. It follows from the above lemma that A/IF is weak∗ -RFD, i.e., for ([fij ]) ∈ Mk (A/IF ) we have ([fij ]) = sup{(ρ([fij ]))} where the supremum is taken over all weak∗ -continuous completely contractive homomorphisms ρ from A/IF into matrix algebras.


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Let (fij ) ∈ Mk (A), then (fij )Mk (A) = supF ([fij ]) since A is local. Recall, that the weak∗ -topology on A requires all the quotient maps of the form πF : A → A/IF , πF (f ) = [f ] to be weak∗ -continuous. Thus, for each finite subset F ⊆ X, πF is a weak∗ -continuous completely contractive homomorphism. The result now follows by considering the composition of the weak∗ -continuous quotient maps with the weak∗ -continuous finite-dimensional representations of each quotient algebra. 2 Corollary 3.5. Every local operator algebra of functions is RFD. Proof. This follows immediately from the fact that every local operator algebra is completely isometrically contained in a BPW complete local operator algebra. 2 4. Quantized function theory on domains Whenever one replaces scalar variables by operator variables in a problem or definition, then this process is often referred to as quantization. It is in this sense that we would like to quantize the function theory on a family of complex domains. In some sense this process has already been carried out for balls in the work of Drury [20], Popescu [30], Arveson [7], and Davidson and Pitts [18] and for polydisks in the work of Agler [1,2], and Ball and Trent [10]. Our work is closely related to the idea of “quantizing” other domains defined by inequalities that occurs in the work of Ambrozie and Timotin [5], Ball and Bolotnikov [9], and Kalyuzhnyi-Verbovetzkii [22], but the terminology is our own. We approach these same ideas via operator algebra methods. We will show that in many cases this process yields local operator algebras of functions to which the results of the earlier sections can be applied. We begin by defining a family of open sets for which our techniques will apply. Definition 4.1. Let G ⊆ CN be an open set. If there exists a set of matrix-valued functions, Fk = (fk,i,j ) : G → Mmk ,nk , k ∈ I , whose components are analytic functions on G, and satisfy Fk (z) < 1, k ∈ I , then we call G an analytically presented domain and we call the set of functions R = {Fk : G → Mmk ,nk : k ∈ I } an analytic presentation of G. The subalgebra A of the algebra of bounded analytic functions on G generated by the component functions {fk,i,j : 1 i mk , 1 j nk , k ∈ I } and the constant function is called the algebra of the presentation. We say that R is a separating analytic presentation provided that the algebra A separates points on G. Remark 4.2. An analytic presentation of G by a finite set of matrix-valued functions, Fk : G → Mmk ,nk , 1 k K, can always be replaced by the single block-diagonal matrix-valued function, F (z) = F1 (z) ⊕ · · · ⊕ FK (z) into Mm,n with m = m1 + · · · + mK , n = n1 + · · · + nK and we will sometimes do this to simplify proofs. But it is often convenient to think in terms of the set, especially since this will explain the sums that occur in Agler’s factorization formula. Definition 4.3. Let G ⊆ CN be an analytically presented domain with presentation R = {Fk = (fk,i,j ) : G → Mmk ,nk , k ∈ I }, let A be the algebra of the presentation and let H be a Hilbert space. A homomorphism π : A → B(H) is called an admissible representation provided that (π(fk,i,j )) 1 in Mmk ,nk (B(H)) = B(Hnk , Hmk ), for every k ∈ I . We call the homomorphism π an admissible strict representation when these inequalities are all strictly less than 1. Given (gi,j ) ∈ Mn (A) we set (gi,j )u = sup{(π(gi,j ))}, where the supremum is

3210


taken over all admissible representations π of A. We let (gi,j )u0 denote the supremum that is obtained when we restrict to admissible strict representations. The theory of [5,9] studies domains defined as above with the additional restrictions that the set of defining functions is a finite set of polynomials. However, they do not need their polynomials to separate points, while we shall shortly assume that our presentations are separating, in order to invoke the results of the previous sections. This latter assumption can, generally, be dropped in our theory, but it requires some additional argument. There are several other places where our results and definitions given below differ from theirs. So while our results extend their results in many cases, in other cases we are using different definitions and direct comparisons of the results are not so clear. Proposition 4.4. Let G have a separating analytical presentation and let A be the algebra of the presentation. Then A endowed with either of the family of norms · u or · u0 is an operator algebra of functions on G. Proof. It is clear that it is an operator algebra and by definition it is an algebra of functions on G. It follows from the hypotheses that it separates points of G. Finally, for every λ = (λ1 , . . . , λN ) ∈ G, we have a representation of A on the one-dimensional Hilbert space given by πλ (f ) = f (λ). Hence, |f (λ)| f u and so A is an operator algebra of functions on G. 2 It will be convenient to say that matrices, A1 , . . . , Am are of compatible sizes if the product, A1 · · · Am exists, that is, provided that each Ai is an ni × ni+1 matrix. Given an analytically presented domain G, we include one extra function, F1 which denotes the constant function 1. By an admissible block-diagonal matrix over G we mean a blockdiagonal matrix-valued function of the form D(z) = diag(Fk1 , . . . , Fkm ) where ki ∈ I ∪ {1} for 1 i m. Thus, we are allowing blocks of 1’s in D(z). Finally, given a matrix B we let B (q) = diag(B, . . . , B) denote the block-diagonal matrix that repeats B q times. Theorem 4.5. Let G be an analytically presented domain with presentation R = {Fk = (fk,i,j ) : G → Mmk ,nk , k ∈ I }, let A be the algebra of the presentation and let P = (pij ) ∈ Mm,n (A), where m, n are arbitrary. Then the following are equivalent: (i) P u < 1, (ii) there exist an integer l, matrices of scalars Cj , 1 j l with Cj < 1 and admissible block-diagonal matrices Dj (z), 1 j l, which are of compatible sizes and are such that P (z) = C1 D1 (z) · · · Cl Dl (z), (iii) there exist a positive, invertible matrix R ∈ Mm and matrices P0 , Pk ∈ Mm,rk (A), k ∈ K, where K ⊆ I is a finite set, such that Im − P (z)P (w)∗ = R + P0 (z)P0 (w)∗ +

(q ) Pk (z) I − Fk (z)Fk (w)∗ k Pk (w)∗

k∈K

where rk = qk mk and z = (z1 , . . . , zN ), w = (w1 , . . . , wN ) ∈ G.


3211

Proof. Although we will not logically need it, we first show that (ii) implies (i), since this is the easiest implication and helps to illustrate some ideas. Note that if π : A → B(H) is any admissible representation, then the norm of π of any admissible block-diagonal matrix is at most 1. Thus, if P has the form of (ii), then for any admissible π , we will have (π(pi,j )) expressed as a product of scalar matrices and operator matrices all of norm at most one and hence, (π(Pi,j )) C1 · · · Cl < 1. Thus, P u C1 · · · Cl < 1. We now prove that (i) implies (ii). The ideas of the proof are similar to [25, Corollary 18.2], [14, Corollary 2.11] and [23, Theorem 1] and use in an essential way the abstract characterization of operator algebras. For each m, n ∈ N, one proves that P m,n := inf{C1 · · · Cl }, defines a norm on Mm,n (A), where the infimum is taken over all l and all ways to factor P (z) = C1 D1 (zi1 ) · · · Cl Dl (zil ) as a product of matrices of compatible sizes with scalar matrices Cj , 1 j l and admissible block-diagonal matrices Dj , 1 j l. Moreover, one can verify that Mm,n (A) with this family { · m,n }m,n of norms satisfies the axioms for an abstract unital operator algebra as given in [15] and hence by the Blecher–Ruan– Sinclair representation theorem [15] (see also [25]) there exist a Hilbert space H and a unital completely isometric isomorphism π : A → B(H). Thus, for every m, n ∈ N and for every P = (pij ) ∈ Mm,n (A), we have that P m,n = (π(pij )). However, π (mk ,nk ) (Fk ) = (π(fk,i,j )) 1 for 1 i K, and so, π is an admissible representation. Thus, P m,n = (π(pij )) P u . Hence, if P u < 1, then P m,n < 1 which implies that such a factorization exists. This completes the proof that (i) implies (ii). We will now prove that (ii) implies (iii). Suppose that P has a factorization as in (ii). Let K ⊆ I be the finite subset of all indices that appear in the block-diagonal matrices appearing in the factorization of P . We will use induction on l to prove that (iii) holds. First, assume that l = 1 so that P (z) = C1 D1 (z). Then, ∗ Im − P (z)P (w)∗ = Im − C1 D1 (z) C1 D1 (w) = Im − C1 C1∗ + C1 I − D1 (z)D1 (w)∗ C1∗ . Since D1 (z) is an admissible block-diagonal matrix the (i, i)-th block-diagonal entry of I − D1 (z)D1 (w)∗ is I − Fki (z)Fki (w)∗ for some finite collection, ki . Let Ek be the diagonal matrix that has 1’s wherever Fk appears (so Ek = 0 when there is no Fk term in D1 ). Hence, C1 I − D1 (z)D1 (w)∗ C1∗ = C1 Ek I − Fk (z)Fk (w)∗ Ek C1∗ . k

Therefore, gathering terms for common values of i, Im − P (z)P (w)∗ = R0 +

Pk I − Fk (z)Fk (w)∗ Pk∗ ,

k∈K

where R0 = Im − C1 C1∗ is a positive, invertible matrix and Pi is, in this case a constant. Thus, the form (iii) holds, when l = 1. We now assume that the form (iii) holds for any R(z) that has a factorization of length at most l − 1, and assume that

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P (z) = C1 D1 (z) · · · Dl−1 (z)Cl Dl (z) = C1 D1 (z)R(z), where R(z) has a factorization of length l − 1. Note that a sum of expressions such as on the right-hand side of (iii) is again such an expression. This follows by using the fact that given any two expressions A(z), B(z), we can write A(z)A(w)∗ + B(z)B(w)∗ = C(z)C(w)∗ where C(z) = (A(z), B(z)). Thus, it will be sufficient to show that Im − P (z)P (w)∗ is a sum of expressions as above. To this end we have that, Im − P (z)P (w)∗ = Im − C1 D1 (z)D1 (w)∗ C1∗ + C1 D1 (z) I − R(z)R(w)∗ D1 (w)∗ C1∗ . The first term of the above equation is of the form as on the right-hand side of (iii) by case l = 1. Also, the quantity (I − R(z)R(w)∗ ) = R0 R0∗ + R0 (z)R0 (w)∗ + k∈K Rk (z)(I − Fk (z)Fk (w)∗ )(qk ) Rk (w)∗ by the inductive hypothesis. Hence, C1 D1 (z) I − R(z)R(w)∗ D1 (w)∗ C1∗ ∗ ∗ = C1 D1 (z)R0 C1 D(w)R0 + C1 D1 (z)R0 (z) C1 D1 (w)R0 (w) (q ) ∗ C1 D1 (z)Rk (z) I − Fk (z)Fk (w)∗ k C1 D1 (w)Rk (w) . + k∈K

Thus, we have expressed (I − P (z)P (w)∗ ) as a sum of two terms both of which can be written in the form desired. Using again our remark that the sum of two such expressions is again such an expression, we have the required form. Finally, we will prove (iii) implies (i). Let π : A → B(H) be an admissible representation and let P = (pi,j ) ∈ Mm,n (A) have a factorization as in (iii). To avoid far too many superscripts we simplify π (m,n) to Π . Now observe that Im − Π(P )Π(P )∗ = Π(R) + Π(P0 )Π(P0 )∗ (q ) ∗ + Π(Pk ) I − Π(Fk )Π(Fk )∗ k Π(Pk ) . k∈K

Clearly the first two terms of the sum are positive. But since π is an admissible representation, Π(Fk ) 1 and hence, (I − Π(Fk )Π(Fk )∗ ) 0. Hence, each term on the right-hand side of the above inequality is positive and since R is strictly positive, say R δIm for some scalar ∗ δ > 0, we have that Im − Π(P √ )Π(P ) δIm . Therefore, Π(P ) 1 − δ. Thus, since π was an arbitrary admissible representation, √ P u 1 − δ < 1, which proves (i). 2 When we require the functions in the presentation to be row vector-valued, then the above theory simplifies somewhat and begins to look more familiar. Let G be an analytically presented domain with presentation Fk : G → M1,nk , k ∈ I . We identify M1,n with the Hilbert space Cn so


3213

that 1 − Fk (z)Fk (w)∗ = 1 − Fk (z), Fk (w), where the inner product is in Cn . In this case we shall say that G is presented by vector-valued functions. Corollary 4.6. Let G be presented by vector-valued functions, Fk = (fk,j ) : G → M1,nk , k ∈ I , let A be the algebra of the presentation and let P = (pij ) ∈ Mm,n (A). Then the following are equivalent: (i) P u < 1, (ii) there exist an integer l, matrices of scalars Cj , 1 j l with Cj < 1 and admissible block-diagonal matrices Dj (z), 1 j l, which are of compatible sizes and are such that P (z) = C1 D1 (z) · · · Cl Dl (z), (iii) there exist a positive, invertible matrix R ∈ Mm and matrices P0 ∈ Mm,r0 (A), Pk ∈ Mm,rk (A), k ∈ K, where K ⊆ I is finite, such that Im − P (z)P (w)∗ = R + P0 (z)P0 (w)∗ +

1 − Fk (z), Fk (w) Pk (z)Pk (w)∗ k∈K

where z = (z1 , . . . , zN ), w = (w1 , . . . , wN ) ∈ G. The following result gives us a Nevanlinna-type result for the algebra of the presentation. Theorem 4.7. Let Y be a finite subset of an analytically presented domain G with separating analytic presentation Fk = (fk,i,j ) : G → Mmk ,nk , k ∈ I , let A be the algebra of the presentation and let P be an Mm,n -valued function defined on a finite subset Y = {x1 , . . . , xl } of G. Then the following are equivalent: (i) there exists P˜ ∈ Mmn (A) such that P˜ |Y = P and P˜ u < 1, (ii) there exist a positive, invertible matrix R ∈ Mm and matrices P0 ∈ Mm,r0 (A), Pk ∈ Mm,rk (A), k ∈ K, where K ⊆ I is a finite set, such that Im − P (z)P (w)∗ = R + P0 (z)P0 (w)∗ +

(q ) Pk (z) I − Fk (z)Fk (w)∗ k Pk (w)∗

k∈K

where rk = qk mk and z = (z1 , . . . , zN ), w = (w1 , . . . , wN ) ∈ Y . Proof. Note that (i) ⇒ (ii) follows immediately as a corollary of Theorem 4.5. Thus it only remains to show that (ii) ⇒ (i). Since A is an operator algebra of functions, therefore, A/IY is a finite-dimensional operator algebra of idempotents and A/IY = span{E1 , . . . , El } where l = |Y |. Thus there exist a Hilbert space HY and a completely isometric representation π of A/IY . By Theorem 2.4, there exists a kernel KY such that π(A/IY ) = M(KY ) completely isometrically under the map ρ : π(A/IY ) → M(KY ) which sends π(B) to Mf , where B = li=1 ai π(Ei ) and f : Y → C is a function defined by f (xi ) = ai . Note that I − Fk (xi )Fk (xj ) ⊗ KY (xi , xj ) = I − π(Fk + IY )(xi )π(Fk + IY )(xj )∗ ⊗ KY (xi , xj ) ij 0

3214


since π(Fk + IY ) Fk u 1 for all k ∈ I . From this it follows that Im − π(P + IY ) (xi ) π(P + IY ) (xj )∗ ⊗ KY (xi , xj ) ij 0. Using that R > 0, we get that π(P + IY ) < 1. This shows that there exists P˜ ∈ A such that P˜ |Y = P and P˜ u < 1. This completes the proof. 2 We now turn towards defining quantized versions of the bounded analytic functions on these domains. For this we need to recall that the joint Taylor spectrum [31] of a commuting N -tuple of operators T = (T1 , . . . , TN ), is a compact set, σ (T ) ⊆ CN and that there is an analytic functional calculus [32,33] defined for any function that is holomorphic in a neighborhood of σ (T ). Definition 4.8. Let G ⊆ CN be an analytically presented domain, with presentation R = {Fk : G− → Mmk ,nk , k ∈ I }. We define the quantized version of G to be the collection of all commuting N -tuples of operators, Q(G) = T = (T1 , T2 , . . . , TN ) ∈ B(H): σ (T ) ⊆ G and Fk (T ) 1, ∀k ∈ I , where H is an arbitrary Hilbert space. We set Q0,0 (G) = T = (T1 , T2 , . . . , TN ) ∈ Mn : σ (T ) ⊆ G and Fk (T ) 1, ∀k ∈ I , where n is an arbitrary positive integer. Note that if we identify a point (λ1 , . . . , λN ) ∈ CN with an N -tuple of commuting operators on a one-dimensional Hilbert space, then we have that G ⊆ Q(G). If T = (T1 , . . . , TN ) ∈ Q(G), is a commuting N -tuple of operators on the Hilbert space H, then since the joint Taylor spectrum of T is contained in G, we have that if f is analytic on G, then there is an operator f (T ) defined and the map π : Hol(G) → B(H) is a homomorphism, where Hol(G) denotes the algebra of analytic functions on G [33]. Definition 4.9. Let G ⊆ CN be an analytically presented domain, with presentation R = ∞ (G) to be the set of functions f ∈ Hol(G), such that {Fk : G− → Mmk ,nk , k ∈ I }. We define HR ∞ (G)), we set (f ) = f R ≡ sup{f (T ): T ∈ Q(G)} is finite. Given (fi,j ) ∈ Mn (HR i,j R sup{(fi,j (T )): T ∈ Q(G)}. ∞ (G) = A, ˜ completely isometrically and whether We are interested in determining when HR or not the · R norm is attained on the smaller set Q0,0 (G). ∞ (G) ⊆ H ∞ (G), and f Note that since each point in G ⊆ Q(G), we have that HR ∞ ∞ f R . Also, we have that A ⊆ HR (G) and for (fi,j ) ∈ Mn (A), (fi,j )R (fi,j )u0 ∞ (G) might not even be isometric. (fi,j )u . Thus, the inclusion of A into HR

Theorem 4.10. Let G be an analytically presented domain with a separating presentation R = {Fk : G → Mmk ,nk : k ∈ I }, let A be the algebra of the presentation and let A˜ be the BPWcompletion of A. Then ∞ (G), completely isometrically, (i) A˜ = HR


3215

∞ (G) is a local weak∗ -RFD dual operator algebra. (ii) HR

˜ with f L < 1. Then there exists a net of functions, fλ ∈ Mn (A), Proof. Let f ∈ Mn (A), such that fλ u < 1 and limλ fλ (z) = f (z) for every z ∈ G. Since fλ ∞ < 1, by Montel’s theorem, there is a subsequence {fn } of this net that converges to f uniformly on compact sets. Hence, if T ∈ Q(G), then limn f (T ) − fn (T ) = 0 and so f (T ) supn fn (T ) 1. ∞ (G)), with f 1. This proves that A˜ ⊆ H ∞ (G) and that Thus, we have that f ∈ Mn (HR R R f L f R . ∞ (G)) with g < 1. Given any finite set Y = {y , . . . , y } ⊆ G, Conversely, let g ∈ Mn (HR 1 t R let A/IY = span{E1 , . . . , Et } be the corresponding t-idempotent algebra and let πY : A → A/IY denote the quotient map. Write yi = (yi,1 , . . . , yi,N ), 1 i t and let Tj = y1,j E1 +· · ·+yt,j Et , 1 j N so that T = (T1 , . . . , TN ) is a commuting N -tuple of operators with σ (T ) = Y . For k ∈ I , we have that Fk (T ) = Fk (y1 ) ⊗ E1 + · · · + Fk (yt ) ⊗ Et = πY (Fk ) Fk u = 1. Thus, T ∈ Q(G), and so, g(T ) = g(y1 ) ⊗ E1 + · · · + g(yt ) ⊗ Et gR < 1. Since A separates points, we may pick f ∈ Mn (A) such that f = g on Y . Hence, πY (f ) = f (T ) = g(T ) and πY (f ) < 1. Thus, we may pick fY ∈ Mn (A), such that πY (fY ) = πY (f ) and fY u < 1. This net of functions, {fY } converges to g pointwise and is bounded. Therefore, ˜ and gL 1. This proves that H ∞ (G) ⊆ A˜ and that gL gR . g ∈ Mn (A) R ∞ (G) and the two matrix norms are equal for matrices of all sizes. The rest of Thus, A˜ = HR the conclusions follows from the results on BPW-completions. 2 ∞ (G), f = sup{π(f )} Remark 4.11. The above result yields that for every f ∈ HR R where the supremum is taken over all finite-dimensional weak∗ -continuous representations, ∞ (G) → M with n arbitrary. For many examples, we can show that π : HR n

f R = sup f (T ): T ∈ Q00 (G) ∞ (G). Also, we can verify these hypotheses are met for most of the algebras given for any f ∈ HR in the example section. In particular, for Examples 5.1, 5.2, 5.3, 5.4, 5.6, 5.7, and 5.8. It would be interesting to know if this can be done in general. ∞ (G). In particular, we wish We now seek other characterizations of the functions in HR to obtain analogues of Agler’s factorization theorem and of the results in [5,9]. By Theorem 2.4, if we are given an analytically presented domain G ⊆ CN , with presentation R = {Fk : G → Mmk ,nk , k ∈ I }, then there exist a Hilbert space H and a positive definite function, K : G × G → B(H) such that A˜ = M(K). We shall denote any kernel satisfying this property by KR .

Definition 4.12. Let G ⊆ CN be an analytically presented domain, with presentation R = {Fk : G− → Mmk ,nk , k ∈ I }. We shall call a function H : G × G → Mm an R-limit, provided that H is the pointwise limit of a net of functions Hλ : G × G → Mm of the form given by Theorem 4.5(iii).

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Corollary 4.13. Let G ⊆ CN be an analytically presented domain, with a separating presentation R = {Fk : G− → Mmk ,nk , k ∈ I }. Then the following are equivalent: ∞ (G)) and f 1, (i) f ∈ Mm (HR R (ii) (Im − f (z)f (w)∗ ) ⊗ KR (z, w) is positive definite, (iii) Im − f (z)f (w)∗ is an R-limit.

In the case when the presentation contains only finitely many functions we can say considerably more about R-limits. Proposition 4.14. Let G be an analytically presented domain with a finite presentation R = {Fk = (fk,i,j ) : G → Mmk ,nk , 1 k K}. For each compact subset S ⊆ G, there exists a constant C, depending only on S, such that given a factorization of the form, Im − P (z)P (w)∗ = R + P0 (z)P0 (w)∗ +

K

(q ) Pk (z) I − Fk (z)Fk (w)∗ k Pk (w)∗ ,

k=1

then Pk (z) C for all k ∈ I and for all z ∈ S. Proof. By the continuity of the functions, there is a constant δ > 0, such that Fk (z) 1 − δ, for all k ∈ I and for all z ∈ S. Thus, we have that I − Fk (z)Fk (z)∗ δI , for all k ∈ I and for all z ∈ S. Also, we have that (q ) Im Im − P (z)P (z)∗ Pk (z) I − Fk (z)Fk (z)∗ k Pk (z)∗ δPk (z)Pk (z)∗ . This shows that Pk (z) 1/δ for all k ∈ I and for all z ∈ S.

2

The proof of the following result is essentially contained in [9, Lemma 3.3]. Proposition 4.15. Let G be a bounded domain in CN and let F = (fi,j ) : G → Mm,n be analytic with F (z) < 1 for z ∈ G. If H : G × G → Mp is analytic in the first variables, coanalytic in the second variables and there exists a net of matrix-valued functions Pλ ∈ Mp,rλ (Hol(G)) which are uniformly bounded on compact subsets of G, such that H (z, w) is the pointwise limit of Hλ (z, w) = Pλ (z)(Im − F (z)F (w)∗ )(qλ ) Pλ (w)∗ where rλ = qλ mk , then there exist a Hilbert space H and an analytic function, R : G → B(H ⊗ CM , Cp ) such that H (z, w) = R(z)[(Im − F (z)F (w)∗ ) ⊗ IH ]R(w)∗ . Proof. We identify (Im − F (z)F (w)∗ )(qλ ) = (Im − F (z)F (w)∗ ) ⊗ ICqλ , and the p × mqλ matrixvalued function Pλ as an analytic function, Pλ : G → B(Cm ⊗ Cqλ , Cp ). Writing Cm ⊗ Cqλ = Cqλ ⊕ · · · ⊕ C qλ (m times) allows us to write Pλ (z) = [P1λ (z), . . . , Pmλ (z)] where each Piλ (z) be the (1, n) vectors that represent the rows of the is p × qλ . Also, if we let f1 (z), . . . , fm (z) ∗ matrix F , then we have that F (z)F (w)∗ = m i,j =1 fi (z)fj (w) Ei,j . Finally, we have that Hλ (z, w) =

m i=1

Piλ (z)Piλ (w)∗ −

m i,j =1

fi (z)fj (w)∗ Piλ (z)Pjλ (w).


3217

Let Kλ (z, w) = (Piλ (z)Pjλ (w)∗ ), so that Kλ : G×G → Mm (Mp ) = B(Cm ⊗Cp ), is a positive definite function that is analytic in z and co-analytic in w. By dropping to a subnet, if necessary, we may assume that Kλ converges uniformly on compact subsets of G to K = (Ki,j ) : G × G → Mm (Mp ). Note that this implies that Piλ (z)Pjλ (w)∗ → Ki,j (z, w) for all i, j and that K is a positive definite function that is analytic in z and coanalytic in w. The positive definite function K gives rise to a reproducing kernel Hilbert space H of analytic Cm ⊗ Cp -valued functions on G. If we let E(z) : H → Cm ⊗ Cp , be the evaluation functional, then K(z, w) = E(z)E(w)∗ and E : G → B(H, Cm ⊗ Cp ) is analytic. Identifying Cm ⊗ Cp = Cp ⊕ · · · ⊕ Cp (m times), yields analytic functions, Ei : G → B(H, Cp ), i = 1, . . . , m, such that (Ki,j (z, w)) = K(z, w) = E(z)E(w)∗ = (Ei (z)Ej (w)∗ ). Define an analytic map R : G → B(H ⊗ Cm , Cp ) by identifying H ⊗ Cm = H ⊕ · · · ⊕ H (m times) and setting R(z)(h1 ⊕ · · · ⊕ hm ) = E1 (z)h1 + · · · + Em (z)hm . Thus, we have that R(z) Im − F (z)F (w)∗ ⊗ IH R(w)∗ =

m

Ei (z)Ei (w)∗ −

=

Ki,i (z, w) −

λ

m

fi (z)fj (w)∗ Ki,j (z, w)

i,j =1

i=1

= lim

fi (z)fj (w)∗ Ei (z)Ej (w)∗

i,j =1

i=1 m

m

m

Piλ (z)Piλ (w)∗ −

fi (z)fj (w)∗ Piλ (z)Pjλ (w)∗ = H (z, w),

i,j =1

i=1

and the proof is complete.

m

2

Remark 4.16. Conversely, any function that can be written in the form H (z, w) = R(z)[(Im − F (z)F (w)∗ ) ⊗ IH ]R(w)∗ can be expressed as a limit of a net as above by considering the directed set of all finite-dimensional subspaces of H and for each finite-dimensional subspace setting HF (z, w) = RF (z)[(Im − F (z)F (w)∗ ) ⊗ IF ]RF (w)∗ , where RF (z) = R(z)(PF ⊗ Im ) with PF the orthogonal projection onto F . Definition 4.17. We shall refer to a function H : G × G → Mm (Mp ) that can be expressed as H (z, w) = R(z)[(Im − F (z)F (w)∗ ) ⊗ H]R(w)∗ for some Hilbert space H and some analytic function R : G → B(H ⊗ Cm , Cp ), as an F -limit. Theorem 4.18. Let G be an analytically presented domain with a finite separating presentation R = {Fk = (fk,i,j ) : G → Mmk ,nk , 1 k K}, let f = (fij ) be an Mm,n -valued function defined on G. Then the following are equivalent: ∞ (G)) and f 1, (1) f ∈ Mmn (HR R (2) there exist an analytic operator-valued function R0 : G → B(H0 , Cm ) and Fk -limits, Hk : G × G → Mm , such that ∗

∗

I − f (z)f (w) = R0 (z)R0 (w) +

K k=1

Hk (z, w),

∀z, w ∈ G,

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(3) there exist Fk -limits, Hk (z, w), such that I − f (z)f (w)∗ =

K

Hk (z, w),

∀z, w ∈ G.

k=1 ∞ (G). Let us first assume that f ∈ M (A) Proof. Recall that A˜ = HR mn ˜ and f Mmn (A) ˜ < 1. Then for each finite set Y , there exists fY ∈ Mmm (A) such that fY converges to f pointwise and fY fY u 1. We may assume that fY u < 1 by replacing fY by 1+1/|Y | , where |Y | denotes the cardinality of the set Y . Thus by Theorem 4.5 there exist a positive, invertible matrix R Y ∈ Mm and matrices Y Pk ∈ Mm,rkY (A), 0 k K, such that

Im − fY (z)fY (w)∗ = R Y + P0Y (z)P0Y (w)∗ +

K

(q ) PkY (z) I − Fk (z)Fk (w)∗ kY PkY (w)∗

k=1

where rkY = qkY mk and z, w ∈ G. If we define a map F0 : G → Mm0 ,n0 as the zero map then the above factorization can be written as Im − fY (z)fY (w)∗ = R Y +

K


k=0

where rkY = qkY mk and z, w ∈ G. Note that the net R Y is uniformly bounded above by 1, thus there exists R ∈ Mm and a subnet R Ys which converges to R. Finally, since the net fY converges to f pointwise we have that the net K


k=1

converges pointwise on G. Also note that for each k, {PkY } is a net of vector-valued holomorphic functions and is uniformly bounded on compact subsets of G by Proposition 4.14. Thus by Proposition 4.15 there exist Fk -limits for each 0 k K, that is, there exist K + 1 Hilbert spaces Hk and K + 1 analytic function, Rk : G → B(Hk ⊗ CM , Cp ) such that H (z, w) = Rk (z)[(Im − Fk (z)Fk (w)∗ ) ⊗ IHk ]Rk (w)∗ and the corresponding subnet of the net

kK

K Y ∗ (qkY ) P Y (w)∗ converges to k=0 Pk (z)(I − Fk (z)Fk (w) ) k=0 Hk (z, w) for all z, w ∈ G. This k completes the proof that (1) implies (2). To show the converse, assume that there exist an analytic operator-valued function R0 : G → M p spaces B(H0 , Cm ) and K analytic functions, Rk : G → B(H

Kk ⊗ C , C ) on some Hilbert ∗ ∗ Hk such that I − f (z)f (w) = R0 (z)R0 (w) + k=1 Rk (z)(I − Fk (z)Fk (w)∗ )(qk ) Rk (w)∗ for z, w ∈ G. ˜ By using Theorem 2.4 there exists a vector-valued kernel K such that Mn (M(K)) = Mn (A) ∗ completely isometrically for every n. It is easy to see that ((I − f (z)f (w) ) ⊗ K(z, w)) 0 for z, w ∈ G. This is equivalent to f ∈ Mm (M(K)) and Mf 1 which in turn is equivalent to (1). Thus, (1) and (2) are equivalent.


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Clearly, (3) implies (2). The argument for why (2) implies (3) is contained in [9] and we recall it. If we fix any k0 , then since Fk0 (z) < 1 on G, we have that |fk0 ,1,1 (z)|2 + · · · + |fk0 ,1,m (z)|2 < 1 on G. From this it follows that H (z, w) = (1 − fk0 ,1,1 (z)f¯k0 ,1,1 (w) − · · · − fk0 ,1,m (z)f¯k0 ,1,m (w)) is an Fk0 -limit and that H −1 (z, w) is positive definite. Now we have that R0 (z)R0 (w)∗ H −1 (z, w) is positive definite and so we may write, R0 (z)R0 (w)∗ H −1 (z, w) = G0 (z)G0 (w)∗ and we have that R0 (z)R0 (w)∗ = G0 (z)H (z, w)G(w)∗ . This shows that R0 (z)R0 (w)∗ is an Fk -limit and so it may be absorbed into the sum. 2 5. Examples and applications In this section we present a few examples to illustrate the above definitions and results. Example 5.1. Let G = DN be the polydisk which has a presentation given by the coordinate functions Fi (z) = zi , 1 i N . Then the algebra of this presentation is the algebra of polynomials and an admissible representation is given by any choice of N commuting contractions, (T1 , . . . , TN ) on a Hilbert space. Given a matrix of polynomials, (pi,j )u = sup (pi,j (T1 , . . . , TN )) where the supremum is taken over all N -tuples of commuting contractions. This is the norm considered by Agler in [1], which is sometimes called the Schur–Agler norm [23]. Our Q(DN ) = {T = (T1 , . . . , TN ): σ (T ) ⊆ DN and Ti 1}. Note that if we replace such a T by rT = (rT1 , . . . , rTN ) then rTi < 1, rT ∈ QR (DN ) and taking suprema over all T ∈ QR (DN ) will be the same as taking a suprema over this smaller set. Thus, the algebra ∞ (DN ) consists of those analytic functions f such that HR f R = sup f (T1 , . . . , TN ): Ti < 1, i = 1, . . . , N < +∞. Our result that this is a weak∗ -RFD algebra shows that this supremum is also attained by considering commuting N -tuples of matrices satisfying Ti < 1, i = 1, . . . , N . By Theorem 4.18 for ∞ (DN )), we have that f 1 if and only if f ∈ Mm,n (HR R Im − f (z)f (w)∗ =

N (1 − zi w¯ i )Ki (z, w), i=1

for some analytic-coanalytic positive definite functions, Ki : DN × DN → Mm . Example 5.2. Let G = BN denote the unit Euclidean ball in CN . If we let F1 (z) = (z1 , . . . , zN ) : BN → M1,N , then this gives us a polynomial presentation. Again the algebra of the presentation is the polynomial algebra. An admissible representation corresponds to an N -tuple of commuting operators (T1 , . . . , Tn ) such that T1 T1∗ + · · · + TN TN∗ I , which is commonly called a row contraction and an admissible strict representation is given when T1 T1∗ + · · · + TN TN∗ < I , which is generally referred to as a strict row contraction. In this case one can again easily see that · u = · u0 by ∞ (B ) if and only if using the same r < 1 argument as in the last example and that f ∈ HR N f R = sup f (T ): T1 T1∗ + · · · + TN TN∗ < I < +∞.

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These are the norms on polynomials considered by Drury [20], Popescu [30], Arveson [7], and Davidson and Pitts [18]. Again our weak∗ -RFD result shows that f R is attained by taking the supremum over commuting N -tuples of matrices satisfying T1 T1∗ + · · · + TN TN∗ < I . ∞ (B )) that f 1 if and only if By Theorem 4.18 we will have for f ∈ Mm,n (HR N R Im − f (z)f (w)∗ = 1 − z, w K(z, w), where K : BN × BN → Mm is an analytic-coanalytic positive definite function. Example 5.3. Let G = BN as above and let F1 (z) = (z1 , . . . , zN )t : BN → MN,1 . Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N -tuple of commuting operators (T1 , . . . , TN ) such that (T1 , . . . , TN )t 1, i.e., such that T1∗ T1 + · · · + TN∗ TN I , which is generally referred to as a ∞ (B ) will be defined by taking suprema over all column contraction. This time the norm on HR N strict column contractions and we will have that f R 1 if and only if Im − f (z)f (w)∗ = R1 (z) IN − (zi w¯j ) ⊗ IH R1 (w)∗ for some R1 : BN → B(Cm , H), analytic. Again, the weak∗ -RFD result shows that f R is attained by taking the supremum over matrices that form strict column contractions. Example 5.4. Let G = BN as above, let F1 (z) = (z1 , . . . , zN ) : BN → M1,N and F2 (z) = (z1 , . . . , zN )t : BN → MN,1 . Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N -tuple of commuting operators (T1 , . . . , TN ) such that T1 T1∗ + · · · + TN TN∗ I and T1∗ T1 + · · · + TN∗ TN I , ∞ (B ) is defined that is, which is both a row and column contraction. This time the norm on HR N as the supremum over all commuting N -tuples that are both strict row and column contractions and again this is attained by restricting to commuting N -tuples of matrices that are strict row and ∞ (B )) with f 1 if and only if column contractions. We will have that f ∈ Mm,n (HR N R Im − f (z)f (w)∗ = 1 − z, w K1 (z, w) + R1 (z) IN − (zi w¯j ) ⊗ IH R1 (w)∗ , where K1 and R1 are as before. The last three examples illustrate that it is possible to have multiple rational representations of G, all with the same algebra, but which give rise to (possibly) different operator algebra norms on A. Thus, the operator algebra norm depends not just on G, but also on the particular presentation of G that one has chosen. We have surpressed this dependence on R to keep our notation simplified. Example 5.5. Let G = D be the open unit disk in the complex plane and let F1 (z) = z2 , F2 (z) = z3 . It is easy to check that the algebra A of this presentation is generated by the component functions and the constant function so that A is the span of the monomials, {1, zn : n 2}. Also, A separates the points of G. In this case a (strict) admissible representation, π : A → B(H), is given by any choice of a pair of commuting (strict) contractions,


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A = π(z2 ), B = π(z3 ), satisfying A3 = B 2 . Again, it is easy to see that · u = · u0 . On the other hand Q(D) = T : σ (T ) ⊆ D and T 2 1, T 3 1 ∞ (D) is defined by and it can be seen that HR

f R = sup f (T ): T 2 < 1, T 3 < 1 < +∞. ∞ (D)) and f 1 if and only if In this case we have that f ∈ Mm,n (HR R

Im − f (z)f (w)∗ = 1 − z2 w¯ 2 K1 (z, w) + 1 − z3 w¯ 3 K2 (z, w). However, our weak∗ -RFD result only guarantees that f R is attained by taking the supremum all finite-dimensional representations π such that π(z2 ) = A and π(z3 ) = B are commuting strict contractions satisfying A3 = B 2 . However, given such a pair there is, in general, no single matrix T such that T 2 = A and T 3 = B. So our results do not guarantee, that f R is attained by taking the supremum over all matrices T satisfying T 2 < 1 and T 3 < 1. Example 5.6. Let L = {z ∈ C: |z − a| < 1, |z − b| < 1}, where |a − b| < 1, then the functions f1 (z) = z − a, f2 (z) = z − b give a polynomial presentation of this “lens”. The algebra of this presentation is again the algebra of polynomials. An admissible representation of this algebra is defined by choosing any operator satisfying T − aI 1 and T − bI 1, with strict inequalities for the admissible strict representations. In this case we easily see that · u = · u0 , since given any operator T satisfying T − aI 1 and T − bI 1, and r < 1, Sr = rT + (1 − r)(a + b) corresponds to the admissible strict representations and for any matrix of polynomials (pi,j (T )) = limr→1 (pij (Sr )). This algebra with this norm was studied in [11]. Their work shows that this norm is completely boundedly equivalent to the usual supremum ∞ (L) = H ∞ (L), as sets, but the norms are different. norm and consequently, HR ∞ (L)) and f 1 if and only if Our results imply that f ∈ Mm,n (HR R Im − f (z)f (w)∗ = 1 − (z − a)(w − b) K1 (z, w) + 1 − (z − b)(w − b) K2 (z, w). Since the coordinate function z belongs to the algebra A, our weak∗ -RFD results again show that f R is attained by choosing matrices satisfying T − aI < 1, T − bI < 1. Example 5.7. Let G = {(zi,j ) ∈ MM,N : (zi,j ) < 1} and let F : G → MM,N be the identity map F (z) = (zi,j ). Then this is a polynomial presentation of G and the algebra of the presentation is the algebra of polynomials in the MN variables {zi,j }. An admissible representation of this algebra is given by any choice of MN commuting operators {Ti,j } on a Hilbert space H, such that (Ti,j ) 1 in MM,N (B(H)) and as above, one can show that · R is achieved by taking suprema over all commuting MN -tuples of matrices for which (Ti,j ) < 1. We have that f ∈ ∞ (G)) and f 1 if and only if Mm,n (HR R Im − f (z)f (w)∗ = R1 (z) IM − (zi,j )(wi,j )∗ ⊗ IH R1 (w)∗ , for some appropriately chosen R1 .

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All of the above examples are also covered by the theory of [5,9], except that their definition of the norm is slightly different and the fact that the suprema are attained over matrices rather than operators, i.e., the weak∗ -RFD consequences, seem to be new. We address the difference between their definition of the norm and ours in a later remark. We now turn to some examples that are not covered by these other theories. Example 5.8. Let 0 < r < 1 be fixed and let Ar = {z ∈ C: r < |z| < 1} be an annulus. Then this has a rational presentation given by F1 (z) = z and F2 (z) = rz−1 , and the algebra of this presentation is just the Laurent polynomials. Admissible representations of this algebra are given by selecting any invertible operator T satisfying T 1 and T −1 r −1 . Admissible strict representations are given by invertible operators satisfying T < 1 and T −1 < r −1 . The al∞ (A ) is also introduced in [3] where it is called the Douglas–Paulsen gebra that we denote HR r algebra. It is no longer quite so clear that · u = · u0 . However, this algebra with these norms is studied by the first author in [24] and among other results the equality of these norms was shown. Consequently, f R is attained by taking the supremum over matrices T satisfying T < 1 and T −1 < r −1 . The formula for the norm is given by f R 1 if and only if ¯ 1 (z, w) + 1 − r 2 z−1 w −1 K2 (z, w). Im − f (z)f (w)∗ = (1 − zw)K The scalar version of this formula is also shown in [3]. Douglas and the second author showed in [19] that · u is completely boundedly equivalent to the usual supremum norm, but that the two norms are not equal. In fact, they exhibit an explicit function for which the norms are different. Since the norms are equivalent, it fol∞ (A ) = H ∞ (A ) as sets. Badea, Beckermann and Crouzeix [8] show that not only lows that HR r r are the norms equivalent, but that there is a universal constant C, independent of r, such that f ∞ f R Cf ∞ . Example 5.9. Let G be a simply connected domain in C and φ : G → D be a biholomorphic map. Then G = {z ∈ C: |φ(z)| < 1} and Q(G) = {T : σ (T ) ⊆ G and φ(T ) 1} where R = {φ}. In this case the algebra A of the presentation is just the algebra of all polynomials in φ, regarded as a subalgebra of the algebra of analytic functions on G. Thus, an admissible representation of this algebra is defined by choosing an operator B ∈ B(H) that satisfies B 1 and defining π : A → B(H) via π(p(φ)) = p(B), where p is a polynomial. A strict admissible representation is defined similarly by first choosing a strict contraction. In this case, it is immediate that · u = ∞ (G) if and only if f ∈ Hol(G) and · u0 and that f ∈ HR f R = sup f (T ): T ∈ QR (G) < +∞. Our results imply that f R 1 if and only if 1 − f (z)f (w)∗ = 1 − φ(z)φ(w)∗ K1 (z, w). In particular, if we take φ(z) = z−1 z+1 then it maps the half plane H = {z: Re(z) > 0} to the unit disk. For this particular φ, we have that QR = {T : σ (T ) ⊆ H and Re(T ) 0}.


3223

Example 5.10. Similarly, if we let G = {z ∈ CN : |φi (z)| < 1, i = 1, . . . , N } where φi (z) = zi −1 zi +1 , then G is an intersection of half planes and QR consists of all commuting N -tuples of operators, (T1 , . . . , TN ) such that σ (Ti ) ⊆ H and Re(Ti ) 0 for all i. Applying our results, we obtain a factorization result for half planes. These algebras have been studied by D. KalyuzhnyiVerbovetzkii in [22]. Example 5.11. Let G ⊆ C be an open convex set and represent it as an intersection of half planes Hθ . Each half plane can be expressed as {z: |Fθ (z)| < 1} for some family of linear fractional maps. If we let R = {Fθ }, then QR (G) = {T : σ (T ) ⊆ G and Fθ (T ) 1, ∀θ }. Moreover, each inequality Fθ (T ) 1 can be re-written as an operator inequality for the real z part of some translate and rotation of T . For example, when G = D, we may take Fθ (z) = z−2e iθ , iθ for 0 θ < 2π . In this case, one checks that Fθ (T ) 1 if and only if Re(e T ) I . Thus, it follows that Q(D) = T : σ (T ) ⊆ D and w(T ) 1 , ∞ (D) becomes the “universal” operator where w(T ) denotes the numerical radius of T . Thus, HR algebra that one obtains by substituting an operator of numerical radius less than one for the variable z and we have a quite different quantization of the unit disk. Our results give a formula for this norm, but only in terms of R-limits, so further work would need to be done to make it explicit.

Example 5.12. There is a second way that one can quantize many convex sets. Let G = {z: |z − ak | < rk , k ∈ I } ⊆ C be an open, bounded convex set that can be expressed as an intersection of a possibly infinite set of open disks. For example, the open unit square cannot be expressed as such an intersection, but any convex set with a smooth boundary with uniformly bounded curvature can be expressed in such a fashion. Then G has a rational presentation given by Fk (z) = rk−1 (z − ak ), k ∈ I the algebra of the presentation is just the polynomial algebra and an admissible representation is given by selecting any operator T satisfying, T − ak I rk , k ∈ I . Thus, we again a factorization result, but only in terms of R-limits. The above definitions allow one to consider many other examples. For example, one could fix 0 < r < 1 and let G = {z ∈ BN : r < |z1 |}, with rational presentation f1 (z) = (z1 , . . . , zN ) ∈ M1,N , and f2 (z) = rz1−1 . An admissible representation would then correspond to a commuting row contraction with T1 invertible and T1−1 r −1 . We now compare and contrast some of our hypotheses with those of [5,9]. Remark 5.13. Let G = {z ∈ CN : Fk (z) < 1, k = 1, . . . , K} where the Fk ’s are matrix-valued polynomials defined on G. Then for f ∈ Hol(G), [5,9] really study a norm given by f s = sup{f (T )} where the supremum is taken over all commuting N -tuples of operators T with Fk (T ) < 1, for 1 k K. We wish to contrast this norm with our f R . In [5] it is shown that the hypotheses Fk (T ) < 1, k = 1, . . . , K implies that σ (T ) ⊆ G. Thus, we have that f s f R . In fact, we have that f s = f R . This can be seen by the fact that they obtain identical factorization theorems to ours. This can also be seen directly in some cases where the algebra A contains the polynomials and when it can be seen that · R is attained by taking the supremum over matrices (see Remark 4.11). Indeed, if f R is attained as the supremum over

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commuting N -tuples of finite matrices T = (T1 , . . . , TN ) satisfying σ (T ) ⊆ G and Fk (T ) 1 then such an N -tuple of commuting matrices, can be conjugated by a unitary to be simultaneously put in upper triangular form. Now it is easily argued that the strictly upper triangular entries can be shrunk slightly so that one obtains new N -tuples T = (T1, , . . . , TN, ) satisfying, Fk (T ) < 1, k = 1, . . . , K and Ti − Ti, < . But we do not have a simple direct argument that works in all cases. Remark 5.14. We do not know how generally it is the case that · u is a local norm. That is, we do not know if f u = f R for f ∈ Mn (A). In particular, we do not know if this is the case for Example 5.5. In this case, the algebra of the presentation is A = span{zn : n 0, n = 1}. If we write a polynomial p ∈ A in terms of its even and odd decomposition, p = pe + po , then pe (z) = q(z2 ) and po = z3 r(z2 ) for some polynomials q, r. In this case it is easily seen that pu = sup q(A) + Br(A): A 1, B 1, AB = BA, A3 = B 2 , while pL = pR = sup p(T ): T 2 1, T 3 1 . References [1] J. Agler, Some interpolation theorems of Nevanlinna–Pick type, preprint, 1988. [2] J. Agler, On the representation of certain holomorphic function defined on a polydisk, in: Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, in: Oper. Theory Adv. Appl., vol. 48, 1990, pp. 47–66. [3] J. Agler, J. McCarthy, Pick Interpolation and Hilbert Function Spaces, Grad. Stud. Math., vol. 44, American Mathematical Society, Providence, RI, 2002. [4] D. Alpay, A. Dijksma, J. Rovnyak, H. de Snoo, Schur functions, operator colligation, and reproducing Kernel Pontryagin spaces, in: Oper. Theory Adv. Appl., vol. 96, Birkhäuser, Basel, 1997, pp. 1–13. [5] C.G. Ambrozie, D. Timotin, A von Neumann type inequality for certain domains in Cn , Proc. Amer. Math. Soc. 131 (2003) 859–869. [6] R.J. Archbold, On residually finite-dimensional C∗ -algebras, Proc. Amer. Math. Soc. 123 (9) (1995) 2935–2937. [7] W. Arveson, Subalgebras of C ∗ -algebras. III. Multivariable operator theory, Acta Math. 181 (2) (1998) 159–228. [8] C. Badea, B. Beckermann, M. Crouzeix, Intersections of several disks of the Riemann sphere as K-spectral sets, preprint. [9] J.A. Ball, V. Bolotnikov, Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining functions in Cn , J. Funct. Anal. 213 (1) (2004) 45–87. [10] J.A. Ball, T.T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna–Pick interpolation in several variables, J. Funct. Anal. 197 (1998) 1–61. [11] B. Beckermann, M. Crouzeix, A lenticular version of a von Neumann inequality, Arch. Math. 86 (2006) 352–355. [12] B. Blackadar, Operator algebras: Theory of C∗ -algebras and von Neumann algebras, in: Operator Algebras and Non-commutative Geometry III, in: Encyclopaedia Math. Sci., vol. 122, Springer-Verlag, Berlin, 2006. [13] D.P. Blecher, C. Le Merdy, Operator Algebras and Their Modules – an Operator Space Approach, London Math. Soc. Monogr., vol. 30, Oxford University Press, 2004. [14] D.P. Blecher, V.I. Paulsen, Explicit construction of universal operator algebras and an application to polynomial factorization, Proc. Amer. Math. Soc. 112 (1991) 839–850. [15] D.P. Blecher, Z.-J. Ruan, A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990) 188–201. [16] J. Burbea, P. Masani, Banach and Hilbert spaces of vector-valued functions, in: Their General Theory and Applications to Holomorphy, in: Res. Notes Math., vol. 90, Pitman Advanced Publishing Program, Boston/London/ Melbourne, 1984. [17] M. Dadarlat, Residually finite dimensional C∗ algebras, in: Operator Algebras and Operator Theory: International Conference on Operator Algebras and Operator Theory, By Liming Ge Shanghai, China, July 4–9, 1997, pp. 45–50.


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[18] K.R. Davidson, D. Pitts, Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (3) (1998) 321–337. [19] R.G. Douglas, V.I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged) 50 (1– 2) (1986) 143–157. [20] S.W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (3) (1978) 300–304. [21] K.R. Goodearl, P. Menal, Free and residually finite-dimensional C∗ -algebras, J. Funct. Anal. 90 (2) (1990) 391–410. [22] D. Kalyuzhnyi-Verbovetzkii, On the Bessmertnyi class of homogeneous positive holomorphic functions of several variables, in: J.A. Ball, J.W. Helton, M. Klaus, L. Rodman (Eds.), Current Trends in Operator Theory and Its Applications, vol. OT 149, Birkhäuser-Verlag, Basel, 2004, pp. 255–289. [23] S. Lata, M. Mittal, V.I. Paulsen, An operator algebraic proof of Agler’s factorization theorem, Proc. Amer. Math. Soc. 137 (11) (2009) 3741–3748. [24] M. Mittal, PhD thesis, University of Houston. [25] V.I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78, 2002. [26] V.I. Paulsen, Operator algebras of idempotents, J. Funct. Anal. 181 (2001) 209–226. [27] V.I. Paulsen, M. Raghupathi, Representations of logmodular algebras, Trans. Amer. Math. Soc., in press. [28] G.B. Pedrick, Theory of reproducing kernels in Hilbert spaces of vector-valued functions, Univ. of Kansas Tech. Rep., vol. 19, Lawrence, 1957. [29] G. Pisier, Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser., vol. 294, Cambridge University Press, 2003. [30] G. Popescu, Interpolation problems in several variables, J. Math. Anal. Appl. 227 (1) (1998) 227–250 (English summary). [31] J.L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970) 172–191. [32] J.L. Taylor, The analytic functional calculus for several commuting contractions, Acta Math. 125 (1970) 1–38. [33] J.L. Taylor, Analytic Functional Calculus and Spectral Decompositions, Math. Appl., vol. 1, D. Reidel Publishing Co. Dordrecht, Holland, 1982.


Endpoint Strichartz estimates for the magnetic Schrödinger equation Piero D’Ancona a , Luca Fanelli b,∗ , Luis Vega b , Nicola Visciglia c a SAPIENZA – Università di Roma, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185 Roma, Italy b Universidad del Pais Vasco, Departamento de Matemáticas, Apartado 644, 48080, Bilbao, Spain c Universitá di Pisa, Dipartimento di Matematica, Largo B. Pontecorvo 5, 56100 Pisa, Italy

Received 24 January 2009; accepted 4 February 2010 Available online 19 February 2010 Communicated by I. Rodnianski

Abstract We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n 3. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the potentials, while the potentials themselves can be large. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials. © 2010 Elsevier Inc. All rights reserved. Keywords: Strichartz estimates; Dispersive equations; Schrödinger equation; Magnetic potential

1. Introduction Recent research on linear and nonlinear dispersive equations is largely focused on measuring precisely the rate of decay of solutions. Indeed, decay and Strichartz estimates are one of the central tools of the theory, with immediate applications to local and global well posedness, existence of low regularity solutions, and scattering. This point of view includes most fundamental equations of physics like the Schrödinger, Klein–Gordon, wave and Dirac equations. Strichartz estimates appeared in [29]; the basic framework for this study was laid out in the two papers * Corresponding author.

E-mail addresses: [email protected] (P. D’Ancona), [email protected] (L. Fanelli), [email protected] (L. Vega), [email protected] (N. Visciglia). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.007

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P. D’Ancona et al. / Journal of Functional Analysis 258 (2010) 3227–3240

[12,18], which examined in an exhaustive way the case of constant coefficient, unperturbed equations. This leads naturally to the possible extensions to equations perturbed with electromagnetic potentials or with variable coefficients; a general theory of dispersive properties for such equations is still under construction and very actively researched. In the present paper we shall focus on the time dependent Schrödinger equation i∂t u(t, x) = H u(t, x),

u(0, x) = ϕ(x),

x ∈ Rn , n 3,

(1.1)

associated with the electromagnetic Schrödinger operator H := −∇A2 + V (x),

∇A := ∇ − iA(x)

(1.2)

where A = (A1 , . . . , An ) : Rn → Rn , V : Rn → R. We recall that in the unperturbed case A ≡ 0, V ≡ 0, dispersive properties are best expressed in terms of the mixed norms on R1+n Lp Lq := Lp Rt ; Lq Rnx as follows: for every n 3, it e ϕ

p q

Lt Lx

cn ϕL2 ,

provided the couple (p, q) satisfies the admissibility condition n n 2 = − , p 2 q

2 p ∞.

(1.3)

These estimates are usually referred to as Strichartz estimates. Our main goal is to find sufficient conditions on the potentials A, V such that Strichartz estimates are true for the perturbed equation (1.1). In the purely electric case A ≡ 0 the literature is extensive and almost complete; we may cite among many others the papers [1,2,13,14,23]. It is now clear that in this case the decay V (x) ∼ 1/|x|2 is critical for the validity of Strichartz estimates; suitable counterexamples were constructed in [15]. In the magnetic case A = 0, the Coulomb decay |A| ∼ 1/|x| is likely to be critical, however no explicit counterexamples are available at the time. An intense research is ongoing concerning Strichartz estimates for the magnetic Schrödinger equation, see e.g. [5,7,8, 11,21]; see also [22] for a more general class of first order perturbations. Due to the perturbative techniques used in the above mentioned papers, an assumption concerning absence of zero-energy resonances for the perturbed operator H is typically required in order to preserve the dispersion. In the case A ≡ 0 it was shown in [2] how this abstract condition can be dispensed with, by directly proving some weak dispersive estimates (also called Morawetz or smoothing estimates) via multipliers methods. Here we shall give a very short proof of Strichartz estimates for the magnetic Schrödinger equation with potentials of almost Coulomb decay, based uniquely on the weak dispersive estimates proved in [10]. The leading theme is that direct multiplier techniques allow to avoid, under suitable repulsivity conditions on V and nontrapping conditions on A (see also [9]), the presence of nondispersive components, and to preserve Strichartz estimates.


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We begin by introducing some notations. Regarding as usual the potential A as a 1-form, we define the corresponding magnetic field as the 2-form B = dA, which can be identified with the anti-symmetric gradient of A: B ∈ Mn×n ,

B = DA − (DA)t ,

(1.4)

where (DA)ij = ∂i Aj , (DA)tij = (DA)j i . In dimension 3, B is uniquely determined by the vector field curl A via the vector product Bv = curl A × v,

∀v ∈ R3 .

(1.5)

We define the trapping component of B as Bτ (x) =

x B(x); |x|

(1.6)

when n = 3 this reduces to Bτ (x) =

x × curl A(x), |x|

n = 3,

(1.7)

thus we see that Bτ is a tangential vector. The trapping component may be interpreted as an obstruction to the dispersion of solutions; some explicit examples of potentials A with Bτ = 0 in dimension 3 are given in [9,10]. Moreover, by ∂r V = ∇V ·

x , |x|

we denote the radial derivative of V , and we decompose it into its positive and negative part ∂r V = (∂r V )+ − (∂r V )− . The positive part (∂r V )+ also represents an obstruction to dispersion, and indeed we shall require it to be small in a suitable sense. To ensure good spectral properties of the operator we shall also assume that the negative part V− is not too large in the sense of the Kato norm: Definition 1.1. Let n 3. A measurable function V (x) is said to be in the Kato class Kn provided lim sup

r↓0 x∈Rn |x−y|r

|V (y)| dy = 0. |x − y|n−2

We shall usually omit the reference to the space dimension and write simply K instead of Kn . The Kato norm is defined as |V (y)| dy. V K = sup n |x − y|n−2 x∈R

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A last notation we shall need is the radial-tangential norm p f Lp L∞ (S ) r r

∞ :=

sup |f |p dr.

|x|=r

0

In our results we always assume that the operators H and −A := −(∇ −iA)2 are self-adjoint and positive on L2 , in order to ensure the existence of the propagator eitH and of the powers H s via the spectral theorem. There are several sufficient conditions for self-adjointness and positivity, which can be expressed in terms of the local integrability properties of the coefficients (see the standard references [4,19]); here we prefer to leave this as an abstract assumption. Our main result is the following: 1 (Rn \ {0}), assume the operators = −(∇ − iA)2 Theorem 1.1. Let n 3. Given A, V ∈ Cloc A and H = −A + V are self-adjoint and positive on L2 . Moreover assume that n

V− K
0 (M + 12 )2 3 2 1 |x| 2 Bτ 2 ∞ + (2M + 1)|x|2 (∂r V )+ L1 L∞ (S ) < , L L (S ) r r r r M 2

(1.11)

while for n 4 we assume that 2 |x| Bτ (x)2 ∞ + 2|x|3 (∂r V )+ (x) ∞ < 2 (n − 1)(n − 3). L L 3

(1.12)

Then, for any Schrödinger admissible couple (p, q), the following Strichartz estimates hold: itH e ϕ p q Cϕ 2 , L L L

2 n n = − , p 2 q

p 2,

p = 2 if n = 3.

(1.13)

In dimension n = 3, we have the endpoint estimate |D| 12 eitH ϕ

L2 L6

1 H 4 ϕ L2 .

(1.14)


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Remark 1.1. By optimizing condition (1.11) with respect to M, we can rephrase the condition as follows: writing for short 3 2 α = |x| 2 Bτ L2 L∞ (S ) , r

r

β = |x|2 (∂r V )+ L1 L∞ (S ) , r

r

we can rewrite it in the following equivalent form: √ √ √ √ α + α + 2β α + α + 2β √ 1 α+β < . √ 2 2 α + 2β Notice that when Bτ ≡ 0 the condition reduces to β < 1/2, and when (∂r V )+ ≡ 0 the condition reduces to α < 1/4. 1 (Rn \ {0}) is actually Remark 1.2. Let us remark that the regularity assumption A, V ∈ Cloc stronger than what we really require. For the validity of the theorem, we just need to give meaning to inequalities (1.11), (1.12).

Remark 1.3. Assumptions (1.10), (1.11), and (1.12) imply the weak dispersion of the propagator eitH (see Theorems 1.9, 1.10, assumptions (1.24), and (1.27) in [10]). Actually assumption (1.24) in [10] seems to be stronger than (1.11), but reading carefully the proof of Theorem 1.9 in [10] it is clear that the real assumption is our (1.11) (see inequality (3.14) in [10]). The strict inequality in (1.11), (1.12) is essential, in order to dispose of the weighted L2 -estimate in the above mentioned Theorems by [10] (see also inequality (3.5) below). Remark 1.4. We recall that, usually, suitable spectral conditions must be required for the dispersion to hold, see e.g. [7,8] where resonances at zero are excluded. In our case, such conditions are implied by the smallness assumptions (1.11), (1.12) which can be checked easily in concrete examples. The derivation of Strichartz estimates from the weak dispersive ones turns out to be remarkably simple if working on the half derivative |D|1/2 u, see Section 3 for details. As a drawback, the final estimates are expressed in terms of fractional Sobolev spaces generated by the perturbed magnetic operator −A . Thus, in order to revert to standard Strichartz norms as in (1.13), we need suitable bounds for the perturbed Sobolev norms in terms of the standard ones. This is provided by the following theorem, which we think is of independent interest. Theorem 1.2. Let n 3. Given A ∈ L2loc (Rn ; Rn ), V : Rn → R, assume the operators A = −(∇ − iA)2 and H = −A + V are self-adjoint and positive on L2 . Moreover, assume that V+ is of Kato class, V− satisfies n

V− K
p − 1 was given without proof by Berezin for classical domains in [4]. H. Upmeier and A. Unterberger extended it to all domains and gave a Jordan theoretic proof [24]. The second problem was solved (for the same parameters) by G. Zhang and (independently) by G. van Dijk and M. Pevzner [30,26], and also by Y. Neretin for classical groups [15]. Those two problems are in fact similar. The restriction map from D to D (resp. from D × D to D) gives rise to the Berezin transform on D (resp. D), which is a kernel operator. The solution then consists in computing the spectral symbol of the Berezin transform, or, if one prefers, in computing the spherical Fourier transform of the Berezin kernel. In [29] and in [26, Section 5] the problem of decompos π α+l where l ∈ N is also solved, by the same method. A similar problem is also studied ing πα ⊗ in [6]. For arbitrary parameters, those problems are more complicated, and no general method seems to apply. In [18] the tensor product problem for G = SU(2, 2) is solved for any parameter. In [31] π d is decomposed for any G (π d is called the minimal holomorphic the representation π d ⊗ 2 2 2 representation). In [17], Y. Neretin solves the restriction problem from U (r, s) to O(r, s) (r s) for any parameter by analytic continuation of the result for large parameters. If r = s the support of the Plancherel formula remains the same for all α > r − 1 (here d = 2) but when s − r is sufficiently large new pieces appear when α crosses p − 1 = 2(r + s) − 1 and the situation gets worse as α approaches to (r − 1) d2 , as he had already explained in [16]. For points in the discrete Wallach set, the situation is not clear. In his thesis the second author manages to decompose the restriction of SO(2, n) to SO(1, n) for any parameter [22], as well as the restriction of the minimal holomorphic representation of SU(p, q) to SO(p, q) [21], and the minimal holomorphic representation of Sp(n, R) (resp. SU(n, n)) to GL+ (n, R) (resp. to GL(n, C)) [23]. Assume that D is of tube type, i.e. that D is biholomorphic to the tube domain TΩ over the symmetric cone Ω. Then the inverse image of Ω is a real bounded symmetric domain. In this paper, generalising [23], we establish, for any parameter in the discrete Wallach set, the branching rule for the restriction of the associated representation of G = G0 (TΩ ) to G = GL0 (Ω). We use the model by Rossi and Vergne which realises the representation given by the l-th point in the Wallach set as L2 (∂l Ω, μl ), where ∂l Ω is the set of positive semidefinite elements in ∂Ω of rank l, and μl is a relatively G-invariant measure on ∂l Ω. A key observation is that for any x in ∂l Ω, the function g → ν (g ∗ x) on G, where ν is the power function of the Jordan algebra (cf. (2) for the definition), transforms like a function in a certain parabolically induced representation. A naive approach to construct an intertwining operator from L2 (∂l Ω, μl ) into a direct

S. Merigon, H. Seppänen / Journal of Functional Analysis 258 (2010) 3241–3265

3243

sum of parabolically induced representations would then to weight the functions above by compactly supported smooth functions, i.e., to consider mappings f → ∂l Ω f (x)ν (g ∗ x) dμl (x), for f in C0∞ (∂l Ω). It will become clear that this approach is in fact fruitful. However, there are two problems that have to be dealt with. First of all, it is not obvious that the natural target spaces are unitarisable. Secondly, and more importantly, the integrals above need not converge for the suitable choice of parameters ν. However, as we shall see, both these problems can be solved. The paper is organised as follows. In Section 2 we recall some facts about Jordan algebras and symmetric cones that will be needed in the paper. In Section 3 we prove an identity between the restriction of a spherical function of the cone Ω to a cone of lower rank in its boundary and the corresponding spherical function of the lower rank cone. In Section 4 we define a class of irreducible unitary spherical representations that provides target spaces for the integral operators discussed above. These are constructed using the Levi decomposition of the group G by twisting parabolically induced unitary representations of the semisimple factor of G by a certain character. In Section 5 we construct the intertwining operator as an analytic continuation of the integral operator above. After this has been taken care of, a polar decomposition for the measure μl due to J. Arazy and H. Upmeier [1] will allow to express the restriction of the intertwining operator to K-invariant vectors in terms of the Fourier transform for a cone of rank l. Using this identification, the inversion formula for the Fourier transform can be used to prove the Plancherel theorem for the branching problem. In Appendix A we provide a framework for certain restrictions of distributions to submanifolds which will be useful for giving an analytic continuation for the integral that should give an intertwining operator. It should be pointed out that the standard theory for restricting distributions (e.g. [8, Corollary 8.2.7]) does not apply to our situation since the condition on the wave front set for the distribution is not satisfied. Instead we have to use restrictions based on extensions of test functions in such a way that they are constant in certain directions from the submanifold (cf. Appendix A). We finally want to mention that branching problems related to holomorphic involutions of D have also been studied in [20,11,2,19]. 2. Jordan theoretic preliminaries Let V be a Euclidean Jordan algebra. It is a commutative real algebra with unit element e such that the multiplication operator L(x) satisfies [L(x), L(x 2 )] = 0, and provided with a scalar product for which L(x) is symmetric. An element is invertible if its quadratic representation P (x) = 2L(x)2 − L(x 2 ) is so. The cone of invertible squares to be denoted Ω is a symmetric cone: it is homogeneous under the identity component, G, of the Lie group GL(Ω) = {g ∈ GL(V ) | gΩ = Ω}, and it is self-dual. It follows that the involution θ (g) := (g ∗ )−1 (where g ∗ is the adjoint of g with respect to the scalar product of V ) preserves G (which is hence reductive). The stabiliser K = Ge of e coincides with the identity component of the group Aut(V ) of automorphisms of V and with the fixed points of G under the involution θ , and hence is compact. Thus Ω is a Riemannian symmetric space. The tube TΩ = V ⊕ iΩ over Ω in the complexification of V is a Hermitian symmetric space of the non-compact type, diffeomorphic via the Cayley transform to a (tube type) bounded symmetric domain. Any element of GL(Ω), when extended complex-linearly, preserves TΩ . In this fashion G is seen as a subgroup of the identity component G of the group of biholomorphisms of TΩ .

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We assume that V is simple. Then there exists a positive integer r, called the rank of V , such that any family of mutually orthogonal minimal idempotents has r elements. Such a family is called a Jordan frame. Let g=k⊕p be the Cartan decomposition of the Lie algebra g of G. Then the map x → L(x) yields an isomorphism V → p. The subspace generated by a Jordan frame (more precisely by the associated multiplication operators) is a maximal abelian subspace of p and conversely, any maximal abelian subspace of p determines (up to permutation) a Jordan frame. From now we fix a choice of a Jordan frame (c1 , . . . , cr ) and let a = L(cj ), j = 1, . . . , r and A = exp a. Any x in V can be written x=k

λj cj ,

1j r

where k ∈ K and the λj are real numbers, and the family (λ1 , . . . , λr ) is unique up to permutation (its members are called the eigenvalues of x). Then x belongs to Ω if and only if for all 1 j r, λj > 0, and this spectral decomposition corresponds to the KAK decomposition of G, the Acomponent in the decomposition being unique up to conjugation by an element of the Weyl group W = Sr of G (cf. [10, Chapter VII.3]). The rank of x is defined to be the number of its nonzero eigenvalues. There exist on V a K-invariant polynomial function (x) (the determinant) and a K-invariant linear function tr(x) (the trace) that satisfy (x) =

r

λj

and

tr(x) =

j =1

r

λj .

j =1

The determinant defines the character (g) := (ge)

(1)

of the group G. A Jordan frame gives rise to the Peirce decomposition. Since multiplications by orthogonal idempotents commute, the space V decomposes into a direct sum of joint eigenspaces for the (symmetric) operators (L(cj ))j =1,...,r . The eigenvalues of L(c) when c is an idempotent, belong to {0, 12 , 1}. Let us denote by V (c, α) the eigenspace corresponding to the value α. The decomposition into joint eigenspaces is then given by V=

1ij r

where

Vij ,


Vii = V (ci , 1) ∩

3245

V (cj , 0),

j =i

and when i = j ,

1 1 Vij = V ci , V (cj , 0). ∩ V cj , ∩ 2 2 k ∈{i,j / }

We have Vii = Rci and the Vij all have the same dimension d, called the degree of the Jordan algebra. We can now describe the roots of (g, a). Let (δj )j =1,...,r be the dual basis of (L(cj ))j =1,...,r in a∗ . Then the roots are αij± = ±

δj − δi , 2

1 i < j r,

and the corresponding root spaces are g+ ij = {a ei | a ∈ Vij }, g− ij = {a ej | a ∈ Vij }, where x y = L(L(x)y) + [L(x), L(y)]. Let N be the nilpotent subgroup

N = exp

g+ ij .

1i<j r

Then G has the Iwasawa decomposition G = N AK. For any idempotent c, the projection on V (1, c) is P (c), and V (1, c) is a Jordan subalgebra, hence a Euclidean Jordan algebra with neutral element c (note that it is simple with rank the one of c). We denote by Ω1 (c) its symmetric cone. In particular for ej =

l

ck ,

k=1

we set V (l) = V (1, el )

and Ω (l) = Ω1 (el ),

and also note G(l) the identity component of G(Ω (l) ), K (l) = Gel and (l) the determinant of V (l) . The principal minors of V are then defined by the formula

(j ) (x) := (j ) P (ej )(x) . Then x is in Ω if and only if for all 1 j r, (j ) (x) > 0. Let ν ∈ Cr and set for x in Ω, ν −ν3

2 1 −ν2 (x)(2) ν (x) = ν(1)

ν

−ν

r r−1 r (x) . . . (r−1) (x)ν(r) (x).

(2)

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∗ . Then if a(g) is the projection of Using the basis (δj ), we can identify ν with an element of aC g on A in the Iwasawa decomposition,

ν (gx) = eν log a(g) ν (x).

(3)

The action of G on the boundary ∂Ω of Ω has r − 1 orbits, which may be parametrised by the rank of their elements. We denote by ∂l Ω the orbit of rank l elements, i.e., ∂l Ω = Gel . There exists on ∂l Ω a unique (up to constant) relatively G-invariant measure μl , which transforms according to ld

dμl (gx) = 2 (g) dμl (x). The Hilbert space associated to the Wallach point l d2 is, up to renormalisation, isometric to L2 (∂l Ω, μl ) [5, Theorem X.III.4], and the representation of G in this picture is then given by ld π l (g)f = 4 (g)f g ∗ · .

(4)

The measures μl were constructed by M. Lassalle [14] and can also be obtain as Riesz distributions, thanks to S. Gindikin’s theorem [5, VII.3]. A major tool for our purpose will be the Arazy–Upmeier polar decomposition of μl [1, Theorem 2.7]. Let Πl = K.el be the set of idempotents of rank l. Then ∂l Ω is the disjoint union ∂l Ω =

Ω(u).

u∈Πl

Since elements of G permute the faces of Ω (which are of the form Ω(u) for idempotents u), an action is induced on Πl , such that the preceding equality defines a G-equivariant fibration ∂l Ω → Πl . For any f in the space C0∞ (∂l Ω) of smooth functions with compact support on ∂l Ω,

f dμl = ∂l Ω

dk

K

rd

2 (l) (x)f (kx) d∗(l) x,

(5)

Ω (l) (l)

where the Haar measure on K is normalised and d∗ x is the unique (up to constant) G(l) -invariant measure on Ω (l) . The set ∂l Ω is not a submanifold of V . However let Vl be the (open) set of elements in V with rank greater or equal than l. To any l-element subset Il ⊂ {1, . . . , r} one can associate the idempotent eIl = j ∈Il cj and the minor Il (x) := (P (eIl )x + e − eIl ). Then


Vl =

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x ∈ V Il (x) = 0 ,

Il ⊂{1,...,r}

and [14, Propositions 3 and 7] show that ∂l Ω is a (closed) submanifold of Vl . 3. An identity between spherical functions The spherical functions on Ω may be defined for ν in a∗C by the formula Φν (x) =

ν (kx) dk. K

When ν satisfies ν1 · · · νr 0, a property that we will denote by ν 0, the generalised power function ν and the spherical function Φν extend continuously to Ω. Now let al = L(cj ), j = 1, . . . , l

(6)

for 1 l r − 1 and assume that ν belongs to (al C )∗ , i.e. that its (r − l) last coordinates van(l) ish. Then ν also defines a spherical function Φν of Ω (l) . Let α be a real number. When ν appears in the argument of an object related to V (l) , we will use the convention that ν + α := (ν1 + α, . . . , νl + α). Recall that Ω (l) ⊂ ∂l Ω. Theorem 3.1. Let ν in al ∗C such that ν 0. Then for all x in Ω (l) , Φν (x) = γν(l) Φν(l) (x),

where γν(l) =

ΓΩ (l) ( rd 2 )ΓΩ (l) (ν + ΓΩ (l) ( ld2 )ΓΩ (l) (ν +

ld 2) . rd 2 )

Here ΓΩ (l) is the Gindikin Gamma function for the cone Ω (l) , ΓΩ (l) (ν) = (2π)

l(l−1)d 4

l j =1

d Γ νj − (j − 1) . 2

The theorem is proved in the case ν ∈ Nl in [1, Proposition 3.7]. We use this result and the following lemma, which is based on Blaschke’s theorem (see [12, Lemma A.1] for a detailed proof). Lemma 3.2. Let f be a holomorphic function defined on the right half-plane {z ∈ C | z > 0}. If f is bounded and f (n) = 0 for n ∈ N, then f is identically zero. Proof of Theorem 3.1. Let us set zj = νj − νj +1 , j = 1, . . . , l − 1, zl = νl , so that zj 0 and νj = lk=j zk . Let x ∈ Ω (l) and let (l)

(l)

F (z1 , . . . , zl ) = Φν(z1 ,...,zl ) (x) − γν(z1 ,...,zl ) Φν(z1 ,...,zl ) (x).

3248


Let us fix zj = mj ∈ N, j = 2, . . . , l. If b a > 0, one can see by Stirling’s formula that is bounded on the right half-plane. It follows that the function z →

Γ (z + Γ (z +

l

k=2 mk

+

k=2 mk

+

l

Γ (z+a) Γ (z+b)

ld 2) rd 2 )

(l) is bounded and hence also z → γν(z,m . Now 2 ,...,ml )

Φν(z,m

2 ,...,ml ) (x)

z (kx)m2 (kx) . . . ml (kx) dk (2) (l) (1)

K

(z) 2

ml sup m (kx) . . . (kx) sup (kx) , (1) (2) (l) K

K

and (z) (l) m2

ml Φ sup (1) (kx) . ν(z,m2 ,...,ml ) (x) sup (2) (kx) . . . (l) (kx) K (l)

K (l)

Let δ > supK (1) (kx) supK (l) (1) (kx). Then the holomorphic function f (z) = F (z, m2 , . . . , ml )δ −z is bounded and vanishes on N, hence on the right half-plane, i.e., for every z ∈ C with z > 0 and mj ∈ N, F (z, m2 , . . . , ml ) = 0. By the same argument one shows that for every z1 ∈ C with z1 > 0 and mj ∈ N, the map z → F (z1 , z, m3 , . . . , ml ) vanishes identically, and the proof follows by induction. 2 4. A series of spherical unitary representations In this section we introduce a family of spherical unitary representations that will occur in the decomposition of L2 (∂l Ω) under the action of G. For 1 l r − 1 let nl =

g− ij .

li<j

Note that it is a (nilpotent) Lie algebra and that (cf. (6)) al ⊕ RL(e) =

ker αij± .

li<j

The closed subgroup ZG (al ) = ZG (al ⊕ RL(e)) normalises N l = exp nl hence Ql = ZG (al )N l


3249

is a subgroup of G. Moreover ZG (al ) ∩ N l = {id} so this decomposition is a semidirect product. The Lie algebra of Ql , ql = zg (al ) ⊕ nl = m ⊕ a ⊕

g± ij ⊕

l 0. 2 We now recall, in order to fix the notations, the definition of the spherical Fourier transform on Ω (l) . If f is a continuous function with compact support on Ω (l) which is K (l) -invariant, its spherical Fourier transform is fˆ(ν) =

(l)

f (x)Φ−ν+ρ (l) (x) d∗(l) x,

Ω (l)

where ν ∈ (al C )∗ and ρ (l) = d4 lj =1 (2j − l − 1)δj . Since f has compact support, the function fˆ is holomorphic on (al C )∗ . For latter use we also recall the inversion formula for (l) f ∈ C0∞ (Ω (l) )K and λ ∈ a∗l (cf. [5, Theorem XIV.5.3] and [7, Chapter III, Theorem 7.4]): (l)

f (x) = c0

a∗l

(l) fˆ(iλ)Φiλ+ρ (l) (x)

dλ , |c(l) (λ)|2

(13)

3256


where dλ is the Lebesgue measure on a∗l Rl , c(l) (λ) is the Harish-Chandra c-function for Ω (l) , and c0(l) is a positive constant. Now let f ∈ C0∞ (∂l Ω)K , and observe that Ω (l) , being a fibre of ∂l Ω → Πl , is closed in ∂l Ω, and hence f |Ω (l) (sometimes still denoted by f ) has compact support in Ω (l) . Moreover, since any k ∈ K (l) extends to an element of K, the function f |Ω (l) is K (l) -invariant. Proposition 5.3. If f ∈ C0∞ (∂l Ω) is K-invariant then (cf. Theorem 3.1)

rd (l) ˆ Tν f (g) = γ−(iν+ρ −(iν+ρ l ) g ∗ e . f iν − ) l 4

(14)

Proof. Let us set ν = −(iν + ρ l ) in the following. Again, by analytic continuation, it suffices to prove the equality for (ν ) 0. Since f and μl are K-invariant one has for all k in K, Tν f (g) =

f k −1 x ν g ∗ x dμl (x) =

∂Ωl

f (x)ν g ∗ kx dμl (x).

∂Ωl

Hence Tν f (g) =

Tν f (g) dk =

K

∗ f (x) ν g kx dk dμl (x).

∂Ωl

K

Writing g ∗ = th, t ∈ N A, h ∈ K, we have ν (thkx) = ν (hkx)ν (te) = ν (hkx)ν g ∗ e , and using the left invariance of the Haar measure of K,

ν g ∗ kx dk =

K

ν (hkx) dk ν g ∗ e

K

=

ν (kx) dk ν g ∗ e

K

= Φν (x)ν g ∗ e , hence, Tν f (g) =

f (x)Φν (x) dμl (x) ν g ∗ e .

Ω (l)

Now Upmeier and Arazy’s polar decomposition (5) for μl yields Tν f (g) = Ω (l)

rd 2 f (x)Φν (x)(l) (x) d∗(l) x ν g ∗ e ,


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and by Theorem 3.1, (l) Tν f (g) = γν

rd (l) 2 f (x)Φν (x)(l) (x) d∗(l) x ν g ∗ e

Ω (l)

= γν(l)

f (x)Φ (l)

ν + rd 2

Ω (l)

ˆ = γν(l) f Since −ρ l + ρ (l) −

rd 2

−ν + ρ

(x) d∗(l) x ν g ∗ e

rd ν g ∗ e . − 2

(l)

= − rd 4 , we eventually get

(l) Tν f (g) = γ−(iν+ρ ) fˆ l

rd −(iν+ρ l ) g ∗ e . iν − 4

2

In the following we set,

rd (l) ˆ , f (ν) = γ−(iν+ρ ) f iν − l 4 and we note that it defines a meromorphic function whose poles are those of ΓΩ (−(iν +ρ l )+ ld2 ). The following lemma will be used in the proof of the Plancherel formula. Lemma 5.4 (Inversion formula). Let f ∈ C0∞ (∂l Ω)K and let λ ∈ a∗l . Then for x ∈ Ω (l) , (l) f (x) = c0

a∗l

(l) f(λ)Φ

iλ+ρ (l) − rd 4

−1 (l) (x) γ−(iλ+ρ ) l

dλ |c(l) (λ)|2

.

Proof. We have

rd (l)

−1 rd 4 = f(λ) γ−(iλ+ρ ) , f (l) (iλ) = fˆ iλ − l 4 rd 4 hence the inversion formula (13) applied to the function f (l) gives the desired identity.

(15) 2

Recall the notations from the end of Section 4. Let p be the measure on a∗l /Wl defined by (l)

dp(λ) =

c0

(l) (l) 2 |γ−(iλ+ρ ) c (λ)|

dλ.

(16)

l

Consider the direct integral Hilbert spaces a∗ Hλ dp(λ) with inner product ((vλ ), (wλ )) := l a∗l vλ , wλ λ dp(λ), where ·,·λ denotes the inner product on Hλ . We now state the main result of the article.

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Theorem 5.5 (Plancherel theorem). There exists an isomorphism of unitary representations

T : π l , L2 (∂l Ω)

πλ dp(λ),

a∗l

Hλ dp(λ) ,

a∗l

such that for every f ∈ C0∞ (∂l Ω), (Tf )λ = Tλ f . Proof. First we prove that for any K-invariant function f in C0∞ (∂l Ω),

f (x)2 dμl (x) =

f(λ)2 dp(λ).

(17)

a∗l

∂l Ω

For this purpose we use the polar decomposition (5) for μl and the inversion formula of Lemma 5.4. Then f (x)2 dμl (x) ∂l Ω

=

rd f (x)2 2 (x) d (l) x ∗

(l)

Ω (l)

=

rd

(l)

(l) f(λ)Φ

2 f (x)(l) (x)c0

a∗l

Ω (l)

=

(l) f(λ) γ−(iλ+ρ ) −1

= a∗l

= a∗l

(l)

(x) γ−(iλ+ρ ) −1 l

dλ |c(l) (λ)|2

rd

(l)

4 f (x)(l) (x)Φ−iλ+ρ (l) (x) d∗(l) x

l

a∗l

iλ+ρ (l) − rd 4

Ω (l)

d∗(l) x

(l)

c0 dλ |c(l) (λ)|2

(l) rd

(l) c0 dλ 4 (iλ) (l) f(λ) γ−(iλ+ρ ) −1 f (l) l |c (λ)|2

f(λ)2

(l)

c0 dλ (l)

|γ−(iλ+ρ ) c(λ)|2

.

l

In the last equality we have used again the formula (15). The next step is to prove that for a dense subset of functions f in C0∞ (∂l Ω), the identity ∂l Ω

f (x)2 dμl (x) =

|Tλ f |2λ dp(λ) a∗l

holds. Recall that L1 (G) is a Banach ∗-algebra when equipped with convolution as multiplication, and ϕ ∗ (g) := ϕ(g −1 ). Let L1 (G)# denote the commutative closed subalgebra of left and right


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K-invariant functions in L1 (G) (recall that (G, K) is a Gelfand pair since it is a Riemannian symmetric pair, cf. [7, Chapter IV, Theorem 3.1]). There is a natural projection L1 (G) → L1 (G)# ,

ϕ k1−1 · k2 dk1 dk2 .

ϕ → ϕ := #

K K

For a unitary representation (τ, H ) of G, there is a ∗-representation (also denoted by τ ) of L1 (G) on H given by τ (ϕ)v :=

ϕ(g)τ (g)v dg,

v∈H .

G

The representations of K and L1 (G) are related by

τ (k1 )τ (ϕ)τ (k2 ) = τ ϕ k1−1 · k2−1 ,

ϕ ∈ L1 (G), k1 , k2 ∈ K.

(18)

The subspace H K of K-invariants is invariant under L1 (G)# . From (18), it follows that for any ϕ ∈ L1 (G), and u, v ∈ H K ,

τ (ϕ)u, v = τ ϕ # u, v .

(19)

Let ξ be the K-invariant cyclic vector in L2 (∂l Ω). We claim that there exists a sequence ∞ K 2 {ξn }∞ n=1 ⊆ C0 (∂l Ω) , such that ξn → ξ in L (∂l Ω). To see this, we can first choose a se∞ ∞ quence {ζn }n=1 ⊆ C0 (∂l Ω) that converges to ξ . Next, observe that the orthogonal projection P : L2 (∂l Ω) → L2 (∂l Ω)# is given by f → K f (k −1 ·) dk. Then P (f ) is smooth if f is smooth. Moreover, supp f is contained in the image of the map K × supp f → ∂l Ω, (k, x) → kx. It follows that P (C0∞ (∂l Ω)) ⊆ C0∞ (∂l Ω)K . Hence, the claim holds with ξn := P (ζn ). The subspace H0 := π l (f )ξn f ∈ C0∞ (G), n ∈ N is then dense in L2 (∂l Ω). For ϕ ∈ C0∞ (G), n ∈ N, we have, by (19) and (17),

π l (ϕ)ξn , π l (ϕ)ξn

L2 (∂l Ω)

= π l ϕ ∗ ∗ ϕ ξn , ξn L2 (∂ Ω) l l ∗

# = π ϕ ∗ ϕ ξn , ξn L2 (∂ Ω) l l ∗

# Tλ π ϕ ∗ ϕ ξn , Tλ (ξn ) λ dp(λ) = a∗l

=

# πλ ϕ ∗ ∗ ϕ Tλ (ξn ), Tλ (ξn ) λ dp(λ)

a∗l

= a∗l

πλ ϕ ∗ ∗ ϕ Tλ (ξn ), Tλ (ξn ) λ dp(λ)

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=

πλ (ϕ)Tλ (ξn ), π(ϕ)Tλ (ξn ) λ dp(λ)

a∗l

=

Tλ π l (ϕ)ξn , Tλ π l (ϕ)ξn λ dp(λ).

a∗l

Hence, the operator T defined on H0 by T (π l (ϕ)ξn ) = (πλ (ϕ)Tλ (ξn )) extends uniquely to a G-equivariant isometric operator 2 T : L (∂l Ω) → Hλ dp(λ). a∗l

It now only remains to prove the surjectivity of T . Assume therefore that there exists a vector (ηλ ) which is orthogonal to the image of T . Then for all ϕ in L1 (G) and h ∈ L1 (G)# , πλ (ϕ ∗ h)(T ξ )λ , ηλ λ dp(λ) = 0, a∗l

i.e.,

ˇ h(λ) πλ (ϕ)(T ξ )λ , ηλ λ dp(λ) = 0,

a∗l

ˇ where h(λ) is the Gelfand transform of h restricted to a∗l . Recall that the character space of L1 (G)# can be parametrised by the quotient of a subset of (aC )∗ modulo the Weyl group (cf. [7, Chapter IV, Theorems 3.3 and 4.3]). Since the image of L1 (G)# under the Gelfand transform separates points in this space, it follows from the Stone–Weierstrass theorem that the functions hˇ are dense in the space of continuous functions on a∗l which are invariant under the action of Wl . Hence πλ (f )(T ξ )λ , ηλ λ = 0 p-almost everywhere. By separability of L1 (G), there is a set U with p(a∗l \ U ) = 0 such that for all f in L1 (G) and λ ∈ U , πλ (f )(T ξ )λ , ηλ λ = 0. By cyclicity of (T ξ )λ (note that (T ξ )λ is nonzero p-almost everywhere), ηλ is zero p-almost everywhere. 2 Remark 5.6. We want to point out that it is actually not necessary to prove the analytic continuation of Tν (and hence to use the theory of Riesz distributions) to derive the decomposition of π l (however, the natural operator T above is then replaced by an abstract one). Indeed, by the Cartan–Helgason theorem [7, Chapter III, Lemma 3.6] we have HλK = Cvλ when λ ∈ a∗l , and hence we can set f → f(λ)vλ , and by (17) we thus obtain an operator T : L2 (∂l Ω)K → a∗ HλK dp(λ). Assume that we can l prove that T intertwines the respective actions of C0∞ (G)# on L2 (∂l Ω)K and a∗ HλK dp(λ). Tλ : C0∞ (∂l Ω)K → HλK ,

Then for ϕ ∈ C0∞ (G)# ,

l


π (ϕ)ξ, ξ = T π (ϕ)ξ, T ξ = l

l

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πλ (ϕ)(T ξ )λ , (T ξ )λ λ dp(λ)

a∗l

2 ϕ(λ) ˇ (T ξ )λ λ dp(λ),

= a∗l

ˇ where ϕˇ is defined by πλ (ϕ)vλ = ϕ(λ)v λ . Here ξ is again a K-invariant cyclic vector in L2 (∂l Ω)K . The proof of [22, Theorem 10] shows that the decomposition of π l then follows. We now prove the intertwining property. It is equivalent to the equality l (ϕ)f (λ) = f(λ)ϕ(λ). π ˇ

(20)

Let ν ∈ Cl . Then for f ∈ C0∞ (∂l Ω) and ϕ ∈ C0∞ (G)# we have, with ν = −(iν + ρ l ),

πν (ϕ) ν (·)∗ e ⊗ 1 (g) =

−1 ld ϕ(h)ν g ∗ h∗ e ⊗ − 4 (he) dh

G

=

− ld4

ϕ(h) G

=

K

∗ −1

− ld4 ϕ(h) (he) ν k h e dk dh ν g ∗ e ⊗ 1

G

=

∗ ∗ −1 (he) ν g k h e dk ⊗ 1 dh

K

−1

ld ϕ(h)− 4 (he)Φν h∗ e dh ν g ∗ e ⊗ 1 ,

G

i.e., πν (ϕ)vν = ϕ(ν)v ˇ ν, where ϕ(ν) ˇ is holomorphic on Cl . If (−(iν + ρ l )) 0, then, by Lemma 5.1, the operator Tν intertwines the actions of C0∞ (G)# , and hence l (ϕ)f (ν) = f(ν)ϕ(ν). π ˇ

Thus (20) follows by analytic continuation. Acknowledgment The authors would like to thank Karl-Hermann Neeb for enlightening discussions and comments that led to substantial improvement of the presentation.

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Appendix A. Restrictions of distributions Let X be a smooth n-dimensional manifold. Let D(X) denote the space of compactly supported smooth functions on X, i.e., the test functions on X. For any chart (V , φ), compact subset := φ(V ), and N ∈ N, consider the seminorm K ⊆V pV ,K,N (f ) :=

supx∈K D α f ◦ φ −1 (x)

(A.1)

α∈Nn , |α|N

on the space of smooth functions on X. Here D α :=

∂ |α| α ∂x1 1 ···∂xnαn

for α = (α1 , . . . , αn ), and |α| =

|α1 | + · · · + |αn |. For a compact set K ⊆ X, let D(K) be the space of smooth functions with support in K equipped with the topology induced by the above seminorms. We recall that a distribution on X is a continuous functional on D(X) = K D(K) equipped with the inductive limit topology. We let D (X) denote the space of distributions on X. Let {Ui } be an open cover of X. Then a linear functional on D(X) is continuous if and only if its restriction to every D(Ui ) is continuous. We will now construct restrictions to a closed submanifold Y of distributions that have support on Y . To have a well-defined notion of restriction one cannot permit arbitrary extensions to X of test functions on Y . Instead, we will require the extension to be locally constant along some predescribed direction. This can be made precise using tubular neighbourhoods. Definition A.1. Let X be a smooth n-dimensional manifold, and let Y be a k-dimensional submanifold. A tubular neighbourhood of Y in X consists of a smooth vector bundle π : E → Y , an open neighbourhood Z of the image, ζE (Y ), of the zero section in E, and a diffeomorphism f : Z → O onto an open set O ⊆ X containing Y , such that the diagram Z f ζE j

Y

X

commutes. Here j : Y → X is the inclusion map. Remark A.2. Any closed submanifold Y of X admits a tubular neighbourhood (cf. [13, Chapter IV, F, Theorem 9]). Any splitting of the tangent bundle of X over Y , T (X)|Y = T (Y ) ⊕ E, gives such a vector bundle E. In particular, given a Riemannian metric on X, E can be chosen as the orthogonal complement to T (Y ) in T (X)|Y . Since the concepts we are dealing with are of a local nature we can without loss of generality assume that X = Z itself is a tubular neighbourhood of Y . Definition A.3. A function f on X is said to be locally vertically constant around Y , l.v.c., if for any x ∈ Y , there exists an open neighbourhood Wx of x in X such that for y ∈ Wx , f (y) = f (π(y)). Moreover, if g is a function on Y , and f |Y = g, f is called an l.v.c. extension of g.


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Lemma A.4. (i) Any test function ϕ on Y admits an l.v.c. extension ϕ ∈ D(X). (ii) An l.v.c. function f that vanishes on Y vanishes on some neighbourhood of Y . Proof. Since supp ϕ is compact, it can be covered by finitely many open neighbourhoods O1 , . . . , ON diffeomorphic to products Ui × Vi ⊂ Rk ⊕ Rn−k , where Ui is open in Rk and Vi is to the projection onto the an open neighbourhood of 0 in Rn−k , in such a way that π corresponds π(O first coordinate. Let (ψi ) be a smooth partition of unity on U = N i ) subordinate to the i=1 cover π(Oi ), i = 1, . . . , N , and let ϕi = ϕψi . Then each ϕi can be identified with a test function on Ui , and by multiplying this with a test functionon Vi which is 1 on some neighbourhood of 0, ϕ= ϕi is an l.v.c. extension of ϕ. This proves (i). we obtain an l.v.c. extension ϕi of ϕi . Then For (ii) just observe that if f is an l.v.c. function that vanishes on Y , then Y is in the complement of the support of f . 2 Assume now that u ∈ D (X) has support on the submanifold Y . Then l.v.c. test functions that vanish on a neighbourhood of Y are in the kernel of u, and the preceding lemma enables us to make the following definition. Definition A.5. Let u ∈ D (X) be a distribution with support on Y . The vertical restriction, u|Y , of u to Y is the unique distribution on Y that satisfies u|Y (ϕ|Y ) = u(ϕ), for any ϕ ∈ D(X) which is l.v.c. around Y . To see that the functional u|Y really is a distribution on Y , it suffices to verify the continuity for test functions with support in trivialising open sets. In this case the verification is straightforward using the l.v.c. extension from the proof of Lemma A.4. Remark A.6. Note that the vertical restriction depends on the choice of tubular neighbourhood, i.e., on a choice of complement E in the splitting of vector bundles in Remark A.2. However, when u is a measure on X with support on Y , the vertical restriction u|Y is u itself, now viewed as a distribution on Y . We now consider holomorphic families of distributions and their properties under restriction. Definition A.7. Let Ω ⊆ Cm be an open set, and let {uz }z∈Ω be a family of distributions on a smooth manifold X. Then this family is called a holomorphic family of distributions if the map z → uz (ϕ) is holomorphic on Ω for every ϕ ∈ D(X). Remark A.8. It follows immediately from Definition A.5 that if {uz }z∈Ω is a holomorphic family of distributions with support on Y , then the family {uz |Y }z∈Ω is a holomorphic family of distributions on Y . Proposition A.9. Let Ω ⊆ Cn be open and connected, and let {uz }z∈Ω be a holomorphic family of distributions on X with support on Y . Assume that there exists an open subset U ⊆ Ω, such

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that uz is a measure with support on Y for z ∈ U . Then the whole family {uz |Y }z∈Ω is independent of E. References [1] J. Arazy, H. Upmeier, Boundary measures for symmetric domains and integral formulas for the discrete Wallach points, Integral Equations Operator Theory 47 (4) (2003) 375–434. [2] S. Ben Saïd, Weighted Bergman spaces on bounded symmetric domains, Pacific J. Math. 206 (1) (2002) 39– 68. [3] F.A. Berezin, Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (2) (1975) 363–402, 472. [4] F.A. Berezin, The connection between covariant and contravariant symbols of operators on classical complex symmetric spaces, Dokl. Akad. Nauk SSSR 241 (1) (1978) 15–17. [5] J. Faraut, A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., The Clarendon Press/Oxford University Press/Oxford Science Publications, New York, 1994. [6] J. Faraut, M. Pevzner, Berezin kernels and analysis on Makarevich spaces, Indag. Math. (N.S.) 16 (3–4) (2005) 461–486. [7] S. Helgason, Groups and geometric analysis, in: Integral Geometry, Invariant Differential Operators, and Spherical Functions, in: Pure Appl. Math., vol. 113, Academic Press, Orlando, FL, 1984. [8] L. Hörmander, The analysis of linear partial differential operators. I, in: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin, 1983. [9] H. Ishi, Positive Riesz distributions on homogeneous cones, J. Math. Soc. Japan 52 (1) (2000) 161–186. [10] A.W. Knapp, Lie Groups Beyond an Introduction, second ed., Progr. Math., vol. 140, Birkhäuser Boston, Boston, MA, 2002. [11] T. Kobayashi, Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal. 152 (1) (1998) 100–135. [12] B. Krötz, Formal dimension for semisimple symmetric spaces, Compos. Math. 125 (2) (2001) 155–191. [13] S. Lang, Differential Manifolds, Addison–Wesley Publishing, Reading, MA/London/Don Mills, ON, 1972. [14] M. Lassalle, Algèbre de Jordan et ensemble de Wallach, Invent. Math. 89 (2) (1987) 375–393. [15] Y.A. Neretin, Matrix analogues of the B-function, and the Plancherel formula for Berezin kernel representations, Mat. Sb. 191 (5) (2000) 67–100. [16] Y.A. Neretin, On the separation of spectra in the analysis of Berezin kernels, Funktsional. Anal. i Prilozhen. 34 (3) (2000) 49–62, 96. [17] Y.A. Neretin, Plancherel formula for Berezin deformation of L2 on Riemannian symmetric space, J. Funct. Anal. 189 (2) (2002) 336–408. [18] B. Ørsted, G. Zhang, Tensor products of analytic continuations of holomorphic discrete series, Canad. J. Math. 49 (6) (1997) 1224–1241. [19] L. Peng, G. Zhang, Tensor products of holomorphic representations and bilinear differential operators, J. Funct. Anal. 210 (1) (2004) 171–192. [20] J. Repka, Tensor products of holomorphic discrete series representations, Canad. J. Math. 31 (4) (1979) 836– 844. [21] H. Seppänen, Branching laws for minimal holomorphic representations, J. Funct. Anal. 251 (1) (2007) 174–209. [22] H. Seppänen, Branching of some holomorphic representations of SO(2, n), J. Lie Theory 17 (1) (2007) 191–227. [23] H. Seppänen, Tube domains and restrictions of minimal representations, Internat. J. Math. 19 (10) (2008) 1247– 1268. [24] A. Unterberger, H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (3) (1994) 563–597. [25] E.P. van den Ban, Induced representations and the Langlands classification, in: Representation Theory and Automorphic Forms, Edinburgh, 1996, in: Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 123–155. [26] G. van Dijk, M. Pevzner, Berezin kernels of tube domains, J. Funct. Anal. 181 (2) (2001) 189–208. [27] M. Vergne, H. Rossi, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1–2) (1976) 1–59. [28] N.R. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc. 251 (1–17) (1979) 19–37.


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Hierarchy of Hamilton equations on Banach Lie–Poisson spaces related to restricted Grassmannian Tomasz Goliński ∗ , Anatol Odzijewicz Institute of Mathematics, University in Białystok, Lipowa 41, 15-424 Białystok, Poland Received 10 September 2009; accepted 14 January 2010 Available online 2 February 2010 Communicated by D. Voiculescu

Abstract We consider the Banach Lie–Poisson space iR ⊕ U L1res and its complexification C ⊕ L1res , where the first one of them contains the restricted Grassmannian Grres as a symplectic leaf. Using the Magri method we define an involutive family of Hamiltonians on these Banach Lie–Poisson spaces. The hierarchy of Hamilton equations given by these Hamiltonians is investigated. The operator equations of Ricatti-type are included in this hierarchy. For a few particular cases we give the explicit solutions. © 2010 Elsevier Inc. All rights reserved. Keywords: Integrable systems; Banach Lie–Poisson spaces; Restricted Grassmannian; Banach Lie groups and algebras

1. Introduction The foundation of Banach Poisson differential geometry was developed in [11]. This theory, in particular, gives us a possibility to formulate geometrically and analytically rigorous language for the theory of infinite dimensional Hamiltonian systems. A special place in this theory is occupied by the Banach Lie–Poisson spaces. Recall that by definition b is a Banach Lie–Poisson space if its dual b∗ is a Banach Lie algebra such that ad∗x b ⊂ b ⊂ b∗∗ for x ∈ b∗ , where ad∗x : b∗∗ → b∗∗ is dual to the adjoint representation adx := [x, ·] : b∗ → b∗ . * Corresponding author.

E-mail addresses: [email protected] (T. Goliński), [email protected] (A. Odzijewicz). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.019

T. Goliński, A. Odzijewicz / Journal of Functional Analysis 258 (2010) 3266–3294

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Many infinite dimensional physical systems can be considered as systems on some Banach Lie–Poisson space b in the Hamilton way d ρ = − ad∗Dh(ρ) ρ, dt

(1.1)

where ρ ∈ b and h ∈ C ∞ (b), e.g. see [13]. The first aim of this paper is to investigate Banach Lie–Poisson spaces related to the restricted Grassmannian Grres , see [2,12]. The restricted Grassmannian Grres has its own long story as one of the most important infinite dimensional Kähler manifolds in mathematical physics. It is a set of Hilbert subspaces W ⊂ H of a polarized Hilbert space H = H+ ⊕ H− such that the projectors P+ : W → H+ and P− : W → H− are Fredholm and Hilbert–Schmidt operators respectively. The geometry of Grres and its symmetry group play an important role in the quantum field theory [14,22,23,25,26], the loop group theory [15,21], and the integration of the KdV and KP hierarchies [8,17,20]. The second aim is to define and investigate a hierarchy of the Hamilton equations on the Banach Lie–Poisson space iR ⊕ UL1res and on its complexification C ⊕ L1res . This hierarchy is obtained from the involutive system of Hamiltonians constructed on iR ⊕ UL1res and C ⊕ L1res by the Magri method [7]. The pair of coupled operator Ricatti equations, see (3.41), belongs to this hierarchy. As we show in Example 4.3 the finite dimensional version of the hierarchy provides the non-trivial example of an integrable Hamiltonian system. In our considerations we use the functional analytical methods as well as Banach differential geometric methods. All the results we have obtained are also valid in finite dimension case. Since the hierarchy consists of Hamilton equations, the flows preserve symplectic leaves of iR ⊕ UL1res and C ⊕ L1res . In particular, they preserve Grres , which is one of the symplectic leaves of iR ⊕ UL1res , see [2]. We show in Example 4.1 that after restriction to Grres the flows linearize in natural complex coordinates. The central place of the paper is occupied by Section 3, where (using the Magri method) we construct the infinite hierarchy under consideration. We also discuss its various realizations, see (3.20), (3.19), (3.38), (3.52) and (3.54). In Section 2 we prepare the material necessary for the application of Magri method, i.e. we res,0 of GLres,0 , the find explicit formulas for coadjoint representation of central extension GL 1 Poisson bracket and Casimirs of the Banach Lie–Poisson spaces C ⊕ Lres and iR ⊕ UL1res . Finally Section 4 gives formulas for solutions in some particular cases. We also include in the paper two appendices, where we present the Magri method and the theory of extensions of Banach Lie–Poisson spaces. 2. Banach Lie–Poisson spaces related to restricted Grassmannian We investigate the extensions of Banach Lie groups, Banach Lie algebras and Banach Lie– Poisson spaces related to the restricted Grassmannian Grres . One of the aims of this section is to obtain explicit formulas for adjoint and coadjoint actions of constructed Banach Lie groups and Banach Lie algebras. To this end we apply the methods described in Appendix A. Before that let us recall the definitions of objects we are going to use and fix the notation. For more information see [15,21,26].

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2.1. Preliminary definitions and notation Let us consider a complex separable Hilbert space with a fixed decomposition into two orthogonal Hilbert subspaces H = H+ ⊕ H− .

(2.1)

Let P+ and P− denote the orthogonal projectors onto H+ and H− respectively. We assume that in general both Hilbert subspaces are infinite dimensional. However we also admit the case when one (or both) of them is finite dimensional. In this case many analytical problems are considerably simplified. In what follows we omit the symbols H and H± in the notation for various operator algebras and groups and put the subscript ± if we mean that the operators act in H± . In this way, for example instead of L2 (H) or L2 (H+ ) we write L2 or L2+ . In order to simplify our notation we use the block decomposition A++ 0 , 0 0 0 0 P− AP+ = , A−+ 0

P+ AP+ =

0 A+− , 0 0 0 0 P− AP− = 0 A−−

P+ AP− =

(2.2)

and we identify the operators A++ : H+ → H+ , A−− : H− → H− , A−+ : H+ → H− and A+− : H− → H+ with P+ AP+ , P− AP− , P− AP+ , P+ AP− respectively when there is no risk of confusion. By Lp we denote the Schatten classes of operators acting in H equipped with the norm · p . The Lp spaces are ideals in associative algebra L∞ of bounded operators in H. In particular L1 denotes the ideal of trace-class operators and L2 is the ideal of Hilbert–Schmidt operators. By L0 ⊂ L∞ one denotes the ideal of compact operators, which is · ∞ -norm closure Lp = L0 of any Lp ideal, see [4,18]. Let GL∞ be the Banach Lie group of invertible bounded operators in H. By UL∞ ⊂ GL∞ we denote the real Banach Lie group of the unitary operators and its Lie algebra is denoted by UL∞ . By GL1+ we denote the group of invertible operators on H+ which have a determinant (i.e. they differ from identity by a trace-class operator) and by SL1+ —its subgroup which consists of operators with the determinant equal to 1. See [5,16] for the definition and properties of the determinant for this case. The unitary restricted group ULres is defined as ULres := u ∈ UL∞ [u, P+ ] ∈ L2 . (2.3) It possesses the Banach Lie group structure given by the embedding ULres u → (u, u−+ ) ∈ UL∞ × L2+− .

(2.4)

This structure is not compatible with Banach Lie group structure of UL∞ . The Banach Lie algebra of ULres is (2.5) ULres := x ∈ L∞ x + = −x, [x, P+ ] ∈ L2


3269

xres := x++ ∞ + x−− ∞ + x−+ 2 + x+− 2 ,

(2.6)

with the norm

where x + is the operator adjoint to x. Note that the topology of ULres is strictly stronger than the operator topology on L∞ . C The complexifications of ULres and ULres are ULC res = GLres and ULres = Lres respectively, where (2.7) GLres = g ∈ GL∞ [g, P+ ] ∈ L2 and Lres = x ∈ L∞ [x, P+ ] ∈ L2 .

(2.8)

By definition the restricted Grassmannian Grres consists of Hilbert subspaces W ⊂ H such that: i) the projection P+ restricted to W is a Fredholm operator; ii) the projection P− restricted to W is a Hilbert–Schmidt operator; see e.g. [15,26]. The restricted Grassmannian is a Hilbert manifold modelled on the Hilbert space L2+− . The groups ULres and GLres act on it transitively. In this way the tangent space to the restricted Grassmannian in the point H+ can be described as follows ∞ (2.9) TH+ Grres ∼ = ULres / UL∞ + × UL− . Both ULres and GLres are disconnected and their connected components are ULres,k and GLres,k , where g ∈ ULres,k and g ∈ GLres,k iff the Fredholm index ind g++ of the upper left block g++ of the operator g is equal to k, see [3,15]. The maximal connected subgroups ULres,0 and GLres,0 will be of special interest. In a similar fashion, connected components of the restricted Grassmannian Grres are the sets Grres,k consisting of elements of Grres such that the index of the orthogonal projection P+ restricted to that element is equal to k. Let us note that ULres,0 acts transitively on Grres,0 . 2.2. Extensions of GLres,0 The central object in the following construction is the group E defined as 1 E := (q, A) ∈ GL∞ + × GLres,0 A++ − q ∈ L+

(2.10)

with pairwise multiplication. The topology and Banach manifold structure on E is given by the embedding (q, A) → (A++ − q, A) ∈ L1+ × GLres , see [15,26].

(2.11)

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Let us consider the Banach Lie group extensions presented in the following commutative diagram: {1}

{1}

C×

{1}

ι

res,0 GL

det

{1}

GL1+

{1}

SL1+

{1}

{1}

π

GLres,0 id

δ ι1

ι1

{1}

E

π2

SL1+ × {1}

GLres,0

π2

{1}

(2.12)

{1}

{1}

The map ι1 is defined by ι1 (q) := (q, 1) and the map π2 is a projection onto the second 1 component of the Cartesian product GL∞ + × GLres,0 . Thus ι1 (SL+ ) is a normal subgroup of E res,0 is defined as the quotient group and the group GL res,0 := E/ι1 SL1+ . GL

(2.13)

The maps ι and π in upper row of diagram (2.12) are given as quotients of ι1 and π2 respectively. res,0 . In this way all rows and columns in The map δ is the quotient map E → E/ι1 (SL1+ ) = GL diagram (2.12) are exact sequences of Banach Lie groups. Using the approach described in Appendix A we define a local section (A.2) of the bundle GL1+ → E → GLres,0 by σ (A) := (A++ , A)

(2.14)

for A ∈ GLres,0 such that A++ is invertible. The Banach Lie group E can be locally identified with GL1+ ×Φ,Ω GLres,0 through the isomorphism Ψ : GL1+ ×Φ,Ω GLres,0 → E given in the properly chosen neighborhood of identity by Ψ (n, A) = (nA++ , A).

(2.15)

The maps Φ : GLres,0 → Aut GL1+ and Ω : GLres,0 × GLres,0 → GL1+ defined in the general case by (A.7) and (A.8) in this case can be expressed locally as follows:


3271

Φ(A)(n) = A++ nA−1 ++ ,

(2.16)

Ω(A1 , A2 ) = A1++ A2++ (A1 A2 )−1 ++

(2.17)

for n ∈ GL1+ , A, A1 , A2 ∈ GLres,0 such that A++ , A1++ , A2++ and (A1 A2 )++ are invertible. The map Φ(A) descends to the trivial automorphism of C× . Thus the Banach Lie group GLres,0 can be identified with C× ×id,Ω˜ GLres,0 for Ω˜ := det ◦ Ω

(2.18)

and it is a central extension of GLres,0 . 2.3. Extensions of Lres The Banach Lie algebra counterpart of diagram (2.12) is the following: {0}

{0}

{0}

C

C ⊕ Lres

Lres

{0}

Tr1

Tr

{0}

L1+

{0}

S L1+

{0}

ι1

L1+ ⊕ Lres

ι1

S L1+ ⊕ {0}

{0}

id π2

π2

Lres

{0}

(2.19)

{0}

{0}

where S L1+ := ρ ∈ L1+ Tr ρ = 0

(2.20)

is the Banach Lie algebra of the group SL1+ . The Banach Lie algebra (L1+ ⊕ Lres )/(S L1+ ⊕ {0}) res,0 is naturally identified by DΨ (1, 1) with C ⊕ Lres . The map Tr1 is of the quotient group GL given by taking trace of the first component of (ρ, X) ∈ L1+ ⊕ Lres . The direct sums in diagram (2.19) are understood as direct sums of Banach spaces. Similarly as in the group case, all rows and columns in diagram (2.19) are exact sequences of Banach Lie algebras.

3272


By using formula (A.14) with functions (2.16) and (2.17) we obtain a local formula for the adjoint action Ad(n,A) (ρ, X) = nA++ (ρ + X++ )(A++ )−1 n−1 − AXA−1 ++ , AXA−1

(2.21)

for (n, A) in some open set in GL1+ ×Φ,Ω GLres,0 and (ρ, X) ∈ L1+ ⊕ Lres . From (A.16) and (A.17) we get that ϕ(X) := [X++ , ·],

(2.22)

ω(X, Y ) := −X+− Y−+ + Y+− X−+ .

(2.23)

Bracket (A.15) for ϕ and ω given by (2.22) and (2.23) assumes the form

(ρ, X), ρ , Y = ρ, ρ + X++ , ρ − [Y++ , ρ] − X+− Y−+ + Y+− X−+ , [X, Y ]

(2.24)

where (ρ, X), (ρ , Y ) ∈ L1+ ⊕ Lres . This Banach Lie algebra was presented in [12] (up to the sign conventions) as an example of extensions of Banach Lie algebras. The structure of Banach Lie algebra on C ⊕ Lres is given by the function ϕ(X) ˜ ≡0

(2.25)

ω(X, ˜ Y ) = −s(X, Y ) = − Tr(X+− Y−+ − Y+− X−+ ),

(2.26)

and the cocycle

where s(X, Y ) is called the Schwinger term, see [19,26]. Thus the adjoint representation of the Lie group C× ×id,Ω˜ GLres,0 on C ⊕ Lres is given by Ad(γ ,A) (λ, X) = λ + Tr P+ − A−1 P+ A , AXA−1

(2.27)

for (γ , A) ∈ C× ×id,Ω˜ GLres,0 , (λ, X) ∈ C ⊕ Lres . Moreover, the Lie bracket for (λ, X), (λ , Y ) ∈ C ⊕ Lres is the following

(λ, X), λ , Y = −s(X, Y ), [X, Y ] .

(2.28)

Let us note that formula (A.14) allows one to express Ad only locally. However the right-hand side of formula (2.27) defines some global representation of C× ×id,Ω˜ GLres,0 , which coincides with Ad on an open neighborhood of (1, 1). However every open neighborhood of the unit element in Banach Lie group generates a connected component, and C× ×id,Ω˜ GLres,0 is connected. Thus the formula (2.27) is valid for all (γ , A) ∈ C× ×id,Ω˜ GLres,0 .


3273

2.4. Extensions of complex Banach Lie–Poisson space L1res In order to find a Banach space predual to the Banach Lie algebra Lres , we define the Banach space L1res := μ ∈ Lres μ++ ∈ L1+ , μ−− ∈ L1−

(2.29)

μ∗ := μ++ 1 + μ−− 1 + μ−+ 2 + μ+− 2 .

(2.30)

with the norm

Moreover we define the restricted trace Trres : L1res → C by Trres μ := Tr(μ++ + μ−− ),

(2.31)

for μ ∈ L1res . The domain of Trres is larger than L1 since L1 ⊂ Lres . However for trace-class operators the restricted trace Trres coincides with the standard trace Tr. The properties of the restricted trace are similar to the properties of the standard trace but one needs to replace L∞ with Lres . Proposition 2.1. The Banach space L1res is an ideal (in the sense of commutative algebras) in the Banach space Lres . Moreover for μ ∈ L1res , ν ∈ Lres we have Trres (μν) = Trres (νμ).

(2.32)

Proof. The conclusion that L1res is an ideal follows from the fact that L1 and L2 are ideals in L∞ and a product of two operators from L2 is trace-class. To prove formula (2.32) we expand its left-hand side Trres (μν) = Tr(μ++ ν++ + μ+− ν−+ + μ−+ ν+− + μ−− ν−− ).

(2.33)

By assumptions of the proposition, we conclude that each term in the right-hand side is a traceclass operator. Since for A ∈ L∞ and B ∈ L1 or for A, B ∈ L2 one has Tr(AB) = Tr(BA),

(2.34)

we conclude that Trres (μν) = Trres (νμ).

2

(2.35)

As a corollary to this proposition we get that for g ∈ GLres and μ ∈ L1res , the operator μg −1 belongs to L1res and Trres gμg −1 = Trres (μ).

(2.36)

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Using the pairing between μ ∈ L1res , A ∈ Lres given by μ, A := Trres (μA) = Tr(μ++ A++ ) + Tr(μ+− A−+ ) + Tr(μ−+ A+− ) + Tr(μ−− A−− ),

(2.37)

we conclude that (L1res )∗ ∼ = Lres , i.e. the Banach space L1res is predual of Lres . This duality can be found in [2,12]. Proposition 2.2. Space L1res with norm · ∗ is Banach ∗-algebra. Proof. The only point yet to be shown is the inequality μρ∗ μ∗ ρ∗

(2.38)

for μ, ρ ∈ L1res . In order to prove it we observe that μρ∗ = (μρ)++ 1 + (μρ)+− 2 + (μρ)−+ 2 + (μρ)−− 1 = μ++ ρ++ + μ+− ρ−+ 1 + μ++ ρ+− + μ+− ρ−− 2 + μ−+ ρ++ + μ−− ρ−+ 2 + μ−− ρ−− + μ−+ ρ+− 1 .

(2.39)

Next, applying the following inequalities ρ2 ρ1 ,

(2.40)

ρμ1 ρ∞ μ1 ρ1 μ1

(2.41)

for ρ, μ ∈ L1 and the inequalities ρμ1 ρ2 μ2 ,

(2.42)

ρμ2 ρ∞ μ2 ρ2 μ2

(2.43)

for ρ, μ ∈ L2 , we obtain μρ∗ μ++ 1 ρ++ 1 + μ+− 2 ρ−+ 2 + μ++ 1 ρ+− 2 + μ+− 2 ρ−− 2 + μ−+ 2 ρ++ 1 + μ−− 1 ρ−+ 2 + μ−− 1 ρ−− 1 + μ−+ 2 ρ+− 2 μ∗ ρ∗

(2.44)

The latter inequality in (2.44) follows directly from (2.30).

2

The Banach space predual to extended Banach Lie algebra L1+ ⊕ Lres is L0+ ⊕ L1res with natural component-wise pairing

(A, μ), (ρ, X) = Tr(Aρ) + Trres (μX).

(2.45)


3275

It follows from the fact that the Banach space dual to the ideal of compact operators L0+ is L1+ , see [24]. Analogously, the predual of C ⊕ Lres is C ⊕ L1res with the pairing given by

(γ , μ), (λ, X) = γ λ + Trres (μX), (2.46) for μ ∈ L1res , X ∈ Lres , γ , λ ∈ C. 1 ∗ The map Tr∗ : C → L∞ + dual to Tr : L+ → C is given by Tr (λ) = λ1. Since the ideal of 0 1 compact operators L+ is the Banach space predual to L+ and Tr∗ does not take values in L0+ , we conclude that Tr∗ cannot be restricted to predual spaces. Therefore only horizontal exact sequences in diagram (2.19) have their predual counterparts {0} {0}

L1res L1res

π2∗

π2∗

C ⊕ L1res L0+ ⊕ L1res

ι∗1

ι∗1

C L0+

{0} {0}

(2.47) (2.48)

where the map π2∗ is an injection into the second argument and the map ι∗1 is the projection onto the first component of the respective direct sums. It follows from (A.25) and from (2.16), (2.17) that the coadjoint representation of the Banach Lie group GL1+ ×Φ,Ω GLres,0 on the predual Banach space L0+ ⊕ L1res is given by Ad∗(n,A) (τ, μ) = (A++ )−1 n−1 τ nA++ , (A++ )−1 n−1 τ nA++ − A−1 τ A + A−1 μA (2.49) for (n, A) ∈ GL1+ ×Φ,Ω GLres,0 and (τ, μ) ∈ L0+ ⊕ L1res . Similarly, the coadjoint representation of C× ×id,Ω˜ GLres,0 on C ⊕ L1res is the following Ad∗(λ,A) (γ , μ) = γ , A−1 μA + γ P+ − A−1 P+ A (2.50) where (λ, A) ∈ C× ×id,Ω˜ GLres,0 and (γ , μ) ∈ C ⊕ L1res . Let us also note that these coadjoint representations preserve the Banach subspaces L0+ ⊕ L1res ⊂ (L1+ ⊕ Lres )∗ and C ⊕ L1res ⊂ C ⊕ L∗res respectively Ad∗(n,A) L0+ ⊕ L1res ⊂ L0+ ⊕ L1res , Ad∗(λ,A) C ⊕ L1res ⊂ C ⊕ L1res .

(2.51) (2.52)

The coadjoint representation of the Banach Lie algebra L1+ ⊕ Lres on its predual L0+ ⊕ L1res is the following (2.53) ad∗(ρ,X) (τ, μ) = [−ρ, τ ] − [X++ , τ ], −[X, μ] − [ρ, τ ] − τ X+− + X−+ τ where (ρ, X) ∈ L1+ ⊕ Lres , (τ, μ) ∈ L0+ ⊕ L1res . The coadjoint representation of the Banach Lie algebra C ⊕ Lres on C ⊕ L1res is given by ad∗(λ,X) (γ , μ) = 0, −[X, μ] − γ (X+− − X−+ ) (2.54) where (λ, X) ∈ C ⊕ Lres , (γ , μ) ∈ C ⊕ L1res .

3276


We observe that the conditions (A.26) are satisfied for both extensions, thus the Banach spaces L0+ ⊕ L1res and C ⊕ L1res are Banach Lie–Poisson spaces. The Poisson bracket for F, G ∈ C ∞ (C ⊕ L1res ) is obtained from the general formula (A.27) and is given by

{F, G}(γ , μ) = μ, D2 F (γ , μ), D2 G(γ , μ) − γ s D2 F (γ , μ), D2 G(γ , μ) ,

(2.55)

where D2 denotes the partial Fréchet derivative with respect to the second variable. 2.5. Extensions of real Banach Lie–Poisson space UL1res In previous subsections we considered the extensions of the complex linear restricted group GLres,0 , its Banach Lie algebra Lres and the complex Banach Lie–Poisson space L1res , which is Banach predual of Lres . As we mentioned above they are complexifications of ULres,0 , ULres and (ULres )∗ ∼ = UL1res := μ ∈ L1res μ+ = −μ

(2.56)

respectively. The pairing between elements of ULres and UL1res as in the complex case is given by (2.37). By a construction similar to those for GLres,0 , we obtain the central extension of ULres,0 by U (1) {1}

U (1)

res,0 UL

ULres,0

{1} .

(2.57)

Namely we define the real Banach Lie group U E := (A, q) ∈ E A ∈ ULres,0 , q ∈ UL∞ + ,

(2.58)

res,0 in (2.57) is defined as which complexification U E C is E . The group UL res,0 := U E/ ι1 SU L1+ , UL

(2.59)

where SU L1+ := SL1+ ∩ UL∞ and ι1 (q) := (q, 1). Restricting the Schwinger term (2.26) to ULres we obtain the central extension {0}

iR

iR ⊕ ULres

ULres

{0}

(2.60)

of the real Banach Lie algebra ULres . The exact sequence of Banach Lie–Poisson spaces predual to (2.60) is {0}

iR

iR ⊕ UL1res

UL1res

{0} .

(2.61)

The complexification of (2.61) gives (2.47). All expressions obtained above, including the ones for the Poisson bracket (2.55) and coadjoint representation (2.50), (2.54) are valid for the real case if one assumes that γ = −γ and μ+ = −μ.


3277

3. Hierarchy of Hamilton equations on Banach Lie–Poisson spaces C ⊕ L1res and R ⊕ U L1res iR In this section we use the Magri method (see [7]), to introduce the hierarchy of the Hamiltonian systems on the Banach Lie–Poisson spaces C ⊕ L1res and iR ⊕ UL1res , which were investigated in Section 2. Short description of Magri method is presented in Appendix B. To this end we define for any k ∈ N the function I k (γ , μ) := Trres (μ − γ P+ )k+1 − (−γ )k (μ − γ P+ )

(3.1)

on C ⊕ L1res . Note that the expression under Trres is a polynomial in variable μ without a free term, thus from Proposition 2.1 it follows that the function I k is well defined. We observe that I k is invariant with respect to the coadjoint representation (2.50) I k Ad∗[n,A] (γ , μ) = I k γ , A−1 μA + γ P+ − A−1 P+ A k+1 A = Trres A−1 μ − γ P+ + γ AP+ A−1 − γ AP+ A−1 − A−1 (−γ )k μ − γ P+ + γ AP+ A−1 − γ AP+ A−1 A = Trres (μ − γ P+ )k+1 − (−γ )k (μ − γ P+ ) = I k (γ , μ).

(3.2)

Thus the functions I k , k ∈ N, are Casimirs

I k, · = 0

(3.3)

for Poisson bracket (2.55). Note that the coordinate function γ is a Casimir too. Observing that Poisson brackets for F, G ∈ C ∞ (C ⊕ L1res ) given by

{F, G}1 (γ , μ) := μ, D2 F (γ , μ), D2 G(γ , μ)

(3.4)

{F, G}2 (γ , μ) := −γ s D2 F (γ , μ), D2 G(γ , μ)

(3.5)

and by

are compatible, we introduce a Poisson pencil {F, G}ε (γ , μ) := {F, G}1 (γ , μ) + ε{F, G}2 (γ , μ)

= μ, D2 F (γ , μ), D2 G(γ , μ) − εγ s D2 F (γ , μ), D2 G(γ , μ)

(3.6)

on C ⊕ L1res . Compatibility of {·,·}1 and {·,·}2 follows from the fact that the Poisson tensor for {·,·}2 is constant with respect to the variable μ and {·,·}1 depends only on the derivations with respect to the variable μ. Due to the latter equality in (3.6) the Casimirs for {·,·}ε are: Iεk (γ , μ) = Trres (μ − εγ P+ )k+1 − (−εγ )k (μ − εγ P+ )

(3.7)

3278


where k ∈ N. According to Magri method, we expand these Casimirs with respect to the parameter −εγ Iεk (γ , μ) =

k−1 (−εγ )n Trres Wnk+1 (μ) + (−εγ )k Trres Wkk+1 (μ) − μ ,

(3.8)

n=0

where the operators Wnk (μ) are polynomials in operator arguments μ and P+ defined by the equality (μ + λP+ )k =

k

λn Wnk (μ),

λ ∈ R.

(3.9)

n=0

In this way from (B.7) it follows that we obtain a family hkn (γ , μ) = γ n Trres Wnk+1 (μ), 0 n k − 1, hkk (γ , μ) = γ k Trres Wkk+1 − μ

(3.10a) (3.10b)

of Hamiltonians in involution

hkn , hlm

ε

=0

(3.11)

with respect to the brackets {·,·}ε for ε ∈ R. In the particular case ε = 1 they are in involution with respect to the bracket {·,·} given by (2.55). Let us now investigate the infinite system of the Hamilton equations on the Banach Lie– Poisson space C ⊕ L1res ∂ (γ , μ) = − ad∗(D hk (γ ,μ),D hk (γ ,μ)) (γ , μ) 1 n 2 n ∂tnk

(3.12)

defined by the hierarchy of Hamiltonians hkn , k ∈ N, n = 0, 1, . . . , k. Using the explicit form of coadjoint action (2.54) one sees that observe that Eq. (3.12) take the form ∂ γ = 0, ∂tnk

∂ μ = − μ, D2 hkn (γ , μ) + γ P+ D2 hkn (γ , μ)P− − P− D2 hkn (γ , μ)P+ . ∂tnk

(3.13a) (3.13b)

In (3.13) the real parameter tnk parametrizes the Hamiltonian flow generated by hkn . In order to compute D2 hkn (γ , μ) we apply the partial derivative operator D2 to both sides of equality (3.8). Since D2 Iεk (γ , μ) = (k + 1)(μ − εγ P+ )k − (−εγ )k 1

(3.14)

D2 hkn (γ , μ) = (k + 1)γ n Wnk (μ)

(3.15a)

we get that


3279

for 0 n k − 1, and D2 hkk (γ , μ) = γ k (k + 1)Wkk (μ) − 1 .

(3.15b)

Substituting (3.15) into (3.13) we obtain

∂ μ = −(k + 1)γ n μ − γ P+ , Wnk (μ) k ∂tn

(3.16)

for 0 n k. From (3.9) we the following recurrence rules k (μ)P+ , Wnk+1 (μ) = Wnk (μ)μ + Wn−1 k Wnk+1 (μ) = μWnk (μ) + P+ Wn−1 (μ)

(3.17)

which yield the commutation relation

k μ, Wnk (μ) + P+ , Wn−1 (μ) = 0,

(3.18)

k := 0. Using (3.18) we can express the Hamilton equations where 0 n k and we put W−1 (3.16) in the following two ways

∂ k μ = −(k + 1)γ n μ, Wnk (μ) + γ Wn+1 (μ) , k ∂tn

∂ k μ = (k + 1)γ n P+ , γ Wnk (μ) + Wn−1 (μ) , k ∂tn

(3.19) (3.20)

where 0 n k. Rewriting (3.20) in the block form (2.2) we obtain ∂ μ++ = 0, ∂tnk

∂ μ−− = 0 ∂tnk

(3.21)

and ⎧ ∂ k n k ⎪ ⎪ ⎨ ∂t k μ+− = (k + 1)γ P+ γ Wn (μ) + Wn−1 (μ) P− , n

⎪ ∂ ⎪ k (μ) P . ⎩ μ−+ = −(k + 1)γ n P− γ Wnk (μ) + Wn−1 + k ∂tn

(3.22)

Let us observe that Eq. (3.19) is a Hamilton equation for the Poisson bracket {·,·}1 and the Hamiltonian hkn + hkn+1 , while Eq. (3.20) is a Hamilton equation for the Poisson bracket {·,·}2 and the Hamiltonian hkn + hkn−1 . From (3.21) we conclude that diagonal blocks μ++ and μ−− are invariants for all Hamiltonian flows under consideration. The symplectic leaves for C ⊕ L1res and iR ⊕ UL1res with Poisson bracket {·,·}2 are the affine spaces obtained by shifting the vector spaces L2+− ⊕L2−+ or L2+− respectively by diagonal blocks μ++ and μ−− . This fact explain why we have obtained additional integrals of motion (3.21).

3280


Eqs. (3.16) and (3.19) are in the Lax form. In the first case it is an equation on μ − γ P+ , while it is on μ in the other. Let us calculate several operators Wnk (μ). We can do this by iterating the recurrence (3.17). The result is: Wkk = P+ ,

(3.23)

k 2, (3.24) k = μ2 P+ + μP+ μ + P+ μ2 + (k − 3) P+ μ2 P+ + P+ μP+ μ + μP+ μP+ Wk−2

k Wk−1

= μP+ + P+ μ + (k − 2)P+ μP+ ,

+

(k − 3)(k − 4) P+ μP+ μP+ , 2

k 4,

(3.25)

.. . W1k = P+ μk−1 + μP+ μk−2 + · · · + μk−1 P+ ,

(3.26)

W0k = μk .

(3.27)

It is obvious that the Hamiltonians hkn are functionally interdependent and it implies the interdependence of tnk -flows given by (3.22). The above formulas suggest the introduction of the homogeneous polynomials 1

Hnl (μ) :=

i0 ,i1 ,...il =0 i0 +···+il =n

i

i

P+0 μP+i1 μ . . . μP+l

(3.28)

of the degree l ∈ N in the operator variable μ ∈ L1res , where n l + 1. These polynomials are linearly independent and they satisfy the recurrences l+1 l (μ) = P+ μHnl (μ) + μHn+1 (μ) Hn+1

(3.29a)

l+1 l (μ) = P+ μHl+1 (μ) Hl+2

(3.29b)

for n l, l ∈ N and

for l ∈ N. Proposition 3.1. Polynomials Wnk are linear combinations of the homogeneous polynomials Hnl k Wk−l (μ) =

l+1

max 0, pnl (k) Hnl (μ)

(3.30)

n=1

for l < k and W0k (μ) = H0k (μ), where pnl ∈ Rn−1 [x] are polynomials of degree n − 1 that are defined by the recurrences:

(3.31)


l pn+1 (k) =

k−1

max 0, pnl (i) ,

3281

(3.32)

i=l+1

pnl+1 (k) = pnl (k − 1)

(3.33)

with initial condition p1l (k) = 1. Proof. We prove formula (3.30) by induction with respect to l. From recurrence (3.32) we infer that p21 (k) = k − 2 and thus from (3.23) and (3.24) we see that formula (3.30) is satisfied for l = 0 and l = 1. From (3.17) we conclude that k−1 k (μ) = μWk−l (μ) + P+ μ Wk−l

k−l

k−i−1 Wk−l−i (μ).

(3.34)

i=1 k assuming that (3.30) is satisfied for l − 1 and obtain We apply (3.34) to Wk−l

k Wk−l (μ) = μ

l

max 0, pnl−1 (k − 1) Hnl−1 (μ)

n=1

+ P+ μ

k−l−1 l

max 0, pnl−1 (k − i − 1) H l−1 (μ) + P+ μH0l−1 (μ).

(3.35)

i=1 n=1

Changing the order of summation and using recurrences (3.32) and (3.33) we get

k Wk−l (μ) = μ

l−1

l−1 l−1 max 0, pn+1 (k − 1) Hn+1 (μ)

n=0

+ P+ μ

l

l−1 max 0, pn+1 (k − 1) Hnl−1 (μ) + P+ μH0l−1 (μ).

(3.36)

n=1

By rearranging terms in the sums and applying (3.29) we end up with (3.30). Direct check shows that relations (3.32) and (3.33) are compatible. The fact that deg pnl = n − 1 follows from (3.32). 2 From Proposition 3.1 we conclude: Corollary 3.2. k }∞ i) The dimension of the complex vector space spanned by {Wk−l k=l+1 is equal to l + 1 and l+1 l {Hn }n=1 is a basis of this space; k , where ii) Using (3.30) one can express Hnl , 0 n l + 1 as a finite linear combination of Wk−l k l + 1.

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k }∞ Proof. i) From (3.30) it follows that all elements of the set {Wk−l k=l+1 are linear combinations l+1 l l of {Hn }n=1 . Let us also note that the polynomials pn assume positive values pnl (k) > 0 for k large enough and that the set {pnl }l+1 n=1 spans an (l + 1)-dimensional vector space. This concludes the proof. ii) This statement is a consequence of i). 2

Introducing new variables τnl ∈ R through the linear combination k tk−l

= (k − 1)γ

k−l

l+1

max 0, pnl (k) τnl ,

(3.37)

n=1

we rewrite the hierarchy (3.16) in the equivalent form

∂ μ = μ − γ P+ , Hnl (μ) l ∂τn

(3.38)

where l ∈ N and n = 1, . . . , l + 1. Let us write out explicitly several equations from hierarchy (3.38). For H0k and H1k in block notation (2.2) we obtain ⎧ k ∂ ⎪ ⎪ μ = −γ μ +− , +− ⎪ k ⎨ ∂τ 0 k ⎪ ∂ ⎪ ⎪ ⎩ k μ−+ = γ μ −+ ∂τ0

(3.39)

⎧ k+1 i k−i ∂ ⎪ ⎪ − γ k−1 , ⎪ i=0 μ ++ μ +− +− ⎨ ∂τ k μ+− = − μ 1 k+1 ⎪ ∂ ⎪ ⎪ + γ ki=1 μi −+ μk−i ++ ⎩ k μ−+ = μ −+ ∂τ1

(3.40)

and

respectively. For k = 1 and k = 2 we get from (3.39) linear equations and for k = 3 we obtain from (3.39) a pair of coupled operator Ricatti-type equations ⎧ ∂ ⎪ 2μ 2 , ⎪ μ = −γ (μ ) + μ μ μ + μ μ μ + μ (μ ) +− ++ +− +− −+ +− ++ +− −− +− −− ⎪ 3 ⎨ ∂τ 0 ⎪ ∂ ⎪ 2 2 ⎪ ⎩ 3 μ−+ = γ μ−+ (μ++ ) + μ−− μ−+ μ++ + μ−+ μ+− μ−+ + (μ−− ) μ−+ . ∂τ0

(3.41)

Let us recall that blocks μ++ and μ−− are constant with respect to all flows. Moreover if we assume that μ++ = 0 or μ−− = 0 then Eqs. (3.38) become linear. After certain modifications we can also consider the hierarchy of Eqs. (3.38) on the real Banach Lie–Poisson iR ⊕ UL1res . To this end we have to modify the Hamiltonians hkn is such a way


3283

that they will take real values when restricted to iR ⊕ UL1res . In consequence we obtain the following equations

∂ μ = i l+1 μ − γ P+ , Hnl (μ) ∂τnl

(3.42)

where μ ∈ UL1res , γ ∈ iR. Now we express the Hamiltonian hierarchy (3.16) in a more compact and elegant form. To this end let us define the “generating” Hamiltonian for the Hamiltonians (3.10) hκ,λ (γ , μ) :=

∞ k=1

1 κk λn hkn (γ , μ), k+1 k+1

(3.43)

n=0

where κ, λ ∈ R. In order to show that the series of functions (3.43) is convergent on some nonempty open subset of C ⊕ L1res we observe that k+1

λn hkn (γ , μ) = Trres (μ + γ λP+ )k+1 − (γ λ)k (μ + γ λP+ ) .

(3.44)

n=0

Equality (3.44) follows from (3.10) and (3.9). Next, let us prove the following lemma. Lemma 3.3. One has (μ + βP+ )k+1 − β k (μ + βP+ ) μ∗ + |β| k+1 − |β|k μ∗ + |β| , ∗

(3.45)

where β ∈ C and μ ∈ L1res . Proof. We expand the left-hand side and apply the triangle inequality and Proposition 2.2. Moreover we note that νP+ ∗ ν∗ for ν ∈ L1res . In this way we get k+1 k+1 (μ + βP+ )k+1 − β k (μ + βP+ ) μi∗ |β|k−i+1 − |β|k μ∗ . (3.46) ∗ i i=1

By adding and subtracting the term |β|k+1 and collecting terms we obtain the right-hand side. 2 From this lemma and Proposition 2.2 we conclude: Proposition 3.4. One has 1 log(1 − κλγ ) hκ,λ (γ , μ) = Trres − log 1 − κ(μ + γ λP+ ) + (μ + γ λP+ ) κ κλγ

(3.47)

and {hκ,λ , hκ ,λ } = 0

(3.48)

for |κ|(μ∗ + |λγ |) < 1 and |κ |(μ∗ + |λ γ |) < 1, where the Poisson bracket in (3.48) is given by (2.55).

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Now we find the explicit form of the Hamilton equation ∂ (γ , μ) = − ad∗Dhκ,λ (γ ,μ) (γ , μ) ∂tκ,λ

(3.49)

generated by Hamiltonian (3.43). Using (2.54) we obtain ∂ γ = 0, ∂tκ,λ

(3.50a)

∂ μ = − μ, D2 hκ,λ (γ , μ) + γ P+ D2 hκ,λ (γ , μ)P− − P− D2 hκ,λ (γ , μ)P+ . ∂tκ,λ

(3.50b)

Since −1 log(1 − κλγ ) D2 hκ,λ (γ , μ) = 1 − κ(μ + λγ P+ ) + κλγ

(3.51)

∂ x = −α P+ , (1 − x)−1 , ∂tκ,λ

(3.52)

x := κ(μ + λγ P+ ),

(3.53)

we get

where

and α := κ(1 + λ)γ . Replacing x in (3.52) by y := (1 − x)−1 we get the hierarchy of equations ∂ y = α[y, yP+ y], ∂tκ,λ

λ, κ ∈ C

(3.54)

equivalent to the hierarchy (3.16). In this paper we don’t intend to address the problem of finding general solutions for the considered Hamiltonian systems but in the next section we will present several examples of solutions. Let us also observe that if H+ is finite dimensional, then the operator μ − γ P+ has a discrete res,0 coincide with spectrum. Formula (2.50) shows that orbits of coadjoint action of group GL ∞ orbits of standard coadjoint action of GLres,0 ⊂ GL shifted by γ P+ . Thus one can use the spectrum spec(μ − γ P+ ) to distinguish partially symplectic leaves of the Banach Lie–Poisson space iR ⊕ UL1res , i.e. if spec(μ1 − γ P+ ) = spec(μ2 − γ P+ ) then they belong to different symplectic leaves. If the dimension of H+ is infinite then the shift μ − γ P+ of the operator μ ∈ L1res by the operator γ P+ is not an element of L1res , but of Lres . However from Weyl’s criterion (see [16]) we deduce that the set spec(μ − γ P+ ) \ {0, −γ } is discrete. Therefore if H+ is infinite dimensional, one can use also elements of spec(μ − γ P+ ) for the partial indexation of the symplectic leaves. The problem of description of these leaves is complicated, see [2] for more information.


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4. Examples of solutions In this section we present several examples of explicit solutions to Eqs. (3.38) in some particular cases. Example 4.1 (Restricted Grassmannian). The connected component Grres,0 of the restricted res,0 in the Grassmannian Grres can be identified with the coadjoint orbit Oγ of the group UL 1 Banach Lie–Poisson space iR ⊕ ULres generated from point (γ , 0), see [2]. Namely from (2.50) we see that Ad∗(λ,g) (γ , 0) = γ , γ P+ − g −1 P+ g

(4.1)

for (λ, g) ∈ U (1) ×id,Ω˜ ULres,0 and γ ∈ iR. This suggests to define the map ιγ : Grres,0 → Oγ in the following way ιγ (W ) := γ , γ (P+ − PW ) .

(4.2)

Since ULres,0 acts transitively on Grres,0 , we see that ιγ maps Grres,0 bijectively on Oγ . Let us introduce homogeneous coordinates on some open subset in Grres . To this end we fix a basis {|n}, n ∈ Z in H, such that |n for n < 0 spans H− and for n 0 spans H+ . Let us fix a basis w1 , w2 , . . . in a subspace W ∈ Grres and put the coefficients of wk in the basis {|n}n∈Z in the matrix form α := n | wk n∈Z,k∈N , (4.3) β where α, β are blocks obtained for n 0 and n < 0 respectively. Let us consider a subspace W ∈ Grres such that there exists an orthonormal basis {wk }k∈N such that α is invertible. Then we define z := βα −1 .

(4.4)

Definition of z is independent of the choice of the basis {wk }k∈N . The matrix of the projector PW is n|PW k n,k∈Z =

α + + α β . β

(4.5)

Thus ιγ (W ) takes the following form ιγ (W ) =

(1 + z+ z)−1 − 1 z(1 + z+ z)−1

(1 + z+ z)−1 z+ z(1 + z+ z)−1 z+

,

(4.6)

where we consider z as an operator z : H+ → H− . Hamilton equations (3.13) can be written in terms of z. Due to (3.21) we note that z+ z is constant. Thus any polynomial of ιγ (W ) has only constant and linear terms in z and Eqs. (3.13) are linear in homogeneous coordinates on Oγ .

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Example 4.2 (Vector case). Let us consider a particular case of the equations given above. We assume that dim H+ = 1. We introduce the block notation for elements μ ∈ L1res μ=

a w

v+ A

(4.7)

,

where a ∈ C, A ∈ L∞ (H− ), v, w ∈ L2 (C, H− ) ∼ = 2 . We consider Eqs. (3.38) as non-linear equations for two vectors v, w coupled by interaction depending on constants a and A. Nonlinear behavior is due to the terms of the type v | Al w. First of all let us remark that in general all functions hkl are linear combinations of funck1 k2 kn tions Trres μk = hk−1 0 (γ , μ) and Trres (μ P+ μ P+ · · · + P+ μ P+ ). However due to the fact that dim H+ = 1 we have k hkn (γ , μ) 1 h 1 (γ , μ) Trres μk1 P+ μk2 P+ · · · + P+ μkn P+ = n 1 ... 1 . γ k1 + 1 kn + 1

(4.8)

Thus all integrals of motion hkl are functionally dependent on hk0 and hk1 . Therefore one has only two independent families of Hamilton equations, i.e. (3.39) and (3.40). In order to solve these families we note that k μ −+ = Mk (γ , μ)w, k μ +− = v + Mk (γ , μ),

(4.9) (4.10)

where Mk (γ , μ) :=

hk−2 (γ , μ) h2 (γ , μ) k−3 hk−1 1 (γ , μ) + 1 A + ··· + 1 A + aAk−2 + Ak−1 γk γ (k − 1) 3γ

(4.11)

is a time independent operator. Thus Eqs. (3.39) take the form ⎧ ∂ ⎪ + ⎪ ⎪ ⎨ ∂τ k v = −γ Mk γ , μ v, 0 ∂ ⎪ ⎪ ⎪ ⎩ k w = γ Mk (γ , μ)w. ∂τ0

(4.12)

In this way we have reduced system (3.39) to a linear system. Thus its solution is ∞ k 2 3 + Mk γ , μ (0, 0, . . .) τ0 v(0, 0, . . .), v τ0 , τ0 , . . . = exp −γ

(4.13)

k=2

∞ 2 3 k w τ0 , τ0 , . . . = exp γ Mk γ , μ(0, 0, . . .) τ0 w(0, 0, . . .) k=2

where v(0, 0, . . .), w(0, 0, . . .) ∈ 2 , A ∈ L∞ (H− ), a ∈ C are initial conditions.

(4.14)


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In the case of Eqs. (3.40) we get ⎧ k−1 j h (γ , μ+ ) ⎪ ∂ ⎪ 1 ⎪ Mk−j γ , μ+ + Mk+1 γ , μ+ v, v=− ⎪ ⎪ k ⎪ j +1 ⎨ ∂τ1 j =1 ⎪ k−1 k−j ⎪ h1 (γ , μ) ⎪ ∂ ⎪ ⎪ Mj (γ , μ) + Mk+1 (γ , μ) w. ⎪ ⎩ ∂τ k w = k−j +1 1

(4.15)

j =1

These equation are also linear and their solution can be obtained by exponentiation. Example 4.3 (4-dimensional case). In this example we solve equation (3.41) in the iR ⊕ UL1res case assuming dim H+ = dim H− = 2. We will use the following notation A Z , (4.16) γ = iχ, μ=i Z+ D where χ ∈ R and A = A+ , D = D + , Z ∈ Mat2×2 (C). Substituting (4.16) into (3.41) we obtain d A = 0, dt

d D=0 dt

(4.17)

and d Z = −iχ A2 Z + ZD 2 + AZD + ZZ + Z , dt d + Z = iχ Z + A2 + D 2 Z + + DZ + A + Z + ZZ + . dt

(4.18) (4.19)

Let us note that Eqs. (4.18) do not change their form with respect to the transformation A → U AU + , D → V DV + , Z → V ZU + , where U U + = 1 and V V + = 1. So without loss of the generality we can assume d1 0 a b a1 0 , D= , Z= , (4.20) A= c d 0 a2 0 d2 where a1 , a2 , d1 , d2 ∈ R are constants and a, b, c, d are complex-valued functions of t ∈ R. From (4.18) we obtain d a = iχ a12 + a1 d1 + d12 + |a|2 + |b|2 + |c|2 a + iχbcd, dt d b = iχ a12 + a1 d2 + d22 + |a|2 + |b|2 + |d|2 b + iχacd, dt d c = iχ a22 + a2 d1 + d12 + |a|2 + |c|2 + |d|2 c + iχabd, dt d c = iχ a22 + a2 d2 + d22 + |b|2 + |c|2 + |d|2 c + iχ abc. dt

(4.21)

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Now we consider the generic case, i.e. a1 = a2 and d1 = d2 . In order to solve this system of equations we calculate explicitly the integrals of motion h10 (μ) = Trres μ2 , h20 (μ) = Trres μ3 , h21 (μ) = γ Tr(μ2 P+ ), h31 (μ) = γ Tr(μ3 P+ ) and h30 (μ) = Trres μ4 . From that we conclude that |a|2 + |b|2 =: p 2 = const, |a|2 + |c|2 =: q 2 = const, |c|2 + |d|2 =: r 2 = const, |b|2 + |d|2 =: s 2 = const,

(4.22)

|a|2 a1 d1 + |b|2 a1 d2 + |c|2 a2 d1 + |d|2 a2 d2 + |a|2 |c|2 + |b|2 |d|2 + 2 Re(abcd) =: = const,

(4.23)

where p 2 + r 2 = q 2 + s 2 . Using (4.22) and (4.21) we find d 2 d d d |a| = − |b|2 = − |c|2 = |d|2 = 2χ Im(abcd). dt dt dt dt

(4.24)

Now from (4.23) and (4.24) we obtain the following equation d x = ± w(x) dt

(4.25)

w(x) := 4x p 2 − x q 2 − x r 2 − q 2 − x − v 2 (x)

(4.26)

v(x) := x(a1 − a2 )(d1 − d2 ) − x p 2 − x − q 2 − x r 2 − q 2 + x − a1 d2 p 2 − a2 d1 q 2 − a2 d2 r 2 − q 2 .

(4.27)

on the function x := |a|2 , where

and

Since w is a polynomial of the fourth degree, this equation is solved by an elliptic integral of the first kind t=

dx . √ w(x)

(4.28)

This allows us to express x(t) as an elliptic function of time parameter t. By (4.22) we may calculate |b|2 , |c|2 and |d|2 in terms of x(t). In order to find the functions a(t), b(t), c(t) and d(t) we substitute their polar decompositions a = |a|eiα , b = |b|eiβ , c = |c|eiγ and d = |d|eiδ into Eqs. (4.21). In this way we obtain


d v(x) α = χ a12 + a1 d1 + d12 + p 2 + q 2 − x + , dt 2x d v(x) 2 2 2 2 2 β = χ a1 + a1 d2 + d 2 + p + r − q + x + , dt 2(p 2 − x) v(x) d , γ = χ a22 + a2 d1 + d12 + r 2 + x + dt 2(q 2 − x) d v(x) δ = χ a22 + a2 d2 + d22 + p 2 + r 2 − x + . dt 2(r 2 − q 2 + x)

3289

(4.29)

Since the right-hand sides of Eqs. (4.29) are known, we can find α(t), β(t), γ (t) and δ(t) by integration. In this way we have solved equations (4.18) in quadratures. The Hamiltonian system solved in this example have applications for example in the nonlinear optics, see [6]. Acknowledgments The authors would like to thank A.B. Tumpach, D. Belti¸taˇ and V. Dragovic for many valuable remarks and comments. This work is partially supported by Polish Government grant 1 P03A 001 29. The authors would like to thank Banach Center for their hospitality during the workshop “Banach Lie–Poisson spaces and integrable systems” (5–10 August 2008, B¸edlewo). Appendix A. Extensions of Banach Lie groups and related Banach Lie–Poisson spaces Let us present an abbreviated description of extensions of Banach Lie groups, Banach Lie algebras and Banach Lie–Poisson spaces associated with them. Some of the results given below can be found in papers [1,9,10,12]. Our main aim is to compute the formulas for the adjoint and coadjoint actions of the extended Banach Lie group. A.1. Extensions of Banach Lie groups Let us consider an exact sequence of Banach Lie groups {eN }

N

ι

G

π

H

{eH } .

(A.1)

We assume that N → G → H is a smooth principal bundle, i.e. the maps ι and π are smooth and there exists a smooth local section σ : U → G, where U ⊂ H is an open neighborhood of identity. Additionally we impose on σ the normalization condition σ (eH ) = eG . One can extend σ to a global section σ : H −→ G,

(A.2)

but in general such extension will not be smooth. Let us define the map Ψ : N × H −→ G by Ψ (n, h) := ι(n)σ (h).

(A.3)

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Since G/N ∼ = H , we get that for g ∈ G there exists a unique n ∈ N such that g = ι(n)σ (π(g)). Thus Ψ is a locally smooth bijection with the inverse Ψ −1 : G −→ N × H given by −1 Ψ −1 (g) := ι−1 g σ π(g) , σ π(g) . Using Ψ one defines the multiplication on N × H by (n1 , h1 ) · (n2 , h2 ) := Ψ −1 Ψ (n1 , h1 )Ψ (n2 , h2 )

(A.4)

(A.5)

and can express it as follows (n1 , h1 ) · (n2 , h2 ) = n1 Φ(h1 )(n2 )Ω(h1 , h2 ), h1 h2 ,

(A.6)

where maps Φ : H → Aut(N ) and Ω : H × H → N are defined by Φ(h)(n) := ι−1 σ (h)ι(n)σ (h)−1 , Ω(h1 , h2 ) := ι−1 σ (h1 )σ (h2 )σ (h1 h2 )−1 .

(A.7) (A.8)

Let us denote by Φ the map Φ : H × N (h, n) → Φ(h)(n) ∈ N.

(A.9)

One has the following properties of Φ and Ω: Φ(eH ) = id,

(A.10a)

Ω(eH , h) = Ω(h, eH ) = eN , Ω(h1 , h2 )Ω(h1 h2 , h3 ) = Φ(h1 ) Ω(h2 , h3 ) Ω(h1 , h2 h3 ),

(A.10b)

Ω(h1 , h2 )Φ(h1 h2 )(n) = Φ(h1 ) ◦ Φ(h2 )(n)Ω(h1 , h2 ).

(A.10d)

(A.10c)

Forgetting about definitions (A.7), (A.8) we can consider Φ and Ω as abstract maps satisfying conditions (A.10). We assume that the map Φ is smooth on U × N and Ω is smooth on some neighborhood of (eH , eH ). Moreover we have to assume that the map H x → Ω(h, x)Ω(hxh−1 , h)−1 is smooth for all h ∈ H on a neighborhood of eH (if H is connected then this condition is automatically satisfied). Under these conditions there exists on N × H a structure of Banach Lie group defined by (A.6), see [10]. One denotes this Banach Lie group by N ×Φ,Ω H . We get that the inverse of (n, h) in N ×Φ,Ω H is given by −1 (n, h)−1 = Ω h−1 , h Φ h−1 n−1 , h−1

(A.11)

and the inner automorphism I(n,h) (m, g) := (n, h) · (m, g) · (n, h)−1 can be expressed in terms of Φ and Ω as −1 I(n,h) (m, g) = nΦ(h)(m)Ω(h, g)Ω hgh−1 , h Φ hgh−1 n−1 , hgh−1 . (A.12) Now, let us pass to the extensions of related Banach Lie algebras.


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A.2. Extensions of Banach Lie algebras We will denote the Banach Lie algebras of G, H , N by g, h, n respectively. Taking derivatives of the maps in (A.1) we obtain the exact sequence of Banach Lie algebras {0}

n

Dι(eN )

g

Dπ(eG )

h

{0} .

(A.13)

The derivative DΨ −1 (eG ) : g → n ⊕ h of Ψ −1 at the point eG allows us to identify the Banach space g with n ⊕ h. The adjoint representation of N ×Φ,Ω H on g ∼ = n ⊕ h can be locally computed from (A.12) and it is the following Ad(n,h) (ζ, η) = Adn D2 Φ(h, eN )(ζ ) + D2 Ω(h, eH )(η) − D1 Ω(eH , h)(Adh η) + DLn n−1 ◦ D1 Φ eH , n−1 ◦ Adh (η), Adh η (A.14) for (n, h) ∈ N ×Φ,Ω U , (η, ζ ) ∈ n ⊕ h, where Φ is defined by (A.9), and Di denotes partial derivative with respect to the ith argument. We denote by Ln the left group action Ln m = nm, n, m ∈ N , on itself. Differentiating (A.14) we obtain the formula for the Lie bracket

(ζ, η), (ν, ξ ) := [ζ, ν] + ϕ(η)(ν) − ϕ(ξ )(ζ ) + ω(η, ξ ), [η, ξ ] ,

(A.15)

for (ζ, η), (ν, ξ ) ∈ n ⊕ h, where ϕ : h → Aut(n) is the linear continuous map and ω : h × h → n is the continuous bilinear skew symmetric map defined by Φ and Ω as follows: ϕ(η)(ζ ) := D1 D2 Φ(eH , eN )(η, ζ ),

(A.16)

ω(η, ξ ) := D1 D2 Ω(eH , eH )(η, ξ ) − D1 D2 Ω(eH , eH )(ξ, η).

(A.17)

In these formulas D1 D2 is the second mixed partial derivative. The maps ϕ and ω satisfy the following infinitesimal version of conditions (A.10):

ω η, η , η

+ ω η , η

, η + ω η

, η , η − ϕ(η) ω η , η

− ϕ η ω η

, η − ϕ η

ω η, η = 0,

(A.18)

adω(η,η ) +ϕ η, η − ϕ(η), ϕ η = 0

(A.19)

and

for all η, η , η

∈ h. If we forget about the underlying Banach Lie groups and consider their Banach Lie algebras only, then the maps ϕ and ω satisfying conditions (A.18)–(A.19) with additional smoothness conditions, define the structure of Banach Lie algebra on n ⊕ h, see [1,12].

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A.3. Extensions of Banach Lie–Poisson spaces According to [11] the Banach Lie–Poisson space is a Banach space b such that its dual b∗ is Banach Lie algebra with the property ad∗x b ⊂ b ⊂ b∗∗

(A.20)

for all x ∈ b∗ . This property allows us to define the Poisson bracket on b

{f, g}(b) = Df (b), Dg(b) , b ,

(A.21)

where Df (b), Dg(b) ∈ b∗ are Fréchet derivatives at point b ∈ b. The bracket makes the Banach space b a Banach Poisson space in the sense of [11]. Let us assume that Banach Lie algebras n, h and g possess predual Banach spaces n∗ , h∗ and g∗ satisfying condition (A.20). We also assume that maps (Dι(eN ))∗ , (Dπ(eG ))∗ (h∗ ) dual to Dι(eN ) and Dπ(eG ) preserve predual spaces, i.e. ∗ ∗ Dι(eN ) (g∗ ) ⊂ n∗ , Dπ(eG ) (h∗ ) ⊂ g∗ . (A.22) In that situation one obtains the exact sequence of predual Banach spaces {0}

h∗

(Dπ(eG ))∗

g∗

(Dι(eG ))∗

n∗

{0} ,

(A.23)

see Lemma 3.7 in [12]. We can identify g∗ with n∗ ⊕ h∗ by the map dual to the derivative DΨ −1 (eG ) at the point eG . This identification allows us to compute coadjoint actions of N ×Φ,Ω H and n ⊕ h on n∗ ⊕ h∗ as follows: ∗ ∗ ∗ Ad∗(n,h) (τ, μ) = D2 Φ(h, eN ) Ad∗n τ, D2 Ω(h, eH ) Ad∗n − Ad∗h D1 Ω(eH , h) Ad∗n ∗ τ + Ad∗h μ , + Ad∗h D1 Φ eH , n−1 (A.24) where (n, h) ∈ N ×Φ,Ω U , (τ, μ) ∈ n∗ ⊕ h∗ and ∗ ∗ ∗ ad∗(ζ,η) (τ, μ) = ad∗ζ τ + ϕ(η) τ, − ϕ( · )(ζ ) τ + ω(η, ·) τ + ad∗η μ

(A.25)

for (ζ, η) ∈ n ⊕ h, (τ, μ) ∈ n∗ ⊕ h∗ . The coadjoint representation (A.25) satisfies condition (A.20) if and only if ∗ ∗ ∗ ϕ(η) (n∗ ) ⊂ n∗ , ϕ(·)(ζ ) (n∗ ) ⊂ h∗ , ω(η, ·) (n∗ ) ⊂ h∗ .

(A.26)

So, under these conditions the Banach space n ⊕ h is a Banach Lie–Poisson space. Using definition (A.21) and Lie bracket (A.15) we obtain the Poisson bracket on n∗ ⊕ h∗ :

{f, g}(τ, μ) = [D1 f, D1 g] + ϕ(D2 f )(D1 g) − ϕ(D2 g)(D1 f ) + ω(D2 f, D2 g), τ

+ [D2 f, D2 g], μ (A.27) for f, g ∈ C ∞ (n∗ ⊕ h∗ ).


3293

Further investigation of extensions of Lie groups and Lie algebras is beyond the scope of this paper, so for more information we refer to [1,9,10,12]. Appendix B. Magri method We briefly recall the Magri method of constructing integrals of motion in involution. For more details see e.g. [7]. Let us consider a pencil of compatible Poisson brackets {·,·}ε := {·,·}1 + ε{·,·}2

(B.1)

where ε ∈ R. Compatibility of Poisson brackets means that {·,·}ε is also a Poisson bracket for any parameter ε. Let Iεk be a family of Casimirs for Poisson bracket {·,·}ε indexed by k ∈ N, i.e. k Iε , · ε = 0.

(B.2)

Assuming that Iεk depends analytically on the parameter ε one expands the equality (B.2) and computes the coefficients in front of ε n . Thus one obtains that {hk0 , ·}1 = 0 and

hkl , ·

1

= hkl+1 , · 2 ,

l = 0, 1, . . .

(B.3)

where hkl are defined by =

Iεk

∞

hkl ε l .

(B.4)

l=0

Due to relation (B.3), the sequence {hkl }l∈N∪{0} is called a Magri chain. By using (B.3) twice one gets that

hkl , hkn

1

= hkl−1 , hkn+1 1 .

(B.5)

Next, by iterating this procedure one concludes that k k

hl , hn 1 = hk0 , hkn+l 1 = 0.

(B.6)

Thus functions hkl are in involution k k

hl , hn ε = hkl , hkn 1 = hkl , hkn 2 = 0 for all Poisson brackets under consideration.

(B.7)

3294


References [1] D. Alekseevsky, P.W. Michor, W. Ruppert, Extensions of Lie algebras, ESI preprint 881. [2] D. Belti¸taˇ , T.S. Ratiu, A.B. Tumpach, The restricted Grassmannian, Banach Lie–Poisson spaces, and coadjoint orbits, J. Funct. Anal. 247 (2007) 138–168. [3] A.L. Carey, C.A. Hurst, D.M. O’Brien, Automorphisms of the canonical anticommutation relations and index theory, J. Funct. Anal. 48 (1982) 360–393. [4] P. de la Harpe, Classical Banach–Lie Algebras and Banach–Lie Groups of Operators in Hilbert Space, SpringerVerlag, 1972. [5] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, vol. 1, Birkhäuser, 1990. [6] D.D. Holm, Geometric Mechanics, Part I: Dynamics and Symmetry, Imperial College Press, 2008. [7] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978) 1156–1162. [8] T. Miwa, M. Jimbo, E. Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge University Press, 2000. [9] K.-H. Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier 52 (2002) 1365–1442. [10] K.-H. Neeb, Non-abelian extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier 56 (2006) 209–271. [11] A. Odzijewicz, T.S. Ratiu, Banach Lie–Poisson spaces and reduction, Comm. Math. Phys. 243 (2003) 1–54. [12] A. Odzijewicz, T.S. Ratiu, Extensions of Banach Lie–Poisson spaces, J. Funct. Anal. 217 (2004) 103–125. [13] A. Odzijewicz, T.S. Ratiu, Induced and coinduced Banach Lie–Poisson spaces and integrability, J. Funct. Anal. 255 (2008) 1225–1272. [14] R.T. Powers, E. Størmer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970) 1–33. [15] A. Pressley, G. Segal, Loop Groups, Oxford Math. Monogr., Clarendon Press, Oxford, 1986. [16] M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, 1978. [17] M. Sato, Y. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, Lect. Notes in Num. Appl. Anal. 5 (1982) 259–271. [18] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, 1960. [19] J. Schwinger, Field theory commutators, Phys. Rev. Lett. 3 (6) (1959) 296–297. [20] G.B. Segal, G. Wilson, Loop groups and equations of KdV type, Publ. Math. Inst. Hautes Études Sci. 61 (1985) 5–65. [21] A. Sergeev, Kähler Geometry of Loop Spaces, MCNMO, Moscow, 2001 (in Russian). [22] D. Shale, W.F. Stinespring, States on the Clifford algebra, Ann. of Math. 80 (1964) 365–381. [23] M. Spera, T. Wurzbacher, Determinants, Pfaffians, and quasi-free representations of the CAR algebra, Rev. Math. Phys. 10 (5) (1998) 705–721. [24] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979. [25] B. Thaller, The Dirac Equation, Springer, 1992. [26] T. Wurzbacher, Fermionic second quantization and the geometry of the restricted Grassmannian, in: A. Huckleberry, T. Wurzbacher (Eds.), Infinite Dimensional Kähler Manifolds, in: DMV Seminar, vol. 31, Birkhäuser, Basel, 2001.


Lane–Emden systems with negative exponents ✩ Marius Ghergu School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland Received 15 September 2009; accepted 3 February 2010 Available online 12 February 2010 Communicated by J. Coron

Abstract We study the elliptic system ⎧ ⎨ −u = u−p v −q −v = u−r v −s ⎩ u=v=0

in Ω, in Ω, on ∂Ω,

in a bounded domain Ω ⊂ RN (N 1) with a smooth boundary, p, s 0 and q, r > 0. We investigate the existence, non-existence, and uniqueness of C 2 (Ω) ∩ C(Ω) solutions in terms of p, q, r and s. A necessary and sufficient condition for the C 1 -regularity of solutions up to the boundary is also obtained. © 2010 Elsevier Inc. All rights reserved. Keywords: Lane–Emden equation; Elliptic system; Negative exponent; Boundary behavior

1. Introduction In this paper we study the elliptic system ⎧ ⎨ −u = u−p v −q , u > 0 −v = u−r v −s , v > 0 ⎩ u=v=0


(1)

✩ This research was partially supported by the ESF Research Network “Harmonic and Complex Analysis and Applications” (HCAA). E-mail address: [email protected].

0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.003

3296

M. Ghergu / Journal of Functional Analysis 258 (2010) 3295–3318

where Ω ⊂ RN (N 1) is a bounded domain with C 2 -boundary, p, s 0 and q, r > 0. By solution of (1) we understand a pair (u, v) with u, v ∈ C 2 (Ω) ∩ C(Ω) such that u, v > 0 in Ω and satisfies (1) pointwise. The first motivation for the study of system (1) comes from the so-called Lane–Emden equation (see [5,8,15]) −u = up

in BR (0), R > 0,

(2)

subject to Dirichlet boundary condition. In astrophysics, the exponent p is called polytropic index and positive radially symmetric solutions of (2) are used to describe the structure of the polytropic stars (we refer the interested reader to the book by Chandrasekhar [2] for an account on the above equation as well as for various mathematical techniques to describe the behavior of the solution to Lane–Emden equation). Systems of type (1) with p, s 0 and q, s < 0 have received considerably attention in the last decade (see, e.g., [1,3,6,7,16,18–23] and the references therein). It has been shown that for such range of exponents system (1) has a rich mathematical structure. Various techniques such as moving plane method, Pohozaev-type identities, rescaling arguments have been developed and suitably adapted to deal with (1) in this case. Recently, there has been some interest in systems of type (1) where not all the exponents are negative. In [10–12] the system (1) is considered under the hypothesis p, r < 0 < q, s. This corresponds to the singular Gierer–Meinhardt system arising in the molecular biology. In [9] the authors provide a nice sub and supersolution device that applies to general systems both in cooperative and non-cooperative setting. This method was then used to discuss singular counterpart of some well-known models such as Gierer–Meinhardt, Lotka–Voltera or predator-prey systems. In this paper, we shall be concerned with system (1) in case p, s 0 and q, r > 0. This corresponds to the prototype equation (2) in which the polytropic index p is negative. For such range of exponents, the above mentioned methods do not apply; another difficulties in dealing with system (1) come from the non-cooperative character of our system and from the lack of a variational structure. In turn, our approach relies on the boundary behavior of solutions to (2) (with p < 0) or more generally, to singular elliptic problems of the type

−u = k δ(x) u−p , u=0

u > 0 in Ω, on ∂Ω,

(3)

where δ(x) = dist(x, ∂Ω),

x ∈ Ω,

and k : (0, ∞) → (0, ∞) is a decreasing function such that limt0 k(t) = ∞. The approach we adopt in this paper can be used to study more general systems in the form

−Lu = f (x, u, v), u > 0 in Ω, −Lv = g(x, u, v), v > 0 in Ω, u=v=0 on ∂Ω,

where L is a second order differential operator not necessarily in divergence form and f (x, u, v) = k1 (x)u−p v −q ,

g(x, u, v) = k2 (x)u−r v −s ,


3297

or f (x, u, v) = k11 (x)u−p + k12 (x)v −q ,

g(x, u, v) = k21 (x)u−r + k22 (x)v −s ,

with ki , kij : Ω → (0, ∞) (i, j = 1, 2) continuous functions that behave like −a

δ(x)

b

log

A δ(x)

near ∂Ω,

(4)

for some A, a > 0 and b ∈ R. Our first result concerning the study of (1) is the following. Theorem 1.1 (Non-existence). Let p, s 0, q, r > 0 be such that one of the following conditions holds: 2−q (i) r min{1, 1+p } 2;

(ii) q min{1, 2−r 1+s } 2; (iii) p > max{1, r − 1}, 2r > (1 − s)(1 + p) and q(1 + p − r) > (1 + p)(1 + s); (iv) s > max{1, q − 1}, 2q > (1 − p)(1 + s) and r(1 + s − q) > (1 + p)(1 + s). Then the system (1) has no solutions. Remark that condition (i) in Theorem 1.1 restricts the range of the exponent q to the interval (0, 2) while in (iii) the exponent q can take any value greater than 2, provided we adjust the other three exponents p, r, s accordingly. The same remark applies for the exponent r from the above conditions (ii) and (iv). The existence of solutions to (1) is obtained under the following assumption on the exponents p, q, r, s: (1 + p)(1 + s) − qr > 0.

(5)

We also introduce the quantities 2−r , α = p + q min 1, 1+s

2−q β = r + s min 1, . 1+p

The above values of α and β are related to the boundary behavior of the solution to the singular elliptic problem (3) as explained in Proposition 2.6 below. Our existence result is as follows. Theorem 1.2 (Existence). Let p, s 0, q, r > 0 satisfy (5) and one of the following conditions: (i) α 1 and r < 2; (ii) β 1 and q < 2; (iii) p, s 1 and q, r < 2. Then, the system (1) has at least one solution.

3298


The proof of the existence is based on the Schauder’s fixed point theorem in a suitable chosen closed convex subset of C(Ω) × C(Ω) that contains all the functions having a certain rate of decay expressed in terms of the distance function δ(x) up to the boundary of Ω. From Theorem 1.1(i)–(ii) and Theorem 1.2(i)–(ii) we have the following necessary and sufficient conditions for the existence of solutions to (1): Corollary 1.3. Let p, s 0, q, r > 0 satisfy (5). (i) Assume p + q 1. Then system (1) has solutions if and only if r < 2; (ii) Assume r + s 1. Then system (1) has solutions if and only if q < 2. A particular feature of system (1) is that it does no posses C 2 (Ω) solutions. Indeed, due to the fact that q, r < 0 and to the homogeneous Dirichlet boundary condition imposed on u and v we have that u−p v −q and u−r v −s are unbounded around ∂Ω, so there are no C 2 (Ω) solutions of (1). In turn, C 2 (Ω) ∩ C 1 (Ω) may exist and our next result provides necessary and sufficient conditions in terms of p, q, r and s for the existence of such solutions. Theorem 1.4 (C 1 -regularity). Let p, s 0, q, r > 0 satisfy (5). Then: (i) System (1) has a solution (u, v) with u ∈ C 1 (Ω) if and only if α < 1 and r < 2; (ii) System (1) has a solution (u, v) with v ∈ C 1 (Ω) if and only if β < 1 and q < 2; (iii) System (1) has a solution (u, v) with u, v ∈ C 1 (Ω) if and only if p + q < 1 and r + s < 1. Another feature of system (1) is that under some conditions on p, q, r, s it has a unique solution (see Theorem 1.5 below). This is a striking difference between our setting and the case p, s 0 and q, r < 0 largely investigated in the literature so far, where the uniqueness does not seem to occur. In our framework, the uniqueness is achieved from the boundary behavior of solution to (1) deduced from the study of the prototype model (3). Theorem 1.5 (Uniqueness). Let p, s 0, q, r > 0 satisfy (5) and one of the following conditions: (i) p + q < 1 and r < 2; (ii) r + s < 1 and q < 2. Then, the system (1) has a unique solution. The rest of the paper is organized as follows. In Section 2 we obtain some useful properties related to the boundary behavior of the solution to (3). Sections 3–6 are devoted to the proofs of the above results. 2. Preliminary results In this section we collect some old and new results concerning problems of type (3). Note that the method of sub and supersolutions is also valid in the singular framework as explained in [13, Theorem 1.2.3]. Our first result is a straightforward comparison principle between subsolutions and supersolutions for singular elliptic equations.


3299

Proposition 2.1. Let p 0 and φ : Ω → (0, ∞) be a continuous function. If u is a subsolution and u is a supersolution of

−u = φ(x)u−p , u=0

u > 0 in Ω, on ∂Ω,

then u u in Ω. Proof. If p = 0 the result follows directly from the maximum principle. Let now p > 0. Assume by contradiction that the set ω := {x ∈ Ω: u(x) < u(x)} is not empty and let w := u − u. Then, w achieves its maximum on Ω at a point that belongs to ω. At that point, say x0 , we have

0 −w(x0 ) φ(x0 ) u(x0 )−p − u(x0 )−p < 0, which is a contradiction. Therefore, ω = ∅, that is, u u in Ω.

2

Proposition 2.2. Let u ∈ C 2 (Ω) ∩ C(Ω) be such that u = 0 on ∂Ω and 0 −u cδ(x)−a

in Ω,

where 0 < a < 2 and c > 0. Then, u ∈ C 0,γ (Ω) for some 0 < γ < 1. Furthermore, if 0 < a < 1, then u ∈ C 1,1−a (Ω). Proof. Let G denote Green’s function for the negative Laplace operator. Thus, for all x ∈ Ω we have u(x) = − G(x, y)u(y) dy. Ω

Let x1 , x2 ∈ Ω. Then u(x1 ) − u(x2 ) −

G(x1 , y) − G(x2 , y)u(y) dy

Ω

c

G(x1 , y) − Gx (x2 , y)δ(y)−a dy.

Ω

Next, using the method in [14, Theorem 1.1] we have u(x1 ) − u(x2 ) C|x1 − x2 |γ

for some 0 < γ < 1.

Hence u ∈ C 0,γ (Ω). Assume now 0 < a < 1. Then, Gx (x, y)u(y) dy

∇u(x) = − Ω

for all x ∈ Ω,

3300


and ∇u(x1 ) − ∇u(x2 ) −

Gx (x1 , y) − Gx (x2 , y)u(y) dy

Ω

c

Gx (x1 , y) − Gx (x2 , y)δ(y)−a dy.

Ω

The same technique as in [14, Theorem 1.1] yields ∇u(x1 ) − ∇u(x2 ) C|x1 − x2 |1−a Therefore u ∈ C 1,1−a (Ω).

for all x1 , x2 ∈ Ω.

2

Proposition 2.3. Let (u, v) be a solution of system (1). Then, there exists a constant c > 0 such that u(x) cδ(x)

and v(x) cδ(x)

in Ω.

(6)

Proof. Let w be the solution of

−w = 1, w=0

w>0

in Ω, on ∂Ω.

(7)

Using the smoothness of ∂Ω, we have w ∈ C 2 (Ω) and by Hopf’s boundary point lemma (see [17]), there exists c0 > 0 such that w(x) c0 δ(x) in Ω. Since −u C = −(Cw) in Ω, for some constant C > 0, by standard maximum principle we deduce u(x) Cw(x) cδ(x) in Ω and similarly v(x) cδ(x) in Ω, where c > 0 is a positive constant. 2 Let (λ1 , ϕ1 ) be the first eigenvalue/eigenfunction of − in Ω. It is well known that λ1 > 0 and ϕ1 ∈ C 2 (Ω) has constant sign in Ω. Further, using the smoothness of Ω and normalizing ϕ1 with a suitable constant, we can assume c0 δ(x) ϕ1 (x) δ(x)

in Ω,

(8)

1 for some 0 < c0 < 1. By Hopf’s boundary point lemma we have ∂ϕ ∂n < 0 on ∂Ω, where n is the outer unit normal vector at ∂Ω. Hence, there exists ω Ω and c > 0 such that

|∇ϕ1 | > c

in Ω \ ω.

(9)

Theorem 2.4. Let p 0, A > diam(Ω) and k : (0, A) → (0, ∞) be a decreasing function such that A tk(t) dt = ∞. 0


3301

Then, the inequality

−u k δ(x) u−p , u=0

u > 0 in Ω, on ∂Ω,

(10)

has no solutions u ∈ C 2 (Ω) ∩ C(Ω). Proof. Suppose by contradiction that there exists a solution u0 of (10). For any 0 < ε < A − diam(Ω) we consider the perturbed problem

−u = k δ(x) + ε (u + ε)−p , u=0

u > 0 in Ω, on ∂Ω.

(11)

Then, u = u0 is a supersolution of (11). Also, if w is the solution of problem (7) it is easy to see that u = cw is a subsolution of (11) provided c > 0 is small enough. Further, by Proposition 2.1 it follows that u u in Ω. Thus, by the sub and supersolution method we deduce that problem (11) has a solution uε ∈ C 2 (Ω) such that cw uε u0

in Ω.

(12)

Multiplying with ϕ1 in (11) and then integrating over Ω we find

uε ϕ1 dx =

λ1 Ω

k δ(x) + ε (uε + ε)−p ϕ1 dx.

Ω

Using (12) we obtain M := λ1

u0 ϕ1 dx λ1

Ω

uε ϕ1 dx Ω

k δ(x) + ε (u0 + ε)−p ϕ1 dx,

ω

for all ω Ω. Passing to the limit with ε → 0 in the above inequality and using (8) we find M

−p −p k δ(x) u0 ϕ1 dx c0 u0 ∞

ω

k δ(x) δ(x) dx.

ω

Since ω Ω was arbitrary, we deduce

k δ(x) δ(x) dx < ∞.

Ω

Using the smoothness of ∂Ω, the above condition yields assumption on k. Hence, (10) has no solutions. 2

A 0

tk(t) dt < ∞, which contradicts our

3302


A direct consequence of Theorem 2.4 is the following result. Corollary 2.5. Let p 0 and q 2. Then, there are no functions u ∈ C 2 (Ω) ∩ C(Ω) such that

−u δ(x)−q u−p , u=0

u > 0 in Ω, on ∂Ω.

Proposition 2.6. Let p 0 and 0 < q < 2. There exists c > 0 and A > diam(Ω) such that any supersolution u of

−u = δ(x)−q u−p , u=0

u > 0 in Ω, on ∂Ω,

(13)

satisfies: (i) u(x) cδ(x) in Ω, if p + q < 1; 1

A (ii) u(x) cδ(x) log 1+p ( δ(x) ) in Ω if p + q = 1; 2−q

(iii) u(x) cδ(x) 1+p in Ω, if p + q > 1. A similar result holds for subsolutions of (13). Proof. If p > 0 then the result follows from Theorem 3.5 in [4] (see also [13, Section 9]). If p = 0 we proceed as in [4, Theorem 3.5], namely, for m > 0 we show that the function ⎧ mϕ (x) 1

if q < 1, ⎪ ⎨ A if q = 1, A > diam(Ω), u(x) = mϕ1 (x) log ϕ1 (x) ⎪ ⎩ mϕ1 (x)2−q if q > 1, satisfies −u δ(x)−q in Ω. Thus, the estimates in Proposition 2.6 follow from (8) and the maximum principle. 2 Theorem 2.7. Let 0 < a < 1, A > diam(Ω), p 0 and q > 0 be such that p + q = 1. Then, the problem

−u = δ(x)−q log−a u=0

A u−p , δ(x)

u > 0 in Ω,

(14)

on ∂Ω,

has a unique solution u which satisfies 1−a

c1 δ(x) log 1+p for some c1 , c2 > 0.

A δ(x)

1−a

u(x) c2 δ(x) log 1+p

A δ(x)

in Ω,

(15)


3303

Proof. Let w(x) = ϕ1 (x) logb where b =

1−a 1+p

A , ϕ1 (x)

x ∈ Ω,

∈ (0, 1). A straightforward computation yields

A A + b |∇ϕ1 |2 − λ1 ϕ12 ϕ1−1 logb−1 ϕ1 (x) ϕ1 (x)

A + b(1 − b)|∇ϕ1 |2 ϕ1−1 logb−2 in Ω. ϕ1 (x)

−w = λ1 ϕ1 logb

Using (9) we can find C1 , C2 > 0 such that C1 ϕ1−1 logb−1

A ϕ1 (x)

−w C2 ϕ1−1 logb−1

A ϕ1 (x)

in Ω,

that is, −q

C1 ϕ1 log−a

A A −q w −p −w C2 ϕ1 log−a w −p ϕ1 (x) ϕ1 (x)

in Ω.

We now deduce that u = mw and u = Mw are respectively subsolution and supersolution of (14) for suitable 0 < m < 1 < M. Hence, the problem (14) has a solution u ∈ C 2 (Ω) ∩ C(Ω) such that 1−a

mϕ1 log 1+p

A ϕ1 (x)

1−a

u M log 1+p

A ϕ1 (x)

in Ω.

(16)

The uniqueness follows from Proposition 2.1 while the boundary behavior of u follows from (16) and (8). This finishes the proof. 2 Corollary 2.8. Let C > 0 and a, A, p, q be as in Theorem 2.7. Then, there exists c > 0 such that any solution u of

−q

−u Cδ(x)

−a

log

u=0

A u−p , δ(x)

u > 0 in Ω, on ∂Ω,

satisfies u(x) cδ(x) log

1−a 1+p

A δ(x)

in Ω.

3304


Proposition 2.9. Let A > 3 diam(Ω) and C > 0. There exists c > 0 such that any solution u ∈ C 2 (Ω) ∩ C(Ω) of

−u Cδ

−1

(x) log

−1

u=0

A , δ(x)

u > 0 in Ω, on ∂Ω,

satisfies

A u(x) cδ(x) log log δ(x)

in Ω.

(17)

Proof. Let

A , w(x) = ϕ1 (x) log log ϕ1 (x)

x ∈ Ω.

An easy computation yields

|∇ϕ1 |2 − λ1 ϕ12 |∇ϕ1 |2 A −w = λ1 ϕ1 log log + + ϕ1 (x) ϕ1 log( ϕ1A(x) ) ϕ1 log2 ( ϕ1A(x) )

c0

in Ω,

ϕ1 log( ϕ1A(x) )

for some c0 > 0. Using (8) we can find m > 0 small enough such that −(mw)

C A δ(x) log( δ(x) )

in Ω.

Now by maximum principle we deduce u mw in Ω and by (8) we obtain that u satisfies the estimate (17). 2 Theorem 2.10. Let p 0, A > diam(Ω) and a ∈ R. Then, problem

−u = δ(x)−2 log−a u=0

A u−p , δ(x)

u > 0 in Ω,

(18)

on ∂Ω,

has solutions if and only if a > 1. Furthermore, if a > 1 then (21) has a unique solution u and there exist c1 , c2 > 0 such that c1 log

1−a 1+p

A δ(x)

u(x) c2 log

1−a 1+p

A δ(x)

in Ω.

(19)


3305

Proof. Fix B > A be such that the function k : (0, B) → R, k(t) = t −2 log−a ( Bt ) is decreasing on (0, A). Then, any solution u of (18) satisfies −u ck δ(x) u−p , u > 0 in Ω, u=0 on ∂Ω, A where c > 0. By virtue of Theorem 2.4 we deduce 0 tk(t) dt < ∞, that is, a > 1. For a > 1, let

B b , x ∈ Ω, w(x) = log ϕ1 (x) where b =

1−a 1+p

< 0. It is easy to see that

B ϕ1 (x)

B in Ω. − b(b − 1)|∇ϕ1 |2 ϕ1−2 logb−2 ϕ1 (x)

−w = −b |∇ϕ1 |2 + λ1 ϕ12 ϕ1−2 logb−1

Choosing B > 0 large enough, we may assume

B log 2(1 − b) ϕ1 (x)

in Ω.

(20)

Therefore, from (9) and (20) there exist C1 , C2 > 0 such that

B B −2 −2 b−1 b−1 −w C2 ϕ1 log in Ω, C1 ϕ1 log ϕ1 (x) ϕ1 (x) that is, C1 ϕ1−2 log−a

B B w −p −w C2 ϕ1−2 log−a w −p ϕ1 (x) ϕ1 (x)

in Ω.

As before, from (8) it follows that u = mw and u = Mw are respectively subsolution and supersolution of (18) provided m > 0 is small and M > 1 is large enough. The rest of the proof is the same as for Theorem 2.7. 2 Corollary 2.11. Let C > 0, p 0, A > diam(Ω) and a > 1. Then, there exists c > 0 such that any solution u ∈ C 2 (Ω) ∩ C(Ω) of

A −2 −a u−p , u > 0 in Ω, −u Cδ(x) log (21) ϕ1 (x) u=0 on ∂Ω, satisfies u(x) c log

1−a 1+p

A δ(x)

in Ω.

3306


3. Proof of Theorem 1.1 Since the system (1) is invariant under the transform (u, v, p, q, r, s) → (v, u, s, r, q, p), we only need to prove (i) and (iii). (i) Assume that there exists (u, v) a solution of system (1). Note that from (i) we have 0 < q < 2. Also, using Proposition 2.3, we can find c > 0 such that (6) holds. Case 1: p + q < 1. From our hypothesis (i) we deduce r 2. Using the estimates (6) in the first equation of the system (1) we find

−u c1 δ(x)−q u−p , u=0

u>0

in Ω, on ∂Ω,

(22)

for some c1 > 0. From Proposition 2.6(i) we now deduce u(x) c2 δ(x) in Ω, for some c2 > 0. Using this last estimate in the second equation of (1) we find

−v c3 δ(x)−r v −s , u=0

v > 0 in Ω, on ∂Ω,

(23)

where c3 > 0. According to Corollary 2.5, this is impossible, since r 2. Case 2: p + q > 1. From hypothesis (i) we also have r(2−q) 1+p 2. In the same manner as above, u satisfies (22). Thus, by Proposition 2.6(iii), there exists c4 > 0 such that 2−q

u(x) c4 δ(x) 1+p

in Ω.

Using this estimate in the second equation of system (1) we obtain

− r(2−q) 1+p v −s ,

−v c5 δ(x) u=0

v > 0 in Ω, on ∂Ω,

for some c5 > 0, which is impossible in view of Corollary 2.5, since

r(2−q) 1+p

2.

Case 3: p + q = 1. From (i) it follows that r 2. As in the previous two cases, we easily find that u is a solution of (22), for some c1 > 0. Using Proposition 2.6(ii), there exists c6 > 0 such that u(x) c6 δ(x) log

1 1+p

A δ(x)

in Ω,

for some A > 3 diam(Ω). Using this estimate in the second equation of (1) we obtain

−v c7 δ(x)−r log u=0

r − 1+p

A v −s , δ(x)

v > 0 in Ω,

where c7 is a positive constant. From Theorem 2.4 it follows that

on ∂Ω,

(24)


1 t

1−r

r − 1+p

log

3307

A dt < ∞. t

0

Since r 2, the above integral condition implies r = 2. Now, using (24) (with r = 2) and Corollary 2.11, there exists c8 > 0 such that p−1 (1+p)(1+s)

v(x) c8 log

A δ(x)

in Ω.

(25)

Using the estimate (25) in the first equation of system (1) we deduce

−u c9 log

q(1−p) (1+p)(1+s)

u=0

A u−p , δ(x)

u > 0 in Ω,

(26)

on ∂Ω,

for some c9 > 0. Fix 0 < a < 1 − p. Then, from (26) we can find a constant c10 > 0 such that u satisfies −u c10 δ(x)−a u−p , u > 0 in Ω, u=0 on ∂Ω. By Proposition 2.6(i) (since a + p < 1) we derive u(x) c11 δ(x) in Ω, where c11 > 0. Using this last estimate in the second equation of (1) we finally obtain (note that r = 2):

−v c12 δ(x)−2 v −s , v=0

v>0

in Ω, on ∂Ω,

which is impossible according to Corollary 2.5. Therefore, the system (1) has no solutions. (iii) Suppose that the system (1) has a solution (u, v) and let M = maxx∈Ω v. From the first equation of (1) we have

−u c1 u−p , u=0

u > 0 in Ω, on ∂Ω, 2

where c1 = M −q > 0. Using Proposition 2.6(iii) there exists c2 > 0 such that u(x) c2 δ(x) 1+p in Ω. Combining this estimate with the second equation of (1) we find

Since

2r 1+p

2r − 1+p v −s ,

−v c3 δ(x) v=0

v > 0 in Ω, on ∂Ω.

+ s > 1, again by Proposition 2.6(iii) we obtain that the function v satisfies 2(1+p−r)

v(x) c4 δ(x) (1+p)(1+s)

in Ω,

for some c4 > 0. Using the above estimate in the first equation of (1) we find c5 > 0 such that

3308


2q(1+p−r) − (1+p)(1+s) u−p ,

−u c5 δ(x) u=0

u>0

in Ω, on ∂Ω,

which contradicts Corollary 2.5 since q(1 + p − r) > (1 + p)(1 + s). Thus, the system (1) has no solutions. This ends the proof of Theorem 1.1. 4. Proof of Theorem 1.2 (i) We divide the proof into six cases according to the boundary behavior of singular elliptic problems of type (3), as described in Proposition 2.6. Case 1: r + s > 1 and α = p + 1 < c2 such that:

q(2−r) 1+s

< 1. By Proposition 2.6(i) and (iii) there exist 0 < c1
0 in Ω, on ∂Ω,

(27)

satisfy u(x) c1 δ(x)

and u(x) c2 δ(x)

in Ω.

(28)

• Any subsolution v and any supersolution v of the problem

−v = δ(x)−r v −s , v=0

v > 0 in Ω, on ∂Ω,

(29)

satisfy 2−r

v(x) c1 δ(x) 1+s

2−r

and v(x) c2 δ(x) 1+s

in Ω.

(30)

We fix 0 < m1 < 1 < M1 and 0 < m2 < 1 < M2 such that q

r

M11+s m2 c1 < c2 M1 m21+p ,

(31)

and q

r

M21+p m1 c1 < c2 M2 m11+s .

(32)

Note that the above choice of mi , Mi (i = 1, 2) is possible in view of (5). Set A = (u, v) ∈ C(Ω) × C(Ω):

m1 δ(x) u(x) M1 δ(x) 2−r

in Ω 2−r

m2 δ(x) 1+s v(x) M2 δ(x) 1+s

in Ω

.

For any (u, v) ∈ A, we consider (T u, T v) the unique solution of the decoupled system


⎧ ⎨ −(T u) = v −q (T u)−p , T u > 0 −(T v) = u−r (T v)−s , T v > 0 ⎩ Tu=Tv =0


3309

(33)

and define F : A → C(Ω) × C(Ω)

by F (u, v) = (T u, T v) for any (u, v) ∈ A.

(34)

Thus, the existence of a solution to system (1) follows once we prove that F has a fixed point in A. To this aim, we shall prove that F satisfies the conditions: F (A) ⊆ A,

F is compact and continuous.

Then, by Schauder’s fixed point theorem we deduce that F has a fixed point in A, which, by standard elliptic estimates, is a classical solution to (1). Step 1: F (A) ⊆ A. Let (u, v) ∈ A. From 2−r

v(x) M2 δ(x) 1+s

in Ω,

it follows that T u satisfies q(2−r) −q −(T u) M2 δ(x)− 1+s (T u)−p , Tu=0

Tu>0

in Ω, on ∂Ω.

q

Thus, u := M21+p T u is a supersolution of (27). By (28) and (32) we obtain q − 1+p

T u = M2

q − 1+p

u c 1 M2

δ(x) m1 δ(x)

in Ω.

2−r

From v(x) m2 δ(x) 1+s in Ω and the definition of T u we deduce that q(2−r) −q −(T u) m2 δ(x)− 1+s (T u)−p , T u > 0 in Ω, Tu=0 on ∂Ω. q

Thus, u := m21+p T u is a subsolution of problem (27). Hence, from (28) and (31) we obtain q − 1+p

T u = m2

q − 1+p

u c2 m2

δ(x) M1 δ(x)

in Ω.

We have proved that T u satisfies m1 δ(x) T u M1 δ(x)

in Ω.

In a similar manner, using the definition of A and the properties of the sub and supersolutions of problem (29) we show that T v satisfies 2−r

2−r

m2 δ(x) 1+s T v M2 δ(x) 1+s Thus, (T u, T v) ∈ A for all (u, v) ∈ A, that is, F (A) ⊆ A.

in Ω.

3310


Step 2: F is compact and continuous. Let (u, v) ∈ A. Since F (u, v) ∈ A, one can find 0 < a < 2 such that 0 −(T u), −(T v) cδ(x)−a

in Ω,

for some positive constant c > 0. Using Proposition 2.2 we now deduce T u, T v ∈ C 0,γ (Ω) (0 < γ < 1). Since the embedding C 0,γ (Ω) → C(Ω) is compact, it follows that F is also compact. It remains to prove that F is continuous. To this aim, let {(un , vn )} ⊂ A be such that un → u and vn → v in C(Ω) as n → ∞. Using the fact that F is compact, there exists (U, V ) ∈ A such that up to a subsequence we have T un → U,

T vn → V

in C(Ω) as n → ∞.

On the other hand, by standard elliptic estimates, the sequences {T un } and {T vn } are bounded in C 2,β (ω) (0 < β < 1) for any smooth open set ω Ω. Therefore, up to a diagonally subsequence, we have T un → U,

T vn → V

in C 2 (ω) as n → ∞,

for any smooth open set ω Ω. Passing to the limit in the definition of T un and T vn we find that (U, V ) satisfies

−U = v −q U −p , U > 0 in Ω, −V = u−r V −s , V > 0 in Ω, U =V =0 on ∂Ω.

By uniqueness of (33), it follows that T u = U and T v = V . Hence T un → T u,

T vn → T v

in C(Ω) as n → ∞.

This proves that F is continuous. We are now in a position to apply the Schauder’s fixed point theorem. Thus, there exists (u, v) ∈ A such that F (u, v) = (u, v), that is, T u = u and T v = v. By standard elliptic estimates, it follows that (u, v) is a solution of system (1). The remaining five cases will be considered in a similar way. Due to the different boundary behavior of solutions described in Proposition 2.6, the set A and the constants c1 , c2 have to be modified accordingly. We shall point out the way we choose these constants in order to apply the Schauder’s fixed point theorem. Case 2: r + s = 1 and α = p + q < 1. According to Proposition 2.6(i)–(ii) there exist 0 < a < 1 and 0 < c1 < 1 < c2 such that: • Any subsolution u of the problem

−u = δ(x)−q u−p , u=0

u > 0 in Ω, on ∂Ω,


3311

satisfies u(x) c2 δ(x)

in Ω.

• Any supersolution u of the problem −u = δ(x)−q(1−a) u−p , u=0

u > 0 in Ω, on ∂Ω,


in Ω.

• Any subsolution v and any supersolution v of problem (29) satisfy the estimates v(x) c2 δ(x)1−a

and v(x) c1 δ(x)

in Ω.

We now define m1 δ(x) u(x) M1 δ(x) A = (u, v) ∈ C(Ω) × C(Ω): m2 δ(x) v(x) M2 δ(x)1−a

in Ω , in Ω

where 0 < mi < 1 < Mi (i = 1, 2) satisfy (31), (32) and

a m2 diam(Ω) < M2 .

(35)

We next define the operator F in the same way as in Case 1 by (33) and (34). The fact that F (A) ⊆ A and that F is continuous and compact follows in the same manner. Case 3: r + s < 1 and α = p + q < 1. In the same manner we define m1 δ(x) u(x) M1 δ(x) A = (u, v) ∈ C(Ω) × C(Ω): m2 δ(x) v(x) M2 δ(x)

in Ω , in Ω

where 0 < mi < 1 < Mi (i = 1, 2) satisfy (31)–(32) for suitable constants c1 and c2 . Case 4: r + s < 1 and α = p + q = 1. The approach is the same as in Case 2 above if we interchange u with v in the initial system (1). Case 5: r + s > 1 and α = p + q = 1. Let 0 < a < 1 be fixed such that ar + s > 1. From Proposition 2.6(i), (iii), there exist 0 < c1 < 1 < c2 such that: • Any subsolution u of the problem

−u = δ(x)− u=0

q(2−ar) 1+s

u−p ,

u > 0 in Ω, on ∂Ω,

satisfies u(x) c2 δ(x)a

in Ω.

3312


• Any supersolution u of the problem

−u = δ(x)− u=0

q(2−r) 1+s

u−p ,

u > 0 in Ω, on ∂Ω,


in Ω.

• Any subsolution v of problem (29) satisfies 2−r

v(x) c2 δ(x) 1+s

in Ω.

• Any supersolution v of the problem

−v = δ(x)−ar v −s , v=0

v > 0 in Ω, on ∂Ω,

satisfies 2−ar

v(x) c1 δ(x) 1+s We now define A = (u, v) ∈ C(Ω) × C(Ω):

in Ω.

m1 δ(x) u(x) M1 δ(x)a 2−ar

in Ω 2−r

m2 δ(x) 1+s v(x) M2 δ(x) 1+s

in Ω

,

where 0 < mi < 1 < Mi (i = 1, 2) satisfy (31)–(32) in which the constants c1 , c2 are those given above and

1−a < M1 , m1 diam(Ω)

r(1−a) m2 diam(Ω) 1+s < M2 .

Case 6: r + s = 1 and α = p + q = 1. We proceed in the same manner as above by considering m1 δ(x) u(x) M1 δ(x)1−a A = (u, v) ∈ C(Ω) × C(Ω): m2 δ(x) v(x) M2 δ(x)1−a

in Ω , in Ω

where 0 < a < 1 is a fixed constant and mi , Mi (i = 1, 2) satisfy (31)–(32) for suitable c1 , c2 > 0 and

a mi diam(Ω) < Mi ,

i = 1, 2.

(iii) Let a=

2(1 + s − q) , (1 + p)(1 + s) − qr

b=

2(1 + p − r) . (1 + p)(1 + s) − qr


3313

Then (1 + p)a + bq = 2,

ar + (1 + s)b = 2.

(36)

Since p + bq > 1 and s + ar > 1, from Proposition 2.6(iii) and (36) above we can find 0 < c1 < 1 < c2 such that: • Any subsolution u and any supersolution u of the problem

−u = δ(x)−bq u−p , u=0

u > 0 in Ω, on ∂Ω,

satisfy u(x) c1 δ(x)a

and u(x) c2 δ(x)a

in Ω.

• Any subsolution v and any supersolution v of the problem

−v = δ(x)−ar v −s , v=0

v > 0 in Ω, on ∂Ω,

satisfy v(x) c1 δ(x)b

and v(x) c2 δ(x)b

in Ω.

As before, we now define m1 δ(x)a u(x) M1 δ(x)a A = (u, v) ∈ C(Ω) × C(Ω): m2 δ(x)b v(x) M2 δ(x)b

in Ω , in Ω

where 0 < m1 < 1 < M1 and 0 < m2 < 1 < M2 satisfy (31)–(32). This concludes the proof of Theorem 1.2. 5. Proof of Theorem 1.4 (i) Assume first that the system (1) has a solution (u, v) with u ∈ C 1 (Ω). Then, there exists c > 0 such that u(x) cδ(x) in Ω. Using this fact in the second equation of (1), we derive that v satisfies the elliptic inequality (23) for some c3 > 0. By Corollary 2.5 this entails r < 2. In order to prove that α < 1 we argue by contradiction. Suppose that α 1 and we divide our argument into three cases. Case 1: r + s > 1. Then, α = p + q(2−r) 1+s 1. From Proposition 2.3 we have u(x) cδ(x) in Ω, for some c > 0. Then v satisfies

−v c1 δ(x)−r v −s , u=0

v > 0 in Ω, on ∂Ω,

(37)

3314

M. Ghergu / Journal of Functional Analysis 258 (2010) 3295–3318 2−r

where c1 > 0. Since r < 2, from Proposition 2.6(iii) we find v(x) c2 δ(x) 1+s in Ω, for some c2 > 0. Using this estimate in the first equation of system (1) we deduce

where c3 > 0. Now, if q(2−r) 1+s

−u c3 δ(x)− u=0

q(2−r) 1+s

q(2−r) 1+s

u−p ,

u > 0 in Ω, on ∂Ω,

(38)

2, from Corollary 2.5 the above inequality is impossible. Assume

next that < 2. If α > 1, from (8), (38) and Proposition 2.6(iii) we find u(x) c4 δ(x)τ c4 ϕ1 (x)τ

in Ω,

(39)

where τ=

2−

q(2−r) 1+s

1+p

∈ (0, 1)

and c4 > 0.

Fix x0 ∈ ∂Ω and let n be the outer unit normal vector on ∂Ω at x0 . Using (39) and the fact that 0 < τ < 1 we have ∂u u(x0 + tn) − u(x0 ) (x0 ) = lim t 0 ∂n t c4 lim

t 0

ϕ1 (x0 + tn) − ϕ1 (x0 ) τ −1 ϕ1 (x0 + tn) t

∂ϕ1 (x0 ) lim ϕ1τ −1 (x0 + tn) t 0 ∂n = −∞. = c4

Hence, u ∈ / C 1 (Ω). If α = 1 we proceed in the same manner. From (38) and Proposition 2.6(ii) we deduce 1

u(x) c5 δ(x) log 1+p

A δ(x)

where c5 , c6 > 0. As before, we obtain

1

c6 ϕ1 (x) log 1+p

∂u ∂n (x0 ) = −∞, x0

A ϕ1 (x)

in Ω,

∈ ∂Ω, which contradicts u ∈ C 1 (Ω).

Case 2: r + s < 1. Then, α = p + q 1. As in Case 1, v fulfills (37) and by Proposition 2.6(i) we find v(x) c7 δ(x) in Ω, for some c7 > 0. Thus, u satisfies

−u c8 δ(x)−q u−p , u=0

u>0

in Ω, on ∂Ω,

where c8 > 0. From Corollary 2.5 it follows q < 2. Since α = p + q 1, it follows that u satisfies either the estimate (ii) (if p + q = 1) or the estimate (iii) (if p + q > 1) in Proposition 2.6. Proceeding in the same way as before we derive that the outer unit normal derivative of u on ∂Ω is −∞, which is impossible.


3315

Case 3: r + s = 1. This also yields α = p + q 1. As before v satisfies (37) and by Proposition 2.6(ii) we deduce v(x) c9 δ(x) log

1 1+s

A δ(x)

in Ω,

where c9 > 0. It follows that u satisfies

q

−u c10 δ(x)−q log− 1+s

u=0

A u−p , δ(x)

u > 0 in Ω,

(40)

on ∂Ω,

where c10 > 0. If q − b 2 the above inequality is impossible in the light of Corollary 2.5. Assume next that q − b < 2. If α = p + q > 1, we fix 0 0 in Ω, u=0 on ∂Ω, for some c11 > 0. Now, since p + q − b > 1, from Proposition 2.6(iii) we find u(x) c12 δ(x)

2−(q−b) 1+p

in Ω,

where c12 > 0. Since 0 < 2−(q−b) 1+p < 1, we obtain as before that the normal derivative of u on ∂Ω is infinite which is impossible. It remains to consider the case α = p + q = 1, that is, p + q = r + s = 1. First, if q < 1 + s, that is, q = 1 and s = 0, by (40) and Corollary 2.8 we deduce 1+s−q

u(x) c13 δ(x) log (1+p)(1+s)

A δ(x)

in Ω,

for some c13 > 0. Proceeding as before we obtain ∂u ∂n = −∞ on ∂Ω, which is impossible. If q = 1 and s = 0 then we apply Proposition 2.9 to obtain

A in Ω, u(x) c14 δ(x) log log δ(x) where c14 > 0. This also leads us to the same contradiction ∂u ∂n = −∞ on ∂Ω. Thus, we have proved that if the system (1) has a solution (u, v) with u ∈ C 1 (Ω) then α < 1 and r < 2. Conversely, assume now that α < 1 and r < 2. By Theorem 1.2(i) (Cases 1, 2 and 3) there exists a solution (u, v) of (1) such that u(x) cδ(x)

in Ω,

and v(x) cδ(x)

in Ω, if r + s 1,

3316


or 2−r

v(x) cδ(x) 1+s

in Ω, if r + s > 1,

for some c > 0. Using the above estimates we find −u = u−p v −q Cδ(x)−α

in Ω,

for some C > 0. By Proposition 2.2, we now deduce u ∈ C 1,1−α (Ω). The proof of (ii) is similar. (iii) Assume first that the system (1) has a solution (u, v) with u, v ∈ C 1 (Ω). Then, there exists c > 0 such that v(x) cδ(x) in Ω. Using this estimate in the first equation of (1) we find that −u Cδ(x)−q u−p , u > 0 in Ω, u=0 on ∂Ω, where C is a positive constant. If p + q 1, then we combine the result in Proposition 2.6(ii)– / C 1 (Ω). Thus, p + q < 1 (iii) with the techniques used above to deduce ∂u ∂n = −∞ on ∂Ω, so u ∈ and in a similar way we obtain r + s < 1. Assume now that p + q < 1 and r + s < 1. By Theorem 1.2(i) (Case 3) we have that (1) has a solution (u, v) such that u(x), v(x) cδ(x) in Ω, for some c > 0. This yields −u Cδ(x)−(p+q)

in Ω,

−(r+s)

in Ω,

−v Cδ(x)

where C > 0. Now Proposition 2.2 implies u, v ∈ C 1 (Ω). This concludes the proof. 6. Proof of Theorem 1.5 We shall prove only (i); the case (ii) follows in the same manner. Let (u1 , v1 ) and (u2 , v2 ) be two solutions of system (1). Using Proposition 2.3 there exists c1 > 0 such that ui (x), vi (x) c1 δ(x)

in Ω, i = 1, 2.

(41)

Hence, ui satisfies

−p

−ui c2 δ(x)−q ui , ui = 0

ui > 0 in Ω, on ∂Ω,

for some c2 > 0. By Proposition 2.6(i) and (41) there exists 0 < c < 1 such that 1 cδ(x) ui (x) δ(x) c

in Ω, i = 1, 2.

This means that we can find a constant C > 1 such that Cu1 u2 and Cu2 u1 in Ω.

(42)


3317

We claim that u1 u2 in Ω. Supposing the contrary, let M = inf{A > 1: Au1 u2 in Ω}. By our assumption, we have M > 1. From Mu1 u2 in Ω, it follows that −s −r −r −s −v2 = u−r 2 v2 M u1 v2

in Ω.

r

Therefore v1 is a solution and M 1+s v2 is a supersolution of

−s −w = u−r 1 w , w=0

w>0

in Ω, on ∂Ω.

By Proposition 2.1 we obtain r

v1 M 1+s v2

in Ω.

The above estimate yields −p −q

qr

−p −q

−u1 = u1 v1 M − 1+s u1 v2

in Ω.

qr

It follows that u2 is a solution and M (1+p)(1+s) u1 is a supersolution of

−q

−w = v2 w −p , w=0

w > 0 in Ω, on ∂Ω.

By Proposition 2.1 we now deduce qr

M (1+p)(1+s) u1 u2

in Ω.

qr Since M > 1 and (1+p)(1+s) < 1, the above inequality contradicts the minimality of M. Hence, u1 u2 in Ω. Similarly we deduce u1 u2 in Ω, so u1 ≡ u2 which also yields v1 ≡ v2 . Therefore, the system has a unique solution. This completes the proof of Theorem 1.4.

References [1] J. Busca, R. Manásevich, A Liouville-type theorem for Lane–Emden system, Indiana Univ. Math. J. 51 (2002) 37–51. [2] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications Inc., New York, 1967. [3] Ph. Clément, J. Fleckinger, E. Mitidieri, F. de Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000) 455–477. [4] L. Dupaigne, M. Ghergu, V. R˘adulescu, Lane–Emden–Fowler equations with convection and singular potential, J. Math. Pures Appl. 87 (2007) 563–581. [5] V.R. Emden, Gaskugeln, Anwendungen der mechanischen Warmetheorie auf kosmologische und meteorologische Probleme, Teubner-Verlag, Leipzig, 1907. [6] P. Felmer, D.G. de Figueiredo, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa 21 (1994) 387–397. [7] D.G. de Figueiredo, B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Math. Ann. 333 (2005) 231–260.

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[8] R.H. Fowler, Further studies of Emden’s and similar differential equations, Q. J. Math. (Oxford Ser.) 2 (1931) 259–288. [9] J. Hernández, F.J. Mancebo, J.M. Vega, Positive solutions for singular semilinear elliptic systems, Adv. Differential Equations 13 (2008) 857–880. [10] M. Ghergu, Steady-state solutions for Gierer–Meinhardt type systems with Dirichlet boundary condition, Trans. Amer. Math. Soc. 361 (2009) 3953–3976. [11] M. Ghergu, V. R˘adulescu, On a class of singular Gierer–Meinhardt systems arising in morphogenesis, C. R. Math. Acad. Sci. Paris 344 (2007) 163–168. [12] M. Ghergu, V. R˘adulescu, A singular Gierer–Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 1215–1234. [13] M. Ghergu, V. R˘adulescu, Singular Elliptic Equations: Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. Math. Appl., vol. 37, Oxford University Press, 2008. [14] C. Gui, F. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 1021–1029. [15] J.H. Lane, On the theoretical temperature of the sun under hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, Amer. J. Sci. 50 (1869) 57–74. [16] M. Naito, H. Usami, Existence of nonoscillatory solutions to second-order elliptic systems of Emden–Fowler type, Indiana Univ. Math. J. 55 (2006) 317–340. [17] M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, NJ, 1967. [18] P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49–81. [19] W. Reichel, H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations 161 (2000) 219–243. [20] J. Serrin, H. Zou, Existence of positive solutions of the Lane–Emden system, Atti Semin. Mat. Univ. Modena XLVI (1998) 369–380. [21] J. Serrin, H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differential Integral Equations 9 (1996) 635–653. [22] Ph. Souplet, The proof of the Lane–Emden conjecture in four space dimensions, Adv. Math. 221 (2009) 1409–1427. [23] H. Zou, A priori estimates for a semilinear elliptic system without variational structure and their applications, Math. Ann. 323 (2002) 713–735.


Joint extensions in families of contractive commuting operator tuples ✩ Stefan Richter ∗ , Carl Sundberg Department of Mathematics, University of Tennessee, Knoxville, TN 37996, United States Received 15 September 2009; accepted 10 January 2010 Available online 2 February 2010 Communicated by N. Kalton

Abstract In this paper we systematically study extension questions in families of commuting operator tuples that are associated with the unit ball in Cd . © 2010 Elsevier Inc. All rights reserved. Keywords: Extension; Dilation; Spherical contraction; Row contraction; Spherical isometry

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Spherical isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Extremality of the adjoint of the d-shift . . . . . . . . . . . . . . . . 4. A proposition about tuples of the type S ∗ ⊕ V . . . . . . . . . . . 5. Finite rank extensions of spherical contractions and isometries 6. Extensions of spherical contractions . . . . . . . . . . . . . . . . . . 7. Rank one extensions of row contractions . . . . . . . . . . . . . . . 8. Extensions of row contractions . . . . . . . . . . . . . . . . . . . . . . 9. An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Work of the authors was supported by the National Science Foundation, grant DMS-0556051.


E-mail addresses: [email protected] (S. Richter), [email protected] (C. Sundberg). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.011

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3320 3326 3328 3332 3336 3339 3341 3343 3346 3346

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S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346

1. Introduction It is fair to say that the Sz.-Nagy dilation theorem is of central importance for the theory of contraction operators on Hilbert spaces. One version of this theorem states that every contraction on a Hilbert space can be extended to a co-isometric operator acting on a larger Hilbert space. Because of the known structure of the co-isometric operators, this means that one can use the function theory of the Hardy space of the unit disc to study arbitrary contractions. Partial extensions of Sz.-Nagy’s theorem are available for the study of tuples of operators. The best known result is Ando’s theorem which says that for any pair of commuting contraction operators S and T acting on a Hilbert space H, there is a pair U, V of commuting co-isometric operators acting on a larger space K ⊇ H such that U extends S and V extends T [2]. It is also known that a direct analogue of Ando’s theorem fails for three or more commuting contractions. Ando’s theorem relates the study of commuting contractions to function theory on the bidisc, while it remains an open problem to find an effective model for three or more commuting contractions. The spherical contractions and the row contractions are collections of operator tuples which have been studied recently and which can be associated with function theory in the unit ball of Cd . A convenient way to approach many such theorems is through J. Agler’s model theory (see [1]). In this note we will present some examples of this model theory for the multivariable context. The following definition is from [1]. We will assume that all our Hilbert spaces are separable. Definition 1.1. Let d 1. A family is a collection F of d-tuples T = (T1 , . . . , Td ) of Hilbert space operators, Ti ∈ B(H), such that: (a) F is bounded, i.e. there exists c > 0 such that for all T = (T1 , . . . , Td ) ∈ F we have Ti c for all i = 1, . . . , d, (b) F is preserved under restrictions to invariant subspaces, i.e. whenever T ∈ F and M ⊆ H such that Ti M ⊆ M for all i, then T |M ∈ F , (c) F is preserved under direct sums, i.e. whenever Tn ∈ F is a sequence of tuples, then n Tn ∈ F , (d) F is preserved under unital ∗-representations, i.e. if π : B(H) → B(K) is a ∗-homomorphism with π(I ) = I and if T = (T1 , . . . , Td ) ∈ F , then π(T ) = (π(T1 ), . . . , π(Td )) ∈ F . For d = 1 some examples are given by the families of contractions, isometries, subnormal contractions, and hyponormal contractions. For d > 1 we will be interested only in families which consist of commuting tuples of operators. The family of commuting contractions has already been mentioned. A tuple (T1 , . . . , Td ) is called a spherical isometry if di=1 Ti x2 = x2 for every x ∈ H. It is immediately clear that the collection of spherical isometries satisfies (a), (b) and (c) of Definition 1.1. Furthermore (d) follows as well, because (T1 , . . . , Td ) is a spherical isometry if and only if di=1 Ti∗ Ti = I . We will write Fsi to denote the family of commuting spherical isometries. The spherical contractions Fsc are those commuting d-tuples T = (T1 , . . ∗. , Td ) of Hilbert consists of the space operators satisfying dj =1 Tj∗ Tj I . The collection of adjoint tuples Fsc row contractions Frc . They satisfy


d 2 d Tj x j xj 2 j =1

3321

for all x1 , . . . , xd

j =1

in the Hilbert space. As for the spherical isometries it is easy to check that both Fsc and Frc = ∗ form a family. Fsc Suppose T is an operator tuple acting on a Hilbert space H and R is a tuple acting on K. We will write R T if R is an extension of T , i.e. if H ⊆ K is a subspace which is invariant for each Ri , and if Ti = Ri |H for all i. In this case we will call dim K H the rank of the extension. If R = T ⊕ B for some operator tuple B, then R is called a trivial extension of T . Definition 1.2. Let F be a family. An operator tuple T ∈ F acting on H is called an extremal for F if T has only trivial extensions in F , i.e. whenever R ∈ F satisfies R T , then H reduces R. We shall write ext(F ) for the extremals of the family F . Theorem (J. Agler). If F is a family and if T ∈ F , then T can be extended to a tuple S ∈ ext(F ). The theorem is stated for families of single operators in [1], but it is mentioned there that the result also holds in the multivariable context. For a proof we refer the reader to [14] or the unpublished note [6]. Thus it is an important question to identify the extremals of families of interest. We note that it is easy to see that the extremals for the family of contractions are the co-isometric operators, the extremals for the isometric operators are the unitary operators, and extremals for the subnormal contractions are the normal contractions. It is unknown what the extremals for the hyponormal contractions are (see [13]). Next we discuss some examples for d > 1. Ando’s theorem can be used to show that the pairs of two commuting co-isometric operators are extremal for the pairs of commuting contractions. Alternatively, one can use a one-step extension as in the proof of the commutant lifting theorem (see [20, p. 65]) to identify the extremals. In this case Ando’s theorem follows from the above theorem of Agler’s. It is an open problem to identify the extremals for the d-tuples of commuting contractions if d > 2. On the other hand the extremals for the family of commuting isometries are easily identified as the tuples of commuting unitary operators. The resulting extension theorem is due to Ito [17] and Brehmer [9]. In this paper we will discuss extremals of families that are associated with the unit ball in Cd . A particular emphasis is placed on identifying which spatial and spectral properties of an operator tuple allow nontrivial extensions. A tuple U = (U1 , . . . , Ud ) of commuting operators is called spherical unitary if di=1 Ui∗ Ui = I and each Ui is a normal operator. Our first theorem is the following. Theorem 1.3. Let Fsi be the family of commuting spherical isometries, then ext(Fsi ) equals the collection of commuting spherical unitaries. The resulting extension theorem says that commuting spherical isometries are jointly subnormal and it is due to Athavale [7].

3322


We now turn to spherical and row contractions. An important example of a row contraction is the d-shift Mz = (Mz1 , . . . , Mzd ) acting on the Drury–Arveson space Hd2 . Hd2 is the reproducing kernel Hilbert space defined by the kernel kλ (z) =

1 , 1 − z, λ

λ, z ∈ Bd , z, λ =

d

zi λi .

i=1

Hd2 consists of analytic functions in Bd , and for d > 1 it is properly contained in the classical 1 Hardy space H 2 (∂Bd ), which has reproducing kernel (1− z,λ) d. Since Mz∗i kλ = λi kλ it follows that

d

Mzi Mz∗i kλ

i=1

(z) =

z, λ = kλ (z) − 1. 1 − z, λ

This implies that d

Mzi Mz∗i = I − 1 ⊗ 1 I,

(1.1)

i=1

thus Mz∗ is a spherical contraction and Mz is a row contraction. We say that S is a direct sum of d-shifts if S = (S1 , . . . , Sd ), Si = Mzi ⊗ I ∈ B(Hd2 ⊗ C) for some Hilbert space C. Theorem 1.4. Let Fsc be the family of commuting spherical contractions, and let T = (T1 , . . . , Td ) be a commuting operator tuple. Then the following are equivalent: (i) T ∈ ext(Fsc ), (ii) T = S ∗ ⊕ U , where U is spherical unitary and S is a direct sum of d-shifts, d Ti∗ Ti = P = a projection, (iii) (a) i=1 d ∗ (b) i=1 Ti Ti I , (c) if x1 , . . . , xd ∈ H with Ti xj = Tj xi , then there is an x ∈ H with xi = Ti x for all i. When we write T = S ∗ ⊕ U , we want to include the possibility that one of the summands is absent. Note that (iii)(c) says that the Koszul complex for T is exact at Λ1 (H) (Section 3 contains a short summary of elementary facts about the Koszul complex). The resulting extension theorem (i.e. that any R ∈ Fsc has an extension T of the type as in (ii)) had been known and is due to Müller–Vasilescu [18] and to Arveson [4]. Arveson also proved that the adjoint of the d-shift is an extremal spherical contraction (see [4, pp. 205, 206]). Among other things his proofs are based on his earlier results [3] and an analysis of the C ∗ -algebra generated by the d-shift and the identity operator. We note that the extremality of S ∗ also follows directly from Agler’s theorem once the implication (i) ⇒ (ii) of Theorem 1.4 has been established. Indeed, by Agler’s theorem the zero tuple 0 = (0, . . . , 0) acting on a nonzero space extends to an extremal spherical contraction. By (i) ⇒ (ii) there must be an extremal of the type S ∗ ⊕ U


3323

such that 0 = S ∗ ⊕ U |M where S is a direct sum of d-shifts, U is a spherical unitary tuple, and M is invariant for S ∗ ⊕ U . If the direct summand S ∗ were absent, then 0 = U |M would have to be a spherical isometry, which is absurd. Thus S ∗ ⊕ U is extremal and definitely has a d-shift as a direct summand. Now it is easy to verify that if X ⊕ Y is extremal for a family F , then both X and Y have to be extremal for F also. Hence the adjoint of the d-shift must be extremal for Fsc . For this paper we decided to present yet another proof of the extremality of S ∗ , this one based on spatial properties of S = Mz as it acts on Hd2 (Section 3). When we apply Theorem 1.4 to the tuple of adjoints we obtain the following corollary. Corollary 1.5 (Wold-decomposition). A d-tuple of commuting operators is of the form T = S ⊕U if and only if d Ti Ti∗ is a projection, (a) i=1 d ∗ (b) i=1 Ti Ti I , and d (c) whenever x1 , . . . , xd ∈ H with there is an antisymmetric matrix i=1 Ti xi = 0, then {yij }1i,j d with entries yij ∈ H such that xi = dj =1 Tj yij for each i (i.e. the Koszul complex for T is exact at Λd−1 (H)). We will give a short proof at the end of Section 6. Note that when d = 1 and condition (a) is satisfied, then T is a partial isometry. In this case each of the conditions (b) or (c) implies that T is 1–1 and thus T must be an isometry. For d > 1 neither conditions (a) and (b) nor conditions (a) and (c) alone will imply that T = S ⊕ U . In fact if T = (Mz∗ , H 2 (∂Bd )) then T ∗ is a spherical isometry, thus it satisfies (a). Furthermore, it is well known that the Koszul complex for (Mz∗ , H 2 (∂Bd )) is exact at all stages except at the first one, so (c) is satisfied provided d > 1 (see e.g. Section 2 of [15]), but (b) is not satisfied. In order to exhibit an example which satisfies (a) and (b) but not (c) we let M0 = {f ∈ Hd2 : f (0) = 0} and T = S|M0 , where S is the d-shift acting on Hd2 . It is clear that T satis fies (b), because S does. Furthermore, di=1 Si Si∗ = P = I − 1 ⊗ 1 is the projection from Hd2 d onto M0 (see (1.1)). Then i=1 Ti Ti∗ = di=1 P Si P Si∗ P = P ( di=1 Si Si∗ − Si (1 ⊗ 1)Si∗ )P = P − di=1 zi ⊗ zi and this is a projection, because zi = 1 for all i and zi ⊥ zj for all i = j (see Eq. (1.2) below). Thus, T satisfies (a). We will now show that for d > 1 Tdoes not satisfy (c). Let f1 (z) = z2 , f2 (z) = −z1 , and f3 = · · · = fd = 0. Then fi ∈ M0 and di=1 Ti fi = 0. If z2 = f1 (z) = dj =1 zj g1j for some g1j ∈ M0 , then we take a partial derivative with respect to z2 and obtain 1 = g12 (z) + dj =1 zj ∂z∂ 2 g1j (z). Evaluating at z = 0 we conclude 1 = g12 (0), but this contradicts g12 ∈ M0 . For the family of row contractions we have partial results. Theorem 1.6. Let Frc be the family of commuting row contractions. Let T ∈ Frc and write D∗ = (I − di=1 Ti Ti∗ )1/2 . (i) If D∗ = 0, then T ∈ ext(Frc ). / ext(Frc ). (ii) If D∗ is onto, then T ∈ / ext(Frc ) if and only if there are x1 , . . . , xd ∈ di=1 ker Ti∗ (iii) If D∗ is a projection, then T ∈ d with i=1 xi 2 > 0 and Ti xj = Tj xi for all i, j .

3324


(iv) If D∗ has rank one, i.e. if D∗ = u ⊗ u for some u = 0, then T ∈ ext(Frc ) if and only if dim span{u, T1 u, . . . , Td u} 3. If d = 1, then part (i) of Theorem 1.6 describes all extremals (the co-isometric operators). For d > 1 the d-shift is an example of an extremal with D∗ = 0. For the d-shift one verifies that D∗ is a projection of rank 1 (see Eq. (1.1)), so its extremality can be derived either from part (iii) or part (iv) of Theorem 1.6. We will see that in all of the above cases, when T is not extremal, then T actually has a nontrivial rank 1 extension in Frc . If S = (Mz , Hd2 ) is the d-shift, and if M Hd2 is invariant for S, then T = PM⊥ S|M⊥ ∈ Frc and D∗ has rank 1. Because of this one can use Theorem 1.6 to verify the following corollary (see Corollary 8.4). Corollary 1.7. Let Frc be the family of commuting row contractions. If M = Hd2 is an invariant subspace for the d-shift S = (Mz , Hd2 ), and if L= a+

d

bi zi : a, b1 , . . . , bd ∈ C

i=1

denotes the collection of polynomials of degree less than or equal to one, then T = PM⊥ S|M⊥ ∈ ext(Frc ) if and only if dim M ∩ L < d − 1. Under the hypothesis of the corollary one easily checks that D∗2 = ϕ ⊗ ϕ, where ϕ = PM⊥ 1 (see the proof of Corollary 8.4). Thus D∗ is a projection if and only if 1 ∈ M⊥ , and the corollary can be used to exhibit many examples of extremal row contractions whose defect operators are not projections. For example, if d = 2 and λi = (λi1 , λi2 ), i = 1, 2, 3 are three distinct points in B2 , then we can let M = f ∈ H22 : f (λ1 ) = f (λ2 ) = f (λ3 ) = 0 . In this case T = (T1 , T2 ) acts on the 3-dimensional space M⊥ = span{kλ1 , kλ2 , kλ3 }, and by the corollary T is extremal if and only if M ∩ L = {0}. From this one deduces with a little bit of elementary algebra that T is extremal if and only if (λ31 − λ11 )(λ22 − λ12 ) = (λ21 − λ11 )(λ32 − λ12 ). Hence there are extremal row contractions on finite dimensional spaces that are not spherical unitaries. We also note that the above examples of extremals where the defect operator is not a projection show that part (iii) of Theorem 1.6 does not cover all extremals. This is in contrast to the family Fsc where for all extremals the defect operator D = (I − di=1 Ti∗ Ti )1/2 must be a projection (see Theorem 1.4). If d = 1 and if F is either the family of contractions or the family of isometries, then any nonextremal operator T ∈ F has a nontrivial rank one extension in F . This is well known and easy to see (compare Lemma 7.2). If d > 1, then for each of the families Fsc , Fsi and Frc there


3325

is a difference between extremals and operator tuples that allow nontrivial finite rank or rank one extensions. We shall show in Corollary 5.4 that a spherical isometry V has no nontrivial rank one extensions in Fsi if and only if the Koszul complex for V −b is exact at Λ1 for all b ∈ Bd . We will review definitions and elementary properties of the Koszul complex in Section 3. Furthermore, in Theorem 6.1 we will show that if T ∈ Fsc , then T has no nontrivial finite rank extensions if and only if T has no nontrivial rank one extensions and this happens if and only if T = S ∗ ⊕ V , where S is a direct sum of d-shifts and V is a spherical isometry with no nontrivial rank one extension. For the row contractions we only have the following technical condition, and we note that all extension results of Theorem 1.6 follow from this result, i.e. if either D∗ is onto, or if T ∈ Frc is nonextremal and D∗ is a projection or a rank one operator, then T has a nontrivial rank 1 extension in Frc . Theorem 1.8. Let Frc be the family of commuting row contractions, and let T ∈ Frc . The following are equivalent: (a) T has a nontrivial rank 1 extension in Frc , (b) T has a nontrivial finite rank extension in Frc , (c) there are a1 , . . . , ad ∈ ran D∗ , i ai > 0, and b = (b1 , . . . , bd ) ∈ Bd such that (Ti − bi )aj = (Tj − bj )ai for all i, j . In Section 9 we will present an example of a nonextremal commuting row contraction which has no nontrivial finite dimensional extensions. The remainder of the paper is structured as follows. In Section 2 we will prove Theorem 1.3 and we will see that spherical unitaries are extremal spherical contractions. Section 3 contains a proof that the adjoint of the d-shift is an extremal among the spherical contractions. A basic proposition about spherical contractions of the form S ∗ ⊕ V , where S is a sum of d-shifts and V is a spherical isometry will be presented and proved in Section 4. Section 5 contains a theorem characterizing the spherical isometries that have nontrivial rank one extensions (Corollary 5.4) and it also has some preliminary results about rank one and finite rank extensions of spherical contractions. Theorem 6.1 characterizes spherical contractions with nontrivial finite rank extensions and Corollaries 6.2 and 6.3 are Theorem 1.4 and Corollary 1.5. In Section 7 we give our results about finite rank extensions of row contractions and in Section 8 we present our main results about extremals of Frc . At various places throughout the paper we will use multinomial notation. If α ∈ Nd0 , then

|α|! = α! . If z ∈ Cd and if α = (α1 , . . . , αd ), |α| = α1 + · · · + αd , α! = α1 · . . . · αd !, and |α| α α α α α T = (T1 , . . . , Td ) ∈ B(H)d , then zα = z1 1 · . . . · zd d and T α = T1 1 . . . Td d . Furthermore, we will use ej = (0, . . . , 0, 1, 0, . . . , 0) where the 1 is in the j th spot. The reproducing kernel for the Drury–Arveson space Hd2 satisfies kλ (z) =

∞ ∞ |α| 1 n α α z λ = zα λα . =

z, λn = α α 1 − z, λ d n=0

n=0 |α|=n

α∈N0

Since we also have |α| |β| kλ (z) = kλ , kz = z β λα w α , w β H 2 α β d α∈Nd0 β∈Nd0

3326


it follows that α β 1 w , w H 2 = δαβ |α| , d

where δαβ = 0 if α = β,

From this one deduces that for f ∈ Hol(Bd ), f (z) = f 2H 2 = d

d

i=1 zi z

α 2

=

α∈Nd0

fˆ(α)zα one has

|fˆ(α)|2 α! fˆ(α)2 .

|α| = |α|! d d α

α∈N0

If α ∈ N0 d , then

δαβ = 1 if α = β.

α

α! |α|+d |α|! |α|+1 , d

(1.2)

α∈N0

hence it follows that

zi f 2 f 2

(1.3)

i=1

for all f ∈ Hd2 . Furthermore, one calculates that for α ∈ Nd0 and 1 i d Mz∗i zα =

αi α−ei z |α|

whenever αi > 0,

(1.4)

and Mz∗i zα = 0 if αi = 0. 2. Spherical isometries In this section we will prove Theorem 1.3 and part of the proof of (ii) ⇒ (i) of Theorem 1.4. The fact that extremals of the spherical isometries must be jointly normal follows easily from the arguments of Attele and Lubin [8], who presented an alternate proof of Athavale’s theorem. In fact, let T = (T1 , . . . , Td ) be a commuting spherical isometry acting on H and assume that T1 is not normal. We must show that T is not extremal, i.e. we have to construct a commuting spherical isometry S that extends T nontrivially. By Corollary 6 of [8] T1 is a subnormal contraction. Thus we can let S1 ∈ B(K) be the minimal normal extension of T1 . Since we assumed that T1 is not normal, it is clear that any extension of T of the form S = (S1 , S2 , . . . , Sd ) will be nontrivial. In order to define S2 , . . . , Sd we use the standard extensions n n k ∗ Si S1 xk = S1∗ k Ti xk , i = 2, . . . , d, x0 , . . . , xn ∈ H, k=0

k=0

see the proof of Proposition 7 of [8]. Since S1 is normal it is easy to verify that n 2 n 2 d S1∗ k Ti xk = S1∗ k xk . i=1 k=0

k=0

This implies that S2 , . . . , Sd are well defined and extend to K and S = (S1 , S2 , . . . , Sd ) forms a spherical isometry. Finally, we see that for all 1 i, j d


Sj Si

n

S1∗ k xk = Sj

k=0

n

3327

S1∗ k Ti xk

k=0

=

n

S1∗ k Tj Ti xk

k=0

= Si Sj

n

S1∗ k xk .

k=0

Thus S forms a tuple of commuting operators, and this proves that the extremals of the spherical isometries must be commuting normals. For later reference we make some simple observations about extremals. Lemma 2.1. Let F and G be families. (a) Let U ∈ ext(F ) and V ∈ F . If R ∈ F with R U ⊕ V , then R = U ⊕ R for some R V , R ∈ F . (b) Finite or infinite direct sums of extremals are extremal. (c) If F ⊆ G, then ext(G) ∩ F ⊆ ext(F ). Proof. (a) is obvious and it easily implies (b) for finite direct sums. In order to prove (b) for infinite direct sums let Un ∈ ext(F ) ∩ B(Hn )d , V = U1 ⊕ U2 ⊕ · · · and R ∈ F ∩ B(K)d with R V . For n ∈ N we let Pn be the projection from K onto H1 ⊕ · · · ⊕ Hn . By the finite case we have Ri Pn = Pn Ri for all i and n. The sequence Pn converges in the strong operator topology to P , the projection from K onto H1 ⊕ H2 ⊕ · · · . It follows that P commutes with R. Hence V must be extremal. (c) is immediate. 2 Theorem 2.2. Commuting spherical unitary tuples are extremal for the families of commuting spherical contractions and commuting spherical isometries. Proof. Let U = (U1 , . . . , Ud ) be a commuting spherical unitary tuple. By Lemma 2.1(c) it suffices to show that U is extremal for the commuting spherical contractions. Thus let S U be a commuting spherical contraction. Then for each 1 i d we have Si =

Ui 0

Ai Bi

∈ B(H ⊕ K)

with Ui Aj + Ai Bj = Uj Ai + Aj Bi

(2.1)

for all 1 i, j d and d i=1

Si∗ Si

=

d

I

∗ i=1 Ai Ui

d

∗ i=1 Ui Ai ∗ ∗ i=1 Ai Ai + Bi Bi

d

I 0

0 . I

3328

It follows that


d

∗ i=1 Ui Ai

= 0 and d

A∗i Ai + Bi∗ Bi I.

(2.2)

i=1

Let j ∈ {1, . . . , d}. We shall establish Lemma 2.2 by showing that Aj = 0. By hypothesis each Ui is normal and Ui Uj = Uj Ui . Hence it follows from Fuglede’s theorem [19] that Ui∗ Uj = Uj Ui∗ . We now apply Ui∗ on the left in Eq. (2.1), sum in i, and obtain d

Ui∗ Ui Aj +

i=1

d

Ui∗ Ai Bj =

i=1

d

Ui∗ Uj Ai +

i=1

d

Ui∗ Aj Bi .

i=1

Since di=1 Ui∗ Ui = I and di=1 Ui∗ Ai = 0 this implies Aj = di=1 Ui∗ Aj Bi . Since U is a spherical contraction it follows that U ∗ = (U1∗ , . . . , Ud∗ ) is a row contraction. Hence for x ∈ K, x 1 we have d 2 d Aj x2 = Ui∗ Aj Bi x Aj Bi x2 i=1

Aj 2

i=1

d

Bi x2 .

i=1

By Eq. (2.2) this implies Aj x Aj 2

2

x − 2

d

Ai x

2

Aj 2 1 − Aj x2 .

i=1

We now rearrange the terms to get Aj x2 (1 + Aj 2 ) Aj 2 and after taking the sup over x 1 we obtain Aj 2 (1 + Aj 2 ) Aj 2 which implies that Aj = 0. 2 3. Extremality of the adjoint of the d-shift Let T = (T1 , . . . , Td ) be a commuting tuple of operators on a Hilbert space H. We will now define the Koszul complex of T . We will follow [5]. For more information of a general type on the Koszul complex and its relationship to invertible and Fredholm tuples, the reader is also referred to [10] and [21]. Let Λ = Λ[e] = Λd [e] be the exterior algebra generated by the d symbols e1 , . . . , ed , along with the identity e0 defined by e0 ∧ ξ = ξ for all ξ . Then Λ is the algebra of forms in e1 , . . . , ed with complex coefficients, subject to the anticommutative property ei ∧ej +ej ∧ei = 0 (1 i, j d). In fact, we can make Λ into a 2d -dimensional Hilbert space with orthonormal basis {e0 } ∪ ei1 ∧ · · · ∧ eik ij ∈ {1, . . . , d}, i1 < i2 < · · · < ik .


3329

For each i = 0, 1, . . . , d let Ei : Λ → Λ be given by Ei ξ = ei ∧ ξ . E0 is thus the identity on Λ. For i = 1, . . . , d the Ei are called the creation operators and they satisfy the following anticommutation relations Ei Ej + Ej Ei = 0 and Ei∗ Ej + Ej Ei∗ = δij E0 . Let Λ(H) := H ⊗C Λ and define ∂T : Λ(H) → Λ(H) by ∂T :=

d

Ti ⊗ E i .

i=1

It follows easily from the anticommutation relationships that ∂T2 = 0. Thus, the Koszul complex of the tuple T can be defined by ∂T ,0

∂T ,1

∂T ,d−1

K(T ) : 0 → Λ0 (H) −−→ Λ1 (H) −−→ · · · −−−−→ Λd (H) → 0 where Λp (H) is the collection of p forms in Λ(H) and ∂T ,p := ∂T |Λp (H). For purposes of notation we also define Λ−1 (H) = 0 and ∂T ,−1 and ∂T ,d to be the zero maps at the two ends of the complex. The identity ∂T2 = 0 implies that for each p = 0, 1, . . . , d ran ∂T ,p−1 ⊆ ker ∂T ,p and ran ∂T∗ ,p ⊆ ker ∂T∗ ,p−1 , and one says that the Koszul complex K(T ) is exact at Λp (H), if ran ∂T ,p−1 = ker ∂T ,p . In particular, if K(T ) is exact at Λp (H), then ran ∂T ,p−1 must be closed, hence ∂T ,p−1 ∂T∗ ,p−1 is 1–1 and onto when restricted to ran ∂T ,p−1 = (ker ∂T∗ ,p−1 )⊥ . Furthermore, in this case one also has that ran ∂T∗ ,p is dense in ker ∂T∗ ,p−1 . It follows that the operator DT ,p : Λp (H) → Λp (H),

DT ,p = ∂T ,p−1 ∂T∗ ,p−1 + ∂T∗ ,p ∂T ,p

(3.1)

is 1–1 and has dense range whenever K(T ) is exact at Λp (H), and DT ,p is invertible if it is also known that ran ∂T ,p , or what is the same, ran ∂T∗ ,p is closed. In order to relate properties of the Koszul complex for T with the Koszul complex for T ∗ we define the Hodge ∗-operator (see [16] for more information on this topic). For p = 0, 1, . . . , d we have dim Λp = dim Λd−p and ∗ establishes a conjugate linear isomorphism between Λp and Λd−p that is compatible with ∂. Indeed, ∗(ei1 ∧ ei2 ∧ · · · ∧ eip ) = (−1)ε ej1 ∧ ej2 ∧ · · · ∧ ejd−p where {ei1 , ei1 , . . . , eip , ej1 , . . . , ejd−p } = {e1 , . . . , ed } and ε ∈ {0, 1} is chosen so that η ∧ ∗ω =

η, ωe1 ∧ · · · ∧ ed for all p-forms η, ω ∈ Λp . If η ∈ Λp−1 and ω ∈ Λp , then

η ∧ ∗Ei∗ ω = η, Ei∗ ω e1 ∧ · · · ∧ ed = Ei η, ωe1 ∧ · · · ∧ ed = Ei η ∧ ∗ω = ei ∧ η ∧ ∗ω = (−1)p−1 η ∧ ei ∧ ∗ω = (−1)p−1 η ∧ Ei (∗ω).

3330


Thus for x ∈ H and ω ∈ Λp we have (I ⊗ ∗)∂T∗ ,p−1 (x ⊗ ω) =

d

Ti∗ x ⊗ ∗Ei∗ ω

i=1

= (−1)p−1

d

Ti∗ x ⊗ Ei (∗ω)

i=1

= (−1)p−1 ∂T ∗ ,d−p (I ⊗ ∗)(x ⊗ ω). Now note that elementary functional analysis results imply that for any p ran ∂T ,p is closed if and only if ran ∂T∗ ,p is closed, and by the above this happens if and only if ran ∂T ∗ ,d−(p+1) is closed. Thus, if it is known for some p that ran ∂T ,p is closed and K(T ) is exact at Λp , then K(T ∗ ) is exact at Λd−p . It is known that if α > 0 and Kα is the Hilbert space of analytic functions on Bd with reproducing kernel kλ (z) = (1 − z, λ)−α , then the Koszul complex for S = (Mz , Kα ) is exact at all stages p = 0, . . . , d − 1 and at the last stage we have dim ker ∂S,d / ran ∂S,d−1 = 1 (see Proposition 2.6 of [15]). Thus, ran ∂S,p is closed for all p. From this and the above remarks about the Hodge ∗-operator it follows that for T = S ∗ we have ran ∂T ,p is closed for all p and K(T ) is exact at Λp (H) for each p 1. In the following let T be a commuting d-tuple of operators on H. We will later take T so that (T ∗ , H) = (Mz , Hd2 ) is the d-shift. Suppose that we have an extension R ∈ B(H ⊕ K)d of T . Then Ti Ai . Ri = 0 Bi Define ∂T : Λ(H) → Λ(H),

∂T =

d

Ti ⊗ E i ,

i=1

∂A : Λ(K) → Λ(H),

∂A =

d

Ai ⊗ Ei ,

i=1

∂B : Λ(K) → Λ(K),

∂B =

d i=1

Lemma 3.1. If R is a tuple of commuting operators, then ∂T ∂A + ∂A ∂B = 0. Proof. Since Λ(H ⊕ K) = Λ(H) ⊕ Λ(K) we can write ∂R =

∂T 0

∂A ∂B

.

Bi ⊗ Ei .


3331

The lemma follows easily, because the expression equals the (1, 2)-entry of the matrix for ∂R2 = 0. 2 Proposition 3.2. Let d 2 and let T = (T1 , . . . , Td ) ∈ B(H)d be a commuting tuple of operators which is graded in the sense that there is a decomposition of H as a direct sum of mutually orthogonal subspaces, H = H0 ⊕ H1 ⊕ · · · such that Tj (H0 ) = (0) and Tj (Hn ) ⊆ Hn−1 for all n 1 and all 1 j d. Assume that the Koszul complex K(T ) is exact at Λp (H) for p = 1 and p = 2. If R ∈ B(H ⊕ K)d is a commuting extension of T of the form Ri = and if

d

∗ j =1 Aj Tj

Ti 0

Ai Bi

,

i = 1, . . . , d,

= 0, then Ai = 0 for all i = 1, . . . , d.

Proof. We start by noting that in terms of the Koszul complex the hypothesis dj =1 A∗j Tj = 0 ∗ ∂ ∗ can be restated as ∂A,0 T ,0 = 0 and that we have to show that ∂A,0 = 0. p p+1 Since Tj (Hn ) ⊆ Hn−1 we see that ∂T ,p (Λ (Hn )) ⊆ Λ (Hn−1 ) for all n 1. Furthermore, one easily checks that for all n 0 and all 1 j d we have Tj∗ (Hn ) ⊆ Hn+1 . Thus for each p the selfadjoint operators ∂T ,p−1 ∂T∗ ,p−1 and DT ,p (see Eq. (3.1)) leave Λp (Hn ) invariant, and the hypothesis implies that

DT ,1 Λ1 (Hn ) = Λ1 (Hn )

(3.2)

and that

DT ,2 Λ2 (Hn )

is dense in Λ2 (Hn )

(3.3)

for each n 0. Define C = ∂T ,1 ∂A,0 , then ∗ C ∗ ∂T ,1 = ∂A,0 DT ,1 ,

(3.4)

∗ ∂ ∗ because ∂A,0 T ,0 ∂T ,0 = 0. We also have

∗ ∗ ∂T∗ ,1 ∂T ,1 ∂T∗ ,1 + ∂T∗ ,2 ∂T ,2 = ∂A,0 ∂T∗ ,1 ∂T ,1 ∂T∗ ,1 C ∗ DT ,2 = ∂A,0 ∗ ∗ = −∂B,0 ∂A,1 ∂T ,1 ∂T∗ ,1 ,

(3.5)

by Lemma 3.1 and because ∂T∗ ,1 ∂T∗ ,2 = 0. We shall now show inductively that

1 ∗ Λ (Hn ) = (0) ∂A,0

and C ∗ Λ2 (Hn ) = (0)

for each n 0.

3332


We start by applying (3.4) to Λ1 (H0 ). Since ∂T ,1 (Λ1 (H0 )) = 0 and DT ,1 (Λ1 (H0 )) = ∗ (Λ1 (H )) = 0. This implies that for each i we have A∗ (H ) = 0, thus Λ1 (H0 ) we have ∂A,0 0 0 i ∗ ∂ ∗ 2 ∗ ∂A,1 T ,1 ∂T ,1 (Λ (H0 )) = 0. In light of (3.5) and (3.3) this means that C = 0 on a dense subset of Λ2 (H0 ), hence C ∗ (Λ2 (H0 )) = (0). ∗ (Λ1 (H )) = (0) and C ∗ (Λ2 (H )) = 0. Then Next suppose that for some n 0 we have ∂A,0 n n 1 2 since ∂T ,1 (Λ (Hn+1 )) ⊆ (Λ (Hn )) we can use (3.4) and the induction hypothesis to see that ∗ D ∗ ∗ 1 1 0 = ∂A,0 T ,1 (Λ (Hn+1 )) = ∂A,0 (Λ (Hn+1 )). Thus, for each i we have Ai (Hn+1 ) = 0 and so

∗ ∂T ,1 ∂T∗ ,1 Λ2 (Hn+1 ) = 0. ∂A,1 In light of (3.5) and (3.3) this means that C ∗ = 0 on a dense subset of Λ2 (Hn+1 ), hence C ∗ (Λ2 (Hn+1 )) = (0). 2 Theorem 3.3. If S = (Mz , Hd2 ), then S ∗ is extremal for the family of spherical contractions. Proof. Set T = S ∗ and let R ∈ B(Hd2 ⊕ K)d be a commuting spherical contraction which extends T . Then R is of the form R = (R1 , . . . , Rd ), Ti Ai . Ri = 0 Bi We shall use Proposition 3.2 to show that each Ai equals 0. To this end let Hn be the homogeneous polynomials of degree n, so that Hd2 = H0 ⊕ H1 ⊕ · · · , Tj (H0 ) = (0) and Tj (Hn ) ⊆ Hn−1 for all n 1 and all 1 j d (see (1.4)). We noted earlier in this section that the Koszul com∗ plex d K(T∗ ) = K(S ) is exact at stages 1 and 2. Thus by Proposition 3.2 it suffices to show that j =1 Aj Tj = 0. The condition I − di=1 Ri∗ Ri 0 implies that for all f ∈ Hd2 and y ∈ K we have f − 2

d i=1

Ti f − 2 Re 2

d

A∗i Ti f, y

+ y2 −

i=1

d

Ai y2 + Bi y2 0.

(3.6)

i=1

Now we recall that I − di=1 Ti∗ Ti = I − di=1 Si Si∗ equals the projection onto H0 (the con stants, see Eq. (1.1)). Thus, if f ⊥ H0 , then (3.6) implies that di=1 A∗i Ti f = di=1 A∗i Si∗ f = 0. If f ∈ H0 , then Si∗ f = 0 for all 1 i d, hence di=1 A∗i Ti = 0. 2 4. A proposition about tuples of the type S ∗ ⊕ V In this section we shall prove the following proposition which will easily imply one of the equivalent statements of Theorem 1.4. Proposition 4.1. Let T ∈ B(H)d be a commuting operator tuple which satisfies the following two conditions: d ∗ (a) i=1 Ti Ti is a projection, and (b) if x1 , . . . , xd ∈ H with Ti xj = Tj xi for all i, j , then there is an x ∈ H with xi = Ti x for all i.


3333

Then T is unitarily equivalent to S ∗ ⊕ V , where S is a direct sum of d-shifts, and V is a spherical isometry. Note that for d = 1 condition (b) says that T is surjective, while condition (a) implies that T is a partial isometry. Thus T must be a co-isometry, and if T ∗ = S ⊕ V ∗ , then V ∗ must be isometric. This means that for d = 1 the operator V in the proposition is automatically unitary. If d > 1, then the operator tuple T = (Mz , H 2 (∂Bd )) provides an example of a d-tuple that satisfies conditions (a) and (b) of the proposition, but is not a spherical unitary tuple. Let E0 = di=1 ker Ti . Inductively define a sequence of positive operators by

P0 = I

and PN +1 =

d

Ti∗ PN Ti

for N = 0, 1, . . . .

(4.1)

i=1

One verifies that for N 1 PN =

N T ∗α T α . α

|α|=N

Hence it follows that ker PN = |α|=N ker T α , and E0 = ker P1 . Note that for N 1 we have PN − PN +1 = di=1 Ti∗ (PN −1 − PN )Ti . Hence part (a) of the hypothesis and an induction argument imply that {PN }N ∈N is a nonincreasing sequence of positive operators which thus converges strongly to a positive operator P . Our first step will be to show that P and each PN are projections, and that Ti P = P Ti for all 1 i d. This means that M = ran P reduces each Ti and we will see that T |M is a spherical isometry and that T ∗ |M⊥ is unitarily equivalent to the d-shift acting on Hd2 (E0 ). Lemma 4.2. Let T ∈ B(H)d be as in Proposition 4.1. Then for each N ∈ N the operator PN is a projection such that Ti PN = PN −1 Ti for all i ∈ {1, . . . , d}. Proof. We will start by using induction on N ∈ N to show that Ti PN = PN −1 Ti for all i ∈ {1, . . . , d}. The hypothesis (a) of Proposition 4.1 implies that P1 is a projection, hence ran(I − P1 ) = ker P1 = di=1 ker Ti . This implies Ti (I − P1 ) = 0 for all i ∈ {1, . . . , d}. Thus P0 Ti = Ti = Ti P1 and the statement is true for N = 1. Next suppose that N > 1 and that Ti PN −1 = PN −2 Ti for all i ∈ {1, . . . , d}. For x ∈ H and j ∈ {1, . . . , d} set zj = PN −1 Tj x. Then for all i and j we have Ti zj = Ti PN −1 Tj x = PN −2 Ti Tj x = PN −2 Tj Ti x = Tj zi . Thus the hypothesis (b) of Proposition 4.1 implies that there exists y ∈ H such that PN −1 Tj x = zj = Tj y for all j . Then for all i we have Ti PN x = Ti

d j =1

Tj∗ PN −1 Tj x = Ti

d j =1

Tj∗ Tj y = Ti P1 y = Ti y = PN −1 Ti x.

3334


Thus Ti PN = PN −1 Ti for all N ∈ N and i ∈ {1, . . . , d}. For N > 1 we can iterate this to obtain Ti Tj PN = Ti PN −1 Tj = PN −2 Ti Tj for all i, j . Continuing the same way and using that P0 = I we see that for all N ∈ N and all multiindices α ∈ Nd0 with |α| = N we have T α PN = T α . Thus PN2 =

N N T ∗α T α PN = T ∗α T α = PN , α α |α|=N

|α|=N

2

which shows that each PN is a projection.

Lemma 4.3. Let T be a commuting operator tuple on H which satisfies condition (b) of Proposition 4.1, and let N ∈ N. Suppose that for each α ∈ Nd with |α| = N we are given an element xα ∈ H. Then there is an x ∈ H such that xα = T α x for all |α| = N if and only if Ti xβ+ej = Tj xβ+ei

for all 1 i, j d and all β ∈ Nd0 with |β| = N − 1.

(4.2)

Proof. It is clear that if xα = T α x for all |α| = N , then Ti xβ+ej = Tj xβ+ei for all 1 i, j d and all β ∈ Nd0 with |β| = N − 1. We will use induction on N to verify the sufficiency of condition (4.2). For N = 1 this is just the hypothesis of the lemma. Suppose that the lemma holds for N 1, and suppose that {xα }|α|=N +1 satisfies (4.2) with N + 1 instead of N . Let |β| = N . For i = 1, . . . , d set zi = xβ+ei . Then we have Tj zi = Ti zj for all i and j . Thus by our hypothesis on T there exists xβ ∈ H with xβ+ei = zi = Ti xβ for all i. The collection {xβ }|β|=N satisfies (4.2). Indeed, let |γ | = N − 1 and 1 i, j d, then Tj xγ +ei = x(γ +ei )+ej = x(γ +ej )+ei = Ti xγ +ej . Hence the induction hypothesis and the construction imply that there is an x ∈ H such that xβ = T β x and xβ+ei = Ti xβ = T β+ei for all |β| = N and 1 i d. Thus, xα = T α x for all |α| = N + 1. 2 Lemma 4.4. Let T ∈ B(H)d be as in Proposition 4.1 and let N ∈ N. Then for all x ∈ E0 and all α, β ∈ Nd0 with |α| = |β| = N we have

N α

N T α T ∗β x = δαβ x, β

where δαβ = 1 if α = β and δαβ = 0 otherwise. Of course, the definition of E0 then immediately implies that T γ T ∗β x = 0 for all x ∈ E0 and β, γ ∈ Nd0 with |γ | > N = |β|. Proof. Define the column operator T (N ) : H → T

(N )

x=

|α|=N

N Tα α

H by

. |α|=N


3335

Then T

(N ) ∗

T

(N )

N T ∗α T α = PN . = α |α|=N

∗

∗

Lemma 4.2 implies that T (N ) T (N ) is a projection and hence it follows that T (N ) T (N ) is the orthogonal projection onto ran T (N ) . d Now let x ∈ E0 , fix β ∈ N0 with |β| = N , and define a column vector z = {xα }|α|=N ∈ |α|=N H by xα = 0 if α = β and xβ = x. Then Ti xγ +ej = 0 = Tj xγ +ei for all 1 i, j d and all |γ | = N − 1. Thus it follows from Lemma 4.3 that there is a w ∈ H such that xα = T α w for all |α| = N and hence z ∈ ran T (N ) and z=T

(N )

T

(N ) ∗

z=

N α

The lemma now follows from the definition of z.

N α ∗β T T x β

. |α|=N

2

Proof of Proposition 4.1. From Lemma 4.2 and the remarks preceding it we know that the sequence {PN }N∈N forms a decreasing sequence of projections. Let P denote the strong limit of this sequence. Then P is a projection and the assertion PN −1 Ti = Ti PN of Lemma 4.2 implies that P Ti = Ti P for all 1 i d. Thus M = ran P reduces T and the identity P = di=1 Ti∗ P Ti shows that T |M is a spherical isometry. Let S denote the d-shift acting on Hd2 (E0 ), then S is unitarily equivalent to Mz ⊗ I acting on Hd2 ⊗ E0 . We will show that Mz ⊗ I is unitarily equivalent to T ∗ |M⊥ . Since E0 ⊆ M⊥ we can define a linear transformation U : Hd2 ⊗ E0 → M⊥ by setting U (p ⊗ x) = p(T ∗ )x for every polynomial p and x ∈ E0 . Note that for x, y ∈ E0 and α, β ∈ Nd0 Lemma 4.4 implies

Thus if p(z) = have

ˆ α p(α)z

α

|α| α

|β| ∗α T x, T ∗β y = δαβ x, y. β

and q(z) =

β

β are polynomials, then for all x, y ∈ E we q(β)z ˆ 0

∗

∗α p T x, q T ∗ y = p(α) ˆ q(β) ˆ T x, T ∗β y α,β

=

p(α) ˆ q(α) ˆ

|α| x, y = p, qH 2 x, y, α

d

α

where the identity for the Hd2 -inner product follows from (1.2). This implies that 2 ∗

∗ U = p T x T xj = p ⊗ x , p

pi , pj H 2 xi , xj j j i i j d j

i,j

i,j

3336


2 = pj ⊗ x j

,

Hd2 ⊗E0

j

thus U extends to be an isometric operator on Hd2 ⊗ E0 . We shall now finish the proof by showing that U has dense range. Let [E0 ]T ∗ denote the smallest common invariant subspace for T1∗ , . . . , Td∗ that contains E0 . We have to show that [E0 ]T ∗ = M⊥ = ran(I − P ). For k 0 we set Qk = Pk − Pk+1 , then each Qk is a projection and I − P = limN →∞ I − PN = ∞ k=0 Qk . Note that ran Q0 = E0 . Thus if we define Ek = ran Qk then we must show that for each k 0 we have Ek ⊆ [E0 ]T ∗ . This is trivially true for k = 0. Thus assume that Ek ⊆ [E0 ]T ∗ for some k 0. Then for x ∈ H we have Qk+1 x =

d

Ti∗ Qk Ti x ∈

i=1

d

Ti∗ Ek ⊆ [E0 ]T ∗ .

i=1

Hence the density of ran U in M⊥ follows by induction.

2

5. Finite rank extensions of spherical contractions and isometries We start out with a trivial lemma that will be used repeatedly and without further mention. Lemma 5.1. Let T be a commuting d-tuple of operators acting on a Hilbert space H and let R = (R1 , . . . , Rd ) be a nontrivial rank one extension of T acting on H ⊕ C i.e. Ti Ai Ri = , 0 Bi where Ai 1 = εxi and Bi 1 = bi for some ε > 0, x1 , . . . , xd ∈ H, di=1 xi 2 = 1, and b = (b1 , . . . , bd ) ∈ Cd . Then R is a commuting d-tuple if and only if for all i, j we have (Ti − bi )xj = (Tj − bj )xi . The following two lemmas are only preliminary results. A more definitive result for spherical contractions will be presented in Theorem 6.1, the result about spherical isometries will follow in Corollary 5.4. family of commuting spherical contractions, let T ∈ Fsc ∩ B(H)d , Lemma 5.2. Let F scd be the ∗ and let D = (I − i=1 Ti Ti )1/2 . Then T has a nontrivial rank one extension in Fsc if and only if there exist b = (b1 , . . . , bd ) ∈ Cd , x1 , . . . , xd ∈ H such that d 2 (i) i=1 xi = 1, (ii) (Ti − bi )xj = (Tj − bj )xi for all i, j , (iii) |b| d< 1, ∗and (iv) i=1 Ti xi ∈ ran D. Proof. Let R be a commuting rank 1 extension of T as in Lemma 5.1. Let x ∈ H and y ∈ C and calculate


3337

2 d d Ri x = Ti x + εxi y2 + |b|2 |y|2 y i=1

i=1

=

d i=1

Ti x + 2ε Re y x, 2

d

Ti∗ xi

+ ε

2

i=1

d

xi + |b| 2

2

|y|2 .

i=1

We now set x0 = di=1 Ti∗ xi and recall di=1 xi 2 = 1. Then we see that R is a spherical contraction if and only if for all x ∈ H and y ∈ C we have

2ε Re y x, x0 + ε 2 + |b|2 |y|2 Dx2 + |y|2 .

(5.1)

Assume now that R is a spherical contraction for some ε > 0, then clearly |b| < 1. By changing the argument of y if necessary, it follows from (5.1) that 2ε|y|| x, x0 | Dx2 + |y|2 for all x ∈ H and y ∈ C. Thinking of this as a quadratic inequality in |y| we conclude that ε 2 | x, x0 |2 Dx2 for all x ∈ H. By the Douglas lemma [12] di=1 Ti∗ xi = x0 ∈ ran D. This proves the necessity of the four conditions. Conversely assume (i)–(iv). In particular, x0 = Dz0 for some z0 ∈ H. Then

2ε Re y x, x0 + ε 2 + |b|2 |y|2 2ε y Dx, z0 + ε 2 + |b|2 |y|2

εDx2 z0 2 + ε + ε 2 + |b|2 |y|2 , which will be Dx2 + |y|2 for sufficiently small ε > 0.

2

Lemma 5.3. Let Fsi be the family of commuting spherical isometries and let T ∈ Fsi ∩ B(H)d . Then the following are equivalent: (a) T has a nontrivial rank one extension in Fsi , (b) T has a nontrivial rank one extension in Fsc , (c) there exist b = (b1 , . . . , bd ) ∈ Cd and x1 , . . . , xd ∈ H such that d 2 (i) i=1 xi = 1, (ii) (Ti − bi )xj = (Tj − bj )xi for all i, j , (iii) |b| d< 1, ∗and (iv) i=1 Ti xi = 0. Proof. (i) ⇒ (ii) is trivial and (ii) ⇒ (iii) follows immediately from Lemma 5.2, because D = 0. The implication (iii) ⇒ (i) follows from the proof of Lemma 5.2. Indeed since D = 0 and x0 = 0 we can take ε = 1 − |b|2 to obtain equality in (5.1). 2 Using the definition as given e.g. in Section 3 one checks that the Koszul complex for a commuting operator tuple R = (R1 , . . . , Rd ) acting on H is exact at Λ1 (H) if and only if whenever x1 , . . . , xd ∈ H are such that Ri xj = Rj xi for all i, j , then there is an x ∈ H such that xi = Ri x for all i. Corollary 5.4. Let T ∈ B(H)d be a commuting spherical isometry. Then the following are equivalent:

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(i) T has a nontrivial rank one extension in Fsi , (ii) T has a nontrivial rank one extension in Fsc , (iii) there exists b ∈ Bd such that the Koszul complex for T − b is not exact at Λ1 (H). For example, if Mz = (Mz , H 2 (∂D)) is the unilateral shift, then T = (Mz , 0) is a commuting spherical isometry. One easily checks that for (b1 , b2 ) ∈ B2 the Koszul complex for T − b is exact at Λ1 (H 2 (∂D)) if and only if b2 = 0, and since Mz has a nontrivial rank 1 extension it is of course clear that T has a nontrivial rank one extension. On the other hand, if d > 1 and if T = Mz = (Mz1 , . . . , Mzd ) acting on H 2 (∂Bd ), then it is known that the Koszul complex for Mz − b is exact at Λ1 (H 2 (∂Bd )) for all b ∈ Bd (see e.g. Proposition 2.6 of [15]). Thus Mz does not have a nontrivial rank one extension in Fsi . Proof. We have already seen the equivalence of (i) and (ii). To prove (i) ⇒ (iii) suppose that T has a nontrivial rank one extension in Fsi , then there exist a b ∈ Bd and x1 , . . . , xd ∈ H such that (i)–(iv) of Lemma 5.3(c) are satisfied. We will show that the Koszul complex for T − b is not exact at Λ1 (H). If it were exact, then by (ii) and the definition of exactness at Λ1 (H) there is an x ∈ H such that xi = (Ti − bi )x for all i. By (i) we have x = 0, and by (iv) we have 0=

d i=1

Ti∗ xi =

d

Ti∗ (Ti − bi )x = x −

i=1

d

Ti∗ bi x.

i=1

Thus x2 = di=1 Ti∗ bi x2 di=1 bi x2 = |b|2 x2 , because the adjoints of spherical isometries must be row contractions. Since x = 0 we conclude |b| 1 which is a contradiction. Hence the Koszul complex for T − b cannot be exact at Λ1 (H). We now prove (iii) ⇒ (i). Suppose that b ∈ Bd such that the Koszul complex for T − b is not exact at Λ1 (H). First we will assume that b = 0. Since T is a spherical isometry, the range of ∂0 : H → H ⊕ · · · ⊕ H, x → (T1 x, . . . , Td x) must be closed. Thus if the Koszul complex is not exact at Λ1 (H) H ⊕ · · · ⊕ H, then there is (x1 , . . . , xd ) ⊥ ran ∂0 such that di=1 xi 2 = 1 and Ti xj = Tj xi for all i, j . Then di=1 Ti∗ xi = ∂0∗ (x1 , . . . , xd ) = 0, so (i)–(iv) of Lemma 5.3(c) are satisfied with b = 0 and hence T has a nontrivial rank one extension in Fsi . If b = 0, then we consider a ball automorphism ϕb that takes b to 0. As in the paragraph preceding Lemma 2.4 of [15] we can define S = ϕb (T ). Then one checks that S is a commuting spherical isometry. Thus it is clear that T has a nontrivial rank one extension in Fsi if and only if S has a nontrivial rank one extension in Fsi . By Lemma 2.4 of [15] the Koszul complex for S is isomorphic to the Koszul complex for T − b. Hence the result follows from the case b = 0. 2 If F is a family and if an operator tuple T ∈ F has a nontrivial finite rank extension R ∈ F acting on H ⊕ K, then the compressions of the Ri to K will have a common eigenvector x0 and R = R|(H ⊕ Cx0 ) will be a rank one extension of T in F . However, it may happen that R is a trivial extension of T . For such situations the following lemma is useful. Lemma 5.5. Let H1 , H2 , H3 be Hilbert spaces, let K = H1 ⊕H2 ⊕H3 , and let R = (R1 , . . . , Rd ) be an operator tuple acting on K with matrix representation of the form Ri =

Ti 0 0

0 Bi 0

Ai Ci Di

.


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If R is a commuting spherical contraction (resp. commuting row contraction), then S = (S1 , . . . , Sd ), Si =

Ti 0

Ai Di

is a commuting spherical contraction (resp. commuting row contraction). Proof. Write H13 = H1 ⊕ H3 and note that S = PH13 R|H13 . The contractiveness assertions thus follow immediately. The commutativity follows from the special form of R. For all i, j we have Si Sj = PH13 Ri PH13 Rj |H13 = PH13 Ri Rj |H13 − PH13 Ri PH2 Rj |H13 = PH13 Ri Rj |H13

since PH13 Ri PH2 = 0

= PH13 Rj Ri |H13 = Sj Si .

2

Corollary 5.6. Let F = Fsc or F = Frc and let T ∈ F . Then T has a nontrivial finite rank extension in F if and only if T has a nontrivial rank one extension in F . Proof. Suppose T acts on a Hilbert space H and let R be a nontrivial finite rank extension of T in F acting on H ⊕ K with 1 < dim K < ∞. Let B = PK R|K be the compression of R to K. Then B is a commuting tuple of linear transformations on a finite dimensional space, thus the transformations Bi will have a common eigenvector x0 = 0. Then either R = R|(H ⊕ Cx0 ) is a nontrivial rank one extension of T in F or R is of the form as in Lemma 5.5 with H1 = H and H2 = Cx0 . In the latter case we can use the lemma to get a nontrivial extension R of T acting on H ⊕ K with R ∈ F and dim K = dim K − 1. Thus the result follows by an induction argument. 2 6. Extensions of spherical contractions Theorem 6.1. Let T be a commuting spherical contraction. Then the following are equivalent: (i) T has only trivial rank one extensions in Fsc , (ii) T has only trivial finite rank extensions in Fsc , (iii) T = S ∗ ⊕ V , where S is a direct sum of d-shifts and V is a spherical isometry such that for all b ∈ Bd the Koszul complex for V − b is exact at Λ1 , d ∗ (iv) (a) i=1 Ti Ti is a projection, and (b) for all b ∈ Bd the Koszul complex for T − b is exact at Λ1 . Proof. (i) and (ii) are equivalent by Corollary 5.6. (iii) ⇒ (i): If R is a rank one extension of T = S ∗ ⊕ V in Fsc , then since S ∗ is extremal in Fsc Lemma 2.1(a) implies that R = S ∗ ⊕ R for some R ∈ Fsc with R V . Then clearly R is a rank one extension of V and (i) follows from the hypothesis and the equivalence of (ii) and (iii) of Corollary 5.4.

3340


(iv) ⇒ (iii): By Proposition 4.1 the conditions (iv)(a) and (b) with b = 0 imply that T is of the form T = S ∗ ⊕ V , where S is a direct sum of d-shifts and V is a spherical isometry. The Koszul complex of S ∗ ⊕ V splits into a direct sum of Koszul complexes. Thus it is clear that (iv)(b) implies that for each b ∈ Bd the Koszul complex for V − b is exact at Λ1 . (i) ⇒ (iv): We will show the contrapositive. First suppose that (iv)(a) is not satisfied, i.e. d ∗ will use Lemma 5.2 with b = 0. i=1 Ti Ti is not a projection. We If E is the spectral measure for di=1 Ti∗ Ti , then there are real numbers r, s such that 0 < r < s < 1 and such that Q = E([r, s]) = 0. Let x0 ∈ ran Q, x0 = 0 and set xi = Ti x0 . Then d

xi = 2

i=1

d

1 Ti x0 =

t d Et x0 , x0 rx0 2 = 0.

2

i=1

0

Thus by scaling x0 we may assume that di=1 xi 2 = 1, and we have (i), (ii), and (iii) of s Lemma 5.2. Furthermore, since s < 1 we have Q = D r √ 1 dE, so ran Q ⊆ ran D. Hence 1−t

d

Ti∗ xi =

i=1

d

Ti∗ Ti x0 =

i=1

d

Ti∗ Ti Qx0 = Q

i=1

d

Ti∗ Ti x0 ∈ ran D

i=1

and (iv) of Lemma 5.2 is also satisfied. Thus T must have a nontrivial rank one extension in Fsc i.e. condition (i) of Theorem 6.1 does not hold. Next suppose that di=1 Ti∗ Ti = P1 is a projection, but that the Koszul complex for T is not exact at Λ1 . This implies that the column operator T (1) : H → Hd defined by ⎛

⎞ T1 x . T (1) x = ⎝ .. ⎠ Td x ∗

satisfies P1 = T (1) T (1) and hence T (1) is a partial isometry and in particular has closed range. x1 . / ran T (1) . Furthermore, there exist x1 , . . . , xd ∈ H such that Ti xj = Tj xi for all i, j , but .. ∈ xd x1 . Since ran T (1) is closed we may assume that .. ⊥ ran T (1) and di=1 xi 2 = 1. But this xd x1 d ∗ .. ∗ (1) or i=1 Ti xi = 0 ∈ ran D. Thus again we can use Lemma 5.2 to means that . ∈ ker T xd

see that condition (i) of Theorem 6.1 does not hold. Finally we suppose that di=1 Ti∗ Ti is a projection, that the Koszul complex for T is exact at Λ1 , but that there is a b ∈ Bd , b = 0 such that the Koszul complex for T − b is not exact at Λ1 . Then Proposition 4.1 implies that T = S ∗ ⊕ V , where S is a direct sum of d-shifts and V is a spherical isometry. Since the Koszul complex of S ∗ − b is exact at Λ1 , it follows that the Koszul complex for V − b cannot be exact at Λ1 . Thus in this case it follows from Corollary 5.4 that T would have a nontrivial rank one extension in Fsc . This concludes the proof of (iv) ⇒ (i). 2 Corollary 6.2. Theorem 1.4 holds.


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Proof. The implication (ii) ⇒ (i) follows from Theorems 3.3 and 2.2 and Lemma 2.1. We now show (iii) ⇒ (ii). If T satisfies the conditions (a), (b), and (c) of part (iii) of Theorem 1.4, then by Proposition 4.1 T is unitarily equivalent to S ∗ ⊕ V , where S is a direct sum of d-shifts and V = (V1 , . . . , Vd ) is a spherical isometry. We will see that V is a spherical unitary. Condition (b) implies that 0

d

Vi Vi∗ − I =

i=1

d

Vi Vi∗ − Vi∗ Vi .

i=1

We already mentioned that each operator in a spherical isometric tuple must be subnormal (also see Theorem 1.3), thus for each i we have Vi Vi∗ − Vi∗ Vi 0. Hence Vi Vi∗ = Vi∗ Vi for all 1 i d. Finally we prove (i) ⇒ (iii). If T is extremal then it has no nontrivial rank one extension, hence conditions (iii)(a) and (c) follow from the equivalence of (i) and (iv) in Theorem 6.1. Then it follows from Proposition 4.1 that T = S ∗ ⊕ V for a spherical isometry V . Since T is extremal Theorem 2.2 implies that V must in fact be a spherical unitary tuple. Then T ∗ = S ⊕ U for some spherical unitary tuple U and a direct sum of d-shifts S. Condition (iii)(b) follows easily (see (1.3)). 2 Corollary 6.3. Corollary 1.5 holds. Proof. Let T = S ⊕ U , where S is a direct sum of d-shifts and U is a spherical unitary tuple. It follows from (1.1) and (1.3) that T satisfies conditions (a) and (b) of Corollary 1.5. Furthermore, in Section 3 we mentioned that the Koszul complex for the d-shift is exact at Λp (H) for all p with 1 p d − 1. The Taylor spectrum of any spherical unitary tuple U must be contained in ∂Bd . Since such U is normal this can easily be deduced from Proposition 7.2 of [11]. Hence the Koszul complex of T = S ⊕ U is exact at Λd−1 (H), i.e. (c) of Corollary 1.5 holds as well. Conversely suppose that T satisfies (a), (b), and (c) of Corollary 1.5. Then the adjoint tuple T ∗ satisfies (iii)(a) and (iii)(b) of Theorem 1.4. Since di=1 Ti Ti∗ is a projection the operator H → Hd defined by x → (T1∗ x, . . . , Td∗ x) is a partial isometry and thus has closed range. This operator is unitarily equivalent to ∂T ∗ ,0 , hence ∂T ∗ ,0 has closed range in Λ1 (H). Thus, as the hypothesis (c) is that K(T ) is exact at Λd−1 (H) the discussion about the Hodge ∗-operator at the beginning of Section 3 implies that K(T ∗ ) is exact at Λ1 (H), i.e. T ∗ satisfies (iii)(c) of Theorem 1.4. Hence Theorem 1.4 implies that T ∗ = S ∗ ⊕ U , where S is a direct sum of d-shifts and U (and hence U ∗ ) is a spherical unitary tuple. This concludes the proof of the corollary. 2 7. Rank one extensions of row contractions of commuting row contractions, Next we consider row contractions. Let Frc denote the family let T = (T1 , . . . , Td ) ∈ Frc ∩ B(H)d and write D∗ = (I − di=1 Ti Ti∗ )1/2 for the defect operator. If R ∈ B(H ⊕ K) is an operator tuple that extends T , then we will use the notation Ri =

Ti 0

Ai Bi

,

i = 1, . . . , d.

(7.1)

3342


Note that R will be a commuting tuple if and only if Ti Aj − Tj Ai = Aj Bi − Ai Bj

and Bi Bj = Bj Bi

for all i, j.

(7.2)

Lemma 7.1. Let T be a commuting row contraction. (a) If b = (b1 , . . . , bd ) ∈ Cd , |b| = 1, then ker(I − di=1 bi Ti ) ⊆ ker D∗ . (b) If R is a row contraction that extends T , then di=1 Ai A∗i D∗2 and for each i we have ran Ai ⊆ ran D∗ . Proof. (a) Let |b| = 1 and let x ∈ ker(I −

d

i=1 bi Ti ).

Then

2 2 d d ∗ x = x, b i Ti x = bi Ti x, x 4

i=1

|b|2

i=1

d

∗ 2

T x x2 = x2 − D∗ x2 x2 . i

i=1

This implies D∗ x = 0. (b) Recall that R is a row contraction if and only if R ∗ is a spherical contraction, i.e. for all x ∈ H, y ∈ K we have d ∗ x 2 2 2 R i y x + y . i=1

A short calculation shows that this happens if and only if d ∗ A x + B ∗ y 2 D∗ x2 + y2 . i

i

(7.3)

i=1

In particular, we see that if R is a row contraction, then di=1 Ai A∗i D∗2 and hence for each i we must have ran Ai ⊆ ran D∗ (by the Douglas lemma [12]). This proves (b). 2 Lemma 7.2.Let Frc be the family of commuting row contractions, let T ∈ Frc ∩ B(H)d , and let D∗ = (I − di=1 Ti Ti∗ )1/2 . Then T has a nontrivial rank one extension in Frc , if and only if there exist b = (b1 , . . . , bd ) ∈ Cd , x1 , . . . , xd ∈ H such that d 2 (i) i=1 xi = 1, (ii) (Ti − bi )xj = (Tj − bj )xi for all i, j , (iii) |b| < 1, and (iv) xi ∈ ran D∗ for each i.


3343

Proof. First assume that we are given b ∈ Cd and x1 , . . . , xd ∈ H such that (i)–(iv) are satisfied. For any ε > 0 we define a rank one extension R of T as in Lemma 5.1. By (i) and (ii) it be nontrivial and commutative. Thus by (7.3) R will be a row contraction if and only if will d |ε x, xi + bi y|2 D∗ x2 + |y|2 for all x ∈ H, y ∈ C. i=1 Since each xi ∈ ran D∗ , there are zi ∈ H such that xi = D∗ zi . Then for x ∈ H and y ∈ C we have d d d ε x, xi + bi y 2 ε 2 D∗ x, zi 2 + 2ε D∗ x, zi |bi ||y| + |b|2 |y|2 i=1

i=1

ε

2

d

i=1

! d ! " 2 zi D∗ x + 2ε D∗ x zi |b||y| + |b|2 |y|2 2

2

i=1

i=1

d

ε2 + ε zi 2 D∗ x2 + (1 + ε)|b|2 |y|2 D∗ x2 + |y|2 , i=1

whenever ε is sufficiently small. Thus T has a nontrivial rank one extension in Frc . Conversely, assume that T has a nontrivial rank one extension R in Frc . Then R can be written as in Lemma 5.1. Thus we have ε > 0, b ∈ Cd and x1 , . . . , xd ∈ H satisfying (i) and (ii). Furthermore, a calculation similar to what was done in the first part of the proof shows that since R is a row contraction we must have d d d ε x, xi + bi y 2 = ε 2 x, xi 2 + 2ε Re

x, xi bi y + |b|2 |y|2 i=1

i=1

i=1

D∗ x + |y| 2

2

for all x ∈ H and all y ∈ C. By taking y = 0 we see that the Douglas lemma [12] implies that (iv) must be satisfied, and by taking x = 0 it follows that |b| 1. We will be done if we can rule out the possibility that |b| = 1. Note that R ∗ is a nontrivial extension in Fsc of the tuple of scalars b : C → C. Hence Theorem 2.2 implies |b| < 1. A somewhat |b| = 1, then more direct argument goes as follows: If the above inequality implies that Re di=1 x, xi bi y = 0 for all x and y. Hence di=1 bi xi = 0. Now we multiply (ii) by bi and sum in i to obtain ( di=1 bi Ti − I )xj = (Tj − bj ) di=1 bi xi = 0 for each j . Thus each xj ∈ ker(I − di=1 bi Ti ) and hence by Lemma 7.1(a) xj ∈ ker D∗ . This contradicts (i) and (iv), which is already known to hold. 2 8. Extensions of row contractions In this section we shall prove Theorems 1.6 and 1.8 and Corollary 1.7. Proposition 8.1. (a) If D∗ = 0, then T ∈ ext(Frc ). (b) If D∗ is onto, then T has a rank one extension in ext(Frc ).

3344


(c) If D∗ is a projection, then the following are equivalent: (i) T has a nontrivial rank one extension in Frc , (ii) T ∈ / ext(Frc ), (iii) there are x1 , . . . , xd ∈ dj =1 ker Tj∗ with di=1 xi 2 > 0 and Ti xj = Tj xi for all i, j . Proof. (a) follows directly from Lemma 7.1(b). (b) Clearly the zero tuple, T = (0, . . . , 0), is not extremal. Thus assume that D∗ is onto and one of the Ti ’s is not zero. Then we can set b = 0 and choose x ∈ H such that the hypothesis of Lemma 7.2 is satisfied with xi = Ti x. (c) (iii) ⇒ (i) follows directly from Lemma 7.2 with b = 0. (i) ⇒ (ii) is trivial. (ii) ⇒ (iii): We assume that D∗ is a projection and that we have a nontrivial extension in Frc . Then with the notation as in (7.1) we set xi = Ai x, where x is chosen so that di=1 xi 2 > 0. Lemma 7.1(b) implies that for all k we have ran Ak ⊆ ran D∗ . Thus x1 , . . . , xd ∈ ran D∗ . Furthermore, since D∗ is a projection we have d #

ran Ak ⊆ ran D∗ =

d $

ker Tj∗

=

j =1

k=1

d #

⊥ ran Tj

.

j =1

Thus, commutativity implies that for all i and j Ti xj − Tj xi = Ti Aj x − Tj Ai x = Aj Bi x − Ai Bj x ∈

d #

ran Ak ∩

k=1

This establishes (iii).

d #

ran Tj = (0).

j =1

2

Proposition 8.2. If T ∈ Frc and if there is a u ∈ ran D∗ , u = 1, such that dim span{u, T1 u, . . . , Td u} 2, then T has a nontrivial rank one extension in Frc . Proof. The hypothesis implies that there is v ∈ H, v ⊥ u and α = (α1 , . . . , αd ), β = (β1 , . . . , βd ) ∈ Cd , β = 0 such that Ti u = αi u + βi v for i = 1, . . . , d. Indeed, if dim span{u, T1 u, . . . , Td u} = 2, then we can find such a unit vector v satisfying this, while if dim span{u, T1 u, . . . , Td u} = 1 we take v = 0 and any β = 0. We set γ = Pβ ⊥ α = α − cβ, where c = α,β . Then β, γ = 0 and α, γ = α, Pβ ⊥ α = |γ |2 . |β|2

βi The conclusion will follow from Lemma 7.2 with xi = |β| u and b = γ . Conditions (i) and (iv) are obvious from the definition of the xi . In order to verify (iii) we calculate

(Ti − γi )xj =

βi βj (v − cu) = (Tj − γj )xi |β|

for all i, j . Finally, di=1 γi Ti u = α, γ u + β, γ v = |γ |2 u. Since T is a row contraction this implies d 2 d |γ | = Ti (γi u) γi u2 = |γ |2 . 4

i=1

Hence |γ | 1.

i=1


3345

If |γ | = 1, then (I − di=1 γi Ti )u = 0, thus Lemma 7.1(a) implies that u ∈ ker D∗ . But since u ∈ ran D∗ this would mean u = 0, which is impossible. Hence |γ | < 1. 2 We shall now prove part (iv) of Theorem 1.6. Theorem 8.3. Let T be a commuting row contraction with D∗ = u ⊗ u for some u ∈ H, u = 0. Then the following are equivalent: (i) T ∈ ext(Frc ), (ii) T has only trivial rank one extensions in Frc , (iii) dim span{u, T1 u, . . . , Td u} 3. Proof. (i) ⇒ (ii) is trivial and (ii) ⇒ (iii) follows directly from Proposition 8.2. (iii) ⇒ (i): Suppose that dim span{u, T1 u, . . . , Td u} 3. Since u = 0 we may without loss of generality assume that the set {u, T1 u, T2 u} is linearly independent. Let R be an extension of T in Frc and assume each Ri is of the form as in (7.1). We must show that each Ai = 0. Since D∗ = u ⊗ u Lemma 7.1(b) implies the existence of x1 , . . . , xd ∈ K such that Ai x =

x, xi u for each i. Then commutativity (see (7.2)) implies for all x ∈ K and all i, j

x, xj Ti u − x, xi Tj u = Bi x, xj u − Bj x, xi u = x, Bi∗ xj − Bj∗ xi u.

(8.1)

Take i = 1 and j = 2. Then the linear independence of {u, T1 u, T2 u} shows that x, x1 =

x, x2 = 0 for all x ∈ K. Thus x1 = x2 = 0. Next consider (8.1) with i = 1 and j > 2. Since x1 = 0 we get

x, xj T1 u = x, B1∗ xj u. Again linear independence implies xj = 0. Thus Aj = 0 for all j , and T must be extremal.

2

Corollary 8.4. Corollary 1.7 holds. Proof. Write P = PM⊥ for the projection of Hd2 onto M⊥ . Recall that if S denotes the d-shift on Hd2 , then I − di=1 Si Si∗ = 1 ⊗ 1 is the projection onto the constants. For i = 1, . . . , d we have Ti = P Si |M⊥ , hence D∗2

= IM⊥ −

d

Ti Ti∗

=P I −

i=1

d

Si Si∗

P = ϕ ⊗ ϕ,

i=1

/ M, thus ϕ = 0 and rank D∗ = 1. where ϕ = P 1. Since we are assuming M = Hd2 we have 1 ∈ Let α0 , . . . , αd ∈ C, then α0 ϕ +

d i=1

αi Ti ϕ = 0

⇔

α0 +

d

αi zi ∈ M.

i=1

This implies that span{ϕ, T1 ϕ, . . . , Td ϕ} is isomorphic to L/M ∩ L. But dim L/M ∩ L = d + 1 − dim(M ∩ L). Thus Corollary 8.4 follows from Theorem 8.3. 2

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9. An example Let S = (Mz , Mw ) be the 2-shift on H22 and let M = {f ∈ H22 : f (z, 0) = 0}. M is invariant for S, thus T = S|M is a nonextremal row contraction. We claim that T has no nontrivial finite rank extensions in Frc . Note that the linear span of monomials of the form zn w m , n 0, m > 0 is dense in M and one computes % 1 n D∗2 zn w m = n+1 z w if m = 1, 0 if m > 1. From this one easily sees that there are no nonzero f, g ∈ ran D∗ and b = (b1 , b2 ) ∈ B2 such that (z − b1 )f = (w − b2 )g. Hence Theorem 1.8 applies to show the claim. References [1] Jim Agler, An abstract approach to model theory, in: Surveys of Some Recent Results in Operator Theory, vol. II, in: Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 1–23. [2] T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963) 88–90. [3] William Arveson, Subalgebras of C ∗ -algebras. II, Acta Math. 128 (3–4) (1972) 271–308. [4] William Arveson, Subalgebras of C ∗ -algebras. III. Multivariable operator theory, Acta Math. 181 (2) (1998) 159– 228. [5] William Arveson, The Dirac operator of a commuting d-tuple, J. Funct. Anal. 189 (1) (2002) 53–79. [6] William Arveson, Notes on the unique extension property, unpublished manuscript, see http://math.berkeley.edu/~ arveson/texfiles.html, 2003. [7] Ameer Athavale, On the intertwining of joint isometries, J. Operator Theory 23 (2) (1990) 339–350. [8] K.R.M. Attele, A.R. Lubin, Dilations commutant lifting for jointly isometric operators—a geometric approach, J. Funct. Anal. 140 (2) (1996) 300–311. [9] S. Brehmer, Über vetauschbare Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. (Szeged) 22 (1961) 106– 111. [10] Raul E. Curto, Fredholm invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1) (1981) 129–159. [11] Raúl E. Curto, Applications of several complex variables to multiparameter spectral theory, in: Surveys of Some Recent Results in Operator Theory, vol. II, in: Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 25–90. [12] R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966) 413–415. [13] Michael A. Dritschel, Scott McCullough, Model theory for hyponormal contractions, Integral Equations Operator Theory 36 (2) (2000) 182–192. [14] Michael A. Dritschel, Scott A. McCullough, Boundary representations for families of representations of operator algebras and spaces, J. Operator Theory 53 (1) (2005) 159–167. [15] Jim Gleason, Stefan Richter, Carl Sundberg, On the index of invariant subspaces in spaces of analytic functions of several complex variables, J. Reine Angew. Math. 587 (2005) 49–76. [16] Phillip Griffiths, Joseph Harris, Principles of Algebraic Geometry, Wiley Classics Lib., John Wiley & Sons Inc., New York, 1994, reprint of the 1978 original. [17] Takasi Itô, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958) 1–15. [18] V. Müller, F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (4) (1993) 979–989. [19] Walter Rudin, Functional Analysis, second edition, Internat. Ser. Pure Appl. Math., McGraw–Hill Inc., New York, 1991. [20] Béla Sz.-Nagy, Ciprian Foia¸s, Harmonic Analysis of Operators on Hilbert Space, translated from the French and revised, North-Holland Publishing Co., Amsterdam, 1970. [21] Joseph L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970) 172–191.


On Schechter’s linking theorems W. Zou a,∗,1 , S. Li b,2 a Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China b Institute of Mathematics, AMSS, Academia Sinica, Beijing 100080, China

Received 16 September 2009; accepted 6 November 2009 Available online 26 November 2009 Communicated by L. Gross

Abstract The existence theorems of critical points due to M. Schechter [M. Schechter, The saddle point alternative, Amer. J. Math. 117 (1995) 1603–1626] are developed to the case of sign-changing critical points. A new relationship between the linking without value-splitting property and the sign-changing (weak) Palais– Smale sequence is established. The new abstract theorems are applied to find nodal solutions to elliptic equations under rather weak hypotheses. © 2009 Elsevier Inc. All rights reserved. Keywords: Schechter’s linking; Sign-changing solution; PS-sequence; Value-splitting

1. Introduction A fundamental method of finding critical points of a C1 functional on a Banach space E consists of finding two subsets A and B of E such that A links B in the sense of Schechter [17] (or Rabinowitz [11]) and satisfy the value-splitting condition a0 := sup G b0 := inf G. A

B


E-mail addresses: [email protected] (W. Zou), [email protected] (S. Li). 1 Supported by NSFC and the program of the Ministry of Education in China for NCET. 2 Supported by NSFC.


(1.1)

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W. Zou, S. Li / Journal of Functional Analysis 258 (2010) 3347–3361

In [17] it was shown the existence of a sequence {uk } in E with G(uk ) → a b0 and G (uk ) → 0. This kind sequence is called Palais–Smale sequence ((PS) sequence, for short). The (PS) sequence usually can deduce a critical point (a solution of the equations) in case that additional hypotheses are imposed. However, if (1.1) is violated, the existence of the (PS) sequence is not assured. In Schechter’s paper [14], he considered this situation in which there is a sequence {Ak } in E such that Ak links B for each k and (1.1) is violated. He got the following alternative: Either (1) a (PS) sequence exists, or (2) there exists a sequence {uk } such that G(uk ) → a,

uk → ∞,

G(uk ) → 0, uk θ

G (uk ) → 0, uk θ−1

where θ 1 is a constant. The usefulness of this alternative theorem lies in the fact that the kind of arguments which are used in showing the boundedness of the (PS) sequences can be used to eliminate option (2) from the alternative. By this way, the existence of critical points can also be deduced. The abstract theorems of [14] are successfully applied to solve the existence of solutions to a series of semilinear elliptic equations with very weak assumptions, where the standard linking methods are usually ineffective. However, when ones want to study the existence of sign-changing (nodal) solutions to elliptic equations, Schechter’s theorems in [14] fail. The aim of the present paper is to develop Schechter’s theory and establish new theorems where (1.1) is not needed and sign-changing critical points can be obtained. Roughly speaking, I introduce some new linking sets A and B. If (1.1) is violated, we either get a sequence such that {uk } is sign-changing,

G(uk ) → a

and G (uk ) → 0,

or show that there exists a sequence {uk } such that {uk } is sign-changing,

G(uk ) → 0 and uk β

G (uk ) → 0, uk β−1

where β 1. The new saddle point alternative theorem may also produce the sign-changing critical points. Moreover, in this paper, a series of new theorems on the existence of sign-changing (PS) (or weak) sequence for various linkings will be established when (1.1) does not hold. Even if (1.1) holds, we shall build new theorems on the existence of sign-changing weak (PS) sequences. We illustrate this method with several applications to semilinear elliptic boundary value problems. 2. On Schechter’s linking theorems Let E be a Hilbert space endowed with an inner product ·,· and the associated norm · . Suppose that there is another norm · ∗ of E such that u∗ C∗ u for all u ∈ E, here C∗ > 0 is a constant. Moreover, we assume that un − u∗ ∗ → 0 whenever un u∗ weakly in (E, · ). For example, in the Sobolev space H01 (Ω), we may choose · ∗ = · p , the usual Lp -norm (see [12,21,22]). A functional G ∈ C1 (E, R) maps bounded sets to bounded sets. Let K := {u ∈ E: G (u) = 0}, E˜ := E\K, K[a, b] := {u ∈ K: G(u) ∈ [a, b]}, Ga := {u ∈ E: G(u) a}. As in


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Schechter [14–16], let ±P be a positive (negative) cone of E. Define a class of contractions of E as follows: ⎫ ⎧ Γ (·,·) ∈ C [0, 1] × E, E , Γ (0, ·) = id, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for each t ∈ [0, 1), Γ (t, ·) is a homeomorphism of E ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −1 onto itself and Γ (t, ·) is continuous on [0, 1) × E; ⎬ . (2.1) Φ := Γ : ⎪ ⎪ there exists an x0 ∈ E such that Γ (1, x) = x0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for each x ∈ E and that Γ (t, x) → x0 as t → 1 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ uniformly on bounded subsets of E Obviously, Γ (t, u) = (1 − t)u ∈ Φ. Let A ⊂ E be a bounded subset of E. Definition 2.1. (See [17,19].) A subset A of E is linked to a subset B of E if A ∩ B = ∅ and, for every Γ ∈ Φ, there is a t ∈ [0, 1] such that Γ (t, A) ∩ B = ∅. Since Φ = ∅ and A is bounded, we may choose a β > 0 such that ΦA,β := Γ ∈ Φ: Γ [0, 1], A ⊂ Gβ = ∅.

(2.2)

Definition 2.2. Subset A of E is called ΦA,β -linked to a subset B of E if A ∩ B = ∅ and, for every Γ ∈ ΦA,β , there is a t ∈ [0, 1] such that Γ (t, A) ∩ B = ∅. Obviously, if A links B in the sense of Definition 2.1, then we may find a β > 0 such that A is ΦA,β -linked to B. The following linking is different from the classical ones in Rabinowitz [11]. Proposition 2.1. Let E = M ⊕ Y , where M, Y are closed subspaces and Y is of finite dimensional. Choose R > 0, ρ > 0 and y0 ∈ M\{0} with yR0 = ρ and y0 ∗ = 1. Set A := u = v + sy0 : v ∈ Y, s 0, u = R ∪ u ∈ Y : u R , B := u ∈ M: u∗ = ρ , then A links B in the sense of Definition 2.1 and hence, of Definition 2.2 for some β > 0. Remark 2.1. The choice of B is essential since in applications, dist(B ∩ Gβ , ±P) > 0 and this plays an important role in getting sign-changing solutions (see [12]). However, in classical case, B = {u ∈ M: u = ρ} and dist(B, ±P) = 0 in applications, hence we can not be applied directly to get sign-changing critical points. Proof. We first let Q = {sy0 + v: v ∈ Y, s 0, sy0 + v R}. Then A = ∂Q in RN +1 . Let u = v + w with v ∈ Y , w ∈ M, we define F u = v + w∗ y0 . Then F |Q = id and B = F −1 (ρy0 ). We can apply Proposition 2.6.2 of Schechter [17] to conclude that A links B in the sense of Definition 2.1. Hence, A is ΦA,β -linked to B for some β > 0. 2 As in [12,22], we make the following assumption. (A1 ) B ∩ Gc = ∅ ⇒ dist(B ∩ Gc , ±P) > 0.

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In most applications, (A1 ) can be satisfied readily. Note that if B = {u0 } and u0 ∈ / ±P, then (A1 ) is satisfied. For μ0 > 0, define Dμ0 := {u ∈ E: dist(u, P) < μ0 }. Set D := Dμ0 ∪ (−Dμ0 ), Sβ = E\D. Noting (A1 ) and B ∩ Gβ = ∅, we may assume μ0 small enough so that B ∩ Gβ ⊂ Sβ . Define a0 := sup G, A∩Sβ

b0 := inf G, B

a :=

inf

sup

Γ ∈ΦA,β Γ ([0,1],A)∩Sβ

G(u).

(2.3)

If A ∩ Sβ = ∅, we set a0 = −∞. Note that a is well defined if A is ΦA,β -linked to B. Further, we make the following hypotheses. (A2 ) Assume G is of the form G (u) = u − G(u), where G : E → E is a continuous operator and G(±Dμ0 ) ⊂ ±Dμ for some μ ∈ (0, μ0 ); K ∩ (E\{−P ∪ P}) ⊂ Sβ . Theorem 2.1. Assume (A1 )–(A2 ) and A is ΦA,β -linked to B. Assume that a0 b0 . Then a ∈ [b0 , β] and there exists a sequence {uk } such that {uk } ∈ Sβ ,

G(uk ) → a,

G (uk ) → 0.

(2.4)

If a = b0 , then we may require that dist(uk , B) → 0. Proof. Obviously, a ∈ [b0 , β]. We first consider the case of a > b0 . √ If (2.4) were false, there would exist a positive constant ε¯ ∈ (0, 1/8) such that G (u) 8¯ε for u ∈ T1 := {u ∈ Sβ : |G(u) − a| 3δ} for some δ ∈ (0, ε¯ /2) ∩ (0, a − b0 ). Note that T1 = ∅. Let V0 : E˜ → E be a ˜ ⊂ (±Dμ ) for some pseudo-gradient vector field of G with V0 (u) = u−L0 (u) and L0 (±Dμ0 ∩ E) V0 (u) μ ∈ (0, μ0 ). The construction of such V0 and L0 can be found in [22]. Let Y0 (u) := 1+V . 0 (u) Then Y0 (u) is locally Lipschitz continuous and G (u), Y0 (u) ε¯ on T1 . Let T2 := {u ∈ Sβ : dist(u,T2 ) |G(u) − a| 2δ}, T3 := {u ∈ Sβ : |G(u) − a| δ}. Define ξ(u) = dist(u,T . We remark 3 )+dist(u,T2 )

(t,u) that T3 = ∅. If T2 = ∅, we set ξ ≡ 1. Consider the Cauchy problem: dωdt = −ξ(ω)Y0 (ω), ω(0, u) = u ∈ E. Then ω(t, ±Dμ0 ) ⊂ ±Dμ0 for all t 0 (cf. [20,22]); and d G(ω(t, u))/dt / T3 for some t1 ∈ [0, 1], then ei−¯ε ξ(ω(t, u)) if ω(t, u) ∈ Sβ . Let u ∈ Ga+δ . If ω(t1 , u) ∈ / Sβ (hence, ω(t, u) ∈ D for all t t1 ). ther G(ω(1, u)) G(ω(t1 , u)) < a − δ or ω(t1 , u) ∈ On the other hand, if σ (t, u) ∈ T3 for all t ∈ [0, 1], then G(ω(1, u)) a − δ. Summing up, we have ω(1, Ga+δ ) ⊂ Ga−δ ∪ D. By the definition of a, we have a Γ ∈ ΦA,β such that Γ ([0, 1], A) ⊂ Ga+δ ∪ D. Set Γ1 (s, u) = ω(2s, u) if s ∈ [0, 1/2] and = ω(1, Γ (2s − 1, u)) for all s ∈ [1/2, 1]. Then Γ1 ∈ ΦA,β . It is easy to check that Γ1 (s, A) ⊂ Ga−δ ∪ D, which contradicts the definition of a. Next we consider the case of a = b0 . Here we have to construct different vector field and need a careful analysis of √ the flow. If it were not true, there would exist positive numbers ε, δ, T such that G (u) 8ε for u ∈ T4 := {u ∈ Sβ : dist(u, B) 4T , |G(u)−a| < 3δ} for some δ < T ε/2. Let T5 := {u ∈ Sβ : dist(u, B) 3T , |G(u)−a| < 2δ}, T6 := {u ∈ Sβ : dist(u, B) 2T , |G(u) − a| δ}. Then we observe that T6 = ∅. Define dist(u,E\T5 ) ζ (u) = dist(u,E\T . Consider the similar Cauchy problem as above, where ξ is replaced 5 )+dist(u,T6 ) by ζ and we get a new flow ω satisfying ω(t, ±Dμ0 ) ⊂ ±Dμ0 for all t 0. Take any u ∈ Ga+δ , / T6 , then either ω(t1 , u) ∈ D or ω(t1 , u) < a − δ if there exists a t1 T such that ω(t1 , u) ∈ or dist(ω(t1 , u), B) > 2T . For the last case, since ω(t, u) − ω(s, u) |t − s|, we see that dist(ω(t, u), B) > T for all t ∈ [0, T ]. On the other hand, if ω(t, u) ∈ T6 for all t ∈ [0, T ],


3351

then G(ω(T , u)) a − δ. Summing up, we see that ω(T , Ga+δ ) ∩ B ∩ Sβ = ∅. We claim that ω([0, T ], A) ∩ B ∩ Sβ = ∅. Otherwise, we have an ω(t, u) ∈ B ∩ Sβ , u ∈ A ∩ Sβ . Note that

t G(ω(t, u)) a0 − ε 0 ζ (ω(s, u)) ds and G(ω(t, u)) b0 , hence ζ (ω(s, u)) = 0 for all s ∈ [0, t] hence ω(s, u) ∈ E\T5 , that is, either G(ω(s, u)) < a −δ for all s ∈ [0, t] or dist(ω(s, u), B) > 2T for all s ∈ [0, t] or ω(s, u) ∈ D for all s ∈ [0, t]. These are absurd. Then our claim is correct. By the definition of a, we have a Γ ∈ ΦA,β such that Γ ([0, 1], A) ⊂ Ga+δ ∪ D. Define Γ1 such that Γ1 = ω(2sT , u) for s ∈ [0, 1/2] and = ω(T , Γ (2s − 1, u)) if s ∈ [1/2, 1]. We have Γ1 ∈ ΦA,β . But Γ1 ([0, 1], A) ∩ B ∩ Sβ = ∅, also a contradiction. 2 Remark 2.2. If we replace (A1 ) by “B ⊂ Sβ ”, then Theorem 2.1 is still true (see [20,22]). Lemma 2.1. Assume (A2 ). If for any Γ ∈ ΦA,β , there holds / A, G(v) a0 = ∅, SΓ := v = Γ (s, u): s ∈ (0, 1], u ∈ A, v ∈ Sβ , v ∈ then there is a sequence {uk } ∈ Sβ such that G(uk ) → a and G (uk ) → 0 as k → ∞. Proof. We set B0 := Γ ∈ΦA,β SΓ . Then A ∩ B0 = ∅, B0 ⊂ Sβ and A is ΦA,β -linked to B0 . In particular a0 infB0 G. By Theorem 2.1 and Remark 2.2, we get the conclusion. 2 Lemma 2.2. Assume (A2 ) and either (A1 ) or B ⊂ Sβ . If a0 = a, then there is a sequence {uk } ∈ Sβ such that G(uk ) → a and G (uk ) → 0 as k → ∞. Proof. Note that A ∩ Sβ ⊂ Γ ([0, 1], A) ∩ Sβ , we see that a > a0 . Hence, for any Γ ∈ ΦA,β , there exists a u ∈ A, s ∈ (0, 1] such that G(Γ (s, u)) > a0 , Γ (s, u) ∈ Sβ , hence SΓ = ∅. By the above Lemma 2.1, we get the conclusion. 2 Theorem 2.2. Let A, a0 be as above. Then the sufficient and necessary condition of SΓ = ∅ for all Γ ∈ ΦA,β is there exists a B0 ⊂ Sβ such that A is ΦA,β -linked to B0 and a0 infB0 G. Proof. If SΓ = ∅, then B0 = Γ ∈ΦA,β SΓ is what we want. If SΓ = ∅ for some Γ ∈ ΦA,β , then for any B0 ⊂ Sβ with A ∩ B0 = ∅ and a0 infB0 G, we must have Γ ([0, 1], A) ∩ B0 = ∅, a contradiction. 2 Theorem 2.3. If for each Γ ∈ ΦA,β , supΓ ([0,1],A)∩Sβ G is attained at w ∈ / A, then there is a sequence {uk } ⊂ Sβ such that G(uk ) → a and G (uk ) → 0 as k → ∞. Proof. For any Γ ∈ ΦA,β , supΓ ([0,1],A)∩Sβ G a0 . If for each Γ ∈ ΦA,β , supΓ ([0,1],A)∩Sβ G is attained at w ∈ / A, then w = Γ (s, e) ∈ Sβ , G(w) a0 . Hence, SΓ = ∅. By Lemma 2.1, we get the conclusion. 2 ˜ B). Set A˜ := {u ∈ A ∩ Sβ : G(u) > b0 }. Then A˜ = ∅ if and only if a0 b0 . Let α := dist(A, If A˜ = ∅, we take α = ∞. Let φ(t) be a positive nondecreasing function on [0, ∞) such that

T φ(t)2 dt for some T α. a0 − b0 < 14 0 1+φ(t)

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Theorem 2.4. Assume (A1 )–(A2 ). If A is ΦA,β -linked to B and b0 ∈ R, then for each ε > 0 there is a uε ∈ E such that uε ∈ Sβ ,

b0 − ε G(uε ) a + ε,

G (uε ) < φ dist(uε , B) .

Proof. By Lemma 2.2, we may assume a = a0 . By negation, we assume that there is an ε > 0 such that ε < φ(dist(u, B)) G (u) for all u ∈ Q := {u ∈ Sβ : b0 − 3ε G(u) a + 3ε}. We note that Q = ∅ for such an ε. We may choose ε < 1 small enough and let T < α such

T φ 2 (t) φ 2 (0) that a − b0 + ε < 14 0 1+φ(t) dt and ε < 4(1+φ(0)) T . Let Q0 := {u ∈ Q: either b0 − 2ε G(u) or G(u) a + 2ε}, Q1 := {u ∈ Q: b0 − ε G(u) a + ε}, and η(u) =

dist(u,Q0 ) dist(u,Q1 )+dist(u,Q0 ) . Then

Q1 = ∅. If Q0 = ∅, we let η ≡ 1. Consider dσ dt = −η(σ )Y0 (σ ), σ (0, u) = u. Then σ (t, u) − u =

t − 0 η(σ )Y0 (σ ) dt, σ ([0, ∞), ±Dμ0 ) ⊂ ±Dμ0 (cf. [22]) and dist(u, B) − t dist σ (t, u), B dist(u, B) + t Further,

d G(σ (t,u)) dt

∀u ∈ E, t 0.

(2.5)

2

φ (dist(σ,B)) − 14 η(σ ) 1+φ(dist(σ,B)) if σ (t, u) ∈ Sβ and

1 G σ (t, u) G(u) − 4

t 0

φ 2 (dist(σ (s, u), B)) ds η σ (s, u) 1 + φ(dist(σ (s, u), B))

if σ (t, u) ∈ Sβ .

Assume for the moment that σ (T , A) ∩ Sβ = ∅ and let W1 := {u ∈ A: σ (T , u) ∈ Sβ }. Then for each u ∈ W1 , we have that σ (t, u) ∈ Sβ for all t ∈ [0, T ]. In particular, Sβ ∩ A = ∅. Now we consider u ∈ W1 . If there is a t1 ∈ [0, T ] such that σ (t1 , u) ∈ / Q1 , then G(σ (T , u)) G(σ (t1 , u)) < b0 − ε. On the other hand, if σ (t, u) ∈ Q1 for all t ∈ [0, T ], then 1 G σ (T , u) G(u) − 4

T 0

1 G(u) − 4

T 0

φ 2 (dist(σ (s, u), B)) ds 1 + φ(dist(σ (s, u), B)) φ 2 (dist(u, B) − s) ds. 1 + φ(dist(u, B) − s)

˜ B) = α > T , then If u ∈ A˜ ∩ W1 , then dist(u, B) dist(A, 1 G σ (T , u) a − 4

dist(u,B)

dist(u,B)−T

φ 2 (τ ) 1 dτ a − 1 + φ(τ ) 4

T 0

φ 2 (τ ) dτ. 1 + φ(τ )

˜ then G(u) b0 , then G(σ (T , u)) < b0 − ε. It implies that G(σ (T , u)) < b0 − ε. If u ∈ W1 \A, Let A1 = σ (T , A), a10 = supA1 ∩Sβ G. If A1 ∩ Sβ = ∅, i.e., W1 = ∅, we set a10 = −∞. Then


3353

a10 < b0 . Denote ΦA1 ,β := {Γ ∈ Φ: Γ ([0, 1], A1 ) ⊂ Gβ }. We claim ΦA1 ,β = ∅. Indeed, take Γ ∈ ΦA,β . Set Γ1 (s, u) =

σ (2sT , u)−1 , Γ (2s − 1, σ (T , u)−1 ),

s ∈ [0, 1/2], s ∈ (1/2, 1].

Then Γ1 ∈ Φ and G(σ (T − 2sT , A)), G Γ1 s, σ (T , A) = G(Γ (2s − 1, A)),

s ∈ [0, 1/2], s ∈ (1/2, 1].

Thus supΓ1 ([0,1],A1 ) G supΓ ([0,1],A) G β. It follows that Γ1 ∈ ΦA1 ,β . Define a1 := infΓ ∈ΦA1 ,β supΓ ([0,1],A1 )∩Sβ G. To show that a1 is well defined, we have to show that A1 is ΦA1 ,β -linked to B. To this aim, we first prove σ ([0, T ], A) ∩ B = ∅. Recall A ⊂ Gβ , then σ ([0, T ], A) ⊂ Gβ . If it were not true, then there would be σ (t0 , u0 ) ∈ B ∩ Gβ ⊂ Sβ for some ˜ by (2.5), we see that dist(σ (t0 , u0 ), B) t0 ∈ [0, T ] and u0 ∈ A (hence, u0 ∈ Sβ ). If u0 ∈ A, α − t0 α − T > 0, a contradiction. Then we must have u0 ∈ A ∩ Sβ \A˜ and then G(u0 ) b0 , ¯ 0 . But then hence G(σ (t0 , u0 )) < b0 . Unless η(σ (s, u0 )) = 0 for all s ∈ [0, t0 ], i.e., σ (s, u0 ) ∈ Q G(σ (t0 , u0 )) b0 − 2ε. Hence G(σ (t0 , u0 )) < b0 , a contradiction. Let Γ be any map in ΦA1 ,β . Set σ (2sT , u), s ∈ [0, 1/2], Γ1 (s, u) = Γ (2s − 1, σ (T , u)), s ∈ (1/2, 1]. Then Γ1 ([0, 1], A) ⊂ Gβ . Then Γ1 ∈ ΦA,β . Since A is ΦA,β -linked to B, there is an s1 ∈ [0, 1] such that Γ1 (s1 , A) ∩ B = ∅. Since σ (2sT , A) ∩ B = ∅ for all s ∈ [0, 1/2], we must have s1 > 1/2. Hence, Γ (2s1 − 1, σ (T , A)) ∩ B = ∅. Then, A1 is ΦA1 ,β -linked to B. Thus, a1 is well defined. Next, we show that a10 < b0 a1 a. Let Γ ∈ ΦA,β . Set Γ1 (s, u) =

σ (2sT , u)−1 , Γ (2s − 1, σ (T , u)−1 ),

s ∈ [0, 1/2], s ∈ (1/2, 1].

Then Γ1 ∈ Φ and G(σ (T − 2sT , u), G Γ1 s, σ (T , u) = G(Γ (2s − 1, u)),

s ∈ [0, 1/2], s ∈ (1/2, 1].

Thus Γ1 ∈ ΦA1 ,β and a1 supΓ1 ([0,1],A1 )∩Sβ G supΓ ([0,1],A)∩Sβ G. It follows that a1 a. Obviously, we have known b0 a1 . Finally, by the previous arguments, we have a10 < b0 . We now apply Theorem 2.1 to A1 and B, there exists a sequence {uk } such that {uk } ⊂ Sβ , G(uk ) → a1 and G (uk ) → 0. If a1 = b0 , then we may require that dist(uk , B) → 0. However, this contradicts the assumptions at the beginning of the proof. 2 Let {Ak , Bk } be a sequence of pairs of subsets of E, where each Ak is bounded. Assume that there is a βk > 0 such that (A3 ) ΦAk ,βk := {Γ ∈ Φ: Γ ([0, 1], Ak ) ⊂ Gβk } = ∅, ∀k ∈ N. Moreover, Ak is ΦAk ,βk -linked to Bk and Bk ∩ Gβk ⊂ Sβk .

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Let ak,0 := supAk ∩Sβ G and bk,0 := infBk G. Define ak = infΓ ∈ΦAk ,βk supΓ ([0,1],Ak )∩Sβ G. Set k k A˜ k = {u ∈ Ak ∩ Sβ : G(u) > bk,0 } and αk = dist(A˜ k , Bk ). Theorem 2.5. Assume (A2 )–(A3 ) and bk,0 ∈ R. Suppose that αk → ∞ and there exists a θ 0 ak,0 −bk,0 such that (α θ+1 → 0 as k → ∞. Then there exists a sequence {uk } such that uk ∈ Sβk ⊂ k) E\(−P ∪ P) and that bk,0 −

G (uk ) → 0, (1 + dist(uk , Bk ))θ

1 1 G(uk ) ak + , k k

k → ∞.

Proof. Let h(t) := 4(θ + 1)(ak,0 − bk,0 )(1 + t) /(αk ) θ

Then we may check that

αk 0

φ 2 (t) 1+φ(t)

θ+1

,

φ(t) :=

h(t) +

h2 (t) + 4h(t) . 2

dt > 4(ak,0 − bk,0 ). We may assume that bk,0 < ak,0 for

each k. We now apply Theorem 2.4 for each k there is a uk ∈ E such that uk ∈ Sβk , bk,0 − G(uk ) ak + k1 and G (uk ) < φ(dist(uk , Bk )), which implies the conclusion. 2

1 k

An immediate consequence of Theorem 2.5 is the following corollary. Corollary 2.1. Assume that αk → ∞ and there exist a ρ1 > −∞ and ρ2 < ∞ such that bk,0 ρ1 , ak,0 ρ2 . Then there exists a sequence {uk } such that uk ∈ Sβk and that G(uk ) → c ∈ [ρ1 , ∞), G (uk ) → 0, k → ∞. Let E, M, Y, A and B be as in Proposition 2.1. Theorem 2.6. Assume (A1 )–(A2 ). Assume there is a constant c0 such that G(u) c0 for all u ∈ Y and G(u) c0 for all u ∈ {u ∈ M: u∗ = ρ}. Further, G(sy0 + v) Ck for s 0, v ∈ Y and sy0 + v = k, here y0 ∈ M with y0 ∗ = 1 and ρ > 0. If there is a θ 0 such that k limk→∞ kCθ+1 = 0, then there exists a sequence {um } such that um ∈⊂ E\(−P ∪ P) and that G(um ) → c ∈ [c0 , ∞],

G (um ) → 0, (1 + um )θ

m → ∞.

Proof. Let Ak := {sw0 + v = u: s 0, v ∈ Y, u = k} ∪ {u ∈ Y : u k} and B := Bk := {u ∈ M: u∗ = ρ}. Then A˜ k ⊂ {sw0 + v = u: s 0, v ∈ Y, u = k}. Note that sw0 + v − u∗ CY sw0 + v − ρ = CY k − ρ, where CY > 0 is a constant depending on Y only. Then we observe that αk = dist(A˜ k , Bk ) (CY k − ρ)/C∗ → ∞. On the other hand, we may assume a −b that ak,0 > bk,0 (otherwise, we are done!), then k,0θ+1k,0 → 0. Then by Theorem 2.5, we get the conclusion.

2

αk

In particular, if Ck is uniformly bounded (this is indeed true for superlinear cases), then we have


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Theorem 2.7. In Theorem 2.6, if Ck < c∗ for all k > 0, then there exists a sequence {um } such that um ∈ E\(−P ∪ P) and that G(um ) → c ∈ [c0 , c∗ ], G (um ) → 0, m → ∞. Next we consider the existence of sign-changing Cerami’s sequence. Theorem 2.8. Assume (A1 )–(A2 ). Assume that A is bounded and A is ΦA,β -linked to B and ∞ a0 b0 . Let φ(t) be a positive nonincreasing function on (0, ∞) such that 1 φ(t) dt = ∞. Then there exists a sequence {uk } such that {uk } ⊂ Sβ ,

G(uk ) → a,

G (uk ) → 0. φ(uk )

Proof. We note that if the theorem were false, there would be an ε > 0 and a φ(t) satisfying G (u) φ(u) when u ∈ Q := {u ∈ Sβ : |G(u) − a| 3ε} = ∅. Assume b0 < a and 3ε < a − b0 . Let Q0 := {u ∈ Sβ : |G(u) − a| 2ε} and Q1 := {u ∈ Sβ : |G(u) − a| (t,u) dist(u,Q2 ) ε}, Q2 := E\Q0 , η(u) = dist(u,Q . Consider d γdt = −η(γ )Y (γ ), γ (0, u) = u, 1 )+dist(u,Q2 )

) where Y (u) = V0 (u)/V0 (u), V0 is as that in the proof of Theorem 2.1. Then d G(γ dt 1 − 4 η(γ (t, u))φ(u + t) and γ ([0, ∞), ±Dμ0 ) ⊂ ±Dμ0 . By the definition of a, there is a Γ ∈ ΦA,β such that G(Γ ([0, 1], A) ∩ Sβ ) < a + ε. Set ξ := sup{Γ (s, u): s ∈ [0, 1], u ∈ A}.

ξ +T Choose a T > 0 such that 8ε < ξ φ(t) dt. Let W := {v: v = Γ (s, u) ∈ Sβ for some s ∈ [0, 1], u ∈ A}. Next, we only consider those v ∈ W satisfying γ (T , v) ∈ Sβ . If there is a t0 T such that γ (t0 , v) ∈ / Q1 , then we must have G(γ (T , v)) < a − ε. If γ (t, v) ∈ Q1 for

T all t ∈ [0, T ], then we have G(γ (T , v)) a + ε − 14 0 φ(ξ + t) dt < a − ε. Summing up, we see that G(γ [T , Γ ([0, 1], A) ∩ Sβ ] ∩ Sβ ) < a − ε. Now we define Γ ∗ (t, u) = γ (2tT , u) if t ∈ [0, 1/2] and Γ ∗ (t, u) = γ (T , Γ (2t − 1, u)) if t ∈ [1/2, 1]. Then Γ ∗ ∈ Φ. Obviously, G(Γ ∗ ([0, 1], A) ∩ Sβ ) a − ε and G(Γ ∗ ([0, 1], A)) β. This contradicts the definition of a. Next we consider the case of b0 = a. For each u ∈ A ∩ Sβ , we note that G(γ (t, u)) a0 −

1 t 4 0 η(γ (s, u))φ(γ (s, u)) ds. We show that γ (t, A) ∩ B ∩ Sβ = ∅ for all t 0. Otherwise, there is a γ (t, u) ∈ B ∩ Sβ with u ∈ A ∩ Sβ . Then G(γ (s, u)) b0 for all s ∈ [0, t], hence η(γ (s, u)) = 0 which implies that γ (s, u) ∈ Q2 and then G(γ (s, u)) a − ε = b0 − ε and thus γ (s, u) ∈ / B for all s ∈ [0, t]. Similar to the first case, γ (T , Γ ([0, 1], A) ∩ Sβ ) ∩ B ∩ Sβ = ∅. Let Γ ∗ be defined as in the first case, we know that Γ ∗ ([0, 1], A) ∩ B = ∅, this contradicts the fact that A is ΦA,β -linked to B. 2

We have the following immediate consequence on the existence of the sign-changing Cerami’s sequence. Theorem 2.9 (Sign-changing Cerami’s sequence). Assume (A1 )–(A2 ). Suppose that A is bounded and A is ΦA,β -linked to B, a0 b0 . Then there exists a sequence {uk } such that {uk } ⊂ Sβ ,

G(uk ) → a,

1 + uk G (uk ) → 0.

Let a0 , b0 , a, A˜ and α be defined as before. We shall establish the following alternative theorems.

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Theorem 2.10. Assume (A1 )–(A2 ). Suppose that A is bounded and A is ΦA,β -linked to B. Assume a, b0 ∈ R and let φ be a positive nondecreasing function on [0, +∞) such that a0 − b0
0 there is a uε ∈ Sβ such that b0 − ε G(uε ) a + ε,

˜ T + ε, G (uε ) < φ dist(uε , B) . dist(uε , A)

Proof. By Theorem 2.1 and Lemma 2.2, we may assume that a = a0 and b0 < a0 . Hence, A˜ = ∅ and α > 0. If option (b) were false, there would be an ε > 0 such that φ(dist(u, B)) G (u) for all u in the set ˜ T + 3ε . Q := u ∈ Sβ : b0 − 3ε G(u) a + 3ε, dist(u, A)

(2.6)

We remark that Q is nonempty as long as ε small enough. We assume that a − b0 + ε
ρ, where u0 ∈ M with u0 ∗ = 1. If lim supk→∞ kmθ+1 0 for some θ 0, then either (1) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [c0 , ∞) with G (um ) → 0 or (2) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [c0 , ∞] and G(um ) = 0, m→∞ um θ+1 lim

G (um ) = 0, m→∞ um θ lim

um → ∞.

Proof. Let Ak := {w = su0 + u: s 0, u ∈ Y, w = k} ∪ {u ∈ Y : u k} and Bk := B := {u ∈ M: u∗ = ρ}. Then Ak links Bk with respect to Φ and A˜ k ⊂ {w = su0 + u: s 0, u ∈ Y,

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w = k}. Then similar to the proof of Theorem 2.6, αk := dist(A˜ k , B) → ∞. Apply Corollary 2.2. 2 Theorem 2.12. Assume (A2 ) and lim supR→∞ (−P ∪ P), either

supu=R G R θ+1

0 for some θ 0, then for any w0 ∈ /

(1) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [G(w0 ), ∞) with G (um ) → 0 or (2) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [G(w0 ), ∞] and G(um ) = 0, m→∞ um θ+1 lim

G (um ) = 0, m→∞ um θ lim

um → ∞.

Proof. Let Ak := {u: u = k} and Bk := B := {w0 }. Then Ak links Bk with respect to Φ (cf. [17]). Then similarly, using Corollary 2.2 we get the conclusion. 2 Remark 2.3. Assume (A2 ) and lim supR→∞ G (u)

G(u) limu→∞ u θ+1

supu=R G R θ+1

0 for some θ 0 and either

0 or limu→∞ uθ = 0. Then for any w0 ∈ = / (−P ∪ P), there is a sequence {um } ⊂ E\(−P ∪ P) such that G(um ) → c ∈ [G(w0 ), ∞) with G (um ) → 0. If moreover, for any such a (PS) sequence {um }, it converges to a sign-changing critical point, then G has infinitely many sign-changing critical points, where G is not necessarily even. Remark 2.4. The above theorems and corollaries can be traced back to the results due to Martin Schechter [14,17], where there is no any information on the sign-changing property of the critical point or of the (PS) sequence. Also some other earlier results on sign-changing solutions have been obtained in [1,6,12,21,22], etc. 3. Applications In this section, we just give two theorems to illustrate the efficacy of the abstract theorems in Section 2. We will observe that it is very convenient to get sign-changing solutions under very weak hypotheses. We believe the abstract theorems may have more deep applications. Consider the sign-changing solutions to the following Dirichlet boundary value problem

−u = f (x, u), u = 0,

in Ω, on ∂Ω,

(3.1)

where Ω ⊂ RN is a bounded domain with the smooth boundary ∂Ω and finite measure meas Ω := |Ω|. Let E := H01 (Ω) be the usual Sobolev space endowed with the inner prod uct u, v := Ω (u · v) dx for u, v ∈ E and the norm u := u, u1/2 . Let 0 < λ1 < · · · < λm < · · · denote the distinct Dirichlet eigenvalues of − on Ω with zero boundary value. Then each λm has finite multiplicity. The principal eigenvalue λ1 is simple with a positive eigenfunction ϕ1 , and the eigenfunctions ϕm corresponding to λm (m 2) are sign-changing. Let Nm denote the eigenspace of λm . Then dim Nm < ∞. We fix m and let Em := N1 ⊕ · · · ⊕ Nm . We first consider the following assumptions:


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(B1 ) f is a Carathèodory function and f (x, t)t 0 for (x, t) ∈ Ω × R; limt→0 f (x,t) = 0 unit formly for x ∈ Ω.

t (B2 ) C(1 + |t|2 ) 2F (x, t) λm t 2 − κ0 for all x ∈ Ω, t ∈ R, where F (x, t) = 0 f (x, s) ds, C, κ0 > 0 is a constant. (x,t)t > 0 a.e. on Ω. (B3 ) lim inf|t|→∞ 2F (x,t)−f |t| The above assumptions include the following cases:

f (x, t)/t → a f (x, t)/t → b

a.e. x ∈ Ω as t → −∞, a.e. x ∈ Ω as t → ∞.

The existence of solutions of (3.1) is deeply related to the equation −u = bu+ − au− , where u± = max{±u, 0}. Conventionally, the set Σ := (a, b) ∈ R2 : −u = bu+ − au− has nontrivial solutions is called the Fuˇcík spectrum of − (see S. Fuˇcík [2] and M. Schechter [13]). It plays a key role in most results on this aspect. However, so far no complete description of Σ has been found. Only partial answers were given. It was shown in M. Schechter [13] that in the square (λl−1 , λl+1 )2 there are decreasing curves Cl1 , Cl2 (which may or may not coincide) passing through the point (λl , λl ) such that all points above or below both curves in the square (the so-called type (I) region) are not in Σ, while points on the curves are in Σ . Usually, the status of points between the curves (referred to as type (II) region, if the curves do not coincide) is unknown. However, it was shown in T. Gallouët, O. Kavian [3] that when λ is a simple eigenvalue, then points of type (II) region are not in Σ. On the other hand, C.A. Margulies, W. Margulies [7] have shown that there are boundary value problems for which many curves in Σ emanate from a point (λl , λl ) when λl is a multiple eigenvalue. Certainly, these curves are contained in the region (II). We refer the readers to [8–10] for further developments. Recently, related applications also can be seen in [5,4] (and the references cited therein). Their results heavily rely on Σ . Our results in this current paper are independent of Σ. By our assumption, F (x, u) λ41 |u|2 + CF |u|p , ∀x ∈ Ω, u ∈ R, where p ∈ (2, 2∗ ). On the other hand, we have a constant Cp > 0 such that u2p Cp u2 for u ∈ E. Set S0 := {u ∈ ⊥ : u = ρ} for ρ := ( 1 )1/(p−2) . Ek−1 p 8CF Cp Theorem 3.1. Assume (B1 )–(B3 ). If κ0 ( 8Cp1CF )2/(p−2) 4Cp1|Ω| , then Eq. (3.1) has a signchanging solution. Proof. For u ∈ Em−1 we see that G(u)

κ0 |Ω| 2 .

For any u ∈ S0 , we have that

1 p G(u) u2p − CF up 4Cp

1 8Cp CF

2/(p−2)

1 . 8Cp

We choose u0 ∈ Nm with u0 ∗ := u0 p = 1 and set mk = sup{G(tu0 + u): t 0, u ∈ Em−1 , tu0 + u = k}. Then lim supk→∞ mkk 0. Let P := {u ∈ E: u(x) 0 for a.e. x ∈ Ω}. Then P is the positive cone of E. We define Dμ0 := {u ∈ E: dist(u, P) < μ0 }. Then by (B1 ), we may check that conditions (A1 )–(A2 ) are satisfied (see for example [22]). By Theorem 2.11, either

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(1) there is a sequence {uk } ∈ E\(−P ∪ P) such that G(uk ) → c ∈ [ κ0 |Ω| 2 , ∞) with G (uk ) → 0 or (2) there is a sequence {uk } ∈ E\(−P ∪ P) such that G(uk ) → c ∈ [ κ0 |Ω| 2 , ∞] and

lim

k→∞

G(uk ) = 0, uk

lim G (uk ) = 0,

k→∞

uk → ∞.

Assume the second alternative is true. Let u¯ k = uk /uk , then u¯ k → u¯ strongly in L2 . By (B3 ) we know that u¯ ≡ 0. But by (B2 ) we may observe that u¯ ≡ 0. This eliminate the case of (2). Hence, we just have case (1). By standard arguments, we know that {uk } has a convergent subsequence whose limit is still sign-changing (by (B1 )) and its energy is κ0 |Ω| 2 . 2 We define the following numbers (cf. M. Schechter [18,17]): αk = max u2 : u ∈ Ek , u 0, u2 = 1 λk , 2 γk (a) = sup u2 − a u− 2 : u ∈ Ek , u+ 2 = 1 λ1 , 2 Γk (a) = inf u2 − a u− 2 : u ∈ Ek⊥ , u+ 2 = 1 . Then α1 = λ1 = γ1 (a) for any a ∈ R. Moreover, (1) (2) (3) (4) (5)

γk (a), Γk (a) are continuous and decreasing, if k 2, αk < λk , if k 2 and a > αk , then γk (a) < ∞ and the supremum is attained, for each k, the infimum in the definition of Γk (a) is attained, if a λk+1 , then Γk (a) λk+1 .

Theorem 3.2. Assume (B1 ). Assume there are numbers m > αk−1 , m1 > αk , p > 2 and β < λk such that 2 2 m t − + γk−1 (m) t + − W1 (x) 2F (x, t) βt 2 + C0 |t|p + W2 (x) and 2 2 m1 t − + γk (m1 ) t + 2F (x, t) + W3 (x) for all x ∈ Ω, t ∈ R. Further, f (x, t) = β+ (x), t→+∞ t

f (x, t) = β− (x), t→−∞ t

lim

lim

uniformly for x ∈ Ω and β± ∈ L∞ (Ω). Assume moreover, B1 + B2
⊥ with w = ρ. By Theorem 2.6, there is a sign-changing sequence {u } ∈ B1 /2 for all u ∈ Ek−1 ∗ k E\(−P ∪ P) and a c ∈ R such that G(uk ) → c ∈ [C1 , B3 /2], G (uk ) → 0. Since the equation −u = β+ u+ − β− u− has the only solution u ≡ 0, it is easy to see that {uk } is bounded, hence it has a convergent subsequence whose limit is still sign-changing (by (B1 )). 2 Acknowledgment The authors thank so much the anonymous referee for his/her generous and kind suggestions. References [1] T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001) 117–152. ˇ [2] S. Fuˇcík, Boundary value problems with jumping nonlinearities, Casopis Pˇest. Mat. 101 (1976) 69–87. [3] T. Gallouët, O. Kavian, Résultats d’existence et de non existence pour certains problèms demi linéaires à l’infini, Ann. Fac. Sci. Toulouse Math. 3 (1981) 201–246. [4] C. Li, S. Li, Z. Liu, Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems, Calc. Var. Partial Differential Equations 32 (2008) 237–251. [5] C. Li, S. Li, Z. Liu, J. Pan, On the Fuck spectrum, J. Differential Equations 244 (2008) 2498–2528. [6] S. Li, Z.Q. Wang, Ljusternik–Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc. 354 (2002) 3207–3227. [7] C.A. Margulies, W. Margulies, An example of the Fuˇcík spectrum, Nonlinear Anal. TMA 29 (1997) 1373–1378. [8] K. Perera, On the Fuˇcík spectrum of the p-Laplacian, NoDEA Nonlinear Differential Equations Appl. 11 (2) (2004) 259–270. [9] K. Perera, M. Schechter, Computation of critical groups in Fuˇcík resonance problems, J. Math. Anal. Appl. 279 (2003) 317–325. [10] K. Perera, M. Schechter, Double resonance problems with respect to the Fuˇcík spectrum, Indiana Univ. Math. J. 52 (2003) 1–17. [11] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., 1986. [12] M. Ramos, H. Tavares, W. Zou, A Bahri–Lions theorem revisted, Adv. Math. 222 (2009) 2173–2195. [13] M. Schechter, The Fuˇcík spectrum, Indiana Univ. Math. J. 43 (1994) 1139–1157. [14] M. Schechter, The saddle point alternative, Amer. J. Math. 117 (1995) 1603–1626. [15] M. Schechter, Critical point theory with weak-to-weak linking, Comm. Pure Appl. Math. 51 (1998) 1247–1254. [16] M. Schechter, Infinite-dimensional linking, Duke Math. J. 94 (1998) 573–595. [17] M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999. [18] M. Schechter, Sandwich pairs in critical point theory, Trans. Amer. Math. Soc. 360 (2008) 2811–2823. [19] M. Schechter, Minimax Systems and Critical Point Theory, Birkhäuser, 2009. [20] M. Schecher, W. Zou, Sign-changing critical points of linking type theorems, Trans. Amer. Math. Soc. 358 (2006) 5293–5318. [21] M. Schechter, W. Zou, On the Brézis–Nirenberg problem, Arch. Ration. Mech. Anal., in press. [22] W. Zou, Sign-changing Critical Points Theory, Springer, New York, 2008.


On cocycle twisting of compact quantum groups Kenny De Commer 1 Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium Received 16 September 2009; accepted 4 November 2009 Available online 12 November 2009 Communicated by D. Voiculescu

Abstract We provide a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also gives examples of cocycle twists where the underlying C∗ -algebra of the quantum group changes. © 2009 Elsevier Inc. All rights reserved. Keywords: Compact quantum group; von Neumann algebraic quantum group; Cocycle twisting; Comonoidal W∗ -Morita equivalence

0. Introduction In the seventies, Kac and Vainerman [11], and independently Enock and Schwartz [6], introduced the notion of (what was called by the latter) a Kac algebra, based on the fundamental work of Kac concerning ring groups in the sixties [10]. Such Kac algebras, which are von Neumann algebras M with a coproduct : M → M ⊗ M satisfying certain conditions, can naturally be made into a category, containing as a full sub-category the category of all locally compact groups, but allowing a duality functor which extends the Pontryagin duality functor on the sub-category of all abelian locally compact groups. Moreover, Kac algebras with a commutative underlying von Neumann algebra arise precisely from locally compact groups, by passing to the L ∞ -space of the latter with respect to the left (or right) Haar measure, and with dual to the multiplication in the group. E-mail address: [email protected]. 1 Research Assistant of the Research Foundation, Flanders (FWO, Vlaanderen).


K. De Commer / Journal of Functional Analysis 258 (2010) 3362–3375

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However, these Kac algebras do not cover all interesting examples of what could be called ‘locally compact quantum groups’. In the eighties, Woronowicz introduced ‘compact matrix quantum groups’ [20], which are to be seen as quantum versions of compact Lie groups. He also constructed in that paper certain compact matrix quantum groups SU q (2), which are deformations of the classical SU(2)-group by some parameter q ∈ R with 0 < |q| < 1. These quantum groups do not fit into the Kac algebra framework. The reason for this is that the antipode of these quantum groups is no longer a ∗ -preserving anti-automorphism, but some unbounded operator on the associated C∗ -algebra of the quantum group. A satisfactory theory, covering both the compact quantum groups, the Kac algebras and some isolated examples, was obtained in 2000, when Kustermans and Vaes introduced their C∗ -algebraic quantum groups [12]. In a follow-up paper, they also introduced von Neumann algebraic quantum groups [13], and proved that the C∗ -algebra approach and the von Neumann algebra approach were just different ways to look at the same structure (in that one can pass from the von Neumann algebra setup to a (reduced or universal) C∗ -algebraic setup and back). We also remark that in [18], an slightly alternative approach to von Neumann algebraic quantum groups was presented. In this paper, we will be mainly using the von Neumann algebraic approach, which simply asks for the existence of a coproduct and invariant weights on a von Neumann algebra (see Definition 1.2). An interesting and important part of the theory consists in finding construction methods for von Neumann algebraic quantum groups. For example, in [1] the double product construction was worked out, while in [16] the bicrossed product construction was treated. In [4], we developed another construction method, namely the generalized twisting of a von Neumann algebraic quantum group (by a Galois object for its dual). This covers in particular the twisting by unitary 2-cocycles, special situations of which had been considered by Enock and Vainerman in [7], and by Fima and Vainerman in [8]. When we have two von Neumann algebraic quantum groups, one of which is obtained from the other by the above generalized twisting construction, we call them comonoidally W∗ -Morita equivalent. The reason for this name is simple: the underlying von Neumann algebras of two such quantum groups are Morita equivalent (in the sense of Rieffel [14]), with the equivalence ‘respecting the coproduct structure’. One can also show then that the representation categories of their associated universal C∗ -algebraic quantum groups are unitarily comonoidally equivalent (so that they have the ‘same’ tensor category of ∗ -representations). The special case of cocycle twisting corresponds to those comonoidal Morita equivalences whose underlying Morita equivalence is (isomorphic to) the identity. It was shown in [5] that comonoidal W∗ -Morita equivalence provides a genuine equivalence relation between von Neumann algebraic quantum groups. It is then a natural question to find out which properties are preserved by this equivalence relation. In [2], it was shown for example that the discreteness of a quantum group is preserved, while its amenability is not. It also follows from the results of that paper that ‘being discrete and Kac’ is not preserved. In the general setting of von Neumann algebraic quantum groups, we showed in [5] that the scaling constant is an invariant for comonoidal W∗ -Morita equivalence. In this article, we will show in a very concrete way that the notion of ‘being compact’ is not an invariant. This implies in particular that the representation category of a locally compact quantum group, as a monoidal W∗ -category, does not necessarily remember the topology of the quantum group. In [5], we have also shown, by more general methods, that ‘being compact and Kac’ is an invariant. This article is divided into two sections. In the first section, we recall some notions concerning compact quantum groups, von Neumann algebraic quantum groups and the twisting by unitary

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2-cocycles. In the second section, we recall the definition of Woronowicz’s SU q (2) quantum groups. We then consider an infinite tensor product of these compact quantum groups over varying q’s, with the condition that the q’s go to zero sufficiently fast. Taking a limit of appropriate coboundaries, we obtain a 2-cocycle Ω on this infinite product quantum group which is no longer a coboundary. By giving an explicit formula for a non-finite but semi-finite invariant weight, we conclude that the Ω-twisted quantum group is no longer compact. Remarks on notation: When A is a set, we denote by ι the identity map A → A. We will need the following tensor products: we denote by the algebraic tensor product of two vector spaces over C, by ⊗min the minimal tensor product of two C∗ -algebras, and by ⊗ the spatial tensor product between von Neumann algebras or Hilbert spaces. We will further use the following notations concerning weights on von Neumann algebras. If M is a von Neumann algebra, and ϕ : M + → [0, +∞] an nsf (= normal semi-finite faithful) weight, we denote by Nϕ ⊆ M the σ -weakly dense left ideal of square integrable elements: Nϕ = x ∈ M ϕ x ∗ x < ∞ . We denote by L 2 (M, ϕ) the Hilbert space completion of Nϕ with respect to the inner product x, y = ϕ y ∗ x , and by Λϕ the canonical embedding Nϕ → L 2 (M, ϕ). We remark that Λϕ is (σ -strong)-(norm) closed. We denote Mϕ+ for the space of elements x ∈ M + for which ϕ(x) < ∞, and Mϕ for the complex linear span of Mϕ+ . One can show then that Mϕ = Nϕ∗ · Nϕ , i.e. any element x of Mϕ can be written as ni=1 xi∗ yi with xi , yi ∈ Nϕ . As far as applicable, we use the same notation when considering, more generally, operator valued weights. We also remark here that if M1 and M2 are von Neumann algebras, and ϕ an nsf weight on M2 , we can make sense of (ι ⊗ ϕ) as an nsf operator valued weight from M1 ⊗ M2 to M1 in a natural way: if x ∈ (M1 ⊗ M2 )+ , we let (ι ⊗ ϕ)(x) be the element ω ∈ (M1 )+ ∗ → ϕ (ω ⊗ ι)(x) ∈ [0, +∞] in the extended positive cone of M1 . For more information concerning the theory of weights and operator valued weights, we refer to the first chapters of [15]. 1. Compact and von Neumann algebraic quantum groups As we mentioned in the Introduction, S.L. Woronowicz developed the notion of a compact matrix quantum group in [20] (there called compact matrix pseudogroup). Later, he also introduced the more general notion of a compact quantum group [22]. Definition 1.1. A compact quantum group consists of a couple (A, ), where A is a unital C∗ -algebra, and a unital ∗ -homomorphism A → A ⊗min A such that 1. the map is coassociative: ( ⊗ ι) = (ι ⊗ ),


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and 2. the linear subspaces (A)(1 ⊗ A) :=

(ai )(1 ⊗ bi ) ai , bi ∈ A

i

and (A)(A ⊗ 1) :=

(ai )(bi ⊗ 1) ai , bi ∈ A

i

are norm-dense in A ⊗min A. The compact quantum group is called a compact matrix quantum group if there exist n ∈ N0 and a unitary u = ni,j =1 uij ⊗ eij ∈ A ⊗ Mn (C), such that (uij ) = nk=1 uik ⊗ ukj and such that the uij generate A as a unital C∗ -algebra. Such a unitary is then called a fundamental unitary corepresentation. It is not difficult to show that any compact quantum group (A, ) with A commutative is of the form (C(G), ) for some compact group G, and dual to the group multiplication. Moreover, (A, ) will then be a compact matrix quantum group iff G is a compact Lie group. It is common practice to denote, by analogy, also a general compact quantum group (A, ) as (C(G), ), although this notation is now of course purely formal, since there is no underlying object G. In [22] (and [17] for the non-separable case), it is proven that to any compact quantum group (C(G), ) one can associate a unique state ϕ satisfying (ι ⊗ ϕ)(a) = ϕ(a)1 = (ϕ ⊗ ι)(a),

∀a ∈ C(G).

This state is called the invariant state of the compact quantum group. It is also proven there that with any compact quantum group, one can associate a Hopf ∗ -algebra (Pol(G), ) such that Pol(G) ⊆ C(G) is a norm-dense sub-∗ -algebra, the comultiplication being the restriction of the comultiplication on C(G). The invariant state is then faithful on Pol(G). Conversely, any Hopf ∗ -algebra (Pol(G), ) possessing an invariant state can be completed to a compact quantum group in essentially two ways. The first construction gives the associated reduced compact quantum group. Its underlying C∗ -algebra Cr (G) is given as the closure of the image of the GNS-representation of Pol(G) with respect to the invariant state. The second construction gives the associated universal compact quantum group. Its underlying C∗ -algebra Cu (G) is now the universal C∗ -envelope of Pol(G) (which can be shown to exist). For coamenable compact quantum groups, which are those compact quantum groups possessing both a bounded counit and a faithful invariant state, the reduced and universal construction for the underlying Hopf ∗ -algebra both coincide with the original compact quantum group, so that in this case, one only has to specify the Hopf ∗ -algebra to determine completely the associated C∗ -algebraic structure. With any compact quantum group C(G), one can also associate a von Neumann algebra, which we will denote as L ∞ (G). It is the σ -weak closure of the image of Pol(G) under the GNS-representation for ϕ. Then can be completed to a normal unital ∗ -homomorphism

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: L ∞ (G) → L ∞ (G) ⊗ L ∞ (G). This makes (L ∞ (G), ) into a von Neumann algebraic quantum group, whose definition we now present. Definition 1.2. (See [13].) A von Neumann algebraic quantum group is a couple (M, ), consisting of a von Neumann algebra M and a normal unital ∗ -homomorphism : M → M ⊗ M, such that 1. the map is coassociative: ( ⊗ ι) = (ι ⊗ ), and 2. there exist normal, semi-finite, faithful (nsf) weights ϕ and ψ on M such that, for any state ω ∈ M∗ , we have ϕ (ω ⊗ ι)(x) = ϕ(x),

∀x ∈ Mϕ+

ψ (ι ⊗ ω)(x) = ψ(x),

∀x ∈ Mψ+ .

and

The weights appearing in this definition turn out to be unique (up to multiplication with a positive non-zero scalar), and are called respectively the left and right invariant weights. The von Neumann algebraic quantum groups (L ∞ (G), ) arising from compact quantum groups can be characterized as those von Neumann algebraic quantum groups (M, ) which have a left invariant normal state. One can also show that from such a (L ∞ (G), ), a σ -weakly dense Hopf ∗ -subalgebra (Pol(G), ) can be reconstructed, providing a one-to-one correspondence between von Neumann algebraic quantum groups with an invariant normal state and Hopf ∗ -algebras with an invariant state. We now introduce the notion of a unitary 2-cocycle. Definition 1.3. Let (M, ) be a von Neumann algebraic quantum group. A unitary 2-cocycle for (M, ) is a unitary element Ω ∈ M ⊗ M satisfying the 2-cocycle equation (Ω ⊗ 1)( ⊗ ι)(Ω) = (1 ⊗ Ω)(ι ⊗ )(Ω). It is then easily seen that if (M, ) is a von Neumann algebraic quantum group, and Ω a unitary 2-cocycle for it, we can define a new coproduct Ω on M by putting Ω (x) := Ω(x)Ω ∗ ,

∀x ∈ M.

(1)

The coassociativity of Ω then follows precisely from the 2-cocycle equation. A non-trivial result from [4] states that (M, Ω ) in fact possesses invariant nsf weights, so that it is a von Neumann algebraic quantum group. The construction of these weights is rather complicated, and relies on some deep theorems from non-commutative integration theory. However, in the concrete example which we develop in the next section, we will be able to construct the weights on our cocycle twisted quantum group in a fairly straightforward way.


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2. Twisting does not preserve compactness The compact quantum groups which we will need will be constructed using the ‘twisted SU(2)’ groups from [20] (see also [21]). We recall their definition. Definition-Proposition 2.1. Let q be a real number with 0 < |q| 1. Define Pol(SU q (2)) as the unital ∗ -algebra, generated (as a unital ∗ -algebra) by two generators a and b satisfying the relations

a ∗ a + b∗ b = 1, ab = qba, aa ∗ + q 2 bb∗ = 1, a ∗ b = q −1 ba ∗ , bb∗ = b∗ b.

Then there exists a Hopf ∗ -algebra structure (Pol(SU q (2)), ) on Pol(SU q (2)) which satisfies

(a) = a ⊗ a − qb∗ ⊗ b, (b) = b ⊗ a + a ∗ ⊗ b.

Moreover, this Hopf ∗ -algebra possesses an invariant state ϕ, and has a unique completion to a compact matrix quantum group (C(SU q (2)), ). We will always use a and b to denote the generators of C(SU q (2)). Later on however, when we will be working with a sequence of SU qn (2)’s, we will index these generators by the corresponding index n of qn . We will follow the same convention for the comultiplication, the invariant state, and the other special elements in C(SU q (2)) which we will later introduce. The following proposition gives us more information about how the underlying C∗ -algebra and the associated invariant state of C(SU q (2)) look like. Proposition 2.2. (See [21].) Let q be a real number with 0 < |q| < 1. Let H be the Hilbert space l 2 (N) ⊗ l 2 (Z), whose canonical basis elements we denote as ξn,k (and with the convention ξn,k = 0 when n < 0). Then there exists a faithful unital ∗ -representation of C(SU q (2)) on H , determined by

π(a)ξn,k = 1 − q 2n ξn−1,k , π(b)ξn,k = q n ξn,k+1 .

The invariant state ϕ on C(SU q (2)) is given by 2n ϕ(x) = 1 − q 2 q π(x)ξn,0 , ξn,0 . n∈N

We now introduce special elements which will be of importance later on. Define, in the notation of the previous proposition, matrix units fmn on l 2 (N)⊗l 2 (Z), by putting fmn ξr,k = δr,n ξm,k , with δ the Kronecker delta, and where m, n take values in N. It is clear then that each fmn is in the unital C∗ -algebra generated by π(a). Hence emn := π −1 (fmn ) is an element of C(SU q (2)). We define elements p, p , w ∈ C(SU q (2)) by the following formulas:

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p := e00 ,

(2)

p := e11 ,

(3)

w := e01 + e12 + e20 +

∞

ekk .

(4)

k=3

Thus, with respect to the matrix units emn , the element w is the unitary ⎛

0 ⎜0 w=⎝ 1 0

1 0 0 0

⎞ 0 0 1 0⎟ ⎠. 0 0 0 I

We need to form infinite tensor products of the above quantum groups (see [19] for more detailed information). Let q := (qn )n∈N0 be a sequence of reals satisfying 0 < |qn | < 1. Then we can form an inductive sequence n Pol SU qk (2) , k k=1

of Hopf ∗ -algebras, by tensoring with 1 to the right at each step. We denote the inductive limit as (Pol(SU q (2)), q ). It is easy to see that this is again a Hopf ∗ -algebra with an invariant state ϕq , which is the pointwise limit of the functionals nk=1 ϕk . The associated compact quantum group will then again be coamenable, with underlying C∗ -algebra C(SU q (2)) the universal infinite tensor product of the C(SU qk (2)). This will equal the reduced C∗ -tensor product with respect to the states ϕk , by nuclearity of the C(SU qk (2)) (see the appendix of [21]). We will show now that those (C(SU q (2)), q ) for which q is square summable possess the property enunciated in the abstract, namely: they possess a unitary 2-cocycle which allows to twist them into a non-compact quantum group. Therefore, we now fix some q satisfying the property of square summability. We need to show some properties of L ∞ (SU q (2)). We begin with some well-known general lemmas. Lemma 2.3. Let M ⊆ B(H ) be a von Neumann algebra, and ξ ∈ H a separating vector for M. Let xn be a bounded sequence in M for which xn ξ converges to a vector η. Then xn converges in the σ -strong topology. Lemma 2.4. (See [9, Proposition 1.ı].) Let (Hn , ξn ) be a sequence of Hilbert spaces with distinguished unit vectors, and let (H , ξ ) be their tensor product. Let ηn ∈ Hn be non-zero infinite n ∞ vectors with

η

1. Then ( η ) ⊗ ( ξ n k k ) ∈ H converges to a non-zero vector k=1 k=n+1 ∞ |1 − ηk , ξk | < ∞. k=1 ηk if Recall the special element p (defined at (2)), for which we will now also use index notation. ∞ Lemma 2.5. The sequence ( nk=1 p k ) ⊗ 1 converges σ -strongly in L (SU q (2)) to an operator ∞ ∞ k=1 pk , while the sequence 1 ⊗ ( k=n+1 pk ) converges σ -strongly to 1.


3369

∞ Proof. Note first that2 the GNS-construction of L (SU q (2)) with respect to ϕq can be iden(L (SU (2)), ξ ), with ξ the separating and tified with ∞ qk k k k=1 cyclic vector associated to ϕk . Combining Lemma 2.3 and Lemma 2.4, we only have to see if (1 − ϕk (pk )) < ∞ to have the first statement of the lemma. Since ϕk (pk ) = 1 − qk2 , this follows by the assumption that qk is a square summable sequence. n ∞ Then we can of course also make sense of x ⊗( ∞ k=n+1 pk ), for any x ∈ k=1 L (SU qk (2)). Since

n 2 ∞ ∞ ∞ ξqk ⊗ pk ξqk ξqk = 1 − ϕqk (pk ) − k=1

k=n+1

k=1

k=n+1

=1−

∞ 1 − qk2 , k=n+1

2 the second statement follows from the convergence of ∞ k=1 (1−qk ) to a non-zero number, which in turn follows easily from the square summability of the qn . 2 ∞ Corollary 2.6. Denote En (x) = sn xsn , where sn = 1 ⊗ ( ∞ k=n+1 pk ) and x ∈ L (SU q (2)). Then En (x) converges to x in the σ -strong topology. Recall the special elements w defined by the formula (4). We again denote this element, when we regard it inside some C(SU qn (2)), as wn . Lemma 2.7. Let 0 < |q| < 1. Let ξ be the cyclic separating vector in the GNS-construction for L ∞ (SU q (2)) w.r.t. the invariant state ϕ. Then ∗ w − a ∗ ξ 3q 2 . Proof. In the matrix representation introduced just after Proposition 2.2, it is easy to calculate that (w − a) w ∗ − a ∗ ⎛ (1 − 1 − q 2 )2 ⎜ 0 ⎜ ⎜ 0 ⎜ =⎜ ⎜ 0 ⎜ ⎜ 0 ⎝ .. . Since 1 −

0 (1 − 1 − q 4 )2 0 0

0 0 2 − q6 − 1 − q6

0 .. .

0 .. .

0 0 − 1 − q6 2 − q8 − 1 − q8 .. .

0 0 0

− 1 − q8 2 − q 10 .. .

⎞ ... ...⎟ ⎟ ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎠ .. .

√ 1 − c c for 0 c 1, we have

ϕ (w − a) w ∗ − a ∗ 2 2 = 1 − q2 1 − 1 − q2 + q2 1 − 1 − q4 + q4 2 − q6 + q6 2 − q8 + · · ·

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2 1 − q 2 q 4 + q 10 + q 4 + q 6 + · · · 9q 4 . So (w ∗ − a ∗ )ξ 3q 2 .

2

∞ ∗ ∞ Theorem 2.8. The σ -strong∗ limit n=1 ((wn ⊗ wn )n (wn )) exists in L (SU q (2)) ⊗ ∞ ∞ L (SU q (2)), and determines a unitary 2-cocycle Ω for L (SU q (2)). Proof. Remark that (L ∞ (SU q (2)) ⊗ L ∞ (SU q (2)), ϕq ⊗ ϕq ) can be identified with

∞ ∞ ∞ ∞ L SU qk (2) ⊗ L SU qk (2) , (ϕk ⊗ ϕk ) . k=1

k=1

Then by Lemmas 2.3 and 2.4, it is enough to prove that ∞ 1 − (ϕn ⊗ ϕn ) (wn ⊗ wn )n w ∗ < ∞ n

n=1

to know that nk=1 ((wk ⊗ wk )k (wk∗ )) converges in the σ -strong∗ -topology. Denoting by ξk the GNS vector associated with ϕk , we estimate ∞ 1 − (ϕn ⊗ ϕn ) (wn ⊗ wn )n w ∗ n

n=1

∞ ∞ 1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ + (ϕn ⊗ ϕn ) (an ⊗ an ) n a ∗ − n w ∗ n

n=1

+

n

n

n=1

∞ (ϕn ⊗ ϕn ) (an ⊗ an − wn ⊗ wn )n w ∗ n

n=1

∞ ∞ 1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ + (ϕn ⊗ ϕn ) n (an − wn )(an − wn )∗ 1/2 n

n=1

+

n=1

∞

(ϕn ⊗ ϕn ) (an ⊗ an − wn ⊗ wn )(an ⊗ an − wn ⊗ wn )∗ 1/2

n=1

=

∞

1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ n

n=1

+

∞ ∞ (an − wn )∗ ξn + (an ⊗ an − wn ⊗ wn )∗ (ξn ⊗ ξn ) n=1

n=1

∞

∞ 1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ + 3 (an − wn )∗ ξn . n

n=1

n=1


3371

By the previous lemma and the square summability of the qn , we have ∞ (an − wn )∗ ξn < ∞. n=1

So we only have to compute if ∞ 1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ < ∞. n

n=1

Now an easy calculation shows that (ϕn ⊗ ϕn ) (an ⊗ an )n an∗ = Since 1 − Thus

1 (1+qn2 )2

1 . (1 + qn2 )2

2qn2 , we can again conclude convergence by square summability of the qn .

Ω=

∞ (wn ⊗ wn )n wn∗ n=1

is a well-defined unitary as a σ -strong∗ limit of unitaries. Since multiplication is jointly continuous on the group of unitaries with the σ -strong∗ topology, Ω will satisfy the 2-cocycle identity since each nk=1 ((wk ⊗ wk )k (wk∗ )) does. 2 Lemma 2.9. Let 0 < |q| < 1, and take w ∈ C(SU q (2)) as defined by the formula (4). Then the invariant state ϕ for C(SU q (2)) satisfies ϕ q −2 ϕ w ∗ · w . Proof. This follows from a straightforward computation, using the concrete form of ϕ as in Proposition 2.2. 2 Hence we can form on L ∞ (SU q (2)) the normal faithful weight ϕΩ := lim

n→∞

n k=1

qk−2

n

∞ ∗ ϕk wk · wk ⊗ ϕk ,

k=1

k=n+1

the limit being taken pointwise on elements of L ∞ (SU q (2))+ . This is a well-defined normal faithful weight, since it is an increasing sequence of normal, faithful, positive functionals. It is clear that ϕΩ is not finite. Proposition 2.10. The weight ϕΩ on L ∞ (SU q (2)) is semi-finite.

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Proof. We prove that the projections 1 ⊗ ( ∞ k=n+1 pk ), introduced in Lemma 2.5, are integrable with respect to ϕΩ . By Corollary 2.6, this will prove the proposition. 2 2 But wn∗ p n wn = pn , using also the notation of (3). Since ϕn (pn ) = qn (1 − qn ), the integrability p ) follows. 2 of all 1 ⊗ ( ∞ k=n+1 k We now end by proving that ϕΩ is an invariant nsf weight for the couple (L ∞ (SU q (2)), Ω ), where Ω was introduced in Theorem 2.8 and ϕΩ just before Lemma 2.10, and where Ω is the twisted coproduct defined by (1). Theorem 2.11. The nsf weight ϕΩ is a left and right invariant nsf weight for (L ∞ (SU q (2)), Ω ). Proof. We only prove left invariance, since the proof for right invariance follows by symmetry. Denote, for n ∈ N0 ,

Pn = x ⊗

∞

pk

n ∞ L SU qk (2) , x∈

k=n+1

k=1

still using the notation (2), and denote P = n∈N0 Pn . By the proof of Lemma 2.10, we know that P consists of integrable elements for ϕΩ . We first show that also Ω (P) ⊆ M(ι⊗ϕΩ ) , and that (ι ⊗ ϕΩ ) Ω (y) = ϕΩ (y)1 for y in P. Choose n ∈ N0 , and choose x = nk=1 xk ∈ nk=1 L ∞ (SU qk (2)) with all xk positive. Put y =x⊗( ∞ k=n+1 pk ). Then (ι ⊗ ϕΩ )(Ω (y)) is an element in the extended positive cone of L ∞ (SU q (2)). As such, it is the pointwise supremum of the elements zm =

n k=1

αk ⊗

n+m k=n+1

∞

βk ⊗

γk ,

k=n+m+1

regarded as semi-linear functionals on L ∞ (SU q (2))+ ∗ , where αk = qk−2 ι ⊗ ϕk wk∗ · wk (wk ⊗ wk )k wk∗ xk wk wk∗ ⊗ wk∗ , βk = qk−2 ι ⊗ ϕk wk∗ · wk (wk ⊗ wk )k wk∗ pk wk wk∗ ⊗ wk∗ and γk = (ι ⊗ ϕk ) (wk ⊗ wk )k wk∗ pk wk wk∗ ⊗ wk∗ , and where the infinite tensor product is to be seen as a σ -strong limit. We can simplify αk and βk to respectively qk−2 ϕk (wk∗ xk wk )1 and qk−2 ϕk (wk∗ pk wk )1, by invariance of ϕk , while γk satisfies γk qk−2 ϕk wk∗ pk wk 1


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by Lemma 2.9 and left invariance of ϕk . This implies that zm ϕΩ (y)1. Since zm is increasing, and the invariant state ϕq on L ∞ (SU q (2)) is a faithful normal state, we only have to prove that ϕq (zm ) → ϕΩ (y) to conclude that (ι ⊗ ϕΩ )(Ω (y)) = ϕΩ (y)1. But it is easily seen that ϕq (zm ) converges to ϕΩ (y) if we can show that ∞

ϕk (γk ) − −−−→ 1. m→∞

k=n+m+1

This is equivalent with proving that ∞

(ϕk ⊗ ϕk ) (wk ⊗ wk )k wk∗ pk wk wk∗ ⊗ wk∗ = 0.

1

Since the left hand side equals (ϕq ⊗ ϕq )(Ωq (s)Ω ∗ ), with s = ∞ k=1 pk , the above product is indeed non-zero, by faithfulness of ϕq . One then easily concludes that, since any y ∈ P is a linear combination of elements of the above form, we have Ω (y) ∈ Mι⊗ϕΩ for y ∈ P, with (ι ⊗ ϕΩ )Ω (y) = ϕΩ (y). Next we prove that P is a σ -strong-norm core for the GNS-map ΛϕΩ associated with the nsf ϕΩ for weight ϕΩ . For this, it is enough to prove that sn = 1 ⊗ ( ∞ k=n+1 pk ) is invariant under σt any t ∈ R, since then, using Corollary 2.6, we can conclude that, whenever y ∈ NϕΩ , we will have sn ysn → y in the σ -strong topology and ΛϕΩ (sn ysn ) = sn JϕΩ sn JϕΩ ΛϕΩ (y) → ΛϕΩ (y) in the norm topology (where JϕΩ is the modular conjugation on L 2 (M, ϕΩ )). ϕ By formula (A1.4) of [20] (or by a direct verification), we have that σt k (ak ) = |q|2it ak for ∗ any t ∈ R. Hence ak ak is in the centralizer of ϕk , and so the same is true of pk and pk . Since sn ∈ MϕΩ , we will have xsn , sn x ∈ MϕΩ for x ∈ NϕΩ ∩ Nϕ∗Ω , and then ϕΩ (sn x) = lim

n+m

m→∞

qk−2

k=1

= lim

m→∞

ϕk (pk ·)

qk−2

k=1

= lim

m→∞

⊗

ϕk (pk ·)

qk−2

k=1 ∞

k=n+m+1

= ϕΩ (xsn ).

n+m ∗ ∗ ϕk wk · wk ⊗ ϕk pk wk · wk k=n+1

k=n+m+1

n+m

n

(x)

k=1

∞

⊗

k=n+1

k=n+m+1

n+m

n+m ∗ ∗ ϕk wk · wk ⊗ ϕk wk pk · wk

k=1

∞

⊗

n

n

(x) n+m ∗ ∗ ϕk wk · wk ⊗ ϕk wk · wk pk

k=1

k=n+1

ϕk (·pk )

(x)

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K. De Commer / Journal of Functional Analysis 258 (2010) 3362–3375 ϕ

From this, the equality σt Ω (sn ) = sn for any t ∈ R follows (for example by Theorem VIII.2.6 of [15]). We can now conclude the proof. By left invariance of ϕΩ on P, we can introduce an isometry ∗ on L 2 (SU (2), ϕ ) ⊗ L 2 (SU (2), ϕ ) by putting WΩ q Ω q Ω ∗ WΩ ΛϕΩ (x) ⊗ ΛϕΩ (y) = (ΛϕΩ ⊗ ΛϕΩ ) Ω (y)(x ⊗ 1) for x ∈ NϕΩ and y ∈ P. Then by the core-property of P, we conclude that for x ∈ NϕΩ and any y ∈ NϕΩ , we have Ω (y)(x ⊗ 1) square integrable for ϕΩ ⊗ ϕΩ , with ∗ WΩ ΛϕΩ (x) ⊗ ΛϕΩ (y) = (ΛϕΩ ⊗ ΛϕΩ ) Ω (y)(x ⊗ 1) . Hence ϕΩ (ω ⊗ ι)Ω (y) = ϕΩ (y) for y ∈ Mϕ+Ω and ω a state of the form ·ΛϕΩ (x), ΛϕΩ (x) with x ∈ NϕΩ . Hence the elements (ι⊗ϕΩ )(Ω (y)) and ϕΩ (y)1 in the extended cone of M are equal on a normdense subset of M∗+ . Since the latter element is bounded, the same is true of the former by lower-semi-continuity, and then their equality everywhere follows. 2 Since the C∗ -algebra underlying a non-compact von Neumann algebraic quantum group is non-unital, we obtain the following corollary. Corollary 2.12. There exist a von Neumann algebraic quantum group (M, ) and a unitary 2-cocycle Ω ∈ M ⊗ M, such that the reduced (resp. universal) C∗ -algebra associated to (M, ) is not isomorphic to the reduced (resp. universal) C∗ -algebra associated to (M, Ω ), the Ω-twisted von Neumann algebraic quantum group. Remark 1. Note that the compact quantum group L ∞ (SU q (2)) is not a compact matrix quantum group. It would therefore be interesting to see if one can also twist compact matrix quantum groups into non-compact locally compact quantum groups. By the results of [5], this is closely related to the question whether a compact matrix quantum group can act ergodically on an infinite-dimensional type I -factor. Added in proof: in the meantime, the author succeeded in proving that this phenomenon is indeed also possible for compact matrix quantum groups, in fact even for the compact quantum group SU q (2). He intends to treat this example in detail in a future paper. Remark 2. It follows from the results of [3] that if (C(G), ) is a compact quantum group, and Ω a unitary 2-cocycle inside Pol(G) Pol(G), then the cocycle twisted von Neumann algebraic quantum group is again compact, with Pol(G) as the ∗ -algebra underlying the associated Hopf ∗ -algebra. Hence also the associated C∗ -algebras remain unaltered. Acknowledgments I would like to thank my thesis advisor Alfons Van Daele for his unwavering support of my work, and Stefaan Vaes for valuable suggestions concerning the presentation of the results in this article.


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References [1] S. Baaj, S. Vaes, Double crossed products of locally compact quantum groups, J. Inst. Math. Jussieu 4 (2005) 135–173. [2] J. Bichon, A. De Rijdt, S. Vaes, Ergodic coactions with large quantum multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (2006) 703–728. [3] K. De Commer, Galois objects for algebraic quantum groups, J. Algebra 321 (6) (2009) 1746–1785. [4] K. De Commer, Galois objects and cocycle twisting for locally compact quantum groups, J. Operator Theory, submitted for publication. [5] K. De Commer, Galois coactions for algebraic and locally compact quantum groups, PhD thesis, K.U. Leuven, 2009. [6] M. Enock, J.-M. Schwartz, Une dualité dans les algèbres de von Neumann, Supp. Bull. Soc. Math. France Mémoire 44 (1975) 1–144. [7] M. Enock, L. Vainerman, Deformation of a Kac algebra by an abelian subgroup, Comm. Math. Phys. 178 (3) (1996) 571–596. [8] P. Fima, L. Vainerman, Twisting and Rieffel’s deformation of locally compact quantum groups. Deformation of the Haar measure, Comm. Math. Phys. 286 (3) (2009) 1011–1050. [9] A. Guichardet, Produits tensoriels infinis et représentations des relations d’anti-commutation, Ann. Sci. Ecole Norm. Sup. 83 (1966) 1–52. [10] G. Kac, Ring groups and the principle of duality, I, II, Trans. Moscow Math. Soc. (1963) 291–339; (1965) 94–126. [11] G. Kac, L. Vainerman, Nonunimodular ring groups and Hopf–von Neumann algebras, Math. USSR Sb. 23 (1974) 185–214. [12] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole Norm. Sup. 33 (6) (2000) 837–934. [13] J. Kustermans, S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (1) (2003) 68–92. [14] M. Rieffel, Morita equivalence for C∗ -algebras and W∗ -algebras, J. Pure Appl. Algebra 5 (1974) 51–96. [15] M. Takesaki, Theory of Operator Algebras II, Springer, Berlin, 2003. [16] S. Vaes, L. Vainerman, Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math. 175 (1) (2003) 1–101. [17] A. Van Daele, The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123 (1995) 3125–3128. [18] A. Van Daele, Locally compact quantum groups. A von Neumann algebra approach, arXiv:math.OA/0602212. [19] S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (3) (1995) 671–692. [20] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987) 613–665. [21] S.L. Woronowicz, Twisted SU(2) group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987) 117–181. [22] S.L. Woronowicz, Compact quantum groups, in: Symétries Quantiques, Les Houches, 1995, North-Holland, 1998, pp. 845–884.


Ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space near BMO−1 Tsuyoshi Yoneda Institute for Mathematics and its Applications, University of Minnesota, 114 Lind Hall 207 Church Street S.E., Minneapolis, MN 55455, USA Received 18 September 2009; accepted 2 February 2010 Available online 11 February 2010 Communicated by J. Bourgain

Abstract The ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space which is smaller than −1 (q > 2) is considered. In 2008, Bourgain–Pavlović proved that the 3D-Navier–Stokes equation is B∞,q

−1 ill-posed in B∞,∞ by showing norm inflation phenomena of the solution for some initial data. On the other −1 hand, in 2008, Germain proved that the flow map is not C 2 in the space B∞,q for q > 2. However he did not treat ill-posed problem in such spaces. Thus our result is an extension of these previous results. © 2010 Elsevier Inc. All rights reserved.

Keywords: Navier–Stokes equations; Ill-posedness; Besov spaces

1. Introduction We consider the nonstationary incompressible Navier–Stokes equations in R3 : ⎧ ⎨ ∂u − u + (u · ∇)u + ∇p = 0, div u = 0 in x ∈ R3 , t ∈ (0, T ), ∂t ⎩ u|t=0 = u0 ,

(1)

where u = u(t) = (u1 (x, t), u2 (x, t), u3 (x, t)) and p = p(t) = p(x, t) denote the velocity vector field and the pressure of fluid at the point (x, t) ∈ R3 × (0, T ), respectively, while u0 = (u10 (x), u20 (x), u30 (x)) is a given initial velocity vector field. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.005

T. Yoneda / Journal of Functional Analysis 258 (2010) 3376–3387

3377

In this paper we are concerned with the ill-posedness of the Cauchy problem for (1). More precisely for a given function space X = X(R3 ) we say that the Cauchy problem is wellposed in X if there exists a space Y ⊂ C([0, T ), X) such that for all u0 ∈ X there exists a unique solution u ∈ Y for (1), and the flow map u0 → u = Φ(u0 ) is continuous from X to C([0, T ), X). Also we say that the Cauchy problem is ill-posed in X if it is not. The classical results on the existence theorem of the mild solution were shown by Kato [9] and Giga and Miyakawa [7]. Making use of the iteration procedure, they constructed a global solution in the class C([0, ∞); Ln (Rn )) ∩ C((0, ∞); Lp (Rn )) for n < p ∞, when an initial data u0 is small enough in Ln (Rn ). To construct a solution in more general classes of initial data is very important problem. Giga and Miyakawa [8], Kato [10] and Taylor [20] proved the well-posedness in certain Morrey spaces. Cannone [3] and Kozono and Yamazaki [13] investigated this problem in Besov spaces. In particular, Koch and Tataru [12] obtained the global solvability for (1), when the initial data u0 is small enough in BMO−1 . BMO−1 includes above function spaces and it has been considered as the largest space of initial data (see Lemarié-Rieusset [14]). On the other hand, Montgomery-Smith [16] introduced an equation similar to Navier–Stokes equation and proved −1 , which is larger than BMO−1 . In 2008, Bourgain and ill-posedness in the Besov space B∞,∞ −1 by showing norm inflation phenomena of the Pavlović [2] showed that (1) is ill-posed in B∞,∞ solution for some initial data. More precisely, they proved that for any δ > 0 there exists an initial data u0 with u0 B −1 < δ such that the corresponding solution u satisfies u(t)B −1 > 1/δ for ∞,∞ ∞,∞ some t < δ. This shows that the flow map Φ is not continuous. On the other hand, Germain [4] −1 for q > 2. However he did not proved that the flow map is not C 2 in the Besov spaces B∞,q treat ill-posed problem in such spaces. The purpose of the present paper is to show ill-posedness of 3D-Navier–Stokes equations in a generalized Besov space V which is strictly smaller than −1 (q > 2). Thus our result is an extension of both Bourgain–Pavlovi´ B∞,q c’s and Germain’s results. Now we give a sketch of the proof. First we introduce a generalized Besov space V which is −1 (q > 2). The idea is proposed by [18]. Second, we introduce initial data which smaller than B∞,q is composed by a sum of r cosine functions. The idea of setting of the initial data is proposed by [2] and [4]. We take a lacunary frequency set, and the norm of initial data in V is controlled by r. Third, we extract an inflation term from second approximation. Fourth, we estimate the remainder term y. The remainder term satisfies certain integral equation composed by first and second approximations including an inflation term. We also control the remainder term by r. Since we set refined initial data from Bourgain–Pavlović’s setting, we can get better estimate of second approximation than their estimate. According to their setting of initial data, using BMO−1 norm to estimate remainder term y is important. Since we got better estimate of second approximation, we can use the bilinear estimate of a class of bounded uniformly continuous functions (equipped with the L∞ norm). 2. Preliminaries Before presenting our results, we define the Besov spaces, a generalized Besov space which −1 (q > 2), the bilinear operator, j -th approximation and the remainder is smaller than B˙ ∞,q term. Now, we recall the Littlewood–Paley decomposition ψ, ϕj ∈ S, j = 0, 1, . . . , such that supp ψˆ ⊂ |ξ | 5/6 ,

supp ϕˆ ⊂ 3/5 |ξ | 5/3 ,

ϕj (x) = 2nj ϕ 2j x ,

3378


ˆ )+ 1 = ψ(ξ

∞

ϕˆ j (ξ )

ξ ∈ Rn ,

j =0 ∞

1=

ξ ∈ Rn \ {0} ,

ϕˆj (ξ )

(2)

j =−∞

where fˆ denotes the Fourier transform of f . Definition 1. (See Besov space cf. [1,19].) The inhomogeneous and homogeneous Besov spaces s and B s are defined as follows: ˙ p,q Bp,q s s Bp,q ≡ f ∈ S ; f Bp,q 2. The idea of the definition is based on [18].


3379

Definition 2 (A generalized Besov space V ). s,α denotes the set of all f ∈ S /P for which the norm (1) B˙ ∞,∞

α js s,α = sup 2 |j | + 1 ϕj ∗ f ∞ < ∞. f B˙ ∞,∞

(6)

j ∈Z

−1,1/2 −1 is the set of all f ∈ S /P that is written as a sum f = f1 + f2 , where (2) V := B˙ ∞,∞ + B˙ ∞,2 −1,1/2 −1 . The norm is defined by f1 ∈ B˙ ∞,∞ and f2 ∈ B˙ ∞,2

f V := f B˙ −1,1/2 +B˙ −1 = ∞,∞

∞,2

inf

f =f1 +f2

f1 B˙ −1,1/2 + f2 B˙ −1 ∞,∞

(7)

∞,2

−1,1/2 −1 for f ∈ B˙ ∞,∞ + B˙ ∞2 , where f1 , f2 run over all admissible representation f = f1 + f2 −1,1/2 −1 ˙ with f1 ∈ B∞,∞ and f2 ∈ B˙ ∞,2 .

Let us investigate the above generalized Besov space. The following proposition is also based on [18]. Proposition 3. −1 −1 for all q > 2. ⊂ V ⊂ B˙ ∞,q (1) B˙ ∞,2

−1,1/2 −1 and B˙ ∞,∞ . (2) There is no inclusion relationship between B˙ ∞,2

Proof of Proposition 3. (1) is easy to check from the definition of the norm. So let us consider (2). Let δz be the Dirac delta function massed at z ∈ R3 . Define f=

∞

aj δ2j

j =1

for {aj }∞ j =1 ⊂ R.

(8)

Then we have f B˙ −1

∞,2

So if we take aj =

j √2 j +1

2−2j |aj |2

1 2

,

j ∈Z

f B˙ −1,1/2 sup 2−j j + 1|aj |. ∞,∞

for j 0, aj = 0 for j < 0, then f B˙ −1,1/2 1, f B˙ −1 = ∞. ∞,∞

−1,1/2 −1 Therefore, B˙ ∞,2 is not included in B˙ ∞,∞ .

Let δj k be Kronecker’s delta. For fixed k ∈ N, if we take aj = j < 0, then we have f B˙ −1,1/2 1, f B˙ −1 = −1 in B˙ ∞,2 .

∞,∞

2

(9)

j ∈Z

∞,2

1 k.

δ 2j √j k j +1

∞,2

for j 0, aj = 0 for −1,1/2

Since k is arbitrary, B˙ ∞,∞ is not included

Now we define the bilinear operator and j -th approximation of the solution to the Navier– Stokes equations.

3380


Definition 4 (Bilinear operator). For w1 , w2 ∈ (L1 (0, T : L∞ ))3 with w1 ⊗ w2 ∈ (L1 (0, T : L∞ ))3×3 , let t B(w1 , w2 ) :=

∇ · e(t−τ ) P w1 (τ ) ⊗ w2 (τ ) dτ,

0

where P is the Helmholtz projection. The bilinear estimate is important to consider ill-posed problem. According to [2], they used the bilinear estimate in BMO−1 provided by Koch and Tataru [12]. In this paper we use the following estimate. See [5,15] for example (for elementary proof, see [6]). Proposition 5. There exists a constant c > 0 such that t ∇e Pf

∞

ct −1/2 f ∞

for t > 0, f ∈ L∞ .

Corollary 6. Let w1 , w2 ∈ (L1 (0, T : L∞ ))3 be such that w1 ⊗ w2 ∈ (L1 (0, T : L∞ ))3×3 . Then we have the following estimate: B(w1 , w2 )

t

∞

0

C w1 (τ ) w2 (τ ) dτ. ∞ ∞ (t − τ )1/2

Now we define j -th approximation of the solution u, and the remainder term y. Definition 7. Let ⎧ u1 = u1 (t) := et u0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ uj = uj (t) :=

B(uk1 , uk2 )

for j 2,

k1 +k2 =j, 1k1 ,k2 j −1

⎪ ⎪ ⎪ ⎪ uk . y = y(t) := ⎪ ⎪ ⎩ k3

For example, u2 = B(u1 , u1 ) and u3 = B(u1 , u2 ) + B(u2 , u1 ). Remark 8. By a formal calculation, we see that u = u1 + u2 + y. Moreover, the remainder term satisfies the following integral equation: y = B(y, y) + B(y, u2 + u1 ) + B(u2 + u1 , y) + B(u2 , u2 ) + B(u2 , u1 ) + B(u1 , u2 ) (10) on (0, ∞) with the initial condition y(0) = 0. Throughout this paper, we only treat periodic functions, since the following embedding inequality is necessary for (13). Let BUC be the space of all bounded uniformly continuous functions equipped with the L∞ -norm.


3381

Remark 9. A fundamental observation shows that for L > 0, if an initial data u0 ∈ BUC is a periodic function in [0, L)3 and its mean value is zero, then the solution u(t) ∈ BUC (t > 0) to Eq. (1) is also periodic in [0, L)3 and its mean value is also zero. Moreover we have the following embedding inequality: u(t) CL u(t) (11) V ∞ for t > 0. According to the above remark, homogeneous and inhomogeneous type function spaces are equivalent in this paper. We always denote by C > 0 universal constants unless no confusion occurs. 3. Main result Recall that BUC is the space of all bounded uniformly continuous functions equipped with the L∞ -norm. The main result is as follows. Theorem 10. For any δ > 0, there exist real-valued initial data u0 ∈ BUC with div u0 = 0 in S

and u0 V < δ such that the corresponding solution u exists in C([0, T ] : BUC) (T < δ) with u(T )V > 1/δ. −1 or F˙ −1 (Triebel–Lizorkin spaces) Remark 11. The same result is true if we replace V by B˙ ∞,q ∞,q for q > 2. The proof is quite similar to the case of V . Thus we omit its detail.

Remark 12. The solution (u, p) is unique in L∞ ((0, T ) × R3 ) × L1loc ((0, T ) : L1,φ ), where L1,φ is generalized Campanato spaces which include BMO (see [11,17]). Proof of Theorem 10. First, we set initial data which bring norm inflation phenomena. In order to set initial data, we need several definitions. For sufficiently small > 0, let Γ1 , Γ2 : N → R be such that Γ1 (m) :=

m

s −1 ,

1− 2

Γ2 (m) := Γ1

(m)

s=1

for m ∈ N. We take sufficiently small C1 = C1 ( ) > 0 and sufficiently large C2 = C2 ( ) > 0, and let us take Γ3 : N → N satisfying C1 Γ13 (m) Γ3 (m) C2 Γ13 (m) for m ∈ N. Let us set coefficients v 0 := (0, 0, 1) and v 1 := (0, 1, 0), and let {ks0 }rs=1 ⊂ N and {ks1 }rs=1 ⊂ N be frequency sets satisfying the following property, ks0 := 23s T −1/2 a0 ,

ks1 := ks0 + a1

(s = 1, . . . , r),

where a0 = (1, 0, 0), a1 = (0, 0, 1) and T = (1/Γ32 (r)) for r ∈ N. The constant T with r 1 will be the existence time. To obtain the embedding inequality (11), we require Γ3 to be integervalued.

3382


0 0 2 −1 , 1 |k 0 | |k 1 | 2|k 0 | for Remark 13. It is easy to see that |ks0 | 4 s−1 s s s s =1 |ks |, |k1 | = 64T s = 1, . . . , r, and ⎧ 0 0 a · v = 0, ⎪ ⎪ ⎨ 1 0 a · v = 1, (12) 0 1 ⎪ ⎪ ⎩ a · v = 0, a 1 · v 1 = 0. If the space dimension is two, we cannot obtain the above properties. Thus, ill-posed result is obtained only for the space dimension n 3. We set the initial data as follows: u0 (x) :=

r 1 0 −1/2 0 ks s v cos ks0 · x + v 1 cos ks1 · x . Γ2 (r) s=1

Remark 14. We state properties of the initial data. • In [2], the coefficients of v 0 and v 1 depend on s. • A direct calculation yields div u0 = 0. • Since {ks0 }rs=1 and {ks1 }rs=1 are lacunary, and homogeneous and inhomogeneous type function spaces are equivalent, we see u0 V u0 B˙ −1,1/2 = sup 2−j j + 1ϕj ∗ u0 ∞ ∞,∞

= sup 2 j ∈N

−j

j ∈Z

j + 1ϕj ∗ u0 ∞

√ r j + 1 |ks0 | ϕj ∗ cos k 0 · x + ϕj ∗ cos k 1 · x s s ∞ ∞ j 1/2 2 s s=1 √ j +1 1 sup 1/2 → 0 as r → ∞. Γ2 (r) j ∈N j

1 sup Γ2 (r) j ∈N

Our strategy is to decompose second approximation u2 as Bourgain and Pavlović [2] did. Since r 1 u2 = B(u1 , u1 ) = 2 Γ2 (r) s,s =1

t

σ e(t−τ ) PUs,s

(τ, x) dτ,

σ ∈{0,1}3 0

where σ ,σ2 ,σ3

σ 1 Us,s

(τ, x) = Us,s

(τ, x) 0 0 −1/2 −(|k σ1 |2 +|k σ2 |2 )τ s

e s := (1/2)ks ks ss × v σ1 sin ksσ1 + (−1)σ3 ksσ 2 · x v σ2 · ksσ1 + (−1)σ3 ksσ 2 ,


3383

we can decompose u2 as u2 = u02 + u12 + u˜ 2 , where ⎧ t r ⎪ ⎪ ⎪ (σ1 ,σ2 ,0) 0 2 ⎪ u2 := 1/Γ2 (r) e(t−τ ) PUs,s (τ, x) dτ, ⎪ ⎪ ⎪ ⎪ s=1 σ ,σ ∈{0,1} ⎪ 1 2 0 ⎪ ⎪ ⎪ ⎪ t ⎪ r ⎨ (σ1 ,σ2 ,1) 1 2 u := 1/Γ2 (r) e(t−τ ) PUs,s (τ, x) dτ, ⎪ 2 ⎪ s=1 σ ,σ ∈{0,1} ⎪ 1 2 0 ⎪ ⎪ ⎪ ⎪ ⎪ t r ⎪ ⎪ ⎪ σ 2 ⎪ u˜ := 1/Γ2 (r) e(t−τ ) P Usσ ,s (τ, x) + Us,s ⎪

(τ, x) dτ. ⎪ ⎩ 2

s=1 s <s σ ∈{0,1}3 0

We note u12 is an inflation term. Remark 15. By (12), we see that 1 for (σ1 , σ2 ) = (1, 0), v σ2 · ksσ1 + (−1)σ3 ksσ 2 = 0 otherwise. Lemma 16. We have the following key estimates of u12 , u02 , u˜ 2 , u1 and y. • The estimate of the inflation term u12 . We have the following inequalities: 1 u (t) 2

∞

C

Γ1 (r) = CΓ1 (r) Γ22 (r)

for t > 0, 1 u (t) C Γ1 (r) = CΓ (r) 2 1 V Γ22 (r) for T /64 t 1 with sufficiently large r. • The estimate of the first approximation and another part of the second approximation (exclude the inflation term). We have the following inequalities: ⎧ u0 (t) C , ⎪ ⎪ 2 ⎪ ∞ ⎪ Γ22 (r) ⎪ ⎪ ⎪ ⎨ u˜ 2 (t) C , ∞ ⎪ Γ22 (r) ⎪ ⎪ ⎪ ⎪ ⎪ C ⎪ ⎩ et u0 ∞ Γ2 (r)t 1/2

3384


for 0 < t < T = (1/Γ32 (r)) and t e u0

7 1 Γ3 (r) − = Γ1 2 2 (r) Γ2 (r)

∞

for T /64 < t < T .

• The estimate of the remainder term. 3

−1

Let ρ1 (r) := Γ1 (r)/Γ23 (r) = Γ1 2 2 (r) and ρ2 (r) := Γ12 (r)/(Γ24 (r)Γ3 (r)) CΓ1− (r). (Note that ρ1 (r), ρ2 (r) → 0 as r → ∞.) There exists the remainder term y ∈ C([0, T ] : BUC), y(0) = 0 satisfying the following estimate: y(t) 2C ρ1 (r) + ρ2 (r) ∞ for 0 < t < T = 1/Γ32 (r) with sufficiently large r. We postpone to prove the above lemma. We now prove the main theorem. By the embedding inequality (11) and Lemma 16, we have the following estimate: u(t) u1 (t) − u0 (t) − u˜ 2 (t) − et u0 − y(t) 2 2 V V ∞ ∞ ∞ ∞ CΓ1 (r) −

C Γ22 (r)

−

Γ3 (r) − 2C ρ1 (r) + ρ2 (r) CΓ1 (r). Γ2 (r)

for T /64 < t < T (T = 1/Γ32 (r)) with sufficiently large r. This is the desired estimate.

(13) 2

4. Proof of the key lemma In this section, we prove Lemma 16. First we estimate the inflation term u12 . It is easy to see that 001 111 ≡ 0 and Us,s ≡ 0 for s = 1, . . . , r. Us,s 011 ≡ 0 for s = 1, . . . , r and Since v 1 · a1 = 0 and v 0 · a1 = 1, we also see that Us,s Pv 1 (v 0 · a1 ) sin(a1 · x) = v 1 sin x3 . Thus we have the following equality:

1 Γ22 (r) s=1 r

u12 (x, t) =

=

t (1,0,1) e(t−τ ) PUs,s (τ, x) dτ 0

r 1 −1 0 2 s ks Γ22 (r) s=1

t

e−(t−τ )|a1 | e−(|ks | 2

1 2 +|k 0 |2 )τ s

dτ Pv 1 v 0 · a1 sin(a1 · x).

0

We now estimate lower and upper bound of u12 . By Remark 13, we have 0 2 k

t

s

0

e−(t−τ )|a1 | e−(|ks | 2

1 2 +|k 0 |2 )τ s

2 1 − e−(|ks1 |2 +|ks0 |2 −1)t 0 2 e−t 1 − e−2|k1 | t C dτ = e−t ks0 0 2 1 2 |ks | + |ks | − 1


3385

for t ∈ [T /64, 1] = [1/|k10 |2 , 1]. Since we easily estimate sin x3 in V , we have 1 u (t) C Γ1 (r) 2 V Γ22 (r) for t ∈ [T /64, 1] = [1/|k10 |2 , 1] with sufficiently large r. We also have u12 (t)∞ B(Γ1 (r)/ Γ22 (r)) for some constant B > C and t > 0. Thus we complete the estimate of the inflation term. Next we consider u˜ 2 ∞ , u02 ∞ and et u0 ∞ . However, we only calculate u˜ 2 ∞ , since the calculations of u02 ∞ and et u0 ∞ are easy. Let t Js,s :=

σ1 2 σ σ σ | +|ks 2 |2 )τ −|ks 1 +(−1)σ3 ks 2 |2 (t−τ )

e−(|ks

dτ

0

=e

σ

σ

−|ks 1 +(−1)σ3 ks 2 |2 t

σ3 σ1 σ2 e2(−1) ks ·ks t − 1 . 2(−1)σ3 ksσ1 · ksσ 2

A direct calculation shows that σ u˜ 2

r 2 −1/2 0 0 ss ks ks |Js,s |. Γ22 (r) s=1 s <s

By Remark 15, we only need to consider just two cases, the case σ = (σ1 , σ2 , σ3 ) = (1, 0, 1) and the case σ = (σ1 , σ2 , σ3 ) = (1, 0, 0). If σ3 = 1, the function

e

σ σ 2(−1)σ3 ks 1 ·k 2 t s −1 σ σ 2(−1)σ3 ks 1 ·ks 2 t

is uniformly

bounded with respect to t > 0. But if σ3 = 0, it is not. Thus we need to distinguish the cases σ3 = 0 and σ3 = 1. The case σ = (1, 0, 1). Since s (s − 1) < s, we see that 2 2

−ksσ1 + (−1)σ3 ksσ 2 = −23s T −1/2 a0 − 23s T −1/2 a0 − a1 2 2

− 23s − 23s T −1/2 a0 − 23s − 23(s−1) T −1/2 a0 49 0 2 ks . =− 64 Thus we have |Js,s | Cte−C|ks | t for t > 0. 0 2

The case σ = (1, 0, 0). A direct calculation yields Js,s = e Thus we have

σ

2

σ

σ

−|ks 1 +ksσ |2 t 2ks 1 ·ks 2 t

e

σ1

σ2 t

1 − e−2ks ·ks

2ksσ1 · ksσ 2

σ1 2 σ | +|ks 2 |2 )t

Cte−(|ks

Cte−|ks | t . 0 2

3386


|u˜ 2 |

r C −1/2 0 0 −C|ks0 |2 t ks ks te ss Γ22 (r) s=1 s <s

C 0 2 −C|ks0 |2 t C ks te 2 , 2 Γ2 (r) s=1 Γ2 (r) r

0 0 2 −C|ks0 |2 t C j j where we used ∞ j ∈Z 2 exp(−2 ) < ∞ for t > 0 and 4 s <s |ks | s=1 t|ks | e |ks0 |. Thus we complete the estimate of the term u˜ 2 . Now we consider the remainder term y which satisfies (10). Recall that ρ1 (r) = 3

−1

Γ1 (r)/Γ23 (r) = Γ1 2 2 (r) and ρ2 (r) = Γ12 (r)/(Γ24 (r)Γ3 (r)) CΓ1− (r). Let ρ3 (r) := Γ1 (r)/(Γ22 (r)Γ3 (r)) CΓ1−2 (r). Note that ρ1 (r) → 0, ρ2 (r) → 0 and ρ3 (r) → 0 as r → ∞. Also we recall that T = 1/Γ32 (r). By the above estimates of u1 ∞ , u2 ∞ and Corollary 6, and since the worst term to estimate in B(u1 , u2 ) or B(u2 , u1 ) is B(u12 , u1 ), we have the following inequality: B u1 (t), u2 (t) + B u2 (t), u1 (t) C B u1 (t), u1 (t) Cρ1 (r). 2 ∞ ∞ ∞ Since the worst term to estimate in B(u2 , u2 ) is B(u12 , u12 ) and B(y, u2 ) is B(y, u12 ), we have 1 B u (t), u1 (t) Cρ2 (r) C 2 2 ∞ ∞

B u2 (t), u2 (t) and B u1 (t) + u2 (t), y(t)

∞

+ B y(t), u1 (t) + u2 (t) ∞ C B y(t), u12 (t) ∞ Cρ3 (r) sup y(τ )∞ 0 1, none of the E0 -semigroups constructed from boundary weight doubles satisfying the conditions of Theorem 5.4 are cocycle conjugate to any of the E0 -semigroups obtained from one-dimensional boundary weights by Powers in [9] (Corollary 5.5). We turn our attention to the unital q-pure maps that are invertible. These maps are best understood through their (conditionally negative) inverses. In Theorem 6.11, we find a necessary and sufficient condition for an invertible unital map φ on Mn (C) to √ be q-pure. In √ this case, however, if ν is a normalized unbounded boundary weight of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ), then the E0 -semigroup induced by the boundary weight double (φ, ν) is entirely determined by ν. This E0 -semigroup is the one induced by ν in the sense of [9]. 2. Background 2.1. Completely positive maps Let φ : U → B be a linear map between C ∗ -algebras. For each n ∈ N, define φn : Mn (U) → Mn (B) by ⎛

A11 . φn ⎝ .. An1

··· .. . ···

⎞ ⎛ φ(A11 ) A1n .. ⎠ ⎝ .. = . . Ann

φ(An1 )

··· .. . ···

⎞ φ(A1n ) .. ⎠. . φ(Ann )

We say that φ is completely positive if φn is positive for all n ∈ N. A linear map φ : B(H1 ) → B(H2 ) is completely positive if and only if for all A1 , . . . , An ∈ B(H1 ), f1 , . . . , fn ∈ H2 , and n ∈ N, we have n fi , φ A∗i Aj fj 0. i,j =1

3416

C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451

Stinespring’s theorem asserts that if U is a unital C ∗ -algebra and φ : U → B(H ) is a unital completely positive map, then φ dilates to a ∗-homomorphism in that there is a Hilbert space K, a ∗-homomorphism π : U → B(K), and an isometry V : H → K such that φ(A) = V ∗ π(A)V for all A ∈ U. From the work of Choi [4] and Arveson [1], we know that a normal linear map φ : B(H1 ) → B(H2 ) is completely positive if and only if it can be written in the form φ(A) =

n

Si ASi∗

i=1

for some n ∈ N ∪ {∞} and maps Si : H1 → H2 which are linearly independent over 2 (N) in rn the sense that if i=1 zi Si = 0 for a sequence {zi }ri=1 ∈ 2 (N), then zi = 0 for all i. With these hypotheses satisfied, the number n is unique. We will use the above conditions for complete positivity interchangeably. 2.2. Conditionally negative maps Wesay a self-adjoint linear map ψ : B(K) → B(K) is conditionally negative if, whenm ever i=1 Ai fi = 0 for A1 , . . . , Am ∈ B(K), f1 , . . . , fm ∈ K, and m ∈ N, we have m ∗ n (f i=1 i , ψ(Ai Aj )fj ) 0. If K = C , then from the literature (see, for example, Theorem 3.1 of [10]) we know that ψ has the form ψ(A) = sA + Y A + AY ∗ −

p

λi Si ASi∗ ,

i=1

where s ∈ R, tr(Y ) = 0, and for all i and j we have λi > 0, tr(Si ) = 0 and tr(Si∗ Sj ) = nδij , where p n2 is independent of the maps Si . This form for ψ is unique in the sense that if ψ is written in the form ∗

ψ(A) = tA + ZA + AZ −

p

μi Ti ATi∗ ,

i=1

where t ∈ R, tr(Z) = 0, and for all i and j we have μi > 0, tr(Ti ) = 0, and tr(Ti∗ Tj ) = nδij , p p μi Ti ATi∗ for all A ∈ Mn (C). Indeed, let {vk }nk=1 then s = t, Z = Y , and i=1 λi Si ASi∗ = i=1 √ n be any orthonormal basis for C , let hk = vk / n for each k, let f ∈ Cn be arbitrary, and for k = 1, . . . , n, define Ak ∈ Mn (C) by Ak = f h∗k . Using the trace conditions, we find n k=1

ψ(Ak )hk =

n n n hk , Y ∗ hk f (hk , hk )sf + (hk , hk )Yf + k=1

−

k=1

n

p

k=1

i=1

λi hk , Si∗ hk Si f

k=1


= sf + Yf + 0 −

n p i=1

= sf + Yf −

p

λi hk , Si∗ hk Si f

3417

k=1

λi (0)Si f = sf + Yf.

i=1

An analogous computation shows that nk=1 ψ(Ak )hk = tf + Zf . Since f ∈ Cn was arbitrary, we conclude (t − s)I = Y − Z. Therefore, tr((t − s)I ) = tr(Y − Z) = 0, so t = s and Y = Z. p p Consequently, i=1 λi Si ASi∗ = i=1 μi Ti ATi∗ for all A ∈ Mn (C). 2.3. CP-flows and Bhat’s theorem Let K be a separable Hilbert space and let H = K ⊗ L2 (0, ∞). We identify H with the space of K-valued measurable functions on (0, ∞) which are square integrable. Under this identification, the inner product on H is L2 ((0, ∞); K),

∞ (f, g) = f (x), g(x) dx. 0

Let U = {Ut }t0 be the right shift semigroup on H , so for all t 0 and f ∈ H we have (Ut f )(x) = f (x − t) for x > t and (Ut f )(x) = 0 otherwise. Let Λ : B(K) → B(H ) be the map defined by (Λ(A)f )(x) = e−x Af (x) for all A ∈ B(K), f ∈ H . Definition 2.1. Assume the above notation. A strongly continuous semigroup α = {αt : t 0} of completely positive contractions of B(H ) into itself is a CP-flow if αt (A)Ut = Ut A for all A ∈ B(H ). A theorem of Bhat in [3] allows us to generate E0 -semigroups from unital CP-flows, and, more generally, from strongly continuous completely positive semigroups of unital maps on B(H ), called CP-semigroups. We give a reformulation of Bhat’s theorem (see Theorem 2.1 of [9]): Theorem 2.2. Suppose α is a unital CP-semigroup of B(H1 ). Then there is an E0 -semigroup α d of B(H2 ) and an isometry W : H1 → H2 such that αt (A) = W ∗ αtd W AW ∗ W and αt (W W ∗ ) W W ∗ for all t > 0. If the projection E = W W ∗ is minimal in that the closed linear span of the vectors αtd1 (EA1 E) · · · αtdn (EAn E)Ef for f ∈ K, Ai ∈ B(H1 ) and ti 0 for all i = 1, 2, . . . , n and n = 1, 2, . . . is H2 , then α d is unique up to conjugacy.

3418


In [8], Powers showed that every spatial E0 -semigroup acting on B(H) (for H a separable Hilbert space) is cocycle conjugate to an E0 -semigroup which is a CP-flow, and that every CPflow over K arises from a boundary weight map over H = K ⊗ L2 (0, ∞). The boundary weight map ρ → ω(ρ) of a CP-flow α associates to every ρ ∈ B(K)∗ a boundary weight, that is, a linear functional ω(ρ) acting on the null boundary algebra A(H ) =

IH − Λ(IK )B(H ) IH − Λ(IK )

which is normal in the following sense: If we define a linear functional (ρ) on B(H ) by

(ρ)(A) = ω(ρ) IH − Λ(IK )A IH − Λ(IK ) , then (ρ) ∈ B(H )∗ . If ω(ρ)(IH − Λ(IK )) = ρ(IK ) for all ρ ∈ B(K)∗ , then α is unital. For the sake of neatness, we will omit the subscripts H and K from the previous sentence ∞ when they are clear. Let δ be the generator of α, and define Γ : B(H ) → B(H ) by Γ (A) = 0 e−t Ut AUt∗ . ∞ The resolvent Rα := (I − δ)−1 of α satisfies Rα (A) = 0 e−t αt (A) dt for all A ∈ B(H ). Its associated predual map Rˆ α is given by ˆ +η Rˆ α (η) = Γˆ ω(Λη)

(1)

for all η ∈ B(H )∗ . A CP-flow α over K is entirely determined by a set of normal completely positive contractions π # = {πt# : t > 0} from B(H ) into B(K), called the generalized boundary representation of α. Its relationship to the boundary weight map is as follows. For each t > 0, denote by πˆ t : B(K)∗ → B(H )∗ the predual map induced by πt# . For the truncated boundary weight maps ρ → ωt (ρ) ∈ B(H )∗ defined by ωt (ρ)(A) = ω(ρ) Ut Ut∗ AUt Ut∗ ,

(2)

ˆ t )−1 and ωt = πˆ t (I − Λˆ πˆ t )−1 for all t > 0. The maps {π # }b>0 have we have πˆ t = ωt (I + Λω b # a σ -strong limit π0 as b → 0 for each A ∈ t>0 Ut B(H )Ut∗ , called the normal spine of α. If α is unital, then the index of α d as an E0 -semigroup is equal to the rank of π0# as a completely positive map (Theorem 4.49 of [8]). Having seen that every CP-flow has an associated boundary weight map, we would like to approach the situation from the opposite direction. More specifically, under what conditions is a map ρ → ω(ρ) from B(K)∗ to weights acting on A(H ) the boundary weight map of a CP-flow over K? Powers has found the answer (see Theorem 3.3 of [9]): Theorem 2.3. If ρ → ω(ρ) is a completely positive mapping from B(K)∗ into weights on B(H ) satisfying ω(ρ)(I − Λ(IK )) ρ(IK ) for all positive ρ ∈ B(K)∗ , and if the maps πˆ t := ωt (I + ˆ t )−1 are completely positive contractions from B(K)∗ into B(H )∗ for all t > 0, then ρ → Λω ω(ρ) is the boundary weight map of a CP-flow over K. The CP-flow is unital if and only if ω(ρ)(I − Λ(IK )) = ρ(IK ) for all ρ ∈ B(K)∗ .


3419

If dim(K) = 1, the boundary weight map is just c ∈ C → ω(c) = cω(1), so we may view our boundary weight map as a single positive boundary weight ω := ω(1) acting on A(L2 (0, ∞)). Since the functional defined on B(H ) by (A) = ω

I − Λ(1)A I − Λ(1)

is positive and normal, it has the form (A) = n∈N∪{∞} , so vectors {fk }k=1 ω

n

k=1 (fk , Afk )

for some mutually orthogonal

n

I − Λ(1)A I − Λ(1) = (fk , Afk ) k=1

for all A ∈ B(H ). If ω is normalized (that is, ω(I − Λ(1)) = 1), then nk=1 fk 2 = 1. In [9], 2 Powers induced E0 -semigroups using normalized boundary weights √ over L (0,√∞). The type d induced by a normalized boundary weight ω( I − Λ(1)A I − Λ(1) ) = of E -semigroup α n 0 k=1 (fk , Afk ) depends on whether ω is bounded in the sense that for some r > 0 we have |ω(B)| rB for all B ∈ A(H ). Results from [8] imply that α d is of type In if ω is bounded and of type II0 if ω is unbounded. If ω is unbounded, then both ωt (I ) and ωt (Λ(1)) approach infinity as t approaches√zero. We will√focus on normalized unbounded boundary weights over L2 (0, ∞) of the form ω( I − Λ(1)A I − Λ(1) ) = (f, Af ). We note that, as discussed in detail in [7], such boundary weights are not normal weights. If α and β are CP-flows, we say that α β if αt − βt is completely positive for all t 0. The subordinates of a CP-flow are entirely determined by the subordinates of its generalized boundary representation (see Theorem 3.4 of [9]): Theorem 2.4. Let α and β be CP-flows over K with generalized boundary representations π # = {πt# } and ξ # = {ξt# }, respectively. Then β is subordinate to α if and only if πt# − ξt# is completely positive for all t > 0. Given two unital CP-flows α and β, it is natural to ask when their minimally dilated E0 semigroups are cocycle conjugate. The following definition from [8] provides us with a key: Definition 2.5. Let α and β be CP-flows over K1 and K2 , respectively, where H1 = K1 ⊗ L2 (0, ∞) and H2 = K2 ⊗ L2 (0, ∞). We say that a family of linear maps γ = {γt : t 0} from B(H2 , H1 ) into itself is a flow corner from α to β if the family of maps Θ = {Θt : t 0} defined by Θt

A11 A21

A12 A22

=

αt (A11 ) γt∗ (A21 )

γt (A12 ) βt (A22 )

is a CP-flow over K1 ⊕ K2 . If γ is a flow corner from α to β, we consider subordinates Θ of Θ that are CP-flows of the form Θt

A11 A21

A12 A22

:=

αt (A11 ) γt∗ (A21 )

γt (A12 ) . βt (A22 )

3420


We say that γ is a hyper maximal flow corner from α to β if, for every such subordinate Θ of Θ, we have α = α and β = β . Our results will involve type II0 E0 -semigroups. These are spatial E0 -semigroups which are not semigroups of ∗-automorphisms and have only one unit V = {Vt }t0 up to scaling by etλ for λ ∈ C. In the case that unital CP-flows α and β minimally dilate to type II0 E0 -semigroups, we have a necessary and sufficient condition for α d and β d to be cocycle conjugate (Theorem 4.56 of [8]): Theorem 2.6. Suppose α and β are unital CP-flows over K1 and K2 and α d and β d are their minimal dilations to E0 -semigroups. Suppose γ is a hyper maximal flow corner from α to β. Then α d and β d are cocycle conjugate. Conversely, if α d is a type II0 and α d and β d are cocycle conjugate, then there is a hyper maximal flow corner from α to β. We will later use this theorem to determine a necessary and sufficient condition for some of the E0 -semigroups we construct to be cocycle conjugate (see Definition 4.4 and Proposition 4.6). 3. Our boundary weight map Recall that a completely positive linear map φ can have negative eigenvalues. Moreover, even if I + tφ is invertible for a given t, it does not necessarily follow that φ(I + tφ)−1 is completely positive. In our boundary weight construction, we will require a special kind of completely positive map: Definition 3.1. A linear map φ : Mn (C) → Mn (C) is q-positive if φ has no negative eigenvalues and φ(I + tφ)−1 is completely positive for all t 0. Henceforth, we naturally identify a finite-dimensional Hilbert space K with Cn and B(K ⊗ with Mn (B(L2 (0, ∞))). Under these identifications, the right shift t units on K ⊗ 2 L (0, ∞) is the matrix whose ij th entry is δij Vt for Vt the right shift on L2 (0, ∞). The map Λn×n : B(K) → B(K ⊗ L2 (0, ∞)) sends an n × n matrix B = (bij ) ∈ Mn (C) to the matrix Λn×n (B) whose ij th entry is bij Λ(1) ∈ B(L2 (0, ∞)). The null boundary algebra A(H ) is simply Mn (A(L2 (0, ∞))). Given a boundary weight ν over L2 (0, ∞), we write Ων,n×k for the map that sends an n × k matrix A = (Aij ) ∈ Mn×k (A(L2 (0, ∞))) to the matrix Ων,n×k (A) ∈ Mn×k (C) whose ij th entry is ν(Aij ). We will suppress the integers n and k when they are clear, writing the above maps as Ων and Λ. In the proposition and corollary that follow, we show how to construct a CP-flow using a q-positive map φ : Mn (C) → Mn (C), a normalized boundary weight ν over L2 (0, ∞), and the map Ων := Ων,n×n : A(H ) → Mn (C). The map Ων is completely positive since ν is positive. L2 (0, ∞))

Proposition 3.2. Let H = Cn ⊗ L2 (0, ∞). Let φ : Mn (C) → Mn (C) be a unital completely positive map with no negative eigenvalues, and let ν be a normalized unbounded boundary weight over L2 (0, ∞). Then the map ρ → ω(ρ) from Mn (C)∗ into boundary weights on A(H ) defined by ω(ρ)(A) = ρ φ Ων (A)


3421

ˆ t )−1 define normal completely is completely positive. Furthermore, the maps πˆ t := ωt (I + Λω positive contractions πt# of B(H ) into Mn (C) for all t > 0 if and only if φ is q-positive. Proof. The map ρ → ω(ρ) is completely positive since it is the composition of two completely positive maps. Before proving either direction, we let st = νt (Λ(1)) for all t > 0 and prove the equality πˆ t (ρ) = ρ φ(I + st φ)−1 Ωνt

(3)

for all ρ ∈ Mn (C)∗ . Denoting by Ut the right shift on H for every t > 0, we claim that (I + ˆ t )−1 = (I + st φ) ˆ −1 . Indeed, for arbitrary t > 0, B ∈ Mn (C), and ρ ∈ Mn (C)∗ , we have Λω ˆ t (ρ)(B) = ρ φ Ων Ut Ut∗ Λ(B)Ut Ut∗ = ρ φ Ωνt Λ(B) = st ρ φ(B) , Λω ˆ t = st φˆ and (I + Λω ˆ t )−1 = (I + st φ) ˆ −1 . hence Λω For any t > 0 and A ∈ B(H ), we have ˆ t )−1 (ρ)(A) = (I + Λω ˆ t )−1 (ρ) φ Ωνt (A) πˆ t (ρ)(A) = ωt (I + Λω ˆ −1 (ρ) φ Ωνt (A) = ρ (I + st φ)−1 φ Ωνt (A) = (I + st φ) = ρ φ(I + st φ)−1 Ωνt (A) , establishing (3). Assume the hypotheses of the backward direction and let t > 0. By construction, πˆ t maps Mn (C)∗ into B(H )∗ . It is also a contraction, since for all ρ ∈ Mn (C)∗ we have πˆ t (ρ) = ρ φ(I + st φ)−1 Ων ρφ(I + st φ)−1 Ων t t = ρφ(I + st φ)−1 Ωνt (I ) = ρφ(I + st φ)−1 νt (I )ICn νt (I ) νt (I ) n = ρ I 1 + s C = ρ 1 + ν (Λ(1)) ρ, t t where the last inequality follows from the fact that νt I − Λ(1) ν I − Λ(1) = 1. Therefore, for every t > 0, πˆ t defines a normal contraction πt# from B(H ) into Mn (C) satisfying πˆ t (ρ) = ρ ◦ πt# for all ρ ∈ Mn (C)∗ . From Eq. (3) we see πt# = φ(I + st φ)−1 Ωνt , so πt# is the composition of completely positive maps and is thus completely positive for all t > 0. Now assume the hypotheses of the forward direction. By unboundedness of ν, the (monotonically decreasing) values {st }t>0 form a set equal to either (0, ∞) or [0, ∞). Choose any t > 0 such that st > 0. Let T ∈ B(H ) be the matrix with ij th entry (1/νt (I ))I , and let κt : Mn (C) → B(H ) be the map that sends B = (bij ) ∈ Mn (C) to the matrix κt (B) ∈ B(H ) whose ij th entry is (bij /νt (I ))I . We note that κt is the Schur product B → B · T , which is completely positive since T is positive. For all B ∈ Mn (C), we have φ(I + st φ)−1 (B) = πt# κt (B) ,

3422


so φ(I + st φ)−1 is the composition of completely positive maps and is thus completely positive. As noted above, the values {st }t>0 span (0, ∞), so φ is q-positive. 2 Corollary 3.3. The map ρ → ω(ρ) in Proposition 3.2 is the boundary weight map of a unital CP-flow α over Cn , and the Bhat minimal dilation α d of α is a type II0 E0 -semigroup. Proof. The first claim of the corollary follows immediately from Theorem 2.3 and Proposition 3.2 since ω(ρ) I − Λ(ICn ) = ρ φ(ICn ) = ρ(ICn )

(4)

for all ρ ∈ Mn (C)∗ . For the second assertion, we note that by Theorem 4.49 of [8], the index of α d is equal to the rank of the normal spine π0# of α, where π0# is the σ -strong limit of the maps {πb# }b>0 for each A ∈ t>0 Ut B(H )Ut∗ . Fix t > 0, and let A ∈ Ut B(H )Ut∗ . From formula (3), −1 Ωνb (A) . πb# (A) = φ I + νb Λ(1) φ For all b < t we have Ωνb (A) = Ωνt (A) < ∞. Since νb (Λ(1)) → ∞ as b → 0, we conclude limb→0 πb# (A) = 0, hence π0# = 0 and the index of α is zero. However, α d is not completely spatial since α is not derived from the zero boundary weight map (see Lemma 4.37 and Theorem 4.52 of [8]), so α d is of type II0 . 2 Given a q-positive φ : Mn (C) → Mn (C) and a normalized unbounded boundary weight ν over L2 (0, ∞), we call (φ, ν) a boundary weight double. As we have seen, if φ is unital then the boundary weight double naturally defines a boundary weight map through the construction of Proposition 3.2, inducing a type II0 E0 -semigroup α d which is unique up to conjugacy by Theorem 2.2. We should note that it is not necessary for φ to be unital in order for the boundary weight double to induce a CP-flow: If φ is any q-positive contraction such that νt (I )φ(I + νt (Λ(1))φ)−1 1 for all t > 0, then the arguments given in the proofs of Proposition 3.2 and Corollary 3.3 show that the boundary weight double (φ, ν) induces a CP-flow α. However, if φ is not unital, then by Eq. (4) and Theorem 2.3, neither is α. Motivated by [8], we make the following definition: Definition 3.4. Suppose α : B(H1 ) → B(K1 ) and β : B(H2 ) → B(K2 ) are normal and completely positive. Write each A ∈ B(H1 ⊕ H2 ) as A = (Aij ), where Aij ∈ B(Hj , Hi ) for each i, j = 1, 2. We say a linear map γ : B(H2 , H1 ) → B(K2 , K1 ) is a corner from α to β if ψ : B(H1 ⊕ H2 ) → B(K1 ⊕ K2 ) defined by ψ

A11 A21

A12 A22

=

α(A11 ) γ ∗ (A21 )

γ (A12 ) β(A22 )

is a normal completely positive map. We will repeatedly use the following lemma, which gives us the form of any corner between normal completely positive contractions of finite index. We believe that this result is already present in the literature, but we present a proof here for the sake of completeness:


3423

Lemma 3.5. Let H1 , H2 , K1 , and K2 be separable Hilbert spaces. Let α : B(H1 ) → B(K1 ) and β : B(H2 ) → B(K2 ) be normal completely positive contractions of the form α(A11 ) =

n

Si A11 Si∗ ,

β(A22 ) =

p

Tj A22 Tj∗ ,

j =1

i=1 p

where n, p ∈ N and the sets of maps {Si }ni=1 and {Tj }j =1 are both linearly independent. A linear map γ : B(H2 , H1 ) → B(K2 , K1 ) is a corner from α to β if and only if for all A12 ∈ B(H2 , H1 ) we have γ (A12 ) =

cij Si A12 Tj∗ ,

i,j

where C = (cij ) ∈ Mn×p (C) is any matrix such that C 1. Proof. For the backward direction, let C = (cij ) ∈ M n×p (C) be any contraction, and define a linear map γ : B(H2 , H1 ) → B(K2 , K1 ) by γ (A) = i,j cij Si ATj∗ . We need to show that the map L

A11 A21

A12 A22

=

α(A11 ) γ ∗ (A21 )

γ (A12 ) β(A22 )

is normal and completely positive. To prove this, we first assume that n p and note that by Polar Decomposition we may write Cn×p = Vn×p Tp×p , where Vn×p is a partial isometry of ∗ D rank p and T is positive. Unitarily diagonalizing T we see Cn×p = Vn×p Wp×p p×p Wp×p . ∗ We may easily add columns to Vn×p Wp×p to form a unitary matrix in Mn (C), which we call U ∗ . Defining D˜ = (dij ) ∈ Mn×p (C) to be the matrix obtained from D by adding n − p rows of ∗ D, so C ∗ ˜ zeroes, we see U ∗ D˜ = Vn×p Wp×p n×p = U DW p×p and ∗ ˜ U Cn×p Wp×p = D.

In other words,

cij uki w j =

i,j

δk dk if k p . 0 if k > p

Next, define {Si }ni=1 : H1 → K1 and {Tj }j =1 : H2 → K2 by p

Si

=

n

uik Sk ,

k=1

so Si =

n

k=1 uki Sk

and Tj =

p

=1 w j Tj

Tj

=

p =1

for all i and j .

w j Tj ,

3424


Since U and W are unitary, it follows that D = C 1 and that the maps {Si }ni=1 are p linearly independent, as are the maps {Tj }j =1 . We observe that for any A11 ∈ B(H1 ) and A22 ∈ B(H2 ), n

Si A11 Si∗ =

i=1

n

∗ Si A11 Si

p

and

Tj A22 Tj∗ =

j =1

i=1

p

∗ Tj A22 Tj .

j =1

Finally, for any A12 ∈ B(H2 , H1 ), we use our above computations to find that i,j

∗ ∗ cij uki w j Sk A12 T = cij uki w j Sk A12 T

cij Si A12 Tj∗ =

i,j,k,

=

(kp),

∗ cij uki w j Sk A12 T

(k>p),

i,j

+

=

k,

∗ T

cij uki w j Sk A

i,j

i,j

∗ ∗ Tk + 0 = dkk Sk A Tk . p

dkk Sk A12

k=1

kp

We have shown that n S A11 (Si )∗ L(A) = p i=1 i ∗ i=1 dii Ti A21 (Si )

p

∗ i=1 dii Si A12 (Ti ) p ∗ i=1 Ti A22 (Ti )

for all A=

A11 A21

A12 A22

∈ B(H1 ⊕ H2 ).

For each i = 1, . . . , p, define Zi : H1 ⊕ H2 → K1 ⊕ K2 by Zi =

dii Si 0

0 Ti

,

so L(A) =

p i=1

Zi AZi∗

p (1 − |dii |2 )Si A11 Si∗ + 0 i=1

0 0

n Si A11 Si∗ + 0 i=p+1

0 . 0

Since D 1, the line above shows that L is the sum of two normal completely positive maps and is thus normal and completely positive. Therefore, γ is a corner from α to β. If, on the other hand, n < p, then the same argument we just used shows that γ ∗ is a corner from β to α, which is equivalent to showing that γ is a corner from α to β. For the forward direction, suppose that γ is a corner from α to β, so the map Υ : B(H1 ⊕ H2 ) → B(K1 ⊕ K2 ) defined by


Υ

A12 A22

A11 A21

n

∗ i=1 Si A11 Si γ ∗ (A21 )

=

p

3425

γ (A12 )

∗ j =1 Tj A22 Tj

is normal and completely positive. Therefore, for some q ∈ N ∪ {∞} and maps Yi : H1 ⊕ H2 → K1 ⊕ K2 for i = 1, 2, . . . , linearly independent over 2 (N), we have ˜ = Υ (A)

q

˜ i∗ Yi AY

i=1

for all A˜ ∈ B(H1 ⊕ H2 ). For i = 1, 2, let Ei ∈ B(H1 ⊕ H2 ) be projection onto Hi , and let Fi ∈ B(K1 ⊕ K2 ) be projection onto Ki . Since α and β are contractions we have Υ (E1 ) F1 and Υ (E2 ) F2 , so Yi Ej Yi∗ Fj for each i and j . It follows that each Yi , i = 1, . . . , q, can be written in the form Yi =

S˜i 0

0 T˜i

for some S˜i ∈ B(H1 , K K2 ). 1 ) and T˜i ∈ B(H2 , q Note that α(A11 ) = ni=1 Si A11 Si∗ = i=1 S˜i A11 S˜i∗ for all A11 ∈ B(H1 ). For each S˜i , define a completely positive map Li by Li (A) = S˜i AS˜i∗ for A ∈ B(H1 ). Since α − Li is completely positive, it follows from the work of Arveson in [1] that S˜i can be written as S˜i =

n

rij Sj

j =1

for some complex coefficients {rij }nj=1 . The same argument shows that for each T˜i we have T˜i =

p

bij Tj

j =1 p

q

for some coefficients {bij }j =1 . It now follows from linear independence of the maps {Yi }i=1 that q n + p. Let R = (rij ) ∈ Mq×n (C) and B = (bij ) ∈ Mq×p (C), and let A ∈ B(H1 ). We calculate n

Si ASi∗

=

i=1

q

S˜i AS˜i∗

i=1

=

n

=

n q i=1

j,k=1

q i=1

rij rik Sj ASk∗

j,k=1

rij rik Sj ASk∗ .

(5)

3426


q Let M = R T (R T )∗ ∈ Mn (C), so its j kth entry is mj k = i=1 rij rik . Unitarily diagonalizing M as U MU ∗ = D for some diagonal D and defining maps {Si }ni=1 by Si = nk=1 uik Sk , we see that Eq. (5) and the same linear algebra technique from the proof of the backward direction yield n

Si ASi∗ =

n

i=1

n

Si ASi∗ =

i=1

mj k Sj ASk∗ =

j,k=1

n

dii Si ASi∗ .

i=1

Therefore D = I and consequently M = I , hence R = 1. An identical argument shows that B = 1. Let A11 A12 ∈ B(H1 ⊕ H2 ) A˜ = A21 A22 be arbitrary. Let C = (cj k ) ∈ Mn×p (C) be the matrix C = (B ∗ R)T , noting that C 1. ˜ = q Yi AY ˜ ∗ yields A straightforward computation of Υ (A) i i=1 γ (A12 ) =

q

Si A12 Ti∗

=

i=1

=

q j,k

n q i=1

aij Sj A12

j =1

aij bik Sj A12 Tk∗ =

p

bik Tk∗

k=1

cj k Sj A12 Tk∗ ,

j,k

i=1

hence γ is of the form claimed.

2

4. Comparison theory for q-positive maps Just as in the general study of various classes of linear operators, it is natural to impose, and examine, an order structure for q-positive maps. If φ and ψ are q-positive maps acting on Mn (C), we say that φ q-dominates ψ (and write φ q ψ ) if φ(I + tφ)−1 − ψ(I + tψ)−1 is completely positive for all t 0. We would like to find the q-positive maps with the least complicated structure of q-subordinates. That last statement is not as simple as it seems. We might think to define a q-positive map φ to be “q-pure” if φ q ψ q 0 implies ψ = λφ for some λ ∈ [0, 1], but there exist q-positive maps φ such that for every λ ∈ (0, 1) we have φ q λφ. One such example is the Schur map φ on M2 (C) given by φ

a11 a21

a12 a22

=

a11 ( 1−i 2 )a21

( 1+i 2 )a12 a22

.

As it turns out, every q-positive map is guaranteed to have a one-parameter family of q-subordinates of a particular form: Proposition 4.1. Let φ q 0. For each s 0, let φ (s) = φ(I + sφ)−1 . Then φ (s) q 0 for all s 0. Furthermore, the set {φ (s) }s0 is a monotonically decreasing family of q-subordinates of φ, in the sense that φ (s1 ) q φ (s2 ) if s1 s2 .


3427

Proof. For all s 0 and t 0, we have −1 −1 = φ(I + sφ)−1 I + tφ(I + sφ)−1 φ (s) I + tφ (s) −1 = φ I + tφ(I + sφ) (I + sφ) −1 = φ I + (s + t)φ , which is completely positive by q-positivity of φ. Therefore, φ (s) q 0 for all s 0. To prove that φ (s1 ) q φ (s2 ) if s1 s2 , we let t 0 be arbitrary and examine the map −1 −1 Φ := φ (s1 ) I + tφ (s1 ) − φ (s2 ) I + tφ (s2 ) . Letting t1 = s1 + t and t2 = s2 + t, we make the following observations: −1 φ (sj ) I + tφ (sj ) = φ (tj ) for j = 1, 2, φ (t1 ) − φ (t2 ) = (I + t2 φ)−1 (I + t2 φ)φ − φ(I + t1 φ) (I + t1 φ)−1 .

(6) (7)

Eqs. (6) and (7) give us Φ = (I + t2 φ)−1 (I + t2 φ)φ − φ(I + t1 φ) (I + t1 φ)−1 = (I + t2 φ)−1 (t2 − t1 )φ 2 (I + t1 φ)−1 = (t2 − t1 ) φ(I + t2 φ)−1 φ(I + t1 φ)−1 . The last line is a non-negative multiple of a composition of completely positive maps and is thus completely positive. We conclude that φ (s1 ) q φ (s2 ) . 2 We now have the correct notion of what it means to be q-pure: Definition 4.2. Let φ : Mn (C) → Mn (C) be unital and q-positive. We say that φ is q-pure if its set of q-subordinates is precisely {0} ∪ {φ (s) }s0 . Lemma 4.3. Let ν be a normalized unbounded boundary weight over L2 (0, ∞) of the form ν

I − Λ(1)B I − Λ(1) = (f, Bf ).

Let φ : Mn (C) → Mn (C) be a q-positive contraction such that νt (I )φ(I + νt (Λ(1))φ)−1 1 for all t > 0, and let α be the CP-flow derived from the boundary weight double (φ, ν), with boundary generalized representation π = {πt# }t>0 . Let β be any CP-flow over Cn , with generalized boundary representation ξ # = {ξt# }t>0 and boundary weight map ρ → η(ρ). Then α β if and only if β is induced by the boundary weight double (ψ, ν), where ψ : Mn (C) → Mn (C) is a q-positive map satisfying φ q ψ.

3428


Proof. As before, for each t > 0 we let st = νt (Λ(1)). Assume the hypotheses of the backward direction. Then ξt# = ψ(I + st ψ)−1 Ωνt , and the direction now follows from Theorem 2.4 since the line below is completely positive for all t > 0: πt# − ξt# = φ(I + st φ)−1 − ψ(I + st ψ)−1 Ωνt . Now assume the hypotheses of the forward direction. Recall that by construction of ν, the set {st }t>0 is decreasing. If st > 0 for all t > 0 we define P = ∞. Otherwise, we define P to be the smallest positive number such that sP = 0. Fix any t0 ∈ (0, P ). Notationally, write each g ∈ H := Cn ⊗ L2 (0, ∞) in its components as g(x) = (g1 (x), . . . , gn (x)), and write ft0 for the function Vt0 Vt∗0 f ∈ L2 (0, ∞), where Vt0 is the right shift t0 units on L2 (0, ∞). Let Ut0 be the right shift t0 units on H. Under our identifications, Ut0 Ut∗0 is the diagonal matrix in Mn (B(L2 (0, ∞))) with iith entries Vt0 Vt∗0 . Define S : H → Cn by Sg = (ft0 , g1 ), . . . , (ft0 , gn ) , noting that Ωνt0 (A) = SAS ∗ for all A ∈ B(H ). Since φ(I + st0 φ)−1 is completely positive, ∗ we know it has the form φ(I + st0 φ)−1 (M) = m i=1 Ri MRi for some R1 , . . . , Rm ∈ Mn (C). Therefore, πt#0 (A) =

m −1 Ωνt0 (A) = φ(I + st0 φ) Ri SAS ∗ Ri∗ . i=1

The map ξt#0 is a subordinate of πt#0 , so from Arveson’s work in metric operator spaces in [1], we know that ξt#0 has the form ξt#0 (A) =

m

cij Ri SAS ∗ Rj∗ ,

i,j =1

for some complex numbers {cij }. Let Lt0 be the map Lt0 (M) = i,j cij Ri MRj∗ , noting that ξt#0 (A) = Lt0 (SAS ∗ ) = Lt0 (Ωνt0 (A)) for all A ∈ B(H ). Defining ψt0 : Mn (C) → Mn (C) by ψt0 = (I − ξt#0 Λ)−1 Lt0 , we find that for arbitrary A ∈ B(H ) and A´ ∈ Mn (C), −1 # ξt0 (A) ηt0 (ρ)(A) = ξˆt0 (I − Λˆ ξˆt0 )−1 (ρ)(A) = ρ I − ξt#0 Λ −1 = ρ I − ξt#0 Λ Lt0 Ωνt0 (A) = ρ ψt0 (Ωνt0 A)

(8)

´ ´ = ηt0 (ρ) Λ(A) ´ = ρ ψt0 Ωνt Λ(A) ´ , ˆ t0 (ρ)(A) = st0 ρ ψt0 (A) Λη 0

(9)

and

ˆ t0 = st0 ψˆ t0 . so Λη ˆ t0 )−1 , we find Using formulas (8) and (9) and the fact that ξˆt0 = ηt0 (I + Λη


3429

ˆ t0 )−1 (ρ) = (I + Λη ˆ t0 )−1 (ρ) (ψt0 Ωνt ) ρ ξt#0 = ξˆt0 (ρ) = ηt0 (I + Λη 0 = (I + st0 ψˆ t0 )−1 (ρ) (ψt0 Ωνt0 ) = ρ (I + st0 ψt0 )−1 ψt0 Ωνt0 = ρ ψt0 (I + st0 ψt0 )−1 Ωνt0 for all ρ ∈ Mn (C)∗ , hence ξt#0 = ψt0 (I + st0 ψt0 )−1 Ωνt0 . We now show that the maps {ψt }t>0 are constant on the interval (0, P ). Let t ∈ [t0 , P ) be arbitrary. For each A´ = (aij ) ∈ Mn (C), let A ∈ B(H ) be the matrix with ij th entry (aij /νt (I ))Vt Vt∗ . Let ρ ∈ Mn (C)∗ . Straightforward computations using formula (2) yield Ωt0 (A) = Ωt (A) = A´ and ηt0 (ρ)(A) = ηt (ρ)(A). Combining these equalities gives us ´ = ρ ψt0 Ωνt (A) = ηt0 (ρ)(A) ρ ψt0 (A) 0 ´ . = ηt (ρ)(A) = ρ ψt Ωνt (A) = ρ ψt (A) Since the above formula holds for every A´ ∈ Mn (C) and ρ ∈ Mn (C)∗ , we have ψt0 = ψt . But both t0 ∈ (0, P ) and t ∈ [t0 , P ) were chosen arbitrarily, so the previous sentence shows that ψt = ψt0 for all t ∈ (0, P ). Letting ψ = ψt0 , we have ξt# = ψ(I + st ψ)−1 Ωνt

(10)

for all t ∈ (0, P ). Defining κt as in the proof of Proposition 3.2, we observe that ψ(I + st ψ)−1 = ξt# κt for all t ∈ (0, P ), where the right hand side is completely positive by hypothesis. Since every t ∈ (0, ∞) can be written as t = st for some t ∈ (0, P ), it follows that ψ(I + tψ)−1 is completely positive for all t > 0. Furthermore, ψ(I + st ψ)−1 → ψ in norm as t → ∞, hence ψ q 0. Similarly, since πt# − ξt# is completely positive for all t > 0 by assumption, it follows from our formula φ(I + st φ)−1 − ψ(I + st ψ)−1 = πt# − ξt# κt that φ(I + st φ)−1 − ψ(I + st ψ)−1 is completely positive for all t > 0, and so its norm limit (as t → ∞) φ − ψ is completely positive. Therefore, φ q ψ . Finally, since the CP-flow β is entirely determined by its generalized boundary representation ξ # , which itself is determined by any sequence {ξt#n } with tn tending to 0 (see the remarks preceding Theorem 4.29 of [8]), it follows from (10) that β is induced by the boundary weight double (ψ, ν). 2 In a manner analogous to that used by Powers in [9] and [8], we define the terms q-corner and hyper maximal q-corner: Definition 4.4. Let φ : Mn (C) → Mn (C) and ψ : Mk (C) → Mk (C) be q-positive maps. A corner γ : Mn×k (C) → Mn×k (C) from φ to ψ is said to be a q-corner from φ to ψ if the map Υ

An×n Ck×n

Bn×k Dk×k

=

φ(An×n ) γ ∗ (Ck×n )

γ (Bn×k ) ψ(Dk×k )

is q-positive. A q-corner γ is called hyper maximal if, whenever

3430


Υ q Υ =

φ γ∗

γ ψ

q 0,

we have Υ = Υ . Proposition 4.5. For any q-positive φ : Mn (C) → Mn (C) and unitary U ∈ Mn (C), define a map φU by φU (A) = U ∗ φ U AU ∗ U. 1. The map φU is q-positive, and there is an order isomorphism between q-positive maps β such that φ q β and q-positive maps βU such that φU βU . In particular, φ is q-pure if and only if φU is q-pure. 2. If φ is unital and q-pure, then there is a hyper maximal q-corner from φ to φU . Proof. To prove the first assertion, we define a completely positive map ζ on Mn (C) by ζ (A) = U ∗ AU , noting that ζ −1 is also completely positive. For every t 0 and A ∈ Mn (C), we find that (I + tφU )−1 (A) = U ∗ (I + tφ)−1 (U AU ∗ )U and φU (I + tφU )−1 (A) = U ∗ φ U U ∗ (I + tφ)−1 U AU ∗ U U ∗ U = U ∗ φ(I + tφ)−1 U AU ∗ U = ζ ◦ φ(I + tφ)−1 ◦ ζ −1 (A),

(11)

so φU q 0. Given any q-positive map β such that φ q β, define βU by βU (A) = U ∗ β(U AU ∗ )U . Then βU is q-positive by (11), and for each t 0 we have φU (I + tφU )−1 − βU (I + tβU )−1 = ζ ◦ φ(I + tφ)−1 − β(I + tβ)−1 ◦ ζ −1 , hence φU q βU . Of course, since φ = (φU )U ∗ , the argument just used gives an identical correspondence between q-subordinates α of φU and q-subordinates αU ∗ of φ. Our first assertion now follows. To prove the second statement, we define γ : Mn (C) → Mn (C) by γ (A) = φ(AU ∗ )U . By Lemma 3.5, γ is a corner from φ to φU , so the map Θ

A11 A21

A12 A22

=

φ(A11 ) γ ∗ (A21 )

γ (A12 ) φU (A22 )

is completely positive. We calculate γ (I + tγ )−1 (A) = φ(I + tφ)−1 (AU ∗ )U , so for each t 0 and A˜ = (Aij ) ∈ M2n (C), we have ˜ = Θ(I + tΘ)−1 (A)

φ(I + tφ)−1 (A11 ) U ∗ φ(I + tφ)−1 (U A21 )

φ(I + tφ)−1 (A12 U ∗ )U φU (I + tφU )−1 (A22 )

.

This shows that γ (I + tγ )−1 is a corner from φ(I + tφ)−1 to φU (I + tφU )−1 for all t 0, so γ is a q-corner. Finally, if


Θ

A11 A21

A12 A22

=

α(A11 ) γ ∗ (A21 )

γ (A12 ) β(A22 )

3431

is q-positive and Θ q Θ , then since φ and φU are q-pure we have α = φ(I + tφ)−1 for some t 0 and β = φU (I + sφU )−1 for some s 0. Complete positivity of Θ implies that Θ

I U∗

U I

=

1 1+t I U∗

U 1 1+s I

so s = t = 0 and Θ = Θ , hence γ is hyper maximal.

0,

2

We have arrived at the key result of the section, which tells us that, under certain conditions, the problem of determining whether two E0 -semigroups induced by boundary weight doubles are cocycle conjugate can be reduced to the much simpler problem of finding hyper maximal q-corners between q-positive maps: Proposition√4.6. Let ν be√a normalized unbounded boundary weight over L2 (0, ∞) which has the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ). Let φ and ψ be unital q-positive maps on Mn (C) and Mk (C), respectively, and induce CP-flows α and β through the boundary weight doubles (φ, ν) and (ψ, ν). Then α d and β d are cocycle conjugate if and only if there is a hyper maximal q-corner from φ to ψ . Proof. Let N = n + k. For the forward direction, suppose α d and β d are cocycle conjugate. Since α d and β d are of type II0 , we know from Theorem 2.6 that there is a hyper maximal flow corner σ from α to β, with associated CP-flow Θ=

α σ∗

σ β

.

Let Π # = {Πt# }, π # = {πt# }, and ξ # = {ξt# } be the generalized boundary representations for Θ, α, and β, respectively. Define st = νt (Λ(1)) for all positive t, so for each t > 0 there is some Zt such that Πt# =

πt# Z∗t

Zt ξt#

=

φ(I + st φ)−1 ◦ Ωνt ,n×n Z∗t

Zt ψ(I + st ψ)−1 ◦ Ωνt ,k×k

.

Since each Zt is a corner from φ(I + st φ)−1 ◦ Ωνt ,n×n to ψ(I + st φ)−1 ◦ Ωνt ,k×k , we have Zt = Lt ◦ Ωνt ,n×k for some Lt . Define Bt for each t > 0 by Bt =

φ(I + st φ)−1 L∗t

Lt ψ(I + st ψ)−1

.

We observe that Πt# = Bt ◦ Ωνt ,N ×N for all t > 0, whereby the same argument given in the proof of Lemma 4.3 shows that each Bt has the form Bt = Wt (I + st Wt )−1 for some

3432


Wt : Mn (C) → Mn (C) and that the maps Wt are independent of t. Therefore, for some γ : Mn×k (C) → Mn×k (C), we have Zt = γ (I + st γ )−1 ◦ Ωνt ,n×k for all t > 0. Define κt,N ×N : MN (C) → B(H ) as in Proposition 3.2. Letting ϑ=

φ γ∗

γ ψ

,

we observe for each t that ϑ(I + st ϑ)−1 = Πt# ◦ κt,N ×N is the composition of completely positive maps and is thus completely positive, hence ϑ q 0. Suppose that for some map ϑ we have

ϑ q ϑ =

φ γ∗

γ ψ

q 0.

As in Proposition 3.2, the boundary weight map ρ ∈ MN (C)∗ → L(ρ) defined by L(ρ)(C) = ρ(ϑ (Ων,N×N (C))) induces a CP-flow Θ over CN , where for some CP-flows α over Cn and β over Ck , we have

Θ =

α σ∗

σ β

.

By Lemma 4.3, we have Θ Θ since ϑ q ϑ . But Θ is a hyper maximal flow corner, so Θ = Θ . Our formulas for the generalized boundary representations imply that φ(I + tφ)−1 = φ (I + tφ )−1 and ψ(I + tψ)−1 = ψ (I + tψ )−1 for all t > 0, hence φ = φ and ψ = ψ . We conclude that γ is a hyper maximal q-corner. For the backward direction, suppose there is a hyper maximal q-corner γ from φ to ψ , so the map Υ : MN (C) → MN (C) defined by Υ

An×n Ck×n

Bn×k Dk×k

=

φ(An×n ) γ ∗ (Ck×n )

γ (Bn×k ) ψ(Dk×k )

is q-positive. By Proposition 3.2, the boundary weight map ρ ∈ MN (C)∗ → Ξ (ρ) defined by Ξ (ρ)(A) = ρ Υ Ων,N×N (A) is the boundary weight map of a CP-flow θ over CN , where for some Σ we have θ=

α Σ∗

Σ β

.

Let θ =

α Σ∗

Σ β


3433

be any CP-flow such that θ θ . Letting Zt = γ (I + st γ )−1 ◦ Ωνt ,n×k for all t > 0, we see the generalized boundary representations Π # = {Πt# } and Π = {Πt } for θ and θ satisfy Πt# =

πt# Zt∗

Zt ξt#

Πt =

πt Zt∗

Zt ξt

for all t > 0. Lemma 4.3 implies that for some φ and ψ with φ q φ q 0 and ψ q ψ q 0 we have πt = φ (I + st φ )−1 ◦ Ωνt ,n×n and ξt = ψ (I + st ψ )−1 ◦ Ωνt ,k×k for all t > 0. Defining Υ : MN (C) → MN (C) by Υ

An×n Ck×n

Bn×k Dk×k

=

φ (An×n ) γ ∗ (Ck×n )

γ (Bn×k ) , ψ (Dk×k )

we observe that Πt ◦ κνt ,N ×N = Υ (I + st Υ )−1 for all st > 0, hence γ is a q-corner from φ to ψ . Hyper maximality of γ implies φ = φ and ψ = ψ , thus θ = θ . Therefore, σ is a hyper maximal flow corner from α to β, so α d and β d are cocycle conjugate by Theorem 2.6. 2 5. E0 -semigroups obtained from rank one unital q-pure maps Any unital linear map φ : Mn (C) → Mn (C) of rank one is of the form φ(A) = τ (A)I for some linear functional τ . If φ is positive, then τ is positive and τ (I ) = 1, so τ is a state. On the other hand, given any state ρ, the map φ defined by φ(A) = ρ(A)I is unital and completely positive. Furthermore, φ is q-positive since φ(I + tφ)−1 = (1/(1 + t))φ for all t > 0. The rank one unital q-positive maps are therefore precisely the maps A → ρ(A)I for states ρ. The goal of this section is to determine when such maps are q-pure, and then to determine when the E0 -semigroups induced by (φ, ν) and (ψ, ν) are cocycle conjugate, where φ and ψ are√rank one unital √ q-pure maps and ν is a normalized unbounded boundary weight of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ) (Theorem 5.4). We also obtain a partial result for comparing E0 -semigroups induced by (φ, ν) and (ψ, μ) for rank one unital q-pure maps φ and ψ and any normalized unbounded boundary weights ν and μ over L2 (0, ∞) (Corollary 5.5). We begin with a lemma: Lemma 5.1. Let ρ be a faithful state on Mn (C), and define a unital q-positive map φ : Mn (C) → Mn (C) by φ(A) = ρ(A)I . For any non-zero positive linear functional τ on Mn (C) and non-zero positive operator C ∈ Mn (C), define ψτ,C : Mn (C) → Mn (C) by ψτ,C (A) = τ (A)C. Then ψτ,C is q-positive, and φ q ψτ,C if and only if ψτ,C = λφ for some λ ∈ (0, 1]. Proof. Note that for all A ∈ Mn (C) and t 0, we have (I + tψτ,C )−1 (A) = A − tτ (A)/(1 + tτ (C))C, so ψτ,C (I + tψτ,C )−1 (A) =

τ (A) C, 1 + tτ (C)

(12)

hence ψτ,C is q-positive. It follows from (12) that φ(I + tφ)−1 (A) = (ρ(A)/(1 + t))I for all A ∈ Mn (C). Assume the hypotheses of the forward direction. Since φ q ψτ,C , we have

3434


τ (A)C ρ(A)I 1+t 1 + tτ (C)

(13)

for all t 0 and A 0. This is impossible if τ (C) = 0, so we may assume τ (C) = 0. Letting t → ∞ in (13) yields ρ(A)I

τ (A)C τ (C)

(14)

for all A 0. Setting A = C in (14), we see ρ(C)I − C 0, yet ρ ρ(C)I − C = ρ(C) − ρ(C) = 0, hence C = ρ(C)I by faithfulness of ρ. Rewriting (14) as ρ(A)I

τ (A) τ (A) ρ(C)I = I τ (ρ(C)I ) τ

for all A 0, we see that ρ − τ/τ is a positive linear functional. Therefore, ρ − τ = ρ(I ) − τ (I ) = 1 − 1 = 0, τ τ hence τ = τ ρ. Setting t = 0 and A = I in (13) gives us τ = τ (I ) = λ/ρ(C) for some λ ∈ (0, 1]. Therefore, ψτ,C (A) = τ (A)C = τ ρ(A)ρ(C)I = λρ(A)I = λφ(A) for all A ∈ Mn (C), proving the forward direction. The backward direction follows from Proposition 4.1 since λφ = φ (−1+1/λ) for every λ ∈ (0, 1]. 2 Remark. Let ψ : Mn (C) → Mn (C) be a non-zero q-positive contraction such that the maps Lψt := tψ(I + tψ)−1 satisfy Lψt < 1 for all t > 0. By compactness of the unit ball of B(Mn (C)), the maps Lψt have some norm limit as t → ∞. This limit is unique: Pick any orthonormal basis with respect to the trace inner product (A, B) = tr(A∗ B) of Mn (C), and let Mt be the n2 × n2 matrix of Lψt with respect to this basis. From the cofactor formula for (I + tψ)−1 , we know that the ij th entry of Mt is a rational function rij (t). Uniqueness of limt→∞ Lψt now follows from the fact that each rij (t) has a unique limit as t → ∞. We call this limit Lψ . Noting that tψ = Lψt (I − Lψt )−1 = Lψt + L2ψt + · · · for each t > 0, we claim that Lψ fixes a positive element T of norm one. To prove this, we first observe for each k ∈ N and t > 0 that


3435

∞ (Lψ )kn (I ) tψ = t ψ(I ) Lψt (I ) + · · · + (Lψt )k−1 (I ) + k t n=1

< (k − 1) + k

∞

(Lψ )k (I )n , t

n=1

hence 1 = lim (Lψt )k (I ) = (Lψ )k (I ). t→∞

Therefore, all elements of the sequence {Tk }k∈N defined by Tk = (Lψ )k (I ) satisfy Tk 0 and Tk = 1. Since Tk − Tk+1 = (Lψ )k (I − T1 ) 0 for all k, the sequence {Tk }k∈N is monotonically decreasing and therefore has a positive norm limit T with T = 1. Finally, Lψ fixes T since Lψ (T ) = limk→∞ Lk+1 ψ (I ) = T . The information at hand suffices in showing that a large class of maps is q-pure: Proposition 5.2. Let ρ be a state on Mn (C), and define a q-positive map φ on Mn (C) by φ(A) = ρ(A)I . Then φ is q-pure if and only if ρ is faithful. Proof. For the forward direction, we prove the contrapositive. If ρ is not faithful, then for some k < n and mutually orthogonal vectors f1 , . . . , fk with ki=1 fi 2 = 1, we have ρ(A) = k i=1 (fi , Afi ) for all A ∈ Mn (C). Let P be the projection onto the k-dimensional subspace of Cn spanned by the vectors f1 , . . . , fk , and define a q-positive map ψ : Mn (C) → Mn (C) by ψ(A) = ρ(A)P . For each t 0 and A ∈ Mn (C), we find (t) φ − ψ (t) (A) =

1 1 φ(A) − ψ(A) = ρ(A)(I − P ), 1+t 1+t

so φ q ψ . Obviously, ψ = φ (s) for any s 0, so φ is not q-pure. To prove the backward direction, suppose φ q ψ q 0 for some ψ = 0, and form Lψ and Lφ . Since Lφt = (t/(1 + t))φ for each t > 0, we have Lφ = φ. The map Lφt − Lψt is completely positive for all t, so by taking its limit as t → ∞ we see φ − Lψ is completely positive. By the remarks preceding this proposition, we know that Lψ fixes a positive T with T = 1. But (φ − Lψ )(T ) = ρ(T )I − T 0, so ρ(T ) = 1, hence T = I by faithfulness of ρ. By complete positivity of φ − Lψ , we have φ − Lψ = φ(I ) − Lψ (I ) = 0, so φ = Lψ . Therefore, I I I +ψ = lim φ + ψ − Lψt +ψ 0 = lim (φ − Lψt ) t→∞ t→∞ t t t φ I φ = lim + φψ − tψ(I + tψ)−1 +ψ = lim + φψ − ψ t→∞ t t→∞ t t = φψ − ψ.

(15)

Letting τ be the positive linear functional τ = ρ ◦ ψ , we conclude from (15) that ψ(A) = ρ(ψ(A))I = τ (A)I for all A ∈ Mn (C). Lemma 5.1 implies that ψ = λφ = φ (−1+1/λ) for some λ ∈ (0, 1]. 2

3436


To prove the main result of the section, we need the following: Lemma 5.3. Let φ : Mn (C) → Mn (C) and ψ : Mk (C) → Mk (C) be rank one unital q-pure maps, and let ν and μ be normalized unbounded boundary weights over L2 (0, ∞). If the boundary weight doubles (φ, ν) and (ψ, μ) induce cocycle conjugate E0 -semigroups α d and β d , then there is a corner γ from φ to ψ such that γ = 1. Proof. By construction, α d and β d are type II0 E0 -semigroups. If they are cocycle conjugate, then by Theorem 2.6, there is a hyper maximal flow corner σ from α to β with associated CPflow α σ . Θ= σ∗ β Let H1 = Cn ⊗ L2 (0, ∞), let H2 = Ck ⊗ L2 (0, ∞), and let H = Cn+k ⊗ L2 (0, ∞). Write the boundary representation Π = {Πt# } for Θ as Πt#

=

1 1+νt (Λ(1)) φ ◦ Ωνt ,n×n Z∗t

Zt 1 1+μt (Λ(1)) ψ

◦ Ωμt ,k×k

for some maps {Zt }t>0 . Let ρ11 → ω(ρ11 ) and ρ22 → η(ρ22 ) denote the boundary weight maps for α and β, respectively. Let ρ → Ξ (ρ) be the boundary weight map for Θ, so for some map ρ12 → (ρ12 ) from Mn×k (C)∗ to weights on B(H2 , H1 ) we have Ξ

ρ11 ρ21

ρ12 ρ22

=

ω(ρ11 ) ∗ (ρ21 )

(ρ12 ) . η(ρ22 )

Denote by Ut the right shift t units on H . For every A = (Aij ) ∈ t>0 Ut B(H )Ut∗ and bounded family of functionals {ρ(t) = (ρij (t))}t>0 in Mn+k (C)∗ , we observe that the argument used in Corollary 3.3 to show that π0# = ξ0# = 0 implies ˆ t )−1 ρ11 (t) (A11 ) = lim ηt (I + Λη ˆ t )−1 ρ22 (t) (A22 ) = 0, lim ωt (I + Λω

t→0

t→0

so by complete positivity of the generalized boundary representation, we have ˆ t )−1 ρ12 (t) (A12 ) = 0. lim t (I + Λ

t→0

(16)

We claim that ρ12 → (ρ12 ) is unbounded. If is bounded, then for each ρ12 ∈ Mn×k (C)∗ , ˆ t )(ρ12 ) is bounded, and it follows from (16) that the family ρ12 (t) := (I + Λ lim t (ρ12 )(A12 ) = 0

t→0

(17)

for each A12 ∈ t>0 Wt B(H2 , H1 )Xt∗ , where Wt and Xt are the right shift t units on H1 and H2 , respectively. Let A12 ∈ t>0 Wt B(H2 , H1 )Xt∗ , so A12 = Ws BXs∗ for some s > 0 and B ∈ B(H2 , H1 ). For all b < s, we have


3437

b (ρ12 )(A12 ) = b (ρ12 ) Ws BXs∗ = (ρ12 ) Wb Wb∗ Ws BXs∗ Xb Xb∗ ∗ = (ρ12 ) Wb Ws−b BXs−b Xb∗ = (ρ12 ) Ws BXs∗ = (ρ12 )(A12 ). Therefore, by Eq. (17) we have (ρ12 )(A12 ) = 0. Let A ∈ B(H2 , H1 ), ρ12 ∈ Mn×k (C)∗ , and t > 0 be arbitrary. From above we have t (ρ12 )(A) = (ρ12 ) Wt AXt∗ = 0, hence t ≡ 0 for all t > 0. We conclude from uniqueness of the generalized boundary representation that ρ12 → (ρ12 ) is the zero map. The boundary weight map ρ → Ξ (ρ) defined by Ξ

ρ11 ρ21

ρ12 ρ22

=

0 0

ω(ρ11 ) 0

gives rise to the CP-flow Θ =

α σ∗

σ β

,

where β is the non-unital CP-flow βt (A22 ) = Xt A22 Xt∗ . Trivially, Θ = Θ and Θ Θ , contradicting hyper maximality of σ . Therefore, the map ρ12 → (ρ12 ) is unbounded. Since Πt# is a contraction for every t > 0, so is Zt , hence the map Zt ◦ Λ : Mn×k (C) → Mn×k (C) is a contraction for each t > 0. A compactness argument shows that Ztn ◦ Λ has a norm limit γ for some sequence {tn } tending to zero, where γ 1. From unboundedness of and the formula t = Zˆ t (I − Λˆ Zˆ t )−1 for all t > 0, it follows that I − γ is not invertible, so γ 1, hence γ = 1. We claim that γ is a corner from φ to ψ. Indeed, for the family of completely positive maps {Rt }t>0 defined by Rt = Πt# ◦ Λ, we have

lim Rtn = lim

n→∞

n→∞

νtn (Λ(1)) 1+νtn (Λ(1)) φ

Ztn ◦ Λ

(Ztn ◦ Λ)∗

μtn (Λ(1)) 1+μtn (Λ(1)) ψ

=

φ γ∗

γ ψ

.

2

If ν is a √normalized unbounded boundary weight over L2 (0, ∞) of the form √ ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ) and if φ : Mn (C) → Mn (C) is unital and q-pure, we know from Propositions 4.5 and 4.6 that the condition ψ = φU is sufficient for the boundary weight doubles (φ, ν) and (ψ, ν) to induce cocycle conjugate E0 -semigroups. In the case that φ is a rank one unital q-pure map, this condition is also necessary: Theorem 5.4. Let φ1 : Mn (C) → Mn (C) and φ2 : Mk (C) → Mk (C) be rank one unital 2 q-pure √ maps. Let √ ν be a normalized unbounded boundary weight over L (0, ∞) of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ). Then the boundary weight doubles (φ1 , ν) and (φ2 , ν) induce cocycle conjugate E0 semigroups if and only if n = k and φ2 = (φ1 )U for some unitary U ∈ Mn (C).

3438


Proof. The backward direction follows immediately from Propositions 4.5 and 4.6. Assume the hypotheses of the forward direction. Since φ1 and φ2 are rank one, unital, and q-pure, there exist faithful states ρ1 on Mn (C) and ρ2 on Mk (C) such that φ1 (M) = ρ1 (M)In×n and φ2 (B) = ρ2 (B)Ik×k for all M ∈ Mn (C), B ∈ Mk (C). By Lemma 5.3, there is a corner γ from φ1 to φ2 such that γ = 1. Therefore, for some A0 ∈ Mn×k (C) of norm one and unit vectors f0 ∈ Cn and g0 ∈ Ck , we have |(f0 , γ (A0 )g0 )| = 1. Define ω ∈ Mn×k (C)∗ by ω(A) = (f0 , γ (A)g0 ), noting that ω = |ω(A0 )| = 1. We claim that the map ψ˜ : Mn+k (C) → M2 (C) defined by ψ˜

A11 A21

A12 A22

=

ρ1 (A11 ) ω∗ (A21 )

ω(A12 ) ρ2 (A22 )

is completely positive. To see this, let {F˜i } i=1 be arbitrary vectors in C2 , writing each F˜i as F˜i =

λ1i λ2i

for some complex numbers {λ1i } i=1 and {λ2i } i=1 . Since the map ψ : Mn+k (C) → Mn+k (C) defined by ψ

A11 A21

A12 A22

=

ρ1 (A11 )I γ ∗ (A21 )

γ (A12 ) ρ2 (A22 )I

is completely positive by assumption, we know that for any A1 , . . . , A ∈ Mn+k (C) and the vectors λ1i f0 ∈ Cn+k , i = 1, . . . , k, Fi = λ2i g0 we have Fi , ψ A∗i Aj Fj 0. i,j =1

However, for each i and j we find that Fi , ψ A∗i Aj Fj Cn+k = λ1i λ1j ρ1 A∗i Aj 11 + λ1i λ2j ω A∗i Aj 12 ∗ + λ2i λj 1 ω A∗i Aj 21 + λ2i λ2j ρ2 A∗i Aj 22 = F˜i , ψ˜ A∗i Aj F˜j C2 . Therefore, for all ∈ N, A1 , . . . , A ∈ Mn+k (C), and F˜1 , . . . , F˜ ∈ C2 , we have ˜ ˜ ∗ ˜ ˜ i,j =1 (Fi , ψ(Ai Aj )Fj ) 0, so ψ : M2n (C) → M2 (C) is completely positive. Since ρ1 and ρ2 are positive linear functionals (hence completely positive maps), ω is a corner from ρ1 to ρ2 . monotonically By faithfulness of ρ1 and ρ2 , there exist increasing sequences of strictly positive numbers {λi }ni=1 and {μj }kj =1 with ni=1 λ2i = kj =1 μ2j = 1, along with orthonor mal sets of vectors {fi }ni=1 and {gj }kj =1 , such that ρ1 (M) = ni=1 λ2i (fi , Mfi ) and ρ2 (B) =


3439

k

for all M ∈ Mn (C), B ∈ Mk (C). Given A ∈ Mn×k (C), let A˜ be the ma˜ = A. Let Dλ and Dμ be the diagonal trix whose j ith entry is (fi , Agj ), observing that A matrices whose iith entries are λi and μi , respectively, for all i, and let Dλ2 and Dμ2 be the diagonal matrices whose iith entries are λ2i and μ2i , respectively, observing that Dλ2 = (Dλ )2 and Dμ2 = (Dμ )2 . By Proposition 3.5, ω has the form 2 j =1 μj (gj , Bgj )

ω(A) =

˜ λ ) = tr CDμ Dλ A˜ ∗ ∗ cij λi μj (fi , Agj ) = tr(CDμ AD

i,j

for some C = (cij ) ∈ Mn×k (C) such that C 1. By the Cauchy–Schwartz inequality for the inner product (B, A) = tr(AB ∗ ) on Mn×k (C), we have 2 ∗ 2 2 1 = ω(A0 ) = tr CDμ Dλ A˜ ∗0 = CDμ , Dλ A˜ ∗0 (CDμ , CDμ ) Dλ A˜ ∗0 , Dλ A˜ ∗0 = tr Dμ C ∗ CDμ tr Dλ A˜ ∗0 A˜ 0 Dλ tr(Dμ2 Ik )tr(Dλ2 In ) 1 ∗ 1 = 1.

(18)

Since equality holds in all the inequalities above, we have mCDμ = Dλ A˜ ∗0 for some m ∈ C. It follows from (18) that |m| = 1 since CDμ tr = Dλ A˜ ∗0 tr = 1. Furthermore, since equality holds in (18) and the trace map is faithful, we have C ∗ C = Ik and A˜ ∗0 A˜ 0 = In . But C ∈ Mn×k (C) and A˜ ∗0 ∈ Mn×k (C), so n = k, hence C and A˜ 0 are unitary. Writing Dλ = mCDμ A˜ 0 = (mC A˜ 0 )(A˜ ∗0 Dμ A˜ 0 ), we observe that mC A˜ 0 is unitary and A˜ ∗0 Dμ A˜ 0 is positive. Uniqueness of the right Polar Decomposition for the invertible matrix Dλ implies Dλ = A˜ ∗0 Dμ A˜ 0 . Since the diagonal entries in Dλ and Dμ are listed in increasing order, it follows that Dλ = Dμ , hence ρ2 is of the form ρ2 (M) = ni=1 λ2i (gi , Mgi ). Defining a unitary U ∈ Mn (C) by letting Ugi = fi for all i and extending linearly, we observe that ρ2 (M) =

n i=1

n λ2i U ∗ fi , MU ∗ fi = λ2i fi , U MU ∗ fi = ρ1 U MU ∗ i=1

for all M ∈ Mn (C). In other words, φ2 = (φ1 )U .

2

In [9], Powers constructed E0 -semigroups using boundary weights over L2 (0, ∞). It is routine to check that in our notation, these are the E0 -semigroups arising from the boundary weight doubles (ıC , η), where ıC is the identity map on C and η is any boundary weight over L2 (0, ∞). Corollary 5.5. Let φ : Mn (C) → Mn (C) and ψ : Mk (C) → Mk (C) be unital rank one q-pure maps, and let ν and η be normalized unbounded boundary weights over L2 (0, ∞). Denote by α d

3440


and β d the Bhat minimal dilations of the CP-flows induced by the boundary weight doubles (φ, ν) and (ψ, μ), respectively. If n = k, then α d and β d are not cocycle conjugate. In particular, if n = 1, then α d is not cocycle conjugate to the E0 -semigroup induced by (ıC , μ). Proof. From the proof of Theorem 5.4, we know that every corner γ from φ to ψ satisfies γ < 1 since n = k. The result now follows from Lemma 5.3. 2 6. Invertible unital q-pure maps Now that we have classified the unital q-pure maps on Mn (C) of rank one, we explore the unital q-pure maps φ which are invertible. In a stark contrast to the√rank one case, √ we find that for a given normalized unbounded boundary weight of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ) on L2 (0, ∞), the doubles (φ, ν) and (ψ, ν) always induce cocycle conjugate E0 semigroups if φ and ψ are unital invertible q-pure maps on Mn (C) and Mk (C), respectively. The following proposition gives us a bijective correspondence between invertible unital q-positive maps φ : Mn (C) → Mn (C) and unital conditionally negative maps Ψ : Mn (C) → Mn (C): Proposition 6.1. If φ : Mn (C) → Mn (C) is an invertible unital q-positive map, then φ −1 is conditionally negative. On the other hand, if Ψ : Mn (C) → Mn (C) is a unital conditionally negative map, then Ψ is invertible and Ψ −1 is q-positive. Proof. Let ψ = φ −1 . Since φ is self-adjoint, so is ψ , and the first statement of the proposition now follows from the fact that for large positive t we have −1

tφ(I + tφ)

= tψ

−1

ψ −1 ψ ψ2 −1 −1 −1 I + tψ = t (ψ + tI ) = I + =I − + − ···. t t t

To prove the second statement, let Ψ : Mn (C) → Mn (C) be any unital conditionally negative map. Since Ψ is conditionally negative, it follows from a result of Evans and Lewis in [5] that −sΨ = e−sΨ (I ) = e−s I = e−s for e−sΨ is completely positive ∞for all s 0. Therefore, e all s 0, and the integral 0 e−sΨ ds converges. Observing that (d/ds)(−e−sΨ ) = Ψ e−sΨ , we find that

∞ Ψ

e

−sΨ

0

∞

ds =

s Ψ e−sΨ ds = lim −e−sΨ 0 = I, s→∞

0

∞ so Ψ is invertible and Ψ −1 = 0 e−sψ ds. Since Ψ −1 is the integral of completely positive maps, it is completely positive. Furthermore, we find that tI + Ψ is invertible for every t > 0 and that Ψ −1 q 0, since the following holds for all t > 0: ∞ e 0

−st −sΨ

e

∞ ds = 0

−1 e−s(tI +Ψ ) ds = (tI + Ψ )−1 = Ψ −1 I + tΨ −1 .

2


3441

Examining the inverse of a unital invertible q-positive map φ is the key to finding the invertible q-subordinates of φ, as we find in the following proposition and corollary: Proposition 6.2. Let φ1 : Mn (C) → Mn (C) be an invertible unital q-positive map, and let ψ1 = φ1−1 . Suppose ψ2 : Mn (C) → Mn (C) is conditionally negative and ψ2 − ψ1 is completely positive. Then ψ2 is invertible, and φ2 := (ψ2 )−1 satisfies φ1 q φ2 q 0. Proof. Assume the hypotheses of the proposition, and let s > 0 be arbitrary. Define a function f 1 on R by f (t) = e−tsψ1 e(t−1)sψ2 . The equality below is f (1) − f (0) = 0 f (t) dt:

e

−sψ1

−e

−sψ2

1

se−tsψ1 (ψ2 − ψ1 )e(t−1)sψ2 dt.

= 0

The inside of the integral above is the composition of completely positive maps, so e−sψ1 − e−sψ2 is completely positive. This implies e−sψ1 (I ) − e−sψ2 (I ) 0, so −sψ −sψ e 2 = e 2 (I ) e−sψ1 (I ) = e−s (I ) = e−s . ∞ Now the argument given in the previous proposition shows that 0 e−sψ2 ds converges and is equal to ψ2−1 . Letting φ2 = ψ2−1 , we observe that φ1 q φ2 since the quantity below is completely positive for every t 0: −1

φ1 (I + tφ1 )

−1

− φ2 (I + tφ2 )

∞ =

e−st e−sψ1 − e−sψ2 ds.

2

0

Corollary 6.3. Let φ1 : Mn (C) → Mn (C) be an invertible unital q-positive map, and let φ2 : Mn (C) → Mn (C) be linear and invertible. Then φ1 q φ2 q 0 if and only if φ2−1 is conditionally negative and φ2−1 − φ1−1 is completely positive. Proof. The backward direction follows from Proposition 6.2. Assume the hypotheses of the forward direction and let ψ1 = φ1−1 and ψ2 = φ2−1 . Since φ2 is self-adjoint, so is ψ2 . For sufficiently large positive t we have ψ2 −1 ψ2 ψ22 + 2 − ··· tφ2 (I + tφ2 )−1 = I + =I − t t t and 2 ψ2 − ψ12 ψ23 − ψ13 −1 −1 t φ1 (I + tφ1 ) − φ2 (I + tφ2 ) = ψ2 − ψ1 + + ··· . − t t2 2

The first equation shows that φ2−1 is conditionally negative, while the second shows that φ2−1 − φ1−1 is completely positive. 2

3442


Now that we know how to find all invertible q-subordinates of an invertible unital q-positive map φ, we ask if there can be any other q-subordinates of φ. We will find that the answer is no (see Proposition 6.9). Proving this will require the use of some machinery (notably Lemma 6.8), which we now build. Definition 6.4. For every φ : Mn (C) → Mn (C) and I + (1 − )φ.

∈ [0, 1], we define a map φ by φ =

If φ is q-positive, then φ is invertible for all ∈ (0, 1]. In the lemmas that follow, we make frequent use of the fact that for all t 0 we have tφ(I + tφ)−1 = I − (I + tφ)−1 .

(19)

We present a quick consequence of (19) for all a 0 and b 0: a(I + btφ)−1 = aI − abtφ(I + btφ)−1 .

(20)

Lemma 6.5. Let φ : Mn (C) → Mn (C) be completely positive. If φ k q 0 for some monotonically decreasing sequence { k } of positive real numbers tending to 0, then φ q 0. Proof. Assume the hypotheses of the lemma. Let k be arbitrary. Since φ k q 0, we know I − (I + tφ k )−1 is completely positive for all t 0. Noting that −1 I − (I + tφ )−1 = I − (1 + t I ) + (1 − )tφ =I − and substituting t = t (1 −

k )/(1 + t k ),

1 1+t

t (1 − ) −1 I+ φ 1+t

we see

I − (I + tφ )−1 = I −

1+

1 k

1− k +t

−1 I + t φ . t k

Varying t throughout [0, ∞), we find that the above equation is completely positive for all t ∈ [0, −1 + 1/ k ). Of course, for any t ∈ [0, −1 + 1/ k ), we have t ∈ [0, −1 + 1/ ) for all k by monotonicity of the sequence { n }. Therefore, we may repeat the same argument to conclude that for any t ∈ [0, −1 + 1/ k ), the map I−

1+

1

1− +t

−1 I + t φ t

is completely positive for all k. Now fix any t > 0, so t ∈ (0, −1 + 1/ k ) for some k ∈ N. A straightforward computation shows that the sequence {cn } defined by cn = n /(1 − n + t n ) monotonically decreases to 0. From the previous paragraph, we know that the map I−

−1 1 I + t φ 1 + c t


3443

is completely positive for all k. Since cn ↓ 0 it follows that −1 I − I + t φ is completely positive. In other words, t φ(I + t φ)−1 is completely positive. Since t > 0 was chosen arbitrarily and φ is completely positive, the lemma follows. 2 Lemma 6.6. If φ : Mn (C) → Mn (C) and φ q 0, then φ q 0 for all ∈ [0, 1). Proof. Suppose that φ q 0, and let ∈ [0, 1) be arbitrary. For each t > 0, we apply formula (20) to a = 1/(1 + t ) and b = t (1 − )/(1 + t ) to find t (1 − ) −1 I+ φ 1+t t (1 − ) 1 t (1 − ) −1 I+ = 1− φ I + , φ 1+t 1+t (1 + t )2

I − (I + tφ )−1 = I −

1 1+t

where both terms on the last line are completely positive by assumption. Furthermore, φ is completely positive, hence φ q 0. 2 Corollary 6.7. Let φ : Mn (C) → Mn (C) be a completely positive map. Then φ q 0 if and only if φ q 0 for all ∈ (0, 1). Lemma 6.8. Let φ : Mn (C) → Mn (C) and ψ : Mn (C) → Mn (C) be q-positive maps. Then φ q ψ if and only if φ q ψ for all ∈ (0, 1). Proof. For any ∈ (0, 1) we have φ − ψ = (φ − ψ), so φ − ψ is completely positive if and only if φ − ψ is completely positive for all ∈ (0, 1). For all t > 0 we have −1 −1 −1 −1 = I + t ψ − ψ I + t ψ − I + t φ , t φ I + t φ

(21)

and for all t > 0 we have t φ (I + tφ )−1 − ψ (I + tψ )−1 = I − (I + tφ )−1 − I − (I + tψ )−1 −1 t (1 − ) 1 t (1 − ) −1 . I+ ψ φ = − I+ 1+t 1+t 1+t

(22)

Assume the hypotheses of the forward direction. Showing that φ q ψ for all ∈ (0, 1) is equivalent to proving that (22) is completely positive for every t ∈ (0, ∞) and ∈ (0, 1). But this follows from complete positivity of (21) since t (1 − )/(1 + t ) ∈ (0, ∞) for every ∈ (0, 1) and t ∈ (0, ∞). Now assume the hypotheses of the backward direction. Any t ∈ (0, ∞) can be written as t (1 − )/(1 + t ) for some ∈ (0, 1) and t ∈ (0, ∞), so complete positivity of (22) for all such and t implies that (21) is completely positive for all t > 0, hence φ q ψ . 2 We are now in a position to prove what is perhaps the most striking result of the section:

3444


Proposition 6.9. Let ξ : Mn (C) → Mn (C) be an invertible unital q-positive map. If φ : Mn (C) → Mn (C) is q-positive and ξ q φ, then φ is either invertible or identically zero. Proof. For every ∈ (0, 1), form ξ and φ as in Definition 6.4, and let ψ := (φ )−1 . By Lemma 6.8 we have ξ q φ for each , so ψ is conditionally negative and ψ − (ξ )−1 is completely positive by Corollary 6.3. We first examine the case when the norms ψ remain bounded as → 0. More precisely, suppose that for all sufficiently small we have ψ < r for some r > 0. By compactness of the closed unit ball of radius r in B(Mn (C)), there is a decreasing sequence { k }k∈N converging to 0 such that {ψ k }k∈N has a (bounded) norm limit ψ as k → ∞. Noting that I − φψ = φ k ψ k − φψ = (φ k − φ)(ψ k − ψ) + φ(ψ k − ψ) + (φ k − φ)ψ and then applying the triangle inequality, we find that I − φψ = φ k ψ k − φψ φ k − φψ k − ψ + φψ k − ψ + φ k − φψ for all k ∈ N. But φ and ψ are bounded maps while ψ k → ψ in norm and φ k → φ in norm, so the above equation tends to 0 as k → ∞. We conclude that φψ = I . Similarly ψφ = I , hence φ is invertible and ψ = φ −1 . If the first case does not hold, then for some decreasing sequence { k } tending to zero, the norms {ψ k }k∈N form an unbounded sequence. For each k ∈ N, we write −1

(ξ k )

(A) = sk A + Yk A + AYk∗

−

mk

Ski ASk∗i

i=1

and ψ k (A) = tk A + Zk A + AZk∗ −

k

Tki ATk∗i ,

i=1

where mk , k n2 , sk ∈ R, tk ∈ R, tr(Yk ) = tr(Zk ) = 0, tr(Ski ) = 0 and tr(Sk∗i Skj ) is non-zero if and only if i = j (i, j mk ), and tr(Tki ) = 0 and tr(Tk∗i Tkj ) is non-zero if and only if i = j (i, j k ). Since ψ k − (ξ k )−1 is completely positive for all k ∈ N, we know that for each k, there pk pk , and maps {Xki }i=1 with tr(Xki ) = 0, such that for all exist pk n2 , complex numbers {xki }i=1 A ∈ Mn (C), k ψ k − (ξ k )−1 (A) = (Xki + xki I )A(Xki + xki I )∗

p

i=1

=

pk i=1

|xki |

2

A+

pk i=1

xki Xki A + A

pk i=1

∗ xki Xki

+

pk

Xki AXk∗i .

i=1

(23)


3445

Simultaneously, for all A ∈ Mn (C) we have ψ k − (ξ k )−1 (A) = (tk − sk )A + (Zk − Yk )A + A(Zk − Yk )∗

m k k ∗ ∗ + Ski ASki − Tki ATki . i=1

(24)

i=1

We claim that m p k k Xki AXk∗i Ski ASk∗i i=1

(25)

i=1

√ for all k ∈ N. To prove this, we let {vj }nj=1 be any orthonormal basis for Cn , let hj = vj / n for each i, let f ∈ Cn be arbitrary, and define maps Aj for j = 1, . . . , n by Aj = f h∗j . Using the trace conditions on the maps Yk , Zk , {Tki }, {Ski }, and {Xki }, we find that n ψ k − (ξ k )−1 (Aj )hj = (tk − sk )f + (Zk − Yk )f j =1

=

pk

|xki |

f+

2

i=1

pk

xki Xki f.

i=1

Since f was arbitrary, it follows that

tk − sk −

pk

|xki |

2

I=

i=1

pk

xki Xki

− (Zk − Yk ).

i=1

Taking the trace of both sides yields

0 = tr

pk

xki Xki

− (Zk − Yk ) = tr

tk − sk −

i=1

so tk − sk =

pk

2 i=1 |xki |

|xki |2 I ,

i=1

pk

and Zk − Yk = pk

i=1 xki Xki .

Xki AXk∗i

=

i=1

Therefore, the map A →

pk

mk

Ski ASk∗i

Formulas (23) and (24) now imply that −

i=1

mk

∗ i=1 Ski ASki

−

k

Tki ATk∗i

.

i=1

k

∗ i=1 Tki ATki

is completely positive, and

m m p k k k k Xki Xk∗i = Ski Sk∗i − Tki Tk∗i Ski Sk∗i , i=1

establishing (25).

i=1

i=1

i=1

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We now show that there exists some M ∈ N such that Xki M

(26)

for all k ∈ N and i ∈ {1, . . . , pk }. To do this, we first note that since the sequence of invertible maps {ξ k }k∈N converges in norm to the invertible map ξ , the sequence {(ξ k )−1 }k∈N converges in norm to ξ −1 . Write ξ −1 in the form ξ −1 (A) = sA + Y A + AY ∗ −

m

Si ASi∗ ,

i=1

where m n2 , s ∈ R, tr(Y ) = 0, and for all i and j , tr(Si ) = 0 and tr(Si Sj∗ ) is non-zero if and only if i = j . Let f ∈ Cn be arbitrary, and define vectors {hj }nj=1 and maps {Aj }nj=1 ex actly as we did earlier in the proof. Then nj=1 (ξ k )−1 (Aj )hj = sk f + Yk f for all k ∈ N n and j =1 ξ −1 (Aj )hj = sf + Yf . Since (ξ k )−1 converges to ξ −1 as k → ∞, we see that (sk − s)f + (Yk − Y )f converges to 0 as k → ∞. But f was arbitrary, so lim (sk − s)I + Yk − Y = 0.

k→∞

The limit of the trace of the above equation must also be zero, so sk converges to s and consequently Yk converges to Y . This implies that not only are the sequences of complex numbers {Yk }∞ {Wk }∞ {sk }∞ k=1 and maps k=1 both bounded, but that the sequence of linear maps k=1 demk ∗ ∗ fined by Wk (A) = i=1 Ski ASki is bounded and converges to the map W (A) = m i=1 Si ASi . Choose M ∈ N so that M 2 n2 supk∈N {Wk }. For every k ∈ N and i ∈ {1, . . . , mk }, we have Ski 2 Wk M 2 /n2 . Combining this fact with (25), we find that for every k ∈ N and i ∈ {1, . . . , pk }, p m m k k k ∗ ∗ ∗ Xki Xki Ski Ski Ski 2 Xki = Xki Xki i=1 i=1 i=1 2 2 2 n max Ski : i = 1, . . . , mk M , 2

proving (26). Since ψ k → ∞ as k → ∞ while (ξ k )−1 → (ξ )−1 < ∞, there is a sequence of maps {A k } of norm one such that (ψ k − (ξ k )−1 )(A k ) → ∞ as k → ∞. However, we also have p

p k k −1 2 ψ − (ξ ) (A ) = |xki | A k + xki Xki A k k k k i=1 i=1

p ∗ pk k +A k xki Xki + Xki A k Xk∗i i=1

pk i=1

|xki |2 + 2M

i=1

pk i=1

|xki | + pk M 2 .

(27)


3447

We note that

pk i=1

2 |xki |

pk i=1

pk pk |xki |)2 ( i=1 |xki |)2 ( i=1 |xki | pk n2 2

(28)

pk for all k. For each k, let λk = i=1 |xki |, noting that λk → ∞ as k → ∞ since Eq. (27) tends to infinity as k → ∞. Let A ∈ Mn (C) be any matrix such that A = 1, and let C = supk∈N (ξ k )−1 < ∞. Using the reverse triangle inequality and (28), we find that for each k ∈ N, ψ (A) ψ − (ξ )−1 (A) − (ξ )−1 (A) k k k k

λ2k − 2Mλk − n2 M 2 − C. n2

(29)

Since limk→∞ λk = ∞, Eq. (29) tends to infinity as k → ∞. For all k large enough that (29) is positive, we have φ k =

1 1 , 2 2 inf{ψ k (A): A = 1} λk /n − 2Mλk − n2 M 2 − C

so limk→∞ φ k = 0. But the sequence {φ k }∞ k=1 converges to φ in norm, hence φ ≡ 0.

2

Proposition 6.10. An invertible unital linear map φ : Mn (C) → Mn (C) is q-pure if and only if φ −1 is of the form φ −1 (A) = A + Y A + AY ∗ for some Y = −Y ∗ ∈ Mn (C) such that tr(Y ) = 0. Proof. Let ψ = φ −1 . Assume the hypotheses of the forward direction. Write ψ(A) = sA + Y A + AY ∗ −

k

λi Xi AXi∗ ,

i=1

where s ∈ R, tr(Y ) = 0, and for each i and j we have λi 0, tr(Xi ) = 0, and tr(Xi∗ Xj ) = nδij . Defining ψ : Mn (C) → Mn (C) by ψ (A) = sA + Y A + AY ∗ , we conditionally negative, and ψ − ψ is completely positive since (ψ − ψ)(A) = knote that ψ is ∗ −1 j =1 λj Xj AXj for all A. By Lemma 6.2, it follows that ψ is invertible and that φ := (ψ ) satisfies φ q φ q 0. Since φ is q-pure, there is some t0 0 such that φ = φ (t0 ) , hence

−1 −1 −1 −1 −1 ψ = φ I + t0 ψ −1 = φ(I + t0 φ)−1 = ψ = (t0 I + ψ)−1 = t0 I + ψ.

3448


Therefore, for all A ∈ Mn (C) we have ψ (A) = ψ(A) +

k

λj Xj AXj∗ = ψ(A) + t0 A,

j =1

so the map L : A → λj Xj AXj∗ satisfies L = t0 I . We repeat a familiar argument: Let f ∈ Cn be √ arbitrary, choose an orthonormal basis {vk }nk=1 of Cn , define hk = vk / n for neach k, and form the maps {X } imply that {Ak }nk=1 by Ak = f h∗k . The trace conditions for j k=1 L(Ak )hk = 0. However, since L = t0 I , we must also have nk=1 L(Ak )hk = t0 f . From arbitrariness of f , we conclude t0 = 0. Therefore, ψ has the form ψ(A) = sA + Y A + AY ∗ . Since ψ(I ) = I = sI + Y + Y ∗ and tr(Y ) = 0, we have s = 1 and consequently Y = −Y ∗ . Now assume the hypotheses of the backward direction. Note that ψ is conditionally negative and unital, hence φ is q-positive by Proposition 6.1. Let Φ be any non-zero q-positive map such that φ q Φ, so by Corollary 6.3 and Proposition 6.9, Φ is invertible and Ψ := (Φ)−1 is a conditionally negative map such that Ψ − ψ is completely positive. Write Ψ in the form Ψ (A) = s A + ZA + AZ ∗ −

m

μi Ti ATi∗ ,

i=1

where s ∈ R and for all i and j , μi > 0, tr(Ti ) = 0, and tr(Ti∗ Tj ) = nδij . Writing C = Z − Y , we have m ∗ μi Ti ATi∗ . (Ψ − ψ)(A) = s − 1 A + CA + AC − i=1

By a familiar argument, complete positivity of Ψ − ψ and the trace conditions for the above maps imply that s 1, C = 0, and Ti = 0 for all i. Therefore Ψ = ψ + (s − 1)I , so Φ = Ψ −1 = φ (s −1) . We conclude that φ is q-pure. 2 Let the matrices {ej k }nj,k=1 denote the standard basis for Mn (C), writing each A = (aj k ) ∈ Mn (C) as A = j,k aj k ej k . The following theorem classifies all unital invertible q-pure maps on Mn (C): Theorem 6.11. An invertible unital linear map φ : Mn (C) → Mn (C) is q-pure if and only if for some unitary U ∈ Mn (C), the map φU is the Schur map ⎧ aj k ⎪ ⎨ 1+i(λj −λk ) ej k φU (aj k ej k ) = aj k ej k ⎪ aj k ⎩ e 1−i(λj −λk ) j k

if j < k, if j = k, if j > k

for all A = (aj k ) ∈ Mn (C) and j, k = 1, . . . , n, where λ1 , . . . , λn ∈ R and λ1 + · · · + λn = 0. Proof. Assume the hypotheses of the forward direction. By the previous proposition, ψ := φ −1 has the form ψ(A) = A + Y˜ A + AY˜ ∗ for some Y˜ ∈ Mn (C) with Y˜ = −Y˜ ∗ and tr(Y˜ ) = 0. Let B = −i Y˜ , so B = B ∗ . Defining Y := (1/2)I + Y˜ = (1/2)I + iB, we find ψ(A) = Y A + AY ∗


3449

for all A ∈ Mn (C). Since B is self-adjoint, there is some unitary U ∈ Mn (C) such that U ∗ BU is a diagonal matrix D. For each k ∈ {1, . . . , n} let λk ∈ R be the kk entry of D. Note that since tr(B) = 0 we have nk=1 λk = 0, and that U ∗ Y U is the diagonal matrix M whose kk entry is 1/2 + iλk . Defining a map ψU by ψU (A) = U ∗ ψ(U AU ∗ )U for all A ∈ Mn (C), we find that ψU (A) = U ∗ Y U AU ∗ + U AU ∗ Y ∗ U ∗ = U ∗ Y U A + A U ∗ Y U = MA + AM ∗ . A quick calculation shows that this is just the Schur map ⎧ ⎨ (1 + i(λj − λk ))aj k ej k ψU (aj k ej k ) = aj k ej k ⎩ (1 − i(λj − λk ))aj k ej k

if j < k, if j = k, if j > k,

and so (ψU )−1 has the form ⎧ aj k ⎪ ⎨ 1+i(λj −λk ) ej k (ψU )−1 (aj k ej k ) = aj k ej k ⎪ aj k ⎩ 1−i(λj −λk ) ej k

if j < k, if j = k, if j > k.

It is straightforward to verify that (ψU )−1 is the map φU (A) = U ∗ φ(U AU ∗ )U . Assume the hypotheses of the backward direction. Let T be the diagonal matrix whose kkth entry is λk for every k = 1, . . . , n. We observe that tr(T ) = 0 and T = T ∗ . Now let C = iT , and let T˜ = (1/2)I + C. We routinely verify that C = −C ∗ and tr(C) = 0, and that (φU )−1 satisfies (φU )−1 (A) = T˜ A + AT˜ ∗ = A + CA + AC ∗ for all A ∈ Mn (C). Proposition 6.10 implies that φU is q-pure, whereby φ is q-pure by Proposition 4.5. 2 As it turns out, boundary weight doubles (φ, ν) for invertible unital q-pure maps φ : 2 M√ n (C) → Mn (C) √ and normalized unbounded boundary weights ν over L (0, ∞) of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ) give us nothing new in terms of E0 -semigroups: Theorem 6.12. Let φ : Mn (C) → Mn (C) be unital, invertible, and√q-pure, and √ let ν be a normalized unbounded boundary weight over L2 (0, ∞) of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ). Then (φ, ν) and (ıC , ν) induce cocycle conjugate E0 -semigroups. Proof. By Theorem 6.11 and Propositions 4.5 and 4.6, we may assume that φ is the Schur map ⎧ aj k ⎪ ⎨ 1+i(λj −λk ) ej k φ(aj k ej k ) = aj k ej k ⎪ aj k ⎩ 1−i(λj −λk ) ej k


for some λ1 , . . . , λn ∈ R with nk=1 λk = 0. By Proposition 4.6, it suffices to find a hyper maximal q-corner from φ to ıC . For this, define γ : Mn×1 (C) → Mn×1 (C) by

3450

C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451 1 ⎞ ⎛ 1+iλ b1 ⎞ b1 1 1 ⎟ ⎜ b2 ⎟ ⎜ ⎜ 1+iλ2 b2 ⎟ ⎟ = γ⎜ ⎜ ⎟. .. ⎝ ... ⎠ ⎝ ⎠ . bn 1 1+iλn bn

⎛

Now define Υ : Mn+1 (C) → Mn+1 (C) by Υ

An×n C1×n

Bn×1 a

=

φ(An×n ) γ ∗ (C1×n )

γ (Bn×1 ) . a

Letting λn+1 = 0, we observe that Υ is the Schur map satisfying ⎧ aj k ⎪ ⎨ 1+i(λj −λk ) ej k Υ (aj k ej k ) = aj k ej k ⎪ aj k ⎩ 1−i(λj −λk ) ej k


n for all j, k = 1, . . . , n + 1 and A = (aj k ) ∈ Mn (C). Since n+1 i=1 λk = 0, it follows i=1 λk = from Theorem 6.11 that Υ is q-positive (in fact, q-pure), hence γ is a q-corner from φ to ıC . Now suppose that Υ q Υ q 0 for some Υ of the form Υ

An×n C1×n

Bn×1 a

=

φ (An×n ) γ ∗ (C1×n )

γ (Bn×1 ) . ı (a)

Since Υ is q-pure and Υ is not the zero map, we know that Υ = Υ (t) for some t 0, and a quick calculation gives us Υ

An×n C1×n

Bn×1 a

=

φ (t) (An×n ) (γ ∗ )(t) (C1×n )

γ (t) (Bn×1 ) . 1 1+t (a)

By inspecting the two formulas for Υ we see γ = γ (t) . But γ (t) has the form 1 ⎞ ⎛ 1+t+iλ b1 ⎞ b1 1 1 ⎜ b2 ⎟ ⎜ b ⎟ ⎟ ⎜ 1+t+iλ2 2 ⎟ γ (t) ⎜ ⎟, . ⎝ ... ⎠ = ⎜ ⎠ ⎝ .. bn 1 1+t+iλn bn

⎛

hence t = 0. Therefore, Υ = Υ , and we conclude the q-corner γ is hyper maximal.

2

In conclusion, we approach the broader question of simply finding all unital q-pure maps φ : Mn (C) → Mn (C), as they provide us with the simplest way to construct E0 -semigroups through boundary weight doubles. We believe that all q-pure maps are invertible or have rank one. For n = 2, we find in [6] that this conjecture holds: There is no unital q-pure map φ : M2 (C) → M2 (C) of rank 2, and there is no unital q-positive map φ : M2 (C) → M2 (C) of rank 3. It seems that for n = 3, the key to classifying unital q-pure maps is through investigation of the limits Lφ = limt→∞ tφ(I + tφ)−1 , though the situation becomes very complicated if n > 3.


3451

Acknowledgments The author very gratefully thanks his thesis advisor, Robert Powers, for his boundless enthusiasm, constant encouragement, and guidance in research. His help in the author’s thesis work has been indispensable. The author would also like to thank Geoff Price for proofreading an earlier draft of the paper and making suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

W.B. Arveson, The index of a quantum dynamical semigroup, J. Funct. Anal. 146 (1997) 557–588. W.B. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80 (409) (1989). B.V.R. Bhat, An index theory for quantum dynamical semigroups, Trans. Amer. Math. Soc. 348 (2) (1996) 561–583. M. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975) 285–290. D.E. Evans, J.T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A 24 (1977). C. Jankowski, Unital q-pure maps on M2 (C), in preparation. D. Markiewicz, R.T. Powers, Local unitary cocycles of E0 -semigroups, J. Funct. Anal. 256 (5) (2009) 1511–1543. R.T. Powers, Continous spatial semigroups of completely positive maps of B(H ), New York J. Math. 9 (2003) 165–269. R.T. Powers, Construction of E0 -semigroups of B(H) from CP-flows, in: Advances in Quantum Dynamics, in: Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 57–97. R.T. Powers, Induction of semigroups of endomorphisms of B(H) from completely positive semigroups of (n × n) matrix algebras, Internat. J. Math. 10 (7) (1999) 773–790. R.T. Powers, New examples of continuous spatial semigroups of ∗-endomorphisms of B(H), Internat. J. Math. 10 (2) (1999) 215–288. R.T. Powers, An index theory for semigroups of ∗-endomorphisms of B(H) and type II1 factors, Canad. J. Math. 40 (1988) 86–114. R.T. Powers, A nonspatial continous semigroup of ∗-endomorphisms of B(H), Publ. Res. Inst. Math. Sci. 23 (6) (1987) 1053–1069. R.T. Powers, G. Price, Continuous spatial semigroups of ∗-endomorphisms of B(H), Trans. Amer. Math. Soc. 321 (1990) 347–361. B. Tsirelson, Non-isomorphic product systems, in: Advances in Quantum Dynamics, in: Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 273–328. E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. of Math. 40 (1939) 149–204.


A renorming in some Banach spaces with applications to fixed point theory Carlos A. Hernandez Linares, Maria A. Japon ∗ Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain Received 24 September 2009; accepted 23 October 2009 Available online 5 November 2009 Communicated by N. Kalton

Abstract We consider a Banach space X endowed with a linear topology τ and a family of seminorms {Rk (·)} which satisfy some special conditions. We define an equivalent norm ||| · ||| on X such that if C is a convex bounded closed subset of (X, ||| · |||) which is τ -relatively sequentially compact, then every nonexpansive mapping T : C → C has a fixed point. As a consequence, we prove that, if G is a separable compact group, its Fourier–Stieltjes algebra B(G) can be renormed to satisfy the FPP. In case that G = T, we recover P.K. Lin’s renorming in the sequence space 1 . Moreover, we give new norms in 1 with the FPP, we find new classes of nonreflexive Banach spaces with the FPP and we give a sufficient condition so that a nonreflexive subspace of L1 (μ) can be renormed to have the FPP. © 2009 Elsevier Inc. All rights reserved. Keywords: Fixed point theory; Renorming theory; Nonexpansive mappings; Fourier algebras; Fourier–Stieltjes algebra; Topology of convergence locally in measure

1. Introduction Let (X, · ) be a Banach space and C a convex closed bounded subset of X. A mapping T : C → C is called nonexpansive if for any x, y ∈ C we have T x − T y x − y. A point x ∈ C is a fixed point of T if T x = x. It is clear that Banach’s Contraction Principle does not * Corresponding author.

E-mail addresses: [email protected] (C.A.H. Linares), [email protected] (M.A. Japon). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.025

C.A.H. Linares, M.A. Japon / Journal of Functional Analysis 258 (2010) 3452–3468

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extend to the setting of nonexpansive mappings. However, some positive results concerning the existence of fixed points for this class of mappings where given in 1965 by F.E. Browder [5] and D. Göhde [13] for uniformly convex Banach spaces and by W. Kirk [16] for reflexive Banach spaces with normal structure. Since then, many authors have studied the problem of the existence of fixed points for nonexpansive mappings and many positive results have been found (see for instance [12,17] and the references therein). It is usually said that a Banach space X has the fixed point property (FPP) if every nonexpansive mapping defined from a closed convex bounded subset onto itself has a fixed point. It is well known that the geometry of the Banach space plays a fundamental role to assure the FPP. In fact, Kirk’s result [16] means that a reflexive Banach space with normal structure has the FPP. Many other geometric properties are known to imply the FPP for reflexive Banach spaces (uniform Kadec Klee property, uniform Opial condition, existence of a monotone unconditional basis, etc.). Moreover the classical nonreflexive Banach spaces 1 , c0 , L1 do not have the FPP (in fact L1 does not satisfy a stronger condition called the weakly fixed point property [2]). For a long time, it was an open question whether all Banach spaces with the FPP were reflexive. In 2008, P.K. Lin [20] found the first known nonreflexive Banach space with the FPP. In fact, the Banach space given by P.K. Lin was the sequence space 1 endowed with an equivalent norm to the usual one. His result raises the question: can any Banach space be renormed to have the FPP? This is not, in general, the case because the Banach spaces 1 (Γ ) and c0 (Γ ), if Γ is uncountable, and the Banach space ∞ cannot be renormed to have the FPP [8]. A positive partial answer was given by T. Domínguez Benavides [6], who proved that every reflexive Banach space can be renormed to have the FPP. This leads to the following question: Which type of nonreflexive Banach spaces can be renormed to have the FPP? In this paper we find some classes of nonreflexive Banach spaces which under an equivalent renorming satisfy the FPP. Our techniques are inspired by those of P.K. Lin’s paper [20] but our applications go beyond the sequence space 1 as we will illustrate with many examples. As particular cases, we will recover P.K. Lin’s result and we will find new renormings in 1 with the FPP. Moreover, we will renorm the Fourier–Stieltjes algebra of a separable compact group to have the FPP. Notice that if G is locally compact, its Fourier–Stieltjes algebra B(G) has the FPP if and only if G is finite [18]. We also find new classes of nonreflexive Banach spaces with the FPP which are nonisomorphic to any subspace of 1 . Finally, we will apply our results to the particular case of subspaces of L1 (μ) for a σ -finite measure. It is known that a closed subspace X of L1 (μ) has the FPP if and only if X is reflexive [23,7]. Nevertheless, we will show that some nonreflexive subspaces of L1 (μ) can still be renormed to have the FPP. This paper is organized as follows: Section 2 is dedicated to the necessary fixed point background and we establish a technical lemma which is basic in our proofs. In Section 3 we will state our main Theorem and we will introduce the first applications: we are able to find new renormings in 1 with the FPP, we renorm B(G) with the FPP if G is a separable compact group and we give new examples of nonreflexive Banach spaces with the FPP that are nonisomorphic to any subspace of 1 . Section 4 is dedicated to the proof of the main Theorem. Finally, in Section 5, we will apply the main Theorem to closed subspaces of L1 (μ) when (Ω, Σ, μ) is a σ -finite measure space. We obtain a sufficient condition to assure that a nonreflexive subspace X of L1 (μ) can be renormed to have the FPP (it is known that, with the usual norm, X fails to have this property [7]). We finish the paper by introducing some examples of nonreflexive subspaces of L1 (μ) which can be renormed to have the FPP.

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2. Fixed point background Let C be a closed convex bounded subset of a Banach space (X, · ) and T : C → C a nonexpansive mapping. Fix any x0 ∈ C. A direct application of the Banach’s Contraction Principle to the sequence of mappings Tn : C → C defined by 1 1 T x; Tn x = x 0 + 1 − n n provides a sequence {xn }n ⊂ C, where xn is the unique point of Tn , such that lim xn − T xn = 0. n

Such sequences are called approximated fixed point sequences (a.f.p.s.). Moreover, if d > 0 and the set D = x ∈ C: lim sup xn − x d n

is nonempty, it is easy to check that D is convex, closed and a T -invariant subset of C. Hence, we can find another a.f.p.s. in D. As an application of Cantor’s Intersection Theorem, we can prove the following: Lemma 1. Let (X, · ) be a Banach space and C a convex, closed, bounded subset of X. Let T : C → C be a nonexpansive mapping and suppose that T is fixed point free. Then there exist some a > 0 and a convex closed T -invariant subset D of C such that for each approximated fixed point sequence (xn ) in D and for any z ∈ D lim sup xn − z a. n

Proof. If the statement is false there exists an a.f.p.s. (xn1 ) in C and z1 ∈ C such that 1 lim supxn1 − z1 < . 2 n Hence 1 D1 = z ∈ C: lim supxn1 − z 2 n is a nonempty, convex, closed, T -invariant subset of C. With the same argument, we deduce the existence of an approximated fixed point sequence (xn2 ) in D1 and z2 ∈ D1 such that 1 lim supxn2 − z2 < 2 . 2 n


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Hence the set 2 1 D2 = z ∈ D1 : lim sup xn − z 2 2 n is again a nonempty, convex, closed, T -invariant subset of D1 . In this way we construct a decreasing sequence (Dn ) of convex closed bounded T -invariant 1 . By the Cantor’s Intersection Theorem, n Dn is a subsets of C such that diam(Dn ) 2n−1 singleton. Since each Dn is T -invariant this point has to be a fixed point of T . Thus, we have obtained a contradiction since T is fixed point free. 2 Remark. Notice that if X is endowed with a topology τ such that every bounded sequence has a τ -convergent subsequence, then the conditions of Lemma 1 also imply that inf lim sup xn − x: (xn ) ⊂ D, (xn ) a.f.p.s. xn → x in τ > 0. n

Indeed, applying the triangular inequality, for every (xn ) ⊂ D a.f.p.s. such that (xn ) converges to x in the topology τ , we have lim sup xn − x n

1 a lim sup lim sup xn − xm . 2 m 2 n

3. Main result and first examples In this section we state the main result of this paper. As a consequence, we obtain the renorming given in [20] in the sequence space 1 , which provided the first known nonreflexive Banach space with the FPP. Also we will give new equivalent norms on 1 with the FPP and we will obtain new classes of nonreflexive Banach spaces with the FPP. In particular, we will prove that the Fourier–Stieltjes algebra B(G) of a separable compact group can be renormed to have the FPP. Notice that B(G) itself has the FPP if and only if G is finite [18] (Theorem 5.8). More applications of the main Theorem will be studied in the last section. Let (X, · ) be a Banach space endowed with a linear topology τ . Assume that there exists a family of seminorms Rk : X → [0, +∞) (k 1) that satisfy the following properties: (I) R1 (x) = x while for k 2, Rk (x) x for all x ∈ X. (II) limk Rk (x) = 0 for all x ∈ X. (III) If xn → 0 in τ and is norm-bounded, then for all k 1 lim sup Rk (xn ) = lim sup xn . n

n

(IV) If xn → 0 in τ , is norm-bounded and x ∈ X, then lim sup Rk (xn + x) = lim sup Rk (xn ) + Rk (x) n

for all k 1.

n

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Then we can state the following: Theorem 1. Let {γk }k ⊂ (0, 1) be any nondecreasing sequence such that limk γk = 1 and define |||x||| = sup γk Rk (x);

x ∈ X.

k1

Then ||| · ||| is an equivalent norm on X such that (X, ||| · |||) satisfies the following property: for every nonempty closed convex bounded subset C which is τ -relatively sequentially compact and for every T : C → C nonexpansive, there exists a fixed point. That ||| · ||| is an equivalent norm on (X, · ) is clear. In fact, γ1 x |||x||| x for all x ∈ X. We will prove Theorem 1 in the next section. Now we give several families of Banach spaces where our results can be applied. Example 1. A first application of Theorem 1 is a generalization of P.K. Lin’s example given in [20], where he proves that if γk = 8k /(1 + 8k ), then the renorming on 1 given by ∞ 8k

|||x||| = sup x e n n k k 1+8 n=k

has the FPP. Notice that Lin’s result can be derived from Theorem 1 defining the seminorms Rk (x) = ∞ n=k xn en and τ the weak-star topology associated to the duality σ (1 , c0 ). Since the unit ball is weak-star compact and c0 is separable, every closed convex bounded subset is σ (1 , c0 )-sequentially compact. Also, we obtain the renorming given in [10] where P.K. Lin’s result is generalized by using any nondecreasing sequence (γk ) ∈ (0, 1) with limk γk = 1 and γ1 > 2/3. Notice that in our approach the condition γ1 > 2/3 can be dropped. Example 2. If we consider again the sequence space 1 and change the family of seminorms, then we can obtain new renormings on 1 with the FPP. For instance, let p > 1 and for k 2 define ∞

x(n) + Rk (x) = n=2k

2k−1

1/p |xn |

p

n=k

and R1 (x) = x1 . It is easy to check that {Rk (·)}k is a family of seminorms that verify properties (I), (II), (III) and (IV), so 1 with the norm generated by the seminorms {Rk (·)}k satisfies the FPP. Corollary 1. Let {Xn }n be a sequence of finite dimensional Banach spaces and consider X = ⊕1

Xn = x = (xn )n : xn ∈ Xn , x = xn Xn < ∞ .

n

Then X can be renormed to have the FPP.

n


Proof. Define the seminorms Rk (x) = the predual of X is

nk xn Xn

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and let τ be the weak star topology where

E = x = (xn ): xn ∈ Xn , lim xn Xn = 0, x = sup xn Xn . n

It is not difficult to check that the family {Rk (·)}k satisfies properties (I), (II), (III) and (IV). Using the renorming given in Theorem 1, the Banach space (X, ||| · |||) has the FPP. 2 A first application of Corollary 1 is the following example: Example 3. Consider Xn = np for some 1 < p +∞ and X = ⊕1

np .

n

Then we obtain a nonreflexive Banach space that can be renormed to have the FPP and that is not isomorphic to any subspace of 1 . Indeed, p is finitely representable in X and the type and the cotype of X is equal to the type and the cotype of p respectively. Notice that for every 1 < p +∞, either the type or the cotype of p is different from that of 1 , since type(p ) = min{2, p} and cotype(p ) = max{2, p} (see [22], p. 73). Thus, X is not isomorphic to any subspace of 1 and we obtain new classes of nonreflexive Banach spaces with the FPP. For the definitions of the Fourier algebra A(G) and the Fourier–Stieltjes algebra B(G) of a locally compact group see [9] or [19] and the references therein. When G is compact, B(G) = A(G). Another application of Corollary 1 is the following. Corollary 2. Let G be a separable compact group and B(G) its Fourier–Stieltjes algebra. Then B(G) can be renormed to have the FPP. Proof. Using the arguments in the proof of Lemma 3.1 of [19], and having in mind that B(G) is norm separable when G is a separable compact group [14, Corollary 6.9], the Banach space B(G) can be written as B(G) = ⊕1

T(Hn )

n

where Hn is a finite dimensional Hilbert space and T(Hn ) is the trass class operators on Hn . Applying Corollary 1 we obtain a renorming of B(G) with the FPP. 2 In the particular case that G = T, the circle group, B(G) is isometric to 1 (Z) via Bochner’s Theorem. Thus, Corollary 2 includes the sequence space 1 and the renorming given by P.K. Lin [20]. Also, recall that B(G) with its usual norm has the FPP if and only if G is finite [18].

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4. Proof of the main result Before proving Theorem 1 we prove two technical lemmas: Lemma 2. Let X be a Banach space endowed with a linear topology τ and a family of seminorms {Rk (·)k } satisfying properties (I), (II), (III) and (IV) stated above. Define the ||| · ||| norm as in Theorem 1 and let (xn ), (yn ) be two bounded sequences in X. Then the following statements are satisfied: (1) If xn → 0 in τ , then lim sup |||xn ||| = lim sup xn . n

n

(2) If xn → x and yn → y in τ then lim sup lim sup |||xn − ym ||| lim sup |||xn − x||| + lim sup |||ym − y|||. m

n

n

m

Proof. (1) For every k 1, using the definition of the ||| · ||| norm and property (III), we have lim sup xn lim sup |||xn ||| γk lim sup Rk (xn ) = γk lim sup xn . n

n

n

n

Taking limit as k goes to infinity we deduce (1). (2) By property (IV) we have lim sup lim sup Rk (xn − ym ) = lim sup lim sup Rk (xn − x) + Rk (x − ym ) m

n

m

n

= lim sup Rk (xn − x) + lim sup Rk (ym − y) + Rk (x − y), n

m

for every k 1. Then, using again the definition of ||| · ||| and property (III), lim sup lim sup |||xn − ym ||| γk lim sup Rk (xn − x) + lim sup Rk (ym − y) + Rk (x − y) m

n

n

m

γk lim sup |||xn − x||| + lim sup |||ym − y||| . n

m

Taking limit as k goes to infinity we get the desired result.

2

The following lemma is the key for the arguments in the proof of Theorem 1: Lemma 3. Consider the Banach space (X, ||| · |||) and let C and T be as in Theorem 1. If T is fixed point free we find D as in Lemma 1. Let K be any closed convex T -invariant subset of D and denote ρ = inf lim sup |||xn − x|||: (xn ) ⊂ K is an a.f.p.s. and xn → x in τ > 0. n


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Then for every a.f.p.s. (xn ) ⊂ K which is τ -convergent and for every z ∈ K we have lim sup |||xn − z||| 2ρ. n

Proof. Assume that there exist a τ -convergent approximate fixed point sequence (xn ) in K and z ∈ K such that r = lim sup |||xn − z||| < 2ρ. n

Then K1 = w ∈ K: lim sup |||xn − w||| r n

is a nonempty, convex, closed, bounded T -invariant subset of K. Choose an approximate fixed τ point sequence (yn ) in K1 such that yn → y. Denote by x the τ -limit of the sequence (xn ). Then by (2) of Lemma 2, we have r lim sup lim sup |||xn − ym ||| m

n

lim sup |||xn − x||| + lim sup |||yn − y||| n

n

ρ + ρ = 2ρ, which is a contradiction.

2

Now we prove Theorem 1. Proof. Assume the contrary, that T has no fixed point. Let D be as in the conclusion of Lemma 1. Define τ c = inf lim sup |||xn − x|||: (xn ) ⊂ D is an a.f.p.s. and xn → x n

which is greater than zero by the remark made after Lemma 1. Without loss generality we can assume that c = 1. Take 0 < 1 < 1/2 and an a.f.p.s. (xn ) ⊂ D τ such that xn → x and lim supn |||xn − x||| < 1 + 1 . Again, by translation, we can assume that x = 0. Let us consider now K = z ∈ D: lim sup |||xn − z||| 2 + 2 1 . n

The set K is closed, convex, T -invariant and nonempty. Indeed, we can find n0 such that xn ∈ K for all n n0 . Define

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τ ρ = inf lim sup |||yn − y|||: (yn ) ⊂ K is an a.f.p.s. and yn → y . n

It is clear that 1 ρ lim supn |||xn ||| < 1 + 1 . We are going to find an a.f.p.s. (yn ) ⊂ K and z ∈ K such that lim sup |||yn − z||| < 2ρ n

and then we obtain a contradiction according to Lemma 3. τ Notice the following: If (yn ) ⊂ K is an a.f.p.s. and yn → y, then for all k, 2 + 2 1 lim sup lim sup |||xn − ym ||| = lim sup lim sup xn − (ym − y) − y m

n

m

n

γk lim sup lim sup Rk xn − (ym − y) − y m

n

= γk lim sup Rk (xn ) + lim sup Rk (ym − y) + Rk (y) n

m

= γk lim sup |||xn ||| + lim sup |||ym − y||| + Rk (y) γk 2 + Rk (y) . n

m

τ

Consequently, if (yn ) ⊂ K is an a.f.p.s. and yn → y, we have Rk (y) 2

1 + 1 −1 . γk

Let p := 1 + 1 + 2

1 + 1 − 1 > ρ, γ1

δ ∈ ( 1 , 1/2),

0 < 2 < ρ − 2δ.

Since, by Lemma 2(1), lim supn xn = lim supn |||xn ||| < 1 + 1 , we can find x ∈ K such that x < 1 + 1 . Also there exists m ∈ N such that if k m Rk (x) < 2

by property (II)

and 1 + 1 < γk 1+δ

since lim γk = 1 . k

We take λ ∈ (0, 1) such that λ
0 such that (i)

(2 − λ)(ρ + 3 ) + λ( 2 + 2δ) < 2ρ

and (ii)

γm (2 − λ)(ρ + 3 ) + λp < 2ρ. τ

Take (yn ) ⊂ K to be an a.f.p.s. such that yn → y and lim sup yn − y = lim sup |||yn − y||| < ρ + 3 n

using Lemma 2(1) .

n

There exists s ∈ N such that yN − y < ρ + 3 for all N s and define z = (1 − λ)ys + λx which belongs to K because K is convex. Let us prove that lim supn |||yn − z||| < 2ρ. In order to do this, we will prove that there exists M > 0 such that for all k and N s we have γk Rk (yN − z) < M < 2ρ. We split the proof into two cases: Case 1: k m: γk Rk (yN − z) = γk Rk yN − (1 − λ)ys − λx Rk yN − y − (1 − λ)(ys − y) − λ(x − y) Rk (yN − y) + (1 − λ)Rk (ys − y) + λRk (x − y) yN − y + (1 − λ)ys − y + λRk (x − y) (ρ + 3 ) + (1 − λ)(ρ + 3 ) + λ Rk (x) + Rk (y) (2 − λ)(ρ + 3 ) + λ 2 + Rk (y) 1 + 1 (2 − λ)(ρ + 3 ) + λ 2 + 2 −1 γk < (2 − λ)(ρ + 3 ) + λ( 2 + 2δ) < 2ρ

by (i).

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Case 2: k m: γk Rk (yN − z) γm Rk yN − (1 − λ)ys − λx γm Rk yN − y − (1 − λ)(ys − y) − λ(x − y) γm Rk (yN − y) + (1 − λ)Rk (ys − y) + λRk (x − y) γm (ρ + 3 ) + (1 − λ)(ρ + 3 ) + λ Rk (x) + Rk (y) γm (2 − λ)(ρ + 3 ) + λ 1 + 1 + Rk (y) 1 + 1 −1 < 2ρ γm (2 − λ)(ρ + 3 ) + λ 1 + 1 + 2 γ1

by (ii).

Take M = max (2 − λ)(ρ + 3 ) + λ( 2 + 2δ), 1 + 1 γm (2 − λ)(ρ + 3 ) + λ 1 + 1 + 2 −1 . γ1 Then, for all N s, |||yN − z||| < M < 2ρ. Thus lim supn |||yn − z||| < 2ρ and this finishes the proof. 2 5. Applications to the Lebesgue function space L1 (μ) In this section we are going to consider the Banach space L1 (μ), where (Σ, Ω, μ) is a σ -finite measure space and we will apply Theorem 1 to this space. As a consequence we will obtain new results about renorming and FPP for nonreflexive subspaces of L1 (μ). In order to do that we will define a family of seminorms {Rk (·)}k1 which satisfies properties (I), (II), (III) and (IV) stated in Section 3. We denote by · the usual norm on L1 (μ), that is x = |x| dμ, for all x ∈ L1 (μ) Ω

x for all x ∈ L1 (μ). and R1 (x) = Let Ω = n Ωn be such that μ(Ωn ) < +∞ for all n ∈ N and denote Ak = kn=1 Ωn . For k 2 define the seminorms 1 + xχAck . Rk (x) = sup |x| dμ: μ(E) < (1) k E∩Ak

Let τ be the topology of locally convergence in measure, which is given by the metric d(x, y) =

∞

1 1 |x − y| dμ; n 2 μ(Ωn ) 1 + |x − y| n=1

Ωn

x, y ∈ L1 (μ).


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This topology is related to the convergence almost everywhere in the following way: every sequence that converges almost everywhere also converges locally in measure to the same function. Moreover, if a sequence converges locally in measure, then it has a subsequence that converges almost everywhere [15], pp. 157–158. Fix any nondecreasing sequence (γk )k ⊂ (0, 1) such that limk γk = 1 and define the equivalent norm on L1 (μ) as |||x||| = sup γk Rk (x). k

Now we have all the ingredients to state the following: Theorem 2. The seminorms {Rk (·)}k defined above satisfy properties (I), (II), (III) and (IV) stated in Section 3. Thus the following holds: Let C be a convex bounded closed subset of L1 (μ) and T : C → C a ||| · |||-nonexpansive mapping. If every sequence in C has a subsequence which is almost everywhere convergent in L1 (μ), then T has a fixed point. Remark 1. Notice that the above fixed point result does not hold for the usual norm in L1 (μ). Indeed, consider (hn ) a disjointly supported normalized sequence in L1 (μ). Let C=

tn hn ; tn 0,

n

tn = 1 = co(hn ).

n

The set C is closed convex bounded and every sequence in C has aconvergent subsequence almost everywhere. Indeed, consider a sequence (fk ) ⊂ C. Then fk = n tn (k)hn where tn (k) 0 and n tn (k) = 1 for every k. Define t k = (tn (k))n . The sequence (t k )k belongs to the unit ball of 1 which is σ (1 , c0 )-compact. So there exists a subsequence, denoted again by t k , which is σ (1 , c0 )-convergent to some t = (tn )n belonging to unit ball of 1 . Now we can easily check that (fk ) is pointwise convergent to f = n tn hn (notice that f is not, in general, in C). Define T : C → C by T

n

tn hn =

tn hn+1 .

n

It is easy to check that T is · -nonexpansive and has no fixed point in C. Remark 2. If the measure space is not σ -finite and we consider the convergence almost everywhere in L1 (μ), we cannot renorm the space L1 (μ) so that Theorem 2 remains true. Indeed, in this case 1 (Γ ) is contained isometrically in L1 (μ) for some uncountable set Γ and every sequence in a bounded subset of 1 (Γ ) has a pointwise convergent subsequence. So if Theorem 2 holds then we would have a renorming in 1 (Γ ) with the FPP. This is impossible since every renorming of 1 (Γ ) contains an asymptotically isometric copy of 1 and then it fails the FPP [8]. Before proving Theorem 2 we give a simpler definition of the ||| · ||| norm in two special cases: when μ is finite and when μ is purely atomic.

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Remark 3. Assume that the measure μ is finite. Consider Ak = Ω for all k. Then Ack = ∅ and xχAck = 0 for all x ∈ L1 (μ). Therefore 1 Rk (x) = sup |x| dμ: μ(E) < k E

for all k ∈ N. In this case, the topology of the convergence locally in measure is given by the metric d(x, y) = Ω

|x − y| dμ; 1 + |x − y|

x, y ∈ L1 (μ),

and the convergence with respect to this topology is the convergence in measure. Remark 4. Assume now that Ω = N and μ is the counting measure defined on the subsets of N. Then the space L1 (μ) becomes the sequence space 1 . Denote Ωk = {n} and Ak = {1, . . . , n}. Then Rk (x) = xχAck =

∞

x(n) . n=k+1

In this case we recover again the ||| · ||| norm defined by P.K. Lin in [20] for the particular case γk = 8k /(1 + 8k ). Notice that the convergence almost everywhere in 1 is the pointwise convergence and every bounded sequence in 1 has a pointwise convergent subsequence because the unit ball of 1 is σ (1 , c0 )-compact. In general, if the measure space is σ -finite and purely atomic, L1 (μ) is isometric to 1 . Thus by Theorem 2, it can be renormed to have the FPP. Now, to prove Theorem 2 we only have to check that properties (I), (II), (III), (IV) are fulfilled for the family of seminorms Rk (·) defined on L1 (μ). Proof. To simplify the notation we let Sk (x) := sup

|x| dμ: μ(E)
0. By the definition of Sk (·), there exists a measurable set A ⊂ Ak with μ(A) < k1 such that

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|x| dμ Sk (xχAk ) − . A

Since

n μ(En ) μ(Ak ) < +∞ there exists n0

such that μ(A) +

nn0

μ(En ) < k1 . Therefore

lim sup Sk (hn + x) lim sup n

n

A∪ nn En

0

= lim sup n

A∪ nn En

lim sup n

|hn + x| dμ |hn | dμ +

0

A∪

|hn | dμ +

En

nn0

|x| dμ En

|x| dμ A

lim sup hn χAk + Sk (xχAk ) −

n

= lim sup Sk (hn χAk ) + Sk (xχAk ) − . n

Since is arbitrary we obtain the desired equality. Therefore: lim sup Rk (xn + x) = lim sup Sk (hn χAk ) + Sk (xχAk ) + lim sup xn χAck + xχAck n

n

n

= lim sup Sk (xn χAk ) + Sk (xχAk ) + lim sup xn χAck + xχAck n

n

= lim sup Rk (xn ) + Rk (x) n

and we obtain (IV).

2

Although it is well known that the space L1 (μ) does not have the FPP (it does not satisfies the weak fixed point property w-FPP [2]), in 1980, B. Maurey [23] proved that all reflexive subspaces of L1 (μ) do have the FPP. In 1997 P. Dowling and C. Lennard [7] proved that the converse holds, that is, a subspace X of L1 (μ) has the FPP if and only if X is reflexive. This leads us to the following question: Can a nonreflexive subspace of L1 (μ) be renormed to have the FPP? Theorem 2 lets us give a partial answer to this question: Corollary 3. Let X be a closed subspace of L1 (μ). If the unit ball of X is relatively sequentially compact for the topology of the convergence locally in measure, then (X, ||| · |||) has the FPP. Corollary 4. Let X be a closed subspace of L1 (μ). If X is a dual space such that the topology of the convergence locally in measure coincides with the weak star topology on the unit ball of X, then (X, ||| · |||) has the FPP. Notice that this is the case of the sequence space 1 . Here we present another example.


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Example 4. Let D denote the open unit disc. The Bergman space La (D) is defined as the subspace of L1 (D) of all analytic functions on D. This space is a dual space and for bounded sequences weak* convergence is equivalent to uniform convergence on compact sets [24]. This shows that the weak* topology is finer than the topology of convergence in measure on the unit ball of La (D) and consequently these two topologies coincide for BLa (D) . Thus the Bergman space endowed with the ||| · ||| norm given in this section has the FPP. Notice that, from P.K. Lin’s paper [20], it is deduced that the Bergman space can be renormed to have the FPP. Indeed, J. Lindenstrauss, A. Pelczynski [21] proved that the Bergman space and the sequence space 1 are isomorphic, although they did not give an explicit definition of the isomorphism. In fact, it turns out to be a difficult problem to find a system of functions which is a basis in La (D) equivalent to the unit vector basis in 1 (see Notes and Remarks in Chapter III.A of [25] and the references therein). However, using Theorem 2 we can give explicitly the renorming on the Bergman space with the FPP. The following example satisfies the hypothesis of Corollary 3 but does not fit in the scope of Corollary 4: Example 5. In [11], Théorème 7, we can find an example of a subspace X of L1 [0, 1] such that its unit ball BX is compact for the topology of convergence in measure but not locally convex for this topology. Then, by Corollary 3, (X, ||| · |||) has the FPP. To finish we show another example of a Banach space which can be renormed to satisfy the FPP: Example 6. In [3] we can find an example of a Banach space E contained in L1 , over a probability space, and such that E fails to have the Radon–Nikodym property and every L1 -bounded sequence in E has a subsequence converging in measure. Applying again Theorem 2, we deduce that E can be renormed to have the FPP and by the failure of the Radon–Nikodym property, E is not isomorphic to any subspace of 1 . Acknowledgments The authors would like to thank Professors T. Domínguez Benavides, B. Sims and A.T.-M. Lau for their valuable suggestions in the preparation of this paper. References [1] F. Albiac, N.J. Kalton, Topics in Banach Space Theory, Grad. Texts in Math., vol. 233, Springer-Verlag, New York, 2006. [2] D.E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981) 423–424. [3] J. Bourgain, H.P. Rosenthal, Martingales valued in certain subspaces of L1 , Israel J. Math. 37 (1–2) (1980) 54–75. [4] M. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (3) (1993) 486–490. [5] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965) 1041– 1044. [6] T. Domínguez Benavides, A renorming of some nonseparable Banach spaces with the Fixed Point Property, J. Math. Anal. Appl. 350 (2) (2009) 525–530. [7] P.N. Dowling, C.J. Lennard, Every nonreflexive subspace of L1 [0, 1] fails the fixed point property, Proc. Amer. Math. Soc. 125 (2) (1997) 443–446.

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[8] P.N. Dowling, C.J. Lennard, B. Turett, Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (1996) 653–662. [9] P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964) 181–236. [10] H. Fetter, personal communication. [11] G. Godefroy, N.J. Kalton, D. Li, Propriété d’approximation métrique inconditionnelle et sous-espaces de L1 dont la boule est compacte en measure, C. R. Acad. Sci. Paris Sér. I 320 (1995) 1069–1073. [12] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990. [13] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965) 251–258. [14] C. Graham, A.T.-M. Lau, M. Leinert, Separable translation-invariant subspaces of M(G) and other dual spaces on locally compact groups, Colloq. Math. 55 (1) (1988) 131–145. [15] E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, 1965. [16] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1964) 1004–1006. [17] W.A. Kirk, B. Sims, Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht, 2001. [18] A.T.-M. Lau, M. Leinert, Fixed point property and the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 360 (12) (2008) 6389–6402. [19] A.T.-M. Lau, P.F. Mah, Fixed point property for Banach algebras associated to locally compact groups, J. Funct. Anal. 258 (2) (2010) 357–372. [20] P.K. Lin, There is an equivalent norm on 1 that has the fixed point property, Nonlinear Anal. 68 (8) (2008) 2303– 2308. [21] J. Lindenstrauss, A. Pelczynski, Contributions of the theory of the classical Banach spaces, J. Funct. Anal. 8 (2) (1971) 225–249. [22] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II, Springer-Verlag, 1979. [23] B. Maurey, Points fixes des contractions sur un convexe ferme de L1 , in Seminaire d’Analyse Fonctionnelle, 1980– 1981, Ecole Polytechnique. [24] M. Nowak, Compact Hankel operators with conjugate analytic symbols, Rend. Circ. Mat. Palermo (2) XLVII (1998) 363–374. [25] P. Wojtaszczyk, Banach Spaces of Analysts, Cambridge Stud. Adv. Math., vol. 25, 1991.


Local solvability for a class of evolution equations ✩ Ferruccio Colombini a , Paulo D. Cordaro b,∗ , Ludovico Pernazza c a Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy b Departamento de Matemática Aplicada, IME–USP, Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil c Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy

Received 29 September 2009; accepted 8 December 2009 Available online 12 January 2010 Communicated by Paul Malliavin

Abstract Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider a class of evolution operators with real-analytic coefficients and study their local solvability both in L2 and in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg–Treves condition (ψ) which is suitable to our study. © 2009 Published by Elsevier Inc. Keywords: Local solvability; Linear PDE; Evolution equations

1. Introduction In 1971 Y. Kannai [7] presented an example of a linear partial differential operator which is hypoelliptic but not locally solvable. More precisely, in R2 , where the coordinates are written as (y, t), Kannai considered the operator

K=

✩

∂2 ∂ +t 2 ∂t ∂y

This research project was partially supported by CNPq and Fapesp.


E-mail addresses: [email protected] (F. Colombini), [email protected] (P.D. Cordaro), [email protected] (L. Pernazza). 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.12.004

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and showed that K is hypoelliptic in any open set Ω ⊆ R2 and is not locally solvable near any point of the form (y0 , 0).1 Since the transpose of a hypoelliptic operator is locally solvable (cf. [10, Theorem 52.2]) it follows in particular that the operator K = −t

∂2 ∂ − , 2 ∂t ∂y

= ±1,

is locally solvable, say, near the origin if and only if = −1. A natural extension of this statement was proved in [2]. In this work the authors considered real operators in the form B = −t d

m j,k=1

m ∂ ∂ ∂ ∂ cj k (y, t) − − hj (y, t) + g(y, t), ∂yj ∂yk ∂t ∂yj

= ±1,

()

j =1

m+1 and in which d ∈ N, the coefficients m are smooth in an open neighborhood of the origin in R the quadratic form ξ → j,k=1 cj k (y, t)ξj ξk is positive definite everywhere. The main result in [2] states that when d is even then B is always locally solvable near the origin whereas when d is odd then B is locally solvable near the origin if and only if = −1. Furthermore, if either d is even or if d is odd and = −1 then the solvability of B occurs in the L2 sense. Following the work of R. Beals and C. Fefferman [1] we can account for this result in an invariant formulation. For this we consider real, smooth operators defined in an open neighborhood of the origin in Rn of the form

Q=−

n j,k=1

n ∂ ∂ ∂ d ϕ(x) aj k (x) − bj (x) + c(x), ∂xj ∂xk ∂xj j =1

where d ∈ N and the following conditions are imposed: ϕ(0) = 0 and dϕ = 0 on ϕ −1 (0); n . ξ → A(x)(ξ ) = aj k (x)ξj ξk

is nonnegative for every x ∈ Ω;

(i) (ii)

j,k=1

A(0) has rank n − 1;

(iii)

A(x)(dϕ) = 0 for all x ∈ Ω; . ϑ= bk ∂ϕ/∂xk (0) = 0.

(iv) (v)

k

Notice that, thanks to (iv), the function k bk ∂ϕ/∂xk is invariantly defined on ϕ −1 {0}. In particular, the sign of ϑ is invariantly defined even after multiplication of Q by a nonvanishing smooth function. √ 2s(∂/∂s + ∂ 2 /∂y 2 ) and consequently K is indeed hypoelliptic and locally solvable near any point (y0 , t0 ) with t0 = 0. 1 In the new variable s = t 2 /2 we can write K =


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If we choose the coordinates (x1 , . . . , xn ) in such a way that ϕ = xn then conditions (ii), (iii) and (iv) imply that aj n = anj = 0 for all j = 1, . . . , n and that ξ → n−1 j,k aj k (x)ξj ξk is positive . definite everywhere. Consequently, if we write yj = xj , j = 1, . . . , m = n − 1, t = xn we easily conclude that a nonvanishing multiple of Q has the form (), where equals the sign of ϑ . We can then state: Theorem 1.1. If d is even then Q is locally solvable near the origin. If d is odd then Q is locally solvable near the origin if and only if ϑ < 0. Furthermore, if either d is even or if d is odd and ϑ < 0 then the solvability of Q occurs in the L2 sense.2 Hence, for this class of operators, the local solvability is dictated by the natural generalization of the so-called Nirenberg–Treves condition (ψ) (cf. [8,2]), which in this case reads n ∂ bj (x) sgn ϕ d 0 as a measure. ∂xj j =1

Here sgn(τ ) denotes the sign function, which is defined by sgn(τ ) = 1 if τ > 0, sgn(τ ) = −1 if τ < 0 and sgn(0) = 0. In the present work we shall deepen our analysis for linear partial differential operators in the form P = X ∗ f X − Y + g,

(1)

where now X and Y are real-valued, smooth vector fields defined in an open, connected neighborhood Ω of the origin in Rn , X ∗ denotes the adjoint of X, f is a real-valued, real-analytic function defined in Ω and g is a real-valued, smooth function also defined in Ω. We assume that f does not vanish identically and also that f (0) = 0,

Y = 0 in Ω.

(2)

For operators in the form (1) we have A(x)(ξ ) = −σX (x, ξ )2 , where σX denotes the symbol of X, and then (ii) is always satisfied (recall that σX is purely imaginary). Noticing furthermore that (iv) is equivalent to Xϕ = 0, (iii) is equivalent to n = 2 and X(0) = 0 and (v) is equivalent to Y ϕ(0) = 0, we conclude Corollary 1.1. Suppose that n = 2 and assume that f = ϕ d , for some d ∈ N, where ϕ satisfies (i). Suppose also that X(0) = 0, Y ϕ(0) = 0 and Xϕ = 0. Then the same conclusion as in Theorem 1.1 holds for P . Our goal in the present work is to extend Corollary 1.1 under hypotheses much less restrictive than (i), (iii), (iv) and (v). Although the characterization of the local solvability of P seems to us a rather difficult problem in full generality, we have been able to prove fairly general results that we believe bring a significant understanding to the question. 2 The weak solvability of Q when d = 1 and ϑ < 0 also follows from [1, Theorem 2] in conjunction with [10, Theorem 52.2].

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We summarize the contents of the paper. After a preliminary section where we prove that any partial derivative of the function sgn(f ) is indeed a Radon measure (a result where the assumption of real-analiticity of f plays a crucial role) and study the main properties of such measures, in Section 3 we tackle the study of the L2 -solvability for the operator P . We prove that P is always L2 -solvable near the origin when either f does not change sign or else when Y sgn(f ) 0 as a measure (a property that will be called condition (ψ) in this work) and a weak version of (v) is satisfied. Conversely, in Section 4, we assume that P is real-analytic and prove the necessity of condition (ψ) for the weak solvability of P now under a suitable version of condition (iv). We emphasize that the necessity and the sufficiency of condition (ψ) are proved under different geometrical assumptions. In Section 6 we analyze this gap by presenting the study of a particular but illustrative example. Section 5, in which we relax the assumptions on P , extends the contents of [2, Section 2.2]. We now assume that P is only smooth and prove, by the so-called method of concatenations [4], the local solvability of P when Xf (0) = 0. Here condition (ψ ) is irrelevant and this result is important to illustrate the connection between condition (ψ ) and property (iv) (needless to recall that A(x)(df ) = −σX (x, df )2 = (Xf )2 ). A related example is discussed in Section 6.2. Finally, the authors take the opportunity to express their gratitude to François Treves for his interest in this work and his constant encouragement. 2. Geometrical preliminaries Let f be a real-analytic real-valued function defined in an open neighborhood Ω of the origin in Rn . Assume that f does not vanish identically and that f (0) = 0. Proposition 2.1. Let L be a real vector field on Ω with C 1 coefficients. Then the distribution . μ[L; f ] = L{sgn(f )} is a real Radon measure. Moreover μ[L; f ] = 0 if f does not change sign and f μ[L; f ] = 0, that is, the support of μ[L; f ] is contained in V , the zero set of f . For the proof we recall the following result (cf. [9]): Lemma 2.1. Let p : E → X be a real-analytic morphism between real-analytic spaces such that all fibers p −1 {x} have dimension equal to one. Let also F : E → R be a real-analytic function. Then, given compact sets K2 ⊂ X, K1 ⊂ E there is N = N (K1 , K2 ) such that, for all x ∈ K2 , the restriction of F to p −1 {x} ∩ K1 changes sign at most N times. Proof of Proposition 2.1. We can assume L = ∂/∂x1 . Write the coordinates as x = (x1 , x ), x = (x2 , . . . , xn ) and consider a neighborhood of the origin of the form U = I × B, where I (resp. B) is an open interval (resp. ball) centered at the origin in R (resp. Rn−1 ). Let also K1 (resp. K2 ) be a compact subset of I (resp. B) and set also K = K1 × K2 ⊂ U . If ϕ ∈ Cc∞ (K) then ∂ ∂ϕ x1 , x dx1 dx . ; f (ϕ) = − sgn(f ) x1 , x μ ∂x1 ∂x1

K2

K1


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Applying Lemma 2.1 with the choices p : U → B, p(x1 , x ) = x and F = f we obtain

μ ∂ ; f (ϕ) 2N |K2 | ∂x 1

sup f −1 {0}∩K

|ϕ|.

This inequality shows that μ[∂/∂x1 ; f ] is indeed a (real) Radon measure satisfying the stated properties. 2 We continue the discussion of the measure μ[L; f ]. By contracting Ω about the origin we can assume that df does not vanish in the complement of V . Thanks to this property we can consider the distributions δ(f − σ ), where δ is the Dirac distribution and σ = 0 is real and close to zero. We have (Lf )δ(f − σ ) (ϕ) =

f −1 {σ }

Lf ϕ dHn−1 , |∇f |

ϕ ∈ Cc∞ (Ω),

(3)

where Hn−1 denotes the (n − 1)-Hausdorff measure on f −1 {σ } (cf. [6, Theorem 6.1.5]). Applying a result in [5, Section 1] we obtain, for K ⊂ Ω compact, (Lf )δ(f − σ ) (ϕ) C sup |ϕ|,

ϕ ∈ Cc∞ (K),

(4)

where C > 0 depends on K but not on σ . This shows that the family {(Lf )δ(f − σ )}, σ = 0 small, is bounded in D 0 (Ω), the space of distributions of order 0 defined in Ω. Furthermore, since sgn(f − σ ) → sgn(f ) pointwise in Ω \ V as σ → 0, by the dominated convergence theorem, we obtain sgn(f − σ ) → sgn(f ) in D (Ω) and hence (4) and the argument in the proof of [6, Theorem 2.1.9] show that . μ = μ[L; f ] = lim 2(Lf )δ(f − σ ) in D 0 (Ω).

(5)

σ →0

We further consider the positive measures μ+ = lim 2(Lf )+ δ(f − σ ), σ →0

μ− = lim 2(Lf )− δ(f − σ ), σ →0

|μ| = lim 2|Lf |δ(f − σ ). σ →0

Notice that μ = μ+ − μ− and that |μ| = μ+ + μ− (in other words, (μ+ , μ− ) is the Hahn decomposition of the signed measure μ). We shall now prove the following result. Proposition 2.2. Let F be a compact, subanalytic subset of V of dimension n − 2. Then |μ|(F ) = 0. Proof. Denote by dF the distance function to F and by F ε the set of points x such that dF (x) ε. It is enough to show that |μ|(F ε ) → 0 as ε → 0. Letting ψ(τ ) be a real-valued,

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continuous function on R, such that ψ(τ ) = 1 for 0 τ 1, ψ(τ ) = 0 for τ 2, we have (with an abuse of notation) |μ| F ε |μ| ψ(dF /ε) = 2 lim

σ →0 f −1 {σ }

|Lf | ψ(dF /ε) dHn−1 . |∇f |

Since clearly we have f −1 {σ }

|Lf | n−1 ψ(dF /ε) dH Hn−1 f −1 {σ } ∩ F 2ε , |∇f |

it suffices to show that Hn−1 f −1 {σ } ∩ F 2ε = O(ε)

(6)

uniformly in σ . For this we shall use an integral-geometric estimate [3, 3.2.27]. For each j = 1, . . . , n let pj denote the orthogonal projection pj (x1 , . . . , xn ) = (x1 , . . . , xˆj , . . . , xn ) and let also

H0 f −1 {σ } ∩ F 2ε ∩ pj−1 {w} dw.

aj = Rn−1

Then we have

n

1/2 aj2

n Hn−1 f −1 {σ } ∩ F 2ε aj .

j =1

(7)

j =1

It then suffices to show that aj = O(ε) uniformly in σ , for all j = 1, . . . , n. For simplicity we assume j = 1. Consider the map G : K → Rn , G = (f, p1 ), where K is a compact subanalytic subset of Ω that contains all sets F 2ε , with ε > 0 small. Applying a result in [5, Section 1] we obtain a bound sup H0 G−1 {x} : G−1 (x) is finite < ∞. This means . M = sup H0 f −1 {σ } ∩ K ∩ p1−1 {w} : f −1 {σ } ∩ K ∩ p1−1 {w} is finite < ∞. Applying (7) we obtain a1 Hn−1 f −1 {σ } ∩ F 2ε


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and consequently H0 (f −1 {σ } ∩ F 2ε ∩ p1−1 {w}) < ∞ for w ∈ Bσ,ε , where p1 (F 2ε ) \ Bσ,ε has (n − 1)-dimensional Lebesgue measure equal to zero. But then H0 (f −1 {σ } ∩ F 2ε ∩ p1−1 {w}) M for w ∈ Bσ,ε and so a1 MLn−1 p1 F 2ε . Finally we have p1 (F 2ε ) ⊂ {w: d(w, p1 (F )) 2ε}. Since p1 (F ) is a compact subanalytic subset of Rn−1 and since F has dimension n − 2 it follows that dim p1 (F ) n − 2 and consequently Ln−1 ({w: d(w, p1 (F )) 2ε}) = O(ε) (cf. [3, 3.2.34] in conjunction with [5, Section 3]). 2 It follows from Proposition 2.2 that |μ| = 0 if dim V n − 2. Moreover if we denote by Rn−1 (V ) the set of all points p ∈ V for which V is an (n − 1)-dimensional manifold in a neighborhood of p, and if we set Sn−1 (V ) = V \ Rn−1 (V ), then |μ|(Sn−1 (V )) = 0. Thus we assume dim V = n − 1 and compute the expression of μ± on Rn−1 (V ). We first remark that if we start with Ω sufficiently contracted around the origin, we can decompose Rn−1 (V ) as a finite, disjoint union of connected, real-analytic (n − 1)-dimensional manifolds Nι , each of them containing the origin in its closure (each one of these manifolds is referred to as an (n − 1)-dimensional stratum of V ). For each ι we can take an open connected set Uι containing Nι as a closed set, with Uι ∩ V = Nι , and a real-analytic function ρι defined on Uι such that Nι is defined by ρι = 0 and dρι = 0 on Nι . On Uι we can write f = fι• ριkι , with kι ∈ N and fι• not identically zero in Nι . Notice that either fι• 0 or fι• 0 in Uι and that fι• = 0 on Uι \ Nι . We have μ|Uι = 0 if kι is even. On the other hand, if kι is odd we have sgn(f ) = γ (f, ρι ) sgn(ρι ), where γ (f, ρι ) = 1 if fι• 0 and γ (f, ρι ) = −1 if fι• 0. We have μ|Uι = 2γ (f, ρι )

Lρι dHn−1 |∇ρι |

where now Hn−1 denotes the Hausdorff measure on Nι . We then obtain μ+ |Uι =

2(γ (f, ρι )Lρι )+ dHn−1 , |∇ρι | |μ||Uι =

μ− |Uι =

2(γ (f, ρι )Lρι )− dHn−1 , |∇ρι |

2|Lρι | dHn−1 . |∇ρι |

We shall now take a closer look at the closure (in Ω) of the set where f changes sign: . V0 = Nι .

(8)

kι odd

Notice that V0 is a semianalytic subset of Ω which has dimension n − 1 when f changes sign (otherwise V0 is empty) and that μ[L; f ] is supported in V0 for any C 1 vector field L. Furthermore, if x ∈ / V0 then it is easily seen that f is either 0 or 0 in a neighborhood of x. After a suitable redefinition of the function sgn(f ) on V \ V0 we obtain a new function sgn (f ) which is continuous on Ω \ V0 and satisfies

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sgn (f ) = sgn(f )

−1 sgn (f ) {0} = V0 .

on (Ω \ V ) ∪ V0 ,

(9)

In particular sgn(f ) = sgn (f ) a.e. In order to prepare for the final result of this section we need a simple lemma. Lemma 2.2. Let U be an open and convex subset of Rn and let s : U → {−1, 0, 1} be such that s −1 {−1} and s −1 {1} are open subsets of U . If D denotes the distance function to the closed set . Z = s −1 {0} and if λ = sD then sgn(λ) = s and λ(x) − λ(y) |x − y|,

x, y ∈ U.

(10)

Proof. That sgn(λ) = s is obvious. For the other statement we first point out that (10) is certainly true if s(x) and s(y) have the same sign or else if either s(x) = 0 or s(y) = 0. Suppose then that s(x) = 1 and s(y) = −1. We consider the segment [x, y] joining x to y. Since s −1 {−1} and s −1 {1} are open subsets of U by connectness it follows that [x, y] ∩ Z is nonempty. Let p the nearest point to x belonging to [x, y] ∩ Z and let pj ∈ [x, p[, pj → p. Then s(pj ) = 1 for all j and thus λ(x) − λ(pj ) = D(x) − D(pj ) |x − pj | |x − p|. Likewise let q be the nearest point to y belonging to [x, y] ∩ Z and let qj ∈ ]q, y], qj → q. By the same argument λ(qj ) − λ(y) |q − y| and consequently λ(x) − λ(y) |x − p| + |q − y| + λ(pj ) − λ(qj ) |x − y| + λ(pj ) − λ(qj ). Letting j → ∞ and noticing that λ(pj ), λ(qj ) → 0 we obtain (10).

2

We can now state Proposition 2.3. Let L be a real, smooth vector field in Ω with no critical points and assume that L is transversal to V0 in the following sense: H1 x ∈ γ : D0 (x) ε = O(ε)

(11)

uniformly for an arbitrary orbit γ of L (here D0 denotes the distance function to V0 and H1 denotes the one-dimensional Hausdorff measure on the orbits of L). Then there is a Lipschitz continuous solution to the equation Lh = sgn(f ) in some neighborhood of the origin. . Proof. By Lemma 2.2 the function λ = sgn (f )D0 satisfies (10). We fix a smooth function ψ : R → R satisfying ψ(τ ) = sgn(τ ) for |τ | 1 and set, for ε > 0, ψε (τ ) = ψ(τ/ε). Then ψε (λ) −→ sgn (f ), as ε → 0+ , in D (U ), by the dominated convergence theorem.


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Thanks to the Ascoli–Arzelà theorem, in order to conclude the proof it suffices to show that, in some open and convex neighborhood U of the origin, we can solve Lhε = ψε (λ), where {hε } is a family of Lipschitz continuous functions defined on U satisfying, for some constant C > 0, sup |hε | + sup |∇hε | C, U

ε > 0.

U

After a smooth change of variables we can assume that L = ∂/∂x1 . We shall write x = (x1 , x ), x = (x2 , . . . , xn ) and work in an open neighborhood of the origin of the form U = ]−a, a[ × B Ω, where a > 0 and B is an open ball centered at the origin in Rn−1 . Our transversality hypothesis reads L1 x1 ∈ ]−a, a[: D0 x1 , x ε = O(ε)

(12)

uniformly for x ∈ B. Here now L1 denotes the Lebesgue measure in R. Thus if we form x1 hε (x) =

ψε λ σ, x dσ

−a

we have sup |hε | 2a sup |ψ|, U

∂hε sup |ψ|, sup U ∂x1

and, moreover, (10) and (12) give a ∂hε 1 ψ λ σ, x /ε dσ C supψ ∂x (x) ε k −a

for almost all x ∈ U , all k = 2, . . . , n and some constant C. The proof is complete.

2

Corollary 2.1. Under the same hypotheses as in Proposition 2.3, there is a sequence {hk } ⊂ Cc∞ (Rn ) which converges uniformly in Rn and satisfies the following properties: (1) {hk } is bounded in Cc1 (Rn ); (2) Lhk → sgn (f ) pointwise a.e. in some neighborhood W of the origin in Rn . Proof. Let h be as in the conclusion of Proposition 2.3. After cutting off h near the origin we can take a neighborhood W of the origin in Rn and a compactly supported and Lipschitz continuous function h˜ in Rn such that h˜ = h in W . If {ρε }ε>0 denotes a family of compactly supported . smooth mollifiers and if we set hε = ρε h˜ then {hε }ε>0 is bounded in Cc1 (Rn ), hε → h˜ as + n + p n ˜ ε → 0 uniformly in R and ∂hε /∂xj → ∂ h/∂x j as ε → 0 in L (R ), for all 1 p < ∞ . + and all j = 1, . . . , n. Hence for a conveniently chosen sequence εk → 0 the functions hk = hεk satisfy the required properties. 2

3478


3. Sufficient conditions for L2 -solvability We recall that we are dealing with a linear partial differential operator in the form P = X ∗ f X − Y + g, where X and Y are real-valued, smooth vector fields defined in an open, connected neighborhood Ω of the origin in Rn , f is a real-valued, real-analytic function defined in Ω and g is a real-valued, smooth function also defined in Ω. We assume that f does not vanish identically and also that f (0) = 0,

Y = 0 in Ω.

The operator P is said to be locally solvable near the origin if there is an open neighborhood U ⊂ Ω of the origin for which P D (U ) ⊃ Cc∞ (U ). A stronger concept is that of L2 -solvability: P is said to be L2 -solvable in the open set U ⊂ Ω if P L2 (U ) ⊃ L2 (U ). In the sequel we shall make use of the following well-known statement: P is L2 -solvable in U if and only if there is a constant C > 0 such that u C P ∗ u,

u ∈ Cc∞ (U ),

where the norm is the L2 -norm and P ∗ denotes the (formal) L2 -adjoint of P . We now start the study of the solvability of P , by first stating a fundamental identity. Proposition 3.1. The following identity holds, for all u ∈ Cc∞ (Ω) and all w ∈ C ∞ (Ω): 1 P ∗ u, wu = (wf Xu, Xu) + 2

P w + g + div(Y ) w |u|2 .

(13)

Here (·,·) denotes the inner product in L2 . Before the proof we first point out that for any real, smooth vector field Z and any real-valued, smooth function ψ we have 2(Zu, ψu) =

∗ 2 Z ψ |u| .

(14)

Proof of Proposition 3.1. It suffices to compute P ∗ u, wu = (wf Xu, Xu) + Xu, (f Xw)u + (Y u, wu) + g + div(Y ) u, wu and observe that (14) yields (Y u, wu) =

1 2

∗ 2 Y w |u| ,

From this (13) follows immediately.

2

1 Xu, (f Xw)u = 2

∗ X (f Xw) |u|2 .


3479

Remark 3.1. By taking limits it is clear that (13) remains true for w ∈ D (Ω), where now the integrals must be interpretated as duality brackets between Cc∞ (Ω) and D (Ω). Let now φ be a smooth function on Ω, also real-valued. If Mφ denotes the operator “multiplication” by exp{φ} then we have . Pφ = M−φ P Mφ = X ∗ f X − Yφ + gφ

(15)

Yφ = Y + 2f (Xφ)X

(16)

gφ = g − Y φ − (Xf )(Xφ) − f div(X)Xφ + X 2 (φ) + (Xφ)2 .

(17)

where

and

All these formulas follow from direct computation. We will now prove the main result of this section. For this, and following Nirenberg and Treves [8], we introduce the following definition: Definition 3.1. We shall say that P satisfies condition (ψ) if μ[Y ; f ] 0 as a measure.

(18)

Remark 3.2. It is easily seen that (ψ) is invariant under real-analytic changes of coordinates. On the other hand, if h is a real-valued, real-analytic function defined in Ω and nowhere vanishing, then hP = X ∗ (hf )X − hY + f (Xh)X + hg. Since μ hY − f (Xh)X; hf = |h|μ[Y ; f ] − sgn(h)f (Xh)μ[X; f ] = |h|μ[Y ; f ], it follows that (ψ) is also invariant after multiplication of P by a real-analytic factor that never vanishes. Theorem 3.1. Assume that P satisfies condition (ψ) and that Y is transversal to V0 in the sense of Proposition 2.3. Then P is L2 -solvable in some neighborhood of the origin. Proof. We apply (13) with Pφ substituted for P and w = sgn(f ). Since Pφ sgn(f ) = −μ[Y, f ] + gφ sgn(f ) (13) and condition (ψ ) imply Pφ∗ u, sgn(f )u

1 gφ + div(Yφ ) sgn(f )|u|2 . 2

(19)

3480


Now we have div(Yφ ) = div(Y ) + 2f X(φ) div(X) + 2f ∇(Xφ) · X + 2X(f )X(φ) and then we obtain gφ +

1 1 div(Yφ ) = g + div(Y ) − Y (φ) + f ∇(Xφ) · X − X 2 (φ) − (Xφ)2 . 2 2

Now we take a smooth solution v to the equation Y v = g + (1/2) div(Y ) and set φk = v − hk , where {hk } is the sequence given by Corollary 2.1 when L is replaced by Y . We can of course assume that v is also defined in V , the neighborhood of the origin described in the conclusion of Corollary 2.1. We then apply (19) after replacing φ by φk . Since gφ k +

1 div(Yφk ) = Y hk + f ∇(Xφk ) · X − X 2 (φk ) − (Xφk )2 2

we derive, for u ∈ Cc∞ (V ), Pφ∗k u, sgn(f )u

(Y hk ) sgn(f )|u| − 2

|f |∇(Xφk ) · X − X 2 (φk ) − (Xφk )2 |u|2 .

We then observe that the following property holds: given ε > 0 there is an open neighborhood Vε ⊂ V of the origin such that

|f | ∇(Xφk ) · X − X 2 (φk ) − (Xφk )2 |u|2 −εu2 ,

u ∈ Cc∞ (Vε ), k ∈ N.

Indeed, since f vanishes at the origin, thanks to Corollary 2.1(1) it suffices to show that φ → ∇(Xφ) · X − X 2 (φ) is a first order operator. If we write X = ∇(Xφ) =

aj ∂j we have

(∂j φ)∇(aj ) + aj ∇(∂j φ)

which gives (∂j φ)∇(aj ) · X + aj ∇(∂j φ) · X = aj X(∂j φ) + first order terms =X aj ∂j φ + first order terms

∇(Xφ) · X =

= X 2 (φ) + first order terms, hence our claim.


3481

Choosing ε = 1/2 we obtain, in some neighborhood W of the origin, Pφ∗k u, sgn(f )u

(Y hk ) sgn(f ) − 1/2 |u|2 ,

u ∈ Cc∞ (W ).

Since the Cauchy–Schwarz inequality and Corollary 2.1(1) give Pφ∗k u, sgn(f )u C P ∗ M−φk uM−φk u, where C > 0 is independent of k, we get

e2φk (Y hk ) sgn(f ) − 1/2 |u|2 CP ∗ uu,

u ∈ Cc∞ (W ).

Letting k → ∞, and taking advantage of the properties (1) and (2) in Corollary 2.1, we finally obtain, with a new constant C > 0, u C P ∗ u, The proof of Theorem 3.1 is complete.

u ∈ Cc∞ (W ).

2

Remark 3.3. It is not difficult to see that Theorem 3.1 remains true even if we replace g by a properly supported pseudo-differential operator of order 0. The argument in the proof requires only small modifications. As a consequence we derive our first positive result. Corollary 3.1. If f keeps constant sign in a neighborhood of the origin (in particular, if the germ of V at the origin has dimension n − 2) then P is L2 -solvable in some neighborhood of the origin. Proof. Indeed, in such situation we have μ[Y ; f ] = 0 and V0 = ∅.

2

As a further consequence we state a generalization of Corollary 1.1: Corollary 3.2. Assume f = ϕ d , where d is odd and ϕ is real-analytic. If Y ϕ < 0 then P is L2 -solvable in some neighborhood of the origin. Proof. Since dϕ(0) = 0 it follows that Y is transversal to V = V0 . Moreover, since also sgn(f ) = sgn(ϕ), we have μ[Y ; f ] = μ[Y ; ϕ] =

Yϕ dHn−1 , |∇ϕ|

where Hn−1 denotes the (n − 1)-dimensional Hausdorff measure on the regular set V . But then condition (ψ) holds and the result follows. 2

3482


4. A partial converse to Theorem 3.1 If (ψ) is not satisfied in an open neighborhood U ⊂ Ω of the origin then there is ϕ ∈ Cc (U ), ϕ 0 such that μ(ϕ) = μ[Y, f ](ϕ) > 0. By Proposition 2.2 we have

ϕ dμ =

V

ϕ dμ =

ι

Rn−1 (V )

ϕ dμ > 0

Nι

and then there must exist an (n − 1)-dimensional stratum Nι such that ϕYρι dHn−1 > 0. ϕ dμ = γ (f, ρι ) |∇ρι | Nι

Nι

If we now assume that condition (ψ) is not satisfied in any neighborhood of the origin then the preceding argument allows us to assert the existence of an (n − 1)-dimensional stratum Nι and of a sequence of nonnegative functions {ϕj } ⊂ Cc (U ) such that supp ϕj → {0} and γ (f, ρι ) Nι

ϕj Yρι dHn−1 > 0, |∇ρι |

∀j.

We define Aι as being the open subset of Nι formed by all points p ∈ Nι such that Y (p) is transversal to Nι and f changes sign at p from − to + along the orbit of Y through p. Of course we have Aι = p ∈ Nι : γ (f, ρι )Yρι (p) > 0 . We can then state Lemma 4.1. If condition (ψ) is not satisfied in any neighborhood of the origin then there is an (n − 1)-dimensional stratum Nι of V such that Aι contains the origin in its closure. We are now in a position to prove a partial converse to Theorem 3.1. Theorem 4.1. Assume that P is real-analytic and that condition (ψ) is not satisfied in any neighborhood of the origin and let Nι be an (n − 1)-dimensional stratum of V such that Aι contains the origin in its closure. If X|Nι is tangent to Nι and not identically zero then P is not solvable near the origin. Proof. Since {p ∈ Nι : X(p) = 0}, being the complement of a proper analytic subset of Nι , is dense in Nι , any arbitrary open neighborhood U of the origin contains a point p0 ∈ Nι such that γ (f, ρι )Yρι (p0 ) > 0,

fι• (p0 ) = 0,

X(p0 ) = 0.


3483

We then choose local coordinates (x1 , . . . , xn ) centered at p0 and defined on an open ball B0 in such a way that we can write on B0 : B0 ∩ Nι = B0 ∩ V = {x1 = 0},

f = x12+1 f • ,

Y = ∂/∂x1 ,

f • > 0.

(20)

We then write X=

aj (x)∂xj .

j 1

Since X is tangent to S, we must have a1 (0, x ) = 0 for every x = (x2 , . . . , xn ) near the origin; then we can write a1 = x1 a˜ 1 . Let us now take a real-analytic function v = exp(h), where h solves Xh = −a˜ 1 . This is possible since X does not vanish near p0 . Then Xv + a˜ 1 v = 0 and the change of variables y1 = x1 v(x),

yj = xj ,

j = 2, . . . , n

takes now ∂x1 into (v + x1 vx1 )∂y1 and ∂xj into ∂yj + x1 vxj ∂y1 . The vector field X then becomes

X = a1 (v + x1 vx1 ) + x1 aj vxj ∂x1 + aj ∂yj j 2

= x1 [a˜ 1 v + Xv]∂y1 + =

j 2

aj ∂yj

j 2

aj ∂yj

j 2

and thus P=

j 2

αj ∂yj

∗ f αj ∂yj − β∂y1 + γ . j 2

Dividing by β, whose value at the origin is equal to one, we then have that ∗ • 1 f γ P = y12+1 αj ∂yj αj ∂yj − ∂y1 + bj ∂yj + , β vβ β j 2

j 2

j 2

which is not solvable near p0 thanks to [2, Theorem 3.1].3 It then follows that P is not solvable in U , and this concludes the proof of Theorem 4.1. 2 Again we derive a particular case in order to compare it with Corollary 1.1. 3 What is needed here is indeed a slightly more general version of this result. An analysis of the proof of Theorem 3.1 in [2] shows that, when = 1, there is nonsolvability with the only assumption that there exists ξ ∈ Rn \ {0} such that Q(0, 0)(ξ, ξ ) > 0.

3484


Corollary 4.1. Assume that P is real-analytic and that f = ϕ d , where d is odd and ϕ is realanalytic. If X(0) = 0, X(ϕ) = O(ϕ) and Y ϕ > 0 then P is not solvable near the origin. Indeed, as in the proof of Corollary 3.2, the condition Y (ϕ) > 0 implies that (ψ) is not satisfied on any neighborhood of the origin. 5. Solvability when (ψ) is not necessarily satisfied In this section we prove Theorem 5.1 below. Since its proof only requires that P be smooth, this result generalizes the contents in [2, Section 2.2]. Theorem 5.1. Assume that X, Y, f and g are smooth in Ω. If X(f )(0) = 0 then P is locally solvable near the origin. Proof. We can choose, thanks to our hypotheses, local coordinates (y, t), y = (y1 , . . . , ym ), n = m + 1, such that f (y, t) = t and X = ∂/∂t. The key ingredient in the proof is the so-called “method of concatenations”. It is convenient to write P in the form P = Dt tDt + iB, where B = B(y, t; Dy , Dt ) is a differential operator of order 1 with real principal part. . We set Λ = (1 + Dt2 )1/2 . Making use of the elementary fact that for χ(Dt ), a pseudodifferential operator in R with symbol χ(τ ), we have t, χ(Dt ) = iχ (Dt ),

(21)

then −1/2 Dt tDt Λ = ΛDt tDt + iDt3 1 + Dt2 and thus, setting T = Dt2 (1 + Dt2 )−1 , we can write P Λ = ΛP1 , with P1 = Dt tDt + iT Dt + iΛ−1 BΛ. Iterating the process we can write P Λk = Λk Pk , where Pk = Dt tDt + ikT Dt + iΛ−k BΛk .

(22)

Since the local solvability of P is equivalent to that of Pk it suffices to show that if k is sufficiently large then Pk is L2 -solvable in a small neighborhood U of the origin. We shall assume that U = W × ]−δ, δ[, where W is an open neighborhood of the origin in Rm , and that the coefficients of B are defined in an open set that contains the closure of U .


3485

Let u ∈ Cc∞ (U ). We write Pk∗ u, tu = (Dt tDt u, tu) − k(iT Dt u, tu) − iΛk B ∗ Λ−k u, tu .

(23)

Now, an elementary computation gives 2(Dt tDt u, tu) = 2tDt u2 − u2

(24)

2 2(−iT Dt u, tu) = i T Dt (tu) − tT Dt (u) , u = ψ(Dt )u ,

(25)

and we also have

where ψ(Dt )2 = i[T Dt , t]. By the remark above ψ(τ ) = τ

(3 + τ 2 )1/2 . (1 + τ 2 )

Lemma 5.1. If δ > 0 is small enough then there is c > 0 such that ψ(Dt )u2 cu2 ,

u ∈ Cc∞ (U ).

(26)

Proof. We can write ψ(Dt ) = Dt ψ1 (Dt ) = ψ1 (Dt )Dt where, for every s, ψ1 (Dt ) defines isomorphisms ψ1 (Dt ) : H s (R) → H s+1 (R). Denoting by | · |s the norm in H s (R), it follows the existence of c1 > 0 such that ψ(Dt )v c1 |Dt v|−1 , 0

v ∈ S(R).

On the other hand, the ellipticity of Dt on R implies that, if δ > 0 and c2 > 0 are small enough, |Dt v|−1 c2 |v|0 ,

v ∈ Cc∞ ] − δ, δ[ .

Hence ψ(Dt )v c1 c2 |v|0 , 0

v ∈ Cc∞ ] − δ, δ[ .

Applying this inequality to v = u(y, ·), where u ∈ Cc∞ (U ), and integrating in y ∈ Rm , imply (26). 2 From (23), (24), (25) and (26) we derive 2 Pk∗ u, tu (ck − 1)u2 − 2 iΛk B ∗ Λ−k u, tu . Let us now write B=

m j =1

aj (y, t)Dyj + a0 (y, t)Dt + g(y, t),

(27)

3486


where aj are real-valued for j = 0, 1, . . . , m. After multiplication by a cut-off function we can assume that aj and b are defined in the whole Rn and have compact support. In order to study the last term in the right-hand side of (27) we first notice that iΛk Dyj aj Λ−k u, tu = (iDyj Sk,j u, u), where Sk,j = Λk (taj + M)Λ−k and M is any real constant4 . Next select M > 0 such that taj + M 1. With such a choice we can write Sk,j = Λk (taj + M)1/2 Λ−k 1/2

and consequently 1/2 1/2 1/2 1/2 iΛk Dyj aj Λ−k u, tu = iDyj Sk,j u, Sk,j u + i Dyj , Sk,j Sk,j u, u 1/2 1/2 = i Dyj , Sk,j Sk,j u, u . Notice that the operator Sk,j (y, t; Dt ) is a pseudo-differential operator in t of order 0, depending 1/2 1/2 smoothly in y, and thus bounded in L2 , and that the same is true for [Dyj , Sk,j ]Sk,j . Moreover, the pseudo-differential calculus gives 1/2 1/2 Dyj , Sk,j Sk,j = hj (y, t) + Rj,k (y, t; Dt ), where hj (y, t) is a smooth function and Rj,k (y, t; Dt ) is a pseudo-differential operator in t of order −1, depending smoothly in y. Hence there are constants C > 0 (which is independent of k and j ) and Ck > 0 (which is independent of j ) such that k iΛ Dy aj Λ−k u, tu Cu2 + Ck u0,−1 u, j

(28)

where · 0,−1 denotes the norm in L2 (Rm , H −1 (R)). Next we have 2 iΛk Dt a0 Λ−k u, tu = i tΛk Dt a0 Λ−k − Λ−k a0 Dt Λk t u, u . By the pseudo-differential calculus, tΛk Dt a0 Λ−k − Λ−k a0 Dt Λk t is a pseudo-differential operator in Dt of order zero which can be written as

∂a0 k −k −k k itΛ Dt a0 Λ − iΛ a0 Dt Λ t = t − a0 + tRk,0 (y, t; Dt ) + Rk,−1 (y, t; Dt ), ∂t where Rk,0 (y, t; Dt ) and Rk,−1 (y, t; Dt ) are pseudo-differential operators in Dt of order zero and −1 respectively, depending on k. Hence increasing the constants C > 0 and Ck > 0 if necessary we can write k iΛ Dt a0 Λ−k u, tu Cu2 + Ck δu2 + u0,−1 u . 4 Here we use the elementary fact that (iD u, u) = 0 for every u ∈ S. yj

(29)


3487

Finally, by a very similar argument, we can write k −k iΛ gΛ u, tu Ck δu2 ,

(30)

where the constants Ck > 0 are again appropriately increased if necessary. Putting together (28), (29) and (30) and giving ε > 0 arbitrary we obtain, with new constants C > 0 and Ck > 0 and δ > 0 appropriately small, 2 iΛk B ∗ Λ−k u, tu Cu2 + εCk u2 ,

u ∈ Cc∞ (U ).

(31)

Inserting (31) in (27) gives 2 Pk∗ u, tu (ck − 1 − C − εCk )u2 .

(32)

Choosing k such that ck − 1 − C 1 and then ε 1/(2Ck ), (32) finally yields 4 Pk∗ u, tu u2 ,

u ∈ Cc∞ (U ),

(33)

for some δ > 0 chosen conveniently small. Hence, (33) together with the Cauchy–Schwarz inequality implies the L2 solvability of Pk . The proof of Theorem 5.1 is now complete. 2 6. Final remarks 1. As already mentioned in the Introduction the necessity and the sufficiency of condition (ψ) are proved under different hypotheses. Since filling this gap seems to be a quite difficult question, we limit ourselves to a much more modest task. We assume that P has smooth coefficients and work under the condition X(0) = 0

and Yf > 0 on V

(34)

(recall that V denotes the zero set of the function f ). We provide an answer to a very particular case. Theorem 6.1. Assume that n = 2 and that condition (34) holds. Then P is locally solvable at the origin if and only if Xf (0) = 0. We sketch the proof. Thanks to Theorem 5.1 it suffices to show that P is not locally solvable near the origin when Xf (0) = 0. We start by selecting local coordinates (y, t) near the origin such that Y = ∂/∂y. We then have fy (0, 0) = 0,

Yf (0, 0) > 0.

If we write Y = b(y, t)∂y + c(y, t)∂t

3488


we must have c(0, 0)ft (0, 0) > 0 and then replacing P by −P /c allows us to reduce the proof of Theorem 6.1 to the proof of nonsolvability for the operator P = f (y, t)∂y2 + ∂t + b(y, t)∂y + g(y, t), where fy (0, 0) = 0, ft (0, 0) > 0. After a factorization we write P in the form P = t − γ (y) q(y, t)∂y2 + ∂t + b(y, t)∂y + g(y, t),

(35)

with γ (0) = γ (0) = 0.

q(0, 0) > 0,

(36)

Notice that the case γ ≡ 0 follows from Corollary 4.1 (and is also a direct consequence of Theorem 3.1 in [2]). In the sequel we show that a slight modification of the arguments in [2, Theorem 3.1], allows us to complete the proof of Theorem 6.1. We first observe that P ∗ = t − γ (y) q(y, t)∂y2 − ∂t + b• (y, t)∂y + g • (y, t) for conveniently defined real-analytic functions b• and g • . We then look for quasi-solutions of P ∗ u = 0 of the form uλ (y, t) = eiλΦ(y,t) K(y, t). We have e−iλΦ(y,t) P ∗ uλ = −Kt − λΦt K + b• (Ky + iλΦy K) + g • K + t − γ (y) q(y, t) Kyy − 2iλΦy Ky + iλΦyy K − λ2 Φy2 K

(37)

and consequently, after performing the change of variables z = λ5/2 y,

s = λ3 t,

we see from (37) that we must have −3 λ s − γ˜ (z) q(z, ˜ s) λ5 K˜ zz − 2iλ6 Φ˜ z K˜ z + iλ6 Φ˜ zz K˜ − λ7 Φ˜ z2 K˜ − λ3 K˜ s − λ4 Φ˜ s K˜ + b˜ • λ5/2 K˜ z + iλ7/2 Φ˜ z K˜ + g˜ • K˜ = 0,

(38)

˜ K˜ etc. to denote the functions in the new variables. Furthermore, as in where we used Φ, [2, p. 116], we shall take K˜ to be a formal series of the form ˜ s) = K(z,

j 0

kj (z, s)λ−j/2 .


3489

Thus, after expanding the coefficients γ˜ , q, ˜ b˜ • , g˜ • as power series in (λ−5/2 z, λ−3 s), and observing that (36) gives γ˜ =

∞

γ (j ) (0)λ−5j/2 zj /j !

j =2

we see that the term in (38) with biggest power in λ is given by (−s q(0, ˜ 0)Φ˜ z2 − iΦs )λ4 . From this we obtain the eikonal equation −s q(0, ˜ 0)Φ˜ z2 − iΦs = 0, which we solve under the initial condition Φ(z, 0) = iz + z2 . It is worth mentioning that this Cauchy problem is exactly the same as that arising in [2, p. 116], where it is proved that, near the origin, its solution has imaginary part of the form Φ(z, s) = z2 +

2 q(0, 0) 2 s + O (z, s) . 2

From now on the argument can be completed, with only minor modifications, as in [2, pp. 116–119], for the coefficients kj can recursively be found by solving the usual “transport equations”. The details are left to the interested reader. 2. Finally we discuss a natural question that can be raised after Theorem 5.1. Assume that V is a regular hypersurface. Does the transversality of X to V near the origin imply local solvability? The answer is negative in general, as it can be seen by the following example. Consider the operator in R2 in coordinates (y, t): M,a = ∂t∗ t 3 ∂t − ∂t + a∂y , where = ±1 and a ∈ R. Since this operator is locally L2 -solvable at the origin when = −1 (cf. Theorem 3.1) we limit ourselves to the case = 1. We have Proposition 6.1. M1,a is locally solvable near the origin if and only if a = 0. Proof. We first observe that M1,0 = −∂t t 3 ∂t + 1 . =L and that, by (14), 2 L∗ u, u = 2u2 + 2 t 3 u, ut = 2u2 − 3 if the diameter of U in the t-direction is

t 2 |u|2 u2 ,

u ∈ Cc∞ (U )

√ 3/3. Hence M1,0 is solvable near the origin.

3490


We now show that M1,a is not solvable near the origin for a a = 0. After scaling the y variable we can reduce to the case when a = 1. We proceed as in the proof of Theorem 6.1, that is, we ∗ u = 0 of the form start by finding formal solutions to the equation M1,1 λ uλ (y, t) = eiλφ(y,t) K(y, t). A simple computation then gives e−iλφ M ∗ uλ = −t 3 Ktt + 2iλφt Kt − (λφt )2 K + iλφtt K + 1 − 3t 2 {iλφt K + Kt } − iλφy K − Ky = 0. We now make the change of variables s = λ1/2 t, z = λ1/2 y. In the new variables we must then have 3/2 ˜ + K˜ s − K˜ z − 3is 2 φ˜ s K˜ − is 3 φ˜ ss K˜ − 2is 3 φ˜ s K˜ s λ1/2 i φ˜ s − i φ˜ z + s 3 φ˜ s2 Kλ − s 3 K˜ ss + 3s 2 K˜ s λ−1/2 = 0, ˜ K˜ denote the functions in the new variables. If we choose a real-analytic funcwhere again φ, ˜ tion φ φ˜ s − φ˜ z − is 3 φ˜ s2 = 0

(39)

and take K˜ of the form ˜ s) = K(z,

kj (z, s)λ−j/2

j 0 ∗ u = 0 can be found by solving recursively the it is easily seen that a formal solution to M1,1 λ equations

Lkj = 0, Lkj = s 3 ∂s2 kj −2

j = 0, 1,

+ 3s ∂s kj −2 , 2

(40) j 2,

(41)

where we have written L = 1 − 2is 3 φ˜ s ∂s − ∂z − i 3s 2 φ˜ s + s 3 φ˜ ss . Equations (40) are solved under the conditions kj (0, s) = 1 (j = 0, 1) whereas equations (41) ˜ s) = under the conditions kj (0, s) = 0, j 2. As far as (39) is concerned we write φ(z, j and solve it under the condition φ (z)s j 0 j ˜ 0) = φ0 (z) = z + iz2 . φ(z, After computing the first terms explicitly φ1 (z) = 1 + 2iz,

φ2 (z) = i,

φ3 (z) = 0,

φ4 (z) = i(1 + 2iz)2 /4,


3491

we conclude that ˜ s) = (z + s)2 + s 4 /4 − z2 s 4 + O s 5 , φ(z, which shows that φ˜ has a local strict minimum at the origin. At this point we can refer again to [2, pp. 116–119], since the conclusion of the argument is now completely standard. 2 References [1] R. Beals, C. Fefferman, On hypoellipticity of second order equations, Comm. Partial Differential Equations 1 (1976) 73–85. [2] F. Colombini, L. Pernazza, F. Treves, Solvability and nonsolvability of second-order evolution equations, in: Hyperbolic Problems and Related Topics, in: Grad. Ser. Anal., Int. Press, Somerville, MA, 2003, pp. 111–120. [3] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss., vol. 153, Springer-Verlag, 1996 (reprint of the 1969 edition). [4] A. Gilioli, F. Treves, An example in the solvability theory of linear PDE’s, Amer. J. Math. 96 (1974) 367–385. [5] R. Hardt, Some analytic bounds for subanalytic sets, in: R. Brockett, R. Millman, H. Sussmann (Eds.), Differential Geometric Control Theory, in: Progr. Math., vol. 27, Birkhäuser, 1983, pp. 259–267. [6] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Grundlehren Math. Wiss., vol. 256, SpringerVerlag, 1983. [7] Y. Kannai, An unsolvable hypoelliptic differential operator, Israel J. Math. 9 (1971) 306–315. [8] L. Nirenberg, F. Treves, On local solvability of linear partial differential equations I, Necessary conditions, Comm. Pure Appl. Math. 23 (1970) 1–38. [9] B. Teissier, Appendice: sur trois questions de finitude en géométrie analytique réelle, Acta Math. 51 (1983) 39–48. [10] F. Treves, Topological Vector Spaces, Distributions and Kernels, Pure Appl. Math., vol. 25, Academic Press, 1967.


Cameron–Martin formula for the σ -finite measure unifying Brownian penalisations Kouji Yano Graduate School of Science, Kobe University, Kobe, Japan Received 30 September 2009; accepted 25 November 2009 Available online 9 December 2009 Communicated by Paul Malliavin

Abstract Quasi-invariance under translation is established for the σ -finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor [J. Najnudel, B. Roynette, M. Yor, A remarkable σ -finite measure on C(R+ , R) related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris 345 (8) (2007) 459–466]. For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes. II, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 225–240 (electronic)], plays a key role. © 2009 Elsevier Inc. All rights reserved. Keywords: Cameron–Martin formula; Quasi-invariance; Penalisation; Wiener integral

1. Introduction Let Ω = C([0, ∞) → R). Let (Xt : t 0) denote the coordinate process and set F∞ = σ (Xt : t 0). We consider the following σ -finite measure on (Ω, F∞ ): ∞ W = 0

du Π (u) • R √ 2πu

where Π (u) • R is given as follows: E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.021

(1.1)

K. Yano / Journal of Functional Analysis 258 (2010) 3492–3516

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(i) Π (u) denotes the law of the Brownian bridge from 0 to 0 of length u; (ii) R denotes the law of the symmetrized 3-dimensional Bessel process; (iii) Π (u) • R denotes the concatenation of Π (u) and R. This measure W has been introduced by Najnudel, Roynette and Yor [11,12] so that it unifies various Brownian penalisations. The Brownian penalisations can be explained roughly as follows (we will discuss details in Section 2): For a “good” family {Γt (X)} of non-negative F∞ functionals such that Γt (X) → Γ (X) as t → ∞, it holds that πt (1.2) W Fs (X)Γt (X) −→ W Fs (X)Γ (X) t→∞ 2 for any bounded Fs -measurable functional Fs (X). The purpose of this paper is to establish quasi-invariance of W under h-translation when h belongs to the Cameron–Martin type space:

t h ∈ Ω: ht =

f (s) ds for some f ∈ L (ds) ∩ L (ds) . 2

1

(1.3)

0

Now we state our main theorem. t Theorem 1.1. Suppose that ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Then, for any nonnegative F∞ -measurable functional F (X), it holds that W F (X + h) = W F (X)E(f ; X)

(1.4)

where ∞ E(f ; X) = exp

1 f (s) dXs − 2

0

∞ 2

f (s) ds .

(1.5)

0

Theorem 1.1 will be proved in Section 4. ∞ Theorem 1.1 involves Wiener integral, i.e., the stochastic integral 0 f (s) dXs of a deterministic function f . (To avoid confusion, we give the following remark: In [3,4], the Wiener integral means the integral with respect to the Wiener measure.) The author has proved in his recent work [18] that this Wiener integral is well defined if f ∈ L2 (ds) ∩ L1 ( 1+ds√s ), i.e., ∞ ∞ f (s)2 ds + f (s) 0

0

ds √ < ∞. 1+ s

(1.6)

Note the obvious inclusion: L1 (ds) ⊂ L1 ( 1+ds√s ). We will discuss details in Section 3. One may t conjecture that Theorem 1.1 is valid for ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 ( 1+ds√s ), but we have not succeeded at this point. We give several remarks which help us to understand Theorem 1.1 deeply.

3494


1◦ ). Rephrasing the main theorem. Let g(X) denote the last exit time from 0 for X: g(X) = sup{u 0: Xu = 0}.

(1.7)

For u 0, let θu X denote the shifted process: (θu X)s = Xu+s , s 0. Then the definition (1.1) says that the measure W can be described as follows: (i) W (g(X) ∈ du) = √du ; 2πu (ii) For (Lebesgue) a.e. u ∈ [0, ∞), it holds that, given g(X) = u, (iia) (Xs : s u) is a Brownian bridge from 0 to 0 of length u; (iib) ((θu X)s : s 0) is a symmetrized 3-dimensional Bessel process. In the same manner as this, Theorem 1.1 can be rephrased as the following corollary. We write Th∗ W for the image measure of X + h under W . For u ∈ [0, ∞), we define u Eu (f ; X) = exp 0

Corollary 1.2. Suppose that ht = Th∗ W

t 0

1 f (s) dXs − 2

u 2

f (s) ds .

(1.8)

0

f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Then it holds that ∞

=

du ρ f (u)Π (u),f • R f (·+u)

(1.9)

0

where

1 ρ f (u) = √ Π (u) Eu (f ; ·) R E f (· + u); · , 2πu Π (u),f (dX) = R f (·+u) (dX) =

(1.10)

Eu (f ; X)Π (u) (dX) , Π (u) [Eu (f ; ·)]

(1.11)

E(f (· + u); X)R(dX) . R[E(f (· + u); ·)]

(1.12)

In other words, the law of the process X + h under W may be described as follows: (i ) W (g(X + h) ∈ du) = ρ f (u) du; (ii ) For a.e. u ∈ [0, ∞), it holds that, given g(X + h) = u, (iia ) (Xs + hs : s u) has law Π (u),f ; (iib ) ((θu (X + h))s : s 0) has law R f (·+u) . 2◦ ). Sketch of the proof. We will divide the proof of Theorem 1.1 into the following steps: Step 1. W [F (X + h·∧T )] = W [F (X)ET (f ; X)] for 0 < T < ∞; Step 2. W [F (X)ET (f ; X)] → W [F (X)E(f ; X)] as T → ∞;


3495

Step 3. W [F (X + h·∧T )] → W [F (X + h)] as T → ∞. Note that, in Steps 2 and 3, we will confine ourselves to certain particular classes of test functions F . One may think that Step 1 should be immediate from the following rough argument using (1.2): For any “good” Fs -measurable functional Fs (X), W Fs (X + h·∧T )Γ (X + h·∧T ) = lim

t→∞

πt W Fs (X + h·∧T )Γt (X + h·∧T ) 2

πt W Fs (X)ET (f ; X)Γt (X) = lim t→∞ 2 = W Fs (X)ET (f ; X)Γ (X) .

(1.13) (1.14) (1.15)

This observation, however, should be justified carefully, because the functional ET (f ; X) is not bounded. We shall utilize Markov property for {(Xt ), W } (see Section 2.4 for the details): W FT (X)G(θT X) = W FT (X)WXT G(·)

(1.16)

where Wx is the image measure of x + X under W (dX). The identity (1.16) suggests, in a way, that {Wx : x ∈ R} is a family of exit laws whose transition up to finite time is the Brownian motion, while the Markov property of the Brownian motion asserts that W FT (X)G(θT X) = W FT (X)WXT G(·) .

(1.17)

This makes a remarkable contrast with Itô’s excursion law n (see [8]), which satisfies the Markov property: n FT (X)G(θT X) = n FT (X)WX0 T G(·)

(1.18)

where {(Xt ), (Wx0 )} denotes the Brownian motion killed upon hitting the origin. In other words, n produces a family of entrance laws whose transition after positive time is the killed Brownian motion. ∞ We remark again that the Wiener integral 0 f (s) dXs is not Gaussian. In order to prove necessary estimates involving Wiener integrals in Step 2, we utilize the theory of Wiener integrals for centered Bessel processes, which is due to Funaki, Hariya and Yor [5]. For the 3-dimensional Bessel process {(Xt ), Ra+ } starting from a 0, we define t(a) = Xt − Ra+ [Xt ] X

(1.19)

t ), Ra+ } the centered Bessel process. We shall apply, to the convex function and call {(X |x| ψ(x) = (e − 1)2 , the following theorem, which was proved by Funaki, Hariya and Yor [5] via Brascamp–Lieb inequality [2], and from which we derive our necessary estimates. (a)

Theorem 1.3. (See [5].) For any f ∈ L2 (ds) and any non-negative convex function ψ on R, it holds that

3496


Ra+

∞

∞

(a) t f (t) dX f (t) dXt . ψ W ψ 0

(1.20)

0

For the proof of Theorem 1.3, see [5, Proposition 4.1]. 3◦ ). Comparison with the Brownian case. Let us recall the well-known Cameron–Martin formula for Brownian motion (see [3,4]). Let W stand for the Wiener measure on Ω with W (X0 = 0) = 1. t It is well known that, if ht = 0 f (s) ds with f ∈ L2 (ds), W F (X + h) = W F (X)E(f ; X)

(1.21)

for any non-negative F∞ -measurable functional F (X). It is also well known that, if h ∈ / H , the to W (dX). image measure of X + h under W (dX) is mutually singular on F ∞ t It is immediate from (1.21) that, if ht = 0 f (s) ds with f ∈ L2loc (ds), then W Ft (X + h) = W Ft (X)Et (f ; X)

(1.22)

for any non-negative Ft -measurable functional Ft (X) where t Et (f ; X) = exp

1 f (s) dXs − 2

0

t

f (s)2 ds .

(1.23)

0

Now we give some remarks about comparison between the two cases of W and W . (i) Let f ∈ L2 (ds). As a corollary of (1.21), we see that W [E(f ; X)] < ∞ and, consequently, that W [E(f ; X)p ] < ∞ for any p 1. This shows that, if F (X) ∈ Lp (W (dX)) for some p > 1, then F (X + h) ∈ L1 (W (dX)). Let f ∈ L2 (ds) ∩ L1 (ds). In the case of W , however, we see immediately by taking F ≡ 1 in (1.4) that W E(f ; X) = ∞,

(1.24)

which we should always keep in mind. Now the following question arises: W E(f ; X)Γ (X) < ∞

(1.25)

holds for what functional Γ (X)? The problem is that we do not know the distribution of the ∞ Wiener integral 0 f (s) dXs under W ; in fact, it is no longer Gaussian! In Theorem 4.2, we will appeal to a certain penalisation result and establish (1.25) for Feynman–Kac functionals Γ (X), the class of which we shall introduce in Section 2.2. (ii) In the Brownian case, we have the following criterion: The h-translation of W is quasi/ L2 (ds), respectively. invariant or singular with respect to W according as h ∈ L2 (ds) or h ∈ In the case of W , however, we do not know what happens on W (dX) when h ∈ / H or when f∈ / L1 (ds).


3497

(iii) Let f ∈ L2loc (ds). In the Brownian case, we have the quasi-invariance (1.22) on each Ft . In the case of W (dX), however, we find a drastically different situation (see Theorem 2.5): For any non-negative Ft -measurable functional Ft (X), W Ft (X + h) = W Ft (X) = 0 or ∞

(1.26)

according as W (Ft (X) = 0) = 1 or W (Ft (X) = 0) < 1. 4◦ ). Integration by parts formulae. From the Cameron–Martin theorem (1.21) in the Brownian case, we immediately obtain the following integration by parts formula: ∞ W ∇h F (X) = W F (X) f (s) dXs

(1.27)

0

t

for ht = 0 f (s) ds with f ∈ L2 (ds) and for any good functional F (X), where ∇ denotes the Gross–Sobolev–Malliavin derivative (see, e.g., [17]). In the case of W , from Theorem 1.1, we may expect the following integration by parts formula: ∞ W ∂h F (X) = W F (X) f (s) dXs

(1.28)

0

t for ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds) and for any good functional F (X), where ∂h is in the Gâteaux sense. We have not succeeded in finding a reasonable class of functionals F for which both sides of (1.28) make sense and coincide. Let us give a remark about 3-dimensional Bessel bridge of length u from 0 to 0, which we denote by {(Xs : s ∈ [0, u]), R +,(u) }. Although we do not have the Cameron–Martin formula for the bridge, there is a remarkable result due to Zambotti [20,21] that the following integration by parts formula holds:

R

+,(1)

∞ +,(1) ∂h F (X) = R F (X) f (s) dXs + (BC)

(1.29)

0

t for ht = 0 f (s) ds with f satisfying a certain regularity condition and for any good functional F (X), where ∂h is in the Gâteaux sense and where 1

(BC) = − 0

du hu 2πu3 (1 − u)3

+,(u)

R • R +,(1−u) F (·) .

(1.30)

The remainder term (BC) may describe the boundary contribution. Indeed, the measure R +,(1) is supported on the set of non-negative continuous paths on [0, 1], while the measure R +,(u) • R +,(1−u) is supported on the subset of paths which hit 0 once and only once; the latter set may be regarded in a certain sense as the boundary of the former. See also Bonaccorsi and

3498


Zambotti [1], Zambotti [22], Hariya [7] and Funaki and Ishitani [6] for similar results about integration by parts formulae. The organization of this paper is as follows. In Section 2, we recall several results of Brownian penalisations. In Section 3, we study Wiener integrals for the processes considered. Section 4 is devoted to the proofs of our main theorems. 2. Brownian penalisations 2.1. Notations Let X = (Xt : t 0) denote the coordinate process of the space Ω = C([0, ∞); R) of continuous functions from [0, ∞) to R. Let Ft = σ (Xs : s t) for 0 < t < ∞ and F∞ = σ ( t Ft ). For 0 < u < ∞, we write X (u) = (Xt : 0 t u) and Ω (u) = C([0, u]; R). 1◦ ). Brownian motion. For a ∈ R, we denote by Wa the Wiener measure on Ω with Wa (X0 = a) = 1. We simply write W for W0 . 2◦ ). Brownian bridge. We denote by Π (u) the law on Ω (u) of the Brownian bridge: Π (u) (·) = W (·|Xu = 0).

(2.1)

The process X (u) under Π (u) is a centered Gaussian process with covariance Π (u) [Xs Xt ] = s − st/u for 0 s t u. As a realization of {X (u) , Π (u) }, we may take s Bs − Bu : s ∈ [0, u] . u

(2.2)

3◦ ). 3-Dimensional Bessel process. For a 0, we denote by Ra+ the law √ on Ω of the 3-dimensional Bessel process starting from a, i.e., the law of the process ( Zt ) where (Zt ) is the unique strong solution to the stochastic differential equation dZt = 2 |Zt | dβt + 3 dt,

Z0 = a 2

(2.3)

with (βt ) a one-dimensional standard Brownian motion. Under Ra+ , the process X satisfies dXt = dBt +

1 dt, Xt

X0 = a

(2.4)

with {(Bt ), Ra+ } a one-dimensional standard Brownian motion. − For a > 0, we denote by R−a the law on Ω of (−Xt ) under Ra+ . We define Ra = and

Ra+ Ra−

if a > 0, if a < 0

(2.5)


R = R0 =

R0+ + R0− ; 2

3499

(2.6)

in other words, R is the law on Ω of (εXt ) under the product measure P (dε) ⊗ R0+ (dX) where P (ε = 1) = P (ε = −1) = 1/2. 4◦ ). The σ -finite measure W . For u > 0 and for two processes X (u) = (Xt : 0 t u) and Y = (Yt : t 0), we define the concatenation X (u) • Y as ⎧ ⎨ Xt (u)

X • Y t = Yt−u ⎩ Xu

if 0 t < u, if t u and Xu = Y0 , if t u and Xu = Y0 .

(2.7)

We define the concatenation Π (u) • R as the law of X (u) • Y under the product measure Π (u) (dX (u) ) ⊗ R(dY ). Then we define ∞ W = 0

du Π (u) • R. √ 2πu

(2.8)

For x ∈ R, we define Wx as the image measure of x + X under W (dX); in other words, Wx F (X) = W F (x + X)

(2.9)

for any non-negative F∞ -measurable functional F (X). 5◦ ). Random times. For a ∈ R, we denote the first hitting time of a by τa (X) = inf{t > 0: Xt = a}.

(2.10)

We denote the last exit time from 0 by g(X) = sup{t 0: Xt = 0}.

(2.11)

2.2. Feynman–Kac penalisations y

Let Lt (X) denote the local time by time t of level y: For Wx (dX)-a.e. X, it holds that

t 1A (Xs ) ds = 0

y

Lt (X) dy,

A ∈ B(R), t 0.

(2.12)

A

For a non-negative Borel measure V on R and a process (Xt ) under W (dX), we write Kt (V ; X) = exp − Lxt (X) V (dx) R

(2.13)

3500


and x K(V ; X) = exp − L∞ (X) V (dx) .

(2.14)

R

The following theorem is due to Roynette, Vallois and Yor [14], [15, Theorem 4.1] and [16, Theorem 2.1]. Theorem 2.1. (See [16, Theorem 2.1].) Let V be a non-negative Borel measure on R and suppose that

0< 1 + |x| V (dx) < ∞. (2.15) R

Then the following statements hold: (i) ϕV (x) := limt→∞ πt 2 Wx [Kt (V ; X)] and the limit exists in R+ ; (ii) ϕV is the unique solution of the Sturm–Liouville equation ϕV (x) = 2ϕV (x) V (dx)

(2.16)

in the sense of distributions (see, e.g., [13, Appendix §8]) subject to the boundary condition: lim ϕ (x) = −1 x→−∞ V

and

lim ϕ (x) = 1; x→∞ V

(2.17)

(iii) For any 0 < s < ∞ and any bounded Fs -measurable functional Fs (X), Wx [Fs (X)Kt (V ; X)] ϕV (Xs ) → Wx Fs (X) Ks (V ; X) as t → ∞; Wx [Kt (V ; X)] ϕV (X0 )

(2.18)

(Xs ) (iv) (Ms(V ) (X) := ϕϕVV (X Ks (V ; X): s 0) is a (Wx , (Fs ))-martingale which converges a.s. to 0 0) as s → ∞; (V ) (v) Under the probability measure Wx on F∞ induced by the relation

Wx(V ) Fs (X) = Wx Fs (X)Ms(V ) (X) ,

(2.19)

the process (Xt ) solves the stochastic differential equation t Xt = x + Bt +

ϕV (Xs ) ds ϕV

(2.20)

0

where (Bt ) is a (Wx(V ) , (Ft ))-Brownian motion starting from 0; in particular, the process (Xt ) is a transient diffusion which admits the following function γV (x) as its scale function:


x γV (x) = 0

3501

dy . ϕV (y)2

(2.21)

Remark 2.2. By (ii) of Theorem 2.1, we see that the function ϕV also enjoys the following properties: (V )

(vi) ϕV (x) ∼ |x| as x → ∞. This suggests that the process {(Xt ), (Wx )} behaves like 3dimensional Bessel process when the value of |Xt | is large. (vii) infx∈R ϕV (x) > 0. This shows that the origin is regular for itself. Example 2.3. (A key example for [14].) Suppose that V = λδ0 with some λ > 0 where δ0 denotes the Dirac measure at 0. That is,

Kt (λδ0 ; X) = exp −λL0t (X) .

(2.22)

Then we can solve Eqs. (2.16)–(2.17) and consequently we obtain 1 + |x|, λ

(λδ ) Mt 0 (X) = 1 + λ|Xt | exp −λL0t (X) ϕλδ0 (x) =

(2.23) (2.24)

and t Xt = x + Bt + 0

sgn(Xs ) 1 λ

+ |Xs |

ds

(V )

under Wx .

(2.25)

2.3. The universal σ -finite measure Najnudel, Roynette and Yor [11,12] introduced the measure W on F∞ defined by (2.8) to give a global view on the Brownian penalisations. It unifies the Feynman–Kac penalisations in the sense of the following theorem, which is due to Najnudel, Roynette and Yor [12, Theorem 1.1.2 and Theorem 1.1.6]; see also Yano, Yano and Yor [19, Theorem 8.1]. See also Najnudel and Nikeghbali [9,10] for careful treatment of augmentation of filtrations. Theorem 2.4. (See [12].) Let x ∈ R and let V be a non-negative measure on R satisfying (2.15). Then it holds that Wx Zt (X)K(V ; X) = Wx Zt (X)ϕV (Xt )Kt (V ; X)

(2.26)

for any t 0 and any non-negative Ft -measurable functional Zt (X), where K(V ; X) has been defined as (2.14). Consequently, it holds that ϕV (x) = Wx K(V ; X) and that

(2.27)

3502


Wx(V ) (dX) =

1 K(V ; X) Wx (dX) ϕV (x)

on F∞ .

(2.28)

The following theorem can be found in [12, p. 6, point v) and Theorem 1.1.6]; see also [19, Theorem 5.1]. Theorem 2.5. (See [12].) The following statements hold: (i) W (g(X) ∈ du) =

√du 2πu

on [0, ∞).

In particular, W is σ -finite on F∞ ; (ii) For A ∈ Ft with 0 < t < ∞,

W (A) =

0 if W (A) = 0, ∞ if W (A) > 0.

In particular, W is not σ -finite on Ft . We give the proof for completeness of this paper. Proof. Claim (i) is obvious by definition (1.1) of W . Let us prove claim (ii). Let 0 < t < ∞. Suppose that A ∈ Ft and W (A) = 0. Then we have W [1A K(δ0 ; X)] = 0 by (2.26), which implies that W (A) = 0. Suppose in turn that A ∈ Ft and W (A) > 0. For λ > 0, we apply (2.26) for V = λδ0 and we have 1 1 0 −λL0∞ −λL0t W 1A e−λLt . + |Xt | e = W 1A (2.29) W (A) W 1A e λ λ Letting λ → 0+, we obtain, by the monotone convergence theorem, that W [1A e−λLt ] → W (A) > 0, and consequently, that W (A) = ∞. 2 0

We also need the following property. Proposition 2.6. For x ∈ R, it holds that

Wx τ0 (X) = ∞ = |x|.

(2.30)

Proof. By symmetry, we have only to prove the claim for x 0. Let V = δ0 and F (X) = 1{τ0 (X)=∞} . Note that L0∞ (X) = 0 if τ0 (X) = ∞. Hence it follows from Example 2.3 and Theorem 2.4 that

Wx τ0 (X) = ∞ = ϕδ0 (x)Wx(δ0 ) τ0 (X) = ∞ . (2.31) Since ϕδ0 (x) = 1 + x and since γδ0 (x) =

x 1+x ,

we have

γδ (x) − γδ0 (0) = x. Wx τ0 (X) = ∞ = (1 + x) · 0 γδ0 (∞) − γδ0 (0) The proof is complete.

2

(2.32)


3503

2.4. Markov property of {(Xt ), (Ft ), (Wx )} We may say that {(Xt ), (Ft ), (Wx )} possesses Markov property in the following sense. Theorem 2.7. (See [11,12].) Let x ∈ R and t 0. Let F be a non-negative F∞ -measurable functional. Then it holds that Wx Zt (X)F (θt X) = Wx Zt (X)WXt F (·)

(2.33)

for any non-negative Ft -measurable functional Zt (X). Moreover, the constant time t in (2.33) may be replaced by any finite (Ft )-stopping time τ . Proof. Let V be as in Theorem 2.1. Then we have Wx Zt (X)Kt (V ; X)F (θt X)K(V ; θt X)

by the multiplicativity property of K(V ; X) = Wx Zt (X)F (θt X)K(V ; X)

by (2.28) = ϕV (x)Wx(V ) Zt (X)F (θt X) (V ) (V ) = ϕV (x)Wx(V ) Zt (X)WXt F (·) by the Markov property of W· ϕV (Xt )

(V ) Kt (V ; X) by (2.19) = ϕV (x)Wx Zt (X)WXt F (·) · ϕV (X0 )

= Wx Zt (X)Kt (V ; X)WXt F (·)K(V ; ·) by (2.28) .

(2.34) (2.35) (2.36) (2.37) (2.38) (2.39)

Taking V = λδ0 and letting λ → 0+, we obtain (2.33) by the monotone convergence theorem. In the same way, we can prove (2.33) also in the case where the constant time t is replaced by a finite stopping time τ . 2 Since the measure Wx has infinite total mass, we cannot consider conditional expectation in the usual sense. But, by the help of Theorem 2.7, we can introduce a counterpart in the following sense. Corollary 2.8. (See [11,12]; see also [19].) Let x ∈ R and t 0. Let F be a F∞ -measurable functional which is in L1 (Wx ). Then there exists a unique {(Ft ), Wx }-martingale Mt [F ; X] such that Wx Zt (X)F (X) = Wx Zt (X)Mt [F ; X]

(2.40)

for any bounded Ft -measurable functional Zt (X). Moreover, it is given as Mt [F ; X] = Ω

WXt (dY )F X (t) • Y ,

Wx (dX)-a.s.

(2.41)

3504


Remark 2.9. If F ∈ L1 (Wx ), then the family of the conditional expectations {Wx [F |Ft ]: t 0} is a uniformly integrable martingale. In contrast with this fact, if F ∈ L1 (Wx ), the martingale {Mt [F ; X]: t 0} under Wx converges to 0 as t → ∞, and consequently, it is not uniformly integrable. Remark 2.10. Since Mt is an operator from L1 (Wx ) to L1 (Wx ), we do not have a counterpart of the tower property for the usual conditional expectation. Example 2.11. Let V be a non-negative measure on R satisfying (2.15). Then (iv) and (v) of Theorem 2.1 may be rewritten as Mt K(V ; ·); X = ϕV (Xt )Kt (V ; X).

(2.42)

From this and from Remark 2.2, we see that Mt K(V ; ·); X ∈ Lp (W )

for any p 1.

(2.43)

In particular, formula (2.24) may be rewritten as Mt K(λδ0 ; ·); X =

1 + |Xt | Kt (λδ0 ; X). λ

(2.44)

3. Wiener integrals Let S denote the set of all step functions f on [0, ∞) of the form: f (t) =

n

ck 1[tk−1 ,tk ) (t),

t 0

(3.1)

k=1

with n ∈ N, ck ∈ R (k = 1, . . . , n) and 0 = t0 < t1 < · · · < tn < ∞. Note that S is dense in L2 (ds). For a function f ∈ S and a process X, we define ∞ f (t) dXt =

n

ck (Xtk − Xtk−1 ).

(3.2)

k=1

0

∞

∞ If 0 f (t) dXt can be defined as the limit in some sense of 0 fn (t) dXt for an approximating sequence {fn } of f , then we will call it Wiener integral of f for the process X. We have the following facts: If a sequence {fn } ⊂ S approximates f in L2 (ds), then it holds that ∞

∞ fn (s) dXs −→

f (s) dXs

n→∞

0

and that, for any u > 0,

0

in W -probability

(3.3)


u

3505

u fn (s) dXs −→

0

in Π (u) -probability.

f (s) dXs

n→∞

(3.4)

0

3.1. Wiener integral for 3-dimensional Bessel process Let pt (x) denote the density of the Brownian semigroup: pt (x) = √

2 x , exp − 2t 2πt 1

t > 0, x ∈ R.

(3.5)

Let a 0 be fixed. It is well known (see, e.g., [13, §VI.3]) that, for t > 0 and x > 0, x

a {pt (x − a) − pt (x 2x 2 t pt (x) dx,

Ra+ (Xt ∈ dx) =

+ a)} dx, a > 0, a = 0.

(3.6)

From this formula, it is straightforward that, for t > 0 and x > 0, φa (t) := Ra+

1 Xt

⎧ a ⎨ a1 −a pt (x) dx, = ⎩ 2pt (0) = 2 , πt

a > 0, a = 0.

(3.7)

Since pt (x) pt (0), it is obvious by definition that φa (t) φ0 (t),

a > 0, t > 0.

(3.8)

Note that φa (t) has the following asymptotics as t → 0+:

1/a φa (t) ∼ √ 2/(πt)

if a > 0, if a = 0.

(3.9)

By the stochastic differential equation (2.4), we see that Ra+ [Xt ] = a

t +

Ra+

t 1 ds = a + φa (s) ds, Xs

0

(3.10)

0

Now the following lemma is obvious. Lemma 3.1. Let f ∈ L2 (ds) ∩ L1 (φa (s) ds). Then, according to the stochastic differential equation (2.4), the Wiener integral may be defined as ∞

∞ f (s) dXs =

0

∞ f (s) dBs +

0

f (s) ds. Xs

0

If a sequence {fn } ⊂ S approximates f both in L2 (ds) and in L1 (φa (s) ds), i.e.,

(3.11)

3506


∞ ∞ fn (s) − f (s)2 ds + fn (s) − f (s)φa (s) ds −→ 0, n→∞

0

(3.12)

0

then it holds that ∞


0

in Ra+ -probability.

f (s) dXs

n→∞

(3.13)

0

Following Funaki, Hariya and Yor [5], we may propose another way of constructing the Wiener integral. We define s(a) = Xs − Ra+ [Xs ] X

(3.14)

s for X s(0) . By applying s(a) ), Ra+ } the centered Bessel process. We simply write X and we call {(X Theorem 1.3 with ψ(x) = x 2 , we obtain the following fact: If a sequence {fn } ⊂ S approximates f in L2 (ds), then it holds that ∞

s(a) −→ fn (s) dX

∞

n→∞

0

s(a) f (s) dX

in Ra+ -probability.

(3.15)

0

We then obtain the following lemma. Lemma 3.2. Let f ∈ L2 (ds) ∩ L1 (φa (s) ds). Then it holds that ∞

∞ f (s) dXs =

0

s(a) f (s) dX

0

∞ +

f (s)φa (s) ds

Ra+ -a.s.

(3.16)

0

3.2. Wiener integral for X under W Define L1+ (W ) = G : Ω → R+ , F -measurable, W (G = 0) = 0, W [G] < ∞ .

(3.17)

For G ∈ L1+ (W ), we define a probability measure W G on (Ω, F ) by W G (A) =

W [1A G] , W [G]

A ∈ F.

(3.18)

We recall the following notion of convergence. Proposition 3.3. Let Z, Z1 , Z2 , . . . be F∞ -measurable functionals. Then the following statements are equivalent:


(i) (ii) (iii) (iv)

3507

For any ε > 0 and any A ∈ F with W (A) < ∞, it holds that W (A ∩ {|Zn − Z| ε}) → 0. Zn → Z in W G -probability for some G ∈ L1+ (W ). Zn → Z in W G -probability for any G ∈ L1+ (W ). One can extract, from an arbitrary subsequence, a further subsequence {n(k): k = 1, 2, . . .} along which Zn(k) → Z W -a.e.

If one (and hence all) of the above statements holds, then we say that Zn → Z

locally in W -measure.

(3.19)

For the proof of Proposition 3.3, see, e.g., [18]. Wiener integral for X under W (dX) may be defined with the help of the following theorem. Theorem 3.4. (See [18].) Let f ∈ L2 (ds) ∩ L1 ( 1+ds√s ). Suppose that a sequence {fn } ⊂ S approximates f both in L2 (ds) and in L1 ( 1+ds√s ), i.e., ∞ ∞ fn (s) − f (s)2 ds + fn (s) − f (s) 0

0

ds √ −→ 0. 1 + s n→∞

(3.20)

(Note that this condition is strictly weaker than the condition (3.12).) Then it holds that ∞


f (s) dXs

n→∞

0

locally in W -measure.

(3.21)

0

Moreover, there exists a functional J (f ; u, X) measurable with respect to the product σ -field B([0, ∞)) ⊗ F∞ such that ∞

f (s) dXs = J f ; g(X), X W -a.e.

(3.22)

0

and that it holds du-a.e. that

J f ; u, X (u) • Y =

u

∞ f (s) dXs +

0

f (s + u) dYs

(3.23)

0

is valid a.e. with respect to Π (u) (dX (u) ) ⊗ R(dY ). The following lemma allows us to use the same notation for Wiener integrals under W (dX) and W (dX). Let us temporarily write I W (f ; X) (resp. I W (f ; X)) for the Wiener integral I (f ; X) under W (dX) (resp. W (dX)).

3508


Lemma 3.5. Suppose that there exist F ∈ L1 (W ) and G ∈ L1 (W ) such that W H (X)F (X) = W H (X)G(X)

(3.24)

holds for any bounded measurable functional H (X). Then, for any f ∈ L2 (ds) ∩ L1 ( 1+ds√s ), it holds that

W ϕ I W (f ; X) H (X)F (X) = W ϕ I W (f ; X) H (X)G(X)

(3.25)

for any bounded Borel function ϕ on R. Proof. This is obvious by Theorem 3.4 and by the dominated convergence theorem.

2

3.3. Integrability lemma For later use, we need the following lemma. Lemma 3.6. Let f ∈ L1 (ds). Define ∞ ∞ ds ds f (s + t) √ = f (s) √ f (t) = , s s −t

t > 0.

(3.26)

t

0

Then the following statements hold: (i) For any a > 0, it holds that a

√ f (t) dt 2 a

0

∞ f (s) ds;

(3.27)

0

(ii) There exists a sequence t (n) → ∞ such that f (t (n)) → 0. Proof. (i) Let a > 0. Then we have a

a f (t) dt =

0

a dt

0

a = 0

a 0

t

f (s) √ ds + s−t

ds f (s)

s 0

a

dt + √ s −t

√ f (s)(2 s) ds +

∞ a

dt 0

∞

∞ f (s) √ ds s −t

(3.28)

a

ds f (s)

a

ds f (s)

a 0

a 0

dt √ s−t

dt √ a−t

(3.29)

(3.30)


√ 2 a

∞ f (s) ds.

3509

(3.31)

0

(ii) Let 0 < a a b b

b f (t) dt 2 a

∞ f (s) ds.

(3.32)

0

√ Since (b − a)/ b → ∞ as b → ∞ with a fixed, we see that inft: t>a f (t) = 0 for any a > 0. This implies that lim inf f (t) = 0.

(3.33)

t→∞

The proof is now complete.

2

4. Cameron–Martin formula For a function ht = x ∈ R, we write

t 0

f (s) ds with f ∈ L2 (ds) ∩ L1 ( 1+ds√s ) and a process (Xs ) under Wx for t

Et (f ; X) = exp

1 f (s) dXs − 2

0

t f (s)2 ds

(4.1)

0

and ∞ E(f ; X) = exp 0

1 f (s) dXs − 2

∞ 2

f (s) ds .

(4.2)

0

In what follows, let V be a non-negative Borel measure satisfying (2.15). 4.1. The first step t Proposition 4.1. Let ht = 0 f (s) ds with f ∈ L2 (ds) and let T > 0. Then, for any non-negative F∞ -measurable functional F (X), it holds that W F (X + h·∧T ) = W F (X)ET (f ; X) .

(4.3)

If, moreover, MT [F ; X] ∈ Lp (W ) for some p > 1, then F (X + h·∧T ) ∈ L1 (W ). Proof. Let t T be fixed. By the multiplicativity property of K(δ0 ; ·) and since h(·+t)∧T = hT , we have K(δ0 ; X + h·∧T ) = Kt (δ0 ; X + h·∧T )K(δ0 ; θt X + hT ).

(4.4)

3510


Let Gt (X) be a non-negative Ft -measurable functional. Then, by the Markov property (2.41), by (2.9) and by (2.23), we have Mt K(δ0 ; · + h·∧T ); X = Kt (δ0 ; X + h·∧T )WXt K(δ0 ; X + hT ) = Kt (δ0 ; X + h·∧T )WXt +hT K(δ0 ; X)

= Kt (δ0 ; X + h·∧T ) 1 + |Xt + hT | .

(4.5) (4.6) (4.7)

Hence we obtain W Gt (X + h·∧T )K(δ0 ; X + h·∧T )

= W Gt (X + h·∧T )Kt (δ0 ; X + h·∧T ) 1 + |Xt + hT | .

(4.8)

By the Cameron–Martin formula (1.21), by formula (2.44), and then by the Markov property (2.33), we have

(4.8) = W Gt (X)Kt (δ0 ; X) 1 + |Xt | ET (f ; X) = W Gt (X)Mt K(δ0 ; ·); X ET (f ; X) = W Gt (X)K(δ0 ; X)ET (f ; X) .

(4.9) (4.10) (4.11)

Since t T is arbitrary, we see that W G(X + h·∧T )K(δ0 ; X + h·∧T ) = W G(X)K(δ0 ; X)ET (f ; X)

(4.12)

holds for any non-negative F∞ -measurable functional G(X). Replacing the functional G(X) by F (X)K(δ0 ; X)−1 , we obtain (4.3). Suppose that MT [F ; X] ∈ Lp (W ) for some p > 1. Since ET (h; X) is FT -measurable, we have W F (X)ET (f ; X) = W MT [F ; X]ET (f ; X) 1/p 1/q W MT [F ; X]p W ET (f ; X)q 0. Then it holds that W K(V ; X)E(f ; X) ϕV (0) exp

1 f L1 (ds) . CV

(4.15)


3511

Proof. By Theorem 2.4, we have 1 W K(V ; X)E(f ; X) = W (V ) E(f ; X) . ϕV (0)

(4.16)

By (v) of Theorem 2.1, we see that (4.16) = W

(V )

∞ E(f ; B) exp

ϕ f (s) V (Xs ) ds ϕV

(4.17)

0

where {(Bt ), W (V ) } is a Brownian motion. Since |ϕV (x)| 1 and ϕV (x) CV for any x ∈ R, we have (4.17) W

(V )

∞ 1 f (s) ds . E(f ; B) exp CV

(4.18)

0

Since W (V ) [E(f ; B)] = 1, we obtain the desired inequality.

2

4.3. The second step We utilize the following lemma. Lemma 4.3. Let ht = that

t 0

f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Then, for any 0 < s < ∞, it holds W Et (f ; X)e−g(X) ; g(X) > t −→ 0.

(4.19)

t→∞

Proof. By the Markov property (2.33), we see that W Et (f ; X)e−g(X) ; g(X) > t = W Et (f ; X)e−t WXt e−g(X) ; τ0 (X) < ∞ .

(4.20)

By the strong Markov property (2.33), we see, for any x ∈ R, that

Wx e

−g(X)

; τ0 (X) < ∞ = Wx e−τ0 (X) W0 e−g(X)

∞ 0

du −u 1 e =√ . √ 2πu 2

(4.21)

Hence we obtain 1 1 (4.20) √ e−t W Et (f ; X) = √ e−t −→ 0. t→∞ 2 2 The proof is now complete.

2

(4.22)

3512


Lemma 4.4. Let ht = holds that

t 0

f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Let V be as in Theorem 2.1. Then it

W E(f ; X)K(V ; X)e−g(X) ; g(X) > t −→ 0. t→∞

(4.23)

Proof. Since W [E(f ; X)K(V ; X)] < ∞ by Theorem 4.2. The desired conclusion is now obvious by the dominated convergence theorem. 2 Lemma 4.5. Let ht =

t 0

f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Set ∞ ds f (t) = f (s + t) √ , s 0

! ! σt = !f (· + t)! =

∞

t > 0,

(4.24)

1/2 2

f (s) ds

,

t > 0,

(4.25)

t

and set 2 1 E(t) = E exp σt |N | + cf (t) + σt2 − 1 , 2 where N stands for the standard Gaussian variable and c =

t >0

√ 2/π . Then it holds that

2

Ra E f (· + t); · − 1 E(t) for any t > 0 and any a ∈ R. Proof. Let us write f, g =

∞ 0

(4.26)

(4.27)

f1 (s)f2 (s) ds for f1 , f2 ∈ L2 (ds). Note that

∞ # 1 2 "

(a) s + f (· + t), φa − σt f (s + t) dX E f (· + t); X = exp under Ra+ . 2

(4.28)

0

Since |eb − 1| e|b| − 1 for any b ∈ R, we have 2 ∞ 2

# " 1 E f (· + t); · − 1 exp f (s + t) dX s(a) + f (· + t), φa − σt2 − 1 . 2

(4.29)

0

Since, for any constant b ∈ R, ψ(x) = (e|x+b| − 1)2 is a convex function, we may apply Theorem 1.3 and obtain Ra+ Since

2

# " E f (· + t); · − 12 E exp σt N + f (· + t), φa − 1 σ 2 − 1 . t 2

(4.30)


# " # " f (· + t), φa f (· + t), φ0 = cf (t),

t 0

(4.31)

2

we obtain the desired result. Lemma 4.6. Let ht = t (n) → ∞ such that

3513

f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Then there exists a sequence

W e−g(X) K(V ; X)E(f ; X) − Et (n) (f ; X) → 0.

(4.32)

Proof. By Lemmas 4.3 and 4.4, it suffices to prove that W e−g(X) K(V ; X)E(f ; X) − Et (f ; X); g(X) t

(4.33)

converges to 0 along some sequence t = t (n) → ∞. By the multiplicativity:

E(f ; X) = Et (f ; X)E f (· + t); θt X ,

(4.34)

(4.33) = W e−g(X) K(V ; X)Et (f ; X)E f (· + t); θt X − 1; g(X) t .

(4.35)

we have

By the Schwarz inequality, (4.35) is dominated by A1/2 B 1/2 where A = W K(V ; X)2 Et (f ; X)2

(4.36)

2

B = W e−2g(X) E f (· + t); θt X − 1 ; g(X) t .

(4.37)

and

By Theorem 4.2, we see that

A W K(2V ; X)E(2f 1[0,t) ; X) exp f 2L2 (ds) 2 2 f L1 (ds) . ϕ2V (0) exp f L2 (ds) + C2V

(4.38) (4.39)

By Lemma 4.5, we see that t B= 0

t = 0

2

du −2u (u) Π • R E f (· + t); θt X − 1 e √ 2πu

(4.40)

2

du −2u e R RXt−u E f (· + t); · − 1 √ 2πu

(4.41)

3514


∞ E(t) 0

du −2u e . √ 2πu

(4.42)

Therefore we see that (4.33) is dominated by E(t) up to a multiplicative constant. The proof is now completed by (ii) of Lemma 3.6. 2 4.4. The third step In what follows, we take and utilize a non-negative, bounded, continuous function v0 on R such that v0 (x) = 1 for |x| 2 and v0 (x) = 0 for |x| 3. We write v1 = 1[−1,1] . We set V0 (dx) = v0 (x) dx and V1 (dx) = v1 (x) dx. For any V , we write Γ (V ; X) = e−g(X) K(V ; X). Lemma 4.7. Let ht =

t 0

(4.43)

f (s) ds with f ∈ L1 (ds). Suppose that ∞ f (s) ds 1

(4.44)

T

for some 0 < T < ∞. Then it holds that K(V0 ; X + h·∧t ) K(V1 ; X + h·∧T ),

t T.

(4.45)

Proof. Note that we have |ht − hT | 1 for any t T . If s T satisfies |Xs + hT | 1, then we have |Xs + hs∧t | 2. Hence we have ∞

∞ v0 (Xs + hs∧t ) ds

0

This completes the proof.

v1 (Xs + hs∧T ) ds,

t T.

(4.46)

0

2

t Lemma 4.8. Let ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Let 0 < r < ∞ and let Gr (X) be a non-negative, bounded, continuous Fr -measurable functional. Then it holds that W Gr (X + h·∧t )Γ (V0 ; X + h·∧t ) −→ W Gr (X + h)Γ (V0 ; X + h) . t→∞

(4.47)

Proof. Note that g(X + h·∧t ) → g(X + h) as t → ∞, because h·∧t → h uniformly. By the continuity assumptions on Gr and v, we have Gr (X + h·∧t )Γ (V0 ; X + h·∧t ) → Gr (X + h)Γ (V0 ; X + h)

(4.48)

for W (dX)-almost every path X. Since Gr (X) is bounded, it suffices to find Z ∈ L1 (W ) such that Γ (V0 ; X + h·∧t ) Z(X), W -a.e. for any large t; in fact, we may obtain (4.47) by the dominated convergence theorem.


3515

Since f ∈ L1 (ds), we may take T > 0 such that (4.44) holds. By Lemma 4.7, we have (4.45), and hence we have Γ (V0 ; X + h·∧t ) K(V1 ; X + h·∧T ),

t T.

(4.49)

Since MT [K(V1 ; ·); X] ∈ L2 (W ) by (2.43), we see, by Proposition 4.1, that K(V1 ; X + h·∧T ) ∈ L1 (W ). Therefore this functional K(V1 ; X + h·∧T ) is as desired.

(4.50)

2

4.5. Proof of Theorem 1.1 We now proceed to prove Theorem 1.1. t Proof of Theorem 1.1. Let ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Let 0 < s < ∞ and let Gs (X) be a non-negative, bounded, continuous Fs -measurable functional. Let T > 0. Then, by Proposition 4.1, we have W Gs (X + h·∧T )Γ (V0 ; X + h·∧T ) = W Gs (X)Γ (V0 ; X)ET (f ; X) .

(4.51)

By Lemma 4.8, we have W Gs (X + h·∧T )Γ (V0 ; X + h·∧T ) −→ W Gs (X + h)Γ (V0 ; X + h) . T →∞

(4.52)

By Lemma 4.6, we have W Gs (X)Γ (V0 ; X)ET (f ; X) → W Gs (X)Γ (V0 ; X)E(f ; X)

(4.53)

along some sequence T = t (n) → ∞. Thus, taking the limit as T = t (n) → ∞ in both sides of (4.51), we obtain W Gs (X + h)Γ (V0 ; X + h) = W Gs (X)Γ (V0 ; X)E(f ; X) .

(4.54)

Hence we obtain W G(X + h)Γ (V0 ; X + h) = W G(X)Γ (V0 ; X)E(f ; X)

(4.55)

for any non-negative F∞ -measurable functional G(X). Replacing G(X) by F (X)Γ (V0 ; X)−1 , we obtain the desired conclusion. 2 Acknowledgments The author would like to thank Professors Marc Yor, Tadahisa Funaki, Shinzo Watanabe and Ichiro Shigekawa for their useful comments which helped improve this paper.

3516


References [1] S. Bonaccorsi, L. Zambotti, Integration by parts on the Brownian meander, Proc. Amer. Math. Soc. 132 (3) (2004) 875–883 (electronic). [2] H.J. Brascamp, E.H. Lieb, On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (4) (1976) 366–389. [3] R.H. Cameron, W.T. Martin, Transformations of Wiener integrals under translations, Ann. of Math. (2) 45 (1944) 386–396. [4] R.H. Cameron, W.T. Martin, Transformations of Wiener integrals under a general class of linear transformations, Trans. Amer. Math. Soc. 58 (1945) 184–219. [5] T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes. II, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 225–240 (electronic). [6] T. Funaki, K. Ishitani, Integration by parts formulae for Wiener measures on a path space between two curves, Probab. Theory Related Fields 137 (3–4) (2007) 289–321. [7] Y. Hariya, Integration by parts formulae for Wiener measures restricted to subsets in Rd , J. Funct. Anal. 239 (2) (2006) 594–610. [8] K. Itô, Poisson point processes attached to Markov processes, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, California, 1970/1971, in: Probab. Theory, vol. III, University California Press, Berkeley, CA, 1972, pp. 225–239. [9] J. Najnudel, A. Nikeghbali, On some universal sigma finite measures and some extensions of Doob’s optional stopping theorem, preprint, arXiv:0906.1782, 2009. [10] J. Najnudel, A. Nikeghbali, A new kind of augmentation of filtrations, preprint, arXiv:0910.4959, 2009. [11] J. Najnudel, B. Roynette, M. Yor, A remarkable σ -finite measure on C(R+ , R) related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris 345 (8) (2007) 459–466. [12] J. Najnudel, B. Roynette, M. Yor, A Global View of Brownian Penalisations, MSJ Mem., vol. 19, Mathematical Society of Japan, Tokyo, 2009. [13] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, third ed., Grundlehren Math. Wiss., vol. 293, Springer-Verlag, Berlin, 1999. [14] B. Roynette, P. Vallois, M. Yor, Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I, Studia Sci. Math. Hungar. 43 (2) (2006) 171–246. [15] B. Roynette, P. Vallois, M. Yor, Some penalisations of the Wiener measure, Jpn. J. Math. 1 (1) (2006) 263–290. [16] B. Roynette, M. Yor, Penalising Brownian Paths, Lecture Notes in Math., vol. 1969, Springer, Berlin, 2009. [17] A.S. Üstünel, An Introduction to Analysis on Wiener Space, Lecture Notes in Math., vol. 1610, Springer-Verlag, Berlin, 1995. [18] K. Yano, Wiener integral for the coordinate process under the σ -finite measure unifying Brownian penalisations, preprint, arXiv:0909.5130, 2009. [19] K. Yano, Y. Yano, M. Yor, Penalising symmetric stable Lévy paths, J. Math. Soc. Japan 61 (3) (2009) 757–798. [20] L. Zambotti, Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Related Fields 123 (4) (2002) 579–600. [21] L. Zambotti, Integration by parts on δ-Bessel bridges, δ > 3 and related SPDEs, Ann. Probab. 31 (1) (2003) 323– 348. [22] L. Zambotti, Integration by parts on the law of the reflecting Brownian motion, J. Funct. Anal. 223 (1) (2005) 147–178.


Gradient estimates for the heat equation under the Ricci flow Mihai Bailesteanu a , Xiaodong Cao a , Artem Pulemotov a,b,∗ a Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853-4201, USA b Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637-1514, USA

Received 10 October 2009; accepted 8 December 2009 Available online 21 December 2009 Communicated by L. Gross

Abstract The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on M. Among other results, we prove Li–Yau-type inequalities in this context. We consider both the case where M is a complete manifold without boundary and the case where M is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on M. © 2009 Elsevier Inc. All rights reserved. Keywords: Ricci flow; Heat equation; Li–Yau inequality; Harnack inequality; Manifold with boundary

1. Introduction The paper deals with a manifold M evolving under the Ricci flow and with positive solutions to the heat equation on M. We establish a series of gradient estimates for such solutions including several Li–Yau-type inequalities. First, we study the case where M is a complete manifold without boundary. Our results contain estimates of both local and global nature. Second, we look at the situation where M is compact and has nonempty boundary ∂M. We impose the condition that ∂M remain convex and umbilic at all times. Our arguments then yield two global estimates. * Corresponding author at: Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637-1514, USA. E-mail addresses: [email protected] (M. Bailesteanu), [email protected] (X. Cao), [email protected] (A. Pulemotov).


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M. Bailesteanu et al. / Journal of Functional Analysis 258 (2010) 3517–3542

Suppose M is a manifold without boundary. Let (M, g(x, t))t∈[0,T ] be a complete solution to the Ricci flow ∂ g(x, t) = −2 Ric(x, t), ∂t

x ∈ M, t ∈ [0, T ].

(1.1)

We assume its curvature remains uniformly bounded for all t ∈ [0, T ]. Consider a positive function u(x, t) defined on M × [0, T ]. In Section 2, we assume u(x, t) solves the equation −

∂ u(x, t) = 0, ∂t

x ∈ M, t ∈ [0, T ].

(1.2)

The symbol here stands for the Laplacian given by g(x, t). It is important to emphasize that depends on the parameter t. Thus, we look at the Ricci flow (1.1) combined with the heat equation (1.2). Note that formula (1.1) provides us with additional information about the coefficients of the operator appearing in (1.2) but is itself fully independent of (1.2). To learn about the history, the intuitive meaning, the technical aspects, and the applications of the Ricci flow, one should refer to the many quality books on the subject such as, for example, [12,35,24,9,10]. Problem (1.1) combined with (1.2) admits a simple interpretation in terms of the process of heat conduction. More specifically, one may think of the manifold M with the initial metric g(x, 0) as an object having the temperature distribution u(x, 0). Suppose we let M evolve under the Ricci flow and simultaneously let the heat spread on M. Then the solution u(x, t) will represent the temperature of M at the point x at time t. The work [2] provides a probabilistic interpretation of (1.1)–(1.2). In particular, it constructs a Brownian motion related to u(x, t). The study of system (1.1)–(1.2) arose from R. Hamilton’s paper [16]. The original idea in [16] was to investigate the Ricci flow combined with the heat flow of harmonic maps. The system we examine in Section 2 may be viewed as a special case. The idea to consider the Ricci flow combined with the heat flow of harmonic maps was further exploited in [30,31] for the purposes of regularizing non-smooth Riemannian metrics. We point out, without a deeper explanation, that looking at the two evolutions together leads to interesting simplifications in the analysis. After its conception in [16], the study of (1.1)–(1.2) was pursued in [14,25,38,2,6]. A large amount of work was done to understand several problems that are similar to (1.1)–(1.2) in one way or another. The list of relevant references includes but is not limited to [38,5,6] and [10, Chapter 16]. For instance, there are substantial results concerning the Ricci flow combined with the conjugate heat equation. The connection of this problem to (1.1)–(1.2) is beyond superficial. Q. Zhang used a gradient estimate for (1.1)–(1.2) to prove a Gaussian bound for the conjugate heat equation in [38]. The results of the present paper may have analogous applications. System (1.1)–(1.2) could serve as a model for researching the Ricci flow combined with the heat flow of harmonic maps. There are other geometric evolutions for which (1.1)–(1.2) plays the same role. One example is the Ricci Yang–Mills flow; see [19,33,37]. The analysis of this evolution is technically complicated. Its properties are not yet well understood. We expect that investigating the simpler model case of system (1.1)–(1.2) will provide insight on the behavior of the Ricci Yang–Mills flow. We also speculate that the results of the present paper may aid in proving relevant existence theorems; cf. [1,26] and also [27,7]. The scalar curvature of a surface which evolves under the Ricci flow satisfies the heat equation with a potential on that surface. In the same spirit, we expect to find geometric quantities on M that obey (1.1)–(1.2). The gradient estimates in this paper would then lead to new knowledge


3519

about the behavior of the metric g(x, t) under the Ricci flow. In particular, we believe that our results will be helpful in classifying ancient solutions of (1.1). L. Ni’s work [25] offers yet another way to use the Ricci flow combined with the heat equation to study the evolution of g(x, t). Section 2.2 discusses space-only gradient estimates for system (1.1)–(1.2). The first predecessor of these results was obtained by R. Hamilton in the paper [15]. It applies to the case where M is a closed manifold, the metric g(x, t) is independent of t, and Eq. (1.1) is not in the picture. New versions of R. Hamilton’s result were proposed in [32,38,5,6]. The beginning of Section 2 describes them thoroughly. For related work done by probabilistic methods, one should consult [18, Chapter 5] and [2]. Theorem 2.2 states a space-only gradient estimate for (1.1)–(1.2). It is a result of local nature. Section 2.3 deals with space–time gradient estimates for (1.1)–(1.2). Our results resemble the Li–Yau inequalities from the paper [22]; see also [28, Chapter IV]. More precisely, the solution u(x, t) of Eq. (1.2) satisfies |∇u|2 ut n , − u 2t u2

x ∈ M, t ∈ (0, T ],

(1.3)

if M is a closed manifold with nonnegative Ricci curvature, the metric g(x, t) does not depend on t, and (1.1) is not assumed. Here, ∇ stands for the gradient, the subscript t denotes the derivative in t, and n is the dimension of M. This result goes back to [22] and constitutes the simplest Li–Yau inequality. It opened new possibilities for the comparison of the values of solutions of (1.2) at different points and led to important Gaussian bounds in heat kernel analysis. Integrating the above estimate along a space–time curve yields a Harnack inequality. A precursory form of (1.3) appeared in [3]. Many variants of (1.3) now exist in the literature; see, e.g., [20,11,4,21]. R. Hamilton proved one in [15] which further extended our ability to compare the values of solutions of (1.2). Li–Yau inequalities served as prototypes for many estimates connected to geometric flows. The list of relevant references includes but is not limited to [8,17,10]. In particular, the Li–Yau-type inequality for the Ricci flow became one of the central tools in classifying ancient solutions to the flow as detailed in [12, Chapter 9]. Analogous results played a significant part in the study of Kähler manifolds; see [9, Chapter 2]. Our Theorems 2.7 and 2.9 establish space–time gradient estimates for (1.1)–(1.2). As an application, we lay down two Harnack inequalities for (1.1)–(1.2). They help compare the values of a solution at different points. We are also hopeful that the techniques in Section 2.3 will lead to the discovery of new informative Li– Yau-type inequalities related to the Ricci flow and other geometric flows. Our investigation of (1.1)–(1.2) would then be a model for the proof of such inequalities. In Section 3, we consider the case where M is a compact manifold and ∂M = ∅. We impose the boundary condition on the Ricci flow (1.1) by demanding that the second fundamental form II(x, t) of the boundary with respect to g(x, t) satisfy II(x, t) = λ(t)g(x, t),

x ∈ ∂M, t ∈ [0, T ],

(1.4)

for some nonnegative function λ(t) defined on [0, T ]. Thus, ∂M must remain convex and umbilic1 for all t ∈ [0, T ]. We then assume u(x, t) solves the heat equation (1.2) and satisfies the 1 There is ambiguity in the literature as to the use of the term “umbilic” in this context. See the discussion in [13].

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Neumann boundary condition ∂ u(x, t) = 0, ∂ν

x ∈ ∂M, t ∈ [0, T ].

(1.5)

∂ The outward unit normal ∂ν is determined by the metric g(x, t) and, therefore, depends on the parameter t. The Ricci flow on manifolds with boundary is not yet deeply understood. We remind the reader that Eq. (1.1) fails to be strictly parabolic. As a consequence, it is not even clear how to impose the boundary conditions on (1.1) to obtain a well-posed problem. Progress in this direction was made by Y. Shen in the paper [29]. He proposed to consider the Ricci flow on a manifold with boundary assuming formula (1.4) holds with λ(t) identically equal to a constant. Furthermore, he managed to prove the short-time existence of solutions to the flow in this case. The work [13] continues the investigation of problem (1.1) subject to (1.4) with λ(t) equal to a constant. It also contains a complete set of references on the subject. In the present paper, we consider a more general situation by allowing λ(t) to depend on the parameter t nontrivially. Note that Y. Shen’s method of proving the short-time existence applies to this case, as well. Section 3.1 ponders on the geometric meaning of the function λ(t). We explain why it is beneficial to let λ(t) depend on t. The discussion is rather informal. Section 3.2 provides gradient estimates for system (1.1)–(1.2) subject to the boundary conditions (1.4)–(1.5). Theorems 3.1 and 3.4 state versions of inequalities from Theorems 2.4 and 2.9. Related work was done in [22,36,4,26]. Note that Theorem 3.1 appears to be new in the case where ∂M is nonempty even if g(x, t) is independent of t (see Remark 3.3 for the details). At the same time, the proof is not particularly complicated. Theorems 3.1 and 3.4 are likely to have applications similar to those of Theorems 2.4 and 2.9. We hope that the material in Section 3 will help shed light on the behavior of the Ricci flow on manifolds with boundary. Last but not least, our results may serve as a model for the investigation of problems similar to (1.1)–(1.2)–(1.4)–(1.5). For example, it is natural to look at the Ricci flow subject to (1.4) combined with the conjugate heat equation. As we previously explained, such problems were actively studied on manifolds without boundary, but the case where ∂M is nonempty remains unexplored.

Note. After this paper was completed, we became aware that space–time gradient estimates for (1.1)–(1.2) were researched independently by Shiping Liu in [23], and Jun Sun in [34]. The results of those works are not identical to ours. 2. Manifolds without boundary Our goal is to investigate the Ricci flow combined with the heat equation. The present section establishes space-only and space–time gradient estimates in this context. 2.1. The setup Suppose M is a connected, oriented, smooth, n-dimensional manifold without boundary. Some of the results in this section, but not all of them, concern the case where M is compact.


3521

Given T > 0, assume (M, g(x, t))t∈[0,T ] is a complete solution to the Ricci flow ∂ g(x, t) = −2 Ric(x, t), ∂t

x ∈ M, t ∈ [0, T ].

(2.1)

Suppose a smooth positive function u : M × [0, T ] → R satisfies the heat equation ∂ u(x, t) = 0, − ∂t

x ∈ M, t ∈ [0, T ].

(2.2)

Here, stands for the Laplacian given by g(x, t). In what follows, we will use the notation ∇ and | · | for the gradient and the norm with respect to g(x, t). It is clear that , ∇, and | · | all depend on t ∈ [0, T ]. We will write XY for the scalar product of the vectors X and Y with respect to g(x, t). Section 2.2 offers space-only gradient estimates for u(x, t). These results require that u(x, t) be a bounded function. A local space-only gradient estimate for solutions of (2.2) was originally proved in the paper [32] in the situation where g(x, t) did not depend on t ∈ [0, T ] and (2.1) was not in the picture. It was further generalized in [38] to hold in the case of the backward Ricci flow combined with the heat equation. Our Theorem 2.2 constitutes a version of this result for u(x, t). A global space-only gradient estimate for solutions of (2.2) was originally established in [15] with g(x, t) independent of t ∈ [0, T ] and (2.1) not assumed. It is now known to hold in the cases of both the backward Ricci flow and the Ricci flow combined with the heat equation; see [38,5,6]. We restate it in Theorem 2.4 for the completeness of our exposition. Section 2.3 contains Li–Yau-type estimates for (2.1)–(2.2). As applications, we obtain two Harnack inequalities. The results in this section prevail, with obvious modifications, if the function u(x, t) is defined on M × (0, T ] instead of M × [0, T ]. In order to see this, it suffices to replace u(x, t) and g(x, t) with u(x, t + ) and g(x, t + ) for a sufficiently small > 0, apply the corresponding formula, and then let go to 0. We thus justify, for example, the application of the theorems in Section 2.3 to heat-kernel-type functions. Two more pieces of notation should be introduced at this point. Let us fix x0 ∈ M and ρ > 0. We write dist(χ, x0 , t) for the distance between χ ∈ M and x0 with respect to the metric g(x, t). The notation Bρ,T stands for the set {(χ, t) ∈ M × [0, T ] | dist(χ, x0 , t) < ρ}. We point out that Theorems 2.2 and 2.7 still hold if u(x, t) is defined on Bρ,T instead of M × [0, T ] and satisfies the heat equation in Bρ,T . The proofs in this section will often involve local computations. Therefore, we assume a coordinate system {x1 , . . . , xn } is fixed in a neighborhood of every point x ∈ M. The notation Rij refers to the corresponding components of the Ricci tensor. In order to facilitate the computations, we often implicitly assume that {x1 , . . . , xn } are normal coordinates at x ∈ M with respect to the appropriate metric. We use the standard shorthand: Given a real-valued function f on the ∂f manifold M, the notation fi stands for ∂x , the notation fij refers to the Hessian of f applied to i ∂ ∂ and , and f is the third covariant derivative applied to ∂x∂ i , ∂x∂ j , and ∂x∂ k . The subscript t ij k ∂xi ∂xj designates the differentiation in t ∈ [0, T ]. The proofs of Theorems 2.2 and 2.7 will involve a cut-off function on Bρ,T . The construction of this function will rely on the basic analytical result stated in the following lemma. This result is well known. For example, it was previously used in the proofs of Theorems 2.3 and 3.1 in [38]; see also [28, Chapter IV] and [32].

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Lemma 2.1. Given τ ∈ (0, T ], there exists a smooth function Ψ¯ : [0, ∞) × [0, T ] → R satisfying the following requirements: 1. The support of Ψ¯ (r, t) is a subset of [0, ρ] × [0, T ], and 0 Ψ¯ (r, t) 1 in [0, ρ] × [0, T ]. ¯ 2. The equalities Ψ¯ (r, t) = 1 and ∂∂rΨ (r, t) = 0 hold in [0, ρ2 ] × [τ, T ] and [0, ρ2 ] × [0, T ], respectively. 1

¯ ¯ ¯ 3. The estimate | ∂∂tΨ | C Ψτ 2 is satisfied on [0, ∞) × [0, T ] for some C¯ > 0, and Ψ¯ (r, 0) = 0 for all r ∈ [0, ∞). 2 ¯ ¯a ¯a ¯ 4. The inequalities − CaρΨ ∂∂rΨ 0 and | ∂∂rΨ2 | CaρΨ2 hold on [0, ∞) × [0, T ] for every a ∈ (0, 1) with some constant Ca dependent on a.

2.2. Space-only gradient estimates Let us begin by stating the local space-only gradient estimate. Theorem 2.2. Suppose (M, g(x, t))t∈[0,T ] is a complete solution to the Ricci flow (2.1). Assume that |Ric(x, t)| k for some k > 0 and all (x, t) ∈ Bρ,T . Suppose u : M × [0, T ] → R is a smooth positive function solving the heat equation (2.2). If u(x, t) A for some A > 0 and all (x, t) ∈ Bρ,T , then there exists a constant C that depends only on the dimension of M and satisfies √ 1 A |∇u| 1 1 + log C +√ + k u ρ u t

(2.3)

for all (x, t) ∈ B ρ ,T with t = 0. 2

We will now establish a lemma of computational character. It will play a significant part in the proof of Theorem 2.2. Lemma 2.3. Let (M, g(x, t))t∈[0,T ] be a complete solution to the Ricci flow (2.1). Consider a smooth positive function u : M × [0, T ] → R satisfying the heat equation (2.2). Assume that |∇f |2 . Then the inequality u(x, t) 1 for all (x, t) ∈ Bρ,T . Let f = log u and w = (1−f )2

∂ 2f − w ∇f ∇w + 2(1 − f )w 2 ∂t 1−f

holds in Bρ,T . Proof. A direct computation demonstrates that n 2f 2 fi fij fj fi fj fij ∂ ij w= − +8 −4 ∂t (1 − f )2 (1 − f )3 (1 − f )2 i,j =1

+6

|∇f |4 |∇f |4 − 2 (1 − f )4 (1 − f )3


3523

and

4

n fi fij fj |∇f |4 ∇f ∇w − 4 = 2 (1 − f ) (1 − f )3 (1 − f )4

i,j =1

at every point (x, t) ∈ Bρ,T ; cf. [32,38]. Using these formulas, we conclude that n 2f 2 fi fij fj fi fj fij ∂ ij − w= +4 −4 ∂t (1 − f )2 (1 − f )3 (1 − f )2 i,j =1

|∇f |4 |∇f |4 ∇f ∇w −2 +2 4 (1 − f ) (1 − f ) (1 − f )3 2 n fi fj fij + =2 1−f (1 − f )2 +2

i,j =1

+2 at (x, t) ∈ Bρ,T .

|∇f |4 ∇f ∇w +2 − 2∇f ∇w (1 − f ) (1 − f )3

2f ∇f ∇w + 2(1 − f )w 2 1−f

2

The preparations required to prove Theorem 2.2 are now completed. Note that we will also make use of arguments from the paper [38]. Proof of Theorem 2.2. Without loss of generality, we can assume A = 1. If this is not the case, one should just carry out the proof replacing u(x, t) with u(x,t) A . Let us pick a number τ ∈ (0, T ] and fix a function Ψ¯ (r, t) satisfying the conditions of Lemma 2.1. We will establish (2.3) at (x, τ ) for all x such that dist(x, x0 , τ ) < ρ2 . Because τ is chosen arbitrarily, the assertion of the theorem will immediately follow. Define Ψ : M × [0, T ] → R by the formula Ψ (x, t) = Ψ¯ dist(x, x0 , t), t . It is easy to see that Ψ (x, t) is supported in the closure of Bρ,T . This function is smooth at (x , t ) ∈ M × [0, T ] whenever x = x0 and x is not in the cut locus of x0 with respect to the |∇f |2 metric g(x, t ). We will employ the notation f = log u and w = (1−f introduced in Lemma 2.3. )2 2f It will also be convenient for us to write β instead of − 1−f ∇f . Our strategy is to estimate

( − ∂t∂ )(Ψ w) and scrutinize the produced formula at a point where Ψ w attains its maximum. The desired result will then follow.

3524


We use Lemma 2.3 to conclude that ∂ − (Ψ w) Ψ −β∇w + 2(1 − f )w 2 + (Ψ )w + 2∇Ψ ∇w − Ψt w ∂t in the portion of Bρ,T where Ψ (x, t) is smooth. This implies 2 ∂ (Ψ w) −β∇(Ψ w) + ∇Ψ ∇(Ψ w) + 2Ψ (1 − f )w 2 − ∂t Ψ + wβ∇Ψ − 2

|∇Ψ |2 w + (Ψ )w − Ψt w. Ψ

(2.4)

The latter inequality holds in the part of Bρ,T where Ψ (x, t) is smooth and nonzero. Now let (x1 , t1 ) be a maximum point for Ψ w in the closure of Bρ,T . If (Ψ w)(x1 , t1 ) is equal to 0, then (Ψ w)(x, τ ) = w(x, τ ) = 0 for all x ∈ M such that dist(x, x0 , τ ) < ρ2 . This yields ∇u(x, τ ) = 0, and the estimate (2.3) becomes obvious at (x, τ ). Thus, it suffices to consider the case where (Ψ w)(x1 , t1 ) > 0. In particular, (x1 , t1 ) must be in Bρ,T , and t1 must be strictly positive. A standard argument due to E. Calabi (see, for example, [28, p. 21]) enables us to assume that Ψ (x, t) is smooth at (x1 , t1 ). Because (x1 , t1 ) is a maximum point, the formulas (Ψ w)(x1 , t1 ) 0, ∇(Ψ w)(x1 , t1 ) = 0, and (Ψ w)t (x1 , t1 ) 0 hold true. Together with (2.4), they yield 2Ψ (1 − f )w 2 −wβ∇Ψ + 2

|∇Ψ |2 w − (Ψ )w + Ψt w Ψ

(2.5)

at (x1 , t1 ). We will now estimate every term in the right-hand side. This will lead us to the desired result. A series of computations implies that |wβ∇Ψ | Ψ (1 − f )w 2 +

c1 f 4 , ρ 4 (1 − f )3

|∇Ψ |2 1 c1 w Ψ w2 + 4 , Ψ 8 ρ 1 c1 −(Ψ )w Ψ w 2 + 4 + c1 k 2 8 ρ at (x1 , t1 ) for some constant c1 > 0; see [32,38]. Here, we have used the inequality for the weighted arithmetic mean and the weighted geometric mean, as well as the properties of the function Ψ¯ (r, t) given by Lemma 2.1. Our next mission is to find a suitable bound for (Ψt w)(x1 , t1 ). It is clear that (Ψt w)(x1 , t1 ) =

∂ Ψ¯ dist(x1 , x0 , t1 ), t1 w(x1 , t1 ) ∂t ∂ ∂ Ψ¯ dist(x1 , x0 , t1 ), t1 dist(x1 , x0 , t1 ) w(x1 , t1 ). + ∂r ∂t

(2.6)


3525

We also observe that ∂ Ψ¯ 1 c2 2 ∂t dist(x1 , x0 , t1 ), t1 w(x1 , t1 ) 16 Ψ w (x1 , t1 ) + τ 2 for a positive constant c2 . Because the function Ψ¯ (r, t) satisfies the conditions listed in Lemma 2.1, the inequality ∂ Ψ¯ C 12 1 ∂r dist(x1 , x0 , t1 ), t1 ρ Ψ 2 (x1 , t1 )

(2.7)

holds with C 1 > 0. It remains to estimate the derivative of the distance. Utilizing the assumptions 2 of the theorem, we conclude that ∂ dist(x1 , x0 , t1 ) sup ∂t

dist(x 1 ,x0 ,t1 )

Ric d ζ (s), d ζ (s) ds ds ds

0

k dist(x1 , x0 , t1 ) kρ.

(2.8)

In this particular formula, Ric designates the Ricci curvature of g(x, t1 ). The supremum is taken over all the minimal geodesics ζ (s), with respect to g(x, t1 ), that connect x0 to x1 and are parametrized by arclength; see, e.g., [12, proof of Lemma 8.28]. It now becomes clear that Ψt w

1 1 c2 1 c2 Ψ w 2 + 2 + C 1 kwΨ 2 Ψ w 2 + 2 + c3 k 2 2 16 8 τ τ

at (x1 , t1 ) for some c3 > 0. We have thus found estimates for every term in the right-hand side of (2.5). We will combine them all, and the assertion of the theorem will shortly follow. Given the preceding considerations, formula (2.5) implies Ψ (1 − f )w 2

c4 f 4 4 ρ (1 − f )3

1 c4 c4 + Ψ w 2 + 4 + 2 + c4 k 2 2 ρ τ

at the point (x1 , t1 ). The constant c4 here equals max{3c1 , c2 , c1 + c3 }. Since f (x, t) 0 and f4 1, we can conclude that (1−f )4 Ψ w2

c4 f 4 4 ρ (1 − f )4

Ψ 2 w2 Ψ w2

1 c4 c4 + Ψ w 2 + 4 + 2 + c4 k 2 , 2 ρ τ

4c4 2c4 + 2 + 2c4 k 2 ρ4 τ

at (x1 , t1 ). Because Ψ (x, τ ) = 1 when dist(x, x0 , τ ) < ρ2 , the estimate

3526


w(x, τ ) = (Ψ w)(x, τ ) (Ψ w)(x1 , t1 )

C2 C2 + + C2k τ ρ2

√ holds with C = 2 c4 for all x ∈ M such that dist(x, x0 , τ ) < ρ2 . Recalling the definition of w(x, t) and the fact that τ ∈ (0, T ] was chosen arbitrarily, we obtain the inequality √ |∇f (x, t)| 1 1 C +√ + k 1 − f (x, t) ρ t for (x, t) ∈ B ρ ,T provided t = 0. The assertion of the theorem follows by an elementary compu2 tation. 2 Our next step is to assume M is compact and state a global gradient estimate for the function u(x, t). This result was previously established in [38,6]. We restate it here for the completeness of our exposition. Moreover, we believe it is appropriate to present the proof, which is quite short. A computation from this proof will be used in Section 3. Theorem 2.4. (See Q. Zhang [38], X. Cao and R. Hamilton [6].) Suppose the manifold M is compact, and let (M, g(x, t))t∈[0,T ] be a solution to the Ricci flow (2.1). Assume a smooth positive function u : M × [0, T ] → R satisfies the heat equation (2.2). Then the estimate |∇u| u

A 1 log , t u

x ∈ M, t ∈ (0, T ],

(2.9)

holds with A = supM u(x, 0). Remark 2.5. The maximum principle implies that A is actually equal to supM×[0,T ] u(x, t). This explains why the right-hand side of (2.9) is well defined. A Proof of Theorem 2.4. Consider the function P = t |∇u| u − u log u on the set M × [0, T ]. It is clear that P (x, 0) is nonpositive for every x ∈ M. A computation shows that 2

∂ ∂ |∇u|2 − P =t − ∂t ∂t u n ui uj 2 t uij − =2 0, u u

x ∈ M, t ∈ [0, T ].

i,j =1

In accordance with the maximum principle, this implies P (x, t) is nonpositive for all (x, t) ∈ M × [0, T ]. The desired assertion follows immediately. 2 2.3. Space–time gradient estimates This subsection establishes Li–Yau-type inequalities for system (2.1)–(2.2). We will obtain a local and a global estimate. The following lemma will be important to our considerations; cf. Lemma 1 in [28, Chapter IV]. It will also reoccur in Section 3.


3527

Lemma 2.6. Suppose (M, g(x, t))t∈[0,T ] is a complete solution to the Ricci flow (2.1). Assume that −k1 g(x, t) Ric(x, t) k2 g(x, t) for some k1 , k2 > 0 and all (x, t) ∈ Bρ,T . Suppose u : M × [0, T ] → R is a smooth positive function satisfying the heat equation (2.2). Given α 1, define f = log u and F = t (|∇f |2 − αft ). The estimate 2 ∂ 2aαt − F −2∇f ∇F + |∇f |2 − ft − |∇f |2 − αft ∂t n αtn max k12 , k22 , (x, t) ∈ Bρ,T , − 2k1 αt|∇f |2 − (2.10) 2b holds for any a, b > 0 such that a + b = α1 . Proof. We begin by finding a convenient bound on F . Observe that

n 2 F = t 2 fij + 2fj fj ii − α(ft ) , x ∈ M, t ∈ [0, T ]. i,j =1

Our assumption on the Ricci curvature of M implies the inequality n

fj fj ii =

i,j =1

n

(fj fiij + Rij fi fj )

i,j =1

= ∇f ∇(f ) + Ric(∇f, ∇f ) ∇f ∇(f ) − k1 |∇f |2 at an arbitrary point (x, t) ∈ Bρ,T . Using (2.1), we can show that (ft ) = (f )t − 2

n

Rij fij .

i,j =1

Consequently, the estimate

n 2 2 F t 2 fij + 2αRij fij + 2∇f ∇(f ) − 2k1 |∇f | − α(f )t i,j =1

holds at (x, t) ∈ Bρ,T . Our next step is to find a suitable bound on those terms in the right-hand side that involve fij . We do so by completing the square. More specifically, observe that n n 2 fij + αRij fij = (aα + bα)fij2 + αRij fij i,j =1

i,j =1

=

n √ Rij 2 α aαfij2 + α b fij + √ − Rij2 4b 2 b i,j =1

n α 2 2 aαfij − Rij 4b i,j =1

3528


at (x, t) ∈ Bρ,T for any a, b > 0 such that a + b = α1 . Employing the standard inequality n i,j =1

fij2

(f )2 n

and the assumptions of the lemma, we obtain the estimate n 2 aα αn (f )2 − max k12 , k22 , fij + αRij fij n 4b

(x, t) ∈ Bρ,T .

i,j =1

It is easy to conclude that

2 2 2aα αn 2 2 F t (f ) + 2∇f ∇(f ) − 2k1 |∇f | − α(f )t − max k1 , k2 n 2b 2 2aαt ft − |∇f |2 + 2t∇f ∇ ft − |∇f |2 = n αtn − 2k1 t|∇f |2 − αt ft − |∇f |2 t − max k12 , k22 (2.11) 2b in the set Bρ,T . Formula (2.11) provides us with a convenient bound on F . Let us now include the derivative of F in t ∈ [0, T ] into our considerations. One easily computes ∂F = |∇f |2 − αft + t |∇f |2 − αft t . ∂t Subtracting this from (2.11), we see that the inequality 2 2aαt ∂ F ft − |∇f |2 + 2t∇f ∇ ft − |∇f |2 − 2k1 t|∇f |2 − ∂t n αtn max k12 , k22 − |∇f |2 − αft + (α − 1)t |∇f |2 t − 2b holds in the set Bρ,T . In order to arrive to (2.10) from here, we need to estimate (|∇f |2 )t . The Ricci flow equation (2.1) and the assumptions of the lemma imply |∇f |2 t = 2∇f ∇(ft ) + 2 Ric(∇f, ∇f ) 2∇f ∇(ft ) − 2k1 |∇f |2 at (x, t) ∈ Bρ,T . As a consequence, −

2 2aαt ∂ F ft − |∇f |2 − |∇f |2 − αft ∂t n αtn max k12 , k22 − 2t∇f ∇ |∇f |2 − αft − 2k1 αt|∇f |2 − 2b

in Bρ,T . The desired assertion follows immediately.

2


3529

With Lemma 2.6 at hand, we are ready to establish the local space–time gradient estimate. We will also make use of arguments from the proof of Theorem 4.2 in [28, Chapter IV]. Recall that n designates the dimension of M. Theorem 2.7. Let (M, g(x, t))t∈[0,T ] be a complete solution to the Ricci flow (2.1). Suppose −k1 g(x, t) Ric(x, t) k2 g(x, t) for some k1 , k2 > 0 and all (x, t) ∈ Bρ,T . Consider a smooth positive function u : M × [0, T ] → R solving the heat equation (2.2). There exists a constant C

that depends only on the dimension of M and satisfies the estimate |∇u|2 α2 1 ut nk1 α 3

2 C + max{k + − α α , k } + 1 2 2 2 u α−1 u ρ (α − 1) t

(2.12)

for all α > 1 and all (x, t) ∈ B ρ ,T with t = 0. 2

Proof. We preserve the notation f = log u and F = t (|∇f |2 − αft ) from Lemma 2.6. Our strategy in this proof will be similar to that in the proof of Theorem 2.2. The role of the function w(x, t) now goes to the function F (x, t). Let us pick τ ∈ (0, T ] and fix Ψ¯ (r, t) satisfying the conditions of Lemma 2.1. Define Ψ : M × [0, T ] → R by setting Ψ (x, t) = Ψ¯ dist(x, x0 , t), t . We will establish (2.12) at (x, τ ) for x ∈ M such that dist(x, x0 , τ ) < ρ2 . This will complete the proof. Our plan is to estimate ( ∂t∂ − )(Ψ F ) and analyze the result at a point where the function Ψ F attains its maximum. The required conclusion will follow therefrom. Lemma 2.6 and some straightforward computations imply

∂ − (Ψ F ) −2∇f ∇(Ψ F ) + 2F ∇f ∇Ψ ∂t 2 2aαt + |∇f |2 − ft − |∇f |2 − αft Ψ n αtn ¯ 2 − 2k1 αt|∇f |2 + k Ψ 2b + (Ψ )F + 2

∇Ψ |∇Ψ |2 ∂Ψ ∇(Ψ F ) − 2 F− F Ψ Ψ ∂t

(2.13)

with k¯ = max{k1 , k2 }. This inequality holds in the part of Bρ,T where Ψ (x, t) is smooth and strictly positive. Let (x1 , t1 ) be a maximum point for the function Ψ F in the set {(x, t) ∈ M × [0, τ ] | dist(x, x0 , t) ρ}. We may assume (Ψ F )(x1 , t1 ) > 0 without loss of generality. Indeed, if this is not the case, then F (x, τ ) 0 and (2.12) is evident at (x, τ ) whenever dist(x, x0 , τ ) < ρ2 . We may also assume that Ψ (x, t) is smooth at (x1 , t1 ) due to a standard trick explained, for example, in [28, p. 21]. Since (x1 , t1 ) is a maximum point, the formulas (Ψ F )(x1 , t1 ) 0, ∇(Ψ F )(x1 , t1 ) = 0, and (Ψ F )t (x1 , t1 ) 0 hold true. Combined with (2.13), they yield

3530


0 2F ∇f ∇Ψ 2 2aαt1 αt1 n ¯ 2 |∇f |2 − ft − |∇f |2 − αft − 2k1 αt1 |∇f |2 − + k Ψ n 2b + (Ψ )F − 2

|∇Ψ |2 ∂Ψ F− F Ψ ∂t

(2.14)

at (x1 , t1 ). We will now use (2.14) to show that a certain quadratic expression in Ψ F is nonpositive. The desired result will then follow. Let us recall Lemma 2.1 and apply the Laplacian comparison theorem to conclude that C 21 |∇Ψ |2 − 22 , − Ψ ρ Ψ −

C1

2

ρ2

1

−

C1 Ψ 2 2

ρ

1 d1 d1 Ψ 2 (n − 1) k1 coth( k1 ρ) − 2 − k1 ρ ρ

at the point (x1 , t1 ) with d1 a positive constant depending on n. There exists C¯ > 0 such that the inequality ¯ 12 1 CΨ ∂Ψ ¯ 2 − − C 1 kΨ − 2 ∂t τ holds true; cf. (2.6), (2.7), and (2.8). Using these observations along with (2.14), we find the estimate 0 −2F |∇f ||∇Ψ | 2 2aαt1 αt1 n ¯ 2 2 2 2 |∇f | − ft − |∇f | − αft − 2k1 αt1 |∇f | − + k Ψ n 2b 1 1 1 1 Ψ2 Ψ2 ¯ 2 F + d2 − 2 − − kΨ k1 − ρ τ ρ ¯ If one further multiplies by tΨ and makes at (x1 , t1 ). Here, d2 is equal to max{3d1 , C 1 , 3C 21 , C}. 2

2

a few elementary manipulations, one will obtain 3

0 −2t1 F

C1 Ψ 2 2

ρ

|∇f |

2 2t12 n2 α ¯ 2 2 aα Ψ |∇f |2 − Ψft − nk1 αΨ 2 |∇f |2 − k Ψ n 4b √ 1 k1 1 ¯ − − k (Ψ F ) − Ψ F + d 2 t1 − 2 − ρ τ ρ

+

(2.15)

at (x1 , t1 ). Our next step is to estimate the first two terms in the right-hand side. In order to do so, we need a few auxiliary pieces of notation.

M. Bailesteanu et al. / Journal of Functional Analysis 258 (2010) 3517–3542 1

Define y = Ψ |∇f |2 and z = Ψft . It is clear that y 2 (y − αz) = yields

3531

3

Ψ 2 F |∇f | t

when t = 0, which

2 n2 α ¯ 2 2 2t 2 aα Ψ |∇f |2 − Ψft − nk1 αΨ 2 |∇f |2 − k Ψ ρ n 4b n2 α ¯ 2 2 nC 12 1 2t 2 aα(y − z)2 − nk1 αy − k Ψ − y 2 (y − αz) . n 4b ρ 3

−2tF

C1 Ψ 2 2

|∇f | +

Let us observe that (y − z)2 =

1 (α − 1)2 2 2(α − 1) 2 (y − αz) + y + y(y − αz) α2 α2 α2

and plug this into the previous estimate. Regrouping the terms and applying the inequality κ1 v 2 − κ2

κ2 v − 4κ21 valid for κ1 , κ2 > 0 and v ∈ R, we obtain 2 n2 α ¯ 2 2 2t 2 aα Ψ |∇f |2 − Ψft − nk1 αΨ 2 |∇f |2 − k Ψ ρ n 4b 2 2 3 2 2 2 n k1 α 2t n d2 α n2 α ¯ 2 2 2t a − . (y − αz)2 − (y − αz) + Ψ k n α n 8aρ 2 (α − 1) 4b 4a(α − 1)2 3

−2tF

C1 Ψ 2 2

|∇f | +

Because t (y − αz) = Ψ F by definition, (2.15) now implies 0

2a α nd2 t1 ρ2 (Ψ F )2 + − 2 + 1 + ρ k¯ + + ρ 2 k¯ − 1 (Ψ F ) nα a(α − 1) τ ρ

nk12 α 3 αn 2 ¯ 2 2 t k Ψ t12 − 2 2b 1 2a(α − 1) α ρ2 2a d 3 t1 2 2¯ (Ψ F ) + − 2 + + ρ k − 1 (Ψ F ) nα a(α − 1) τ ρ −

−

nk12 α 3 αn 2 ¯ 2 2 t k Ψ t12 − 2 2b 1 2a(α − 1)

at (x1 , t1 ) with d3 = 4nd2 . The expression in the last two lines is a polynomial in Ψ F of degree 2. Consequently, in accordance with the quadratic formula,

α nα d3 t1 a ¯ k1 α ρ2 2¯ ΨF + +ρ k +1+ t1 + t1 kΨ 2a ρ 2 a(α − 1) τ α−1 b at (x1 , t1 ). We will now use this conclusion to obtain a bound on F (x, τ ) for an appropriate range of x ∈ M. Recall that Ψ (x, τ ) = 1 whenever dist(x, x0 , τ ) < ρ2 . Besides, (x1 , t1 ) is a maximum point for Ψ F in the set {(x, t) ∈ M × [0, τ ] | dist(x, x0 , t) ρ}. Hence

3532


F (x, τ ) = (Ψ F )(x, τ ) (Ψ F )(x1 , t1 )

α ρ2 nk1 α 2 ατ nk¯ 1 nα nαd3 τ 2¯ + +ρ k + + τ+ τ 2a 2a(α − 1) 2 ab 2aρ 2 a(α − 1) for all x ∈ M such that dist(x, x0 , τ ) < ρ2 . Since τ ∈ (0, T ] was chosen arbitrarily, this formula implies α ρ2 αd4 + + ρ 2 k¯ |∇f |2 − αft (x, t) 2 t aρ a(α − 1)

αnk¯ 1 nk1 α 2 + , (x, t) ∈ B ρ ,T , + 2 2a(α − 1) 2 ab 1 with d4 = max{nd3 , n} as long as t = 0. If we set a = 2α , note that b = α1 − a, and define the

constant C appropriately, estimate (2.12) will follow by a straightforward computation. 2 1 Remark 2.8. The value 2α for the parameter a in the proof of the theorem might not be optimal. It is not unlikely that a different a will lead to a sharper estimate.

Let us now consider the case where the manifold M is compact. We will present a global estimate on u(x, t) demanding that the Ricci curvature of M be nonnegative. A related inequality for (2.1)–(2.2) may be found in [14]. Theorem 2.9. Suppose the manifold M is compact and (M, g(x, t))t∈[0,T ] is a solution to the Ricci flow (2.1). Assume that 0 Ric(x, t) kg(x, t) for some k > 0 and all (x, t) ∈ M × [0, T ]. Consider a smooth positive function u : M × [0, T ] → R satisfying the heat equation (2.2). The estimate n |∇u|2 ut kn + − u 2t u2

(2.16)

holds for all (x, t) ∈ M × (0, T ]. Proof. As before, we write f instead of log u. It will be convenient for us to denote F1 = t (|∇f |2 − ft ). Fix τ ∈ (0, T ] and choose a point (x0 , t0 ) ∈ M × [0, τ ] where F1 attains its maximum on M × [0, τ ]. Our first step is to show that n F1 (x0 , t0 ) t0 kn + . 2

(2.17)

The assertion of the theorem will follow therefrom. If t0 = 0, then F1 (x, t0 ) is equal to 0 for every x ∈ M and estimate (2.17) becomes evident. Consequently, we can assume t0 > 0 without loss of generality. Lemma 2.6 and our conditions on the Ricci curvature of M imply the inequality 2a F12 F1 t0 n ∂ k2 − − − F1 −2∇f ∇F1 + ∂t n t0 t0 2(1 − a)


3533

for all a ∈ (0, 1) at the point (x0 , t0 ). Now recall that F1 attains its maximum at (x0 , t0 ). This tells us that F1 (x0 , t0 ) 0, ∂t∂ F1 (x0 , t0 ) 0, and ∇F1 (x0 , t0 ) = 0. In consequence, the estimate 2a F12 F1 t0 n k2 0 − − n t0 t0 2(1 − a) holds at (x0 , t0 ), and the quadratic formula yields n F1 (x0 , t0 ) 1+ 4a

4at02 2 k . 1+ 1−a

1+kt0 . The expression in the right-hand side is minimized in a ∈ (0, 1) when a is equal to 1+2kt 0 Plugging this value of a into the above inequality, we arrive at (2.17). Only a simple argument is now needed to complete the proof. The fact that (x0 , t0 ) is a maximum point for F1 on M × [0, τ ] enables us to conclude that

F1 (x, τ ) F1 (x0 , t0 ) t0 kn +

n n τ kn + 2 2

for all x ∈ M. Therefore, the estimate n |∇u|2 ut kn + − u 2τ u2 holds at (x, τ ). Because the number τ ∈ (0, T ] can be chosen arbitrarily, the assertion of the theorem follows. 2 Our last goal in this section is to state two Harnack inequalities for (2.1)–(2.2). These may be viewed as applications of Theorems 2.7 and 2.9; cf., for example, [28, Chapter IV]. One can find other Harnack inequalities for (2.1)–(2.2) in the papers [14,25]. We first introduce a piece of notation. Given x1 , x2 ∈ M and t1 , t2 ∈ (0, T ) satisfying t1 < t2 , define 2 t2 d Γ (x1 , t1 , x2 , t2 ) = inf γ (t) dt. dt t1

The infimum is taken over the set Θ(x1 , t1 , x2 , t2 ) of all the smooth paths γ : [t1 , t2 ] → M that connect x1 to x2 . We remind the reader that the norm | · | depends on t. Let us now present a lemma. It will be the key to the proof of our results. Lemma 2.10. Suppose (M, g(x, t))t∈[0,T ] is a complete solution to the Ricci flow (2.1). Let u : M × [0, T ] → R be a smooth positive function satisfying the heat equation (2.2). Define f = log u and assume that 1 ∂f A3 |∇f |2 − A2 − , ∂t A1 t

x ∈ M, t ∈ (0, T ],

3534


for some A1 , A2 , A3 > 0. Then the inequality − A3 A1 t2 A1 A2 u(x2 , t2 ) u(x1 , t1 ) exp − Γ (x1 , t1 , x2 , t2 ) − (t2 − t1 ) t1 4 A1 holds for all (x1 , t1 ) ∈ M × (0, T ) and (x2 , t2 ) ∈ M × (0, T ) such that t1 < t2 . Proof. The method we use is rather traditional; see, for example, [28, Chapter IV] and [6]. Consider a path γ (t) ∈ Θ(x1 , t1 , x2 , t2 ). We begin by computing d d ∂ f γ (t), t = ∇f γ (t), t γ (t) + f γ (t), s s=t dt dt ∂s d 2 1 A3 ∇f γ (t), t − A2 − −∇f γ (t), t γ (t) + dt A1 t 2 A1 d 1 A3 A2 + , t ∈ [t1 , t2 ]. − γ (t) − 4 dt A1 t κ2

The last step is a consequence of the inequality κ1 v 2 − κ2 v − 4κ21 valid for κ1 , κ2 > 0 and v ∈ R. The above implies t2 f (x2 , t2 ) − f (x1 , t1 ) =

d f γ (t), t dt dt

t1

A1 − 4

2 t2 d γ (t) dt − A2 (t2 − t1 ) − A3 ln t2 . dt A1 A1 t1

t1

The assertion of the lemma follows by exponentiating.

2

We are ready to formulate our Harnack inequalities for (2.1)–(2.2). The first one applies on noncompact manifolds. The second one does not, but it provides a more explicit estimate. Theorem 2.11. Let (M, g(x, t))t∈[0,T ] be a complete solution to the Ricci flow (2.1). Assume that −k1 g(x, t) Ric(x, t) k2 g(x, t) for some k1 , k2 > 0 and all (x, t) ∈ M × [0, T ]. Suppose a smooth positive function u : M × [0, T ] → R satisfies the heat equation (2.2). Given α > 1, the estimate −C α t2 u(x2 , t2 ) u(x1 , t1 ) t1 α nk1 α 2

(t2 − t1 ) × exp − Γ (x1 , t1 , x2 , t2 ) − C α max{k1 , k2 } + 4 α−1 holds for all (x1 , t1 ) ∈ M × (0, T ) and (x2 , t2 ) ∈ M × (0, T ) such that t1 < t2 . The constant C

comes from Theorem 2.7.


3535

Proof. Letting ρ go to infinity in (2.12), we conclude that nk1 α 3 ut 1 |∇u|2 C α 2

2 − − C α max{k1 , k2 } + u α t α−1 u2 on M × (0, T ]. The desired assertion is now a consequence of Lemma 2.10.

2

Theorem 2.12. Suppose M is compact and (M, g(x, t))t∈[0,T ] is a solution to the Ricci flow (2.1). Assume that 0 Ric(x, t) kg(x, t) for some k > 0 and all (x, t) ∈ M × [0, T ]. Consider a smooth positive function u : M × [0, T ] → R satisfying the heat equation (2.2). The estimate − n 2 t2 1 u(x2 , t2 ) u(x1 , t1 ) exp − Γ (x1 , t1 , x2 , t2 ) − kn(t2 − t1 ) t1 4 holds for all (x1 , t1 ) ∈ M × (0, T ) and (x2 , t2 ) ∈ M × (0, T ) as long as t1 < t2 . Proof. Theorem 2.9 implies ut |∇u|2 n − kn − , 2 u 2t u

x ∈ M, t ∈ (0, T ].

One may now use Lemma 2.10 to complete the proof.

2

3. Manifolds with boundary This section considers a compact manifold with boundary evolving under the Ricci flow and offers heat equation estimates on this manifold. We will present variants of Theorems 2.4 and 2.9. The proofs are largely based on the Hopf maximum principle. 3.1. The Ricci flow Suppose M is a compact, connected, oriented, smooth manifold with nonempty boundary ∂M. Consider a Riemannian metric g(x, t) on M that evolves under the Ricci flow. The parameter t runs through the interval [0, T ]. We investigate the case where the boundary ∂M remains umbilic for all t ∈ [0, T ]. More precisely, given a smooth nonnegative function λ(t) on [0, T ], we assume that (M, g(x, t))t∈[0,T ] is a solution to the problem ∂ g(x, t) = −2 Ric(x, t), x ∈ M, t ∈ [0, T ], ∂t II(x, t) = λ(t)g(x, t), x ∈ ∂M, t ∈ [0, T ].

(3.1)

In the second line, g(x, t) is understood to be restricted to the tangent bundle of ∂M. The notation II(x, t) here stands for the second fundamental form of ∂M with respect to g(x, t). That is, ∂ II(X, Y ) = DX Y ∂ν

3536


Fig. 1. The Ricci flow (3.1) with λ(t) = λ0 after the normalization.

Fig. 2. The Ricci flow (3.1) with λ(t) = λ1 (t) before the normalization.

if X and Y are tangent to the boundary at the same point. The letter D refers to the Levi-Civita ∂ is the outward unit normal vector field on ∂M with connection corresponding to g(x, t), and ∂ν respect to g(x, t). We should explain that problem (3.1) has different geometric meanings for different choices of the function λ(t). E.g., let us assume that λ(t) is equal to the same constant λ0 for all t ∈ [0, T ]. The papers [29,13] discuss this case in detail. Theorem 3 in [13] suggests that the Ricci flow (3.1), if normalized so as to preserve the volume of M, takes a sufficiently well-behaved Riemannian metric on M to a metric with totally geodesic boundary. An example of such an evolution is shown in Fig. 1. By letting λ(t) be a nontrivial function of t, we allow our results to include several cases which are, in a sense, more natural than the one just described. For instance, suppose we apply the Ricci flow to the sphere S in Fig. 1. The manifold M will then evolve along with S. This evolution will be described by Eqs. (3.1) with λ(t) equal to some nonconstant function λ1 (t). We provide an illustration in Fig. 2. Let us normalize the Ricci flow on the sphere S so as to preserve the volume of S. It is well known that S will then remain unchanged for all t. Analogously, we can normalize the Ricci flow (3.1) with λ(t) = λ1 (t) so as to preserve the volume of M. This will allow a better comparison with the situation shown in Fig. 1. After such a normalization, the flow will keep M unchanged for all t. 3.2. Gradient estimates Let us recollect some notation. The operator is the Laplacian given by the metric g(x, t). We write ∇ and | · | for the gradient and the norm with respect to g(x, t). Our attention will be centered round the heat equation ∂ u(x, t) = 0, x ∈ M, t ∈ [0, T ], (3.2) − ∂t


3537

with the Neumann boundary condition ∂ u(x, t) = 0, ∂ν

x ∈ ∂M, t ∈ [0, T ].

(3.3)

The results in this section still hold, with obvious modifications, if the solution u(x, t) is only defined on M × (0, T ]. In this case, one just has to replace u(x, t) and g(x, t) with u(x, t + ) and g(x, t + ) for a sufficiently small > 0, apply the corresponding theorem, and then let go to 0. Our first result is a space-only estimate. It is analogous to (2.9). Theorem 3.1. Let (M, g(x, t))t∈[0,T ] be a solution to the Ricci flow (3.1). Suppose u(x, t) : M × [0, T ] → R is a smooth positive function satisfying the heat equation (3.2) with the Neumann boundary condition (3.3). Then the estimate |∇u| u

A 1 log , t u

x ∈ M, t ∈ (0, T ],

(3.4)

holds with A = supM u(x, 0). Remark 3.2. Using the strong maximum principle and the Hopf maximum principle, one can show that A is actually equal to supM×[0,T ] u(x, t). Consequently, the right-hand side of (3.4) is well defined. We emphasize that the Laplacian , the normal vector field | · | appearing above depend on the parameter t ∈ [0, T ].

∂ ∂ν ,

the gradient ∇, and the norm

A Proof of Theorem 3.1. Introduce the function P = t |∇u| u − u log u . One may repeat the computation from the proof of Theorem 2.4 and conclude that 2

∂ − P 0 ∂t

(3.5)

for all (x, t) ∈ M × (0, T ]. Employing this inequality, we will demonstrate that P must be nonpositive. The assertion of the theorem will immediately follow. Fix τ ∈ (0, T ]. Let us prove that the function P is nonpositive on M × [0, τ ]. If P attains its largest value at the point (x, 0) for some x ∈ M, then P is less than or equal to −u log Au computed at (x, 0). In this case, P must be nonpositive. Suppose this function attains its largest value at the point (x, t) for some x in the interior of M and some t in the interval (0, τ ]. We then use estimate (3.5) and the strong maximum principle. They imply P must also assume its largest value at (x, 0). As a consequence, P is nonpositive. Thus, we only have to consider the situation where this function has no maxima on M × [0, τ ] away from ∂M × (0, τ ]. Unless this is the case, P cannot become strictly greater than 0 anywhere. Let (x0 , t0 ) ∈ ∂M × (0, τ ] be a point where the function P attains its largest value on M × [0, τ ]. The Hopf maximum principle tells us that the inequality ∂ P (x0 , t0 ) > 0 ∂ν

3538


holds true. But the Neumann boundary condition (3.3) and the second line of (3.1) imply ∂ ∂ ∂ |∇u|2 ∂ A ∂ 2 1 P =t |∇u| −t 2 u− u log + u ∂ν ∂ν u ∂ν u ∂ν u ∂ν 1 t t ∂ |∇u|2 = 2 D ∂ (∇u) ∇u = −2 II(∇u, ∇u) =t ∂ν u u ∂ν u t = −2 λ(t)|∇u|2 0 u for all (x, t) ∈ ∂M × [0, τ ] (related computations appear in [26] and [28, Chapter IV]). Consequently, P must have a maximum on M × [0, τ ] away from ∂M × (0, τ ]. We conclude that P is nonpositive on M × [0, τ ]. Since the number τ ∈ (0, T ] can be chosen arbitrarily, the same assertion holds on M × [0, T ]. The theorem follows at once. 2 Remark 3.3. Consider the case where the metric g(x, t) does not depend on t and Eqs. (3.1) are not assumed. Suppose the Ricci curvature of M is nonnegative and ∂M is convex in the sense that the second fundamental form of ∂M is nonnegative definite. Then the solution u(x, t) of problem (3.2)–(3.3) satisfies (3.4). This fact can be established by the same argument we used to prove the theorem. The computation leading to (3.5) in this case may be found in [15]. Our next estimate is similar to (2.16). Henceforth, the subscript t denotes the derivative in t. The number n is the dimension of the manifold M. Theorem 3.4. Let (M, g(x, t))t∈[0,T ] be a solution to the Ricci flow (3.1). Consider a smooth positive function u(x, t) : M × [0, T ] → R satisfying the heat equation (3.2) with the Neumann boundary condition (3.3). If 0 Ric(x, t) kg(x, t) for a fixed k > 0 and all (x, t) ∈ M × [0, T ], then the estimate n |∇u|2 ut kn + − u 2t u2

(3.6)

holds for all (x, t) ∈ M × (0, T ]. Remark 3.5. We will make use of Lemma 2.6 in the arguments below. The proof of this lemma relies on local computations. Therefore, it prevails on manifolds with boundary. Proof of Theorem 3.4. Fix τ ∈ (0, T ]. Introduce the functions f = log u and F1 = t (|∇f |2 − ft ). Let us pick a point (x0 , t0 ) ∈ M × [0, τ ] where F1 attains its maximum on M × [0, τ ]. We will demonstrate that the inequality F1 (x0 , t0 ) t0 kn +

n 2

(3.7)

holds true. The assertion of the theorem will follow therefrom. If t0 = 0, then F1 (x, t0 ) = 0 for every x ∈ M and estimate (3.7) is evident. Consequently, we assume t0 > 0. In accordance with Lemma 2.6 and our conditions on the Ricci curvature of M,


3539

the inequality −

∂ 2a F12 F1 t0 n F1 −2∇f ∇F1 + k2 − − ∂t n t0 t0 2(1 − a)

holds for all a ∈ (0, 1) at the point (x0 , t0 ). Setting a = and using the quadratic formula, we see that

1+kt0 1+2kt0

like in the proof of Theorem 2.9

∂ nkt0 (1 + 2kt0 ) n − F1 + 2∇f ∇F1 F1 + F1 − t0 kn − . ∂t 2(1 + kt0 ) 2

(3.8)

If (3.7) fails to hold, then the right-hand side of (3.8) must be strictly positive. We will now show this is impossible. Suppose x0 lies in the interior of M. The fact that (x0 , t0 ) is a maximum point then yields F1 (x0 , t0 ) 0, ∂t∂ F1 (x0 , t0 ) 0, and ∇F1 (x0 , t0 ) = 0. Hence the right-hand side of (3.8) cannot be strictly positive. Suppose now x0 lies in the boundary of M. If the right-hand side of (3.8) is indeed positive, then the Hopf maximum principle tells us that the inequality ∂ F1 > 0 ∂ν

(3.9)

holds at (x0 , t0 ). We will make a computation to show this cannot be the case. Fix a system {y1 , . . . , yn } of local coordinates in a neighborhood U of the point x0 demanding that U ∩ ∂M = {x ∈ U | yn (x) = 0}. We write gij and Rij for the corresponding components of the metric and the Ricci tensor. Clearly, they depend on the parameter t. Without loss of generality, assume ∂y∂ 1 , . . . , ∂y∂n−1 are all orthogonal to ∂y∂ n on the boundary with respect to g(x, t0 ). It is easy to see that g in ∂ ∂ =− 1 ∂ν (g nn ) 2 ∂yi n

(3.10)

i=1

in U ∩ ∂M. Here, g ij are the components of the matrix inverse to (gij )ni,j =1 . ∂ The Neumann boundary condition (3.3) implies ∂ν f = 0. Utilizing this fact, we obtain ∂ ∂ ∂ ∂ 2 F1 = t |∇f | − ft = t 2 D ∂ (∇f ) ∇f − ft ∂ν ∂ν ∂ν ∂ν ∂ν ∂ ∂ ∂ ∂ f− = t −2 II(∇f, ∇f ) + f ∂t ∂ν ∂t ∂ν ∂ ∂ f . = t −2 II(∇f, ∇f ) + ∂t ∂ν For related computations, see [26] and [28, Chapter IV]. According to (3.10) and the first formula in (3.1), the equality

3540


n ∂ nn ∂ in nn 12 ∂ 1 ∂ ∂ ∂t g in g = − nn g − g 1 ∂t ∂ν g ∂t ∂y i 2(g nn ) 2 i=1 n 2Rj l g j i g nl Rj l g j n g nl g in ∂ =− − 1 1 ∂yi (g nn ) 2 g nn (g nn ) 2 i,j,l=1

holds in U ∩ ∂M. A calculation based on the Codazzi equation and the second line in (3.1) then implies ∂ ∂ ∂ = Rnn g nn ∂t ∂ν ∂ν near x0 at time t0 . Here, we make use of that fact that ∂y∂ 1 , . . . , ∂y∂n−1 are orthogonal to boundary with respect to g(x, t0 ). Combining the above equalities, we conclude that

∂ ∂yn

on the

∂ ∂ F1 = t0 −2 II(∇f, ∇f ) + Rnn g nn f ∂ν ∂ν = −2t0 II(∇f, ∇f ) = −2t0 λ(t0 )|∇f |2 0 at the point (x0 , t0 ). But this contradicts (3.9). Thus, the right-hand side of (3.8) cannot be strictly positive, and our assumption that (3.7) failed to hold must have been false. Because (x0 , t0 ) is a maximum point for F1 on M × [0, τ ], it is easy to see that F1 (x, τ ) F1 (x0 , t0 ) t0 kn +

n n τ kn + 2 2

for any x ∈ M. Consequently, n |∇u|2 ut kn + − 2 u 2τ u at (x, τ ). Since the number τ ∈ (0, T ] can be chosen arbitrarily, this yields the assertion of the theorem. 2 Acknowledgments Xiaodong Cao wishes to thank Professor Qi Zhang for useful discussions. X.C.’s research is partially supported by NSF grant DMS 0904432. Artem Pulemotov is grateful to Professor Leonard Gross for helpful conversations. A large portion of this paper was written when A.P. was a graduate student at Cornell University. At that time, he was partially supported by Professor Alfred Schatz’s NSF grant DMS 0612599. References [1] M. Arnaudon, R.O. Bauer, A. Thalmaier, A probabilistic approach to the Yang–Mills heat equation, J. Math. Pures Appl. 81 (2002) 143–166. [2] M. Arnaudon, K.A. Coulibaly, A. Thalmaier, Brownian motion with respect to a metric depending on time: definition, existence and applications to Ricci flow, C. R. Math. Acad. Sci. Paris 346 (2008) 773–778.


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[3] D.G. Aronson, P. Bénilan, Régularité des solutions de l’équation des milieux poreux dans RN , C. R. Acad. Sci. Paris Sér. A–B 288 (1979) A103–A105. [4] D. Bakry, Z.M. Qian, Harnack inequalities on a manifold with positive or negative Ricci curvature, Rev. Mat. Iberoamericana 15 (1999) 143–179. [5] X. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow, J. Funct. Anal. 255 (2008) 1024–1038. [6] X. Cao, R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Anal. 19 (2009) 989–1000. [7] N. Charalambous, L. Gross, The Yang–Mills heat semigroup on three-manifolds with boundary, in preparation. [8] B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math. 45 (1992) 1003–1014. [9] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects, American Mathematical Society, Providence, RI, 2007. [10] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications. Part II. Analytic Aspects, American Mathematical Society, Providence, RI, 2008. [11] B. Chow, R.S. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math. 129 (1997) 213–238. [12] B. Chow, P. Lu, L. Ni, Hamilton’s Ricci Flow, American Mathematical Society/Science Press, Providence, RI/New York, 2006. [13] J.C. Cortissoz, Three-manifolds of positive curvature and convex weakly umbilic boundary, Geom. Dedicata 138 (2009) 83–98. [14] C.M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal. 12 (2002) 425–436. [15] R.S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993) 113–126. [16] R.S. Hamilton, The Formation of Singularities in the Ricci Flow, Surv. Differ. Geom., vol. II, International Press, Cambridge, MA, 1995, pp. 7–136. [17] R.S. Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995) 215–226. [18] E.P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Society, Providence, RI, 2002. [19] J.D.T. Jane, The effect of the Ricci flow on geodesic and magnetic flows, and other related topics, PhD Dissertation, University of Cambridge, 2008. [20] J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal. 100 (1991) 233–256. [21] J. Li, X. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, arXiv:0901.3849v1 [math.DG]. [22] P. Li, S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986) 153–201. [23] S. Liu, Gradient estimates for solutions of the heat equation under Ricci flow, Pacific J. Math. 243 (2009) 165–180. [24] J. Morgan, G. Tian, Ricci Flow and the Poincaré Conjecture, American Mathematical Society/Clay Mathematics Institute, Providence, RI/Cambridge, MA, 2007. [25] L. Ni, Ricci flow and nonnegativity of sectional curvature, Math. Res. Lett. 11 (2004) 883–904. [26] A. Pulemotov, The Li–Yau–Hamilton estimate and the Yang–Mills heat equation on manifolds with boundary, J. Funct. Anal. 255 (2008) 2933–2965. [27] L.A. Sadun, Continuum regularized Yang–Mills theory, PhD Dissertation, University of California, Berkeley, 1987. [28] R. Schoen, S.-T. Yau, Lectures on Differential Geometry, International Press, Cambridge, MA, 1994. [29] Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 (1996) 203–221. [30] M. Simon, Deformation of C 0 Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002) 1033–1074. [31] M. Simon, Deforming Lipschitz metrics into smooth metrics while keeping their curvature operator non-negative, in: Geometric Evolution Equations, American Mathematical Society, Providence, RI, 2005, pp. 167–179. [32] P. Souplet, Q. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. Lond. Math. Soc. 38 (2006) 1045–1053. [33] J. Streets, Ricci Yang–Mills flow on surfaces, Adv. Math. 223 (2) (2010) 454–475. [34] J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, preprint. [35] P. Topping, Lectures on the Ricci Flow, Cambridge University Press, Cambridge, 2006.

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[36] J. Wang, Global heat kernel estimates, Pacific J. Math. 178 (1997) 377–398. [37] A. Young, Stability of the Ricci Yang–Mills flow at Einstein Yang–Mills metrics, submitted for publication, arXiv: 0812.1823v1 [math.DG]. [38] Q. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Not. (2006), Art. ID 92314, 39 pp.


Dynamics of stochastic 2D Navier–Stokes equations Salah Mohammed a,∗,1 , Tusheng Zhang b a Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA b Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England,

United Kingdom Received 11 October 2009; accepted 9 November 2009 Available online 18 November 2009 Communicated by Paul Malliavin

Abstract In this paper, we study the dynamics of a two-dimensional stochastic Navier–Stokes equation on a smooth domain, driven by linear multiplicative white noise. We show that solutions of the 2D Navier–Stokes equation generate a perfect and locally compacting C 1,1 cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium/stationary solution. We give sufficient conditions on the parameters of the Navier–Stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space. © 2009 Elsevier Inc. All rights reserved. Keywords: Stochastic Navier–Stokes equation; Cocycle; Lyapunov exponents; Stable manifolds; Invariant manifolds

1. Introduction Two-dimensional stochastic Navier–Stokes equations (SNSE’s) are often used to describe the time evolution of an incompressible fluid in a smooth bounded planar domain. In this article, we characterize the long-time asymptotics of the following two-dimensional stochastic Navier–Stokes equation with Dirichlet boundary conditions: * Corresponding author.

E-mail addresses: [email protected] (S. Mohammed), [email protected] (T. Zhang). 1 The research of this author is supported by NSF award DMS-0705970.


3544

S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591

du − νu dt + (u · ∇)u dt + ∇p dt = γ u dt +

∞

σk u(t) dWk (t),

k=1

(div u)(t, x) = 0,

x ∈ D, t > 0,

⎫ ⎪ ⎪ t > 0, ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

u(t, x) = 0, x ∈ ∂D, t > 0, u(0, x) = u0 (x), x ∈ D,

(1.1)

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 ∈ D, p(t, x) denotes the pressure field, and ν > 0 the viscosity coefficient. Moreover, the random force field is driven by independent one-dimensional standard Brownian motions Wk , k 1, and a deterministic linear drift term γ u dt with a fixed parameter γ . The Brownian motions Wk , k 1, are defined on a complete filtered Wiener space F , (Ft )t0 , P ). We assume that the noise parameters σk , k 1, are such that ∞ (Ω, 2 < ∞. σ k=1 k To formulate the dynamics of the above stochastic Navier–Stokes equation, we introduce the following standard spaces: Consider the Hilbert space

V := v ∈ H01 D, R2 : ∇ · v = 0 a.e. in D , with the norm vV :=

1 2

|∇v| dx 2

D

and inner product ·,·. Denote by H the closure of V in the L2 -norm |v|H :=

1 |v|2 dx

2

.

D

The inner product on H will be denoted by ·,·. Denote by PH the Helmholtz–Hodge projection from the Hilbert space L2 (D, R2 ) onto H . Define the (Stokes) operator A in H by the formula Au := −νPH u,

u ∈ H 2 D, R2 ∩ V ,

and the nonlinear operator B by

B(u, v) := PH (u · ∇)v , whenever u, v are such that (u · ∇v) belongs to the space L2 . We will often use the short notation B(u) := B(u, u). By applying the operator PH to each term of the above stochastic Navier–Stokes equation (SNSE), we can rewrite the equation in the following abstract form:

S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591 ∞

du(t) + Au(t) dt + B u(t) dt = γ u(t) dt + σk u(t) dWk (t)

3545

(1.2)

k=1

in L2 (0, T ; V ) with the initial condition u(0) = u0 ∈ H,

(1.3)

where V is the dual of V . Finally, and for the remainder of the article, we will adopt the following convention: Definition 1.1 (Perfection). A family of propositions {P (ω): ω ∈ Ω} is said to hold perfectly in ω if there is a sure event Ω ∗ ∈ F such that θ (t, ·)(Ω ∗ ) = Ω ∗ for all t ∈ R and P (ω) is true for every ω ∈ Ω ∗ . There is a large amount of literature on the stochastic Navier–Stokes equation. We will only refer to some of it. A good reference for stochastic Navier–Stokes equations driven by additive noise is the book [4] and the references therein; see also [5]. The existence and uniqueness of solutions of stochastic 2D Navier–Stokes equations with multiplicative noise were obtained in [9,17]. Ergodic properties, invariant measures, asymptotic compactness and absorbing sets of stochastic 2D Navier–Stokes equations are studied in [8,12,11,2]. The results in [12] address important aspects of the ergodic theory and invariant measures for 2D stochastic Navier–Stokes equations with (additive) periodic random “kicks”. The small noise large deviation of the stochastic 2D Navier–Stokes equations was established in [17] and the large deviation of occupation measures was considered in [10]. Related results on the dynamics of semilinear stochastic partial differential equations are given in [6,7,3]. The purpose of this paper is to study the dynamics of the two-dimensional stochastic Navier– Stokes equation (1.1) driven by multiplicative noise. In particular, we will establish the following: • Existence of a perfect locally compacting C 1,1 cocycle (semiflow) generated by all solutions of the stochastic Navier–Stokes equation; • Long-time asymptotics for the stochastic semiflow given by a countable non-random Lyapunov spectrum of the linearized cocycle at the equilibrium (viz. stationary solution); • Existence of countable families of C 1,1 local and global flow-invariant submanifolds through the equilibrium (when γ = 0); • Sufficient conditions for hyperbolicity of the equilibrium; viz. existence of flow-invariant local stable/unstable manifolds in the neighborhood of the equilibrium; • Sufficient conditions on the parameters ν, γ , σi , i 1, and the geometry of the domain D to guarantee uniqueness of the (zero) equilibrium. We believe that it is possible to modify the arguments in this article so as to cover the case of additive noise, white in time and sufficiently smooth in space. 2. Preliminaries Let us identify the Hilbert space H in Section 1 with its dual H . We then consider the stochastic Navier–Stokes equation (1.1) in the framework of the Gelfand triple: V ⊂H ∼ = H ⊂ V

3546


where V is the dual of V . Thus, we may consider the Stokes operator A as a bounded linear map from V into V . Moreover, we also denote by ·,· : V × V → R, the canonical bilinear pairing between V and V . Hence, using integration by parts, we have Au, w = ν

2

∂i uj ∂i wj dx = νu, w

(2.1)

i,j =1 D

for u = (u1 , u2 ) ∈ V , w = (w1 , w2 ) ∈ V . Define the real-valued trilinear form b on H × H × H by setting b(u, v, w) :=

2

(2.2)

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) = B(u, v), w = (u · ∇)v, w =

ui ∂i vj wj dx.

i,j D

Using integration by parts, it is easy to see that b(u, v, w) = −b(u, w, v),

(2.3)

b(u, v, v) = 0

(2.4)

for all u, v, w ∈ V . Thus,

for all u, v ∈ V . Throughout the paper, we will denote various generic positive constants by the same letter c, although the constants may differ from line to line. We now list some well-known estimates for b which will be used frequently in the sequel (see [19,15] for example): b(u, v, w) cuV · vV · wV , u, v, w ∈ V , b(u, v, w) c|u|H · vV · |Aw|H , u ∈ H, v ∈ V , w ∈ D(A), b(u, v, w) cuV · |v|H · |Aw|H , u ∈ V , v ∈ H, w ∈ D(A), 1 1 1 1 b(u, v, w) 2u 2 · |u| 2 · w 2 · |w| 2 · vV , u, v, w ∈ V . V H V H

(2.5) (2.6) (2.7) (2.8)

Moreover, combining (2.3) and (2.8), we obtain B(u, w) = sup b(u, w, v) = sup b(u, v, w) V vV 1 1 2

vV 1

1 2

1 2

1

2uV · |u|H · wV · |w|H2 for all u, w ∈ V .

(2.9)


3547

3. Existence of the cocycle In this section, we will show that strong solutions of the stochastic NSE generate a Fréchet C 1,1 locally compacting cocycle (viz. stochastic semiflow) u : R+ × H × Ω → H on the Hilbert space H . Our approach is to use a variational technique which transforms the SNSE into a random NSE that we then analyze using a combination of Galerkin approximations and a priori estimates (cf. [19,15]). Consider the SNSE ⎧ ∞ ⎪ ⎨ du(t, f ) + Au(t, f ) dt + B u(t, f ) dt = γ u(t, f ) dt + σ u(t, f ) dW (t), t > 0, k

⎪ ⎩

k

k=1

u(0, f ) = f ∈ H. (3.1)

It is known that for each f ∈ H , the SNSE (3.1) admits a unique strong solution u(·, f ) ∈ L2 (Ω; C([0, T ]; H )) ∩ L2 (Ω × [0, T ]; V ) [1]. Writing (3.1) in integral form, we have

t

t

u(t, f ) = f −

Au(s, f ) ds − 0

+

B u(s, f ) ds + γ

0

∞ t

t u(s, f ) ds 0

(3.2)

σk u(s, f ) dWk (s),

k=1 0

for all t ∈ [0, T ]. Let Q : [0, ∞) × Ω → R be the solution of the one-dimensional linear stochastic ordinary differential equation dQ(t) = γ Q(t) dt +

∞

σk Q(t) dWk (t),

⎫ ⎪ t 0, ⎬

k=1

Q(0) = 1.

⎪ ⎭

(3.3)

t 0.

(3.4)

By Itô’s formula, we have Q(t) = exp

∞ k=1

∞ t 2 σk Wk (t) − σk + γ t , 2 k=1

This implies that EQ∞ < ∞, where Q∞ ≡ Q(·, ω)∞ := sup Q(t, ω), 0tT

ω ∈ Ω,

3548


for any finite positive T . Define v(t, f ) := u(t, f )Q−1 (t),

t 0.

(3.5)

Applying Itô’s formula to the relation u(t, f ) = v(t, f )Q(t), t 0, and using (3.3), it is easy to see that v(t) ≡ v(t, f ) satisfies the random NSE

dv(t) = −Av(t) dt − Q(t)B v(t) dt, v(0) = f ∈ H.

t 0,

(3.6)

Our next proposition gives a priori bounds on solutions of the random NSE (3.6). Proposition 3.1. For f ∈ H and ω ∈ Ω, let v(·, f, ω) ∈ C([0, T ], H ) ∩ L2 ([0, T ], V ) be a solution of (3.6) on [0, T ] for some T > 0. Then for each ω ∈ Ω and any f ∈ H , the following estimates hold sup v(t, f, ω)H |f |H

(3.7)

v(t, f, ω)2 dt 1 |f |2 . V 2ν H

(3.8)

0tT

and

T 0

Moreover, for each ω ∈ Ω, the map H f → v(·, f, ω) ∈ C([0, T ], H ) ∩ L2 ([0, T ], V ) is Lipschitz on bounded sets in H . Proof. Let f ∈ H and v(t) ≡ v(t, f, ω), t ∈ [0, T ], be a solution of (3.6). We fix and suppress ω ∈ Ω throughout this proof. Employing the divergence free condition, B(v), v = 0, we obtain v(t, f )2 = |f |2 − 2 H H

t

Av(s, f ), v(s, f ) ds − 2

0

Q(s) B v(s, f ) , v(s, f ) ds

0

t = |f |2H − 2

t

Av(s, f ), v(s, f ) ds

0

t = |f |2H − 2ν 0

for all t ∈ [0, T ]. Hence,

v(s, f )2 ds V

(3.9)


v(t, f )2 + 2ν H

t

v(s, f )2 ds = |f |2 , H V

3549

t ∈ [0, T ].

0

This immediately gives (3.7) and (3.8). It remains to prove the last assertion of the proposition. Let f, g ∈ H , and t ∈ [0, T ] for the rest of the proof. Using the identity b(u, v, v) = 0,

u, v ∈ V ,

and the chain rule we obtain v(t, f ) − v(t, g)2 = |f − g|2 − 2 H H

t

A v(s, f ) − v(s, g) , v(s, f ) − v(s, g) ds

0

t −2

Q(s) B v(s, f ) − B v(s, g) , v(s, f ) − v(s, g) ds

0

t = |f

− g|2H

− 2ν

v(s, f ) − v(s, g)2 ds V

0

t −2

Q(s) b v(s, f ), v(s, f ), v(s, f ) − v(s, g)

0

− b v(s, g), v(s, g), v(s, f ) − v(s, g) ds

t = |f

− g|2H

− 2ν

v(s, f ) − v(s, g)2 ds V

0

t −2

Q(s)b v(s, f ) − v(s, g), v(s, g), v(s, f ) − v(s, g) ds.

0

(3.10) Thus, we have v(t, f ) − v(t, g)2 |f − g|2 − 2ν H H

t

v(s, f ) − v(s, g)2 ds V

0

t + 4Q∞ 0

v(s, f ) − v(s, g) v(s, g) v(s, f ) − v(s, g) ds V V H

3550


t |f

− g|2H

−ν

v(s, f ) − v(s, g)2 ds V

0

+4

Q2∞ ν

t

v(s, g)2 v(s, f ) − v(s, g)2 ds. V H

(3.11)

0

Applying Gronwall’s lemma (Lemma 5.1) to the above inequality and using (3.8), we get v(t, f ) − v(t, g)2 + ν H

t

v(s, f ) − v(s, g)2 ds V

0

Q2∞ |f − g|2H exp 4 ν

T

v(s, g)2 ds V

0

2 |f − g|2H exp 2 Q2∞ |g|2H . ν

(3.12)

This implies that, for each ω ∈ Ω, the solution map

H f → v(·, f, ω) ∈ C [0, T ], H ∩ L2 [0, T ], V (when it exists) is Lipschitz on bounded sets in H .

2

Our next proposition proves the existence of a unique strong global solution to the random NSE (3.6). Proposition 3.2. Let f ∈ H , ω ∈ Ω. Then for each T > 0, there exists a unique solution v(·, f, ω) ∈ C([0, T ], H ) ∩ L2 ([0, T ], V ) to Eq. (3.6). Furthermore, the solution map R+ × H × Ω (t, f, ω) → v(t, f, ω) ∈ H is jointly measurable, and for each f the process v(·, f, ·) : R+ × Ω → H is (Ft )t0 -adapted. Proof. The proof is based on Galerkin approximations coupled with a priori estimates (cf. [19,18]). Let f ∈ H , fix and suppress ω ∈ Ω. We use Galerkin approximations to prove existence of a solution to the random NSE

⎫ dv(t) = −Av(t) dt − Q(t)B v(t) dt, t > 0, ⎪ ⎬

(3.13) v(0) = f ∈ L2 D, R2 = H, ⎪ ⎭ v(t)|∂D = 0, t > 0. Let {ei }∞ i=1 be a complete orthonormal basis of H that consists of eigenvectors of the operator −A under Dirichlet boundary conditions with corresponding eigenvalues {μi }∞ i=1 ; that is A(ei ) = −μi ei , ei |∂D = 0, i 1. Let Hn denote the n-dimensional subspace of H spanned by {e1 , e2 , . . . , en }. Define fn ∈ Hn by


fn :=

n

3551

f, ej ej .

j =1

Clearly, the sequence {fn }∞ n=1 converges to f in H . Now for every integer n 1, we seek a solution vn of the random NSE

dvn (t) = −Avn (t) dt − Q(t)B vn (t) dt, vn (0) = fn , vn (t)|∂D = 0, t > 0,

⎫ t > 0, ⎪ ⎬ ⎪ ⎭

(3.13n )

such that vn (t) :=

n

gj n (t)ej ,

t 0,

j =1

for appropriate choice of the real-valued random processes gj n . We will show that the Fourier coefficients gj n (t) solve a system of random ordinary differential equations with locally Lipschitz coefficients. To see this, we proceed as follows. Since vn satisfies the NSE (3.13n ), then for each 1 j n, we have dgj n (t) = d vn (t), ej

= − Avn (t), ej dt − Q(t) vn (t) · ∇ vn (t), ej dt

= − vn (t), Aej dt − Q(t) vn (t) · ∇ vn (t), ej dt

= μj gj n (t) dt − Q(t) vn (t) · ∇ vn (t), ej dt

(3.14)

for all t > 0. Consider

vn (t) · ∇ vn (t) =

n i=1

=

n

gin (t)(ei · ∇)

n

gkn (t)ek

k=1

gin (t)gkn (t)(ei · ∇)(ek ).

i,k=1

Hence n

vn (t) · ∇ vn (t), ej = gin (t)gkn (t) (ei · ∇)(ek ), ej i,k=1

=

n

gin (t)gkn (t)b(ei , ek , ej ),

1 j n.

(3.15)

i,k=1

Substituting (3.15) into (3.14) gives the following random system of ode’s for gj n (t), 1 j n,

3552


dgj n (t) = μj gj n (t) dt − Q(t)

n

gin (t)gkn (t)b(ei , ek , ej ) dt,

⎫ ⎪ ⎪ t > 0, ⎬ ⎪ ⎪ ⎭

i,k=1

gj n (0) = f, ej .

(3.16)

The vector fields in (3.16) are locally Lipschitz and so the system (3.16) of random ode’s admits a unique local solution defined on a local time interval [0, T0 ), where T0 is possibly random. Hence the system (3.13n ) has a unique local solution defined on [0, T0 ). Since Q is jointly measurable and (Ft )t0 -adapted, then so are the gj n ’s. To show that T0 = ∞ a.s., we first derive a priori estimates on vn (or the gj n ). Suppose T0 ≡ T0 (ω) < ∞ for some ω ∈ Ω. Multiply both sides of (3.13n ) by vn (t), integrate over D and use the relation

B vn (t) , vn (t) H = 0,

n 1, t ∈ [0, T0 ),

to obtain 2 d vn (t)H = −2 Avn (t), vn (t) H dt,

t ∈ [0, T0 ).

Therefore,

2 d vn (t) + 2ν

∇vn (t)2 dt = 0,

0 < t < T0 .

D

Hence, vn (t)2 − vn (0)2 + 2ν H

t

vn (s)2 ds = 0, V

0 < t < T0 ;

0

and so vn (t)2 + 2ν H

t

vn (s)2 ds = |fn |2 |f |2 , H H V

0

for all n 1 and all t ∈ (0, T0 ). In particular, 2 sup vn (t)H |f |2H

(3.17)

vn (s)2 ds 1 |f |2 V 2ν H

(3.18)

0t 0. This contradicts the maximality of T0 . Hence T0 = ∞ a.s. We next show that the sequence {vn }∞ n=1 converges to a weak solution v of the random NSE (3.13). As before, we view Eq. (3.13n ) as an equation in V :

dv n (t) = −Av n (t) − Q(t)B v n (t) . dt Therefore, using (2.9), (3.17), (3.18) and the fact that A|V : V → V is continuous linear, we have

T n 2

T

T dv (t)

dt 2 Av n (t)2 dt + 2Q∞ B v n (t) 2 dt dt

V V V 0

0

T C 0

0

n 2 v (t) dt + C V

T 0

n 2 n 2 v (t) v (t) dt V H


C |f |2H 1 + |f |2H , 2ν

3555

(3.27)

where C is a positive random constant independent of f . Since the embedding V → H is compact, by the proof of Theorem 2.1 in [19, pp. 111–113], it follows from (3.17), (3.18) and (3.27) that there exists a subsequence v nk (t), k 1, and v(t) such that v nk (·) → v(·) in the weak star topology of C([0, T ], H ), v nk (·) → v(·) weakly in L2 ([0, T ], V ) and moreover v nk (·) → v(·) strongly in L2 ([0, T ], H ) as k → ∞. For w ∈ Hm , if nk m we have

v (t), w = v nk (0), w − nk

t

t

Av (s), w ds + nk

0

= v nk (0), w −

t

Q(s)b v nk (s), w, v nk (s) ds

0

Av nk (s), w ds

0

+

2 t

Q(s)vink (s)(∂i wj )vjnk (s) dx ds

(3.28)

i,j =1 0 D

for all t ∈ [0, T ]. Letting k → ∞ in the above relation, we obtain

t

v(t), w = f, w −

Av(s), w ds +

t

Av(s), w ds −

= (f, w) − 0

Q(s)vi (s)(∂i wj )vj (s) dx ds

i,j =1 0 D

0

t

2 t

Q(s) B v(s) , w ds

(3.29)

0

for all t ∈ [0, T ] and all m 1. Since m is arbitrary and ∞ m=1 Hm is dense in V , it follows that v is a solution to Eq. (3.6). Uniqueness follows by setting f = g in (3.12). Since the Galerkin approximations vn are jointly measurable and (Ft )t0 -adapted, it follows that the limiting process v must have the same measurability properties. 2 Our next result addresses the issue of local compactness of the solution map H f → v(t, f, ω) ∈ H for t > 0, ω ∈ Ω. Proposition 3.3. For t > 0 and ω ∈ Ω, the solution map H f → v(t, f, ω) ∈ H of (3.6) sends bounded sets into relatively compact sets in H . Proof. Fix ω ∈ Ω throughout this proof. √ First, we show that the map [0, T ] t → tv(t, f, ω) ∈ V is L∞ , and provide a bound for v in L∞ ([0, T ], V ). Let f ∈ V . Then by an argument similar to the proof of Theorem 3.10

3556


in [18, p. 314], one can show that v(·, f, ω) ∈ L2 ([0, T ], H 2 (D)) ∩ L∞ ([0, T ], V ) and the following energy equation holds:

d v(t, f, ω)2 + 2Av(t, f, ω)2 = −Q(t) B v(t, f, ω) , Av(t, f, ω) V H dt

(3.30)

for all t ∈ (0, T ). Consequently, 2

d t v(t, f, ω)V dt 2 2

= −2t Av(t, f, ω)H − tQ(t) B v(t, f, ω) , Av(t, f, ω) + v(t, f, ω)V 2 1 2 3 −2t Av(t, f, ω)H + ctQ(t)v(t, f, ω)H2 v(t, f, ω)V Av(t, f, ω)H2 + v(t, f, ω)V 2 2 4 ctQ(t)4 v(t, f, ω) v(t, f, ω) + v(t, f, ω) , t ∈ (0, T ], (3.31) H

V

V

where the following Young’s inequality “with ”: ab

3 4/3 4/3 a4 b , + 4 4 4

a, b 0,

and the fact that 1 1 B(u) c|u| 2 u|Au| 2 , H H H

u ∈ V,

have been used. By Gronwall’s inequality it follows from (3.31) that √ ·v(·, f, ω)2 ∞

L ([0,T ],V )

T

T v(t, f, ω)2 dt exp cQ4 v(t, f, ω)2 v(t, f, ω)2 dt ∞ V H V

0

0

1 2 4 1 4 |f |H exp cQ∞ |f |H . 2ν 2ν

(3.32)

Since the right side of (3.32) depends only on the H -norm of f , by a limiting procedure it is easy to see that (3.32) also holds for all f ∈ H . Fix t > 0, f ∈ H . Then, for 0 < δ < t, v(t, f, ω) = Tδ v(t − δ, f, ω) +

t

Tt−s B v(s, f, ω) ds.

(3.33)

t−δ

Suppose δk 0, 0 < δk < t, and let {fn }∞ n=1 ⊂ H be a bounded sequence; i.e., there exists M > 0 such that |fn | M for all n 1.


3557

∞ Claim. There exists a subsequence {fñ }∞ n=1 of {fn }n=1 such that for each k 1, the sequence ∞ ˜ {Tδk [v(t − δk , fn , ω)]}n=1 converges in H .

Proof. We use a diagonalization argument. The set {v(t − δ1 , fn , ω): n 1} is bounded in H because |v(t − δ1 , fn , ω)|H |fn |H M for all n 1. So by compactness of Tδ1 : H → H , the sequence {Tδ1 [v(t − δ1 , fn , ω)]}∞ n=1 has a convergent subsequence. Therefore, there is a ∞ such that the sequence {T [v(t − δ , f 1 , ω)]}∞ converges. subsequence {fn1 }∞ of {f } n n=1 δ1 1 n n=1 n=1 Similarly, by compactness of the map H f → Tδ2 v(t − δ2 , f, ω) ∈ H ∞ 1 ∞ 2 there is a subsequence {fn2 }∞ n=1 ⊂ {fn }n=1 such that {Tδ2 [v(t − δ2 , fn , ω)]}n=1 converges in H . ∞ By induction, there are subsequences {fnk }n=1 , k 1, such that {Tδk [v(t − δk , fnk , ω)]}∞ n=1 con∞ ∞ k+1 k n ˜ verges and {fn }n=1 ⊂ {fn }n=1 for each k 1. Let fn := fn , n 1, be the diagonal subse∞ ˜ quence of {fn }∞ n=1 . Then the sequence {Tδk [v(t − δk , fn , ω)]}n=1 converges in H for each k 1. This proves the claim. 2

We will now show that the map H f → v(t, f, ω) ∈ H is compact i.e., takes bounded sets in H into relatively compact sets. It is sufficient to show that for the bounded sequence {fn }∞ n=1 ⊂ H ∞ ∞ ˜ ˜ there exists a subsequence {fn }n=1 such that {v(t, fn , ω)}n=1 converges. Pick the subsequence {fñ } ⊂ {fn } as in the claim with each sequence {Tδk [v(t − δk , fñ , ω)]}∞ n=1 convergent. Consider v(t, fñ , ω) − v(t, f˜m , ω) Tδ v(t − δk , fñ , ω) − Tδ v(t − δk , f˜m , ω) k k H H

t

Tt−s B v(s, fñ , ω) − Tt−s B v(s, f˜m , ω) ds H

+ t−δk

Tδk v(t − δk , fñ , ω) − Tδk v(t − δk , f˜m , ω) H 1 + 2CM √ δk , t − δk for all k 1 and all m, n 1, where the following estimate has been used

Tt−s B v(s, fñ , ω) − Tt−s B v(s, f˜m , ω) 1 C√ t −s

B v(s, fñ , ω)

V

H

˜

+ B v(s, fm , ω) V

1 v(s, fñ , ω)H v(s, fñ , ω)V + v(s, f˜m , ω)H v(s, f˜m , ω)V

C√ t −s √ 1 sup v(s, fñ , ω)H sup sv(s, fñ , ω)V

C√ √ t − s s 0sT 0sT √ + sup v(s, f˜m , ω) sup sv(s, f˜m , ω)

0sT

H

0sT

V

(3.34)

3558


CM (ω) √

1 √ , t −s s

because of (3.32). For fixed k 1, the claim implies lim sup Tδk v(t − δk , fñ , ω) − Tδk v(t − δk , f˜m , ω) H = 0.

m,n→∞

Take lim supm,n→∞ in (3.34) (with k 1 fixed) and use the above relation to get √ δk ˜ ˜ lim sup v(t, fn , ω) − v(t, fm , ω) H 2CM √ , t − δk m,n→∞

(3.35)

for all k 1. Now let k → ∞ in (3.35) to obtain √ δk ˜ ˜ lim sup v(t, fn , ω) − v(t, fm , ω) H 2CM lim √ = 0. k→∞ t − δk m,n→∞ Therefore, {v(t, fñ , ω)}∞ n=1 is a Cauchy sequence and hence converges in H . This proves compactness of the map H f → v(t, f, ω) ∈ H . 2 Theorem 3.1. The solution map H f → v(t, f, ω) ∈ H is C 1,1 for ω ∈ Ω and all t 0, and has bounded Fréchet derivatives on bounded sets in H . Furthermore, the Fréchet derivative Dv(t, f, ω)(·) : H → H is compact for any t > 0, f ∈ H and ω ∈ Ω. Proof. Let f, g ∈ H . Fix and suppress ω ∈ Ω in this proof. Consider the following random integral equation

t z(t, f )(g) = g −

Az(s, f )(g) ds − 0

t −

t

Q(s) z(s, f )(g) · ∇ v(s, f ) ds

0

Q(s) v(s, f ) · ∇ z(s, f )(g) ds,

t ∈ [0, T ].

(3.36)

0

The existence and uniqueness of the solution z(t, f )(g) of (3.36) can be proved similarly as for Eq. (3.6), using Galerkin approximations (cf. proof of Proposition 3.2). Furthermore, uniqueness of the solution to (3.36) implies that the solution z(t, f )(g) is linear in g. We will now derive some useful estimates for the solution z(t, f )(g) of (3.36). Using the chain rule in (3.36), we obtain


z(t, f )(g)2 = |g|2 − 2ν H

t

3559

z(s, f )(g)2 ds V

0

t −2

Q(s)b z(s, f )(g), v(s, f ), z(s, f )(g) ds

0

t −2

Q(s)b v(s, f ), z(s, f )(g), z(s, f )(g) ds

0

t |g|2H − 2ν

z(s, f )(g)2 ds V

0

t

z(s, f )(g) v(s, f ) z(s, f )(g) ds H V V

+ cQ∞ 0

t |g|2H

− 2ν

z(s, f )(g)2 ds + cQ∞

V

0

t

z(s, f )(g)2 ds V

0

+ cQ∞ −1

t

z(s, f )(g)2 v(s, f )2 ds H V

0

t = |g|2H

−ν

z(s, f )(g)2 ds + cQ∞ −1 V

0

t

v(s, f )2 z(s, f )(g)2 ds V H

0

+ cQ∞ − ν

t

z(s, f )(g)2 ds, V

t ∈ [0, T ],

(3.37)

0

where we have used the following “Young inequality with ”: ab a 2 + −1 b2 ,

a, b > 0,

for any > 0. Now in (3.37), choose sufficiently small (and random) such that cQ∞ < ν. So (3.37) implies z(t, f )(g)2 |g|2 − ν H H

t 0

for all t ∈ [0, T ].

z(s, f )(g)2 ds + cQ ˜ ∞ V

t 0

v(s, f )2 z(s, f )(g)2 ds V H

3560


By Gronwall’s lemma (Lemma 5.1), the above inequality gives 2 sup z(t, f )(g)H + ν

0tT

T 0

T z(s, f )(g)2 ds |g|2 exp cQ2 v(s, f )2 ds ∞ H V V 0

1 |g|2H exp cQ2∞ |f |2H , 2ν

(3.38)

where (3.8) has been used for the last inequality. Since z(t, f )(g) is linear in g, (3.38) implies that z(t, f )(·) ∈ L(H ) for each t ∈ [0, T ], and z(·, f )(·) ∈ L(H, L2 ([0, T ], V )). Furthermore, 1 2 1 2 ˜ |f | . sup z(t, f ) L(H ) exp cQ ∞ 2 2ν H 0tT

(3.39)

Next we will show that the map H f → v(t, f, ω) ∈ H has a continuous Fréchet derivative given by Dv(t, f, ω) = z(t, f, ω)(·). To this end, it suffices to prove that v(t, f + hg, ω) − v(t, f, ω) lim sup − z(t, f )(g) = 0 h→0 h |g|H 1

(3.40)

H

and the map H f → z(t, f, ω) ∈ L(H ) is continuous. First, we prove lim sup

h→0 |g|H 1

2 sup v(t, f + hg) − v(t, f )H + ν

0tT

T

v(s, f + hg) − v(s, f )2 ds = 0. V

0

(3.41) Using the equations satisfied by v(t, f ) and v(t, f + hg), the chain rule and “Young’s inequality with ”, it follows that v(t, f + hg) − v(t, f )2 H

t =h

2

|g|2H

− 2ν 0

v(t, f + hg) − v(t, f )2 ds V


t −2

3561

Q(s)b v(s, f + hg) − v(s, f ), v(s, f ), v(s, f + hg) − v(s, f ) ds

0

t h

2

|g|2H

− 2ν

v(t, f + hg) − v(t, f )2 ds V

0

t + cQ∞

v(s, f + hg) − v(s, f ) v(s, f ) v(s, f + hg) − v(s, f ) ds H V V

0

t h

2

|g|2H

−ν

v(t, f + hg) − v(t, f )2 ds V

0

t + cQ2∞

v(s, f + hg) − v(s, f )2 v(s, f )2 ds H V

(3.42)

0

for all t ∈ [0, T ]. By Gronwall’s inequality, we deduce that 2 sup v(t, f + hg) − v(t, f )H + ν

T

0tT

v(s, f + hg) − v(s, f )2 ds V

0

h

2

|g|2H

exp

t

cQ2∞

v(s, f )2 ds V

(3.43)

0

for all f, g ∈ H and h ∈ R. This immediately implies (3.41). Set v(t, f + hg, ω) − v(t, f, ω) , h X(t, f, g, h) = U (t, f, g, h) − z(t, f )(g),

U (t, f, g, h) =

for t ∈ [0, T ] and h ∈ R\{0}. Then,

t X(t, f, g, h) = −

t AX(s, f, g, h) ds −

0

t −

Q(s) v(s, f ) · ∇ X(s, f, g, h) ds

0

Q(s) X(s, f, g, h) · ∇ v(s, f + hg) ds

0

t + 0

Q(s) z(s, f )(g) · ∇ v(s, f ) − v(s, f + hg) ds,

(3.44)

3562


for all t ∈ [0, T ]. By the chain rule, X(t, f, g, h)2 = −2ν H

t

X(s, f, g, h)2 ds V

0

t −2

Q(s)b X(s, f, g, h), v(s, f + hg), X(s, f, g, h) ds

0

t +2

Q(s)b z(s, f )(g), v(s, f ) − v(s, f + hg), X(s, f, g, h) ds,

0

(3.45) where b(u, v, v) = 0 has been used. Hence, X(t, f, g, h)2 −2ν H

t

X(s, f, g, h)2 ds V

0

t +c

Q(s)X(s, f, g, h)H v(s, f + hg)V X(s, f, g, h)V ds

0

t +c

1 1 1 Q(s)z(s, f )(g)V2 z(s, f )(g)H2 v(s, f + hg) − v(s, f )V2

0

1 × v(s, f + hg) − v(s, f )H2 X(s, f, g, h)V ds,

(3.46)

for t ∈ [0, T ]. We next prove the following estimate X(t, f, g, h)2 −ν H

t 0

t +c

X(s, f, g, h)2 ds + c V

t

2 2 Q(s)X(s, f, g, h)H v(s, f + hg)V ds

0

2 2 Q(s)z(s, f )(g)V v(s, f + hg) − v(s, f )H ds

0

t +c

2 2 Q(s)z(s, f )(g)H v(s, f + hg) − v(s, f )V ds,

0

for t ∈ [0, T ]. Use (3.46) and “Young’s inequality with ” to see that

(3.47)


X(t, f, g, h)2 −2ν H

t

3563

X(s, f, g, h)2 ds V

0

t +c

Q(s)X(s, f, g, h)H v(s, f + hg)V X(s, f, g, h)V ds

0

t +c

1/2 1/2 1/2 Q(s)z(s, f )(g)V z(s, f )(g)H v(s, f + hg) − v(s, f )V

0

1/2 × v(s, f + hg) − v(s, f )H X(s, f, g, h)V ds

t −2ν

X(s, f, g, h)2 ds V

0

+ c

−1

t


0

t + cQ∞

X(s, f, g, h)2 ds + cQ∞ V

0

+ −1 c

t

t

X(s, f, g, h)2 ds V

0

Q(s)z(s, f )(g)V z(s, f )(g)H v(s, f + hg) − v(s, f )V

0

× v(s, f + hg) − v(s, f )H ds

t −2ν

X(s, f, g, h)2 ds V

0

+ c

−1

t


0

t + cQ∞ 0

1 + −1 c 2

t

X(s, f, g, h)2 ds + cQ∞ V

t

X(s, f, g, h)2 ds V

0


0

1 + −1 c 2

t 0

2 2 Q(s)z(s, f )(g)H v(s, f + hg) − v(s, f )V ds,

(3.48)

3564


for t ∈ [0, T ]. Now choose small enough such that 2 cQ∞ < ν. This gives X(t, f, g, h)2 −ν H

t

X(s, f, g, h)2 ds V

0

+ c

−1

t


0

1 + −1 c 2

t


0

1 + −1 c 2

t

2 2 Q(s)z(s, f )(g)H v(s, f + hg) − v(s, f )V ds

(3.49)

0

for all t ∈ [0, T ], which implies (3.47). By (3.47) and Gronwall’s inequality (Lemma 5.1), it follows that 2 sup X(t, f, g, h)H + ν

0tT

T

X(s, f, g, h)2 ds V

0

! c Q∞

+ Q∞

2 sup v(t, f + hg) − v(t, f )H

0tT

z(s, f )(g)2 ds V

0

2 sup z(t, f )(g)H

0tT

T

T

v(t, f + hg) − v(s, f )2 ds V

"

0

T

× exp cQ∞

v(s, f + hg)2 ds V

(3.50)

0

for all f, g ∈ H and h ∈ R\{0}. By virtue of (3.41) and (3.38), (3.50) implies that lim sup

h→0 |g|H 1

2 sup X(t, f, g, h)H + ν

0tT

T

X(s, f, g, h)2 ds = 0. V

(3.51)

0

The equality (3.40) follows immediately from the above relation. To complete the proof that the map H f → v(t, f ) ∈ H is C 1,1 (Fréchet), observe first that the Gateaux derivative z(t, f ) ∈ L(H ). So for the map H f → v(t, f ) ∈ H to be Fréchet continuously differentiable, it is sufficient to prove that the map H f → z(t, f ) ∈ L(H ) is Lipschitz continuous on bounded sets.


3565

In what follows, let g, f1 , f2 ∈ H be such that |fi |H M, i = 1, 2, |g|H 1 and t ∈ [0, T ]. From (3.36), we obtain z(t, f1 )(g) − z(t, f2 )(g)

t =−

A z(s, f1 )(g) − z(s, f2 )(g) ds −

0

t

Q(s) z(s, f1 )(g) · ∇ v(s, f1 ) ds

0

t

Q(s) z(s, f2 )(g) · ∇ v(s, f2 ) ds −

+ 0

t

Q(s) v(s, f1 ) · ∇ z(s, f1 )(g) ds

0

t

Q(s) v(s, f2 ) · ∇ z(s, f2 )(g) ds

+ 0

t =−

A z(s, f1 )(g) − z(s, f2 )(g) ds

0

t −

Q(s) z(s, f1 )(g) − z(s, f2 )(g) · ∇ v(s, f1 ) ds

0

t −

Q(s) z(s, f2 )(g) · ∇ v(s, f1 ) − v(s, f2 ) ds

0

t −

Q(s) v(s, f1 ) · ∇ z(s, f1 )(g) − z(s, f2 )(g) ds

0

t −

Q(s) v(s, f1 ) − v(s, f2 ) · ∇ z(s, f2 )(g) ds.

(3.52)

0

Differentiating both sides of (3.52) with respect to t, taking inner products of the resulting differential equation with z(t, f1 )(g) − z(t, f2 )(g), using the chain rule and integrating over t gives z(t, f1 )(g) − z(t, f2 )(g)2 H

t = −2ν

z(s, f1 )(g) − z(s, f2 )(g)2 ds V

0

t −2 0

Q(s)b z(s, f1 )(g) − z(s, f2 )(g), v(s, f1 ), z(s, f1 )(g) − z(s, f2 )(g) ds

3566


t −2

Q(s)b z(s, f2 )(g), v(s, f1 ) − v(s, f2 ), z(s, f1 )(g) − z(s, f2 )(g) ds

0

t −2

Q(s)b v(s, f1 ), z(s, f1 )(g) − z(s, f2 )(g), z(s, f1 )(g) − z(s, f2 )(g) ds

0

t −2

Q(s)b v(s, f1 ) − v(s, f2 ), z(s, f2 )(g), z(s, f1 )(g) − z(s, f2 )(g) ds

0

t = −2ν

z(s, f1 )(g) − z(s, f2 )(g)2 ds + I1 + I2 + I3 , V

(3.53)

0

where we have used the fact that b(u, w, w) = 0, and where

t I1 := −2

Q(s)b z(s, f1 )(g) − z(s, f2 )(g), v(s, f1 ), z(s, f1 )(g) − z(s, f2 )(g) ds,

(3.54)

Q(s)b z(s, f2 )(g), v(s, f1 ) − v(s, f2 ), z(s, f1 )(g) − z(s, f2 )(g) ds,

(3.55)

Q(s)b v(s, f1 ) − v(s, f2 ), z(s, f2 )(g), z(s, f1 )(g) − z(s, f2 )(g) ds.

(3.56)

0

t I2 := −2 0

t I3 := −2 0

We now estimate each of the terms on the right-hand side of (3.54), (3.55) and (3.56). Using “Young’s inequality with ” in (3.54), we obtain

t |I1 | : 4Q∞

v(s, f1 ) z(s, f1 )(g) − z(s, f2 )(g) z(s, f1 )(g) − z(s, f2 )(g) ds V H V

0

t 4 Q∞

z(s, f1 )(g) − z(s, f2 )(g)2 ds V

0

+ 4

−1

t Q∞

v(s, f1 )2 z(s, f1 )(g) − z(s, f2 )(g)2 ds. V H

(3.57)

0

From (3.55), (3.38), “Young’s inequality with ”, Hölder’s inequality and (3.43), it follows that


t |I2 | : 4Q∞

3567

z(s, f2 )(g)1/2 z(s, f2 )(g)1/2 v(s, f1 ) − v(s, f2 ) V

H

V

0

1/2 1/2 × z(s, f1 )(g) − z(s, f2 )(g)V z(s, f1 )(g) − z(s, f2 )(g)H ds

t 2cQ∞

z(s, f2 )(g) v(s, f1 ) − v(s, f2 )2 ds V H

0

t + 2cQ∞

z(s, f1 )(g) − z(s, f2 )(g) z(s, f2 )(g) V V

0

× z(s, f1 )(g) − z(s, f2 )(g)H ds t 2 v(s, f1 ) − v(s, f2 ) ds 2cQ∞ sup z(s, f2 )(g)H · V 0sT

t + 2cQ∞

0

z(s, f1 )(g) − z(s, f2 )(g)2 ds V

0

+ 2c

−1

t Q∞

z(s, f2 )(g)2 z(s, f1 )(g) − z(s, f2 )(g)2 ds V H

0

$ # # $ 1 1 1 2cQ∞ |g|2H exp cQ2∞ |f2 |2H √ |f1 − f2 |H exp Q2∞ c 2ν ν ν

t + 2cQ∞

z(s, f1 )(g) − z(s, f2 )(g)2 ds V

0

+ 2c

−1

t Q∞

z(s, f2 )(g)2 z(s, f1 )(g) − z(s, f2 )(g)2 ds. V H

(3.58)

0

In (3.56), we use Hölder inequality and “Young’s inequality with ”, together with (3.38), (3.43) and (3.8), to get the following estimates

t |I3 | 4Q∞

v(s, f1 ) − v(s, f2 )1/2 v(s, f1 ) − v(s, f2 )1/2 V

H

0

1/2 1/2 × z(s, f2 )(g)V z(s, f1 )(g) − z(s, f2 )(g)V z(s, f1 )(g) − z(s, f2 )(g)H ds

t 2cQ∞ 0

v(s, f1 ) − v(s, f2 ) v(s, f1 ) − v(s, f2 ) z(s, f2 )(g) ds V H V

3568


t + 2Q∞

z(s, f1 )(g) − z(s, f2 )(g)

V

0

× z(s, f1 )(g) − z(s, f2 )(g)H z(s, f2 )(g)V ds t 2cQ∞

z(s, f2 )(g)2 ds V

1/2 t ·

0

v(s, f1 ) − v(s, f2 )2 ds V

1/2

0

× sup v(s, f1 ) − v(s, f2 )H + 2 Q∞ 0sT

t

z(s, f1 )(g) − z(s, f2 )(g)2 ds V

0

+ 2 −1 Q∞

t

z(s, f2 )(g)2 z(s, f1 )(g) − z(s, f2 )(g)2 ds V H

0

$ $ # # 1 1 2 1 2 2 2 2cQ∞ |g|H exp cQ∞ |f2 |H √ |f1 − f2 |H exp cQ∞ 2ν ν ν

t + 2 Q∞

z(s, f1 )(g) − z(s, f2 )(g)2 ds V

0

+ 2

−1

t Q∞

z(s, f2 )(g)2 z(s, f1 )(g) − z(s, f2 )(g)2 ds. V H

(3.59)

0

Choose > 0 sufficiently small such that 6 Q∞ + 2c Q∞ < ν.

(3.60)

Using (3.57), (3.58), (3.59), and (3.60) in (3.53), we get z(t, f1 )(g) − z(t, f2 )(g)2 + ν H

t

z(s, f1 )(g) − z(s, f2 )(g)2 ds V

0

t c|f1 − f2 |2H

+c

v(s, f1 )2 + z(s, f2 )(g)2 V V

0

2 × z(s, f1 )(g) − z(s, f2 )(g)H ds,

(3.61)

for all f1 , f2 , g ∈ H such that |fi |H M, i = 1, 2, |g|H 1, with c a random constant (dependent on M, ν and T ). Applying Gronwall’s lemma (Lemma 5.1) to (3.61) and using (3.8) and (3.38), we get


z(t, f1 )(g) − z(t, f2 )(g)2 + ν H

t

3569

z(s, f1 )(g) − z(s, f2 )(g)2 ds V

0

t c|f1 − f2 |2H exp

v(s, f1 )2 + z(s, f2 )(g)2 ds V V

0

& %

|g|2 1 c|f1 − f2 |2H exp |f1 |2H + |f2 |2H × H exp cQ2∞ |f2 |2H 2ν ν & % 1 2 1 2 2 2 2M · exp cQ∞ M c|f1 − f2 |H exp 2ν ν c|f1 − f2 |2H .

(3.62)

Therefore, z(t, f1 ) − z(t, f2 )

1/2

L(H )

c|f1 − f2 |H

(3.63)

for all f1 , f2 ∈ H with |f1 |H M, |f2 |H M and all t ∈ [0, T ]. This proves that the map H f → v(t, f, ω) ∈ H is C 1 for each ω ∈ Ω and each t ∈ [0, T ]. Furthermore, its Fréchet derivative H f → Dv(t, f, ω) = z(t, f, ω) ∈ L(H ) is Lipschitz continuous on bounded sets in H . The compactness of the Fréchet derivative Dv(t, f, ω); H → H , t > 0, follows immediately from the fact that the map H f → v(t, f, ω) ∈ H , t > 0, is C 1 and carries bounded sets into relatively compact ones (Proposition 3.3). See the proof of Theorem 3.1 and Lemma 3.1 in [13, Part I]. This completes the proof of Theorem 3.1. 2 We are now ready to state the main result in this section. Theorem 3.2 (The cocycle). Let u(t, f, ·) be the unique global solution of the stochastic Navier– Stokes equation (3.1) for t 0 and f ∈ H . Denote by θ : R+ × Ω → Ω the standard Brownian shift θ (t, ω)(s) := ω(t + s) − ω(t),

t, s 0, ω ∈ Ω,

(3.64)

on Wiener space (Ω, F , P ). Then there is a version u : R+ × H × Ω → H of the solution of (3.1) with the following properties: (i) The map u : R+ × H × Ω → H is jointly measurable, and for each f ∈ H , the process u(·, f, ·) : R+ × Ω → H is (Ft )t0 -adapted. (ii) For each t > 0 and ω ∈ Ω, the map u(t, ·, ω) : H → H takes bounded sets into relatively compact sets. (iii) (u, θ ) is a C 1,1 perfect cocycle; viz.

u t2 , u(t1 , f, ω), θ (t1 , ω) = u(t1 + t2 , f, ω) for all t1 , t2 0, f ∈ H , ω ∈ Ω.

(3.65)

3570


(iv) For each (t, f, ω) ∈ R+ × H × Ω, the Fréchet derivative Du(t, f, ω) ∈ L(H ) of the map u(t, ·, ω) is compact linear, and the map R+ × H × Ω → L(H ), (t, f, ω) → Du(t, f, ω) is strongly measurable. (v) For fixed ρ, a > 0, E log+

sup

0t1 ,t2 a |f |H ρ

u t2 , f, θ (t1 , ·) + Du t2 , f, θ (t1 , ·) < ∞. H L(H )

(3.66)

Proof. To prove assertion (i) of the theorem, define the required version u : R+ × H × Ω → H by setting u(t, f, ω) := Q(t, ω)v(t, f, ω),

t 0, ω ∈ Ω, f ∈ H.

(3.67)

Note first that Q is jointly measurable and (Ft )t0 -adapted. In view of Proposition 3.2, it follows from (3.67) that u satisfies assertion (i). Assertion (ii) of the theorem follows immediately from (3.67) and Proposition 3.3. Next, we establish the perfect cocycle property (iii). To see this, observe that Q has the cocycle property

Q(t1 + t2 , ω) = Q t2 , θ (t1 , ω) Q(t1 , ω),

t1 , t2 0, ω ∈ Ω.

(3.68)

This follows directly from (3.4). Thus, (3.65) will follow if we prove the following identity

v(t1 + t2 , f, ω) = Q(t1 , ω)−1 v t2 , Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω)

(3.69)

for t1 , t2 0, ω ∈ Ω, f ∈ H . Indeed, assume that (3.69) holds. Fix ω ∈ Ω and t1 0 throughout this proof. Then, for t 0, we have

u t, u(t1 , f, ω), θ (t1 , ω) = Q t, θ (t1 , ω) v t, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω)

= Q(t1 + t, ω)Q(t1 , ω)−1 v t, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω) = Q(t1 + t, ω)v(t1 + t, f, ω) = u(t1 + t, f, ω). Hence the perfect cocycle property (3.65) holds. We now show (3.69). To do this, define the processes

z(t, ω) := Q(t1 , ω)−1 v t, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω) , z(t, ω) := v(t + t1 , f, ω),

(3.70)


3571

for all t 0 and all ω ∈ Ω. Shifting the time-variable t by t1 in the integral equation for v, it is easy to see that z(t, ω) satisfies the following equation

t z(t, ω) = v(t1 , f, ω) −

t Az(s, ω) ds −

0

Q(t1 + s, ω)B z(s, ω) ds

(3.71)

0

for all t 0, ω ∈ Ω. On the other hand,

v t, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω)

t = Q(t1 , ω)v(t1 , f, ω) −

Av s, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω) ds

0

t −

Q s, θ (t1 , ω) B v s, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω) ds

(3.72)

0

for all t 0, ω ∈ Ω. Multiplying the above equation by Q(t1 , ω)−1 and using the bilinear property of B, we easily see that

t z(t, ω) = v(t1 , f, ω) −

t Az(s, ω) ds −

0

Q(t1 + s, ω)B z(s, ω) ds

(3.73)

0

for all t 0, ω ∈ Ω. Subtract z(t, ω) from z(t, ω) and use a similar calculation as for (3.12) to deduce that z(t, ω) = z(t, ω) for all t 0 and ω ∈ Ω. This proves the identity (3.69), and so the cocycle property (iii) holds for the solution map u : R+ × H × Ω → H of the SNSE (3.1). Assertion (iv) of the theorem is a consequence of Theorem 3.1, Proposition 3.2 and relation (3.67). (The strong measurability of Du(t, f, ω) follows from Eq. (3.40).) Let us now prove the integrability estimate in assertion (v) of the theorem. Let 0 t1 , t2 a and f ∈ H with |f |H ρ. It follows from (3.7) that

u t2 , f, θ (t1 , ω) = Q t2 , θ (t1 , ω) v t2 , f, θ (t1 , ω) H H

Q t2 , θ (t1 , ω) |f |H = Q(t1 + t2 , ω)Q−1 (t1 , ω)|f |H ρQ∞ Q−1 ∞ ,

(3.74)

where Q−1 ∞ := sup0t2a Q−1 (t). Using (3.39) we have

Du t2 , f, θ (t1 , ω)

L(H )

= Q t2 , θ (t1 , ω) Dv t2 , f, θ (t1 , ω) L(H ) $ # −1 2 −1 2 1 2 |f |H Q∞ Q exp cQ∞ Q ∞ 2ν ∞ # $ 2 1 Q∞ Q−1 ∞ exp cQ2∞ Q−1 ∞ ρ 2 . 2ν

(3.75)

3572


Combining (3.74) and (3.75), we get E log+

sup

0t1 ,t2 a |f |H ρ

u t2 , f, θ (t1 , ·) + E log+ H

sup

0t1 ,t2 a |f |H ρ

Du t2 , f, θ (t1 , ·)

L(H )

& %

2 1 E 2 log+ Q∞ Q−1 ∞ + cQ2∞ Q−1 ∞ ρ 2 + log+ ρ < ∞. 2ν (3.76) The above relation implies the integrability condition (3.66). The proof of Theorem 3.2 is now complete. 2 It is easy to see from (3.76) and (3.63) that the following stronger estimate holds E log+

sup

0t1 ,t2 a

u t2 , ·, θ (t1 , ·)

C 1,1

< ∞,

where · C 1,1 denotes the C 1,1 norm on the ball B(0, ρ) in H . 4. The multiplicative ergodic theory Our objective in this section is to characterize the local behavior of solutions of the SNSE (3.1) near an equilibrium or a stationary point/solution. We next describe the concepts of equilibrium or a stationary point for the SNSE (3.1). Definition 4.1 (Equilibrium/stationary point). An F -measurable random variable Y : Ω → H is said be an equilibrium or a stationary random point for the cocycle (u, θ ) if

u t, Y (ω), ω = Y θ (t, ω)

(4.1)

perfectly in ω ∈ Ω for all t ∈ R+ . A trivial equilibrium or stationary solution of the SNSE (3.1) is u(t, 0, ω) ≡ 0 corresponding to the zero initial function f ≡ 0 ∈ H . 4.1. Dynamics near a general equilibrium In order to analyze the dynamics of the SNSE (3.1) near a general equilibrium or stationary point Y : Ω → H , we linearize the C 1,1 cocycle u : R+ × H × Ω → H at Y . This gives a linear cocycle of Fréchet derivatives Du(t, Y (ω), ω) ∈ L(H ) satisfying the following random equations Du(t, Y ) = Q(t, ·)Dv(t, Y ), and

t 0,

(4.2)


t Dv(t, Y )(g) = g −

t ADv(s, Y )(g) ds −

0

t −

3573

Q(s) Dv(s, Y )(g) · ∇ v(s, Y ) ds

0

Q(s) v(s, f ) · ∇ Dv(s, Y )(g) ds,

t 0, g ∈ H.

(4.3)

0

We next apply the Oseledec–Ruelle spectral theorem to the compact linear cocycle (Du(t, Y (ω), ω), θ (t, ω)), t 0, ω ∈ Ω ([16, Theorem 2.1.1], [14]). This gives Theorem 4.1 (The Lyapunov spectrum: General equilibrium). Let (u(t, ·, ω), θ (t, ω)) be the C 1,1 cocycle on H generated by the stochastic Navier–Stokes equation (3.1). Suppose that Y : Ω → H is a stationary random point for the cocycle (u, θ ) of the SNSE (3.1) with E log+ |Y | < ∞. Then the following limit Λ(ω) := lim

∗

1/2t Du t, Y (ω), ω ◦ Du t, Y (ω), ω

t→∞

(4.4)

exists in the uniform operator norm in L(H ), perfectly in ω. The Oseledec operator Λ(ω) in (4.4) is compact, self-adjoint and non-negative with discrete non-random spectrum eλ1 > eλ2 > eλ3 > · · · > eλn > · · · .

(4.5)

The Lyapunov exponents {λn }∞ n=1 correspond to values of the limit

1 log Du t, Y (ω), ω (g)H ∈ {λn }∞ n=1 t→∞ t lim

for any g ∈ H , perfectly in ω. Each eigenvalue eλj has a fixed finite multiplicity mj with a corresponding finite-dimensional eigenspace Fj (ω) such that mj := dim Fj (ω), j 1, ω ∈ Ω. If we set ! E1 (ω) := H,

En (ω) :=

n−1 '

"⊥ Fj (ω)

,

n > 1,

j =1

then for each n 1, codim En (ω) =

n−1

j =1 mj

< ∞, and the following assertions are true:

En (ω) ⊂ En−1 (ω) ⊂ · · · ⊂ E2 (ω) ⊂ E1 (ω) = H,

1 lim log Du t, Y (ω), ω (g)H = λn t→∞ t

n > 1; (4.6)

for g ∈ En (ω)\En+1 (ω);

1 log Du t, Y (ω), ω L(H ) = λ1 ; t→∞ t lim

and

(4.7)

3574


Du t, Y (ω), ω En (ω) ⊆ En θ (t, ω)

(4.8)

for all t 0, perfectly in ω ∈ Ω, for all n 1. Proof. Recall the Oseledec integrability condition E log+

sup

0t1 ,t2 a

Du t2 , Y θ (t1 , ·) , θ (t1 , ·)

L(H )

eλ2 > eλ3 > · · · > eλn > · · ·

(4.11)

due to the ergodicity of the Brownian shift θ . The assertions (4.7) and (4.8) of the theorem follow from the Oseledec–Ruelle spectral theorem [14, Theorem 2.1.1]. 2 Remark. If the cocycle is linearized at the zero equilibrium Y ≡ 0, the Oseledec–Ruelle operator Λ(ω) is non-random. Consequently, the Oseledec spaces {En : n 1} are also non-random. This will be shown later in the section. Definition 4.2 (Hyperbolicity). A stationary point Y : Ω → H for the SNSE (3.1) is hyperbolic if the linearized cocycle (Du(t, Y (ω), ω), θ (t, ω)) has a non-vanishing Lyapunov spectrum: λi = 0 for all i 1. Theorem 4.2 below is a consequence of the nonlinear multiplicative ergodic theorem [14, Theorem 2.2.1]. It describes the saddle-point behavior of the random flow of the SNSE (3.1) in the neighborhood of any equilibrium. ¯ For any ρ > 0 and any f ∈ H , we will denote by B(f, ρ) the closed ball in H center f and radius ρ. Theorem 4.2 (The local stable manifold theorem: General equilibrium). Assume that Y : Ω → H is a hyperbolic stationary random point for the cocycle (u, θ ) of the SNSE (3.1) with E log+ |Y | < ∞. Denote by {· · · < λi+1 < λi < · · · < λ2 < λ1 } the Lyapunov spectrum of the linearized cocycle (Du(t, Y (ω), ω), θ (t, ω), t 0) as given in Theorem 4.1. Define i0 := min{i: λi < 0}. Fix 1 ∈ (0, −λi0 ) and 2 ∈ (0, λi0 −1 ). Then there exist (i) a sure event Ω ∗ ∈ F with θ (t, ·)(Ω ∗ ) = Ω ∗ for all t ∈ R;


3575

(ii) F -measurable random variables ρi , βi : Ω ∗ → (0, 1), βi > ρi > 0, i = 1, 2, such that for each ω ∈ Ω ∗ , the following is true: ¯ (ω), ρ1 (ω)) and B(Y ¯ (ω), ρ2 (ω)) (resp.) There are C 1,1 submanifolds S(ω), U(ω) of B(Y with the following properties: ¯ (ω), ρ1 (ω)) such that (a) For λi0 > −∞, S(ω) is the set of all f ∈ B(Y

u(n, f, ω) − Y θ (n, ω) β1 (ω) exp (λi + 1 )n 0 H ¯ (ω), ρ1 (ω)) such for all integers n 0. If λi0 = −∞, then S(ω) is the set of all f ∈ B(Y that

u(n, f, ω) − Y θ (n, ω) β1 (ω)eλn H for all integers n 0 and any λ ∈ (−∞, 0). Furthermore, lim sup t→∞

1 log u(t, f, ω) − Y θ (t, ω) H λi0 t

(4.12)

for all f ∈ S(ω). The stable subspace S 0 (ω) of the linearized cocycle (Du(t, Y (ω), ·), θ (t, ·)) is tangent at Y (ω) to the submanifold S(ω), viz. TY (ω) S(ω) = S 0 (ω). In partici −1 ular, codim S(ω) = codim S 0 (ω) = j0=1 dim Fj (ω) is fixed and finite. (b)

$& % # |u(t, f1 , ω) − u(t, f2 , ω)|H 1 lim sup log sup : f1 = f2 , f1 , f2 ∈ S(ω) λi0 . |f1 − f2 |H t→∞ t

(c) (Cocycle-invariance of the stable manifolds): There exists τ1 (ω) 0 such that

u(t, ·, ω) S(ω) ⊆ S θ (t, ω)

(4.13)

for all t τ1 (ω). Also

Du t, Y (ω), ω S 0 (ω) ⊆ S 0 θ (t, ω) ,

t 0.

(4.14)

¯ (ω), ρ2 (ω)) with the property that there is a discrete(d) U(ω) is the set of all f ∈ B(Y time “history” process y(·, ω) : {−n: n 0} → H such that y(0, ω) = f and for each integer n 1, one has u(1, y(−n, ω), θ (−n, ω)) = y(−(n − 1), ω) and

y(−n, ω) − Y θ (−n, ω) β2 (ω) exp −(λi −1 − 2 )n . 0 H ¯ (ω), ρ2 (ω)) with the property that there is If λi0 −1 = ∞, U(ω) is the set of all f ∈ B(Y a discrete-time “history” process y(·, ω) : {−n: n 0} → H such that y(0, ω) = f and for each integer n 1,

y(−n, ω) − Y θ (−n, ω) β2 (ω) exp{−λn}, H

3576


for any λ ∈ (0, ∞). Furthermore, for each f ∈ U(ω), there is a unique continuoustime “history” process also denoted by y(·, ω) : (−∞, 0] → H such that y(0, ω) = f , u(t, y(s, ω), θ (s, ω)) = y(t + s, ω) for all s 0, 0 t −s, and lim sup t→∞

1 log y(−t, ω) − Y θ (−t, ω) H −λi0 −1 . t

Each unstable subspace U 0 (ω) of the linearized cocycle (Du(t, Y (·), ·), θ (t, ·)) is tangent at Y (ω) to U(ω), viz. TY (ω) U(ω) = U 0 (ω). In particular,

dim U(ω) =

i 0 −1

dim Fj (ω)

j =1

is finite and non-random. (e) Let y(·, fi , ω), i = 1, 2, be the history processes associated with fi = y(0, fi , ω) ∈ U(ω), i = 1, 2. Then % # $& |y(−t, f1 , ω) − y(−t, f2 , ω)|H 1 lim sup log sup : f1 = f2 , fi ∈ U(ω), i = 1, 2 |f1 − f2 |H t→∞ t −λi0 −1 . (f) (Cocycle-invariance of the unstable manifolds): There exists τ2 (ω) 0 such that

U(ω) ⊆ u t, ·, θ (−t, ω) U θ (−t, ω)

(4.15)

for all t τ2 (ω). Also

Du t, ·, θ (−t, ω) U 0 θ (−t, ω) = U 0 (ω),

t 0;

and the restriction

Du t, ·, θ (−t, ω) U 0 θ (−t, ω) : U 0 θ (−t, ω) → U 0 (ω),

t 0,

is a linear homeomorphism onto. (g) The submanifolds U(ω) and S(ω) are transversal, viz. H = TY (ω) U(ω) ⊕ TY (ω) S(ω). We will only give an outline of the proof of Theorem 4.2. Full details of the proof may be obtained by adapting the arguments in [13,14].


3577

An outline of the proof of Theorem 4.2. • Develop perfect continuous-time versions of Kingman’s subadditive ergodic theorem as well as the ergodic theorem [14, Lemma 2.3.1(ii), (iii)]. The linearized cocycle (Du(t, Y (ω), ·), θ (t, ·)) at the equilibrium Y can be shown to satisfy the hypotheses of these perfect ergodic theorems. As a consequence of the perfect ergodic theorems, one obtains stable/unstable subspaces for the linearized cocycle, which will constitute tangent spaces to the local stable and unstable manifolds of the nonlinear cocycle (u, θ ). • We use hyperbolicity of the equilibrium Y , the continuous-time integrability condition (3.66) on the cocycle and perfect versions of the ergodic and subadditive ergodic theorems to show the existence of local stable/unstable manifolds for the discrete cocycle (u(n, ·, ω), θ (n, ω)) near Y (ω) (cf. [16, Theorems 5.1 and 6.1]). These manifolds are random objects and are perfectly defined for ω ∈ Ω. Using interpolation between discrete times and the (continuoustime) integrability condition (3.66), it can be shown that the above manifolds for the discretetime cocycle (u(n, ·, ω), θ (n, ω)), n 1, also serve as perfectly defined local stable/unstable manifolds for the continuous-time cocycle (u(t, ·, ω), θ (t, ω)), t 0, near the equilibrium Y (see [13,14,16]). • Again, by using the integrability condition (3.66) on the nonlinear cocycle and its Fréchet derivatives, it is possible to control the excursions of the continuous-time cocycle (u(t, ·, ω), θ (t, ω)), t 0, between discrete times. In view of the perfect subadditive ergodic theorem, these estimates show that the local stable manifolds are asymptotically invariant under the nonlinear cocycle. The asymptotic invariance of the unstable manifolds is obtained via the concept of a stochastic history process for the cocycle. The existence of a stochastic history process is needed because the (locally compact) cocycle is not invertible. This completes the outline of the proof of Theorem 4.2.

2

We next discuss the behavior of the cocycle (u, θ ) near the zero equilibrium. 4.2. Dynamics near the zero equilibrium For the rest of the article, we will focus on the dynamics of the SNSE (3.1) relative to its zero equilibrium Y ≡ 0. In this special case, we are able to express the Lyapunov spectrum of the linearized cocycle (Du(t, 0, ω), θ (t, ω)) explicitly in terms of the parameters ν, γ , σi , i 1, in the SNSE (3.1). Let {Tt }t0 be the strongly continuous semigroup of the operator −A = ν with a Dirichlet boundary condition on ∂D. The operator −A has a discrete spectrum of eigenvalues {μn : n 1} and a complete orthonormal system of corresponding eigenfunctions {en : n 1}. We assume μ1 < μ2 < · · · < μn < · · · . Let Fn0 be the finite-dimensional eigenspace of −A corresponding to the eigenvalue μn for n 1. Next, we linearize the cocycle u : R+ × H × Ω → H at the zero equilibrium. To do this, put Y ≡ 0 in Eq. (4.3) to obtain

t Dv(t, 0, ω)(g) = g −

ADv(s, 0, ω)(g) ds,

(4.16)

0

for all g ∈ H , t ∈ [0, T ] and ω ∈ Ω. This implies that Dv(t, 0, ω) = Tt , t 0, ω ∈ Ω, and thus

3578


Du(t, 0, ω) = Q(t, ω)Dv(t, 0, ω) = Q(t, ω)Tt ,

t 0, ω ∈ Ω.

(4.17)

The next result is a special case (Y ≡ 0) of the Oseledec–Ruelle spectral theorem (Theorem 4.1). Theorem 4.3 (The Lyapunov spectrum: Zero equilibrium). Let (u(t, ·, ω), θ (t, ω)) be the C 1,1 cocycle on H generated by the stochastic Navier–Stokes equation (3.1). Then all the assertions of Theorem 4.1 hold perfectly in ω ∈ Ω subject to the following: (i) Y ≡ 0;

∞

(ii) λn = λ0n := −μn + γ −

1 2 σk , 2

n 1;

k=1

1

(iii) The Oseledec–Ruelle operator is deterministic and given by Λ0 (ω) = eγ − 2 (iv) The Oseledec spaces are deterministic and given by ! E10

:= H,

En0

:=

n−1 '

∞

2 k=1 σk

T1 ;

"⊥ Fj0

,

n > 1.

j =1

(v) The Lyapunov exponents satisfy the relations ∞ 1 1 2 log Du(t, 0, ω)(g)H = −μn + γ − σk t→∞ t 2

lim

k=1

0 ; and for g ∈ En0 \En+1 ∞ 1 1 2 log Du(t, 0, ω)L(H ) = −μ1 + γ − σk . t→∞ t 2

lim

k=1

Proof. In order to evaluate the Lyapunov spectrum {λ0n : n 1} of the linearized cocycle (Du(t, 0, ω), θ (t, ω)), we first compute the Oseledec–Ruelle operators Λ0 (ω) associated with this cocycle. To do this, use relation (4.17) to obtain Λ0 (ω) := lim

∗ 1/2t Du(t, 0, ω) ◦ Du(t, 0, ω)

t→∞

"1/2t ∞

∗ 2 2σk Wk (t) − σk t = lim exp 2γ t + Tt ◦ Tt !

t→∞

k=1

1/2t Wk (t) σk2 σk − = lim exp γ + lim T ∗ ◦ Tt t→∞ t→∞ t t 2 k=1 ∞

1/2t 1 2 = exp γ − σk lim Tt∗ ◦ Tt . t→∞ 2 ∞

k=1

(4.18)


3579

Now, it is easy to see that ∗

Tt ◦ Tt (en ) = exp{2μn t}en for all n 1. Therefore,

Tt∗ ◦ Tt

1/2t

= T1

(4.19)

for all t > 0. By (4.18) and (4.19),assertion (iii) of the theorem holds. In particular, the Oseledec– ∞ 1 2 Ruelle operator Λ0 (ω) = eγ − 2 k=1 σk T1 is non-random. Consequently, the Oseledec spaces {En : n 1} are also non-random. Assertions (iv) and (v) of the theorem follow directly from Theorem 4.1. 2 The next corollary is an immediate consequence of the above theorem: It gives necessary and sufficient conditions for hyperbolicity of the zero equilibrium Y ≡ 0. Corollary 4.3.1 (Hyperbolicity of the zero equilibrium). In the SNSE (3.1), the zero equilibrium is hyperbolic if and only if the following conditions hold 2 (i) −μ1 + γ − 12 ∞ k=1 σk > 0; 1 ∞ (ii) −μn + γ − 2 k=1 σk2 = 0 for all n 2. Theorem 4.4 below is a consequence of Theorem 4.2. It describes saddle-point behavior of the random flow of the SNSE (3.1) in the neighborhood of the zero equilibrium. Theorem 4.4 (The local stable manifold theorem: Zero equilibrium). In the SNSE (3.1), assume that the zero equilibrium is hyperbolic. Then all the assertions of the local stable manifold theorem (4.2) hold under the same choice of parameters as in Theorem 4.3. Our next result gives sufficient conditions on the parameters of the SNSE (3.1) to guarantee that the zero equilibrium is its only stationary point. Theorem 4.5 (Uniqueness of the stationary solution). Suppose that ∞

μ1 + γ +

1 2 σk < 0. 2

(4.20)

k=1

Then the zero equilibrium Y ≡ 0 is the only equilibrium (stationary point) of the SNSE (3.1). Proof. Assume that the SNSE (3.1) admits a non-zero stationary solution u0 (t). By stationarity, a := E[|u0 (t)|2H ] > 0 and b := E[u0 (t)2V ] > 0 are independent of t. Suppose t > s > 0. Then from (3.1) and Ito’s formula, we have

3580


u0 (t)2 = u0 (s)2 − 2ν H H

t

∞ u0 (r)2 dr + 2 V

t

2 σk u0 (r)H dWk (r)

k=1 s

s

+ 2γ

t

∞ u0 (r)2 dr + H

t

2 σk2 u0 (r)H dr.

(4.21)

k=1 s

s

Taking expectations on both sides of the above identity, we obtain a = a − 2ν(t − s)b +

∞

σk2 (t − s)a + 2γ (t − s)a.

(4.22)

k=1

Hence ! 2νb =

∞

" σk2

+ 2γ a.

(4.23)

k=1

Combining the above equality with the Poincare inequality: a

νb , −μ1

(4.24)

it follows that ∞

−μ1 γ +

1 2 σk . 2

(4.25)

k=1

This proves the theorem.

2

We conclude this section by stating the Local and Global Invariant Manifold Theorems for the SNSE (3.1) when γ = 0 (Theorems 4.6 and 4.7 below). The Local Invariant Manifold Theorem (Theorem 4.6) characterizes the almost sure asymptotic stability of the random flow of the SNSE (3.1) in the neighborhood of the zero equilibrium, in the special case when the linear drift vanishes (γ = 0). On the other hand, the Global Invariant Manifold Theorem (Theorem 4.7) gives a random cocycle-invariant foliation of the energy space H . The leaves of the foliation are 2 characterized by the Lyapunov exponents {λ0i = −μi + γ − 12 ∞ k=1 σk : i 1} of the linearized cocycle (Du(t, 0, ω), θ (t, ω)). Theorem 4.6(Local invariant manifolds). Consider the SNSE (3.1) with γ = 0. Fix 1 ∈ 2 (0, −μ1 + 12 ∞ k=1 σk ). Then there exist (i) a sure event Ω ∗ ∈ F with θ (t, ·)(Ω ∗ ) = Ω ∗ for all t ∈ R; (ii) F -measurable random variables ρi , βi : Ω ∗ → (0, 1), βi > ρi ρi+1 > 0, i 1, such that for each ω ∈ Ω ∗ , the following is true: ¯ ρi (ω)) with the following properties: There are C 1,1 submanifolds Si (ω), i 1, of B(0,


3581

¯ ρi (ω)) such that (a) Si (ω) is the set of all f ∈ B(0, u(n, f, ω) βi (ω) exp H

∞ 1 2 μi − σ k + 1 n 2 k=1

for all integers n 0. Furthermore, lim sup t→∞

∞ 1 1 2 σk log u(t, f, ω)H μi − t 2

(4.26)

k=1

for all f ∈ Si (ω). Each Oseledec space Ei0 of the linearized cocycle (Du(t, 0, ·), θ (t, ·)) is tangent at 0 to the submanifold Si (ω), viz. T0 Si (ω) = Ei0 . In particular, codim Si (ω)= 0 codim Ei0 = i−1 j =1 dim Fj ( fixed and finite). (b) % # $& |u(t, f1 , ω) − u(t, f2 , ω)|H 1 : f1 = f2 , f1 , f2 ∈ Si (ω) lim sup log sup |f1 − f2 |H t→∞ t ∞

μi −

1 2 σk . 2 k=1

(c) (Cocycle-invariance): There exists τi (ω) 0 such that

u(t, ·, ω) Si (ω) ⊆ Si θ (t, ω)

(4.27)

Du(t, 0, ω) Ei0 ⊆ Ei0 ,

(4.28)

for all t τi (ω). Also t 0.

2 ∗ ∗ ∗ Proof. Let 1 ∈ (0, −μ1 + 12 ∞ k=1 σk ). Then there exist Ω ∈ F such that θ (t, ·)(Ω ) = Ω for ∗ all t ∈ R, and F -measurable random variables ρi , βi : Ω → (0, 1) such that βi (ω) > ρi (ω) > 0, ¯ ρi (ω)) such that and C 1,1 local stable submanifolds Si (ω) ⊂ B(0,

1 ∞ 2 Si (ω) := f ∈ B¯ 0, ρi (ω) : u(n, f, ω)H βi (ω)e(μi − 2 k=1 σk + 1 )n for all n 1 . (4.29) Furthermore, ∞ 1 1 2 lim sup log u(t, f, ω) H μi − σk 2 t→∞ t k=1

for all f ∈ Si (ω).

(4.30)

3582


Also, T0 Si (ω) = Ei0 , the Oseledec space for the linearized cocycle (Du(t, 0, ω), θ (t, ω)) cor 2 responding to the Lyapunov exponent λi := μi − 12 ∞ k=1 σk , i 1. Following the argument in [13,16], the random variables ρi (ω), βi (ω) may be selected such that

ρi (ω)e(λi + 1 )t ρi θ (t, ω)

(4.31)

βi (ω)e(λi + 1 )t βi θ (t, ω)

(4.32)

and

for all t 0 and ω ∈ Ω ∗ . We now show that there exists τi (ω) > 0 such that

u(t, ·, ω) Si (ω) ⊆ Si θ (t, ω)

(4.33)

for all t τi (ω) and all ω ∈ Ω ∗ . Let f ∈ Si (ω), t 0 and let n 0 be any integer. Then (by the cocycle property),

u n, u(t, f, ω), θ (t, ω) = u(n + t, f, ω) . H H

(4.34)

From [13,16], we have 1 lim sup log t→∞ t

% sup

f ∈Si (ω) f =0

|u(t, f, ω)|H |f |H

& λi .

From the above estimate, for any ∈ (0, 1 ), there exists N0 = N0 ( ) > 0 such that % 1 sup log tN t

sup

f ∈Si (ω) f =0

|u(t, f, ω)|H |f |H

&

λi + ,

for all N N0 . Thus sup

f ∈Si (ω) f =0

|u(t, f, ω)|H

e(λi + )t , |f |H

for all t N0 . Define

βi (ω) := sup

sup

0tN0 f ∈Si (ω) f =0

|u(t, f, ω)|H −(λi + )N0 ·e . |f |H

Therefore, sup

f ∈Si (ω) f =0

|u(t, f, ω)|H

βi (ω) · e(λi + )t |f |H

(4.35)


3583

for all t 0. Since f ∈ Si (ω) ⊂ B(0, 1), then |f |H 1 and u(t, f, ω) β (ω)e(λi + )t i H

(4.36)

for all t 0. Therefore, from (4.34) and (4.36),

u n, u(t, f, ω), θ (t, ω) β (ω)e(λi + )(n+t) i H

βi (ω)e(λi + )t · e(λi + 1 )n

(4.37)

for t 0 and all integers n 0. Since < 1 , it follows that

β (ω)e(λi + )t βi (ω) ( − 1 )t ·e = lim = 0. lim i t→∞ βi (ω)e(λi + 1 )t t→∞ βi (ω) Hence there exists τ˜i (ω) > 0 so that

βi (ω)e(λi + )t βi (ω)e(λi + 1 )t ,

(4.38)

for all t τ˜i (ω). By (4.37), (4.38) and (4.32), we get

u n, u(t, f, ω), θ (t, ω) βi (ω)e(λi + 1 )t · e(λi + 1 )n H

βi θ (t, ω) e(λi + 1 )n

(4.39)

for all t τ˜i (ω) and all n 1. Again, because < 1 , we have

βi (ω)e(λi + )t β (ω)e(λi + )t lim i = 0. t→∞ ρi (θ (t, ω)) t→∞ ρi (ω)e(λi + 1 )t lim

(4.40)

Therefore, there exists τ˜˜ i (ω) > 0 such that

βi (ω)e(λi + )t ρi θ (t, ω)

(4.41)

¯ ρi (θ (t, ω))) for all t τ˜˜ i (ω). for all t τ˜˜ i (ω). Hence u(t, f, ω) ∈ B(0, ˜ ¯ ρi (θ (t, ω))) and satisfies (4.39) for all Set τi (ω) := τ˜i (ω) ∨ τ˜ i (ω). Then u(t, f, ω) ∈ B(0, n 0 and all t τi (ω). By definition of Si (θ (t, ω)), it follows that u(t, f, ω) ∈ Si (θ (t, ω)) for all t τi (ω). Thus u(t, ·, ω)(Si (ω)) ⊆ Si (θ (t, ω)) for all t τi (ω). Note that τi (ω), τ˜i (ω), τ˜˜ i (ω)

are all independent of f ∈ Si (ω) because βi (ω), βi (ω) and ρi (ω) are independent of f ∈ Si (ω). This completes the proof of asymptotic invariance of Si (ω), i 1. 2 Our final result (Theorem 4.7 below) gives the existence of a global invariant flag for the cocycle (u, θ ). The foliation is induced by the Lyapunov spectrum {λi }∞ i=1 of the linearized cocycle.

3584


Theorem 4.7 (Global invariant flag). Consider the SNSE (3.1) with γ = 0. Define the random family of sets {Mi (ω): ω ∈ Ω ∗ , i 1} by $ # 1 Mi (ω) := f ∈ H : lim log u(t, f, ω) H λi t→∞ t

(4.42)

for i 1, ω ∈ Ω ∗ . For fixed i 1, ω ∈ Ω ∗ , define the sequence {Sin (ω)}∞ n=1 , inductively by: Si1 (ω) := Si (ω), u(n, ·, ω)−1 [Si (θ (n, ω))], n Si (ω) := Sin−1 (ω),

(4.43) if

Sin−1 (ω) ⊆ u(n, ·, ω)−1 [Si (θ (n, ω))],

otherwise,

(4.44)

for all n 2. In (4.43) and (4.44), the Si (ω) are the local invariant C 1,1 Hilbert submanifolds of H constructed in Theorem 4.6. Then the following is true for each i 1 and ω ∈ Ω ∗ : (i) The sets {Mi (ω): ω ∈ Ω ∗ , i 1} are cocycle-invariant:

u(t, ·, ω) Mi (ω) ⊆ Mi θ (t, ω)

(4.45)

for all t 0. (ii) Sin (ω) ⊆ Sin+1 (ω) for all n 1, and Mi (ω) =

∞ (

Sin (ω),

i 1,

(4.46)

n=1

( perfectly in ω). (iii) Mi+1 (ω) ⊆ Mi (ω). (iv) For any f ∈ Mi (ω)\Mi+1 (ω), 1 log u(t, f, ω)H ∈ (λi+1 , λi ]. t→∞ t lim

(4.47)

Proof. Fix ω ∈ Ω ∗ , where Ω ∗ is defined as in Theorem 4.6. (i) To prove the cocycle invariance property (4.45), let f ∈ Mi (ω) and t1 > 0. Then by definition (4.42) of Mi (ω), we have 1 log u(t, f, ω)H λi . t→∞ t lim

(4.48)

By the cocycle property of (u, θ ), we have

1 1 log u t, u(t1 , f, ω), θ (t1 , ω) H = lim log u(t + t1 , f, ω)H t→∞ t t→∞ t 1 t + t1 = lim log u(t + t1 , f, ω)H · lim t→∞ t + t1 t→∞ t lim


= lim

1

t→∞ t

3585

log u(t, f, ω)H

λi . The above inequality implies that u(t1 , f, ω) ∈ Mi (θ (t1 , ω)). Hence u(t1 , ·, ω)(Mi (ω)) ⊆ Mi (θ (t1 , ω)) and so (4.45) holds for all t 0. (ii) To prove assertion (ii) of the theorem, observe first that (4.44) implies that Sin (ω) ⊆ n+1 Si (ω) for all n 1. Next, we show that Sin (ω) ⊂ Mi (ω)

(4.49)

for all n 1. We prove (4.49) by induction on n 1. Let f ∈ Si1 (ω) = Si (ω). By Theorem 4.6 and assertion (4.26), it follows that 1 log u(t, f, ω)H λi t→∞ t lim

(4.50)

perfectly in ω. Therefore, f ∈ Mi (ω). Hence Si1 (ω) = Si (ω) ⊂ Mi (ω). Assume, by induction, that Sik (ω) ⊂ Mi (ω) for all 1 k n. If Sin+1 (ω) u(n + 1, ·, ω)−1 [Si (θ (n + 1, ω))], then Sin+1 (ω) = Sin (ω) ⊂ Mi (ω), by inductive hypothesis. Otherwise, Sin+1 (ω) = u(n + 1, ·, ω)−1 [Si (θ (n + 1, ω))]. Let f ∈ Sin+1 (ω) = u(n+1, ·, ω)−1 [Si (θ (n+1, ω))]. Then by the cocycle property and the definition of Si (θ (n + 1, ω)), it follows that

u n + n + 1, f, ω βi θ (n + 1, ω) en λi H

(4.51)

for all n 1. This implies that

1 u n + n + 1, f, ω λi . log n →∞ n

lim

Hence lim

n

→∞

1 1 log u n + n + 1, f, ω H log u n

, f, ω H = lim

n n →∞ n + n + 1

n

1 · lim log u n + n + 1, f, ω H = lim

n →∞ n + n + 1 n →∞ n λi .

Therefore, f ∈ Mi (ω), and Sin+1 (ω) ⊂ Mi (ω). So, by induction, it follows that Sin (ω) ⊂ Mi (ω) for all n 1. Thus

(4.52)

3586

S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591 ∞ (

Sin (ω) ⊆ Mi (ω).

(4.53)

n=1

In order to prove the converse inclusion Mi (ω) ⊆

∞ (

Sin (ω),

(4.54)

n=1

we establish the following: Claim. There exist an increasing (random) sequence of integers nk ↑ ∞ such that

−1 k Sin (ω) = u nk , ·, ω Si θ n, ω for all k 1. Proof. Define n1 := inf{n > 1: Sin−1 (ω) ⊆ u(n, ·, ω)−1 [Si (θ (n, ω))]}. Then 1 −1

Sin

−1 1

Si θ n , ω , (ω) ⊆ u n1 , ·, ω

and by definition (4.44),

Sin1 (ω) = u(n1 , ·, ω)−1 Si θ (n1 , ω) .

(4.55)

Furthermore, Sin−1 (ω) u(n, ·, ω)−1 [Si (θ (n, ω))] for all 1 < n < n1 , and so by definition (4.44), Sin (ω) = Sin−1 (ω) = Sin−2 (ω) = · · · = Si1 (ω) = Si (ω) for all 1 < n < n1 . In particular, 1 −1

Si (ω) = Sin

−1 1

Si θ n , ω . (ω) ⊆ u n1 , ·, ω

Therefore,

u n1 , ·, ω Si (ω) ⊆ Si θ n1 , ω . Hence

n1 = inf n > 1: u(n, ·, ω) Si (ω) ⊆ Si θ (n, ω) .

(4.56)

Since Si (ω) is asymptotically cocycle invariant (Theorem 4.6(c), (4.27)), it follows from (4.56) that 1 < n1 < ∞. Next, define n2 > n1 by

n2 := inf n > n1 : Sin−1 (ω) ⊆ u(n, ·, ω)−1 Si θ (n, ω) . As before, the definition (4.44) implies that

(4.57)

S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591 2 −1

Sin

1 +1

(ω) = Sin

−1 1

Si θ n , ω (ω) = u n1 , ·, ω

3587

(4.58)

and

−1 2

2 Si θ n , ω . Sin (ω) = u n2 , ·, ω

(4.59)

Since 2 −1

Sin

−1 2

Si θ n , ω , (ω) ⊆ u n2 , ·, ω

(4.60)

it follows from (4.58) that

−1 1

−1 2

Si θ n , ω ⊆ u n2 , ·, ω Si θ n , ω . u n1 , ·, ω Therefore,

−1 1

−1 1

2 Si θ n , ω ⊆ Si θ n2 , ω . ◦ u n , ·, ω u n , ·, ω

(4.61)

Using the cocycle property (Theorem 3.2(iii)), (4.61) implies

u n2 − n1 , ·, θ n1 , ω Si θ n1 , ω ⊆ Si θ n2 − n1 , θ n1 , ω .

(4.62)

By the asymptotic cocycle invariance of Si (θ (n1 , ω)), it follows from (4.62) that n1 < n2 < k ∞. Hence by induction, there exists an increasing sequence of integers {nk }∞ k=1 such that n ↑ ∞ as k → ∞ and

−1 k

k Si θ n , ω Sin (ω) = u nk , ·, ω for all integers k 1. This completes the proof of our claim.

(4.63)

2

We now proceed to prove the inclusion (4.54). Let f ∈ Mi (ω). Then by definition of Mi (ω), we have 1 log u(t, f, ω)H λi . t→∞ t lim

(4.64)

Fix 1 ∈ (0, −λ1 ) as in Theorem 4.6. Let 0 < < 1 . Then, using (4.64), there exists a positive integer n0 such that 1 log u(t, f, ω)H < λi +

tn t

sup for all n n0 . In particular,

u(n, f, ω) < en(λi + ) H for all n n0 . Define

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K(ω) := max u(k, f, ω)H . 1k 0 such that ¯ ρi (ω)) for all n m2 . Thus (4.69) implies that u(n, f, ω) ∈ B(0,

u(n, f, ω) ∈ Si θ (n, ω) for all n max(m1 , m2 ); i.e. f ∈ u(n, ·, ω)−1 [Si (θ (n, ω))], for all n max(m1 , m2 ). Now pick nk k k −1 k k sufficiently large such ∞that nn max(m1 , m2 ) and f ∈ u(n , ·, ω) [Si (θ (n , ω))] = Si (ω). This proves that f ∈ n=1 Si (ω); and so the inclusion (4.54) holds. The proof of assertion (ii) of the theorem is complete. Assertions (iii) and (iv) of the theorem follow directly from the definition (4.42) of the flag Mi (ω), i 1. 2 Remark. It is not clear if the Mi (ω) in Theorem 4.7 are C 1,1 immersed submanifolds in H . This would require transversality of the global semiflow u(n, ·, ω) and the local stable manifold Si (θ (n, ω)).


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5. Appendix The following version of Gronwall’s lemma is used throughout Section 3. Lemma 5.1. Suppose α : [0, T ] → R is C 1 and h, ψ : [0, T ] → R0 are continuous. Assume that

t α(t) +

t ψ(s) ds α(0) +

h(s)α(s) ds

0

0

t

t

(5.1)

for all t ∈ [0, T ]. Then ψ(s) ds α(0) exp

α(t) +

h(s) ds

0

(5.2)

0

for all t ∈ [0, T ]. Proof. Suppose (5.1) holds. Since α is C 1 , then (5.1) implies

t

t h(s)α(s) ds −

0

t

α (s) ds − 0

ψ(s) ds 0 0

for all t ∈ [0, T ]. Hence

t

h(s)α(s) − α (s) − ψ(s) ds 0

(5.3)

0

for all t ∈ [0, T ]. The above relation implies h(t)α(t) − α (t) − ψ(t) 0 for all t ∈ [0, T ]; i.e., α (t) − h(t)α(t) −ψ(t)

(5.4)

for) all t ∈ [0, T ]. We now multiply both sides of (5.4) by the “integrating factor” μ(t) := t e− 0 h(s) ds . This gives d μ(t)α(t) −μ(t)ψ(t) dt for all t ∈ [0, T ]. Integrating both sides of (5.5), we get

(5.5)

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t μ(t)α(t) − α(0) −

μ(s)ψ(s) ds 0

for all t ∈ [0, T ]. Therefore, −1

t

α(t) + μ(t)

μ(s)ψ(s) ds α(0)μ(t)−1

(5.6)

0

for all t ∈ [0, T ]. So α(t) + e

t

)t

0 h(s) ds

e−

)s 0

h(u)du

ψ(s) ds α(0)e

)t 0

h(s) ds

;

0

i.e.,

t α(t) +

e

)t s

h(u)du

ψ(s) ds α(0)e

)t 0

h(s) ds

(5.7)

0

for all t ∈ [0, T ]. Since h(u) 0 for all u ∈ [0, T ], (5.7) implies that

t α(t) +

t ψ(s) ds α(t) +

0

for all t ∈ [0, T ]. Therefore (5.2) holds.

e

)t s

h(u)du

ψ(s) ds α(0)e

)t 0

h(s) ds

0

2

References [1] Z. Brze´zniak, M. Capinski, F. Flandoli, Stochastic Navier–Stokes equations with multiplicative noise, Stoch. Anal. Appl. 10 (1992) 523–532. [2] Z. Brze´zniak, Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier–Stokes equations on some unbounded domains, Trans. Amer. Math. Soc. 358 (12) (2006) 5587–5629. [3] T. Caraballo, P.E. Kloeden, B. Schmalfuss, Exponentially stable stationary solution for stochastic evolution equations and their perturbation, Appl. Math. Optim. 50 (2004) 183–207. [4] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [5] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. [6] J. Duan, K. Lu, B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab. 31 (2003) 2109–2135. [7] J. Duan, K. Lu, B. Schmalfuss, Stable and unstable manifolds for stochastic partial differential equations, J. Dynam. Differential Equations 16 (4) (2004) 949–972. [8] F. Flandoli, Stochastic Flows for Nonlinear Second-Order Parabolic SPDE’s, Stoch. Monogr., vol. 9, Gordon and Breach Science Publishers, Yverdon, 1995. [9] F. Flandoli, Stochastic differential equations in fluid dynamics, Rend. Sem. Mat. Fis. Milano 66 (1996) 121–148. [10] M. Gourcy, A large deviation principle for 2D stochastic Navier–Stokes equation, Stochastic Process. Appl. 117 (7) (2007) 904–927.


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[11] M. Hairer, J.C. Mattingly, Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Ann. of Math. (2) 164 (3) (2006) 993–1032. [12] N. Masmoudi, L.-S. Young, Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs, Comm. Math. Phys. 227 (2002) 461–481. [13] S.-E.A. Mohammed, M.K.R. Scheutzow, The stable manifold theorem for nonlinear stochastic systems with memory, Part I: Existence of the semiflow, J. Funct. Anal. 205 (2003) 271–305, Part II: The local stable manifold theorem J. Funct. Anal. 206 (2004) 253–306. [14] S.-E.A. Mohammed, T.S. Zhang, H.Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Part 1: The stochastic semiflow, Part 2: Existence of stable and unstable manifolds, Mem. Amer. Math. Soc. 196 (2008) 105. [15] J.C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2001. [16] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. 115 (1982) 243–290. [17] S.S. Sritharan, P. Sundar, Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise, Stochastic Process. Appl. 116 (11) (2006) 1636–1659. [18] R. Temam, Navier–Stokes Equations, North-Holland Amsterdam, New York, Oxford, 1979. [19] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.


The role of BMOA in the boundedness of weighted composition operators ✩ Eva A. Gallardo-Gutiérrez a,∗ , Jonathan R. Partington b a Departamento de Matemáticas, Universidad de Zaragoza e IUMA, Plaza San Francisco s/n, 50009 Zaragoza, Spain b School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

Received 22 April 2009; accepted 27 February 2010

Communicated by K. Ball

Abstract Boundedness (resp. compactness) of weighted composition operators Wh,ϕ acting on the classical Hardy space H2 as Wh,ϕ f = h(f ◦ ϕ) are characterized in terms of a Nevanlinna counting function associated to the symbols h and ϕ whenever h ∈ BMOA (resp. h ∈ VMOA). Analogous results are given for Hp spaces and the scale of weighted Bergman spaces. In the latter case, BMOA is replaced by the Bloch space (resp. VMOA by the little Bloch space). © 2010 Elsevier Inc. All rights reserved. Keywords: Weighted composition operators; BMOA and VMOA; Nevanlinna counting functions

1. Introduction Let D denote the open unit disk of the complex plane, T its boundary and m the normalized arc-length measure on T. Let Hp , 1 p < ∞ be the classical Hardy space, that is, the space of holomorphic functions f on D for which the norm

✩

The authors are partially supported by Plan Nacional I+D grant No. MTM2007-61446 and Gobierno de Aragón research group Análisis Matemático y Aplicaciones, Ref. DGA E-64. * Corresponding author. E-mail addresses: [email protected] (E.A. Gallardo-Gutiérrez), [email protected] (J.R. Partington). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.019

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E.A. Gallardo-Gutiérrez, J.R. Partington / Journal of Functional Analysis 258 (2010) 3593–3603

f p =

sup 0r 1, from a reduction argument to this case. First, we slice C with the family of affine hyperplanes Ht = {x · e = t}, where t ∈ [h− , h+ ], which are parallel to H0 . In this way, by Fubini–Tonelli theorem, the Lebesgue measure L n of C can be recovered by integrating the (n − 1)-dimensional Hausdorff measures of the sections of C ∩ Ht over the segment [h− , h+ ] which parameterizes the parallel hyperplanes. Then, as the faces in C are transversal to H0 , one can see each point in C ∩ Ht as the image of a map σ t defined on C ∩ H0 which couples the points lying on the same face. Suppose that the (n − 1)-dimensional Hausdorff measure H n−1 (C ∩ Ht ) is absolutely continuous w.r.t. the pushforward measure σ#t (H n−1 (C ∩ H0 )) with Radon–Nikodym derivative α t . Then we can reduce each integral over the section C ∩ Ht to an integral over the section C ∩ H0 : dL n = H n−1 (C ∩ Ht ) dt = α t σ t (z) dH n−1 (z) dt.

[h− ,h+ ]

C

[h− ,h+ ] C ∩H0

Exchanging the order of the last iterated integrals, we obtain the following: n dL = α t σ t (z) dt dH n−1 (z). C −

+

C ∩H0 [h− ,h+ ]

Since the sets {σ [h ,h ] (z)}z∈C ∩H0 are exactly the elements of our partition, i.e. the 1dimensional faces of f , the last equality provides the explicit disintegration we are looking − + for: in particular, the conditional measure concentrated on σ [h ,h ] (z) is absolutely continuous − + w.r.t. H 1 σ [h ,h ] (z).

L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661

3607

The core of the proof is then to show that H n−1

(C ∩ Ht ) σ#t H n−1 (C ∩ H0) .

We prove this fact as a consequence of the following quantitative estimate: for all 0 t h+ and S ⊂ C ∩ H0 , H n−1 σ t (S)

t − h− −h−

n−1 H n−1 (S).

(1.1)

This fundamental estimate, as in [7,8], is proved approximating the 1-dimensional faces with a sequence of finitely many cones with vertex in C ∩ Hh− and basis in C ∩ Ht . At this step of the technique, the construction of such approximating sequence heavily depends on the nature of the partition one has to deal with. In this case, our main task is to find the suitable cones relying on the fact that we are approximating the faces of a convex function. One can also derive an estimate symmetric to the above one, showing that σ#t (H n−1 (C ∩ H0 )) is absolutely continuous w.r.t. H n−1 (C ∩ Ht ): as a consequence, α t is strictly positive and therefore the conditional measures are not only absolutely continuous w.r.t. the proper Hausdorff measure, but equivalent to it. The fundamental estimate (1.1) implies moreover a Lipschitz continuity and BV regularity of α t (z) w.r.t. t: this yields an improvement of the regularity of the partition that now we are going to describe. Consider a vector field v which at each point x is parallel to the face through that point x. If we restrict the vector field to an open Lipschitz set Ω which does not contain points in the relative boundaries of the faces, then we prove that its distributional divergence is the sum of two terms: an absolutely continuous measure, and an (n − 1)-rectifiable measure representing the flux of v through the boundary of Ω. The density (div v)a.c. of the absolutely continuous part is related to the density of the conditional measures defined by the disintegration above. In the case of the set C previously considered, if the vector field is such that v · e = 1, the expression of the density of the absolutely continuous part of the divergence satisfies

∂t α t = (div v)a.c. α t . Up to our knowledge, no piecewise BV regularity of the vector field v of the directions of the faces is known. Therefore, it is a remarkable fact that a divergence formula holds. The divergence of the whole vector field v is the limit, in the sense of distributions, of the sequence of measures which are the divergence of truncations of v on the elements {K }∈N of a suitable partition of Rn . However, in general, it fails to be a measure. In the last part, we change point of view: instead of looking at vector fields constrained to the faces of the convex function, we describe the faces as an (n + 1)-uple of currents, the k-th one corresponding to the family of k-dimensional faces, for k = 0, . . . , n. The regularity results obtained for the vector fields can be rewritten as regularity results for these currents. More precisely, we prove that they are locally flat chains. When truncated on a set Ω as above, they are locally normal, and we give an explicit formula for their border; the (n + 1)-uple of currents is the limit, in the flat norm, of the truncations on the elements of a partition.

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An application of this kind of further regularity is presented in Section 8 of [7]. Given a vector field v constrained to live on the faces of f , the divergence formula we obtain allows to reduce the transport equation div ρv = g to a PDE on the faces of the convex function. We do not pursue this issue in the paper. 1.1. Outline of the article In the following we describe the structure of the paper. In Section 2 we first give the definition of disintegration of a measure consistent with a fixed partition of the ambient space. Then, we report an abstract disintegration theorem which guarantees the existence and uniqueness of the disintegration under quite general assumptions on the σ -algebra of the ambient space and on the partition; under these hypotheses, the conditional probabilities given by the disintegration are concentrated on the sets of the partition. In Section 3, after giving the basic definitions and notations we will be working with, we apply the disintegration theorem recalled in Section 2 and get the existence and uniqueness of the disintegration of the Lebesgue measure on the faces of a convex function. For notational convenience, we work with the projections of the faces on Rn and we neglect the set where the convex function is not differentiable. In Section 3.2 we state our main theorem on the equivalence between the conditional probabilities and the k-Hausdorff measure on the k-dimensional faces where they are concentrated. As the conditional measures on the 0-dimensional and n-dimensional faces are already determined (they must be respectively given by Dirac deltas and by the H n -measure on the corresponding faces), we focus on the disintegration of the Lebesgue measure on the k-dimensional faces for k = 1, . . . , n − 1. In Section 4 we prove the explicit disintegration theorem. In Section 4.1 we explain the first idea of our disintegration technique, which consists in the reduction to countably many model sets like C and in the application to these sets of the Fubini– Tonelli technique, which has been briefly sketched in the introduction. In Section 4.2 we address the Borel measurability of the multivalued function D which assigns to each point x ∈ Rn the directions of the face passing through x. This is needed in the following in order to reduce the ambient space into countably many model sets. In Section 4.3 we define the partition of Rn into the model sets (called D-cylinders) on which we will first prove our disintegration theorem. When k = 1, the k-dimensional faces are partitioned into sets like C . When k > 1, each model set C k is defined taking a collection of k-dimensional faces which are transversal to a fixed (n − k)-dimensional affine plane (as, e.g., H = {x ∈ Rn : x · e1 = · · · = x · ek = 0}) and considering the points of these faces whose projection on the perpendicular k-plane (H ⊥ = {x ∈ Rn : x · ek+1 = · · · = x · en = 0}) is contained + in a fixed rectangle (as, e.g., {x ∈ Rn : x · ek+1 = · · · = x · en = 0, x · ei ∈ [h− i , hi ] for i = ± 1, . . . , k and hi ∈ R}). Section 4.4 is devoted to the proof of the quantitative estimate (1.1). Actually, in Lemma 4.7 we prove that an estimate like (1.1) holds for the pushforward of the H n−k -dimensional measure on the sections of a model set C k which are obtained cutting it with transversal (n − k)dimensional affine planes. The core of the proof, which is the construction of a suitable sequence of approximating cones for the 1-dimensional faces of f , is contained in Lemma 4.14.


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In Section 4.5 we study some regularity properties of the Radon–Nikodym derivative α t ; they will be used in Section 5 to study the regularity of the divergence of vector fields parallel to the faces. In Section 4.6 we prove the explicit disintegration theorem on the model sets C k of the partition. In Section 4.7 we collect the disintegrations obtained on the model set and we obtain the global result, namely the disintegration of the Lebesgue measure on all Rn . Section 5 deals with the divergence of the directions of the faces, with two equivalent approaches. In Section 5.1 we consider the divergence of any vector field which at each point x is parallel to the face of f through x. In Section 5.1.1 we truncate this vector field to k-dimensional D-cylinders. The divergence of these truncated vector fields turns out to be a Radon measure. The density of the absolutely continuous part of this distributional divergence involves the density α defined in (4.67). The precise statement is given in Corollary 5.4. In Section 5.1.2 we consider the vector field v in the whole Rn . Its divergence is the limit, in the sense of distributions, of finite sums of the Radon measures corresponding to the divergence of the truncations of v on a family of D-cylinders as above, which constitute a partition of Rn . In general, the divergence of v fails to be a measure. In Section 5.2 we consider the k-dimensional currents associated to k-faces, where each kface is thought as a k-covector field, for k = 0, . . . , n. We rephrase the results of Section 5.1 in this formalism. Section 5.2.1 is devoted to recalls on tensors and currents in order to fix the notation. In Section 5.2.2 we fix the attention on the current associated to a k-vector field that gives the directions of the k-faces on C k and vanishes elsewhere. The border of this current is the sum of two currents, both representable by integration. One is the integral on C k of the divergence of the k-vector field truncated to C k and it is again related to α. The other one is concentrated on an (n − 1)-dimensional set dC k (see definition (4.23)) and arises from the truncation of the faces to the D-cylinder. The statement is given in Lemma 5.9. In Section 5.2.3 we consider the (n + 1)-uple of currents associated to the faces of f , the k-th one acting on k-forms on Rn . By means of a partition of Rn into D-cylinders as above, we recover each of them as the limit, in the flat norm, of the normal currents defined as truncations of this (n + 1)-uple to the elements of the partition. The last section contains a long Table of Notations, for the reader’s convenience. 2. An abstract disintegration theorem A disintegration of a measure over a partition of the space on which it is defined is a way to write that measure as a “weighted sum” of probability measures which are possibly concentrated on the elements of the partition. Let (X, Σ, μ) be a measure space (which will be called the ambient space of the disintegration), i.e. Σ is a σ -algebra of subsets of X and μ is a measure with finite total variation on Σ, and let {Xα }α∈A ⊂ X be a partition of X. After defining the following equivalence relation on X x∼y

⇔

∃α ∈ A: x, y ∈ Xα ,

we make the identification A = X/∼ and we denote by p the quotient map p : x ∈ X → [x] ∈ A.

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Moreover, we endow the quotient space A with the measure space structure given by the largest σ -algebra that makes p measurable, i.e. A := F ⊂ A: p −1 (F ) ∈ Σ , and by the pushforward measure ν = p# μ, defined as ν(F ) := μ(p −1 (F )). Definition 2.1 (Disintegration). A disintegration of μ consistent with the partition {Xα }α∈A is a family {μα }α∈A of probability measures on X such that 1. ∀E ∈ Σ , α → μα (E) is ν-measurable; 2. μ = μα dν, i.e. μ E ∩ p −1 (F ) =

μα (E) dν(α),

∀E ∈ Σ, F ∈ A .

(2.1)

F

The disintegration is unique if the measures μα are uniquely determined for ν-a.e. α ∈ A. The disintegration is strongly consistent with p if μα (X\Xα ) = 0 for ν-a.e. α ∈ A. The measures μα are also called conditional probabilities of μ w.r.t. ν. Remark 2.2. When a disintegration exists, formula (2.1) can be extended by Beppo Levi’s theorem to measurable functions f : X → R as f dμ = f dμα dν(α). The existence and uniqueness of a disintegration can be obtained under very weak assumptions which concern only the ambient space. Nevertheless, in order to have the strong consistency we need structural assumptions also on the quotient measure algebra, otherwise in general μα (Xα ) = 1 (i.e. the disintegration is consistent but not strongly consistent). The more general result of existence of a disintegration which is consistent with a given partition is contained in [18], while a weak sufficient condition in order that a consistent disintegration is also strongly consistent is given in [12]. In the following we recall an abstract disintegration theorem, in the form presented in [6]. It guarantees, under suitable assumptions on the ambient and on the quotient measure spaces, the existence, uniqueness and strong consistency of a disintegration. Before stating it, we recall that a measure space (X, Σ) is countably-generated if Σ coincides with the σ -algebra generated by a sequence of measurable sets {Bn }n∈N ⊂ Σ . Theorem 2.3. Let (X, Σ) be a countably-generated measure space and let μ be a measure on X with finite total variation. Then, given a partition {Xα }α∈A of X, there exists a unique consistent disintegration {μα }α∈A . Moreover, if there exists an injective measurable map from (A, A ) to (R, B(R)), where B(R) is the Borel σ -algebra on R, the disintegration is strongly consistent with p. Remark 2.4. If the total variation of μ is not finite, a disintegration of μ consistent with a given partition as defined in (2.1) in general does not exists, even under the assumptions on the ambient


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and on the quotient space made in Theorem 2.3 (take for example X = Rn , Σ = B(Rn ), μ = L n and Xα = {x: x · e = α}, where e is a fixed vector in Rn and α ∈ R). Nevertheless, if μ is σ -finite and (X, Σ), (A, A ) satisfy the hypotheses of Theorem 2.3, as soon as we replace the possibly infinite-valued measure ν = p# μ with an equivalent σ -finite measure m on (A, A ), we can find a family of σ -finite measures {μ˜ α }α∈A on X such that μ = μ˜ α dm(α) (2.2) and μ˜ α (X\Xα ) = 0 for m-a.e. α ∈ A.

(2.3)

For example, we can take m = p# θ , where θ is a finite measure equivalent to μ. We recall that two measures μ1 and μ2 are equivalent if and only if μ1 μ2

and μ2 μ1 .

(2.4)

Moreover, if λ and {λ˜ α }α∈A satisfy (2.2) and (2.3) as well as m and {μ˜ α }α∈A , with m and λ equivalent to p# μ (and therefore to each other), then the following relation holds λ˜ α =

dm (α)μ˜ α , dλ

where dm dλ is the Radon–Nikodym derivative of m w.r.t. λ. In the following, whenever μ is a σ -finite measure with infinite total variation, by disintegration of μ strongly consistent with a given partition we will mean any family of σ -finite measures {μ˜ α }α∈A which satisfies the above properties; in fact, whenever μ has finite total variation we will keep the definition of disintegration given in (2.1). Finally, we recall that any disintegration of a σ -finite measure μ can be recovered by the disintegrations of the finite measures {μ Kn }n∈N , where {Kn }n∈N ⊂ X is a partition of X into sets of finite μ-measure.

3. Statement of the main theorem In this section, after setting the notation and some basic definitions, we apply Theorem 2.3 to get the existence, uniqueness and strong consistency of the disintegration of the Lebesgue measure on the faces of a convex function. Then, we give a rigorous formulation of the problem we are going to deal with and we state our main theorem. 3.1. Setting Let us consider the ambient space

Rn , B Rn , L n

K ,

where L n is the Lebesgue measure on Rn , B(Rn ) is the Borel σ -algebra, K is any set of finite Lebesgue measure and L n K is the restriction of the Lebesgue measure to the set K. Indeed,

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by Remark 2.4, the disintegration of the Lebesgue measure w.r.t. a given partition is determined by the disintegrations of the Lebesgue measure restricted to finite measure sets. Then, let f : Rn → R be a convex function. We recall that the subdifferential of f at a point x ∈ Rn is the set ∂ − f (x) of all r ∈ Rn such that f (w) − f (x) r · (w − x),

∀w ∈ Rn .

From the basic theory of convex functions, as f is real-valued and is defined on all Rn , ∂ − f (x) = ∅ for all x ∈ Rn and it consists of a single point if and only if f is differentiable at x. Moreover, in that case, ∂ − f (x) = {∇f (x)}, where ∇f (x) is the differential of f at the point x. We denote by dom ∇f a σ -compact set where f is differentiable and such that Rn \ dom ∇f is Lebesgue negligible. The σ -compactness is just a secondary technical requirement for simplifying measurability arguments. ∇f : dom ∇f → R denotes the differential map and Im ∇f the image of dom ∇f under the differential map. The partition of Rn on which we want to decompose the Lebesgue measure is given by the sets ∇f −1 (y) = x ∈ Rn : ∇f (x) = y ,

y ∈ Im ∇f,

along with the set Σ 1 (f ) = Rn \ dom ∇f . By the convexity of f , we can moreover assume w.l.o.g. that the intersection of ∇f −1 (y) with dom ∇f is convex. Since ∇f is a Borel map and Σ 1 (f ) is an L n -negligible Borel set (see e.g. [2,1]), we can assume that the quotient map p of Definition 2.1 is given by ∇f and that the quotient space is given by (Im ∇f, B(Im ∇f )), which is measurably included in (Rn , B(Rn )). Then, this partition satisfies the hypotheses of Theorem 2.3 and there exists a family {μy }y∈Im ∇f of probability measures on Rn such that L

n

K B ∩ ∇f

−1

(A) =

μy (B) d∇f# L n

K (y),

∀A, B ∈ B Rn .

A

In the following we give a formal definition of face of a convex function and we relate this object to the sets ∇f −1 (y) of our partition. Definition 3.1. A tangent hyperplane to the graph of a convex function f : Rn → R is a subset of Rn+1 of the form Hy = where x ∈ ∇f −1 (y).

z, hy (z) : z ∈ Rn , hy (z) = f (x) + y · (z − x) ,

(3.1)


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We note that, by convexity, the above definition is independent of x ∈ ∇f −1 (y). Definition 3.2. A face of a convex function f : Rn → R is a set of the form Hy ∩ graph f|dom ∇f .

(3.2)

It is easy to check that, ∀y ∈ Im ∇f and ∀z such that (z, f (z)) ∈ Hy ∩ graph f|dom ∇f , we have that y = ∇f (z). If we denote by πRn : Rn+1 → Rn the projection map on the first n coordinates, one can see that, for all y ∈ Im ∇f , ∇f −1 (y) = πRn (Hy ∩ graph f|dom ∇f ). For notational convenience, the set ∇f −1 (y) will be denoted as Fy and called projected face. We also write Fyk instead of Fy whenever we want to emphasize the fact that the latter has dimension k, for k = 0, . . . , n (where the dimension of a convex set C is the dimension of its affine hull aff(C)) and we denote the set of k-dimensional projected faces as

Fk =

(3.3)

Fy .

{y: dim(Fy )=k}

3.2. Absolute continuity of the conditional probabilities Since the measure we are disintegrating (L n ) has the same Hausdorff dimension as the space on which it is concentrated (Rn ) and since the sets of the partition on which the conditional probabilities are concentrated have a well defined linear dimension, we address the problem of whether this absolute continuity property of the initial measure is still satisfied by the conditional probabilities produced by the disintegration: we want to see if dim(Fy ) = k

⇒

μy H k

Fy .

(3.4)

The answer to this question is not trivial. Indeed, when n 3 one can construct sets of full Lebesgue measure in Rn and Borel partitions of those sets into convex sets such that the conditional probabilities of the corresponding disintegration do not satisfy property (3.4) for k = 1 (see e.g. [4]). However, we show that the absolute continuity property is preserved by the disintegration w.r.t. the partition given by the faces of a convex function. Our main result is the following: Theorem 3.3. Let {μy }y∈Im ∇f be the family of probability measures on Rn such that L

n

K B ∩ ∇f

−1

(A) =

μy (B) d∇f# L n

K (y),

∀A, B ∈ B Rn .

(3.5)

A

Then, for ∇f# (L n K)-a.e. y ∈ Im ∇f , the conditional probability μy is equivalent to the k-dimensional Hausdorff measure H k restricted to Fyk ∩ K, i.e.

3614


μy H k

k Fy ∩ K

and H k

k Fy ∩ K μ y .

(3.6)

Remark 3.4. The result for k = 0, n is trivial. Indeed, for all y such that Fy ∩ K = ∅ and dim(Fy ) = 0 we must put μy = δ{Fy } , where δx0 is the Dirac mass supported in x0 , whereas L nF if dim(Fy ∩ K) = n we have that μy = |L nFyy | . Remark 3.5. Since the map id × f : Rn → Rn+1 , x → x, f (x) is locally Lipschitz and preserves the Hausdorff dimension of sets, Theorem 3.3 holds also for the disintegration of the n-dimensional Lebesgue measure over the partition of the graph of f given by the faces defined in (3.2). We have chosen to deal with the disintegration of the Lebesgue measure over the projections of the faces on Rn only for notational convenience. Theorem 3.3 will be proved in Section 4.7, where we provide also an explicit expression for the conditional probabilities. If we knew some Lipschitz regularity for the field of directions of the faces of a convex function, we could try to apply the Area or Coarea formula in order to obtain within a single step the disintegration of the Lebesgue measure and the absolute continuity property (3.6). However, such regularity is presently not known and for this reason we have to follow a different approach. 4. The explicit disintegration 4.1. A disintegration technique In this paragraph we give an outline of the technique we use in order to prove Theorem 3.3. This kind of strategy was first used in order to disintegrate the Lebesgue measure on a collection of disjoint segments in [7], and then in [8]. For simplicity, we focus on the disintegration of the Lebesgue measure on the 1-dimensional faces and, in the end, we give an idea of how we will extend this technique in order to prove the absolute continuity of the conditional probabilities on the faces of higher dimension. The disintegration on model sets: Fubini–Tonelli theorem and absolute continuity estimates on affine planes which are transversal to the faces. First of all, let us suppose that the projected 1-dimensional faces of f are given by a collection of disjoint segments C whose projection on a fixed direction e ∈ Sn−1 is equal to a segment [h− e, h+ e] with h− < 0 < h+ , more precisely C=

a(z), b(z) ,

(4.1)

z∈Zt

where Zt is a compact subset of an affine hyperplane of the form {x · e = t} for some t ∈ R and a(z) · e = h− , b(z) · e = h+ . Any set of the form (4.1) will be called a model set (see also Fig. 1).


3615

Fig. 1. A model set of 1-dimensional projected faces. Given a subset Z = Z0 of the hyperplane {x · e = 0}, the above model set is made of the 1-dimensional faces of f passing through some z ∈ Z0 , truncated between {x · e = h− }, {x · e = h+ } and projected on Rn .

Then, we want to find the conditional probabilities of the disintegration of the Lebesgue measure on the segments which are contained in the model set C and see if they are absolutely continuous w.r.t. the H 1 -measure. The idea of the proof is to obtain the required disintegration by a Fubini–Tonelli argument, that reverts the problem of absolute continuity w.r.t. H 1 of the conditional probabilities on the projected 1-dimensional faces to the absolute continuity w.r.t. H n−1 of the pushforward by the flow induced by the directions of the faces of the H n−1 -measure on transversal hyperplanes. First of all, we cut the set C with the affine hyperplanes which are perpendicular to the segment [h− e, h+ e], we apply Fubini–Tonelli theorem and we get h+

ϕ(x) dL (x) = n

C

h−

ϕ dH n−1 dt

∀ϕ ∈ Cc0 Rn .

(4.2)

{x·e=t}∩C

Then we observe the following: for every s, t ∈ [h− , h+ ], the points of {x · e = t} ∩ C are in bijective correspondence with the points of the section {x · e = s} ∩ C and a bijection is obtained by pairing the points that belong to the same segment [a(z), b(z)], for some z ∈ Zt . For example, a map which sends the transversal section Z = {x · e = 0} ∩ C into the section Zt = {x · e = t} ∩ C (for any t ∈ [h− , h+ ]) is given by σ t : Z → σ t (Z) = {x · e = t} ∩ C , z → z + t

ve (z) = {x · e = t} ∩ a(z), b(z) , |ve (z) · e|

b(z)−a(z) where [a(z), b(z)] is the segment of C passing through the point z and ve (z) = |b(z)−a(z)| . Therefore, as soon as we fix a transversal section of C , say e.g. Z = {x · e = 0} ∩ C , we can try to rewrite the inner integral in the r.h.s. of (4.2) as an integral of the function ϕ ◦ σ t w.r.t. the H n−1 -measure on the fixed section Z. This can be done if

t −1 n−1 σ # H

σ t (Z)

H n−1

Z.

(4.3)

3616


Indeed,

ϕ(y) dH n−1 (y) = σ t (Z)

−1 ϕ σ t (z) d σ t # H n−1

σ t (Z) (z)

(4.4)

Z

and if (4.3) is satisfied for all t ∈ [h− , h+ ], then h+ (4.2) =

ϕ σ t (z) α(t, z) dH n−1 (z) dt,

h− Z

n−1 where α(t, z) is the Radon–Nikodym derivative of (σ t )−1 σ t (Z)) w.r.t. H n−1 Z. # (H Having turned the r.h.s. of (4.2) into an iterated integral over a product space isomorphic to Z + [h− e, h+ e], the final step consists in applying Fubini–Tonelli theorem again so as to exchange the order of the integrals and get

h+ ϕ(x) dL n (x) = ϕ σ t (z) α(t, z) dt dH n−1 (z).

C

Z

(4.5)

h−

This final step can be done if α is Borel-measurable and locally integrable in (t, z). By the uniqueness of the disintegration stated in Theorem 2.3 we have that dμz (t) =

α(t, z) dH 1 [a(z), b(z)](t) h+ h− α(s, z) ds

for H n−1 -a.e. z ∈ Z.

(4.6)

The same reasoning can be applied to the case k > 1. Indeed, let us consider a collection C k of faces whose projection on a certain k-plane e1 , . . . , ek is given by a rectangle kk-dimensional − + + [h e , h e ], with h− i i i=1 i i i < 0 < hi for all i = 1, . . . , k (see Fig. 2, p. 3617). Then, as soon as we fix an affine (n − k)-dimensional plane which is perpendicular to the k-plane e1 , . . . , ek , as for example H k = ki=1 {x · ei = 0}, and we denote by πe1 ,...,ek : Rn → e1 , . . . , ek the projection map on the k-plane e1 , . . . , ek , the k-dimensional faces in C k can be parameterized with the map σ te (z) = z + t

ve (z) , |πe1 ,...,ek (ve (z))|

(4.7)

where z ∈ Z k = H k ∩ C k , e is a unit vector in the k-plane e1 , . . . , ek , t ∈ R satisfies te · ei ∈ + [h− i , hi ] for all i = 1, . . . , k and ve (z) is the unit direction contained in the face passing through πe1 ,...,ek (ve (z)) |πe1 ,...,ek (ve (z))| = e. C k with affine hyperplanes which are perpendicular to

z which is such that

If we cut the set ei for i = 1, . . . , k and apply k-times the Fubini–Tonelli theorem, the main point is again to show that, for every e and t as above, te −1 n−k σ # H

σ te Zk

H n−k

Zk

(4.8)


3617

Fig. 2. Sheaf sets and D-cylinders (Definitions 4.3, 4.5). Roughly, a sheaf set Z k is a collection of relative interiors of kplane. faces of f , projected on Rn , which intersect exactly at one point some set Z k contained in an (n − k)-dimensional k −1 − + k k A D-cylinder C k is the intersection of a sheaf set with πe ,...,e (C ), for some rectangle C = i=1 [ti ei , ti ei ], 1

k

where {e1 , . . . , en } constitutes an orthonormal basis of Rn . Such sets Z k are called sections, while the k-plane e1 , . . . , ek is the axis of C k .

and, after this, that the Radon–Nikodym derivative between the above measures satisfies proper measurability and integrability conditions. Then, to prove Theorem 3.3 on model sets that are, up to translations and rotations, like the set C k , it is sufficient to prove (4.8) and some weak properties of the related density function, such as Borel-measurability and local integrability. Actually, the properties of this function will follow immediately from our proof of (4.8), which is given in a stronger form in Lemma 4.31. Partition of Rn into model sets and the global disintegration theorem. In the next section we show that the set F k defined in (3.3), for k = 1, . . . , n − 1, can be partitioned, up to a negligible set, into a countable collection of Borel-measurable model sets like C k . After proving the disintegration theorem on the model sets we will see how to glue the “local” results in order to obtain a global disintegration theorem for the Lebesgue measure over the whole faces of the convex function (restricted to a set of finite L n -measure). 4.2. Measurability of the directions of the k-dimensional faces The aim of this subsection is to show that the set of the projected k-dimensional faces of a convex function f can be parameterized by an L n -measurable (and multivalued) map. This will allow us to decompose Rn into a countable family of Borel model sets on which to prove Theorem 3.3. First of all we give the following definition, which generalizes Definition 3.1. Definition 4.1. A supporting hyperplane to the graph of a convex function f : Rn → R is an affine hyperplane in Rn+1 of the form H = w ∈ Rn+1 : w · b = β ,

3618


where b = 0, w · b β for all w ∈ epi f = {(x, t) ∈ Rn × R: t f (x)} and w · b = β for at least one w ∈ epi f . As f is defined and real-valued on all Rn , every supporting hyperplane is of the form Hy =

z, hy (z) : z ∈ Rn , hy (z) = f (x) + y · (z − x) ,

(4.9)

for some y ∈ ∂ − f (x). Whenever y ∈ Im ∇f , Hy is a tangent hyperplane to the graph of f according to Definition 3.1. Then we define the map x → P(x) = z ∈ Rn : ∃y ∈ ∂ − f (x) such that f (z) − f (x) = y · (z − x) . By definition, P(x) = Moreover, the map

y∈∂ − f (x) πRn (Hy

(4.10)

∩ graph f ).

dom ∇f x → R(x) := P(x) ∩ dom ∇f, gives precisely the set Fy of our partition that passes through the point x. As the disintegration over the 0-dimensional faces is trivial, we will restrict our attention to the set T = x ∈ dom ∇f : R(x) = {x} . For all such points there is at least one segment [w, z] ⊂ R(x) such that w = z. We can also define the multivalued map giving the unit directions contained in the faces passing through the set T, that is

T x → D(x) =

z−x : z ∈ R(x), z = x . |z − x|

(4.11)

We recall that a multivalued map is defined to be Borel measurable if the inverse image of any open set is Borel. The measurability of the above maps is proved in the following lemma: Lemma 4.2. The graph of the multivalued function P is a closed set in Rn × Rn . As a consequence, P, R and D are Borel measurable multivalued maps and T is a Borel set. Proof. The closedness of the graph of P follows immediately from the continuity of f and from the upper-semicontinuity of its subdifferential. Then, the graph of P is σ -compact in Rn × Rn and, due to the continuity of the projections from Rn × Rn to Rn , it follows that the map P is Borel. Moreover, since we chose dom ∇f to be σ -compact, also the graph of R is σ -compact, thus R is a Borel map. The same reasoning that is made for the map P can be applied to the multifunction R\id (where id denotes the identity map id(x) := x), thus giving the measurability of the set T, since


3619

T = π graph(R\id) ∩ dom ∇f, where π : Rn × Rn → Rn denotes the projection on the first n coordinates. The measurability of D follows by the continuity of the map Rn × Rn (x, z) → the diagonal. 2

z−x |z−x|

out of

4.3. Partition into model sets First of all, we introduce some preliminary notation. If C ⊂ Rd is a convex set and aff(C) is its affine hull, we denote by ri(C) the relative interior of C, which is the interior of C in the topology of aff(C), and by rb(C) its relative boundary, which is the boundary of C in aff(C). In order to find a countable partition of F k into model sets like the set C k which was defined in Section 4.1, we have to neglect the points that lie on the relative boundary of the k-dimensional faces. More precisely, from now onwards we look for the disintegration of the Lebesgue measure over the sets Ey = ri(Fy ),

y ∈ Im ∇f.

(4.12)

As we did for the sets Fy , we set Eyk = Ey

if dim(Ey ) = k

and

Ek =

Eyk .

(4.13)

{y∈Im ∇f : dim(Ey )=k}

This restriction will not affect the characterization of the conditional probabilities because, as we will prove in Lemma 4.19, the set

T\

n

Ek

k=1

is Lebesgue negligible. Now we can start to build the partition of E k into model sets. Definition 4.3. For all k = 1, . . . , n, we call sheaf set a σ -compact subset of E k of the form Zk=

z∈Z k

ri R(z) ,

(4.14)

3620


where Z k is a σ -compact subset of E k which is contained in an affine (n − k)-plane in Rn and is such that ri R(z) ∩ Z k = {z},

∀z ∈ Z k .

(4.15)

We call sections of Z k all the sets Y k that satisfy the same properties of Z k in the definition. A subsheaf of a sheaf set Z k is a sheaf set W k of the form Wk=

ri R(w) ,

w∈W k

where W k is a σ -compact subset of a section of the sheaf set Z k . Similarly to Lemma 2.6 in [8], we prove that the set E k can be covered with countably many disjoint sets of the form (4.14). First of all, let us take a dense sequence {Vi }i∈N ⊂ G(k, n), where G(k, n) is the compact set of all the k-planes in Rn passing through the origin, and fix, ∀i ∈ N, an orthonormal set {ei1 , . . . , eik } in Rn such that Vi = ei1 , . . . , eik .

(4.16)

Denote by Sn−1 ∩ V the k-dimensional unit sphere of a k-plane V ⊂ Rn w.r.t. the Euclidean norm and by πi = πVi : Rn → Vi the projection map on the k-plane Vi . For every fixed 0 < ε < 1 the following sets form a disjoint covering of the k-dimensional unit spheres in Rn : Sk−1 = Sn−1 ∩ V : V ∈ G(k, n), i

inf

x∈Sn−1 ∩V

i−1

πi (x) 1 − ε \ Sk−1 , j

j =1

i = 1, . . . , I,

(4.17)

where I ∈ N depends on the ε we have chosen. In order to determine a countable partition of E k into sheaf sets we consider the k-dimensional rectangles in the k-planes (4.16) whose boundary points have dyadic coordinates. For all l = (l1 , . . . , lk ),

m = (m1 , . . . , mk ) ∈ Zk

with lj < mj ∀j = 1, . . . , k

(4.18)

k and for all distinct i1 , . . . , ik ⊂ {1, . . . , I }, p ∈ N, let Ciplm be the rectangle

k = 2−p Ciplm

k

[lj eij , mj eij ].

(4.19)

j =1

Lemma 4.4. The following sets are sheaf sets covering E k : for i = 1, . . . , I, p ∈ N, and S ⊂ Zk take


3621

k ZipS = x ∈ E k : D(x) ⊂ Sk−1 and S ⊂ Zk is the maximal set such that i

k Cipl(l+1)

⊂ πi ri R(x) .

(4.20)

l∈S

Moreover, a disjoint family of sheaf sets that cover E k is obtained in the following way: in case k as above, whereas for all p > 1 we take a set Z k if and only p = 1 we consider all the sets ZipS ipS k k does not contain any rectangle of the form Cip if the set l∈S Cipl(l+1) l(l+1) for every p < p. As soon as a nonempty sheaf set Z k belongs to this partition, it will be denoted by Z¯ k . ipS

ipS

For the proof of this lemma we refer to the analogous Lemma 2.6 in [8]. Then, we can refine the partition into sheaf sets by cutting them with sections which are perpendicular to fixed k-planes. Definition 4.5. (See Fig. 2.) A k-dimensional D-cylinder is a σ -compact set of the form k −1 C , C k = Z k ∩ πe 1 ,...,ek

(4.21)

where Z k is a k-dimensional sheaf set, e1 , . . . , ek is any fixed k-dimensional subspace which is perpendicular to a section of Z k and C k is a rectangle in e1 , . . . , ek of the form

Ck =

k − ti ei , ti+ ei , i=1

with −∞ < ti− < ti+ < +∞ for all i = 1, . . . , k, such that C k ⊂ πe1 ,...,ek ri R(z)

k −1 ∀z ∈ Z k ∩ πe C . 1 ,...,ek

(4.22)

We set C k = C k (Z k , C k ) when we want to refer explicitly to a sheaf set Z k and to a rectangle C k that can be taken in the definition of C k . The k-plane e1 , . . . , ek is called the axis of the D-cylinder and every set Z k of the form −1 (w), C k ∩ πe 1 ,...,ek

for some w ∈ ri C k

is called a section of the D-cylinder. We also define the border of C k transversal to D and its outer unit normal as k −1 rb C , dC k = C k ∩ πe ,...,e k 1

k −1 nˆ |dC k (x) = outer unit normal to πe C at x, for H n−1 -a.e. x ∈ dC k . 1 ,...,ek

(4.23)

3622


Lemma 4.6. The set E k can be covered by the D-cylinders k k C k ZipS , , Cipl(l+1)

(4.24)

k , Ck where S ⊂ Zk , l ∈ S and ZipS ipl(l+1) are the sets defined in (4.19), (4.20). Moreover, there exists a countable covering of E k with D-cylinders of the form (4.24) such that

k k k k k πi C k ZipS ∩ C k Zipk S , Cip ⊂ rb Cipl(l+1) ∩ rb Cip , Cipl(l+1) l (l +1) l (l +1)

(4.25)

for any couple of D-cylinders which belong to this countable family (if i = i , it follows from the k , Ck k k k definition of sheaf set that C k (ZipS ipl(l+1) ) ∩ C (Zi p S , Ci p l (l +1) ) must be empty). Proof. If x ∈ E k , then x ∈ Eyk for y = ∇f (x). By definition Eyk is the relative interior of a convex k-dimensional set. Moreover, by construction, its projection on some k-plane Vi is an open, k . In particuconvex set and therefore it can be covered by rectangles of dyadic coordinates Ciplm k k k k lar, the projection of x on Vi belongs to some Ciplm and this means that x ∈ C (ZipS , Cip(+1) ) k for some . This proves that E can be covered by the D-cylinders defined in (4.24). Our aim is then to construct a countable covering of E k with D-cylinders which satisfy property (4.25). k which belongs to the countable partition of E k First of all, let us fix a nonempty sheaf set Z¯ipS given in Lemma 4.4. In the following we will determine the D-cylinders of the countable covering which are conk ; the others can be selected in the same way starting from a different sheaf set of tained in Z¯ipS the partition given in Lemma 4.4. Then, the D-cylinders that we are going to choose are of the form C k Z k ˆ , Ck i pˆ S

i pˆ ˆl(ˆl+1)

,

k . where Z k ˆ is a subsheaf of the sheaf set Z¯ipS i pˆ S

The construction is done by induction on the natural number pˆ which determines the diamk on the axis obtained projecting the D-cylinders contained in Z¯ipS eter of the squares C k ˆ ˆ i pˆ l(l+1) ei1 , . . . , eik . Then, as the induction step increases, the diameter of the k-dimensional rectangles associated to the D-cylinders that we are going to add to our countable partition will be smaller and smaller (see Fig. 3). k is a nonempty element of the partition defined By definition (4.20) and by the fact that Z¯ipS in Lemma 4.4, the smallest natural number pˆ such that there exists a k-dimensional rectangle of k ) is exactly p; then, w.l.o.g., we can assume in which is contained in πi (Z¯ipS the form C k ˆ ˆ i pˆ l(l+1) our induction argument that p = 1. For all pˆ ∈ N, we call Cylpˆ the collection of the D-cylinders which have been chosen up to step p. ˆ When pˆ = 1 we set k k Cyl1 = C k Z¯i1S : l∈S . , Ci1l(l+1) Now, let us suppose to have determined the collection of D-cylinders Cylpˆ for some pˆ ∈ N.


3623

Fig. 3. Partition of E k into D-cylinders (Lemma 4.6).

Then, we define k k : Zi(kp+1) Cylp+1 = Cylpˆ ∪ C k = C k Zi(kp+1) , C is a subsheaf of Z¯ipS and ˆ ˜ ˜ ˜ ˆ S ˆ S˜ i(p+1) ˆ l(l+1) k k k C k C k Zipk S , Cip Zip S , Cik p l (l +1) ∈ Cylpˆ . l (l +1) for all C

2 (4.26)

As we did in (4.7), any k-dimensional D-cylinder C k = C k (Z k , C k ) can be parameterized in the following way: if we fix w ∈ ri(C k ), then −1 C k = σ w+te (z): z ∈ Z k = πe (w) ∩ C k , e ∈ Sn−1 ∩ e1 , . . . , ek 1 ,...,ek and t ∈ R is such that (w + te) · ej ∈ tj− , tj+ ∀j = 1, . . . , k ,

(4.27)

where σ w+te (z) = z + t

ve (z) , |πe1 ,...,ek (ve (z))|

πe

(4.28)

,...,e (ve (z))

and ve (z) ∈ D(z) is the unit vector such that |πe1 ,...,ek (ve (z))| = e. k 1 We observe that, according to our notation, w+te −1 σ = σ (w+te)−te .

(4.29)

4.4. An absolute continuity estimate According to the strategy outlined in Section 4.1, in order to prove Theorem 3.3 for the disintegration of the Lebesgue measure on the D-cylinders we have to show that, for every D-cylinder

3624


C k parameterized as in (4.27) w+te −1 n−k H σ #

σ w+te Zk

H n−k

Zk .

(4.30)

This will allow us to make a change of variables from the measure space (σ w+te (Z k ), H n−k (σ w+te (Z k ))) to (Z k , αH n−k Z k ), where α is an integrable function w.r.t. H n−k Z k (see Section 4.1). It is clear that the domain of the parameter t, which can be interpreted as a time parameter for a flow σ w+te that moves points along the k-dimensional projected faces of a convex function, depends on the section Z k which has been chosen for the parameterization of C k and on the direction e. Then, if e1 , . . . , ek is the axis of a D-cylinder C k , for every w ∈ ri(C k ) and for every e ∈ n−1 S ∩ e1 , . . . , ek , we define the numbers

h− (w, e) = inf t ∈ R: w + te ∈ C k ,

h+ (w, e) = sup t ∈ R: w + te ∈ C k .

We observe that, since w ∈ ri(C k ), h− (w, e) < 0 < h+ (w, e). We obtain (4.30) in Corollary 4.15 as a consequence of the following fundamental lemma. Lemma 4.7 (Absolutely continuous pushforward). Let C k be a k-dimensional D-cylinder parameterized as in (4.27). Then, for all S ⊂ Z k the following estimate holds:

h+ (w, e) − t h+ (w, e) − s

n−k

H n−k σ w+se (S) H n−k σ w+te (S)

t − h− (w, e) s − h− (w, e)

n−k

H n−k σ w+se (S) ,

(4.31)

where h− (w, e) < s t < h+ (w, e). Moreover, if s = h− (w, e) the left inequality in (4.31) still holds and if t = h+ (w, e) the right one. Lemma 4.7 will be proven at p. 3634. The idea to prove this lemma, as in [7] and [8], is to get the estimate (4.31) for the flow σjw+te induced by simpler vector fields {vj }j ∈N and then to show that they approximate the initial vector field ve in such a way that the inequalities in (4.31) pass to the limit. The main problem in our proof is then to find a suitable sequence of vector fields {vj }j ∈N that approximate, in a certain region, the geometry of the projected k-dimensional faces of a convex function in the direction e, which is described by the vector field ve . For the construction of this family of vector fields we strongly rely on the fact that the sets on which we want to disintegrate the Lebesgue measure are, other than disjoint, the projections of the k-dimensional faces of a convex function. For simplicity, we first prove the estimate (4.31) for 1-dimensional D-cylinders. In this case, if e is the axis of a 1-dimensional D-cylinder C , there are only two possible directions ±e that can be chosen to parameterize it. Up to translations by a multiple of the same vector, we can assume that w = 0 and s = 0. Moreover, since the choice of −e instead of e in


3625

the definition of the parameterization map (4.28) simply reverses the order of s and t in (4.31), in order to prove (4.31) it is sufficient to show that, for all 0 t h+ and for all S ⊂ σ t (Z), H n−1 (S)

t − h− −h−

n−1

−1 H n−1 σ t (S) ,

(4.32)

where σ t = σ 0+te and h± = h± (0, e). In our construction we first approximate the 1-dimensional faces that lie on the graph of f restricted to the given D-cylinder and then we get the approximating vector fields {vj }j ∈N simply projecting the directions of those approximations on the first n coordinates. Before giving the details we recall and introduce some useful notation: Sn−1 = x ∈ Rn : x = 1 ; e ∈ Sn−1 a fixed vector; Hs := x ∈ Rn : x · e = s , where s ∈ h− , h+ and h− , h+ ∈ R: h− < 0 < h+ ; Bn−1 R (x) = z ∈ H{x·e} : z − x R ; Z is the σ -compact section of the 1-dimensional D-cylinder C which is contained in H0 ; ve (x) = D(x), πe ve (x) = ve (x) · e e; C = σ t (z): z ∈ Z, t ∈ h− , h+ , Ct =

σ t (z) = z + t

ve (z) ; |πe (ve (z))|

Hs ∩ C ;

s∈[h− ,t]

lt (x) = R(x) ∩ Ct , ∀x ∈ Ct ; ∀x ∈ Rn , x˜ := x, f (x) ∈ Rn+1

and ∀A ⊂ Rn ,

A˜ := graph f|A .

Moreover, we recall the following definitions: Definition 4.8. The convex envelope of a set of points X ⊂ Rn is the smallest convex set conv(X) that contains X. The following characterization holds: conv(X) =

J j =1

λj xj : xj ∈ X, 0 λj 1,

J

λj = 1, J ∈ N .

(4.33)

j =1

Definition 4.9. The graph of a compact convex set C ⊂ Rn+1 , that we denote by graph(C), is the graph of the function g : πRn (C) → R which is defined by g(x) = min t ∈ R: (x, t) ∈ C .

(4.34)

Definition 4.10. A supporting k-plane to the graph of a convex function f : Rn → R is an affine k-dimensional subspace of a supporting hyperplane to the graph of f (see Definition 4.1) whose intersection with graph f is nonempty.

3626


Definition 4.11. An R-face of a convex set C ⊂ Rd is a convex subset C of C such that every closed segment in C with a relative interior point in C has both endpoints in C . The 0-dimensional R-faces of a convex set are also called extreme points and the set of all extreme points in a convex set C will be denoted by ext(C). The definition of R-face corresponds to the definition of face of a convex set given in [20]. We also recall the following propositions, for which we refer to Section 18 of [20]. Proposition 4.12. Let C = conv(D), where D is a set of points in Rd , and let C be a nonempty R-face of C. Then C = conv(D ), where D consists of the points in D which belong to C . Proposition 4.13. Let C be a bounded closed convex set. Then C = conv(ext(C)). The key to get the fundamental estimate (4.32) is contained in the following lemma: Lemma 4.14 (Construction of regular approximating vector fields). For all 0 t h+ , there exists a sequence of H n−1 -measurable vector fields t vj j ∈N ,

vjt : σ t (Z) → Sn−1

such that 1. 2.

vjt converges H n−1 -a.e. to ve on σ t (Z); t − h− n−1 n−1 t −1 n−1 σv t (S) H (S) , H j −h−

(4.35) ∀S ⊂ σ t (Z),

where σvt t is the flow map associated to the vector field vjt .

(4.36)

j

Indeed, if we have such a sequence of vector fields, the proof of the estimate (4.32) follows as in [8]. Proof. Step 1. Preliminary considerations. First of all, let us fix t ∈ [0, h+ ]. Partitioning, if necessary, C into a countable collection of sets, we can assume that σ t (Z) and − n−1 h h− σ (Z) are bounded, with σ t (Z) ⊂ Bn−1 R1 (x1 ) ⊂ Ht and σ (Z) ⊂ BR2 (x2 ) ⊂ Hh− . Then, if n−1 we call Kt the convex envelope of Bn−1 R1 (x1 ) ∪ BR2 (x2 ), the function f|Kt is uniformly Lipschitz with a certain Lipschitz constant Lf . Step 2. Construction of approximating functions (see Fig. 4). Now we define a sequence of functions {fj }j ∈N whose 1-dimensional faces approximate, in a certain sense, the pieces of the 1-dimensional faces of f which are contained in Ct . The directions of a properly chosen subcollection of the 1-dimensional faces of fj will give, when projected on the first n coordinates, the approximate vector field vjt . h− ˜ First of all, take a sequence {y˜i }i∈N ⊂ σ˜ (Z) such that the collection of segments {lt (yi )}i∈N ˜ is dense in y∈σ h− (Z) lt (y).


3627

Fig. 4. Illustration of a vector field approximating the 1-dimensional faces of f (Lemma 4.14). One can see in the picture the graph of f4 , which is the convex envelope of {y˜i }i=1,...,4 and f |H . The faces of fj connect H n−1 -a.e. point of t Ht to a single point among the {y˜i }i=1,...,j , while the remaining points of Ht correspond to some convex envelope conv({y˜i } )—here represented by the segments [y˜i , y˜i+1 ]. The region where the map Dt4 , giving the directions of the faces of f4 , is multivalued is basically the projection of the k-faces of f4 with k > 1. If x belongs to this region, i.e. Dt4 (x) consists of the directions of a ‘planar’ face of f4 , intersecting the affine hull of this ‘planar’ face with Ht × R one finds a supporting hyperplane for f|H passing through the point (x, f (x)). These supporting hyperplanes are represented by t

the tangent lines in the right side of the picture. The intersection of σ t (Z) ⊂ Ht with any supporting plane to the graph of f |H must contain just one point, otherwise D would be multivalued at some point of σ t (Z). t

For all j ∈ N, let Cj be the convex envelope of the set j

{y˜i }i=1 ∪ graph f|

(4.37)

Bn−1 R1 (x1 )

and call fj : πRn (Cj ) → R the function whose graph is the graph of the convex set Cj . j j = graph(conv({y˜i }i=1 )). We note that πRn (Cj )∩Hh− = conv({yi }i=1 ) and graph fj | j conv({yi }i=1 )

We claim that the graph of fj is made of segments that connect the points of j (indeed, by convexity and by the fact that graph(conv({y˜i }i=1 )) to the graph of f| n−1 BR

1

(x1 )

y˜i = (yi , f (yi )), fj = f on Bn−1 R1 (x1 )). In order to prove this, we first observe that, by definition, all segments of this kind are contained in the set Cj . On the other hand, by (4.33), all the points in Cj are of the form w=

J i=1

λi wi ,

(4.38)

3628


where

J

i=1 λi

j

= 1, 0 λi 1 and wi ∈ {y˜i }i=1 ∪ graph f|

Bn−1 R1 (x1 )

w = αz + (1 − α)r,

. In particular, we can write

j where 0 α 1, z ∈ conv {y˜i }i=1 and r ∈ epi f|

Bn−1 R1 (x1 )

Moreover, if we take two points z ∈ graph(conv({y˜i }i=1 )), r ∈ graph f| j

πRn (z ) = πRn (z)

and

πRn (r ) = πRn (r),

(4.39)

.

Bn−1 R1 (x1 )

such that

we have that the point

w = αz + (1 − α)r

(4.40) j

belongs to Cj , lies on a segment which connects graph(conv({y˜i }i=1 )) to graph f|

Bn−1 R1 (x1 )

and its

(n + 1)-coordinate is less than the (n + 1)-coordinate of w. j The graph of fj contains also all the pieces of 1-dimensional faces {l˜t (yi )}i=1 , since by construction it contains their endpoints and it lies over the graph of f|π n (Cj ) . R

Step 3. Construction of approximating vector fields (see Fig. 4). j Among all the segments in the graph of fj that connect the points of graph(conv({y˜i }i=1 )) j to the graph of f| n−1 we select those of the form [x, ˜ y˜k ], where x ∈ σ t (Z), yk ∈ {yi }i=1 , and BR

1

(x1 )

we show that for H n−1 -a.e. x ∈ σ t (Z) there exists only one segment within this class which passes through x. ˜ The approximating vector field will be given by the projection on the first n coordinates of the directions of these segments. First of all, we claim that for all x ∈ Bn−1 R1 (x1 ) the graph of fj contains at least a segment of the form [x, ˜ y˜i ] for some i ∈ {1, . . . , j }. Indeed, we show that if x˜ is the endpoint of a segment of the form [x, ˜ (y, fj (y))] where y j j / ext(conv({yi }i=1 )), then there are at least two segments belongs to conv({yi }i=1 ) but (y, fj (y)) ∈ j j of the form [x, ˜ y˜k ] with y˜k ∈ ext(conv({y˜i }i=1 )) ⊂ {y˜i }i=1 (here we assume that j 2). ˜ (y, fj (y))) and a supIn order to prove this, take a point (z, fj (z)) in the open segment (x, porting hyperplane H (z) to the graph of fj that contains that point. By definition, H (z) contains the whole segment [x, ˜ (y, fj (y))] and the set H (z) ∩ (Hh− × R) is a supporting hyperplane to j the set graph(conv({y˜i }i=1 )) that contains the point (y, fj (y)). j j Now, take the smallest R-face C of conv({y˜i }i=1 ) which is contained in graph(conv({y˜i }i=1 )) and contains the point (y, fj (y)), that is given by the intersection of all R-faces which contain (y, fj (y)). j By Propositions 4.12 and 4.13, C = conv[ext(conv({y˜i }i=1 )) ∩ C] and as (y, fj (y)) does not j j belong to ext(conv({y˜i }i=1 )), dim(C) 1 and the set ext(conv({y˜i }i=1 )) ∩ C contains at least two points y˜k , y˜l . In particular, since both C and x˜ belong to H (z) ∩ graph fj , by definition of supporting hy˜ y˜k ], [x, ˜ y˜l ] and our claim is perplane we have that the graph of fj contains the segments [x, proved. n Now, for each j ∈ N, we define the (possibly multivalued) map Dtj : Bn−1 R1 (x1 ) → R as follows:

yi − x : [x, ˜ y˜i ] ⊂ graph fj (4.41) Dtj : x → |yi − x|


3629

and we prove that the set t Bj := σ t (Z) ∩ x ∈ Bn−1 R1 (x1 ): Dj (x) is multivalued

(4.42)

is H n−1 -negligible, ∀j ∈ N. Thus, if we neglect the set B = j ∈N Bj , we can define our approximating vector field as Dtj (x) = vjt (x) ,

∀x ∈ σ t (Z)\B, ∀j ∈ N.

(4.43)

In order to show that H n−1 (Bj ) = 0 we first prove that, for H n−1 -a.e. x ∈ Bn−1 R1 (x1 ), whent ever Dj (x) contains the directions of two segments, fj must be linear on their convex envelope. ˜ y˜ik ], where ik ∈ {1, . . . , j } and Indeed, suppose that the graph of fj contains two segments [x, k = 1, 2, and consider two points (zk , fj (zk )) ⊂ [x, ˜ y˜ik ] such that z1 = x + se + a1 v1 ,

s ∈ [h− − t, 0), v1 ∈ H0 ;

z2 = x + se + a2 v2 ,

s ∈ [h− − t, 0), v2 ∈ H0 .

(4.44)

As fj is linear on [x, yik ], we have that fj (zk ) = fj (x) + rk · (se + ak vk ),

(4.45)

where rk ∈ ∂ − fj (x), k = 1, 2. Moreover, since πH0 ∂ − fj (x) = ∂ − f|

Bn−1 R1 (x1 )

and the set where ∂ − f|

Bn−1 R1 (x1 )

(x)

(4.46)

is multivalued is H n−2 -rectifiable (see for e.g. [23,1]), we have

that, for H n−1 -a.e. x ∈ Bn−1 R1 (x1 ) r · v = ∇(f|

Bn−1 R1 (x1 )

Then, if we put w = ∇(f|

Bn−1 R1 (x1 )

)(x) · v,

∀r ∈ ∂ − fj (x), ∀v ∈ H0 .

(4.47)

)(x), (4.45) becomes

fj (zk ) = fj (x) + rk · se + w · ak vk .

(4.48)

If zλ = (1 − λ)z1 + λz2 , we have that fj (zλ ) (1 − λ)fj (z1 ) + λfj (z2 ) (4.48) = fj (x) + s (1 − λ)r1 + λr2 · e + w · (1 − λ)a1 v1 + λa2 v2 .

(4.49)

3630


As ((1 − λ)r1 + λr2 ) ∈ ∂ − fj (x), we also obtain that fj (zλ ) fj (x) + s (1 − λ)r1 + λr2 · e + (1 − λ)r1 + λr2 · (1 − λ)a1 v1 + λa2 v2 = fj (x) + s (1 − λ)r1 + λr2 · e + w · (1 − λ)a1 v1 + λa2 v2 (4.48)

= (1 − λ)fj (z1 ) + λfj (z2 ).

(4.50)

Thus, we have that fj ((1 − λ)z1 + λz2 ) = (1 − λ)fj (z1 ) + λfj (z2 ) and our claim is proved. In particular, there exists a supporting hyperplane to the graph of fj which contains the affine hull of the convex envelope of {[x, ˜ y˜ik ]}k=1,2 and then this affine hull must intersect Ht × R into which is parallel to the segment [y˜i1 , y˜i2 ]. a supporting line to the graph of f| n−1 BR

1

(x1 )

Thus, if all the supporting lines to the graph of f|

Bn−1 R1 (x1 )

which are parallel to a segment

[y˜k , y˜m ] (with k, m ∈ {1, . . . , j }, k = m) are parameterized as lk,m + w,

(4.51)

where lk,m is the linear subspace of Rn+1 which is parallel to [y˜k , y˜m ] and w ∈ Wk,m ⊂ Ht × R is perpendicular to lk,m , we have that Bj = σ (Z) ∩ t

π (lk,m + w) .

(4.52)

Rn

k,m∈{1,...,j } w∈Wk,m k<m

By this characterization of the set Bj and by Fubini theorem on Ht w.r.t. the partition given by the lines which are parallel to πRn (lk,m ) for every k and m, in order to show that H n−1 (Bj ) = 0 it is sufficient to prove that, ∀w ∈ Wk,m , H n−1 σ t (Z) ∩ πRn (lk,m + w) = 0.

(4.53)

Finally, (4.53) follows from the fact that a supporting line to the graph of f|

Bn−1 R1 (x1 )

cannot

contain two distinct points of σ˜ t (Z), because otherwise they would be contained in a higher dimensional face of the graph of f contradicting the definition of σ˜ t (Z). Then, the vector field defined in (4.43) is defined H n−1 -a.e. Step 4. Convergence of the approximating vector fields. Here we prove the convergence property of the vector field defined in (4.43) as stated in (4.35). This result is obtained as a consequence of the uniform convergence of the approximating functions fj to the function fˆ which is the graph of the set Cˆ = conv l˜t (yi ) i∈N .

(4.54)

ˆ First of all we observe that, since Cj C, ˆ dom fj = πRn (Cj ) dom fˆ = πRn (C) and fj (x) fˆ(x) where fj (x) is defined ∀j j0 such that x ∈ πRn (Cj0 ).

ˆ , ∀x ∈ ri πRn (C)

(4.55)


3631

In order to prove that fj (x) fˆ(x) uniformly, we show that the functions fj are uniformly Lipschitz on their domain, with uniformly bounded Lipschitz constants. We recall that the graph of fj is made of segments that connect the points of graph f| n−1 BR

j

1

(x1 )

to the points of graph(conv({y˜i }i=1 )). In order to find an upper bound for the incremental ratios between points z, w ∈ dom fj , we distinguish two cases. j Case 1: [z, w] ⊂ [x, yk ], where x ∈ Bn−1 ˜ y˜k ] ⊂ graph fj . R1 (x1 ), yk ∈ {yi }i=1 and [x, In this case we have that |fj (z) − fj (w)| |fj (x) − fj (yk )| |f (x) − f (yk )| = = Lf , |z − w| |x − yk | |x − yk |

(4.56)

where Lf is the Lipschitz constant of f on Kt . Case 2: Otherwise we observe that, since fj is convex, fj (z) − fj (w)

sup

r∈∂ − fj (z)∪∂ − fj (w)

r · (z − w).

(4.57)

Let then r ∈ ∂ − fj (z) ∪ ∂ − fj (w) be a maximizer of the r.h.s. of (4.57) and let us suppose, without loss of generality, that r ∈ ∂ − fj (z). If x ∈ Bn−1 ˜ ⊂ R1 (x1 ) is such that (z, fj (z)) ⊂ [(y, fj (y)), x] j

graph fj for some y ∈ conv({yi }i=1 ), we have the following unique decomposition x−z + γj (z, w)q, w − z = βj (z, w) |x − z|

(4.58)

where q ∈ Sn−1 ∩ H0 and βj (z, w), γj (z, w) ∈ R. Then, x−z r · (w − z) = βj (z, w) r · + γj (z, w)(r · q). |x − z|

(4.59)

The first scalar product in (4.59) can be estimated as in Case 1. As for the second term, we note that the supporting hyperplane to the graph of fj given by the graph of the affine function h(p) = fj (z) + r · (p − z) contains the segment [(z, fj (z)), x] ˜ and its intersection with the hyperplane Ht × R is given by a supporting hyperplane to the graph of f| n−1 which contains the point x. ˜ BR

1

(x1 )

Moreover, as q ∈ H0 , we have that r · q = πH0 (r) · q, and we know that πH0 (r) ∈ ∂ − f|

Bn−1 R1 (x1 )

(x).

By definition of subdifferential, for all s ∈ ∂ − f| x − λq ∈ Bn−1 R1 (x1 ),

(4.60)

Bn−1 R1 (x1 )

(x) and for all λ > 0 such that x + λq,

3632


f (x + λq) − f (x) f (x) − f (x − λq) s ·q λ λ

(4.61)

and so the term |r · q| is bounded from above by the Lipschitz constant of f . As the scalar products βj (z, w), γj (z, w) are uniformly bounded w.r.t. j on dom fj ⊂ dom fˆ, we conclude that the functions {fj }j ∈N are uniformly Lipschitz on the sets {dom fj }j ∈N and ˆ their Lipschitz constants are uniformly bounded by some positive constant L. ˆ ˆ If we call fj a Lipschitz extension of fj to the set dom f which has the same Lipschitz constant (Mac Shane lemma), by Ascoli–Arzelá theorem we have that fˆj → fˆ uniformly on dom fˆ. Now we prove that, for H n−1 -a.e. x ∈ σ t (Z)\B, vjt (x) → ve (x). j

Given a point x ∈ σ t (Z)\B, we call y˜j (x) , where j ∈ N, the unique point y˜k ∈ {y˜i }i=1 such that vjt (x) =

yk − x . |yk − x|

By compactness of graph(conv({y˜i }i∈N )), there is a subsequence {jn }n∈N ⊂ N such that y˜jn (x) → yˆ ∈ graph f, hence vjt n (x) → vˆ =

yˆ − x . |yˆ − x|

As the functions fj converge to fˆ uniformly, the point yˆ and the whole segment [x, ˜ y] ˆ belong to the graph of fˆ. ˜ y] ˆ which belong to the graph of fˆ and pass through So, there are two segments l˜t (x) and [x, the point x. ˜ Since fˆ| n−1 = f| n−1 , we can apply the same reasoning we made in order to prove that BR

1

(x1 )

BR

1

(x1 )

the set (4.42) was H n−1 -negligible to conclude that the set ˆ σ t (Z) ∩ x ∈ Bn−1 R1 (x1 ): ∃ more than two segments in the graph of f that connect x˜ to a point of graph conv {y˜i }i∈N has zero H n−1 -measure. Then, [x, ˜ y] ˆ = l˜t (x) and vˆ = ve (x) for H n−1 -a.e. x ∈ σ t (Z), so that property (4.35) is proved. Step 5. Proof of the estimate (4.36) (see Fig. 5). The estimate for the map σvt t induced by the approximating vector fields vjt follows as in [7] j

and [8] from the fact that the collection of segments with directions given by vjt and endpoints in −

j

dom vjt , σ h (Z) form a finite union of cones with bases in dom vjt and vertex in {yi }i=1 .


3633

Fig. 5. The vector field ve (that one can see in the picture between {x · e = t} and {x · e = h+ }) is approximated by directions of approximating cones (in the picture one can see the first approximating cone between {x · e = h− } and {x · e = t}). At the same time, Z is approximated by the pushforward of σ t (Z) with the approximating vector field: compare the blue area (outer) with the yellow one (inner). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Indeed, if we define the sets

yi − x , Ωij = x ∈ σ t (Z): Dtj (x) = vjt (x) and vjt (x) = |yi − x|

j ∈ N, i = 1, . . . , j, (4.62)

for all S ⊂ σ t (Z)\B we have that H n−1 (S) =

j

H n−1 (S ∩ Ωij )

i=1

and t −1 n−1 t −1 σv t σv t (S) = H (S ∩ Ωij ) . j

H

n−1

j

j

i=1

Then it is sufficient to prove (4.36) when the vector field vjt is defined as vjt (x) =

yi − x . |yi − x|

After these preliminary considerations, (4.36) follows from the fact that the set

−1 σvst ◦ σvt t (S)

s∈[h− ,t]

j

(4.63)

j

is a cone with basis S ⊂ Ht and vertex yi ∈ Hh− and (σvt t )−1 (S) is the intersection of this cone with the hyperplane H0 .

2

j

3634


Proof of Lemma 4.7. Given a k-dimensional D-cylinder C k parameterized as in (4.27), the collection of segments

σ w+te (z): t ∈ h− (w, e), h+ (w, e)

(4.64)

z∈Z k

is a 1-dimensional D-cylinder of the convex function f restricted to the (n − k + 1)-dimensional set −1 πe w + te: t ∈ h− (w, e), h+ (w, e) . 1 ,...,ek

(4.65)

Then, as in Lemma 4.14, we can construct a sequence of approximating vector fields also for the directions of the segments (4.64). The only difference with respect to the approximation of the 1-dimensional faces of f is that the domain of the approximating vector fields will be a subset −1 of an (n − k)-dimensional affine plane of the form πe (w) and so the measure involved in 1 ,...,ek n−k n−1 the estimate (4.32) will be H instead of H . Finally, we pass to the limit as with the approximating vector fields given in Lemma 4.14 and we obtain the fundamental estimate (4.31) for the k-dimensional D-cylinders. 2 4.5. Properties of the density function In this subsection, we show that the quantitative estimates of Lemma 4.14 allow not only to derive the absolute continuity of the pushforward with σ w+te , but also to find regularity estimates on the density function. This regularity properties will be used in Section 5. Corollary 4.15. Let C k be a k-dimensional D-cylinder parameterized as in (4.27) and let σ w+se (Z k ), σ w+te (Z k ) be two sections of C k with s and t as in (4.31). Then, if we put s = w + se and t = w + te, we have that t−|s−t|e

H n−k

σ#

σ t Zk

H n−k

σ s Zk

(4.66)

and by the Radon–Nikodym theorem there exists a function α(t, s, ·) which is H n−k -a.e. defined on σ s (Z k ) and is such that t−|s−t|e

σ#

H n−k

σ t Zk

= α(t, s, ·)H n−k

σ s Zk .

(4.67)

Proof. Without loss of generality we can assume that s = 0. If H n−k (A) = 0 for some A ⊂ Z k , by definition of pushforward of a measure we have that w+te −1 n−k H σ #

σ w+te Zk

(A) = H n−k σ w+te (A)

(4.68)

and taking s = 0 in (4.31) we find that H n−k (A) = 0 implies that H n−k (σ w+te (A)) = 0.

2


3635

Remark 4.16. The function α = α(t, s, y) defined in (4.67) is measurable w.r.t. y and, for H n−k a.e. y ∈ σ w+te (Z k ), we have that −1 . α s, t, y = α t, s, σ t−|s−t|e y

(4.69)

Moreover, from Lemma 4.7 we immediately get the uniform bounds:

h+ (t, e) − u h+ (t, e) u − h− (t, e) −h− (t, e)

n−k

α(t + ue, t, ·)

n−k α(t + ue, t, ·)

u − h− (t, e) −h− (t, e) h+ (t, e) − u h+ (t, e)

n−k n−k

if u ∈ 0, h+ (t, e) , if u ∈ h− (t, e), 0 .

(4.70)

We conclude this section with the following proposition: Proposition 4.17. Let C k (Z k , C k ) be a k-dimensional D-cylinder parameterized as in (4.27) and assume without loss of generality that w = πe1 ,...,ek (Z k ) = 0. Then, the function α(t, 0, z) defined in (4.67) is locally Lipschitz in t ∈ ri(C k ) (and so jointly measurable in (t, z)). Moreover, for H n−k -a.e. y ∈ σ t (Z) the following estimates hold: 1. Derivative estimate d n−k α(t + ue, t, y) α(t + ue, t, y) − + h (t, e) − u du n−k α(t + ue, t, y); u − h− (t, e)

(4.71)

2. Integral estimate

|h+ (t, e) − u| |h+ (t, e)|

n−k (−1)1{u (1 − ε)r k , such that

H n−k A ∩ πV−1 (t) ∩ [0, r]n (1 − ε)r n−k Moreover, there is a set Q ⊂ [0, re], with H 1 (Q) > (1 −

for all t ∈ T .

(4.89)

√ ε)r, such that

√ −1 (q) > (1 − ε )r k−1 H k−1 T ∩ πe

for q ∈ Q.

(4.90)

−1 Consider two points q, s := q + 2εre ∈ Q, and take t ∈ T ∩ πe (q). By the fundamental estimate (4.88), one has

H n−k σ t+2εre (St,r ) (1 − ε)n−k H n−k (St,r )

where St,r := A ∩ πV−1 (t) ∩ [0, r]n .

Furthermore, condition (4.86) implies that x + 2εre − σ t+2εre (x) 2εr for each x ∈ A ∩ πV−1 (t). Moving points of πV−1 (t) ∩ [0, r]n by means of the map σ t+2εre , they can therefore reach

3642


Fig. 6. Illustration of the construction in the proof of Lemma 4.19. A is the set of points on the relative boundary of the k-faces of f , projected on Rn , having directions close to V = e1 , . . . , ek and such that, for each point x ∈ A, k k πV (F∇f (x) ) contains a fixed k-cone centered at x with direction e1 . T is a subset of the square i=1 [0, rei ] such that, for every t ∈ T , πV−1 (t) ∩ A is ‘big’. Finally, q, s = q + 2εre1 are points on [0, re1 ] such that the intersection of T with

−1 −1 the affine hyperplanes πe (q), πe (s) is ‘big’. The absurd arises from the following. Due to the fundamental estimate, 1

1

−1 −1 (q), one finds points in the complementary of T . Since T ∩ πe (q) translating by 2εre1 the points in the set T ∩ πe 1 1 −1 −1 was ‘big’, then T \ πe (s) should be ‘big’, contradicting the fact that T ∩ πe (s) is ‘big’. 1

1

only the square πV−1 (t + 2εre) ∩ [−2εr, (1 + 2ε)r]n . Notice that for ε small, since our proof is needed for n 3 and k 1, n−k H n−k −2εr, (1 + 2ε)r \ [0, r]n−k = (1 + 4ε)n−k r n−k − r n−k 4(n − k)εr n−k + o(ε) < n2n εr n−k . As a consequence, the portion which stays inside πV−1 (t + 2εre) ∩ [0, r]n can be estimated as follows: H n−k σ t+2εre (St,r ) ∩ [0, r]n H n−k σ t+2εre (St,r ) − n2n εr n−k . As noticed before, condition (4.87) implies that the points σ t+2εre (St,r ) ∩ [0, r]n belong to the complementary of A. By the above inequalities we obtain then H n−k Ac ∩ πV−1 (t + 2εre) ∩ [0, r]n H n−k σ t+2εre (St,r ) ∩ [0, r]n (1 − ε)n−k H n−k (St,r ) − n2n εr n−k


3643

(4.89)

(1 − ε)n−k+1 r n−k − n2n εr n−k 1 n−k . r 2

−1 The last estimate shows that, for each t ∈ T ∩ πe (q), the point t + 2εre does not satisfy the −1 (q)) + 2εre lies in the complementary of T . In particular inequality in (4.89): thus (T ∩ πe

−1 −1 H k−1 T ∩ πe (s) < r k−1 − H k−1 T ∩ πe (q) . However, by construction both t and s belong to Q. This yields the contradiction, by definition of Q: 1 k−1 (4.90) k−1 −1 −1 (4.90) 1 k−1 r r . T ∩ πe < H (s) < r k−1 − H k−1 T ∩ πe (t) < 2 2

2

Proof of Theorem 3.3. As we observed in Remark 3.4, it is sufficient to prove the theorem for the disintegration of the Lebesgue measure on the set F k when k ∈ {1, . . . , n − 1}. Thanks to Lemma 4.19, we can further restrict the disintegration to the set E k defined in (4.13); moreover, by (4.25), for all k = 1, . . . , n − 1 there exists an L n -negligible set N k such that E k \N k =

Cjk \dCjk ,

j ∈N

where {Cjk }j ∈N is the countable collection of k-dimensional D-cylinders covering E k which was constructed in Lemma 4.6, so that the sets Cˆjk := Cjk \dCjk are disjoint. The fundamental observation is the following:

j ∈N

Cˆjk =

j ∈N y∈Im ∇f|

k Ey,j =

y∈Im ∇f|

Ek

Ek

k Ey,j =

y∈Im ∇f|

j ∈N

Eyk \N k ,

(4.91)

Ek

k = E k ∩ Cˆk . where Ey,j y j For all j ∈ N, we set

k Yj = y ∈ Im ∇f|E k : Ey,j = ∅ = ∇f Cˆjk ,

(4.92)

we denote by pj : Cˆjk → Yj the quotient map corresponding to the partition Cˆjk =

y∈Im ∇f|

k Ey,j = Ek

k Ey,j

y∈Yj

and we set νj = pj # (L n (Cˆjk ∩ K)). Since the quotient space (Yj , B(Yj )) is isomorphic to (Zjk , B(Zjk )), where Zjk is a section of Cjk , by Theorem 4.18 we have that

3644


L

n

k Cj ∩ K Ej ∩ pj−1 (Fj ) =

j

μy (Ej ) dνj (y), Fj

∀Ej ∈ B Cjk , Fj ∈ B(Yj ),

(4.93)

j

k ∩ K) for ν -a.e. y ∈ Y . where μy is equivalent to H k (Ey,j j j n k Moreover, for every E ∈ B(R ) ∩ E there exist sets Ej ∈ B(Cjk ) such that

E=

Ej

j ∈N

and for all F ∈ B(Y ), where Y = F=

j ∈N Yj

= Im ∇f|E k , setting F := F ∩ Yj we have that

and ∇f −1 (F ) =

Fj

j ∈N

pj−1 (Fj ).

j ∈N

Then, Ln

K E ∩ ∇f −1(F )

=

+∞

Ln

j =1

=

(4.93)

=

+∞ j =1F

Cjk Ej ∩ pj−1(Fj ) j

μy (Ej ) dνj (y)

j

+∞

j

1Fj (y)μy (Ej ) dνj (y)

j =1Y

j

=

+∞

j

1Fj (y)μy (Ej )fj (y) dν(y),

(4.94)

j =1 Y

where fj is the Radon–Nikodym derivative of νj w.r.t. the measure ν = ∇f# (L n K) on Y . Since, as we proved in Section 3.1, there exists a unique disintegration {μy }y∈Im ∇f| k such E that L n K E ∩ ∇f −1 (F ) = μy (E) dν(y) for all E ∈ B Rn , F ∈ B(Y ),

F

we conclude that the last term in (4.94) converges and μy =

+∞ j =1

so that the theorem is proved.

2

j

fj (y) μy

for ν-a.e. y ∈ Y,

(4.95)


3645

5. A divergence formula The previous section led to a definition of a function α, on any D-cylinder C k = C k (Z k , C k ), as the Radon–Nikodym derivative in (4.67). In the present section we find that on C k the function α satisfies the system of ODEs ∂t α t = πe1 ,...,ek (x), 0, x −

k

x · ei vi (x)

i=1

= (div v )a.c. (x)α πe1 ,...,ek (x), 0, x −

k

x · ei vi (x)

i=1

for = 1, . . . , k, where we assume w.l.o.g. that 0 ∈ C k , e1 , . . . , ek is the axis of C k , vi (x) is the vector field −1 x → 1C k (x) D(x) ∩ πe (e ) 1 ,...,ek i and (div vi )a.c. (x) is the density of the absolutely continuous part of the divergence of vi , that we prove to be a measure. This is a consequence of the absolute continuity of the conditional measures of the disintegration, stated in Theorem 4.18, and of the regularity estimates on the density α proved in Proposition 4.17. Notice that even the fact that the divergence of vi is a measure is not trivial, since the vector field is just Borel. Heuristically, the ODEs above can be formally derived as follows. In Section 4 we saw that C k is the image of the product space C k + Z k , where Z k = C k ∩ −1 πe1 ,...,ek (0) is a section of C k , under the change of variable Φ(t + z) = z +

k

ti vi (z) = σ t (z)

for all t =

i=1

k

ti e i ∈ C k , z =

i=1

n

zi ei ∈ Z k .

(5.1)

i=k+1

In Theorem 4.18 we found that the weak Jacobian of this change of variable is defined, and given by J(t + z) = α(t, 0, z).

(5.2)

From (5.1) one finds that, if vi was smooth instead of only Borel, this Jacobian would be ! J(t + z) = det

[vj · ei ] i=1,...,n

j =1,...,k

! k " t ∂zj v (z) · ei + δi,j =1

" .

(5.3)

i=1,...,n j =k+1,...,n

Moreover, by direct computations with Cramer rule and the multilinearity of the determinant, starting from (5.2)-(5.3) one would prove the relation

3646


−1 J(t + z), ∂t J(t + z) = trace J v (z) J Φ(t + z) where J g denotes the Jacobian matrix of a function g. By the Lipschitz regularity of α w.r.t. the {ti }ki=1 variables given in Proposition 4.17, one could then expect that ∂t α(t, 0, z) =

n j =1

−1 ∂xj vi Φ (x) · ej x=Φ(t+z) α(t, 0, z).

(5.4)

Notice that j ∂xj (vi (Φ −1 (x)) · ej )|x=Φ(t+z) is the pointwise divergence of the vector field vi (Φ −1 (x)) evaluated at x = Φ(t + z). In this article, we denote it with (div(vi ◦ Φ −1 ))a.c. . Finally, given a regular domain Ω ⊂ Rn , by the Green–Gauss–Stokes formula one should have div vi ◦ Φ −1 a.c. dL n (x) = vi Φ −1 (x) · nˆ dH n−1 (x), (5.5) Ω

∂Ω

where nˆ is the outer normal to the boundary of Ω. The analogue of formulas (5.4) and (5.5) is the additional regularity we prove in this section, in a weak context, for vector fields parallel to the faces and for the current of k-faces. Actually, for simplicity of notations we will continue working with the projection of the faces on Rn instead of with the faces themselves. We give now the idea of the proof, in the case of 1-dimensional faces. −1 (0). Fix the attention on a 1-dimensional D-cylinder C with axis e and section Z = C ∩ πe Consider the distributional divergence of the vector field v giving pointwise on C the direction of projected faces, normalized with v · e = 1, and vanishing elsewhere. The explicit disintegration in Theorem 4.18 decomposes integrals on C to integrals first on the projected faces, with the additional density factor α, then on Z. By means of it, one then reduces the integral − C ∇ϕ · v, defining the distributional divergence, to the following integrals on the projected faces: −

∇ϕ x = z + t1 v(z) · v(z)α(t, 0, z) dH 1 (t)

where z varies in Z.

[h− e,h+ e]

Since α is Lipschitz in t and ∇ϕ(x = σ w+t1 e (z)) · v(z) = ∂t1 (ϕ ◦ σ w+t (z)), by integrating by parts one arrives to t=h+ e ϕ ◦ σ w+t (z)∂t1 α(t, 0, z) dH 1 (t) − ϕ ◦ σ w+t (z)α(t, 0, z) t=h− e . [h− e,h+ e]

Applying again the disintegration theorem in the other direction, by the invertibility of α, one comes back to integrals on the D-cylinder, where in the first addend ϕ is now integrated with the factor ∂t1 α/α. An argument of this kind yields an explicit representation of the distributional divergence of the truncation to C k of a vector field v parallel at each point x to the projected face through x. This divergence is a Radon measure: the absolutely continuous part is basically given by (5.4)


3647

and, as in (5.5), there is an additional singular term representing the flux of v through dC k , the border of C k transversal to D, which we have already defined as k −1 rb C . dC k = C k ∩ πe 1 ,...,ek

(5.6)

−1 (C k ). We also remind that nˆ |dC k denotes the outer unit normal to πe 1 ,...,ek k As C are not regular sets, but just σ -compact, there is a loss of regularity for the divergence of v in the whole Rn . In general, the distributional divergence of v will be just a series of measures.

5.1. Vector fields parallel to the faces In the present subsection, we study the regularity of a vector field parallel, at each point, to the corresponding face through that point. 5.1.1. Study on model sets As a preliminary step, fix the attention on the D-cylinder C k = C k Z k , Ck . One can assume w.l.o.g. that the axis of C k is identified by vectors {e1 , . . . , ek } which are the first k coordinate vectors of Rn and that C k is the square

Ck =

k

[−ei , ei ].

i=1 −1 Denote with Z k the section Z k ∩ πe (0). We also assume w.l.o.g. that C k is bounded. 1 ,...,ek

Definition 5.1 (Coordinate vector fields). We define on Rn k-coordinate vector fields for C k as follows:

vi (x) =

0 v ∈ D(x)

if x ∈ / C k, such that πe1 ,...,ek v = ei if x ∈ C k .

The k-coordinate vector fields are a basis for the module on the algebra of measurable functions from Rn to R constituted by the vector fields with values in D(x) at each point x ∈ C k , and vanishing elsewhere. Consider the distributional divergence of vi , denoted by div vi . As a consequence of the absolute continuity of the pushforward with σ , and by the regularity of the density α, one gains more regularity for the divergence. Let us fix a notation. Given any vector field v : Rn → Rn whose distributional divergence is a Radon measure, we will denote by (div v)a.c. the density of the absolutely continuous part of the measure div v.

3648


Lemma 5.2. The distribution div vi is a Radon measure. Its absolutely continuous part has density ∂ti α(t = πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) (div vi )a.c. (x) = 1C k (x). α(πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) Its singular part is H n−1

(5.7)

(C k ∩ {x · ei = −1}) − H n−1 (C k ∩ {x · ei = 1}).

n Proof. Consider any test function ϕ ∈ C∞ c (R ) and apply the explicit disintegration of Theorem 4.18:

div vi , ϕ := −

∇ϕ(x) · vi (x) dL n (x)

Ck

=−

α(t, 0, z) ∇ϕ σ t (z) · vi (z) dH k (t) dH n−k (z),

(5.8)

Zk C k

where we used that vi is constant on the faces, i.e. vi (z) = vi (σ t (z)). Being σ t (z) = z + k i=1 ti vi (z), one has ∇x ϕ x = σ t (z) · vi (z) = ∇x ϕ x = σ t (z) · ∂ti σ t (z) = ∂ti ϕ σ t (z) . The inner integral is thus

∇ϕ σ t (z) · vi (z)α(t, 0, z) dH k (t) =

Ck

∂ti ϕ σ t z α(t, 0, z) dH k (t).

Ck

Since Proposition 4.17 ensures that α is Lipschitz in t, for t ∈ C k , one can integrate by parts: Ck

∂ti ϕ σ t (z) α(t, 0, z) dH k (t) = −

Ck

ϕ σ t (z) ∂ti α(t, 0, z) dH k (t)

+

ϕ σ t (z) α(t, 0, z) dH k−1 (t)

C k ∩{ti =1}

−

ϕ σ t (z) α(t, 0, z) dH k−1 (t).

C k ∩{ti =−1}

Substitute now the last formula in the first expression (5.8). Recall moreover the definition of α in (4.67), as a Radon–Nikodym derivative of a push-forward measure, and its invertibility and Lipschitz estimates (Remark 4.16, Proposition 4.17), among which in particular the L1 estimate on the function ∂ti α/α. Then, pushing the measure from t = 0 to a generic t, one comes back to the integral on the D-cylinder


div vi , ϕ =

3649

ϕ σ t (z) ∂ti α(t, 0, z) dH k (t) dH n−k (z)

Zk C k

−

ϕ σ t (z) α(t, 0, z) dH k−1 (t) dH n−k (z)

Z k C k ∩{ti =1}

ϕ σ t (z) α(t, 0, z) dH k−1 (t) dH n−k (z)

+ Z k C k ∩{ti =−1}

=

ϕ(x)(div vi )a.c. (x) dL (x) − n

Ck

ϕ(x) dH n−1 (x)

C k ∩{x·ei =1}

+

ϕ(x) dH n−1 (x),

C k ∩{x·ei =−1}

where (div vi )a.c. 1C k is the function proved by the last formula. 2

∂ti α α

precisely written in the statement. Thus our thesis is

Remark 5.3. Consider a function λ ∈ L1 (C k ; R) constant on each face, meaning that λ(σ t (z)) = λ(z) for t ∈ C k and z ∈ Z k . One can regard this λ as a function of ∇f (x). Then the same statement of Lemma 5.2 applies to the vector field λvi , but the divergence is clearly div(λvi ) = λ div vi . The proof is the same, observing that div(λvi ), ϕ

:=

−

∇ϕ(x) · λ(x)vi (x) dL n (x)

Ck Th. 4.18

=

−

λ(z)∇ϕ σ t (z) · vi (z)α(t, 0, z) dH k (t) dH n−k (z)

Zk C k

=

−

λ(z)∂ti ϕ σ t (z) α(t, 0, z) dH k (t) dH n−k (z)

Zk C k

=

Zk C k

λ(z)ϕ σ t (z) ∂ti α(t, 0, z) dH k (t) dH n−k (z)

−

λ(z)ϕ σ t (z) α(t, 0, z) dH k−1 (t) dH n−k (z)

Z k C k ∩{ti =1}

+ Z k C k ∩{ti =−1}

λ(z)ϕ σ t (z) α(t, 0, z) dH k−1 (t) dH n−k (z)

3650

L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661 Th. 4.18

=

ϕ(x)λ(x)(div vi )a.c. (x) dL (x) − n

Ck

ϕ(x)λ(x) dH n−1 (x)

C k ∩{x·ei =1}

+

ϕ(x)λ(x) dH n−1 (x).

C k ∩{x·e

(5.9)

i =−1}

Suitably adapting the integration by parts in the above equality (5.9) with

λ σ t (z) ∂ti ϕ σ t (z) α(t, 0, z) dH k (t)

Ck

=− Ck

λ σ t (z) ϕ σ t (z) ∂ti α(t, 0, z) dH k (t) −

∂ti λ σ t (z) ϕ σ t (z) α(t, 0, z) dH k (t)

Ck

λ σ t (z) ϕ σ t (z) α(t, 0, z) dH k−1 (t)

+ C k ∩{ti =1}

−

λ σ t (z) ϕ σ t (z) α(t, 0, z) dH k−1 (t)

C k ∩{ti =−1}

one finds moreover that for all λ ∈ L1 (Rn ; R) continuously differentiable along vi with integrable directional derivative ∂vi λ, the following relation holds: div(λvi ) = λ div vi + ∂vi λ L n .

(5.10)

Notice that in (5.10) there is the addend λH n−1 (C k ∩ {x · ei = 1}), hidden in the term λ div vi , which would make no sense for a general λ ∈ L1 (Rn ; R). Now we prove that the restriction to k k n−k -a.e. z ∈ Z k , C k ∩ {x · ei = 1} of each representative of λ which is C1 (F∇f (z) ∩ C ), for H 1 k identifies the same function in L (C ∩ {x · ei = 1}). ˜ λˆ of the L1 -class of λ can differ only on an L n -negligible Indeed, any two representatives λ, set N . By the explicit disintegration in Theorem 4.18, and using moreover Fubini theorem for reducing the integral on C k to integrals on lines parallel to ei , one has that the intersection of N with each of the lines on the projected faces with projection on e1 , . . . , ek parallel to ei is almost always negligible: H 1 N ∩ q + vi (q) = 0

for q ∈ C k ∩ {x · ei = 0} \ M, with H n−1 (M)=0.

˜ λˆ are continuously differentiable along vi , one can have that N ∩ (q + vi (q)) = ∅ for Since λ, all q ∈ C k ∩ {x · ei = 0} \ M. As a consequence N ∩ {x · ei = t} is a subset of τ tei (M), where τ tei is the map moving along each projected face with tvi : C k ∩ {x · ei = 0} q → τ tei (q) := q + tvi = σ πe1 ,...,ek (q)+te1 (q).


3651

By the pushforward formula (4.67), denoting wq := πe1 ,...,ek (q) and zq := πek+1 ,...,en (q), H n−1

te te τ i (S) = α(wq , wq + tei , zq )τ# i H n−1 (q)

S

for S ⊂ C k ∩ {x · e1 = 0} and t ∈ [−1, 1]. Therefore, as H n−1 (M) = 0, one has that λ˜ and λˆ identify the same integrable function on each section of C k perpendicular to ei , showing that the measure λH n−1 ({x · ei = 1}) is well defined. Actually, the same argument as above should be used in (5.9) in order to show that λ(z) is integrable on Z k , so that one can separate the three integrals as we did. Indeed, being constant on each face by assumption, the restriction of λ to a section is trivially well defined, since it associates to a point the value of λ corresponding to the face of that point, but the integrability w.r.t. H n−1 on each slice is a consequence of the pushforward estimate.

As a direct consequence of (5.10), by linearity, one gets a divergence formula for any sufficiently regular vector field which, at each point of C k , is parallel to the corresponding projected face of f , and vanishes elsewhere. The precise statement is given in the following: Corollary 5.4. Consider any vector field v = ki=1 λi vi with λi ∈ L1 (C k ; R) continuously differentiable along vi , with directional derivative ∂vi λi integrable on C k . Then the divergence of v is a Radon measure and for every ϕ ∈ C1c (Rn ), div v, ϕ =

ϕ(x)(div v)a.c. (x) dL (x) − n

Ck

ϕ(x) v(x) · n(x) ˆ dH n−1 (x),

(5.11)

dC k

where dC k , the border of C k transversal to D, and n, ˆ the outer unit normal, are defined in formula (5.6). In particular, the density (div v)a.c. (x) of the absolutely continuous part of the divergence vanishes out of C k , while for x ∈ C k one has the expression ∂ti α(t = πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) (div v)a.c. (x) = λi (x) α(πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) i=1 k

+

k

∂vi λi (x).

(5.12)

i=1

Remark 5.5. The result is essentially based on the application of the integration by parts formula when the integral on C k is reduced, by our explicit disintegration theorem, to integrals on C k : this is why we assume the C1 regularity of the λi , w.r.t. the directions of the k-face passing through each point of C k . Such regularity could be further weakened, however we do not pursue this issue here. As a consequence, one can easily extend the statement of the previous corollary −1 to sets of the form CΩk = F k ∩ πe (Ω), for an open set Ω ⊂ e1 , . . . , ek with piecewise 1 ,...,ek −1 (rb(Ω)). Lipschitz boundary, defining dCΩk := F k ∩ πe 1 ,...,ek

3652


5.1.2. Global version We study now the distributional divergence of an integrable vector field v on T, as we did in Section 5.1.1 for such a vector field truncated on D-cylinders. Corollary 5.6. Consider a vector field v ∈ L1 (T; Rn ) such that v(x) ∈ D(x) for x ∈ Rn , where we define D(x) = {0} for x ∈ / T. Suppose moreover that the restriction to every face Ey , for y ∈ Im ∇f , is continuously differentiable with integrable derivatives. Then, for every ϕ ∈ C1c (Rn ) one can write div v, ϕ = lim

→∞

ϕ(x) div(1Ci v) a.c. (x) dL n (x) −

i=1 C i

ϕ(x)v(x) · nˆ i (x) dH

n−1

(x) ,

dCi

(5.13) where {C }∈N is the countable partition of T in D-cylinders given in Lemma 4.6, while (div(1Ci v))a.c. is the one of Corollary 5.4 and dCi , nˆ i are defined in formula (5.6). Remark 5.7. By construction of the partition, each of the second integrals in the r.h.s. of (5.13) appears two times in the series, with opposite sign. Intuitively, the finite sum of these border terms is the integral on a perimeter which tends to the singular set of points in the relative boundary of projected k-faces. Remark 5.8. Suppose that div v is a Radon measure. Then Corollary 5.6 implies that 1C k (div v)a.c. ≡ div(1C k v) a.c. . Proof of Corollary 5.6. The partition of nk=1 E k into the sets {C }∈N is given by Lemma 4.6, as stated in the corollary. Moreover, Lemma 4.19 shows that the set T \ nk=1 Ek is Lebesgue negligible. Therefore, by dominated convergence theorem one finds that

v(x) · ∇ϕ(x) dL (x) = − lim

div v, ϕ = −

n

→∞

T

i=1 C

v(x) · ∇ϕ(x) dL n (x).

i

The addends in the r.h.s. are, by definition, the distributional divergence of the vector fields v1Ci applied to ϕ. In particular, by Corollary 5.4, they are equal to −

v(x) · ∇ϕ(x) dL n (x)

Ci

=−

1Ci (x)v(x) · ∇ϕ(x) dL n (x)

Rn Cor. 5.4

=

Rn

ϕ(x)1Ci (x) div(1Ci v) a.c. (x) dL n (x) +

dCik

ϕ(x)v(x) · nˆ i (x) dH n−1 (x)


=

3653

ϕ(x)(div v)a.c. (x) dL (x) +

ϕ(x)v(x) · nˆ i (x) dH n−1 (x),

n

Ci

dCik

proving the thesis.

2

5.2. The currents of k-faces In the present subsection, we change point of view. Instead of looking at vector fields constrained to the faces of f , we regard the k-dimensional faces of f as a k-dimensional current. We establish that this current is a locally flat chain, providing a sequence of normal currents converging to it in the flat norm.The border of these normal currents has formally the same representation one would have in a smooth setting. Before proving it, we devote Section 5.2.1 to recalls on this argument, in order to fix the notations. They are taken mainly from Chapter 4 of [17] and Sections 1.5.1, 4.1 of [10]. 5.2.1. Recalls Let {e1 , . . . , en } be a basis of Rn . The wedge product between vectors is multilinear and alternating, i.e.:

n

λi ei ∧ u1 ∧ · · · ∧ um =

i=1

n

λi (ei ∧ u1 ∧ · · · ∧ um ),

m ∈ N, λ1 , . . . , λn ∈ R,

i=1

u0 ∧ · · · ∧ ui ∧ · · · ∧ um = (−1)i ui ∧ u0 ∧ · · · ∧ u#i ∧ · · · ∧ um ,

0 < i m, u0 , . . . , um ∈ Rn ,

where the element under the hat is missing. The space of all linear combinations of ei1 ...im := ei1 ∧ · · · ∧ eim : i1 < · · · < im in {1, . . . , n} is the space of m-vectors, denoted by Λm Rn . The space Λ0 R is just R. Λm Rn has the inner product given by ei1 ...im · ej1 ...jm =

m

δi k j k

where δij =

k=1

1 if i = j , 0 otherwise.

The induced norm is denoted by · . An m-vector field is a map ξ : Rn → Λm Rn . The dual Hilbert space to Λm Rn , denoted by Λm Rn , is the space of m-covectors. The element dual to ei1 ...im is denoted by dxi1 ...im . A differential m-form is a map ω : Rn → Λm Rn . We denote by ·, · the duality pairing between m-vectors and m-covectors. Moreover, the same symbol denotes in this paper the bilinear pairing, which is a map Λp Rn × Λq Rn → Λp−q Rn for p > q and Λp Rn × Λq Rn → Λq−p Rn for q > p whose non-vanishing images on a basis are dxi1 ...i = dxi1 ...i ∧ dxi+1 ...i+m , ei+1 ...i+m , ei+1 ...i+m = dxi1 ...i , ei1 ...i ∧ ei+1 ...i+m ,

if p = + m > m = q, if p = < + m = q.

3654


Consider any differential m-form ω=

ωi1 ...im dxi1 ...im

i1 ...im

which is differentiable. The exterior derivative dω of ω is the differential (m + 1)-form dω =

n ∂ωi

1 ...im

∂xj

i1 ...im j =1

dxj ∧ dxi1 ...im .

If ω ∈ Ci (Rn ; Λm Rn ), the i-th exterior derivative is denoted by d i ω. Consider any m-vector field ξ=

ξi1 ...im ei1 ...im

which is differentiable. The pointwise divergence (div ξ )a.c. of ξ is the (m − 1)-vector field (div ξ )a.c. :=

n ∂ξi i1 ...im j =1

1 ...im

∂xj

dxj , ei1 ...im .

Consider the space D m of C∞ -differential m-form with compact support. The topology is generated by the seminorms i νK (φ) =

sup

x∈K, 0j i

j d φ(x)

with K compact subset of Rn , i ∈ N.

The dual space to D m , endowed with the weak topology, is called the space of m-dimensional currents and it is denoted by Dm . The support of a current T ∈ Dm is the smallest closed set K ⊂ Rn such that T (ω) = 0 whenever ω ∈ D m vanishes out of K. The mass of a current T ∈ Dm is defined as M(T ) = sup T (ω): ω ∈ D m , sup ω(x) 1 . x∈Rn

The flat norm of a current T ∈ Dm is defined as F(T ) = sup T (ω): ω ∈ D m , sup ω(x) 1, sup dω(x) 1 . x∈Rn

x∈Rn

An m-dimensional current T ∈ Dm is representable by integration, and we denote it by T = μ ∧ ξ , if there exists a Radon measure μ over Rn and a μ-locally integrable m-vector field ξ such that T (ω) = ω, ξ dμ ∀ω ∈ D m . Rn


3655

If m 1, the boundary of an m-dimensional current T is defined as (∂T )(ω) = T (dω) whenever ω ∈ D m−1 .

∂T ∈ Dm−1 ,

If either m = 0, or both T and ∂T are representable by integration, then we will call T locally normal. If T is locally normal and compactly supported, then T is called normal. The F-closure, in Dm , of the normal currents is the space of locally flat chains. Its subspace of currents with finite mass is the M-closure, in Dm , of the normal currents. To each L n -measurable m-vector field ξ such that ξ is locally integrable there corresponds the current L n ∧ ξ ∈ Dm (Rn ). If ξ is of class C1 , then this current is locally normal and the divergence of ξ is related to the boundary of the corresponding current by −∂ L n ∧ ξ = L n ∧ div ξ. ˆ Moreover, if Ω is an open set with C1 boundary, nˆ is its outer unit normal and d nˆ the dual of n, then ∂ L n ∧ (1Ω ξ ) = − L n

Ω

∧ div ξ + H n−1

∂Ω

∧ d n, ˆ ξ .

(5.14)

In the next subsection, we are going to find the analogue of the Green–Gauss formula (5.14) for the k-dimensional current associated to k-faces, restricted to D-cylinders. It will involve the distributional divergence of a less regular k-vector field (see below). 5.2.2. Divergence of the current of k-faces on model sets As a preliminary study, restrict again the attention to a D-cylinder as in Section 5.1.1, and keep the notation we had there. The k-faces, restricted to C k , define a k-vector field ξ = v1 ∧ · · · ∧ vk . In general, this vector field does not enjoy much regularity. Nevertheless, as a consequence of the study of Section 4, already exploited in Section 5.1 with a different formalism, one can find a representation of ∂(L n ∧ ξ ) like the one in a regular setting given by (5.14). This involves the density α of the push-forward with σ which was studied before, see (4.67) and (5.11). Lemma 5.9. Consider a function λ∈ L1 (C k ) such that it is continuously differentiable on each face and assume C k bounded. Then, the k-dimensional current (L n ∧ λξ ) is normal and the following formula holds ∂ L n ∧ λξ = −L n ∧ (div λξ )a.c. + H n−1

dC k

∧ d n, ˆ λξ ,

where dC k , nˆ are defined in (5.6), d nˆ is the differential 1-form at each point dual to the vector field n, ˆ and (div λξ )a.c. is defined here as (div λξ )a.c. :=

k (−1)i+1 (div λvi )a.c. v1 ∧ · · · ∧ vˆ i ∧ · · · ∧ vk i=1

3656


where the functions (div λvi )a.c. are given in (5.7): ∂t α(t = πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) (div λvi )a.c. (x) = λ(x) i + ∂vi λ(x) 1C k (x). k α(πe1 ,...,ek (x), 0, x − i=1 x · ei vi (x)) Proof. Actually, reducing to computations in coordinates, this follows from Corollary 5.4 in Section 5.1.1. One has to verify the equality of the two currents on a basis. For simplicity, consider first ω = φ dx2 ∧ · · · ∧ dxk . with φ ∈ C1 (Rn ). Then dω = ∂x1 φ dx1 ∧ · · · ∧ dxk +

∂xi φ dxi ∧ dx2 ∧ · · · ∧ dxk ,

i=k+1

dω, ξ = ∇φ · v1

n

ω, (div λξ )a.c. = (div λv1 )a.c. φ,

ω, d n, ˆ ξ = φ nˆ · e1

and the thesis reduces exactly to Lemma 5.2 and Remark 5.3:

∂ L n ∧ λξ (ω)

:=

dω, λξ dL n Ck

Lemma 5.2

=

−

Ck

=:

ω, d n, ˆ λξ dH n−1

ω, (div λξ )a.c. dL + n

dC k

−L n ∧ (div λξ )a.c. + H n−1

dC k

∧ d n, ˆ λξ .

(5.15)

Consider then $i ∧ · · · ∧ dxk . ω = φ dx1 ∧ · · · ∧ dx The equality analogous to (5.15) can be deduced again by Lemma 5.2, as above, observing that the following formulas hold: dω, ξ = (−1)i+1 ∇φ · vi ,

ω, (div λξ )a.c. = (−1)i+1 (div λvi )a.c. φ, ω, d n, ˆ ξ = (−1)i+1 φ nˆ · ei .

Let us show the equality more in general. By a direct computation, one can verify that v1 ∧ · · · ∧ vˆ i ∧ · · · ∧ vk =

k−1

h=0 k −∞. For more details, we also refer the reader to the survey [18] for a list of properties as well as applications of free entropies in the theory of von Neumann algebras. The aim of this note is to prove the easiest result in that direction i.e. under the assumption that the free Fisher Information Φ ∗ (X1 , . . . , Xn ) < ∞ (an assumption stronger than χ ∗ (X1 , . . . , Xn ) > −∞ by a logarithmic Sobolev inequality of [17]), we intend to prove that W ∗ (X1 , . . . , Xn ) doesn’t have property Γ (especially is not amenable) (cf. [16] for the corresponding result in the case of microstates free entropy). Let us note that this especially implies that for any X1 , . . . , Xn , in a W ∗ -probability space, and S1 , . . . , Sn free semicircular elements free with X1 , . . . , Xn , then W ∗ (X1 + tS1 , . . . , Xn + tSn ) doesn’t have property Γ (a result not known, to the best of our knowledge, at least when W ∗ (X1 , . . . , Xn ) is not known to satisfy Connes’ embedding conjecture into an ultrapower of the hyperfinite II 1 factor). We will also prove factoriality under finiteness of non-microstates entropy, especially proving the same kind of degenerate convexity as the one of microstates entropy, i.e. all non-extremal states have χ ∗ (X1 , . . . , Xn ) = −∞. More precisely, let us recall that Φ ∗ (X1 , . . . , Xn ) is defined (in [17]) thanks to Hilbert– Schmidt-valued derivations, the so-called partial free difference quotients δi := ∂Xi : CX1 ,...,Xˆ i ,...,Xn : CX1 , . . . , Xn → H S L2 W ∗ (X1 , . . . , Xn ) , δi (Xj ) := δij 1 ⊗ 1, 2 ∗ H S L W (X1 , . . . , Xn ) L2 W ∗ (X1 , . . . , Xn ), τ ⊗ L2 W ∗ (X1 , . . . , Xn ), τ . If Φ ∗ (X1 , . . . , Xn ) = ni=1 δi∗ 1 ⊗ 1 22 < ∞, these derivations are closable, thanks to a result of Voiculescu. And, having in mind of proving first factoriality, if an element, say Z, of the center of W ∗ (X1 , . . . , Xn ) were in the domain of δi , we would write 0 = δi ([Z, Xj ]) = [δi (Z), Xj ] for j = i thanks to Leibniz rule and center property. And thus we would obtain that δi (Z), seen as a Hilbert–Schmidt operator, thus a compact operator, commutes with a diffuse operator, and thus is zero. A free Poincaré inequality (due to Voiculescu [19] and recalled later) would imply our result, that is Z is a scalar times the unit of the von Neumann algebra. At that point, we have thus to remove the domain assumption assumed valid on the element Z in the center. In Section 1, we prove factoriality under a slightly more general assumption for Fisher information relative to a subalgebra B. We will then, in Section 2, using a variant of Free Poincaré inequality and new boundedness results of (unbounded) dual systems, show our main result according to which W ∗ (X1 , . . . , Xn ) does not have property Γ . Let us mention that a previous preprint version of this paper used deeply the notion of L2 -rigidity introduced in [12] to get the same result under a supplementary non-amenability assumption. Here, we thus get non-amenability as a byproduct. Moreover, Section 3 applies the same tools to prove factoriality

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Y. Dabrowski / Journal of Functional Analysis 258 (2010) 3662–3674

under finite non-microstates entropy. We also give a corresponding quantitative inequality in terms of one variant of non-microstates free entropy dimension. 1. Factoriality under finite Fisher information Let us fix some notations (close to those of [12]). We consider M a finite von Neumann algebra with normal faithful tracial state τ , and H an M–M-bimodule. D(δ) a weakly dense ∗-subalgebra of M. We suppose here that δ : D(δ) → H is a real closable derivation (real means δ(x), yδ(z) = δ(z∗ )y ∗ , δ(x ∗ )). = δ ∗ δ¯ the corresponding generator of a completely Dirichlet form, as proved in [14] (see this paper for the non-commutative definition of a Dirichlet form, here the Dirichlet form is E(x) = δ(x), δ(x), D(E) = D(1/2 ), completely means that ⊗ In is also the generator of a Dirichlet form on Mn (M)). Let us introduce a deformation of resolvent maps (a multiple of a so-called strongly continuous contraction resolvent, cf. e.g. [10] for the terminology) ηα = α(α + )−1 , which are unital, tracial (τ ◦ ηα = τ ), positive, completely positive maps, and moreover contractions on L2 (M, τ ) and normal contractions on M, such that

x − ηα (x) 2 x and x − ηα (x) 2 →α→∞ 0 (as recalled e.g. in Proposition 2.5 of [2]). We will also consider φt = e−t the semigroup of generator −. Let us recall two relations of the resolvent maps (see [10] for the first and [12] for the second, the integrals are understood as pointwise Riemann integral): ∞ ∀α > 0,

ηα = α

e−αt φt dt,

0

∀α > 0,

ζα := ηα1/2

=π

−1

∞ 0

t −1/2 ηα(1+t)/t dt. 1+t

¯ and Range(ηα1/2 ) = D(1/2 ) = D(δ) ¯ so that The point is that Range(ηα ) = D() ⊂ D(δ) ¯δ ◦ ζα is bounded (remark that this way to pre-compose with ηα1/2 to extend a map to the whole space is usual in classical Dirichlet form theory (especially in the relation with Malliavin calculus), in that way, for instance, the gradient operator of Malliavin calculus is extended to a distribution valued operator (after post-composition with another operator)). We now prove the first theorem of that note: Theorem 1. Let (M, τ ) a tracial W ∗ -probability space (i.e. M a von Neumann algebra with τ a faithful tracial normal state). Let (X1 , . . . , Xn ) an n-tuple (of self-adjoints, n 2) such that the microstates free Fisher information Φ ∗ (X1 , . . . , Xn ) < ∞, then W = W ∗ (X1 , . . . , Xn ) is a factor. Proof. Let δi = ∂Xi : CX1 ,...,Xˆ i ,...,Xn following the notation of Voiculescu for the non-commutative difference quotient. We see δi : CX1 , . . . , Xn → H S(L2 (W )) L2 (W, τ ) ⊗ L2 (W, τ ). First, thanks to a result of Voiculescu, Φ ∗ (X1 , . . . , Xn ) < ∞ implies that all the derivations δi are closable as unbounded operators L2 (W, τ ) → H S(L2 (W )) and they are even real closable derivations. But let us now fix i and consider Y ∈ CX1 , . . . , Xˆ i , . . . , Xn . By definition, we have δi Y = 0, so that if i = δi∗ δ¯i , we have especially i Y = 0 (and Y ∈ D(i )).


3665

Using a complete positivity argument (or an easy differential equation argument on the corresponding semigroup) one can easily show that ζα,i (ZY ) = ζα,i (Z)Y and ζα,i (Y Z) = Y ζα,i (Z). Thus, ζα,i ([Z, Y ]) = [ζα,i (Z), Y ], and if we note δ˜α,i = α −1/2 δi ◦ ζα (a bounded map as already noted), we have, using Leibniz rule and δ(Y ) = 0: δ˜α,i [Z, Y ] = δ˜α,i (Z), Y . Consequently, if Z is in the center of W ∗ (X1 , . . . , Xn ), we have proved [δ˜α,i (Z), Y ] = 0. But now, if Y = Xj (j = i), Y is diffuse (inasmuch as Φ ∗ (X1 , . . . , Xn ) < ∞ implies χ ∗ (X1 ) + · · · + χ ∗ (Xn ) χ ∗ (X1 , . . . , Xn ) > −∞, and if ξ ∈ L2 (W ) were an eigenvector of Xj with eigenvalue λ, the projector on ξ in B(L2 (W )) were not zero, implying the spectral projection 1Xj =λ to be not zero, and by faithfulness τ (1Xj =λ ) = 0 a contradiction, since χ ∗ (Xj ) > −∞ implies that the distribution of Xj has no point masses). But now, a Hilbert–Schmidt (thus compact) operator commuting with a diffuse one is zero (using the spectral theorem for compact operators, the diffuse one should have an eigenvector!). We have eventually proved δ˜α,i (Z) = 0 for all i (and all α > 0) as soon as Z is in the center of ∗ W (X1 , . . . , Xn ) and thus, by closability, knowing Z − ζα,i (Z) 2 →α→∞ 0, we obtain the fact that Z ∈ D(δ¯i ) and δ¯i (Z) = 0. Then, we conclude with the following lemma, due to Voiculescu (unpublished [19]). 2 Lemma 2 (Free Poincaré inequality). (See [19].) Consider δi the partial free difference quotient with respect to X1 , . . . , Xn , and Y a self-adjoint variable in the domain of all the operators δ¯i (as unbounded operators L2 (W ∗ (X1 , . . . , Xn )) → H S(L2 (W ∗ (X1 , . . . , Xn )))), then, there exists a positive constant C depending on the Xi but not on Y such that:

C

n j =1

δ¯j Y HS Y − τ (Y ) 2 .

We refer the reader to [19] for a proof, but we note that the key tool is the following remark, that for a polynomial Y = P (X1 , . . . , Xn ) ∈ CX1 , . . . , Xn , we verify immediately by linearity and monomial case that: n (δj P )(Xj ⊗ 1) − (1 ⊗ Xj )(δj P ) = P ⊗ 1 − 1 ⊗ P .

(1)

j =1

Remark 3. As Jesse Peterson pointed out to us, after reading an earlier version of this note, once we have shown δ¯1 (Z) = 0 (using only commutation with X2 , . . . , Xn ), we can conclude by writing down 0 = δ¯1 ([Z, X1 ]) = [δ¯1 (Z), X1 ] + [Z, 1 ⊗ 1] = [Z, 1 ⊗ 1] and conclude taking the

. 2 norm. (We have used our original proof inasmuch as a variant of free Poincaré inequality will be essential in the next part. But somehow, the following result is the only one not provable under weakened assumptions in what follows. The counterpart of the powerfulness of free Poincaré like technique being its non-applicability in the case Φ ∗ (X : B) < ∞, up to now.) We have thus also proved the following result:

3666


Theorem 4. Let X a self-adjoint element in M a tracial W ∗ -probability space, and B a subalgebra of M, algebraically free with X and containing a diffuse element. Suppose moreover that the free Fisher information of X relative to B: Φ ∗ (X : B) < ∞, then W ∗ (X, B) is a factor. Let us end this part with a corollary. Consider, e.g. as in [7], the full (universal) free-product C ∗ -algebra C([−R, R])N and note T the space of tracial states on this C ∗ -algebra. It is (elementarily known to be) a compact convex set for the weak-∗ topology. It is moreover known by the reduction theory for von Neumann algebras that this is a Choquet simplex. It is known (see the beginning of the proof of Theorem 3.1.18 of [13] using mainly Proposition 3.1.10 and the discussion before the theorem) that factorial states (i.e. states for which bicommutants in the GNS construction give factors) are exactly extreme points of this convex set. We will show that T is a Poulsen simplex (see [9]), i.e. a metrizable Choquet simplex in which the extreme points form a dense set. Since, as a weak-∗ compact of the dual of a separable space, T is metrizable, we have merely to prove the last statement about density of the set of extreme points. Let us prove this in the following: Corollary 5. The set of tracial states T on C = C([−R, R])N is a Poulsen simplex. Proof. To conclude the proof, consider thus a tracial state τ on C, consider Xi , the function f0 (t) = t in the i-th copy of C([−R, R]), the usual self-adjoint generators (as a C ∗ -algebra) of C. Let W the weak closure of πτ (C), the GNS construction associated to τ , we always note τ the associated (faithful normal tracial) state on W . We can consider the von Neumann algebra M generated by W and a free family of semicircular elements M = W ∗ (W, {Si }), and get another faithful tracial state on M (the last one noted τ , cf. [5] for faithfulness). Consider ∗ i +tSi ) Yi,t = R(X R+2t . By Corollary 3.9 in [17], Φ (Y1,t , . . . , YN,t ) < ∞ and thus by our Theorem 1, ∗ W (Y1,t , . . . , YN,t ) ⊂ M is a factor. But since Yi,t R, we have a ∗-homomorphism C → M sending Xi to Yi,t (e.g. Proposition 2.1 in [7]). This defines by composition with the state on M, a state τt on C. Since the state considered on M is faithful, the kernel of the ∗-homomorphism is nothing but the ideal of elements with τt (Z ∗ Z) = 0 by which we quotient C in the GNS construction for τt on C, we thus get a ∗-isomorphism, from this quotient on its image which preserves the trace, and thus, L2 (W ∗ (Y1,t , . . . , YN,t ), τ ) is isomorphic to L2 (C, τt ) (by the induced map). And a step further we get the ∗-isomorphism between W ∗ (Y1,t , . . . , YN,t ) and πτt (C). Thus, τt is a factorial, thus an extremal tracial state in T . Now, to get weak-∗ convergence of τ1/n to τ in T , and thus the claimed density, we have merely to consider convergence on monomials on which a usual trick shows the concluding inequality:

τ (Xi . . . Xi ) − τt (Xi . . . Xi ) = τ (Xi . . . Xi ) − τ (Yi ,t . . . Yi ,t )

p p p p 1 1 1 1 R(Xi + tSi ) − X R p−1 p sup i R + 2t i R p−1 p

4Rt . R + 2t

2

2. Non-Γ In the preceding part, we used the semigroup and resolvent maps associated to a derivation δi = ∂Xi : CX1 ,...,Xˆ i ,...,Xn . The drawback is that, if we doesn’t have exactly a commutator equal to


3667

zero, as this is the case when we want to prove non-Γ , we cannot move the estimate on the commutator (giving an estimate on the Hilbert–Schmidt operator), to an estimate on Z − τ (Z) using something like free Poincaré inequality, inasmuch as we have not the same resolvent maps for different derivations δi . We have thus searched to move (somehow) the preceding reasoning in case we consider δ := (δ1 , . . . , δn ), the resolvent map associated to it ηα (Z), and then δi ◦ ηα (Z). This was our approach in a previous version of this paper where we assumed non-amenability and used then an L2 -rigidity technique to conclude. However, working a little bit more from the following variant of free Poincaré inequality will be much more efficient, enabling us to prove non-Γ without any other assumption, and thus proving non-amenability instead of assuming it. 2.1. Two preliminaries First in order to get our inequality, we will use the following general result about derivations in von Neumann algebras, which can be thought of as a “Kaplansky’s density theorem for derivations” (the proof also confirms this), which is of independent interest and really likely known to specialists but for which we have not found any reference. Proposition 6. Let δ a symmetric derivation defined on a weakly dense ∗-algebra D(δ) of the tracial W ∗ -probability space (M, τ ), closable as an operator D(δ) ⊂ L2 (M, τ ) → H, H an involutive M–M Hilbert W ∗ -bimodule (with isometric involution, as usual we assume σ -weak continuity of both actions). Then the following properties are equivalent: ¯ ∩ M is a ∗-algebra on which δ| ¯ D(δ)∩M (i) D(δ) is a (symmetric) derivation. ¯ ¯ (ii) For any Z ∈ D(δ) ∩ M, there exists a sequence Zn ∈ D(δ) with Zn Z , Zn − Z 2 , ¯

δ(Zn ) − δ(Z)

2 → 0. Proof. The fact that (ii) implies (i) is clear since taking Zn , Yn for Z and Y , as in (ii), Zn Yn − ZY 2 → 0, and for any ξ ∈ H, we get successively ξ(Yn∗ − Y ∗ ) H → 0, by coincidence of L2 and σ -∗-strong topologies on bounded sets in M and σ -weak (thus σ -∗-strong) continuity of the action, and thus δ(Zn ), ξ Yn∗ → δ(Z), ξ Y ∗ , which gives at the end weak convergence of ¯ ) + δ(Z)Y ¯ ¯ we get ZY ∈ D(δ) ¯ ∩M , and by (weak) closability of the graph of δ, δ(Zn Yn ) to Z δ(Y with the derivation property. The proof of the converse follows verbatim the proof of Kaplansky’s density theorem. We may first assume that δ : D(δ) → L2 (M) is closed as a derivation M → L2 (M), since it is closable and then obtaining Zn in this enlarged D(δ) is harmless. We can also assume Z 1. Consider X = (1 + (1 − ZZ ∗ )1/2 )−1 Z ∈ M, then X ∈ D(δ) by closability of any closed derivation on a C ∗ -algebra by C 1 -functional calculus. Look at f (x) = 2(1 + xx ∗ )−1 x = 2x(1 + x ∗ x)−1 so that ¯ n − X) 2 → 0. Then consider Zn = Z = f (X). Take Xn ∈ D(δ) converging to X in L2 with δ(X 2 f (Xn ) so that Zn 1 and Zn converge to Z in L (even if this is not a consequence of ∗-strong continuity of f since we don’t know whether Xn converge to X ∗-strongly since it is not bounded as a sequence in M, the proof is however standard, like in the proof of Kaplansky’s density ¯ D(δ)∩M theorem, see bellow for an example for the derivative of f ). Since, by hypothesis, δ and δ| ¯ are derivations, closed seen as derivation M → H, we get δ(Zn ) = 2δ(Xn )(1 + Xn∗ Xn )−1 − ¯ using appropriate series 2Xn (1 + Xn∗ Xn )−1 δ(Xn∗ Xn )(1 + Xn∗ Xn )−1 and the analog for δ(Z), expansions. Now the boundedness as sequences in M of Xn (1 + Xn∗ Xn )−1 Xn∗ , (1 + Xn∗ Xn )−1 ¯ and Xn (1 + Xn∗ Xn )−1 shows that it suffices to show the convergence of 2δ(X)(1 + Xn∗ Xn )−1 −

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¯ ∗ )Xn + Xn∗ δ(X))(1 ¯ 2Xn (1 + Xn∗ Xn )−1 (δ(X + Xn∗ Xn )−1 . Likewise, by coincidence of L2 and σ -∗-strong topologies on bounded sets in M and σ -∗-strong continuity of the action, it suffices to show convergence in L2 (M) of Xn (1 + Xn∗ Xn )−1 Xn∗ , (1 + Xn∗ Xn )−1 and Xn (1 + Xn∗ Xn )−1 . Let us for instance prove the first one, let us write: −1 −1 Xn 1 + Xn∗ Xn Xn∗ − X 1 + X ∗ X X ∗ −1 ∗ −1 ∗ Xn − X ∗ + Xn 1 + Xn∗ Xn X − Xn∗ X = Xn 1 + Xn∗ Xn −1 −1 + Xn∗ (X − Xn ) 1 + X ∗ X X ∗ + (Xn − X) 1 + X ∗ X X ∗ . Since for each term, both sides of X − Xn or its adjoint are bounded by functional calculus, this concludes. 2 Remark 7. Let us note that in order to apply the previous proposition to the free difference quotient in case of finite Fisher information, we prove (i) using Proposition 3.4 in [4] to get ¯ ∩ M is an algebra (knowing that D(δ) ¯ is the domain of a Dirichlet form, as already recalled, D(δ) thanks to [14]), and then, for instance use the formula for δ ∗ given by Corollary 4.3 in [17] to show δ¯ is closable as an operator valued in L1 (M ⊗ M), and we prove there the derivation property, deducing it for the derivation valued in L2 as a consequence. Second, we recall for the reader convenience some results about bounded and unbounded dual systems in the sense of Voiculescu and Shlyakhtenko respectively. Even if we will not use their results explicitly, this will enable to express some assumptions and results in terms of these standard objects. Let us recall the following result of [15] (deduced from Theorem 1 and its proof), δi the i-th partial difference quotient as earlier. Proposition 8. (See [15].) δi∗ 1 ⊗ 1 exists (in L2 (W ), W = W ∗ (X1 , . . . , Xn )) if and only if there exists a closable unbounded operator Yi : L2 (W ) → L2 (W ) with CX1 , . . . , Xn ⊂ D(Yi ), Yi 1 = 0, 1 ∈ D(Yi∗ ) such that [Yi , Xj ] = δi (Xj ). Moreover, necessarily such a Yi = 1 ⊗ τ ◦ δi (or is an extension of it beyond CX1 , . . . , Xn ). Then Corollary 1 of the same paper noticed that Y˜i = 12 (Yi − Yi∗ ) is an anti-symmetric closable dual system. Moreover the proof of Theorem 1 also shows that Yi∗ X = Xδi∗ 1 ⊗ 1 − Yi X (also a consequence of Corollary 4.3 in [17] in the free difference quotient case we are interested in here), so that Y˜i (X) = Yi (X) − 12 Xδi∗ 1 ⊗ 1 i.e. Y˜i 1 = − 12 δi∗ 1 ⊗ 1. Moreover, it is easily seen that such a Y˜i gives in inverting the above relation to get a Yi similar to the one in the previous proposition, we will thus be later interested in bounded dual systems in the sense of Voiculescu verifying this relation for the specific relation they have with the canonical dual system of the previous proposition (for which we will get latter e.g. nice boundedness properties). 2.2. A mixed Poincaré-non-Γ (in)equality Our main tool will be a lemma based on the same argument as free Poincaré inequality. After proving it, we develop several consequences under (more or less) stronger assumptions for further use.


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Lemma 9. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple (of selfadjoints, n 2 in order to have a non-trivial result) such that the microstates free Fisher ¯ (δ the free difference quoinformation Φ ∗ (X1 , . . . , Xn ) < ∞. Let Z ∈ W ∗ (X1 , . . . , Xn ) ∩ D(δ) tient), then we have the following equality: n 2 2(n − 1) Z − τ (Z) 2 = [Z, Xi ], Z, (Xi ) i=1

+ 2 (τ ⊗ 1 − 1 ⊗ τ ) δ¯i (Z) , [Z, Xi ] .

Proof. It suffices to show the result for Z ∈ CX1 , . . . , Xn (using Proposition 6). Inasmuch as δi is a derivation, we have δi [Z, Xi ] = [δi (Z), Xi ] + [Z, 1 ⊗ 1] and we have already noticed that [Z, 1 ⊗ 1] 2HS = 2 Z − τ (Z) 22 . Everything will be based on the equality on which is based free Poincaré inequality. Let us compute [Z, 1 ⊗ 1] 2HS = δi [Z, Xi ] − [δi (Z), Xi ], [Z, 1 ⊗ 1]: δi (Z), Xi , [Z, 1 ⊗ 1] = δi (Z), [Z, 1 ⊗ 1], Xi = δi (Z), Z, [1 ⊗ 1, Xi ] − δi (Z), 1 ⊗ 1, [Z, Xi ] . At that point, we notice that: Z, [1 ⊗ 1, Xi ] = [Z, 1 ⊗ Xi − Xi ⊗ 1] = Z ⊗ Xi − ZXi ⊗ 1 − 1 ⊗ Xi Z + Xi ⊗ Z = (1 ⊗ Xi )[Z, 1 ⊗ 1] − [Z, 1 ⊗ 1](Xi ⊗ 1). But now, we have in fact written an “inner” commutant of Xi and [Z, 1 ⊗ 1] (i.e. a commutant with the action of the von Neumann algebra on M ⊗ M on the side of the tensor product not on the outer side, remark that the preceding equation is just commutation of the two actions after writing [1 ⊗ 1, Xi ] in terms of an inner commutant). We will merely now use that the scalar product of Hilbert Schmidt operators is compatible with this inner commutant (which is nothing but an extension of traciality of τ ⊗ τ on M ⊗ M):

n δi (Z), Z, [1 ⊗ 1, Xi ] = (1 ⊗ Xi )δi (Z) − δi (Z)(Xi ⊗ 1), [Z, 1 ⊗ 1]

n i=1

i=1

= (1 ⊗ Z − Z ⊗ 1), [Z, 1 ⊗ 1] 2 = − [Z, 1 ⊗ 1] .

We have used Eq. (1) on which is based the proof of free Poincaré inequality. Thus, we have obtained: n n 2 δi (Z), Xi , [Z, 1 ⊗ 1] = − [Z, 1 ⊗ 1] − δi (Z), 1 ⊗ 1, [Z, Xi ] . i=1

i=1

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We have now to compute δi [Z, Xi ] , [Z, 1 ⊗ 1] = Z ∗ , δi [Z, Xi ] , 1 ⊗ 1 = δi Z ∗ , [Z, Xi ] , 1 ⊗ 1 − δi Z ∗ , [Z, Xi ] , 1 ⊗ 1 = [Z, Xi ], Z, (Xi ) + δi Z ∗ , 1 ⊗ 1, Z ∗ , Xi . We can now conclude using that 1 ⊗ τ (δi (Z ∗ )) = τ ⊗ 1(δi (Z))∗ : n 2 2 n [Z, 1 ⊗ 1] HS = [Z, 1 ⊗ 1] + [Z, Xi ], Z, (Xi ) i=1 n +2 δi (Z), 1 ⊗ 1, [Z, Xi ] .

2

i=1

For our purpose, the following lemma is only an intermediary step to the next lemma, but, as the remark after it shows, it can have an independent interest. Lemma 10. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple of n 2 self-adjoints such that the microstates free Fisher information Φ ∗ (X1 , . . . , Xn ) < ∞. Let Z ∈ ¯ then the following inequality holds: W ∗ (X1 , . . . , Xn ) ∩ D(δ),

(1 ⊗ τ ) δ¯i (Z) − Z(Xi ) 2 Z(Xi ) 2 + δ¯i Z ∗ Z , 1 ⊗ (Xi ) . 2

2

Thus, if we assume moreover we have second order conjugate variables J2,j = J2 (Xj : CX1 , . . . , Xˆ j , . . . , Xn ) (in L1 (M, τ ), as defined in [17, Definition 3.1]). Then the following inequality holds:

(1 ⊗ τ ) δ¯i (Z) 2 Z(Xi ) + Z ∗ Z, J2,i 1/2 . 2 2 As a consequence, (1 ⊗ τ ) ◦ δ¯i extends as a bounded map M → L2 (M, τ ). Remark 11. If we assume moreover we have bounded first and second order conjugate variables (i.e. (Xi ), J2,i ∈ M or more generally bounded conjugate variable and dual system Y˜i in the sense of Voiculescu), then the previous lemma shows that (1 ⊗ τ ) ◦ δ¯i extends as a bounded map on L2 (M) and moreover the inequality above implies that X1 , . . . , Xn is a non-Γ ¯ in the sense of [11]. As a consequence of Corollary 3.3 in [11] this shows that set (for D(δ)) ∗ W (X1 , . . . , Xn ) doesn’t have property (T). A study under less restrictive assumptions will need new investigations, but we can already note (using well-known results of [17]) that this implies that for any X1 , . . . , Xn , if S1 , . . . , Sn is a free semicircular system free with X1 , . . . , Xn , then W ∗ (X1 + S1 , . . . , Xn + Sn ) doesn’t have property (T) (for any > 0). This result was proved in [8] assuming moreover W ∗ (X1 , . . . , Xn ) embeddable in R ω . Proof. The only non-trivial statement is the first one (using the previous lemma for proving consequences). Moreover we can assume Z ∈ CX1 , . . . , Xn as usual (take the limit in the fifth


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¯ ∩ M). Let us compute (using line below using Proposition 6 and then compute with Z ∈ D(δ) the formula for δi∗ , Corollary 4.3 in [17], and coassociativity in the third line): (1 ⊗ τ ) δ¯i (Z) 2 = (1 ⊗ τ ) δ¯i (Z) ⊗ 1, δ¯i (Z) 2 = (1 ⊗ τ ) δ¯i (Z) (Xi ), Z − (1 ⊗ τ )δ¯i (1 ⊗ τ ) δ¯i (Z) , Z = (1 ⊗ τ ) δ¯i (Z) (Xi ), Z − (1 ⊗ τ ⊗ τ ) 1 ⊗ δ¯i ◦ δ¯i (Z) , Z = (1 ⊗ τ ) δ¯i (Z) (Xi ) − (1 ⊗ τ ) δ¯i (Z)(Xi ) , Z = δ¯i (Z), Z (Xi ), 1 ⊗ 1 = δ¯i (Z), Z(Xi ) ⊗ 1 − δ¯i Z ∗ Z − δ¯i Z ∗ Z, 1 ⊗ (Xi ) = δ¯i (Z), Z(Xi ) ⊗ 1 + δ¯i Z ∗ , 1 ⊗ (Xi )Z ∗ − δ¯i Z ∗ Z , 1 ⊗ (Xi ) . Now note we can use (1 ⊗ τ )(δ¯i (Z))∗ = (τ ⊗ 1)(δ¯i (Z ∗ )) (using δ¯i is a real derivation), to conclude: (1 ⊗ τ ) δ¯i (Z) − Z(Xi ) 2 = Z(Xi ) 2 − δ¯i Z ∗ Z , 1 ⊗ (Xi ) . 2

2

2

Lemma 12. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple of n 2 self-adjoints such that the microstates free Fisher information Φ ∗ (X1 , . . . , Xn ) < ∞. Let Z ∈ ¯ a self-adjoint or a unitary, then we have the following inequality: W ∗ (X1 , . . . , Xn ) ∩ D(δ) (1 ⊗ τ ) δ¯i (Z) − Z(Xi ) (Xi ) Z . 2 2 As a consequence, (1 ⊗ τ ) ◦ δ¯i extends as a bounded map M → L2 (M, τ ) and: n 2 1 (n − 1) Z − τ (Z) 2 − [Z, Xi ], Z, (Xi ) +2 [Z, Xi ] 2 (Xi ) 2 Z . 2 i=1

Proof. Take Z of norm less than 1 ( Z < 1). If it is self-adjoint, we can write Z as a half ¯ using stability by C 1 functional calculus (e.g. sum of two unitaries in W ∗ (X1 , . . . , Xn ) ∩ D(δ) Lemma 7.2 in [2]), we have only to prove the inequality for any unitary U . This follows at once from the previous lemma. The second statement is a direct consequence. 2 2.3. The main result Now Lemma 12 contains immediately the non-Γ result we wanted: Theorem 13. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple (of self-adjoints, n 2) such that the microstates free Fisher information Φ ∗ (X1 , . . . , Xn ) < ∞, then W ∗ (X1 , . . . , Xn ) doesn’t have property Γ , i.e. all central sequences Zm (i.e. bounded in W ∗ (X1 , . . . , Xn ) and such that ∀Y ∈ W ∗ (X1 , . . . , Xn ) [Zm , Y ] 2 → 0) are trivial: Zm − τ (Zm ) 2 → 0. As a consequence, W ∗ (X1 , . . . , Xn ) is not amenable.

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3. Factoriality under finite non-microstates entropy We will now prove factoriality under the weaker assumption χ ∗ (X1 , . . . , Xn ) > −∞ as another consequence of Lemma 12. In this part, we thus let X1 , . . . , Xn , n 2, √ self-adjoints, S1 , . . . , Sn , a free semicircular system, free with X1 , . . . , Xn . Let Yjt = Xj + tSj and Et the (trace preserving) conditional expectation onto W ∗ (Y1t , . . . , Ynt ) (seen as a sub-von Neumann algebra of W ∗ ({Xi , Si })). Then recall that the non-microstates entropy is defined by the following integral:

1 χ (X1 , . . . , Xn ) = 2 ∗

∞ 0

√ √ n n ∗ − Φ (X1 + tS1 , . . . , Xn + tSn ) dt + log 2πe. 1+t 2

√ √ It is readily seen that if lim inft→0 tΦ ∗ (X1 + tS1 , . . . , Xn + tSn ) = 0 we have necessarily χ ∗ (X1 , . . . , Xn ) = −∞, we will thus use the assumption in that way. More generally, a variant √ of (X , . . . , X ) = n − lim inf ∗ (X + tS , free entropy dimension was defined in [3] by δ tΦ 1 n t→0 1 1 √ . . . , Xn + tSn ), we will thus express our result in function of this entropy dimension. We can now prove our claimed result: Theorem 14. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple (of selfadjoints, n 2) then the following inequality holds for any central self-adjoint Z: Z − τ (Z) 2 2 n − δ (X1 , . . . , Xn ) Z 2 . 2 n−1

√ √ As a consequence, if n − δ ∗ (X1 , . . . , Xn ) := lim inft→0 tΦ ∗ (X1 + tS1 , . . . , Xn + tSn ) = 0, then W ∗ (X1 , . . . , Xn ) is a factor. ∗ As another example, if δ ∗ (X1 , . . . , Xn ) > n+1 2 , W (X1 , . . . , Xn ) has no central projection of trace one half, especially, doesn’t have diffuse center. Proof. Let Z in the center of W ∗ (X1 , . . . , Xn ), with Z 1 and apply Lemma 12 to Et (Z) ∈ W ∗ (Y1t , . . . , Ynt ) to get: 2 (n − 1) Et (Z) − τ Et (Z) 2

n

−

1 Et (Z), Yit , Et (Z), Yit +2 Et (Z), Yit 2 Yit 2 Z

2

−

1 1 1 Et Z, Yit , Et (Z), √ Et (Si ) + 2 Et Z, Yit 2 (S ) E √ t i Z

2 t t 2

−

1 Et [Z, Si ] , Et (Z), Et (Si ) +2 Et [Z, Si ] 2 Et (Si ) 2 Z , 2

i=1

=

n i=1

=

n i=1


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2 (n − 1) Et (Z) − τ Et (Z) 2

n

−

i=1

1 Et Z − Et (Z), Si , Et (Z), Et (Si ) 2

1 Et (Z), Et (Si ) , Et (Z), Et (Si ) 2 + 2 Et Z − Et (Z), Si 2 + Et (Z), Et (Si ) 2 Et (Si ) 2 Z

−

n i=1

2 1 12 Z − Et (Z) 2 Et (Si ) 2 Z − Et (Z), Et (Si ) 2 2

2 2 + Et (Z), Et (Si ) 2 + Et (Si ) 2 Z

n 1 Et (Z), Et (Si ) 2 + Et (Si ) Z 2 12n Z − Et (Z) 2 Z + 2 2 2 i=1

n Et (Si ) 2 . 12n Z − Et (Z) 2 Z + 2 Z 2 2 i=1

We used at the second line the result of [17] about the conjugate variable in the algebra generated by Yit : (Yit ) = √1t Et (Si ). We also used conditional expectation property and then at line 3 that Z commutes with Xi . In the fourth line we used Z = Z − Et (Z) + Et (Z) and then we only compute using Si = 2, Si 2 = 1 and arithmetico geometric inequality. At the end, we thus get using the definition of free Fisher information and the result of [17] above: 2 (n − 1) Et (Z) − τ Et (Z) 2 12n Z − Et (Z) 2 Z

√ √ + 2 Z 2 tΦ ∗ (X1 + tS1 , . . . , Xn + tSn ). It is thus sufficient to notice that Et (Z) − Z 2 goes to 0 with t to get the inequality stated by taking a lim inf. Et (Z) − Z 2 → 0 follows from Kaplansky’s density theorem, and from the remark that for P a non-commutative polynomial Et (P (X1 , . . . , Xn )) − P (X1 , . . . , Xn ) 2

P (Y1t , . . . , Ynt ) − P (X1 , . . . , Xn ) 2 . The consequences are trivial: for instance for the second, apply the inequality to Z = 1 − 2P the corresponding central self-adjoint unitary of trace 0 if P a central projection of trace 1/2. 2 Acknowledgments The author would like to thank Professor Dan Voiculescu for allowing him to use the argument of his free Poincaré inequality in another context, Professor Dimitri Shlyakhtenko for showing him references [14,2,12], and Professor Jesse Peterson for pointing out to his attention his result Corollary 3.3 in [11]. The author would also like to thank D. Shlyakhtenko, J. Peterson and P. Biane for useful discussions and comments.

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References [1] P. Biane, A. Guionnet, M. Capitaine, Large deviation bounds for matrix brownian motion, Invent. Math. 152 (2003) 433–459. [2] F. Cipriani, J.-L. Sauvageot, Derivations as square roots of Dirichlet forms, J. Funct. Anal. 201 (2003) 78–120. [3] A. Connes, D. Shlyakhtenko, L2 -homology for von Neumann algebras, J. Reine Angew. Math. 586 (2005) 125–168. [4] E. Davis, J.M. Lindsay, Non-commutative symmetric Markov semigroups, Math. Z. 210 (1992) 379–411. [5] K. Dykema, D. Voiculescu, A. Nica, Free Random Variables, CRM Monogr. Ser., AMS, 1992. [6] L. Ge, Applications of free entropy to finite von Neumann algebras, II, Ann. of Math. 147 (1998) 143–157. [7] F. Hiaï, Free analog of pressure and its Legendre transform, Comm. Math. Phys. 255 (1) (2005) 229–252. [8] K. Jung, D. Shlyakhtenko, All generating sets of all property T von Neumann algebras have free entropy dimension 1, preprint, arXiv:math.OA/0603669, 2006. [9] J. Lindenstrauss, G. Olsen, Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble) 28 (1) (1978) 91–114. [10] Z. Ma, M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Universitext, Springer, Berlin, 1992. [11] J. Peterson, A 1-cohomology characterization of property (T) in von Neumann algebras, preprint, arXiv:math.OA/ 0409527, 2004. [12] J. Peterson, L2 -rigidity in von Neumann algebras, preprint, arXiv:math.OA/0605033, 2006. [13] S. Sakai, C ∗ -Algebras and W ∗ -Algebras, Springer, 1971. [14] J.-L. Sauvageot, Strong Feller semigroups on C ∗ -algebras, J. Operator Theory 42 (1999) 83–102–120. [15] D. Shlyakhtenko, Remarks on free entropy dimension, arXiv:math.OA/0504062, 2005. [16] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, III: The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1) (1996) 172–199. [17] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, V: Non commutative Hilbert Transforms, Invent. Math. 132 (1998) 189–227. [18] D. Voiculescu, Free entropy, Bull. Lond. Math. Soc. 34 (3) (2002) 257–278. [19] D. Voiculescu, A free Poincaré inequality.


Functions of operators under perturbations of class Sp ✩ A.B. Aleksandrov a , V.V. Peller b,∗ a St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-Petersburg, Russia b Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Received 29 August 2009; accepted 16 February 2010 Available online 26 February 2010 Communicated by N. Kalton

Abstract This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα (R) and 1 < p < ∞, the operator f (A) − f (B) belongs to S p/α , whenever A and B are self-adjoint operators with A − B ∈ S p . We also obtain sharp estimates for the Schatten–von Neumann norms f (A) − f (B)S p/α in terms of A − BS p and establish similar results for other operator ideals. We also estimate Schatten– m−j m f (A + j K). We prove that analogous von Neumann norms of higher order differences m j j =0 (−1) results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f (A) − f (B) to belong to S q under the assumption that A − B ∈ S p . We also obtain Schatten–von Neumann estimates for quasicommutators f (A)R − Rf (B), and introduce a spectral shift function and find a trace formula for operators of the form f (A − K) − 2f (A) + f (A + K). © 2010 Elsevier Inc. All rights reserved. Keywords: Operator ideals; Schatten–von Neumann classes; Self-adjoint operators; Unitary operators; Contractions; Perturbations; Functions of operators

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3676 Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3679

✩

The first author is partially supported by RFBR grant 08-01-00358-a and by Russian Federation presidential grant NSh-2409.2008.1; the second author is partially supported by NSF grant DMS 0700995 and by ARC grant. * Corresponding author. E-mail address: [email protected] (V.V. Peller). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.011

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3. Ideals of operators on Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Multiple operator integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Self-adjoint operators. Sufficient conditions . . . . . . . . . . . . . . . . . . . . . 6. Unitary operators. Sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . . 7. The case of contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Finite rank perturbations and necessary conditions. Unitary operators . . . . 9. Finite rank perturbations and necessary conditions. Self-adjoint operators . 10. Spectral shift function for second order differences . . . . . . . . . . . . . . . . 11. Commutators and quasicommutators . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3681 3687 3691 3698 3702 3704 3711 3718 3720 3723

1. Introduction This paper is a continuation of our paper [2]. In [2] we obtained sharp estimates for the norms of f (A) − f (B) in terms of the norm of A − B for various classes of functions f . Here A and B are self-adjoint operators on Hilbert space and f is a function on the real line R. We also obtained in [2] sharp estimates for the norms of higher order differences m m def K f (A) = (−1)m−j j =0

m f (A + j K), j

(1.1)

where A and K are self-adjoint operators. Similar results were obtained in [2] for functions of unitary operators and for functions of contractions. In this paper we are going to obtain sharp estimates for the Schatten–von Neumann norms of first order differences f (A) − f (B) and higher order differences (m K f )(A) for functions f that belong to a Hölder–Zygmund class Λα (R), 0 < α < ∞, (see Section 2 for the definition of these spaces). In particular we study the question, under which conditions on f the operator f (A) − f (B) (or (m K f )(A)) belongs to the Schatten–von Neumann class S q , whenever A − B (or K) belongs to S p . We also obtain related results for more general ideals of operators on Hilbert space (see Section 3 for the introduction to operator ideals on Hilbert space). In connection with the Lifshits–Krein trace formula, M.G. Krein asked in [16] the question whether f (A) − f (B) ∈ S 1 , whenever f is a Lipschitz function (i.e., |f (x) − f (y)| const |x − y|, x, y ∈ R) and A − B ∈ S 1 . Functions f satisfying this property are called trace class perturbations preserving. Farforovskaya constructed in [13] an example that shows that the answer to the Krein question is negative. Later in [27] and [29] necessary conditions and sufficient conditions for f to be trace class perturbations preserving were found. It was shown in [27] and [29] that if f belongs to the Besov 1 (R) (see Section 2), then f is trace class perturbations preserving. On the other hand, space B∞1 it was shown in [27] that if f is trace class perturbations preserving, then it belongs to the Besov space B11 (R) locally. This necessary condition also proves that a Lipschitz function does not have to be trace class perturbations preserving. Moreover, in [27] and [29] a stronger necessary


3677

condition was also found. Note that a function is trace class perturbations preserving if and only if it is operator Lipschitz (see [27] and [18]). We also mention here the paper [28], in which analogs of the above results were obtained for perturbations of class S p with p ∈ (0, 1). On the other hand, Birman and Solomyak in [9] proved that a Lipschitz function f must preserve Hilbert–Schmidt class perturbations: f (A) − f (B) ∈ S 2 , whenever A − B ∈ S 2 and f (A) − f (B)

S2

sup x=y

|f (x) − f (y)| A − BS 2 . |x − y|

To prove that result, Birman and Solomyak developed in [7,8], and [9] their beautiful theory of double operator integrals and established a formula for f (A) − f (B) in terms of double operator integrals (see Section 4). Note also that the paper [18] studies functions that preserve perturbations belonging to operator ideals. We mention here two recent results. In [22] it was proved that if f is a Lipschitz function and rank(A − B) < ∞, then f (A) − f (B) belongs the weak space S 1,∞ (see Section 3 for the definition). It was also shown in [22] that if A − B ∈ S 1 , then f (A) − f (B) belongs to the ideal S Ω , i.e., n sj f (A) − f (B) const log(2 + n) j =0

(here sj is the j th singular value). This allowed the authors of [22] to deduce that for p 1 and ε > 0, the operator f (A) − f (B) belongs to S p+ε , whenever f is a Lipschitz function and A − B ∈ Sp . The epsilon was removed later in [34] in the case 1 < p < ∞. It was shown in [34] that for p ∈ (1, ∞), the operator f (A) − f (B) belongs to S p , whenever A − B ∈ S p and f is a Lipschitz function. Note that similar results also hold for functions on the unit circle T and unitary operators. It was shown in [6] that if A and B are positive self-adjoint operators and I is a normed ideal of operators on Hilbert space with majorization property, then for α ∈ (0, 1), the following inequality holds: α A − B α |A − B|α . I I In this paper we study the problem under which conditions on a function f and a (quasi)normed ideal I of operators on Hilbert space the following inequality holds: f (A) − f (B) const|A − B|α . I I In Section 5 of this paper among other results we show that if f belongs to the Hölder class Λα , 0 < α < 1, and 1 < p < ∞, then f (A) − f (B) ∈ S p/α and f (A) − f (B)

S p/α

constf Λα (R) A − BαS p .

On the other hand, this is not true for p = 1 (a counter-example is given in Section 9). Nevertheless, for p = 1, under the assumptions that f ∈ Λα (R) and A − B ∈ S 1 , we

3678


prove that f (A) − f (B) belongs to the weak space S 1/α,∞ . To make the conclusion that f (B) − f (A) ∈ S 1/α under the assumption that A − B ∈ S 1 , we need the stronger condition: α . We also obtain similar results for other ideals of operators f belongs to the Besov space B∞1 on Hilbert space. In particular, we show that for every p ∈ (1, ∞) and every l 0, the following inequality holds l l 1/α p p p/α sj f (A) − f (B) sj (A − B) , constf Λα (R) j =0

j =0

where the constant does not depend on l. We also establish in Section 5 similar results for higher order differences (m K f )(A) and functions f ∈ Λα (R) with α ∈ [m − 1, m). In Section 6 we obtain analogs of the result of Section 5 for functions on T and unitary operators, while in Section 7 we establish similar results for functions analytic in the unit disk and contractions. In Section 8 we obtain refinements of some results of Section 6 in the case of finite rank perturbations of unitary operators. We also give some necessary conditions on a function f for f (U ) − f (V ) to belong to S q , whenever U − V ∈ S p . Analogs of the results of Section 8 for self-adjoint operators are given in Section 9. In Section 10 we consider the problem of evaluating the trace of f (A − K) − 2f (A) + 2 (R). We f (A + K) under the assumptions that K ∈ S 2 and f belongs to the Besov class B∞1 introduce a spectral shift function ς associated with the pair (A, K) and establish the following trace formula:

trace f (A − K) − 2f (A) + f (A + K) = f (x)ς(x) dx. R

We also show that similar results hold in the case of unitary operators. The final Section 11 is devoted to estimates of commutators and quasicommutators in the norm of Schatten–von Neumann classes (as well as in the norms of more general operator ideals). We consider a bounded operator R, self-adjoint operators A and B and for a function f ∈ Λα (R), we prove that f (A)R − Rf (B) ∈ S p/α , whenever p > 1 and AR − RB ∈ S p . We also obtain norm estimates for f (A)R − Rf (B) that are similar to the estimates obtained in Section 5 for first order differences f (A) − f (B). In Section 2 we give a brief introduction to Besov spaces and, in particular, we discuss Hölder– Zygmund classes Λα (R), 0 < α < ∞. In Section 3 we introduce quasinormed ideals of operators on Hilbert space and define the upper Boyd index of a quasinormed ideal. In Section 4 we give an introduction to double and multiple operator integrals which will be used in the paper to obtain desired estimates. We also define multiple operator integrals with respect to semi-spectral measures. Note that in this paper we give detailed proofs in the case of bounded self-adjoint operators and explain briefly that the main results also hold in the case of unbounded self-adjoint operators. We are going to consider in detail the case of unbounded self-adjoint operators in [3]. Note also that we are going to consider separately in [4] similar problems for perturbations of dissipative operators and improve earlier results of [20]. The main results of this paper have been announced without proofs in [1].


3679

2. Besov spaces The purpose of this section is to give a brief introduction to Besov spaces that play an important role in problems of perturbation theory. We start with Besov spaces on the unit circle. s of functions (or distributions) on T can be Let 0 < p, q ∞ and s ∈ R. The Besov class Bpq defined in the following way. Let w be an infinitely differentiable function on R such that 1 supp w ⊂ , 2 , 2

x and w(x) = 1 − w for x ∈ [1, 2]. 2

w 0,

(2.1)

Consider the trigonometric polynomials Wn , and Wn defined by k Wn (z) = w n zk , 2

n 1,

W0 (z) = z¯ + 1 + z,

and Wn (z) = Wn (z),

n 0.

k∈Z

Then for each distribution f on T, f=

f ∗ Wn +

n0

f ∗ Wn .

n1

s consists of functions (in the case s > 0) or distributions f on T such that The Besov class Bpq

ns 2 f ∗ Wn p ∈ q L n1

and

ns q 2 f ∗ W p n L n1 ∈ .

(2.2)

Besov classes admit many other descriptions. In particular, if s > 0 and s > 1/p − 1, the space s admits the following characterization. A function f ∈ Lp belongs to B s , s > 0, if and only Bpq pq if

T

q

nτ f Lp dm(τ ) < ∞ for q < ∞ |1 − τ |1+sq

and nτ f Lp < ∞ for q = ∞, s τ =1 |1 − τ | sup

(2.3)

where m is normalized Lebesgue measure on T, n is an integer greater than s, and τ , τ ∈ T, is the difference operator: (τ f )(ζ ) = f (τ ζ ) − f (ζ ),

ζ ∈ T.

s . We use the notation Bps for Bpp def

α form the Hölder–Zygmund scale. If 0 < α < 1, then f ∈ Λ if and The spaces Λα = B∞ α only if

f (ζ ) − f (τ ) const|ζ − τ |α ,

ζ, τ ∈ T,

3680


while f ∈ Λ1 if and only if f is continuous and f (ζ τ ) − 2f (ζ ) + f (ζ τ¯ ) const|1 − τ |,

ζ, τ ∈ T.

By (2.3), α > 0, f ∈ Λα if and only if f is continuous and def f Λα = sup |1 − τ |−α nτ f L∞ < ∞, τ

where n is the positive integer such that n − 1 α < n. Note that the f Λα is equivalent to f − fˆ(0) ∗ W0 ∞ + sup 2nα f ∗ Wn ∞ + f ∗ W ∞ , n L L L n1

where for a function or a distribution f on T, fˆ(n) is the nth Fourier coefficient of f . It is easy to see from the definition of Besov classes that the Riesz projection P+ , P+ f =

fˆ(n)zn ,

n0 def

s . Functions (or distributions) in (B s ) = P B s admit a natural extension is bounded on Bpq + pq pq + s ) admit the to analytic functions in the unit disk D. It is well known that the functions in (Bpq + following description:

s f ∈ Bpq +

1 ⇐⇒

q (1 − r)q(n−s)−1 fr(n) p dr < ∞,

q < ∞,

0

and s f ∈ Bp∞ +

⇐⇒

sup (1 − r)n−s fr(n) p < ∞,

0 0, can be described as the classes of continuous functions f on R such that B∞ m f (x) const|t|α , t

t ∈ R,

where the difference operator t is defined by (t f )(x) = f (x + t) − f (x),

x ∈ R,

and m is the integer such that m − 1 α < m. As in the case of functions on the unit circle, we consider the following (semi)norm on Λα (R): sup 2nα f ∗ Wn L∞ + f ∗ Wn L∞ ,

f ∈ Λα (R).

n∈Z

We refer the reader to [24] and [30] for more detailed information on Besov spaces. 3. Ideals of operators on Hilbert space In this section we give a brief introduction to quasinormed ideals of operators on Hilbert space. First we recall the definition of quasinormed vector spaces.

3682


Let X be a vector space. A functional · : X → [0, ∞) is called a quasinorm on X if (i) x = 0 if and only if x = 0; (ii) αx = |α| · x, for every x ∈ X and α ∈ C; (iii) there exists a positive number c such that x + y c(x + y) for every x and y in X. We say that a sequence {xj }j 1 of vectors of a quasinormed space X converges to x ∈ X if limj →∞ xj − x = 0. It is well known that there exists a translation invariant metric on X which induces an equivalent topology on X. A quasinormed space is called quasi-Banach if it is complete. To proceed to operator ideals on Hilbert space, we also recall the definition of singular values of bounded linear operators on Hilbert space. Let T be a bounded linear operator. The singular values sj (T ), j 0, are defined by

sj (T ) = inf T − R: rank R j . Clearly, s0 (T ) = T and T is compact if and only if sj (T ) → 0 as j → ∞. For a bounded operator T on Hilbert space we also introduce the sequence {σn (T )}n0 defined by def

σn (T ) =

1 sj (T ). n+1 n

(3.1)

j =0

Definition. Let H be a Hilbert space and let I be a nonzero linear manifold in the set B(H ) of bounded linear operators on H that is equipped with a quasi-norm · I that makes I a quasi-Banach space. We say that I is a quasinormed ideal if for every A and B in B(H ) and T ∈ I, AT B ∈ I

and AT BI A · B · T I .

(3.2)

A quasinormed ideal I is called a normed ideal if · I is a norm. Note that we do not require that I = B(H ). It is easy to see that if T1 and T2 are operators in a quasinormed ideal I and sj (T1 ) = sj (T2 ) for j 0, then T1 I = T2 I . Thus there exists a function Ψ = ΨI defined on the set of nonincreasing sequences of nonnegative real numbers with values in [0, ∞] such that T ∈ I if and only if Ψ (s0 (T ), s1 (T ), s2 (T ), . . .) < ∞ and T I = Ψ s0 (T ), s1 (T ), s2 (T ), . . . ,

T ∈ I.

If T is an operator from a Hilbert space H1 to a Hilbert space H2 , we say that T belongs to I if Ψ (s0 (T ), s1 (T ), s2 (T ), . . .) < ∞. For a quasinormed ideal I and a positive number p, we define the quasinormed ideal I{p} by

p/2 ∈I , I{p} = T : T ∗ T

p/2 1/p def . T I{p} = T ∗ T I


3683

If T is an operator on a Hilbert space H and d is a positive integer, we denote by [T ]d the operator on the orthogonal sum dj =1 Tj , where Tj = T , 1 j d. It is easy to see that sn [T ]d = s[n/d] (T ),

n 0,

where [x] denotes the largest integer that is less than or equal to x. We denote by βI,d the quasinorm of the transformer T → [T ]d on I. Clearly, the sequence {βI,d }d1 is nondecreasing and submultiplicative, i.e., βI,d1 d2 βI,d1 βI,d2 . It is well known that the last inequality implies that lim

d→∞

log βI,d log βI,d = inf . d2 log d log d

(3.3)

An analog of (3.3) for submultiplicative functions on (0, ∞) is proved in [17], Ch. 2, Theorem 1.3. To reduce the case of sequences to the case of functions, one can proceed as follows. Suppose that {βn }n1 is a nondecreasing submultiplicative sequence such that β1 = 1. We can define the function v on (0, ∞) by v(t) = min{βn : n t}. Then v(n) = βn and to prove (3.3), it suffices to apply Theorem 1.3 of Ch. 2 of [17] to the function v. Definition. If I is a quasinormed ideal, the number def

βI = lim

d→∞

log βI,d log βI,d = inf d2 log d log d

is called the upper Boyd index of I. It is easy to see that βI 1 for an arbitrary normed ideal I. It is also clear that βI < 1 if and only if limd→∞ d −1 βI,d = 0. Note that the upper Boyd index does not change if we replace the initial quasinorm in the quasinormed ideal with an equivalent one that also satisfies (3.2). It is also easy to see that βI{p} = p −1 βI . Theorem 3.1 below is known to experts. Its analog for rearrangement invariant spaces can be found in [17], Ch. 2, Theorem 6.6 (see also [19], Theorem 2(i)). A similar method can be used to prove Theorem 3.1. We give a proof here for reader’s convenience. Theorem 3.1. Let I be a quasinormed ideal. The following are equivalent: (i) βI < 1; (ii) for every nonincreasing sequence {sn }0 of nonnegative numbers, ΨI {σn }n0 const ΨI {sn }n0 , def

where σn = (1 + n)−1

n

j =0 sj .

(3.4)

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In the proof of Theorem 3.1 we are going to use an elementary fact that if n1 xn is a series of vectors in a quasi-Banach space X such that xn const γ n for some γ < 1, then the series converges in X. This is obvious if cγ < 1, where c is the constant in the definition of quasinorms. In the general case we can partition the series n1 xn in several series, after which each resulting series satisfies the above assumption. Proof of Theorem 3.1. Let us first show that (i) ⇒ (ii). Suppose that βI < δ < 1. Then there exists C > 0 such that βI,d Cδ d for all positive d. Let {sn }n0 be a nonincreasing sequence of positive numbers such that Ψ ({sn }n0 ) < ∞. Let {ej }j 0 be an orthonormal basis in a Hilbert space H . For k 0, we consider the operator Ak ∈ B(H ) defined by Ak ej = s[2−k j ] ej , j 0. It is easy to see that Ak is unitarily equivalent to the operator [A0 ]2k and Ak I C2δk A0 I = C2δk Ψ {sn }n0 . −k It follows that the series A = ∞ in I and AI cΨ ({sn }n0 ), where c is k=0 2 ∞Ak converges a positive number. Clearly, sn (A) = k=0 2−k s[ nk ] . 2 We have n

sj = sn +

j =0

1+[log2 n] [2−k+1 n]−1

sj

j =[2−k n]

k=1

2−k+1 n − 2−k n s[2−k n]

1+[log2 n]

sn +

k=1 ∞ 2−k n + 1 s[2−k n] sn + 3n 2−k s[2−k n]

1+[log2 n]

sn +

k=1

3(n + 1)

∞

k=1

2−k s[2−k n] .

k=0

Hence, σn 3sn (A), n 0, and so ΨI {σn }n0 3Ψ σn (A) n0 = 3AI 3cΨI {sn }n0 . Let us prove now that (ii) ⇒ (i). Let {sn }n0 be a nonincreasing sequence of nonnegative numbers. Put n n n k n 1 1 1 1 1 ξn = σn = sj = sj . n+1 n+1 k+1 n+1 k+1 def

k=0

k=0

j =0

j =0

k=j


3685

For an arbitrary positive integer d, we have n [n/d] s[n/d] 1 ξn n+1 k+1 j =0 k=j n s[n/d] 1 [n/d] + 1 n+1 k+1 k=[n/d]

([n/d] + 1)s[n/d] n+1

n [n/d]

dx x +1

s[n/d] n+2 log d [n/d] + 1 log d s[n/d] . d This together with inequality (3.4) applied twice yields ΨI {s[n/d] }n0

d d ΨI {ξn }n0 const ΨI {sn }n0 log d log d

for d 2. Thus βI,d < d for sufficiently large d, and so βI < 1.

2

Remark. Suppose that I is a normed ideal and let C I be the best possible constant in inequality (3.4). It is easy to see from the proof of Theorem 3.1 that CI 3

∞

2−k βI,2k .

(3.5)

k=0

Let S p , 0 < p < ∞, be the Schatten–von Neumann class of operators T on Hilbert space such that p 1/p def T S p = sj (T ) . j 0

This is a normed ideal for p 1. We denote by S p,∞ , 0 1. It is easy to see that βS p = βS p,∞ =

1 , p

0 1.

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It follows easily from (3.5) that for p > 1, −1 C S p 3 1 − 21/p−1 . Suppose now that I is a quasinormed ideal of operators on Hilbert space. With a nonnegative integer l we associate the ideal (l) I that consists of all bounded linear operators on Hilbert space and is equipped with the norm Ψ(l) I (s0 , s1 , s2 , . . .) = Ψ (s0 , s1 , . . . , sl , 0, 0, . . .). It is easy to see that for every bounded operator T ,

T (l) I = sup RT I : R 1, rank R l + 1

= sup T RI : R 1, rank R l + 1 . The following fact is obvious. Lemma 3.2. Let I be a quasinormed ideal. Then for all l 0, C (l) I C I . def (l)

Note that if I = S p , p 1, then S lp = linear operators equipped with the norm

def

T S l = p

S p is the normed ideal that consists of all bounded

l p sj (T )

1/p .

j =0

It is well known that · S l is a norm for p 1 (see [10]). p We need the following well-known inequality: T1 T2 S l T1 S l T2 S l , r

p

(3.6)

q

where T1 and T2 bounded operator on Hilbert space and 1/p + 1/q = 1/r. Inequality (3.6) can be deduced from the corresponding inequality for S p norms. Indeed, let R be an operator of rank l such that T1 T2 S l = T1 T2 RS r . There exists an orthogonal projection P of rank l such r that T1 T2 RS r = P T1 T2 RS r . Then T1 T2 S l = P T1 T2 RS r P T1 S p T2 RS q T1 S l T2 S l . r

p

q

We refer the reader to [14] and [10] for further information on singular values and normed ideals of operators on Hilbert space.


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4. Multiple operator integrals 4.1. Multiple operator integrals with respect to spectral measures In this subsection we review some aspects of the theory of double and multiple operator integrals. Double operator integrals appeared in the paper [12] by Daletskii and S.G. Krein. In that paper the authors obtained the following formula d f (A + tK) − f (A) = t=0 dt

(Df )(x, y) dEA (x) K dEA (y)

for a function f of class C 2 (R), and bounded self-adjoint operators A and K (EA stands for the spectral measure of A). Here we use the notation def

(Df )(x, y) =

f (x) − f (y) , x −y

(Df )(x, x) = f (x), def

x = y,

x, y ∈ R.

However, the beautiful theory of double operator integrals was developed later by Birman and Solomyak in [7–9], see also their survey [11]. We are not going to define double operator integrals Φ(x, y) dE1 (x) Q dE2 (y) in the case of Hilbert–Schmidt operators Q that was the starting point in [7–9]. We use the approach based on (integral) projective tensor products. In the case of bounded or trace class operators Q this approach is equivalent to the approach of Birman and Solomyak, see [27]. Let (X , E1 ) and (Y , E2 ) be spaces with spectral measures E1 and E2 on Hilbert spaces H1 and H2 . Suppose that a function Φ on X × Y belongs to the projective tensor product ˆ L∞ (E2 ) of L∞ (E1 ) and L∞ (E2 ), i.e., Φ admits a representation L∞ (E1 ) ⊗ Φ(x, y) =

ϕn (x)ψn (y),

(4.1)

ϕn L∞ ψn L∞ < ∞.

(4.2)

n0

where ϕn ∈ L∞ (E1 ), ψn ∈ L∞ (E2 ), and n0

Then for an arbitrary bounded linear operator Q : H2 → H1 we put

def

Φ(x, y) dE1 (x) Q dE2 (y) = X Y

ϕn dE1 Q ψn dE2 .

n0 X

Y

ˆ ∞ (E2 ), its norm in L∞ (E1 ) ⊗ ˆ Note that if Φ belongs to the projective tensor product L∞ (E1 ) ⊗L ∞ L (E2 ) is, by definition, the infimum of the left-hand side of (4.2) over all representations (4.1). It is easy to see that

Φ ∞ Φ(x, y) dE (x) Q dE (y) 1 2 ˆ ∞ (E2 ) Q. L (E1 )⊗L X Y

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ˆ L∞ (E2 ), then Moreover, if Q belongs to a normed ideal I and Φ ∈ L∞ (E1 ) ⊗

Φ(x, y) dE1 (x) Q dE2 (y) ∈ I X Y

and

Φ(x, y) dE1 (x) Q dE2 (y) ΦL∞ (E1 )⊗L ˆ ∞ (E2 ) QI . I

X Y

We can enlarge the class of functions Φ, for which double operator integrals can be defined by considering integral projective tensor products. We do this in the case of multiple operator integrals. This approach for multiple operator integrals was given in [32]. To simplify the notation, we consider here the case of triple operator integrals; the case of arbitrary multiple operator integrals can be treated in the same way. Let (X , E1 ), (Y , E2 ), and (Z, E3 ) be spaces with spectral measures E1 , E2 , and E3 on Hilbert spaces H1 , H2 , and H3 . Suppose that Φ belongs to the integral projective tensor product ˆ i L∞ (E2 ) ⊗ ˆ i L∞ (E3 ), i.e., Φ admits a representation L∞ (E1 ) ⊗

Φ(x, y, z) =

ϕ(x, ω)ψ(y, ω)χ(z, ω) dσ (ω),

(4.3)

Ω

where (Ω, σ ) is a measure space with a σ -finite measure, ϕ is a measurable function on X × Ω, ψ is a measurable function on Y × Ω, χ is a measurable function on Z × Ω, and

ϕ(·, ω)

L∞ (E1 )

ψ(·, ω)

L∞ (E2 )

χ(·, ω)

L∞ (E3 )

dσ (ω) < ∞.

(4.4)

Ω

Suppose now that T1 is a bounded linear operator from H2 to H1 and T2 is a bounded linear ˆ i L∞ (E2 ) ⊗ ˆ i L∞ (E3 ) of the form (4.3), operator from H3 to H2 . For a function Φ in L∞ (E1 ) ⊗ we put

Φ(x, y, z) dE1 (x)T1 dE2 (y)T2 dE3 (z) X Y Z def

=

Ω

X

ϕ(x, ω) dE1 (x) T1 ψ(y, ω) dE2 (y) T2 χ(z, ω) dE3 (z) dσ (ω). Y

(4.5)

Z

It was shown in [32] (see also [5] for a different proof) that the above definition does not depend on the choice of a representation (4.3). The norm ΦL∞ ⊗ˆ i L∞ ⊗ˆ i L∞ is defined as the infimum of the left-hand side of (4.4) over all representations (4.3).


3689

It is easy to see that the following inequality holds

Φ(x, y, z) dE1 (x) T1 dE2 (y) T2 dE3 (z) ΦL∞ ⊗ˆ i L∞ ⊗ˆ i L∞ T1 · T2 . X Y Z

In particular, the triple operator integral on the left-hand side of (4.5) can be defined if Φ ˆ L∞ (E2 ) ⊗ ˆ L∞ (E3 ). It is easy to see that if belongs to the projective tensor product L∞ (E1 ) ⊗ T1 ∈ S p and T2 ∈ S q , and 1/p + 1/q 1, then the triple operator integral (4.5) belongs to S r and

Φ(x, y, z) dE1 (x) T1 dE2 (y) T2 dE3 (z) ΦL∞ ⊗ˆ i L∞ ⊗ˆ i L∞ T1 S p · T2 S q , Sr

X Y Z

where 1/r = 1/p + 1/q. Note tat similar inequalities hold for multiple operator integrals and for S lp norms. Recall that multiple operator integrals were considered earlier in [23] and [36]. However, in those papers the class of functions Φ for which the left-hand side of (4.5) was defined is much narrower than in the definition given above. It follows from the results of Birman and Solomyak [9] that if A is a self-adjoint operator (not necessarily bounded), K is a bounded self-adjoint operator, and f is a continuously differentiable ˆ i L∞ (EA ), then function on R such that Df ∈ L∞ (EA+K ) ⊗

f (A + K) − f (A) =

(Df )(x, y) dEA+K (x) K dEA (y)

(4.6)

R×R

and if K belongs to a normed ideal I, then f (A + K) − f (A) ∈ I and f (A + K) − f (A) constDf ∞ ˆ ∞ (EA ) KI . L (EA+K )⊗L I In case I = S p or I = S lp , p 1, the last inequality admits the following improvement: f (A + K) − f (A) constDf ∞ ˆ i L∞ (EA ) KI . L (EA+K )⊗ I

(4.7)

Similar results also hold for unitary operators, in which case we have to integrate the divided difference Df of a function f on the unit circle with respect to the spectral measures of the corresponding operator integrals. It was shown in [27] that if f is a trigonometric polynomial of degree d, then Df C(T)⊗C(T) const df L∞ . ˆ

(4.8)

On the other hand, it was shown in [29] that if f is a bounded function on R whose Fourier transform is supported on [−σ, σ ] (in other words, f is an entire function of exponential type at ˆ i L∞ and most σ that is bounded on R), then Df ∈ L∞ ⊗ Df L∞ ⊗ˆ i L∞ const σ f L∞ (R) .

(4.9)

3690


In this paper we are going to integrate divided differences of higher orders to estimate the norms of higher order operator differences (1.1). For a function f on R, the divided difference Dk f of order k is defined inductively as follows: def

D0 f = f ; if k 1, then in the case when x1 , x2 , . . . , xk+1 are distinct points in R, k−1 f )(x , . . . , x k−1 f )(x , . . . , x k def (D 1 k−1 , xk ) − (D 1 k−1 , xk+1 ) D f (x1 , . . . , xk+1 ) = xk − xk+1

(the definition does not depend on the order of the variables). Clearly, Df = D1 f. If f ∈ C k (R), then Dk f extends by continuity to a function defined for all points x1 , x2 , . . . , xk+1 . It can be shown that n+1 n+1 k−1 n f (xk ) (xk − xj )−1 (xk − xj )−1 . D f (x1 , . . . , xn+1 ) = j =1

k=1

j =k+1

Similarly, one can define the divided difference of order k for functions on T. It was shown in [32] that if f is a trigonometric polynomial of degree d, then k D f

ˆ ⊗C(T) ˆ C(T)⊗···

const d k f L∞ .

(4.10)

It was also shown in [32] that if f is an entire function of exponential type at most σ and is bounded on R, then k D f

ˆ i ···⊗ ˆ i L∞ L∞ ⊗

const σ k f L∞ (R) .

(4.11)

4.2. Multiple operator integrals with respect to semi-spectral measures Let H be a Hilbert space and let (X , B) be a measurable space. A map E from B to the algebra B(H ) of all bounded operators on H is called a semi-spectral measure if E () 0,

∈ B,

E (∅) = 0 and E (X ) = I,

and for a sequence {j }j 1 of disjoint sets in B, E

∞ j =1

j

= lim

N →∞

N

E (j )

in the weak operator topology.

j =1

Multiple operator integrals with respect to semi-spectral measures were defined in [33] (see also [28]); the definition is based on integral projective tensor products of L∞ spaces.


3691

If T is a contraction on a Hilbert space H , then by the Sz.-Nagy dilation theorem (see [37]), T has a unitary dilation, i.e., there exist a Hilbert space K such that H ⊂ K and a unitary operator U on K such that T n = PH U n |H ,

n 0,

(4.12)

where PH is the orthogonal projection onto H . Let EU be the spectral measure of U . Consider the operator set function E defined on the Borel subsets of the unit circle T by E () = PH EU ()|H ,

⊂ T.

Then E is a semi-spectral measure. It follows immediately from (4.12) that

Tn =

ζ n dE (ζ ) = PH

T

ζ n dEU (ζ )|H ,

n 0.

(4.13)

T

Such a semi-spectral measure E is called a semi-spectral measure of T. Note that it is not unique. To have uniqueness, we can consider a minimal unitary dilation U of T , which is unique up to an isomorphism (see [37]). The following analog of the Birman–Solomyak formula holds:

f (R) − f (T ) =

(Df )(ζ, τ ) dER (ζ ) (R − T ) dET (τ ). T×T

Here T and R contractions on Hilbert space, ET and ER are their semi-spectral measures, and f 1 ) (see [28] and [33]). is an analytic function in D of class (B∞1 + 5. Self-adjoint operators. Sufficient conditions Recall that for l 0 and p > 0, the normed ideal S lp consists of all bounded linear operators equipped with the norm def

T S l = p

l p sj (T )

1/p .

j =0

Theorem 5.1. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every l 0, p ∈ [1, ∞), f ∈ Λα (R), and for arbitrary self-adjoint operators A and B on Hilbert space with bounded A − B, the following inequality holds: sj f (A) − f (B) cf Λα (R) (1 + j )−α/p A − BαS l

p

for every j l.

3692

A.B. Aleksandrov, V.V. Peller / Journal of Functional Analysis 258 (2010) 3675–3724 def

Proof. Put fn = f ∗ Wn + f ∗ Wn , n ∈ Z, and fix an integer N . We have by (4.7) and (4.9), N fn (A) − fn (B)

S lp

n=−∞

N fn (A) − fn (B)

n=−∞

const

N

S lp

2n fn L∞ A − BS l

p

n=−∞ N

constf Λα (R)

2n(1−α) A − BS l

p

n=−∞

const 2N (1−α) f Λα (R) A − BS l . p

On the other hand, 2 f (A) − f (B) fn L∞ n n n>N

n>N

constf Λα (R)

2−nα

n>N

const 2

−N α

f Λα (R) .

Put def

RN =

N fn (A) − fn (B)

def

and QN =

n=−∞

fn (A) − fn (B) . n>N

Clearly, for j l, sj f (A) − f (B) sj (RN ) + QN (1 + j )−1/p f (A) − f (B)S l + QN p −1/p N (1−α) const (1 + j ) 2 f Λα (R) A − BS l + 2−N α f Λα (R) . p

To obtain the desired estimate, it suffices to choose the number N so that 2−N < (1 + j )−1/p A − BS l 2−N +1 . p

2

Theorem 5.2. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα (R) and arbitrary self-adjoint operators A and B on Hilbert space with A − B ∈ S 1 , the operator f (A) − f (B) belongs to S 1/α,∞ and the following inequality holds: f (A) − f (B)

S 1/α,∞

cf Λα (R) A − BαS 1 .


Proof. This is an immediate consequence of Theorem 5.1 in the case p = 1.

3693

2

Note that the assumptions of Theorem 5.2 do not imply that f (A) − f (B) ∈ S 1/α . In Section 9 we obtain a necessary condition on f for f (A) − f (B) ∈ S 1/α whenever A − B ∈ S 1 . The following result ensures that the assumption that A − B ∈ S 1 implies that f (A) − f (B) ∈ S 1/α under a slightly more restrictive condition on f . Theorem 5.3. Let 0 < α 1. Then there exists a positive number c > 0 such that for every α (R) and arbitrary self-adjoint operators A and B on Hilbert space with A − B ∈ S , f ∈ B∞1 1 the operator f (A) − f (B) belongs to S 1/α and the following inequality holds: f (A) − f (B)

S 1/α

α α (R) A − B . cf B∞1 S1

Note that in the case α = 1 this was proved earlier in [29].

Proof of Theorem 5.3. Put fn = f ∗ Wn + f ∗ Wn . Clearly, fn is trace class perturbations preserving and it is easy to see that fn (A) − fn (B) Since f (A) − f (B) =

S 1/α

α 1−α fn (A) − fn (B)S fn (A) − fn (B) . 1

n∈Z (fn (A) − fn (B)),

(5.1)

it suffices to prove that

fn (A) − fn (B)

S 1/α

< ∞.

n∈Z

We have by (5.1) and (4.7), fn (A) − fn (B) n∈Z

S 1/α

fn (A) − fn (B)α · fn (A) − fn (B)1−α S 1

n∈Z

const

1−α α 2nα fn αL∞ · 21−α fn L ∞ A − BS 1

n∈Z

const

2nα fn L∞ A − BαS 1

n∈Z α α (R) A − B . constf B∞1 S1

2

Theorem 5.4. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα (R) and arbitrary self-adjoint operators A and B on Hilbert space with bounded A − B, the following inequality holds: 1/α 1/α cf Λα (R) σj (A − B), sj f (A) − f (B) Proof. It suffices to apply Theorem 5.1 with l = j and p = 1.

j 0.

2

Now we are in a position to obtain a general result in the case f ∈ Λα (R) and A − B ∈ I for an arbitrary quasinormed ideal I with upper Boyd index less than 1.

3694


Theorem 5.5. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα (R), for an arbitrary quasinormed ideal I with βI < 1, and for arbitrary self-adjoint operators A and B on Hilbert space with A − B ∈ I, the operator |f (A) − f (B)|1/α belongs to I and the following inequality holds: f (A) − f (B) 1/α cC I f 1/α A − BI . Λα (R) I Proof. Theorem 3.1 implies that ΨI σn (A − B) n0 constA − BI , and Theorem 5.4 implies that f (A) − f (B) 1/α cf 1/α ΨI σn (A − B) . Λα (R) I n0

2

We can reformulate Theorem 5.5 in the following way. Theorem 5.6. Under the hypothesis of Theorem 5.5, the operator f (A) − f (B) belongs to I{1/α} and f (A) − f (B)

I{1/α}

cα C αI f Λα (R) A − BαI .

We deduce now some more consequences of Theorem 5.5. Theorem 5.7. Let 0 < α < 1 and 1 < p < ∞. Then there exists a positive number c such that for every f ∈ Λα (R), every l ∈ Z+ , and arbitrary self-adjoint operators A and B with bounded A − B, the following inequality holds: l l 1/α p p p/α sj f (A) − f (B) sj (A − B) . cf Λα (R) j =0

j =0

Proof. The result immediately follows from Theorem 5.5 and Lemma 3.2.

2

Theorem 5.8. Let 0 < α < 1 and 1 < p < ∞. Then there exists a positive number c such that for every f ∈ Λα (R) and for arbitrary self-adjoint operators A and B with A − B ∈ S p , the operator f (A) − f (B) belongs to S p/α and the following inequality holds: f (A) − f (B)

S p/α

cf Λα (R) A − BαS p .

Proof. The result is an immediate consequence of Theorem 5.7.

2

Suppose now that m−1 α < m and f ∈ Λα (R). For a self-adjoint operator A and a bounded self-adjoint operator K, we consider the finite difference m m def K f (A) = (−1)m−j j =0

m f (A + j K). j


3695

In the case when A is unbounded, by the right-hand side we mean the following operator m m−j m fn (A + j K), (−1) j n∈Z j =0

where as usual, fn = f ∗Wn +f ∗Wn . It has been proved in [2] that under the above assumptions, m m−j m (−1) fn (A + j K) < ∞. j n∈Z j =0

(We refer the reader to [3], where the situation with unbounded A will be discussed in detail.) We are going to use the following representation for (m K f )(A) in terms of multiple operator integrals: m K f (A)

m = m! · · · D f (x1 , . . . , xm+1 ) dEA (x1 )K dEA+K (x2 )K · · · K dEA+mK (xm+1 ), (5.2) m+1 m (R). Forwhere A is a self-adjoint operator, K is a bounded self-adjoint operator, and f ∈ B∞1 mula (5.2) was obtained in [2]. It follows from (5.2), (4.11), and (3.6) that if p m 1, l 0, and f is an entire function of exponential type at most σ that is bounded on R, then

m f (A) K

S lp

m

const σ m f L∞ Km . Sl

(5.3)

p

Moreover, the constant in (5.3) does not depend on p. Theorem 5.9. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every l 0, p ∈ [m, ∞), f ∈ Λα (R), and for arbitrary self-adjoint operator A and bounded self-adjoint operator K, the following inequality holds: −α/p sj m KαS l K f (A) cf Λα (R) (1 + j )

p

for j l. Proof. As in the proof of Theorem 5.1, we put def

RN =

m K fn (A) nN

def

and QN =

m K fn (A). n>N

3696


It follows from (5.3) that RN S l

p/m

const 2mn fn L∞ Km Sl

p

nN

Km f Λα (R) Sl p

2

2(m−α)n

nN

(m−α)N

f Λα (R) Km . Sl p

On the other hand, it is easy to see that QN const

fn L∞ f Λα (R)

n>N

2−nα 2−αN f Λα (R) .

n>N

Hence, sj m K fn (A) sj (RN ) + QN (1 + j )−m/p RN S l + QN p/m −αN constf Λα (R) (1 + j )−m/p 2(m−α)N Km . + 2 l S p

To complete the proof, it suffices to choose N such that 2−N < (1 + j )−1/p KS l 2−N +1 . p

2

The following result is an immediate consequence from Theorem 5.9. Theorem 5.10. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα (R), and for an arbitrary self-adjoint operator A and an arbitrary self-adjoint m operator K of class S m , the operator (m K f )(A) belongs to S α ,∞ and the following inequality holds: m f (A) K

S m ,∞ α

cf Λα (R) KαS m .

As in the case 0 < α < 1 (see Theorem 5.3), we are going to improve the conclusion of Theorem 5.10 under a slightly more restrictive assumption on f . Note that in the following theorem α is allowed to be equal to m. Theorem 5.11. Let α > 0 and m − 1 α m. There exists a positive number c such that for α (R), and for an arbitrary self-adjoint operator A and an arbitrary self-adjoint every f ∈ B∞1 m operator K of class S m , the operator (m K f )(A) belongs to S α and the following inequality holds: m f (A) K

Sm α

α α (R) K cf B∞1 Sm .


3697

Proof. Clearly, m fn (A) K

Sm α

α/m m 1−α/m fn (A) m . K fn (A) S K 1

By (5.3), m fn (A) K

S1

const 2mn fn L∞ Km Sm .

Thus m fn (A) K

Sm

α

n∈Z

m fn (A)α/m m fn (A)1−α/m K K S 1

n∈Z

const

1−α/m

α/m

2αn fn L∞ KαS m fn L∞

n∈Z

constKαS m

2αn fn L∞

n∈Z α α (R) K constf B∞1 Sm .

2

Recall that for a bounded linear operator T the numbers, σj (T ) are defined by (3.1). Theorem 5.12. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα (R), and for arbitrary self-adjoint operator A and bounded self-adjoint operator K, the following inequality holds: m/α m/α sj m cf Λα (R) σj |K|m , K f (A)

j 0.

Proof. The result follows immediately from Theorem 5.9 in the case j = l and p = m.

2

Theorem 5.13. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα (R), every quasinormed ideal I with βI < m−1 , and for arbitrary self-adjoint operator A and bounded self-adjoint operator K, the following inequality holds: m 1/α 1/α f (A) cC 1/m f Λα (R) KI . K I I{1/m} Proof. Clearly, |K|m ∈ I{1/m} and βI{1/m} = mβI < 1. Therefore, by Theorem 5.12, m m/α f (A)

I{1/m}

K

which implies the result.

2

m/α cC I{1/m} f Λα (R) |K|m I{1/m}

3698


Theorem 5.14. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λα (R), every l ∈ Z+ , and for arbitrary self-adjoint operator A and bounded self-adjoint operator K, the following inequality holds: l l m 1/α p p p/α sj K f (A) sj (K) . cf Λα (R) j =0

j =0

Proof. The result follows from Theorem 5.13 and Lemma 3.2.

2

The last theorem of this section is an immediate consequence of Theorem 5.14. Theorem 5.15. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λα (R), for an arbitrary self-adjoint operator A, and an arbitrary selfadjoint operator K of class S p , the following inequality holds: m f (A) K

S p/α

cf Λα (R) KαS p .

6. Unitary operators. Sufficient conditions In this section we are going to obtain analogs of the results of the previous section for functions of unitary operators. In the case of first order differences we can use the Birman–Solomyak formula for functions of unitary operators and the proofs are the same as in the case of functions of self-adjoint operators. However, in the case of higher order differences, formulae that express a difference of order m involves not only multiple operator integrals of multiplicity m + 1, but also multiple operator integrals of lower multiplicities, see [2]. This makes proofs more complicated than in the self-adjoint case. We start with first order differences. If U and V are unitary operators, then by the Birman– Solomyak formula,

f (U ) − f (V ) = T×T

f (ζ ) − f (τ ) dEU (ζ ) (U − V ) dEV (τ ), ζ −τ

(6.1)

ˆ L∞ . Here EU and EV are the spectral whenever the divided difference Df belongs to L∞ ⊗ 1 . measures of U and V . Recall that it was shown in [27] that (6.1) holds if f ∈ B∞1 It follows from (4.8) that if I is a normed ideal, U −V ∈ I and f is a trigonometric polynomial of degree d, then f (U ) − f (V ) ∈ I and f (U ) − f (V ) const df L∞ U − V I . I

(6.2)

Moreover, the constant does not depend on I. Theorem 6.1. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every l 0, p ∈ [1, ∞), f ∈ Λα , and for arbitrary unitary operators U and V on Hilbert space, the following inequality holds:


3699

sj f (U ) − f (V ) cf Λα (1 + j )−α/p U − V αS l

p

for every j l. Theorem 6.2. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα and arbitrary unitary operators U and V on Hilbert space with U −V ∈ S 1 , the operator f (U ) − f (V ) belongs to S 1/α,∞ and the following inequality holds: f (U ) − f (V )

S 1/α,∞

cf Λα U − V αS 1 .

As in the self-adjoint case, the assumptions of Theorem 6.2 do not imply that f (U ) − f (V ) ∈ S 1/α . In Section 8 we obtain a necessary condition on f for f (U ) − f (V ) ∈ S 1/α , whenever U − V ∈ S1. Theorem 6.3. Let 0 < α 1. Then there exists a positive number c > 0 such that for every α and arbitrary unitary operators U and V on Hilbert space with U − V ∈ S , the f ∈ B∞1 1 operator f (U ) − f (V ) belongs to S 1/α and the following inequality holds: f (U ) − f (V )

S 1/α

α α U − V . cf B∞1 S1

Note that in the case α = 1 this was proved earlier in [27]. Theorem 6.4. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα and arbitrary unitary operators U and V on Hilbert space, the following inequality holds: 1/α 1/α cf Λα σj (U − V ), sj f (U ) − f (V )

j 0.

Recall that the numbers σj (U − V ) are defined in (3.1). Theorem 6.5. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα , for an arbitrary quasinormed ideal I with βI < 1, and for arbitrary unitary operators U and V on Hilbert space with U − V ∈ I, the operator |f (U ) − f (V )|1/α belongs to I and the following inequality holds: f (U ) − f (V ) 1/α cC I f 1/α U − V I . Λα (R) I Theorem 6.6. Let 0 < α < 1 and 1 < p < ∞. Then there exists a positive number c such that for every f ∈ Λα , every l ∈ Z+ , and arbitrary unitary operators U and V , the following inequality holds: l l 1/α p p p/α sj f (U ) − f (V ) sj (U − V ) . cf Λα j =0

j =0

3700


Theorem 6.7. Let 0 < α < 1 and 1 < p < ∞. Then there exists a positive number c such that for every f ∈ Λα and for arbitrary unitary operators U and V with U − V ∈ S p , the operator f (U ) − f (V ) belongs to S p/α and the following inequality holds: f (U ) − f (V )

S p/α

cf Λα U − V αS p .

The proofs of the above results are almost the same as in the self-adjoint case. The only difference is that we have to use (6.2) instead of the corresponding inequality for self-adjoint operators. We proceed now to higher order differences. Let U be a unitary operator and A a self-adjoint operator. We are going to study properties of the following higher order differences m k m f eikA U . (−1) k

(6.3)

k=0

As we have already mentioned in the introduction to this section, such finite differences can be expressed as a linear combination of multiple operator integrals of multiplicity at most m + 1. We refer the reader to [2], Theorem 5.2. For simplicity, we state the formula in the case m = 3. 2 . Let U , U , and U be unitary operators. Then Let f ∈ B∞1 1 2 3 f (U1 ) − 2f (U2 ) + f (U3 )

2 =2 D f (ζ, τ, υ) dE1 (ζ ) (U1 − U2 ) dE2 (τ ) (U2 − U3 ) dE3 (υ)

+ (Df )(ζ, τ ) dE1 (ζ ) (U1 − 2U2 + U3 ) dE3 (τ ).

(6.4)

Let U1 = U , U2 = eiA U , and U3 = e2iA U . Lemma 6.8. Let I be a normed ideal such that I{1/2} is also a normed ideal. If f is a trigonometric polynomial of degree d and A ∈ I, then f (U ) − 2f (eiA U ) + f (e2iA U ) ∈ I{1/2} and f (U ) − 2f eiA U + f e2iA U

I{1/2}

const · d 2 f L∞ A2I .

Moreover, the constant does not depend on I. Proof. Let U1 = U , U2 = eiA U , and U3 = e2iA U . By (4.10), we have

2 D f (ζ, τ, υ) dE1 (ζ ) (U1 − U2 ) dE2 (τ ) (U2 − U3 ) dE3 (υ) const · d f L∞ U1 − U2 I U2 − U3 I . 2

Clearly, U1 − U2 I = U2 − U3 I = I − eiA I constAI .

I{1/2}


On the other hand, by (4.8),

(Df )(ζ, τ ) dE1 (ζ ) (U1 − 2U2 + U3 ) dE3 (τ )

I{1/2}

3701

const · dU1 − 2U2 + U3 I{1/2}

and 2 U1 − 2U2 + U3 I{1/2} = I − eiA I{1/2} constA2I . The result follows now from (6.4).

2

In the general case, for a trigonometric polynomial f of degree d, the following inequality holds: m m f eikA U (−1)k const · d m f L∞ Am (6.5) I, k {1/m} k=0

I

whenever I is a normed ideal such that I{1/m} is also a normed ideal. This follows from an analog of formula (6.4) for higher order differences, see [2], Theorem 5.2. We state the remaining results in this section without proofs. The proofs are practically the same as in the self-adjoint case. The only difference is that instead of inequality (5.3), one has to use inequality (6.5) with I = S lp , p m. Theorem 6.9. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every l 0, p ∈ [m, ∞), f ∈ Λα , and for arbitrary unitary operator U self-adjoint operator A, the following inequality holds: m ikA k m f e U cf Λα (1 + j )−α/p AαS l (−1) sj k p k=0

for j l. Theorem 6.10. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα , and for an arbitrary unitary operator U and an arbitrary self-adjoint operator A of class S m , the operator (6.3) belongs to S mα ,∞ and the following inequality holds: m ikA k m f e U (−1) k k=0

cf Λα AαS m .

S m ,∞ α

Theorem 6.11. Let α > 0 and m − 1 α m. There exists a positive number c such that for α , and for an arbitrary unitary operator U and an arbitrary self-adjoint operator every f ∈ B∞1 A of class S m , the operator (6.3) belongs to S mα and the following inequality holds: m ikA k m f e U (−1) k k=0

α α A cf B∞1 Sm .

Sm α

3702


Theorem 6.12. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα , and for arbitrary unitary operator U and bounded self-adjoint operator A, the following inequality holds: m/α m m m/α f eikA U sj (−1)k cf Λα σj |A|m , k

j 0.

k=0

Theorem 6.13. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα , every quasinormed ideal I with βI < m−1 , and for arbitrary unitary operator U and bounded self-adjoint operator A, the following inequality holds: m 1/α m 1/m 1/α f eikA U cC I{1/m} f Λα AI . (−1)k k k=0

I

Theorem 6.14. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λα , every l ∈ Z+ , and for arbitrary unitary operator U and bounded self-adjoint operator A, the following inequality holds: 1/α p m l l p m p/α f eikA U sj (A) . (−1)k cf Λα sj k j =0

j =0

k=0

Theorem 6.15. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λα , for an arbitrary unitary operator U , and an arbitrary self-adjoint operator A of class S p , the following inequality holds: m m f eikA U (−1)k k k=0

cf Λα AαS p .

S p/α

7. The case of contractions In this section we obtain analogs of the results of Sections 5 and 6 for contractions. To obtain desired estimates, we use multiple operator integrals with respect to semi-spectral measures. Suppose that T and R are contractions on Hilbert space and f is a function in the diskalgebra CA (i.e., f is analytic in D and continuous in clos D). We are going to study properties of differences m k=0

(−1)k

k m f T + (R − T ) , k m

m 1.

(7.1)

In particular, when m = 1, we obtain first order differences f (T ) − f (R). In this section we are not going to state separately results for first order differences. They can be obtained from the general results by putting m = 1.


3703

It was shown in [2] that

k m f T + (R − T ) (−1) k m k=0

m m! D f (ζ1 , . . . , ζm+1 ) dE1 (ζ1 )(R − T ) · · · (R − T ) dEm+1 (ζm+1 ), (7.2) = m ··· m

m

k

m+1

where Ek is a semi-spectral measure of T + mk (R − T ). Suppose now that I is a normed ideal such that I{1/m} is also a normed ideal. It follows from (7.2) and (4.11) that for an arbitrary trigonometric polynomial f of degree d, m k k m f T + (R − T ) (−1) k m k=0

const · d m f L∞ T − Rm I,

(7.3)

I{1/m}

where the constant can depend only on m. We state the results without proofs. The proofs are almost the same as in the self-adjoint case. The only difference is that to estimate higher order differences, we should use inequality (7.3). Theorem 7.1. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every l 0, p ∈ [m, ∞), f ∈ (Λα )+ , and for arbitrary contractions T and R on Hilbert space, the following inequality holds: sj

m k k m f T + (R − T ) (−1) cf Λα (1 + j )−α/p T − RαS l k p m k=0

for j l. Theorem 7.2. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ (Λα )+ , and for arbitrary contractions T and R with T − R ∈ S m , the operator (7.1) belongs to S mα ,∞ and the following inequality holds: m k m f T + (R − T ) (−1)k k m k=0

cf Λα T − RαS m .

S m ,∞ α

Theorem 7.3. Let α > 0 and m − 1 α m. There exists a positive number c such that for α ) , and for arbitrary contractions T and R with T − R ∈ S , the operator (7.1) every f ∈ (B∞1 + m belongs to S mα and the following inequality holds: m k m f T + (R − T ) (−1)k k m k=0

α α T − R cf B∞1 Sm .

Sm α

3704


Theorem 7.4. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ (Λα )+ , and for arbitrary contractions T and R, the following inequality holds: m m/α k m/α k m f T + (R − T ) sj (−1) cf Λα σj |T − R|m , k m

j 0.

k=0

Theorem 7.5. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ (Λα )+ , every quasinormed ideal I with βI < m−1 , and for arbitrary contractions T and R, the following inequality holds: m 1/α k 1/m 1/α k m f T + (R − T ) cC I{1/m} f Λα T − RI . (−1) k m k=0

I

Theorem 7.6. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ (Λα )+ , every l ∈ Z+ , and for arbitrary contractions T and R, the following inequality holds: m 1/α p l l p k m p/α f T + (R − T ) sj (T − R) . (−1)k cf Λα sj k m j =0

j =0

k=0

Theorem 7.7. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ (Λα )+ , for arbitrary contractions T and R with T − R ∈ S p , the following inequality holds: m k m f T + (R − T ) (−1)k k m k=0

cf Λα T − RαS p . S p/α

8. Finite rank perturbations and necessary conditions. Unitary operators In this sections we study the case of finite rank perturbations of unitary operators. We also obtain some necessary conditions. In particular we show that the assumptions that rank(U − V ) = 1 and f ∈ Λα , 0 < α < 1, do not imply that f (U ) − f (V ) ∈ S 1/α . Let us introduce the notion of Hankel operators. For ϕ ∈ L∞ (T), the Hankel operator Hϕ def

from the Hardy class H 2 to H−2 = L2 H 2 is defined by Hϕ g = P− ϕg,

g ∈ H 2,

where P− is the orthogonal projection from L2 onto H−2 . Note that the operator Hϕ has Hankel matrix def Γϕ = ϕ(−j ˆ − k) j 1, k0 with respect to the orthonormal bases {zk }k0 and {¯zj }j 1 of H 2 and H−2 .


3705

We need the following description of Hankel operators of class S p that was obtained in [25] for p 1 and [26] and [35] for p < 1 (see also [30], Ch. 6): Hϕ ∈ S p

⇐⇒

1/p

P− ϕ ∈ Bp ,

0 < p < ∞.

(8.1)

The following result gives us a necessary condition on f for the assumption U − V ∈ S 1 to imply that f (U ) − f (V ) ∈ S 1/α . Theorem 8.1. Suppose that 0 < p < ∞. Let f be a continuous function on T such that f (U ) − f (V ) ∈ S p , whenever U and V are unitary operators with rank(U − V ) = 1. Then 1/p f ∈ Bp . Proof. Consider the operators U and V on the space L2 (T) with respect to normalized Lebesgue measure on T defined by Uf = z¯ f

and Vf = z¯ f − 2(f, 1)¯z,

f ∈ L2 .

It is easy to see that both U and V are unitary operators and rank(V − U ) = 1. It is also easy to verify that for n 0, ⎧ j −n ⎪ ⎨z , n j V z = −zj −n , ⎪ ⎩ j −n z ,

j n, 0 j < n, j < 0.

It follows that for f ∈ C(T), we have f (V ) − f (U ) zj , zk = fˆ(n) V n zj , zk − zj −n , zk + fˆ(n) V n zj , zk − zj −n , zk n>0

⎧ ⎪ ⎨ fˆ(j − k), = −2 fˆ(j − k), ⎪ ⎩ 0,

n 0. Theorem 8.2. Let 0 < α < ∞ and let U and V be unitary operators such that rank(U − V ) < +∞. Then f (U ) − f (V ) ∈ S 1 ,∞ for every function f ∈ Λα (T). α

Proof. Let m be a positive integer and let f ∈ Λα . By Bernstein’s theorem, we can represent f in the form f = f1 + f2 , where f1 is a trigonometric polynomial of degree at most m and f2 L∞ const m−α (this can be deduced easily from (2.2)). It is easy to see that Um − V m =

m−1

U j (U − V )V m−1−j .

j =0

Hence, m Range f1 (U ) − f1 (V ) ⊂ Range U j (U − V ) + Range U −j U ∗ − V ∗ , j =1

and so rank f1 (U ) − f1 (V ) 2m rank(U − V ), while f2 (U ) − f2 (V ) 2f2 L∞ const m−α . It follows that s2m rank(U −V ) f (U ) − f (V ) const m−α .

2


3707

We can compare Theorem 8.2 with the following result obtained in [28]: if 0 < p 1, and U 1/p and V are unitary operators such that U − V ∈ S p , then f (U ) − f (V ) ∈ S p for every f ∈ B∞p . The following result allows us to estimate the singular values of Hankel operators with symbols in Λα . Lemma 8.3. Let 0 < α < ∞. Then there exists a positive number c such that for every f ∈ Λα (T), the following inequality holds: sm (Hf ) cf Λα (1 + m)−α . Proof. We can represent f in the form f = f1 + f2 , where f1 is a trigonometrical polynomial of degree at most m and f2 const(1 + m)−α . Then rank Hf1 m and Hf2 const(1 + m)−α which implies the result. 2 The following theorem shows that Theorems 8.2 and 6.2 cannot be improved. Theorem 8.4. Let α > 0. There exist unitary operators U and V and a real function h in Λα such that rank(U − V ) = 1 and sm h(U ) − h(V ) (1 + m)−α ,

m 0.

Proof. Let U and V be the unitary operators defined in the proof of Theorem 8.1. Consider the function g defined by ∞ def −αn 4n

g(ζ ) =

4

ζ

n + ζ¯ 4 ,

ζ ∈ T.

(8.4)

n=1

It follows easily from (2.2) that g ∈ Λα (T). By Lemma 8.3, sm (Hg ) const(1 + m)−α , m 0. Let us obtain a lower estimate for sm (Hg ). Consider the matrix Γg of the Hankel operator Hg with respect to the standard orthonormal bases:

Γg = g(−j ˆ − k) j 1, k0 = g(j ˆ + k) j 1, k0 . Let n 1. Define the 3 · 4n−1 × 3 · 4n−1 matrix Tn by

Tn = gˆ j + k + 4n−1 + 1 0j, k q. Proof. It is easy to see that f ∈ Uc (S p , S q ) if and only if for every two sequences {ζn } and {τn } in T,

|ζn − τn |p < ∞

⇒

f (ζn ) − f (τn ) q < ∞.

(8.5)

Clearly, the condition |f (ζ ) − f (ξ )| const|ζ − ξ |p/q implies (8.5). Consider the modulus of continuity ωf associated with f :

def ωf (δ) = sup f (x) − f (y) : |x − y| < δ ,

δ > 0.

Condition (8.5) obviously implies that ωf (δ) < ∞ for some δ > 0, and so it is finite for all δ > 0. We have to prove that (8.5) implies that ωf (δ) const · δ p/q . Assume the contrary. Then there exist two sequences {ζn } and {τn } in T such that ζn = τn for all n, lim |ζn − τn |p = 0 and

n→∞

|f (ζn ) − f (τn )|q = ∞. n→∞ |ζn − τn |p lim

Now the result is a consequence of the following elementary fact: If {αk } and {βk } are sequences of positive numbers such that limk→∞ βk = 0 and −1 lim k→∞ αk βk = +∞, then there exists a sequence {nk } of nonnegative integers such that nk βk < +∞ and nk αk = +∞. 2


3709

Corollary 8.6. Let 0 < p, q < +∞. Then ! U(S p , S q ) ⊂

Λp/q , Lip,

p < q, p = q.

The space U(S p , S q ) is trivial if p > q. It has been shown recently in [34] that if f is a Lipschitz function on R and 1 < p < ∞, then f (A) − f (B)S p constA − BS p , whenever A and B are self-adjoint operators such that A − B ∈ Sp . Remark 1. The Potapov–Sukochev theorem implies an analogous result for unitary operators: if U and V are unitary operators such that U − V ∈ S p , 1 < p < ∞, and f is a Lipschitz function on T, then f (U ) − f (V ) ∈ S p and f (U ) − f (V )S p constU − V S p . Indeed, it is easy to reduce the general case to the case when the support of f is contained in an arc I with m(I ) = 1/4. Without loss of generality, we may assume that −I = I¯. Then f (ζ ) + f (ζ¯ ) = 2g(ζ + ζ¯ )

and f (ζ ) − f (ζ¯ ) = 2(ζ − ζ¯ )h(ζ + ζ¯ ),

ζ ∈ T,

for some functions g, h ∈ Lip(R) ∩ L∞ (R). Hence f (ζ ) = g(ζ + ζ¯ ) + ζ h(ζ + ζ¯ ) − ζ¯ h(ζ + ζ¯ ). It remains to apply the Potapov–Sukochev theorem to the self-adjoint operators U + U ∗ , V + V ∗ , and the functions g, h. We are going to use this analog of the Potapov–Sukochev theorem for unitary operators in the following result. Theorem 8.7. Let 1 < q p < +∞. Then ! U(S p , S q ) =

Λp/q , Lip,

p < q, p = q.

Proof. By Corollary 8.6, it suffices to show that Λp/q ⊂ U(S p , S q ) for p < q and Lip ⊂ U(S p , S q ) for p = q. The fact that Λp/q ⊂ U(S p , S q ) for q < p is a consequence of Theorem 6.7. The inclusion Lip ⊂ U(S p , S q ) for q = p is the analog of the Potapov–Sukochev theorem mentioned above. 2 Remark 2. There exists a function f of class Lip such that f ∈ / U(S p , S q ) for any p > 0 and q ∈ (0, 1]. Indeed, if U and V are the unitary operators constructed in the proof of Theorem 8.1, then rank(U − V ) = 1 and f (U ) − f (V ) ∈ S 1 if and only if f ∈ B11 . It suffices to take a Lipschitz function f that does not belong to B11 . Remark 3. Let α > 0. There exists a function f in Λα such that f ∈ / U(S p , S q ) for any p > 0 and q ∈ (0, 1/α]. Indeed, it suffices to consider the unitary operators U and V constructed in the α . proof of Theorem 8.1 and take a function f ∈ Λα that does not belong to B1/α

3710


Theorem 8.8. Let 0 < p, q < +∞. Then Λp/q ⊂ U(S p , S q ) if and only if 1 q, the result follows from Corollary 8.6 and Theorem 8.7. On the other hand, if p q and p 1, then Λp/q ⊂ U(S p , S q ) by Remark 2. 2 Theorem 8.9. Let 0 < p, q < +∞. Then Lip ⊂ U(S p , S q ) if and only if 1 < p q or p 1 < q. Proof. As in the proof of Theorem 8.8, it suffices to consider the case p 1. It was shown in [22] that Lip ⊂ U(S 1 , S q ) ⊂ U(S p , S q ) if p 1 < q. It remains to apply Remark 1. 2 Now we are going to obtain a quantitative refinement of Corollary 8.6. Let f ∈ C(T). Put

def Ωf,p,q (δ) = sup f (U ) − f (V )S : U − V S p δ, U, V are unitary operators . q

Lemma 8.10. Let U1 and U2 be a unitary operators with U1 − U2 ∈ S p . Then there exists a unitary operator V such that U1 − V S p

πU1 − U2 S p

and U2 − V S p

4

πU1 − U2 S p 4

.

Proof. Clearly, there exists a self-adjoint operator A such that exp(iA) = U1−1 U2 and A π . Note that π|eiθ − 1| 2|θ | for |θ | π . Hence, AS p π2 U1 − U2 S p . It remains to put V = U1 exp( 2i A). 2 Corollary 8.11. Let 0 < q < +∞. Then there exists a positive number cq such that for every p ∈ (0, ∞), Ωf,p,q (2δ) cq Ωf,p,q (δ),

δ > 0.

Lemma 8.12. Let 0 < p, q < ∞ and let f ∈ U(S p , S q ). Then Ωf,p,q n1/p δ n1/q Ωf,p,q (δ) for every positive integer n. Proof. The result is trivial if Ωf,p,q (δ) = 0 or Ωf,p,q (δ) = ∞. Suppose now that p 0 < Ωf,p,q (δ) < ∞. Fix ε ∈ (0, 1). Let U and V be unitary operators such that U − V S δ and f (U ) − f (V )S q (1 − ε)Ωf,p,q (δ). Put U = nj=1 U and V = nj=1 V (the orthogonal sum of n copies of U and V ). Clearly, U − VS p δn1/p , and f (U) − f (V)

Sq

(1 − ε)n1/q Ωf,p,q (δ)

Hence, Ωf,p,q (n1/p δ) (1 − ε)n1/q Ωf,p,q (δ) for every ε ∈ (0, 1).

2


3711

Theorem 8.13. Let 0 < p, q < ∞ and let f ∈ U(S p , S q ). Then Ωf,p,q (δ) < +∞ for all δ > 0 and Ωf,p,q (δ) Ωf,p,q (δ) Ωf,p,q (δ) Ωf,p,q (δ) = inf sup = lim , p/q p/q p/q δ→∞ δ→0 δ>0 δ δ δ δ p/q δ>0 lim

where both limits exist in [0, +∞]. In particular, if f is a nonconstant function, then Ωf,p,q (δ) c1 δ p/q for every δ ∈ (0, 1] and Ωf,p,q (δ) c2 δ p/q for every δ ∈ [1, ∞), where c1 and c2 are positive numbers. Proof. Since Ωf,p,q is nondecreasing, Corollary 8.11 implies that either Ωf,p,q (δ) is finite for all δ > 0 or Ωf,p,q (δ) = ∞ for all δ > 0. The latter is impossible because we would be able to find sequences of unitary operators {Uj } and {Vj } such that "

(Uj − Vj ) ∈ S p ,

but

j

" f (Uj ) − f (Vj ) ∈ / Sq . j

Hence, Ωf,p,q (δ) < +∞ for all δ > 0. We can find a sequence {δj }∞ j =1 of positive numbers such −p/q

that δj → 0 and limj →∞ δj

def

Ωf,p,q (δj ) = lim supδ→0 δ −p/q Ωf,p,q (δ) = a. Fix ε ∈ (0, 1). −p/q

Ωf,p,q (δj ) (1 − ε)a for all j > N . Then there exists a positive integer N such that δj Lemma 8.12 implies Ωf,p,q (n1/p δj ) (1 − ε)a(n1/p δj )p/q for all j > N and n > 0. Hence, Ωf,p,q (δ) (1 − ε)aδ p/q for all δ > 0 and ε ∈ (0, 1). Thus Ωf,p,q (δ) aδ p/q for all δ > 0 and limδ→0 δ −p/q Ωf,p,q (δ) = a. In the same way we can prove that Ωf,p,q (δ) Ωf,p,q (δ) = lim . p/q δ→∞ δ δ p/q δ>0

sup

2

9. Finite rank perturbations and necessary conditions. Self-adjoint operators We are going to obtain in this section analogs of the results of the previous section in the case of self-adjoint operators. We obtain estimates for f (A) − f (B) in the case when rank(A − B) < ∞. We also obtain some necessary conditions. In particular, we show that f (A) − f (B) does not have to belong to S 1/α under the assumptions rank(A − B) = 1 and f ∈ Λα (R). However, there is a distinction between the case of unitary operators and the case of selfadjoint operators. To describe the class of functions f on R, for which f (A) − f (B) ∈ S q , whenever A − B ∈ S p , we have to introduce the space Λα of functions on R that satisfy the Hölder condition of order α uniformly on all intervals of length 1. We are going to deal in this section with Hankel operators on the Hardy class H 2 (C+ ) of functions analytic in the upper half-plane C+ . Recall that the space L2 (R) can be represented as L2 (R) = H 2 (C+ ) ⊕ H 2 (C− ), where H 2 (C− ) is the Hardy class of functions analytic in the lower half-plane C− . We denote by P + and P − the orthogonal projections onto H 2 (C+ ) and H 2 (C− ). For a function ϕ in L∞ (R), the Hankel operator Hϕ : H 2 (C+ ) → H 2 (C− ) is defined by def

Hϕ g = P − ϕg,

g ∈ H 2 (C+ ).

3712


As in the case of Hankel operators on the Hardy class H 2 of functions analytic in D, the Hankel operators Hϕ of class S p can be described in terms of Besov spaces: Hϕ ∈ S p

⇐⇒

1/p

P− ϕ ∈ Bp (R),

0 < p < ∞,

(9.1)

where the operator P− on L∞ (R) is defined by def P− ϕ = P− (ϕ ◦ ω) ◦ ω−1 ,

ϕ ∈ L∞ (R),

def

and ω(ζ ) = i(1 + ζ )(1 − ζ )−1 , ζ ∈ T. This was proved in [25] for p 1, and in [27] and [35] for 0 < p < 1, see also [30], Ch. 6. Note also that by Kronecker’s theorem, Hϕ has finite rank if and only if P− ϕ is a rational function (see [30], Ch. 1). Recall that the Hilbert transform H is defined on L2 (R) by H g = −ig+ + ig− , where we use def

def

the notation g+ = P + g and g− = P − g. Theorem 9.1. Let A and B be bounded self-adjoint operators on Hilbert space such that rank(A − B) < ∞ and let α > 0. Then f (A) − f (B) ∈ S 1 ,∞ for every function f in Λα (R). α

Proof. Consider the Cayley transforms of A and B: U = (A − iI )(A + iI )−1

and V = (B − iI )(B + iI )−1 .

It is well known that U and V are unitary operators. Moreover, it is easy to see that rank(U − V ) < ∞. Indeed, (A − iI )(A + iI )−1 − (B − iI )(B + iI )−1 = 2i (B + iI )−1 − (A + iI )−1 = 2i(A + iI )−1 (A − B)(B + iI )−1 , and so rank(U − V ) rank(A − B). Without loss of generality, we may assume that f has compact support. Otherwise, we can multiply f by an infinitely smooth function with compact support that is equal to 1 on an interval containing the spectra of A and B. Consider the function h on T defined by h(ζ ) = f (−i(ζ + i)(ζ − i)−1 ). Obviously, h ∈ Λα . By Theorem 8.2, h(U ) − h(V ) ∈ S 1/α,∞ . It remains to observe that h(U ) = f (A) and h(V ) = f (B). 2 In Section 5 we have proved Theorem 5.2 that says that the condition f ∈ Λα (R), 0 < α < 1, implies that f (A) − f (B) ∈ S 1/α,∞ , whenever A − B ∈ S 1 . On the other hand, by Theorem 5.3, α (R), 0 < α < 1, implies that f (A) − f (B) ∈ S the stronger condition f ∈ B∞1 1/α , whenever A−B ∈ S 1 . The following result gives a necessary condition on f for the assumption A−B ∈ S 1 to imply that f (A) − f (B) ∈ S 1/α . It shows that the condition f ∈ Λα (R) does not ensure that f (A) − f (B) ∈ S 1/α even under the much stronger assumption that A − B has finite rank.


3713

Theorem 9.2. Let f be a continuous function on R and let p > 0. Suppose that f (A) − f (B) ∈ S p , whenever A and B are bounded self-adjoint operators such that 1/p rank(A − B) < ∞. Then f ◦ h ∈ Bp (R) for every rational function h that is real on R and has no pole at ∞. Proof. Let ϕ ∈ L∞ (R) and let Mϕ denote multiplication by ϕ. For g ∈ L2 (R), we have Mϕ g − H −1 Mϕ H g = ϕg + H ϕ(H g) = ϕ − (ϕg+ )+ + (ϕg− )+ + (ϕg+ )− − (ϕg− )− = ϕ(g+ + g− ) − (ϕg+ )+ + (ϕg− )+ + (ϕg+ )− − (ϕg− )− = 2(ϕg+ )− + 2(ϕg− )+ = 2Hϕ g+ + 2Hϕ∗ g− .

(9.2)

Hence, by (9.1), Mϕ − H −1 Mϕ H ∈ S p if and only if ϕ ∈ Bp (R). Moreover, by Kronecker’s theorem, rank(Mϕ − H −1 H ϕ H ) < +∞ if and only if ϕ is a rational function. Suppose now that h is a rational function that takes real values on R and has no pole at ∞. Define the bounded self-adjoint operators A and B by 1/p

def

A = Mh ,

B = H −1 Mh H . def

and

By (9.2), rank(A − B) < ∞. Again, by (9.2) with ϕ = f ◦ h, the f (A) − f (B) = Mf ◦h − H −1 Mf ◦h H 1/p

belongs to S p if and only if f ◦ h ∈ Bp (R).

2 1/p

Note that the conclusion of Theorem 9.2 implies that f belongs to Bp (R) locally, i.e., the 1/p restriction of f to an arbitrary finite interval can be extended to a function of class Bp (R). Now we are going to show that Theorem 5.2 cannot be improved even under the assumption that rank(A − B) = 1. Denote by L2e (R) the set of even functions in L2 (R) and by L2o (R) the set of odd functions in 2 L (R). Clearly, L2 (R) = L2e (R) ⊕ L2o (R). Let ϕ be an even function in L∞ (R). Then L2e (R) and L2o (R) are invariant subspaces of the operators Mϕ and H −1 Mϕ H . The orthogonal projections Pe and Pe onto L2e (R) and L2o (R) are given by (Pe g)(x) =

1 g(x) + g(−x) 2

and (Po g)(x) =

1 g(x) − g(−x) . 2

Lemma 9.3. Let ϕ(x) = (x 2 + 1)−1 , x ∈ R. Then (H ϕ)(x) = x(x 2 + 1)−1 and Mϕ f − H −1 Mϕ H f =

1 1 (f, ϕ)ϕ − (f, H ϕ)H ϕ. π π

3714


In particular, # Mϕ f − H

−1

Mϕ H f =

Proof. It is easy to see that ϕ+ (x) = x ∈ R. Hence,

1 π (f, ϕ)ϕ, − π1 (f, H ϕ)H ϕ,

i 2(x+i) ,

f is even, f is odd.

i ϕ− (x) = − 2(x−i) , and (H ϕ)(x) = x(x 2 + 1)−1 ,

Mϕ f − H −1 Mϕ H f = ϕf + H ϕH f = 2(ϕf+ )− + 2(ϕf− )+ = 2(ϕ− f+ )− + 2(ϕ+ f− )+ f− (−i) f+ f− f+ (i) +i = −i +i = −i x −i − x+i + x−i x +i 1 1 1 1 1 1 f, − f, =− 2π x + i x − i 2π x −i x +i 2 2 (f, ϕ+ )ϕ− + (f, ϕ− )ϕ+ π π 1 1 (f, ϕ + iH ϕ)(ϕ − iH ϕ) + (f, ϕ − iH ϕ)(ϕ + iH ϕ) = 2π 2π 1 1 = (f, ϕ)ϕ − (f, H ϕ)H ϕ. 2 π π =

Corollary 9.4. Let ϕ(x) = (x 2 + 1)−1 , x ∈ R. Then rank(Mϕ − H −1 Mϕ H ) = 2, rank(Pe (Mϕ − H −1 Mϕ H )Pe ) = 1 and rank(Po (Mϕ − H −1 Mϕ H )Po ) = 1. Lemma 9.5. Let ϕ be an even function in L∞ (R). Then √ sn Mϕ − H −1 Mϕ H Pe 2sn (Hϕ ) and √ sn Mϕ − H −1 Mϕ H Po 2sn (Hϕ ). Proof. Note that P − (Mϕ − H −1 Mϕ H )|H 2 (C+ ) = 2Hϕ . It remains to observe that isometrically from L2e (R) onto H 2 (C+ ) and from L2o (R) onto H 2 (C+ ). 2

√ 2P + acts

Lemma 9.6. There exists a function ρ ∈ C ∞ (T) such that ρ(ζ ) + ρ(iζ ) = 1, ρ(ζ ) = ρ(ζ¯ ) for all ζ ∈ T, and ρ vanishes in a neighborhood of the set {−1, 1}. Proof. Fix a function ψ ∈ C ∞ (T) such that ψ vanishes in a neighborhood of the set {−1, 1}, ψ 0, and ψ(ζ ) + ψ(iζ ) > 0 for all ζ ∈ T. Put ρ0 (ζ ) =

ψ(ζ ) + ψ(−ζ ) . ψ(ζ ) + ψ(iζ ) + ψ(−ζ ) + ψ(−iζ )


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Clearly, ρ0 vanishes in a neighborhood of the set {−1, 1}, ρ0 0, and ρ0 (ζ ) + ρ0 (iζ ) = 1 for all ζ ∈ T. It remains to put ρ(ζ ) = 12 (ρ0 (ζ ) + ρ0 (ζ¯ )). 2 In what follows we fix such a function ρ. Lemma 9.7. Let g be a function in Λα such that g(iζ ) = g(ζ ) for all ζ ∈ T. Suppose that infn0 (n + 1)α sn (Hg ) > 0. Then infn0 (n + 1)α sn (Hρg ) > 0. Proof. Fix a positive p such that αp < 1. Clearly, there exists a positive number c1 such that p p Hg S n c1 n1−αp for all n 0. Note that Hρ(z)g(z) S np = Hρ(iz)g(z) S np . Hence, Hρg S n p

p

1 1−αp for all n 0. By Lemma 8.3, there exists a positive number c such that H p 2 ρg S n 2 c1 n p p p c2 n1−αp for all n 1. Hence, there exists an integer M such that Hρg Mn − Hρg S n n1−αp S

Hρg Thus (sn (Hρg ))p

p

p

for all n 1. Note that

1 −αp M−1 n

p S Mn p

p p − Hρg S n (M − 1)n sn (Hρg ) .

for all n 1.

p

2

Lemma 9.8. There exists a real function g0 ∈ Λα that vanishes in a neighborhood of the set {−1, 1} and such that g0 (ζ ) = g0 (ζ¯ ), ζ ∈ T, and sn (Hg0 ) (n + 1)−α for all n 0. def

Proof. Let g is the function given by (8.4). We can put g0 = Cρg for a sufficiently large number C. 2 Theorem 9.9. Let α > 0. Let ϕ(x) = (x 2 + 1)−1 . Consider the operators A and B on L2e (R) defined by Ag = H −1 Mϕ H g and Bg = ϕg. Then (i) rank(A − B) = 1, (ii) there exists a real bounded function f ∈ Λα (R) such that sn f (A) − f (B) (n + 1)−α ,

n 0.

Proof. The equality rank(A − B) = 1 is a consequence of Corollary 9.4. Let g0 be the function obtained in Lemma 9.8. It is easy to see that there exists a real bounded function f ∈ Λα (R) such that f (ϕ(x)) = g0 ( x−i x+i ). It is well known (see [30], Ch. 1, Section 8) that Hf ◦ϕ can be obtained from Hg0 by multiplying on the left and on the right by unitary operators. Hence, by Lemma 9.5, √ √ √ sn f (B) − f (A) 2sn (Hf ◦ϕ ) = 2sn (Hg0 ) 2(n + 1)−α .

2

Remark. The same result holds if we consider operators A and B on L2o (R) defined in the same way. In Section 5 we have obtained sufficient conditions on a function f on R for the condition A − B ∈ S p to imply that f (A) − f (B) ∈ S q for certain p and q. We are going to obtain here necessary conditions and consider other values p and q.

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As in the case of functions on T, we consider the space Lip(R) of Lipschitz functions on R such that $ ! |f (x) − f (y)| def f Lip(R) = sup : x, y ∈ R, x = y < +∞. |x − y| For α ∈ (0, 1], we denote by Λα the set of all functions defined on R and satisfying the Hölder condition of the order α uniformly on all intervals of a fixed length: f Λα

$ |f (x) − f (y)| = sup : x, y ∈ R, x = y, |x − y| 1 < +∞. |x − y|α

def

!

Clearly, f ∈ Λα if and only if ωf (δ) const δ α for δ ∈ (0, 1]. Note that Λ1 = Lip(R). Lemma 9.10. Let 0 < α < 1. Then Λα = Λα (R) + Lip(R). Proof. The inclusion Λα (R) + Lip(R) ⊂ Λα is evident. Let f ∈ Λα . We can consider the piecewise linear function f0 such that f0 (n) = f (n) and f0 |[n, n + 1] is linear for all n ∈ Z. Clearly, f0 ∈ Lip(R) and f − f0 ∈ Λα (R). 2 Denote by SA(S p , S q ) the set of all continuous functions f on R such that f (B)−f (A) ∈ S q , whenever A and B are self-adjoint operators such that B − A ∈ S p . We also denote by SAc (S p , S q ) the set of all continuous functions f on R such that f (B) − f (A) ∈ S q , whenever A and B are commuting self-adjoint operators such that B − A ∈ Sp . As in the case of unitary operators we say that the class SA(S p , S q ) (or SAc (S p , S q )) is trivial if it contains only constant functions. Theorem 9.11. Let 0 < p, q < +∞. Then SAc (S p , S q ) = Λp/q for p q and the space SAc (S p , S q ) is trivial for p > q. Proof. To prove the inclusion Λp/q ⊂ SAc (S p , S q ), it suffices to observe that f ∈ SAc (S p , S q ) if and only if for every two sequences {xn } and {yn } in R

|xn − yn |p < +∞

⇒

f (xn ) − f (yn ) q < +∞.

(9.3)

Condition (9.3) easily implies that ωf (δ) < +∞ for some δ > 0, and so for all δ > 0. To complete the proof, we have to prove that (9.3) implies that ωf (δ) Cδ p/q for δ ∈ (0, 1]. This can be done in exactly the same way as in the case of unitary operators, see the proof of Theorem 8.5. 2 The following result is an immediate consequence of Theorem 9.11. Theorem 9.12. Let 0 < p, q < +∞. Then SA(S p , S q ) ⊂ Λp/q for p q and SA(S p , S q ) is trivial for p > q.


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Theorem 9.13. Let 1 < p q < +∞. Then SA(S p , S q ) = Λp/q . Proof. In view of Theorem 9.12, we have to prove that Λp/q ⊂ SA(S p , S q ). In the case p = q this was proved by Potapov and Sukochev [34]. Suppose now that p < q. By Lemma 9.10, it is sufficient to verify that Lip(R) ⊂ SA(S p , S q ) and Λp/q (R) ⊂ SA(S p , S q ). The first inclusion follows from the results of [34] as well as from the results of [22]. Indeed, Lip(R) ⊂ SA(S p , S p ) ⊂ SA(S p , S q ). The inclusion Λp/q (R) ⊂ SA(S p , S q ) follows from Theorem 5.8. 2 Theorem 9.14. If 0 < p, q 1, then Lip(R) ⊂ SA(S p , S q ). If 0 < α < 1, 0 < p 1, and 0 < q 1/α, then Λα (R) ⊂ SA(S p , S q ). Proof. The result follows from Theorem 9.2. Indeed, there exists a function in Lip(R) which does not belong to B11 (R) locally, and for each α ∈ (0, 1) there exists a function in Λα (R) that α (R) locally. 2 does not belong to B1/a Theorem 9.15. Let 0 < q, p < +∞. Then Λp/q (R) ⊂ SA(S p , S q ) if and only if 1 < p < q. Proof. If 1 < q, p < +∞ or q < p, the result follows from Theorems 9.13 and 9.12. If p q and p 1, then Λp/q (R) ⊂ SA(S p , S q ) by Theorem 9.14. 2 Theorem 9.16. Let 0 < p, q < +∞. Then Lip(R) ⊂ SA(S p , S q ) if and only if 1 < p q or p 1 < q. Proof. In the same way as in the proof of Theorem 9.15, we see that it suffices to consider the case p 1. From the results of [22] or the results of [34] it follows that Lip(R) ⊂ SA(S 1 , S q ) ⊂ SA(S p , S q ) if p 1 < q. The converse follows from Theorem 9.14. 2 Now we are going to obtain a quantitative refinement of Theorem 9.12. Let f ∈ C(R). Put

def Ωf,p,q (δ) = sup f (A) − f (B)S : A − BS p δ, A, B are self-adjoint operators . q

It is easy to see that given q > 0, there exists a positive number cq such that Ωf,p,q (2δ) cq Ωf,p,q (δ). Theorem 9.17. Let 0 < p, q < ∞ and let f ∈ SA(S p , S q ). Then Ωf,p,q (δ) < +∞ for all δ > 0 and Ωf,p,q (δ) Ωf,p,q (δ) Ωf,p,q (δ) Ωf,p,q (δ) = inf sup = lim , p/q p/q p/q δ→+∞ δ→0 δ>0 δ δ δ δ p/q δ>0 lim

(both limits exist and take values in [0, ∞]). In particular, if f is a nonconstant function, then Ωf,p,q (δ) c1 δ p/q for every δ ∈ (0, 1] and Ωf,p,q (δ) c2 δ p/q for every δ ∈ [1, +∞), where c1 and c2 are positive numbers. The proof of Theorem 9.17 is the same as that of Theorem 8.13.

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Theorem 9.18. Let f ∈ C(R) and p ∈ [1, +∞). Then either Ωf,p,p (δ) = +∞ for all δ > 0 or Ωf,p,p is a linear function. Proof. If f ∈ / SA(S p , S p ), then Ωf,p,p (δ) = +∞ for all δ > 0. Suppose now that f ∈ SA(S p , S p ). By the analog of Lemma 8.12 for self-adjoint operators (it is easy to see that it holds for self-adjoint operators), Ωf,p,p (nδ) nΩf,p,p (δ) for all positive integer n. On the other hand, clearly, Ωf,p,p (nδ) nΩf,p,p (δ) for all positive integer n. Hence, Ωf,p,p is a linear function. 2 10. Spectral shift function for second order differences In this section we obtain trace formulae for second order differences in the case of self-adjoint operators and unitary operators. By Theorem 5.11, if A is a self-adjoint operator, K is a self-adjoint operator of class S 2 and 2 (R), then f (A + K) − 2f (A) + f (A − K) ∈ S . We are going to obtain a formula for f ∈ B∞1 1 the trace of this operator. In this section m denotes Lebesgue measure on the real line. Theorem 10.1. Let A be a self-adjoint operator and K a self-adjoint operator of class S 2 . Then 2 (R), there exists a unique function ς ∈ L1 (R) such that for every f ∈ B∞1 trace f (A + K) − 2f (A) + f (A − K) =

f (x)ς(x) dm(x).

(10.1)

R

Moreover, ς(x) 0, x ∈ R. Definition. The function ς satisfying (10.1) is called the second order spectral shift function associated with the pair (A, K). We are going to use the spectral shift function of Koplienko. Koplienko proved in [15] that with each pair of a self-adjoint operator A and a self-adjoint operator K of class S 2 , there exists a function η ∈ L1 (R) such that η 0 and for every rational function f with poles off R, the following trace formula holds

d = f (x)η(x) dm(x). trace f (A + K) − f (A) − f (A + tK) t=0 dt

(10.2)

R

The function η is called the Koplienko spectral shift function associated with the pair (A, K). 2 (R). Note that Note that later in [31] it was proved that trace formula (10.2) holds for f ∈ B∞1 the derivative d f (A + tK) t=0 dt


3719

1 (R) (see [29] and [32]) and does not have to exist under the exists under the condition f ∈ B∞1 2 2 (R), by condition f ∈ B∞1 (R). However, in the case f ∈ B∞1

f (A + K) − f (A) −

d f (A + tK) t=0 dt

we can understand n∈Z

d fn (A + K) − fn (A) − fn (A + tK) , t=0 dt def

and the series converges absolutely, see [31]. Here, as usual, fn = f ∗ Wn + f ∗ Wn . Proof of Theorem 10.1. Let η1 be the spectral shift function associated with the pair (A, K) and let η2 be the spectral shift function associated with the pair (A, −K). We have

d = f (x)η1 (x) dm(x) trace f (A + K) − f (A) − f (A + tK) t=0 dt R

and

d = f (x)η2 (x) dm(x) trace f (A − K) − f (A) − f (A − tK) t=0 dt R

2 (R). Taking the sum, we obtain for f ∈ B∞1

trace f (A + K) − 2f (A) + f (A − K) =

f (x) η1 (x) + η2 (x) dm(x).

R def

It remains to put ς = η1 + η2 . Uniqueness is obvious. 2 We proceed now to the case of unitary operators. Suppose that U is a unitary operator and V 2 , then is a unitary operator such that I − V ∈ S 2 . It follows from Theorem 6.11 that if f ∈ B∞1 ∗ f (V U ) − 2f (U ) + 2(V U ) ∈ S 1 . We are going to obtain a trace formula for this operator. Theorem 10.2. Let U be a unitary operator and let V be a unitary operator such that I −V ∈ S 2 . Then there exists an integrable function ς on T such that trace f (V U ) − 2f (U ) + 2 V ∗ U =

T

f ς dm.

(10.3)

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It is easy to see that ς is determined by (10.3) modulo a constant. It is called a second order spectral shift function associated with the pair (U, V ). We are going to use a trace formula of Neidhardt [21], which is an analog of the Koplienko trace formula for unitary operators. Suppose that U and V be unitary operators such that U − V ∈ S 2 . Then V can be represented as V = eiA U , where A is a self-adjoint operator of class S 2 whose spectrum σ (A) is a subset of (−π, π]. It was shown in [21] that one can associate with the pair (U, V ) a function η in L1 (T) (a Neidhardt spectral shift function) such that if the second derivative f of a function f has absolutely converging Fourier series, then

d isA = f η dm. f e U trace f (V ) − f (U ) − s=0 ds

(10.4)

T

2 . Later it was shown in [31] that formula (10.4) holds for an arbitrary function f in B∞1

Proof of Theorem 10.2. We can represent V as V = eiA , where A is a self-adjoint operator of def

class S 2 such that the spectrum σ (A) of A is a subset of (−π, π]. Let V1 = V U . Clearly, V1 is a def

unitary operator and U − V1 ∈ S 2 . We have V1 = eiA U . Let V2 = V ∗ U . Then U − V2 ∈ S 2 and V2 = e−iA U . Let η1 be the Neidhardt spectral shift function associated with (U, V1 ) and let η2 be the Neidhardt spectral shift function associated with (U, V2 ). We have

d isA = f η1 dm f e U trace f (V1 ) − f (U ) − s=0 ds T

and

d −isA f e U trace f (V2 ) − f (U ) − = f η2 dm s=0 ds T

2 . Taking the sum, we obtain for f ∈ B∞1

trace f (V U ) − 2f (U ) + 2 V ∗ U = trace f (V1 ) − 2f (U ) + f (V2 )

= f (η1 + η2 ) dm. T def

It remains to put ς = η1 + η2 .

2

11. Commutators and quasicommutators In this section we obtain estimates for the norm of quasicommutators f (A)R − Rf (B) in terms of AR − RB for self-adjoint operators A and B and a bounded operator R. We assume for simplicity that A and B are bounded. However, we obtain estimates that do not depend on


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the norms of A and B. In [3] we will consider the case of not necessarily bounded operators A and B. Note that in the special case A = B, this problem turns into the problem of estimating the norm of commutators f (A)R − Rf (A) in terms of AR − RA. On the other hand, in the special case R = I the problem turns into the problem of estimating f (A) − f (B) in terms A − B. Similar results can be obtained for unitary operators and for contractions. Birman and Solomyak (see [11]) discovered the following formula

f (A)R − Rf (B) =

f (x) − f (y) dEA (x) (AR − RB) dEB (y), x−y

(11.1)

whenever f is a function, for which Df is a Schur multiplier of class M(EA , EB ) (see Section 4). Theorem 11.1. Let 0 < α < 1. There exists a positive number c > 0 such that for every l 0, p ∈ [1, ∞), f ∈ Λα (R), for arbitrary bounded self-adjoint operators A and B and an arbitrary bounded operator R, the following inequality holds: sj f (A)R − Rf (B) cf Λα (R) (1 + j )−α/p R1−α AR − RBαS l

p

for every j l.

Proof. Clearly, we may assume that R = 0. As usual, fn = f ∗ Wn + f ∗ Wn , n ∈ Z. Fix an integer N . We have by (11.1) and (4.9), N fn (A)R − Rfn (B) n=−∞

S lp

N fn (A)R − Rfn (B) n=−∞

const

N

S lp

2n fn L∞ AR − RBS l

p

n=−∞

const 2N (1−α) f Λα (R) AR − RBS l . p

On the other hand, fn (A)R − Rfn (B) 2R fn L∞ n>N

n>N

constf Λα (R) R

2−nα

n>N

const 2−N α f Λα (R) R.

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Put N fn (A)R − Rfn (B)

def

XN =

def

and YN =

n=−∞

fn (A)R − Rfn (B) . n>N

Clearly, for j l, −1 sj f (A)R − Rf (B) sj (XN ) + YN (1 + j ) p AR − RBS l + YN p

−1 constf Λα (R) (1 + j ) p 2N (1−α) AR − RBS l + 2−N α R . p

To obtain the desired estimate, it suffices to choose the number N so that 2−N < (1 + j )−1/p AR − RBS l R−1 2−N +1 . p

2

The proofs of the remaining results of this section are the same as those of the results of Section 5 for first order differences. Theorem 11.2. Let 0 < α < 1. There exists a positive number c > 0 such that for every f ∈ Λα (R), for arbitrary bounded self-adjoint operators A and B with AR − RB ∈ S 1 and an arbitrary bounded operator R, the operator f (A)R − Rf (B) belongs to S 1/α,∞ and the following inequality holds: f (A)R − Rf (B)

S 1/α,∞

cf Λα (R) R1−α AR − RBαS 1 .

Theorem 11.3. Let 0 < α 1. There exists a positive number c > 0 such that for every f ∈ α (R), for arbitrary bounded self-adjoint operators A and B with AR − RB ∈ S and an B∞1 1 arbitrary bounded operator R, the operator f (A)R − Rf (B) belongs to S 1/α and the following inequality holds: f (A)R − Rf (B)

S 1/α

1−α α (R) R cf B∞1 AR − RBαS 1 .

Theorem 11.4. Let 0 < α < 1. There exists a positive number c > 0 such that for every f ∈ Λα (R), for arbitrary bounded self-adjoint operators A and B and an arbitrary bounded operator R on Hilbert space, the following inequality holds: 1/α 1−α 1/α sj f (A)R − Rf (B) cf Λα (R) R α σj (AR − RB),

j 0.

Theorem 11.5. Let 0 < α < 1. There exists a positive number c > 0 such that for every f ∈ Λα (R), for an arbitrary quasinormed ideal I with βI < 1, for arbitrary bounded self-adjoint operators A and B with AR − RB ∈ I, the operator |f (A)R − Rf (B)|1/α belongs to I and the following inequality holds: f (A)R − Rf (B) 1/α cC I f 1/α R 1−α α AR − RBI . Λα (R) I


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Theorem 11.6. Let 0 < α < 1 and 1 < p < ∞. There exists a positive number c such that for every f ∈ Λα (R), every l ∈ Z+ , for arbitrary bounded self-adjoint operators A and B and an arbitrary bounded operator R, the following inequality holds: l l 1/α p p 1−α p/α sj f (A)R − Rf (B) sj (AR − RB) . cf Λα (R) Rp α j =0

j =0

Theorem 11.7. Let 0 < α < 1 and 1 < p < ∞. There exists a positive number c such that for every f ∈ Λα (R), for arbitrary bounded self-adjoint operators A and B, and for an arbitrary bounded operator R, the operator f (A)R − Rf (B) belongs to S p/α and the following inequality holds: f (A)R − Rf (B)

S p/α

cf Λα (R) R1−α AR − RBαS p .

References [1] A.B. Aleksandrov, V.V. Peller, Functions of perturbed operators, C. R. Math. Acad. Sci. Paris Sér. I 347 (2009) 483–488. [2] A.B. Aleksandrov, V.V. Peller, Operator Hölder–Zygmund functions, Adv. Math. (2010), doi:10.1016/j.aim. 2009.12.018, in press. [3] A.B. Aleksandrov, V.V. Peller, Functions of perturbed unbounded self-adjoint operators, in preparation. [4] A.B. Aleksandrov, V.V. Peller, Functions of perturbed dissipative operators, in preparation. [5] N.A. Azamov, A.L. Carey, P.G. Dodds, F.A. Sukochev, Operator integrals, spectral shift and spectral flow, Canad. J. Math. 61 (2009) 241–263. [6] M.S. Birman, L.S. Koplienko, M.Z. Solomyak, Estimates of the spectrum of a difference of fractional powers of selfadjoint operators, Izv. Vyssh. Uchebn. Zaved. Mat. 154 (3) (1975) 3–10 (in Russian); English translation: Soviet Math. (Iz. VUZ) 19 (3) (1975) 1–6. [7] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals, in: Problems Math. Phys., vol. 1, Leningrad. Univ., 1966, pp. 33–67 (in Russian); English translation in: Topics Math. Phys., vol. 1, Consultants Bureau Plenum Publishing Corporation, New York, 1967, pp. 25–54. [8] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals. II, in: Problems Math. Phys., vol. 2, Leningrad. Univ., 1967, pp. 26–60 (in Russian); English translation in: Topics Math. Phys., vol. 2, Consultants Bureau Plenum Publishing Corporation, New York, 1968, pp. 19–46. [9] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals. III, in: Problems Math. Phys., vol. 6, Leningrad. Univ., 1973, pp. 27–53 (in Russian). [10] M.Sh. Birman, M.Z. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Math. Appl. (Sov. Ser.), D. Reidel Publishing Co., Dordrecht, 1987. [11] M.S. Birman, M.Z. Solomyak, Double operator integrals in Hilbert space, Integral Equations Operator Theory 47 (2003) 131–168. [12] Yu.L. Daletskii, S.G. Krein, Integration and differentiation of functions of Hermitian operators and application to the theory of perturbations, Trudy Sem. Funktsion. Anal. Voronezh. Gos. Univ. 1 (1956) 81–105 (in Russian). [13] Yu.B. Farforovskaya, An example of a Lipschitzian function of selfadjoint operators that yields a nonnuclear increase under a nuclear perturbation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30 (1972) 146–153 (in Russian). [14] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Nauka, Moscow, 1965; English translation: Amer. Math. Soc., Providence, RI, 1969. [15] L.S. Koplienko, The trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (5) (1984) 62–71 (in Russian); English translation: Sib. Math. J. 25 (1984) 735–743. [16] M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597–626 (in Russian). [17] S.G. Krein, Yu.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English translation: Transl. Math. Monogr., vol. 54, American Mathematical Society, Providence, RI, 1982.

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[18] E. Kissin, V.S. Shulman, Classes of operator-smooth functions. III. Stable functions and Fuglede ideals, Proc. Edinb. Math. Soc. (2) 48 (1) (2005) 175–197. [19] S.J. Montgomery-Smith, The Hardy operator and Boyd indices, in: Interaction Between Functional Analysis, Harmonic Analysis, and Probability, Columbia, MO, 1994, in: Lect. Notes Pure Appl. Math., vol. 175, Dekker, New York, 1996, pp. 359–364. [20] S.N. Naboko, Estimates in operator classes for the difference of functions from the Pick class of accretive operators, Funktsional. Anal. i Prilozhen. 24 (3) (1990) 26–35 (in Russian); English translation: Funct. Anal. Appl. 24 (1990) 187–195. [21] H. Neidhardt, Spectral shift function and Hilbert–Schmidt perturbation: extensions of some work of L.S. Koplienko, Math. Nachr. 138 (1988) 7–25. [22] F.L. Nazarov, V.V. Peller, Lipschitz functions of perturbed operators, C. R. Math. Acad. Sci. Paris Sér. I 347 (2009) 857–862. [23] B.S. Pavlov, On multiple operator integrals, in: Linear Operators and Operator Equations, in: Problems Math. Anal., vol. 2, Izdat. Leningrad. Univ., Leningrad, 1969, pp. 99–122 (in Russian). [24] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Press, Durham, NC, 1976. [25] V.V. Peller, Hankel operators of class Sp and their applications (rational approximation, Gaussian processes, the problem of majorizing operators), Mat. Sb. 113 (1980) 538–581; English translation: Math. USSR Sb. 41 (1982) 443–479. [26] V.V. Peller, A description of Hankel operators of class Sp for p > 0, an investigation of the rate of rational approximation, and other applications, Mat. Sb. 122 (1983) 481–510; English translation: Math. USSR Sb. 50 (1985) 465–494. [27] V.V. Peller, Hankel operators in the theory of perturbations of unitary and self-adjoint operators, Funktsional. Anal. i Prilozhen. 19 (2) (1985) 37–51 (in Russian); English translation: Funct. Anal. Appl. 19 (1985) 111–123. [28] V.V. Peller, For which f does A − B ∈ Sp imply that f (A) − f (B) ∈ Sp ?, in: Oper. Theory Adv. Appl., vol. 24, Birkhäuser, Basel, 1987, pp. 289–294. [29] V.V. Peller, Hankel operators in the perturbation theory of unbounded self-adjoint operators, in: Analysis and Partial Differential Equations, in: Lect. Notes Pure Appl. Math., vol. 122, Dekker, New York, 1990, pp. 529–544. [30] V.V. Peller, Hankel Operators and Their Applications, Springer-Verlag, New York, 2003. [31] V.V. Peller, An extension of the Koplienko–Neidhardt trace formulae, J. Funct. Anal. 221 (2005) 456–481. [32] V.V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct. Anal. 233 (2006) 515–544. [33] V.V. Peller, Differentiability of functions of contractions, in: Linear and Complex Analysis, in: Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 109–131. [34] D. Potapov, F. Sukochev, Operator-Lipschitz functions in Schatten–von Neumann classes, arXiv:0904.4095, 2009. [35] S. Semmes, Trace ideal criteria for Hankel operators and applications to Besov classes, Integral Equations Operator Theory 7 (1984) 241–281. [36] V.V. Sten’kin, Multiple operator integrals, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (79) (1977) 102–115 (in Russian); English translation: Soviet Math. (Iz. VUZ) 21 (4) (1977) 88–99. [37] B.Sz.- Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, Akadémiai Kiadó, Budapest, 1970.


Characterization of the critical Sobolev space on the optimal singularity at the origin Sei Nagayasu a , Hidemitsu Wadade b,∗ a Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan b Department of Mathematics, Taida Institute for Mathematical Sciences, National Taiwan University,

Taipei 106, Taiwan Received 10 September 2009; accepted 25 February 2010 Available online 10 March 2010 Communicated by H. Brezis

Abstract In the present paper, we investigate the optimal singularity at the origin for the functions belonging to the n

critical Sobolev space H p Nirenberg type inequality:

,p

(Rn ), 1 < p < ∞. With this purpose, we shall show the weighted Gagliardo–

1 uLq (Rn ; dx ) C n−s |x|s

1 + 1 q

p

(n−s)p 1 n 1− (n−s)p nq nq q p uLp (Rn ) (−) 2p uLp (Rn ) ,

(GN)

where C depends only on n and p. Here, 0 s < n and p˜ q < ∞ with some p˜ ∈ (p, ∞) determined only n p < ∞, we can prove the growth orders for s as by n and p. Additionally, in the case n 2 and n−1 s ↑ n and for q as q → ∞ are both optimal. (GN) allows us to prove the Trudinger type estimate with the homogeneous weight. Furthermore, it is obvious that (GN) cannot hold with the weight |x|n itself. However, 1 )r |x|n at the origin, we cover this critical weight. with a help of the logarithmic weight of the type (log |x| Simultaneously, we shall give the minimal exponent r = q+p p so that the continuous embedding can hold. © 2010 Elsevier Inc. All rights reserved.

Keywords: Sobolev embedding theorem; Gagliardo–Nirenberg type inequality; Trudinger type inequality; Caffarelli–Kohn–Nirenberg inequality


E-mail addresses: [email protected] (S. Nagayasu), [email protected] (H. Wadade). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.015

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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757

1. Introduction and main results n

,p

In the present paper, we give some characterization of the functions in H p (Rn ) with n ∈ N and 1 < p < ∞ called the critical Sobolev space in the sense that the continuous embedding n n ,p ,p H p (Rn ) → Lq (Rn ) holds for all p q < ∞, but H p (Rn ) ⊂ L∞ (Rn ) which implies that n ,p n n H p (R ) possibly has a singularity at some point. Indeed, at least in the case n 2 and n−1 1 τ 1 p < ∞, a compactly supported function such as [log( |x| )] with 0 < τ < p at the origin implies the failure of the embedding in the case q = ∞, see Lemma 2.6 in Section 2. More precisely, Ozawa [11] gave the Gagliardo–Nirenberg type estimate of the following type: p 1 1− pq n uLq (Rn ) Cq p uLq p (Rn ) (−) 2p uLp (R n) n

(1.1)

p holds for all u ∈ H p (Rn ) and p q < ∞, where C depends only on n and p, and p := p−1 denotes the Hölder conjugate exponent of p. The inequality (1.1) was originally obtained by n Ogawa [9] and Ogawa and Ozawa [10] in the case p = 2, i.e., H 2 ,2 (Rn ). Moreover, we refer n ,p to Kozono and Wadade [6] which treats the marginal case of (1.1) as p → ∞ in H p (Rn ). In fact, the functions having bounded mean oscillation BMO can be expressed as the limit case of n n ,p H p (Rn ) with p = ∞ in some sense, and [6] proved (1.1) with (−) 2p uLp (Rn ) replaced by uBMO . In addition, Wadade [18] is also a generalization of (1.1) in terms of the Besov and the Triebel–Lizorkin spaces. Our purpose in this article is to generalize (1.1) with the weighted Lebesgue space. In general, dx ) as the function space endowed for a measurable weight function w(x), we define Lq (Rn ; w(x) with the norm: ,p

uLq (Rn ;

dx w(x) )

:= Rn

u(x)q dx w(x)

1 q

for 1 < q < ∞.

We shall show the following inequality with the homogeneous weight w(x) = |x|s : Theorem 1.1. Let n ∈ N and 1 < p < ∞. Then there exist positive constants p˜ ∈ (p, ∞) and C which both depend only on n and p such that the inequality

1 uLq (Rn ; dxs ) C |x| n−s n

1 + 1 q

p

1 1−θ n q p uθLp (Rn ) (−) 2p uLp (Rn )

,p

holds for all u ∈ H p (Rn ), 0 s < n and p˜ q < ∞, where θ := 1

1

(n−s)p nq

(1.2)

∈ (0, 1). Further1

1 q + p p < ∞, the growth orders ( n−s ) as s ↑ n and q p as q → ∞ are more, if n 2 and 1 1 1 1 1 1 −ε 1 q + p 1 q + p −ε both optimal in the sense that we cannot replace ( n−s ) and q p by ( n−s ) and q p n n−1

for any small ε > 0, respectively.

Remark 1.2. (i) If we do not pay attention to the growth orders of s and q, the inequality (1.2) itself is shown by Caffarelli, Kohn and Nirenberg [1] with the first order derivative, i.e., pn = 1. However, we aim to obtain the optimal growth orders of s and q, and in fact we can prove that


3727

n those orders are optimal in the case n 2 and n−1 p < ∞. Unfortunately, we do not know n because of some the optimality in the cases n = 1 and 1 < p < ∞, or n 2 and 1 < p < n−1 technical reason, see Lemma 2.6 in Section 2. Moreover, we shall prove a weighted Trudinger 1

type estimate as an effect of this growth order q p as q → ∞, which will be stated below. (ii) The exponent p˜ actually can be chosen as p˜ := max{p + 1, p + 1, n + 1}. This restriction for the range of q will be used to prove Lemma 2.4. As stated in Remark 1.2(i), we can prove a weighted Trudinger type estimate as an application of Theorem 1.1: Corollary 1.3. Let n ∈ N, 1 < p < ∞, and define the function Φn,p by Φn,p (t) := exp t −

j 0 −1 j t j =0

j!

for t ∈ R with j0 := min j ∈ N; p j p˜ ,

where p˜ ∈ (p, ∞) is the positive constant given by Theorem 1.1. Then there exist two positive constants α and β which depend only on n and p such that Rn n

,p

(n−s)p p dx

β n u Φn,p α(n − s)u(x) p (Rn ) L |x|s n−s

n

holds for all u ∈ H p (Rn ) with (−) 2p uLp (Rn ) 1 and 0 s < n. Remark 1.4. The procedure to get the Trudinger type estimate from the Gagliardo–Nirenberg type estimate was originally seen in [9,10,12,11]. Especially, [12] clarified the relationship between the positive constants in the Trudinger and the Gagliardo–Nirenberg type estimates with the exact formula, which shows these two inequalities are actually equivalent each other. Next, we shall state the result which deals with the critical weight s = n in Theorem 1.1. Obviously, the inequality (1.1) cannot hold with the weight |x|n itself. However, with a help of the logarithmic weight, we shall show the following inequality: Theorem 1.5. Let n ∈ N, 1 < p < r < ∞ and p q (r − 1)p . Then there exists a positive constant C which depends only on n, p, q and r such that uLq (Rn ; n

dx wr (x) )

Cu

n ,p

Hp

(Rn )

(1.3)

,p

holds for all u ∈ H p (Rn ), where the weight function wr (x) is given by r 1 wr (x) := log e + |x|n . |x|

(1.4)

n Furthermore, if n 2 and n−1 p < ∞, the bound (r − 1)p is sharp in the sense that the inequality (1.3) no longer holds provided q > (r − 1)p .

3728


Remark 1.6. (i) There are more general results of such embeddings in case of Besov and Triebel– Lizorkin spaces including the Sobolev scale, cf. [5] and [7,8], but restricted to Muckenhoupt weights or so-called admissible weights, the former allows the weight to have a local singularity, while the latter is some class of smooth functions. We emphasize that these classes of weight functions do not cover the above limiting situation. Indeed, it is well known that the weight w1r as in (1.4) no longer belongs to even the class of Muckenhoupt weights. (ii) Since there exists an upper bound (r − 1)p with respect to q so that the inequality (1.3) holds, we cannot deduce the Trudinger type estimate from the inequality (1.3) unlike the case with the subcritical weight |x|s with 0 s < n. We additionally note that the critical exponent q = (r − 1)p comes from the following computation:

{|x|< 12 }

q 1 q p 1 dx log =∞ |x| wr (x)

provided that q (r − 1)p . Here, note that the marginal case q = (r − 1)p is included in the above observation. However, we shall overcome this difficulty to get (1.3) by using the generalized Young inequality by O’Neil [13], see Theorem B in Section 2. Finally let us describe the organization of this article. Section 2 is devoted to prepare the several lemmas for the proof of main theorems, and we shall show our theorems in Section 3. 2. Preliminaries This section is devoted to prepare several lemmas for the proof of main theorems. First, let us introduce the higher-dimensional Hardy inequality proved by Drábek, Heinig and Kufner [2]: Theorem A. (i) Let U1 and V1 be non-negative weight functions in Rn , and 1 R}

−(p −1)

V1 (x)

1 dx

p

< ∞.

{|x||x|}

Rn

3729

1

p

p

f (x) V2 (x) dx Rn

holds for all f 0 a.e. in Rn if and only if

1 q

A2 := sup

−(p −1)

U2 (x) dx

R>0

V2 (x)

{|x|R}

Moreover, the constant C2 can be taken as

1 1 C2 = p p p q A2 . By scaling and changing a variable, we have the following variant of Theorem A: Theorem A . (i) Let U1 and V1 be non-negative weight functions in Rn , and 1 < p q < ∞. Then the inequality

q

f (y) dy

Rn

1 q

U1 (x) dx

C˜ 1

{2|y|0

V1 (x)

1 dx

p

< ∞.

{|x|2R}

Moreover, the constant C˜ 1 can be taken as

1 1 C˜ 1 = p p p q A˜ 1 . (ii) Let U2 and V2 be non-negative weight functions in Rn , and 1 2|x|}

1 f (x)p V2 (x) dx

p

Rn

holds for all f 0 a.e. in Rn if and only if A˜ 2 := sup

R>0

1 q

U2 (x) dx {|x|2R}

1 dx

p

< ∞.

3730


Moreover, the constant C˜ 2 can be taken as

1 1 C˜ 2 = p p p q A˜ 2 . In what follows, C denotes a positive constant which may vary from line to line. We shall show key lemmas by applying Theorem A below. The idea of this procedure was inspired by Rakotondratsimba [14,15], who proved the weighted Young inequalities for convolutions towards the functions behaving like the Riesz potential |x|−(n−α) with 0 < α < n. However, we need to consider not only the Riesz potential but also ϕ, ψ and the Bessel potential Gα which are defined below, and for the purpose to get exact growth orders concerning s and q, we investigate these individual kernels precisely. Lemmas 2.1–2.4 will be used to prove Theorem 1.1 in Section 3. Lemma 2.1. Let n ∈ N, 1 < p < ∞ and ϕ(x) = e−π|x| . Then there exists a positive constant C which depends only on n and p such that 2

1

1 ϕ ∗ uLq (Rn ; dxs ) C |x| n−s

q

uLp (Rn )

(2.1)

holds for all u ∈ Lp (Rn ), 0 s < n and p q < ∞. Proof. Obviously, it is enough to show the inequality (2.1) for non-negative functions. First, we decompose the integral into three parts:

dx (ϕ ∗ u)(x) 3q |x|s

q

Rn

dx |x|s

ϕ(x − y)u(y) dy

q

Rn

{|y|< |x| 2 }

q

+

ϕ(x − y)u(y) dy

Rn

{ |x| 2 |y|2|x|}

q

+

ϕ(x − y)u(y) dy

Rn

{|y|>2|x|}

dx |x|s

dx |x|s

=: 3 (S1 + S2 + S3 ). q

We first estimate S1 . Note that |y| < ϕ(x − y)u(y) dy {|y|< |x| 2 }

|x| 2

implies

|x| 2

sup ϕ(z) { |x| 2 a, x, y ∈ X, T (t1 )y = x, f ∈ L1 (R), f L1 = 1 and supp (F f ) ⊆ [0, a]. Proof. Let n 1 and a > 1 be fixed, and let t1 , t2 , x and y be as in the statement of the theorem. Let g(τ ) =

(1 + tτ1 )n (1 −

τ n t2 )

(−t1 τ < 0), (0 τ t2 ).

Let s ∈ R and t2 yr =

g (r) (τ )e−isτ T (τ + t1 )y dτ

(r = 0, 1, . . . , n).

−t1

Then yr ∈ D(A) and (A − is)yr = yr+1 + g (r) (0−) − g (r) (0+) x

(r = 0, 1, . . . , n − 1), (A − is)yn = g (n) (0+)e−ist2 T (t2 )x − g (n) (0−)eist1 y + g (n) (0−) − g (n) (0+) x. By the assumption (1.5), yr+1 cyr − xg (r) (0−) − g (r) (0+) (r = 0, 1, . . . , n − 1), (n) g (0+)e−ist2 T (t2 )x − g (n) (0−)eist1 y cyn − xg (n) (0−) − g (n) (0+). Noting that g (r) (0−) = obtain

n! (n−r)!t1r

and g (r) (0+) =

(−1)r n! (n−r)!t2r ,

and in particular g(0−) = g(0+), we

T (t2 )x y c 1 1 y + − x + n t2n t1n n! t1n t2n c2 n n 1 1 yn−1 − x n−1 + n−1 + n + n n! t1 t2 t1 t2 .. .

n 1 n! 1 cn . y0 − x + n! (n − r)! t1r t2r r=1

Now let f ∈ L1 (R) with f L1 = 1 and supp (F f ) ⊆ [0, a]. Then

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C.J.K. Batty, Y. Tomilov / Journal of Functional Analysis 258 (2010) 3855–3878

a t2 cn cn g(τ )(F f )(τ )T (τ )x dt = f (s) g(τ )e−isτ T (τ + t1 )y dτ ds n! n! −t1

R

0

n T (t2 )x y 1 n! 1 + n + x + r . t2n t1 (n − r)! t1r t2 r=1

However, for τ > 0, g(τ ) = 1 +

n (−1)r n! r τ . r!(n − r)!t2r r=1

Hence a cn (F f )(τ )T (τ )x dt n! 0

a a n cn n! r g(τ )(F f )(τ )T (τ )x dt + τ (F f )(τ )T (τ )x dτ n! r!(n − r)! r=1

0

T (t2 )x y + n + x t2n t1

n

0

n! a r+1 Ma 1+ (n − r)! r! r=1 1 1 T (t2 )x y + + α x + n t2n t1n t1 t2 where Ma = sup{T (t): 0 t a} and αn is suitably chosen.

1 1 + r t1r t2

2

Corollary 4.2. Let A be the generator of a C0 -semigroup (T (t))t0 on a Banach space X, and assume that (1.5) holds. Let x ∈ t>0 Ran(T (t)) with x = 0. Then at least one of the following holds: (i) For each n ∈ N, T (t)x/t n → ∞ as t → ∞; (ii) For each n ∈ N, inf{y: T (t)y = x} →∞ tn

as t → ∞.

Choose f ∈ L1 (R) with f L1 = 1, supp (F f ) ⊆ [0, a] for some a > 0, and Proof. a (F f )(t)T (t)x dt = 0. This is possible by strong continuity of (T (t))t0 . 0 If neither (i) nor (ii) held, then one could choose values of n ∈ N, t1 > a and t2 > a such that (4.1) failed. 2 Corollary 4.3. Let A be the generator of a C0 -group (T (t))t∈R on a Banach space X, and assume that (1.5) holds. Then (T (t))t0 is expansive.


3875

Remark 4.4. We can introduce a norm · T ,a on X defined by a xT ,a = sup (F f )(τ )T (τ )x dτ : f ∈ L1 (R), f 1 = 1, supp (F f ) ⊆ [0, a] . 0

The different norms for different values of a are equivalent to each other. Now (4.1) can be written as T (t2 )x y cn 1 1 . + n xT ,a − αn x + t2n t1 n! t1 t2 This suggests that the norm · T ,a is a possible candidate to show that T (t) satisfies (HE) , but it would be necessary to obtain an inequality of this type with this norm on the left-hand side. Our second partial result indicates exponential growth in an integral sense. A very similar result is given in [16, Theorem 2.5]. However the proof there seems to be incomplete (in general, boundedness of the Carleman transform of u near z0 ∈ iR does not imply that z0 is not in the Carleman spectrum of u). So we give a complete proof here. Recall that a function g : R → X is a complete trajectory of T if g(t + s) = T (t)g(s) (t 0, s ∈ R). Any complete trajectory is continuous on R. When T is a C0 -group, the complete trajectories are of the form g(t) = T (t)x (t ∈ R) for some x ∈ X. Theorem 4.5. Let (T (t))t0 be a C0 -semigroup on a Banach space X with generator A, and assume that (1.5) holds. If g : R → X is a non-zero complete trajectory of (T (t))t0 then there exists ε > 0 such that g(·) ∈ / L1 (R, e−ε|t| dt). Proof. Let c > 0 be as in (1.5). Let λ = a + is ∈ C, where |a| c/2. Then (A − λ)x (A − is)x − |a|x c x x ∈ D(A) . 2

(4.2)

Let g be a complete trajectory and suppose that g(·) ∈ L1 (R, e−ε|t| dt) for every ε > 0. Let x = g(0). The Carleman transform gˆ of g is defined for Re λ = 0 by: ∞ 0

g(λ) ˆ =

−

e−λt g(t) dt

∞ λt 0 e g(−t) dt

(Re λ > 0), (Re λ < 0).

Then gˆ is holomorphic, and bounded on {λ: | Re λ| > ε} for each ε > 0. In particular, gˆ is bounded on {λ: | Re λ| c/2}. k Consider λ with Re λ > 0, and let k > 0. Then 0 e−λt g(t) dt ∈ D(A) and k (A − λ)

e

0

−λt

g(t) dt = e−λk g(k) − x.

3876


Since (A − λ) is closed, letting k → ∞ (through a suitable sequence) shows that g(λ) ˆ ∈ D(A)

and (A − λ)g(λ) ˆ = −x.

(4.3)

A variation of this argument shows that (4.3) also holds for Re λ < 0. It follows from (4.2) that g(λ) ˆ 2x/c for 0 < | Re λ| < c/2. Thus gˆ is bounded on C \ iR. Now suppose that 0 < | Re λ| c/2 and 0 < | Re μ| c/2. Then 2 (A − λ) g(λ) g(λ) ˆ − g(μ) ˆ ˆ − g(μ) ˆ c 2 4 |λ − μ|x. = −x − −x + (μ − λ)g(μ) ˆ c c2 Now by Cauchy’s criterion, gˆ extends continuously to iR, and then gˆ is a bounded entire function. Since g(a) ˆ → 0 as a → ∞, gˆ is identically zero. Then g = 0 by the uniqueness theorem for Laplace transforms. 2 Finally we consider the situation where (1.5) holds but no trajectory grows exponentially in forward time. Recall that the growth bound ω(T ) of (T (t))t0 is the unique quantity in R ∪ {−∞} such that the spectral radius of T (t) is exp(tω(T )) for some/all t > 0. If ω(T ) < 0, then (1.5) holds, and T is exponentially stable and therefore trivially hyperbolic. If ω(T ) = 0, then no trajectory grows exponentially in forward time, and we now show that (1.5) implies that all non-zero trajectories grow exponentially in reverse time, with the same exponent. Example 3.3 is an illustration of this situation. Theorem 4.6. Let (T (t))t0 be a C0 -semigroup with generator A and with growth bound 0, and assume that the lower bound (1.5) holds. Then (i) (T (t))t0 is not quasi-hyperbolic, (ii) A is invertible, (iii) there exist constants ε > 0 and κ > 0 such that y κeεt A−1 x whenever t > 0, x, y ∈ X and T (t)y = x. Proof. Since the growth bound is 0, T contains a point in the boundary of σ (T (1)) and hence in σap (T (1)), so T is not quasi-hyperbolic. Moreover, R(λ, A) exists whenever Re λ > 0 and R(λ, A) is bounded for Re λ a for any a > 0. It follows from (4.2) that R(λ, A) is bounded for 0 < Re λ < a for some a > 0, and then it follows from the Neumann series for the resolvent that R(λ, A) exists and is uniformly bounded for Re λ > −ε for some ε > 0. By a theorem of Weis and Wrobel [1, Theorem 5.1.9], there exists a constant C such that T (t)A−1 Ce−εt (t > 0). Hence y C −1 eεt A−1 x if T (t)y = x. 2 References [1] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2001.


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[2] A.G. Baskakov, Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators, Sovrem. Mat. Fundam. Napravl. 9 (2004) 3–151 (in Russian); translation in J. Math. Sci. (N. Y.) 137 (2006) 4885–5036. [3] C.J.K. Batty, S. Srivastava, The non-analytic growth bound of a C0 -semigroup and inhomogeneous Cauchy problems, J. Differential Equations 194 (2003) 300–327. [4] C. Chicone, Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr., vol. 70, Amer. Math. Soc., Providence, RI, 1999. [5] C. Chicone, R.C. Swanson, The spectrum of the adjoint representation and the hyperbolicity of dynamical systems, J. Differential Equations 36 (1980) 28–39. [6] C. Chicone, R.C. Swanson, Spectral theory for linearisations of dynamical systems, J. Differential Equations 40 (1981) 155–167. [7] J. Cooper, H. Koch, The spectrum of a hyperbolic evolution operator, J. Funct. Anal. 133 (1995) 301–328. [8] G. Couper, Characterization of quasi-Anosov diffeomorphisms, Bull. Austral. Math. Soc. 17 (1977) 321–334. [9] M. Eisenberg, J.H. Hedlund, Expansive automorphisms of Banach spaces, Pacific J. Math. 34 (1970) 647–656. [10] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, Berlin, 2000. [11] J. Franks, C. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc. 223 (1976) 267–278. [12] M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006. [13] P.R. Halmos, G. Lumer, J.J. Schäffer, Square roots of operators, Proc. Amer. Math. Soc. 4 (1953) 142–149. [14] B. Hasselblatt, Hyperbolic dynamical systems, in: Handbook of Dynamical Systems, vol. 1A, North-Holland, Amsterdam, 2002, pp. 239–319. [15] J.H. Hedlund, Expansive automorphisms of Banach spaces. II, Pacific J. Math. 36 (1971) 671–675. [16] S.Z. Huang, Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems, Studia Math. 116 (1995) 23–41. [17] R. Johnson, Analyticity of spectral subbundles, J. Differential Equations 35 (1980) 366–387. [18] M.A. Kaashoek, S.M. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Differential Equations 112 (1994) 374–406. [19] Yu. Latushkin, A. Pogan, R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory 58 (2007) 387–414. [20] Yu. Latushkin, F. Räbiger, Operator valued Fourier multipliers and stability of strongly continuous semigroups, Integral Equations Operator Theory 51 (2005) 375–394. [21] Yu. Latushkin, R. Shvydkoy, Hyperbolicity of semigroups and Fourier multipliers, in: Systems, Approximation, Singular Integral Operators, and Related Topics, Bordeaux, 2000, in: Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 341–363. [22] Yu. Latushkin, Yu. Tomilov, Fredholm properties of evolution semigroups, Illinois J. Math. 48 (2004) 999–1020. [23] Yu. Latushkin, Yu. Tomilov, Fredholm differential operators with unbounded coefficients, J. Differential Equations 208 (2005) 388–429. [24] Y. Latushkin, M. Vishik, Linear stability in an ideal incompressible fluid, Comm. Math. Phys. 233 (2003) 439–461. [25] K.B. Laursen, M. Neumann, An Introduction to Local Spectral Theory, London Math. Soc. Monogr., vol. 20, Oxford University Press, New York, 2000. [26] B.M. Levitan, V.V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, 1982. [27] O. Lopes, On the structure of the spectrum of a linear time periodic wave equation, J. Anal. Math. 47 (1986) 55–68. [28] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977) 351–370. [29] J. Mather, Characterization of Anosov diffeomorphisms, Indag. Math. 30 (1968) 479–483. [30] T.L. Miller, V.G. Miller, M. Neumann, Local spectral properties of weighted shifts, J. Operator Theory 51 (2004) 71–88. [31] V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Oper. Theory Adv. Appl., vol. 139, Birkhäuser, Basel, 2003. [32] S.Y. Pilyugin, Generalizations of the notion of hyperbolicity, J. Difference Equ. Appl. 12 (2006) 271–282. [33] F. Räbiger, R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup Forum 52 (1996) 225–239. [34] C.J. Read, Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element, Trans. Amer. Math. Soc. 286 (1986) 715–725. [35] C.J. Read, Spectrum reducing extension for one operator on a Banach space, Trans. Amer. Math. Soc. 308 (1988) 413–429.

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[36] W.C. Ridge, Approximate point spectrum of a weighted shift, Trans. Amer. Math. Soc. 147 (1970) 349–356. [37] R. Schnaubelt, Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations, in: Evolution Equations, Semigroups and Functional Analysis, Milano, 2000, in: Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2002, pp. 311–338. [38] R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equations, in: Functional Analytic Methods for Evolution Equations, in: Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 401–472. [39] R. Shvydkoy, The essential spectrum of advective equations, Comm. Math. Phys. 265 (2006) 507–545. [40] R. Shvydkoy, Cocycles and Mane sequences with an application to ideal fluids, J. Differential Equations 229 (2006) 49–62. [41] R.C. Swanson, The spectrum of vector bundle flows with invariant subbundles, Proc. Amer. Math. Soc. 83 (1981) 141–145. [42] V.M. Tyurin, Invertibility of linear differential operators in some function spaces, Sibirsk. Mat. Zh. 32 (1991) 160– 165 (in Russian); translation in Siberian Math. J. 32 (1991) 485–490. [43] M. Wolff, A remark on the spectral bound of the generator of positive operators with applications to stability theory, in: Functional Analysis and Applications, Proc. Oberwolfach, 1980, Birkhäuser, 1981, pp. 39–50.


Particle approximation of the Wasserstein diffusion Sebastian Andres ∗ , Max-K. von Renesse Technische Universität Berlin, Germany Received 26 October 2009; accepted 27 October 2009 Available online 2 December 2009 Communicated by Paul Malliavin

Abstract We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein diffusion (von Renesse and Sturm (2009) [25]), assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya (2003) [19]. © 2009 Elsevier Inc. All rights reserved. Keywords: Measure-valued processes; Hydrodynamic limit; Nonlinear fluctuations; McKean–Vlasov limit

1. Introduction The large scale behaviour of stochastic interacting particle systems is often described by (possibly nonlinear) deterministic evolution equations in the hydrodynamic limit, a fact which can be understood as a dynamic version of the law of large numbers, cf. e.g. [15]. The fluctuations around such deterministic limits usually lead to linear Ornstein–Uhlenbeck type SPDE on the diffusive time scale. In view of this typical two-step scaling hierarchy it is no surprise that only few types of SPDE with nonlinear drift operator admitting a rigorous particle approximation are known (cf. [13] for a survey on lattice models, e.g. [23,17] for exchangeable diffusions and e.g. [5,9] for interactive population models). According to [13] the appearance of a nonlinear SPDE as the scaling limit of some microscopic system is an indication of criticality, i.e. a situation * Corresponding author.

E-mail addresses: [email protected] (S. Andres), [email protected] (M.-K. von Renesse). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.029

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S. Andres, M.-K. von Renesse / Journal of Functional Analysis 258 (2010) 3879–3905

when microscopic fluctuations become too large for an averaging principle as in the law of large numbers to become effective. In this work we add one more example to the collection of (in this case conservative) interacting particle systems with a nonlinear stochastic evolution in the hydrodynamic limit. We study a sequence of Langevin-type SDEs with reflection for the positions 0 ≡ xt0 xt1 xt2 · · · xtN xtN +1 ≡ 1 of N moving particles on the unit interval 1 β 1 dt −1 − N +1 xti − xti−1 xti+1 − xti √ + 2 dwti + dlti−1 − dlti , i = 1, . . . , N,

dxti =

(1)

driven by independent real Brownian motions {w i } and local times l i satisfying t dlti

0,

lti

=

1{x i =x i+1 } dlsi . s

s

(2)

0

At first sight Eq. (1) resembles familiar Dyson-type models of interacting Brownian motions with electrostatic interaction, for which the convergence towards (deterministic) McKean– Vlasov equations under various assumptions is known since long, cf. e.g. [22,6]. Except from the fact that (1) models a nearest-neighbour and not a mean-field interaction, the most important difference towards the Dyson model is however, that in the present case for N β − 1 the drift is attractive and not repulsive. One technical consequence is that the system (1) and (2) has to be understood properly because it can no longer be defined in the class of Euclidean semi-martingales (cf. [4] for a rigorous analysis). The second and more dramatic consequence is a clustering of hence strongly correlated particles such that fluctuations are seen on large hydrodynamic scales. More precisely, assuming Markov uniqueness for the corresponding infinite dimensional Kolmogorov operator, cf. Definition 2.2 below, we show that for properly chosen initial condition the empirical probability distribution of the particle system in the high density regime μN t

N 1 = δx i N·t N i=1

converges for N → ∞ to the Wasserstein diffusion (μt ) on the space of probability measures P([0, 1]). This process was introduced in [25] as a conservative model for a diffusing fluid when its heat flow is perturbed by a kinetically uniform random forcing. In particular (μt ) is a solution in the sense of an associated martingale problem to the SPDE dμt = βμt dt + (μt ) dt + div( 2μt dBt ),

(3)

where is the Neumann Laplace operator and dBt is space–time white noise over [0, 1] and, for μ ∈ P([0, 1]), (μ) ∈ D ([0, 1]) is the Schwartz distribution acting on f ∈ C ∞ ([0, 1]) by

(μ), f =

f (I− ) + f (I+ ) f (I+ ) − f (I− ) f (0) + f (1) − − , 2 |I | 2

I ∈gaps(μ)


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where gaps(μ) denotes the set of intervals I = (I− , I+ ) ⊂ [0, 1] of maximal length with μ(I ) = 0 and |I | denotes the length of such an interval. The SPDE (3) has a familiar structure. For instance, the Dawson–Watanabe (‘super-Brownian √ motion’) process solves dμt = βμt dt + 2μt dBt whereas the empirical measure of a countable family of independent Brownian motions satisfies the equation dμt = μt dt + √ div( 2μt dBt ), both again in the weak sense of the associated martingale problems, cf. e.g. [7]. The additional nonlinearity introduced through the operator into (3) is crucial for the construction of (μt ) by Dirichlet form methods because it guarantees the existence of a reversible measure Pβ on P([0, 1]) which plays a central role for the convergence result, too. For β > 0, Pβ can be defined as the law of the random probability measure η ∈ P([0, 1]) defined by 1

f, η =

β f Dt dt,

∀f ∈ C [0, 1] ,

0 β

γ

is the real-valued Dirichlet process over [0, 1] with parameter β and γ where t → Dt = γt·β β denotes the standard Gamma subordinator. It is argued in [25] that Pβ admits the formal Gibbsean representation Pβ (dμ) =

1 −β Ent(μ) 0 e P (dμ) Z

with the Boltzmann entropy Ent(μ) = [0,1] log(dμ/dx) dμ as Hamiltonian and a particular uniform measure P0 on P([0, 1]), which illustrates the non-Gaussian character of Pβ . For instance, Pβ is neither log-concave nor does it project nicely to linear subspaces. However an appropriate version of the Girsanov formula holds true for Pβ , see also [26], which implies the L2 (P([0, 1]), Pβ )-closability of the quadratic form w 2 β ∇ F P (dμ), F ∈ Z, E(F, F ) = μ P ([0,1])

on the class F (μ) = f φ , μ, φ , μ, . . . , φ , μ , 2 k 1 Z = F : P [0, 1] → R f ∈ Cc∞ Rk , {φi }ki=1 ⊂ C ∞ [0, 1] , k ∈ N where w ∇ F = (D|μ F ) (·) 2 μ L ([0,1],μ) and (D|μ F )(x) = ∂t|t=0 F (μ + tδx ). The corresponding closure, still denoted by E, is a local regular Dirichlet form on the compact space (P([0, 1]), τw ) of probabilities equipped with the weak topology. This allows to construct a unique Hunt diffusion process ((Pη )η∈P ([0,1]) , (μt )t0 ) properly associated with E, cf. [11]. Starting (μt ) from equilibrium Pβ yields what shall be called in the sequel exclusively the Wasserstein diffusion. Note that the SPDE (3) may be called singular in several respects. Apart from the curious nonlinear structure of the drift component (μ), the noise is multiplicative non-Lipschitz and de-

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generate elliptic in all states μ which are not fully supported on [0, 1], i.e. the infinite dimensional Kolmogorov operator associated to (3) is not nicely behaved. Our approach for the approximation result is therefore again based on Dirichlet form methods. We use that the convergence of symmetric Markov semigroups is equivalent to an amplified notion of Gamma-convergence [20] of the associated sequence quadratic forms E N . In our situation it suffices to verify that Eqs. (1) and (2) define a sequence of reversible finite dimensional particle systems whose equilibrium distributions converge nicely enough to Pβ . In particular we show that also the logarithmic derivatives converge in an appropriate L2 -sense which implies the Mosco-convergence of the Dirichlet forms. (The pointwise convergence of the same sequence E N to E has been used in a recent work by Döring and Stannat to establish the logarithmic Sobolev inequality for E, cf. [8].) Since the approximating state spaces are finite dimensional we employ [19] for a generalized framework of Mosco-convergence of forms defined on a scale of Hilbert spaces. In case of a fixed state space with varying reference measure the criterion of L2 -convergence of the associated logarithmic derivatives has been studied in e.g. [16]. However, in our case this result does not directly apply because in particular the metric and hence also the divergence operation itself is depending on the parameter N . However, only little effort is needed to see that things match up nicely, cf. Section 4.3. For the sake of a clearer presentation in the proofs we will work with a parametrization of a probability measure on [0, 1] by the generalized right continuous inverse of its distribution function, which however is mathematically inessential. A side result of this parametrization is a diffusive scaling limit result for a (1 + 1)-dimensional gradient interface model with non-convex interaction potential (cf. [12]), see Section 6. 2. Set up and main results Construction of (XtN ). Following [4], for β > 0, the generalized solution to the system (1) and (2) is obtained rigorously by Dirichlet form methods, observing that in the regular case β N +1 the solution XtN = (xt1 , . . . , xtN ) defines a reversible Markov process on ΣN := {x ∈ RN , 0 x 1 x 2 · · · x N 1} ⊂ RN with equilibrium distribution qN dx 1 , . . . , dx N =

(β)

N +1

(( Nβ+1 ))N +1

i=1

i β −1 x − x i−1 N+1 dx 1 . . . dx N .

The generator LN of X N satisfies L f (x) = N

N 1 ∂ β 1 −1 − f (x) N +1 x i − x i−1 x i+1 − x i ∂x i i=1

+ f (x)

for x ∈ Int(ΣN ),

(4)

and E (f, g) = − N

ΣN

f (x)L g(x) qN (dx) = N

∇f (x) · ∇g(x) qN (dx),

ΣN

for all f, g ∈ C ∞ (ΣN ), ∇g · ν = 0 on ∂ΣN , where ν denotes the inner normal field on ∂ΣN .


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For general β > 0, N ∈ N the measure qN ∈ P(ΣN ) is well defined and satisfies the Hamza condition, because it has a strictly positive density with locally integrable inverse, cf. e.g. [1]. This implies that the form E N (f, f ) with domain f ∈ C ∞ (ΣN ) is closable on L2 (ΣN , qN ). The closure of E N , denoted by the same symbol, defines a local regular Dirichlet form on L2 (ΣN , qN ), from which a properly associated qN -symmetric Hunt diffusion ((Px )x∈ΣN , (XtN )t0 ) is obtained. Now we can state the main results of this work as follows. Theorem 2.1. Let E 0 be the limit of any -convergent subsequence of (N · E N )N . Then, there ˜ and exists a Dirichlet form E˜ , not depending on E 0 , which is extending E , i.e. D(E) ⊆ D(E) ˜ E(u) = E(u) for u ∈ D(E), such that ˜ E(u) E 0 (u) E(u)

˜ for all u ∈ D(E).

In particular, every -limit of (N · E N ) coincides with the Wasserstein Dirichlet form E on D(E). For the precise definition of -convergence of Dirichlet forms, see Section 3 below. By a general compactness result (see Theorem 3.3 below) every sequence of Dirichlet forms contains -convergent subsequences. As a corollary to Theorem 2.1 we obtain that the associated sequence of Hunt processes on P([0, 1]) converges weakly, provided E is Markov unique in the following sense. Definition 2.2. The Wasserstein Dirichlet form E is Markov unique if there is no proper extension ˜ where ZN = {F ∈ Z | E˜ of E on L2 (P([0, 1]), Pβ ) with generator L˜ such that ZN ⊂ D(L), F (μ) = f ( φ1 , μ, . . . , φk , μ), φi (0) = φi (1) = 0, i = 1, . . . , k}. Corollary 2.3. For β > 0, assume that the Wasserstein Dirichlet form E is Markov unique. Let (XtN ) denote the qN -symmetric diffusion on ΣN induced from the Dirichlet form E N , starting 1 N from equilibrium X0N ∼ qN , and let μN t = N i=1 δx i ∈ P([0, 1]), then the sequence of proN·t

cesses (μN . ) converges weakly to (μ.) in CR+ ((P([0, 1]), τw )) for N → ∞. Remark 2.4 (On the Markov uniqueness assumption). Condition 2.2 is a very subtle assumption. By general principles, cf. [2, Theorem 3.4], it is weaker than the essential self-adjointness of the generator of (μt )t0 on ZN = {F ∈ Z | F (μ) = f ( φ1 , μ, . . . , φk , μ), φi (0) = φi (1) = 0, i = 1, . . . , k} and stronger than the well-posedness, i.e. uniqueness, in the class of Hunt processes on P([0, 1]) of the martingale problem defined by Eq. (3) on the set of test functions ZN . In analytic terms, condition 2.2 is closely related to the Meyers–Serrin (weak = strong) property of the corresponding Sobolev space, cf. Corollary 5.2 and [10,18]. Variants of this assumption appear in several quite similar infinite dimensional contexts as well [14,16] and the verification depends crucially on the integration by parts formula which in the present case of Pβ has a very peculiar structure. None of the standard arguments for Gaussian or even log-concave or regular measures with smooth logarithmic derivative is applicable here. However, in the accompanying paper [4] we have managed to prove Markov uniqueness for E N , using the fact that the reference measure qN admits an extension qˆN lying in the Muckenhoupt class A2 (RN ).

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Remark 2.5. For illustration we present some numerical simulation results, using a regularized version of (1) and (2), by courtesy of Theresa Heeg, Bonn, in the case of N = 4 particles with β = 10, β = 1 and β = 0.3 respectively, at large times.

3. Finite dimensional approximation of Dirichlet forms in Mosco and Gamma sense In this section we recall the concept of Mosco- and -convergence of a sequence of Dirichlet forms in the generalized sense of Kuwae and Shioya, allowing for varying base L2 -spaces, developed in [19]. Our main results will follow by applying these concepts to the sequence of generating Dirichlet forms N · E N of (g·N ) on L2 (ΣN , qN ) and E on L2 (G, Qβ ) introduced in Section 4.1 below. Definition 3.1 (Convergence of Hilbert spaces). A sequence of Hilbert spaces H N converges to a Hilbert space H if there exists a family of linear maps {Φ N : H → H N }N such that limΦ N uH N = uH N

for all u ∈ H.

A sequence (uN )N with uN ∈ HN converges strongly to a vector u ∈ H if there exists a sequence (u˜ N )N ⊂ H tending to u in H such that lim lim supΦ M u˜ N − uM H M = 0, N

M

and (uN ) converges weakly to u if lim uN , vN H N = u, vH , N

for any sequence (vN )N with vN ∈ H N tending strongly to v ∈ H . Moreover, a sequence (BN )N of bounded operators on H N converges strongly (resp. weakly) to an operator B on H if BN uN → Bu strongly (resp. weakly) for any sequence (uN ) tending to u strongly (resp. weakly). Definition 3.2 (-convergence). A sequence (E N )N of quadratic forms E N on H N -converges to a quadratic form E on H if the following two conditions hold:


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(i) If a sequence (uN )N with uN ∈ H N strongly converges to a u ∈ H , then E(u, u) lim inf E N (uN , uN ). N

(ii) For any u ∈ H there exists a sequence (uN )N with uN ∈ H N which converges strongly to u such that E(u, u) = lim E N (uN , uN ). N

The main interest of -convergence relies on the following general compactness theorem (see Theorem 2.3 in [19]). Theorem 3.3. Any sequence (E N )N of quadratic forms E N on H N has a -convergent subsequence whose -limit is a closed quadratic form on H . For the convergence of the corresponding semigroup operators the appropriate notion is Mosco-convergence. Definition 3.4 (Mosco-convergence). A sequence (E N )N of quadratic forms E N on H N converges to a quadratic form E on H in the Mosco sense if the following two conditions hold: (Mosco I) If a sequence (uN )N with uN ∈ H N weakly converges to a u ∈ H , then E(u, u) lim inf E N (uN , uN ). N

(Mosco II) For any u ∈ H there exists a sequence (uN )N with uN ∈ H N which converges strongly to u such that E(u, u) = lim E N (uN , uN ). N

Extending [20] it is shown in [19] that Mosco-convergence of a sequence of Dirichlet forms is equivalent to the strong convergence of the associated resolvents and semigroups. However, we shall prove that the sequence N · E N converges to E in the Mosco sense in a slightly modified fashion, namely the condition (Mosco II) will be replaced by (Mosco II ) There is a core K ⊂ D(E) such that for any u ∈ K there exists a sequence (uN )N with uN ∈ D(E N ) which converges strongly to u such that E(u, u) = limN E N (uN , uN ). Theorem 3.5. Under the assumption that H N → H the conditions (Mosco I) and (Mosco II ) are equivalent to the strong convergence of the associated resolvents. Proof. We proceed as in the proof of Theorem 2.4.1 in [20]. By Theorem 2.4 of [19] strong convergence of resolvents implies Mosco-convergence in the original stronger sense. Hence we need

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to show only that our weakened notion of Mosco-convergence also implies strong convergence of resolvents. Let {RλN , λ > 0} and {Rλ , λ > 0} be the resolvent operators associated with E N and E, respectively. Then, for each λ > 0 we have to prove that for every z ∈ H and every sequence (zN ) tending strongly to z the sequence (uN ) defined by uN := RλN zN ∈ H N converges strongly to u := Rλ z as N → ∞. The vector u is characterized as the unique minimizer of E(v, v) + λ v, vH − 2 z, vH over H and a similar characterization holds for each uN . Since for each N the norm of RλN as an operator on H N is bounded by λ−1 , by Lemma 2.2 in [19] there exists a subsequence of (uN ), still denoted by (uN ), that converges weakly to some u˜ ∈ H . By (Mosco II ) we find for every v ∈ K a sequence (vN ) tending strongly to v such that limN E N (vN , vN ) = E(v, v). Since for every N E N (uN , uN ) + λ uN , uN H N − 2 zN , uN H N E N (vN , vN ) + λ vN , vN H N − 2 zN , vN H N , using the condition (Mosco I) we obtain in the limit N → ∞ ˜ H E(v, v) + λ v, vH − 2 z, vH , E(u, ˜ u) ˜ + λ u, ˜ u ˜ H − 2 z, u which by the definition of the resolvent together with the density of K ⊂ D(E) implies that u˜ = Rλ z = u. This establishes the weak convergence of resolvents. It remains to show strong convergence. Let uN = RλN zN converge weakly to u = Rλ z and choose v ∈ K with the respective strong approximations vN ∈ H N such that E N (vN , vN ) → E(v, v), then the resolvent inequality for RλN yields E N (uN , uN ) + λuN − zN /λ2H N E N (vN , vN ) + λvN − zN /λ2H N . Taking the limit for N → ∞, one obtains lim sup λuN − zN /λ2H N E(v, v) − E(u, u) + λv − z/λ2H . N

Since K is a dense subset we may now let v → u ∈ D(E), which yields lim sup uN − zN /λ2H N u − z/λ2H . N

Due to the weak lower semicontinuity of the norm this yields limN uN − zN /λH N = u − z/λH . Since strong convergence in H is equivalent to weak convergence together with the convergence of the associated norms the claim follows (cf. Lemma 2.3 in [19]). 2 4. Proof of Theorem 2.1 4.1. The G-parameterization We parameterize the space P([0, 1]) in terms of right continuous quantile functions, cf. e.g. [24,25]. The set G = g : [0, 1) → [0, 1] g càdlàg nondecreasing ,


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equipped with the L2 ([0, 1], dx) distance dL2 is a compact subspace of L2 ([0, 1], dx). It is homeomorphic to (P([0, 1]), τw ) by means of the map ρ : G → P [0, 1] ,

g → g∗ (dx),

which takes a function g ∈ G to the image measure of dx under g. The inverse map κ = ρ −1 : P([0, 1]) → G is realized by taking the right continuous quantile function, i.e. gμ (t) := inf s ∈ [0, 1]: μ[0, s] > t . Let now (g· ) = (κ(μ. )) be the G-image of the Wasserstein diffusion under the map κ with invariant initial distribution Qβ , where Qβ denotes the law of the real-valued Dirichlet process with parameter β > 0 as described in the introduction. In [25, Theorem 7.5] it is shown that (g· ) is generated by the Dirichlet form, again denoted by E , which is obtained as the L2 (G, Qβ )-closure of E(u, v) =

∇u|g (·), ∇v|g (·) L2 ([0,1]) Qβ (dg),

u, v ∈ C1 (G),

G

on the class C1 (G) = u : G → R u(g) = U f1 , gL2 , . . . , fm , gL2 , U ∈ Cc1 Rm , 2 {fi }m i=1 ⊂ L [0, 1] , m ∈ N , where ∇u|g is the L2 ([0, 1], dx)-gradient of u at g. We are now going to apply the results of Kuwae and Shioya, summarized in the last section, when H N = L2 (ΣN , qN ), H = L2 (G, Qβ ) and Φ N is defined to be the conditional expectation operator ΦN : H → H N ,

N Φ u (x) := E u|gi/(N+1) = x i , i = 1, . . . , N .

To that purpose, we need to show that this sequence of Hilbert spaces is convergent in the sense of Definition 3.1. Proposition 4.1. H N converges to H along Φ N , for N → ∞. Proof. We have to show that Φ N uH N → uH for each u ∈ H . Let F N be the σ -algebra on G generated by the projection maps {g → g(i/(N + 1)) | i = 1, . . . , N }. By abuse of notation we identify Φ N u ∈ H N with E(u|F N ) of u, considered as an element of L2 (Qβ , F N ) ⊂ H . Since the measure qN coincides with the respective finite dimensional distributions of Qβ on ΣN we have Φ N uH N = Φ N uH . Hence the claim will follow once we show that Φ N u → u in H . For the latter we use the following abstract result, whose proof can be found, e.g. in [3, Lemma 1.3].

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Lemma 4.2. Let (Ω, D, μ) be a measure space and (Fn )n∈N a sequence of σ -subalgebras of D. Then E(f |Fn ) → f for all f ∈ Lp , p ∈ [1, ∞) if and only if for all A ∈ D there is a sequence An ∈ Fn such that μ(An A) → 0 for n → ∞. In order to apply this lemma to the given case (G, B(G), Qβ ), where B(G) denotes the Borel σ -algebra on G, let FQβ ⊂ B(G) denote the collection of all Borel sets F ⊂ G which can be approximated by elements FN ∈ F N with respect to Qβ in the sense above. Note that FQβ is again a σ -algebra, cf. the appendix in [3]. Let M denote the system of finitely based open cylinder sets in G of the form M = {g ∈ G | gti ∈ Oi , i = 1, . . . , L} where ti ∈ [0, 1] and Oi ⊂ [0, 1] open. From the almost sure right continuity of g and the fact that g. is continuous at t1 , . . . , tL for Qβ -almost all g it follows that MN := {g ∈ G | g(ti ·(N +1)/(N+1)) ∈ Oi , i = 1, . . . , L} ∈ F N is an approximation of M in the sense above. Since M generates B(G) we obtain B(G) ⊂ FQβ such that the assertion holds, due to Lemma 4.2. 2 Remark 4.3. It is much simpler to prove Proposition 4.1 for a dyadic subsequence N = 2m − 1, m ∈ N when the sequence Φ N uH N is nondecreasing and bounded, because Φ N is a projec tion operator in H with increasing range im(Φ N ) as N grows. Hence, Φ N uH N is Cauchy and thus N Φ u − Φ M u2 = Φ N u2 − Φ M u2 → 0 for M , N → ∞, H H H

i.e. the sequence Φ N u converges to some v ∈ H . Since obviously Φ N u → u weakly in H it follows that u = v such that the claim is obtained from |Φ N uH − uH | Φ N u − uH . 4.2. The upper bound In this subsection we shall prove that the Wasserstein Dirichlet form E is an upper bound for any -limit of (N · E N ) as stated in Theorem 2.1. To that aim we essentially need to show that the condition (Mosco II ) holds for (N · E N ) and E. To simplify notation for f ∈ L2 ([0, 1], dx) denote the functional g → f, gL2 ([0,1]) on G by lf . We introduce the set K of polynomials defined by n ki lfi (g), ki ∈ N, fi ∈ C [0, 1] . K = u ∈ C(G) u(g) =

i=1

Lemma 4.4. The linear span of K is a core of E. Proof. Recall that by [25, Theorem 7.5] the set C1 (G) = u : G → R u(g) = U f1 , gL2 , . . . , fm , gL2 , U ∈ Cc1 Rm , 2 {fi }m i=1 ⊂ L [0, 1] , m ∈ N , is a core for the Dirichlet form E in the G-parametrization. The boundedness of G ⊂ L2 ([0, 1], dx) implies that U is evaluated on a compact subset of Rm only, where U can be


3889

approximated by polynomials in the C 1 -norm. From this, the chain rule for the L2 -gradient operator ∇ and Lebesgue’s dominated convergence theorem in L2 (G, Qβ ) it follows that the linear span of polynomials of the form u(g) = ni=1 lfkii (g) with ki ∈ N, fi ∈ C([0, 1]), ki ∈ N, is also a core of E . 2 Lemma 4.5. For a polynomial u ∈ K with u(g) = H N , then uN → u strongly.

n

ki i=1 lfi (g)

let uN :=

n

i=1 (Φ

N (l

ki fi ))

∈

Proof. Let u˜ N := ni=1 (Φ N (lfi ))ki ∈ H be the respective product of conditional expectations, where as above Φ N also denotes the projection operator on H = L2 (G, Qβ ). Note that by Jensen’s inequality for any measurable functional u : G → R, |Φ N (u)|(g) Φ N (|u|)(g) for Qβ -almost all g ∈ G, such that in particular Φ N (lfi )L∞ (G ,Qβ ) lfi L∞ (G ,Qβ ) f C([0,1]) . Hence each of the factors Φ N (lfi ) ∈ H is uniformly bounded and converges strongly to lfi in L2 (G, Qβ ), such that the convergence also holds true in any Lp (G, Qβ ) with p > 0. This implies u˜ N → u in H . Furthermore, n n M k k lim limΦ u˜ N − uM H M = lim limΦ M Φ N (lfi ) i − Φ M (lfi ) i N M N M i=1 i=1 H n n k k Φ N (lfi ) i − = lim lfii = 0. 2 N i=1

i=1

H

For the proof of (Mosco II ) we will also need that the conditional expectation of the random variable g w.r.t. Qβ given finitely many intermediate points {g(ti ) = x i } yields the linear interpolation. Lemma 4.6. For X ∈ ΣN define gX ∈ G by gX (t) = x + (N + 1) · t − i x i+1 − x i i

i i +1 , if t ∈ , i = 0, . . . , N, N +1 N +1

then E g|F N (X) = gX . Proof. This quite classical claim follows essentially from the bridge representation of the Dirichlet process, i.e. Qβ is the law of (γ (β · t))t∈[0,1] on G conditioned on γ (β) = 1 where γ is the standard Gamma subordinator, cf. e.g. [25]. Together with the elementary property that EQβ (g(t)) = t for t ∈ [0, 1] the claim follows from the homogeneity of γ together with simple scaling and iterated use of the Markov property. 2 Proposition 4.7. For all u ∈ K there is a sequence uN ∈ D(E N ) converging strongly to u in H and N · E N (uN , uN ) → E(u, u). In particular, condition (Mosco II ) is satisfied. From Proposition 4.7 one can immediately derive the upper bound in Theorem 2.1. To see this fix a subsequence, still denoted by N , such that (N · E N ) has a -limit E 0 . Then, we need

3890


to show that E 0 E on D(E) (note that by definition E = ∞ on the complement of D(E)). Fix now any u ∈ K. Using (Mosco II ) we find a sequence uN ∈ H N converging strongly to u in H such that E(u) = lim N · E N (uN ) E 0 (u), N

where the last inequality follows from the first property of -convergence. Hence, E 0 (u) E(u) for all u ∈ K and, since K is a core of D(E), by approximation we get that this also holds for all u ∈ D(E). Proof of Proposition 4.7. For u ∈ K let uN := ni=1 (Φ N (lfi ))ki ∈ H N as above then the strong convergence of uN to u is assured by Lemma 4.5. From Lemma 4.6 we obtain that Φ N (lf )(X) =

f, gX . In particular i ∇Φ N (lf )(X) =

1 · ηN ∗ f (ti ), N +1

(5)

where ti := i/(N + 1), i = 1, . . . , N + 1 and ηN denotes the convolution kernel t → ηN (t) = (N + 1) · (1 − min(1, |(N + 1) · t|)). For the convergence of N · EN (uN , uN ) to E(u, u) note first that f ∗ ηN → f in C([0, 1]) as N → ∞. Hence, using (5) we also get 2 N · ∇Φ N (lf )(X) =

N N 2 N η ∗ f (ti ) → f, f L2 ([0,1]) . 2 (N + 1) i=1

Similarly, for arbitrary f1 , f2 ∈ C([0, 1]) N · ∇Φ N (lf1 ), ∇Φ N (lf2 ) RN → f1 , f2 L2 ([0,1]) . Consider now u ∈ K with u(g) = ∇

L2

n

ki i=1 lfi (g).

The chain rule for the L2 -gradient operator ∇ =

yields n

∇u, ∇uL2 (0,1) =

i=j

j,s=1

and analogously for uN =

∇uN , ∇uN RN

n

i=1 (Φ

N (l

k lfii

ki fi ))

r=s

lfkrr

k −1 ks −1 lfs fj , fs L2 (0,1) ,

kj ks lfjj

with ∇ = ∇ R

N

n N N ki kr Φ (lfi ) Φ (lfr ) = j,s=1

i=j

(6)

r=s

k −1 N k −1 × kj ks Φ N (lfj ) j Φ (lfs ) s ∇Φ N (lfj ), ∇Φ N (lfs ) RN .


3891

Since N · E (uN , uN ) = N ·

∇uN , ∇uN RN dqN

N

ΣN

=N ·

∇uN , ∇uN g(t1 ), . . . , g(tN ) Qβ (dg)

G

and for Qβ -a.e. g Φ N (lf ) g(t1 ), . . . , g(tN ) → lf (g),

as N → ∞,

if f ∈ C([0, 1]) the first assertion of the proposition holds by (6) and dominated convergence. The second assertion follows now from linearity and polarisation together with Lemma 4.4. 2 For later use we make an observation which follows easily from the proof of the last proposition. Lemma 4.8. For u and uN as in the proof of Proposition 4.7 and for Qβ -a.e. g we have (N + 1)ιN ∇uN g(t1 ), . . . , g(tN ) − ∇u|g

→ 0,

L2 (0,1)

as N → ∞,

with ιN : RN → D([0, 1), R) defined as above and tl := l/(N + 1), l = 0, . . . , N + 1. Proof. By the definitions we have for every x ∈ ΣN (N + 1)ιN ∇uN (x) n N N N ki kj −1 l kj Φ (lfj )(x) Φ (lfi )(x) (N + 1) ∇ Φ N (lfj )(x) 1[tl ,tl+1 ] = j =1

=

i=j

n j =1

l=1

N k k −1 Φ (lfi )(x) i kj Φ N (lfj )(x) j

i=j

N

N η ∗ fj (tl )1[tl ,tl+1 ] ,

l=1

where we have used again (5). Furthermore, for every j , 2 1 N N η ∗ fj (tl )1[tl ,tl+1 ] (t) − fj (t) dt 0

l=1 N N 2 η ∗ fj (tl ) − fj (t) dt = tl+1

l=1 tl N N N 2 2 η ∗ fj (tl ) − fj (tl ) dt + 2 fj (tl ) − fj (t) dt, 2 tl+1

l=1 tl

tl+1

l=1 tl

3892


where the first term tends to zero as N → ∞ since ηN ∗ fj → fj in the sup-norm and the second term tends to zero by the uniform continuity of fj . From this we can directly deduce the claim because Φ N (lf )(g(t1 ), . . . , g(tN )) → lf (g) as N → ∞ for Qβ -a.e. g. 2 4.3. The lower bound For the lower bound, which will be based on the (Mosco I) condition, we exploit that the respective integration by parts formulas of E N and E converge. In case of a fixed state space a similar approach is discussed abstractly in [16]. However, here also the state spaces of the processes change which requires some extra care for the varying metric structures in the Dirichlet forms. Let T N := {f : ΣN → RN } be equipped with the norm f 2T N :=

1 N

f (x)2 N qN (dx), R

ΣN

then the corresponding integration by parts formula for qN on ΣN reads

∇u, ξ T N = −

1

u, divqN ξ H N . N

(7)

To state the corresponding formula for E we introduce the Hilbert space of vector fields on G by T = L2 G × [0, 1], Qβ ⊗ dx , and the subset Θ ⊂ T Θ = span ζ ∈ T ζ (g, t) = w(g) · ϕ g(t) , w ∈ K, ϕ ∈ C ∞ [0, 1] : ϕ(0) = ϕ(1) = 0 . Lemma 4.9. Θ is dense in T . Proof. Let us first remove the condition φ(0) = φ(1), i.e. let us show that the T -closure of Θ coincides with that of Θ¯ = span{ζ ∈ T | ζ (g, t) = w(g)·ϕ(g(t)), w ∈ K, ϕ ∈ C ∞ ([0, 1])}. Since supg∈G w(g) < ∞ for any w ∈ K, it suffices to show that any ζ ∈ Θ¯ of the form ζ (g, t) = ϕ(g(t)) can be approximated in T by functions ζk (g, t) = ϕk (g(t)) with ϕk ∈ C ∞ ([0, 1]) and ϕk (0) = ϕk (1) = 0. Choose a sequence of functions ϕk ∈ C0∞ ([0, 1]) such that supk ϕk C([0,1]) < ∞ and ϕk (s) → ϕ(s) for all s ∈ ]0, 1[. Now for Qβ -almost all g it holds that {s ∈ [0, 1] | g(s) = 0} = {0} and {s ∈ [0, 1] | g(s) = 1} = {1}, such that φk (g(s)) → φ(g(s)) for Qβ ⊗ dx-almost all (g, s). The uniform boundedness of the sequence of functions ζk : G × [0, 1] → R together with dominated convergence w.r.t. the measure Qβ ⊗ dx the convergence is established. In order to complete the proof of the lemma note that Qβ -almost every g ∈ G is a strictly increasing function on [0, 1]. This implies that the set Θ¯ is separating the points of a full measure subset of G × [0, 1]. Hence the assertion follows from the following abstract lemma. 2


3893

Lemma 4.10. Let μ be a probability measure on a Polish space (X, d) and let A be a subalgebra of C(X) containing the constants. Assume that A is μ-almost everywhere separating on X, i.e. ˜ = 1 and for all x, y ∈ X˜ there is an a ∈ A such there exists a measurable subset X˜ with μ(X) that a(x) = a(y). Then A is dense in any Lp (X, μ) for p ∈ [1, ∞). Proof. We may assume w.l.o.g. that A is stable w.r.t. the operation of taking the pointwise inf and sup. Let u ∈ Lp (X), then we may also assume w.l.o.g. that u is continuous and bounded on X. By the regularity of μ we can approximate X˜ from inside by compact subsets Km such that μ(Km ) 1 − m1 . On each Km the theorem of Stone–Weierstrass tells that there is some am ∈ A such that u|Km − am |Km C(Km ) m1 , and by truncation am C(X) uC(X) . In particular, for > 0, μ(|am − u| > ) μ(X \ Km ) m1 , if m 1/, i.e. am converges to u on X in μprobability. Hence some subsequence am converges to u pointwise μ-a.s. on X, and hence the claim follows from the uniform boundedness of the am by dominated convergence. 2 The L2 -derivative operator ∇ defines a map ∇ : C1 (G) → T which by [25, Proposition 7.3], cf. [26], satisfies the integration by parts formula

∇u, ζ T = − u, divQβ ζ H ,

u ∈ C1 (G), ζ ∈ Θ,

where, for ζ (g, t) = w(g) · ϕ(g(t)), divQβ ζ (g) = w(g) · Vϕβ (g) + ∇w(g)(.), ϕ g(.) L2 (dx) with 1 Vϕβ (g) := Vϕ0 (g) + β

ϕ (0) + ϕ (1) ϕ g(x) dx − 2

0

and Vϕ0 (g) :=

ϕ (g(a+)) + ϕ (g(a−)) a∈Jg

2

−

δ(ϕ ◦ g) (a) . δg

Here Jg ⊂ [0, 1] denotes the set of jump locations of g and ϕ(g(a+)) − ϕ(g(a−)) δ(ϕ ◦ g) (a) := . δg g(a+) − g(a−) By formula (8) one can extend ∇ to a closed operator on D(E) such that E(u, u) = ∇u2T .

(8)

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4.3.1. Definition of E˜ Lemma 4.11. The functional ˜ u)1/2 := sup E(u,

ζ ∈Θ

− u, divQβ ζ H ζ T

˜ ˜ = u ∈ L2 G, Qβ E(u) 1. β

Proof. We rewrite VN,ϕ (g(t1 ), . . . , g(tN )) as N ϕ(g(ti+1 )) − ϕ(g(ti )) β (ti+1 − ti ) VN,ϕ g(t1 ), . . . , g(tN ) = β g(ti+1 ) − g(ti ) i=0

N −1 ϕ(g(ti+1 )) − ϕ(g(ti )) ϕ(g(t1 )) − ϕ(g(t0 )) ϕ g(ti ) − + − g(t1 ) − g(t0 ) g(ti+1 ) − g(ti ) i=1

ϕ(g(tN +1 )) − ϕ(g(tN )) . + ϕ g(tN ) − g(tN +1 ) − g(tN )

(11)


3897

Note that all terms are uniformly bounded in g with a bound depending on the supremum norm β of ϕ and ϕ , respectively. Since the same holds for Vϕ (g) (cf. Section 5 in [25]), it is sufficient to show convergence Qβ -a.s. By the support properties of Qβ g is continuous at tN +1 = 1, so that the last line in (11) tends to zero. Using Taylor’s formula we obtain that the first term in (11) is equal to N

β

i=0

N 1 ϕ g(ti ) (ti+1 − ti ) + ϕ (γi ) g(ti+1 ) − g(ti ) (ti+1 − ti ), 2 i=0

1 for some γi ∈ [g(ti ), g(ti+1 )]. Obviously, the first term tends to β 0 ϕ (g(s)) ds and the second one to zero as N → ∞. Thus, it remains to show that the second line in (11) converges to ϕ (g(a+)) + ϕ (g(a−)) 2

a∈Jg

δ(ϕ ◦ g) ϕ (0) + ϕ (1) − (a) − . δg 2

(12)

Note that by the right-continuity of g the first term in the second line in (11) tends to −ϕ (0). Let now a2 , . . . , al−1 denote the l − 2 largest jumps of g on ]0, 1[. For N very large (compared with l) we may assume that a2 , . . . , al−2 ∈ ] N 2+1 , 1 − N 2+1 [. Put a1 := N 1+1 , al := 1 − N 1+1 . For j = 1, . . . , l let kj denote the index i ∈ {1, . . . , N}, for which aj ∈ [ti , ti+1 [. In particular, k1 = 1 and kl = N . Then i∈{k2 ,...,kl−1 }

l−1 ϕ(g(ti+1 )) − ϕ(g(ti )) δ(ϕ ◦ g) −→ (aj ) ϕ g(ti ) − ϕ g(aj −) − N →∞ g(ti+1 ) − g(ti ) δg j =2

−→

l→∞

a∈Jg

δ(ϕ ◦ g) (a). (13) ϕ g(a−) − δg

Provided l and N are chosen so large that g(ti+1 ) − g(ti ) C l for all i ∈ {0, . . . , N} \ {k1 , . . . , kl }, where C = sups |ϕ (s)|/6, again by Taylor’s formula we get for every j ∈ {1, . . . , l − 1} kj +1 −1

i=kj +1

ϕ(g(ti+1 )) − ϕ(g(ti )) ϕ g(ti ) − g(ti+1 ) − g(ti ) kj +1 −1

=−

1 1 2 ϕ g(ti ) g(ti+1 ) − g(ti ) + ϕ (γi ) g(ti+1 ) − g(ti ) 2 6

i=kj +1

1 −→ − N →∞ 2

a j +1 −

g(aj +1 −)

aj +

g(aj +)

1 ϕ g(s) dg(s) + O l −2 = − 2

ϕ (s) ds + O l −2 .

3898


Summation over j leads to +1 −1 l−1 kj

j =1 i=kj +1

ϕ(g(ti+1 )) − ϕ(g(ti )) ϕ g(ti ) − g(ti+1 ) − g(ti )

1 −→ − N →∞ 2 l−1

g(aj +1 −)

ϕ (s) ds + O l −1

j =1 g(a +) j

1 =− 2

1

1 ϕ (s) ds + 2

0

−→ −

l→∞

l−1

g(a j +)

ϕ (s) ds + O l −1

j =2 g(a −) j

1 1 ϕ (1) − ϕ (0) + ϕ g(a+) − ϕ g(a−) . 2 2 a∈Jg

Combining this with (13) yields that the second line of (11) converges in fact to (12), which completes the proof. 2 Since wN (g(t1 ), . . . , g(tN )) converges to w in Lp (G, Qβ ), p > 0 (cf. proof of Lemma 4.5 above), the last lemma ensures that the first term of dN ζ converges to the first term of divQβ ζ in H , while the following lemma deals with the second term. Lemma 4.14. For Qβ -a.s. g we have ∇wN g(t1 ), . . . , g(tN ) , ϕ g(t1 ), . . . , g(tN ) RN → ∇w|g , ϕ g(.) L2 (0,1) ,

as N → ∞,

and we have also convergence in H . Proof. As in the proof of the last lemma it is enough to prove convergence Qβ -a.s. Note that

∇wN ( g ) L2 (0,1) , g ), ϕ( g ) RN = (N + 1) ιN ∇wN ( g ) , ιN ϕ(

writing g := (g(t1 ), . . . , g(tN )) and using the extension of ιN on RN . By triangle and Cauchy– Schwarz inequality we obtain (N + 1)ιN ∇wN ( g ) L2 (0,1) − ∇w|g , ϕ g(.) L2 (0,1) g ) , ιN ϕ( (N + 1)ιN ∇wN ( g ) L2 (0,1) + ∇w|g , ιN ϕ( g ) − ϕ g(.) L2 (0,1) g ) − ∇w|g , ιN ϕ( (N + 1)ιN ∇wN ( g ) L2 (0,1) g ) − ∇w|g L2 (0,1) ιN ϕ( g ) − ϕ g(.) L2 (0,1) , + ∇w|g L2 (0,1) ιN ϕ( which tends to zero by Lemma 4.8 and by the definition of ιN .

2


3899

4.3.3. Condition (Mosco I) and the lower bound Proposition 4.15. Let uN ∈ D(E N ) converge weakly to u ∈ H , then ˜ u) lim inf N · E N (uN , uN ). E(u, N →∞

Proof. Let u ∈ H and uN ∈ H N converge weakly to u. Let ζ ∈ Θ and ζ N be as in Lemma 4.12, then − u, divQβ ζ H = − lim uN , divqN ζN H N = lim N · ∇uN , ζN T N 1/2 lim inf N · ∇uN T N · ζN TN = lim inf N · E N (uN , uN ) · ζ T , such that

1/2 − u, divQβ ζ H ˜ u) 1/2 = sup E(u, lim inf N · E N (uN , uN ) . ζ T ζ ∈Θ

2

Finally we prove the lower bound in Theorem 2.1. Let as before N · E N be -convergent to E 0 and let u ∈ H be arbitrary. Then, using the second property of -convergence we find a sequence uN ∈ H N tending strongly to u such that E 0 (u) = limN N · E N (uN ). In particular, uN converges weakly to u and Proposition 4.15 implies that ˜ E(u) lim inf N · E N (uN ) = lim N · E N (uN ) = E 0 (u), N

N

and the claim follows. 5. Weak convergence of processes This section is devoted to the proof of Corollary 2.3. As usual we show compactness of the laws of (μN . ) and in a second step the uniqueness of the limit. 5.1. Tightness Proposition 5.1. The sequence (μN . ) is tight in CR+ ((P([0, 1]), τw )). Proof. According to Theorem 3.7.1 in [7] it is sufficient to show that the sequence ( f, μN . )N ∈N is tight, where f is taken from a dense subset in F ⊂ C([0, 1]). Choose F := {f ∈ C 3 ([0, 1]) | N N f (0) = f (1) = 0}, then f, μN t = F (XN ·t ) with F N (x) =

N 1 i f x . N i=1

3900


The condition f (0) = f (1) = 0 implies F N ∈ D(LN ) and N +1 N +1 N β f (x i ) − f (x i−1 ) f (x i ) − f (x i−1 ) i − + f x , N +1 x i − x i−1 x i − x i−1

N · LN F N (x) =

i=1

i=1

i=1

which can be written as N · LN F N (x) =

N +1 β f (x i ) − f (x i−1 ) N +1 x i − x i−1 i=1

+

N i=1

i f (x i ) − f (x i−1 ) f (x N +1 ) − f (x N ) f x − − . x i − x i−1 x N +1 − x N

Finally, this can be estimated as follows: N · LN F N (x) f (β + 1) + f =: C β, f 3 C ([0,1]) . ∞ ∞

(14)

This implies a uniform in N Lipschitz bound for the BV part in the Doob–Meyer decompoN ). The process X N has continuous sample paths with square field operator sition of F N (XN. N N (F, F ) = L (F 2 ) − 2F · LN F = |∇F |2 . Hence the quadratic variation of the martingale part N ) satisfies of F N (XN ·

F

N

N XN ·

t

N − F N XN · s =N ·

t

∇F N 2 X N dr = 1 r N

s

t N 2 i f xr dr s

2 (t − s)f ∞ .

i=1

(15)

Since N N 1 β f Di/(N+1) → f Dsβ ds F X0 = N 1

N

i=1

Qβ -a.s.,

0

N ) the law of F N (X0N ) is convergent and by stationarity we conclude that also the law of F N (XN ·t is convergent for every t. Using now Aldous’ tightness criterion the assertion follows once we have shown that

N N N XN ·τN → 0, E F N X N ·(τN +δN ) − F

as N → ∞,

(16)

for any given δN ↓ 0 and any given sequence of bounded stopping times (τN ). Now, the Doob– Meyer decomposition reads as N N N F XN XN ·τN = MτNN +δN − MτNN + ·(τN +δN ) − F

τN+δN

N

τN

N N · LN F X N ·s ds,


3901

N )] . Using (14) and (15) where M N is a martingale with quadratic variation [M N ]t = [F N (XN · t we get

N N N E F N X N XN ·τN E MτNN +δN − MτNN + C1 δN ·(τN +δN ) − F 2 1/2 E MτNN +δN − MτNN + C 1 δN 1/2 = C2 E M N τ +δ − M N τ + C 1 δN N N N 1/2 C3 δN + δN , for some positive constants Ci , which implies (16).

2

The argument above shows the balance of first and second order parts of N · LN as N tends to infinity. Alternatively one could use the symmetry of (X·N ) and apply the Lyons–Zheng decomposition for the same result. 5.2. Identification of the limit Throughout this section we will assume that Markov-uniqueness holds for the Wasserstein Dirichlet form E. An immediate consequence is that E coincides with the Dirichlet form E˜ defined in the previous section. ˜ Corollary 5.2 (Meyers–Serrin property). Assume E is Markov-unique, then E = E. Proof. The assumption means that E has no proper extension in the class of Dirichlet forms on L2 (G, Qβ ). By Lemma 4.11 we obtain E˜ = E which is the claim. 2 In order to identify the limit of the sequence (μN . ) we will work again with G-parametrization introduced in Section 4.1. For technical reasons we introduce the following modification of (μN . ) which is better behaved in terms of the map κ. Lemma 5.3. For N ∈ N define the Markov process νtN :=

N 1 μN δ0 ∈ P [0, 1] , t + N +1 N +1

then (ν.N ) is convergent on CR+ ((P([0, 1]), τw )) if and only if (μN . ) is. In this case both limits coincide. Proof. Due to Theorem 3.7.1 in [7] it suffices to consider the sequences of real-valued process N

f, μN . and f, ν. for N → ∞, where f ∈ C([0, 1]) is arbitrary. From 2 1 f, μN f ∞ t − f, δ0 N + 1 N +1 t0

N = sup sup f, μN t − f, νt t0

3902


N it follows that dC ( f, μN . , f, ν. ) → 0 almost surely, where dC is any metric on C([0, ∞), R) inducing the topology of locally uniform convergence. Hence for a bounded and uniformly dC continuous functional F : C([0, ∞), R) → R

E F f, μN − E F f, ν.N → 0 .

for N → ∞.

Since weak convergence on metric spaces is characterized by expectations of uniformly continuous bounded functions (cf. e.g. [21, Theorem 6.1]) this proves the claim. 2

by

Let (g·N ) := (κ(ν·N )) be the process (ν·N ) in the G-parameterization. It can also be obtained N gtN = ι XN ·t

with the imbedding ι = ιN ι : ΣN → G,

ι(x) =

N

x i · 1[ti ,ti+1 ) ,

i=0

with ti := i/(N + 1), i = 0, . . . , N + 1. The convergence of (μN · ) to (μ· ) in CR+ (P([0, 1]), τw ) is thus equivalent to the convergence of (g·N ) to (g· ) in CR+ (G, dL2 ). By Proposition 5.1 and Lemma 5.3 (g·N )N is a tight sequence of processes on G. The following statement identifies (g· ) as the unique weak limit. Proposition 5.4. For any f ∈ C(G l ) and 0 t1 < · · · < tl , N →∞ −→ E f (gt1 , . . . , gtl ) . E f gtN1 , . . . , gtNl Proposition 5.4 will essentially be implied by the following statement, which follows immediately by combining the Markov-uniqueness of E and Corollary 5.2 with Propositions 4.15 and 4.7. Theorem 5.5. (N · E N , H N ) converges to (E, H ) along Φ N in Mosco sense. By the abstract results in [19] Mosco-convergence is equivalent to the strong convergence of the associated semigroup operator, from which we will now derive the convergence of the finite dimensional distributions stated in Proposition 5.4. Lemma 5.6. For u ∈ C(G) let uN ∈ H N be defined by uN (x) := u(ιx), then uN → u strongly. Moreover, for any sequence fN ∈ H N with fN → f ∈ H strongly, uN · fN → u · f strongly. Proof. Let u˜ N ∈ H be defined by u˜ N (g) := u(g N ), where g N := i/(N + 1), then u˜ N → u in H strongly. Moreover,

N

i=1 g(ti )1[ti ,ti+1 ) , ti

lim limΦ M u˜ N − uM H M = lim limΦ M u˜ N − u˜ M H = lim u˜ N − uH = 0, N

M

N

M

N

:=


3903

where as above we have identified Φ M with the corresponding projection operator in L2 (G, Qβ ). For the proof of the second statement, let H fÑ → f in H such that limN lim supM Φ M fÑ − fM H M = 0. From the uniform boundedness of u˜ N it follows that also u˜ N · fÑ → u · f in H . In order to show H M uM · fM → u · f write M Φ (u˜ N · fÑ ) − uM · fM M Φ M (u˜ N · fÑ ) − uM · Φ M (f˜M ) M H H M ˜ + uM · fM − uM · Φ (fM ) H M . Identifying the map Φ M with the associated conditional expectation operator, considered as an orthogonal projection in H , the claim follows from M Φ (u˜ N · fÑ ) − uM · Φ M (f˜M ) M = Φ M (u˜ N · fÑ ) − u˜ M · Φ M (f˜M ) H H M = Φ (u˜ N · fÑ ) − Φ M (u˜ M · f˜M )H u˜ N · fÑ − u˜ M · f˜M H and uM · fM − uM · Φ M (f˜M ) M u∞ fM − Φ M (fÑ ) M H H M + u∞ Φ (fÑ ) − Φ M (f˜M )H M = u∞ fM − Φ M (fÑ )H M + u∞ Φ M (fÑ ) − Φ M (f˜M )H u∞ fM − Φ M (fÑ ) M H

+ u∞ fÑ − f˜M H such that in fact limN lim supM Φ M (u˜ N · fÑ ) − uM · fM H M = 0.

2

Proof of Proposition 5.4. It suffices to prove the claim for functions f ∈ C(G l ) of the form f (g1 , . . . , gl ) = f1 (g1 ) · f2 (g2 ) · · · · · fl (gl ) with fi ∈ C(G). Let PtN : H N → H N be the semiN )] = P N f, g group on H N induced by g N via Eg·qN [f (XN H N . From Theorem 5.5 and the t ·t N abstract results in [19] it follows that Pt converges to Pt strongly, i.e. for any sequence uN ∈ H N converging to some u ∈ H strongly, the sequence PtN uN also strongly converges to Pt u. Let fiN := fi ◦ ιN , then inductive application of Lemma 5.6 yields N N PtN fl · PtN fl−1 · PtN . . . f2N · PtN fN ... l −tl−1 l−1 −tl−2 l−2 −tl−3 1 1 −→ Ptl −tl−1 fl · Ptl−1 −tl−2 (fl−1 · Ptl−2 −tl−3 . . . f2 · Pt1 f1 ) . . .

N →∞

strongly,

which in particular implies the convergence of inner products. Hence, using the Markov property of g N and g we may conclude that

3904


lim E f1 gtN1 . . . fl gtNl N

= lim E f1N XtN1 . . . flN XtNl N

N N N N N N N = lim 1, PtN −tl−1 fl · Ptl−1 −tl−2 fl−1 · Ptl−2 −tl−3 . . . f2 · Pt1 f1 . . . H N l N = 1, Ptl −tl−1 fl · Ptl−1 −tl−2 (fl−1 · Ptl−2 −tl−3 . . . f2 · Pt1 f1 ) . . . H = E f1 (gt1 ) . . . fl (gtl ) . 2 6. A non-convex (1 + 1)-dimensional ∇φ-interface model We conclude with a remark on a link to stochastic interface models, cf. [12]. Consider an interface on the one-dimensional lattice N := {1, .√ . . , N}, whose location at time t is represented by the height variables φt = {φt (x), x ∈ N } ∈ N · ΣN with dynamics determined by√the generator L˜ N defined below and with the boundary conditions φt (0) = 0 and φ(N + 1) = N at ∂N := {0, N + 1}. L˜ N f (φ) :=

β −1 N +1

x∈N

1 1 ∂ − f (φ) + f (φ) φ(x) − φ(x − 1) φ(x + 1) − φ(x) ∂φ(x)

√ √ for φ ∈ Int( N √ · ΣN ) and with φ(0) := 0 and φ(N + 1) := N . L˜ N corresponds to LN as an operator on C 2 ( N · ΣN ) with Neumann boundary conditions. Note that this system involves a non-convex interaction potential function V on (0, ∞) given by V (r) = (1 − Nβ+1 ) log(r) and the Hamiltonian HN (φ) :=

N V φ(x + 1) − φ(x) ,

φ(0) := 0,

φ(N + 1) :=

√

N.

x=0

Then, the natural stationary distribution of the interface is the Gibbs measure μN conditioned on √ N · ΣN : μN (dφ) :=

1 exp −HN (φ) 1{(φ(1),...,φ(N ))∈√N ·ΣN } dφ(x), ZN x∈N

where ZN is √a normalization constant. Note that μN is the corresponding measure of qN on the state space N · ΣN . Suppose now that (φt )t0 is the stationary process generated by L˜ N . Then the space–time scaled process 1 Φ˜ tN (x) := √ φN 2 t (x), N

x = 0, . . . , N + 1,

taking values in ΣN , is associated with the Dirichlet form N · E N . Introducing the G-valued fluctuation field Φ˜ tN (x)1[x/(N+1),(x+1)/(N +1)) (ϑ), ΦtN (ϑ) := ιN Φ˜ tN (ϑ) = x∈N

ϑ ∈ [0, 1),


3905

by our main result we have weak convergence for the law of the equilibrium fluctuation field Φ N to the law of the nonlinear diffusion process κ(μ. ) on G, which is the G-parametrization of the Wasserstein diffusion. References [1] S. Albeverio, M. Röckner, Classical Dirichlet forms on topological vector spaces—closability and a Cameron– Martin formula, J. Funct. Anal. 88 (2) (1990) 395–436. [2] S. Albeverio, M. Röckner, Dirichlet form methods for uniqueness of martingale problems and applications, in: Stochastic Analysis, Ithaca, NY, 1993, Amer. Math. Soc., Providence, RI, 1995, pp. 513–528. [3] A. Alonso, F. Brambila-Paz, Lp -continuity of conditional expectations, J. Math. Anal. Appl. 221 (1) (1998) 161– 176. [4] S. Andres, M.-K. von Renesse, Uniqueness and regularity for a system of interacting Bessel processes via the Muckenhoupt condition, 2009, submitted for publication. [5] D. Blount, Diffusion limits for a nonlinear density dependent space–time population model, Ann. Probab. 24 (2) (1996) 639–659. [6] E. Cépa, D. Lépingle, Diffusing particles with electrostatic repulsion, Probab. Theory Related Fields 107 (4) (1997) 429–449. [7] D.A. Dawson, Measure-valued Markov processes, in: École d’Été de Probabilités de Saint-Flour XXI—1991, in: Lecture Notes in Math., vol. 1541, Springer, Berlin, 1993, pp. 1–260. [8] M. Döring, W. Stannat, The logarithmic Sobolev inequality for the Wasserstein diffusion, Probab. Theory Related Fields 145 (1–2) (2009) 189–209. [9] R. Durrett, L. Mytnik, E. Perkins, Competing super-Brownian motions as limits of interacting particle systems, Electron. J. Probab. 10 (35) (2005) 1147–1220. [10] A. Eberle, Uniqueness and Non-uniqueness of Semigroups Generated by Singular Diffusion Operators, SpringerVerlag, Berlin, 1999. ¯ [11] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter & Co., Berlin, 1994. [12] T. Funaki, Stochastic interface models, in: Lectures on Probability Theory and Statistics, in: Lecture Notes in Math., vol. 1869, Springer, Berlin, 2005, pp. 103–274. [13] G. Giacomin, J.L. Lebowitz, E. Presutti, Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems, in: Stochastic Partial Differential Equations: Six Perspectives, Amer. Math. Soc., Providence, RI, 1999, pp. 107–152. [14] M. Grothaus, Y.G. Kondratiev, M. Röckner, N/V -limit for stochastic dynamics in continuous particle systems, Probab. Theory Related Fields 137 (1–2) (2007) 121–160. [15] C. Kipnis, C. Landim, Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin, 1999. [16] A.V. Kolesnikov, Mosco convergence of Dirichlet forms in infinite dimensions with changing reference measures, J. Funct. Anal. 230 (2) (2006) 382–418. [17] P. Kotelenez, A class of quasilinear stochastic partial differential equations of McKean–Vlasov type with mass conservation, Probab. Theory Related Fields 102 (2) (1995) 159–188. [18] A.M. Kulik, Markov uniqueness and Rademacher theorem for smooth measures on infinite-dimensional space under successful filtration condition, Ukraïn. Mat. Zh. 57 (2) (2005) 170–186. [19] K. Kuwae, T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom. 11 (4) (2003) 599–673. [20] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (2) (1994) 368–421. [21] K.R. Parthasarathy, Probability Measures on Metric Spaces, Probab. Math. Statist., vol. 3, Academic Press Inc., New York, 1967. [22] A.-S. Sznitman, Topics in propagation of chaos, in: École d’Été de Probabilités de Saint-Flour XIX—1989, in: Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 165–251. [23] J. Vaillancourt, On the existence of random McKean–Vlasov limits for triangular arrays of exchangeable diffusions, Stoch. Anal. Appl. 6 (4) (1988) 431–446. [24] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., vol. 58, Amer. Math. Soc., Providence, RI, 2003. [25] M.-K. von Renesse, K.-T. Sturm, Entropic measure and Wasserstein diffusion, Ann. Probab. 37 (3) (2009) 1114– 1191. [26] M.-K. von Renesse, M. Yor, L. Zambotti, Quasi-invariance properties of a class of subordinators, Stochastic Process. Appl. 118 (11) (2008) 2038–2057.


On the existence of an extremal function in critical Sobolev trace embedding theorem ✩ A.I. Nazarov ∗ , A.B. Reznikov Saint-Petersburg State University, Universitetskii pr. 28, St. Petersburg, Russian Federation Received 28 October 2009; accepted 26 February 2010 Available online 16 March 2010 Communicated by H. Brezis

Abstract Sufficient conditions for the existence of extremal functions in the trace Sobolev inequality and the trace Sobolev–Poincaré inequality are established. It is shown that some of these conditions are sharp. © 2010 Elsevier Inc. All rights reserved. Keywords: Critical trace Sobolev inequality; Critical trace Sobolev–Poincaré inequality; Sharp constants

1. Introduction Let n 2, and let Ω be a domain in Rn with strictly Lipschitz boundary. For 1 < p < n denote by p = (n−1)p n−p the trace Sobolev exponent for p, that is the critical exponent for the trace embedding Wp1 (Ω) → Lq (∂Ω). Since the embedding operator Wp1 (Ω) → Lp (∂Ω) is non-compact, the problem of attainability of the norm of this operator (i.e. the problem of existence of an extremal function in the trace embedding theorem) is non-trivial. Corresponding problem for conventional embedding np is the Sobolev conjugate of p) was treated in many papers, Wp1 (Ω) → Lp∗ (Ω) (here p ∗ = n−p see, e.g., the recent survey [9] and further references therein. ✩

This paper was partially supported by grant NSh.227.2008.1. The first author was also supported by RFBR grant 08-01-00748. * Corresponding author. E-mail address: [email protected] (A.I. Nazarov). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.018

A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921

3907

The problem for the trace embedding is considerably less investigated. Let us consider the inequality K(n, p) ≡

∇vp,Rn+

inf

v∈C˙∞ (Rn+ )\{0}

v(·, 0)p ,Rn−1

> 0,

(1)

where C˙∞ (Rn+ ) is the set of functions on Rn+ with bounded support. Obviously, the functional in (1) is invariant with respect to translations and dilations of v. It is well known that the infimum in (1) is attained on some function with an unbounded support. Escobar [4] conjectured that the minimizer in (1) is − n−p wε (x) = x − x ε p−1 ,

(2)

with x ε = (0, . . . , 0, −ε), and proved it for p = 2. Later his conjecture was proved in full generality in the remarkable paper [8]. The result of [8] implies K(n, p) =

n−p p−1

1 p

1 (n−1)p ωn−2 n−1 n−1 ·B , . 2 2 2(p − 1)

In this paper we consider the critical trace embedding in bounded domains, i.e. the inequality λ1 (n, p, Ω) =

inf

v∈Wp1 (Ω)\{0}

vWp1 (Ω) vp ,∂Ω

p

p

>0

(I) p

(the norm of the numerator is defined as vW 1 (Ω) = ∇vp,Ω + vp,Ω ). p

Our first result reads as follows. Theorem 1. Let n 2, and let Ω be a bounded domain in Rn with ∂Ω ∈ C 2 . Then for some β(Ω) > 0 and for 1 < p < n+1 2 + β, the infimum in (I) is attained. Remark 1. By standard argument it follows that under suitable normalization the extremal function in (I), if it exists, is a positive solution to the non-linear Neumann problem − p u + up−1 = 0 in Ω;

|∇u|p−2

∂u = up −1 ∂n

on ∂Ω

(here p u = div(|∇u|p−2 ∇u)). Our main tool is the concentration-compactness principle of Lions ([6]; see also [7]). It is used in various forms; for the problem (I) it can be reformulated as follows. Proposition 1. Let Ω be a bounded domain in Rn with ∂Ω ∈ C 1 . Let the infimum in (I) satisfy the inequality λ1 (n, p, Ω) < K(n, p). Then the infimum is attained.

3908


Proof. Let us consider a minimizing sequence {vk }, normalized in Lp (∂Ω). Without loss of generality one can assume that vk v in Wp1 (Ω). By the Lions theorem ([6, Part 2]; see also [7, Lemma 2.2]), |vk |p |v|p + |∇vk |p μ |∇v|p + cj δ x − x j , Cj δ x − x j j

j

(the convergence is in the sense of measures on ∂Ω and on Ω, respectively), where {x j } is at most countable set of distinct points in ∂Ω while cj and Cj are positive constants. Since {vk } is a minimizing sequence, by verbal repetition of the proof of Theorem 2.2 [7] we obtain the alternative — either v is a minimizer of the corresponding problem and the set {x j } is empty, or v = 0 and the set {x j } contains a single point x 0 . In the second case c0 = 1 and p C0 = λ1 (n, p, Ω). Note that until now all the arguments do not use the smoothness of ∂Ω. Let the second case occur. Then, similarly to Corollary 2.1 [7], multiplying vk by a cutoff function we can assume that supports of vk are sufficiently small. Due to the assumption ∂Ω ∈ C 1 , the neighborhood of x 0 in the large scale looks like a half-space. Hence λ1 (n, p, Ω) K(n, p). This contradiction proves the desired statement. 2 Thus, to prove the attainability of the infimum in (I) it is sufficient to present a function such that the quotient in (I) is less than K(n, p). Following [2], see also [7], we succeed, constructing a function with a small support, simulating the behavior of wε (x). Remark 2. The attainability of the infimum in (I) was proved in [5] under some additional assumption on Ω. This assumption means that the quotient in (I) taken on constant function is less than K(n, p). In particular, this is the case for “small” domains. Namely, for a domain with a smooth boundary define Ω as a “dilation” of Ω with the coefficient . Then for any 1 0 such that the infimum in (I) is attained on Ω for < ∗ (note that, in contrast with (1), the quotient in (I) is not homogeneous with respect to dilations). Further, we slightly strengthen the statement of [5] and show that the infimum in (I) is attained for “small” domains without assumption of smoothness. Theorem 2. Let n 2. Suppose Ω is a bounded domain in Rn with strictly Lipschitz boundary. Then for any 1 0 such that for < ∗ , the infimum in (I) is attained on Ω. On the other hand, for “large” domains the assumption of smoothness in Theorem 1 is essential. Theorem 3. Let n 2. Suppose Ω is a polyhedron in Rn , and 1 0 such that for > ∗ , the infimum in (I) is not attained on Ω. Finally, we consider the trace Sobolev–Poincaré inequality λ2 (n, p, Ω) = (here we use the notation v = |∂Ω|−1

inf

v∈Wp1 (Ω)\{c}

∂Ω

v dΣ).

∇vp,Ω >0 v − vp ,∂Ω

(II)


3909

Theorem 4. Let n 3, and let Ω be a bounded domain in Rn with ∂Ω ∈ C 2 . Then for some β(Ω) > 0 and for 1 < p < n+1 2 + β, the infimum in (II) is attained. The structure of our paper is the following. In Section 2 we establish required integral estimates and prove Theorem 1. Theorems 2 and 3 are proved in Section 3; inequality (II) is considered in Section 4. In Section 5 we generalize Theorems 1 and 4 to the case of non-euclidean norm of gradient. Let us recall some notation. A point in Rn is denoted by x = (x , xn ) where x ∈ Rn−1 ; r stands for |x |. Rn+ = {x ∈ Rn : xn > 0}. Qρ = {x: r < ρ, 0 < xn < ρ} is a cylinder in Rn , Bρ (x 0 ) = {x: |x − x 0 | < ρ}. n/2 p ωn−1 = 2π is the area of the unit sphere in Rn . p = p−1 is the Hölder conjugate exponent Γ ( n2 ) to p. B is the Euler beta-function. We denote by oρ (1) a quantity which tends to zero as ρ → 0. We use letter C to denote various positive constants. To indicate that C depends on some parameters, we write C(. . .). 2. Proof of Theorem 1 Consider the least ball containing Ω. Let x 0 ∈ ∂Ω be a point of contact of Ω with this ball. Then all the principal curvatures of ∂Ω at x 0 are positive, and therefore H (x 0 ) > 0. We introduce a local coordinate system such that x 0 is the origin, x lies in the tangent plane and the axis xn is directed into Ω. Since ∂Ω ∈ C 2 , in some neighborhood of the origin ∂Ω is the graph of a function xn = F (x ), and F (x ) = (Ax , x ) + o(r 2 ) as r → 0, where A is a positive definite matrix. For sufficiently small ε > 0 and ρ > 0 we introduce the function u(x) = u(r, xn ) = ϕ(r, xn )wε (r, xn ),

(3)

where wε is defined in (2), while ϕ is a smooth cut-off function such that ϕ = 1 in Q ρ ; 2

and |∇ϕ|

ϕ = 0 in Rn \ Qρ ,

C ρ.

2.1. Estimate of ∇up,Ω for 1 < p

n+1 2

Let us apply the estimate ∇(f g)p |f ∇g|p + C |∇f g|p + |∇f g| · |f ∇g|p−1 to ∇u. Since ∇ϕ does not vanish only in Qρ \ Q ρ , one obtains 2

Ω

|∇ϕ · wε |p + |∇ϕ · wε | · |ϕ∇wε |p−1 dx

(4)

3910


ρ/2 ρ

ρ ρ

C

ρ −p r n−2 dxn dr

+ 0 ρ/2

p(n−p)

(r 2 + xn2 ) 2(p−1)

ρ/2 0

p−1 ρ − n−p Cρ p−1 . · 1+ 2 2 r + xn

(5)

Furthermore, one has |ϕ∇wε |p |∇wε |p =

n−p p−1

p

x − x ε −(n−1)p .

The inequalities (4) and (5) imply

|∇u| dx p

Qρ ∩Ω

Ω

n−p p−1

p

n−p n−p x − x ε −(n−1)p dxn dx + Cρ − p−1 = I1 − I2 + Cρ − p−1 , (6)

where

ρ

I1 = |x | 0, when b is e.g. bounded and measurable. This remarkable fact was first observed by Zvonkin in his celebrated paper [31]. An extension of this important result to the multidimensional case was given by Veretennikov [28]. The authors first employ estimates of solutions of parabolic partial differential equations to construct weak solutions. Then a pathwise uniqueness argument is applied to ensure a unique strong solution.

3924

T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953

We mention that the latter results, which can be considered a milestone of the theory of SDE’s, have been generalized by Gyöngy and Martínez [8] and Krylov and Röckner [14], who impose deterministic integrability conditions on the drift coefficient to guarantee existence and uniqueness of strong solutions. As in [31] and [28], the authors first derive weak solutions. Here the approach of Gyöngy and Martínez rests on the Skorohod embedding, whereas the method of Krylov and Röckner is based on an argument of Portenko [23]. Finally, the authors resort to pathwise uniqueness and obtain strong solutions. Another technique which relies on an Euler scheme of approximation and pathwise uniqueness can be found in Gyöngy and Krylov [7]. See also [6], where a modified version of Gronwall’s lemma is invoked. We repeat that the common idea in all the references above is the construction of weak solutions together with the subsequent use of the celebrated pathwise uniqueness argument obtained by Yamada and Watanabe [30] which can be formulated as weak existence + pathwise uniqueness

⇒

strong uniqueness.

(5)

In this paper we devise a new method to study strong solutions of SDEs of type (1) with irregular coefficients. Our method does not rely on the pathwise uniqueness argument but gives a direct construction of strong solutions. Further, to conclude strong uniqueness, we derive the following result that in some sense is diametrically opposed to (5): strong existence + uniqueness in law

⇒

strong uniqueness.

Our technique is mainly based on Malliavin calculus. More precisely, we use a compactness criterion based on Malliavin calculus combined with “local time variational calculus” and an approximation argument to show that a certain generalized process in the Hida distribution space satisfies the SDE. As our main result we derive stochastic (and deterministic) integrability conditions on the coefficients to guarantee the existence of strong solutions. Moreover, our method yields the insight that these strong solutions are Malliavin differentiable. As a special case of our technique we obtain the result of Zvonkin [31] for SDE’s with merely bounded and measurable drift coefficients together with the additional and rather surprising information that these solutions are Malliavin differentiable. The latter sheds a new light on solutions of SDE’s and raises the question to which extent the “nature” of strong solutions is tied to the property of Malliavin differentiability. We emphasize that, although in this paper we focus on SDE’s driven by Brownian motion, our technique exhibits the capacity to cover stochastic equations for a broader class of driving noises. In particular, it can be applied to SDE’s driven by Lévy processes, fractional Brownian motion or more general by fractional Lévy processes. Other applications are evolution equations on Hilbert spaces, SPDE’s and anticipative SDE’s. In particular, the applicability of our technique to infinite dimensional equations on Hilbert spaces seems very interesting since very little is known in this case. Such equations cannot be captured by the framework of the above authors. The latter is due to the fact that the authors’ techniques utilize specific estimates, which work in the Euclidean space, but fail in infinite dimensional spaces. See e.g. [8, Lemma 3.1], where an estimate of Krylov [13] for semimartingales is used. Our paper is inspired by ideas initiated in [19] and [25]. In [25] the author constructs strong solutions of SDE’s with functional discontinuous coefficients by using certain estimates for the n-th homogeneous chaos of chaos expansions of solutions. The paper [19] exploits a comparison theorem to derive solutions of SDE’s.


3925

The paper is organized as follows: In Section 2 we give the framework of our paper. Here we review basic concepts of Malliavin calculus and Gaussian white noise theory. Then in Section 3 we illustrate our method by studying the SDE (1). Our main results are Theorems 4, 5 and 22. Section 4 concludes with a discussion of our method. 2. Framework In this section we recall some facts from Gaussian white noise analysis and Malliavin calculus, which we aim at employing in Section 3 to construct strong solutions of SDE’s. See [9,22,15] for more information on white noise theory. As for Malliavin calculus the reader is referred to [21,17,18,4]. 2.1. Basic elements of Gaussian white noise theory As mentioned in the Introduction we want to use generalized stochastic processes on a certain stochastic distribution space to analyze strong solutions of SDE’s. In the sequel we give the construction of the stochastic distribution space, which goes back to T. Hida (see [9]). From now on we fix a time horizon 0 < T < ∞. Consider a (positive) self-adjoint operator A on L2 ([0, T ]) with Spec(A) > 1. Let us require that A−r is of Hilbert–Schmidt type for some r > 0. Denote by {ej }j 0 a complete orthonormal basis of L2 ([0, T ]) in Dom(A) and let λj > 0, j 0 be the eigenvalues of A such that 1 < λ0 λ1 · · · → ∞. Let us assume that each basis element ej is a continuous function on [0, T ]. Further let Oλ , λ ∈ Γ be an open covering of [0, T ] such that −α(λ)

sup λj

j 0

sup ej (t) < ∞ t∈Oλ

for α(λ) 0. In what follows let S([0, T ]) denote the standard countably Hilbertian space constructed from (L2 ([0, T ]), A). See [22]. Then S([0, T ]) is a nuclear subspace of L2 ([0, T ]). We denote by S ([0, T ]) the corresponding conuclear space, that is, the topological dual of S([0, T ]). Then the Bochner–Minlos theorem provides the existence of a unique probability measure π on B(S ([0, T ])) (Borel σ -algebra of S ([0, T ])) such that eiω,φ π(dω) = e

− 12 φ2 2

L ([0,T ])

S ([0,T ])

holds for all φ ∈ S([0, T ]), where ω, φ is the action of ω ∈ S ([0, T ]) on φ ∈ S([0, T ]). Set Ωi = S [0, T ] ,

Fi = B S [0, T ] ,

for i = 1, . . . , d. Then the product measure

μi = π,

3926


×μ d

μ=

(6)

i

i=1

on the measurable space (Ω, F ) :=

d

i=1

Ωi ,

d

Fi

(7)

i=1

is referred to as d-dimensional white noise probability measure. Consider the Doléans–Dade exponential

1 2 e(φ, ˜ ω) = exp ω, φ − φL2 ([0,T ];Rd ) , 2 d (1) (d) d for d ω = (ω1 , . . . , ωd ) ∈ (S ([0, T ])) and φ = (φ , . . . , φ ) ∈ (S([0, T ])) , where ω, φ := i=1 ωi , φi . n be the n-th completed symmetric tensor product of In the following let ((S([0, T ]))d )⊗ ˜ ω) is holomorphic in φ around zero. Hence there (S([0, T ]))d with itself. One verifies that e(φ, n ) such that exist generalized Hermite polynomials Hn (ω) ∈ (((S([0, T ]))d )⊗

e(φ, ˜ ω) =

1 Hn (ω), φ ⊗n n!

(8)

n0

for φ in a certain neighbourhood of zero in (S([0, T ]))d . It can be shown that d ⊗ n Hn (ω), φ (n) : φ (n) ∈ S [0, T ] , n ∈ N0

(9)

is a total set of L2 (μ). Further one finds that the orthogonality relation

Hn (ω), φ (n) Hm (ω), ψ (m) μ(dω) = δn,m n! φ (n) , ψ (n) L2 ([0,T ]n ;(Rd )⊗n )

(10)

S ([0,T ]) n m is valid for all n, m ∈ N0 , φ (n) ∈ ((S([0, T ]))d )⊗ , ψ (m) ∈ ((S([0, T ]))d )⊗ where

δn,m

1 if n = m, 0 else.

Define Lˆ 2 ([0, T ]n ; (Rd )⊗n ) as the space of square integrable symmetric functions f (x1 , . . . , xn ) with values in (Rd )⊗n . Then the orthogonality relation (10) implies that the mappings φ (n) → Hn (ω), φ (n) n to L2 (μ) possess unique continuous extensions from (S([0, T ])d )⊗

⊗n In : Lˆ 2 [0, T ]n ; Rd → L2 (μ)


3927

for all n ∈ N. We remark that In (φ (n) ) can be viewed as an n-fold iterated Itô integral of φ (n) ∈ Lˆ 2 ([0, T ]n ; (Rd )⊗n ) with respect to a d-dimensional Wiener process (1) (d) Bt = Bt , . . . , Bt

(11)

(Ω, F , μ).

(12)

on the white noise space

It turns out that square integrable functionals of Bt admit a Wiener–Itô chaos representation which can be regarded as an infinite dimensional Taylor expansion, that is, L2 (μ) =

⊗n . In Lˆ 2 [0, T ]n ; Rd

(13)

n0

We construct the Hida stochastic test function and distribution space by using the Wiener–Itô chaos decomposition (13). For this purpose let Ad := (A, . . . , A),

(14)

where A was the operator introduced in the beginning of the section. We define the Hida stochastic test function space (S) via a second quantization argument, that is, we introduce (S) as the space of all f = n0 Hn (·), φ (n) ∈ L2 (μ) such that f 20,p :=

⊗n p 2 n! Ad φ (n) 2

L ([0,T ]n ;(Rd )⊗n )

0 the set G = G ∈ D1,2 : GL2 (Ω) + DGL2 (Ω;H ) c is relatively compact in L2 (Ω). We also need the following technical result, which can be found in [3].


3935

Lemma 8. Let vs , s 0, be the Haar basis of L2 ([0, 1]). For any 0 < α < 1/2 define the operator Aα on L2 ([0, 1]) by Aα vs = 2kα vs

if s = 2k + j

for k 0, 0 j 2k and Aα 1 = 1. Then for all β with α < β < (1/2), there exists a constant c1 such that 1 1

Aα f c1 f L2 ([0,1]) +

0 0

|f (t) − f (t )|2 dt dt |t − t |1+2β

1/2 .

In the sequel we will make use of the concept of stochastic integration over the plane with respect to Brownian local time. See [5]. Consider elementary functions f : [0, 1] × R → R given by

f (s, x) =

fij χ(sj ,sj +1 ] (s).χ(xi ,xi+1 ] (x),

(43)

(sj ,xi )∈

where (xi )1in , (fij )1in,1j m are finite sequences of real numbers, (sj )1j m a partition of [0, 1] and = {(sj , xi ), 1 i n, 1 j m}. Denote by {L(t, x)}0t1,x∈R the local time of a 1-dimensional Brownian motion B. Then integration of elementary functions with respect to L can be defined by 1 f (s, x) L(ds, dx) 0 R

=

fij L(sj +1 , xi+1 ) − L(sj , xi+1 ) − L(sj +1 , xi ) + L(sj , xi ) .

(44)

(sj ,xi )∈

The latter integral actually extends to integrands of the Banach space (H, · ) of measurable functions f such that 1 f = 2 0 R

1 + 0 R

1/2

2 2 ds dx x f (s, x) exp − √ 2s 2πs

2 ds dx xf (s, x) exp − x < ∞. √ 2s s 2πs

The proof of the following theorem is given in [5, Theorem 3.1, Corollary 3.2].

(45)

3936


Theorem 9. Let f ∈ H. (i) Then t f (s, x) L(ds, dx),

0 t 1,

0 R

exists and t f (s, x) L(ds, dx) f E 0 R

for 0 t 1. (ii) If f is such that f (t, ·) is locally square integrable and f (t, ·) continuous in t as a map from [0, 1] to L2loc (R), then t

f (s, x) L(ds, dx) = − f (·, B), B t ,

0 t 1,

0 R

where [·,·] stands for the quadratic covariation of processes. (iii) In particular, if f (t, x) is differentiable in x, then t

t f (s, x) L(ds, dx) = −

0 R

f (s, B) ds,

0 t 1,

0

where f (s, x) denotes the derivative in x. The proof of Theorem 4 relies on the following lemma. Lemma 10. Retain the conditions of Theorem 4. Then the sequence Xt(n) = Ytbn , n 1 (see (33)) is relatively compact in L2 (μ; Rd ), 0 t 1. (n)

Proof. We want to prove the relative compactness of the sequence of SDE solutions Xt n 1, by applying Theorem 7. Because of Lemma 8 it is sufficient to show that (n) 2 (n) 2 E X1 + E Aα D· X1 L2 ([0,1];Rd ) M

= Ytbn ,

(46)

for all n 1, where 0 M < ∞. (n) Without loss of generality we study the sequence X1 , n 1. Then by the conditions of (n) Theorem 4 the solutions Xt , n 1, are Malliavin differentiable (see [21,4,29]). Moreover the


3937

chain rule with respect to the Malliavin derivative Dt implies that

(n) Dt X 1

1 =

bn s, Xs(n) Ds Xs(n) ds + Id ,

0 t 1, n 1.

(47)

t

It follows from our assumptions that the solution to the linear equation (47) for each n 1 is given by

(n) Dt X 1

1 = exp

bn

(n) s, Xs ds ∈ Rd×d ,

0 t 1.

(48)

t

See e.g. [1]. Fix 0 t t 1. Then we find that (n) (n) 2 E Dt X1 − Dt X1 1 1 2 (n) (n) = E exp bn s, Xs ds − exp bn s, Xs ds . t

t

So it follows from Girsanov’s theorem, the properties of supermartingales and (36) that (n) (n) 2 E D t X 1 − D t X 1 1 1 2 1/2 C · E exp bn (s, Bs ) ds − exp bn (s, Bs ) ds t

t

for a constant C which is independent of n. The latter in connection with the properties of evolution operators for linear systems of ODE’s, the mean value theorem and Hölder’s inequality give (n) (n) 2 E Dt X1 − Dt X1 4 1/2 1 4 t const.E exp bn (s, Bs ) ds · exp bn (s, Bs ) ds − Id t

t

4 1 4 const.E exp bn (s, Bs ) ds bn (s, Bs ) ds t

t

t

t 4 1/2 · sup exp λ bn (s, Bs ) ds 0λ1 t

3938


1 12 1/6 const.E exp bn (s, Bs ) ds t

t 12 1/6 t 12 1/6 · E bn (s, Bs ) ds E sup exp λ bn (s, Bs ) ds . 0λ1 t

(49)

t

In the next step of our proof we want to invoke a “local time variational calculus” argument based on Theorem 9 to get rid of the derivatives bn in (49). In order to simplify notation let us assume that the initial value x of the SDE (29 ) is zero. In the sequel we introduce the notation Li (ds, dx) (i) to indicate local time–space integration in the sense of Theorem 9 with respect to Bt (i.e. the i-th component of Bt ) on (Ωi , μi ), i = 1, . . . , d (see (7)). So using Theorem 9, point (iii), we infer from (49) that (n) (n) 2 E Dt X1 − Dt X1 1 (j ) const.E exp − bn (s, x) Li (ds, dx) t R

t (j ) ·E − bn (s, x) Li (ds, dx) t

1i,j d

R

1i,j d

12 1/6

12 1/6

(j ) · E sup exp λ − bn (s, x) Li (ds, dx) 0λ1

t

t

1i,j d

R

12 1/6

= const.I1 · I2 · I3 ,

(50)

where 1 (j ) I1 := E exp − bn (s, x) Li (ds, dx)

1i,j d

t R

t (j ) I2 := E − bn (s, x) Li (ds, dx) t

1i,j d

R

12 1/6 ,

12 1/6

and t (j ) bn (s, x) Li (ds, dx) I3 := E sup exp λ − 0λ1

t

R

1i,j d

12 1/6 .


3939

Now we want to use the following decomposition of local time–space integrals (see the proof of Theorem 3.1 in [5]): t

t

fi s, Bs(i) dBs(i) +

fi (s, x) Li (ds, dx) = 0

0

1

fi 1 − s, Bˆ s(i) d W˜ s(i)

1−t

1 − 1−t

Bˆ s(i) fi 1 − s, Bˆ s(i) ds, 1−s

(51)

0 t 1, a.e. for fi ∈ H, i = 1, . . . , d (see Theorem 9). Here Bˆ t is the i-th component of the (i) time-reversed Brownian motion and W˜ t is a Brownian motion on (Ωi , μi ) with respect to the ˆ (i) filtration FtB , i = 1, . . . , d (see (35)). Let us apply (51) to establish some upper bounds for the factors I1 , I2 , I3 . (1) Estimate for I1 : (51) and Hölder’s inequality entail that (i)

1 (j ) (i) I1 = E exp − bn (s, Bs ) dBs t

· exp

1i,j d

1−t

(j ) bn (1 − s, Bˆ s ) d W˜ s(i)

−

1i,j d

0

1−t ˆ s(i) B (j ) ds · exp bn (1 − s, Bˆ s ) 1−s

1i,j d

0

1 (j ) E exp − bn (s, Bs ) dBs(i) t

1i,j d

1−t (j ) (i) · E exp − bn (1 − s, Bˆ s ) d W˜ s 0

12 1/6

36 1/18

1i,j d

(i) Bˆ s (j ) ˆ ds · E exp bn (1 − s, Bs ) 1−s

36 1/18

1−t

1i,j d

0

36 1/18 .

(2) Estimate for I3 : Similarly to (1) we find t (j ) (i) I3 E sup exp −λ bn (s, Bs ) dBs 0λ1

t

1i,j d

36 1/18

(52)

3940


1−t (j ) (i) ˆ ˜ · E sup exp −λ bn (1 − s, Bs ) d Ws 0λ1

1−t

1i,j d

1−t ˆ s(i) B (j ) ds · E sup exp λ bn (1 − s, Bˆ s ) 1−s 0λ1

36 1/18

1−t

1i,j d

36 1/18 .

(53)

(3) Estimate for I2 : The decomposition (51), Burkholder’s and Hölder’s inequality imply that

I2 const.

d i,j =1

12 1/6 t (j ) E bn (s, x) Li (ds, dx) R

t

12 1/6 12 1/6 1−t t d (j ) (j ) const. +E bn (1 − s, Bˆ s ) d W˜ s(i) E bn (s, Bs ) dBs(i) i,j =1

1−t

t

12 1/6 1−t ˆ s(i) B (j ) ds +E bn (1 − s, Bˆ s ) 1−s 1−t

const.

d i,j =1

6 1/6 1−t 6 1/6 t (j ) (j ) 2 2 bn (s, Bs ) ds bn (1 − s, Bˆ s ) ds +E E 1−t

t

12 1/6 1−t ˆ s(i) B (j ) ds +E bn (1 − s, Bˆ s ) 1−s 1−t

t 12/(2+ε) 1/6 d ε/(2+ε) (j ) 2+ε t − t bn (s, Bs ) E ds const. i,j =1

t

1−t 12/(2+ε) 1/6 ε/(2+ε) (j ) 2+ε bn (1 − s, Bˆ s ) + t − t E ds 1−t

2ε/(2+ε) + t − t E

1−t 24/(2+ε) 1/6 ˆ (i) 1+ε/2 (j ) bn (1 − s, Bˆ s ) Bs ds 1−s 1−t

for > 0. Hence by the estimates in (1), (2), (3) we conclude from (50) that ε/(2+ε) (n) (n) 2 E Dt X1 − Dt X1 C t − t for all 0 t t 1, n 1, where C 0 is a universal constant. Then using Theorem 7 and Lemma 8 completes the proof. 2


3941

We are coming to Step 2 of our programme. Under the conditions of Theorem 4 we want to gradually prove the following: • Ytb in (33) is a well-defined object in the Hida distribution space (S)∗ , 0 t 1 (Lemma 11). • Ytb ∈ L2 (μ), 0 t 1 (Lemma 13). (n) • We invoke Theorem 9 and show that Xt = Ytbn converges to Ytb in L2 (μ; Rd ) for a subsequence, 0 t 1. • We apply a transformation property for Ytb (Lemma 15) and identify Ytb as a solution to (29). The first lemma gives a criterion under which the process Ytb belongs to the Hida distribution space. Lemma 11. Suppose that

1

Eμ exp 36

b(s, Bs )2 ds

< ∞,

(54)

0

where the drift b : [0, 1] × Rd → Rd is measurable. Then the coordinates of the process Ytb , defined in (33), that is, Yti,b = Eμ˜ B˜ t(i) ET (b) ,

(55)

are elements of the Hida distribution space. Proof. Without loss of generality we consider the case d = 1. Set Φ(ω, ˜ ω) = ϕ B˜ t (ω) ˜ ET (b)(ω, ω) ˜ and 1 1 φ 2 1 E Mt = exp b(t, B˜ t ) + φ(t) d B˜ t − b(t, B˜ t ) + φ(t) dt 2 0

0

for φ ∈ SC ([0, 1]), where s Msφ (ω) ˜ =

b t, B˜ t (ω) ˜ + φ(t) d B˜ t (ω). ˜

0

Using the Kubo–Yokoi delta function (see [15, Theorem 13.4]), the characterization theorem of Hida distributions and the concept of G-entire functions (see [24]) we find that Φ is a well-defined map from Ω˜ to (S)∗ (up to equivalence). Further it follows from our assumption, Hölder’s inequality and the supermartingale property of Doléans–Dade exponentials that

3942


φ ˜ ·) (φ) = Eμ˜ ϕ(B˜ t )E Mt Eμ˜ S Φ(ω, 1 1 1 2 2 K · Eμ˜ E 2 b(t, B˜ t ) + Re φ(t) d B˜ t exp a φ(t) dt 0

0

K exp a|φ|20 , for φ ∈ SC ([0, 1]), where a, K 0 are constants and |φ|20 := rem 13.4] yields the result. 2

1 0

|φ(t)|2 dt. Then [15, Theo-

The next auxiliary result gives a justification for the factor Rn in (42) of Theorem 4. Lemma 12. Let bn : [0, 1] × Rd → Rd be a sequence of Borel measurable functions with b0 = b such that the integrability condition (36) is valid. Then 1 2 i,bn i,b S Yt − Yt (φ) const · Rn · exp 34 φ(s) ds 0

for all φ ∈ (SC ([0, 1]))d , i = 1, . . . , d, with the factor Rn as in (42). Proof. For i = 1, . . . , d we obtain by Proposition 1 and (19) that i,bn S Yt − Yti,b (φ) Eμ˜

d 1 (i) (j ) (j ) B˜ t exp b (s, B˜ s ) + φ (j ) (s) d B˜ s Re j =1

1 − 2

1

0

2 b(j ) (s, B˜ s ) + φ (j ) (s) ds

0

d 1 (j ) (j ) · exp bn (s, B˜ s ) − b(j ) (s, B˜ s ) d B˜ s j =1 0

1 + 2

1

(j ) b(j ) (s, B˜ s )2 − bn (s, B˜ s )2 ds

0

1 +

φ

(j )

(j ) (j ) (s) b (s, B˜ s ) − bn (s, B˜ s ) ds − 1 .

0

Since exp(z) − 1 |z| exp |z|


3943

it follows from Hölder’s inequality that i,bn 1 S Yt − Yti,b (φ) Eμ˜ |Qn |2 2 · Eμ˜


1 − 2

1

(j ) 2 b (s, B˜ s ) + φ (j ) (s) ds

0

2

1 2 exp 2|Qn | ,

0

where d (j ) (j ) 1 (j ) (j ) Qn = bn (s, B˜ s ) − b(j ) (s, B˜ s ) d B˜ s + b (s, B˜ s )2 − bn (s, B˜ s )2 ds 2 1

1

j =1 0

1 +

0

(j ) φ (j ) (s) b(j ) (s, B˜ s ) − bn (s, B˜ s ) ds.

0

We find that 1 d 1 2 2 (j ) (j ) φ(s) ds · Eμ˜ bn (s, B˜ s ) − b(j ) (s, B˜ s ) d B˜ s Eμ˜ |Qn |2 9d 2 exp j =1

0

1 +

(j ) (j ) b (s, B˜ s )2 − bn (s, B˜ s )2 ds

0

2

1 +

0

(j ) 2 (j ) b (s, B˜ s ) − bn (s, B˜ s ) ds

0

1 = 3 exp

φ(s)2 ds Eμ˜ [Jn ],

0

where 2 1 1 d (j ) (j ) 2 (j ) (j ) 2 2 b (s, B˜ s ) − bn (s, B˜ s ) ds . Jn = bn (s, B˜ s ) − b (s, B˜ s ) ds + 2 j =1

0

0

Further we get that d 1 (i) (j ) B˜ t exp Eμ˜ b (s, B˜ s ) + φ (j ) (s) d B˜ s Re j =1

1 − 2

1 0

0

(j ) 2 b (s, B˜ s ) + φ (j ) (s) ds

2

exp 2|Qn |

3944


Eμ˜


1 − 2

1

b

(j )

0

(s, B˜ s ) + φ

(j )

2 (s) ds

4 1 2

0

1 1 1 · √ Eμ˜ exp{−8 Re Qn } 2 + Eμ˜ exp{8 Re Qn } 2 2 1 1 + Eμ˜ exp{−8 Im Qn } 2 + Eμ˜ exp{8 Im Qn } 2 . By Hölder’s inequality again and the supermartingale property of Doléans–Dade exponentials we get the estimate Eμ˜ exp{−8 Re Qn } d 1 1 (j ) (j ) 2 (j ) (j ) Eμ˜ exp b (s, B˜ s ) − bn (s, B˜ s ) ds − 8 b (s, B˜ s )2 − bn (s, B˜ s )2 ds 128 j =1

1 +8

0

0

2 Re φ (j ) (s) ds +

0

1

(j ) 2 (j ) b (s, B˜ s ) − bn (s, B˜ s ) ds

1 2

0

1 2 Ln exp 4 φ(s) ds , 0

where

Ln = Eμ˜ exp

d j =1

1 +8

1 128

(j ) 2 (j ) b (s, B˜ s ) − bn (s, B˜ s ) ds

0

(j ) b (s, B˜ s )2 − bn(j ) (s, B˜ s )2 ds

1 2

.

0

Similarly we conclude that 1 2 1 2 φ(s) ds . Eμ˜ exp{8 Re Qn } Ln exp 4 0 1

1

Also Eμ˜ [exp{−8 Im Qn }] 2 and Eμ˜ [exp{8 Im Qn }] 2 have the same upper bound as in the previous inequality.


3945

Finally we see that d 1 (i) (j ) (j ) Eμ˜ B˜ t exp b (s, B˜ s ) + φ (j ) (s) d B˜ s Re j =1

1 − 2

1

0

(j ) 2 b (s, B˜ s ) + φ (j ) (s) ds

4 1 2

0

1 1 2 2 (i) 8 1 18 4 ˜ ˜ b(s, Bs ) ds φ(s) ds . Eμ˜ Bt Eμˆ exp 512 exp 64 0

0

Altogether we get 1 2 i,bn i,b S Yt − Yt (φ) const · Rn · exp 34 φ(s) ds 0

2

with Rn as in (42).

The next lemma, which also is of independent interest, shows that under some conditions Ytb is square integrable. Lemma 13. Let bn : [0, 1] × Rd → Rd with b0 = b be a sequence of Borel measurable functions such that the conditions (36) and (42) are fulfilled. Further impose on each bn , n 1, Lipschitz continuity and a linear growth condition. Then the process Ytb given by (55) is square integrable for all t. Proof. Because of our assumptions on bn , n 1, we conclude from Lemma 11 that Ytbn are square integrable unique solutions to (29). Further, Hölder’s inequality and the supermartingale property of Doléans–Dade exponentials imply that i,bn 2 Yt 2

L (μ)

= Eμ˜

(i) 2 E B˜ t

1

bn (s, B˜ s ) d Bˆ s

0

1 1 4 2 M < ∞. exp 6 bn (s, B˜ s ) ds

const · sup Eμ˜ n1

(56)

0

Thus the sequence Ytbn is relatively compact in L2 (μ; Rd ) in the weak sense. From this we see that there exists a subsequence of Ytbn which converges to an element Zt ∈ L2 (μ; Rd ) weakly. Without loss of generality we assume that Ytbn → Zt

weakly for n → ∞.

3946


In particular, since 1 E

φ(s) dBs ∈ Lp (μ),

p > 0,

0

one finds that Eμ Yti,bn E

1

φ(s) dBs

(i) → Eμ Zt E

1

φ(s) dBs

0

for n → ∞.

0

On the other hand the estimate as given in Lemma 12 yields Eμ Yti,bn E

1

φ(s) dBs

=

(i) Eμ˜ B˜ t E

1

0

bn (s, B˜ s ) + φ(s) d B˜ s

0

→ Eμ˜

B˜ t(i) E

1

= S Ytb (φ),

b(s, B˜ s ) + φ(s) d B˜ s

0

d φ ∈ SC [0, 1] .

Hence S Yti,b (φ) = S Zt(i) (φ),

d φ ∈ SC [0, 1] .

Since the S-transform is a monomorphism we get that Yt = Zt ∈ L2 (μ; Rd ).

2

We shall introduce a class M of approximating functions. Definition 14. We denote by M the class of Borel measurable functions b : [0, 1] × Rd → Rd for which there exists a sequence of approximating functions bn : [0, 1] × Rd → Rd in the sense of Rn → 0 in (42) such that the conditions of Theorem 4 hold. We are ready to state the following “transformation property” for Ytb . Lemma 15. Assume that b : [0, 1] × Rd → Rd belongs to the class M. Then ϕ (i) t, Ytb = Eμ˜ ϕ (i) (t, B˜ t )ET (b) a.e. for all 0 t 1, i = 1, . . . , d and ϕ = (ϕ (1) , . . . , ϕ (d) ) such that ϕ(Bt ) ∈ L2 (μ; Rd ). Proof. See [25, Lemma 16] or [19] for a proof.

2

Using the above auxiliary results we can finally give the proof of Theorem 4.

(57)


3947

Proof of Theorem 4. We aim at employing the transformation property (57) of Lemma 15 to verify that Ytb is a unique strong solution of the SDE (29). To shorten notation we set t d t (j ) (j ) j =1 0 ϕ (s, ω) dBs and x = 0. 0 ϕ(s, ω) dBs := We first remark that Y·b has a continuous modification. The latter can be checked as follows: Since each Ytbn is a strong solution of the SDE (29) with respect to the drift bn we obtain from Girsanov’s theorem and (36) that 1 i,bn (i) i,bn 2 (i) 2 ˜ ˜ ˜ ˜ = Eμ˜ Bt − Bu E bn (s, Bs ) d Bs Eμ Yt − Yu 0

const · |t − u| for all 0 u, t 1, n 1, i = 1, . . . , d. By Lemma 10 we know that (bn )

Yt

(b)

→ Yt

in L2 μ; Rd

for a subsequence, 0 t 1. So we get that 2 Eμ Yti,b − Yui,b const · |t − u|

(58)

for all 0 u, t 1, i = 1, . . . , d. Then Kolmogorov’s lemma provides a continuous modification of Ytb . Since B˜ t is a weak solution of (29) for the drift b(s, x) + φ(s) with respect to the measure 1 dμ∗ = E( 0 (b(s, B˜ s ) + φ(s)) d B˜ s ) dμ we obtain that 1 i,b (i) ˜ ˜ ˜ S Yt (φ) = Eμ˜ Bt E b(s, Bs ) + φ(s) d Bs (i) = Eμ∗ B˜ t 1 = Eμ∗

0

(i) b (s, B˜ s ) + φ (i) (s) ds

0

t =

Eμ˜

b (s, B˜ s )E

1

(i)

0

b(u, B˜ u ) + φ(u) d B˜ u

ds + S Bt(i) (φ).

0

Hence the transformation property (57) applied to b gives S Yti,b (φ) = S

t

(i) i,b b u, Yu du (φ) + S Bt (φ). (i)

0

3948


Then the injectivity of S implies that t Ytb

=

b s, Yub ds + Bt .

0

The Malliavin differentiability of Ytb follows from the fact that sup Yti,bn 1,2 M < ∞

n1

for all i = 1, . . . , d and 0 t 1. See e.g. [21]. Finally, if the solution Xt = Ytb is unique in law then our conditions allow the application of Girsanov’s theorem to any other strong solution. Then the proof of Proposition 1 (see e.g. [25, Proposition 1]) shows that any other solution necessarily takes the form Ytb . 2 As a consequence of Theorem 4 we obtain the following result. Corollary 16. Replace in Theorem 4 the conditions (36)–(39) by

1

sup E exp 2592d n0

5

bn (s, Bs )2 ds

|f | ∗ ξ(x) = =

f (y)ξ y −1 x dy =

G

ΔG y −1 f xy −1 ξ(y) dy

G

ΔH,G h−1 ΔG h−1 y −1 f xh−1 y −1 ξ(yh) dh dy

G/H H

=

−1/2 ΔH,G (h)ΔG h−1 ΔG y −1 f xh−1 y −1 ξ(y) dh dy

G/H H

=

1/2 ΔH,G (h)ΔG (h)ΔG y −1 f xhy −1 ΔH h−1 ξ(y) dh dy

G/H H

= G/H

− 12

ΔH,G (h)

−1

−1 f xhy dh ξ(y) dy. ΔG y

H

Therefore, by the theorem of Fubini, there exists for every x ∈ G\N a set Mx ⊂ G of measure 0, such that

1 ΔH,G (h)− 2 ΔG y −1 f xhy −1 dh < ∞ H

3960

M. Elloumi et al. / Journal of Functional Analysis 258 (2010) 3955–3976

1 for every y ∈ / Mx and such that the function y → H ΔH,G (h)− 2 ΔG (y −1 )|f (xhy −1 )| dh is integrable. Whence for x ∈ / N and η ∈ Cc (G/H, ρ),

π(f )η (x) = f (y)η y −1 x dy = ΔG y −1 f xy −1 η(y) dy G

=

G

−1/2 ΔH,G (h)ΔG h−1 ΔG y −1 f xh−1 y −1 ρ h−1 η(y) dh dy

G/H H

= G/H

1 ΔH,G (h)− 2 ΔG y −1 f xhy −1 ρ(h) dh η(y) dy.

(2)

H

We deduce from (2) that f ∈ ker(indG H ρ) if and only if for every x ∈ G \ N , there exists a set Nx ⊃ Mx of measure 0 in G such that the linear operator

1 ΔH,G (h)− 2 ΔG y −1 f xhy −1 ρ(h) dh H

is 0 for every y ∈ / Nx .

2

3. Flat orbits In this section we characterize the flat orbits of a completely solvable Lie group of endomorphisms of a finite dimensional real vector space V. Let D = exp(D) be an exponential Lie group of linear endomophisms of V . We assume that D is completely solvable. This means that the eigenvalues of every D ∈ D, considered as an endomorphism of the complexification VC of V , are real numbers. We denote by D the associative hull in the endomorphism ring of the vector space V generated by D. Then the group D is contained in the algebra RIV + D. Note that D is linearly generated by the set {D j : D ∈ D, j ∈ N}. For l ∈ V ∗ , we define:

ND (l) = x ∈ V : l, D(x) = 0, ∀D ∈ D ,

D(l) = D ∈ D: D t (l) = 0 ,

AD (l) = x ∈ V : l, T (x) = 0, ∀T ∈ D . Here D t denotes the transpose of D: D t l, X := l, D(X), X ∈ V , l ∈ V ∗ . It follows from the definitions, that AD (l) ⊂ ND (l) and that

AD (l) = X ∈ ND (l): T (X) ∈ ND (l), ∀T ∈ D . Definition 3.1. We say that an orbit O(l) = Dt l ⊂ V ∗ of the exponential completely solvable group D is flat, if the subspace ND (l) of V (and hence ND (q) of every element q ∈ O(l)) is D-invariant.


3961

Theorem 3.2. Let D = exp(D) be an exponential completely solvable Lie group of endomorphisms of the real finite dimensional vector space V . Let l ∈ V ∗ and O = O(l) = Dt l be the D-orbit of l. The following statements are equivalent: 1) O is flat, i.e. ND (l) is D-invariant ⇔ ND (l) = AD (l). 2) Dt · l|ND (l) = l|ND (l) . 3) There exists an analytic function P : R → R; P (ξ ) = 1 + a2 ξ 2 + a3 ξ 3 + · · · for small ξ , with a2 = 0, such that for every q in the orbit O(l), P (D t )q ∈ O(l) for D ∈ D small enough. Proof. 1) ⇒ 2) Let X ∈ ND (l) = AD (l). Since D j (X) ∈ ND (l) for every j ∈ N∗ , it follows that l, D j (X) = 0,

j ∈ N∗ , X ∈ ND (l), D ∈ D,

and so exp(D t )l, X = l, X. 2) ⇒ 1) Let X ∈ ND (l). For all D ∈ D, s ∈ R, we have then that

l, X = exp sD t (l), X (sD t )k (l), X = k! k0

s 2 t 2 D + · · · (l), X = IV + sD t + 2! s 2 t 2 D (l), X + · · · . = l, X + s D t (l), X + 2! It follows that, s 2 t 2 D (l), X + · · · = 0 s D t (l), X + 2! and therefore for all j 1, (D t )j (l), X = 0. Hence, l, T (X) = 0 for all T ∈ D and thus ND (l) ⊂ AD (l), which completes the proof in this case. 3) ⇒ 1) We proceed by induction on d = dim(V ) + dim(D). The result is obviously true if d = 1. Let d 2. We take V0 = ker(l) ∩ AD (l). We have to treat the following cases: Case 1. V0 = {0}. Let p : V −→ V˜ = V /V0 be the canonical projection and j the transposed map of p. Take ˜ by ˜ = l. We define the Lie algebra D ˜l ∈ V˜ ∗ such that j (l)

D˜ p(x) = p D(x) ,

D ∈ D and x ∈ V .

˜ = P (D t )(q), q ∈ O and D small enough, the induction hypothesis applied to As j (P (D˜ t )(q)) ˜ ˜ = A ˜ (l). ˜ Hence ND (l) = p −1 (N ˜ (l)) ˜ is D-invariant. ˜ V and D implies that ND˜ (l) D D

3962


Case 2. V0 = {0}. This implies that dim(AD (l)) = 0 or 1. Subcase 2.1. AD (l) = RZ, for some Z ∈ V \ {0}, D(Z) = {0} and l(Z) = 0. In this case, there exists a non-zero vector Y ∈ V and two linear functionals α, β = 0 from D to R such that D(Y ) = α(D)Y + β(D)Z,

∀D ∈ D,

since D is completely solvable. It follows that α is a homomorphism of the Lie algebra D. Therefore, we can suppose that α and β are linearly independent, if α = 0. In the case where α and β are linearly dependent and α = 0 there exists c ∈ R such that β = cα. Thus we have D(Y + cZ) = α(D)(Y + cZ),

∀D ∈ D,

and we are in the same situation as subcase 2.2 which is detailed later. Without loss of generality, we can assume that l(Y ) = 0 and l(Z) = 1. Let D0 be the kernel of β. Then D0 is a subalgebra of G. Let D0 := exp(D0 ). Then D0 is a closed connected subgroup of D. Assume first that α = 0. The D0 -orbit O0 of l is given by:

O0 = q ∈ O(l): q(Y ) = 0 . ˙ = 1, α(D) ˙ = 0 and D = D0 ⊕RD. ˙ Thus D(Y ˙ )=Z In fact, there exists D˙ ∈ D\D0 such that β(D) ˙ ) = 0, for all D ∈ D. So we have and D D(Y ˙ D = D0 exp(RD). Let q ∈ O(l) such that q(Y ) = 0. There exists D0 ∈ D0 , and s ∈ R such that q = exp(D0t ) × exp(s D˙ t )(l). Since q(Y ) = 0, we have

0 = exp D0t exp s D˙ t (l), Y ˙ exp(D0 )(Y ) = l, exp(s D)

˙ eα(D0 ) Y = l, exp(s D) = l, eα(D0 ) (Y + sZ) = seα(D0 ) . This implies that, s = 0 and so q ∈ (D0 )l. Thus {q ∈ O(l): q(Y ) = 0} ⊂ O0 . On the other hand, we evidently have (Dt0 )l(Y ) = 0. As P (Dt0 )(q), Y = {0} for every q ∈ O0 , it follows for D ∈ D0 , q ∈ O0 , that P (D t )q ∈ O0 whenever P (D t )q ∈ O. We can apply the induction hypothesis to D0 and O0 . Hence RY ⊕ ND (l) = ND0 (l) = AD0 (l). We show now that ND (l) is D-invariant. Let v ∈ ND (l). We have l, D 2 (v) = 0, for all D ∈ D. In fact, for all D0 ∈ D0 and s ∈ R small enough,


3963

P s(D˙ + D0 )t l, Y = l, Y + a2 s 2 (D˙ + D0 )2 (Y ) + o s 3

˙ 0 + D0 D˙ (Y ) + o s 3 = l, Y + a2 s 2 D˙ 2 + D02 + DD

˙ 0 (Y ) + o s 3 = a2 s 2 l, DD

= a2 α(D0 )s 2 + o s 3 =: Q(s). On the other hand, we have

exp −Q(s)D˙ t P s(D˙ + D0 )t l, Y = P s(D˙ + D0 )t l, Y − Q(s)Z

= P s(D˙ + D0 )t l, Y − Q(s) = 0.

It follows that for s ∈ R small enough,

exp −Q(s)D˙ t P s(D˙ + D0 )t l ∈ D0 . Since v ∈ ND (l) ⊂ ND0 (l) = AD0 (l), we have

l, v = exp −Q(s)D˙ t P s(D˙ + D0 )t l, v

˙ = P s(D˙ + D0 )t l, v − Q(s)D(v) + o s3

˙ = l, v − Q(s)D(v) + a2 s 2 (D˙ + D0 )2 (v) + o s 3

= l, v + a2 s 2 (D˙ + D0 )2 (v) + o s 3 . This implies that a2 l, (D˙ + D0 )2 (v) = 0. As a2 = 0, we have l, (D˙ + D0 )2 (v) = 0. Hence

l, D 2 (v) = 0 for all D ∈ D. Now, for D1 , D2 ∈ D and v ∈ ND (l),

0 = l, (D1 + D2 )2 (v) = l, D12 + D22 + 2D1 D2 + [D1 , D2 ] (v) = 2 l, D1 D2 (v) (since [D1 , D2 ] ∈ D). This shows that D(v) ⊂ ND (l) and so ND (l) is D-invariant. The subcase α = 0 is similar. Subcase 2.2. AD (l) = {0}. Then there exists a non-zero Y ∈ V and non-zero homomorphism α on D such that D(Y ) = α(D)Y,

∀D ∈ D.

Without loss of generality, we can assume that l(Y ) = 1. Let D0 be the kernel of α and D0 := ˙ = 1 and D = D0 ⊕ RD. ˙ It is easy to see that exp(D0 ). There exists D˙ ∈ D\D0 such that α(D)

O0 := Dt0 l = q ∈ O: q(Y ) = 1 . On the other hand, for all s ∈ R small enough and D0 ∈ D0

P s(D˙ + D0 )t (l), Y = l, Y + a2 s 2 (D˙ + D0 )2 (Y ) + a3 s 3 (D˙ + D0 )3 (Y ) + · · · = 1 + a2 s 2 + a3 s 3 + · · · = 1 + Q(s) > 0.

3964


˙ (s(D˙ + D0 )t )(l) ∈ O0 for s small enough Then, for q(s) = ln(1 + Q(s)) we get exp(−q(s)D)P in R. In addition, by the same reasoning as above, using the induction hypothesis, we see that ND0 (l) is D0 -invariant. Let v ∈ ND (l), we compute

l, v = exp −q(s)D˙ P s(D˙ + D0 )t (l), v

˙ = P s(D˙ + D0 )t (l), v − a2 s 2 D(v) + o s3

˙ = l, v − a2 s 2 D(v) + a2 s 2 (D˙ + D0 )2 (v) + o s 3

= l, v + a2 s 2 (D˙ + D0 )2 (v) + o s 3 . It follows that

a2 s 2 l, (D˙ + D0 )2 (v) + θ s 3 = 0,

for all s ∈ R.

Hence, we have l, D 2 (v) = 0 for all D ∈ D and so l, D1 D2 (v) = 0 for all D1 , D2 ∈ D i.e. ND (l) is D-invariant. 1) ⇒ 3) Since now ND (l) is D-invariant, the D-orbit O of l is contained in l + ND (l)⊥ . The dimension of this orbit O is equal to the dimension of V /ND (l) because the dimension of O is equal to the dimension of D modulo the stabilizer D(l) := {D ∈ D, D t (l) = 0} of l and since the bilinear map

D/D(l) × V /ND (l) : D + D(l), v + ND (l) → l, D(v) establishes a duality between the two quotient spaces. Hence O is an open subset of l + ND (l)⊥ . We take the function P (ξ ) := 1 + ξ 2 , ξ ∈ R. Then the mapping

D × l + ND (l)⊥ → l + ND (l)⊥ ;

(D, q) → P (D)q

is continuous and so for every q ∈ O we can find a small neighbourhood U of 0 in D, such that P (D)q ∈ O for every D ∈ U . 2 Corollary 3.3. Let D be an exponential completely solvable Lie group of endomorphisms of the real finite dimensional vector space V . Let l ∈ V ∗ and let O(l) be the D-orbit of l in V ∗ . If O(l) is closed, then the following statements are equivalent: 1) ND (l) is D-invariant: ND (l) = AD (l). 2) Dt · l|ND (l) = l|ND (l) . 3) a) O(l) is affine linear. b) O(l) = l + AD (l)⊥ . 4) There exists an analytic function P : R → R; P (ξ ) = 1 + a2 ξ 2 + a3 ξ 3 + · · · for small ξ , with a2 = 0, such that for every q in the orbit O(l), P (D t )q ∈ O(l) for D ∈ D small enough. Proof. It suffices to proof the implications 1) ⇒ 3)a) and 3)a) ⇒ 3)b). 1) ⇒ 3)a) For X ∈ ND (l) and D ∈ D, we have

1 exp D t (l), X = l, X + l, D(X) + l, D 2 (X) + · · · = l, X. 2!


3965

Hence O(l) ⊂ l + ND (l)⊥ . On the other hand, reasoning as in the proof of the preceding theorem, we see that O(l) is open in l + ND (l)⊥ . Since by hypothesis it is also closed, it follows that O = l + ND (l)⊥ . 3)a) ⇒ 3)b) We evidently have O(l) ⊂ l + AD (l)⊥ . On the other hand, let W be a subspace of V such that O(l) = l + W ⊥ . For all D ∈ D and s ∈ R, we have

1 t

exp sD (l) − l ∈ O(l) − l ⊂ W ⊥ . s Hence for X ∈ W ,

1 t

exp sD (l) − l , X = 0 s

and so

s t 2 s 2 t 3 D t (l), X + D l, X + D l, X + · · · = 0 2! 3!

and therefore for all j 1, D ∈ D, X ∈ W , D j (X) ∈ ker(l), j 1, i.e. W ⊂ AD (l). Whence, W = AD (l). 2 Corollary 3.4. Let G = exp(g) be a completely solvable Lie group and let l ∈ g∗ . If the G-orbit O(l) of l is closed, then the following statements are equivalent: 1) g(l) is an ideal in g. 2) Ad∗ (G)l|g(l) = l|g(l) . 3) O(l) = l + g(l)⊥ . 4. Representations associated to flat orbits Let l ∈ g∗ and pl be a polarization for l satisfying the Pukanszky condition. Let Pl = exp(pl ) ˆ be the representation indG χl , where χl is the unitary character of Pl defined by and πl ∈ G Pl χl (x) := e−i l,log(x) , x ∈ Pl . Let as in Section 2.1 J = (gi )ni=1 be a Jordan–Hölder sequence and Z = {Z1 , . . . , Zn } be a Jordan–Hölder basis of g adapted to J . We denote by I pl the index set I pl := {i ∈ {1, . . . , n}; pl ∩ gi = pl ∩ gi+1 }. Then for i ∈ I pl , we can take the vector Zi in pl . Let also I g/pl be the index set {1, . . . , n} \ I pl = {i ∈ {1, . . . , n}; gi ∩ pl = gi+1 ∩ pl }. We consider the function ψpl defined on G by ψpl (x) =

i∈I g/pl

. ρi (log(x)) −

ρi (log(x)) e

ρi (log(x)) 2

−e

2

The function ψpl is bounded and Ad(G)-invariant. For p ∈ Pl we have the following identity:

3966


ΔPl ,G (p)

−1 2

jpl (log p) = ψpl (p). jg (log p)

(3)

Indeed, ΔPl ,G (p)

−1 2

ρi (log(p)) jpl (log p) −ρi (log(p))/2 e = 1 − e−ρi (log(p)) jg (log p) g/p g/p i∈I l i∈I l ρi (log(p)) = ρi (log(p)) ρi (log(p)) − 2 2 g/p e − e i∈I l = ψpl (p).

Let I (l, pl ) be the closed subspace of L1 (G), given by f (uxv)ψpl (x)e−i l,log x dx = 0 . I (l, pl ) = f ∈ L1 (G): ∀u, v ∈ G, G

Then I (l, pl ) is in fact a twosided ideal of the algebra L1 (G), since for every f ∈ I (l, pl ) the left and right translates of f are all contained in I (l, pl ). Proposition 4.1. I (l, pl ) is contained in ker(πl ). Proof. Let g ∈ I (l, pl ) and α ∈ Cc (G) and let f := g ∗ α. Then f ∈ I (l, pl ) too and the function p → f (uxpv) is contained in L1 (Pl ) for every x, u, v ∈ G since

f (uxpv) dp =

Pl

g(y)α y −1 uxpv dp dy

G Pl

ΔG y −1 g uxy −1 α(ypv) dp dy

= G Pl

g ux(yq)−1

−1

=

ΔG (yq)

G/Pl Pl

l

Pl

ΔG (yq)−1 ΔPl ,G (q)−1 g ux(yq)−1

= G/Pl Pl

The function y → Pl

Pl

α(yqpv) dpΔP ,G (q)−1 dq dμG/P (y)

l

α(ypv) dp dq dμG/P (y). l

Pl

|α(ypv)| dp =: α˜ v (y) is uniformly bounded in y and so

f (uxpv) dp =

ΔG (yq)−1 ΔPl ,G (q)−1 g ux(yq)−1 α˜ v (y) dq dμG/Pl (y)

G/Pl Pl

= G

g(uxy)α˜ v y −1 dy < ∞.


Now, for all u, v, x ∈ G and all p ∈ Pl we have that 0=

f upxp −1 v ψpl (x)e−i l,log x dx

G

=

f upxp −1 v ψpl pxp −1 e−i l,log x dx

G

= ΔG (p)

f (uxv)ψpl (x)e−i Ad

∗ (p)l,log x

dx

G

= ΔG (p)

∗ f u exp(Y )v ψpl (exp Y )e−i Ad (p)l,Y jg (Y ) dY.

g ⊥ As Ad∗ (Pl )l = l + p⊥ l , we get for u, v, x ∈ G, q ∈ pl , that:

0=

f u exp(Y )v ψpl exp(Y ) e−i l+q,Y jg (Y ) dY

g

=

e

−i q+l,Y

f u exp(Y + U )v ψpl exp(Y + U ) e−i l,U jg (Y + U ) dU d Y˙ .

pl

g/pl

Hence, for every Y ∈ g, u, v ∈ G, 0=

f u exp(Y + U )v ψpl exp(Y + U ) e−i l,U jg (Y + U ) dU.

pl

Therefore, for Y = 0, u, v ∈ G, 0=

f u exp(U )v ψpl exp(U ) e−i l,U jg (U ) dU.

pl

Hence, by (3), for all u, v ∈ G, 0=

−1

f u exp(U )v ΔPl ,G exp(U ) 2 jpl (U )e−i l,U dU

pl

=

f (upv)ΔPl ,G (p)

−1 2

e−i l,log(p) dp.

Pl

Thus, by Lemma 2.1, f ∈ ker(πl ) and finally g ∈ ker(πl ).

2

3967

3968


Theorem 4.2. Let G = exp(g) be a completely solvable Lie group. Let l ∈ g∗ such that the Gorbit O(l) is closed. If O(l) is affine linear then

ker(πO(l) ) = f ∈ L1 (G): (f ◦ exp)jg ˆ O(l) = {0} . ⊥ Proof. Let τl := indG G(l) χl , where G(l) := exp(g(l)). As O(l) is affine linear, O(l) = l + g(l) and g(l) is an ideal of g (by Corollary 3.4). Hence G(l) is a closed connected normal subgroup of G. Furthermore, we have

ker(τl ) =

ker(πq ) = ker(πl )

q∈l+g(l)⊥

(see [4]). We show that ker(τl ) = {f ∈ L1 (G): [(f ◦ exp)jg ]ˆ(O(l)) = {0}}. Let f ∈ Cc (G) ∗ ker(τl ) ∗ Cc (G). Then, by Lemma 2.1 we get

f exp(s)h χl (h) dh =

G(l)

f ◦ exp s + ϕs (h) χl (h)jg(l) (h) dh = 0,

g(l)

for all s ∈ g, where ϕs (h) = s ·g h − s 1 1 1 s, [s, h] + h, [h, s] + · · · = h + [s, h] + 2 12 12

for small s ∈ g, h ∈ g(l).

We see that the mapping ϕs : g(l) → g(l) is a diffeomorphism, whose inverse ψs is given by: ψs (h) = (−s) ·g (h + s),

h ∈ g(l) (s ∈ g).

On the other hand, for all f ∈ L1 (G) we have

f ◦ exp(s + h)jg (s + h) dh ds

g/g(l) g(l)

f (g) dg =

= G

=

f (sh) dh

G/G(l) G(l)

f ◦ exp s + ϕs (h) jg(l) (h)jg/g(l) (s) dh ds

g/g(l) g(l)

= g/g(l) g(l)

This proves that

f ◦ exp(s + h)jg(l) ψs (h) jg/g(l) (s)Jac(ψs )(h) dh ds.

(4)


Jac(ψs )(h) =

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jg (s + h) . jg(l) (ψs (h))jg/g(l) (s)

We deduce from Eq. (4) that

f ◦ exp(s + h)jg (s + h)e−il(h) dh = 0,

s ∈ g,

g(l)

since l, ψs (h) = l, h for all h ∈ g(l), because g(l) is an ideal of g. Therefore f ◦ exp(Y )jg (Y )e−i l+q,Y dY = 0, q ∈ g(l)⊥ . g

As Cc (G) ∗ ker(τl ) ∗ Cc (G) is dense in ker(τl ), it follows that ker(τl ) ⊂ {f ∈ L1 (G): [(f ◦ exp)jg ]ˆ(l + g(l)⊥ ) = 0}. Let now f ∈ L1 (G) such that [(f ◦ exp)jg ]ˆ(l + g(l)⊥ ) = 0, then by the same computation as above, we can show that G(l) f (sh)χl (tht −1 ) dh = 0 for all s, t ∈ G. That means by Lemma 2.1 that f ∈ ker(τl ) and thus

ker(τl ) = f ∈ L1 (G): (f ◦ exp)jg ˆ O(l) = 0 .

2

We show now the converse direction. We take an exponential solvable Lie group G = exp(g) and an exponential completely solvable Lie group D = exp(D) of automorphisms of g containing the group Ad(G). We also suppose that there is an analytic mapping P : D × g → g such that for small (D, X) P (D, X) = X + aD 2 (X) +

ki+nj 2

a k1 ...kr D k1 adn1 (X) . . . D kr adnr (X)D kr+1 (X), (n1 ...nr )

with a = 0. We write P (D) : g → g for the mapping: P (D)(X) = P (D, X) (D ∈ D) and we suppose that P (D) is a diffeomorphism of g for every D ∈ D. Define for D ∈ D the linear bijection Pˇ (D) of L1 (g) defined by

Pˇ (D)f (X) := f P (D)X JP (D) (X),

X ∈ g,

where JP (D) (X) denotes the Jacobian of P (D) at X ∈ g. Definition 4.3. For an ideal I in the algebra L1 (g), let h(I ) be the set of characters ∗ −i q,Y h(I ) := q ∈ g , 0 = χq (f ) = f (Y )e dY, f ∈ I . g

Then h(I ) is a closed (possibly empty) subset of g∗ . Lemma 4.4. Let g, D and P as above. Let l ∈ g∗ and let I be a closed ideal in L1 (g), so that h(I ) is the closure O(l) of the D-orbit O(l) of l. If I is invariant under the maps Pˇ (D), D ∈ D, then ND (l) is D-invariant.

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Proof. We proceed by induction on the number d := dim(D) + dim(g). If d = 1, the result is obviously true. Suppose now that d 2. Let g0 = ker(l) ∩ AD (l). Then g0 is D-invariant. We first assume that g0 = {0}. Let p be the canonical projection of g onto g˜ = g/g0 and let j be ˜ = l. We define a Lie algebra of exponential the transpose of p. Let l˜ ∈ g˜ ∗ be defined by j (l) ˜ derivations on g˜ in the following way: for all D ∈ D, let D˜ be defined by D(p(x)) = p(D(x)), ˜ l) ˜ ˜ and ad g˜ ⊂ D. x ∈ g. So we have evidently O(l) = j ((exp D) Let I˜ = π(I ), where π : L1 (g) → L1 (˜g) is the canonical surjection: π(f )(x) ˜ :=

f (x + z) dz,

f ∈ L(g), x˜ = x + g0 .

g0

˜ l. ˜ Thus I˜ is a closed ideal in L1 (˜g) (see [7], p. 177) and the hull of I˜ is h(I˜) = (exp D) ˜ ˜ ˜ Define the maps P (D) (D ∈ D) by: ˜ x) P (D)( ˜ = P (D)x + g0 ,

x˜ = x + g0 ,

˜2

= x˜ + a D (x) ˜ + ···,

for small x˜ ∈ g˜ .

Then

˜ p(x) = p P (D)(x) , P (D)

for all x ∈ g.

˜ ˜ and x˜ = p(x) It is easy to see that I˜ is Pˇ (D)-invariant; Indeed, for all f˜ = π(f ) ∈ I˜, D˜ ∈ D

˜ f˜(x) Pˇ (D) ˜ = f˜ p P (D)(x) JP (D) ˜ p(x)

= f P (D)(x + h) JP (D) (x + h) dh g0

= g0

JP (D) (x + Q(D, x)−1 (h)) f P (D)(x) + h dh, JQ(D,x)

where Q(D, x) is a diffeomorphism of g0 given by Q(D, x)(h) = P (D)(x + h) − P (D)(x)

(h ∈ g0 , x ∈ g, D ∈ D).

On the other hand, for all ϕ ∈ L1 (g), we have

ϕ(x) dx =

g

g/g0 g0

= g/g0 g0

and

ϕ P (D)(x + h) JP (D) (x + h) dh d x˜

JP (D) (x + Q(D, x)−1 (h)) ϕ P (D)(x) + h dh d x, ˜ JQ(D,x)(h)


ϕ(x) dx = g

ϕ(x + h) dh d x˜ =

g/g0 g0

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ϕ P (D)(x) + h dh JP (D) ˜ d x. ˜ ˜ (x)

g/g0 g0

This implies that JP (D) ˜ = ˜ (x)

JP (D) (x + Q(D, x)−1 (h)) , JQ(D,x)(h)

x ∈ g, h ∈ g0 .

˜ f˜(x) We obtain thus that Pˇ (D) ˜ = π(Pˇ (D)f (x)). ˜ = A ˜ (l) ˜ and thus ND (l) is D-invariant. By the induction hypothesis we get ND˜ (l) D Suppose now that g0 = {0}. Then dim(AD (l)) = 0 or 1. Case 1. AD (l) = RZ, for some non-zero Z in g. Since AD (l) is D-invariant and l, Z = 0, it follows that D(Z) = 0 for all D ∈ D. In this case, there exists a non-zero Y ∈ g and two linear functionals α, β on D such that D(Y ) = α(D)Y + β(D)Z,

∀D ∈ D

since D is completely solvable with β = 0. If α = 0, then we can assume that α and β are linearly independent. The case where α and β are linearly dependent and α = 0 is tackled similarly as in the Case 2 which is treated later. The linear functional α is a homomorphism of D. Without loss of generality, we can assume that l(Y ) = 0 and l(Z) = 1. We can find an element D˙ ∈ ker(α), ˙ = 1. such that β(D) We apply now the technique of restriction of an ideal to an invariant subgroup developed in [3]. Let g1 be any D-invariant subspace of g containing AD (l), such that dim(g/g1 ) = 1. l = 0 and let χt Such a g1 always exists since D is completely solvable. Let l0 ∈ g⊥ 1 with 0 be the character of g defined by: χt (v) = ei tl0 ,v , v ∈ g. The ideal J = t∈R χt I is Pˇ (D)invariant and h(J ) = h(I ) + Rl0 . Define for x ∈ g and a function f : g → C the function x f by 1 x f (y) := f (x + y), y ∈ g. Let J be the set of all functions f ∈ J , so that x f|g1 ∈ L (g1 ) for all 1 x ∈ g and so that the maps x → x f|g1 from g1 to L (g1 ) are continuous. J is dense in J and Pˇ (D)-invariant, since I is a Pˇ (D)-invariant ideal of L1 (g). We define the ideal I1 of L1 (g1 ) as the closure in L1 (g1 ) of the functions f|g1 f ∈ J . Let D1 be the restriction of the D on g1 and let l1 = l|g1 . The hull of I1 is exactly the restriction of the hull of J on g1 . Thus h(I1 ) = (exp D1 t )l1 . We still have ad g1 ⊂ D1 . The ideal I1 is Pˇ (D1 )-invariant. Indeed, since P (D) maps g1 into g1 , we have that JP (D) (h) is a constant times JP (D1 ) (h), h ∈ g1 . The induction hypothesis applied to I1 , l1 , D1 implies that ND (l) ⊃ ND (l) ∩ g1 = ND1 (l1 ) = AD1 (l1 ) = AD (l). Hence we obtain that either AD (l) = ND (l) (if ND (l) ⊂ g1 ) or dim(AD (l)) + 1 = dim(ND (l)) 2. Let now D0 be the kernel of β. Then D0 is a subalgebra of D. Let D0 = exp(D0 ). We have that ND0 (l) = RY ⊕ ND (l) and the closure of the D0 -orbit of l is given by: t

D0 l = q ∈ Dt l: q(Y ) = 0 . We look at the ideal I1 defined to be the closure in L1 (g) of the sum of the ideal I and the kernel KY of the surjective homomorphism

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πY : L (g) → L (g/RY ), 1

1

π(f )(x + RY ) :=

f (x + yY ) dy. R

It is easy to check that KY is Pˇ (D)-invariant for every D ∈ D0 , since P (D)(RY ) ⊂ RY , D ∈ D0 . The hull of I1 is the subset h(I ) ∩ {q ∈ g∗ ; q, Y = 0} = Dt0 l =: O0 (l). By the induction hypothesis for I1 , D0 and g, we get ND0 (l) = AD0 (l) and so ND0 (l) is D0 -invariant. This implies that dim(ND0 (l)) 3 since dim(ND0 (l)) = dim(ND (l))+1. If dim(ND0 (l)) = 2 then ND0 (l) = AD0 (l) = RY +RZ. Whence ND (l) = AD (l). We prove now that the case dim(ND0 (l)) = 3 cannot happen. Suppose otherwise. If we have a D-invariant subspace g1 of co-dimension 1 containing ND0 (l) ⊃ ND (l), then we have seen that, AD (l) = ND (l) and so ND0 (l) is of dimension 2. Hence no D-invariant subspace can contain ND0 (l). In other terms, either g = AD0 (l) or the smallest D-invariant subspace of g containing AD0 (l) equals g. We show that in the two cases g is abelian. If g = AD0 (l) = RY0 + RY + RZ, then for a D˙ ∈ D ˙ = 1 and α(D) ˙ = 0, we have that with β(D) t

˙ ˙ 0 = l, exp s D(g), exp s D(g) = exp s D˙ l, [g, g] for all s ∈ R. It follows that if we write [Y0 , Y ] = cY for some c, then ˙ = c l, Y + sZ = cs ˙ 0 , Y ] = l, c exp s DY 0 = l, exp s D[Y and so [Y0 , Y ] = 0 and g is abelian. In the second case take again D˙ ∈ D \ D0 and let ˙ 0, Y1 = DY

...,

Yk = D˙ k Y0

(k = 2, 3, . . . , n),

where n is the largest integer such that the set {Y1 , . . . , Yn } is linearly independent modulo ˙ span{Y, Z}. The subspace h of g, spanned by Y0 , Y1 , . . . , Yn , Y and Z is by definition D-invariant. It is also D0 -invariant. Indeed AD0 (l) is D0 -invariant, it follows that D0 (Y0 ) ⊂ AD0 (l) ⊂ h. If the functional α = 0, then D0 is an ideal in D and so we see that inductively on k = 1, . . . , for D ∈ D0 ,

˙ k−1 ) = D(Y ˙ k−1 ) + [D, D](Y ˙ k−1 ) ∈ h. D(Yk ) = D D(Y If α = 0, we can take D˙ in ker(α) ∩ [D, D]. In particular D˙ is a nilpotent endomorphism. Take now D00 = ker(α) ∩ ker(β), which is an ideal of D contained in D0 . The subspace h is therefore D00 -invariant by the argument above. There exists an element D˙ 0 ∈ D0 , such that α(D˙ 0 ) = 1. ˙ = −D˙ modulo D00 , we have that Again, by induction on k, as [D˙ 0 , D] ˙ k−1 + D˙ D˙ 0 Yk−1 ∈ h, D˙ 0 Yk = [D˙ 0 , D]Y k = 1, 2, . . . , n. Thus h is D-invariant and so h = g.


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We show first now that Y is central in g; we have that, since D˙ is a derivation of g and since Y ∈ AD0 (l),

˙ 0 ), Y = l, D˙ [Y0 , Y ] − Y0 , D(Y ˙ ) = α ad(Y0 ) . 0 = l, [Y1 , Y ] = l, D(Y =0

It follows that [Y0 , Y ] = 0 and so by induction on k, ˙ k−1 ), Y [Yk , Y ] = D(Y

˙ ) = D˙ [Yk−1 , Y ] − Yk−1 , D(Y = 0 − [Yk−1 , Z] = 0. Hence Y is central in g. We prove now that [Y0 , g] = 0. We remark that for all j 1, ad(Yj ) is nilpotent, since for these j ’s, ad(Yj ) ∈ [D, D]. This implies that [Y0 , Yj ] = aj Y ∈ RY for some aj ∈ R, because Y0 ∈ AD0 (l) and so [Yj , Y0 ] ∈ RY . On the other hand, by induction on k, we can check that [Yk , Y ] =

k j (−1)j Ck D˙ k−j [Y0 , Y+j ],

∀k, = 1, 2, . . . , n.

j =0

Using the formula 0 = [Yk , Yk ] =

k j (−1)j Ck D˙ k−j [Y0 , Yk+j ] j =0

= (−1)k [Y0 , Y2k ] − (−1)k−1 D˙ [Y0 , Y2k−1 ] = (−1)k a2k Y − (−1)k−1 a2k−1 Z,

k = 1, . . . , n,

we deduce that for any k = 1, 2, . . . , n, ak = 0, and hence ad(Y0 ) = 0. Whence Y0 is contained ˙ 0 ) and inductively all the Yk ’s, k = 2, . . . , n. Finally g in the center of g and so is then Y1 = D(Y is abelian. Then the polynomial maps P (D), D ∈ D, are reduced to the linear maps given by P (D)(x) = x + aD 2 (x) +

bk D 2+k (x)

k0

for some bk ∈ R and for D ∈ D. As I is invariant under these linear maps, the hull h(I ) of I is invariant under the corresponding linear maps P (D)t , which have the form P (D)t = 1 + aD 2 +

k0

bk D 2+k ,

D ∈ D small.

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Since the orbit O(l) is open in its closure (see [1]), we have that for every q ∈ O(l), P (D)t (q) ∈ O(l) for D small enough. Applying now Theorem 3.2 we have that ND (l) = AD (l), but this contradicts the assumption that dim(ND0 (l)) = 3. Case 2. dim(AD (l)) = 0. In this case, there exists a non-zero Y ∈ g and a homomorphism α = 0 on D such that D(Y ) = α(D)Y,

∀D ∈ D.

Without loss of generality, we can assume that l(Y ) = 1. Let D0 be the kernel of α and suppose that ad(g) ⊂ D0 . There exists X ∈ g so that [X, Y ] = Y . Then, the D-orbit is saturated with respect to the D-invariant subspace g1 = {U ∈ g: [U, Y ] = 0}. Let D1 be the restriction of D on g1 and

I1 = h ∗ f|g1 : f ∈ I, h ∈ Cc (G) .1 . Then the ideal I1 is Pˇ (D1 )-invariant and h(I1 ) = h(I )|g1 . Furthermore h(I1 ) = (exp Dt1 )l|g1 and by the induction hypothesis for I1 , D1 , g1 , we have that ND (l) = ND1 (l|g1 ) = AD1 (l|g1 ) = AD (l) = {0}. Assume now that α(ad g) = 0. In this case Y is central in g and we have for D0 := ker(α)

exp Dt0 l = q ∈ O(l): q(Y ) = 1 . Let g1 be a D-invariant subspace of g of co-dimension 1. We define J=

χq I

and I1 = h ∗ f|g1 : f ∈ J, h ∈ Cc (G) .1 .

q∈g⊥ 1 t Then I1 is Pˇ (D|g1 )-invariant, h(J ) = h(I ) + g⊥ 1 and h(I1 ) = D l1 . Hence, by the induction hypothesis applied to I1 , D1 , g1 we obtain:

ND (l) ∩ g1 = ND|g1 (l|g1 ) = AD|g1 (l|g1 ) = {0}. Let now y := RY and let K := {f ∈ L1 (g):

y f (u + y) dy = 0, u almost everywhere}. subspace l + y⊥ and K is Pˇ (D0 )-invariant for

Then the hull of the ideal K is the affine D0 ∈ D0 small enough, since we can write for u ∈ g and y ∈ y:

every

P (D)(u + y) = uD + P (D)y = uD + q(D)y for some uD ∈ g depending only on u and D and some real number q(D). Hence the closure J0 of the ideal I + K is also Pˇ (D0 )-invariant and its hull is equal to the closure of the D0 -orbit of l. Applying the induction hypothesis to D0 and J0 , we see that ND (l) + RY = ND0 (l) = AD0 (l). We have seen above that for any D-invariant co-one dimensional subspace g1 of g the dimension of


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ND (l) ∩ g1 = 1. Hence the dimension of ND0 (l) is less or equal to 2. If this dimension is one, then ND (l) = {0}. If ND (l) is contained in a proper D-invariant subspace, then we have also finished by the argument above. It remains the case where ND0 (l) is of dimension 2 and contained in no Dinvariant proper subspace. We can write ND0 (l) = RU + RY , where l(U ) = 0. Since U ∈ ND0 (l) and ND0 (l) is D0 -invariant, it follows that RU must be itself D0 -invariant. Hence there exists a character γ of g, such that [T , U ] = γ (T )U , T ∈ g. Hence ND0 (l) is contained in the nilradical of g. But then g itself is nilpotent, since the smallest D-invariant subspace containing U and Y is equal to g and the elements of D are derivations of g. Then necessarily α = 0 and so ND0 (l) is contained in the center of g and finally g itself is abelian. Since the hull of I is the closure of a D-orbit, which is P (D)t -invariant for small D in D, we can now apply as before Theorem 3.2 and we have that ND (l) is D-invariant. 2 Theorem 4.5. Let G = exp(g) be a completely solvable Lie group and let l ∈ g∗ . Suppose that ˆ be associated to O(l). The following the coadjoint orbit O(l) of l is closed in g∗ . Let πl ∈ G statements are equivalent: 1) ker(πl ) = {f ∈ L1 (G): [(f ◦ exp)jg ]ˆ(O(l)) = 0}, 2) The orbit O(l) is affine linear. Proof. 1 ⇒ 2) It is clear that Il = {(f ◦ exp)jg : f ∈ ker(πl )} is invariant under the linear maps Pˇ (ad(X)), X ∈ g, defined by:

1 P ad(X) (Y ) = X ·g Y ·g X − 2X = Y + ad(X)2 Y + · · · higher brackets in X, Y ∈ g, 6 since ker(πl ) is translation-invariant by elements of G and Il ⊂ L1 (g) is translation-invariant by elements of g. Furthermore we have that

jg (Y ) ΔG (exp X) dY = f (Y ) dY, X ∈ g, f ∈ L1 (g). f P ad(X)Y jg (Y + 2X) g

g

Finally the hull of h(Il ) is the orbit O(l), because Il = {h ∈ L1 (g), hˆ = 0 on O(l)}. Indeed, we have that L1 (G) = {f : G → C; jg (f ◦ exp) ∈ L1 (g)} and L1 (g) = {h : g → C; (jg−1 h) ◦ log ∈ L1 (G)} and therefore by our assumption 1)

Il = (f ◦ exp)jg : f ∈ ker(πl )

= (f ◦ exp)jg : (f ◦ exp)jg ˆ = 0 on O(l)

= h ∈ L1 (g): hˆ = 0 on O(l) . Then by Lemma 4.4, O(l) is an affine linear orbit (we take D = ad g). 2) ⇒ 1) Theorem 4.2. 2 References [1] M.P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard, M. Vergne, Représentations des groupes de Lie résolubles, Monogr. Soc. Math. Fr., vol. 4, Dunod, Paris, 1972.

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[2] J. Boidol, ∗-Regularity of exponential Lie groups, Invent. Math. 56 (1980) 231–238. [3] W. Hauenschild, J. Ludwig, The injection and the projection theorem for spectral sets, Monatsh. Math. 92 (1981) 167–177. [4] H. Leptin, J. Ludwig, Unitary Representation Theory of Exponential Lie Groups, de Gruyter Exp. Math., vol. 18, 1994. [5] J. Ludwig, Good ideals in the group algebra of a nilpotent Lie group, Math. Z. 161 (1978) 195–210. [6] J. Ludwig, On the Hilbert–Schmidt semi-norms of L1 of a nilpotent Lie group, Math. Ann. 273 (1986) 383–395. [7] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Clarendon Press, Oxford, 1968. [8] O. Ungermann, The Jacobson topology of Prim∗ L1 (G) for exponential Lie groups, Thesis, 2007. [9] N.R. Wallach, Harmonic Analysis on Homogeneous Spaces, Pure Appl. Math., vol. 19, Marcel Dekker, Inc., New York, 1973.


The Heyde theorem for locally compact Abelian groups G.M. Feldman Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47, Lenin ave., Kharkov, 61103, Ukraine Received 8 April 2009; accepted 10 March 2010 Available online 20 March 2010 Communicated by S. Vaes

Abstract We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj , βj ∈ Aut(X), j = 1, 2, . . . , n, n 2, such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Let ξj be independent random variables with values in X and distributions μj with nonvanishing characteristic functions. If the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn is symmetric, then each μj = γj ∗ ρj , where γj are Gaussian measures, and ρj are distributions supported in G. © 2010 Elsevier Inc. All rights reserved. Keywords: Gaussian measure; Locally compact Abelian group

1. Introduction By the well-known Skitovich–Darmois theorem a Gaussian measure on the real line can be characterized by the independence of two linear forms of independent random variables. A similar result was obtained by C.C. Heyde, where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. He proved the following theorem ([9], see also [10, §13.4]). E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.005

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G.M. Feldman / Journal of Functional Analysis 258 (2010) 3977–3987

Theorem A. Let ξj , j = 1, 2, . . . , n, n 2, be independent random variables, let αj , βj be nonzero constants such that βi αi−1 ± βj αj−1 = 0 for all i = j . If the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn is symmetric, then all ξj are Gaussian. In the last years much attention has been devoted to generalizing of the classical characterization theorems into various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, symmetric spaces (see e.g. [7,11], and [5] where one can find additional references). The present article continues these researches. Let X be a locally compact second countable Abelian group and let Y = X ∗ be its character group. Denote by (x, y) the value of a character y ∈ Y at an element x ∈ X. Let Aut(X) be the set of all topological automorphisms of X. Denote by T the circle group (the one-dimensional torus) T = {z ∈ C: |z| = 1}. Let M 1 (X) be the convolution semigroup of probability distributions on X. For μ ∈ M 1 (X) denote by μ(y) ˆ =

(x, y) dμ(x),

y∈Y

X

its characteristic function (the Fourier transform). If μ = μ1 ∗ μ2 , μj ∈ M 1 (X), then μj are called factors of μ. A probability measure μ ∈ M 1 (X) is called Gaussian (in the sense of Parthasarathy) [12, Ch. IV, §6] if its characteristic function can be represented in the form μ(y) ˆ = (x, y) exp −ϕ(y) ,

y ∈ Y,

where x ∈ X and ϕ is a continuous nonnegative function satisfying the equation ϕ(u + v) + ϕ(u − v) = 2 ϕ(u) + ϕ(v) ,

u, v ∈ Y.

(1)

Taking into account that in the article we will deal only with Gaussian measures in the sense of Parthasarathy we will name them Gaussian. Denote by Γ (X) the set of Gaussian measures on the group X. Let ξj , j = 1, 2, . . . , n, n 2, be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. Let αj , βj ∈ Aut(X) such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Consider linear forms L1 = α1 ξ1 + · · · + αn ξn and L2 = β1 ξ1 + · · · + βn ξn . It was proved in [4] that the symmetry of the conditional distribution of L2 given L1 implies that all μj ∈ Γ (X) if and only if X contains no elements of order 2. If a group X contains elements of order 2, then the following natural problem arises: Problem 1. Let X be a locally compact second countable Abelian group, and assume that X contains elements of order 2. Let ξj , j = 1, 2, . . . , n, n 2, be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. Let αj , βj ∈ Aut(X) such that βi αi−1 ±βj αj−1 ∈ Aut(X) for all i = j . What distributions μj are characterized by the symmetry of the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn ? This problem was solved in [4] for the case when X is the two-dimensional torus T2 . Namely, the following theorem holds: Let ξ1 , ξ2 be independent random variables with values in T2


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and distributions μj with non-vanishing characteristic functions. Consider linear forms L1 = α1 ξ1 +α2 ξ2 and L2 = β1 ξ1 +β2 ξ2 , where αj , βj ∈ Aut(T2 ) such that β1 α1−1 ±β2 α2−1 ∈ Aut(T2 ). If the conditional distribution of L2 given L1 is symmetric, then μj = γj ∗ρj , where γj ∈ Γ (T2 ), ρj ∈ M 1 (G), j = 1, 2, and G is the subgroup of T2 generated by all elements of order 2. The aim of this article is to solve Problem 1 for groups X containing no subgroup topologically isomorphic to T. It turned out that the answer is the same as in case when X is the two-dimensional torus T2 . We will prove the following theorem. Theorem 1. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to T. Let G be the subgroup of X generated by all elements of order 2. Let αj , βj ∈ Aut(X), j = 1, 2, . . . , n, n 2, such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn is symmetric, then each μj = γj ∗ ρj , where γj ∈ Γ (X), ρj ∈ M 1 (G). The proof of Theorem 1 can be reduced to solving of a functional equation in the class of continuous positive definite functions on a locally compact Abelian group. To prove Theorem 1 we shall use some results of the structure theory for locally compact Abelian groups and the duality theory (see [8]). If L is a subgroup of Y , then denote by A(X, L) = {x ∈ X: (x, y) = 1 for all y ∈ L} its annihilator. Denote by cX the connected component of zero of X and by bX the subgroup of X consisting of all compact elements of X. For each natural number n let fn : X → X be a homomorphism fn (x) = nx, and X (n) = Im fn , X(n) = Ker fn . For μ ∈ M 1 (X) we define the distribution μ¯ ∈ M 1 (X) by the formula μ(E) ¯ = ˆ¯ μ(−E) for all Borel sets E ⊂ X. Observe that μ(y) = μ(y). ˆ We denote by σ (μ) the support of μ ∈ M 1 (X). It should be noted that if μ(y) ˆ = 1 for all L, then σ (μ) ⊂ A(X, L). If α ∈ Aut(X), then the adjoint automorphism α˜ ∈ Aut(Y ) is defined by the formula (x, αy) ˜ = (αx, y) for all x ∈ X, y ∈ Y . A subgroup G of X is called characteristic if α(G) = G for any α ∈ Aut(X). Let ψ : Y → C be an arbitrary function, and let h ∈ Y . We denote by h the finite difference operator h ψ(y) = ψ(y + h) − ψ(y),

y ∈ Y.

A continuous function ψ on Y is called a polynomial if for some natural m the equality m+1 ψ(y) = 0 h is fulfilled for all y, h ∈ Y . The minimal m for which this equality holds is called the degree of ψ. Observe that a nonzero function satisfying (1) is a polynomial of degree 2. 2. Proof of lemmas To prove Theorem 1 we need some lemmas. Lemma 1. (See [4].) Let Y be a locally compact Abelian group, ψj , j = 1, 2, . . . , n, n 2, be functions on Y satisfying the equation

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G.M. Feldman / Journal of Functional Analysis 258 (2010) 3977–3987 n ψj (u + δ˜j v) − ψj (u − δ˜j v) = 0,

u, v ∈ Y,

(2)

j =1

where δˆj ∈ Aut(Y ) such that δî ± δˆj ∈ Aut(Y ) for i = j . Then each function ψj satisfies the equation ψj (y) = 0, 2k 2n−2 h

(3)

where k, h and y are arbitrary elements of Y . Lemma 2. Let n be a natural number and X be a locally compact Abelian group satisfying the where X is a subgroup of X such that X (n) = X. condition X (n) = cX . Then X = cX × X, Proof. Assume first that X is a compact group. Then Y = X ∗ is a discrete group. Set L = X/cX . It follows from the condition of the lemma that L(n) = L. This implies that (L∗ )(n) = L∗ . Taking into account that L∗ ∼ = A(Y, cX ) = bY , we see that bY is a bounded subgroup of Y . Since bY is a pure subgroup of Y , by the Kulikov theorem (see [6, §27]) bY is a direct factor of Y , i.e. Y = D × bY ,

(4)

∼ Y/bY ∼ where D = = (cX )∗ . Taking into account that cX = A(X, bY ), it follows from (4) that where X ∼ X = cX × X, = (bY )∗ . Thus, the statement of the lemma is proved for a compact group X. Let X be an arbitrary locally compact Abelian group. By the structure theorem X ∼ = Rm × G, where m 0 and G contains a compact open subgroup. Without restricting the generality, we may assume that X = Rm × G. This implies that cX = Rm × cG . Let K be a compact open sub group of G. It is obvious that cK = cG and K (n) = cK . As has been shown above K = cK × K, (n) = K. Let π be a continuous homomorphism π : K → cK , such that the restriction of where K π to cK is the identity automorphism. Since the group cK is divisible, the homomorphism π can be extended to a homomorphism πˆ : G → cK = cG (see [8, §A.7]). Taking into account that the homomorphism πˆ is continuous on an open subgroup K, we conclude that πˆ is continuous on G, [8, §6.22]. Hence, X = Rm × cG × G = cX × X. Lemma 2 is proved. 2 and then G = cG × G Lemma 3. (See [4].) Let X be a locally compact second countable Abelian group. Let ξ1 , . . . , ξn , n 2, be independent random variables with values in X and distributions μj . Assume that δj ∈ Aut(X). The conditional distribution of L2 = δ1 ξ1 + · · · + δn ξn given L1 = ξ1 + · · · + ξn is symmetric if and only if the characteristic functions of the distributions μj satisfy the equation n j =1

μˆ j (u + δ˜j v) =

n

μˆ j (u − δ˜j v),

u, v ∈ Y.

(5)

j =1

Lemma 4. Let X be a locally compact second countable Abelian group. Let αj , βj ∈ Aut(X), j = 1, 2, . . . , n, n 2, such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. Assume that the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given


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L1 = α1 ξ1 + · · · + αn ξn is symmetric. Then the random variables ξj can be substituted by their shifts ξj in such a way that the conditional distribution of L 2 = β1 ξ1 + · · · + βn ξn given L 1 = α1 ξ1 + · · · + αn ξn is symmetric and all supports σ (μ j ) ⊂ {x ∈ X: 2x ∈ cX }. Proof. Passing to the random variables ξ˜j = αj ξj we can assume without loss of generality that L1 = ξ1 + · · · + ξn and L2 = δ1 ξ1 + · · · + δn ξn , where δj ∈ Aut(X) such that δi ± δj ∈ Aut(X) for i = j . By Lemma 3, the characteristic functions μˆ j (y) satisfy Eq. (5). It is clear that the characteristic functions of the distributions μ¯ j satisfy Eq. (5) too. Hence, the characteristic functions of the distributions νj = μj ∗ μ¯ j also satisfy Eq. (5). Observe that νˆ j (y) = |μˆ j (y)|2 > 0, and set ψj (y) = − ln νˆ j (y). It follows from (5) that the functions ψj satisfy Eq. (2) and hence by Lemma 1, the functions ψj satisfy Eq. (3). This implies that the function ψj is a polynomial on the subgroup Y (2) . Particularly, the function ψj is a polynomial on the subgroup bY (2) . It is well known (see [2]) that any polynomial on the subgroup consisting of all compact elements of a group is a constant. So, we have ψj (y) = ψj (0) = 0 for y ∈ bY (2) . Hence, νˆ j (y) = 1 for all y ∈ bY (2) and for this reason σ (νj ) ⊂ K = A(X, bY (2) ). Since cX = A(X, bY ), we have K = {x ∈ X: (x, 2y) = 1 for all y ∈ bY } = {x ∈ X: (2x, y) = 1 for all y ∈ bY } = {x ∈ X: 2x ∈ cX }. Since νˆ j (y) = 1 for y ∈ bY (2) , the restrictions of the characteristic functions μˆ j (y) to the subgroup bY (2) are some characters of the subgroup bY (2) . We can extend these characters to some characters of the group Y . By the duality theorem, there exist elements xj ∈ X such

that μˆ j (y) = (xj , y) for y ∈ bY (2) . Set x1 = −δ1−1 nj=2 δj xj and xj = xj , j = 2, . . . , n.

n Since j =1 δj xj = 0, the functions fj (y) = (xj , y) satisfy Eq. (15) on the group Y . Put λˆ j (y) = μˆ j (y)(xj , y), j = 1, 2, . . . , n. It is clear that the characteristic functions λˆ j (y) possess the following properties: (i) (ii) (iii)

λˆ j (y) satisfy Eq. (5) on the group Y ; λˆ j (y) = 1 for y ∈ bY (2) and hence, σ (λj ) ⊂ K, j = 2, . . . , n;

λˆ 1 (y) = (x , y), where x = x1 + δ −1 nj=2 δj xj , for y ∈ b (2) . 1

1

1

Y

Inasmuch as bY (2) is a characteristic subgroup of the group Y , it follows from (i) and (ii) that the restriction of Eq. (5) to the subgroup bY (2) is of the form λˆ 1 (u + δ˜1 v) = λˆ 1 (u − δ˜1 v),

u, v ∈ bY (2) .

Putting here u = δ˜1 v we conclude that λˆ 1 (y) = 1 for y ∈ bY (4) . This implies that σ (λ1 ) ⊂ F = A(X, bY (4) ) = {x ∈ X: 4x ∈ cX }. Since K ⊂ F and taking into account that F is a characteristic subgroup of the group X, we can assume from the beginning that the group X satisfies the where X (4) = X. It follows from this that Y = condition X (4) = cX . By Lemma 2, X = cX × X, ∗ ∗ ∼ ∼ D ×bY , where D = cX , bY = X . Represent the element x1 in the form x1 = c + x, ˜ where c ∈ cX , This implies that (x , y) = (x, ˜ y) for y ∈ b . Set g (y) = ( x, ˜ y), g (y) = 1, y ∈ Y , x˜ ∈ X. Y 1 j 1 j = 2, . . . , n, and μˆ j (y) = λˆ j (y)gj (y), j = 1, 2, . . . , n. It follows from (iii) that μˆ 1 (y) = 1 for y ∈ bY (2) . Moreover μˆ j (y) = 1 for y ∈ bY (2) , j = 2, . . . , n, and hence, σ (μ j ) ⊂ A(X, bY (2) ) = K, j = 1, 2, . . . , n. Since μˆ j (y) = 1 for y ∈ bY (2) , j = 1, 2, . . . , n, the characteristic functions μˆ j (y) are invariant with respect to the subgroup bY (2) and hence, they satisfy Eq. (5) on the subgroup bY . Taking into account that the characteristic functions μˆ j (y) and λˆ j (y) satisfy Eq. (5) on the sub-

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group bY we conclude that the functions gj (y) also satisfy Eq. (5) on the subgroup bY . It follows from this that (x, ˜ u + δ˜1 v) = (x, ˜ u − δ˜1 v),

u, v ∈ bY ,

and hence, 2x˜ = 0. This implies that the characteristic functions μˆ j (y) satisfy Eq. (5) on the group Y . By Lemma 3, if independent random variables ξj take values in X and have distributions μ j , then the conditional distribution of L 2 = β1 ξ1 +· · ·+βn ξn given L 1 = α1 ξ1 +· · ·+αn ξn is symmetric. Lemma 4 is proved. 2 Remark 1. Taking into account that K = {x ∈ X: 2x ∈ cX } is a characteristic subgroup of X, we draw the following conclusion from Lemma 4. Let ξj , j = 1, 2, . . . , n, n 2, be independent random variables with values in a locally compact second countable Abelian group X and distributions μj with non-vanishing characteristic functions. Let αj , βj ∈ Aut(X) such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Assume that the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn is symmetric. Studying the possible distributions μj we can suppose, without restricting of generality, that X (2) = cX . Lemma 5. (See [2].) If a locally compact second countable Abelian group X contains no subgroup topologically isomorphic to T and the characteristic function of a distribution μ ∈ M 1 (X) is of the form μ(y) ˆ = exp −ψ(y) , where ψ is a polynomial, then μ ∈ Γ (X). Lemma 6. Let Y be a locally compact Abelian group, let ψ be a continuous function on Y satisfying the equation 2k 2h ψ(y) = 0,

h, k, y ∈ Y,

(6)

and ψ(−y) = ψ(y), ψ(0) = 0. Then the function ψ can be represented in the form ψ(y) = ϕ(y) + rα ,

y ∈ yα + Y (2) ,

where ϕ is a continuous function on Y satisfying Eq. (1), and Y = decomposition of the group Y with respect to the subgroup Y (2) .

(7)

α (yα

+ Y (2) ), y0 = 0, is a

Proof. It should be noted that any function ϕ on the subgroup Y (2) satisfying Eq. (1) can be extended to the function ϕ˜ on the group Y by the formula 1 ϕ(y) ˜ = ϕ(2y), 4

y ∈ Y.

(8)

The function ϕ˜ also satisfies Eq. (1). Analogously, any additive function l, i.e. any function satisfying the Cauchy equation


l(u + v) = l(u) + l(v)

3983

(9)

on the subgroup Y (2) can be extended to an additive function l˜ on the group Y by the formula ˜ = 1 l(2y), l(y) 2

y ∈ Y.

(10)

Observe that any polynomial f on Y of degree 2 can be represented as a sum f (y) = ϕ(y) + l(y) + c, where ϕ is a continuous function satisfying Eq. (1), l is a continuous function satisfying Eq. (9), and c is a constant. Note that there exists one-to-one correspondence between functions ϕ satisfying (1) and biadditive functions Φ(u, v). Namely, Φ(u, v) = 12 [ϕ(u + v) − ϕ(u) − ϕ(v)], ϕ(y) = Φ(y, y). Consider an arbitrary coset yα + Y (2) of the group Y with respect to the subgroup Y (2) . The function ψ(yα + y), as a function of y satisfies Eq. (6). For this reason its restriction to the subgroup Y (2) is a polynomial of degree 2. Hence, we have the following representation ψ(yα + y) = ϕα (y) + lα (y) + cα ,

y ∈ Y (2) ,

(11)

where the function ϕα satisfies Eq. (1), the function lα satisfies Eq. (9), and cα is a constant. Extend functions ϕα and lα from the subgroup Y (2) to the functions ϕ˜ α and l˜α on the group Y by formulas (8) and (10) respectively. Passing from the function ϕ˜α to the corresponding biadditive function Φα we find from (11)

ψ(−yα − y) = ψ yα + (−2yα − y) = ϕ˜ α (−2yα − y) + l˜α (−2yα − y) + cα = 4Φα (yα , yα ) + 4Φα (yα , y) + Φα (y, y) − 2l˜α (yα ) − l˜α (y) + cα ,

y ∈ Y (2) .

(12)

Since ψ(yα + y) = Φα (y, y) + l˜α (y) + cα ,

y ∈ Y (2) ,

(13)

and ψ(−y) = ψ(y), we conclude from equalities (12) and (13) that 2Φα (yα , y) = l˜α (y),

y ∈ Y (2) .

(14)

It follows from (13) and (14) that ψ(yα + y) = Φα (yα + y, yα + y) − 2Φα (yα , y) − Φα (yα , yα ) + l˜α (y) + cα = ϕ˜ α (yα + y) + rα , where rα is a constant.

y ∈ Y (2) ,

(15)

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Putting k = h in (6) we obtain the equation ψ(y + 4h) − 2ψ(y + 3h) + 2ψ(y + h) − ψ(y) = 0,

y, h ∈ Y.

(16)

Substituting y = 0, h = yα + 2u, u ∈ Y in (16) we arrive at ψ(4yα + 8u) − 2ψ(3yα + 6u) + 2ψ(yα + 2u) = 0,

u ∈ Y.

Taking into account (15) we infer that ϕ˜ 0 (4yα + 8u) − 2ϕ˜ α (3yα + 6u) + 2ϕ˜ α (yα + 2u) = 0,

u ∈ Y.

Passing here from the functions ϕ˜0 and ϕ˜α to the corresponding biadditive functions Φ0 and Φα we obtain

4 Φ0 (u, u) − Φα (u, u) + 4 Φ0 (yα , u) − Φα (yα , u)

+ Φ0 (yα , yα ) − Φα (yα , yα ) = 0, u ∈ Y. This implies that ϕ˜ α (y) ≡ ϕ˜ 0 (y), and representation (7) follows from (15). Lemma 6 is proved. 2 Denote by Rℵ0 the space of all sequences of real numbers equipped with the product topology. We note that the topological group Rℵ0 is not locally compact. Denote by Rℵ0 ∗ the space of all finitary sequences of real numbers with the topology of strictly inductive limit of spaces Rn . The topological group Rℵ0 ∗ is not locally compact either. Let t = (t1 , . . . , tn , . . .) ∈ Rℵ0 and s = (s1 , . . . , sn , 0, . . .) ∈ Rℵ0 ∗ . Set t, s = ∞ j =1 tj sj , and put (t, s) = exp{i t, s }. ℵ 0 Let μ be a distribution on the group R . We define the characteristic function of μ by the formula μ(s) ˆ = Rℵ0 (t, s) dμ(t), s ∈ Rℵ0 ∗ . Let A = (αij )∞ i,j =1 be a symmetric positive semidefi

ℵ0 ∗ . We can nite matrix, i.e. the square form As, s = ∞ α s s i,j =1 ij i j is nonnegative for all s ∈ R define now a Gaussian measure on the group Rℵ0 (see, e.g. in [3, §5]). A probability measure μ on the group Rℵ0 is called Gaussian if its characteristic function is represented in the form μ(s) ˆ = (t, s) exp − As, s ,

s ∈ Rℵ0 ∗ ,

where t ∈ Rℵ0 , and A = (αij )∞ i,j =1 is a symmetric positive semidefinite matrix. Denote by ℵ 0 Γ (R ) the set of Gaussian measures on the group Rℵ0 . Lemma 7. (See [1].) Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to T, let E be either Rm or Rℵ0 , let γ be a symmetric Gaussian measure on X. Then there exists a continuous monomorphism π : E → X, such that γ = π(N ), N ∈ Γ (E). Lemma 8. (See [4].) Assume that a group X is of the form X = E × G, where either E = Rm or E = Rℵ0 and G is a locally compact second countable Abelian group such that all nonzero elements of G have order 2. If τ ∈ M 1 (X), τ = λ ∗ ω, where λ ∈ Γ (E), ω ∈ M 1 (G), and the


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characteristic function of ω does not vanish, then each factor τ1 of τ can be represented in the form τ1 = λ1 ∗ ω1 , where λ1 ∈ Γ (E) and ω1 ∈ M 1 (G). Remark 2. Denote by Z(2) the two-element cyclic group. It is relevant to remark that if G is a locally compact Abelian group and all nonzero elements of G have order 2, then by the structure theorem for such groups [8, §25.29] we have

N∗

M × Z(2) , G∼ = Z(2) where M and N are cardinal numbers, the group (Z(2))M is a direct product of the group Z(2) ∗ considering in the product topology, and the group (Z(2))N is a weak direct product of the group Z(2) considering in the discrete topology. 3. Proof of Theorem 1 Proof of Theorem 1. Passing to the random variables ξj = αj ξj we can assume without loss of generality that L1 = ξ1 + · · · + ξn and L2 = δ1 ξ1 + · · · + δn ξn , where δj ∈ Aut(X) such that δi ± δj ∈ Aut(X) for i = j . Put νj = μj ∗ μ¯ j and ψj (y) = − ln νˆ j (y). By Lemma 3, the symmetry of the conditional distribution of L2 given L1 implies that the characteristic functions μˆ j (y) satisfy Eq. (5). Hence, the characteristic functions νˆ j (y) also satisfy Eq. (5). It follows from this that the functions ψj satisfy Eq. (2). By Lemma 1, equality (3) holds true. It follows from this that the restriction of the function ψj to the subgroup Y (2) is a polynomial. We will prove at first that any distribution νj is represented as a convolution of a Gaussian measure and a distribution supported in G. By Lemma 4 and Remark 1, we can suppose that the group X possesses the property: X (2) = cX , i.e. the mapping π : X → cX , π(x) = 2x is an epimorphism, Ker π = G and cX ∼ = X/G. It follows from this that (cX )∗ ∼ = A(Y, G) = Y (2) . We can assume that cX = {0}, for otherwise the statement of Theorem 1 follows directly from Lemma 4. Taking into account that cX contains no subgroup topologically isomorphic to T and (cX )∗ ∼ = Y (2) we can apply Lemma 5 to the restriction of the function exp{−ψj (y)} to Y (2) . We obtain that this restriction is the characteristic function of a Gaussian measure on X/G ∼ = cX . This implies that the restriction of the function ψj to Y (2) satisfies Eq. (1). If ψj (y) = 0 for y ∈ Y (2) , then σ (νj ) ⊂ A(X, Y (2) ) = G, and the statement of Theorem 1 for the distribution μj follows from the fact that μj is a factor of νj . Therefore we can assume that ψj (y) ≡ 0 on Y (2) . Thus, the restriction of the function ψj to Y (2) is a polynomial of degree 2. Hence, for any fixed k ∈ Y the restriction of the function 2k ψj (y) to Y (2) is a polynomial of degree 1. On the other hand (3) implies that the function 2k ψj (y) is a polynomial on Y . It follows from this that 2k ψj (y) is a polynomial of degree 1 on Y . So, the function ψj satisfies Eq. (6). Applying Lemma 6 we obtain representation (7) for the function ψj , where the function ϕ and rα depend on ψj . Let γ be a Gaussian measure on the group X with the characteristic function γˆ (y) = exp −ϕ(y) ,

y ∈ Y.

(17)

Consider on the group Y the function g(y) = exp{−rα },

y ∈ yα + Y (2) .

(18)

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It follows from g(y) = νˆ j (y)/γˆ (y) that the function g is continuous. Observe also that g is invariant with respect to the subgroup Y (2) . We will verify that g is a positive definite function. For this purpose we take arbitrary elements t1 , . . . , tn ∈ Y , such that tj ∈ yαj + Y (2) and verify that for arbitrary z1 , . . . , zn ∈ C the inequality n

g(ti − tj )zi z¯ j 0

(19)

i,j =1

holds true. Consider a subgroup H in Y generated by cosets yαj + Y (2) , j = 1, . . . , n. Note ∼ (A(X, Y (2) ))∗ = G∗ . We conclude from this that H consists of a finite number of that Y/Y (2) = cosets. Set K = H ∗ , L = A(X, H ). Then K ∼ = X/L, L ⊂ G. Consider the restriction g|H of the function g to H . This restriction is invariant with respect to the subgroup Y (2) and hence, it can be considered as a function on the factor-group H /Y (2) . Note that the factor-group H /Y (2) is finite and all nonzero elements of it have order 2. This implies that any real valued function on H /Y (2) is the characteristic function of a signed measure. From what has been said it follows that g|H is the characteristic function of a signed measure β on a finite subgroup F = A(K, Y (2) ). It follows from (7), (17) and (18) that the restriction of the function νˆ j (y) to H is the characteristic function of a convolution of a Gaussian measure α on K and a signed measure β on F . We will verify that β is a probability distribution and hence, (19) will be proved. We remind that we assume that X (2) = cX . This property implies that cX ∼ = X/G and hence, cX ∼ = (X/L)/(G/L). Taking into account that cX contains no subgroup topologically isomorphic to T it follows from this that the group K also contains no subgroup topologically isomorphic to T. By Lemma 7, there exists a continuous monomorphism p : E → K, where either E = Rm or E = Rℵ0 such that α = p(N), N ∈ Γ (E). Thus, the Gaussian measure α is concentrated on a Borel subgroup B = p(E) of K. Note that all nonzero elements of subgroup F have order 2 and hence, B ∩ F = {0}. Since α ∗ β is a probability distribution, we conclude that β is a probability distribution too. Thus, g is a positive definite function. By the Bochner theorem there exists a distribution ρ ∈ M 1 (X) such that ρ(y) ˆ = g(y). Since g(y) = 1 for all y ∈ Y (2) , we see 1 (2) that σ (ρ) ⊂ A(X, Y ) = G, i.e. ρ ∈ M (G). We inferred that νj = γ ∗ ρ, where γ ∈ Γ (X), ρ ∈ M 1 (G). Prove now that the distribution μj has the required representation. By Lemma 7, there exists a continuous monomorphism q : E → X, where either E = Rm or E = Rℵ0 , such that γ = q(N ), N ∈ Γ (E). Since q(E) ∩ G = {0}, the monomorphism q can be extended to a continuous monomorphism q˜ : E × G → X by the formula q(t, ˜ ζ ) = q(t) + ζ , t ∈ E, ζ ∈ G. Then νj = q(N ˜ ∗ ρ) and hence, the distribution νj is concentrated on the Borel subgroup q(E) + G of X. The distribution μj is a factor of νj . Substituting, if it is necessary, the distribution μj by its shift, we can assume that μj is also concentrated on the Borel subgroup q(E)+G. For this reason μj = q(M ˜ j ), where Mj ∈ M 1 (E × G) and Mj is a factor of N ∗ ρ. By Lemma 8, Mj = Nj ∗ ρj , where Nj ∈ Γ (E), ρj ∈ M 1 (G). Hence, μj = q(M ˜ j ) = q(N ˜ j ∗ ρj ) = q(N ˜ j ) ∗ q(ρ ˜ j ) = γj ∗ ρ j , where γj = q(N ˜ j ). Since γj ∈ Γ (X), Theorem 1 is proved. 2 We note that the following statement results from Lemma 6 and the proof of Theorem 1.


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Proposition 1. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to T, G be the subgroup of X generated by all elements of ¯ and the characteristic function of the distribution ν be order 2. Let μ ∈ M 1 (X), ν = μ ∗ μ, of the form νˆ (y) = exp −ψ(y) , where the function ψ satisfies the equation 2k m h ψ(y) = 0,

k, h, y ∈ Y.

Then μ = γ ∗ ρ, where γ ∈ Γ (X), and ρ ∈ M 1 (G). Proposition 1 can be regarded as one more group analogue of the well-known Marcinkiewicz theorem (compare with Lemma 5). Acknowledgment I thank the referee for carefully reading the manuscript and useful remarks. References [1] G.M. Feldman, Gaussian distributions on locally compact Abelian groups, Theory Probab. Appl. 23 (1978) 529– 542. [2] G.M. Feldman, Marcinkiewicz and Lukacs theorems on Abelian groups, Theory Probab. Appl. 34 (1989) 290–297. [3] G.M. Feldman, Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups, Amer. Math. Soc. Transl. Math. Monogr., vol. 116, Amer. Math. Soc., Providence, RI, 1993. [4] G.M. Feldman, On a characterization theorem for locally compact Abelian groups, Probab. Theory Related Fields 133 (2005) 345–357. [5] G.M. Feldman, Functional Equations and Characterization Problems on Locally Compact Abelian Groups, EMS Tracts Math., vol. 5, European Math. Soc., Zurich, 2008. [6] L. Fuchs, Infinite Abelian Groups. 1, Academic Press, New York/London, 1970. [7] P. Graczyk, J.-J. Loeb, A Bernstein property of measures on groups and symmetric spaces, Probab. Math. Statist. 20 (1) (2000) 141–149. [8] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis. 1, Springer-Verlag, Berlin/Gottingen/Heildelberg, 1963. [9] C.C. Heyde, Characterization of the normal low by the symmetry of a certain conditional distribution, Sankhya Ser. A 31 (1969) 115–118. [10] A.M. Kagan, Yu.V. Linnik, C.R. Rao, Characterization Problems of Mathematical Statistics, Wiley, New York, 1973. [11] D. Neuenschwander, R. Schott, The Bernstein and Skitovich–Darmois characterization theorems for Gaussian distributions on groups, symmetric spaces, and quantum groups, Expo. Math. 15 (4) (1997) 289–314. [12] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York/London, 1967.


Homological properties of modules over semigroup algebras Paul Ramsden Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom Received 22 April 2009; accepted 10 March 2010 Available online 23 March 2010 Communicated by S. Vaes

Abstract Let S be a semigroup. In this paper we investigate the injectivity of 1 (S) as a Banach right module over 1 (S). For weakly cancellative S this is the same as studying the flatness of the predual left module c0 (S). For such semigroups S, we also investigate the projectivity of c0 (S). We prove that for many semigroups S for which the Banach algebra 1 (S) is non-amenable, the 1 (S)-module 1 (S) is not injective. The main result about the projectivity of c0 (S) states that for a weakly cancellative inverse semigroup S, c0 (S) is projective if and only if S is finite. © 2010 Elsevier Inc. All rights reserved. Keywords: Banach algebra; Homology; Cohomology; Module; Projective; Injective; Flat; Amenable; Semigroup

1. Introduction Let S be a semigroup, and let 1 (S) be its associated Banach convolution algebra. In this paper we study certain homological properties of modules over 1 (S). The aim is to characterize homological properties of the Banach algebra 1 (S) (and its modules) in terms of the underlying semigroup S. Homological properties of Banach algebras associated with groups and semigroups have been studied by many authors. Some recent papers are [1,6–8]. The notions of projectivity, injectivity, and flatness are fundamental in homology theory. The theory of these concepts in the category of Banach modules is expounded by A.Ya. Helemskii in [10]. In [4], H.G. Dales and M.E. Polyakov undertook a study of these properties for variE-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.007

P. Ramsden / Journal of Functional Analysis 258 (2010) 3988–4009

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ous canonical modules associated with the group algebra L1 (G). For example, they proved the following theorem. Here we regard C0 (G) as a submodule of the dual module L∞ (G) = L1 (G) . Theorem 1.1. (See [4, Theorems 3.1, 4.7 and 4.9].) Let G be a locally compact group. Then: (i) L1 (G) is an injective right L1 (G)-module if and only if G is discrete and amenable. (ii) C0 (G) is a flat left L1 (G)-module if and only if G is amenable. (iii) C0 (G) is a projective left L1 (G)-module if and only if G is compact. In this paper we shall investigate the same questions for the Banach algebra 1 (S), where S is a (discrete) semigroup. We cannot hope for such simple characterizations for a general semigroup. This is partly due to the complexities of describing the amenability of the Banach algebra 1 (S) in terms of the semigroup S; see [3, Theorem 10.12]. 1.1. Overview of contents • Section 2 contains all the background material and definitions about Banach modules and semigroups that we shall need. • In Section 3 we prove some general results about injective Banach modules. • In Section 4 we show how amenability of the underlying semigroup enters the picture. Here we show that the injectivity of certain modules over 1 (S) implies that S is amenable. This follows from a general result which also applies to modules over L1 (G) and M(G) for a locally compact group G. In Theorem 4.10, under the additional assumption that S is cancellative, we give a characterization of the injectivity of 1 (S) (S must be an amenable group). At the end of this section we give an example (Example 4.12) of a finite semigroup S such that 1 (S) is a right injective module, but 1 (S) is not amenable. • In Section 5 we investigate the flatness of the left 1 (S)-predual module c0 (S) for a weakly right cancellative semigroup S. Our main theorem (Theorem 5.5) gives a necessary combinatorial condition on the set of idempotents. This condition is not satisfied by the bicyclic semigroup or (N, max). • In Section 6 we move on to investigating the semigroups S such that c0 (S) has the stronger property of being projective. Here we prove, in Theorem 6.5 that, if S is a weakly cancellative semigroup such that c0 (S) is projective, then S must be finite. An immediate corollary of this result (Theorem 6.6) gives a characterization for the class of inverse semigroups (c0 (S) is projective if and only if S is finite). 2. Preliminaries For n ∈ N, we set Nn = {1, 2, . . . , n}. The indicator function of a subset T of a set S is denoted by χT . Let f : S → E be a function from a set S to a vector space E. For T ⊂ S we define χT f : S → E by (χT f )(s) = f (s) (s ∈ T ) and (χT f )(s) = 0 (s ∈ S \ T ). Let E be a Banach space. We denote the dual space by E ; the action of λ ∈ E on an element x ∈ E is written as x, λ. For a subspace F of a Banach space E we set F 0 = λ ∈ E : x, λ = 0 (x ∈ F ) . Then F 0 is a closed subspace of E . We set F 00 = (F 0 )0 ⊂ E .

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, and B(E, F ) denotes the space of all bounded The projective tensor product is denoted by ⊗ linear operators between Banach spaces E and F . 2.1. Banach modules Throughout this section A is a Banach algebra. We denote by A-mod, and by mod-A the categories of Banach left A-modules, and of Banach right A-modules, respectively. We shall give definitions only for one category of module; similar definitions apply for modules in other categories. Let E ∈ A-mod. Then the dual space E has a natural right A-module structure given by x, λ · a = a · x, λ

a ∈ A, x ∈ E, λ ∈ E .

This module is called the dual module of E. Let S ⊂ A be a subset, and let E ∈ A-mod. We set S · E = {a · x: a ∈ S, x ∈ E} and SE = lin S · E, the linear span of S · E. If S is a left ideal of A then SE is a submodule of E. The essential part of E is the closed submodule AE, and E is essential if AE = E. Now we describe the ‘dual’ concept. Let F ∈ mod-A. For a subset S ⊂ A we set F ⊥S = x ∈ F : x · S = {0} . If S is a left ideal of A then F ⊥S is a closed submodule of F . We set F ⊥ = F ⊥A , which is the annihilator submodule of F . The A-module F is faithful if F ⊥ = {0}, and F is an annihilator module if F ⊥ = F . Now suppose that F = E for some E ∈ A-mod. Then F ⊥S = (SE)0 , and so we have (SE) = F /(SE)0 = F /F ⊥S . Hence, if SE = E, then F ⊥S = {0}. In particular the dual of an essential module is faithful. Similarly, for E ∈ A-mod we set S⊥

E = x ∈ E: S · x = {0} .

For E, F ∈ A-mod we denote by A B(E, F ) the subspace of B(E, F ) consisting of bounded left A-module morphisms. Similarly, BA (E, F ) denotes the space of bounded right A-module morphisms when E, F ∈ mod-A. The unitization of A is denoted by A . If E ∈ A-mod, then we consider E ∈ A -mod in the obvious way. E ∈ A-mod with module operation specified by Let E ∈ A-mod. Then A ⊗ a · (b ⊗ x) = ab ⊗ x

a ∈ A, b ∈ A , x ∈ E .


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E → E by We define the canonical morphism π : A ⊗ a ∈ A , x ∈ E .

π(a ⊗ x) = a · x

Let E ∈ mod-A. Then B(A , E) ∈ mod-A with the module operation a ∈ A, b ∈ A , T ∈ B A , E .

(T · a)(b) = T (ab)

We define the canonical embedding Π : E → B(A , E) by the formula a ∈ A , x ∈ E .

Π(x)(a) = x · a 2.2. Banach homology

We now recall the definitions and basic relationships from Banach homology that this paper is concerned with. For full details see [10] and [11]. We eschew the original homological definitions and give the standard characterizations which have proved most useful for checking projectivity and injectivity in specific cases. Because of the duality involved it is convenient to study injective right modules and projective and flat left modules. Proposition 2.1. Let A be a Banach algebra. (i) Let F ∈ mod-A. Then F is injective if and only if there exists ρ ∈ BA (B(A , F ), F ) with ρ ◦ Π = IF . E) with (ii) Let E ∈ A-mod. Then E is projective if and only if there exists ρ ∈ A B(E, A ⊗ π ◦ ρ = IE . E) ) such that (iii) Let E ∈ A-mod. Then E is flat if and only if there exists ρ ∈ A B(E, (A ⊗ the following diagram commutes ρ

E ιE

E) (A ⊗ π

E . (iv) If either E is essential or F is faithful, then we can replace A by A in the above characterizations. Part (i) is proved in [10, III.1.31] and the case where E is faithful is [4, Proposition 1.7]; part (ii) is [10, IV.1.1, IV.1.2]; part (iii) is similar to [10, Exercise VII2.8], see also [15, Lemma 4.3.22]. The following facts are elementary: a module E is flat if and only if the dual module E is injective; every projective module is flat. The classes of amenable and contractible Banach algebras are particularly important and well studied (see [10, Chapter IV], [12] or [2, §2.8]). The following proposition gives one nice property of these Banach algebras.

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Proposition 2.2. Let A be a Banach algebra, and let E ∈ A-mod or E ∈ mod-A. (i) If A is contractible, then E is projective. (ii) If A is amenable, then E is flat. The following is a long standing open problem. Question 2.3. Let A be a Banach algebra such that every E ∈ A-mod and every E ∈ mod-A is flat [ projective]. Is A amenable [contractible]? The answer is known to be positive for many classes of Banach algebras, in particular for the class of group algebras. Part of the motivation for this work was to answer this question in the class of semigroup algebras. Although we come short of a complete answer, our results strongly suggest that the answer is ‘yes’ in this class. 2.3. Semigroups and semigroup algebras Let S be a semigroup. The set of idempotents in S is denoted by E(S). A semigroup S is a semilattice if S is commutative and E(S) = S. The canonical partial order on E(S) is given by pq

⇐⇒

p = pq = qp

p, q ∈ E(S) .

Let S be a semigroup. The semigroup algebra 1 (S) is the completion in the 1 -norm of the algebra CS. It is the Banach algebra generated by the semigroup. The convolution product

on 1 (S) is uniquely defined by requiring that δs δt = δst (s, t ∈ S). We identify 1 (S) with 1 (S ), where S is the semigroup formed by adjoining an identity to S. These Banach algebras have been studied by many authors. A recent exposition is the memoir [3]. 2.3.1. Cancellativity We shall use the following notation introduced by Grønbæk in [9]. For s, t ∈ S we define the sets −1 = {u ∈ S: ut = s} st

and

−1 t s = {u ∈ S: tu = s}.

Let S be a semigroup. An element t ∈ S is left cancellable if u = v whenever tu = tv. Equivalently we require that |[t −1 s]| 1 (s ∈ S). Right cancellable elements are defined similarly. The semigroup S is cancellative if each element is both left and right cancellative. The semigroup S is weakly left (respectively, right) cancellative if [t −1 s] (respectively, [st −1 ]) is finite for each s, t ∈ S, and weakly cancellative if it is both weakly left cancellative and weakly right cancellative. Lemma 2.4. Let S be a semigroup such that the Banach algebra 1 (S) has a left identity. Suppose that S contains a right cancellable element. Then: (i) S has a left identity eS ; (ii) for each right cancellable element t we have teS = t.


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Proof. (i) Let eA = s∈S es δs be a left identity for 1 (S), and let t be a right cancellable element, so that |[tt −1 ]| 1. We have δt =

s∈S

Hence

s∈[tt −1 ] es

es δst =

r∈S

e s δr .

s∈[rt −1 ]

= 1 and in particular the set [tt −1 ] is non-empty, say [tt −1 ] = {u}. Then δu δt = δt = eA δt .

Since t is right cancellable, it follows that δu = eA , and so u is a left identity for S. (ii) Let t be a right cancellable element and eS a left identity for S. Then t 2 = t (eS t) = (teS )t, which implies that t = teS . 2 2.3.2. Module actions Let S be a semigroup, and let E ∈ 1 (S)-mod or E ∈ mod-1 (S). We shall use the following more compact notation for the module actions t · x = δt · x,

x · t = x · δt

(x ∈ E, t ∈ S).

The standard actions of 1 (S) on 1 (S) are given by (t · a)(s) =

a(u),

(a · t)(s) =

u∈[t −1 s]

a(u)

s, t ∈ S, a ∈ 1 (S) ,

u∈[st −1 ]

where we define the sum over an empty set to be zero. The dual actions of 1 (S) on ∞ (S) are given by (t · λ)(s) = λ(st),

(λ · t)(s) = λ(ts)

s, t ∈ S, λ ∈ ∞ (S) .

For s ∈ S, the indicator function of the set {s} will be denoted by δs when considered as an element of 1 (S) and by λs when considered as an element of ∞ (S). This notation implies that the left module actions satisfy t · δs = δts

and t · λs = χ[st −1 ]

(t ∈ S).

Proposition 2.5. (See [3, Theorem 4.6].) Let S be a semigroup. Then c0 (S) is a left [right] 1 (S)submodule of ∞ (S) if and only if S is weakly right [left] cancellative. 3. Some general results on injective modules In this section we prove some basic intrinsic properties of injective modules. We begin with a generalization of [4, Proposition 1.8].

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Proposition 3.1. Let A be a Banach algebra, and let E ∈ mod-A be injective. Suppose that there exist Q ∈ B(A, E) and a subset S ⊂ A such that Q(ba) = Q(b) · a

(b ∈ S, a ∈ A).

Then there is an element x ∈ E with Q(b) = x · b

(b ∈ S).

Proof. There exists a morphism ρ ∈ BA (B(A , E), E) such that ρ ◦ Π = IE . Extend Q to A by setting Q(e ) = 0. Then Q ∈ B(A , E) and Q · b = Π Q(b)

(b ∈ S).

Set x = ρ(Q) ∈ E. Then we have Q(b) = (ρ ◦ Π) Q(b) = ρ(Q · b) = x · b as required.

(b ∈ S),

2

Corollary 3.2. Let A be a Banach algebra, let I be a complemented ideal in A, and let E ∈ mod-A be injective. Then the map j : E → BA (I, E) given by j (x) : a → x · a : I → E is a Banach A-module epimorphism with kernel E ⊥I . Proof. Take T ∈ BA (I, E), and set Q = T ◦ P , where P : A → I is a projection. Then Q ∈ B(A , E) and satisfies Q(ba) = Q(b) · a (b ∈ I, a ∈ A). By Proposition 3.1, there exists x ∈ E such that T (b) = Q(b) = x · b (b ∈ I ), i.e., T = j (x). The rest is clear. 2 Corollary 3.3. Let A be a Banach algebra. (i) Let A be a subalgebra of a Banach algebra B. Suppose that B is injective in mod-A. Then there exists p ∈ B with pa = a (a ∈ A). (ii) Let I be a closed right ideal in A. Suppose that A/I is injective in mod-A. Then I has a left modular identity. (iii) Let I be a closed, complemented right ideal in A. Suppose that I is injective in mod-A. Then there exists p ∈ I with pa = a (a ∈ I ). (iv) Let I be a closed, complemented right ideal in A. Suppose that I is injective in mod-A. Then I has a bounded left approximate identity. Proof. These all follow from Proposition 3.1 by choosing specific maps Q. (i) Take Q : A → B to be the inclusion map. (ii) Take Q : A → A/I to be the quotient map. (iii) Take Q : A → I to be a projection.


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(iv) Take the map Q : A → I to be a projection onto I followed by the natural embedding into I . Then there exists Φ ∈ I with Φ · a = a (a ∈ I ). The result now follows by a standard argument involving the weak-∗ topology on I , see [2, Proposition 2.9.14(iii)]. 2 Proposition 3.4. Let A be a Banach algebra, and let E ∈ mod-A be injective. Suppose that there exists a0 ∈ A \ {0} with a0 A = 0. Then E ⊥ = E · a0 . In particular, the subspace E · a0 is closed. Proof. The inclusion E · a0 ⊂ E ⊥ is clear. Let x ∈ E ⊥ . Take μ ∈ (A ) with a0 , μ = 1, and let T ∈ B(A , E) be the rank-1 operator given by T (a) = a, μx

a ∈ A .

Let PA , PCe ∈ B(A ) be projections on to the subspaces A and Ce , respectively. Then we have (Πx) ◦ PCe = (T · a0 ) ◦ PCe . Clearly, for any S ∈ B(A , E), we have ρ(S) = ρ(S ◦ PA ) + ρ(S ◦ PCe ). Combining this with the fact that ρ((Πx) ◦ PA ) = ρ((T · a0 ) ◦ PA ) = 0, we have x = ρ (Πx) ◦ PCe = ρ (T · a0 ) ◦ PCe = ρ(T · a0 ) = ρ(T ) · a0 . Therefore E ⊥ = E · a0 .

2

Example 3.5. (See [16, Example 4.4].) Let X be a Banach space with dim X 2, and take ϕ ∈ X \ {0}. Define a product on X by ab = ϕ(a)b

(a, b ∈ X).

With this product X is a Banach algebra which, following [16], we denote by Aϕ (X). By [16], Aϕ (X) is a biprojective Banach algebra. Since Aϕ (X) does not have a right identity, Aϕ (X) is not injective in Aϕ (X)-mod. The Banach algebra Aϕ (X) is faithful in Aϕ (X)-mod, but not faithful in mod-Aϕ (X). Proposition 3.6. The Banach algebra Aϕ (X) is injective in mod-Aϕ (X) if and only if dim X = 2. Proof. Set A = Aϕ (X). Suppose that A is injective in mod-A. Choose a0 ∈ ker ϕ \ {0}. By Proposition 3.4 we have ker ϕ = A⊥ = Aa0 = Ca0 . By the rank-nullity theorem, dim X = 2.

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Conversely, suppose that dim X = 2. Let {a0 , a1 } be a basis of X with a0 , ϕ = 0 and a1 , ϕ = 1. Take λ ∈ X with a0 , λ = 1 and a1 , λ = 0. We define a map ρ : B(A , A) → A by

ρ(T ) = T a0 , λa1 − T a1 , λa0 + T e , λ a0

T ∈ B A , A .

This is a right A-module morphism with ρ ◦ Π = IA . The easiest way to see this is to check that ρ(T · a0 ) = ρ(T ) · a0 , ρ(T · a1 ) = ρ(T ) · a1 (T ∈ B(A , A)), and ρ(Πa0 ) = a0 , ρ(Πa1 ) = a1 . Therefore A is injective in mod-A. 2 4. Amenability and injectivity In this section we show how the injectivity of certain modules over a semigroup algebra implies that the underlying semigroup must be amenable. This is a generalization of the argument in [4, §4]. In contrast to the group case, for a general semigroup, amenability is only part of the story. 4.1. General results Definition 4.1. Let A be a Banach algebra, and let E ∈ mod-A. Then an element λ ∈ E \ {0} is an augmentation-invariant functional if there exists a character ϕ on A, with a · λ = ϕ(a)λ for each a ∈ A. The triple (E, λ, ϕ) is an augmentation-invariant Banach right A-module. Example 4.2. (i) Let A be a Banach algebra with a character ϕ, and let I be a closed left ideal of A with I ⊂ ker ϕ. Then (I, ϕ|I, ϕ) is an augmentation-invariant right A-module. (ii) If E ∈ mod-A is augmentation-invariant, then so is E . Lemma 4.3. Let A be a Banach algebra, let I be a complemented left ideal of A, and let (E, λ, ϕ) be an augmentation-invariant Banach right A-module with I ⊂ ker ϕ. Suppose that E is injective in mod-A. Then there exists Λ ∈ B(I, E) such that: (i) a · Λ = ϕ(a)Λ for each a ∈ A; (ii) Π(x), Λ = x, λ for each x ∈ E. Proof. Since E is injective there is a right A-module morphism ρ : B(A , E) → E with ρ ◦ Π = IE . Set Λ0 = ρ (λ) ∈ B(A , E) . Since ρ is a left A-module morphism, we have a · Λ0 = ϕ(a)Λ0 (a ∈ A). Take T ∈ B(A , E) such that T |I = 0. Pick a ∈ I with ϕ(a) = 1. Then T · a = 0, and 0 = T · a, Λ0 = T , Λ0 . Now take T ∈ B(I, E). We can extend T to T ∈ B(A , E) by setting T = T ◦ P , where P : A → I is a projection. Set T , Λ := T, Λ0

T ∈ B(I, E) .


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Since (T · a − T · a)|I = 0 (a ∈ A), it follows that a · Λ = ϕ(a)Λ (a ∈ A). Similarly, since (Π(x) − Π(x))|I = 0 (x ∈ E), it follows that Π(x), Λ = x, λ. 2 F ) → E = F . If F is a left In the following theorem we set π = Π : B(I, E) = (I ⊗ π |I ⊗ F ⊂ F , and we can replace π by π . A-submodule of F , then Theorem 4.4. Let A be a Banach algebra, let I be a complemented left ideal of A, and let (E, λ, ϕ) be an augmentation-invariant Banach right A-module with I ⊂ ker ϕ. Suppose that E is the dual of a Banach space F , and that E is injective in mod-A. Then there exists a bounded F such that: net (vα ) ⊂ I ⊗ (i) limα a · vα − ϕ(a)vα π = 0 for each a ∈ A; π (vα ) = x, λ for each x ∈ E. (ii) limα x, F , and let σ = σ (X, X ) be the weak topology on X. Proof. Set X = I ⊗ First, a net (uα ) is indexed by the family of all finite subsets of B(I, E), with the ordering specified by inclusion. For each such α = {T1 , . . . , Tk }, choose uα ∈ X such that Ti , uα = Ti , Λ (i = 1, . . . , k), where Λ ∈ X was specified in Lemma 4.3. For each a ∈ A and T ∈ B(I, E), we have T , a · uα = T · a, uα = T · a, Λ = ϕ(a)T , Λ = ϕ(a)T , uα , for each sufficiently large α, and so limα (a · uα − ϕ(a)uα ) = 0 in (X, σ ). Also for each x ∈ E, we have x, π (uα ) = Π(x), uα = Π(x), Λ = x, λ,

for each sufficiently large α, and so limα Π(x), uα − x, λ = 0. Let {a1 , . . . , ak } and {x1 , . . . , x } be finite subsets of A and E, respectively, and let ε > 0. Let C = C {x1 , . . . , x }, ε = z ∈ X: Π(xi ), z − xi , λ < ε (i = 1, . . . , ) , and consider the Banach space Y = X1 ⊕ · · · ⊕ Xk , where each Xi = X (i = 1, . . . , k) and we are taking the 1 -sum. Also consider the linear operator W : z → a1 · z − ϕ(a1 )z, . . . , ak · z − ϕ(ak )z ,

X → Y.

The set C is convex in X, and so W (C) is convex in Y . We have shown that 0 belongs to the σ (Y, Y )-closure of W (C) in Y . By Mazur’s theorem, it follows that 0 belongs to the · -closure of W (C) in Y . The existence of the required net (vα ) follows. 2 4.2. The Banach algebras L1 (G) and M(G) Let G be a locally compact group, and let M(G) denote the measure algebra on G. There is always one character on M(G); this is the augmentation character ϕG , defined by ϕG (μ) = μ(G)

μ ∈ M(G) .

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The restriction of ϕG to L1 (G) regarded as a closed ideal in M(G) has the form ϕG (f ) =

f (s) dm(s)

f ∈ L1 (G) .

G

Let G be a locally compact group. We set P (G) = f ∈ L1 (G): f 0, f = 1 . We shall use the following characterization of amenability, known as Reiter’s condition. Proposition 4.5. (See [13, Proposition (0.8)].) Let G be a locally compact group. Then G is amenable if and only if there is a net (fα ) ⊂ P (G) such that lim t · fα − fα = 0 for each t ∈ G. Theorem 4.6. Let G be a locally compact group, and let (E, λ, ϕG ) be an augmentationinvariant Banach right M(G)-module. Suppose that E is a dual space, and that E is injective in mod-M(G). Then G is amenable. F Proof. Let E have a Banach space predual F . Set A = M(G) and I = L1 (G). Let (vα ) ⊂ I ⊗ be the net given by Theorem 4.4. Take x ∈ E[1] with x, λ = 1. We have π (vα )F x, π (vα ) 1/2 vα π for large enough α. Hence by passing to a subnet we may suppose that, for each α, vα π 1/2. F = L1 (G, F ) to define a net (kα ) in I by We use the identification I ⊗ kα (s) =

vα (s)F vα π

(s ∈ G).

Then kα 0, and kα 1 =

kα (s) dm(s) =

G

G

vα (s)F dm(s) = 1. vα π

Now take t ∈ G. We have 1 t · kα − kα 1 vα π

−1 vα t s − vα (s) dm(s) 2t · vα − vα π . F

G

Hence limα t · kα − kα 1 = 0. Therefore by Proposition 4.5, G is amenable.

2

The same result holds under the hypothesis that E is injective in mod-L1 (G). This combines with Johnson’s theorem and Proposition 2.2 to give the following result, which was proved under additional hypothesis in [4, Theorem 4.6].


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Theorem 4.7. Let G be a locally compact group, and let (E, λ, ϕG ) be an augmentationinvariant Banach right L1 (G)-module. Suppose that E is a dual space. Then E is injective in mod-L1 (G) if and only if G is amenable. These theorems apply to the L1 (G) or M(G)-modules; L1 (G), M(G) and their second duals. Some further results about modules over M(G) are contained in [14]. 4.3. The Banach algebra 1 (S) Theorem 4.8. Let S be a semigroup such that 1 (S) is injective in mod-1 (S). Then S is a left-amenable semigroup and 1 (S) has a left identity. Proof. That S is left-amenable follows in the same way as the proof of Theorem 4.6. That 1 (S) has a left identity is Corollary 3.3(i). 2 Let A be a Banach algebra with a left identity eA . Then eA is an identity for A if and only if {a ∈ A: aA = 0} = {0}, i.e., A is a faithful right A-module. Lemma 4.9. Let S be a semigroup such that 1 (S) is injective in mod-1 (S). Let t ∈ S be a left cancellable element. Then there exists at ∈ 1 (S) with a t δt e A = e A for each left identity eA ∈ 1 (S). Proof. Set A = 1 (S). There is a map T : tS → S : ts → s which extends to a bounded linear operator T : 1 (tS) → A. Set U = T ◦ P , where P : A → 1 (tS) is a projection. Then U satisfies U (b a) = U (b) a

b ∈ 1 (tS), a ∈ A .

By Proposition 3.1, there exists at ∈ A with U (b) = at b (b ∈ 1 (tS)). In particular, eA = U (δt eA ) = at δt eA . 2 Theorem 4.10. Let S be a cancellative semigroup. Then 1 (S) is injective in mod-1 (S) if and only if S is an amenable group. Proof. Sufficiency is obvious; we shall prove necessity. Suppose that 1 (S) is an injective right module. By Theorem 4.8, the Banach algebra 1 (S) has a left identity. By Lemma 2.4, S has an identity eS . By Lemma 4.9, for each s ∈ S, there exists as ∈ 1 (S) with as δs = δeS . It follows that there is an element s −1 ∈ S with s −1 s = eS . From the equation ss −1 s = eS s and right cancellativity we have ss −1 = eS . Therefore S is a group. It is amenable by Theorem 4.8. 2

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Corollary 4.11. The Banach algebra 1 (N) is not injective in mod-1 (N), equivalently c0 (N) is not flat in 1 (N)-mod. Example 4.12. Let S be the right-zero semigroup. The product is given by st = t

(s, t ∈ S).

The Banach algebra 1 (S) belongs to the class of Banach algebras of the form Aϕ (X) defined in Example 3.5. By Proposition 3.6, 1 (S) is right injective if and only if |S| = 2. Since in this case 1 (S) is finite dimensional, this is equivalent to the predual module c0 (S) = ∞ (S) being projective in 1 (S)-mod. Let S2 be the right-zero semigroup on 2 elements. We note that 1 (S2 ) is not amenable. Indeed S2 is left-amenable but not right-amenable; further, S2 has a left identity, but 1 (S2 ) does not have a right identity. Hence the plausible conjecture that 1 (S) is injective in mod-1 (S) only if 1 (S) is amenable is false. 5. Flatness of the predual module c0 (S) Let S be a weakly right cancellative semigroup. Then, by Proposition 2.5 c0 (S) ∈ 1 (S)-mod, and we can identify the dual right 1 (S)-module c0 (S) with 1 (S). Hence for weakly right cancellative semigroups, injectivity of 1 (S) in mod-1 (S) is the same as flatness of c0 (S) in 1 (S)-mod. In this section we shall study this problem for the class of weakly right cancellative semigroups. 5.1. A necessary combinatorial condition For the next two lemmas, we suppose that S is a semigroup, and that E is a Banach space. E. For an element t ∈ S, and F ∈ 1 (S)-mod, we set For a subset T ⊂ S, we set XT = 1 (T ) ⊗ t⊥F = {δt }⊥F . Lemma 5.1. For each t ∈ S, we have t⊥

XS

0

= XS · t.

Proof. Let t ∈ S. The inclusion XS · t ⊂ (t⊥XS )0 is clear. Take λ ∈ (t⊥XS )0 and u, v ∈ S with tu = tv. Then for each x ∈ E we have δu ⊗ x − δv ⊗ x ∈ t⊥XS . Hence, under the identification XS = ∞ (S, E ), we have

0 = δu ⊗ x − δv ⊗ x, λ = x, λ(u) − λ(v)

(x ∈ E).

It follows that λ(u) = λ(v). Hence, for each s ∈ S, λ is constant on the set [t −1 s]. Therefore we can define


ϕ(s) = Then ϕ ∈ XS and λ = ϕ · t.

if there exists u ∈ [t −1 s], if [t −1 s] = ∅

λ(u), 0,

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(s ∈ S).

2

For a subset T ⊂ S and an element t ∈ S, we define the following subset of T :

G(T , t) =

−1 t s ∩ T = {u ∈ T : ∃v ∈ T , v = u, tu = tv}.

{s∈S: |[t −1 s]∩T |2}

The complement of G(T , t) in T is perhaps simpler to describe: we have T \ G(T , t) = s ∈ T : t −1 (ts) ∩ T = {s} . For example, if t is a left cancellable element, then G(T , t) = ∅. For a subset T ⊂ S, we identify XT with the closed, complemented subspace of XS consisting of functions on S whose support is contained in T . We can then identify XT with XT00 in XS . Lemma 5.2. For each t ∈ S and T ⊂ S, we have: (i) t · XS ⊂ (XtS ) ; (ii) t⊥ (XT ) ⊂ XG(T ,t) . Proof. (i) Let t ∈ S and ϕ ∈ (XtS )0 . Then ϕ · t = 0, and so for each Λ ∈ XS , we have ϕ, t · Λ = ϕ · t, Λ = 0. 00 = X . Therefore t · Λ ∈ XtS tS (ii) Let t ∈ S and T ⊂ S. Take z = s∈S δs ⊗ xs ∈ t⊥XT . The equation t · z = 0 gives

xu = 0

(s ∈ S).

u∈[t −1 s]∩T

Take u ∈ supp z. Then u ∈ [t −1 (tu)] ∩ T . Hence |[t −1 (tu)] ∩ T | 2, and z ∈ XG(T ,t) . We have proved that t⊥XT ⊂ XG(T ,t) . Now by Lemma 5.1 we have (XG(T ,t) )0 ⊂

t⊥

XT

0

Hence t⊥ (XT ) = (XT · t)0 ⊂ (XG(T ,t) )00 = XG(T ,t) .

= XT · t. 2

Theorem 5.3. Let S be a weakly right cancellative semigroup such that, for each N ∈ N, there exist elements (s1 , r1 , t1 ), . . . , (sN , rN , tN ) in S × S × S with the following properties:

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(i) sn ∈ Srn \ Stn rn (n ∈ NN ); (ii) the sets [s1 r1−1 ], . . . , [sN rN−1 ] are pairwise disjoint; (iii) the sets G(r1 S , t1 ), . . . , G(rN S , tN ) are pairwise disjoint. Then c0 (S) is not flat in 1 (S)-mod. Proof. We set E = c0 (S), and for a subset T ⊂ S , we set AT = 1 (T ). Assume towards a contradiction that E is flat in 1 (S)-mod. Then there exists a left 1 (S) E) with π ◦ ρ = iE . Fix N ∈ N, and let (s1 , r1 , t1 ), . . . , module morphism ρ : E → (AS ⊗ (sN , rN , tN ) ∈ S × S × S be the elements given by the hypothesis. For each n ∈ NN , set xn = rn · λsn = χ[sn r −1 ] . n

E) . Since sn ∈ By Lemma 5.2(i), ρ(xn ) ∈ (Arn S ⊗ / Stn rn , we have tn · ρ(xn ) = ρ(tn · xn ) = ρ(χ[sn (tn rn )−1 ] ) = 0. E) . Hence by Lemma 5.2(ii), ρ(xn ) ∈ (AG(rn S ,tn ) ⊗ N E) . By (ii), N Set Φ = ρ( n=1 xn ) ∈ (AS ⊗ n=1 xn ∞ = 1. Hence there is a net (zα ) E)[ρ] such that limα zα = Φ in the weak-∗ topology. For each n ∈ NN , let in (AS ⊗ E → AG(rn S ,tn ) ⊗ E be a projection. By (iii), we have Pn (ρ(xm )) = 0 (n = m). Pn : AS ⊗ Hence limα Pn (zα ) = Pn (Φ) = ρ(xn ) in the weak-∗ topology. For each i ∈ NN , by (i) we can pick un ∈ [sn rn−1 ]. For large enough α, we have Π(δu ), Pn (zα ) − Π(δu ), ρ(xn ) < 1/2. n n Now for each n ∈ NN , we have

Π(δun ), ρ(xn ) = δun , π ◦ ρ(xn ) = xn , δun = 1.

Hence, for large enough α, we have Π(δu ), Pn (zα ) > 1/2, n and so Pn (zα )π 1/2. But now using (iii), for sufficiently large α,

ρ zα π

N

Pn (zα ) N/2. π n=1

This holds for each N ∈ N, the required contradiction.

2


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5.2. A special case Here we describe a special case of the condition in Theorem 5.3, which is easier to apply to certain examples. The key is the following description of the sets G(qS, p). Lemma 5.4. Let S be a semigroup, and let p, q ∈ E(S) with p < q. Then pS ⊂ qS and G(qS, p) = (qS \ pS) ∪ p(qS \ pS) is a disjoint union of sets. Proof. It is clear that pS ⊂ qS and (qS \ pS) ∩ p(qS \ pS) = ∅. Let π1 : S × S → S be the projection onto the first coordinate. Consider the sets G = (u, v) ∈ qS × qS: u = v, pu = pv and F = (u, pu): u ∈ qS \ pS ∪ (pu, u): u ∈ qS \ pS . We make the identifications π1 (G) = G(qS, p)

and π1 (F ) = (qS \ pS) ∪ p(qS \ pS).

Since F ⊂ G we have π1 (F ) ⊂ π1 (G). Now take u ∈ π1 (G) with u = π1 ((u, v)) for some (u, v) ∈ G. If u ∈ / pS, then (u, pu) ∈ F . If u ∈ pS, then v ∈ qS \ pS and (u, v) = (pv, v) ∈ F . In either case u ∈ π1 (F ). 2 Theorem 5.5. Let S be a weakly right cancellative semigroup. Suppose that there exists an infinite chain of idempotents r1 > s1 > t1 > · · · > rn > sn > tn > rn+1 > sn+1 > tn+1 > · · · such that for each n ∈ N, tn (rn S \ tn S) ∩ rn+1 S = ∅. Then c0 (S) is not flat in 1 (S)-mod. Proof. We shall apply Theorem 5.3; we verify clauses (i)–(iii). Clearly sn ∈ Srn \ Stn = Srn \ Stn rn (n ∈ N), so that clause (i) holds. −1 ]. Then Take n < m. Assume towards a contradiction that there exists u ∈ [sn rn−1 ] ∩ [sm rm urn = sn and urm = sm . Multiplying the first of these equations on the right by rm gives urm = rm . Hence rm = sm , which is a contradiction. Therefore clause (ii) holds. For each l m we have tk S ⊃ rm S. Hence we have (rn S \ tn S) ∩ G(rm S, tm ) ⊂ (S \ tn S) ∩ rm S = ∅.

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Using Lemma 5.4, we have G(rn S, tn ) ∩ G(rm S, tm ) = tn (rn S \ tn S) ∩ G(rm S, tm ) ⊂ tn (rn S \ tn S) ∩ rm S ⊂ tn (rn S \ tn S) ∩ rn+1 S = ∅ (by hypothesis). Therefore condition (iii) of Theorem 5.3 also holds, and hence c0 (S) is not flat in 1 (S)-mod.

2

Example 5.6. We give some examples of semigroups which satisfy the hypothesis of Theorem 5.5. (i) Let N∨ = (N, max). The canonical partial order on N∨ is the reverse of the natural order on N. We set rn = 3n − 2,

sn = 3n − 1,

tn = 3n (n ∈ N).

For any n ∈ S, we have nN∨ = [n, ∞). Hence for each n ∈ N we have tn (rn N∨ \tn N∨ ) = 3n[3n− 2, 3n − 1] = {3n}, which is disjoint from the set rn+1 N∨ = [3n + 1, ∞). Therefore c0 (N∨ ) is not flat in 1 (N∨ )-mod. (ii) Let B be the bicyclic semigroup. Then B = N0 × N0 with the multiplication (m, n)(p, q) = m − n + max{n, p}, q − p + max{n, p} (m, n), (p, q) ∈ B . We set rn = (3n − 2, 3n − 2),

sn = (3n − 1, 3n − 1),

tn = (3n, 3n) (n ∈ N).

For any (m, n) ∈ B, we have (m, n)B = [m, ∞) × N0 . Hence for each n ∈ N we have tn (rn B \ tn B) = (3n, 3n) [1, 3n − 1] × N0 = {3n} × N0 , which is disjoint from the set rn+1 B = [3n + 1, ∞) × N0 . Therefore c0 (B) is not flat in 1 (B)-mod. The next example is ‘at the opposite extreme’ to those above. This semigroup does not satisfy the hypothesis of Theorem 5.3 (and hence also of Theorem 5.5). Example 5.7. Let X be a set, let PX = P(X) be the power set of X, and let FX be the set of all finite subsets of X. Then PX is a semilattice with the multiplication st = s ∪ t

(s, t ∈ PX ),

and FX is a subsemilattice of PX called the free semilattice over X. The empty set is the identity of PX . For s, t ∈ PX , we have −1 −1 ∅ if t ⊂ s, st = t s = {u: ut = s} = {s \ u: u ⊂ t} if t ⊂ s.


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For t ⊂ s ∈ FX we have |[t −1 s]| = 2|t| , and hence FX is weakly cancellative. For each t ∈ PX , we have [tt −1 ] = {u: u ⊂ t}. The canonical partial order on PX is given by st

⇐⇒

t ⊂s

(s, t ∈ PX ).

Take r, t ∈ PX with r t, i.e., t \r = ∅. Take u ∈ rPX . We can write u = r ∪s where s ∩r = ∅. Set u ∪ (t \ u) if t \ u = ∅, v= u \ (s ∩ t) if s ∩ t = ∅. The condition t \ r = ∅ ensures that one of these cases must occur. Then u = v and t ∪ u = t ∪ v. Hence u ∈ G(rPX , t) and G(rPX , t) = rPX . Hence for r1 t1 and r2 t2 we have G(r1 PX , t1 ) ∩ G(r2 PX , t2 ) = r1 r2 PX . Therefore Theorem 5.5 gives no information about the injectivity of the module 1 (PX ). It is proved in [14] that if X is an infinite set, then 1 (PX ) and 1 (FX ) are not right injective Banach algebras. The proof of this result involves a long technical combinatorial calculation and will be presented elsewhere. 6. Projectivity of the predual module c0 (S) Again for a weakly right cancellative semigroup S we now consider when c0 (S) has the stronger property of being projective in 1 (S)-mod. 6.1. Some technical ‘smallness’ results Lemma 6.1. Let S be an infinite, weakly right cancellative semigroup such that, for every finite set F ⊂ S, there exists r ∈ S with rS ∩ F = ∅. Suppose that c0 (S) is projective in 1 (S)-mod c0 (S). Then for each N ∈ N, there exist elements with splitting morphism ρ : c0 (S) → 1 (S ) ⊗ x1 , . . . , xN in c0 (S) and a partition {F1 , . . . , FN } of S with the properties: (i) N i=1 xi ∞ = 1, (ii) ρ(xi )π 1 for each i ∈ NN , and (iii) χFi ρ(xi ) − ρ(xi )π < 1/3i for each i ∈ NN . Proof. We set E = c0 (S), and for a subset T ⊂ S, we set AT = 1 (T ). Fix N ∈ N. To begin, choose r1 , t1 ∈ S with [t1 r1−1 ] = ∅ and set x1 = r1 · λt1 = χ[t1 r −1 ] . 1

E = 1 (r1 S , E), and 1 = x1 ∞ = π ◦ ρ(x1 )∞ ρ(x1 )π . Take a Then ρ(x1 ) ∈ Ar1 S ⊗ finite set F1 ⊂ r1 S with χF1 ρ(x1 ) − ρ(x1 )π < 1/3. Now suppose that x1 , . . . , xk and {F1 , . . . , Fk } are already constructed. Choose rk+1 ∈ S with rk+1 S ∩ ki=1 Fi = ∅. Set G=

k −1 −1 ti ri and H = (srk+1 )rk+1 . i=1

s∈G

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The set H is finite, so we can choose an element u in the complement. Set tk+1 = urk+1

and xk+1 = rk+1 · λtk+1 = χ[tk+1 r −1 ] . k+1

−1 Since u ∈ [tk+1 rk+1 ] we have ρ(xk+1 )π 1. −1 ] is disjoint from G. Assume towards a contradiction We shall show that the set [tk+1 rk+1 −1 −1 ] ⊂ H. that there exists v ∈ [tk+1 rk+1 ] ∩ G. Then vrk+1 = tk+1 = urk+1 , and so u ∈ [(vrk+1 )rk+1 k+1 −1 This is a contradiction, and therefore [tk+1 rk+1 ] ∩ G = ∅, whence i=1 xi ∞ = 1. We have E. Take a finite set Fk+1 ⊂ rk+1 S with χFk+1 ρ(xk+1 ) − ρ(xk+1 )π < ρ(xk+1 ) ∈ Ark+1 S ⊗ 1/3k+1 . −1 In the final stage we may take FN = S \ N i=1 Fi , so that the sets (Fi ) form a partition of S. 2

Theorem 6.2. Let S be an infinite, weakly right cancellative semigroup. Suppose that c0 (S) is projective in 1 (S)-mod. Then there exists a finite set F ⊂ S such that, for each r ∈ S, rS ∩ F = ∅. Proof. We set AS = 1 (S ) and E = c0 (S). Since E is projective in 1 (S)-mod, there exists a left 1 (S)-module morphism ρ : E → E with π ◦ ρ = IE . Assume towards a contradiction that the condition is not satisfied, so AS ⊗ that we can apply Lemma 6.1. Fix N ∈ N, and let x1 , . . . , xN and {F1 , . . . , FN } be the elements corresponding to ρ given by Lemma 6.1. Firstly, for each m ∈ NN we have

1 χF ρ χF ρ(xi ) χS\F ρ(xi ) xi m i m π π 3i π

i=m

and χFm ρ(xm )π 1 −

1 3m .

i=m

i=m

i=m

Hence

N

1 1 1 1 χ − 1 = , ρ x 1 − − 1 − Fm i 3m 3i 1 − 1/3 2 i=1

i=m

π

and so, N N N

N ρ ρ xi = χF1 ρ xi + · · · + χFN ρ xi . 2 i=1

π

i=1

i=1

π

This holds for each N ∈ N, the required contradiction.

π

2

Corollary 6.3. Let S be a weakly right cancellative semigroup. Suppose that c0 (S) is projective in 1 (S)-mod. Then the set


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tt −1 t∈S

is finite. Hence the set E(S) is also finite. Proof. Let F be the finite set given by Theorem 6.2. Take t ∈ S and u ∈ [tt −1 ]. Then either t ∈ F or f = ts for some s ∈ S and f ∈ F . In the latter case we have uf = uts = ts = f , and so u ∈ [ff −1 ]. Hence −1 tt −1 ⊂ ff , f ∈F

t∈S

and so the set t∈S [tt −1 ] is finite. For each p ∈ E(S) we have p ∈ [pp −1 ], hence the set E(S) is finite.

2

6.2. Main result for weakly cancellative semigroups We shall adapt the argument that works for groups [4, Theorem 3.1], to show that if S is a weakly cancellative semigroup and c0 (S) is projective, then S must be finite. The following technical looking lemma is a key to adapting the group argument. It is trivial that every element t in an infinite group has the property given in the lemma. Lemma 6.4. Let S be an infinite, weakly right cancellative semigroup. Suppose that c0 (S) is projective in 1 (S)-mod. Then there exists an element t ∈ S such that, for every finite set F ⊂ S, there exists r ∈ S \ F with [tr −1 ] = ∅. Proof. Assume towards a contradiction that the conclusion is false. Then, for every t ∈ S, there exists a finite set F (t) ⊂ S such that [tr −1 ] = ∅ for all r ∈ S \ F (t). Let F be the finite set given by Theorem 6.2. Take s ∈ S \ F . Then su = f for some f ∈ F and u ∈ S. Since s ∈ [f u−1 ] it must be that u ∈ F (f ), and hence S⊂

f u−1 ∪ F.

f ∈F u∈F (f )

But the set on the right-hand side is finite, and so S is finite. This is a contradiction. Therefore the conclusion holds. 2 Theorem 6.5. Let S be a weakly cancellative semigroup such that c0 (S) is projective in 1 (S)-mod. Then S is finite. c0 (S) be a left 1 (S)-module morphism with π ◦ ρ = Ic0 (S) . Proof. Let ρ : c0 (S) → 1 (S ) ⊗ Assume towards a contradiction that S is infinite. Let t ∈ S be the element specified in Lemma 6.4. Fix N ∈ N, and take a finite set F ⊂ S with χF ρ(λt ) − ρ(λt )π < 1/N . We shall construct elements r1 , . . . , rN ∈ S with the following properties: (i) the sets [tr1−1 ], . . . , [trN−1 ] are pairwise disjoint and non-empty, and (ii) the sets r1 F, . . . , rN F are pairwise disjoint.

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To begin choose any r1 ∈ S with [tr1−1 ] = ∅. Now suppose that r1 , . . . , rk are already constructed. Set k (ri f )g −1 , X(k) =

k −1 Y (k) = tri ,

i=1 f,g∈F

Z(k) =

i=1

u−1 t . u∈Y (k)

Since the sets X(k) and Z(k) are finite, we can use Lemma 6.4 to choose an element rk+1 ∈ −1 ] = ∅. We now show that clauses (i) and (ii) are satisfied. S \ X(k) ∪ Z(k) with [trk+1 Take 1 i < j N . Assume that there exists u ∈ [tri−1 ] ∩ [trj−1 ]. Then urj = t, and so

rj ∈ [u−1 t] for u ∈ [tri−1 ] ⊂ Y (j − 1). Hence rj ∈ Z(j − 1), which is a contradiction, giving clause (i). Next assume that there exists v ∈ ri F ∩ rj F so that ri f = rj g for some f, g ∈ F . But then rj ∈ [(ri f )g −1 ] ⊂ X(j − 1), which is a contradiction, and so clause (ii) holds. For each i ∈ NN , since [tri−1 ] = ∅, we have ri · ρ(λt )π 1 and we have the norm estimate ri · χF ρ(λt ) ri · ρ(λt ) − ri · χF ρ(λt ) − ρ(λt ) 1 − 1/N. π π π Now, we have N N N

ρ ρ ri · λt ri · χF ρ(λt ) − ri · ρ(λt ) − χF ρ(λt ) i=1 i=1 i=1 π π π N N

ri · χF ρ(λt ) − = ri · ρ(λt ) − χF ρ(λt ) π i=1

i=1

π

N (1 − 1/N) − N/N = N − 2. This holds for each N ∈ N, the required contradiction. Therefore S is finite.

2

Let S be a finite inverse semigroup. Then 1 (S) is contractible [5, Theorem 8], and so every E ∈ 1 (S)-mod is projective. Hence we have the following. Theorem 6.6. Let S be a weakly cancellative inverse semigroup. Then c0 (S) is projective in 1 (S)-mod if and only if S is finite. Acknowledgments This work forms part of the authors PhD thesis at the University of Leeds, which was supported by an EPSRC grant. The author wishes to thank his supervisor Professor H.G. Dales for his advice. References [1] Y. Choi, Hochschild homology and cohomology of 1 (Zk ), published online October 2008, by Quart. J. Math. (Oxford) (2008). [2] H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. Ser., vol. 24, Clarendon Press, Oxford, 2000.


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[3] H.G. Dales, A.T.-M. Lau, D. Strauss, Banach algebras on semigroups and on their compactifications, Mem. Amer. Math. Soc. 205 (966) (2010). [4] H.G. Dales, M.E. Polyakov, Homological properties of modules over group algebras, Proc. Lond. Math. Soc. (2) 63 (2004) 390–426. [5] J. Duncan, I. Namioka, Amenability of inverse semigroups and their semigroup algebras, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978) 309–321. [6] Brian E. Forrest, Hun Hee Lee, Ebrahim Samei, Projectivity of modules over Fourier algebras, preprint, see arXiv: 0807.4361. [7] M. Ghandehari, H. Hatami, N. Spronk, Amenability constants for semilattice algebras, Semigroup Forum 79 (2) (2009) 279–297. [8] F. Gourdeau, A. Pourabbas, M.C. White, Simplicial cohomology of some semigroup algebras, Canad. Math. Bull. 50 (1) (2007) 50–70. [9] N. Grønbæk, Amenability of weighted discrete convolution algebras on cancellative semigroups, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988) 351–360. [10] A.Ya. Helemskii, The Homology of Banach and Topological Algebras, Kluwer Academic Publishers, Dordrecht, 1986. [11] A.Ya. Helemskii, Banach and Locally Convex Algebras, Clarendon Press, Oxford, 1993. [12] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). [13] A.L.T. Paterson, Amenability, Math. Surveys Monogr., vol. 29, American Mathematical Society, Providence, RI, 1988. [14] P. Ramsden, Homological properties of semigroup algebras, thesis, University of Leeds, 2009. [15] V. Runde, Lectures on Amenability, Lecture Notes in Math., vol. 1774, Springer Verlag, 2002. [16] Yu.V. Selivanov, Coretraction problems and homological properties of Banach algebras, in: A.Ya. Helemskii (Ed.), Topological Homology, Nova Science, Huntington, New York, 2000, pp. 145–200.


Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents ✩ David Damanik, Zheng Gan ∗ Department of Mathematics, Rice University, Houston, TX 77005, USA Received 9 June 2009; accepted 4 March 2010 Available online 21 March 2010 Communicated by L. Gross

Abstract We investigate the spectral properties of discrete one-dimensional Schrödinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dimension and all spectral measures are purely singular continuous. The associated Lyapunov exponent is a continuous strictly positive function of the energy. It is possible to include a coupling constant in the model and these results then hold for every non-zero value of the coupling constant. © 2010 Elsevier Inc. All rights reserved. Keywords: Limit-periodic Schrödinger operators; Singular continuous spectrum; Lyapunov exponent

1. Introduction This paper is a part of a sequence of papers devoted to the study of spectral properties of discrete one-dimensional limit-periodic Schrödinger operators. The first paper in this sequence [7] contains results in the regime of zero Lyapunov exponents, while the present paper investigates the regime of positive Lyapunnov exponents. Our general aim is to exhibit as rich a spectral picture as possible within this class of operators. In particular, we want to show that all basic ✩

D.D. & Z.G. were supported in part by NSF grant DMS-0800100.


E-mail addresses: [email protected] (D. Damanik), [email protected] (Z. Gan). URLs: http://www.ruf.rice.edu/~dtd3 (D. Damanik), http://math.rice.edu/~zg2 (Z. Gan). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.002

D. Damanik, Z. Gan / Journal of Functional Analysis 258 (2010) 4010–4025

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spectral types are possible and, in addition, in the case of singular continuous spectrum and pure point spectrum, we are interested in examples with positive Lyapunov exponents and examples with zero Lyapunov exponents. From this point of view, the present paper will, to the best of our knowledge for the first time, exhibit limit-periodic Schrödinger operators with purely singular continuous spectrum and positive Lyapunov exponents (whereas [7] had the first examples of limit-periodic Schrödinger operators with purely singular continuous spectrum and zero Lyapunov exponents). Examples with purely absolutely continuous spectrum have been known for a long time, dating back to works of Avron and Simon [2], Chulaevsky [4], and Pastur and Tkachenko [15,16] in the 1980s. These examples (must) have zero Lyapunov exponents. Examples with pure point spectrum (and positive Lyapunov exponents at least at many energies in the spectrum) can be found in Pöschel’s paper [17]; compare also the work of Molchanov and Chulaevsky [13] (who have examples with zero Lyapunov exponents). In the third paper of this sequence we use Pöschel’s general theorem from [17] to construct limit-periodic examples with uniform pure point spectrum within our framework (actually these examples have uniform localization of eigenfunctions); see [8]. Our study is motivated by the recent paper [1], in which Avila disproves a conjecture raised by Simon; see [19, Conjecture 8.7]. That is, he has shown that it is possible to have ergodic potentials with uniformly positive Lyapunov exponents and zero-measure spectrum. The examples constructed by Avila are limit-periodic. In fact, the paper [1] proposes a novel way of looking at limit-periodic potentials. In hindsight, this way is quite natural and provides one with powerful technical tools. Consequently, we feel that a general study of limit-periodic Schrödinger operators may be based on this new approach and we have implemented this in [7,8] and the present paper. We anticipate that further results may be obtained along these lines. It has been understood since the early papers on limit-periodic Schrödinger operators, and more generally almost periodic Schrödinger operators, that these operators belong naturally to the class of ergodic Schrödinger operators, where the potentials are obtained dynamically, that is, by iterating an ergodic map and sampling along the iterates with a real-valued function; see [3,5, 14] for general background. Indeed, taking the closure in ∞ of the set of translates of an almost periodic function on Z (i.e., the hull of the function), one obtains a compact Abelian group with a unique translation invariant probability measure (Haar measure). In particular, the shift on the hull is ergodic with respect to Haar measure and each element of the hull may be obtained by continuous sampling (using the evaluation at the origin, for example). As pointed out by Avila, it is quite natural to take this one step further. That is, once a compact Abelian group and a minimal translation have been fixed, one is certainly not bound to sample along the orbits merely with functions that evaluate a sequence at one point. Rather, every continuous real-valued function on the group is a reasonable sampling function. While this is quite standard in the quasi-periodic case, we are not aware of any systematic use of it in the context of limit-periodic potentials before Avila’s work [1]. The ability to fix the base dynamics and independently vary the sampling functions is very useful in constructing examples of potentials and operators that exhibit a certain desired spectral feature. This has been nicely demonstrated in [1] and is also the guiding principle in our present work. As mentioned above, our main motivation is to find examples of limit-periodic Schrödinger operators with prescribed spectral type. From this point of view, the singular continuity result we prove here is the main result of the paper. However, there was additional motivation to improve the zero measure result of Avila to a zero Hausdorff dimension result. Recent work of Damanik and Gorodetski [9,10] focused on the weakly coupled Fibonacci Hamiltonian. This is an ergodic model that is not (uniformly) almost periodic. Among the results obtained in [9,10], there is a

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theorem that states that the Hausdorff dimension of the spectrum, as a function of the coupling constant, is continuous at zero. That is, as the coupling constant approaches zero, the Hausdorff dimension of the spectrum approaches dimH ([−2, 2]) = 1. When presenting this result, the authors of [9,10] were asked whether this is a universal feature, which holds for all potentials. Thus, our purpose here is to show that there are indeed limit-periodic potentials such that continuity at zero coupling fails in the worst way possible, that is, the Hausdorff dimension of the spectrum is zero for all non-zero values of the coupling constant.1 Let us now describe the models and results in detail. We consider discrete one-dimensional ergodic Schrödinger operators acting in 2 (Z) given by ω Hf,T ψ (n) = ψ(n + 1) + ψ(n − 1) + Vω (n)ψ(n)

(1)

with Vω (n) = f T n (ω) , where ω belongs to a compact space Ω, T : Ω → Ω is a homeomorphism preserving an ergodic Borel probability measure μ and f : Ω → R is a continuous sampling function. It is often benω } eficial to study the operators {Hf,T ω∈Ω as a family, as opposed to a collection of individual ω are always μ-almost surely indepenoperators, since the spectrum and the spectral type of Hf,T dent of ω due to ergodicity. Moreover, if T is in addition minimal (i.e., all T -orbits are dense), ω are independent of ω. then both the spectrum and the absolutely continuous spectrum of Hf,T The Lyapunov exponent is defined as 1 n→∞ n

L(E, T , f ) = lim

(E,T ,f ) logAn (ω) dμ(ω),

(2)

Ω (E,T ,f )

where E ∈ R is called the energy and An

is the n-step transfer matrix of (1) defined as

(E,T ,f ) An (ω) = Sn−1 · · · S0 ,

where Si =

E − f (T i (ω)) 1

−1 . 0

(3)

ω is By the Ishii–Pastur–Kotani theorem, the almost sure absolutely continuous spectrum of Hf,T given by the essential closure of the set of energies where the Lyapunov exponent vanishes. Next we make the spaces and homeomorphisms of especial interest to us explicit.

Definition 1.1. Ω is called a Cantor group if it is an infinite totally disconnected compact Abelian topological group. Definition 1.2. Let Ω be a Cantor group. For ω1 ∈ Ω, let T : Ω → Ω be the translation by ω1 , that is, T (ω) = ω1 · ω. T is called minimal if {T n (ω): n ∈ Z} is dense in Ω for every ω ∈ Ω. 1 Our work was carried out right after the preprint leading to the publication [1] had been released. That version proved zero-measure and did not discuss the Hausdorff dimension issue. After we informed Avila about our results, we learned from him that he had added a remark to the final version of [1] stating that a suitable modification of his proof of zero measure yields zero Hausdorff dimension; see [1, Remark 1.1].


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We will restrict our attention to the case where Ω is a Cantor group and T is a minimal ω have a common spectrum which we will translation. As mentioned above, the operators Hf,T denote by Σ(f ). Here is our main result: Theorem 1.3. Suppose Ω is a Cantor group and T is a minimal translation on Ω. Then there exists a dense set F ⊂ C(Ω, R) such that for every f ∈ F and every λ = 0, the following stateω ments hold true: Σ(λf ) has zero Hausdorff dimension, Hλf,T has purely singular continuous spectrum for every ω ∈ Ω, and E → L(E, T , λf ) is a positive continuous function. The proof of this theorem is based on the constructions in [1]. We make several modifications to these constructions to better control the size of the spectrum and to ensure that the potentials we construct are Gordon potentials. The latter property then implies the absence of point spectrum, which in turn yields singular continuity since the absence of absolutely continuous spectrum already follows from zero measure spectrum. Let us state the Gordon property as a separate result. Definition 1.4. A bounded map V : Z → R is called a Gordon potential if there exist positive integers qi → ∞ such that

max V (n) − V (n ± qi ) i −qi

1nqi

for every i 1. Clearly, if V is a Gordon potential, so is λV for every λ ∈ R. A key part in proving Theorem 1.3 is to establish the following result: Theorem 1.5. Suppose Ω is a Cantor group. Then there exists a dense set F ⊂ C(Ω, R) such that for every f ∈ F , every minimal translation T : Ω → Ω, every ω ∈ Ω, and every λ = 0, λf (T n (ω)) is a Gordon potential. 2. Preliminaries 2.1. Hausdorff measures and dimensions For our relatively restricted purposes, we will simply recall the definition of Hausdorff measures and Hausdorff dimension in this subsection. We refer the reader to [18] for more information. Definition 2.1. Let A ⊆ R be a subset. A countable collection of intervals {bn }∞ n=1 is called a ∞ δ-cover of A if A ⊂ n=1 bn with |bn | < δ for all n’s. (Here, | · | denotes Lebesgue measure, and we will adopt this notation throughout the paper.)

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Definition 2.2. Let α ∈ R. For any subset A ⊆ R, the α-dimensional Hausdorff measure of A is defined as hα (A) = lim inf

∞

δ→0 δ -covers

|bn |α .

(4)

n=1

α The quantity hα (A) is well defined as an element of [0, ∞] since infδ -covers ∞ n=1 |bn | is monotonically increasing as δ decreases to zero and therefore the limit in (4) exists. Restricted to the Borel sets, h1 coincides with Lebesgue measure and h0 is the counting measure. If α < 0, we always have hα (A) = ∞ for any A = ∅, while if α > 1, hα (R) = 0. It is not hard to see that for every A ⊆ R, there is a unique α ∈ [0, 1], called the Hausdorff dimension dimH (A) of A, such that hβ (A) = ∞ for every β < α and hβ (A) = 0 for every β > α. In particular, every A ⊆ R with |A| > 0 must have dimH (A) = 1. 2.2. Minimal translations of Cantor groups and limit-periodic potentials In this subsection we recall how the one-to-one correspondence between hulls of limitperiodic sequences and potential families generated by minimal translations of Cantor groups and continuous sampling functions exhibited by Avila in [1] arises. Definition 2.3. Let S : ∞ (Z) → ∞ (Z) be the shift operator, (SV )(n) = V (n + 1). A two-sided sequence V ∈ ∞ (Z) is called periodic if its S-orbit is finite and it is called limit-periodic if it belongs to the closure of the set of periodic sequences. If V is limit-periodic, the closure of its S-orbit is called the hull and denoted by hullV . The first lemma (see [1, Lemma 2.1]) shows how one can write the elements of the hull of a limit-periodic function in the form Vω (n) = f T n (ω) ,

ω ∈ Ω, n ∈ Z,

(5)

with a minimal translation T of a Cantor group and a sampling function f ∈ C(Ω, R): Lemma 2.4. Suppose V is limit-periodic. Then, Ω := hullV is compact and has a unique topological group structure with identity V such that Z k → S k V ∈ hullV is a homomorphism. Moreover, the group structure is Abelian and there exist arbitrarily small compact open neighborhoods of V in hullV that are finite index subgroups. In particular, Ω = hullV is a Cantor group, T = S|Ω is a minimal translation, and every element of Ω may be written in the form (5) with the continuous function f (ω) = ω(0). The second lemma (see [1, Lemma 2.2]) addresses the converse: Lemma 2.5. Suppose Ω is a Cantor group, T : Ω → Ω is a minimal translation, and f ∈ C(Ω, R). Then, for every ω ∈ Ω, the element Vω of ∞ (Z) defined by (5) is limit-periodic and . we have hullVω = {Vω˜ }ω∈Ω ˜ These two lemmas show that a study of limit-periodic potentials can be carried out by considering potentials of the form (5) with a minimal translation T of a Cantor group Ω and a


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continuous sampling function f . As shown for the first time in the context of limit-periodic potentials by Avila in [1], it is often advantageous to fix Ω and T and to vary f . 2.3. Periodic sampling functions, potentials, and Schrödinger operators In this subsection we discuss the periodic case. For example, which sampling functions f ∈ C(Ω, R) give rise to periodic potentials for some or all (ω, T )? Moreover, what can then be said about the associated Schrödinger operators? Definition 2.6. Suppose Ω is a Cantor group and T : Ω → Ω is a minimal translation. We say that a sampling function f ∈ C(Ω, R) is n-periodic with respect to T if f (T n (ω)) = f (ω) for every ω ∈ Ω. Proposition 2.7. Let f ∈ C(Ω, R). If f (T n0 +m (ω0 )) = f (T m (ω0 )) for some ω0 ∈ Ω, some minimal translation T : Ω → Ω and every m ∈ Z, then for every minimal translation T˜ : Ω → Ω, f is n0 -periodic with respect to T˜ . Proof. Let ϕ : Ω → ∞ (Z), ϕ(ω) = (f (T n (ω)))n∈Z . Since T is minimal, the closure of {T n (ω0 ): n ∈ Z} is Ω. By Lemma 2.5 we have ϕ(Ω) = hull(ϕ(ω0 )). Since f (T n0 +m (ω0 )) = f (T m (ω0 )) for any m ∈ Z, hull(ϕ(ω0 )) is a finite set. Then for any ω ∈ Ω, (f (T n (ω)))n∈Z is some element in hull(ϕ(ω0 )). Since every element in hull(ϕ(ω0 )) is n0 -periodic, (f (T n (ω)))n∈Z is n0 -periodic. This shows that f is n0 -periodic with respect to T . That is, we have f (T n0 +m (ω)) = f (T m (ω)) for every ω ∈ Ω and m ∈ Z. Assume T is the minimal translation by ω1 and let T˜ be another minimal translation by ω2 . n +m By the previous analysis, we have f (ω1 0 · ω) = f (ω1m · ω) for every m ∈ Z and every ω ∈ Ω. q q If ω2 is equal to ω1 for some integer q, obviously we have f (T˜ n0 (ω)) = f ((ω1 )n0 · ω) = f (ω) n for any ω ∈ Ω. If not, since {ω1 : n ∈ Z} is dense in Ω (this follows from the minimality of T ), n we have limk→∞ ω1nk = ω2 , and then f (ω2 0 · ω) = limk→∞ f ((ω1nk )n0 · ω) = f (ω). The result follows. 2 The above proposition tells us that the periodicity of f is independent of T . Thus we may say f is n-periodic without making a minimal translation explicit. Next we recall from [1] how periodic sampling functions in C(Ω, R) can be constructed. Given a Cantor group Ω, a compact subgroup Ω0 with finite index (such subgroups can be found in any neighborhood of the identity element; see above), and f ∈ C(Ω, R), we can define a periodic fΩ0 ∈ C(Ω, R) by fΩ0 (ω) =

f (ω · ω) ˜ dμΩ0 (ω). ˜

Ω0

Here, μΩ0 denotes Haar measure on Ω0 . This shows that the set of periodic sampling functions is dense in C(Ω, R). Moreover, as already noted in[1], there exists a decreasing sequence of Cantor subgroups Ωk with finite index nk such that Ωk = {e}, where e is the identity element of Ω. Let Pk be the set of sampling functions defined on Ω/Ωk , that is, the elements in Pk are nk -periodic potentials. Denote by P the set of all periodic sampling functions. Then, we have Pk ⊂ Pk+1 (which implies nk |nk+1 ) and P = Pk .

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Proposition 2.8. Let f be p-periodic. Then, for every ω ∈ Ω, (E,T ,f ) 1 (ω) logAm m→∞ m (E,T ,f ) 1 = log ρ Ap (e) , p

L(E, T , f ) = lim

(E,T ,f )

(6)

(E,T ,f )

(e)) is the spectral radius of Ap (e). In particular, if restricted to periodic where ρ(Ap sampling functions, the Lyapunov exponent is a continuous function of both the energy E and the sampling function. Proof. If f is p-periodic, as in the proof of Proposition 2.7, for every ω, (f (T n (ω)))n∈Z is some element of the orbit of (f (T n (e)))n∈Z , and so its monodromy matrix (i.e., the transfer matrix over one period) is a cyclic permutation of the monodromy matrix associated with f (T n (e)). (E,T ,f ) (E,T ,f ) (ω) is independent of ω, and since det Ap (ω) = 1, we can conclude that Thus Tr Ap (E,T ,f ) (ω) are independent of ω. So the logarithm of the spectral radius of the eigenvalues of Ap (E,T ,f ) Ap (ω) is independent of ω and (6) follows. The continuity statement follows readily. 2 Lemma 2.9. Let fn ∈ C(Ω, R) be a sequence of periodic sampling functions converging uniformly to f∞ ∈ C(Ω, R). Assume limn→∞ L(E, T , fn ) exists for every E and the convergence is uniform. Then we have that L(E, T , f∞ ) coincides with limn→∞ L(E, T , fn ) everywhere. Proof. Since limn→∞ L(E, T , fn ) exists everywhere, from [1, Lemma 2.5], we have L(E, T , fn ) → L(E, T , f∞ ) in L1loc . So L(E, T , f∞ ) coincides with limn→∞ L(E, T , fn ) almost everywhere. From Proposition 2.8, L(E, T , fn ) is a continuous function, and by uniform convergence, we have that limn→∞ L(E, T , fn ) is also a continuous function. Since L(E, T , f∞ ) is a subharmonic function (cf. [6, Theorem 2.1]), we get that L(E, T , f∞ ) = limn→∞ L(E, T , fn ) for every E. The statement follows. 2 To conclude this subsection on the periodic case, we state two lemmas. The first is well known and the second is [1, Lemma 2.4]. Lemma 2.10. Let f ∈ C(Ω, R) be p-periodic. ω is purely absolutely continuous for every ω ∈ Ω and Σ(f ) is made of (i) The spectrum of Hf,t p bands (compact intervals whose interiors are disjoint). (ii) Σ(f ) = {E ∈ R: L(E, T , f ) = 0}.

Lemma 2.11. Let f ∈ C(Ω, R) be p-periodic. (i) The Lebesgue measure of each band of Σ(f ) is at most 2π p . (ii) Let C 1 be such that for every E ∈ Σ(f ), there exist ω ∈ Ω and k 1 such that (E,T ,f ) Ak (ω) C. Then, |Σ(f )| 4πp C .


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3. Proof of the theorems (E,f )

(E,f,T )

(E,f )

(ω) = An (ω), An = Assume Ω and T are given. For convenience, we write An and L(E, f ) = L(E, T , f ). Since T : Ω → Ω is a minimal translation, the homomorphism Z → Ω, n → T n e is injective with dense image in Ω, and we can write f (n) = f (T n (e)) without any conflicts. We need two more lemmas before proving our theorems. More precisely, we will make further use of the constructions which play central roles in the proof of these two lemmas. (E,f,T ) (e), An

Lemma 3.1. Let B be an open ball in C(Ω, R), let F ⊂ P ∩ B be finite, and let 0 < ε < 1. Then there exists a sequence FK ⊂ P ∩ B such that (i) L(E, λFK ) > 0 whenever ε |λ| ε −1 , E ∈ R. (ii) L(E, λFK ) → L(E, λF ) uniformly on compacts (as functions of (E, λ) ∈ R2 ). This is [1, Lemma 3.1]. As in [1], we use the notation L(E, λF ) =

1 L(E, T , λf ), #F f ∈F

where F is a finite family of sampling functions (with multiplicities!) and λ ∈ R. The proof of this lemma is constructive. We will describe this construction explicitly in the proof of Theorem 1.3 for the reader’s convenience. Lemma 3.2. Suppose B is an open ball in C(Ω, R) and F ⊂ P ∩ B is a finite family of sampling functions. Then for every N 2 and K sufficiently large, there exists FK ⊂ PK ∩ B such that (i) L(E, λFK ) → L(E, λF ) uniformly on compacts (as functions of (E, λ) ∈ R2 ). −N/2 (ii) The diameter of FK is at most nK . This lemma is a variation of [1, Lemma 3.2]. We will prove this lemma using suitable modifications of Avila’s arguments. Some of these modifications, which will later enable us to prove the Gordon property, are not apparent from the statement of the lemma. We will give detailed arguments for the modified parts of the proof and refer the reader to [1] for the parts that are analogous. Proof of Lemma 3.2. Assume that F = {f1 , f2 , . . . , fm } ⊂ C(Ω, R) is a finite family of nk periodic sampling functions with nk 2, and let K > k be large enough. We construct FKt as follows. Let nK = mnk r + d, 0 d mnk − 1. Let Ij = [j nk , (j + 1)nk − 1] ⊂ Z and let 0 = j0 < j1 < · · · < jm−1 < jm = nK /nk be a sequence such that ji+1 − ji = r + 1 when 0 i < d/nk and ji+1 − ji = r when d/nk i m − 1. Define an nK -periodic f as follows. For 0 l nK − 1, let j be such that l ∈ Ij and let i be such that ji−1 j < ji and then let f (l) = fi (l). Next, for any sequence t = (t1 , t2 , . . . , tm ) with ti ∈ {0, 1, . . . , r − 1}, we define an nK -periodic fKt as follows. If j = ji − 1 for some 1 i < m, we let fKt (l) = f (l) + r −N ti , and if j = jm − 2, we let fKt (l) = f (l) + r −N tm . Otherwise we let fKt (l) = f (l). Let FKt be the family consisting of all fKt ’s. The statement (ii) is clear for large K. (Note: in [1], Avila’s

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construction is such that if j = ji − 1 for some 1 i m, then fKt (l) = f (l) + r −20 ti ; otherwise, fKt (l) = f (l).) (E,λf t )

For fixed E and λ, we let AnK K = C (tm ,m) B (m) · · · C (t1 ,1) B (1) , where B (i) = (E,λfi ) ji −ji−1 −1 (E,λf ) (Ank ) , 1 i m − 1 and B (m) = (Ank m )jm −jm−1 −2 , and C (ti ,i) = (E−λr −N t ,λf )

(E,λf )

(E−λr −N t ,λf )

m m i i Ank , 1 i m − 1 and C (tm ,m) = Ank m Ank . When E and λ are (t ,i) i in a compact set, the norm of the C -type matrices is bounded as r grows, while the norm of the B (i) -type matrices may get large. Notice that our perturbation here is r −N t (as opposed to Avila’s r −20 t perturbation in [1, Lemma 3.2 ]), so [1, Claim 3.7] should be replaced by the following version: “Let sj be the most contracted direction of Bˆ (j ) and let uj be the image under Bˆ (j ) of the most expanded direction. Call t j -nice, 1 j d, if the angle between Cˆ (j ) uj and Sj +1 (less than π ) is at least r −3N with the convention that j + 1 = 1 for j = d. Let r be sufficiently large, and let t be j -nice. If z is a non-zero vector making an angle at least r −4N with sj , then z = Cˆ (j ) Bˆ (j ) z makes an angle at least r −4N with Sj +1 and z Bˆ (j ) r −5N z .” The proof of [1, Claim 3.7] can be applied to get the above version of the claim with the corresponding quantitative modification. Moreover, we have also made a little shift in the perturbation,

(E,λf )

(E−λr −N t ,λf )

(E−λr −20 t ,λf )

m m m m , while Avila’s C (tm ,m) = Ank . [1, Claim 3.8] so C (tm ,m) = Ank m Ank still holds, but Avila’s proof of [1, Claim 3.8] cannot be applied directly. To this end we prove the following claim:

Claim 3.3. For any M ∈ SL(2, R), there are m1 , m2 ∈ (0, ∞) with the following property. Suppose A and B are two vectors in R2 , and θ is the angle between A and B with 0 < θ π . Let θ˜ be the angle between MA and MB (again so that 0 < θ˜ π ). Then, m1 θ θ˜ m2 θ. Proof. By the singular value decomposition (see [11, Theorem 2.5.1]), there exist O1 and O2 in SO(2, R) such that M = O1 SO2 , where S is a diagonal matrix. Since O1 and O2 are rotations on R2 , it is sufficient to consider S=

μ1

0

0

μ−1 1

.

Without loss of generality, assume μ1 1. Let A = (a, b)t (t denotes the transpose of vectors) and B = (c, d)t be two normalized vectors, and let θA and θB be the argument of A and the argument of B respectively. Let A˜ = SA = (aμ1 , b/μ1 )t with the argument θA˜ and B˜ = SB = (cμ1 , d/μ1 )t with the argument θB˜ . We adopt the following notation for convenience. Let I , II, III, IV denote one of two vectors in the first quadrant (including {(x, 0): x 0}), the second quadrant (including {(0, y): y > 0}), the third quadrant (including {(x, 0): x < 0}) and the fourth quadrant (including {(0, y): y < 0}), respectively. Then (I, I ) denotes that both two vectors are in the first quadrant, (I, II) denotes that one vector is in the first quadrant while the other is in the second quadrant, and so on. We will need the following observation: 0 < θ1 , θ2 < π/2 and

tan θ1

tan θ2 μ21

⇒

θ1

θ2 . 4μ21

(7)


Indeed, since tan θ1

1 μ21

tan θ2

1 θ 2μ21 2

and 0
δ1 for some 0 < δ1 < 1 whenever ε1 < |λ| < ε11 and E ∈ R (note that L(E, λfi ) 1 if |E| λfi + 4). Our constructions start with F1 and we will divide them into several steps. Construction 1. First, we will apply Lemma 3.1 to F1 in order to enlarge the range 1 ,δ1 } . Then, there exists a finite family of p˜ 1 -periodic potentials F˜1 = of λ’s. Let ε2 = min{ε 10 ˜ ˜ ˜ {f1 , f2 , . . . , fm˜ 1 } ⊂ Bε0 (f ) such that L(E, λF˜1 ) > δ˜1 for some 0 < δ˜1 < 1 whenever ∀ε2 < |λ|
ε24π hp˜ 1 . 4πl (i,l) (i) For 0 l N2 (p˜ 1 ), let f˜ = f˜ + . Then F˜1 is just the family obtained by colε2 p˜ 1 N2 (p˜ 1 )

lecting the f˜(i,l) for different fi ∈ F1 and 0 l N2 (p˜ 1 ). Order F˜1 as F˜1 = {f˜1 , f˜2 , . . . , f˜m˜ 1 } such that f˜1 = f˜(1,0) and f˜m˜ 1 = f˜(1,1) . We can also assume that N2 (p˜ 1 ) was chosen large enough, so that we have f˜m˜ 1 − f˜1 = ε2 p˜14π N2 (p˜ 1 ) < 1/3 (this will be used to conclude that our limit-periodic potentials are Gordon potentials). Construction 2. Applying Lemma 3.2 to F˜1 , there exists a finite family of p2 -periodic poten

tm

tials F2 = {f2t1 , f2t2 , . . . , f2 2 } such that F2 ⊂ Bp−2 ⊂ Bε2 ⊂ Bε0 (f ) 2

and L(E, λF2 ) > δ2 for some 0 < δ2 < 1 whenever ε2 < |λ|
δ˜1 . If r2 large enough, (E,λf˜ )

˜

we have A(r2 −2)i p˜1 > eδ1 (r2 −2)p˜1 . Then we have (E,λf2tk ) tk (E,λf˜i ) δ˜1 (r2 −2)p˜ 1 = A A . (r2 −2)p˜ 1 > e (r2 −2)p˜ 1 f2 (ji−1 p˜ 1 ) Since E is arbitrary, we can apply Lemma 2.11 to conclude that the total Lebesgue measure of

˜

1/2

Σ(λf2tk ) is at most 4πp2 e−δ1 (r2 −2)p˜1 < e−p˜1 p2 when r2 sufficiently large. (Here f2tk can be any element from F2 .) Construction 3. Repeating the above procedures. Once we have constructed Fi−1 ⊂ Bp−(i−1) ⊂ i−1

Bεi−1 , by Lemma 3.1, we can get a finite family of p˜ i−1 -periodic potentials F˜i−1 ⊂ Bp−(i−1) satisfying the following. Let εi =

min{εi−1 ,δi−1 } , 10

i−1

and we have

L(E, λF˜i−1 ) > δ˜i−1 for some 0 < δ˜i−1 < 1 whenever ∀εi < |λ|
δi for some 0 < δi < 1 and all E ∈ R and εi < |λ| < εi−1 . (ii) |L(E, λFi ) − L(E, λFi−1 )| < εi for |E| < ε1i and |λ| < ε1i . (iii) Fi ⊂ Bp−i ⊂ Bεi ⊂ Bεi−1 ⊂ Bε0 (f ), i > 2. (Note: Bε2 may not be in Bε1 .)

i

1/2

(iv) ∀fitk ∈ Fi , |Σ(λfitk )| e−p˜i−1 pi when εi < |λ| < εi−1 (here | · | denotes the Lebesgue measure). (v) pi−i < 13 (i − 1)−p˜i−1 since we can let pi be sufficiently large.

tm

1 −p˜ i−1 . Here N (p˜ (vi) fit1 − fi i = εi p˜i−1 4π 2 i−1 ) appears as in Construction 1, N2 (p˜ i−1 ) < 3 (i − 1) and we can ensure that this inequality holds since p˜ i−1 is fixed while N2 (p˜ i−1 ) can be taken as large as needed.

Then we will get a limit-periodic potential f∞ ∈ Bε0 (f ), whose Lyapunov exponent is a positive continuous function of energy E and the Lebesgue measure of the spectrum is zero (Lemma 2.9 implies that L(E, λfit) → L(E, λf∞ )). Moreover, we have the following two claims. Claim 3.5. f∞ is a Gordon potential.


4023

Proof. Let qi = p˜ i . Obviously, qi → ∞ as i → ∞. For i 1, we have

t1 t1 max f∞ (n) − f∞ (n ± qi ) f∞ (n) − fi+1 (n) + f∞ (n ± qi ) − fi+1 (n ± qi )

1nqi

t1

t1 + fi+1 (n) − fi+1 (n ± qi )

−(i+1)

pi+1

−(i+1)

+ pi+1

+

4π εi+1 p˜ i N2 (p˜ i )

1 1 2 (i)−p˜i + (i)−p˜i 3 3 i −qi .

t1 is an element of Fi+1 .) So f∞ is a Gordon potential. (Here fi+1

2

Claim 3.6. Σ(λf∞ ) has zero Hausdorff dimension for every λ = 0. Proof. Let λ = 0 and 0 < α 1 be given. Without loss of generality, assume λ > 0. Choose i large enough so that εi < λ < 1/εi and 1/i < α. For every fitk ∈ Fi , λf∞ − λfitk < λpi−i

implies2 dist(Σ(λf∞ ), Σ(λfitk )) < λpi−i . Since λfitk is pi -periodic, we have i t Σ λfi k = I˜z(tk ,i) ,

p

z=1

where I˜z(tk ,i) = [az , bz ] is a closed interval. (t ,i) t Let Iz k = [az − λpi−i , bz + λpi−i ] and since dist(Σ(λf∞ ), Σ(λfi k )) λpi−i , we have Σ(λf∞ ) ⊂

pi

Iz(tk ,i) .

z=1 1/2

Moreover, bz − az e−p˜i−1 pi

t

1/2

since |Σ(λfi k )| e−p˜i−1 pi . Then we have

pi −p˜ p1/2 α h Σ(λf∞ ) lim e i−1 i + 2λpi−i α

i→∞

z

1/2 α = lim pi e−p˜i−1 pi + 2λpi−i

i→∞

1/2 1/α −i+1/α α = lim pi e−p˜i−1 pi + 2λpi .

i→∞

2 It is well known that for V , W : Z → R bounded, we have dist(σ ( + V ), σ ( + W )) V − W , where ∞ dist(A, B) denotes the Hausdorff distance of two compact subsets A, B of R.

4024


Since 1/i < α, we have −i + 1/α < 0, and it follows that 1/2 1/α −i+1/α α lim pi e−pi−1 pi + 2λpi = 0.

i→∞

So we have hα (Σ(λf∞ )) = 0 (note: when i → ∞, λ belongs to (εi , ε1i ) for all i large enough since this interval is expanding). So the Hausdorff dimension of the spectrum is less than α. Since α was arbitrary, the Hausdorff dimension must be zero. 2 This implies all the assertions in Theorem 1.3 except for the absence of eigenvalues for every ω. Given the Gordon lemma (see Lemma 3.4 above), this last statement will follow once Theorem 1.5 is established. 2 Remark 3.7. Since δi δi−1 /10, i 1, it is true that when εi < |λ| < ε1i , L(E, λf∞ ) 89 δi for any E ∈ R. This gives information about the range of the Lyapunov exponent on certain intervals. Clearly, δi → 0 when i → ∞ since the Lyapunov exponent will go to zero when λ goes to zero. Proof of Theorem 1.5. Let ω = e first. Relative to any minimal translation T˜ , the selected f in the proof of Theorem 1.3 is still n0 -periodic by Proposition 2.7, so we can start with the same ball Bε0 (f ) and choose the same periodic potentials in Bε0 (f ). Then we get the same f∞ . For the finite family Fi from Construction 3, though the Lyapunov exponent may change, the following properties hold (note that fit1 does not change). (i) Fi ⊂ Bp−i ⊂ Bεi ⊂ Bε0 (f ). i

(ii) pi−i < 13 (i − 1)−p˜i−1 .

tm

(iii) fit1 − fi i =

4π εi p˜ i−1 N2 (p˜ i−1 )

< 13 (i − 1)−p˜i−1 .

˜ if we repeat Then Claim 3.5 holds true, and so f (T˜ n (e)) is a Gordon potential. For arbitrary ω, the same procedures, (i)–(iii) above still hold as stated (since none of them are related to ω), and Theorem 1.5 follows. 2 References [1] A. Avila, On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators, Comm. Math. Phys. 288 (2009) 907–918. [2] J. Avron, B. Simon, Almost periodic Schrödinger operators. I. Limit periodic potentials, Comm. Math. Phys. 82 (1981) 101–120. [3] R. Carmona, J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990. [4] V. Chulaevsky, Perturbations of a Schrödinger operator with periodic potential, Uspekhi Mat. Nauk 36 (1981) 203– 204. [5] V. Chulaevsky, Almost Periodic Operators and Related Nonlinear Integrable Systems, Manchester University Press, Manchester, 1989. [6] W. Craig, B. Simon, Subharmonicity of the Lyapunov index, Duke Math. J. 50 (1983) 551–560. [7] D. Damanik, Z. Gan, Spectral properties of limit-periodic Schrödinger operators, Discrete Contin. Dyn. Syst. Ser. S, in press. [8] D. Damanik, Z. Gan, Limit-periodic Schrödinger operators with uniformly localized eigenfunctions, preprint, arXiv:1003.1695. [9] D. Damanik, A. Gorodetski, The spectrum of the weakly coupled Fibonacci Hamiltonian, Electron. Res. Announc. Math. Sci. 16 (2009) 23–29.


4025

[10] D. Damanik, A. Gorodetski, Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian, preprint, arXiv:1001.2552. [11] G. Golub, C. van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, 1989. [12] A. Gordon, On the point spectrum of the one-dimensional Schrödinger operator, Uspekhi Mat. Nauk 31 (1976) 257–258. [13] S. Molchanov, V. Chulaevsky, The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator, Funct. Anal. Appl. 18 (1984) 343–344. [14] L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer-Verlag, Berlin, 1992. [15] L. Pastur, V. Tkachenko, On the spectral theory of the one-dimensional Schrödinger operator with limit-periodic potential, Soviet Math. Dokl. 30 (1984) 773–776. [16] L. Pastur, V. Tkachenko, Spectral theory of a class of one-dimensional Schrödinger operators with limit-periodic potentials, Trans. Moscow Math. Soc. 51 (1984) 115–166. [17] J. Pöschel, Examples of discrete Schrödinger operators with pure point spectrum, Comm. Math. Phys. 88 (1983) 447–463. [18] C. Rogers, Hausdorff Measure, Cambridge University Press, London, 1970. [19] B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging 1 (2007) 713–772.


Symmetrization of Lévy processes and applications Rodrigo Bañuelos a,∗,1 , Pedro J. Méndez-Hernández b,2 a Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States b Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica

Received 29 June 2009; accepted 26 February 2010

Communicated by Daniel W. Stroock

Abstract It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Lévy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabilistic structure of the processes plays a crucial role in the proofs. © 2010 Elsevier Inc. All rights reserved. Keywords: Symmetrization; Lévy processes

1. Introduction Let D be an open connected set in Rd of finite Lebesgue measure. Henceforth we shall refer to such sets simply as domains. We will denote by D ∗ the open ball in Rd centered at the origin 0 with the same Lebesgue measure as D, and |D| will denote the Lebesgue measure of D. There is a large class of quantities which are related to Brownian motion killed upon leaving D that are maximized, or minimized, by the corresponding quantities for D ∗ . Such results often go by the name of generalized isoperimetric inequalities. They include the celebrated Rayleigh–Faber– Krahn inequality on the first eigenvalue of the Dirichlet Laplacian, inequalities for transition densities (heat kernels), Green functions, and electrostatic capacities (see [4,15–18]). * Corresponding author.

E-mail addresses: [email protected] (R. Bañuelos), [email protected] (P.J. Méndez-Hernández). 1 Supported in part by NSF Grant # 0603701-DMS. 2 Supported in part by project No. 821-A7-177 of Centro de Investigación en Matemática Pura y Aplicada (CIMPA).


R. Bañuelos, P.J. Méndez-Hernández / Journal of Functional Analysis 258 (2010) 4026–4051

4027

Many of these isoperimetric inequalities can be beautifully formulated in terms of exit times of the Brownian motion Bt from the domain D. For example, if τD is the first exit time of Bt from D, then for all x ∈ D P x {τD > 0} P 0 {τD ∗ > 0},

(1.1)

where 0 is the origin of Rd . Inequality (1.1) contains not only the classical Rayleigh–Faber– Krahn inequality but inequalities for heat kernels and Green functions as well. This inequality is now classical and can be found in many places in the literature. For one of its first occurrences, using the Brascamp–Lieb–Luttinger multiple integrals techniques, please see Aizenman and Simon [1]. Similar inequalities can be obtained by these methods for domains of fixed inradius rather than fixed volume. For more on this, we refer the reader to [5] and [12]. Also, versions of some of these results hold for Brownian motion on spheres and hyperbolic spaces, see [8] and references therein. Once these isoperimetric-type inequalities are formulated in terms of exit times of Brownian motion, it is completely natural to enquire as to their validity for other stochastic processes, and particularly for more general Lévy processes whose generators, as pseudo differential operators, are natural extensions of the Laplacian. Such extensions have been obtained in recent years for the so-called “symmetric stable processes” in Rd and for more general processes obtained from subordination of Brownian motion. We refer the reader to [5,6,12,21]. The purpose of this paper is to show that many of these results continue to hold for very general Lévy processes. At the heart of these extensions are the rearrangement inequalities of Brascamp, Lieb and Luttinger [7]. However, the probabilistic structure of Lévy processes enters in a very crucial way. Of particular importance for our method is the fact, derived from the Lévy– Khintchine formula, that our processes are weak limits of sums of a compound Poisson process and a Gaussian process. We begin with a general description of Lévy processes. A Lévy process Xt in Rd is a stochastic process with independent and stationary increments which is “stochastically” continuous. That is, for all 0 < s < t < ∞, A ⊂ Rd , P x {Xt − Xs ∈ A} = P 0 {Xt−s ∈ A}, for any given sequence of ordered times 0 < t1 < t2 < · · · < tm < ∞, the random variables Xt1 − X0 , Xt2 − Xt1 , . . . , Xtm − Xtm−1 are independent, and for all ε > 0, lim P x |Xt − Xs | > ε = 0.

t→s

The celebrated Lévy–Khintchine formula [20] guarantees the existence of a triple (b, A, ν) such that the characteristic function of the process is given by E x eiξ ·Xt = e−tΨ (ξ )+iξ ·x , where 1 Ψ (ξ ) = −ib, ξ + A · ξ, ξ + 2

Rd

1 + iξ, yIB − eiξ ·y dν(y).

(1.2)

4028


Here, b ∈ Rd , A is a nonnegative d × d symmetric matrix, IB is the indicator function of the ball B centered at the origin of radius 1, and ν is a measure on Rd such that |y|2 dν(y) < ∞ and ν {0} = 0. (1.3) 2 1 + |y| Rd

The triple (b, A, ν) is called the characteristic of the process and the measure ν is called the Lévy measure of the process. Conversely, given a triple (b, A, ν) with such properties there is Lévy processes corresponding to it. We will use the fact that any Lévy process has a version with paths that are right continuous with left limits, so-called “càdlàg” paths. Next we recall the basic facts on symmetrization needed to state our results; more details on the properties of symmetrization used in this paper can be found in Section 4. Given a positive measurable function f , its symmetric decreasing rearrangement f ∗ is the unique function satisfying f ∗ (x) = f ∗ (y),

if |x| = |y|,

f ∗ (x) f ∗ (y),

if |x| |y|,

lim

|x|→|y|+

∗

f (x) = f ∗ (y),

and m{f > t} = m f ∗ > t ,

(1.4)

for all t 0. Following [14], under the assumption that f vanishes at infinity, an explicit expression for this function is ∗

∞

f (x) =

∗ χ{|f |>t} (x) dt.

0

For symmetrization purposes, in this paper we will only consider Lévy measures ν that are absolutely continuous with respect to the Lebesgue measure m. It may be that some of the results in this paper hold for more general Lévy processes but at this stage we are not able to go beyond the absolute continuity case. Let φ be the density of ν and φ ∗ be its symmetric decreasing rearrangement. Since the function ψ(y) = 1 −

|y|2 1 + |y|2

is positive, decreasing and radially symmetric, that is, ψ ∗ = ψ , it follows that (see Theorem 3.4 in [14]) Rd

|y|2 φ ∗ (y) dy 1 + |y|2

Rd

|y|2 φ(y) dy < ∞. 1 + |y|2

Hence the measure φ ∗ (y) dy satisfies (1.3) and it is also a Lévy measure.

(1.5)


4029

We denote the d × d identity matrix by Id and the determinant of A by det A. Set A∗ = (det A)1/d Id and define Xt∗ to be the rotationally invariant Lévy process in Rd associated to the triple (0, A∗ , φ ∗ (y) dy). We will often refer to Xt∗ as the symmetrization of Xt . Notice that ∗ ∗ E x eiξ ·Xt = e−tΨ (ξ )+iξ ·x ,

(1.6)

where

1 Ψ (ξ ) = A∗ · ξ, ξ + 2 ∗

1 − eiξ ·y φ ∗ (y) dy

Rd

=

1 ∗ A · ξ, ξ + 2

1 − cos(ξ · y) φ ∗ (y) dy,

Rd

where the last inequality follows from the fact that φ ∗ is symmetric and y → sin(ξ · y) is antisymmetric. The next two theorems are the main results of this paper. Theorem 1.1. Suppose Xt is a Lévy process with Lévy measure absolutely continuous with respect to the Lebesgue measure and let Xt∗ be the symmetrization of Xt constructed as above. Let f1 , . . . , fm be nonnegative lower semicontinuous functions. Then for all z ∈ Rd , E

z

m

fi (Xti ) E

i=1

0

m

fi∗

∗ X ti ,

(1.7)

i=1

for all 0 t1 · · · tm . One easily proves that this result is not valid when the functions f1 , . . . , fm are not lower semicontinuous. For example, applying the results with compound Poisson processes, it would follow that for all nonnegative bounded measurable functions f , f (z) f ∗ (0) = f L∞ for all z ∈ Rd . This inequality is true for lower semicontinuous functions but not for measurable functions simply by changing the function on a set of measure zero. However, if we assume further that the distributions of Xt and Xt∗ are absolutely continuous with respect to the Lebesgue measure, we can extend Theorem 1.1 to measurable functions. Theorem 1.2. Suppose Xt is a Lévy process with Lévy measure absolutely continuous with respect to the Lebesgue measure and let Xt∗ be the symmetrization of Xt as constructed above. Assume further that for all t > 0 the distributions of Xt and Xt∗ are absolutely continuous with respect to the Lebesgue measure. That is, for all t > 0, P x {Xt ∈ A} =

p(t, x, y) dy A

and

4030


P Xt∗ ∈ A =

x

p ∗ (t, x, y) dy,

A

for any Borel set A ⊂ Rd . Let f1 , . . . , fm , m 1, be nonnegative measurable functions. Then for all z ∈ Rd , E

z

m

fi (Xti ) E

i=1

0

m

fi∗

∗ X ti ,

i=1

for all 0 t1 · · · tm . Remark 1.3. A sufficient condition for the absolute continuity of the law of a Lévy process is given in [20, page 177]. In our case this is satisfied by both Xt and Xt∗ whenever det(A) > 0 or φ∈ / L1 (Rd ). As we shall see below, Theorem 1.1 together with the now standard Brownian motion argument of Aizenman and Simon [1] implies a generalization of (1.1) to Lévy processes whose Lévy measure is absolutely continuous with respect to the Lebesgue measure. In fact, we will obtain a more general result which applies to Schrödinger perturbations of Lévy semigroups. Let D ⊂ Rd be a domain of finite measure, and consider τDX = inf{t > 0: Xt ∈ / D}, ∗

the first exit time of Xt from D. We also have the corresponding quantity τDX∗ for Xt∗ in D ∗ . As explained in Section 5, the following isoperimetric-type inequality is a consequence of Theorem 1.1. Theorem 1.4. Let D be a domain in Rd of finite measure and f and V be nonnegative continuous functions. Suppose Xt is a Lévy process with Lévy measure absolutely continuous with respect to the Lebesgue measure and Xt∗ is the symmetrization of Xt . Then for all z ∈ Rd and all t > 0, E

z

t X f (Xt ) exp − V (Xs ) ds ; τD > t 0

E

0

t ∗ ∗ ∗ X∗ f Xt exp − V Xs ds ; τD ∗ > t . ∗

(1.8)

0

Our symmetrization results are based on the following now classical rearrangement inequality of Brascamp, Lieb and Luttinger [7]. Theorem 1.5. Let f1 , . . . , fm be nonnegative functions in Rd and denote by f1∗ , . . . , fm∗ their symmetric decreasing rearrangements. Then


··· Rd

m

Rd

Rd

fj

k

j =1

···

bj i xi dx1 · · · dxk

i=1

m

Rd

4031

j =1

fj∗

k

bj i xi dx1 · · · dxk ,

i=1

for all positive integers k, m, and any m × k matrix B = [bj i ]. In Section 5, we also show that Theorem 1.2 implies several isoperimetric inequalities for the transition density of the process Xt . There is a similar result for capacities of Lévy symmetric processes. Let CX (A) be the capacity, associated to Xt , of the set A. If Xt is a symmetric processes, that is, P (Xt ∈ A) = P (Xt ∈ −A), for all t > 0, then T. Watanabe [23] proved that, CX (A) CX∗ A∗ .

(1.9)

This inequality can be obtained, from the methods used in this paper only in the case that Xt is an isotropic unimodal Lévy process, see for example [13]. As explained in [5] and [12], in the case that the process Xt is isotropic unimodal, Theorem 1.1 is an immediate consequence of Theorem 1.5. Recall that Xt is isotropic unimodal if it has transition densities p(t, x, y) of the form p(t, x, y) = qt |x − y| ,

(1.10)

where qt is a function such that qt (r1 ) qt (r2 ), for all r1 r2 and all t > 0. Thus for such Lévy processes (with y fixed)

∗ p(t, ·, y) = p(t, ·, 0),

and Xt = Xt∗ . This class of Lévy processes includes the Brownian motion, rotational invariant symmetric α-stable processes, relativistic stable processes and any other subordinations of the Brownian motion. Notice that in our more general setting, and under the assumption that the distribution of Xt is absolutely continuous relative to the Lebesgue measure, we cannot even ensure that [p(t, ·, y)]∗ is the transition density of a Lévy processes. The rest of the paper is organized as follows. In Section 2 we will prove Theorem 1.1 for Compound Poisson processes. We will consider the case of Gaussian Lévy processes in Section 3. Theorem 1.1 and Theorem 1.2 are proved in Section 4, using a weak approximation of Xt and Xt∗ by Lévy processes of the form Gt + Ct , where Gt is a nondegenerate Gaussian process and Ct is an independent compound process. We will then show some of the applications in Section 5.

4032


2. Symmetrization of compound Poisson processes In this section we prove a version of the inequality (1.7) for compound Poisson processes. This result, combined with the results in Section 3, will lead to a proof of Theorem 1.1. We start by recalling the structure of compound Poisson processes in terms of random walks. If Ct is a compound Poisson process, starting at x, then its characteristic function is given by E x eiξ ·Ct = eix·ξ −tΨC (ξ ) ,

(2.1)

where ΨC (ξ ) = c

1 − eiξ ·y φ(y) dy,

Rd

and φ is a probability density. We now use the fact that Ct can be written in terms of sums of independent random variables. That is, by Theorem 4.3 [20] there exist a Poisson process Nt with parameter c > 0, and a sequence of i.i.d. random variables {Xn }∞ n=1 such that (1) {Nt }t>0 and {Xn }∞ n=1 are independent, (2) φ(y) is the density of the distribution of Xi , i 1, (3) Ct = SNt + x, where Sn = X1 + · · · + Xn and S0 = 0. Hence if f is a nonnegative Borel function, then E x f (Ct ) = E x f (SNt ) =

∞

P [Nt = n]E f (x + Sn ) .

(2.2)

n=0

Let φ ∗ be the symmetric decreasing rearrangement of φ. Since Rd

φ ∗ (y) dy =

φ(y) dy = 1, Rd

we can consider a new sequence of i.i.d. random variables {Xn∗ }∞ n=1 independent of Nt such that φ ∗ (y) is the density of Xn∗ . Define Sn∗ = X1∗ + · · · + Xn∗ to be the corresponding random walk and Ct∗ the compound Poisson process given by ∗ Ct∗ = SN . t

Notice that the distribution μt of Ct is not absolutely continuous with respect to Lebesgue measure. However, if C0 = x we have the following representation μt = P [Nt = 0]δx +

∞ k=1

P [Nt = k]μk (x),

(2.3)


4033

with μk the distribution of Sk . That is,

E f (Sk ) =

f (x + y) dμk (y)

x

Rd

=

···

Rd

f

k

xj

j =0

Rd

k

φ(xi ) dx1 · · · dxk .

i=1

Thus if f is a bounded measurable function we have that f ∗ S0∗ = f ∗ (0) = f L∞ and the inequality f (S0 + x) = f (x) f ∗ S0∗ can only be asserted to hold almost everywhere. The next result is a version of inequality (1.7) for random walks where the functions are only assumed to be measurable but the conclusion is only a.e. with respect to the Lebesgue measure. We label it as “Theorem” because it may be of some independent interest. Theorem 2.1. Let f1 , . . . , fm nonnegative functions and k1 · · · km nonnegative integers. Then

m m ∗ ∗ (2.4) fi (x0 + Ski ) E fi Ski , E i=1

i=1

almost everywhere in x0 , with respect to Lebesgue measure. In the case that f1 , . . . , fm are continuous, (2.4) holds pointwise. Proof. Given that X1 , . . . , Xkm are i.i.d we can apply Theorem 1.5 to obtain that E

m

fi (x0 + Ski ) = E

i=1

m

fi (x0 + X1 + · · · + Xki )

i=1

=

··· Rd

Rd

··· Rd

m

fi

fi∗

i=1

m ∗ ∗ =E fi Ski .

i=1

ki

xj

j =0

i=1

m

Rd

ki

km

φ(xi ) dx1 · · · dxm

i=1

xj

j =1

km

φ ∗ (xi ) dx1 · · · dxm

i=1

2

(2.5)

4034


We can now prove the inequality (1.7) for the compound Poisson process Ct . Let f1 , . . . , fm be nonnegative continuous functions. Since Nt is independent of Sk and Sk∗ , we can combine (2.2) and Theorem 2.1 to obtain

m

m ∞ x fi (SNti ) = P [Nt1 = k1 , . . . , Ntm = km ]E fi (x + Ski ) E k1 k2 ···km

i=1

∞

P [Nt1 = k1 , . . . , Ntm = km ]E

k1 k2 ···km

= E0

m

∗ fi∗ SN t

i

i=1 m

fi∗

∗ Ski

i=1

(2.6)

.

i=1

Thus E

x

m

fi (Cti ) E

0

i=1

m

fi∗

∗ Cti ,

(2.7)

i=1

which is desired result. 3. Symmetrization of Gaussian processes Let Gt be a nondegenerate Gaussian process. Then there exist b ∈ Rd and a strictly positive definite symmetric d × d matrix A such that the density of Gt is given by fA,b (t, x) =

1 exp − (x − tb), A−1 · (x − tb) , √ 2t [2tπ]d/2 det A 1

for all x ∈ Rd and all t > 0. Let us first assume that b = 0. Let u > 0, then −1/2

d d −1/2 · x, A · x < t ln x ∈ R : fA,0 (t, x) > u = x ∈ R : A

1 d (2tπ) u2 det A

A change of variables implies that m x ∈ Rd : fA,0 (t, x) > u =

1 m B(rA,d,u,t ) , 1/2 [det A]

where rA,d,u,t = t ln

1 . (2tπ)d u2 det A

Consider the diagonal matrix 1

A∗ = (det A) d Id .

.


4035

Then m x ∈ Rd : fA,0 (t, x) > u = m x ∈ Rd : fA∗ ,0 (t, x) > u , for all u > 0. Given that fA∗ ,0 (t, x) is rotational invariant and radially decreasing, we conclude that ∗ ∗ fA,b (t, x) = fA,0 (t, x − tb) = fA∗ ,0 (t, x).

(3.1)

If Gt is a degenerate Gaussian process, then t E eiξ ·Gt = exp itb · ξ − i A · ξ, ξ , 2

(3.2)

where A is a positive definite d × d matrix such that det A = 0. Let {v1 , . . . , vd } be the orthonormal eigenvectors of A with eigenvalues λ1 , . . . , λd . We can assume that {λ1 , . . . , λk }, 1 k < d, are the nonzero eigenvalues of A. Let W be the subspace spanned by v1 , . . . , vk . Then Gt can be identified with a nondegenerate Gaussian process in the lower dimension space W and P z [Gt ∈ D] = P z Gt ∈ PW (D) , where PW (D) is the projection of D on the space W . Define A∗ to be the symmetric positive defined matrix with eigenvectors v1 , . . . , vd such that A∗ vi = 0,

k < i d,

and A∗ vi = λvi ,

1 i k,

where λ = (λ1 · · · λk )1/k . The arguments of this section imply that P z [Gt ∈ D] = P z Gt ∈ PW (D) ∗ = P 0 G∗t ∈ DW , ∗ is the ball in W , centered at the origin, with the same k-dimension measure as P (D). where DW W Hence the corresponding symmetrization for this processes should be done in lower dimensions.

4036


4. Symmetrization of Lévy processes: proof of Theorem 1.1 We will now consider general Lévy processes whose Lévy measures are absolutely continuous with respect to the Lebesgue measure. Our proof uses the fact that the symmetrization of functions preserves order, continuity and almost everywhere convergence. For the convenience of the reader, and for completeness, we briefly describe how to prove these properties of the symmetrization. We recall once again that the symmetric decreasing rearrangement f ∗ of f is the function satisfying f ∗ (x) = f ∗ (y),

if |x| = |y|,

f ∗ (x) f ∗ (y),

if |x| |y|,

lim

∗

|x|→|y|+

f (x) = f ∗ (y),

and m{f > t} = m f ∗ > t ,

(4.1)

m{f > t} = m B 0, r(f, t) .

(4.2)

for all t 0. Define r(f, t) as

Whenever f is a radially symmetric nonincreasing function such that f is right continuous at |x0 |, we have that r f, f (x0 ) = sup r > 0: f (r) > f (x0 ) = |x0 |.

(4.3)

Using this properties of r(f, t), one easily proves the following result, see page 81 of [14]. Lemma 4.1. Let f be a nonnegative function. If f is continuous, then f ∗ is continuous. In addition, if g is a nonnegative function such that g(x) f (x) almost everywhere with respect to the Lebesgue measure, then g ∗ (x) f ∗ (x),

(4.4)

for all x ∈ R. We will also need the following result on almost everywhere convergence and symmetrization. Lemma 4.2. Let {φn }∞ n=1 be a sequence of bounded functions such that lim φn = φ,

n→∞

almost everywhere with respect to the Lebesgue measure. If x0 is a point of continuity of φ ∗ then


4037

lim φ ∗ (x0 ) = φ ∗ (x0 ). n→∞ n In particular lim φ ∗ n→∞ n

= φ∗,

(4.5)

almost everywhere with respect to the Lebesgue measure. Proof. Assume there exists x0 a continuity point of φ ∗ such that lim φ ∗ (x0 ) = φ ∗ (x0 ). n→∞ n Then there exist > 0 and a subsequence nk such that either φn∗k (x0 ) > φ ∗ (x0 ) + ,

(4.6)

φn∗k (x0 ) < φ ∗ (x0 ) − .

(4.7)

or

Let us assume that (4.6) holds. Since x0 is a continuity point of φ ∗ , there exist 0 < δ < and y0 a continuity point of φ ∗ such that φ ∗ (x0 ) + δ = φ ∗ (y0 ),

and |y0 | < |x0 |.

However, thanks to (4.1) and (4.3), m B 0, |x0 | = lim sup m φn∗k > φn∗k (x0 ) nk →∞

lim sup m φn∗k > φ ∗ (x0 ) + δ nk →∞

= m φ ∗ > φ ∗ (y0 ) = m B 0, |y0 | , which is a contradiction. A similar argument shows that φn∗k (x0 ) < φ ∗ (x0 ) − , yields a contradiction.

2

We recall that under our assumptions E x eiξ ·Xt = e−tΨ (ξ )+iξ ·x , where

4038


1 Ψ (ξ ) = −ib, ξ + A · ξ, ξ + 2

1 + iξ, yIB − eiξ ·y φ(y) dy,

Rd

B is the unit ball centered at the origin and φ is such that Rd

|y|2 φ(y) dy < ∞. 1 + |y|2

(4.8)

Consider the sequence φn (y) = φ(y)I{t∈R:

1 n 0: Xt ∈ is the first exit time of Xt from a domain D. Let Dk be a sequence of bounded domains with smooth boundaries such that Dk ⊂ Dk+1 , and ∞ k=1 Dk = D. Since any Lévy process has a version with right continuous paths, we have E

z0

t X f (Xt ) exp − V (Xs ) ds ; τD > t 0

=E

z0

t f (Xt ) exp − V (Xs ) ds ; Xs ∈ D, ∀s ∈ [0, t] 0

m t = lim lim E z0 f (Xt ) exp − V (X it ) ; X it ∈ Dk , i = 1, . . . , m m→∞ k→∞ m m m i=1 m t z0 = lim lim E f (Xt ) exp − V (X it ) IDk (X it ) . m→∞ k→∞ m m m

(5.1)

i=1

Since ∗ exp −sV (x) = exp −sV ∗ (x) , for all s > 0 and all x ∈ Rd , and the functions IDk are lower semicontinuous, Theorem 1.1 implies that m t E z0 f (Xt ) exp − V (X it ) IDk (X it ) m m m i=1 m ∗ ∗ t ∗ ∗ 0 ∗ E f Xt exp − V X it IDk∗ X it . m m m i=1

4044


Hence we have the following Ez

t f (Xt ) exp − V (Xs ) ds ; τDX > t 0

E

0

t ∗ ∗ ∗ X∗ f Xt exp − V Xs ds ; τD ∗ > t , ∗

(5.2)

0

which is Theorem 1.4. Taking V = 0 and f = 1, gives ∗ P z τDX > t P 0 τDX∗ > t ,

(5.3)

which is a generalization of inequality (1.1). Integrating this inequality with respect to t gives the following result. Corollary 5.1. If ψ is a nonnegative increasing function, then ∗ E z ψ τDX E 0 ψ τDX∗ ,

(5.4)

p ∗ p E z τDX E 0 τDX∗ ,

(5.5)

for all z ∈ D. In particular

for all 0 < p < ∞. Our results imply many isoperimetric inequalities for the potentials and the eigenvalues of Schrödinger operators of the form X HD,V = HDX + V ,

where HDX is the pseudo differential operator associated to Xt with Dirichlet boundary conditions on D. For the convenience of the reader we will give a brief description of the operators and semigroups associated to Lévy processes. For purposes of our formulae below we define the Fourier transform of an L2 (Rd ) function as f(ξ ) =

1 (2π)d/2

e−ix·ξ f (x) dx,

Rd

with 1 f (x) = (2π)d/2

eix·ξ f(ξ ) dξ.

Rd

We define the semigroup associated to the Lévy process Xt by


Tt f (x) = E x f (Xt ) = =

1 E0 (2π)d/2

1 (2π)d/2

ei(Xt +x)·ξ f(ξ ) dξ

4045

Rd

eix·ξ E 0 eiXt ·ξ f(ξ ) dξ

Rd

=

1 (2π)d/2

eix·ξ e−tΨ (ξ ) f(ξ ) dξ.

Rd

This semigroup takes C0 (Rd ) into itself. That is, it is a Feller semigroup. From this we see that, at least formally for f ∈ S(Rd ), the infinitesimal generator is H X f (x) = −

∂Tt f (x) 1 = eix·ξ Ψ (ξ )fˆ(ξ ) dξ. ∂t t=0 (2π)d/2 Rd

Then the Lévy–Khintchine formula implies that the operator associated to Xt is given by H X f (x) =

d

bj ∂j f (x) −

j =1

+

d 1 aj k ∂j ∂k f (x) 2 j,k=1

f (x + y) − f (x) − y · ∇f (x)I{|y| t ,

(5.7)

0

defined for t > 0, z ∈ D, and f ∈ L2 (D). Recall our assumption that V is nonnegative and continuous. Thus, t z D,V X Tt f (z) = E f (Xt ) exp − V (Xs ) ds ; τD > t 0

E f (Xt ); τDX > t = TtD |f |(z). z

(5.8)

For the rest of the paper we shall assume that the distributions of Xt and Xt∗ have densities ∗ and p X (t, z, w), respectively, which are continuous in both z and w for all t > 0. X (t, z, w) satisfying The killed semigroup has a heat kernel pD,V p X (t, z, w)

TtD,V f (z) =

X pD,V (t, z, w)f (w) dw.

(5.9)

D

Inequality (5.2) is equivalent to

X f (w)pD,V (t, z, w) dw

∗

X f ∗ (w)pD ∗ ,V ∗ (t, 0, w) dw,

(5.10)

D∗

D

for all z ∈ D and all t > 0, and this in fact holds for all nonnegative Borel functions f by Theorem 1.2. Since f is arbitrary, the continuity assumption of the kernels together with (5.8) gives that for all z, w ∈ D, ∗

∗

X X X (t, z, w) pD pD,V ∗ ,V ∗ (t, 0, 0) pD ∗ (t, 0, 0) < ∞.

(5.11)

If in addition Xt is transient, we can integrate (5.10) in time to obtain the following isoperimetric inequality for the potentials associated to Xt and Xt∗ . Corollary 5.2. Suppose both Xt and Xt∗ are transient and have continuous densities for all t > 0. Then for all z ∈ D,

f (w)GX D,V (z, w) dw

∗

f ∗ (w)GX D ∗ ,V ∗ (0, w) dw,

(5.12)

D∗

D ∗

X ∗ where GX D,V (z, w) and GD ∗ ,V ∗ (0, w) are the Green’s functions corresponding to Xt and Xt , respectively.


4047

By inequalities (5.10), (5.12), and Proposition 2.1 in [2] (see also page 671 of [8]), we have Corollary 5.3. Suppose both Xt and Xt∗ are symmetric, transient and have continuous densities for all t > 0. Then for all increasing convex functions Φ : R+ → R+ ,

X Φ pD,V (t, z, w) dw

X∗ Φ pD ∗ ,V ∗ (t, w, 0) dw,

(5.13)

∗ Φ GX D ∗ ,V ∗ (w, 0) dw,

(5.14)

D∗

D

and

Φ GX D,V (z, w) dw

D∗

D

for all z ∈ D, t > 0. These corollaries extend several results in C. Bandle [4], see for example page 214. X (t, z, w) can also be represented in terms of the multidimensional distriThe heat kernel pD,V butions. One easily proves, see [12], that X pD,V (t, z, w) = p X (t, z, w)E z

t exp − V (Xs ) ds ; τDX > t Xt = w .

(5.15)

0

If 0 = t0 < t1 < · · · < tm < t, the conditional finite dimensional distribution P z0 {Xt1 ∈ dz1 , . . . , Xtm ∈ dzm | Xt = w} is given by p X (t − tm , zm , w) X p (ti − ti−1 , zi , zi−1 )dz1 · · · dzm . p X (t, z0 , w) m

i=1

Combining (5.15) with the arguments used in (5.1) we have that

X pD,V (t, z, w) = lim m→∞

lim

k→∞ Dk

···

e

− mt

m

i=1 V (X it ) m

m+1 i=1

Dk

p

X

t , zi , zi−1 dz1 · · · dzm , m

(5.16)

where z0 = z and zm+1 = w. The proof of Theorem 1.1 can be adapted to obtain

X pD,V (t, w, w) dw

D

D∗

∗

X pD ∗ ,V ∗ (t, w, w) dw < ∞,

(5.17)

4048


where the last inequality follows from (5.11) and the fact that |D ∗ | < ∞. That is, the trace X is maximized by the trace of the Schrödinger semiof the Schrödinger semigroup for HD,V ∗ X group HD ∗ ,V ∗ . As explained in [22], the amount of heat contained in the domain D at time t, when D has temperature 1 at t = 0 and the boundary of D is kept at temperature 0 at all times, is given by Qt (D) =

B pD (t, z, w) dz dw, D D

where B is a Brownian motion. Also the torsional rigidity of D is given by

∞ Qt (D)dt = 0

GB D (z, w) dz dw. D D

Using the representation (5.16), we obtain the following results for the heat content and torsional rigidity of Lévy processes. Corollary 5.4. Suppose both Xt and Xt∗ are transient and have continuous densities for all t > 0. Then for all z ∈ D and t > 0,

X pD,V (t, z, w) dz dw

D∗

D D

∗

X pD ∗ ,V ∗ (t, z, w) dz dw,

(5.18)

D∗

and

GX D,V (z, w) dz dw

∗

GX D ∗ ,V ∗ (z, w) dz dw.

(5.19)

D∗ D∗

D D

We recall that the semigroup of the process Xt is self-adjoint in L2 if and only if the process Xt is symmetric. That is, for any Borel set A ⊂ Rd , P 0 {Xt ∈ A} = P 0 {Xt ∈ −A}. In terms of the exponent in the Lévy–Khintchine formula this leads to the representation (see [3]) 1 Ψ (ξ ) = A · ξ, ξ − 2

cos(x · ξ ) − 1 dν(x),

Rd

where A is a symmetric matrix and ν is a symmetric Lévy measure. That is, [ν(A) = ν(−A)] for all Borel sets A. In this case the general theory of Dirichlet forms (see [9]) guarantees that the Markovian semigroup generated by Xt gives rise to the self-adjoint generator H X . Recall that HVX,D is the operator obtained by imposing Dirichlet boundary conditions on D to the Schrödinger operator H X + V . That is, the generator of the killed semigroup {TtD,V }t0 . By (5.11) we have that


X pD,V (t, w, w) dw

4049

∗

X pD ∗ ,V ∗ (t, 0, 0) dw

D∗

D

∗ X∗ < ∞. = pD ∗ ,V ∗ (t, 0, 0) D

(5.20)

That is, the semigroup of the killed process has finite trace. Whenever D is of finite volume, the operator TtD,V maps L2 (D) into L∞ (D) for every t > 0. This follows from (5.11) and the general theory of heat semigroups as described on page 59 of [9]. In fact, under these assumptions it follows from [9] that there exists an orthonormal basis n ∞ ∞ 2 of eigenfunctions {ϕD,V ,X }n=1 for L (D) and corresponding eigenvalues {λn (D, V , X)}n=1 for the semigroup {TtD,V }t0 satisfying 0 < λ1 (D, V , X) < λ2 (D, V , X) λ3 (D, V , X) · · · with λn (D, V , X) → ∞ as n → ∞. That is, the pair n ϕD,V ,X , λn (D, V , X) satisfies n −λn (D,V ,X)t n ϕD,V ,X (z), TtD ϕD,V ,X (z) = e

z ∈ D, t > 0.

n Notice that λn (D, V , X) is a Dirichlet eigenvalue of H X +V on D with eigenfunction ϕD,V ,X (z). Under such assumptions we have

X (t, z, w) = pD,V

∞

n n e−λn (D,V ,X)t ϕD,V ,X (z)ϕD,V ,X (w).

(5.21)

n=1 X (t, z, w) implies that This eigenfunction expansion for pD,V

t 1 z X −λ1 (D, V , X) = lim log E exp − V (Xs ) ds ; τD > t , t→∞ t

(5.22)

0

for all domains D of finite volume. This gives the following corollary. Corollary 5.5 (Faber–Krahn inequality for Lévy processes). Suppose both Xt and Xt∗ are symmetric, transient and have continuous densities for all t > 0. Then λ1 D ∗ , V ∗ , X ∗ λ1 (D, V .X). More generally, we also have the trace inequality ∞ n=1

valid for all t > 0.

e−tλn (D,X,V )

∞ n=1

e−tλn (D

∗ ,X ∗ ,V ∗ )

(5.23)

4050


Finally, denote by CX (A) the capacity of the set A for the process Xt . In [23], T. Watanabe proved that CX (A) CX∗ A∗ .

(5.24)

(This question, for Riesz capacities of all orders was raised by P. Mattila in [11].) As explained in [13], this inequality can be obtained from the existing rearrangement inequalities of multiple integrals only in the case that Xt is isotropic unimodal. For general Lévy processes we have the following representation of the capacity due to Port and Stone [19] 1 t→∞ t

lim

P z0 τAXc t dz0 = CX (A).

(5.25)

Since

P z0 τAXc t dz0

= lim lim

k→∞ m→∞

···

1−

m j =1

IAck (zj )

m j =1

p

X

t , zj , zj −1 dz0 · · · dzm , m

(5.26)

where Ak is a decreasing sequence of compact sets such that the interior of Ak contains A for all k and ∞ k=1 Ak = A. We expect that (5.24) can be obtained using a result similar to Theorem 1.5 for more general Lévy processes. However, the corresponding rearrangement inequality for this type of multiple integrals is only known for radially symmetric decreasing functions, see Corollary 2 in [10]. That is, only when Xt is an isotropic unimodal Lévy process. We believe, in the case that the sets Ak are compact, the more general rearrangement inequality for multiple integrals needed for this application should be true but at present we are not able to prove it. Acknowledgments We are very grateful to an anonymous referees for the very careful reading of this paper and the many suggestions which improved the presentation and readability of the paper. References [1] M. Aizenman, B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (2) (1982) 209–273. [2] A. Alvino, G. Trombetti, P.-L. Lions, On optimization problems with prescribed rearrangements, Nonlinear Anal. 13 (1989) 209–273. [3] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Stud. Adv. Math., vol. 93, Cambridge University Press, Cambridge, 2004. [4] C. Bandle, Isoperimetric Inequalities and Applications, Monogr. Stud. Math., Pitman, 1980. [5] R. Bañuelos, R. Latała, P.J. Méndez-Hernández, A Brascamp–Lieb–Luttinger-type inequality and applications to symmetric stable processes, Proc. Amer. Math. Soc. 129 (2001) 2997–3008. [6] D. Betsakos, Symmetrization, symmetric stable processes, and Riesz capacities, Trans. Amer. Math. Soc. 356 (2004) 735–755, 3821. [7] H.J. Brascamp, E.H. Lieb, J.M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974) 227–237. [8] A. Burchard, M. Schmuckenschläger, Comparison theorems for exit times, Geom. Funct. Anal. 11 (2001) 651–692.


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[9] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. [10] R. Friedberg, J.M. Luttinger, Rearrangement inequality for period functions, Arch. Ration. Mech. 61 (1976) 35–44. [11] P. Mattila, Orthogonal projections, Riesz capacities, and Minkowski content, Indiana Univ. Math. J. 39 (1990) 185– 198. [12] P.J. Méndez-Hernández, Brascamp–Lieb–Luttinger inequalities for convex domains of finite inradius, Duke Math. J. 113 (2002) 93–131. [13] P.J. Méndez-Hernández, An isoperimetric inequality for Riesz capacities, Rocky Mountain J. Math. 36 (2) (2006) 675–682. [14] E. Lieb, M. Loss, Analysis, Grad. Stud. Math., vol. 14, American Mathematical Society, Providence, RI, 2001. [15] J.M. Luttinger, Generalized isoperimetric inequalities, I, J. Math. Phys. 14 (1973) 586–593. [16] J.M. Luttinger, Generalized isoperimetric inequalities, II, J. Math. Phys. 14 (1973) 1444–1447. [17] J.M. Luttinger, Generalized isoperimetric inequalities, III, J. Math. Phys. 14 (1973) 1448–1450. [18] G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951. [19] S.C. Port, C.J. Stone, Infinite divisible processes and their potential theory I, Ann. Inst. Fourier (Grenoble) 21 (1971) 157–275. [20] K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. [21] B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, 1979. [22] M. van den Berg, J.F. Le Gall, Mean curvature and the heat equation, Math. Z. 215 (1994) 437–464. [23] T. Watanabe, The isoperimetric inequality for isotropic unimodal Lévy processes, Z. Wahrscheinlichkeitstheorie 63 (1983) 487–499.


A transference theorem for the Dunkl transform and its applications ✩ Feng Dai a,∗ , Heping Wang b a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada b School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received 28 August 2009; accepted 10 March 2010 Available online 29 March 2010 Communicated by N. Kalton

Abstract For a family of weight functions invariant under a finite reflection group, we show how weighted Lp multiplier theorems for Dunkl transform on the Euclidean space Rd can be transferred from the corresponding results for h-harmonic expansions on the unit sphere Sd of Rd+1 . The result is then applied to establish a Hörmander type multiplier theorem for the Dunkl transform and to show the convergence of the Bochner–Riesz means of the Dunkl transform of order above the critical index in weighted Lp spaces. © 2010 Elsevier Inc. All rights reserved. Keywords: Hörmander type multiplier theorem; Dunkl transform; Transference theorem; h-Harmonics

1. Introduction Let R be a reduced root system in Rd normalized so that α, α = 2 for all α ∈ R, where ·,· denotes the standard Euclidean inner product. Given a nonzero vector α ∈ Rd , we denote by σα the reflection with respect to the hyperplane perpendicular to α; that is, σα x = x − 2(x, α/α2 )α for all x ∈ Rd , where · denotes the usual Euclidean norm. Let G denote ✩ The first author was partially supported by the NSERC Canada under grant G121211001. The second author was supported by the National Natural Science Foundation of China (Project No. 10871132) and the Beijing Natural Science Foundation (Project No. 1102011). * Corresponding author. E-mail addresses: [email protected] (F. Dai), [email protected] (H. Wang).


F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074

4053

the finite subgroup of the orthogonal group O(d) generated by the reflections σα , α ∈ R. Let κ : R → R+ be a nonnegative multiplicity function on R with the property κ(gα) = κ(α) for all α ∈ R and g ∈ G. Associated with the reflection group G and the function κ is the weight function hκ defined by hκ (x) :=

x, ακ(α) ,

x ∈ Rd ,

(1.1)

α∈R+

where R+ is an arbitrary but fixed positive subsystem of R. The function hκ is a homogeneous function of degree γκ := α∈R+ κ(α), and is invariant under the reflection group G. For convenience, we shall set λκ = d−1 2 + γκ for the rest of the paper. Given 1 p ∞, we denote by Lp (Rd ; h2κ ) the weighted Lebesgue space endowed with the norm f κ,p :=

f (y)p h2 (y) dy

1

κ

p

,

Rd

with the usual change when p = ∞. The Dunkl transform, a generalization of the classical Fourier transform, is defined, for f ∈ L1 (Rd ; h2κ ), by Fκ f (x) = cκ

f (y)Eκ (−ix, y)h2κ (y) dy,

x ∈ Rd ,

(1.2)

Rd

x2 where cκ = ( Rd h2κ (x)e− 2 dx)−1 , and Eκ (ix, y) = Vκ [eix,· ](y) is the weighted analogue of the character eix,y . Here Vκ is the Dunkl intertwining operator, whose precise definition will be given in Section 2. The Dunkl transform plays the same role as the Fourier transform in classical Fourier analysis, and enjoys properties similar to those of the classical Fourier transform (see [11]). Several important results in classical Fourier analysis have been extended to the setting of Dunkl transform by Thangavelu and Yuan Xu [18,17]. The problem, however, turns out to be rather difficult in general. One of the difficulties comes from the fact that the generalized translation operator τy , which plays the role of the usual translation f → f (· − y), is not positive in general (see, for instance, [17, Proposition 3.10]). In fact, even the Lp boundedness of τy is not established in general (see [18,17]). In this paper, we shall first prove a transference theorem (Theorem 3.1) between the Lp multiplier of h-harmonic expansions on the unit sphere and that of the Dunkl transform. This theorem, combined with the corresponding results on h-harmonic expansions on the unit sphere recently established in [2–4], is then applied to establish a Hörmander type multiplier theorem for the Dunkl transform (Theorem 4.1), and to show the convergence of the Bochner–Riesz means in the weighted Lp spaces (Theorem 4.3). The paper is organized as follows. In Section 2, we describe briefly some known results on Dunkl transform and h-harmonic expansions, which will be needed in later sections. The transference theorem, Theorem 3.1, is proved in Section 3. As applications, we prove Theorems 4.1 and 4.3 in the final section, Section 4.

4054


2. Preliminaries In this section, we shall present some necessary material on the Dunkl transform and the h-harmonic expansions, most of which can be found in [8,11,13,14,17]. 2.1. The Dunkl transform Let R, R+ , G, κ and hκ be as defined in Section 1. Recall that a reduced root system is a finite subset R of Rd \ {0} with the properties σα R = R and R ∩ {tα: t ∈ R} = {±α} for all α ∈ R. The Dunkl operators associated with G and κ are defined by Dκ,i f (x) = ∂i f (x) +

κ(α)

α∈R+

f (x) − f (σ (α)x) α, ei , x, α

1 i d,

(2.1)

where e1 , . . . , ed are the standard unit vectors of Rd . Those operators mutually commute, and map Pdn to Pdn−1 , where Pdn is the space of homogeneous polynomials of degree n in d variables (see [5]). We denote by Π d := Π(Rd ) the C-algebra of polynomial functions on Rd . An important result in Dunkl theory states that there exists a linear operator Vκ : Π d → Π d determined uniquely by

Vκ Pdn ⊂ Pdn ,

Vκ (1) = 1,

and Dκ,i Vκ = Vκ ∂i ,

1 i d.

(2.2)

Such an operator is called the intertwining operator. A very useful explicit formula for Vκ was obtained by C.F. Dunkl [6] in the case of G = Z2 , and was later extended to the more general case of G = Zd2 (d ∈ N) by Xuan Xu [21]. In general, one has the following important result of Rösler [13]: Lemma 2.1. (See [13, Theorem 1.2 and Corollary 5.3].) For every x ∈ Rd , there exists a unique probability measure μκx on the Borel σ -algebra of Rd such that Vκ P (x) =

P (ξ ) dμκx (ξ ),

P ∈ Πd.

(2.3)

Rd

Furthermore, the representing measures μκx are compactly supported in the convex hull C(x) := co{gx: g ∈ G} of the orbit of x under G, and satisfy

μκrx (E) = μκx r −1 E and μκgx (E) = μκx g −1 E

(2.4)

for all r > 0, g ∈ G and each Borel subset E of Rd . In particular, the above lemma shows that the intertwining operator Vκ is positive. We point out that Lemma 2.1 will play a crucial role in the analysis of our paper. By means of (2.3), Vκ can be extended to the space C(Rd ) of continuous functions on Rd . We denote this extension by Vκ again. The Dunkl transform associated with G and κ is defined by (1.2) with

Eκ (−ix, y) := Vκ e−ix,· (y),

x, y ∈ Rd .

(2.5)


4055

If κ = 0 then Vκ = id and the Dunkl transform coincides with the usual Fourier transform, whereas if d = 1 and G = Z2 then it is closely related to the Hankel transform on the real line. We list some of the known properties of the Dunkl transform in the following lemma. Lemma 2.2. (See [7,11].) (i) If f ∈ L1 (Rd ; h2κ ) then Fκ f ∈ C(Rd ) and limξ →∞ Fκ f (ξ ) = 0. (ii) The Dunkl transform Fκ is an isomorphism of the Schwartz class S(Rd ) onto itself, and Fκ2 f (x) = f (−x). (iii) The Dunkl transform Fκ on S(Rd ) extends uniquely to an isometric isomorphism on L2 (Rd ; h2κ ), i.e., f κ,2 = Fκ f κ,2 . (iv) If f and Fκ f are both in L1 (Rd ; h2κ ) then the following inverse formula holds: f (x) = cκ

Fκ f (y)Eκ (ix, y)h2κ (y) dy,

x ∈ Rd .

Rd

(v) If f, g ∈ L2 (Rd ; h2κ ) then

Fκ f (x)g(x) h2κ (x) dx =

Rd

f (x)Fκ g(x) h2κ (x) dx. Rd

(vi) Given ε > 0, let fε (x) = ε −2−2γκ f (ε −1 x). Then Fκ fε (ξ ) = Fκ f (εξ ). (vii) If f (x) = f0 (x) is radial, then Fκ f (ξ ) = Hλκ − 1 f0 (ξ ) is again a radial function, 2 where Hα denotes the Hankel transform defined by Hα g(s) =

1 Γ (α + 1)

∞ g(r)

Jα (rs) 2α+1 r dr, (rs)α

0

and Jα denotes the Bessel function of the first kind. Statements (i)–(vi) of Lemma 2.2 above were proved by M.F.E. de Jeu [11], and were, in fact, contained in Corollaries 4.7 and 4.22, Theorems 4.26 and 4.20, Lemmas 4.13 and 4.3(3) of [11], respectively. Statement (vii) of Lemma 2.2 was proved by Dunkl [7] and was stated more explicitly in [17, Proposition 2.4]. Given y ∈ Rd , the generalized translation operator f → τy f is defined on L2 (Rd ; h2κ ) by Fκ (τy f )(ξ ) = Eκ (−iξ, y)Fκ f (ξ ),

ξ ∈ Rd .

It is known that τy f (x) = τx f (y) for a.e. x ∈ Rd and a.e. y ∈ Rd . In general, the operator τy is not positive (see, for instance, [17, Proposition 3.10]), and it is still an open problem whether τy f can be extended to a bounded operator on L1 (Rd ; h2κ ). On the other hand, however, it was shown in [17, Theorem 3.7] that the generalized translation operator τy can be extended to all p radial functions in Lp (Rd ; h2κ ), 1 p 2, and τy : Lrad (Rd ; h2κ ) → Lp (Rd ; h2κ ) is a bounded p operator, where Lrad (Rd ; h2κ ) denotes the space of all radial functions in Lp (Rd ; h2κ ).

4056


The generalized convolution of f, g ∈ L2 (Rd ; h2κ ) is defined by f ∗κ g(x) =

f (y)τx g (y)h2κ (y) dy,

(2.6)

Rd p

where g (y) = g(−y). Since τy is a bounded operator from Lrad (Rd ; h2κ ) to Lp (Rd ; h2κ ) for 1 p p 2, it follows that the definition of f ∗κ g can be extended to all g ∈ Lrad (Rd ; h2κ ) and f ∈ 1 1 p d 2 L (R ; hκ ) with 1 p 2 and p + p = 1. The generalized convolution satisfies the following basic property: Fκ (f ∗κ g)(ξ ) = Fκ f (ξ )Fκ g(ξ ).

(2.7)

More properties on the generalized translation operator and the generalized convolution can be found in [17]. 2.2. h-Harmonic expansions Let Sd−1 = {x ∈ Rd : x = 1} denote the unit sphere of Rd equipped with the usual Lebesgue measure dσ (x). For the weight function hκ given in (1.1), we consider the weighted Lebesgue space Lp (h2κ ; Sd−1 ) of functions on Sd−1 endowed with the finite norm f Lp (h2κ ;Sd−1 ) ≡ f κ,p :=

1/p f (y)p h2 (y) dσ (y) , κ

1 p < ∞,

Sd−1

and for p = ∞ we assume that L∞ is replaced by C(Sd−1 ), the space of continuous functions on Sd−1 with the usual uniform norm f ∞ . We shall use the notation · κ,p to denote the weighted norm for functions defined either on Rd or on Sd−1 whenever it causes no confusion. A homogeneous polynomial is called an h-harmonic if it is orthogonal to all polynomials of lower degree with respect to the inner product of L2 (h2κ ; Sd−1 ). Let Hnd (h2κ ) denote the space of all h-harmonics of degree n, and let projκn : L2 (h2κ ; Sd−1 ) → Hnd (h2κ ) denote the orthogonal projection operator. The projection projκn has an integral representation projκn f (x) :=

f (y)Pnκ (x, y)h2κ (y) dσ (y),

x ∈ Sd−1 ,

(2.8)

Sd−1

where Pnκ (x, y) is the reproducing kernel of Hnd (h2κ ) which can be written in terms of the interwining operator Vκ as (see [20, Theorem 3.2, (3.1)]) Pnκ (x, y) =

n + λk λk

Vκ Cn x, · (y), λκ

x, y ∈ Sd−1 ,

(2.9)

λ with λκ := λκ − 12 = γκ + d−2 2 . Here Cn denotes the standard Gegenbauer polynomial of degree n and index λ as defined in [16]. By means of (2.8) and (2.9), the projection projκn f can be extended to all f ∈ L1 (h2κ ; Sd−1 ).


4057

The following Marcinkiewicz type multiplier theorem was proved recently in [2, Theorem 2.3]: Theorem 2.3. Let {μj }∞ j =0 be a sequence of real numbers that satisfies (i) supj |μj | c < ∞, j +1 (ii) supj 1 2j (r−1) 2l=2j | r μl | c < ∞, with r being the smallest integer

d 2

+ γκ ,

where μl = μl − μl+1 and j +1 μl = j μl − j μl+1 . Then {μj } defines an Lp (h2κ ; Sd−1 ) multiplier for all 1 < p < ∞; that is, ∞ μj projκj f j =0

Ap cf κ,p ,

1 −1, the Cesàro (C, δ) means of the h-harmonic expansion are defined by n

−1 Snδ h2κ ; f, x := Aδn Aδn−k projκk f (x),

Aδn−k =

k=0

In the case when G = Zd2 and hκ (x) = recently in [3]:

d

κ(ei ) , i=1 |x, ei |

the following result was proved

Theorem 2.4. Let G = Zd2 and let 1 p ∞ satisfy | p1 − 12 | σκ :=

d−2 2

n−k+δ . n−k

1 2σκ +2

with

+ γκ − min κ(ei ). 1id

Then

supSnδ h2κ ; f κ,p cf κ,p ,

n∈N

for all f ∈ Lp h2κ ; Sd−1

if and only if 1 1 1 δ > δκ (p) := max (2σκ + 1) − − , 0 . p 2 2

(2.10)

4058


3. A transference theorem The main goal in this section is to establish a transference theorem between the Lp multipliers of h-harmonic expansions on the unit sphere Sd := {x ∈ Rd+1 : x = 1} and those of the Dunkl transform in Rd . Let G, R, hκ be as defined in Section 1. Given g ∈ G, we denote by g the orthogonal transformation on Rd+1 determined uniquely by g x = (gx, xd+1 )

for x = (x, xd+1 ) with x ∈ Rd and xd+1 ∈ R.

Then G := {g : g ∈ G} is a finite reflection group on Rd+1 with a reduced root system R := {(α, 0): α ∈ R}. Let κ denote the nonnegative multiplicity function defined on R with the property κ (α, 0) = κ(α). We denote by Vκ the intertwining operator on C(Rd+1 ) associated with the reflection group G and the multiplicity function κ . Define the weight function hκ (x, xd+1 ) := hκ (x) =

x, ακ(α) ,

x ∈ Rd , xd+1 ∈ R.

α∈R+

Recall that projκn : L2 (Sd ; h2κ ) → Hnd+1 (h2κ ) denotes the orthogonal projection onto the space of h-harmonics. Our main result is the following. Theorem 3.1. Let m : [0, ∞) → R be a continuous and bounded function, and let Uε , ε > 0, be a family of operators on L2 (Sd ; h2κ ) given by

projκn (Uε f ) = m(εn) projκn f,

n = 0, 1, . . . .

(3.1)

Assume that sup Uε f Lp (Sd ;h2 ) Af Lp (Sd ;h2 ) , ε>0

κ

κ

∀f ∈ C Sd ,

(3.2)

for some 1 p ∞. Then the function m( · ) defines an Lp (Rd ; h2κ ) multiplier; that is, Tm f Lp (Rd ;h2κ ) cd,κ Af Lp (Rd ;h2κ ) ,

∀f ∈ S Rd ,

where Tm is an operator initially defined on L2 (Rd ; h2κ ) by

Fκ (Tm f )(ξ ) = m ξ Fκ f (ξ ),

f ∈ L2 Rd ; h2κ , ξ ∈ Rd .

(3.3)

In the case of ordinary spherical harmonics (i.e., κ = 0), Theorem 3.1 is due to Bonami and Clerc [1, Theorem 1.1].


4059

3.1. Lemmas The proof of Theorem 3.1 relies on several lemmas. Lemma 3.2. If f ∈ Π d+1 then for any x ∈ Rd and xd+1 ∈ R,

Vκ f (x, xd+1 ) = Vκ f (·, xd+1 ) (x) =

f (ξ, xd+1 ) dμκx (ξ ),

(3.4)

Rd

where dμκx is given in (2.3). Proof. Clearly, the second equality in (3.4) follows directly from (2.3). To show the first equalκ f (x, xd+1 ) = Vκ [f (·, xd+1 )](x) for f ∈ C(Rd+1 ) and x ∈ Rd . Since Vκ is a linear ity, we set V operator uniquely determined by (2.2), it suffices to show that the following conditions are satisfied:

κ Pd+1 ⊂ Pd+1 V n n ,

κ (1) = 1, V

κ ∂i , κ = V and Dκ ,i V

1 i d + 1.

Indeed, these conditions can be easily verified using the properties of Vκ in (2.2), and the following identities, which follow directly from (2.1):

Dκ ,i g(x, xd+1 ) = Dκ,i g(·, xd+1 ) (x), Dκ ,d+1 g(x, xd+1 ) = ∂d+1 g(x, xd+1 ),

1 i d,

for g ∈ Π d+1 , x ∈ Rd and xd+1 ∈ R.

2

This completes the proof of Lemma 3.2.

To formulate the next lemma, we define the mapping ψ : Rd → Sd by

ψ(x) := ξ sinx, cosx for x = xξ ∈ Rd and ξ ∈ Sd−1 . Given N 1, we denote by NSd := {x ∈ Rd+1 : x = N } the sphere of radius N in Rd+1 , and define the mapping ψN : Rd → N Sd by ψN (x) := N ψ

x N

x x = N ξ sin , N cos N N

(3.5)

with x = xξ ∈ Rd and ξ ∈ Sd−1 . Lemma 3.3. If f : NSd → R is supported in the set {x ∈ N Sd : arccos(N −1 xd+1 ) 1}, then

f (Nx)h2κ (x) dσ (x) = N −2λκ −1

Sd

where B(0, N) = {y ∈ Rd : y N }.

B(0,N )

sin(x/N ) 2λκ f ψN (x) h2κ (x) dx, x/N

4060


Proof. First, using the polar coordinate transformation (ξ, θ ) ∈ Sd−1 × [0, π] → x := (ξ sin θ, cos θ ) ∈ Sd , and the fact that dσ (x) = sind−1 θ dθ dσ (ξ ), we obtain π

f (N x)h2κ (x) dσ (x) =

f (N ξ sin θ, N

Sd

0

cos θ )h2κ (ξ

sin θ, cos θ ) dσ (ξ ) (sin θ )d−1 dθ

Sd−1

1 =

sin θ d−1+2γκ d−1 f (N ξ sin θ, N cos θ )h2κ (θ ξ ) dσ (ξ ) θ dθ, θ

0 Sd−1

where the last step uses the identity hκ (y, yd+1 ) = hκ (y), the fact that h2κ is a homogeneous function of degree 2γκ , and the assumption that f is supported in the set {x ∈ N Sd : arccos(N −1 xd+1 ) 1}. Using the usual spherical coordinate transformation in Rd , the last double integral equals Ny siny siny 2λκ 2 , N cosy hκ (y) f dy y y

y1

= N −d−2γκ

xN

= N −2λκ −1

B(0,N )

x x 2 x sin(x/N ) 2λκ sin , N cos hκ (x) f N dx x N N x/N sin(x/N ) 2λκ f (ψN x)h2κ (x) dx, x/N

where the first step uses the homogeneity of the weight hκ and the change of variables y = This proves the desired formula. 2

x N.

Remark 3.1. It is easily seen that the restriction ψN |B(0,N ) of the mapping ψN on B(0, N ) is a bijection from B(0, N ) to {x ∈ N Sd : arccos(N −1 xd+1 ) 1}. Thus, given a function f : B(0, N) → R, there exists a unique function fN supported in {x ∈ N Sd : arccos(N −1 xd+1 ) 1} such that fN (ψN x) = f (x),

∀x ∈ B(0, N ).

(3.6)

On the other hand, using Lemma 3.3, we have Sd

fN (Nx)h2κ (x) dσ (x) = N −2λκ −1

B(0,N )

f (x)h2κ (x)

sin(x/N ) x/N

The formula (3.7) will play an important role in our proof of Theorem 3.1.

2λκ dx.

(3.7)


4061

We also need a small observation on a formula of Rösler [14] for τy f (x): Lemma 3.4. If f (x) = f0 (x) is a continuous radial function in L2 (Rd ; h2κ ), then for a.e. y ∈ Rd and a.e. x ∈ Rd , y 2 2 . τy f (x) = Vκ f0 x + y − 2yx, · y

(3.8)

Formula (3.8) was first proved in [14] under the assumption that f is a radial Schwartz function. Thangavelu and Yuan Xu [17, Proposition 3.3] later observed that it also holds for radial functions f ∈ L(Rd ; h2κ ) with Fκ f ∈ L(Rd ; h2κ ). Clearly, our assumption in Lemma 3.4 is slightly weaker than that of [17, Proposition 3.3]. Lemma 3.4 can be deduced from the result of Rösler [14], using a density argument. Proof. We first choose a sequence of even, C ∞ functions gj on R satisfying sup gj (t) − f0 (t) 2−j

− 1

2j

|t|2j +1

2

s 2λκ ds

.

0

Let ϕj be an even, C ∞ function on R such that χ[2−j ,2j ] (|t|) ϕj (t) χ[2−j −1 ,2j +1 ] (|t|), and let fj (x) ≡ fj,0 (x) := gj (x)ϕj (x) for x ∈ Rd . Then it’s easily seen that {fj } is a sequence of radial Schwartz functions on Rd satisfying lim

sup

j →∞ 2−j |t|2j

fj,0 (t) − f0 (t) = 0

(3.9)

and lim fj − f κ,2 = 0.

j →∞

(3.10)

Since each fj is a radial Schwartz function, by Lemma 2.1 and the already proven case of Lemma 3.4 (see [14]), we obtain

τy (fj )(x) = fj,0 x2 + y2 − 2yx, ξ dμκy/y (ξ ). (3.11) ξ 1

Next, we fix y ∈ Rd , and set An ≡ An (y) := x ∈ Rd : 2−n x − y x + y 2n for n ∈ N and n n0 (y) := [log y/ log 2] + 1. Since

2

2 x − y x2 + y2 − 2yx, ξ x + y for all ξ 1, it follows by (3.9) that

4062


lim fj,0

j →∞

x2 + y2 − 2yx, ξ = f0 x2 + y2 − 2yx, ξ

uniformly for x ∈ An (y) and ξ 1. This together with (3.11) and Lemma 2.1 implies lim τy (fj )(x) =

j →∞

f0

x2 + y2 − 2yx, ξ dμκy/y (ξ )

ξ 1

y 2 2 = Vκ f0 x + y − 2yx, · y for every x ∈ An (y) \ {0} and n n0 (y). On the other hand, however, by (3.10), we have lim τy (fj ) − τy f κ,2 = 0

j →∞

for all y ∈ Rd . Thus, y 2 2 τy (f )(x) = Vκ f0 x + y − 2yx, · y for a.e. x ∈ An (y) and all n n0 (y). Finally, observing that the set R \ d

∞

An (y) = x ∈ Rd : x = y

n=n0 (y)

has measure zero in Rd , we deduce the desired conclusion.

2

Remark 3.2. By (2.4) and the supporting condition of the measure dμκx , we observe that Vκ F (rx) =

F (rξ ) dμκx (ξ ),

for all F ∈ C Rd , x ∈ Rd , and r > 0.

(3.12)

Rd

Thus, (3.8) can be rewritten more symmetrically as τy f (x) = Vκ f0 x2 + y2 − 2x, · (y).

(3.13)

Lemma 3.5. Let Φ ∈ L1 (R, |x|2λκ ) be an even, bounded function on R, and let TΦ be an operator L2 (Rd ; h2κ ) → L2 (Rd ; h2κ ) defined by

Fκ (TΦ f )(ξ ) := Fκ f (ξ )Φ ξ , Then TΦ has an integral representation TΦ f (x) = f (y)K(x, y)h2κ (y) dy, Rd

f ∈ L2 Rd ; h2κ .

for f ∈ S Rd and a.e. x ∈ Rd ,


4063

where

K(x, y) = c

Φ(s)Vκ 0

(s x2 + y2 − 2x, · ) (y)s 2λκ ds. λκ − 12 2 2 (s x + y − 2x, · )

J

∞

λκ − 12

(3.14)

Furthermore, K(x, y) = K(y, x) for a.e. x ∈ Rd and a.e. y ∈ Rd . Proof. Let g(x) = Hλκ − 1 Φ(x), where x ∈ Rd and Hα denotes the Hankel transform. Since Φ 2

is an even function in L1 (R, |x|2λκ )∩L∞ (R), it follows by the properties of the Hankel transform that g is a continuous radial function in L2 (Rd ; h2κ ) and Fκ g(ξ ) = Φ(ξ ). Thus, using (2.7), we have TΦ f (x) = f ∗κ g(x) =

f (y)τy g(x)h2κ (y) dy Rd

for f ∈ L2 (Rd ; h2κ ). Since g is a continuous radial function in L2 (Rd ; h2κ ), by Lemma 3.4 and Remark 3.2 it follows that

K(x, y) := τy g(x) = Vκ Hλκ − 1 Φ x2 + y2 + 2x, · (y) 2 J ∞ 2 2 λκ − 12 (s x + y − 2x, · ) = c Φ(s)Vκ (y)s 2λκ ds, λκ − 12 2 2 (s x + y − 2x, · ) 0

where the last step uses (2.3), the inequality J 1 (rs) Φ(s) λκ − 2 cΦ(s) λκ − 12 (rs) and Fubini’s theorem. This proves the desired equation (3.14). That K(x, y) = K(y, x) follows from the fact that τx g(y) = τy g(x). 2 Our final lemma is a well-known result for the ultraspherical polynomials: Lemma 3.6. (See [16, (8.1.1), p. 192].) For z ∈ C and μ 0, 1 Γ (μ + 12 ) z −μ+ 2 z μ Jμ− 1 (z). lim k 1−2μ Ck cos = 2 k→∞ k Γ (2μ) 2

This formula holds uniformly in every bounded region of the complex z-plane.

(3.15)

4064


3.2. Proof of Theorem 3.1 We follow the idea of the proof of Theorem 1.1 of [1]. We first prove the theorem under the additional assumption |m(t)| c1 e−c2 t for all t > 0 and some c1 , c2 > 0. By Lemma 3.5, the operator Tm has an integral representation Tm f (x) =

f (y)K(x, y)h2κ (y) dy, Rd

where K(x, y) is given by (3.14) with Φ = m. Thus, it is sufficient to prove that I := f (y)g(x)K(x, y)h2κ (x)h2κ (y) dx dy cA

(3.16)

Rd Rd

whenever f ∈ Lp (Rd ; h2κ ) and g ∈ Lp (Rd ; h2κ ) both have compact supports and satisfy f Lp (Rd ;h2κ ) = gLp (Rd ;h2 ) = 1. κ To this end, we choose a positive number N to be sufficiently large so that the supports of f and g are both contained in the ball B(0, N ). By Remark 3.1, there exist functions fN and gN both supported in {x ∈ N Sd : arccos(N −1 xd+1 ) 1} and satisfying

fN ψN (x) = f (x),

gN ψN (x) = g(x),

x ∈ Rd ,

(3.17)

where ψN is defined by (3.5). It’s easily seen from (3.7) that 2λκ +1 fN (N·) p d 2 cN − p , L (S ;h )

gN (N ·)

κ

Lp (Sd ;h2κ )

Thus, using (2.8), (2.9), (3.1) and the assumption (3.2) with ε =

IN := N

1 N,

cN

− 2λpκ +1

we obtain

∞

κ −1 2 2 m N n Pn (x, y) fN (Ny)gN (N x)hκ (x)hκ (y) dσ (x) dσ (y)

2λκ +1

Sd Sd

n=0

cA,

.

(3.18)

where Pnκ (x, y) =

λκ n+λκ λκ Vκ [Cn (x, ·)](y).

HN (x, y) = N

−2λκ −1

Setting

∞

−1 κ x y ,ψ , m N n Pn ψ N N n=0

and invoking (3.17) and Lemma 3.3, we obtain


4065

sin(x/N ) 2λκ 2 2 IN = HN (x, y)f (y)g(x)hκ (x)hκ (y) x/N Rd Rd

sin(y/N) × y/N

2λκ

dx dy .

(3.19)

On the other hand, setting −1 N ∞ x y n −2λκ −1 κ 2λκ bN (ρ, x, y) = N Pn ψ ,ψ m t dt χ[ n , n+1 ) (ρ), N N N N N n+1

n=0

n N

we have ∞ HN (x, y) =

bN (ρ, x, y)ρ 2λκ dρ. 0

Hence, by (3.19), ∞ 2λκ IN = bN (ρ, x, y)ρ dρ f (y)g(x)h2κ (x)h2κ (y) Rd Rd

0

sin(x/N) × x/N

2λκ

sin(y/N ) y/N

2λκ

dx dy .

(3.20)

The key ingredient in our proof is to show that limN →∞ IN = cI , where c is a constant depending only on d and κ. In fact, once this is proven, then the desired estimate (3.16) will follow immediately from (3.18). To show limN →∞ IN = cI , we make the following two assertions: Assertion 1. For any N > 0 and x, y ∈ Rd , bN (ρ, x, y) ce−c2 ρ , where c is independent of x, y and N . Assertion 2. For any fixed x, y ∈ Rd and ρ > 0, lim bN (ρ, x, y) = cm(ρ)Vκ

N →∞

where u(x, y, ξ ) =

J

λκ − 12 (ρu(x, y, ·)) 1

(ρu(x, y, ·))λκ − 2

(y),

(3.21)

x2 + y2 − 2x, ξ , and c is a constant depending only on d and κ.

4066


For the moment, we take the above two assertions for granted, and proceed with the proof of Theorem 3.1. By Assertion 1 and Hölder’s inequality, we can apply the dominated convergence theorem to the integrals in (3.20), and obtain ∞ 2λκ 2 2 lim IN = lim bN (ρ, x, y)ρ dρ f (y)g(x)hκ (x)hκ (y) dx dy , N →∞ N →∞ Rd Rd

0

which, using Assertion 2, equals ∞ J λκ − 12 (ρu(x, y, ·)) 2λκ 2 2 (y)ρ m(ρ)Vκ dρ f (y)g(x)h (x)h (y) dx dy = c κ κ λκ − 12 (ρu(x, y, ·)) Rd Rd

0

2 2 = c K(x, y)f (y)g(x)hκ (x)hκ (y) dx dy = cI, Rd Rd

where the second step uses (3.14). Thus, we have shown the desired relation limN →∞ IN = cI , assuming Assertions 1 and 2. Now we return to the proofs of Assertions 1 and 2. We start with the proof of Assern −c2 Nn tion 1. Assume that Nn ρ < n+1 ce−c2 ρ , and N for some n ∈ Z+ . Then |m( N )| c1 e n+1 N t 2λκ dt cN −1 ρ 2λκ . Hence, n N

−1 N n x y −2λ −1 κ 2λ bN (ρ, x, y) = N κ m Pn ψ ,ψ t κ dt N N N n+1

n N

cN

−2λκ −2λκ −c2 ρ n + λκ

ρ

e

λκ

y Vκ C λκ ψ x , · ψ n N N

c(Nρ)−2λκ e−c2 ρ n2λκ ce−c2 ρ , where we used (2.9) in the second step, and the positivity of Vκ and the estimate |Cnλκ (t)| cn2λκ −1 in the third step. This proves Assertion 1. Next, we show Assertion 2. A straightforward calculation shows that for Nn ρ n+1 N and ρ > 0, n+1 N

−1 t

n N

2λκ

dt

=

N

1 + oρ (1) , 2λ ρ κ

as N → ∞.


This implies that for

n N

ρ

n+1 N

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and ρ > 0,

x y n2λκ −2λκ κ bN (ρ, x, y) = m(ρ) ,ψ 1 + oρ (1) n Pn ψ 2λ κ N N (Nρ) x y y y ,· sin , cos + oρ (1), = cm(ρ)n−2λκ +1 Vκ Cnλκ ψ N y N N

where we used the continuity of m in the first step, and the estimate n−2λκ |Pnκ (ψ( Nx ), ψ( Ny ))| c, as well as the fact that limN →∞ ing Lemma 3.2 and (3.12), we obtain bN (ρ, x, y) = cm(ρ)n

−2λκ +1

Cnλκ

y y sin N

= cm(ρ)n

(Nρ)2λκ

= 1 in the last step (see [4]). Thus, us-

x x y 1 sin cos xj ξj + cos x N N N d

j =1

Rd

× dμκ y

n2λκ

(ξ ) + oρ (1)

−2λκ +1

Cnλκ cos θN (x, y, ξ ) dμκy (ξ ) + oρ (1),

(3.22)

ξ y

where θN (x, y, ξ ) ∈ [0, π] satisfies cos θN (x, y, ξ ) =

d 1 y x y x sin + cos cos . xj ξj sin xy N N N N j =1

Since d

1 2 2 xj ξj + Ox,y N −4 cos θN (x, y, ξ ) = 1 − x + y − 2 2 2N j =1

=1−

1 u(x, y, ξ )2 + Ox,y N −4 , 2N 2

it follows that

1 2 −2 θN (x, y, ξ ) = 2 arcsin u(x, y, ξ ) + Ox,y N 2N

1 u(x, y, ξ )2 + Ox,y N −2 + Ox,y N −2 = N ρu(x, y, ξ ) + ox,y,ρ (1) , = n √ where the last step uses the uniform continuity of the function t ∈ [0, M] → t for any M > 0, n = 1. and the relation limN →∞ Nρ

4068


Thus, by (3.22) and (3.15), we have lim bN (ρ, x, y) = cm(ρ) lim

N →∞

N →∞ ξ y

ρu(x, y, ξ ) + ox,y,ρ (1) n−2λκ +1 Cnλκ cos dμκy (ξ ) n

−λ + 1

ρu(x, y, ξ ) κ 2 Jλκ − 1 ρu(x, y, ξ ) dμκy (ξ )

= cm(ρ)

2

ξ y

−λ + 1

= cm(ρ)Vκ ρu(x, y, ·) κ 2 Jλκ − 1 ρu(x, y, ·) (y), 2

where we used the fact that Cnλκ ∞ cn2λκ −1 , the bounded convergence theorem and (3.15) in the last step. This proves Assertion 2. In summary, we have shown the theorem with the additional assumption |m(t)| c1 e−c2 t . Finally, we prove that the conclusion of Theorem 3.1 remains true without the additional assumption |m(t)| c1 e−c2 t . To this end, let mδ (t) = m(t)e−δt for δ > 0, and define Tmδ : L2 (Rd , h2κ ) → L2 (Rd ; h2κ ) by

f ∈ L2 Rd ; h2κ .

Fκ (Tmδ f )(ξ ) = mδ (ξ )Fκ f (ξ ), It is known (see [8, p. 191]) that given any ε > 0, f → on Lp (Sd ; h2κ ) that satisfies ∞ −nε κ sup e projn f ε>0 n=0

2) Lp (Sd ;h κ

∞

n=0 e

κ −nε proj nf

is a positive operator

f Lp (Sd ;h2 ) . κ

Indeed, this follows from [6, Theorem 4.2] and the fact that Vκ is positive, which was proved in [13]. Thus, applying Theorem 3.1 for the already proven case, we have supTmδ f Lp (Rd ;h2κ ) cAf Lp (Rd ;h2κ ) .

(3.23)

δ>0

On the other hand, from the definition we can decompose the operator Tmδ as Tmδ f = Pδ (Tf ),

(3.24)

where Fκ (Tf )(ξ ) = m(ξ )Fκ f (ξ ) and Fκ (Pδ f )(ξ ) = e−δξ Fκ f (ξ ). The function Pδ f is called the Poisson integral of f , and it can be expressed as a generalized convolution (see [3]) Pδ f (x) := (f ∗κ Pδ )(x)


4069

with d

Pδ (x) := 2γκ + 2

Γ (γκ + d+1 ) δ . √ 2 d+1 2 π (δ + x2 )γκ + 2

It was shown in [3, Theorem 6.2] that lim Pδ f (x) = f (x),

δ→0+

a.e. x ∈ Rd

for any f ∈ Lq (Rd ; h2κ ) with 1 q < ∞. Since m is bounded, Tf ∈ L2 (Rd ; h2κ ) for f ∈ L2 (Rd ; h2κ ). Thus, for any f ∈ S, using (3.24), lim Tmδ f (x) = lim Pδ (Tf )(x) = Tf (x),

δ→0+

δ→0+

a.e. x ∈ Rd ,

(3.25)

which combined with (3.23) and the Fatou theorem implies the desired estimate Tf Lp (Rd ;h2κ ) cAf Lp (Rd ;h2κ ) . This completes the proof of the theorem. 4. Applications 4.1. Hörmander’s multiplier theorem and the Littlewood–Paley inequality As a first application of Theorem 3.1, we shall prove the following Hörmander type multiplier theorem: Theorem 4.1. Let m : (0, ∞) → R be a bounded function satisfying m∞ A and Hörmander’s condition 1 R

2R (r) m (t) dt AR −r ,

for all R > 0,

R

where r is the smallest integer λκ + 1. Let Tm be an operator on L2 (Rd ; h2κ ) defined by

Fκ (Tm f )(ξ ) = m ξ Fκ f (ξ ), Then Tm f κ,p Cp Af κ,p for all 1 0 and = 0, 1, . . . . Then r μ = ε r

m (εt1 + · · · + εtr + ε) dt1 · · · dtr (r)

[0,1]r

(r) m (t1 + · · · + tr + ε) dt1 · · · dtr ε r−1

[0,ε]r

ε(r+)

(r) m (t) dt.

ε

This implies that for 2j r,

2

j (r−1)

j +1 2

2 r μl 2j (r−1) ε r−1

j +1

l=2j

l=2j

ε(r+)

(r) m (t) dt

ε j +1 +r) ε(2

(r − 1)2

(r) m (t) dt

j (r−1) r−1

ε

2j ε 2j +2 ε

2

j (r−1)

(r − 1)ε

(r) m (t) dt cr A,

r−1 2j ε

where the last step uses (4.1). On the other hand, however, for 2j r, we have

2

j (r−1)

j +1 2

r μl cr max|μj | cr A.

l=2j

j

Thus, using Theorem 2.3, we deduce ∞ κ sup m(εn) projn f ε>0 n=0

Lp (Sd ;h2κ )

The desired conclusion then follows by Theorem 3.1.

cf Lp (Sd ;h2 ) . κ

2

Remark 4.1. Hörmander’s condition is normally stated in the following form

1 R

1 2R 2 (r) 2 m (t) dt AR −r ,

for all R > 0.

(4.2)

R

See, for instance, [10, Theorem 5.2.7]. Clearly, the condition (4.1) in Theorem 4.1 is weaker than (4.2). On the other hand, however, Theorem 4.1 is applicable only to radial multiplier m( · ).


4071

Corollary 4.2. Let Φ be an even C ∞ -function that is supported in the set {x ∈ R: and satisfies either

Φ 2−j ξ = 1,

9 10

|x|

21 10 }

ξ ∈ R \ {0},

j ∈Z

or

Φ 2−j ξ 2 = 1,

ξ ∈ R \ {0}.

j ∈Z

Let j be an operator defined by

Fκ ( j f )(ξ ) = Φ 2−j ξ Fκ f (ξ ),

ξ ∈ Rd .

Then we have 1 2 2 | j f | f κ,p ∼κ,p j ∈Z

κ,p

holds for all f ∈ Lp (Rd ; h2κ ) and 1 −1, the Bochner–Riesz means of order δ for the Dunkl transform are defined by BRδ

2 hκ ; f (x) = c

yR

y2 1− 2 R

δ Fκ f (y)Eκ (ix, y)h2κ (y) dy,

R > 0.

(4.3)

Convergence of the Bochner–Riesz means in the setting of Dunkl transform was studied recently by Thangavelu and Yuan Xu [17, Theorem 5.5], who proved that if δ > λκ := d−1 2 + γκ and 1 p ∞ then

sup BRδ h2κ ; f κ,p cf κ,p .

(4.4)

R>0

Our next result concerns the critical indices for the validity of (4.4) in the case of G = Zd2 : Theorem 4.3. Suppose that G = Zd2 , f ∈ Lp (Rd ; h2κ ), 1 p ∞, and | p1 − 12 | (4.4) holds if and only if 1 1 1 δ > δκ (p) := max (2λκ + 1) − − , 0 . p 2 2

1 2λκ +2 .

Then

(4.5)

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It should be pointed out that the result of [17, Theorem 5.5] is applicable to the case of a general finite reflection group G, while Theorem 4.3 above applies to the case of Zd2 only. Proof. We start with the proof of the sufficiency. Assume that κ := (κ1 , . . . , κd ) and hκ (x) := d κj d j =1 |xj | . Let κ = (κ, 0) and hκ (x, xd+1 ) = hκ (x) for x ∈ R and xd+1 ∈ R. Set m(t) = 2 δ (1 − t )+ . By the equivalence of the Riesz and the Cesàro summability methods of order δ 0 (see [9]), we deduce from Theorem 2.4 ∞ m(εn) projκn f sup ε>0 n=0

Lp (Sd ;h2κ )

cf Lp (Sd ;h2

κ

)

whenever | p1 − 12 | 2σ 1 +2 and δ > δκ (p), where σκ = λκ and δκ (p) = δκ (p). Thus, invoking κ Theorem 3.1, we conclude that for δ > δκ (p), δ 2 B h ; f κ

1

κ,p

cf κ,p .

The estimate (4.4) then follows by dilation. This proves the sufficiency. The necessity part of the theorem follows from the corresponding result for the Hankel transform. To see this, let f (x) = f0 (x) be a radial function in Lp (Rd , h2κ ). Using (4.3) and Lemma 2.2(vii), we have (vii), we have

BRδ

2 hκ ; f (x) =

R

r2 δ 1 − 2 Hλκ − 1 f0 (r)r 2λκ Eκ ix, ry h2κ y dσ y dr. 2 R Sd−1

0

However, by [17, Proposition 2.3] applied to n = 0 and g = 1, we have

1

rx −λκ + 2 2 Eκ ix, ry hκ y dσ y = c Jλκ − 1 rx . 2 2

Sd−1

It follows that

BRδ

2 hκ ; f (x) = c

R

r2 1− 2 R

δ

1

rx −λκ + 2 Hλκ − 1 f0 (r) Jλκ − 1 rx r 2λκ dr 2 2 2

0

Rδ f0 x , = cB δ denotes the Bockner–Riesz mean of order δ for the Hankel transform H where B R λκ − 12 . However, δ p 2λ , 0 < δ < λκ , is bounded on L ((0, ∞), t κ ) if and only if it is known (see [19]) that B R

2λκ + 1 2λκ + 1 1. For any i = j we introduce off-diagonal subspaces of H of the form H(i, j ) =

∞

m=1 (j,i2 ) =··· =(im ,im )

0 Hj,i ⊗ Hi02 ,i3 ⊗ · · · ⊗ Hi0m ,im 2

and the associated off-diagonal partial isometries Vi,j : Hi,j ⊗ H(i, j ) → H for i = j : 0 0 ⊗ · · · ⊗ Hj0m ,jm → Hj,j ⊗ · · · ⊗ Hj0m ,jm , ξi,j ⊗ Hj,j 1 1 0 0 0 0 Hi,j ⊗ Hj,j ⊗ · · · ⊗ Hj0m ,jm → Hi,j ⊗ Hj,j ⊗ · · · ⊗ Hj0m ,jm , 1 1 where m 1. 0 and Each H(i, j ) is spanned by simple tensors which do not begin with vectors from Hi,j for that reason it is suitable for the left free action of the operators creating such vectors. Thus, roughly speaking, both types of partial isometries jointly replace the unitary maps used in free probability. It is the diagonal subordination property which is responsible for distinguishing two types of isometries. Consider an array of C ∗ -algebras (Ai,j ), each with a unit 1i,j and a state ϕi,j , and let (Hi,j , πi,j , ξi,j ) be the associated GNS triples, so that ϕi,j (a) = πi,j (a)ξi,j , ξi,j for any a ∈ Ai,j . For any i, j , let λi,j be the ∗-representation of Ai,j given by ∗ λi,j (a) = Vi,j πi,j (a) ⊗ IH(i,j ) Vi,j

for a ∈ Ai,j ,

where IH(i,j ) denotes the identity on H(i, j ). Note that these representations are, in general, non-unital. In fact, ∗ = ri,j + si,j , λi,j (1i,j ) = Vi,j Vi,j

where ri,j and si,j are canonical projections in B(H) given by ri,j = PH(i,j )

and si,j = PK(i,j ) ,

0 ⊗ H(i, j ). For given i, j , the projections r where K(i, j ) = Hi,j i,j and si,j are orthogonal and their sum is the canonical projection onto the subspace of H onto which λi,j (Ai,j ) acts nontrivially.

R. Lenczewski / Journal of Functional Analysis 258 (2010) 4075–4121

4081

The λi,j ’s remind the representations λi of free probability [24,28], but the corresponding operators λi,j (a) have larger kernels. Using λi,j ’s, we shall define product representations on A :=

Ai,j ,

i,j

the free product without identification of units, equipped with the unit 1A , and products of states which are analogs of the free product representation and the free product of states, respectively. Definition 2.3. The matricially free product representation πM = ∗M i,j πi,j is the unital ∗-homomorphism λ : A → B(H) given by the linear extension of λ(1A ) = 1 and λ(a1 a2 . . . an ) = λi1 ,j1 (a1 )λi2 ,j2 (a2 ) . . . λin ,jn (an ) for any ak ∈ Aik ,jk , k = 1, . . . , n, with (i1 , j1 ) = (i2 , j2 ) = · · · = (in , jn ). The associated state ϕ = ∗M i,j ϕi,j : A → C is given by

ϕ(a) = πM (a)ξ, ξ and will be called the matricially free product of (ϕi,j ). Basic properties of the product state ϕ are collected in the proposition given below. Roughly speaking, they show that this state (on the free product of C ∗ -algebras without identification of units) has similar properties as the free product of states (on the free product of C ∗ -algebras with identification of units) except that the units of these algebras act as units only on ‘matricial’ tensor products and otherwise they act as null projections. For that purpose, it will be useful to introduce sets of indices associated with ‘matricial’ tensor products: Λn =

(i1 , i2 ), (i2 , i3 ), . . . , (in , in+1 ) : (i1 , i2 ) = (i2 , i3 ) = · · · = (in , in+1 )

and their union Λ = ∞ n=1 Λn . Finally, I stands for the unital subalgebra of A generated by the units 1i,j . In analogy to the notion of marginal laws in classical probability, by marginal moments we shall understand moments of the form ϕi,j (a1 . . . ak ), where a1 . . . ak ∈ Ai,j and i, j are arbitrary. Proposition 2.2. Let ϕ be the matricially free product of states (ϕi,j ) and let ak ∈ Aik ,jk , where k ∈ [n] and (i1 , j1 ) = · · · = (in , jn ). 1. If ak ∈ Ker ϕik ,jk for k ∈ [n], then ϕ(a1 a2 . . . an ) = 0. 2. If ar = 1ir ,jr and am ∈ Ker ϕim ,jm for r < m n, then ϕ(a1 . . . an ) =

ϕ(a1 . . . ar−1 ar+1 . . . an ) 0

if ((ir , jr ), . . . , (in , jn )) ∈ Λ, otherwise.

4082


3. For any a ∈ A, u1 , u2 ∈ I and i, j ∈ I , it holds that ϕ(u1 au2 ) = ϕ(u1 )ϕ(a)ϕ(u2 ) and ϕ(1i,j ) = δi,j . 4. The restriction of ϕ to Aj,j is ϕj,j for any j ∈ I . 5. The mixed moments ϕ(a1 a2 . . . an ) are uniquely expressed in terms of marginal moments. Proof. If ak ∈ Ker ϕik ,jk for k ∈ [n], where (i1 , j1 ) = · · · = (in , jn ), then it follows from the definition of the λi,j that / Λ or in = jn , 1. πM (a1 . . . an )ξ = 0 if ((i1 , j1 ), . . . , (in , jn )) ∈ 2. πM (a1 . . . an )ξ ∈ Hi01 ,j1 ⊗ · · · ⊗ Hi0n ,jn if ((i1 , j1 ), . . . , (in , jn )) ∈ Λ and in = jn . In both cases we obtain a vector orthogonal to ξ on the RHS, which proves (1). Suppose now that the assumptions of (2) hold. If ((ir , jr ), . . . , (in , jn )) ∈ Λ, then λir ,jr (1ir ,jr ) acts as a unit on Hi0r+1 ,jr+1 ⊗ · · · ⊗ Hi0n ,jn by the definition of the representations λi,j . On the other hand, λir ,jr (1ir ,jr ) kills any simple tensor beginning with h ∈ Hi0r+1 ,jr+1 if jr = ir+1 or ((ir+1 , jr+1 ), . . . , (in , jn )) ∈ / Λ since Vir ,jr does, which completes the proof of (2). In turn, (3) follows from the action of the λ(1i,j ) onto ξ . That ϕ agrees with ϕj,j on Aj,j for any j ∈ I follows from the action of the λj,j (a), a ∈ Aj,j , onto ξ , namely πM (a)ξ = (πj,j (a)ξ )0 + ϕj,j (a)ξ , which gives (4). Finally, (5) is a consequence of (1)–(2). 2 In a similar way we can define states associated with other unit vectors from H. We shall 0 , j ∈ I , which are in consider the simplest case of states associated with unit vectors ej ∈ Hj,j the ranges of πj,j (Aj,j ), respectively, namely ϕj : A → C defined by the formulas

ϕj (a) = πM (a)ej , ej , called conditions associated with ϕ, which will be used for computing normalized traces. Most properties of the states ϕj are inherited from ϕ as the proposition given below demonstrates. However, ϕj |I is quite different than ϕ|I due to different normalization conditions. Proposition 2.3. Let ϕj , j ∈ I , be the conditions associated with ϕ and let ak ∈ Aik ,jk , where k ∈ [n] and (j, j ) = (i1 , j1 ) = · · · = (in , jn ) = (j, j ). 1. If ak ∈ Ker ϕik ,jk for k ∈ [n], then ϕj (a1 a2 . . . an ) = 0 for each j . 2. If ar = 1ir ,jr and am ∈ Ker ϕim ,jm for r < m n, then ϕj (a1 . . . an ) =

ϕj (a1 . . . ar−1 ar+1 . . . an ) 0

if ((ir , jr ), . . . , (in , jn )) ∈ Λ, otherwise.

3. For any a ∈ A, u1 , u2 ∈ I and i, j, k ∈ I , it holds that ϕj (u1 au2 ) = ϕj (u1 )ϕj (a)ϕj (u2 )

and ϕj (1i,k ) = δj,k .

4. The restriction of ϕj to Ai,j is ϕi,j for any i = j . 5. The mixed moments ϕj (a1 a2 . . . an ) are uniquely expressed in terms of marginal moments.


4083

Proof. Properties (1) and (2) follow from (1) and (2) of Proposition 2.2. The normalization in 0 . In this context, notice that the unit (3) follows directly from the action of λi,k (1i,k ) onto Hj,j vectors ej play the same role with respect to the action of the λi,j (a) for any i = j as ξi,j plays with respect to the action of πi,j (a), where a ∈ Ai,j , and thus ϕj (a) = λi,j (a)ej , ej =

πi,j (a)ξi,j , ξi,j = ϕi,j (a), which gives (4) for i = j . Finally, there exists bj ∈ Aj,j ∩ Ker ϕ such that ϕj (w) = ϕ bj∗ wbj for any w ∈ i,j Ai,j , which reduces computations of mixed moments in each state ϕj to computations of mixed moments in the state ϕ. This proves (5). 2 Remark 2.1. The states ϕ and (ϕj ) share together the property of extending the array of states (ϕi,j ). Thus, ϕ extends the diagonal states ϕj,j for all j , but it does not extend the off-diagonal states ϕi,j for i = j since πM (a)ξ = 0 for any a ∈ Ai,j . In turn, ϕj extends the off-diagonal states ϕi,j , where i = j , but it does not extend ϕj,j . This is a natural consequence of differences in the definitions of the diagonal and off-diagonal partial isometries. Finally, let us denote by λ(I) the unital commutative ∗-subalgebra of B(H) generated by the λ(1i,j ), where i, j ∈ I . By abuse of notation, λ(1i,j ) will also be denoted by 1i,j (in general, these projections are not mutually orthogonal). 3. Strongly matricially free products Of special importance is the subspace of the matricially free Fock space, called the ‘strongly matricially free Fock space’, in which the diagonal Hilbert spaces appear only at the end of tensor products. The main reason is that it is related to both free and monotone Fock spaces. We also study the associated product states which can be viewed as direct generalizations of both free and monotone products of states. := (Hi,j ) we understand the Definition 3.1. By the strongly matricially free Fock space over H of the form subspace of M(H) = CΩ ⊕ R(H)

∞

m=1 i1 =··· =im n1 ,...,nm ∈N

m 1 2 Hi⊗n ⊗ Hi⊗n ⊗ · · · ⊗ Hi⊗n , m ,im 1 ,i2 2 ,i3

with the canonical inner product. A justification for the word ‘strong’ is that in this case the words i1 i2 . . . im which label the tensor products in the above definition satisfy i1 = i2 = · · · = im . Example 3.1. The simplest space of this type is associated with a two-dimensional square ar Then ray H. = R(H)

∞ m=0

R(m) (H),

4084


where the first few summands are of the form = CΩ, R(0) (H) = H1,1 ⊕ H2,2 , R(1) (H) = H⊗2 ⊕ H⊗2 ⊕ (H1,2 ⊗ H2,2 ) ⊕ (H2,1 ⊗ H1,1 ), R(2) (H) 1,1 2,2 ⊗3 = H ⊕ H⊗3 ⊕ H2,1 ⊗ H⊗2 ⊕ H1,2 ⊗ H⊗2 ⊕ H⊗2 ⊗ H1,1 R(3) (H) 1,1 2,2 1,1 2,2 2,1 ⊗2 ⊕ H1,2 ⊗ H2,2 ⊕ (H1,2 ⊗ H2,1 ⊗ H1,1 ) ⊕ (H2,1 ⊗ H1,2 ⊗ H2,2 ), we do not have tensor products like H2,2 ⊗ H2,1 ⊗ H1,1 and H1,1 ⊗ etc. In contrast to M(H), H1,2 ⊗ H2,2 in the summand of the third order. = (Hi,j ), we have inclusions Remark 3.1. For a given array of Hilbert spaces H ⊆ M(H) ⊆F R(H)

Hi,j

i,j

which, in most cases, are proper. Moreover, if we have a square array and Hi,j ∼ = Hi for any i, j ∈ I , where (Hi )i∈I is a family of Hilbert spaces, then there is a natural isomorphism ∼ R(H) =F

Hi

i∈I ⊗n ⊗n ⊗n m ∼ 1 2 since Hi⊗n ⊗ Hi⊗n ⊗ · · · ⊗ Hi⊗n = Hi1 1 ⊗ Hi2 2 ⊗ · · · ⊗ Him m for any i1 , i2 , . . . , im ∈ I , m ,im 1 ,i2 2 ,i3 n1 , . . . , nm , m ∈ N. Similarly, if we have a lower-triangular array and Hi,j ∼ = Hi for any i j , is isomorphic to the monotone Fock space. then R(H)

Moreover, as expected, there is a product of Hilbert spaces related to the strongly matricially free Fock space, and an analog of Proposition 2.1 holds. Definition 3.2. By the strongly matricially free product of (Hi,j , ξi,j ) we understand the pair (G, ξ ), where G is the subspace of H of the form G = Cξ ⊕

∞

m=1 i1 =··· =im

Hi01 ,i2 ⊗ Hi02 ,i3 ⊗ · · · ⊗ Hi0m ,im .

We denote it (G, ξ ) = ∗Si,j (Hi,j , ξi,j ). If we consider a family of unital C ∗ -algebras (Ai )i∈I , each equipped with a family of states (ϕi,j )j ∈I , then we can look at this product space as follows. If (Hi,j , πi,j , ξi,j ) is the GNS triple associated with the pair (Ai , ϕi,j ), then the Hilbert space Hi,j (as well as the corresponding state and representation) taken as the representation space for the algebra Ai at some given tensor site depends on the algebra Aj represented at the following tensor site on the space Hj,k for some k. It is worth noting that in this framework we can assume that for fixed i ∈ I all vectors ξi,j ,


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j ∈ I , are identified since we can take the tensor product of Hilbert spaces j Hi,j and set ξi = j ∈I ξi,j for each i ∈ I . It is not hard to see that in this framework our model is related to freeness with infinitely many states [8,9]. The construction of the product state is similar to that of the matricially free product. The only difference in all definitions is that the sets Λn are replaced by Γn =

(i1 , i2 ), (i2 , i3 ), . . . , (in , in+1 ) : i1 = i2 = · · · = in

and their union Λ by Γ = ∞ n=1 Γn . Note that the conditions which define Γn ’s are stronger than those which define Λn ’s and therefore all objects constructed in the strong case are obtained from the standard ones by a projection-type operation. The partial isometries in the strong case, denoted by Wi,j ’s, remind Vi,j ’s except that they refer to G rather than H. In particular, the diagonal partial isometries Wj,j : Hj,j ⊗ G(j, j ) → G are given by ξj,j ⊗ ξ → ξ

0 0 and Hj,j ⊗ ξ → Hj,j ,

where G(j, j ) = Cξ for any j , whereas the off-diagonal partial isometries Wi,j and the associated subspaces G(i, j ) are similar to those in the standard case. Definition 3.3. Let ρi,j be the ∗-representation of Ai,j on G given by the formula ∗ ρi,j (a) = Wi,j πi,j (a) ⊗ IG (i,j ) Wi,j

where a ∈ Ai,j ,

for any (i, j ) ∈ J . The corresponding strongly matricially free product representation πS = ∗Si,j πi,j and strongly matricially free product of states ∗Si,j ϕi,j are defined in terms of the ρi,j as in the matricially free case. Using appropriate direct sums of these representations, we can reproduce products of C ∗ probability spaces in free probability of Voiculescu and in monotone probability of Muraki. If (Hi , ξi ) is a family of Hilbert spaces with distinguished unit vectors, then in the decomposition theorem given below, ∗i∈I (Hi , ξi ), i∈I (Hi , ξi ) and i∈I (Hi , ξi ), respectively, stand for free, monotone and boolean products of Hilbert spaces. Theorem 3.1 (Decomposition theorem). Let (Ai,j , ϕi,j ) = (Ai , ϕi ) for any i, j and let (πi , Hi , ξi ) be the GNS triple associated with (Ai , ϕi ) for any i ∈ I . 1. If (Ai,j , ϕi,j ) is a square array, then (G, ξ ) ∼ = ∗i∈I (Hi , ξi ) and each λi = j ∈I ρi,j is the canonical ∗-representation of Ai on ∗i∈I (Hi , ξi ). ∼ 2. If (Ai,j , ϕi,j ) is a lower-triangular array, then (G, ξ ) = i∈I (Hi , ξi ) and each τi = ij ρi,j is the canonical ∗-representation of Ai on i∈I (Hi , ξi ). 3. If (Ai,j , ϕi,j ) is a diagonal array, then (G, ξ ) ∼ = i∈I (Hi , ξi ) and each ρi,i is the canonical ∗-representation of Ai on i∈I (Hi , ξi ). Proof. First, let us remark that therepresentations λi and τi are well defined since the corresponding direct sums of operators j ρi,j (a) are convergent in the strong-operator topology on

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B(G) for any a ∈ Ai and i ∈ I . Now, since Hi,j ∼ = Hi for any i, j , we have a natural isomorphism Hi01 ,i2 ⊗ Hi02 ,i3 ⊗ · · · ⊗ Hi0m ,im ∼ = Hi01 ⊗ Hi02 ⊗ · · · ⊗ Hi0m for any i1 = i2 = · · · = im or any i1 > i2 > · · · > im , in the case of square or lower-triangular arrays, respectively, which leads to the corresponding isomorphisms (G, ξ ) ∼ = ∗i∈I (Hi , ξi )

or

(G, ξ ) ∼ = i∈I (Hi , ξi )

(recall our tacit assumption that I is linearly ordered when dealing with triangular arrays). Then partial isometries which lie in the same row of the array (Wi,j ) are orthogonal in the sense that Wi,j Wi,k = δj,k Wi,j for any i, j , k (in the monotone case the array of partial isometries is lower-triangular and we have here i > j > k). Therefore, we have an orthogonal direct sum decomposition

Wi,j = Vi ,

i ∈ I,

j ∈I

of the unitaries Vi used in the definition of the free product representation [21,24], and an analogous decomposition in the monotone case. The direct sum decompositions of λi (a) and τi (a) in terms of ρi,j (a)’s, where a ∈ Ai , follow then immediately from Definition 3.3, which completes the proof of (1) and (2). In particular, it follows that each λi is unital, but τi are, in general, non-unital. Since the case of a diagonal array is rather elementary, the proof is completed. 2 Consequently, if A is the C ∗ -algebra generated by the family {λi (Ai )} of subalgebras of B(G) and ϕ(.) = .ξ, ξ , then (A, ϕ) is the free product of C ∗ -probability spaces. Similar statements hold for the monotone and boolean products of C ∗ -probability spaces, except that the identity I ∈ B(G) has to be added to the generators. In the natural way this leads to properties (A), (B) and (C) of the introduction, of which the first two can be viewed as decompositions of free and monotone independent random variables in terms of strongly matricially free ones. The above theorem allows us to view the strongly matricially free product of states as a deformation of the free product of states, obtained by a natural direct sum decomposition of the free product of Hilbert spaces. The matricially free product of states is then obtained from its strong counterpart by an extension to a slightly larger Hilbert space which includes all products which arise naturally in matrix multiplication. It is clear that basic results on the strongly matricially free product of states are similar to those for the matricially free product and therefore we state them in an abbreviated form without a proof. Proposition 3.1. Let ϕ be the strongly matricially free product of states (ϕi,j ) and let ϕj , j ∈ I , be the associated conditions. Then the statements of Propositions 2.2–2.3 remain true, with Λ replaced by Γ . As we have mentioned earlier, one of the advantages of using the strong structures is that they are straightforward generalizations of those in free probability (in the case of square arrays) and monotone probability (in the case of lower-triangular arrays). However, it is the matricial freeness which gives a connection with random matrices.


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Remark 3.2. Slightly more general is the case when the diagonal and off-diagonal states differ, but the latter stay the same within each row, namely (Ai,i , ϕi,i ) = (Ai , ϕi )

and (Ai,j , ϕi,j ) = (Ai , ψi )

for i = j,

where each Ai is equipped with two states, ϕi and ψi , respectively. Similar reasoning to that in the free case leads then to the conditionally free product of states (for square arrays) and conditionally monotone products of states (for lower-triangular arrays). 4. Matricial freeness Guided by the notion of the matricially free product of states, we shall introduce now the associated concept of independence called ‘matricial freeness’, and closely related to it, ‘strong matricial freeness’. They involve arrays of noncommutative probability spaces and to some extent they remind models with many states [6,8,9], but they cannot be reduced in a natural way to any of these (freeness with infinitely many states has some non-empty intersection with ‘strong matricial freeness’ and conditional freeness is its special case). Moreover, we will study discrete (strong) matricially free Fock spaces. Let A be a unital algebra with an array (Ai,j ) of subalgebras of A. We will assume that each Ai,j has an internal unit 1i,j which may be different from the unit of A, and we assume that the unital subalgebra I generated by all internal units is commutative. Let ϕ be a distinguished state on A and let {ϕj : j ∈ I } be a family of additional states on A, where by a state we understand a normalized linear functional. If A is a unital ∗-algebra, then we assume that Ai,j ’s are ∗subalgebras and all states are positive functionals. Further, Definition 4.1. States ϕj , j ∈ I , will be called conditions associated with ϕ if for any j ∈ I there exist bj , cj ∈ Aj,j ∩ Ker(ϕ) such that ϕj (a) = ϕ(cj abj ) for any a ∈ A. The array of states on A given by ϕj,j = ϕ

and ϕi,j = ϕj

for any i = j

will be said to be defined by ϕ and the associated conditions ϕj . In addition, if A is a ∗-algebra, we assume that cj = bj∗ for any j . Throughout this paper we will assume the following normalization conditions: ϕ(1i,j ) = δi,j

and ϕj (1i,k ) = δj,k

for any i, j , k. Although they are natural in view of the Hilbert space formulation presented before and will be assumed in this paper, we prefer to exclude them from the definition given below. In particular, the normalization of each ϕj implies that if a ∈ Ai,k and j = k, then ϕj (a) = 0. Moreover, it is equivalent to scaling the variables cj , bj according to ϕ(cj bj ) = 1. Definition 4.2. Let (ϕi,j ) be the array defined by ϕ and the associated conditions ϕj . We say that (1i,j ) is a matricially free array of units associated with (Ai,j ) and (ϕi,j ) if

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1. ϕ(u1 au2 ) = ϕ(u1 )ϕ(a)ϕ(u2 ) for any a ∈ A and u1 , u2 ∈ I, 2. for ak ∈ Aik ,jk ∩ Ker ϕik ,jk , where r < k n, ϕ(a1ir ,jr ar+1 . . . an ) =

ϕ(aar+1 . . . an ) 0

if ((ir , jr ), . . . , (in , jn )) ∈ Λ, otherwise,

where a ∈ A is arbitrary and (ir , jr ) = · · · = (in , jn ). The array (1i,j ) is called a strongly matricially free array of units if Λ is replaced by Γ . The above definition enables us to define the concepts of matricial freeness and its strong version called strong matricial freeness. They both bear some resemblance to freeness, but the main difference is that the identified unit in the context of freeness is replaced by the (strongly) matricially free array of units. In fact, as we have already remarked, it is the strong matricial freeness which can be viewed as a direct generalization of freeness. Definition 4.3. We say that (Ai,j ) is matricially free with respect to the array (ϕi,j ) defined by ϕ and the associated conditions ϕj if 1. for any ak ∈ Ker ϕik ,jk ∩ Aik ,jk , where k ∈ [n] and (i1 , j1 ) = · · · = (in , jn ), ϕ(a1 a2 . . . an ) = 0, 2. (1i,j ) is a matricially free array of units associated with (Ai,j ) and (ϕi,j ). In an analogous manner we define strongly matricially free arrays of subalgebras. Definition 4.4. The array of variables (ai,j ) in a unital algebra A will be called (strongly) matricially free with respect to the array (ϕi,j ) of states on A if the array (C[ai,j , 1i,j ]) is (strongly) matricially free with respect to (ϕi,j ) for some array of elements (1i,j ) of A which is a (strongly) matricially free array of units. If A is a unital ∗-algebra, then, in addition, we require that the functionals ϕi,j are positive, the Ai,j are ∗-subalgebras and the 1i,j are projections. Then an array of variables (ai,j ) will be called ∗-(strongly) matricially free if the array of ∗-algebras ∗ , 1 ) is (strongly) matricially free. (C ai,j , ai,j i,j Using the above definitions, we can uniquely express mixed moments under ϕ and thus under ϕj ’s of arbitrary matricially free random variables in terms of marginal moments under ϕi,j ’s. To see how this computation works, it is convenient to assume that neighboring variables come from different algebras and use the recurrence given below. Remark 4.1. Suppose that (Ai,j ) is matricially free with respect to the array (ϕi,j ) defined by ϕ and the associated conditions ϕj . Writing ak = ak0 + ϕik ,jk (ak )1ik ,jk ∈ Aik ,jk for any 1 k n, we obtain the recurrence


ϕ(a1 . . . an ) =

ϕik ,jk (ak )ϕ a10 . . . 1ik ,jk . . . an0

1kn

+

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ϕik ,jk (ak )ϕil ,jl (al )ϕ a10 . . . 1ik ,jk . . . 1il ,jl . . . an0

1k 0,

and the projection 1i,j . However, the corresponding moments vanish: q ∗p ϕi,j i,j i,j = 0 for any i, j since p + q > 0. Moreover, ϕi,j (1i,j ) = 1 for any i, j . Therefore, in order to show that condition (1) of Definition 4.2 holds, it is enough to show that q ∗p q ∗p ϕ i11,j1 i1 ,j11 . . . inn,jn in ,jnn = 0 whenever (i1 , j1 ) = · · · = (in , jn ) and p1 + q1 > 0, . . . , pn + qn > 0. The same argument as in [27] allows us to reduce the proof to the case when q1 = · · · = qn = 0, which implies that p1 > 0, . . . , pn > 0. But then the moment clearly vanishes. This proves condition (1) of Definition 4.2. Condition (2) follows easily from the definition of the projections 1i,j in view of the relations given in Proposition 4.1. This completes the proof. 2 Example 4.2. For i, j ∈ I , let Gi,j ∼ = F (1) be the free group on one generator gi,j with unit i,j . 2 of l 2 (∗ G ) spanned by vectors of the form δ(g), where g is either Consider the subspace lM i,j i,j the unit e of the free product ∗i,j Gi,j , or a product of the form g1 g2 . . . gm , where gk ∈ G0ik ,ik+1 := 2 , which Gik ik+1 \ {ik ,ik+1 } for each k, with (i1 , i2 ) = · · · = (im , im+1 ) and im+1 = im . The space lM is a simple example of the matricially free Fock space, can be viewed as the space of square integrable functions on the ‘matricially free product of groups’ (which is not a group). Let 1i,j 2 spanned by vectors δ(g), where denote the projection from l 2 (∗i,j Gi,j ) onto the subspace of lM 0 0 g begins with an element from Gi,j or Gj,k for some k or, in the case of i = j , also g = e. We can now define λi,j (g) = λi,j (g)1i,j for g ∈ Gi,j , where by λi,j we denote the left regular representation of Gi,j on l 2 (∗i,j Gi,j ). Then the array (Ai,j ), where Ai,j is the ∗-subalgebra of B(l 2 (∗i,j Gi,j )) generated by λi,j (gi,j ) and 1i,j , with the standard involution, is matricially free with respect to the array (ϕi,j ), where the diagonal states coincide with ϕ(.) = .δ(e), δ(e) and the associated conditions are ϕj (.) = .δ(gj,j ), δ(gj,j ), where j ∈ I . In a similar way we proceed in the case of functions on the ‘strongly matricially free product of groups’. For classical results on random walks on free groups, see [13]. Example 4.3. To be more concrete, we consider a two-dimensional square array of free groups on one generator. Then our Hilbert space is 2 lM = span δ(e), δ(g): g ∈ H1 ∪ H2 ,


4095

where k k l k l m k l m , g1,2 g2,2 , g1,1 g1,2 g2,2 , g1,2 g2,1 g1,1 , . . . : k, l, m ∈ Z0 H1 := g1,1 and k k l k l m k l m H2 := g2,2 , g2,1 g1,1 , g2,2 g2,1 g1,1 , g2,1 g1,2 g2,2 , . . . : k, l, m ∈ Z0 , respectively, with Z0 denoting the set of non-zero integers. The diagonal units 11,1 and 12,2 are projections onto subspaces spanned by δ(e) and δ(g), where g ∈ H1 and g ∈ H2 , respectively. The off-diagonal units 11,2 and 12,1 are defined in a similar way, so that we have decompositions 2 . 11,1 + 11,2 = 12,1 + 12,2 = 1 of the unit 1 on lM Example 4.4. In the case of square integrable functions on the ‘strongly matricially free product of groups’, the Hilbert space is smaller, namely lS2 = span δ(e), δ(g): g ∈ K1 ∪ K2 , where k k l k l m K1 := g1,1 , g1,2 g2,2 , g1,2 g2,1 g1,1 , . . . : k, l, m ∈ Z0 and k k l k l m K2 := g2,2 , g2,1 g1,1 , g2,1 g1,2 g2,2 , . . . : k, l, m ∈ Z0 , which reflects the fact that the diagonal generators act non-trivially only on δ(e). Internal units are reduced accordingly. Note that in this case lS2 ∼ = l 2 (F (2)), which conforms with Theorem 3.1 and the resulting decomposition of the left regular representation. Example 4.5. In the case of an n-dimensional square array of copies of F (1), we form a tree (a subtree of the homogeneous tree H2n2 ) which corresponds to the ‘matricially free product of n2 free groups’. Suppose the root e corresponds to the ‘father’. We distinguish ‘sons’ and −1 , and gi,j ‘daughters’ in each ‘generation’ which correspond to the left action of gj,j or gj,j

−1 or gi,j , respectively, where i = j . The rules of drawing the tree follow from matricial freeness and are the following: each ‘son’ has 1 ‘son’ and 2n − 2 ‘daughters’, whereas each ‘daughter’ has 2 ‘sons’ and 2n − 1 ‘daughters’. Therefore, ‘sons’ and ‘daughters’ correspond to vertices of valencies 2n and 2n + 2, respectively. In Fig. 1 we draw such a tree for n = 2 (black and empty circles are assigned to ‘sons’ and ‘daughters’, respectively). If we make an additional assumption, for instance that ‘daughters’ cannot have ‘sons’ (this fact corresponds to strong matricial freeness, where diagonal generators kill words beginning with the off-diagonal ones), we recover H2n of free probability.

be the discrete strongly matricially free Fock space and let R(H) ∼ Example 4.6. Let R(H) = F ( j Cej ) be the natural isomorphism of Remark 3.1, where {ej : j ∈ N} is an orthonormal basis of some Hilbert space. If (wi,j ) is an infinite matrix with non-negative parameters p and q

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Fig. 1. Matricially free analog of H4 .

above and below the main diagonal, respectively, and 1’s on the diagonal, then it is easy to see that the (p, q)-creation operators studied in [18] can be identified (with the use of this isomorphism) with the strongly convergent sums Ai =

wi,j ki,j ,

j

where i ∈ N and the (p, q)-annihilation operators are their adjoints. A similar approach can be applied to square arrays of arbitrary Hilbert spaces and ∗-representations, which leads to some notion of ‘(p, q)-independence’. Moreover, it can be carried out for more general matrices (wi,j ) within the framework of the strong matricial freeness, which generalizes notions of independence of this type. 5. Traces In this section we introduce some real-valued functions on the set of non-crossing pairpartitions. These functions are obtained by computing traces of a square real-valued matrix. We will assume later that this matrix has non-negative entries which represent variances of probability measures on the real line MR and we will demonstrate that the functions introduced in this section describe the asymptotics of matricially free random variables in central limit theorems. Let N C m denote the set of non-crossing partitions of the set [m], i.e. if π = {π1 , π2 , . . . , πk } ∈ N C m , then there are no numbers i < p < j < q such that i, j ∈ πr and p, q ∈ πs for r = s. The block πr is inner with respect to πs if p < i < q for any i ∈ πr and p, q ∈ πs (then πs is outer with respect to πr ). It is clear that if πr has outer blocks, then there exists a unique block among them, say πs , which is nearest to πr , i.e. if another block, say πt , is outer with respect to πr , then we must have a < p < b for any a, b ∈ πt and p ∈ πs . In that case we shall write πs = o(πr ) and call the pair (πr , o(πr )) the nearest inner–outer pair of blocks. If πi does not have an outer block, it is called a covering block. It is convenient to extend each partition π ∈ N C 2m to the


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partition π obtained from π by adding one block, say π0 = {0, m + 1}, called the imaginary block. If B(π) is the set of blocks of π , we shall denote by Fr (π) the set of all mappings f : B(π) → [r] called colorings of the blocks of π by the set [r] := {1, 2, . . . , r}. Then the pair (π, f ) plays the role of a colored partition. Let N CC m denote the set of non-crossing covered partitions of [m], by which we understand the subset of N C m consisting of those partitions in which 1 and m belong to the same block (if m = 1, we understand that the partition consists of one block). In terms of diagrams, all blocks of π ∈ N CC m , where m > 1, are covered by the block containing 1 and m. We denote by N C 2m and of [m] and non-crossing pair-partitions N CC 2m the sets of non-crossing pair-partitions

∞ covered 2 2 2 N C and N CC = N CC . of [m], respectively, and we set N C 2 = ∞ m m m=1 m=1 It is easy to see that each π ∈ N C m can be decomposed as π = π (1) ∪ π (2) ∪ · · · ∪ π (p) ,

(5.1)

where π (1) , . . . , π (p) are non-crossing covered partitions of subintervals I1 , I2 , . . . , Ip of [m] whose union gives [m]. By a partition of a set I consisting of r elements we understand the corresponding partition of [r]. On the other hand, each π ∈ N CC m can be decomposed as π = π (0) ∪ π (1) ∪ · · · ∪ π (r) ,

(5.2)

where π (0) is the block containing 1 and m and π (1) , π (2) , . . . , π (r) are non-crossing covered partitions of subintervals I1 , I2 , . . . , Ir of [m] \ π0 . Consider now a square real-valued matrix V = (vi,j ) ∈ Mn (R), where n ∈ N. The usual trace and the normalized trace will be denoted Tr(V ) =

n

1 vj,j , n n

vj,j

and tr(V ) =

j =1

j =1

respectively. For each n, we define the ‘diagonalization mapping’ τ : Mn (R) → Dn (R),

τ (V ) = diag vj,1 , . . . , vj,n , j

j

where Dn (R) is the set of square diagonal real-valued matrices of dimension n (by abuse of notation, the same symbol τ is used for all n). In other words, τ computes the sum of all elements of V in each column separately and puts this value on the diagonal. Using the above trace operations, we shall define two real-valued functions on the set N C 2 , denoted v and v0 , associated with given V ∈ Mn (R). Although there is a close similarity between these functions when restricted to N CC 2 (in particular, they are defined as traces of certain matrix-valued quasi-multiplicative functions), note that they are extended to N C 2 in two different ways. Definition 5.1. For a given matrix V ∈ Mn (R), we define a mapping from N C 2 to Dn (R) by assigning to each π ∈ N C 2 the matrix V (π) by the following recursion:

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Fig. 2. Colored partitions with imaginary blocks.

1. if π consists of one block, we set V (π) = τ (V ), 2. if π ∈ N CC 2 consists of more than one block, then V (π) = τ V π (1) . . . V π (r) V

(5.3)

according to the decomposition (5.2), 3. if π ∈ N C 2 , then V (π) = V π (1) V π (2) . . . V π (p)

(5.4)

according to the decomposition (5.1). Let v : N C 2 → R be the function defined by v(π) = tr(V (π)). Example 5.1. For some V ∈ Mn (R), consider three partitions given in Fig. 2. Color each block by a number from the set [n]. Computation of the corresponding values of the function v gives 1 vi,j vj,k , v(π) = tr τ τ (V )V = n i,j,k

1 v(η) = tr τ τ (V )V τ (V ) = vi,j vj,l vk,l , n i,j,k,l

1 v(ζ ) = tr τ τ (V )τ (V )V = vi,k vj,k vk,l . n i,j,k,l

One can see that to each nearest inner–outer pair of blocks (πr , πs ) we assign the matrix element vp,q , where p and q are the colors of πr and πs respectively. Moreover, if a block does not have any outer blocks and is colored by q, then we assign to it the matrix element vq,t , where t is assumed to be the same for all such blocks (one can imagine that we have an additional ‘imaginary block’ colored by t which covers all other blocks). At the end we sum over all colorings. The function v0 is defined on the set N CC 2 in a very similar manner, except that we replace the right multiplier V in (5.3) by its main diagonal V0 := diag(v1,1 , . . . , vn,n ),


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which corresponds to changing only the contribution of the covering block. Then we extend v0 to all of N C 2 by multiplicativity of v0 . Definition 5.2. For a given matrix V0 ∈ Mn (R), we define a mapping from N CC 2 to Dn (R) by assigning to each π ∈ N CC 2 the matrix V0 (π) by the following recursion: 1. if π consists of one block, we set V0 (π) = V0 , 2. if π ∈ N CC 2 consists of more than one block, then V0 (π) = τ V π (1) . . . V π (r) V0

(5.5)

according to the decomposition (5.2). Let v0 : N C 2 → R be the function defined by v0 (π) = Tr(V0 (π)) for π ∈ N CC 2 , extended to N C 2 by multiplicativity v0 (π) = v0 (π (1) ) . . . v0 (π (p) ) according to the decomposition (5.1). Example 5.2. Let us compute the values of v0 corresponding to the partitions in Fig. 2. We have vi,j vj,j , v0 (π) = Tr τ (V )V0 = i,j

v0 (η) = Tr τ (V )V0 Tr(V0 ) = vi,j vj,j vk,k , i,j,k

v0 (ζ ) = Tr τ τ (V )τ (V )V0 = vi,k vj,k vk,k . i,j,k

Note that the main difference (apart from normalization) between v0 (π) and v(π) concerns the blocks which do not have outer blocks. Here, if such a block is colored by q, we assign to it the matrix element vq,q and we do not use ‘imaginary blocks’. Below we shall prove a lemma which gives explicit combinatorial formulas for v(π) and v0 (π). For that purpose, to the blocks B(π, f ) = (π1 , f ), (π2 , f ), . . . , (πk , f ) of each colored non-crossing partition (π, f ), where f ∈ Fr (π), we assign entries of a given real-valued matrix V ∈ Mr (R) according to the definition given below. If the imaginary block is used, it is convenient to assume that it is also colored by a number from the set [r]. Definition 5.3. Let (π, f ) be a colored non-crossing partition with blocks as above, where f ∈ Fr (π) and let V ∈ Mr (R) be given. For any 0 j r we define vj (π, f ) = vj (π1 , f )vj (π2 , f ) . . . vj (πk , f ), where the functions vj : B(π, f ) → R are given by the following rules:

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1. vj (πi , f ) = vp,q if f (πi ) = p and f (o(πi )) = q, where 0 j r, 2. vj (πi , f ) = vp,j if f (πi ) = p and πi does not have outer blocks, where 1 j r, 3. v0 (πi , f ) = vp,p if f (πi ) = p and πi does not have outer blocks. We are ready to prove purely combinatorial formulas for vj (π) for any 0 j n and any π ∈ N C 2 . In particular, if V is a square n-dimensional matrix with all entries equalto 1/n and f runs over Fn (π), then v0 (π) = 1 and vj (π) = 1/n for any π . In that case we get π∈N C 2 v0 (π) = 2m n j =1 π∈N C 22m vj (π) = cm , the m-th Catalan number. Limit theorems studied in Section 6 will give matricial deformations of Catalan numbers induced by the formulas given by the lemma proven below. Lemma 5.1. For any V ∈ Mn (R), where n ∈ N, and π ∈ N C 2 , it holds that v0 (π) =

v0 (π, f )

and v(π) =

f ∈Fn (π)

1 n

vj (π, f ),

f ∈Fn (π) j ∈[n]

where the summation over j corresponds to all colorings of the imaginary block. Proof. We provide an induction proof for the function v (the proof for v0 is similar). The main induction step will be carried out on the level of the (diagonal) matrices V (π). We claim that its diagonal entries are of the form V (π) q,q = vq (π, f ) f ∈Fn (π)

for any π ∈ N CC 2 and q ∈ [n]. In view of (5.4), the required formula for v(π) is then a straightforward consequence of the claim. Of course, if π consists of one block, then V (π) q,q = vj,q j

and thus our assertion easily follows. Assume now that π has k 2 blocks and suppose the assertion holds for non-crossing covered partitions which have less than k blocks. Since π has a decomposition of type (5.2), the assertion holds for π (1) , π (2) , . . . , π (r) in this decomposition. We know that the matrix assigned to π has the form (5.3) and therefore the product of (diagonal) matrices corresponding to these subpartitions has (diagonal) matrix elements of the form V (π) q,q = τ (W V ) q,q = Wj,j vj,q j

where q is the color of the imaginary block of π and Wj,j =

f1 ∈Fn (π (1) )

vj π (1) , f1 . . .

fr ∈Fn (π (r) )

vj π (r) , fr ,


4101

by the inductive assumption. Now, since the blocks of π (1) , . . . , π (r) are colored independently, we have vj π (1) , f1 . . . vj π (r) , fr vj,q = vq (π, f ) for a uniquely determined coloring f of the blocks of π in which j can be interpreted as the color of π (0) since π (0) covers π (1) , . . . , π (r) and q can be viewed as the color of the imaginary block of π . This and the above formula for Wj,j gives the desired formula V (π)q,q =

vq (π, f ),

f ∈Fn (π)

which proves our claim and thus completes the proof of the theorem.

2

Under suitable assumptions on V , there is a simple connection between functions v and v0 if V0 = aIn /n, where In is a unit matrix and a is a positive number. We shall express this relation in terms of the corresponding formal Laurent series, which turn out to be Cauchy transforms of (compactly supported) probability measures on the real line associated with appropriately constructed random variables. Proposition 5.1. Let V ∈ M n (R) be such that V0 = aIn /n, where a > 0. If G(z) = ∞ ∞ −2k−2 and G (z) = −2k−2 are formal Laurent series, where 0 k=0 a2k z k=0 b2k z a2k =

v(π)

and b2k =

π∈N C 22k

v0 (π),

π∈N C 22k

and both v(π) and v0 (π) are associated with V , then G0 (z) = 1/(z − aG(z)). Proof. Observe that in the case when vj,j = a/n for all j ∈ [n], we have v(π) =

v0 (π ) a

for any π ∈ N C 2m , where the partition π ∈ N CC 2m+2 is obtained from π by adding to π the block that covers all blocks of π , say {0, m + 1}. This leads to

A(z) :=

∞

a2m z2m = 1 +

v(π)z2m

m=1 π∈N C 2 2m

m=0

= 1+

∞

∞ 1 a

m=1 π∈N CC 2 2m+2

v0 (π)z2m =

C(z) − 1 , az2

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where C(z) = we obtain


∞

m=0 c2m z

2m

B(z) :=

and c2m =

∞

π∈N CC 22m v0 (π). Now, using the multiplicativity of v0 ,

b2m z2m = 1 +

m=0

∞ m C(z) − 1 = m=1

1 , 2 − C(z)

which leads to 1 1 1 A(z) = 2 1 − and G0 (z) = , B(z) z − aG(z) az where 1 1 1 1 G(z) = A and G0 (z) = B , z z z z which completes the proof.

2

6. Random pseudomatrices In this section we will study the asymptotic behavior of random pseudomatrices for two sequences of states: the sequence of distinguished states, with respect to which our variables will be matricially free, and the sequence of traces which are normalized sums of conditions in the definition of matricial freeness. Under suitable assumptions, we will later obtain two types of limit theorems: the ‘standard’ central limit theorem as well as the ‘tracial’ central limit theorem for matricially free random variables, the latter being related to random matrix models. Suppose we have a sequence (A(n)) of unital ∗-algebras, each equipped with a distinguished state φ(n) and associated conditions {φj (n): 1 j n}. For each n, let (Xi,j (n))1i,j n be an array of self-adjoint random variables in A(n). We are going to study the asymptotic behavior of random pseudomatrices S(n) =

n

Xi,j (n),

(6.1)

i,j =1

where we assume that each array (Xi,j (n)) is matricially free with respect to the array (φi,j (n)) defined by φ(n) and φj (n)’s. In the first setting we shall compute the asymptotic moments of random pseudomatrices with respect to φ(n). This corresponds to the ‘standard CLT’ reminding the CLT for free (or, monotone) random variables [24]. In the second setting, we shall compute the asymptotic moments of random pseudomatrices with respect to the convex linear combinations of conditions 1 φj (n) n n

ψ(n) :=

(6.2)

j =1

which play the role of traces in the context of random pseudomatrices, with φj (n) corresponding the classical expectation E composed with the state associated with vector ei of the canonical


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orthonormal basis (ei )1in in Cn . This will lead to the ‘tracial CLT’ for matricially free random variables reminding the limit theorem for random matrices [25]. Thanks to the properties of the matricially free product of states which essentially boil down to properties (P1) and (P3) of Section 2, the normalization of square root type works in both cases since the number of summands in S(n) which give a non-zero contribution to the limits is in both cases of order n. We partition the set [n] := {1, 2, . . . , n} into disjoint non-empty intervals, [n] = N1 ∪ N2 ∪ · · · ∪ Nr , where r ∈ N, such that their relative sizes nj /n → dj as n → ∞, where nj is the cardinality of Nj for any j ∈ [r]. Then the numbers dj form a diagonal matrix D of trace one called the dimension matrix. Our results will involve the following assumptions: (A1) (Xi,j (n)) is matricially free with respect to (φi,j (n)) for any n ∈ N, (A2) the variables have zero expectations, φi,j (n) Xi,j (n) = 0 for all i, j ∈ [n] and n ∈ N, (A3) their moments are uniformly bounded, i.e. ∀m ∃Mm 0 such that φi,j (n) X m (n) Mm i,j nm/2 for all i, j ∈ [n] and n ∈ N, (A4) their variances are block-identical and are of order 1/n, namely 2 up,q φi,j (n) Xi,j (n) = n for any i ∈ Np , j ∈ Nq , where each up,q is a non-negative real number. Of course, √ the uniform boundedness assumption is satisfied if we take variables of type Xi,j (n) = Xi,j / n, where (Xi,j ) is an infinite array of random variables whose distributions in the states φi,j (n), respectively, are identical and do not depend on n. Although it is convenient to think of the Xi,j (n) as if they were of this form, we want to study similar variables, whose variances are of order 1/n and stay the same within blocks whose sizes become infinite as n → ∞. If the tuple ((i1 , j1 ), . . . , (im , jm )) ∈ (I × I )m , where I is an index set, defines a partition π = {π1 , . . . , πk } of the set [m], i.e. (ip , jp ) = (iq , jq ) if and only if there exists r such that p, q ∈ πr , we will write P (i1 , j1 ), . . . , (im , jm ) = π. Of course, if π is a non-crossing pair-partition, then each πr is a two-element set. By a k-block we will understand a block consisting of k elements. In particular, a 1-block will be called a singleton. We will also adopt the convention that if m is odd, then N C 2m = ∅ and the summation

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over π ∈ N C 2m gives zero. This allows us to state results for moments of the S(n) of all orders without distinguishing even and odd moments. Lemma 6.1. Let (A(n), ψ(n)) be a sequence of noncommutative probability spaces, each with an array (φi,j (n)) defined by φ(n) and φj (n)’s, for which (A1)–(A3) hold, where ψ(n) is given by (6.2). Then 1 ψ(n) S m (n) = v(π, n) + O √ , n 2 π∈N C m

where v(π, n) = tr(V (π, n)) is given by Definition 5.1 and corresponds to the variance matrix 2 (n)). V (n) = (vi,j (n)), where vi,j (n) = φ(n)(Xi,j Proof. We have ψ(n) S m (n) =

i1 ,j1 ,...,im ,jm

=

ψ(n) Xi1 ,j1 (n) . . . Xim ,jm (n)

π∈Pm

i1 ,j1 ,...,im ,jm P ((i1 ,j1 ),...,(im ,jm ))=π

ψ(n) Xi1 ,j1 (n) . . . Xim ,jm (n) ,

where Pm denotes the set of all partitions of [m]. Arguments presented below allow us to conclude that for large n only non-crossing pair-partitions give relevant contributions. 1. If π has a singleton associated with some (ik , jk ) = (j, j ), then the moment φj (n) Xi1 ,j1 (n) . . . Xim ,jm (n) vanishes by Lemma 4.1 since in that case the variable Xik ,jk is the only variable from Aik ,jk in the corresponding moment under φ(n), by which we mean φ(n)(bj∗ abj ), where a = Xi1 ,j1 (n) . . . Xim ,jm (n) and bj ∈ C[Xj,j (n), 1j,j ] ∩ Ker φ(n). For that reason, in the remaining cases we can assume that there are no such singletons. 2. If π has exactly one singleton associated with some (ik , jk ) = (j, j ), then the extended partition π associated to bj∗ abj does not have singletons and has at least one 3-block (this is the imaginary block colored with j ). Therefore, π has the same number of blocks as π and s (m + 1)/2. Since with each block we can associate at most one independent index to sum over (by Lemma 4.1), the sum of the above moments over i1 , j√ 1 , . . . , im , jm and j for the given π (with the 1/n normalization coming from ψ(n)) is O(1/ n ) by (A3). 3. If π has no singletons and is not a pair-partition, then the number of blocks of π is either s (if (j, j ) is present among (i1 , j1 ), . . . , (im , jm )) or s + 1 (in the opposite case). In both cases, we have that s (m + 1)/2 and the summation of the considered moments over √ i1 , j1 , . . . , im , jm and j for the given π is O(1/ n ). 4. If π is a crossing pair-partition, then the associated π is a crossing partition for any j . It suffices to consider the case when π is a pair-partition since otherwise it has a 4-block and a similar argument to that in (3) works. In turn, if π is a pair-partition for some j , then by Lemma 4.1 and the mean zero assumption, the corresponding moment under φj (n) vanishes, which implies that the corresponding moment under ψ(n) also vanishes.


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The above arguments imply that only non-crossing pair-partitions contribute to the limit and we can write m 1 ψ(n) Xi1 ,j1 (n) . . . Xim ,jm (n) + O √ . ψ(n) S (n) = n 2 π∈N C m

i1 ,j1 ,...,im ,jm P ((i1 ,j1 ),...,(im ,jm ))=π

Now, suppose that m is even, π ∈ N C 2m and the sequence of pairs ((i1 , j1 ), . . . , (im , jm )) is compatible with the matricial multiplication, i.e. such that if (ik , jk ) and (ir , jr ) label an inner– outer pair of blocks, then jk = ir . If πj = {r, r + 1} is the last block which has no inner blocks, then we can pull out the variance corresponding to that block, namely: ψ(n) Xi1 ,j1 (n) . . . Xim ,jm (n) = vir ,jr ψ(n) Xi1 ,j1 (n) . . . Xir−1 ,jr−1 (n)Xir+2 ,jr+2 (n) . . . Xim ,jm (n) , where we use the proof of Lemma 4.1. Continuing this procedure with other blocks which have no inner blocks, and summing over indices i1 , j1 , . . . , im , jm , for which it holds that P ((i1 , j1 ), . . . , (im , jm )) = π , we arrive at 1 n

k0 ,k1 ...,km

1 vk1 ,ko(1) (n) . . . vkm ,ko(m) (n) + O √ , n

where o(r) = 0 if πr has no outer blocks (k0 labels the imaginary block) and o(r) = j if the nearest outer block of πr is labelled by kj . Let us observe that we included in the above sum all possible labellings of the blocks of π . This is done for convenience since it enables us to express the final result in terms of v(π). More explicitly, we allow k0 , k1 , . . . , km to assume arbitrary values from the set [m] (in particular, they can all be equal), which produces certain terms which cannot be obtained from the summation over all ((i1 , j1 ), . . . , (im , jm )) which define π . For example, no nearest inner–outer pair of blocks can √ contribute vj,j vj,j , which appears in the above sum. However, all such terms are of order 1/ n due to insufficient number of different summation indices (there are fewer than m/2 independent indices) and therefore they can be included in the sum without changing the asymptotics. Using Lemma 5.1, we obtain our assertion. 2 Lemma 6.2. Let (A(n), φ(n)) be a sequence noncommutative probability spaces, each with an array (φi,j (n)) of states defined by φ(n) and conditions φj (n), for which (A1)–(A3) hold. Then m 1 v0 (π, n) + O √ , φ(n) S (n) = n 2 π∈N C m

where v0 (π, n) = Tr(V0 (π, n)) is given by Definition 5.2 and corresponds to the variance matrix V (n) = (vi,j (n)). Proof. The proof is similar to that of Lemma 6.1 (in fact, it is simpler since we only need to compute the moments under φ(n) and thus the complications involving moments under φj (n)’s do not appear). 2

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In order to ensure existence of the limits of v(π, n) and v0 (π, n) as n → ∞, we need to specify the sequence of matrices (V (n))n∈N more closely. We shall assume that (A4) holds. Assuming that the variance matrices V (n) are of this form, we can now state the standard and tracial central limit theorems, with limit distributions described in terms of traces of Section 5. Theorem 6.1 (Tracial central limit theorem). Under the assumptions of Lemma 6.1, if V (n) is of the block form (A4) for each n ∈ N, then lim ψ(n) S m (n) = b(π),

n→∞

(6.3)

π∈N C 2m

for any m ∈ N, where b(π) = Tr(B(π)D) and B(π) is the diagonal matrix of Definition 5.1 corresponding to π and the matrix B = DU . Proof. Clearly, if m is odd, we get zeros on both sides of the above formula (we use our convention that in this case N C 2m = ∅). The proof for m = 2k, where k ∈ N, is based on Lemmas 5.1 and 6.1. If, in the combinatorial expression for v(π, n), we substitute for the matrix elements of V (n) the assumed block form, then, using the partition of the set of colors [n] = N1 ∪ N2 ∪ · · · ∪ Nr , we can perform summations over the colorings which belong to each interval Nj separately. Thus, the contributions of various [n]-colorings of π to the limit laws reduce to those corresponding to [r]-colorings and are described in terms of numbers uj (πi , f ), where i ∈ [k], j ∈ [r] and f ∈ Fr (π) (the number j is the color of the imaginary block). We have lim v(π, n) = lim

n→∞

1

nf (1) uj (π1 , f ) . . . nf (k) uj (πk , f )

nj nk+1 j ∈[r] f ∈Fr (π) dj df (1) uj (π1 , f ) . . . df (k) uj (πk , f ) = Tr B(π)D , = n→∞

j ∈[r]

f ∈Fr (π)

where π → B(π) is the matrix-valued function which corresponds to the matrix B = DU in accordance with Definition 5.1. In terms of matrix multiplication, the expression on the righthand side is obtained from that of Lemma 5.1 corresponding to matrix U by multiplying U from the left by the dimension matrix D and multiplying the whole product of matrices from the right by D. This proves our assertion. 2 Theorem 6.2 (Central limit theorem). Under the assumptions of Lemma 6.2, if V (n) is of the block form (A4) for each n ∈ N, then b0 (π), lim φ(n) S m (n) =

n→∞

(6.4)

π∈N C 2m

for any m ∈ N, where π → b0 (π) is the real-valued function of Definition 5.2 corresponding to the matrix B = DU . Proof. The proof is similar to that of Theorem 6.1 and is based on Lemmas 5.1 and 6.2. The only difference is that we do not use imaginary blocks to describe the colorings of all blocks of π . Thus, the contributions of various [n]-colorings of π to the limit laws reduce to those


4107

corresponding to [r]-colorings and are described in terms of numbers u0 (πi , f ), where i ∈ [k] and f ∈ Fr (π). Namely, we have lim v0 (π, n) = lim

n→∞

n→∞

=

1 nk

nf (1) u0 (π1 , f ) . . . nf (k) u0 (πk , f )

f ∈Fr (π)

df (1) u0 (π1 , f ) . . . df (k) u0 (πk , f ) = Tr B0 (π)

f ∈Fr (π)

as n → ∞, where π → B0 (π) is the function defined by Definition 5.2, which proves our assertion. 2 7. Matricial semicircle distributions The results of Section 6 lead to combinatorial formulas for the asymptotic moments in the corresponding central limit theorems. In this section we are going to express the limits in terms of their Cauchy transforms represented in the form of continued fractions. They play the role of the (standard and tracial) ‘matricial semicircle distributions’. For that purpose let us recall definitions of certain convolutions of distributions, or more generally, of probability measures. If Fμ is the reciprocal Cauchy transform of some probability measure μ ∈ MR , then the K-transform of μ is given by Kμ (z) = z − Fμ (z). The boolean additive convolution μ ν can be defined by the equation Kμν (z) = Kμ (z) + Kν (z), where μ, ν ∈ MR and z ∈ C+ , respectively. In fact, this equation shows that the K-transform is the boolean analog of the logarithm of the Fourier transform [23]. We will also need another convolution, which reminds the monotone convolution [20], called the orthogonal additive convolution and defined by the equation Kμν (z) = Kμ Fν (z) , where μ, ν ∈ MR and z ∈ C+ . It was introduced in [14], where we showed that the above formula defines a unique probability measure on the real line. Moreover, if μ and ν are compactly supported, both μ ν and μ ν are compactly supported. Using these convolutions, we will now define certain important continued fractions (or, ‘continued multifractions’) which converge uniformly on the compact subsets of C+ to the Ktransforms of some probability measures μi,j ∈ MR . Lemma 7.1. For given B ∈ Mr (R) with non-negative entries, continued fractions of the form Ki,j (z) =

z−

k

bi,j bk,i bp,k z − p z − ···

,

where i, j ∈ [r], converge uniformly on the compact subsets of C+ to the K-transforms of some μi,j ∈ MR with compact supports.

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R. Lenczewski / Journal of Functional Analysis 258 (2010) 4075–4121 (0)

Proof. Let us define a sequence of functions which approximate the Ki,j . Namely, set Ki,j (z) = bi,j /z for any i, j ∈ [r], which are the K-transforms of probability measures on R for any i, j (Bernoulli measures if bi,j > 0 and δ0 if bi,j = 0). In order to use an inductive argument, let us establish the recurrence (m) Ki,j (z) =

z−

bi,j

(m−1) (z) k Kk,i

(m−1)

(m−1)

for m 1. If the Kk,i are the K-transforms of some μk,i ∈ MR for any i and k, respec (m−1) tively, then the sums k Kk,i are the K-transforms of their boolean convolution (m−1)

μi

(m−1)

:= μ1,i

(m−1)

μ2,i

(m−1)

· · · μr,i

∈ MR ,

(m)

and next, each Ki,j is the K-transform of the orthogonal convolution (m)

(m−1)

μi,j := κi,j μi

∈ MR ,

where the κi,j ’s are the Bernoulli measures with K-transforms Ki,j (z) = bi,j /z, respectively. It is easy to see that all these measures are compactly supported. Moreover, the properties of the orthogonal additive convolution (Corollary 5.3 in [14]) say that the moments of μ(n) i,j of orders (m)

2m agree with the corresponding moments of μi,j for any n > m and any given i, j . More precisely, in the formal Laurent series expansion (m)

Ki,j (z) =

∞

c−2n−1 z−2n−1 ,

n=0

the coefficients c−1 , . . . , c−2m−1 are uniquely determined by the constants which appear in the (m) continued fraction of the corresponding Cauchy transform Gi,j at depths 2m, and these de(m)

termine the moments of μi,j of orders 2m. The recurrence for the K-transform given above is (m)

(m−1)

agree down to depth m − 1 for such that the corresponding Cauchy transforms Gi,j and Gi,j any given i, j , which proves our assertion. Therefore, we have weak convergence (m)

w − lim μi,j = μi,j m→∞

to some μi,j ∈ MR for any i, j . These measures are also compactly supported since supi,j bi,j is finite. In turn, this implies that the corresponding Cauchy transforms (and thus K-transforms) converge uniformly to the Cauchy transform (K-transforms) of μi,j on compact subsets of C+ . This completes the proof. 2 We are ready to state a theorem, which is another version of the tracial central limit theorem for matricially free random variables.


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Theorem 7.1. Under the assumptions of Theorem 6.1, the ψ(n)-distributions of Sn converge weakly to the distribution given by the convex linear combination μ=

r

dj μj ,

j =1

where μj = μ1,j μ2,j · · · μr,j for each j = 1, . . . , r and μi,j is the distribution defined by Ki,j for any i, j . Proof. By Theorem 6.1, we have combinatorial formulas for the moments Mm of the limit law in the tracial central limit theorem. The associated distribution extends to a unique compactly supported probability measure μ on the real line since its moments are bounded by the moments of the Wigner semicircle distribution σa with variance a = supi,j bi,j . Using the multiplicative formula (5.4) for B(π), we can formally write the Cauchy transform of μ in the form Gμ (z) =

∞

M2k z−2k−1

k=0

=

∞ 1 + Tr B(π)D z−2k−1 z 2 k=1

π∈N C 2k

−1 = Tr z − K(z) D , which can be called the ‘trace formula’ for Gμ , where K(z) =

∞

B(π)z−2k+1

(7.1)

k=1 π∈N CC 2

2k

is a diagonal-matrix-valued formal power series. Moreover, we will show below that each function Kj on its diagonal is, in fact, the K-transform of some μj ∈ MR . Then, the formal power series given by the trace formula is the Cauchy transform of μ as a convex linear combination of Cauchy transforms of probability measures. In fact, using the definition of B(π) and (5.3), we obtain the equation −1 K(z) = τ z − K(z) B ,

(7.2)

where K(z) = diag(K1 (z), . . . , Kr (z)). By analogy with the scalar-valued case, we can find its solution in the form of a continued fraction. Namely, observe that each Kj (z) has the form of a formal Laurent series Kj (z) =

∞ n=0

c−2n−1 z−2n−1

4110


for some c−1 , c−3 , . . . , and therefore the above vector equation can be solved by successive approximations. Namely, we set μj tobe the (compactly supported) probability measure associated with the K-transform Kj (z) = i Ki,j (z) for each j ∈ [r], where the Ki,j are given by Lemma 7.1 for B = DU . These K-transforms solve (7.2). This, together with the trace formula for Gμ , gives Gμ =

r

dj Gμj ,

j =1

where Gμj (z) = 1/(z − Kj (z)) is the Cauchy transform of μj for j = 1, . . . , r. That completes the proof. 2 Remark 7.1. The Cauchy transform of each μj can be written as a continued fraction of the form Gμj (z) =

z−

1

i

z−

k

bi,j bk,i bp,k z − p z − ···

which converges on the compact subsets of C+ . Next, we state a theorem, which is another version of the standard central limit theorem for matricially free random variables. Theorem 7.2. Under the assumptions of Theorem 6.2, the φ(n)-distributions of Sn converge weakly to the distribution μ0 = μ1,1 μ2,2 · · · μr,r , where μj,j is the distribution defined by Kj,j for each j . Proof. By Theorem 6.2, we have combinatorial expressions for the limit moments Mm . The proof is similar to that of Theorem 7.1 and is based on the trace formula for the K-transform of μ0 −1 Kμ0 (z) = Tr z − K(z) B0 derived from the definition of the function b0 , which leads to the equation for the Cauchy transform Gμ0 (z) = which completes the proof.

2

z−

1 , j Kj,j (z)


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Remark 7.2. The Cauchy transform Gμ0 can be written as a continued fraction of the form Gμ0 (z) =

z−

1

j

z−

bj,j

i

bi,j bk,i z − k z − ···

which gives a matricial extension of the continued fraction of the Wigner semicircle distribution. 8. Decompositions in terms of subordinations In the one-dimensional case, the limit distributions μ0 and μ are related by Proposition 5.1. In particular, if each variance matrix V (n) has identical entries equal to one, both central limit theorems (standard and tracial) give the Wigner semicircle distribution with variance 1 (of course, the standard case also follows from free probability, whereas the tracial case is related to random matrices). In this section we will analyze in more detail the limit distributions for the two-dimensional case, namely when each variance matrix V (n) consists of four blocks. They will be expressed in terms of two-dimensional arrays of distributions. Finding simple analytic formulas for the corresponding four-parameter Cauchy transforms and densities does not seem possible in the general case. However, we shall derive decomposition formulas for those measures in terms of s-free additive convolutions [14], which gives some insight into their structure (see also [21] for recent results on the multivariate case). The s-free additive convolution refers to the subordination property for free additive convolution, discovered by Voiculescu [26] and generalized by Biane [4]. As shown in [14] and [15], there is a notion of independence, called freeness with subordination, or simply s-freeness, associated with the s-free additive convolution and its multiplicative counterpart. Recall that the s-free additive convolution of μ, ν ∈ MR is the unique probability measure μ ν ∈ MR defined by the subordination equation ν μ = ν (μ

ν),

where denotes the monotone additive convolution [20]. Equivalently, the above subordination property can be written in terms of Cauchy transforms or their reciprocals. Using s-free additive convolutions and the boolean convolution, we obtain a decomposition of the free additive convolution of the form μ ν = (μ

ν) (ν

μ),

which allows us to interpret both s-free additive convolutions appearing here as (in general, non-symmetric) halves of μ ν. We find it interesting that the limit distributions in the twodimensional case will turn out to be deformations of the free additive convolution of semicircle laws implemented by this decomposition. In other words, the subordination property and the associated convolutions give a natural framework for studying matricial generalizations of the semicircle law. For simplicity, it will be convenient to use the indices-free notation for the two-dimensional matrix of K-transforms:

4112


a(z) c(z)

b(z) d(z)

=

K1,1 (z) K2,1 (z)

K1,2 (z) K2,2 (z)

and A=

α γ

β δ

b1,1 = b2,1

√ b 1,2 = B b2,2

where the square root is interpreted entry-wise. Moreover, we will distinguish two laws by special notations: we denote by σα the Wigner semicircle distribution with the Cauchy transform √ z2 − 4α 2 , Gσα (z) = 2α 2 √ √ where the branch of z2 − 4α 2 is chosen so that z2 − 4α 2 > 0 if z ∈ R and z ∈ (2α, ∞), and by κγ we denote the Bernoulli law with the Cauchy transform z−

Gκγ (z) =

1 , z − γ 2 /z

i.e. σγ = 1/2(δ−γ + δγ ). Finally, we will also use the boolean compressions of μ ∈ MR , where t 0, defined by multiplying its K-transform by t, namely we define Tt μ to be the (unique) probability measure on R, for which KTt μ = tKμ . These transformations were introduced and studied in [7] and called ‘t-transformations’ of μ. We allow t = 0, in which case T0 μ = δ0 . In particular, we shall use two-parameter boolean compressions of semicircle distributions, σα,β = Tt σα for t = (β/α)2 , with √ (2α 2 − β 2 )z − β 2 z2 − 4α 2 Gσα,β (z) = (2α 2 − 2β 2 )z2 + 2β 4 being their Cauchy transforms, where the branch of the square root is the same as in the case of Gσα . Theorem 8.1. If α, β, γ , δ = 0, then the diagonal measures μj,j defined by Kj,j , where 1 j 2, have the form μ1,1 = T1/t (σα,β

σδ,γ ),

μ2,2 = T1/s (σδ,γ

σα,β ),

with the off-diagonal measures given by μ1,2 = Tt μ1,1 and μ2,1 = Ts μ2,2 , where t = (β/α)2 and s = (γ /δ)2 ,


4113

Proof. It is easy to see that the following algebraic relations hold: a(z) =

α2 z − a(z) − c(z)

and d(z) =

δ2 , z − b(z) − d(z)

b(z) =

β2 z − a(z) − c(z)

and c(z) =

γ2 . z − b(z) − d(z)

Thus, b(z) = ta(z) and c(z) = sd(z), which gives μ1,2 = Tt μ1,1 and μ2,1 = Ts μ2,2 . In turn, from the equation for a(z), we get a(z) =

z − c(z) −

(z − c(z))2 − 4α 2 = Kσα z − c(z) 2

and thus μ1,1 = σα μ2,1 . In a similar manner we obtain μ2,2 = σδ μ1,2 . Therefore, we arrive at the equations μ1,1 = σα (Ts σδ Tt μ1,1 ), μ2,2 = σδ (Tt σα Ts μ2,2 ) since Tt (μ ν) = (Tt μ) ν. In order to express the μj,j in terms of s-free additive convolutions, we need to use the properties of the orthogonal convolution. We have shown in [14] that the moment of order k of μ ν depends on the moments of orders k of μ and the moments of orders k − 2 of ν. This leads to the conclusion that for any compactly supported μ, ν ∈ MR and the associated sequence of measures (μ m ν), defined recursively by μ m ν = μ (ν m−1 μ)

with μ 1 ν = μ ν,

we have weak convergence w − limm→∞ μ m ν = μ ν. If we take μ = Tt σα and ν = Ts σδ (these measures are compactly supported), we get the desired formulas. 2 Corollary 8.1. The measures μ0 , μ1 , μ2 can be decomposed as μ0 = T1/t (σα,β

σδ,γ ) T1/s (σδ,γ

μ1 = T1/t (σα,β

σδ,γ ) (σδ,γ

σα,β ),

μ2 = T1/s (σδ,γ

σα,β ) (σα,β

σδ,γ ),

σα,β ),

where the assumptions and notations are the same as in Theorem 8.1. Proof. These decompositions follow immediately from Theorems 7.1, 7.2 and 8.1.

2

Remark 8.1. The formulas for the diagonal measures μj,j in the proof of Theorem 8.1 remind those for two-periodic continued fractions if we take t = s = 1. The latter are of the same form, except that the semicircle distributions are replaced by much simpler Bernoulli laws. Nevertheless, if t = s = 1, the formulas for the μj take a simple form μj = σ α σ δ ,

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where j = 0, 1, 2. By Theorem 7.1 and Corollary 8.1, the same formula holds for μ. Therefore, all measures μ0 , μ1 , μ2 , μ can be viewed as deformations of the free additive convolution of two semicircle distributions, implemented by means of boolean compressions. Let us consider now the situation in which some of the numbers α, β, γ , δ vanish. Suitable formulas can be derived algebraically, as we did in the proof of Theorem 8.1. However, one can also obtain the same results by taking weak limits in the formulas for the measures μi,j , using the fact that all measures involved have compact supports. For that purpose, let us state a few useful facts about weak limits which will be of interest to us. Then, we consider eight cases, to which the remaining cases are similar (for instance, α = β = 0 is similar to γ = δ = 0). Proposition 8.1. Let t = β 2 /α 2 , where α, β > 0, and let μ ∈ MR be compactly supported. 1. If α → 0+ , then (a) w − lim σα = δ0 , (b) w − lim(σα,β ) = κβ , (c) w − lim(T1/t (σα,β μ)) = δ0 . 2. If β → 0+ , then (a) w − lim σα,β = δ0 , (b) w − lim(Tt μ) = δ0 , (c) w − lim(T1/t (σα,β μ)) = σα μ. Proof. If α → 0+ , then Kσα (z) → 0, which proves 1(a). Here, as well as in the remaining cases, convergence is uniform on compact subsets of C+ . Moreover, Kσα,β = β 2 /(z − α 2 Kσα ) → β 2 /z = Kκβ , which gives 1(b). In turn, if β → 0+ , then Kσα,β (z) = β 2 /(z − Kσα (z)) → 0, thus also w − lim σα,β = δ0 , which proves 2(a). If, in addition, t → 0+ , then Tt μ → δ0 for any μ ∈ MR , which proves 2(b). Finally, T1/t (σα,β

μ) = T1/t Tt σα (μ

σα,β ) = σα (μ

σα,β )

and thus the right-hand side tends weakly to δ0 as α → 0+ , which gives 1(c), and tends weakly to σα (μ δ0 ) = σα μ as β → 0+ , which gives 2(c). This holds for any μ ∈ MR , and we also use the right unit property of δ with respect to the s-free additive convolution, namely μ δ0 = μ. 2 Corollary 8.2. If some of the entries of the matrix A vanish, we can distinguish eight different cases, for which the distributions μi,j are given by Table 1. Proof. If ai,j = 0, then μi,j = δ0 , which easily follows from the algebraic equations for the corresponding K-transforms. An alternative proof can be given by taking weak limits of the formulas of Theorem 8.1, as we proceed with the remaining measures. Thus, if δ → 0+ , then μ1,1 = w − lim T1/t (σα,β

σδ,γ ) = T1/t (σα,β

μ1,2 = w − lim(σα,β

σδ,γ ) = σα,β

μ2,1 = w − lim(σδ,γ

σα,β ) = κγ

κγ , σα,β ,

κγ ),


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Table 1 Distributions μi,j in the case A has zero entries. a1,1

a1,2

a2,1

a2,2

μ1,1

α 0 α 0 0 α α 0

β β 0 β 0 0 0 β

γ γ γ 0 γ 0 0 0

0 0 δ δ δ δ 0 0

T1/t (σα,β δ0 σα σδ,γ δ0 δ0 σα σα δ0

κγ )

μ1,2

μ2,1

μ2,2

σα,β κγ κβ κγ δ0 κβ δ0 δ0 δ0 κβ

κγ σα,β κγ κβ σδ,γ δ0 σδ,γ δ0 δ0 δ0

δ0 δ0 σδ σδ κβ σδ σδ δ0 δ0

by 1(b) of Proposition 8.1, which proves the first case in Table 1. The remaining cases are proved in a similar manner. 2 In the case of arbitrary matrix A, finding the four-parameter densities of μ0 and μ is unwieldy. Below we shall just consider two special cases, in which we can find nice formulas for these measures for matrices A of arbitrary dimension. These two cases are of special interest since they are associated with (asymptotic) freeness and (asymptotic) monotone independence. Proposition 8.2. If A is a square r-dimensional matrix with identical positive entries αj in the j -th row, then μj = σα1 σα2 · · · σαr for each j ∈ [r], and the measures μj coincide with μ and μ0 . Proof. Since the columns of A are identical, the functions Ki,j are the same for all j ’s. Denote them Li = Ki,j , where i, j ∈ [r]. Moreover, Li (z) =

bi

z−

for any i ∈ [r]. Therefore,

r

j =1 Lj (z)

r

i=1 Li (z)

and

r i=1

r bi ri=1 Li (z) = z − j =1 Lj (z)

is the K-transform of the measure

σα1 σα2 · · · σαr and since Kμj (z) = ri=1 Ki,j (z) = ri=1 Li (z), the proof for μj is completed. It is then easy to see that we get the same result for μ and μ0 . 2 Proposition 8.3. If A is a lower-triangular r-dimensional matrix with identical positive entries αj in the j -th row below and on the main diagonal, then μj = σαj σαj +1 · · · σαr for each j ∈ [r]. Moreover, μ0 = μ1 and μ is the convex linear combination of the measures μj as in Theorem 7.1.

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Proof. As in the proof of Proposition 8.2, note that the measures μi,j do not depend on j and thus we can set Li = Ki,j for any i j . If i = r, we have Lr (z) =

br , z − Lr (z)

using the continued fraction for the Kr,j of Lemma 7.1. Therefore, μr,j = σαr for any j r. Next, we have Lk (z) =

z−

bk r

i=k Li (z)

,

which leads to Lk (z) = Kσαk z −

r

Li (z) ,

i=k+1

giving the orthogonal decomposition of μk,j , μk,j = σαk (μk+1,j · · · μr,j ) for any j k < r. Now, we claim that μi,j · · · μr,j = σαi σαi+1 · · · σαr for any 1 j i r. Clearly, it holds for i = r and any j r since we have already shown that μr,j = σαr for any j r. Suppose now that this formula holds for i > k and any j i. We will show that it holds for i = k and any j k. Using the orthogonal decomposition of μk,j given above and the inductive assumption, we obtain μk,j = σαk (σαk+1 · · · σαr ). However, for any μ, ν ∈ MR , we have a simple relation (μ ν) ν = μ ν, which gives μk,j · · · μr,j = σαk σαk+1 · · · σαr and the desired expression for μj . In a similar manner we obtain μ0 and μ.

2

Example 8.1. If α = δ = 0 and β = γ = 0, then we can use√Table 1 to obtain μ0 = σα σα , which is the arcsine law with Cauchy transform Gμ0 (z) = 1/ z2 − α 2 whereas μ is the Wigner semicircle distribution σα . In turn, if α = δ = 0 and β = γ = 0, then μ0 = δ0 , whereas μ = σβ .


4117

9. Weighted binary trees and Catalan paths In this section we show how to express the limit distributions in terms of walks on weighted binary trees, or equivalently, in terms of weighted Catalan paths. The binary tree serves here as an example of the strongly matricially free Fock space. The usual framework which gives a description of distributions in terms of walks on graphs is the following. Let W (n) denote the set of root-to-root walks of length n on a rooted graph (G, e) and let μ be the spectral distribution of (G, e), i.e. the distribution given by the moments of the adjacency matrix A(G) in the state ϕ associated with the vector δe on the space of square integrable functions on the set V (G) of the vertices of G. Then the n-th moment of A(G) in the state ϕ is equal to the cardinality of the set W (n). In particular, it is well known that the moments of the Wigner semicircle distribution of variance 1 can be expressed in terms of walks on the half-line (T1 , e) with the first vertex denoted by e and chosen as the root. For many distributions we have to use a more general framework, in which the moments of these distributions are expressed in terms of root-to-root (random or, more generally, weighted) walks on some rooted graph, except that to each walk w on this graph we have to assign a realvalued weight ξ(w). Then we can write Mμ (n) =

ξ(w)

w∈W (n)

for any n 1, where μ is the considered distribution. In particular, we obtain the moments of σα for any α > 0 by putting ξ(w) = α n , where n = |w| is the length of w. In the cases which are of interest to us, the

weight function ξ is first defined on the set of edges E(G) of G and then is extended to W = n1 W (n) by multiplicativity. Namely, if we are given a mapping ξ : E(G) → R, we set ξ(w) = ξ(E1 )ξ(E2 ) . . . ξ(En ), where w = (E1 , E2 , . . . , En ) and E1 , E2 , . . . , En are the edges of w (we choose to describe walks on graphs as sequences of edges). Such extension, by abuse of notation denoted also by ξ , will be called multiplicative. For instance, it is easy to see that the moments of μ = σα σδ can be expressed in this form. It is enough to take the free product of two half-lines, which is the binary tree (T2 , e) with root e. Let us color this graph in the natural way, namely each edge which belongs to a copy of the first half-line is colored by 1 and each edge which belongs to a copy of the second half-line is colored by 2. Then the above formula holds for the moments of μ if we take ξ(E) = α whenever E is colored by 1 and ξ(E) = δ whenever E is colored by 2. We will demonstrate below that we can express our distributions μ0 , μ1 , μ2 in a similar form (in particular, we can use the binary tree), except that the weight function ξ will depend on all four parameters which appear in the matrix A. Before formulating the theorem, let us introduce special weight functions related to matricial freeness. Definition 9.1. Let Tr be an r-ary rooted tree with root e and let A ∈ Mr (R). The weight function ξ : E(Tr ) → R which assigns the entries of A to the edges of Tr is called matricial if, for any pair of edges E1 , E2 ∈ E(Tr ), incident on the same vertex and such that E1 is the ‘father’ of E2 ,

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Fig. 3. Binary tree with a matricial weight function.

the following implication holds: ξ(E1 ) = ai,j

for some i, j

⇒

ξ(E2 ) = ak,i

for some k.

The unique multiplicative extension of this weight function to the set of all walks on Tr will also be called matricial. We specialize to p = 2 and the binary tree. Note that any matricial weight function ξ on the binary tree is uniquely determined (up to equivalence) by the set of those entries of the matrix A which are assigned to the set {E1 , E2 } of two edges incident on the root of the tree, called the initial weights. In order to establish a connection with our limit distributions, we will assume, as in Section 8, that A is the ‘square root’ of B = DU of Section 6. Then, in particular, the binary tree with the matricial weight function associated with A and the initial weights {α, δ}, shown in Fig. 3, describes μ0 as we show below. Finally, recall after [1,14] that if (G1 , e1 ) and (G2 , e2 ) are two locally finite simple graphs and μ1 and μ2 are the associated spectral distributions, then the s-free product of G1 and G2 (in that order) can be interpreted as this half of the free product G1 ∗ G2 which ‘begins’ (starting from the root) with a copy G1 . Moreover, the associated spectral distribution is given by μ1 μ2 . Let us add that a similar result holds in the multiplicative case: both the s-free multiplicative convolution and the associated s-free loop product of graphs were introduced in [15]. Below we shall use the s-free product of half-lines, which are (left and right) halves of the binary tree. Theorem 9.1. Let ξ0 , ξ1 , ξ2 be the multiplicative matricial weight functions on the set of walks on T2 associated with matrix A and the initial weights {α, δ}, {β, δ} and {α, γ }, respectively. Then Mμj (n) =

ξj (w)

(9.1)

w∈W (n)

for j = 0, 1, 2 and any n ∈ N, where W (n) denotes the set of root-to-root walks on T2 of length n. Proof. We need to translate the result of Corollary 8.1 to the language of graphs. It is well known that the moments of σα can be interpreted in terms of walks on the half-line with the weight α


4119

assigned to each edge. The boolean compression Tt of √ σα changes only the weight assigned to the edge incident on the root, namely it multiplies it by t = β/α, which can be illustrated as . Now, the s-free additive convolutions of compressed semicircle distributions which appear in the decompositions of Corollary 8.1, namely σα,β

σδ,γ

and σδ,γ

σα,β

are the spectral distributions of the s-free products of these half-lines (taken in two different orders). These turn out to be the spectral distributions of the two halves of the binary tree T2 with the weight function defined by the initial set {β, γ }. In order to obtain μ0 , we still need to apply T1/t and T1/s , respectively, to the left and right halves of the tree, which amounts to changing the initial weights from β and γ , respectively, to α and δ. As a result, we obtain the weight function associated with A and the initial set {α, δ}. This proves the statement concerning μ0 (the corresponding weight function on the binary tree is shown in Fig. 3). The cases of μ1 and μ2 are very similar (at the end, a boolean compression is applied to only one half of the binary tree). 2 Another geometric interpretation of Corollary 8.1 can be given in terms of Catalan paths. In order to define a Catalan path, we begin with a function f : [2n] → [n], such that f (0) = f (2n) and |f (i) − f (i − 1)| = 1 for any 1 i 2n, and then we define a Catalan path as its unique extension f : [0, 2n] → [0, n] (by abuse of notation, denoted by the same symbol) obtained by connecting each (i − 1, f (i − 1)) with (i, f (i)) with a segment, where 1 i 2n. Clearly, each Catalan path consists of segments of two types: ‘rises’ R1 , R2 , . . . , Rn , and ‘falls’ F1 , F2 , . . . , Fn . Moreover, to each ‘rise’ Rj there corresponds the closest ‘fall’ Fτ (j ) lying to the right of Rj and on the same (vertical) level. There is a natural mapping from the set W (2n) of walks of length 2n on the binary tree and the set C(n) of Catalan paths of length 2n. In order to rephrase Theorem 8.1 in terms of Catalan paths, we need to take the sets of weighted Catalan paths, by which we understand pairs (f, ξ ), where f is a Catalan path and ξ is a real-valued weight function defined on the set of segments of f . The multiplicative formula ξ(f ) = ξ(R1 ) . . . ξ(Rn )ξ(F1 ) . . . ξ(Fn ) assigns the corresponding weight to f . Weighted Catalan paths of special type defined below allow us to rephrase Theorem 8.1. Definition 9.2. A weighted Catalan path (f, ξ ) of length 2n is called matricially weighted if ξ assigns entries of A ∈ Mp (R) to the segments of f in such a way that the following implications hold: ξ(R1 ) = ai,j

for some i, j

⇒

ξ(R2 ) = ak,i

for some k,

ξ(F1 ) = ai,j

for some i, j

⇒

ξ(F2 ) = aj,k

for some k,

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Fig. 4. A weighted Catalan path associated with A.

for any two consecutive ‘rises’ R1 , R2 and two consecutive ‘falls’ F1 , F2 , and the same weights are assigned to Ri and Fτ (i) for each i ∈ [n]. We will consider below the set of matricially weighted Catalan paths of length 2n associated with the matrix A. In order to rephrase Theorem 9.1, using weighted Catalan paths, we need to restrict the set of weights which can be assigned to the first segment of each path (by analogy with trees, we call them initial weights). An example of a weighted Catalan path contributing to μ0 is given in Fig. 4. Corollary 9.1. Let C0 (n), C1 (n) and C2 (n) be the sets of matricially weighted Catalan paths associated with A, with the initial weights {α, δ}, {β, δ} and {γ , α}, respectively. Then Mμj (2n) =

ξ(f )

(f,ξ )∈Cj (n)

for j = 0, 1, 2 and any n ∈ N. Proof. This is an easy consequence of Theorem 9.1 since there is a natural bijection between each Cj (n) and the pair (W (2n), ξj ) for any n. 2 Acknowledgments The remarks and suggestions of the anonymous referees were very helpful in the preparation of the revised version of the manuscript. References [1] L. Accardi, R. Lenczewski, R. Sałapata, Decompositions of the free product of graphs, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007) 303–334. [2] M. Anshelevich, Free Meixner states, Comm. Math. Phys. 276 (2007) 863–899. [3] D. Avitzour, Free products of C ∗ -algebras, Trans. Amer. Math. Soc. 271 (1982) 423–465. [4] Ph. Biane, Processes with free increments, Math. Z. 227 (1998) 143–174. [5] M. Bo˙zejko, Uniformly bounded representations of free groups, J. Reine Angew. Math. 377 (1987) 170–186. [6] M. Bo˙zejko, R. Speicher, ψ -independent and symmetrized white noises, in: L. Accardi (Ed.), Quantum Probability and Related Topics VI, World Scientific, Singapore, 1991, pp. 170–186. [7] M. Bo˙zejko, J. Wysoczański, Remarks on t -transformations of measures and convolutions, Ann. Inst. H. Poincaré Probab. Statist. 37 (6) (2001) 737–761. [8] T. Cabanal-Duvillard, Probabilités libres et calcul stochastique. Application aux grandes matrices aléatoires, PhD thesis, l’Université Pierre et Marie Curie, 1998. [9] T. Cabanal-Duvillard, V. Ionescu, Un théorème central limite pour des variables aléatoires non-commutatives, C. R. Acad. Sci. Paris, Sér. I 325 (1997) 1117–1120. [10] W.M. Ching, Free products of von Neumann algebras, Trans. Amer. Math. Soc. 178 (1993) 147–163. [11] K. Dykema, On certain free product factors via an extended matrix model, J. Funct. Anal. 112 (1993) 31–60.


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Confluent operator algebras and the closability property ✩ H. Bercovici a,∗ , R.G. Douglas b , C. Foias b , C. Pearcy b a Department of Mathematics, Indiana University, Bloomington, IN 47405, United States b Department of Mathematics, Texas A&M University, College Station, TX 77843, United States

Received 18 September 2009; accepted 10 March 2010 Available online 26 March 2010 Communicated by N. Kalton Dedicated to the memory of our good friends and mentors, Paul R. Halmos and Béla Sz˝okefalvi-Nagy

Abstract Certain operator algebras A on a Hilbert space have the property that every densely defined linear transformation commuting with A is closable. Such algebras are said to have the closability property. They are important in the study of the transitive algebra problem. More precisely, if A is a two-transitive algebra with the closability property, then A is dense in the algebra of all bounded operators, in the weak operator topology. In this paper we focus on algebras generated by a completely nonunitary contraction, and produce several new classes of algebras with the closability property. We show that this property follows from a certain strict cyclicity property, and we give very detailed information on the class of completely nonunitary contractions satisfying this property, as well as a stronger property which we call confluence. © 2010 Elsevier Inc. All rights reserved. Keywords: Confluent algebra; Closability property; Completely nonunitary contraction; Rationally strictly cyclic vector; Quasisimilarity

✩

H.B. and R.G.D. were supported in part by grants from the National Science Foundation.


E-mail addresses: [email protected] (H. Bercovici), [email protected] (R.G. Douglas), [email protected] (C. Pearcy). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.009

H. Bercovici et al. / Journal of Functional Analysis 258 (2010) 4122–4153

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1. Introduction Probably the best known problem in operator theory is the question of whether every bounded linear operator on a complex, separable, infinite dimensional Hilbert space H has a nontrivial invariant subspace. Despite considerable effort by many researchers for more than half a century, the general problem remains open. A generalization, still unresolved, asks whether every transitive algebra of operators must be dense in the weak operator topology. (Recall that an algebra is said to be transitive if there is no common nontrivial invariant subspace for the operators in it.) In the sixties, Arveson approached this problem iteratively, starting from an observation going back essentially to von Neumann. Namely, assume that A is an algebra of operators on a Hilbert space H, and n 1 is an integer. The algebra A is said to be n-transitive if every invariant subspace for A(n) = X (n) = X ⊕ X ⊕ · · · ⊕ X: X ∈ A n times

is invariant for every operator of the form Y (n) where Y is an operator on H. Then A is dense, in the weak operator topology, if and only if it is n-transitive for every n 1. Arveson observed that 2-transitivity is equivalent to the following statement: every closed linear transformation commuting with A is a scalar multiple of the identity operator. For n 3, n-transitivity is implied by a similar statement for densely defined linear transformations commuting with A. Thus, provided that every densely defined linear transformation commuting with A is closable, 2-transitivity implies n-transitivity for all n. As a consequence, Arveson was able to prove that transitive algebras containing certain kinds of subalgebras are indeed dense in the weak operator topology. His results apply to algebras on an L2 -space, containing the algebra L∞ of all bounded measurable multipliers, and to algebras on the Hardy space H 2 (D), containing the algebra H ∞ (D). A few similar results were obtained by others for closely related algebras in the following years; see for instance [19, Chapter 8]. A year ago, Haskell Rosenthal became interested in the question of which algebras of operators on Hilbert space had what he called the closability property which means that every densely defined linear transformation in its commutant is closable. A key step in Arveson’s proofs was to show that the algebras L∞ acting on L2 , and H ∞ (D) acting on H 2 (D), have the closability property. Rosenthal showed that various algebras have the closability property and asked the authors a specific followup question. In finding the answer, the question piqued our interest and resulted in a series of questions related to this topic. Our investigation took us in some unexpected directions, making surprising connections with other topics in operator theory. After some preliminaries in Section 2, we begin in Section 3 by investigating the closability property and determining some algebras which have it, as well as some that do not. In Section 4 we introduce the concept of a rationally strictly cyclic vector, and show that the existence of such a vector for a commutative algebra A implies the closability property. In Section 5 we discuss the invariance of the closability property, and of the existence of rationally strictly cyclic vectors, under an appropriate notion of quasisimilarity. We deduce, for instance, that the commutant of any contraction of class C0 has the closability property. In the course of our study, the importance of something like a functional calculus for quotients became clear. To make this idea precise, in Section 6 we study the related notion of confluence (introduced in Section 4) as it applies to the algebra obtained by applying the H ∞ functional calculus to a completely nonunitary contraction. Confluence implies the existence of a rationally strictly cyclic vector, and therefore the closability

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property as well. Section 7 contains a thorough study of confluence in the context of functional models for contractions. In particular, a characterization is obtained for those contractions which are quasisimilar to the unilateral shift of multiplicity one. This characterization involves the ‘size’ of the analytic functions in the reproducing kernel representative for the operator. The analysis of confluence is somewhat subtle and rests on the harmonic analysis of contractions [23], the theory of the class C0 [4], the theory of dual algebras [5], and results about the class B1 (D) [10]. We thank Haskell Rosenthal for the questions which led to this research. We are also grateful to the referee, who pointed out several errors and numerous misprints in our original manuscript. 2. Preliminaries We will work with operators on Hilbert spaces over the complex numbers C. The algebra of bounded linear operators on a Hilbert space H is denoted L(H). Given T ∈ L(H), PT denotes the smallest unital algebra containing T ; that is, the set of all polynomials in T . The closure of PT in the weak operator topology (also known as WOT) is denoted WT . The norm closure of a subset M ⊂ H is denoted M. The orthogonal projection of H onto a closed linear subspace M ⊂ H is denoted PM . Several special operators play an important role. The space L2 is the space of functions defined on the unit circle T which are square integrable relative to arclength measure. The bilateral shift U ∈ L(L2 ) is the unitary operator defined by (Uf )(ζ ) = ζf (ζ ) for f ∈ L2 and a.e. ζ ∈ T. The Hardy space H 2 ⊂ L2 is the cyclic subspace for U generated by the constant function 1, and S ∈ L(H 2 ) is the unilateral shift of multiplicity 1 defined as S = U | H 2 . More generally, denote by H ∞ = H ∞ (D), the algebra of bounded analytic functions in the unit disk D. For each u ∈ H ∞ one defines an analytic Toeplitz operator Tu ∈ L(H 2 ) as the operator of pointwise multiplication by u. Here one takes advantage of the fact that functions in H ∞ have a.e. defined radial limits on T. Given a subset A ⊂ L(H), A denotes the set of operators commuting with every element of A. The set A is called the commutant of A, and it is a unital algebra, closed in the weak operator topology. An important example is {S} = WS = Tu : u ∈ H ∞ . A function m ∈ H ∞ is inner if |m(ζ )| = 1 for a.e. ζ ∈ T. For every inner function m ∈ H ∞ , the space mH 2 = Tm H 2 is closed and invariant for S. The compression of S to H(m) = H 2 mH 2 is denoted S(m). In other words, S(m) = PH(m) S | H(m) or, equivalently, S(m)∗ = S ∗ | H(m). Another important example of a commutant is

S(m)∗ = WS(m)∗ = Tu∗ H(m): u ∈ H ∞ .

This was proved by Sarason [20]. An operator T ∈ L(H) is a contraction if T 1. A contraction T is completely nonunitary if it has no invariant subspace on which it acts as a unitary operator. For completely nonunitary contractions T , there is a homomorphism u → u(T ) ∈ L(H) which extends the polynomial functional calculus to functions u ∈ H ∞ . This is called the Sz.-Nagy–Foias functional calculus. For instance, u(S) = Tu , and u(S(m)) = PH(m) Tu | H(m).


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A completely nonunitary contraction T ∈ L(H) is of class C0 if u(T ) = 0 for some u ∈ H ∞ \ {0}. The ideal {u ∈ H ∞ : u(T ) = 0} ⊂ H ∞ is principal, and it is generated by an inner function, uniquely determined up to a constant factor of absolute value 1. This function is called the minimal function of T . The most basic example is S(m), whose minimal function is m. We refer the reader to [23] for further background on the analysis of contractions, to [5] for dual algebras, and to [4] for the class C0 . We will refer as needed to these and other original sources for specific results. 3. The closability property Consider a unital subalgebra A of the algebra L(H) of bounded operators on the complex Hilbert space H. The algebra A is not assumed to be norm closed. Definition 3.1. A linear transformation X : D(X) → H is said to commute with A if for every h ∈ D(X) and every T ∈ A we have T h ∈ D(X) and XT h = T Xh. We define now the main concept we study in this paper. Definition 3.2. The algebra A is said to have the closability property if every linear transformation X which commutes with A, and whose domain D(X) is dense in H, is closable. We recall that a linear transformation X is closable if the closure of its graph G(X) = h ⊕ Xh: h ∈ D(X) is again the graph of a linear transformation, usually denoted X and called the closure of X. Equivalently, X is closable if given a sequence hn ∈ D(X) such that limn→∞ hn = 0 and the limit k = limn→∞ Xhn exists, it follows that k = 0. The following observation is a trivial consequence of the fact that a linear transformation commuting with an algebra also commutes with smaller algebras. Lemma 3.3. Assume that A ⊂ B ⊂ L(H) are unital algebras. If A has the closability property then so does B. In particular, if A is commutative and has the closability property, then its commutant A also has the closability property. We start with some examples of algebras which do not have the closability property. The arguments are based on the following simple fact. Lemma 3.4. Let A be a unital subalgebra of L(H). Assume that there exist linear manifolds M, N ⊂ H such that (1) T M ⊂ M and T N ⊂ N for every T ∈ A; (2) M ∩ N = {0} and M + N = H; (3) M ∩ N = {0}. Then A does not have the closability property.

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Proof. Define a linear transformation with domain D(X) = M + N by setting Xh = 0 for h ∈ M and Xh = h for h ∈ N . If X were closable, its closure would satisfy Xh = 0 and Xh = h for any h ∈ M ∩ N , and this is absurd for h = 0. 2 Proposition 3.5. The following algebras do not have the closability property: (1) (2) (3) (4) (5)

The algebra PS generated by the unilateral shift S. The algebra PS(m) , where m is an inner function which is not a finite Blaschke product. The WOT-closed algebra WS ∗ . The WOT-closed algebra WU generated by the bilateral shift U on L2 . Any algebra of the form A ⊗ IK , where A ⊂ L(H) is a unital algebra, and K is an infinite dimensional Hilbert space.

Proof. For the first example, choose an outer function f ∈ H 2 which is not rational, and define M to consist of all polynomials and N = {pf : p a polynomial}. The hypotheses of Lemma 3.4 are verified trivially since both of these spaces are dense in H 2 . Next, assume that m is an inner function but not a finite Blaschke product, and consider a factorization m = m1 m2 such that the inner functions mj are not finite Blaschke products. We can define then subspaces M, N ⊂ H(m) by M = {PH(m) p: p a polynomial} and N = {PH(m) (pm2 ): p a polynomial}. The space M is dense in H(m), so to verify the hypotheses of Lemma 3.4 it suffices to show that M ∩ N = {0}. Consider indeed two polynomials p, q such that PH(m) p = PH(m) (qm2 ). In other words, we have p = qm2 + rm1 m2 for some r ∈ H 2 . If p = 0, this equality implies that the inner factor of p (obviously a finite Blaschke product) is divisible by m2 , contrary to our choice of factors. For example (3), we choose M = {p: p a polynomial} ⊂ H 2 , and we denote by N the linear manifold generated by the functions kλ (z) = (1 − λz)−1 , λ ∈ D \ {0}. These spaces are dense in H 2 , and the identities ∗

p(z) − p(0) S p (z) = , z

S ∗ kλ = λkλ

easily imply that they are invariant under WS ∗ . Finally, a function p in their intersection must be both a polynomial, and a rational function vanishing at ∞, hence p = 0. For example (4), define two sets ω± = {e±it : 0 < t < 3π/2} ⊂ T, denote by χ± their characteristic functions, and set M = χ+ H 2 and N = χ− H 2 . Since M = χ+ L2 and N = χ− L2 , we clearly have M + N = L2 and M ∩ N = χω+ ∩ω− L2 . The fact that M ∩ N = {0} follows easily from the F. and M. Riesz theorem. Finally, assume that K is an infinite dimensional Hilbert space, and let M0 , N0 ⊂ K be two dense linear manifolds such that M0 ∩ N0 = {0}. Then M = H ⊗ M0 and N = H ⊗ N0 will satisfy the hypotheses of Lemma 3.4 for the algebra A ⊗ IK . 2 The first two examples above indicate that an algebra with the closability property must be reasonably large, while the last one shows that it should not have uniform infinite multiplicity. In this paper we will focus on algebras which have multiplicity one. The first example of an algebra with the closability property was of this kind: any maximal abelian selfadjoint subalgebra of L(H) has the closability property, as shown in [3]. This, along with the examples described in Proposition 3.7 (the first of which also appeared in [3], while the second was proved indepen-


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dently in [21]), will be treated in a unified manner in Section 4. The proofs of these particular cases do in fact suggest the more general methods. First, a useful observation about bounded outer functions. This is certainly known, but we could not find a reference. We use the notation u 2 for the norm of a function u ∈ L2 . ∞ with Lemma 3.6. Let v ∈ H ∞ be an outer function. There exist outer functions (wn )∞ n=1 ⊂ H 2 the property that limn→∞ u − wn vu 2 = 0 for every u ∈ L . In particular, if T is a completely nonunitary contraction, the sequence (wn (T )v(T ))∞ n=1 converges to I in the strong operator topology.

Proof. The functions wn will be specified by the requirements that (wn v)(0) > 0, and (wn v)(ζ ) = 1 |v(ζ )|

if |v(ζ )| 1/n, if |v(ζ )| < 1/n

for a.e. ζ ∈ T, so that wn ∞ n. Observe that 1 (wn v)(0) = 2π

logv(ζ ) |dζ | → 1

|v(ζ )| 0 and vf (ζ ) = min 1,

1 |f (ζ )|

for almost every ζ ∈ T. The functions vf and uf = f vf belong to H ∞ , and in fact uf (ζ ) = min 1, f (ζ )

a.e. ζ ∈ T.

(3.1)

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Consider first the algebra WS which consists precisely of the analytic Toeplitz operators Tu with u ∈ H ∞ . Let X be a densely defined linear transformation commuting with this algebra, and let f, g ∈ D(X). Observe first that uf = vf f = Tvf f ∈ D(X), and therefore we can write vg uf Xg = Tvg uf Xg = XTvg uf g = X(vg uf g) = X(uf ug ) = XTug uf = Tug Xuf = ug Xuf . Let now gn ∈ D(X) be a sequence converging to zero such that the limit h = limn→∞ Xgn exists. Passing if necessary to a subsequence, we may assume that gn (ζ ) → 0 for almost every ζ ∈ T. By virtue of (3.1) we also have |vgn (ζ )| → 1 and ugn (ζ ) → 0 for a.e. ζ , and therefore ugn Xuf 22

1 = 2π

2π it 2 ug e (Xuf ) eit 2 dt → 0 n 0

as n → ∞ by the dominated convergence theorem. The identity vgn uf Xgn = ugn Xuf proved earlier, along with the fact that |vgn | → 1 a.e., implies that uf h = 0 for every f ∈ D(X). Choosing a function f which is not identically zero, we deduce that h = 0, thus proving that X is closable. Consider now an inner function m, and define a map J : H(m) → H(m) by setting (Jf )(ζ ) = ζf (ζ )m(ζ ),

ζ ∈ T.

(3.2)

This is a conjugate linear surjective isometry on H(m) satisfying the equation J S(m) = S(m)∗ J . More generally, we have the identity J A = A∗ J,

A ∈ WS(m) .

This identity is easily verified when A is a polynomial in S(m), and it extends to arbitrary A using the continuity properties of the functional calculus with H ∞ functions. Let us denote by ξ = 1 − m(0)m ∈ H(m) the projection of 1 onto H(m). This is a separating vector for H(m). That is, the equality Aξ = 0 implies A = 0 for A ∈ H(m). Indeed, if A = u(S(m)), we have Aξ = PH(m) u, and this function is zero if and only if u divides m, in which case u(S(m)) = 0. Consider now a densely defined linear transformation X commuting with WS(m) . We will show that X is closable by proving the identity Xh1 , J h2 = h1 , J Xh2 ,

h1 , h2 ∈ D(X),

which shows that J X ⊂ X ∗ J , and hence X ∗ is densely defined. Indeed, fix h1 , h2 ∈ D(X), and choose an outer function v ∈ H ∞ such that the functions a1 = vh1 ,

a2 = vh2 ,

b1 = vXh1 ,

b2 = vXh2


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are bounded. Set V = v(S(m)), Aj = aj (S(m)), Bj = bj (S(m)) so that V hj = Aj ξ and V Xhj = Bj ξ for j = 1, 2. Observe first that A1 B2 ξ = A1 V Xh2 = XA1 V h2 = XA1 A2 ξ, and a similar calculation shows that A1 B2 ξ = A2 B1 ξ . We conclude that A1 B2 = A2 B1 because ξ is separating. Next we use Lemma 3.6 to find operators Wn ∈ WS(m) such that Wn V converges to I in the strong operator topology. We have then Xh1 , J h2 = lim Wn V Xh1 , J Wn V h2 n→∞

= lim Wn B1 ξ, J Wn A2 ξ n→∞ = lim Wn ξ, B1∗ J Wn A2 ξ n→∞

= lim Wn ξ, J Wn B1 A2 ξ n→∞

= lim Wn ξ, J Wn A1 B2 ξ n→∞ = lim Wn ξ, A∗1 J Wn B2 ξ n→∞

= lim Wn A1 ξ, J Wn B2 ξ = h1 , J Xh2 , n→∞

where we used the identity A1 B2 = A2 B1 and the fact that Wn commutes with Aj and Bj . The theorem is proved. 2 Note incidentally that the example of WS shows that closability of an algebra A is not generally inherited by the algebra {T ∗ : T ∈ A}. We conclude this section with a simple fact which will be used in thestudy of closability for quasisimilar algebras. Let Ai ⊂ L(Hi ), i ∈ I , be algebras. The algebra i∈I Ai ⊂ L( i∈I Hi ) consists of those operators of the form i∈I Ti , where Ti ∈ Ai for each i, and sup{ Ti : i ∈ I } < ∞. Lemma 3.8. A direct sum A = property for every i ∈ I .

i∈I

Ai has the closability property if and only if Ai has this

Proof. Assume first that A has the closability property, and Xi0 is a densely defined linear transformation on Hi0 commuting with Ai0 for some i0 ∈ I . We define a linear transformationX with dense domain D(X) = i∈I Di , where Di0 = D(Xi0 ), Di = Hi for i = i0 , and X( hi ) = ki , where ki0 = Xi0 hi0 and ki = 0 for i = i0 . The linear transformation X commutes with A, hence it is closable. It follows that Xi0 must be closable as well. Conversely, assume that each Ai has the closability property, and let X be a densely defined linear transformation commuting with A. If Pj ∈ A denotes the orthogonal projection onto the j th component of i∈I Hi , we have then Pj X ⊂ XPj , and the linear transformation Xj : Dj = Pj D(X) → Hj defined by Xj = X | Dj commutes with Aj . It follows that each Xj is closable, and then it is easy to verify that X is closable as well. 2

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4. Rationally strictly cyclic vectors and confluence The examples in Proposition 3.7, as well as maximal abelian selfadjoint subalgebras (also known as masas), can actually be treated in a unified manner. For this purpose we need a new concept. Definition 4.1. Let A ⊂ L(H) be a unital algebra. A vector h0 ∈ H is called a rationally strictly cyclic vector for A if for every h ∈ H there exist A, B ∈ A such that Bh = Ah0 and ker B = {0}. Recall that h0 is said to be strictly cyclic for A if Ah0 = H. Thus, a strictly cyclic vector is rationally strictly cyclic, but not conversely. The examples considered in this paper do not have strictly cyclic vectors except in trivial cases. Lemma 4.2. The following algebras have rationally strictly cyclic vectors: (1) WS . (2) WS(m) . (3) Any masa on a separable Hilbert space. More generally, any masa with a cyclic vector. Proof. The vector 1 ∈ H 2 is rationally strictly cyclic for WS , while 1 − m(0)m = PH(m) 1 is rationally strictly cyclic for WS(m) . For (3), we may assume that H = L2 (μ), where μ is a Borel probability measure on some compact topological space, and A = {Mu : u ∈ L∞ (μ)}, where Mu f = uf,

u ∈ L∞ (μ), f ∈ L2 (μ).

Since every function in L2 (μ) is the quotient of two bounded functions, the constant function 1 is rationally strictly cyclic for A. 2 Here are two useful properties of algebras with rationally strictly cyclic vectors. Lemma 4.3. Let A ⊂ L(H) be a unital algebra with a rationally strictly cyclic vector h0 . (1) If T ∈ A \ {0} then T h0 = 0. (2) If A is commutative and D ⊂ H is a dense linear manifold, invariant for A, then {ker T : T ∈ A, T h0 ∈ D} = {0}. Proof. Assume that T ∈ A and T h0 = 0. Given x ∈ H, choose Ax , Bx ∈ A such that Bx x = Ax h0 and ker Bx = {0}. We have then Bx T x = T Bx x = T Ax h0 = Ax T h0 = 0, and therefore T x = 0. This implies that T = 0 since x is arbitrary. Assume now that A is commutative and D ⊂ H is a dense linear manifold, invariant for A. Let h ∈ H be a vector such that Ah = 0 whenever A ∈ A and Ah0 ∈ D. Using the notation above, we have Ax h0 = Bx x ∈ D whenever x ∈ D, and therefore Ax h = 0 for x ∈ D. Thus


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0 = Bh Ax h = Ax Bh h = Ax Ah h0 = Ah Ax h0 = Ah Bx x = Bx Ah x for x ∈ D, which implies Ah x = 0 for such vectors x. From the density of D we deduce that Ah = 0, and thus Bh h = Ah h0 = 0 and h = 0, as desired. 2 We can now prove a generalization of Proposition 3.7. Theorem 4.4. Any unital commutative algebra A ⊂ L(H) which has a rationally strictly cyclic vector has the closability property. Proof. Let h0 ∈ H be a rationally strictly cyclic vector for the algebra A, and let X be a linear transformation with dense domain D(X) commuting with A. Consider a sequence {xn } ⊂ D(X) such that xn → 0 and Xxn → h as n → ∞. By Lemma 4.3(2), it will suffice to show that T h = 0 whenever T ∈ A and T h0 ∈ D(X). Fix such an operator T , and choose operators An , Bn , B, A ∈ A satisfying Bn xn = An h0 , BXT h0 = Ah0 and ker Bn = ker B = {0} for all n 1. Using the commutativity of A, and the fact that X commutes with A we deduce that Bn (BT Xxn − Axn ) = BT XBn xn − ABn xn = BT XAn h0 − AAn h0 = BXAn T h0 − AAn h0 = An (BXT h0 − Ah0 ) = 0. Since Bn is one-to-one, we have BT Xxn = Axn . Letting n → ∞ we obtain BT h = 0 and hence T h = 0, as desired.

2

Observe that if an algebra B ⊂ L(H) contains a unital commutative algebra A with the closability property, then B also has the closability property. Therefore Theorem 4.4 and the result of Arveson mentioned in the introduction have the following consequence. Corollary 4.5. There exists no proper subalgebra of L(H) which is 2-transitive, closed in the strong operator topology and contains a unital commutative subalgebra with a rationally strictly cyclic vector. The calculations in the preceding proof can be used to relate closed, densely defined linear transformations commuting with A with linear transformations of the form B −1 A with A, B ∈ A and ker B = {0}. Note that

G B −1 A = {h ⊕ k ∈ H ⊕ H: Ah = Bk}, and this is generally larger than

G AB −1 = {Bh ⊕ Ah: h ∈ H}.

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Also observe that two linear transformations of this form, say B −1 A, B1−1 A1 , which agree on a dense linear manifold D, must in fact be equal. Indeed, the equality on D implies that BA1 h = B1 Ah for h ∈ D, and therefore B1 A = BA1 . Thus for h ⊕ k ∈ G(B −1 A) we have B(B1 k − A1 h) = B1 (Bk − Ah) = 0, and hence h ⊕ k ∈ G(B1−1 A1 ) because B is injective. Proposition 4.6. Let A be a commutative algebra with a rationally strictly cyclic vector h0 . For every densely defined linear transformation X commuting with A such that h0 ∈ D(X), there exist A, B ∈ A such that ker B = {0} and X ⊂ B −1 A. If X ∈ L(H), we have X = B −1 A. In particular, the commutant A is a commutative algebra. Proof. As in the preceding proof, we choose for each h ∈ H operators Ah , Bh ∈ A such that ker Bh = {0} and Bh h = Ah h0 . Assume now that h0 ∈ D(X) and X commutes with A. We have then for h ∈ D(X), Bh BXh0 Xh = BXh0 XBh h = BXh0 XAh h0 = Ah BXh0 Xh0 = Ah AXh0 h0 = AXh0 Bh h = Bh AXh0 h, −1 from which we conclude that X ⊂ BXh AXh0 because Bh is injective. The remaining assertions 0 follow easily from this. 2

Sometimes an algebra with a rationally strictly cyclic vector has the stronger property defined below. Definition 4.7. Let A ⊂ L(H) be a unital algebra. We will say that A is confluent if for every two vectors h1 , h2 ∈ H \ {0} there exist injective operators B1 , B2 ∈ A such that B1 h1 = B2 h2 . Proposition 4.8. For a commutative unital algebra A ⊂ L(H), the following two assertions are equivalent: (1) A has a rationally strictly cyclic vector and ker B = {0} for every B ∈ A \ {0}; (2) A is confluent. If these equivalent conditions are satisfied, then every nonzero vector is rationally strictly cyclic for A; moreover, every densely defined linear transformation commuting with A is contained in B −1 A for some A, B ∈ A such that ker B = {0}. Proof. Assume first that (1) holds, and h1 , h2 ∈ H \ {0}. With the notation used earlier, we have Ah2 Bh1 h1 = Ah2 Ah1 h0 = Ah1 Bh2 h2 . The operators Ah1 , Ah2 are not zero, and therefore Ah2 Bh1 , Ah1 Bh2 are injective by hypothesis.


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Conversely, assume that A is confluent. Clearly, every nonzero vector is then rationally strictly cyclic. It remains to show that every B ∈ A \ {0} is injective. Assume to the contrary that Bh1 = 0 / ker B. If B1 , B2 are as in Definition 4.7, we obtain for some h1 = 0, and choose h2 ∈ 0 = B1 Bh1 = BB1 h1 = BB2 h2 = B2 Bh2 . This implies Bh2 = 0, contrary to the choice of h2 . The last assertion follows from Proposition 4.6. 2 As an application of the results in this section, we show that some other algebras of Toeplitz operators have the closability property. Consider a bounded, connected open set Ω ⊂ C bounded by n + 1 analytic simple Jordan curves, and fix a point ω0 ∈ Ω. The algebra H ∞ (Ω) consists of the bounded analytic functions on Ω, while Hω20 (Ω) is defined as the space of analytic functions f on Ω with the property that |f |2 has a harmonic majorant in Ω. The norm on Hω20 (Ω) is defined as f 22 = inf u(ω0 ): u a harmonic majorant of |f |2 ,

f ∈ Hω20 (Ω).

Multiplication by a function u ∈ H ∞ (Ω) determines a bounded operator Tu on Hω20 (Ω). Proposition 4.9. The constant function 1 ∈ Hω20 (Ω) is a rationally strictly cyclic vector for the algebra {Tu : u ∈ H ∞ (Ω)}. In particular, this algebra has the closability property. The statement is equivalent to the following result. We refer to [1] and [13] for the function theoretical background. Lemma 4.10. For every function f ∈ Hω20 (Ω) there exist u, v ∈ H ∞ (Ω) such that v ≡ 0 and vf = u. Proof. Denote by π : D → Ω a (universal) covering map such that π(0) = ω0 , and denote by Γ the corresponding group of deck transformations. Thus, Γ consists of those analytic automorphisms ϕ of D with the property that π ◦ ϕ = π . The map f → f ◦ π is an isometry from Hω20 (Ω) onto the space of those functions g ∈ H 2 such that g ◦ ϕ = g for every ϕ ∈ Γ . Fix now f ∈ Hω20 (Ω) \ {0}, and construct an outer function w ∈ H 2 such that |w(ζ )| = min{1, 1/|f ◦ π(ζ )|} for almost every ζ ∈ T. The function w is obviously modulus automorphic in the sense that |w ◦ ϕ| = |w| for every ϕ ∈ Γ . It follows that there is a group homomorphism γ : Γ → T such that w ◦ ϕ = γ (ϕ)w for every ϕ ∈ Γ . Choose a modulus automorphic Blaschke product b ∈ H ∞ such that b ◦ ϕ = γ (ϕ)b for γ ∈ Γ ; see [13, Theorem 5.6.1] for the construction of such products. Then there exist functions u, v ∈ H ∞ (Ω) such that v ◦ π = bw and u ◦ π = bw(f ◦ π). These functions satisfy the requirements of the lemma. 2 5. Quasisimilar algebras We will now study the effect of quasisimilarity on the closability property and the existence of rationally strictly cyclic vectors. Recall that an operator Q ∈ L(H1 , H2 ) is called a quasiaffinity if it is injective and has dense range.

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Definition 5.1. An algebra A1 ⊂ L(H1 ) is a quasiaffine transform of an algebra A2 ⊂ L(H2 ) if there exists a quasiaffinity Q ∈ L(H1 , H2 ) such that for every T2 ∈ A2 , we have QT1 = T2 Q for some T1 ∈ A1 . We write A1 ≺ A2 if A1 is a quasiaffine transform of A2 . The relation A1 ≺ A2 can simply be written as Q−1 A2 Q ⊂ A1 for some quasiaffinity Q. Proposition 5.2. Assume that A1 ⊂ L(H1 ) and A2 ⊂ L(H2 ) are unital algebras such that A1 ≺ A2 . Then (1) If A1 is commutative, then A2 is commutative as well. (2) If A2 has the closability property, then so does A1 . (3) If A2 is confluent, then so is A1 . Proof. Let Q be as in Definition 5.1. Since the map T → Q−1 T Q is obviously an injective algebra homomorphism on A2 , part (1) is immediate. To prove (2), let X be a densely defined linear transformation commuting with A1 . Define the linear transformation Y = QXQ−1 on the dense subspace D(Y ) = QD(X). Since all the operators T2 ∈ A2 have the property that Q−1 T2 Q is in A1 , it follows easily that Y commutes with A2 . Assume now that A2 has the closability property, so that Y is closable. We will verify that X is closable as well. Assume that hn ∈ D(X) are such that hn → 0 and Xhn → k as n → ∞. Obviously then D(Y ) Qhn → 0 and Y Qhn → Qk. We deduce that Qk = 0, and therefore k = 0 since Q is a quasiaffinity. Finally, assume that A2 is confluent and let h1 , h2 ∈ H1 \ {0}. We choose injective C1 , C2 ∈ A2 so that C1 Qh1 = C2 Qh2 , and observe that B1 h1 = B2 h2 , where Bj = Q−1 Cj Q ∈ A1 are injective. 2 Definition 5.3. An algebra A1 ⊂ L(H1 ) is quasisimilar to an algebra A2 ⊂ L(H2 ) if there exist quasiaffinities Q ∈ L(H1 , H2 ) and R ∈ L(H2 , H1 ) such that Q−1 A2 Q ⊂ A1 , R −1 A1 R ⊂ A2 , QR ∈ A2 , and RQ ∈ A1 . We write A1 ∼ A2 if A1 is quasisimilar to A2 . Using the proofs of parts (1) and (2) of the following result, it is easy to see that quasisimilarity is an equivalence relation. Proposition 5.4. Assume that A1 and A2 are commutative quasisimilar algebras, and Q, R satisfy the conditions of Definition 5.3. Then (1) We have Q−1 A2 Q = A1 and R −1 A1 R = A2 . (2) The maps T2 → Q−1 T2 Q and T1 → R −1 T1 R are mutually inverse algebra isomorphisms between A1 and A2 . (3) The commutant A1 is commutative if and only if A2 is commutative. (4) If h1 ∈ H1 is rationally strictly cyclic for A1 then Qh1 is rationally strictly cyclic for A2 . (5) The algebra A1 is confluent if and only if A2 is confluent. (6) The algebra A1 is confluent if and only if A2 is confluent. (7) The algebra A1 has the closability property if and only if A2 does.


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Proof. Define Φ : A2 → A1 and Ψ : A1 → A2 by setting Φ(T2 ) = Q−1 T2 Q and Ψ (T1 ) = R −1 T1 R. We have

Ψ Φ(T2 ) = R −1 Q−1 T2 QR = R −1 Q−1 QRT2 = T2 ,

T2 ∈ A 2 ,

and similarly Φ(Ψ (T1 )) = T1 for T1 ∈ A1 . This proves (2), and (1) follows from (2). Assume now that A1 is commutative and A, B ∈ A2 . We claim that RAQ and RBQ belong to A1 . Indeed,

T1 RAQ = R R −1 T1 R AQ = RA R −1 T1 R Q = RAR −1 T1 (RQ) = RAR −1 (RQ)T1 = RAQT1 for T1 ∈ A1 . We deduce that RAQRBQ = RBQRAQ and hence AQRB = BQRA. Taking A or B to be the identity operator, we deduce that QR commutes with B and A, and therefore QRAB = QRBA, and finally the desired equality AB = BA. To prove (4), assume that h1 is rationally strictly cyclic for A1 . Proposition 4.6 implies the existence of A1 , B1 ∈ A1 such that ker B1 = {0} and RQ = B1−1 A1 . Set A2 = R −1 A1 R, B2 = R −1 B1 R ∈ A2 , and observe that B2 QR = R −1 (B1 RQ)R = R −1 A1 R = A2 , that is QR = B2−1 A2 . Since QR is a quasiaffinity, it follows that ker A2 = {0}. To show that Qh1 is rationally strictly cyclic for A2 , fix a vector h2 ∈ H2 , and choose S1 , T1 ∈ A1 such that ker T1 = {0} and T1 Rh2 = S1 h1 . Set now T2 = R −1 T1 R, S2 = R −1 S1 R ∈ A2 , and note that ker T2 = {0}. We have RQRT2 h2 = RQT1 Rh2 = RQS1 h1 = S1 RQh1 = RS2 Qh1 , so that QRT2 h2 = S2 Qh1 . Applying B2 to both sides we obtain A2 T2 h2 = B2 S2 Qh1 , and strict cyclicity follows because A2 T2 , B2 S2 ∈ A2 and ker(A2 T2 ) = {0}. Assertion (5) follows easily from (4) and Proposition 4.8, or directly from Proposition 5.2(3). Assume now that A1 is confluent, and let h, k ∈ H2 be two nonzero vectors. Then there exist injective operators A1 , B1 ∈ A1 such that A1 Rh = B1 Rk. Thus we have A2 h = B2 k, where A2 = QA1 R and B2 = QB1 R are injective operators in A2 . This proves (6). Finally, assume that A2 has the closability property, and let X be a densely defined linear transformation commuting with A1 . As in the proof of Proposition 5.2(2), to prove (7) it will suffice to show that the linear transformation Y0 = QXQ−1 defined on the dense space D(Y0 ) = QD(X) is closable. To show this, we will define a linear transformation Y ⊃ Y0 which commutes with A2 . Its domain D(Y ) consists of all the finite sums of the form n Tn Qhn , where Tn ∈ A2 and hn ∈ D(X), and Y

n

Tn Qhn =

n

Tn QXhn .

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Toshow that Y is well defined, it will suffice to prove that n Tn Qhn = 0 implies R n Tn QXhn = 0. Indeed, since RTn Q ∈ A1 , we have RTn Qhn ∈ D(X) and

RTn QXhn =

n

XRTn Qhn = XR

n

The fact that Y commutes with every T ∈ A2 is easily verified. If n T Tn Qhn ∈ D(Y ), and YT

n

Tn Qh =

Tn Qhn = 0.

n

T Tn QXhn = T Y

n

The inclusion Y ⊃ Y0 is obvious since A2 is unital.

n Tn Qhn

∈ D(Y ) then clearly

Tn Qhn .

n

2

We will be using the results in this section for the special case of algebras generated by a completely nonunitary contraction T ∈ L(H). For such a contraction we will write H ∞ (T ) = u(T ): u ∈ H ∞ . Parts (1) and (2) of the following lemma are easily verified; in fact Definition 5.3 was formulated so as to make part (2) correct. Lemma 5.5. Let T1 and T2 be two completely nonunitary contractions. (1) (2) (3) (4)

If T1 ≺ T2 then H ∞ (T1 ) ≺ H ∞ (T2 ). If T1 ∼ T2 then H ∞ (T1 ) ∼ H ∞ (T2 ). If H ∞ (T1 ) ∼ H ∞ (T2 ) and T1 is of class C0 , then T2 is also of class C0 . If H ∞ (T1 ) ∼ H ∞ (T2 ) and T1 is not of class C0 , then T1 ∼ ϕ(T2 ) for some conformal automorphism ϕ of D.

Proof. To prove (3), observe that H ∞ (T1 ) ∼ H ∞ (T2 ) implies that H ∞ (T2 ) is isomorphic to H ∞ (T1 ). Assume that T1 is of class C0 . If T1 is a scalar multiple of the identity, then H ∞ (T1 ) = CI , and therefore H ∞ (T2 ) = CI and then T2 must be a scalar multiple of the identity, hence of class C0 . If T1 is not a scalar multiple of the identity, then H ∞ (T1 ) has zero divisors. Indeed, in this case the minimal function m of T1 can be factored into a product m = m1 m2 of two nonconstant inner functions, and then m1 (T1 ) = 0 = m2 (T1 ), while m1 (T1 )m2 (T1 ) = 0. We conclude that H ∞ (T2 ) must also have zero divisors, and this obviously implies that T2 is of class C0 . Finally, assume that H ∞ (T1 ) ∼ H ∞ (T2 ) and T1 (as well as T2 by part (3)) is not of class C0 . Let Q and R be quasiaffinities satisfying the conditions of Definition 5.3 for the algebras A1 = H ∞ (T1 ) and A2 = H ∞ (T2 ). The hypothesis implies that the maps u → u(T1 ) and u → u(T2 ) are algebra isomorphisms from H ∞ to H ∞ (T1 ) and H ∞ (T2 ), respectively. Thus, for every u ∈ H ∞ there exists a unique v ∈ H ∞ satisfying v(T2 ) = R −1 u(T1 )R. The map Φ : u → v is an algebra automorphism of H ∞ . In particular, the function ϕ = Φ(idD ) must have spectrum (in H ∞ ) equal to D, so that ϕ(D) = D. We claim that Φ(u) = u ◦ ϕ for every u ∈ H ∞ . Indeed, given λ ∈ D, we can factor u(z) − u(ϕ(λ)) = (z − ϕ(λ))w for some w ∈ H ∞ , so that Φ(u) − u(ϕ(λ)) = (ϕ − ϕ(λ))Φ(w). The equality (Φ(u))(λ) = u(ϕ(λ)) follows immediately.


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Since Φ is an automorphism, it follows that ϕ is a conformal automorphism of D, and clearly T1 ∼ ϕ(T2 ). 2 Corollary 5.6. Let T be a completely nonunitary contraction. If T ∼ S then H ∞ (T ) is confluent. If T ∼ S(m) then H ∞ (T ) has a rationally strictly cyclic vector. Proof. It suffices to observe that H ∞ (S) = WS , H ∞ (S(m)) = WS(m) , and to apply Proposition 5.4(2), (5) and (4). 2 For operators of class C0 , the converse of the preceding result is also true. The case of confluent algebras of the form H ∞ (T ) will be discussed more thoroughly in the remaining two sections of the paper. Proposition 5.7. Assume that T is a completely nonunitary contraction such that H ∞ (T ) has a rationally strictly cyclic vector. (1) If there exists f ∈ H ∞ \ {0} such that ker f (T ) = {0}, then T is of class C0 and T ∼ S(m), where m is the minimal function of T . (2) If ker f (T ) = {0} for every f ∈ H ∞ \ {0}, then H ∞ (T ) is confluent. Proof. Part (2) follows immediately from Proposition 4.8. To verify (1), assume that f ∈ H ∞ \ {0}, ker f (T ) = {0}, and H ∞ (T ) has a rationally strictly cyclic vector h0 ∈ H. Choose a nonzero vector h1 ∈ ker f (T ), and functions u1 , v1 ∈ H ∞ such that v1 (T ) is injective and v1 (T )h1 = u1 (T )h0 . The function u1 is not zero since v1 (T )h1 = 0. We claim that f (T )u1 (T ) = 0. Indeed, let h be an arbitrary vector in H. Choose u, v ∈ H ∞ such that v(T ) is injective and v(T )h = u(T )h0 . We have then v(T ) f (T )u1 (T )h = f (T )u1 (T ) v(T )h = f (T )u1 (T )u(T )h0 = f (T )u(T )u1 (T )h0 = f (T )u(T )v1 (T )h1 = 0, and therefore f (T )u1 (T )h = 0. Thus T is of class C0 because (f u1 )(T ) = 0 and f u1 ∈ H ∞ \ {0}. Finally, let m be the minimal function of T , denote by M the cyclic space for T generated by h0 , and set N = M⊥ . Let T = PN T | N be the compression of T to N . Clearly we have m(T ) = 0. Now let h ∈ N be a vector, and pick u, v ∈ H ∞ such that v(T ) is injective and v(T )h = u(T )h0 . In particular, we have v(T )h = 0. The injectivity of v(T ) is equivalent to the condition v ∧ m = 1, and this implies that v(T ) is injective as well, so that h = 0 We proved therefore that M = H. In other words, T has a cyclic vector, and thus T ∼ S(m) by the results of [25] (see also [4, Theorem III.2.3]). 2 We conclude this section with a result about arbitrary operators of class C0 . Proposition 5.8. For any operator T of class C0 , the commutant {T } has the closability property. Proof. The operator T is quasisimilar to an operator of the form T = i∈I S(mi ), where each mi is an inner function; see [4, Theorem III.5.1]. By Proposition 5.4(7), it suffices to show that

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{T } has the closability property. Now, {T } ⊃ i∈I {S(mi )} , and Lemma 3.8 shows that it suffices to show that {S(m)} has the closability property for each inner function m. This follows from Proposition 3.7 because {S(m)} = WS(m) . 2 6. Confluent algebras of the form H ∞ (T ) Consider a completely nonunitary contraction T ∈ L(H) such that H ∞ (T ) has a rationally strictly cyclic vector. According to Proposition 5.7, we have T ∼ S(m) if some nonzero operator in H ∞ (T ) has nonzero kernel. Therefore we will restrict ourselves now to operators T such that f (T ) is injective for every nonzero element of H ∞ . In other words, we will assume that H ∞ (T ) is a confluent algebra (cf. Proposition 4.8) and dim H > 1. In this case, the space H can be identified with a space of meromorphic functions. Let us denote by N the Nevanlinna class consisting of those meromorphic functions in D which can be written as u/v, with u, v ∈ H ∞ . Lemma 6.1. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent. Let h, h0 be two vectors such that h0 = 0, and choose u, v ∈ H ∞ , v = 0, such that v(T )h = u(T )h0 . Then the function u/v ∈ N is uniquely determined by h and h0 . We have u/v = 0 if and only if h = 0. Proof. Choose another pair of functions u1 , v1 ∈ H ∞ , v1 = 0, satisfying v1 (T )h = u1 (T )h0 . We have

v1 (T )u(T ) − v(T )u1 (T ) h0 = v1 (T )v(T ) − v(T )v1 (T ) h = 0, and therefore h0 ∈ ker(v1 u − vu1 )(T ). The hypothesis implies that v1 u = vu1 and hence u/v = u1 /v1 . 2 The function u/v will be denoted h/ h0 . It is clear that the map h → h/ h0 is an injective linear map from H to N , and u(T )h/u(T )h0 = h/ h0 if u ∈ H ∞ \ {0}. We also have h h h1 = · h0 h1 h0 provided that h0 , h1 ∈ H \ {0}. Now let h, h0 ∈ H \ {0}. There exists a unique integer n such that the nonzero function h/ h0 can be written as h u(z) (z) = zn h0 v(z) with u, v ∈ H ∞ and u(0) = 0 = v(0). The number n will be denoted ord0 (h/ h0 ). It will be convenient to write ord0 (h/ h0 ) = ∞ if h = 0. Lemma 6.2. Let T be a completely nonunitary contraction such that H ∞ (T ) is confluent. Then 0 inf{ord0 (h/ h0 ): h ∈ H} > −∞ for every h0 ∈ H \ {0}. Proof. Clearly ord0 (h0 / h0 ) = 0. For each integer n, the set Hn = h ∈ H: ord0 (h/ h0 ) −n


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is a linear manifold, and n0 Hn = H. Given integers m, k such that k 1, we denote by Dm,k the set of all vectors h ∈ H for which h/ h0 can be written as h u(z) (z) = z−m h0 v(z) with u ∞ , v ∞ k and |u(0)|, |v(0)| 1. Observe that Dm,k = Hn \ {0}. mn,k1

The proposition will follow if we can show that one of the sets Dm,k has an interior point, and this will follow from the Baire category theorem once we prove that each Dm,k is closed. Assume indeed that (hi )∞ i=0 ⊂ Dm,k is a sequence such that hi → h as i → ∞. For each i write hi ui (z) = z−m h0 vi with ui ∞ , vi ∞ k and |ui (0)|, |vi (0)| 1. By the Vitali–Montel theorem we can assume, after dropping to a subsequence, that there exist functions u, v ∈ H ∞ such that ui (z) → u(z) and vi (z) → v(z) uniformly for z in each compact subset of D. Clearly u ∞ , v ∞ k and |u(0)|, |v(0)| 1. Moreover, we have ui (T )h0 → u(T )h0 and vi (T )hi → v(T )h in the weak topology. (For the second sequence we need to write

vi (T )hi − v(T )h = vi (T )(hi − h) + vi (T ) − v(T ) h, and use the fact that the first term tends to zero in norm, while the second tends to zero weakly by [23, Theorem III.2.1].) The identities T m vi (T )hi = ui (T )h0 for m 0 (resp., vi (T )hi = T −m ui (T )h0 for m < 0) therefore imply T m v(T )h = u(T )h0 (resp., v(T )h = T −m u(T )h0 ) so that h/ h0 = z−m u/v, and thus h ∈ Dm,k , as desired. 2 Lemma 6.3. Let T be a completely nonunitary contraction such that H ∞ (T ) is confluent. Then T is injective and T H is a closed subspace of codimension 1. Thus T is a Fredholm operator with index(T ) = −1. Proof. The operator T belongs to a confluent algebra, hence it is injective. Note next that ord0 (T h/ h0 ) = ord0 (h/ h0 ) + 1 and hence inf ord0 (h/ h0 ): h ∈ H + 1 = inf ord0 (h/ h0 ): h ∈ T H . Since these numbers are finite, we cannot have T H = H. To conclude the proof, it will suffice to show that T H has codimension one, since this implies that it is closed as well. Choose h0 ∈ H \ T H, and note that ord0 (h/ h0 ) 0 for every h. Indeed, ord0 (h/ h0 ) = −n < 0 implies an identity of the form T n v(T )h = u(T )h0

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with u(0) = 0. Factoring u(z) − u(0) = zw(z), we obtain h0 =

1 n−1 v(T )h − w(T )h0 ∈ T H, T T u(0)

a contradiction. Thus the function h/ h0 is analytic at 0, and we can therefore define a linear functional Φ : H → C by setting Φh = (h/ h0 )(0). We will show that ker Φ ⊂ T H. Indeed, h ∈ ker Φ implies that v(T )h = T u(T )h0 for some u, v ∈ H ∞ with v(0) = 0. Factoring again v(z) − v(0) = zw(z), we obtain h=

1 T u(T )h0 − w(T )h ∈ T H, v(0)

as claimed. Thus T H has codimension 1, and the lemma is proved.

2

The preceding results allow us to describe completely the spectral picture of T , as well as its commutant. The argument for (3) already appears in [10], and is included for the reader’s convenience. Theorem 6.4. Let T ∈ L(H) be a completely nonunitary contraction such that H ∞ (T ) is confluent. Then (1) We have σ (T ) = D and σe (T ) = T. (2) For each λ ∈ D, λI − T is injective and has closed range of codimension 1. (3) {ker(λI − T ∗ ) : λ ∈ D} = H. More generally, {ker(λI − T ∗ ): λ ∈ S} = H whenever the set S ⊂ D has an accumulation point in D. (4) For every nonzero invariant subspace M of T , there exists an inner function m ∈ H ∞ such that m(T )H = M and the compression TM⊥ of T to M⊥ is quasisimilar to S(m). Conversely, for every inner function m, the minimal function of T(m(T )H)⊥ is m. (5) {T } = H ∞ (T ). (6) The operator T is of class C10 . Thus, the powers T ∗n converge strongly to zero and limn→∞ T n h = 0 for h ∈ H \ {0}. In particular, properties (2) and (3) say that T ∗ belongs to the class B1 (D) defined in [10]. Proof. For λ ∈ D, the operator Tλ = (I − λT )−1 (T − λI ) is also a completely nonunitary contraction, and H ∞ (Tλ ) = H ∞ (T ) is confluent. Thus Lemma 6.3 implies immediately (2). In turn, (1) follows from (2) since T is a contraction. Next we prove (4). Let M = {0} be invariant for T , set N = M⊥ , and choose h0 ∈ M \ {0}. Denote by T = PN T | N the compression of T to N . Given h ∈ N , an equality of the form v(T )h = u(T )h0 implies v(T )h ∈ M, and therefore v(T )h = 0. The fact that h0 is rationally strictly cyclic for H ∞ (T ) implies that T is locally of class C0 , and hence of class C0 by [24] (see also [4, Theorem III.3.1]). Denote by m the minimal function of T . Note that in particular PN m(T ) | N = m(T ) = 0, so that m(T )N ⊂ M.

(6.1)


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We show next that T has a cyclic vector, and hence it is quasisimilar to S(m). Assume to the contrary that T does not have a cyclic vector, and let N1 , N2 be cyclic spaces for T generated by two nonzero vectors h1 , h2 such that T | N1 ∼ S(m) and N1 ∩ N2 = {0} (see [25] or [4, Theorem III.2.13]). There exist nonzero functions u1 , u2 ∈ H ∞ such that u1 (T )h1 = u2 (T )h2 . Dividing these functions by their greatest common inner divisor, we may assume that u1 and u2 do not have any (non constant) common inner factor. We also have u1 (T )h1 = u2 (T )h2 ∈ N1 ∩ N2 , and hence these vectors are equal to zero. We deduce that m divides u1 , and hence m ∧ u2 = 1. This last equality implies that u2 (T ) is a quasiaffinity, hence u2 (T )h2 = 0, a contradiction. Thus T is indeed cyclic. Using (6.1), we observe that m(T )H = m(T )M + m(T )N ⊂ M. Denote now M1 = m(T )H, N1 = M⊥ 1 , and T1 = PN1 T | N1 . Clearly m(T1 ) = 0, and T ∗ = T1∗ | N . It follows that the minimal function of T1 is also m. Applying to T1 the argument showing that T has a cyclic vector yields the same for T1 . Hence M = M1 by the results of [25] (see also [4, Theorem III.2.13]). We start next with a given inner function m, and denote by m1 the minimal function of T(m(T )H)⊥ . The function m1 must divide m, so that m = m1 m2 for some other inner function m2 . With the notation H1 = m1 (T )H = m(T )H, T1 = T | H1 , the algebra H ∞ (T1 ) is confluent, and m2 (T1 )H1 = m2 (T )m1 (T )H = m(T )H = H1 , so m2 (T1 ) has dense range. We claim that m2 (T1 )M = M for every invariant subspace M for T1 . Indeed, from the first part of (4) we know that M = m3 (T1 )H1 for some inner function m3 . Hence m2 (T1 )M = m2 (T1 )m3 (T1 )H1 = m3 (T1 )m2 (T1 )H1 = m3 (T1 )H1 = M, as claimed. Since H ∞ (T1 ) is confluent, we have σ (T1 ) = D by part (1) of the theorem. This implies that T1 belongs to the class A defined in [5]. By the results of [8], there exist vectors x, y ∈ H1 such that

1 u(T1 )x, y = 2π

2π

1 − m2 (0)m2 eit u eit dt 0

for all u ∈ H ∞ . In particular, v(T1 )m2 (T1 )x, y = 0 for v ∈ H ∞ . Set M = {T1n x: n 0}, and observe now that y ⊥ m2 (T1 )M, and therefore y ⊥ M as well. In particular, 1 0 = x, y = 2π

2π 2

1 − m2 (0)m2 eit dt = 1 − m2 (0) , 0

and this implies that m2 is a constant function. We reach the desired conclusion that the minimal function of T(m(T )H)⊥ is m. To prove (3),∗ assume that S ⊂ D has an ∗accumulation point in⊥D, and note that the space N = {ker(λI − T ): λ ∈ S} is invariant for T . Therefore M = N is invariant for T . If M = {0}, we have then m(T )H ⊂ M for some inner function m, and therefore ker m(T )∗ ⊃ N . Given λ ∈ S, choose a nonzero vector fλ ∈ ker(λI − T ∗ ), and observe that 0 = m(T )∗ fλ = m(λ)fλ .

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Thus m(λ) = 0 for λ ∈ S, and we conclude that m = 0, which is impossible. This contradiction implies that M = {0}, thus verifying (6.1). Consider next an operator X ∈ {T } = H ∞ (T ) . By Proposition 4.6, there exist u, v ∈ H ∞ , v = 0, so that v(T )X = u(T ). With fλ as above, we have

∗ v(λ)X ∗ fλ = v(T )X fλ = u(T )∗ fλ = u(λ)f (λ), and thus u(λ) X ∗ fλ ∗ v(λ) = f X . λ We deduce that w = u/v ∈ H ∞ and X = w(T ). The fact that the powers of T ∗ tend strongly to zero follows from (3) because T ∗n fλ = n λ fλ → 0 as n → ∞ for λ ∈ D. It remains to prove that the space M = h ∈ H: lim T n h = 0 n→∞

is equal to {0}. Assume to the contrary that M = {0}, and observe that H ∞ (T | M) is also confluent. In particular, σ (T | M) = D and T | M is of class C00 . According to [7] and [5, Theorem 6.6], T | M belongs to the class Aℵ0 , and, by [5, Corollary 5.5], T has a further invariant subspace N ⊂ M such that N T N has infinite dimension. This space must however have dimension 1 because H ∞ (T | N ) is confluent. This contradiction shows that we must have M = {0}, as claimed. 2 Recall that N+ ⊂ N denotes the collection of functions of the form u/v, where u, v ∈ H ∞ and v is outer. ∞ Corollary 6.5. Let T ∈ L(H) be a completely nonunitary contraction n such that H (T ) is con∗ fluent, and fix a vector h0 ∈ ker T , h0 = 0. Assume that H = {T h0 : n 0}; that is, h0 is cyclic for T . Then h/ h0 ∈ N+ for every h ∈ H.

Proof. We can assume that h = 0. Now choose functions u, u0 ∈ H ∞ \ {0} such that u0 /u = h/ h0 . Thus we have u0 (T )h0 = u(T )h. Consider the factorizations u = mv and u0 = m0 v0 , where m, m0 are inner and v, v0 are outer. By [23, Proposition III.3.1], the operator v0 (T ) is a quasiaffinity, and therefore m0 (T )H =

n0

T n v0 (T )m0 (T )h0 =

T n v(T )m(T )h ⊂ m(T )H.

n0

It follows that (m(T )H)⊥ ⊂ (m0 (T )H)⊥ , and thus m divides m0 by Theorem 6.4(4). It follows that h u0 v0 (m0 /m) = ∈ N+ , = h0 u v as claimed.

2


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We will denote by A the disk algebra. This consists of those functions in H ∞ which are restrictions of continuous functions on D. If T is a completely nonunitary contraction, we set A(T ) = {u(T ): u ∈ A}. Corollary 6.6. Consider an operator T ∈ L(H), where H is an infinite dimensional Hilbert space. Then (1) The algebra PT is not confluent. (2) If T is a completely nonunitary contraction, then A(T ) is not confluent. Proof. In proving (1), there is no loss of generality in assuming that T < 1 since PT = PαT for any α > 0. Under this assumption, we have PT ⊂ A(T ), so it suffices to prove part (2). Assume therefore that T is a completely nonunitary contraction and A(T ) is confluent. The larger algebra H ∞ (T ) is confluent as well, and Proposition 4.6 implies that for every f ∈ H ∞ , the operator f (T ) ∈ {T } = A(T ) can be written as f (T ) = v(T )−1 u(T ) with u, v ∈ A, v = 0. We have then v(T )f (T ) = u(T ), and thus f = u/v. It is known, however, that there are functions in H ∞ which cannot be represented as quotients of elements of A. An example is provided by any singular inner function f (λ) = e−

ζ +λ T ζ −λ

dμ(ζ )

,

λ ∈ D,

such that the closed support of the singular measure μ is the entire circle T.

2

The assertion in Proposition 4.6, concerning unbounded linear transformations can be improved when H ∞ (T ) is confluent. Proposition 6.7. Let T ∈ L(H) be a completely nonunitary contraction such that H ∞ (T ) is confluent. Then every closed, densely defined linear transformation commuting with T is of the form v(T )−1 u(T ), where u, v ∈ H ∞ and v is an outer function. Proof. Let X be a closed, densely defined linear transformation commuting with T . Since X is closed, it must also commute with every operator in H ∞ (T ). By Proposition 4.6, there exist u, v ∈ H ∞ such that v ≡ 0 and X ⊂ v(T )−1 u(T ). Let us set

T1 = (T ⊕ T ) | G v(T )−1 u(T ) , and observe that the quasiaffinity Q : h ⊕ k → h from G(v(T )−1 u(T )) to H satisfies QT1 = T Q. Thus H ∞ (T1 ) ≺ H ∞ (T ), and therefore H ∞ (T1 ) is confluent by Proposition 5.2(3). The subspace G(X) is invariant for T1 , so

G(X) = m(T1 )G v(T )−1 u(T ) for some inner function m. To prove the equality X = v(T )−1 u(T ), it suffices to show that m is in fact constant. Indeed, we have

m(T )H = m(T )QG v(T )−1 u(T ) = Qm(T1 )G v(T )−1 u(T ) = QG(X) = D(X) = H,

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and the desired conclusion follows from the second assertion in Theorem 6.4(4). There is no loss of generality in assuming that u and v do not have any nonconstant common inner divisor. We conclude the proof by showing that in this case v must be outer. Let m be an inner divisor of v, and note that for every h ⊕ k ∈ G(X) we have u(T )h = v(T )k ∈ m(T )H, and therefore u(T )D(X) ⊂ m(T )H. Since D(X) is dense in H, we conclude that u(T(m(T )H)⊥ ) = 0, and therefore m divides u. Thus m is constant, and hence v is outer. 2 It follows from the results of [10] that the one-dimensional spaces ker(λI − T )∗ depend analytically on λ and, in fact, there exists an analytic function f : D → H such that ker(λI − T )∗ = Cf (λ) for λ ∈ D. A local version of this result is easily proved. Indeed, set L = (T ∗ T )−1 T ∗ . Given a unit vector f0 ∈ ker T ∗ , the function ∞

−1 λn L∗n f0 f (λ) = I − λL∗ f0 =

(6.2)

n=0

is analytic for |λ| < 1/ L , and obviously T ∗ f (λ) = λf (λ). This calculation is valid for any left inverse of T . The operator L has the advantage that L∗ H = T H, and therefore T n f0 , f0 = f0 , L∗n f0 = 0 for n 1. These relations, along with LT = I , obviously imply

T n f0 , L∗m f0 = δnm ,

n, m 0.

(6.3)

Proposition 6.8. Let T ∈ L(H) be a completely nonunitary contraction such that H ∞ (T ) is confluent. Define L = (T ∗ T )−1 T ∗ and fix a unit vector f0 ∈ ker T ∗ . Then (1) The f0 is cyclic for L∗ . vector n (2) {T H: n 0} = {0}. (3) {L∗n H: n 0} = H [ {T n f0 : n 0}]. Proof. We have seenthat ker(λI − T ∗ ) = Cf (λ) for λ close to zero, where f (λ) is given by (6.2) and belongs to {L∗n f0 : n 0}. Thus (1) follows from Theorem 6.4(3). To prove (2), let f be a nonzero element in the intersection, and set k = inf ord0 (h/f0 ): h ∈ H ,

m = ord0 (f/f0 ) < ∞.

For each integer n we can write f = T n g for some g = 0, and therefore m = n + ord0 (g/f0 ) n + k. This yields a contradiction for large n. The orthogonality relations (6.3) imply the inclusion n0

L∗n H ⊂ H

n0

T n f0 .


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Conversely, consider a vector h ∈ H [ {T n f0 : n 0}]. Given n 1, we have h = L∗n T ∗n h +

n−1

L∗k I − L∗ T ∗ T ∗k h.

k=0

Since I − L∗ T ∗ is the orthogonal projection onto Cf0 , and

T ∗k h, f0 = h, T k f0 = 0,

we deduce that h = L∗n T ∗n h ∈ L∗n H, thus proving the opposite inclusion.

2

7. Confluence and functional models The results in Section 6 show that completely nonunitary contractions T for which H ∞ (T ) is confluent share many of the properties of the unilateral shift S. In this section we will describe some quasiaffine transforms of such operators T . These quasiaffine transforms are in fact functional models associated with purely contractive inner functions of the form Θ=

θ1 , θ2

where θ1 , θ2 ∈ H ∞ . The condition that Θ be inner amounts to the requirement that θ1 (ζ )2 + θ2 (ζ )2 = 1,

a.e. ζ ∈ T,

while pure contractivity means simply that θ1 (0)2 + θ2 (0)2 < 1. We recall the construction of the functional model associated with such a function Θ. The subspace θ1 u ⊕ θ2 u: u ∈ H 2 ⊂ H 2 ⊕ H 2 is obviously invariant for S ⊕ S, and thus the orthogonal complement H(Θ) = H 2 ⊕ H 2 θ1 u ⊕ θ2 u: u ∈ H 2 is invariant for S ∗ ⊕ S ∗ . The operator S(Θ) ∈ L(H(Θ)) is the compression of S ⊕ S to this space or, equivalently, S(Θ)∗ = (S ∗ ⊕ S ∗ ) | H(Θ). Observe that I − S(Θ)∗ S(Θ) has rank one, while I − S(Θ)S(Θ)∗ has rank two. It follows that σ (S(Θ)) = D, and in particular S(Θ) is not of class C0 . Lemma 7.1. Let Θ = θθ21 be a purely contractive inner function. The algebra H ∞ (S(Θ)) is confluent if and only if the functions θ1 and θ2 do not have a nonconstant common inner factor.

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Proof. If either of the functions θj is equal to zero, the other one must be inner. The lemma is easily verified in this case. Indeed, assume that θ1 is inner and θ2 = 0. If θ1 is not constant then ker θ1 (S(Θ)) = {0}, so that H ∞ (S(Θ)) is not confluent. Also, θ1 is a common inner divisor of θ1 and θ2 , so that both conditions in the statement are false. On the other hand, if θ1 is constant then Θ is not pure, so this case does not arise. For the remainder of this proof, we consider the case in which both functions θj are different from zero. Assume first that θj = mϕj , where m is a nonconstant inner function and ϕj ∈ H ∞ for j = 1, 2. The nonzero vector h ∈ H(Θ) defined by h = PH(Θ) (ϕ1 ⊕ ϕ2 ) satisfies m(S(Θ))h = 0, and therefore m(S(Θ)) has nontrivial kernel. Thus H ∞ (S(Θ)) is not confluent. Assume now that θ1 and θ2 do not have a nonconstant common inner factor. We verify first that ker u(S(Θ)) = {0} for u ∈ H ∞ \ {0}. It suffices to consider the case of an inner function u. A vector f1 ⊕ f2 ∈ ker u(S(Θ)) must satisfy uf1 = θ1 g and uf2 = θ2 g for some g ∈ H 2 . The fact that θ1 ∧ θ2 = 1 implies that u divides g, and therefore f1 ⊕ f2 = θ1 (g/u) ⊕ θ2 (g/u) belongs to H(Θ)⊥ ; the equality f1 ⊕ f2 = 0 follows. To conclude the proof, we will show that h = PH(Θ) (1 ⊕ 0) is a rationally strictly cyclic vector for H ∞ (S(Θ)). Indeed, assume that f = f1 ⊕ f2 ∈ H(Θ) \ {0}, and write f1 = a1 /b and f2 = a2 /b, where a1 , a2 , b ∈ H ∞ and b is outer. Define functions u = −bθ2 , v = θ1 a2 − θ2 a1 , and note that

v S(Θ) h − u S(Θ) f = PH(Θ) (v ⊕ 0 − uf1 ⊕ uf2 ) = PH(Θ) (θ1 a2 ⊕ θ2 a2 ) = 0. The lemma follows because u ≡ 0, and hence u(S(Θ)) is injective.

2

Let us remark that the condition θ1 ∧ θ2 = 1 is equivalent to the fact that the function Θ is ∗-outer. In other words, the operators S(Θ) described in the preceding lemma are of class C10 . This is in agreement with Theorem 6.4(6). Proposition 7.2. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent. Then (1) Either S ≺ T or there exists a purely contractive inner function Θ = S(Θ) ≺ T and H ∞ (S(Θ)) is confluent. (2) We have S ≺ T if and only if T has a cyclic vector.

θ1 θ2

such that

Proof. Denote by U+ ∈ L(K+ ) the minimal isometric dilation of T . Thus H ⊂K+ and T PH = PH U+ . Since T ∈ C10 , the operator U+ is a unilateral shift. Let us set M = {T n h1 : n 0}, where h1 ∈ H \ {0}, and let h2 ∈ H M be a cyclic vector for the compression of T to this subspace. Such a vector exists by Theorem 6.4(4). Observe that H = {T n h1 , T n h2 : n 0}. We define now a space E=

U+n h1 , U+n h2 : n 0

and an operator Y ∈ L(E, H) by setting Y = PH | E. The space E is invariant for U+ , Y (U+ | E) = T Y , and Y has dense range. Moreover, the restriction U+ | E is a unilateral shift of multiplicity 1 or 2. Finally, set H = E ker Y , X = Y | H , and denote by T the compression


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of U+ | E to the space H . Then XH = Y E so that X is a quasiaffinity, and XT = T X. Thus we have T ≺ T and hence H ∞ (T ) is confluent by Proposition 5.2(3). We will now prove that at least one of the following alternatives must θ1 hold: either S ≺ T or T is unitarily equivalent to an operator of the form S(Θ), where Θ = θ2 is a purely contractive inner function. For this, we first note that U+ | E is the minimal isometric lifting of T . Therefore T is of class C•0 and its functional model is the compression of the canonical shift on H 2 (DT ∗ ) to H(ΘT ) = H 2 (DT ∗ ) ΘT H 2 (DT ), where ΘT (z) : DT → DT ∗ ,

z ∈ D,

is the characteristic function of T . Therefore ΘT is a purely contractive inner function and according to [23, Chapter VI] dim DT dim DT ∗ = dim E. Thus we must consider the following possibilities: (i) (ii) (iii) (iv) (v)

dim DT dim DT dim DT dim DT dim DT

= dim DT ∗ = 2; = 1, dim DT ∗ = 2; = 0, dim DT ∗ = 2; = dim DT ∗ = 1; and = 0, dim DT ∗ = 1.

In cases (i) and (iv) T is of class C00 , hence of class C0 (see [23, Proposition VI.3.5 and Theo rem VI.5.2]), thus H ∞ (T ) is not θ1 confluent. In case (ii) T is unitarily equivalent to an operator of the form S(Θ), where Θ = θ2 is a purely contractive inner function. In case (iii) T is unitarily equivalent to S ⊕ S and H ∞ (S ⊕ S) is not confluent; to see this consider the vectors 1 ⊕ 0 and 0 ⊕ 1. Finally, in case (v) T is unitarily equivalent to S, and thus S ≺ T . If T has a cyclic vector h1 , we can take h2 = 0, and then U+ | E is a shift of multiplicity 1. In this case, we must have ker Y = {0} so that U+ | E ≺ T . Conversely, S ≺ T implies that T has a cyclic vector since S has one. 2 The argument in the preceding proof appeared earlier in the classification of contractions of class C•0 [26,27], and even earlier in [14] and in the study of the class C0 [22]. When T has a cyclic vector, it is natural to ask under what conditions we actually have T ∼ S. Lemma 7.3. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent. Then the following assertions are equivalent: (1) T ≺ S. (2) T | M ≺ S for some invariant subspace M of T . (3) T | M ≺ S for every nonzero invariant subspace M of T .

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Proof. The implications (3) ⇒ (1) ⇒ (2) are obvious. Next we show that T ≺ T | M for every nonzero invariant subspace M of T . By Theorem 6.4(4), there is an inner function m such that m(T )H = M. Then the operator X : H → M defined by Xh = m(T )h, h ∈ H, is a quasiaffinity and XT = (T | M)X. Using this fact, it is easy to show that (2) ⇒ (1). Indeed, if (2) holds we have T | M ≺ S for some M, and the relations T ≺ T | M ≺ S imply the desired conclusion T ≺ S. Finally, we prove that (1) ⇒ (3). Assume that (1) holds, so that Y T = SY for some quasiaffinity Y . If M is a nonzero invariant subspace for T , the operator Z = Y | M : M → Y M is a quasiaffinity realizing the relation T | M ≺ S | Y M. We conclude that (3) is true since S | Y M is unitarily equivalent to S. 2 We can now state some conditions equivalent to the relation T ∼ S. Theorem 7.4. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent and has a cyclic vector. Let f : D → H be an analytic function such that f (0) = 1 and ker(λI − T ∗ ) = Cf (λ) for every λ ∈ D, and denote H0 = {T n f (0): n 0}. Then the following conditions are equivalent: (1) (2) (3) (4)

T ∼ S. T | H0 ≺ S. There exists an outer function b ∈ H ∞ such that b(h/f (0)) ∈ H 2 for every h ∈ H0 . There exists an outer function b ∈ H ∞ such that b

¯ h, f (λ) ∈ H2 f (0), f (λ¯ )

for every h ∈ H0 . Proof. Since T has a cyclic vector, we have S ≺ T by Proposition 7.2(2). Therefore T ∼ S is equivalent to T ≺ S, and this is equivalent to condition (2) by Lemma 7.3. This establishes the equivalence (1) ⇔ (2). For an arbitrary h ∈ H \ {0}, write the function h/f (0) as a quotient u/v of functions in H ∞ . We have then ¯ = h, v(λ)f (λ) ¯ = v(λ) h, f (λ) ¯ , ¯ = h, v(T )∗ f (λ) v(T )h, f (λ)

and analogously u(T )f (0), f (λ¯ ) = u(λ)f (0), f (λ¯ ). Since v(T )h = u(T )f (0), we conclude that b(λ)

¯ h, f (λ) h (λ) = b(λ) ¯ f (0) f (0), f (λ)

(7.1)

for those λ for which the denominators do not vanish. This proves the equivalence (3) ⇔ (4). ¯ cannot be identically zero since it takes the value 1 Note that the analytic function f (0), f (λ) for λ = 0. It remains to prove the equivalence (2) ⇔ (3), and for this purpose we may as well assume that case. Thus, H = H0 . We apply the construction in the proof of Proposition 7.2 for this particular consider the minimal isometric dilation U+ ∈ L(K+ ) of T , and denote E = {U+n f (0): n 0}.


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Since U+∗ f (0) = T ∗ f (0) = 0, there exists a unitary operator W : H 2 → E such that W 1 = f (0) and W S = (U+ | E)W . One can then construct a quasiaffinity Y : H 2 → H, namely Y = PH W , such that T Y = Y S and Y 1 = f (0). Since an equality of the form v(S)x = u(S)1 for x ∈ H 2 is equivalent to v(T )Y x = u(T )f (0), we deduce that Yx = x, f (0)

x ∈ H 2,

and, conversely, that any vector h ∈ H such that k = h/f (0) ∈ H 2 must belong to Y H 2 ; namely h = Y k. With this preparation, assume that (2) holds, and let X ∈ L(H, H 2 ) be a quasiaffinity such that XT = SX. Then the operator XY is a quasiaffinity in the commutant of S, and therefore XY = b(S) for some outer function b ∈ H ∞ . The equality

X b(T ) − Y X = b(S)X − (XY )X = 0 implies that we also have Y X = b(T ). For any h ∈ H \ {0} we have then b

b(T )h Y Xh h = = = Xh ∈ H 2 , f (0) f (0) Y1

thus proving (3). Conversely, if (3) holds, we can define a linear map X : H → H 2 by setting Xh = b(h/f (0)) for h ∈ H, and this map obviously satisfies XT = SX. Using (7.1) it is easy to verify that X is a closed linear transformation, and hence X is continuous. It is also immediate that XY = b(S) and Y X = b(T ), and this implies that X is a quasiaffinity since b is outer. 2 Corollary 7.5. Assume that T ∈ L(H) is a completely nonunitary contraction such that T ∼ S. ∗ Let f : D → H be an analytic function nsuch that f (0) = 1 and ker(λI − T ) = Cf (λ) for every λ ∈ D, and assume that H = {T f (0): n 0}. Then f (0), f (λ) = 0 for every λ ∈ D. Proof. Let b be an outer function satisfying condition (4) of Theorem 7.4. Assume that ¯ = 0 for some λ ∈ D. Since b(λ) = 0, it follows that h, f (λ) ¯ = 0 for every h ∈ H, f (0), f (λ) and therefore f (λ¯ ) = 0, which is impossible since this vector generates ker(λI − T )∗ . 2 The relation T ≺ S can also be studied in terms of the minimal unitary dilation of T . We will denote by R∗ ∈ L(R∗ ) the ∗-residual part of this minimal unitary dilation; see [23, Section II.3] for the relevant definitions. The facts we require about this operator are as follows: (a) R∗ is a unitary operator with absolutely continuous spectral measure relative to arclength measure on T. (b) There exists an operator Z : H → R∗ (namely, the orthogonal projection onto R∗ ) such that ZT = R∗ Z and Zh = lim T n h. n→∞

In particular, Z is injective if and only if T is of class C1· . (c) The smallest reducing subspace for R∗ containing ZH is R∗ .

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Proposition 7.6. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent. Then (1) The ∗-residual part R∗ of the minimal unitary dilation of T has spectral multiplicity at most 1. (2) We have T ≺ S if and only if R∗ is a bilateral shift of multiplicity 1. (3) We have T ≺ R∗ | ZH, and T ≺ S if and only if ZH = R∗ . (4) T ∗ has a cyclic vector. Proof. Given h1 , h2 ∈ H \ {0}, select u1 , u2 ∈ H ∞ \ {0} such that u1 (T )h1 = u2 (T )h2 . Then we have u1 (R∗ )Zh1 = u2 (R∗ )Zh2 . Since u1 (ζ ) and u2 (ζ ) are different from zero a.e. relative to the spectral measure of R∗ , it follows that the vectors Zh1 and Zh2 generate the same reducing space for R∗ . Therefore R∗ has a ∗-cyclic vector, and this implies (1). Next we prove (3). The fact that T ≺ R∗ | ZH is immediate. If ZH is not reducing, then R∗ | ZH is unitarily equivalent to S and hence T ≺ S. Conversely, if T ≺ S, let W be a quasiaffinity such that W T = SW . For any h ∈ H we have W h = lim S n W h = lim W T n h W Zh , n→∞

n→∞

so there exists an operator X : ZH → H 2 such that X W and XZ = W . Since the range of X contains the range of W , we have X = 0. Pick a vector f ∈ H 2 such that X ∗ f = 0, and observe that lim (R∗ ZH)∗n X ∗ f = lim X ∗ S ∗n f = 0. n→∞

n→∞

Therefore R∗ | ZH is not unitary, and consequently ZH = R∗ . Assume now that T ≺ S. The fact that R∗ is a bilateral shift follows from (3) because the only absolutely continuous unitary operator of multiplicity 1 which has nonreducing invariant subspaces is the bilateral shift. Conversely, if R∗ is a bilateral shift, the results of [17] imply the existence of an invariant subspace M for T such that T | M ≺ S. We deduce that T ≺ S by Lemma 7.3. This proves (2). Finally, (4) also follows from (3) because (R∗ | ZH)∗ has a cyclic vector. 2 Corollary 7.7. Assume that Θ = θθ21 is inner, ∗-outer, and purely contractive. Then S(Θ) ≺ S. More precisely, the operator Q : H(Θ) → H 2 defined by Q(f1 ⊕ f2 ) = θ1 f2 − θ2 f1 , f1 ⊕ f2 ∈ H(Θ), is a quasiaffinity and QS(Θ) = SQ. Proof. We will show that PR∗ H(Θ) = R∗ . To do this, we observe first that the minimal unitary dilation of S(Θ) is the operator U ⊕ U on L2 ⊕ L2 . The space R∗ is the orthogonal complement of the smallest reducing space for U ⊕ U containing {θ1 u ⊕ θ2 u: u ∈ H 2 }. Thus

R∗ = L2 ⊕ L2 θ1 u ⊕ θ2 u: u ∈ L2 , and it follows that PR∗ is the operator of pointwise multiplication by the matrix |θ2 |2 −θ2 θ1 I − ΘΘ ∗ = . −θ2 θ1 |θ1 |2


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Finally, we have PR∗ H(Θ) = PR∗ (H 2 ⊕ H 2 ), and therefore PR∗ H(Θ) is the invariant subspace for U generated by PR∗ (1 ⊕ 0) and PR∗ (0 ⊕ 1). These two vectors are precisely |θ2 |2 ⊕ (−θ2 θ1 ) = (−θ2 u) ⊕ θ1 u, (−θ2 θ1 ) ⊕ |θ1 |2 = (−θ2 v) ⊕ θ1 v, with u = −θ2 and v = θ1 . Since θ1 and θ2 do not have nonconstant common inner divisors, the invariant subspace for S they generate is the entire H 2 . It follows that PR∗ H(Θ) = (−θ2 w) ⊕ θ1 w: w ∈ H 2 , and R∗ | PR∗ H(Θ) is unitarily equivalent to S. The final assertion is verified by noting that (see (b) above)

Z(f1 ⊕ f2 ) = PR∗ (f1 ⊕ f2 ) = −θ2 Q(f1 ⊕ f2 ) ⊕ θ1 Q(f1 ⊕ f2 ) for f1 ⊕ f2 ∈ H(Θ).

2

The preceding result can be extended considerably. As seen in the proof below, the assumption that I − T ∗ T has finite rank can be replaced by the requirement that I − T T ∗ have finite rank. Corollary 7.8. Let T be a completely nonunitary contraction such that H ∞ (T ) is confluent and I − T ∗ T has finite rank. Then (1) We have T ≺ S. (2) If in addition T has a cyclic vector, then T ∼ S. Proof. Denote by n the rank of I − T ∗ T , and observe that the characteristic function ΘT is inner, ∗-outer, and coincides with an (n + 1) × n matrix over H ∞ . Indeed, ΘT (0) is a Fredholm operator of index −1. It follows that I − ΘT (ζ )ΘT (ζ )∗ has rank 1 for a.e. ζ ∈ T, and therefore R∗ is a bilateral shift by [23, Section VI.6]. Thus (1) follows from Proposition 7.6(2). Part (2) follows from (1) and Proposition 7.2(2). 2 The result shows that there exist some purely contractive inner functions of the form following Θ = θθ21 with the property that S(Θ) ≺ S. Thus part (1) of Proposition 7.2 could be restated as follows: ∞ there ex(1) If T is a completely nonunitary contraction such θ1 that H (T ) is confluent, then ists a purely contractive inner function Θ = θ2 such that S(Θ) ≺ T , and H ∞ (S(Θ)) is confluent.

Corollary 7.9. Assume that Θ =

θ1 θ2

is purely contractive, inner and ∗-outer.

(1) If f1 ⊕ f2 ∈ H(Θ) is cyclic for S(Θ), then θ1 f2 − θ2 f1 is an outer function. (2) Conversely, if θ1 f2 − θ2 f1 is outer for some f1 , f2 ∈ H 2 , then PH(Θ) (f1 ⊕ f2 ) is cyclic for S(Θ).

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(3) There exists Θ such that S(Θ) does not have a cyclic vector. (4) We have S(Θ) ∼ S if and only if S(Θ) has a cyclic vector. Proof. With the notation of Corollary 7.7, Q(f1 ⊕ f2 ) must be cyclic for S if f1 ⊕ f2 is cyclic for S(Θ). This proves (1). Conversely, assume that u = θ1 f2 − θ2 f1 is outer for some f1 , f2 ∈ H 2 . Upon multiplying f1 , f2 by some outerfunction, we may assume that f1 , f2 ∈ H ∞ . Let g1 ⊕ g2 ∈ H(Θ) be a vector orthogonal to {S(Θ)n PH(Θ) (f1 ⊕ f2 ): n 0}. We have then g1 ⊕ g2 , θ1 p ⊕ θ2 p = g1 ⊕ g2 , f1 p ⊕ f2 p = 0 for every polynomial p. Equivalently, θ1 g1 + θ2 g2 and f1 g1 + f2 g2 belong to L2 H 2 , and therefore the functions ug1 = f2 (θ1 g1 + θ2 g2 ) − θ2 (f1 g1 + f2 g2 ), ug2 = θ1 (f1 g1 + f2 g2 ) − f1 (θ1 g1 + θ2 g2 ) are also in L2 H 2 . Thus gj , up = 0 for all polynomials p, and hence gj = 0, j = 1, 2, because u is outer. Assertion (2) follows. To prove (3), let m1 and m2 be two relatively prime inner functions, and set θ1 = 35 m1 and θ2 = 45 m2 . Nordgren [18] showed that it is possible to choose m1 and m2 so that no function of the form m1 f2 − m2 f1 is outer if f1 , f2 ∈ H 2 . The corresponding operator S(Θ) does not have a cyclic vector. Finally (4) follows from Corollary 7.7 and Proposition 7.2(2). 2 Let us also note a related result which follows easily from [28]. Proposition 7.10. Assume that Θ = θθ21 is inner and ∗-outer. Then the operator S(Θ) is similar to S if and only if there exist f1 , f2 ∈ H ∞ such that θ1 f2 − θ2 f1 = 1. Proof. It was shown in [28] that S(Θ) is similar to an isometry if and only if Θ is left invertible. To conclude, one must observe that the only possible isometry is a unilateral shift of multiplicity 1. 2 The proof of the following proposition follows easily from the above arguments, along with the corresponding properties of S. Proposition 7.11. Let T be a completely nonunitary contraction such that T ∼ S. Then T is of class C10 , both T and T ∗ have cyclic vectors, and the ∗-residual part R∗ of the minimal unitary dilation of T is a bilateral shift of multiplicity 1. The converse of this proposition is not true. Indeed, it was shown in [6] (see also [16]) that there exist operators T of class C10 , with a cyclic vector, such that R∗ is a bilateral shift of multiplicity 1, and σ (T ) ⊃ D. For such operators we will have R∗∗ ≺ T ∗ , so T ∗ also has a cyclic vector, but T ⊀ S. The following partial converse follows from Propositions 7.2(2) and 7.6(2).


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Proposition 7.12. Let T be a completely nonunitary contraction such that H ∞ (T ) is confluent, T has a cyclic vector, and the ∗-residual part R∗ of the minimal unitary dilation of T is a bilateral shift of multiplicity 1. Then T ∼ S. Remark 7.13. For more information about which operators are or can be quasisimilar to the unilateral shift, see [2,9,11,12,15,29] and the references therein. References [1] M.B. Abrahamse, R.G. Douglas, A class of subnormal operators related to multiply connected domains, Adv. Math. 19 (1976) 106–148. [2] J. Agler, J.E. Franks, D.A. Herrero, Spectral pictures of operators quasisimilar to the unilateral shift, J. Reine Angew. Math. 422 (1991) 1–20. [3] W.B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967) 635–647. [4] H. Bercovici, Operator Theory and Arithmetic on H ∞ , Amer. Math. Soc., Providence, RI, 1988. [5] H. Bercovici, C. Foias, C. Pearcy, Dual Algebras with Applications to Invariant Subspaces and Dilation Theory, Amer. Math. Soc., Providence, RI, 1985. [6] H. Bercovici, L. Kérchy, Spectral behaviour of C10 -contractions, preprint. [7] S.W. Brown, Contractions with spectral boundary, Integral Equations Operator Theory 11 (1988) 49–63. [8] S.W. Brown, B. Chevreau, C. Pearcy, On the structure of contraction operators. II, J. Funct. Anal. 76 (1) (1988) 30–55. [9] S. Clary, Quasisimilarity and subnormal operators, PhD thesis, University of Michigan, 1973. [10] M.J. Cowen, R.G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978) 187–261. [11] R.E. Curto, Raúl, L.A. Fialkow, Similarity, quasisimilarity, and operator factorizations, Trans. Amer. Math. Soc. 314 (1989) 225–254. [12] L.A. Fialkow, Quasisimilarity and closures of similarity orbits of operators, J. Operator Theory 14 (1985) 215–238. [13] S.D. Fisher, Function Theory on Planar Domains. A Second Course in Complex Analysis, John Wiley & Sons, New York, 1983. [14] C. Foias, A classification of doubly cyclic operators, in: Hilbert Space Operators, Tihany, Colloquia Math. Soc. J. Bolyai 5 (1970) 155–161. [15] D.A. Herrero, On multicyclic operators, Integral Equations Operator Theory 1 (1978) 57–102. [16] L. Kérchy, On the spectra of contractions belonging to special classes, J. Funct. Anal. 67 (1986) 153–166. [17] L. Kérchy, Shift-type invariant subspaces of contractions, J. Funct. Anal. 246 (2007) 281–301. [18] E.A. Nordgren, The ring N+ is not adequate, Acta Sci. Math. (Szeged) 36 (1974) 203–204. [19] H. Radjavi, P. Rosenthal, Invariant Subspaces, second ed., Dover Publications, Mineola, NY, 2003. [20] D. Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. 127 (1967) 179–203. [21] D. Sarason, Unbounded operators commuting with the commutant of a restricted backward shift, Oper. Matrices 4 (2010) 293–300. [22] B. Sz.-Nagy, C. Foias, Modèle de Jordan pour une classe d’opérateurs de l’espace de Hilbert, Acta Sci. Math. (Szeged) 31 (1970) 91–115. [23] B. Sz.-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. [24] B. Sz.-Nagy, C. Foias, Local characterization of operators of class C0 , J. Funct. Anal. 8 (1971) 76–81. [25] B. Sz.-Nagy, C. Foias, Compléments l’étude des opérateurs de classe C0 . II, Acta Sci. Math. (Szeged) 33 (1972) 113–116. [26] B. Sz.-Nagy, C. Foias, Jordan model for contractions of class C·0 , Acta Sci. Math. (Szeged) 36 (1974) 305–322. [27] B. Sz.-Nagy, C. Foias, Injections of shifts into strict contractions , in: Linear Operators and Approximation. II, Birkhäuser, Basel, 1974, pp. 29–37. [28] B. Sz.-Nagy, C. Foias, On contractions similar to isometries and Toeplitz operators, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976) 553–564. [29] M. Uchiyama, Curvatures and similarity of operators with holomorphic eigenvectors, Trans. Amer. Math. Soc. 319 (1990) 405–415.


Landesman–Lazer type results for second order Hamilton–Jacobi–Bellman equations Patricio Felmer a,∗ , Alexander Quaas b , Boyan Sirakov c,d a Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile b Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla: V-110, Avda. España 1680,

Valparaíso, Chile c UFR SEGMI, Université de Paris 10, 92001 Nanterre Cedex, France d CAMS, EHESS, 54 bd. Raspail, 75006 Paris, France

Received 22 September 2009; accepted 11 March 2010

Communicated by H. Brezis

Abstract We study the boundary-value problem

F (D 2 u, Du, u, x) + λu = f (x, u) in Ω, u=0 on ∂Ω,

where the second order differential operator F is of Hamilton–Jacobi–Bellman type, f is sub-linear in u at infinity and Ω ⊂ RN is a regular bounded domain. We extend the well-known Landesman–Lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator. We provide conditions under which the solution branches extend globally along the eigenvalue gap. We also present examples illustrating the results and hypotheses. © 2010 Elsevier Inc. All rights reserved. Keywords: Hamilton–Jacobi–Bellman equation; Landesman–Lazer condition; Bifurcation from infinity; Principal eigenvalues


E-mail addresses: [email protected] (P. Felmer), [email protected] (A. Quaas), [email protected] (B. Sirakov). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.012

P. Felmer et al. / Journal of Functional Analysis 258 (2010) 4154–4182

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1. Introduction We study the boundary-value problem

F (D 2 u, Du, u, x) + λu = f (x, u) u=0

in Ω, on ∂Ω,

(1.1)

where the second order differential operator F is of Hamilton–Jacobi–Bellman (HJB) type, that is, F is a supremum of linear elliptic operators, f is sub-linear in u at infinity, and Ω ⊂ RN is a regular bounded domain. HJB operators have been the object of intensive study during the last thirty years – for a general review of their theory and applications we refer to [25,31,40,14]. Well-known examples include the Fucik operator u + bu+ + au− [26], the Barenblatt operator max{au, bu} 2 [10,30], and the Pucci operator M+ λ,Λ (D u) [35,15]. To introduce the problem we are interested in, let us first recall some classical results in the case when F is the Laplacian and λ ∈ (−∞, λ2 ) (we shall denote with λi the i-th eigenvalue of the Laplacian). If f is independent of u, the solvability of (1.1) is a consequence of the Fredholm alternative, namely, if λ = λ1 , problem (1.1) has a solution for each f , while if λ = λ1 (resonance) it has solutions if and only if f is orthogonal to ϕ1 , the first eigenfunction of the Laplacian. The existence result in the non-resonant case extends to nonlinearities f (x, u) which grow sub-linearly in u at infinity, thanks to Krasnoselski–Leray–Schauder degree and fixed point theory, see [1]. A fundamental result, obtained by Landesman and Lazer [32] (see also [29]), states that in the resonance case λ = λ1 the problem u + λ1 u = f (x, u)

in Ω,

u = 0 on ∂Ω

is solvable provided f is bounded and, setting f ± (x) := lim sup f (x, s), s→±∞

f± (x) := lim inf f (x, s) s→±∞

(1.2)

(this notation will be kept from now on), one of the following conditions is satisfied:

−

f ϕ1 < 0 < Ω

f− ϕ1 > 0 >

f+ ϕ1 , Ω

Ω

f+ ϕ1 .

(1.3)

Ω

This result initiated a huge amount of work on solvability of boundary-value problems in which the elliptic operator is at, or more generally close to, resonance. Various extensions of the results in [32] for resonant problems were obtained in [2,4,12]. Further, Mawhin and Schmitt [34] – see also [19,18] – considered (1.1) with F = for λ close to λ1 , and showed that the first (resp. the second) condition in (1.3) implies that for some δ > 0 problem (1.1) has at least one solution for λ ∈ (λ1 − δ, λ1 ] and at least three solutions for λ ∈ (λ1 , λ1 + δ) (resp. at least one solution for λ ∈ [λ1 , λ1 + δ) and at least three solutions for λ ∈ (λ1 , λ1 + δ)). These results rely on degree theory and, more specifically, on the notion of bifurcation from infinity, studied by Rabinowitz in [37].

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The same results naturally hold if the Laplacian is replaced by any uniformly elliptic operator in divergence form. Further, they do remain true if a general linear operator in non-divergence form L = aij (x)∂x2i xj + bi (x)∂xi + c(x)

(1.4)

is considered, but we have to change ϕ1 in (1.3) by the first eigenfunction of the adjoint operator of L. This fact is probably known to the experts, though we are not aware of a reference. Its proof – which will also easily follow from our arguments below – uses the Donsker–Varadhan [21] characterization of the first eigenvalue of L and the results in [11] which link the positivity of this eigenvalue to the validity of the maximum principle and to the Alexandrov–Bakelman–Pucci inequality (the degree theory argument remains the same as in the divergence case). The interest in this type of problems has remained high in the PDE community over the years. Recently a large number of works have considered the extensions of the above results to quasilinear equations (for instance, replacing the Laplacian by the p-Laplacian), where somewhat different phenomena take place, see [6,20,22,23]. There has also been a considerable interest in refining the Landesman–Lazer hypotheses (1.3) and finding general hypotheses on the nonlinearity which permit to determine on which side of the first eigenvalue the bifurcation from infinity takes place, see [3,5,27]. It is our goal here to study the boundary-value problem (1.1) under Landesman–Lazer conditions on f , when F is a Hamilton–Jacobi–Bellman (HJB) operator, that is, the supremum of linear operators as in (1.4): F [u] := F D 2 u, Du, u, x = sup tr Aα (x)D 2 u + bα (x).Du + cα (x)u , α∈A

(1.5)

where A is an arbitrary index set. The following hypotheses on F will be kept throughout the paper: Aα ∈ C(Ω), bα , cα ∈ L∞ (Ω) for all α ∈ A and, for some constants 0 < λ Λ, we have λI Aα (x) ΛI , for all x ∈ Ω and all α ∈ A. We stress however that all our results are new even for operators with smooth coefficients. Let us now describe the most distinctive features of HJB operators – with respect to the operators considered in the previous papers on Landesman–Lazer type problems – which make our work and results different. The HJB operator F [u] defined in (1.5) is nonlinear, yet positively homogeneous (that is, F [tu] = tF [u] for t 0), thus one may expect it has eigenvalues and eigenfunctions on the cones of positive and negative functions, but they may be different to each other. This fact was established by Lions in 1981, in the case of operators with regular coefficients, see [33]. In that paper he proved F [u] has two real “demi”- or “half”-principal − + − eigenvalues λ+ 1 , λ1 ∈ R (λ1 λ1 ), which correspond to a positive and a negative eigenfunction, respectively, and showed that the positivity of these numbers is a sufficient condition for the solvability of the related Dirichlet problem. Recently in [36] the second and the third author extended these results to arbitrary operators and studied the properties of the eigenvalues and the eigenfunctions, in particular the relation between the positivity of the eigenvalues and the validity of the comparison principle and the Alexandrov–Bakelman–Pucci estimate, thus obtaining extensions to nonlinear operators of the results of Berestycki, Nirenberg, and Varadhan in [11]. − In what follows we always assume that F is indeed nonlinear in the sense that λ+ 1 < λ1 – note + − the results in [36] easily imply that λ1 = λ1 can occur only if all linear operators which appear in (1.5) have the same principal eigenvalues and eigenfunctions.


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In the subsequent works [39,24] we considered the Dirichlet problem (1.1) with f independent of u, and we obtained a number of results on the structure of its solution set, depending − on the position of the parameter λ with respect to the eigenvalues λ+ 1 and λ1 . In particular, we − p proved that for each λ in the closed interval [λ+ 1 , λ1 ] and each h ∈ L , p > N , which is not a + ∗ multiple of the first eigenfunction ϕ1 , there exists a critical number tλ,F (h) such that the equation F [u] + λu = tϕ1+ + h

in Ω,

u=0

on ∂Ω

(1.6)

∗ (h) and has no solutions for t < t ∗ (h). We remark this is in sharp has solutions for t > tλ,F λ,F contrast with the case of linear F , say F = , when (1.6) has a solution if and only if t = ∗ (h) = − 2 tλ, Ω (hϕ1 ) (we shall assume all eigenfunctions are normalized so that their L -norm is one). Much more information on the solutions of (1.6) can be found in [39] and [24]. The value of tλ∗+ ,F (h) in terms of F and h was computed by Armstrong [7], where he obtained an extension 1

to HJB operators of the Donsker–Varadhan minimax formula. We now turn to the statements of our main results. A standing assumption on the function f will be the following (F0) f : Ω × R → R is continuous and sub-linear in u at infinity: lim

|s|→∞

f (x, s) = 0 uniformly in x ∈ Ω. s

Remark 1.1. For continuous f it is known [16,41,42] that all viscosity solutions of (1.1) are actually strong, that is, in W 2,p (Ω), for all p < ∞. Without serious additional complications we could assume that the dependence of f in x is only in Lp , for some p > N . Remark 1.2. Some of the statements below can be divided into subcases by supposing that f is sub-linear in u only as u → ∞ or as u → −∞ (such results for the Laplacian can be found in [19,18]). We have chosen to keep our theorems as simple as possible. Now we introduce the hypotheses which extend the Landesman–Lazer assumptions (1.3) for the Laplacian to the case of general HJB operators. From now on we write the critical t-values ∗ = t ∗ (h) = t ∗ ∗ = t ∗ (h) = t ∗ at resonance as t+ (h) and t− (h), and p > N is a fixed number. + − λ+ ,F λ− ,F We assume there are (F+ ) (F− ) (F+r ) (F−r )

1

1

∗ (c ) < 0, a function c+ ∈ Lp (Ω), such that c+ (x) f+ (x) in Ω and t+ + − p − − ∗ (c− ) > 0, a function c ∈ L (Ω), such that c (x) f (x) in Ω and t− ∗ (c+ ) > 0, a function c+ ∈ Lp (Ω), such that c+ (x) f + (x) in Ω and t+ p ∗ a function c− ∈ L (Ω), such that c− (x) f− (x) in Ω and t− (c− ) < 0.

Remark 1.3. Note that, decomposing h(x) = ( Ω hϕ1+ )ϕ1+ (x) + h⊥ (x), where ϕ1+ is the eigenfunction associated to λ+ 1 , we clearly have tλ∗ (h) = tλ∗ h⊥ −

Ω

+ hϕ1

−

for each λ ∈ λ+ 1 , λ1 .

(1.7)

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So when F = hypotheses (F+ )–(F− ) and (F+r )–(F−r ) reduce to the classical Landesman– Lazer conditions (1.3), since for the Laplacian the critical t-value of a function orthogonal to ϕ1 is always zero, by the Fredholm alternative. We further observe that whenever one of the limits f± , f ± is infinite, the functions c± , c± with the required in (F+ )–(F− ), (F+r )–(F−r ) property always exist, while if any of f± , f ± is in Lp (Ω), we take the corresponding c to be equal to this limit. Note also that the strict inequalities in (F+ )–(F− ) and (F+r )–(F−r ) are important, see Section 7. Throughout the paper we denote by S the set of all pairs (u, λ) ∈ C(Ω) × R which satisfy Eq. (1.1). For any fixed λ we set S(λ) = {u | (u, λ) ∈ S} and if C ⊂ S we denote C(λ) = C ∩ S(λ). − Our first result gives a statement of existence of solutions for λ around λ+ 1 and λ1 , under the above Landesman–Lazer type hypotheses. We recall that for some constant δ0 > 0 (all constants − in the paper will be allowed to depend on N , λ, Λ, γ , diam(Ω)), λ+ 1 , λ1 are the only eigenvalues of F in the interval (−∞, λ− 1 + δ0 ) – see Theorem 1.3 in [36]. Theorem 1.1. Assume (F0) and (F+ ) hold. Then there exist δ > 0 and two disjoint closed connected sets of solutions of (1.1), C1 , C2 ⊂ S such that (1) C1 (λ) = ∅ for all λ ∈ (−∞, λ+ 1 ], + (2) C1 (λ) = ∅ and C2 (λ) = ∅ for all λ ∈ (λ+ 1 , λ1 + δ). The set C2 is a branch of solutions “bifurcating from plus infinity to the right of λ+ 1 ”, + that is, C2 ⊂ C(Ω) × (λ1 , ∞) and there is a sequence {(un , λn )} ∈ C2 such that λn → λ+ 1 and un ∞ → ∞. Moreover, for every sequence {(un , λn )} ∈ C2 such that λn → λ+ 1 and un ∞ → ∞, un is positive in Ω, for n large enough. If we assume (F− ) holds, then there is a branch of solutions of (1.1) “bifurcating from minus − infinity to the right of λ− 1 ”, that is, a connected set C3 ⊂ S such that C3 ⊂ C(Ω) × (λ1 , ∞) for − which there is a sequence {(un , λn )} ∈ C3 such that λn → λ1 and un ∞ → ∞. Moreover, for every sequence {(un , λn )} ∈ C3 such that λn → λ− 1 and un ∞ → ∞, un is negative in Ω for n large. Under the sole hypothesis (F+ ) it cannot be guaranteed that the sets of solutions C1 and C2 extend much beyond λ+ 1 . This important fact will be proved in Section 7, where we find δ0 > 0 such that for each δ ∈ (0, δ0 ) we can construct a nonlinearity f (x, u) which satisfies (F0), (F+ ) and (F− ), but for which S(λ+ 1 + δ) is empty. It is clearly important to give hypotheses on f under which we can get a global result, that is, existence of continua of solutions which extend over the gap between the two principal eigenvalues (this gap accounts for the nonlinear nature of the HJB operator!). The next theorems deal with that question, and use the following additional assumptions. (F1) f (x, 0) 0 and f (x, 0) ≡ 0 in Ω. (F2) f (x, ·) is locally Lipschitz, that is, for each R ∈ R there is CR such that |f (x, s1 ) − f (x, s2 )| CR |s1 − s2 | for all s1 , s2 ∈ (−R, R) and x ∈ Ω. A discussion on these hypotheses, together with examples and counterexamples, will be given in Section 7.


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Theorem 1.2. Assume (F0), (F1), (F2), (F+ ) and (F− ) hold. Then there exist a constant δ > 0 and three disjoint closed connected sets of solutions C1 , C2 , C3 ⊂ S, such that (1) C1 (λ) = ∅ for all λ ∈ (−∞, λ+ 1 ], − (2) Ci (λ) = ∅, i = 1, 2, for all λ ∈ (λ+ 1 , λ1 ], − − ∅, i = 1, 2, 3, for all λ ∈ (λ1 , λ1 + δ). (3) Ci (λ) = The sets C2 end C3 have the same “bifurcation from infinity” properties as in the previous theorem. While Theorem 1.2 deals with bifurcation branches going to the right of the corresponding eigenvalues, the next theorem takes care of the case where the branches go to the left of the eigenvalues. Theorem 1.3. Assume (F0), (F1), (F2), (F+r ) and (F−r ) hold. Then there exist δ > 0 and disjoint closed connected sets of solutions C1 , C2 ⊂ S such that (1) C1 (λ) = ∅ for all λ ∈ (−∞, λ+ 1 − δ], + (2) C1 (λ) = ∅, C2 (λ) contains at least two elements for all λ ∈ (λ+ 1 − δ, λ1 ), and C2 is a branch + “bifurcating from plus infinity to the left of λ1 ”, − (3) C1 (λ) = ∅ and C2 (λ) = ∅ for all λ ∈ [λ+ 1 , λ1 ), and either: (i) C1 is the branch “bifurcating from minus infinity to the left of λ− 1 ”, (ii) there is a closed connected set of solutions C3 ⊂ S, disjoint of C1 and C2 , “bifurcating from minus infinity to the left of λ− 1 ” such that C3 (λ) has at least two elements for all − − λ ∈ (λ1 − δ, λ1 ), − (4) C2 (λ) = ∅ for all λ ∈ [λ− 1 , λ1 + δ]. In case (ii) in (3), C2 (λ) = ∅ and C3 (λ) = ∅ for all − − λ ∈ [λ1 , λ1 + δ]. Note alternative (3)(ii) in this theorem is somewhat anomalous. While we are able to exclude it in a number of particular cases (in particular for the model nonlinearities which satisfy the hypotheses of the theorem), we do not believe it can be ruled out in general. See Proposition 6.1 in Section 6. Going back to the case when F is linear, a well-known “rule of thumb” states that the number of expected solutions of (1.1) changes by two when the parameter λ crosses the first eigenvalue of F . A heuristic way of interpreting our theorems is that, when F is a supremum of linear operators, crossing a “half”-eigenvalue leads to a change of the number of solutions by one. The following graphs illustrate our theorems.

The paper is organized as follows. The next section contains some definitions, known results, and continuity properties of the critical values t ∗ . In Section 3 we obtain a priori bounds for the

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solutions of (1.1), and construct super-solutions or sub-solutions in the different cases. In Section 4 bifurcation from infinity for HJB operators is established through the classical method of Rabinowitz, while in Section 5 we construct and study a bounded branch of solutions of (1.1). These results are put together in Section 6, where we prove our main theorems. Finally, a discussion on our hypotheses and some examples which highlight their role are given in Section 7. 2. Preliminaries and continuity of t ∗ First, we list the properties shared by HJB operators of our type. The function F : SN × RN × R × Ω → R satisfies (with S, T ∈ SN × RN × R): (H0) F is positively homogeneous of order 1: F (tS, x) = tF (S, x) for t 0. (H1) There exist λ, Λ, γ > 0 such that for S = (M, p, u), T = (N, q, v) M− λ,Λ (M − N ) − γ |p − q| + |u − v| F (S, x) − F (T , x)

M+ λ,Λ (M − N ) + γ |p − q| + |u − v| .

(H2) The function F (M, 0, 0, x) is continuous in SN × Ω. (DF) We have −F (T − S, x) F (S, x) − F (T , x) F (S − T , x) for all S, T . + + In (H1) M− λ,Λ and Mλ,Λ denote the Pucci extremal operators, defined as Mλ,Λ (M) = supA∈A tr(AM), M− λ,Λ (M) = infA∈A tr(AM), where A ⊂ SN denotes the set of matrices whose eigenvalues lie in the interval [λ, Λ], see for instance [15]. Note under (H0) assumption (DF) is equivalent to the convexity of F in S – see Lemma 1.1 in [36]. Hence for each φ, ψ ∈ W 2,p (Ω) we have the inequalities F [φ + ψ] F [φ] + F [ψ] and F [φ − ψ] F [φ] − F [ψ]. We recall the definition of the principal eigenvalues of F from [36]

+ λ+ 1 (F, Ω) = sup λ Ψ (F, Ω, λ) = ∅ ,

− λ− 1 (F, Ω) = sup λ Ψ (F, Ω, λ) = ∅ ,

where Ψ ± (F, Ω, λ) = {ψ ∈ C(Ω) | ±(F [ψ] + λψ) 0, ±ψ > 0 in Ω}. Many properties of the eigenvalues (simplicity, isolation, monotonicity and continuity with respect to the domain, relation with the maximum principle) are established in Theorems 1.1–1.9 of [36]. We shall repeatedly use these results. We shall also often refer to the statements on the solvability of the Dirichlet problem, given in [36] and [24]. We recall the following Alexandrov–Bakelman–Pucci (ABP) and C 1,α estimates, see [28,17, 42]. Theorem 2.1. Suppose F satisfies (H0), (H1), (H2), and u is a solution of F [u] + cu = f (x) in Ω, with u = 0 on ∂Ω. Then there exist α ∈ (0, 1) and C0 > 0 depending on N , λ, Λ, γ , c and Ω such that u ∈ C 1,α (Ω), and uC 1,α (Ω) C0 uL∞ (Ω) + f Lp (Ω) . Moreover, if one chooses c = −γ (so that by (H1) F − γ is proper) then this equation has a unique solution which satisfies uC 1,α (Ω) C0 f L∞ (Ω) . More precisely, any solution of F [u] − γ u f (x) satisfies supΩ u sup∂Ω u + Cf LN .


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For readers’ convenience we state a version of Hopf’s lemma (for viscosity solutions it was proved in [9]). Theorem 2.2. Let Ω ⊂ RN be a regular domain and let γ > 0, δ > 0. Assume w ∈ C(Ω) 2 is a viscosity solution of M− λ,Λ (D w) − γ |Dw| − δw 0 in Ω, and w 0 in Ω. Then either w ≡ 0 in Ω or w > 0 in Ω and at any point x0 ∈ ∂Ω at which w(x0 ) = 0 we have 0) lim supt 0 w(x0 +tν)−w(x < 0, where ν is the interior normal to ∂Ω at x0 . t The next theorem is a consequence of the compact embedding C 1,α (Ω) → C 1 (Ω), Theorem 2.1, and the convergence properties of viscosity solutions (see Theorem 3.8 in [17]). Theorem 2.3. Let λn → λ in R and fn → f in Lp (Ω). Suppose F satisfies (H1) and un is a viscosity solution of F [un ] + λn un = fn in Ω, un = 0 on ∂Ω. If {un } is bounded in L∞ (Ω) then a subsequence of {un } converges in C 1 (Ω) to a function u, which solves F [u] + λu = f in Ω, u = 0 on ∂Ω. As a simple consequence of this theorem, the homogeneity of F and the simplicity of the eigenvalues we obtain the following proposition. p Proposition 2.1. Let λn → λ± 1 in R and fn be bounded in L (Ω). Suppose F satisfies (H1) and un is a viscosity solution of F [un ] + λn un = fn in Ω, un = 0 on ∂Ω. If {un } is unbounded in L∞ (Ω) then a subsequence of uunn converges in C 1 (Ω) to ϕ1± . In particular, un is positive (negative) for large n, and for each K > 0 there is N such that |un | Kϕ1+ for n N .

For shortness, from now on the zero boundary condition on ∂Ω will be understood in all differential (in)equalities we write, and · will refer to the L∞ (Ω)-norm. We devote the remainder of this section to the definition and some basic continuity properties of the critical t-values for (1.6). These numbers are crucial in the study of existence of solutions − p at resonance and in the gap between the eigenvalues. For each λ ∈ [λ+ 1 , λ1 ] and each d ∈ L , + which is not a multiple of the first eigenfunction ϕ1 , the number tλ∗ (d) = inf t ∈ R F [u] + λu = sϕ1+ + d has solutions for s t − is well defined and finite. The non-resonant case λ ∈ (λ+ 1 , λ1 ) was considered in [39], while the + − resonant cases λ = λ1 and λ = λ1 were studied in [24]. − In what follows we prove the continuity of tλ∗ : Lp (Ω) → R for any fixed λ ∈ [λ+ 1 , λ1 ]. Ac− tually, in [39] the continuity of this function is proved for all λ ∈ (λ+ 1 , λ1 ), so we only need to + − ∗ ∗ . In doing so, it is take care of the resonant cases λ = λ1 and λ = λ1 , that is, to study t+ and t− ∗ ∗ convenient to use the following equivalent definitions of t+ and t− (see [24])

∗ (d) = inf t ∈ R for each s > t and λn λ+ t+ 1 there exists un

such that F [un ] + λn un = sϕ1+ + d and un is bounded

and

(2.1)

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∗ t− (d) = inf t ∈ R for each s > t and λn λ− 1 there exists un

such that F [un ] + λn un = sϕ1+ + d and un is bounded .

(2.2)

∗ , t ∗ : Lp (Ω) → R are continuous. Proposition 2.2. The functions t+ − ∗ is not continuous, then there is d ∈ Lp (Ω), ε > 0 and a sequence d → d Proof. If we assume t+ n p ∗ (d ) t ∗ (d) + 3ε for all n ∈ N or t ∗ (d ) t ∗ (d) − 3ε for all n ∈ N. in L (Ω) such that either t+ n + + n + ∗ (d ) t ∗ (d) + 3ε for all n ∈ N. Then, for any sequence λ λ+ we First we suppose that t+ n m + 1 find solutions um n of the equation

∗ + m F um n + λm un = t+ (dn ) − 2ε ϕ1 + d

in Ω,

and the sequence {um n } is bounded as m → ∞, for each fixed n – see (2.1). We also consider the solutions wn of F [wn ] − γ wn = dn − d. By Theorem 2.1 we know that wn → 0 in C 1 (Ω). Then by the structural hypotheses on F (recall F [u + v] F [u] + F [v]) we have

m ∗ + F um n + wn + λm un + wn t+ (dn ) − 2ε ϕ1 + dn + (γ + λm )wn ∗ t+ (dn ) − ε ϕ1+ + dn , where the last inequality holds if n is large, independently of m. Fix one such n. On the other hand we can take solutions znm of

∗ F znm + λm znm = t+ (dn ) − ε ϕ1+ + dn

m F um n + wn + λm un + wn .

By (2.1) for any n we have znm → ∞ as m → ∞. By the comparison principle (valid by m m m λm < λ+ 1 and Theorem 1.5 in [36]) we obtain zn un + wn in Ω, hence zn is bounded above + m as m → ∞. Since zn is bounded below, by Theorem 1.7 in [36] and λm λ1 < λ− 1 , we obtain a contradiction. ∗ (d ) t ∗ (d) − 3ε. Let u be a solution of Assume now that t+ n n + ∗ + F [un ] + λ+ 1 un = t+ (d) − 2ε ϕ1 + dn

in Ω,

∗ (d ) < t ∗ (d) − 2ε – Theorem 1.2 in [24]. Let w be the solution of F [w ] + which exists since t+ n n n + cwn = d − dn in Ω, with wn → 0 in C 1 (Ω). Then there exists n0 large enough so that (λ+ 1 + γ )wn0 < εϕ1+ , and consequently un0 + wn0 is a super-solution of

∗ + F [u] + λ+ 1 u = t+ (d) − ε ϕ1 + d

in Ω.

(2.3)

Now, if w is the solution of F [w] − γ w = −d in Ω, by defining vk = kϕ1− − w we obtain + + ∗ + − − F [vk ] + λ+ 1 vk k λ1 − λ1 ϕ1 + d − λ1 + γ w > t+ (d) − ε ϕ1 + d, for k large enough. By taking k large we also have vk < un0 − wn0 in Ω, thus Eq. (2.3) possesses ordered super- and sub-solutions. Consequently it has a solution (by Perron’s method – see for


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∗ (d). This completes the instance Lemma 4.3 in [36]), a contradiction with the definition of t+ ∗. proof of the continuity of the function t+ ∗ . Assuming t ∗ is not conThe rest of the proof is devoted to the analysis of continuity of t− − ∗ (d ) t ∗ (d) + 3ε or tinuous, there is ε > 0 and a sequence dn → d in Lp (Ω) such that either t− n − ∗ (d ) t ∗ (d) − 3ε. In the first case, let us consider a sequence λ λ− , and a solution v of t− n m m − 1 the equation

∗ F [vm ] + λm vm = t− (d) + ε ϕ1+ + d

in Ω.

∗ (d ) t ∗ (d) + We recall vm exists, by the results in [7] and [24]. We have shown in [24] that t− n − ∗ 3ε > t− (d) + ε implies that vm can be chosen to be bounded as m → ∞ (see (2.2)). Let wn be the solution to F [wn ] − γ wn = dn − d in Ω, as above. Then znm0 = vm + wn0 satisfies for some large n0

∗ ∗ F znm0 + λm znm0 t− (d) + ε ϕ1+ + (λm + γ )wn + dn t− (d) + 2ε ϕ1+ + dn , since again wn → 0 in C 1 (Ω). On the other hand, we consider a solution of

∗ + m F um n + λm un = t− (d) + 2ε ϕ1 + dn

in Ω.

∗ (d ) t ∗ (d) + 3ε > t ∗ (d) + 2ε for all n, the sequence um is not bounded (again by (2.2) As t− n − − n − m and [24]) and um n /un ∞ → ϕ1 as m → ∞, for each fixed n. Therefore for large m the function Ψ = um n0 − (vm + wn0 ) < 0 and F [Ψ ] + λm Ψ 0, which is a contradiction with the definition − of λ1 , since λm > λ− 1. ∗ (d ) Let us assume now that for ε > 0 and the sequence dn → d in Lp (Ω) we have t− n − ∗ t− (d) − 3ε, for all n. Let λm λ1 and vm be a solution of the equation F [vm ] + λm vm = ∗ (d) − ε)ϕ + + d in Ω (by (2.2) v is unbounded), and let w be the solution to F [w ] − γ w = (t− m n n n 1 d − dn in Ω. Then, vm /vm ∞ → ϕ1− and wn → 0 in C 1 (Ω). We take a solution um n to

∗ + m F um n + λm un = t− (d) − 2ε ϕ1 + dn

in Ω,

∗ (d ) t ∗ (d) − 3ε < t ∗ (d) − 2ε, for any given n there exists a constant c and note that, since t− n n − − m such that un ∞ cn , for all m. Now, as above, we define Ψ = vm − (um + w ), and see that n n

∗ ∗ F [Ψ ] + λm Ψ t− (d) − ε ϕ1+ + d − t− (d) − 2ε ϕ1+ − dn − F [wn ] − λm wn εϕ1+ − (λm + γ )wn . We choose n large enough to have (λm + γ )wn < εϕ1+ in Ω. Then, keeping n fixed, we can choose m large enough to have Ψ < 0 in Ω, a contradiction with the definition of λ− 1 , since − λm > λ 1 . 2 Finally we prove that the function tλ∗ (d) is also continuous in λ at the end points of the interval when d is kept fixed. This fact will be needed in Section 7.

− [λ+ 1 , λ1 ],

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Proposition 2.3. For every d ∈ Lp (Ω) ∗ lim tλ∗ (d) = t+ (d) and

λ λ+ 1

∗ lim tλ∗ (d) = t− (d).

λλ− 1

∗ ∗ Proof. Let us assume that there are ε > 0 and a sequence λn λ+ 1 such that tλn < t+ − ε (since ∗ d is fixed, we do not write it explicitly). Then by the definition of tλn there is a function un satisfying

∗ F [un ] + λn un = t+ − ε ϕ1+ + d

in Ω.

Since λn λ+ 1 , un cannot be bounded, for otherwise we get a contradiction with the defini∗ by finding a solution with t < t ∗ – from Theorem 2.3. Then by Proposition 2.1 tion of t+ + un /un ∞ → ϕ1+ , un is positive for large n, and ∗ + + ∗ + F [un ] + λ+ 1 un = t+ − ε ϕ1 + d + λ1 − λn un < t+ − ε ϕ1 + d, ∗ , let u be a solution of F [u] + λ+ u = that is, un is a super-solution. On the other hand, for t > t+ 1 + ∗ − ε)ϕ + + d as a right-hand tϕ1 + d, in Ω, then u is a sub-solution for this equation with (t+ 1 side. By taking n large enough, we have un u, so that the equation

∗ + F [u] + λ+ 1 u = t+ − ε ϕ1 + d

in Ω

∗. has a solution, a contradiction with the definition of t+ ∗ ∗ Now we assume that there are ε > 0 and a sequence λn λ+ 1 such that tn = tλn > t+ + 2ε. Let v be a solution to

∗ + F [v] + λ+ 1 v = t+ + ε/2 ϕ1 + d

in Ω,

∗ + ε)ϕ + + d − ε/2ϕ + + (λ − λ+ )v. Since t ∗ + ε < t ∗ − ε/2, by choosing then F [v] + λn v = (t+ n + λn 1 1 1 n large we find F [v] + λn v < (tλ∗n − ε/2)ϕ1+ + d, so that v is a super-solution of

F [u] + λn u = tλ∗n − ε/2 ϕ1+ + d.

(2.4)

− Next we consider a solution uK of F [u] + (λ+ 1 + ν)u = K (where we have set ν = (λ1 − + λ1 )/2 > 0), for each K > 0. Such a solution exists by Theorem 1.9 in [36], and it further satisfies uK < 0 in Ω and uK ∞ → ∞ as K → ∞, so |uK | C(K)ϕ1+ , where C(K) → ∞ as K → ∞. Let w be the (unique) solution of F [w] − γ w = −d in Ω. Since F [uK − w] F [uK ] − F [w], we easily see that the function uK − w is a sub-solution of (2.4) and uK − w < v, for large K. Then Perron’s method leads again to a contradiction with the definition of tλn . This shows tλ∗ is right-continuous at λ+ 1. Now we prove the second statement of Lemma 2.3. Assume there are ε > 0 and a sequence ∗ ∗ λn λ− 1 such that tλn < t− − ε. Let un be a solution to

∗ − ε ϕ1+ + d F [un ] + λn un = t−

in Ω.


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− Since λn → λ− 1 , un cannot be bounded (as before) and then un /un ∞ → ϕ1 . Thus, for large n we have un < 0 and

∗ + − ∗ + F [un ] + λ− 1 un = t− − ε ϕ1 + d + λ1 − λn un < t− − ε ϕ1 + d, so that uk is a super-solution for some large (fixed) k. Consider now a sequence λ˜ n λ− 1 and let vn be the solution to ∗ − ε ϕ1+ + d F [vn ] + λ˜ n vn = t−

in Ω,

whose existence was proved in [7] and [24]. Then vn cannot be bounded, so vn /vn ∞ → ϕ1− , and for large n we have ∗ + − ∗ + ˜ F [vn ] + λ− 1 vn = t− − ε ϕ1 + d + λ1 − λn vn > t− − ε ϕ1 + d, that is, vn is a sub-solution. For the already fixed uk , we can find n sufficiently large so that uk > vn , which implies that the equation ∗ + F [u] + λ− 1 u = t− − ε ϕ1 + d

in Ω

∗. has a solution, a contradiction with the definition of t− ∗ ∗ ∗ Finally, assume that there are ε > 0 and a sequence λn λ− 1 such that tλn > tλn − 2ε > t− + ε. By Theorem 1.4 in [24] we can find a function u which solves the equation F [u] + λ− 1u= + ∗ (t− + ε)ϕ1 + d in Ω. Then

+ ∗ + F [u] + λn u < tλ∗n − ε ϕ1+ + d − εϕ1+ + λn − λ− 1 ϕ1 < tλn − ε ϕ1 + d, so that u is a super-solution of F [u] + λn u = (tλ∗n − ε)ϕ1+ + d, for some large fixed n. As we explained above, since λn < λ− 1 , by Theorem 1.9 in [36] we can construct an arbitrarily negative sub-solution of this problem, hence a solution as well, contradicting the definition of tλ∗n . 2 3. Resonance and a priori bounds In this section we assume that the nonlinearity f (x, s) satisfies the one-sided Landesman– Lazer conditions at resonance, that is, one of (F+ ), (F− ), (F+r ) and (F−r ). Under each of these conditions we analyze the existence of super-solutions, sub-solutions and a priori bounds when − λ is close to the eigenvalues λ+ 1 and λ1 . This information will allow us to obtain existence of solutions by using degree theory and bifurcation arguments. In particular we will get branches bifurcating from infinity which curve right or left depending on the a priori bounds obtained here. We start with the existence of a super-solution and a priori bounds at λ+ 1 , under hypothesis (F+ ). Proposition 3.1. Assume f satisfies (F0) and (F+ ). Then there exists a super-solution z such that + F [z] + λz < f (x, z) in Ω, for all λ ∈ (−∞, λ+ 1 ]. Moreover, for each λ0 < λ1 there exist R > 0 + and a super-solution z0 such that if u is a solution of (1.1) with λ ∈ [λ0 , λ1 ], then u R and u z0 in Ω.

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Proof. We first replace c+ by a more appropriate function: we claim that for each ε > 0 there exist R > 0 and a function d ∈ Lp (Ω) such that d − c+ Lp (Ω) ε

and u Rϕ1+

implies f x, u(x) d(x) in Ω.

In fact, setting σ =

ε , we can find s0 such that f (x, s) c+ (x) − σ in Ω, for all s s0 . Let 2|Ω|1/p + R Ω = {x ∈ Ω | Rϕ1 (x) > s0 } and define the function dR as dR (x) = c+ (x) − σ if x ∈ Ω R , and dR (x) = −M for x ∈ Ω \ Ω R , where M is such that f (x, s) −M, for all s ∈ [0, s0 ]. It is then trivial to check that the claim holds for d = dR , if R is taken such that |Ω \ Ω R | < (ε/2M)p . ∗ (Proposition 2.2) we can fix ε so small that the function Now, by (F+ ) and the continuity of t+

d chosen above satisfies

∗ t+ (d) < 0.

(3.1)

Let zn be a solution to + F [zn ] + λ+ 1 zn = tn ϕ1 + d

in Ω,

∗ (d) < 0, t t ∗ (d), is a sequence such that z can be chosen to be unbounded – where tn → t+ n n + such a choice of tn and zn is possible thanks to Theorem 1.2 in [24]. Then zn /zn → ϕ1+ , which implies that for large n

F [zn ] + λ+ 1 zn < d

and zn Rϕ1+ ,

by (3.1), where R is as in the claim above. Thus zn is a strict super-solution and, since zn is positive, F [zn ] + λzn < f (x, zn ), for all λ ∈ (−∞, λ+ 1 ]. From now on we fix one such n0 and drop the index, calling the super-solution z. Suppose there exists an unbounded sequence un of solutions to F [un ] + λn un = f (x, un )

in Ω,

+ + with λn ∈ [λ0 , λ+ 1 ] and λn → λ. If λ < λ1 then a contradiction follows since λ1 is the first + eigenvalue (divide the equation by un and let n → ∞). If λ = λ1 then un /un → ϕ1+ , so that for n large we have un > z and un Rϕ1+ , consequently f (x, un ) d(x) in Ω. Thus, setting w = un − z we get, by λn λ+ 1 , w > 0, + F [w] + λ+ 1 w F [un ] − F [z] + λ1 (un − z) > f (x, un ) − d 0.

Since w > 0, Theorem 1.2 in [36] implies the existence of a constant k > 0 such that w = kϕ1+ , a contradiction with the last strict inequality. Now that we have an a priori bound for the solutions, we may choose an appropriate n0 for the definition of z0 = zn0 , which makes it larger than all solutions. 2 Next we state an analogous proposition on the existence of a sub-solution to our problem at

λ− 1 under hypothesis (F− ).


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Proposition 3.2. Assuming that f satisfies (F0) and (F− ), there exists a strict sub-solution z such that F [z] + λz > f (x, z) in Ω for all λ ∈ (−∞, λ− 1 ]. Moreover, for each δ > 0 there exist R > 0 − and a sub-solution z such that if u solves (1.1) with λ ∈ [λ+ 1 + δ, λ1 ] then u∞ R and u z in Ω. Proof. By using essentially the same proof as in Proposition 3.1, we can find R > 0 and a func∗ (d) > 0, and u −Rϕ + implies f (x, u(x)) d(x) in Ω. Consider tion d ∈ Lp (Ω) such that t− 1 ∗ a sequence tn t− (d) and solutions zn to + F [zn ] + λ− 1 zn = tn ϕ1 + d

in Ω,

(3.2)

chosen so that zn is unbounded and zn /zn ∞ → ϕ1− – see Theorem 1.4 in [24]. Hence for n large enough F [zn ] + λ− 1 zn > d

and zn −Rϕ1+ .

(3.3)

Thus zn is a strict sub-solution and, since zn is negative for sufficiently large n, F [zn ] + λzn > f (x, zn ), for all λ ∈ (−∞, λ− 1 ]. Fix one such n0 and set z = zn0 . If un is an unbounded sequence of solutions to F [un ] + λn un = f (x, un ), in Ω, with λn ∈ − [λ+ 1 + δ, λ1 ] and λn → λ we obtain a contradiction like in the previous proposition. Namely, if − λ ∈ [λ+ 1 + δ, λ1 ) then the conclusion follows since there are no eigenvalues in this interval. If − λ = λ1 then un /un → ϕ1− , so that for n large un < z and un −Rϕ1+ , hence f (x, un ) d(x), which leads to the contradiction F [z − un ] + λ− 1 (z − un ) 0 and z − un > 0. Then, given the a priori bound, we can choose n0 such that zn0 is smaller than all solutions. 2 The next two propositions are devoted to proving a priori bounds under hypotheses (F+r ) and (F−r ). Proposition 3.3. Under assumptions (F0) and (F+r ) for each δ > 0 the solutions to (1.1) with − λ ∈ [λ+ 1 , λ1 − δ] are a priori bounded. Proof. As in the proof of Proposition 3.1, we may choose R > 0 and a function d so that + + ∗ (d) > 0, that is, ∗ ⊥ dϕ t+ Ω 1 < t+ (d ) (recall (1.7)), and whenever u Rϕ1 then f (x, u) d. + ∗ (d ⊥ ). If the proposition were not true, then there would Let t˜ be fixed such that Ω dϕ1 < t˜ < t+ + be sequences λn λ1 and un of solutions to F [un ] + λn un = f (x, un ), such that un is unbounded. Then un /un → ϕ1+ , in particular, un is positive for large n. Then ⊥ ˜ + F [un ] + λ+ 1 un f (x, un ) d < t ϕ1 + d , ⊥ ˜ + that is, un is a super-solution of F [un ] + λ+ 1 un = t ϕ1 + d . Next, take the solution w of F [w] − ⊥ γ w = −d in Ω, where, as before, γ is the constant from (H1), so that F − γ is proper. For α > 0 we define v = αϕ1− − w, then

+ + − − ⊥ F [v] + λ+ 1 v α λ1 − λ1 ϕ1 − λ1 + γ w + d ,

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exactly like in the proof of Proposition 2.2. If we choose α large enough, we see that v is a sub⊥ ˜ + solution for F [un ] + λ+ 1 un = t ϕ1 + d , and v is smaller than the super-solution we constructed ∗ (d ⊥ ) and before. The existence of a solution to this equation contradicts the definition of t+ ∗ ⊥ t˜ < t+ (d ). 2 Proposition 3.4. Under assumptions (F0) and (F−r ) there exists δ > 0 such that the solutions to − (1.1) with λ ∈ [λ− 1 , λ1 + δ] are a priori bounded. ∗ (d ⊥ ), and Proof. We proceed like in the proof of the previous proposition. Now Ω dϕ1+ > t− + + ∗ ⊥ ˜ ˜ whenever u −Rϕ1 then f (x, u) d. If t is such that Ω dϕ1 > t > t+ (d ), and we assume there are sequences λn λ− 1 and un of solutions to F [un ] + λn un = f (x, un ) in Ω, such that un is unbounded, we get un /un → ϕ1− , consequently ⊥ ˜ + F [un ] + λ− 1 un > t ϕ1 + d . ⊥ ˜ + On the other hand if z solves F [z] + λ− 1 z = t ϕ1 + d in Ω (such z exists by Theorem 1.4 − in [24]), then F [un − z] + λ1 (un − z) > 0, and un − z < 0 in Ω, for large n. Thus, we may apply Theorem 1.4 in [36] to obtain k > 0 so that un − z = kϕ1− , a contradiction with the strict inequality. 2 − 4. Bifurcation from infinity at λ+ 1 and λ1

In this section we prove the existence of unbounded branches of solutions of (1.1), bifurcating − from infinity at the eigenvalues λ+ 1 and λ1 . Then, thanks to the a priori bounds obtained in Section 3, for the two types of Landesman–Lazer conditions (see Propositions 3.1–3.4), we may determine to which side of the eigenvalues these branches curve. We recall that F (M, q, u, x) + cu is decreasing in u for any c −γ , in other words, F + c is a proper operator. Given v ∈ C 1 (Ω) we consider the problem F [u] + cu = (c − λ)v + f (x, v)

in Ω,

u = 0 on ∂Ω,

(4.1)

see Theorem 2.1. We define the operator K : R × C 1 (Ω) → C 1 (Ω) as follows: K(λ, v) is the unique solution u ∈ C 1,α (Ω) of (4.1). The operator K is compact in view of Theorem 2.1 and the compact embedding C 1,α (Ω) → C 1 (Ω). With these definitions, our Eq. (1.1) is transformed into the fixed point problem u = K(λ, u), u ∈ C 1 (Ω), with λ ∈ R as a parameter. We are going to show that the sub-linearity of the function f (x, ·), given by assumption (F0), implies bifurcation at infinity at the eigenvalues of F . The proof follows the standard procedure for the linear case, see for example [38] or [8], so we shall be sketchy, discussing only the main differences. We define

v , G(λ, v) = v2C 1 K λ, v2C 1 for v = 0, and G(λ, 0) = 0. Finding u = 0 such that u = K(λ, u) is equivalent to solving the fixed point problem v = G(λ, v), v ∈ C 1 (Ω), for v = u/u2C 1 . The important observation is that bifurcation from zero in v is equivalent to bifurcation from infinity for u.


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Let u = G0 (λ, v) be the solution of the problem F [u] + cu = (c − λ)v

in Ω,

u = 0 on ∂Ω,

(4.2)

and set G1 = G − G0 , so that G(λ, v) = G0 (λ, v) + G1 (λ, v). Lemma 4.1. Under the hypothesis (F0) we have limvC 1 →0

G1 (λ,v) vC 1

= 0.

Proof. Let g = G(λ, v) and g0 = G0 (λ, v). Then we have

1 v F [g] − F [g0 ] + c(g − g0 ) = vC 1 f x, . vC 1 v2C 1 The right-hand side here goes to zero as vC 1 → 0, by (F0). Then by (DF)

v 1 , F |g − g0 | + c|g − g0 | −vC 1 f x, vC 1 v2 1 C

so the ABP inequality (Theorem 2.1) implies

v 1 , sup |g − g0 | CvC 1 f x, vC 1 v2 1 Lp Ω

C

and the result follows.

2

The next proposition deals with the equation v = G0 (λ, v), v ∈ C 1 (Ω) (recall we want to solve v = G0 (λ, v) + G1 (λ, v)), which is equivalent to F D 2 v, Dv, v, x = −λv

in Ω,

v = 0 on ∂Ω.

(4.3)

+ − Proposition 4.1. There exists δ > 0 such that for all r > 0 and all λ ∈ (−∞, λ− 1 + δ) \ {λ1 , λ1 }, the Leray–Schauder degree deg(I − G0 (λ, ·), Br , 0) is well defined. Moreover

⎧ if λ < λ+ 1, ⎨1 deg I − G0 (λ, ·), Br , 0 = 0 < λ < λ− if λ+ 1 1, ⎩ − −1 if λ1 < λ < λ− 1 + δ. Proof. We recall it was proved in [36] that problem (4.3) has only the zero solution in + − (−∞, λ− 1 + δ) \ {λ1 , λ1 }, for certain δ > 0. The compactness of G0 follows from Theorem 2.1, so the degree is well defined in the given ranges for λ. Suppose λ < λ+ 1 and consider the operator I − tG0 (λ, ·) for t ∈ [0, 1]. Since tλ is not an eigenvalue of (4.3), we have for t ∈ [0, 1] deg I − G0 (λ, ·), Br , 0 = deg I − tG0 (λ, ·), Br , 0 = deg(I, Br , 0) = 1.

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− The case λ+ 1 < λ < λ1 was studied in [39]. Consider the problem

F [u] + cu = (c − λ)v − tϕ1+

in Ω,

u = 0 on ∂Ω

(4.4)

0 (λ, v, t). It follows from the results in for t ∈ [0, ∞), whose unique solution is denoted by G [36,39] that for t > 0 the equation F [u] + cu = (c − λ)u − tϕ1+

in Ω,

u=0

on ∂Ω

(4.5)

does not have a solution. On the other hand, since λ is not an eigenvalue, there is R > 0 such that the solutions of (4.5), for t ∈ [0, t], are a priori bounded, consequently 0 (λ, ·, 0), BR , 0 deg I − G0 (λ, ·), Br , 0 = deg I − G 0 (λ, ·, t), BR , 0 = 0. = deg I − G − If λ− 1 < λ < λ1 + δ we proceed as in [24], where the computation of the degree was done by making a homotopy with the Laplacian (see the proof of Lemma 4.2 in that paper). 2

Now we are in position to apply the general theory of bifurcation to v = G(λ, v), see for instance the surveys [38] and [8], and obtain bifurcation branches emanating from (λ+ 1 , 0) and + , 0), exactly like in [13]. In short, from (λ , 0) bifurcates a continuum of solutions of v = (λ− 1 1 G(λ, v), which is either unbounded in λ, or unbounded in u, or connects to (λ, 0), where λ = λ+ 1 − − is an eigenvalue (recall λ+ and λ are the only eigenvalues in (−∞, λ + δ), for some δ > 0). 1 1 1 A similar situation occurs at (λ− 1 , 0). Inverting the variables we obtain bifurcation at infinity for our problem (1.1): Theorem 4.1. Under the hypotheses of Theorem 1.1 there are two connected sets C2 , C3 ⊂ S such that: 1) There is a sequence (λn , un ) with un ∈ C2 (λn ) (un ∈ C3 (λn )), and un ∞ → ∞, λn → λ+ 1 (λ− 1 ). − 2) If (λn , un ) is a sequence such that un ∈ C2 (λn ) (C3 (λn )), un ∞ → ∞ and λn → λ+ 1 (λ1 ), then un is positive (negative) for large n. 3) The branch C2 satisfies one of the following alternatives, for some δ > 0: (i) C2 (λ) = ∅ for − − all λ ∈ (λ+ 1 , λ1 + δ); (ii) there is λ ∈ (−∞, λ1 + δ] such that 0 ∈ C2 (λ); (iii) C2 (λ) = ∅ for + all λ ∈ (−∞, λ1 ); (iv) there is a sequence (λn , un ) such that un ∈ C2 (λn ), un ∞ → ∞, − λn → λ− 1 , and λn λ1 . 4) The branch C3 satisfies one of the following alternatives, for some δ > 0: (i) C3 (λ) = ∅ for − − all λ ∈ (λ− 1 , λ1 + δ); (ii) there is λ ∈ (−∞, λ1 + δ] such that 0 ∈ C3 (λ); (iii) C3 (λ) = ∅ for all λ ∈ (−∞, λ− 1 ); (iv) there is a sequence (λn , un ) such that un ∈ C3 (λn ), un ∞ → ∞, . and λn → λ+ 1 We remark that (F1) excludes alternatives 3)(ii) and 4)(ii) in this theorem.


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5. A bounded branch of solutions In this section we prepare for the proof of our main theorems by establishing the existence of a continuum of solutions of (1.1) which is not empty for all λ ∈ (−∞, λ+ 1 + δ), for some δ > 0. Our first proposition concerns the behavior of solutions of (1.1) when λ → −∞. Proposition 5.1. (1) Assume f satisfies (F0). Then there exists a constant C0 > 0, depending only on F , f , and Ω, such that any solution of (1.1) satisfies u∞ C0 λ−1 as λ → −∞. (2) If in addition f is Lipschitz at zero, that is, for some ε > 0 and some C > 0 we have |f (x, s1 ) − f (x, s2 )| C|s1 − s2 | for s1 , s2 ∈ (−ε, ε), then (1.1) has at most one solution when λ is sufficiently large and negative. Proof. (1) Let uλ be a sequence of solutions of (1.1), with λ → −∞. We first claim that uλ ∞ is bounded. Suppose this is not so, and say u+ λ ∞ → ∞ (with the usual notation for the positive , on the set Ωλ+ = {uλ > 0} we have the inequality part of u). Then, setting vλ = uλ /u+ λ ∞ F [vλ ] − γ vλ

f (x, uλ ) → 0, u+ λ ∞

as λ → −∞.

The ABP estimate (see Theorem 2.1) then implies supΩ + vλ → 0, which is a contradiction with λ

supΩ + vλ = 1. In an analogous way we conclude that u− λ ∞ is bounded. λ

Hence there exists a constant C such that |f (x, uλ (x))| C in Ω, so λ , F [uλ ] − γ uλ −(λ + γ )uλ − C 0 on the set Ω λ = {uλ > C/(|λ| + γ )}. Applying the maximum principle or the ABP inequality in this where Ω set implies it is empty, which means uλ C/(|λ| + γ ) in Ω. By the same argument we show uλ is bounded below, and (1) follows. (2) From statement (1) we conclude that for λ small, all solutions of (1.1) are in (−ε, ε). If u1 , u2 are two solutions of (1.1) then for |λ| > γ + C we have F [u1 − u2 ] − γ (u1 − u2 ) 0 on {u1 > u2 } which means this set is empty. 2 The next result is stated in the framework of Theorem 1.1 and gives a bounded family of solutions (uλ , λ), for λ ∈ (−∞, λ+ 1 + δ). No assumption of Lipschitz continuity on f is needed. Proposition 5.2. Assume f satisfies (F0) and (F+ ). Then there is a connected subset C1 of S such that C1 (λ) = ∅, for all λ ∈ (−∞, λ+ 1 + δ). Proof. According to Proposition 3.1, given λ0 < λ+ 1 , there is R > 0 so that all solutions of (1.1) ] belong to the ball B . In particular, the equation does not have a solution with λ ∈ [λ0 , λ+ R 1 ] × ∂B . Moreover, there is δ > 0 such that (1.1) does not have a solution in (λ, u) ∈ [λ0 , λ+ R 1 + + [λ1 , λ1 + δ] × ∂BR – otherwise we obtain a contradiction by a simple passage to the limit. Consequently the degree deg(I − K(λ, ·), BR , 0) is well defined for all λ ∈ [λ0 , λ+ 1 + δ] (K(λ, ·) is defined in the previous section). We claim that its value is 1.

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To compute this degree, we fix λ < λ+ 1 and analyze the equation F [u] + λu = sf (x, u)

in Ω,

for s ∈ [0, 1]. Since λ is not an eigenvalue of F in Ω, the solutions of this equation are a priori bounded, uniformly in s ∈ [0, 1], that is, there is R1 R, such that no solution of the equation exists outside of the open ball BR1 . Given v ∈ C 1 (Ω) we denote by Ks (λ, v) the unique solution of the equation F [u] + λu = sf (x, v) in Ω. Then we have deg I − K(λ, ·), BR , 0 = deg I − K1 (λ, ·), BR1 , 0 = deg I − K0 (λ, ·), BR1 , 0 = 1, where the last equality is given by Proposition 4.1. Hence, again by the homotopy invariance of the degree, we have deg(I − K(λ, ·), BR , 0) = 1, for all λ ∈ (λ0 , λ+ 1 + δ). The last fact together with standard degree theory implies that for every λ ∈ [λ0 , λ+ 1 + δ] there is at least one (λ, u), solution of (1.1), and, moreover, there is a connected subset C1 of S such that C1 (λ) = ∅ for all λ in the interval [λ0 , λ+ 1 + δ]. Since λ0 is arbitrary, we can use the same argument for each element of a sequence {λn0 }, with λn0 → −∞. Then, by a limit argument (like the one in the proof of Theorem 1.5.1 in [24]), we find a connected set C1 with the desired properties. 2 Next we study a branch of solutions driven by a family of super- and sub-solutions, assuming that f is locally Lipschitz continuous. In this case the statement of the previous proposition can be made more precise. Specifically, we assume that f satisfies (F2), and there exist u, u ∈ C 1 (Ω), such that u is a super-solution and u is a sub-solution of (1.1), for all λ λ, where λ is fixed. We further assume that u and u are not solutions of (1.1), and u 0. Lemma 5.1. With the definitions given above, we have deg I − K(λ, ·), O ∩ BR , 0 = 1,

for all λ ∈ [λ0 , λ].

Proof. First we have to prove that the degree is well defined. We just need to show that there are no fixed points of K(λ, ·) on the boundary of O ∩ BR . For this purpose it is enough to prove that, given v ∈ C 1 (Ω) such that u v u in Ω, we have u < K(λ, v) < u in Ω. In what follows we write u = K(λ, v).


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By (F2) we can assume that the negative number c, chosen in Section 4, is such that the function s → f (x, s) + (c − λ)s is decreasing, for s ∈ (−τ, τ ), where τ = max{u∞ , u∞ }. Then F [u] = F [u] − f (x, u) − (c − λ)u + f (x, u) + (c − λ)u F [u] − f (x, v) − (c − λ)v + f (x, u) + (c − λ)u = −cu + f (x, u) + (c − λ)u = c(u − u) + f (x, u) − λu F [u] + c(u − u). By (H1) this implies M+ (D 2 (u − u)) + γ |Du − Du| + (γ − c)(u − u) > 0 in Ω. It follows from ∂u Theorem 2.2 that u < u in Ω and ∂u ∂ν < ∂ν on ∂Ω. The other inequality is obtained similarly. By using its homotopy invariance, the degree we want to compute is equal to the degree at λ0 . But the latter was shown to be one in the proof of Proposition 5.2, which completes the proof of the lemma. 2 Now we can state a proposition on the existence of a branch of solutions for λ ∈ (−∞, λ], whose proof is a direct consequence of Lemma 5.1 and general degree arguments. Proposition 5.3. Assume f satisfies (F0) and (F2). Suppose there are functions u, u ∈ C 1 (Ω) such that u is a super-solution and u is a sub-solution of (1.1) for all λ λ, these functions are not solutions of (1.1) and satisfy (5.1). Then there is a connected subset C1 of S such that C1 (λ) = ∅ for all λ ∈ (−∞, λ) and each u ∈ C1 (λ) is such that u u u. Remark 5.1. In the next section we use this proposition with appropriately chosen sub-solutions and super-solutions. Remark 5.2. The branch C1 is isolated of other branches of solutions by the open set O, since we know there are no solutions on ∂O. 6. Proofs of the main theorems In this section we put together the bifurcation branches emanating from infinity obtained in Theorem 4.1 with the bounded branches constructed in Section 5, and study their properties. Proof of Theorem 1.1. This theorem is a consequence of Proposition 5.2, for the definition of C1 , and of Theorem 4.1, 1)–2), for the definition of C2 and C3 . Both C2 and C3 curve to the right − of λ+ 1 and λ1 , respectively – as a consequence of the a priori bounds obtained in Propositions 3.1 and 3.2. 2 Proof of Theorem 1.2. We first construct the branch C1 , through Proposition 5.3. In view of (F1) we may take as a super-solution the function u ≡ 0. In order to define the corresponding sub-solution we use Proposition 3.2, where a sub-solution is constructed for all λ ∈ (−∞, λ− 1 ]. We can rewrite inequality (3.2) in the following way ˜ + F [zn ] + λ− 1 + δ zn = tn ϕ1 + d,

with tñ (x) =

δzn (x) + tn . ϕ1+ (x)

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Since zn /zn ∞ → ϕ1− < 0 in C 1 (Ω) we find that for some c > 0 |zn (x)| c, zn ∞ ϕ1+ (x)

∀x ∈ Ω.

Consequently, once n is chosen so that (3.3) holds, we can fix δ > 0 such that tñ (x) −δc + ∗ (d) > 0, for all x ∈ Ω, which means that z is a sub-solution also for F [u] + (λ− + δ)u = t− n 1 f (x, u), as in the proof of Proposition 3.2. Now we define u = zn , chosen as above, and take λ = λ− 1 + δ in Proposition 5.3. Clearly u and u satisfy also (5.1), so the existence of the branch C1 (with the properties stated in Theorem 1.2) follows from Proposition 5.3. Further, the branches C2 and C3 are given by Theorem 4.1 and both of them curve to the right − of λ+ 1 and λ1 , respectively. Neither C2 or C3 connects to C1 , since C1 is isolated from the exterior of the open set O, see Remark 5.2. Observe that the elements of C2 (resp. C3 ) are outside O for − λ close to λ+ 1 (resp. λ1 ). Therefore the uniqueness statement of Proposition 5.1 excludes the alternatives in Theorem 4.1, 3)(iii) and 4)(iii), since we already know that C1 contains solutions for arbitrary small λ. We already noted cases 3)(ii) and 4)(ii) are excluded by (F1). Finally, case 3)(iv) in Theorem 4.1 is excluded by the a priori bound in Proposition 3.2, so only case 3)(i) remains. 2 Proof of Theorem 1.3. We fix a small number ε > 0 and for each K > 0 consider a solution uK of F [uK ] + (λ− 1 − ε)uK = K in Ω, uK = 0 on ∂Ω, uK < 0 in Ω. We know such a function uK exists, by Theorem 1.9 in [36]. By (F0) we can fix K0 such that K0 > f (x, K0 ) in Ω, hence u = uK0 is a sub-solution of (1.1), for all λ ∈ (−∞, λ− 1 − ε). The super-solution to consider is u ≡ 0, as given by hypothesis (F1). Then Proposition 5.3 yields the existence of a branch C1ε such that C1ε (λ) = ∅ for all λ ∈ (−∞, λ− 1 − ε). Next we pass to the limit as ε → 0, like in the proofs of Proposition 5.2 and Theorem 1.5.1 in [24], and obtain either a connected component of S which bifurcates from infinity to the left − of λ− 1 , or a bounded branch of solutions which “survives” up to λ1 , and hence “continues” in − some small right neighborhood of λ1 , again like in the proof of Proposition 5.2. The first of these alternatives is (3)(i). In case the second alternative is realized there is a connected set of solutions C3 bifurcating from minus infinity towards the left of λ− 1 , as predicted in Theorem 4.1. We claim this branch contains only negative solutions. To prove this, we set , max A = (λ, u) ∈ C3 λ ∈ −∞, λ− u > 0 . 1 Ω

The set A is clearly open in C3 , and A = C3 . Hence if A is not empty, then A is not closed in C3 , by the connectedness of C3 . This means there is a sequence (λn , un ) ∈ A such that λn → λ, un → u, and the limit function u satisfies u 0 in Ω, u vanishes somewhere in Ω, and solves the equation F [u] + (λ − c)u = f (x, u) − cu 0 in Ω, for some large c. Hence by Hopf’s lemma u ≡ 0, a contradiction with (F1). Therefore C3 cannot connect with the branch bifurcating from plus infinity at λ+ 1 . It is not connected to C1 either – by the isolation property of C1 (λ), see Remark 5.2. Further, C3 cannot contain solutions for arbitrarily small λ, since C1 does, and we know solutions are unique for sufficiently small λ. Hence C3 must eventually curve to the right, so extra solutions appear, proving (3)(ii) and (4).


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Finally, a branch C2 bifurcating from plus infinity towards the left of λ+ 1 exists thanks to Theorem 4.1. This branch is kept away from C1 and C3 , as we already saw, and, again by the uniqueness of solutions for sufficiently small λ, C2 has to curve to the right. This completes the proof. 2 The occurrence of alternative (3)(ii) in Theorem 1.3 can be avoided if f satisfies some further hypotheses. Proposition 6.1. Under the hypotheses of Theorem 1.3, if in addition we make one of the following assumptions (1) f (x, s) is concave in s for s < 0, (2) for each a0 > 0 there exists k0 > 0 such that f (x, −kϕ1+ ) < f x, −aϕ1+ , k

for all a ∈ (0, a0 ), k > k0 ,

(6.1)

then alternative (3)(ii) in Theorem 1.3 does not occur. Remark. Note the model example of a sub-linear nonlinearity which satisfies the hypotheses of Theorem 1.3 f (x, s) = −s|s|α−1 + h(x),

α ∈ (0, 1), h 0,

satisfies both hypotheses in the above proposition. Proof of Proposition 6.1. We are going to prove the following stronger claim: under the hypotheses of the proposition, there cannot exist sequences λn , un , vn , such that λn < λn+1 , λn → λ − 1 , un , vn < 0 in Ω, un is bounded, vn → ∞ and un and vn are solutions of (1.1) with λ = λn . Assume this is false and (1) holds. Then (passing to subsequences if necessary) un is convergent in C 1 (Ω), and vn /vn → ϕ1− in C 1 (Ω), so there is n0 such that for all n n0 we have vn < un+1 in Ω. The negative function un+1 is clearly a strict sub-solution of F [u] + λn u = f (x, u), and, since the zero function is a strict super-solution of this equation, it has a negative solution which is above un+1 . We define v n = inf v un+1 < v < 0, v is a super-solution of F [u] + λn u = f (x, u) . Then v n is a solution of F [u]+λn u = f (x, u) such that between un+1 and v n no other solution of this problem exists. Indeed, v n is a super-solution (as an infimum of super-solutions), so between un+1 and v n there is a minimal solution, with which v n has to coincide, by its definition. Note Hopf’s lemma trivially implies that for some ε > 0 we have vn < un+1 − εϕ1+ < v n − 2εϕ1+ . Next, by the convexity of F and the concavity of f we easily check that the function uα = αvn + (1 − α)v n is a super-solution of F [u] + λn u = f (x, u), for each α ∈ [0, 1]. This gives a contradiction with the definition of v n , for α small enough but positive.

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Assume now our claim is false and (2) holds. We again have −C0 ϕ1+ un −c0 ϕ1+ < 0 and vn /vn → ϕ1− , so the numbers εn := sup{ε > 0 | un εvn in Ω} clearly satisfy εn > 0 and εn → 0. Hypothesis (6.1) implies that for sufficiently large n we have εn f (x, vn ) < f (x, un ), that is, F [εn vn ] + λn εn vn < F [un ] + λn un , and Hopf’s lemma yields a contradiction with the definition of εn . 2 7. Discussion and examples The main point of this section is to provide some examples showing that when (F1) or (F2) fails, then the bifurcation diagram for (1.1) may look very differently from what is described in Theorems 1.2–1.3. However, we begin with some general comments on our hypotheses and their use. Hypothesis (F0) is classical sub-linearity for f , which guarantees bifurcation from infinity and also ensures the solutions of (1.1) tend to zero as λ → −∞. Condition (F1) guarantees the existence of a strict super-solution of (1.1) for all λ, while (F2) is used in some comparison statements and to prove uniqueness of solutions of (1.1) for sufficiently negative λ. Further, conditions (F+ )–(F− ) and (F+r )–(F−r ) are the Landesman–Lazer type hypotheses which give a priori bounds when λ stays on one side of the eigenvalues, and thus provide a solution at resonance and determine on which side of each eigenvalue the bifurcation from infinity takes place. The strict inequalities in (F+ )–(F−r ) are important and cannot be relaxed in √ general – for instance the problem F [u] + λ+ 1 u = − max{1 − u, 0} has no solutions (and hence Theorem 1.1 fails), as Theorems 1.6 and 1.4 in [36] show, even though the nonlinearity satisfies the hypotheses of Theorem 1.1, except for the strict inequality in (F+ ). On the other hand, for F = it is known that in the case of equalities in (F+ )–(F−r ) one can give supplementary assumptions on f and the rate of convergence of f to its limits f± , f ± , so that results like Theorem 1.1 still hold, see for instance Remark 21 in [5]. Extensions of these ideas to HJB operators are out of the scope of this work and could be the basis of future research. Now we discuss examples where (F1) or (F2) fails. Example 1. Our first example shows that for all sufficiently small δ > 0 we can construct a nonlinearity f which does not satisfy (F1) and for which the set S(λ+ 1 + δ) is empty. This means that, in the framework of Theorems 1.1–1.2, the branch bifurcating from infinity to the right of + λ+ 1 “turns back” before it reaches λ1 + δ. A similar situation can be described for the branch − bifurcating from minus infinity to the left of λ− 1 that “turns right”, before reaching λ1 − δ. In + − particular there cannot be a continuum of solutions along the gap between λ1 and λ1 . Consider the Dirichlet problem F [u] + λu = tϕ1+ + h

in Ω,

u = 0 on ∂Ω,

(7.1)

∗ at resonance, that is, for λ = λ+ 1 . When t = t+ (h) Eq. (7.1) may or may not have a solution, depending on F and h. An example of such a situation was given in [7] and we recall it here. Take F [u] = max{u, 2u}, and h ∈ C(Ω) such that Ω hϕ1 = 0 and h changes sign on ∂Ω. Here − + − λ+ 1 = λ1 , λ1 = 2λ1 , ϕ1 = −ϕ1 = ϕ1 , where λ1 and ϕ1 are the first eigenvalue and eigenfunction


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∗ =0 of the Laplacian. Then (see Example 4.3 in [7]) under the above hypotheses on h we have t+ + ∗ and problem (7.1) has no solutions if λ = λ1 and t = t+ . By exactly the same reasoning it is ∗ possible to show that problem (7.1) has no solutions if λ = λ− 1 and t = t− . ∗ Lemma 7.1. If Eq. (7.1) with λ = λ+ 1 and t = t+ does not have a solution then there exists δ0 + + ∗ ∗ ∗ such that tλ > t+ provided λ ∈ (λ1 , λ1 + δ0 ). Similarly, if (7.1) with λ = λ− 1 and t = t− does − − ∗ ∗ not have a solution then there exists δ0 such that tλ > t− whenever λ ∈ (λ1 − δ0 , λ1 ).

Before proving the lemma, we use it to construct a nonlinearity with the desired properties. ∗ ∗ ∗ ∗ For λ sufficiently close to λ+ 1 we have t+ < tλ so that we can choose t ∈ (t+ , tλ ). We then define ⎧ + if u −M, ⎪ ⎨ tϕ1 ∗+ h t−t+ +ε f (x, u) = ( (u + M) + t)ϕ1+ + h if −2M u −M, ⎪ ⎩ ∗M (t+ − ε)ϕ1+ + h if u −2M,

(7.2)

where ε and M are some positive constants. We readily see that f satisfies (F0) and (F+ ), the hypotheses of Theorem 1.1, but S(λ) is empty. Indeed, if u ∈ S(λ), then u is a super-solution for (7.1). On the other hand, by Theorem 1.9 in [36], the equation F [u] + λu = Ktϕ1+ + hL∞ (Ω) with λ < λ− 1 has a solution uK , for each K > 0. Moreover, for large K, uK is a sub-solution of (7.1) and uK < u. Then by Perron’s method (7.1) has a solution, a contradiction. − ∗ ∗ Similarly, for λ < λ− 1 sufficiently close to λ1 we choose t ∈ (t− , tλ ) and define f (x, u) being + + ∗ equal to tϕ1 + h if u M and to (t− − ε)ϕ1 + h if u 2M. By the same reasoning we find that S(λ) is empty. We summarize: with these choices of λ and f there is a region of non-existence in the gap − between λ+ 1 and λ1 . In other words, the connected sets of solutions of (1.1) C2 (resp. C3 ), predicted in Theorem 1.1, do not extend to the right (resp. to the left) of λ. The first graph at the end of this section is an illustration of this situation. We observe that if we take M sufficiently large then all solutions of (7.1) and (1.1), with f as given in (7.2), coincide. In fact, we can take −M to be a lower bound for all solutions of the inequality F [u] + λu c + h, where c is such that f (x, u) c + h in Ω. Such an M exists by the one-sided ABP inequality given in Theorem 1.7 in [36]. Now we see that (1.1) with this f has a unique solution for λ < λ+ 1 , as an application of Theorem 1.8 in [36], and then the branch of solutions bifurcating from plus infinity must turn left and go towards infinity near the λ-axis, as drawn on the picture. − ∗ Proof of Lemma 7.1. Given λ ∈ (λ+ 1 , λ1 ), let vλ be a solution of

F [u] + λu = tλ∗ ϕ1+ + h in Ω,

(7.3)

whose existence is guaranteed by the results in [39]. We notice that vλ∗ is unbounded as + ∗ ∗ λ λ+ 1 , as otherwise vλ a subsequence of vλ would converge to a solution of (7.1) with λ = λ1 + ∗ ∗ , which is excluded by assumption. That t → t ∗ as λ → λ was proved in Proposiand t = t+ + λ 1 + + ∗ ∗ tion 2.3. Then, by the simplicity of λ+ 1 , we find that vλ /vλ ∞ → ϕ1 as λ → λ1 , in particular, + ∗ vλ becomes positive in Ω, for λ larger than and close enough to λ1 . Suppose for contradiction

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∗ t ∗ , then v ∗ 0 satisfies that t+ λ λ

∗ ∗ ∗ ∗ + ∗ + F vλ∗ + λ+ 1 vλ F vλ + λvλ = tλ ϕ1 + h t+ ϕ1 + h, ∗ so vλ∗ is a super-solution for (7.1) with λ = λ+ 1 and t = t+ . As we already showed above, (7.1) ∗ has a sub-solution below vλ , providing a contradiction. ∗ In the same way, we see that vλ∗ /vλ∗ ∞ → ϕ1− as λ λ− 1 and then vλ becomes negative in Ω, − − ∗ ∗ ∗ for λ < λ1 and close enough to λ1 . Then t− tλ would imply that vλ 0 satisfies

∗ ∗ ∗ ∗ + ∗ + F vλ∗ + λ− 1 vλ F vλ + λvλ = tλ ϕ1 + h t− ϕ1 + h, ∗ so vλ∗ is a super-solution for (7.1) with λ = λ− 1 and t = t− . To construct a sub-solution we − + ∗ consider vε a solutions of F [vε ] + λ1 vε = (t− + ε)ϕ1 + h, with ε > 0. By our assumption, vε /vε ∞ → ϕ1− as ε → 0 (see also Theorem 1.4 in [24]). Hence there exists ε = ε(λ) such that ∗ + vε < vλ∗ and F [vε ] + λ− 1 vε t− ϕ1 + h and then Perron’s method gives a contradiction again. 2

Remark 7.1. The claim gives an idea of the behavior of tλ∗ , with respect to λ, near the extremes of − ∗ the interval [λ+ 1 , λ1 ]. However we do not have any idea about the global behavior of tλ , actually ∗ ∗ we do not even know how t+ and t− compare. For completeness we give a direct proof of the fact that in the above examples condition (F1) is not satisfied by nonlinearities like in (7.2). In this direction we have the following lemma, which is of independent interest. Lemma 7.2. For any h ∈ Lp (Ω), p > N , which is not a multiple of ϕ1+ , ∗ (h) < 0 and t ∗ (h) < 0; (a) if h 0 and h ≡ 0 then t+ − ∗ (h) > 0 and t ∗ (h) > 0; (b) if h 0 and h ≡ 0 then t+ − ∗ (h)ϕ + + h and t ∗ (h)ϕ + + h change sign in Ω. (c) the functions t+ − 1 1 ∗ (h) 0 then, as h 0, by Theorem 1.9 in [36] the problem F [u] + λ+ u = Proof. (a) If t+ 1 ∗ (h)ϕ + + h has a solution. Then by Theorem 1.2 in [24] u + kϕ + is a solution of the same t+ 1 1 problem, for all k > 0. Since u + kϕ1+ is positive for sufficiently large k, by Theorem 1.2 in [36] we get that u is a multiple of ϕ1+ , a contradiction, since h = 0. ∗ (h) 0, by Theorem 1.5 in [24] either there exist sequences ε → 0 and u of solutions If t− n n + ∗ of the problem F [un ] + λ− 1 un = (t− (h) + εn )ϕ1 + h such that un is unbounded and un is negative − + ∗ for large n, or F [u + kϕ1− ] + λ− 1 (u + kϕ1 ) = t− (h)ϕ1 + h for some u and all k > 0. In both cases − we get a negative solution of F [u] + λ1 u 0, which by Theorem 1.4 in [36] is then a multiple of ϕ1− , a contradiction. ∗ (h) 0 then F [u] + λ+ u = t ∗ (h)ϕ + + h has no solution by Theorems 1.6 and 1.4 (b) If t+ + 1 1 ∗ (h)ϕ + + h 0. If t ∗ (h) 0 we again have t ∗ (h)ϕ + + h 0, then F [u] + λ− u = in [36], since t+ − − 1 1 1 ∗ (h)ϕ + + h has no solutions by the anti-maximum principle, see for instance Proposition 4.1 t− 1 − in [24]. Hence by Theorems 1.2 and 1.4 in [24] there exist sequences εn → 0, u+ n and un of ± ± + ± ∗ ± ± ± solutions of F [un ] + λ1 un = (t± (h) + εn )ϕ1 + h such that un /un ∞ → ϕ . Fix w to be the solution of the Dirichlet problem F (w) − γ w = −h in Ω. This problem is uniquely solvable,


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with w < 0 in Ω, since by (H1) the operator F − γ is decreasing in u (see for instance [17] and [36]). Then by the maximum principle and Hopf’s lemma εn ϕ1+ + (λ+ 1 + γ )w < 0 in Ω, + + + + if n is sufficiently large. Hence un + w is positive and F [un + w] + λ1 (un + w) < 0 in Ω, which is a contradiction with Theorem 1.4 in [36]. Similarly, u− n + w is negative and satisfies − − F [u− n + w] + λ1 (un + w) < 0 in Ω, which is a contradiction with Theorem 1.2 in [36]. (c) This is an immediate consequence of (a) and (b). Indeed, if (c) is false we just replace h ∗ (h)ϕ + + h in (a) or (b). 2 by t± 1 ∗ in the preceding lemma also follow from Theorem 1.1 and Remark 7.2. The statements on t+ formula (1.12) in [7].

The following example illustrate the role of hypothesis (F2), which allows the use of the method of sub- and super-solutions, and prevents the branches which bifurcate from infinity to survive for arbitrarily negative λ. Example 2. Consider the function ω(u) =

√u , |u|

ω(0) = 0 and the problem

u + λu = −ω(u)

in Ω.

(7.4)

This problem is variational and its associated functional is J (u) = |∇u|2 − λu2 − |u|3/2 dx, Ω

which is even, bounded below, takes negative values and attains its minimum on H01 (Ω), for each λ < λ1 . The same is valid for J+ (u) = J (u+ ) and J− (u) = J (u− ), whose minima are then a positive and a negative solutions of (7.4). In the context of nonlinear HJB operators we may consider max{u, 2u} + λu = −ω(u),

in Ω,

u = 0 on ∂Ω.

(7.5)

For this problem we have bifurcation from plus infinity to the left of λ1 and from minus infinity to the left of 2λ1 . These branches cannot reach the trivial solution set R × {0}, since bifurcation of positive or negative solutions from the trivial solution does not occur for (7.4). Exactly as in the proof of Theorem 1.3 (see the definition of the set A in the previous section) we can show that they contain only positive or negative solutions. Actually these branches are curves which can never turn, since positive and negative solutions of (7.5) are unique – this can be proved in the same way as Proposition 7.1 below. Example 3. Finally, let us look at an example of a sub-linear nonlinearity f which satisfies (F2) but f (x, 0) ≡ 0. For any HJB operator F satisfying our hypotheses consider F [u] + λu = f˜(u) :=

−u if |u| 1, −ω(u) if |u| 1.

(7.6)

In this situation we have positive (resp. negative) bifurcation from zero at λ = λ+ 1 − 1 (resp. + + − − − 1), more precisely, (λ − 1, kϕ ) and (λ − 1, kϕ ) are solutions for k ∈ [0, 1] (for λ = λ− 1 1 1 1 1

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more general results on bifurcation from zero see [13]). Further, note that there are only positive (resp. negative) solutions on these branches, as well on the branches which bifurcate from plus (resp. minus) infinity, given by Theorem 1.3. This is a simple consequence of the strong maximum principle and the fact that the right-hand side of (7.6) is positive (resp. negative) if u is negative (resp. positive), so if u () 0 and u vanishes at one point then u is identically zero. The bifurcation branches connect, as shown by the following uniqueness result. Proposition 7.1. If u and v are two solutions of (7.6) having the same sign and u > 1 or v > 1 then u ≡ v. If u 1 and v 1 then (by the simplicity of the eigenvalues) λ = λ± 1 −1 and u = v + kϕ1± for some k ∈ [0, 1]. Proof. Say u > 0, v > 0, v > 1. Set τ := sup{μ > 0 | u μv in Ω}. By Hopf’s lemma τ > 0 and we have u τ v. First, suppose τ < 1. By the definition of f˜ in (7.6) and v > 1 we easily see that f˜(u) f˜(τ v) τ f˜(v)

in Ω.

Hence (7.6) and the hypotheses on F imply 2 M− λ,Λ D (u − τ v) − γ D(u − τ v) − (γ + λ)(u − τ v) 0 and u − τ v 0 in Ω, so Hopf’s lemma implies u (τ + ε)v for some ε > 0, a contradiction with the definition of τ . Second, if τ 1 we repeat the above argument with u and v interchanged. This leaves u v and v u as the only case not excluded. 2 The following picture summarizes the above examples.

Acknowledgments P.F. was partially supported by Fondecyt Grant #1070314, FONDAP and BASAL-CMM projects and Ecos-Conicyt project C05E09. A.Q. was partially supported by Fondecyt Grant #1070264 and USM Grant #12.09.17 and Programa Basal, CMM, U. de Chile.


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Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes Nicholas Michalowski a , David J. Rule b , Wolfgang Staubach b,∗ a School of Mathematics and the Maxwell Institute of Mathematical Sciences, University of Edinburgh,

James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom b Department of Mathematics and the Maxwell Institute of Mathematical Sciences, Heriot–Watt University, Colin Maclaurin Building, Edinburgh, EH14 4AS, United Kingdom Received 24 October 2009; accepted 12 March 2010

Communicated by I. Rodnianski

Abstract We prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables. The result is sharp, even for smooth amplitudes. Nevertheless, in the case 1−ρ when the amplitude contains the oscillatory factor ξ → ei|ξ | , the result can be substantially improved. p We extend the L -boundedness of pseudo-pseudodifferential operators to certain weights. End-point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable. Our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation. © 2010 Elsevier Inc. All rights reserved. Keywords: Weighted norm inequality; Pseudodifferential operator; Pseudo-pseudodifferential operator; BMO commutator


E-mail addresses: [email protected] (N. Michalowski), [email protected] (D.J. Rule), [email protected] (W. Staubach). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.013

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1. Introduction Weighted norm inequalities have a long history in harmonic analysis and partial differential equations. Since the introduction of Ap weights by B. Muckenhoupt [21] (see Definition 2.4), there has been a large amount of activity establishing weighted Lp estimates with weights belonging to Ap . Such estimates have concerned singular integral operators of Calderón–Zygmund type, strongly singular integral operators (or singular integrals of Hirschman–Wainger type), maximal functions, standard pseudodifferential operators, and mildly regular pseudodifferential operators. However, there are still some gaps in the knowledge of weighted norm inequalities for pseudodifferential operators, and the aim of this paper is to fill in some of these. We do this first in the context of symbols introduced by C.E. Kenig and W. Staubach in [18] which are only measurable in the spatial variable. Our methods also extend to amplitudes and we obtain some sharp results in this direction. Most of these weighted boundedness results are new even in the smooth case. However, in the smooth case we can go on to consider end-point cases by using an interpolation technique. The results of this paper extend the applications derived in [18] and as a separate application we prove new boundedness results for the commutators of pseudodifferential operators with functions of bounded mean oscillation (written BMO, see Definition 4.1). Recall that for a function u ∈ C0∞ (Rn ) a pseudodifferential operator is an operator given formally by Ta u(x) :=

1 (2π)n

Rn

a(x, y, ξ )eix−y,ξ u(y) dy dξ,

(1.1)

Rn

whose amplitude (x, y, ξ ) → a(x, y, ξ ) is assumed to satisfy certain growth conditions. The most common class of amplitudes are those introduced by L. Hörmander in [15] and we refer the reader to Definitions 2.1–2.3 below for the specific amplitudes we will consider here and the standard notation we will use. By the weighted boundedness of a pseudodifferential operator Ta or a weighted norm inequality for a pseudodifferential operator we mean the existence of an estimate of the form Ta u Lpw C u Lpw ,

(1.2)

p

for some weight w and 1 p ∞. As usual Lw denotes the weighted Lp space with weight w: u Lpw =

u(x)p w(x) dx

1

p

.

Rn

All our results will concern w ∈ Aq (see Definition 2.4) for some q which may not always be equal to p. In the remainder of this section we summarize the results to follow. Section 2 sets out some definitions, fixes some notation and records some well-known results we will need later. In Section 3 our first main result is Theorem 3.3. It is a pointwise bound of operators in OPL∞ Sρm by a maximal function and corresponds to the Lp -boundedness of OPL∞ Sρm in the region C ∪ D of Fig. 1 obtained in [18]. Such an estimate immediately leads to weighted boundedness results of the form (1.2).


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Fig. 1. Regions of Lp -boundedness.

In Theorem 3.4 we go on to consider symbols of the form (x, ξ ) → ei|ξ | σ (x, ξ ), with σ ∈ L∞ S1m and m < n(ρ − 1)/2. We build on the methods developed by S. Chanillo and A. Torchinsky [7] to prove that operators corresponding to these symbols can be bounded p pointwise by certain maximal functions and consequently are bounded on Lw for w ∈ Ap and 1 < p < ∞. This estimate is obtained when m < n(ρ − 1)/2 and corresponds to the region B ∪ C ∪ D of Fig. 1. In Corollary 3.8 we observe the result can be generalized to amplitudes 1−ρ (x, y, ξ ) → ei|ξ | σ (x, y, ξ ), with σ ∈ L∞ Am 1 . In the case of symbols of the aforementioned form, if we make the additional assumption that the symbol is homogeneous in the ξ variables, we can extend the weighted boundedness result to the end-point value m = n(ρ − 1)/2. This is formulated as Theorem 3.6. These results are particularly interesting in connection to weighted norm inequalities for maximal functions associated to strongly singular integrals. In the case of ρ = 1, Theorem 3.6 yields the weighted version of an Lp -boundedness result due to R. Coifman and Y. Meyer [10]. The weighted boundedness of operators corresponding to symbols can be used to prove pointwise bounds by a maximal function for operators corresponding to amplitudes. This idea is first used to prove Theorem 3.7, where weighted boundedness results are shown for OPL∞ Am ρ in region D of Fig. 1. This is shown to be sharp. The amplitudes, and in particular the rough amplitudes, considered here are interesting because of the connection to the Weyl quantization of rough symbols, and the potential applications of the latter in semiclassical microlocal analysis. The Weyl quantization is when the amplitude takes the form (x, y, ξ ) → a((x + y)/2, ξ ). We can then use a fairly general method of interpolation, based on complex interpolation, to obtain end-point results for smooth amplitudes. The idea is to use properties of Ap -weights to enable us to interpolate between the weighted boundedness for values of m less than the end-point and the unweighted boundedness which is known for values of m greater than the endpoint. An example of this is Theorem 3.10, where we show weighted boundedness for operators n(ρ−1) with 0 < ρ 1 and 0 δ < 1. The interpolation technique can also, for example, in OPAρ,δ be used to prove weighted boundedness results for smooth symbols in the region A ∪ B ∪ C ∪ D of Fig. 1 when δ < ρ (see Theorem 3.11). In Section 4 we use the weighted norm inequalities to prove a variety of boundedness results for the commutators of pseudodifferential operators with BMO functions. When we have the p Lw -boundedness of pseudodifferential operators for all weights w ∈ Ap , we are actually able to show weighted boundedness of k-th commutators as a direct consequence of a general result due to J. Álvarez, R.J. Bagby, D.S. Kurtz and C. Pérez [1], see Section 4 for the definition of these commutators. For these results see Theorem 4.5. However, in Theorem 4.4, we also obtain 1−ρ

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unweighted Lp -boundedness of a commutator knowing only the weighted Lw -boundedness of the original operator for w in a much smaller class of weights. Theorems 4.4 and 4.5 extend the Lp boundedness of BMO commutators with OPS01,0 due to R. Coifman, R. Rochberg and G. Weiss [11], and the Lp boundedness of commutators of bmo functions with rough pseudodifferential operators arising from homogeneous symbols of order zero (the symbol class L∞ Scl0 ), proved by F. Chiarenza, M. Frasca and P. Longo [8]. The space of bmo functions is a localized version of the class BMO. To our knowledge, there are no results in the existing literature concerning the boundedness of BMO commutators of operators with rough amplitudes, even in the case of the Weyl quantization. We would like to thank Andrew Hassell for prompting us to consider the problem of weighted norm inequalities for pseudodifferential operators arising from amplitudes. 2. Definitions, notation and preliminaries First we introduce a standard Littlewood–Paley partition of unity {ϕk }k0 . Let ϕ0 : Rn → R be a smooth radial function which is equal to one on the unit ball centred at the origin and supported on its concentric double. Set ϕ(ξ ) = ϕ0 (ξ ) − ϕ0 (2ξ ) and ϕk (ξ ) = ϕ(2−k ξ ). Then ϕ0 (ξ ) +

∞

ϕk (ξ ) = 1 for all ξ ∈ R,

k=1

and supp(ϕk ) ⊂ {ξ | 2k−1 |ξ | 2k+1 } for k 1. One also has, for all multi-indices α and N 0, α ∂ ϕ0 (ξ ) cα,N ξ −N , ξ 1

where ξ := (1 + |ξ |2 ) 2 , and α ∂ ϕk (ξ ) cα 2−k|α| ξ

for some cα > 0 and all k 1.

(2.1)

We now fix some common notation and terminology. Definition 2.1. A function a : Rn × Rn × Rn → Rn is called an amplitude when it belongs to any one of the following sets. Let m ∈ R, ρ ∈ [0, 1] and δ ∈ [0, 1]. (a) We say a ∈ Am ρ,δ when for each triple of multi-indices α, β and γ there exists a constant Cα,β,γ such that α β γ ∂ ∂ ∂y a(x, y, ξ ) Cα,β,γ ξ m−ρ|α|+δ|β+γ | . ξ x

(b) We say a ∈ L∞ Am ρ when for each multi-index α there exists a constant Cα such that α ∂ a(·,·, ξ ) ξ

L∞ (Rn ×Rn )

Cα ξ m−ρ|α| .

Therefore, here we are only assuming measurability in the (x, y)-variables.


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Given an amplitude a we define the pseudodifferential operator Ta associated to a formally by (1.1), although it is not clear that the integral in (1.1) is well-defined. This can be made rigorous by considering the partial sums of the Littlewood–Paley pieces as follows. We define ak = aϕk . The integral in (1.1) converges absolutely when a is replaced with ak , so in this way we obtain a family of operators {Tak }k . The methods of proof we use will show in each case that the partial sums N

Tak

k=0

converge in operator norm as N → ∞. Consequently, we obtain our desired operator as Ta := limN →∞ N k=0 Tak . An important special case of these operators is when there is no dependency on the y-variable. It is convenient to have a slightly different terminology in this case. Definition 2.2. A function a : Rn × Rn → Rn is called a symbol when it belongs to any one of the following sets. Let m ∈ R, ρ ∈ [0, 1] and δ ∈ [0, 1]. m when for each pair of multi-indices α and β there exists a constant C (a) We say a ∈ Sρ,δ α,β such that

α β ∂ ∂ a(x, ξ ) Cα,β ξ m−ρ|α|+δ|β| . ξ x (b) We say a ∈ L∞ Sρm when for each multi-index α there exists a constant Cα such that α ∂ a(·, ξ ) ξ

L∞

Cα ξ m−ρ|α| .

Therefore, here we are only assuming measurability in the x-variable. m ⊂ Am , L∞ S m ⊂ L∞ Am , S m ⊂ L∞ S m and Am ⊂ L∞ Am . The Obviously, we have Sρ,δ ρ ρ ρ ρ ρ,δ ρ,δ ρ,δ amplitudes of (a) in both Definitions 2.1 and 2.2 were first introduced in [14,15]. If the amplitude is smooth and δ < ρ, then operators arising from amplitudes in Am ρ,δ can be rewritten as a sum m , see, for example, [23]. However, if δ ρ, then the of operators arising from symbols in Sρ,δ boundedness results are known to be different [15,19], as we will see played out in this paper. The symbols of (b) in Definition 2.2 were introduced in [18], and Definition 2.2(b) is the natural generalization of this to amplitudes. They are, for example, much rougher than those considered by S. Nishigaki [22] and K. Yabuta [25] in their investigations of weighted norm inequalities for pseudodifferential operators.

Definition 2.3. Given a class X of symbols or amplitudes, operators which arise from elements in X are denoted by OPX, that is, we say T ∈ OPX when there exists a ∈ X such that T = Ta , defined as in (1.1). When the amplitude of an operator is only measurable in the spatial variables, that is the operator belongs to OPL∞ Sρm or OPL∞ Am ρ , then following [18], we say it is a pseudopseudodifferential operator. It is well known that for standard pseudodifferential operators

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the singular support of the Schwartz kernel of the operator is contained in the diagonal —this is called the pseudo-local property. Singularities in the Schwartz kernel of pseudopseudodifferential operators will in general go beyond the diagonal, so they do not have the pseudo-local property. p Given u ∈ Lloc , the Lp maximal function Mp (u) is defined by

1 Mp (u)(x) = sup B x |B|

u(y)p dy

1

p

(2.2)

B

where the supremum is taken over balls B in Rn containing x. Clearly then, the Hardy– Littlewood maximal function is given by M(u) := M1 (u). An immediate consequence of Hölder’s inequality is that M(u)(x) Mp (u)(x) for p 1. We shall use the notation uB :=

1 |B|

u(y) dy

B

for the average of the function u over B. One can then define the class of Muckenhoupt Ap weights as follows. Definition 2.4. Let w ∈ L1loc be a positive function. One says that w ∈ A1 if there exists a constant C > 0 such that Mw(x) Cw(x),

for almost all x ∈ Rn .

(2.3)

One says that w ∈ Ap for p ∈ (1, ∞) if sup

B balls in Rn

− 1 p−1 wB w p−1 B < ∞.

(2.4)

The Ap constants of a weight w ∈ Ap are defined by [w]A1 :=

sup

wB w −1 L∞ (B)

(2.5)

sup

− 1 p−1 wB w p−1 B .

(2.6)

B balls in Rn

and [w]Ap :=

B balls in Rn

The following results are well known and can be found in, for example, [17,23].


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Theorem 2.5. Suppose p > 1 and w ∈ Ap . There exists an exponent q < p, which depends only on p and [w]Ap , such that w ∈ Aq . There exists ε > 0, which depends only on p and [w]Ap , such that w 1+ε ∈ Ap . q

Theorem 2.6. For 1 < q < ∞, the Hardy–Littlewood maximal operator is bounded on Lw if and p only if w ∈ Aq . Consequently, for 1 p < ∞, Mp is bounded on Lw if and only if w ∈ Aq/p . Theorem 2.7. Suppose that φ : Rn → R is integrable non-increasing and radial. Then, for u ∈ L1 , we have φ(y)u(x − y) dy φ L1 M(u)(x) for all x ∈ Rn . It is useful to record here results of J. Álvarez and J. Hounie [2], see also [15] and [16]. These will be used in Section 3. Theorem 2.8. Let 1 < q < ∞, 0 < ρ 1, 0 δ < 1 and suppose either: 1 1 (a) a ∈ Am ρ,δ and m n(ρ − 1)| q − 2 | + min{0, n(ρ − δ)}; or m and m n(ρ − 1)| 1 − 1 | + min{0, n(ρ − δ)/2}. (b) a ∈ Sρ,δ q 2

Then Ta is bounded on Lq . Part (a) is a consequence of remark (d) on page 11 of [2], together with straightforward adjustments of Theorem 2.2, Lemma 3.1 and Theorem 3.4 therein to the case of amplitudes. Part (b) is explicitly stated in [2] as Theorem 3.4 on page 13. As is common practice, we will denote constants which can be determined by know parameters in a given situation, but whose value is not crucial to the problem at hand, by C. Such parameters in this paper would be, for example, m, ρ, δ, p, n, [w]Ap , and the constants Cα,β,γ , Cα,β and Cα in Definitions 2.1 and 2.2. The value of C may differ from line to line, but in each instance could be estimated if necessary. We sometimes write a b as shorthand for a Cb. 3. Pseudodifferential operators and their weighted Lp boundedness The first fairly simple lemma yields a classical kernel estimate, which in particular implies the rapid decrease off the diagonal of the Schwartz kernel of pseudodifferential operators with certain amplitudes. Lemma 3.1. Let a ∈ L∞ Am ρ with m ∈ R and ρ ∈ [0, 1]. Let ak (x, y, ξ ) = a(x, y, ξ )ϕk (ξ ), for k 0 with ϕk as in the above Littlewood–Paley decomposition. Then, for each l 0, iz,ξ |z| ak (x, y, ξ )e dξ 2k(n+m−ρl) , l

for all x, y, z ∈ Rn .

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Proof. Using the definition of L∞ Am ρ , inequality (2.1) and the Leibniz rule we see that α ∂ ak (x, ξ ) cα 2k(m−ρ|α|) , ξ

for some cα > 0 and k = 1, 2, . . . ,

(3.1)

where we have also used the assumption ρ 1. First suppose l is an integer. For a multi-index α with |α| = l, integration by parts then yields α

iz,ξ α iz,ξ z e dξ (x, y, ξ )e dξ = a (x, y, ξ )∂ a k k ξ = ∂ξα ak (x, y, ξ )eiz,ξ dξ 2k(n+m−ρl) . Summing over all α with |α| = l proves this special case of the lemma. The general result of non-integer values of l follows by interpolation of the inequality for k and k + 1, where k < l < k + 1. 2 The kernel of operators of the form (1.1) is K(x, x − y) = Lemma 3.1 it is easy to conclude K(x, x − y) |x − y|−N

1 (2π)n

a(x, y, ξ )eix−y,ξ dξ . From

for N > 0, |x − y| 1,

(3.2)

provided either ρ > 0 and m ∈ R, or ρ = 0 and m < −n. It was shown in [18] that under certain conditions on the order m pseudo-pseudodifferential operators are bounded on Lp spaces. We record that result here. Theorem 3.2. Fix p ∈ [1, 2] and let a ∈ L∞ Sρm with 0 ρ 1 and m < pn (ρ − 1). Then Ta is a bounded operator on Lq for each q p. Now, our goal is to show that under the same assumptions, OPL∞ Sρm are also bounded on weighted Lp spaces. More precisely, we prove the following theorem. The proof uses a similar method to [18]. Theorem 3.3. Fix p ∈ [1, 2] and let a ∈ L∞ Sρm with 0 ρ 1 and m < pn (ρ − 1). Then there exists a constant C, depending only on n, p, m, ρ and a finite number of the constants Cα in Definition 2.1, such that Ta (u)(x) CMp u(x), q

for all x ∈ Rn . Consequently, Ta is a bounded operator on Lw for each q > p and w ∈ Aq/p . q

Proof. The Lw -boundedness follows immediately from the pointwise estimate, by Theorem 2.6. To prove the pointwise estimate we use the Littlewood–Paley partition of unity introduced in Section 2, we decompose the symbol as a(x, ξ ) = a0 (x, ξ ) +

∞ k=1

ak (x, ξ )


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with ak (x, ξ ) = a(x, ξ )ϕk (ξ ), k 0. First we consider the operator Ta0 . We have 1 Ta0 (u)(x) = (2π)n

a0 (x, ξ )e

ix−y,ξ

u(y) dy dξ =

K0 (x, y)u(x − y) dy,

with K0 (x, y) =

1 (2π)n

a0 (x, ξ )eiy,ξ dξ.

Lemma 3.1 gives us the estimate K0 (x, y) y−M , for each M > n. Theorem 2.7 yields Ta (u)(x) 0

y−M u(x − y) dy Mu(x) Mp u(x),

(3.3)

for all 1 p 2. 1 ak (x, ξ )u(ξ ˆ )eix,ξ dξ for k 1. We note, just as Now let us analyse Tak (u)(x) = (2π) n before, that Tak (u)(x) can be written as Tak (u)(x) =

Kk (x, y)u(x − y) dy

with Kk (x, y) =

1 (2π)n

ak (x, ξ )eiy,ξ dξ = aˇ k (x, y),

where aˇ k here denotes the inverse Fourier transform of ak (x, ξ ) with respect to ξ . One observes that p p Ta (u)(x)p = Kk (x, y)u(x − y) dy = Kk (x, y)σk (y) 1 u(x − y) dy , k σk (y) with weight functions σk (y) which will be chosen momentarily. Therefore, Hölder’s inequality yields Ta (u)(x)p k where

1 p

+

1 p

Kk (x, y)p σk (y)p dy

p p

|u(x − y)|p dy , |σk (y)|p

= 1. Now for an l > pn , we define σk by σk (y) =

2 2

−kρn p

|y| 2−kρ ,

,

−kρ( pn −l)

|y|l ,

|y| > 2−kρ .

(3.4)

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By Hausdorff–Young’s theorem and the estimate (3.1), first for α = 0 and then for |α| = l, we have 2

−kp ρn p

−kp ρn Kk (x, y)p dy 2 p

2

−kp ρn p

ak (x, ξ )p dξ

p p

p 2

pmk

p

dξ

2

kp (m− pn (ρ−1))

,

|ξ |∼2k

and 2

−kρp ( pn −l)

p

−kρp ( pn −l) Kk (x, y) |y|p l dy 2 2

−kρp ( pn −l)

l ∇ ak (x, ξ )p dξ

p p

ξ

p 2

kp(m−ρl)

p

dξ

2

kp (m− pn (ρ−1))

.

|ξ |∼2k

Hence, splitting the integral into |y| 2−kρ and |y| > 2−kρ yields

Kk (x, y)p σk (y)p dy

p p

n kp (m− n (ρ−1)) p p p = 2kp(m− p (ρ−1)) . 2

Furthermore, once again using Theorem 2.7, we have

p |u(x − y)|p dy Mp u(x) p |σk (y)|

with a constant that only depends on the dimension n. Thus (3.4) yields n

Ta u(x)p 2k(m− p (ρ−1)) Mp u(x) p . k

(3.5)

Summing in k using (3.3) and (3.5), we obtain ∞ ∞ p p p n

p k(m− (ρ−1)) Ta u(x) a0 (x, D)u(x) + ak (x, D)u(x) Mp u(x) p 2 1+ . k=1

k=1

We observe that the series above converges if m < pn (ρ − 1). This ends the proof of the theorem. 2 For symbols in L∞ Sρm which contain the oscillatory factor ξ → ei|ξ | , one can improve the p Lw -boundedness result of Theorem 3.3 to all weights w ∈ Ap for m < n2 (ρ − 1). 1−ρ


Theorem 3.4. Let σ ∈ L∞ S1m , with m < n2 (ρ − 1) and set a(x, ξ ) = ei|ξ | ρ < 1. For each 1 < p < ∞, one has the pointwise estimate

1−ρ

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σ (x, ξ ), with 0
(21−k dj )ν }. The number ν will be fixed depending only on n, m, ρ and p (as we shall see below) and will be such that ν < ρ. We consider −k −n

−k −n Ta (x,·) uk,j (x − y) dy Tak (x,·) (u)(x − y) dy = 2 dj 2 dj k 1 Qkj

Qkj

−n + 2−k dj

Ta

k,j

u2 (x − y) dy

Ta

k,j

u3 (x − y) dy

k (x,·)

Qkj

−n + 2−k dj

k (x,·)

Qkj

= J1 + J2 + J3 . We can estimate J1 using the L1 boundedness of multipliers of this form: Indeed, since |∂ξα ak (·, ξ )| 2k(m−ρ|α|) , following [18], if Kk (x, y) denotes the kernel of Tak (x,·) (recall again that x is fixed here) then Kk (x, y) dy = Kk (x, y) dy + Kk (x, y) dy := J1,1 + J1,2 . |y|2−kρ

|y|>2−kρ

To estimate J1,1 we use the Cauchy–Schwarz inequality and the kernel estimates prior to the proof of (3.5) for the case p = p = 2. Therefore


1

2

J1,1

dy

Kk (x, y)2 dy

1 2

4195

n

2k(m− 2 (ρ−1)) .

|y|2−kρ

To estimate J1,2 we use again the Cauchy–Schwarz inequality and once again the kernel estimates prior to the proof of (3.5) for the case p = p = 2. This yields

−2l

J1,2

|y|

1 2

dy

Kk (x, y)2 |y|2l dy

1 2

n

2k(m− 2 (ρ−1)) .

|y|>2−kρ

Thus, by Young’s inequality, the L1 norm of the multiplier Tak (x,·) has the size 2−εk where ε = n2 (ρ − 1) − m, which in turn yields

−n J1 2−k dj C2

−εk

Ta

k (x,·)

−k −n 2 dj

k,j

u1 (x − y) dy

k,j u (x − y) dy 1

C2−εk Mu(x). To estimate J3 we use Lemma 3.1 with l so large that n2 (1 − ρ) n(1 − ν) + l(ν − ρ) to obtain the estimate Kk (x, z) C2k(n+m−ρl) |z|−l

(3.10)

and so conclude, for any y ∈ Qkj ,

Ta (x,·) uk,j (x − y) = Kk (x, z)uk,j (x − y − z) dz k 3 3

C2k(n+m−ρl)

|z|−l u(x − y − z) dz

|y+z|>(21−k dj )ν

C2k(n+m−ρl)

|z|−l u(x − y − z) dz

|z|>(2ν −1)2−kν djν

(n−l)

C2k(n+m−ρl) 2ν − 1 2−kν djν Mu(x − y)

(n−l) Mu(x − y) C2k(n+m−ρl) 2−kν C2−εk Mu(x − y) since dj 1 and 21−k dj < 1/4, so |z| |y + z| − |y| > (21−k dj )ν − 2−k dj (2ν − 1)2−kν djν . Consequently, J3 C2−εk M(Mu)(x).

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For the case d < 1, it remains only to estimate J2 . We will use the arguments in Theorem 1.2 1−ρ of [7]. We observe that we may write ak (x, ξ ) = 2−εk bk (x, ξ ) where bk (x, ξ ) = ei|ξ | τk (x, ξ ) n

(ρ−1)

and τk ∈ L∞ S12 , with constants Cα independent of k. Suppressing the indices j and k, we γ k,j further partition the function u2 = γ0=2 gγ , with gγ supported in {y | 2γ −1 d < |y − x| 2γ d} and 2γ0 d ∼ d ν , where d = 21−k dj . We obtain, then, the estimate analogous to (3.1) in [7]: J2 C2

−εk

Mu(x) + 2

−εk

γ0

d

−n

γ =2

Sγ (gγ )(y) dy

Qkj

where Sγ is defined as follows. Let δ = n(1 + ρ)( p1 − 12 ) and set

−1/ρ

ξ ηγ (ξ ) = η (1 − ρ)/ 2γ d where η is radial, smooth and such that ⎧ 1/ρ ⎨ 0, if |ξ | < (1/8) , 1/ρ η(ξ ) = 1, if (1/7) < |ξ | < 301/ρ , ⎩ 0, if |ξ | > 401/ρ . Let θ be a smooth cut-off function which is equal to one near infinity and zero near the origin, then define Sγ as Sγ (u)(y) =

ei(y·ξ +|ξ |

1−ρ )

Rn

θ (ξ )τk (x, ξ ) n(1−ρ)/2 |ξ | ηγ (ξ )|ξ |δ+n(ρ−1)/2 u(ξ ˆ ) dξ. |ξ |δ

Following the reasoning of [7], we obtain for fixed x that Sγ (gγ )

Lp

−(δ+ n (ρ−1))/ρ

−(δ+ n (ρ−1))/ρ+n/p 2 2 C 2γ d gγ Lp C 2γ d Mp u(x).

Therefore, observing that −(δ + n2 (ρ − 1))/ρ + n/p = n/(ρp ), we have γ0 γ =2

d

−n

γ0 γ n/(ρp ) Sγ (gγ )(y) dy Cd −n/p Mp u(x) 2 d γ =2

Qkj

ν

= Cd n( ρ −1)/p Mp u(x). So we choose ν so that n(1 − ρν )/p = ε/2, and then ν

−ε/2

2−εk d n( ρ −1)/p = 2−εk/2 dj and

2−εk/2


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J2 C2−εk/2 Mp u(x). k,j

k,j

k,j

Now consider the case d 1/4. We decompose u = u1 + (u − u1 ), where u1 is defined as before. We have −k −n 2 dj

Ta

k (x,·) (u)(x

−n − y) dy = 2−k dj

Qkj

Ta

k (x,·)

Qkj

−n + 2−k dj

Ta

k,j

u1 (x − y) dy

k (x,·)

k,j

u − u1 (x − y) dy

Qkj

= L1 + L2 . The term L1 is identical to J1 , and in fact the same analysis works for d 1/4 as did for d < 1/4. For L2 , by (3.10) with l sufficiently large, we see that, for y ∈ Qkj , k,j

k,j

K = u − u (x − y) (x − y − z) dz (x, z) u − u k k (x,·) 1 1

Ta

C2

k(n+m−ρl)

|z|−l u(x − y − z) dz

|y+z|>22−k dj

C2

k(n+m−ρl)

|y + z|−l u(x − y − z) dz

|y+z|>22−k dj

C2k(n+m−ρl) Mu(x) C2−εk Mu(x), so, L2 C2−εk Mu(x). Collecting all these estimates together, we find ∞ Ta (u)(x) Ta (u)(x) k k=0

M(u)(x) +

αj (J1 + J2 + J3 ) +

d 0. For σ0 (x, ξ ) one has σ0 (x, ξ )ei|ξ | supports of σ0 and θ , therefore we have Ta2 u(x) =

fk (x)

n

=:

k

1−ρ

|ξ | 2 (ρ−1) ei|ξ |

k

ξ 1 − χ(ξ ) , |ξ |

fk (x)Uk (D)u(x).

1−ρ

θ (ξ )wk

θ (ξ ) = 0 because of the disjoint

ξ 1 − χ(ξ ) eix,ξ u(ξ ˆ ) dξ |ξ |


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Now it is clear that Uk (D)u(x) = (Sk (D) ◦ T (D)u)(x) with

Sk (D)u(x) :=

wk

ξ 1 − χ(ξ ) eix,ξ u(ξ ˆ ) dξ |ξ |

and T (D)u(x) :=

n

|ξ | 2 (ρ−1) ei|ξ |

1−ρ

θ (ξ )eix,ξ u(ξ ˆ ) dξ.

0 , a result of N. Miller [20] concerning weighted Lp boundNow since wk ( |ξξ | )(1 − χ(ξ )) ∈ S1,0 0 yields edness of operators with symbols in S1,0

Uk (D)u

p

Lw

T (D)uLp , w

where the constants of the inequality only depend on finite number of derivatives of wk ( |ξξ | )(1 − χ(ξ )), and are of polynomial growth in k. Furthermore, a result of S. Chanillo concerning weighted Lp boundedness of strongly singular integral operators of Hirschman–Wainger type [5], yields T (D)u Lpw u Lpw from which it follows that Uk (D)u Lpw u Lpw with polynomial bounds in k and hence summing in k and using the fact that fk L∞ CN k−N , for all N > 0 yields Ta2 u Lpw u Lpw , which concludes the proof of the theorem.

2

Now we turn to the problem of weighted boundedness of operators arising from amplitudes. To this end we have the following result, which is sharp modulo the end-point m = n(ρ − 1). Theorem 3.7. Suppose 0 ρ 1, m < n(ρ − 1) and a ∈ L∞ Am ρ , then, for each p > 1, we have Ta u(x) Mp u(x), and consequently Ta u Lpw u Lpw . Proof. The weighted norm inequality follows from the pointwise estimate by Theorems 2.5 and 2.6. 1 a(x, y, ξ )eiz,ξ dξ , then we have Let K(x, y, z) := (2π) n Ta u(x) = |x−y|1

K(x, y, x − y)u(y) dy +

K(x, y, x − y)u(y) dy = I + II.

|x−y|>1

However since m < n(ρ − 1), we also know from estimate (3.2) that |K(x, y, x − y)| C|x − y|−N for sufficiently large N and |x − y| 1. So II can be easily majorized by M(u)(x).

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As mentioned earlier we will again use the Littlewood–Paley partition of unity {ϕk }k just as we did in Theorem 3.4. Using that partition of unity and setting Kk (x, y, z) :=

1 (2π)n

ak (x, y, ξ )eiz,ξ dξ

yields I=

∞

Kk (x, y, x − y)u(y) dy =

k=0 |x−y|1

∞

Ik .

k=0

Now once again for k = 0 we observe that |K0 (x, y, x − y)| x − y−N for all N > 0, hence |I0 | Mu(x). If we consider an individual term with k 1, we have |Ik | =

Kk (x, y, x − y)u(y) dy

|x−y|1

=

r Kk (x, y, x − y)b(x − y)

|x−y|1

1 , u(y) dy |b(x − y)|r

where b and r are parameters to be chosen later. Therefore, Hölder’s inequality yields

Kk (x, y, x − y)p b(x − y)rp dy

|Ik |

1

p

|x−y|1

|x−y|1

|u(y)|p dy |b(x − y)|rp

1

p

.

By Theorem 2.7, for r < pn , we have

|x−y|1

|u(y)|p dy |b(x − y)|rp

1

p

Cb−r Mp u(x),

therefore |Ik | C

Kk (x, x − z, z)p |bz|rp dz

1 p

b−r Mp u(x).

(3.11)

Considering the remaining integral, setting σkx (z, ξ ) := ak (x, x − z, ξ ) and using (3.9) we have


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Kk (x, x − z, z) =

ak (x, x − z, ξ )eiz·ξ dξ

=

σkx (z, ξ )eiz·ξ dξ

= =

σkx (z, ξ ) φk−1 (ξ ) + φk (ξ ) + φk+1 (ξ ) eiz·ξ dξ σkx (z, ξ )ψˆ k (ξ )eiz·ξ dξ = Tσkx (ψk )(z).

Therefore, taking b = 2k ,

Kk (x, x − z, z)p |bz|rp dy

1 p

=

Tσ x (ψk )(z)p 2k zrp dz

1

k

p

.

Now we observe that since x is fixed, σkx belongs to the symbol class L∞ Sρm with semi-norms that

are uniform in x. Furthermore, the weight z → |z|rp is in Ap if and only if −n/p < r < n/p.

Now since p > 2, we may apply (3.5) with the p in that estimate taken equal to 1, and use Theorem 2.6 to obtain

Tσ x (ψk )(z)p 2k zrp dz

1 p

k

C2k(m−n(ρ−1)) C2

k(m−n(ρ−1))

M(ψk )(z)p 2k zrp dz ψk (z)p 2k zrp dz

= C2k(m−n(ρ−1)+n/p) .

1 p

1 p

(3.12)

Combining this with (3.11) we obtain |Ik | C2k(m−n(ρ−1)−(r−n/p)) Mp (u)(x). Therefore choosing r such that r − n/p = (m − n(ρ − 1))/2 and summing in k proves the theorem. 2 Theorem 3.7 is sharp in m up to the end-point n(ρ − 1). Indeed, suppose operators in p OPL∞ Am ρ were bounded on L for some p and m > n(ρ − 1). Since taking the adjoint of an operator in OPL∞ Am gives an operator which is also in OPL∞ Am ρ ρ , such operators would also

p be bounded on L . By interpolation they would be bounded on L2 . But this would contradict ∞ m Theorem 3 in [15] as Am ρ,1 ⊂ L Aρ . We also have a generalization of the result in Theorem 3.4 to the settings of amplitudes. Corollary 3.8. Let a(x, y, ξ ) = ei|ξ | σ (x, y, ξ ) with 0 < ρ 1 and assume that σ (x, y, ξ ) ∈ p n L∞ Am 1 with m < 2 (ρ − 1), then Ta is bounded on Lw , for all p ∈ (1, ∞) and w ∈ Ap . 1−ρ

4202


Proof. In the case ρ = 1, we are exactly in the situation of Theorem 3.7. For the case 0 < ρ < 1 we first observe that in the proof of Theorem 3.4 we have, in fact, proved for the Littlewood– Paley pieces that Ta (u)(x) 2−εk M(Mu)(x) + 2−εk/2 Mp u(x), k so Theorem 2.6 gives us the estimate Ta (u) p 2−εk/2 u p k Lw L w

(3.13)

for 1 < p < ∞ and w ∈ Ap . Now we may repeat the proof of Theorem 3.7, the bound (3.13) enabling us to prove an analogue of (3.12). We leave the details to the interested reader. 2 We now prove an interpolation result that will allow us to extend some of our previous results when we also assume the amplitudes are smooth. Lemma 3.9. Let 0 ρ 1, 0 δ 1, 1 < p < ∞ and m1 < m2 . Suppose that ∞ m1 1 (a) operators in OPAm ρ,δ (or OPL Aρ ) are bounded on Lw for a fixed w ∈ Ap , and ∞ m2 2 p (b) operators in OPAm ρ,δ (or OPL Aρ ) are bounded on L , p

where the bounds depend only on a finite number of Cα,β,γ (or Cα ) in Definition 2.1. Then, for p ∞ m ν each m ∈ (m1 , m2 ), operators in OPAm ρ,δ (or OPL Aρ ) are bounded on Lμ , where μ = w and ν=

m2 − m . m2 − m1

∞ m Proof. For a ∈ Am ρ,δ (or a ∈ L Aρ ) we introduce a family of symbols az (x, y, ξ ) := z ξ a(x, y, ξ ), where z ∈ Ω := {z ∈ C; m1 − m Re z m2 − m}. It is easy to see that, for |α + β| C1 with C1 large enough and z ∈ Ω,

α β γ

∂ ∂ ∂y az (x, y, ξ ) 1 + | Im z| C2 ξ Re z+m−ρ|α|+δ|β+γ | , ξ x for some C2 . (We only require β = γ = 0 if a ∈ L∞ Am ρ .) We introduce the operator m2 −m−z − m2 −m−z

Tz u := w p(m2 −m1 ) Taz w p(m2 −m1 ) u .

First we consider the case of p ∈ [1, 2]. In this case, Ap ⊂ A2 which in turn implies that both 1

−1

p

w p and w p belong to Lloc and therefore for z ∈ Ω, Tz is an analytic family of operators in the sense of Stein and Weiss [24]. Now we claim that for z1 ∈ C with Re z1 = m1 − m, the operator (1 + | Im z1 |)−C2 Taz1 p is bounded on Lw with bounds uniform in z1 . Indeed the amplitude of this operator is 1 −C ∞ m1 2 (1 + | Im z1 |) az1 (x, y, ξ ) which belongs to Am ρ,δ (or L Aρ ) with constants uniform in z1 . Thus, the claim follows from assumption (a).


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Consequently, we have − m2 −m−z1 p

pC

−C2 m2 −m−z1 p w p(m2 −m1 ) Taz1 w p(m2 −m1 ) u Lp Tz1 u Lp = 1 + | Im z1 | 2 1 + | Im z1 |

pC − m2 −m−z1 p 1 + | Im z1 | 2 w p(m2 −m1 ) uLp

w

pC p = 1 + | Im z1 | 2 u Lp , m2 −m−z1

where we have used the fact that |w (m2 −m1 ) u| = w.

m2 −m−z2

Similarly if z2 ∈ C with Re z2 = m2 − m, then |w (m2 −m1 ) u| = 1 and the amplitude of the oper2 ∞ m2 ator (1 + | Im z2 |)−C2 Taz2 belongs to Am ρ,δ (or L Aρ ) with constants uniform in z2 . Assumption (b) therefore implies that

pC p p Tz2 u Lp 1 + | Im z2 | 2 u Lp . Therefore the complex interpolation of operators in [4] implies that for z = 0 we have m2 −m − m2 −m p p p T0 u Lp = w p(m2 −m1 ) Ta w p(m2 −m1 ) u Lp C u Lp . Hence, setting v = w

m −m 2 1)

− p(m2 −m

u this reads Ta (v)p p C u p p , L L μ

μ

where μ = w ν and ν = (m2 − m)/(m2 − m1 ). This ends the proof in the case 1 p 2. p At this point we recall the fact that if a linear operator T is bounded on Lw , then its adjoint T ∗ p is bounded on L 1−p . Therefore, in the case p > 2, we apply the above proof to Ta ∗ , with w

p

p ∈ [1, 2) and v = w 1−p , which yields that Ta ∗ is bounded on Lv ν and since w ∈ Ap , we have p p p v ∈ Ap and so Ta is bounded on Lv (1−p)ν = L (1−p )(1−p)ν = Lμ , which concludes the proof of w the theorem. 2 For smooth amplitudes we can use Lemma 3.9 to show the following end-point versions of Theorems 3.7 and 3.3. n(ρ−1)

Theorem 3.10. If a ∈ Aρ,δ

with 0 < ρ 1, 0 δ < 1, then for all 1 < p < ∞ and all w ∈ Ap , p

n(ρ−1)/2

the corresponding pseudodifferential operator Ta is bounded on Lw . If a ∈ Sρ,δ with 0 < ρ < 1, 0 δ < 1, then for all 2 p < ∞ and all w ∈ Ap/2 , the corresponding pseudodifferential p operator Ta is bounded on Lw . Proof. We begin by proving the first statement for 0 < ρ < 1. By the Extrapolation Theorem of J. Rubio De Francia (see [12]) it is sufficient to show the boundedness of Ta on L2w spaces for each w ∈ A2 , with constants depending only on [w]A2 . Fix m2 such that n(ρ − 1) < m2 < min{0, n(ρ − δ)}. By Theorem 2.5, given w ∈ A2 we can find ε > 0 so that w 1+ε ∈ A2 . For

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this ε take m1 < n(ρ − 1) in such a way that the straight line L that joins points with coordinates (m1 , 1 + ε) and (m2 , 0), passes through the point (n(ρ − 1), 1). Clearly this is possible due to the fact that we can choose the point m1 as close as we like to n(ρ − 1). 1 2 By Theorem 3.7, OPAm ρ,δ are bounded operators on Lw 1+ε for w ∈ A2 and, by Theorem 2.8(a), n(ρ−1)

2 2 are bounded operators on L2w . OPAm ρ,δ are bounded on L . Therefore, by Lemma 3.9, OPAρ,δ 0 In the case a ∈ A1,δ , one just uses the fact that Ta = Ta0 + Ta1 , where Taj , j = 0, 1, are

−j (1−δ)

pseudodifferential operators belonging to OPS1,δ . Now a straightforward modification of the proof of the weighted boundedness of operators in OPS01,0 in [20], yields that Taj are both p bounded on Lw for w ∈ Ap , and therefore the same is true for Ta . The proof of the second statement is similar. By the extrapolation theorem of P. Auscher and J.M. Martell (see [3, Thm. 4.9]) it is sufficient to show the boundedness of Ta on L2w spaces for each w ∈ A1 , with constants depending only on [w]A1 . We now repeat the argument above with n(ρ − 1)/2 instead of n(ρ − 1), n(ρ − δ)/2 instead of n(ρ − δ), and Theorem 3.3 and Theorem 2.8(b) replacing Theorem 3.7 and 2.8(a), respectively. 2 In connection to Theorem 3.10, it should be mentioned that the second part is the extension of the result in [7] to the case of δ ρ. The first part of the theorem extends the weighted Lp n(ρ−1) boundedness of Ta ∈ OPSρ,δ proved in [2] for the range 0 < ρ 12 to the range 0 < ρ 1. However, for symbols we can extend the first part of the theorem to ρ = 0. Indeed, Theorem 3.7 m with m < −n. Furthermore yields the L2w1+ε boundedness of operators with symbols in S0,δ −n m by Theorem 3.2, operators with symbols in S0,δ with m < 2 are bounded on L2 . Hence, an p

n(ρ−1)

for interpolation procedure as above yields the boundedness in Lw of operators in OPSρ,δ 0 ρ < 1, 0 δ < 1. We also mention in passing that the first part of the above theorem yields in particular a sharp weighted boundedness result for the Weyl quantization of symbols. Also using Lemma 3.9, we can extend the range of m at the price of obtaining boundedness for fewer weights. Theorem 3.11. Let 1 < q < ∞, 0 < ρ < 1, 0 δ < 1 and suppose either: 1 1 (a) a ∈ Am ρ,δ and m < n(ρ − 1)| q − 2 | + min{0, n(ρ − δ)}; or m and m < n(ρ − 1)| 1 − 1 | + min{0, n(ρ − δ)/2}. (b) a ∈ Sρ,δ q 2

Then, for all w ∈ Aq , there exists α ∈ (0, 1), depending on m, ρ, δ, q and [w]Ap , such that, for q all ε ∈ [0, α], Ta is bounded on Lwε . Proof. If m < n(ρ −1), then, by Theorem 3.7, there is nothing to prove, so assume m > n(ρ −1). First assuming (a) we fix m2 such that m < m2 < n(ρ − 1)| q1 − 12 | + min{0, n(ρ − δ)} and m1 < p 1 n(ρ − 1). Then since OPAm ρ,δ are a bounded operators on Lw for all w ∈ Ap by Theorem 3.7, and m2 p OPAρ,δ are bounded on L , by Theorem 2.8(a). We see that all the assumptions of Lemma 3.9 are fulfilled and therefore we obtain the desired result. The proof under assumption (b) is the same, except we replace Theorem 2.8(a) by Theorem 2.8(b). 2


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4. Some applications in harmonic analysis In this section we show how our weighted norm inequalities can be used to derive the Lp boundedness of commutators of functions of bounded mean oscillation with a wide range of pseudodifferential operators. We start with the precise definition of a function of bounded mean oscillation. Definition 4.1. A locally integrable function u is of bounded mean oscillation if f BMO := sup B

1 |B|

f (x) − fB dx < ∞,

(4.1)

B

where the supremum is taken over all balls in Rn . We denote the set of such functions by BMO. For u ∈ BMO it is well known that er|u(x)| is locally integrable for r < 1. This is a consequence of the John–Nirenberg theorem, see, for example, [13, p. 524]. Furthermore, for all γ < 21n e , there exits a constant Cn,γ so that for u ∈ BMO and all balls B, 1 |B|

eγ |u(x)−uB |/ u BMO dx Cn,γ .

(4.2)

B

For this see [13, p. 528]. The following abstract lemma will enable us to prove the Lp boundedness of the BMO commutators of pseudodifferential operators. p

Lemma 4.2. For 1 < p < ∞, let T be a linear operator which is bounded on Lwα for all w ∈ Ap for some fixed α ∈ (0, 1]. Then given a function f ∈ BMO, if Φ(z) := ezf (x) T (e−zf (x) u)(x) × v(x) dx is holomorphic in a disc |z| < λ, then the commutator [f, T ] is bounded on Lp . Proof. Without loss of generality we can assume that f BMO = 1. We take u and v in C0∞ with u Lp 1 and v Lp 1, and an application of Hölder’s inequality to the holomorphic function Φ(z) together with the assumption on v yield Φ(z)p

p ep Re zf (x) T e−zf (x) u dx.

Our first goal is to show that the function Φ(z) defined above is bounded on a disc with centre at the origin and sufficiently small radius. At this point we recall a lemma due to Chanillo [6] which states that if f BMO = 1, then for 2 < s < ∞, there is an r0 depending on s such that for all r ∈ [−r0 , r0 ], erf (x) ∈ A 2s . Taking s = 2p in Chanillo’s lemma, we see that there is some r1 depending on p such that for |r| < r1 , erf (x) ∈ Ap . Then we claim that if R := min(λ, αrp1 ) and |z| < R then |Φ(z)| 1. Indeed since R
0 sufficiently small, it follows from Definition 2.4 ε that w ∈ Aq/p when w ∈ Aq and 0 ε α. Therefore, from Lemma 4.2, we conclude that [f, a(x, D)]u Lp u Lp . 2 Theorem 4.5. Suppose either: (a) a ∈ L∞ Am ρ with m < n(ρ − 1) and 0 ρ 1; or (b) a(x, y, ξ ) = ei|ξ |

1−ρ

n σ (x, y, ξ ) and σ ∈ L∞ Am 1 with 0 < ρ 1 and m < 2 (ρ − 1); or

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N. Michalowski et al. / Journal of Functional Analysis 258 (2010) 4183–4209 n(ρ−1)

(c) a ∈ Aρ,δ

with 0 δ < 1 and 0 < ρ 1; or

(d) a(x, ξ ) = ei|ξ |

1−ρ

n

σ (x, ξ ) and σ ∈ L∞ Scl2

(ρ−1)

with 0 < ρ 1.

Then, for f ∈ BMO and k a positive integer, the k-th commutator defined by

k

Ta,f,k u(x) := Ta f (x) − f (·) u (x) q

is bounded on Lw for each w ∈ Aq and q ∈ (1, ∞). Proof. The claims follow from Theorem 2.13 in [1], and Theorem 3.7, Corollary 3.8, Theorem 3.10 and Theorem 3.6, respectively. Once again, part (c) of the theorem establishes in particular the boundedness of the commutator of a BMO function and the Weyl quantization of symbols. 2 References [1] Josefina Álvarez, Richard J. Bagby, Douglas S. Kurtz, Carlos Pérez, Weighted estimates for commutators of linear operators (English summary), Studia Math. 104 (2) (1993) 195–209. [2] J. Álvarez, J. Hounie, Estimates for the kernel and continuity properties of pseudo-differential operators, Ark. Mat. 28 (1) (1990) 1–22. [3] J. Álvarez, M. Milman, Vector valued inequalities for strongly singular Calderón–Zygmund operators, Rev. Mat. Iberoamericana 2 (4) (1986) 405–426. [4] C. Bennett, R.C. Sharpley, Interpolation of Operators, Pure Appl. Math., vol. 129, Academic Press, 1988. [5] S. Chanillo, Weighted norm inequalities for strongly singular integral operators, Trans. Amer. Math. Soc. 281 (1) (1984) 77–107. [6] S. Chanillo, Remarks on commutators of pseudo-differential operators, in: Multidimensional Complex Analysis and Partial Differential Equations, São Carlos, 1995, in: Contemp. Math., vol. 205, Amer. Math. Soc., Providence, RI, 1997, pp. 33–37. [7] S. Chanillo, A. Torchinsky, Sharp function and weighted Lp estimates for a class of pseudodifferential operators, Ark. Mat. 24 (1) (1986) 1–25. [8] F. Chiarenza, M. Frasca, P. Longo, Interior W 2,p estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1) (1991) 149–168. [9] R. Coifman, Y. Meyer, Au dela des operateurs pseudo-différentiels, Asterisque, vol. 57, Societe Mathematique de Paris, France, 1978. [10] R. Coifman, Y. Meyer, Wavelets. Calderón–Zygmund and multilinear operators, translated from the 1990 and 1991 French originals by David Salinger, Cambridge Stud. Adv. Math., vol. 48, Cambridge University Press, Cambridge, 1997. [11] R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (3) (1976) 611–635. [12] J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities and related topics, in: Notas de Matemática [Mathematical Notes], 104, in: North-Holland Math. Stud., vol. 116, North-Holland Publishing Co., Amsterdam, 1985. [13] Loukas Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. [14] L. Hörmander, Pseudo-differential operators and hypoelliptic equations, in: Singular Integrals, Chicago, IL, 1966, in: Proc. Sympos. Pure Math., vol. X, 1967, pp. 138–183. [15] L. Hörmander, On the L2 continuity of pseudo-differential operators, Comm. Pure Appl. Math. 24 (1971) 529–535. [16] J. Hounie, On the L2 continuity of pseudo-differential operators, Comm. Partial Differential Equations 11 (1986) 765–778. [17] J.-L. Journe, Calderón–Zygmund Operators, Pseudodifferential Operators and the Cauchy Integral of Calderón, Lecture Notes in Math., vol. 994, Springer-Verlag, Berlin, 1983. [18] C.E. Kenig, W. Staubach, Ψ -pseudodifferential operators and estimates for maximal oscillatory integrals, Studia Math. 183 (3) (2007) 249–258.


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[19] D.S. Kurtz, R.L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979) 343–362. [20] N. Miller, Weighted Sobolev spaces and pseudodifferential operators with smooth symbols, Trans. Amer. Math. Soc. 269 (1) (1982) 91–109. [21] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (March 1972) 207–226. [22] S.-I. Nishigaki, Weighted norm inequalities for certain pseudodifferential operators, Tokyo J. Math. 7 (1) (1984) 129–140. [23] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. [24] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser., vol. 32, Princeton University Press, Princeton, NJ, 1971, x+297 pp. [25] K. Yabuta, Weighted norm inequalities for pseudodifferential operators, Osaka J. Math. 23 (3) (1986) 703–723.


Classification of generalized multiresolution analyses ✩ Lawrence W. Baggett a , Veronika Furst b , Kathy D. Merrill c,∗ , Judith A. Packer a a Department of Mathematics, University of Colorado, Boulder, CO 80309, USA b Department of Mathematics, Fort Lewis College, Durango, CO 81301, USA c Department of Mathematics, Colorado College, Colorado Springs, CO 80903, USA

Received 2 November 2009; accepted 1 December 2009 Available online 6 January 2010 Communicated by N. Kalton

Abstract We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions m and matrix-valued filter functions H . Given a natural number valued function m and a system of functions encoded in a matrix H satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function m and filter system H . An equivalence relation on GMRAs is defined and described in terms of their associated pairs (m, H ). This classification system is applied to MRAs and other classical examples in L2 (Rd ) as well as to previously studied abstract examples. © 2009 Elsevier Inc. All rights reserved. Keywords: Wavelet; Generalized multiresolution analysis; Filter; Ruelle operator

1. Introduction A generalized multiresolution analysis (GMRA) is a Hilbert space structure traditionally associated with classical wavelets, that is, functions whose dilates of translates provide an orthonormal basis for L2 (Rd ). Given a wavelet, the nested sequence of subspaces Vj that result ✩

This research was supported by the National Science Foundation through grant DMS-0701913.


E-mail addresses: [email protected] (L.W. Baggett), [email protected] (V. Furst), [email protected] (K.D. Merrill), [email protected] (J.A. Packer). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.001

L.W. Baggett et al. / Journal of Functional Analysis 258 (2010) 4210–4228

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from taking only dilation powers less than j are dense and have trivial intersection, with Vj +1 the dilate of Vj , and with V0 invariant under translation. Such a structure is called a GMRA [6], and was developed to understand such wavelets as the famous example given by Journé, whose V0 space does not have an orthonormal basis given by translates of a single function called a scaling function. When V0 has this stronger property, the nested sequence {Vj } is called a multiresolution analysis (MRA) [22,23]. Both MRAs and GMRAs have been extensively exploited to produce and understand wavelets, which in turn have proven useful for applications such as image and signal processing. While wavelets and multiresolution structures were first studied in the Hilbert space L2 (Rd ), analogous definitions make sense in other Hilbert spaces that have appropriate dilation and translation operators. Dutkay and Jorgensen [16] pioneered the study of wavelets in function spaces on fractals, with later work by D’Andrea et al. [14]. Larsen, Raeburn and coworkers then showed that these and other interesting examples can be constructed via direct limits [19,4,5]. Dutkay et al. [9,17] constructed MRAs and super-wavelets in Hilbert spaces formed by direct sums of L2 (Rd ) to orthonormalize examples such as the Cohen wavelet. Tensor products of known examples lead to more exotic specimens (see Section 5). Our purpose in this paper is to construct a set of classifying parameters for GMRAs in order to unify and allow comparison of all these disparate examples. We also provide an explicit construction of a canonical GMRA equivalent to each of them. Accordingly, we will consider GMRA structures in an abstract Hilbert space H, equipped with “translations” given by a unitary representation π of a countable abelian group Γ acting in H, and a “dilation” given by a unitary operator δ. We assume that these operators are related by δ −1 πγ δ = πα(γ )

(1)

for all γ ∈ Γ , where αis an isomorphism of Γ into itself such that the index of α(Γ ) in Γ equals N > 1, and such that α n (Γ ) = {0}. These definitions generalize the classical case of ordinary translation by the integer lattice in L2 (Rd ), given √ by πn f (x) = f (x − n), and dilation by an expansive integer matrix A, given by δf (x) = |det A|f (Ax). The structure of a GMRA, and thus the parameters that uniquely identify it, are revealed via Stone’s Theorem on unitary representations of abelian groups. Using this theorem, we know that the representation π restricted to V0 is completely determined by a measure μ on the dual group Γ and a Borel multiplicity function m : Γ → {0, 1, 2, . . . , ∞}, which essentially describes a unitary equivhow many times each character occurs in the decomposition of π|V0 . There is L2 (σi ), where alence J between the action of π on V0 and multiplication by characters on σi = {ω: m(ω) i}. Because of this, we think of J as a partial alternative Fourier transform. For simplicity, in this paper we will restrict our attention to the commonly studied case where μ is Haar measure, and m is finite a.e. The multiplicity function m is one of the parameters that determine a GMRA. As we will see in Section 4, the other parameter is a “filter” that shows how the operator J interacts with dilation. Classical filters were periodic functions h and g in L2 (Rd ) that described inverse dilates of Fourier transforms of bases of V1 in terms of those of V0 . Starting with an MRA in L2 (Rd ), such functions could be shown to satisfy certain orthogonality relations. Mallat, Meyer and Daubechies [22,23,15] turned this process around by using functions h and g satisfying orthogonality together with additional low-pass and non-vanishing conditions to construct MRAs and wavelets. Lawton [20] and Bratteli and Jorgensen [12] were able to relax the non-vanishing

4212


condition by allowing Parseval frames in place of orthonormal bases, and Baggett, Courter, Jorgensen, Merrill, Packer [1,3] generalized this work to the GMRA setting by replacing h and g by matrix-valued functions H and G. In [11], Bratteli and Jorgensen related filters h and g to Ruelle operators Sh and Sg , which satisfy relations similar to those of Cuntz operators, and can be used to represent inverse dilations. This work was extended to generalized filters in [3] and later [4]. In the next section, we recall the relationship between abstract GMRAs, multiplicity functions and generalized filters. In particular, we describe conditions on a multiplicity function m and a filter H that guarantee that they will produce a GMRA. It turns out that these conditions are considerably more relaxed in an abstract Hilbert space than in L2 (Rd ). In Section 3 we describe a construction procedure that produces an abstract GMRA from any m and H meeting the required conditions. This construction gives an explicit realization of the abstract direct limit GMRAs built in [4]. While the procedure relies on first choosing a filter G complementary to H , we show in Section 4 that the equivalence between GMRAs does not depend on the choice of G. Thus, the classifying set described there depends only on the pair m and H . In this section, we also give a necessary and sufficient condition on the equivalence class of a filter so that SH is a pure isometry and thus associated with a GMRA, in the case of a finite multiplicity function. We conclude in Section 5 with a variety of examples that illustrate our main theorems, including an example of a GMRA where the translation group is not isomorphic to Zd . 2. GMRAs, multiplicity functions and filters Let H be an abstract, separable Hilbert space, equipped with operators πγ and δ satisfying Eq. (1). Definition 1. A collection {Vj }∞ −∞ of closed subspaces of H is called a generalized multiresolution analysis (GMRA) relative to π and δ if (1) (2) (3) (4)

Vj ⊆ Vj +1 for all j. all j. V j +1 = δ(Vj ) for Vj = {0}, and Vj is dense in H. V0 is invariant under the representation π.

The subspace V0 is called the core subspace of the GMRA {Vj }. For each j , write Wj for the orthogonal comLet {Vj } be a GMRA in a Hilbert space H. plement to Vj in Vj +1 . It follows that H = ∞ j =−∞ Wj . Also, for each j 0, Wj is an invariant subspace for the representation π. We apply Stone’s Theorem on unitary representations of abelian groups to the subrepresentations of π acting in V0 and W0 . Accordingly, there exists a finite, Borel measure μ (unique up to equivalence of measures) on Γ, Borel subsets σ1 ⊇ σ2 ⊇ · · · of Γ(unique up to sets of μ measure 0), and a (not necessarily unique) unitary operator J : V0 → i L2 (σi , μ) satisfying J πγ (f ) (ω) = ω(γ ) J (f ) (ω)


4213

for all γ

∈ Γ , all f ∈ V0 , and μ almost all ω ∈ Γ. We write m for the function on Γ given by m(ω) = i χσi (ω), and call it the multiplicity function associated to the representation π|V0 . The GMRA {Vj } is an MRA if and only if m ≡ 1. Analogously, there exists a finite, Borel measure μ, Borel subsets σk , and an operator J: W0 → k L2 ( σk , μ) satisfying J πγ (f ) (ω) = ω(γ ) J(f ) (ω) for all γ ∈ Γ , f ∈ W0 , and μ almost all ω. We write m for the function on Γ given by m (ω) =

χ (ω), and call it the multiplicity function associated to the representation π| . σk W0 k In this paper, we will assume that the measures μ and μ are absolutely continuous with respect to Haar measure, and thus take μ and μ to be the restrictions of Haar measure to the subsets σ1 and σ1 , respectively. We also assume that the multiplicity function m associated to the representation π|V0 is finite almost everywhere. Let α ∗ be the dual endomorphism of Γ onto itself defined by [α ∗ (ω)](γ ) = ω(α(γ )), and note that the kernel of α ∗ contains exactly N elements and that α ∗ is ergodic with respect to the Haar measure μ on Γ. Indeed, suppose that for some γ ∈ Γ and some n > 0, we have γ ◦ (α ∗ )n = γ . Since the characters of Γ separate the points of Γ , it follows that α n γ = γ , and hence γ ∈ α n Γ = {0}. That is, the trivial character is the only γ ∈ Γ that satisfies γ ◦ (α ∗ )n = γ for some n > 0, and it follows that any function f ∈ L2 (Γ) that satisfies f ◦ α ∗ = f must be a constant function. The last statement is equivalent to the definition of the ergodicity of α ∗ . Using α ∗ to relate the representations π|V1 and π|V0 , it is shown in [6] and more generally in [4] that multiplicity functions for a GMRA must satisfy the following consistency equation:

m(ω) + m (ω) =

m(ζ ).

(2)

α ∗ (ζ )=ω

It follows, since the function m is finite a.e., that the sets σi and σk are completely determined by the multiplicity function m. It also follows that a multiplicity function m associated with a GMRA must satisfy the consistency inequality: m(ω)

m(ζ ).

(3)

α ∗ (ζ )=ω

We will see in the next section that the consistency inequality is a sufficient as well as necessary condition for a function m : Γ → {0, 1, 2, . . .} to be a multiplicity function associated to an abstract GMRA. Accordingly, we make the following definition. Definition 2. A multiplicity function is a Borel function m : Γ → {0, 1, 2, . . .} that satisfies the consistency inequality (3). In contrast, Bownik, Rzeszotnik and Speegle [10] and Baggett and Merrill [7] showed that an additional technical condition related to dilates of the translates of the support of m is required for m to be a multiplicity function for a GMRA in L2 (Rd ). We will need the following observation about multiplicity functions.

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Proposition 3. Suppose m : Γ → {0, 1, 2, . . .} satisfies the consistency inequality. If m is not identically 0, then there exists a set F of positive measure in Γ such that

m(ω)
k. Then, for almost every ω ∈ α ∗ −n (E), we have

m(ωζ ) = m α ∗ n (ω) k,

ζ ∈ker(α ∗n )

implying that there exists some ζ ∈ ker(α ∗ n ), and a subset E ⊆ α ∗ −n (E) of positive measure, such that m(ωζ ) = 0 for all ω ∈ E . Hence, m(ω) = 0 on a set F of positive measure. But, from the equation m α ∗ (ω) = m(ωζ ), α ∗ (ζ )=1

it follows that the sequence {m(α ∗ n (ω))} is nondecreasing. Because α ∗ is ergodic, we must have that the sequence {α ∗ n (ω)} intersects the set F infinitely often for almost all ω. Hence m(ω) = 0 a.e. 2 The other ingredients we will need for our GMRA construction are filters, which are defined in terms of a multiplicity function m as follows: Definition 4. Let m be a multiplicity function, and write σi = {ω: m(ω) i}. Set m (ω) =

m(ζ ) − m(ω),

α ∗ (ζ )=ω

and set σk = {ω: m (ω) k}. Let H = [hi,j ] and G = [gk,j ] be (possibly infinite) matrices of Borel, complex-valued functions on Γ such that for every j , hi,j and gk,j are supported in σj . Suppose further that H and G satisfy the following “filter equations”:


α ∗ (ζ )=ω

4215

hi,j (ζ )hi ,j (ζ ) = N δi,i χσi (ω),

(4)

gk,j (ζ )gk ,j (ζ ) = N δk,k χ σk (ω),

(5)

j

α ∗ (ζ )=ω j

and

gk,j (ζ )hi,j (ζ ) = 0.

(6)

α ∗ (ζ )=ω j

Then H is called a filter relative to m and α ∗ , and G is called a complementary filter to H . We note that it will sometimes be useful to consider filters and complementary filters to be matrix valued functions on Γ rather than a matrix of complex valued functions. It is then a consequence of the definition above that the nonzero portion of the matrix H (ω) is contained in the upper left block of dimensions m(α ∗ (ω)) × m(ω), while the nonzero portion of G(ω) is contained in the upper left block of dimensions m (α ∗ (ω)) × m(ω). Given a filter H relative to m and α ∗ , we may define a “Ruelle” operator SH on i L2 (σi ) by SH (f ) (ω) = H t (ω)f α ∗ (ω) . Similarly, a complementary filter G defines a Ruelle operator SG from by

i

L2 ( σi ) to

i

L2 (σi )

SG (f ) (ω) = Gt (ω)f α ∗ (ω) . The filter equations satisfied by H and G translate to the following Cuntz-like conditions for the Ruelle operators (see [3,4]): Lemma 5. If H is a filter relative to m and α ∗ , and G is a complementary filter to H , then the Ruelle operators they define satisfy ∗ S = I , S∗ S = I , (1) SH H G G ∗ (2) SH SG = 0, and ∗ + S S∗ = I , (3) SH SH G G

where I is the identity operator on

i

L2 (σi , μ) and Iis the identity operator on

kL

2 ( σk , μ).

Filters, like multiplicity functions, arise naturally out of GMRAs. Let {Vj } be a GMRA (with finite multiplicity function and associated measure absolutely continuous with respect to Haar), μ, { σk }, and J be as in the Stone’s Theorem discussion above. Write Ci for and let μ, {σi }, J , the element of the direct sum space j L2 (σj , μ) whose ith coordinate is χσi and whose other 2 k for the element in coordinates are 0, and C σl , μ) whose kth coordinate σk and l L ( is χ whose other coordinates are 0. Let j hi,j be the element J (δ −1 (J −1 (Ci ))) and j gk,j be the k ))), both in j L2 (σj , μ). It was shown in [2] that the matrix H = [hi,j ] element J (δ −1 (J−1 (C

4216


is then a filter relative to m and α ∗ , and the matrix G = [gk,j ] is a complementary filter to H . We call these filters constructed from a GMRA, and note that they are not unique, but rather depend on the choice of the maps J and J. The operators J ◦ δ −1 ◦ J −1 and J ◦ δ −1 ◦ J−1 are the corresponding Ruelle operators SH and SG respectively. It follows directly from their definitions that SH and SG are isometries, and the GMRA requirement that Vj = {0} implies that SH = J ◦ δ −1 ◦ J −1 is a pure isometry. Just as with multiplicity functions, this necessary condition on a filter to be associated with a GMRA turns out to be sufficient 2 as well. In Theorem 5 of [4], it is shown that if SH is a pure isometry on a Hilbert space L (σi ), then it is possible to construct a generalized multiresolution analysis via a direct limit process. Our construction in the next section will give a concrete realization under the same hypotheses. Again, as with multiplicity functions, we see that this necessary and sufficient condition on the filter H is much weaker than what is required for a filter to be associated with a GMRA in L2 (Rd ). For example, in that context, the “refinement equation”, (ω) = √ φ

−1 t −1 1 A ω , H At ω φ |det A|

suggests some sort of convergence of the infinite product

∞

j =1

√ 1 H ((At )−j ω), |det A| √

(7) which in

turn requires that the filter H satisfies some low-pass condition of being close to |det A| times a partial identity near the origin [3,4]. Theorems from [2,5] indicate that in the abstract setting, a much weaker condition is sufficient to guarantee that SH is a pure isometry. In the case where the matrix H is 1 × 1, the simple condition that |H (ω)| = 1 on a set of positive measure is sufficient to show that SH is a pure isometry [11,5]. In particular, filters traditionally labeled “high-pass” can be used as H . Proposition 19 in Section 4 of this paper gives a new, more general result of this type. 3. Explicit construction of GMRAs on abstract Hilbert spaces Let m be a multiplicity function on Γ, as in Definition 2 and let H be a filter relative to m and α ∗ . Using Proposition 3, define m (ω) =

m(ζ ) − m(ω),

(8)

α ∗ (ζ )=ω

σk } as in the preceding section. As is shown in [1], given a filter and define the sets {σi } and { H relative to m and α ∗ , there always exists a complementary filter G. For the purposes of this construction, let G be any filter complementary to H . a J map guaranteed by Stone’s Theorem takes the core subspace V0 of any GMRA to Because L2 (σi ), this direct sum of L2 spaces is a natural candidate for the core subspace of an abstract GMRA built out of a multiplicity function and a filter. The group Γ acts on this space in a natural way via multiplication by characters. Similarly, the space L2 ( σk ) is an obvious candidate for −1 the abstract W0 = V1 V0 , and the relationships J ◦ δ ◦ J −1 = SH and J ◦ δ −1 ◦ J−1 = SG suggest that the Ruelle operators SH and SG provide natural abstract inverse dilations on these spaces. Thus the main task remaining in building a GMRA given m and H is to describe positive dilates of W0 ; such subspaces could then be used to fill out the rest of the Hilbert space.


4217

If the group Γ = Zd , then embedding Γ = Td as [− 12 , 12 ]d in Rd provides us with a sim 2 L ( σk ). Since in this case, α is an ple candidate for the dilate of our constructed W0 = d , we must have α(n) = An, for a matrix A. We define positive dilations isomorphism of Z σk ) → L2 (At σk ) by Dj : L2 ( D

j

k

−j 1 fk (ω) = f At ω . √ j k |det A| k

For more general Γ , we will use an abstract construction to define the positive dilation D in terms of a cross section for the map α ∗ . Just as in√the case of Γ = Zd , our dilated space will be a direct sum of L2 spaces such that the map f → N f ◦ α ∗ determines an isometry of the dilated space onto the original one. Let c be a regular Borel section for the map α ∗ ; i.e., c is a Borel map from Γ ≡ α ∗ (Γ) ≡ Γ / ker(α ∗ ) into Γ, for which α ∗ (c(ω)) = ω for all ω ∈ Γ. (See, e.g. [21, Lemma 1.1].) Define τ : Γ → ker(α ∗ ) by τ (ω) = c α ∗ (ω) ω−1 . For example, in the simple case associated with dilation by 2 in L2 (R), where Γ = T ≡ [− 12 , 12 ), with α ∗ (ω) = 2ω, we can take c(ω) = ω2 . Thus, here τ (ω) =

0

if ω ∈ [− 14 , 14 ),

1 2

otherwise.

(9)

Now, let ν be a finite Borel measure on Γ. Let E be a Borel subset of Γ, let ζ be an element of the kernel of α ∗ , and set Eζ = ω ∈ E: τ (ω) = ζ . Proposition 6. The set E is the disjoint union

ζ

Eζ , and α ∗ is 1–1 on each Eζ into Γ.

Proof. The first statement is clear. For ω ∈ Eζ we have c α ∗ (ω) = τ (ω)ω = ζ ω, which shows that α ∗ must be 1–1 on Eζ .

2

Now let E1 , E2 , . . . be a (countable) collection of Borel subsets of Γ. For each i let νi be the of νi to the restriction to Ei of the measure ν. Write Ei,ζ for [Ei ]ζ , and let νi,ζ be the restriction = α ∗ (E ), and subset Ei,ζ of Ei . Write K = i L2 (Ei , νi ). For each ζ ∈ ker(α ∗ ), define Ei,ζ i,ζ equal to the measure N α ∗ (ν ) that is defined on E by set νi,ζ ∗ i,ζ i,ζ νi,ζ (F ) = N νi,ζ α ∗ −1 (F ) .

4218


For example, in the simple setting described by Eq. (9), if we take E1 = [− 38 , 38 ) and E2 = [ 18 , 12 ), then 1 3 1 1 3 1 E1,0 = − , , E1, 1 = − , − ∪ , , 2 4 4 8 4 4 8 1 1 1 1 E2,0 = , , E2, 1 = , , (10) 2 8 4 4 2 so that E1,0

1 1 1 1 1 1 , E1, 1 = − , − ∪ , , = − , 2 2 2 4 4 2 2 1 1 1 E2,0 , E2, , 0 . = , = − 1 4 2 2 2

here is just Haar measure restricted to E . The measure νi,ζ i,ζ Using our newly defined sets and measures, we let

K =

. L2 Ei,ζ , νi,ζ

i,ζ

Proposition 7. For each f ∈ K, set D(f ) equal to the element of K given by 1 D(f ) i,ζ (ω) = √ fi ζ −1 c(ω) . N Then the operator D is an isometry of K onto K . Proof. D(f )2 = ζ

i

=

Ei,ζ

ζ

Ei,ζ

fi ζ −1 c α ∗ (η) 2 dνi,ζ (η) i

=

D(f ) (ω)2 dν (ω) i,ζ i,ζ

2 1 −1 fi ζ c(ω) dνi,ζ (ω) N i

=

ζ E i,ζ

fi ζ −1 τ (η)η 2 dνi,ζ (η) i

ζ E i,ζ

fi (η)2 dνi,ζ (η) = i

ζ E i,ζ


4219

fi (η)2 dνi (η) = i E i

= f 2 , where the second to last step is justified because, for η ∈ Ei,ζ , we have τ (η) = ζ . Thus, D is an isometry. To see that D is onto K , it suffices to note that the inverse of D is given by √ −1 D (f ) i (ω) = N fi,τ (ω) α ∗ (ω) .

2

We will refer to the space K = D(K) as a dilation by α ∗ of K. Note that this general definition of D is consistent with the definition given at the beginning of this section for the special case of Γ = Zd . For example, in the situation defined by Eq. (10), we can see this equivalence by using integer translation in R to identify L2 ([− 12 , 12 )) ⊕ L2 ([− 12 , − 14 ) ∪ [ 14 , 12 )) with L2 ([− 34 , 34 )) and L2 ([ 14 , 12 )) ⊕ L2 ([− 12 , 0)) with L2 ([ 14 , 1)). We are now ready to construct explicitly a GMRA from the parameters m, H , and G. Theorem 8. Suppose m : Γ → {0, 1, 2, . . .} is a Borel function that satisfies the consistency ∗ inequality, and that H = [h i,j ] is a filter relative to m and α such that the Ruelle operator 2 SH is a pure isometry on be defined from m by the consistency equation i L (σi ). Let m 2 (as in Eq. (8)), and let G be a complementary filter to H . Define V = 0 i L (σi ) and W0 = 2 ∞ σk ). For n 1, inductively set Wn = D(Wn−1 ), and set H = V0 ⊕ n=0 Wn . Define a k L ( representation π of Γ , acting in H, by πγ (f ) (ω) = ω(γ )f (ω). Finally, define an operator T on H by SH (fV0 ) + SG (fW0 ), T (f ) a = D−1 (fWn+1 ),

a = V0 , a = Wn , n 0,

(11)

where we represent an element f of H by {fV0 , fW0 , fW1 , . . .}. Then (1) T is a unitary operator on H. (2) T πγ T −1 = πα(γ ) for all γ ∈ Γ . (3) If Vj is defined to be T −j (V0 ), then the collection {Vj } is a GMRA relative to π and δ, where δ = T −1 . (4) The multiplicity function associated to the core subspace V0 is the given function m, and the given H is a filter constructed from the GMRA {Vj }. Proof. To prove the first claim, note that by Proposition 7, D−1 is an isometry from Wn+1 onto Wn . By Lemma 5, we also have that the definition of T above gives an isometry from V0 ⊕ W0 onto V0 . The second claim follows immediately from the definitions, since the operators SH , SG and D−1 all change the argument of the function from ω to α ∗ (ω).

4220


Next, we show that the collection {Vj } is a GMRA. The fact that Vj ⊆ Vj +1 follows from T (V0 ) ⊂ V0 , which is immediate from the definition of T . That Vj +1 = δ(Vj ) follows immediately from the definition of δ = T −1 . The trivial intersection property follows from our assumption that SH is a pure isometry, and the dense union from the fact noted in the previous paragraph that T −1 V0 = V0 ⊕ W0 and T −1 Wn = Wn+1 . As a component of the direct sum space, V0 is clearly invariant under the multiplication operators ω(γ ) that define the representation π . The given function m is clearly the multiplicity function of that representation. To establish that H is a corresponding filter, we note that we can take J to be the identity for this V0 , and calculate J δ −1 J −1 (Ci )(ω) = T (Ci )(ω) = SH (Ci )(ω) = hi,j (ω)χσi α ∗ (ω) j

=

hi,j (ω),

j

where the last equality follows from the fact that by the filter equation, hi,j is supported on α ∗ −1 (σi ). 2 Remark 9. We will denote the GMRA {Vj } constructed above by {Vjm,H,G } and refer to it as the canonical GMRA having these parameters. In Section 5 we will construct canonical GMRAs related to classical examples, as well as new ones. First, we establish in the next section conditions under which two GMRAs are the same. While our construction procedure requires the choice of a complementary filter G, we will see that the equivalence classes depend only on the two parameters m and H . 4. A classifying set for GMRAs Let {Vj } be a GMRA in a Hilbert space H, relative to a representation π of Γ and a unitary operator δ, and let {Vj } be a GMRA in a Hilbert space H , relative to a representation π of Γ and a unitary operator δ . Definition 10. We say that the GMRAs {Vj } and {Vj } are equivalent if there exists a unitary operator U : H → H that satisfies: (1) U (Vj ) = Vj for all j. (2) U ◦ πγ = πγ ◦ U for all γ ∈ Γ. (3) U ◦ δ = δ ◦ U. For classical examples in L2 (Rd ), the Fourier transform F gives an equivalence between any j }. Further, if an operator U gives an equivalence between {Vj } and {V }, GMRA {Vj } and {V j = F ◦ U ◦ F −1 is two GMRAs for dilation by A and translation by Zd in L2 (Rd ), then U multiplication by a function u with absolute value 1, and such that u(A∗j ω) = u(ω) for all


4221

integers j [8]. Thus equivalence between GMRAs for the same dilation in L2 (Rd ) generalizes the notion of different MSF wavelets attached to the same wavelet set. Recall that we consider only GMRAs with a finite multiplicity function m and with the associated measure μ absolutely continuous with respect to Haar measure. Our first aim is to prove that every such GMRA is equivalent to one of the canonical GMRAs constructed in the preceding section. We will then describe the equivalence relation among these GMRAs in terms of the parameters m, H and G. We will need the following lemma. Lemma 11. The GMRAs {Vj } and {Vj } are equivalent if and only if there exists a unitary operator P mapping V0 onto V0 that satisfies: (1) P ◦ πγ = πγ ◦ P for all γ ∈ Γ. (2) P ◦ δ −1 = δ −1 ◦ P .

Proof. We first assume that the conditions above are satisfied and show that {Vj } and {Vj } are equivalent. For each n 0, define an operator Qn : Wn → Wn by Qn = δ

n+1

◦ P ◦ δ −(n+1) .

∞ Now, define U = P ⊕ ∞ n=0 Qn on H = V0 ⊕ n=0 Wn . One checks directly that U satisfies the required conditions. For the converse, assume that {Vj } and {Vj } are equivalent, with U : H → H implementing the equivalence. Define P = U |V0 . By the definition of equivalence, P maps V0 to V0 , and conditions (1) and (2) follow. 2 Theorem 12. Let {Vj } be a GMRA. Let m be its (finite) associated multiplicity function, and let H = [hi,j ] be a filter constructed from the GMRA using the map J . Let G be a complementary filter to H . Then the GMRA {Vj } is equivalent to the canonical GMRA {Vjm,H,G }. Proof. Define P : V0 → L2 (σi ) by P = J . Condition (1) of Lemma 11 follows immediately. The fact that J ◦ δ −1 ◦ J −1 = SH proves the second condition of that lemma. 2

Theorem 13. The canonical GMRAs {Vjm,H,G } and {Vjm ,H ,G } are equivalent if and only if m = m , and there exists a matrix-valued function A on Γ such that (1) A(ω) = A10(ω) 00 , where A1 (ω) is a unitary matrix of dimension m(ω). (2) H (ω)At (ω) = At (α ∗ (ω))H (ω). Proof. Suppose first that m = m and that there exists a matrix-valued function A satisfying the conditions. Let τr be the subset of Γ on which m(ω) = m (ω) = r. Then both subspaces V0m,H,G

and V0m ,H

,G

are equal to i

L2 (σi ) ≡

r

L2 τ r , C r .

4222


Define P : V0m,H,G → V0m ,H

,G

by [P (f )](ω) = A(ω)f (ω). It follows directly that P satisfies

the conditions of Lemma 11, and hence {Vjm,H,G } and {Vjm ,H ,G } are equivalent. Conversely, suppose an operator P exists and satisfies the conditions of Lemma 11. The first condition on P implies that the two representations of Γ on V0m,H,G and V0m ,H ,G are unitarily equivalent, whence m must equal m , and V0m,H,G = V0m ,H ,G = i L2 (σi ) = r L2 (τr , Cr ). It is known (e.g. [8]) that any unitary operator P on the direct sum of vector-valued L2 spaces that commutes with all the multiplication operators γ (ω), is itself a multiplication operator of the form P (f ) (ω) = A(ω)f (ω), where A(ω) = A10(ω) 00 , and A1 (ω) is a unitary matrix whose dimension is r = m(ω) for ω ∈ τr . The second condition of Lemma 11 then implies that A satisfies condition (2) of the theorem. 2 Corollary 14. Let m be a multiplicity function and let H be a filter relative to m and α ∗ for which SH is a pure isometry. If G and G are any two complementary filters to H , then the GMRAs {Vjm,H,G } and {Vjm,H,G } are equivalent. The preceding theorem introduces a notion of equivalence among filters that we will use to build a set of classifying parameters for the equivalence classes of GMRAs. In the following definition, we use our knowledge of the form of A to rewrite the equivalence using the conjugate transpose A∗ . ∗ Definition 15. Let m be a multiplicity function. Filters H and H relative to m and α are called A1 (ω) 0 equivalent if there exists a matrix-valued function A on Γ , with A(ω) = , where A1 (ω) 0 0 is a unitary matrix of dimension m(ω), and such that

H (ω) = A α ∗ (ω) H (ω)A∗ (ω) for almost all ω ∈ Γ. Remark 16. If H and H are two filters constructed from the same GMRA using different Stone’s Theorem operators J and J , then H and H are equivalent according to this definition. Here the matrix-valued function A comes from the multiplication operator J J −1 . Lemma 17. Let H be a filter relative to m and α ∗ , and let A be a matrix-valued function of the form described in the preceding theorem. Define the matrix-valued function H by H (ω) = A α ∗ (ω) H (ω)A∗ (ω). Then H is a filter relative to m and α ∗ , i.e., H satisfies the filter equation. Proof. We note that if we write H1 (ω) for the upper left m(α ∗ (ω)) × m(ω) block of H (ω), and let Λ(ω) be N times the m(ω) × m(ω) identity, then the filter equation (4) can be rewritten as


H1 (ζ )H1∗ (ζ ) = Λ(ω).

4223

(12)

α ∗ (ζ )=ω

We must show that if H satisfies Eq. (12), then so does H . We have α ∗ (ζ )=ω

∗

H1 (ζ )H 1 (ζ ) =

A1 (ω)H1 (ζ )A∗1 (ζ )A1 (ζ )H1∗ (ζ )A∗1 (ω)

α ∗ (ζ )=ω

= A1 (ω)Λ(ω)A∗1 (ω) = Λ(ω).

2

Let H be a filter relative to m and α ∗ . In [2] it was shown that the operator SH fails to be a pure isometry if and only if it has an eigenvector, i.e., if and only if there exists an element F ∈ L2 (σi ) and a complex number λ for which H t (ω)F (α ∗ (ω)) = λF (ω), where |λ| = 1 = F (ω) for almost all ω. Motivated by this result, we make the following definition: Definition 18. A filter H is called an eigenfilter if there exists a constant λ with |λ| = 1 such that for almost all ω, H1,1 (ω) = λ and H1,j (ω) = 0 for j > 1. Using this definition, we have the following restatement of the result from [2]: Proposition 19. SH fails to be a pure isometry if and only if H is equivalent to an eigenfilter. Proof. If there exists a matrix-valued function A such that H (ω)A(ω) = A α ∗ (ω) H (ω), where H (ω) is an eigenfilter, then, computing the first rows of both sides, we see that the first row of A is the desired eigenvector F . Conversely, if SH has an eigenvector F , build a unitary-valued matrix A(ω) having F (ω) as its first row. Set H (ω) = A(α ∗ (ω))H (ω)A∗ (ω). By the previous lemma, H is a filter relative = λ. Because H is a filter, it follows that the to m and α ∗ . Moreover, one can see that H1,1 elements H1,j (ω) are all 0 for j > 1. Hence, H has the desired form. 2 Now, let S be the set of all pairs (m, H ), where m is a multiplicity function and H is a filter relative to m and α ∗ . Let S0 be the subset of S comprising those pairs (m, H ) for which H is equivalent to an eigenfilter, and let S1 = S \ S0 . Finally let E = S1 / ≡ be the set of equivalence classes of S1 with respect to the equivalence relation (m1 , H1 ) ≡ (m2 , H2 ) if m1 = m2 and H1 is equivalent to H2 . Theorem 20. The set E is a classifying set for the equivalence classes of GMRAs (with finite multiplicity functions and associated measures absolutely continuous with respect to Haar measure), in the sense that there is a 1–1 correspondence between E and the classes of GMRAs, and this correspondence can be described explicitly.

4224


Proof. Given an element s ∈ E, let (m, H ) be a representative of the equivalence class s. Let G be a filter complementary to H , and define κ(s) to be the equivalence class of the GMRA {V m,H,G }. By Theorem 13, the map κ is both well defined and one-to-one, and by Theorem 12, it is onto. 2 5. Examples We will now use the technique outlined in Section 3 to construct examples of canonical GMRAs, and apply the ideas of Section 4 to discuss their equivalence. We work first in the classical setting of MRAs (so m ≡ 1) with single wavelets (so m ≡ 1) for dilation by 2 in L2 (R). Since m is determined for MRAs, their equivalence depends only on the filter H . Example 21. Any MRA for dilation by 2 in L2 (R) with m = m ≡ 1 has canonical Hilbert space L (T) ⊕ L (T) ⊕ 2

2

∞

∞ m,H,G j m,H,G m,H,G L 2 T = V0 ⊕ W0 ⊕ Wj 2

j =1

(13)

j =1

with πn ( fl ) = en · fl , where en (x) = e2πinx , and δ

−1

fV0 ⊕ fW0 ⊕

∞

fWj

√ (ω) = h(ω)fV0 (2ω) + g(ω)fW0 (2ω) ⊕ 2fW1 (2ω)

j =1

⊕

∞ √ 2fWj (2ω) .

(14)

j =2

Equivalence for two different MRAs with single wavelets is equivalent to the existence of a period 1 function a such that |a(ω)| = 1 and h (ω) = a(2ω)h(ω)a(ω), where h and h are filters constructed from the two MRAs. Thus, in particular, equivalence requires that |h| = |h |. However, this is not sufficient, as we will see below. Determining which filters give equivalent MRAs requires determining exactly which functions on the 1-torus are coboundaries where cohomological equivalence is given by Definition 15. √ √ 0 = L2 ([− 1 , 1 ]), we have h = 2χ 1 1 and g = 2χ 1 1 For the Shannon MRA, with V [− , ] ±[ , ] 2 2 4 4

4 2

in the above formula. By mapping Wjm,H,G → L2 (±2j [ 12 , 1]), we can map this canonical GMRA to the Fourier transform of the Shannon GMRA. For the Haar MRA, with V0 spanned by translates of χ[0,1] , we have h = √1 (1 + e−1 ), g = 2

− 1) in the above formula. Here there is no obvious mapping between the canonical GMRA and either the original or its Fourier transform. However, we know all three are equivalent by Theorem 12. In either the Shannon or Haar examples, we can switch the roles of h and g to get a new MRA that cannot be realized in L2 (R) (since iterating the refinement equation (7) leads to a scaling function that must be identically 0). The canonical Hilbert space will still be given by Eq. (13), and the operators πn will be as above. However, in the dilation formula (14), we will now have h given by the old g, and g by the old h. Proposition 19 shows that we still have Sh a pure isometry, m,h,g m,h,g so that the canonical construction does produce a GMRA. Looking at the V−j and W−j that √1 (e−1 2


4225

result in the case of the reversed Shannon GMRA shows how this example differs from Shannon MRA itself: 1 1 1 1 3 1 2 2 − , → L ± , → L ± , → · · · , δ =L 2 2 4 2 8 2 1 1 1 1 1 3 m,h,g δ −1 : W0 → L2 − , → L2 ± , → · · · . = L2 − , 2 2 4 4 4 8 −1

m,h,g : V0

2

Since the absolute values of the filters in the three examples discussed here are all different on sets of positive measure, the three are seen to be inequivalent MRAs. To see that for MRAs with wavelets, the filters having equal absolute value almost everywhere is not sufficient for equivalence, consider the MRA built from h = −h, where h is the filter for the Haar example. A simple Fourier analysis argument shows that there is no solution to h (ω) = a(2ω)h(ω)a(ω), so this MRA must be inequivalent to the Haar MRA. We note that it has the same canonical Hilbert space as Haar, and the same subspaces Vj , but its dilation on V0 is the negative of the Haar dilation. This negative sign causes problems in the iteration of the refinement equation, so this example cannot be realized in L2 (R). A fourth example in this setting begins with the Cohen filters h = √1 (1 + e−3 ) and g = 2 √1 (1 − e−3 ). The infinite product construction which follows from the refinement equation 2 in L2 (R) yields the functions φ = 13 χ[0,3) and ψ = 13 (χ[0, 3 ) − χ[ 3 ,3) ), which fail to be an or2

2

thonormal scaling function and orthonormal wavelet, respectively, since neither has orthonormal translates. However, it can be shown that the negative dilate space (for dilation by 2) of the Cohen Parseval wavelet coincides with that of the Haar orthonormal wavelet. Hence, the Cohen GMRA equals the Haar MRA. We may apply Theorem 8 to the Cohen filters and multiplicity functions m ≡ 1, m ≡ 1. The canonical Hilbert space will be that given by Eq. (13), on which the integers act by multiplication by exponentials. We see that the spaces Vj for the canonical Cohen GMRA are the same as those for the canonical Haar MRA whenever j 0. However, since √1 (1 + e−3 ) and √1 (1 + e−1 ) 2 2 have different moduli, the two filters must be inequivalent. Therefore the two canonical GMRAs must be inequivalent. Lastly, we remark that while the Cohen wavelet ψ = 13 (χ[0, 3 ) − χ[ 3 ,3) ) is only a Parseval 2

2

wavelet in L2 (R), the element (0, χ[− 1 , 1 ) , 0, 0, 0, . . .) is an orthonormal wavelet for the canon2 2 ical Hilbert space (13), with respect to πn and δ defined by Eq. (14) using the Cohen filters. Dutkay et al. [9,17] also produced an orthonormal wavelet from the Cohen filter, using a “superwavelet” construction. The associated GMRA in L2 (R) ⊕ L2 (R) ⊕ L2 (R) can be seen to be equivalent to our canonical Cohen GMRA by defining the map P in Lemma 11 in the natural way to take the nth translate of the Dutkay scaling function to e2πinx in the canonical V0 . Of course, this example cannot be realized in L2 (R). Next, we consider two non-MRA examples for dilation by 2 in L2 (R): the Journé GMRA, and the example for the Journé multiplicity function with low-pass filter of rank a = 2 described in [4, Example 13]. As is noted there, a GMRA cannot be constructed for this second example using the infinite product construction. However, by Proposition 19, the construction of this paper can be carried out to give such a GMRA.

4226


Example 22. Let m be the multiplicity function corresponding to the Journé wavelet: ⎧ ⎪ ⎨2 m(x) = 1 ⎪ ⎩ 0

if x ∈ [− 17 , 17 ), if x ∈ ±[ 17 , 27 ) ∪ ±[ 37 , 12 ), otherwise,

so σ1 = [− 12 , − 37 ] ∪ [− 27 , 27 ] ∪ [ 37 , 12 ] and σ2 = [− 17 , 17 ]. Since we know the Journé GMRA has an associated single orthonormal wavelet, m ≡ 1. Filters that give rise to the Journé wavelet via the infinite product construction are described in [13,1,3]. In particular, we may take √ 2χ[− 2 ,− 1 ]∪[− 1 , 1 )∪[ 1 , 2 ] 7 7 4 7 √ 7 4 H= 2χ[− 1 ,− 3 ]∪[ 3 , 1 ] 2

7

0

0

7 2

and G=

√

2χ[− 1 ,− 1 ]∪[ 1 , 1 ] 4

7

7 4

√

2χ[− 1 , 1 ] . 7 7

Here V0m,H,G = L2 (σ1 ) ⊕ L2 (σ2 ), and Wjm,H,G = L2 (2j T), j 0. This canonical GMRA can be mapped to the usual Journé GMRA by integrally translating σ1 and σ2 to the scaling set to form V0 , and T ≡ [− 12 , 12 ] to the wavelet set to form W0 . In [4], an alternative filter H for the same multiplicity function, but which satisfies the lowpass condition of rank a = 2 is constructed: h1,1 =

√

h1,2 = h2,1 = 0,

2χ[− 2 ,− 1 )∪[− 1 , 1 )∪[ 1 , 2 ) , 7

4

7 7

4 7

and h2,2 =

√ 2χ[− 1 , 1 ) . 14 14

By partitioning R/Z as described in [13,1], we can build the following complementary filter G : g1,1 =

√ 2χ[− 1 ,− 3 )∪[− 1 ,− 1 )∪[ 1 , 1 )∪[ 3 , 1 ) , 2

7

4

7

7 4

7 2

g1,2 =

√ 2χ[− 1 ,− 1 )∪[ 1 , 1 ) . 7

14

14 7

The spaces V0m,H ,G and Wjm,H ,G for j 0 are the same as those for the canonical GMRA corresponding to the standard Journé filters. However, the different filters in the rank 2 exam ple will change the dilation, and thus change the spaces Vjm,H ,G and Wjm,H ,G for j < 0. m,H For example, we have V−1

,G

1 1 = L2 ([− 17 , 17 ] ∪ ±[ 14 , 27 ]) ⊕ L2 ([− 14 , 14 ]), while the standard

m,H,G m,H ,G = L2 ([− 17 , 17 ] ∪ ±[ 14 , 27 ] ∪ ±[− 37 , 12 ]) ⊕ 0. The fact that all the V−j allow nonzero V−1 second components with support overlapping that of the first component suggests the impossibility of mapping the rank 2 example into L2 (R) as we mapped the standard example. Indeed, since iterating the refinement equation would lead to a scaling function with a degenerate multiplicity function [4], the rank 2 example cannot be realized in L2 (R). Thus, these two examples must not be equivalent.

For our next example, we consider dilation by 3, both in L2 (R) and in the Dutkay/Jorgensen enlarged Cantor fractal space [16].


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Example 23. The MRA Haar 2-wavelet for dilation by 3 in L2 (R) has canonical Hilbert space L (T) ⊕ L2 (T) ⊕ L2 (T) ⊕

2

∞

j 2 j L 3 T ⊕L 3 T . 2

j =1

−j The canonical δ −1 = Sh ⊕ (Sg1 ⊕ Sg2 ) ⊕ ( ∞ j =1 D ), where 1 h = √ (1 + e1 + e2 ), 3

1 g1 = √ (e1 − e2 ) 2

1 and g2 = √ (−2 + e1 + e2 ), 6

√ j and D−j (f1 ⊕ f2 )(ω) = 3 (f1 ⊕ f2 )(3j ω). The Cantor set MRA has the same canonical GMRA except with 1 h = √ (1 + e2 ), 2

g1 = e1

1 and g2 = √ (1 − e2 ). 2

These two examples must be inequivalent since √ their h’s have different absolute values. The latter cannot be realized in L2 (R), since h(0) = 2, so that the iterated refinement equation (7) would again force the scaling function to be identically 0. Our final example uses a group Γ different from Zd . ∞ Example 24. Let Γj = ∞ i=j [Z2 ]i = i=j {1, −1}i , embedded as a subgroup of D = Γ−∞ = ∞ j −1 ∞ i=−∞ [Z2 ]i by Γj = i=j [Z2 ]i . Let α be defined on Γ0 by α(γ )n = γn−1 for i=−∞ {1}i ⊕ n > 0 and α(γ )0 = 1. Let H = l 2 (D), and let π be the restriction to Γ0 of the regular representation of D. Define S on D by [S(d)]n = dn−1 , and note that S(γ ) ≡ α(γ ) for γ ∈ Γ0 . Define δ f (S(d)), and note that δ −1 πγ δ = πα(γ ) . on H = l 2 (D) by [δ(f )](d) = We have Γj +1 ⊆ Γj , and ∞ j =−∞ Γj = {eD }, where eD = (. . . , 1, 1, 1, . . .) denotes the 2 multiplicative identity element of D = Γ−∞ , so that l 2 (Γj +1 ) ⊆ l 2 (Γj ) and ∞ j =−∞ l (Γj ) = l 2 ({eD }). If we let Vj = l 2 (Γ−j ), then {Vj } is almost a GMRA. It fails only because constant multiples of the function χ{eD } belong to Vj . We will make it into a GMRA by tensoring it with the dilation by 2 Haar GMRA. It is known (as in [18]) that the tensor product of two GMRAs gives a GMRA. By tensoring our almost GMRA with an actual one, we will preserve all the properties of the almost GMRA, and eliminate the non-trivial intersection. 2 √ Accordingly, let Γ = Z act in H = L (R) by πn f (x) = f (x − n), and let δ f (x) = 2f (2x). We have α acting on Γ by α (n) = 2n. Write {Vj } for the usual Haar GMRA that results from taking V0 to be the closed linear span of translates of χ[0,1] . Set H = H ⊗ H , equipped with the representation π × π of Γ = Γ0 × Γ and the operator δ ⊗ δ . We let ∗ α = α × α and note that α ∗ acts on Γ = ∞ i=0 [Z2 ]i × T by α ((ω0 , ω1 , ω2 , . . .) × x) = 1 1 (ω1 , ω2 , . . .) × 2x, where we parameterize T by [− 2 , 2 ). We have N = 4, and ker(α ∗ ) =

1 ({−1, 1} × ∞ i=1 [{1}]i ) × {0, 2 }. To build the dilation described in Section 3, we can take the ≡ 3, so cross section c((ω0 , ω 1 , ω2 , . . .) × x) = (1, ω0 , ω1 , . . .) × x2 . We have m ≡ 1 and m σ1 = σ2 = σ3 = ∞ σ1 = i=0 [Z2 ]i × T.

∞ √ We define our filter H = h1 ⊗ h2 , where h1 is the filter on i=0 [Z2 ]i given by h1 = 2χ{1}0 × ∞ , and h2 is the low-pass filter for the Haar GMRA described in Example 21, i=1 [Z2 ]i

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√

√1 (1+e−1 ). For our filter complementary to H , we define g1 = 2χ{−1} × ∞ [Z ] 0 i=1 2 i 2 and let g2 be the high-pass filter for the Haar GMRA, g2 = √1 (e−1 − 1). We then take our com2 plementary filter G to be the matrix whose rows are h1 ⊗ g2 , g1 ⊗ h2 , and g1 ⊗ g2 . − χ{−1}0 × ∞ and For an alternative GMRA, we can replace h1 by h1 = χ{1}0 × ∞ i=1 [Z2 ]i i=1 [Z2 ]i

∞ + χ . These could be viewed as more fractal-like when g1 by g1 = −χ{1}0 × ∞ {−1}0 × i=1 [Z2 ]i i=1 [Z2 ]i

that is, h2 =

combined with the h2 and g2 in the standard tensor product construction. References [1] L.W. Baggett, J.E. Courter, K.D. Merrill, The construction of wavelets from generalized conjugate mirror filters in L2 (Rn ), Appl. Comput. Harmon. Anal. 13 (2002) 201–223. [2] L.W. Baggett, V. Furst, K.D. Merrill, J.A. Packer, Generalized filters, the low-pass condition, and connections to multiresolution analysis, J. Funct. Anal. 257 (2009) 2760–2779. [3] L.W. Baggett, P.E.T. Jorgensen, K.D. Merrill, J.A. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys. 46 (2005), #083502, 28 pp. [4] L.W. Baggett, N.S. Larsen, K.D. Merrill, J.A. Packer, I. Raeburn, Generalized multiresolution analyses with given multiplicity functions, J. Fourier Anal. Appl. 15 (2009) 616–633. [5] L.W. Baggett, N.S. Larsen, J.A. Packer, I. Raeburn, A. Ramsay, Direct limits, multiresolution analyses, and wavelets, J. Funct. Anal. 258 (8) (2010) 2714–2738. [6] L.W. Baggett, H.A. Medina, K.D. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rn , J. Fourier Anal. Appl. 5 (1999) 563–573. [7] L.W. Baggett, K.D. Merrill, Abstract harmonic analysis and wavelets in Rn , in: The Functional and Harmonic Analysis of Wavelets and Frames, in: Contemp. Math., vol. 247, Amer. Math. Soc., Providence, 1999, pp. 17–27. [8] R. Beals, Operators in function spaces which commute with multiplications, Duke Math. J. 35 (1968) 353–362. [9] S. Bildea, D. Dutkay, G. Picioroaga, MRA super-wavelets, New York J. Math. 11 (2005) 1–19. [10] M. Bownik, Z. Rzeszotnik, D. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal. 10 (2001) 71–92. [11] O. Bratteli, P.E.T. Jorgensen, Isometries, shifts, Cuntz algebras and multiresolution analyses of scale N , Integral Equations Operator Theory 28 (1997) 382–443. [12] O. Bratteli, P. Jorgensen, Wavelets Through a Looking Glass, Birkhäuser, Boston/Basel/Berlin, 2002. [13] J. Courter, Construction of dilation-d wavelets, in: The Functional and Harmonic Analysis of Wavelets and Frames, in: Contemp. Math., vol. 247, Amer. Math. Soc., Providence, 1999, pp. 183–205. [14] J. D’Andrea, K.D. Merrill, J.A. Packer, Fractal wavelets of Dutkay–Jorgensen type for the Sierpinski gasket space, in: Frames and Operator Theory in Analysis and Signal Processing, in: Contemp. Math., vol. 451, Amer. Math. Soc., Providence, 2008, pp. 69–88. [15] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Lecture Notes, vol. 61, SIAM, 1992. [16] D. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, Rev. Math. Iberoamericana 22 (2006) 131–180. [17] D. Dutkay, P.E.T. Jorgensen, Fourier series on fractals: a parallel with wavelet theory, in: Radon Transforms, Geometry and Wavelets, in: Contemp. Math., vol. 464, Amer. Math. Soc., Providence, 2008, pp. 75–101 (English summary). [18] S. Jaffard, Y. Meyer, Les ondelettes, in: Harmonic Analysis and Partial Differential Equations, El Escorial, 1987, in: Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 182–192. [19] N.S. Larsen, I. Raeburn, From filters to wavelets via direct limits, in: Operator Theory, Operator Algebras and Applications, in: Contemp. Math., vol. 414, Amer. Math. Soc., Providence, 2006, pp. 35–40. [20] W. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990) 1898–1901. [21] G.W. Mackey, Induced representations of locally compact groups I, Ann. of Math. 55 (1952) 101–139. [22] S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2 (R), Trans. Amer. Math. Soc. 315 (1989) 69–87. [23] Y. Meyer, Wavelets and Operators, Cambridge Stud. Adv. Math., vol. 37, Cambridge University Press, Cambridge, England, 1992.


Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces Guozheng Cheng, Kunyu Guo ∗ , Kai Wang School of Mathematical Sciences, Fudan University, Shanghai, 200433, China Received 8 November 2009; accepted 24 January 2010 Available online 2 February 2010 Communicated by N. Kalton

Abstract In this paper, we consider the well-known transitive algebra problem and reductive algebra problem on vector valued reproducing analytic Hilbert spaces. For an analytic Hilbert space H(k) with complete Nevanlinna–Pick kernel k, it is shown that both transitive algebra problem and reductive algebra problem on multiplier invariant subspaces of H(k) ⊗ Cm have positive answer if the algebras contain all analytic multiplication operators. This extends several known results on the problems. © 2010 Elsevier Inc. All rights reserved. Keywords: Transitive algebra property; Reductive algebra property; Fiber dimension; Complete NP kernel

1. Introduction Throughout this paper, let H denote a complex separable Hilbert space, and B(H ) the algebra of all bounded linear operators on H . For a set A of operators in B(H ), we write Lat A for all invariant subspaces of A. Let N be all of the positive integers. For N ∈ N, let IN be the identity matrix acting on CN . Set A ⊗ IN = {A ⊗ IN : A ∈ A}, a set of diagonal operators acting on H ⊗ CN . A unital subalgebra A of B(H ) is called transitive if it is closed in weak operator topology and Lat A = {{0}, H }. * Corresponding author.

E-mail addresses: [email protected] (G. Cheng), [email protected] (K. Guo), [email protected] (K. Wang). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.021

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G. Cheng et al. / Journal of Functional Analysis 258 (2010) 4229–4250

The well-known transitive algebra problem is: if A is a transitive algebra on a Hilbert space H , is A equal to B(H )? This problem was firstly explicitly stated by Arveson [4], and it remains open up to now. The basic results in [4] led to research on the transitive algebra problem by a number of authors [1,3,7,16,20,23]. Recently, some interesting progress on the connection with other topics in operator theory has been made in [6]. A unital subalgebra A of B(H ) is called reductive if it is closed in weak operator topology and its invariant subspaces are reducing, that is, if M ∈ Lat A, then M⊥ ∈ Lat A. The reductive algebra problem, as stated in [22], is the question: if A is a reductive algebra, must A be self-adjoint? In short, the reductive algebra problem asks if a reductive algebra is a von Neumann algebra. It is a long-standing problem in operator theory. As observed in [23], an affirmative answer to the reductive algebra problem would imply an affirmative answer to the transitive algebra problem. The reductive algebra problem is also an unsolved problem, but a few partial results have been obtained [2,3,21,25,23,22]. In this paper, we study both the transitive algebra and reductive algebra problems on reproducing analytic Hilbert spaces. Let Ω be a domain (an open and connected subset) in Cn , and suppose H is a Hilbert space consisting of some analytic functions on Ω. We call H a reproducing analytic Hilbert space provided that the evaluation functional Eλ : f → f (λ) is continuous for each λ ∈ Ω. It follows that there exists a unique function kλ ∈ H such that f (λ) = f, kλ for each f ∈ H. We call the function kλ (z) = k(z, λ), defined on Ω × Ω, the reproducing kernel of H. Let H(k) denote a reproducing analytic Hilbert space on a domain Ω ⊆ Cn with a reproducing kernel k. An analytic function ϕ on Ω is a multiplier of H(k) if ϕf ∈ H(k) for every f ∈ H(k). We shall write M(k) for the algebra of all multipliers on H(k). The closed graph theorem implies that each ϕ ∈ M(k) induces a bounded multiplication operator Mϕ : f → ϕf on H(k). Throughout the paper, Mϕ ⊗ IN acting on H(k) ⊗ CN is written as Mϕ for simplicity. Write W (k) for the weakly closed algebra {Mϕ : ϕ ∈ M(k)}. A subspace M of H(k) ⊗ CN is called multiplier invariant if it is invariant for W (k). Write WM (k) for the restriction of W (k) to M. To continue we need the following definition. Definition 1.1. Let H(k) be a reproducing analytic Hilbert space on a domain Ω ⊆ Cn . If M is a linear manifold of H(k) ⊗ CN , the fiber dimension f .d. M of M, is defined by f .d. M = sup dim f (z): f ∈ M . z∈Ω

It is straightforward to check that f .d. M = f .d. M, where M is the closure of M in H(k) ⊗ CN . Using the same method as in the proof of Theorem 4.3 [18], one can prove that for almost every z ∈ Ω, f .d. M = dim f (z): f ∈ M . Fiber dimension is a useful invariant in both function theory and operator theory, and paid more and more attention by many authors [11,13,14,12,17,18]. For any element f in H⊗ CN , we shall write it as f = (f1 , . . . , fN ) with respect to the N decomposition H ⊗ CN = N i=1 H ⊗ ei for a given orthonormal basis e1 , . . . , eN of C .


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Definition 1.2. A linear manifold M of H ⊗ CN (N 2) is called a graph if there exist linear transformations T1 , . . . , TN −1 on a linear manifold D of H distinct from {0}, such that M = (x, T1 x, . . . , TN −1 x): x ∈ D . When M is closed, it is called a graph subspace. For a subalgebra A of B(H ), when a graph subspace M of H ⊗ CN is invariant for A ⊗ IN , then M is called an invariant graph subspace for A ⊗ IN and each Ti is called a graph transformation for A. Moreover, the corresponding graph transformations satisfy Ti A = ATi on D for each A ∈ A. If A is a transitive algebra, then it is easy to see that the domain D is dense in H since the closure of D is an invariant subspace for A. Our first result is the following, whose proof is given in Section 2. Theorem A. Let H(k) be a reproducing analytic Hilbert space and M be a subspace of H(k) ⊗ Cm . If A is a transitive algebra on M, then A = B(M) if and only if for each positive integer N 2, and each invariant graph subspace N for A ⊗ IN , f .d. M = f .d. N . Let Ω be a domain containing the origin in Cn . A function k : Ω × Ω → C is called a complete Nevanlinna–Pick kernel [18] (a complete NP kernel for short) if k0 ≡ 1 and 1 − 1/kλ (z) is positive definite on Ω × Ω. For example, let Bn be the open unit ball in Cn , and k(z, w) =

1 , 1 − z, w

z, w ∈ Bn ,

it is easy to check that k is a complete NP kernel. The analytic Hilbert space defined by this kernel is usually called Drury–Arveson space [8,5]. When n = 1, it coincides with the Hardy space on the unit disk D. While the Dirichlet kernel k : D × D → C, k(z, w) =

∞ (zw)n n=0

n+1

is also a complete NP kernel [26]. A subset A of B(H ) is said to have the transitive algebra property if the only transitive algebra containing A is B(H ). An operator T ∈ B(H ) is said to have the transitive algebra property if the singleton {T } has the transitive algebra property. Based on Theorem A, in Section 3 we will prove that W (k) acting on H(k) ⊗ Cm has the transitive algebra property when k is a complete NP kernel, and more generally, we prove that WM (k) acting on a multiplier invariant subspace M of H(k) ⊗ Cm also has the transitive algebra property. Theorem B. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. (1) If A is a transitive algebra on H(k) ⊗ Cm which contains W (k) then A = B H(k) ⊗ Cm .

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(2) If M is a multiplier invariant subspace of H(k) ⊗ Cm , and A is a transitive algebra on M which contains WM (k), then A = B(M). Compared with transitive algebra property, we say that a subset A of B(H ) has the reductive algebra property if any reductive algebra containing A is self-adjoint. Theorem 8.7 [23] implies that if A has reductive algebra property, then it also has transitive algebra property. Based on Theorem B, in Section 4 we get the following stronger result. Theorem C. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. (1) If A is a reductive algebra on H(k) ⊗ Cm which contains W (k) then A is self-adjoint. (2) If M is a multiplier invariant subspace of H(k) ⊗ Cm , and A is a reductive algebra on M which contains WM (k), then A is self-adjoint. 2. Transitive algebra problem via fiber dimension This section is mainly devoted to prove Theorem A. Some additional terminology is needed. Definition 2.1. A linear transformation T is said to have a compression spectrum if there exists λ ∈ C such that the range of T − λ is not dense in H , that is, if D is the domain of T , then closure of {(T − λ)x: x ∈ D} is not H . Now we present the Arveson’s lemma, which can be inferred from [4] and plays an important role in the proof of Theorem A. Lemma 2.2. (See [23, Lemma 8.15].) If A is a transitive algebra such that every graph transformation for A has a compression spectrum, then we have A = B(H ). Based on the above lemma, we turn to the proof of Theorem A. Theorem A. Let H(k) be a reproducing analytic Hilbert space and M be a subspace of H(k) ⊗ Cm . If A is a transitive algebra on M, then A = B(M) if and only if for each positive integer N 2, and each invariant graph subspace N for A ⊗ IN , f .d. M = f .d. N . Proof. Let N = (f, T1 f, . . . , TN −1 f ): f ∈ D ⊆ M be an invariant graph subspace for A ⊗ IN . If A = B(M), then D = M since D is invariant for B(M). Define T : M → M ⊗ CN −1 by Tf = (T1 f, . . . , TN −1 f ),

f ∈ M,


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then the graph of T is N . The closed graph theorem implies that T is bounded, which implies that each Ti is bounded. Applying the fact Ti A = ATi for each A ∈ A = B(M) shows that each Ti is a multiple of the identity operator. In other words, N = (f, λ1 f, . . . , λN −1 f ): f ∈ M for some λ1 , . . . , λN −1 ∈ C. By a simple verification, one has f .d. M = f .d. N . Conversely, we assume that f .d. N = f .d. M for each invariant graph subspace N , and we write d = f .d. M. Since D is dense in M, we have f .d. D = f .d. N = d. This means for almost every z ∈ Ω, dim f (z): f ∈ D = dim f (z), (T1 f )(z), . . . , (TN −1 f )(z) : f ∈ D = d.

(2.1)

Now fix such a z0 . Since dim{f (z0 ): f ∈ D} = d, then there exist f1 , . . . , fd ∈ D, such that these vectors f1 (z0 ), . . . , fd (z0 ) are linearly independent in Cm and for each f ∈ D, f (z0 ) =

d

λj (z0 , f )fj (z0 )

(2.2)

j =1

for some λ1 (z0 , f ), . . . , λd (z0 , f ) ∈ C. The d vectors f1 (z0 ), (T1 f1 )(z0 ), . . . , (TN −1 f1 )(z0 ) , .. . fd (z0 ), (T1 fd )(z0 ), . . . , (TN −1 fd )(z0 ) are linearly independent in CmN . By (2.1), the above d vectors form a basis for the space f (z0 ), (T1 f )(z0 ), . . . , (TN −1 f )(z0 ) : f ∈ D . Combining this with the formula (2.2), one has (Ti f )(z0 ) =

d j =1

λj (z0 , f )(Ti fj )(z0 ),

1 i N − 1.

(2.3)

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Set E = {f (z0 ): f ∈ D} and Fi = {(Ti f )(z0 ): f ∈ D}, then both E and Fi are subspaces in Cm . Furthermore, since D is dense in M, we have E = {f (z0 ): f ∈ M} and hence Fi ⊆ E. For 1 i N − 1, define Ai : E → E by Ai f (z0 ) = (Ti f )(z0 ),

f ∈ D.

(2.4)

Using formulas (2.2) and (2.3), each Ai is a well-defined linear transformation on E. Next we will show that Ti has a compression spectrum. Since each Ai is a linear transformation on the finite dimensional vector space E, we can take an eigenvalue μi of Ai , then μi is a compression spectrum of Ti . Otherwise, {Ti f − μi f : f ∈ D} is dense in M and hence {(Ti f − μi f )(z0 ): f ∈ D} is dense in E. From the fact that E is finite dimensional, we see that

(Ti f − μi f )(z0 ): f ∈ D = E.

By (2.4) the range of Ai − μi I is E, contradicting that μi is an eigenvalue of Ai . It follows that every graph transformation for A has a compression spectrum, hence by Lemma 2.2, A = B(M), as desired.

2

The following corollary is a consequence of Theorem A. Corollary 2.3. Let H(k) be a reproducing analytic Hilbert space. If A is a transitive algebra on H(k) ⊗ Cm , then A = B(H(k) ⊗ Cm ) if and only if for each positive integer N 2, and each invariant graph subspace M for A ⊗ IN , f .d. M = m. In the remainder of this section, we establish two properties of the fiber dimension. The following proposition shows that the fiber dimension can reflect the structure of the linear manifolds. Proposition 2.4. Let H(k) be a reproducing analytic Hilbert space over Ω. If a linear manifold M of H(k) ⊗ CN has f .d. M = 1, then M is a graph. Proof. Without loss of generality, take f = (f1 , . . . , fN ) ∈ M with f1 = 0. Let Z(f1 ) be the zeros of f1 in Ω. Since f .d. M = 1, then for each g = (g1 , . . . , gN ) ∈ M and z ∈ Ω \ Z(f1 ), there exists λ(z, g) ∈ C such that g(z) = λ(z, g)f (z). In this case, λ(z, g) = z ∈ Ω,

g1 (z) f1 (z) .

Since the measure of Z(f1 ) is zero, one has that for almost every

gi (z) = λ(z, g)fi (z) =

fi (z) g1 (z), f1 (z)

i = 2, . . . , N.

(2.5)


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Let D be the set of the first entries of all elements in M. Consequently, by (2.5) for 2 i N , define Ti−1 : D → H(k) by Ti−1 g1 = gi . Then each Ti−1 is a well-defined linear transform. Thus M can be written as M = (g1 , T1 g1 , . . . , TN −1 g1 ): g1 ∈ D . This finishes the proof.

2

We end this section with the following proposition, which may be useful to calculate fiber dimension. Proposition 2.5. Let H(k) be a reproducing analytic Hilbert space. If M1 , M2 are multiplier invariant subspaces of H(k) ⊗ CN with M2 ⊆ M1 and dim M1 /M2 < ∞, then f .d. M1 = f .d. M2 . Proof. It is clear that f .d. M2 f .d. M1 , so we need only to prove that f .d. M1 f .d. M2 . Recall that in this paper we assume that M(k) is nontrivial, so we can take a nonconstant multiplier ϕ and define Sϕ on R = M1 M2 by Sϕ f = PR (ϕf ),

f ∈ R,

here PR is the projection from M1 onto R. Since M2 is a multiplier invariant subspace, it is easy to check that for each positive integer m, (Sϕ )m = Sϕ m , and hence the condition dim R = dim M1 /M2 < ∞ implies there exists a nonzero polynomial in ϕ, denoted by ψ, such that Sψ = 0, which means that ψR ⊆ M2 , and hence ψM1 ⊆ M2 . We have f .d.(ψM1 ) f .d. M2 . From the definition of fiber dimension, we see f .d.(ψM1 ) = f .d. M1 . Combining this fact and (2.6), one has f .d. M1 f .d. M2 , as desired.

2

(2.6)

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3. Transitive algebras on complete NP kernel spaces The aim of this section is to prove Theorem B, the transitive algebra property of both W (k) and WM (k) when k is a complete NP kernel. From Theorem A we see that the transitive algebra problem on H(k) ⊗ Cm can be reduced to the calculation of fiber dimension. So we firstly investigate further properties of the fiber dimension. In this paper, for a subspace M ⊆ H(k) ⊗ CN , let PM denote the projection onto M. Now we present the “occupy” invariant, which first was introduced by Fang [13] for the multiplier invariant subspaces of vector valued Drury–Arveson space. Definition 3.1 is a natural extension of Fang’s definition. Definition 3.1. Let H(k) be a reproducing analytic Hilbert space. For a linear manifold M ⊆ H(k) ⊗ CN , define the “occupy” invariant of M, denoted by lM , to be the maximal dimension of a subspace E of CN with the following property: there exist an orthonormal basis e1 , . . . , el (l = lM ) of E and h1 , . . . , hl ∈ M such that PH(k)⊗E hi ( = 0) ∈ H(k) ⊗ ei ,

i = 1, . . . , l.

When E has the above property we say that M occupies H(k) ⊗ E in H(k) ⊗ CN . There are many important properties of the “occupy” invariant, which were originally established by Fang for the vector valued Drury–Arveson space [13, Lemmas 22 and 23]. The conclusion of Lemma 22 [13] was obtained in a more general setting by Gleason, Richter and Sundberg (Lemma 3.2 [17]). Proposition 3.2. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ CN and it occupies H(k) ⊗ E for some E ⊆ CN , then for any nonzero vector e ∈ E, there exists an element h ∈ M such that PH(k)⊗E h( = 0) ∈ H(k) ⊗ e. Proposition 3.3. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. For any multiplier invariant subspace M of H(k) ⊗ CN , we have lM = f .d. M. One can prove the above two propositions by the same methods as in [13], which we omit here. We point out that Fang’s proof depends on the fact that any multiplier invariant subspace of vector valued Drury–Arveson space is generated by elements having multiplier entries. For complete NP kernel spaces, this is true. Lemma 3.4. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ CN , then the subset {(f1 , . . . , fN ): fi ∈ M(k)} ∩ M is dense in M. We say that f = (f1 , . . . , fN ) ∈ H(k) ⊗ CN has multiplier entries if each fi ∈ M(k) with respect to the decomposition H(k) ⊗ CN =

N i=1

H(k) ⊗ ei


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for some orthonormal basis e1 , . . . , eN of CN . It is easy to see that the definition is independent of the choice of the orthonormal basis of CN . Proof. Recall that functions in H(k) are defined on a domain Ω ⊆ Cn . Theorem 0.7 [19] implies that there exist an auxiliary Hilbert space F and an inner multiplier Φ : Ω → L(F , CN ) (all bounded linear operators from F to CN ) such that ∗ PM = MΦ MΦ ,

M = Φ H(k) ⊗ F ,

here (MΦ f )(λ) = Φ(λ)f (λ) for f ∈ H(k) ⊗ F , λ ∈ Ω. For ϕ ∈ M(k), x ∈ F , it is easy to check that MΦ (ϕ ⊗ x) has multiplier entries. Using the same argument as in the proof of Lemma 2.2(a) [18], one has that kλ ∈ M(k),

λ ∈ Ω.

Therefore, for e ∈ CN and λ ∈ Ω, ∗ M Φ MΦ (kλ ⊗ e) = MΦ kλ ⊗ Φ(λ)∗ e has multiplier entries. Then the conclusion follows from the fact that {kλ : λ ∈ Ω} is dense in H(k). 2 Remark 3.5. By the above lemma and the same argument in [13], we can get a stronger version of Proposition 3.2. That is, under the condition of Proposition 3.2, for any nonzero vector e ∈ E, there exists an element f ∈ M such that PH(k)⊗E f ( = 0) ∈ H(k) ⊗ e, here f has multiplier entries. Indeed, from Proposition 3.2, there exists h ∈ M such that PH(k)⊗E h( = 0) ∈ H(k) ⊗ e. By Lemma 3.4, the multiplier invariant subspace generated by h contains a nonzero function f which has multiplier entries. This implies that PH(k)⊗E f ( = 0) ∈ H(k) ⊗ e, as desired. In fact, this result has been obtained in [17]. Let H(k) be a reproducing analytic Hilbert space. A linear manifold M of H(k)⊗CN (N 2) is called an m-graph (m N ) if there exists an orthonormal basis e1 , . . . , eN of CN such that with respect to this basis, M has the form M = (f1 , . . . , fm , T1 f, . . . , TN −m f ): f = (f1 , . . . , fm ) ∈ D where D is the linear manifold of the first m entries of elements in M with f .d. D = m, and each Ti is a linear transform from D to H(k). M is called an m-graph subspace if it is closed. We need the following extension of Proposition 2.4 for further discussion, which is of independent interest. Firstly, we point out that the sufficiency of the following theorem is also observed by Gleason, Richter and Sundberg in Lemma 3.3 [17]. Theorem 3.6. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. For a multiplier invariant subspace M ⊆ H(k) ⊗ CN , then M is an m-graph subspace if and only if f .d. M = m.

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Proof. Suppose f .d. M = m, then the “occupy” invariant lM = m by Proposition 3.3. This means that there exists a subspace E ⊆ CN with an orthonormal basis e1 , . . . , em , and h1 , . . . , hm ∈ M such that PH(k)⊗E hi ( = 0) ∈ H(k) ⊗ ei ,

i = 1, . . . , m.

N N Extend {ei }m i=1 to an orthonormal basis {ei }i=1 of C . With respect to this basis, for any f ∈ H(k) ⊗ CN , we write f = (f1 , . . . , fN ). In particular, h1 , . . . , hm can be written as

h1 = h11 , 0, . . . , 0, h1m+1 , . . . , h1N , .. . m m hm = 0, . . . , 0, hm m , hm+1 , . . . , hN , where hii = 0 since PH(k)⊗E hi = 0 for 1 i m. i 1 Let Z(hii ) be the zeros of hii in Ω. Then for z ∈ Ω \ m i=1 Z(hi ), the m vectors h (z), . . . , m N h (z) are linearly independent in C . From the equality f .d. M = m, these vectors consist of a basis of {f (z): f ∈ M}. Then for each f = (f1 , . . . , fN ) ∈ M, there exist λ1 (z, f ), . . . , λm (z, f ) ∈ C such that f (z) = λ1 (z, f )h1 (z) + · · · + λm (z, f )hm (z). Obviously, λi (z, f ) =

fi (z) , hii (z)

i = 1, . . . , m.

It follows that fj (z) =

h1j (z) h11 (z)

f1 (z) + · · · +

hm j (z) hm m (z)

fm (z),

j = m + 1, . . . , N.

Note that the above identities hold for almost every z ∈ Ω since the measure of zero. For (f1 , . . . , fN ) ∈ M, define πm (f1 , . . . , fm , fm+1 , . . . , fN ) = (f1 , . . . , fm ),

(3.1) m

i i=1 Z(hi )

is

(3.2)

and let D = πm (M), the set of first m entries of elements in M. Consequently, by (3.1), for m + 1 j N , define Tj −m : D → H(k) by Tj −m (f1 , . . . , fm ) = fj . Then each Tj −m is a well-defined linear transform on D. Since for almost every z ∈ Ω, the m vectors πm h1 (z), . . . , πm hm (z)


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are linearly independent, and hence f .d. D = m. The above reasoning shows that M is an mgraph subspace. Conversely, suppose M = (f1 , . . . , fm , T1 f, . . . , TN −m f ): f = (f1 , . . . , fm ) ∈ D is an m-graph subspace with respect to an orthonormal basis e1 , . . . , eN of CN . By Proposition 3.3 it is enough to show that the “occupy” invariant lM = m. From the definition of m-graph subspace, f .d. D = m, and hence lM = f .d. M f .d. D m. It remains to show that lM m. We prove that by contradiction. Suppose M occupies H(k) ⊗ F in H(k) ⊗ CN with dim F = lM > m, then there exists an orthonormal basis e1 , . . . , el (l = lM ) of F and h1 , . . . , hl ∈ M such that PH(k)⊗F hi ( = 0) ∈ H(k) ⊗ ei ,

i = 1, . . . , l,

(3.3)

here by Remark 3.5 we can assume that each hi has multiplier entries. Write hi = hi1 , . . . , him , T1 hi , . . . , TN −m hi , i = 1, . . . , l,

where hi = (hi1 , . . . , him ) ∈ D. Let d = f .d.( li=1 {hi }), then d m and there exists z ∈ Ω such

that dim( li=1 {hi (z)}) = d. Without loss of generality, we assume that h1 (z), . . . , hd (z) are lin

early independent in Cm , which implies that f .d.( di=1 {hi }) = d. Furthermore, we may assume that the determinant of the d × d matrix Θ = (hij )di,j =1 is not zero at z. Note that the determinant det(Θ) is a nonzero multiplier on H(k). Using Cramer’s rule shows that there exist not all zero multipliers g1 , . . . , gl satisfying the following system of equations ⎧ d+1 d l 1 ⎪ ⎨ g1 h1 + · · · + gd h1 + gd+1 h1 + · · · + gl h1 = 0, .. . ⎪ ⎩ + · · · + gl hld = 0. g1 h1d + · · · + gd hdd + gd+1 hd+1 d Hence g1 h1 + · · · + gl hl = ( 0, . . . , 0, r1 , . . . , rm−d ). d

Moreover, since f .d.

d i=1

d hi , g1 h1 + · · · + gl hl = f .d. hi = d, i=1

we have that for almost every z ∈ Ω, the rank of the (d + 1) × m matrix ⎞ h11 (z) · · · h1d (z) h1d+1 (z) · · · h1m (z) ⎜· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ⎟ ⎠ ⎝ d h1 (z) · · · hdd (z) hdd+1 (z) · · · hdm (z) · · · rm−d (z) 0 ··· 0 r1 (z) ⎛

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does not exceed d. This implies that each ri = 0 since det(Θ) is a nonzero analytic function, and hence g1 h1 + · · · + gl hl = 0.

(3.4)

Combining (3.4) and the fact that for 1 i N − m, Ti Mg = Mg Ti on D for each g ∈ M(k), we have g1 h1 + · · · + gl hl = 0. Therefore, by (3.3) and the fact that e1 , . . . , el are orthogonal in F , for each 1 i l, 0 = PH(k)⊗ei g1 h1 + · · · + gl hl = PH(k)⊗ei PH(k)⊗F g1 h1 + · · · + gl hl = gi PH(k)⊗ei hi , and hence each gi = 0. This leads to a contradiction since g1 , . . . , gl are not all zero multipliers. Therefore, we have lM m. Based on the above discussion, it follows that f .d. M = lM = m. 2 The following corollary is a direct consequence of the above theorem and is used in the proof of Theorem B. Corollary 3.7. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If a graph subspace M = (f, T1 f, . . . , TN −1 f ): f ∈ D ⊆ H(k) ⊗ Cm is multiplier invariant, and f .d. D = m, then f .d. M = m. Theorem B(1). Let H(k) be a reproducing analytic Hilbert space with complete NP kernel k. If A is a transitive algebra on H(k) ⊗ Cm which contains W (k), then A = B H(k) ⊗ Cm . Proof. We will combine Corollaries 2.3 and 3.7 to complete the proof. Let M = (f, T1 f, . . . , TN −1 f ): f ∈ D ⊆ H(k) ⊗ Cm be an invariant graph subspace for A ⊗ IN . It is easy to see that D is dense in H(k) ⊗ Cm since it is invariant for the transitive algebra A. Therefore, f .d. D = f .d. H(k) ⊗ Cm = m. Since A contains W (k), M is multiplier invariant and hence by Corollary 3.7, f .d. M = m.


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The required identity A = B H(k) ⊗ Cm follows from Corollary 2.3.

2

Remark 3.8. If H(k) is a reproducing analytic Hilbert space on the open unit ball Bn with a complete NP kernel k and the coordinate functions z1 , . . . , zn are multipliers of H(k). Suppose A is a transitive algebra on H(k) ⊗ Cm which contains {Mzi : 1 i n}, then Lemma 4.1 [18] implies that A contains W (k). Therefore, by Theorem B(1), we have that A = B(H(k) ⊗ Cm ). As pointed in the Introduction, both the Hardy kernel and Dirichlet kernel are complete NP kernels. Therefore, as an immediate consequence, both the unilateral shift and the Dirichlet shift of finite multiplicity have the transitive algebra property. These results were obtained by Arveson [4], Radjavi, Rosenthal [23] and Richter [24]. Now we finish the proof of Theorem B(2). Theorem B(2). Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , and A is a transitive algebra on M which contains WM (k), then A = B(M). Proof. By Theorem A, it is sufficient to prove that for each positive integer N 2, and each invariant graph subspace N = (f, T1 f, . . . , TN −1 f ): f ∈ D ⊆ M for A ⊗ IN , f .d. M = f .d. N . Assume f .d. M = d(d m), then by Theorem 3.6, M is a d-graph subspace, that is, there exists a linear manifold D1 with f .d. D1 = d such that M = (g1 , . . . , gd , S1 g, . . . , Sm−d g): g = (g1 , . . . , gd ) ∈ D1 with respect to an orthonormal basis e1 , . . . , em of Cm , where each Sj is a linear transform from D1 to H(k). We turn to the invariant graph subspace N . Then by representation of M, the linear manifold D can be written as D = g1 , . . . , gd , S1 g , . . . , Sm−d g : g = g1 , . . . , gd ∈ D1 with D1 ⊆ D1 . Moreover, since D is dense in M, one has that D1 is dense in D1 , and hence f .d. D1 = d. Because both S1 , . . . , Sm−d and T1 , . . . , TN −1 are all linear transforms, there exist linear transforms T1 , . . . , TmN −d from D1 to H(k) such that N=

g1 , . . . , gd , T1 g , . . . , TmN −d g : g = g1 , . . . , gd ∈ D1 .

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Therefore, N is a d-graph subspace. Since A contains WM (k), N is multiplier invariant. Hence by Theorem 3.6, f .d. M = d = f .d. N , as required.

2

4. Reductive algebras on complete NP kernel spaces In this section we mainly consider the reductive algebra property of analytic multiplication operators on complete NP kernel spaces. Firstly, fix some notation. For a subset A of B(H ), let A∗ = {A∗ : A ∈ A}, and A = {B ∈ B(H ): BA = AB, A ∈ A}, the commutant of A. Proposition 4.1. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. Suppose A is a reductive algebra on H(k) which contains W (k), then A = B(H(k)). In particular, A is self-adjoint. Proof. Suppose M is an invariant subspace for A, then it is reducing because A is a reductive algebra. Hence one has PM A = APM

for A ∈ A.

(4.1)

As observed in the proof of Lemma 3.4, M(k) is dense in H(k). So by a routine verification, one has W (k) = W (k).

(4.2)

By (4.2) and the hypothesis that A contains W (k), the PM in (4.1) must be equal to Mg for some multiplier g in M(k). PM = Mg forces g = 0 or 1, which implies M = {0} or H(k). In other words, A has no nontrivial invariant subspace, hence A is a transitive algebra. The required identity A = B H(k) follows from Theorem B(1).

2

The above proposition is a special case of Theorem C(1). In general, in the case of the vector valued H(k) ⊗ Cm (m > 1), a reductive algebra which contains W (k) is properly contained in B(H(k) ⊗ Cm ). An example is the diagonal von Neumann algebra generated by W (k). To continue we need the following lemma. Lemma 4.2. (See [23, Theorem 9.11].) If A is a reductive algebra on a Hilbert space H , and if there exists a collection of invariant subspaces {Mi }ni=1 of A such that H = ni=1 Mi and A|Mi = B(Mi ) for each i, then A is self-adjoint. Now we complete the proof of Theorem C(1).


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Theorem C(1). Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If A is a reductive algebra on H(k) ⊗ Cm which contains W (k), then A is self-adjoint. Proof. Suppose A is a reductive algebra on H(k) ⊗ Cm which contains W (k). Firstly we have the following claim. Claim. The von Neumann algebra V = {A, A∗ } is finite dimensional. To see this, for each projection P ∈ V, one has that P M ϕ = Mϕ P ,

ϕ ∈ M(k).

(4.3)

Write PM = (Aij )m×m for some Aij ∈ B(H(k)). Then by (4.3), Aij Mϕ = Mϕ Aij ,

ϕ ∈ M(k).

Consequently, Aij = Mϕij for some ϕij ∈ M(k) by (4.2). Since P is a projection, this implies that Mϕij = Mϕ∗j i . Noticing k0 = 1, and hence for any λ ∈ Ω, Mϕij 1, kλ = Mϕ∗j i 1, kλ . A direct calculus shows that ϕij (λ) = ϕj i (0), which implies that each ϕij is a constant. Combining this observation and the fact that any von Neumann algebra is generated by projections in it, we have that V is finite dimensional, as desired. Based on the Claim, there exist finitely many mutually orthogonal minimal projections P1 , . . . , Ps ∈ V such that P1 + · · · + Ps = I, here I is the identity operator on H(k) ⊗ Cm . Let Mi be the range of minimal projection Pi for i = 1, . . . , s. Then each Mi is a minimal reducing subspace of A, thus the operator algebra A|Mi acting on Mi is transitive. Therefore, by Theorem B(2), A|Mi = B(Mi ), and the desired conclusion follows from Lemma 4.2.

2

Let H(k) be a reproducing analytic Hilbert space on the open unit ball Bn with a complete NP kernel k such that the coordinate functions z1 , . . . , zn are multipliers. If A is a reductive algebra on H(k) ⊗ Cm which contains {Mzi : 1 i n}, as pointed in Remark 3.8, then A contains W (k) and hence A is self-adjoint by Theorem C(1). As an immediate consequence, the unilateral shift of finite multiplicity has the reductive algebra property. This result was obtained by Nordgren, Rosenthal [21] and Ansari [2] respectively. In what follows we will consider the reductive algebra problem on a multiplier invariant subspace M of H(k) ⊗ Cm . We need to develop some new techniques. In the remainder of this section, let e1 , . . . , em be any given orthonormal basis of Cm . For a subset E of a Hilbert space H ,

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then the span of E, denoted by E, is the intersection of all subspaces containing E. For any subspace M ⊆ H(k) ⊗ Cm , set M(λ) = {f (λ): f ∈ M}, λ ∈ Ω. We begin with the following lemmas. Lemma 4.3. Let H(k) be a reproducing analytic Hilbert space over a domain Ω ⊆ Cn . For any subspace M ⊆ H(k) ⊗ Cm , one has that m PM (kλ ⊗ ei )(λ) . M(λ) = i=1

Proof. It is clear that M(λ) ⊇ m i=1 {PM (kλ ⊗ ei )(λ)}, thus we need only to prove the opposite inclusion. To this end we write f ∈ H(k) ⊗ Cm as f = (f1 , . . . , fm ) with respect to the basis e1 , . . . , em , then f (λ) = f, kλ ⊗ e1 , . . . , f, kλ ⊗ em . Therefore, for any f ∈ M, one also has f (λ) = f, PM (kλ ⊗ e1 ) , . . . , f, PM (kλ ⊗ em ) .

(4.4)

Moreover, for each f ∈ M, it can be expressed as f =f +

m

ci PM (kλ ⊗ ei )

i=1

with ci ∈ C and f ⊥PM (kλ ⊗ ei ), i = 1, . . . , m. By (4.4), f (λ) = 0, and hence f (λ) =

m

ci PM (kλ ⊗ ei )(λ),

i=1

which means that f (λ) ∈

m

i=1 {PM (kλ

⊗ ei )(λ)}, as desired.

2

Lemma 4.4. Let H(k) be a reproducing analytic Hilbert space and M be any subspace of H(k) ⊗ Cm . For each subset {ei1 , . . . , eid } ⊆ {e1 , . . . , em } and λ ∈ Ω, d vectors PM (kλ ⊗ ei1 ), . . . , PM (kλ ⊗ eid ) are linearly independent in M if and only if the d vectors PM (kλ ⊗ ei1 )(λ), . . . , PM (kλ ⊗ eid )(λ) are linearly independent in Cm .


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d Proof. In fact, if j =1 cj PM (kλ ⊗ eij ) = 0 for some c1 , . . . , cd ∈ C, then obviously d d j =1 cj PM (kλ ⊗ eij )(λ) = 0. Conversely, we assume that j =1 cj PM (kλ ⊗ eij )(λ) = 0, then for any 1 i m,

d

cj PM (kλ ⊗ eij ), kλ ⊗ ei = 0,

j =1

and hence ! d !2 d d ! ! ! ! c P (k ⊗ e ) = c P (k ⊗ e ), cs PM (kλ ⊗ eij ) ! j M λ ij ! j M λ ij ! ! j =1

=

j =1

d j =1

s=1

cj PM (kλ ⊗ eij ),

d

cs (kλ ⊗ ej )

s=1

= 0, which implies that holds. 2

d

j =1 cj PM (kλ

⊗ eij ) = 0. Based on the above discussion, the lemma

Next we present a key proposition, which is of independent interest. For λ ∈ Ω, let Mλ (k) = {ϕ ∈ M(k): ϕ(λ) = 0}. If M is a multiplier invariant subspace of H(k) ⊗ Cm , let Mλ = Mλ (k)M. It is well known that dim(M Mλ ) is an important invariant in operator theory [13, 10,9,15,17]. Proposition 4.5. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , then for every λ ∈ Ω with f .d. M = dim M(λ), m PM (kλ ⊗ ei ) = M Mλ . i=1

In fact, the above result has been observed by Gleason, Richter and Sundberg [17]. Now we present a different proof here. Proof. Firstly, it is easy to verify that "

# ker PM Mϕ∗ #M = M Mλ .

ϕ∈Mλ (k)

From the above identity, it is easy to check that the inclusion “⊆” holds in Proposition 4.5. Next we show the opposite inclusion. The following proof is essentially the same as that of the necessity of Theorem 3.6.

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Assume f .d. M = d (d m). By Lemma 4.3 and the property of fiber dimension, for almost every λ ∈ Ω, d = f .d. M = dim M(λ) = dim

m

PM (kλ ⊗ ei )(λ) .

i=1

Fix such a λ. Write f ∈ H(k) ⊗ Cm as f = (f1 , . . . , fm ) with respect to the basis e1 , . . . , em , and write i PM (kλ ⊗ ei ) = g1i , . . . , gm , i = 1, . . . , m. (4.5) From the proof of Lemma 3.4, we see that each gji ∈ M(k). Without loss of generality, we assume that the vectors PM (kλ ⊗ e1 )(λ), . . . , PM (kλ ⊗ ed )(λ) are linearly independent in Cm . Furthermore, we assume the determinant of the d × d matrix Θ = (gij )di,j =1 , denoted by det(Θ), is not zero at λ. Note that det(Θ) is a nonzero multiplier. Now for any g = (g1 , . . . , gm ) ∈ M Mλ , using Cramer’s rule shows that there exist h1 , . . . , hm ∈ H(k) satisfying the following system of equations ⎧ d+1 d m 1 ⎪ ⎨ h1 g1 + · · · + hd g1 + hd+1 g1 + · · · + hm g1 = det(Θ)g1 , .. . ⎪ ⎩ h1 gd1 + · · · + hd gdd + hd+1 gdd+1 + · · · + hm gdm = det(Θ)gd . Hence by (4.5) h1 PM (kλ ⊗ e1 ) + · · · + hm PM (kλ ⊗ em ) − det(Θ)g = ( 0, . . . , 0, r1 , . . . , rm−d ). d

Set h = h1 PM (kλ ⊗ e1 ) + · · · + hm PM (kλ ⊗ em ) − det(Θ)g. Since g ∈ M, then by Lemmas 4.3 and 4.4 d d PM (kλ ⊗ ei ), h = f .d. PM (kλ ⊗ ei ) = d. f .d. i=1

i=1

Thus for almost every z ∈ Ω, the rank of the (d + 1) × m matrix 1 (z) · · · g 1 (z) ⎞ g11 (z) · · · gd1 (z) gd+1 m ⎜· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ⎟ ⎝ d d (z) · · · d (z) ⎠ gm g1 (z) · · · gdd (z) gd+1 · · · rm−d (z) 0 ··· 0 r1 (z)

⎛

does not exceed d. This implies that each ri = 0 since det(Θ) is a nonzero multiplier function, and hence h1 PM (kλ ⊗ e1 ) + · · · + hm PM (kλ ⊗ em ) = det(Θ)g.

(4.6)


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Write hi PM (kλ ⊗ ei ) = hi − hi (λ) PM (kλ ⊗ ei ) + hi (λ)PM (kλ ⊗ ei ), det(Θ)g = det(Θ) − det(Θ)(λ) g + det(Θ)(λ)g. It is easy to see that (det(Θ) − det(Θ)(λ))g ∈ Mλ . Since M(k) is dense in H(k) and each PM (kλ ⊗ ei ) has multiplier entries, it is easy to check that (hi − hi (λ))PM (kλ ⊗ ei ) ∈ Mλ for i = 1, . . . , m. Therefore, we apply the projection PMMλ to two sides of (4.6) and have m

hi (λ)PM (kλ ⊗ ei ) = det(Θ)(λ)g,

i=1

which means that g ∈

m

i=1 {PM (kλ

⊗ ei )} since det(Θ)(λ) = 0. This completes the proof.

2

The above proposition means that for every λ ∈ Ω with f .d. M = dim M(λ), dim

m PM (kλ ⊗ ei ) = dim(M Mλ ). i=1

Combining this observation as well as Lemmas 4.3 and 4.4, we immediately have the following corollary. Corollary 4.6. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , then for every λ ∈ Ω with f .d. M = dim M(λ), dim(M Mλ ) = f .d. M. Corollary 4.7. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M1 and M2 are mutually orthogonal invariant subspaces of H(k) ⊗ Cm for W (k), then f .d.(M1 ⊕ M2 ) = f .d. M1 + f .d. M2 . Proof. Let M = M1 ⊕ M2 . By Lemma 4.3 and the definition of fiber dimension, we can take λ ∈ Ω such that for j = 1, 2, the following hold: m f .d. M = dim M(λ) = dim PM (kλ ⊗ ei )(λ) i=1

and f .d. Mj = dim Mj (λ) = dim

m PMj (kλ ⊗ ei )(λ) . i=1

4248


Thus combining Lemma 4.4 and Proposition 4.5, one has f .d. M = dim

m PM (kλ ⊗ ei ) = dim(M Mλ )

(4.7)

i=1

and f .d. Mj = dim

m PMj (kλ ⊗ ei ) = dim(Mj Mj λ ).

(4.8)

i=1

By the fact that both M1 and M2 are invariant subspaces for W (k), it follows that for each ϕ ∈ M(k), M ϕM = (M1 ϕM1 ) ⊕ (M2 ϕM2 ) and hence M Mλ = (M1 M1λ ) ⊕ (M2 M2λ ). Combining this fact with (4.7) and (4.8), we have f .d. M = f .d. M1 + f .d. M2 .

2

In the proof of Theorem C(1), we use the fact that the commutant of the von Neumann algebra generated by W (k) is finite dimensional. In what follows we generalize this fact to multiplier invariant subspaces, which plays an important role in the proof of Theorem C(2). Recall that for a multiplier invariant subspace M of H(k) ⊗ Cm , WM (k) = {Mϕ |M : ϕ ∈ M(k)}. Proposition 4.8. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , then the von Neumann algebra W = WM (k), WM (k)∗ is finite dimensional. Proof. By Lemma 4.3, we can take λ ∈ Ω such that f .d. M = dim M(λ) = dim

m PM (kλ ⊗ ei )(λ) .

(4.9)

i=1

Let E = m i=1 {PM (kλ ⊗ ei )}, then it is a finite dimensional subspace of M which is reducing for W by Proposition 4.5. Now we define a C ∗ -homomorphism τ : W → W|E


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by τ (A) = A|E . Now we show that it is injective. Let A ∈ ker τ , equivalently, E ⊆ ker A. Since A commutes with both Mϕ |M and PM Mϕ∗ |M , M ker A and ker A are invariant for WM (k), hence they are also invariant for W (k). Therefore, M has a decomposition as M = (M ker A) ⊕ ker A, then by Corollary 4.7, f .d. M = f .d.(M ker A) + f .d.(ker A).

(4.10)

m f .d.(ker A) f .d. E dim PM (kλ ⊗ ei )(λ) = f .d. M,

(4.11)

Meanwhile,

i=1

where the last equality follows from (4.9). Combining (4.10) and (4.11), one has that f .d.(M ker A) = 0, and hence M ker A = {0}. It follows that ker A = M, that is A = 0, as desired. The above argument shows that τ is a C ∗ -isomorphism, and hence W is finite dimensional. 2 Now we give the proof of Theorem C(2). Theorem C(2). Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , and A is a reductive algebra on M which contains WM (k), then A is self-adjoint. Proof. Let V = {A, A∗ } . Since A contains WM (k), then the von Neumann algebra V ⊆ W, which is given in Proposition 4.8, and hence V is finite dimensional by Proposition 4.8. Based on this fact, the conclusion follows from the same argument as in the proof of Theorem C(1). 2 Acknowledgments The authors thank the referee for helpful suggestions which make this paper more readable. This work is partially supported by NKBRPC (2006CB805905) and NSFC (10525106), and Foundation of Fudan University (EYH1411039) and Laboratory of Mathematics for Nonlinear Science, Fudan University. References [1] M. Ansari, Transitive algebra containing triangular operator matrics, J. Operator Theory 14 (1985) 173–180. [2] M. Ansari, Reductive algebras containing a direct sum of the unilateral shift and a certain other operators are self adjoint, Proc. Amer. Math. Soc. 93 (1985) 284–286. [3] M. Ansari, S. Richter, A strong transitive algebra property of operators, J. Operator Theory 22 (1989) 267–276. [4] W. Arveson, A density theorem for operator algebras, Duke Math. J. 24 (1967) 635–647.

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[5] W. Arveson, Subalgebras of C ∗ algebras III: Multivariable operator theory, Acta Math. 181 (1998) 159–228. [6] H. Bercovici, R. Douglas, C. Foias, C. Pearcy, Confluent operators algebras and the closability property, arXiv:0908. 0729v2. [7] R. Douglas, C. Pearcy, Hyperinvariant subspaces and transitive algebras, Michigan Math. J. 19 (1972) 1–12. [8] S. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978) 300–304. [9] X. Fang, Hilbert polynomials and Arveson’s curvature invariant, J. Funct. Anal. 198 (2003) 445–464. [10] X. Fang, Samuel multiplicity and the structure of semi-Fredholm operators, Adv. Math. 186 (2004) 411–437. [11] X. Fang, Invariant subspaces of the Dirichlet space and commutative algebra, J. Reine Angew. Math. 569 (2004) 189–211. [12] X. Fang, The Fredholm index of quotient Hilbert modules, Math. Res. Lett. 12 (2005) 911–920. [13] X. Fang, The Fredholm index of a pair of commuting operators, Geom. Funct. Anal. 16 (2006) 367–402. [14] X. Fang, Additive invariant on the Hardy space over the polydisc, J. Funct. Anal. 253 (2007) 359–372. [15] X. Fang, The Fredholm index of a pair of commuting operators. II, J. Funct. Anal. 256 (2009) 1669–1692. [16] J. Fang, D. Hadwin, M. Ravichandran, On transitive algebras containing a standard finite von Neumann subalgebra, J. Funct. Anal. 252 (2007) 581–602. [17] J. Gleason, S. Richter, C. Sundberg, On the index of invariant subspaces in spaces of analytic function of several complex variables, J. Reine Angew. Math. 587 (2005) 49–76. [18] D. Greene, S. Richter, C. Sundberg, The structure of inner multipliers on spaces with complete Nevanlinna Pick kernels, J. Funct. Anal. 194 (2002) 311–331. [19] S. McCullough, T. Trent, Invariant subspaces and Nevanlinna–Pick kernel kernels, J. Funct. Anal. 178 (2000) 226– 249. [20] E. Nordgren, H. Radjavi, P. Rosenthal, On density of transitive algebras, Acta Sci. Math. (Szeged) 30 (1969) 175– 179. [21] E. Nordgren, P. Rosenthal, Algebras containing unilateral shifts or finite rank operators, Duke Math. J. 40 (1973) 419–424. [22] H. Radjavi, P. Rosenthal, A sufficient condition that an operator algebra be self-adjoint, Canad. J. Math. 23 (1971) 588–597. [23] H. Radjavi, P. Rosenthal, Invariant Subspaces, Springer-Verlag, Berlin, 1973. [24] S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 286 (1988) 205–220. [25] P. Rosenthal, On reductive algebras containing compact operators, Proc. Amer. Math. Soc. 47 (1975) 338–340. [26] H. Shapiro, A. Shields, On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962) 217–229.


On the global existence and wave-breaking criteria for the two-component Camassa–Holm system Guilong Gui a,b , Yue Liu c,∗ a Department of Mathematics, Jiangsu University, Zhenjiang 212013, China b Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China c Department of Mathematics, University of Texas, Arlington, TX 76019, United States

Received 9 November 2009; accepted 5 February 2010 Available online 24 February 2010 Communicated by H. Brezis

Abstract Considered herein is a two-component Camassa–Holm system modeling shallow water waves moving over a linear shear flow. A wave-breaking criterion for strong solutions is determined in the lowest Sobolev space H s , s > 32 by using the localization analysis in the transport equation theory. Moreover, an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this Sobolev space H s . © 2010 Elsevier Inc. All rights reserved. Keywords: Global solutions; Transport equation theory; Two-component Camassa–Holm system; Wave breaking

1. Introduction We consider here the coupled two-component Camassa–Holm shallow water system [12,23, 30,31], namely, ⎧ ⎨ mt + umx + 2ux m − Aux + ρρx = 0, t > 0, x ∈ R, m = u − uxx , t > 0, x ∈ R, ⎩ ρt + (uρ)x = 0, t > 0, x ∈ R, * Corresponding author.

E-mail addresses: [email protected] (G. Gui), [email protected] (Y. Liu). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.008

(1.1)

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G. Gui, Y. Liu / Journal of Functional Analysis 258 (2010) 4251–4278

where the variable u(t, x) represents the horizontal velocity of the fluid, and ρ(t, x) is related to the free surface elevation from equilibrium (or scalar density) with the boundary assumptions, u → 0 and ρ → 1 as |x| → ∞. The parameter A > 0 characterizes a linear underlying shear flow so that (1.1) models wave-current interactions (see the discussions in [15,25,26] and see also [4,24]). All of those are measured in dimensionless units. Recently, Ivanov [23] gave a rigorous justification of the derivation of the system (1.1) which is a valid approximation to the governing equations for water waves in the shallow water regime with nonzero constant vorticity, where the nonzero vorticity case arises for example in situations with underlying shear flow [24]. Set g(x) = 12 e−|x| , x ∈ R. Then (I − ∂x2 )−1 f = g ∗ f for f ∈ L2 (R), where ∗ denotes the spatial convolution. Let η := ρ − 1, (1.1) can be rewritten as a quasi-linear nonlocal evolution system of the type ⎧ ⎨ u + uu = −∂ g ∗ u2 + 1 u2 − Au + 1 η2 + η , t x x 2 x 2 ⎩ ηt + uηx + ηux + ux = 0,

t > 0, x ∈ R,

(1.2)

t > 0, x ∈ R,

or equivalently, ⎧ ut + uux + ∂x P = 0, ⎪ ⎪ ⎨ 1 1 −∂x2 P + P = u2 + u2x − Au + η2 + η, ⎪ 2 2 ⎪ ⎩ ηt + uηx + ηux + ux = 0,

t > 0, x ∈ R, t > 0, x ∈ R, t > 0, x ∈ R.

For A = ρ = 0 in (1.1), one obtains the classical Camassa–Holm model [5], whose relevance for water waves was established in [10,27]. The system (1.1) is formally integrable [19,23,31] as it can be written as a compatibility condition of two linear systems (Lax pair) with a spectral parameter ζ , that is, 1 A 2 2 + Ψ, Ψxx = −ζ ρ + ζ m − 2 4 1 1 − u Ψx + ux Ψ Ψt = 2ζ 2 and has a bi-Hamiltonian structure corresponding to the Hamiltonian H1 =

1 2

mu + (ρ − 1)2 dx

R

with m = u − uxx and the Hamiltonian H2 =

1 2

u(ρ − 1)2 + 2u(ρ − 1) + u3 + uu2x − Au2 dx.

R

There are two Casimirs, i.e. ρ → 1 as |x| → ∞.

ρ − 1 and

m with boundary conditions are taken as u → 0 and


4253

The system (1.1) without vorticity, i.e. A = 0, was also rigorously justified by Constantin and Ivanov [12] to approximate the governing equations for shallow water waves. M. Chen, S. Liu and Y. Zhang [8] established a reciprocal transformation between the two-component Camassa– Holm system and the first negative flow of the AKNS hierarchy. More recently, Holm, Nraigh and Tronci [22] proposed a modified two-component Camassa–Holm system which possesses singular solutions in component ρ. Mathematical properties of (1.1) with A = 0 have been also studied further in many works. For example, Escher, Lechtenfeld and Yin [18] investigated local wellposedness for the two-component Camassa–Holm system with initial data (u0 , ρ0 ) ∈ H s × H s−1 with s 2 and derived some precise blow-up scenarios for strong solutions to the system. Constantin and Ivanov [12] provided some conditions of wave breaking and small global solutions. Gui and Liu [21] recently obtained results of local well-posedness in the Besov spaces (especially in the Sobolev space H s × H s−1 with s > 32 ) and wave breaking for certain initial profiles. More recently, Guan and Yin [20] studied global existence and blow-up phenomena for the system (1.2) with initial data (u0 , ρ0 − 1) ∈ H s × H s−1 with s 52 . It is known that different from the Korteweg–de Vries (KdV) equation, the Camassa–Holm (CH) equation has a remarkable property, that is, the presence of breaking waves [5,11], which means, the solution remains bounded while its slope becomes unbounded in finite time [9,11]. After wave breaking the solutions of the CH equation can be continued uniquely as either global conservative [2] or global dissipative solutions [3]. The goal of the present paper is to investigate whether or not the two-component Camassa–Holm system has the similar wave-breaking phenomena as the classical Camassa–Holm equation in a lower Sobolev space H s × H s−1 for s > 32 . In other words, whether or not both of two components u and ρ of the solution remain bounded while their slopes become unbounded in finite time. As we know, a crucial ingredient to obtain wave breaking in finite time or global solution for the CH equation is the following invariant property [9].

m t, q(t, x) qx2 (t, x) = m0 (x),

(t, x) ∈ [0, T ) × R,

where m(t, x) = u(t, x) − uxx (t, x) and the function q ∈ C 1 is an increasing diffeomorphism of R and satisfies the following differential equation, ⎧ ⎨ ∂q = u(t, q), 0 < t < T , ∂t ⎩ q(0, x) = x, x ∈ R. This is related to the geodesic equation which is on the diffeomorphism group of the circle [14] or on the Bott–Virasoro group [13,28,29]. Without such a nice invariant property of the CH equation, the issue of whether or not particular initial data of the two-component Camassa–Holm system generate a global solution or wave breaking is more subtle. Our work is motivated in the study of nonlinear models, especially of the transport equation, that is, ∂t f + v∂x f = g, (t, x) ∈ R+ × R, f |t=0 = f0 . It is well known that most of estimates are available when v has enough regularity. Roughly speaking, the regularity of the initial data is expected to be preserved as soon as v belongs to L1loc (R+ ; Lip).

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We give the following remark to explain the meaning of the lowest Sobolev space corresponding to the system (1.1) or (1.2). Remark 1.1. We say H s × H s−1 with s > 32 is the lowest Sobolev space for the two-component Camassa–Holm system based on the following facts. (a) For every Sobolev index s 32 , the Sobolev space H s cannot be embedded in the Lipschitz space (Lip), which is the lowest condition for preserving the regularity of the strong solution to the two-component Camassa–Holm system according to the localized analysis in the transport equation theory. (b) Without the effect of linear dispersion, i.e. A = 0, the system (1.1) has peakon solitons of the form, u(x, t) = ce−|x−ct| , c = 0 with ρ ≡ 0 as the solution of the Camassa–Holm equation. It is noted that the peakon soliton ce−|x−ct| is the weak solution in the Sobolev space H s only for s < 32 . (c) Following the proof of Proposition 4 in [17], one can see that (1.1) is not locally well posed 3 2 in B2,∞ in the following sense. 3

2 ) and ρ ≡ 0 to (1.1) such that for any positive There exists a global solution u ∈ L∞ (R+ ; B2,∞ 3

2 ) and ρ ≡ 0 with T > 0 and > 0 there exists a solution v ∈ L∞ (0, T ; B2,∞

u(0) − v(0)

3

2 B2,∞

Therefore, the exponent s =

3 2

and u − v

3

2 ) L∞ (0,T ;B2,∞

1.

s for r ∈ [1, ∞]. is critical in the range of Besov spaces B2,r

Inspired by [12], we use the properties of invariance of the component ρ associated to a transport equation with more delicate localization analysis in the transport equation theory to derive a new wave-breaking criterion for solutions for the system (1.1) in the lowest Sobolev spaces H s × H s−1 with s > 32 . In this case, due to the Hamiltonian H1 , the horizontal velocity component u is uniformly bounded by the Sobolev imbedding of H 1 into L∞ . It is shown that the slope of u is bounded below, then the slope of the component ρ cannot break in finite time. This implies that the wave breaking of the solution is determined only by the slope of the component u of the solution definitely. Note in [12,18] that the wave breaking in finite time is determined by either the slope of the first component u or the slope of the second component ρ in the Sobolev space H s × H s−1 with s 52 . It is, however, established in Theorem 4.1 and Theorem 4.2 that the wave breaking in finite time only depends on the slope of the first component u in the Sobolev space H s × H s−1 with s > 32 . In other words, the wave breaking in the first component u must occur before that in the second component ρ in finite time. On the other hand, we find a sufficient condition for global solutions which determined only by a nonzero initial profile of the free surface component ρ of the system in H s × H s−1 with s > 32 . This can be done because the slope of the component u can be controlled by the component ρ in finite time provided the sign of ρ does not change. These of improved results of global solutions and wave breaking indeed reveal more important features of wave propagation to the system (1.1). Our main results of the present paper are Theorem 4.1 (Wave-breaking criterion), Theorem 4.2 (Precise wave-breaking criterion) and Theorem 5.1 (Global solution). The remainder of the paper is organized as follows. In Section 2, we recall some basic facts on the Littlewood–Paley theory, which the localization technique is constantly used in the whole


4255

paper. Section 3 is devoted to the transport equation theory, where Theorem 3.2 is specially interesting to the system (1.2). Using the transport equation theory in the Besov spaces, two wave-breaking criteria to solutions in the lowest Sobolev space H s × H s−1 with s > 32 are demonstrated in Section 4. Finally, a result of global existence of solution in the lowest Sobolev space H s × H s−1 with s > 32 is obtained in the last section, Section 5. Notation. Let A, B be two operators, we denote [A; B] = AB − BA, the commutator between A and B; a b means that there is a uniform constant C that may be different on different lines, such that a Cb. We denote (cj )j ∈N (or (cj (t))j ∈N ) to be a sequence in 2 with norm 1. All of different positive constants might be denoted by the uniform constant C which may depend only on initial data. 2. Littlewood–Paley analysis For convenience of the reader, we shall recall some basic facts on the Littlewood–Paley theory, one may check [1,6,7,16,32] for more details. def

Proposition 2.1 (Littlewood–Paley decomposition). (See [6].) Let B = {ξ ∈ Rd , |ξ | 43 } and def

C = {ξ ∈ Rd , that

3 4

|ξ | 83 }. There exist two radial functions χ ∈ Cc∞ (B) and ϕ ∈ Cc∞ (C) such χ(ξ ) +

∀ξ ∈ Rd ,

q0

q − q 2 q 1

ϕ 2−q ξ = 1,

⇒ ⇒

Supp ϕ 2−q · ∩ Supp ϕ 2−q · = ∅,

Supp χ(·) ∩ Supp ϕ 2−q · = ∅,

and

2 1 χ(ξ )2 + ϕ 2−q ξ 1, 3

∀ξ ∈ Rd .

q0

def def Let h = F −1 ϕ and h˜ = F −1 χ . Then the dyadic operators q and Sq can be defined as follows

def −q

qd h 2q y f (x − y) dy, for q 0, q f = ϕ 2 D f = 2 Rd

def Sq f = χ 2−q D f =

q f = 2qd

−1kq−1 def

−1 f = S0 f

h˜ 2q y f (x − y) dy,

Rd def

and q f = 0 for q −2.

(2.1)

Lemma 2.1 (Bernstein’s inequality). (See [6].) Let B be a ball with center 0 in Rd and C a ring with center 0 in Rd . A constant C exists so that, for any positive real number λ, any nonnegative

4256


integer k, any smooth homogeneous function σ of degree m, and any couple of real numbers (a, b) with b a 1, there hold Supp uˆ ⊂ λB Supp uˆ ⊂ λC

⇒

⇒

C

Supp uˆ ⊂ λC

1 1 sup ∂ α u Lb C k+1 λk+d( a − b ) uLa ,

|α|=k

λ uLa sup ∂ α u La C 1+k λk uLa ,

−1−k k

⇒

σ (D)u

Lb

|α|=k

1

1

Cσ,m λm+d( a − b ) uLa ,

(2.2)

for any function u ∈ La . Definition 2.1 (Besov spaces). (See [6].) Let s ∈ R, 1 p, r ∞. The inhomogeneous Besov s (Rd ) (B s for short) is defined by space Bp,r p,r d def

s s 12 , the following statement holds:

log e + f H s , f L∞ C 1 + f B∞,∞ 0 where the constant C = C(s) is independent of f . The proof of this proposition is trivial, which can be found in [6], and we omit it. The following proposition is devoted to dealing with the pseudo-differential operator ∂x (1 − ∂x2 )−1 (or ∂x g∗). Proposition 2.3. (See [6].) Let m ∈ R and f be an S m -multiplier (that is, f : Rd → R is smooth and satisfies that for all multi-index α, there exists a constant Cα such that ∀ξ ∈ Rd , |∂ α f (ξ )| Cα (1 + |ξ |)m−|α| ). Then for all s ∈ R and 1 p, r ∞, the operator f (D) is cons to B s−m . tinuous from Bp,r p,r


4257

In this paper, we are going to use Bony’s decomposition which consists of writing uv = Tu v + Tv u + R(u, v),

(2.4)

where

Tu v =

Sq−1 u q v

and R(u, v) =

q u q v =

|q−q |1

q−1

q v, q u

q−1

q := q−1 + q + q+1 . where Proposition 2.4 (1-D Moser-type estimates). The following estimates hold. (i) For s 0,

f gH s (R) C f H s (R) gL∞ (R) + f L∞ (R) gH s (R) .

(2.5)

f ∂x gH s (R) C f H s+1 (R) gL∞ (R) + f L∞ (R) ∂x gH s (R) .

(2.6)

(ii) For s > 0,

(iii) For s1 12 , s2 >

1 2

and s1 + s2 > 0, f gH s1 (R) Cf H s1 (R) gH s2 (R) ,

(2.7)

where C’s are constants independent of f and g. Proof. The proof of this lemma is rather classical, and similar estimates can be found in [6]. (2.5) is a standard Moser-type estimate, and (2.7) was used in [16] and [21]. For completeness, we present the detailed proof of (2.6) here. Thanks to Bony’s decomposition (2.4), we decompose f ∂x g as follows: f ∂x g = Tf ∂x g + T∂x g f + R(f, ∂x g). Thanks to Bernstein’s inequalities (2.2), we have

q (Tf ∂x g)

L2

q (Sq −1 f q ∂x g) |q−q |5

Cf L∞

|q−q |5

and

L2

Sq −1 f L∞ q ∂x gL2

|q−q |5

cq 2−sq ∂x gH s Ccq 2−sq f L∞ ∂x gH s

(2.8)

4258


q (T∂ g f ) 2 x L

q (Sq −1 ∂x g q f ) |q−q |5

Cf H s+1 gL∞

L2

Sq −1 ∂x gL∞ q f L2

|q−q |5

cq 2−(s+1)q 2q

|q−q |5

Ccq 2−sq f H s+1 gL∞ .

(2.9)

While for s > 0, using Bernstein’s inequalities (2.2) again and Young’s inequality, we get

q R(f, ∂x g)

L2

q ( q f q ∂x g) q q−5

C

L2

q ∂x gL∞ q f L2

q q−5

q gL∞ q f L2

q =−1, q q−5

+C

q gL∞ q ∂x f L2

q 0, q q−5

Ccq 2−sq f H s+1 gL∞ , which, together with (2.8), (2.9) and (2.3), completes the proof of (2.6).

2

3. Transport equation theory To study the well-posedness problem of the system (1.2), we need the following theorem on the transport equation (especially taking the space dimension d = 1), which has been used in [21]. Theorem 3.1. (See [16].) Suppose that s > − d2 . Let v be a vector field such that ∇v belongs d

to L1 ([0, T ]; H s−1 ) if s > 1 + d2 or to L1 ([0, T ]; H 2 ∩ L∞ ) otherwise. Suppose also that f0 ∈ H s , F ∈ L1 ([0, T ]; H s ) and that f ∈ L∞ ([0, T ]; H s ) ∩ C([0, T ]; S ) solves the d-dimensional linear transport equations (T )

∂t f + v · ∇f = F, f |t=0 = f0 .

Then f ∈ C([0, T ]; H s ). More precisely, there exists a constant C depending only on s, p and d, and such that the following statements hold: (1) If s = 1 + d2 , t f

Hs

f0

Hs

+ 0

or hence,

F (τ )

t Hs

dτ + C 0

V (τ ) f (τ ) H s dτ,

(3.1)


t

f H s eCV (t) f0 H s +

e−CV (τ ) F (τ )

4259

Hs

dτ

(3.2)

0

t 0

∇v(τ )

H

d 2

∩L∞

dτ if s < 1 +

d 2

and V (t) =

t

∇v(τ )H s−1 dτ else. t (2) If f = v, then for all s > 0, the estimates (3.1) and (3.2) hold with V (t) = 0 ∂x u(τ )L∞ dτ . with V (t) =

0

The following theorem (Theorem 3.2) is crucial to prove wave-breaking criterion (Theorem 4.1 in Section 4). Compared with Theorem 3.1, the following theorem is also specially interesting to the regularity propagation of the solution to the second equation of the twocomponent Camassa–Holm system (1.2) (where ρ − 1 = η = u), since only one derivative of u is involved in V (t) in (3.3) below. It is noted that the estimate (3.3) is quite different from (3.1) in Theorem 3.1, because there is (1 + 12 )-order derivative of u involved. This then makes the problem more difficult to deal with. The proof actually needs more delicate localization analysis in details. Theorem 3.2. Let 0 < σ < 1. Suppose that f0 ∈ H σ , g ∈ L1 ([0, T ]; H σ ), v, ∂x v ∈ L1 ([0, T ]; L∞ ) and that f ∈ L∞ ([0, T ]; H σ ) ∩ C([0, T ]; S ) solves the 1-dimensional linear transport equation (T )

∂t f + v∂x f = g, f |t=0 = f0 .

Then f ∈ C([0, T ]; H σ ). More precisely, there exists a constant C depending only on σ and such that the following statement holds:

f (t) σ f0 H σ + C H

t

g(τ )

t Hσ

dτ + C

0

f (τ )

Hσ

V (τ ) dτ

(3.3)

0

or hence,

f (t)

Hσ

e

CV (t)

t

f0 H σ + C

g(τ )

Hσ

dτ

0

with V (t) =

t

0 (v(τ )L

∞

+ ∂x v(τ )L∞ ) dτ .

Proof. The proof of this theorem is motivated by the one of Theorem 3.1 (see [16]). Applying the localization operator q to the transport equation (T ), we transform the transport equation (T ) along the flow of v, in the following equation (Tq ) on q f , which is a transport equation along the flow of Sq v (Tq )

∂t q f + Sq v∂x q f = q g − Rq , q f |t=0 = q f0 ,

where Rq = Rq (v, f ) := q (v∂x f ) − v q ∂x f + (v − Sq v) q ∂x f .

4260


To deal with Rq , we need to use the following lemma, which we admit for the time being. Lemma 3.1. For all 0 < σ < 1,

Rq (t)

L2

cq (t)2−qσ f (t) H σ vL∞ + ∂x vL∞

with cq (t) ∈ l 2 and cq (t)l 2 ≡ 1. With Lemma 3.1 in hand, we can continue the proof of Theorem 3.2. Taking the inner product between the first equation of (Tq ) and q f in L2 , we have 1 d 1 q f 2L2 = − 2 dt 2

Sq v∂x | q f |2 + ( q g| q f )L2 − Rq (v, f )| q f L2

R

∂x Sq vL∞ q f 2L2 + Rq L2 + q gL2 q f L2 , which implies

1 d q f 2L2 ∂x vL∞ q f L2 + Rq L2 + q gL2 q f L2 2 dt

Ccq (t)2−qσ f (t) σ vL∞ + ∂x vL∞ + g(t) H

Hσ

q f L2 .

Therefore, one has

q f (t)

t L2

q f0 L2 + C

cq (τ )2−qσ f (τ ) H σ vL∞ + ∂x vL∞ + g(τ ) H σ dτ.

0

(3.4) Multiplying (3.4) by 2qσ , then taking the l 2 norm over q and applying Minkowski’s inequality, we reach

f (t)

t Hσ

f0 H σ + C

g(τ )

t Hσ

dτ + C

0

This ends the proof of Theorem 3.2.

f (r)

Hσ

vL∞ + ∂x vL∞ dτ.

0

2

We now are in a position to prove Lemma 3.1. Proof of Lemma 3.1. Firstly, using Bony’s decomposition, we decompose the term Rq as follows


4261

Rq = q (v∂x f ) − v q ∂x f + (v − Sq v) q ∂x f = [ q ; Tv ]∂x f + q T∂x f v − T q ∂x f v + q R(v, ∂x f ) − R(v, q ∂x f ) + (v − Sq v) q ∂x f :=

6

Rqi .

(3.5)

i=1

For Rq1 =

|q−q |5 [ q ; Sq −1 v]∂x q f ,

[ q ; Sq −1 v]∂x q f = 2q

thanks to (2.1), one has

h 2q (x − y) Sq −1 v(y) − Sq −1 v(x) ∂x q f (y) dy.

R

Hence, (2.2) applied ensures

1

R (t) q

L2

|q −q|5

∂x Sq −1 vL∞ 2−q ∂x q f L2 cq (t)2−qσ f (t) H σ ∂x vL∞ . (3.6)

For Rq2 = q T∂x f v = q

2

R (t) q

L2

q 0, |q−q |5 Sq −1 ∂x f q v,

Sq −1 ∂x f L2 q vL∞

q 0, |q −q|5

(2.2) applied again implies

k ∂x f L2 2−q q ∂x vL∞

q 0, |q −q|5 kq −2

2k k f L2 2−q q ∂x vL∞ ,

q 0, |q −q|5 kq −2

which yields that

2

R (t) q

L2

2k ck 2−kσ f H σ 2−q q ∂x vL∞

q 0, |q −q|5 kq −2

2(k−q )(1−σ ) ck f H σ 2−qσ ∂x vL∞

q 0, |q −q|5 kq −2

cq (t)2−qσ f H σ ∂x vL∞ ,

(3.7)

where we used the assumption σ < 1. Similarly, for Rq3 = −T q ∂x f v = − q 0, q q−5 Sq −1 q ∂x f q v, we have

3

R (t) q

L2

Sq −1 q ∂x f L2 q vL∞

q 0, q q−5

q f L2 2q−q q ∂x vL∞

q 0, q q−5

q 0, q q−5

2q−q cq 2−qσ f H σ ∂x vL∞ cq (t)2−qσ f H σ ∂x vL∞ .

(3.8)

4262


Since Rq4 = q R(v, ∂x f ) = q

4

R (t) q

L2

q q−5 q v q ∂x f ,

we get from (2.2) that

q ∂x f L2 q vL∞

q q−5

=

q =−1, q q−5

q ∂x f L2 + q vL∞

q ∂x f L2 q vL∞

q 0, q q−5

q f L2 + vL∞

q =−1, q q−5

q f L2 , q ∂x vL∞

q 0, q q−5

which gives rise to

4

R (t) q

L2

cq 2(q−q )σ vL∞ 2−qσ f H σ

q =−1, q q−5

+

cq 2(q−q )σ ∂x vL∞ 2−qσ f H σ

q 0, q q−5

cq (t)2−qσ f H σ vL∞ + ∂x vL∞ ,

(3.9)

where we used the assumption σ > 0. q q ∂x f , we have While for Rq5 = R(v, q ∂x f ) = |q−q |5 q v

5

R (t) q

L2

q q ∂x f L2 q vL∞

|q−q |5

=

q q ∂x f L2 q vL∞

q =−1, |q−q |5

+

q q ∂x f L2 , q vL∞

q 0, |q−q |5

from which and (2.2), we get

5

R (t) q

L2

2−q σ cq 2(q−q )σ vL∞ 2−qσ f H σ

q =−1, |q−q |5

+

cq 2(q−q )σ ∂x vL∞ 2−qσ f H σ

q 0, |q−q |5

cq (t)2−qσ f H σ vL∞ + ∂x vL∞ .

(3.10)

Finally, for Rq6 = (v − Sq v) q ∂x f ,

6

R (t) 2 q vL∞ q ∂x f L2 q L q q

=

q =−1, q q

q vL∞ q ∂x f L2 +

q 0, q q

q vL∞ q ∂x f L2 ,


4263

from which and (2.2), we reach

6

R (t) q

L2

vL∞ q f L2 +

q =−1, q q

2q−q q ∂x vL∞ q f L2

q 0, q q

cq (t)2−qσ f H σ vL∞ + ∂x vL∞ ,

which together with (3.5)–(3.10) completes the proof of Lemma 3.1.

2

4. Wave-breaking criteria Let us first state the following local well-posedness result of (1.2), which was obtained in [21] (up to a slight modification). Lemma 4.1. Suppose that u0 = (u0 , η0 ) ∈ H s × H s−1 , s > 32 . Then there exist T = T (u0 H s ×H s−1 ) > 0 and a unique solution u = (u, η) ∈ C([0, T ); H s × H s−1 ) ∩ C 1 ([0, T ); H s−1 × H s−2 ) of (1.2) with u(0) = u0 . Moreover, the solution u depends continuously on the initial value u0 and the maximal time of existence T > 0 is independent of s. In addition, the Hamiltonian

1 2 u + u2x + η2 dx (4.1) H = H (u, η) = 2 R

is independent of the existence time T . With Lemma 4.1 in hand, we establish the associated Lagrangian scale of (1.2) the initialvalue problem ⎧ ⎨ ∂q = u(t, q), 0 < t < T , (4.2) ∂t ⎩ q(0, x) = x, x ∈ R, where u ∈ C([0, T ), H s ) is the first component of the solution (u, η) of (1.2) with initial data (u0 , ρ0 − 1) ∈ H s × H s−1 with s > 32 , and T > 0 being the maximal time of existence. A direct calculation also yields qtx (t, x) = ux (t, q(t, x))qx (t, x). Hence for t > 0, x ∈ R, we have qx (t, x) = e

t 0

ux (τ,q(τ,x)) dτ

> 0,

which implies that q(t, ·) : R → R is a diffeomorphism of the line for every t ∈ [0, T ). This is inferred that the L∞ norm of any function u(t, ·) ∈ L∞ (R), t ∈ [0, T ) is preserved under the family of diffeomorphisms q(t, ·) with t ∈ [0, T ), that is,

u(t, ·)

L∞ (R)

= u t, q(t, ·) L∞ (R) ,

Similarly, one gets

sup u(t, x) = sup u t, q(t, x) . x∈R

x∈R

t ∈ [0, T ).

(4.3)

4264


The following wave-breaking criterion shows that the wave breaking only depends on the slope of u but not the slope of ρ. This improves the wave-breaking criterion in [21] and [20], where the slopes of both components u and ρ must be considered. The proof of the following result strongly depends on Theorem 3.2 on the localization analysis for the transport equation. Theorem 4.1. Let u0 = (u0 , η0 ) ∈ H s × H s−1 be as in Lemma 4.1 with s > 32 and u = (u, η) being the corresponding solution to (1.2). Assume Tu∗0 > 0 is the maximal time of existence. Then T

Tu0 < ∞

⇒

u0

∂x u(τ )

L∞

dτ = ∞.

0

Proof. We shall prove this theorem by an inductive argument with respect to the index s. To this end, let us first give a control on η(t)L∞ . In fact, applying the maximal principle to the transport equation about ρ, ρt + uρx + ρux = 0, we have

ρ(t)

t

L∞

ρ0 L∞ + C

∂x uL∞ ρL∞ dτ. 0

A simple application of Gronwall’s inequality implies

ρ(t)

L∞

ρ0 L∞ eC

t 0

∂x uL∞ dτ

,

which gives rise to

t

η(t) ∞ ρ(t) ∞ + 1 1 + 1 + η0 L∞ eC 0 ∂x uL∞ dτ . L L

(4.4)

Now let us concentrate our attentions to the proof of Theorem 4.1. This can be achieved as follows. Step 1. For s ∈ ( 32 , 2), applying Theorem 3.2 to the transport equation with respect to η, ηt + uηx + ηux + ux = 0,

(4.5)

we have (for every 1 < s < 2, indeed)

η(t)

t H s−1

η0 H s−1 + C

t η∂x u + ∂x uH s−1 dτ + C

0

Thanks to the Moser-type estimate (2.5), one has

0

ηH s−1 uL∞ + ∂x uL∞ dτ.


η∂x u + ∂x uH s−1 ∂x uH s−1 + C ∂x uH s−1 ηL∞ + ηH s−1 ∂x uL∞ .

4265

(4.6)

Therefore, we have t

η(t)

H s−1

η0 H s−1 + C

∂x u(τ )

H s−1

1 + η(τ ) L∞ dτ

0

t +C

η(τ )

H s−1

u(τ )

L∞

+ ∂x u(τ ) L∞ dτ.

(4.7)

0

On the other hand, Theorem 3.1 applied to the equation about u, 1 2 1 2 2 ut + uux + ∂x g ∗ u + ux + η + η − Au = 0, 2 2 implies (for every s > 1, indeed)

u(t) s u0 H s + C H

t

∂x g ∗ u2 + 1 u2 + 1 η2 + η − Au (τ ) dτ x

s 2 2 H 0

t +C

u(τ )

Hs

∂x u(τ )

L∞

dτ.

0

Thanks to the Moser-type estimate (2.5) and Proposition 2.3, one has

∂x g ∗ u2 + 1 u2 + 1 η2 + η − Au C u2 + 1 u2 + 1 η2 + η − Au x

s

s−1 2 x 2 2 2 H H C uH s−1 uL∞ + ∂x uH s−1 ∂x uL∞

+ ηH s−1 ηL∞ + uH s−1 + ηH s−1 . From this, we reach

u(t)

Hs

t u0 H s + C

u(τ )

Hs

u(τ ) ∞ + ∂x u(τ ) ∞ + 1 dτ L L

0

t +C

η(τ )

0

which together with (4.7) ensures that

H s−1

η(τ )

L∞

+ 1 dτ,

(4.8)

4266


u(t)

Hs

+ η(t)

t H s−1

u0 H s + η0 H s−1 + C

η(τ )

H s−1

+ u(τ ) H s

0

× u(τ ) L∞ + ∂x u(τ ) L∞ + η(τ ) L∞ + 1 dτ.

(4.9)

Thanks to the Gronwall’s inequality again, one can see

u(t)

Hs

t + η(t) H s−1 u0 H s + η0 H s−1 eC 0 (uL∞ +∂x uL∞ +ηL∞ +1) dτ . (4.10)

Using the Sobolev embedding theorem H s → L∞ (for s > 12 ), we get from (4.1) that

u(t)

L∞

C u0 H 1 + η0 L2 ,

(4.11)

which together with (4.4) and (4.10) implies that

u(t)

Hs

t

+ η(t) H s−1 u0 H s + η0 H s−1 eC1 (t+1) exp{ 0 C∂x u(τ )L∞ dτ } ,

where C1 = C1 (u0 H 1 , η0 L2 , η0 L∞ ). Therefore, if the maximal existence time Tu0 < ∞ satisfies obtain from (4.12) that

Tu0 0

(4.12)

∂x u(τ )L∞ dτ < ∞, we

lim sup u(t) H s + η(t) H s−1 < ∞

(4.13)

t→Tu

0

contradicts the assumption on the maximal existence time Tu0 < ∞. This completes the proof of Theorem 4.1 for s ∈ ( 32 , 2). Step 2. For s ∈ [2, 52 ), applying Theorem 3.1 to the transport equation (4.5), we have

η(t)

t H s−1

η0 H s−1 + C

(η∂x u + ∂x u)(τ )

t H s−1

dτ + C

0

ηH s−1 ∂x u

1

dτ.

1

dτ,

L∞ ∩H 2

0

(4.6) applied implies that t

η(t)

H s−1

η0 H s−1 + C

∂x uH s−1 1 + ηL∞ dτ + C

0

t ηH s−1 ∂x u

L∞ ∩H 2

0

which together with (4.8) yields

u(t)

Hs

+ η(t) H s−1 u0 H s + η0 H s−1 t +C 0

η(τ )

H s−1

+ u(τ ) H s u

H

3 + 2 0

+ η(τ ) L∞ + 1 dτ

G. Gui, Y. Liu / Journal of Functional Analysis 258 (2010) 4251–4278 1

4267

1

with 0 < 0 < 12 , where we used the fact H 2 +0 → L∞ ∩ H 2 . Gronwall’s inequality applied gives that

u(t)

Hs

+ η(t)

u0 H s

H s−1

C + η0 H s−1 e

t

0 (u 3 +0 +η(τ )L∞ +1) dτ H2

.

(4.14)

Therefore, thanks to the uniqueness of solution in Lemma 4.1, (4.1) and (4.13), we get that: if the Tu maximal existence time Tu0 < ∞ satisfies 0 0 ∂x u(τ )L∞ dτ < ∞, then (4.14) implies that

lim sup u(t) H s + η(t) H s−1 < ∞ t→Tu

0

contradicts the assumption on the maximal existence time Tu0 < ∞. This completes the proof of Theorem 4.1 for 2 s < 52 . Step 3. For 2 < s < 3, by differentiating once (4.5) with respect to x, we have ∂t ηx + u∂x (ηx ) + 2ux ηx + ηuxx + uxx = 0.

(4.15)

Theorem 3.2 applied to (4.15) implies that

ηx (t)

t η0x H s−2 + C

H s−2

(2ηx ux + η∂x ux + ∂xx u)(τ )

H s−2

dτ

0

t +C

ηx (τ )

H s−2

u(τ )

L∞

+ ∂x u(τ ) L∞ dτ

0

t η0x H s−2 + C

ηH s−1 + uH s uL∞ + ∂x uL∞ + ηL∞ + 1 dτ,

0

(4.16) where we used the following Moser-type estimates (from (2.6)):

ηx ux H s−2 C ∂x uH s−1 ηL∞ + ∂x ηH s−2 ux L∞ and

η∂x ux H s−2 C ηH s−1 ∂x uL∞ + uxx H s−2 ηL∞ . (4.16), together with (4.8) and (4.7) (where s − 1 is replaced by s − 2), implies that

η(t)

H

s−1 + u(t)

t Hs

η0 H s−1 + u0 H s + C

× u(τ )

η(τ )

0

∞ + ∂x u(τ )

L

L∞

H s−1

+ u(τ ) H s

+ η(τ ) L∞ + 1 dτ.

4268


Gronwall’s inequality applied again gives (4.10). Hence, using arguments as in Step 1, it completes the proof of Theorem 4.1 for 2 < s < 3. Step 4. For s = k ∈ N, k 3, by differentiating (4.5) k − 2 times with respect to x, we have

∂t ∂xk−2 η + u∂x ∂xk−2 η +

C1 ,2 ∂x1 +1 u∂x2 +1 η + η∂x ∂xk−2 u + ∂xk−1 u = 0.

1 +2 =k−3, 1 ,2 0

(4.17) Applying Theorem 3.1 to the transport equation (4.17), we have

k−2

∂ η(t) x

H1

t

∂ k−2 η0 x

H1

+C

k−2

∂ η(τ ) x

H1

∂x u(τ )

1

L∞ ∩H 2

dτ

0

t

+C

0

C1 ,2 ∂x1 +1 u∂x2 +1 η + η∂x

1 +2 =k−3, 1 ,2 0

k−2

k−1 ∂x u + ∂x u (τ )

dτ. H1

Since H 1 is an algebra, we have

k−2

η∂x ∂ u 1 Cη 1 ∂ k−1 u 1 Cη 1 uH s H H x x H H and

C1 ,2 ∂x1 +1 u∂x2 +1 η

1 H 1 +2 =k−3, 1 ,2 0

C

1 +2 =k−3, 1 ,2 0

C1 ,2 ∂x1 +1 u H 1 ∂x2 +1 η H 1 CuH s−1 ηH s−1 .

Hence,

k−2

∂ η(t) x

H1

t

∂ k−2 η0 x

H1

+C

ηH s−1 + uH s uH s−1 + ηH 1 + 1 dτ. (4.18)

0

(4.18), together with (4.8) and (4.7) (where s − 1 is replaced by 1), implies that

η(t)

H s−1

+ u(t) H s t

η0 H s−1 + u0 H s + C

η(τ )

0

Gronwall’s inequality applied yields that

H s−1

+ u(τ ) H s u(τ ) H s−1 + η(τ ) H 1 + 1 dτ.


u(t)

Hs

4269

t + η(t) H s−1 u0 H s + η0 H s−1 eC 0 (uH s−1 +ηH 1 +1) dτ .

Therefore, if the maximal existence time Tu0 < ∞ satisfies the uniqueness of solution in Lemma 4.1, we get that

u(t)

H s−1

Tu0 0

(4.19)

∂x u(τ )L∞ dτ < ∞, thanks to

+ η(t) H 1

is uniformly bounded by the induction assumption, which together with (4.19) implies

lim sup u(t) H s + η(t) H s−1 < ∞. t→Tu

0

This leads to a contradiction. Step 5. For k < s < k + 1 with k ∈ N, k 3, by differentiating (4.5) k − 1 times with respect to x, we have

∂t ∂xk−1 η + u∂x ∂xk−1 η +

C1 ,2 ∂x1 +1 u∂x2 +1 η + η∂x ∂xk−1 u + ∂xk u = 0.

1 +2 =k−2, 1 ,2 0

Theorem 3.2 applied again implies that

k−1

∂ η(t) x

H s−k

t

∂ k−1 η0 x

H s−k

+C

k−1

∂ η(τ ) x

H s−k

u(τ )

L∞

+ ∂x u(τ ) L∞ dτ

0

t

+C 0

1 +2 =k−2, 1 ,2 0

C1 ,2 ∂x1 +1 u∂x2 +1 η + η∂x ∂xk−1 u + ∂xk u (τ )

dτ. H s−k

Using the Moser-type estimate (2.6) and the Sobolev embedding inequality, we have for ∀0 < 0 < 12

k−1

η∂x ∂ u x

H s−k

C ηL∞ ∂xk u H s−k + ηH s−k+1 ∂xk−1 u L∞

C ηL∞ uH s + ηH s−k+1 u k− 1 +0 H

and

1 +2 =k−2, 1 ,2 0

C1 ,2 ∂x1 +1 u∂x2 +1 η

C

1 +2 =k−2, 1 ,2 0

C u

H

k− 21 +0

2

H s−k

C1 ,2 ∂x1 +1 u L∞ ∂x2 +1 η H s−k + ∂x2 η L∞ ∂x1 +1 u H s−k+1

ηH s−1 + η

H

k− 23 +0

uH s .

4270


Hence,

k−1

∂ η(t) x

H s−k

t

∂ k−1 η0 x

H s−k

× u

+C

η(τ )

0 H

k− 21 +0

+ η

H

k− 23 +0

H s−1

+ u(τ ) H s

+ 1 dτ.

(4.20)

(4.20), together with (4.8) and (4.7) (where s − 1 is replaced by s − k), implies that

η(t) s−1 + u(t) s η0 s−1 + u0 H s + C H H H × u

t

η(τ )

0 H

k− 21 +0

+ η

H

k− 23 +0

H s−1

+ u(τ ) H s

+ 1 dτ.

Applying Gronwall’s inequality then gives that

u(t)

Hs

+ η(t)

H s−1

C u0 H s + η0 H s−1 e

t

0 (u k− 1 +0 +η k− 3 +0 +1) dτ 2 2 H H

In consequence, if the maximal existence time Tu0 < ∞ satisfies thanks to the uniqueness of solution in Lemma 4.1, then we get that

u(t)

H

k− 21 +0

+ η(t)

H

Tu0 0

.

∂x u(τ )L∞ dτ < ∞,

k− 23 +0

is uniformly bounded by the induction assumption, which implies

lim sup u(t) H s + η(t) H s−1 < ∞, t→Tu

0

which leads to a contradiction. Therefore, from Step 1 to Step 5, we complete the proof of Theorem 4.1.

2

Theorem 4.2. Let (u0 , η0 ) be as in Lemma 4.1 with s > 32 and u = (u, η) being the corresponding solution to (1.2). Then the corresponding solution blows up in finite time if and only if lim inf ux (t, x) = −∞.

t→Tu x∈R 0

(4.21)

Lemma 4.2. Let (u, η) (with η := ρ − 1) be the solution of (1.2) with initial value (u0 , ρ0 − 1) ∈ H s (R) × H s−1 (R), s > 32 , and T the maximal existence time. If there is M1 0 such that inf

(t,x)∈[0,T )×R

then

ux (t, x) −M1 ,

(4.22)


ρ(t, ·)

L∞

ρ0 L∞ eM1 t ,

4271

(4.23)

sup ux (t, x) u0,x L∞ + C1 + ρ0 L∞ eM1 t

(4.24)

x∈R

hold for t ∈ [0, T ), with C1 =

1 + A2 2

1 2

(u0 , ρ0 − 1)

H 1 ×L2

,

(4.25)

and C a positive constant depending only on A, M1 and the norm (u0 , ρ0 − 1)H s ×H s−1 . Proof. By Lemma 4.1 and a simple density argument, it is needed only to show the desired results are valid when s 3. So in the sequel of this section s = 3 is taken for simplicity of notation. Differentiating both sides of the first equation of (1.2) with respect to x and using the identity −∂x2 g ∗ f = f − g ∗ f lead to 1 1 1 1 utx + uuxx + u2x = A∂x g ∗ ux + u2 + ρ 2 − g ∗ u2 + u2x + ρ 2 . 2 2 2 2

(4.26)

Given x ∈ R, let

M(t) = ux t, q(t, x) ,

γ (t) = ρ t, q(t, x) ,

(4.27)

t ∈ [0, T ), with q(t, x) determined in (4.2). Using these notations, Eq. (4.26) and the second one of (1.2) can be rewritten, respectively, as

1 1 M (t) = − M 2 + γ 2 + f t, q(t, x) , 2 2

γ (t) = −γ M,

(4.28)

for t ∈ [0, T ), where the notation denotes the derivative with respect to t and f represents the function 1 1 f = A∂x g ∗ ux + u2 − g ∗ u2 + u2x + ρ 2 . 2 2

(4.29)

Hence, we have 1 1 1 f = Agx ∗ ux + u2 − g ∗ u2 + u2x − g ∗ 1 − g ∗ (ρ − 1) − g ∗ (ρ − 1)2 2 2 2 1 1 |A||gx ∗ ux | + u2 − + g ∗ (ρ − 1), 2 2 with the help of g ∗ (u2 + 12 u2x ) 12 u2 (cf. [9]). Applying Young’s inequality and g(x) = 12 e−|x| leads to

4272


1 1 1 |A||gx ∗ ux |(t, x) |A|gx L2 ux L2 |A|ux L2 + A2 ux 2L2 , 2 4 4 1 1 g ∗ (ρ − 1)(t, x) g 2 ρ − 1 2 ρ − 1 2 + 1 ρ − 12 2 , L L L L 2 4 4

(4.30) (4.31)

for (t, x) ∈ [0, T ) × R. On the other hand, the continuous embedding of H 1 (R) into L∞ (R) gives (cf. [11], for example, for the best embedding constant) 2u (t, x) 2

2

u R

+ u2x

2

u

+ u2x

+ (ρ − 1) =

R

u20 + u20x + (ρ0 − 1)2 ,

2

(4.32)

R

for all (t, x) ∈ [0, T ) × R, where we used the fact that H (u, ρ − 1) is the conservation law of the system (1.2) in the last identity. Combining (4.30), (4.31), and (4.32) together gives

2

2

1 1 1 1 + A2 (u, ρ − 1) H 1 ×L2 = 1 + A2 (u0 , ρ0 − 1) H 1 ×L2 = C12 , (4.33) 4 4 2 2 where the conservation law H (u, η) = R u + u2x + η2 of (1.2) was used again in the second identity and C1 was introduced in (4.25). Similarly, we have f

1 1 1 −f |A||gx ∗ ux | + g ∗ u2 + u2x + + g ∗ (ρ − 1) + g ∗ (ρ − 1)2 2 2 2

1 1 1 2 1 2 1 1 1 1 u + ux + A2 ux 2L2 + + + + ρ − 12L2 + ρ − 12L2 , 4 4 2 2 L1 2 4 4 4 where we used the estimate g ∗ (u2 + 12 u2x )(t, x) gL∞ u2 + 12 u2x L1 12 u2H 1 . Therefore, we get −f 1 +

2

1 + A2

(u, ρ − 1) 2 1 2 1 + 1 + A (u0 , ρ0 − 1) 2 1 2 1 + C 2 . 1 H H ×L ×L 2 2

(4.34)

In view of the definition of M(t) in (4.27), the assumption (4.22) is now expressed as, for each x ∈ R, M(t) −M1 ,

for t ∈ [0, T ).

In view of this condition, it then follows from the second equation of (4.28) that, for each x ∈ R, t

ρ t, q(t, x) = γ (t) = γ (0)e 0 −M(τ ) dτ ρ0 L∞ eM1 t ,

(4.35)

for t ∈ [0, T ). Hence combining this with (4.3) leads to (4.23). Given any x ∈ R, let us define P (t) = M(t) − u0,x L∞ − C1 − ρ0 L∞ eM1 t , with M(t) = ux (t, q(t, x)) and C1 in (4.25). Observe that P (t) is a C 1 -differentiable function in [0, T ) and satisfies


4273

P (0) = M(0) − u0,x L∞ − C1 − ρ0 L∞ u0,x (x) − u0,x L∞ 0. We now claim P (t) 0,

for all t ∈ [0, T ).

(4.36)

Assume the contrary that there is t0 ∈ [0, T ) such that P (t0 ) > 0. Let t1 = max t < t0 ; P (t) = 0 . Then P (t1 ) = 0 and P (t1 ) 0, or equivalently, M(t1 ) = u0,x L∞ + C1 + ρ0 L∞ eM1 t1 ,

(4.37)

M (t1 ) M1 ρ0 L∞ eM1 t1 > 0.

(4.38)

and

From (4.32), (4.35), (4.37), and the first equation of (4.28), it follows that

1 1 M (t1 ) = − M 2 (t1 ) + γ 2 (t1 ) + f t1 , q(t1 , x) 2 2

2 1 1 1 − u0,x L∞ + C1 + ρ0 L∞ eM1 t1 + ρ0 2L∞ e2M1 t1 + C12 2 2 2 0, a contradiction to (4.38), so the claim (4.36) is valid. Therefore, the arbitrarily chosen of x and (4.3) imply (4.24). 2 Proof of Theorem 4.2. Assume (4.21) is not valid. Then there is some positive number M1 > 0 such that ux (t, x) −M1 holds for (t, x) ∈ [0, T ) × R. It now follows from (4.24) in Lemma 4.2 that ux (t, x) CeM1 t , with C a positive constant depending only on A, M1 and the norm (u0 , ρ0 − 1)H s ×H s−1 . Theorem 4.1 applied implies that the maximal existence time Tu0 = ∞, which contradicts the assumption on the maximal existence time Tu0 < ∞. Conversely, the Sobolev embedding theorem H s (R) → L∞ (R) (with s > 12 ) implies that if (4.21) holds, the corresponding solution blows up in finite time, which completes the proof of Theorem 4.2. 2

4274


Theorem 4.3. Assume that the initial value (u0 , η0 ) ∈ H s × H s−1 with s > 32 . Let Tu0 > 0 be the maximal time of existence for the corresponding solution (u, η) to the system (1.2). Then we have T

Tu0 < ∞

⇒

u0

∂x u(τ )

0 B∞,∞

+ ρ(τ ) − 1 B 0

∞,∞

dτ = ∞.

0

Proof. We only need to prove this theorem for the case 32 < s < 2. For s 2, the induction argument as in the proof of Theorem 4.1 will complete the proof of Theorem 4.3. Thanks to Proposition 2.2, we have for s > 32 ,

log e + ∂x uH s−1 ∂x uL∞ C 1 + ∂x uB∞,∞ 0

(4.39)

ηL∞ C 1 + ηB∞,∞ log e + ηH s−1 . 0

(4.40)

and

Plugging (4.39) and (4.40) into (4.10), and using the fact (4.11), we get

u(t)

Hs

+ η(t) H s−1

Ct+C u0 H s + η0 H s−1 e

t

+η(τ )B 0 ) log(e+u(τ )H s +η(τ )H s−1 ) dτ 0 0 (∂x u(τ )B∞,∞ ∞,∞

Therefore,

log e + u(t) H s + η(t) H s−1

log e + u0 H s + η0 H s−1 + Ct t +C

∂x u(τ )

0 B∞,∞

+ η(τ ) B 0

∞,∞

log e + u(τ ) H s + η(τ ) H s−1 dτ.

0

Applying Gronwall’s inequality yields

log e + u(t) H s + η(t) H s−1 e

C

t

+η(τ )B 0 ) dτ 0 0 (∂x u(τ )B∞,∞ ∞,∞

Hence, the proof of Theorem 4.3 is complete.

log(e + u0 H s + η0 H s−1 ) + Ct . 2

.


4275

5. Global existence In view of the criterion for wave breaking (Theorem 4.1), a sufficient condition of global solutions can be obtained in the following. Theorem 5.1 (Global solution). Let (u0 , ρ0 − 1) ∈ H s (R) × H s−1 (R) with s > 32 , and T > 0 being the maximal time of existence of the solution (u, ρ) to the system (1.1) with initial data (u0 , ρ0 ). If inf ρ0 (x) > 0,

(5.1)

x∈R

then T = +∞, and the solution (u, ρ) is global. Remark 5.1. Theorem 5.1 improves the result of the global solutions in [20], where the special case s = 2 is required. To prove Theorem 5.1, we need the following lemma. Lemma 5.1. Assume (u, ρ) is the local solution of (1.1) with the initial value (u0 , ρ0 − 1) ∈ H s (R) × H s−1 (R), s > 32 , and T the maximal existence time. If infx∈R ρ0 (x) > 0, then

ux t, q(t, x) , ρ t, q(t, x)

1 C5 eC4 t |ρ0 (x)|

(5.2)

hold for all t ∈ [0, T ), with (see (4.25) for C1 ) C4 =

2

3 3 1 + C12 = + 1 + A2 (u0 , ρ0 − 1) H 1 ×L2 , 2 2 2

C5 = 1 + u0,x 2L∞ + ρ0 2L∞ , and positive constant C depending only on A and the norm (u0 , ρ0 − 1)H s ×H s−1 . Proof. In view of the proof of Lemma 4.2, by Lemma 4.1 and a simple density argument, it suffices to show that the desired results are valid when s 3. So in the sequel of this section s = 3 is taken for simplicity of notation. Observe that the system (1.2) leads to the following ordinary differential equations (see Lemma 4.2 for derivation) for a fixed x ∈ R,

1 1 M (t) = − M 2 + γ 2 + f t, q(t, x) , 2 2

γ (t) = −γ M,

(5.3)

for t ∈ [0, T ) with notation M(t) = ux (t, q(t, x)), γ (t) = ρ(t, q(t, x)) defined in (4.27) and f in (4.29). The second equation of (5.3) implies that γ (t) and γ (0) are of the same sign. For every x ∈ R satisfying γ (0) = ρ0 (x) > 0, define the Lyapunov function (cf. [12]), w(t) := γ (0)γ (t) +

γ (0) 1 + M 2 (t) , γ (t)

4276


which is a positive function of t ∈ [0, T ). By (5.3), it yields

2 γ (0) γ 1 + M 2 + γ (0)MM

2 γ γ

1 2 = γ (0)M f t, q(t, x) + γ 2

1 γ (0) 2 1+M C4 w(t), f t, q(t, x) + γ 2

w (t) = γ (0)γ −

in [0, T ), where |f | 1 + C12 is derived from (4.33) and (4.34). The preceding differential inequality gives w(t) w(0)eC4 t = C5 eC4 t ,

t ∈ [0, T )

(5.4)

with the help of w(0) = ρ02 (x) + 1 + u20,x (x) 1 + u0,x 2L∞ + ρ0 2L∞ = C5 . Recalling that γ (t) and γ (0) are of the same sign, the definition of w implies γ (0)γ (t) w(t) and |γ (0)||M(t)| w(t). By (5.4),

ux t, q(t, x) = M(t)

ρ t, q(t, x) = γ (t)

1 1 w(t) C5 eC4 t , |γ (0)| |ρ0 (x)| 1 1 w(t) C5 eC4 t |γ (0)| |ρ0 (x)|

are valid for t ∈ [0, T ). Thus the conclusions of (5.2) are obtained.

2

Proof of Theorem 5.1. Assume the contrary that T < ∞ and the solution blows up in finite time. It then transpires from Theorem 4.1 that T

ux (t, x)

L∞

dt = ∞.

(5.5)

0

Note that infx∈R ρ0 (x) > 0. By (5.2) in Lemma 5.1, we have ux (t, x)

1 1 CeCt CeCT < ∞ |ρ0 (x)| infx∈R ρ0 (x)

for all (t, x) ∈ [0, T ) × R, a contrary to (5.5). So T = +∞, and the solution (u, ρ) is global.

2


4277

Acknowledgments The work of G. Gui is partly supported by Morningside Center of Mathematics, Chinese Academy of Sciences and the work of Y. Liu is partially supported by the NSF grant DMS0906099. The authors would like to thank the referees for constructive suggestions and comments. References [1] J.M. Bony, Calcul symbolique et propagation des singularités pour les q´ uations aux drivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981) 209–246. [2] A. Bressan, A. Constantin, Global conservative solutions of the Camassa–Holm equation, Arch. Ration. Mech. Anal. 183 (2007) 215–239. [3] A. Bressan, A. Constantin, Global dissipative solutions of the Camassa–Holm equation, Appl. Anal. 5 (2007) 1–27. [4] J.C. Burns, Long waves on running water, Math. Proc. Cambridge Philos. Soc. 49 (1953) 695–706. [5] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661– 1664. [6] J.Y. Chemin, Localization in Fourier space and Navier–Stokes system, in: Phase Space Analysis of Partial Differential Equations, in: CRM Series, Scuola Norm. Sup., Pisa, 2004, pp. 53–136. [7] J.Y. Chemin, Perfect Incompressible Fluids, Oxford Univ. Press, New York, 1998. [8] M. Chen, S. Liu, Y. Zhang, A 2-component generalization of the Camassa–Holm equation and its solutions, Lett. Math. Phys. 75 (2006) 1–15. [9] A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000) 321–362. [10] A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186. [11] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998) 229–243. [12] A. Constantin, R.I. Ivanov, On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A 372 (2008) 7129–7132. [13] A. Constantin, T. Kappeler, B. Kolev, P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom. 31 (2007) 155–180. [14] A. Constantin, B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv. 78 (2003) 787–804. [15] A. Constantin, W.A. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004) 481–527. [16] R. Danchin, A few remarks on the Camassa–Holm equation, Differential Integral Equations 14 (2001) 953–988. [17] R. Danchin, A note on well-posedness for Camassa–Holm equation, J. Differential Equations 192 (2003) 429–444. [18] J. Escher, O. Lechtenfeld, Z.Y. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation, Discrete Contin. Dyn. Syst. 19 (2007) 493–513. [19] G. Falqui, On a Camassa–Holm type equation with two dependent variables, J. Phys. A 39 (2006) 327–342. [20] C. Guan, Z.Y. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system, J. Differential Equations 248 (8) (2010) 2003–2014. [21] G. Gui, Y. Liu, On the Cauchy problem for the two-component Camassa–Holm system, Math. Z. (2010) doi:10.1007/s00209-009-0660-2. [22] D.D. Holm, L. Nraigh, C. Tronci, Singular solutions of a modified two-component Camassa–Holm equation, Phys. Rev. E 79 (2009) 016601. [23] R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion 46 (2009) 389–396. [24] R.S. Johnson, The Camassa–Holm equation for water waves moving over a shear flow, Fluid Dynam. Res. 33 (2003) 97–111. [25] R.S. Johnson, Nonlinear gravity waves on the surface of an arbitrary shear flow with variable depth, in: Nonlinear Instability Analysis, in: Adv. Fluid Mech., vol. 12, Comut. Mech. Southampton, 1997, pp. 221–243. [26] R.S. Johnson, On solutions of the Burns condition, Geophys. Astrophys. Fluid Dyn. 57 (1991) 115–133.

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