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Journal of Functional Analysis 260 (2011) 1257–1284 www.elsevier.com/locate/jfa An exact estimate result for a class of...

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Journal of Functional Analysis 260 (2011) 1257–1284 www.elsevier.com/locate/jfa

An exact estimate result for a class of singular equations with critical exponents Sun Yijing a,∗ , Wu Shaoping b a School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, PR China b Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, PR China

Received 18 June 2009; accepted 26 November 2010 Available online 14 December 2010 Communicated by G. Godefroy

Abstract We consider the singular boundary value problem −u =

h(x) + λup−1 uγ

in Ω,

u = 0 on ∂Ω

with p = 2N/(N − 2), γ ∈ (0, 1). It is well known that there exists λ∗ > 0 such that the problem has a solution for all λ ∈ (0, λ∗ ) and no solution for λ > λ∗ . We obtain an exact result for λ∗ (Ω, p, γ , h). © 2010 Elsevier Inc. All rights reserved. Keywords: An exact estimate result; Extremal value; Singular nonlinearity; Critical exponent

1. Introduction Let Ω be a smooth bounded domain in RN , N 3, and p = 2N/(N − 2). We consider the range of λ in the singular problem u +

h(x) + λup−1 = 0 in Ω, uγ

u > 0 in Ω,

* Corresponding author.

E-mail addresses: [email protected], [email protected] (Y.J. Sun). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.018

u=0

on ∂Ω

(1λ )

1258

Y.J. Sun, S.P. Wu / Journal of Functional Analysis 260 (2011) 1257–1284

where h ∈ L∞ (Ω) is like distα (x, ∂Ω) with α − γ 0 (i.e. there exist two positive constants m, M such that m distα (x, ∂Ω) h(x) M distα (x, ∂Ω), ∀x ∈ Ω), γ ∈ (0, 1), and λ > 0 is a parameter. Equations of the type (1λ ) have been intensively studied for both bounded and unbounded domains because of its wide applications to physical models in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogenous catalysts, glacial advance, etc. (cf. [2,8–11,13,15,16,19–26,28]). In [10] Coclite and Palmieri proved that there exists λ∗ > 0 such that (1λ ) has a solution for all λ ∈ (0, λ∗ ) and no solution for λ > λ∗ . Furthermore, our previous work [26] and Yang [28] showed the multiplicity of (1λ ). We are now interested in the dependence of λ∗ on Ω, p, γ and h (i.e. how large is λ∗ ?). This is precisely the aim of this paper. As we shall see in Section 3, for λ in an exact range (see Section 2), (1λ ) has at least two solutions. To see this, we give a complete description of a constraint set associated to the action functional and use careful estimates inspired by these in [26,27]. We emphasize that there is no restriction on the shape of Ω. Thus we obtain uniform lower bounds for λ∗ = λ∗ (Ω, p, γ , h). There, it must be said that the method of sub and supersolutions does not adapt for dealing with estimates of this type, since for general Ω (without symmetric property, say) precise information about sub/supersolutions is no longer possible and explicit calculations for λ∗ cannot be actually carried out. The distance condition on h(x) has already been introduced in the study of regularity of pure singular problem (i.e. λ = 0) (cf. [12,16,18]). Gomes [16], del Pino [12] proved that the unique solution of (10 ) belongs to C 1,β (Ω), ∀β ∈ [0, 1). Moreover, Gui and Lin [18] established the following estimate for the unique solution c1 dist(x, ∂Ω) u(x) c2 dist(x, ∂Ω),

∀x ∈ Ω.

As it turns out, the condition also plays an important role in the combined effect of singular and critical nonlinearities, which contributes to the boundedness of the gradient of desired minimizers. Actually, from our arguments the behavior near the boundary (i.e. h(x) ∼ distα (x, ∂Ω) for all x near ∂Ω) is sufficient to guarantee the boundedness. To state our results we first introduce some notations and definitions. Throughout the paper we assume that Ω ∈ L, where L = Ω ⊂ RN ; Ω bounded open and regular say C 1,β . measure of A. C deFor a measurable set A ⊂ RN denote with |A| the N -dimensional Lebesgue note (possibly different) positive constants. Furthermore, u2 = Ω |∇u|2 dx denotes the usual norm of u in H01 (Ω), while for any other function space X, we denote its norm by · X . We denote by the first eigenfunction e1 with e1 + λ1 e1 = 0 in Ω, e1 |∂Ω = 0, 0 e1 1, and we know that 0 < d0 e1 (x) dist(x, ∂Ω)−1 d1 on Ω for some constants d0 , d1 . We assume N 3, let p = 2N/(N − 2) and set |∇u|2 dx 1 u ∈ H (Ω), u =

0 S = inf Ω p 0 ( Ω |u| dx)2/p the best Sobolev constant. It is well known that S is independent of Ω and depends only on N . The infimum can be achieved by the function


U ∗ (x) =

1259

1 (1 + |x|2 )(N −2)/2

that is, ∗ 2 N |∇U | dx S= R . ∗ p ( RN |U | dx)2/p The functional associated to (1λ ) is 1 Iλ (u) = 2

1 |∇u| dx − 1−γ

2

Ω

1−γ

h(x)|u|

λ dx − p

Ω

|u|p dx,

∀u ∈ H01 (Ω).

Ω

Clearly Iλ is only a continuous functional on H01 (Ω). Define the constraint set Nλ = t (u)u: u ∈ H01 (Ω)\{0} where t (u) are the zeros of the map 1

d Iλ (tu) dt

2−p 2 −γ −p+1 1−γ |∇u| dx − t h(x)|u| dx − λ |u|p dx. =t

t → φ(t, u) =

t p−1

Ω

Ω

Ω

Let Nλ+ (resp. Nλ− ) be the subset of Nλ corresponding to t (u) with d dt |t=t (u) φ(t, u) < 0), that is

d dt |t=t (u) φ(t, u)

> 0 (resp.

Nλ± = v = t (u)u ∈ Nλ : (2 − p) |∇v|2 dx + (p + γ − 1) h(x)|v|1−γ dx > ( 0 a.e. in Ω and

∇u · ∇ϕ dx −

Ω

h(x) ϕ dx − λ uγ

Ω

up−1 ϕ dx = 0,

∀ϕ ∈ H01 (Ω).

Ω

Our main result is as follows: Theorem. Let λ∗ be the extremal value for problem (1λ ). Then ∗

λ (Ω, p, γ , h)

1+γ p+γ −1

p−2 p+γ −1

p−2 1+γ

S |Ω|2/N

p+γ −1 1+γ

1 h∞

p−2 1+γ

:= Λ.

1260


For general domains without symmetric properties it is difficult to derive an exact result for λ∗ . Still few general results are known except in [14] Gazzola and Malchiodi provide uniform lower bounds of λ∗ for the problem −u = λ(1 + u)p , 1 < p (N + 2)/(N − 2) and our recent paper [24,25] for singular-subcritical and nonsingular-critical cases. The outline of the paper is the following. In Section 2 we obtain the value Λ through the connection between Nλ and the fibrering maps (i.e., maps of the form t → Iλ (tu); see Alves and El Hamidi [1], Brown and Zhang [7]). In Section 3, we discuss infN + Iλ under λ ∈ (0, Λ). λ First, we provide an estimate for u0 as a weak limit of a minimizing sequence for infN + Iλ , λ which will influence a series of estimates of critical case since ∇u turns rather delicate in singular case (see Lazer and Mckenna [21]). Then, with the help of the estimate and the family Uε,a (x) := η(x)ε −(N −2)/2 U ∗ ( x−a ε ), we manage to locate that u0 ∈ Nλ . Finally, using the ideas of Graham-Eagle [17] and the location information we prove that u0 is a solution of (1λ ). By taking advantage of the structure of Nλ under λ ∈ (0, Λ), we discuss the problem infN − Iλ and λ obtain the multiplicity of (1λ ). In Section 4 we provide uniform bounds for λ∗ (Ω, p, γ , h). 2. The number Λ Lemma 1. Suppose that λ ∈ (0, Λ). Then for any u ∈ H01 (Ω)\{0}, φ(t, u) has exactly two zeros t ∓ (u) which satisfy 0 < t − (u) < t + (u),

t − (u)u ∈ Nλ+ , t + (u)u ∈ Nλ− .

Proof. Define φ : (0, ∞) × {H01 (Ω)\{0}} → R by

φ(t, u) = t 2−p

|∇u|2 dx − t −γ −p+1

Ω

h(x)|u|1−γ dx − λ Ω

|u|p dx. Ω

Since φ(t, u) is increasing/decreasing along t > 0, it is easily derived that

(p − 2)∇u22 tmax,u = (p + γ − 1) Ω h(x)|u|1−γ dx

−1/(1+γ ) ,

φ(tmax,u , u) (p−2)/(1+γ )

2(p+γ −1)/(1+γ ) ∇u2 p−2 1+γ = − λ |u|p dx p+γ −1 p+γ −1 ( Ω h(x)|u|1−γ dx)(p−2)/(1+γ ) Ω

>

p−2 1+γ p+γ −1 p+γ −1 p 1 p ∇u2 −λ √ S p

:= D(λ)∇u2 , and

(p−2)/(1+γ )

1 h∞

(p−2)/(1+γ )

√ |Ω|

S

(p−2)(1−γ ) (1+γ )

p−(1−γ ) p(1−γ )

(2)


D(λ) = 0 iff

1261

λ = Λ,

where we have used Hölder’s and Sobolev inequalities, and the following two relations (p − 2)(1 − γ ) 2(p + γ − 1) +p= , 1+γ 1+γ 2 p+γ −1 p − (1 − γ ) p − 2 p − 2 (p + γ − 1) · = · = . p 1+γ p (1 + γ ) N 1+γ Since λ < Λ, it follows D(λ) > 0 and φ(tmax,u , u) > 0, therefore φ(t, u) has exactly two zeros 0 < t − (u) < t + (u), that is

∇v22 −

h(x)|v|1−γ − λ

Ω

|v|p = 0,

where v = t ∓ (u)u

Ω

such that t − (u)u ∈ Nλ+ , This completes the proof of Lemma 1.

t + (u)u ∈ Nλ− .

2

Set E0 =

p+γ −1 p−1

1 1+γ

1

1+γ h∞

1+γ E(λ) = λ(p + γ − 1)

1

+1

1−γ

|Ω| 2 N 1+γ , √ 1−γ 1+γ S

(N −2)/4

√ N/2 S .

Lemma 2. Suppose that λ ∈ (0, Λ). Then Nλ has a gap structure in the sense that ∇u2 < E0 , ∀u ∈ Nλ+ ; ∇U 2 > E(λ) > E0 , ∀U ∈ Nλ− . Clearly, E(λ) → ∞ as λ → 0. Proof. If u ∈ Nλ+ then necessarily (p − 2)∇u22 − (p + γ − 1) other hand, for all U ∈ Nλ− (⊂ Nλ )

Ω

h(x)|u|1−γ dx < 0. On the

p

(1 + γ )∇U 22 − λ(p + γ − 1)U p

2 1−γ = − (p − 2)∇U 2 − (p + γ − 1) h(x)|U | dx < 0. Ω

Consequently, ∇U 2 > E(λ), ∇u2 < E0 ,

∀U ∈ Nλ− , ∀u ∈ Nλ+ .

(3) (4)

1262


Surprisingly enough, E(λ) = E0

λ = Λ,

iff

we conclude that ∇U 2 > E(λ) > E0 > ∇u2 ,

∀u ∈ Nλ+ , U ∈ Nλ−

(5)

for all λ ∈ (0, Λ), where we have used the following two relations 1 1−γ p − (1 − γ ) 1 = + , p(1 + γ ) 2 N 1+γ 1−γ 4 2N 1−γ 2(p + γ − 1) + = (p − 2) + p = . 1+γ N −2 N −2 1+γ 1+γ This completes the proof of Lemma 2.

2

Lemma 3. Suppose that λ ∈ (0, Λ). Then Nλ− is a closed set in H01 -topology. Proof. From the arguments of Lemma 1 we derive that if u ∈ H01 (Ω)\{0} satisfies the following two equalities

∇u22 −

h(x)|u|1−γ dx − λ

Ω

(p − 2)∇u22 − (p + γ − 1)

|u|p dx = 0, Ω

h(x)|u|1−γ dx = 0, Ω

then p

D(λ)∇u2 (p−2)/(1+γ )

2(p+γ −1)/(1+γ ) ∇u2 p−2 1+γ < − λ |u|p dx p+γ −1 p+γ −1 ( Ω h(x)|u|1−γ dx)(p−2)/(1+γ ) Ω

=

1+γ p+γ −1

−

p−2 p+γ −1

(p−2)/(1+γ )

( Ω

2(p−2)/(1+γ ) ∇u2 ∇u22 h(x)|u|1−γ dx)(p−2)/(1+γ )

1+γ ∇u22 = 0, p+γ −1

which is impossible as D(λ) > 0 for all λ ∈ (0, Λ). This fact, together with (3) implies that Nλ− is closed. This completes the proof of Lemma 3. 2 After these preliminaries, let us give Section 3.


1263

3. Solutions of (1λ ) for all λ ∈ (0, Λ) Theorem 1. Suppose that λ ∈ (0, Λ). Then the singular problem (1λ ) has a solution u0 ∈ H01 (Ω) ∩ C 1,β (Ω), ∀0 < β < 1, satisfying Iλ (u0 ) < 0 and ∇u0 2 E0 (E0 defined in Lemma 2). Proof. Note that for u ∈ Nλ it is clear that

1 Iλ (u) = 2 =

1 |∇u| dx − 1−γ

2

Ω

1 1 − 2 p

1−γ

h(x)|u| Ω

|∇u|2 dx − Ω

λ dx − p

1 1 − 1−γ p

1 1 1−γ − ∇u22 − C∇u2 , 2 p

|u|p dx Ω

h(x)|u|1−γ dx Ω

∀u ∈ Nλ .

Therefore Iλ is coercive and bounded below in Nλ . So, two immediate candidates for solutions of the singular problem (1λ ) would be that found by considering the following minimization problems inf Iλ ,

Nλ+

inf Iλ .

Nλ−

d Iλ (tu) has the same sign with φ(t, u), Iλ (tu) is increasing in [t − (u), t + (u)] for Observe that dt 1 each u ∈ H0 (Ω)\{0}. In particular, if u ∈ Nλ− (i.e., t + (u) = 1) we clearly have Iλ (t − (u)u) Iλ (t + (u)u) = Iλ (u), and consequently infN + Iλ infN − Iλ . Also, infNλ Iλ = infN + Iλ . λ

λ

λ

In view of the arguments in Lemma 3, Nλ+ ∪ {0} and Nλ− are two closed sets in H01 (Ω) provided λ ∈ (0, Λ). This allows us to select “best” minimizing sequences by means of Ekeland’s principle (see [3]). First, consider (un ) ⊂ Nλ+ ∪ {0} with the properties: (i) Iλ (un ) < infN + ∪{0} Iλ + n1 ; λ

(ii) Iλ (u) Iλ (un ) − n1 u − un , ∀u ∈ Nλ+ ∪ {0}. Since I (|u|) = I (u), we may assume un 0. Clearly, (un ) is bounded in H01 (Ω), so (a subsequence of) un u0 weakly in H01 (Ω) and Lp (Ω), strongly in L1−γ (Ω), and pointwise a.e. in Ω, with u0 0. Write un = u0 + wn with wn 0 weakly in H01 (Ω). Now, taking into account that, Iλ (u) =

1 1 1 1 2 ∇u2 − h(x)|u|1−γ dx − − 2 p 1−γ p

< that is,

1 p−2 1 − p 2 1−γ

∇u22 < 0,

Ω

for all u ∈ Nλ+

1264


inf Iλ = inf Iλ < 0

Nλ+ ∪{0}

(6)

Nλ+

while by the weak lower semi-continuity of norm Iλ (u0 ) lim inf Iλ (un ) = infN + ∪{0} Iλ , we see λ

that u0 ≡ 0 and (un ) ⊂ Nλ+ . Now, using techniques developed in our previous work [26], we investigate further properties of (un ) which yield the important estimate for u0 : Claim 1. There exists ε0 > 0 such that u0 (x) ε0 e1 (x), ∀x ∈ Ω. We start by observing that lim inf (p − 2)∇un 22 < (p + γ − 1)

1−γ

h(x)u0

n→∞

(7)

dx.

Ω

In fact, arguing by contradiction and assume that lim infn→∞ [(p − 2)∇un 22 ] = 1−γ (p + γ − 1) Ω h(x)u0 dx. Since un ∈ Nλ+ , then lim inf (p − 2)∇un 22 lim sup (p − 2)∇un 22 (p + γ − 1) n→∞

1−γ

h(x)u0

n→∞

dx

Ω

and thus lim ∇un 22 =

n→∞

p+γ −1 p−2

1−γ

h(x)u0

(8)

dx.

Ω

Consequently,

1+γ 1−γ 1−γ p h(x)u0 dx. lim λun p = lim ∇un 22 − h(x)un dx = n→∞ n→∞ p−2 Ω

Ω

Note that D(λ) > 0. This provides the necessary contradiction, as (8) and (9) imply that p

0 < D(λ)∇un 2 (p−2)/(1+γ ) 2(p+γ −1)/(1+γ ) ∇un 2 1+γ p−2 p < − λun p 1−γ p+γ −1 p+γ −1 ( Ω h(x)un dx)(p−2)/(1+γ ) (p−2)/(1+γ ) ( p+γ −1 h(x)u1−γ dx)(p+γ −1)/(1+γ ) 1+γ p−2 0 Ω p−2 n→∞ −−−− → 1−γ p+γ −1 p+γ −1 ( Ω h(x)u0 dx)(p−2)/(1+γ )

1+γ 1−γ − h(x)u0 dx = 0 p−2 Ω

that is, un → 0 strongly in H01 (Ω) while Iλ (un ) → infN + Iλ < 0. λ

(9)


1265

By (7), we may extract a subsequence such that

(p − 2)∇un 22

1−γ

− (p + γ − 1)

h(x)un

dx −C

(10)

Ω

for suitable constant C > 0. Let ϕ ∈ H01 (Ω) with ϕ(x) 0. From Lemma 1 we know that, for each un there exists a continuous function fn (t) such that fn (t)(un + tϕ) ∈ Nλ+ (⊂ Nλ ) for all sufficiently small t 0. Clearly, fn (0) = 1. Therefore,

2 1−γ p p 2 0 = fn (t) un + tϕ − fn (t) h(x)(un + tϕ)1−γ dx − λ fn (t) un + tϕp , Ω

0 = un 2 −

1−γ

h(x)un

p

dx − λun p ,

Ω

for t > 0 small, that is, 0 = fn2 (t) − 1 un + tϕ2 + un + tϕ2 − un 2

1−γ 1−γ h(x)(un + tϕ)1−γ − h(x)(un + tϕ)1−γ − h(x)un − fn (t) − 1 Ω

Ω

Ω

p p p p − λ fn (t) − 1 un + tϕp − λun + tϕp − λun p fn2 (t) − 1 un + tϕ2 + un + tϕ2 − un 2

1−γ p p − fn (t) − 1 h(x)(un + tϕ)1−γ − λ fn (t) − 1 un + tϕp , Ω

dividing by t > 0 and passing to the limit for t → 0, we derive

1−γ p 0 2fn (0)∇un 22 + 2 ∇un · ∇ϕ − (1 − γ )fn (0) h(x)un − λpfn (0)un p Ω

Ω

1−γ 2 + 2 ∇un · ∇ϕ = fn (0) (2 − p)∇un 2 + (p + γ − 1) h(x)un Ω

Ω

where fn (0) ∈ [−∞, +∞] denotes the right derivate of fn (t) at zero (for the sake of simplicity, we assume henceforth that the right derivate of fn at t = 0 exists. Indeed, if it isn’t real, we let tk → 0 (instead of t → 0), tk > 0 is chosen in such a way that fn satisfies qn := limk→∞ fn (ttkk)−1 , where qn ∈ [−∞, +∞], and then replace fn (0) by qn ). Since un ∈ Nλ+ , fn (0) = −∞. Furthermore, from (10) we conclude that fn (0) is uniformly bounded from below. On the other hand, using (ii) we clearly have 1 Iλ (un ) Iλ fn (t)(un + tϕ) + fn (t)(un + tϕ) − un n

(11)

1266


for t > 0 small, that is, 1 fn (t) − 1un + tfn (t)ϕ n 1 fn (t)(un + tϕ) − un n Iλ (un ) − Iλ fn (t)(un + tϕ) 2 1 1 1 1 1 1 p 2 − un + λ − un p − − fn (t) un + tϕ2 = 2 1−γ 1−γ p 2 1−γ 1 1 p p − fn (t) un + tϕp , −λ 1−γ p dividing by t > 0 and passing to the limit as t → 0, we get

fn (0) 1 1−γ 2 f (0) un + ϕ (p + γ − 1) h(x)un − (p − 2)∇un 2 n n 1−γ

1+γ + 1−γ

Ω

∇un · ∇ϕ − λ Ω

p+γ −1 1−γ

p−1

un

ϕ.

(12)

Ω

But by (10), for n large enough

un 1 1−γ 2 − (p − 2)∇un 2 − (p + γ − 1) h(x)un C − 1−γ n

(13)

Ω

with C > 0 a suitable constant. Putting together (12) and (13), we see that fn (0) is uniformly bounded from above. In conclusion, fn (0) is uniformly bounded in n. Now, applying (11) again, 1 fn (t) − 1un + tfn (t)ϕ n 1 fn (t)(un + tϕ) − un n Iλ (un ) − Iλ fn (t)(un + tϕ)

2 1 1 λ 1 1−γ p = un 2 − h(x)un − un p − fn (t) un + tϕ2 2 1−γ p 2 Ω

+

1 1−γ

1−γ fn (t)

h(x)(un + tϕ)1−γ + Ω

p λ p fn (t) un + tϕp , p

(14)


1267

dividing by t > 0 and passing to the limit as t → 0, we obtain 1 fn (0) un + ϕ n

1−γ p p−1 −fn (0)un 2 − ∇un · ∇ϕ + λfn (0)un p + λ un ϕ + fn (0) h(x)un Ω

+ lim inf t→0

Ω

1 1−γ

h(x)

Ω

t

Ω

∇un · ∇ϕ + λ

=−

1−γ (un + tϕ)1−γ − un

Ω

p−1 un ϕ

+ lim inf t→0

Ω

1 1−γ

1−γ

(un + tϕ)1−γ − un h(x) t

,

Ω

which gives,

lim inf t→0

1 1−γ

1−γ

(un + tϕ)1−γ − un h(x) t

Ω

∇un · ∇ϕ dx − λ Ω

p−1

un

ϕ dx +

dx

|fn (0)|un + ϕ . n

Ω

1−γ −u1−γ n

Since h(x) (un +tϕ) t

0 in Ω, by Fatou’s Lemma we know that

1−γ (un + tϕ)1−γ − un lim inf h(x) t→0 t

is integrable, and

lim inf

1−γ

t→0

1 (un + tϕ)1−γ − un h(x) 1−γ t

Ω

∇un · ∇ϕ dx − λ Ω

p−1

un

ϕ dx +

dx

|fn (0)|un + ϕ . n

Ω

Note that 1 (un h(x) 1−γ

+ tϕ)1−γ t

⎧ 0, un (x) = 0, ϕ(x) = 0, ⎪ ⎨ +∞, un (x) = 0, ϕ(x) > 0, −−−→ ⎪ ⎩ h(x) un (x) > 0, ϕ(x) 0. γ ϕ, u

1−γ − un t→0

n

Now, if we use ϕ = e1 as a test-function in (15), we see un (x) > 0 a.e. in Ω, then

(15)

1268


Ω

h(x) γ ϕ dx un

∇un · ∇ϕ dx − λ

Ω

p−1

un

ϕ dx +

|fn (0)|un + ϕ , n

Ω

and in view of (14), we can proceed as above to conclude that u0 (x) > 0 a.e. in Ω, and

Ω

h(x) γ ϕ dx u0

∇u0 · ∇ϕ dx − λ

Ω

p−1

u0

∀ϕ ∈ H01 (Ω), ϕ 0.

ϕ dx,

(16)

Ω

At this point the conclusion of Claim 1 follows by a result of Brezis and Nirenberg [6, Theorem 3] which shows that u0 (x) c dist(x, ∂Ω),

∀x ∈ Ω.

(17)

Let uε (x) =

ε (N −2)/2 , (ε 2 + |x|2 )(N −2)/2

ε > 0, x ∈ RN

be an extremal function for the Sobolev inequality in RN . For a ∈ Ω let η ∈ C0∞ (Ω) such that 0 η(x) 1 in Ω and η(x) = 1, ∀x ∈ B r (a) ⊂ Ω for a suitable r > 0. Set p Uε,a (x) = η(x)uε (x − a) ∈ H01 (Ω). It is well known that ∇Uε,a 22 = B + O(ε N −2 ), Uε,a p = A + O(ε N ), and S = AB2/p , where

B=

∇U ∗ 2 dx,

RN

A= RN

1 dx. (1 + |x|2 )N

The crucial step in our proof is the following: Claim 2. u0 ∈ Nλ with λ ∈ (0, Λ). Denote by

a0 = ∇u0 22 −

h(x)|u0 |1−γ dx − λ

Ω

|u0 |p dx. Ω

Let ϕ = u0 in (16), we know that a0 0. Let us argue by contradiction and assume that a0 > 0. In the following we will concentrate on a contradiction. By the (contradictory) assumption a0 > 0, there exists a unique c0 > 0 such that p c02 B −λc0 A = −a0 , i.e. S(c0 A1/p )2 −λ(c0 A1/p )p = −a0 . But, as Iλ (un ) → μ0 := infN + ∪{0} Iλ = infN + Iλ with un ∈ Nλ+ (⊂ Nλ ), by the Brezis–Lieb Lemma [5] we have λ

λ


μ0 + o(1) = Iλ (un ) = =

1 1 − 2 1−γ

1 1 − 2 1−γ

h(x)|un |1−γ dx + λ

Ω

h(x)|u0 |1−γ dx + λ

1269

1 1 p − un p 2 p

1 1 p − u0 p 2 p

Ω

1 1 p − wn p + o(1) +λ 2 p

and

0 = ∇un 22 −

h(x)|un |1−γ dx − λ Ω

|un |p dx Ω

= a0 + ∇wn 22

p − λwn p

p

+ o(1) a0 + Swn 2p − λwn p + o(1)

which would imply that limn→∞ wn p exists and limn→∞ wn p c0 A1/p . In other words, u0 satisfies μ0

1 1 − 2 1−γ

1 1 1 1 p p − u0 p + λ − c A. dx + λ 2 p 2 p 0

h(x)|u0 |

1−γ

(18)

Ω

p On the other hand, for any u ∈ H01 (Ω) with au = ∇u22 − Ω h(x)|u|1−γ dx − λup > 0, p p we can find Ru > 0 such that ∇u22 − Ω h(x)|u|1−γ dx − λup + Ru2 B − λRu A < 0, and thus ∇(u + Ru Uε,a )2 −

p

h(x)|u + Ru Uε,a |1−γ dx − λu + Ru Uε,a p

2

Ω

= ∇u22

+ Ru2 ∇Uε,a 22

+ 2Ru

∇u · ∇Uε,a dx − Ω

h(x)|u + Ru Uε,a |1−γ dx Ω

p p p − λ up + Ru Uε,a p + o(1) p

= au + Ru2 B − λRu A + o(1) < 0 for ε > 0 small, where we have used

Ω

∇u · ∇Uε,a dx = o(1) and the fact

h(x)|u + cUε,a |1−γ dx − h(x)|u|1−γ dx Ω

h∞

Ω

(N−2)(1−γ ) 2 . (cUε,a )1−γ dx = h∞ c1−γ O ε

Ω

This allows us to take 0 < cε,u < Ru to satisfy

1270


∇(u + cε,u Uε,a )2 −

p

h(x)|u + cε,u Uε,a |1−γ dx − λu + cε,u Uε,a p = 0

2

(19)

Ω

that is, u + cε,u Uε,a ∈ Nλ . p

Furthermore, since au > 0, let cu > 0 be the unique such that cu2 B − λcu A = −au . Then, clearly B 1/(p−2) 2 B − λcp A + o(1), and hence cu > ( λA ) . From (19) it follows that 0 = au + cε,u ε,u cε,u → cu

as ε → 0,

(20)

which yields ∇(u + cε,u Uε,a )2 = ∇u2 + c2 B + o(1) > c2 B > u u 2 2

B λA

2/(p−2)

(N −2)/2 1 B= S N/2 λ

for ε > 0 small. Necessarily, ∇(u + cε,u Uε,a ) > 2

(N −2)/4 (N −2)/4

√ N/2 √ N/2 1 1+γ S > S = E(λ). λ λ(p + γ − 1)

The gap structure of Nλ (see (5)) then guarantees u + cε,u Uε,a ∈ Nλ− . This information will be useful in the proof of Theorem 2. Thus, in view of the fact that infN + Iλ = infNλ Iλ , we derive λ that μ0 Iλ (u + cε,u Uε,a )

1 1 1 1 p 1−γ − − u + cε,u Uε,a p h(x)|u + cε,u Uε,a | dx + λ = 2 1−γ 2 p =

Ω

1 1 − 2 1−γ

1 1 1 1 p p − up + λ − cε,u A + o(1), dx + λ 2 p 2 p

1−γ

h(x)|u| Ω

that is, μ0

1 1 − 2 1−γ

1 1 1 1 p p − up + λ − cu A. dx + λ 2 p 2 p

1−γ

h(x)|u|

(21)

Ω

Now, putting together (18) and (21), we see that μ0 =

1 1 − 2 1−γ

h(x)|u0 |1−γ dx + λ

Ω

1 1 1 1 p p u0 p + λ c A. − − 2 p 2 p 0

(22)


1271

This implies that, necessarily u0 is a local minimizer for the functional:

1 1 − 2 1−γ

1 1 1 1 p p − up + λ − cu A. dx + λ 2 p 2 p

1−γ

h(x)|u|

(23)

Ω

For the functional cu , let ϕ ∈ C0∞ (Ω) and evaluate in a small neighborhood of t = 0

g(t) := cu0 +tϕ that is,

p 2 p g(t) B − λ g(t) A = − ∇u0 + tϕ22 − h(x)|u0 + tϕ|1−γ dx − λu0 + tϕp . Ω

By a0 > 0, we know that g(t) exists, with g(0) = c0 . Moreover, since u0 (x) ε0 e1 (x) in Ω (see Claim 1), by dominated convergence,

h(x)|u0 + tϕ|1−γ dx − h(x)|u0 |1−γ dx t

h(x)(1 − γ )(u0 + θ tϕ)−γ ϕ dx =

Ω

supp ϕ

−γ h(x)(1 − γ )u0 ϕ dx

t→0

−−−→

=

supp ϕ

−γ

h(x)(1 − γ )u0 ϕ dx, Ω

and consequently p−1 g(t) − g(0) B g(t) + g(0) − λAp g(0) + θ g(t) − g(0) t 2 p 2 p [g(t)] B − λ[g(t)] A − [g(0)] B + λ[g(0)] A = t

1 p = − ∇u0 + tϕ22 − h(x)|u0 + tϕ|1−γ dx − λu0 + tϕp t Ω

− ∇u0 22

+

h(x)|u0 |

1−γ

dx

p + λu0 p

Ω

h(x) p−1 t→0 − −−→ − 2 ∇u0 · ∇ϕ dx − (1 − γ ) ϕ dx − λp u ϕ dx γ 0 u0 Ω

Ω

Ω

which implies that g (0) exists and g (0) =

h(x) p−1 2 ∇u · ∇ϕ dx − (1 − γ ) ϕ dx − λp u ϕ dx . 0 γ 0 p−1 u0 2c0 B − λpc A −1

0

Ω

Ω

Ω

1272


Resuming from (23) we see that

1 1 1 1 d p − − u0 + tϕp h(x)|u0 + tϕ|1−γ + λ dt 2 1−γ 2 p Ω

p 1 1 − g(t) A +λ =0 2 p t=0 that is,

1 1 h(x) 1 1 1 1 p−1 p−1 − (1 − γ ) − p − pc0 A ϕ dx + λ u ϕ dx + λ γ 0 2 1−γ 2 p 2 p u0 Ω

Ω

−1 h(x) p−1 2 × ∇u · ∇ϕ dx − (1 − γ ) ϕ dx − λp u ϕ dx =0 0 γ 0 p−1 u0 2c0 B − λpc A

0

Ω

Ω

Ω

for all ϕ ∈ C0∞ (Ω). Since h(x) is like distα (x, ∂Ω) with α − γ 0, from Claim 1 follows −γ immediately that h(x)u0 ∈ L∞ (Ω). Hence, for all ϕ ∈ H01 (Ω) we conclude that

1 1 h(x) 1 1 1 1 p−1 p−1 − (1 − γ ) − p u0 ϕ + λ − Apc0 0= γ ϕ+λ 2 1−γ 2 p 2 p u0 Ω

Ω

h(x) p−1 2 × ∇u · ∇ϕ − (1 − γ ) ϕ − λp u ϕ . 0 γ 0 p−1 u0 2c0 B − λpc A

−1

0

Ω

Ω

(24)

Ω

Thus, we can use Eq. (24) to derive that u0 ∈ C 1,β (Ω), ∀0 < β < 1 by usual bootstrap argument, and so the famous estimates follow (see [27]):

∇u0 · ∇Uε,a dx = O ε (N −2)/2 ,

Ω p−1 Uε,a u0 dx

= u0 (a) RN

Ω

|u0 |

p−2

u0 Uε,a dx =

Ω

Ω

1 (1 + |x|2 )(N +2)/2

dx ε (N −2)/2 + o ε (N −2)/2 ,

|u0 |p−2 u0 η dx ε (N −2)/2 + o ε (N −2)/2 . (|x − a|2 )(N −2)/2

−γ

In particular, as h(x)u0 ∈ L∞ (Ω), we can reevaluate

h(x)(u0 + cε,u Uε,a )1−γ dx − h(x)u1−γ dx 0 0 Ω

=

Ω

h(x)(1 − γ )(u0 + θ cε,u0 Uε,a )−γ cε,u0 Uε,a dx

Ω

−γ = ε (N −2)/2 (1 − γ )c0 h(x)u0 Ω

η(x) dx + o(1) (|x − a|2 )(N −2)/2


1273

that is,

h(x)(u0 + cε,u0 Uε,a )1−γ dx Ω

=

1−γ h(x)u0

dx + (1 − γ )c0

Ω

−γ

h(x)u0 Ω

η(x) dx ε (N −2)/2 + o ε (N −2)/2 . (|x − a|2 )(N −2)/2

Write cε,u0 = c0 + δε . By (20), δε → 0. Inserting all the above estimates into (19), we obtain 2 0 = ∇(u0 + cε,u0 Uε,a )2 −

p

h(x)|u0 + cε,u0 Uε,a |1−γ dx − λu0 + cε,u0 Uε,a p Ω

− λp

−γ

h(x)u0 Ω

p−1

u0

∇u0 · ∇Uε,a dx − Ω

− (1 − γ )c0

2 = ∇u0 22 + cε,u ∇Uε,a 22 + 2cε,u0 0

η(x) p p p dx ε (N −2)/2 − λu0 p − λcε,u0 Uε,a p (|x − a|2 )(N −2)/2 p−1

Ω

p−1 Uε,a u0 dx + o ε (N −2)/2

Ω

p p 2 B − λcε,u0 A + 2c0 = − c02 B − λc0 A + cε,u 0 − (1 − γ )c0 p−1

− λpc0

∇u0 · ∇Uε,a dx Ω

−γ h(x)u0

Ω

dx

Ω

cε,u0 Uε,a dx − λpcε,u0

1−γ

h(x)u0

η(x) dx ε (N −2)/2 − λpc0 |x − a|N −2

p−1

u0

Uε,a dx

Ω

p−1 Uε,a u0 dx + o ε (N −2)/2 ,

Ω

which gives p−1 2c0 B − λpc0 A + o(1) (−δε )

−γ = 2c0 ∇u0 · ∇Uε,a dx − (1 − γ )c0 h(x)u0 Uε,a dx Ω

− λpc0

Ω

p−1

u0

p−1

Uε,a dx − λpc0

Ω

Furthermore, from (24) follows that

Ω

p−1 Uε,a u0 dx + o ε (N −2)/2 .

1274


(−δε ) =

c0 p−1

[2c0 B − λpc0 − −

−

Ω

c0 p−1

[2c0 B − λpc0 p−1

[2c0 B − c0

A]

−

Ω p−1

u0

)(1 − γ )

λp Ω

h(x) Ω uγ Uε,a dx 0

p−1

c0 [2c0 B

Uε,a dx

p−1 Uε,a u0 dx + o ε (N −2)/2

λ( 12

−

h(x)u0 Uε,a dx

Ω

p−1 − λpc0 A]

1 1−γ

−γ

(1 − γ )

λp A]

p−1 c0

[2c0 B

c0

( 12

= c0

∇u0 · ∇Uε,a dx

2 A]

p−1 − λpc0 A]

λp

−

+ λ( 12 − p1 )p

Ω

p−1

u0

Uε,a dx

p−1 1 p )Apc0

see (24)

p−1 Uε,a u0 dx + o ε (N −2)/2 .

(25)

Ω

Also, δε = O(ε (N −2)/2 ). Now, we can proceed to get the contradiction. Since a0 > 0, clearly 2c0 B

p−1 − λpc0 A =

2 2 p p 2 2 2 p c B − λ c0 A < c B − λc0 A = − a0 < 0. c0 0 2 c0 0 c0

Subsequently, in virtue of u0 + cε,u0 Uε,a ∈ Nλ , applying (22) and (25) we obtain Iλ (u0 + cε,u0 Uε,a )

1 1 1 1 p − − u0 + cε,u0 Uε,a p h(x)(u0 + cε,u0 Uε,a )1−γ dx + λ = 2 1−γ 2 p =

Ω

1 1 − 2 1−γ

1−γ

h(x)u0

dx + λ

Ω

1 1 − + 2 1−γ

1 1 1 1 p p − u0 p + λ − cε,u0 A 2 p 2 p

−γ (1 − γ )c0 h(x)u0 Ω

η(x) dx ε (N −2)/2 |x − a|N −2

1 1 1 1 p−1 p−1 p−1 − p u0 cε,u0 Uε,a dx + λ − pcε,u0 Uε,a u0 dx + o ε (N −2)/2 +λ 2 p 2 p

=

1 1 − 2 1−γ

+λ

Ω 1−γ

h(x)u0

Ω

1 1 1 1 p p − u0 p + λ − c A dx + λ 2 p 2 p 0

Ω

1 1 1 1 −γ p−1 − pc0 δε A + − (1 − γ )c0 h(x)u0 Uε,a dx 2 p 2 1−γ Ω


+λ

1275

1 1 1 1 p−1 p−1 p−1 − pc0 u0 Uε,a dx + λ − pc0 Uε,a u0 dx + o ε (N −2)/2 2 p 2 p

Ω

Ω

see (22)

1 1 p−1 − pc0 Aδε 2 p

1 1 1 1 −γ p−1 − − + (1 − γ )c0 h(x)u0 Uε,a dx + λ pc0 u0 Uε,a dx 2 1−γ 2 p

= μ0 + λ

Ω

Ω

1 1 p−1 p−1 − pc0 +λ Uε,a u0 dx + o ε (N −2)/2 2 p

Ω

= μ0 + λ

p−1 c0 1 1 1 1 p−1 p−1 p−1 − pc0 Aδε + λ − Apc0 U u dx λp 0 ε,a p−1 2 p 2 p 2c0 B − λpc A

1 1 p−1 + (−δε ) λ − pc0 A 2 p

1 1 p−1 p−1 +λ − pc0 Uε,a u0 dx + o ε (N −2)/2 2 p

0

Ω

see (25)

Ω

1 1 2c0 B p−1 p−1 − = μ0 + λ Uε,a u0 dx + o ε (N −2)/2 < μ0 pc0 p−1 2 p 2c0 B − λpc A 0

Ω

which is clearly impossible. This concludes the proof of Claim 2. Claim 3. u0 is a solution of (1λ ). The proof is inspired by Graham-Eagle in [17]. For ϕ ∈ H01 (Ω), ε > 0 define Ψ := (u0 + εϕ)+ ∈ H01 (Ω). Using Claim 2 and inserting Ψ into (16), we see that

∇u0 · ∇Ψ −

0 Ω

h(x) p−1 γ Ψ − λu0 Ψ dx u0

∇u0 · ∇(u0 + εϕ) −

= [u0 +εϕ>0]

−

= Ω

[u0 +εϕ0]

= ∇u0 22 −

h(x) p−1 ∇u0 · ∇(u0 + εϕ) − γ (u0 + εϕ) − λu0 (u0 + εϕ) dx u0 1−γ

h(x)u0 Ω

h(x) p−1 γ (u0 + εϕ) − λu0 (u0 + εϕ) dx u0

dx − λ

p

u0 dx + ε Ω

∇u0 · ∇ϕ − Ω

h(x) p−1 γ ϕ − λu0 ϕ dx u0

1276


−

∇u0 · ∇(u0 + εϕ) −

[u0 +εϕ0]

=ε

Ω

−

h(x) p−1 ∇u0 · ∇ϕ − γ ϕ − λu0 ϕ dx − u0

∇u0 · ∇(u0 + εϕ)

[u0 +εϕ0]

h(x) p−1 γ (u0 + εϕ) − λu0 (u0 + εφ) dx u0 ∇u0 · ∇ϕ −

ε

h(x) p−1 γ (u0 + εϕ) − λu0 (u0 + εϕ) dx u0

h(x) p−1 γ ϕ − λu0 ϕ dx − ε u0

∇u0 · ∇ϕ dx.

[u0 +εϕ0]

Ω

Since the measure of the domain of integration [u0 + εϕ 0] tends to zero as ε → 0, it follows that [u0 +εϕ0] ∇u0 · ∇ϕ dx → 0. Dividing by ε and letting ε → 0 therefore shows

∇u0 · ∇ϕ dx −

Ω

Ω

h(x) γ ϕ dx − λ u0

p−1

u0

ϕ dx 0

Ω

and since this holds equally well for −ϕ, it follows that u0 is a solution of the singular problem (1λ ). By Claim 1 we derive that u0 ∈ C 1,β (Ω), ∀0 < β < 1. Since un u0 weakly in H01 (Ω), by the weak lower semi-continuity of · we conclude that u0 lim infn→∞ un E0 , Claim 2 and the gap structure of Nλ in turn imply that u0 ∈ Nλ+ . At this point, from Iλ (un ) → infN + Iλ λ we see that inf Iλ

Nλ+

1 1 1 1 1−γ 2 − ∇u0 2 − − h(x)u0 dx = Iλ (u0 ) 2 p 1−γ p Ω

that is, Iλ (u0 ) = infN + Iλ . This completes the proof of Theorem 1.

2

λ

Theorem 2. Suppose that λ ∈ (0, Λ). Then the singular problem (1λ ) has a solution U0 ∈ H01 (Ω) ∩ C 1,β (Ω), ∀0 < β < 1, satisfying ∇U0 2 E(λ) > E0 with E(λ) → +∞ as λ → 0. Proof. We provide only a sketch, as the arguments are by now familiar. Then, consider (Un ) ⊂ Nλ− the “best” minimizing sequence (i.e., satisfying Ekeland’s principle) for infN − Iλ . λ

Since (Un ) is bounded in H01 (Ω), after passing to a subsequence, we may assume that Un U0 weakly in H01 (Ω), and pointwise a.e. Write Un = U0 + Wn with Wn 0 weakly in H01 (Ω). 1−γ The result lim infn→∞ [(p − 2)∇Un 22 ] > (p + γ − 1) Ω h(x)U0 follows easily with an argument by contradiction. In fact suppose that lim inf (p − 2)∇Un 22 = (p + γ − 1)

Ω

that is,

1−γ

h(x)U0

n→∞

dx,


lim inf n→∞

1277

(p − 2)∇Un 22 = 1, 1−γ (p + γ − 1) Ω h(x)Un dx

then there exists a subsequence of Un , called Unk , such that (p − 2)∇Unk 22 → 1, 1−γ (p + γ − 1) Ω h(x)Unk dx

k → ∞.

Therefore ∇Unk 22

p+γ −1 → p−2

p

λUnk p = ∇Unk 22 −

1−γ

h(x)U0 Ω

1−γ

h(x)Unk

dx →

Ω

1+γ p−2

dx,

1−γ

h(x)U0

dx,

Ω

and (recalling u E(λ) for all u ∈ Nλ− ) consequently, p

D(λ)E p (λ) < D(λ)∇Unk 2
0, E(λ) > 0 for all λ ∈ (0, Λ). Thus, we can proceed as in the proof of Theorem 1 to obtain U0 (x) > 0 e1 (x), ∀x ∈ Ω, and

Ω

h(x) γ ϕ U0

∇U0 · ∇ϕ − λ

Ω

p−1

U0

ϕ,

∀ϕ ∈ H01 (Ω), ϕ 0.

Ω

1−γ p By taking ϕ = U0 we know that ∇U0 22 − h(x)U0 − λU0 p 0. All that remains is to 1−γ prove that U0 ∈ Nλ . Arguing by contradiction and assume that a˜ 0 = ∇U0 22 − h(x)U0 − p p λU0 p > 0. Then there would exist a unique point c˜0 > 0 such that c˜02 B − λc˜0 A = −a˜ 0 . Since Iλ (Un ) → π0 := infN − Iλ with Un ∈ Nλ− (⊂ Nλ ), we have λ

π0 + o(1) = Iλ (Un ) = =

and

1 1 − 2 1−γ

1 1 − 2 1−γ

Ω

1−γ

h(x)Un Ω

1−γ h(x)U0

dx + λ

1 1 p − Un p 2 p

1 1 1 1 p p − U0 p + λ − Wn p + o(1), dx + λ 2 p 2 p

1278


0 = ∇Un 22

1−γ

−

h(x)Un

p

dx − λUn p

Ω p

p

= a˜ 0 + ∇Wn 22 − λWn p + o(1) a˜ 0 + SWn 2p − λWn p + o(1), which would imply that limn→∞ Wn p exists and limn→∞ Wn p c˜0 A1/p , and consequently, π0

1 1 − 2 1−γ

1−γ

h(x)U0

dx + λ

1 1 1 1 p p − U0 p + λ − c˜ A. 2 p 2 p 0

(26)

Ω

As shown in the proof of Theorem 1, for any u ∈ H01 (Ω) with au = ∇u22 − h(x)|u|1−γ − p λup > 0, we can always find 0 < cε,u < Ru such that u + cε,u Uε,a ∈ Nλ− for ε > 0 small. Subsequently, π0 Iλ (u + cε,u Uε,a )

1 1 1 1 p − − h(x)|u + cε,u Uε,a |1−γ dx + λ = u + cε,u Uε,a p 2 1−γ 2 p =

Ω

1 1 − 2 1−γ

1 1 1 1 p p − up + λ − cε,u A + o(1), dx + λ 2 p 2 p

1−γ

h(x)|u| Ω

which yields, π0

1 1 − 2 1−γ

h(x)|u|1−γ dx + λ

1 1 1 1 p p − up + λ − cu A. 2 p 2 p

(27)

Ω

Putting together (26) and (27), we obtain: π0 =

1 1 − 2 1−γ

h(x)|U0 |1−γ dx + λ

1 1 1 1 p p − U0 p + λ − c˜ A, 2 p 2 p 0

Ω

and that for every ϕ ∈ C0∞ (Ω), d dt

1 1 − 2 1−γ

h(x)|U0 + tϕ|1−γ + λ

1 1 p − U0 + tϕp 2 p

Ω

p 1 1 − A G(t) +λ =0 2 p t=0

where

2 p p G(t) B − λ G(t) A = − ∇U0 + tϕ22 − h(x)|U0 + tϕ|1−γ dx − λU0 + tϕp , Ω


1279

and we can proceed as in (24), (25) to reach a contradiction. The desired result that U0 is a solution of the singular problem (1λ ) follows. Still no location information can be obtained for U0 . In the sequel we prove that U0 ∈ Nλ− . Claim 4. There exist ε1 > 0 and C > 0 such that

Ω

u0 (x) + RUε,a (x) u0 + RUε,a

p dx C,

∀ε ∈ (0, ε1 ), ∀R 1.

We consider two cases. First if R 1 is such that R 2 B u0 2 , we have p

p

p

u0 + RUε,a p = u0 p + R p Uε,a p + pR

+ pR p−1

p−1

u0

Uε,a dx

Ω

p−1 Uε,a u0 dx + o ε (N −2)/2

Ω

p = u0 p

p + R p A + O ε (N −2)/2 > u0 p ,

u0 + RUε,a

2

= ∇u0 22

+R

2

∇Uε,a 22

+ 2R

∇u0 · ∇Uε,a Ω

= ∇u0 22 + R 2 B + O ε (N −2)/2 < 2∇u0 22 + 1 for ε > 0 small, so

Ω

u0 + RUε,a u0 + RUε,a

p

p

dx >

u0 p (2∇u0 22 + 1)p/2

.

On the other hand, if R 1 is such that R 2 B > u0 2 , then p

p

p

u0 + RUε,a p = u0 p + R p Uε,a p + pR

+ pR p−1

p−1

u0

Uε,a

Ω

p−1 Uε,a u0 + R β o ε (N −2)/2

Ω

p = u0 p

(N −2)/2 1 p p A +O ε where β ∈ (0, p) see [4] + R A+R 2 2

1 > R p A, 2

u0 + RUε,a 2 = ∇u0 22 + R 2 ∇Uε,a 22 + 2R

∇u0 · ∇Uε,a < 2R 2 (B + 1) Ω

1280


for ε > 0 small, and therefore

Ω


p dx >

[2(B

1 p 2R A + 1)]p/2 R p

=

1 2A

[2(B + 1)]p/2

.

Thus there exist constants C > 0 and ε1 > 0 such that for all ε ∈ (0, ε1 ) and R 1,

Ω


p dx C.

Define u 1 + Σ1 = u ∈ H0 (Ω)\{0}: u < t , u u Σ2 = u ∈ H01 (Ω)\{0}: u > t + . u It is easily verified that u . Nλ− = u ∈ H01 (Ω)\{0}: u = t + u

Nλ+ ⊂ Σ1 ,

Claim 5. There exist ε2 > 0 and R0 > 1 so that u0 + R0 Uε,a ∈ Σ2

for all ε < ε2 .

u +RU

Note that t + ( u00 +RUε,a ) satisfies ε,a

λ Ω


p

= t

+

− t


+

2−p


−γ −p+1

Ω

u0 + RUε,a 1−γ h(x) . u0 + RUε,a

u +RU

By Claim 4, t + ( u00 +RUε,a ) is forced to be uniformly bounded from above, that is, ε,a t+


C,

∀ε ∈ (0, ε1 ), ∀R 1.

On the other hand, for sufficiently large R

B 2 1 1 + O ε (N −2)/2 > R 2 B > C 2 u0 + RUε,a 2 = ∇u0 22 + R 2 B + R 2 2 2 R 2


1281

provided ε > 0 small. Hence, there exist 0 < ε2 (< ε1 ) and R0 > 1 such that for all ε ∈ (0, ε2 ) and R R0 , u0 + RUε,a . u0 + RUε,a > t + u0 + RUε,a This readily gives u0 + RUε,a ∈ Σ2 , ∀ε ∈ (0, ε2 ), ∀R R0 . Claim 6. There exists ε3 > 0 such that ∀ε ∈ (0, ε3 ) there holds 1 N/2 1 (N −2)/2 Iλ (u0 + tR0 Uε,a ) < Iλ (u0 ) + S , N λ

∀t ∈ [0, 1].

Since u0 is a solution of (1λ ), we derive Iλ (u0 + tR0 Uε,a )

2 1 λ 1 p h(x)(u0 + tR0 Uε,a )1−γ dx − u0 + tR0 Uε,a p = ∇(u0 + tR0 Uε,a ) 2 − 2 1−γ p Ω

1 1 = ∇u0 22 + (tR0 )2 ∇Uε,a 22 + (tR0 ) 2 2

− (tR0 )

Ω

−γ h(x)u0 Uε,a dx

Ω

− λ(tR0 )

p−1

1 ∇u0 · ∇Uε,a dx − 1−γ

1−γ

h(x)u0 Ω

λ λ p p − u0 p − (tR0 )p Uε,a p − λ(tR0 ) p p

p−1

u0

dx

Uε,a dx

Ω

p−1 Uε,a u0 dx + o ε (N −2)/2

Ω

1 λ = Iλ (u0 ) + (tR0 )2 B − (tR0 )p A − λ(tR0 )p−1 u0 (a)Dε (N −2)/2 + o ε (N −2)/2 2 p with D ≡ RN (1+|x|21)(N+2)/2 dx, where we have used

h(x)(u0 + tR0 Uε,a )1−γ dx −

Ω

= (1 − γ )(tR0 )

dx

Ω −γ

h(x)u0 Ω

1−γ

h(x)u0

η(x) dx ε (N −2)/2 + o ε (N −2)/2 . N −2 |x − a|

Define q(s) =

B 2 λA p s − s − λu0 (a)Dε (N −2)/2 s p−1 , 2 p

∀s 0.

Following the argument in [27], we can estimate q(tR0 ). We provide a short proof for the reader’s B 1/(p−2) ) and sε > 0 be the unique such that q(sε ) = max∀s0 q(s). convenience. Let s0 = ( λA Clearly, sε → s0 as ε → 0. Write sε = s0 + lε with lε → 0. Since q (sε ) = 0 it follows that

1282


3−p B(s0 + lε )3−p − λA(s0 + lε ) − Bs0 − λAs0 = λ(p − 1)u0 (a)Dε (N −2)/2 , so lε = O(ε (N −2)/2 ). Consequently, B 2 λA p s − s − λu0 (a)Dε (N −2)/2 sεp−1 2 ε p ε λA p B 2 p−1 s0 + 2s0 lε + o ε (N −2)/2 − s0 + ps0 lε + o ε (N −2)/2 = 2 p p−1 p−2 − λu0 (a)Dε (N −2)/2 s0 + (p − 1)s0 lε + o ε (N −2)/2 1 N/2 1 (N −2)/2 p−1 = S − λu0 (a)Ds0 ε (N −2)/2 + o ε (N −2)/2 . N λ

q(tR0 ) q(sε ) =

Therefore, for all t ∈ [0, 1] we have 1 N/2 1 (N −2)/2 p−1 Iλ (u0 + tR0 Uε,a ) Iλ (u0 ) + S − λu0 (a)Ds0 ε (N −2)/2 + o ε (N −2)/2 N λ then there exists 0 < ε3 (< ε2 ) such that ∀ε ∈ (0, ε3 ) there holds p−1 1 N/2 1 (N −2)/2 λu0 (a)Ds0 ε (N −2)/2 , Iλ (u0 + tR0 Uε,a ) < Iλ (u0 ) + S − N λ 2

∀t ∈ [0, 1].

Now we locate U0 . Since from Theorem 1 and Claim 5 we have that u0 ∈ Nλ+ ⊂ Σ1 and u0 + R0 Uε,a ∈ Σ2 , there must exists tε ∈ (0, 1) such that u0 + tε R0 Uε,a ∈ Nλ− , and from Claim 5 we derive that inf Iλ < Iλ (u0 ) +

Nλ−

1 N/2 1 (N −2)/2 S . N λ

(28)

Moreover, since Un , U0 ∈ Nλ , we clearly have

0 = ∇U0 22 −

1−γ

h(x)U0

p

p

dx − λU0 p + ∇Wn 22 − λWn p + o(1)

Ω

= ∇Wn 22

p

− λWn p + o(1).

(29)

The desired result Un → U0 strongly in H01 (Ω) now follows with an argument by contradiction. In fact suppose that there exists a subsequence {Unk } with ∇Wnk 2 C > 0, and from (29) that p Wnk p C. Then, (29) yields

Wnk p

1/(p−2) S + o(1) , λ


1283

and ∇Wnk 22 λ

N/2 S + o(1). λ

(30)

Combining (28) and (30) we obtain 1 N/2 1 (N −2)/2 Iλ (u0 ) + S N λ > Iλ (Unk ) since Iλ (Un ) → inf Iλ =

Nλ−

1 1 1 1 1−γ 2 − ∇Unk 2 − − h(x)Unk dx 2 p 1−γ p Ω

1 1 − ∇Wnk 22 + o(1) (here it is essential that U0 ∈ Nλ !) 2 p 1 1 1 1 − ∇Wnk 22 + o(1) = Iλ (u0 ) + − ∇Wnk 22 + o(1) inf Iλ + 2 p 2 p Nλ N/2 S 1 1 1 N/2 1 (N −2)/2 − λ + o(1) = Iλ (u0 ) + S + o(1) Iλ (u0 ) + 2 p λ N λ

= Iλ (U0 ) +

a contradiction. The gap structure of Nλ ensures that U0 ∈ Nλ− ; therefore u0 and U0 define two different solutions for the singular problem (1λ ). This completes the proof of Theorem 2. 2 4. Estimate for λ∗ (Ω, p, γ , h) Combining the above results we provide the estimate for λ∗ (N, Ω, γ , h): Theorem 3. For all Ω ∈ L, all γ ∈ (0, 1), p = distα (x, ∂Ω) with α − γ 0 we have λ∗ (Ω, p, γ , h)

1+γ p+γ −1

p−2 p+γ −1

2N N −2 ,

p−2 1+γ

and all functions h ∈ L∞ (Ω) like

S |Ω|2/N

p+γ −1 1+γ

1 h∞

p−2 1+γ

.

Acknowledgments This work was supported by the National Science Foundation of China grants 10601063 and 10971238. The first author thanks Dr. Duanzhi Zhang for insightful discussions. The authors thank the referee for useful suggestions. References [1] C.O. Alves, A. El Hamidi, Nehari manifold and existence of positive solutions to a class of quasilinear problems, Nonlinear Anal. 60 (2005) 611–624.

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[2] R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarendon Press, Oxford, 1975. [3] J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis, Pure Appl. Math., Wiley–Interscience Publications, 1984. [4] M. Badiale, G. Tarantello, Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities, Nonlinear Anal. 29 (1997) 639–677. [5] H. Brezis, E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals, Proc. Amer. Math. Soc. 28 (1983) 486–490. [6] H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris 317 (1993) 465–472. [7] K.J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations 193 (2003) 481–499. [8] A. Canino, Minimax methods for singular elliptic equations with an application to a jumping problem, J. Differential Equations 221 (2006) 210–223. [9] Y.S. Choi, A.C. Lazer, P.J. Mckenna, Some remarks on a singular elliptic boundary value problem, Nonlinear Anal. 3 (1998) 305–314. [10] M.M. Coclite, G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations 14 (1989) 1315–1327. [11] M.G. Crandall, P.H. Rabinowitz, L. Tatar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977) 193–222. [12] M. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992) 341–352. [13] J.I. Diaz, J.M. Morel, L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987) 1333–1344. [14] F. Gazzola, A. Malchiodi, Some remarks on the equation −u = λ(1 + u)p for varying λ, p and varying domains, Comm. Partial Differential Equations 27 (2002) 809–845. [15] J. Giacomoni, K. Saoudi, Multiplicity of positive solutions for a singular and critical problem, Nonlinear Anal. 71 (2009) 4060–4077. [16] S.M. Gomes, On a singular nonlinear elliptic problem, SIAM J. Math. Anal. 17 (1986) 1359–1369. [17] J. Graham-Eagle, A variational approach to upper and lower solutions, IMA J. Appl. Math. 44 (1990) 181–184. [18] C. Gui, F. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 1021–1029. [19] J. Hernández, F.J. Mancebo, J.M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 41–62. [20] N. Hirano, C. Saccon, N. Shioji, Existence of multiple positive solutions for singular elliptic problems with a concave and convex nonlinearities, Adv. Differential Equations 9 (2004) 197–220. [21] A.C. Lazer, P.J. Mckenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc. 111 (1991) 720–730. [22] W.L. Perry, A monotone iterative technique for solution of pth order (p < 0) reaction–diffusion problems in permeable catalysis, J. Comput. Chem. 5 (1984) 353–357. [23] J.P. Shi, M.X. Yao, On a singular semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 1389– 1401. [24] Y.J. Sun, S.J. Li, Some remarks on a superlinear-singular problem: Estimates for λ∗ , Nonlinear Anal. 69 (2008) 2636–2650. [25] Y.J. Sun, S.J. Li, A nonlinear elliptic equation with critical exponent: estimates for extremal values, Nonlinear Anal. 69 (2008) 1856–1869. [26] Y.J. Sun, S.P. Wu, Y.M. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations 176 (2001) 511–531. [27] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992) 281–304. [28] H.T. Yang, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations 189 (2003) 487–512.


Operators whose dual has non-separable range ✩ Pandelis Dodos Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece Received 17 November 2009; accepted 7 December 2010 Available online 18 December 2010 Communicated by Gilles Godefroy

Abstract Let X and Y be separable Banach spaces and T : X → Y be a bounded linear operator. We characterize the non-separability of T ∗ (Y ∗ ) by means of fixing properties of the operator T . © 2010 Elsevier Inc. All rights reserved. Keywords: Operators; Trees; Schauder bases

1. Introduction The study of fixing properties of certain classes of operators1 between separable Banach spaces is a heavily investigated part of Banach Space Theory which is closely related to some central questions, most notably with the problem of classifying, up to isomorphism, all complemented subspaces of classical function spaces (see [28] for an excellent exposition). Typically, one has an operator T : X → Y which is “large” in a suitable sense and tries to find a concrete object that the operator T preserves. Various versions of this problem have been studied in the literature and several satisfactory answers have been obtained; see, for instance, [1,4,5,13–16,23,24]. Among them, there are two fundamental results that deserve special attention. The first one is due to A. Pełczyński and asserts that every non-weakly compact operator T : C[0, 1] → Y must fix a copy2 of c0 . The second result is due to H. P. Rosenthal and asserts ✩

Research supported by NSF grant DMS-0903558. E-mail address: [email protected]. 1 Throughout the paper by the term operator we mean bounded, linear operator. 2 An operator T : X → Y is said to fix a copy of a Banach space E if there exists a subspace Z of X which is isomorphic to E and is such that T |Z is an isomorphic embedding. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.004

1286

P. Dodos / Journal of Functional Analysis 260 (2011) 1285–1303

that every operator T : C[0, 1] → Y whose dual T ∗ has non-separable range must fix a copy of C[0, 1]. The present paper is a continuation of this line of research and is devoted to the study of the following problem. Problem 1. Let X and Y be separable Banach spaces and T : X → Y be an operator such that T ∗ has non-separable range. What kind of fixing properties does the operator T have? To state our main results we need to fix some pieces of notation and introduce some terminology. By 2 0 is given by μt (S) = inf SE: E is a projection in N with φ(1 − E) t . 1 The ideal √ L (N , φ) consists of those operators T ∈ N such that T 1 := φ(|T |) < ∞ where ∗ |T | = T T . In the type I setting this is the usual trace class ideal. We will denote the norm on L1 (N , φ) by · 1 . An alternative definition in terms of singular values is that T ∈ L1 (N , φ) if ∞ T 1 := 0 μt (T ) dt < ∞. When N = B(H), L1 (N , φ) need not be complete in this norm but it is complete in the norm · 1 + · ∞ (where · ∞ is the uniform norm). We use the notation

L

(1,∞)

(N , φ) = T ∈ N : T L(1,∞)

1 := sup t>0 log(1 + t)

t

μs (T ) ds < ∞ .

0

The reader should note that L(1,∞) (N , φ) is often taken to mean an ideal in the algebra N˜ of φ-measurable operators affiliated to N . Our notation is however consistent with that of [14] in the special case N = B(H). With this convention the ideal of φ-compact operators, K(N ), consists of those T ∈ N (as opposed to N˜ ) such that μ∞ (T ) := limt→∞ μt (T ) = 0. Definition 3.2. A semifinite spectral triple (A, H, D) relative to (N , φ) with A unital is (1, ∞)summable if (D − λ)−1 ∈ L(1,∞) (N , φ) for all λ ∈ C \ R. It follows that if (A, H, D) is (1, ∞)-summable then it is n-summable (with respect to the trace φ) for all n > 1. We next need to briefly discuss Dixmier traces. For more information on semifinite Dixmier traces, see [9,11]. For T ∈ L(1,∞) (N , φ), T 0, the function

A.L. Carey et al. / Journal of Functional Analysis 260 (2011) 1637–1681

1 FT : t → log(1 + t)

1669

t μs (T ) ds 0

∗ is bounded. There are certain ω ∈ L∞ (R+ ∗ ) [9,14], which define (Dixmier) traces on (1,∞) (N , φ) by setting L

φω (T ) = ω(FT ),

T 0

and extending to all of L(1,∞) (N , φ) by linearity. For each such ω we write φω for the associated Dixmier trace. Each Dixmier trace φω vanishes on the ideal of trace class operators. Whenever the function FT has a limit at infinity, all Dixmier traces return that limit as their value. This leads to the notion of a measurable operator [14,24], that is, one on which all Dixmier traces take the same value. 3.2. The Kasparov module and modular spectral triple We have seen that the algebras Qλ do not possess a faithful gauge invariant trace but that there is a KMSβ where β = − log(λ) for the gauge action, γ , namely ψ := τ ◦ Φ : Qλ → C, where Φ : Qλ → F λ is the expectation and τ : F λ → C is a faithful normalised trace. In fact, ψ is the only KMS state for the gauge action (for any β), by Proposition 2.30. We show below that the generator of the gauge action D acting on a suitable C ∗ -F λ -module X gives us a Kasparov module (X, D) whose class lies in KK 1,T (Qλ , F λ ). In some examples, including the case λ ∈ Q, we have K1 (Qλ ) = {0} and so pairing with ordinary K1 would be fruitless. However, following [8,5] we may compute a numerical pairing using a ‘modular spectral triple’ constructed from the Kasparov module. We now review this construction adapted to the present situation. Let H = L2 (Qλ ) be the GNS Hilbert space given by the faithful state ψ with the inner product on Qλ defined by a, b = ψ(a ∗ b) = (τ ◦ Φ)(a ∗ b). Then D is a self-adjoint unbounded operator on H [8]. The representation of Qλ on H by left multiplication (which we now denote by π in place of π0 ) is bounded and nondegenerate: the left action of an element a ∈ Qλ by π(a) satisfies π(a)b = ab for all b ∈ Qλ . This distinction between elements of Qλ as vectors in L2 (Qλ ) and operators on L2 (Qλ ) is sometimes crucial. The dense subalgebra Qλc := eAλc e which is the finite span of elements in Qλ of the form X[a,b) · δg is in the smooth domain of the derivation δ = ad(|D|). We remind the reader that the KMS condition on the modular automorphism group of the state ψ [32] (for t = i) is: ψ(xy) = ψ(σi (π(y))x) = ψ(σ (y)x) for x, y ∈ π(Qλ ), where σ (y) = −1 (y). Lemma 3.3. The group of modular automorphisms of the von Neumann algebra π(Qλ ) is given on the generators by

σt π(f · δg ) := it π(f · δg )−it = π it (f · δg ) = |g|it π(f · δg ) = det(g)it π(f · δg ). Proof. This is immediate from Lemma 2.36 if we note that |g| = det(g).

(2) 2

Corollary 3.4. With Qλ acting on H := L2 (Qλ ) and with D the generator of the natural unitary implementation of the gauge action of T1 on Qλ , we have = λD or eit D = it/ log λ .

1670


To simplify notation, we let A = Qλ and F = F λ = Aγ , the fixed point algebra for the T1 gauge action, γ . For convenience we will suppress the notations D ⊗ 1k and so on. The algebras Ac , Fc are defined as the finite linear span of the generators. Right multiplication makes A into a right F -module, and similarly Ac is a right module over Fc . We define an F -valued inner product (·|·)R on both these modules by (a|b)R := Φ(a ∗ b). Definition 3.5. Let X be the right F C ∗ -module obtained by completing A (or Ac ) in the norm

x2X := (x|x)R F = Φ x ∗ x F . The algebra A acting by left multiplication on X provides a representation of A as adjointable operators on X. Let Xc be the copy of Ac ⊂ X. The T1 action on Xc is unitary and extends to X [5,25]. For all k ∈ Z, the projection operator onto the k-th spectral subspace of the T1 action is also denoted (somewhat carelessly) Φk on X: Φk (x) =

1 2π

z−k uz (x) dθ,

z = eiθ , x ∈ X.

T1

Observe that Φ0 restricts to Φ on A and on generators of Qλ we have Φk (f · δg ) =

f · δg 0

if |g| = λk , otherwise.

(3)

Of course L2 (Qλ ) and X have a common dense subspace Qλc on which these projections are identical. Let Ak = Φk (A) and observe from (3) that A∗k Ak = F = Ak A∗k so that the gauge action γ on Qλ has full spectral subspaces. We quote the following result from [25], the proof in our case is the same. Lemma 3.6. The operators Φk are adjointable endomorphisms of the F -module X such that Φk∗ = Φk = Φk2 and Φk Φl = δk,l Φk . If K ⊂ Z then the sum k∈K Φk converges strictly to a projection in the endomorphism algebra. The sum Φ converges to the identity operator k∈Z k on X. For all x ∈ X, the sum x = k∈Z Φk x = k∈Z xk converges in X. The unbounded operator of the next proposition is of course the generator of the T1 action on X. We refer to Lance’s book [23, Chapters 9, 10], for information on unbounded operators on C ∗ -modules. Proposition 3.7. (See [25].) Let X be the right C ∗ -F -module of Definition 3.5. Define D : XD ⊂ X to be the linear space xk ∈ X: XD = x = k∈Z

2 < ∞ . k (x |x ) k k R k∈Z


For x ∈ XD define D(x) =

k∈Z kxk . Then D

1671

: XD → X is self-adjoint, regular operator on X.

This should be compared to the following Hilbert space version. Proposition 3.8. The generator D of the one-parameter unitary group {uz | z ∈ T1 } on L2 (Qλ , ψ) has eigenspaces given by the ranges of the Φk and D(x) = kx iff Φk (x) = x. In particular 2 2 dom(D) = x = xk Φk (xk ) = xk and k xk < ∞ , k

k

and D( k xk ) = k kxk . Remark. On generators in Qλ regarded as elements of either X or L2 (Qλ , ψ) we have D(f · δg ) = (logλ (|g|))f · δg . To continue, we recall the underlying right C ∗ -F λ -module, X, which is the completion of Qλ R by Θ R z = for the norm x2X = Φ(x ∗ x)F λ . Introduce the rank one operators on X: Θx,y x,y x(y|z)R . Then using the operators Sk,m defined above, we obtain formulas for the projections Φk similar to those of [25, Lemma 4.7] with some important differences. First recall [8, Lemma 3.5]. Lemma 3.9. Any F λ -linear endomorphism T of the module X which preserves the copy of Qλ inside X, extends uniquely to a bounded operator on the Hilbert space H = L2 (Qλ ). In particular, the finite rank endomorphisms of the pre-C ∗ module Qλc (acting on the left) λ satisfy this condition, and we denote the algebra of all these endomorphisms by End00 F (Qc ). Lemma 3.10. (Compare [25, Lemma 4.7].) The following formulas hold in both L(X) and in B(H). (1) For k 0, we have

R Φ0 = Θe,e

while for k > 0, Φk =

mk

ΘSRk,m ,Sk,m .

m=0

(2) For −k < 0, we have Φ−k = ΘSR∗

∗ k,m ,Sk,m

f

or any m = 0, 1, . . . , mk − 1 and also for mk if λ−k = mk + 1.

Proof. Since both Φk and the finite rank endomorphisms satisfy the hypotheses of the previous lemma, the first statement of this lemma will follow from calculations done on generators. The following calculations are based on the formulas in Lemma 2.15.

1672


(1) Let k > 0 and let x = mk

ΘSRk,m ,Sk,m (x) =

l xl

be a finite sum of generators, xl satisfying Φl (xl ) = xl . Then

mk

ΘSRk,m ,Sk,m (xl ) =

mk

l m=0

m=0

=

mk

l

∗ Sk,m Φ Sk,m xk =

m=0

mk

∗ Sk,m Φ Sk,m xl

m=0 ∗ Sk,m Sk,m xk = exk = xk = Φk (x).

m=0

For k = 0 this is a similar but far easier calculation. (2) Let −k < 0 and let x = l xl be a finite sum of generators as above. Then, for 0 m < mk ∗ (x) = k,m ,Sk,m

ΘSR∗

l

∗ (xl ) = k,m ,Sk,m

ΘSR∗

∗ ∗ Sk,m Φ(Sk,m xl ) = Sk,m Φ(Sk,m x−k )

l

∗ = Sk,m Sk,m x−k = ex−k = x−k = Φ−k (x).

2

We recall the following result discussed in Section 3 of [5] (a ‘bare hands’ proof can be given by the method in [8]). λ Proposition 3.11. Let N be the von Neumann algebra N = (End00 F (Qc )) , where we take the commutant inside B(H). Then N is semifinite, and there exists a faithful, semifinite, normal trace R of Qλ , τ˜ : N → C such that for all rank one endomorphisms Θx,y c

R

= (τ ◦ Φ) y ∗ x , τ˜ Θx,y

x, y ∈ Qλc .

In addition, D is affiliated to N and π(Qλ ) is a subalgebra of N . The fact that τ˜ (Φk ) = λ−k implies that with respect to the trace τ˜ we cannot expect D to satisfy a finite summability criterion. We solve this problem exactly as in [8]. Definition 3.12. We define a new weight on N + : let T ∈ N + then τ (T ) := supN τ˜ (N T ) where N = ( |k|N Φk ). Remarks. Since N is τ˜ -trace-class, we see that T → τ˜ (N T ) is a normal positive linear functional on N and hence τ is a normal weight on N + which is easily seen to be faithful and semifinite. As in [8], we now give another way to define τ which is not only conceptually useful but also makes a number of important properties straightforward to verify. Many proofs require only trivial notation changes and the substitution of n± with λ∓ . Notation. Let M be the relative commutant in N of the operator . Equivalently,M is the relative commutant of the set of spectral projections {Φk | k ∈ Z} of D. Clearly, M = k∈Z Φk N Φk . trace with τ˜ (Φk )= λ−k we deDefinition 3.13. As τ˜ restricted to each Φk N Φk is a faithful finite k fine τˆk on Φk N Φk to be λ times the restriction of τ˜ . Then, τˆ := k τˆk on M = k∈Z Φk N Φk is a faithful normal semifinite trace τˆ with τˆ (Φk ) = 1 for all k.


1673

We use τˆ to give an alternative expression for τ below: Lemma 3.14. An element m ∈ N is in M if and only if it is in the fixed point algebra of the −it . Both π(F λ ) and the projections Φ action, σtτ on N defined for T ∈ N by σtτ (T ) = it T k belong to M. The map Ψ : N → M defined by Ψ (T ) = k Φk T Φk is a conditional expectation onto M and τ (T ) = τˆ (Ψ (T )) for all T ∈ N + . That is, τ = τˆ ◦ Ψ so that τˆ (T ) = τ (T ) for all T ∈ M+ . Finally, if one of A, B ∈ M is τˆ -trace-class and T ∈ N then τ (AT B) = τ (AΨ (T )B) = τˆ (AΨ (T )B). Proof. The proof is the same as the proof of [8, Lemma 3.9] with λk in place of n−k .

2

Lemma 3.15. The modular automorphism group σtτ of τ is inner and given by σtτ (T ) = it T −it . The weight τ is a KMS weight for the group σtτ , and σtτ |Qλ = σtτ ◦Φ . Proof. This follows from: [21, Theorem 9.2.38], which gives us the KMS properties of τ : the modular group is inner since is affiliated to N . The final statement about the restriction of the modular group to Qλ is clear. 2 We now have the key lemma: Lemma 3.16. Suppose g is a function on R such that g(D) is τ trace-class in M, then for all f ∈ F λ we have

g(k). τ π(f )g(D) = τ g(D) τ (f ) = τ (f ) k∈Z

Proof. First note that τ (g(D)) = τˆ ( k∈Z g(k)Φk ) = k∈Z g(k)τˆ (Φk ) = k∈Z g(k). We first do the computation for f ∈ Fcλ so that all the sums are finite. Now,

τ π(f )g(D) = τˆ π(f ) g(k)Φk = g(k)τˆ π(f )Φk =

k∈Z

k∈Z

g(k)τˆk π(f )Φk = g(k)λk τ˜ π(f )Φk .

k∈Z

k∈Z

So it suffices to see for each k ∈ Z, we have τ˜ (π(f )Φk ) = λ−k τ (f ). Now, by Theorem 2.35 π(F λ ) is a type II1 factor on H whose unique trace say Tr (with norm one) extends the trace τ on F λ in the sense that Tr(π(f )) = τ (f ). Since the projection Φk is in the commutant of the factor π(F λ ) the map T ∈ π F λ → T Φk = Φk T Φk is a normal isomorphism by [17, Chapter 1, Section 2, Proposition 2] and so it has a unique normalised trace also given by Trace(T Φk ) = Tr(T ). But τ˜ (T Φk ) is a trace on Φk π(F λ ) Φk = π(F λ ) Φk and so must be τ˜ (Φk ) = λ−k times the unique norm one trace. That is, we get the

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required formula:

τ˜ π(f )Φk = λ−k Trace π(f )Φk = λ−k Tr π(f ) = λ−k τ (f ). So for f ∈ Fcλ , we have the formula:

g(k)τ (f ). τ π(f )g(D) = τ g(D) τ (f ) = k∈Z

Now, the right-hand side is a norm-continuous function of f . To see that the left side is normcontinuous we do it in more generality. Let T ∈ N , then since τˆ is a trace on M we get

τ T g(D) = τˆ Ψ T g(D) = τˆ Ψ (T )g(D) Ψ (T )τˆ g(D)

T τˆ g(D) = T τ g(D) . That is the left-hand side is norm-continuous in T and so we have the formula:

τ π(f )g(D) = τ g(D) τ (f ) = g(k)τ (f ) k∈Z

for all f ∈ F λ .

2

Proposition 3.17. (i) We have (1 + D2 )−1/2 ∈ L(1,∞) (M, τ ). That is, τ ((1 + D2 )−s/2 ) < ∞ for all s > 1. Moreover, for all f ∈ F λ

−s/2

= 2τ (f ) lim (s − 1)τ π(f ) 1 + D2

s→1+

so that π(f )(1 + D2 )−1/2 is a measurable operator in the sense of [24]. (ii) For π(a) ∈ π(Qλ ) ⊂ N the following (ordinary) limit exists and

1

−s/2

= τ ◦ Φ(a), τˆω π(a) = lim (s − 1)τ π(a) 1 + D2 2 s→1+ the original KMS state ψ = τ ◦ Φ on Qλ . Proof. (i) This proof is identical to [8, Proposition 3.12]. (ii) This proof is the same as [8, Proposition 3.14] with Qλ , F λ replacing On , F .

2

Definition 3.18. The triple (A, H, D) along with γ , ψ, N , τ satisfying properties (0) to (3) below is called the modular spectral triple of the dynamical system (Qλ , γ , ψ) (0) The ∗-subalgebra A = Qλc of the algebra Qλ is faithfully represented in N with the latter acting on the Hilbert space H = L2 (Qλ , ψ).


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(1) There is a faithful normal semifinite weight τ on N such that the modular automorphism group of τ is an inner automorphism group σt (for t ∈ C) of (the Tomita algebra of) N with σi |A = σ in the sense that σi (π(a)) = π(σ (a)), where σ is the automorphism σ (a) = −1 (a) on A. (2) τ restricts to a faithful semifinite trace τˆ on M = N σ , with a faithful normal projection Ψ : N → M satisfying τ = τˆ ◦ Ψ on N . (3) With D the generator of the one parameter group implementing the gauge action of T on H we have: [D, π(a)] extends to a bounded operator (in N ) for all a ∈ A and for λ in the resolvent set of D, (λ − D)−1 ∈ K(M, τ ), where K(M, τ ) is the ideal of compact operators in M relative to τ . In particular, D is affiliated to M. For matrix algebras A = Qλc ⊗ Mk over Qλc , (Qλc ⊗ Mk , H ⊗ Mk , D ⊗ Idk ) is also a modular spectral triple in the obvious fashion. We need some technical lemmas for the discussion in the next section. A function f from a complex domain Ω into a Banach space X is called holomorphic if it is complex differentiable in norm on Ω. The following is proved in [8, Lemma 3.15]. Lemma 3.19. (1) Let B be a C ∗ -algebra and let T ∈ B + . The mapping z → T z is holomorphic (in operator norm) in the half-plane Re(z) > 0. (2) Let B be a von Neumann algebra with faithful normal semifinite trace φ and let T ∈ B + be in L(1,∞) (B, φ). Then, the mapping z → T z is holomorphic (in trace norm) in the half-plane Re(z) > 1. (3) Let B, and T be as in item (2) and let A ∈ B then the mapping z → φ(AT z ) is holomorphic for Re(z) > 1. Lemma 3.20. In these modular spectral triples (A, H, D) for matrices over the algebras Qλ we have (1 + D2 )−s/2 ∈ L1 (M, τ ) for all s > 1 and for x ∈ N , τ (x(1 + D2 )−r/2 ) is holomorphic for Re(r) > 1 and we have for a ∈ Qλc , τ ([D, π(a)](1 + D2 )−r/2 ) = 0, for Re(r) > 1. Proof. We include a brief proof since there are some small but important differences from [8, Lemma 3.16]. Since the eigenvalues for D are precisely the set of integers, and the projection Φk on the eigenspace with eigenvalue k satisfies τ (Φk ) = 1, it is clear that (1 + D2 )−s/2 ∈ L1 (M, τ ). Now, τ (x(1 + D2 )−r/2 ) = τˆ (Ψ (x)(1 + D2 )−r/2 ) is holomorphic for Re(r) > 1 by item (3) of the previous lemma. To see the last statement, we observe that τ ([D, π(a)](1 + D2 )−r/2 ) = τ (Ψ ([D, π(a)]) × (1 + D2 )−r/2 ), so it suffices to see that Ψ ([D, π(a)]) = 0 for a ∈ A = Qλc . To this end, let a = f · δg where det(g) = λn is one of the linear generators of Qλc . Then by calculating the action of the operator Dπ(f · δg ) on the linear generators fi · δhi of the Hilbert space, H, we obtain

Dπ(f · δg ) = nπ(f · δg ) + π(f · δg )D that is D, π(f · δg ) = logλ |g| π(f · δg ). More generally,

#

D, π

m i=1

$ ci fi · δhi

=

m

ci logλ |hi | π(fi · δhi ). i=1

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If we apply Ψ to this equation, we see that Ψ (π(fi · δhi )) = π(Φ(fi · δhi )) = 0 whenever logλ (|hi |) = 0, and so the whole sum is 0. We also observe that [D, π(a)] ∈ π(Qλc ) for all a ∈ Qλc . This is not too surprising since D is the generator of the action γ of T on Qλ . 2 3.3. Modular K1 We now make appropriate modifications to [8, Section 4] using [5] introducing elements of these modular spectral triples (A, H, D) (where A is a matrix algebra over Qλc ) that will have a well-defined pairing with our Dixmier functional τˆω . Let A = Qλ . Following [20] we say that a unitary (invertible, projection, . . .) in the n × n matrices over Qλ for some n is a unitary (invertible, projection, . . .) over Qλ . We write σt for the automorphism σt ⊗ Idn of A. Definition 3.21. Let v be a partial isometry in the ∗-algebra A. We say that v satisfies the modular condition with respect to σ if the operators vσt (v ∗ ) and v ∗ σt (v) are in the fixed point algebra F ⊂ A for all t ∈ R. Of course, any partial isometry in F is a modular partial isometry. Lemma 3.22. (See [8, Lemma 4.8].) Let v ∈ A be a modular partial isometry. Then we have

uv =

1 − v∗v v

v∗ 1 − vv ∗

is a modular unitary over A. Moreover there is a modular homotopy uv ∼ uv ∗ . Note that in [8] we used a different approach which is implied by the one given here. In [8] we defined modular unitaries in terms of the regular automorphism:

π σ (a) = π −1 (a) = −1 π(a) = σi π(a) . That is we said that a unitary in A is modular if uσ (u∗ ) and u∗ σ (u) are in the fixed point algebra. Examples. (1) For k, j > 0 recall Sk,m ∈ Qλc with m < mk (see Definition 2.14) we write Pk,m = ∗ =X λ Sk,m Sk,m [mλk ,(m+1)λk ) · δ1 which is in clearly F . Then for each {k, m}, {j, n} we have a unitary

u{k,m},{j,n} =

1 − Pk,m ∗ Sj,n Sk,m

∗ Sk,m Sj,n 1 − Pj,n

.

It is simple to check that this a self-adjoint unitary satisfying the modular condition, and that τ (Pk,m ) = λk and τ (Pj,n ) = λj . These examples behave very much like the Sμ Sν∗ examples of [8]. (2) For k, j > 0 consider the “leftover” partial isometries Sk,mk and Sj,mj of Definition 3.13 which we will denote by Sk and Sj to lighten the notation. We let vj,k = Sj Sk∗ and calculate its range and initial projections which are both in F λ : Pj = Sj Sk∗ Sk Sj∗ = X[mj λj ,mj λj +λj (λ−k −mk )) · δ1 ,


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and Pk = Sk Sj∗ Sj Sk∗ = X[mk λk ,mk λk +λk (λ−j −mj )) · δ1 . We note for future reference that

τ (Pj ) = λj λ−k − mk

and τ (Pk ) = λk λ−j − mj .

We also note that we have a modular unitary uj,k :

uj,k =

1 − Pk Sj Sk∗

Sk Sj∗ 1 − Pj

.

Define the modular K1 group as follows. Definition 3.23. Let K1 (A, σ ) be the abelian group with one generator [v] for each partial isometry v over A satisfying the modular condition and with the following relations: (1) [v] = 0 if v is over F , (2) [v] + [w] = [v ⊕ w], (3) if vt , t ∈ [0, 1], is a continuous path of modular partial isometries in some matrix algebra over A then [v0 ] = [v1 ]. One could use modular unitaries as in [8] in place of these modular partial isometries. The following can now be seen to hold. Lemma 3.24. (Compare [8, Lemma 4.9].) Let (A, H, D) be our modular spectral triple relative to (N , τ ) and set F = Aσ and σ : A → A. Let L∞ () = L∞ (D) be the von Neumann algebra generated by the spectral projections of then L∞ () ⊂ Z(M). Let v ∈ A be a partial isometry with vv ∗ , v ∗ v ∈ F . Then π(v)Qπ(v ∗ ) ∈ M and π(v ∗ )Qπ(v) ∈ M for all spectral projections Q of D, if and only if v is modular. That is, π(v)π(v ∗ ) and π(v ∗ )π(v) (or π(v)Dπ(v ∗ ) and π(v ∗ )Dπ(v)) are both affiliated to M if and only if v is modular. Thus we see that modular partial isometries conjugate to an operator affiliated to M, and so vv ∗ commutes with (and vDv ∗ commutes with D). We will next show that there is an analytic pairing between (part of) modular K1 and modular spectral triples. To do this, we are going to use the analytic formulae for spectral flow in [6]. 3.4. The mapping cone algebra Our aim in the remainder is to calculate an index pairing explicitly for the matrix algebras A over the smooth subalgebra Qλc of Qλ . In the following few pages we will sometimes abuse notation and write a in place of π(a) for a ∈ A in order to make our formulae more readable. Whenever we do this, however, we will use σi (·) = −1 (·) the spatial version of the algebra homomorphism, σ . We will generally use the spatial version σi when in the presence of operators not in π(A).

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We briefly review some results from [5], that provide an interpretation of the modular index pairing given by the spectral flow. If F ⊂ A is a sub-C ∗ -algebra of the C ∗ -algebra A, then the mapping cone algebra for the inclusion is M(F, A) = f : R+ = [0, ∞) → A: f is continuous and vanishes at infinity, f (0) ∈ F . When F is an ideal in A it is known that K0 (M(F, A)) ∼ = K0 (A/F ) [28]. In general, K0 (M(F, A)) is the set of homotopy classes of partial isometries v ∈ Mk (A) with range and source projections vv ∗ , v ∗ v in Mk (F ), with operation the direct sum and inverse −[v] = [v ∗ ]. All this is proved in [28]. It is shown in [5] that there is a natural map that injects K1 (A, σ ) into K0T (M, F ), the equivariant K-theory of the mapping cone algebra. Note that the T action on A lifts in the obvious way to the mapping cone. Now, it was shown in [7] that certain Kasparov A, F -modules extend to Kasparov M(F, A), F -modules, and this was extended to the equivariant case in [5]. Importantly the theory applies to the equivariant Kasparov module coming from a circle action. The extenˆ which is a graded unbounded Kasparov module for ˆ D) sion is explicit, namely there is a pair (X, the mapping cone algebra M(F, A) constructed using a generalised APS construction [2]. If v is a partial isometry in Mk (A), setting

ev (t) =

vv ∗ 1+t 2 t iv ∗ 1+t 2

1−

t −iv 1+t 2 v∗ v 1+t 2

,

defines ev as a projection over M(F, A). Then in [5] we showed that if v ∈ A is a modular partial isometry we have %

[ev ] −

1 0

0 0

&

ˆ ˆ D) , (X, = − Index P vP : v ∗ vP (X) → vv ∗ P (X) ∈ K0 (F )

= Index P v ∗ P : vv ∗ P (X) → v ∗ vP (X) ∈ K0T (F ).

(4)

We thus obtain an index map K1 (A, σ ) → K0T (F ). The latter may be thought of as the ring of Laurent polynomials K0 (F )(χ, χ −1 ) where we think of χ, χ −1 as generating the representation ring of T. We may obtain a real valued invariant from this map by evaluating χ at e−β where β is the inverse temperature of our KMS state and applying the trace to the resultant element of K0 (F ). Then one of the main results of [5] is that the real valued invariant so obtained is identical with the spectral flow invariant of the next subsection. However the general theory of [5] does not tell us the range of this index map and it is the latter that is of interest for these explicit calculations. 3.5. A local index formula for the algebras Qλ Using the fact that we have full spectral subspaces we know from [5] that there is a formula for spectral flow which is analogous to the local index formula in noncommutative geometry. We remind the reader that τ = τˆ ◦ Ψ where Ψ : N → M is the canonical expectation, so that τ restricted to M is τˆ .


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Theorem 3.25. (Compare [8, Theorem 5.5].) Let (A, H, D) be the (1, ∞)-summable, modular spectral triple for the algebra Qλ we have constructed previously. Then for any modular partial isometry v and for any Dixmier trace τˆω˜ associated to τˆ , we have spectral flow as an actual limit

1

−s/2

sfτˆ vv ∗ D, vDv ∗ = lim (s − 1)τˆ v D, v ∗ 1 + D2 2 s→1+

−1/2

1 = τˆω˜ v D, v ∗ 1 + D2 = τ ◦ Φ v D, v ∗ . 2 The functional on A ⊗ A defined by a0 ⊗ a1 → 12 lims→1+ (s − 1)τ (a0 [D, a1 ](1 + D2 )−s/2 ) is a σ -twisted b, B-cocycle (see the proof below for the definition). Remark. Spectral flow in this setting is independent of the path joining the endpoints of unbounded self adjoint operators affiliated to M however it is not obvious that this is enough to show that it is constant on homotopy classes of modular unitaries. This latter fact is true but the proof is lengthy so we refer to [5]. Theorem 3.26. We let (Qλc ⊗ M2 , H ⊗ C2 , D ⊗ 12 ) be the modular spectral triple of (Qλc ⊗ M2 ). (1) Let u be a modular unitary defined in Section 5 of the form

∗ 1 − Pk,m Sk,m Sj,n . u{k,m},{j,n} = ∗ Sj,n Sk,m 1 − Pj,n Then the spectral flow is positive being given by

sfτ D, uDu∗ = (k − j ) λj − λk ∈ Z[λ] ⊂ Γλ . (2) Let u be a modular unitary defined in Section 5 of the form:

1 − Pk Sk Sj∗ uj,k = , Sj Sk∗ 1 − Pj ∗ and Pk and Pj are its range and initial projections, respectively. where Sk Sj∗ = Sk,mk Sj,m j Then the spectral flow is given by

sfτ D, uDu∗ = (k − j ) λj λ−k − mk − λk λ−j − mj ∈ Γλ . Proof. We have already observed that these are, in fact modular unitaries. For the computations we use a calculation from the proof of Lemma 3.20 to get in example (1):

∗ ∗ ] 0 [D, Sk,m Sj,n 1 − Pk,m Sk,m Sj,n u[D ⊗ 12 , u] = ∗ ∗ ] Sj,n Sk,m 1 − Pj,n 0 [D, Sj,n Sk,m

∗ ∗ 0 (k − j )Sk,m Sj,n 1 − Pk,m Sk,m Sj,n = ∗ ∗ Sj,n Sk,m 1 − Pj,n (j − k)Sj,n Sk,m 0

−Pk,m 0 . = (k − j ) 0 Pj,n

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So using Theorem 3.25 and our previous computation of the Dixmier trace, Proposition 3.17, ∗ =X k and the fact that Pk,m = Sk,m Sk,m [mλk ,(m+1)λk ) · δ1 so that τ (Pk,m ) = λ we have

sfτ (D, uk,m Duk,m ) = (k − j )τ (Pj,n − Pk,m ) = (k − j ) λj − λk . This number is always positive as the reader may check, and is contained in Z[λ]. The computations in example (2) are similar and use the fact that Pk = X[mk λk ,mk λk +λk (λ−j −mj )) · δ1 , so that τ (Pk ) = λk (λ−j − mj ) ∈ Γλ . In these examples, the spectral flow is not contained in the smaller polynomial ring, Z[λ]. 2 Remarks. The observation of [8] that the twisted residue cocycle formula for spectral flow is calculating Araki’s relative entropy of two KMS states [1] also applies to the examples in this subsection. Acknowledgments We would like to thank Nigel Higson, Ryszard Nest, Sergey Neshveyev, Marcelo Laca, Iain Raeburn and Peter Dukes for advice and comments. The first and fourth named authors were supported by the Australian Research Council. The second and third named authors acknowledge the support of NSERC (Canada). References [1] H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci. 11 (1976) 809–833; H. Araki, Relative entropy for states of von Neumann algebras II, Publ. Res. Inst. Math. Sci. 13 (1977) 173–192. [2] M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99. [3] O. Bratteli, D. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, second ed., Springer-Verlag, 1987. [4] O. Bratteli, D. Robinson, Operator Algebras and Quantum Statistical Mechanics 2, second ed., Springer-Verlag, 1987. [5] A.L. Carey, R. Nest, S. Neshveyev, A. Rennie, Twisted cyclic theory, equivariant KK-theory and KMS states, J. Reine Angew. Math., in press. [6] A.L. Carey, J. Phillips, Spectral flow in θ -summable Fredholm modules, eta invariants and the JLO cocycle, KTheory 31 (2004) 135–194. [7] A.L. Carey, J. Phillips, A. Rennie, A noncommutative Atiyah–Patodi–Singer index theorem in KK-theory, J. Reine Angew. Math. 643 (2010) 59–109. [8] A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras, K-Theory 6 (2010) 339–380. [9] A.L. Carey, J. Phillips, F. Sukochev, Spectral flow and Dixmier traces, Adv. Math. 173 (2003) 68–113. [10] A.L. Carey, J. Phillips, A. Rennie, F. Sukochev, The local index formula in semifinite von Neumann algebras I: Spectral Flow, Adv. Math. 202 (2006) 451–516. [11] A.L. Carey, A. Rennie, A. Sedaev, F. Sukochev, The Dixmier trace and asymptotics of zeta functions, J. Funct. Anal. 249 (2007) 253–283. [12] A.L. Carey, A. Rennie, K. Tong, Spectral flow invariants and twisted cyclic theory from the Haar state on SU q (2), J. Geom. Phys. 59 (2009) 1431–1452. [13] A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. 6 (4) (1973) 18–252. [14] A. Connes, Noncommutative Geometry, Academic Press, 1994.


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[15] J. Cuntz, Simple C ∗ -algebras generated by isometries, Comm. Math. Phys. 57 (1977) 173–189. [16] J. Cuntz, C ∗ -algebras associated with the ax + b-semigroup over N, in: G. Cortinas, J. Cuntz, M. Karoubi, R. Nest, C.A. Weibel (Eds.), K-Theory and Noncommutative Geometry, in: EMS Series of Congress Reports, vol. 2, 2008. [17] J. Dixmier, Von Neumann Algebras, North-Holland, 1981. [18] G. Elliott, Some simple C ∗ -algebras constructed as crossed products with discrete outer automorphism groups, Publ. Res. Inst. Math. Sci. 16 (1980) 299–311. [19] T. Fack, H. Kosaki, Generalised s-numbers of τ -measurable operators, Pacific J. Math. 123 (1986) 269–300. [20] N. Higson, J. Roe, Analytic K-Homology, Oxford University Press, 2000. [21] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II: Advanced Theory, Academic Press, 1986. [22] M. Laca, J. Spielberg, Purely infinite C ∗ -algebras from boundary actions of discrete groups, J. Reine Angew. Math. 480 (1996) 125–139. [23] E.C. Lance, Hilbert C ∗ -Modules, Cambridge University Press, Cambridge, 1995. [24] S. Lord, A. Sedaev, F.A. Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators, J. Funct. Anal. 224 (1) (2005) 72–106. [25] D. Pask, A. Rennie, The noncommutative geometry of graph C ∗ -algebras I: The index theorem, J. Funct. Anal. 233 (2006) 92–134. [26] G.K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, London Math. Soc. Monogr., vol. 14, Academic Press, London, 1979. [27] J. Phillips, I. Raeburn, Semigroups of isometries, Toeplitz algebras and twisted crossed products, Integral Equations Operator Theory 17 (1993) 579–602. [28] I. Putnam, An excision theorem for the K-theory of C ∗ -algebras, J. Operator Theory 38 (1997) 151–171. [29] I. Putnam, On the K-theory of C ∗ -algebras of principal groupoids, Rocky Mountain J. Math. 28 (4) (1998) 1483– 1518. [30] M. Rørdam, E. Størmer, Classification of Nuclear C ∗ -Algebras. Entropy in Operator Algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002. [31] C. Schochet, Topological methods for C ∗ -algebras II: geometric resolutions and the Künneth formula, Pacific J. Math. 98 (2) (1982) 443–458. [32] M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and Its Applications, Lecture Notes in Math., vol. 128, Springer, Berlin, 1970.


Geometric analysis on small unitary representations of GL(N, R) Toshiyuki Kobayashi a,b,∗ , Bent Ørsted c , Michael Pevzner d a Graduate School of Mathematical Sciences, IPMU, The University of Tokyo, 3-8-1 Komaba, Meguro,

Tokyo 153-8914, Japan b Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France 1 c Matematisk Institut, Byg. 430, Ny Munkegade, 8000 Aarhus C, Denmark d Laboratoire de Mathématiques, Université de Reims, 51687 Reims, France

Received 15 February 2010; accepted 9 December 2010 Available online 28 December 2010 Communicated by P. Delorme

Abstract The most degenerate unitary principal series representations πiλ,δ (λ ∈ R, δ ∈ Z/2Z) of G = GL(N, R) attain the minimum of the Gelfand–Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction πiλ,δ |H (branching law) with respect to all symmetric pairs (G, H ). For N = 2n with n 2, the restriction πiλ,δ |H remains irreducible for H = Sp(n, R) if λ = 0 and splits into two irreducible representations if λ = 0. The branching law of the restriction πiλ,δ |H is purely discrete for H = GL(n, C), consists only of continuous spectrum for H = GL(p, R) × GL(q, R) (p + q = N ), and contains both discrete and continuous spectra for H = O(p, q) (p > q 1). Our emphasis is laid on geometric analysis, which arises from the restriction of ‘small representations’ to various subgroups. © 2010 Elsevier Inc. All rights reserved. Keywords: Small representation; Branching law; Symmetric pair; Reductive group; Phase space representation; Symplectic group; Degenerate principal series representations

* Corresponding author at: Graduate School of Mathematical Sciences, IPMU, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan. E-mail addresses: [email protected] (T. Kobayashi), [email protected] (B. Ørsted), [email protected] (M. Pevzner). 1 Current address.

0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.008

T. Kobayashi et al. / Journal of Functional Analysis 260 (2011) 1682–1720

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1. Introduction The subject of our study is geometric analysis on ‘small representations’ of GL(N, R) through branching problems to non-compact subgroups. Here, by a branching problem, we mean a general question on the understanding how irreducible representations of a group decompose when restricted to a subgroup. A classic example is studying the irreducible decomposition of the tensor product of two representations. Branching problems are one of the most basic problems in representation theory, however, it is hard in general to find explicit branching laws for unitary representations of non-compact reductive groups. For reductive symmetric spaces G/H , the multiplicities in the Plancherel formula of L2 (G/H ) are finite [1,6], whereas the multiplicities in the branching laws for the restriction G ↓ H are often infinite even when (G, H ) are symmetric pairs (see e.g. [16] for recent developments and open problems in this area). Our standing point is that ‘small representations’ of a group should have ‘large symmetries’ in the representation spaces, as was advocated by one of the authors from the perspectives in global analysis [17]. In particular, considering the restrictions of ‘small representations’ to reasonable subgroups, we expect that their breaking symmetries should have still fairly large symmetries, for which geometric analysis would deserve finer study. Then, what are ‘small representations’? For this, the Gelfand–Kirillov dimension serves as a coarse measure of the ‘size’ of infinite dimensional representations. We recall that for an irreducible unitary representation π of a real reductive Lie group G the Gelfand–Kirillov dimension DIM(π) takes the value in the set of half the dimensions of nilpotent orbits in the Lie algebra g. We may think of π as one of the ‘smallest’ infinite dimensional representations of G, if DIM(π) equals n(G), half the dimension of the minimal nilpotent orbit. For the metaplectic group G = Mp(m, R), the connected two-fold covering group of the symplectic group Sp(m, R) of rank m, the Gelfand–Kirillov dimension attains its minimum n(G) = m at the Segal–Shale–Weil representation. For the indefinite orthogonal group G = O(p, q) (p, q > 3), there exists π such that DIM(π) = n(G) (= p + q − 3) if and only if p + q is even according to an algebraic result of Howe and Vogan. See e.g. a survey paper [11] for the algebraic theory of ‘minimal representations’, and [10,17–22] for their analytic aspects. In general, a real reductive Lie group G admits at most finitely many irreducible unitary representations π with DIM(π) = n(G) if the complexified Lie algebra gC does not contain a simple factor of type A (see [11]). In contrast, for G = GL(N, R), there exist infinitely many irreducible unitary representations π with DIM(π) = n(G) (= N − 1). For example, the unitarily induced representations GL(N,R)

πiλ,δ

GL(N,R)

:= IndPN

(χiλ,δ )

(1.1)

from a unitary character χiλ,δ of a maximal parabolic subgroup PN := GL(1, R) × GL(N − 1, R) RN −1

(1.2)

are such representations with parameter λ ∈ R and δ ∈ Z/2Z. GL(N,R) In this paper, we find the irreducible decomposition of these ‘small representations’ πiλ,δ with respect to all symmetric pairs. We recall that a pair of Lie groups (G, H ) is said to be a symmetric pair if there exists an involutive automorphism σ of G such that H is an open subgroup of Gσ := {g ∈ G: σ g = g}.

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According to M. Berger’s classification [4], the following subgroups H = K, Gj (1 j 4) and G exhaust all symmetric pairs (G, H ) for G = GL(N, R) up to local isomorphisms and the center of G: K := O(N )

(maximal compact subgroup),

G1 := Sp(n, R)

(N = 2n),

G2 := GL(n, C)

(N = 2n),

G3 := GL(p, R) × GL(q, R) (N = p + q), G4 := O(p, q)

(N = p + q). GL(N,R)

It turns out that the branching laws for the restrictions of πiλ,δ with respect to these subgroups behave nicely in all the cases, and in particular, the multiplicities of irreducible representations in the branching laws are uniformly bounded. GL(N,R) To be more specific, the restriction of πiλ,δ to K splits discretely into the space of spherN ical harmonics on R , and the resulting K-type formula is multiplicity-free and so-called of ladder type. For the non-compact subgroups Gj (1 j 4), we prove the following irreducible decompositions in Theorems 8.1, 9.1, 10.1 and 11.1: GL(N,R)

Theorem 1.1. For λ ∈ R and δ ∈ Z/2Z, the irreducible unitary representation πiλ,δ poses when restricted to symmetric pairs as follows: 1) GL(2n, R) ↓ Sp(n, R) (n 2): GL(2n,R) πiλ,δ G1

(λ = 0),

Irreducible Sp(n,R) + (π0,δ )

Sp(n,R) − ⊕ (π0,δ )

(λ = 0).

2) GL(2n, R) ↓ GL(n, C): ⊕ GL(n,C) GL(2n,R) πiλ,m . πiλ,δ G 2

m∈2Z+δ

3) GL(p + q, R) ↓ GL(p, R) × GL(q, R): GL(p+q,R) πiλ,δ G3

⊕

δ ∈Z/2Z R

GL(p,R)

πiλ ,δ

πi(λ−λ ),δ−δ dλ . GL(q,R)

4) GL(p + q, R) ↓ O(p, q): GL(p+q,R) πiλ,δ G4

⊕ ν∈Aδ+ (p,q)

O(p,q) π+,ν

⊕

⊕ ν∈Aδ+ (q,p)

O(p,q) π−,ν

⊕ ⊕2 R+

O(p,q)

πiν,δ

dν.

decom-


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Here, each summand in the right-hand side stands for (pairwise inequivalent) irreducible representations of the corresponding subgroups which will be defined explicitly in Sections 8, 9, 10 and 11. GL(2n,R) As indicated above, we see that the representation πiλ,δ remains generically irreducible when restricted to the subgroup G1 = Sp(n, R) and splits into a direct sum of two irreducible subrepresentations for λ = 0 and n > 1. The case n = 1 is well known (cf. [3]): the group Sp(1, R) is isomorphic to SL(2, R), and πiλ,δ are irreducible except for (λ, δ) = (0, 1), while π0,1 splits into the direct sum of two irreducible unitary representations i.e. the (classical) Hardy space and its dual. GL(2n,R) The representation πiλ,δ is discretely decomposable in the sense of [15] when restricted to the subgroup G2 = GL(n, C). In other words, the non-compact group G2 behaves in the repGL(2n,R) resentation space of πiλ,δ as if it were a compact subgroup. In contrast, the restriction of GL(p+q,R)

πiλ,δ

to another subgroup G3 = GL(p, R) × GL(q, R) decomposes without discrete specGL(p+q,R)

trum, while both discrete and continuous spectra appear for the restriction of πiλ,δ to G4 = O(p, q) if p, q 1 and (p, q) = (1, 1). Finally, in Theorem 12.1 we give an irreducible decomposition of the tensor product of the Segal–Shale–Weil representation with its dual, giving another example of explicit branching laws of small representations with respect to symmetric pairs. We have stated Theorem 1.1 from representation theoretic viewpoint. However, our emphasis is not only on results of this nature but also on geometric analysis of concrete models via branching laws of small representations, which we find surprisingly rich in its interaction with various domains of classical analysis and their new aspects. It includes the theory of Hilbert-space valued Hardy spaces (Section 2), the Weyl operator calculus (Section 3), representation theory of Jacobi and Heisenberg groups, the Segal–Shale–Weil representation of the metaplectic group (Section 4), (complex) spherical harmonics (Section 5), the K-Bessel functions (Section 7), and global analysis on space forms of indefinite-Riemannian manifolds (Section 11). Further, we introduce a non-standard L2 -model for the degenerate principal series representations of Sp(n, R) where the Knapp–Stein intertwining operator becomes an algebraic operator (Theorem 6.1). In this model the minimal K-types are given in terms of Bessel functions (Propo± sition 7.1). The two irreducible components π0,δ at λ = 0 in Theorem 1.1 1) will be presented in three ways, that is, in terms of Hardy spaces based on the Weyl operator calculus as giving the P -module structure, complex spherical harmonics as giving the K-module structure, and the eigenspaces of the Knapp–Stein intertwining operators (see Theorem 8.3). The authors are grateful to an anonymous referee for bringing the papers of Barbasch [2] and Farmer [9] to our attention. Notation. N = {0, 1, 2, . . .}, N+ = {1, 2, 3, . . .}, R± = {ρ ∈ R: ±ρ 0}, R× = R \ {0}, and C× = C \ {0}. 2. Hilbert space valued Hardy space Let W be a (separable) Hilbert space. Then, we can define the Bochner integrals of weakly measurable functions on R with values in W . For a measurable set E in R, we denote by L2 (E, W ) the Hilbert space consisting of W -valued square integrable functions on E. Clearly, it is a closed subspace of L2 (R, W ).

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Suppose F is a W -valued function defined on an open subset in C. We say F is holomorphic if the scalar product (F, w)W is a holomorphic function for any w ∈ W . Let Π+ be the upper half plane {z = t + iu ∈ C: u = Im z > 0}. Then, the W -valued Hardy space is defined as 2 (W ) := F : Π+ → W : F is holomorphic and F H2 (W ) < ∞ , H+ +

(2.1)

where the norm F H2 (W ) is given by +

F H2 (W ) := +

sup u>0

F (t + iu)2 dt W

1 2

.

R

2 (W ) is defined by replacing Π with the lower half plane Π . Notice that Similarly, H− + − is the classical Hardy space, if W = C. Next, we define the W -valued Fourier transform F as

2 (W ) H+

F : L2 (R, W ) → L2 (R, W ),

f (t) → (F f )(ρ) :=

f (t)e−2πiρt dt.

R

Here, the Bochner integral converges for f ∈ (L1 ∩ L2 )(R, W ) with obvious notation. Then, F extends to the Hilbert space L2 (R, W ) as a unitary isomorphism. Example 2.1. Suppose W = L2 (Rk ) for some k. Then, we have a natural unitary isomorphism L2 (R, W ) L2 (Rk+1 ). Via this isomorphism, the L2 (Rk )-valued Fourier transform F is identified with the partial Fourier transform Ft with respect to the first variable t as follows: L2 (R, L2 (Rk ))

∼

F

L2 (R, L2 (Rk ))

L2 (Rk+1 )

∼

(2.2)

L2 (Rk+1 )

Ft

2 ≡ H2 (C), we As in the case of the classical theory on the (scalar-valued) Hardy space H+ + 2 (W ) by means of the Fourier transform: can characterize H± 2 (W ) the W -valued Hardy spaces Lemma 2.2. Let W be a separable Hilbert space, and H± (see (2.1)). 2 (W ), the boundary value 1) For F ∈ H±

F (t ± i0) := lim F (t ± iu) u↓0

exists as a weak limit in the Hilbert space L2 (R, W ), and defines an isometric embedding:

T. Kobayashi et al. / Journal of Functional Analysis 260 (2011) 1682–1720 2 H± (W ) → L2 (R, W ).

1687

(2.3)

2 (W ) as a closed subspace of L2 (R, W ). From now, we regard H± 2) The W -valued Fourier transform F induces the unitary isomorphism: ∼ 2 F : H± (W ) − → L2 (R± , W ). 2 (W ) ⊕ H2 (W ) (direct sum). 3) L2 (R, W ) = H+ − 2 (W ) satisfies F (t + i0) = F (−t + i0) then F ≡ 0. 4) If a function F ∈ H+

Proof. The idea is to reduce the general case to the classical one by using a uniform estimate on norms as the imaginary part u tends to zero. 2 (W ). Then we have Let {ej } be an orthonormal basis of W . Suppose F ∈ H+

F 2H2 (W ) = sup +

u>0

= sup

F (t + iu)2 dt W

R

(2.4)

Ij (u),

u>0 j

where we set Ij (u) :=

F (t + iu), ej 2 dt. W

R

Then, it follows from (2.4) that for any j supu>0 Ij (u) < ∞ and therefore Fj (z) := Fj (z), ej W

(z = t + iu ∈ Π+ )

2 . By the classical Paley–Wiener theorem for the belongs to the (scalar-valued) Hardy space H+ 2 (scalar-valued) Hardy space H+ , we have

weak limit in L2 (R) ,

Fj (t + i0) := lim Fj (t + iu) u↓0

F Fj (t + i0) ∈ L2 (R+ ), F Fj (t + iu) (ρ) = e−2πuρ F Fj (t + i0) (ρ)

(2.6) for u > 0,

Ij (u) is a monotonely decreasing function of u > 0, 2 2 lim Ij (u) = Fj (t + iu)H2 = Fj (t + i0)L2 (R) . +

u↓0

(2.5)

(2.7) (2.8) (2.9)

The formula (2.7) shows (2.8), which is crucial in the uniform estimate as below. In fact by (2.8) we can exchange supu>0 and j in (2.4). Thus, we get

F 2H2 (W ) = +

j

lim Ij (u) = u↓0

Fj (t + i0)2 2

L (R)

j

.

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Hence we can define an element of L2 (R, W ) as the following weak limit: F (t + i0) :=

Fj (t + i0)ej .

j

Equivalently, F (t + i0) is the weak limit of F (t + iu) in L2 (R, W ) as u → 0. Further, (2.6) implies supp F F (t + i0) ⊂ R+ because F F (t + i0) =

F Fj (t + i0)ej

(weak limit).

j

In summary we have shown that F (t + i0) ∈ L2 (R, W ), F F (t + i0) ∈ L2 (R+ , W ), and

F H2 (W ) = F (t + i0)L2 (R,W ) = F F (t + i0)L2 (R +

+ ,W )

2 (W ). Thus, we have proved that the map for any F ∈ H+ 2 F : H+ (W ) → L2 (R+ , W )

is well defined and isometric. 2 (W ) is proved in a similar way. Conversely, the opposite inclusion F −1 (L2 (R+ , W )) ⊂ H+ Hence the statements 1), 2) and 3) follow. The last statement is now immediate from 2) because F F (t +i0)(ρ) = F F (−t +i0)(−ρ). 2 3. Weyl operator calculus In this section, based on the well-known construction of the Schrödinger representation and the Segal–Shale–Weil representation, we introduce the action of the outer automorphisms of the Heisenberg group on the Weyl operator calculus (see (3.11), (3.13), and (3.14)), and discuss carefully its basic properties, see Proposition 3.2 and Lemma 3.4. In particular, the results of this GL(2n,R) section will be used in analyzing of the ‘small representation’ πiλ,δ , when restricted to a certain maximal parabolic subgroup of Sp(n, R), see e.g. the identity (4.12). Let R2m be the 2m-dimensional Euclidean vector space endowed with the standard symplectic form ω(X, Y ) ≡ ω (x, ξ ), (y, η) := ξ, y − x, η.

(3.1)

The choice of this non-degenerate closed 2-form gives a standard realization of the symplectic group Sp(m, R) and the Heisenberg group H 2m+1 . Namely, Sp(m, R) := T ∈ GL(2m, R): ω(T X, T Y ) = ω(X, Y ) and H 2m+1 := g = (s, A) ∈ R × R2m


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equipped with the product

1 g · g ≡ (s, A) · s , A := s + s + ω A, A , A + A . 2 Accordingly, the Heisenberg Lie algebra h2m+1 is then defined by (s, X), (t, Y ) = ω(X, Y ), 0 . Finally we denote by Z the center {(s, 0): s ∈ R} of H 2m+1 . The Heisenberg group H 2m+1 admits a unitary representation, denoted by ϑ , on the configuration space L2 (Rm ) by the formula 1

ϑ(g)ϕ(x) = e2πi(s+x,α− 2 a,α) ϕ(x − a),

g = (s, a, α).

(3.2)

This representation, referred to as the Schrödinger representation, is irreducible and unitary [25]. The symplectic group, or more precisely its double covering, also acts on the same Hilbert space L2 (Rm ). In order to track the effect of Aut(H 2m+1 ), we recall briefly its construction. The group Sp(m, R) acts by automorphisms of H 2m+1 preserving the center Z pointwise. Composing ϑ with such automorphisms T ∈ Sp(m, R) one gets a new representation ϑ ◦ T of H 2m+1 on L2 (Rm ). Notice that these representations have the same central character, namely ϑ ◦ T (s, 0, 0) = e2πis id = ϑ(s, 0, 0). According to the Stone–von Neumann theorem (see Fact 3.3 below) the representations ϑ and ϑ ◦ T are equivalent as irreducible unitary representations of H 2m+1 . Thus, there exists a unitary operator Met(T ) acting on L2 (Rm ) in such a way that (ϑ ◦ T )(g) = Met(T )ϑ(g) Met(T )−1 ,

g ∈ H 2m+1 .

(3.3)

Because ϑ is irreducible, Met is defined up to a scalar and gives rise to a projective unitary representation of Sp(m, R). It is known that this scalar factor may be chosen in one and only one way, up to a sign, so that Met becomes a double-valued representation of Sp(m, R). The resulting unitary representation of the metaplectic group, that we keep denoting Met, is referred to as the Segal–Shale–Weil representation and it is a lowest weight module with respect to a fixed Borel subalgebra. Notice that choosing the opposite sign of the scalar factor in the definition of Met one gets a highest weight module which is isomorphic to the contragredient representation Met∨ . The unitary representation Met splits into two irreducible and inequivalent subrepresentations Met0 and Met1 according to the decomposition of the Hilbert space L2 (Rm ) = L2 (Rm )even ⊕ L2 (Rm )odd . The Weyl quantization, or the Weyl operator calculus, is a way to associate to a function S(x, ξ ) the operator Op(S) on L2 (Rm ) defined by the equation Op(S)u (x) =

Rm ×Rm

x +y , η e2πix−y,η u(y) dy dη. S 2

Such a linear operator sets up an isometry

(3.4)

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∼ Op : L2 R2m − → HS L2 Rm , L2 Rm ,

(3.5)

from the phase space L2 (Rm × Rm ) onto the Hilbert space consisting of all Hilbert–Schmidt operators on the configuration space L2 (Rm ). Introducing the symplectic Fourier transformation Fsymp by: (Fsymp S)(X) :=

S(Y )e−2iπω(X,Y ) dY,

(3.6)

Rm ×Rm

one may give another, fully equivalent, definition of the Weyl operator by means of the equation Op(S) =

(Fsymp S)(Y )ϑ(0, Y ) dY,

(3.7)

R2m

where the right-hand side is a Bochner operator-valued integral. The Heisenberg group H 2m+1 acts on R2m H 2m+1 /Z, by R2m → R2m ,

X → X + A for g = (s, A),

and consequently it acts on the phase space L2 (R2m ) by left translations. The symplectic group Sp(m, R) also acts on the same Hilbert space L2 (R2m ) by left translations. (This representation is reducible. See Section 12 for its irreducible decomposition.) In fact, both representations come from an action on L2 (R2m ) of the semidirect product group GJ := Sp(m, R) H 2m+1 which is referred to as the Jacobi group. Let us recall some classical facts in a way that we shall use them in the sequel: Fact 3.1. 1) The representations ϑ and Met form a unitary representation of the double covering Mp(m, R) H 2m+1 of GJ on the configuration space L2 (Rm ). This action induces a representation of the Jacobi group GJ on the Hilbert space of Hilbert–Schmidt operators HS(L2 (Rm ), L2 (Rm )) by conjugations. 2) The Weyl quantization map Op intertwines the action of GJ on L2 (R2m ) with the representation Met ϑ on the Hilbert space HS(L2 (Rm ), L2 (Rm )) defined in 2). Namely, ϑ(g) Op(S)ϑ g −1 = Op S ◦ g −1 , g ∈ H 2m+1 , Met(g) Op(S) Met−1 (g) = Op S ◦ g −1 , g ∈ Sp(m, R).

(3.8) (3.9)

3) Any unitary operator satisfying (3.8) and (3.9) is a scalar multiple of the Weyl quantization map Op. Proof. Most of these statements may be found in the literature (e.g. [10, Chapter 2] for the second statement), but we give a brief explanation of some of them for the convenience of the reader. Namely, the first statement follows from (3.3). Consequently, the semi-direct product


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Mp(m, R) H 2m+1 also acts by conjugations on the space HS(L2 (Rm ), L2 (Rm )), and this action is well defined for the Jacobi group GJ = Sp(m, R) H 2m+1 because the kernel of the metaplectic cover Mp(m, R) → Sp(m, R) acts trivially on HS(L2 (Rm ), L2 (Rm )). The third statement follows from the fact that L2 (R2m ) is already irreducible by the codimension one subgroup Sp(m, R) R2m of GJ . Indeed, any translation-invariant closed subspace of L2 (R2m ) is a Wiener space, i.e. the pre-image by the Fourier transform of L2 (E) for some measurable set E in R2m . On the other hand, the symplectic group acts ergodically on R2m , in the sense that the only Sp(m, R)-invariant measurable subsets of R2m are either null or conull with respect to the Lebesgue measure. Hence, the whole group Sp(m, R) R2m+1 acts irreducibly on L2 (R2m ). 2 Now we consider the ‘twist’ of the metaplectic representation by automorphisms of the Heisenberg group. The group of automorphisms of the Heisenberg group H 2m+1 , to be denoted by Aut(H 2m+1 ), is generated by – symplectic maps: (s, A) → (s, T (A)), where T ∈ Sp(m, R); – inner automorphisms (s, A) → I(t,B) (s, A) := (t, B)(s, A)(t, B)−1 = (s − ω(A, B), A), where (t, B) ∈ H 2m+1 ; – dilations (s, A) → d(r)(s, A) := (r 2 s, rA), where r > 0; – inversion: (s, A) → i(s, A) := (−s, α, a), where A = (a, α). In the sequel we shall pay a particular attention to the rescaling map τρ which is defined for every ρ = 0 by

τρ : H 2m+1 → H 2m+1 ,

(s, a, α) →

ρ ρ s, a, α . 4 4

(3.10)

Here we have adopted the parametrization of τρ in a way that it fits well into Lemma 4.2. We note that (τ−4 )2 = id and τ4 = id. The whole group Aut(H 2m+1 ) of automorphisms is generated by GJ and {τρ : ρ ∈ R× }. We denote by Aut(H 2m+1 )o the identity component of Aut(H 2m+1 ). Then we have Aut H 2m+1 = {1, τ−4 } · Aut H 2m+1 o . For any given automorphism τ ∈ Aut(H 2m+1 ), we denote by τ the induced linear operator on R2m and by π(τ ) its pull-back π(τ )f := f ◦ (τ )−1 . We notice that π(τ ) is a unitary operator on L2 (R2m ) if τ ∈ GJ . Further, we define the τ -twist Opτ of the Weyl quantization map Op by

H 2m+1 /Z

Opτ := Op ◦ π(τ ).

(3.11)

In particular, it follows from (3.4) and (3.10) that Opτρ (S)u (x) =

Rm ×Rm

S

x+y 4 , ξ e2πix−y,ξ u(y) dy dη. 2 ρ

(3.12)

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Similarly, we define the τ -twist ϑτ of the Schrödinger representation ϑ by ϑτ := ϑ ◦ τ −1 .

(3.13)

Finally, we define the τ -twist Metτ of the Segal–Shale–Weil representation Met. For this, we begin with the identity component Aut(H 2m+1 )o . We set Metτ := A−1 ◦ Met ◦ A, where ⎧ ⎨ Met(τ ), for τ ∈ Sp(m, R), A = ϑ(τ ), for τ ∈ H 2m+1 , ⎩ Id, for τ = d(r).

(3.14)

It follows from Fact 3.1 1) that Metτ is well defined for τ ∈ Aut(H 2m+1 )o . For the connected component containing τ−4 , we set Metτ := (Metτ )∨

(3.15)

for τ = τ−4 τ , τ ∈ Aut(H 2m+1 )o . Thereby, Metτ is a unitary representation of Mp(m, R) on L2 (Rm ) characterized for every T ∈ Sp(m, R) by Metτ (T )ϑτ (g) Metτ (T )−1 = ϑτ T (g) . Hence, the group Aut(H 2m+1 ) acts on L2 (R2m ) in such a way that the following proposition holds. Proposition 3.2. 1) The τ -twisted Weyl calculus is covariant with respect to the Jacobi group: ϑτ (g) Opτ (S)ϑτ g −1 = Opτ S ◦ g −1 , g ∈ H 2m+1 , −1 , g ∈ Sp(m, R). Metτ (g) Opτ (S) Met−1 τ (g) = Opτ S ◦ g

(3.16) (3.17)

2) For any τ ∈ Aut(H 2m+1 ) the representation Metτ is equivalent either to Met or to its contragredient Met∨ . The special case of the τ -twist, namely, the τ -twist associated with the rescaling map τρ (3.10) deserves our attention for at least the following two reasons. First, the parameter ρ4 has a concrete physical meaning – this is the inverse of the Planck constant h (see [10, Theorem 4.57], where a slightly different notation was used. Namely, the Schrödinger representations that we denote by ϑτρ correspond therein to ρh with h = ρ4 ). Secondly, dilations do not preserve the center Z of the Heisenberg while the symplectic automorphisms of H 2m+1 do. More precisely, the whole Jacobi group GJ fixes Z pointwise. The last observation together with the Stone– von Neumann theorem (see below) shows that the action of Aut(H 2m+1 )/GJ {τρ : ρ ∈ R× }


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( R× ) is sufficient in order to obtain all infinite dimensional irreducible unitary representations of the Heisenberg group. We set ϑρ := ϑτρ ,

(3.18)

to which we refer as the Schrödinger representations with central character ρ. Fact 3.3 (Stone–von Neumann theorem [12,25]). The representations ϑρ constitute a family of irreducible pairwise inequivalent unitary representations with real parameter ρ. Any infinite dimensional irreducible unitary representation of H 2m+1 is uniquely determined by its central character and thus equivalent to one of the ϑρ ’s. To end this section, we give yet another algebraic property of the Weyl operator calculus. GL(2n,R) We shall see in Lemma 4.5 that the irreducible decomposition of πiλ,δ , when restricted to a maximal parabolic subgroup of Sp(n, R), is based on an involution of the phase space coming from the parity preserving involution on the configuration space. Consider on L2 (Rm ) an involution defined by u(x) ˇ := u(−x) and induce through the map Opτρ : L2 (R2m ) → HS(L2 (Rm ), L2 (Rm )) two involutions on L2 (R2m ), denoted by S → †ρ S and S → S†ρ , by the following identities: †

S (u) = Opτρ (S)(u), ˇ † Opτρ S ρ (u) = Opτρ (S)(u) ˇ.

Opτρ

ρ

Then †ρ S and S†ρ are characterized by their partial Fourier transforms defined by

S(x, ξ )e−2πiξ,η dξ

(Fξ S)(x, η) :=

for S ∈ L2 R2m .

Rm

Lemma 3.4.

† ρ 2 ρ Fξ S (x, η) = (Fξ S) − η, − x , ρ 2

2 ρ Fξ S†ρ (x, η) = (Fξ S) η, x . ρ 2 Proof. By (3.12) the first equality (3.19) amounts to

†ρ Rm ×Rm

x +y 4 S , ξ e2iπx−y,ξ u(y) dy dξ 2 ρ

= Rm ×Rm

S

x +y 4 , ξ e2iπx−y,ξ u(−y) dy dξ. 2 ρ

(3.19) (3.20)

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The right-hand side equals

|ρ| 4

n

Rm ×Rm

ρ x−y S , ξ e2iπ 4 (x+y),ξ u(y) dy dξ. 2

This equality holds for all u ∈ L2 (Rm ), and therefore,

†ρ Rm

n

ρ x +y 4 |ρ| x−y 2iπx−y,ξ S dξ = S , ξ e , ξ e2iπ 4 (x+y),ξ dξ. 2 ρ 4 2 Rm

Namely,

† x +y ρ ρ x −y ρ , (y − x) = (Fξ S) , − (x + y) . Fξ S 2 4 2 4 Thus the first statement follows and the second may be proved in the same way.

2

4. Restriction of πiλ,δ to a maximal parabolic subgroup Let n = m + 1. Consider the space of homogeneous functions ∞ Vμ,δ := f ∈ C ∞ R2n \ {0} : f (r·) = (sgn r)δ |r|−n−μ f (·), r ∈ R× ,

(4.1)

for δ = 0, 1 and μ ∈ C. It may be seen as the space of even or odd smooth functions on the unit sphere S 2n−1 according to δ = 0 or 1, since homogeneous functions are determined by their restriction to S 2n−1 . Let Vμ,δ denote its completion with respect to the L2 -norm over S 2n−1 . Likewise, by restricting to the hyperplane defined by the first coordinate to be 1, we can identify the space Vμ,δ with the Hilbert space L2 (R2n−1 ) up to a scalar multiple on the inner product. GL(2n,R) induced from the charThe normalized degenerate principal series representations πμ,δ acter χμ,δ of a maximal parabolic subgroup P2n of GL(2n, R) corresponding to the partition 2n = 1 + (2n − 1) may be realized on these functional spaces. The realization of the same representation on Vμ,δ will be referred to as the K-picture, and on L2 (R2n−1 ) as the N -picture. GL(2n,R) , we shall use another model In addition to these standard models of πμ,δ 2 2 m 2 m L (R, HS(L (R ), L (R ))), which we call the operator calculus model. It gives a strong machinery for investigating the restriction to the maximal parabolic subgroup of Sp(n, R) (see (4.3) below). Let us denote by Ft (f )(ρ, X) =

f (t, X)e−2iπtρ dt

R

the partial Fourier transform of f (t, X) ∈ L2 (R1+2m ) with respect to the first variable. Applying the direct integral of the operators Opτρ and using (2.2), we obtain the unitary isomor-


1695

standard model Vμ,δ = L2 (S 2n−1 )δ

K-picture

restrict ∞ = {f ∈ C ∞ (R2n \ {0}): f (rX) = |r|−μ−n (sgn r)δ f (X), r ∈ R× } Vμ,δ restrict

L2 (H 2m+1 ) = L2 (R, L2 (R2m ))

N -picture Ft

L2 (R, L2 (R2m )) L2 (R2m+1 )

Fξ

R Opτρ dρ

Uμ,δ = L2 (R2m+1 ) (see Section 6)

L2 (R, HS(L2 (Rm ), L2 (Rm ))) (see Section 5)

non-standard model

operator calculus model

Fig. 4.1.

phisms Vμ,δ L2 R1+2m L2 R, L2 R2m L2 R, L2 R2m Ft

2 m 2 m ∼ −−−−− − → 2 . Opτρ dρ L R, HS L R , L R

(4.2)

According to situations we shall use the following geometric models for the induced representations: see Fig. 4.1. The group G1 = Sp(n, R) (= Sp(m + 1, R)) acts by linear symplectomorphisms on R2n and thus it also acts on the real projective space P2m+1 R. Fix a point in P2m+1 R and denote by P its stabilizer in G1 . This is a maximal parabolic subgroup of G1 with Langlands decomposition P = MAN R× · Sp(m, R) H 2m+1 .

(4.3)

Let g1 = n + m + a + n be the Gelfand–Naimark decomposition for the Lie algebra g1 = Lie(G1 ). We identify the standard Heisenberg Lie group H 2m+1 with the subgroup N = exp n through the following Lie groups isomorphism: ⎛

1 ⎜x ⎜ (s, x, ξ ) → ⎜ ⎝ 2s ξ

0 Im tξ 0

⎞ 0 0 0 0 ⎟ ⎟ ⎟. t 1 − x⎠ 0 Im

(4.4)

∞ → L2 (H 2m+1 ) is given by Thus, in the coordinates (t, x, ξ ) ∈ H 1+2m , the restriction map Vμ,δ

f → f (1, 2t, x, ξ ).

(4.5)

1696


The action of G1 on P2n−1 R is transitive, and all such isotropy subgroups are conjugate to each other. Therefore, we may assume that P = Sp(n, R) ∩ P2n . Then, the natural inclusion Sp(n, R) ⊂ GL(2n, R) induces the following isomorphisms ∼ Sp(n, R)/P − → GL(2n, R)/P2n P2n−1 R. Sp(n,R)

Sp(n,R)

Hence, the (normalized) induced representation πμ,δ ≡ πμ,δ := IndP χμ,δ can (cf. Section 8) also be realized on the Hilbert space Vμ,δ . Therefore, πμ,δ is equivalent to the restriction GL(2n,R) of πμ,δ with respect to Sp(n, R). Notice that πμ,δ is unitary for μ = iλ, λ ∈ R. It is noteworthy that the unipotent radical N of P is the Heisenberg group H 2n−1 which is not abelian if n 2, although the unipotent radical of P2n clearly is. Notice also that the automorphism group Aut(H 2n−1 ) contains P /{±1} as a subgroup of index 2. Denote by Mo Sp(m, R) the identity component of M O(1) × Sp(m, R). The subgroup Mo N is isomorphic to the Jacobi group GJ introduced in Section 3. We have then the following inclusive relations for subgroups of symplectomorphisms: G1

⊃ MAN ⊃ GJ = Mo N ⊃

Symplectic group

Jacobi group

N. Heisenberg group

Our strategy of analyzing the representations πiλ,δ of G1 (see Theorem 8.3) will be based on their restrictions to these subgroups (see Lemmas 4.1 and 4.5). We recall from (3.18) that ϑρ is the Schrödinger representation of the Heisenberg group H 2m+1 with central character ρ. While the abstract Plancherel formula for the group N H 2m+1 : L2 (N ) = ϑρ ⊗ ϑρ∨ dρ, R

underlines the decomposition with respect to left and right regular actions of the group N , we shall consider the decomposition of this space with respect to the restriction of the principal series representation πiλ,δ to the Jacobi group GJ = Sp(m, R) H 2m+1 (see Lemma 4.1). Let us examine how the restriction πiλ,δ |GJ defined on the Hilbert space Viλ,δ on the left-hand side of (4.2) is transferred to L2 (R, L2 (R2m )) via the partial Fourier transform Ft . The restriction πiλ,δ |N coincides with the left regular representation of N on L2 (R1+2m ) given by 1 πiλ,δ (g)f (t, X) = f t − s − ω(A, X), X − A 2

1 = f t − s + ξ, a − x, α , x − a, ξ − α , 2

(4.6)

for f (t, X) ∈ L2 (R1+2m ) and g = (s, A) ≡ (s, a, α) ∈ H2m+1 . Taking the partial Fourier transform Ft of (4.6), we get 1 Ft πiλ,δ (g)f (ρ, x, ξ ) = e−2πiρ(s− 2 (ξ,a−x,α)) (Ft f )(ρ, x − a, ξ − α).

(4.7)


1697

Now, for each ρ ∈ R, we define a representation ρ of N on L2 (R2m ) by 1

ρ (g)h(x, ξ ) := e−2πiρ(s− 2 (ξ,a−x,α)) h(x − a, ξ − α),

(4.8)

for g = (s, a, α) ∈ N and h ∈ L2 (R2m ). Then, ρ is a unitary representation of N for any ρ, and the formula (4.7) may be written as: Ft πiλ,δ (g)f (ρ, x, ξ ) = ρ (g)(Ft f )(ρ, x, ξ ),

(4.9)

for g ∈ N . Here, we let ρ (g) act on Ft f seen as a function of (x, ξ ). For each ρ ∈ R, we can extend the representation ρ of N to a unitary representation of the Jacobi group GJ by letting Mo act on L2 (R2m ) by ρ (g)h(x, ξ ) = h(y, η),

with (y, η) = g −1 (x, ξ ), g ∈ Mo Sp(m, R).

Then, clearly the identity (4.9) holds also for g ∈ Mo . Thus, we have proved the following decomposition formula: Lemma 4.1. For any (λ, δ) ∈ R × Z/2Z, the restriction of πiλ,δ to the Jacobi group is unitarily equivalent to the direct integral of unitary representations ρ via Ft (see (4.2)): ⊕ πiλ,δ |GJ Ft

ρ dρ.

(4.10)

R

Next we establish the link between the representations (ρ , L2 (R2m )) and (ϑρ , L2 (Rm )) of the Heisenberg group N H 2m+1 . For this we note that the representation ρ brings us to the changeover of one parameter families of automorphisms of H 2m+1 , from {τρ : ρ ∈ R× } to {ψρ : ρ ∈ R× } defined by

ψρ (s, a, α) :=

1 1 2 s, a, α . ρ 2 ρ

(4.11)

Then we state the following covariance relation given by Opτρ : Lemma 4.2. For every g ∈ H 2m+1 the following identity in End(L2 (Rm )) holds for any S ∈ L2 (R2m ): Opτρ ρ (g)S = Opτρ (S) ◦ ϑψρ g −1 .

(4.12)

Proof. Let g = (s, a, α) ∈ H 2m+1 and take an arbitrary function u ∈ L2 (Rm ). Using the integral formula (3.12) for Opτρ , we get

1698


Opτρ ρ (g)S u(x)

x +y 4 , ξ e2πix−y,ξ u(y) dy dξ ρ (g)S = 2 ρ Rm ×Rm

1

4

e−2πiρ(s− 2 ( ρ ξ,a−

= Rm ×Rm

=

e

−2πiB

Rm ×Rm

x+y 2 ,α))

S

x +y 4 − a, ξ − α e2πix−y,ξ u(y) dy dξ 2 ρ

x+y 4 , ξ u(y + 2a) dy dξ, S 2 ρ

where

x + y + 2a ρ ρ 4 ξ + α, a − , α − x − y − 2a, ξ + α 2 ρ 2 4 ρ = ρs + a + y, α − x − y, ξ . 2

B = ρs −

In view of the definitions (3.13) and (4.11),

−1 −1 −1 ρ ϑψρ g = ϑ ψρ g = ϑ −ρs, −2a, − α . 2 Thus, by the definition (3.2) of the Schrödinger representation ϑ , we have ρ ϑψρ s −1 u (y) = e−2πi(ρs+ 2 a+y,α) u(y + 2a). Hence, the last integral equals

S Rm ×Rm

x +y 4 , ξ ϑψρ g −1 u (y)e2πix−y,ξ dy dξ 2 ρ

= Opτρ (S)ϑψρ g −1 u (x).

2

Then, it turns out that the decomposition (4.10) is not irreducible, but the following lemma holds: Lemma 4.3. For any ρ ∈ R× , ρ is a unitary representation of the Jacobi group GJ on L2 (R2m ), which splits into a direct sum ρ0 ⊕ ρ1 of two pairwise inequivalent unitary irreducible representations. Proof. Consider the rescaling map τρ introduced by (3.10) and recall that the τρ -twisted Weyl quantization map induces a GJ equivariant isomorphism ∼ → HS L2 Rm , L2 Rm Opτρ : L2 R2m − intertwining the ρ and ϑψρ actions (4.12).

(4.13)


1699

The irreducibility of the Schrödinger representation ϑρ of the group N (Fact 3.3) implies therefore that any N -invariant closed subspace in HS(L2 (Rm ), L2 (Rm )) must be of the form HS(L2 (Rm ), U ) for some closed subspace U ⊂ L2 (Rm ). In view of the covariance relation (3.17) of the Weyl quantization, the subspace HS(L2 (Rm ), U ) is Sp(m, R)-invariant if and only if U itself is Mp(m, R)-invariant (see Proposition 3.2), and the latter happens only if U is one of {0}, L2 (Rm )even , L2 (Rm )odd or L2 (Rm ). Thus, we have the following irreducible decomposition of ρ , seen as a representation of GJ on L2 (R2m ): L2 R2m = W+ ⊕ W− 2 m 2 m 2 m 2 m − −∼−→ Opτρ HS L R , L R even ⊕ HS L R , L R odd .

(4.14)

From Proposition 3.2 2) we deduce that the corresponding representations, to be denoted by ρδ , of GJ , where δ labels the parity, are pairwise inequivalent, i.e. ρδ = ρδ if and only if ρ = ρ and δ = δ for all ρ, ρ ∈ R and δ, δ ∈ Z/2Z. 2 The following lemma is straightforward from the definition of the involution S → S†ρ (see (3.19)). Lemma 4.4. The subspaces W+ and W− introduced above are the +1 and −1 eigenspaces of the involution S → S†ρ , respectively. Eventually, we take the A-action into account, and give the branching law of the (degenerate) principal series representation πiλ,δ of G1 when restricted to the maximal parabolic subgroup MAN . Lemma 4.5 (Branching law for G1 ↓ MAN ). For every (λ, δ) ∈ R × Z/2Z the space Viλ,δ acted upon by the representation πiλ,δ |MAN splits into the direct sum of four irreducible representations: 2 2 2 2 Viλ,δ H+ (W+ ) ⊕ H+ (W− ) ⊕ H− (W+ ) ⊕ H− (W− ).

(4.15)

Proof. We shall prove first that each summand in (4.15) is already irreducible as a representation of Mo AN GJ A. Then we see that it is stable by the group MAN and thus irreducible because M is generated by Mo and −I2n , which acts on Viλ,δ by the scalar (−1)δ . In light of the GJ -irreducible decomposition (4.10), any GJ -invariant closed subspace U of Viλ,δ must be of the form U = Ft−1 L2 (E+ , W+ ) ⊕ Ft−1 L2 (E− , W− ) , for some measurable sets E± in R. Suppose furthermore that U is A-invariant. Notice that the group A acts on Viλ,δ L2 (R2m+1 ) by πiλ,δ (a)f (t, X) = a −1−m−iλ f a −2 t, a −1 X .

1700


In turn, their partial Fourier transforms with respect to the t ∈ R variable are given by Ft πiλ,δ (a)f (ρ, X) = a 1−m−iλ (Ft f ) a 2 ρ, a −1 X . Therefore, Ft f is supported in E± if and only if Ft πiλ,δ (a)f is supported in a −2 E± as a W± valued function on R. In particular, U is an A-invariant subspace if and only if E± is an invariant measurable set under the dilation ρ → a 2 ρ (a > 0), namely, E± = {0}, R− , R+ , or R (up to measure zero sets). Since Mo AN GJ A, Mo AN -invariant proper closed subspaces must be of the form −1 2 Ft (L (R± , Wε )) with ε = + or −. We recall from Lemma 2.2 that the Hilbert space L2 (R, Wε ) is a sum of Wε -valued Hardy spaces: ∼

2 2 2 → 2 (Wε ) ⊕ H− (Wε ) − L2 (R, Wε ) = H+ Ft L (R+ , Wε ) ⊕ L (R− , Wε ).

Now Lemma 4.5 has been proved.

(4.16)

2

Lemma 4.5 implies that the representation πiλ,δ of G1 has at most four irreducible subrepresentations. The precise statement for this will be given in Theorem 8.3. 5. Restriction of πiλ,δ to a maximal compact subgroup As the operator calculus model L2 (R, HS(L2 (Rm ), L2 (Rm ))) was appropriate for studying the P -structure of πiλ,δ , we use complex spherical harmonics for the analysis of the K-structure of these representations. We retain the convention n = m + 1. Identifying the symplectic form ω on R2n with the imaginary part of the Hermitian inner product on Cn we realize the group of unitary transformations K = U (n) as a subgroup of G1 = Sp(n, R). Then the group K is a maximal compact subgroup of G1 . Analogously to the classical spherical harmonics on Rn , consider harmonic polynomials on Cn as follows. For α, β ∈ N, let Hα,β (Cn ) denote the vector space of polynomials p(z0 , . . . , zm , z¯ 0 , . . . , z¯ m ) on Cn which (1) are homogeneous of degree α in (z0 , . . . , zm ) and of degree β in (¯z0 , . . . , z¯ m );

∂2 (2) belong to the kernel of the differential operator m i=0 ∂zi ∂ z¯ i . Then, Hα,β (Cn ) is a finite dimensional vector space. It is non-zero except for the case where n = 1 and α, β 1. The natural action of K on polynomials, p(z0 , . . . , zm , z¯ 0 , . . . , z¯ m ) → p g −1 (z0 , . . . , zm ), g −1 (z0 , . . . , zm ) (g ∈ K), leaves Hα,β (Cn ) invariant. The resulting representations of K on Hα,β (Cn ), which we denote by the same symbol Hα,β (Cn ), are irreducible and pairwise inequivalent for any such α, β. 2 The restriction of Hα,β (Cn ) to the unit sphere S 2m+1 = {(z0 , . . . , zm ) ∈ Cn : m j =0 |zj | = 1} is injective and gives a complete orthogonal basis of L2 (S 2m+1 ), and we have a discrete sum decomposition


⊕ α,β n C S 2m+1 L2 S 2m+1 H

(m 1).

1701

(5.1)

α,β∈N

The case m = 0 collapses to ⊕ ⊕ α,0 1 C S 1 ⊕ H H0,β C1 S 1 . L2 S 1 α∈N

β∈N+

Fixing a μ ∈ C we may extend functions on S 2m+1 to homogeneous functions of degree −(m + 1 + μ). The decomposition (5.1) gives rise to the branching law (K-type formula) with respect to the maximal compact subgroup. Lemma 5.1 (Branching law for G1 ↓ K). The restriction of πμ,δ to the subgroup K of G1 is decomposed into a discrete direct sum of pairwise inequivalent representations: πμ,δ |K

⊕

Hα,β (Cn )

(m 1),

α,β∈N α+β≡δ mod 2

πμ,δ |K

⊕

Hα,0 (C) ⊕

α∈N α≡δ mod 2

⊕

H0,β (C)

(m = 0).

β∈N+ β≡δ mod 2

We shall refer to Hα,β (Cn ) as a K-type of the representation πμ,δ . The restriction G1 ↓ K is multiplicity free. Therefore any K-intertwining operator (in particular, any G1 -intertwining operator) acts as a scalar on every K-type by Schur’s lemma. We give an explicit formula of this scalar for the Knapp–Stein intertwining operator: Tμ,δ : V−μ,δ → Vμ,δ , which is defined as the meromorphic continuation of the following integral operator

−μ−n δ sgn ω(ξ, η) dσ (ξ ). f (ξ )ω(ξ, η)

(Tμ,δ f )(η) := S 2n−1

Here dσ is the Euclidean measure on the unit sphere. Further, we normalize it by Tμ,δ :=

1 Tμ,δ , C2n (μ, δ)

(5.2)

where

C2n (μ, δ) := 2π

μ+n− 12

×

⎧ 1−μ−n Γ( 2 ) ⎪ ⎪ ⎨ μ+n Γ(

2

(δ = 0),

)

⎪ Γ ( 2−μ−n ) ⎪ 2 ⎩ −i μ+n+1 Γ(

2

)

(δ = 1).

1702


Proposition 5.2. For α, β ∈ N, we set δ ≡ α + β mod 2. The normalized Knapp–Stein intertwining operator Tμ,δ acts on Hα,β (Cn ) as the following scalar (−1)β π −μ

Γ ( α+β+μ+n ) 2 Γ ( α+β−μ+n ) 2

.

Proof. See [5, Theorem 2.1] for δ = 0. The proof for δ = 1 works as well by using Lemma 5.4. 2 Remark 5.3. Without normalization, the Knapp–Stein intertwining operator Tμ,δ acts on Hα,β (Cn ) as Tμ,δ |Hα,β (Cn ) = (−1)β Aα+β (μ) id, where δ ≡ α + β mod 2 and

1

Ak (μ) := 2π n− 2

Γ ( k+μ+n ) 2 Γ ( k−μ+n ) 2

×

⎧ 1−μ−n Γ( 2 ) ⎪ ⎪ ⎨ μ+n Γ(

(k ∈ 2N),

)

2

⎪ Γ ( 2−μ−n ) ⎪ 2 ⎩ −i μ+n+1 Γ(

2

)

(k ∈ 2N + 1).

The symplectic Fourier transform Fsymp , defined by (3.6), may be written as: (Fsymp f )(Y ) =

f (X)e−2πiω(X,Y ) dX = (FR2n f )(J Y ),

R2n

where J : R2n → R2n is given by J (x, ξ ) := (−ξ, x). ∞ of homoFor generic complex parameter μ (e.g. μ = n, n + 2, . . . for δ = 0), the space Vμ,δ 2n 2n geneous functions on R \ {0} may be regarded as a subspace of the space S (R ) of tempered distributions, and we have the following commutative diagram: ∼

Fsymp : S (R2n ) ∪ V−μ,δ

S (R2n )

∼

∪ Vμ,δ

Lemma 5.4. As operators that depend meromorphically on μ, Tμ,δ satisfy the following identity: Tμ,δ = Fsymp |V−μ,δ . Proof. The proof parallels that of [5, Proposition 2.3]. For h ∈ C ∞ (S 2n−1 )δ , we define a homo∞ by geneous function hμ−n ∈ V−μ,δ hμ−n (rξ ) := r μ−n h(ξ )

r > 0, ξ ∈ S 2n−1 .


1703

Then we recall from [5, Proposition 2.2] the following formula: πi

Γ (μ + n)e− 2 (μ+n) FR2n hμ−n (sη) = (2π)μ+n s μ+n

−μ−n ξ, η − i0 h(ξ ) dσ (ξ ),

S 2n−1

where (ξ, η − i0)λ is a distribution of ξ, η, obtained by the substitution of t = ξ, η into the distribution (t − i0)λ of one variable t. To conclude, we use

π π(μ + n) −μ−n π(μ + n) −μ−n |t| |t| (t − i0)−μ−n = e 2 i(μ+n) cos − i sin sgn t 2 2

π |t|−μ−n |t|−μ−n sgn t . 2 = πe 2 i(μ+n) − i 1−μ−n μ+n 2−μ−n Γ ( 1+μ+n )Γ ( ) Γ ( )Γ ( ) 2 2 2 2 We note that the Knapp–Stein intertwining operator induces a unitary equivalence of representations πiλ,δ and π−iλ,δ of G1 = Sp(n, R): πiλ,δ π−iλ,δ ,

for any λ ∈ R and δ ∈ Z/2Z.

(5.3)

6. Algebraic Knapp–Stein intertwining operator We introduce yet another model Uμ,δ L2 (R2m+1 ), referred to as the non-standard model, of the representation πμ,δ as the image of the partial Fourier transform ∼ 2 1+m+m , →L R Fξ : L2 R1+m+m − where ξ denotes the last variable in Rm . Then the space Uμ,δ inherits a G1 -module structure from (πμ,δ , Vμ,δ ) through Fξ ◦ Ft (see Fig. 4.1). The advantage of this model is that the Knapp–Stein intertwining operator becomes an algebraic operator (see Theorem 6.1 below). The price to pay is that the Lie algebra k acts on Uμ,δ by second-order differential operators. We can still give an explicit form of minimal K-types on the model Uμ,δ when it splits into two irreducible components (μ = 0, δ = 0, 1) by means of K-Bessel functions (Section 7). We define an endomorphism of L2 (R2m+1 ) by −μ

ρ 2 ρ δ (Tμ,δ H )(ρ, x, η) := (sgn ρ) H ρ, η, x . 2 ρ 2 Regarding Tμ,δ as an operator on the N -picture, we have

(6.1)

1704


Theorem 6.1 (Algebraic Knapp–Stein intertwining operator). For any μ ∈ C and δ ∈ Z/2Z, the following diagram commutes:

V−μ,δ

Tμ,δ

Vμ,δ

Fξ Ft

U−μ,δ

Fξ Ft Tμ,δ

Uμ,δ

To prove Theorem 6.1, we work on the ambient space R2n (= R2m+2 ). Let FRn denote the partial Fourier transform of the last n coordinates in R2n . Lemma 6.2. 1) For f ∈ V−μ,δ , the function FRn f satisfies (FRn f ) rx, r −1 η = |r|μ (sgn r)δ (FRn f )(x, η),

r ∈ R× , x, η ∈ Rn .

2) For f ∈ S (R2n ), x, η ∈ Rn , we have FRn ◦ Fsymp ◦ FR−1 n f (x, ξ ) = f (ξ, x). Proof. 1) This is a straightforward computation. 2) For f (x, ξ ) ∈ S(R2n ), FRn ◦ Fsymp ◦ FR−1 y, η n f = f x, ξ e2πiξ,ξ e−2πi(ξ,y−x,η) e−2πiη,η dξ dx dξ dη Rn R2n Rn

=

f x, ξ e2πiξ −y,ξ e−2πiη −x,η dξ dx dξ dη

Rn ×R2n ×Rn

=

f x, ξ δ ξ − y δ η − x dx dξ

Rn

= f η , y .

2

From now x, ξ, η will stand again for elements of Rm , where m = n − 1. Proof of Theorem 6.1. According to the choice of the isomorphism (4.4) between the Lie group N and the standard Heisenberg Lie group, for f ∈ V−μ,δ , we set F (t, x, ξ ) := f (1, x, 2t, ξ ), H (ρ, x, η) := (Ft Fξ F )(ρ, x, η),


1705

where t, ρ ∈ R and x, ξ ∈ Rm . Then H (ρ, x, η) = 12 (FRn f )(1, x, ρ2 , η). Thus, according to Lemma 6.2,

1 ρ Ft Fξ (Fsymp f )(ρ, x, η) = FRn (Fsymp f ) 1, x, , η 2 2

1 ρ = (FRn f ) , η, 1, x 2 2

2 ρ ρ 1 ρ −μ δ n = (sgn ρ) (FR f ) 1, η, , x 2 2 ρ 2 2 −μ

ρ 2 ρ δ = (sgn ρ) H ρ, η, x . 2 ρ 2 Now Theorem follows from Lemma 5.4.

2

7. Minimal K-type in a non-standard model We give an explicit formula for two particular K-finite vectors of π0,δ (in fact, minimal K± types of irreducible components π0,δ of π0,δ ; see Theorem 8.3 1)) in the non-standard L2 -model U0,δ ( L2 (R2m+1 )). The main results (see Proposition 7.1) show that minimal K-types are represented in terms of K-Bessel functions in this model. Although we do not use these results in the proof of Theorem 8.3, we think they are interesting of their own from the view point of geometric analysis of small representations. It is noteworthy that similar feature to Proposition 7.1 has been observed in the L2 -model of minimal representations of some other reductive groups (see e.g. [22]). We begin with the identification ∼ → H0,0 Cm+1 , C−

1 → 1

(constant function),

and extend it to a homogeneous function on R2n belonging to V0,0 (see (4.1)). Using the formula (4.5) in the N -picture, we set − m+1 2 . h+ (t, x, ξ ) := 1 + 4t 2 + |x|2 + |ξ |2 Notice that h+ (t, x, ξ ) ∈ V0,0 ∩ H0,0 (Cm+1 ) in the K-type formula of π0,0 (see Lemma 5.1). Let

2 1 2 1 2 2 ρ 2 ψ(ρ, x, η) := 1 + |x| + |η| . 4 Likewise we identify ∼ Cm+1 − → H0,1 Cm+1 ,

b →

m j =0

bj z j ,

(7.1)

1706


and set 2 h− b (t, x, ξ ) := 1 + 4t

m+2 2 − 2

+ |x| + |ξ | 2

ϕb (ρ, x, η) := ω

1

(1 + |x|2 ) 2 2

1

( ρ4 + |η|2 ) 2

b0 (1 − 2it) +

m

bj (xj − iξj ) ,

j =1

, b0

1 ρ 2

+

m

bj

j =1

xj ηj

,

(7.2)

(7.3)

where ω denotes the standard symplectic form on C2 defined as in (3.1). Then h− b ∈ V0,1 ∩ H0,1 (Cm+1 ) in the K-type formula of π0,1 (see Lemma 5.1). Let Kν (z) denote the modified Bessel function of the second kind (K-Bessel function for m+1 ) in the standard model (N -picture) are short). Then the K-finite vectors h+ and h− b (b ∈ C of the following form in the non-standard model U0,δ . Proposition 7.1. m+2

1) (Ft Fξ h+ )(ρ, x, η) =

π 2 K0 (2πψ(ρ, x, η)). Γ ( m+1 2 )

2) (Ft Fξ h− b )(ρ, x, η) =

ϕb (ρ,x,η) π 2 ψ(ρ,x,η) 2Γ ( m+2 2 )

m+2

exp(−2πψ(ρ, x, η)).

The rest of this section is devoted to the proof of Proposition 7.1. In order to get simpler ν (z) := ( z )−ν Kν (z) [19, Section 7.2]. formulas we also use the following normalization K 2 Lemma 7.2. For every μ ∈ R let us define the following function on R × Rm : 2 −μ −2iπξ,η Iμ ≡ Iμ (a, η) := a + |ξ |2 e dξ. Rm

Then, m

2π 2 m−2μ a Iμ (a, η) = K m2 −μ 2πa|η| . Γ (μ) Proof. Recall the classical Bochner formula m e−2iπsξ,ξ dσ (ξ ) = 2πs 1− 2 J m2 −1 (2πs),

for ξ ∈ S m−1 ,

S m−1

where Jν (z) denotes the Bessel function of the first kind. Then, ∞

2 −μ −2iπr|η|ξ, η m−1 |η| r a + r2 e dr dσ (ξ )

Iμ (a, η) = 0 S m−1

1− m 2

∞

= 2π|η|

0

−μ m r 2 J m2 −1 2πr|η| r 2 + a 2 dr.

(7.4)


1707

According to [7, 8.5(20)] we have ∞ 0

1

−μ−1 1 1 a ν−μ y μ+ 2 Kν−μ (ay) , x ν+ 2 x 2 + a 2 Jν (xy)(xy) 2 dx = 2μ Γ (μ + 1)

for Re a > 0 and −1 < Re ν < 2 Re μ + 32 , which implies

m 2π μ a 2 −μ K m2 −μ 2πa|η| Iμ (a, η) = Γ (μ) |η| m

2π 2 m−2μ a = K m2 −μ 2πa|η| . Γ (μ)

2

In particular, we have I m+1 (a, η) = 2

I m+2 (a, η) = 2

1 (z) = Here we used K − 2

π

m+2 2

Γ ( m+1 2 ) 2π

Γ ( m+2 2 )

√

− and h− (1) for h(1,0,...,0) and

m+2 2

exp(−2πa|η|) , a |η| K1 2πa|η| . a

π −z in the first identity. 2 e − h(0,1,0,...,0) , respectively.

By a little abuse of notation, we write h− (0)

Lemma 7.3. For (t, x) ∈ R × Rm , we set a ≡ a(t, x) :=

1 + 4t 2 + |x|2 .

Then, Fξ h+ (t, x, η) = I m+1 a(t, x), η , 2

1 ∂ I m+2 a(t, x), η . (t, x, η) = x + Fξ h− 1 (1) 2 2π ∂η1 Proof. By definition Fξ h+ (t, x, η) = Fξ h− (1) (t, x, η) =

− m+1 −2πiξ,η 2 e 1 + 4t 2 + |x|2 + |ξ |2 dξ,

Rm

− m+2 2 (x − iξ )e −2πiξ,η dξ 1 + 4t 2 + |x|2 + |ξ |2 1 1

Rm

1 ∂ = x1 + I m+2 1 + 4t 2 + |x|2 , η . 2 2π ∂η1 Hence Lemma 7.3 is proved.

2

1708


Proof of Proposition 7.1. We recall from [7, vol. I, 1.4(27); 1.13(45); 2.13(43)] the following formulas: For Re d > 0, Re c > 0 and s > 0, ∞

1

exp(−d(t 2 + c2 ) 2 ) (t 2 + c2 )

0

∞

(t 2 + c2 ) 2

1 cos(st) dt = K0 c s 2 + d 2 2 , !

1

Kν (d(t 2 + c2 ) 2 ) ν

0

1 2

cos(st) dt =

1

2 2 π Kν− 12 (c(s + d ) 2 ) 2 d ν cν− 12 (s 2 + d 2 ) 14 − 12 ν

2 1 √ 2 2 , = 2ν−1 π d −ν c1−2ν K −ν+ 1 c s + d 2

∞

1

tK1 (d(t 2 + c2 ) 2 ) 1

(t 2 + c2 ) 2

0

(7.5)

(7.6)

1

πs exp(−c(s 2 + d 2 ) 2 ) sin(st) dt = . 1 2d (s 2 + d 2 ) 2

(7.7)

1

We apply the formulas (7.5) and (7.6) with d = 4π|η|, c = 12 (1 + |x|2 ) 2 and s = 2πρ. In view 1 2

1 2

that a ≡ a(t, x) = 2(t 2 + c2 ) and 2πψ(ρ, x, η) = c(s 2 + d 2 ) , we get ∞ −∞

∞ −∞

exp(−2πa|η|) −2πitρ e dt = K0 2πψ(ρ, x, η) , a

K1 (2πa|η|) −2πitρ 1 e dt = exp −2πψ(ρ, x, η) . 1 a 4|η|(1 + |x|2 ) 2 √

1 (z) = π e−z for the second equation. Thus the first statement has Here, we have used again K −2 2 been proved. To see the second statement, it is sufficient to treat the following two cases: b = (1, 0, . . . , 0) and b = (0, 1, 0, . . . , 0). We use Ft Fξ h− (1)

1 ∂ I m+2 a(t, x), η = F t x1 + 2 2π ∂η1

1 ∂ Ft I m+2 a(t, x), η . = x1 + 2 2π ∂η1

Now use

1 ∂ exp(−2πψ(ρ, x, η)) ϕ(1) (ρ, x, η) exp(−2πψ(ρ, x, η)) x1 + . = 1 2π ∂η1 ψ(ρ, x, η) (1 + |x|2 ) 2

The case b = (1, 0, . . . , 0) goes similarly by using the formula (7.7).

2


1709

R) ↓ Sp(n,R R) 8. Branching law for GL(2n,R From now we give a proof of Theorem 1.1 with emphasis on geometric analysis involved. Our strategy is the following. Suppose P is a closed subgroup of a Lie group G, χ : P → C× a unitary character, and L := G ×P χ a G-equivariant line bundle over G/P . We write L2 (G/P , L) for the Hilbert space consisting of L2 -sections for the line bundle L ⊗ 1 (Λtop T ∗ (G/P )) 2 . Then the group G acts on L2 (G/P , L) as a unitary representation, to be denoted by πχG , by translations. If (G, H ) is a reductive symmetric pair and P is a parabolic subgroup"of G, then there exist finitely many open H -orbits O(j ) on the real flag variety G/P such that j O(j ) is open dense in G/P . (In our cases below, the number of open H -orbits is at most two.) Applying the Mackey theory, we see that the restriction of the unitary representation πχG to the subgroup H is unitarily equivalent to a finite direct sum: # L2 O(j ) , L|O(j ) . πχG H j

Thus the branching problem is reduced to the irreducible decomposition of L2 (O(j ) , L|O(j ) ), equivalently, the Plancherel formula for the homogeneous line bundle L|O(j ) over open H orbits O(j ) . In our specific setting, where G = GL(N, R) and P = PN (see (1.2)), the base space G/P is the real projective space PN −1 R. For (λ, δ) ∈ R × Z/2Z, we define a unitary character χiλ,δ of PN by

χiλ,δ

a

tb

0

C

:= |a|λ (sgn a)δ ,

a ∈ GL(1, R), C ∈ GL(N − 1, R), b ∈ RN −1 ,

G in previous notation. In this and in the matrix realization of PN . Then πχGiλ,δ coincides with πiλ,δ the next three sections, we find the explicit irreducible decomposition of L2 (O(j ) , L|O(j ) ) with G . respect to πiλ,δ We begin with the case H = G1 , i.e.

(G, H ) ≡ GL(2n, R), Sp(n, R) . As we have already seen in Section 4 the group G1 acts transitively on G/PN , and we have the following unitary equivalence of unitary representations of G1 = Sp(n, R): G1 G πiλ,δ πiλ,δ . G 1

Sp(n,R)

Here πiλ,δ is a unitary representation of Sp(n, R) induced from the maximal parabolic subgroup P = G1 ∩ PN (GL(1, R) × Sp(n − 1, R)) H 2n−1 . Thus the following two statements are equivalent. GL(2n,R)

from GL(2n, R) to Sp(n, R) stays irreducible for any Theorem 8.1. The restriction of πiλ,δ λ ∈ R× and δ ∈ {0, 1}. It splits into two irreducible components for λ = 0, δ = 0, 1 and n 2.

1710


Theorem 8.2. Let P be a maximal parabolic subgroup of G1 whose Levi part is isomorphic to GL(1, R) × Sp(n − 1, R), and denote by πiλ,δ (λ ∈ R, δ = 0, 1) the corresponding unitary (degenerate) principal series representation of G1 . Then for n 2, πiλ,δ is irreducible for any (λ, δ) ∈ R× × Z/2Z, and splits into a direct sum of two irreducible components for λ = 0, δ = 0, 1. Theorem 8.2 itself was proved in [23, Theorem 7.3]. The case of δ = 0 was studied by different methods earlier in [9] and also very recently in [2] (λ = 0 and δ = 0) in the context of special unipotent representations of the split group Sp(n, R). We give yet another proof of Theorem 8.2 in the most interesting case, i.e. in the case λ = 0 and δ = 0, 1 below. Theorem 8.3 describes a finer structure of the irreducible summands. The novelty here (even for the δ = 0 case) is that we characterize explicitly the two irreducible summands by their Kmodule structure, and also by their P -module structure. The former is given in terms of complex spherical harmonics (cf. Lemma 5.1) and the latter in terms of Hardy spaces (cf. Lemma 4.5), as follows: Theorem 8.3. Let n 2 and δ ∈ Z/2Z. The unitary representation π0,δ of G1 = Sp(n, R) splits into the direct sum of two irreducible representations of G1 : + − π0,δ = π0,δ ⊕ π0,δ .

(8.1)

1) (Characterization by K-type.) Each irreducible summand in (8.1) has the following K-type formula: + π0,δ

⊕

Hα,β Cn ,

β∈2N α≡β+δ mod 2 − π0,δ

⊕

Hα,β Cn ,

β∈2N+1 α≡β+δ mod 2

where ⊕ denotes the Hilbert completion of the algebraic direct sum. ± 2) (Characterization by Hardy spaces.) The irreducible summands π0,δ consist of two Hardy spaces via the isomorphism (4.15): + 2 2 π0,0 H+ (W+ ) ⊕ H− (W+ ),

− 2 2 π0,0 H+ (W− ) ⊕ H− (W− ),

+ 2 2 H+ (W+ ) ⊕ H− (W− ), π0,1

− 2 2 π0,1 H+ (W− ) ⊕ H− (W+ ).

2 (W ) are the W -valued Hardy Here, W± are the subspaces of L2 (R2m ) defined in (4.14), and H± ε ε spaces. 3) (Characterization by the Knapp–Stein intertwining operator.) The irreducible sum± mands π0,δ are the ±1 eigenspaces of the normalized Knapp–Stein intertwining operator T0,δ (see (5.2)).


1711

Proof. 1) and 3). The normalized Knapp–Stein intertwining operator T0,δ has eigenvalues either 1 or −1 according to the parity of the K-type Hα,β (Cn ), namely β ≡ 0 or β ≡ 1 mod 2 by Proposition 5.2. Hence the statements 1) and 3) are proved. 2) In the model U0,δ L2 (R2m+1 ) (see Section 6), the Knapp–Stein intertwining operator T0,δ is equivalent to the algebraic operator

2 ρ T0,δ : H (ρ, x, η) → (sgn ρ)δ H ρ, η, x , ρ 2 by Theorem 6.1. In turn, it follows from Lemma 3.4 that T0,δ is transfered to the operator S(ρ, ∗) → (sgn ρ)δ S†ρ (ρ, ∗)

(8.2)

in the operator calculus model L2 (R, HS(L2 (Rm ), L2 (Rm ))) (see Fig. 4.1). In view of the ±1 eigenspaces of the transform (8.2), we see that the statement 2) follows from the characterization ∼ 2 (W ) − 2 of W± (see Lemma 4.4) and the isomorphism Ft : H± ε → L (R± , Wε ) given in Lemma 2.2. ± Finally, we need to prove that the summands π0,δ are irreducible G1 -modules. This is de± by means of Hardy spaces in 2) and from the following duced from the decomposition of π0,δ lemma. 2 2 (W ) (ε = ±) is G -stable with Lemma 8.4. For any δ ∈ Z/2Z, none of the Hardy spaces H± ε 1 respect to π0,δ .

Proof. For Z := (z1 , . . . , zm ) = x + iξ ∈ Cm R2m (see (4.5)), we set − m+1 2 , f0,0 (t, x, ξ ) := 1 + 4t 2 + |x|2 + |ξ |2 − m+2 2 (x − iξ ), f0,1 (t, x, ξ ) := 1 + 4t 2 + |x|2 + |ξ |2 1 1 m+2 − 2 (x1 + iξ1 ), f1,0 (t, x, ξ ) := 1 + 4t 2 + |x|2 + |ξ |2 − m+3 2 1 + 4t 2 − x12 − ξ12 . f1,1 (t, x, ξ ) := 1 + 4t 2 + |x|2 + |ξ |2 − We note that f0,0 = h+ and f0,1 = h− (0,1,0,...,0) = h(1) in the notation of Section 7. Then we have fα,β ∈ Hα,β (Cn ) for any α, β ∈ {0, 1}. In view of Theorem 8.3 1), we get

+ , f0,0 (t, x, ξ ) ∈ H0,0 Cn ⊂ V0,0 − f0,1 (t, x, ξ ) ∈ H0,1 Cn ⊂ V0,1 , + f1,0 (t, x, ξ ) ∈ H1,0 Cn ⊂ V0,1 , − f1,1 (t, x, ξ ) ∈ H1,1 Cn ⊂ V0,0 , ± ± stands for the representation space in the N -picture corresponding to π0,δ in Thewhere V0,δ 2 orem 8.3. Suppose now that one of the Hardy spaces H± (Wε ) were G1 -stable with respect

1712


to π0,δ . Then its orthogonal complementary subspace for the decomposition in Theorem 8.3 2) 2 (W ) would be also G1 -stable. Since K-type is multiplicity-free in π0,δ by Lemma 5.1, either H± ε or its complementary subspace should contain the K-type Hα,β (Cn ) for some α, β = 0 or 1. But this never happens because fα,β (t, x, ξ ) = fα,β (−t, x, ξ ) and thus supp Ft fα,β R± (see Lemma 2.2 4)). Thus lemma is proved. 2 Remark 8.5. The case n = 1 is well known. Here the group Sp(1, R) is isomorphic to SL(2, R), and πiλ,δ are irreducible except for (λ, δ) = (0, 1), while π0,1 splits into the direct sum of two irreducible unitary representations: Sp(1,R)

π0,1

2 2 H+ (C) ⊕ H− (C)

⊕ ⊕ Hα,0 (C) ⊕ H0,β (C) . α∈2N+1

β∈2N+1

The spaces Hα,0 (C) and H0,β (C) are one-dimensional, and − α+1 2 ∈ Hα,0 (C) ∩ V (t + i)α t 2 + 1 0,1 , β+1 − 2 (t − i)β t 2 + 1 ∈ H0,β (C) ∩ V0,1 . The former function extends holomorphically to the upper half plane Π+ , and the latter one extends holomorphically to Π− if α, β ≡ 1 mod 2, namely, if δ ≡ 1. As formulated in Theorem 8.2, our result may be compared with general theory on (degenerate) principal series representations of real reductive groups. For instance, according to Harish-Chandra and Vogan and Wallach [27], such representations are at most a finite sum of irreducible representations and are ‘generically’ irreducible. A theorem of Kostant [24] asserts that spherical unitary principal series representations (induced from minimal parabolic subgroups) are irreducible. There has been also extensive research on the structure of (degenerate) principal series representations in specific cases, in particular, in the case where the unipotent radical of P is abelian by A.U. Klimyk, B. Gruber, R. Howe, E.-T. Tan, S.-T. Lee, S. Sahi and others by algebraic and combinatorial methods (see e.g. [13] and references therein). We have not adopted here the aforementioned methods, but have used the idea of branching laws to non-compact subgroups (see [16]) primarily because of the belief that the latter approach to very small representations will open new aspects of the theory of geometric analysis. R) ↓ GL(n,C C) 9. Branching law for GL(2n,R C Let PnC = LC n Nn be the standard maximal parabolic subgroup of GL(n, C) corresponding to C the partition n = 1 + (n − 1), namely, the Levi subgroup LC n of Pn is isomorphic to GL(1, C) × GL(n − 1, C) and the unipotent radical NnC is the complex abelian group Cn−1 . Inducing from 1 a unitary character (ν, m) ∈ R × Z of the first factor of LC n , GL(1, C) R+ × S we define a GL(n,C) degenerate principal series representation πiν,m of GL(n, C). They are pairwise inequivalent, irreducible unitary representations of GL(n, C) (see [13, Corollary 2.4.3]).


1713

We identify Cn with R2n , and regard G2 := GL(n, C) as a subgroup of G = GL(2n, R). Theorem 9.1 (Branching law GL(2n, R) ↓ GL(n, C)). GL(2n,R)

πiλ,δ

GL(n,C)

⊕

GL(n,C)

πiλ,m

.

(9.1)

m∈2Z+δ

Proof. The group G2 = GL(n, C) acts transitively on the real projective space P2n−1 R, and the unique (open) orbit O2 := P2n−1 R is represented as a homogeneous space G2 /H2 where the isotropy group H2 is of the form H2 O(1) × GL(n − 1, C) NnC . Since PnC /H2 S 1 /{±1}, we have a G2 -equivariant fibration: S 1 /{±1} → P2n−1 R → GL(n, C)/PnC . Further, if we denote by Cδ the one-dimensional representation of H2 obtained as the following compositions: δ H2 → H2 /GL(n − 1, C)NnC − → C×,

then the G-equivariant line bundle Liλ,δ = G ×P Ciλ,δ is represented as a G2 -equivariant line bundle simply by Lδ := Liλ,δ |O2 GL(n, C) ×H2 Cδ . Therefore, we have an isomorphism as unitary representations of G2 : GL(2n,R)

Hiλ,δ

G2

L2 (O2 , Lδ ).

Taking the Fourier series expansion of L2 (O2 , Lδ ) along the fiber S 1 /{±1}, we get the irreducible decomposition (9.1). 2 An interesting feature of Theorem 9.1 is that the degenerate principal series representation GL(2n,R) is discretely decomposable with respect to the restriction GL(2n, R) ↓ GL(n, C). We πiλ,δ have seen this by finding explicit branching law, however, discrete decomposability of the reGL(2n,R) striction πiλ,δ |GL(n,C) can be explained also by the general theory [15] as follows: Let t be a Cartan subalgebra of o(2n), and we take a standard basis {f1 , . . . , fn } in it∗ such that the dominant Weyl chamber for the disconnected group K = O(2n) is given as it∗+ = (λ1 , . . . , λn ): λ1 λ2 · · · λn 0 .

1714


For K2 := G2 ∩ K U (n) the Hamiltonian action of K on the cotangent bundle T ∗ (K/K2 ) has the momentum map T ∗ (K/K2 ) → ik∗ . The intersection of its image with the dominant Weyl chamber it∗+ is given by it∗+ ∩ Ad∨ (K) ik⊥ 2 % &' $ n ∗ . = (λ1 , . . . , λn ) ∈ it+ : λ2i−1 = λi for 1 i 2 On the other hand, it follows from Lemma 5.1 that the asymptotic K-support of πiλ,δ amounts to ASK (πiλ,δ ) = R+ (1, 0, . . . , 0). Hence, the triple (G, G2 , πiλ,δ ) satisfies ASK (πiλ,δ ) ∩ Ad∨ (K) ik⊥ 2 = {0}.

(9.2)

This is nothing but the criterion for discrete decomposability of the restriction of the unitary representation πiλ,δ |G2 [15, Theorem 2.9]. GL(2n,R) For G1 = Sp(n, R), we saw in Theorem 8.1 that the restriction πiλ,δ |G1 stays irreducible. Thus, this is another (obvious) example of discretely decomposable branching law. We can see this fact directly from the observation that G1 and G2 have the same maximal compact subgroups, (K1 :=) K ∩ G1 = K ∩ G2 (=: K2 ). In fact, we get from (9.2) ASK (πiλ,δ ) ∩ Ad∨ (K) ik⊥ 1 = {0}. Therefore, the restriction πiλ,δ |G1 is discretely decomposable, too. Remark 9.2. In contrast to the restriction of the quantization of elliptic orbits (equivalently, of Zuckerman’s Aq (λ)-modules), it is rare that the restriction of the quantization of hyperbolic orbits (equivalently, unitarily induced representations from real parabolic subgroups) is discretely decomposable with respect to non-compact reductive subgroups. Another discretely decomposable case was found by Lee–Loke in their study of the Jordan–Hölder series of a certain degenerate principal series representations. R) ↓ GL(p,R R) × GL(q,R R) 10. Branching law for GL(N,R Let N = p + q (p, q 1), and consider a subgroup G3 := GL(p, R) × GL(q, R) in G := GL(N,R) GL(N, R). The restriction of πiλ,δ with respect to the symmetric pair (G, G3 ) = GL(N, R), GL(p, R) × GL(q, R) is decomposed into the same family of degenerate principal series representations of G3 :


1715

Theorem 10.1 (Branching law GL(p + q, R) ↓ GL(p, R) × GL(q, R)). GL(p+q,R)

πiλ,δ

G3

⊕ δ =0,1 R

GL(p,R)

πiλ ,δ

πi(λ−λ ),δ−δ dλ . GL(q,R)

Outline of the proof. The proof is similar to that of Theorem 9.1. The group G3 = GL(p, R) × GL(q, R) acts on Pp+q−1 R with an open dense orbit O3 which has a G3 -equivariant fibration R× → O3 → GL(p, R)/Pp × GL(q, R)/Pq . Hence, taking the Mellin transform by the R× -action along the fiber, we get Theorem 10.1. 2 R) ↓ O(p, q) 11. Branching law for GL(N,R For N = p + q, we introduce the standard quadratic form of signature (p, q) by 2 2 Q(x) := x12 + · · · + xp2 − xp+1 − · · · − xp+q

for x ∈ Rp+q .

Let G4 be the indefinite orthogonal group defined by O(p, q) := g ∈ GL(N, R): Q(gx) = Q(x) for any x ∈ Rp+q . For q = 0, G4 is nothing but a maximal compact subgroup K = O(N ) of G, and the branching GL(N,R) law πiλ,δ |G4 is so-called the K-type formula. In order to describe the branching law G ↓ G4 for general p and q, we introduce a family O(p,q) (ν ∈ A+ (p, q) below), of irreducible unitary representations of G4 , to be denoted by π+,ν O(p,q)

O(p,q)

π−,ν (ν ∈ A+ (q, p)), and πiν,δ (ν ∈ R) as follows. Let t be a compact Cartan subalgebra of g4 , and we take a standard dual basis {ej } of t such that the set of roots for k4 := o(p) ⊕ o(q) is given by $ % & % & % & % &' p p p q (k4 , t4 ) = ±(ei ± ej ): 1 i < j or +1i <j + 2 2 2 2 % &' $ p (p: odd) ∪ ±ei : 1 i 2 $ % & % &' % & p q p ∪ ±ei : +1i + (q: odd). 2 2 2 Then, attached to the coadjoint orbits Ad∨ (G4 )(νei ) for ν ∈ A+ (p, q) and Ad∨ (G4 )(νe[ p ]+1 ) 2

O(p,q)

for ν ∈ A+ (q, p), we can define unitary representations of G4 , to be denoted by π+,ν

O(p,q) π−,ν

and

as their geometric quantizations. These representations are realized in Dolbeault cohomologies over the corresponding coadjoint orbits endowed with G4 -invariant complex structures, and their underlying (gC , K)-modules are obtained also as cohomologically induced representations from characters of certain θ -stable parabolic subalgebras (see [21, §5] for details).

1716

T. Kobayashi et al. / Journal of Functional Analysis 260 (2011) 1682–1720 O(p,q)

such that its infinitesimal character is given by

% & p+q p+q p+q p+q − 2, − 3, . . . , − 2 2 2 2

We normalize π+,ν

ν,

O(p,q)

in the Harish-Chandra parametrization. The parameter set that we need for π+,ν A+ (p, q) := A0+ (p, q) ∪ A1+ (p, q) where ⎧ {ν ∈ 2Z + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ {ν ∈ 2Z + δ A+ (p, q) := ∅ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 {2}

p−q 2 p−q 2

+ 1 + δ: ν > 0} + 1 + δ: ν >

p 2

is

(p > 1, q = 0); − 1} (p > 1, q = 0); (p = 1, (q, δ) = (0, 1)) or (p = 0); (p = 1, (q, δ) = (0, 1)). O(p,q)

O(q,p)

π+,ν . Notice that the identification O(p, q) O(q, p) induces the equivalence π−,ν For p, q > 0 the group G4 = O(p, q) is non-compact and there are continuously many hyperbolic coadjoint orbits. Attached to (minimal) hyperbolic coadjoint orbits, we can define another O(p,q) for ν ∈ R and family of irreducible unitary representations of G4 , to be denoted by πiν,δ O(p,q)

be the unitary representation of G4 induced from a unitary charδ ∈ {0, 1}. Namely, let πiν,δ acter (iν, δ) of a maximal parabolic subgroup of G4 whose Levi part is O(1, 1)×O(p −1, q −1). We note that the Knapp–Stein intertwining operator gives a unitary isomorphism O(p,q)

πiν,δ

O(p,q)

π−iν,δ

(ν ∈ R, δ = 0, 1).

Theorem 11.1 (Branching law GL(p + q, R) ↓ O(p, q)). GL(p+q,R) πiλ,δ O(p,q)

⊕ ν∈Aδ+ (p,q)

O(p,q) π+,ν

⊕

⊕ ν∈Aδ+ (q,p)

O(p,q) π−,ν

⊕ ⊕2

O(p,q)

πiν,δ

dν.

R+

Notice that in case when q = 0 the latter two components of the above decomposition do not occur and one gets the K-type formula GL(n, R) ↓ O(n). As a preparation of the proof, we formalize the Plancherel formula on the hyperboloid from a modern viewpoint of representation theory. Let X(p, q)± be a hypersurface in Rp+q defined by 2 2 X(p, q)± := x = x , x ∈ Rp+q : x − x = ±1 . We endow X(p, q)± with pseudo-Riemannian structures by restricting ds 2 = dx12 + · · · + 2 2 − dxp+1 − · · · − dxp+q on Rp+q . Then, X(p, q)± becomes a space form of pseudoRiemannian manifolds in the sense that its sectional curvature κ is constant. To be explicit, X(p, q)+ has a pseudo-Riemannian structure of signature (p − 1, q) with sectional curvature κ ≡ 1, whereas X(p, q)− has a signature (p, q − 1) with κ ≡ −1. Clearly, G4 acts on X(p, q)± as isometries. dxp2


1717

We denote by L2 (X(p, q)± ) the Hilbert space consisting of square integrable functions on X(p, q)± with respect to the induced measure from ds 2 |X(p,q) . The irreducible decomposition of the unitary representation of G4 on L2 (X(p, q)± ) is equivalent to the spectral decomposition of the Laplace–Beltrami operator on X(p, q)± with respect to the G4 -invariant pseudo-Riemannian structures. The latter viewpoint was established by Faraut [8] and Strichartz [26]. As we saw in [21, §5], the discrete series representations on hyperboloids X(p, q)± are isoO(p,q) with parameter set A± (p, q). morphic to π±,ν L X(p, q)+ δ = 2

O(p,q) π+,ν

⊕ ⊕

ν∈Aδ+ (p,q)

L X(p, q)− δ = 2

O(p,q)

dν,

(11.1)

O(p,q)

dν.

(11.2)

πiν,δ

R+ O(p,q) π−,ν

⊕ ⊕

ν∈Aδ+ (q,p)

πiν,δ

R+

Here we note that each irreducible decomposition is multiplicity free, the continuous spectra in both decompositions are the same and the discrete ones are distinct. Proof of Theorem 11.1. According to the decomposition Rp+q ⊃

dense

x ∈ Rp+q : Q(x) > 0 ∪ x ∈ Rp+q : Q(x) < 0 ,

the group G4 = O(p, q) acts on Pp+q−1 R with two open orbits, denoted by O4+ and O4− . A distinguishing feature for G4 is that these open G4 -orbits are reductive homogeneous spaces. To be explicit, let H4+ and H4− be the isotropy subgroups of G4 at [e1 ] ∈ O4+ and [ep+q ] ∈ O4− , respectively, where {ej } denotes the standard basis of Rp+q . Then we have O4+ G4 /H4+ = O(p, q)/ O(1) × O(p − 1, q) , O4− G4 /H4− = O(p, q)/ O(p, q − 1) × O(1) . Correspondingly, the restriction of the line bundle Liλ,δ = G ×P χiλ,δ to the open sets O4± of the base space G/P is given by G4 ×H ± Cδ , 4

where Cδ is a one-dimensional representation of H4± defined by O(1) × O(p − 1, q) → C× ,

(a, A) → a δ ,

O(p, q − 1) × O(1) → C× ,

(B, b) → bδ ,

respectively. It is noteworthy that unlike the cases G2 = GL(n, C) and G3 = GL(p, R) × GL(q, R), the continuous parameter λ is not involved in (11.1).

1718


Since the union O4+ ∪ O4− is open dense in Pp+q−1 R, we have a G4 -unitary equivalence (independent of λ): GL(p+q,R)

Hiλ,δ

G4

L2 G4 ×H4 Cδ , O4+ ⊕ L2 G4 ×H4 Cδ , O4− .

Sections for the line bundle G4 ×H ± Cδ over O4± are identified with even functions (δ = 0) 4 or odd functions (δ = 1) on hyperboloids X(p, q)± because X(p, q)± are double covering manifolds of O4± . According to the parity of functions on the hyperboloid X(p, q)± , we decompose L2 X(p, q)± = L2 X(p, q)± 0 ⊕ L2 X(p, q)± 1 . Hence, we get Theorem 11.1.

2

12. Tensor products Met∨ ⊗ Met The irreducible decomposition of the tensor product of two representations is a special example of branching laws. It is well understood that the tensor product of the same Segal–Shale–Weil representation (e.g. Met ⊗ Met) decomposes into a discrete direct sum of lowest weight representations of Sp(n, R) (see [14]). In this section, we prove Theorem 12.1. Let Met be the Segal–Shale–Weil representation of the metaplectic group Mp(n, R), and Met∨ its contragredient representation. Then the tensor product representation Met∨ ⊗ Met is well defined as a representation of Sp(n, R), and decomposes into the direct integral of irreducible unitary representations as follows:

Met∨ ⊗ Met

⊕ δ=0,1 R

Sp(n,R)

2πiλ,δ

dλ.

(12.1)

+

Remark 12.2. The branching formula in Theorem 12.1 may be regarded as the dual pair correspondence O(1, 1) · Sp(n, R) with respect to the Segal–Shale–Weil representation of Mp(2n, R). We note that the Lie group O(1, 1) is non-abelian, and its finite dimensional irreducible unitary representations are generically of dimension two, which corresponds the multiplicity two in the right-hand side of (12.1). Proof of Theorem 12.1. By Proposition 3.2, the Weyl operator calculus ∼ → HS L2 Rn , L2 Rn Op : L2 R2n −

(12.2)

gives an intertwining operator as unitary representations of Mp(n, R). We write L2 (Rn )∨ for the dual Hilbert space, and identify ∨ ( L2 R n , HS L2 Rn , L2 Rn L2 Rn ⊗

(12.3)


1719

( denotes the completion of the tensor product of Hilbert spaces. Composing (12.2) where ⊗ and (12.3), we see that the tensor product representation Met∨ ⊗ Met of Mp(n, R) is unitarily equivalent to the regular representation on L2 (R2n ). This representation on the phase space L2 (R2n ) is well defined as a representation of Sp(n, R). We consider the Mellin transform on R2n , which is defined as the Fourier transform along the radial direction: 1 f→ 4π

∞ |t|n−1+iλ (sgn t)δ f (tX) dt, −∞

with λ ∈ R, δ = 0, 1, X ∈ R2n . Then, the Mellin transform gives a spectral decomposition of the Hilbert space L2 (R2n ). Therefore, the phase space representation L2 (R2n ) is decomposed as a direct integral of Hilbert spaces: ⊕ 2n L R Viλ,δ dλ. 2

(12.4)

δ=0,1 R Sp(n,R)

Since πiλ,δ

Sp(n,R)

π−iλ,δ

(see (5.3)), we get Theorem 12.1.

2

Acknowledgments The authors are grateful to the Institut des Hautes Études Scientifiques, the Institute for the Physics and Mathematics of the Universe of the Tokyo University, the Universities of Århus and Reims where this work was done. References [1] E.P. van den Ban, H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space. I, II, Invent. Math. 161 (2005) 453–566, 567–628. [2] D. Barbasch, The unitary spherical spectrum for split classical groups, J. Inst. Math. Jussieu 9 (2010) 265–356. [3] V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. 48 (1947) 568–640. [4] M. Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957) 85–177. [5] J.-L. Clerc, T. Kobayashi, M. Pevzner, B. Ørsted, Generalized Bernstein–Reznikov integrals, Math. Ann. (2010), doi:10.1007/s00208-010-0516-4. [6] P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs, Ann. of Math. (2) 147 (2) (1998) 417– 452. [7] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Tables of Integral Transforms, vols. I, II, McGraw–Hill, New York, 1954. [8] J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl. 58 (1979) 369–444. [9] T.A. Farmer, Irreducibility of certain degenerate principal series representations of Sp(n, R), Proc. Amer. Math. Soc. 83 (1981) 411–420. [10] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, 1989. [11] W.-T. Gan, G. Savin, On minimal representations definitions and properties, Represent. Theory 9 (2005) 46–93. [12] R. Howe, On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. (N.S.) 3 (1980) 821– 843. [13] R. Howe, S.-T. Lee, Degenerate principal series representations of GLn (C) and GLn (R), J. Funct. Anal. 166 (1999) 244–309.

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[14] M. Kashiwara, M. Vergne, On the Segal–Shale–Weil representations and harmonic polynomials, Invent. Math. 44 (1978) 1–47. [15] T. Kobayashi, Discrete decomposability of the restriction of Aq (λ) with respect to reductive subgroups II: Microlocal analysis and asymptotic K-support, Ann. of Math. 147 (1998) 709–729. [16] T. Kobayashi, Branching problems of unitary representations, in: Proc. of ICM 2002, vol. 2, Beijing, 2002, pp. 615– 627. [17] T. Kobayashi, Algebraic analysis on minimal representations, Publ. Res. Inst. Math. Sci. 47 (2011), Special Issue in Commemoration of the Golden Jubilee of Algebraic Analysis, in press, arXiv:1001.0224. [18] T. Kobayashi, G. Mano, Integral formula of the unitary inversion operator for the minimal representation of O(p, q), Proc. Japan Acad. Ser. A 83 (2007) 27–31. [19] T. Kobayashi, G. Mano, The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p, q), Mem. Amer. Math. Soc. 212 (1000) (2011), in press, available at arXiv:0712.1769. [20] T. Kobayashi, B. Ørsted, Analysis on the minimal representations of O(p, q), I – Realization and conformal geometry, Adv. Math 180 (2003) 486–512. [21] T. Kobayashi, B. Ørsted, Analysis on the minimal representations of O(p, q), II – Branching laws, Adv. Math 180 (2003) 513–550. [22] T. Kobayashi, B. Ørsted, Analysis on the minimal representations of O(p, q), III – Ultra-hyperbolic equations on Rp−1,q−1 , Adv. Math 180 (2003) 551–595. [23] T. Kobayashi, B. Ørsted, M. Pevzner, A. Unterberger, Composition formulas in the Weyl calculus, J. Funct. Anal. 257 (2009) 948–991. [24] B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969) 627–642. [25] J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931) 570–578. [26] R.S. Strichartz, Harmonic analysis on hyperboloids, J. Funct. Anal. 12 (1973) 341–383. [27] D.A. Vogan Jr., N.R. Wallach, Intertwining operators for real reductive groups, Adv. Math. 82 (1990) 203–243.


Level sets and composition operators on the Dirichlet space O. El-Fallah a,1 , K. Kellay b,∗,1,2 , M. Shabankhah b,2,3 , H. Youssfi b,1,2 a Département de Mathématiques, Université Mohamed V, B.P. 1014 Rabat, Morocco b CMI, LATP, Université de Provence, 39, rue F. Joliot-Curie, 13453 Marseille, France

Received 24 March 2010; accepted 21 December 2010 Available online 3 January 2011 Communicated by Gilles Godefroy

Abstract We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert–Schmidt class membership are established. Some of these criteria are shown to be optimal. © 2010 Elsevier Inc. All rights reserved. Keywords: Dirichlet space; Composition operators; Capacity

1. Introduction In this note we consider composition operators in the Dirichlet space of the unit disc. A comprehensive study of composition operators in function spaces and their spectral behavior could be found in [3,11,16]. See also [6–8,12,13,17] for a treatment of some of the questions addressed in this paper. * Corresponding author.

E-mail addresses: [email protected] (O. El-Fallah), [email protected] (K. Kellay), [email protected], [email protected] (M. Shabankhah), [email protected] (H. Youssfi). 1 Research partially supported by a grant from AI PHC Volubilis MA 09209. 2 Research partially supported by ANR Dynop. 3 Current address: Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada H3A 2K6. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.023

1722

O. El-Fallah et al. / Journal of Functional Analysis 260 (2011) 1721–1733

Let D be the unit disc in the complex plane and let T = ∂D be its boundary. We denote by D the classical Dirichlet space. This is the space of all analytic functions f on D such that D(f ) :=

2 f (z) dA(z) < ∞,

D

where dA(z) = dx dy/π stands for the normalized area measure in D. We call D(f ) the Dirichlet integral of f . The space D is endowed with the norm 2 f 2D := f (0) + D(f ). It is standard that a function f (z) = only if

∞

n=0 f (n)z

n,

holomorphic on D, belongs to D if and

f(n)2 (1 + n) < ∞, n0

and that this series defines an equivalent norm on D. Since the Dirichlet space is contained in the Hardy space H2 (D), every function f ∈ D has non-tangential limits f ∗ almost everywhere on T. In this case, however, more can be said. Indeed, Beurling [2] showed that if f ∈ D then f ∗ (ζ ) = limr→1 f (rζ ) exists for ζ ∈ T outside of a set of logarithmic capacity zero. Let ϕ be a holomorphic self-map of D. The composition operator Cϕ on D is defined by Cϕ (f ) = f ◦ ϕ,

f ∈ D.

We are interested herein in describing the spectral properties of the composition operator Cϕ , such as compactness and Hilbert–Schmidt class membership, in terms of the size of the level set of ϕ. For s ∈ (0, 1), the level set Eϕ (s) of ϕ is given by Eϕ (s) = ζ ∈ T: ϕ(ζ ) s . We give new characterizations of Hilbert–Schmidt class membership in the case of the Dirichlet space. We also establish the sharpness of these results. 2. A general criterion For α > −1, dAα will denote the finite measure on D given by

α dAα (z) := (1 + α) 1 − |z|2 dA(z). p

For p 1 and α > −1, the weighted Bergman space Aα consists of the holomorphic functions f on D for which f p,α := D

1/p f (z)p dAα (z) < ∞.


1723

p

We denote by Dα the space consisting of analytic functions f on D such that p p p f Dp := f (0) + f p,α < ∞. α

Appropriate choices of the parameter α give, with equivalent norm, all the standard holomorphic function spaces. Indeed, the Hardy space H2 can be identified with D12 . The classical Besov p p p space is precisely Dp−2 , and if p < α + 1, Dα = Aα−2 . Finally, the classical Dirichlet space D is identical to D02 . p We recall that, by the reproducing formula [16], for every f ∈ Aα , f (z) = D

f (w) dAα (w), (1 − wz)2+α

z ∈ D.

(1)

Lemma 2.1. Let p 1 and let σ > −1. Then, there exists a constant C depending only on p and p σ such that for every f ∈ Aσ , f (z)p C

D

|f (λ)|p |1 − λz|2+σ

z ∈ D.

dAσ (λ),

Proof. By the above reproducing formula, f (z) = 1 − zw

D

dAσ (λ) f (λ) , 1 − λw (1 − λz)2+σ

z, w ∈ D,

p

for every f ∈ Aσ . By Hölder’s inequality, with q = p/(p − 1), |f (z)|p |1 − zw|p

|f (λ)|p dAσ (λ) |1 − λz|2+σ

D

× D

dAσ (λ) |1 − λw|q |1 − λz|(2+σ )p

p q

.

Taking w = z, and using the standard estimate, [16, Lemma 3.10] D

dAc (λ) |1 − zλ|2+c+d

we get the desired conclusion.

1 , (1 − |z|2 )d

if d > 0, c > −1,

2

For λ ∈ D, consider the test function Fλ,β (z) = (1 − λz)−(1+β) ,

z ∈ D.

If β 0 is chosen such that δ := δ(p, α, β) = 2 + β − (2 + α)/p > 0, by (2), we have

−pδ p Fλ,β Dp 1 − |λ|2 . α

(2)

1724


The following theorem unifies and generalizes the previously known results of MacCluer [3, Theorem 3.12], Tjani [12, Theorem 3.5] and Wirths and Xiao [13, Theorem 3.2] on Hardy, Besov and weighted Dirichlet spaces, respectively. The techniques required in the proof are known, for the completeness, we give here the proof. p

Theorem 2.2. Let p > 1. Suppose ϕ ∈ Dα satisfies ϕ(D) ⊂ D. Fix β 0 such that δ := δ(p, α, β) = 2 + β − (2 + α)/p > 0. Then: p

(a) Cϕ is bounded on Dα ⇐⇒ supλ∈D (1 − |λ|2 )δ Fλ,β ◦ ϕDαp < ∞; p (b) Cϕ is compact on Dα ⇐⇒ lim|λ|→1 (1 − |λ|2 )δ Fλ,β ◦ ϕDαp = 0. Proof. Without loss of generality we assume that ϕ(0) = 0. To prove (a), we observe that if Cϕ is bounded, then

−δ

Fλ,β ◦ ϕDαp = O 1 − |λ|2 . p

For the converse, it follows from Lemma 2.1 that, for f ∈ Dα ,

p

ϕ (z) f ϕ(z) p dAα (z)

D

C D

=C

p ϕ (z)

D

|f (λ)|p |1 − λϕ(z)|(2+β)p

dA2p+βp−2 (λ) dAα (z)

p

f (λ) 1 − |λ|2 pδ (Fλ,β ◦ ϕ) p dAα (λ). p,α

D

Therefore part (a) follows. (b) Assume that lim|λ|→1 (1 − |λ|2 )δ Fλ,β ◦ ϕDαp = 0. Let (fn )n be a bounded sequence of p Dα such that fn → 0 uniformly on compact sets. Since fn → 0 uniformly on compact sets, it follows from the proof of part (a) and the hypothesis that, for r close enough to 1,

Cϕ (fn ) p p − fn (0)p Dα

p p

pδ fn (λ) 1 − |λ|2 (Fλ,β ◦ ϕ) p,α dAα (λ) rD

+

p

f (λ) 1 − |λ|2 pδ (Fλ,β ◦ ϕ) p dAα (λ) → 0, n p,α

D\rD

and Cϕ is compact. The converse is obvious.

2

The following result is an immediate consequence of Theorem 2.2.

n → ∞,


1725

Corollary 2.3. Let ϕ : D → D such that ϕ ∈ D. (a) If supn1 D(ϕ n ) < ∞, then Cϕ is bounded. (b) If limn→∞ D(ϕ n ) = 0, then Cϕ is compact. Proof. We consider the test function Fλ,0 with β = α = 0 and p = 2. Both (a) and (b) follow from the following inequality:

2 D(Fλ,0 ◦ ϕ) 2 1 − |λ|2

D

|ϕ (z)|2 dA(z) (1 − |λ|2 ϕ(z)|2 )4

2 c 1 − |λ|2 (n + 1)3 |λ|2n n0

2 n 2 ϕ (z) ϕ (z) dA(z)

D

2

= c 1 − |λ|2 (1 + n)|λ|2n D ϕ n+1 n0

c lim sup D ϕ n+1 . n→∞

2

Remark 2.4. The compactness criterion for Cϕ in the Bloch space is equivalent to ϕ n B → 0 as was shown in [15] (see also [10,12]). In the case of the Hardy space H2 , however, we know that if Cϕ is compact on H2 then ϕ n H2 → 0 but the converse does not hold [3]. √ Note that as before in the proof of Corollary 2.3 (β = 0, α = 1 and p = 2) if ϕ n H2 = o(1/ n ), then Cϕ is compact on H2 . 3. Hilbert–Schmidt membership In the case of the Hardy space H2 , one can completely describe the membership of Cϕ in the Hilbert–Schmidt class in terms of the size of the level sets of the inducing map ϕ. Indeed, Cϕ is Hilbert–Schmidt in H2 if and only if 2

ϕ n 2 = H n0

T

|dζ | < ∞. 1 − |ϕ(ζ )|2

Given an arbitrary measurable function f on T, consider the associated distribution function mf defined by mf (λ) = ζ ∈ T: f (ζ ) > λ ,

λ > 0.

It then follows that Cϕ is in the Hilbert–Schmidt class of H2 if and only if T

|dζ | = 1 − |ϕ(ζ )|2

∞

1 m(1−|ϕ|2 )−1 (λ) dλ

1

0

|Eϕ (s)| ds < ∞. (1 − s)2

1726


It was shown by Gallardo-Gutiérrez and González [8, Main Theorem] that there is a mapping ϕ taking D to itself such that Cϕ is compact in H2 , and that the level set Eϕ (1) has Hausdorff dimension equal to one. Recall that the Hausdorff dimension of E: d(E) = inf α: Λα (E) = 0 where Λα (E) is the α-Hausdorff measure of E given by α Λα (E) = lim inf | i | : E ⊂

i , | i | < . →0

i

i

Given E ⊂ T and t > 0, let us write Et = ζ ∈ T: d(ζ, E) t where d denotes the arclength distance and |Et | denotes the Lebesgue measure of E. Let E be a closed subset of T with |Et | = O((log(e/t))−3 ) and E has Hausdorff dimension one (such examples can be given by generalized Cantor sets [2]). Let ω(t) = (log(e/t))−2 , and ∗ is given by consider the outer function fω,E such that its radial limit fω,E ∗ f (ζ ) = e−w(d(ζ,E)) , ω,E

a.e. on T.

Since ω satisfies the Dini condition

ω(t) dt < ∞, t

0

it follows that fω,E ∈ A(D) := Hol(D) ∩ C(D), disc algebra (see [9, pp. 105–106]) and so Efω,E (1) = E. On the other hand T

|dζ | 1 − |fω,E (ζ )|2

T

|dζ | ω(d(ζ, E))

|Et |

ω (t) dt ω(t)2

0

(see [4, Proposition A.1] for the last equality). Since the last integral converges, Cϕ is a Hilbert– Schmidt operator in H2 . We have the following more precise result. Theorem 3.1. Let E be a closed subset of T with Lebesgue measure zero. There exists a mapping ϕ : D → D, ϕ ∈ A(D), such that Cϕ is a Hilbert–Schmidt operator on H2 and that Eϕ (1) = E. Proof. The proof is based a well-known construction of peak functions in the disc algebras. Let T \ E = n1 (eian , eibn ). For t ∈ (an , bn ), we define

g eit = τn

(bn − an )1/2 , [(bn − an )2 − (2t − (bn + an ))2 ]1/4

where (τn )n ⊂ (0, ∞) will be chosen later, and g(eit ) := +∞ if eit ∈ E.


1727

Note that 2π

2 g eit dt = τn2 (bn − an ) n1

0

bn [(bn − an

an

1

1 2 = τn (bn − an ) 2 n1

= Since

∞

n=1 (bn

π 2

∞

−1

)2

dt − (2t − (bn + an ))2 ]1/2

du [1 − u2 ]1/2

τn2 (bn − an ).

n=1

− an ) = 2π , there exists a sequence (τn )n such that ∞

lim τn = +∞ and

n→+∞

τn2 (bn − an ) < ∞.

n=1

Let U denote the harmonic extension of g on the unit disc given by

1 U reiθ = 2π

2π 0

1 − r2 g eit dt = g (n)r |n| einθ . it iθ 2 |e − re | n∈Z

Since τn → ∞, one can easily verify that limt→θ g(eit ) = +∞, for eiθ ∈ E. Hence, limr→1− U (reiθ ) = +∞, for eiθ ∈ E. Let V be the harmonic conjugate of U , with V (0) = 0. It is given by

n g (n)r |n| einθ . V reiθ = |n| n=0

Now, since g is a C 1 function on T \ E, we see that the holomorphic function f = U + iV is continuous on D \ E. Knowing that limr→1− U (reit ) = +∞, for eit ∈ E, we get that ϕ = f f+1 ∈ A(D), disc algebra, and Eϕ (1) = E. Finally 1 2π

2π 0

dt 1 = it 2 2π 1 − |ϕ(e )| 1 2π

2π 0

(U (eit ) + 1)2 + V 2 (eit ) dt (U (eit ) + 1)2 − U 2 (eit )

2π it

2

U e + 1 + V 2 eit dt 0

1+2

2 g (n) , n∈Z

which shows that Cϕ is Hilbert–Schmidt because g ∈ L2 (T).

2

1728


Let E be a closed subset of the unit circle T. Fix a non-negative function w ∈ C 1 (0, π] such that

w d(ζ, E) |dζ | < ∞, T

where d denotes the arclength distance. Now, let fw,E be the outer function given by ∗ f (ζ ) = e−w(d(ζ,E)) , w,E

a.e. on T.

(3)

The following lemma gives an estimate for the Dirichlet integral of fw,E in terms of w and the distance function on E. The proof is based on Carleson’s formula, and can be achieved by slightly modifying the arguments used in [5, Theorem 4.1]. Lemma 3.2. Assume that the function ω is nondecreasing and ω(t γ ) is concave for all γ > 2. Then

2 D(fw,E ) ω d(ζ, E) e−2w(d(ζ,E)) d(ζ, E) |dζ |. T

√ Since the sequence {zn / n + 1 }∞ n=0 is an orthonormal basis of D, the operator Cϕ is Hilbert– Schmidt on the Dirichlet space if and only if 1 π

D

D(ϕ n ) |ϕ (z)|2 < ∞. dA(z) = n (1 − |ϕ(z)|2 )2 n1

Theorem 3.3. Assume that the function ω is nondecreasing and ω(t γ ) is concave for some γ > 2. Then Cfw,E is in the Hilbert–Schmidt class in D if and only if T

ω (d(ζ, E))2 d(ζ, E) |dζ | < ∞. w(d(ζ, E))2

n =f Proof. We first note that fw,E nw,E . Therefore, by Lemma 3.2, we have

D

(z)|2 |fw,E

(1 − |fw,E (z)|2 )2

dA(z) =

∞ D(fnw,E ) n=1

n

∞

2 ω d(ζ, E) d(ζ, E) ne−2nw(d(ζ,E)) |dζ | n=1

T

T

ω (d(ζ, E))2 d(ζ, E) |dζ |. [1 − e−2w(d(ζ,E)) ]2

Since 1 − e−2w(d(ζ,E)) w(d(ζ, E)), the result follows.

2


1729

Given a (Borel) probability measure μ on T, we define its α-energy, 0 α < 1, by Iα (μ) =

∞ | μ(n)|2 n=1

n1−α

.

For a closed set E ⊂ T, its α-capacity capα (E) is defined by capα (E) := 1/ inf Iα (μ): μ is a probability measure on E . If α = 0, we simply note cap(E) and this means the logarithmic capacity of E. The weak-type inequality for capacity [2] states that, for f ∈ D and t 4f 2D , 16f 2D cap ζ : f (ζ ) t . t2 As a result of this inequality, we see that if lim inf ϕ n D = 0, then cap(Eϕ (1)) = 0. Indeed, since Eϕ (1) = Eϕ n (1), the weak capacity inequality implies that

2

cap Eϕ (1) = cap Eϕ n (1) 16 ϕ n D . Now let n → ∞. Hence, in particular, if the operator Cϕ is in the Hilbert–Schmidt class in D, then cap(Eϕ (1)) = 0. This result was first obtained by Gallardo-Gutiérrez and González [6,7] using a completely different method. Theorems 3.4 and 3.6 give quantitative versions of this result. Theorem 3.4. If Cϕ is a Hilbert–Schmidt operator in D, then 1 0

cap(Eϕ (s)) 1 log ds < ∞. 1−s 1−s

Proof. Fix λ ∈ T and let ϕλ (ζ ) = log Re

1 + λϕ(ζ ) , 1 − λϕ(ζ )

ζ ∈ T.

Since D

|ϕ (z)|2 dA(z) < ∞, (1 − |ϕ(z)2 |)2

it follows that ϕλ ∈ D(T), see [6], where 2

2 2 D(T) := f ∈ L (T): f D(T) = f (n) 1 + |n| < ∞ . n∈Z

(4)

1730


Setting λ := {ζ ∈ T: |1 − λϕ(ζ )| 1}, we see that ϕλ (ζ ) log

1 , 1 − |ϕ(ζ )|2

∀ζ ∈ λ .

Applying the strong capacity inequality [14, Theorem 2.2] to ϕλ , we get ∞ ∞ > ϕλ D(T) c 2

cap ζ ∈ T: ϕλ (ζ ) > s ds 2 2 log 1 − |ϕ(ζ )| > s ds 2 |1 − λϕ(ζ )|2

∞

cap ζ ∈ T:

∞

cap ζ ∈ T ∩ λ :

∞

cap ζ ∈ T ∩ λ : log

=c c c

1 c1

2 log 1 − |ϕ(ζ )| > s ds 2 |1 − λϕ(ζ )|2 1 > 4s ds 2 1 − |ϕ(ζ )|2

cap ζ ∈ T ∩ λ : ϕ(ζ ) > u d log

1 1−u

2 .

Since T = 1 ∪ −1 , the subadditivity of the capacity implies that 1 ∞ > ϕ1 D(T) + ϕ−1 D(T) c2 2

2

and hence the theorem follows.

cap ζ ∈ T: ϕ(ζ ) > u d log

1 1−u

2 ,

2

1−α

Remark 3.5. Since {zn /(1+n) 2 }∞ n=0 is an orthonormal basis in Dα , α ∈ (0, 1), Cϕ is a Hilbert– Schmidt operator in Dα if and only if ∞ Dα (ϕ n ) n=1

n1−α

D

|ϕ (z)|2 dAα (z) < ∞. (1 − |ϕ(z)|2 )2+α

Therefore, for fixed λ ∈ T, the function 1 + λϕ(ζ ) −α/2 ϕλ (ζ ) = Re , 1 − λϕ(ζ )

ζ ∈ T,

belongs to the weighted harmonic Dirichlet space 2

1−α 2 2 2). In connection with the algebraic approach sketched above, we would like to emphasize the previous work of J. Bellissard, who was one of the first to stress the advantage of considering C ∗ -algebras generated by Hamiltonians in the context of solid state physics [2,3], and that of H.O. Cordes, who studied C ∗ -algebras of pseudo-differential operators on manifolds and their quotients with respect to the ideal of compact operators [9] already in the seventies. 1.4. Now let’s get back to our problem. Assuming we have chosen the “correct” algebra E (X), we must find the relevant ideals. In the group case, this is easy, because there is a natural class of ideals associated to translation invariant filters [15]. Proposition 6.6 gives a characterization of these filters which involves only the metric structure of X (in fact, only the coarse structure associated to it [30]). Thus what we call coarse filters in a metric space are analogs of the invariant filters in a group. To each coarse filter ξ we then associate an ideal Jξ defined in terms of the behavior of the operators at a certain region at infinity defined by ξ , cf. (2.6). These are the geometric ideals which play the main role in or analysis. Recall that the set of ultrafilters finer than the Fréchet filter is a compact subset δ(X) of the ˇ Stone–Cech compactification β(X) of X. Any filter ξ finer than Fréchet can be thought as a closed subset of δ(X) by identifying it with the set ξ † of ultrafilters finer than it, and then the sets F ∈ ξ can be thought as traces on X of neighborhoods of this closed set in β(X). The sets ξ † with ξ coarse will be called coarse subsets of δ(X) (they are closed). If X is a group then X acts on δ(X), the coarse subsets are the closed invariant subsets of δ(X), and the small invariant sets are parametrized as follows: to each ∈ δ(X) we associate the smallest closed invariant set containing (i.e. the closure of the orbit which passes through it). But this can be easily expressed in group independent terms: if ∈ δ(X) let co() be the finer coarse filter included in and let := co()† be the smallest coarse set containing . Then the co() are the large coarse filters, the the small coarse sets, and the E() := Jco() are the large coarse ideals which

1738

V. Georgescu / Journal of Functional Analysis 260 (2011) 1734–1765

should allow us to compute the essential spectrum of the operators in E . Heuristically speaking, . For example, if X is discrete, so E contains E() consists of the operators in E which vanish at the bounded functions ϕ on X, we have ϕ ∈ E() if and only if the continuous extension of ϕ to β(X) is zero on . We stress that this strategy denotes a certain bias toward the role played by the behavior at infinity in X (thought as physical or configuration space): we think that it has a dominant role since we hope that our choices of ideals are sufficient to describe the quotient E /K . There is no a priori reason for this to be true: there are physically natural situations in which ideals defined in terms of behavior at infinity in momentum or phase space must be taken into account [15]. However, it does not seem so clear to us how to define such physically meaningful objects in the present context (there is no natural phase space). Anyway, the situation is not simple even at the level of geometrically defined ideals. Indeed, the ideals E() are defined in terms of the behavior of the operators in E at , but it is not completely clear how to express the intuitive idea that an operator T vanishes on . Our choice is the most restrictive one, but there is a second one which is also quite natural and leads to a distinct class of ideals G , cf. (5.27) and (5.28). One has E() ⊂ G strictly in general but equality holds if the space X has the Property A. An interesting point is that in general the ideals G do not suffice to compute E /K , i.e. we do not have ∈δ G = K . In fact an ideal G which contains the compacts appears naturally in the algebra E , the so-called ghost ideal, and this ideal could contain a projection of infinite rank, hence be strictly larger than the compacts. The construction of such a projection is due to Higson, Laforgue, and Skandalis [20] and is important in the context of the Baum–Connes conjecture. They consider the simplest case of discrete metric spaces with bounded geometry (the number of points in a ball of radius r is bounded independently of the center of the ball) when E is the uniform Roe C ∗ -algebra [30]. More information concerning this question may be found in the papers [7,8,34] by Chen and Wang where the ideal structure of the uniform Roe algebra is studied in detail. Their idea of using kernel truncations with the help of positive type functions in case X has Yu’s Property A plays an important role in our proofs, as we shall see in Section 3. But before going into details on these matters we shall describe in the next section in precise terms the framework and the main results of this paper. As explained before, a representation like (1.1) involving ideals which are as large as possible will provide the most detailed information on the structure of the essential spectrum of the observables affiliated to E . Thus the fact that ∈δ G = K shows that in general the large ideals are not sufficient to compute the essential spectrum. We leave open the question whether E = K holds even if ∈δ () ∈δ G = K . 2. Main results A metric space X = (X, d) is proper if each closed ball Bx (r) = {y | d(x, y) r} is a compact set. This implies the local compactness of the topological space X but is much more because local compactness means only that the small balls are compact. In particular, if X is not compact, then the metric cannot be bounded. We are interested in proper non-compact metric spaces equipped with Radon measures μ with support equal to X, so μ(Bx (r)) > 0 for all x ∈ X and all r > 0, and which satisfy (at least) the following condition V (r) := sup μ Bx (r) < ∞ for all real r > 0. x∈X

(2.3)


1739

We shall always assume that a metric measure space (X, d, μ) satisfies these conditions. On the other hand, for the proof of our main results we need the following supplementary condition: inf μ Bx (1/2) > 0. x

(2.4)

The choice of 1/2 in (i) is, of course, rather arbitrary, and an assumption of the form infx μ(Bx (r)) > 0 for all r > 0 would be more natural. Each time we use (2.4) we shall mention it explicitly. To simplify the notations we set dμ(x) = dx, L2 (X) = L2 (X, μ), and Bx = Bx (1). We denote B(X) the C ∗ -algebra of all bounded operators on L2 (X) and K (X) the ideal of B(X) consisting of compact operators. For A ⊂ X we denote 1A its characteristic function and if A is measurable then we use the same notation for the operator of multiplication by 1A in L2 (X). Several versions of Yu’s Property A appear in the literature (see [30, Definition 11.35] and [33] for the discrete case), we have chosen that which was easier to state and use in our context. Later on we shall state and use a more abstract version which can easily be reformulated in terms of positive type functions on X 2 . See p. 1760 here and [30, Chapter 3] for the relation with amenability in the group case. Definition 2.1. We say that the metric measure space (X, d, μ) has Property A if for each ε, r > 0 there is a Borel map φ : X → L2 (X) with φ(x) = 1, supp φ(x) ⊂ Bx (s) for some number s independent of x, and such that φ(x) − φ(y) < ε if d(x, y) < r. Definition 2.2. We say that X = (X, d, μ) is a class A space if (X, d) is a proper non-compact metric space and μ is a Borel measure on X such that: (i) μ(Bx (r)) > 0 and supx μ(Bx (r)) < ∞ for each r > 0, (ii) infx μ(Bx (1/2)) > 0, (iii) (X, d, μ) has Property A. Since X is locally compact the spaces Co (X) and Cc (X) of continuous functions on X which tend to zero at infinity or have compact support respectively are well defined. We use the slightly unusual notation C(X) for the set of bounded uniformly continuous functions on X equipped with the sup norm. Then C(X) is a C ∗ -algebra and Co (X) is an ideal in it. We embed C(X) ⊂ B(X) by identifying ϕ ∈ C with the operator ϕ(Q) of multiplication by ϕ (this is an embedding because the support of μ is equal to X). We shall however use the notation ϕ(Q) if we think that this is necessary for the clarity of the text. Functions k : X 2 → C on the product space X 2 = X × X are also called kernels on X. We say that k is a controlled kernel if there is a real number r such that d(x, y) > r ⇒ k(x, y) = 0. With the terminology of [21], a kernel is controlled if it is supported by an entourage of the bounded coarse structure on X coming from the metric. We denote Ctrl (X 2 ) the set of bounded 2 uniformly continuous controlled kernels and to each k ∈ Ctrl (X ) we associate an operator Op(k) 2 on L (X) by (Op(k)f )(x) = X k(x, y)f (y) dy. It is easy to check (see Section 3) that the set of such operators is a ∗-subalgebra of B(X). Hence

E (X) ≡ E (X, d, μ) = norm closure of Op(k) k ∈ Ctrl X 2 is a C ∗ -algebra of operators on L2 (X). We shall say that E (X) is the elliptic algebra of X.

(2.5)

1740


Remark 2.3. The following alternative presentation of the framework clarifies the role of the metric. Fix a couple X = (X, μ) consisting of a locally compact non-compact topological space X equipped with a Radon measure μ with support equal to X. This fixes the Hilbert space L2 (X). Then to each proper metric compatible with the topology of X and such that supx μ(Bx (r)) < ∞ for all r we associate a C ∗ -algebra E (X, d) of operators on L2 (X) which contains K (X). It is interesting to note that E (X, d) depends only on the coarse equivalence class of the metric. Recall that two metrics d, d are coarse equivalent if there are positive increasing functions u, v such that d u(d ) and d v(d). This can also be expressed in terms of coarse structures on X [32, p. 810]. There is an obvious C(X)-bimodule structure on E (X) and we have K (X) = Co (X)E (X) = E (X)Co (X) ⊂ E (X). As explained in the introduction we are interested in a “geometrically meaningful” representation of the quotient C ∗ -algebra E (X)/K (X). For this we introduce the class of “coarse ideals” described below. If F ⊂ X and r > 0 is real we denote F (r) the set of points x which belong to the interior of F and are at distance larger than r from the boundary, more precisely infy ∈F / d(x, y) > r. A filter ξ of subsets of X will be called coarse if F ∈ ξ ⇒ F (r) ∈ ξ for all r. Note that the set of complements of a coarse filter is a coarse ideal of subsets of X in the sens of [21]. The Fréchet filter, i.e. the set of sets with relatively compact complement, is clearly coarse, we denote it ∞. There is a trivial coarse filter, namely ξ = {X}, which is of no interest for us. All the other coarse filters are finer that ∞. To each coarse filter ξ on X we associate an ideal of E (X) by defining

Jξ (X) = T ∈ E (X) inf 1F T = 0 = T ∈ E (X) inf T 1F = 0 F ∈ξ

F ∈ξ

(2.6)

where the inf is taken only over measurable F ∈ ξ . We shall see that the set Iξ (X) of ϕ ∈ C(X) such that limξ ϕ = 0 is an ideal of C(X) and Jξ (X) = Iξ (X)E (X) = E (X)Iξ (X). ˇ Let β(X) be the set of all ultrafilters of X (this is the Stone–Cech compactification of the discrete space X) and let δ(X) be the set of ultrafilters finer than the Fréchet filter. For each ∈ β(X) we denote co() the largest coarse filter contained in and we set C() (X) = Ico() (X) and E() (X) = Jco() (X). These are ideals in C(X) and E (X) respectively and we have E() (X) = C() (X)E (X) = E (X)C() (X).

(2.7)

If X is of class A then from Theorem 5.9 we get a second description of these ideals. Proposition 2.4. If X is a space of class A then for any ∈ δ(X) we have

E() (X) = T ∈ E (X) lim 1Bx (r) T = 0, ∀r > 0 . x→

(2.8)

Then to each ultrafilter ∈ δ(X) we associate the quotient C ∗ -algebra E (X) = E (X)/E() (X)

(2.9)


1741

and call it localization of E (X) at . We denote .T the image of T ∈ E (X) through the canonical morphism E (X) → E (X) and we say that .T is the localization of T at . Our main result is: Theorem 2.5. If X is a class A space then

∈δ(X) E() (X) = K

E (X)/K (X) →

E .

(X), hence (2.10)

∈δ(X)

In particular, the essential spectrum of any normal operator T ∈ E (X) is equal to the closure of the union of the spectra of its localizations at infinity: Spess (T ) =

∈δ(X) Sp(.T ).

(2.11)

In view of applications to self-adjoint operators affiliated to E (X), we recall [1] that an observable affiliated to a C ∗ -algebra A is a morphism H : Co (R) → A . We set ϕ(H ) := H (ϕ). If P : A → B is a morphism between two C ∗ -algebras then ϕ → P(ϕ(H )) is an observable affiliated to B denoted P(H ). So P(ϕ(H )) = ϕ(P(H )). If A and B are realized on Hilbert spaces Ha , Hb , then any self-adjoint operator H on Ha affiliated to A defines an observable affiliated to A , but the observable P(H ) is not necessarily associated to a self-adjoint operator on Hb because the natural operator associated to it could be non-densely defined (in our context, it often has domain equal to {0}). The spectrum and essential spectrum of an observable are defined in an obvious way [1]. Now clearly, if H is an observable affiliated to E (X) then .H defined by ϕ(.H ) = .ϕ(H ) is an observable affiliated to E (X). This is the localization of H at and we have Spess (H ) =

∈δ(X) Sp(.H ).

(2.12)

We shall not give in this paper affiliation criteria specific to the algebra E (X) but the results of Section 6 and the examples form [17] should convince the reader that the class of operators affiliated to E (X) is very large. On the other hand, if H is a positive self-adjoint operator such that e−H ∈ E (X) then H is affiliated to E (X). Or this condition is certainly satisfied by the Laplace operator associated to a large class of Riemannian manifolds due to known estimates on the heat kernel of the manifold. We thank Thierry Coulhon for an e-mail exchange on this question. In connection with Proposition 2.4 we mention that in Section 5 we consider a second class of ideals G (X) in E (X) which are similar to the E() (X). More precisely, let G (X) be defined as the right-hand side of (2.8) for any ∈ δ(X). Then G (X) is an ideal of E (X) and E() (X) ⊂ G (X) where equality holds if X is a space of class A but the inclusion is strict in general. We say that G is the ghost envelope of E() . Thus for each ultrafilter ∈ δ(X) we may have two distinct contributions to the essential spectrum of H associated to : first the spectrum of the localization .H = H /E() at and second the spectrum of H /G , which is a subset of the first one. In particular, besides the smallest ideal K (X) of E (X) there is a second “small” ideal which appears quite naturally in the theory. This is the ghost ideal defined by

(2.13) G (X) = T ∈ E (X) lim 1Bx (r) T = 0 for all r > 0 . x→∞

1742


The operators T ∈ G (X) vanish everywhere at infinity in the configuration space X but could be not compact. The role of the Property A is to ensure that G (X) = K (X). For discrete metric spaces of bounded geometry, this phenomenon is studied in detail by Chen and Wang, see [7,8,34] and references therein. Proposition 5.10 shows, among other things, that our definition of the ghost ideal in the discrete case coincides with theirs. Observe that in general, if H is an observable affiliated to E (X) then the ghost spectrum of H , i.e. the spectrum of the quotient observable H /G (X), is strictly included in the essential spectrum of H . 3. The elliptic C ∗ -algebra In this section X = (X, d, μ) is a metric space (X, d) equipped with a measure μ and such that: • (X, d) is a locally compact not compact metric space and each closed ball is a compact set, • μ is a Radon measure on X with support equal to X and supx μ(Bx (r)) = V (r) < ∞, ∀r > 0. If k is a controlled kernel let d(k) be the least number r such that d(x, y) > r ⇒ k(x, y) = 0. Recall that

Ctrl X 2 = k : X 2 → C k is a bounded uniformly continuous controlled kernel . (3.14) If k ∈ Ctrl (X 2 ) then Op(k) is the operator on L2 (X) given by (Op(k)f )(x) = From Op(k)2 sup k(x, y) dy · sup k(x, y) dx, x

X k(x, y)f (y) dx.

(3.15)

y

which is the Schur estimate, we get Op(k) V d(k) sup |k|.

(3.16)

¯ x) and (k l)(x, y) = k(x, z)l(z, y) dz. Clearly If k, l ∈ Ctrl (X 2 ) then we denote k ∗ (x, y) = k(y, Op(k)∗ = Op(k ∗ ) and Op(k) Op(l) = Op(k l). The following simple fact is useful. Lemma 3.1. If k, l ∈ Ctrl (X 2 ) then k l ∈ Ctrl (X 2 ), we have d(k l) d(k) + d(l), and

sup |k l| sup |k| · sup |l| · min V d(k) , V d(l) . Proof. If we set s = d(k) and t = d(l) then clearly (k l)(x, y) sup |k| · sup |l| · μ Bx (s) ∩ By (t) which gives both estimates from the statement of the lemma. To prove the uniform continuity we use


(k l)(x, y) − (k l) x , y supk(x, z) − k x , z

1743

l(z, y) dz

z

supk(x, z) − k x , z · sup |l| · V (t) z

and a similar inequality for |(k l)(x, y) − (k l)(x, y )|.

2

Thus Ctrl (X 2 ), when equipped with the usual linear structure and the operations k ∗ and k l, becomes a ∗-algebra and k → Op(k) is a morphism into B(X) hence its range is a ∗-subalgebra of B(X). Hence the elliptic algebra E (X) defined in (2.5) is a C ∗ -algebra of operators on L2 (X). The uniform continuity assumption involved in the definition (3.14) of Ctrl (X) hence in that of E (X) is important because thanks to it we have E (X) = C(X) r X if X is a unimodular locally compact group, cf. Sections 6 and 7. Here C(X) is the C ∗ -algebra of right uniformly continuous functions on X on which X acts by left translations and r denotes the reduced crossed product. In particular, the equality C(X) r X = E (X) gives a description of the crossed product independent of the group structure of X. We say that T ∈ B(X) is a controlled operator if there is r > 0 such that if F, G are closed subsets of X with d(F, G) > r then 1F T 1G = 0; let d(T ) be the smallest r for which this holds (see [30]; this class of operators has also been considered in [12] and in [14]). Observe that the Op(k) with k ∈ Ctrl (X 2 ) are controlled operators but if X is not discrete then there are many others and most of them do not belong to E (X). The norm closure of the set of controlled operators will be discussed in Section 7. Since the kernel of ϕ(Q) Op(k) is ϕ(x)k(x, y) and that of Op(k)ϕ(Q) is k(x, y)ϕ(y), we clearly have C(X)E (X) = E (X)C(X) = E (X). This defines a C(X)-bimodule structure on E (X). We note that, as a consequence of the Cohen– Hewitt theorem, if A is a C ∗ -subalgebra of C(X) then the set AE(X) consisting of products AT of elements A ∈ A and T ∈ E (X) is equal to the closed linear subspace of E (X) generated by these products. Proposition 3.2. We have K (X) = Co (X)E (X) = E (X)Co (X) ⊂ E (X). Proof. If ϕ ∈ Cc and k ∈ Ctrl then the operator ϕ Op(k) has kernel ϕ(x)k(x, y) which is a continuous function with compact support on X 2 , hence ϕ Op(k) is a Hilbert–Schmidt operator. Thus we have Co (X)E (X) ⊂ K (X) and by taking adjoints we also get E (X)Co (X) ⊂ K (X). Conversely, an operator with kernel in Cc (X 2 ) clearly belongs to Cc (X)E (X) for example. 2 E (X) is a non-degenerate Co (X)-bimodule and there is a natural topology associated to such a structure, we call it the local topology on E (X). Its utility will be clear from Section 6. Definition 3.3. The local topology on E (X) is the topology associated to the family of seminorms T θ = T θ (Q) + θ (Q)T with θ ∈ Co (X). This is the analog of the topology of local uniform convergence on C(X). Obviously one may replace the θ with 1Λ where Λ runs over the set of compact subsets of X. If T ∈ E (X) and

1744


{Tα } is a net of operators in E (X) we write Tα → T or limα Tα = T locally if the convergence takes place in the local topology. Since X is σ -compact there is θ ∈ Co (X) with θ (x) > 0 for all x ∈ X and then · θ is a norm on E (X) which induces on bounded subsets of E (X) the local topology. The local topology is finer than the ∗-strong operator topology inherited from the embedding E (X) ⊂ B(X). We may also consider on E (X) the (intrinsically defined) strict topology associated to the smallest essential ideal K (X); this is weaker than the local topology and finer than the ∗-strong operator topology, but coincides with the last one on bounded sets. Lemma 3.4. The involution T → T ∗ is locally continuous on E (X). The multiplication is locally continuous on bounded sets. Proof. Since T ∗ θ = T θ¯ the first assertion is clear. Now assume Sα → S locally and Sα C and Tα → T locally. If θ ∈ Co then T θ is a compact operator so there is θ ∈ Co such that T θ = θ K for some compact operator K. Then we write (Sα Tα − ST )θ = Sα (Tα − T )θ + (Sα − S)θ K. 2 The ghost ideal is defined as follows:

G (X) := T ∈ E (X) lim 1Bx (r) T = 0, ∀r x→∞

= T ∈ E (X) lim T 1Bx (r) = 0, ∀r . x→∞

(3.17)

The fact that G is an ideal of E follows from the equality stated above which in turn is proved as follows: for each ε > 0 there is a controlled kernel k such that T − Op(k) < ε hence if R = r + d(k) we have T 1Bx (r) < ε + Op(k)1Bx (r) = ε + 1Bx (R) Op(k)1Bx (r) < 2ε + 1Bx (R) T which is less than 3ε for large x. We have K (X) ⊂ G (X) because limx→∞ 1Bx (r) = 0 strongly on L2 . It is known that the inclusion is strict in general [20, p. 349]. In the rest of this section we prove that equality holds if X is of class A. We begin with some general useful remarks. Lemma 3.5. If (2.4) holds then there a subset Z ⊂ X with X = z∈Z Bz and a function N : R → N such that: for any x ∈ X and r 1 the number of z ∈ Z such that Bz (r) ∩ Bx (r) = ∅ is at most N(r). Proof. Let Z be a maximal subset of X such that d(a, b) > 1 if a, b are distinct points in Z. Then we have X = z∈Z Bz (the contrary would contradict the maximality of Z). Now fix r 1, let x ∈ X, denote Zx the set of z ∈ Z such that Bz (r) ∩ Bx (r) = ∅, and let Nx be the number of elements of Zx . Choose a ∈ Z such that x ∈ Ba . Then Bx (r) ⊂ Ba (r + 1) hence if z ∈ Zx then Bz (r) ∩ Ba (r + 1) = ∅ so d(z, a) 2r + 1. Since the balls Bz (1/2) corresponding to these z are pairwise disjoint and included in Ba (2r + 2), the volume of their union is larger than νNx , where ν = infy∈X μ(By (1/2)), and smaller than V (2r + 2), hence Nx V (2r + 2)/ν. Thus we may take N(r) = V (2r + 2)/ν. 2


1745

From now on, if (2.4) is satisfied, the set Z and the function N will be as in Lemma 3.5. Lemma 3.6. If (2.4) is satisfied and T is a controlled operator, then 1/2 T N d(T ) + 1 sup 1Bx T .

(3.18)

x∈X

Proof. Set R = d(T ) + 1. Then for any f ∈ L2 we have Tf 2

1Bz Tf 2 =

z∈Z

1Bz T 1Bz (R) f 2 sup 1Bz T 2 z∈Z

z∈Z

and from Lemma 3.5 we get

z∈Z 1Bz (R)

N (R).

1Bz (R) f 2

z∈Z

2

Lemma 3.7. Assume that (2.4) is satisfied and let T ∈ B(X). If limx→∞ 1Bx (r) T = 0 holds for r = 1 then it holds for all r > 0. In particular, we have

G (X) = T ∈ E (X) lim 1Bx T = 0 = T ∈ E (X) lim T 1Bx = 0 . x→∞

x→∞

(3.19)

Proof. Let r > 1, ε > 0 and let F be a finite subset of Z such that 1Bz T < ε/N (r) if z ∈ Z \ F . We consider points x such that d(x, F ) > r + 1 and denote Z(x, r) the set of z ∈ Z such that Bz ∩ Bx (r) = ∅. Then Z(x, r) has at most N (r) elements and Bx (r) ⊂ z∈Z(x,r) Bz hence 1Bx (r) T N(r) maxz∈Z(x,r) 1Bz T < ε because F ∩ Z(x, r) = ∅. 2 An operator T ∈ B(X) is called locally compact if for any compact set K the operators 1K T and T 1K are compact. Clearly any operator in E (X) is locally compact. Lemma 3.8. Assume that (2.4) is satisfied. If T ∈ B(X) is a controlled locally compact operator such that 1Bx T → 0 as x → ∞ then T is compact. Proof. Choose o ∈ X and let 1R be the characteristic function of the ball Bo (R). Then 1R T is compact so it suffices to show that 1R T converges in norm to T as R → ∞. Clearly T − 1R T is controlled with d(T − 1R T ) d(T ) hence from Lemma 3.6 we get T − 1R T C sup 1Bx (1 − 1R )T C x∈X

which proves the lemma.

sup

1Bx T

d(x,o)>R−1

2

Now we use an idea from [7] (truncation of kernels with the help of functions of positive type) and the technique of the proof of Theorem 5.1 from [26]. Let H be an arbitrary separable Hilbert space (in Definition 2.1 we took H = L2 (X)) and let φ : X → H be a Borel function such that φ(x) = 1 for all x. Define Mφ : L2 (X) → L2 (X; H) = L2 (X)⊗H by (Mφ f )(x) = f (x)φ(x). Then Mφ is a linear operator with Mφ = 1 and its adjoint Mφ∗ : L2 (X; H) → L2 (X) acts as follows: (Mφ∗ F )(x) = φ(x)|F (x). Let T → Tφ be the linear continuous map on B(X) given by Tφ = Mφ∗ (T ⊗ 1)Mφ . Clearly Tφ T .

1746


Let k : X 2 → C be a locally integrable function. We say that an operator T ∈ B(X) has integral kernel k if f |T g = X2 k(x, y)f¯(x)g(y) dx dy for all f, g ∈ Cc (X). If k is a Schur kernel, i.e. supx X (|k(x, y)| + |k(y, x)|) dy < ∞, then we say that T is a Schur operator and we have the estimate (3.15) for its norm. And T is a Hilbert–Schmidt operator if and only if k ∈ L2 (X 2 ). From the relation f |Tφ g = f φ|T ⊗ 1gφ valid for f, g ∈ Cc (X) we easily get: Lemma 3.9. If T has kernel k then Tφ has kernel kφ (x, y) = φ(x)|φ(y)k(x, y). In particular, if T is a Schur, Hilbert–Schmidt, or compact operator, then Tφ has the same property. Lemma 3.10. Assume that φ(x)|φ(y) = 0 if d(x, y) > r. Then for each T ∈ B(X) the operator Tφ is controlled, more precisely: if F, G are closed subsets of X with d(F, G) > r then 1F Tφ 1G = 0. Proof. We have to prove that 1F f |Tφ 1G g = 0 for all f, g ∈ L2 (X) and T ∈ B(X). The map T → Tφ is continuous for the weak operator topology and the set of finite range operators is dense in B(X) for this topology. Thus it suffices to assume that T is Hilbert–Schmidt (or even of rank one) and then the assertion is clear by Lemma 3.9. 2 Observe that if θ : X → C is a bounded Borel function then Mφ θ (Q) = (θ (Q) ⊗ 1)Mφ hence θ Tφ = (θ T )φ and Tφ θ = (T θ )φ with the usual abbreviation θ = θ (Q). In particular, Lemma 3.9 implies: Lemma 3.11. Let T ∈ B(X). If T is locally compact then Tφ is locally compact. If 1Bx (r) T → 0 as x → ∞, then 1Bx (r) Tφ → 0 as x → ∞. Theorem 3.12. If X is a class A space then K (X) = G (X). Proof. Let T ∈ G (X) and φ as above. Then T is locally compact hence Tφ is locally compact, and we have 1Bx Tφ → 0 as x → ∞ by Lemma 3.11. Moreover, if φ is as in Lemma 3.10 then Tφ is controlled so, by Lemma 3.8, Tφ is compact. Thus it suffices to show that any T ∈ E (X) is a norm limit of operators Tφ with φ of the preceding form. Since T → Tφ is a linear contraction, it suffices to show this for operators of the form T = Op(k) with k ∈ Ctrl (X 2 ). But then T − Tφ is an operator with kernel k(x, y)(1 − φ(x)|φ(y)) hence, if we denote M = sup |k|, d = d(k), from (3.15) we get

1 − φ(x)φ(y) dy.

T − Tφ M sup x

Bx (d)

Until now we did not use the fact that H = L2 (X) in Definition 2.1. If we are in this situation note that we may replace φ(x) by |φ(x)| and then φ(x)|φ(y) is real. More generally, assume that the φ(x) belong to a real subspace of the (abstract) Hilbert space H so that φ(x)|φ(y) is real for all x, y. Then 1 − φ(x)|φ(y) = φ(x) − φ(y)2 /2 so we have T − Tφ (M/2) sup x

Bx (d)

φ(x) − φ(y)2 dy.


1747

Since X has Property A, one may choose φ such that this be smaller than any given number. 2 4. Coarse filters on X and ideals of C(X) 4.1. Filters We recall some elementary facts; for the moment X is an arbitrary set. A filter on X is a nonempty set ξ of subsets of X which is stable under finite intersections, does not contain the empty set, and has the property: G ⊃ F ∈ ξ ⇒ G ∈ ξ . If Y is a topological space and φ : X → Y then limξ φ = y or limx→ξ φ(x) = y means that y ∈ Y and if V is a neighborhood of y then φ −1 (V ) ∈ ξ . The set of filters on X is equipped with the order relation given by inclusion. Then the trivial filter {X} nonempty set F of filters exists: is the smallest filter and the lower bound of any inf F = ξ ∈F ξ . A set F of filters is called admissible if ξ ∈F Fξ = ∅ if Fξ ∈ ξ for all ξ and ξ . If F is admissible then the upper bound sup F exists: Fξ = X but for a finite number of indices this is the set of sets of the form ξ ∈F Fξ where Fξ ∈ ξ for all ξ and Fξ = X but for a finite number of indices ξ . Let β(X) be the set of ultrafilters on X. If ξ is a filter let ξ † be the set of ultrafilters finer than it. Then ξ = inf ξ † . We equip β(X) with the topology defined by the condition: a nonempty subset of β(X) is closed if and only if it is of the form ξ † for some filter ξ . Note that for the trivial filter consisting of only one set we have {X}† = β(X). Then β(X) becomes a compact topological ˇ space, this is the Stone–Cech compactification of the discrete space X, and is naturally identified with the spectrum of the C ∗ -algebra of all bounded complex functions on X. There is an obvious dense embedding X ⊂ β(X), any bounded function ϕ : X → C has a unique continuous extension β(ϕ) to β(X), and any map φ : X → X has a unique extension to a continuous map β(φ) : β(X) → β(X). More generally, if Y is a compact topological space, each map φ : X → Y has a unique extension to a continuous map β(φ) : β(X) → Y . The following simple fact should be noticed: if ξ is a filter and o is a point in Y then limξ φ = o is equivalent to β(φ)|ξ † = o. Indeed, limξ φ = o is equivalent to lim φ = o for any ∈ ξ † (for the proof, observe that if this last relation holds then for each neighborhood V of o the set φ −1 (V ) belongs to for all ∈ ξ † , hence φ −1 (V ) ⊂ ∈ξ † = ξ ). Now assume that X is a locally compact non-compact topological space. Then the Fréchet filter is the set of complements of relatively compact sets; we denote it ∞, so that limx→∞ φ(x) = y has the standard meaning. Let δ(X) = ∞† be the set of ultrafilters finer than it. Thus δ(X) is a compact subset of β(X) and we have δ(X) ⊂ β(X) \ X (strictly in general):

/ . δ(X) = ∈ β(X) if K ⊂ X is relatively compact then K ∈ Indeed, if is an ultrafilter then for any set K either K ∈ or K c ∈ . If we interpret as a character of ∞ (X) then ∈ δ(X) means (ϕ) = 0 for all ϕ ∈ Co (X). 4.2. Coarse filters Now assume that X is a metric space. If F ⊂ X then F¯ is its closure and F c = X \ F its complement. We set dF (x) := infy∈F d(x, y). Note that dF = dF¯ and |dF (x) − dF (y)| d(x, y). If r > 0 let F(r) := {x | d(x, F ) r} = x∈F Bx (r) be the neighborhood “of order r” of F .

1748


If r > 0 we denote F (r) the set of points x such that d(x, F c ) > r. This is an open subset of X included in F and at distance r from the boundary of F (so if F is too thin, F (r) is empty). In other terms, x ∈ F (r) means that there is r > r such that Bx (r ) ⊂ F . In particular, (F (r) )(r) ⊂ F and for an arbitrary pair of sets F, G we have (F ∩ G)(r) = F (r) ∩ G(r) and F ⊂ G ⇒ F (r) ⊂ G(r) . We say that a filter ξ is coarse if for any F ∈ ξ and r > 0 we have F (r) ∈ ξ . We emphasize that this should hold for all r > 0. If for each F ∈ ξ there is r > 0 such that F (r) ∈ ξ then the filter is called round. Equivalently, ξ is coarse if for each F ∈ ξ and r > 0 there is G ∈ ξ such that G(r) ⊂ F and ξ is round if for each F ∈ ξ there are G ∈ ξ and r > 0 such that G(r) ⊂ F . Our terminology is related to the notion of coarse ideal introduced in [21] (our space X being equipped with the bounded metric coarse structure). More precisely, a coarse ideal is a set I of subsets of X such that B ⊂ A ∈ I ⇒ B ∈ I and A ∈ I ⇒ A(r) ∈ I for all r > 0. Clearly I → I c := {Ac | A ∈ I} is a one-one correspondence between coarse ideals and filters. Coarse filters on groups are very natural objects: if X is a group, then a round filter is coarse if and only if it is translation invariant (Proposition 6.6). The Fréchet filter is coarse because if K is relatively compact then K(r) is compact for any r (the function dK is proper under our assumptions on X). The trivial filter {X} is coarse. More general examples of coarse filters are constructed as follows [12,15]. Let L ⊂ X be a set such that L(r) = X for all r > 0. Then the filter generated by the sets Lc(r) = {x | d(x, L) > r} when r runs over the set of positive real numbers is coarse (indeed, it is clear that the L(r) generate a coarse ideal). If L is compact the associated filter is ∞. If X = R and L = ]−∞, 0] then the corresponding filter consists of neighborhoods of +∞ and this example has obvious n-dimensional versions. If L is a sparse set (i.e. the distance between a ∈ L and L \ {a} tends to infinity as a → ∞) then the ideal in C(X) associated to it (cf. below) and its crossed product by the action of X (if X is a group) are quite remarkable objects, cf. [15]. It should be clear however that most coarse filters are not associated to any set L. Let X be an Euclidean space and let G(X) be the set of finite unions of strict vector subspaces of X. The sets Lc(r) when L runs over G(X) and r over R+ form a filter basis and the filter generated by it is the Grassmann filter γ of X. This is a translation invariant hence coarse filter which plays a role in a general version of the N -body problem, see [17, Section 6.5]. The relation limγ ϕ = 0 means that the function ϕ vanishes when we are far from any strict affine subspace. Lemma 4.1. If F is a nonempty set of coarse filters then inf F is a coarse filter. If F is admissible then sup F is a coarse filter. Proof. If F ∈ inf F = ξ ∈F ξ then for any r > 0 and ξ we have F (r) ∈ ξ and so F (r) ∈ ξ ∈F ξ . Now assume for example that F ∈ ξ and G ∈ η with ξ, η ∈ F and let r > 0. Then there are ⊂ F and G ⊂ G hence (F ∩ G ) F ∈ ξ and G ∈ η such that F(r) (r) ⊂ F(r) ∩ G(r) ⊂ F ∩ G. (r) The argument for sets of the form ξ Fξ with Fξ = X but for a finite number of indices ξ is similar. 2 Lemma 4.2. A coarse filter is either trivial, and then ξ † = β(X), or finer than the Fréchet filter, and then ξ † ⊂ δ(X). Proof. Assume that ξ is not finer than the Fréchet filter. Then there is a compact set K such that Kc ∈ / ξ . Hence for any F ∈ ξ we have F ⊂ K c so F ∩ K = ∅. Note that the closed sets in ξ form a basis of ξ (if F ∈ ξ then the closure of F (2) belongs to ξ and is included in F (1) hence in F ).


1749

The set {F ∩ K | F ∈ ξ and is closed} is a filter basis consisting of closed sets in the compact set K hence there is a ∈ K such that a ∈ F for all F ∈ ξ . Then if F ∈ ξ and r > 0 there is G ∈ ξ such that G(r) ⊂ F and since a ∈ G we have Ba (r) ⊂ G(r) ⊂ F . But X = r Ba (r) so X ⊂ F . 2 4.3. Coarse ideals of C(X) We now recall some facts concerning the relation between filters on X and ideals of C(X). To each filter ξ on X we associate an ideal Iξ (X) of C(X):

Iξ (X) := ϕ ∈ C(X) lim ϕ = 0 . ξ

(4.20)

If ξ is the Fréchet filter then limξ ϕ = 0 means limx→∞ ϕ(x) = 0 in the usual sense and so the corresponding ideal is Co (X). The ideal associated to the trivial filter clearly is {0}. We also have: ξ ⊂η

⇒

Iξ (X) ⊂ Iη (X),

Iξ ∩η (X) = Iξ (X) ∩ Iη (X) = Iξ (X)Iη (X).

(4.21) (4.22)

The round envelope ξ ◦ of ξ is the finer round filter included in ξ . Clearly this is the filter generated by the sets F(r) when F runs over ξ and r over R+ . Note that Iξ (X) = Iξ ◦ (X), i.e. for ϕ ∈ C(X) we have limξ ϕ = 0 if and only if limξ ◦ ϕ = 0. Indeed, if ε > 0 let F be the set of points were |ϕ(x)| < ε/2 and let r > 0 be such that |ϕ(x) − ϕ(y)| < ε/2 if d(x, y) r. Then |ϕ(x)| < ε if x ∈ F(r) . We recall a well-known description of the spectrum of the algebra C(X) in terms of round filters. Proposition 4.3. The map ξ → Iξ (X) is a bijection between the set of all round filters on X and the set of all ideals of C(X). An ideal I of C(X) will be called coarse if for each positive ϕ ∈ I and r > 0 there is a positive ψ ∈ I such that d(x, y) r

and ψ(y) < 1

⇒

ϕ(x) < 1.

(4.23)

Lemma 4.4. Let F, G be subsets of X such that G(r) ⊂ F . Then the function θ = dF c (dF c + dG )−1 belongs to C(X) and satisfies the estimates 1G θ 1F and |θ (x) − θ (y)| 3r −1 d(x, y). In particular, a filter ξ is coarse if and only if for any F ∈ ξ and any ε > 0 there is G ∈ ξ and a function θ such that 1G θ 1F and |θ (x) − θ (y)| εd(x, y). Proof. If a ∈ G and b ∈ / F then r < d(a, b) d(x, a) + d(x, b) for any x. By taking the lower bound of the right-hand side over a, b we get r dG (x) + dF c (x) ≡ D(x). Hence if d(x) ≡ dF c (x) then θ (x) − θ (y) |d(x) − d(y)| + d(y) |D(x) − D(y)| D(x) D(x)D(y) d(x, y) d(x, y) + D(x) − D(y) . r 3r

1750


To prove the last assertion, notice that if such a θ exists for some ε < 1/r and if x ∈ G and d(x, y) r then θ (x) = 1 and |θ (x) − θ (y)| < 1 hence θ (y) > 0 so y ∈ F . Thus G(r) ⊂ F . 2 Proposition 4.5. The filter ξ is coarse if and only if the ideal Iξ (X) is coarse. Proof. Assume ξ is not trivial and coarse and let ϕ ∈ Iξ positive and r > 0. Then Oϕ := {ϕ < 1} ∈ ξ hence there is G ∈ ξ such that G(2r) ⊂ Oϕ . By using Lemma 4.4 we construct ψ ∈ C such that 0 ψ 1, ψ|G = 0, and ψ|Gc(r) = 1. Clearly ψ ∈ Iξ . If ψ(y) < 1 then y ∈ G(r) hence if d(x, y) r then x ∈ G(2r) so ϕ(x) < 1. Thus Iξ is coarse. Reciprocally, assume that Iξ is a coarse ideal and let F ∈ ξ and r > 0. There is ϕ ∈ Iξ positive such that Oϕ ⊂ F and there is a positive function ψ ∈ Iξ such that (4.23) holds. But then Oψ ∈ ξ and (Oψ )(r) ⊂ Oϕ so ξ is coarse. 2 4.4. Coarse envelope If ξ is a filter then the family of coarse filters included in ξ is admissible, hence there is a largest coarse filter included in ξ . We denote it co(ξ ) and call it coarse envelope (or cover) of ξ . A set F belongs to co(ξ ) if and only if F (r) ∈ ξ for any r > 0 (the set of such F is a filter, see p. 1748). By Lemma 4.2 we have only two possibilities: either co(ξ ) = {X} or co(ξ ) ⊃ ∞. Since co(ξ ) ⊂ ξ , we see that either ξ is finer than Fréchet, and then co(ξ ) ⊃ ∞, or not, and then co(ξ ) = {X}. To each ultrafilter ∈ β(X) we associate a compact subset ⊂ β(X) by the rule := co()† = set of ultrafilters finer than the coarse envelope of .

(4.24)

Thus we have either ⊂ δ(X), or ∈ / δ(X) and then = β(X). On the other ∈ δ(X) and then hand, we have ∈δ(X) = δ(X) because ∈ . More explicitly, if , χ ∈ δ(X) then χ ∈ means: if F is a set such that F (r) ∈ for all r, then F ∈ χ (which is equivalent to F ∩ G = ∅ for all G ∈ χ ). If is an ultrafilter on X then C() (X) is the coarse ideal of C(X) defined by

C() (X) = Ico() = ϕ ∈ C(X) lim ϕ = 0 . co()

(4.25)

The quotient C ∗ -algebra C (X) = C(X)/C() (X) will be called localization of C(X) at . If ϕ ∈ C(X) then its image in the quotient is denoted .ϕ and is called localization of ϕ at . The next comments give another description of these objects and will make clear that localization means extension followed by restriction. Observe that ϕ ∈ C(X) belongs to C() (X) if and only if the restriction of β(ϕ) to is zero. Hence two bounded uniformly continuous functions are equal modulo C() (X) if and only if their restrictions to are equal. Thus ϕ → β(ϕ)| induces an embedding C (X) → C( ) which allows us to identify C (X) with an algebra of continuous functions on . From this we deduce ∈δ(X)

C() (X) = Co (X).

(4.26)


1751

Indeed, ϕ belongs to the left-hand side if and only if β(ϕ)| = 0 for all ∈ δ(X). But the union of the sets is equal to δ(X) hence this means β(ϕ)|δ(X) = 0 which is equivalent to ϕ ∈ Co (X). A maximal coarse filter is a coarse filter which is maximal in the set of coarse filters equipped with inclusion as order relation. This set is inductive (the union of an increasing set of coarse filters is a coarse filter) hence each coarse filter is majorated by a maximal one. Dually, we say that a subset T ⊂ δ(X) is coarse if it is of the form T = † for some coarse filter . Note that if T is a minimal coarse set then T = for any ultrafilter ∈ T . In general the coarse sets of the form with ∈ δ(X) are not minimal. 5. Ideals of E (X) There are two classes of ideals in E (X) which can be defined in terms of the behavior at infinity of the operators. For any filter ξ on X we define

Jξ (X) = T ∈ E (X) inf 1F T = 0 ,

(5.27)

Gξ (X) = T ∈ E (X) lim 1Bx (r) T = 0, ∀r .

(5.28)

F ∈ξ

x→ξ

Here infF ∈ξ 1F T is the lower bound of the numbers 1F T when F runs over the set of measurable F ∈ ξ and we define infF ∈ξ T 1F similarly. Note that 1F T 1G T and T 1F T 1G if F ⊂ G are measurable. Recall also that limx→ξ 1Bx (r) T = 0 means: for each ε > 0 there is G ∈ ξ such that 1Bx (r) T < ε for all x ∈ G. Observe that for the Fréchet filter ξ = ∞ we have K = J∞

and K ⊂ G∞ = G

(5.29)

where G (X) is the ghost ideal introduced in (3.17). That J∞ = K follows from the fact that 1K T is compact if K is compact (or use (5.30) and Proposition 3.2). The equality G∞ (X) = G (X) is just a change of notation. Lemma 5.1. If T ∈ E and ξ is a coarse filter then infF ∈ξ 1F T = infF ∈ξ T 1F . Proof. If infF ∈ξ 1F T = a and ε > 0 then there is F ∈ ξ such that 1F T < a + ε. We may choose k ∈ Ctrl such that T − Op(k) < ε and then 1F Op(k) < a + 2ε. Assume that k(x, y) = 0 if d(x, y) r and let G ∈ ξ such that G(r) ⊂ F . Then k(x, y)1G (y) = 1G(r) (x)k(x, y)1G (y) hence Op(k)1G = 1G(r) Op(k)1G = 1G(r) 1F Op(k)1G so Op(k)1G 1F Op(k) < a + 2ε and so T 1G < a + 3ε. 2 Lemma 5.2. For any filter ξ the set Gξ is an ideal of E and we have Jco(ξ ) ⊂ Gξ . If ξ is coarse then Jξ is also an ideal of E and Jξ ⊂ Gξ . Proof. Gξ is obviously a closed right ideal in E so it will be an ideal if we show that limx→ξ T 1Bx (r) = 0 for all T ∈ Gξ . Choose ε > and let S be a controlled operator such that S − T < ε. Then there is R such that S1Bx (r) = 1Bx (R) S1Bx (r) and there is F ∈ ξ such that

1752


1Bx (R) T < ε for x ∈ F , hence T 1Bx (r) < ε + S1Bx (r) ε + 1Bx (R) S < 2ε + 1Bx (R) T < 3ε. If T ∈ Jco(ξ ) then for any ε > 0 there is F such that F (r) ∈ ξ for all r such that 1F T < ε. So if we fix r and take G = F (r) ∈ ξ then G ∈ ξ and 1Bx (r) T < ε for all x ∈ G. Thus T ∈ Gξ . Clearly Jξ is a closed right ideal in E . That it is an ideal if ξ is coarse follows from Lemma 5.1. 2 Proposition 5.3. If ξ is a coarse filter on X then Jξ is an ideal of E and we have Jξ = Iξ E = E Iξ .

(5.30)

Proof. We prove the first equality in (5.30) (the second one follows by taking adjoints). Clearly ϕ ∈ Iξ if and only if for each ε > 0 there is F ∈ ξ such that 1F ϕ < ε hence if and only if infF ∈ξ 1F ϕ = 0. This implies Iξ E ⊂ Jξ and so it remains to be shown that for each T ∈ Jξ there are ϕ ∈ Iξ and S ∈ E such that T = ϕS. If ξ is trivial this is clear, so we may suppose that ξ is finer than ∞. Choose a point o ∈ X and let Kn = Bo (n) for n 1 integer. We get an increasing sequence of compact sets such that n Kn = X and Knc ∈ ξ . We construct by induction a sequence F1 ⊃ G1 ⊃ F2 ⊃ G2 ⊃ · · · of sets in ξ such that: Fn ⊂ Knc ,

1Fn T n−2 ,

d Gn , Fnc > 1,

d Fn+1 , Gcn > 1.

We start with F1 ∈ ξ such that 1F1 T 1, we set F1 = F1 ∩ K1c and then we choose G1 ∈ ξ such that d(G1 , F1c ) > 1. Next, we choose F2 ∈ ξ with 1F2 T 1/4 and G 1 ∈ ξ with G 1 ⊂ G1 and d(G 1 , Gc1 ) > 1. We take F2 = F2 ∩ G 1 ∩ K2c , so d(F2 , Gc1 ) > 1, and then we choose G2 ∈ ξ with G2 ⊂ F2 such that d(G2 , F2c ) > 1, and so on. Now we use Lemma 4.4 and for each n we construct a function θn ∈ C such that 1Gn θn 1Fn and |θn (x) − θn (y)| 3d(x, y). Then either Ba ∩ F1 = ∅ or there is a unique m such that Ba ∩ Fm = ∅ and Ba ∩ Fm+1 = ∅ and in this case θn = 1 on Ba if n < m and θn = 0 on Ba if n > m. Let θ (x) = n θn (x). Then θ (x) = 0 on F1c and if Ba ∩ Fm = ∅ and Ba ∩ Fm+1 = ∅ we get θ (x) =

θn (x) = m − 1 + θm (x).

(5.31)

nm

¯ + is well defined and for d(x, y) < 1 and a conveniently chosen m we have Thus θ : X → R θ (x) − θ (y) = θm (x) − θm (y) 3d(x, y). On the other hand θn T 1Fn T n−2 . Thus if θ0 = 1 then the limit of m → ∞ exists in norm and defines an element S of E . Then T=

nm

θn

−1 nm

θn T → (1 + θ )−1 S

nm θn T

as


1753

because ( nm θn )−1 → (1 + θ )−1 strongly on L2 (X). If ϕ := (1 + θ )−1 then 0 ϕ 1 and ϕ(x) − ϕ(y) θ (x) − θ (y) 3d(x, y) if d(x, y) < 1. Thus ϕ ∈ C. If x ∈ Ba with Ba ∩ Fm = ∅ and Ba ∩ Fm+1 = ∅ then (5.31) gives −1 1/m ϕ(x) = 1 + m − 1 + θm (x) hence ϕ(x) 1/m on Fm . Thus limξ ϕ = 0 and T = ϕS with ϕ ∈ I and S ∈ E .

2

We make now more precise the relation between Jξ and Gξ . Lemma 5.4. If (2.4) holds, T ∈ E is controlled, ξ is coarse, and limx→ξ 1Bx T = 0, then T ∈ Jξ . Proof. Assume (2.4) is satisfied and let T ∈ B(X) be a controlled operator. Let Z be as in Lemma 3.5 and let us set a = d(T ) + 1, so that 1Bx T = 1Bx T 1Bx (a) for all x. If F is a measurable set and if we denote Z(F ) the set of z ∈ Z such that Bz ∩ F = ∅ then for any f ∈ L2 (X) we have 1F Tf 2

1Bz Tf 2 =

z∈Z(F )

1Bz T 1Bz (a) f 2

z∈Z(F )

sup 1Bz T

2

z∈Z(F )

1Bz (a) f 2 sup 1Bx T 2 N (a)f 2 x∈F(1)

z∈Z(F )

so 1F T N (a)1/2 supx∈F(1) 1Bx T . Thus for any controlled operator we have infF ∈ξ 1F T = 0 if limx→ξ 1Bx T = 0. If T ∈ E (X) this means T ∈ Jξ . 2 Proposition 5.5. If X is a class A space then for any filter ξ finer than Fréchet we have Jco(ξ ) ⊂ Gξ . If ξ is coarse and T ∈ E then T ∈ Jξ

⇔

lim T 1Bx = 0

x→ξ

⇔

lim 1Bx T = 0.

x→ξ

(5.32)

Proof. We use the same techniques as in the proof of Theorem 3.12. Let T ∈ E (X) and let us assume that limx→ξ T 1Bx = 0. Then as we saw in Section 3 we have (T 1Bx )φ = Tφ 1Bx hence for conveniently chosen φ the operator Tφ ∈ E (X) is controlled and limx→ξ Tφ 1Bx = 0. From Lemma 5.4 we get Tφ ∈ Jξ (X) which is closed, so since Tφ → T in norm as φ → 1, we get T ∈ Jξ (X). 2 Remark 5.6. The relation (5.32) is not true in general if Property A is not satisfied. Indeed, if we take ξ = ∞ then this would mean K = G , which does not hold in general. We now seek for a more convenient description of Jco(ξ ) for not coarse filters. Remark 5.7. The following observations are easy to prove and will be useful below. Let F be any subset of X and let r, s > 0. Then F (r+s) ⊂ (F (r) )(s) and if 0 < r < s then F (s) ⊂ F (r) and F ⊂ (F(s) )(r) .

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Proposition 5.8. Assume that (2.4) is satisfied and let T be a controlled operator and ξ a filter finer than the Fréchet filter. Then infF ∈co(ξ ) 1F T = 0 if and only if limx→ξ 1Bx (r) T = 0 for all r > 0. Proof. If T ∈ B(X) and infF ∈co(ξ ) 1F T = 0 then the first few lines of the proof of Lemma 5.4 give limx→co(ξ ) 1Bx (r) T = 0 for all r > 0, which is more than required. Now let T be a controlled operator and let us set a = d(T ) + 1. If F is a measurable set and Z(F ) is as in the proof of Lemma 5.4 then d(F, Z(F )) 1 hence for any r > 0 we have Bz (r + 1) F(r) ⊂ Z(F )(r+1) = z∈Z(F )

hence for any f ∈ L2 we have 1Bz (r+1) Tf 2 = 1Bz (r+1) T 1Bz (r+a) f 2 1F(r) Tf 2 z∈Z(F )

z∈Z(F )

sup 1Bz (r+1) T 2 z∈Z(F )

1Bz (r+a) f 2 sup 1Bx (r+1) T 2 N (r + a)f 2 . x∈F(1)

z∈Z(F )

If x ∈ F(1) and y ∈ F is such that d(x, y) 1 then Bx (r + 1) ⊂ By (r + 2) hence we obtain 1F(r) T N (r + a)1/2 sup 1Bx (r+2) T .

(5.33)

x∈F

Observe also that for an arbitrary measurable set G we have the estimate 1G T N (a)1/2 sup 1G∩Bx T .

(5.34)

x∈X

This follows from Lemma 3.6 after noticing that d(1G T ) d(T ). Now assume that limx→ξ 1Bx (r) T = 0 for all r > 0 and let us fix ε > o. Then for each r > 0 there is F r ∈ ξ such that 1Bx (r+2) T εN (r + a)−1/2 N (a)−1/2 ,

∀x ∈ F r .

For each f ∈ L2 and each number s > 0 the map x → 1Bx (s) f ∈ L2 is strongly continuous, hence the function x → 1Bx (r+2) T is lower semi-continuous, so we may assume that F r is r ∈ ξ is closed and 1 T εN (a)−1/2 because closed, hence measurable. Then the Gr := F(r) Gr (α)

(α)

of (5.33). Moreover, if α < r then Gr ≡ (Gr )(α) ⊃ F r hence Gr ∈ ξ . Now fix α > 1 and let (α) (α) G = r>α Gr . This is a union of open set hence it is open and contains all the Gr , which (s) ⊃ belong to ξ , hence belongs to ξ . If s > 0 and we choose some r > s + α then G(s) ⊃ (G(α) r ) (α+s) (s) Gr ∈ ξ (Remark 5.7). Thus we see that G ∈ ξ for all s > 0, which means that G ∈ co(ξ ). In order to estimate the norm of 1G T we use (5.34) and observe that if G ∩ Bx = ∅ the there is (α) (α) (α) r > α such that Gr ∩ Bx = ∅ hence Bx ⊂ (Gr )(1) . But it is easy to check that (Gr )(1) ⊂ Gr because α > 1, hence Bx ⊂ Gr , and then 1G∩Bx T 1Bx T 1Gr T εN (a)−1/2 . Finally, from (5.34) we get 1G T ε.

2


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Theorem 5.9. Let X be a class A space and let ξ be a filter finer than Fréchet on X. If T ∈ E then T ∈ Jco(ξ )

⇔ ⇔

lim T 1Bx (r) = 0,

∀r > 0,

lim 1Bx (r) T = 0,

∀r > 0.

x→ξ x→ξ

(5.35)

Proof. This is a repetition of the proof of Proposition 5.5. For example, let limx→ξ 1Bx (r) T = 0 for all r > 0. Since (1Bx (r) T )φ = 1Bx (r) Tφ for all r, we see that for conveniently chosen φ the operator Tφ ∈ E (X) is controlled and limx→ξ Tφ 1Bx (r) = 0 for all r. From Proposition 5.8 we clearly get Tφ ∈ Jco(ξ ) which is closed. So T ∈ Jco(ξ ) because Tφ → T in norm as φ → 1. 2 The ideals of E (X) which are of real interest in our context are defined as follows ∈ δ(X)

⇒

E() (X) := Jco() (X) = T ∈ E (X)

inf 1F T = 0 .

F ∈co()

(5.36)

By Proposition 5.3 this can be expressed in terms of the ideals of C(X) introduced in (4.25) as follows: E() (X) = C() (X)E (X) = E (X)C() (X).

(5.37)

Prof of Theorem 2.5. Assume that T ∈ E() for all ∈ δ(X); we have to show that T is a compact operator (the converse being obvious). If ∈ δ(X) and r > 0 then for any ε > 0 there is F ∈ co() such that 1F T < ε and there is G ∈ such that G(r) ⊂ F , hence for any x ∈ G we have 1Bx (r) T < ε. This proves that limx→ 1Bx (r) T = 0. Now define θ (x) = 1Bx (r) T , we obtain a bounded function on X such that lim θ = 0 for any ∈ δ(X). The continuous extension β(θ ) : β(X) → R has the property β(θ )() = lim θ thus β(θ ) is zero on the compact subset δ(X) = ∞† of β(X) hence we have lim∞ θ = 0 according to a remark from Section 4.1. Thus we have limx→∞ 1Bx (r) T = 0, which means that T belongs to the ghost ideal G . Now the compactness of T follows from Theorem 3.12. 2 We end this section with some remarks on the case of discrete spaces with bounded geometry. Assume that X is an infinite set equipped with a metric d such that the number of points in a ball is bounded by a number independent of the center of the ball. We equip X with the counting measure, so L2 (X) = 2 (X), and embed X ⊂ 2 (X) by identifying x = 1{x} ≡ 1x , so X becomes the canonical orthonormal basis of 2 (X). Then any operator T ∈ B(X) has a kernel kT (x, y) = x|T y and E (X) is the closure of set of T such that x|T y = 0 if d(x, y) > r(T ) (this is the uniform Roe algebra). Observe that for each T ∈ E and each ε > 0 there is an r such that |x|T y| < ε if d(x, y) > r. If ξ is a filter on X and f : X 2 → C we write limx,y→ξ f (x, y) = 0 if for each ε > 0 there is F ∈ ξ such that |f (x, y)| < ε if x, y ∈ F .

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Proposition 5.10. Let X be discrete with bounded geometry. Then if ξ is a filter and T ∈ E we have T ∈ Gξ

⇔

lim

sup y|T z = 0,

x→ξ y,z∈Bx (r)

∀r > 0.

(5.38)

Moreover, if ξ is coarse then

Gξ = T ∈ E lim x|T y = 0 .

(5.39)

x,y→ξ

Proof. By definition, we have T ∈ Gξ if and only if limx→ξ T 1Bx (r) = 0 for all r. Since the norm of the operator T 1y is equal to the norm of the vector T y, we have sup T y T 1Bx (r)

y∈Bx (r)

T y V (r) sup T y.

y∈Bx (r)

y∈Bx (r)

Thus T ∈ Gξ is equivalent to limx→ξ supy∈Bx (r) T y = 0 for all r, in particular the property from the right-hand side of (5.38) is satisfied. Conversely, let T ∈ E satisfying this condition and let ε > 0. Choose an operator S such that S− T < ε and such that x|Sy = 0 if d(x, y) > R for some fixed R. Then we have |Sy|a| z |Sy|z||z|a| S z∈By (R) |z|a| hence z|T y T y2 = y T ∗ T y εT + Sy|T y εT + S εT + SV (R) sup z|T y.

z∈By (R)

z∈By (R)

So for each ε > 0 there are C, R < ∞ with T y2 εT + C supz∈By (R) |z|T y| for all y. Hence:

sup T y2 εT + C z|T y y ∈ Bx (r), z ∈ By (R) y∈Bx (r)

εT + C sup z|T y y, z ∈ Bx (r + R) .

This proves the converse implication in (5.38). Now assume that ξ is coarse. If T is as in the right-hand side of (5.39) then for each ε > 0 there is F ∈ ξ such that |y|T z| < ε if y, z ∈ F and for each r there is G ∈ ξ such that G(r) ⊂ F . Then if x ∈ G we have Bx (r) ⊂ F hence supy,z∈Bx (r) |y|T z| ε so T ∈ Gξ by (5.38). Reciprocally, let T ∈ Gξ and let ε, r > 0. By (5.38), there is F ∈ ξ such that if y, z ∈ Bx (r) for some x ∈ F then |y|T z| ε. Let us choose r such that |y|T z| < ε if d(y, z) > r and let G ∈ ξ such that G(r) ⊂ F . If y, z ∈ G then either d(y, z) > r and then |y|T z| < ε, or d(y, z) r and then |y|T z| < ε because y, z ∈ By (r) and y, z ∈ G ⊂ F . Thus we found G ∈ ξ such that |y|T z| < ε if y, z ∈ G. 2 Finally, for the convenience of the reader we sketch the construction of the ghost projection of Higson, Laforgue, and Skandalis. Note that G (X) is a C ∗ -algebra of operators on 2 (X) independent of the metric of X. Assume that X is a disjoint union of finite sets Xn with 1 n ∞


1757

2 such that the number vn of elements of Xn tends to infinity with n.Then 2 (X) = n 2 (Xn ), the vector en = x∈Xn x/vn is a unit vector in 2 (Xn ), and π := n |en en | is an orthogonal projection in 2 (X) such that x|πy = 0 if x, y belong to different sets Xn and x|πy = vn−2 if x, y ∈ Xn . Thus π is an infinite rank projection and π ∈ G (X). All this is easy, but the choice of the metric is not: for this we refer to p. 348 in [20]. 6. Locally compact groups 6.1. Crossed products In this section we assume that X is a locally compact topological group with neutral element e and μ is a left Haar measure. We write dμ(x) = dx and denote the modular function defined by d(xy) = (y) dx or dx −1 = (x)−1 dx (with slightly formal notations). There are left and right actions of X on functions ϕ defined on X given by (a.ϕ)(x) = ϕ(a −1 x) and (ϕ.a)(x) = ϕ(xa). √ The left and right regular representation of X are defined by λa f = a.f and ρa f = (a)f.a for f ∈ L2 (X). Then λa and ρa are unitary operators on L2 (X) which induce unitary representation of X on L2 (X). These representations commute: λa ρb = ρb λa for all a, b ∈ X. Moreover, for ϕ ∈ L∞ (X) we have λa ϕ(Q)λ∗a = (a.ϕ)(Q) and ρa ϕ(Q)ρa∗ = (ϕ.a)(Q). The convolution of two functions f, g on X is defined by (f ∗ g)(x) =

f (y)g y

−1

x dy =

f xy −1 (y)−1 g(y) dy.

For ψ ∈ L1 (X) let λψ = ψ(y)λy dy ∈ B(X). Then λψ ψL1 and ψ ∗ g = λψ g for g ∈ L2 . We recall the definition of the ∗-algebra L1 (X): the product is the convolution product f ∗ g and the involution is given by f ∗ (x) = (x)−1 f¯(x −1 ); the factor −1 ensures that f ∗ L1 = f L1 . The enveloping C ∗ -algebra of L1 (G) is the group C ∗ -algebra C ∗ (X). The norm closure in B(X) of the set of operators λψ with ψ ∈ L1 (X) is the reduced group C ∗ algebra Cr∗ (X). There is a canonical surjective morphism C ∗ (X) → Cr∗ (X) which is injective if and only if X is amenable. Lemma 6.1. If T ∈ Cr∗ (X) then ρa T = T ρa , ∀a ∈ X. If X is not compact then Cr∗ (X) ∩ K (X) = {0}. Proof. The first assertion is clear because ρa λb = λb ρa . If X is not compact, then ρa → 0 weakly on L2 (X) hence if T ∈ Cr∗ (X) is compact Tf = T ρa f → 0 hence Tf = 0 for all f ∈ L2 (X). 2 In what follows by uniform continuity we mean “right uniform continuity”, so ϕ : X → C is uniformly continuous if for any ε > 0 there is a neighborhood V of e such that xy −1 ∈ V ⇒ |ϕ(x) − ϕ(y)| < ε (see p. 60 in [29]). Let C(X) be the C ∗ -algebra of bounded uniformly continuous complex functions. If ϕ : X → C is bounded measurable then ϕ ∈ C(X) if and only if λa ϕ(Q)λ∗a − ϕ(Q) → 0 as a → e. We consider now crossed products of the form A X where A ⊂ C(X) is a C ∗ -subalgebra stable under (left) translations (so a.φ ∈ A if φ ∈ A; only the case A = C(X) is of interest later).

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We refer to [35] for generalities on crossed products. The C ∗ -algebra A X is the enveloping C ∗ -algebra of the Banach ∗-algebra L1 (X; A), where the algebraic operations are defined as follows: (f ∗ g)(x) = f (y)y.g y −1 x dy, f ∗ (x) = (x)−1 x.f¯ x −1 . Thus C ∗ (X) = C X. If we define Λ : L1 (X; A) → B(X) by Λ(φ) = φ(a)λa da it is easy to check that this is a continuous ∗-morphism hence it extends uniquely to a morphism A X → B(X) for which we keep the same notation Λ. A short computation gives for φ ∈ Cc (X; A) and f ∈ L2 (X) Λ(φ)f (x) = φ x, xy −1 (y)−1 f (y) dy where for an element φ ∈ Cc (X; A) we set φ(x, a) = φ(a)(x). Thus Λ(φ) is an integral operator with kernel k(x, y) = φ(x, xy −1 )(y)−1 or Λ(φ) = Op(k) with our previous notation. The next simple characterization of Λ follows from the density in Cc (X; A) of the algebraic tensor product A ⊗alg Cc (X): there is a unique morphism Λ : A X → B(X) such that Λ(ϕ ⊗ ψ) = ϕ(Q)λψ for ϕ ∈ A and ψ ∈ Cc (X). Here we take φ = ϕ ⊗ ψ with ϕ ∈ A and ψ ∈ Cc (X), so φ(a) = ϕψ(a). Note that the kernel of the operator ϕ(Q)λψ is k(x, y) = ϕ(x)ψ(xy −1 )(y)−1 . The reduced crossed product A r X is a quotient of the full crossed product A X, the precise definition is of no interest here. Below we give a description of it which is more convenient in our setting. As usual, we embed A ⊂ B(X) by identifying ϕ = ϕ(Q) and if M , N are subspaces of B(X) then M · N is the closed linear subspace generated by the operators MN with M ∈ M and N ∈ N . Theorem 6.2. The kernel of Λ is equal to that of A X → A r X, hence Λ induces a canonical embedding A r X ⊂ B(X) whose range is A · Cr∗ (X). This allows us to identify A r X = A · Cr∗ (X). We thank Georges Skandalis for showing us that this is an easy consequence of results from the thesis of Athina Mageira. Indeed, it suffices to take A = A and B = Co (X) in [23, Proposition 1.3.12] by taking into account that the multiplier algebra of Co (X) is Cb (X), and then to use Co (X) X = K (X) (Takai’s theorem, cf. [23, Example 1.3.4]) and the fact that the multiplier algebra of K (X) is B(X). The crossed product of interest here is C(X) r X = C(X) · Cr∗ (X). Obviously we have K (X) = Co (X) r X ⊂ C(X) r X, the first equality being a consequence of Takai’s theorem but also of the following trivial argument: if ϕ, ψ ∈ Cc (X) then the kernel ϕ(x)ψ(xy −1 )(y)−1 of the operator ϕ(Q)λψ belongs to Cc (X 2 ) hence ϕ(Q)λψ is a Hilbert–Schmidt operator. We recall that the local topology on C(X) r X (see Definition 3.3 here and [17, p. 447]) is defined by the family of seminorms of the form T Λ = 1Λ T + T 1Λ with Λ ⊂ X compact. The following is an extension of [17, Proposition 5.9] in the present context (see also pp. 30–31 in the preprint version of [15] and [31]). Recall that any bounded function ϕ : X → C extends to a continuous function β(ϕ) on β(X). If ∈ β(X) we define ϕ : X → C by ϕ (x) = β x −1 ϕ () = lim ϕ(xa). (6.40) a→


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Lemma 6.3. If ϕ ∈ C(X) then for any θ ∈ Co (X) the set {θ ϕ.a | a ∈ X} is relatively compact in Co (X) and the map a → θ ϕa ∈ Co (X) is norm continuous. In particular, for any ∈ β(X) the limit in (6.40) exists locally uniformly in x and we have ϕ ∈ C(X). Proof. By the Ascoli–Arzela theorem, to show the relative compactness of the set of functions of the form θ ϕ.a it suffices to show that the set is equicontinuous. For each ε > 0 there is a neighborhood V of e such that |ϕ(x) − ϕ(y)| < ε if xy −1 ∈ V . Then |ϕ(xa) − ϕ(ya)| < ε for all a ∈ X, which proves the assertion. In particular, lima→ θ ϕ.a exists in norm in Co (X), hence the limit in (6.40) exists locally uniformly in x. Moreover, we shall have |ϕ (x) − ϕ (y)| < ε so ϕ belongs to C(X). Finally, we show that for any compact set K and any ε > 0 there is a neighborhood V of e such that supK |ϕ(xa) − ϕ(x)| < ε for all a ∈ V . For this, let U be an open cover of K such that the oscillation of ϕ over any U ∈ U is < ε and note that there is a neighborhood V of e such that for any x ∈ K there is U ∈ U such that xV ⊂ U (use the Lebesgue property for the left uniform structure). 2 Proposition 6.4. For each T ∈ C(X) r X and each a ∈ X we have τa (T ) := ρa T ρa∗ ∈ C(X) r X and the map a → τa (T ) is locally continuous on X and has locally relatively compact range. For each ultrafilter ∈ β(X) and each T ∈ C(X) r X the limit τ (T ) := lima→ τa (T ) exists in the local topology of C(X) r X. The so defined map τ : C(X) r X → C(X) r X is a morphism uniquely determined by the property τ (ϕ(Q)λψ ) = ϕ (Q)λψ . Proof. If T = ϕ(Q)λψ then ρa T ρa∗ = (ϕ.a)(Q)λψ is an element of C(X) r X and so τa is an automorphism of C(X) r X. If we take ψ with compact support and Λ is a compact set then λψ 1Λ = 1K λψ 1Λ where K = (supp ψ)Λ is also compact. Then τa (T )1Λ = (ϕ.a)(Q)1K λψ 1Λ and the map a → (ϕ.a)(Q)1K is norm continuous, cf. Lemma 6.3. This implies that a → τa (T ) is locally continuous on X for any T . To show that the range is relatively compact, it suffices again to consider the case T = ϕ(Q)λψ with ψ with compact support and to use τa (T )1Λ = (ϕ.a)(Q)1K λψ 1Λ and the relative compactness of the {(ϕ.a)(Q)1K | a ∈ X} established in Lemma 6.3. The other assertions of the proposition follow easily from these facts. 2 6.2. Elliptic C ∗ -algebra Let X be a locally compact non-compact topological group. Since we do not require that X be metrizable, we have to adapt some of the notions used in the metric case to this context. Of course, we could use the more general framework of coarse spaces [30] to cover both situations, but we think that the case of metric groups is already sufficiently general. So the reader may assume that X is equipped with an invariant proper distance d. Our leftist bias in Section 6.1 forces us to take d right invariant, i.e. d(x, y) = d(xz, yz) for all x, y, z. If we set |x| = d(x, e) then we get a function | · | on X such that |x −1 | = |x|, |xy| |x| + |y|, and d(x, y) = |xy −1 |. The balls B(r) defined by relations of the form |x| r are a basis of compact neighborhoods of e, a function on X is d-uniformly continuous if and only if it is right uniformly continuous, etc. Note that Bx (r) = B(r)x so in the non-metrizable case the role of the balls Bx (r) is played by the sets V x with V compact neighborhoods of e. Recall that the range of the modular function is a subgroup of the multiplicative group ]0, ∞[ hence it is either {1} or unbounded. Since μ(V x) = μ(V )(x) our assumption (2.3) is satisfied only if X is unimodular and in this case we have μ(V x) = μ(V ) for all x.

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We emphasize the importance of the condition that the metric be proper. Fortunately, it has been proved in [18] that a locally compact group is second countable if and only if its topology is generated by a proper right invariant metric. For coherence, in the non-metrizable case we are forced to say that a kernel k : X 2 → C is / K. The symbol d(k) controlled if there is a compact set K ⊂ X such that k(x, y) = 0 if xy −1 ∈ should be defined now as the smallest compact set K with the preceding property. On the other hand, k is uniformly continuous if it is right uniformly continuous, i.e. if for any ε > 0 there is a neighborhood V of e such that |k(ax, by) − k(x, y)| < ε for all a, b ∈ V and x, y ∈ X. Then the Schur estimate (3.15) gives Op(k) sup |k| supa μ(Ka) so only if X is unimodular we have a simple estimate Op(k) μ(K) sup |k|. To summarize, if X is unimodular then Ctrl (X 2 ) is well defined and Lemma 3.1 remains valid if we set V (d(k)) = μ(d(k)) so we may define the elliptic algebra E (X) as in (2.5). But in fact, what we get is just a description of the crossed product C(X) r X independent of the group structure of X: Proposition 6.5. If X is unimodular then E (X) = C(X) r X = C(X) · Cr∗ (X). Proof. From the results presented in Section 6.1 and the fact that = 1 we get that C(X) X is the closed linear space generated by the operators Op(k) with kernels k(x, y) = ϕ(x)ψ(xy −1 ), where ϕ ∈ C(X) and ψ ∈ Cc (X). Thus C(X) X ⊂ E (X). To show the converse, let k ∈ Ctrl (X 2 ) and let k(x, y) = k(x, y −1 x) hence k(x, y) = k(x, xy −1 ). If K = K −1 ⊂ X is a compact set such −1 that k(x, y) = 0 ⇒ xy ∈ K then supp k ⊂ X × K. Fix ε > 0 and let V be a neighborhood of the originsuch that | k(x, y) − k(x, z)| < ε if yz−1 ∈ V . Then let Z ⊂ K be a finite set such that K ⊂ z∈Z V z and let {θz } be a partition of unity subordinated to this covering. If l(x, y) = (y) or l = then k(x, z)θ k(·, z) ⊗ θ z z z∈Z z∈Z k(x, y) − l(x, y) = k(x, y) − k(x, z) θz (y) k(x, y) − k(x, z)θz (y) ε z∈Z

z∈Z

l(x, xy −1 )= z∈Z because supp θz ⊂ V z. Now let us set l(x, y) = k(x, z)θz (xy −1 ). If l(x, y) = 0 then θz (xy −1 ) = 0 for some z hence xy −1 ∈ V z ⊂ V K. In this construction we may choose V ⊂ U where U is a fixed compact neighborhood of the origin. Then we will have l(x, y) = 0 ⇒ xy −1 ⊂ U K which is a compact set independent of l and from (3.16) we get Op(k) − Op(l) C sup |k − l| Cε for some constant C independent of ε. But clearly Op(l) ∈ C(X) r X. 2 Thus if X is a unimodular group then we may apply Proposition 6.4 and get endomorphisms τ of E (X) indexed by ∈ δ(X). These will play an important role in the next subsection. We make now some comments on the relation between amenability and Property A in the case of groups. First, the Property A is much more general than amenability, cf. the discussion in [24] for the case of discrete groups. To show that amenability implies Property A we choose from the numerous known equivalent descriptions that which is most convenient in our context [25, p. 128]: X is amenable if and only if for any ε > 0 and any compact subset K of X there is a positive function ϕ ∈ Cc (X) with ϕ = 1 such that ρa ϕ − ϕ < ε for all a ∈ K. Now let us set φ(x) = ρx∗ ϕ, so φ(x)(z) = (x)−1/2 ϕ(zx −1 ). We get a strongly continuous function φ : X → L2 (X) such that φ(x) = 1, supp φ(x) = (supp ϕ)x, and φ(x) − φ(y) = ρxy −1 ϕ − ϕ ε if xy −1 ∈ K. In the metric case we get a function as in Definition 2.1, so the metric version of the Property A is satisfied.


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6.3. Coarse filters in groups A filter ξ on a locally compact non-compact group X is called round if the sets of the form V G = {xy | x ∈ V , y ∈ G}, where V runs over the set of neighborhoods of e and G over ξ , are a basis of ξ . And ξ is (left) invariant if x ∈ X, F ∈ ξ ⇒ xF ∈ ξ . Naturally, ξ is coarse if for any F ∈ ξ and any compact set K ⊂ X there is G ∈ ξ such that KG ⊂ F . The simplicity of the next proof owes much to a discussion with H. Rugh. In our initial argument Proposition 6.6 was a corollary of Proposition 4.5. Proposition 6.6. A filter is coarse if and only if it is round and invariant. Proof. Note first that ξ is invariant if and only if for each H ∈ ξ and each finite N ⊂ X there is G ∈ ξ such that H ⊃ N G. This is clear because N G ⊂ H is equivalent to G ⊂ x∈N x −1 H . Now assume that ξ is also round. Then for any F ∈ ξ there is a neighborhood V of e and a set H ∈ ξ such that F ⊃ V H . If K is any compact set then there is a finite set N such that V N ⊃ K. Then there is G ∈ ξ such that H ⊃ N G. So F ⊃ V N G ⊃ KH . 2 Proposition 6.7. Let X be unimodular and let ξ be a coarse filter. Then for any T ∈ Jξ (X) we have lima→ξ τa (T ) = 0 locally. If X is amenable then the converse assertion holds, so

Jξ (X) = T ∈ E (X) lim τa (T ) = 0 locally a→ξ

= T ∈ E (X) τ (T ) = 0, ∀ ∈ ξ † .

(6.41)

Moreover, if X is amenable then for any compact neighborhood V of e and any T ∈ E (X) we have T ∈ Jξ (X)

⇔

lim T 1V a = 0

a→ξ

⇔

lim τa (T )1V = 0.

a→ξ

(6.42)

Proof. We have 1V a (Q) = ρa∗ 1V (Q)ρa hence T 1V a = T ρa∗ 1V (Q)ρa = τa (T )1V (Q) hence for T ∈ Jξ (X) we have lima→ξ τa (T ) = 0 locally. If X is amenable then Proposition 5.5 in the metric case and a suitable modification in the non-metrizable group case gives (6.41). Then (6.42) is easy. 2 Theorem 6.8. Let X be a unimodular amenable locally compact group. Then for each ∈ δ(X) and for each T ∈ E (X) the limit τ (T ) := lima→ ρa T ρa∗ exists in the local topology of E (X), in particular in the strong operator topology of B(X). The maps τ are endomorphisms of E (X) and χ∈δ(X) ker τχ = K (X). In particular, the map T → (τ (T )) is a morphism E (X) → ∈δ(X) E (X) with K (X) as kernel, hence the essential spectrum of any normal oper ator H ∈ E (X) or any observable H affiliated to E (X) is given by Spess (H ) = Sp(τ (H )). Proof. We have seen in Section 4.4 that E() (X) =

χ∈

= δ(X) ∈δ(X)

ker τχ

and from (6.41) we get

for each ∈ δ(X).

(6.43)

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On the other hand, we have shown before that Property A, hence of amenability. 2

∈δ(X) E() (X)

= K (X) is a consequence of

Remark 6.9. Recall that after (2.9) we defined the localization .T at ∈ δ(X) of some T ∈ E as the quotient of T in E = E /E() . If T is normal then from (6.43) we get Sp(.T ) = χ∈ Sp(τχ (T )) but many of the operators τχ (T ) which appear here are unitary equivalent, in particular have the same spectrum. Indeed, note that there is a natural (left) action of X on β(X) which leaves δ(X) invariant and is the minimal closed invariant subset of δ(X) which contains . And if χ ∈ δ(X) and a ∈ X then by using aχ = limb→χ ab we get τaχ (T ) = ρa τχ (T )ρa∗ . 7. Quasi-controlled operators In this section we describe briefly other C ∗ -algebras of operators which are analogs of E (X). We emphasize that our choice of E (X) was determined by our desire to mimic the crossed product C(X) X which is a very natural object in the abelian group case, but there are of course many other possibilities. For example, we could allow bounded Borel (instead of uniformly continuous) kernels in (3.14). The C ∗ -algebra generated by such kernels is strictly larger than E (even if we require the kernels to be continuous, see Example 7.2) but an analogue of Theorem 2.5 remains true. It is not clear to us if this algebra is really significant in applications, the set of observables affiliated to E being already very large. We now consider the C ∗ -algebra obtained as norm closure of the set of controlled operator. This notion has been introduced in the metric case in Section 3 but in fact it makes sense in the general framework of coarse spaces X and geometric Hilbert X-modules [30]. In particular, if X is a locally compact group an operator T ∈ B(X) is controlled if there is a compact set Λ ⊂ X such that if F, G are closed subsets of X with F ∩ (ΛG) = ∅ then 1F T 1G = 0. If X is a metric group with a metric as in Section 6.2 this is equivalent to the definition of Section 3. We denote C (X) the norm closure of the set of controlled operators and we call quasi-controlled operators its elements. If X is a proper metric space this is the “standard algebra” from [12]. If X is a discrete metric space with bounded geometry then C (X) = E (X) is the “uniform Roe C ∗ -algebra” from [30,7,8,34]. Clearly C (X) ⊃ E (X). One may define analogs of the ideals Jξ and Gξ . Indeed, form the proof of Lemma 5.1 it follows that if ξ is a coarse filter on X then the set Jξ (X) of T ∈ C (X) such that infF ∈ξ 1F T = 0 is an ideal of C (X). And if ξ is an arbitrary filter then the set Gξ (X) of T ∈ C (X) such that limx→ξ 1Λx T = 0 for each compact set Λ is also an ideal of C (X). But if X is not discrete this class of ideals is too small to allow one to describe the quotient C (X)/K (X) even in simple cases. For example, if X = R then the operators in C may have an anisotropic behavior in momentum space (see Proposition 7.4 and [16]). In order to clarify the difference between E (X) and C (X) we consider the case when X is an abelian group. We first recall a result from [17]. Let X ∗ be the dual group and for p ∈ X ∗ let νp be the unitary operator on L2 (X) given by (νp f )(x) = p(x)f (x). To any Borel function ψ on X ∗ we associate an operator ψ(P ) = F −1 Mψ F on L2 (X), where Mψ is the operator of multiplication by ψ on L2 (X ∗ ) and F is the Fourier transformation. Proposition 7.1. If X is an abelian group then E (X) = C(X) X = C(X) r X is the set of operators T ∈ B(X) such that νp T νp∗ − T → 0 and (λa − 1)T (∗) → 0 if p → e in X ∗ and a → e in X.


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The equality E (X) = C(X) X has been proved before in a more general setting. Proposition 7.1 gives in fact a description of the crossed product C(X) X if X is abelian. If we accept it, then we get the following easy proof of the inclusion E (X) = C(X) X. The operators νp Op(k)νp∗ and λa Op(k) have kernels p(x)k(x, y)p(y) ¯ = p(xy −1 )k(x, y) and k(xa −1 , y). Hence from (3.16) we get νp Op(k)ν ∗ − Op(k) sup p xy −1 − 1k(x, y)μ(K) p xy −1 ∈K

which tends to zero as p → e in X ∗ . Similarly (λa − 1) Op(k) → 0 as a → e in X. Hence Op(k) ∈ C(X) X for each k ∈ Ctrl (X 2 ). The next example shows the role played by the uniform continuity condition in the definition of E (X). Example 7.2. If X = R then we identify X ∗ = R by setting p(x) = eipx . Then the elliptic algebra can be described in very simple terms. Indeed, if λa , νa are the unitary operators on L2 (R) given by (λa f )(x) = f (x − a) and (νa f )(x) = eiax f (x), we have

E (R) = T ∈ B(R) (λa − 1)T (∗) → 0 and νa T νa∗ − T → 0 as a → 0 . Here T (∗) means that the relation holds for T and T ∗ . If we take k(x, y) = ϕ(x)θ (x − y) with ϕ ∈ C(R) and θ ∈ Cc (R) then Op(k) = ϕ(Q)ψ(P ) ∈ E (R) with ψ the Fourier transform (conveniently normalized) of θ . The advantage now is that we can see what happens if ϕ is only bounded and continuous. Then it is easy to check that ϕ(Q)ψ(P ) ∈ E (R) if and only if 2 (ϕ(Q + a) − ϕ(Q))ψ(P ) → 0 when a → 0. For example, if ϕ(x) = eix the last condition is equivalent to (eiaQ − 1)ψ(P ) → 0, which is equivalent to ψ(P ) = η(Q)S for some η ∈ Co (R) and S ∈ B(R). But then ψ(P ) is compact as a norm limit of operators of the form ζ (Q)ψ(P ) with ζ ∈ Co (R), which is not true if ψ = 0. Thus, the operator associated to a kernel of the form 2 k(x, y) = eix θ (x − y) with θ ∈ Cc∞ (R) and not zero does not belong to E (R). To describe C (X), we need an analogue of Lemma 3.5 in the group context. Lemma 7.3. Let ω be a compact neighborhood of e and Z a maximal ω-separated subset of X (i.e. if a, b are distinct elements of Z then (ωa) ∩ (ωb) = ∅). Then for any compact set K ⊃ ω−1 ω we have KZ = X and for any a ∈ Z the number of z ∈ Z such that (Kz) ∩ (Ka) = ∅ is at most μ(ωK −1 K)/μ(ω). Proof. That such maximal Z exist follows from Zorn lemma. By maximality, (ωx) ∩ (ωZ) = ∅ for any x, hence x ∈ ω−1 ωZ, so X = KZ if K ⊃ ω−1 ω. Now fix a ∈ Z and let N be the number of points z ∈ Z such that (Kz) ∩ (Ka) = ∅. For each such z we have z ∈ K −1 Ka hence ωz ⊂ ωK −1 Ka. But the sets ωz are pairwise disjoint and have the same measure μ(ω) so Nμ(ω) μ(ωK −1 Ka) = μ(ωK −1 K). 2 If X is an abelian group then a Q-regular operator is an operator T ∈ B(X) which satisfies only the first condition from Proposition 7.1, i.e. is such that the map p → νp T νp∗ is norm continuous. These operators form a C ∗ -algebra which contains E (X), strictly if X is not discrete,

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which seems to depend on the group structure of X. But in fact this is not the case, it depends only on the coarse structure of X. Proposition 7.4. If X is an abelian group then C (X) = {T ∈ B(X) | limp→e νp T νp∗ − T = 0}. For the proof, it suffices to use [14, Propositions 4.11 and 4.12] (arXiv version) and Lemma 7.3. Now let L C (X) be the set of locally compact operators in C (X). Obviously L C is a C ∗ algebra and E ⊂ L C ⊂ C strictly in general. Indeed, let X be an abelian group, ϕ a bounded continuous function on X, and ψ ∈ C(X ∗ ). Then φ(Q)ψ(P ) belongs to C but not to L C in general, and if ψ ∈ Co (X ∗ ) it belongs to L C but not to C in general, cf. Example 7.2. Note that an operator T ∈ C is locally compact if and only if lima→e λa T (∗) = T (∗) in the local topology of C . Finally, we mention another C ∗ -algebra which is of a similar nature to C (X) and makes sense and is useful in the context of arbitrary locally compact spaces X and arbitrary geometric Hilbert X-modules, see [14,30]. Let us say that S ∈ B(H) is quasilocal (or “decay preserving”) if for each ϕ ∈ Co (X) there are operators S1 , S2 ∈ B(H) and functions ϕ1 , ϕ2 ∈ Co (X) such that Sϕ(Q) = ϕ1 (Q)S1 and ϕ(Q)S = S2 ϕ2 (Q). The set of quasilocal operators is a C ∗ -algebra which contains strictly C (X) if X is a locally compact non-compact abelian group. Indeed, if ψ ∈ L∞ (X ∗ ) has compact support then ψ(P ) is quasilocal (because ψ(P )ϕ(Q) and ϕ(Q)ψ(P ) are compact) but it belongs to C (X) if and only if ψ is continuous. Acknowledgments I am grateful to Hans-Henrik Rugh, Armen Shirikyan and Georges Skandalis, several discussions with them were very helpful. References [1] W. Amrein, A. Boutet de Monvel, V. Georgescu, C0 -Groups, Commutator Methods and Spectral Theory of N -Body Hamiltonians, Birkhäuser, 1996. [2] J. Bellissard, Gap labelling theorems for Schrödinger operators, in: J.M. Luck, P. Moussa, M. Waldschmidt (Eds.), From Number Theory to Physics, Les Houches, 1989, Springer, 1993, pp. 538–630. [3] J. Bellissard, Non Commutative Methods in Semiclassical Analysis, Lecture Notes in Math., vol. 1589, Springer, 1994. [4] A. Boutet de Monvel, V. Georgescu, Graded C ∗ -algebras in the N -body problem, J. Math. Phys. 32 (1991) 3101– 3110. [5] A. Boutet de Monvel, V. Georgescu, Graded C ∗ -algebras associated to symplectic spaces and spectral analysis of many channel Hamiltonians, in: Dynamics of Complex and Irregular Systems, Bielefeld, 1991, in: Bielefeld Encount. Math. Phys., vol. 8, Oxford Science Publications, River Edge, NJ, 1993, pp. 22–66. [6] S.N. Chandler-Wilde, M. Lindner, Limit operators, collective compactness, and the spectral theory of infinite matrices, available at http://www.reading.ac.uk/maths/research/maths-preprints.aspx. [7] X. Chen, Q. Wang, Ideal structure of uniform Roe algebras of coarse spaces, J. Funct. Anal. 216 (2004) 191–211. [8] X. Chen, Q. Wang, Ghost ideal in uniform Roe algebras of coarse spaces, Arch. Math. 84 (2005) 519–526. [9] H.O. Cordes, Spectral Theory of Linear Differential Operators and Comparison Algebras, Cambridge University Press, 1987. [10] M. Damak, V. Georgescu, Self-adjoint operators affiliated to C ∗ -algebras, Rev. Math. Phys. 16 (2004) 257–280, this is part of 99–481 at http://www.ma.utexas.edu/mp_arc/. [11] M. Damak, V. Georgescu, On the spectral analysis of many-body systems, J. Funct. Anal. (February 2010), preprint, available at arXiv:0911.5126v1, http://arxiv.org.


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[12] E.B. Davies, Decomposing the essential spectrum, J. Funct. Anal. 257 (2009) 506–536, http://arxiv.org/abs/ 0809.5130. [13] V. Georgescu, On the spectral analysis of quantum field Hamiltonians, J. Funct. Anal. 245 (2007) 89–143, preprint, available at arXiv:math-ph/0604072v1, http://arXiv.org. [14] V. Georgescu, S. Golénia, Decay preserving operators and stability of the essential spectrum, J. Operator Theory 59 (2008) 115–155, a more detailed version is http://arxiv.org/abs/math/0411489. [15] V. Georgescu, A. Iftimovici, Crossed products of C ∗ -algebras and spectral analysis of quantum Hamiltonians, Comm. Math. Phys. 228 (2002) 519–560, see also 00-521 at http://www.ma.utexas.edu/mp_arc/. [16] V. Georgescu, A. Iftimovici, C ∗ -algebras of quantum Hamiltonians, in: J.-M. Combes, J. Cuntz, G.A. Elliot, G. Nenciu, H. Siedentop, S. Stratila (Eds.), Operator Algebras and Mathematical Physics, Proceedings of the Conference Operator Algebras, Mathematical Physics, Constanta, 2001, Theta, 2003, pp. 123–167, and preprint 02-410 at http://www.ma.utexas.edu/mp_arc/. [17] V. Georgescu, A. Iftimovici, Localizations at infinity and essential spectrum of quantum Hamiltonians: I. General theory, Rev. Math. Phys. 18 (2006) 417–483, see also http://arxiv.org/abs/math-ph/0506051. [18] U. Haagerup, A. Przybyszewska, Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces, preprint, available at http://www.imada.sdu.dk/haagerup/, 2006. [19] B. Helffer, A. Mohamed, Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique, Ann. Inst. Fourier (Grenoble) 38 (2) (1988) 95–112. [20] N. Higson, V. Laforgue, G. Skandalis, Counterexamples to the Baum–Connes conjecture, Geom. Funct. Anal. 12 (2002) 330–354. [21] N. Higson, E.K. Pedersen, J. Roe, C ∗ -algebras and controlled topology, K-Theory 11 (1997) 209–239. [22] Y. Last, B. Simon, The essential spectrum of Schrödinger, Jacobi, and CMV operators, J. Anal. Math. 98 (2006) 183–220, and preprint 05-112 at http://www.ma.utexas.edu/mp_arc/. [23] A. Mageira, C ∗ -algèbres graduées par un semi-treillis, thèse Université Paris 7, Février 2007, also available as preprint number arXiv:0705.1961v1 at http://arxiv.org. [24] P. Nowak, G. Yu, What is Property A? Notices Amer. Math. Soc. 55 (2008) 474–475. [25] A.T. Paterson, Amenability, Math. Surveys Monogr., vol. 29, Amer. Math. Soc., Providence, RI, 1988. [26] G. Pisier, Similarity Problems and Completely Bounded Maps, second ed., Lecture Notes in Math., vol. 1618, Springer, 2001. [27] V.S. Rabinovich, S. Roch, J. Roe, Fredholm indices of band-dominated operators, Integral Equations Operator Theory 49 (2004) 221–238. [28] V.S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory, Oper. Theory Adv. Appl., vol. 150, Birkhäuser, 2004. [29] H. Reiter, J. Stegman, Classical Harmonic Analysis and Locally Compact Groups, Oxford Science Publications, 2000. [30] J. Roe, Lectures on Coarse Geometry, Am. Math. Soc., 2003. [31] J. Roe, Band-dominated Fredholm operators on discrete groups, Integral Equations Operator Theory 51 (2005) 411–416. [32] G. Skandalis, J.-L. Tu, G. Yu, The coarse Baum–Connes conjecture and groupoids, Topology 41 (2002) 807–834. [33] J.-L. Tu, Remarks on Yu’s Property A for discrete metric spaces and groups, Bull. Soc. Math. France 129 (2001) 115–139. [34] Q. Wang, Remarks on ghost projections and ideals in the Roe algebras of expander sequences, Arch. Math. 89 (2007) 459–465. [35] D.P. Williams, Crossed Products of C ∗ -Algebras, Amer. Math. Soc., 2007. [36] G. Yu, The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert spaces, Invent. Math. 139 (2000) 201–240.


Two-body threshold spectral analysis, the critical case Erik Skibsted a,∗ , Xue Ping Wang b,1 a Institut for Matematiske Fag, Aarhus Universitet, Ny Munkegade 8000 Aarhus C, Denmark b Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, 44322 Nantes Cedex, France

Received 9 June 2010; accepted 15 December 2010 Available online 22 December 2010 Communicated by J. Bourgain

Abstract We study in dimension d 2 low-energy spectral and scattering asymptotics for two-body d-dimensional Schrödinger operators with a radially symmetric potential falling off like −γ r −2 , γ > 0. We consider angular momentum sectors, labelled by l = 0, 1, . . . , for which γ > (l + d/2 − 1)2 . In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift. © 2010 Elsevier Inc. All rights reserved. Keywords: Threshold spectral analysis; Schrödinger operator; Critical potential; Phase shift

Contents 1. 2.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Model operator and construction of model resolvent . . . . . . 2.2. Asymptotics of model resolvent . . . . . . . . . . . . . . . . . . . . Asymptotics for full Hamiltonian, compactly supported perturbation 3.1. Construction of resolvent . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Asymptotics of resolvent . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. d-dimensional Schrödinger operator . . . . . . . . . . . . . . . . .

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E-mail addresses: [email protected] (E. Skibsted), [email protected] (X.P. Wang). 1 Supported in part by the French National Research Agency under the project No. ANR-08-BLAN-0228-01.


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4. Asymptotics for full Hamiltonian, more general perturbation . . . . . . . . 5. Regular positive energy solutions and asymptotics of phase shift . . . . . . 6. Asymptotics of physical phase shift for a potential like −γ χ (r > 1)r −2 . Appendix A. Regular zero energy solutions . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Case (d, l) = (2, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The low-energy spectral and scattering asymptotics for two-body Schrödinger operators depend heavily on the decay of the potential at infinity. The most well-studied class is given by potentials decaying faster than r −2 (see for example [6] and references there). The expansion of the resolvent is in this case in terms of powers of dimension-dependent modifications of the spectral parameter and it depends on possible existence of zero-energy bound states and/or zeroenergy resonance states. Classes of negative potentials decaying slower than r −2 were studied in [5,7,16]. In that case the resolvent is more regular at zero energy. It has an expansion in integer powers of the spectral parameter and there are no zero-energy bound states nor resonance states. Moreover, the nature of the expansion is “semi-classical”. For general perturbations of critical decay of the order r −2 and with an assumption related to the Hardy inequality, the threshold spectral analysis is carried out in [13,14]. It is shown that for this class of potentials the zero resonance may appear in any space dimension with arbitrary multiplicity. Recall that for potentials decaying faster than r −2 , the zero resonance is absent if the space dimension d is bigger than or equal to five and its multiplicity is at most one when d is equal to three or four. The goal of this paper is to treat a class of radially symmetric potentials decaying like −γ r −2 at infinity, where γ > 0 is big such that the condition used in [13,14] is not satisfied. In this case, there exist infinitely many negative eigenvalues (see (1.3) for a precise condition). We will give a resolvent expansion as well as an asymptotic formula for the phase shift. These expansions are to our knowledge not semi-classical even though there are common features with the more slowly decaying case. Consider for d 2 the d-dimensional Schrödinger operator H v = (− + W )v = 0, for a radial potential W = W (|x|) obeying Condition 1.1. 1) 2) 3) 4)

W (r) = W1 (r) + W2 (r); W1 (r) = − rγ2 χ(r > 1) for some γ > 0, W2 ∈ C(]0, ∞[, R), ∃1 , C1 > 0: |W2 (r)| C1 r −2−1 for r > 1, ∃2 , C2 > 0: |W2 (r)| C2 r 2 −2 for r 1.

Here the function χ(r > 1) is a smooth cutoff function taken to be 1 for r 2 and 0 for r 1 (see the end of this introduction for the precise definition). Under Condition 1.1 H is self-adjoint as defined in terms of the Dirichlet form on H 1 (Rd ). Let Hl , l = 0, 1, . . . , be the corresponding

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reduced Hamiltonian on L2 (R+ ) corresponding to the eigenvalue l(l + d − 2) of the Laplace– Beltrami operator on S d−1 Hl u = −u + (V∞ + V )u.

(1.1)

Here ν 2 − 1/4 V∞ (r) = χ(r > 1); r2 V (r) = W2 (r) +

(l +

d 2

d ν = l+ −1 2 2

2 − γ,

− 1)2 − 1/4 1 − χ(r > 1) . 2 r

(1.2a) (1.2b)

Notice that V is small at infinity compared to V∞ . We are interested in spectral and scattering properties of Hl at zero energy in the case 2 d γ > l+ −1 . 2

(1.3)

This condition is equivalent to having ν in (1.2a) purely imaginary (for convenience we fix it in this case as ν = −iσ , σ > 0), and it implies the existence of a sequence of negative eigenvalues of Hl accumulating at zero energy. Our first main result is on the expansion of the resolvent −1 Rl (k) := Hl − k 2

for k ∈ Γθ± ,

where here (for any θ ∈ ]0, π/2[) Γθ+ = {k = 0 | 0 < arg k θ },

(1.4a)

Γθ− = {k = 0 | π − θ arg k < π}.

(1.4b)

We say that a solution u to the equation −u (r) + V∞ (r) + V (r) u(r) = 0

(1.5)

is regular if the function r → χ(r < 1)u(r) belongs to D(Hl ). For any t ∈ R we introduce the weighted L2 -space Ht := r−t L2 (R+ ); r = (1 + r 2 )1/2 . Theorem 1.2. Suppose Condition 1.1 and (1.3) for some (fixed) l ∈ N ∪ {0}. Let θ ∈ ]0, π/2[. There exist (finite) rational functions f ± in the variable k 2ν for k ∈ Γθ± for which lim

Γθ± k→0

Im f ± k 2ν do not exist,

(1.6)


1769

there exist Green’s functions for Hl at zero energy, denoted by R0± , and there exists a real nonzero regular solution to (1.5), denoted by u, such that the following asymptotics hold. For all s > s > 1, s 1 + 1 /2, s 3: lim sup |k|1−s Rl (k) − R0± − f ± k 2ν |u u|B(H ,H ) < ∞. s −s ±

(1.7)

Γθ k→0

Due to (1.6) the rank-one operators f ± (k 2ν )|u u| in (1.7) are non-trivially oscillatory. This phenomenon does not occur for low-energy resolvent expansions for potentials either decaying faster or slower than r −2 (cf. [6] and [5,16], respectively), nor for sectors where (1.3) is not fulfilled (cf. [14]). Combining Theorem 1.2 and the results of [14], we can deduce the resolvent asymptotics near threshold for d-dimensional Schrödinger operators with critically decaying, spherically symmetric potentials, see Theorem 3.7. An advantage to work with spherically symmetric potentials is that we can diagonalise the operator in spherical harmonics and explicitly calculate some subtle quantities. For example, one can easily show that if zero is a resonance of H , then its multiplicity is equal to (m + d − 3)! (m + d − 2)! + (d − 2)!(m − 1)! (d − 2)!m! where m ∈ N ∪ {0} is such that (m + d−2 2

d 2

− 1)2 − γ ∈ ]0, 1]. This shows that multiplicity of zero

resonance grows like γ when γ is big and d 3. To study the resolvent asymptotics for nonspherically symmetric potential W (x) behaving like q(θ) at infinity (x = rθ with r = |x|), one r2 is led to analyse the interactions between different oscillations and resonant states. This is not carried out in the present work. Our second main result is on the asymptotics of the phase shift. Let ul be a regular solution to the reduced Schrödinger equation −u + (V∞ + V )u = λu;

λ > 0.

Write √ lim ul (r) − C sin( λr + Dl ) = 0.

r→∞

The standard definition of the phase shift (coinciding with the time-depending definition) is phy

σl

(λ) = Dl +

d − 3 + 2l π. 4

The notation σ per = σ per (t) signifies below the continuous real-valued 2π -periodic function determined by

σ per (0) = 0, per eπσ e−it − eit = r(t)ei(σ (t)−t) ;

r(t) > 0, t ∈ R.

1770


Theorem 1.3. Suppose Condition 1.1 and (1.3) for some l ∈ N ∪ {0}. Let σ=

2 d γ − l+ −1 2

(recall ν = −iσ ). There exist C1 , C2 ∈ R such that phy

σl

√ √ (λ) + σ ln λ − σ per (σ ln λ + C1 ) → C2

for λ ↓ 0.

(1.8)

√ Whence the leading term in the asymptotics of the phase shift is linear in ln λ while the next term is oscillatory in the same quantity. The (positive) sign agrees with the well-known Levinson theorem (cf. [8, (12.95) and (12.156)]) valid for potentials decaying faster than r −2 . Also the qualitative behaviour of these terms as σ → 0 (i.e. finiteness in the limit) is agreeable to the case where (1.3) is not fulfilled (studied in [1] from a different point of view). The bulk of this paper concerns somewhat more general one-dimensional problems than discussed above. In particular we consider for (d, l) = (2, 0) a model with a local singularity at r = 0 that is more general than specified by Condition 1.1 4) and (1.2b). This extension does not contribute by any complication and is therefore naturally included. It would be possible to extend our methods to certain types of more general local singularities, however this would add some extra complication that we will not pursue. Our methods rely heavily on explicit properties of solutions to the Bessel equation as well as ODE techniques. These properties compensate for the fact that, at least to our knowledge, semi-classical analysis is not doable in the present context (for instance the semi-classical formula (6.8) for the asymptotics of the phase shift for slowly decaying potentials is not correct under Condition 1.1). See however [2] in the case the potential is positive. One of our motivations for studying a potential with critical fall off comes from an N -body problem: Consider a 2-cluster N -body threshold under the assumption of Coulomb pair interactions, this could be given by two atoms each one being confined in a bound state. Suppose one atom is charged while the other one is neutral. The effective intercluster potential will in this case in a typical situation (given by nonzero moment of charge of the bound state of the neutral atom) have r −2 decay although with some angular dependence (the so-called dipole approximation). Whence we expect (due to the present work) that the N -body resolvent will have some oscillatory behaviour near the threshold in question. Proving this (and related spectral and scattering properties) would, in addition to material from the present paper, rely on a reduction scheme not to be discussed here. We plan to study this problem in a separate future publication. In this paper we consider parameters ±ν, z ∈ C satisfying ν = −iσ where σ > 0 and z ∈ C \ {0} with Im z 0. Powers of z are throughout the paper defined in terms of the argument function fixed by the condition arg z ∈ [0, π]. We shall use the standard notation z := (1 + |z|2 )1/2 . For any given c > 0 we shall use the notation χ(r > c) to denote a given real-valued function χ ∈ C ∞ (R+ ) with χ(r) = 0 for r c and χ(r) = 1 for r 2c. We take it such that there exists a real-valued function χ< ∈ C ∞ (R+ ), denoted by χ< = χ(· < c), such that χ 2 + χ 0

Γθ, = {k = 0 | 0 arg k θ or π − θ arg k π} ∩ |k| , ± Γθ, = Γθ, ∩ {± Re k > 0}.

(1.9) (1.10)


1771

2. Model asymptotics In this section, we give the resolvent asymptotics at zero for a model operator under the condition (1.3). See [13] when (1.3) is not satisfied. Recall firstly some basic formulas for Bessel and Hankel functions from [11, pp. 228–230] and [12, pp. 126–127, 204] (or see [15]): (z/2)ν Jν (z) = Γ (1/2)Γ (ν + 1/2) 1 −1

1

ν−1/2 izt 1 − t2 e dt,

(2.1a)

−1

ν−1/2 Γ (1/2)Γ (ν + 1/2) , 1 − t2 dt = Γ (ν + 1)

J−ν (z) − e−iνπ Jν (z) , i sin(νπ) 1/2 i(z−νπ/2−π/4) ∞ t ν−1/2 e 2 (1) Hν (z) = e−t t ν−1/2 1 − dt. πz Γ (ν + 1/2) 2iz Hν(1) (z) =

(2.1b)

(2.1c)

(2.1d)

0

(1)

The functions Jν and Hν

solve the Bessel equation

d2 ν 2 − 1/4 − 1 z1/2 u(z) = 0. z−1/2 − 2 + dz z2

(2.2)

We have Jν (z) = eiνπ Jν¯ (−¯z),

(2.3a)

Hν(1) (z) = e−iνπ H−ν (z) = −H−¯ν (−¯z). (1)

(1)

(2.3b)

2.1. Model operator and construction of model resolvent Consider HD = −

d2 ν 2 − 1/4 + dr 2 r2

on HD = L2 [1, ∞[

(2.4)

with Dirichlet boundary condition at r = 1. Let for any ζ ∈ C, φ = φζ be the (unique) solution to ⎧ ν 2 − 1/4 ⎪ ⎪ φ(r) = ζ φ(r), ⎨ −φ (r) + r2 φ(1) = 0, ⎪ ⎪ ⎩ φ (1) = 1.

(2.5)

1772


This solution φζ is entire in ζ , and r 1/2+ν − r 1/2−ν . 2ν

(2.6)

π r 1/2 Jν¯ (k)Jν (kr) − Jν (k)Jν¯ (kr) . 2 sin(νπ)

(2.7)

φ0 (r) = In fact, cf. [11, (3.6.27)], φk 2 (r) =

(1)

Let for k ∈ C \ {0} with Im k 0 and Hν (k) = 0 φk+ (r) = r 1/2

(1)

Hν (kr) Hν(1) (k)

.

(2.8)

Due to (2.3b) the dependence of ν in φk+ is through ν 2 only, i.e. replacing ν → ν¯ yields the same expression (obviously this is also true for φk 2 ). Notice also that φk 2 and φk+ solve the equation −φ (r) +

ν 2 − 1/4 φ(r) = k 2 φ(r). r2

(2.9)

The kernel RkD (r, r ) of (H D − k 2 )−1 for k with Im k > 0 and Hν(1) (k) = 0 is given by RkD r, r = φk 2 (r< )φk+ (r> );

(2.10)

here and henceforth r< := min(r, r ) and r> := max(r, r ). (The fact that the right-hand side of (2.10) defines a bounded operator on HD follows from the Schur test and the bounds (2.15) (1) and (2.21) given below.) The condition Hν (k) = 0 is fulfilled for k ∈ {Im k > 0} \ iR+ since 2 otherwise k would be a non-real eigenvalue of H D . The zeros in iR+ correspond to the negative eigenvalues of H D . They constitute a sequence accumulating at zero. We have the properties, cf. (2.3b), D r, r = R D r , r . RkD r, r = R− k ¯k

(2.11)

In the regime where |k| is very small and stays away from the imaginary axis, more precisely in (1) Γθ, for any θ ∈ [0, π/2[ and > 0, we can derive a lower bound of |Hν (k)| as follows: From (2.1a) and (2.1b) we obtain that Jν (z) =

(z/2)ν 1 + O z2 . Γ (ν + 1)

(2.12)

Whence (recall that ν = −iσ where σ > 0) we obtain with Cν := |Γ (ν + 1) sin(νπ)| (1) −σ arg k H (k) e − e−σ π eσ arg k − O |k|2 Cν ν e−σ θ 1 − e−σ (π−2θ) Cν − O |k|2 for all k ∈ Γθ, .

(2.13)


In particular for > 0 small enough (depending on θ ) ∀k ∈ Γθ, : Hν(1) (k) e−σ π/2 1 − e−σ (π−2θ) Cν .

1773

(2.14)

Note that the bound (2.14) implies that there is a limiting absorption principle at all real E = k 2 with k ∈ Γθ, . In particular H D does not have small positive eigenvalues. 2.2. Asymptotics of model resolvent Let us note the following global bound (cf. (2.1d)) + φ (r) C

k

r

kr

1/2

e−(Im k)r

for all k ∈ Γθ, and r 1.

(2.15)

Let Dν = 2−ν /Γ (ν + 1).

(2.16)

Notice that D¯ ν = D−ν . By (2.1c) and (2.12) we obtain the following asymptotics of φk+ as k → 0 in Γθ, : φk+ (r) = r 1/2

D¯ ν r −ν k −ν − e−σ π Dν r ν k ν + O((kr)2 ) . D¯ ν k −ν − e−σ π Dν k ν + O(k 2 )

(2.17)

Introducing ζ (k) =

2iσ e−σ π Dν k 2ν , D¯ ν − Dν e−σ π k 2ν

(2.18)

we can slightly modify (2.17) (in terms of (2.6) and by using (2.15)) as φk+ (r) = r 1/2−ν

+ ζ (k)φ0 (r) + r

1/2

2

O (kr)

r +

kr

1/2

e−(Im k)r O k 2 .

(2.19)

There is a “global” bound of the third term (due to (2.15)): 2 1/2 r O (kr)2 Cr 1/2 |kr|

kr2


(2.20)

As for φk 2 we first note the following global bound (cf. (2.1a), (2.7) and [9, Theorem 4.6.1]) φ 2 (r) C k

r

kr

1/2 e(Im k)r


(2.21)

Using (2.21) we obtain similarly φk 2 (r) = φ0 (r) + r 1/2 O (kr)2 +

r

kr

1/2

e(Im k)r O k 2 .

(2.22)

1774


There is a global bound of the second term: 2 1/2 r O (kr)2 Cr 1/2 |kr| e(Im k)r

kr2


(2.23)

Whence in combination with (2.10) we obtain uniformly in k ∈ Γθ, and r, r 1 1/2 Ek r, r ; RkD r, r = R0D r, r + ζ (k)T r, r + r 1/2 r 1/2−ν R0D r, r = φ0 (r< )r> , T r, r = φ0 (r)φ0 r , 2 Ek r, r C |k|r> .

kr>

(2.24a) (2.24b) (2.24c) (2.24d)

Clearly T = |φ0 φ0 | is a rank-one operator and the function ζ has a non-trivial oscillatory behaviour. The error estimate can be replaced by: (2.25) ∃C > 0 ∀δ ∈ [0, 2]: Ek r, r C|kr> |δ for all k ∈ Γθ, and r, r 1. In particular introducing weighted spaces HsD = r−s HD , we obtain ∀s > 1:

lim

D R − R D − ζ (k)T

Γθ, k→0

k

0

D ) B(HsD ,H−s

= 0.

(2.26)

In fact we deduce from (2.24a)–(2.24d) the following more precise result: Lemma 2.1. For all s > s > 1, s 3, there exists C > 0: D s −1 H − i R D − R D − ζ (k)T k 0 B(HD ,HD ) C|k| s

−s

for all k ∈ Γθ, .

(2.27)

3. Asymptotics for full Hamiltonian, compactly supported perturbation Consider with V∞ (r) :=

ν 2 −1/4 χ(r r2

H =−

> 1)

d2 + V∞ + V dr 2

on H := L2 ]0, ∞[

(3.1)

with Dirichlet boundary condition at r = 0. As for the potential V we impose in this section Condition 3.1. 1) V ∈ C(]0, ∞[, R), 2) ∃R > 3: V (r) = 0 for r R, 3) ∃C1 , C2 > 0 ∃κ > 0: C1 (r −2 + 1) V (r) (κ 2 − 1/4)r −2 − C2 .


1775

Notice that the operator H is defined in terms of the (closed) Dirichlet form on the Sobolev space H01 (R+ ) (i.e. H is the Friedrichs extension), cf. [3, Lemma 5.3.1]. For the limiting cases C1 = ∞ and/or κ = 0 in 3) it is still possible to define H as the Friedrichs extension of the action on Cc∞ (R+ ) however the form domain of the extension might be different from H01 (R+ ) and some arguments of this paper would be more complicated. An example of this type (with κ = 0) is discussed in Appendix B. If V (r) 3/4r −2 − C the operator H is essentially self-adjoint on Cc∞ (R+ ), cf. [10, Theorem X.10]. In terms of the resolvent RkD considered in Section 2 and cutoffs χ1 = χ1 (r < 7) and χ2 = χ2 (r > 7) we introduce for k ∈ Γθ, Re k −1 i Gk = χ1 H − χ1 + χ2 RkD χ2 . |Re k|

(3.2)

Let −1 D G± 0 = χ1 (H ∓ i) χ1 + χ2 R0 χ2

(3.3)

K ± = H G± 0 − I.

(3.4)

and

Notice that the operators K ± are compact on Hs := r−s H for s > 1. Due to Lemma 2.1 we have the following expansions in B(Hs ) (with s > s > 1 and s 3) ± : ∀k ∈ Γθ,

H − k 2 Gk = I + K ± + ζ (k)|ψ0 χ2 φ0 | + O |k|s −1 ;

ψ0 := H χ2 φ0 .

(3.5)

± Lemma 3.2. For all k ∈ Γθ, the following form inequality holds (on Hs for any s > 1)

± Im Gk χ1 (H ± i)−1 (H ∓ i)−1 χ1 . Proof. This is obvious from the fact that ± Im RkD 0.

(3.6)

2

Proposition 3.3. For all s > 1 the operators I + K ± ∈ B(Hs ) have zero null space, i.e. Ker I + K ± = {0}.

(3.7)

Proof. We prove only (3.7) for the superscript “+ case”. The “− case” is similar. Suppose 0 = + H G+ 0 f for some f ∈ Hs . We shall show that f = 0. Let u0 = G0 f . Integrating by parts yields 0 = Im u0 , −H u0 = lim Im u¯ 0 u0 (r) r→∞

2 = lim Im (1/2 − ν)|u0 |2 (r)/r = σ χ2 φ0 , f . r→∞

(3.8)

So

χ2 φ0 , f = 0,

(3.9)

1776


and therefore (seen again by using the explicit kernel of R0D and by estimating by the Cauchy– Schwarz inequality) u0 = O r 3/2−s

and u0 = O r 1/2−s for r → ∞.

(3.10)

From (3.10) we can conclude that u0 = 0;

(3.11)

this can be seen by writing u0 as a linear combination of r 1/2+ν and r 1/2−ν at infinity, deduce that u0 vanishes at infinity and then invoke unique continuation. For a more general result (with detailed proof) see Lemma 4.2. Using Lemmas 2.1 and 3.2, (3.9) and (3.11) we compute 2 Im f, Gk f (H − i)−1 χ1 f .

(3.12)

χ1 f = 0.

(3.13)

R0D χ2 f = 0 on supp(χ2 ).

(3.14)

0 = Im f, u0 =

lim

+ Γθ,

k→0

We conclude that

D So 0 = G+ 0 f = χ2 R0 χ2 f , and therefore

We apply H D to (3.14) and conclude that χ2 f = 0, so indeed f = 0.

(3.15)

2

3.1. Construction of resolvent Due to Proposition 3.3 we can write, cf. (3.5), −1 H − k 2 Gk I + K ± = I + ζ (k)|ψ0 φ ± + O |k|s −1 ,

(3.16)

± for k ∈ Γθ, , where

φ ± :=

−1 ∗ I + K± χ2 φ0 .

(3.17)

We have −1 ζ (k) = I − ± |ψ0 φ ± ; I + ζ (k)|ψ0 φ ± η (k)

η± (k) := 1 + ζ (k) φ ± , ψ0 .

(3.18)


1777

Of course this is under the condition that η± (k) = 0.

(3.19)

± Lemma 3.4. For all k ∈ Γθ, the condition (3.19) is fulfilled.

Proof. Let us prove (3.19) for the superscript “+ case”. The “− case” is similar. + Suppose on the contrary that η+ (k) = 0 for some k ∈ Γθ, . Then k 2ν =

eσ π D¯ ν . Dν 1 − 2iσ φ + , ψ0

(3.20)

+ k 2ν is oscillatory, the set of all solutions of (3.20) in Γθ, constitutes a sequence converging to + zero. In particular we can pick a sequence Γθ, kn → 0 with 0 = η+ (kn ) → 0. We apply (3.16) and (3.18) to this sequence {kn }. Substituting (3.18) into (3.16) and multiplying the equation obtained by η(kn ), we get

−1 + H − kn2 Gkn I + K + η (kn ) − ζ (kn )|ψ0 φ + = η+ (kn ) 1 + +O |kn |s −1 . Taking the limit n → ∞, this leads to + −1 −ζ (∞)H G+ |ψ0 φ + = 0. ∞ I +K

(3.21)

Here ζ (∞) := limn→∞ ζ (kn ) can be computed by substituting k 2ν given by (3.20) in the expression for ζ (k) (this is the limit and one sees that it is nonzero), and similarly for G+ ∞ := limn→∞ Gkn . We learn that H u+ = 0;

+ + + −1 u+ := G+ ψ0 . ∞ f , f := I + K

(3.22)

Now, the argument of integration by parts used in (3.8) applied to u+ leads to 2 ζ (∞) 2 ζ (∞) 2 χ2 φ0 , f + . − 0 = σ 1 − 2ν 2ν

(3.23)

We claim that

χ2 φ0 , f + = 0.

(3.24)

2 2 1 − ζ (∞) ζ (∞) = eσ π k −2ν 2 = e2σ (π−2 arg k) > 1, 2ν 2ν

(3.25)

+ In fact for any k ∈ Γθ, obeying (3.20),

whence indeed (3.24) follows from (3.23). Using (3.24) we can mimic the rest of the proof of Proposition 3.3 and eventually conclude that f + = 0. This is a contradiction since ψ0 = 0. 2

1778


Combining (3.16)–(3.19) we obtain (possibly by taking > 0 smaller) ± : ∀k ∈ Γθ,

−1 ζ (k) I − ± |ψ0 φ ± I + O |k|s −1 = I. H − k 2 Gk I + K ± η (k)

(3.26)

In particular we have derived a formula for the resolvent. 3.2. Asymptotics of resolvent Let u be any nonzero regular solution to the equation −u (r) + V∞ (r) + V (r) u(r) = 0.

(3.27)

By regular solution, we mean that the function r → χ(r < 1)u(r) belongs to D(H ). It will be shown in Appendix A that the regular solution u is fixed up to a constant and can be chosen real-valued. See (3.33c) for a formula and for further elaboration. Let −1 R(k) := H − k 2

± for all k ∈ Γθ, ∩ {Im k > 0}.

(3.28)

± for which Theorem 3.5. There exist (finite) rational functions f ± in the variable k 2ν for k ∈ Γθ,

lim

± Γθ,

k→0

Im f ± k 2ν do not exist,

(3.29)

there exist Green’s functions for H at zero energy, denoted by R0± , and there exists a real nonzero regular solution to (3.27), denoted by u, such that the following asymptotics hold. For all s > s > 1, s 3, there exists C > 0: ± ∀k ∈ Γθ, ∩ {Im k > 0}: s −1 (H − i) R(k) − R ± − f ± k 2ν |u u| . 0 B(Hs ,H−s ) C|k|

(3.30)

Here ∀s > 1:

(H − i)R0± = I − iR0± ∈ B(Hs , H−s )

and (H − i)u = −iu ∈ H−s .

(3.31)

Proof. By (3.26) −1 ζ (k) I − ± |ψ0 φ ± I + O |k|s −1 R(k) = Gk I + K ± η (k)

± for k ∈ Γθ, . We expand the product yielding up to errors of order O(|k|s −1 )

(3.32)


ζ (k) ± ± u u ; η± (k) 1 2 ± −1 R0± = G± , 0 I +K

R(k) ≈ R0± +

where

1779

(3.33a) (3.33b)

± u± 1 = −R0 ψ0 + χ2 φ0 , ± ± −1 ∗ u± χ2 φ0 . 2 =φ = I +K

(3.33c) (3.33d)

± ± Clearly, u± 2 = 0. According to (3.5), H u1 = −ψ0 + H (χ2 φ0 ) = 0. In addition, u1 = 0. In fact for r > 14 (ensuring that χ2 (r) = 1) one has D ± u± 1 = −R0 f + φ0 ,

−1 with f ± = χ2 1 + K ± ψ0 ∈ Hs , s > 1.

Using then (2.24b) and (2.6) we compute

r

−1/2−ν

∞ 1 d ± r − (1/2 − ν) u1 (r) = 1 − τ 2 −ν f ± (τ ) dτ, dr r

showing that u± 1 (r) = 0 for all r large enough. By the uniqueness of regular solutions, there ± exist constants b± = 0 such that u± 1 = b u, where u is a real-valued nonzero regular solution to (3.27). Combining the duality relation R(k)∗ = R(k) and (3.33a), we obtain that ± ∓ ± ∓ u± 2 = c u1 = c b u

for some constants c± = 0.

(3.34)

ζ (k) and f ± k 2ν = C ± ± , η (k)

(3.35)

Whence indeed (3.30) holds with ± −1 R0± = G± 0 I +K

where the constants C ± = c± b∓ b± are nonzero. Whence indeed (3.29) holds. The properties (3.31) follow from the expressions (3.33b) and (3.33c). 2 Corollary 3.6. There is a limiting absorption principle at energies in ]0, 2 ]: ∀k ∈ [−, ] \ {0} ∀s > 1:

R k :=

lim

Γθ, ∩{Im k>0} k→k

R(k) exists in B(Hs , H−s ).

(3.36)

In particular ]0, 2 ] ∩ σpp (H ) = ∅. ± Moreover the bounds (3.30) extend to Γθ, . Introducing the spectral density as an operator in B(Hs , H−s ), s > 1,

R(k) − R(−k) δ H − k 2 := 2πi

for 0 < k ,

(3.37)

1780


we have lim δ H − k 2 does not exist.

k0

(3.38)

Proof. Only (3.38) needs a comment: We represent R(−k) = R(k)∗ and use (3.30) yielding ∗ Im f + (k 2ν ) δ H − k 2 ≈ (2πi)−1 R0+ − R0+ + |u u|. π The right-hand side does not converge, cf. (3.29).

2

3.3. d-dimensional Schrödinger operator As another application of Theorem 3.5, we consider a d-dimensional Schrödinger operator with spherically symmetric potential of the form H = − + W |x| γ in L2 (Rd ), d 2, where W is continuous and W (|x|) = − |x| 2 for x outside some compact set

and γ > ( d2 − 1)2 . Assume that

2 d γ = l + − 1 , 2

l ∈ N.

(3.39)

Denote Nγ = {l ∈ N ∪ {0} | (l + d2 − 1)2 < γ }. Let πl denote the spectral projection associated to the eigenvalue l(l + d − 2), l ∈ N ∪ {0}, of the Laplace–Beltrami operator on Sd−1 (and also its natural extension as operator on H = L2 (Rd )). Then H can be decomposed into a direct sum H=

∞

l πl , H

l=0

where 2 l = − d − d − 1 d + l(l + d − 2) + W (r) H r dr dr 2 r2

:= L2 (R+ ; r d−1 dr). When l ∈ Nγ , we can apply Theorem 3.5 with ν = νl , ν 2 = (l + on H l d 2 − γ < 0, to expand the resolvent (H l − k 2 )−1 up to O(|k| ) (see Section 6 for a relevant − 1) 2 l − k 2 )−1 may have singularities at zero, l used here). For l ∈ / Nγ , the resolvent (H reduction of H l (defined below). according to whether zero is an eigenvalue and/or a resonance of H −s −s Denote Hs = r H and Hs = x H, s ∈ R. Under the condition (3.39), we say that 0 is a resonance of H if there exists u ∈ H−1 \ H such that H u = 0. We call such function u a resonance function. (If the condition (3.39) is not satisfied, the definition of zero resonance has to be modified.) The number 0 is called a regular point of H if it is neither an eigenvalue nor Clearly Lemma 4.2 stated below l on H. a resonance of H . The same definitions apply for H


1781

shows that for any resonance function u necessarily πl u = 0 for all l ∈ Nγ . In fact Lemma 4.2 l , l ∈ Nγ . shows that 0 is a regular point of H If H u = 0 and u ∈ H−1 , then by expanding u in spherical harmonics, one can show that (cf. Theorem 4.1 of [13]) u(rθ ) =

ψ(θ ) r

d−2 2 +μ

+ v,

(3.40a)

where v ∈ L2 (|x| > 1),

2 d m + − 1 − γ , m = min N \ Nγ , μ= 2 nμ 1 γ − d−2 +μ (j ) (j ) 2 W + 2 u, |y| ψ(θ ) = − ϕμ ϕμ (θ ). 2μ |y|

(3.40b)

(3.40c)

j =1

(j )

Here {ϕμ , 1 j nμ } is an orthonormal basis of the eigenspace of −Sd−1 with eigenvalue m(m + d − 2) and nμ its multiplicity (cf. [12]): nμ =

(m + d − 3)! (m + d − 2)! + . (d − 2)!(m − 1)! (d − 2)!m!

(3.40d)

The expansion (3.40a) implies that a solution u to H u = 0 with u ∈ H−1 is a resonance function of H if and only if μ ∈ ]0, 1] and ψ = 0 and that if zero is a resonance, its multiplicity (cf. [6,13] for the definition) is at most nμ . Conversely, if the equation H u = 0 has a solution u ∈ H−1 \ H, then the equation m g = 0 H −1 decaying like 1/(r 2 +μ ) at infinity. It follows that has a nonzero regular solution g ∈ H (j ) uj = g ⊗ ϕμ , 1 j nμ , are all resonance functions of H . This proves that if 0 is a resonance of H its multiplicity is equal to nμ . Now let us come back to the asymptotics of the resolvent R(k) = (H − k 2 )−1 near 0. If 0 is a regular point of H (this is a generic condition and concerns by the discussion above only sectors l with l ∈ l with l ∈ / Nγ ), then it is a regular point for all H / Nγ . One deduces easily that there H (l) s , H −s ) for all s > 1, such that for any such s there exists > 0: exists R0 ∈ B(H d−2

l − k 2 −1 = R (l) + Ol |k| H 0

s , H −s ) for |k| small and k 2 ∈ in B(H / [0, ∞[.

(3.41)

The error term can be uniformly estimated in l as in [13], yielding an expansion for R(k). If 0 is l with l ∈ a resonance but not an eigenvalue of H , then 0 is a regular point for all H / Nγ ∪ {m} and m − k 2 )−1 contains a singularity the expansion (3.41) remains valid for such l. When l = m, (H at 0 which can be calculated as in [14]. Let kμ =

k 2μ ,

if μ ∈ ]0, 1[,

k 2 ln(k 2 ),

if μ = 1.

(3.42)

1782


−1 \ H verifying H m g = 0, a rank-one operator-valued entire function Then there exist g ∈ H (m) s , H −s ), s > 3, such ζ → Fm (ζ ) ∈ B(Hs , H−s ), s > 1, verifying Fm (0) = 0 and R0 ∈ B(H that for any s > 3 2 2 k eiμ π 1 |k| (m) 2 −1 s , H −s ), + R0 + O in B(H Hm − k = |g g| + Fm kμ kμ kμ |kμ |

(3.43)

where μ is the fractional part of μ: μ = μ if μ ∈ ]0, 1[ and μ = 0 if μ = 1. Note that the sign “−” is missing in the constant c1 corresponding to μ = 1 given in (4.19) of [14]. In particular if μ ∈ ]0, 12 ] one has iμπ m − k 2 −1 = e |g g| + R (m) + O |k| H 0 kμ

s , H −s ), s > 3, in B(H

while in the “worse case”, μ = 1, the error term in (3.43) is of order O(|ln k|−1 ). Summing up we have proved the following γ Theorem 3.7. Assume that W (|x|) is continuous and W (|x|) = − |x| 2 outside some compact set

with γ > ( d2 − 1)2 satisfying (3.39).

i) Suppose that zero is a regular point of H . Then there exist R0± ∈ B(Hs , H−s ) and vl ∈ −s \ {0} for all s > 1 and l ∈ Nγ , such that for any s > 1 there exists > 0: H fl± k 2νl |vl vl | ⊗ πl R(k) = l∈Nγ

+ R0± + O |k| in B(Hs , H−s ) for k ∈ Γθ± .

(3.44)

Here fl± (k 2νl ) are the oscillatory functions given in Theorem 3.5 with ν = νl = −i γ − (l + d2 − 1)2 , l ∈ Nγ . ii) Suppose that zero is a resonance of H . Let m and μ be defined by (3.40b). Then μ ∈ ]0, 1] and the multiplicity of the zero resonance of H is equal to (m + d − 3)! (m + d − 2)! + . (d − 2)!(m − 1)! (d − 2)!m! −1 \ H with Suppose in addition that zero is not an eigenvalue of H . Then there exist g ∈ H Hm g = 0, a rank-one operator-valued analytic function ζ → Fm (ζ ) ∈ B(Hs , H−s ), s > 1, defined for ζ near 0 verifying Fm (0) = 0, and R1± ∈ B(Hs , H−s ), s > 3, such that for any s>3 2 iμ π k 1 e ⊗ πm + |g g| + Fm fl± k 2νl |vl vl | ⊗ πl R(k) = kμ kμ kμ + R1± + O |ln k|−1 Here fl± and vl are the same as in i).

l∈Nγ

in B(Hs , H−s ) for k ∈ Γθ± .

(3.45)


1783

The case that 0 is an eigenvalue of H can be studied in a similar way. The zero eigenfunctions of H may have several angular momenta l > m and the asymptotics of R(k) up to o(1) as k → 0 contains many terms and we do not give details here. Note that if (3.39) is not satisfied and l − k 2 )−1 may contain a term of the order ln k as γ = (l + d2 − 1)2 for some l ∈ N ∪ {0}, (H k → 0. 4. Asymptotics for full Hamiltonian, more general perturbation We shall “solve” the equation −u (r) + V∞ (r) + V (r) u(r) = 0

(4.1)

on the interval I = ]0, ∞[ for a class of potentials V with faster decay than V∞ at infinity (re2 call V∞ (r) = ν −1/4 χ(r > 1)). In particular we shall show absence of zero eigenvalue for a r2 more general class of perturbations than prescribed by Condition 3.1. Explicitly we keep Conditions 3.1 1) and 3) but modify Condition 3.1 2) as 2) V (r) = O(r −2− ), > 0. This means that we now impose Condition 4.1. 1) V ∈ C(]0, ∞[, R), 2) V (r) = O(r −2− ), > 0, 3) ∃C1 , C2 > 0 ∃κ > 0: C1 (r −2 + 1) V (r) (κ 2 − 1/4)r −2 − C2 . Lemma 4.2. Under Condition 4.1 suppose u is a distributional solution to (4.1) obeying one of the following two conditions: 1) u ∈ L2−1 (at infinity). √ √ 2) u(r)/ r → 0 and u (r) r → 0 for r → ∞. Then u = 0.

(4.2)

Proof. Let φ ± (r) = r 1/2±ν . Then φ ± are linear independent solutions to the equation −u (r) + V∞ (r)u(r) = 0;

r > 2.

(4.3)

First we shall show that u = O r 1/2− and u = O r −1/2− .

(4.4)

Note that under the condition 1) in fact u ∈ L2 (at infinity) due to a standard ellipticity argument.

1784


We shall apply the method of variation of parameters. Specifically, introduce “coefficients” a2+ and a2− of the ansatz u = a+φ+ + a−φ−.

(4.5)

Using the differential equations for a + and a − we shall derive estimates of these quantities. The equations read

φ+ d + dτ φ

φ− d − dτ φ

d dτ

a+ a−

=V

0 φ+

0 φ−

a+ a−

.

(4.6)

Note that the Wronskian W (φ − , φ + ) = φ − drd φ + − φ + drd φ − = 2ν. (4.6) can be transformed into d dr

a+

a−

=N

a+

a−

,

where V N= 2ν

φ−φ+ −(φ + )2

(φ − )2 −φ − φ +

.

Clearly for V obeying Condition 4.1 the quantity N = O(r −1− ) and whence it can be integrated to infinity. Whence there exist a ± (∞) = lim a ± (r); r→∞

in fact a ± (∞) − a ± (r) = O r − .

(4.7)

a ± (∞) = 0.

(4.8)

We need to show that

Note that

φ+

φ−

d + dτ φ

d − dτ φ

a+ a−

=

u u

.

(4.9)

We solve for (a + , a − ) and multiply the result by r −1/2 . Under the condition 1) each component of the right-hand side of the resulting equation is in L2 . Whence also a ± (∞)/r 1/2 ∈ L2 and (4.8) and therefore (4.4) follow. We argue similarly under the condition 2). To show (4.2) note that the considerations preceding (4.8) hold for all solutions distributional u (not only a solution u obeying 1) or 2)) yielding without 1) nor 2) the bounds (4.4) with = 0. In particular for a solution u˜ with W (u, u) ˜ = 1 (assuming conversely that u = 0) we have


r

1785

−1 W (u, u)(x)x ˜ dx = ln r.

1

The right-hand side diverges while the left-hand side converges due to (4.4), and (4.2) follows. 2 2

Using Lemma 4.2 we can mimic Section 3 and obtain similar results for H = − drd 2 + V∞ + V with V satisfying (the more general) Condition 4.1. In particular Theorem 3.5 and Corollary 3.6 hold under Condition 4.1 provided that we in Theorem 3.5 impose the additional condition s 1 + /2.

(4.10)

This is here needed to guarantee that the operators K ± of (3.4) are compact on Hs . Also Theorem 3.7 has a similar extension. We leave out further elaboration. 5. Regular positive energy solutions and asymptotics of phase shift Under Condition 3.1, or in fact more generally under Condition 4.1, we can define the notion of regular positive energy solutions as follows: Let k ∈ R+ . A solution u to the equation −u (r) + V∞ (r) + V (r) u(r) = k 2 u(r)

(5.1)

is called regular if the function r → χ(r < 1)u(r) belongs to D(H ). Notice that this definition naturally extends the one applied in Section 3 in the case k = 0. Again we claim that the regular solution u is fixed up to a constant (and hence in particular can be taken real-valued): For the uniqueness we may proceed exactly as in Appendix A (uniqueness at zero energy). For the existence part we use the zero energy Green’s function R0+ and the regular zero energy solution u appearing in Theorem 3.5. Consider the equation uk 2 = u + k 2 R0+ χ(· < 1)uk 2 .

(5.2)

Notice that a solution to (5.2) indeed is a solution to (5.1) for r < 1 and hence it can be extended to a global solution u˜ k 2 . Clearly χ(· < 1)u˜ k 2 ∈ D(H ) so u˜ k 2 is a regular solution. It remains to solve (5.2) for some nonzero uk 2 . For that we let K = R0+ χ(· < 1) and note that K is compact on H−s for any s > 1. Whence we have −1 u, uk 2 = I − k 2 K

(5.3)

Ker I − k 2 K = {0}.

(5.4)

provided that

We are left with showing (5.4). So suppose u0 = k 2 Ku0 for some u0 ∈ s>1 H−s , then we need to show that u0 = 0. Notice that (H − k 2 χ(· < 1))u0 = 0 and that here the second term can be absorbed into the potential V . The computation (3.8) shows that also in the present context

1786


0 = lim Im u¯ 0 u0 (r) = lim Im (1/2 − ν)|u0 |2 (r)/r . r→∞

r→∞

(5.5)

From (5.5) we deduce the condition of Lemma 4.2 1) with u → u0 and whence from the conclusion of Lemma 4.2 that indeed u0 = 0. Now let uk 2 denote any nonzero real regular solution. By using the variation of parameters formula, more specifically by replacing the functions φ ± in the proof of Lemma 4.2 by cos(k·) and sin(k·) and repeating the proof (see Step I of the proof of Theorem 5.3 stated below for the details), we find the asymptotics lim uk 2 (r) − C sin kr + σ sr = 0.

r→∞

(5.6)

Here C = C(k) = 0. Assuming (without loss of generality) that C > 0 the (real) constant σ sr = σ sr (k) is determined modulo 2π . Definition 5.1. The quantity σ sr = σ sr (k) introduced above is called the phase shift at energy k 2 . Definition 5.2. The notation σ per = σ per (t) signifies the continuous real-valued 2π -periodic function determined by

σ per (0) = 0, per eπσ e−it − eit = r(t)ei(σ (t)−t) ;

t ∈ R, r(t) > 0.

(5.7)

Theorem 5.3. Suppose Condition 4.1. The phase shift σ sr (k) can be chosen continuous in k ∈ R+ . Any such choice obeys the following asymptotics as k ↓ 0: There exist C1 , C2 ∈ R such that σ sr (k) + σ ln k − σ per (σ ln k + C1 ) → C2

for k ↓ 0.

(5.8)

Proof. Step I. We shall show the continuity. From (5.2) and (5.3) we see that for any r > 0 the functions ]0, ∞[ k → uk 2 (r) and ]0, ∞[ k → uk 2 (r) are continuous. Similar statements hold upon replacing uk 2 → Re uk 2 and uk 2 → Im uk 2 which are both real-valued regular solutions (solving (5.1) for r < 1). Since uk 2 = 0 one of these functions must be nonzero. Without loss of generality we can assume that uk 2 is a real-valued nonzero regular solution obeying that for r = 1/2 the functions ]0, ∞[ k → uk 2 (r) and ]0, ∞[ k → uk 2 (r) are continuous. By a standard regularity result for linear ODE’s with continuous coefficients these results then hold for any r > 0 too. Moreover (to used in Step II) we have (again for r > 0 fixed) uk 2 (r) − u(r) = O k 2 uk 2 (r) − u (r) = O k 2

for k ↓ 0,

(5.9a)

for k ↓ 0.

(5.9b)

We introduce φ + (r) = cos kr Mimicking the proof of Lemma 4.2 we write

and φ − (r) = sin kr.

(5.10)


uk 2 = a + φ + + a − φ − .

1787

(5.11)

Noting that the Wronskian W (φ − , φ + ) = −k we have d dr

a+

=N

a−

a+

(5.12)

,

a−

where N = −k −1 (V∞ + V )

φ−φ+ −(φ + )2

(φ − )2 −φ − φ +

.

(5.13)

Since N = O(r −2 ) there exist a ± (∞) = lim a ± (r).

(5.14)

r→∞

By the same argument as before either a + (∞) = 0 or a − (∞) = 0. We write + a (∞), a − (∞) a + (∞)2 + a − (∞)2 = sin σ sr , cos σ sr

(5.15)

and conclude the asymptotics (5.6) with some C = 0. It remains to see that a ± (∞) are continuous in k (then by (5.15) σ sr can be chosen continuous too). For that we use the “connection formula”

uk 2 uk 2

=

φ+

φ−

φ−

φ+

a+

a−

which is “solved” by

a+ a−

= −k −1

φ− −φ +

−φ − φ+

uk 2 uk 2

.

(5.16)

We use (5.16) at r = 1/2. By the comments at the beginning of the proof the right-hand side is continuous in k and therefore so is the left-hand side. Solving (5.12) by integrating from r = 1/2 and noting that (5.13) is continuous in k we then conclude that a ± (r) are continuous in k for any r > 1/2. Since the limits (5.14) are taken locally uniformly in k > 0 we consequently deduce that indeed a ± (∞) are continuous in k. Step II. We shall show (5.8) under Condition 3.1. We shall mimic Step I with (5.10) replaced by φ + (r) = r 1/2 Hν(1) (kr)

and φ − (r) = r 1/2 H−ν (kr). (1)

(5.17)

For completeness of presentation note that in terms of another Hankel function, cf. [11, (3.6.31)], (2) φ − (r) = r 1/2 Hν (kr). We compute the Wronskian W (φ − , φ + ) = 4i/π , cf. (2.1c) and [11, (3.6.27)]. Since V (r) = 0 for r R a ± (r) = a ± (∞)

for r R.

(5.18)

1788


Moreover (5.16) reads

a+ a−

π = 4i

φ− −φ +

−φ − φ+

uk 2 uk 2

(5.19)

.

We will use (5.19) at r = R. Clearly the right-hand side is continuous in k > 0 and therefore so is the left-hand side. From the asymptotics

2 1/2 ikr e → 0 for r → ∞, πk 1/2 2 φ − (r) − C−ν e−ikr → 0 for r → ∞; πk φ + (r) − Cν

(5.20)

Cν := e−iπ(2ν+1)/4 ,

(5.21)

we may readily rederive the continuity statement shown more generally in Step I. The point is that now we can “control” the limit k → 0. To see this we need to compute the asymptotics of the matrix in (5.19) as k → 0 (with r = R). Using (2.1c) we compute ν 1 −ν 1 2 R2 2−ν R 2 +ν ν 1 (5.22a) k −ν − e−σ π k + O k2 , i sin(νπ) Γ (1 − ν) Γ (1 + ν) ν 1 −ν −ν 21 +ν 2 2 R2 1 − −ν σπ 2 R ν φ (R) = k −e k +O k , (5.22b) −i sin(νπ) Γ (1 − ν) Γ (1 + ν) 1 1 −1 2ν R − 2 −ν −ν 2−ν R − 2 +ν ν 1 2 −ν k − e−σ π 2−1 + ν k + O k2 , φ + (R) = i sin(νπ) Γ (1 − ν) Γ (1 + ν) φ + (R) =

φ − (R) =

1 −i sin(νπ)

2−1 − ν

1 2ν R − 2 −ν

Γ (1 − ν)

k −ν − eσ π 2−1 + ν

1 2−ν R − 2 +ν

Γ (1 + ν)

(5.22c) kν + O k2 . (5.22d)

We combine (5.9a) and (5.9b) for r = R with (5.18)–(5.22d) and obtain 1/2

π Cν σ π ν e Dk − Dk −ν + O k 2 eikr + h.c. + o r 0 ; 4i i sin(νπ) 1 2ν R 2 −ν 2−1 − ν u(R) − u (R) . D := (5.23) Γ (1 − ν) R

uk 2 (r) =

2 πk

Here the term O(k 2 ) depends on R but not on r and the term o(r 0 ) depends on k. The second term, denoted by h.c., is given as the hermitian (or complex) conjugate of the first term. Note that D = 0. We write D = |D|eiθ0 yielding per eσ π Dk ν − Dk −ν = eσ π Dk ν − Dk −ν ei(σ (σ ln k+θ0 )−(σ ln k+θ0 )) .

(5.24)


1789

Next we substitute (5.24) into (5.23), use that Cν = |Cν |e−iπ/4 and conclude (5.8) with C1 = θ 0

and C2 = π/4 − θ0 + 2πp

for some p ∈ Z.

(5.25)

Step III. We shall show (5.8) under Condition 4.1. This is done by modifying Step II using the proof of Step I too. Explicitly using again the functions φ ± of (5.17) “the coefficients” a ± need to be constructed. Since V is not assumed to be compactly supported these coefficients will now depend on r. We first construct them at any large R, this is by the formula (5.19) (at r = R). Then the modification of (5.12) + a d a+ = N , (5.26) dr a − a− with N=

π V 4i

φ−φ+ −(φ + )2

(φ − )2 −φ − φ +

(5.27)

,

is invoked. We integrate to infinity using that N = O(r −1− ) uniformly in k > 0. This leads to a ± (r) = a ± (∞) + O r − , a ± (R) = a ± (∞) + O R − ,

(5.28a) (5.28b)

with the error estimates being uniform in k > 0. In particular for r R a ± (r) = a ± (R) + O R − + O r −

(5.29)

uniformly in k > 0. From (5.29) we obtain the following modification of (5.23) 1/2

2 − ikr π Cν σ π ν −ν uk 2 (r) = e Dk − Dk +O k +O R e 4i i sin(νπ) + h.c. + o r 0 ; 1 2ν R 2 −ν 2−1 − ν u(R) − u (R) . D = D(R) := Γ (1 − ν) R

2 πk

The term O(k 2 ) depends on R, and the term O(R − ) depends on k but it is estimated uniformly in k > 0. By Lemma 4.2 there exist δ > 0 and a sequence Rn → ∞ such that D(Rn ) δ

for all n.

(5.30)

Using these values of D in (5.24) we can write per eσ π Dk ν − Dk −ν = eσ π Dk ν − Dk −ν ei(σ (σ ln k+θ)−(σ ln k+θ)) ; D = D(Rn ),

θ = θn ∈ [0, 2π[.

(5.31)

1790


We can assume that for some θ0 ∈ [0, 2π] θn → θ0

for n → ∞.

Using this number θ0 we obtain again (5.8) with C1 and C2 given as in (5.25).

(5.32) 2

6. Asymptotics of physical phase shift for a potential like −γ χ(r > 1)r −2 We shall reduce a d-dimensional Schrödinger equation to angular momentum sectors and discuss the asymptotics of the “physical” phase shift for small angular momenta in the low energy regime. We consider for d 2 the stationary d-dimensional Schrödinger equation H v = (− + W )v = λv;

λ > 0,

for a radial potential W = W (|x|) obeying Condition 6.1. 1) 2) 3) 4)

W (r) = W1 (r) + W2 (r); W1 (r) = − rγ2 χ(r > 1) for some γ > 0, W2 ∈ C(]0, ∞[, R), ∃1 , C1 > 0: |W2 (r)| C1 r −2−1 for r > 1, ∃2 , C2 > 0: |W2 (r)| C2 r 2 −2 for r 1.

Under Condition 6.1 H = − + W is self-adjoint as defined in terms of the Dirichlet form on H 1 (Rd ), cf. [4]. Let Hl , l = 0, 1, . . . , be the corresponding reduced Hamiltonian corresponding to an eigenvalue l(l + d − 2) of the Laplace–Beltrami operator on Sd−1 Hl u = −u + (V∞ + V )u.

(6.1)

Here V∞ (r) =

ν 2 − 1/4 χ(r > 1); r2

V (r) = W2 (r) +

(l +

d 2

2 d ν2 = l + − 1 − γ , 2

− 1)2 − 1/4 1 − χ(r > 1) , 2 r

(6.2a) (6.2b)

and the stationary equation reads −u + (V∞ + V )u = λu.

(6.3)

Notice that for

d γ > l+ −1 2 and

2 ,

(6.4)


(d, l) = (2, 0),

1791

(6.5)

indeed Condition 4.1 is fulfilled and Hl coincides with the Hamiltonian given by the construction of Section 4. The case (d, l) = (2, 0) needs a separate consideration which is given in Appendix B. Under the conditions (6.4) and (6.5) let ul be a regular solution to the reduced Schrödinger equation (6.3). Write √ lim ul (r) − C sin( λr + Dl ) = 0.

(6.6)

r→∞

The standard definition of the phase shift (coinciding with the time-depending definition) is phy

σl

(λ) = Dl +

d − 3 + 2l π. 4

(6.7)

It is known from [16,4] that for a potential W (r) behaving at infinity like −γ r −μ with γ > 0 and μ ∈ ]1, 2[ ∃σ0 ∈ R:

phy σl (λ) −

∞ √ λ − λ − W (r) dr → σ0

for λ ↓ 0.

(6.8)

R0

Here R0 is any sufficiently big positive number, and the integral does not have a (finite) limit as λ ↓ 0. In the present case, μ = 2, (6.8) indicates a logarithmic divergence. This is indeed occurring although (6.8) is incorrect for μ = 2. The correct behaviour of the phase shift under the conditions (6.4) and (6.5) follows directly from Section 5: Theorem 6.2. Suppose Condition 6.1 and (6.4) for some l ∈ N ∪ {0}. Let σ=

2 d γ − l+ −1 . 2

(6.9)

phy

The phase shift σl (λ) can be chosen continuous in λ ∈ R+ . Any such choice obeys the following asymptotics as λ ↓ 0: There exist C1 , C2 ∈ R such that phy

σl

√ √ (λ) + σ ln λ − σ per (σ ln λ + C1 ) → C2

for λ ↓ 0.

(6.10)

Note that we have included the case (d, l) = (2, 0) in this result. The necessary modifications of Section 5 for this case are outlined in Appendix B. Appendix A. Regular zero energy solutions We shall elaborate on the notion of regular solutions as used in Sections 3 and 4. Recall from the discussion around (3.27) that we call a solution u to (3.27) for regular if r → χ(r < 1)u(r) belongs to D(H ) where H is defined in terms of a potential V satisfying Condition 3.1 (or Condition 4.1). The existence of a (nonzero) regular solution is shown explicitly by the formula (3.33c).

1792


We shall show that the regular solution is unique up to a constant. Notice that as a consequence of this uniqueness result a regular solution is real-valued up to constant. Suppose conversely that all solutions are regular. Due to [10, Theorem X.6(a)] there exists a nonzero solution v to −v (r) + V∞ (r) + V (r) v(r) = iv(r)

(A.1)

which is in L2 at infinity. By the variation of parameter formula now based on the basis of regular solutions to (3.27), cf. the proof of [10, Theorem X.6(b)], we conclude that v ∈ D(H ) and that (H − i)v = 0. This violates that H is self-adjoint. Appendix B. Case (d, l) = (2, 0) For (d, l) = (2, 0) Condition 4.1 fails for the operator Hl of Section 6 (this example would require κ = 0 in Condition 4.1 3)). The form domain is not H01 (R+ ) in this case. The form is given as follows: 1 D(Q) = f ∈ L2 (R+ ) g ∈ L2 (R+ ) where g(r) = f (r) − f (r) , 2r ∞ f (r) − 1 f (r)2 + W (r)f (r)2 dr; f ∈ D(Q). Q(f ) = 2r

(B.1a)

(B.1b)

0

This is a closed semi-bounded quadratic form and the domain D(H ) of the corresponding operator H (cf. [3,10]) is characterised as the subset of f ’s in D(Q) for which h ∈ L2 (R+ )

d2 1 where h(r) := − 2 − 2 + W (r) f (r) as a distribution on R+ , dr 4r

(B.2)

and for f ∈ D(H ) we have d2 1 (Hf )(r) = − 2 − 2 + W (r) f (r). dr 4r

(B.3)

To see the connection to the two-dimensional Hamiltonian of Section 6 defined with form domain H 1 (R2 ) let us note the alternative description of Q:

D(Q) = f ∈ L2 (R+ ) g˜ | · | ∈ H 1 R2 where g(r) ˜ = r −1/2 f (r) , −1/2 2 2 ∇ |x| f |x| + W |x| |x|−1/2 f |x| dx Q(f ) = (2π)−1

(B.4a) for f ∈ D(Q).

R2

(B.4b) Clearly the integral to the right in (B.4b) is the form of the two-dimensional Hamiltonian (applied to radially symmetric functions).


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We also note that H01 (R+ ) ⊆ D(Q) and that Cc∞ (R+ ) + span(f0 );

f0 (r) := r 1/2 χ(r < 1)

/ H01 (R+ ), the set Cc∞ (R+ ) is actually a core for Q. Whence is a core for Q. In fact, although f0 ∈ H is the Friedrichs extension of the action (B.3) on Cc∞ (R+ ). Due to (B.1b) and the description in (B.2) of the domain D(H ) we can show the uniqueness of regular solutions exactly as in Appendix A. The existence of (nonzero) regular solutions follows from the previous scheme too. Indeed the basic operators K ± of (3.4) are again compact on B(Hs ). To see this we need to see that various terms are compact. Let us here consider the contribution from the first term of (3.3) d (H ∓ i)−1 χ1 + χ1 (±i)(H ∓ i)−1 χ1 =: K1± + K2± . − χ1 + 2χ1 dr (The contribution from the second term of (3.3) is treated in the same way as before.) We decompose using any C > 0 such that H 0 C + 1 K1± = B ± K; 1 d 1 ± (H ∓ i)−1 (H − C)1/2 , B = − χ1 (r) + 2χ1 (r) + 2χ1 (r) − 2r dr 2r K = (H − C)−1/2 χ1 . The operator B ± is bounded and the operator K is compact (the latter may be seen easily by going back to the space L2 (R2 ) and there invoking standard Sobolev embedding); whence K1± is compact. Clearly also K2± is compact. References [1] G. Carron, Le saut en zero de la fonction de decalage spectral, J. Funct. Anal. 212 (2004) 222–260. [2] O. Costin, W. Schlag, W. Staubach, S. Tanveer, Semiclassical analysis of low and zero energy scattering for onedimensional Schrödinger operators with inverse square potentials, J. Funct. Anal. 255 (2008) 2321–2362. [3] E.B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, 1995. [4] J. Dereziński, E. Skibsted, Scattering at zero energy for attractive homogeneous potentials, Ann. Henri Poincaré 10 (2009) 549–571. [5] S. Fournais, E. Skibsted, Zero energy asymptotics of the resolvent for a class of slowly decaying potentials, Math. Z. 248 (2004) 593–633. [6] A. Jensen, G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys. 13 (6) (2001) 717–754. [7] S. Nakamura, Low energy asymptotics for Schrödinger operators with slowly decreasing potentials, Comm. Math. Phys. 161 (1) (1994) 63–76. [8] R.G. Newton, Scattering Theory of Waves and Particles, Springer, New York, 1982. [9] F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York/London, 1974. [10] M. Reed, B. Simon, Methods of Modern Mathematical Physics I–IV, Academic Press, New York, 1972–1978. [11] M. Taylor, Partial Differential Equations, Basic Theory, Springer, New York, 1996; corrected second printing 1999. [12] M. Taylor, Partial Differential Equations II, Qualitative Studies of Linear Equations, Springer, New York, 1996; corrected second printing 1997.

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[13] X.P. Wang, Threshold energy resonance in geometric scattering, Mat. Contemp. 26 (2004) 135–164. [14] X.P. Wang, Asymptotic expansion in time of the Schrödinger group on conical manifolds, Ann. Inst. Fourier (Grenoble) 56 (6) (2006) 1903–1945. [15] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1952. [16] D. Yafaev, The low energy scattering for slowly decreasing potentials, Comm. Math. Phys. 85 (2) (1982) 177–196.


Sums of Laplace eigenvalues—rotationally symmetric maximizers in the plane R.S. Laugesen ∗ , B.A. Siudeja Department of Mathematics, University of Illinois, Urbana, IL 61801, USA Received 24 June 2010; accepted 20 December 2010 Available online 28 December 2010 Communicated by Gilles Godefroy

Abstract The sum of the first n 1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio (area)3 /(moment of inertia) for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schrödinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues. © 2010 Elsevier Inc. All rights reserved. Keywords: Isoperimetric; Membrane; Tight frame

1. Introduction Eigenvalues of the Laplacian represent frequencies in wave motion, rates of decay in diffusion, and energy levels in quantum mechanics. Eigenvalues are challenging to understand: they are known in closed form on only a handful of domains. This difficulty has motivated considerable work on estimating eigenvalues in terms of simpler, geometric quantities such as area and perimeter.


E-mail addresses: [email protected] (R.S. Laugesen), [email protected] (B.A. Siudeja). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.018

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R.S. Laugesen, B.A. Siudeja / Journal of Functional Analysis 260 (2011) 1795–1823

We will obtain a sharp bound on the sum of the first n 1 eigenvalues of linear images of rotationally symmetric domains. Our methods apply equally well to Dirichlet, Robin, and Neumann boundary conditions. Write λ1 , λ2 , λ3 , . . . for the Dirichlet eigenvalues of the Laplacian on a plane domain. Let A be the area of the domain and I be its moment of inertia about the center of mass. We will prove that for each n 1, the normalized, scale invariant eigenvalue sum (λ1 + · · · + λn )

A3 I

(1.1)

is maximal among triangular domains for the equilateral triangle. Among parallelograms the maximizer is the square, and the disk is the maximizer among ellipses. The only case known previously was the fundamental tone, n = 1, due to Pólya [45]. An analogous result will be shown for the sum of Neumann eigenvalues, (μ1 + · · · + μn )

A3 , I

(1.2)

and then for Robin and Schrödinger eigenvalues too. These latter results are new even for n = 1. See Section 3. Our work suggests conjectures for general convex domains. Is the Dirichlet eigenvalue sum (1.1) maximal for the disk? Not when n = 1, curiously, because any rectangle or equilateral triangle gives a larger value for the fundamental tone. We conjecture that those domains maximize λ1 A3 /I . For the Neumann eigenvalue sum (1.2) it does seem plausible to conjecture maximality for the disk, as we discuss in Section 4. Central to the paper is a new technique we call the “Method of Rotations and Tight Frames”. The idea is to linearly transplant the eigenfunctions of the extremal domain and then average with respect to allowable rotations of that domain. This averaging of the Rayleigh quotient is accomplished using a “tight frame” or Parseval identity for the root-of-unity vectors. The Hilbert–Schmidt norm of the linear transformation arises naturally in such averaging, and is then represented in terms of the moment of inertia about the centroid. 1.1. Intuition The eigenvalue sum (1.1) can be written as a product of two scale invariant factors, as (λ1 + · · · + λn )A ·

A2 . I

The first factor, (λ1 + · · · + λn )A, is normalized by the area of the domain, and so can be thought of as a generalized “Faber–Krahn” term (although the Faber–Krahn theorem says λ1 A is minimal for the disk, not maximal). The second factor, A2 /I , is purely geometric and measures the “deviation from roundness” of the domain. This factor is small when the domain is elongated, and hence it balances the largeness of the first factor on such domains.


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The motivating intuition is that for a domain with characteristic length scales a and b, we have λ1

1 1 ab(a 2 + b2 ) I + = 3. 2 2 3 a b (ab) A

This rough calculation is exact for rectangles, up to constant factors. The first inequality in the calculation extends readily to higher dimensions, but the algebraic identity in the middle becomes more complicated. Thus in higher dimensions it seems that the moment of inertia should be evaluated instead on some kind of “reciprocal” domain that has length scales 1/a and 1/b and so on. Higher dimensional results of this nature will be developed in a later paper [36]; the maximizing domains are regular tetrahedra, cubes, and other Platonic solids. 1.2. Related work The major contribution of this paper is that it proves upper bounds that are geometrically sharp, on eigenvalue sums of arbitrary length. We do not know any similar results in the literature. Some results of a different type are known, as we now describe. A bound due to Kröger [27] for Neumann eigenvalues says that (μ1 + · · · + μn )A/n2 2π . The inequality is asymptotically sharp for each domain, because μn A ∼ 4πn by the Weyl asymptotics. But Kröger’s bound is not geometrically sharp for fixed n, because there are no domains for which equality holds. Kröger’s result should be viewed as a weak version of the Pólya conjecture. That conjecture asserts that the Weyl asymptotic estimate is in fact a strict upper bound on each Neumann eigenvalue. It has been proved for tiling domains by Kellner [25], and up through the third eigenvalue for simply connected plane domains by Girouard, Nadirashvili and Polterovich [19], but it remains open in general. Kröger also proved an upper bound on Dirichlet eigenvalue sums, involving ε-neighborhoods of the boundary [28]. This bound is again not geometrically sharp. Kröger’s estimates were generalized to domains in homogeneous spaces by Strichartz [54]. Weak versions of Pólya’s conjectured lower bound for Dirichlet eigenvalues [47] are due to Berezin [9] and Li and Yau [38], with later developments by Laptev [30] and others using Riesz means and “universal” inequalities, as surveyed by Ashbaugh [5]. Useful upper bounds on eigenvalue sums in terms of other eigenvalue sums have lately been obtained this way, by Harrell and Hermi [21, Corollary 3.1]. Note Pólya’s lower bound has been investigated also for eigenvalues under a constant magnetic field, by Frank, Loss and Weidl [13]. There is considerable literature on low eigenvalues of domains constrained by perimeter, inradius, or conformal mapping radius, rather than moment of inertia. We summarize this literature in Section 8. Eigenvalues of triangular domains have been studied a lot, in recent years [1–3,14–16,33,34, 39,51,52], and this paper extends the theory of their upper bounds. Lower bounds on Dirichlet eigenvalues of triangles are proved in a companion paper [35]: there the triangles are normalized by diameter (rather than area and moment of inertia) and equilateral triangles are shown to minimize (rather than maximize) the eigenvalue sums. For broad surveys of isoperimetric eigenvalue inequalities, one can consult the monographs of Bandle [8], Henrot [22], Kesavan [26] and Pólya and Szeg˝o [49], and the survey paper by Ashbaugh [4].

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2. Assumptions and notation 2.1. Eigenvalues For a bounded plane domain D, we denote the Dirichlet eigenvalues of the Laplacian by λj (D), the Robin eigenvalues by ρj (D; σ ) where the constant σ > 0 is the Robin parameter, and the Neumann eigenvalues by μj (D). In the Robin and Neumann cases we make the standing assumption that the domain has Lipschitz boundary, so that the spectra are well defined. Denoting the eigenfunctions by uj in each case, we have

−uj = λj uj uj = 0

in D, on ∂D,

−uj = ρj uj in D, ∂uj + σ uj = 0 on ∂D, ∂n

−uj = μj uj ∂uj =0 ∂n

in D, on ∂D

and 0 < λ1 < λ 2 λ 3 · · · ,

0 < ρ1 < ρ2 ρ3 · · · ,

0 = μ1 < μ2 μ3 · · · .

The Robin case reduces to Neumann when σ = 0, and formally reduces to the Dirichlet case when σ = ∞. The corresponding Rayleigh quotients are

|∇u|2 dx 2 D u dx

Dirichlet: R[u] = D Robin: R[u] =

for u ∈ H01 (D),

+ σ ∂D u2 ds 2 D u dx

2 D |∇u| dx

|∇u|2 dx 2 D u dx

Neumann: R[u] = D

for u ∈ H 1 (D),

for u ∈ H 1 (D).

The Rayleigh–Poincaré principle [8, p. 98] characterizes the sum of the first n 1 eigenvalues as λ1 + · · · + λn = min R[v1 ] + · · · + R[vn ]: v1 , . . . , vn ∈ H01 (D) are pairwise orthogonal in L2 (D) in the Dirichlet case, and similarly in the Robin and Neumann cases (using trial functions in H 1 instead of H01 ). 2.2. Geometric quantities Let A = area, I = moment of inertia (about the centroid).


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Fig. 1. A domain D with rotational symmetry, and its image under a linear map T .

That is, |x − x|2 dx

I (D) = D

1 where the centroid is x = A(D) D x dx. Given a matrix M, write its Hilbert–Schmidt norm as MHS =

1/2 Mj2k

1/2 = tr MM †

j,k

where M † denotes the transposed matrix. 3. Sharp upper bounds on eigenvalue sums 3.1. Dirichlet and Neumann eigenvalues Our first result examines the effect on eigenvalues of linearly transforming a rotationally symmetric domain, like in Fig. 1. Theorem 3.1. If D has rotational symmetry of order greater than or equal to 3, then

2 1 (λ1 + · · · + λn )|T (D) T −1 HS (λ1 + · · · + λn )|D 2

(3.1)

for each n 1 and each invertible linear transformation T of the plane. The same inequality holds for the Neumann eigenvalues. Equality holds in (3.1) for the first Dirichlet eigenvalue (n = 1) if and only if either (i) T is a scalar multiple of an orthogonal matrix, or (ii) D is a square and T (D) is a rectangle ( possibly with sets of capacity zero removed). Equality holds for the second Neumann eigenvalue (n = 2) if and only if T is a scalar multiple of an orthogonal matrix.

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The proof is in Section 6. Notice equality does hold in the theorem when T is a scalar multiple of an orthogonal matrix, because if T = rS where S is orthogonal, then λj (T (D)) = r −2 λj (D) by rescaling and rotation, and 12 T −1 2HS = r −2 . The rotationally symmetric domain D in the theorem need not be convex, need not be a regular polygon, and need not have any axis of symmetry. For example, it could be shaped like a three-bladed propeller. Pólya obtained the theorem for n = 1 (the Dirichlet fundamental tone), although with no equality statement. He stated this result in [45], and Pólya and Schiffer proved it along with results for torsional rigidity and capacity in [48, Chapter IV]. Our method differs subtly from theirs, as we explain in Section 6, and this difference allows us to handle higher eigenvalue sums and Neumann eigenvalues too. To express the theorem more geometrically, we observe that the Hilbert–Schmidt norm of the transformation T −1 can be expressed in terms of moment of inertia and area (Lemma 5.3). Hence in particular: Corollary 3.2. Among triangles, the normalized Dirichlet eigenvalue sum (λ1 + · · · + λn )

A3 I

(3.2)

is maximal for the equilateral triangle, for each n 1. When n = 1, every maximizer is equilateral. Among parallelograms, the quantity (3.2) is maximal for the square. When n = 1, every maximizer is a rectangle and every rectangle is a maximizer. Among ellipses, the quantity (3.2) is maximal for the disk. When n = 1, every maximizer is a disk. The normalized Neumann eigenvalue sum (μ2 + · · · + μn )

A3 I

is maximal among triangles for the equilateral triangle, among parallelograms for the square, and among ellipses for the disk, for each n 2. When n = 2, every maximizer is an equilateral triangle, square, or disk, respectively. The Neumann case with n = 1 is not interesting, because the first eigenvalue equals 0 for each domain. Remarks. 1. The method extends to linear images of regular N -gons for any N , but the most interesting cases are triangles and parallelograms (N = 3 and N = 4), as considered in the corollary. 2. For triangles, the moment of inertia can be calculated in terms of the side lengths l1 , l2 , l3 as I=

A2 l + l22 + l32 . 36 1

(3.3)


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For a parallelogram with adjacent side lengths l1 , l2 , the moment of inertia equals I=

A2 l1 + l22 . 12

(3.4)

3. The eigenvalues and eigenfunctions of the extremal domains (the equilateral triangle, square and disk) are not used in our proofs. The eigenvalues are stated anyway in Appendix A, for reference. It is interesting to substitute them into Corollary 3.2 and obtain explicit estimates on eigenvalue sums. For example, for the Dirichlet fundamental tone (n = 1) one obtains that ⎧ 12π 2 A3 ⎨ 12π 2 λ1 ⎩ 2 2 I 2j0,1 π 11.5π 2

for triangles, with equality for equilaterals, for parallelograms, with equality for rectangles, for ellipses, with equality for disks.

(3.5)

All three inequalities were obtained by Pólya [45], [48, pp. 308, 328]. The first inequality, for triangles, was rediscovered with a different proof by Freitas [14, Theorem 1]. The second inequality was rediscovered for the special case of rhombi by Hooker and Protter [24, §5] and for all parallelograms by Hersch [23, formula (5)], again with different proofs. These authors stated their results in terms of side lengths, as (l12 + l22 + l32 )π 2 /3A2 for triangles, (3.6) λ1 (l12 + l22 )π 2 /A2 for parallelograms. These inequalities are equivalent to Pólya’s by formulas (3.3) and (3.4) for the moment of inertia. For the first nonzero Neumann eigenvalue we find from Corollary 3.2 and Appendix A that ⎧ 2 4π A3 ⎨ 2 6π μ2 ⎩ I )2 π 2 6.8π 2 2(j1,1

for triangles, with equality for equilaterals, for parallelograms, with equality for squares, for ellipses, with equality for disks.

(3.7)

These inequalities too can be stated in terms of side lengths. For stronger inequalities on μ2 , see Section 8. For n = 3, the corollary says (μ2 +μ3 )A3 /I is maximal for the equilateral triangle. This result was proved recently by the authors using a different method with explicit trial functions [33, Theorem 3.5]. 4. Corollary 3.2 becomes false when applied to individual eigenvalues instead of eigenvalue sums. For example, λ3 A3 /I is not maximal for the square among rectangles: to the contrary, it is locally minimal. The underlying reason is that a square has a double eigenvalue λ2 = λ3 that “splits” when the square is deformed into a rectangle of the same area; the second eigenvalue decreases and the third increases, while the moment of inertia varies only at second order. 5. The corollary also holds for moment of inertia about an arbitrary center, provided the moment of inertia of the equilateral triangle is taken about its centroid, since the moment is minimal when taken about the centroid. 3.2. Robin eigenvalues In the next theorem we normalize the Robin parameter in terms of T −1 , in order to obtain a scale invariant expression.

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Theorem 3.3. If D has rotational symmetry of order greater than or equal to 3, then (ρ1 + · · · + ρn )

A3 A3 (ρ + · · · + ρ ) 1 n I σ T −1 HS /√2,T (D) I σ,D

for each n 1 and each invertible linear transformation T of the plane. Equality holds for the first Robin eigenvalue (n = 1) if and only if T is a scalar multiple of an orthogonal matrix. The subscript “σ, D” on the right side of the inequality specifies the domain where the eigenvalues and geometric quantities are to be evaluated, and also the value of the Robin parameter to be used. The subscript on the left √ side of the inequality similarly specifies the domain T (D) and the Robin parameter σ T −1 HS / 2 to be used there. Corollary 3.4. Fix the Robin parameter σ > 0. Among all triangles of the same area, the quantity (ρ1 + · · · + ρn )

A3 I

is maximal for the equilateral triangle. When n = 1, every maximizer is equilateral. Analogous results hold among parallelograms and ellipses, with squares and disks being the maximizers, respectively. 3.3. Schrödinger eigenvalues Consider the Schrödinger eigenvalue problem −h¯ 2 u + W u = Eu 2 in the plane, with Planck constant h¯ > 0 and real-valued potential W ∈ L∞ loc (R ) that tends to +∞ as |x| → ∞. The spectrum is discrete [50, Theorem XIII.67], and the eigenvalues Ej are characterized in the usual way by the Rayleigh quotient

R[u] =

h¯

R2

|∇u|2 dx + R2 W u2 dx , 2 R2 u dx

u ∈ H 1 R2 ∩ L2 (W ).

Here L2 (W ) denotes the weighted space with measure |W | dx. Once more we show that a rotationally symmetric situation maximizes the sum of eigenvalues. Theorem 3.5. If W has rotational symmetry of order greater than or equal to 3, then (E1 + · · · + En )|√2h¯ /T −1 HS ,W ◦T −1 (E1 + · · · + En )|h¯ ,W for each n 1 and each invertible linear transformation T of the plane. When n = 1, equality holds if and only if T is a scalar multiple of an orthogonal matrix.


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The subscript “h¯ , W ” on the right side of the inequality specifies the potential to be used and the value of the Planck constant, and similarly for the subscript on the left side of the inequality. This Schrödinger result formally implies the Dirichlet√ result in Theorem 3.1, by taking W = 0 on D and W = +∞ off D, and choosing h¯ = T −1 HS / 2. 3.4. More general quadrilaterals Among quadrilaterals we have so far handled only the special class of parallelograms. Now we show how to handle a larger class of quadrilaterals having two “halves” of equal area. To construct such domains, first write the upper and lower halfplanes as R2+ = (x1 , x2 ): x2 > 0 , R2− = (x1 , x2 ): x2 < 0 . Choose two linear transformations T+ and T− that agree on the x1 -axis, with T± mapping R2± onto itself. Then the map T (x) =

T+ x

if x ∈ R2+ ,

T− x

if x ∈ R2− ,

defines a piecewise linear homeomorphism of the plane mapping the upper and lower halfplanes to themselves. Assume also det T+ = det T− , so that T distorts areas by the same factor in the upper and lower halfplanes. We will not need explicit formulas for the linear transformations T+ and T− , but for the sake of concreteness we present them anyway:

a c± T± = 0 b where a = 0, b > 0 and c± ∈ R. Let D be the square with vertices at (±1, 0), (0, ±1). Our goal is to show that this square maximizes eigenvalue sums among quadrilaterals of the form E = T (D). These quadrilaterals have two vertices on the x1 -axis and have upper and lower halves of equal area. Write I0 (E)= |x|2 dx E

for the moment of inertia about the origin, for a domain E. Theorem 3.6 (Quadrilaterals with equal-area halves). Let D be the square with vertices at (±1, 0), (0, ±1). Then for every map T constructed as above, (λ1 + · · · + λn )

A3 A3 (λ + · · · + λ ) 1 n I0 T (D) I 0 D

for each n 1. The inequality holds also for Neumann eigenvalues.

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The moment I0 is generally greater than the moment of inertia I for the domain T (D), because the centroid of T (D) need not be at the origin. For example, if both T+ and T− are shear transformations towards the right, then the centroid of T (D) will lie on the positive x1 -axis, to the right of the origin. Meanwhile, the centroid of the rotationally symmetric domain D will always lie at the origin, so that I (D) = I0 (D). We conjecture that Theorem 3.6 can be strengthened to use I instead of I0 . For the first eigenvalue (n = 1), Freitas and Siudeja [16] showed recently with a computerassisted proof that

λ1

l12

+ l22

A2 + l32 + l42

is maximal for rectangles among all quadrilaterals, not just among quadrilaterals with halves of equal area. For parallelograms, this result and Theorem 3.6 give the same information (see formula (3.6)). For general quadrilaterals we cannot easily compare the two results, because the relationship between the sum of squares of side lengths and the moment of inertia is unclear. 4. Open problems for general convex domains For the Dirichlet fundamental tone we raise: Conjecture 4.1. Suppose Ω is a bounded convex plane domain. Then 9 2 A3 π < λ1 12π 2 2 I Ω with equality on the right for equilateral triangles and all rectangles, and asymptotic equality on the left for degenerate acute isosceles triangles and sectors. The convexity assumption is necessary on the right side of the conjecture because otherwise one could drive the eigenvalue to infinity without affecting the area or moment of inertia, by removing sets of measure zero (such as curves) from the domain. The maximizer cannot be the disk in the last conjecture because triangles and rectangles yield a larger value, as we observed already in (3.5). As evidence for the conjecture, we note that λ1 A3 /I is bounded above and below on convex domains by an Inclusion Lemma, as was shown by Pólya and Szeg˝o [49, §1.19, 5.11b]. They further evaluated λ1 A3 /I for a variety of triangles, sectors, degenerate ellipses and degenerate sectors [49, p. 267]. Asymptotic expansions can be obtained also for degenerate triangles [15]. We examine the family of isosceles triangles in Fig. 2. All this evidence is consistent with Conjecture 4.1. For the first nonzero Neumann eigenvalue, we know μ2 A3 /I is definitely maximal for the disk among all bounded domains, by an inequality of Szeg˝o and Weinberger (see Section 8). This quantity has no minimizer because it approaches zero for a degenerate rectangle.


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Fig. 2. Numerical plot of the normalized Dirichlet fundamental tone λ1 A3 /I for isosceles triangles of aperture α ∈ (0, π ). The maximizer is equilateral (α = π/3), and the minimizer is degenerate acute (α → 0).

Now consider sums of eigenvalues. Conjecture 4.2. Suppose Ω is a bounded convex plane domain. Then for the Neumann eigenvalues, A3 (μ2 + · · · + μn ) I Ω is maximal when Ω is a disk, for each n 2. The conjecture is true for the special case of ellipses by Corollary 3.2. For Dirichlet eigenvalues, the conjecture fails because the square gives a larger value than the disk for (λ1 + · · · + λn )A3 /I when n = 1, 2, 3, 5, 6, 9, 10, 12; the disk does give a larger value for all other n 50, and we suspect for n > 50 as well. 5. Consequences of symmetry: tight frames, and moment matrices In this section we recall the tight frame property of rotationally symmetric systems of vectors, and develop a moment of inertia formula for the linear image of a rotationally symmetric domain. These elementary consequences of symmetry will be used in proving Theorem 3.1. 5.1. Tight frames Let N 3 and write Um for the matrix representing rotation by angle 2πm/N , for m = 1, . . . , N . For each nonzero y ∈ R2 , the rotations generate a rotationally symmetric system {U1 y, . . . , UN y} in the plane. For example, the system consists of the N th roots of unity when y = 10 . We start with a well-known Plancherel-type identity for such systems. Lemma 5.1. Let N 3. For all column vectors x, y ∈ R2 one has N 2 1 1 x · (Um y) = |x|2 |y|2 . N 2 m=1

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Fig. 3. The “Mercedes–Benz” tight frame (N = 3) in the plane.

Proof. We may suppose x and y have length 1 and lie at angles θ and φ to the positive horizontal axis, respectively. Then N N 2 1 1 x · (Um y) = cos2 (θ − φ − 2πm/N ) N N m=1

m=1

N 1 1 + cos 2(θ − φ − 2πm/N ) 2 2N m=1 N −i4π/N m 1 1 i2(θ−φ) Re e e = + 2 2N

=

m=1

1 = 2 as desired. The assumption N 3 ensures that e−i4π/N = 1 when summing the geometric series, in the last step. 2 Fig. 3 illustrates the lemma for N = 3: it shows the projection formula 3m=1 (x · Um y)Um y = 0 3 2 2 x for a typical x ∈ R , where y = 1 is the vertical unit vector and Um denotes rotation by 2πm/3. Dotting the projection formula with x yields the Parseval-type identity in Lemma 5.1. The lemma says that the rotationally symmetric system {U1 y, . . . , UN y} forms a tight frame. Readers who want to learn about frames and their applications in Hilbert spaces may consult the monograph by Christensen [11] or the text by Han et al. [20]. We next deduce a tight frame identity in which the vector y is replaced by a matrix. Lemma 5.2. Let N 3, K 1. For all row vectors x ∈ R2 and all 2 × K real matrices Y one has N 1 1 |xUm Y |2 = |x|2 Y 2HS . N 2 m=1

Proof. Write y1 , . . . , yK for the column vectors of Y , so that |xUm Y |2 = apply Lemma 5.1. 2

K

2 k=1 |xUm yk | .

Now


1807

5.2. Hilbert–Schmidt norms and moment of inertia When proving Corollary 3.2, we will need to evaluate the Hilbert–Schmidt norm of T −1 in terms of moment of inertia and area. Lemma 5.3. If the bounded plane domain D has rotational symmetry of order N 3, and T is an invertible 2 × 2 matrix, then

1

T −1 2 = I (T D)/ I (D). HS 2 A3 A3 Proof. The centroid of D lies at the origin, in view of the rotational symmetry of D. Thus the centroid of T D also lies at the origin. The moment matrix of D is defined to be M(D) = [ D xj xk dx]j,k . We show it equals a scalar multiple of the identity, as follows. Let U denote the matrix for rotation by 2π/N . The rotational invariance of D under U implies that M(D) = U M(D)U † , so that if x is an eigenvector of M(D) then so is U x, with the same eigenvalue. Since x and U x span R2 (using here that N 3), we conclude every vector in R2 is an eigenvector with that same eigenvalue. Thus M(D) is a multiple of the identity. In particular, the diagonal entries in M(D) are equal. Since they sum to the moment of inertia I (D), we have

1 1 0 M(D) = I (D) . 0 1 2

(5.1)

The moment of inertia of T (D) can now be computed as I (T D) = tr M(T D) = tr T M(D)T † |det T | 1 = I (D) tr T T † |det T | by (5.1) 2 1 = I (D)T 2HS |det T |. 2

(5.2)

This formula gives us the Hilbert–Schmidt norm of T , whereas we want the Hilbert–Schmidt norm of T −1 . Fortunately, the two are related, with

−1 2

T = T 2 /|det T |2 HS HS

(5.3)

by the explicit formula for T −1 in terms of the matrix entries, in two dimensions. Hence

2

1 I (T D) = I (D) T −1 HS |det T |3 , 2 from which the lemma follows easily.

2

(5.4)

1808


An interesting consequence of the last lemma is that the moment of inertia of a linear image of a rotationally symmetric domain equals the moment of inertia of its inverse image, after normalizing by the area. Lemma 5.4. If the bounded plane domain D has rotational symmetry of order N 3, and T is an invertible 2 × 2 matrix, then I I (T D) = 2 T −1 D . 2 A A Proof. By (5.2), and then using (5.4) with T replaced by T −1 , we find I (T D) I (T −1 D) 1 I (D) T 2HS = = . A(T D)2 2 A(D)2 |det T | A(T −1 D)2

2

The lemma holds also with T −† D instead of T −1 D, since T † and T have the same Hilbert– Schmidt norm and determinant. 6. Proof of Theorem 3.1 We prove the Dirichlet case of the theorem. The idea is to construct trial functions on the domain T (D) by linearly transplanting eigenfunctions of D, and then to average with respect to the rotations of D. The Neumann proof is identical, except using Neumann eigenfunctions. Let u1 , u2 , u3 , . . . be orthonormal eigenfunctions on D corresponding to the Dirichlet eigenvalues λ1 , λ2 , λ3 , . . . . Consider an orthogonal matrix U ∈ O(2) that fixes D, so that U (D) = D. Define trial functions vj = uj ◦ U ◦ T −1 on the domain E = T (D), noting vj ∈ H01 (E) because uj ∈ H01 (D). The functions vj are pairwise orthogonal, since vj vk dx = uj uk dx · det T U −1 = 0 E

D

when j = k. Thus by the Rayleigh–Poincaré principle, we have n

λj (E)

j =1

n

2 E |∇vj | dx 2 . v dx j E j =1

(6.1)

For each function v = vj we evaluate the Rayleigh quotient as

2 E |∇v| dx 2 E v dx

= = D

T D |(∇u)(x)U 2 D u dx

−1 |2 dx

· |det T U −1 | · |det T U −1 |

(∇u)U T −1 2 dx,

(6.2)


1809

where the gradient ∇u is regarded as a row vector. In the last line we used that u = uj is normalized in L2 (D). Since D has N -fold rotational symmetry for some N 3, we may choose U to be the matrix Um representing rotation by angle 2πm/N , for m = 1, . . . , N . By averaging (6.1) and (6.2) over these rotations we find n

λj (E)

j =1

n j =1 D

=

m=1

n j =1 D

N 1 2 (∇uj )Um T −1 dx N

1 2 −1 2 dx |∇uj | T HS 2

by Lemma 5.2

n

2 1 = T −1 HS λj (D), 2 j =1

which proves the inequality in Theorem 3.1. 6.0.1. Equality statement for Dirichlet fundamental tone, n = 1 Suppose equality holds in the theorem for the first Dirichlet eigenvalue. That is, suppose

2 1 λ1 T (D) = T −1 HS λ1 (D). 2

(6.3)

We reduce to T being diagonal, as follows. The singular value decomposition of T can be written T = QRS where Q and S are orthogonal matrices with det S = 1 (so that S is a rotation matrix) and R = r01 r02 is diagonal with r1 , r2 > 0. If r1 = r2 then T is a scalar multiple of an orthogonal matrix. So suppose from now on that r1 = r2 . and = S(D), so that D has rotational symmetry of order N . Note λ1 (D) = λ1 (D), Write D that T (D) = QR(D) so that . λ1 T (D) = λ1 R(D) Also

−1 2

T = R −1 2 = r −2 + r −2 . 1 2 HS HS Hence equality in (6.3) implies = 1 r −2 + r −2 λ1 (D), λ1 R(D) 1 2 2 under the diagonal linear transformawhich means that equality holds in (3.1) for the domain D tion R. so that Write u = u1 for a first Dirichlet eigenfunction on D, ux1 x1 + ux2 x2 = −λ1 (D)u.

(6.4)

1810


is Inspecting the proof of the theorem, above, we see that one of the trial functions on R(D) v = u ◦ R −1 , in other words v(x1 , x2 ) = u(x1 /r1 , x2 /r2 ). Since equality holds in the Rayleigh principle (6.1) with n = 1, we deduce that this trial function must actually be a first eigenfunction That is, on R(D). v, v = −λ1 R(D) which means r1−2 ux1 x1 + r2−2 ux2 x2 = −

1 −2 r + r2−2 λ1 (D)u. 2 1

(6.5)

By solving the simultaneous linear equations (6.4) and (6.5) (which is possible since r1 = r2 ) we find that 1 ux1 x1 = ux2 x2 = − λ1 (D)u. 2

(6.6)

This last formula must apply also if we rotate u through angle 2π/N , because that rotate of u was used in one of the trial functions in the proof of the theorem above. Hence the second directional That second derivative is derivative of u in direction θ = 2π/N must equal − 12 λ1 (D)u. 2 cos θ ux1 x1 + 2(cos θ sin θ )ux1 x2 + sin2 θ ux2 x2 , + sin(2θ )ux1 x2 by (6.6). We conclude sin(2θ )ux1 x2 = 0. which equals − 12 λ1 (D)u Then u = F1 (x1 ) + Suppose N = 4. Then sin(2θ ) = sin(4π/N) = 0, and so ux1 x2 = 0 in D. F2 (x2 ) for some functions F1 and F2 , and substituting this formula into (6.6) gives that F1 (x1 ) = 1 (x1 ) + F2 (x2 )]. Taking the x2 derivative shows that F2 is constant. Similarly F1 is − 12 λ1 (D)[F constant, and so u is constant, animpossibility. Eqs. (6.6) say ux1 x1 = ux2 x2 = −ω2 u, and so Therefore N = 4. Write ω = λ1 (D)/2. u(x1 , x2 ) = A cos(ωx1 ) cos(ωx2 ) + B sin(ωx1 ) sin(ωx2 ) + C cos(ωx1 ) sin(ωx2 ) + D∗ sin(ωx1 ) cos(ωx2 ) for some constants A, B, C, D∗ . The 4-fold rotational symmetry of the domain further on D, implies that each of the four terms A cos(ωx1 ) cos(ωx2 ),

(6.7)

B sin(ωx1 ) sin(ωx2 ), 2 C + D∗2 cos(ωx1 ) sin(ωx2 ), 2 C + D∗2 sin(ωx1 ) cos(ωx2 )

(6.8) (6.9) (6.10)

with eigenvalue 2ω2 = λ1 (D), or else is identically zero, as is by itself a Dirichlet mode for D we will now show. First, by adding and subtracting u(x1 , x2 ) and u(−x1 , −x2 ) (its rotation by π ) we find that the functions f (x1 , x2 ) = A cos(ωx1 ) cos(ωx2 ) + B sin(ωx1 ) sin(ωx2 )


1811

and g(x1 , x2 ) = C cos(ωx1 ) sin(ωx2 ) + D∗ sin(ωx1 ) cos(ωx2 ) (or else are identically zero). By adding and subtracting f (x1 , x2 ) are each eigenfunctions on D and f (−x2 , x1 ) (rotation by π/2) we find that (6.7) and (6.8) are each eigenfunctions (or else are identically zero). By considering Cg(x1 , x2 ) − D∗ g(−x2 , x1 ) and D∗ g(x1 , x2 ) + Cg(−x2 , x1 ) we learn that (6.9) and (6.10) are each eigenfunctions (or else are identically zero). The fundamental Dirichlet mode does not change sign. The nodal domains for each of the must lie within one of those squares. For (6.8), functions (6.7)–(6.10) are squares, and so D (6.9) and (6.10), rotation by angle π maps each nodal square to a completely disjoint square, cannot have 2-fold rotational symmetry, let alone 4-fold symmetry. Hence which means that D (6.8), (6.9) and (6.10) must not be eigenfunctions, and so necessarily B = C = D∗ = 0. Thus the eigenfunction is (6.7). Taking A = 1, we have u = cos(ωx1 ) cos(ωx2 ). Rotation by π rules out every nodal square except the one centered at the origin, which is is contained in this square. (−π/2ω, π/2ω)2 . Hence D Thus D must fill the whole The square has first Dirichlet eigenvalue 2ω2 , which equals λ1 (D). square (except perhaps omitting a set of capacity zero, which does not affect the fundamental is a square and T (D) = QR(D) is a is a rectangle, and D = S −1 (D) tone [17]). Then R(D) rectangle. This completes the proof of the “only if” part of the proof of the equality statement. For the “if” part of the equality statement, suppose D is a square and T (D) is a rectangle (possibly with sets of capacity zero removed). By rotating and reflecting D and T (D) suitably, we can suppose they have sides parallel to the coordinate axes and that T = r01 r02 for some r1 , r2 > 0. Writing L for the side length of the square, we have λ1 (D) = 2(π/L)2 ,

2 1 λ1 T (D) = (π/r1 L)2 + (π/r2 L)2 = T −1 HS λ1 (D), 2

so that equality holds in (3.1) with n = 1. 6.0.2. Equality statement for second Neumann eigenvalue, n = 2 Suppose equality holds in the theorem for the second Neumann eigenvalue. Most of the preceding argument in the Dirichlet equality case applies without change, simply replacing λ1 with μ2 and the Dirichlet eigenfunction u1 with the Neumann eigenfunction u2 , and replacing the word “Dirichlet” with “Neumann”. The argument works because the first Neumann eigenvalue is zero, with constant eigenfunction u1 ≡ const., and so the trial function v1 = u1 ◦ U ◦ R −1 of D Thus if equality holds in the Rayleigh is also constant and hence is a first eigenfunction on R(D). principle (6.1) for n = 2 then the trial function v2 = u2 ◦ U ◦ R −1 is a second eigenfunction on R(D). The significant difference from the Dirichlet proof begins at the sentence “The fundamental Dirichlet mode does not change sign”. The second Neumann eigenfunction u = u2 does change it has exactly two nodal domains {u > 0} and {u < 0}, each of which is connected. (We sign on D: know the eigenfunction has at least two nodal domains because u is orthogonal to the constant eigenfunction; it has at most two by Courant’s nodal domain theorem [8, p. 112].)

1812


Consider each of the four possible forms of u in turn, namely (6.7)–(6.10). Each one has must be subsets of such squares. square nodal domains, and the two nodal domains of u in D intersects exactly two of the squares. At the same time, D has 4-fold rotational symmeHence D because if D intersected try. These requirements prevent (6.7) from being an eigenfunction for D, two of the nodal squares, then it would have to intersect at least five of them. Hence A = 0. Similarly (6.8) cannot be an eigenfunction, and so B = 0. Next we deal with (6.9). (The argument is similar for (6.10).) Suppose C 2 + D∗2 > 0 in (6.9), so that we may take u = cos(ωx1 ) sin(ωx2 ).

(6.11)

to intersect exactly two of the nodal squares, they must be the squares Then in order for D adjacent to the origin, so that ⊂ (−π/2ω, π/2ω) × (−π/ω, π/ω). D

(6.12)

We will deduce a contradiction below, so that necessarily C 2 + D∗2 = 0. Hence none of the functions (6.7)–(6.10) is an eigenfunction, and so the case N = 4 cannot occur. Therefore the only way for equality to hold is to have r1 = r2 , so that T is a scalar multiple of an orthogonal matrix. To obtain the desired contradiction, we will examine how the Neumann boundary condition is has Lipschitz boundary, there exists an affected by the linear transformation. Since the domain D outward normal vector (n1 , n2 ) at almost every boundary point (with respect to arclength mea we know u satisfies the Neumann (or natural) boundary sure). At each such point (x1 , x2 ) ∈ ∂ D, condition 0 = ∇u · (n1 , n2 ) = ux1 n1 + ux2 n2 ; here we used that u as defined by (6.11) is globally smooth. Further, the point (r1 x1 , r2 x2 ) ∈ has an outward normal vector (n1 /r1 , n2 /r2 ). Since v(x1 , x2 ) = u(x1 /r1 , x2 /r2 ) is a ∂R(D) it satisfies the Neumann boundary condition: smooth eigenfunction on the closure of R(D), 0 = ∇v · (n1 /r1 , n2 /r2 ) = ux1 n1 /r12 + ux2 n2 /r22 . Recalling that r1 = r2 , these simultaneous equations imply ux1 n1 = 0 and ux2 n2 = 0. Recalling the formula (6.11) for u and the conHence either ux1 = 0 or ux2 = 0, a.e. on ∂ D. straint (6.12) on D, we deduce that almost every boundary point is contained in the lines {x1 = 0, ±π/2ω}, {x2 = 0, ±π/2ω, ±π/ω}. Furthermore, we can rule out the vertical lines {x1 = ±π/2ω} because on those lines ux1 = 0 and so n1 = 0, which means the normal would be vertical and the tangent horizontal, so that the boundary would depart the given vertical lines. must lie in Similarly we rule out the horizontal lines {x2 = 0, ±π/ω}. Hence the boundary of D the union of the lines {x1 = 0}, {x2 = ±π/2ω}. Since these lines fail to bound a domain, we have arrived at a contradiction, as desired.


1813

6.1. Why did Pólya not prove our theorem? Pólya proved Theorem 3.1 for the first Dirichlet eigenvalue λ1 , except that he proved no equality statement. His result appeared in [45] and its proof in [48, Chapter IV]. Why did he not prove the theorem for sums of eigenvalues, or for Neumann eigenvalues, as we do in this paper? Or for higher dimensions as we do in a forthcoming paper [36]? A possible reason is that our method is subtly different from Pólya’s. We use rotational symmetry at a later stage in the argument. This delay permits us to handle more than just the first eigenvalue, and to handle Neumann eigenvalues too. Let us explain in more detail. Pólya began by using rotational symmetry of the domain to obtain rotational symmetry of the fundamental Dirichlet eigenfunction u1 : he observed that the rotate of u1 is itself a positive eigenfunction and so must equal u1 . Then Pólya deduced that D

∂u1 ∂x1

2

dx = D

∂u1 ∂x2

2 dx =

1 2

|∇u1 |2 dx, D

D

∂u1 ∂u1 dx = 0. ∂x1 ∂x2

Hence his linearly transplanted trial function u1 ◦ T −1 has Rayleigh quotient −1 )|2 dx −1 |2 dx 2

1 −1 2 E |∇(u1 ◦ T D |(∇u1 )(x)T D |∇u1 | dx

T = = HS 2 2 −1 )2 dx 2 E (u1 ◦ T D u1 dx D u1 dx as desired. The difficulty when trying to extend Pólya’s approach to sums of eigenvalues is that the higher eigenfunctions are usually not symmetric under rotations, because of sign changes. The insight that permits us to prove Theorem 3.1 is that while the rotate of a higher eigenfunction need not equal itself, it must still be an eigenfunction with the same eigenvalue, and thus can still be used to generate trial functions by linear transplantation. Our proof uses the whole family of rotations to generate many trial functions, and then averages over the resulting family of inequalities. This approach applies (without change!) to the Neumann eigenvalues too. 7. Proofs of other results 7.1. Proof of Corollary 3.2 Every triangle can be written (after translation) as the image under a linear transformation T of an equilateral triangle centered at the origin. Hence the inequality for triangles in Corollary 3.2 follows from Theorem 3.1 and Lemma 5.3. The statements about parallelograms and ellipses are proved similarly. 7.1.1. Remark on Dirichlet maximizers when n 2 It is not clear how to determine all maximizing domains for sums of eigenvalues beyond the first. For example, some (but not all) non-square rectangles can maximize (λ1 + · · · + λn )A3 /I when n 2, as we now show. Consider a rectangle with side lengths l1 , l2 , so that the area is A = l1 l2 , the moment of inertia is I = l1 l2 (l12 + l22 )/12 and 12 A3 = −2 . I l1 + l2−2

1814


The fundamental tone is λ1 = π 2 l1−2 + l2−2 . Notice λ1 A3 /I = 12π 2 for every rectangle (not just for the square), as we have observed before. Thus every rectangle is a maximizer when n = 1. Now fix n 2, and fix the side length l2 . For l1 sufficiently large, we have eigenvalues λj = π 2 (j 2 l1−2 + l2−2 ) for j = 1, . . . , n, and so lim (λ1 + · · · + λn )

l1 →∞

A3 = 12π 2 n. I

Meanwhile, the square satisfies (λ1 + · · · + λn )

A3 A3 > nλ1 = 12π 2 n. I I

Hence for sufficiently large l1 , the rectangle with side lengths l1 and l2 is not a maximizer. Nonetheless, the rectangle can be a maximizer for some values of l1√and n. For example, let n = 3 and suppose the side lengths of the rectangle satisfy l2 l1 8/3l2 . Then by simple comparisons we find λ1 = π 2 l1−2 + l2−2 , λ2 = π 2 22 l1−2 + l2−2 , λ3 = π 2 l1−2 + 22 l2−2 , and so (λ1 + λ2 + λ3 )

A3 = 72π 2 . I

This value is the same as achieved for the square (l1 = l2 ), and so there are many non-square maximizers when n = 3. The idea behind this construction is to identify a range of (l1 , l2 ) values for which the eigenvalues λ1 , λ2 , λ3 have values π 2 (j 2 l1−2 + k 2 l2−2 ) for (j, k) = (1, 1), (2, 1), (1, 2). This set of index pairs in Z2 is invariant with respect to interchanging j and k. Hence λ1 + λ2 + λ3 is proportional to l1−2 + l2−2 , which allows us to cancel the denominator in A3 /I and obtain an expression independent of the side lengths. This construction can of course be extended to arbitrarily large values of n, if desired. 7.2. Proof of Theorem 3.3 The proof goes exactly as for the Dirichlet and Neumann cases in the proof of Theorem 3.1, except that for the Robin eigenvalues we must take account also of a boundary integral in the Rayleigh quotient:


v ∂E

2 ds(x)

Ev

2 dx

= =

u(U T ∂E

1815

−1 x)2 ds(x)

E u(U T

−1 x)2 dx

u(U x)2 |T τ (x)| ds(x) ∂D 2 D u(U x) dx · |det T |

by x → T x, where τ (x) denotes the unit tangent vector to ∂D at x. Geometrically, |T τ (x)| is the factor by which T stretches the tangent direction to ∂D at x. The symmetry of D implies that the tangent vectors rotate according to τ (U −1 x) = U −1 τ (x), and so replacing x with U −1 x in the last integral gives v 2 ds(x) ∂E −1 2 −1 ds(x). T U = |det T | u(x) τ (x) 2 E v dx ∂D

Choose U to be the matrix Um representing rotation by angle 2πm/N , for m = 1, . . . , N . Averaging the preceding quantity over m and applying Cauchy–Schwarz gives the upper estimate −1

|det T |

∂D

2

u(x)

N 2 1 T Um−1 τ (x) N

1/2 ds(x)

m=1

1 = |det T |−1 √ T HS 2

u(x)2 ds(x)

(7.1)

∂D

√ by Lemma 5.2, since |τ (x)| = 1. Multiplying by σ T −1 HS / 2 gives

1

T −1 2 σ u(x)2 ds(x) HS 2 ∂D

by (5.3). With the aid of this last estimate we can straightforwardly adapt the proof of Theorem 3.1 to the Robin situation, and then call on Lemma 5.3 to interpret the Hilbert–Schmidt norm of T −1 in terms of moment of inertia. 7.2.1. Equality statement for Robin fundamental tone, n = 1 The proof of the equality statement follows the Dirichlet case in Theorem 3.1 up to the point where N = 4 and u = cos(ωx1 ) cos(ωx2 ), contained in the open square S = (−π/2ω, π/2ω)2 . We want to deduce a contradiction, and D so that the only way for equality to hold when n = 1 is for T to be a scalar multiple of an orthogonal matrix. Equality must hold in the application of Cauchy–Schwarz at (7.1), except using R instead instead of D. Hence of T and using D RU −1 τ (x) = RU −1 τ (x) = RU −1 τ (x) = RU −1 τ (x) (7.2) 1 2 3 4

1816


here we use that u(x)2 > 0 for almost every (with respect to arclength measure) x ∈ S ∩ ∂ D; on S. Consider x-value and write τ1 and τ2 for the components of the tangent vector τ (x). τ such an −τ Then |R τ12 | = |R τ12 | by (7.2), or (r1 τ1 )2 + (r2 τ2 )2 = (−r1 τ2 )2 + (r2 τ1 )2 . Since r12 = r22 , we can simplify to τ12 = τ22 . Thus the tangent line at x has slope ±1, and hence so does the normal vector. √ The four possible normal vectors are n(x) = (ε1 , ε2 )/ 2 where ε1 , ε2 ∈ {−1, 1}. Thus the Robin boundary condition ∂u ∂n + σ u = 0 says ε1 ux1 + ε2 ux2 +

√ 2σ u = 0.

Substituting u = cos(ωx1 ) cos(ωx2 ) yields that ε1 tan(ωx1 ) + ε2 tan(ωx2 ) =

√ 2σ/ω.

lies on one of these four curves. We conclude that every point x ∈ S ∩ ∂ D These curves have slope ±1 at only finitely many points in the square S, and so we conclude lie in that square. Hence ∂ D lies entirely in the boundary of the square S. that no points of ∂ D The Robin condition fails on ∂S, though, because u = 0 there while ∂u ∂n = 0. This contradiction completes the proof. 7.3. Proof of Corollary 3.4 Take D to be a domain with rotational symmetry of order at least 3 and assume the linear transformation T has |det T | = 1, so that T (D) has the same area as D. The corollary now follows from Theorem 3.3 and the elementary inequality T −1 2HS 2|det T −1 |. 7.4. Proof of Theorem 3.5 The proof proceeds as for the Dirichlet case in Theorem 3.1, except that we must consider also the potential term in the numerator of the Rayleigh quotient. The key observation is that

−1 )v 2 dx E (W ◦ T 2 E v dx

= =

−1 2 −1 D (W ◦ U )u dx · |det T U | 2 −1 D u dx · |det T U |

W u2 dx D

by the rotational symmetry of W . The proof is now easily completed. Incidentally, the assumption in the theorem that the potential W should grow at infinity can be significantly weakened [37].


1817

7.4.1. Equality statement for Schrödinger fundamental tone, n = 1 Just like in the Dirichlet case, the singular value decomposition allows us to reduce to T being diagonal. The analogues of Eqs. (6.4) and (6.5) are that − E1 (W ) u, h¯ 2 (ux1 x1 + ux2 x2 ) = W

2h¯ 2

r1−2

+ r2−2

− E1 (W ) u. r1−2 ux1 x1 + r2−2 ux2 x2 = W

(These equations hold pointwise a.e. by elliptic regularity theory, since the potential is locally bounded [18, Theorem 8.8].) Solving these simultaneous equations, we deduce (since r1 = r2 ) that ux1 x1 = ux2 x2 =

1 ) u. W − E1 (W 2 2h¯

(7.3)

(x) is assumed to tend to ∞ as |x| → ∞, and so W − E1 > 0 whenever |x| The potential W is sufficiently large. Multiplying (7.3) by u and integrating in the x1 direction, we deduce that − R u2x1 dx1 0 when |x2 | is sufficiently large, so that u(x1 , x2 ) = 0 for almost every x1 . Since (7.3) says that u satisfies the one-dimensional wave equation with x2 playing the role of time variable and x1 playing the role of space variable, we conclude that u = 0 a.e. in R2 . This contradiction completes the proof. 7.5. Proof of Theorem 3.6 We prove a generalization of Theorem 3.1, namely that (λ1 + · · · + λn )(T D)

1

T −1 2 + T −1 2 (λ1 + · · · + λn )(D) + − HS HS 4

(7.4)

for any bounded D having rotational symmetry of order N 4 with N even. The proof of Theorem 3.1 requires some modifications. First we show that pairwise orthogonality of the vj remains valid. Decomposing D and E = T (D) into their upper and lower halves D± = D ∩ R2± and E± = E ∩ R2± , we compute

uj uk dx · det T± U −1 .

vj vk dx = E±

D±

These upper and lower terms sum to zero because det T+ = det T− and sumption, when j = k. Thus E vj vk dx = 0. Next we consider the Rayleigh quotient of v. We decompose it as

2 E |∇v| dx 2 E v dx

=

±

D uj uk dx

= 0 by as-

(∇u)(x)U T −1 2 dx, ±

U (D± )

where in this calculation we use once more that the determinants of T+ and T− agree.

(7.5)

1818


Since N is even, UN/2 represents rotation by π , so that Um+N/2 (D± ) = Um (D∓ ) and Um+N/2 = −Um . Hence when we average (7.5) over the rotations U = Um we obtain

N 1 N ± m=1

=

(∇u)(x)Um T −1 2 dx ±

Um (D± )

N/2 1 N ±

+

m=1 U (D ) m ±

(∇u)(x)Um T −1 2 dx ±

Um (D∓ )

N/2 1 (∇u)(x)Um T −1 2 dx = ± N ±

=

± D

since Um (D) = D

m=1

D

N 2 1 (∇u)(x)Um T±−1 dx 2N m=1

1

−1 2

= T± HS |∇u|2 dx 4 ± D

by Lemma 5.2. Now complete the proof of (7.4) by recalling u = uj and summing over j . Then the theorem follows from (7.4) and the evaluation of the Hilbert–Schmidt norms in the next lemma. Lemma 7.1. Let T be the piecewise linear homeomorphism in Theorem 3.6. If the bounded plane domain D has rotational symmetry of order N 4, with N even, then

1

T −1 2 + T −1 2 = I0 (T D)/ I0 (D). + − HS HS 4 A3 A3 Recall I0 denotes the moment of inertia about the origin. Proof. The moment integrals over the upper and lower halves of D agree, with

xj xk dx = D+

xj xk dx,

j, k = 1, 2,

D−

because D+ maps to D− under rotation by π (that is, x → −x). Here we use evenness of the order of rotation. Hence moment matrices satisfy M(D+ ) = M(D− ) = M(D)/2. Since M(D) = 1 0the 1 2 I0 (D) 0 1 , as shown in the proof of Lemma 5.3, we deduce

1 1 0 M(D± ) = I0 (D) . 0 1 4

(7.6)


1819

Now the moment of inertia of T D about the origin can be computed as I0 (T D) = tr M(T D) = tr M(T+ D+ ) + tr M(T− D− ) tr T± M(D± )T±† · |det T± | = ±

1 = I0 (D) tr T± T±† · |det T± | 4 ±

by (7.6)

1 = I0 (D) T+ 2HS + T− 2HS |det T± |, 4 where in the last step we used that det T+ = det T− . The Hilbert–Schmidt norm of the inverse T±−1 is related to the Hilbert–Schmidt norm of T± by (5.3), and so

2

2 1 I0 (T D) = I0 (D) T+−1 HS + T−−1 HS |det T± |3 , 4 from which the lemma follows.

2

8. Literature on maximizing low eigenvalues under area, perimeter, inradius or conformal mapping normalization This paper gives sharp upper bounds on the sum of the first n 1 eigenvalues, normalized by A3 /I . To help put these results in context, we now describe results and conjectures that apply to the low eigenvalues (n = 1, 2, 3). 8.1. Dirichlet eigenvalues The quantity λ1 A2 /L2 (where L is the perimeter) is maximal among triangles for the equilateral triangle, by work of Siudeja [51]. This result is stronger than Pólya’s upper bound (3.5) on λ1 A3 /I , because AL2 /I = 36(l1 + l2 + l3 )2 /(l12 + l22 + l33 ) by (3.3) and this ratio is maximal for the equilateral triangle (when l1 = l2 = l3 ). Further, the normalized spectral gap (λ2 − λ1 )A2 /L2 is maximal among triangles for the equilateral, by more recent work of Siudeja [52], and thus λ2 A2 /L2 is maximal for the equilateral also. Hence (λ2 − λ1 )A3 /I and λ2 A3 /I are maximal for the equilateral, which improves on Corollary 3.2 for n = 2. Among general convex domains, λ1 A2 /L2 is maximal for degenerate rectangles by work of Pólya [46]. That result differs from our Conjecture 4.1 on λ1 A3 /I , where the equilateral triangle should also be a maximizer. Turning now to the inradius R, it is easy to see for triangles (or any polygon with an inscribed circle) that A/L is proportional to R. Hence the preceding upper bounds for eigenvalues of triangles using A2 /L2 can be restated using a normalizing factor of inradius squared. In particular, λ1 R 2 is maximal for the equilateral triangle. A more general result is due to Solynin [53]:

1820


among all N -gons with an inscribed circle, λ1 R 2 is maximal for the regular N -gon. Of course, among general domains the maximizer of λ1 R 2 is simply the disk, by domain monotonicity. For area normalization, Antunes and Freitas [2, Conjecture 6.1] conjecture that the Faber– Krahn lower bound on λ1 A has a sharp upper analogue that includes an isoperimetric correction term: they conjecture that among simply connected plane domains, 2 λ1 A πj0,1

π 2 L2 − 4π + 4 A

with equality for the disk and (in a limiting sense) for degenerate rectangles. Under a conformal mapping normalization, Pólya and Schiffer proved lower bounds on sums of reciprocal eigenvalues 1/λ1 + · · · + 1/λn , with the disk being extremal [48]. Extensions to surfaces with bounded curvature were proved by Bandle [8, p. 120], and to spectral zeta functions and doubly connected surfaces by Laugesen and Morpurgo [31,32]. Lastly, the scale invariant ratio λ2 /λ1 is maximal for the equilateral triangle among acute triangles, by work of Siudeja [52]. The conjecture remains open for obtuse triangles. For general domains, this Payne–Pólya–Weinberger functional is known to be maximal for the disk, by Ashbaugh and Benguria [6]. 8.2. Neumann eigenvalues Stronger inequalities are known than the one we found for μ2 A3 /I in (3.7) (which is the case n = 2 of Corollary 3.2). In fact, μ2 A is maximal for the equilateral triangle among triangles, and for the square among parallelograms, and for the disk among all bounded plane domains. The first of these stronger inequalities was proved recently by the authors [33, Theorem 3.1]. The second, for parallelograms, is unpublished work of the authors. The third inequality is a result of Szeg˝o and Weinberger [55,56]. These inequalities for μ2 A are stronger because A2 /I is maximal for the equilateral triangle among triangles, for the square among parallelograms, and for the disk among all domains. Our inequalities in Corollary 3.2 hold for all n 2. In contrast, the stronger inequalities fail to extend to n = 3. The maximizing domains are instead somewhat elongated: the “arithmetic mean” (μ2 + μ3 )A seems to be maximal among isosceles triangles for an aperture slightly greater than π/6 (according to numerical work), rather than for the equilateral triangle with aperture π/3; and (μ2 +μ3 )A seems to be maximal among parallelograms for the 2:1 rectangle rather than the square (see [7, §5] for comments on rectangles). The maximizer among convex domains is apparently not known. The only positive result is that the disk is maximal among 4-fold sym√ metric domains [7, §4]. Incidentally, it is open to maximize the geometric mean μ2 μ3 A. The disk is conjectured to be extremal, by I. Polterovich. Among convex plane domains, it is open to maximize μ2 L2 . The disk is not the maximizer, because the equilateral triangle and the square have a larger value (in fact, the same value). The maximizer for μ2 D 2 , where D is diameter, is known to be the degenerate obtuse isosceles triangle by work of Cheng [10, Theorem 2.1], [34, Proposition 3.6]. For the problems mentioned above, and for related conjectures on triangles, see [33, §IX]. Sums of reciprocal Neumann eigenvalues were minimized by Dittmar [12], under conformal mapping normalization.


1821

8.3. Lower bounds Sharp lower bounds on Dirichlet eigenvalue sums for triangles are proved in a companion paper [35], under diameter normalization. Lower bounds for the Neumann eigenvalue μ2 are found in an earlier work [34]. References to other lower bounds can be found in those papers. Acknowledgment We are grateful to Mark Ashbaugh for guiding us to relevant literature. Appendix A. Eigenvalues of equilateral triangles, rectangles, disks The Dirichlet eigenfunctions of equilateral triangles were derived about 150 years ago by Lamé [29, pp. 131–135]. (See the treatment in the text of Mathews and Walker [40, pp. 237–239] or in the paper by Pinsky [44]. Note also the recent exposition by McCartin [41].) Dirichlet eigenfunctions of rectangles and disks are well known too [8]. The eigenvalues are: 16π 2 /9 j12 + j1 j2 + j22 : j1 , j2 1 for an equilateral triangle of side 1, 2 π (j1 / l1 )2 + (j2 / l2 )2 : j1 , j2 1 for a rectangle of side lengths l1 , l2 , 2 jm,p : m 0, p 1 for the unit disk, where jm,p is the pth zero of the Bessel function Jm . The Neumann eigenvalues are: 16π 2 /9 j12 + j1 j2 + j22 : j1 , j2 0 for an equilateral triangle of side 1, 2 π (j1 / l1 )2 + (j2 / l2 )2 : j1 , j2 0 for a rectangle of side lengths l1 , l2 , 2 jm,p : m 0, p 1 for the unit disk, where jm,p is the pth zero of the Bessel derivative Jm . See [8,42]. The Robin eigenvalues of rectangles and disks can be found by separation of variables. The eigenvalues are known also for the equilateral triangle [43].

References [1] P. Antunes, P. Freitas, New bounds for the principal Dirichlet eigenvalue of planar regions, Experiment. Math. 15 (2006) 333–342. [2] P. Antunes, P. Freitas, A numerical study of the spectral gap, J. Phys. A 41 (2008) 055201, 19 pp. [3] P. Antunes, P. Freitas, On the inverse spectral problem for Euclidean triangles, Proc. Roy. Soc. A (2010), online. [4] M.S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in: Spectral Theory and Geometry, Edinburgh, 1998, in: London Math. Soc. Lecture Note Ser., vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 95–139. [5] M.S. Ashbaugh, The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile–Protter, and H.C. Yang, in: Spectral and Inverse Spectral Theory, Goa, 2000, Proc. Indian Acad. Sci. Math. Sci. 112 (2002) 3–30. [6] M.S. Ashbaugh, R.D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. of Math. (2) 135 (1992) 601–628. [7] M.S. Ashbaugh, R.D. Benguria, Universal bounds for the low eigenvalues of Neumann Laplacians in n dimensions, SIAM J. Math. Anal. 24 (1993) 557–570.

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[8] C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, MA, 1979. [9] F. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR 37 (1972) 1134–1167 (in Russian); English transl. in: Math. USSR-Izv. 6 (1972) (1973) 1117–1151. [10] S.Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975) 289–297. [11] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003. [12] B. Dittmar, Free membrane eigenvalues, Z. Angew. Math. Phys. 60 (2009) 565–568. [13] R.L. Frank, M. Loss, T. Weidl, Pólya’s conjecture in the presence of a constant magnetic field, J. Eur. Math. Soc. (JEMS) 11 (2009) 1365–1383. [14] P. Freitas, Upper and lower bounds for the first Dirichlet eigenvalue of a triangle, Proc. Amer. Math. Soc. 134 (2006) 2083–2089. [15] P. Freitas, Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi, J. Funct. Anal. 251 (2007) 376–398. [16] P. Freitas, B. Siudeja, Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals, ESAIM Control Optim. Calc. Var. 16 (3) (2010) 648–676. [17] F. Gesztesy, Z. Zhao, Domain perturbations, Brownian motion, capacities, and ground states of Dirichlet Schrödinger operators, Math. Z. 215 (1994) 143–150. [18] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer-Verlag, Berlin, 2001, reprint of the 1998 edition. [19] A. Girouard, N. Nadirashvili, I. Polterovich, Maximization of the second positive Neumann eigenvalue for planar domains, J. Differential Geom. 83 (2009) 637–662. [20] D. Han, K. Kornelson, D. Larson, E. Weber, Frames for Undergraduates, Stud. Math. Libr., vol. 40, American Mathematical Society, Providence, RI, 2007. [21] E.M. Harrell II, L. Hermi, Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Anal. 254 (2008) 3173–3191. [22] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Front. Math., Birkhäuser Verlag, Basel, 2006. [23] J. Hersch, Contraintes rectilignes parallèles et valeurs propres de membranes vibrantes, Z. Angew. Math. Phys. 17 (1966) 457–460. [24] W. Hooker, M.H. Protter, Bounds for the first eigenvalue of a rhombic membrane, J. Math. Phys. 39 (1960/1961) 18–34. [25] R. Kellner, On a theorem of Pólya, Amer. Math. Monthly 73 (1966) 856–858. [26] S. Kesavan, Symmetrization & Applications, Ser. Anal., vol. 3, World Scientific Publishing, Hackensack, NJ, 2006. [27] P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal. 106 (1992) 353–357. [28] P. Kröger, Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal. 126 (1994) 217–227. [29] M.G. Lamé, Leçons sur la Théorie Mathématique de L’Élasticité des Corps Solides, Deuxième édition, Gauthier– Villars, Paris, 1866. [30] A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997) 531–545. [31] R.S. Laugesen, Eigenvalues of the Laplacian on inhomogeneous membranes, Amer. J. Math. 120 (1998) 305–344. [32] R.S. Laugesen, C. Morpurgo, Extremals for eigenvalues of Laplacians under conformal mapping, J. Funct. Anal. 155 (1998) 64–108. [33] R.S. Laugesen, B.A. Siudeja, Maximizing Neumann fundamental tones of triangles, J. Math. Phys. 50 (2009) 112903. [34] R.S. Laugesen, B.A. Siudeja, Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality, J. Differential Equations 249 (2010) 118–135. [35] R.S. Laugesen, B.A. Siudeja, Dirichlet eigenvalue sums on triangles are minimal for equilaterals, Preprint, 2010, arXiv:1008.1316. [36] R.S. Laugesen, B.A. Siudeja, Sums of Laplace eigenvalues—rotations and tight frames in higher dimensions, Preprint, 2010, http://www.math.uiuc.edu/~laugesen/. [37] D. Lenz, P. Stollman, D. Wingert, Compactness of Schrödinger semigroups, Math. Nachr. 283 (2010) 94–103. [38] P. Li, S.-T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983) 309–318. [39] Z. Lu, J. Rowlett, The fundamental gap conjecture on polygonal domains, Preprint, arXiv:0810.4937, 2008. [40] J. Mathews, R.L. Walker, Mathematical Methods of Physics, second ed., W.A. Benjamin, New York, 1970. [41] B.J. McCartin, Eigenstructure of the equilateral triangle. I. The Dirichlet problem, SIAM Rev. 45 (2003) 267–287. [42] B.J. McCartin, Eigenstructure of the equilateral triangle. II. The Neumann problem, Math. Probl. Eng. 8 (2002) 517–539.


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[43] B.J. McCartin, Eigenstructure of the equilateral triangle. III. The Robin problem, Int. J. Math. Math. Sci. (2004) 807–825. [44] M.A. Pinsky, Completeness of the eigenfunctions of the equilateral triangle, SIAM J. Math. Anal. 16 (1985) 848– 851. [45] G. Pólya, Sur le rôle des domaines symétriques dans le calcul de certaines grandeurs physiques, C. R. Acad. Sci. Paris 235 (1952) 1079–1081. [46] G. Pólya, Two more inequalities between physical and geometrical quantities, J. Indian Math. Soc. (N.S.) 24 (1960) (1961) 413–419. [47] G. Pólya, On the eigenvalues of vibrating membranes, Proc. Lond. Math. Soc. (3) 11 (1961) 419–433. [48] G. Pólya, M. Schiffer, Convexity of functionals by transplantation, J. Anal. Math. 3 (1953–1954) 245–345, reprinted in: George Pólya: Collected Papers, vol. 3, MIT Press, Cambridge, MA, 1984, pp. 290–390. [49] G. Pólya, G. Szeg˝o, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, NJ, 1951. [50] M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press (Harcourt Brace Jovanovich Publishers), New York, 1978. [51] B. Siudeja, Sharp bounds for eigenvalues of triangles, Michigan Math. J. 55 (2007) 243–254. [52] B. Siudeja, Isoperimetric inequalities for eigenvalues of triangles, Indiana Univ. Math. J. 59 (2010) 1087–1120. [53] A.Yu. Solynin, Isoperimetric inequalities for polygons and dissymetrization, Algebra i Analiz 4 (1992) 210–234 (in Russian); translation in: St. Petersburg Math. J. 4 (1993) 377–396. [54] R.S. Strichartz, Estimates for sums of eigenvalues for domains in homogeneous spaces, J. Funct. Anal. 137 (1996) 152–190. [55] G. Szeg˝o, Inequalities for certain eigenvalues of a membrane of given area, J. Ration. Mech. Anal. 3 (1954) 343– 356. [56] H.F. Weinberger, An isoperimetric inequality for the N -dimensional free membrane problem, J. Ration. Mech. Anal. 5 (1956) 633–636.


On a probabilistic approach to the Schrödinger equation with a time-dependent potential Halim Doss Université de Paris-Dauphine CEREMADE, UMR CNRS no 7534, Place du Maréchal de Lattre de Tassigny, 75775, Paris cedex 16, France Received 7 July 2010; accepted 1 December 2010 Available online 22 December 2010 Communicated by Daniel W. Stroock

Abstract We study, by probabilistic methods, some classes of Schrödinger equations related to time-dependent potentials, analytic with respect to the space variable. © 2010 Elsevier Inc. All rights reserved. Résumé On étudie, par des méthodes probabilistes, certaines classes d’Équations de Schrödinger associées à des potentiels dépendant du temps, analytiques en la variable d’espace. © 2010 Elsevier Inc. All rights reserved. Keywords: Schrödinger equation; Time-dependent potentials; Analytic functions; Brownian motion; Stochastic differential equations

Consider the Schrödinger equation: ⎧ ⎨

h2 ∂Ψ (t, x) = − Ψ (t, x) + V (t, x)Ψ (t, x), 2m ⎩ ∂t Ψ (0, x) = f (x) ıh

(1)

where t ∈ R+ , x ∈ Rn , ı 2 = −1, V (resp. f ) is a sufficiently regular map from R+ × Rn E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.007

H. Doss / Journal of Functional Analysis 260 (2011) 1824–1835

(resp. Rn ), to the complex numbers C, h > 0, m > 0 are constants, =

1825

n

∂2 l=1 ∂x 2 l

is the usual

Laplacian. First of all, we introduce, for each p ∈ R∗ , a small perturbation p of the operator (p “tends” to 0 = , when p → 0) such that, if the potential V = (V (t, x)) and the initial condition f = (f (x)) admit continuous extensions to R+ × Cn and Cn respectively, holomorphic with respect to the space variable x, without any growth condition at infinity, then problem (1) (associated to p ), admits a unique strong solution Ψ = (Ψ (t, x)) which can be extended as a C 1 map from R+ × Cn to C, analytic with respect to x ∈ Cn . Moreover we have, under these conditions, a Feynman–Kac type stochastic representation of the solution Ψ . When the parameter p is equal to zero, we remark that problem (1) admits similarly a unique “regular” solution, by the same method and under some analyticity and growth conditions at infinity, also expressed in [7]. The present work is therefore a natural extension of the results established in [7,8] (see also [13,14]) to the case where the potential is time-dependent. It may also be related to the numerous studies devoted to the same problem and using the Feynman Integral developed, for instance, in [1–3,9–12] and references therein. However and to the best of our knowledge, the class of time-dependent potentials that we are able to handle here is totally new. In the sequel, we assume, without loss of generality, that the constant m, in problem (1), is equal to 1. Assumption (I). from R+ × Cn to C such that, for each t ∈ R+ , the partial 1. There exists a continuous map V n |R+ ×Rn = V . map: x ∈ C → V (t, x) ∈ C is holomorphic and V n 2. There exists a holomorphic map f from C to C such that f|Rn = f . We introduce a filtered probability space: (Ω, F , (Ft )t0 , P ) satisfying the usual conditions, and an Ft -Brownian motion (B t )t0 = (B1 (t), . . . , Bn (t), Bn+1 (t))t0 with values in Rn+1 such that B 0 = 0 a.s. All the proofs in the sequel are based on the following Itô formula, valid for holomorphic functions. Lemma 1. Consider (Z1 (t))t0 , . . . , (Zd (t))t0 , d complex-valued continuous Ft -semi-martingales and ϕ = (ϕ(t, x))t0, x∈Cd a C 1 map from R+ × Cd to C such that, for each t 0, the map: x ∈ Cd → ϕ(t, x) ∈ C is holomorphic. If Zt = (Z1 (t), . . . , Zd (t)), then we have, a.s., for every t 0: t ϕ(t, Zt ) = ϕ(0, Z0 ) + 0

∂ϕ (s, Zs ) ds + ∂t d

t

l=1 0

∂ϕ (s, Zs ) dZl (s) ∂xl

d t

∂ 2ϕ 1 (s, Zs ) d Z l , Z k s + 2 ∂xl ∂xk l,k=1 0

where

(2)

1826


Z l , Z k = Z l + ı Z l , Z k + ı(Zk )

= Z l , Z k − Z l , (Zk ) + ı Z l , Z k + Z l , (Zk ) .

Proof. Immediate, using the standard Itô formula and the analyticity condition, cf. [7].

2

1. Perturbations p , p = 0, of the Laplacian For each p ∈ R∗ and x = (x1 , x2 , . . . , xn ) ∈ Cn , we define n √ ∂2 (1 + ıpxl )2 2 p (x) = ∂xl l=1

(∗)

√ √ , ı 2 = −1. where ı = 1+ı 2 First, we study problem (1) by substituting by p . Theorem 1. Let p ∈ R∗ . Consider the Cauchy problem: ⎧ ⎨

h2 ∂ Ψ (t, x) = − p Ψ (t, x) + V (t, x)Ψ (t, x), 2 ⎩ ∂t Ψ (0, x) = f (x), t 0, x ∈ Rn ıh

(3)

where the data V and f satisfy Assumption (I). The problem (3) admits then a unique strong solution Ψ = (Ψ (t, x))t0,x∈Rn which can be extended as a C 1 map from R+ × Cn to C, analytic, for each t 0, with respect to the space variable x ∈ Cn . In addition, one has the following representation, for each (t, x) ∈ R+ × Cn :

t x 1 x Ψ (t, x) = E f Xt exp V t − s, Xs ds ıh

(4)

0

where (Xsx )s0 = (X1 (s), . . . , Xn (s))s0 is a diffusion process with values in Cn , given by ⎧ √ 1 1 2 1 ⎨ X (s) = exp ıp p hB (s) + hs − √ , + x √ l l l 2 p ı p ı ⎩ n l ∈ {1, 2, . . . , n}, s 0, x = (x1 , x2 , . . . , xn ) ∈ C .

(5)

Proof. Let x ∈ Cn , y ∈ C and 0 s u be fixed. Consider the solution Zt = Zts (x, y), s t u, of the following S.D.E. in Cn × C Cn+1 : t t x Zt = + σ (Zv ) dB v + b(v, Zv ) dv, y s

t ∈ [s, u]

(6)

s

where, for each z = (z1 , . . . , zn , zn+1 ) ∈ Cn+1 , v ∈ [s, u], σ (z) is the (n + 1, n + 1) complex diagonal matrix given by


σl,l (z) =

√ √ ıh 1 + p ızl ,

1827

l = 1, 2, . . . , n,

σn+1,n+1 (z) = 0 and ⎧ ⎨ b(v, z) = bl (v, z) l=1,2,...,n+1 , ⎩ bn+1 (v, z) = 1 V (u − v, z), bl (v, z) = 0 if l ∈ {1, 2, . . . , n}. ıh A simple computation shows that the Cn+1 -valued process Zt = Zts (x, y) (s t u), is given by the explicit formulae: ⎧ Zt = Z1 (t), . . . , Zn (t), Zn+1 (t) , where for l ∈ {1, 2, . . . , n}: ⎪ ⎪ ⎪ ⎪ √ 1 2 ⎪ 1 1 ⎪ ⎪ √ + xl exp ıp h Bl (t) − Bl (s) + p h(t − s) − √ , ⎨ Zl (t) = 2 p ı p ı ⎪ t ⎪ ⎪ 1 ⎪ ⎪ u − v, Z1 (v), . . . , Zn (v) dv. ⎪ Zn+1 (t) = y + V ⎪ ⎩ ıh

and (7)

s

Note that the diffusion process (Zts [(x, y), •])t∈[s,u] given by (6) and (7) is such that, for almost all ω ∈ Ω, the map (x, y) ∈ Cn+1 → Zts [(x, y), ω] ∈ Cn+1 is analytic, and we have, furthermore, the following remarkable property: ⎧ for each compact set K ⊆ Cn+1 , T > 0, (α, β) ∈ N × N: ⎪ ⎪ ⎪ α+β ⎪ ⎨ ∂ s C (x, y), ω Z Sup ω∈Ω t 0stuT ∂x α ∂y β ⎪ ⎪ ⎪ K (x,y)∈ ⎪ ⎩ where C = CK,T ,(α,β) is a positive constant

(8)

and | • | is the usual norm on Cn+1 . Now, let us denote by H the set of holomorphic maps from Cn+1 to C. For each r ∈ H, 0 s t u and (x, y) ∈ Cn × C, we put

Π(s,t) r(x, y) = E r Zts (x, y), • .

(9)

Remark, thanks to property (8), that Π(s,t) r(x, y) is well defined and holomorphic with respect to (x, y) ∈ Cn+1 . Lemma 2. Given the preceding definitions, Π(s,t) r(x, y) has a continuous derivative with respect to the variables s and t, under the condition 0 s t u. Proof. We know, by the Markov property, that for every 0 s t u and (x, y) ∈ Cn × C,

E r Zus (x, y) /Ft = Π(t,u) r Zts (x, y) ,

a.s.

(10)

1828


and so

Π(s,u) r(x, y) = E Π(t,u) r Zts (x, y) .

(11)

Let us denote by Lt the “infinitesimal generator” of the diffusion process (Zts (x, y)): ⎧ for every (x, y) ∈ Cn × C, t ∈ [s, u], and r ∈ H: ⎪ ⎨ n 2 √ ıh 1 ∂ 2 ∂ r(x, y). L r(x, y) = (1 + ıpx ) + V (u − t, x) ⎪ l ⎩ t 2 ∂y ∂xl2 ıh

(12)

l=1

Applying Lemma 1, we see, first, that for every t ∈ [s, u]:

Π(s,t) r(x, y) = E r Zts (x, y) = r(x, y) +

t

E Lv r Zvs (x, y) dv

(13)

s

since E

n t √ ∂r s √ Z ıh 1 + p ıZl (v) dBl (v) = 0 ∂xl v l=1 s

the integrability conditions justifying these computations are satisfied, by the estimations (8). We conclude, by (13), that the map: t ∈ [s, u] → Π(s,t) r(x, y) has a continuous derivative on [s, u]. Let us now study the derivative with respect to s, using formula (11) and putting θ (s) = Π(s,u) r(x, y)

for s ∈ [0, u[.

It follows from Lemma 1 and (8), that if t ∈ [s, u],

θ (s) = E Π(t,u) r Zts (x, y) t = Π(t,u) r(x, y) +

E Lv Π(t,u) r Zvs (x, y) dv.

s

Therefore, if 0 s < t < u: θ (s) − θ (t) 1 =− (s − t) (t − s) and we have, by continuity:

t s

E Lv [Π(t,u) r] Zvs (x, y) dv,

(14)


lim

θ (s) − θ (t) = −Lt [Π(t,u) r](x, y) (s − t)

lim

θ (s) − θ (t) = −Ls [Π(s,u) r](x, y). (s − t)

s→t ss

So, the left and right derivatives of the function θ , defined by (14), exist and are equal and continuous on [0, u[. We conclude, by the preceding computations that for every 0 t u, (x, y) ∈ Cn × C and r ∈ H, the map s ∈ [0, t] → Π(s,t) r(x, y) has a continuous derivative on [0, t] given by ∂ Π(s,t) r(x, y) = −Ls [Π(s,t) r](x, y). ∂s

2

(15)

End of the proof (Theorem 1): For every (x, y) ∈ Cn × C, let us denote r(x, y) = f(x) exp(y) where f satisfy Assumption (I). Consider, for each t ∈ [0, u] t (x, y) = Π(u−t,u) r(x, y). Ψ

(16)

t (x, y) is holomorphic with respect to The preceding considerations (Lemma 2) show that Ψ (x, y), C 1 with respect to t and we see, substituting in formula (15), t by u and s by u − t, is a solution of the following problem: respectively, that Ψ ⎧ ⎨ ∂Ψ t (x, y) = Lu−t [Ψ t ](x, y), ∂t ⎩ Ψ0 (x, y) = r(x, y) = f(x) exp(y);

(17) (x, y) ∈ Cn × C, t ∈ [0, u];

but t ](x, y) = Lu−t [Ψ

n 2 ∂ √ ıh 1 2 ∂ t (x, y) Ψ (1 + ıpxl ) + V u − (u − t), x 2 ∂y ∂xl2 ıh l=1

and

t (x, y) = E r Zuu−t (x, y) = E f Z1 (u), . . . , Zn (u) exp Zn+1 (u) Ψ

(18)

1830


where, for each l ∈ {1, 2, . . . , n} and v u − t: Zl (v) is given by formula (7) where we substitute t by v and s by u − t respectively; furthermore 1 Zn+1 (u) = y + ıh

u

u − v, Z1 (v), . . . , Zn (v) dv V

u−t

1 =y+ ıh

t

t − w, Z1 (w + u − t), . . . , Zn (w + u − t) dw. V

0

We remark then, by homogeneity, that the processes (Z1 (w + u − t), . . . , Zn (w + u − t))w0 x) x and (Xw w0 have the same law, where (Xw )w0 is given by formula (5). So, if Ψ (t, x) and Ψt (x, y) are defined by formulae (4) and (16) respectively, we have the following relation: t (x, y) = Ψ (t, x) exp(y) Ψ

(19)

and, coming back to Eqs. (17) and (18), we see that (Ψ (t, x))t0, x∈Cn gives us a strong solution of the Cauchy problem (3), satisfying the conditions imposed in Theorem 1. Uniqueness: Let (X (t, x)) be a strong solution of the Cauchy problem (3), satisfying the conditions imposed in Theorem 1. For each (x, y) ∈ Cn × C and t 0 define t (x, y) = X (t, x) exp(y). X

(20)

Let u ∈ R∗+ and consider (Zt )t∈[0,u] the process solving S.D.E. (6) (where we fix s = 0), given by (7). Introduce, for every t ∈ [0, u]: u−t (Zt ) = X u − t, Z1 (t), . . . , Zn (t) exp Zn+1 (t) . ϕ(t) = X We see, thanks to Lemma 1 and using the notation of Theorem 1, that ϕ(u) = f Z1 (u), . . . , Zn (u) exp Zn+1 (u) u x 1 x = f Xu exp y + V u − s, Xs ds ıh 0

u ∂ = ϕ(0) + Mu + − Xu−v (Zv ) + Lv Xu−v (Zv ) dv ∂u 0

u (x, y) + Mu =X

(21)


1831

where Mu is a square integrable stochastic integral, such that E{Mu } = 0 (thanks to estimations (8)) and Lv is the “infinitesimal generator” given by (12) (substituting t by v in that formula). Taking expectations on both sides of (21), we conclude that for every u > 0 and (x, y) ∈ Cn × C, we have u x 1 x Xu (x, y) = E f Xu exp y + V u − s, Xs ds ıh 0

= Ψ (u, x) exp(y) = X (u, x) exp(y) and hence, for every u 0 and x ∈ Cn : X (u, x) = Ψ (u, x).

2

Remark 1. Let us come back to formula (10) (Markov property); we see that the process:

Π(t,u) r Zts (x, y) , t ∈ [s, u]

is a continuous Ft -martingale on [s, u]. Moreover, knowing that Π(t,u) r(x, y) has a continuous derivative with respect to t (on [s, u]), and is analytic with respect to (x, y) ∈ Cn × C, we can apply again Lemma 1: ⎧ a.s., for every t ∈ [s, u], ⎪ ⎪ s ⎪ ⎪ ⎪ ⎨ Π(t,u) r Zt (x, y) = Π(s,u) r(x, y) + Nt t ⎪ s s ∂ ⎪ ⎪ Π(v,u) r Zv (x, y) + Lv r Zv (x, y) dv + ⎪ ⎪ ⎩ ∂v

(22)

s

where Lv is given in (12) and (Nt )t∈[s,u] is a continuous Ft -martingale. Appealing to classical theory (cf. e.g. [7]), we deduce from (22) that ⎧ ⎨ a.s., for every v ∈ [s, u], ∂ ⎩ Π(v,u) r Zvs (x, y) + Lv r Zvs (x, y) = 0 ∂v

(23)

and, if v = s: ∂ Π(s,u) r(x, y) + Ls r(x, y) = 0, ∂s recovering formula (15). Remark 2. Let us clarify, from a probabilistic point of view, in what sens the operator p , p = 0, given by (∗), converges to , when p → 0.

1832


For each p ∈ R∗ and x ∈ Cn , consider x p,x Xs s0 = Xs s0 = X1 (s), . . . , Xn (s) s0 the diffusion process with values in Cn , given by (5), appearing in the Feynman–Kac formula (4) (see Theorem 1). We see that, for each l ∈ {1, 2, . . . , n}, (Xl (s))s0 is a solution of the following S.D.E.: √ √ Xl (s) = xl + ıh 1 + ıpXl (s) dBl (s) s

(24)

0

(cf. [7,6]) and we deduce from (24), using classical estimations, that: for each T > 0 and compact K ⊆ Cn , there exists a constant C = CT ,K > 0 such that for every p ∈ ([−1, +1]\{0}): √ 2 p,x

Supx∈K E Supt∈[0,T ] Xt − (x + ıhBt ) p 2 C − −−→ 0 p→0

(25)

where (Bt )t0 = (B1 (t), . . . , Bn (t))t0 . p,x

Therefore, the diffusion process (Xt )t0 , related to the operator ıh 2 p (via Theorem 1), √ (cf. Theorem 2, below). “converges”, when p → 0, to the process (x + ıhBt )t0 , related to ıh 2 2. Study of the limit case p = 0 Let be a non-empty open set of Rn and V (resp. f ), a sufficiently regular map from R+ × (resp. ), to the complex numbers C. As in [7], we study here, under analyticity conditions, the Schrödinger equation: ⎧ ⎨

∂Ψ h2 (t, x) = − Ψ (t, x) + V (t, x)Ψ (t, x), 2 ⎩ ∂t Ψ (0, x) = f (x) ıh

(26)

where (t, x) ∈ R+ × , ı 2 = −1. Consider D = D the open set of Cn given by

√ D = x + ıy where x ∈ , y ∈ Rn .

(27)

Assumption (I ). from R+ × D to C such that, for each t ∈ R+ , the partial 1. There exists a continuous map V (t, •) : x ∈ D → V (t, x) ∈ C is holomorphic and V |R+ × = V . map V 2. There exists a holomorphic map f from D to C such that f| = f . We are going to formulate a theorem analogous to Theorem 1, and which corresponds to the p,x limit case p = 0. Note, however, that the diffusion process (Xtx )t0 = (Xt )t0 related to the


1833

operator ıh 2 p , which satisfies estimations (8), when p = 0, must be substituted, when p = 0 by the simple process (Xt0,x )t0 where Xt0,x = x +

√ ıhBt

(28)

and since the latter process doesn’t satisfy estimations (8), we shall introduce, as in [7], for and f. integrability reasons, some growth conditions at infinity concerning the data V The open set D = D ⊆ Cn being given by (27), we introduce the space: (∗∗) T = {ϕ : D → C satisfying: for each x o ∈ , there exists an open neighbourhood Ux o ⊆ of x o and positive√constants αx o and βx o such that, if (x, y) ∈ Ux o × Rn , we have the inequality |ϕ(x + ıy)| exp(αx o + βx o .|y|)}. Assumption (II). Assumption (I ) being satisfied, we assume furthermore that (t, •)} ∈ T where V is the imaginary part of V . 1. For every T > 0: Supt∈[0,T ] exp{V 2. f ∈ T . Theorem 2. Under Assumption (II) about the data f and V , the problem (26) admits a unique strong solution Ψ = (Ψ (t, x))(t,x)∈R+ × which can be extended as a C 1 map from R+ × D to C, analytic with respect to x, x ∈ D, and such that, for every T > 0: Supt∈[0,T ] |Ψ (t, •)| ∈ T . Moreover, one has the following representation, for each (t, x) ∈ R+ × D: Ψ (t, x) = E f(x +

√

1 ıhBt ) exp ıh

t

√ (t − s, x + ıhBs ) ds V

.

(29)

0

Proof. It follows, step by step, the proof of Theorem 1, as the random variables involved in this context are integrable, thanks to Assumption (II); cf. [7]. 2 Examples. Let v be continuous a map from R+ × C to C such that, for every t 0, the partial map: x ∈ C → v(t, x) ∈ C is holomorphic. (1) Given l = (l1 , l2 , . . . , ln ) ∈ Rn \{0} and c ∈ R, we consider the open set of Rn :

= x = (x1 , x2 , . . . , xn ) ∈ Rn such that l.x + c = 0 where l.x =

n

j =1 lj xj ,

and the map V : R+ × → C defined by

V (t, x) = v t,

1 , |l.x + c|r

for every (t, x) ∈ R+ ×

(30)

where r > 0 is fixed. We introduce the open set of Cn D = D , given by (27). Let f be an analytic map from D to C, belonging to the space T , defined by (∗∗).

1834


Proposition 3. With the preceding notations and assumptions about the data V and f , consider the Schrödinger equation: ⎧ ⎨

h2 1 ∂Ψ Ψ (t, x), (t, x) = − Ψ (t, x) + v t, ıh ∂t 2 |l.x + c|r ⎩ Ψ (0, x) = f (x), where (t, x) ∈ R+ × .

(31)

The problem (31) admits a unique strong solution, Ψ = (Ψ (t, x))(t,x)∈R+ × , which can be extended as a C 1 map from R+ × D to C, analytic with respect to x, x ∈ D, and such that for every T > 0, Supt∈[0,T ] |Ψ (t, •)| ∈ T . For (t, x) ∈ R+ × D, the solution is given by the formula Ψ (t, x) = E f (x +

√

1 ıhBt ) exp ıh

t √ r 2 ds v t − s, exp − log (l.x + c + ıhl.Bs ) 2 0

(32) where the symbol log[ ] represents the principal complex determination of the logarithmic function extended to C\J , where J = ıR+ . Proof. One can easily prove, using the estimations established in [7, Proposition 4], that the potential V given in (30), and the initial function f satisfy Assumption (II). Therefore Proposition 3 follows from a direct application of Theorem 2. 2 (2) With the notations of Example 1, one can handle similarly other potentials given, for instance, by ⎧ ⎪ ⎨ V (t, x) = v t,

1 , t 0; r ∈ N∗ fixed, (sin(l.x + c))r ⎪ ⎩ x ∈ = x ∈ Rn : l.x + c = kπ, for every k ∈ Z, r V (t, x) = v t, tan(l.x + c) , t 0; r ∈ N∗ fixed,

x ∈ = x ∈ Rn : l.x + c = π2 + kπ, for every k ∈ Z , V (t, x) = v t, P (x) , t 0, x ∈ Rn

(33)

(34) (35)

where P (•) is a polynomial function on Cn of degree d ∈ N∗ , P (x) = a0 + a1 .x + · · · + ad .x d , with the following conditions: • al , l = 0, 1, . . . , d is a symmetric multilinear map from (Cn )l to C such that al [(Rn )l ] ⊆ R; • d = 2 + 4k, k ∈ N and (−1)k .ad < 0. The previous examples illustrate the fact that, by Theorem 2, one can solve the Schrödinger equation with time-dependent potentials admitting strong singularities.


1835

Remark 3. Consider the solution Ψ = (Ψh (t, x)) of the Schrödinger equation (3) (given by Theorem 1), or (26) (given by Theorem 2). We can prove, using the results established in [4,8], for “small” t, when h → 0, semi-classical asymptotic expansions, for every N ∈ N∗ :

ı Ψh (t, x) = exp S(t, x) b0 (t, x) + b1 (t, x)h + · · · + bN (t, x)hN + ◦ hN h

(36)

where the functions S(t, x) and bl (t, x), l ∈ N, have explicit expressions with the help of Wiener integrals “around” a “critical” trajectory γ (t,x) . P.S. Our motivation for studying Schrödinger equation stems from the work of [5]. References [1] S. Albeverio, Z. Brzezniak, Z. Haba, On the Schrödinger equation with potentials which are Laplace transforms of measures, Potential Anal. 9 (1) (1998) 65–82. [2] S. Albeverio, R. Høegh-Krohn, S. Mazzuchi, Mathematical Theory of Feynman Path Integrals. An Introduction, second and enlarged edition, Lecture Notes in Math., vol. 523, Springer-Verlag, Berlin/Heidelberg, 2008. [3] S. Albeverio, S. Mazzuchi, The time-dependent quartic oscillator—a Feynman path integral approach, J. Funct. Anal. 238 (2) (2006) 471–488. [4] R. Azencott, H. Doss, L’Équation de Schrödinger quand h tend vers zéro : une approche probabiliste, in: Stochastic Aspects of Classical and Quantum Systems, Marseille, 1983, in: Lecture Notes in Math., vol. 1109, Springer, Berlin, 1985, pp. 1–17. [5] L. de Broglie, Recherches d’un demi siècle, Albin Michel, Paris, 1976. [6] H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré XIII (2) (1977) 99–125. [7] H. Doss, Sur une Résolution Stochastique de l’Équation de Schrödinger à Coefficients Analytiques, Comm. Math. Phys. 73 (1980) 247–264. [8] H. Doss, Démonstration probabiliste de certains développements asymptotiques quasi-classiques, Bull. Sci. Math. (2) 109 (1985) 179–208. [9] M. Grothaus, L. Streit, A. Vogel, Feynman integrals as Hida distributions – the case of non-perturbative potentials, Asterisque 327 (2010) 55–68. [10] G.W. Johnson, M.L. Lapidus, The Feynman integral and Feynman’s operational calculus, Oxford Math. Monogr. (2002). [11] T. Kuna, L. Streit, W. Westerkamp, Feynman integrals for a class of exponentially growing potentials, J. Math. Phys. 39 (9) (1998) 4476–4491. [12] S. Mazzuchi, Functional integral solution for the Schrödinger equation with polynomial potential: a white noise approach, Technical Report UTM736, Mathematica, Dept. Math., Univ. of Trento, 2010. [13] H. Thaler, Solutions of Schrödinger equations on compact Lie groups via probabilistic methods, Potential Anal. 18 (2003) 119–140. [14] H. Thaler, The Doss trick on symmetric spaces, Lett. Math. Phys. 72 (2005) 115–127.


Resonance free regions in magnetic scattering by two solenoidal fields at large separation Ivana Alexandrova a,∗,1 , Hideo Tamura b a 124 Austin Building, Department of Mathematics, East Carolina University, Greenville, NC 27858, USA b Department of Mathematics, Okayama University, Okayama, 700-8530, Japan

Received 11 July 2010; accepted 6 December 2010 Available online 22 December 2010 Communicated by L. Gross

Abstract We consider the problem of quantum resonances in magnetic scattering by two solenoidal fields at large separation in two dimensions. This system has trapped trajectories oscillating between two centers of the fields. We give a sharp lower bound on resonance widths when the distance between the two centers goes to infinity. The bound is described in terms of backward amplitudes calculated explicitly for scattering by each solenoidal field. The study is based on a new type of complex scaling method. As an application, we also discuss the relation to semiclassical resonances in scattering by two solenoidal fields. © 2010 Elsevier Inc. All rights reserved. Keywords: Resonances; Magnetic scattering; Solenoidal fields; Aharonov–Bohm effect

1. Introduction In the present paper we study the problem of quantum resonances in magnetic scattering by two solenoidal fields at large separation. We work in two dimensions R 2 throughout the entire discussion. We write H (A) = (−i∇ − A)2 =

2 (−i∂j − aj )2 ,

∂j = ∂/∂xj

j =1


E-mail addresses: [email protected] (I. Alexandrova), [email protected] (H. Tamura). 1 Current address: 1400 Washington Avenue, ES 110, State University of New York, Albany, NY 12222, USA.


I. Alexandrova, H. Tamura / Journal of Functional Analysis 260 (2011) 1836–1885

1837

for the magnetic Schrödinger operator with potential A = (a1 , a2 ) : R 2 → R 2 . The magnetic field b : R 2 → R associated with the vector potential A is defined by b(x) = ∇ × A(x) = ∂1 a2 − ∂2 a1 and the magnetic flux of b is defined by α = (2π)−1 b(x) dx, where the integration with no domain attached is taken over the whole space. Let Φ : R 2 → R 2 be the potential defined by Φ(x) = −x2 /|x|2 , x1 /|x|2 = −∂2 log |x|, ∂1 log |x| ,

(1.1)

which generates the point-like field (solenoidal field) ∇ × Φ = ∂1 ∂1 log |x| + ∂2 ∂2 log |x| = log |x| = 2πδ(x) with center at the origin. The quantum particle moving in the solenoidal field 2παδ(x) with α as a magnetic flux is governed by the energy operator Pα = H (αΦ).

(1.2)

This is symmetric over C0∞ (R 2 \ {0}), but it is not necessarily essentially self-adjoint in the space L2 = L2 (R 2 ) because of the strong singularity at the origin of Φ. We know [1,8] that it is a symmetric operator with type (2, 2) of deficiency indices. The self-adjoint extension is realized by imposing a boundary condition at the origin. Its Friedrichs extension denoted by the same notation Pα has the domain D(Pα ) = u ∈ L2 : (−i∇ − αΦ)2 u ∈ L2 , lim u(x) < ∞ , |x|→0

(1.3)

where (−i∇ − αΦ)2 u is understood in D (R 2 \ {0}) (in the sense of distribution). The energy operator which governs quantum particles moving in a solenoidal field is often called the Aharonov–Bohm Hamiltonian in the physics literatures. This model was employed by Aharonov and Bohm [4] in 1959 in order to convince us theoretically that a magnetic potential itself has a direct significance in quantum mechanics. This phenomenon, unexplainable from a classical mechanical point of view, is now called the Aharonov–Bohm effect, which is known as one of the most remarkable quantum phenomena. The scattering by one solenoidal field is also known as one of the exactly solvable quantum systems. We give a quick review of it in Section 2. In particular, the amplitude fα (θ → ω; E) for scattering from the initial direction ω ∈ S 1 to the final direction θ at energy E > 0 is explicitly calculated as fα (ω → θ ; E) = (2/π)1/2 eiπ/4 E −1/4 sin(απ)ei[α](θ−ω)

ei(θ−ω) , 1 − ei(θ−ω)

(1.4)

where the Gauss notation [α] denotes the greatest integer not exceeding α and the coordinates over the unit circle S 1 are identified with the azimuth angles from the positive x1 axis. We also

1838


note that there are no resonances in the case of scattering by one solenoidal field, as seen in Section 2 below. We formulate the problem which we want to discuss in this paper. We consider the energy operator Hd = H (Φd ),

Φd (x) = α1 Φ(x − d1 ) + α2 Φ(x − d2 ),

(1.5)

which describes the quantum particle moving in the two solenoids 2πα1 δ(x − d1 ) and 2πα2 δ(x − d2 ). The operator Hd becomes self-adjoint under the boundary conditions lim|x−dj |→0 |u(x)| < ∞ for j = 1, 2, and the resolvent R(ζ ; Hd ) = (Hd − ζ )−1 : L2 → L2 ,

ζ = E + iη, E > 0, η > 0,

is meromorphically continued over the lower half of the complex plane across the positive real axis where the spectrum of Hd is located. Then R(ζ ; Hd ) with Im ζ 0 is well defined as an operator from L2comp to L2loc in the sense that χR(ζ ; Hd )χ : L2 → L2 is bounded for every χ ∈ C0∞ (R 2 ), where L2comp and L2loc denote the spaces of square integrable functions with compact support and of locally square integrable functions over R 2 , respectively. We refer to [14, Section 7] for the spectral properties of Hd : Hd has no bound states and the spectrum is absolutely continuous on [0, ∞). The meromorphic continuation of R(ζ ; Hd ) over the unphysical sheet (the lower-half plane) follows as an application of the analytic perturbation theory of Fredholm for compact operators. For completeness, we shall show it in Appendix A. The resonances of Hd are defined as the poles of the meromorphic function with values in operators from L2comp to L2loc . Our aim is to study to what extent R(ζ ; Hd ) can be analytically extended across the positive real axis as the distance |d| = |d2 − d1 | goes to infinity. We give a sharp lower bound on the resonance widths (imaginary parts of resonances) in terms of the backward amplitude fj (ω → −ω; E) for scattering by each solenoidal field 2παj δ(x). As is seen from (1.4), the backward amplitude takes the form fj (ω → −ω; E) = (2π)−1/2 eiπ/4 E −1/4 (−1)[αj ]+1 sin(αj π), which is independent of the direction ω. The main theorem is as follows. Theorem 1.1. Let the notation be as above and let E > 0. Assume that neither the flux α1 nor α2 is an integer. Set dˆ = d/|d| for d = d2 − d1 . Then, for any ε > 0 small enough, there exists dε (E) 1 large enough such that ζ = E − iη with 0 < η < ηεd (E) is not a resonance of Hd for |d| > dε (E), where ηεd (E) =

E 1/2 ˆ E)f2 (dˆ → −d; ˆ E) − ε . log |d| − logf1 (−dˆ → d; |d|

Remark 1.1. If either of the two fluxes α1 and α2 is an integer, Hd is easily seen to be unitarily equivalent to the Hamiltonian with one solenoidal field, and hence Hd has no resonances. Since the scattering amplitude vanishes for an integer flux, Theorem 1.1 remains true in this special case also.


1839

Remark 1.2. A slightly modified argument applies to magnetic Schrödinger operators with fields with compact supports at large separation. For example, such an argument applies to the operator Hd = (−i∇ − Bd )2 ,

Bd (x) = A1 (x − d1 ) + A2 (x − d2 ),

where Aj ∈ C ∞ (R 2 → R 2 ) has the fields bj = ∇ × Aj ∈ C0∞ (R 2 → R). The result of Theorem 1.1 remains true with the backward amplitude for scattering by the fields bj . Corollary 1.1. Assume that the same assumptions as in Theorem 1.1 are fulfilled. If ζd (E) = E + i Im ζd (E) is a resonance of Hd , then, for any ε > 0 small enough, there exists dε (E) 1 such that the resonance width − Im ζd (E) satisfies − Im ζd (E) > ηεd (E) for |d| > dε (E). We make a comment on how to determine the constant ηεd (E) in the theorem. It is determined so that 2ik|d| e ˆ ˆ ˆ ˆ |d| f1 (−d → d; E)f2 (d → −d; E) < 1 − ε/2,

k = ζ 1/2

(1.6)

for |d| 1, provided that ζ = E − iη satisfies 0 < η < ηεd (E). We shall explain here from a heuristic point of view how sharp the bound in the theorem is and how reasonable ρ0 =

e2ik|d| ˆ E)f2 (dˆ → −d; ˆ E) = 1 f1 (−dˆ → d; |d|

(1.7)

is as an approximate relation to determine the location of the resonances near the real axis. We first consider the scattering by the solenoidal filed 2παδ(x). As stated in Proposition 5.1 in Section 5, the Green function Rα (x, y; ζ ) of the resolvent R(ζ ; Pα ) = (Pα − ζ )−1 with ζ = E − iη in the lower-half plane behaves like −1/2 Rα (x, y; ζ ) ∼ eik|x−y| |x − y|−1/2 + eik(|y|+|x|) |y||x| fα (−yˆ → x; ˆ E)

(1.8)

with yˆ = y/|y| and xˆ = x/|x| when |x|, |y| 1 and |x − y| 1, where k = ζ 1/2 and some numerical factors are ignored for brevity. The first term on the right side corresponds to the free trajectory which goes from y to x directly without being scattered at the origin, while the second term comes from the scattering trajectory which starts from y and arrives at x after being scattered by 2παδ(x). We now turn to scattering by the two solenoidal fields 2πα1 δ(x) and 2πα2 δ(x − d) with the origin and d ∈ R 2 as centers. We denote by fj (ω → θ ) the amplitude for scattering from the direction ω to θ by 2παj δ(x), and in particular, we write simply f1 and f2 for the backward ˆ and f2 (dˆ → −d), ˆ respectively. According to the asymptotic formula amplitudes f1 (−dˆ → d) (1.8), the quantity associated with the trajectory starting from the origin and coming back to the origin after being scattered by 2πα2 δ(x − d) takes the form (e2ik|d| /|d|)f2 , which is seen by setting x = y = −d in the second term on the right side of (1.8). Let τ0 (x, y) be the trajectory which starts from y, hits the origin and arrives at x from the origin after oscillating between the origin and d several times. Then the contribution from τ0 (x, y) to the asymptotic form of the Green function is formally given by the series

1840


−1/2 eik|x−y| |x − y|−1/2 + eik(|y|+|x|) |y||x| f1 (−yˆ → x) ˆ ∞

2ik|d| ˆ + eik|y| |y|−1/2 f1 (−yˆ → d) ρn /|d| f2 f1 (−dˆ → x)e ˆ ik|x| |x|−1/2 , e 0

n=0

where ρ0 is defined by (1.7). For example, the term with ρ0n describes the contribution from the trajectory oscillating n + 1 times. Thus the location of the resonance is approximately determined by the relation ρ0 = 1, and this intuitive idea clarifies the mechanism by which trapping trajectories generate the resonances near the real axis. The rigorous proof of Theorem 1.1 is based on a new type of complex scaling method. The details are explained in Section 3 where we prove the theorem, accepting some lemmas as proved, and Sections 4, 5 and 6 are devoted to proving those lemmas. One of the difficulties in the resonance problem is that we have to control quantities growing exponentially at infinity. Such quantities cannot be controlled simply by integration by parts using oscillatory properties. We use a new method of complex scaling to avoid these difficulties. We discuss the relation to the semiclassical theory for quantum resonances in scattering by two solenoidal fields. We now consider the self-adjoint operator H˜ h = (−ih∇ − Ψ )2 ,

Ψ (x) = α1 Φ(x − p1 ) + α2 Φ(x − p2 ),

0 < h 1,

under the boundary conditions lim|x−pj |→0 |u(x)| < ∞ at the two centers p1 and p2 . We denote by γ (x) the azimuth angle from the positive x1 axis to xˆ = x/|x| and define the two unitary operators (U1 f )(x) = h−1 f h−1 x ,

(U2 f )(x) = exp igh (x) f (x)

acting on L2 , where gh = [α1 / h]γ (x − d1 ) + [α2 / h]γ (x − d2 ) with dj = pj / h. Since ∇γ (x) = Φ(x), gh (x) satisfies ∇gh = [α1 / h]Φ(x − d1 ) + [α2 / h]Φ(x − d2 ), and exp(igh (x)) is well defined as a single valued function. Then H˜ h turns out to be unitarily equivalent to H (Ψd ) = (U1 U2 )∗ H˜ h (U1 U2 ), where Ψd (x) = β1 Φ(x − d1 ) + β2 Φ(x − d2 ),

βj = αj / h − [αj / h],

dj = pj / h.

Thus the semiclassical resonance problem in scattering by two solenoidal fields is reduced to the resonance problem for magnetic Schrödinger operators with two solenoidal fields with centers at large separation |d| = |d2 − d1 | = |p2 − p1 |/ h = |p|/ h 1. We denote by f˜j (ω → −ω; E), j = 1, 2, the amplitude for the backward scattering by the field 2πβj δ(x) at energy E > 0 and by f˜hj (ω → −ω; E) the semiclassical amplitude for the scattering by the field 2παj δ(x). The two amplitudes are related through f˜hj (ω → −ω; E) =


1841

h1/2 f˜j (ω → −ω; E) by (1.4) with E and α replaced by E/ h2 and α/ h, respectively, and hence it follows that ˆ E)f˜2 (pˆ → −p; ˆ E) = logf˜h1 (−pˆ → p; ˆ E)f˜h2 (pˆ → −p; ˆ E) − log h. logf˜1 (−pˆ → p; The fluxes β1 and β2 vary with h. If at least one of the two fluxes β1 and β2 is an integer, then ˆ E)f˜2 (pˆ → −p; ˆ E) = ∞, − logf˜1 (−pˆ → p;

pˆ = p/|p|,

because the scattering amplitude vanishes for an integer flux. The choice of dε (E) in Theorem 1.1 depends on the fluxes α1 and α2 as well as on the energy E > 0. We require the additional assumption that β1 and β2 stay away from 0 and 1 uniformly in h; c < β1 , β2 < 1 − c for some 0 < c < 1/2. Then we obtain the following result as an immediate consequence of Theorem 1.1. Corollary 1.2. Let the notation be as above. Assume that βj = αj / h − [αj / h], j = 1, 2, fulfills the flux condition above. Then, for any ε > 0 small enough, there exists hε (E) > 0 such that ζ = E − iη with 0 0 is calculated as ϕ+ (x; ω, E) =

exp(−iνπ/2) exp ilγ (x; −ω) Jν E 1/2 |x|

(2.2)

l

with ν = |l − α|, where Jμ (z) denotes the Bessel function of order μ. The eigenfunction ϕ+ behaves like ϕ+ (x; ω, E) ∼ ϕ0 (x; ω, E) = exp iE 1/2 x · ω as |x| → ∞ in the direction −ω (x = −|x|ω), and the difference ϕ+ − ϕ0 satisfies the outgoing radiation condition at infinity. We decompose ϕ+ (x; ω, E) into the sum ϕ+ = ϕin + ϕsc of incident and scattering waves to calculate the scattering amplitude through the asymptotic behavior at infinity of the scattering wave ϕsc (x; ω, E). The idea is due to Takabayashi [16]. If we set σ = σ (x; ω) = γ (x; ω) − π , then ϕ+ =

l

e−iνπ/2 eilσ Jν E 1/2 |x| ,

ν = |l − α|.


1843

If we further make use of the formula e−iμπ/2 Jμ (iw) = Iμ (w) for the Bessel function π Iμ (w) = (1/π)

∞ ew cos ρ cos(μρ) dρ − sin(μπ)

0

e−w cosh p−μp dp

(2.3)

0

with Re w 0 [22, p. 181], then ϕ+ (x; ω, E) takes the form ϕ+ = (1/π)

π e

l

− (1/π)

e−i

ilσ

√

E|x| cos ρ

cos(νρ) dρ

0

∞

e

ilσ

ei

sin(νπ)

l

√ E|x| cosh p −νp

e

(2.4)

dp.

0

We take ϕin (x; ω, E) as ϕin = eiασ ϕ0 (x; ω, E) = eiασ ei

√ E|x| cos γ (x;ω)

= eiασ e−i

√

E|x| cos σ

,

which is different from the usual plane wave ϕ0 (x; ω, E). The modified factor eiασ appears because of the long-range property of the potential Φ(x) defined by (1.1). The incident wave admits the Fourier expansion ϕin (x; ω, E) = (1/π)

π l

e

√ −i E|x| cos ρ

cos(νρ) dρ eilσ (x;ω) .

0

This, together with (2.4), yields ϕsc (x; ω, E) = −(1/π)

∞ e

ilσ

ei

sin(νπ)

l

√ E|x| cosh p −νp

e

dp.

0

We compute the series

e

ilσ −νp

e

sin(νπ) =

l

l[α]

eilσ e−νp sin(νπ) + l[α]+1

= sin(απ)(−1)[α]

e−αp (eiσ ep )[α] eαp (eiσ e−p )[α] + 1 + e−iσ e−p 1 + e−iσ ep

for |σ | < π . Thus we have sin(απ) ϕsc = − (−1)[α] ei[α]σ (x;ω) π

∞

−∞

ei

√ E|x| cosh p

e−βp dp 1 + e−iσ e−p

1844


with β = α − [α]. We apply the stationary phase method to the integral on the right side. Since eiσ (x;ω) = ei(γ (x;ω)−π) = −ei(θ−ω) by identifying θ = x/|x| = xˆ ∈ S 1 with the azimuth angle θ , we see that ϕsc (x; ω, E) obeys ˆ E)ei ϕsc = fα (ω → x;

√ E|x|

|x|−1/2 + o |x|−1/2 ,

|x| → ∞.

Here fα (ω → θ ; E) defined by (1.4) for θ = ω is called the amplitude for scattering from the initial direction ω ∈ S 1 to the final one θ at energy E > 0. If, in particular, α is an integer, then fα (ω → θ ; E) vanishes. We calculate the Green function of the resolvent R(ζ ; Pα ) = (Pα − ζ )−1 with Im ζ > 0. Let k = ζ 1/2 , Im k > 0, and let Plα be as in (2.1). Then the equation (Plα − ζ )u = 0 has {r 1/2 Jν (kr), r 1/2 Hν (kr)} with Wronskian 2i/π as a pair of linearly independent solutions, (1) where Hμ (z) = Hμ (z) denotes the Hankel function of the first kind. Thus (Plα − ζ )−1 has the integral kernel Rlα (r, ρ; ζ ) = (iπ/2)r 1/2 ρ 1/2 Jν k(r ∧ ρ) Hν k(r ∨ ρ) ,

ν = |l − α|,

where r ∧ ρ = min(r, ρ) and r ∨ ρ = max(r, ρ). Hence the Green function Rα (x, y; ζ ) of R(ζ ; Pα ) is given by Rα (x, y; ζ ) = (i/4)

eil(θ−ω) Jν k |x| ∧ |y| Hν k |x| ∨ |y| ,

(2.5)

l

where x = (|x| cos θ, |x| sin θ ) and y = (|y| cos ω, |y| sin ω) in the polar coordinates. This makes sense even for ζ in the lower half of the complex plane by analytic continuation. Then R(ζ ; Pα ) with Im ζ 0 is well defined as an operator from L2comp to L2loc . Thus R(ζ ; Pα ) does not have any poles as a function with values in operators from L2comp to L2loc . We can say that Pα with one solenoidal field 2παδ(x) has no resonances. We do not discuss the possibility of resonances at zero energy. 3. Proof of Theorem 1.1 by the complex scaling method The proof of Theorem 1.1 is based on the complex scaling method initiated by [3,5] and further developed by [18,20] (see [12] also). In this section we complete the proof of Theorem 1.1, accepting the five lemmas (Lemmas 3.1–3.5) formulated in the course of the proof as proved. We first reformulate the problem to which the complex scaling method can be applied in a more convenient way and fix some basic notation used throughout the entire discussion in the sequel. We work in the coordinate system in which the two centers d1 and d2 are represented as d1 = d− = (−d/2, 0),

d2 = d+ = (d/2, 0),

d 1,

and we set α− = α1 and α+ = α2 for two given fluxes α1 and α2 . Then the operator Hd = H (Φd ) under consideration is self-adjoint with domain D = u ∈ L2 : (−i∇ − Φd )2 u ∈ L2 ,

lim

|x−d± |→0

u(x) < ∞ at d− and d+

(3.1)


1845

and the potential Φd (x) takes the form Φd (x) = Φ−d (x) + Φ+d (x) = α− Φ(x − d− ) + α+ Φ(x − d+ ).

(3.2)

We denote by H0 = − the free Hamiltonian with domain H 2 (R 2 ) (Sobolev space of order two) and define the auxiliary operators by H±d = H (Φ±d ),

(3.3)

which are self-adjoint with domain D± = u ∈ L2 : (−i∇ − Φ±d )2 u ∈ L2 ,

lim

|x−d± |→0

u(x) < ∞ .

(3.4)

We fix E0 > 0. We always assume that ζ is restricted to the complex neighborhood

1/2 Dd = ζ = E + iη ∈ C: |E − E0 | < δE0 , |η| < 2E0 (log d)/d

(3.5)

with 0 < δ 1 small enough, and we set D±d = Dd ∩ {ζ ∈ C: ± Im ζ > 0}. We also introduce smooth cut-off functions χ0 , χ∞ and χ± over the real line R = (−∞, ∞) with the following properties: 0 χ0 , χ∞ , χ± 1 and χ0 (t) = 1 for |t| 1,

χ0 (t) = 0

for |t| 2,

χ∞ (t) = 1 − χ0 (t),

χ+ (t) = 1 for t 1,

χ+ (t) = 0

for t −1,

χ− (t) = 1 − χ+ (t).

We often use these functions without further references throughout the future discussion. We define jd (x) : R 2 → C 2 by jd (x1 , x2 ) = x1 , x2 + iηd (x2 )x2 , −1/2

with η0d = 5E0

ηd (t) = η0d χ∞ (t/d),

(3.6)

(log d)/d and consider the complex scaling mapping 1/2 f jd (x) (Jd f )(x) = det(∂jd /∂x)

associated with jd (x). The Jacobian det(∂jd /∂x) of jd (x) does not vanish for d 1, and therefore Jd is invertible. Since the coefficients of Hd are analytic in R 2 \ {d− , d+ }, we can define the operator Kd = Jd Hd Jd−1 .

(3.7)

This becomes a closed operator under the same boundary condition as in (3.1), but it is not necessarily self-adjoint. The domain of Kd coincides with D. We do not require the explicit form of Kd in the future discussion.

1846


We define the complex scaled operator as above for the auxiliary operators H±d defined by (3.3). Recall that γ (x; ω) denotes the azimuth angle from ω ∈ S 1 to xˆ = x/|x|. The potential Φ(x) defined by (1.1) satisfies the relation Φ(x) = ∇γ (x; ω). Hence it follows that Φ±d (x) = α± Φ(x − d± ) = α± ∇γ (x − d± ; ω± ),

ω± = (±1, 0).

The angle function γ (x; ω+ ) is represented as γ (x; ω+ ) = −(i/2) log (x1 + ix2 )/(x1 − ix2 ) + π, so that it is well defined for complex variables also. We take arg z, 0 arg z < 2π , to be a single valued function over the complex plane slit along the direction ω+ and define 1/2 1 γ jd (x); ω+ = arg b+d (x) − arg b−d (x) + π − i logbd (x) , 2

(3.8)

where bd (x) = b+d (x)/b−d (x) and b+d (x) = x1 − ηd (x2 )x2 + ix2 ,

b−d (x) = x1 + ηd (x2 )x2 − ix2 .

The function γ (jd (x); ω− ) is similarly defined by taking arg z to be a single valued function over the complex plane slit along the direction ω− . We define g±d by g±d (x) = α± χ∓ 32(x1 ∓ 13d/32)/d γ jd (x) − d± ; ω±

(3.9)

g0d (x) = χ0 (4x1 /d) α− γ jd (x) − d− ; ω− + α+ γ jd (x) − d+ ; ω+ .

(3.10)

and g0d by

By definition, supp g−d ⊂ {x: x1 > −7d/16} and g−d = α− γ (jd (x) − d− ; ω− ) on Π+ = {x: x1 > −3d/8}. Hence exp(ig−d ) acts as exp(ig−d )f (x) = Jd exp iα− γ (x − d− ; ω− ) Jd−1 f (x) on functions f (x) with support in Π+ . On the other hand, g+d (x) has support in {x: x1 < 7d/16} and g+d = α+ γ (jd (x) − d+ ; ω+ ) on Π− = {x: x1 < 3d/8}, so that exp(ig+d ) acts as exp(ig+d )f (x) = Jd exp iα+ γ (x − d+ ; ω+ ) Jd−1 f (x) on functions f (x) with support in Π− . We take into account these relations to define the following closed operator K±d = exp(ig∓d ) Jd H±d Jd−1 exp(−ig∓d ) with the same boundary condition as in (3.4). Since K+d = Jd H α− ∇γ (x − d− ; ω− ) + Φ+d Jd−1

(3.11)


1847

on Π+ , we have K+d = Kd

on Π+ = {x: x1 > −3d/8}.

(3.12)

on Π− = {x: x1 < 3d/8}.

(3.13)

Similarly we have K−d = Kd

The function g0d (x) defined by (3.10) has support in {x: |x1 | < d/2} and satisfies g0d = α− γ jd (x) − d− ; ω− + α+ γ jd (x) − d+ ; ω+ on Π0 = {x: |x1 | d/4}. If we define the operator K0d by K0d = exp(ig0d ) Jd H0 Jd−1 exp(−ig0d ),

(3.14)

then we obtain K0d = K±d = Kd

on Π0 = x: |x1 | d/4 .

(3.15)

We make some comments on the complex scaling mapping Jd defined above before going into the proof of the theorem. This mapping takes a form different from the standard mapping 1/2 (J˜θ f )(x) = det 1 + iθ dF (x) f x + iθ F (x) ,

θ > 0,

used in the existing complex scaling method (for example see [12]), where F : R 2 → R 2 is a smooth vector field satisfying F (x) = x for |x| 1. If we define K˜ dθ = J˜θ Hd J˜θ−1 , then it follows by the Weyl perturbation theorem that the essential spectrum of K˜ dθ is given by

σess (K˜ dθ ) = ζ ∈ C: arg ζ = −2 arg(1 + iθ ) , and the resonances of Hd in question are defined as eigenvalues near the positive real axis of the distorted operator K˜ dθ . The spectrum σ (K˜ dθ ) is discrete in the sector

Sθ = ζ ∈ C: Re ζ > 0, −2 arg(1 + iθ ) < arg ζ 0 and it is known that σ (K˜ dθ ) ∩ Sθ is independent of the vector field F and of θ . On the other hand, the distorted operator Kd = Jd Hd Jd−1 defined by the mapping Jd has its essential spectrum in the region

σess (Kd ) = ζ ∈ C: −2 arg(1 + iη0d ) arg ζ 0 ,

−1/2

η0d = 5E0

(log d)/d,

and has no discrete eigenvalues in this sector. This follows from the Weyl perturbation theorem, if we consider Kd as a perturbation of the operator −∂12 − (1 + iη0d )−2 ∂22 . Hence we have to define the resonances of Hd directly as the poles of the resolvent R(ζ ; Hd ) continued analytically over the unphysical sheet and not as the eigenvalues of Kd . It seems to be difficult to apply the

1848


standard complex scaling method to our resonance problem in scattering by two solenoidal fields with centers at large separation. In particular, it is difficult to separate the two centers from each other without introducing auxiliary operators such as K±d with one solenoidal field. For this reason, we develop the new type of complex scaling method which changes only the variable x2 into the complex variable to separate the two centers from each other. We note that Wang [21] has already studied resonances in strong uniform magnetic fields in three dimensions by making use of a complex scaling method depending only on one variable (direction perpendicular to the magnetic field). However it seems that the motivation in the background is different from that in the present work. In particular, our complex scaled operator has a quite different structure in the essential spectrum. With the notation above, we are now in a position to prove the main theorem. Proof of Theorem 1.1. The proof is divided into five steps. Throughout the proof, we use the notation R(ζ ; K) to denote the resolvent (K − ζ )−1 of K, where K is not necessarily assumed to be self-adjoint. We also denote by the same notation R(ζ ; K) the resolvent obtained by analytic continuation. Step 1. At first we assume that ζ = E + iη ∈ D+d . Let H± = H (α± Φ) be the self-adjoint operator with the boundary condition (1.3) at the origin and let R± (x, y; ζ ) be the kernel of the resolvent R(ζ ; H± ). Then the kernel of the resolvent R(ζ ; H±d ) is given by R± (x − d± , y − d± ; ζ ). We now consider the integral operator R˜ ±d (ζ ) with the kernel R˜ ±d (x, y; ζ ) = j˜d (x, y)R± jd (x) − d± , jd (y) − d± ; ζ ,

(3.16)

where 1/2 1/2 det ∂jd (y)/∂y . j˜d (x, y) = det ∂jd (x)/∂x If we set H˜ ±d = Jd H±d Jd−1 , then H˜ ±d becomes a closed operator with the boundary condition as in (3.4) and a formal argument using a change of variables shows that R˜ ±d (ζ ) = Jd R(ζ ; H±d )Jd−1 = R(ζ ; H˜ ±d ). The rigorous justification is based on the density of analytic vectors in L2 . The first step is to show the following lemma. Lemma 3.1. Assume that ζ ∈ D+d . Let H˜ ±d and R˜ ±d (ζ ) be as above. Then R˜ ±d (ζ ) : L2 → L2 is bounded, and ζ belongs to the resolvent set of H˜ ±d with R˜ ±d (ζ ) as a resolvent. Remark 3.1. We can show that the adjoint operator R˜ ±d (ζ )∗ : L2 → L2 is similarly obtained from the resolvent R(ζ ; H±d ) : L2 → L2 with ζ ∈ D+d and coincides with the resolvent ∗ ). R(ζ ; H˜ ±d Since g±d (x) defined by (3.9) is a bounded function, the lemma, together with (3.11), implies that ζ ∈ D+d belongs to the resolvent set of K±d and the resolvent R(ζ ; K±d ) is given by


1849

R(ζ ; K±d ) = exp(ig∓d )R˜ ±d (ζ ) exp(−ig∓d ) : L2 → L2 for ζ ∈ D+d . Step 2. The second step is to show that ζ ∈ D+d is also in the resolvent set of Kd and to derive the representation for the resolvent R(ζ ; Kd ) in terms of R(ζ ; K±d ). To see this, we define Λd (ζ ) by Λd (ζ ) = χ−d R(ζ ; K−d ) + χ+d R(ζ ; K+d ) : L2 → L2 , where χ±d (x) = χ± (16x1 /d). Since Kd = K±d on supp χ±d by (3.12) and (3.13), we compute (Kd − ζ )Λd (ζ ) = (K−d − ζ )χ−d R(ζ ; K−d ) + (K+d − ζ )χ+d R(ζ ; K+d ) = Id + [K−d , χ−d ]R(ζ ; K−d ) + [K+d , χ+d ]R(ζ ; K+d ). has support in The function χ±d depends on x1 only, and the derivative χ±d

Σ0 = x = (x1 , x2 ): |x1 | < d/16 .

(3.17)

By (3.15), K±d = K0d on Π0 , so that the two commutators [K−d , χ−d ] and [K+d , χ+d ] on the right side equal [K0d , χ−d ] and −[K0d , χ−d ], respectively. Hence we have (Kd − ζ )Λd (ζ ) = Id + X R(ζ ; K−d ) − R(ζ ; K+d ) ,

(3.18)

where X = [K0d , χ−d ],

χ−d = χ− (16x1 /d).

(3.19)

We further compute the operator on the right side of (3.18). If we set χ0d (x) = χ0 (8x1 /d), then χ0d = 1 on Σ0 and K±d = K0d on supp χ0d by (3.15). Hence it equals Td (ζ ) := X R(ζ ; K−d ) − R(ζ ; K+d ) = XR(ζ ; K+d )Y R(ζ ; K−d )

(3.20)

as an operator acting on L2 (Σ0 ), where Y = [K0d , χ0d ],

χ0d = χ0 (8x1 /d).

(3.21)

Then we can prove the following lemma. Lemma 3.2. Assume that ζ ∈ D+d . If Td (ζ ) is considered as an operator from L2 (Σ0 ) into itself, then Id + Td (ζ ) : L2 (Σ0 ) → L2 (Σ0 ) has a bounded inverse.

1850


We shall show that ζ ∈ D+d belongs to the resolvent set of Kd . Let L2comp (Σ0 ) denote the set of L2 functions with support in Σ0 . We often identify L2comp (Σ0 ) with L2 (Σ0 ), including its topology. It follows from (3.18) and (3.20) that (Kd − ζ )Λd (ζ ) = Id + Td (ζ ) on L2comp (Σ0 ). Hence Lemma 3.2 implies that −1 (Kd − ζ )Λd (ζ ) Id + Td (ζ ) f = f for f ∈ L2comp (Σ0 ), so that the operator R(ζ ) defined by −1 R(ζ ) = Λd (ζ ) − Λd (ζ ) Id + Td (ζ ) X R(ζ ; K−d ) − R(ζ ; K+d ) : L2 → L2 satisfies (Kd − ζ )R(ζ )f = f on L2 . Thus we have that the range Ran(Kd − ζ ) of Kd − ζ coincides with L2 . Similarly we can prove that Ran(Kd∗ − ζ ) = L2 (see Remark 3.1). This shows that ζ ∈ D+d belongs to the resolvent set of Kd , and R(ζ ; Kd ) is represented as −1 R(ζ ; Kd ) = Λd (ζ ) − Λd (ζ ) Id + Td (ζ ) X R(ζ ; K−d ) − R(ζ ; K+d ) .

(3.22)

Step 3. We still assume that ζ ∈ D+d . Let Ω0 = {x: |x1 | < d, |x2 | < r0 }

(3.23)

for r0 1 fixed large enough but independently of d. If f ∈ L2comp (Ω0 ) is an L2 function with support in Ω0 , then R(ζ ; Hd )f is analytic outside Ω0 , because the coefficients of Hd are analytic there. We can prove the following lemma. Lemma 3.3. Assume that ζ ∈ D+d . If f ∈ L2comp (Ω0 ), then Jd R(ζ ; Hd )f ∈ L2 . Since Jd acts as the identity operator on L2comp (Ω0 ), Jd R(ζ ; Hd )f with f ∈ L2comp (Ω0 ) satisfies the boundary conditions in (3.4) and solves the equation (Kd − ζ )Jd R(ζ ; Hd )f = Jd (Hd − ζ )R(ζ ; Hd )f = f by (3.7). Since such a solution is unique in L2 , we have Jd R(ζ ; Hd ) = R(ζ ; Kd ) on L2comp (Ω0 ) for ζ ∈ D+d . Thus we obtain −1 R(ζ ; Hd ) = Λd (ζ ) − Λd (ζ ) Id + Td (ζ ) X R(ζ ; K−d ) − R(ζ ; K+d )

(3.24)

from (3.22), when considered as an operator from L2comp (Ω0 ) into itself. Step 4. The relation (3.24) plays a basic role in studying the analytic continuation of R(ζ ; Hd ) as a function of ζ with values in operators from L2comp (Ω0 ) into itself over the lower-half plane. As stated above, L2comp (Ω0 ) is identified with L2 (Ω0 ) together with its topology, and similarly for L2comp (Σ0 ). We can prove the following two lemmas.


1851

Lemma 3.4. Let ζ ∈ Dd . Then R(ζ ; K±d ) is bounded when it is considered as an operator from L2comp (Ω0 ) into L2 (Σ0 ) or from L2comp (Σ0 ) into L2 (Ω0 ), and it depends analytically on ζ ∈ Dd . Lemma 3.5. Let Td (ζ ) be defined by (3.20). Assume that ζ = E − iη fulfills the assumption 0 η < ηεd (E) in the theorem. Then Id + Td (ζ ) : L2 (Σ0 ) → L2 (Σ0 ) has a bounded inverse. The operator R(ζ ; K±d ) depends analytically on ζ when considered as an operator from L2comp (Ω0 ) into itself. The two lemmas above, together with (3.24), imply that R(ζ ; Hd ) is analytically continued as a function of ζ with values in operators from L2comp (Ω0 ) into itself over the region

Dεd = ζ = E − iη ∈ Dd : 0 η < ηεd (E) in the lower half-plane. Step 5. The proof is completed in this step. Once the analytic continuation of R(ζ ; Hd ) : L2comp (Ω0 ) → L2comp (Ω0 ) is established, we can show that R(ζ ; Hd ) is analytically continued as a function of ζ with values in operators from L2comp to L2loc over the above region Dεd . To see this, we introduce the auxiliary operator P0 = H (α0 Φ) with α0 = α− + α+ , the self-adjoint extension (Friedrichs extension) of which is realized by imposing the boundary condition lim|x|→0 |u(x)| < ∞ at the origin. We use the same notation P0 to denote this self-adjoint realization. As is easily seen, the line integral

Φd (x) − α0 Φ(x) · dx = 0

C

vanishes along any curve C outside Ω0 by the Stokes formula. This makes it possible to construct a smooth real function g(x) in such a way that Φd (x) = α0 Φ(x) + ∇g(x)

(3.25)

outside Ω0 . In fact, it is given by the line integral ∞ Φd (tx) − α0 Φ(tx) · xˆ dt, g(x) = −

xˆ = x/|x|,

1

for |x| 1 and obeys g(x) = O(|x|−1 ) as |x| → ∞. This function g(x) is also analytic outside Ω0 , because g solves g = ∇ · (Φd − α0 Φ)

1852


and the function on the right side is analytic there. Let {ψ0 , ψ1 } be a smooth partition of unity over R 2 such that ψ0 + ψ1 = 1,

supp ψ0 ⊂ Ω0 ,

and let ψ2 be a smooth function such that it has a slightly wider support than ψ1 and satisfies ψ2 ψ1 = ψ1 . We may assume that (3.25) remains true on supp ψ2 (and hence on supp ψ1 also). If we define Pˆ0 = eig P0 e−ig , then it follows that Hd = Pˆ0

on supp ψ2 .

(3.26)

This relation enables us to decompose R(ζ ; Hd ) = R(ζ ; Hd )(ψ0 + ψ1 ) into the sum of three terms as follows: R(ζ ; Hd ) = R(ζ ; Hd )ψ0 + ψ2 R(ζ ; Pˆ0 )ψ1 − R(ζ ; Hd )[Pˆ0 , ψ2 ]R(ζ ; Pˆ0 )ψ1 . Since R(ζ ; Pˆ0 ) : L2comp → L2loc depends analytically on ζ and since the commutator [Pˆ0 , ψ2 ] vanishes outside Ω0 , we see that R(ζ ; Hd ) : L2comp → L2comp (Ω0 ) depends analytically on ζ . Similarly we obtain the relation R(ζ ; Hd ) = ψ0 R(ζ ; Hd ) + ψ1 R(ζ ; Pˆ0 )ψ2 + ψ1 R(ζ ; Pˆ0 )[Pˆ0 , ψ2 ]R(ζ ; Hd ) on L2comp . This yields the analytic dependence on ζ of R(ζ ; Hd ) : L2comp → L2loc and the proof of the theorem is now complete. 2 The proofs of the five lemmas which remain unproved are all based on the asymptotic analysis of the behavior at infinity of the Green function for the Schrödinger operator with one solenoidal field. In particular, the proof of Lemma 3.5, which has played an essential role in proving the theorem, occupies the main body of the paper. 4. Proof of Lemmas 3.1 and 3.3 The present section is devoted to proving Lemmas 3.1 and 3.3 among the five lemmas. 4.1. Preliminary proposition and lemmas We begin by introducing the new notation 2 2 rd (x, y)2 = jd (x) − jd (y) , rd (x)2 = rd (x, 0)2 = jd (x) , θd (x, y) = γ jd (x); ω+ − γ jd (y); ω+ , ω+ = (1, 0),

(4.1) (4.2)

where |z|2 = x12 + (x2 + iy2 )2 for z = (x1 , x2 + iy2 ) ∈ R × C. The branch rd (x, y) of rd (x, y)2 is taken in such a way that Re rd (x, y) > 0. We recall that the kernel Rα (x, y; ζ ) of the resolvent R(ζ ; Pα ) with Im ζ > 0 is given by (2.5) for the self-adjoint operator Pα = H (αΦ) defined by (1.2) with domain (1.3). The argument here is based on the following proposition.


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Proposition 4.1. Assume that ζ ∈ D+d . Let Rα (x, y; ζ ) be the kernel of the resolvent R(ζ ; Pα ). Set k = ζ 1/2 with Im k > 0. If x2 > c and y2 > c for some c > 1, then −L Rα jd (x), jd (y); ζ = (i/4)eiαθd (x,y) H0 krd (x, y) + O |x| + |y| (1)

as |x| + |y| → ∞ for any L 1, where H0 (z) = H0 (z) denotes the Hankel function of the first kind, and the order estimate depends on ζ . A similar relation holds true in the case where x2 < −c and y2 < −c. We prove the proposition at the end of this section. We complete the proof of the two lemmas in question after showing two preliminary lemmas. We define R˜ αd (ζ ) = Jd R(ζ ; Pα )Jd−1 as the integral operator with kernel R˜ αd (x, y; ζ ) = j˜d (x, y)Rα jd (x), jd (y); ζ , where j˜d (x, y) is defined in (3.16). Lemma 4.1. If q± ∈ C ∞ (R 2 ) is a bounded function with support in {x: ±x2 > c} for some c > 1, then q+ R˜ αd (ζ )q+ ,

q− R˜ αd (ζ )q− : L2 → L2

is bounded. Proof. Let ηd (t) be defined in (3.6) and set η˜ d (t) = ηd (t)t. We may assume that η˜ d (t) 0. According to (4.1), we calculate 2 rd (x, y)2 = (x1 − y1 )2 + 1 + iηd (x2 , y2 ) (x2 − y2 )2 , where 1 ηd (x2 , y2 ) =

η˜ d y2 + s(x2 − y2 ) ds 0

0

and ηd (x2 , y2 ) = O((log d)/d). Hence we have Im krd (x, y) cη1/2 |x − y|,

|x − y| 1,

for some c > 0, so that the Hankel function H0 (krd (x, y)) falls off exponentially as |x −y| → ∞. This, together with Proposition 4.1, proves the lemma. 2 We denote by P˜αd = Jd Pα Jd−1 the complex scaled operator obtained from Pα . The coefficients of Pα are analytic in R 2 \ {0}. Hence P˜dα has coefficients smooth in R 2 \ {0} and becomes a closed operator under the same boundary condition as in (1.3). Let A be the dense space in L2 spanned by all functions of the form

1854


f (x1 , x2 ) = h(x1 )p(x2 ) exp −cx22 ,

c > 0,

where h ∈ C0∞ (R) and p(x2 ) is a polynomial. According to [12, Proposition 17.10], we know that Jd A is also dense in L2 . If f ∈ Jd A, then R˜ αd (ζ )f satisfies the boundary condition in (1.3) and the relation (P˜αd − ζ )R˜ αd (ζ ) = Id

(4.3)

holds on the dense set Jd A. This is shown by making a change of variables and by deforming the contour by analyticity. Lemma 4.2. Assume that ζ ∈ D+d . Let P˜αd and R˜ αd (ζ ) be as above. Then R˜ αd (ζ ) is bounded on L2 , and ζ belongs to the resolvent set of P˜αd with R˜ αd (ζ ) as a resolvent. Proof. Let {u− , u0 , u+ } be a nonnegative smooth partition of unity such that u− (x2 ) + u0 (x2 ) + u+ (x2 ) = 1 and supp u0 ⊂ (−2c, 2c),

supp u+ ⊂ (3c/2, ∞),

supp u− ⊂ (−∞, −3c/2)

for c > 1 fixed. We shall show that R˜ αd (ζ )u0 and R˜ αd (ζ )u± are bounded on L2 . We first consider R˜ αd (ζ )u0 . If v0 ∈ C0∞ (R) has support in {|x2 | < 4c}, then we have v0 R˜ αd (ζ )u0 = v0 R(ζ ; Pα )u0 , and v0 R˜ αd (ζ )u0 is bounded. Let v+ and v˜+ be smooth functions of x2 such that they have support in (3c, ∞), and v+ = 1 on [4c, ∞), v˜+ = 1 on supp v+ . Then u0 vanishes on supp v+ and Jd−1 v+ Jd = v+ for d 1. Thus we can calculate v+ R˜ αd (ζ )u0 = v˜+ R˜ αd (ζ )(P˜αd − ζ )v+ R˜ αd (ζ )u0 = v˜+ R˜ αd (ζ )[P˜αd , v+ ]R˜ αd (ζ )u0 on the dense set Jd A, and it follows from Lemma 4.1 that v+ R˜ αd (ζ )u0 is bounded on L2 . A similar argument applies to v− R˜ αd (ζ )u0 , where v− is supported in (−∞, 3c) and has properties similar to v+ . Hence we obtain that R˜ αd (ζ )u0 is bounded. Next we show that R˜ αd (ζ )u+ is bounded. The boundedness of R˜ αd (ζ )u− is shown in a similar way. Let {w− , w0 , w+ } be a nonnegative smooth partition of unity such that w− (x2 ) + w0 (x2 ) + w+ (x2 ) = 1 and supp w0 ⊂ (−c/2, c/2),

supp w+ ⊂ (c/3, ∞),

supp w− ⊂ (−∞, −c/3).

By Lemma 4.1, w+ R˜ αd (ζ )u+ is bounded. Let u˜ + ∈ C ∞ (R) be a function such that supp u˜ + ⊂ (3c/4, ∞) and it satisfies u˜ + u+ = u+ . Then we have the relation w0 R˜ αd (ζ )u+ = w0 R˜ αd (ζ )u+ (P˜αd − ζ )R˜ αd (ζ )u˜ + = w0 R˜ αd (ζ )[u+ , P˜αd ]R˜ αd (ζ )u˜ + on Jd A. This, together with Lemma 4.1, implies that w0 R˜ αd (ζ )u+ is bounded. We repeat the commutator calculus on Jd A to obtain w− R˜ αd (ζ )u+ = w˜ − R˜ αd (ζ )[P˜αd , w− ]R˜ αd (ζ )[u+ , P˜αd ]R˜ αd (ζ )u˜ + , where w˜ − ∈ C ∞ (R) has support in (−∞, −c/4) and satisfies w˜ − w− = w− . Hence Lemma 4.1 again shows that w− R˜ αd (ζ )u+ is bounded. Thus we have shown that R˜ αd (ζ ) is bounded on L2 .


1855

Since P˜αd is a closed operator, it follows from (4.3) that the range Ran(P˜αd − ζ ) coincides ∗ − ζ ) = L2 for the adjoint operator P˜ ∗ (see Remark 3.1). with L2 . We can also obtain Ran(P˜αd αd This shows that ζ is in the resolvent set of P˜αd and that the resolvent R(ζ ; P˜αd ) equals R˜ αd (ζ ), and the proof is complete. 2 4.2. Proof of Lemmas 3.1 and 3.3 We prove Lemmas 3.1 and 3.2. Proof of Lemma 3.1. If we apply Lemma 4.2 to H±d = H (Φ±d ) with Φ± (x) = α± Φ(x − d± ), then R˜ ±d (ζ ) = Jd R(ζ ; H±d )Jd−1 with ζ ∈ D+d is bounded on L2 . Since g∓d (x) defined by (3.9) is bounded, it follows from (3.11) that R(ζ ; K±d ) = exp(ig∓d )R˜ ±d (ζ ) exp(−ig∓d ) turns out to be the resolvent of K±d for ζ ∈ D+d . This proves the lemma.

2

Proof of Lemma 3.3. We use the notation with the same meaning as ascribed in Step 5 of the proof of Theorem 1.1. In particular, g satisfies (3.25). In addition, we introduce a smooth function ψ3 ∈ C ∞ (R 2 ) such that ψ3 ψ2 = ψ2 . We may assume that (3.25) remains true on supp ψ3 also. We decompose R(ζ ; Hd )f = (ψ0 + ψ1 )R(ζ ; Hd )f with f ∈ L2comp (Ω0 ) into the sum of three terms in the following way: R(ζ ; Hd )f = ψ0 R(ζ ; Hd )f + ψ1 R(ζ ; Hd )ψ2 f + ψ1 R(ζ ; Hd )[Hd , ψ2 ]R(ζ ; Hd )f. The first term on the right side fulfills Jd ψ0 R(ζ ; Hd )f = ψ0 R(ζ ; Hd )f ∈ L2 . If we take the relation (3.26) into account, then the second term on the right side is further calculated as ψ1 R(ζ ; Hd )ψ2 f = ψ1 R(ζ ; Pˆ0 )ψ2 f + ψ1 R(ζ ; Pˆ0 )[Pˆ0 , ψ3 ]R(ζ ; Hd )ψ2 f. We note that Jd exp ±ig(x) Jd−1 = exp ±ig jd (x) : L2 → L2 is bounded and [Pˆ0 , ψ3 ]R(ζ ; Hd )ψ2 f ∈ L2comp (Ω0 ). Since Jd ψ1 = ψ1 Jd and Jd ψ2 f = ψ2 f for f ∈ L2comp (Ω0 ), Lemma 4.2 with Pα = P0 yields that Jd ψ1 R(ζ ; Hd )ψ2 f is in L2 . Since ψ3 = 1 both on supp ψ1 and on supp ∇ψ2 , a similar argument applies to the third term, and we obtain Jd ψ1 R(ζ ; Hd )[ψ2 , Hd ]R(ζ ; Hd )f ∈ L2 . Thus the proof is complete.

2

1856


4.3. Proof of Proposition 4.1 Before going into the proof, we derive the integral representation for the kernel Rα (x, y; ζ ). The derivation is based on the following formula

Hμ (Z)Jμ (z) =

1 iπ

Zz dt Z 2 + z2 t − Iμ , exp 2 2t t t

κ+i∞

|z| |Z|,

0

for the product of Bessel functions [22, p. 439], where the contour is taken to be rectilinear with corner at κ + i0, κ > 0 being fixed arbitrarily. We apply to (2.5) this formula with Z = k(|x| ∨ |y|) and z = k(|x| ∧ |y|), where Im k = Im ζ 1/2 > 0. If we write x = (|x| cos θ, |x| sin θ ) and y = (|y| cos ω, |y| sin ω) in polar coordinates, then Rα (x, y; ζ ) is represented as κ+i∞ ζ |x||y| dt ζ (|x|2 + |y|2 ) 1 ilψ t − Iν Rα = e exp 4π 2 2t t t l

(4.4)

0

with ν = |l − α|, where ψ = θ − ω. If, in particular, α = 0, then the resolvent (H0 − ζ )−1 of the free Hamiltonian H0 has the kernel (i/4)H0 (k|x − y|) represented as the integral κ+i∞ ζ |x||y| dt ζ (|x|2 + |y|2 ) i 1 ilψ t H0 k|x − y| = − Il , e exp 4 4π 2 2t t t l

0

π where Il (w) = I|l| (w) is defined by Il (ω) = (1/π) 0 ew cos ρ cos(lρ) dρ (see (2.3)). Since the

ilψ series l e Il (w) converges to ew cos ψ by the Fourier expansion and since |x − y|2 = |x|2 + |y|2 − 2|x||y| cos ψ, the kernel (i/4)H0 (k|x − y|) has the integral representation i 1 H0 k|x − y| = 4 4π

ζ |x − y|2 dt t − . exp 2 2t t

κ+i∞

(4.5)

0

We are now in a position to prove the proposition. Proof of Proposition 4.1. We consider only the case when x2 > c and y2 > c and assume throughout the proof that ζ ∈ D+d . The proof is divided into three steps. (i) Let w = Zz/t = ζ |x||y|/t with Z = k(|x| ∨ |y|) and z = k(|x| ∧ |y|). Then Re w 0 for t on the contour in the integral (4.4), and the integral representation (2.3) for Iν (w) is well defined.

We make use of this representation to calculate the series l eilψ Iν (w) in the integral. Then it admits the decomposition l

eilψ Iν (w) =

l

eilψ Ifr,ν (w) +

l

eilψ Isc,ν (w),

(4.6)


1857

where Ifr,ν (w) and Isc,ν (w) are defined by 1 Ifr,ν (w) = π

π e

w cos ξ

sin(νπ) Isc,ν (w) = − π

cos(νξ ) dξ,

0

∞

e−w cosh p−νp dp

0

with ν = |l − α|. A simple calculation yields Ifr,ν (w) = (2π)−1

π

ew cos ξ eiαξ e−ilξ dξ

−π

and hence we have Ifr (w, ψ) =

eilψ Ifr,ν (w) = ew cos ψ eiαψ ,

|ψ| < π,

(4.7)

l

by the Fourier expansion. On the other hand, the second series on the right side of (4.6) is computed in the same way as in Section 2, and we see that it converges to sin(απ) i[α](ψ+π) e Isc (w, ψ) = − π

∞

−∞

e−w cosh p

e(1−β)p dp ep + e−iψ

(4.8)

with β = α − [α], 0 < β < 1. By assumption, x2 > c and y2 > c, so that 0 < θ, ω < π . This implies that −π < ψ = θ − ω < π , and hence the denominator ep + e−iψ in (4.8) never vanishes even for p = 0. Thus Rα (x, y; ζ ) admits the decomposition Rα (x, y; ζ ) = Rfr,α (x, y; ζ ) + Rsc,α (x, y; ζ ), where Rfr,α and Rsc,α are defined by eiαψ Rfr,α (x, y; ζ ) = 4π

ζ |x − y|2 dt ieiαψ t − = H0 k|x − y| , exp 2 2t t 4

κ+i∞

0

1 Rsc,α (x, y; ζ ) = 4π

dt t ζ |x||y| ζ (|x|2 + |y|2 ) − ,ψ . exp Isc 2 2t t t

κ+i∞

0

The function Rα (jd (x), jd (y); ζ ) in question also admits the corresponding decomposition Rα jd (x), jd (y); ζ = Rfr,α jd (x), jd (y); ζ + Rsc,α jd (x), jd (y); ζ .

(4.9)

If we recall the notation in (4.1) and (4.2), the functions on the right side are defined with |x|, |y| and ψ replaced by rd (x), rd (y) and θd (x, y), respectively. In fact, if x2 > c > 0 and y2 > c > 0, then ψ equals

1858


ψ = γ (x; −y) ˆ − π = γ (x; ω+ ) − γ (y; ω+ ) and it is changed into θd (x, y). In particular, we have Rfr,α jd (x), jd (y); ζ = (i/4)eiαθd (x,y) H0 krd (x, y) .

(4.10)

(ii) We prove that Rsc,α (jd (x), jd (y); ζ ) obeys Rsc,α jd (x), jd (y); ζ = O |x| + |y| −L ,

|x| + |y| → ∞.

(4.11)

By definition, Rsc,α (jd (x), jd (y); ζ ) is written as 1 4π

ζ (rd (x)2 + rd (y)2 ) t dt − Isc (t, x, y; ζ ) , exp 2 2t t

κ+i∞

(4.12)

0

and Isc (t, x, y; ζ ) = Isc (ζ rd (x)rd (y)/t, θd (x, y)) takes the form Isc (t, x, y; ζ ) = −

sin(απ) i[α](θd (x,y)+π) e Lsc (t, x, y; ζ ) π

by (4.8), where Lsc (t, x, y; ζ ) is defined by ∞ Lsc (t, x, y; ζ ) =

e−(ζ rd (x)rd (y)/t) cosh p

−∞

e(1−β)p dp. ep + e−iθd (x,y)

(4.13)

We prove the two lemmas below after completing the proof of the proposition. Lemma 4.3. Assume that x2 > c and y2 > c for some c > 0. If x1 1 and y1 −1 or if x1 −1 and y1 1, then there exists c1 > 0 such that Im e−iθd (x,y) c1 |x1 | + |y1 | −1 ,

|x1 | + |y1 | 1.

Lemma 4.4. If 0 < t < κ, then exp −ζ rd (x)2 + rd (y)2 /2t exp −c2 |x|2 + |y|2 /t ,

|x| + |y| 1,

for some c2 > 0, and if 0 < s < M(|x| + |y|) for t = κ + is, M 1 being fixed, then exp −ζ rd (x)2 + rd (y)2 /2t exp −c3 |x| + |y| , for some c3 > 0, where c3 may depend on η.

|x| + |y| 1,


1859

The denominator ep + e−iθd (x,y) in the integral (4.13) does not vanish, but it can take values close to zero around p = 0, provided that θd (x, y) ∼ ±π . This is the case where x1 1 and y1 −1 or where x1 −1 and y1 1. However, Lemma 4.3 implies that |Lsc (t, x, y; ζ )| = O(|x| + |y|), and hence it follows from Lemma 4.4 that κ+iM ζ (rd (x)2 + rd (y)2 ) t dt − Isc (t, x, y; ζ ) exp 2 2t t

−L . = O |x| + |y|

0

(iii) The proof is completed in this step by showing that the integral

κ+i∞

χM (t, x, y) exp

ζ (rd (x)2 + rd (y)2 ) t dt − Isc (t, x, y; ζ ) 2 2t t

κ+i0

obeys O((|x| + |y|)−L ), where χM (t, x, y) = χ∞ s/ M |x| + |y| ,

|x| + |y| 1,

for s = Im t. To see this, we decompose Lsc (t, x, y; ζ ) defined by (4.13) into the sum Lsc (t, x, y, ; ζ ) =

χ0 (p) + χ∞ (p) e−(ζ rd (x)rd (y)/t) cosh p

e(1−β)p dp. ep + e−iθd (x,y)

If we set a0 (t, x, y) = t/2 − ζ (rd (x)2 + rd (y)2 )/2t and a1 (t, x, y, p) = a0 (t, x, y) − ζ rd (x)rd (y)/t cosh p,

|p| < 2,

then we can take M 1 so large that |∂t a0 | c and |∂t a1 | c for some c > 0. The desired bound is obtained by partial integration. We use |∂t a1 | c for the integral with χ0 (p). On the other hand, we make use of |∂t a0 | c and of the relation ∂t e−(ζ rd (x)rd (y)/t) cosh p = −t −1 (cosh p/ sinh p)∂p e−(ζ rd (x)rd (y)/t) cosh p ,

|p| > 2,

to evaluate the integral with χ∞ (p). Thus (4.11) is obtained, and the proposition follows from (4.9), (4.10) and (4.11). 2 We end the section by proving Lemmas 4.3 and 4.4. Proof of Lemma 4.3. We consider only the case when x1 1 and y1 −1, so that θd (x, y) behaves like θd ∼ −π . We write Im e−iθd (x,y) = eIm θd (x,y) sin Re θd (x, y) . We recall the representation (3.8) for γ (jd (x); ω+ ). We note that ηd (t) defined in (3.6) satisfies ηd (t) 0 and ηd (t) = O((log d)/d) uniformly in t. If x2 > c and y2 > c and if x1 1 and

1860


y1 −1, then it follows that Re γ (jd (x); ω+ ) c1 /x1 and Re γ (jd (y); ω+ ) π + c1 /y1 for some c1 > 0. Hence we have −1 Re θd (x, y) = Re γ jd (x); ω+ − γ jd (y); ω+ −π + c1 |x1 | + |y1 | . This shows that | sin(Re θd (x, y))| c1 (|x1 | + |y1 |)−1 . As is easily seen from (3.8), Im γ jd (x); ω+ = O (log d)/d

(4.14)

uniformly in x with |x| > c2 > 0, and hence we have eIm θd (x,y) c3 for some c3 > 0. Thus the lemma is verified. 2 Proof of Lemma 4.4. By (4.1), we have rd (x)2 = x12 + 1 + 2iηd (x2 ) − ηd (x2 )2 x22 . Since ηd (t) = O((log d)/d), we can easily see that Re rd (x)2 + rd (y)2 /t c |x|2 + |y|2 /t,

c > 0,

for 0 < t < κ. Thus the first statement is obtained. If we compute −1 ζ /t = κ 2 + s 2 (Eκ + ηs) + i(ηκ − Es) by setting t = κ + is and ζ = E + iη, then we have Re (ζ /t) rd (x)2 + rd (y)2 c (1 + ηs)/ 1 + s 2 |x|2 + |y|2 for some c > 0. This proves the second statement for 0 < s < M(|x| + |y|), and the proof is complete. 2 5. Proof of Lemmas 3.2 and 3.5 In this section we prove Lemmas 3.2 and 3.5. The proof of both the lemmas is based on the same idea, but Lemma 3.5 is much more difficult to prove than Lemma 3.2. We give a detailed proof for Lemma 3.5 and only a sketch for Lemma 3.2. 5.1. Preliminary proposition and lemmas We begin by formulating the proposition which plays an important role in proving Lemma 3.5. Proposition 5.1. Let Rα (x, y; ζ ) be the kernel of the resolvent R(ζ ; Pα ) with ζ ∈ D −d , D −d being the closure of D−d , and let N 1 be fixed arbitrarily but large enough. Set k = ζ 1/2 with Im k 0. Assume that −3d/4 < x1 , y1 < −d/4, for some c > 0. Then we have the following statements:

|x1 − y1 | > cd


1861

(1) If |x2 | + |y2 | N d, then Rα (jd (x), jd (y); ζ ) behaves like −σ N Rα jd (x), jd (y); ζ = (i/4)eiαθd (x,y) H0 krd (x, y) + O |x| + |y| for some σ > 0 independent of N . (2) Let c(E) be the constant defined by c(E) = (8π)−1/2 eiπ/4 E −1/4 .

(5.1)

If |x2 | + |y2 | N d, then Rα (jd (x), jd (y); ζ ) admits the decomposition Rα jd (x), jd (y); ζ = (i/4)eiαθd (x,y) H0 krd (x, y) + Gα (x, y; ζ ) + O d −N and Gα (x, y; ζ ) takes the asymptotic form −1/2 Gα = c(E)eik(rd (x)+rd (y)) rd (x)rd (y) fα (−ω → θ ; E) + eN (x, y; ζ ) , where fα (−ω → θ ; E) is the amplitude defined by (1.4) for scattering from −ω = −y/|y| to θ = x/|x| at energy E by the field 2παδ(x), and eN (x, y; ζ ) obeys ∂xn ∂ym eN = O (log d)2 d −1−|n|−|m| uniformly in x, y and ζ . (3) Similar asymptotic formulas remain true for the derivatives ∂Rα jd (x), jd (y); ζ /∂xj ,

∂Rα jd (x), jd (y); ζ /∂yj ,

j = 1, 2,

with natural modification in both the cases (1) and (2) above. Remark 5.1. If x1 and y1 satisfy d/4 < x1 , y1 < 3d/4, then the same results remain true with θd (x, y) replaced by θ˜d (x, y) = γ (jd (x); ω− ) − γ (jd (y); ω− ), where ω− = (−1, 0). We prove the proposition at the end of the section. We proceed with the argument, accepting the proposition as proved. We apply this proposition to the kernel F± (x, y; ζ ) of the resolvent R(ζ ; K±d ) with ζ ∈ D −d for the operator K±d defined by (3.11). Let H± = H (α± Φ) be the selfadjoint operator with the boundary condition (1.3) at the origin and let R± (x, y; ζ ) be the kernel of the resolvent R(ζ ; H± ) analytically continued over D −d . Then the kernel of R(ζ ; H±d ) is given by R± (x − d± , y − d± ; ζ ) with d± = (±d/2, 0) for the auxiliary operator H±d = H (Φ±d ), and it follows from (3.11) that F± (x, y; ζ ) = j˜d (x, y)ei(g∓d (x)−g∓d (y)) R±d (x, y; ζ ), where R±d (x, y; ζ ) = R± jd (x) − d± , jd (y) − d± ; ζ

1862


and g±d (x) is defined by (3.9). According to (4.1), it is obvious that rd (x − d± , y − d± ) = rd (x, y). Let X and Y be the commutators defined by (3.19) and (3.21), respectively. The coefficients of X have support in Σ0 (see (3.17)). On the other hand, the support of the coefficients of Y is divided into the two regions Σ− = {x: −d/4 < x1 < −d/8},

Σ+ = {x: d/8 < x1 < d/4}.

(5.2)

We assume that x ∈ Σ0 and y ∈ Σ = Σ− ∪ Σ+ . Then −9d/16 < x1 − d/2 < −7d/16,

−3d/4 < y1 − d/2 < −d/4

and d/16 < |x1 − y1 | < 5d/16. If |x2 | + |y2 | N d for N 1 fixed, then we have −σ N R+d (x, y; ζ ) = (i/4)eiα+ θ+d (x,y) H0 krd (x, y) + O |x| + |y| by Proposition 5.1(1), where θ+d (x, y) = γ jd (x) − d+ ; ω+ − γ jd (y) − d+ ; ω+ . We write r±d (x) for rd (x, d± ) and xˆ±d for (x − d± )/|x − d± |. Let f± (ω → θ ; E) denote the amplitude for scattering from ω to θ by the field 2πα± δ(x). If |x2 | + |y2 | N d, then it follows from Proposition 5.1(2) that R+d (x, y; ζ ) behaves like R+d (x, y; ζ ) = (i/4)eiα+ θ+d (x,y) H0 krd (x, y) + G+d (x, y; ζ ) + O d −N and G+d (x, y; ζ ) takes the form −1/2 f+ (−yˆ+d → xˆ+d ; E) + e+N , G+d = c(E)eik(r+d (x)+r+d (y)) r+d (x)r+d (y) where e+N = e+N (x, y; ζ ) obeys the same bound as eN in Proposition 5.1. We can derive a similar asymptotic form for R−d (x, y; ζ ). Assume that x ∈ Σ = Σ+ ∪ Σ− and y ∈ Σ0 . If |x2 | + |y2 | Nd, then −σ N R−d (x, y; ζ ) = (i/4)eiα− θ−d (x,y) H0 krd (x, y) + O |x| + |y| , where θ−d (x, y) = γ jd (x) − d− ; ω− − γ jd (y) − d− ; ω− . If |x2 | + |y2 | N d, then R−d (x, y; ζ ) = (i/4)eiα− θ−d (x,y) H0 krd (x, y) + G−d (x, y; ζ ) + O d −N and G−d (x, y; ζ ) takes the form −1/2 f− (−yˆ−d → xˆ−d ; E) + e−N . G−d = c(E)eik(r−d (x)+r−d (y)) r−d (x)r−d (y)


1863

We summarize the asymptotic properties of the kernel F± (x, y; ζ ) of R(ζ ; K±d ) with ζ ∈ D −d in the lemma below. Lemma 5.1. Define F±0 (x, y; ζ ) = (i/4)j˜d (x, y)ei(g∓d (x)−g∓d (y)) eiα± θ±d (x,y) H0 krd (x, y) and set F±1 (x, y; ζ ) = j˜d (x, y)ei(g∓d (x)−g∓d (y)) G±d (x, y; ζ ) for G±d (x, y; ζ ) as above. (1) Assume that x ∈ Σ0 and y ∈ Σ = Σ− ∪ Σ+ . If |x2 | + |y2 | N d for N 1, then −σ N , F+ (x, y; ζ ) = F+0 (x, y; ζ ) + O |x| + |y| and if |x2 | + |y2 | N d, then F+ (x, y; ζ ) = F+0 (x, y; ζ ) + F+1 (x, y; ζ ) + O d −N . These relations hold true in the C 1 topology. (2) Assume that x ∈ Σ and y ∈ Σ0 . If |x2 | + |y2 | N d for N 1, then −σ N F− (x, y; ζ ) = F−0 (x, y; ζ ) + O |x| + |y| , and if |x2 | + |y2 | N d, then F− (x, y; ζ ) = F−0 (x, y; ζ ) + F−1 (x, y; ζ ) + O d −N . We prove two preliminary lemmas. Lemma 5.2. Assume that |x1 | d/2 and |y1 | d/2. If ζ ∈ D −d , then there exist μ > 0 and c > 0 such that ikr (x,y) e d = O d μ exp −c (log d)/d |x2 − y2 | for k = ζ 1/2 , and in particular, one has |eikr−d (x) | + |eikr+d (x) | = O(d μ ). Proof. Let ηd (t) be defined in (3.6). We set η˜ d (t) = ηd (t)t. For brevity, we assume that y2 x2 . Then we compute rd (x, y)2 = |x − y|2 + 2i η˜ d (z2 ) + O (log d)2 /d 2 (x2 − y2 )2

1864


for some z2 with y2 z2 x2 , where η˜ d (z2 ) = O((log d)/d). If ζ = E − iη ∈ D −d , then 0 1/2 η 2E0 (log d)/d and k = ζ 1/2 = E 1/2 − iE −1/2 η/2 + O (log d)2 /d 2 ,

d 1.

(5.3)

Hence we have 2 Im krd (x, y) ∼ E 1/2 η˜ d (z2 ) (x2 − y2 )/|x − y| − E −1/2 η/2 |x − y| for d 1. We can take c1 > 0 in such a way that −1/2 η˜ d (z2 ) = η˜ d (x2 ) − η˜ d (y2 ) /(x2 − y2 ) 4E0 (log d)/d when |x2 | + |y2 | > c1 d. If |x2 − y2 | > d and |x2 | + |y2 | > c1 d, then 1/2 E η˜ d (z2 ) − E −1/2 η /2 2(E/E0 )1/2 − (E0 /E)1/2 (log d)/d c(log d)/d for some c > 0. This implies that ikr (x,y) e d = e−Im(krd (x,y)) e−c((log d)/d)|x2 −y2 | for x and y as above. On the other hand, if |x2 − y2 | < d or if |x2 | + |y2 | < c1 d, then we have |Im(krd (x, y))| = O(log d), and hence the estimate in the lemma is obtained. Thus the proof is complete. 2 Lemma 5.3. Assume that ζ ∈ D −d . Let u(x) be a smooth function such that u has support in {d/8 < |x1 | < d/4} and satisfies ∂xn u = O(d −|n| ). Define U (x, y) by U (x, y) =

eikrd (x,ξ ) u(ξ )eikrd (ξ,y) dξ

for k = ζ 1/2 . If x and y are in Σ0 , then there exists c > 0 such that U (x, y) = O d −L exp −c (log d)/d |x2 − y2 | for any L 1. Proof. The proof is based on the property that exp(ikrd (x, y)) oscillates highly in the x1 variable and falls off exponentially in the x2 variable. By Lemma 5.2, we have ikr (x,ξ ) ikr (ξ,y) = O d 2μ exp −c (log d)/d |x2 − ξ2 | + |ξ2 − y2 | e d e d for some c > 0. In particular, if |x2 − ξ2 | > Ld for L 1 fixed arbitrarily, then it follows from Lemma 5.2 that ikr (x,ξ ) = O d −σ L exp −c (log d)/d |x2 − ξ2 | e d


1865

with some σ > 0 independent of L. Since |x2 − ξ2 | + |ξ2 − y2 | > |x2 − y2 | and since

exp −c (log d)/d |ξ2 − y2 | dξ2 = O(d/ log d),

the desired bound is obtained for the integral over the interval |ξ2 − x2 | > Ld. A similar argument applies to the integral over the interval |ξ2 − y2 | > Ld. We assume that |ξ2 − x2 | < Ld and |ξ2 − y2 | < Ld. If x and y are in Σ0 , then |x1 | < d/16 and |y1 | < d/16, and hence it follows that |x1 − ξ1 | > d/16 and |y1 − ξ1 | > d/16 for ξ ∈ supp u. We consider the function ξ1 → rd (x, ξ ) + rd (ξ, y) = |x − ξ | + |ξ − y| + O(log d). Since |x2 − y2 | < 2Ld and since (∂/∂ξ1 ) |x − ξ | + |ξ − y| > c > 0 for ξ ∈ supp u, we make repeated use of partial integration to obtain the desired bound for the integral over the interval where |ξ2 − x2 | < Ld and |ξ2 − y2 | < Ld. This completes the proof. 2 5.2. Proof of Lemmas 3.2 and 3.5 The proof of Lemma 3.5 is done through a series of lemmas. We begin by recalling that Td (ζ ) is defined by Td (ζ ) = XR(ζ ; K+d )Y R(ζ ; K−d ) : L2 (Σ0 ) → L2 (Σ0 ) for ζ = E − iη ∈ D −d (see (3.20)). The commutators X and Y are defined by X = [K0d , χ−d ] with χ−d = χ− (16x1 /d) and by Y = [K0d , χ0d ] with χ0d = χ0 (8x1 /d), where K0d = eig0d (Jd H0 Jd−1 )e−ig0d (see (3.14), (3.19) and (3.21)). By definition, the map Jd commutes with operators depending only on the x1 variable. Hence X is calculated as X = eig0d Jd −∂12 , χ−d Jd−1 e−ig0d = eig0d −∂12 , χ−d e−ig0d

(5.4)

and similarly we have Y = eig0d [−∂12 , χ0d ]e−ig0d . We may write χ0d as the product χ0d (x1 ) = χ+ (16x1 + 3d)/d χ− (16x1 − 3d)/d = χ˜ +d (x1 )χ˜ −d (x1 ), so that Y takes the form Y = eig0d

2

−∂1 , χ˜ +d + −∂12 , χ˜ −d e−ig0d = Y − + Y + ,

(5.5)

where the coefficients of Y ± have support in Σ± defined by (5.2). We note that the function g0d (x) defined by (3.10) satisfies ∂xn g0d = O(d −|n| ) and similarly for g±d (x) and det(∂jd (x)/∂x). We now consider the equation ϕ + Td (ζ )ϕ = h,

(5.6)

1866


for a given h ∈ L2 (Σ0 ). We show that this equation is solvable in L2 (Σ0 ), provided that η satisfies the assumption 0 η < ηεd (E) in Theorem 1.1. We fix N 1 large enough and take ρ > 1/2 close enough to 1/2. Let {u1 , u2 , u3 } be the partition of unity defined by u1 = χ0 x2 /d ρ ,

u2 = χ∞ x2 /d ρ χ0 (x2 /N d),

u3 = χ∞ (x2 /N d).

(5.7)

We further introduce smooth functions u˜ j such that u˜ j has a slightly larger support than uj and satisfies the relation u˜ j uj = uj for j = 1, 2, 3 and that all their derivatives obey the same bounds as those of uj for d 1. We decompose ϕ into ϕ = ϕ1 + ϕ2 + ϕ3 = (u1 + u2 + u3 )ϕ and similarly for h. Then (5.6) is written in the matrix form

Id + S11 S21 S31

S12 Id + S22 S32

S13 S23 Id + S33

ϕ1 ϕ2 ϕ3

=

h1 h2 h3

,

where Sj k = Sj k (ζ ) = uj Td (ζ )u˜ k , 1 j, k 3. We use the notation · to denote the norm of a bounded operator acting on L2 (Σ0 ). Lemma 5.4. We have S33 (ζ ) = O(d −σ N ) for some σ > 0 independent of N . Proof. We show that the kernel S33 (x, y; ζ ) of S33 (ζ ) satisfies S33 (x, y; ζ ) = O d −N exp −c (log d)/d |x2 − y2 | −cN + O |x2 | + d exp −c (log d)/d |y2 | −cN + O |y2 | + d exp −c (log d)/d |x2 | −cN + O |x2 | + |y2 | + d for some c > 0. If x ∈ Σ0 and ξ ∈ Σ = Σ+ ∪ Σ− , then |x − ξ | > d/16. Hence the Hankel function H0 (krd (x, ξ )) takes the asymptotic form H0 krd (x, ξ ) =

1/2 −iπ/4 ikrd (x,ξ ) e e 2 1 + O |rd (x, ξ )|−1 π (krd (x, ξ ))1/2

(5.8)

for |rd (x, ξ )| 1 by formula, and similarly for H0 (krd (ξ, y)) with ξ ∈ Σ and y ∈ Σ0 . Thus the first bound on the right side is obtained by applying Lemma 5.3 to the integral u3 (x2 ) XF+0 (x, ξ )Y F−0 (ξ, y) dξ u˜ 3 (y2 ). The other bounds are obtained by evaluating integrals such as

−cN |x| + |ξ | + d Y F−0 (ξ, y)u˜ 3 (y2 ) dξ,

−cN −cN |ξ | + |y| + d |x| + |ξ | + d dξ.


1867

If |y2 − ξ2 | < |y2 |/2, then |x2 | + |ξ2 | ∼ |x2 | + |y2 |, and if |y2 − ξ2 | > |y2 |/2, then ikr (ξ,y) = O d μ exp −c (log d)/d |y2 | e d by Lemma 5.2. If we take these facts into account, then we can establish the above bound on S33 (x, y; ζ ), and hence the lemma is proved. 2 Lemma 5.5. The operators S32 (ζ ), S23 (ζ ), S31 (ζ ) and S13 (ζ ) obey S32 + S23 + S31 + S13 = O d −σ N for some σ > 0 independent of N . Proof. The lemma is verified in almost the same way as Lemma 5.4. For example, we consider the kernel S32 (x, y; ζ ) of S32 (ζ ). Let {vN 0 , vN ∞ } be the partition of unity defined by vN 0 = χ0 (4x2 /Nd) and vN ∞ = χ∞ (4x2 /Nd). Then the integral u3 (x2 ) XF+ (x, ξ ; ζ )vN ∞ Y F− (ξ, y; ζ ) dξ u˜ 2 (y2 ) is shown to obey the same bound as S33 (x, y; ζ ) in the proof of Lemma 5.4. Let F±0 (x, y; ζ ) and F±1 (x, y; ζ ) be as in Lemma 5.1. We apply Lemma 5.3 to the integral V0 (x, y; ζ ) = u3 (x2 ) XF+0 (x, ξ ; ζ )vN 0 Y F−0 (ξ, y; ζ ) dξ u˜ 2 (y2 ) and Lemma 5.2 to the integral V1 (x, y; ζ ) = u3 (x2 ) XF+0 (x, ξ ; ζ )vN 0 Y F−1 (ξ, y; ζ ) dξ u˜ 2 (y2 ). Since |x2 − ξ2 | > N d/2 for ξ2 ∈ supp vN 0 , it follows from Lemma 5.2 that u3 (x2 )XF+0 (x, ξ ; ζ ) |x2 | + d −σ N ,

ξ2 ∈ supp vN 0 ,

for some σ > 0 independent of N , and we also have vN 0 (ξ2 )Y F−1 (ξ, y; ζ )u˜ 2 (y2 ) = O d μ for some μ > 0 independent of N . Thus we make use of these lemmas to obtain V0 (x, y; ζ ) = O d −σ N exp −c (log d)/d |x2 − y2 | and V1 (x, y; ζ ) = O((|x2 | + d)−σ N )u˜ 2 (y2 ). This yields S32 = O(d −σ N ). The other operators are also dealt with in a similar way. We skip the details. 2

1868


By Lemmas 5.4 and 5.5, the problem is now reduced to the solvability of equation

Id + S11 S21

S12 Id + S22

ϕ1 ϕ2

=

h1 h2

(5.9)

with another h1 and h2 in L2 (Σ0 ). Lemma 5.6. We have S22 (ζ )2 = O(d −L ) for any L 1. Proof. We present only an outline of the proof. A similar but more refined argument is used for proving Lemma 5.8 below. We evaluate the kernel of the operator S22 (ζ )2 = u2 XR(ζ ; K+d )Y R(ζ ; K−d )u2 XR(ζ ; K+d )Y R(ζ ; K−d )u˜ 2 . The idea is based on the fact that a particle which starts from supp u˜ 2 and passes over supp u2 again after being scattered by the fields 2πα± δ(x − d± ) never returns to supp u2 . Let Y ± be as in (5.5) and let F±0 and F±1 be as in Lemma 5.1. Then S22 admits the decomposition + − + S22 , S22 = S22

± S22 = u2 XR(ζ ; K+d )Y ± R(ζ ; K−d )u˜ 2

and it is shown in almost the same way as in the proof of Lemma 5.4 that the asymptotic form of ± (x, y; ζ ) is determined by the sum of the integrals the kernel S22 Uj±k (x, y; ζ ) = u2 (x2 ) XF+j (x, ξ ; ζ )vL0 Y ± F−k (ξ, y; ζ ) dξ u˜ 2 (y2 ) with 0 j, k 1, where vL0 (x2 ) is defined by vL0 = χ0 (x2 /Ld) for L 1 fixed arbitrarily. We − + ± , U01 and U00 . If make use of partial integration in the ξ1 variable to evaluate the integrals U10 ξ = (ξ1 , ξ2 ) ∈ Σ− with ξ2 ∈ supp vL0 and y = (y1 , y2 ) ∈ Σ0 with y2 ∈ supp u˜ 2 , then y1 > ξ1 and (∂/∂ξ1 ) |d+ − ξ | + |ξ − y| > c > 0 − + ± and hence it follows that U10 (x, y; ζ ) = O(d −L ). A similar argument applies to U01 and U00 . On the other hand, we make use of the stationary phase method in the ξ2 variable to evaluate the − (x, y; ζ ) for x = (x1 , x2 ) ∈ Σ0 with x2 ∈ supp u2 . We recall the other integrals. We consider U01 behavior of F+0 (x, ξ ; ζ ) and F−1 (ξ, y; ζ ) from Lemma 5.1. The phase function takes the form

ξ2 → r−d (ξ ) + rd (ξ, x) = |ξ − d− | + |x − ξ | + O(log d) for ξ and x as above. For each ξ1 fixed, the stationary point is attained at ξ = (ξ1 , ξ2 ) on the segment joining x and d− . We see that the stationary point ξ2 is non-degenerate and |x − ξ | + − (x, y; ζ ) takes the asymptotic form |ξ − d− | = |x − d− | at the point ξ = (ξ1 , ξ2 ). Thus U01 − U01 (x, y; ζ ) ∼ eik(|x−d− |+|y−d− |) u− 01 (x, y; ζ ) n m − μ−ρ(|n|+|m|) ) for some μ > 0, where ρ > 1/2 is as in and u− 01 (x, y; ζ ) obeys ∂x ∂y u01 = O(d (5.7). The explicit representation for the leading term of u− 01 does not matter in the proof of the


1869

+ ± + lemma. A similar argument applies to U10 and U11 . For the integral U10 (x, y; ζ ), the stationary point is attained at the point ξ = (ξ1 , ξ2 ) ∈ Σ+ on the segment joining the two points y and d+ for each ξ1 fixed, and the integral takes the asymptotic form + U10 (x, y; ζ ) ∼ eik(|x−d+ |+|y−d+ |) u+ 10 (x, y; ζ ), − where u+ 10 (x, y; ζ ) satisfies the same type of estimates as u01 (x, y; ζ ). For the integral ± U11 (x, y; ζ ), the stationary point is attained at the point ξ = (ξ1 , 0) ∈ Σ± , and we have ± (x, y; ζ ) ∼ eik(|x−d+ |+|y−d− |) u± U11 11 (x, y; ζ ).

evaluate the kernel of the iterated operator S22 (ζ )2 . For example, we consider the integral We − + U01 (x, ξ ; ζ )U10 (ξ, y; ζ ) dξ . If ξ2 ∈ supp u2 , then |ξ2 | > d ρ , and we have (∂/∂ξ2 ) |ξ − d− | + |ξ − d+ | > cd −1+ρ for some c > 0. Since ρ > 1 − ρ, we see by partial integration that the integral obeys the bound O(d −L ). A similar argument applies to other terms, and the proof is complete. 2 It follows from Lemma 5.6 that Id + S22 is invertible on L2 (Σ0 ), and we have 2 −1 (Id + S22 )−1 = Id − S22 (Id − S22 ). Hence the first component ϕ1 of Eq. (5.9) solves Id + S11 − S12 (Id + S22 )−1 S21 ϕ1 = h˜ 1 , where h˜ 1 = h1 − S12 (Id + S22 )−1 h2 . Lemma 5.7. S12 (Id + S22 )−1 S21 = O d −L for any L 1. Proof. We write (Id + S22 )−1 = Id − S22 + S22 (Id + S22 )−1 S22 . Then we have S12 (Id + S22 )−1 S21 = S12 S21 − S12 S22 S21 + S12 S22 (Id + S22 )−1 S22 S21 . For the same reason as in the proof of Lemma 5.6, we can show that S12 S21 + S12 S22 = O d −L . This proves the lemma.

2

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By Lemma 5.7, the solvability of (5.9) is obtained from the lemma below. Lemma 5.8. Let ηεd (E) be as in Theorem 1.1. If ζ = E − iη ∈ D −d satisfies 0 η < ηεd (E), then Id + S11 (ζ ) : L2 (Σ0 ) → L2 (Σ0 ) has a bounded inverse for d 1. Proof. As in the proof of Lemma 5.6, we decompose S11 into the sum + − + S11 , S11 = S11

± S11 = u1 XR(ζ ; K+d )Y ± R(ζ ; K−d )u˜ 1 .

± Then the asymptotic form of the kernel S11 (x, y; ζ ) is determined by the sum of the integrals

Q± j k (x, y; ζ ) = u1 (x2 )

XF+j (x, ξ ; ζ )vL0 Y F−k (ξ, y; ζ ) dξ u˜ 1 (y2 ) ±

with 0 j, k 1, where vL0 is again defined by vL0 (x2 ) = χ0 (x2 /Ld). Among these kernels, + ± Q− 10 , Q01 and Q00 obey − Q (x, y; ζ ) + Q+ (x, y; ζ ) + Q+ (x, y; ζ ) + Q− (x, y; ζ ) = O d −L . 10

01

00

00

The asymptotic behaviors as d → ∞ of the other kernels are analyzed by use of the stationary phase method in the ξ2 variable. We analyze the behavior of Q− 01 (x, y; ζ ) in some detail. Assume that x = (x1 , x2 ) ∈ Σ0 with x2 ∈ supp u1 and y = (y1 , y2 ) ∈ Σ0 with y2 ∈ supp u˜ 1 . Let ξ = (ξ1 , ξ2 ) ∈ Σ− with ξ2 ∈ supp vL0 . For each ξ1 fixed, the stationary point of the phase function ξ2 → |ξ − d− | + |x − ξ | is attained at ξ = (ξ1 , ξ2 ) on the segment joining x and d− . We note that |ξ1 + d/2|/|ξ − d− | = |x1 − ξ1 |/|x − ξ | = |x1 + d/2|/|x − d− | and |ξ − d− | + |x − ξ | = |x − d− | at ξ = (ξ1 , ξ2 ) with the stationary point ξ2 . Thus Q− 01 (x, y; ζ ) takes the asymptotic form ik(|x−d− |+|y−d− |) − q01 (x, y; ζ ). Q− 01 (x, y; ζ ) ∼ e − (x, y; ζ ). The Hessian is calculated as We analyze the behavior as d → ∞ of q01

(ξ1 + d/2)2 (x1 − ξ1 )2 + = |ξ − d− |3 |x − ξ |3

x1 + d/2 |x − d− |

so that the contribution from the Hessian turns out to be

2

|x − d− | , |ξ − d− ||x − ξ |


1/2 iπ/4 −1/2

(2π)

e

k

1871

|ξ − d− ||x − ξ | 1/2 |x − d− |/(x1 + d/2) |x − d− |

according to the stationary phase method [13, Theorem 7.7.5]. Since k = ζ 1/2 = E 1/2 + O((log d)/d) and since |x − d− |/(x1 + d/2) = 1 + O d −2+2ρ , the above quantity behaves like 1/2 (2π)1/2 eiπ/4 E −1/4 |ξ − d− ||x − ξ | |x − d− |−1/2 1 + O d −1+ρ .

(5.10)

We recall the behaviors of F+0 (x, ξ ; ζ ) and of F−1 (ξ, y; ζ ) from Lemma 5.1 to calculate XF+0 (x, ξ ; ζ ) and Y − F−1 (ξ, y; ζ ) when ξ is on the segment joining d− and x. We have j˜d (x, ξ ) = 1 and eiα+ θ+d (x,ξ ) = 1 + O d −1+ρ ,

ei(g−d (x)−g−d (ξ )) = 1 + O d −1+ρ .

We further have F+0 (x, ξ ; ζ ) = c(E)eik|x−ξ | |x − ξ |−1/2 1 + O d −1+ρ (x )∂ by (5.1) and (5.8). It follows from (5.4) and (5.5) that X and Y ± take the forms X ∼ −2χ−d 1 1 ± and Y ∼ −2χ˜ ∓d (x1 )∂1 . Since x1 > ξ1 for x ∈ Σ0 and ξ ∈ Σ− , we have ∂1 |x − ξ | = 1 + O(d −1+ρ ). Thus XF+0 (x, ξ ; ζ ) behaves like

XF+0 = −2iE 1/2 c(E)eik|x−ξ | |x − ξ |−1/2 χ−d (x1 ) + O d −2+ρ .

(5.11)

We consider Y − F−1 (ξ, y; ζ ). Let ξ ∈ Σ− be as above. Assume that y ∈ Σ0 with y2 ∈ supp u˜ 1 . Then the amplitude f− (−yˆ−d → ξˆ−d ; E) for the scattering by the field 2πα− δ(x) satisfies the relation f− (−yˆ−d → ξˆ−d ; E) = f− (ω− → ω+ ; E) + O d −1+ρ ,

ω± = (±1, 0).

Since ∂1 |ξ − d− | = 1 + O(d −1+ρ ), we repeat a similar computation to obtain that Y − F−1 (ξ, y; ζ ) behaves like −1/2 −2iE 1/2 c(E)eik(|ξ −d− |+|y−d− |) |ξ − d− ||y − d− | χ˜ +d (ξ1 )f− + O d −2+ρ with f− = f− (ω− → ω+ ; E). We now note that −2iE 1/2 c(E)(2π)1/2 eiπ/4 E −1/4 = 1 by the definition (5.1) of c(E) and that χ˜ +d (ξ1 ) dξ1 = 1. Then we combine the above behavior of Y − F−1 with (5.10) and (5.11) to see that Q− 01 (x, y; ζ ) takes the form

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I. Alexandrova, H. Tamura / Journal of Functional Analysis 260 (2011) 1836–1885 ik(|x−d− |+|y−d− |) − Q− q01 (x, y; ζ ), 01 = e

− (x, y; ζ ) behaves like where q01

−1/2 − q01 = −2iE 1/2 c(E)u1 (x2 ) |x − d− ||y − d− | u˜ 1 (y2 ) χ−d (x1 )f− + O d −2+ρ . ± The other integrals Q+ 10 (x, y; ζ ) and Q11 (x, y; ζ ) are dealt with in a similar way. Since χ˜ −d (ξ1 ) dξ1 = −1 and ∂1 |x − d+ | = −1 + O(d −1+ρ ), Q+ 10 (x, y; ζ ) takes the form ik(|x−d+ |+|y−d+ |) + Q+ q10 (x, y; ζ ), 10 = e + (x, y; ζ ) behaves like where q10

−1/2 + = −2iE 1/2 c(E)u1 (x2 ) |x − d+ ||y − d+ | u˜ 1 (y2 ) χ−d (x1 )f+ + O d −2+ρ q10 if we take into account the relations with f+ = f+ (ω+ → ω− ; E). On the other hand, χ˜ ±d (ξ1 ) dξ1 = ±1 and ∂1 |x − d+ | = −1 + O(d −1+ρ ), we can show that Q± 11 (x, y; ζ ) takes the form ik(|x−d+ |+|y−d− |) ± q11 (x, y; ζ ), Q± 11 = e ± where q11 (x, y; ζ ) behaves like

−1/2 ∓2iE 1/2 c(E)eikd d −1/2 u1 |x − d+ ||y − d− | u˜ 1 χ−d (x1 )f− f+ + O d −2+ρ . + Hence it follows that the sum Q− 11 (x, y; ζ ) + Q11 (x, y; ζ ) behaves like

−1/2 eik(|x−d+ |+|y−d− |) u1 (x2 )χ−d (x1 ) |x − d+ ||y − d− | u˜ 1 (y2 )O d −2+ρ , because |eikd d −1/2 | = O(1) is bounded uniformly in d 1 when ζ = E − iη satisfies 0 η < ηεd (E) (see (1.6)). We evaluate the norm of the integral operator with the remainder term −1/2 u˜ 1 (y2 )O d −2+ρ r(x, y; ζ ) = eik(|x−d− |+|y−d− |) u1 (x2 ) |x − d− ||y − d− | 2ikd /d| = O(1), it follows of Q− 01 (x, y; ζ ) as a kernel. Since 7d/16 < |x − d− | < 9d/16 and |e 2ik|x−d | 1−μ − that |e | = O(d ) for some μ > 0. If we note that |x2 | = O(d ρ ) on the support of u1 , then we have r(x, y; ζ )2 dx dy = O d −2−2μ+4ρ . Σ0 Σ0

We can take ρ > 1/2 so close to 1/2 that the norm of the integral operator under consideration obeys the bound o(1) as d → ∞. A similar argument applies to the remainder term


1873

− of Q+ 10 (x, y; ζ ), and also the norm of the integral operator with the kernel Q11 (x, y; ζ ) + Q+ 11 (x, y; ζ ) obeys the bound o(1) as d → ∞. We now combine all the results obtained above to see that the kernel of the operator S11 behaves like

S11 (x, y; ζ ) ∼ −2iE 1/2 c(E) f− s− (x) × s˜− (y) + f+ s+ (x) × s˜+ (y) + R(x, y; ζ ) with f− = f− (ω− → ω+ ; E) and f+ = f+ (ω+ → ω− ; E), where s± = χ−d (x1 )eik|x−d± | |x − d± |−1/2 u1 (x2 ),

s˜± = eik|x−d± | |x − d± |−1/2 u˜ 1 (x2 )

and the error term O((|x| + |y|)−N ) is negligible. The remainder term R(x, y; ζ ) on the right side takes the form −1/2 R(x, y; ζ ) = u1 (x2 ) eik(|x−d− |+|y−d− |) |x − d− ||y − d− | r− (x, y; ζ ) −1/2 + eik(|x−d+ |+|y−d+ |) |x − d+ ||y − d+ | r+ (x, y; ζ )

−1/2 + eik(|x−d+ |+|y−d− |) |x − d+ ||y − d− | r0 (x, y; ζ ) u˜ 1 (y2 ), where r0 (x, y; ζ ) satisfies ∂xm ∂yn r0 = O(d −2+ρ−ρ(|m|+|n|) ) and similarly for r± . We denote by R the integral operator with the kernel R(x, y; ζ ) and consider the operator S0 : L2 (Σ0 ) → L2 (Σ0 ) with the kernel S0 (x, y; ζ ) = −2iE 1/2 c(E) f− sˆ− (x) × s˜− (y) + f+ sˆ+ (x) × s˜+ (y) , where sˆ± = (Id + R)−1 s± = s± − (Id + R)−1 e± ,

e± = Rs± .

We claim that Id + S0 has a bounded inverse. Then we obtain that the operator Id + S11 in question also has a bounded inverse. We analyze the behavior of sˆ+ (x)˜s+ (x) dx and sˆ+ (x)˜s− (x) dx. As stated above, |e2ik|x−d± | | = O(d 1−μ ) for some μ > 0. This implies that the L2 norms of s± and s˜± obey s± 2 = O d −μ/2+ρ/2−1/2 ,

˜s± 2 = O d −μ/2+ρ/2+1/2 .

We also have

e+ (x) = u1 (x2 ) eik|x−d− | |x − d− |−1/2 + eik|x−d+ | |x − d+ |−1/2 O d −2+ρ + O d −L by making use of the stationary phase method for the integral with respect to the x2 variable, and hence it follows that e+ 2 = O d −μ/2+ρ/2−1/2 O d −1+ρ

1874


and similarly for e− . We can take ρ > 1/2 so close to 1/2 that

sˆ+ (x)˜s+ (x) dx = O d −L + O d −μ O d 2ρ−1 = o(1),

d → ∞,

and the stationary phase method applied to the integral with respect to the x2 variable yields

−1/2 ikd −1/2 e d + o(1). sˆ+ (x)˜s− (x) dx = − E 1/2 /2πi

A similar argument applies to the integrals sˆ− (x)˜s+ (x) dx and sˆ− (x)˜s− (x) dx. The eigenfunction of S0 takes the form c− sˆ− + c+ sˆ+ with |c− | + |c+ | = 0. Since −1/2 −2iE 1/2 c(E) E 1/2 /2πi = 1, t (c

− , c+ )

is approximately calculated as an eigenvector of the matrix

o(1) −eikd d −1/2 f+ + o(1) . o(1) −eikd d −1/2 f− + o(1)

When ζ = E − iη satisfies 0 η < ηεd (E), we can take dε (E) 1 so large that 2ikd −1 e d f− f+ < 1 − ε/2 for d > dε (E) (see (1.6)). This implies that Id + S0 is invertible, and the proof of the lemma is now complete. 2 We are now in a position to complete the proof of Lemma 3.5 in question. Proof of Lemma 3.5. We combine Lemmas 5.4–5.8 to conclude that Id + Td (ζ ) has a bounded inverse on L2 (Σ0 ) for d 1, provided that ζ = E − iη ∈ D −d satisfies 0 η < ηεd (E). This completes the proof. 2 We make only a brief comment on the proof of Lemma 3.2. Proof of Lemma 3.2. Proposition 5.1 remains true for ζ = E + iη ∈ D+d . Since Im k = Im ζ 1/2 > 0, |e2ikd /d| → 0 as d → ∞. Hence it can be shown that Id + Td (ζ ) has a bounded inverse on L2 (Σ0 ) for d 1. This proves the lemma. 2 5.3. Proof of Proposition 5.1 We end the section by proving Proposition 5.1 which has played a central role in the proof of Lemma 3.5. Proof of Proposition 5.1. (1) We prove the first statement. By assumption, |x1 | 3d/4,

|y1 | 3d/4,

|x2 | + |y2 | N d


1875

for N 1. Let t be on the contour of the line integral in (4.4). Then the following three lemmas enable us to prove the statement in almost the same way as Proposition 4.1. Lemma 5.9. One has Re(ζ rd (x)rd (y)/t) > 0 for ζ = E − iη ∈ D −d . Lemma 5.10. One has | Im e−iθd (x,y) | c(|x2 | + |y2 |)−1 for some c > 0. Lemma 5.11. If 0 < t < κ, then Re ζ rd (x)2 + rd (y)2 /t c |x|2 + |y|2 /t,

c > 0,

and if 0 < s < M(|x2 | + |y2 |) for t = κ + is, M 1 being fixed arbitrarily, then there exists σ > 0 independent of N such that Re ζ rd (x)2 + rd (y)2 /t σ N log |x2 | + |y2 | . We complete the proof of statement (1), accepting these lemmas as proved. Lemma 5.9 makes it possible for us to decompose Rα (jd (x), jd (y); ζ ) into the sum Rα jd (x), jd (y); ζ = (i/4)eiαθd (x,y) H0 krd (x, y) + Rsc,α jd (x), jd (y); ζ as in (4.9), and Lemmas 5.10 and 5.11 enable us to show in almost the same way as in the proof of Proposition 4.1 that Rsc,α jd (x), jd (y); ζ = O |x2 | + |y2 | −σ N . Thus (1) is obtained. Proof of Lemma 5.9. We set w = ζ rd (x)rd (y)/t. We compute 2 2 rd (x)2 = |x|2 1 + 2iηd (x2 ) |x2 |/|x| + O (log d)/d

(5.12)

and similarly for rd (y)2 , where ηd (t) obeys ηd (t) = O((log d)/d). Hence we have 2 2

rd (x)rd (y) ∼ |x||y| 1 + i ηd (x2 ) x2 /|x| + ηd (y2 ) y2 /|y|

(5.13)

for d 1. If 0 < t < κ, then it is easy to see that Re w > 0. If t = κ + is with s > 0, then we have −1 ζ /t = κ 2 + s 2 (Eκ − ηs) − i(Es + ηκ) , and hence Re w behaves like |x||y| Re w ∼ 2 κ + s2

2 2 x2 y2 E ηd (x2 ) − η s + Eκ + ηd (y2 ) |x| |y|

for d 1. It follows from the definition of ηd (t) that

(5.14)

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I. Alexandrova, H. Tamura / Journal of Functional Analysis 260 (2011) 1836–1885 −1/2

ηd (x2 ) = 5E0

(log d)/d

or

−1/2

ηd (y2 ) = 5E0

(5.15)

(log d)/d

1/2

for |x2 | + |y2 | N d. Since 0 η 2E0 (log d)/d for ζ = E − iη ∈ D −d , we have that Re w > 0 for t = κ + is also. 2 Proof of Lemma 5.10. The denominator ep + e−iθd (x,y) of the integrand in (4.13) never vanishes but takes values close to 0 around p = 0, provided that θd (x, y) ∼ ±π . This is the case when x2 1 and y2 −1 or when x2 −1 and y2 1. We consider only the former case. We compute Re θd (x, y) as in the proof of Lemma 4.3. If x1 and y1 fulfill the assumption in the proposition and if x2 1 and y2 −1, then Re γ jd (x); ω+ π/2 + c1 /x2 ,

Re γ jd (y); ω+ 3π/2 + c1 /y2

for some c1 > 0, so that Re θd (x, y) −π + c1 (|x2 | + |y2 |)−1 . This, together with (4.14), implies that Im e−iθd (x,y) c |x2 | + |y2 | −1 for some c > 0. Hence the desired bound is obtained.

2

Proof of Lemma 5.11. We set w = (ζ /t)(rd (x)2 + rd (y)2 ). If 0 < t < κ, then it is easy to see that Re w > c(|x|2 + |y|2 )/t for some c > 0. Assume that t = κ + is with 0 < s < M(|x2 | + |y2 |). If we take (5.12), (5.14) and (5.15) into account, then a simple computation yields Re w > c((log d)/d)(|x2 | + |y2 |) for another c > 0. Since (log p)/p is decreasing for p 1, we have log |x2 | + |y2 | / |x2 | + |y2 | log(Nd)/(N d) (2/N ) × (log d)/d for |x2 | + |y2 | N d. This implies that Re w σ N log(|x2 | + |y2 |) for some σ > 0, and hence the lemma follows at once. 2 (2) We proceed to the second statement. We assume that |x2 | + |y2 | N d for N 1 fixed above. The kernel Rα (x, y; ζ ) is represented by the line integral (4.4) even for ζ = E −iη ∈ D −d . However, the integral representation (2.3) for Iν (ζ |x||y|/t) with ν = |l − α| does not make sense any longer. In fact, Re ζ |x||y|/t ∼ −η|x||y|/s < 0 for t = κ + is with s 1. For this reason, we make use of the different representation formula for Iν (ζ |x||y|/t) when Re(ζ |x||y|/t) < 0. The proof of the statement is divided into four steps. (i) We begin by decomposing Rα (jd (x), jd (y); ζ ) into the sum of three terms. To do this, we take κ as κ = M 2 log d,

M 1, 2

in the line integral (4.4), so that et is at most of polynomial growth |et | = O(d M ) as d → ∞ on the contour (0, κ) ∪ (κ + i0, κ + i∞). We set χM0 (t) = χ0 (s/Md) and χM∞ (t) = χ∞ (s/Md) for s = Im t 0 and decompose Rα (x, y; ζ ) into the sum


1877

Rα (x, y; ζ ) = R(x, y; ζ ) + R∞ (x, y; ζ ), where R=

κ+i∞ ζ |x||y| dt ζ (|x|2 + |y|2 ) t 1 ilψ − Iν , e χM0 (t) exp 4π 2 2t t t l

0

κ+i∞ ζ |x||y| dt ζ (|x|2 + |y|2 ) 1 ilψ t R∞ = − Iν e χM∞ (t) exp 4π 2 2t t t l

0

with ν = |l − α|, and ψ is defined by ψ = θ − ω for x = (|x| cos θ, |x| sin θ ) and y = (|y| cos ω, |y| sin ω). We note that the choice of M depends on N and that Re(ζ |x||y|/t) > 0 for 0 < s < 2Md, which is seen from (5.14). Hence, by formula (2.3), the first term R(x, y; ζ ) on the right side is further decomposed into the sum of two terms R(x, y; ζ ) = Rfr (x, y; ζ ) + Rsc (x, y; ζ ) after calculating the series

eiαψ Rfr (x, y; ζ ) = 4π

le

ilψ I

ν (ζ |x||y|/t)

as in the proof of Proposition 4.1, where

ζ |x − y|2 dt t − , χM0 (t) exp 2 2t t

κ+i∞

0

Rsc (x, y; ζ ) =

1 4π

ζ |x||y| ζ (|x|2 + |y|2 ) dt t − Isc ,ψ χM0 (t) exp 2 2t t t

κ+i∞

0

and Isc (w, ψ) is defined by (4.8). 1/2 We make a similar decomposition for Rα (jd (x), jd (y); ζ ). Since 0 η 2E0 (log d)/d for ζ = E − iη ∈ D −d , we can take M so large that −1 Re w = Re ζ rd (x)rd (y)/t ∼ κ 2 + s 2 (Eκ − ηs)|x||y| > 0

(5.16)

for 0 < s < 2Md. Thus the integral representation (2.3) still makes sense for w as above, and we have Rα jd (x), jd (y); ζ = Gfr (x, y; ζ ) + Gsc (x, y; ζ ) + G∞ (x, y; ζ ), where Gfr (x, y; ζ ) = Rfr (jd (x), jd (y); ζ ) and similarly for Gsc and G∞ . If we use the new notation pd (x) = rd (x)2 + rd (y)2 ,

qd (x, y) = rd (x)rd (y),

then these three terms have the following representations:

1878


t ζ rd (x, y)2 dt − , χM0 (t) exp 2 2t t

κ+i∞

eiαθd (x,y) Gfr = 4π

0

Gsc =

1 4π

κ+i∞

χM0 (t) exp

ζ qd (x, y) ζpd (x, y) t dt − Isc , θd (x, y) , 2 2t t t

0 κ+i∞ ζ qd (x, y) dt ζpd (x, y) 1 ilθd (x,y) t G∞ = − Iν . e χM∞ (t) exp 4π 2 2t t t l

0

Statement (2) is obtained by showing that: G∞ = O d −N , Gfr = (i/4)eiαθd (x,y) H0 krd (x, y) + O d −N , Gsc = c(E)eik(rd (x)+rd (y)) qd (x, y)−1/2 (fα + eN ) + O d −N ,

(5.17) (5.18) (5.19)

where fα = fα (−ω → θ ; E) with θ = x/|x| and ω = y/|y|, and eN = eN (x, y; ζ ) satisfies the estimate in the proposition. (ii) To prove (5.17), we employ the formula e−iμπ/2 Iμ (w) = π

π cos(μρ − iw sin ρ) dρ − sin(μπ) 0

∞ e

−iw sinh p−μp

dp

0

for Im w 0, which follows as an immediate consequence of the relation Iμ (w) = e−iμπ/2 Jμ (iw) [22, p. 176]. We note that Im(ζ qd (x, y)/t) < 0 for t = κ + is with s > Md, M 1, which is seen from (5.13) and (5.14). We insert Iν (ζ qd (x, y)/t) into the integral representation for G∞ (x, y; ζ ) and evaluate the resulting integral by partial integration for each l with |l| < d. If M 1, then ∂t t − ζpd (x, y)/t ± ζ qd (x, y)/t sin ρ > c > 0, ∂t t − ζpd (x, y)/t − 2i ζ qd (x, y)/t sinh p > c > 0 for t = κ + is with s > Md uniformly in ρ, 0 < ρ < π , and in p, 0 < p < 1. If p > 1, then we use |∂t (t − ζpd (x, y)/t)| > c > 0 and ∂t e−i(ζ qd (x,y)/t) sinh p = −t −1 (sinh p/ cosh p)∂p e−i(ζ qd (x,y)/t) sinh p . We take into account these relations to repeat the integration by parts. Since Im θd (x, y) = O((log d)/d) as is seen from (4.14), the sum of the integrals with |l| < d obeys O(d −N ). To see that the sum over l with |l| > d is of order O(d −N ), we make use of the other representation formula


(w/2)μ Iμ (w) = Γ (μ + 1/2)Γ (1/2)

1

μ−1/2 e−wρ 1 − ρ 2 dρ

1879

(5.20)

−1

for Iμ (w) with μ 0 [22, p. 172]. Since |x| + |y| = O(d), we have |ζ qd (x, y)/t| = M −1 O(d) for s = Im t > Md and −wρ = O e|Re(ζ qd (x,y)/t)| = O ed , e

|ρ| < 1,

for w = ζ qd (x, y)/t. Since Γ (μ) behaves like Γ (μ) ∼ (2π)1/2 e−μ μμ−(1/2) for μ 1 by the Stirling formula, we can take M 1 so large that ilθ (x,y) ν e d w /Γ (ν) (1/2)|l| ,

|l| > d.

Hence the sum of integrals with l with |l| > d also obeys O(d −N ), and (5.17) is proved. (iii) (5.18) is easy to prove. By (4.5), we have Gfr (x, y; ζ ) = (i/4)eiαθd (x,y) H0 krd (x, y) + G(x, y; ζ ), where κ+i∞

eiαθd (x,y) G= 4π

χM∞ (t) exp

ζ rd (x, y)2 dt t − . 2 2t t

0

Since |∂t (t − ζ rd (x, y)2 /t)| > c > 0 for s > Md, we have G(x, y; ζ ) = O(d −N ) by partial integration, and hence (5.18) is established. (iv) The proof of (5.19) uses the stationary phase method. By (4.8), we have Isc ζ qd (x, y)/t, θd (x, y) = −Cα ei[α](θd (x,y)+π) Lsc (t, x, y; ζ ), where Cα = sin(απ)/π and ∞ Lsc (t, x, y; ζ ) = −∞

ei(iζ qd (x,y)/t) cosh p

e(1−β)p dp ep + e−iθd (x,y)

with β = α − [α], 0 < β < 1. By (5.16), Re(ζ qd (x, y)/t) > 0. Since d/4 |x1 |, |y1 | 3d/4 and |x2 | + |y2 | N d by assumption, θd (x, y) stays away from ±π uniformly in x, y and ζ ∈ D−d , so that Isc (ζ qd (x, y)/t, θd (x, y)) is bounded uniformly in x, y and ζ as above. If 0 < t < κ, then exp t/2 − ζpd (x, y)/2t exp −cd 2 /t ,

c > 0,

and if 0 < s < d/M for t = κ + is, then it follows from (5.14) that Re(ζ /t) behaves like Re(ζ /t) ∼ M 4 (log d)/d 2 for d 1. Hence we have exp t/2 − ζpd (x, y)/2t = O exp M 2 − cM 4 log d = O d −N −1

1880


for M 1. Thus the integral over the intervals (0, κ) and (κ +i0, κ +id/M) is negligible. We assume that d/M < s < 2Md. We apply the stationary phase method [13, Theorem 7.7.5] to the integral Lsc (t, x, y; ζ ) above. The stationary point is given by p = 0, and Isc (ζ qd (x, y)/t, θd (x, y)) is seen to take the asymptotic form Isc = e−ζ qd (x,y)/t b0 (t, x, y; ζ ) + bL (t, x, y; ζ ) + O d −L for any L 1, where −1 −1/2 b0 = −Cα (2π)1/2 ei[α](θd (x,y)+π) ζ qd (x, y)/t 1 + e−iθd (x,y) and bL obeys |∂xn ∂ym ∂t bL | = O(d −3/2−|n|−|m|−j ). The phase term is calculated as j

2 t/2 − ζ pd (x, y) + 2qd (x, y) /2t = t/2 − ζ rd (x) + rd (y) /2t. If s = Im t satisfies d/M < s < 2d/M or Md < s < 2Md, then ∂t t/2 − ζ rd (x) + rd (y) 2 /2t > c > 0 for 0 < Re t < κ. Hence we deform the contour to the imaginary axis by analyticity and repeat integration by parts to obtain that the leading term comes from the integral ∞ a0 (x, y; ζ ) =

s ζ (rd (x) + rd (y))2 ds + b0 (is, x, y; ζ ) , χM (s) exp i 2 2s s

0

where χM (s) = χ∞ (2Ms/d)χ0 (2s/Md). We now set λd (x, y) = Re k rd (x) + rd (y) ,

μd (x, y) = Im k rd (x) + rd (y)

for k = ζ 1/2 . Then λd (x, y) behaves like λd (x, y) ∼ d and μd (x, y) obeys μd (x, y) = O(log d). If we make a change of variable s = λd (x, y)τ , then a0 (x, y; ζ ) takes the form ∞ a0 =

dτ 1 τ + eiσd (τ,x,y) χ˜ M (τ )b0 iλd (x, y)τ, x, y; ζ exp iλd (x, y) 2 2τ τ

0

where χ˜ M (τ ) = χM (λd (x, y)τ ) and σd (τ, x, y) =

ζ (rd (x) + rd (y))2 − λd (x, y)2 iμd (x, y) = + O (log d)2 /d . 2λd (x, y)τ τ

We apply the stationary phase method to the above integral with τ = 1 as a stationary point to derive the asymptotic form of a0 (x, y; ζ ). We have λd (x, y) + σd (1, x, y) = k rd (x) + rd (y) + O (log d)2 /d


1881

and b0 (iλd (x, y)τ, x, y; ζ ) takes the value −1 −1/2 λd (x, y)1/2 1 + e−iθd (x,y) b0 = −Cα (2πi)1/2 ei[α](θd (x,y)+π) ζ qd (x, y) at τ = 1. We also have ζ = E + O((log d)/d) and θd (x, y) = ψ + O (log d)/d = θ − ω + O (log d)/d . We recall that the amplitude fα (ω → θ ; E) is defined by (1.4) and the constant c(E) is defined by (5.1). We take into account the contribution (λd (x, y)/2πi)−1/2 from the Hessian at the stationary point τ = 1 and compute −Cα (2πi)ei[α](ψ+π) E −1/2 eiψ / 1 + eiψ

= 4πc(E)(2/π)1/2 eiπ/4 E −1/4 sin(απ)ei[α](ψ+π) ei(ψ+π) / 1 − ei(ψ+π) = 4πc(E)fα (−ω → θ ; E).

Since σd (1, x, y) satisfies n m −iσ j iσ ∂ ∂ e d ∂ e d = O (log d)j d −|n|−|m| , x y

τ

we see that a0 (x, y; ζ ) takes the asymptotic form a0 = 4πc(E)eik(rd (x)+rd (y)) qd (x, y)−1/2 fα (−ω → θ ; E) + eN + O d −N , where eN (x, y; ζ ) satisfies the remainder estimate in the proposition. A similar argument applies to the integral associated with bL (t, x, y; ζ ). It takes the form 4πc(E)eik(rd (x)+rd (y)) qd (x, y)−1/2 O d −1 + O d −N and is regarded as a remainder term. Thus (5.19) is established. (3) Finally we make only a brief comment on the asymptotic form of derivatives such as ∂Rα (jd (x), jd (y); ζ )/∂xj . If we take a careful look at the proof of statements (1) and (2), then we see that the asymptotic forms obtained in (1) and (2) remain true in the C 1 topology. We skip the details. The proof of the proposition is now complete. 2 6. Proof of Lemma 3.4 The last section is devoted to proving Lemma 3.4. The proof is based on the following proposition. Proposition 6.1. Let Dd be defined by (3.5) and let Rα (x, y; ζ ) be the kernel of the resolvent R(ζ ; Pα ) with ζ ∈ Dd . If d/c < |x1 | < cd, for some c > 1 or if

|x2 | > d/c,

|y1 | < cd,

|y2 | < c

(6.1)

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|x1 | < cd,

|x2 | < c,

d/c < |y1 | < cd,

|y2 | > d/c,

then −L , Rα jd (x), jd (y); ζ = O |x2 | + |y2 |

|x2 | + |y2 | 1,

for any L 1, where the order estimate depends on d but is uniform in ζ . The derivative ∂Rα (jd (x), jd (y); ζ )/∂ζ also obeys a similar bound. Proof of Proposition 6.1. The proof uses formula (5.20) to evaluate the kernel. We consider only the case when x and y fulfill (6.1). In this case, we have that Rα (jd (x), jd (y); ζ ) equals Rα (jd (x), y; ζ ). The dependence on d does not matter in the proof of the proposition. We only look at the dependence on x2 with |x2 | 1. We write y = (|y| cos ω, |y| sin ω) and set θd (x) = γ (jd (x); ω+ ). Then we can represent Rα (jd (x), y; ζ ) as 1 Rα = 4π

ζ rd (x)2 ζ |y|2 dt t − exp − S(t, x, y; ζ ) exp 2 2t 2t t

κ+i∞

0

by the line integral (4.4), where S(t, x, y; ζ ) =

eil(θd (x)−ω) Iν

l

ζ rd (x)|y| , t

ν = |l − α|.

Since Im θd (x) = O((log d)/d) uniformly in x with |x2 | 1, we make use of (5.20) to obtain that S(t, x, y; ζ ) has the following properties: S(t, x, y; ζ ) exp c|x2 |/|t| , m ∂ S(t, x, y; ζ ) |t|−m exp σm |x2 |/|t| , t

(6.2) (6.3)

where c > 0 and σm > 0 depend on d but are independent of ζ ∈ Dd and x2 . If ζ = E + iη ∈ Dd , then −1 (Eκ + ηs) − i(Es − ηκ) ζ /t = κ 2 + s 2 1/2

for t = κ + is on the contour of the line integral above. Since |η| 2E0 (log d)/d for ζ ∈ Dd and since rd (x)2 behaves like rd (x)2 ∼ 1 − O (log d)2 /d 2 x22 + 2iη0d x22 , −1/2

with η0d = 5E0

|x2 | 1,

(log d)/d, we have Re ζ rd (x)2 /t c1 |x2 |2 /|t|

(6.4)

for some c1 > 0. We divide the line integral into the sum of two parts by use of the cutoff functions χM0 (t) = χ0 (s/M|x2 |) and χM∞ (t) = χ∞ (s/M|x2 |). We can take M 1 so


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large that |∂t (t − ζ rd (x)2 /t)| > c2 > 0 for s > M|x2 |/2. This, together with (6.3), enables us to repeat the partial integration, and we can obtain the bound O(|x2 |−L ) for the line integral cut off by χM∞ (t). On the other hand, we see from (6.2) and (6.4) that the line integral cut off by χM0 (t) also obeys the desired bound. A similar argument applies to the derivative ∂Rα (jd (x), jd (y); ζ )/∂ζ . Thus the proof is complete. 2 Lemma 6.1. Let

Σ = x: d/c < |x1 | < cd ,

Ω = x: |x1 | < cd, |x2 | < c

for some c > 1. Then the resolvent R(ζ ; P˜αd ) of the closed operator P˜αd = Jd Pα Jd−1 is analytic as a function of ζ ∈ Dd with values in bounded operators from L2 (Σ) to L2 (Ω) or from L2 (Ω) to L2 (Σ). Proof. We know that R(ζ ; Pα ) : L2comp → L2loc is well defined. If we set Σ = {x ∈ Σ: |x2 | < d/c}, then R(ζ ; P˜αd ) = R(ζ ; Pα ) : L2 Σ → L2 (Ω),

L2 (Ω) → L2 Σ

is bounded. Hence the lemma is obtained as an immediate consequence of Proposition 6.1.

2

Proof of Lemma 3.4.. The operator H±d has the solenoidal field 2πα± δ(x − d± ) with d± = (±d/2, 0) as a center. Since the relations Σ0 = {7d/16 < x1 + d/2 < 9d/16} = {−9d/16 < x1 − d/2 < −7d/16} and Ω0 ⊂ {|x1 ± d/2| < 3d/2, |x2 | < r0 } hold true for Σ0 and Ω0 , the lemma is obtained by applying Lemma 6.1 to R(ζ ; K±d ) = exp(ig∓d )R˜ ±d (ζ ) exp(−ig∓d ) with R˜ ±d (ζ ) = Jd R(ζ ; H±d )Jd−1 .

2

Acknowledgment The first author gratefully acknowledges the partial support from NSF grant DMS 0801158. Appendix A We shall prove that the resolvent of the magnetic Schrödinger operator with two solenoidal fields has the meromorphic continuation over the lower-half plane as a function of the spectral parameter ζ with values in operators from L2comp to L2loc . The resolvent is shown to be meromorphically continued over the complex plane C \ (−∞, 0] slit along the negative real axis across the positive real axis where the spectrum of the operator is located. The argument here extends to the case of several solenoidal fields without any essential changes.

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We consider the operator H = H (Ψ ) = (−i∇ − Ψ )2 with two solenoidal fields 2πα+ δ(x − p+ ) and 2πα− δ(x − p− ), where Ψ (x) = Φ+ (x) + Φ− (x) = α+ Φ(x − p+ ) + α− Φ(x − p− ). The operator H becomes self-adjoint on L2 = L2 (R 2 ) under the boundary condition lim|x−p± |→0 |u(x)| < ∞ at both the centers p+ = (p, 0) and p− = (−p, 0) for p > 0. We cover the whole space with the three regions

X± = x: |x1 ∓ p| < 3p/2, |x2 | < 3p/2 ,

X0 = R 2 \ x: |x1 | < 2p, |x2 | < p

and approximate H by operators with one solenoidal field over each region. To do this, we define the three operators P± = H (Φ± ) and P = H (Φ0 ), where Φ0 (x) = α0 Φ(x) with α0 = α+ + α− . These auxiliary operators become self-adjoint by imposing the boundary condition as in (1.3) at the center of the solenoidal field. We can construct real smooth bounded functions g± (x) and g0 (x) such that H = eig± P± e−ig± over X± and H = eig0 P0 e−ig0 over X0 . We set Pˆ± = eig± P± e−ig± ,

Pˆ0 = eig0 P0 e−ig0

and introduce a smooth nonnegative partition of unity {u+ , u− , u} such that u± and u have support in X± and X0 , respectively. Then we define the bounded operator G(ζ ) = u+ R(ζ ; Pˆ+ ) + u− R(ζ ; Pˆ− ) + u0 R(ζ ; Pˆ0 ) : L2 → L2 for ζ ∈ C + = {ζ ∈ C: Im ζ > 0}. This operator satisfies (H − ζ )G(ζ ) = Id + Q(ζ ), where Q(ζ ) = [Pˆ+ , u+ ]R(ζ ; Pˆ+ ) + [Pˆ− , u− ]R(ζ ; Pˆ− ) + [Pˆ0 , u0 ]R(ζ ; Pˆ0 ). The commutators above vanish outside the region X = {x: |x1 | < 3p, |x2 | < 3p}. For the Hamiltonians P± and P0 with one solenoidal field, the resolvents R(ζ ; Pˆ± ) and R(ζ ; Pˆ0 ) are continued as analytic functions of ζ with values in operators from L2comp to L2loc over the lowerhalf plane across the positive real axis. If we consider Q(ζ ) as an operator from L2 (X) into itself, then Q(ζ ) turns out to be an analytic function of ζ ∈ C \ (−∞, 0] with values in compact operators. Hence it follows from the analytic perturbation theory of Fredholm that the inverse (Id + Q(ζ ))−1 : L2 (X) → L2 (X) has the meromorphic continuation over C \ (−∞, 0]. Thus R(ζ ; H ) is represented as −1 R(ζ ; H ) = G(ζ ) − G(ζ )Q(ζ ) Id + Q(ζ ) and is defined as a meromorphic function over C \ (−∞, 0] with values in operators from L2comp (X) to L2loc . Once this is established, we can show in almost the same way as in the proof of Theorem 1.1 (Step 5) that R(ζ ; H ) becomes a meromorphic function over C \ (−∞, 0] with values in operators from L2comp to L2loc .


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References [1] R. Adami, A. Teta, On the Aharonov–Bohm Hamiltonian, Lett. Math. Phys. 43 (1998) 43–53. [2] G.N. Afanasiev, Topological Effects in Quantum Mechanics, Kluwer Academic Publishers, 1999. [3] J. Aguilar, J.M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971) 269–279. [4] Y. Aharonov, D. Bohm, Significance of electromagnetic potential in the quantum theory, Phys. Rev. 115 (1959) 485–491. [5] E. Balslev, J.M. Combes, Spectral properties of many body Schrödinger operators with dilation analytic interactions, Comm. Math. Phys. 22 (1971) 280–294. [6] N. Burq, Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math. 124 (2002) 677–735. [7] J.M. Combes, P. Duclos, M. Klein, R. Seiler, The shape resonance, Comm. Math. Phys. 110 (1987) 215–236. [8] L. Dabrowski, P. Stovicek, Aharonov–Bohm effect with δ-type interaction, J. Math. Phys. 39 (1998) 47–62. [9] C. Fernández, R. Lavine, Lower bounds for resonances width in potential and obstacle scattering, Comm. Math. Phys. 128 (1990) 263–284. [10] S. Fujiié, A. Lahmar-Benbernou, A. Martinez, Width of shape resonances for non globally analytic potentials, J. Math. Soc. Japan, in press. [11] B. Helffer, J. Sjöstrand, Résonances en limite semi-classique, Mém. Soc. Math. Fr. (N.S.) 24/25 (1986). [12] P.D. Hislop, I.M. Sigal, Introduction to Spectral Theory. With Applications to Schrödinger Operators, SpringerVerlag, 1996. [13] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, 1983. [14] H.T. Ito, H. Tamura, Aharonov–Bohm effect in scattering by point-like magnetic fields at large separation, Ann. H. Poincaré 2 (2001) 309–359. [15] A. Martinez, Resonance free domains for non globally analytic potentials, Ann. H. Poincaré 3 (2002) 739–756, Erratum; Ann. H. Poincaré 8 (2007) 1425–1431. [16] Y. Ohnuki, Aharonov–Bohm k¯oka, Butsurigaku saizensen, vol. 9, Ky¯oritsu syuppan, 1984 (in Japanese). [17] S.N.M. Ruijsenaars, The Aharonov–Bohm effect and scattering theory, Ann. Physics 146 (1983) 1–34. [18] B. Simon, The definition of molecular resonance curves by the method of exterior complex scaling, Phys. Lett. A 71 (1979) 211–214. [19] J. Sjöstrand, Quantum resonances and trapped trajectories, in: Long Time Behaviour of Classical and Quantum Systems, Bologna, 1999, in: Ser. Concr. Appl. Math., vol. 1, World Sci. Publ., River Edge, NJ, 2001, pp. 33–61. [20] J. Sjöstrand, M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4 (1991) 729–769. [21] X.P. Wang, Barrier resonances in strong magnetic fields, Comm. Partial Differential Equations 17 (1992) 1539– 1566. [22] G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge University Press, 1995.


Extrapolation of weights revisited: New proofs and sharp bounds Javier Duoandikoetxea Departamento de Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Apartado 644, 48080 Bilbao, Spain Received 14 July 2010; accepted 16 December 2010 Available online 24 December 2010 Communicated by Gilles Godefroy

Abstract We use an appropriate factorization of the Ap weights to give another proof of the extrapolation theorem of Rubio de Francia. It provides sharp bounds in terms of the Ap -constant of the weights. Then we extend the result to more general settings including off-diagonal and partial range extrapolation. Among the applications, we prove by iteration a multivariable extrapolation theorem and give a sharp bound for Calderón–Zygmund operators on Lp (w) for weights in Aq (q < p). © 2010 Elsevier Inc. All rights reserved. Keywords: Weighted inequalities; Extrapolation; Sharp bounds; Multilinear operators; Muckenhoupt bases

1. Introduction A weight is a nonnegative locally integrable function. A weight is in Ap (Rn ) for p > 1 if

[w]Ap

p−1 1 1 1−p := sup w w < +∞, |Q| |Q| Q Q

(1.1)

Q

where the supremum is taken over all cubes Q in Rn . The value [w]Ap is the Ap -constant of w. For p = 1, we say that w is in A1 (Rn ) if Mw(x) Cw(x) a.e., where M is the Hardy–Littlewood E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.015

J. Duoandikoetxea / Journal of Functional Analysis 260 (2011) 1886–1901

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maximal function. The A1 -constant of w, [w]A1 , is the essential supremum of Mw/w. In what follows we write simply Ap instead of Ap (Rn ). These classes of weights were introduced by B. Muckenhoupt [23], who proved the following well-known fundamental result: M is bounded on Lp (w) if and only if w ∈ Ap (1 < p < ∞), and is of weak-type (p, p) with respect to the measure w(x) dx if and only if w ∈ Ap (1 p < ∞). For more information on Ap weights the reader can consult [13,9] or [14], among other references. An important property of the Ap weights is the extrapolation theorem of Rubio de Francia, announced in [25] and given with a detailed proof in [26]. In its first version the extrapolation theorem says that if for some p0 a sublinear operator is bounded from Lp0 (w) to Lp0 (w) for all w ∈ Ap0 /λ with 1 λ < ∞ and λ p0 < ∞, then it is bounded from Lp (w) to Lp (w) for all w ∈ Ap/λ and λ < p < ∞. The second proof of the theorem was later supplied by J. GarcíaCuerva in [12], reproduced in [13, Chapter 4, Theorem 5.19]. Another version of the proof is in [9, Theorem 7.8]. In all those proofs there are two cases depending on whether p is smaller or greater than p0 . A unified approach treating both cases together is due to Cruz-Uribe, Martell and Pérez (see [5]). More recently, due to the interest in studying the sharp dependence of the norms of several operators in terms of the Ap -constant of the weights, O. Dragiˇcević, L. Grafakos, M.C. Pereyra, and S. Petermichl gave in [7] a version of the extrapolation theorem with sharp bounds following the approach of García-Cuerva. A different version of the proof is in [14, Theorem 9.5.3]. Although originally given for sublinear operators, it was realized that sublinearity was not necessary. Actually even the operator itself does not play any role and all the statements can be given in terms of families of pairs of nonnegative measurable functions. This was observed in [6, Remark 1.11] and is the setting adopted in [4]. In this paper we also stick to this point of view and write the theorems for pairs of functions. The fact that uv 1−p is an Ap weight for u, v ∈ A1 has been used in a crucial way in the proofs of the extrapolation theorems. We use a different way of factorizing weights to give in Section 3 a proof that provides sharp bounds. Previously, in Section 2, we introduce the three basic ingredients of the proof: factorization, construction of A1 weights and sharp bounds for the Hardy–Littlewood maximal function. Since Aq ⊂ Ap for q < p with [w]Ap [w]Aq , we can expect a better exponent in the bound of the norm in terms of [w]Aq . We consider this question in Section 4 in two ways, depending on whether the basic estimate is for p0 > p or for p0 < p. In particular, for Calderón–Zygmund operators, we extend to Aq weights an estimate for A1 weights due to Lerner, Ombrosi and Pérez [22]. In Section 5 we generalize the extrapolation to the off-diagonal case in which the inequalities are from Lp (w p ) to Lq (w q ) for appropriate w and possibly different values for p and q. It seems that only the case p q appears in the literature but there is no reason for such a restriction. Even more, we show that any q > 0 is acceptable and that in fact q does not play any role in the statement, except for defining the right exponent in terms of p. The interest of the generalization to the off-diagonal case for any q > 0 will be apparent in Section 6 where it is used to obtain by iteration a multivariable extrapolation theorem of Grafakos and Martell [15]. In Section 7 we consider another version of the extrapolation theorem, the limited-range extrapolation considered in [1] (and also to some extent in [10] and [18]). Although the classical extrapolation theorem is a particular case of both Theorems 5.1 and 7.1, we consider that it is worth writing its proof independently because even without taking care of

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the size of the bounds, the proof we propose seems simpler and more direct than the previous ones. Rubio de Francia’s original paper already proposed a setting more general than the Ap weights of Muckenhoupt. An account of several extensions is in [5]. In Section 8, we shall focus on two of the possible extensions. 2. Preliminaries The proofs of the theorems in Sections 3, 5 and 7 will be based on three results: factorization, construction of A1 weights, and sharp bounds for the Hardy–Littlewood maximal operator on Lp (w). They are contained in the three lemmas we present in this section. The (usual) factorization theorem for Ap weights states that w is in Ap if and only if w = uv 1−p for some u, v ∈ A1 . The “if” part is easily obtained from the definition (1.1), while the “only if” part is harder and was first proved by P. Jones. From this factorization theorem it is easy to deduce that one can multiply Ar and As weights each one raised to an appropriate power to get Ap weights. In the following lemma we give the two factorization results needed in our proofs. We remark that only the easy part of the factorization is used to prove the extrapolation theorems. Lemma 2.1 (Factorization). (a) Let 1 p < p0 < ∞. If w ∈ Ap and u ∈ A1 , then wup−p0 is in Ap0 and p−p 0 wu A

p0

p −p

[w]Ap [u]A01

(2.1)

. 1

(b) Let 1 < p0 < p < ∞. If w ∈ Ap and u ∈ A1 , then (w p0 −1 up−p0 ) p−1 is in Ap0 and p −1 p−p 1/(p−1) 0 w 0 u A

p0

p0 −1

p−p0

[w]Ap−1 [u]Ap−1 . p 1

Proof. Use the definition, Hölder’s inequality and the fact that 1 u Mu(x) [u]A1 u(x) for almost every x ∈ Q. |Q|

(2.2)

2

Q

J.L. Rubio de Francia introduced in [26] a construction of A1 weights, now known as Rubio de Francia algorithm. Lemma 2.2 (Rubio de Francia algorithm). Let p > 1. Let f be a nonnegative function in Lp (w) and w ∈ Ap . Let M k be the k-th iterate of M, M 0 f = f , and MLp (w) be the norm of M as a bounded operator on Lp (w). Define Rf (x) =

∞ k=0

M k f (x) . (2MLp (w) )k

(2.3)

Then f (x) Rf (x) a.e., Rf Lp (w) 2f Lp (w) , and Rf is an A1 weight with constant [Rf ]A1 2MLp (w) .


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Proof. The first property is immediate. For the second one, use the fact that M k Lp (w) MkLp (w) and sum a geometric series. For the last one observe that ∞ ∞ M k+1 f (x) M k f (x) p M Rf (x) 2M . L (w) k (2MLp (w) ) (2MLp (w) )k k=0

2

k=1

Lemma 2.3. Let f ∈ Lp (w) with p > 1 and w ∈ Ap . Then 1/(p−1)

Mf Lp (w) C[w]Ap

f Lp (w) ,

(2.4)

where C depends only on p and the dimension. If q < p and w ∈ Aq , then 1/p

Mf Lp (w) C[w]Aq f Lp (w) ,

(2.5)

where C depends only on p, q and the dimension. The bound (2.4) was obtained by S. Buckley in [2]; a simple proof was given by A. Lerner in [20]. The improvement of the exponent for w ∈ Aq also appears in [2]. It is a consequence of 1/p the fact that the weak-type inequality corresponding to (2.4) holds with constant C[w]Ap [23,2] together with the Marcinkiewicz interpolation theorem. 3. The extrapolation theorem of Rubio de Francia with sharp bounds Theorem 3.1. Assume that for some family of pairs of nonnegative functions, (f, g), for some p0 ∈ [1, ∞), and for all w ∈ Ap0 we have g p0 w

1/p0

1/p0 CN [w]Ap0 f p0 w ,

Rn

(3.1)

Rn

where N is an increasing function and the constant C does not depend on w. Then for all 1 < p < ∞ and all w ∈ Ap we have

1/p gp w

1/p CK(w) f pw ,

Rn

Rn

where K(w) =

⎧ ⎨ N ([w]Ap (2MLp (w) )p0 −p ), ⎩ N ([w]

p0 −1 p−1

Ap

(2MLp (w1−p ) )

max(1,

In particular, K(w) C1 N (C2 [w]Ap

p0 −1 p−1 )

p−p0 p−1

) for w ∈ Ap .

if p < p0 ; (3.2) ),

if p > p0 .

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Proof. Case p < p0 . For f ∈ Lp (w) we built the A1 weight Rf given by the Rubio de Francia algorithm as in (2.3). Then

p

g w= p

Rn

g p w(Rf ) p 0

(p−p0 )

p

(Rf ) p 0

(p0 −p)

Rn

p

p0

p0

p−p0

g w(Rf )

1− p

p0

p

(Rf ) w

Rn

Rn

p CN w(Rf )p−p0 A

p f

p0

p −p p CN [w]Ap [Rf ]A01

p0

1− p

p0

p−p0

w(Rf )

Rn

|f | w p

p0

Rn

|f |p w Rn

p −p p CN [w]Ap 2MLp (w) 0

|f |p w, Rn

where we applied Hölder’s inequality, part (a) of the Factorization Lemma 2.1, the inequality f (x) Rf (x) (in the form Rf (x)−1 f (x)−1 ), (2.1), and [Rf ]A1 2MLp (w) . This gives (3.2) for p < p0 . We can use then (2.4) from Lemma 2.3 to obtain the bound p0 −1

). C1 N (C2 [w]Ap−1 p Case p > p0 . We use duality to write

p = sup g p0 hw : h ∈ L p−p0 (w) with norm 1 .

p0 p

p

g w Rn

Rn

Fix such a function h, which we can assume nonnegative, and define H such that H p w 1−p = p

h p−p0 w. Then H is in Lp (w 1−p ) with norm 1. Building the A1 weight RH given by the Rubio de Francia algorithm and using the pointwise inequality H (x) RH(x) a.e., we have

g p0 hw Rn

g p0 w

p0 −1 p−1

(RH)

p−p0 p−1

Rn p−p0 p0 p0 −1 CN w p−1 (RH) p−1 A

f p0 w

p0

p0 −1 p−1

(RH)

p−p0 p−1

Rn p0 −1 p−p0 p0 p−1 2M CN [w]Ap−1 p 1−p L (w ) p

f w Rn

·

(RH)p w 1−p Rn

p 1− p0

,

p0 p

p


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where we applied part (b) of the Factorization Lemma 2.1, the A1 constant of RH given by Lemma 2.2, (2.2), and Hölder’s inequality. Note that the case p0 = 1 is simpler because several terms disappear. This gives the case p > p0 of (3.2) and we obtain from (2.4) the bound in terms of [w]Ap . 2 The sharp form of the extrapolation theorem in terms of [w]Ap is useful to deduce the sharp dependence of the Ap -constants for the norms of several classical operators like the Hilbert, Riesz and Beurling transforms. See [7] for examples and references. 4. Lp (w) bounds for w ∈ Aq with q < p p /p

Let q < p. When we insert the bound (2.5) into (3.2) we get K(w) C1 N (C2 [w]A0q ). Nevertheless, in the case p < p0 , we can get a better bound by modifying the previous proof. For this, we shall consider another way of building A1 weights, namely, that if Mf (x) is finite almost everywhere and s > 1, then (Mf )1/s is an A1 weight with constant depending on s but not on f (see [9, Theorem 7.7]). Theorem 4.1. Under the hypotheses of Theorem 3.1, if w ∈ Aq for some q < p < p0 we have

1/p gp w

1 1 p−p C1 N C2 [w]Aq [w]Aq 0

Rn

1/p f pw

.

Rn

Proof. In the first part of the proof of Theorem 3.1 instead of Rf we use Mf to get

g w p

Rn

p g0 w(Mf )p−p0

p p 0

1− p p

(Mf ) w

Rn

p0

(4.1)

.

Rn

Set u = (Mf )(p0 −p)/(p0 −q) . Since p0 − p < p0 − q, u is an A1 weight with constant independent of f . Then wuq−p0 is an Ap0 weight with constant bounded by C(p, q)[w]Aq . Inserting this into (4.1) we get

p g w CN wuq−p0 A

p f0 w(Mf )p−p0

p

p0

Rn

Rn

p CN C(p, q)[w]Aq MLp (w) 0

p(1− pp )

p CN C(p, q)[w]Aq [w]

1− pp 0 Aq

p p0

1− p |Mf | w p

p0

Rn

|f |p w

Rn

|f |p w.

2

Rn

When the function N appearing in (3.1) is of the form N (t) = t α , then from (3.2) we get αp /p C[w]Aq0 , while Theorem 4.1 gives the exponent α + 1/p − 1/p0 , which is better if αp0 > 1. This condition is satisfied at least in all the examples for which the sharp bound is known.

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Since Ap is the union of Aq for q < p, the proof of Theorem 4.1 also serves as a proof of the extrapolation theorem for Ap weights, regardless of the size of the constants. The (apparently) more general statement of Rubio de Francia in [25] in which (3.1) is assumed only for w ∈ Ap0 /λ for some λ 1 can be obtained as a corollary to Theorem 3.1. Corollary 4.2. Let 1 λ < ∞ and λ p0 < ∞. Assume that for w ∈ Ap0 /λ we have p0

1/p0

g w

1/p0 p0 CN [w]Ap0 /λ f w .

Rn

(4.2)

Rn

Then for λ p < ∞ and w ∈ Ap/λ we have

1/p gp w

1/p p −λ max(1, 0 ) C1 N C2 [w]Ap/λ p−λ f pw .

Rn

Rn

To prove this corollary it is enough to write g p0 = (g λ )p0 /λ and f p0 = (f λ )p0 /λ in (3.1), and apply Theorem 3.1 with p0 /λ as the starting exponent. As an application of Corollary 4.2 we obtain an estimate for Lp (w) norms with Aq weights, q < p. Applied to Calderón–Zygmund operators, this answers a conjecture of Lerner and Ombrosi in [21] (Conjecture 1.3). Corollary 4.3. Let T be an operator such that Tf Lp (w) CN [w]A1 f Lp (w) ,

(4.3)

for all w ∈ A1 and all 1 < p < ∞, with C independent of w. Then we have Tf Lp (w) CN [w]Aq f Lp (w)

(4.4)

for all w ∈ Aq and 1 q < p < ∞, with C independent of w. In particular, (4.4) holds with N (t) = t if T is a Calderón–Zygmund operator. Proof. Given q ∈ (1, p), set p0 = p/q. Then (4.2) holds for λ = p0 . Applying Corollary 4.2 we obtain (4.4). For Calderón–Zygmund operators, the estimate (4.3) was proved in [22] with N (t) = t. Then (4.4) holds linearly in [w]Aq . 2 5. Off-diagonal extrapolation Muckenhoupt and Wheeden proved in [24] that the fractional integral Iα (convolution with |x|α−n ) is bounded from Lp (w p ) to Lq (w q ) for 1/q = 1/p − α/n if and only if w satisfies [w]Ap,q

q/p 1 1 q −p := sup w w < +∞, |Q| |Q| Q Q

Q

(5.1)


1893

where the supremum is taken over all cubes in Rn . The class of weights satisfying this condition is called Ap,q and [w]Ap,q given by (5.1) is the Ap,q constant of w. It is convenient to observe that w ∈ Ap,q is equivalent to w q ∈ A1+q/p with the same constants. The extrapolation theorem of Rubio de Francia was extended by Harboure, Macías and Segovia to these classes of weights in [16]: if an operator is bounded from Lp0 (w p0 ) to Lq0 (w q0 ) for some couple (p0 , q0 ) and all w ∈ Ap0 ,q0 , then it is bounded from Lp (w p ) to Lq (w q ) whenever 1 < p, q < ∞, 1/q − 1/p = 1/q0 − 1/p0 and w ∈ Ap,q . The sharp dependence on the Ap,q constant of the weight for the boundedness of Iα was settled by Lacey, Moen, Pérez and Torres in [19]. In their approach they needed and proved the extrapolation theorem of Harboure, Macías and Segovia with sharp bounds. We shall write the proof of the extrapolation theorem for Ap,q classes using the method of the previous section. Actually, we prove it in a more general setting than that of [16] and [19], in the sense that it holds for any q > 0 and we allow different values for the exponent in the target space and the second index in the weight class. The advantage of this generalization is that we want to apply the theorem in two different situations: fractional integrals and multivariable extrapolation (see the next section). Theorem 5.1. Let 1 p0 < ∞ and 0 < q0 , r0 < ∞. Assume that for some family of nonnegative couples (f, g) and for all w ∈ Ap0 ,r0 we have q0

g w

q0

1/q0

1/p0 p0 p0 CN [w]Ap0 ,r0 f w ,

Rn

(5.2)

Rn

where N is an increasing function and the constant C does not depend on w. Set γ = 1/r0 + 1/p0 . Then for all 1 < p < ∞ and 0 < q, r < ∞, such that 1 1 1 1 1 1 − = − = − , q q0 r r0 p p0

(5.3)

and all w ∈ Ap,r we have 1/q

gq wq

1/p CK(w) f p wp ,

Rn

Rn

where K(w) =

⎧ ⎨ N ([w]Ap,r (2MLγ r (wr ) )γ (r−r0 ) ), ⎩ N ([w]

γ r0 −1 γ r−1

Ap,r

(2MLγp (w−p ) )

max(1,

In particular, K(w) C1 N (C2 [w]Ap,r

r0 p ) rp0

γ (r−r0 ) γ r−1

if q < q0 ; ),

if q > q0 .

) for w ∈ Ap,r .

The condition 1/q − 1/p = 1/q0 − 1/p0 imposes some restrictions on the values of p when 1/q0 − 1/p0 is positive.

1894


Proof of Theorem 5.1. With γ as defined, [w]Ap0 ,r0 = [w r0 ]Ar0 γ . The couples (p, r) obtained in the theorem also satisfy γ = 1/r + 1/p , and [w]Ap,r = [w r ]Arγ . Case q < q0 (p < p0 , r < r0 ). Let f be in Lp (w p ). Define H so that H rγ w r = f p w p . Then H is in Lrγ (w r ). Built the weight RH given by the Rubio de Francia algorithm and use Hölder’s inequality to write

g q w q (RH)qγ (r−r0 )/r0 (RH)qγ (r0 −r)/r0

g w = q

q

Rn

Rn

q0 g w r (RH)γ (r−r0 ) r0 q0

q/q0

γr

(RH) w

Rn

r

1−q/q0 (5.4)

.

Rn

(In the last term we use the first equality of (5.3).) Part (a) of the Factorization Lemma 2.1 implies that w r (RH)γ (r−r0 ) is in Ar0 γ and r w (RH)γ (r−r0 ) A

r0 γ

γ (r −r) w r A [RH]A1 0 . rγ

Using the hypothesis (5.2) we obtain

q0 g q0 w r (RH)γ (r−r0 ) r0

1/q0

CN w r (RH)γ (r−r0 ) A

r0 γ

Rn

·

f

p0

r p0 w (RH)γ (r−r0 ) r0

1/p0 .

Rn

We insert this bound into (5.4) and use the properties of the Rubio de Francia algorithm: p/rγ (RH)−1 H −1 , RHLγ r (wr ) 2H Lγ r (wr ) = 2f Lp (wp ) , [RH]A1 2MLγ r (wr ) . Thus we get

1/q gq wq

1/p γ (r−r0 ) CN w r A 2MLγ r (wr ) f p wp . rγ

Rn

Rn

r0 γ −1

Using (2.4) we obtain the bound

N ([w r ]Arγrγ−1

r0 p

rp ) = N ([w r ]Arγ0

) for w r ∈ Arγ . q

Case q > q0 (p > p0 , r > r0 ). To use duality, consider a nonnegative function h in L q−q0 (w q ) with unit norm. p γ −1 Observe that w ∈ Ap,r is also equivalent to w −p ∈ Ap γ and [w]Ap,r = [w −p ]A . Define

pγ

H ∈ Lp γ (w −p ) by setting H p γ w −p = hq/(q−q0 ) w q . Then we can built the A1 weight RH given by the Rubio de Francia algorithm and use H RH to get


q−q0 g q0 H p γ w −(p +q) q w q

g hw = q0

q

Rn

1895

Rn

rγ (q−q0 )

g q0 w q0 p /p0 (RH) q(rγ −1) . Rn

The weight in the last integral is the q0 /r0 power of an Ar0 γ weight. Indeed, from (5.3) and the definition of γ we can see that it is the q0 /r0 power of

w

p r0 p0

(RH)

p γ (r−r0 ) r

γ (r−r0 ) r0 γ −1 = w r rγ −1 (RH) rγ −1 ,

which is in Ar0 γ according to part (b) of the Factorization Lemma 2.1. Moreover, (r−r0 )γ r r0 γ −1 w rγ −1 (RH) rγ −1 A

r0 γ

(r−r0 )γ r0 γ −1 −1 w r Arγrγ−1 [RH]Arγ . 1

(5.5)

Then we can apply the hypothesis (5.2) to get g q0 hw q CN

(r−r0 )γ r r0 γ −1 w rγ −1 (RH) rγ −1 A

q0

r0 γ

Rn

·

f

p0

w

p (p0 −1)

(RH)

p0 γ (r−r0 ) r0 (rγ −1)

q0

p0

.

Rn

Using Hölder’s inequality with exponents p/p0 and its dual, inserting the bound from (5.5), and taking into account that [RH]A1 MLp γ (w−p ) we get the second part of the theorem. Finally, we can use Lemma 2.3 to get K(w) C1 N (C2 [w r ]Arγ ). 2 In the theorem of Harboure, Macías and Segovia, r = q > p. In the next section we shall need the case r = p > q. 6. Multivariable weighted inequalities L. Grafakos and J.M. Martell proved in [15] an extrapolation theorem for multivariable operators (see also [3] for a two variable version). We give another proof of the theorem by iterating the off-diagonal extrapolation of the previous section. Theorem 6.1. Let T be an operator defined on m-tuples of functions. Let 1 r1 , . . . , rm < ∞ and 1/r = 1/r1 + · · · + 1/rm . Assume that T (f1 , f2 , . . . , fm )

r ) Lr (w1r ···wm

C

m j =1

fj Lrj (wrj ) j

(6.1)

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J. Duoandikoetxea / Journal of Functional Analysis 260 (2011) 1886–1901 r

rm holds for all functions fj ∈ Lrj (wjj ) and for all tuples of weights (w1r1 , . . . , wm ) ∈ (Ar1 , r

. . . , Arm ), with a constant C depending on the values of [wjj ]Arj , but not otherwise on the weights. Then, for every 1 < p1 , . . . , pm < ∞, there exists a constant K such that T (f1 , f2 , . . . , fm )

p p Lp (w1 ···wm )

m

K

j =1

fj Lpj (wpj )

(6.2)

j

p

p

holds with 1/p = 1/p1 + · · · + 1/pm , for all tuples of weights (w1 , . . . , wm ) ∈ (Ap1 , . . . , Apm ) p and all functions fj ∈ Lpj (wj j ). Proof. Fix the functions f2 , . . . , fm , the exponents r2 , . . . , rm , and the weights w2 , . . . , wm . Define the operator T (1) as follows: T (1) (g) = T (g, f2 , . . . , fm )w2 · · · wm

m j =2

fj −1rj

rj

L (wj )

.

Then (6.1) says that T (1) satisfies (1) T (g)

Lr (w1r )

for w1r1 ∈ Ar1 and for that T (1) is bounded

CgLr1 (wr1 ) 1

[w1r1 ]Ar1 .

some constant C depending on We apply Theorem 5.1 to deduce p p p from Lp1 (w1 1 ) to Lp (w1 ) when 1 < p1 < ∞ and w1 1 is in Ap1 , with 1/p = 1/p1 + 1/r2 + · · · + 1/rm . Iterating this process for the other components we get the full range of exponents 1 < p1 , . . . , pm < ∞. 2

Remark 6.1. We can use the bound given by Theorem 5.1 to estimate K of (6.2) in terms of C of (6.1) in the following sense: if there exist functions Nj such that C C0

m j =1

r N j w jj A , rj

with C0 independent of the weights, then K K0

m j =1

r −1

pj max(1, pjj −1 ) . Nj Kj wj Ap j

7. Limited range extrapolation For operators that are unbounded outside a range of the form (p− , p+ ) with 1 < p− < p+ < ∞ we cannot expect the assumptions (3.1) or (4.2) to hold. Instead, we could have weighted inequalities for weights satisfying conditions of the type w α0 ∈ Aq0 for some α0 > 1, for instance. We treat such situation in the following extrapolation theorem, although the statement is more general and even values of p smaller than 1 are allowed.


1897

Theorem 7.1. Let p0 ∈ (0, ∞), q0 ∈ [1, ∞) and α0 1. Assume that for some family of nonnegative couples (f, g) and for all w such that w α0 is in Aq0 we have p0

1/p0

g w

C

f

Rn

p0

1/p0 w

(7.1)

,

Rn

where C depends on [w α0 ]Aq0 , but not otherwise on w. Let qp and αp be defined by p0 q0 −1 and − 1 = α0 qp p

αp α0 p0 . = qp q0 p

(7.2)

α0 p0 . α0 − 1

(7.3)

Set p− =

α0 p0 α0 + q0 − 1

and p+ =

Then for p− < p < p+ and all w such that w αp is in Aqp we have 1/p

p

g w

C

1/p p

f w

Rn

(7.4)

.

Rn

Proof. We shall not assume the values of αp and qp of (7.2) as given, but as part of the conclusion of the theorem. The proof will show how they are deduced. Case p < p0 . Let qp > 1 and let w be a weight such that w αp is in Aqp . Let f be in Lp (w). Define H as H qp w αp = f p w so that H is in Lqp (w αp ). Thus we can built the A1 weight RH using the Rubio de Francia algorithm. On the other hand, since w αp is in Aqp , Lemma 2.1 implies that w αp (RH)qp −q0 is in Aq0 , so that this weight to the power 1/α0 can be used in (7.1). Thus we write qp −q0 p pα q0 −qp p αp 1− p g p w = g p w α0 (RH) α0 p0 w p0 α0 (RH) α0 p0 Rn

Rn

αp

g p0 w α0 (RH)

qp −q0 α0

p

Rn

p0

(q0 −qp )p

(RH) α0 (p0 −p) w

α0 p0 −pαp α0 (p0 −p)

1− p

p0

Rn

where we applied Hölder’s inequality. We use (7.1), RH −1 H −1 , and RHLqp (wαp ) 2H Lqp (wαp ) . This gives (7.4) if the exponents match. Since H = (f p w 1−αp )1/qp , we see that this holds if p qp − q0 = p − p0 qp α0

and

αp 1 − αp qp − q0 + = 1, α0 qp α0

which are the conditions in (7.2). On the other hand, we need qp > 1 and this is achieved if p > p− , for the value of p− given in (7.3).

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J. Duoandikoetxea / Journal of Functional Analysis 260 (2011) 1886–1901 p

Case p > p0 . Let h be a nonnegative function in L p−p0 (w) with unit norm, for w αp ∈ Aqp . αp

αp

p

Since w 1−qp is in Aqp , we define H such that H qp w 1−qp = h p−p0 w, so that the Rubio de Francia algorithm yields the A1 weight RH. Then

αp −1 p−p0 g p0 w 1−qp (RH)qp p w.

g hw p0

Rn

(7.5)

Rn

Let us observe that according to part (b) of Lemma 2.1, q −1

w

αp qp0 −1

qp −q0

(RH) qp −1 ∈ Aq0 .

We identify this weight raised to the power 1/α0 with the weight appearing in (7.5) so that (7.1) can be applied. This requires p − p0 qp p

1 qp − q0 = α0 qp − 1

and

αp p − p0 αp q0 − 1 = , −1 1 − qp p α0 qp − 1

which gives again (7.2). After applying (7.1), use Hölder’s inequality and the properties of RH to end the proof. The condition qp < ∞ demands p < p+ for the value of p+ appearing in (7.3). 2 Remark 7.1. This theorem was proved by P. Auscher and J.M. Martell in [1, Theorem 4.9], although their statement looks different because it is given in terms of reverse Hölder inequalities. The equivalence with powers of weights in the corresponding Ap classes is mentioned in [1] and was obtained by R. Johnson and C. Neugebauer in [18]. It reads Asr = A1+ r−1 ∩ RH s , s

where Asr = {w: w s ∈ Ar }. Here we say that w ∈ RH s for s > 1 if there exists C such that for every cube Q

1 |Q|

1

w

s

s

Q

C |Q|

α

α

w. Q

We easily check that Aq00 = A p0 ∩ RH ( p+ ) and Aqpp = A p−

p0

The particular case p0 = q0 = 2 was proved in [10].

p p−

∩ RH ( p+ ) , as stated in [1]. p

Remark 7.2. It is apparent from the proof that we can keep track of the bounds in terms of the constants of the weights as in the previous theorems.


1899

8. Weights from Muckenhoupt bases and rough operators The extrapolation theorems obtained in the paper are based on the lemmas given in Section 2, but Lemma 2.3 is only needed to give precise bounds in terms of the Ap -constant. The extrapolation results can be adapted to any situation in which factorization and the Rubio de Francia algorithm are available. Several possible extensions can be found in [5]. We show in this section two examples: weights associated to Muckenhoupt bases and to rough operators. 8.1. Muckenhoupt bases Let B be a collection of open sets in Rn (a basis). Define the maximal operator associated to B as 1 |f | MB f (x) = sup x∈B∈B |B| B

if x belongs to some set in B, and MB f (x) = 0 otherwise. The theory of weights for maximal operators associated to bases was studied by B. Jawerth in [17]. The weights associated to B are defined as the usual Ap weights: w ∈ Ap,B if

[w]Ap,B

1 := sup |B| B∈B

w B

1 |B|

w

1−p

p−1 < +∞,

(8.1)

B

and w ∈ A1,B if MB w(x) Cw(x) a.e. In this case, [w]A1,B is the smallest constant fulfilling the inequality. With these definitions it is immediate to see that the Factorization Lemma 2.1 holds. We say that a basis is a Muckenhoupt basis if MB is bounded on Lp (w) whenever w ∈ Ap,B , for all 1 < p < ∞. Then the Rubio de Francia algorithm (Lemma 2.2) can be carried out for Muckenhoupt bases. With both lemmas we can obtain the extrapolation theorem corresponding to Theorem 3.1 with bounds similar to (3.2). We cannot use the bounds of Lemma 2.3, because they are specific to the usual Hardy–Littlewood maximal operator. Similarly, Theorems 5.1, 6.1 and 7.1 can be written in terms of Ap,B weights. 8.2. Rough operators Weighted inequalities for rough operators such as homogeneous singular integrals, Hilbert transforms and maximal functions along curves, and the dyadic spherical maximal function were studied by D. Watson in [28]. An abstract formulation is in [8]. The extrapolation theorems given by D. Watson follows the method of [12]; in [8], the theorem is stated but no proof is supplied. The setting can be described as follows. There exist sublinear positive operators M and M∗ , bounded on L∞ and such that 1/p p 1/p M(uv) M up M v for 1 < p < ∞, with a similar property for M∗ . Define W1 = w: M∗ w(x) Cw(x) a.e. ,

1900


and, for 1 < p < ∞, Wp = w: M is bounded on Lp (w) , and similarly Wp∗ (1 p < ∞). Note that M∗ appears in the definition of W1 , and conversely. Let AW p be the subset of non-extremal weights in Wp , that is, AW p = w ∈ Wp : w s ∈ Wp for some s > 1 . The class AW p can be strictly contained in Wp , as is the case for the dyadic spherical maximal operator [11]. For the rough operators considered in [28] and [8], M is the sum of the Hardy–Littlewood maximal operator and a rough maximal operator given as the supremum of convolutions with positive measures, and M∗ is analogous to M but with the adjoints of such convolutions. Then the following factorization result holds: w is in AW p if and only if there exist w0 ∈ AW ∗1 and w1 ∈ 1−p AW 1 such that w = w0 w1 . Even without a description like (8.1), this is enough to factorize AW p weights as in Lemma 2.1 (without considering (2.1) and (2.2), of course). On the other hand, we need the Rubio de Francia algorithm to build AW 1 and AW ∗1 weights rather than W1 and W1∗ weights. This can be done as follows. From the non-extremality of the weights it follows that if w ∈ AW p , then w ∈ AW p/s for some s > 1. Given f ∈ Lp (w), define Ms f = (Mf s )1/s and built the weight Rf given by the algorithm when applied with the iterations of Ms . We get Ms (Rf ) C Rf , that is, Rf ∈ AW ∗1 . Once the lemmas are available, we can prove the extrapolation theorem for AW p weights (Theorem 3 of [28]). Theorems like those in Sections 5 and 7, for instance, are also feasible. Remark 8.1. For other examples of the presence of two positive operators in the factorization and extrapolation of weights, the reader can consult [5, Section 2.5]. The weights associated to the one-sided Hardy–Littlewood maximal operator (see [27]) yield a familiar example. Acknowledgments Research supported in part by the grant MTM2007-62186 of the Ministerio de Ciencia e Innovación (Spain) and FEDER. References [1] P. Auscher, J.M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights, Adv. Math. 212 (2007) 225–276. [2] S. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1) (1993) 253–272. [3] M.J. Carro, J. Soria, R.H. Torres, Rubio de Francia extrapolation theory: estimates for distribution functions, preprint. [4] D. Cruz-Uribe, J.M. Martell, C. Pérez, Extrapolation from A∞ weights and applications, J. Funct. Anal. 213 (2004) 412–439. [5] D. Cruz-Uribe, J.M. Martell, C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Oper. Theory Adv. Appl., vol. 215, Birkhäuser Verlag, Basel, 2011. [6] D. Cruz-Uribe, C. Pérez, Two weight extrapolation via the maximal operator, J. Funct. Anal. 174 (2000) 1–17. [7] O. Dragiˇcević, L. Grafakos, M.C. Pereyra, S. Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005) 73–91.


1901

[8] J. Duoandikoetxea, Almost-orthogonality and weighted inequalities, Contemp. Math. 189 (1995) 213–226. [9] J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., vol. 29, American Mathematical Society, Providence, RI, 2001. [10] J. Duoandikoetxea, A. Moyua, O. Oruetxebarria, E. Seijo, Radial Ap weights with applications to the disc multiplier and the Bochner–Riesz operators, Indiana Univ. Math. J. 57 (2008) 1261–1281. [11] J. Duoandikoetxea, L. Vega, Spherical means and weighted inequalities, J. Lond. Math. Soc. (2) 53 (1996) 343–353. [12] J. García-Cuerva, An extrapolation theorem in the theory of Ap weights, Proc. Amer. Math. Soc. 87 (1983) 422– 426. [13] J. García-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. [14] L. Grafakos, Modern Fourier Analysis, second ed., Grad. Texts in Math., vol. 250, Springer, New York, 2009. [15] L. Grafakos, J.M. Martell, Extrapolation of weighted norm inequalities for multivariable operators and applications, J. Geom. Anal. 14 (2004) 19–46. [16] E. Harboure, R. Macías, C. Segovia, Extrapolation results for classes of weights, Amer. J. Math. 110 (1988) 383– 397. [17] B. Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 108 (1986) 361–414. [18] R. Johnson, C.J. Neugebauer, Change of variable results for Ap and reverse Hölder RH r -classes, Trans. Amer. Math. Soc. 328 (1991) 639–666. [19] M.T. Lacey, K. Moen, C. Pérez, R.H. Torres, Sharp weighted bounds for fractional integral operators, J. Funct. Anal. 259 (2010) 1073–1097. [20] A. Lerner, An elementary approach to several results on the Hardy–Littlewood maximal operator, Proc. Amer. Math. Soc. 136 (2008) 2829–2833. [21] A. Lerner, S. Ombrosi, An extrapolation theorem with applications to weighted estimates for singular integrals, preprint. [22] A. Lerner, S. Ombrosi, C. Pérez, A1 bounds for Calderón–Zygmund operators related to a problem of Muckenhoupt and Wheeden, Math. Res. Lett. 16 (2009) 149–156. [23] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226. [24] B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974) 261–274. [25] J.L. Rubio de Francia, Factorization and extrapolation of weights, Bull. Amer. Math. Soc. (N.S.) 7 (1982) 393–395. [26] J.L. Rubio de Francia, Factorization theory and Ap weights, Amer. J. Math. 106 (1984) 533–547. [27] E. Sawyer, Weighted inequalities for the one-sided Hardy–Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986) 53–61. [28] D.K. Watson, Vector-valued inequalities, factorization and extrapolation for a family of rough operators, J. Funct. Anal. 121 (1994) 389–415.


Time-frequency partitions and characterizations of modulation spaces with localization operators ✩ Monika Dörfler, Karlheinz Gröchenig ∗ Institut für Mathematik, Universität Wien, Alserbachstrasse 23 A-1090 Wien, Austria Received 7 December 2009; accepted 21 December 2010 Available online 6 January 2011 Communicated by G. Godefroy

Abstract We study families of time-frequency localization operators and derive a new characterization of modulation spaces. This characterization relates the size of the localization operators to the global time-frequency distribution. As a by-product, we obtain a new proof for the existence of multi-window Gabor frames and extend the structure theory of Gabor frames. © 2011 Elsevier Inc. All rights reserved. Keywords: Phase-space localization; Short-time Fourier transform; Modulation space; Localization operator; Gabor frame

1. Introduction A time-frequency representation transforms a function f on Rd into a function on the timefrequency space Rd × Rd . The goal is to obtain a description of f that is local both in time and in frequency [5,20]. The standard time-frequency representations, such as the short-time Fourier transform and its various modifications known as Wigner distribution, radar ambiguity function, Gabor transform, all encode time-frequency information. However, the pointwise interpretation of such a time-frequency representation meets difficulties because, by the uncertainty principle, a small region in the time-frequency plane does not possess a physical meaning. Therefore ✩

M.D. was supported by the FWF Grant T 384-N13. K.G. was supported by the Marie-Curie Excellence Grant MEXTCT-2004-517154 and in part by the National Research Network S106 SISE of the Austrian Science Foundation (FWF). * Corresponding author. E-mail addresses: [email protected] (M. Dörfler), [email protected] (K. Gröchenig). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.021

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M. Dörfler, K. Gröchenig / Journal of Functional Analysis 260 (2011) 1903–1924

the question arises in which sense the short-time Fourier transform describes the local properties of a function and its Fourier transform. Following Daubechies [10], we use time-frequency localization operators to give meaning to the local time-frequency content. By investigating a whole family of localization operators and glueing together the local pieces, we are able to characterize the global time-frequency distribution of a function. In more technical terms, our main result provides a new characterization of modulation spaces. We define the short-time Fourier transform (STFT) of a function f ∈ L2 (Rd ) with respect to a window function ϕ ∈ L2 (Rd ) as Vϕ f (x, ω) =

f (t)ϕ(t ¯ − x)e−2πiω·t dt,

for all z = (x, ω) ∈ R2d .

(1)

Rd

The STFT Vϕ f (z) is a measure of the time-frequency content near the point z in the timefrequency plane R2d . However, the STFT cannot be supported on a set of finite measure by results in [27,29,37]. This fact complicates the interpretation of local information obtained from the STFT. In particular, it is impossible to construct a projection operator that satisfies Vϕ (PΩ f ) = χΩ · Vϕ f . As a remedy one resorts to the following definition of localization operators. We denote translation operators by Tx f (t) = f (t − x) and time-frequency shifts by π(z)f (t) = e2πiω·t f (t − x) for x, ω, t ∈ Rd . Fix a non-zero function ϕ ∈ L2 (R2d ) (a so-called window function) and a symbol σ ∈ L1 (R2d ). Then the time-frequency localization operator Hσ acting on a function f is defined as Hσ f =

σ (z)Vϕ f (z)π(z)ϕ dz.

R2d

The integral is defined strongly on many function spaces, in particular on L2 (Rd ). A useful alternative definition of Hσ is the weak definition Hσ f, gL2 (Rd ) = σ Vϕ f, Vϕ gL2 (R2d ) .

(2)

This definition can be easily extended to distributional symbols σ ∈ S (R2d ). The subtleties of the definition and boundedness properties between various spaces have been investigated in many papers, see [7,36,38] for a sample of results. If σ is non-negative and has compact support in Ω ⊆ Rd , then Hσ f can be interpreted as the part of f that lives essentially on Ω in the time-frequency plane, and so Hσ may be taken as a substitute for the non-existing projection onto the region Ω in the time-frequency plane. In this paper we investigate the behavior of an entire collection of localization operators. Namely, given a lattice Λ ⊆ R2d of the time-frequency plane, we consider the collection of operators {HTλ σ : λ ∈ Λ} and the mapping f → {HTλ σ f }. If the supports of Tλ σ cover R2d , then {HTλ σ f, λ ∈ Λ} should contain enough information to recover f from its local components. In particular, the set {HTλ σ f : λ ∈ Λ} should carry the complete information about the global time-frequency properties of f . We make this intuition precise and derive a new characterization of modulation spaces from it. Similar to Besov spaces, modulation spaces are smoothness spaces, but the smoothness is measured by means of time-frequency distribution rather than


1905

differences and derivatives. Here, we establish a correspondence between the behavior of the sequence HTλ σ f 2 , λ ∈ Λ, and the membership of f in a modulation space. As a special case of our main theorem we formulate the following result. Theorem 1. Fix a non-zero function ϕ in the Schwartz space S(Rd ) and a weight function m on R2d that satisfies m(z1 + z2 ) C(1 + |z1 |)N m(z2 ) for some constants C, N 0 and all z1 , z2 ∈ R2d . Then a tempered distribution f satisfies

Vϕ f (z)p m(z)p dz

1/p < ∞,

(3)

R2d

if and only if

p HTλ σ f 2 m(λ)p

1/p < ∞.

(4)

λ∈Λ p

The expression in (3) is just the norm of f in the modulation space Mm (Rd ). Our main result shows that the expression in (4) (using the time-frequency components of f ) is an equivalent p norm on the modulation space Mm (Rd ). In pseudodifferential calculus one often defines spaces by conditions on their time-frequency components. For instance, Bony, Chemin, and Lerner [3,4] introduced a Sobolev-type space H (m) by using Weyl operators instead of localization operators. For the (extremely simplified) case of a constant Euclidean metric on the time-frequency plane, a distribution f belongs to H (m), whenever for some test function ψ on R2d f 2H (m) =

(TY ψ)w f 2 m(Y ) dY, 2

(5)

R2d

is finite, where σ w is the Weyl operator corresponding to the symbol σ . The only difference between (5) and (4) is the use of Weyl calculus instead of time-frequency localization operators and a continuous definition instead of a discrete one. It was understood only recently that H (m) 2 (Rd ) and that (5) is an equivalent norm on M 2 (Rd ) [25]. coincides with the modulation space Mm m Thus Theorem 1 can be interpreted as an extension of [3] to Lp -like spaces. Let us also mention that in the language of [35], the operators {HTλ σ , λ ∈ Λ} form a g-frame for L2 (Rd ). Our construction seems to be one of the few non-trivial examples of g-frames that are not frames. In this paper we prove the norm equivalence of Theorem 1 for a large class of modulation spaces and arbitrary time-frequency lattices. For a rather restricted class of lattices, namely lattices with integer oversampling, an analogous result was derived in [12] for unweighted modulation spaces. The main arguments for the integer lattice were based on Zak transform methods and interpolation. For a general lattice, these methods are no longer available, and we have to develop a completely new approach to some of the key arguments.

1906


As a by-product of the new techniques we have found several results of independent interest. • We formulate several structural results and characterizations of Gabor frames for multiwindow Gabor frames. • We prove a finite intersection property for time-frequency invariant subspaces of the distribution space M∞ (Rd ). This property resembles the finite intersection property that characterizes compact sets. • We give a new, independent proof for the existence of multi-window Gabor frames with well-localized windows. Previous proofs were based on coorbit theory [15] and the theory of projective modules [32]. Our proof provides additional insight how the windows can be chosen. • We derive precise estimates for the localization of the eigenfunctions of a localization operator. This paper is organized as follows. In Section 2 we recall necessary facts from time-frequency analysis. On the one had, we introduce modulation spaces and explain their characterization by means of multi-window Gabor frames. On the other hand, we state and prove several properties of localization operators. In Section 3, we formulate and prove our main result (Theorem 8). In Section 3.4 we analyze some of the consequences of Theorem 8 and its proof. In Appendix A we collect and sketch the proofs of some of the structural results on Gabor frames. 2. Time-frequency analysis of functions and operators 2.1. Modulation spaces Modulation spaces are a class of function spaces associated to the short-time Fourier transform (1). Note that for a suitable test function ϕ, the short-time Fourier transform can be extended to distribution spaces by duality and Vϕ f (z) = f, π(z)ϕ. For the standard definition of modulation spaces, we fix a non-zero “window function” g ∈ S(Rd ) and consider moderate weight functions m of polynomial growth, i.e., m satisfies m(z1 + z2 ) C(1 + |z1 |)s m(z2 ), z1 , z2 ∈ R2d for some C, s 0. Given a moderate weight m p,q and 1 p, q ∞, the modulation space Mm (Rd ) is defined as the space of all tempered disp,q tributions f ∈ S (Rd ) with Vg f ∈ Lm (R2d ), with norm f Mp,q d = Vg f Lp,q (R2d ) . m (R ) m p

(6)

If p = q, we write Mm (Rd ). For weight functions of faster growth we have to resort to different spaces of test functions and distributions. Let g(t) = e−πt·t be the Gaussian window and H0 = span{π(z)g: z ∈ R2d } be the linear space of all finite linear combinations of time-frequency shifts of the Gaussian. Let ν be a submultiplicative even weight function on R2d and m be a ν-moderate function; this means that ν(z1 + z2 ) ν(z1 )ν(z2 ), ν(z) = ν(−z) and m(z1 + z2 ) ν(z1 )m(z2 ) for all p,q z, z1 , z2 ∈ R2d . For 1 p, q < ∞ the modulation space Mm (Rd ) is then defined as the closure ∗ of H0 in the norm f Mp,q d as in (6). If p = ∞ or q = ∞, we take a weak -closure of H0 . m (R ) These general modulation spaces possess the following properties. Assume that m is ν-moderate


1907

and 1 p, q ∞, then d ∗ p,q ∞ M1ν Rd ⊆ Mm Rd ⊆ M1/ν R = M1ν Rd .

(7)

Further, if ϕ ∈ M1ν (Rd ), then Vϕ f Lp,q Vg f Lp,q = f Mp,q , m m m

(8)

p,q

thus different windows in M1ν (Rd ) yield equivalent norms on Mm . d The embedding (7) says that M1ν (Rd ) may serve as a space of test functions and M∞ 1/ν (R ) as p,q a space of distributions for all modulation spaces Mm with a ν-moderate weight m. If νs (z) = (1 + |z|)s , s 0 and m is νs -moderate, then we have d d p,q S Rd ⊆ M1νs Rd ⊆ Mm Rd ⊆ M∞ 1/νs R ⊆ S R , b

in agreement with the standard definition, but for ν(z) = ea|z| with a > 0 and 0 < b 1 we have d M1ν Rd ⊆ S Rd ⊆ S Rd ⊆ M∞ 1/ν R . d In the sequel we will start with a submultiplicative weight ν and take M∞ 1/ν (R ) as the appropriate distribution space. Our results hold for arbitrary submultiplicative weights ν. For the detailed theory of modulation spaces we refer to [21, Chapters 11–13], for a discussion of weights and possible distribution spaces see [23].

Sequence space norms. Recall that a time-frequency lattice Λ is a discrete subgroup of R2d of the form Λ = AZ2d for some invertible real-valued 2d × 2d-matrix A. Given a lattice Λ ⊆ R2d with relatively compact fundamental domain Q, the discrete space p,q m (Λ) consists of all sequences a = (aλ )λ∈Λ for which the norm a p,q m

= |aλ |χλ+Q λ∈Λ

(9)

p,q

Lm

p,q

is finite. If Λ = aZd × bZd , then this definition reduces to the usual mixed-norm space m (Z2d ) with norm a p,q = m

n∈Zd

q/p 1/q |akn |p m(ak, bn)p

.

k∈Zd

As a technical tool we will need amalgam spaces (in one place only). A measurable function p,q F on R2d belongs to the (Wiener) amalgam space W (Lm ), if the sequence of local suprema akn = ess supx,w∈[0,1]d F (x + k, ω + n) = F · T(k,n) χ ∞ p,q

p,q

belongs to m (Z2d ). The norm on W (Lm ) is F W (Lp,q = a p,q . See [26] for an introducm ) m tory article. We need their behavior under convolution and their properties under sampling.

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(a) Convolution in Wiener amalgam spaces: Let 1 p, q ∞ and let m be a ν-moderate weight. Then C F W (Lp,q G L1ν . F ∗ G W (Lp,q m ) m )

(10)

p,q

(b) Sampling in Wiener amalgam spaces: For F ∈ W (Lm ) the following sampling property holds: CΛ F W (Lp,q . F |Λ p,q m m )

(11)

These statements are proved in [26] or [21, Proposition 11.1.4, Theorem 11.1.5]. 2.2. Gabor frames Gabor frames are closely linked to modulation spaces. They constitute “basis-like” sets for modulation spaces and are used to characterize the membership in a modulation space by the magnitude of coefficients in the corresponding series expansion. For a given lattice Λ ⊆ R2d and a window function ϕ ∈ L2 (Rd ), let G(ϕ, Λ) denote the set of functions {π(λ)ϕ: λ ∈ Λ} in L2 (Rd ). The operator Sϕ f =

f, π(λ)ϕ π(λ)ϕ

λ∈Λ

is the frame operator corresponding to G(ϕ, Λ). If Sϕ is bounded and invertible on L2 (Rd ), then G(ϕ, Λ) is called a Gabor frame for L2 (Rd ). This property is equivalent to the existence of two constants A, B > 0 such that A f 22

f, π(λ)g 2 = Sϕ f, f B f 2

for all f ∈ L2 Rd .

2

(12)

λ∈Λ

Using several windows ϕ = (ϕ1 , . . . , ϕn ), we say that the union window Gabor frame, if the associated frame operator given by Sϕ f =

n

j =1 G(ϕj , Λ)

n

n f, π(λ)ϕj π(λ)ϕj = Sϕj f

j =1 λ∈Λ

is a multi-

(13)

j =1

is invertible on L2 (Rd ). The frame operator can be expressed as the composition of the analysis operator Cϕ,Λ defined by

Cϕ,Λ (f )(λ, j ) = f, π(λ)ϕj ,

λ ∈ Λ, j = 1, . . . , n

and the synthesis operator Dϕ,Λ defined by Dϕ,Λ (c) = Dϕ,Λ ◦ Cϕ,Λ .

λ∈Λ

n

j =1 cλ,j π(λ)ϕj .

Then Sϕ,Λ =


1909

2.3. Characterization of modulation spaces with Gabor frames The following characterization of modulation spaces by means of multi-window Gabor frames is a central result in time-frequency analysis and useful in many applications. It is crucial for the proof of our main theorem (Theorem 8). Theorem 2. Let ν be a submultiplicative weight on R2d satisfying the condition limn→∞ ν(nz)1/n = 1 for all z ∈ R2d and let m be a ν-moderate weight and 1 p, q ∞. Assume further that nj=1 G(ϕj , Λ) is a multi-window Gabor frame and that ϕj ∈ M1ν (Rd ) for j = 1, . . . , n. p

p

(i) A distribution f belongs to Mm (Rd ), if and only if Cϕj f ∈ m for j = 1, . . . , n. In this case p there exist constants A, B > 0, such that, for all f ∈ Mm (Rd ), A f Mpm

λ∈Λ

n

f, π(λ)ϕj 2

p/2

1/p B f Mpm .

m(λ)p

j =1

(ii) Assume in addition that Λ = aZd × bZd is a separable lattice. Then a distribution f bep,q longs to Mm (Rd ) if and only if each sequence Cϕj f (ak, bl) = f, π(ak, bl)ϕj belongs p,q to m (Z2d ). In this case there exist constants A and B depending on p, q, m such that, for p,q all f ∈ Mm A f Mp,q m

n p/2 q/p 1/q

p f, π(ak, bl)ϕj 2 m(ak, bl) l∈Z

k∈Z

j =1

B f Mp,q . m

(14)

(iii) Let Λ ⊆ R2d be an arbitrary lattice and Q be a relatively compact fundamental dop,q main of Λ. Then a distribution f belongs to Mm (Rd ), if and only if the function

p,q n 2 1/2 χλ+Q belongs to Lm (R2d ). In this case there exist conλ∈Λ ( j =1 |f, π(λ)ϕj | ) p,q stants A, B > 0, such that, for all f ∈ Mm (R2d ), A f Mp,q m

n 1/2

2 f, π(λ)ϕj χλ+Q λ∈Λ

j =1

p,q Lm

B f Mp,q . m

d d

Note that (ii) follows from (iii), since for Q = [0, a] × [0, b] the norm equivalence k,l∈Z2d akl χ(ak,bl)+Q Lp,q a p,q holds. m m Theorem 2 has a long history. It extends the basic characterizations of modulation spaces by Gabor frames to multi-window Gabor frames. For Gabor frames with a single window and lattices of the form Λ = aZd × bZd with ab ∈ Q Theorem 2 was proved in [16]. For general lattices it follows from the main result in [24] and the techniques in [16]. See also the discussion in [21, Chapter 13]. The proofs for multi-window Gabor frames require only few modifications, we therefore postpone a discussion to Appendix A.

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2.4. A new characterization of multi-window Gabor frames The proof of our main statement relies on a characterization of multi-window Gabor frames without using inequalities. The following lemma is a generalization of [22] from Gabor frames to multi-window Gabor frames. Lemma 3. Assume that ϕj ∈ M1 (Rd ) for j = 1, . . . , n. Then the following properties are equivalent. (i) nj=1 G(ϕj , Λ) is a multi-window Gabor frame for L2 (Rd ). (ii) The analysis operator Cϕ,Λ is one-to-one from M∞ (Rd ) to ∞ (Λ, Cn ). The idea of the proof will be given in Appendix A, where we will also list many more equivalent conditions. 2.5. Properties of localization operators We next recall some elementary properties of the localization operators HTλ σ . Time-frequency localization operators have been introduced and studied by Daubechies [11,10] and Ramanathan and Topiwala [33], and are also called STFT multipliers, time-frequency Toeplitz operators, Wick operators, time-frequency filters, etc. They are a popular tool in signal analysis for timefrequency filtering or nonstationary filtering [31,34], in quantization procedures in physics [1], or in the approximation of pseudodifferential operators [9,30]. For a detailed account of the early theory we refer to Wong’s book [38], for a study of boundedness and Schatten class properties to [7,8,18,36]. Lemma 4 (Intertwining property). If σ ∈ L∞ (R2d ), ϕ ∈ L2 (Rd ), and λ ∈ Λ, then π(λ)Hσ π(λ)∗ = HTλ σ . The proof consists of a simple calculation, see [12, Lemma 2.6]. For estimates of the STFT of Hσ f we introduce the formal adjoint of Vϕ , namely Vϕ∗ F =

F (z)π(z)ϕ dz,

R2d

which maps functions on R2d to functions or distributions on Rd . With this notation we can write the localization operator Hσ as Hσ f = Vϕ∗ (σ Vϕ f ). The STFT of Vϕ∗ F satisfies a fundamental pointwise estimate [21, Proposition 11.3.2]: ∗ Vϕ V F (z) |Vϕ ϕ| ∗ |F | (z), ϕ

∀z ∈ R2d .

(15)


1911

We note that for F = σ Vϕ f this estimate becomes Vϕ (Hσ f )(z) = Vϕ V ∗ (σ Vϕ f ) (z) |Vϕ ϕ| ∗ σ |Vϕ f | (z). ϕ

(16)

Thus the short-time Fourier transform of Hσ is a so-called product-convolution operator. The standard boundedness results for localization operators can be easily deduced from the well established results for product convolution operators [6]. Estimate (16) is quite useful for the derivation of norm estimates. In the following, we fix a non-negative symbol σ and investigate the set of operators {HTλ σ : λ ∈ Λ}. To simplify notation we will write Hλ instead of HTλ σ , and sometimes H0 = Hσ by some abuse of notation. Lemma 5. (i) Assume that σ ∈ L1 (R2d ), σ 0 and that ϕ ∈ L2 (Rd ). Then each Hλ , λ ∈ Λ, is a positive trace-class operator. (ii) If, in addition, ϕ ∈ M1ν (Rd ) and σ ∈ L1ν (R2d ), then each Hλ is bounded from M∞ (Rd ) into M1ν (Rd ). In particular, all eigenfunctions ϕj of Hσ belong to M1ν (Rd ). d (iii) Furthermore, if ϕ ∈ M1ν (Rd ) and σ ∈ L1ν (R2d ), then each Hλ is bounded from M∞ 1/ν (R ) into L2 (Rd ). Proof. Statement (i) is well known, see, e.g., [2,17,38]. To show (ii), we use (16) to obtain, for f ∈ M∞ (Rd ), Hσ f M1ν = Vϕ (Hσ f )L1 = Vϕ Vϕ∗ (σ Vϕ f )L1 ν ν |Vϕ ϕ| ∗ |σ Vϕ f | L1 ν

Vϕ ϕ L1ν σ Vϕ f L1ν ,

(17)

where we have used Young’s inequality. Since ϕ ∈ M1ν (Rd ) if and only if Vϕ ϕ ∈ L1ν (R2d ) by [21, Proposition 12.1.2], we find that Hσ f M1ν Vϕ ϕ L1ν σ L1ν Vϕ f L∞ C σ L1ν f M∞ , and thus Hσ is bounded from M∞ (Rd ) to M1ν (Rd ). d The proof of (iii) is similar. Again, we apply (16) to obtain for f ∈ M∞ 1/ν (R ): Hσ f L2 |Vϕ ϕ| ∗ |σ Vϕ f |L2 Vϕ ϕ L2 σ Vϕ f L1 . Hence, the result follows from σ Vϕ f L1 =

1 dz σ (z)Vϕ f (z)ν(z) ν(z)

R2d

σ L1ν f M∞ . 1/ν

2

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The spectral theorem for compact self-adjoint operators provides the following spectral representation of Hλ . Corollary 6. Assume ϕ ∈ M1ν (Rd ) and σ ∈ L1ν (R2d ). Then there exists a positive sequence of eigenvalues c = (cj ) ∈ 1 and an orthonormal system of eigenfunctions ϕj ∈ M1ν (Rd ), such that Hσ f =

∞

cj f, ϕj ϕj .

(18)

j =1

It follows that Hλ f = HTλ σ f = π(λ)Hσ π(λ)∗ f =

∞

cj f, π(λ)ϕj π(λ)ϕj ,

(19)

j =1

and {π(λ)ϕj , j ∈ N} is an orthonormal system of eigenfunctions of Hλ . A priori, the spectral representation of Hλ holds only for f ∈ L2 (Rd ). The next corollary d extends the spectral representation to all of M∞ 1/ν (R ). d Corollary 7. The expansion for Hλ f given in (19) is well defined on M∞ 1/ν (R ) and converges d to Hλ f in L2 for all f ∈ M∞ 1/ν (R ).

Proof. Without loss of generality, we assume λ = 0 and set H = Hσ . Since Hf ∈ L2 (Rd ) for d every f ∈ M∞ 1/ν (R ) by Lemma 5(iii), we can expand Hf with respect to the orthonormal system of eigenfunctions of H and obtain that Hf =

∞ Hf, ϕj ϕj + r

(20)

j =1

for some r ∈ L2 (Rd ) in the orthogonal complement of span{ϕj : j ∈ N}. As H is self-adjoint on L2 (Rd ), we also have Hf, ϕj = f, H ϕj = cj f, ϕj , and consequently Hf =

∞

cj f, ϕj ϕj + r.

(21)

j =1

We need to show that r = 0. Since r ∈ L2 (Rd ) is orthogonal to all eigenfunctions ϕj , we find that Hf, r = r 22 . To show r = 0, we first observe that H h, r = 0 for all h ∈ L2 (Rd ) by (18). Since L2 (Rd ) d 2 d is w ∗ -dense in M∞ 1/ν (R ), we may choose an approximating sequence fn ∈ L (R ) such that ∗

w d −→ f ∈ M∞ fn − 1/ν (R ). For instance, fn may be chosen as

fn = R2d

χBn (z)Vg f (z)π(z)g dz,


1913

∗

w where Bn is the ball with radius n and centered at 0. Furthermore, since fn − −→ f , we obtain in particular that Vϕ fn converges to Vϕ f uniformly on compact sets [13, Theorem 4.1]. Consequently

0 = Hfn , r =

σ (z)Vϕ fn (z)Vϕ r(z) dz →

R2d

σ (z)Vϕ f (z)Vϕ r(z) dz = Hf, r = r 22 .

R2d

d This shows that r = 0 and so the series (19) represents Hf for all f ∈ M∞ 1/ν (R ).

2

3. From local information to global information p

We first state and prove the main result for the modulation spaces Mm (Rd ). The generalizap,q tions to Mm (Rd ) will be discussed later. As always, ν denotes a submultiplicative, even weight function on R2d satisfying the condition limn→∞ ν(nz)1/n = 1 for all z ∈ R2d . Theorem 8. Let σ ∈ L1ν (R2d ) be a non-negative symbol satisfying the condition A

Tλ σ B,

a.e.

(22)

λ∈Λ

for two constants A, B > 0. Assume that ϕ ∈ M1ν (Rd ). Then for every ν-moderate weight m and p d d 1 p < ∞ the distribution f ∈ M∞ 1/ν (R ) belongs to Mm (R ), if and only if

p Hλ f 2 m(λ)p

1/p < ∞,

(23)

λ∈Λ p

and the expression in (23) is an equivalent norm on Mm (Rd ). Similarly, for p = ∞ we obtain the norm equivalence sup Hλ f 2 m(λ). f M∞ m

(24)

λ∈Λ

The norm equivalence supports the interpretation that Hλ f carries the local time-frequency information about f near λ ∈ R2d . By combining the local pieces Hλ f , one obtains the global time-frequency information as it is measured by modulation space norms. The proof of Theorem 8 requires some preparations. We first show that finitely many eigenfunctions of H0 = Vϕ∗ σ Vϕ generate a multi-window Gabor frame for L2 (Rd ). With this crucial step in place, Theorem 8 can then be deduced from the characterization of modulation spaces by means of Gabor frames. 3.1. Multi-window Gabor frames

Lemma 9. Assume that σ ∈ L1 (R2d ) and λ∈Λ Tλ σ 1, and that ϕ ∈ M1ν (Rd ). Let {ϕj : j ∈ N} be the orthonormal system of eigenfunctions of H0 . Then there exists n ∈ N, such that the finite union nj=1 G(ϕj , Λ) is a multi-window Gabor frame for L2 (Rd ).

1914


An analogous statement was proved and used in [12] for the lattice Λ = Z2d and rational lattices by means of Zak transform methods. In the case of general lattices we cannot apply Zak-transform methods. As a substitute, we will use a finite intersection property for Λ-invariant subspaces of M∞ . The following statement may be of interest in its own right. Lemma 10. Assume that Wn is a sequence of w -closed subspaces in M∞ (Rd ) such that (i) Wn ⊇ Wn+1 = {0} for all n ∈ N, and (ii) Wn is invariant under all operators π(λ) for λ ∈ Λ. Then

n1 Wn

= {0}.

Proof. Let Q be the closure of a relatively compact fundamental domain of Λ, for instance, if Λ = AZ2d , then Q = A[0, 1]2d . We first choose a sequence hn ∈ Wn with hn M∞ = supz∈R2d |Vϕ hn (z)| = 1. Then there exists a sequence of points λn in Λ, such that sup Vϕ π(λn )hn (z) = 1. z∈Q

Since Wn is invariant under all π(λ), λ ∈ Λ, the distribution fn = π(λn )hn is in Wn . Next we show that the set of restrictions {Vϕ fn |Q } is equicontinuous. We have Vϕ fn (z) − Vϕ fn (ξ ) = fn , π(z) − π(ξ ) ϕ fn M∞ · π(z) − π(ξ ) ϕ 1 . M

(25)

Since fn M∞ = π(λn )hn M∞ = 1, the equicontinuity follows from the strong continuity of time-frequency shifts on M1 (Rd ). We next choose zn ∈ Q with |Vϕ fn (zn )| 12 . Since the unit ball in M∞ (Rd ) is w -compact, there exists a subsequence fnk that converges to some f ∈ M∞ (Rd ) in the w -sense. Furthermore, by compactness of Q, there also exists a subsequence z of znk , such that z → z ∈ Q. Hence, by equicontinuity, Vϕ f (z ) → Vϕ f (z). Since |Vϕ f (z )| 1/2, we conclude that also |Vϕ f (z)| 1/2, and consequently f = 0. By construction, f ∈ Wm for every m, hence we obtain f = w ∗ − lim →∞ f ∈ Wm for all m, because Wm is w -closed. To summarize, we have constructed a non-zero f ∈ M∞ (Rd ) that is in Wm for all m. 2 Proof of Lemma 9. To prove that finitely many eigenfunctions generate a multi-window Gabor frame with respect to the lattice Λ, we assume on the contrary that nj=1 G(ϕj , Λ) is not a frame for every n ∈ N. Using Lemma 10 and Lemma 3, we will derive a contradiction to the assumption that A λ∈Λ Tλ σ B. We use the criterion of Lemma 3. Let ϕ n = (ϕ1 , . . . , ϕn ) be the vector-valued function consisting of the first n eigenfunctions of H0 , and

Wn = ker(Cϕ n ,Λ ) = f ∈ M∞ Rd : f, π(λ)ϕj = 0, ∀λ ∈ Λ, j = 1, . . . , n be the kernel of the coefficient operator Cϕ n ,Λ in M∞ (Rd ).


1915

If nj=1 G(ϕj , Λ) is not a frame, then Wn is a non-trivial subspace of M∞ (Rd ) by Lemma 3. By construction, the Wn ’s form a nested sequence of w ∗ -closed subspaces of M∞ (Rd ), and they are also invariant and we under π(λ), λ ∈ Λ. Thus the assumptions of Lemma 10 are∞satisfied, d ), such that W = {0}. This means that there exists a non-zero f ∈ M (R conclude that ∞ n n=1

f, π(λ)ϕj = 0 for all λ ∈ Λ and all j ∈ N.

(26)

We now consider Hλ f . Since Hλ f ∈ M1 (Rd ) by Lemma 5, the bracket Hλ f, f is well defined and given by Hλ f, f =

2 σ (z − λ)Vϕ f (z) dz.

(27)

R2d

On the other hand, the extended spectral representation of Lemma 7 and (26) imply that Hλ f =

∞

cj f, π(λ)ϕj π(λ)ϕj = 0.

(28)

j =1

2 vanishes on Consequently Hλ f, f = 0 for all λ ∈ Λ, and |Vϕ f (z)| λ∈Λ supp Tλ σ . T σ A > 0 almost everywhere, According to the crucial assumption (22) we have λ∈Λ λ and thus λ∈Λ supp(Tλ σ ) = R2d . Therefore, (27) and (28) imply thatVϕ f = 0, from which f = 0 follows. This is a contradiction to f being a non-zero element in ∞ n=1 Wn . This contradiction shows that there exists an n ∈ N, such that nj=1 G(ϕj , Λ) is a multiwindow Gabor frame, and we are done. 2 Remark 1. Note that for finite-rank operators H0 , it can be seen directly that the finite set of eigenvectors generates a multi-window Gabor frame for Λ. 3.2. Proof of Theorem 8 d We are now ready to prove the main theorem. We observe that for f ∈ M∞ 1/ν (R ), Hλ f ∈ L2 (Rd ) by Lemma 5(iii). Thus the terms in (23) are well defined. p d 1 d First assume that p < ∞ and f ∈ Mm (Rd ) ⊆ M∞ 1/ν (R ). Using the embedding M (R ) → L2 (Rd ) and the estimate (17) with ν ≡ 1, we majorize Hλ f 2 as follows

Hλ f 2 Cϕ Hλ f M1 Cϕ (Tλ σ ) · Vϕ f 1 Vϕ ϕ 1 = Cϕ C σ (z − λ) · Vϕ f (z) dz R2d

= Cϕ C |Vϕ f | ∗ σ ∨ (λ),

(29)

where σ ∨ (z) = σ ∨ (−z). Thus Hλ f 2 is majorized by a sample of |Vϕ f | ∗ σ ∨ . To proceed furp ther, we use the fact that Vϕ f ∈ W (Lm ) and Vϕ f W (Lpm ) C0 ϕ M1ν f Mpm for ϕ ∈ M1ν (Rd )

1916

M. Dörfler, K. Gröchenig / Journal of Functional Analysis 260 (2011) 1903–1924 p

and f ∈ Mm (Rd ) by [21, Theorem 12.2.1]. Now the convolution relation (10) and the sampling inequality (11) imply that

p p Hλ f 2 m(λ)p Cϕ C σ ∨ ∗ |Vϕ f | Λ p

m

λ

p Cϕ CCΛ σ ∨ ∗ |Vϕ f |W (Lp ) m

p

p

p

p

Cϕ CCΛ σ L1 Vϕ f W (Lp ) Cϕ CCΛ σ L1 f Mp . m

ν

ν

m

(30)

The same argument yields supλ∈Λ Hλ f 2 m(λ) C f M∞ . m p p Hence, for 1 p ∞, the mapping f → ( Hλ f 2 )λ∈Λ is bounded from Mm (Rd ) to m (Λ). Conversely, assume that p < ∞ and

p

Hλ f 2 m(λ)p < ∞.

λ p

We need to show that f ∈ Mm (Rd ). Since Hλ f 2 = sup g 2 =1 |Hλ f, g|, we have the inequality p Hλ f, gλ p m(λ)p Hλ f m(λ)p < ∞ 2

λ

λ

for arbitrary sequences gλ ∈ L2 (Rd ) with gλ 2 = 1. Applying the eigenfunction expansion of Corollary 6, we obtain ∞ p

p cj f, π(λ)ϕj π(λ)ϕj , gλ m(λ)p Hλ f 2 m(λ)p < ∞. λ

j =1

(31)

λ

Now fix j0 ∈ N and set gλ = π(λ)ϕj0 for λ ∈ Λ. Since the eigenfunctions of Hλ are orthonormal, the sum over j collapses to a single term, and (31) becomes

p Hλ f, gλ p m(λ)p = cj f, π(λ)ϕj p m(λ)p Hλ f 2 m(λ)p < ∞. 0 0 λ

λ

λ

The last inequality holds for every j0 ∈ N. After summing over finitely many j0 and switching to the 2 -norm on Cn , we obtain the inequality n 1/2 n

f, π(λ)ϕj 2 f, π(λ)ϕj p m(λ)p m(λ)p λ

j =1

j =1 λ

n 1 p c j =1 j

λ

p

Hλ f 2 m(λ)p < ∞.

(32)


1917

We now apply Lemma 9 and choose an n ∈ N, such that nj=1 G(ϕj , Λ) is a multi-window Gabor frame for L2 (Rd ). Since all ϕj are in M1ν (Rd ), the fundamental characterization of modulation p spaces (Section 2.3) is valid. Thus Theorem 2(i) implies that f ∈ Mm (Rd ). If p = ∞ and supλ∈Λ Hλ f 2 m(λ) < ∞, then, by choosing gλ as before, we find

cj0 sup f, π(λ)ϕj0 m(λ) sup Hλ f 2 m(λ) < ∞ λ

λ

for every j0 . Arguing as above, Theorem 2 says that

f M∞ C max sup f, π(λ)ϕj m(λ) m j =1,...,n λ

1 j =1,...,n cj

sup Hλ f 2 m(λ) < ∞,

max

λ

2d and f ∈ M∞ m (R ).

p Combining (30) and (32), we have shown that f Mpm and ( λ∈Λ Hλ f 2 m(λ)p )1/p for p 1 p < ∞ (or supλ∈Λ Hλ f 2 m(λ) for p = ∞) are equivalent norms on Mm (Rd ).

3.3. Variations of Theorem 8 In order to formulate our main result for mixed-norm spaces and arbitrary lattices, we have to resort to the theory of coorbit spaces, as introduced in [13,14]. In particular, for

arbitrary lattices, p,q a sequence (cλ )λ∈Λ is in the sequence spaces associated with Lm (R2d ), if λ∈Λ cλ χλ+Q is in p,q Lm (R2d ) for some fundamental domain Q of Λ. With this definition, we may give the following characterization. Theorem 11. Let Λ be an arbitrary lattice in R2d and Q be a relatively compact fundamental domain Q. Assume the same conditions on σ and ϕ as in Theorem 8. Then a distribution p,q d d f ∈ M∞ 1/ν (R ) belongs to Mm (R ), 1 p, q ∞, if and only if

p,q 2d

Hλ f 2 χλ+Q ∈ Lm

R

(33)

,

λ∈Λ

and

λ∈Λ Hλ f 2 χλ+Q Lm

p,q

f Mp,q . m

Proof. The proof is almost identical to the proof of Theorem 8. The only modifications occur in (30), which has to be replaced by H f χ λ 2 λ+Q λ∈Λ

p,q

Lm

Vϕ f ∗ σˇ (λ)χλ+Q

p,q

Lm

λ∈Λ p

C Vϕ f ∗ σˇ W(Lp,q . m ) p,q

Likewise, in (32) we replace the weighted Lm -norm by the general Lm -norm. p,q

2

For a separable lattice Λ = aZd × bZd the norm in (33) is just the m˜ -norm on Z2d with m(k, ˜ n) = m(ak, bn). In this case, λ = (ka, nb), k, n ∈ Zd and we may write Hλ f = Hk,n f .

1918


Corollary 12. Let Λ = aZd × bZd be a separable lattice and assume the same conditions p,q d d on σ and ϕ as in Theorem 8. Then a distribution f ∈ M∞ 1/ν (R ) belongs to Mm (R ) for 1 p, q < ∞, if and only if n∈Zd

p Hk,n f 2 m(ka, nb)p

q/p 1/q < ∞,

(34)

k∈Zd p,q

and (34) defines an equivalent norm on Mm (Rd ). The result holds for p = ∞ or q = ∞ with the usual modifications. 3.4. Existence of multi-window Gabor frames and properties of the eigenfunctions ϕj We finally point out some immediate consequences of our results and methods. The intermediate results leading to Theorem 8 also imply the existence of multi-window Gabor frames for general lattices. Theorem 13. Let Λ be an arbitrary lattice and ν a submultiplicative weight on R2d . Then there n 1 d exist finitely many functions ϕj ∈ Mν (R ), such that j =1 G(ϕj , Λ) is a multi-window Gabor frame for L2 (Rd ).

Proof. Choose σ ∈ L1ν (R2d ) such that λ∈Λ Tλ σ 1 and fix a window ϕ ∈ M1ν (Rd ). For instance, one may choose the characteristic function χQ of a (relatively compact) fundamental domain of Λ and the Gaussian window ϕ(t) = e−πt·t . Now consider the localization operator H0 = Vϕ∗ σ Vϕ . According to Lemma 5(ii), all eigenfunctions ϕj of H0 belong to M1ν (Rd ). Lemma 9 states that for some finite n ∈ N the set n 2 d j =1 G(ϕj , Λ) is a multi-window Gabor frame for L (R ). 2 The existence of multi-window Gabor frames for general lattices was known before. On the one hand, it is an immediate consequence of coorbit theory applied to the Heisenberg group. To be more precise, according to [15, Theorem 7] for every lattice Λ and every non-zero g ∈ M1ν (Rd ) there exists n ∈ N, such that the set G(g, n1 Λ) is a Gabor frame for L2 (Rd ). Us 1 1 ing a coset decomposition n Λ = (μ + Λ) for suitable μ ∈ Λ, one sees that G(g, n Λ) = G(π(μ)g, Λ) is a multi-window Gabor frame with all windows π(μ)g derived from a single window g. Recently Luef [32] proved the existence of multi-window Gabor frames by exploiting a connection between Gabor analysis and non-commutative geometry. Our methods provide a third, independent proof for this interesting result. The construction of multi-window Gabor frames in Proposition 13 yields more detailed information about the frame generators, since they are eigenfunctions of a localization operator. Intuitively the eigenfunctions corresponding to the largest eigenvalues of a localization operator concentrate their energy on the essential support of the symbol σ of H0 . For the special case of compactly supported σ , this intuition is made precise by the following result. Proposition 14. Let the non-negative function σ ∈ L1 (R2d ) be supported in a compact set Ω in R2d with 0 σ (z) Cσ < ∞ for z ∈ Ω. Consider the localization operator given by Hσ f = Vϕ∗ σ Vϕ f with ϕ ∈ M1 (Rd ), ϕ 2 = 1 and spectral representation as in Corollary 6.


1919

Then the eigenfunctions ϕj of Hσ satisfy the following time-frequency concentration Ω

Vϕ ϕj (z)2 dz cj . Cσ

(35)

Equality holds, if and only if σ (z)/Cσ = χΩ (z) is the characteristic function of Ω. Proof. Using the weak interpretation of Hσ from (2), we obtain Ω

Vϕ ϕj (z)2 dz 1 Cσ

2 σ (z)Vϕ ϕj (z) dz

Ω

cj cj 1 = Hσ ϕj , ϕj = ϕj 22 = . Cσ Cσ Cσ

2

Appendix A. Characterizations of modulation spaces and multi-window Gabor frames In the appendix, we will sketch the proof of Theorem 2 and formulate a series of new characterizations of multi-window Gabor frames. These statements generalize well-known facts from Gabor analysis and the results about Gabor frames without inequalities in [22]. For the investigation of multi-window Gabor frames we need the dual concept of vectorvalued Gabor systems. In this case we consider the Hilbert space H = L2 (Rd , Cn ) consisting of all vector-valued functions f(t) = (f1 (t), . . . , fn (t)) with the inner product f, ϕL2 (Rd ,Cn ) =

n

fj (t)ϕj (t) dt =

j =1

n

fj , ϕj L2 (Rd ) .

(A.1)

j =1

Time-frequency-shifts act coordinate-wise on f. The vector-valued Gabor system G(ϕ, Λ) = {π(λ)ϕ: λ ∈ Λ} is a Riesz sequence in L2 (Rd , Cn ), if there exist constants 0 < A, B < ∞ such that for all finitely supported sequences c, A c 22

2 cμ π(λ)ϕ 2 λ∈Λ

L (Rd ,Cn )

B c 22 .

(A.2)

We now proceed to the proof of Theorem 2. The crucial step is to show the invertibility of the frame operator on M1ν (Rd ). This step requires a special representation of the frame operator due to Janssen [28] and at its core uses “Wiener’s lemma for twisted convolution” [24]. For ϕj , φj in M1 (Rd ), j = 1, . . . , n, we denote frame-type operators by Sϕ,ψ f =

n

n f, π(λ)ϕj π(λ)ψj = Sϕj ,ψj .

λ∈Λ j =1

j =1

The frame operator of the Gabor system nj=1 G(ϕj , Λ) is S = Sϕ,ϕ . We usually omit the reference to the lattice Λ and the windows ϕj .

1920


The volume s(Λ) of a lattice Λ = AZ2d is defined as the measure of a fundamental domain of Λ and is |det(A)|. The adjoint lattice of Λ is Λ◦ = {μ ∈ R2d : π(λ)π(μ) = π(μ)π(λ) for all λ ∈ Λ}. Lemma 15 (Janssen’s representation). Assume that ϕj , ψj ∈ M1 (Rd ) for all j = 1, . . . , n. Then n the frame type operator associated to j =1 G(ϕj , Λ) and nj=1 G(ψj , Λ) can be written as n

ϕj , π(μ)ψj π(μ)f

Sϕ,ψ f = s(Λ)−1

(A.3)

μ∈Λ◦ j =1

with unconditional convergence in the operator norm on L2 . Proof. By Janssen’s result [28] the representation holds for a single Sϕj ,ψj and (A.3) follows by taking a sum. 2 The canonical dual frame is defined to be γj,λ = π(λ)S −1 ϕj . Since the frame operator S = Sϕ,ϕ commutes with time-frequency shifts on Λ, we obtain the reconstruction formulas f = S −1 Sf =

n

f, π(λ)ϕj π(λ)γj

λ∈Λ j =1

= SS −1 f =

n

f, π(λ)γj π(λ)ϕj

λ∈Λ j =1

= Dϕ,Λ Cγ ,Λ f = Dγ ,Λ Cϕ,Λ f. As a general principle the localization of a frame is inherited by the dual frame [19]. The following statement is a generalization of [24, Theorem 9] to multi-window Gabor frames on general lattices. 2d satisfying Lemma 16. Assume that ν is a submultiplicative, even nweight on R 1/n 2d limn→∞ ν(nz) = 1 for all z ∈ R . Assume further that j =1 G(ϕj , Λ) is a frame for 2 d L (R ) and that ϕj ∈ M1ν (Rd ). Then the frame operator S is invertible on M1ν (Rd ) and γj = S −1 ϕj ∈ M1ν (Rd ) for j = 1, . . . , n.

Proof. Janssen’s representation (A.3) implies that S = Sϕ,ϕ = s(Λ)−1

cμ π(μ),

(A.4)

μ∈Λ◦

with a coefficient sequence cμ = nj=1 ϕj , π(μ)ϕj . The hypothesis ϕj ∈ M1ν (Rd ) guarantees that μ∈Λ◦ |ϕj , π(μ)ϕj |ν(μ) < ∞ for each j , see [21, Corollary 12.1.12], and therefore the coefficient sequence (cμ ) is in 1ν (Λ◦ ). Since nj=1 G(ϕj , Λ) is a frame, the frame operator Sϕ,ϕ is invertible on L2 (Rd ). It follows from [24, Theorem 3.1] that the inverse frame operator


1921

S −1 is again of the form S −1 = μ∈Λ◦ dμ π(μ) with a coefficient sequence d in 1ν (Λ◦ ). This representation implies that S −1 is bounded on M1ν (Rd ) and that γj M1ν = S −1 ϕj M1 C ϕj M1ν .

(A.5)

ν

Therefore the dual windows γj , j = 1, . . . , n are in M1ν (Rd ) as claimed.

2

Once the invertibility of the multi-window frame operator on M1ν (Rd ) is established, the proof of Theorem 2 is straight-forward by using the following boundedness properties of the coefficient operator Cϕ,Λ and Dϕ,Λ from [21, Theorems 12.2.3 and 12.3.4]. If ϕj ∈ M1ν (Rd ) p,q p,q and γj ∈ M1ν (Rd ), then both Cϕ,Λ and Cγ ,Λ are bounded from Mm (Rd ) into m (Λ, Cn ) for 1 p, q ∞ and for every ν-moderate weight m. Likewise Dϕ,Λ and Dγ ,Λ are bounded from p,q p,q p,q For the m (Λ, Cn )-norm we use the Euclidean norm on Cn , so m (Λ, Cn ) into Mm (Rd ). n 2 1/2 χ . that c p,q n = λ+Q Lp,q λ∈Λ ( j =1 |cλ,j | ) m (Λ,C ) m As a consequence, the reconstruction formula f = Dϕ,Λ Cγ ,Λ f = Dγ ,Λ Cϕ,Λ f holds for p,q f ∈ Mm (Rd ) with the correct norm estimates. The norm equivalence stated in Theorem 2 then follows from f Mp,q d = Dγ ,Λ Cϕ,Λ f Mp,q (Rd ) Dγ ,Λ op Cϕ,Λ f p,q (Λ,Cn ) m (R ) m m Dγ ,Λ op Cϕ,Λ op f Mp,q d . m (R ) Next we come to the characterization of multi-window Gabor frames (Lemma 3) and extend the list of equivalent conditions. For the formulation of the dual conditions on the adjoint lattice Λ◦ we need the vector-valued versions of the analysis and synthesis operators. For f = (f1 , . . . , fn ) ∈ M∞ (Rd , Cn ) and ϕ = (ϕ1 , . . . , ϕn ) ∈ M1 (Rd , Cn ) the coefficient op◦ ◦ erator is defined

to be Cϕ,Λ (f)(μ) = (f, π(μ)ϕ), μ ∈ Λ , and the synthesis operator is ϕ,Λ◦ D ϕ,Λ◦ is defined on seDϕ,Λ◦ (c) = μ∈Λ◦ cμ π(μ)ϕ. The Gramian operator Gϕ,Λ◦ = C ◦ quences indexed by Λ . Lemma 17. Assume that ϕj ∈ M1 (Rd ) for j = 1, . . . , n. The following are equivalent for the multi-window Gabor system nj=1 G(ϕj , Λ): (i) nj=1 G(ϕj , Λ) is a frame for L2 (Rd ). (ii) Wexler–Raz biorthogonality: There exist γj ∈ M1 (Rd ), j = 1, . . . , n, such that s(Λ)−1

n

ϕj , π(μ)γj = δμ,0

for μ ∈ Λ◦ .

(A.6)

j =1

Ron–Shen duality: G(ϕ, Λ◦ ) is a Riesz sequence in L2 (Rd , Cn ). Sϕ,ϕ is invertible on M1 (Rd ). Sϕ,ϕ is invertible on M∞ (Rd ). Sϕ,ϕ is one-to-one on M∞ (Rd ). The analysis operator Cϕ,Λ : M∞ (Rd ) → ∞ (Λ, Cn ) is one-to-one from M∞ (Rd ) to ∞ (Λ, Cn ). (viii) The synthesis operator Dϕ,Λ defined on 1 (Λ, Cn ) has dense range in M1 (Rd ). (iii) (iv) (v) (vi) (vii)

1922


(ix) Dϕ,Λ is surjective from 1 (Λ, Cn ) onto M1 (Rd ). ϕ,Λ◦ defined on ∞ (Λ◦ ) is one-to-one from ∞ (Λ◦ ) to (x) The synthesis operator D M∞ (Rd , Cn ). ϕ,Λ◦ defined on M1 (Rd , Cn ) has dense range in 1 (Λ◦ ). (xi) The analysis operator C ϕ,Λ◦ is surjective from M1 (Rd , Cn ) onto 1 (Λ◦ ). (xii) C (xiii) Gϕ,Λ◦ is invertible on 1 (Λ◦ ). (xiv) Gϕ,Λ◦ is invertible on ∞ (Λ◦ ). (xv) Gϕ,Λ◦ is one-to-one on 1 (Λ◦ ). The equivalence (i) ⇔ (vii) is claimed in Lemma 3 and is all we need for the main results of our paper. Proof. The implication (i) ⇒ (iv) was sketched in Lemma 16. (i) ⇔ (ii): Time-frequency

shifts on a lattice are linearly independent in the following sense: if c = (cμ )μ∈Λ◦ ∈ ∞ and μ∈Λ◦ cμ π(μ) = 0 (as an operator from M1 (Rd ) to M∞ (Rd )), then cμ = 0 for all μ ∈ Λ◦ , see [22]. Now, if f = Sϕ,γ f for all f ∈ M1 (Rd ), then by Janssen’s representation (A.3) we have f = s(Λ)−1

n

ϕj , π(μ)γj π(μ)f.

μ∈Λ◦ j =1

The linear independence of time-frequency shifts implies (A.6). The converse is obvious. (ii) ⇔ (iii): Assume first that nj=1 G(ϕj , Λ) is a multi-window Gabor frame for L2 (Rd ). The ϕ on L2 (Rd ). To upper bound in (A.2) follows from the boundedness of the synthesis operator D show the existence of a lower bound, we apply the Wexler–Raz relations. Since nj=1 G(ϕj , Λ) is a frame with dual nj=1 G(γj , Λ) and γj ∈ M1 (Rd ) for all j , we have ϕ, π(μ)γ =

n ◦ ◦ j =1 ϕj , π(μ)γj = s(Λ)δμ,0 , and G(ϕ, Λ ) and therefore G(γ , Λ ) are biorthogonal systems

in L2 (Rd , Cn ). If f = μ∈Λ◦ cμ π(μ)ϕ, then cμ = s(Λ)−1 f, π(μ)γ L2 (Rd ,Cn ) and ϕ,Λ◦ f, c = s(Λ)−1 C from which the lower bound in (A.2) follows. Conversely, assume that G(ϕ, Λ◦ ) is a Riesz sequence in L2 (Rd , Cn ). Then there exists a biorthogonal basis of the form {π(μ)γ : μ ∈ Λ◦ } contained in K = span(G(ϕ, Λ◦ )). It can be shown that γ ∈ M1 (Rd , Cn ). The frame property of G(ϕj , Λ) follows from the Wexler–Raz relations (A.6). With three classical statements (A.3) and (ii), (iii) for multi-window Gabor frames the remaining equivalences follow exactly as in [22]. 2 References [1] F.A. Berezin, Wick and anti-Wick symbols of operators, Mat. Sb. (N.S.) 86 (128) (1971) 578–610. [2] P. Boggiatto, E. Cordero, Anti-Wick quantization with symbols in Lp spaces, Proc. Amer. Math. Soc. 130 (9) (2002) 2679–2685 (electronic). [3] J.-M. Bony, J.-Y. Chemin, Functional spaces associated with the Weyl–Hörmander calculus (Espaces fonctionnels associés au calcul de Weyl–Hörmander), Bull. Soc. Math. France 122 (1) (1994) 77–118.


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[4] J.-M. Bony, N. Lerner, Quantification asymptotique et microlocalisations d’ordre supérieur. I, Ann. Sci. École Norm. Sup. (4) 22 (3) (1989) 377–433. [5] K. Brandenburg, M. Kahrs (Eds.), Applications of Digital Signal Processing to Audio and Acoustics, Engineering and Computer Science/Kluwer Academic Publishers, 2003. [6] R.C. Busby, H.A. Smith, Product-convolution operators and mixed-norm spaces, Trans. Amer. Math. Soc. 263 (2) (1981) 309–341. [7] E. Cordero, K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (1) (2003) 107– 131. [8] E. Cordero, K. Gröchenig, L. Rodino, Localization operators and time-frequency analysis, in: N.M. Chong, et al. (Eds.), Harmonic, Wavelet and p-Adic Analysis, World Scient. Publ., 2007, pp. 83–109, 2006. [9] A. Córdoba, C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (11) (1978) 979–1005. [10] I. Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory 34 (4) (1988) 605–612. [11] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (5) (1990) 961–1005. [12] M. Dörfler, H.G. Feichtinger, K. Gröchenig, Time-frequency partitions for the Gelfand triple (S0 , L2 , S0 ), Math. Scand. 98 (1) (2006) 81–96. [13] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (2) (1989) 307–340. [14] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (2–3) (1989) 129–148. [15] H.G. Feichtinger, K. Gröchenig, Gabor wavelets and the Heisenberg group: Gabor expansions and short time fourier transform from the group theoretical point of view, in: C.K. Chui (Ed.), Wavelets: A Tutorial in Theory and Applications, Academic Press, Boston, MA, 1992, pp. 359–398. [16] H.G. Feichtinger, K. Gröchenig, Gabor frames and time-frequency analysis of distributions, J. Funct. Anal. 146 (2) (1997) 464–495. [17] H.G. Feichtinger, K. Nowak, A first survey of Gabor multipliers, in: Advances in Gabor Analysis, in: Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2003, pp. 99–128. [18] C. Fernandez, A. Galbis, Compactness of time-frequency localization operators on L2 (R), J. Funct. Anal. 233 (2) (2006) 335–350. [19] M. Fornasier, K. Gröchenig, Intrinsic localization of frames, Constr. Approx. 22 (3) (2005) 395–415. [20] S.J. Godsill, P.J.W. Rayner, Digital Audio Restoration, Springer, 1998. [21] K. Gröchenig, Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2001. [22] K. Gröchenig, Gabor frames without inequalities, Int. Math. Res. Not. (2007), 2007, Article ID rnm111, 21 pp. [23] K. Gröchenig, Weight functions in time-frequency analysis, in: L. Rodino, M.-W. Wong, et al. (Eds.), Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis, in: Fields Inst. Commun., vol. 23, 2007, pp. 343–366. [24] K. Gröchenig, M. Leinert, Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004) 1–18. [25] K. Gröchenig, J. Toft, Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces, J. Anal. Math., in press. [26] C. Heil, An introduction to weighted Wiener amalgams, in: M. Krishna, R. Radha, S. Thangavelu (Eds.), Wavelets and Their Applications, Chennai, January 2002, Allied Publishers, 2003, pp. 183–216. [27] P. Jaming, Principe d’incertitude qualitatif et reconstruction de phase pour la transformée de Wigner, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 249–254. [28] A.J.E.M. Janssen, Duality and biorthogonality for Weyl–Heisenberg frames, J. Fourier Anal. Appl. 1 (4) (1995) 403–436. [29] A.J.E.M. Janssen, Proof of a conjecture on the supports of Wigner distributions, J. Fourier Anal. Appl. 4 (6) (1998) 723–726. [30] N. Lerner, The Wick calculus of pseudo-differential operators and some of its applications, Cubo Mat. Educ. 5 (1) (2003) 213–236. [31] P. Louizou, Speech Enhancement: Theory and Practice, CRC Press, 2007. [32] F. Luef, Projective modules over non-commutative tori are multi-window Gabor frames for modulation spaces, J. Funct. Anal. 257 (6) (2009) 1921–1946. [33] J. Ramanathan, P. Topiwala, Time-frequency localization via the Weyl correspondence, SIAM J. Math. Anal. 24 (5) (1993) 1378–1393.

1924


[34] C. Roads, The Computer Music Tutorial, MIT Press, 1998. [35] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (October 2006) 437–452. [36] J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I, J. Funct. Anal. 207 (2) (2004) 399–429. [37] E. Wilczock, Zur Funktionalanalysis der Wavelet- und Gabortransformation, thesis, TU München, 1998. [38] M.-W. Wong, Wavelet Transforms and Localization Operators, Oper. Theory Adv. Appl., vol. 136, Birkhäuser, Basel, 2002.


On the double commutant of Cowen–Douglas operators Li Chen a,∗ , Ronald G. Douglas b,1 , Kunyu Guo c,2 a Department of Mathematics, Tianjin University, Tianjin 300072, China b Department of Mathematics, Texas A&M University, College Station, TX 77843, United States c School of Mathematics Science, Fudan University, Shanghai 200433, China

Received 27 December 2009; accepted 28 December 2010 Available online 8 January 2011 Communicated by Gilles Godefroy

Abstract Let T be a Cowen–Douglas operator. In this paper, we study the von Neumann algebra V ∗ (T ) consisting of operators commuting with both T and T ∗ from a geometric viewpoint. We identify operators in V ∗ (T ) with connection-preserving bundle maps on E(T ), the holomorphic Hermitian vector bundle associated to T . By studying such bundle maps, the structure of V ∗ (T ) as well as information on reducing subspaces of T can be determined. © 2010 Elsevier Inc. All rights reserved. Keywords: Cowen–Douglas operator; Von Neumann algebra; Reducing subspace; Connection

1. Introduction Let H be a separable Hilbert space. Given a domain (connected open subset) Ω in C and a positive integer n, M.J. Cowen and the second author [2] introduced the operator class Bn (Ω), consisting of operators T on H satisfying: * Corresponding author.

E-mail addresses: [email protected] (L. Chen), [email protected] (R.G. Douglas), [email protected] (K. Guo). 1 Research supported in part by a grant from the National Science Foundation (US). 2 This work is partially supported by Laboratory of Mathematics for Nonlinear Science, Fudan University, and NSFC (10525106), NKBRPC (2006CB805905). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.030

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(i) (ii) (iii) (iv)

L. Chen et al. / Journal of Functional Analysis 260 (2011) 1925–1943

Ω ⊆ σ (T ); ran(T − w) = H for w in Ω; w∈Ω ker(T − w) = H; dim ker(T − w) = n for w in Ω.

Given an operator T in Bn (Ω), the mapping w → ker(T − w) defines a rank n holomorphic Hermitian vector bundle over Ω, which we denote by E(T ). An important observation in [2] is that invariants of T can be revealed by investigating their geometric counterparts in E(T ). Our main aim in this paper is to study the von Neumann algebra V ∗ (T ) of operators commuting with both T and T ∗ for T in Bn (Ω). There are several motivations for our investigation. For one thing, Bn (Ω) contains many important classes of operators and characterizing reducing subspaces of these operators is an interesting topic in operator theory. Our investigation arises from the study of multiplication operators on Hilbert spaces consisting of holomorphic functions. For a typical example we mention the multiplication operator MB on the Bergman space where B is a finite Blaschke product. In this case, the adjoint of Mφ is a Cowen–Douglas operator. An open conjecture is that if B is a Blaschke product of order n, then MB has at most n distinct minimal reducing subspaces, or in language of operator algebra, the von Neumann algebra V ∗ (MB ) has at most n minimal projections. The algebra V ∗ (MB ) is finite dimensional (see [4]), and using general theory of finite dimensional von Neumann algebras, one can show that the conjecture is equivalent to the statement that V ∗ (MB ) is abelian (see [4,7] for detailed discussions). Progress along this line can also be found in [9,10,14]. For further discussion on the relation between operator theory on function spaces and von Neumann algebras, see [6] and [8]. We will not go any further on concrete problems, which however, suggest that it is worthwhile to have a conceptual understanding of V ∗ (T ) for an arbitrary Cowen–Douglas operator T . Another reason for studying V ∗ (T ) lies in its close relation to the differential geometry of the bundle E(T ). Recall that if S is an operator commuting with T , then S ker(T − w) ⊆ ker(T − w), and hence S induces a holomorphic bundle map on E(T ) which we denote by Γ (S). If S lies in V ∗ (T ), then Γ (S) is not only holomorphic, but also connection-preserving, as we shall see later. Projections in V ∗ (T ), or reducing subspaces of T , are in one-to-one correspondence with reducing subbundles of E(T ). (We say a subbundle F of a holomorphic Hermitian vector bundle E is a reducing subbundle if both F and its orthogonal complement F ⊥ in E are holomorphic subbundles.) Now we briefly describe this correspondence (see [2] for details): If H1 is a reducing subspace for T in Bn (Ω) and H2 = H1⊥ , then T |H1 and T |H2 are both Cowen–Douglas operators. In this case, E(T |H1 ) and E(T |H2 ) are mutually orthogonal holomorphic subbundles such that E(T ) = E(T |H1 ) ⊕ E(T |H2 ). Conversely, if E(T ) can be decomposed into an orthogonal direct sum of two holomorphic exist reducing subspaces H1 and H2 such that H = H1 ⊕ H2 subbundles E1 and E2 , then there with H1 = w∈Ω E1w and H2 = w∈Ω E2w , where Ei w denotes the fibre of Ei at w. Two reducing subspaces H1 and H2 for T are said to be unitarily equivalent if there exists a unitary operator U : H1 → H2 such that U T |H1 = T |H2 U . A key result in [2], which we restate in the following, asserts that H1 and H2 are unitarily equivalent if and only if there exists an isomorphic holomorphic bundle map between E(T |H1 ) and E(T |H2 ).


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Theorem 1.1. (See [2].) Two Cowen–Douglas operators T1 and T2 in Bn (Ω) are unitarily equivalent if and only if there exists a local isometric holomorphic bundle map Φ from E(T1 ) to E(T2 ). In this case, Φ = Γ (U ) where U is the intertwining unitary operator. Remark 1.2. We say that two holomorphic Hermitian vector bundles over Ω are locally equivalent if there exists an isometric holomorphic bundle map Φ defined on an open subset in Ω between them. The theorem says that the bundle map Φ defined on can be extended to a globally defined map Γ (U ); in other words, local equivalence implies global equivalence. This arises from the uniqueness of analytic continuation and the well-known spanning property (see [2] for a proof) that

ker(T − w) = H,

w∈

for any open subset in Ω. Given a Cowen–Douglas operator T , Theorem 1.1 asserts that holomorphic isometric bundle maps on E(T ) are in one-to-one correspondence with unitary operators in V ∗ (T ). In Section 3, we generalize this correspondence to connection-preserving bundle maps on E(T ) and V ∗ (T ) (holomorphic isometric bundle maps are necessarily connection-preserving, as we shall see in the next section). Our result is stated as follows: Theorem 1.3. Let T be a Cowen–Douglas operator in Bn (Ω) and Φ be a bundle map on E(T ). There exists an operator TΦ in V ∗ (T ) such that Φ = Γ (TΦ ) if and only if Φ is connectionpreserving. Consequently, the map Γ is a ∗-isomorphism from V ∗ (T ) to connection-preserving bundle maps on E(T ). In Section 4, by studying connection-preserving bundle maps on E(T ), we show that V ∗ (T ) is isomorphic to the commutant of a matrix algebra. This matrix algebra represents the algebra of bundle maps on E(T ) generated by curvature and its covariant derivatives to all orders. Our discussions are based on a result called “block diagonalization of connections” established by Cowen and the second author [3] where they studied the equivalence problem of C ∞ Hermitian vector bundles. We will also use this result to study reducing subbundles of E(T ), which provides a canonical decomposition of H into the direct sum of minimal reducing subspaces. As a complementary example, we discuss a typical kind of Cowen–Douglas operators, called the bundle shifts, which represent a large class of subnormal operators related to multiply-connected domains [1]. 2. Preliminaries on Hermitian vector bundles In this section, we provide necessary preliminaries on Hermitian vector bundles, which are mainly extracted from [2]. General references can be found in [11,13]. Given a domain Ω in C, a rank n holomorphic vector bundle over Ω is a complex manifold E with a holomorphic map π from E onto Ω such that each fibre Eλ = π −1 (λ) is a copy of Cn and for each λ0 in Ω, there exists a neighborhood of λ0 and holomorphic functions s1 , . . . , sn from to E such that Eλ = {s1 (λ), . . . , sn (λ)}. The n-tuple of functions {s1 , . . . , sn } is called

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a holomorphic frame over . A section is a map s from an open subset of Ω to E such that π(s(λ)) = λ. A bundle map between two bundles E1 and E2 defines a linear transformation from E1λ to E2λ for λ in Ω. Locally a bundle map can be represented by a matrix-valued function relative to the local frames of the two bundles. A bundle map between two holomorphic vector bundles is holomorphic if its representing matrix function relative to holomorphic frames is holomorphic. A holomorphic bundle map is determined by its restriction on any open subset in Ω. A Hermitian vector bundle is a vector bundle E such that each fibre Eλ is an inner product space. Given a bundle map Φ from a Hermitian vector bundle E1 to E2 , we can define its adjoint to be a bundle map Φ ∗ from E2 to E1 satisfying Φs(λ), t (λ) E = s(λ), Φ ∗ t (λ) E

2λ

1λ

for any sections s and t of E1 and E2 , respectively. For a separable Hilbert space H and a positive integer n, let Gr(n, H) denote the Grassmann manifold of all n-dimensional subspaces of H. A map f : Ω → Gr(n, H) is called a holomorphic neighborhood of λ0 and n holomorphic H-valued curve if for any point λ0 in Ω, there exists a functions s1 , . . . , sn on such that f (λ) = {s1 (λ), . . . , sn (λ)}. A holomorphic curve naturally gives a Hermitian holomorphic vector bundle Ef over Ω. The fibre of Ef at a point λ is f (λ) and the metric at each fibre is inherited from the inner product on H. The local holomorphic functions s1 , . . . , sn form a holomorphic frame over . It is shown in [2] that if T is a Cowen– Douglas operator in Bn (Ω), the map w → ker(T − w) is a holomorphic curve and the resulting bundle is E(T ). In this paper, we concentrate on unitary invariants of holomorphic curves, while we would like to mention the work of Jiang and Ji [12], who studied the similarity questions rather than unitary ones and some of their methods are related to ours. Let E(Ω) denote the algebra of C ∞ functions on Ω and let E p (Ω) denote the C ∞ differential forms of degree p on Ω. Then we have E 0 (Ω) = E(Ω), E 1 (Ω) = {f dz + g dz: f, g ∈ E(Ω)} and E 2 (Ω) = {f dz dz: f ∈ E(Ω)}. For a C ∞ vector bundle E over Ω, let E p (Ω, E) denote the differential forms of degree p with coefficients in E, then E 0 (Ω, E) are just C ∞ sections of E on Ω. A connection on E is a first order differential operator D : E 0 (Ω, E) → E 1 (Ω, E) such that D(f σ ) = df ⊗ σ + f D(σ ) for f in E(Ω) and σ in E 0 (Ω, E). The connection D is called metric-preserving if dσ1 , σ2 = Dσ1 , σ2 + σ1 , Dσ2 , for σ1 , σ2 in E 0 (Ω, E). Locally, D can be represented by a connection matrix. Let s = {s1 , . . . , sn } be a local frame on , then the connection matrix Θ(s) = [Θij ] relative to the frame s is a matrix with 1-form entries Θij defined on such that D(si ) = Σjn=1 Θij ⊗ sj . The connection D can be extended to a differential operator from E 1 (Ω, E) to E 2 (Ω, E) so that


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D(σ ⊗ α) = Dσ ∧ α + σ ⊗ dα for σ in E(Ω, E) and α in E 1 (Ω). It is well known that D 2 is C ∞ linear so we have for any σ in E(Ω, E), that D 2 σ = Kσ dz dz, where K is a bundle map on E which is uniquely determined by D 2 . Thus D 2 can be identified with K and we call K the curvature of (E, D). For a Hermitian vector bundle on a domain in C, the curvature K is always self-adjoint provided that its defining connection D is metric-preserving (Section 2.15, [2]). The matrix of D 2 relative to a frame s is given by D 2 (s) = dΘ(s) + Θ(s) ∧ Θ(s).

(2.1)

Note that D is not a bundle map since it is not C ∞ linear, while it can be shown that the commutator of D with a bundle map is still a bundle map (Lemma 2.10, [2]). Thus for the bundle map Φ on E, there exists bundle maps Φz and Φz satisfying [D, Φ] = DΦ − ΦD = Φz ⊗ dz + Φz ⊗ dz. Then Φz and Φz are called covariant derivative of Φ relative to the connection D. Since covariant derivatives are also bundle maps, we can continue this procedure to define higher order covariant derivatives Φzi zj for all positive integers i, j . The covariant derivatives of Φ and Φ ∗ are related as follows (Lemma 2.12, [2]): (Φz )∗ = Φ ∗ z , (Φz )∗ = Φ ∗ z .

(2.2)

A bundle map Φ is called connection preserving if [D, Φ] = 0 or equivalently, Φz = Φz = 0. By an easy computation (or see [3]), the matrix of [D, Φ] relative to a local frame s is dΦ(s)+ [Θ(s), Φ(s)]. Thus a bundle map is connection-preserving if and only if its matrix satisfies dΦ(s) + Θ(s), Φ(s) = 0.

(2.3)

An induction argument shows that a connection-preserving bundle map Φ necessarily preserves curvature as well as its covariant derivatives to all orders, i.e. ΦKzi zj = Kzi zj Φ for all 0 i, j < ∞ (Remark 2.16, [2]).

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Now we turn to holomorphic vector bundles. If E is a holomorphic Hermitian vector bundle, it is well known that there exists a unique canonical connection Θ on E, called the Chern connection, which is metric-preserving and compatible with the holomorphic structure. Locally, given a holomorphic frame s = {s1 , . . . , sn } with metric matrix h = (si , sj ), Θ(s) = ∂hh−1 .

(2.4)

D 2 (s) = ∂ ∂hh−1 .

(2.5)

The matrix of D 2 is given by

The matrix of the covariant derivatives of a bundle map Φ relative to this canonical connection is given by (sec 2.18, [2]): Φz (s) = ∂Φ(s) + ∂hh−1 , Φ(s)

and Φz (s) = ∂Φ(s).

Thus the matrix of Φz is just the usual ∂ derivative of its matrix Φ(s) relative to the holomorphic frame s. Hence a bundle map Φ is holomorphic if and only if Φz = 0. Recall that Φ is connection-preserving if Φz = Φz = 0, and combining this with (2.2), we have: Proposition 2.1. A bundle map Φ on a holomorphic Hermitian vector bundle E over a domain in C preserves the canonical connection if and only if both Φ and Φ ∗ are holomorphic. An isometric holomorphic bundle map Φ is connection-preserving since Φ ∗ = Φ −1 , which is necessarily holomorphic. Given two Hermitian vector bundles E1 and E2 with connections D1 and D2 ; respectively, let Φ be a bundle map from E1 to E2 . We say Φ is connection-preserving if D2 Φ = ΦD1 . Fix local frames s1 and s2 for E1 and E2 ; respectively. Then Φ is connection-preserving if its matrix Φ relative to the two frame satisfies dΦ + Θ2 (s2 )Φ − ΦΘ1 (s1 ) = 0,

(2.6)

where Θi (si ) is the connection matrix of Di with respect to the frame si . For a bundle map between two Hermitian vector bundles, one can define its covariant derivative analogously, and Proposition 2.1 still holds (see [2] for details). 3. Geometric realization of V ∗ (T ) This section is devoted to establishing Theorem 1.3. Throughout this section, a connection means the canonical connection on a given holomorphic Hermitian vector bundle. The following technical lemma (Proposition 1, [5]) is useful in this section.


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Lemma 3.1. Let Ω be a domain in C and f (z, w) be a function on Ω×Ω which is holomorphic in z and anti-holomorphic in w. Then f (z, z) = 0 for all z in Ω if and only if f vanishes identically on Ω × Ω. Corollary 3.2. Let S1 , S2 be two operators commuting with T , then Γ S1 = (Γ S2 )∗ if and only if S1 = S2∗ . Proof. Sufficiency follows from the definition of Γ and it remains to show Γ S1 = (Γ S2 )∗ implies S1 = S2∗ . Takea holomorphic frame {σi (z)} for E(T ) over an open subset . Then by the spanning property λ∈ E(T )λ = H, it suffices to show that S1 σi (z), σj (w) = σi (z), S2 σj (w)

for all i, j and z, w in . Since the frame is holomorphic, both sides of the identity above is holomorphic in z and anti-holomorphic in w, and Γ S1 = (Γ S2 )∗ implies that the identity holds for z = w, so Lemma 3.1 can be applied and we are done. 2 In general, a holomorphic Hermitian bundle does not admit a holomorphic orthonormal frame, but in the special case of holomorphic curves, there always exists a local holomorphic frame which is “normal” at one point (see Lemma 2.4 in [2]). Lemma 3.3. (See [2].) Given a holomorphic curve f over Ω and a point z0 in Ω, there exists a holomorphic frame {σi (z)} for Ef in a neighborhood of z0 such that (σi (z), σj (z0 ) ) is the identity matrix for all z in . The local frame {σi } given by Lemma 3.3 is called a normal frame. The matrix of a connection-preserving bundle map relative to a normal frame is very well behaved. Proposition 3.4. Let f be a holomorphic curve over a domain Ω in C and {σi } be a normal frame over an open subset at a point z0 . If Φ is a connection-preserving bundle map on Ef , then its matrix relative to {σi } is a constant matrix which commutes with the metric matrix (σi (z), σj (w) ) for all z, w in . Proof. By Proposition 2.1, both Φ and Φ ∗ are holomorphic. If we denote by Φ(z) and Ψ (z) the matrix of Φ and Φ ∗ relative to base {σi (z)} of the fibre at z, then Φ(z) and Ψ (z) are both holomorphic matrix-valued functions. If we set h(z, w) = (σi (z), σj (w) ), then h is holomorphic in z and anti-holomorphic in w such that h(z, z0 ) = I . By elementary linear algebra we have Ψ (z) = h(z, z)Φ ∗ (z)h−1 (z, z). Combining this with Lemma 3.1, we have Ψ (z) = h(z, w)Φ ∗ (w)h−1 (z, w).

(3.1)

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Let w = z0 , we see that Ψ (z) = Φ ∗ (z0 ). Thus Ψ (z) is constant which we denote by Ψ . Our original identity becomes Ψ = h(z, z)Φ ∗ (z)h−1 (z, z), taking adjoints we get Ψ ∗ = h−1 (z, z)Φ(z)h(z, z). Another application of Lemma 3.1 yields Ψ ∗ = h−1 (z, w)Φ(z)h(z, w). Taking w = z0 again, we have Ψ ∗ = Φ(z). Thus Φ(z) is constant (which we also denote by Φ) and Φ ∗ = Ψ . By (3.1), both Φ and Ψ commute with h(z, z), and thus commutes with h(z, w) as well, in light of Lemma 3.1. 2 Remark 3.5. From the proof of the above proposition, we see that if we fix a normal frame and a connection-preserving bundle map Φ, the matrix of Φ ∗ is just the adjoint of the matrix of Φ. Recall that a connection-preserving bundle map is necessarily holomorphic, and thus is determined by its restriction to any open subset . Therefore the mapping defined by sending a connection-preserving bundle map to its matrix relative to a local normal frame is an injective ∗-homomorphism. Now we complete the proof of Theorem 1.3. Proof of Theorem 1.3. One direction is easy. For an operator S in V ∗ (T ), both S and S ∗ commutes with T and (Γ (S))∗ = Γ (S ∗ ). Thus the condition of Proposition 2.1 is satisfied and Γ (S) is connection-preserving. We now establish the other direction: any connection preserving bundle map is induced by an operator in V ∗ (T ). As before, we fix an open subset and a local holomorphic frame {σi (z)} for the holomorphic curve E(T ) normalized at a point z0 in . By the previous proposition, the matrix of the connection-preserving bundle map relative to this frame is a constant matrix which we also denote by Φ such that Φh(z, w) = h(z, w)Φ, where h(z, w) = (σi (z), σj (w) ). For any z in , the bundle map defines a linear operator on the fibre ker(T − z) whose matrix relative to the base {σi (z)} is Φ. Since eigenvectors belonging to different eigenvalues are linearly


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independent, these fibre maps together give a well-defined linear transform TΦ on their algebraic linear span H0 = spanz∈ ker(T − z), which is a dense subspace of H. For any z in , TΦ ker(T − z) ⊆ ker(T − z) by our construction, which implies that TΦ commutes with T on ker(T − z), and thus on H0 as well. We claim that TΦ is bounded. To this end, let us take an arbitrary vector f in H0 . For such an f , there exist vectors f1 , . . . , fm with fi ∈ ker(T − zi ) for some z1 , . . . , zm in such that f = f1 + · · · + fm . Since {σi } is a frame, there exist mn complex numbers aij , 1 i m, 1 j n, such that fi =

n

aij σj (zi )

j =1

for any 1 i m. To simplify notation, we write ai = (ai1 , ai2 , . . . , ain ) and T σ (z) = σ1 (z), σ2 (z), . . . , σn (z) . Then fi = ai σ (zi ) and f = a1 σ (z1 ) + · · · + am σ (zm ). Now f 2 = a1 σ (z1 ) + · · · + am σ (zm ), a1 σ (z1 ) + · · · + am σ (zm ) =

m

ai h(zi , zj )a∗j

i,j =1

= (a1 , . . . , am ) h(zi , zj ) (a1 , . . . , am )∗ . Here (a1 , . . . , am ) is a row of mn complex numbers and [h(zi , zj )] is an mn × mn matrix whose n × n block at the (i, j ) place is the matrix h(zi , zj ). For example, if m = 2, there are only two points z1 and z2 involved and h(zi , zj ) = On the other hand,

h(z1 , z1 ) h(z2 , z1 )

h(z1 , z2 ) h(z2 , z2 )

.

1934


TΦ f 2 = a1 Φσ (z1 ) + · · · + am Φσ (zm ), a1 Φσ (z1 ) + · · · + am Φσ (zm ) =

m

ai Φh(zi , zj )Φ ∗ a∗j

i,j =1

= (a1 , . . . , am )(Φ ⊗ Im ) h(zi , zj ) Φ ∗ ⊗ Im (a1 , . . . , am )∗ where Φ ⊗ Im is a block-diagonal matrix with Φ repeated m times on the diagonal. Recall that Φh(zi , zj ) = h(zi , zj )Φ, which implies (Φ ⊗ Im ) h(zi , zj ) = h(zi , zj ) (Φ ⊗ Im ). Note that [h(zi , zj )] is a positive matrix, so we have 1 1 (Φ ⊗ Im ) h(zi , zj ) 2 = h(zi , zj ) 2 (Φ ⊗ Im ), and ∗ 1 1 Φ ⊗ Im h(zi , zj ) 2 = h(zi , zj ) 2 Φ ∗ ⊗ Im . Consequently 1 1 (Φ ⊗ Im ) h(zi , zj ) Φ ∗ ⊗ Im = h(zi , zj ) 2 (Φ ⊗ Im ) Φ ∗ ⊗ Im h(zi , zj ) 2 , thus (Φ ⊗ Im ) h(zi , zj ) Φ ∗ ⊗ Im Φ ⊗ Im 2 h(zi , zj ) = Φ2 h(zi , zj ) which implies that TΦ f Φf . Here Φ is the standard matrix norm of Φ which dose not depend on f , hence the claim is proved. Since H0 is dense, TΦ extends to a bounded operator on H and the extended operator still commutes with T . By our construction, Φ = Γ (TΦ ) for the extended TΦ . We further claim that (TΦ )∗ commutes with T , which means TΦ is in V ∗ (T ) and the proof of the theorem will be complete. Let Ψ be the adjoint of the bundle map Φ, then as in the proof of Proposition 3.4, its matrix relative to the normal frame is also a constant matrix Ψ and Ψ h(z, w) = h(z, w)Ψ for all z, w in . Therefore, using the same argument, there exists a bounded operator TΨ commuting with T such that Ψ = Γ (TΨ ). By Corollary 3.2, (TΦ )∗ = TΨ , hence TΦ is in V ∗ (T ). 2 Just as in Remark 1.2, we have: Remark 3.6. For a Cowen–Douglas operator T , a local connection-preserving bundle map on E(T ) can be extended to a global connection-preserving bundle map induced by an operator in V ∗ (T ).


1935

For a C ∞ vector bundle E on a planar domain with a given connection, bundle maps on E can be seen as sections of the tensor bundle E ⊗ E ∗ and a bundle map Φ is connectionpreserving if and only if it is a parallel section of E ⊗ E ∗ . Thus a connection-preserving bundle map is determined by its action on any fibre. In the case of holomorphic curves with canonical connection, this follows immediately from Proposition 3.4. Consequently, for a Cowen–Douglas operator T in Bn (Ω) and any point w0 in Ω, an operator in V ∗ (T ) is determined by its action on ker(T − w0 ). In particular, we have: Corollary 3.7. For a Cowen–Douglas operator T , V ∗ (T ) is finite dimensional. We end this section with a straightforward proof of this corollary in operator theory, which is of independent interest. Proof. Without loss of generality, we assume w = 0. Since ker T is finite dimensional, it suffices to show that if S is an operator in V ∗ (T ) such that S|ker T = 0, then S = 0. In fact, since H = ker T ⊕ ran T ∗ (note that ran T ∗ is closed in this case), we have ∞

2 k SH = ST ∗ H = T ∗ SH = T ∗ ST ∗ H = T ∗ SH = · · · ⊆ ran T ∗ . k=1

Note that the spanning property implies that ∞ ∗ k k=1 ran(T ) = 0, as desired. 2

∞

k=1 ker T

k

= H (Section 1.7, [2]), hence

4. Connection-preserving bundle maps on E(T ) In this section, we study connection-preserving bundle maps on E(T ) and provide a characterization of V ∗ (T ) in terms of geometric invariants. Before proceeding, we would like to say more about reducing subbundles of holomorphic Hermitian vector bundles. Let E be a holomorphic Hermitian vector bundle with canonical connection D. Given a reducing subbundle E of E, we can chose holomorphic frames s and s of E and E ⊥ ; respectively such that s = {s , s } forms a holomorphic frame for E. Relative to this frame, the metric matrix of E decomposes into two blocks. Therefore by (2.4), (2.5) the matrices of the canonical connection D and curvature K also decompose into two blocks. By the following representation of covariant derivatives: Kz (s) = ∂K(s) + ∂hh−1 , K(s) , Kz (s) = ∂K(s), we see that the matrices of the covariant derivatives of the curvature to all orders also decompose into two blocks relative to this frame. In particular, reducing subbundles are D-invariant. The following result (see Proposition 4.18, Chapter 1 in [11]) asserts that the converse is also true, which can be used to identify reducing subbundles of holomorphic Hermitian vector bundles. We include the proof for the convenience of the readers. Proposition 4.1. (See [11].) Let E be a holomorphic Hermitian vector bundle and D the canonical connection. Let E be a C ∞ subbundle and E be the orthogonal complement of E in E.

1936


If E is invariant under D, both E and E are D-invariant holomorphic subbundles of E and they give a holomorphic orthogonal decomposition: E = E ⊕ E . Proof. As is well known, the canonical connection D can be decomposed as D = D + ∂ with D : E 0 (Ω, E) → E 1,0 (Ω, E), ∂ : E 0 (Ω, E) → E 0,1 (Ω, E). Since E is invariant under D, so is E because D is metric preserving. Let s be a holomorphic section of E and s = s + s be its C ∞ decomposition with respect to E = E ⊕ E . It suffices to show s and s are holomorphic sections. Since D = D + ∂ and s is holomorphic, we have Ds = D s. On the other hand, Ds = Ds + Ds and D s = D s + D s , which implies Ds = D s and Ds = D s . Therefore ∂s = 0 and ∂s = 0, as desired. 2 Since the canonical connection on a holomorphic Hermitian vector bundle is unique, the canonical connection on a reducing subbundle is just the restriction of the original one. As stated in the introduction, our investigation is based on a representation theorem of Cowen and the second author, called the C ∞ block diagonalization of connections. We begin with some necessary terminologies before introducing this result. Let E be a C ∞ Hermitian vector bundle of rank n over a domain Ω in C with metricpreserving connection D and curvature K. We denote by A the algebra of bundle maps generated by the curvature K and its covariant derivatives Kzi zj to all orders. Since K is self-adjoint and the identity (2.2) holds, A is self-adjoint. Let s be a C ∞ orthonormal frame of E over an open subset of Ω. For a bundle map Φ on E and z in , let Φ(z) be the induced fibre map on the fibre Ez and Φ(s)(z) be the matrix of Φ(z) relative to the base s(z). We denote by A (z) the set of linear transforms on the fibre Ez induced by bundle maps in A and A (s)(z) the matrix algebra generated by the matrices Φ(s)(z) for Φ in A , then A (s)(z) is a self-adjoint matrix algebra in Mn (C) since s is orthonormal. It is well known that any self-adjoint matrix algebra is the direct sum of full matrix algebras with multiplicity. More precisely, for any self-adjoint matrix algebra, there exist two tuples of positive integers M = (m1 , . . . , mr ) and N = (n1 , . . . , nr ), such that the algebra consists of matrices of the form A1 ⊗ Im1 ⊕ · · · ⊕ Ar ⊗ Imr , where Ai is an ni × ni matrix repeated mi times on the diagonal, we denote such an algebra by M(N , ⊗M ). For example, M((n1 , n2 ), ⊗(2, 1)) is the algebra of matrices of the form ⎛

A1 A1 ⊗ I2 ⊕ A2 ⊗ I1 = ⎝ 0 0

0 A1 0

where A1 is an n1 × n1 matrix and A2 is an n2 × n2 matrix.

⎞ 0 0 ⎠, A2


1937

Now we can state the theorem on block diagonalization of connections (see Proposition 2.5 in [3], also see Lemma 3.2 in [2] for a special case). Theorem 4.2. (See [3].) Let E be a C ∞ Hermitian vector bundle of rank n over an open subset Ω in C, with metric-preserving connection D. For any point z0 off a non-where dense subset of Ω, there exist two tuples of integers M = (m1 , . . . , mr ), N = (n1 , . . . , nr ), a neighborhood Ω0 of z0 and a C ∞ orthonormal frame s for E over Ω0 with the properties: A (s)(z) = M(N , ⊗M ) for all z in Ω0 , where A is the algebra of bundle maps generated by the curvature K, and its covariant derivatives Kzi zj to all orders. Moreover, Θ(s) = Θ1 ⊗ Im1 ⊕ · · · ⊕ Θr ⊗ Imr , where Θ(s) is the matrix of connection 1-forms of D relative to the frame s and Θi are C ∞ ni × ni matrices with 1-form entries defined on Ω0 . Remark 4.3. There are various ways to understand this theorem. (i) The algebra A (s)(z) does not depend on the point z in Ω0 . (ii) The theorem asserts that the connection matrix has a block diagonal form, thus each block corresponds to a subbundle invariant under D. Explicitly, for any 1 i r, the block Θi ⊗ Imi corresponds to mi D-invariant subbundles of rank ni . We denote these subbundles by Ei1 , . . . , Eimi . With respect to this decomposition, the frame s can be written as s = {sij } where sij is an orthonormal frame for Eij . (iii) By definitions, the curvature as well as its partial derivatives are determined by the connections, while the theorem implies that the connection can be determined by the curvature in some sense. If E is a holomorphic Hermitian vector bundle with canonical connection D, then D-invariant subbundles are actually reducing subbundles for E by Proposition 4.1. Therefore we can apply Theorem 4.2 to obtain a collection of mutually orthogonal reducing subbundles {Eij }, 1 i r, 1 j mi with rank Eij = ni , such that E = E11 ⊕ · · · ⊕ E1m1 ⊕ · · · ⊕ Er1 ⊕ · · · ⊕ Ermr . If we apply the theorem to the bundle E(T ) with canonical connection for a Cowen–Douglas operator T , then {Eij } correspond to reducing subspaces {Hij } such that H = H11 ⊕ · · · ⊕ H1m1 ⊕ · · · ⊕ Hr1 ⊕ · · · ⊕ Hrmr .

(4.1)

We will show that Hij are minimal and (4.1) gives a canonical decomposition of H into minimal reducing subspaces. To get a full understanding of that, we first recall some elementary facts on von Neumann algebras. In light of Corollary 3.7, we concentrate on the finite dimensional case.

1938


Given a von Neumann algebra M, we denote its center by Z(M) and its identity by 1M . Two projections p and q in M are said to be equivalent if there exists an element u in M such that u∗ u = p, uu∗ = q. A projection p in M is said to be minimal if for any projection q in M, q p implies q = 0 or q = p. If M is a finite dimensional von Neumann algebra, there exists finitely many mutually orthogonal minimal projections q1 , . . . , qk in M such that 1M = q1 + · · · + qk .

(4.2)

The center Z(M) is also finite dimensional, thus there are finitely many mutually orthogonal minimal central projections (i.e. minimal projections in Z(M)) p1 , . . . , pr , such that 1 M = p 1 + · · · + pr . One can show, as a routine exercise, that (i) for any minimal projection q in M, there exist exactly one index i such that qpi = q (equivalently, q pi ) and qpj = 0 for j = i, (ii) two minimal projections in M are equivalent if and only if they are dominated by the same minimal central projection. By (i), we can rearrange the minimal projections in (4.2) such that 1M = q11 + · · · + q1m1 + · · · + qr1 + · · · + qrmr

(4.3)

with qi1 + · · · + qimi = pi , and by (ii), qij and qi j are equivalent if and only if i = i . We call (4.3) a canonical decomposition. Now we go back to the Cowen–Douglas operator T . Reducing subspaces of T can be identified with projections in V ∗ (T ) and it is easy to check that two reducing subspaces H1 and H2 are unitarily equivalent if and only if the their corresponding projections in V ∗ (T ) are equivalent. The following theorem says (4.1) is a canonical decomposition in the sense we discussed above, while the proof is geometric. Theorem 4.4. Let T be a Cowen–Douglas operator and E(T ) be its associated holomorphic Hermitian vector bundle with reducing subbundles {Eij } given by the block diagonalization of the canonical connection. Let {Hij } be the corresponding reducing subspaces. Then: (i) The reducing subspaces {Hij } are minimal. (ii) Hij and Hi j are unitarily equivalent if and only if i = i . Proof. (i) It suffices to show the bundle Eij is irreducible. Suppose conversely that Eij = F1 ⊕ F2 for two orthogonal holomorphic subbundles F1 and F2 , then Eij admits a holomorphic frame s = { s1 , s2 }, where si is a holomorphic frame for Fi . Let A (Eij ) be the restriction of A on Eij , then as mentioned in the beginning of the section, for z in Ω0 , the matrix of any linear map in s(z) = { s1 (z), s2 (z)} for the A (Eij )(z) should take a block diagonal form relative to the base fibre Eij z . While on the other hand, A (Eij )(z) contains all linear transformations on the fibre


1939

Eij z since by Theorem 4.2, A (Eij )(sij )(z) is the full matrix algebra Mni (C) relative to the frame sij mentioned in Remark 4.3, a contradiction. (ii) Without loss of generality, we prove the statement for r = 2, m1 = 2, m2 = 1. In this case, relative to the frame s in Theorem 4.2, we have a decomposition E(T ) = E11 ⊕ E12 ⊕ E21 . Here s = {s11 , s12 , s21 } where s11 , s12 , s21 are orthonormal frames for E11 , E12 , E21 respectively as in Remark 4.3. The matrix algebra A (s)(z) contains all matrices of the form ⎛

A1 ⎝ 0 0

0 A1 0

⎞ 0 0 ⎠ A2

and the connection matrix is of the form ⎛ Θ1 ⎝ 0 0

0 Θ1 0

⎞ 0 0 ⎠. Θ2

In light of Theorem 1.1, it suffices to show that E11 and E12 are equivalent while E12 and E21 are not equivalent. That E11 and E12 are equivalent is straightforward. We define an isometric bundle map by sending the orthonormal frame s11 to s12 . We claim that this bundle map is holomorphic, and hence implements an equivalence of the two bundles. In fact, by Proposition 2.1, it suffices to show this bundle map is connection preserving. Since the matrix of this bundle map relative to the frames s11 and s12 is the constant identity matrix and the connection matrices relative to the two frames are the same, we see that (2.6) holds. Hence the claim follows. Next we show that there exists no isometric connection-preserving bundle map from E12 to E21 . If there exists such a bundle map Φ, then E12 and E21 are of the same rank and by the discussions in Section 2, Φ preserves the curvatures as well as their covariant derivatives to all orders. Hence Φ commutes with the restriction of A to E12 and E21 . Suppose rank E12 = rank E21 = k, then by Theorem 4.2, for any fixed z in Ω0 and any two k × k matrices A1 and A2 , there exists a bundle map in A such that the matrices of its restriction to E12 (z) and E21 (z) relative to the base s11 (z) and s12 (z) are A1 and A2 respectively. So if Φ(z) is the matrix of Φ relative to the bases s11 (z) and s12 (z), then Φ(z)A1 = A2 Φ(z), which forces Φ(z) to be zero since A1 and A2 can be arbitrarily chosen.

2

We give the promised geometrical characterization of V ∗ (T ). Theorem 4.5. For a Cowen–Douglas operator T in Bn (Ω), the von Neumann algebra V ∗ (T ) is isomorphic to the commutant of the matrix algebra M(N , ⊗M ) in Mn (C), where M(N , ⊗M ) is given by the block diagonalization of the canonical connection on E(T ).

1940


Proof. By Theorem 1.3, it suffices to identify connection-preserving bundle maps on E(T ) with the commutant of M(N , ⊗M ). Denote the algebra of connection-preserving bundle maps on E(T ) by V. We claim that for any bundle map Φ in V, the matrix of Φ relative to the orthonormal frame s given in Theorem 4.2 is a constant matrix and lies in M (N , ⊗M ). In fact, for a fixed z in Ω0 , let Φ(z) be the matrix of Φ relative to the base s(z), then since Φ is connection-preserving, it commutes with every bundle map in A , thus by Theorem 4.2, Φ(z) commutes with every matrix in M(N , ⊗M ). Moreover, Φ(z) commutes with the connection matrix Θ(z) since Θ(z) is just a matrix in M(N , ⊗M ) tensored with a 1-form, so Θ(z), Φ(z) = 0. Recall that the matrix of a connection-preserving bundle map satisfies dΦ(z) + Θ(z), Φ(z) = 0, therefore dΦ(z) = 0, and Φ(z) is constant. Now we have a map Λ from V to M (N , ⊗M ) sending a connection-preserving bundle map to its matrix relative to the frame s, which is well defined. Note that since the frame is orthonormal, Λ is a ∗-homomorphism. Moreover, Λ is injective since a connection-preserving bundle map is determined by its action on any open subset. Λ is surjective since any constant matrix in M (N , ⊗M ) satisfies (2.3). Thus a local bundle map given by such a matrix relative to the frame s is connection-preserving on Ω0 . By Remark 3.6, this local bundle map can be extended to a connection-preserving bundle map on all Ω, completing the proof. 2 The commutant of M(N , ⊗M ) consists of matrices of the form In1 ⊗ B1 ⊕ · · · ⊕ Inr ⊗ Br , where Bi is an mi × mi matrix. We see that M (N , ⊗M ) is abelian if and only if mi = 1 for all i. Thus we have the following Corollary 4.6. The von Neumann algebra V ∗ (T ) is abelian if and only if there is no multiplicity in the block diagonalization of the canonical connection on E(T ). In general, it is not easy to compute the matrix algebra M(N , ⊗M ) explicitly for an arbitrary Cowen–Douglas operator. We discuss a special kind of operator TE , called the bundle shift. The adjoint of TE lies in the Cowen–Douglas class and V ∗ (TE ) can be identified via the topological construction of a certain flat unitary bundle E. The bundle shift TE was introduced in [1] and we give a quick review of its definition. Let Ω be a bounded domain in C whose boundary consists of finitely many analytic Jordan curves. A flat unitary bundle over Ω is a holomorphic Hermitian vector bundle which locally admits orthonormal holomorphic frames (or equivalently, the transition functions are constant unitary matrices). It is well known that any flat unitary bundle over Ω is equivalent to a canonical flat bundle arising from a unitary representation of the fundamental group π1 (Ω). Let us briefly recall this construction.


1941

By the uniformization theorem, there is a holomorphic covering map π : D → Ω where D is the unit disc. Let U (n) be the group of unitary operators on Cn and a unitary representation of π1 (Ω) is a homomorphism α : π1 (Ω) → U (n). Define an action of π1 (Ω) on D × Cn by A : (z, ξ ) → Az, α(A)ξ for A ∈ π1 (Ω), z ∈ D, and ξ ∈ Cn . (We identify π1 (Ω) with the covering transformation group acting on D.) Then the quotient space D × Cn /π1 (Ω) of this action with the obvious projection onto Ω gives a flat unitary bundle of rank n over Ω. Given a flat unitary bundle E over Ω, one can construct a Hilbert space HE2 consisting of holomorphic sections f of E such that f (z)2Ez has a harmonic majorant. The bundle shift TE is defined on HE2 by TE (f ) = zf . One can show that TE∗ lies in Bn (Ω ∗ ), where Ω ∗ is the complex conjugate of Ω (Theorem 3, [1]). A fundamental result on the bundle shift is the following (Theorem 6, [1]). Theorem 4.7. (See [1].) If E and F are flat unitary bundles over Ω, then the bundle shifts TE and TF are unitarily equivalent if and only if E and F are equivalent. Remark 4.8. Any two flat unitary bundles of the same rank are locally equivalent since they admit local orthonormal holomorphic frames, while the theorem requires that the isometric holomorphic bundle map can be defined globally. Moreover, we have a characterization of the von Neumann algebra V ∗ (TE ) (Theorem 7, [1]): Theorem 4.9. (See [1].) For a rank n flat unitary bundle E over Ω arising from a unitary representation α of π1 (Ω), the von Neumann algebra V ∗ (TE ) is isomorphic to the commutant of C ∗ (α) in Mn (C), where C ∗ (α) is the C ∗ algebra generated by the range of α. A geometric interpretation of Theorem 4.9 in terms of bundle maps, which is related to our investigation, is the following: Corollary 4.10. For a rank n flat unitary bundle E over Ω arising from a unitary representation α of π1 (Ω), any operator in V ∗ (TE ) is induced by a (global)connection-preserving bundle map on E. Proof. For one thing, the connection matrix Θ(s) is zero for any local orthonormal holomorphic frame s by (2.4). Thus for a fixed matrix Φ in Mn (C), a local bundle map defined by this constant matrix relative to the frame s satisfies (2.3). On the other hand, one can check that the transition matrix of two different local orthonormal holomorphic frames whenever their defining domains overlap is nothing but α(A) for some A ∈ π1 (Ω) (in fact, the local holomorphic orthonormal frames arise from branches of local inverses of the covering map), therefore when Θ lies in the commutant of C ∗ (α), Φ = α −1 (A)Φα(A), which is exactly the condition assuring that the locally defined connection-preserving bundle maps glue to a global one. That such a bundle map induces an operator in V ∗ (TE ) follows by tracing back the original proof of Theorem 4.9 and is omitted here. 2 The following consequence of Theorem 4.9 can be seen as a complement of our main results.

1942


Corollary 4.11. For any self-adjoint subalgebra A of Mn (C), there exists a Cowen–Douglas operator T such that V ∗ (T ) A. Proof. It follows from general theory of self-adjoint matrix algebras that the commutant algebra A of A can be generated by finitely many, say, k unitary matrices. Take a planar domain Ω with k holes so that π1 (Ω) is a free group of k generators. The map α defined by taking each generator of π1 (Ω) to one of the unitary matrices generating A extends to a unitary representation α of π1 (Ω) with C ∗ (α) = A . By Theorem 4.9, V ∗ (TE ) A = A, where E is the flat unitary bundle arising from α. 2 Appendix A To better understand the block diagonalization theorem, we describe an alternative proof of the sufficiency part of Theorem 1.3, which is based on the discussions in Section 4. The idea is to replace the normal frame given by Lemma 3.3 by the orthonormal frame given in Theorem 4.2. If we can verify Proposition 3.4 and Remark 3.5 for this orthonormal frame, then all the arguments in the proof of Theorem 1.3 remain valid and we have the same conclusion. By the proof of Theorem 4.5, the matrix Φ of the connection-preserving bundle map relative to the orthonormal frame s in Theorem 4.2 is constant and lies in the commutant of M(N , ⊗M ). We write s = {s1 , . . . , sn } for n C ∞ sections s1 , . . . , sn . To verify Proposition 3.4, we only need to check that for any z, w in , the matrix (si (z), sj (w) ) lies in M(N , ⊗M ). Note that we cannot apply Lemma 3.1 for the non-holomorphic frame s. Without loss of generality, we assume r = 2, m1 = 2, m2 = 1 as in the proof of Theorem 4.4 so that the bundle E(T ) has the decomposition E(T ) = E11 ⊕ E12 ⊕ E21 . We need to show that the matrix (si (z), sj (w) ) is of the form ⎛

A1 ⎝ 0 0

0 A1 0

⎞ 0 0 ⎠. A2

Write {si } = {μi } ∪ {ηi } ∪ {νi } where {μi }, {ηi } and {νi } are orthonormal frames for E11 , E12 and E21 respectively. Take arbitrary sections f1 , f1 and f3 of E11 , E12 and E21 ; respectively. Since f1 (z) ∈ H11 , f2 (w) ∈ H12 and H11 and H12 are mutually orthogonal reducing subspaces, f1 (z), f2 (w) = 0. Similarly, we have f1 (z), f3 (w) = 0 and f2 (z), f3 (w) = 0. This implies that (si (z), sj (w) ) is of the form ⎛

A1 ⎝ 0 0

0 A2 0

⎞ 0 0 ⎠. A3

We claim that A1 = A2 , which gives the desired form. In fact, by Theorem 4.4, the bundle map defined by sending {μi } to {ηi } is induced by a unitary operator from H11 to H12 , and thus


1943

μi (z), μj (w) = ηi (z), ηj (w)

as desired. Since the frame is orthonormal and the commutant of M(N , ⊗M ) is a self-adjoint algebra, Remark 3.5 is trivial for this frame. 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] M.J. Cowen, R.G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978) 187–261. [3] M.J. Cowen, R.G. Douglas, Equivalence of connections, Adv. Math. 56 (1985) 39–61. [4] R.G. Douglas, S. Sun, D. Zheng, Multiplication operators on the Bergman space via analytic continuation, arXiv:0901.3787v1. [5] M. Englis, Density of algebras generated by Toeplitz operator on Bergman spaces, Ark. Mat. 30 (1992) 227–243. [6] K. Guo, Operator theory and von Neumann algebras, preprint. [7] K. Guo, H. Huang, On multiplication operators of the Bergman space: Similarity, unitary equivalence and reducing subspaces, J. Operator Theory, in press. [8] K. Guo, H. Huang, Multiplication operators defined by covering maps on the Bergman space: The connection between operator theory and von Neumann algebras, J. Funct. Anal. 260 (4) (2011) 1219–1255. [9] K. Guo, S. Sun, D. Zheng, C. Zhong, Multiplication operators on the Bergman space via the Hardy space of the bidisk, J. Reine Angew. Math. 628 (2009) 129–168. [10] J. Hu, S. Sun, X. Xu, D. Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space, Integral Equations Operator Theory 49 (2004) 387–395. [11] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton University Press and Iwanami Shoten, 1987. [12] C. Jiang, K. Ji, Similarity classification of holomorphic curves, Adv. Math. 215 (2007) 446–468. [13] R. Wells, Differential Analysis on Complex Manifolds, Springer, New York, 1973. [14] K. Zhu, Reducing subspaces for a class of multiplication operators, J. Lond. Math. Soc. 62 (2000) 553–568.


Continuity of magnetic Weyl calculus Ingrid Belti¸ta˘ , Daniel Belti¸ta˘ ∗ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, Bucharest, Romania Received 11 June 2010; accepted 4 January 2011

Communicated by Gilles Godefroy

Abstract We investigate continuity properties of the operators obtained by the magnetic Weyl calculus on nilpotent Lie groups, using modulation spaces associated with unitary representations of certain infinite-dimensional Lie groups. © 2011 Elsevier Inc. All rights reserved. Keywords: Weyl calculus; Magnetic field; Lie group; Modulation spaces

1. Introduction There are three main themes that occur in the present paper: – The pseudo-differential Weyl calculus that takes into account a magnetic field on Rn , which has been recently developed by techniques of hard analysis, with motivation coming from quantum mechanics; some references in this connection include [24,22,25]. – The modulation spaces from the time-frequency analysis, which have become an increasingly useful tool in the classical pseudo-differential calculus on Rn ; see for instance the seminal papers [11] and [19]. – The theory of locally convex Lie groups and their representations, recently surveyed in [28]. See also [29]. * Corresponding author.

E-mail addresses: [email protected] (I. Belti¸ta˘ ), [email protected] (D. Belti¸ta˘ ). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.004

I. Belti¸ta˘ , D. Belti¸ta˘ / Journal of Functional Analysis 260 (2011) 1944–1968

1945

There is a huge literature devoted to various aspects of magnetic fields. From the point of view of the present paper, it is relevant to mention that some spectral properties of Schrödinger operators with magnetic fields were established by using representation theory for nilpotent Lie groups; see for instance [23,21,26]. However, the magnetic Weyl calculus has been rather recently developed. It gives a functional calculus for the operators of position and magnetic momentum in just the same way in which the classical Weyl calculus is an operator calculus for the positions and momenta, and its key feature is that it is gauge covariant. It follows by our earlier papers [1] and [3] that some of the very basic ideas of infinite-dimensional Lie theory prove to be very useful for understanding the aforementioned magnetic Weyl calculus as a Weyl quantization of a certain coadjoint orbit of a semi-direct product group M = F Rn . Here F is a suitable translation-invariant space of smooth functions on Rn and the coadjoint orbit is associated with a natural unitary representation of M on L2 (Rn ). This representation theoretic approach to the magnetic Weyl calculus is further developed in the present paper by using the second of the themes mentioned above. Specifically, we introduce appropriate versions of modulation spaces and use them for describing the continuity properties of the magnetic pseudo-differential operators. We recall from [1] that our approach to the magnetic Weyl calculus actually allows us to extend the constructions of [24] from the abelian group (Rn , +) to any simply connected nilpotent Lie group, and this will also be the setting of some of the main results of the present paper. However, the proofs are greatly helped by a more general framework that we develop, in the first sections of the paper, for the so-called localized Weyl calculus for representations of locally convex Lie groups that satisfy suitable smoothness conditions. In order to develop this abstract setting we provide infinite-dimensional extensions of some ideas and constructions related to irreducible representations of finite-dimensional nilpotent Lie groups, which we had developed in [2]. These extensions may also be interesting on their own, however their importance consists in pointing out that the magnetic Weyl calculus of [24] and the Weyl–Pedersen calculus initiated in [30] are merely different shapes of the same phenomenon. We now briefly present the structure of the paper. The aim of Sections 2 and 3 is to give general conditions on representations of locally convex Lie groups that ensure good properties of a Weyl calculus and related objects, as Wigner distributions and modulation spaces. In fact, in this way we set up a rather general and systematic procedure for constructing spaces of symbols associated with a group representation and eventually proving continuity of the operators obtained by the Weyl calculus, and of the Weyl calculus itself. The main technical result of the paper could thus be considered the continuity property of the cross-Wigner distributions (Theorem 3.16). A special case of this procedure, that motivated the present paper, appeared in our earlier work [2] on Weyl–Pedersen calculus for irreducible representations of finite-dimensional nilpotent Lie groups. The developments in this paper allow us to treat the magnetic Weyl calculus as a particular case. In Section 4 we show that the conditions in Sections 2 and 3 are met in this case, and continuity/trace-class results are thus derived. 1.1. Notation Throughout the paper we denote by S(V) the Schwartz space on a finite-dimensional real vector space V. That is, S(V) is the set of all smooth functions that decay faster than any polynomial together with their partial derivatives of arbitrary order. Its topological dual—the space ∞ (V) for the space of tempered distributions on V—is denoted by S (V). We use the notation Cpol of smooth functions that grow polynomially together with their partial derivatives of arbitrary

1946


order; the natural locally convex topology of this function space along with some of its special properties are discussed in [31]. For every complex vector space Y we denote by Y the complex vector space defined by the conditions that Y and Y have the same underlying real vector space, and the identity mapping Y → Y is antilinear. If Y is a topological vector space, then Y will always denote the weak topological dual of Y, that is, the space of continuous linear functionals on Y endowed with the topology of uniform convergence on the compact subsets. · the completed projective tensor product of locally convex We shall always denote by · ⊗ ¯ · the natural tensor product of Hilbert spaces. Our references for topological spaces and by · ⊗ tensor products are [10,32,35]. We shall also use the convention that the Lie groups are denoted by upper case Latin letters and the Lie algebras are denoted by the corresponding lower case Gothic letters. 2. Smooth unitary representations of locally convex Lie groups Let M be a locally convex Lie group with a smooth exponential mapping expM : L(M) = m → M (see [28]). Assume that π : M → B(H) is a unitary representation. We denote by H∞ the space of smooth vectors for the representation π , that is, H∞ := φ ∈ H π(·)φ ∈ C ∞ (M, H) . We note that π(M)H∞ = H∞ and, as proved in [27, Sect. IV], the derived representation dπ : m → End(H∞ ) is well defined and is given by (∀X ∈ m)(∀φ ∈ H∞ )

d dπ(X)φ = π expM (tX) φ. dt t=0

Remark 2.1. If we denote by U(mC ) the universal enveloping algebra of the complexified Lie algebra mC , then the homomorphism of Lie algebras dπ extends to a unique homomorphism of unital associative algebras dπ : U(mC ) → End(H∞ ). The space of smooth vectors H∞ will always be considered endowed with the locally convex topology defined by the family of seminorms {pu }u∈U(mC ) , where for every u ∈ U(mC ) we define pu : H∞ → [0, ∞),

pu (φ) = dπ(u)φ .

The inclusion mapping H∞ → H is continuous and, for all u ∈ U(mC ) and m ∈ M, the linear operators dπ(u) : H∞ → H∞ and π(m) : H∞ → H∞ are continuous as well. Definition 2.2. Assume the above setting. If the linear subspace of smooth vectors H∞ is dense in H, then the unitary representation π : M → B(H) is said to be smooth. If this is the case, then π is necessarily continuous, in the sense that the group action M × H → H, (m, f ) → π(m)f , is continuous. The representation π is said to be nuclearly smooth if the following conditions are satisfied:


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(1) π is a smooth representation; (2) H∞ is a nuclear Fréchet space; (3) both mappings M × H∞ → H∞ , (m, φ) → π(m)φ, and m × H∞ → H∞ , (X, φ) → dπ(X)φ are continuous. Let B(H)∞ be the space of smooth vectors for the unitary representation π ⊗ π¯ : M × M → B S2 (H) ,

−1 (π ⊗ π)(m ¯ 1 , m2 )T = π(m1 )T π(m2 ) .

We shall say that the representation π : M → B(H) is twice nuclearly smooth if it satisfies the following conditions: (1) The representation π is nuclearly smooth. (2) There exists the commutative diagram H∞ H∞ ⊗

¯ H H⊗

B(H)∞

S2 (H)

(2.1)

where the vertical arrow on the left is a linear topological isomorphism, while the vertical arrow on the right is the natural unitary operator defined by (φ1 , φ2 ) → φ1 ⊗ φ¯ 2 := (· | φ2 )φ1 . Remark 2.3. Note that there can exist at most one Fréchet topology on H∞ such that the inclusion H∞ → H be continuous, as a direct consequence of the closed graph theorem. Remark 2.4. Let π be a smooth representation and denote by H−∞ the strong dual of H∞ . Equivalently, H−∞ can be described as the space of continuous antilinear functionals on H∞ endowed with the topology of uniform convergence on the bounded subsets of H∞ . Then there exist the dense embeddings H∞ → H → H−∞ , and the duality pairing (· | ·) : H−∞ × H∞ → C extends the scalar product of H. Proposition 2.5. If the unitary representation π : M → B(H) is twice nuclearly smooth, then it also has the following properties: (1) The representation π ⊗ π¯ : M × M → B(S2 (H)) is nuclearly smooth. (2) We have L(H−∞ , H∞ ) B(H)∞ → S1 (H) and there exists the commutative diagram B(H)

L(H∞ , H−∞ )

S1 (H)

L(H−∞ , H∞ )

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where the vertical arrow on the left is the natural linear topological isomorphism defined by the trace duality, and the vertical arrow on the right is also a linear topological isomorphism. Proof. (1) The representation π is twice nuclearly smooth, hence H∞ is a nuclear Fréchet space H∞ B(H)∞ . Then B(H)∞ is in turn a nuclear Fréchet space (see for instance [35, and H∞ ⊗ Props. 50.1 and 50.6]). Moreover, since H∞ is dense in H, it follows that B(H)∞ is dense in S2 (H). To complete the proof of the fact that π ⊗ π¯ is twice nuclearly smooth, we still have to check that the mappings M × M × B(H)∞ → B(H)∞ ,

(m1 , m2 , T ) → π(m1 )T π(m2 )−1

and m × m × B(H)∞ → B(H)∞ ,

(X1 , X2 , T ) → dπ(X1 )T − T dπ(X2 )

H∞ B(H)∞ and both mappings are continuous. To this end use again the fact that H∞ ⊗ M × H∞ → H∞ , (m, φ) → π(m)φ, and m × H∞ → H∞ , (X, φ) → dπ(X)φ are continuous. (2) Since H∞ is a nuclear Fréchet space, we get H∞ B(H)∞ L(H−∞ , H∞ ) = L H∞ , H∞ H∞ ⊗ (see [35, Eq. (50.17)]). Moreover, for every T ∈ B(H)∞ we have T H ⊆ H∞ . Therefore one can prove (as in [4], for instance) that B(H)∞ ⊆ S1 (H). Moreover, by considering the duals of the above topological linear isomorphisms, we get H∞ ) L H∞ , H∞ L(H∞ , H−∞ ) L(H−∞ , H∞ ) (H∞ ⊗ (see [35, Eqs. (50.19) and (50.16)]), and these isomorphisms agree with the isomorphism S1 (H) B(H) in the sense of the commutative diagram in the statement. 2 Remark 2.6. For every f1 , f2 ∈ H we denote by f1 ⊗ f¯2 ∈ B(H) the rank-one operator f → (f | f2 )f1 . If the representation π ⊗ π¯ is twice nuclearly smooth, then for any f1 , f2 ∈ H−∞ we can use Proposition 2.5 to define the continuous antilinear functional f1 ⊗ f¯2 : B(H)∞ → C by (f1 ⊗ f¯2 )(T ) = (f1 | Tf2 ) for every T ∈ B(H)∞ . 2.1. Group square Definition 2.7. The group square of M, denoted by M M, is the semi-direct product defined by the action of M on itself by inner automorphisms. That is, M M is a locally convex Lie group whose underlying manifold is M × M and the group operation is (m1 , m2 )(n1 , n2 ) = m1 n1 , n−1 1 m2 n1 n2 for all m1 , m2 , n1 , n2 ∈ M.


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Lemma 2.8. The following assertions hold: (1) The mapping μ : M M → M × M,

(m1 , m2 ) → (m1 m2 , m1 )

is an isomorphism of Lie groups with tangent map L(μ) : m m → m × m,

(X, Y ) → (X + Y, X).

(2) The Lie group M M has a smooth exponential map expMM : m m → M M,

(X, Y ) → expM X, expM (−X) expM (X + Y ) .

Proof. The arguments of Ex. 2.3 in [2] carry over to the present setting.

2

Definition 2.9. We introduce the continuous unitary representation π : M M → B S2 (H) ,

π (m1 , m2 )T = π(m1 m2 )T π(m1 )−1 .

To see that π is a representation, one can use a direct computation or the fact that so is π ⊗ π¯ and we have π = (π ⊗ π¯ ) ◦ μ,

(2.2)

where μ : M M → M × M is the group isomorphism of Lemma 2.8. 3. Localized Weyl calculus and modulation spaces The localized Weyl calculus (see Definition 3.10 below) was introduced in [1] as a tool for dealing with the magnetic Weyl calculus on nilpotent Lie groups. In the present section we further develop that circle of ideas by introducing the modulation spaces and extending some related techniques of [2] to the general framework provided by the localized Weyl calculus for representations of infinite-dimensional Lie groups. Here we single out fairly general conditions that allow for a Weyl calculus to be defined, modulation spaces to be considered and continuity properties in these spaces to hold as in the classical time-frequency analysis; see [11,17,18] and the references therein. All of these conditions are satisfied in at least two important situations: the Weyl–Pedersen calculus for irreducible representations of finite-dimensional nilpotent Lie groups (see [2]) and the magnetic Weyl calculus of [1] to be treated in the last section. 3.1. Ambiguity functions and Wigner distributions Setting 3.1. Throughout this section we keep the following notation: (1) M is a locally convex Lie group (see [28]) with a smooth exponential mapping expM : L(M) = m → M.

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(2) π : M → B(H) is a nuclearly smooth unitary representation. (3) Ξ and Ξ ∗ are real finite-dimensional vector spaces with a duality pairing ·,· : Ξ ∗ ×Ξ → R and with Lebesgue measures on Ξ and Ξ ∗ suitably normalized for the Fourier transform · : L (Ξ ) → L Ξ ∗ , ∞

1

b(·) → b(·) =

e−i ·,x b(x) dx

Ξ

to give a unitary operator L2 (Ξ ) → L2 (Ξ ∗ ). The inverse of this transform will be denoted by a → a. ˇ Definition 3.2. Let θ : Ξ → m be a linear mapping. (a) Orthogonality relations. If either φ ∈ H∞ and f ∈ H−∞ , or φ, f ∈ H, then we define the ambiguity function along the mapping θ , Aπ,θ φ f : Ξ → C,

π,θ Aφ f (·) = f π expM θ (·) φ .

Note that this is a continuous function on Ξ . We say that the representation π satisfies the orthogonality relations along the mapping θ if π,θ π,θ Aφ1 f1 Aφ2 f2 L2 (Ξ ) = (f1 | f2 )H · (φ2 | φ1 )H

(3.1)

2 for arbitrary φ1 , φ2 , f1 , f2 ∈ H. In particular, Aπ,θ φ f ∈ L (Ξ ) for all φ, f ∈ H. (b) Modulation spaces. Consider any direct sum decomposition Ξ = Ξ1 Ξ2 and r, s ∈ [1, ∞]. For arbitrary f ∈ H−∞ define

f M r,s (π,θ) =

π,θ A f (X1 , X2 )r dX1 φ

φ

Ξ2

s/r

1/s dX2

∈ [0, ∞]

Ξ1

with the usual conventions if r or s is infinite. The space Mφr,s (π, θ ) := f ∈ H−∞ f M r,s (π,θ) < ∞ φ

is called a modulation space for the unitary representation π : M → B(H) with respect to the linear mapping θ : Ξ → m, the decomposition Ξ Ξ1 × Ξ2 , and the window vector φ ∈ H∞ \ {0}. In connection with the above definition, we note that more general “co-orbit spaces” Xφ (π, θ ) can be defined in H−∞ by using any Banach space X of functions on Ξ instead of the mixednorm Lebesgue spaces Lr,s (Ξ1 × Ξ2 ). More specifically, one can define for any window vector φ ∈ H∞ , Xφ (π, θ ) = f ∈ H−∞ Aπ,θ φ f ∈X . A systematic investigation of these spaces can be done in a broader context (see [5]). However, the modulation spaces Mφr,s (π, θ ) introduced in Definition 3.2 above will suffice for the purposes


1951

of the present paper. See [12–14] for these constructions in the case of representations of locally compact groups. There could be two sources for the intuition underlying the direct sum decomposition Ξ = Ξ1 + Ξ2 : Firstly, the spaces of symbols are associated to a representation of the group G G, which gives rise to such a decomposition with Ξ1 = Ξ2 is a linear subspace of the Lie algebra of G; this is the case in both examples in the paper. Secondly, there is the case of the modulation spaces of functions on which the operators act, and which are defined in terms of the representation of the group G. In this case the decomposition Ξ = Ξ1 + Ξ2 corresponds to canonical coordinates for the symplectic structure on the coadjoint orbit associated with the representation. This is the phase space decomposition, on which we did not focus in the present paper; see however [1] for some more details on the coadjoint orbits relevant for the magnetic case. Another natural question concerns the independence of the modulation spaces on the choice of a window vector. We have discussed this issue in [2, Subsect. 3.1] for square-integrable representations of nilpotent Lie groups, which covers the case of time-frequency analysis. So far we have no such result in any other case. Remark 3.3. If the representation π satisfies the orthogonality relations along the linear mapping θ : Ξ → m, then for any decomposition Ξ = Ξ1 Ξ2 and any choice of the window vector φ ∈ H∞ \ {0}, we have Mφ2,2 (π, θ ) = H. Remark 3.4. Let V : H → H1 be a unitary operator and consider the unitary representation π1 : M → B(H1 ) such that V π(m) = π1 (m)V for every m ∈ M. Denote by H1,∞ the space of smooth vectors for π1 and let H1,−∞ be the strong dual of H1,∞ . Then there exist the linear topological isomorphisms V |H∞ : H∞ → H1,∞ and V−∞ : H−∞ → H1,−∞ , where V−∞ f = f ◦ V ∗ |H1,∞ for every f ∈ H−∞ . It is easy to check that for every linear mapping θ : Ξ → m π1 ,θ and arbitrary φ ∈ H∞ and f ∈ H−∞ we have Aπ,θ φ f = AV φ (V−∞ f ). Therefore V−∞ naturally gives rise to isometric isomorphisms from the modulation spaces of the representation π onto the corresponding modulation spaces of the representation π1 . Definition 3.5 (Growth condition). We say that the representation π satisfies the growth condition along the linear mapping θ : Ξ → m if Aπ,θ φ2 φ1 ∈ S(Ξ ),

for all φ1 , φ2 ∈ H∞ .

(3.2)

Note that (3.2) implies that the sesquilinear map Aπ,θ : H∞ × H∞ → S(Ξ ),

(φ1 , φ2 ) → Aπ,θ φ2 φ1

is separately continuous as a straightforward application of the closed graph theorem, and then it is jointly continuous by [32, Cor. 1 to Th. 5.1 in Ch. III]. If the representation π satisfies the orthogonality relations along the mapping θ , and φ, f ∈ H, 2 2 ∗ then Aπ,θ φ f ∈ L (Ξ ), hence we can define the cross-Wigner distribution W(f, φ) ∈ L (Ξ ) by φ) := Aπ,θ f . the condition W(f, φ

Definition 3.6 (Density condition). The representation π is said to satisfy the density condition 2 along the linear mapping θ : Ξ → m if {Aπ,θ φ f | φ, f ∈ H} is a total subset of L (Ξ ), in the sense that it spans a dense linear subspace.

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Remark 3.7. If the representation π satisfies the orthogonality relations along θ , then it follows 2 in particular that {Aπ,θ φ f | φ, f ∈ H} ⊆ L (Ξ ), however it is not clear in general that this subset of L2 (Ξ ) is total. Similarly, if π satisfies the growth condition along θ , then {Aπ,θ φ f | φ, f ∈ 2 2 H∞ } ⊆ S(Ξ ) ⊆ L (Ξ ), however in this way we may not get a total subset of L (Ξ ). Lemma 3.8. If the representation π satisfies the orthogonality relations along the linear mapping θ : Ξ → m, then the following assertions hold: (1) The representation π ⊗ π¯ satisfies the orthogonality relations along the linear mapping θ × θ : Ξ × Ξ → m × m. (2) The representation π satisfies the orthogonality relations along each of the linear mappings L(μ)−1 ◦ (θ × θ ) : Ξ × Ξ → m m and θ × θ : Ξ × Ξ → m m. Proof. To see that assertion (1) holds, first prove the orthogonality relations for rank-one operators in S2 (H), then extend them by sesquilinearity to the finite-rank operators, and eventually extend them by continuity to arbitrary Hilbert–Schmidt operators. Then assertion (2) on L(μ) ◦ (θ × θ ) follows by assertion (1) along with Eq. (2.2). Then, to see that also the representation π satisfies the orthogonality relations along the mapping θ × θ : Ξ × Ξ → m m, just note that L(μ)−1 ◦ (θ × θ ) (X, Y ) = θ (Y ), θ (X) − θ (Y ) = (θ × θ )(Y, X − Y ) and the linear mapping Ξ × Ξ → Ξ × Ξ , (X, Y ) → (Y, X − Y ), has the Jacobian identically equal to 1. 2 Lemma 3.9. If the representation π satisfies the growth condition along the linear mapping θ : Ξ → m, then the following assertions hold: (1) The representation π ⊗ π¯ satisfies the growth condition along the linear mapping θ ×θ : Ξ × Ξ → m × m. (2) The representation π satisfies the growth condition along each of the linear mappings L(μ)−1 ◦ (θ × θ ) : Ξ × Ξ → m m and θ × θ : Ξ × Ξ → m m. Proof. The growth condition for the representation π along θ implies that the bilinear map Aπ,θ : H∞ × H∞ → S(Ξ ) is continuous, hence extends to a continuous linear map H∞ → S(Ξ ). Aπ,θ : H∞ ⊗ By complex conjugation we also have H∞ = H∞ ⊗ H∞ → S(Ξ ). Aπ,θ : H∞ ⊗ Thus we get the continuous mapping S(Ξ ) = S(Ξ × Ξ ). Aπ,θ : H∞ ⊗ H∞ ⊗ H∞ ⊗ H∞ → S(Ξ ) ⊗ Aπ,θ ⊗


1953

By composing this with the permutation (f1 , φ1 , f2 , φ2 ) → (f1 , f2 , φ1 , φ2 ) and using the iso H∞ B(H)∞ , we get a continuous operator B(H)∞ ⊗ B(H)∞ → S(Ξ × Ξ ) morphism H∞ ⊗ ¯ , since which extends Aπ⊗π,θ×θ π,θ×θ ¯ π,θ Aπ⊗ (f1 ⊗ f¯2 ) = Aπ,θ φ1 f1 ⊗ Aφ2 f2 . φ ⊗φ¯ 1

2

The second part in the growth condition can be checked similarly, by using that H−∞ is nuclear, H−∞ (H∞ ⊗ H∞ ) B(H)∞ . like H∞ (see [32, Ch. IV, Th. 9.6]), and noting that H−∞ ⊗ Assertion (2) on L(μ) ◦ (θ × θ ) follows by assertion (1) along with Eq. (2.2). Then, to see that also the representation π satisfies the growth condition along θ × θ , just note that L(μ)−1 ◦ (θ × θ ) (X, Y ) = θ (Y ), θ (X) − θ (Y ) = (θ × θ )(Y, X − Y ) and the linear mapping Ξ × Ξ → Ξ × Ξ , (X, Y ) → (Y, X − Y ), is invertible.

2

3.2. Localized Weyl calculus and its continuity properties Definition 3.10. Let θ : Ξ → m be a linear mapping. 1 (Ξ ) → B(H) given by The localized Weyl calculus for π along θ is the mapping Opθ : L Op (a) = θ

a(X)π ˇ expM θ (X) dX

(3.3)

Ξ 1 (Ξ ) where we use weakly convergent integrals. for a ∈ L The localized Weyl calculus for π along θ is said to be regular if

• π satisfies the growth condition along the mapping θ , • π is twice nuclearly smooth, and • Opθ (a) ∈ B(H)∞ whenever a ∈ S(Ξ ∗ ). Note that the closed graph theorem then implies that Opθ : S(Ξ ∗ ) → B(H)∞ is a continuous linear mapping. If the representation π satisfies the growth condition along the mapping θ , then one can think of (3.3) in the distributional sense in order to define the localized Weyl calculus Opθ : S (Ξ ∗ ) → L(H∞ , H−∞ ). More specifically, for every a ∈ S (Ξ ∗ ) and φ, ψ ∈ H∞ we have θ

ˇ Aπ,θ Op (a)φ ψ = a, φ ψ

(3.4)

where ·,· : S (Ξ ) × S(Ξ ) → C is the usual duality pairing. Remark 3.11. If the localized Weyl calculus for π along θ is regular and moreover defines a linear topological isomorphism Opθ : S(Ξ ∗ ) → B(H)∞ (see Proposition 3.12 for sufficient conditions), then we also have the linear topological isomorphism Opθ : S (Ξ ∗ ) → L(H∞ , H−∞ )

1954


by Proposition 2.5(2). Therefore, by using Remark 2.6, we see that there exist the sesquilinear mappings Aπ,θ : H−∞ × H−∞ → S (Ξ )

and W : H−∞ × H−∞ → S Ξ ∗

(3.5)

such that Opθ W(f1 , f2 ) = f1 ⊗ f¯2 π,θ and W(f 1 , f2 ) = Af2 f1 for all f1 , f2 ∈ H−∞ . In addition, it follows by (3.4) and the definition of the Fourier transform for tempered distributions that for every a ∈ S (Ξ ∗ ) and φ, ψ ∈ H∞ we have

Opθ (a)φ ψ = a W(ψ, φ) .

(3.6)

If moreover the representation π satisfies the orthogonality relations along the linear mapping θ , then it follows by Proposition 3.12 below that the mappings (3.5) agree with the ambiguity functions and the cross-Wigner distributions (see Definition 3.5). Proposition 3.12. If π satisfies the orthogonality relations along the linear mapping θ : Ξ → m, then the following assertions are equivalent: (1) The representation π satisfies the density condition along θ . (2) There exists a unique unitary operator Opθ : L2 (Ξ ∗ ) → S2 (H) which agrees with the localized Weyl calculus for π along θ . If these assertions hold true, then we have (∀f, φ ∈ H)

¯ Opθ W(f, φ) = f ⊗ φ.

(3.7)

If moreover the localized Weyl calculus for π along θ is regular, then the mapping Opθ : S(Ξ ∗ ) → B(H)∞ is a linear topological isomorphism. Proof. We begin with some general remarks. Since we have a unitary Fourier transform L2 (Ξ ) → L2 (Ξ ∗ ), it follows by the orthogonality relations along with (3.4) that for arbitrary f, φ ∈ H we have Opθ W(f, φ) = f ⊗ φ¯

¯ S (H ) . and W(f, φ)L2 (Ξ ∗ ) = f · φ = f ⊗ φ 2

(3.8)

Moreover, span {f ⊗ φ¯ | f, φ ∈ H} is dense in S2 (H). We now come back to the proof.

(3.9)


1955

“(1) ⇒ (2)” Let π satisfy the density condition along θ . Since the Fourier transform L2 (Ξ ) → L2 (Ξ ∗ ) is unitary, it follows that span({W(f, φ) | f, φ ∈ H}) is a dense linear subspace of L2 (Ξ ∗ ). Therefore, by using (3.8) and (3.9), we see that Opθ uniquely extends to a unitary operator L2 (Ξ ∗ ) → S2 (H). “(2) ⇒ (1)” If the operator Opθ : L2 (Ξ ∗ ) → S2 (H) is unitary, then it follows by (3.8) and (3.9) that span({W(f, φ) | f, φ ∈ H}) is a dense linear subspace of L2 (Ξ ∗ ). Then, by using again the fact that the Fourier transform L2 (Ξ ) → L2 (Ξ ∗ ) is unitary, we can see that span({Aπ,θ φ f | 2 f, φ ∈ H}) is a dense linear subspace of L (Ξ ), that is, π satisfies the density condition along θ . Now assume that the assertions (1) and (2) in the statement are satisfied and the localized Weyl calculus for π along θ is regular. Then π satisfies the growth condition along θ , hence the ambiguity function defines a continuous sesquilinear mapping Aπ,θ : H∞ × H∞ → S(Ξ ) (see Definition 3.5). Since the Fourier transform is a linear topological isomorphism S(Ξ ) → S(Ξ ∗ ), the cross-Wigner distributions also define a continuous sesquilinear mapping W : H∞ × H∞ → S(Ξ ∗ ). H∞ → S(Ξ ∗ ), which further induces a continuous linear mapping W : H∞ ⊗ On the other hand, the condition that the localized Weyl calculus for π along θ is regular (see Definition 3.10) includes the assumption that the representation π is twice nuclearly smooth, H∞ B(H)∞ . hence we have a topological linear isomorphism H∞ ⊗ We thus eventually get a continuous linear mapping W : B(H)∞ → S(Ξ ∗ ) which, by (3.8), has the property Opθ ◦ W = id on B(H)∞ . In other words, W = (Opθ )−1 |B(H)∞ . Thus the unitary operator Opθ : L2 (Ξ ∗ ) → S2 (H) restricts to a continuous linear map S(Ξ ∗ ) → B(H)∞ (since the localized Weyl calculus for π along θ is regular), while its inverse (Opθ )−1 restricts to a continuous linear map W : B(H)∞ → S(Ξ ∗ ). It then follows that Opθ : S(Ξ ∗ ) → B(H)∞ is a linear topological isomorphism (whose inverse is W). 2 Definition 3.13. Assume that the localized Weyl calculus for π along the linear mapping θ : Ξ → m is regular and the representation π satisfies both the density condition and the orthogonality relations along θ . It follows by Proposition 3.12 that the localized Weyl calculus Opθ defines a unitary operator L2 (Ξ ∗ ) → S2 (H), and also linear topological isomorphisms S(Ξ ∗ ) → B(H)∞ L(H−∞ , H∞ ) and S (Ξ ∗ ) → L(H∞ , H−∞ ). Hence we can introduce the following notions: (1) If a, b ∈ S (Ξ ∗ ) and the operator product Opθ (a)Opθ (b) ∈ L(H∞ , H−∞ ) is well defined, then Remark 3.11 shows that the Moyal product a#θ b ∈ S (Ξ ∗ ) is uniquely determined by the condition Opθ a#θ b = Opθ (a)Opθ (b). Thus the Moyal product defines bilinear mappings S(Ξ ∗ )×S(Ξ ∗ ) → S(Ξ ∗ ) and L2 (Ξ ∗ )× L2 (Ξ ∗ ) → L2 (Ξ ∗ ). (2) We define the unitary representation π # : M M → B(L2 (Ξ ∗ )) such that for every m ∈ M M there exists the commutative diagram L2 (Ξ ∗ )

π # (m)

Opθ

S2 (H)

L2 (Ξ ∗ ) Opθ

π (m)

S2 (H)

1956


These constructions provide extensions of some notions introduced in [2]. Remark 3.14. In the setting of Definition 3.13 we note the following facts: (1) For every m1 , m2 ∈ M and f ∈ L2 (Ξ ∗ ) we have −1 −1 −1 π # (m1 , m2 )f = Opθ π(m1 m2 ) #θ f #θ Opθ π(m1 ) . (2) For every X1 , X2 ∈ Ξ we have Opθ (ei ·,Xj ) = π(expM (θ (Xj ))) for j = 1, 2, whence by Lemma 2.8(2) π # expMM θ (X1 ), θ (X2 ) f = π # expM θ (X1 ) , expM −θ (X1 ) expM θ (X1 + X2 ) f = ei ·,X1 +X2 #θ f #θ e−i ·,X1 whenever f ∈ L2 (Ξ ∗ ). Proposition 3.15. Assume that the representation π is twice nuclearly smooth. If we have either φ1 , φ2 , f1 , f2 ∈ H, or φ1 , φ2 ∈ H∞ and f1 , f2 ∈ H−∞ , then (∀X, Y ∈ Ξ )

π,θ π ,θ×θ Aφ ⊗φ¯ (f1 ⊗ f¯2 ) (X, Y ) = Aπ,θ φ1 f1 (X + Y ) · Aφ2 f2 (X). 1

2

If moreover the localized Weyl calculus for π along θ is regular and the representation π satisfies both the density condition and the orthogonality relations along θ , then (∀X, Y ∈ Ξ )

π # ,θ×θ π,θ AW (φ1 ,φ2 ) W(f1 , f2 ) (X, Y ) = Aπ,θ φ1 f1 (X + Y ) · Aφ2 f2 (X).

Proof. It follows at once by definition that π,θ×θ ¯ π,θ Aπ⊗ (f1 ⊗ f¯2 ) = Aπ,θ φ1 f1 ⊗ Aφ2 f2 . φ ⊗φ¯ 1

2

On the other hand, we easily get by (2.2) (∀X, Y ∈ Ξ )

π ,θ×θ π¯ ,θ×θ Aφ ⊗φ¯ (f1 ⊗ f¯2 ) (X, Y ) = Aπ⊗ (f1 ⊗ f¯2 ) (X + Y, X). φ ⊗φ¯ 1

2

1

2

For the second part of the statement, just recall that Opθ (W(f1 , f2 )) = f1 ⊗ f¯2 and use Proposition 3.12 along with Remark 3.4. 2 Now we are ready to give one of the main technical results of the present paper. It extends a result in [2] (which is recovered for representations of finite-dimensional nilpotent Lie groups). The general lines of the proof go back to [34] (which is recovered for Heisenberg groups); see also [7].


1957

Theorem 3.16. Let φ1 , φ2 ∈ H∞ \ {0}, and assume the following hypotheses: (1) The representation π satisfies both the density condition and the orthogonality relations along the linear mapping θ : Ξ → m. (2) The localized Weyl calculus for the representation π along θ is regular. Let Ξ = Ξ1 Ξ2 be any direct sum decomposition. If 1 r s ∞ and r1 , r2 , s1 , s2 ∈ [r, s] satisfy the equations r11 + r12 = s11 + s12 = 1r + 1s , then the cross-Wigner distribution defines a continuous sesquilinear map # r,s W(·,·) : Mφr11,s1 (π, θ ) × Mφr22,s2 (π, θ ) → MW (φ1 ,φ2 ) π , θ × θ . Proof. The assertion follows from Proposition 3.15 along the same lines as in the proof of [2, Th. 2.22]. 2 The next corollary records a standard consequence of the continuity of cross-Wigner distributions; see [19,34,18]. Corollary 3.17. Let φ1 , φ2 ∈ H∞ \ {0}, and assume the following hypotheses: (1) The representation π satisfies both the density condition and the orthogonality relations along the linear mapping θ : Ξ → m. (2) The localized Weyl calculus for the representation π along θ is regular. Now let Ξ = Ξ1 Ξ2 be any direct sum decomposition. If r, s, r1 , s1 , r2 , s2 ∈ [1, ∞] satisfy the conditions r s,

r2 , s2 ∈ [r, s],

and

1 1 1 1 1 1 − = − =1− − , r1 r2 s1 s2 r s

r,s # then for every symbol a ∈ MW (φ1 ,φ2 ) (π , θ × θ ) we have a bounded linear operator

Opθ (a) : Mφr11,s1 (π, θ ) → Mφr22,s2 (π, θ ). Moreover, the linear mapping # r1 ,s1 r2 ,s2 r,s Opθ : MW (φ1 ,φ2 ) π , θ × θ → B Mφ1 (π, θ ), Mφ2 (π, θ ) is continuous. Proof. The assertion follows from Theorem 3.16 along the same lines as in the proof of [2, Cor. 2.24]. We just recall that the conditions on the parameters come from Hölder’s inequality and duality theory for the mixed-norm Lebesgue spaces; see [19,34,18] again. 2

1958


Corollary 3.18. Let φ1 , φ2 ∈ H∞ \ {0}, and assume the following hypotheses: (1) The representation π satisfies both the density condition and the orthogonality relations along the linear mapping θ : Ξ → m. (2) The localized Weyl calculus for the representation π along θ is regular. ∞,1 θ θ # Then for every a ∈ MW (φ1 ,φ2 ) (π ) we have Op (a) ∈ B(H), and the linear mapping Op : ∞,1 # MW (φ1 ,φ2 ) (π , θ × θ ) → B(H) is continuous.

Proof. This is the special case of Corollary 3.17 with r1 = s1 = r2 = s2 = 2, r = 1, and s = ∞, (π, θ ) = H for j = 1, 2. 2 since Remark 3.3 shows that Mφ2,2 j ∞,1 # We note that MW (φ1 ,φ2 ) (π ) is precisely Sjöstrand’s algebra introduced in [33] in the case of the Heisenberg groups and their Schrödinger representations; see [2, Sect. 4].

3.3. Trace-class operators obtained by localized Weyl calculus In this subsection we give a standard sufficient condition for a pseudo-differential operator to belong to the trace class. In the special case of the Schrödinger representation of a Heisenberg group, this result goes back to [33]. A proof for this result was also provided in [16], was extended to arbitrary nilpotent Lie groups in [2], and will be adapted below to the present setting. Lemma 3.19. Let the representation π : M → B(H) satisfy the orthogonality relations along the linear mapping θ : Ξ → m, and pick φ0 ∈ H∞ with φ0 = 1. Then the following assertions hold: π,θ 2 (1) The operator Aπ,θ φ0 : H → L (Ξ ), f → Aφ0 f , is an isometry whose image is the reproducing kernel Hilbert space associated with the reproducing kernel

K : Ξ × Ξ → C,

K(X1 , X2 ) = π expM θ (X1 ) φ0 π expM θ (X2 ) φ0 .

The orthogonal projection from L2 (Ξ ) onto Ran Aπ,θ φ0 is just the integral operator defined by the integral kernel K. (2) For every φ, f ∈ H we have

π,θ Aφ0 f (X) · π expM θ (X) φ dX = (φ | φ0 )f.

Ξ

In particular, for every f ∈ H we have

π,θ Aφ0 f (X) · π expM θ (X) φ0 dX = f,

Ξ

where the integral is weakly convergent in H.

(3.10)


1959

Assume that the representation π satisfies the growth condition along θ . Also, assume that for every u ∈ U(mC ) the function dπ(u)π(expM (θ (·)))φ0 has polynomial growth, then moreover we have: (3) If f ∈ H∞ , then the integral in (3.10) is convergent with respect to the topology of H∞ . (4) If f ∈ H−∞ , then (3.10) holds with the integral convergent in the w ∗ -topology. (5) We have H∞ = {f ∈ H−∞ | Aπ,θ φ0 f ∈ S(Ξ )}. Proof. Assertion (1) follows at once by the orthogonality relations along with [15, Prop. 2.12]. Then assertion (2) follows by an application of [15, Prop. 2.11]. The proof for assertions (3)–(5) can be supplied by adapting the method of proof of [2, Cor. 2.9]. We omit the details. 2 Remark 3.20. We note here that in the setting of Lemma 3.19, the condition that for all u ∈ U(mC ) and φ ∈ H∞ the function dπ(AdU(mC ) (expM (θ (·)))u)φ has polynomial growth on Ξ implies that for all f ∈ H−∞ , φ ∈ H∞ , the function Aπ,θ φ f has polynomial growth as well. In fact, if f ∈ H−∞ , then there exists u ∈ U(mC ) such that for every ψ ∈ H∞ we have |(f | ψ)| dπ(u)ψ. (See Remark 2.1.) Then we have π,θ A f (·) = f π exp θ (·) φ dπ(u)π exp θ (·) φ M M φ = dπ AdU(mC ) expM θ (·) u φ and the latter function has polynomial growth by assumption. By using the method of proof of [2, Prop. 2.27] we can now obtain the following sufficient condition for a symbol to give rise to a trace-class operator. Proposition 3.21. Let φ1 , φ2 ∈ H∞ such that φj = 1 and for every u ∈ U(mC ) the function dπ(u)π(expM (θ (·)))φj has polynomial growth, for j = 1, 2, and assume the following hypotheses: (1) The representation π satisfies both the density condition and the orthogonality relations along the linear mapping θ : Ξ → m. (2) The localized Weyl calculus for the representation π along θ is regular. 1,1 θ # Then for every a ∈ MW (φ1 ,φ2 ) (π , θ × θ ) we have Op (a) ∈ S1 (H), and the linear mapping 1,1 # Opθ : MW (φ1 ,φ2 ) (π , θ × θ ) → S1 (H) is continuous.

Proof. It follows by Lemmas 3.8(2), 3.9(2) and Remark 3.4 that the representation π # : M M → B(L2 (Ξ ∗ )) satisfies both the orthogonality relations and the growth condition along the linear mapping θ × θ : Ξ × Ξ → m m. Moreover, it is easily seen that the function Φ0 := W(φ1 , φ2 ) ∈ S(Ξ ∗ ) has the property that for every u ∈ U((m m)C ) the norm of dπ # (u)π # (expMM ((θ × θ )(·)))Φ0 has polynomial growth on Ξ × Ξ , since a similar property has the rank-one operator Opθ (Φ0 ) = (· | φ2 )φ1 ∈ S2 (H) with respect to the representation π , as a direct consequence of the calculation (3.12) below. Therefore we can use Lemma 3.19(4) for the representation π # to see that for arbitrary a ∈ S (Ξ ∗ ) we have

1960


a=

π # ,θ×θ AΦ0 a (X, Y ) · π # expMM θ (X), θ (Y ) Φ0 dX dY,

Ξ ×Ξ

whence by (3.6) we get π # ,θ×θ AΦ0 a (X, Y ) · Opθ π # expMM θ (X), θ (Y ) Φ0 dX dY Opπ (a) =

(3.11)

Ξ ×Ξ

where the latter integral is weakly convergent in L(H∞ , H−∞ ) ( L(H−∞ , H∞ ) by Proposition 2.5(2)). On the other hand, for arbitrary X, Y ∈ Ξ we get by Remarks 3.14 and 3.11 Opθ π # expMM θ (X), θ (Y ) Φ0 −1 = π expM θ (X) + θ (Y ) ◦ Opθ (Φ0 ) ◦ π expM θ (X) = · π expM θ (X) φ2 π expM θ (X + Y ) φ1 .

(3.12)

In particular, Opθ (π # (expMM (θ (X), θ (Y )))Φ0 ) ∈ S1 (H) and θ # Op π exp MM θ (X), θ (Y ) Φ0 1 = π expM θ (X + Y ) φ1 · π expM θ (X) φ2 = 1. It then follows that the integral in (3.11) is absolutely convergent in S1 (H) for every symbol 1,1 # (π , θ × θ ) and moreover we have a ∈ MΦ 0 θ Op (a) 1

π # ,θ×θ A a (X, Y ) dX dY = aM 1,1 (π # ,θ×θ) Φ0 Φ

Ξ ×Ξ

which concludes the proof.

2

4. Applications to the magnetic Weyl calculus We proved in [1] that the magnetic Weyl calculus on Rn constructed in [24] can be alternatively described as the localized Weyl calculus for a suitable representation. This point of view actually allowed us to construct magnetic Weyl calculi on any simply connected nilpotent Lie group G, by using an appropriate representation π : M = F G → B(L2 (G)) and linear mappings θ A : g × g∗ → m. We shall see in the present section that all of the conditions studied in Sections 2 and 3 are met by π and θ A (see Corollary 4.7 below), provided the coefficients of the magnetic potential A ∈ Ω 1 (G) have polynomial growth. Therefore, the abstract results of the previous sections can be used for obtaining continuity and nuclearity properties for the magnetic Weyl calculus (see Corollaries 4.8–4.10 below). Notation 4.1. For any Lie group G we denote by λ : G → End(C ∞ (G)), g → λg , the left regular representation defined by (λg φ)(x) = φ(g −1 x) for every x, g ∈ G and φ ∈ C ∞ (G). Moreover, we denote by 1 the constant function which is identically equal to 1 on G. (This should not be confused with the unit element of G, which is denoted in the same way.)


1961

We now recall the following notion from [1]. Definition 4.2. Let G be a finite-dimensional Lie group. A linear space F of real functions on G is said to be admissible if it is endowed with a sequentially complete, locally convex topology and satisfies the following conditions: (1) The linear space F is invariant under the representation of G by left translations, that is, if φ ∈ F and g ∈ G then λg φ ∈ F . (2) We have a continuous inclusion mapping F → C ∞ (G). (3) The mapping G × F → F , (g, φ) → λg φ is smooth. For every φ ∈ F we denote by ˙ λ(·)φ : g → F the differential of the mapping g → λg φ at the point 1 ∈ G. (4) For every g1 , g2 ∈ G with g1 = g2 there exists φ ∈ F with φ(g1 ) = φ(g2 ). (5) We have {φg | φ ∈ F } = Tg∗ G for every g ∈ G. For instance, the function space CR∞ (G) is admissible. Proposition 4.3. Let G be a finite-dimensional simply connected nilpotent Lie group with the inverse of the exponential map denoted by logG : G → g. If we define FG := spanR λg (ξ ◦ logG ) ξ ∈ g∗ , g ∈ G ,

(4.1)

then the following assertions hold: (1) FG is a finite-dimensional linear subspace of C ∞ (G) which is invariant under the left regular representation and contains the constant functions. (2) The semi-direct product M0 := FG λ G is a finite-dimensional simply connected nilpotent Lie group. Proof. Since G is a simply connected nilpotent Lie group, we may assume that G = (g, ∗). (1) It is clear that the linear space FG is invariant under the left regular representation. On the other hand, for every V , X ∈ g and ξ ∈ g∗ we have

1 (λV ξ )(X) = ξ, (−V ) ∗ X = ξ, −V + X + [−V , X] + · · · . 2 Thus, if we denote by N the nilpotency index of g, then we see that FG consists of polynomial functions on g of degree N , hence dim FG < ∞. Moreover, if z denotes the center of g and we pick V ∈ z and ξ ∈ g∗ , then λV ξ = − ξ, V 1 + ξ . We thus see that the constant functions belong to FG . (2) On the Lie algebra level we have m0 := FG λ˙ g, and both FG and g are nilpotent Lie algebras. Therefore Engel’s theorem shows that, for proving that m0 is nilpotent, it is enough to check that the adjoint action adm0 gives a representation of g on FG by nilpotent endomorphisms. This representation is just λ˙ : g → End(FG ) hence, by the theorem on weight space decompositions for representations of nilpotent Lie algebras (see for instance [6, Th. 2.9]), it suffices to prove ˙ the following fact: If α ∈ g∗ , φ ∈ FG \ {0}, and for every X ∈ g we have λ(X)φ = α(X)φ, then α = 0.

1962


˙ 0 )φ = α(X0 )φ, it follows that for every Y ∈ g and To this end, let X0 ∈ g arbitrary. Since λ(X t ∈ R we have φ((−tX0 )∗Y ) = etα(X0 ) φ(Y ). We have seen above that FG consists of polynomial functions on g of degree N , therefore for every Y ∈ g there exists a constant Cφ,Y > 0 such that N 2 (∀t ∈ R) etα(X0 ) φ(Y ) = φ (−tX0 ) ∗ Y Cφ,Y 1 + |t| . On the other hand, since φ ∈ FG \ {0}, there exists Y ∈ g such that φ(Y ) = 0, and then the above inequality shows that α(X0 ) = 0. This holds for arbitrary X0 ∈ g, hence α = 0, as we wished for. 2 Theorem 4.4. Let G be a finite-dimensional simply connected nilpotent Lie group with an admis∞ (G), sible function space F such that there exist the continuous inclusion maps g∗ → F → Cpol where the embedding g∗ → F is given by ξ → ξ ◦ logG . Denote M = F λ G, fix ∈ R \ {0}, and consider the unitary representation π : M → B(L2 (G)), π(φ, g)f = ei φ λg f for all φ ∈ F , g ∈ G, and f ∈ L2 (G). Then π is a nuclearly smooth representation and its space of smooth vectors is the Schwartz space S(G). Proof. Let us denote H = L2 (G) and let H∞ be the space of smooth vectors for the representation π . We first check that S(G) = H∞ . ∞ (G), it follows at once For proving that S(G) ⊆ H∞ , let f ∈ S(G) arbitrary. Since F → Cpol ∞ that for every φ ∈ F and g ∈ G we have π(φ, ·)f ∈ C (G, H) and π(·, g)f ∈ C ∞ (F , H). It then follows by [27, Sect. I] (see also [20, Th. 3.4.3]) that π(·)f ∈ C ∞ (M, H), hence f ∈ H∞ . To prove the converse inclusion S(G) ⊆ H∞ we need the function space FG defined in (4.1). Since F contains {ξ ◦ logG | ξ ∈ g∗ } and is invariant under the left regular representation of G, we get FG → F . Now Proposition 4.3 shows that M0 := FG G is a finite-dimensional nilpotent Lie group. Since g∗ → FG , it is easily seen that the unitary representation π0 := π|M0 : M0 → B(H) is irreducible. Let H∞,π0 be its space of smooth vectors. If δ1 : C ∞ (G) → C is the Dirac distribution at 1 ∈ G, then the discussion in [1, Subsect. 2.4] shows that FG × {0} is a polarization for the functional (δ1 |FG , 0) ∈ m∗0 , and the corresponding induced representation is just π0 . Now H∞,π0 = S(G) by [9, Cor. to Th. 3.1]. Therefore we get the continuous inclusion H∞ → S(G), which completes the proof for the equality S(G) = H∞ . Furthermore, it easily follows by [8, Cor. A.2.4] that H∞ = S(G) = S(g) as locally convex spaces. On the other hand, it is well known that S(g) is a nuclear Fréchet space; see for instance [35]. Finally, both mappings M × S(G) → S(G), (m, φ) → π(m)φ, and m × S(G) → S(G), (X, φ) → dπ(X)φ are continuous as a direct consequence of [8, Th. A.2.6], and this concludes the proof of the fact that π is a nuclearly smooth representation. 2 We now prove that the conclusion of Theorem 4.4 actually holds under a much stronger form. Corollary 4.5. In the setting of Theorem 4.4, the unitary representation π is twice nuclearly smooth. Proof. The proof has two stages. For the sake of simplicity we assume = 1, however it is clear that the following reasonings carry over to the general case.


1963

1◦ We first make the following remark: For j = 1, 2, let Gj be a finite-dimensional simply connected nilpotent Lie group with an admissible function space Fj such that g∗j → Fj → ∞ (G ) as in Theorem 4.4. Also define the group M = F G and the unitary representation Cpol j j j λ j j −1

πj : Mj → B(L2 (Gj )), πj (φ, g)f = ei(−1) φ λg f for all φ ∈ Fj , g ∈ Gj , and f ∈ L2 (Gj ). Now consider the direct product group G0 := G1 × G2 , the function space ∞ F0 := (F1 ⊗ 1) + (1 ⊗ F2 ) → Cpol (G0 ),

and the representation π0 : M0 → B(L2 (G0 )), π0 (φ, g)f = eiφ λg f for all φ ∈ F0 , g ∈ G0 , and f ∈ L2 (G0 ), where M0 := F0 λ G0 . Then F0 is an admissible function space on G0 and there exists a 1-dimensional central subgroup N ⊆ M1 × M2 such that N ⊆ Ker(π1 ⊗ π2 ), and we have M0 = (M1 × M2 )/N . Moreover, the representation π0 is equal to π1 ⊗ π2 factorized modulo N . In fact, let us define the linear map : F1 × F2 → F0 ,

(φ1 , φ2 ) → φ1 ⊗ 1 − 1 ⊗ φ2 .

Then Ran = F0 and Ker = {(t1, t1) | t ∈ R} R, hence we get a linear isomorphism F0

(F1 × F2 )/ Ker , and this can be used to define the topology of F0 . Moreover, it is clear that Ker is contained in the center of m1 × m2 m0 and Ker ⊆ Ker(d(π1 ⊗ π2 )), hence the above remark holds for N = expM0 (Ker ). 2◦ We now come back to the proof of the corollary. We already know from Theorem 4.4 that the representation π is nuclearly smooth. Moreover, by using the remark of stage 1◦ for G1 = G2 = G along with Theorem 4.4 for the group G × G, we easily see that the space of smooth vectors for the representation π ⊗ π¯ is linear and topologically isomorphic to S(G × G), S(G) (see for instance [35]). On the other hand, S(G) is which in turn is isomorphic to S(G) ⊗ the space of smooth vectors for π , by Theorem 4.4. Thus the representation π also satisfies the second condition in the definition of a twice nuclearly smooth representation (see Definition 2.2), and we are done. 2 Notation 4.6. Let G be any Lie group with the Lie algebra g and with the space of globally defined smooth vector fields (that is, global sections in its tangent bundle) denoted by X(G) and the space of globally defined smooth 1-forms (that is, global sections in its cotangent bundle) denoted by Ω 1 (G). Then there exists a natural bilinear map

·,· : Ω 1 (G) × X(G) → C ∞ (G) defined as usually by evaluations at every point of G. Moreover, for arbitrary g ∈ G, we denote the corresponding right-translation mapping by Rg : G → G, h → hg. Then we define the injective linear mapping ιR : g → X(G) by (ιR X)(g) = (T1 (Rg ))X ∈ Tg G for all g ∈ G and X ∈ g.

1964


Corollary 4.7. Assume the setting of Theorem 4.4. If we have A ∈ Ω 1 (G) such that A, ιR X ∈ F whenever X ∈ g, then we define the linear mapping

(X, ξ ) → ξ ◦ logG + A, ιR X , X .

θ A : g × g∗ → m = F λ˙ g,

Then for every ∈ R \ {0} the representation π : M → B(L2 (G)) has the following properties: (1) The representation π satisfies the orthogonality relations along the mapping θ A . (2) The representation π satisfies the growth condition along θ A . (3) The localized Weyl calculus for π along θ A is regular and defines a unitary operator A Opθ : L2 (g × g∗ ) → S2 (L2 (G)). (4) If u ∈ U(mC ) and φ ∈ S(G), the function dπ(AdU(mC ) (expM (θ A (·)))u)φ has polynomial growth on g × g∗ . Proof. Throughout the proof we assume = 1 and we denote π1 = π for the sake of simplicity. The case of an arbitrary ∈ R \ {0} can be handled by a similar method. Since G is simply connected, we may assume G = (g, ∗). Then the space of smooth vectors for π is equal to S(g) by Theorem 4.4. (1) The assertion follows by [3, Th. 2.8(1)]. (2) To check the growth condition (3.2) we shall denote for every X ∈ g, 1 ΨX : g → g,

ΨX (Y ) =

Y ∗ (sX) ds 0

and also 1

R τA (X, Y ) = exp i A, ι X (−sX) ∗ Y ds 0

for X, Y ∈ g. It then follows by [3, Prop. 2.9(1)] that for every f, φ ∈ S(g) we have π,θ A Aφ f (X, ξ ) =

ei ξ,Y τA X, −ΨX−1 (Y ) f −ΨX−1 (Y ) φ (−X) ∗ −ΨX−1 (Y ) dY.

g A

f : g × g∗ → C is a partial inverse Fourier transform of the function Therefore the function Aπ,θ φ defined on g × g by (X, Y ) → τA X, −ΨX−1 (Y ) f −ΨX−1 (Y ) φ (−X) ∗ −ΨX−1 (Y ) : g → C. On the other hand, it was noted in the proof of [1, Th. 4.4(4)] that each of the mappings Σ1 , Σ2 : g × g → g × g defined by Σ1 (Y, Z) = −Y, Y ∗ (−Z)

and Σ2 (V , W ) = −ΨW (V ), W


1965

is a polynomial diffeomorphism whose inverse is a polynomial. Since Σ2−1 (Y, X) = ΨX−1 (−Y ), X ∞ (g × g), it then easily follows by [8, Lemma A.2.1(a)] that we have a well-defined and τA ∈ Cpol continuous sesquilinear mapping

S(g) × S(g) → S g × g∗ ,

A

(f, φ) → Aπ,θ f. φ

Thus the representation π satisfies the growth condition along the mapping θ A . (3) Use the above assertion (3) along with [1, Th. 4.4(4)]. (4) The assertion follows as a direct consequence of [3, Lemma 2.5]. 2 In the next corollaries we denote by π the representation π in Theorem 4.4 for = 1. Recall that we work with a finite-dimensional simply connected nilpotent Lie group G with an admis∞ (G), sible function space F such that there exist the continuous inclusion maps g∗ → F → Cpol ∗ where the embedding g → F is given by ξ → ξ ◦ logG . Moreover M = F λ G, and the aforementioned unitary representation π : M → B(L2 (G)) is defined by π(φ, g)f = eiφ λg f for all φ ∈ F , g ∈ G, and f ∈ L2 (G). If we have A ∈ Ω 1 (G) such that A, ιR X ∈ F whenever X ∈ g, and we define the linear mapping θ A : g × g∗ → m = F λ˙ g,

(X, ξ ) → ξ ◦ logG + A, ιR X , X

as in Corollary 4.7, then one can consider the modulation spaces of symbols for the localized Weyl calculus for the representation π along the linear mapping θ A . These are just the modulation spaces for the representation π # : M M → B(L2 (g × g∗ )) with respect to the linear mapping (θ A , θ A ) : (g × g∗ ) × (g × g∗ ) → m m. It follows by Remark 3.14 that for arbitrary Φ ∈ S(g × g∗ ) and F ∈ S (g × g∗ ) the corresponding ambiguity function # A A AπΦ ,θ ×θ F : (g × g∗ ) × (g × g∗ ) → C is given by the formula π # ,θ A ×θ A AΦ F (X1 , ξ1 ), (X2 , ξ2 ) = π # expMM θ A (X1 , ξ1 ), θ A (X2 , ξ2 ) F Φ L2 (g×g∗ ) i ·,(X +X ,ξ +ξ ) θ A θ A −i ·,(X ,ξ ) 1 2 1 2 # 1 1 e = F# e Φ(·) g×g∗

where #θ stands for the Moyal product on g × g∗ defined by means of the magnetic potential A. For r, s ∈ [1, ∞] and the window function Φ ∈ S(g × g∗ ) we have the modulation space of symbols A

# A A r,s # A MΦ π , θ × θ A = F ∈ S g × g∗ AπΦ ,θ ×θ F ∈ Lr,s g × g∗ × g × g∗ . Corollary 4.8. In the above setting, pick φ1 , φ2 ∈ S(G) \ {0}. If r, s, r1 , s1 , r2 , s2 ∈ [1, ∞] satisfy the conditions

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r s,

r2 , s2 ∈ [r, s],

and

1 1 1 1 1 1 − = − =1− − , r1 r2 s1 s2 r s

r,s # A A then for every symbol a ∈ MW (φ1 ,φ2 ) (π , θ × θ ) we have a bounded linear operator

A Opθ (a) : Mφr11,s1 π, θ A → Mφr22,s2 π, θ A . Moreover, the linear mapping # A A r,s A Opθ : MW → B Mφr11,s1 π, θ A , Mφr22,s2 π, θ A (φ1 ,φ2 ) π , θ × θ is continuous. Proof. It follows by Theorem 4.4 that the space of smooth vectors for the representation π is the Schwartz space S(G). Moreover, Corollary 4.7 shows that we can apply Corollary 3.17 for the representation π . Now the conclusion follows by using the latter corollary. 2 Corollary 4.9. Assume the setting of Corollary 4.7, let φ1 , φ2 ∈ S(G) \ {0}, and r, s ∈ [1, ∞] r,s θA # A A 2 such that 1r + 1s = 1. Then for every a ∈ MW (φ1 ,φ2 ) (π , θ × θ ) we have Op (a) ∈ B(L (G)). A

r,s # A A 2 Moreover, Opθ : MW (φ1 ,φ2 ) (π , θ × θ ) → B(L (G)) is a continuous linear mapping.

Proof. This is the special case of Corollary 4.8 with r1 = s1 = r2 = s2 = 2, since Remark 3.3 (π, θ A ) = L2 (G) for j = 1, 2. 2 shows that Mφ2,2 j Corollary 4.10. Assume the setting of Corollary 4.7 and let φ1 , φ2 ∈ S(G) \ {0}. Then for 1,1 θ # A A 2 every a ∈ MW (φ1 ,φ2 ) (π , θ × θ ) we have Op (a) ∈ S1 (L (G)), and the linear mapping A

1,1 # A A 2 Opθ : MW (φ1 ,φ2 ) (π , θ × θ ) → S1 (L (G)) is continuous.

Proof. Recall from Theorem 4.4 that the space of smooth vectors for the representation π is the Schwartz space S(G). Moreover, Corollary 4.7 shows that we can use Proposition 3.21, and the conclusion follows. 2 Remark 4.11. In the special case when G is the abelian group (Rn , +) and we have the magnetic potential A ∈ Ω 1 (Rn ), the magnetic Weyl calculus ∗ A Opθ : S Rn × Rn → L S Rn , S Rn is just the one constructed in [24]. In this setting, we note the following: (1) In the case when the coefficients of the magnetic field B := dA ∈ Ω 2 (Rn ) belong to the Fréchet space BC∞ (Rn ) of smooth functions on Rn which are bounded along with all of their partial derivatives, one established in [22] some sufficient conditions on a symbol A a ∈ S (Rn × (Rn )∗ ) that ensure that the magnetic pseudo-differential operator Opθ (a) is bounded on L2 (Rn ). In this connection, we note that the previous Corollary 4.9 provides another type of sufficient conditions for L2 -boundedness when the coefficients of the mag∞ (Rn ) of smooth functions on Rn that grow netic field B belong to the larger LF-space Cpol


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polynomially together with their partial derivatives of arbitrary order. This follows since for ∞ (Rn ), one can construct every closed 2-form B ∈ Ω 2 (Rn ) whose coefficients belong to Cpol ∞ (Rn ) again such that 1 n in the usual way a 1-form A ∈ Ω (R ) whose coefficients belong to Cpol dA = B. (2) It follows by the comments preceding Corollary 4.8 that the modulation spaces of symbols r,s MΦ (π # , θ A × θ A ) can be alternatively described in terms of the modulation mapping which was introduced in [25] in the case of the abelian group G = (Rn , +) by using the magnetic Moyal product #A . It had been already noted in [24] that the magnetic Moyal product on (Rn , +) actually depends only on the magnetic field B = dA. This assertion holds true for the two-step nilpotent Lie groups, as an easy consequence of the formula established in Th. 4.7 in [1]. Acknowledgment We wish to thank the referee for interesting comments and suggestions that helped us to improve the presentation. The second-named author acknowledges partial financial support from Project MTM2010-16679, DGI-FEDER, of the MCYT, Spain, and from the CNCSIS grant PNII – Programme “Idei” (code 1194). References [1] I. Belti¸ta˘ , D. Belti¸ta˘ , Magnetic pseudo-differential Weyl calculus on nilpotent Lie groups, Ann. Global Anal. Geom. 36 (3) (2009) 293–322. [2] I. Belti¸ta˘ , D. Belti¸ta˘ , Modulation spaces of symbols for representations of nilpotent Lie groups, J. Fourier Anal. Appl., doi:10.1007/s00041-010-9143-4, in press; preprint arXiv:0908.3917v2 [math.AP]. [3] I. Belti¸ta˘ , D. Belti¸ta˘ , Uncertainty principles for magnetic structures on certain coadjoint orbits, J. Geom. Phys. 60 (1) (2010) 81–95. [4] I. Belti¸ta˘ , D. Belti¸ta˘ , Smooth vectors and Weyl–Pedersen calculus for representations of nilpotent Lie groups, An. Univ. Bucure¸sti Mat. 1 (LIX) (1) (2010) 17–46. [5] I. Belti¸ta˘ , D. Belti¸ta˘ , Operator Calculus for Lie Group Representations, monograph, forthcoming. [6] R.W. Carter, Lie Algebras of Finite and Affine Type, Cambridge Stud. Adv. Math., vol. 96, Cambridge University Press, Cambridge, 2005. [7] E. Cordero, K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (1) (2003) 107– 131. [8] L.J. Corwin, F.P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications. Part I (Basic Theory and Examples), Cambridge Stud. Adv. Math., vol. 18, Cambridge University Press, Cambridge, 1990. [9] L. Corwin, F.P. Greenleaf, R. Penney, A general character formula for irreducible projections on L2 of a nilmanifold, Math. Ann. 225 (1) (1977) 21–32. [10] R. Douady, Produits tensoriels topologiques et espaces nucléaires, in: A. Douady, J.-L. Verdier (Eds.), Quelques Problèmes de Modules, Sém. Géom. Anal. École Norm. Sup., Paris, 1971–1972, in: Astérisque, vol. 16, Soc. Math. France, Paris, 1974, pp. 7–32. [11] H.G. Feichtinger, Modulation spaces on locally compact abelian groups, in: Proceedings of “International Conference on Wavelets and Applications” 2002, Chennai, India, Allied Publishers, 2003, pp. 99–140; Updated version of a technical report, University of Vienna, 1983. [12] H.G. Feichtinger, K. Gröchenig, A unified approach to atomic decompositions via integrable group representations, in: Function Spaces and Applications, Lund, 1986, in: Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 52–73. [13] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (2) (1989) 307–340. [14] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (2–3) (1989) 129–148.

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[15] H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Lecture Notes in Math., vol. 1863, Springer-Verlag, Berlin, 2005. [16] K. Gröchenig, An uncertainty principle related to the Poisson summation formula, Studia Math. 121 (1) (1996) 87–104. [17] K. Gröchenig, Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Inc., Boston, MA, 2001. [18] K. Gröchenig, A pedestrian’s approach to pseudodifferential operators, in: Harmonic Analysis and Applications, in: Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2006, pp. 139–169. [19] K. Gröchenig, C. Heil, Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory 34 (4) (1999) 439–457. [20] R.S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1) (1982) 65– 222. [21] B. Helffer, A. Mohamed, Sur le spectre essentiel des opérateurs de Schrödinger avec champ magnétique, Ann. Inst. Fourier (Grenoble) 38 (2) (1988) 95–112. [22] V. Iftimie, M. M˘antoiu, R. Purice, Magnetic pseudodifferential operators, Publ. Res. Inst. Math. Sci. 43 (3) (2007) 585–623. [23] P.E.T. Jorgensen, W.H. Klink, Quantum mechanics and nilpotent groups. I. The curved magnetic field, Publ. Res. Inst. Math. Sci. 21 (5) (1985) 969–999. [24] M. M˘antoiu, R. Purice, The magnetic Weyl calculus, J. Math. Phys. 45 (4) (2004) 1394–1417. [25] M. M˘antoiu, R. Purice, The modulation mapping for magnetic symbols and operators, Proc. Amer. Math. Soc. 138 (8) (2010) 2839–2852. [26] A. Mohamed, J. Nourrigat, Encadrement du N (λ) pour un opérateur de Schrödinger avec un champ magnétique et un potentiel électrique, J. Math. Pures Appl. (9) 70 (1) (1991) 87–99. [27] K.-H. Neeb, Infinite-dimensional groups and their representations, in: A. Huckleberry, T. Wurzbacher (Eds.), Infinite Dimensional Kähler Manifolds, Oberwolfach, 1995, in: DMV Sem., vol. 31, Birkhäuser, Basel, 2001, pp. 131–178. [28] K.-H. Neeb, Towards a Lie theory of locally convex groups, Jpn. J. Math. 1 (2) (2006) 291–468. [29] K.-H. Neeb, On differentiable vectors for representations of infinite dimensional Lie groups, J. Funct. Anal. 259 (11) (2010) 2814–2855. [30] N.V. Pedersen, Matrix coefficients and a Weyl correspondence for nilpotent Lie groups, Invent. Math. 118 (1) (1994) 1–36. [31] B. Roider, Die metrisierbaren linearen Teilräume des Raumes OM von L. Schwartz, Monatsh. Math. 79 (4) (1975) 325–332. [32] H.H. Schaefer, Topological Vector Spaces, The Macmillan Co., New York, 1966, Collier–Macmillan Ltd., London, 1966. [33] J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (2) (1994) 185–192. [34] J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I, J. Funct. Anal. 207 (2) (2004) 399–429. [35] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967.


Composition operators on the polydisk induced by affine maps Frédéric Bayart Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448, F-63000 Clermont-Ferrand – CNRS, UMR 6620, Laboratoire de Mathématiques, F-63177 Aubiere, France Received 12 July 2010; accepted 20 December 2010 Available online 7 January 2011 Communicated by Gilles Godefroy

Abstract We study the continuity of composition operators on the classical Hardy and weighted Bergman spaces of the polydisk. We show that this problem involves some delicate properties of the derivative of the symbol. In particular, we characterize continuity when the symbol is a linear self-map of the polydisk. © 2011 Elsevier Inc. All rights reserved. Keywords: Composition operators; Polydisk; Carleson measures

1. Introduction If X is a Banach space of holomorphic functions on a domain U and if φ is a (holomorphic) self-map of U , the composition operator associated to φ is defined by Cφ (f ) = f ◦ φ for any f ∈ X. The study of composition operators consists in the comparison of the properties of the operator Cφ with that of the function φ itself, which is called the symbol of Cφ . It is a very active field in analysis (at time of writing, MathSciNet refers more than 1100 papers with “composition operators” in their title). The first problem to tackle is often that of continuity: given X, for which symbols φ the composition operator Cφ defines a bounded operator on X? The answer is rather easy when X is the Hardy space or a weighted Bergman space of the disk: every symbol defines a bounded composition operator. This is the Littlewood subordination principle, which goes back to 1925. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.019

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F. Bayart / Journal of Functional Analysis 260 (2011) 1969–2003

In several complex variables, the situation is much more involved. Let Bn be the euclidean ball of Cn and let Sn be its boundary, namely the unit sphere. Let also dz be the Lebesgue measure on Bn and let dσ be the normalized surface measure on Sn . The Hardy space H p (Bn ), 1 p < +∞ consists of the holomorphic functions f in Bn such that

p

f H p (Bn ) = sup

0 |Dτ Dτ φη (ξ )| for all ξ and τ ∈ ∂Bn with ξ, τ = 0 and φ(ξ ) = η. This theorem was later extended to weighted Bergman spaces in [7]. In particular, when φ ∈ C 3 (Bn ), the continuity of Cφ does not depend on the Hardy or on the weighted Bergman space where we work: Cφ is continuous on H 2 (Bn ) iff Cφ is continuous on A2β (Bn ) for some β > −1 iff Cφ is continuous on A2β (Bn ) for all β > −1. In this paper, we investigate similar statement for composition operators on the polydisk. We use dA to denote the normalized area measure on the unit disk D and for β > −1, we write β dAβ (z) = (β + 1) 1 − |z|2 dA(z). More generally, let Dn be the unit polydisk in Cn and for β > −1, let dVβ (z) = dAβ (z1 ) . . . dAβ (zn ),

z = (z1 , . . . , zn ). p

For p 1 and β > −1, the weighted Bergman space Aβ (Dn ) consists of all holomorphic func-


1971

tions f in Dn such that p f Ap β

=

f (z)p dVβ (z) < ∞.

Dn

Sometimes, the unweighted Bergman space will be more simply denoted by Ap (Dn ) = p A0 (Dn ). We also consider the Hardy spaces H p (Dn ), p 1, which is the space of all g ∈ H (Dn ) for which p g(rξ )p dσ (ξ ) < ∞, gH p = sup 0 β2 > −1 we find and integer n and a linear map φ : Dn → Dn such that Cφ is continuous on A2β1 (Dn ) and such that Cφ is not continuous on A2β2 (Dn ). This is completely different from what happens on the unit ball. In Section 9, we turn to the study of the continuity of composition operators on the Hardy space of the polydisk. The difficulty is that our main tool, Carleson measures, is less tractable in H 2 (Dn ) than in A2β (Dn ). Nevertheless, we will be able to obtain a similar statement on H 2 (Dn ), using an indirect strategy, and a very close look to the constants which appear when we are proving the continuity on the Bergman spaces. The arguments used throughout this paper are rather classical: Carleson measures, Julia– Caratheodory theorem, . . . . A short survey of what is needed is the content of the next section. Notations. All proofs in this paper will rely on volume estimation arguments. To avoid unuseful complications, the sentence “the volume of E(x) is less than f (x)” will always mean that V E(x) Cf (x) for some constant C which does not depend on x. Moreover, e will denote e = (1, . . . , 1). For φ : Dn → Dn and I = {i1 , . . . , iq } ⊂ {1, . . . , n}, φI denotes (φi1 , . . . , φiq ) and |I | denotes the cardinal number of I . For w ∈ C and δ > 0, R(w, δ) and D(w, δ) mean respectively

R(w, δ) = z ∈ D; e(1 − wz) ¯ −1 and let φ : Dn → Dn be holomorphic. Then Cφ : Aβ (Dn ) → Aβ (Dn ) is bounded if and only if there exists C > 0 such that Vβ φ −1 S(ξ, δ) CVβ S(ξ, δ)

(1)

for all δ ∈ (0, 2)n and all ξ ∈ Tn . Let us comment this statement. Our results in this paper will all ultimately rely on it. It was also used by the authors of [8] when they tried to prove Theorem 1.2. The mistake that they have made there is that they proved (1) only when all δk are small. They do not prove it when some δk are small and some are large. Let us also mention that the characterization of Carleson measures on the Hardy space of the polydisk is more difficult. See Section 9 for details, as well as for a proof of Lemma 2.1 with a precise study of the constants which are involved there. To apply Lemma 2.1, we shall need to estimate the volume of subsets of Dn . Here are the results that we need. Lemma 2.2. Let u ∈ T and δ ∈ (0, 2). Then C −1 δ 3/2 V

z ∈ D; e(1 − uz) ¯ 0. Then

z ∈ D; e(1 − uz) ¯ < δ and |v − z| < δ α has volume less than min(δ 3/2 , δ 1+α ). Proof. The first estimation is already contained in the previous lemma. For the second one we have just to observe that our set is contained in a rectangle whose sides have respective length δ and δ α . 2 Lemma 2.4. There exists ε > 0 such that, for every δ ∈ (0, 2), for every w ∈ C satisfying 1 − εδ e(w) 1 + εδ

√ and m(w) εδ

the set {z ∈ D; |z − w| < δ} has volume greater than δ 2 . Proof. Let w0 = w − δ, w1 = w − 2δ , w2 = w − 2δ + i 4δ . It is easy to verify that w0 , w1 and w2 1 all belong to D ∩ D(w, δ), provided ε is small enough (for instance, ε = 10 works). This triangle 2 has volume greater than δ . 2

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In particular, √ one can apply Lemma 2.4 for w ∈ D satisfying e(w) 1 − εδ, since in that case | m(w)| 2εδ. At this stage, we can point out a crucial idea. Linear forms ψ on Dn such that supz∈Dn |ψ(z)| = 1 may approach an extreme point on sets with different volumes. Consider for instance z → z1 and z → (z1 + z2 )/2. By the above lemma, we get

z ∈ D; |z1 − 1| < δ Cδ 2 ,

C −1 δ 2+3/2 V z ∈ D; |z1 − 1| < δ Cδ 2+3/2 , C −1 δ 2 V

the last inequality coming from the fact that e(z) 1 − δ. 2.2. The Julia–Caratheodory theorem The Julia–Caratheodory theorem is a geometric statement explaining the behaviour of an analytic self-map of D near a boundary fixed point (see [10] for a beautiful exposition). Since we will consider only smooth maps, we just need it in the following form: Lemma 2.5. Let φ : D → D be holomorphic which in C 1 in D ∪ {ξ } with ξ ∈ T. Suppose moreover that φ(ξ ) = ξ . Then φ (ξ ) ∈ (0, +∞). This result was later extended to the polydisk by Abate [1] in the following form: Lemma 2.6. Let f : Dn → D be a holomorphic function and let ξ ∈ ∂Dn . Assume there is α > 0 such that lim inf w→ξ

1 − |f (w)| = α. 1 − w∞

(2)

Then there exists τ ∈ T such that f has K-limit τ at ξ and df (z)(ξ ) has restricted K-limit ατ at ξ . It is not hard to check that, if f is C 1 at ξ and if f maps ξ onto a point of T, then (2) is fulfilled (it suffices to consider radial limits). In particular, the linear map df (ξ ) is non-zero. We use these versions of the Julia–Caratheodory theorem in order to obtain informations on the Taylor expansion of maps from Dn to D: Corollary 2.7. Let u : Dn → D be holomorphic in a neighbourhood of Dn . Suppose that u(e) = 1. Then aj (zj − 1) + O |zj − 1|2 . u(z) = 1 + j n

j n aj =0

Proof. Since u is C 2 at e, we can write directly u(z) = 1 +

j n

aj (zj − 1) + O

j n

|zj − 1|

2

.


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Our task is to delete certain terms in the last sum. Let J = {j ; aj = 0} and suppose for simplicity that J = {1, . . . , m}. We can write around e u(z) = 1 +

aj (zj − 1) + H (zm+1 , . . . , zn ) +

j m

+O

(zj − 1)Gj (zm+1 , . . . , zn )

j m

|zj − 1|

2

j m

with dH (e) = 0 and Gj (e) = 0 for j m. We want to prove that H = Gj = 0. Consider first v(zm+1 , . . . , zn ) = u(1, . . . , 1, zm+1 , . . . , zn ) = 1 + H (zm+1 , . . . , zn ). v is a holomorphic map from Dn−m into D. If v were not constant, it would map Dn−m into D with v(e) = 1 and dv(e) = 0, in contradiction with Lemma 2.6. Hence H is zero. Suppose now that some Gj is not zero. It is a non-constant holomorphic map, hence it is open and one can find ξm+1 , . . . , ξn such that Gj (ξm+1 , . . . , ξn ) ∈ / R. Consider now w(zj ) = u(1, . . . , zj , . . . , 1, ξm+1 , . . . , ξn ) = 1 + aj + Gj (ξm+1 , . . . , ξn ) (zj − 1) + O |zj − 1|2 . w maps D into D and satisfies w (1) ∈ / R. This contradicts again the Julia–Caratheodory theorem. 2 3. A general sufficient condition In this section, we give a general sufficient condition to ensure that a smooth map φ : Dn → Dn induces a bounded composition operator on A2 (Dn ). This will give a necessary and sufficient statement for the bidisk. Theorem 3.1. Let φ : Dn → Dn be such that φ ∈ C 2 (Dn ). Suppose that, for any q 1, for any I ⊂ {1, . . . , n}, |I | = q, for any ξ ∈ Dn with φI (ξ ) ∈ Tq , the derivative dφI (ξ ) has rank q. Then φ maps continuously Ap (Dn ) into itself. Proof. Let I = {i1 , . . . , iq } ⊂ {1, . . . , n} and let FI = {ξ ∈ Dn ; φI (ξ ) ∈ Tq }. Let also ξ ∈ FI . Since dφI (ξ ) has rank q, one can find J = {j1 , . . . , jq } ⊂ {1, . . . , n} such that Mφ (I, J, ξ ) =

∂φi (ξ ) ∂zj i∈I, j ∈J

is invertible. Since φ is C 1 , this property remains true for any z in a neighbourhood VI (ξ ) = VI (ξ ) × VI (ξ ) of ξ . Here, we write Cn = Cq × Cn−q , the q first coordinates corresponding to (j1 , . . . , jq ).

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FI is compact and FI ⊂ ξ ∈FI VI (ξ ). By compactness, we can extract a finite covering

mI FI ⊂ GI := l=1 VI (ξl ). By the compactness of Dn \GI , one can find εI > 0 such that ξ∈ / GI

⇒

max 1 − φi (ξ ) εI . i∈I

We then define: • • • •

ε := minI εI ; MI := maxz∈GI det(Mφ−1 (I, J, z)); M := maxI MI ; m = maxI mI ,

where I runs over all subsets of {1, . . . , n} such that there exists ξ ∈ Dn with φI (ξ ) ∈ Tq . We now pick any ζ ∈ Tn and δ ∈ (0, +∞)n and let us estimate V (φ −1 (S(ζ, δ))). We set I := {1 i n; δi < ε}, |I | = q. It is enough to show that V φ −1 S(ζ, δ) C δi2 . i∈I −1 −1 Now, the definition

mI of ε ensures that either φ (S(ζ, δ)) is empty or that φ (S(ζ, δ)) is contained in GI = l=1 VI (ξl ). Let us denote

Ul = z ∈ Dn ; φi (z) − ζi < δi for all i ∈ I ∩ VI (ξl ). One has to control V (Ul ). Let J be coordinates such that Mφ (I, J, ξl ) is invertible. We write any z ∈ Dn as z = (zJ , zJ ) so that Fubini’s theorem yields V (Ul )

1VI (ξl ) zJ V zJ ∈ Dq ; φi zJ , zJ − ζi < δi for all i ∈ I dzJ .

Dn−q

For a fixed zJ , because of the change of variables formula, the volume inside the integral is less than M i∈I δi2 . This implies V (φ −1 (S(ζ, δ))) mM i∈I δi2 which allows us to conclude. 2 Unfortunately, the condition which appears in Theorem 3.1 is not necessary. Indeed, if we consider φ(z) = ((z1 + z2 + z3 )/3, (z1 + z2 + z3 )/3, 0), then it is clear that φ does not satisfy the assumptions of the theorem. However, we will see later that Cφ is continuous on A2 (D3 ). Observe that this example is given on D3 . Such an example cannot exist on the bidisk: Theorem 3.2. Let φ : D2 → D2 be such that φ defines a holomorphic map on a neighbourhood of D2 . Then Cφ is continuous on A2 (D2 ) if and only if, for any ξ ∈ T2 with φ(ξ ) ∈ T2 , dφ(ξ ) is invertible. That is Theorem 1.2 is correct on the bidisk (except that our assumption on the regularity of φ is slightly stronger).


1977

Proof of Theorem 3.2. The condition is necessary by Theorem 1.2 (we apply its correct part). To prove that the condition is sufficient, we shall apply Theorem 3.1. So, let q ∈ {1, 2}, I ⊂ {1, 2} with |I | = q and ξ ∈ D2 such that φI (ξ ) ∈ Tq . When |I | = 1, we may suppose I = {1}. By the maximum modulus principle, ξ ∈ ∂D2 . By Lemma 2.6, dφ1 (ξ ) is a non-zero linear functional, hence has rank 1. When |I | = 2, ξ belongs to T2 otherwise φ would not depend on one of the two variables, say z2 and this would contradict φ(ξ1 , 1) ∈ T2 and dφ(ξ1 , 1) is invertible. Thus the assumptions of Theorem 3.2 imply that of Theorem 3.1, and Cφ is continuous. 2 Remark 3.3. We can weaken the assumptions of the previous theorem. A look at the proof of it shows that we just need that φ ∈ C 2 (D2 ) and the maps φ(u, ·) and φ(·, u) are holomorphic for any u ∈ T. 4. Two examples In this section, which is purely expository, we intend to study completely the continuity on A2 (D7 ) of two composition operators. Our aim is twofold. Firstly, we want to convince the reader that continuity is a difficult problem: the two symbols will have a very similar definition. One will induce a bounded composition operator while the other one will induce an unbounded composition operator. Secondly, we think that it is a good idea to exhibit on a particular example the methods which will also work in the general case. Example 4.1. Let u(z) = (z1 + · · · + z5 )/5 and let φ(z) = u(z), u(z), u(z), u(z), (z6 + z7 )/2, (z1 + 2z6 + z7 )/4, 0 , ψ(z) = u(z), u(z), u(z), u(z), (z6 + z7 )/2, (2z1 + z6 + z7 )/4, 0 . Then Cφ is continuous on A2 (D7 ) whereas Cψ is not continuous on A2 (D7 ). Proof. We first show that Cψ is not continuous. Let δ > 0 and let δ = (δ, δ, δ, δ, δ 1/2 , δ 1/2 , 2). The volume of S(e, δ) behaves like δ 10 . We will show that the volume of ψ −1 (S(e, δ)) is greater than δ 9+3/4 , showing by Lemma 2.1 that Cψ is not continuous. z belongs to ψ −1 (S(e, δ)) iff ⎧ ⎨ |z1 + z2 + z3 + z4 + z5 − 5| < 5δ, |z6 + z7 − 2| < 2δ 1/2 , ⎩ |2z1 + z6 + z7 − 4| < 4δ 1/2 . Let ε > 0 be small but independent of δ, let R = {z ∈ D; e(z) 1 − εδ}, R 1/2 = {z ∈ D; e(z) 1 − εδ 1/2 } and R = R × R × R × R. Let also, for z = (z1 , . . . , z4 ) ∈ R and for z7 ∈ R 1/2 , Uz = z5 ∈ D; Uz7 = z6 ∈ D;

z5 − (5 − z1 − z2 − z3 − z4 ) < δ ,

z6 − (2 − z7 ) < δ 1/2 .

By Fubini’s theorem, Lemma 2.2 and Lemma 2.4, the volume of the elements z ∈ D7 satisfying

1978


(z1 , . . . , z4 ) ∈ R, 3

3

z5 ∈ Uz , z7 ∈ R 1/2 , z6 ∈ Uz7

(3)

3

is greater than δ 2 ×4 δ 2 δ 4 δ = δ 9+ 4 , provided ε is small enough. Moreover, any z ∈ D7 satisfying (3) belongs to ψ −1 (S(e, δ)). The only non-trivial inequality is |2z1 + z6 + z7 − 4| < 4δ 1/2 . By the triangle inequality, |2z1 + z6 + z7 − 4| 2|z1 − 1| + |z6 + z7 − 2|. √ Now, because 1 e(z1 ) 1 − εδ for any z1 ∈ R, one also deduces | m(z1 )| 2εδ and thus |z1 − 1| 3εδ 1/2 showing that z ∈ ψ −1 (S(e, δ)). This proves that Cψ is not continuous, the main reason being that, when we look at z ∈ ψ −1 (S(e, δ)), the last line |2z1 + z6 + z7 − 4| < 4δ 1/2 does not add any constraint on z1 , z6 or z7 . It is a consequence of the two previous inequalities. We also point out that the choice of δ is crucial. For instance, for δ = (δ, δ, δ, δ, 1, 1, 2) or for δ = (δ, δ, δ, δ, δ, δ, 2), the estimation V (ψ −1 (S(e, δ))) CV (S(e, δ)) is valid. Let us now explain why Cφ is continuous. If we look at the z ∈ D7 belonging to φ −1 (S(e, δ)) for the same value of δ, then the last line becomes |z1 + 2z6 + z7 − 4| < 4δ 1/2 and it now adds informations. Indeed, if we combine it with |z6 + z7 − 2| < 2δ 1/2 , they imply |z1 − z7 | < 6δ 1/2 . In particular, we can replace the condition z7 ∈ R 1/2 (which gives a volume δ 3/4 ) by the condition z7 belongs to some disk of radius δ 1/2 (which gives a volume δ). Thus, we gain a factor δ 1/4 which was exactly the missing part to ensure continuity. Of course, we have just prove (1) for a very particular value of ξ and of δ. We have to proceed with the general case. Let ξ ∈ T7 and let δ ∈ (0, 2)7 . Without loss of generality, we may assume that δ1 = min(δ1 , δ2 , δ3 , δ4 ). Furthermore, for φ −1 (S(ξ, δ)) to be non-empty, it is necessary that δ7 is far away from zero. Hence, we just need to prove that V (φ −1 (ξ, δ)) is less than δ18 δ52 δ62 . We try to find an upper bound for the volume of the set of z ∈ D7 with φ(z) ∈ S(ξ, δ), namely satisfying the three following conditions: ⎧ ⎨ |z1 + z2 + z3 + z4 + z5 − 5ξ1 | < 5δ1 , |z6 + z7 − 2ξ5 | < 2δ5 , ⎩ |z1 + 2z6 + z7 − 4ξ6 | < 4δ6 . We split the proof into several cases. Case 1: δ1 δ5 δ6 . We first analyze the condition |z1 + · · · + z5 − 5ξ1 | < 5δ1 . It implies zi ∈ R(ξ1 , 5δ1 ) for i = 1, . . . , 4 and, when z = (z1 , . . . , z4 ) has been fixed, z5 belongs to some disk D(C(z ), 5δ1 ). The two last conditions imply

|z6 + z7 − 2ξ5 | < 2δ5 , z1 − z7 − 4(ξ6 − ξ5 ) < 8δ6 .


1979

This implies that z7 belongs to some disk D(C(z1 ), 8δ6 ) and that z6 belongs to some disk D(C(z7 ), 2δ5 ). Hence, by Fubini’s theorem and Lemma 2.2, we get that the volume of 3

φ −1 (S(ξ, δ)) is less than δ12

×4 2 2 2 δ1 δ6 δ5 ,

which is exactly equal to δ18 δ53 δ62 .

Case 2: δ1 δ6 δ5 . We do not change anything for z1 , z2 , z3 , z4 . The two last conditions now imply

|2z + z1 + z7 − 4ξ6 | < 4δ6 , 6 z1 − z7 − 4(ξ6 − ξ5 ) < 8δ5

showing that z7 ∈ D(C(z1 ), 6δ5 ) and z6 ∈ D(C(z1 , z7 ), 4δ6 ). The estimation of the volume remains unchanged. Case 3 and Case 4: δ5 δ1 δ6 and δ5 δ6 δ1 . We can proceed exactly like in Case 1, the crucial point being δ5 δ6 . Case 5 and Case 6: δ6 δ1 δ5 and δ6 δ5 δ1 . We can proceed exactly like in Case 2, the crucial point being δ6 δ5 . 2 All the technics of our forthcoming general theorem (estimation of the volumes, triangularization of the conditions, well-ordering of the variables) are already present in this example. We have now to “abstract” them. The difficulty will come from variables which are present in several lines (typically, like z1 in the previous examples). Moreover, the main difference between φ and ψ above is that the restriction of ψ5 and ψ6 to the variables z6 and z7 are equal, which is not the case of φ5 and φ6 . We have to introduce quantities which take this kind of informations into account. This is the content of the next section, where we present our main theorem. 5. Statement of the main result Let φ : Dn → Dn be holomorphic and suppose that φ extends holomorphically in a neighbourhood of Dn . Let ξ ∈ Tn and I ⊂ {1, . . . , n}, |I | = q be such that φI (ξ ) ∈ Tq and I is maximal with respect to this property. Let s be the rank of dφI (ξ ). Let also J = (j (1), . . . , j (s)) be a sequence of I s = I × · · · × I such that (dφj (1) (ξ ), . . . , dφj (s) (ξ )) are independent. For each k ∈ {1, . . . , s}, we introduce the following definitions (we always take the derivatives at ξ ): • rξ,I,J (k) is the number of linear forms in {dφi ; i ∈ I } which are in the subspace span(dφj (1) , . . . , dφj (k) ) and not in span(dφj (1) , . . . , dφj (k−1) ). • qξ,I,J (k) is the number of “new” variables which appear in dφj (k) that is qξ,I,J (k) is the ∂φ ∂φ = 0 whereas ∂zj (m) = 0 for cardinal number of the integers l ∈ {1, . . . , n} such that ∂zj (k) l l m < k. • Eξ,I,J (k) is the set of new variables in dφj (k) . In particular, the cardinal number of Eξ,I,J (k) is qξ,I,J (k). • tξ,I,J (k) is equal to the supremum of the integers t k such that the restriction of dφj (k) to the variables appearing from step t, namely dφj (k)|Eξ,I,J (t)∪···∪Eξ,I,J (k) , does not belong to the span of dφj (t)|Eξ,I,J (t)∪···∪Eξ,I,J (k) , . . . , dφj (k−1)|Eξ,I,J (t)∪···∪Eξ,I,J (k) .

1980


More precisely, t being fixed, we can write z = (z , z ) where z corresponds to the variables which appear for the first time in dφj (t) , dφj (t+1) , . . . and z corresponds to the other variables. We write ψj (m) (z ) = dφj (m) (z , 0) and we ask that ψj (k) does not belong to the span of ψj (t) , . . . , ψj (k−1) . If we look at the examples of Section 4, this function tξ,I,J will be that which will quantify that the restriction of ψ5 and ψ6 to the variables z6 and z7 are equal, which is not the case of φ5 and φ6 . Before going further, we comment these definitions. They look rather complicated (at least tξ,I,J ). However, they can be easily computed for each specific choice of φ and ξ using a variant of Gauss algorithm: see the forthcoming examples for detailed computations (the lecture of these examples can also help to understand the definitions). It would be nice if they could be expressed using the Jordan decomposition of dφ(ξ ). Unfortunately, this is not the case. For instance, the maps φ and ψ of Section 4 have similar Jordan reduction, but the functions r, q, E and t take different values. An important point to keep in mind in that the functions r, q, E and t do not depend only on linear algebra properties of dφ(ξ ). Combinatorial properties of the numbers of variables which come in each linear functional are also very important to compute their values. It is worth to notice that tξ,I,J (k) is well-defined. Indeed, E = Eξ,I,J (1) ∪ · · · ∪ Eξ,I,J (k) corresponds exactly to all the variables appearing in dφj (1) , . . . , dφj (k) . Thus, dφj (k)|E does not belong to the span of dφj (1)|E , . . . , dφj (k−1)|E because the linear forms are independent. Observe also that qξ,I,J (k) > 0 implies that tξ,I,J (k) = k since in that case dφj (k) (ξ )|Eξ,I,J (k) is a non-zero linear form. Moreover, if tξ,I,J (k) is equal to t, then qξ,I,J (t) is positive. Indeed, if qξ,I,J (t) is equal to 0, then Eξ,I,J (t) ∪ · · · ∪ Eξ,I,J (k) = Eξ,I,J (t + 1) ∪ · · · ∪ Eξ,I,J (k) =: F and / span(dφj (t)|F , . . . , dφj (k−1)|F ) dφj (k)|F ∈ ⇒

dφj (k)|F ∈ / span(dφj (t+1)|F , . . . , dφj (k−1)|F ).

Finally, we can also control the number of solutions of tξ,I,J (l) = k: Lemma 5.1. Let k ∈ {1, . . . , s} and let F (k) = {l s; tξ,I,J (l) = k}. Then card(F (k)) qξ,I,J (k). Proof. Let l ∈ F (k). By assumption, there exist coefficients βi,l such that ψl = dφj (l) − l−1 i=k+1 βi,j dφj (i) does not depend on the variables appearing in Eξ,I,J (k + 1) ∪ · · · . Moreover, again by the definition of t, the linear forms ψl|Eξ,I,J (k)∪··· , l ∈ F (k) are linearly independent. This implies, because they all vanish on Eξ,I,J (k + 1) ∪ · · · , that the linear forms ψl|Eξ,I,J (k) , l ∈ F (k), are linearly independent. Thus, card(F (k)) card(Eξ,I,J (k)) = qξ,I,J (k). 2 We then define two finite trees Rξ,I,J and Lξ,I,J as follows. A node will be indexed by a finite sequence (l1 , . . . , ls ) with • l1 = 0; • there exists m s such that li+1 − li ∈ {0, 1} for i < m and li = ∞ for i > m. The integer m which appears above can be seen as the depth of the node in the tree. In particular, the node (0, ∞, . . . , ∞) is the root of the tree. To each node, we associate two values (we are defining two trees) as follows. For the root, we define


1981

1 3 + qξ,I,J (1), 2 2 Rξ,I,J (0, ∞, . . . , ∞) = 2rξ,I,J (1). Lξ,I,J (0, ∞, . . . , ∞) =

If the values of Lξ,I,J (0, l2 , . . . , lm , ∞, . . .) and Rξ,I,J (0, l2 , . . . , lm , ∞, . . .) have been set, then we define the values at the two sons of the node (0, l2 , . . . , lm , ∞, . . .) as follows (recall that lm+1 − lm ∈ {0, 1}): • Rξ,I,J (l1 , . . . , lm+1 , ∞, . . .) = Rξ,I,J (l1 , . . . , lm , ∞, . . .) + • If qξ,I,J (m + 1) > 0, then

1 2lm+1

Lξ,I,J (l1 , . . . , lm+1 , ∞, . . .) = Lξ,I,J (l1 , . . . , lm , ∞, . . .) +

× 2rξ,I,J (m + 1).

1 2lm+1

1 3 + qξ,I,J (m + 1) . 2 2

• If qξ,I,J (m + 1) = 0, then Lξ,I,J (l1 , . . . , lm+1 , ∞, . . .) = Lξ,I,J (l1 , . . . , lm , ∞, . . .) 0 if lm+1 > ltξ,I,J (m+1) , + 1 if lm+1 = ltξ,I,J (m+1) . lm+1 +1 2

It is not very difficult to compute the value of Rξ,I,J at a node (l1 , . . . , ls ). It is exactly Rξ,I,J (l1 , . . . , ls ) =

s 2rξ,I,J (k) k=1

2 lk

.

This is slightly more difficult for Lξ,I,J . The idea is to group together the lines where tξ,I,J (m) = k for the same value of k (observe that qξ,I,J (k) > 0). We then find 1 1 3 q . (k) + Lξ,I,J (l1 , . . . , ls ) = ξ,I,J 2 lk 2 2 k; qξ,I,J (k)>0

t (m)=k lm =lk

Given two trees Lξ,I,J and Rξ,I,J , we say that Lξ,I,J Rξ,I,J if, for each node (l1 , . . . , lm , ∞, . . .), the inequality Lξ,I,J (l1 , . . . , lm , ∞) Rξ,I,J (l1 , . . . , lm , ∞) holds. Our main theorem now reads Theorem 5.2. Suppose that Cφ is continuous on A2 (Dn ). Then for any ξ, I, J as above, one has Lξ,I,J Rξ,I,J . When φ is linear, Cφ is continuous on A2 (Dn ) if and only if, for any ξ, I, J as above, Lξ,I,J Rξ,I,J . The statement of Theorem 5.2 gives an effective algorithm to determine if a linear map induces a bounded composition operator on A2 (Dn ). Here is how it works with the examples of Section 4. Example 5.3. Let u(z) = (z1 + · · · + z5 )/5 and let ψ(z) = u(z), u(z), u(z), u(z), (z6 + z7 )/2, (2z1 + z6 + z7 )/4, 0 . Then Cψ is not continuous on A2 (D7 ).

1982


Proof. Let ξ = e, I = {1, 2, 3, 4, 5, 6}. The rank of dφI (ξ ) is equal to 3. We can define J by j (1) = 1, j (2) = 5, j (3) = 6. One can easily compute r(1) = 4, q(1) = 5,

r(2) = 1, q(2) = 2,

EJ (1) = {z1 , z2 , z3 , z4 , z5 }, t (1) = 1,

r(3) = 1, q(3) = 0,

EJ (2) = {z6 , z7 }, t (2) = 2.

EJ (3) = ∅,

For the value of t (3), observe that EJ (2) = {z6 , z7 } and that dψj (3)|EJ (2) = 12 dψj (2)|EJ (2) = z6 +z7 4 . Thus t (3) < 2 and necessarily t (3) = 1. The computation of R is easy. For L, is not hard to show that 1 3 + × 5 = 8, 2 2 1 3 L(0, 0, ∞) = 8 + + × 2 = 11.5, 2 2 1 1 3 L(0, 1, ∞) = 8 + + × 2 = 9.75. 2 2 2

L(0, ∞, ∞) =

To compute the value at the last nodes, we observe that lt (3) = l1 so that L(0, 0, 0) = 11.5 + 0.5 = 12, L(0, 1, 1) = 9.75,

L(0, 0, 1) = 11.5, L(0, 1, 2) = 9.75.

We then get the two following trees: 8

8

11.5

12

9.75

11.5

9.75

10

9.75

12

9

11

10

9.5

Cφ is not continuous because one node of R is greater than the corresponding node of L. Example 5.4. Let u(z) = (z1 + · · · + z5 )/5 and let φ(z) = u(z), u(z), u(z), u(z), (z6 + z7 )/2, (z1 + 2z6 + z7 )/4, 0 . Then Cφ is continuous on A2 (D7 ).

2


1983

Proof. The only choice for ξ and I is ξ = e and I = {1, 2, 3, 4, 5, 6}. Suppose first that j (1) = 1, j (2) = 5, j (3) = 6 (of course, j (1) ∈ {2, 3, 4} would not change anything). The values of the functions r, q, E, t are now r(1) = 4,

r(2) = 1,

q(1) = 5, EJ (1) = {z1 , z2 , z3 , z4 , z5 }, t (1) = 1,

r(3) = 1,

q(2) = 2, EJ (2) = {z6 , z7 }, t (2) = 2,

q(3) = 0, EJ (3) = ∅, t (3) = 2

(the only change is for the value of t (3) which is now equal to 2 because dφj (3)|EJ (2) = (2z6 + 27 )/4 is not proportional to dφj (2)|EJ (2) = (z6 + z7 )/2). The corresponding trees are now 8

8

11.5

12

9.75

11.5

10

10

9.75

12

9

11

10

9.5

In that case LJ RJ . To conclude, one should (as in Section 4) consider the five other possibilities for J . The easy verifications are left to the reader. 2 Our last example was introduced to point out that the condition which appears in Theorem 3.1 is not necessary for Cφ to be continuous. Example 5.5. Let φ : D3 → D3 be defined by φ(z) = ((z1 + z2 + z3 )/3, (z1 + z2 + z3 )/3, 0). Then Cφ is continuous on A2 (D3 ). Proof. We have to take ξ = e and I = {1, 2}, so that dφI (ξ ) has rank 1. Hence, our trees have only one node! Now, r(1) = 2, q(1) = 3, t (1) = 4 so that R(0) = 4 and L(0) = 32 × 3 + 12 = 5. Thus, L R and Cφ is continuous on A2 (D3 ). 2 6. Proof of the sufficient part In this section, we intend to prove the “sufficient part” of our main theorem. Namely, we start with φ(z) = Az for some matrix A = (ai,j ) ∈ M n (C) satisfying the assumptions of Theorem 5.2. φ maps Dn into Dn iff, for any i ∈ {1, . . . , n}, j |ai,j | 1. We divide the proof into two steps. The first one is to understand how the conditions of Theorem 5.2 can be read on the matrix A. 6.1. Rows with the same direction Definition 6.1. We say that two vectors u ∈ Cn and v ∈ Cn have the same direction if there exists some θ ∈ R such that, for any j ∈ {1, . . . , n}, either vj = 0 or uj = rj eiθ vj for some rj 0.

1984


This notion of vectors with the same direction will be relevant for us when applied to the rows of the matrix A. For i ∈ {1, . . . , n}, ai will denote the row vector ai = (ai,1 , . . . , ai,n ). The statement that we need is the following: Proposition 6.2. There exists δ0 > 0 such that the following properties hold: (a) If j |ai,j | < 1, then j |ai,j | < 1 − δ0 . which do not have the same direction, then for any z ∈ Tn , either (b) If ai and al are two rows | j ai,j zj | < 1 − δ0 or | j al,j zj | < 1 − δ0 . (c) If I ⊂ {1, . . . , n} is such that • all rows in I have the same direction, • | j ai,j | = 1 for any i ∈ I , then one can find z ∈ Tn such that | j ai,j zj | = 1 for any i ∈ I . Proof. The proof of the proposition is very easy. (a) is clear, (c) can be proved by induction. n with | To prove (b), pick a and a two rows such that there exists z ∈ T i l j ai,j zj | = 1 and | j al,j zj | = 1. Since j |ai,j zj | 1 and j |al,j zj | 1, one may find θ, φ ∈ R such that, for any j , ai,j zj = |ai,j zj |eiθ , al,j zj = |al,j zj |eiφ . Thus, ai and al c have the same direction.

2

Let us now comment how Theorem 5.2 can be expressed for linear maps. For ξ ∈ Tn with φI (ξ ) ∈ Tq , and I maximal with this property, then (i) two rows in I always have the same direction; (ii) for any i ∈ I , j |ai,j | = 1, and I is maximal with respect to (i) and (ii). Conversely, if I ⊂ {1, . . . , n} satisfies properties (i) and (ii) above and is maximal with respect to these properties, then one can find ξ ∈ Tn with φI (ξ ) ∈ Tq and I is maximal with respect to this last property. Moreover, for a linear map, the derivative is constant. Summarizing this, we have just to compare the trees RI,J and LI,J for those subsets I ⊂ {1, . . . , n} satisfying (i) and (ii) and maximal for ⊂, and for the associated subsets J . Observe in particular that we have to compare a finite number of trees and that this gives rise to a valuable algorithm: given a linear map φ : Dn → Dn , we can decide in a finite number of steps whether Cφ is continuous on A2 (Dn ) or not. 6.2. The proof We start with some ξ ∈ Tn and some δ ∈ (0, +∞)n and we suppose that φ −1 (S(ξ, δ)) is nonempty. δ0 is defined as in Proposition 6.2. We denote by I˜ = {i; δi δ0 }. All vectors ai , i ∈ I˜, have the same direction. Let I ⊂ {1, . . . , n} containing I˜, satisfying (i) and (ii), and maximal with respect to ⊂. We then define J = (j (1), . . . , j (s)), where s is the rank of dφI , as follows: • j (1) is such that δj (1) = min{δi ; i ∈ I };


1985

• j (2) is such that δj (2) = min{δi ; i ∈ I and φi ∈ / span(φj (1) )}; • more generally, for k s, j (k) is defined by

/ span(φj (1) , . . . , φj (k−1) ) . δj (k) = min δi ; i ∈ I and φi ∈ When several choices are possible, we take arbitrarily one of them. Among our assumptions on the trees, we know that for this specific choice of I and J , LI,J RI,J . For notational simplicity, we will assume that I = {1, . . . , q}, J = {1, . . . , s} and we will set δ1 = δ, δi = δ αi for i ∈ {1, . . . , q}, 1 = α1 α2 · · · αn > 0. From now on, throughout this section, we will forget the subscripts ξ, I, J on all the functions defined in Section 5. Observe that the volume of S(ξ, δ) is comparable to δ12 . . . δq2 (since δi > δ0 for all i > q, and δ0 is defined independently of ξ and δ). Thus, by the very definition of r and J , we just need to prove that V (φ −1 (S(ξ, δ))) is less than δ 2α1 r1 +2α2 r2 +···+2αs rs . To do that, we will just use that

φ −1 S(ξ, δ) ⊂ z ∈ Dn ; φl (z) − ξl < δ αl , l = 1, . . . , s := A1 . Our intention is to estimate the volume by Fubini’s theorem. We will separate the variables as follows. Let l ∈ {1, . . . , s} and let k = t (l). Define ψl exactly as in Lemma 5.1, namely ψl = φl −

l−1

βi,l φi

i=k+1

does not depend on the variables appearing in E(k + 1) ∪ · · · ∪ E(l). Moreover, the linear forms ψl|E (k) , l ∈ F (k), are linearly independent (recall that Fk = {l s; t (l) = k}). The coefficients βi,l do not depend on ξ or on δ and the sequence (αi ) is non-increasing. Thus, one can find C > 0 such that

A1 ⊂ ψl (z) − ωl < Cδ αl , l = 1, . . . , s := A2 for ωl = ξl − l−1 i=t (l)+1 βi,l ξi . Observe that, if q(l) > 0 (or, equivalently, t (l) = l), then ψl = φl and ωl = ξl . Let us now write E(k) = {zk,1 , . . . , zk,p }. Since the linear forms ψl|E (k) , t (l) = k, are independent, we can triangularize them with respect to the variables in E(k). Namely, up to a reordering of the variables in E(k), if we write F (k) = {m1 , . . . , mu } with u p, there exist coefficients γi,j such that u∗mj := ψmj −

γi,j ψmi

i<j

can be written ∗ , u∗mj = θmj zk,j + ∗zk,j +1 + · · · + ∗zk,p + vm j

1986


∗ just belong to E(1) ∪ · · · ∪ E(k − 1) and θ where the variables appearing in vm mj is non-zero. j As above,

A2 ⊂ z ∈ Dn ; u∗l (z) − ζl < Cδ αl , l = 1, . . . , s := A3 for some well-chosen constant C > 0 and, as above, when q(l) > 0, then u∗l = φl and ζl = ξl . We will consider A3 in the following form:

A3 =

z ∈ Dn ; u∗l (z) − ζl < Cδ αl .

k; q(k)>0 l∈F (k)

We begin with k = 1 and we write F (1) = {m1 , . . . , mu }, m1 = 1. For notational simplicity, we write E1 = {z1 , . . . , zp }, p = q(1). For z to belong to l∈F (1) {z ∈ Dn ; |u∗l (z) − ζl | < Cδ αl }, it is necessary that ⎧ |a1,1 z1 + · · · + a1,p zp − ζ1 | < Cδ α1 , ⎪ ⎪ ⎪ ⎪ ⎨ |θm2 z2 + ∗| < Cδ αm2 , ⎪ ... ⎪ ⎪ ⎪ ⎩ |θmu zu + ∗| < Cδ αmu . Since

j

|ai,j | = 1 and |ζ1 | = 1, the first line implies that e(1 − ζ1 zj ) Cδ α1 for all j ∈ (j )

{1, . . . , p}. We then define sets V1 by induction: (0)

• V1 = {(zu+1 , . . . , zp ) ∈ Dp−u ; e(1 − ζ1 zj ) Cδ α1 for j = u + 1, . . . , p} (we use here the information given by the first line for the last variables). • Using the first line and the last line, we get information on zu when zu+1 , . . . , zp are fixed: (1) (0) V1 = (zu , . . . , zp ) ∈ Dp−u+1 ; (zu+1 , . . . , zp ) ∈ V1 , e(1 − ζ1 zu ) Cδ α1

and |θmu zu + ∗zu+1 + · · · + ∗zp − ζmu | < Cδ αmu . (2)

(u−1)

• Inductively, we define V1 , . . . , V1 (j +1)

V1

by

(0) = (zu−j , . . . , zp ) ∈ Dp−u+j +1 ; (zu−j +1 , . . . , zp ) ∈ V1 , e(1 − ζ1 zu−j ) Cδ α1

α and |θmu−j zu−j + ∗zu−j +1 + · · · | < Cδ mu−j . (u)

• We finally define V1

by

(u) (u−1) V1 = (z1 , . . . , zp ) ∈ Dp ; (z2 , . . . , zp ) ∈ V1 , e(1 − ξ1 z1 ) Cδ α1

and a1,1 z1 − (· · · + ζ1 ) < Cδ α1 .


In particular,

l∈F (1) {z ∈ D

n;

1987

|u∗l (z) − ζl | < Cδ αl } is contained in V1 . Now, by Lemma 2.3 (u)

(u)

and by Fubini’s theorem, it is not difficult to see that the volume of V1 3

δ α1 2 (p−u)

u

is less than

αm 3 α (1+ α i ) 1 . min δ 2 α1 , δ 1

i=1

With the notations introduced in Section 5, this is also equal to 3

δ 2 α1 q(1)

u

α1 min 1, δ αmi − 2

i=1

which is also equal to δ

( 32 α1 q(1)+ t (m)=1,

α αm 21

(αm −

α1 2 ))

(4)

.

Thus, the volume of l∈F (1) {z ∈ Dn ; |u∗l (z) − ζl | < Cδ αl } is less than (4). Let now k2 be the least integer k > 1 with q(k) > 0. We turn to the computation of the volume of l∈F (k2 ) {z ∈ Dn ; |u∗l (z) − ζl | < Cδ αl }. We write E(k2 ) = {zp+1 , . . . , zp }, p − p = q(k2 ), and F (2) = {mp+1 , . . . , mu }, mp+1 = k2 . We want that ⎧ |ak2 ,p+1 zp+1 + · · · + ak2 ,p zp + ak2 ,1 z1 + · · · + ak2 ,p zp − ζk2 | < Cδ αk2 , ⎪ ⎪ ⎪ αm ⎪ ∗ ⎪ ⎨ θmp+2 zp+2 + ∗ + vmp+2 (z1 , . . . , zp ) − ζmp+2 < Cδ p+2 , .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ θ z + ∗ + v ∗ (z , . . . , z ) − ζ < Cδ αmu . m u u p m u m u 1 We will proceed exactly as before, except that we will assume now that the variables (z1 , . . . , zp ) (u) are fixed in V1 . Precisely, we set:

• V2 = {(z1 , . . . , zp , zu +1 , . . . , zp ) ∈ Dp+p −u ; (z1 , . . . , zp ) ∈ V1 and e(1 − ζk2 zj ) Cδ αk2 for j = u + 1, . . . , p }. • Using the first line and the last line, we get informations on zu when zu +1 , . . . , zp are fixed: (0)

(u)

(1) (0) V2 = (z1 , . . . , zp , zu , . . . , zp ) ∈ Dp+p −u +1 ; (z1 , . . . , zp , zu +1 , . . . , zp ) ∈ V2 , e(1 − ζk2 zu ) Cδ αk2 and

θm zu + ∗zu +1 + · · · + ∗zp + v ∗ (z1 , . . . , zp ) − ζm < Cδ αmu . m u u u (u −p−1)

(2)

• In the same vein and inductively, we define V2 , . . . , V2

(u −p)

and finally V2

.

Using Fubini’s theorem and Lemma 2.3, we obtain that

(u ) ( 32 α1 q(1)+ δ V V2

α t (m)=1, αm 21

(αm −

α1 2 ))

×δ

( 32 αk2 q(k2 )+

αk t (m)=k2 , αm 22

(αm −

αk 2 2

))

.

1988


The argument is exactly similar for the other values of k with q(k) > 0 and we finally find that V φ −1 S(ξ, δ) δ

3 k; q(k)>0 ( 2 αk q(k)+ t (m)=k, αm αk 2

(αm −

αk 2

))

.

Thus, we would like to prove that 3 αk αm − αk q(k) + 2α1 r1 + · · · + 2αs rs . 2 2

(5)

t (m)=k α αm 2k

k; q(k)>0

It is time to look carefully at the assumption L R. Suppose that αi = 21li where (l1 , . . . , ls ) is the index of a node in these trees, namely l1 = 0, li+1 − li ∈ {0, 1} for i less than or equal to some m, and li = ∞ for i > m + 1. As observed above, L(l1 , . . . , ls ) =

3 αk αk q(k) + , 2 2 t (m)=k lm lk

q; q(k)>0

R(l1 , . . . , ls ) = 2α1 r1 + · · · + 2αs rs . Now, lm = lk iff αm αk /2. Indeed, if lm is not equal to lk , then lm lk + 1 and αm αk /2 (recall that m k if t (m) = k). Moreover, when lm = lk , αk /2 is equal to αm − αk /2. Hence, L(l1 , . . . , ls ) =

3 αk αm − αk q(k) + 2 2

k; q(k)>0

t (m)=k α αm 2k

and the condition L R means that (5) is true when αi = 21li and (l1 , . . . , ls ) is the index of a node in the trees. The general case can be deduced by applying a convexity argument. Precisely, let R ∈ {,}s be an s-uple of or and define

FR = 1 = α1 α2 · · · αs > 0; ∀m ∈ {1, . . . , s}, αm Rm αt (m) /2 . It is clear that {1 = α1 α2 · · · αs 0} is the union of the sets FR , for R describing all choices of inequality signs, { , }s . Thus, it suffices to verify (5) on each FR . So, we may fix some R. FR is convex, and condition (5) is linear in α1 , . . . , αs . Thus, by the Krein–Milman theorem, it suffices to verify (5) at the extreme points of FR . Now, Lemma 6.3 below shows that the extreme points of FR are among the s-uple (α1 , . . . , αs ) with αi = 21li and (l1 , . . . , ls ) is the index of a node in the trees. Since (5) has already been proved in this case and is trivial in the second one, this ends the proof of the sufficient part of Theorem 5.2, provided the proof of the forthcoming lemma. Lemma 6.3. Let s 1, E ⊂ {(i, j ) ∈ {1, . . . , s}2 ; i j } and R = (Ri,j )(i,j )∈E with Ri,j ∈ { , , =}. Let also G be a partition of {1, . . . , s}. Define


1989

FE,R,G = 1 = α1 · · · αs = 0; ∀(i, j ) ∈ E, αj Ri,j αi /2

∀G ∈ G, ∀(u, v) ∈ G 2 , αu = αv . Then the extreme points of FE,R,G are among the points (α1 , . . . , αs ) with αi = li+1 − li ∈ {0, 1, +∞} for i = 1, . . . , s − 1.

1 2li

, l1 = 0,

Proof. We argue by induction of s, the result being clear for s = 1 or s = 2. Let (α1 , . . . , αs ) be ˜ and G ˜ R ˜ as follows: an extremal point of FE,R,G . Suppose first that αs = αs−1 . We define E, ˜ is defined from G by deleting s in the set which contains it; • G • E˜ = {(i, j ) ∈ E; 1 i j s − 1} ∪ {(t, s − 1); (t, s) ∈ E}; ˜ i,j is defined by • R ˜ i,j = Ri,j if 1 i j s − 2; – R ˜ t,s−1 = Rt,s−1 if (t, s − 1) ∈ E and (t, s) ∈ – R / E or if (t, s − 1) ∈ E, (t, s) ∈ E and Rt,s = Rt,s−1 ; ˜ t,s−1 = Rt,s if (t, s) ∈ E and (t, s − 1) ∈ / E; – R ˜ – Rt,s−1 = “ = ” otherwise, namely when (t, s − 1) and (t, s) belong to E, and Rt,s and Rt,s−1 have different values. The (s − 1)-uple (α1 , . . . , αs−1 ) belong to FE, ˜ ,G ˜ R ˜ (the constraints that we add are automatically satisfied since αs = αs−1 ). Suppose that it is not an extreme point of FE, ˜ ,G ˜ R ˜ . Then (α1 , . . . , αs−1 ) =

1 1 α1 , . . . , αs−1 + α1 , . . . , αs−1 , 2 2

. Then (α , . . . , α ) and (α , . . . , α ) do belong to and αs = αs−1 with α = α. Define αs = αs−1 s s 1 1 ˜ ˜ FE, ˜ ,G ˜ R ˜ (because of the definition of E and R) and (α1 , . . . , αs ) is not extremal, a contradiction. Thus, (α1 , . . . , αs−1 ) is an extreme point of FE, ˜ ,G ˜ R ˜ which implies, by induction hypothesis, that

α1 = 21li for i s − 1 with li+1 − li ∈ {0, 1, +∞} and l1 = 0. Since αs = αs−1 , this proves the lemma in that case. Suppose now that αs = αs−1 , namely that αs < αs−1 . Let

M = t < s; (t, s) ∈ E and αs = αt /2 . ˜ and G ˜ R ˜ as follows: We define E,

˜ is defined from G by gluing together the sets G ∈ G such that there exists t ∈ M ∩ G. Of • G course, we also delete {s}, which appears in G since αs < αs−1 ; • E˜ = {(i, j ) ∈ E; i j s − 1}; ˜ i,j = Ri,j for any (i, j ) ∈ E. ˜ • R As before, we intend to show that (α1 , . . . , αs−1 ) is an extreme point of FE, ˜ ,G ˜ R ˜ (observe that it really belongs to this set). If this is not the case, one can write (α1 , . . . , αs−1 ) =

1 1 α , . . . , αs−1 + α1 , . . . , αs−1 2 1 2

1990


) = (α , . . . , α with (α1 , . . . , αs−1 1 s−1 ). If M is empty, the constraints αs Rs,u αu /2 are strictly satisfied for any u with (u, s) ∈ E (namely αs < αu /2 or αs > αu /2). Thus if we choose ) and (α , . . . , α ) above very close to (α , . . . , α (α1 , . . . , αs−1 1 s−1 ), the conditions 1 s−1 α sRs,u αu /2 and α sRs,u αu /2 keep being satisfied. Thus, (α1 , . . . , αs ), (α1 , . . . , αs ) ∈ FE,R,G and

(α1 , . . . , αs ) =

1 1 α1 , . . . , αs + α1 , . . . , αs , 2 2

a contradiction. If M is non-empty, we set αs = αt /2 and αs = αt /2 for any t ∈ M. As above, one can ensure ) that (α1 , . . . , αs ) and (α1 , . . . , αs ) belong to FE,R,G , provided we have chosen (α1 , . . . , αs−1 and (α1 , . . . , αs−1 ) very close to (α1 , . . . , αs−1 ). Thus, in both cases, αi = 21li for i s − 1 with li+1 − li ∈ {0, 1, +∞} and l0 ∈ {0, +∞}. To conclude, we observe that if M is non-empty, αs = αt /2 has the desired form. On the contrary, when M = ∅, every condition on αs is strictly satisfied. This implies that αs = 0. Otherwise, we could write 1 1 (α1 , . . . , αs ) = (α1 , . . . , αs + ε) + (α1 , . . . , αs − ε) 2 2 with (α1 , . . . , αs ± ε) ∈ FE,R,G for ε small enough.

2

Remark 6.4. The sufficient part of Theorem 5.2 remains valid for an affine map with a similar proof. 7. Proof of the necessary part In this section, we intend to prove the “necessary part” of our main theorem. So we start with some φ : Dn → Dn holomorphic in a neighbourhood of Dn . Let ξ ∈ Tn , I, J ⊂ {1, . . . , n} with φI (ξ ) ∈ Tq and such that the condition of our main theorem fails for ξ, I, J . Namely, there exists a node (l1 , . . . , ls ) such that Lξ,I,J (l1 , . . . , ls ) < Rξ,I,J (l1 , . . . , ls ). For notational convenience, we suppose that ξ = e, I = {1, . . . , q}, dφI (ξ ) has rank s, J = {1, . . . , s} and φI (e) = (1, . . . , 1). From now on, we will forget throughout this section 1

the subscript ξ, I, J . Let δ > 0 and set, for i = 1, . . . , s, δi = δ 2li . For k ∈ {s + 1, . . . , q}, δk is defined by 1

δk = δ 2li

provided dφk belongs to span(dφ1 , . . . , dφi ) and does not

belong to span(dφ1 , . . . , dφi−1 ) (observe that dφk belongs to span(dφ1 , . . . , dφs ) for s + 1 k q). For k > q, we set δk = 2 and as usual, δ = (δ1 , . . . , δn ). This part of the proof will be done if we are able to show that lim

δ→0

V (φ −1 (S(e, Cδ))) V (S(e, δ))

= +∞


1991 s

2ri

for some fixed constant C > 0. It can be observed that V (S(e, δ)) behaves exactly like δ i=1 2li = δ R(l1 ,...,ls ) . Let us now give a lower bound for V (φ −1 (S(e, Cδ))). For i q, using Corollary 2.7, φi writes φi (z) = 1 +

ai,j (zj − 1) + O

j 1

|zj − 1|2

j 1, ai,j =0

with ai,j 0. The condition ai,j = 0 implies that zj belongs to E(1) ∪ · · · ∪ E(i). We then deduce that z ∈ φ −1 (S(e, Cδ)) as soon as, for all i q, δi and a (z − 1) |zj − 1|2 δi i,j j j ∈E (1)∪···∪E (i)

j 1

namely dφi (e)(z1 − 1, . . . , zn − 1) δi

and

|zj − 1|2 δi .

j ∈E (1)∪···∪E (i)

We now triangularize these inequalities like in the “sufficient part” of the proof. Precisely, ψl and u∗l are defined from the linear forms dφi (e), 1 i, l s, like in Section 6 (here, they were defined using φi but φ was linear!). Let us also set

A1 := z ∈ Dn ; dφi (z1 − 1, . . . , zn − 1) δi for all i s . Since we just triangularize the system (each dφi is a linear combination of the linear forms u∗j , with j i), and since the sequence (δi ) is non-decreasing, A1 contains the set

A2 := z ∈ Dn ; u∗i (z1 − 1, . . . , zn − 1) εδi for all i s for ε > 0 small enough (and independent of δ). As before, we write A2 =

z ∈ Dn ; u∗l (z1 − 1, . . . , zn − 1) εδl .

k; q(k)>0 l∈F (k)

Webegin with k = 1 and we write F (1) = {m1 , . . . , mu } and E(1) = {z1 , . . . , zp }. For z to belong to l∈F (1) {z ∈ Dn ; |u∗l (z1 − 1, . . . , zn − 1)| εδl }, it suffices that ⎧ 1 ⎪ l ⎪ ⎪ a1,1 (z1 − 1) + · · · + a1,p (zp − 1) < εδ 2 1 , ⎪ ⎪ ⎪ 1 ⎪ ⎨ θ (z − 1) + ∗ < εδ 2lm2 , m2

2

⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ 1 ⎪ ⎩ θmu (zu − 1) + ∗ < εδ 2lmu . We will separate these conditions in two different cases. Let v be the biggest integer such that lmv = l1 . v is equal to the cardinal number of {m; lm = l1 }. For j v + 1 (and j p) we just

1992

F. Bayart / Journal of Functional Analysis 260 (2011) 1969–2003 1

1

impose that 1 − ηδ 2l1 e(zj ) 1 for some small η > 0. Then |zj − 1| η1/2 δ 2l1 +1 . Hence, the conditions 1 θm (zj − 1) + ∗(zj +1 − 1) + · · · + ∗(zp − 1) < εδ 2lmj j

are automatically satisfied for v + 1 j p. For a fixed j in that interval, observe also that zj 3

×

1

can live in a subset of D of volume δ 2 2l1 . For the other lines, we go backward. We first study 1 θm (zv − 1) + ∗(zv+1 − 1) + · · · + ∗(zp − 1) < εδ 2l1 . v

(6)

1

When zv+1 , . . . , zp have been fixed with the condition 1 − ηδ 2l1 e(zj ) 1, (6) will be satis1

1

fied as soon as zv belongs to some disk of center ω satisfying 1 − Cηδ 2l1 e(ω) 1 + Cηδ 2l1 1

1

and | m(ω)| (Cηδ 2l1 )1/2 and of radius like εδ 2l1 . By Lemma 2.4, these zv can live in a sub2

set of D of volume δ 2l1 , provided η > 0 is small enough. Moreover, it is worth noting that 1

1

1 − Cεδ 2l1 e(zv ) 1. Restricting the radius of the disk to ε δ 2l1 (which does not change the 1

order of growth of its volume), we can always assume that 1 − ε δ 2l1 e(zv ) 1 with ε much smaller than ε. This allows us to do exactly the same thing for the previous line θ m

v−1

1 (zv−1 − 1) + ∗(zv − 1) + · · · + ∗(zp − 1) < εδ 2l1 ,

1

1 e(z ) 1 for j = v, . . . , p. We can carry on this the crucial point being only 1 − ε δ 2l j process to conclude that z belongs to l∈F (1) {z ∈ Dn ; |u∗l (z1 − 1, . . . , zn − 1)| < εδl } as soon as (z1 , . . . , zp ) belongs to some set V(1) satisfying

⎧ 2v 3 p−v 1 3 v 1 3 1 ⎨ V V(1) δ 2l1 δ 2 × 2l1 = δ 2l1 ( 2 p+ 2 ) = δ 2l1 ( 2 q(1)+ t (m)=1, lm =l1 2 ) , 1 ⎩ 1 − ε δ 2l1 e(zj ) 1 for any (z1 , . . . , zp ) ∈ V(1) and any j ∈ {1, . . . , p}. From now on, (z1 , . . . , zp ) will always be considered as fixed in V(1). We now consider k2 the least integer k > 1 with q(k) > 0 and we turn to give a lower bound for the volume of

z ∈ Dn ; u∗l (z1 − 1, . . . , zn − 1) εδl ∩ V(1). l∈F (k2 )

We write E(k2 ) = {zp+1 , . . . , zp } and F (k2 ) = {mp+1 , . . . , mu }. We are looking for (zp+1 , . . . , zp ) such that


1993

⎧ 1 ⎪ ⎪ ak ,p+1 (zp+1 − 1) + · · · + ak ,p (zp − 1) + ak ,1 (z1 − 1) + · · · + ak ,p (zp − 1) < Cδ 2lk2 , ⎪ 2 2 2 ⎪ 2 ⎪ ⎪ 1 ⎪ ⎪ l ⎨ θmp+2 (zp+2 − 1) + ∗ + ∗(z1 − 1) + · · · + ∗(zp − 1) < Cδ 2 mp+2 , ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ 1 ⎪ l ⎩ θmu zu + ∗ + ∗(z1 − 1) + · · · + ∗(zp − 1) < Cδ 2 mu . We can argue exactly as before. Indeed, the terms (z1 − 1), . . . , (zp − 1) are unimportant because 1 l

1

we already know that 1 − ε δ 2 k2 1 − ε δ 2l1 e(zm ) 1 for 1 m p. They do not change anything for the last lines (those for which lmj > lk2 l1 ): if we restrict zv +1 , . . . , zu so that they 1 l

satisfy 1 − e(zj ) ηδ 2 k2 , the last inequalities will be satisfied given any (z1 , . . . , zp ) ∈ V(1) 1

1

since |zm − 1| ε δ 2l1 +1 ε δ 2 mj for 1 m p. They just change slightly the center for the first inequalities (those for which lmj = lk2 l1 ). However, the center ω keeps on satisfying l

1 l

1 l

1 l

1 − Cηδ 2 k2 e(ω) 1 + Cηδ 2 k2 and | m(ω)| (Cηδ 2 k2 )1/2 so that this does not affect the volume (Lemma 2.4 remains valid). We can continue this process inductively for each k with q(k) > 0. At the end, we prove that |dφi (e)(z1 − 1, . . . , zn − 1)| δi for all i s provided (z1 , . . . , zn ) belongs to some set V whose volume is greater than

1

δ 2lk

( 32 q(k)+ t (m)=1,

1 lm =l1 2 )

= δ L(l1 ,...,lm ) .

k; q(k)>0

Moreover, for any z ∈ V, one has 1

1 − εδ 2lk e(zj ) 1 provided zj ∈ E(k). In particular, this implies 1

|zj − 1|2 Cδ 2lk

provided zj ∈ E(k).

(7)

(S(e, Cδ)) for some C > 0 when z belongs to V. We have now to deduce that z belongs to φ −1 First of all, when i > s, we know that dφi (e) = j s ∗dφj (e). Since (δi ) is non-decreasing, we obtain immediately that |dφi (e)(z1 − 1, . . . , zn − 1)| Cδi for all i ∈ {1, . . . , q}. Second, we have to verify that, for any i ∈ {1, . . . , q},

|zj − 1|2 Cδi .

j ∈E (1)∪···∪E (i)

Now, if j ∈ E(k) with k i, then by (7), |zj − 1|2 Cδk Cδi . This shows the desired fact and concludes the “necessary part” of Theorem 5.2.

1994


8. Weighted Bergman spaces To avoid complications in the statement and in the proof of Theorem 5.2, we just gave it for the unweighted Bergman space. Corresponding theorems are valid for the weighted Bergman spaces. If their proofs are completely similar, they have interesting consequences. As we might think, the crucial point is to estimate the Vβ -measure of some subsets of Dn . We will be more precise than before. This will be very useful in the next section. Lemma 8.1. Let β > −1. There exists Cβ > 0 such that, for any ζ ∈ Tn , for any δ ∈ (0, 2)n , Cβ−1 (δ1 . . . δn )2+β Vβ S(ζ, δ) Cβ (δ1 . . . δn )2+β . Moreover, when β ∈ (−1, 0], the constant Cβ may be choosen independently of β. Proof. By Fubini’s theorem and rotational invariance, one just need to estimate Vβ (Sδ ) where Sδ = {z ∈ D; |1 − z| < δ}. This can be done by polar integration: 1

π

Vβ (Sδ ) (β + 1)

β 1Sδ reiθ 1 − r 2 r dr dθ

r=1/2 θ=−π

1

π

(β + 1)Dβ

1Sδ reiθ (1 − r)β dr dθ

r=1/2 θ=−π

with Dβ = ( 32 )β 23 if β ∈ (−1, 0] and Dβ = ( 12 )β if β 0. Moreover, Sδ contains the set √ √ {reiθ ; 1 − r < δ/ 2 and |θ | < δ/ 2 }. Thus, δ Vβ (Sδ ) (β + 1)Dβ √ 2

1 (1 − r)β dr √ r=min(1/2,1−δ/ 2 )

1 δ β+1 δ Dβ √ min , √ 2 2 2 which proves one inequality. On the other hand, it is clear that there exists some C > 0 such that reiθ ∈ Sδ implies |θ | Cδ. Thus, 1

Cδ

Vβ (Sδ ) (β + 1)Bβ

(1 − r)β dr dθ √ r=1−δ/ 2 θ=−Cδ

with Bβ = ( 12 )β if β < 0 and Bβ = ( 32 )β if β 0. We end up the proof as before.

2


1995

Lemma 8.2. Let β > −1. There exists Cβ > 0 such that, for any u ∈ T, for any v ∈ C, for any δ > 0 and for any α > 0,

3 ¯ < δ and |v − z| < δ α Cβ min δ 2 +β , δ 1+α+β . Vβ z ∈ D; e(1 − uz) Moreover, when β ∈ (−1, 0], the constant Cβ may be choosen independently of β. Proof. Without loss of generality, we may suppose u = 1. The biggest volume is obtained for v ∈ [0, +∞). In that case, our set is contained in

D ∩ z ∈ D; 1 − δ e(z) 1 and m(z) δ α . This last set is contained in iθ

re ; r > 1 − δ and |θ | Cδ α for some C > 0. We now conclude like in the proof of Lemma 8.3.

2

Our last estimate does not need to be uniform for β ∈ (−1, 0]. We omit its proof which is easy. Lemma 8.3. Let β > −1. There exist Cβ > 0 and ε > 0 such that, for every δ > 0, for every w ∈ C satisfying 1 − εδ e(w) 1 + εδ

√ and m(w) εδ,

then Vβ ({z ∈ D; |z − w| < δ}) Cβ δ 2+β . We have to introduce the trees corresponding to A2β (Dn ). Let φ : Dn → Dn be holomorphic and suppose that φ extends holomorphically in a neighbourhood of Dn . Let ξ, I, J and s like in Section 5. The function r, q, E and t are also defined like in Section 5. We just need to modify β β the definition of the trees to take into account Lemmas 8.1, 8.2 and 8.3. Lξ,I,J and Rξ,I,J are now defined by β

Rξ,I,J (l1 , . . . , ls ) =

s (2 + β)rξ,I,J (k) k=1

β

Lξ,I,J (l1 , . . . , ls ) =

k; q(k)>0

2 lk

1 2 lk

,

1 3 + β qξ,I,J (k) + . 2 2 t (m)=k lm =lk

Our main theorem becomes Theorem 8.4. Suppose that Cφ is continuous on A2β (Dn ). Then for any ξ, I, J as above, one has β

β

Lξ,I,J Rξ,I,J . When φ is linear, Cφ is continuous on A2β (Dn ) if and only if, for any ξ, I, J

1996

F. Bayart / Journal of Functional Analysis 260 (2011) 1969–2003 β

β

as above, Lξ,I,J Rξ,I,J . More precisely, in that case, there exists Cβ > 0 such that, for any ξ ∈ Tn , for any δ ∈ (0, 2)n , Vβ φ −1 S(ξ, δ) Cβ Vβ S(ξ, δ) . The constant Cβ may be choosen independently of β when β ∈ (−1, 0]. Proof. If we forget the last part of the statement, the proof of Theorem 5.2 can be repeated “mutatis mutandis” here. However, the last assertion needs some comments. We follow the notations of Section 6 and we fix β ∈ (−1, 0]. By Lemma 8.1, we know that Vβ S(ξ, δ) Cδ 2α1 r1 +···+2αs rs +β(r1 +···+rs ) for some C > 0 which does not depend on β ∈ (−1, 0] (of course, C depends on δ0 which is fixed by φ). Next, we find an upper bound for Vβ (φ −1 (S(ξ, δ))). The triangularization process does not depend on β. In particular, the inclusion φ −1 (S(ξ, δ)) ⊂ A3 remains true for every β > −1. Finally, when we compute Vβ (A3 ), we replace everywhere Lemma 2.3 by Lemma 8.2 and we find that

Vβ (A3 ) Cδ

3 k; q(k)>0 ( 2 +β)αk q(k)+ t (m)=k, αm αk 2

(αm −

αk 2

)

for some constant C which does not depend on β. Thus, Vβ φ −1 S(ξ, δ) CVβ S(ξ, δ) 1

when δi = δ 2li , (l1 , . . . , ls ) being the index of a node in the trees and C is independent of β. We can now apply the convexity argument, exactly as before, and the constants which are involved do not depend on β ∈ (−1, 0]. 2 As we mentioned in the introduction, when φ : Bn → Bn is smooth on Bn , Cφ is continuous on some A2β (Bn ) if and only if it is continuous on any A2β (Bn ). This property is far from being true on the polydisk, even if Jafari has proven in [5] that continuity on A2β1 (Dn ) implies continuity on A2β2 (Dn ) for any β2 β1 . The converse does not hold. Example 8.5. Let β1 , β2 ∈ (−1, +∞) be such that β1 < β2 . There exist n 2 and a linear map φ : Dn → Dn such that Cφ is continuous on A2β2 (Dn ) and Cφ is not continuous on A2β1 (Dn ). 3

+β

2 Proof. The function β → 2+β is increasing on (0, +∞). Thus, we may find two integers r and q, with q r, such that

3 + β1 + β2 r −1 < 2 < . 2 + β1 q − 1 2 + β2 3 2


1997

This translates into

1 3 + β2 q + > (2 + β2 )r 2 2

and

1 3 + β1 q + < (2 + β1 )r. 2 2

(8)

We now consider φ : Dq → Dq defined by φ(z) = u(z), . . . , u(z), 0, . . . , 0 r times

with u(z) = (z1 + · · · + zq )/q. Here, the trees associated to φ are very easy. They have just one node and Lβ (0) =

1 3 +β q + 2 2

whereas R β (0) = (2 + β)r.

The conclusion follows immediately from Theorem 8.4 and inequality (8).

2

In the same vein, it is not hard to check that the composition operator Cφ studied in Example 5.4 is continuous on A20 (D7 ) but is not continuous on any A2β (D7 ) for any β < 0. We may also observe that Example 8.5 cannot be proved for a fixed n ∈ N. For instance, Corollary 8.6. Let φ : D2 → D2 be a linear map. Then Cφ is continuous on some A2β (Dn ) if and only if it is continuous on any A2β (Dn ), β > −1. Proof. We distinguish several cases. When φ1 ∞ < 1 and φ2 ∞ < 1, there is nothing to prove. When φ1 ∞ = 1 and φ2 ∞ < 1, our trees will have just one node, with r(1) = 1, q(1) ∈ {1, 2} and t (1) = 1. The condition of continuity becomes 2+β

1 3 +β q + 2 2

and this condition is always satisfied. When φ1 ∞ = 1 and φ2 ∞ = 1, two subcases may occur. On the one hand, we may have s = 1 (namely φ2 is a multiple of φ1 ). Our trees have also one node, with r(1) = 2, q(1) ∈ {1, 2} and t (1) = 1. The condition now reads 4 + 2β

3 1 +β q + . 2 2

This is never satisfied! On the other hand, we may have s = 2. This implies r(1) = 1, r(2) = 2, q(1) ∈ {1, 2}, q(2) = 2 − q(1), t (1) = 1, t (2) = 2 when q(1) = 1 or t (2) = 1 when q(2) = 2. Thus we get one of the following 3-uple of conditions:

1998


⎧ 3 1 ⎪ ⎪ 2 + β + β + , ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎨ 1 3 1 3 +β + + +β + , (2 + β) + (2 + β) ⎪ 2 2 2 2 ⎪ ⎪ ⎪ ⎪ 3 1 1 3 1 1 ⎪ ⎪ ⎩ (2 + β) + (2 + β) +β + + +β + , 2 2 2 2 2 2 ⎧ 3 1 ⎪ ⎪ 2+β +β 2+ , ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎨ 1 1 3 +β 2+ + , (2 + β) + (2 + β) ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ 3 1 1 ⎪ ⎪ ⎩ (2 + β) + (2 + β) +β 2+ . 2 2 2 Both 3-uple of conditions are always satisfied! In particular, this proof shows that Cφ is always continuous on A2β (D2 ), φ being linear, except if φ1 ∞ = φ2 ∞ = 1 and φ2 is a multiple of φ1 . Thus, Theorem 3.2 remains valid on A2β (D2 ) for any β > −1 when we just consider linear maps. 2 The last corollary of this section indicates the strategy of the next one! Corollary 8.7. Let β0 ∈ (−1, +∞) and φ : Dn → Dn be linear. Suppose that Cφ is continuous β on A2 (Dn ) for any β > β0 . Then Cφ is continuous on A2β0 (Dn ). β

β

Proof. It suffices to let β to β0 in the inequalities Lξ,I,J Rξ,I,J , valid for β β0 .

2

9. Hardy spaces We conclude this paper by showing that an appropriate version of Theorem 5.2 remains true on the Hardy space H 2 (Dn ). There is one more difficulty in that context: we cannot testify if a measure is a Carleson measure by testing it only on rectangles. Precisely, let I be an interval of T of length δ and center ei(θ0 +δ/2) . S(I ) is defined by S(I ) = {z ∈ D; 1 − δ < r < 1, θ0 < θ < θ0 + δ}. If R = I1 × · · · × In ⊂ Tn is a rectangle of Tn , namely each Ij is an interval of T, S(R) is defined by S(R) = S(I1 ) × · · · × S(In ).

If V is any open set in Tn , S(V ) is equal to S(V ) = α S(Rα ) where (Rα ) runs through all rectangles in V . Let also μ be a Borel measure on Dn . Then Chang [3] has proven that the identity map H 2 (Dn ) → L2 (μ), f → f , is bounded iff there exists C > 0 such that μ S(V ) Cσ (V )

for all connected open sets V ⊂ Tn

(9)


1999

where σ is the Lebesgue measure on Tn . Moreover, Carleson [2] has given an example of a measure satisfying (9) for all rectangles R and not for all connected open sets V ⊂ Tn . Keeping this in mind we realize that the difficult part will be to adapt the sufficient part of the proof, because we need to control σ (φ −1 (S(V ))) for any V connected and open and not only for rectangles. However, a similar statement remains true. We keep the notations of Section 8, β β except that we allow our trees Lξ,I,J and Rξ,I,J to be defined also for β = −1. More precisely, −1 (l1 , . . . , ls ) = Rξ,I,J

s rξ,I,J (k) k=1

L−1 ξ,I,J (l1 , . . . , ls ) =

2 lk

k; q(k)>0

1 2 lk

,

1 qξ,I,J (k) + . 2 2 t (m)=k lm =lk

Our main theorem now reads Theorem 9.1. Suppose that Cφ is continuous on H 2 (Dn ). Then for any ξ, I, J as above, one has −1 2 n L−1 ξ,I,J Rξ,I,J . When φ is linear, Cφ is continuous on H (D ) if and only if, for any ξ, I, J as −1 above, L−1 ξ,I,J Rξ,I,J .

The proof of the necessary condition carries on without any new difficulties, replacing Lemma 2.4 by an appropriate analogue, whose proof is left to the reader: Lemma 9.2. There exists ε > 0 such that, for every δ > 0, for every w ∈ C satisfying √ and m(w) εδ,

1 − εδ e(w) 1 + εδ then σ ({z ∈ T; |z − w| < δ}) Cδ.

To prove the sufficient condition, we do not use directly Carleson measures on the Hardy space. We follow an indirect method with two steps: −1 Step 1. We show that L−1 ξ,I,J Rξ,I,J implies Lξ,I,J Rξ,I,J for any β > −1. β

β

Step 2. We fix β ∈ (−1, 0]. By Theorem 8.4, Cφ is continuous on A2β (Dn ). More precisely, we know that there exists C > 0 which does not depend on β such that, for any ξ ∈ Tn , for any δ ∈ (0, 2)n , Vβ φ −1 S(ξ, δ) CVβ S(ξ, δ) . We will prove later that this implies Cφ (f )A2 (Dn ) Df A2 (Dn ) for any f ∈ A2β (Dn ), β β for some constant D > 0 which does not depend on β. Letting β to −1, this implies Cφ (f )H 2 (Dn ) Df H 2 (Dn ) for any f ∈ H 2 (Dn ). Hence, it remains to verify the two above claims to close the proof of Theorem 9.1.

2000


9.1. Assumptions on H 2 (Dn ) vs. assumptions on A2β (Dn ) −1 We suppose that L−1 ξ,I,J Rξ,I,J and we try to prove that Lξ,I,J Rξ,I,J for any β > −1. For the sake of clarity, we forget the subscript ξ, I, J . Let (l1 , . . . , ls ) be the index of a node in the trees. One can write 2r(m) 3 1 − Lβ (l1 , . . . , ls ) − R β (l1 , . . . , ls ) = q(k) + 2 2 lm 2lm +1 β

t (m)=k lm =lk

k;q(k)>0

+β

q(k) −

β

t (m)=k

r(m) . 2 lm

t (m)=k

k;q(k)>0

The condition L−1 (l1 , . . . , ls ) − R −1 (l1 , . . . , ls ) 0 will imply Lβ (l1 , . . . , ls ) − R β (l1 , . . . , ls ) 0 as soon as r(m) q(k) − 0. (10) 2 lm t (m)=k

k;q(k)>0

Expanding L−1 (l1 , . . . , ls ) − R −1 (l1 , . . . , ls ) 0 we get

q(k)

2r(m) − δl ,l m k . 2 lm

k;q(k)>0 t (m)=k

k;q(k)>0

Now, r(m) 1 so that 2r(m) − δlm ,lk r(m) which yields immediately (10). 9.2. A precise version of Carleson embedding theorem In Lemma 2.1, we have already recalled that, when φ : Dn → Dn satisfies Vβ φ −1 S(ξ, δ) CVβ S(ξ, δ) for all δ ∈ (0, 2)n and all ξ ∈ Tn , then Cφ (f )

A2β (Dn )

Df A2 (Dn ) β

for every f ∈ A2β (Dn ). It is well known that the constant D may be controlled by C. However, this dependance with respect of β is not clarified. Our strategy requires that D may be controled uniquely by C and n, and in particular that it does not depend on β ∈ (−1, 0]. This is the content of the next proposition. Proposition 9.3. Let β ∈ (−1, 0] and let μ be a finite nonnegative Borel measure on Dn . Suppose that there exists C > 0 such that, for any ξ ∈ Tn and any δ ∈ (0, 2)n , μ S(ξ, δ) Cμ Vβ S(ξ, δ) .

(11)


2001

Then for any f ∈ A2β (Dn ),

f (z1 , . . . , zn )2 dμ

1/2 C(n)Cμ f A2 (Dn ) . β

Dn

Proof. We follow the argument of [6] except at the very beginning. Let z ∈ Dn and let δj = 1 − |zj |2 . Consider Wz the polydisk centered at z and with radius δj /2 in the zj -coordinate. Let also Sz be the Carleson box S(ξ, δ) with ξj = zj /|zj |. Then Wz ⊂ Sz . Moreover, for any f ∈ A2β (Dn ), the sub-mean value property for |f | gives, for any γ = (γ1 , . . . , γn ) with γj ∈ (0, δj /2), f (z)

1 (2π)n

f (z1 + γ1 u1 , . . . , zn + γn un ) dσ (u).

u∈Tn

On the other hand, by polar integration,

β f (w) 1 − |wj |2 dA(w) j

Wz δ1 /2

=

δn /2

n β f (z + γ u) γj 1 − |zj + γj uj |2 dσ (u) dγ1 . . . dγn .

...

γ1 =0

j =1

γn =0 u∈Tn

Now, 1 − |zj + γj uj | δj + γj . Taking into account that β 0 and that 1 − |wj |2 2(1 − |wj |), we get

β f (w) 1 − |wj |2 dA(w) j

Wz

δ1 /2

C(n)

...

γ1 =0

C(n)

δn /2

γn =0

δj /2 n j =1 γ =0 j

Now,

j

γj (γj + δj )β

f (z + γ u) dσ (u) dγ1 . . . dγn

u∈Tn

γj (γj + δj )β dγj f (z).

2002

F. Bayart / Journal of Functional Analysis 260 (2011) 1969–2003 δj /2

δj /2

γj (γj + δj ) dγj

γj (γj + δj )β dγj

β

γj =0

γj =δj /4

δj 1 × × 4 β +1

β+1 β+1 3 5 β+1 δj − 2 4

C β+2 δ β +1 j

where C does not depend on β. We then get

n β f (w) 1 − |wj |2 dA(w) j =1

Wz

n C(n) β+2 f (z) δ j (β + 1)n j =1

so that f (z) C(n) Vβ (Sz )

|f | dVβ Sz

where we have used Lemma 8.1 in its precise formulation. This inequality improves the work done in [6], since we know that the constant C(n) which appears above does not depend on β. From now on, we can follow exactly the proof of Jafari. Let 1 B(f )(z) = sup V S=S(ξ,δ); z∈S(ξ,δ) β (S)

|f | dVβ . S

We have obtained |f (z)| C(n)B(f )(z). In [6], it is shown that, under assumption (11), B defines a bounded operator from L2 (dVβ ) into L2 (μ) with B Cμ D(n), where D(n) just depends on n. This gives exactly what we need. 2 10. Concluding remarks 10.1. Compactness At least for the weighted Bergman spaces, the work that we have done for continuity can be modified to study the compactness of composition operators. Using the fact that in Lemma 2.1, the big-oh condition which characterizes continuity has to be replaced by a little-oh condition to characterize compactness, we obtain: Theorem 10.1. Suppose that Cφ is compact on A2β (Dn ). Then for any ξ, I, J as above, one has β

β

Lξ,I,J > Rξ,I,J . When φ is linear, Cφ is compact on A2β (Dn ) if and only if, for any ξ, I, J as β

β

above, Lξ,I,J > Rξ,I,J .


2003

In particular, we obtain linear symbols φ : Dn → Dn generating non-trivial compact composition operators. The term “non-trivial” means here that some coordinate function φi satisfies φi ∞ = 1. For instance, the composition operator which appears in Example 5.5 is compact. Looking at the proof of Wogen’s theorem, it can be shown that continuous composition operators on the ball with a smooth symbol are never compact, except the trivial ones. 10.2. Open questions Our work leads to several interesting questions on composition operators on the polydisk. We just quote two of them. Does Theorem 3.1 remains true for the Hardy space H 2 (Dn )? Is the necessary condition of Theorem 5.2 also sufficient for a larger class of maps than affine self-maps? References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

M. Abate, The Julia–Wolff–Carathéodory theorem in polydisks, J. Anal. Math. 74 (1998) 275–306. L. Carleson, A counterexample for measures bounded on H p for the bi-disc, Mittag-Leffler Report 7 (1974). A. Chang, Carleson measure on the bi-disc, Ann. of Math. 109 (1979) 613–620. J. Cima, C. Stanton, W. Wogen, On boundedness of composition operators on H 2 (B2 ), Proc. Amer. Math. Soc. 91 (1984) 217–222. F. Jafari, On bounded and compact composition operators in polydiscs, Canad. J. Math. 42 (1990) 869–889. F. Jafari, Carleson measures in Hardy and weighted Bergman spaces of polydiscs, Proc. Amer. Math. Soc. 112 (1991) 771–781. H. Koo, W. Smith, Composition operators induced by smooth self-maps of the unit ball in CN , J. Math. Anal. Appl. 329 (2007) 617–633. H. Koo, M. Stessin, K. Zhu, Composition operators on the polydisc induced by smooth symbols, J. Funct. Anal. 254 (2008) 2911–2925. B. MacCluer, Spectra of automorphism-induced composition operators on H p (BN ), J. London Math. Soc. 30 (1984) 95–104. J.H. Shapiro, Composition Operators and Classical Function Theory, Universitext, Springer, New York, 1993. W. Wogen, The smooth mappings which preserve the Hardy space H 2 (Bn ), in: Contributions to Operator Theory and Its Applications, vol. 35, Mesa, AZ, 1987, Birkhäuser, 1988, pp. 249–263.


Regularity estimates of solutions to complex Monge–Ampère equations on Hermitian manifolds Xi Zhang a,∗,1 , Xiangwen Zhang b a Department of Mathematics, Zhejiang University, PR China b Department of Mathematics and Statistics, McGill University, Canada

Received 15 July 2010; accepted 21 December 2010 Available online 5 January 2011 Communicated by I. Rodnianski

Abstract In this paper, we obtain the Bedford–Taylor interior C 2 estimate and local Calabi C 3 estimate for the solutions to complex Monge–Ampère equations on Hermitian manifolds. © 2010 Elsevier Inc. All rights reserved. Keywords: Complex Monge–Ampère equation; Hermitian manifold; Regularity estimates

1. Introduction The complex Monge–Ampère equation is one of the most important partial differential equations in complex geometry. The proof of the Calabi conjecture given by S.T. Yau [18] in 1976 yields significant applications of the Monge–Ampère equation in Kähler geometry. After that, many important geometric results, especially in Kähler geometry, were obtained by studying this equation. It is natural and also interesting to study the complex Monge–Ampère equations in a more general form and in different geometric settings. There are many modifications and generalizations in the existing literature. In [17], Tosatti, Weinkove and Yau gave a partial affirmative answer to a conjecture of Donaldson in symplectic geometry by solving (under additional curvature assumption) the complex Monge–Ampère * Corresponding author.

E-mail addresses: [email protected] (X. Zhang), [email protected] (X. Zhang). 1 The first named author was supported in part by NSF in China, Nos. 10831008 and 11071212.


X. Zhang, X. Zhang / Journal of Functional Analysis 260 (2011) 2004–2026

2005

equation in an almost Kähler geometric setting. By studying a more general form of the Monge– Ampère equation on non-Kähler manifolds, Fu and Yau [8] gave a solution to the Strominger system which is motived by superstring theory. Another direction worth studying is the corresponding equation on Hermitian manifolds. In such a case the equation is not so geometric, since Hermitian metrics do not represent positive cohomology classes. On the other hand the estimates for Hermitian manifolds are more complicated than the Kähler case because of the non-vanishing torsion. In the eighties and nineties, some results regarding the Monge–Ampère equation in the Hermitian setting were obtained by Cherrier [3,4] and Hanani [10]. For the next few years there was no activity on the subject until very recently, when the results were rediscovered and generalized by Guan and Li [9]. Under additional conditions they generalized the a priori estimates due to Yau [18] from the Kähler case and got some existence results for the solution of the complex Monge–Ampère equation. At the same time, Zhang [19] independently proved similar a priori estimates in the Hermitian setting and he also considered a general form of the complex Hessian equation. Later, Tosatti and Weinkove [15,16] gave a more delicate a priori C 2 -estimate and removed the conditions in [9]. Moreover, Dinew and Kolodziej [6] also studied the equation in the weak sense and obtained the L∞ estimates via suitably constructed pluripotential theory. In this paper, we want to study some other regularity properties of the complex Monge–Ampère equation on Hermitian manifolds: the Bedford–Taylor interior C 2 -estimate and Calabi C 3 -estimate. The interior estimate for the second order derivatives is an important and difficult topic in the study of complex Monge–Ampère equation. It has many fundamental applications in complex geometric problems. In the cornerstone work of Bedford and Taylor [1], by using the transitivity of the automorphism group of the unit ball B ⊂ Cn , they obtained the interior C 2 -estimate for the following Dirichlet problem:

det(ui j¯ ) = f in B, u = φ on ∂B,

1

where φ ∈ C 1,1 (∂B) and 0 f n ∈ C 1,1 (B). Unfortunately for generic domains Ω ⊂ Cn , due to the non-transitivity of the automorphism group of Ω, Bedford and Taylor’s method is not applicable and the analogous estimate is still open. Here, we exploit the method of Bedford and Taylor to study the interior estimate for the Dirichlet problem of the complex Monge–Ampère equation in the unit ball in the Hermitian setting (notice that for local arguments the shape of the domain is immaterial and hence it suffices to consider the balls). We consider the following equation

√ ¯ n = f ωn (ω + −1∂ ∂u) u = φ on ∂B,

in B,

(1)

1

where 0 f n ∈ C 1,1 (B) and ω is a smooth positive (1, 1)-form (not necessarily closed) defined ¯ We denote by PSH(ω, Ω) the set of all integrable, upper semicontinuous functions satisfyon B. √ ¯ 0 in the current sense on the domain Ω. Since ω is not necessarily Kähler, ing (ω + −1∂ ∂u) there are no local potentials for ω, and thus Bedford–Taylor’s method cannot be applied directly in our case. Theorem 1. Let B be the unit ball on Cn and ω be a smooth positive (1, 1)-form (not nec¯ Let u ∈ C(B) ¯ ∩ PSH(ω, B) ∩ C 2 (B) solve the Dirichlet problem (1) with essary closed) on B.

2006


φ ∈ C 1,1 (∂B). Then, for arbitrary compact subset B B, there exists a constant C dependent only on ω and dist{B , ∂B} such that 1 u C 2 (B ) C φ C 1,1 (∂B) + C f n C 1,1 (B) . Remark 1. Observe that this estimate is scale and translation invariant i.e. the same constant will work if we consider the Dirichlet problem in any ball with arbitrary small radius (and suitably rescaled set B ). As we have already mentioned, another goal of this paper is to get a local version of the C 3 estimate of the complex Monge–Ampère equation on Hermitian manifolds. Calabi’s C 3 -estimate for the real Monge–Ampère equation was first proved by Calabi himself in [2]. After that many mathematicians paid a lot of attention to this estimate. In Yau’s celebrated work [18] about the Calabi conjecture, he gave a detailed proof of the C 3 -estimate for the complex Monge–Ampère equation on Kähler manifolds, which was generalized to the Hermitian case by Cherrier [3]. All these C 3 -estimates are global. However, in some situations, a local C 3 -estimate is needed. For example Riebesehl and Schulz [14] gave a local version of Calabi’s estimate in order to study the Liouville property of Monge–Ampère equations on Cn . In a recent work by Dinew and the authors [7], aimed to study the C 2,α regularity of solutions to complex Monge–Ampère equation, the local result in [14] also played an important role to get the optimal value of α. Thus, it is also natural to generalize this local estimate to Hermitian manifolds and find some interesting applications. Let (M, g) be a Hermitian manifold. We consider the following complex Monge–Ampère equation (ω +

√

¯ n = ef ωn , −1∂ ∂φ)

(2)

where f (z) ∈ C ∞ (M) and ω is the Hermitian form associated with the metric g. Theorem 2. Let φ(z) ∈ PSH(ω, M) ∩ C 4 (M) be a solution of the Monge–Ampère equation (2), satisfying ¯ ω K. ∂ ∂φ

(3)

Let Ω Ω ⊂ M. Then the third derivatives of φ(z) of mixed type can be estimated in the form ¯ ω C |∇ω ∂ ∂φ|

for z ∈ Ω ,

where C is a constant depending on K, dω ω , R ω , ∇R ω , T ω , ∇T ω , dist(Ω , ∂Ω) and ∇ s f ω , s = 0, 1, 2, 3. Here ∇ is the Chern connection with respect to the Hermitian metric ω, T and R are the torsion tensor and curvature form of ∇. From the detailed proof in Yau’s paper [18] (see also [13]), in the Kähler case, we know that the quantity considered by Calabi ¯

¯

S = g˜ j r¯ g˜ s k g˜ ml φj km ¯ φr¯ s l¯


2007

satisfies the following elliptic inequality: ˜ −C1 S − C2 . S

(4)

Here φ is a√smooth solution of Eq. (2), g˜ denotes the Hermitian metric with respect to the form ¯ φ ¯ denotes the covariant derivative with respect to the Chern connection ωφ = ω + −1∂ ∂φ, ij k ∇. Riebesehl and Schulz [14] used the above elliptic inequality to get the Lp estimate for S. Then, a standard theorem for linear elliptic equations gave the L∞ estimate. For the Hermitian case, due to the non-vanishing torsion term, the estimates are more complicated. In [3], Cherrier proved the elliptic inequality corresponding to (4) on Hermitian manifolds: ˜ −C1 S 2 − C2 , S 3

(5) ¯

˜ is the canonical Laplacian with respect to the Hermitian metric g˜ (i.e. f ˜ = 2g˜ i j fi j¯ ), where positive constants C1 and C2 depend on K, R ω , ∇R ω , T ω , ∇T ω , and ∇ s f ω , s = 0, 1, 2, 3. By a similar method to that in [14], we obtain the Lp estimate for S, and then use Moser iteration to get the L∞ estimate. The estimates obtained in this paper should be useful for the study of problems on Hermitian manifolds. As a simple application, following the lines of [7], one has the following corollary: Corollary 1. Let Ω be a domain in Cn and ω be a Hermitian form defined on Ω. Let φ(z) ∈ PSH(ω, Ω) ∩ C 2 (Ω) be a solution of the Monge–Ampère equation (ω +

√ ¯ n = ef ωn . −1∂ ∂φ)

Suppose that f ∈ C α (Ω) for some 0 < α < 1. Then φ ∈ C 2,α (Ω). Remark 2. In the proof of Corollary 1, we don’t apply the local Calabi’s C 3 estimate to the original function φ ∈ C 1,1 (Ω) directly. Instead of that, for any point x0 ∈ Ω Ω Ω, we consider an approximation solution

√

¯ k )n = ef (x0 ) ωn −1∂ ∂u uk = φ on ∂B x0 , dρ k ,

(ω +

in B x0 , dρ k ,

where ρ = 12 and d = 12 dist(Ω , ∂Ω ). Since φ is only C 1,1 , we first consider the above Dirichlet problem with smooth boundary condition, i.e. instead of φ by its mollification φ ( ) for small enough and φ ( ) 1,1 → φ 1,1 as → 0. By the main theorem in [9] the solutions uk (we suppress the indice for the sake of readability) with the new boundary data coming from φ ( ) are smooth. Now, by Bedford–Taylor’s interior C 2 estimate, one can get uk C 2 (Bk+1 ) c˜1 φ C 1,1 (Ω ) + sup ef (x) , x∈Ω

where c˜1 is a positive constant depending only on ω. This allows one to apply the complex version of Calabi estimate to the above Dirichlet problem. Thus, for any γ ∈ (0, 1), we have

2008


uk C 2,γ (Bk+2 ) c˜2 /ρ kγ , where c˜1 is a positive constant depending only on ω, d, n, φ C 1,1 (Ω ) and supx∈Ω ef (x) . Letting now → 0+ , we obtain that this estimate remains true for the original function uk . Then, using the C α condition on f and following the lines in Ref. [7], we use the regularity of uk to approximate the original φ and obtain a C 2,α estimate of φ. The paper is organized as follows. In Section 2, we prove the interior C 2 -estimate for the complex Monge–Ampère equation. The proof for Calabi’s C 3 -estimate is given in Section 3. In Appendix A, we give a new proof of (5) which follows the idea in Phong, Sesum and Sturm [13], where the authors gave a simpler proof of Calabi’s estimate on Kähler manifolds. 2. Proof of the interior estimates In the proof of interior C 2 -estimates, the comparison theorem will play the key role. Following the same idea as in [5], it’s easy to see that the comparison theorem is still true for the complex Monge–Ampère equation on Hermitian manifold (M, ω). 0 ¯ 2 Lemma √ 1. Let Ω ⊂ M be a bounded set and u, v ∈ C (Ω) ∩ C (Ω), with ω + ¯ > 0 be such that ω + −1∂ ∂v

(ω +

√ ¯ 0, −1∂ ∂u

√ √ ¯ n (ω + −1∂ ∂u) ¯ n −1∂ ∂v)

and vu

on ∂Ω,

¯ then v u in Ω. Proof of Theorem 1. As mentioned above, we will follow the idea of Bedford and Taylor from [1]. For a ∈ B n , let Ta ∈ Aut(B n ) be defined by Ta (z) = Γ (a)

z−a , 1 − a¯ t z

a t a¯ where Γ (a) = 1−v(a) − v(a)I and v(a) = 1 − |a|2 . Note that Ta (a) = 0, T−a = Ta−1 , and Ta (z) is holomorphic in z, and a smooth function in a ∈ B n . For any a ∈ B(0, 1 − η) = {a: |a| < 1 − η} set −1 Ta (z) L(a, h, z) = Ta+h

and U (a, h, z) = L∗1 u(z),

U (a, −h, z) = L∗2 u(z),

Φ(a, h, z) = L∗1 φ(z),

Φ(a, −h, z) = L∗2 φ(z),

for z ∈ ∂B n ,


2009

where L∗i means the pull-back of Li for i = 1, 2 and L1 = L(a, h, z), L2 = L(a, −h, z). Since U (a, h, z) = Φ(a, h, z) for z ∈ ∂B n , it follows that U ∈ C 1,1 (B(0, 1 − η) × B(0, η) × ∂B n ). Consequently, for a suitable constant K1 , depending on η > 0, we have 1 U (a, h, z) + U (a, −h, z) − K1 |h|2 U (a, 0, z) = φ(z) 2

(6)

for all |a| 1 − η, |h| 12 η, and z ∈ ∂B n . If it can be shown that v(a, h, z) satisfies (ω +

√

¯ n f (z)ωn , −1∂ ∂v)

(7)

where v(a, h, z) =

1

U (a, h, z) + U (a, −h, z) − K1 |h|2 + K2 |z|2 − 1 |h|2 , 2

(8)

then it follows from the comparison theorem in the Hermitian case that v(a, h, z) u(z). Thus, if we set a = z, we conclude that 1

u(z + h) + u(z − h) u(z) + (K1 + K2 )|h|2 2 which would prove the theorem. Let now √ ¯ = F (ω + −1∂ ∂v)

(ω +

√ ¯ n −1∂ ∂v)

√ ( −1)n dz1 ∧ d z¯ 1 ∧ · · · ∧ dzn ∧ d z¯ n 1 = det(gi j¯ + vi j¯ ) n ,

1

n

(9)

where gi j¯ is the local expression of ω under the standard coordinate {zi }ni=1 in Cn . By the concavity of F , we have F (ω +

√

√

¯ ∗1 u + ∂ ∂L ¯ = F ω + −1 ∂ ∂L ¯ ∗2 u + 2K2 |h|2 ∂ ∂|z| ¯ 2 −1∂ ∂v) 2 √ 1 1 ¯ 2 ω − L∗1 ω + ω − L∗2 ω + K2 |h|2 −1∂ ∂|z| =F 2 2

√ √ 1 ¯ ∗1 u + 1 L∗2 ω + −1∂ ∂L ¯ ∗2 u + L∗1 ω + −1∂ ∂L 2 2 √ √ 1 ¯ ∗1 u + 1 F L∗2 ω + −1∂ ∂L ¯ ∗2 u F L∗1 ω + −1∂ ∂L 2 2 √ 1 ¯ 2 . + F ω − L∗1 ω + ω − L∗2 ω + 2K2 |h|2 −1∂ ∂|z| 2

(10)

Since the Hermitian metric ω is smooth, one can find K2 large enough, such that √ ¯ 2 0. ω − L∗1 ω + ω − L∗2 ω + K2 |h|2 −1∂ ∂|z|

(11)

2010


On the other hand, since L(a, h, z) is holomorphic in z, it follows from Eq. (1) that √ √ ¯ ∗1 u = F L∗1 (ω + −1∂ ∂u) ¯ F L∗1 ω + −1∂ ∂L √

1 ¯ n n L∗1 (ω + −1∂ ∂u) = √ ( −1)n dz1 ∧ d z¯ 1 ∧ · · · ∧ dzn ∧ d z¯ n

1 n L∗1 (f (z)ωn ) = √ n 1 1 n n ( −1) dz ∧ d z¯ ∧ · · · ∧ dz ∧ d z¯ 1 ∗ 1 = F L1 f n ω = L∗1 f n F L∗1 (ω) .

(12)

Similarly, we can get √ 1 1 ¯ ∗2 u = F L∗2 f n ω = L∗2 f n F L∗2 (ω) . F L∗2 ω + −1∂ ∂L Thus, F (ω +

√ √ 1 1 ¯ 1 F L∗1 f n ω + F L∗2 f n ω + 1 F K2 |h|2 −1∂ ∂|z| ¯ 2 −1∂ ∂v) 2 2 1 1 1 1 1 = F f n ω + F L∗1 f n ω + F L∗2 f n ω − 2F f n ω 2 √ 1 ¯ 2 . + F K2 |h|2 −1∂ ∂|z| 2

(13)

Again, since ω is smooth and f 1/n ∈ C 1,1 , choosing K2 large enough, we have √ 1 1 1 ¯ 2 . F L∗1 f n ω + F L∗2 f n ω − 2F f n ω F K2 |h|2 −1∂ ∂|z|

(14)

Finally, we obtain F (ω + and thus, the inequality (7) follows.

√ 1 ¯ F f nω , −1∂ ∂v)

(15)

2

3. Proof of the Calabi estimate Let (M, J, ω) be a Hermitian manifold and ∇ denote the Chern connection with respect to the √ ¯ metric ω. Let locally ω = −1gi j¯ dzi ∧ dzj , then the local formula for the connection 1-form reads θ = ∂g · g −1 . We also denote θα = ∂α g · g −1 , The torsion tensor of ∇ is defined by

γ

θαβ =

∂gβ δ¯ ∂zα

¯

gγ δ .


T

∂ ∂ , β α ∂z ∂z

i.e.,

γ Tαβ

=

=∇

∂gβ δ¯ ∂zα

2011

∂ ∂ ∂ ∂ − ∇ − , ∂ α ∂zβ ∂zα ∂zβ ∂zβ ∂z

∂g ¯ ¯ − αβδ g γ δ . ∂z

∂ ∂zα

Notice that T = 0 ⇐⇒ ω is Kähler (and ∇ is the Levi-Civita connection on M). ¯ = dθ − θ ∧ θ = ∂(∂g ¯ The curvature form of ∇ is defined by R = ∂θ · g −1 ). In local coordinates, we have 2 j ∂g ¯ ∂gt s¯ ¯ ¯ ∂ g ¯ j Riα β¯ = −∂¯β ∂α g · g −1 i = −g j k α i kβ + iαk g j s¯ β g t k , ∂z ∂ z¯ ∂z ∂ z¯ k . Ri j¯α β¯ = gk j¯ Riα β¯

Note that R (2,0) = R (0,2) = 0 and T (1,1) = 0, since the almost complex structure J is integrable and ∇ is the Chern connection. Proof of Theorem 2. By the assumption (3) for the solution of Eq. (2), we know that 1 g gφ λg λ

for some constant λ,

where λ depends√only on K and f C 0 , and gφ denotes the Hermitian metric with respect to the ¯ Thus, form ωφ = ω + −1∂ ∂φ. ¯

¯

¯

¯

j r¯ s k ml S = (gφ )j r¯ (gφ )s k (gφ )ml φj km ¯ φr¯ s l¯ λ(gφ ) (gφ ) g φj km ¯ φr¯ s l¯.

(16)

On the other hand, we have j k¯ ml¯ j k¯ ml¯ ¯ j k¯ gφ g ml φj km ¯ l¯ = gφ g φj km ¯ l¯ − gφ l¯g φj km ¯ ¯

j s¯

¯

¯

= g ml fml¯ + gφ φt s¯l¯gφt k g ml φj km ¯ , where we used Eq. (2) in the last equality above. Thus

j k¯ ¯ S λ gφ g ml φj km ¯ l¯ − f .

(17)

√ ¯ j k¯ ml¯ ¯ is a globally defined quantity, where Λgφ Notice that gφ g ml φj km ¯ l¯ = Λgφ (g ∇l¯∇m ( −1∂ ∂φ)) n−1 n √ ω ωφ is the contraction with ωφ , i.e. (Λgφ θ ) n!φ = θ ∧ (n−1)! for any (1, 1) form θ = −1θi j¯ dzi ∧ d z¯ j , j k¯

in local coordinates, we have Λgφ θ = gφ θj k¯ . Therefore we can estimate for every sufficiently large exponents ρ, σ, and every nonnegative test function η(z) ∈ C01 (Ω): S σ ηp+1 Ω

ωn λ n!

Ω

ωn

j k¯ ¯ . S σ −1 ηp+1 gφ g ml φj km ¯ l¯ − f n!

(18)

2012


Now, use the following identity: s t¯ φj km ¯ l¯ = φj k¯ lm ¯ + φs k¯ Rj ml¯ − φj t¯Rkm ¯ l¯ s s t¯ t¯ = φj lm ¯ k¯ + φs l¯Rj mk¯ + φs k¯ Rj ml¯ − φj t¯Rlm ¯ k¯ − φj k¯ Rkm ¯ l¯

= φmlj ¯ k¯ + C1 , where C1 is a constant depending on K and |R|ω . Therefore, we have S σ ηp+1

ωn λ n!

Ω

j k¯

¯

S σ −1 ηp+1 gφ g ml φmlj ¯ k¯

ωn + n!

Ω

S σ −1 ηp+1 (C1 − f )

ωn n!

Ω j k¯

S σ −1 ηp+1 gφ (φ)j k¯

λ

ωn n!

+ C2

Ω

S σ −1 ηp+1

ωn , n!

(19)

Ω

where C2 is a constant depending on C1 and f . Now, using integration by parts, it is easy to see that

ω j k¯ S σ −1 ηp+1 gφ (φ)j k¯

n

n!

=

Ω

j k¯

e−f S σ −1 ηp+1 gφ (φ)j k¯

ωφn n!

Ω

= Ω

= Ω

ωφn−1 √ ¯ e−f S σ −1 ηp+1 −1∂ ∂(φ) ∧ (n − 1)! ωφn−1 √ ¯ −1d e−f S σ −1 ηp+1 ∂(φ) ∧ (n − 1)!

− Ω

ωφn−1 √ ¯ ∧ −1d e−f S σ −1 ηp+1 ∧ ∂(φ) (n − 1)!

=: I − II. Next, we will estimate |I | and |II|. First, I= Ω

ωφn−1 √ ¯ −1d e−f S σ −1 ηp+1 ∂(φ) ∧ (n − 1)!

=− Ω

√

¯ −1e−f S σ −1 ηp+1 ∂(φ) ∧ dωφ ∧

ωφn−2 (n − 2)!

.

By the equivalence of two forms ω and ωφ (i.e., the assumption (3) on φ), we know

(20)


2013

ωφn−2 ωφn−2 ∂(φ) ¯ ¯ = ∂(φ) ∧ dω ∧ ∧ dωφ ∧ (n − 2)! (n − 2)! n |dω|g ω ¯ C3 ∂(φ) φ gφ n! n 1 ω , C4 S 2 n!

(21)

where C4 is a constant depending on |dω|g , f C 0 and K (for the justification of the last inequality we refer to the formula of S given in Appendix A). This estimate yields |I | C5

1

S σ − 2 ηp+1

ωn n!

(22)

Ω

for some constant C5 dependent on ω, f C 0 and K. Let us now estimate the second term: II = Ω

ωφn−1 √ −f σ −1 p+1 ¯ S −1d e η ∧ ∂(φ) ∧ (n − 1)!

+ (σ − 1) Ω

+ (p + 1) Ω

ωφn−1 √ ¯ −1e−f S σ −2 ηp+1 dS ∧ ∂(φ) ∧ (n − 1)! ωφn−1 √ ¯ −1e−f S σ −1 ηp dη ∧ ∂(φ) ∧ (n − 1)!

Thus, |II| C6

1

S σ − 2 ηp+1

Ω

+ (p + 1)

ωn + (σ − 1) n!

1

S σ − 2 ηp |∇η|

ωn n!

3

S σ − 2 |∇S|ηp+1

ωn n!

Ω

,

Ω

where C6 is a constant depending on f C 1 (ω) and K. By the estimates (22), (23) and using Cauchy’s inequality 3

(σ − 1)ηp+1 S σ − 2 |∇S| we have, for > 0 small enough,

(σ − 1)2 p+1 σ −3 η S |∇S|2 + ηp+1 S σ 4

(23)

2014


S σ ηp+1

ωn ωn ωn C7 (σ − 1)2 S σ −3 |∇S|2 ηp+1 + S σ −1 ηp+1 n! n! n!

Ω

+ (p + 1)

Ω

S

σ − 12 p

η |∇η|

ωn n!

Ω

+

Ω

S

σ − 12 p+1 ω

η

n

n!

,

(24)

Ω

where C7 is a constant depending on |dω|ω , |R|ω , K, f C 1 (ω) and f . Now we are in the place to use the elliptic inequality (5) in the introduction. Recall that 3

φ S −CS 2 − C0 .

(25)

Multiplying by S σ −2 ηp+1 on both sides of the above inequality and integrating over Ω, we have

1

S σ − 2 ηp+1

−C

ωn − C0 n!

Ω

S σ −2 ηp+1

ωn n!

Ω

S σ −2 ηp+1 φ S

Ω

The right-hand side of above inequality can be estimated as follows

S σ −2 ηp+1 φ S

ωn n!

Ω

ωφn−1 √ ¯ ∧ e−f S σ −2 ηp+1 −1∂ ∂S (n − 1)!

= Ω

ωφn−1 √ ¯ ∧ −1d e−f S σ −2 ηp+1 ∂S (n − 1)!

= Ω

− Ω

=− Ω

√

ωφn−1 −f σ −2 p+1 ¯ ∧ ∂S ∧ −1d e S η (n − 1)!

ωφn−2 √ ¯ ∧ dω ∧ −1e−f S σ −2 ηp+1 ∂S (n − 2)!

√ − −1

Ω

ωφn−1 −f σ −2 p+1 ¯ S d e η ∧ ∂S ∧ (n − 1)!

− (σ − 2)

√

¯ ∧ −1e−f S σ −3 ηp+1 ∂S ∧ ∂S

Ω

− (p + 1) Ω

√

¯ ∧ −1e−f S σ −2 ηp ∂η ∧ ∂S

ωφn−1 (n − 1)!

ωφn−1 (n − 1)!

ωn . n!

(26)


S σ −3 ηp+1 |∇S|2

−C8 (σ − 2)

ωn + C9 n!

Ω

S σ −2 ηp+1 |∇S|

2015

ωn n!

Ω

ωn S σ −2 ηp |∇η||∇S| , n!

+ C9 (p + 1) Ω

for C8 a positive constant. From above inequality, we obtain (σ − 2)

S σ −3 ηp+1 |∇S|2

ωn n!

Ω

ωn ωn + S σ −2 ηp+1 |∇S| C10 (p + 1) S σ −2 ηp |∇η||∇S| n! n! Ω

+

S

σ − 21

ηp+1

ωn n!

+

Ω

S σ −2 ηp+1

Ω

n ω

(27)

.

n!

Ω

Now, by Cauchy’s inequality again, S σ −2 ηp+1 |∇S| |∇S|2 S σ −3 ηp+1 + (p + 1)S σ −2 ηp |∇η||∇S| |∇S|2 S σ −3 ηp+1 +

1 p+1 σ −1 S η 4 (p + 1)2 p−1 σ −1 η S |∇η|2 . 4

These two inequalities, together with (27) and (24) yield S σ ηp+1

ωn n!

Ω

C11 σ (p + 1) 2

2

η

Ω

+

S

σ − 12 p+1 ω

S

σ − 12 p

η |∇η|

Ω

ωn n!

n!

+

1

S σ − 2 ηp+1

Ω

+

n

S

σ −2 p+1 ω

η

Ω

n

S

n!

ωn n!

σ −1 p−1

η

2ω

|∇η|

n

n!

(28)

Ω

for p 2, σ 4. Now, let BR0 (z) Ω be a ball, and let 0 < R r < t R0 , R0 − R 1. By choosing an C appropriate testing function η(z), with 0 η 1, η|Br = 1, η|M/Bt = 0, |∇η| t−r , and putting p = σ − 1, we conclude that (Sη)σ

ωn C12 σ 4 n!

Bt (z)

Bt (z)

+

1 (Sη)σ −2 S (t − r)2

n 1 1 1 ω 1 (Sη)σ −1 S 2 + (Sη)σ − 2 η 2 + (Sη)σ −1 η + (Sη)σ −2 η2 . (29) t −r n!

2016


By Young’s inequality ab

aα 1 bβ + β/α , α β

1 1 + = 1. α β

for > 0,

It follows that, σ 1 1 1 (Sη)σ −1 S 2 σ (Sη)σ −1 σ −1 + σ −1 t −r σ σ −1 σ 1 1 σ −2 σ −2 σ −2 (Sη) (Sη) S + σ −2 σ (t − r)2 2 σ2 σ −2 (Sη)σ −2 (Sη)σ −1 1

(Sη)σ − 2

σ (Sη)σ −4 σ −4 +

σ σ −4

σ (Sη)σ −2 σ −2 +

σ σ −2

σ (Sη)σ −1 σ −1 +

σ σ −1

1

σ −4 4

σ −2 2

σ 4

1

1 1 S2 t −r

1 σ −1 σ

;

1 S (t − r)2

σ (Sη)2 4 ; σ

σ 2

σ

(Sη) 2 ;

σ , β = σ, σ −1

σ

2

;

α=

α=

1 σ (Sη) 2 ;

α=

α=

σ σ , β= , σ −2 2

σ σ , β= , σ −4 4

σ σ , β= , σ −2 2

α=

σ , β = σ. σ −1

All the above inequalities combined with (29), lead to

n σ ω 1 ωn 1 σ 2 C13 B( ) S + + 1 S σ n! (t − r)σ n! (t − r) 2 σ

Br (z)

Bt (z)

C13

B( )σ t n (t − r)σ

Sσ

ωn n!

1 2

,

(30)

Bt (z)

where B( ) is a constant depending on which comes from the coefficients in Young’s inequalities above. Now we can apply Meyers’ lemma: Lemma 2. (See [12].) If u = u(x) is a nonnegative, non-decreasing continuous function in the interval [0, d), which satisfies the functional inequality: u(s)

1−α c u(r) , r −s

for any 0 s < r < d,

with α and c being constants (0 < α < 1), then u(0)

2α+1 c (2α − 1)d

1

α

.


2017

Using (30) and applying Meyers’ lemma with d = R0 − R, s = r − R and Φ(s) = n 1 ( BR+s (z) S σ ωn! ) σ , one can obtain 1

1

C σ B( )R0σ Φ(0) , (R0 − R)2 and thus Sσ

ωn n!

1

BR (z)

σ

1

(CR0 ) σ B( ). (R0 − R)2

(31)

From this, we obtain the Lp estimate of S for arbitrary p. However, by tracking the constant B( ), one can find that B( ) ∼ σ 4 . Thus, we cannot get the estimate for supΩ S by letting σ → ∞. Instead of that, we will use the standard Moser iteration to finish the L∞ estimate for S. Recall that by inequality (27) we have (σ − 2)

S σ −3 ηp+1 |∇S|2

ωn n!

Ω

ωn ωn σ −2 p + S σ −2 ηp+1 |∇S| C10 (p + 1) S η |∇η||∇S| n! n! Ω

+

S

σ − 21

ωn + ηp+1 n!

Ω

Ω

ωn . S σ −2 ηp+1 n!

Ω

Coupling this with Young inequalities S σ −2 ηp+1 |∇S| |∇S|2 S σ −3 ηp+1 + (p + 1)S σ −2 ηp |∇η||∇S| |∇S|2 S σ −3 ηp+1 +

1 p+1 σ −1 η S , 4 (p + 1)2 p−1 σ −1 η S |∇η|2 4

we have (σ − 2)

S σ −3 ηp+1 |∇S|2

ωn n!

Ω

C14 Ω

1 (p + 1)2 p−1 σ −1 ωn 1 S |∇η|2 + η S σ −1 ηp+1 + S σ − 2 ηp+1 + S σ −2 ηp+1 . σ −2 σ −2 n!

(32) Let now q = σ − 1 2, and p = 1, then one obtains

2018


S q−2 η2 |∇S|2 Ω

ωn n!

1 1 ωn 1 1 1 S q+ 2 η2 + S q−1 η2 . (33) S q |∇η|2 + S q η2 + 2 2 q −1 q −1 n! (q − 1) (q − 1)

C15 Ω

By the Sobolev inequality

2m

v m−1

ωn n!

m−1

2m

C

|∇v|2

Ω

ωn n!

1 2

+C

Ω

v2

ωn n!

1 2

Ω

q

applied to v = ηS 2 , we conclude that

q 2m ωn ηS 2 m−1 n!

m−1 2m

Ω

q 2 ωn ∇ ηS 2 n!

C16

1 2

+

Ω

ηS

q 2

2 ωn

1 2

n!

Ω

S q |∇η|2 +

C17

1

1 2 ωn 2 ωn 2 q . S q−2 η2 |∇S|2 + η2 S q 2 n! n!

Ω

(34)

Ω

Using the inequality (33), we have

2 q m ωn η S m−1 n!

Ω

m−1 m

C18

|∇η|2 S q + η2 S q + Ω

q2 q2 q 2 S |∇η| + S q η2 (q − 1)2 (q − 1)2

q 2 q+ 1 2 q 2 q−1 2 ωn 2 S S + η + η q −1 q −1 n!

(35)

for any q > 4. Again, let BR0 (z) Ω be a ball, and let 0 < R r1 < r2 R0 , R0 − R 1. By choosing C an appropriate testing function η(z), with 0 η 1, η|Br1 = 1, η|M/Br2 = 0, |∇η| r2 −r , we 1 conclude that

m

S q m−1 Br1 (z)

ωn n!

m−1 m


1+

C19 Br2 (z)

qC20 qC21

q2 (q − 1)2

1 +1 (r2 − r1 )2 1 +1 (r2 − r1 )2

2019

1 q 2 q+ 1 q 2 q−1 ωn q 2 + + 1 S + S S q −1 q −1 n! (r2 − r1 )2

1

S q + S q−1 + S q+ 2

ωn n!

Br2 (z)

1

S q+ 2

ωn . n!

(36)

Br2 (z)

Thus,

1 Cq +1 S qm m−1 L (Br1 (z)) (r2 − r1 )2

1 q

S

q+ 21 q q+ 21

L

(37) (Br2 (z))

for any 0 < R r1 < r2 R0 . qk m Let m−1 = qk+1 + 12 and rk = R + (R0 − R)2−k . Then, qk =

m m−1

k +

m−1 , 2

and |rk − rk−1 | = (R0 − R)2−k .

By (37), we have Cqk 1 +

S

q +1 L k+1 2 (Brk+1 (z))

1 qk

qk where ak :=

qk + 12 qk

S

1 (R0 − R)2

qk

S akq

1 k + 2 (B (z)) rk

L

1

qk

2k

2 qk S akq L

1 k + 2 (B (z)) rk

,

(38)

qk+1 + 21 (Brk+1 (z))

L

k

qi C 1 + 1 qi

i=1 qk + 12 qk

=

qk−1 m m−1 qk

k i=1

and thus

C 1+

1

. By iteration, it follows from (38) that

Notice that ak =

1 (rk+1 − rk )2

=

ai =

1 (R0 − R)2

m qk−1 m−1 qk ,

m m−1

k

1

qi

2

2i qi

ki=1 ai S

k

i=1 ai q1 + 21 (Br1 (z))

L

so

q0 qk−1 ··· = q1 qk

m m−1

k

q0 qk

.

(39)

2020


lim

k

k→∞

ai = q0 =

i=1

m+1 . 2

Moreover, k

qi C 1 + 1 qi

i=1

1 (R0 − R)2

1

qi

2

2i qi

=

lim

k→∞

qi C 1 + 1 qi

i=1

∞

When k → ∞, it is easy to show that 1 qi log( ∞ i=1 qi ) < ∞. Thus, k

k

1 i=1 qi

qi C 1 + 1 qi

i=1

1 (R0 − R)2 ∞

< ∞ and

1 (R0 − R)2

2i i=1 qi

1

qi

ki=1

1 qi

k

2

2i i=1 qi

.

< ∞. Notice also that

2i

2 qi < ∞.

It follows from (39), by letting k → ∞, S L∞ C S Choosing now σ = q1 +

1 2

=

m m−1

+

m 2

m+1 2 q1 + 21

L

.

(40)

(BR0 (z))

in (31), we finally obtain S L∞ C,

(41)

where C is a positive constant depending on K, |dω|ω , |R|ω , |∇R|ω , |T |ω , |∇T |ω , dist(Ω , ∂Ω) and |∇ s f |ω , s = 0, 1, 2, 3. 2 Acknowledgments The authors would like to thank Prof. Pengfei Guan and Slawomir Dinew for the numerous helpful discussions on this problem. The note was written while the first named author was visiting McGill University. He would like to thank this institution for the hospitality. Finally we wish to thank the referee for his/her valuable comments. Appendix A As mentioned in the introduction, using the idea from [13], we give a new proof for the elliptic inequality (5) in this section. Proof of the elliptic inequality (5).√Let ∇ and ∇˜ denote the Chern connections corresponding ¯ respectively. Define to the Hermitian metrics ω and ω + −1∂ ∂φ h = g˜ · g −1 and j

¯

hi = g˜ i k¯ g j k ,

−1 j ¯ h i = gi k¯ g˜ j k .

(42)


2021

In fact, h can be thought to be an endomorphism h : T 1,0 (M) → T 1,0 (M), such that g(X, ˜ Y) = g(h(X), Y ). Set ¯

¯

S = g˜ j r¯ g˜ s k g˜ ml φj km ¯ φr¯ s l¯,

(43)

where φj km ¯ = ∇m ∇k¯ ∇j φ. By (42), we have θ˜ = ∂ g˜ · g˜ −1 = ∂(h · g) · g −1 h−1 = ∂h · g · g −1 · h−1 + h · ∂g · g −1 · h−1 = ∂h · h−1 + h · θ · h−1 = ∂h · h−1 + h · θ · h−1 − θ · h · h−1 + θ = θ + ∇ 1,0 h · h−1 . R˜ = ∂¯ θ˜ = ∂¯ θ + ∇ 1,0 h · h−1 = R + ∂¯ ∇ 1,0 h · h−1 .

(44)

(45)

By similar computation, we can get θ = ∂g · g −1 = θ˜ − h−1 ∇˜ 1,0 h , R = R˜ − ∂¯ h−1 · ∇˜ 1,0 h .

(46) (47)

Now, using the definitions, one can see that φj km ˜ j , ∂¯k ) = g˜ j k,m ¯ = (∇m g)(∂ ¯ . Thus, 1,0 2 ¯ ¯ S = g˜ j r¯ g˜ s k g˜ ml φj km ¯ φr¯ s l¯ = ∇ g˜ g˜ . On the other hand, ∇m g˜ = ∇m (h · g) = ∇m h · g =

∂ h + h · θ − θ · h · g, m m ∂zm

so ∂ ∇˜ m h = m h + h · θ˜m − θ˜m · h ∂z ∂ = m h + h · θm − θm · h + h · (∇m h) · h−1 − ∇m h ∂z = h · (∇m h) · h−1 .

(48)

2022


Thus, ∇m g˜ = ∇m h · g = h−1 · (∇˜ m h) · h · g = h−1 · (∇˜ m h) · g. ˜ Finally we end up with the formula 2 2 S = ∇ 1,0 g˜ g˜ = h−1 · ∇˜ 1,0 h g˜ = |θ˜ − θ |2g˜

(49)

i.e. S can be thought as the g-norm ˜ of the difference between the two connection 1-forms. Now, we can deduce the elliptic inequality: ˜ = ˜ h−1 · ∇˜ 1,0 h 2 S g˜

¯ = g˜ i j ∂i ∂j¯ h−1 · ∇˜ 1,0 h , h−1 · ∇˜ 1,0 h g˜

¯ = g˜ i j ∂i ∇˜ j¯ h−1 · ∇˜ 1,0 h , h−1 · ∇˜ 1,0 h g˜ + h−1 · ∇˜ 1,0 h , ∇˜ j h−1 · ∇˜ 1,0 h g˜

¯ = g˜ i j ∇˜ i ∇˜ j¯ h−1 · ∇˜ 1,0 h , h−1 · ∇˜ 1,0 h g˜

¯ + g˜ i j h−1 · ∇˜ 1,0 h , ∇˜ i¯ ∇˜ j h−1 · ∇˜ 1,0 h g˜ 2 2 + ∇˜ 1,0 h−1 · ∇˜ 1,0 h g˜ + ∇˜ 0,1 h−1 · ∇˜ 1,0 h g˜ .

(50)

¯ −1 · (∇˜ 1,0 h)), we have Using the relation R = R˜ − ∂(h l ¯ ¯ l l . − Rmt g˜ i j ∇˜ i ∇˜ j¯ h−1 · ∇˜ t1,0 h m = g˜ i j ∇˜ i R˜ mt j¯ j¯

(51)

Recall the Bianchi identities of curvature forms which can be found in [11] (p. 135): R(X, Y )Z = T T (X, Y ), Z + (∇X T )(Y, Z); ∇X R(Y, Z) + R T (X, Y ), Z = 0,

(52) (53)

where X, Y, Z ∈ T M and T is the torsion of the connection ∇ (recall that ∇ is not necessarily the Levi-Civita connection), while denotes the cyclic sum with respect to X, Y , Z. By the first Bianchi identity (52), one obtains ˜ i , ∂ ¯ )∂m + R(∂ ˜ ¯ , ∂m )∂i + R(∂ ˜ m , ∂i )∂ ¯ R(∂ j j j = T˜ T˜ (∂i , ∂j¯ ), ∂m + T˜ T˜ (∂j¯ , ∂m ), ∂i + T˜ T˜ (∂m , ∂i ), ∂j¯ + (∇˜ i T˜ )(∂j¯ , ∂m ) + (∇˜ j¯ T˜ )(∂m , ∂i ) + (∇˜ m T˜ )(∂i , ∂j¯ ). Recall the fact that R˜ 2,0 = R˜ 0,2 = 0, T˜ 1,1 = 0 (since ∇˜ is the Chern connection) and T˜ (∂m , ∂i ) ∈ T 1,0 (M). Also


2023

T˜ (∂i , ∂j¯ ) = T˜ (∂j¯ , ∂m ) = (∇˜ i T˜ )(∂j¯ , ∂m ) = (∇˜ m T˜ )(∂i , ∂j¯ ) = 0, ˜ m , ∂i )∂ ¯ = 0. R(∂ j Thus, ˜ i , ∂ ¯ )∂m + R(∂ ˜ ¯ , ∂m )∂i = (∇˜ ¯ T˜ )(∂m , ∂i ). R(∂ j j j ˜ i , ∂ ¯ )∂m = R˜ l ∂l and R˜ l = −R˜ l , so we get By definition R(∂ j mi j¯ mi j¯ mj¯i l l l = R˜ im + T˜mi, . R˜ mi j¯ j¯ j¯

(54)

l¯ ˜ l¯ ˜ l¯ . R˜ ki ¯ j¯ = Rj¯i k¯ + Tj¯k,i ¯

(55)

Similarly, one can also obtain

Moreover, by the second Bianchi identity (53) and following the same step as above we have l ˜ l ¯ + R˜ l ¯ = −R˜ T˜ (∂i , ∂t ), ∂ ¯ − R˜ T˜ (∂t , ∂ ¯ ), ∂i − R˜ T˜ (∂ ¯ , ∂i ), ∂t + R R˜ mt ¯ j j j j ,i mj i,t mit,j l and R˜ mit, = 0, T˜ (∂t , ∂j¯ ) = T˜ (∂j¯ , ∂i ) = 0. Thus, j¯ l l l = R˜ mt + T˜its R˜ ms . R˜ mi j¯,t j¯,i j¯

(56)

Now, using the identities (54), (55) and (56), we obtain ¯

¯

¯

¯

l l l l g˜ i j ∇˜ i R˜ mt = g˜ i j R˜ mt = g˜ i j R˜ mi − g˜ i j T˜its R˜ ms j¯ j¯,i j¯,t j¯ ¯

¯

¯

lk ij s ˜ l = g˜ i j R˜ mki ¯ j¯,t g˜ − g˜ T˜it R ms j¯

l k¯ ¯ s i j¯ s ˜ l = g˜ i j R˜ i km ¯ j¯,t + T˜mi,j¯t g˜ s k¯ g˜ − g˜ T˜it R ms j¯ ¯ l k¯ i j¯ l i j¯ s ˜ l = −g˜ i j R˜ kim ¯ j¯,t g˜ + g˜ T˜mi,j¯t − g˜ T˜it R ms j¯ ¯ ¯ ¯ l ¯ l k¯ i j¯ l¯ l = −g˜ i j R˜ j¯imk,t g˜ i l¯g˜ l k + g˜ i j T˜mi, − g˜ i j T˜its R˜ ms ¯ g˜ − g˜ T˜j¯k,mt ¯ j¯t j¯ ¯ ¯ ¯ l ¯ l k¯ i j¯ l¯ l = g˜ i j R˜ i j¯mk,t g˜ i l¯g˜ l k + g˜ i j T˜mi, − g˜ i j T˜its R˜ ms ¯ g˜ − g˜ T˜j¯k,mt ¯ j¯t j¯ ¯

¯

¯

¯

¯

¯

i lk ij ˜ l l l = R˜ im g˜ i l¯g˜ l k + g˜ i j T˜mi, − g˜ i j T˜its R˜ ms . ¯ ¯ g˜ − g˜ Tj¯k,mt k,t j¯t j¯

(57)

From the Monge–Ampère equation (2), it follows that i ˜ i ˜ R˜ im ¯ = ∇t Rimk¯ − ∇t fmk¯ . k,t

(58)

2024


In the following, we denote = O(S α ) if there is a constant C depending only on K, |dω|ω , 1 |R|ω , |∇R|ω , |T |ω , |∇T |ω and |∇ s f |ω , s = 0, 1, 2, 3, such that CS α . Note that ∇˜ is O(S 2 ), so 1 i l k¯ 2 + O(1). R˜ im ¯ g˜ = O S k,t

(59)

T˜j¯s¯k,mt = (∂j¯ gnk¯ − ∂k¯ gnj¯ )g˜ n¯s mt ¯ n¯s n¯s = Tj¯kn = ∇˜ t ∇˜ m Tj¯kn ¯ g˜ ¯ g˜ mt n¯s l = ∇˜ t ∇m Tj¯kn ¯ − (θ˜m − θm )n Tj¯kl ¯ g˜ l ˜ t (θ˜m − θm )ln T ¯ ¯ = ∇t (∇m Tj¯kn ¯ ) − (θ˜t − θt )m ∇l Tj¯kn ¯ −∇ j kl n¯s l l s g˜ . − (θ˜t − θt )n ∇m Tj¯kl ¯ − (θ˜m − θm )n ∇t Tj¯kl ¯ − (θ˜t − θt )l Tj¯ks ¯

(60)

For the second term in (57)

Again, by the fact that ∇˜ is O(S 2 ) and |h−1 · (∇˜ 1,0 h)|g˜ is also O(S 2 ), we have 1

i j¯ l¯ g˜ T˜

g˜ i l¯g˜ ¯ j¯k,mt

l k¯

1

1 O S 2 + O(S) + C ∇˜ 1,0 h−1 · ∇˜ 1,0 h + O(1).

(61)

Similarly, we can get the estimate for the last two terms in (57) i j¯ l g˜ T˜

mi,j¯t

O S 12 + O(S) + C ∇˜ 0,1 h−1 · ∇˜ 1,0 h + O(1),

i j¯ s l g˜ T˜ R˜ C ∇˜ 0,1 h−1 · ∇˜ 1,0 h + O(1). it ms j¯

(62) (63)

Putting the above estimates (57)–(63) into (51), we can conclude that i j¯ g˜ ∇˜ i ∇˜ ¯ h−1 · ∇˜ t1,0 h l j m 1 O S 2 + O(S) + C ∇˜ 1,0 h−1 · ∇˜ 1,0 h + C ∇˜ 0,1 h−1 · ∇˜ 1,0 h . One the other hand, ¯ l −1 1,0 ¯ ¯ g˜ i j ∇˜ i¯ ∇˜ j h−1 · ∇˜ 1,0 h = g˜ i j ∇˜ j ∇˜ i¯ h−1 · ∇˜ 1,0 h − g˜ i j R˜ mi # h · ∇˜ h j¯ where i j¯ l −1 1,0 g˜ R˜ mi j¯ # h · ∇˜ h

s l l l s ¯ − h−1 · ∇˜ s1,0 h m R˜ tis j¯ − h−1 · ∇˜ t1,0 h s R˜ mi = g˜ i j h−1 · ∇˜ t1,0 h m R˜ si j¯ j¯ × dzt ⊗ dzm ⊗

∂ ∂zl

(64)


2025

and ¯ l ¯ l ¯ l ¯ ¯ ¯ l k¯ i j¯ ˜ l = g˜ i j R˜ im + g˜ i j T˜mi, = g˜ i j R˜ i j¯mk¯ g˜ l k + g˜ i j T˜j¯s¯k,m g˜ i j R˜ mi ¯ g˜ i s¯ g˜ + g˜ Tmi,j¯ . j¯ j¯ j¯

Thus i j¯ l g˜ R˜

mi j¯

O S 12 + O(1).

Hence we conclude that i j¯ g˜ ∇˜ ¯ ∇˜ j h−1 · ∇˜ 1,0 h i ¯ ¯ g˜ i j ∇˜ j ∇˜ ¯ h−1 · ∇˜ 1,0 h + g˜ i j R˜ l i

mi j¯

−1 1,0 # h · ∇˜ h

1 O S 2 + O(S) + C ∇˜ 1,0 h−1 · ∇˜ 1,0 h + C ∇˜ 0,1 h−1 · ∇˜ 1,0 h .

(65)

Finally, by (50) and (64), (65), we obtain the elliptic inequality: ˜ −C1 S 2 − C2 S 3

(66)

where C1 , C2 are positive constants depending only on K, |dω|ω , |R|ω , |∇R|ω , |T |ω , |∇T |ω and |∇ s f |ω , s = 0, 1, 2, 3. 2 References [1] E. Bedford, B.A. Taylor, The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math. 37 (1976) 1–44. [2] E. Calabi, Improper affine hyperspheres and a generalization of a theorem of K. Jörgens, Michigan Math. J. 5 (1958) 105–126. [3] P. Cherrier, Équations de Monge–Ampère sur les variétés Hermitiennes compactes, Bull. Sci. Math. (2) 111 (1987) 343–385. [4] P. Cherrier, Le probléme de Dirichlet pour des équations de Monge–Ampère complexes modifiées, J. Funct. Anal. 156 (1998) 208–251. [5] L. Caffarelli, J. Kohn, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, II: Complex Monge–Ampère and uniformly elliptic equations, Comm. Pure Appl. Math. 38 (1985) 209–252. [6] S. Dinew, S. Kolodziej, Pluri-potential estimates on compact Hermitian manifolds, arXiv:0910.3937. [7] S. Dinew, X. Zhang, X.W. Zhang, The C 2,α estimate of complex Monge–Ampere equation, Indiana Univ. Math. J., in press. [8] J. Fu, S.T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge–Ampère equation, J. Differential Geom. 78 (2008) 369–428. [9] B. Guan, Q. Li, Complex Monge–Ampère equations and totally real submanifolds, Adv. Math. 225 (3) (2010) 1185–1223. [10] A. Hanani, Equations du type de Monge–Ampère sur les variétés hermitiennes compactes, J. Funct. Anal. 137 (1996) 49–75. [11] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I, Wiley Classics Lib., John Wiley & Sons, 1963. [12] N.G. Meyers, On a class of non-uniformly elliptic quasi-linear equations in the plane, Arch. Ration. Mech. Anal. 12 (1963) 367–391. [13] D.H. Phong, N. Sesum, J. Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow, Comm. Anal. Geom. 15 (2007) 613–632.

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[14] D. Riebesehl, F. Schulz, A priori estimates and a Liouville theorem for complex Monge–Ampère equations, Math. Z. 186 (1984) 57–66. [15] V. Tosatti, B. Weinkove, Estimates for the complex Monge–Ampère equation on Hermitian and balanced manifolds, arXiv:0909.4496. [16] V. Tosatti, B. Weinkove, The complex Monge–Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (4) (2010) 1187–1195. [17] V. Tosatti, B. Weinkove, S.T. Yau, Taming symplectic forms and the Calabi–Yau equation, Proc. Lond. Math. Soc. (3) 97 (2008) 401–424. [18] S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, Comm. Pure Appl. Math. 31 (1978) 339–411. [19] X.W. Zhang, A priori estimates for complex Monge–Ampère equation on Hermitian manifolds, Int. Math. Res. Not. 2010 (19) (2010) 3814–3836.


An additive formula for Samuel multiplicities on Hilbert spaces of analytic functions Guozheng Cheng a,∗ , Xiang Fang b,1 a School of Mathematics, Wenzhou University, Wenzhou, Zhejiang, 325035, China b Department of Mathematics, Kansas State University, Manhattan, KS 6650, United States

Received 21 July 2010; accepted 29 September 2010 Available online 8 October 2010 Communicated by D. Voiculescu

Abstract We establish a short exact sequence to relate the germ model of invariant subspaces of a Hilbert space of vector-valued analytic functions and the sheaf model of the corresponding coinvariant subspaces. As a consequence we obtain an additive formula for Samuel multiplicities. As an application, we give a different proof for a formula relating the fibre dimension and the Samuel multiplicity which is first proved in Fang (2005) [11]. The feature of the new proof is that the analytic arguments in Fang (2005) [11] are now subsumed by algebraic machinery. © 2010 Elsevier Inc. All rights reserved. Keywords: Samuel multiplicity; Fibre dimension; Sheaf model; Germ model

1. Introduction In this paper we prove an additive formula (5) for Samuel multiplicities on Hilbert spaces of analytic functions. To prove the formula we establish a short exact sequence (4) which enables us to capture the information in the much-studied sheaf model [7,16] of a quotient module, by the germ model of a submodule, a model which has received less attention in the past (see Section 4). * Corresponding author.

E-mail addresses: [email protected] (G. Cheng), [email protected] (X. Fang). 1 Partially supported by National Science Foundation Grant DMS 0801174 and Laboratory of Mathematics for

Nonlinear Science, Fudan University. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.015

2028

G. Cheng, X. Fang / Journal of Functional Analysis 260 (2011) 2027–2042

In commutative algebra, the additivity of Samuel multiplicities [6, p. 273, p. 279], [14, p. 52] is of fundamental importance for applications in algebraic geometry and a parallel version in operator theory (see (2)) is proved, say, for the Hardy space H 2 (D) and the Dirichlet space D over the unit disk, but not possible for the Bergman space L2a (D) [10]. The obstacle for L2a (D) is largely the fact that the codimension dim(M zM) of an invariant subspace can be arbitrary [1,3]. In several variables, the formula (2) is true for the symmetric Fock space Hd2 [13], but the problem remains open for the Hardy space over the ball or over the polydisc in Cd , d 2. For more details on Samuel additivity on these function spaces, see Section 3. The purpose of this paper is to show that a modification of the Samuel additivity formula holds for natural Hilbert spaces of analytic functions, such as those related to weighted shifts. Namely, we define Samuel multiplicities on coinvariant subspaces by the sheaf model, while on invariant subspaces we use the germ model to define these multiplicities. Then we show that they naturally add up to the total multiplicity. Our motivation is to obtain a conceptual understanding of the following formula (1) from [11], which holds for invariant subspaces M ⊂ H ⊗ CN of a large class of Hilbert spaces of analytic functions, such as the Hardy space or the Bergman space over the unit ball or the polydisc: f d(M) + e M ⊥ = N.

(1)

For explanation of notations, see Theorem 6. In this formula the fibre dimension f d(M), an analytic invariant, is added to the Samuel multiplicity e(M⊥ ), an algebraic invariant. The novelty in our new proof of (1) is probably that many of the analytic and computational arguments in [11] are now replaced by algebraic ones. Because the arguments in this paper have a heavy algebraic and sheaf-theoretic flavor, in order to see the relevance to other problems in operator theory, we recall that the case of the symmetric Fock space allows one to show that the curvature of a pure d-contraction is equal to the Samuel multiplicity [12]. Also formula (1) can be used to calculate Fredholm indices of many Hilbert modules [11]. 2. Definition of Samuel multiplicity in operator theory For a single operator T ∈ B(H ), the Samuel multiplicity is defined to be dim(H /T k H ) , k→∞ k

e(T , H ) = lim

which is well defined and is indeed a finite integer if dim(H /T H ) < ∞ [9]. In general, by a Hilbert module H over the polynomial ring A = C[z1 , . . . , zd ] [4] we mean a complex, separable Hilbert space H which admits an A-module structure such that the action of each zi induces a bounded operator Ti on H . Then let T = (T1 , . . . , Td ). The assumption we will need is that dim(H /T H ) < ∞, where T H T1 H + · · · + Td H. Let I = (z1 , . . . , zd ) ⊂ A be the maximal ideal at the origin. According to results on Hilbert polynomials [5,6,14], the function φH,T (k) = dim H /I k H


2029

becomes a polynomial when k 0. Moreover, dim(H /I k H ) k→∞ kd

e(H ) = d! · lim

exists, and is an integer, which we define to be the Samuel multiplicity of H with respect to I [8]. This is an important invariant in algebraic geometry and its Hilbert space version has found many connections with operator theory in recent years. Examples of e(·). Let H 2 be the Hardy space or the Bergman space over the unit ball or the polydisc in Cd , and let H = H 2 ⊗ CN , where N ∈ N. Assume that M ⊂ H is a submodule and M⊥ is the associated quotient module, with module actions induced by the multiplication of coordinate functions z1 , . . . , zd . (1) e(H ) = N , which can be checked directly from the definition of e(·). (2) e(M⊥ ) is always finite since we have M⊥ /I M⊥ ∼ = H /(I H + M), which is finite dimensional. Indeed, we have M⊥ /I k M⊥ ∼ = H/ I kH + M ,

∀k ∈ N.

Since the dimension of H /(I k H + M) is at most that of H /I k H , it follows that e M⊥ e(H ) = N. (3) e(M) = dim(M zM) when d = 1. In general, e(M) < ∞ if and only if dim(M/I M) < ∞. (4) When H 2 is the symmetric Fock space Hd2 and M ⊂ H 2 ⊗ CN , e(M) is either at most N or equal to ∞ [13]. Notations and Conventions. In this paper we mainly work with Hilbert modules of analytic functions, as well as their submodules and quotient modules. We always assume that the module actions are induced by the multiplication of coordinate functions. Moreover, we use I = (z1 , . . . , zd ) to denote the maximal idea at the origin, either in the polynomial ring A = C[z1 , . . . , zd ] or in O0 , the local ring of germs of analytic functions around the origin. Samuel multiplicities are always taken with respect to I , unless otherwise specified. Let Iλ = (z1 − λ1 , . . . , zd − λd ) be the maximal ideal of A at λ = (λ1 , . . . , λd ) ∈ Cd . 3. Additivity of Samuel multiplicities Let H be a Hilbert module of analytic functions over a domain Ω ⊂ Cd containing the origin. Let M ⊂ H ⊗ CN be a submodule, and M⊥ be the associated quotient module. Then the Samuel additivity formula concerns whether the following equation holds: e(M) + e M⊥ = e H ⊗ CN .

(2)

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In case of the Hardy space H = H 2 (D) or the Dirichlet space H = D over the unit disc, where the formula does hold, one has e H ⊗ CN = N, and e(M) = dim(M zM). In particular, the codimension-N property dim(M zM) N follows since e(M⊥ ) 0, Indeed the study of (2) has led the second author to show in [10] that dim(M zM) = sup dim M(λ) N, λ∈D

where M(λ) f (λ): f ∈ M is a subspace of CN . On the other hand, for H = Hd2 , the symmetric Fock space over the unit ball in Cd , the additivity formula (2) holds if and only if dim(M/I M) < ∞ [13]. Lastly, for the Bergman space H = L2a (D), it is well known that e(M) = dim(M zM) can be arbitrarily large [1,3], hence the additivity formula is far from being true. In summary, it appears that the failure of the Samuel additivity (2) is largely due to the fact that dim(M I M) can be too large. In order to rescue the formula, one needs to modify the definition of e(M) on M to ensure its finiteness. 4. Sheaf model vs. germ model The (rather successful) idea of sheafifying a Hilbert module H encodes information about H by algebraic modules and in this vein the standard procedure is to consider the so-called sheaf model H˜ [7,16], H˜ = O(H )/(T − w)O(H ). Here O(H ) denotes the sheaf of H -valued analytic functions, the tuple T = (T1 , . . . , Td ) denotes the module actions of multiplication of z1 , . . . , zd on H , w = (w1 , . . . , wd ) the coordinate functions for the sheave O, and (T − w)O(H ) = (T1 − w1 )O(H ) + · · · + (Td − wd )O(H ). Moreover, we are interested in the stalk H˜ λ at a point λ ∈ Cd , H˜ λ = Oλ (H )/(T − w)Oλ (H ), which is a module over Oλ . For further discussion more notations and conventions are needed. We will write a basic tensor f ∈ O(H ) or O0 (H ) as h⊗g(w), where h ∈ H , w being the variable for the sheaf O, and g being


2031

an analytic function in w. Then we use f˜ to denote the class of f , either in H˜ or H˜ 0 . Moreover, an element f ∈ O0 (H ) is represented by an H -valued analytic function over a neighborhood of the origin, so we have a power series expansion f=

fi ⊗ w i ,

i0

where fi ∈ H. In several variables, one just replaces the index i by a multi-index I = (i1 , . . . , id ). (There seems to be no danger of confusing it with the ideal I = (z1 , . . . , zd ).) On the other hand, there is another rather naive way to sheafify a submodule M if we assume that elements of M are E-valued analytic functions over a domain Ω ⊂ Cd . Here E is another ˆ generated by elements of M as analytic funcHilbert space. Namely, we consider the sheaf M tions. For the stalk of a function f ∈ M at a point λ ∈ Ω, we send the function directly to its germ f ∈ M → fλ ∈ Oλ (E), ˆ the germ model of M over Ω. Note that this is not a functorial and call the resulting sheaf M ˆ λ is a finite linear combination of elements of the form operation. Also note that an element in M r · fλ , where r ∈ Oλ and f ∈ M. ˆ The sheaf model H˜ has been thoroughly studied [7], while the germ ˜ and M. Comparison of H ˆ ˆ is not a functorial construction, while model M receives less attention, probably because M ˆ from the viewpoint of the sheaf model H˜ is. In particular, H˜ is a right-exact functor. For M, homological algebra, an operation with no exactness and no functoriality is usually of less value. ˆ is that it applies only to submodules. Another serious drawback of M ˆ however, is clearly much easier to define, and one of the main findings The germ model M, ˆ encodes essentially all of this paper is to show that under natural conditions the germ model M ⊥ of the associated quotient module. In particular, for Samuel information in the sheaf model M additivity, we have the following result. Theorem 1. Let H be the Hardy space or the Bergman space over the unit ball or the polydisc in Cd (d ∈ N), and M ⊂ H ⊗ CN (N ∈ N) be an invariant subspace. Then ⊥ = N. ˆ 0) + e M e(M 0

(3)

So this version of Samuel additivity circumvents the difficulty associated with the largeness of dim(M zM). The proof of Theorem 1 is given after Proposition 5. Remarks. (1) Both Samuel multiplicities are taken with respect to I = (z1 , . . . , zd ) ⊂ O0 . (2) Note that e((H ⊗ CN )0 ) = e((H ⊗ CN )0 ) = N . (3) The theorem is not stated in the most general form and follows from Theorem 2.

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5. Main result Let E be a Hilbert space and let H be a Hilbert module of E-valued analytic functions over a domain Ω ⊂ Cd . We say that H is regular at a point λ ∈ Ω if dim(H /Iλ H ) < ∞. By a natural Oλ -module homomorphism jλ : Hˆ λ → H˜ λ we mean an Oλ -module homomorphism such that jλ (fλ ) = (f ⊗ 1)λ ,

∀f ∈ H.

Recall that fλ ∈ Oλ (E) is the germ at λ of f ∈ H as an E-valued analytic function, and f ⊗ 1 is ˜ denotes a constant function in the space O(H ) of H -valued analytic functions. Moreover, the (·) the class in the sheaf model O(H )/(T − w)O(H ). Remark. To connect the sheaf model H˜ and the germ model Hˆ it is natural to construct homomorphisms between them. We do not know reasonable conditions to guarantee the existence of morphisms in the reverse directions H˜ λ → Hˆ λ , other than isomorphisms. We also do not know the implications that the existence of such morphisms has. Theorem 2. If a Hilbert module H of vector-valued analytic functions over a domain Ω ⊂ Cd satisfies that (1) H is regular at λ ∈ Ω, that is, dim(H /Iλ H ) < ∞, (2) there is a natural Oλ -module homomorphism jλ : Hˆ λ → H˜ λ , extending jλ (fλ ) = (f ⊗ 1)λ for f ∈ H , then for any submodule M ⊂ H , one has a short exact sequence of finitely generated Oλ modules qλ kλ ⊥ → 0. ˆ λ −→ H˜ λ −→ M 0→M λ

(4)

It follows the Samuel additivity formula ⊥ = e(H˜ ). ˆ λ) + e M e(M λ λ

(5)

Here the Samuel multiplicities are taken with respect to Iλ . The map kλ in (4) is the composition ˆ λ → Hˆ λ , which is induced by the inclusion i : M → H , and the natural map of the map iλ : M jλ : Hˆ λ → H˜ λ . Lastly, qλ is induced by the quotient map q : H → H /M ∼ = M⊥ . The existence of jλ is discussed in Section 6. The proof of Theorem 2 is in Section 7. It is ⊥ λ ) admit more operator-theoretic interpretaˆ λ ) and e(M probably natural to ask whether e(M ˆ λ ), as we will see in tion such as that e(M) is just dim(M zM) in one variable. For e(M the proof of Theorem 6, it is equal to the fibre dimension under fairly natural conditions. For ⊥ λ ), although it is not obvious from definition, it is indeed always equal to the Samuel mule(M tiplicity e(M⊥ ) defined by spatial actions directly [13]. These observations form the idea of our


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new proof of formula (1) in Theorem 6. So, in a sense, the short exact sequence (4), as well as (6) below, can be regarded as a lifting of formula (1) to the sheave level. Theorem 2 admits a sheaf version. Theorem 3. If a Hilbert module H of vector-valued analytic functions over a domain Ω ⊂ Cd satisfies that (1) H is regular at any λ ∈ Ω, dim(H /Iλ H ) < ∞, (2) there is a natural O|Ω -module homomorphism of analytic sheaves j : Hˆ |Ω → H˜ |Ω , extend ing jΩ (f |Ω ) = (f ⊗ 1)|Ω for f ∈ H and any open Ω ⊂ Ω, then for any submodule M ⊂ H , one has a short exact sequence of coherent analytic sheaves ⊥ |Ω → 0. ˆ Ω → H˜ |Ω → M 0 → M|

(6)

Proof. By standard results in sheaf theory, the exactness of the above sheave sequence is equivalent to the exactness of the sequence of stalks at each point, which is the conclusion of Theorem 2. 2 ⊥ of M⊥ can So under the conditions of Theorems 2 and 3, the study of the sheaf model M ˆ of M. Next we show that be, in principle, transformed into the study of the germ model M conditions (1) and (2) of Theorem 2 are satisfied for many natural Hilbert modules. 6. On the existence of the natural map jλ : Hˆ λ → H˜ λ and the proof of Theorem 1 Once we know that Hˆ 0 ∼ = H˜ 0 ∼ = O0 when H is the Hardy space or the Bergman space over the unit ball or the polydisc in Cd , Theorem 1 will follow from Theorem 2. Here we prove a more general result. Lemma 4. Let H be a Hilbert module of scalar-valued analytic functions over a domain Ω ⊂ Cd , d ∈ N, obtained by completing the polynomials A = C[z1 , . . . , zd ] with respect to a Hilbert space ¯ norm. That is, H = A. If H is regular at λ ∈ Ω, that is, dim(H /Iλ H ) < ∞, and λ ∈ int(bpe(H )), the interior of the set of bounded point evaluations of H , then Hˆ λ ∼ = H˜ λ ∼ = Oλ . Moreover, the isomorphism between Hˆ λ and H˜ λ can be chosen to be an Oλ -module homomor phism jλ such that jλ (fλ ) = (f ⊗ 1)λ for any f ∈ H . Proof. Without loss of generality we assume that λ = 0, so Iλ = I . The natural isomorphism between Hˆ 0 and O0 is easy: since Hˆ 0 is a submodule of O0 generated by germs f0 , f ∈ H , and Hˆ 0 contains a generator 10 , the germ of the constant function f (z) = 1, the two modules are indeed equal. For H˜ 0 = O0 (H )/(T − w)O0 (H ), we claim that (1) any element x ∈ H˜ 0 can be represented by 1 ⊗ f0 ∈ O0 (H ) for some f0 ∈ O0 ;

2034


(2) {1 ⊗ f0 : f0 ∈ O0 } is naturally isomorphic to O0 ; ⊗ 1)0 ∈ H˜ 0 for any f ∈ H . (3) 1 ⊗ f0 = (f For (1), we first show that dim(H /I H ) < ∞ implies dim(H /I H ) = 1. Notice that dim(H /I H ) < ∞ implies that I H is a closed subspace, hence 1C + I H is still closed. On the other hand, 1C + I H contains all polynomials, so 1C + I H = H . dim(H /I H ) be such that {hi + I H } span the space H /I H . By a result of Let {hi ∈ H }i=1 Markoe [15], when H is regular at the origin, the O0 -module H˜ 0 is finitely generated and ⊗ 1)0 is indeed generated by (h i ⊗ 1)0 . Since dim(H /I H ) = 1, 1 + I H spans H /I H . So (1 ˜ forms a generator of H0 and the submodule generated by (1 ⊗ 1)0 is of the form O0 · (1 ⊗ 1)0 = {1 ⊗ f0 (w), f0 ∈ O0 }. Now (1) is verified. ⊗ f0 = 0 for any nonzero f0 ∈ O0 . To show (2), that is, H˜ 0 ∼ = O0 , it suffices to show that 1 f0 (w) = 0 For convenience, we use 1z to denote the constant function 1 in H . Suppose that 1z ⊗ for some f0 ∈ O0 , that is, there are x (1) , . . . , x (d) ∈ O0 (H ) such that 1z ⊗ f0 (w) =

d (Tj − wj )x (j ) .

(7)

j =1

Then it is sufficient to show that f0 = 0. Since f0 and x (j ) are analytic functions around the origin, we can expand them into power series f0 (w) =

and x (j ) (w) =

cI w I

I

(j )

xI ⊗ w I .

I

(j )

Note that cI ∈ C and xI ∈ H . By comparing the coefficients of each w I in (7), we have (1)

(d)

(8)

Tj xI − xI −ej .

(9)

c 0 1 z = T1 x 0 + · · · + Td x 0 , and for each I = (i1 , . . . , id ), cI 1z =

d j =1

(j )

(j )

For each k 0, let Sk =

d j =1 I : |I |=k

(j )

T I xI −ej .

Claim One: I : |I |=k

cI zI = Sk+1 − Sk .


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Proof of Claim One. The case k = 0 is just the above (8) because (1)

(d)

S 1 = T1 x 0 + · · · + Td x 0 (j )

and S0 = 0, since xI −ej are automatically zero. In general, we look at those I ’s such that |I | = k,

cI z I =

I : |I |=k

T I · cI 1z =

I : |I |=k

d (j ) (j ) T I Tj xI − T I xI −ej . j =1 I : |I |=k

The second term yields Sk . The first term is equal to d

T I +ej xI . (j )

j =1 I : |I |=k

For each J with |J | = k + 1, there are d ways to rewrite it J = I + ej for j = 1, . . . , d. For each (j ) rewriting, the vector xI is determined by j , so we have d

T I +ej xI = (j )

j =1 I : |I |=k

d

j =1 J : |J |=k+1

(j )

T J xJ −ej Sk+1 .

Claim Two: Sk = 0 for each k 1. Proof of Claim Two. We will use induction. First, for k = 1, we have (1)

(d)

c0 = S1 = z1 x0 + · · · + zd x0 . Since, as analytic functions around the origin, the left side is a constant and the right side vanishes at the origin, we have both sides to be zero. Hence c0 = 0 and S1 = 0. Now assume that Sk = 0. To deal with Sk+1 , by Claim One, we have

cI zI = Sk+1 .

I : |I |=k

As analytic functions around the origin, the left side is a homogeneous polynomial of degree k and the right side is of vanishing order at least k + 1. It follows that both sides are zero. So Claim Two is proved and so is (2). For (3), by the definition of the sheaf model, one has zf ⊗ r = Tf ⊗ r = f ⊗ wr ⊗ 1)0 = 1 ⊗ p0 for any polynomial p ∈ H . For for any f ∈ H and r ∈ O0 . In particular, (p any f ∈ H , by the polynomial density assumption in the lemma, we can choose a sequence of polynomials {pi } such that pi − f H → 0

as i → ∞.

2036


Then, as constant functions in O0 (H ), we have pi ⊗ 1 → f ⊗ 1 as i → ∞, where the convergence is the convergence of analytic functions on any small neighborhood of the origin. It follows (f ⊗ 1)0 = lim (p i ⊗ 1)0 , i→∞

which is equal to (pi )0 . lim 1 ⊗

i→∞

Since pi → f in H -norm and 0 ∈ int (bpe(H )), we have pi → f as functions on some neighborhood of 0. That is, (pi )0 → f0 in O0 . It follows that (pi )0 = 1 ⊗ f0 . lim 1 ⊗

i→∞

2

Next we show that if we drop the polynomial density condition in Lemma 4, then it is possible that there exists no required natural map jλ : Hˆ λ → H˜ λ . Proposition 5. Let H be a C[z]-Hilbert module of scalar-valued analytic functions over a domain Ω ⊂ C. If H is regular at a point λ ∈ Ω, but dim(H /Iλ H ) = 1, then there can be no Oλ -module homomorphism jλ : Hˆ λ → H˜ λ such that jλ (f0 ) = (f ⊗ 1)0 , f ∈ H . Proof. Without loss of generality we assume that λ = 0. Let t = dim(H /I H ) ∈ N. Note that the operator T = Mz , the multiplication by z, is a Fredholm operator on H with a trivial kernel; that is, ker(T ) = {0}. By basic properties of Fredholm operators one has dim H /I k H = kt. It follows that the Samuel multiplicity of H is e(H ) = t. According to Theorem 1 in [13], e(H˜ 0 ) = e(H ) = t > 1. Since H is regular at 0, by [15] H˜ 0 is generated (f 1 ⊗ 1)0 , . . . , (f t ⊗ 1)0 , where f1 + I H, . . . , ft + I H forms a basis for H /I H . If the map j0 exists, then by the assumption on j0 , one has (f i ⊗ 1)0 = j0 ((fi )0 ). It follows that j0 is surjective; that is, j0 (Hˆ 0 ) = H˜ 0 . For Noetherian modules, the Samuel multiplicity of a module is at least the Samuel multiplicity of its image under a module homomorphism, so e(Hˆ 0 ) e(H˜ 0 ).


2037

Note that this is indeed a consequence of the Samuel additivity formula in algebra. Now one has e(Hˆ 0 ) t. But Hˆ 0 is just a submodule of O0 , so e(Hˆ 0 ) e(O0 ) = 1. Contradiction.

2

Now Theorem 1 follows from Theorem 2, which is proved in the next section. Proof of Theorem 1. We just need to show the existence of the natural module homomorphism j0 which is the identify map according to Lemma 4. Now the proof follows from Theorem 2. 7. Proof of Theorem 2 We first collect some facts about the I -adic topology on a module from [17]. Let R be a Noetherian ring, I ⊂ R be an ideal, and M be an R-module, with a natural filtration {I k M}k1 . Define the I -adic topology on M by declaring the closure of a subset S ⊂ M to be S + I kM . S¯ = k1

This topology is Hausdorff on M if and only if following fact.

k1 I

kM

= {0}. More importantly, we need the

Fact. (See [17, Corollary 4, p. 18].) If I is contained in the radical of R, that is, the intersection of all the maximal ideals of R, then any submodule N of a finitely-generated R-module M is closed under the I -adic topology on M; that is, N¯ = N . For the proof of Theorem 2, we assume without loss of generality that λ = 0. First we show that the map j0 must be injective. Claim. When H is regular at 0, there are finitely many f1 , . . . , fr ∈ H such that (f1 )0 , . . . , (fr )0 generate Hˆ 0 . In particular, Hˆ 0 is finitely generated. Proof of Claim. First note that I Hˆ 0 = (I

H )0 since each is the submodule of Hˆ 0 generated by I H . Next, since H is regular at 0, we can choose f1 , . . . , fr ∈ H , where r = dim(H /I H ), such that H = span{f1 , . . . , fr } + I H. It follows that H )0 . Hˆ 0 = span (f1 )0 , . . . , (fr )0 + (I

Hence the representatives of f1 , . . . , fr span Hˆ 0 /I Hˆ 0 . By Nakayama’s lemma [6, p. 124], the germs of f1 , . . . , fr generate Hˆ 0 . The claim is proved. 2 Now we can write any element of Hˆ 0 as x = s1 · (f1 )0 + · · · + sr · (fr )0 ,

2038


for some s1 , . . . , sr ∈ O0 . In particular, the image of si · (fi )0 under j0 is si · f i ⊗ 1 = f i ⊗ si since we assume j0 ((fi )0 ) = fi ⊗ 1 and j0 is an O0 -module homomorphism. If x = 0, we can expand x into a power series around the origin x = x0 + x1 + · · · , where we assume that xj is a homogeneous polynomial of degree j . Assume that x0 = · · · = xK−1 = 0 and xK is the first nonzero term in the expansion. Then we call K = ord(x) the order of x at the origin. Let PK (sj ) denote the Taylor polynomial of sj of degree K. Then we write x = PK (s1 ) · (f1 )0 + · · · + PK (sr ) · (fr )0 + x , and note that ord(x ) K + 1 and j0 (x ) ∈ I K+1 H˜ 0 . Suppose that j0 (x) = 0, that is, there exists h(1) , . . . , h(d) ∈ O0 (H ) such that − w)h, j0 (x) = (T (j ) where (T − w)h = dj =1 (Tj − wj )h(j ) . We can expand h(j ) at the origin to get where hI ∈ H . Since fi ⊗ PK (si ) = PK (s i )f ⊗ 1, if we let f = f1 PK (s1 ) + · · · + fr PK (sr ) ∈ H, then we have d f ⊗1+x = (Tj − wj )h(j ) ,

j =1

where x

∈ I K+1 O0 (H ). In particular, the order of x

at the origin satisfies ord(x

) K + 1. Note that d

(Tj − wj )h(j ) =

j =1

d

(j )

T h0 +

j =1

d (j ) (j ) Tj hI − hI −ej ⊗ w I .

I =(0,...,0) j =1

j th

Here I = (i1 , . . . , id ) ∈ denotes the multi-index and ej = (0, . . . , 0, 1 , 0, . . . , 0) denotes the j th coordinate index for j = 1, . . . , d. If any component it of I is negative, that is, it < 0, (j ) then we assume that hI = 0 for any j . By comparing the coefficients of elements in O0 (H ) in terms of power series of w, one has Zd

f=

d j =1

(j )

Tj h0 ∈ H,

(10)


2039

and d j =1

(j )

(j )

Tj hI − hI −ej = 0

(11)

for all 1 |I | K. Here |I | = i1 + · · · + id . Claim. For each 1 t K + 1, one has f=

d j =1 |I |=t

(j )

T I hI −ej .

The case t = 1 is just (10). Now assume that the claim is proved for t K and, in order to prove the statement for t + 1, we look at f=

TI

|I |=t

d

(j )

Tj hI =

j =1

T I +ej hI . (j )

I,j

(j )

Note that for each hI , its sup-index and sub-index determines the power of T in each term. Rewrite I = J − ej , we have f=

d j =1 |J |=t+1

(j )

T J hJ −ej .

Since each h ∈ H and each Tj is the multiplication by zj , it follows that ord(f ) K + 1, hence ord(x) K + 1. Contradiction. The injectivity of j0 is proved. ⊗ 1)0 in H˜ 0 . Next we consider the image of j0 ◦ i0 , which is the submodule generated by (f We denote this submodule by N1 ⊂ H˜ 0 . Let N2 ⊂ H˜ 0 be the submodule generated by representatives of O0 (M) ⊂ O0 (H ) in H˜ 0 , and we want to show that N1 = N2 when H is regular at 0. Clearly, N1 ⊂ N2 . Next we prove N1 = N2 by considering their I -adic topology closures in H˜ 0 . When H is regular, H˜ 0 is finitely generated [15]. So any submodule of H˜ 0 is closed in this topology, by the fact at the beginning of this section, since I is equal to the radical of O0 . It follows that, in order to show N1 = N2 , it suffices to show N1 + I k H˜ 0 = N2 + I k H˜ 0 for each k 1 by the definition of the I -adic closure. It is clear that N1 + I k H˜ 0 ⊂ N2 + I k H˜ 0 . For the other direction, we just need to show that f˜0 ∈ N1 + I k H˜ 0

2040


for any f ∈ O0 (M). Expand f =

∞

l=0 fl

g=

⊗ w l , with fl ∈ M. Let k−1

zl fl ∈ M,

l=0

then f˜0 = (g ⊗ 1)0 +

∞ fl ⊗ w l ∈ (g ⊗ 1)0 + I k H˜ 0 . l=k

0

Now we have proved N1 = N2 . Next we consider the left exactness of the sheaf model over the short exact sequence 0 → M → H → M⊥ → 0. From this we obtain an exact sequence l ⊥ → 0. → O0 (M)/(T − w)O0 (M) → O0 (H )/(T − w)O0 (H ) → M

We can complete the left end of the above exact sequence by observing that image(l) = O0 (M) + (T − w)O0 (H ) /(T − w)O0 (H ). So we have ⊥ → 0. 0 → O0 (M) + (T − w)O0 (H ) /(T − w)O0 (H ) → O0 (H )/(T − w)O0 (H ) → M Note that image(l) is equal to the submodule N2 of H˜ 0 generated by O0 (M), hence it is also ⊗ 1)0 . equal to the submodule N1 generated by (f ˆ 0 )). Since both j0 and i0 are injective O0 On the other hand, we observe that N1 = j0 (i0 (M module homomorphisms, ˆ0∼ M = image(l), and we have the desired exact sequence j0 ◦i0

⊥ 0 → 0. ˆ 0 −→ H˜ 0 → M 0→M Now the Samuel additivity formula ⊥ = e(H˜ ) ˆ 0) + e M e(M 0 0 follows from standard results on Samuel additivity in algebra [6] since all involved modules are now finitely generated over a Noetherian ring O0 .


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8. An application In this section we give another proof of the following Theorem 6 which, under slightly different technical assumptions, is proved by rather different methods in [11]. Our proof here is shorter because the bulk of the argument is absorbed by the algebraic version of Samuel additivity. Theorem 6. Let H be a Hilbert module of scalar-valued analytic functions over a domain Ω ⊂ Cd containing the origin, d ∈ N, obtained by completing the polynomials A = C[z1 , . . . , zd ] ¯ with respect to a Hilbert space norm. That is, H = A. If H is regular at 0 ∈ Ω, that is, dim(H /I H ) < ∞, and 0 ∈ int (bpe(H )), the interior of the set of bounded point evaluations of H , then for any submodule M ⊂ H ⊗ CN , N ∈ N, one has f d(M) + e M⊥ = N. Here the fibre dimension is defined as f d(M) = sup dim M(λ) , λ∈Ω

with M(λ) = {f (λ), f ∈ M} ⊂ CN for any λ ∈ Ω. Moreover, the Samuel multiplicity e(M⊥ ) is still taken with respect to I = (z1 , . . . , zd ), the maximal ideal at the origin. ⊥ )0 ), hence by Theorem 2 and Proof. By Theorem 1 in [13] we know that e(M⊥ ) = e((M ˆ 0 ) = f d(M). This is basically a consequence Lemma 4 in this paper, it suffices to show that e(M of properties of coherent analytic sheaves. By the upper-semicontinuity of the codimension function λ → dim(H /Iλ H ), we know that H is regular on a neighborhood of the origin. Without loss of generality, we assume that H is regular on Ω. By the claim at the beginning of the proof of Theorem 2, we know that Hˆ is ˆ as a subsheaf of Hˆ is also coherent. Now we need to recall two coherent on Ω. It follows that M natural ways to localize the coherent analytic sheave Mˆ at a point λ ∈ Ω [2]. First, let C = Cλ be an Oλ -module with the module action given by (f ∈ Oλ , c ∈ C) → f (λ)c, and consider the tensor ˆ λ ⊗O C λ , M λ which is the first localization we need. For the second localization, we consider ˆ M(λ) = g(λ), g ∈ Mλ ˆ with the Oλ -module action given by (f ∈ Oλ , g(λ)) → f (λ)g(λ). Note that M(λ) is a subspace N of C . We claim that there are nowhere dense, analytic subsets S1 , S2 , S3 of Ω such that (1) for λ ∈ Ω \ S1 , f d(M) = dim(M(λ)); ˆ λ ⊗O C λ ∼ ˆ (2) for λ ∈ Ω \ S2 , M = M(λ); λ ˆ ˆ λ. ˆ (3) for λ ∈ Ω \ S3 , e(Mλ ) = dimC Mλ /Iλ M

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For (1), this is a property of fibre dimension. For (2), this is a property of subsheaves of the locally free sheave O(N ) [2]. For (3), this is a property of any coherent analytic sheaves. For instance, we can choose S3 ˆ is locally free on Ω \ S3 . Then e(Mˆ λ ) is equal to the rank of the locally free such that M ˆ λ /Iλ M ˆ λ . Moreover, e(Mˆ λ ) is independent of λ on sheave, where the rank is equal to dimC M the entire Ω. In particular, e(Mˆ λ ) = e(Mˆ 0 ). Since S1 , S2 , S3 are nowhere dense in Ω, we can choose λ ∈ Ω \ (S1 ∪ S2 ∪ S3 ). Now the ˆ 0 ) = f d(M) can be completed once we observe that proof of e(M ˆ ˆ (a) dim M(λ) = dim M(λ) for any λ ∈ Ω; indeed, we have M(λ) = M(λ) ⊂ CN . ˆ λ , which is a general algebraic fact, true even for non-Noetherian ˆ λ ⊗O C λ ∼ ˆ λ /Iλ M (b) M =M λ modules. 2 Acknowledgments The authors thank Bob Burckel and the referee for helpful suggestions to greatly improve the readability of this paper. References [1] C. Apostol, H. Bercovici, C. Foias, C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I, J. Funct. Anal. 63 (1985) 369–404. [2] C. Banica, O. Stanasila, Algebraic Methods in the Global Theory of Complex Spaces, John Wiley & Sons Ltd., 1976. [3] H. Bercovici, C. Foias, C. Pearcy, Dual Algebras with Applications to Invariant Subspaces and Dilation Theory, CBMS Reg. Conf. Ser. Math., vol. 56, Amer. Math. Soc., Providence, RI, 1985. [4] R. Douglas, V. Paulsen, Hilbert Modules over Function Algebras, Pitman Res. Notes Math. Ser., vol. 217, Longman, New York, 1989. [5] R. Douglas, K. Yan, Hilbert–Samuel polynomials for Hilbert modules, Indiana Univ. Math. J. 42 (1993) 811–820. [6] D. Eisenbud, Commutative Algebra. With a View toward Algebraic Geometry, Grad. Texts in Math., vol. 150, Springer-Verlag, New York, 1995. [7] J. Eschmeier, M. Putinar, Spectral Decompositions and Analytic Sheaves, London Math. Soc. Monogr. New Ser., vol. 10, Oxford University Press, New York, 1996. [8] X. Fang, Hilbert polynomials and Arveson’s curvature invariant, J. Funct. Anal. 198 (2003) 445–464. [9] X. Fang, Samuel multiplicity and the structure of semi-Fredholm operators, Adv. Math. 186 (2004) 411–437. [10] X. Fang, Invariant subspaces of the Dirichlet space and commutative algebra, J. Reine Angew. Math. 569 (2004) 189–211. [11] X. Fang, The Fredholm index of quotient Hilbert modules, Math. Res. Lett. 12 (2005) 911–920. [12] X. Fang, The Fredholm index of a pair of commuting operators, Geom. Funct. Anal. 16.2 (2006) 367–402. [13] X. Fang, The Fredholm index of a pair of commuting operators, II, J. Funct. Anal. 256 (2009) 1669–1692. [14] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math., vol. 52, Springer-Verlag, New York, 1977. [15] A. Markoe, Analytic families of differential complexes, J. Funct. Anal. 9 (1972) 181–188. [16] M. Putinar, Spectral theory and sheaf theory, II, Math. Z. 192 (1986) 473–490. [17] J.-P. Serre, Local Algebra, Springer Monogr. Math., Springer-Verlag, Berlin, 2000.


C 1 linearization for planar contractions ✩ Wenmeng Zhang, Weinian Zhang ∗ Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China Received 26 July 2010; accepted 23 December 2010 Available online 5 January 2011 Communicated by D. Voiculescu

Abstract C 1 linearization is of special interests because it can distinguish characteristic directions of dynamical systems. It is known that planar C 1,α contractions with a fixed point at the origin O admit C 1,β linearization with sufficiently small β > 0 if α = 1 and admit C 1,α linearization if (log |λ1 |/ log |λ2 |) − 1 < α 1, where λ1 and λ2 are eigenvalues of the linear parts of the contractions at O with 0 < |λ1 | |λ2 | < 1. In this paper we improve the lower bound of α to lower the condition of C 1 linearization for planar contractions. Furthermore, we prove that the derivatives of transformations in our C 1 linearization are Hölder continuous and give estimates for the Hölder exponent. Finally, we give a counter example to show that those estimates cannot be improved anymore. © 2010 Elsevier Inc. All rights reserved. Keywords: C 1 linearization; Contractions; Hölder continuous; Regularity; Invariant curve

1. Introduction In the theory of dynamical systems, one of the most fundamental and important problems is linearization. Usually, the C r linearization of a C k diffeomorphism F : Ω → X, where 1 r k ∞, X is a Banach space and Ω ⊂ X is an open set, is to find a C r diffeomorphism Φ from an open set U ⊂ Ω into X such that ✩

Supported by NSFC and MOE research grants.


E-mail addresses: [email protected], [email protected] (W.N. Zhang). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.029

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W.M. Zhang, W.N. Zhang / Journal of Functional Analysis 260 (2011) 2043–2063

Φ F (x) = SΦ(x),

∀x ∈ U,

(1.1)

for a linear mapping S. It has a long history to study linearization. For analytic diffeomorphisms on Cn , the idea of linearization goes back to Poincaré [10], who proved that F can be analytically conjugated to its linear part near a fixed point if all eigenvalues λi (i = 1, . . . , n) of the linear part inside lie m the unit circle S 1 (or outside S 1 ) and satisfy the nonresonant condition, i.e., λi = nj=1 λj j for all i and all (m1 , . . . , mn ) ∈ Zn+ such that m1 + · · · + mn 2. When all eigenvalues lie on S 1 , Siegel [1,14] proved that conclusion also holds if the point (λ1 , . . . , λn ) ∈ Cn is of n Poincaré’s n mj −ν type (c, ν), i.e., |λi − j =1 λj | c( j =1 mj ) for all i and all (m1 , . . . , mn ) ∈ Zn+ such that m1 + · · · + mn 2, where c > 0 and ν > 0. Later, Brjuno [4] extended Siegel’s result in C and proved that all germs of analytic diffeomorphisms with linear part λ = e2πiθ are linearizable if θ log qn+1 < +∞, where (pn /qn )n0 is a Brjuno number, i.e., an irrational number such that ∞ n=0 qn is the sequence of the convergents of θ ’s continued fraction expansion. In 1988 Yoccoz [18] proved that the condition is necessary. Concerning linearization on Rn , a well-known result is the Hartman–Grobman Theorem [8], saying that C 1 diffeomorphisms can be C 0 linearized near the hyperbolic fixed points. This result was generalized to Banach space by Pugh in [11]. Sometimes C 0 linearization is not effective to discuss more details of dynamics, for example, to distinguish a node from a focus. For smooth linearization, Sternberg [15,16] proved that C k (k 1) diffeomorphisms can be C r linearized near the hyperbolic fixed points, where the integer r depends on k and the nonresonant condition. In particular, r = ∞ if k = ∞ and nonresonant conditions of all orders hold. Further efforts were also made to the class C k,α (k 0 is an integer and 0 < α 1 is a real), i.e., the class of all C k mappings F ’s whose derivatives F (k) ’s satisfy that

F (k) (x) − F (k) (y) < ∞. sup x − y α x,y Belitskii [2,3] proved that C k,1 (k 1) diffeomorphisms can be C k linearized locally if the q (1 q n) distinct norms p1 < · · · < pq of their eigenvalues satisfy that the union + − k + k − i<j [pi pj , pj pi ], where pi := max{pi , 1} and pi := min{pi , 1}, does not contain any one of p1 , . . . , pq . His result particularly implies that C 1,1 diffeomorphisms can be C 1 linearized locally if the eigenvalues λ1 , . . . , λn satisfy |λi | · |λj | = |λk | (k = 1, . . . , n) if |λi | < 1 < |λj |. In 1985 Sell [13] extended Sternberg and Belitskii’s results for k 2 and gave some more delicate conditions for C r linearization. In the study of dynamical systems, C 1 linearization is of special interests. Hartman [7] showed that all C 1,1 contractions on Rn admit local C 1,β linearization with small β > 0 depending on the eigenvalues of their linear parts at the fixed point. In the early years of 2000s ElBialy [6] and Rodrigues and Solà-Morales [12] generalized this result to Banach space independently. On the other hand, it is proved in Corollary 1.3.3 in [5] that a C 1,α contraction F on Banach space can be C 1,α linearized near the origin O, which is the fixed point of F , if the constant α ∈ (0, 1] satisfies that −1 1 + α > log ρ F (O)−1 / log ρ F (O) ,

(1.2)

where ρ(F (O)) denotes the spectral radius of F (O). Thus, in either 1-dimensional cases or 2dimensional cases with |λ1 | = |λ2 |, where λ1 and λ2 are eigenvalues of F (O), we can conclude


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that C 1,α contractions admit C 1,α linearization for all α ∈ (0, 1]. In 2-dimensional cases with |λ1 | = |λ2 | condition (1.2) can be written as α > α1 := log |λ1 |/ log |λ2 | − 1.

(1.3)

Further results on the plane were given by Stowe [17] in 1986. He investigated a C k expansion F (x) := Sx + o( x k ), where k 2 is an integer and S := F (O). Let a and b be the modulus of the eigenvalues of S with 1 < a b. Stowe proved that (i) if loga b < k, then the sequence (S n F −n )n∈N converges uniformly in C k norm and that (ii) if loga b k, then F can be trans , which satisfies that formed to another C k expansion F − S)(j ) (x1 , 0) = 0, (F

∀0 j k,

(1.4)

−n )n∈N converges uniformly in C k−1 norm. Acfor all small x1 ∈ R, so that the sequence (S n F k cordingly, he concluded that F admits C and C k−1 linearization in cases (i) and (ii) respectively and estimated the Hölder exponent of Φ (k−1) in case (ii). There are also some results concerning the cases that O is a saddle and r = ∞ in [17]. Since expansions can be discussed similarly, in this paper we investigate C 1 linearization for planar C 1,α (0 < α 1) contractions F near the fixed point O. We discuss under the assumption 0 < |λ1 | < |λ2 | < 1

(1.5)

because the case |λ1 | = |λ2 | was solved in [5] as mentioned above. We give a number α0 which is smaller than min{1, α1 } given by Hartman [7] and Chaperon [5] and prove that F can be linearized by a C 1 transformation provided α > α0 . In the proof the method used in [17] has to be modified because result (1.4) cannot be obtained for C 1,α mappings. Furthermore, we prove that those transformations Φ for the C 1 linearization are not only C 1 but also C 1,β for some β > 0 depending on α. We give estimates for β in various cases and show with a counter example that those estimates cannot be improved anymore. 2. C 1 linearization of F In this section, we aim to find a C 1 diffeomorphism Φ near O such that Eq. (1.1) holds. Thus, replacing Φ in Eq. (1.1) with {Φ (O)}−1 Φ, we may assume that S = F (O). Since λ1 = λ2 by (1.5), we further assume that S is of the diagonal form, i.e.,

S = F (O) =

λ1 0

0 λ2

.

In the following let x := (x1 , x2 ) ∈ R2 and define the norm · as

x := max |x1 |, |x2 | ,

∀x ∈ R2 .

Theorem 1. Let Ω be a neighborhood of the origin O and F : Ω → R2 be a C 1,α (0 < α 1) contraction having O as its a fixed point. Assume that the two eigenvalues λ1 and λ2 of F (O)

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satisfy (1.5) and that α > α0 := 1 − log |λ2 |/ log |λ1 |.

(2.6)

Then there exist a neighborhood U ⊂ Ω of O and a C 1 diffeomorphism Φ : U → R2 such that Eq. (1.1) holds, i.e., F admits C 1 linearization near O. The number α0 given in Theorem 1 is a lower bound of the Hölder exponent α for C 1 linearization. Obviously, α0 < 1. It follows immediately that planar C 1,1 contractions admit local C 1 linearization, as indicated in [7]. Moreover, one can check that α0 < α1 by (1.5), which implies that our condition (2.6) is also weaker than the condition (1.3). Therefore, our Theorem 1 extends the results given in [5,7] in the case of planar systems. In order to prove the theorem, we need the following lemma on invariant curves of F . Lemma 1. Suppose that F : Ω → R2 is a C 1,ζ (0 < ζ 1) contraction and that the two eigenvalues λ1 and λ2 of F (O) satisfy (1.5). Then there exists a closed disk V ⊂ Ω centered at O such that F has a C 1,ζ invariant curve

Γ := (x1 , x2 ) ∈ V : x1 = g(x2 ) , where g : V ∩ R → R is C 1,ζ and g(0) = g (0) = 0, if the constant ζ satisfies that 0 < ζ < α1 , where α1 is given in (1.3). Proof. This lemma is actually a corollary of Theorem 2.1 in [9]. Let the norm · C 1,ζ be defined as ϕ(x + ) − ϕ(x) − ϕ (x) ϕ C 1,ζ := sup ϕ(x) + sup ϕ (x) + sup 1+ζ x∈R2 x∈R2 x,∈R2 for all C 1,ζ mappings ϕ : R2 → R2 . Let S1 and S2 denote x1 -axis and x2 -axis respectively, which are obviously closed subspaces of R2 invariant under F (O) and satisfy that R2 = S1 ⊕ S2 . Then, one can check that there is a constant a > 1 such that the expansion F −1 satisfies the following: (i) (ii) (iii) (iv)

F −1 (O) = O, F (O)|S1 = |λ1 | < a −(1+ζ ) , (F −1 ) (O)|S2 = |λ2 |−1 < a, and there exists a mapping H : R2 → R2 such that H (x) = F −1 (x) near O and H (x) − H (O)x C 1,ζ is sufficiently small.

In fact, (i) is obvious. Note that |λ1 | < |λ2 |1+ζ since 0 < ζ < α1 . There is a sufficiently small constant ε > 0 such that 1+ζ |λ1 | < |λ2 | − ε

−1 and |λ2 |−1 < |λ2 | − ε .

Then we get (ii) and (iii) by putting a := (|λ2 | − ε)−1 . Conclusion (iv) can be proved obviously, as indicated in the fourth remark in [9], by choosing an appropriate bump function (a smooth


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Fig. 1. Straight up an invariant curve.

function with compact support) on R2 . Thus, by Theorem 2.1 in [9], the mapping H has a C 1,ζ invariant curve on R2 and therefore F −1 has a C 1,ζ invariant curve on a closed disk V ⊂ Ω centered at O, which is tangent to x2 -axis at O. So does F . The proof is completed. 2 Proof of Theorem 1. Since C 1 linearization was proved in [5] for all α1 < α 1 as indicated in (1.3), it suffices to consider α ∈ (α0 , α1 ] ∩ (0, 1]. Noting that C 1,α implies C 1,α˜ near O for α > α, ˜ we only need to prove the C 1 linearization for α = α˜ in (α0 , α1 ) ∩ (0, 1]. In order to simplify the deduction, we flatten F along the x2 -axis by straightening up an invariant curve of F tangent to the x2 -axis, as shown in Fig. 1. By Lemma 1, there is a closed disk V ⊂ Ω centered at O such that F has a C 1,α˜ invariant curve Γ := {(x1 , x2 ) ∈ V : x1 = g(x2 )}, where g : V ∩ R → R is C 1,α˜ and g(0) = g (0) = 0. This curve enables us to make the C 1,α˜ transformation Θ : V → R2 defined by Θ(x) := x1 − g(x2 ), x2

(2.7)

:= Θ ◦ F ◦ Θ −1 instead. Once we can prove the C 1 so that we can consider the mapping F linearization for F , we naturally know the C 1 linearization of F because Θ is a local C 1 diffeomorphism. 1 := π1 F and F 2 := π2 F , where π1 and π2 denote the projecLet F1 := π1 F , F2 := π2 F , F tions onto the x1 -axis and x2 -axis respectively. Direct calculation gives

1 (x1 , x2 ) = F1 x1 + g(x2 ), x2 − g F2 x1 + g(x2 ), x2 , F 2 (x1 , x2 ) = F2 x1 + g(x2 ), x2 . F

(2.8)

Obviously, 1 (0, x2 ) = F1 g(x2 ), x2 − g F2 g(x2 ), x2 = 0, F

∀x2 ∈ V ∩ R,

(2.9)

leaves x2 -axis invariant, since the graph of g is invariant with respect to F . Moreover, F i.e., F has the following properties:

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Lemma 2. (O) = O and F (O) = S. is C 1,α˜ in V such that F (i) F 1 (x) ∂F α ˜ (ii) |F1 (x)| (|λ1 | + L x )|x1 | and | ∂x2 | L|x1 |α˜ for all x ∈ V , where L > 0 is a constant independent of x. We leave the proof of Lemma 2 after the proof of this theorem. In what follows we claim n ) (x))n∈N near O. If the claim is true then the the uniform convergence of the sequence (S −n (F −n n sequence (S F (x))n∈N also converges uniformly near O because for any m = n ∈ N −m m

S F (x) − S −n F n (x) = S −m F m (x) − S −n F n (x) − S −m F m (O) − S −n F n (O) m

n

(τ1 x) − S −n F (τ1 x) · x , S −m F n defines a := limn→∞ S −n F where τ1 ∈ (0, 1) is a number depending on x. Thus, the limit Φ 1 C mapping near O, which satisfies Eq. (1.1) because = S lim S −(n+1) F n+1 = S Φ. ◦ F Φ n→∞

Moreover, n

(O) = id, (O) = lim S −n F Φ n→∞

is a C 1 diffeomorphism near O. Therewhere id denotes the identity mapping. It implies that Φ 1 admits C linearization near O. fore, F n ) (x))n∈N , we note that F (x) < x In order to prove the uniform convergence of (S −n (F maps a sufficiently small for small x by Lemma 2(i) because S = |λ2 | < 1. It implies that F closed disk U ⊂ V centered at O into itself. For each n ∈ N, let n

(x) := F

an (x) cn (x)

bn (x) , dn (x)

(2.10)

where an , bn , cn , dn : U → R are functions. Moreover, those entries have the following properties: Lemma 3. There is a closed disk U centered at O such that n n a 1 F b 1 F (x) − λ1 M|λ2 |nα˜ , (x) M|λ1 |nα˜ , n n c 1 F d 1 F (x) M|λ2 |nα˜ , (x) − λ2 M|λ2 |nα˜ , bn (x) M|λ1 |n , an (x) M|λ1 |n , dn (x) M|λ2 |n cn (x) M|λ2 |n , for all n ∈ N and for all x ∈ U , where M > 0 is a constant independent of n and x.

(2.11) (2.12) (2.13) (2.14)


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We leave the proof of Lemma 3 after the proof of this theorem. Note that S

−n

n

(x) = F

an (x)/λn1 cn (x)/λn2

bn (x)/λn1 dn (x)/λn2

and n+1

n n

(x) F F (x) = F (x) = F

=

n (x)) a1 (F c1 (F n (x))

n (x))an (x) + b1 (F n (x))cn (x) a1 (F n (x))an (x) + d1 (F n (x))cn (x) c1 (F

n (x)) b1 (F n (x)) d1 (F

an (x) cn (x)

bn (x) dn (x)

n (x))bn (x) + b1 (F n (x))dn (x) a1 (F n (x))bn (x) + d1 (F n (x))dn (x) , c1 (F (2.15)

where (2.10) is used. We apply Lemma 3 to compute n (x)) − λ1 )an (x) + b1 (F n (x))cn (x) M 2 n an+1 (x) an (x) (a1 (F n n+1 − λn = |λ | μ1 + μ2 , n+1 λ1 λ1 1 1 n n (x)) − λ1 )bn (x) + b1 (F (x))dn (x) M 2 n bn+1 (x) bn (x) (a1 (F = μ1 + μn2 , − n n+1 n+1 λ1 |λ1 | λ1 λ1 n (x))an (x) + (d1 (F n (x)) − λ2 )cn (x) M 2 n cn+1 (x) cn (x) c1 (F n n+1 − λn = |λ | μ3 + μ1 , n+1 λ2 λ2 2 2 n n (x))bn (x) + (d1 (F (x)) − λ2 )dn (x) M 2 n dn+1 (x) dn (x) c1 (F = μ3 + μn1 − n n+1 n+1 λ2 |λ2 | λ2 λ2 for all n ∈ N and for all x ∈ U , where μ1 := |λ2 |α˜ ,

μ2 :=

|λ1 |α˜ |λ2 | , |λ1 |

μ3 :=

|λ1 ||λ2 |α˜ . |λ2 |

It follows that −(n+1) n+1

n M 2 S (x) ηn , F (x) − S −n F |λ1 |

∀n ∈ N, ∀x ∈ U,

(2.16)

where ηn := max{μn1 + μn2 , μn3 + μn1 }. We claim that μ1 , μ2 , μ3 ∈ (0, 1).

(2.17)

In fact, μ3 < μ1 < 1 is obvious by (1.5). From the definition (2.6) of α0 and the choice ˜ log |λ1 |, i.e., |λ2 | < |λ1 |1−α˜ , which implies that μ2 < 1. of α˜ we see that log |λ2 | < (1 − α) This the claim (2.17). Therefore, it follows from (2.16) and (2.17) that the series ∞ proves −(n+1) (F n+1 ) (x) − S −n (F n ) (x)} converges uniformly in U , namely, the sequence {S n=1 −n n

(S (F ) (x))n∈N converges uniformly in U . As shown above, the convergence guarantees the ˜ The proof is completed. 2 C 1 linearization for α = α.

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Proof of Lemma 2. For (i), noting that F and Θ (together with Θ −1 ) are C 1,α˜ since g is C 1,α˜ , we get

F (y) (x) − F

= Θ ◦ F ◦ Θ −1 (x) − Θ ◦ F ◦ Θ −1 (y)

= Θ F ◦ Θ −1 (x) F Θ −1 (x) Θ −1 (x) − Θ F ◦ Θ −1 (y) F Θ −1 (y) Θ −1 (y) Θ F ◦ Θ −1 (x) − Θ F ◦ Θ −1 (y) · F Θ −1 (x) · Θ −1 (x) + Θ F ◦ Θ −1 (y) · F Θ −1 (x) − F Θ −1 (y) · Θ −1 (x)

+ Θ F ◦ Θ −1 (y) · F Θ −1 (y) · Θ −1 (x) − Θ −1 (y) α˜ α˜ L1 F ◦ Θ −1 (x) − F ◦ Θ −1 (y) + L2 Θ −1 (x) − Θ −1 (y) + L3 x − y α˜ L x − y α˜

(2.18)

for all x, y ∈ V , where L1 , L2 , L3 and L are positive constants independent of x and y. This is C 1,α˜ in V . Noting that g(0) = 0 and g (0) = 0 as indicated before (2.7), one implies that F (O) = O and F (O) = S. can verify that F 1 (0, x2 ) = 0 by (2.9). By Lemma 2(i) we have In order to prove (ii), we first note that F 1 (0,0) ∂F = λ1 . Thus, ∂x1 1 (τ2 x1 , x2 ) ∂F F · |x1 | 1 (x1 , x2 ) − F 1 (x) = F 1 (0, x2 ) ∂x1

1 (τ2 x1 , x2 ) ∂ F 1 (0, 0) ∂F |x1 | |λ1 | + − ∂x1 ∂x1 |λ1 | + L x α˜ |x1 |, ∀x ∈ V , where τ2 ∈ (0, 1) is a number depending on x and (2.18) is employed. The first inequality in result (ii) is proved. Furthermore, we have 1 (0, x2 ) ∂F = 0, ∂x2 Otherwise, the continuity of

1 (0,x2 ) ∂F ∂x2

∀x2 ∈ V ∩ R.

(2.19)

in x2 implies that there exists x2∗ ∈ V ∩ R such that ∗

x2 ∂ F1 (0, t) dt = 0. ∂t

(2.20)

0

1 (x1 , x2 ) = x2 ∂ F 1 (x1 ,t) dt + h(x1 ), where h : V ∩ R → R is a function such On the other hand, F 0 ∂t that h(0) = 0 since F1 (0, 0) = 0. It follows from (2.20) that


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∗

1 0, x2∗ = F

x2 ∂ F1 (0, t) dt = 0, ∂t 0

a contradiction to (2.9). Thus, from (2.18) and (2.19) we get ∂F 1 (x1 , x2 ) = ∂ F1 (x1 , x2 ) − ∂ F1 (0, x2 ) L|x1 |α˜ , ∂x2 ∂x2 ∂x2 which proves the second formula given in (ii). The proof is completed.

∀x ∈ V , 2

Proof of Lemma 3. By (2.18) and the second inequality given in Lemma 2(ii), a1 (x) − λ1 = ∂ F1 (x) − ∂ F1 (O) L x α˜ , ∂x ∂x 1 1 b1 (x) = ∂ F1 (x) L|x1 |α˜ , ∂x 2 c1 (x) = ∂ F2 (x) − ∂ F2 (O) L x α˜ , ∂x ∂x 1 1 d1 (x) − λ2 = ∂ F2 (x) − ∂ F2 (O) L x α˜ ∂x ∂x2 2

(2.21) (2.22) (2.23) (2.24)

for all x ∈ V . On the other hand, there is a closed disk U ⊂ V centered at O such that n F (x) M1 |λ2 |n ,

π1 F n (x) M2 |λ1 |n ,

n F (x) M1 |λ2 |n

(2.25)

for all n ∈ N and for all x ∈ U , where M1 , M2 are positive constants independent of n and x. In fact,

F (τ3 x) − F (τ3 x) · x S + F (O) x (x) F |λ2 | + L x α˜ x ,

(2.26)

where τ3 ∈ (0, 1) is a number depending on x. Thus we can inductively prove that there is a closed disk V1 ⊂ V centered at O such that i F (x) |λ2 | + δ i ,

∀i ∈ N, ∀x ∈ V1 ,

where δ := (1 − |λ2 |)/2. Substituting (2.27) in (2.26), we get i+1 i F (x), (x) |λ2 | + L |λ2 | + δ i α˜ F

∀i ∈ N ∪ {0},

(2.27)

2052


implying the first inequality given in (2.25) by induction, where ∞ i α˜ L |λ2 | + δ 1+ < ∞. M1 := |λ2 | i=0

Similarly, by the first inequality in Lemma 2(ii) we see that there is a closed disk V2 ⊂ V center at O such that π1 F i+1 (x) |λ1 | + L |λ2 | + δ i α˜ π1 F i (x), ∀i ∈ N ∪ {0}, ∀x ∈ V2 , implying the second inequality given in (2.25) by induction, where ∞ i α˜ L 1+ < ∞. |λ2 | + δ M2 := |λ1 | i=0

We further consider

F (x) − F (x) S + F (O) |λ2 | + L x α˜ . It follows from (2.27) that n n−1 n−1 i α˜ F (x) F F (x) (x) · · · F |λ2 | + L |λ2 | + δ i=0

M1 |λ2 |

n

(2.28)

for all n ∈ N and for all x ∈ V1 , which gives the third inequality given in (2.25). Thus the three inequalities hold in the neighborhood U := V1 ∩ V2 . Let M > 0 be a constant larger than max{LM1α˜ , LM2α˜ , M1 }. Substituting the first two inequalities given in (2.25) in (2.21)–(2.24) correspondingly, we obtain the four inequalities given in (2.11) and (2.12). From the third inequality given in (2.25) we immediately obtain the two inequalities given in (2.14). In order to prove the inequalities given in (2.13), note that i i ai+1 (x) = a1 F (x) ai (x) + b1 F (x) ci (x) i |λ1 | + M|λ2 |i α˜ ai (x) + M 2 |λ1 |α˜ |λ2 | , ∀i ∈ N, (2.29) by (2.11) and (2.14). Choose M > 0 sufficiently large such that |a1 (x)| M 2 . Then, by (2.29), n−1 n−1 an (x) M 2 |λ1 | + M|λ2 |i α˜ + M 2 |λ1 | + M|λ2 |i α˜ |λ1 |α˜ |λ2 | + · · · i=1

+ M2

i=2 n−1

k n−1 |λ1 | + M|λ2 |i α˜ |λ1 |α˜ |λ2 | + · · · + M 2 |λ1 |α˜ |λ2 |

i=k+1

∞

∞

M |λ1 |α˜ |λ2 | j M2 i α˜ 1 |n , 1+ |λ2 | |λ1 |n = M|λ |λ1 | |λ1 | |λ1 | i=1

j =0

∀n ∈ N,


where

|λ1 |α˜ |λ2 | |λ1 |

2053

= μ2 < 1, known from (2.17), and therefore ∞

∞

2 M |λ1 |α˜ |λ2 | j := M 1+ < ∞. |λ2 |i α˜ M |λ1 | |λ1 | |λ1 | i=1

j =0

we can prove the first inequality given in (2.13). Thus, without loss of generality, putting M := M The second one given in (2.13) can be proved similarly. The proof is completed. 2 Remark that the equality (1.4), given in Theorem 1 in [17], should be written as is C k , but it cannot be obtained − S)(j ) (0, x2 ) = 0 for all 0 j k in our case, where F (F 1,α (0 < α 1) mappings. In order to overcome the difficulty, we proved the inequality for C n (x))| M|λ1 |nα˜ given in Lemma 3, which has a delicate difference that the constant |b1 (F |λ1 | is smaller than the corresponding constant |λ2 | in other three inequalities given in (2.11) n+1 ) (x) − S −n (F n ) (x)) to be controlled and (2.12). This guarantees the sequence (S −(n+1) (F by the sequence (ηn )n∈N given in (2.16). 3. Regularity of linearization In this section we give the smoothness of the transformation Φ obtained in Theorem 1 and show the regularity of linearization. Theorem 2. Let F be given in Theorem 1 and let the two eigenvalues λ1 and λ2 of F (O) satisfy (1.5). Then, the following assertions hold: (i) If α1 < α 1, then F can be linearized by a transformation of class C 1,α near O. (ii) If α = α1 1, then F can be linearized by a transformation of class C 1,β1 near O for any β1 ∈ (0, α). −1 (iii) If α = 1 but α1 > 1, then F can be linearized by a transformation of class C 1,α1 near O. (iv) If α0 < α < min{1, α1 }, then F can be linearized by a transformation of class C 1,β2 near O, where β2 := α1−1 + 1 α − 1 = α0−1 α − 1 ∈ (0, 1). Proof. The result (i) was proved in [5]. We only need to prove (ii), (iii) and (iv) by estimating the Hölder exponent β of the derivative of the C 1 transformation Φ obtained in Theorem 1. As in the proof of Theorem 1, we first assume that α = α˜ in (α0 , α1 ) ∩ (0, 1] and investigate the , which is defined just below (2.7). Let Φ n and claim that := limn→∞ S −n F reduced mapping F

n

n Φ (x) − S −n F (y) (x) − Φ (y) = lim S −n F K x − y ω n→∞

(3.30)

Then, for some positive constant ω and K, which implies C 1,ω smoothness of the mapping Φ. 1, α ˜ smoothness of Θ given in (2.7) and the relation Φ = Φ ◦ Θ we can use the same from C arguments as in (2.18) to see that the mapping Φ has C 1,β smoothness, where β = min{α, ˜ ω}.

(3.31)

2054


In order to compute ω given in (3.30), we have n

n (x) − S −n F (y) lim S −n F n→∞ ∞ n

−n n

−(n+1) n+1

−(n+1) n+1

−n (x) − S (y) S F F F F (x) − S (y) − S n=1

(x) − S −1 F (y) + S −1 F

∞ −(n+1) n+1

n+1 −n n

n S (x) − S −n F (y) F F (x) − S −(n+1) F (y) − S n=1

+ |λ1 |−1 L x − y α˜ for all x, y ∈ V by (2.18), in which ∞ −(n+1) n+1

n+1 −n n

n S (x) − S −n F (y) F F (x) − S −(n+1) F (y) − S n=1

=

an+1 (x) ∞ |( n+1 − λ 1

n=1

(x) |( cn+1 n+1 − λ2

an+1 (y) ) − ( anλ(x) n λn+1 1 1 cn+1 (y) cn (x) ) − ( λn2 λn+1 2

− −

an (y) )| λn1 cn (y) )| λn2

(x) |( bn+1 n+1 − λ1

(x) |( dn+1 n+1 − λ2

bn+1 (y) ) − ( bnλ(x) n λn+1 1 1 dn+1 (y) dn (x) ) − ( λn2 λn+1 2

− −

bn (y) )| λn1 . dn (y) )| λn2

(3.32) Here an , bn , cn , dn are given in (2.10). For the first entry in (3.32), by (2.15) we have

∞

an+1 (x) an+1 (y) an (x) an (y) − − n+1 − n λn1 λ1 λn+1 λ1 1 n=1 =

∞

n

1 n (x) cn (x) a1 F (x) − λ1 an (x) + b1 F n+1 |λ1 | n=1 n

n (y) − λ1 an (y) + b1 F (y) cn (y) − a1 F

Ξ1 (x, y) + Ξ2 (x, y) + Ξ3 (x, y), where Ξ1 (x, y) :=

∞ n=1

Ξ2 (x, y) :=

∞ n=1

1 n a1 F (x) − λ1 · an (x) − an (y), n+1 |λ1 | n 1 n (y) · cn (x), b F F (x) − b 1 1 |λ1 |n+1


2055

∞

n 1 n (y) · an (y) a1 F (x) − a1 F n+1 |λ | n=1 1 n (y) · cn (x) − cn (y) . + b 1 F

Ξ3 (x, y) :=

Lemma 4. There exist a neighborhood U ⊂ V of O and positive constants K1 , K2 and K3 such that −1 2 α˜

Ξ1 (x, y) K1 x − y α1 Ξ2 (x, y) K2 x − y

,

(α1−1 +1)α−1 ˜

,

Ξ3 (x, y) K3 x − y α˜ for all x, y ∈ U . Remark that the exponent (α1−1 + 1)α˜ − 1 is a positive constant since α˜ > α0 . This lemma is proved by Lemma 3 and will be given after the completion of the proof of this theorem. By Lemma 4,

∞

an+1 (x) an (x) an+1 (y) an (y) − − − λn1 λn1 λn+1 λn+1 1 1 n=1 −1 2 α˜

K1 x − y α1

−1

+ K2 x − y (α1

+1)α−1 ˜

+ K3 x − y α˜ .

(3.33)

The estimate for the second entry in (3.32) is almost the same and therefore we get

∞

bn+1 (x) bn (x) bn+1 (y) bn (y) − − − λn1 λn1 λn+1 λn+1 1 1 n=1 −1 2 α˜

K1 x − y α1

−1

+ K2 x − y (α1

+1)α−1 ˜

+ K3 x − y α˜ .

(3.34)

For the third entry in (3.32), note that for all k ∈ N k k k F y + σk (x − y) α˜ x − y α˜ F (x) − F (y) L F F LM1α˜ |λ2 |k α˜ x − y α˜

(3.35)

by (2.18) and the third inequality in (2.25), where each σk ∈ (0, 1) is a number depending on x and y. Then, by (2.28) and (3.35), n

n F (x) − F (y) n−1 n−1 n−i n−i−1 n−i−1 F F F (y) · · · F (y) · F F F (x) − F (y) i=0

n−i−2 (x) F · F (x) · · · F

2056


n−1 n−1 (F n−i (y)) · F (F n−i−1 (x)) · · · F (x) F (F (y)) · · · F

(F n−i−1 (x)) F n−i−1

n−i−1 · F F (x) − F F (y)

i=0

n−1

i=0

M1 |λ2 |n · LM1α˜ |λ2 |(n−i−1)α˜ x − y α˜ (F n−i−1 (x)) F

2 |n x − y α˜ , L|λ

(3.36)

where ∞ −1 (x) LM 1+α˜ := sup F |λ2 |i α˜ < ∞ L 1 x∈U

i=0

is guaranteed by the is a positive constant independent of x, y and n and the boundedness of L

fact that F (O) = |λ2 | = 0. Thus, by (2.12) in Lemma 3, (2.15), (3.35) and (3.36),

∞

cn+1 (x) cn (x) cn+1 (y) cn (y) − − − λn2 λn2 λn+1 λn+1 2 2 n=1

∞

n n 1 n (x) − d1 F (y) · cn (x) c1 F (x) · an (x) − an (y) + d1 F |λ2 |n+1 n=1 n n n (x) − c1 F (y) · an (y) + d1 F (y) − λ2 · cn (x) − cn (y) + c1 F

∞

n

n 1 n n (x) − F (y) c1 F (x) + d1 F (y) − λ2 F |λ2 |n+1 n=1 n n F (x) − F (y) F + cn (x) + an (y) F

K4 x − y α˜ ,

(3.37)

n α˜ ∞ μn }|λ |−1 > 0 is a constant indepen where K4 := {LMM1α˜ ∞ 2 n=1 μ3 + (2LM + LMM1 ) n=1 4 dent of x, y and n. Here μ3 , μ4 ∈ (0, 1), as indicated in (2.17), which guarantees the convergence of the two series in the definition of K4 . The estimate for the fourth entry in (3.32) is almost the same as the third one and therefore we obtain

∞

dn+1 (x) dn (x) dn+1 (y) dn (y) − − n − n K4 x − y α˜ . n+1 n+1 λ λ2 λ λ 2 2 2 n=1

(3.38)

Having estimates (3.33), (3.34), (3.37) and (3.38) for entries in (3.32), we get (3.30), where K = max{K1 , K2 , K3 , K4 } and

ω = min α1−1 α˜ 2 , α1−1 + 1 α˜ − 1, α˜ = α1−1 + 1 α˜ − 1


2057

since α˜ < α1 . This proves (iii) and (iv) by replacing α˜ with α. For (ii), i.e., α = α1 1, the C 1,α implies that F is C 1,α˜ for any α˜ ∈ (α0 , α1 ). Thus we can apply (iv) to prove (ii) smoothness of F because lim α1−1 + 1 α˜ − 1 = α −1 + 1 α − 1 = α.

α→α ˜

Therefore, the proof is completed.

2

Proof of Lemma 4. Before estimating Ξ1 , we note that either an (x) − an (y) an (x) + an (y) 2M|λ1 |n

(3.39)

by the first inequality given in (2.13) or n

n an (x) − an (y) F (x) − F 2 |n x − y α˜ , (y) L|λ

(3.40)

by (3.36). For each fixed x, y in a sufficiently small U we choose n1 (x, y) :=

− y α˜ )} log{2M/(L x > 1, log(|λ2 |/|λ1 |)

where |λ2 |/|λ1 | > 1 by (1.5). Clearly, n1 is a real number depending on x and y such that the right-hand sides of (3.39) and (3.40) are equal, i.e., 2 |n1 x − y α˜ . 2M|λ1 |n1 = L|λ

(3.41)

It implies that

2 |n x − y α˜ 2M|λ1 |n L|λ 2 |n x − y α˜ 2M|λ1 |n L|λ

if 1 n [n1 ], if n [n1 ] + 1,

(3.42)

where [n1 ] denotes the largest integer not exceeding n1 . On the other hand, by the choice (1.3) −1 of α1 we have |λ2 | = |λ1 |(α1 +1) . It follows from (3.41) that −1

|λ1 |n1 = C1 x − y (1+α1

)α˜

−1

and |λ2 |n1 = C2 x − y α1

α˜

,

(3.43)

where C1 and C2 are both positive constants independent of x and y. Having those preparations, we can estimate Ξ1 . By the definition, Ξ1 (x, y) = Ξ11 (x, y) + Ξ12 (x, y), 1] where Ξ11 (x, y) denotes the sum [n n=1 of the first [n1 ] terms in the sum of Ξ1 (x, y) and Ξ12 (x, y) denotes the remaining sum ∞ n=[n1 ]+1 . Noting (3.42) and applying inequalities (3.39) and (3.40), we obtain

2058


Ξ11 (x, y)

[n1 ]

|λ2 |1+α˜ n ML x − y α˜ |λ1 | |λ1 | n=1

=

2 |1+α˜ M L|λ |λ1 |(|λ2 |1+α˜ − |λ1 |)

|λ2 |1+α˜ |λ1 |

[n1 ]

− 1 x − y α˜

and Ξ12 (x, y)

2M 2 |λ1 |

∞

|λ2 |nα˜ =

n=[n1 ]+1

2M 2 |λ2 |([n1 ]+1)α˜ |λ1 |(1 − |λ2 |α˜ )

respectively by the first inequality given in (2.11), where |λ2 |1+α˜ /|λ1 | > 1 since α˜ < α1 . Furthermore, by (3.43), we get

2 |1+α˜ −1 2 M L|λ |λ2 |1+α˜ n1 Ξ11 (x, y) x − y α˜ = K11 x − y α1 α˜ , 1+ α ˜ |λ1 | |λ1 |(|λ2 | − |λ1 |) Ξ12 (x, y)

−1 2 2M 2 |λ2 |n1 α˜ = K12 x − y α1 α˜ , α ˜ |λ1 |(1 − |λ2 | )

where K11 :=

2 |1+α˜ C21+α˜ M L|λ > 0, C1 |λ1 |(|λ2 |1+α˜ − |λ1 |)

K12 :=

2C2 M 2 > 0. |λ1 |(1 − |λ2 |α˜ )

Thus, putting K1 := K11 + K12 we can prove the first inequality in Lemma 4. In order to estimate Ξ2 , we note that either n n n n b 1 F (x) − b1 F (y) b1 F (x) + b1 F (y) 2M|λ1 |nα˜

(3.44)

by the second inequality given in (2.11) or n n n n b 1 F F (x) − b1 F (y) F (x) − F (y) LM α˜ |λ2 |nα˜ x − y α˜ (3.45) F 1 by (3.35). The following procedure is totally similar to the above for Ξ1 . For each fixed x, y in a sufficiently small U we choose n2 (x, y) :=

log{(2M)1/α˜ /(L1/α˜ M1 x − y )} > 1, log(|λ2 |/|λ1 |)

where we note that |λ2 |/|λ1 | > 1 by (1.5). Clearly, n2 is a real number depending on x and y such that the right-hand sides of (3.44) and (3.45) are equal, i.e., 2M|λ1 |n2 α˜ = LM1α˜ |λ2 |n2 α˜ x − y α˜ . Then,

W.M. Zhang, W.N. Zhang / Journal of Functional Analysis 260 (2011) 2043–2063 −1

|λ1 |n2 = C3 x − y α1

(α1 +1)

−1

|λ2 |n2 = C4 x − y α1 ,

,

2059

(3.46)

where C3 and C4 are two positive constants independent of x and y. It follows from the first formula in (2.14), (3.44), (3.45) and (3.46) that [n2 ]

MLM1α˜ |λ2 |1+α˜ n 2M 2 Ξ2 (x, y) x − y α˜ + |λ1 | |λ1 | |λ1 | n=1

∞ n=[n2 ]+1

|λ1 |α˜ |λ2 | |λ1 |

n

˜ n2 MLM1α˜ |λ2 |1+α˜ 2M 2 |λ2 |(1+α) |λ1 |α˜ |λ2 | n2 α˜ x − y + |λ1 | |λ1 | |λ1 |(|λ2 |1+α˜ − |λ1 |) |λ1 | − |λ1 |α˜ |λ2 |

−1

K2 x − y (α1

+1)α−1 ˜

,

where K2 :=

C41+α˜ MLM1α˜ |λ2 |1+α˜ 2C4 M 2 + >0 C3 |λ1 |(|λ2 |1+α˜ − |λ1 |) C31−α˜ (|λ1 | − |λ1 |α˜ |λ2 |)

because |λ2 |1+α˜ /|λ1 | > 1 as mentioned before and |λ1 |α˜ |λ2 |/|λ1 | = μ2 < 1 by (2.17). This proves the second inequality in Lemma 4. The estimate for Ξ3 can be given directly from the first inequality in (2.13), the second one in (2.11), (3.35) and (3.36). One can obtain that Ξ3 (x, y) K3 x − y α˜ , where K3 :=

MLM1α˜

∞

μn1

+ ML

n=1

∞

μn2

|λ1 |−1 ∈ (0, +∞),

n=1

since μ1 , μ2 ∈ (0, 1) as shown in (2.17). The proof is completed.

2

4. Sharpness of estimates Our Theorem 2 gives estimates for the regularity of linearization in various cases when the considered contraction has two different eigenvalues in absolute value. If the considered contraction has two eigenvalues with the same absolute value, Chaperon [5] proves that the linearization is of the same regularity as the contraction. In this section, we give a counter example to show that in the case of two different eigenvalues in absolute value our estimates for the regularity of linearization are the best. Suppose that λ1 , λ2 are real numbers satisfying (1.5), i.e., 0 < |λ1 | < |λ2 | < 1, and that x := (x1 , x2 ) ∈ R2 . Let Ω := {x ∈ R2 : −1 < x1 < 1, −1 < x2 < 1} be an open neighborhood of O. Let the function p : R → R be defined by p(s) :=

s 1+α , 0,

s 0, s 0, x2 = 0, where q(x) :=

1

e t (t−1) , 0,

0 < t < 1, other cases.

One can check that p is C 1,α on R and that u is C ∞ on R2 \{O} such that (U1) u(x1 , x2 ) = 1 if x1 |x2 |, (U2) u(x1 , x2 ) = 0 if x1 0, and (U3) ∂u(x)/∂x1 0 and u(r) (x) A x −r for r = 1, 2 and for all x ∈ R2 \{O}, where A is a positive constant. Define a planar mapping F : Ω → R2 by F (x) :=

(λ1 x1 + u(λ1 x1 , λ2 x2 )p(λ2 x2 ), λ2 x2 ),

x ∈ Ω\{O},

O,

x = O.

(4.48)

According to (U3), one can verify that F (x) − diag(λ1 , λ2 )x = o x ,

lim F (x) = diag(λ1 , λ2 )

x→O

and

F (x) − F (y) L x − y α , for a constant L > 0 in a small neighborhood U ⊂ Ω of O, i.e., F is a C 1,α diffeomorphism in U . We claim the following. Fact. For α ∈ (α0 , 1], the mapping F given in (4.48) cannot be linearized near O by a transformation smoother than as provided in cases (i)–(iv) in Theorem 2. For α ∈ (0, α0 ], the mapping F cannot be linearized near O by C 1,β transformations for any β ∈ (0, 1]. Proof. For α ∈ (α0 , 1], the fact is obvious in case (i). In order to prove the fact in cases (ii)–(iv), we suppose that 0 < λ1 < λ2 < 1. When at least one of eigenvalues of F (O) is negative, we can consider the quadratic iterate F 2 instead of F to obtain the same conclusion. Since (4.47) and (U2) imply that the mapping F defined in (4.48) is linear in the second, third and forth quadrants, our discussion will be proceeded in the first quadrant. Fix a real constant ξ ∈ U ∩ (0, +∞) and choose n0 (x2 ) := α1−1 logλ2 (x2 /ξ ) > 1

(4.49)


2061

for each sufficiently small x2 ∈ U ∩ (0, +∞). We simply let n0 denote n0 (x2 ) when there is no confusion. Clearly, the integer [n0 ], the largest integer not exceeding n0 , is the largest integer in the set {n ∈ N: λn1 ξ λn2 x2 }. Observing (4.48), we get π1 F (x) λ1 x1 ,

∀x ∈ U \{O},

(4.50)

because p(s) 0 for all s ∈ R and u(x) 0 for all x ∈ Ω\{O} by (U2) and the first inequality in (U3). A straightforward computation shows that π2 F n (ξ, x2 ) = λn2 x2 for all n ∈ N and that n−[n0 ]

π1 F [n0 ] (ξ, x2 ) [n 0 ]−1 1+α n−[n0 ] [n ] [n ]−i λ1 0 u λi1 ξ + Ri (x), λi2 x2 λi2 x2 = λ1 λ1 0 ξ +

π1 F n (ξ, x2 ) λ1

i=1

[n ] [n ] [n ] 1+α + u λ1 0 ξ + R[n0 ] (x), λ2 0 x2 λ2 0 x2

(4.51)

for all n [n0 ] by (4.50), where Ri (x) 0 for all x ∈ U \{O} and for all i = 1, . . . , [n0 ] since p(s) 0 and u(x) 0 as mentioned before. Furthermore, by (U1) and the choice of the number n0 given in (4.49), we obtain u λi1 ξ + Ri (x), λi2 x2 = 1, ∀x ∈ U, ∀i = 1, . . . , [n0 ], since λi1 ξ + Ri (x) λi2 x2 . Thus, by (4.51), π1 F

n

n−[n ] (ξ, x2 ) λ1 0

+

[n 0 ]−1

[n ]−i i 1+α λ2 x2 λ1 0

+

[n ] 1+α λ2 0 x2

i=1

= λn1

[n ] λ1 0 ξ

ξ+

[n 0 ]−1

−1 1+α i 1+α −1 1+α [n0 ] 1+α λ1 λ2 x2 + λ 1 λ2 x2

(4.52)

i=1 1+α = 1 by (1.3). Then, by (4.49) for all n [n0 ]. In the case that α = α1 1, we have λ−1 1 λ2 and (4.52),

(4.53) π1 F n (ξ, x2 ) λn1 ξ + [n0 ]x21+α λn1 ξ + α1−1 logλ2 (x2 /ξ ) − 1 x21+α . 1+α In the case of either α = 1 but α1 > 1 or 0 < α < min{1, α1 }, we have λ−1 > 1 by (1.3) 1 λ2 [n0 ]−1 −1 1+α i 1+α and the sum i=1 (λ1 λ2 ) x2 in the last row in (4.52) is positive. Then, using (4.49) and (4.52) again, we get 1+α [n0 ] 1+α π1 F n (ξ, x2 ) λn1 ξ + λ−1 x2 1 λ2 −1 1+α α1 logλ2 (x2 /ξ )−1 1+α x2 λn1 ξ + λ−1 1 λ2

(α −1 +1)α = λn1 ξ + Cx2 1 −(1+α) 1−α −1 α 1

1 because λ1 = λ1+α by (1.3), where C := λ1 λ2 2

ξ

(4.54) > 0.

2062


Having those preparations, we can estimate β. For a fixed number β ∈ (0, 1], suppose that Φ : U → R2 is a C 1,β diffeomorphism satisfying Eq. (1.1) in U . Without loss of generality, we assume that Φ(0, x2 ) = (0, x2 ),

Φ(x1 , 0) = (x1 , 0),

Φ (x1 , 0) = Φ (0, x2 ) = id.

(4.55)

: U → R2 defined by Otherwise, we consider another C 1,β diffeomorphism Φ

:= Θ ◦ Φ (O) −1 Φ, Φ ˆ 2 ), x2 ) for all x ∈ U and gˆ is a C 1,β function on U ∩ R whose graph where Θ(x) := (x1 − g(x Γ := {(x1 , x2 ) ∈ U : x1 = g(x ˆ 2 )} is just the image of the x2 -axis under {Φ (O)}−1 Φ. One can commutes with F (O) and check that Φ is a solution of Eq. (1.1) because the transformation Θ that Φ satisfies (4.55). By (4.55), the Taylor expansion of π1 Φ at (ξ, 0) gives π1 Φ(ξ, x2 ) = 1+β ξ + O(x2 ). Substituting x2 with λk2 for all sufficiently large k ∈ N and taking k as variable, we get k(1+β) . (4.56) π1 Φ ξ, λk2 = ξ + O λ2 Let N ∈ N such that N > max{α1−1 , (1 + β)(βα1 + β)−1 }. By (1.1) and (4.56), k(1+β) k k Nk ξ + O λ2 . π1 Φ F N k ξ, λk2 = λN 1 π1 Φ ξ, λ2 = λ1

(4.57)

On the other hand, we can see that F given in (4.48) satisfies the equality (2.9), i.e., π1 F (0, x2 ) = 0 for all x2 ∈ U ∩ (0, +∞). It follows that the second inequality in (2.25) holds for F . Thus, by (4.50), k Nk k λN ξ, λk2 M2 λN (4.58) 1 ξ π1 F 1 ξ, ∀k ∈ N. (N +1)k

The Taylor expansion of π1 Φ at (0, λ2

) gives

+1)k π1 Φ F N k ξ, λk2 = π1 Φ π1 F N k ξ, λk2 , λ(N 2 1+β = π1 F N k ξ, λk2 + O π1 F N k ξ, λk2 N k(1+β) = π1 F N k ξ, λk2 + O λ1 by (4.55) and (4.58). Since N k > n0 (λk2 ) by the choice of N , it follows from (4.53) and (4.54) that k λk(1+α) + O λN kβ ξ + α1−1 k − C π1 Φ F N k ξ, λk2 λN 1 1 2

(4.59)

:= α −1 logλ ξ + 1, and that when α = α1 1, where C 1 2 N kβ k(α −1 +1)α k π1 Φ F N k ξ, λk2 λN ξ + Cλ2 1 + O λ1 1 when either α = 1 but α1 > 1 or 0 < α < min{1, α1 }.

(4.60)


2063

For the cases (ii)–(iv) in Theorem 2, comparing (4.57) with (4.59) and (4.60) respectively and Nβ 1+β noting that λ1 < λ2 by the choice of N , we get

if α = α1 < 1,

β < α,

β (α1−1 + 1)α − 1, if either α = 1 but α1 > 1 or α0 < α < min{1, α1 }. For α ∈ (0, α0 ], we assume that F can be C 1,β linearized near O for a number β ∈ (0, 1]. Then (4.57) contradicts to (4.60) because 1 + β > 1 α0−1 α = (α1−1 + 1)α, which implies that 1+β

λ2

(α −1 +1)α

< λ2 1 . This completes the proof.

2

Remark. For α ∈ (0, α0 ], the fact only indicates that C 1 linearization will be the best for C 1,α (α ∈ (0, α0 ]) contractions F . It remains interesting to know whether C 1 linearization can be realized for such contractions. Acknowledgment The authors are grateful to the referee for his/her helpful comments and suggestions. References [1] V.I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations, Springer, New York, 1983. [2] G.R. Belitskii, Functional equations and the conjugacy of diffeomorphism of finite smoothness class, Funct. Anal. Appl. 7 (1973) 268–277. [3] G.R. Belitskii, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys 33 (1978) 107–177. [4] A.D. Brjuno, Analytical form of differential equations, Trans. Moscow Math. Soc. 25 (1971) 131–288. [5] M. Chaperon, Invariant manifolds revisited, Tr. Mat. Inst. Steklova 236 (2002) 428–446, dedicated to the 80th anniversary of Academician Evgenii Frolovich Mishchenko, Suzdal, 2000 (in Russian). [6] M.S. ElBialy, Local contractions of Banach spaces and spectral gap conditions, J. Funct. Anal. 182 (2001) 108–150. [7] P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana 5 (1960) 220–241. [8] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964. [9] R. de la Llave, C.E. Wayne, On Irwin’s proof of the pseudo-stable manifold theorem, Math. Z. 219 (1995) 301–321. [10] H. Poincaré, Sur le problème des trois corps et les équations de la dyanamique, Acta Math. 13 (1890) 1–270. [11] C. Pugh, On a theorem of P. Hartman, Amer. J. Math. 91 (1969) 363–367. [12] H.M. Rodrigues, J. Solà-Morales, Linearization of class C 1 for contractions on Banach spaces, J. Differential Equations 201 (2004) 351–382. [13] G.R. Sell, Smooth linearization near a fixed point, Amer. J. Math. 107 (1985) 1035–1091. [14] C.L. Siegel, Iteration of analytic functions, Ann. Math. 43 (1942) 607–612. [15] S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957) 809–824. [16] S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space, Amer. J. Math. 80 (1958) 623–631. [17] D. Stowe, Linearization in two dimensions, J. Differential Equations 63 (1986) 183–226. [18] J.-C. Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de (C, 0), C. R. Acad. Sci. Paris 36 (1988) 55–58.


Relative index pairing and odd index theorem for even dimensional manifolds Zhizhang Xie 1 Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA Received 27 July 2010; accepted 6 October 2010 Available online 20 October 2010 Communicated by Alain Connes

Abstract We prove an analogue for even dimensional manifolds of the Atiyah–Patodi–Singer twisted index theorem for trivialized flat bundles. We show that the eta invariant appearing in this result coincides with the eta invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional counterpart for manifolds with boundary of the relative index pairing by Lesch, Moscovici and Pflaum. © 2010 Elsevier Inc. All rights reserved. Keywords: APS twisted index theorem; Manifolds with boundary; Relative index pairing

0. Introduction In this article, we will prove an analogue for even dimensional manifolds of the Atiyah– Patodi–Singer twisted index theorem for trivialized flat bundles over odd dimensional closed manifolds [3, Proposition 6.2], and some related results. For notational simplicity, we will restrict the discussion mainly to spin manifolds. However all results can be straightforwardly extended to general manifolds. Unless we specify otherwise, we always fix the Riemannian metric for each manifold in this article and use the associated Levi-Civita connection to define its characteristic classes. E-mail address: [email protected]. 1 The author was partially supported by the US National Science Foundation awards No. DMS-0652167.


Z. Xie / Journal of Functional Analysis 260 (2011) 2064–2085

2065

To motivate the subject matter of this paper, we begin by recalling the APS twisted index theorem for odd dimensional closed manifolds in the following form, cf. [12, Corollary 7.9]. For (ps )0s1 ∈ Mk (C ∞ (N )), s ∈ [0, 1], a smooth path of projections over N , one has 1

1 d η(ps Dps ) ds = 2 ds

) ∧ Tch• (ps ). A(N

N

0

) the A-genus Here ps Dps is the Dirac operator twisted by ps , η(ps Dps ) its η-invariant, A(N form of N and Tch• (ps ) is the Chern–Simons transgression form of (ps )0s1 , cf. Section 3. To prove our analogue for even dimensional closed manifolds, we shall replace a path of projections by a path of unitaries. The more interesting issue is what should replace the η-invariant appearing on the left hand side of the above formula. To answer this, let us first consider the case where the manifold in question bounds, that is, it is the boundary of some spin manifold. In this case, the η-invariant by Dai and Zhang [9, Definition 2.2] is the right candidate, cf. Section 6 below. Indeed, suppose the even dimensional manifold Y is the boundary of a spin manifold X and (Us )0s1 is the restriction to Y of a smooth path of unitaries over X. Denote the η-invariant of Dai and Zhang by η(Y, Us ) for each s ∈ [0, 1], then 1 0

1 d η(Y, Us ) ds = 2 ds

) ∧ Tch• (Us ). A(Y

(0.1)

Y

When Y bounds, it follows from the cobordism invariance of the index of Dirac operators that Ind(D + ) = 0, where D + is the restriction of the Dirac operator over Y to the even half of the spinor bundle according to its natural Z2 -grading. The condition Ind(D + ) = 0 is crucial for the definition of the η-invariant by Dai and Zhang, however is often not satisfied by even dimensional closed spin manifolds in general. To cover the general case, we shall use another approach where we lift the data to S1 × Y . The main ingredient of the method of proof is using an explicit formula of the cup product K 1 (S1 ) ⊗ K 1 (Y ) → K 0 (S1 × Y ), inspired by the Powers–Rieffel idempotent construction, cf. [15]. In fact, the formula given for the case when Y = S1 by Loring in [14] also works for all manifolds in general, cf. Section 2 below. Our analogue for even dimensional closed spin manifolds of the APS twisted index theorem (Theorem 4.1 below) is as follows. Theorem (I). Let Y be an even dimensional closed spin manifold and (Us )0s1 ∈ Uk (C ∞ (Y )) a smooth path of unitaries over Y . For s ∈ [0, 1], es ∈ M2k (C ∞ (S1 × Y )) is the projection defined as the cup product of Us with the generator e2πiθ of K 1 (S1 ). Let DS1 ×Y be the Dirac operator over S1 × Y . Then 1 0

1 d η(es DS1 ×Y es ) ds = 2 ds

) ∧ Tch• (Us ). A(Y

(0.2)

Y

The formula of es is given in Section 2. A priori, the η-invariants in the formulas (0.1) and (0.2) appear to be different, we however will show that they are equal to each other modulo Z (Theorem 5.7 below) in the case where Y bounds.

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Theorem (II). Suppose Y is the boundary of an odd dimensional spin manifold. For U ∈ Uk (C ∞ (Y )) and eU the cup product of U with e2πiθ ∈ K 1 (S1 ), one has η(Y, U ) = η(eU DS1 ×Y eU )

mod Z.

The method of proof is based on a slight generalization of a theorem by Brüning and Lesch [6, Theorem 3.9], see Proposition 5.6 below. In this sense, η(eU DS1 ×Y eU ) can be thought of as the extension to general even dimensional manifolds of the definition of the η-invariant by Dai and Zhang. The same technique used above also allows us to prove the following analogue (Theorem 6.3 below) for odd dimensional manifolds with boundary of the relative index pairing formula by Lesch, Moscovici and Pflaum [12, Theorem 7.6]. Suppose M is an odd dimensional spin manifold with boundary ∂M. By a relative K-cycle [U, V , us ] ∈ K 1 (M, ∂M), we mean U, V ∈ Un (C ∞ (M)) are two unitaries over M with us ∈ Un (C ∞ (∂M)), s ∈ [0, 1], a smooth path of unitaries over ∂M such that u0 = U |∂M and u1 = V |∂M . We denote by TU , resp. TV , the Toeplitz operator on M with respect to U , resp. V (see Section 5 for details). Theorem (III). Let [U, V , us ] be a relative K-cycle in K 1 (M, ∂M). If U and V are constant along the normal direction near the boundary, then us Ind[D] [U, V , us ] = Ind(TV ) − Ind(TU ) + SF u−1 s D[0,1] us ; P0 us −1 where SF(u−1 s D[0,1] us ; P0 ) is the spectral flow of the path of elliptic operators (us D[0,1] us ; us P0 ), s ∈ [0, 1], with Atiyah–Patodi–Singer type boundary conditions determined by P0us as in (5.4).

This uses Dai and Zhang’s Toeplitz index theorem for odd dimensional manifolds with boundary [9]. We shall give the details in Section 6. It should be mentioned that, although the objects we work with are from classical geometry, the spirit of the proofs is very much inspired by methods from noncommutative geometry, cf. [8]. A brief outline of the article is as follows. In Section 1, we recall some results about index pairings for manifolds with boundary. Section 2 is devoted to the explicit formula of the cup product in K-theory mentioned earlier. This allows us to carry out explicit calculations for Chern characters in Section 3. With these preparations, we prove an analogue for even dimensional manifolds of the APS twisted index theorem in Section 4. In Section 5, we show the equality of the two a priori different eta invariants. In the last section, we prove the odd dimensional counterpart of the relative index pairing formula by Lesch, Moscovici and Pflaum [12, Theorem 7.6]. 1. Relative index pairing Let M be a compact smooth manifold with boundary ∂M = ∅. Following [4, Section 2], consider an elliptic first order differential operator D : Cc∞ (M \ ∂M, E) → Cc∞ (M \ ∂M, E) where Cc∞ (M \ ∂M, E) is the space of compactly supported smooth sections of the Hermitian vector bundle E. Such an operator has a number of extensions to become a closed unbounded


2067

operator on H = L2 (M \ ∂M, E), e.g. Dmin and Dmax the minimum extension and the maximum extension respectively. Consider De a closed extension of D such that Dmin ⊂ De ⊂ Dmax ,

(1.1)

that is, D(Dmin ) ⊂ D(De ) ⊂ D(Dmax ). Let B=

0 De

De∗ 0

and −1/2 Fe = B B 2 + 1 =

0 T

T∗ 0

with T = De (De∗ De + 1)−1/2 and T ∗ = De∗ (De De∗ + 1)−1/2 . Denote by C0 (M \ ∂M) the space of continuous functions vanishing at infinity. Then the ∗-representation of C0 (M \ ∂M) on H ⊕ H given by scalar multiplication, together with Fe , defines an element in KK(C0 (M \ ∂M), C), see [4] for the precise construction. Such a K-homology class turns out to be independent of the choice of a closed extension of D [4, Proposition 2.1], and will be denoted [D]. Similarly for each formally symmetric elliptic operator, one constructs a cycle in KK(C0 (M \ ∂M), Cl1 ) [4, Section 2], where Cl1 is the Clifford algebra with one generator. For each [D] ∈ KK(C0 (M \ ∂M), Cl• ), one has the index pairing map Ind[D] : K • (M \ ∂M) → Z. An element in K 0 (M \ ∂M) is represented by a triple (E, F, α) with E, F vector bundles over M \ ∂M and α : E → F a bundle homomorphism whose restriction near infinity is an isomorphism, cf. [1]. Moreover, we can choose connections over the bundles E and F such that the forms Ch• (E) and Ch• (F ) coincide near infinity. Under this assumption, one can write down an explicit formula for the index pairing map: Ind[D] [E, F, α] =

ωD ∧ Ch• (E) − Ch• (F ) .

M even (M \ ∂M) is the dual of the Chern character of the K-homology class [D], as Here ωD ∈ HdR explained in the introduction of Chap. I in [7]. In the case where M is a spin manifold and D the Dirac operator over M, one has ωD = A(M). Similarly, in the odd case, an element in K 1 (M \ ∂M) consists of two unitaries U and V over M \ ∂M and a homotopy h between U and V near infinity. Moreover, we can assume that U and V are identical near infinity and the homotopy h becomes the identity map near infinity, cf. e.g.

2068


[10, Proposition 4.3.14], in which case the index pairing map has the following cohomological expression2 : Ind[D] [V , U, h] = − ωD ∧ Ch• (V ) − Ch• (U ) . M

Note that the boundary data are conspicuously absent in the above formulas. Indeed, by definition, K • (M \ ∂M) is essentially the (reduced) K-group of the one point compactification of M \ ∂M. The information from the boundary is therefore completely eliminated from the picture. In order to recover that, we shall turn to the relative K theory of the pair (M, ∂M), denoted K • (M, ∂M), cf. [12]. A relative K-cycle [p, q, hs ] ∈ K 0 (M, ∂M) is a triple where p, q ∈ Mn (C ∞ (M)) are two projections over M and hs ∈ Mn (C ∞ (∂M)), s ∈ [0, 1], is a path of projections over ∂M such that h0 = p|∂M and h1 = q|∂M . Similarly, a relative K-cycle [U, V , us ] ∈ K 1 (M, ∂M) is a triple where U, V ∈ Un (C ∞ (M)) are two unitaries over M with us ∈ Un (C ∞ (∂M)), s ∈ [0, 1], a smooth path of unitaries over ∂M such that u0 = U |∂M and u1 = V |∂M . First notice that K • (M, ∂M) ∼ = K • (M \∂M). Hence the above index pairing induces • a map Ind[D] : K (M, ∂M) → Z. The issue now is to find an explicit formula which incorporates geometric information of the boundary. For even dimensional manifolds with boundary, this is done by Lesch, Moscovici and Pflaum [12, Theorem 7.6]. We shall give an analogous formula for odd dimensional manifolds with boundary in Section 6. 2. Cup product in K-theory Let A and B be local Fréchet algebras. The cup product between K1 (A) and K1 (B) is defined by × : K1 (B) ⊗ K1 (A) = K0 (SB) ⊗ K0 (SA) → K0 (SB ⊗ SA) ∼ = K0 (B ⊗ A) where SA (resp. SB) is the suspension of A (resp. SB), the isomorphism is the Bott isomorphism and K0 (SB) ⊗ K0 (SA) → K0 (SB ⊗ SA) is given by [p] × [q] = [p ⊗ q].

(2.1)

In the case where B = C ∞ (S1 ), we shall give an explicit formula of this cup product. Since is a generator of K1 (C ∞ (S1 )) ∼ = Z, it suffices to give this formula for [e2πiθ ] × [U ] with U ∈ Uk (A).

e2πiθ

Lemma 2.1. (See also [14].) With the above notation, 2πiθ × [U ] = [eU ] e 2 We adopt the negative sign here in order to be consistent with our sign convention throughout the article.


where eU = functions on

f g+hU ∈ M2k (C ∞ (S1 ) ⊗ A) hU ∗ +g 1−f 1 S satisfying the following conditions

2069

is a projection with f, g and h nonnegative

(1) 0 f 1, (2) f (0) = f (1) = 1 and f (1/2) = 0, (3) g = χ[0,1/2] (f − f 2 )1/2 and h = χ[1/2,1] (f − f 2 )1/2 . Proof. It is not difficult to see that × : K1 C ∞ S1 ⊗ K1 (A) → K0 C ∞ S1 ⊗ A is the same as the standard isomorphism [16, Section 7.2] ΘA : K1 (A) → K0 (SA) ⊂ K0 C ∞ S1 ⊗ A after identifying K1 (C ∞ (S1 )) with Z. The inverse of this map is constructed as follows, cf. [10, Proposition 4.8.2], [16, Section 7.2]. The group K0 (SA) is generated by formal differences of normalized loops of projections over A. Such a loop is a projection-valued maps p : [0, 1] → Mn (A) with p(0) = p(1) ∈ Mn (C). For each loop, there is a path of unitaries u : [0, 1] → Un (A) ∗ such that p(t) = u(t)p(1)u(t) and u(0) = 1n . Without loss of generality, we can assume p(0) = 10 p(1) = 0 0 . This implies that u(1) is of the form v0 w0 . Then one checks that [p] → [v] is a well-defined inverse to ΘA . −1 (eU ) = U . To see that our formula agrees with the usual definition, it suffices to show that ΘA 1 0 First notice that eU (0) = eU (1) = 0 0 and eU (θ ) is a projection over A for each θ ∈ S1 = R/Z, hence eU is a normalized loop of projections. Now consider the following path of unitaries over A, U(θ ) =

f1 (θ ) + f2 (θ )U (1 − f )1/2 (θ )

(1 − f )1/2 (θ ) −f1 (θ ) − f2 (θ )U ∗

where f1 = χ[0,1/2] f 1/2 and f2 = χ[1/2,1] f 1/2 . In particular, U(0) = By a direct calculation, one verifies

1 0 01

and U(1) =

U

0 . 0 −U ∗

1 0 eU (θ ) = U(θ ) U(θ )∗ 0 0 from which the lemma follows.

2

We will also make use of the following lemma in Section 3, cf. [14, Lemma 2.2]. Lemma 2.2. For f, g and h nonnegative functions on S1 = R/Z satisfying the following conditions (1) 0 f 1, (2) f (0) = f (1) = 1 and f (1/2) = 0, (3) g = χ[0,1/2] (f − f 2 )1/2 and h = χ[1/2,1] (f − f 2 )1/2 ,

2070


we have 1 0

(k − 1)!(k − 1)! . (2 − 4f )h h2k−1 + 4f h2k dθ = (2k − 1)!

Proof. Notice that 1

1

2k

f h dθ = 0

k f (θ ) − f 2 (θ ) df (θ ) =

1/2

1

k x − x 2 dx =

0

k!k! , (2k + 1)!

and integration by parts gives 1

2k−1

(2 − 4f )h h

2 dθ = k

0

1

f h2k dθ.

2

0

3. Chern characters and transgression formulas Throughout this section, although we deal with commutative algebras, we shall use the similar formalism for the Chern character in K-theory as in cyclic homology [7, Chap. II], [13, Chap. VIII]. Let M be a compact smooth manifold with or without boundary. The even (resp. odd) Chern characters of projections (resp. unitaries) in Mn (C ∞ (M)) can be expressed as follows. For p ∈ Mn (C ∞ (M)) such that p 2 = p and p ∗ = p, Ch• (p) := tr(p) +

∞

(−1)k k=1

1 1 even tr p(dp)2k ∈ HdR (M). k (2πi) k!

(3.1)

For U ∈ Un (C ∞ (M)), Ch• (U ) :=

∞

k=0

2k+1 1 k! odd tr U −1 dU ∈ HdR (M). k+1 (2k + 1)! (2πi)

(3.2)

For each U ∈ Un (C ∞ (M)), let eU be the projection as in Lemma 2.1. If no confusion is likely to arise, we also write e instead of eU . Lemma 3.1. Ch• (eU ) = −

∞

k=1

2k−1 1 k 2k 4f h + (2 − 4f )h h2k−1 dθ ∧ tr U −1 · dU . k (2πi) k!

Proof. Notice that de =

f h U ∗ + g

g + h U −f

dθ +

0 h dU ∗

h dU 0

,


2071

which implies tr e(de)2k = tr

g + hU 1−f

hU ∗ + g j

=2k f + tr hU ∗ + g

0 h dU h dU ∗ 0 0 g + hU 1−f h dU ∗

f

j =1

f g + h U ∗

g + h U −f

dθ

0 h dU ∗

2k

h dU 0

h dU 0

j −1 (3.3)

(2k−j ) .

(3.4)

Since most of the matrices appearing in the above summation only have off diagonal entries, a straightforward calculation gives the following equalities 2k (3.3) = h2k tr U −1 · dU , 2k−1 . (3.4) = −(−1)k k (2 − 4f )h h2k−1 + 4f h2k dθ ∧ tr U −1 · dU On the other hand, 2k 2k−1 tr U −1 · dU = − tr U −1 · dU U −1 · dU from which it follows that (3.3) vanishes. This finishes the proof.

2

As a consequence of Lemma 2.2 and Lemma 3.1, one has the following corollary. From now on, integration along the fiber S1 will be denoted by π∗ . Corollary 3.2. π∗ Ch• (eU ) = −Ch• (U ). Consider a smooth path of unitaries Us ∈ Un (C ∞ (M)) with s ∈ [0, 1], or equivalently U ∈ • (Us ) is given by the formula Un (C ∞ ([0, 1] × M)). The secondary Chern character Ch • (Us ) := Ch

∞

(−1)k k=0

2k 1 k! −1 ˙ tr Us Us Us dU −1 . k+1 (2k)! (2πi)

Then Ch• (U) can be decomposed as • (Us ) Ch• (U) = Ch• (Us ) + ds ∧ Ch • (Us ) do not contain ds. Applying de Rham differential where Ch(Us ) (see (3.2) above) and Ch to both sides gives us the following transgression formula ∂ • (Us ). Ch• (Us ) = d Ch ∂s Similarly, if es ∈ Mm (C ∞ (M)) is a smooth path of projections, or equivalently a projection e ∈ Mm (C ∞ ([0, 1] × M)), then

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• (es ) Ch• (e) = Ch• (es ) + ds ∧ Ch with • (es ) := Ch

∞

k=0

(−1)k+1

1 1 tr (2es − 1)e˙s (des )2k+1 . k+1 k! (2πi)

Applying Corollary 3.2 to Ch• (U) and Ch• (eU ), one has π∗ Ch• (eU ) = −Ch• (U), which implies • (es ) = ds ∧ Ch • (Us ). ds ∧ π∗ Ch Denote the Chern–Simons transgression forms by 1 Tch• (es )0s1 :=

• (es ), ds ∧ Ch

0

1 Tch• (Us )0s1 :=

• (Us ). ds ∧ Ch

0

We summarize the results of this section in the following proposition. Proposition 3.3. Consider U ∈ Un (C ∞ (M)) and Us ∈ Un (C ∞ (M)) for s ∈ [0, 1]. Let e, resp. es , be the cup product of U , resp. Us , with e2πiθ a generator of K 1 (S1 ) as in Lemma 2.1. Then π∗ Ch• (e) = −Ch• (U ) and π∗ Tch• (es )0s1 = Tch• (es )0s1 . 4. Odd index theorem on even dimensional manifolds In this section, we shall prove our analogue for even dimensional closed manifolds of the APS twisted index theorem. Let us first recall the APS twisted index theorem and fix some notation. Let N be closed odd dimensional spin manifold and D / its Dirac operator. If p is a projection in Mn (C ∞ (N )), then p induces a Hermitian vector bundle, denoted Ep , over N . With the Grassmannian connection on Ep , let p(D ⊗ In )p be the twisted Dirac operator with coefficients in Ep . For notational


2073

simplicity, we also write pDp instead of p(D ⊗ In )p. Then by [12, Corollary 7.9], for ps ∈ Mk (C ∞ (N )) a smooth path of projections over N , one has ξ(p1Dp / 1 ) − ξ(p0Dp / 0) =

/ s )0s1 A(N) ∧ Tch• (ps ) + SF(psDp

(4.1)

N

where ξ(piDp / i) =

/ i ) + dim ker(piDp / i) η(piDp 2

/ i . Here SF(psDp / s )0s1 is the spectral flow of (psDp / s )0s1 . the reduced eta invariant of piDp Notice that the vector bundle on which psDp / s acts may vary as s moves along [0, 1]. To make sense of the definition of such a spectral flow, we introduce a path of unitaries us ∈ Un (C ∞ (N)) over N with us p0 u∗s = ps so that p0 u∗sDu / s p0 acts on the same vector bundle Ep0 . SF(psDp / s )0s1 is then defined to be SF(p0 u∗sDu / s p0 )0s1 the spectral flow of the family (p0 u∗sDu / s p0 )0s1 . Now by [11, Lemma 3.4], formula (4.1) is equivalent to 1

1 d η(psDp / s ) ds = 2 ds

) ∧ Tch• (ps ). A(N

(4.2)

N

0

Theorem 4.1. Let Y be a closed even dimensional spin manifold and (Us )0s1 ∈ Uk (C ∞ (Y )) a smooth path of unitaries over Y . For s ∈ [0, 1], es ∈ M2k (C ∞ (Y )) the projection defined as the cup product of Us with the generator e2πiθ of K 1 (S1 ). Let DS1 ×Y be the Dirac operator over S1 × Y . Then 1


) ∧ Tch• (Us ). A(Y

(4.3)

Y

0

Proof. Applying formula (4.1) to S1 × Y , one has ξ(e1 DS1 ×Y e1 ) − ξ(e0 DS1 ×Y e0 ) =

S1 × Y ∧ Tch• (es ) + SF(es DS1 ×Y es ). A

S1 ×Y

1 ) ∧ π ∗ A(M) 1 ) = 1, where π1 : S1 × M → S1 , resp. 1 × M) = π ∗ A(S and A(S Notice that A(S 1 2 from S1 × M to S1 , resp. M. By Proposition 3.3, the integral π2 : S1 × M → M, is the projection on the right side is equal to Y A(Y ) ∧ Tch• (Us ). Now the formula 1 0


Y

) ∧ Tch• (Us ) A(Y

2074


follows from the equality [11, Lemma 3.4] 1 ξ(e1 DS1 ×Y e1 ) − ξ(e0 DS1 ×Y e0 ) = SF(es DS1 ×Y es ) +

1 d η(es DS1 ×Y es ) ds. 2 ds

2

0

Remark 4.2. Mod Z, the reduced η-invariant ξ(es DS1 ×Y es ) is equal to the reduced η-invariant ξ(Y, Us ) defined by Dai and Zhang, cf. [9, Definition 2.2], at least when Y bounds. See Theorem 5.7 below. 5. Equivalence of eta invariants Throughout this section, we assume M is an odd dimensional spin manifold with boundary ∂M. Denote by SM the spinor bundle over M. Let D be the Dirac operator over M, then near the boundary d D = c(d/dx) + D∂ dx where D ∂ is the Dirac operator over ∂M and c(d/dx) is the Clifford multiplication by the normal vector d/dx. Then D ⊗ In is the Dirac operator acting on SM ⊗ Cn , when we use the trivial connection on the bundle M × Cn over M. If no confusion is likely to arise, we shall write D instead of D ⊗ In . Now a subspace L of ker D ∂ is Lagrangian if c(d/dx)L = L⊥ ∩ ker D ∂ . In our case, since ∂M bounds M, the existence of such a Lagrangian subspace follows from the cobordism invariance of the index of Dirac operators. Let L2>0 (SM ⊗ Cn |∂M ) be the positive eigenspace of D ∂ , i.e. the L2 -closure of the direct sum of eigenspaces with positive eigenvalues of D ∂ . Then the projection P ∂ := P∂M (L) = P∂M + PL imposes an APS type boundary condition for D, where P∂M , resp. PL , is the orthogonal projection L2 (SM ⊗ Cn |∂M ) → L2>0 (SM ⊗ Cn |∂M ), resp. L2 (SM ⊗ Cn |∂M ) → L. Let us denote the corresponding self-adjoint elliptic operator by DP ∂ . Let L20 (SM ⊗ Cn ; P ∂ ) be the nonnegative eigenspace of DP ∂ and PP ∂ the orthogonal projection PP ∂ : L2 SM ⊗ Cn → L20 SM ⊗ C n ; P ∂ . More generally, for each unitary U ∈ Un (C ∞ (M)) over M, the projection U P ∂ U −1 imposes an APS type boundary condition for D and we shall denote the corresponding elliptic self-adjoint operator by DU P ∂ U −1 . Similarly, let PU P ∂ U −1 be the orthogonal projection PU P ∂ U −1 : L2 SM ⊗ Cn → L20 SM ⊗ Cn ; U P ∂ U −1 where L20 (SM ⊗ Cn ; U P ∂ U −1 ) is the nonnegative eigenspace of DU P ∂ U −1 . With the above notation, we define the Toeplitz operator on M with respect to U as follows, cf. [9, Definition 2.1].


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Definition 5.1. TU := PU P ∂ U −1 ◦ U ◦ PP ∂ . Dai and Zhang’s index theorem for Toeplitz operators on odd dimensional manifolds with boundary [9, Theorem 2.3] states that Ind(TU ) = − A(M) (5.1) ∧ Ch• (U ) − ξ(∂M, U ) + τμ U P ∂ U −1 , P ∂ , PM M

where PM is the Calderón projection associated to the Dirac operator D on M (cf. [5]) and τμ (U P ∂ U −1 , P ∂ , PM ) is the Maslov triple index [11, Definition 6.8]. The reduced η-invariant ξ(∂M, U ) will be defined after the remarks. Remark 5.2. Notice that the integral in (5.1) differs from Dai and Zhang’s by a constant coefficient (2πi)−(dim M+1)/2 . This is due to the fact that our definition of characteristic classes follows 1 k/2 ) are already included. topologists’ convention, i.e. factors such as ( 2πi Remark 5.3. The Maslov triple index τμ (U P ∂ U −1 , P ∂ , PM ) is an integer. For unitaries U, V ∈ Un (C ∞ (M)), if there is a path of unitaries us ∈ Un (C ∞ (∂M)) with s ∈ [0, 1] such that u0 = U |∂M and u1 = V |∂M , one has τμ U P ∂ U −1 , P ∂ , PM = τμ V P ∂ V −1 , P ∂ , PM , cf. [11, Lemma 6.10]. To define ξ(∂M, U ), let us first consider D[0,1] the Dirac operator over [0, 1] × ∂M. If no confusion is likely to arise, we shall write U for both U |∂M and the trivial lift of U |∂M from ∂M to [0, 1] × ∂M. Let D[0,1] := D[0,1] + (1 − ψ)U −1 [D[0,1] , U ] ψ,U

(5.2)

over [0, 1] × ∂M, where ψ is a cut-off function on [0, 1] with ψ ≡ 1 near {0} and ψ ≡ 0 near {1}. With APS type boundary conditions determined by P ∂ on {0} × ∂M and Id −U −1 P ∂ U on {1} × ψ,U ψ,U ∂M, D[0,1] becomes a self-adjoint elliptic operator, denoted (D[0,1] ; P0U ). See Proposition 5.6 for an explanation of the choice of notation. Similarly, D[0,1] (t) := D[0,1] + (1 − tψ)U −1 [D[0,1] , U ]. ψ,U

ψ,U

ψ,U

(5.3)

Denote by (D[0,1] (t); P0U ) the elliptic operator D[0,1] (t) with boundary condition P0U . Note that ψ,U

ψ,U

D[0,1] (1) = D[0,1] . Definition 5.4. (See [9, Definition 2.2].) ψ,U ψ,U η(∂M, U ) := ξ D[0,1] ; P0U − SF D[0,1] (t); P0U 0t1

2076


where ψ,U

ψ,U

ψ,U dim ker(D[0,1] ; P0U ) + η(D[0,1] ; P0U ) . ξ D[0,1] = 2 Remark 5.5. η(∂M, U ) is independent of the cut-off function ψ [9, Proposition 5.1]. In order to show the equality ξ(∂M, U ) = ξ(eU DS1 ×∂M eU ) mod Z, we need to relate the ψ,U d operator eU DS1 ×∂M eU to D[0,1] , where DS1 ×∂M = c(d/dθ)( dθ + D ∂ ) is the Dirac operator over S1 × ∂M and eU is the cup product of U with e2πiθ ∈ K 1 (S1 ). Recall that

g + hU f DS1 ×∂M 0 hU ∗ + g 1 − f hU ∗ + g 0 DS1 ×∂M 1 0 1 0 0 ∗ DS1 ×∂M U U∗ =U U 0 0 0 0 0 DS1 ×∂M

eU DS1 ×∂M eU =

f

g + hU 1−f

where U=

1/2

f1

1/2

+ f2 U

(1 − f )1/2 1/2

(1 − f )1/2

−f1

− f2 U ∗ 1/2

with f1 = χ[0,1/2] f and f2 = χ[1/2,1] f . Then viewed as an operator over [0, 1] × ∂M, U ∗ (eU DS1 ×∂M eU )U = D[0,1] + f2 U −1 [D[0,1] , U ] with the boundary condition β(0, x) = Uβ(1, x),

for ∀x ∈ ∂M

and β ∈ Γ [0, 1] × ∂M; S ⊗ Cn .

Let H ∂ := L2 ({0} × ∂M; S ⊗ Cn ) ⊕ L2 ({1} × ∂M; S ⊗ Cn ), then the above boundary condition can be written as 1 1 −U β = 0, for ∀β ∈ H ∂ . 1 2 −U −1 From now on, let us assume ψ = 1 − f2 . In particular, one has U ∗ (eU DS1 ×∂M eU )U = D[0,1] . ψ,U

Now consider PtU =

cos2 tP ∂ + sin2 t (I − P ∂ ) − cos t sin tU −1

− cos t sin tU cos2 t (Id −U −1 P ∂ U ) + sin2 tU −1 P ∂ U

(5.4)

for 0 t π/4 (cf. [11, Equation 5.13], [6, Section 3]). This is a path of projections in B(H ∂ ) such that


P0U =

P∂ 0

2077

0

Id −U −1 P ∂ U

and U Pπ/4

1 = 2

1 −U −1

−U 1

.

ψ,U

For each t ∈ [0, π/4], the Dirac operator D[0,1] , with the boundary condition PtU , is a self-adjoint ψ,U

elliptic operator, denoted by (D[0,1] ; PtU ). With the above notation, we have the following slight generalization of a theorem by Brüning and Lesch [6, Theorem 3.9]. Proposition 5.6. d ψ,U U η D[0,1] ; Pt = 0. dt Proof. Following [6, Section 3], we define

0 U 0 U = , U −1 0 U∗ 0 c(d/dθ ) 0 γ˜ := , 0 − c(d/dθ ) ∂ 0

:= D A 0 −U −1 D ∂ U

τ :=

is determined by D ψ,U near the boundary, by noticing that where A [0,1] ψ,U D[0,1]

d ∂ +D = c(d/dθ) dθ

near {0} × ∂M and ψ,U D[0,1]

d −1 ∂ +U D U = c(d/dθ) dθ

near {1} × ∂M. Since c(d/dθ )U = U c(d/dθ ) ∈ End(S ⊗ Cn ), it follows that

+ Aτ

= 0 = τ γ˜ + γ˜ τ, τA

τ 2 = 1, τ = τ ∗ .

Moreover, one verifies by calculation (cf. [6, Eqs. (3.11) to (3.13)]) γ˜ PtU = I − PtU γ˜ ; U 2

= 0; Pt , A

tU .

tU = cos(2t)|A|P PtU AP

2078


Then by [6, Theorem 3.9], it suffices to find a unitary μ : H ∂ → H ∂ such that μ2 = −I,

+ Aμ

= 0. μτ + τ μ = μγ˜ + γ˜ μ = μA

Let μ := This finishes the proof.

0 −U −1

U 0

.

2

Now the equality ξ(∂M, U ) = ξ(eU DS1 ×∂M eU ) mod Z follows as a corollary. To be slightly more precise, we have the following result. Theorem 5.7. ψ,U ψ,U ξ(∂M, U ) = ξ(eU DS1 ×∂M eU ) − SF D[0,1] ; PtU − SF D[0,1] (t); P0U 0t1 . In particular, ξ(∂M, U ) = ξ(eU DS1 ×∂M eU )

mod Z.

Proof. By [11, Lemma 3.4], ψ,U ψ,U ξ(eU DS1 ×∂M eU ) − ξ D[0,1] ; P0U = SF D[0,1] ; PtU 0tπ/4 +

π/4

d 1 ψ,U U η D[0,1] ; Pt dt. dt 2

0

The formula now follows from the definition of η(∂M, U ) and the proposition above.

2

6. Relative index pairing for odd dimensional manifolds with boundary In this section, we shall use the Toeplitz index theorem for odd dimensional manifolds with boundary by Dai and Zhang to prove our analogue of the index pairing formula by Lesch, Moscovici and Pflaum [12, Theorem 7.6]. First let us recall the even case. Let X be an even dimensional spin manifold with boundary ∂X. We assume its Riemannian metric has product structure near the boundary. The associated Dirac operator takes of the following form DX =

D+

D−

=

d dx

+ D∂X

d + D∂X − dx

near the boundary, where D∂X is the Dirac operator over ∂X, cf. Appendix A. Definition 6.1. Let P0 = χ[0,∞) (D∂X ) and DP+0 be the elliptic operator D + with the APS boundary condition P0 , cf. [2]. Then IndAPS (D + ) := Ind(DP+0 ).


2079

Recall that a relative K-cycle in K 0 (X, ∂X) is a triple [p, q, hs ] such that p, q ∈ Mn (C ∞ (X)) are two projections over X and hs ∈ Mn (C ∞ (∂X)), s ∈ [0, 1], is a path of projections over ∂X such that h0 = p|∂X and h1 = q|∂X . If p and q are constant along the normal direction near the boundary, then the relative index pairing by Lesch, Moscovici and Pflaum [12, Theorem 7.6] states that Ind[DX ] [p, q, hs ] = IndAPS qD + q − IndAPS pD + p + SF(hs D∂X hs )0s1 . Now let M be an odd dimensional spin manifold with boundary ∂M. We assume its Riemannian metric has product structure near the boundary. The Dirac operator D over M naturally induces an element in KK(C0 (M \ ∂M), c1 ) ∼ = K1 (M, ∂M), cf. [4, Section 2], from which one has the relative index pairing map Ind[D] : K 1 (M, ∂M) → Z.

(6.1)

As an intermediate step, let us first show a pairing formula by using the lifted data on S1 × M. The method of proof is similar to the one used in proving Theorem 4.1. Denote the Dirac operator and its restriction to the half-spinor bundles by D + . We shall explain in detail over S1 × M by D the structure of D near the boundary in Appendix A. Lemma 6.2. For a relative K-cycle [U, V , us ] ∈ K 1 (M, ∂M), that is, U, V ∈ Un (C ∞ (M)) are two unitaries over M with us ∈ Un (C ∞ (∂M)), s ∈ [0, 1], a smooth path of unitaries over ∂M such that u0 = U |∂M and u1 = V |∂M . If U and V are constant along the normal direction near the boundary, then + eV − IndAPS eU D + eU + SF(eus DS1 ×∂M eus )0s1 . Ind[D] [U, V , us ] = IndAPS eV D Proof. A relative K-cycle [U, V , us ] ∈ K 1 (M, ∂M) naturally induces a relative K-cycle [eU , eV , eus ] ∈ K 0 (M, ∂M). By [12, Theorem 7.6], [U, V , us ] (6.2) + eV ) − IndAPS (eU D + eU ) + SF(eus DS1 ×∂M eus )0s1 IndAPS (eV D is a well-defined map from K 1 (M, ∂M) to Z. We need to show that it does agree with the relative index pairing induced by that of K 1 (M \ ∂M). As before (cf. Section 1 above), we can assume U |[0,)×∂M = V |[0,)×∂M and us = U |∂M = V |∂M , for all s ∈ [0, 1]. It suffices to prove the lemma for representatives of relative K-cycles of this special type. Notice that such a representative also defines an element in K 1 (M \ ∂M) by its restriction to M \ ∂M and recall from Section 1 that the index map (6.1) has the following explicit formula: ∧ Ch• (V ) − Ch• (U ) . Ind[D] [V , U, us ] = − A(M) M

Now by the APS index theorem for manifolds with boundary,

2080


+ eU = IndAPS eU D

S1 × M ∧ Ch• (eU ) − ξ(eU DS1 ×∂M eU ) A

(6.3)

A(M) ∧ Ch• (U ) − ξ(eU DS1 ×∂M eU )

(6.4)

S1 ×M

=− M

where the second equality follows from Proposition 3.3. There is a similar equation where we replace U by V . It follows that the image of a representative of the special type as above, under the map (6.2), is equal to

A(M) ∧ Ch• (V ) − Ch• (U ) .

− M

This agrees with the relative index map (6.1).

2

Using this lemma and another two lemmas below, we shall now prove our main result in this section. Theorem 6.3. For a relative K-cycle [U, V , us ] ∈ K 1 (M, ∂M), that is, U, V ∈ Un (C ∞ (M)) are two unitaries over M with us ∈ Un (C ∞ (∂M)), s ∈ [0, 1], a smooth path of unitaries over ∂M such that u0 = U |∂M and u1 = V |∂M . If U and V are constant along the normal direction near the boundary, then us Ind[D] [U, V , us ] = Ind(TV ) − Ind(TU ) + SF u−1 s D[0,1] us ; P0 us −1 where SF(u−1 s D[0,1] us ; P0 ) is the spectral flow of the path of elliptic operators (us D[0,1] us ; us us P0 ), s ∈ [0, 1], with APS type boundary conditions P0 as in (5.4).

Proof. By formula (5.1), we have Ind(TV ) − Ind(TU ) = −

A(M) ∧ Ch• (V ) − ξ(∂M, V ) + τμ V P ∂ V −1 , P ∂ , PM

M

+

A(M) ∧ Ch• (U ) + ξ(∂M, U ) − τμ U P ∂ U −1 , P ∂ , PM

M

=−

A(M) ∧ Ch• (V ) − Ch• (U ) + ξ(∂M, U ) − ξ(∂M, V )

M

since τμ (U P ∂ U −1 , P ∂ , PM ) = τμ (V P ∂ V −1 , P ∂ , PM ) by [11, Lemma 6.10]. Notice that us ξ(∂M, U ) − ξ(∂M, V ) + SF u−1 s D[0,1] us ; P0 0s1 ψ,U ψ,U = ξ(eU DS1 ×∂M eU ) − SF D[0,1] ; PtU − SF D[0,1] (t); P0U − ξ(eV DS1 ×∂M eV ) ψ,V ψ,V us + SF D[0,1] ; PtV + SF D[0,1] (t); P0V + SF u−1 s D[0,1] us ; P0 0s1


2081

which is equal to ξ(eU DS1 ×∂M eU ) − ξ(eV DS1 ×∂M eV ) + SF(eus DS1 ×∂M eus )0s1 by the lemmas below. Hence us Ind(TV ) − Ind(TU ) + SF u−1 s D[0,1] us ; P0 0s1 = − A(M) ∧ Ch• (V ) − Ch• (U ) M

− ξ(eV DS1 ×∂M eV ) + ξ(eU DS1 ×∂M eU ) + SF(eus DS1 ×∂M eus )0s1 + eV − IndAPS eU D + eU + SF(eus DS1 ×∂M eus )0s1 = IndAPS eV D which is equal to Ind[D] ([U, V , us ]) by Lemma 6.2.

2

Lemma 6.4. ψ,u ψ,V ψ,U SF D[0,1]s ; P0us 0s1 = SF D[0,1] ; PtU − SF D[0,1] ; PtV + SF(eus DS1 ×∂M eus )0s1 . Proof. Consider the (t, s)-parametrized family of operators ψ,us us D[0,1] ; Pt (0tπ/4; 0s1) where Ptus is defined as in Eq. (5.4). Note that P0us =

P∂ 0

0

∂ Id −u−1 s P us

us and Pπ/4 =

1 2

1 −u−1 s

−us 1

.

Hence ψ,u us eus DS1 ×∂M eus = D[0,1]s ; Pπ/4 . Consider the following diagram ψ,V

ψ,V

(D[0,1] ; P0V )

(D[0,1] ;PtV )

ψ,V

V ) (D[0,1] ; Pπ/4

ψ,u

ψ,u

us (D[0,1]s ;Pπ/4 )

(D[0,1]s ;P0us )

ψ,U

ψ,U

(D[0,1] ; P0U )

ψ,U

(D[0,1] ;PtU )

U ) (D[0,1] ; Pπ/4

where the arrows stand for smooth paths connecting the corresponding vertices. Now the lemma follows from the homotopy invariance of the spectral flow. 2

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Now let D[0,1]s (t) := D[0,1] + (1 − tψ)u−1 s [D[0,1] , us ], ψ,u

then the same argument above proves the following lemma. Lemma 6.5. us SF u−1 s D[0,1] us ; P0 0s1 ψ,V ψ,u ψ,U = SF D[0,1] (t); P0U − SF D[0,1] (t); P0V + SF D[0,1]s ; P0us 0s1 . Proof. Consider the (s, t)-parametrized family of operators ψ,us D[0,1] (t), P0us 0t,s1 , cf. the following diagram ψ,V

(D[0,1] (t);P0V )

ψ,V

(D[0,1] (0); P0V )

ψ,V

(D[0,1] (1); P0V )

ψ,u

ψ,u

(D[0,1]s ,P0us )

(D[0,1]s ;P0us )

ψ,U

ψ,U

(D[0,1] (0); P0U )

ψ,V

(D[0,1] (t);P0U )

(D[0,1] (1); P0U )

Notice that D[0,1]s (1) = D[0,1]s and D[0,1]s (0) = u−1 s D[0,1] us . The lemma follows by the homotopy invariance of the spectral flow. 2 ψ,u

ψ,u

ψ,u

Acknowledgments I am greatly indebted to Henri Moscovici for his continuous support and advice. This paper grew out of numerous conversations with him. I want to thank Nigel Higson for helpful suggestions. I am grateful to Alexander Gorokhovsky for a careful reading of the first version of this paper as well as for many helpful comments. I started working on this problem during my visit at the Hausdorff Center for Mathematics in Bonn, Germany. I want to express my thanks to the center for its hospitality and to Matthias Lesch for the invitation, as well as for generously sharing with me his insights into the subject. Appendix A. Spinor bundles and Dirac on manifolds with boundary The material in this appendix is well known. The purpose is to clarify the relations among various Dirac operators arising in this article for the convenience of the reader. Suppose M is an odd dimensional spin manifold with boundary. Its Riemannian metric assumes a product structure near the boundary. Let S (resp. SM ) be the spinor bundle over S1 × ∂M ( resp. M). Then


2083

Cl(T∂M ) the Clifford algebra over ∂M is identified with the even part of Cl(TS1 ×∂M ) the Clifford algebra over S1 × ∂M by c∂ (ei ) → c(ei ) · c(d/dθ ) where c∂ (·), resp. c(·), is the Clifford multiplication on S ∂ , resp. S. This way S|∂M , the restriction of S to {0} × ∂M, is identified with S ∂ = S ∂,+ ⊕ S ∂,− the spinor bundle over ∂M. the spinor bundle over [0, 1) × S1 × ∂M, is naturally isomorphic to C2 ⊗ S∂ . Notice that S, 2 + − stands for graded tensor product. Denote the Dirac operator over Here C = C ⊕ C and ⊗ Then [0, 1) × S1 × ∂M by D. = D

d d + i dθ − dx

0 d + i dθ

d dx

0

D∂ . IS ∂ + IC2 ⊗ ⊗

We identify Cl(TS1 ×∂M ) the Clifford algebra over S1 × ∂M with the even part of Cl(TS1 ×M ) the Clifford algebra over S1 × M by c(ei ) → cˆ (ei ) · cˆ (d/dx) From this, one has for ei ∈ TS1 ×∂M , where cˆ (·) is Clifford multiplication on S. S+ = C+ ⊗ S ∂,+ ⊕ C− ⊗ S ∂,− ∼ = S ∂,+ ⊕ S ∂,− ≡ S, S− = C− ⊗ S ∂,+ ⊕ C+ ⊗ S ∂,− ∼ = c(d/dx)S+ . Lemma A.1. With the idenfications of spinor bundles as above, = D

⎛

⎜ ⎜ =⎜ ⎝

d − dx + DS1 ×∂M d dx

+ DS1 ×∂M d d − dx + i dθ

−iD ∂ |S ∂,+ d dx

d + i dθ

iD ∂ |S ∂,+

−iD ∂ |S ∂,+

d dx

iD ∂ |S ∂,−

⎞

d d ⎟ − dx − i dθ ⎟ ⎟ ⎠

d − i dθ

where DS1 ×∂M (resp. D ∂ ) is the Dirac operator over S1 × ∂M (resp. ∂M). In particular, d DS1 ×∂M = c(d/dθ) + D∂ dθ with c(d/dθ ) =

i −i

and D = ∂

D ∂ |S ∂,+

D ∂ |S ∂,−

.

2084


Proof. With the identification S− = cˆ (d/dx)S+ , one has

d d −ˆc(d/dx)D|S + = −ˆc(d/dx) cˆ (d/dx) + cˆ (d/dθ ) + cˆ (ei )∇ei dx dθ i

d d − cˆ (d/dx) · cˆ (d/dθ ) − = cˆ (d/dx) · cˆ (ei )∇ei dx dθ i

d d + c(d/dθ ) + c(ei )∇ei dx dθ i

d d ∂ = c (ei )∇ei . + c(d/dθ ) + dx dθ =

i

Similarly,

S − cˆ (d/dx) = cˆ (d/dx) d + cˆ (d/dθ ) d + cˆ (ei )∇ei cˆ (d/dx) D| dx dθ i

d d ∂ =− c (ei )∇ei . + c(d/dθ ) + dx dθ i

d Notice that c(d/dθ)( dθ + D ∂ ) is the Dirac operator over S1 × ∂M, hence

= D

d dx

d − dx + DS1 ×∂M

+ DS1 ×∂M

.

To finish the proof, one notices that c(d/dθ ) = cˆ (d/dθ ) · cˆ (d/dx) = =

0 i i 0

i 0 0 −i

IS ∂ · ⊗

IS ∂ . ⊗

0 −1 1 0

IS ∂ ⊗

2

References [1] M.F. Atiyah, K-Theory, Lecture Notes by D.W. Anderson, W.A. Benjamin, Inc., New York, Amsterdam, 1967. [2] M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69. [3] M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99. [4] P. Baum, R.G. Douglas, M.E. Taylor, Cycles and relative cycles in analytic K-homology, J. Differential Geom. 30 (1989) 761–804. [5] B. Booß-Bavnbek, K.P. Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Math. Theory Appl., Birkhäuser Boston Inc., Boston, MA, 1993. [6] J. Brüning, M. Lesch, On the η-invariant of certain nonlocal boundary value problems, Duke Math. J. 96 (1999) 425–468. [7] A. Connes, Noncommutative differential geometry, Publ. Math. Inst. Hautes Études Sci. (1985) 257–360. [8] A. Connes, Noncommutative Geometry, Academic Press Inc., San Diego, CA, 1994.


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[9] X. Dai, W. Zhang, An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary, J. Funct. Anal. 238 (2006) 1–26. [10] N. Higson, J. Roe, Analytic K-Homology, Oxford Math. Monogr., Oxford University Press, Oxford, 2000, Oxford Sci. Publ. [11] P. Kirk, M. Lesch, The η-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary, Forum Math. 16 (2004) 553–629. [12] M. Lesch, H. Moscovici, M.J. Pflaum, Connes–Chern character for manifolds with boundary and eta cochains, http://arxiv.org/abs/0912.0194. [13] J.-L. Loday, Cyclic Homology, Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol. 301, Springer-Verlag, Berlin, 1992, Appendix E by María O. Ronco. [14] T.A. Loring, K-theory and asymptotically commuting matrices, Canad. J. Math. 40 (1988) 197–216. [15] M.A. Rieffel, C ∗ -algebras associated with irrational rotations, Pacific J. Math. 93 (1981) 415–429. [16] N.E. Wegge-Olsen, K-Theory and C ∗ -Algebras, Oxford Sci. Publ., The Clarendon Press/Oxford University Press, New York, 1993, a friendly approach.


Factorization of Blaschke products and ideal theory in H ∞ Kei Ji Izuchi a,∗,1 , Yuko Izuchi b a Department of Mathematics, Niigata University, Niigata 950-2181, Japan b Aoyama-shinmachi 18-6-301, Nishi-ku, Niigata 950-2006, Japan

Received 28 July 2010; accepted 19 August 2010 Available online 17 September 2010 Communicated by J. Bourgain

Abstract Let H ∞ be the Banach algebra of bounded analytic functions on the open unit disk D. Let G be the union set of all nontrivial Gleason parts in the maximal ideal space of H ∞ . Let E be a nonvoid compact and totally disconnected subset of G and nE be a bounded numbering function on E. We characterize nE for which there is a closed ideal I in H ∞ such that Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E. Let I1 , I2 , . . . , Ik be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for 1 i k. We prove that ki=1 Ii = { ki=1 fi : fi ∈ Ii , 1 i k} is a closed ideal. A local ideal theory in H ∞ plays an important role to prove our results. © 2010 Elsevier Inc. All rights reserved. Keywords: Interpolating Blaschke product; Carleson–Newman Blaschke product; Algebra of bounded analytic functions; Gleason part; Ideal theory; Big disk algebra

1. Introduction Let H ∞ be the Banach algebra of bounded analytic functions on the open unit disk D with the supremum norm · ∞ . We denote by M(H ∞ ) the maximal ideal space of H ∞ , i.e., M(H ∞ ) is the family of nonzero multiplicative linear functionals on H ∞ with the weak∗ -topology. We * Corresponding author.

E-mail addresses: [email protected] (K.J. Izuchi), [email protected] (Y. Izuchi). 1 Partially supported by Grant-in-Aid for Scientific Research (No. 21540166), Japan Society for the Promotion of

Science. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.012

K.J. Izuchi, Y. Izuchi / Journal of Functional Analysis 260 (2011) 2086–2147

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identify a function f in H ∞ with its Gelfand transform f(m) = m(f ), m ∈ M(H ∞ ), so we think of f a continuous function on M(H ∞ ). By the Carleson corona theorem [3], D is dense in M(H ∞ ). For 0 < r < 1, we write Dr = {|z| < r}. Let {an }n be a sequence in D satisfying ∞ n=1 (1 − |an |) < ∞. Associated with it, we have a Blaschke product b(z) =

∞ −a n z − an , |an | 1 − a n z

z ∈ D,

n=1

where if an = 0, we consider that −a n /|an | = 1. We call {an }n and b(z) interpolating if for every bounded sequence {cn }n , there exists f in H ∞ such that f (an ) = cn for every n 1. In [2], Carleson also proved that {an }n is an interpolating sequence in D if and only if ak − an inf 1 − a a k n;n=k

n k

> 0.

We write Z(b) = {x ∈ M(H ∞ ): b(x) = 0} and

|b| < r = x ∈ M H ∞ : b(x) < r ,

0 < r < 1.

A Blaschke product B is called Carleson–Newman if B = ki=1 bi for finitely many interpolating Blaschke products b1 , b2 , . . . , bk . In this case, there are many ways to give such factorization. If k is the smallest number of interpolating Blaschke products, B is called a Carleson–Newman Blaschke product of order k. In this paper, we write a CN Blaschke product instead of a Carleson– Newman Blaschke product. For Blaschke products B1 and B2 , if B1 is a subproduct of B2 , we write B1 ≺ B2 . For x, y ∈ M(H ∞ ), the pseudo-hyperbolic distance is defined by ρ(x, y) = f (x): f (y) = 0, f ∈ H ∞ , f ∞ 1 . The set

P (x) = y ∈ M H ∞ : ρ(x, y) < 1 is called the Gleason part containing x ∈ M(H ∞ ). If P (x) = {x}, P (x) is called nontrivial. We denote by G the union set of all nontrivial Gleason parts in M(H ∞ ). In [14], Hoffman studied the structure of Gleason parts extensively. He proved the following facts (see also [7]). (a) Let x ∈ M(H ∞ ). Then x ∈ G if and only if there is an interpolating Blaschke product b satisfying b(x) = 0, and G is an open subset of M(H ∞ ). (b) For a nontrivial Gleason part P (x), there exists a pseudo-hyperbolic distance preserving continuous, one-to-one and onto map Lx : D → P (x) such that Lx (0) = x and (f ◦ Lx )(z) ∈ H ∞ for every f ∈ H ∞ . The map Lx : D → P (x) is called the Hoffman map at x ∈ G. (c) Let b be an interpolating Blaschke product. For small positive numbers η and ε satisfying some additional conditions, we may define the map γ : Z(b) × D (ξ, z) → γ (ξ, z) ∈ |b| < ε

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by γ (ξ, z) ∈ Lξ (Dη ) satisfying (b/ε)(γ (ξ, z)) = z. Then γ is a biholomorphically homeomorphic and onto map, and γ can be extended γ : Z(b) × D (ξ, z) → γ (ξ, z) ∈ |b| ε homeomorphically. After Hoffman’s work, interpolating Blaschke products have played an important role in the study of the structure of H ∞ (see [4,7,15,18,22,23]), especially in the study of ideal theory in H ∞ (see [9–11,17,19,21]). It is known that for a Blaschke product B, B is a CN Blaschke product if and only if Z(B) ⊂ G (see [8,12,24]). Our aim is to study closed ideals I in H ∞ satisfying Z(I ) ⊂ G, where Z(I ) =

Z(f ).

f ∈I

It is extremely difficult to make clear the structure of closed ideals I in H ∞ satisfying Z(I ) ⊂ G (see [1]). For x ∈ G and f ∈ H ∞ , by (b) we may define zero’s order of f at x, we write ord(f, x), by zero’s order of the analytic function f ◦ Lx at 0 ∈ D. For x ∈ M(H ∞ ) \ G and f ∈ H ∞ with f (x) = 0, we define ord(f, x) = ∞. We put ord(I, x) = min ord(f, x): f ∈ I ,

x ∈ M H∞ .

For a compact subset E of G, let I (E) = f ∈ H ∞ : f (x) = 0, x ∈ E , which is called the associated primary ideal of E. Generally we have E ⊂ Z(I (E)). In [11], Gorkin, Mortini, and the first author proved the following two theorems for closed ideals I satisfying Z(I ) ⊂ G. In this case, we note that mI := maxx∈Z(I ) ord(I, x) < ∞. The following is given in Theorem 2.2 in [11]. Theorem A. Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ G. Then I coincides with the set of f in H ∞ satisfying ord(f, x) ord(I, x) for every x ∈ Z(I ). This is a fairly crucial theorem in ideal theory of H ∞ . By this theorem, for closed ideals I1 , I2 satisfying Z(I1 ) = Z(I2 ) ⊂ G, we have that I1 = I2 if and only if ord(I1 , x) = ord(I2 , x) for every x ∈ Z(I1 ). The following is essentially given in Theorem 3.4 in [11]. Theorem B. Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ G and x ∈ Z(I ). Then there is a CN Blaschke product B of order mI in I satisfying ord(B, x) = ord(I, x). In ideal theory of H ∞ , one of the main problems is what is the function ord(I, x) in x ∈ Z(I ). We have also several questions about a closed ideal I in H ∞ satisfying Z(I ) ⊂ G (see [11]). Question 1. Characterize nonvoid compact and totally disconnected subsets E of G satisfying Z(I (E)) = E to use geometrical words in E.


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Question 2. Let E be a nonvoid compact and totally disconnected subset of G. For which bounded numbering function nE : E → {1, 2, . . .}, does there exist a closed ideal I in H ∞ satisfying Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E? Let I1 , I2 be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for i = 1, 2, and I1 ⊗ I2 be the tensor product of I1 and I2 . Question 3. Is I1 ⊗ I2 a closed ideal in H ∞ ? In [19], the authors studied these questions for I (E) the associated primary ideal of E. We have another question. Question 4. Is I1 I2 = {f1 f2 : f1 ∈ I1 , f2 ∈ I2 } a closed ideal in H ∞ ? The purpose of this paper is to answer these questions. To study these questions generally, we need to see local versions of these questions. Hoffman’s results (a)–(c) give us many informations about local properties of functions in H ∞ on G. Let x ∈ G. By (a), there is an interpolating Blaschke product b satisfying b(x) = 0. Then the set {|b| < ε} is an open neighborhood of x in G for small ε > 0. And by (c), we may identify {|b| < ε} with the more simpler space Z(b) × D. It ˇ is known that Z(b) is homeomorphic to the Stone–Cech compactification βN of the set of natural numbers N (see [13]). So using the same notation γ in (c), we have a homeomorphic map γ : βN × D → |b| ε which satisfies some additional conditions. Let C(βN × D) be the space of continuous functions on βN × D. For f ∈ C(βN × D) and ξ ∈ βN, we put fξ (z) = f (ξ, z) for z ∈ D. Let A = f ∈ C(βN × D): fξ (z) ∈ A(D), ξ ∈ βN , where A(D) is the disk algebra on D, i.e., A(D) is the space of continuous functions on D which are analytic in D. We call A the big disk algebra. For a closed ideal J in A, let Z(J ) = x ∈ βN × D: f (x) = 0, f ∈ J . For f ∈ A and x = (ξ, z) ∈ βN × D, we may define zero’s order at x, which we write ord(f, x). We put ord(J, x) = min ord(f, x): f ∈ J ,

x ∈ βN × D.

By condition (c), for each ξ ∈ βN, γ maps {ξ } × D biholomorphically onto an open subset γ ({ξ } × D) in P (λ) for some λ ∈ Z(b). Hence we have H ∞ ◦ γ ⊂ A, and for f ∈ H ∞ we have ord(f, x) = ord(f ◦ γ , γ −1 x) for every x ∈ {|b| < ε}.

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Our strategy of the study is the following. Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ G. It is known that Z(I ) is a totally disconnected set (see [11]). Let x ∈ Z(I ). Then there is an interpolating Blaschke product b satisfying b(x) = 0. Let ε > 0 be sufficiently small. There is an open and closed subset E of Z(I ) such that x ∈ E ⊂ {|b| < ε}. Then there is a closed ideal I1 in H ∞ such that Z(I1 ) = E and ord(I1 , x) = ord(I, x) for every x ∈ E. We have I1 ◦ γ ⊂ A. Here I1 ◦ γ may not be a closed ideal in A. Let J1 be a closed ideal in A generated by I1 ◦ γ . Then we have Z(J1 ) = γ −1 (E) ⊂ βN × D and ord(J1 , γ −1 (x)) = ord(I1 , x) for every x ∈ E. So in Sections 2–7, we study a closed ideal J in A satisfying Z(J ) ⊂ βN × D. And we shall answer the A-versions of Questions 1–3. In Section 8, applying the results in A we get some local ideal theory in H ∞ . In Section 9, we shall answer Questions 1–4 completely. Without using A we can prove Questions 1–4. But their proofs will be heavily complicated. It is more understandable via the space A. In Section 2, we study basic properties of A. In Section 3, we deal with closed ideals I in A satisfying Z(I ) ⊂ βN × D. In Section 4, for a nonvoid compact and totally disconnected subset E of βN × D we define a tilde function nE of a numbering function nE : E → {1, 2, . . .}. Also we define the associated numbering function NE,∞ which represents a geometrical property of E. We put I (E) = {f ∈ A: f (x) = 0, x ∈ E}. In Section 5, we prove that Z(I (E)) ⊂ βN × D if and only if supx∈E NE,∞ (x) < ∞, which answers the A-version of Question 1. In Section 6, we nE = nE on E if and only if there is a prove that nE is a bounded numbering function satisfying closed ideal I in A satisfying Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E, which answers the A-version of Question 2. In Section 7, we prove that I1 ⊗ I2 is a closed ideal in A, which answers the A-version of Question 3. For topological properties of βN, see [5,25]. 2. The big disk algebra Let X = βN × D. Then X = βN × D and ∂X = βN × ∂D. Let π : X → βN be the projection defined by π(ξ, z) = ξ for (ξ, z) ∈ X. For z1 ∈ D and 0 < r < 1 − |z1 |, let Dr (z1 ) = {|z − z1 | < r} and D r (z1 ) = {|z − z1 | r}. If z1 = 0, we write Dr = Dr (0). Let C(X) be the space of continuous functions on X with the supremum norm f X = maxx∈X |f (x)|. For f ∈ C(X) and ξ ∈ βN, we put fξ (z) = f (ξ, z) for z ∈ D, which we call the slice function of f at ξ . We put fξ D = maxz∈D |fξ (z)|. The following is an elementary fact. Lemma 2.1. Let f ∈ C(X) and ξα → ξ in βN. Then fξα − fξ D → 0 as α → ∞. Let A = f ∈ C(X): fξ ∈ A(D), ξ ∈ βN , where A(D) is the disk algebra on D. It is easy to see that A is a closed subalgera of C(X), which we call the big disk algebra on X. Let W be an open and closed subset of βN and χπ −1 (W ) be the characteristic function for π −1 (W ). Then χπ −1 (W ) ∈ A. Let f (z) ∈ A(D). Identifying f (z) with f (ξ, z) = f (z) for (ξ, z) ∈ βN × D, we may consider f (z) ∈ A, so χπ −1 (W ) f (z) ∈ A. For each f ∈ A, let Z(f ) = x ∈ X: f (x) = 0 .


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For x = (ξ, z) ∈ X, we define the order of zero of f at x by ⎧ ⎨ ord(fξ , z), ord(f, x) = ∞, ⎩ 0,

x ∈ Z(f ) ∩ X, x ∈ Z(f ) ∩ ∂X, x ∈ X \ Z(f ),

where ord(fξ , z) is the usual order of zero of the analytic function fξ at z ∈ D. The following is known as the Hurwitz theorem. Lemma 2.2. Let f ∈ A(D) satisfy Z(f ) = {z0 } ⊂ D. Let {fα }α∈Λ be a net in A(D) such that fα − f D → 0 as α → ∞. Then there exists α0 ∈ Λ such that ord(f, z0 ) =

ord(fα , z): z ∈ Z(fα )

for every α α0 , and the net of sets {Z(fα )}α converges to the one point set {z0 }. By Lemmas 2.1 and 2.2, we have the following. Lemma 2.3. Let f ∈ A and x1 = (ξ1 , z1 ) ∈ Z(f ) ∩ X satisfy ord(f, x1 ) < ∞. Take 0 < r < 1 such that Dr (z1 ) ⊂ D and |fξ1 | > 0 on ∂Dr (z1 ). Then there exists an open and closed neighborhood Wξ1 of ξ1 in βN such that

ord(f, x): x ∈ Z(f ) ∩ {ξ1 } × Dr (z1 )

= ord(f, y): y ∈ Z(f ) ∩ {λ} × Dr (z1 ) for every λ ∈ Wξ1 . Corollary 2.4. For f ∈ A, ord(f, x) is upper semicontinuous on X. For f ∈ A, we put Ordξ (f ) =

ord(f, x): x ∈ Z(f ) ∩ π −1 (ξ ) ,

ξ ∈ βN.

Then Ordξ (f ) is also upper semicontinuous on βN. Corollary 2.5. If f ∈ A and |f | > 0 on ∂X, then sup Ordξ (f ) = sup Ordn (f ) < ∞.

ξ ∈βN

n∈N

Proof. By the assumption, Ordξ (f ) < ∞ for every ξ ∈ βN. Suppose that Ordξα (f ) → ∞ for some net {ξα }α such that ξα → ξ in βN. Then Ordξ (f ) = ∞. But this is a contradiction. The equality follows from Lemma 2.3. 2 The following corollaries also follow from Lemma 2.3.

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Corollary 2.6. If f ∈ A and |f | > 0 on ∂X, then {ξ ∈ βN: Ordξ (f ) = j } is an open and closed subset of βN for every j 1. Corollary 2.7. If f ∈ A and |f | > 0 on ∂X, then Z(f ) is a totally disconnected set. A function ϕ in A is called inner if |ϕ| = 1 on ∂X. If ϕ is inner and Z(ϕ) = ∅, then ϕξ is a unimodular constant for each ξ ∈ βN. If Z(ϕ) = ∅, then by Corollary 2.6 π(Z(ϕ)) is an open and closed subset of βN and π(Z(ϕ)) ∩ N is dense in π(Z(ϕ)). We shall study the structure of inner functions. An inner function b in A is called an IBP (interpolating Blaschke product) if ord(b, x) = 1 for every x ∈ Z(b). An IBP b is called simple if Ordξ (b) = 1 for every ξ ∈ π(Z(b)). If b1 , b2 are IBPs, then b1 b2 is an IBP if and only if Z(b1 ) ∩ Z(b2 ) = ∅. First, we study a simple IBP in A. Let N be a subset of N and η be a unimodular function on N. Let a(n) be a function on N satisfying sup a(n) < 1.

(2.1)

n∈N

We define the function q(n, z) on N × D by q(n, z) =

η(n)

z−a(n) , 1−a(n)z

η(n),

n ∈ N, n ∈ N \ N.

(2.2)

We put W = N βN , where N βN is the closure of N in βN. Then W is an open and closed suba ∈ C(W ) satisfying set of βN and W ∩ N = N . There are η ∈ C(βN) satisfying η|N = η and η| = 1 on βN and maxξ ∈W | a (ξ )| = supn∈N |a(n)| < 1. We define the func a |N = a. Then | tion b(ξ, z) on βN × D by η(ξ ) z− a (ξ ) , ξ ∈ W, 1− a (ξ )z b(ξ, z) = η(ξ ), ξ ∈ βN \ W. Then b is a simple IBP in A and b|N×D = q. We have also Z(b) =

X n, a(n) : n ∈ N .

Conversely, suppose that b is a simple IBP. Let q = b|N×D . Then q has a form in (2.2) and its zeros satisfy condition (2.1). By the above fact, there is a simple IBP b1 in A such that b1 |N×D = b|N×D . Since N × D is dense in X, we have b1 = b. Since a product of finitely many simple IBPs is an inner function, we have the following by the above observation. Theorem 2.8. (i) Let {qn (z)}n be a sequence of finite Blaschke products on D such that

ord qn (z), ζ : ζ ∈ Z qn (z) < ∞ sup n∈N


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D and ∞ n=1 Z(qn (z)) ⊂ D. Then there exists an inner function ϕ in A such that ϕ(n, z) = qn (z) for every (n, z) ∈ N × D. Moreover X

∞

Z(ϕ) = (n, ζ ): ζ ∈ Z qn (z) . n=1

(ii) Let ϕ be an inner function in A satisfying Z(ϕ) = ∅ and m = max Ordξ (ϕ) < ∞. ξ ∈βN

Then there are simple IBPs b1 , b2 , . . . , bm in A such that ϕ =

m

i=1 bi .

Let ϕ1 and ϕ2 be inner functions in A. If there is another inner function ϕ3 such that ϕ2 = ϕ1 ϕ3 , we write ϕ3 = ϕ2 /ϕ1 and ϕ1 is called a subfactor of ϕ2 , and we write ϕ1 ≺ ϕ2 . Corollary 2.9. Let ϕ1 , ϕ2 be inner functions in A. If ord(ϕ1 , x) ord(ϕ2 , x) for every x ∈ Z(ϕ1 ) ∩ (N × D), then ϕ1 ≺ ϕ2 . Proof. For each n ∈ N, ϕ2 (n, z)/ϕ1 (n, z) is a finite Blaschke product on D. By Theorem 2.8(i), there is an inner function ψ in A such that ψ(n, z) = ϕ2 (n, z)/ϕ1 (n, z) for every (n, z) ∈ N × D. Then (ϕ1 ψ)(n, z) = ϕ2 (n, z), so ϕ1 ψ = ϕ2 . 2 Let ϕ be an inner function in A. By Corollary 2.7, Z(ϕ) is a totally disconnected set. Let U be an open subset of X such that Z(ϕ) ∩ U is an open and closed subset of Z(ϕ). Let N = π(Z(ϕ) ∩ U ) ∩ N. We note that the slice function ϕn is a finite Blaschke product for every n ∈ N . For each n ∈ N , let q(n, z) be the subproduct of ϕn with zeros {z ∈ D: (n, z) ∈ Z(ϕ) ∩ U } counting multiplicities. We define the function ψ(n, z) on N × D by ψ(n, z) =

q(n, z), 1,

n ∈ N, n ∈ N \ N.

By Theorem 2.8(i), there is an inner function ϕU in A such that ϕU |N×D = ψ . By Corollary 2.9, we have ϕU ≺ ϕ, Z(ϕU ) = Z(ϕ) ∩ U , ϕU = 1 on X \ π −1 (π(Z(ϕ) ∩ U )) and |ϕ/ϕU | > 0 on U . We call ϕU the subfactor of ϕ with zeros Z(ϕ) ∩ U . If Z(ϕ) ∩ U = ∅, we put ϕU = 1. For f ∈ A and an inner function ψ in A, we write also ψ ≺ f if there is h ∈ A such that f = ψh. In this case, we write h = f/ψ , too. Similarly we have the following. Lemma 2.10. Let f ∈ A and U be an open subset of X such that Z(f ) ∩ U is an open and closed subset of Z(f ). Then there is an inner function ϕ in A such that ϕ ≺ f , Z(ϕ) = Z(f ) ∩ U , ϕ = 1 on X \ π −1 (π(Z(f ) ∩ U )) and |f/ϕ| > 0 on U . We have the following as a corollary of Lemma 2.10. Corollary 2.11. Let f ∈ A satisfy |f | > 0 on ∂X. Then there are an inner function ϕ in A and an invertible function h ∈ A such that f = ϕh.

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Lemma 2.12. Let ϕ be an inner function in A satisfying Z(ϕ) = ∅ and x1 = (ξ1 , z1 ) ∈ Z(ϕ). Let k = ord(ϕ, x1 ). Then there are an open and closed neighborhood Wξ1 of ξ1 in βN, r > 0 satisfying Dr (z1 ) ⊂ D and an inner function ψ such that Z(ψ) ⊂ Wξ1 × Dr (z1 ), ψ ≺ ϕ, Z(ϕ/ψ) ∩ (Wξ1 × D r (z1 )) = ∅ and Ordλ (ψ) = k for every λ ∈ Wξ1 . Proof. Take r > 0 satisfying Z(ϕξ1 ) ∩ D r (z1 ) = {z1 }. Then by Lemma 2.3, there is an open and closed neighborhood Wξ1 of ξ1 in βN such that |ϕ| > 0 on Wξ1 × ∂Dr (z1 ) and k=

ord(ϕ, x): x ∈ Z(ϕ) ∩ {λ} × Dr (z1 ) ,

λ ∈ Wξ1 .

Let ψ be the subfactor of ϕ with zeros Z(ϕ) ∩ (Wξ1 × Dr (z1 )). Then we get the assertion.

2

Let ϕ be an inner function in A satisfying Z(ϕ) = ∅. Let mϕ = max ord(ϕ, x)

and Mϕ = max Ordξ (ϕ). ξ ∈βN

x∈Z(ϕ)

By Corollary 2.5, mϕ Mϕ < ∞. By Theorem 2.8(ii), there are simple IBPs b1 , b2 , . . . , bMϕ Mϕ such that ϕ = i=1 bi . We may write ϕ = kj =1 qj for some IBPs q1 , q2 , . . . , qk satisfying k Mϕ . We say that ϕ is an inner function of order k, which we write order(ϕ) = k, if k is the smallest positive integer giving such factorization of ϕ. Theorem 2.13. Let ϕ be an inner function in A satisfying Z(ϕ) = ∅. Then ϕ is an inner function of order mϕ . Proof. By the definition of order(ϕ), we have mϕ order(ϕ). We shall show the reverse inequality. Let ξ ∈ π(Z(ϕ)) and Z(ϕ) ∩ π −1 (ξ ) = {xξ,1 , xξ,2 , . . . , xξ,jξ },

xξ,i = xξ,

(i = ).

We write xξ,i = (ξ, zξ,i ) for 1 i jξ . We put tξ,i = ord(ϕ, xξ,i ) mϕ . By Lemma 2.12, there are 0 < rξ < 1, an open and closed neighborhood Wξ of ξ in βN and inner functions ψxξ,1 , ψxξ,2 , . . . , ψxξ,jξ such that Drξ (zξ,i ) ∩ Drξ (zξ, ) = ∅,

i = ,

(2.3)

1 i jξ ,

(2.4)

λ ∈ Wξ , 1 i jξ ,

(2.5)

Z(ψxξ,i ) ⊂ Wξ × Drξ (zξ,i ), Ordλ (ψxξ,i ) = tξ,i , jξ

ψxξ,i ≺ ϕ

(2.6)

i=1

and Z j ξ

ϕ

i=1 ψxξ,i

∩ π −1 (Wξ ) = ∅.

(2.7)


2095

By (2.5) and Theorem 2.8(ii), there are simple IBPs bξ,i,1 , bξ,i,2 , . . . , bξ,i,tξ,i such that ψxξ,i = tξ,i mϕ =1 bξ,i, . If we put bξ,i, = 1 for tξ,i + 1 mϕ , then ψxξ,i = =1 bξ,i, . For each fixed jξ 1 mϕ , by (2.3) and (2.4) i=1 bξ,i, is an IBP, so order

jξ

ψxξ,i

= order

i=1

m ϕ jξ

bξ,i, mϕ .

(2.8)

=1 i=1

ξ1 , ξ2 , . . . , ξs in π(Z(ϕ)) satisfying π(Z(ϕ)) ⊂ sBy the compactness of π(Z(ϕ)), thereexist k−1 W . Let V = W and V = W \ W {V : 1 k s} is a set 1 ξ1 k ξk k=1 ξk j =1 ξj for 2 k s. Then s k of mutually disjoint open and closed subsets of βN and π(Z(ϕ)) ⊂ k=1 Wξk = sk=1 Vk . For jξk each 1 k s, let ϕk be the subfactor of i=1 ψxξk ,i with zeros

Z

jξ k

ψxξk ,i

∩ π −1 (Vk ).

i=1

By (2.8), order(ϕk ) mϕ for every 1 k s. By (2.6) and (2.7), we have ϕ = ϕ0 sk=1 ϕk for some inner function ϕ0 satisfying Z(ϕ0 ) = ∅. Since Z(ϕk ) ∩ Z(ϕj ) = ∅ for k = j , it is easy to see that order(ϕ) mϕ . 2 For a set A, we denote by #(A) the number of elements in A. Let E be a nonvoid compact subset of X. If E ⊂ Z(b) for some simple IBP b, E is called a simple interpolation set. In this case, #(E ∩ π −1 (ξ )) 1 for every ξ ∈ βN. If E ⊂ Z(b) for some IBP b, E is called an interpolation set, and this is an unusual definition. In Theorem 2.19, we shall prove that this is equivalent to the usual definition for an interpolation set. If E is an interpolation set, then E is totally disconnected by Corollary 2.7. We shall study an interpolation set. Lemma 2.14. Let E be a nonvoid compact subset of X such that #(E ∩ π −1 (ξ )) = 1 for every ξ ∈ π(E). Then E is a simple interpolation set. Proof. For each ξ ∈ π(E), let E ∩ π −1 (ξ ) = {(ξ, f (ξ ))}. Then f (ξ ) is a continuous function on π(E) and E = {(ξ, f (ξ )): ξ ∈ π(E)}. Let r = maxξ ∈π(E) |f (ξ )|. Then r < 1. By the Tietze extension theorem, there is a continuous function f on βN such that f |π(E) = f and f (βN) ⊂ Dr . Let b(ξ, z) = (z − f (ξ ))/(1 − f (ξ )z) for every (ξ, z) ∈ βN × D. Then b is a simple IBP and Z(b) =

ξ, f (ξ ) : ξ ∈ βN ⊃ ξ, f (ξ ) : ξ ∈ π(E) = E.

2

For x1 , x2 ∈ X, let ρ(x1 , x2 ) = sup f (x2 ): f ∈ A, f (x1 ) = 0, f X 1 . We put x1 = (ξ1 , z1 ) and x2 = (ξ2 , z2 ). If ξ1 = ξ2 , then ρ(x1 , x2 ) = 1. If ξ1 = ξ2 , then ρ(x1 , x2 ) = |z1 − z2 |/|1 − z2 z1 |. A subset E of X is called ρ-separated if there exists δ > 0 such that ρ(x, y) > δ for every x, y ∈ E with x = y.

2096


Lemma 2.15. Let E be a nonvoid compact ρ-separated subset of X. Then E is an interpolation set. Proof. By the assumption, m := maxξ ∈π(E) #(E ∩ π −1 (ξ )) < ∞. Since E is ρ-separated, by Lemma 2.12 for each ξ ∈ π(E) there are an open and closed neighborhood W ξ of ξ in βN and continuous maps fξ,1 , fξ,2 , . . . , fξ,m on Wξ to X such that E ∩ π −1 (Wξ ) ⊂ m i=1 fξ,i (Wξ ) and fξ,i (Wξ ) ∩ fξ,j (Wξ ) = ∅ for i = j . By the compactness of π(E), there are ξ1 , ξ2 , . . . , ξk in π(E) j −1 such that π(E) ⊂ kj =1 Wξj . Let V1 = Wξ1 and Vj = Wξj \ =1 Wξ for 2 j k. Then Vj ∩ V = ∅ for j = and W := kj =1 Wj = kj =1 Vj . For each 1 i m, we define a map fi on W by fi (λ) = fmξj ,i (λ) for λ ∈ Vj . Then fi is a continuous map on W , fj (W ) ∩ f (W ) = ∅ for j = and E ⊂ i=1 fi (W ). By Lemma 2.14, there is a simple IBP bi such that Z(bi ) = fi (W ). b . Then b is an IBP and E ⊂ Z(b). 2 Let b = m i=1 i Lemma 2.16. Let E be a nonvoid simple interpolation set in X and ϕ be an IBP in A satisfying E ⊂ Z(ϕ). Then there is a simple IBP b such that E ⊂ Z(b) and b ≺ ϕ. Proof. Let q be a simple IBP satisfying E ⊂ Z(q). Let N = N ∩ π(Z(q)). We have Z(q) ∩ π −1 (n) = {(n, an )} for every n ∈ N and supn∈N |an | < 1. Also we have E ⊂ Z(q) = {(n, an ): n ∈ N}X . Let N1 = N ∩ π(Z(ϕ)). We have π(E) ⊂ N1 βN . For each n ∈ N1 , take (n, cn ) in Z(ϕ) ∩ π −1 (n) such that |cn − an | = min |c − an |: (n, c) ∈ Z(ϕ) ∩ π −1 (n) .

(2.9)

We define the function f (n, z) on N × D by f (n, z) =

z−cn 1−cn z ,

1,

n ∈ N1 , n ∈ N \ N1 .

By Theorem 2.8(i), there exists a simple IBP b such that b|N×D = f . We have b ≺ ϕ. To show E ⊂ Z(b), let x ∈ E. There is a net {nα }α in N1 satisfying (nα , anα ) → x in X. Since ϕ(x) = 0, ϕ(nα , anα ) → 0. By (2.9), we have |cnα − anα | → 0, so (nα , cnα ) → x. Since f (nα , cnα ) = 0, we have x ∈ Z(b). 2 The following is an A-version of Theorem 2.2 in [21]. Lemma 2.17. Let E be a nonvoid interpolation set in X and U be an open and closed subset satisfying E ⊂ U ⊂ X. If ϕ is an inner function satisfying E ⊂ Z(ϕ), then there is an IBP b such that E ⊂ Z(b) ⊂ U and b ≺ ϕ. Proof. By the definition, there is an IBP q such that E ⊂ Z(q). By Theorem 2.8(ii), there are Mq simple IBPs q1 , q2 , . . . , qMq such that q = i=1 qi . Let Ei = E ∩Z(qi ). Since Z(qi )∩Z(qj ) = ∅ for i = j , we have Ei ∩ Ej = ∅ for i = j . Take open subsets Ui , 1 i Mq , such that Ei ⊂ Ui ⊂ U , Z(ϕ) ∩ Ui is open and closed in Z(ϕ) and Ui ∩ Uj = ∅ for i = j . Let ϕi be Mq ϕi ≺ ϕ. By Lemma 2.16, there are simple the subfactor of ϕ with zeros Z(ϕ) ∩ Ui . Then i=1 Mq IBPs b1 , b2 , . . . , bMq such that Ei ⊂ Z(bi ) and bi ≺ ϕi for every 1 i Mq . Let b = i=1 bi . Then b is an IBP, E ⊂ Z(b) ⊂ U and b ≺ ϕ. 2


2097

Lemma 2.18. If E is a nonvoid simple interpolation set in X, then A|E = C(E). Proof. Let f ∈ C(E) satisfy f E < 1. By the Tietze extension theorem, there is f ∈ C(X) such that f |E = f and f X = f E < 1. Let q be a simple IBP satisfying E ⊂ Z(q). Let N = N ∩ π(Z(q)). For each n ∈ N , there is a unique an ∈ D such that q(n, an ) = 0. We have supn∈N |an | < 1 and E ⊂ {(n, an ): n ∈ N }X . Then there exists cn ∈ D such that (an − cn )/(1 − cn an ) = f (n, an ) for n ∈ N and supn∈N |cn | < 1. We define the function g(n, z) on N × D by g(n, z) =

z−cn 1−cn z ,

1,

n ∈ N, n ∈ N \ N.

By Theorem 2.8(i), there exists a simple IBP b in A such that b|N×D = g. Let x ∈ E. Then there is a net {nα }α in N such that (nα , anα ) → x in X. We have b(nα , anα ) = g(nα , anα ) =

anα − cnα = f (nα , anα ). 1 − cnα anα

Hence b(x) = f (x), so b|E = f |E = f . Thus we get the assertion.

2

The following is an A-version of Theorem 3.1 in [17]. Theorem 2.19. Let E be a nonvoid compact subset of X. Then the following conditions are equivalent. (i) E is an interpolation set. (ii) There are simple interpolation sets E1 , E2 , . . . , Em such that E = m i=1 Ei and Ei ∩Ej = ∅ for i = j . (iii) A|E = C(E). (iv) E is ρ-separated. Proof. (i) ⇒ (ii) follows from Theorem 2.8(ii). (ii) ⇒ (iii). Since Ei is a simple interpolation set, there is a simple IBP bi satisfying Ei ⊂ Z(bi ) for 1 i m. Let Ui , 1 i m, be open subsets of X satisfying Ei ⊂ Ui and Ui ∩Uj = ∅ for i = j . We may assume that Z(bi ) ∩ Ui is open and closed in Z(bi ). Taking the subfactor of bi with zeros Z(bi ) ∩ Ui , we may assume that Z(bi ) ⊂ Ui for 1 i m. Let ϕj = ( m i=1 bi )/bj for 1 j m. Then |ϕj | > 0 on Ej . Since ϕj = 0 on E \ Ej , by Lemma 2.18 we have A|E ⊃

m j =1

(iii) ⇒ (iv) is trivial. (iv) ⇒ (i) follows from Lemma 2.15.

2

ϕj A = C(E). E

2098


3. Closed ideals and factorization theorems Let I be a closed ideal in A. We assume that I = {0}. Let Z(I ) =

Z(f ).

f ∈I

For each x ∈ X, we put ord(I, x) = min ord(f, x): f ∈ I . By Corollary 2.4, ord(I, x) is upper semicontinuous on X. In this section, we shall study the structure of closed ideals I in A satisfying Z(I ) ⊂ X. Lemma 3.1. Let I be a closed ideal in A. Then Z(I ) ⊂ X if and only if supx∈Z(I ) ord(I, x) < ∞. Proof. Suppose that Z(I ) ⊂ X. To show supx∈Z(I ) ord(I, x) < ∞, suppose not. Since ord(I, x) is upper semicontinuous, ord(I, x1 ) = ∞ for some x1 = (ξ1 , z1 ) ∈ Z(I ). Then ord(fξ1 , z1 ) = ∞ for every f ∈ I . Since z1 ∈ D, we have fξ1 = 0 on D. Hence π −1 (ξ1 ) ⊂ Z(I ). This contradicts that Z(I ) ⊂ X. Suppose that Z(I ) ⊂ X. Then there is a point x2 in Z(I ) ∩ ∂X, so ord(f, x2 ) = ∞ for every f ∈ I . Thus ord(I, x2 ) = ∞. 2 It is not difficult to see that X coincides with the maximal ideal space of A (see [6]). The following comes from Lemma 1.1 in [11]. I for every 1 i m. If fi = ϕi hi for some Lemma 3.2. Let I bea closed ideal in A and fi ∈ m ϕi , hi ∈ A satisfies ( m Z(h )) ∩ Z(I ) = ∅, then i i=1 i=1 ϕi ∈ I . Let I be a closed ideal in A satisfying Z(I ) ⊂ X and mI = sup ord(I, x). x∈Z(I )

By Lemma 3.1, we have mI < ∞. If ϕ is an inner function in I , then by Theorem 2.13 we have order(ϕ) mI . The following is an A-version of Theorem 2.3 in [9]. Theorem 3.3. Let I be a closed ideal in A satisfying Z(I ) ⊂ X and U be an open subset satisfying Z(I ) ⊂ U ⊂ X. Then I contains an inner function ϕ of order mI satisfying Z(ϕ) ⊂ U , and Z(I ) is a totally disconnected set. Proof. For x ∈ Z(I ), there is fx in I such that ord(fx , x) mI . By Lemma 2.10, there are an open neighborhood Ux of x in U and a factorization fx = ψx hx , where ψx is inner and hx ∈ A satisfying |hx | > 0 on Ux . Since ord(ψx , y) is upper semicontinuous in y, we may assume that ord(ψx , y) mI for every y ∈ Ux . By the compactness of Z(I ), there are x1 , x2 , . . . , xn ∈ Z(I )


2099

n n such n that Z(I ) ⊂ i=1 Uxi ⊂ U . We have ( i=1 Z(hxi )) ∩ Z(I ) = ∅. By Lemma 3.2, we get 2.7, Z(I ) is totally disconnected. i=1 ψxi ∈ I . By Corollary Let A1 = Z(I ) \ ni=2 Uxi . Then A1 is compact and A1 ⊂ Ux1 . Since Z(I ) is totally disconnected, there is an open and closed subset E1 of Z(I ) such that A1 ⊂ E1 ⊂ Ux1 . We have Z(I ) \ E1 ⊂ ni=2 Uxi . Similarly there is an open and closed subset E2 of Z(I ) \ E1 such that E2 ⊂ Ux2 and Z(I ) \ (E1 ∪ E2 ) ⊂ ni=3 Uxi . Repeating the same argument, we have open and closed subsets E1 , E2 , . . . , En of Z(I ) such that Z(I ) = ni=1 Ei , Ei ⊂ Uxi and Ei ∩ Ej = ∅ for i = j . Hence there exist open 1 , V2 , . . . , Vn such that Ei ⊂ Vi ⊂ Uxi and V i ∩ V j = ∅ subsets V for i = j . We have Z(I ) ⊂ ni=1 Vi ⊂ ni=1 Uxi . We may assume that Z(ψxi ) ∩ Vi is open and closed in Z(ψxi ) for every 1 i n. Let ϕi be the subfactor of ψxi with zeros Z(ψxi ) ∩ Vi . Then

n

Z (ψxi /ϕi )hxi ∩ Z(I ) = ∅. i=1

n Since n fxi = ϕi (ψxi /ϕi )hxi , by Lemma 3.2 n again we get ϕ := i=1 ϕi ∈ I and Z(ϕ) ⊂ i=1 Vi ⊂ U . Let y ∈ Z(ϕ). Since Z(ϕ) ⊂ i=1 Vi , there is a unique i such that y ∈ Vi . Hence ord(ϕ, y) = ord(ϕi , y) = ord(ψxi , y) mI .

2

Corollary 3.4. Let I be a closed ideal in A satisfying Z(I ) ⊂ X. Then I is algebraically generated by inner functions in I . Proof. By Theorem 3.3, there is an inner function ϕ in I . Let f ∈ I satisfy f X < 1. By Corollary 2.11, there are an inner function ψ and an invertible function h ∈ A such that ϕ − f = ψh. Then ψ ∈ I and we get the assertion. 2 The following is essentially an A-version of Theorem B. Corollary 3.5. Let I be a closed ideal in A satisfying Z(I ) ⊂ X. For each x ∈ Z(I ) and an open subset U satisfying Z(I ) ⊂ U ⊂ X, there is an inner function ϕ of order mI in I such that ord(ϕ, x) = ord(I, x) and Z(ϕ) ⊂ U . Proof. By Theorem 3.3, there is an inner function ψ of order mI in I satisfying Z(ψ) ⊂ U . We have also f ∈ I such that ord(f, x) = ord(I, x) and f X < 1. Let r = infy∈X\U |ψ(y)|. By Corollary 2.11, there are an inner function ϕ in A and an invertible function h ∈ A such that ψ − rf = ϕh. We have that ϕ ∈ I , Z(ϕ) ⊂ U and ord(I, x) ord(ϕ, x) min ord(ψ, x), ord(f, x) = ord(I, x). Hence ord(ϕ, x) = ord(I, x). For y ∈ Z(I ), we have ord(I, y) ord(ϕ, y) ord(ψ, y) mI . Hence order(ϕ) = mI .

2

2100


For a closed ideal I in A satisfying Z(I ) ⊂ X, let Ordξ (I ) =

ord(I, x): x ∈ Z(I ) ∩ π −1 (ξ ) ,

ξ ∈ βN.

Repeating the same argument as in the proof of Corollary 3.5, we have the following. Corollary 3.6. Let I be a closed ideal in A satisfying Z(I ) ⊂ X. For each ξ1 ∈ βN and an open subset U satisfying Z(I ) ⊂ U ⊂ X, there is an inner function ϕ of order mI in I such that Ordξ1 (ϕ) = Ordξ1 (I ) and Z(ϕ) ⊂ U . Proof. By Theorem 3.3, there is an inner function ψ of order mI in I satisfying Z(ψ) ⊂ U . Put Z(I ) ∩ π −1 (ξ1 ) = {x1 , x2 , . . . , xk }. There is f1 ∈ I satisfying ord(f1 , x1 ) = ord(I, x1 ) and f1 X < 1. Let r1 = infy∈X\U |ψ(y)|. By Corollary 2.11, there are an inner function ψ1 in A and an invertible function h1 ∈ A such that ψ − r1 f1 = ψ1 h1 . We have that ψ1 ∈ I , order(ψ1 ) = mI , Z(ψ1 ) ⊂ U and ord(ψ1 , x1 ) = ord(I, x1 ). Also there is an inner function ψ2 ∈ I such that order(ψ2 ) = mI , Z(ψ2 ) ⊂ U , ord(ψ2 , x1 ) = ord(I, x1 ) and ord(ψ2 , x2 ) = ord(I, x2 ). Repeating the same argument, there is an inner function ψk ∈ I such that order(ψk ) = mI , Z(ψk ) ⊂ U and ord(ψk , xi ) = ord(I, xi ) for every 1 i k. There is an open subset V of X such that Z(I ) ⊂ V , Z(ψk ) ∩ V is open and closed in Z(ψk ), and V ∩ π −1 (ξ1 ) = Z(I ) ∩ π −1 (ξ1 ). Let ϕ be the subfactor of ψk with zeros Z(ψk ) ∩ V . By Lemma 3.2, we have ϕ ∈ I . Also we have order(ϕ) = mI , Z(ϕ) ⊂ U and Ordξ1 (ϕ) =

k i=1

ord(ϕ, xi ) =

k

ord(I, xi ) = Ordξ1 (I ).

2

i=1

Let I be a closed ideal in A. Let {Wi : 1 i m} be open and closed subsets of βN and {fi : 1 i m} ⊂ I . Then we have m i=1 χπ −1 (Wi ) fi ∈ I . For ξ ∈ βN, we put Iξ = {fξ : f ∈ I }. Lemma 3.7. Let I be a closed ideal in A. If f is a function in A satisfying fξ ∈ Iξ for every ξ ∈ βN, then f ∈ I . Proof. For each ξ ∈ βN, by the assumption there is g(ξ ) ∈ I such that fξ = g(ξ )ξ on D. Let ε > 0. By Lemma 2.1, there is an open and closed neighborhood Wξ of ξ in βN such that fλ − fξ D < ε,

λ ∈ Wξ

(3.1)

and g(ξ )λ − g(ξ )ξ < ε, D

λ ∈ Wξ .

(3.2)

By the compactness of βN, there are ξ1 , ξ2 , . . . , ξt in βN such that βN = ti=1 Wξi . Let V1 = Wξ1 j −1 and Vj = Wξj \ i=1 Wξi for 2 j t. Then Vj is open and closed in βN, βN = tj =1 Vj and t Vj ∩ V = ∅ for j = . Let F = j =1 χπ −1 (Vj ) g(ξj ). Then we have F ∈ I . For λ ∈ βN, there is a unique j satisfying λ ∈ Vj ⊂ Wξj , and we have


2101

fλ − Fλ D = fλ − g(ξj )λ D fλ − fξj D + fξj − g(ξj )λ D < ε + g(ξj )ξj − g(ξj )λ D by (3.1) < 2ε

by (3.2).

Hence f − F X < 2ε. Thus we get f ∈ I .

2

Lemma 3.8. Let I be a closed ideal in A satisfying Z(I ) ⊂ X and ξ ∈ βN. Then Iξ is a closed ideal in A(D). Proof. Trivially Iξ is an ideal in A(D). By Corollary 3.6, there exists an inner function ϕ in I such that Ordξ (ϕ) = Ordξ (I ). Then we have ϕξ A(D) ⊂ Iξ ⊂ I ξ = ϕξ A(D). Hence Iξ is closed.

2

The following is an A-version of Theorem A. Theorem 3.9. Let I be a closed ideal in A satisfying Z(I ) ⊂ X. Then I coincides with the set of f in A such that ord(f, x) ord(I, x) for every x ∈ Z(I ). Proof. Let J be the set of f in A such that ord(f, x) ord(I, x) for every x ∈ Z(I ). Then J is a closed ideal in A, I ⊂ J , Z(J ) = Z(I ) and ord(I, x) = ord(J, x) for x ∈ X. Let ξ ∈ βN. We have ord(Iξ , z) = ord(Jξ , z) for z ∈ D. By Lemma 3.8, we have Iξ = Jξ . By Lemma 3.7, we get I = J. 2 The proof of Theorem A given in [11] is more complicated than the one of Theorem 3.9. Lemma 3.10. Let I be a closed ideal in A satisfying Z(I ) ⊂ X and U be an open subset satisfying Z(I ) ⊂ U ⊂ X. If B is an inner function in I , then there is an inner function ϕ of order mI in I such that ϕ ≺ B and Z(ϕ) ⊂ U . Proof. By Theorem 3.3, thereis an inner function q of order mI in I . Then there are IBPs I q1 , q2 , . . . , qmI such that q = m i=1 qi . Let x1 ∈ Z(I ) satisfy ord(I, x1 ) = mI . Then qi (x1 ) = 0 for every 1 i mI . By Lemma 2.17, there is an IBP ψ1 such that Z(q1 ) ∩ Z(I ) ⊂ Z(ψ1 ) ⊂ U

and ψ1 ≺ B.

Similarly, there is an IBP ψ2 such that Z(q2 ) ∩ Z(B/ψ1 ) ∩ Z(I ) ⊂ Z(ψ2 ) ⊂ U

and ψ2 ≺ B/ψ1 .

Repeating the same argument, for each 3 j mI there is an IBP ψj such that

2102


B Z(qj ) ∩ Z j −1 =1

ψ

∩ Z(I ) ⊂ Z(ψj ) ⊂ U

B and ψj ≺ j −1 =1

. ψ

I Let ψ = m j =1 ψj . We note that ψj (x1 ) = 0 for every 1 j mI . Then ψ ≺ B, Z(ψ) ⊂ U and ψ is an inner function of order mI . To show ψ ∈ I , let x ∈ Z(I ). If x ∈ Z(q1 ) ∩ Z(I ), then x ∈ Z(ψ1 ). When x ∈ Z(qj ) ∩ Z(I ) for some 2 j mI , we have two cases. j −1 Case 1. If x ∈ Z(B/ =1 ψ ), then x ∈ Z(ψj ). j −1 Case 2. If x ∈ / Z(B/ =1 ψ ), then ord(B, x) = ord

j −1

ψ , x = ord(ψ, x).

=1

Hence if ord(B, x) = ord(ψ, x), then ord(q, x) ord(ψ, x). Thus we get min ord(B, x), ord(q, x) ord(ψ, x). Since q and B are contained in I , ord(I, x) min ord(B, x), ord(q, x) . Hence ord(I, x) ord(ψ, x) for every x ∈ Z(I ). By Theorem 3.9, we have ψ ∈ I .

2

Theorem 3.11. Let I be a closed ideal in A satisfying Z(I ) ⊂ X, x ∈ Z(I ) and U be an open subset satisfying Z(I ) ⊂ U ⊂ X. If B is an inner function in I , then there is an inner function ϕ of order mI in I such that Z(ϕ) ⊂ U , ϕ ≺ B and ord(ϕ, x) = ord(I, x). Proof. By Theorem 3.3, Z(I ) is totally disconnected. Since ord(I, y) is upper semicontinuous in y, there are open and closed subsets E1 and E2 of Z(I ) such that Z(I ) = E1 ∪ E2 , E1 ∩ E2 = ∅, x ∈ E1 and ord(I, y) ord(I, x) for every y ∈ E1 . Let Ii = f ∈ A: ord(f, y) ord(I, y), y ∈ Ei ,

i = 1, 2.

Then Ii is a closed ideal in A such that Z(Ii ) = Ei and ord(Ii , y) = ord(I, y) for every y ∈ Ei . We have mI1 = ord(I, x) and mI2 mI . Take an open subset Ui satisfying Ei ⊂ Ui ⊂ U for i = 1, 2 and U1 ∩ U2 = ∅. Since B ∈ Ii , by Lemma 3.10 there is an inner function ϕi of order mIi in Ii such that ϕi ≺ B and Z(ϕi ) ⊂ Ui . Let ϕ = ϕ1 ϕ2 . Then Z(ϕ) ⊂ U . Since U1 ∩ U2 = ∅, we have ϕ ≺ B and order(ϕ) = max order(ϕ1 ), order(ϕ2 ) = max{mI1 , mI2 } = mI . We have that ord(ϕ, y) = ord(ϕi , y) ord(Ii , y) for y ∈ Ei . Hence ord(ϕ, y) ord(I, y) for every y ∈ Z(I ). By Theorem 3.9, we have ϕ ∈ I . We have also ord(I, x) ord(ϕ, x) = ord(ϕ1 , x) mI1 = ord(I, x). Hence ord(ϕ, x) = ord(I, x).

2


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The following is a generalized A-version of Theorem 3.5 in [19]. Corollary 3.12. Let I be a closed ideal in A satisfying Z(I ) ⊂ X and B ∈ I be an inner function. Let {ϕα }α be the set of inner functions in I such that ϕα ≺ B. Then I is generated by {ϕα }α as a closed ideal. Proof. Let J be the closed ideal in A generated by {ϕα }α . Then J ⊂ I , so ord(I, x) ord(J, x) for every x ∈ Z(I ). By Theorem 3.11, we have Z(J ) = Z(I ), and for each x ∈ Z(I ) there exists ϕβ ∈ {ϕα }α satisfying ord(ϕβ , x) = ord(I, x). Hence ord(J, x) = ord(I, x) for every x ∈ Z(I ). By Theorem 3.9, we get J = I . 2 Corollary 3.13. Let I be a closed ideal in A satisfying Z(I ) ⊂ X and ξ ∈ π(Z(I )). If B is an inner function in I , then there is an inner function ϕ of order mI in I such that ϕ ≺ B and Ordξ (ϕ) = Ordξ (I ). Proof. Let Z(I ) ∩ π −1 (ξ ) = {x1 , x2 , . . . , xn }. By Theorem 3.11, there is an inner function ϕ1 of order mI in I such that ϕ1 ≺ B and ord(ϕ1 , x1 ) = ord(I, x1 ). By Theorem 3.11 again, there is an inner function ϕ2 in I such that ϕ2 ≺ ϕ1 and ord(ϕ2 , x2 ) = ord(I, x2 ). We note that ϕ2 ≺ B, order(ϕ2 ) = mI and ord(ϕ2 , x1 ) = ord(I, x1 ). Repeating the same argument, there is an inner function ϕn of order mI in I such that ϕn ≺ B and ord(ϕn , xi ) = ord(I, xi ) for every 1 i n. We have Ordξ (ϕn ) =

n

ord(ϕn , xi ) =

i=1

n

ord(I, xi ) = Ordξ (I ).

2

i=1

Corollary 3.14. Let I be a closed ideal in A satisfying Z(I ) ⊂ X. Then Ordξ (I ) is upper semicontinuous in ξ ∈ π(Z(I )). Proof. By Theorem 3.3, I contains an inner function. Let ξ ∈ π(Z(I )). By Corollary 3.13, there is an inner function ϕ in I such that Ordξ (ϕ) = Ordξ (I ). By Corollary 2.6, there is an open neighborhood Wξ of ξ in βN such that Ordλ (ϕ) = Ordξ (ϕ) for every λ ∈ Wξ . Since ϕ ∈ I , we have Ordλ (I ) Ordλ (ϕ) for every λ ∈ π(Z(I )). Hence Ordλ (I ) Ordξ (I ) for λ ∈ Wξ . Therefore we get the assertion. 2 4. Numbering functions Let I be a closed ideal in A satisfying Z(I ) ⊂ X. By Theorem 3.3, Z(I ) is a compact and totally disconnected set, and I contains an inner function. Then maxξ ∈π(Z(I )) #(Z(I ) ∩ π −1 (ξ )) < ∞, where #(A) denotes the number of elements in A. We are interested in the bounded numbering function ord(I, x) for x ∈ Z(I ). So in this section we assume that E is a nonvoid compact and totally disconnected subset of X, and

m := max # E ∩ π −1 (ξ ) < ∞. ξ ∈π(E)

(4.1)

For each ξ ∈ π(E), let k = #(E ∩ π −1 (ξ )) and E ∩ π −1 (ξ ) = {x1 , x2 , . . . , xk }. We put xi = (ξ, zi ), zi ∈ D. Take r0 > 0 such that Dr0 (zi ) ∩ Dr0 (zj ) = ∅ for i = j . Let {Wα (ξ )}α be a set of

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fundamental open and closed neighborhood of ξ in βN. We define α1 α2 by Wα2 (ξ ) ⊂ Wα1 (ξ ). For each 0 < r r0 , there exists Wα (ξ ) such that E∩π

−1

(λ) ⊂ {λ} ×

k

Dr (zi ) ,

λ ∈ Wα (ξ ).

i=1

Take λα ∈ Wα (ξ ). For each 1 i k, the net of sets E ∩ ({λα } × Dr (zi )) converges to the point (ξ, zi ) in X as α → ∞. Let nE : E → {1, 2, . . .} be a bounded numbering function. Since sup

nE (ζ ): ζ ∈ E ∩ {λ} × Dr (zi )

λ∈Wα (ξ )

decreases as α → ∞, we can define nE (xi ) = lim

sup

α→∞ λ∈W (ξ ) α

nE (ζ ): ζ ∈ E ∩ {λ} × Dr (zi ) ,

and there exists α0 such that

nE (xi ) = sup

nE (ζ ): ζ ∈ E ∩ {λ} × Dr (zi )

λ∈Wα (ξ )

for every α α0 . More generally, for x = (ξ, z) ∈ E we can define nE (x) = lim lim

sup

r→0 α→∞ λ∈Wα (ξ )

nE (ζ ): ζ ∈ E ∩ {λ} × Dr (z) ,

and by the above observation nE (x) = sup

nE (ζ ): ζ ∈ E ∩ {λ} × Dr (z)

λ∈Wα (ξ )

nE on E. Since nE is bounded for every 0 < r r0 and α α0 . By the definition, we have nE on E, by (4.1) nE is bounded on E. Also we have the following. Lemma 4.1. nE (x) is upper semicontinuous on E. Theorem 4.2. Let I be a closed ideal in A satisfying Z(I ) ⊂ X and nZ(I ) (x) = ord(I, x) for nZ(I ) = nZ(I ) on Z(I ). every x ∈ Z(I ). Then nZ(I ) is a bounded numbering function satisfying Proof. By Lemma 3.1, nZ(I ) is bounded on Z(I ). By Theorem 3.3, Z(I ) is totally disconnected. Let x = (ξ, z) ∈ Z(I ). By Corollary 3.5, there is an inner function ϕ in I satisfying ord(ϕ, x) = ord(I, x). By Lemma 2.3, there exist r0 > 0 and α0 such that ord(ϕ, x) =

ord(ϕ, ζ ): ζ ∈ Z(ϕ) ∩ {λ} × Dr (z) ,

λ ∈ Wα (ξ )


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for every 0 < r r0 and α α0 . Then nZ(I ) (x) = ord(ϕ, x)

ord(ϕ, ζ ): ζ ∈ Z(I ) ∩ {λ} × Dr (z)

= nZ(I ) (ζ ): ζ ∈ Z(I ) ∩ {λ} × Dr (z) for every λ ∈ Wα (ξ ), 0 < r r0 and α α0 . By the definition of a tilde function, we have nZ(I ) (x). Thus we get nZ(I ) = nZ(I ) on Z(I ). 2 nZ(I ) (x) nE = nE on E. Let We shall give how to get bounded numbering functions nE satisfying nE,1 is also bounded on E. For a nE,1 be an arbitrary bounded numbering function on E. Then positive integer j with j 2, inductively we can define nE,j (x) = nE,j −1 (x),

x ∈ E.

We have nE,j −1 nE,j , so we may define nE,∞ (x) = lim nE,j (x), j →∞

x ∈ E.

By (4.1), we have nE,1 (x) m max nE,1 (y) < ∞, nE,2 (x) = y∈E

x ∈ E.

Similarly, nE,j (x) mj −1 max nE,1 (y) < ∞, y∈E

x ∈ E.

But it may happen that nE,∞ (x) = ∞ for some x ∈ E. As a special case, let NE,1 (x) = 1 for every x ∈ E. We may define NE,j and NE,∞ (x) = lim NE,j (x), j →∞

x ∈ E.

We call NE,∞ the associated numbering function of the set E. By the definition, NE,2 = NE,1 on E if and only if E is ρ-separated. But generally, the condition nE,2 = nE,1 on E does not imply that E is ρ-separated. The following is a generalized A-version of Lemma 7.4 in [19]. Lemma 4.3. Let x ∈ E and j be a positive integer with j 2. If nE,j (x) < j , then there is an open neighborhood Ux of x in E such that nE,j = nE,j −1 on Ux . Proof. Let j0 = min nE,1 (y) 1. y∈E

(4.2)

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Since j0 nE,1 (x) nE,j (x) < j , we may assume that j j0 + 1. We shall prove the assertion by induction on j with j j0 + 1. Let j = j0 + 1. Suppose that nE,j0 +1 (x) < j0 + 1. Then nE,j0 +1 (x) = j0 . By Lemma 4.1, there is an open neighborhood Ux of x in E such that nE,j0 +1 (y) nE,j0 +1 (x) for every y ∈ Ux . By (4.2), j0 nE,j0 (y) nE,j0 +1 (y) nE,j0 +1 (x) = j0 ,

y ∈ Ux .

Thus we get the assertion for the case j = j0 + 1. Let k j0 + 2. Suppose that the assertion holds for j = k − 1. We shall prove for the case j = k. Suppose that nE,k (x) < k. We have nE,k−1 (x) nE,k (x) k − 1.

(4.3)

We consider two cases separately. Case 1. Suppose that nE,k−1 (x) < k − 1. By the assumption of induction, there exists an open neighborhood Ux of x in E such that nE,k−1 = nE,k−2 on Ux . By the definition of a tilde function, we have that nE,k = nE,k−1 on Ux . Case 2. Suppose that nE,k−1 (x) = k − 1. By Lemma 4.1, Ek := {y ∈ E: nE,k (y) k} is closed. By (4.3), there exists an open neighborhood Ux of x in E such that Ek ∩ Ux = ∅, that is, Ux ⊂ {y ∈ E: nE,k (y) < k}. Since nE,k−1 nE,k k − 1 on Ux , we have Ux = y ∈ Ux : nE,k−1 (y) = k − 1 ∪ y ∈ Ux : nE,k−1 (y) < k − 1 . We have nE,k = nE,k−1 on {y ∈ Ux : nE,k−1 (y) = k − 1}. Let y ∈ Ux satisfy nE,k−1 (y) < k − 1. By the assumption of induction, there exists an open neighborhood Uy of y in E such that nE,k−1 = nE,k−2 on Uy . Hence we have nE,k = nE,k−1 on Uy . Therefore we get nE,k = nE,k−1 on Ux . 2 Corollary 4.4. Suppose that nE,∞ is a bounded function on E. Then there is a positive integer j nE,∞ = nE,∞ on E. such that nE,j +1 = nE,j , so nE,∞ = nE,j and Proof. Let j = maxx∈E nE,∞ (x). Then nE,j +1 (x) < j + 1 for every x ∈ E. By Lemma 4.3, there is an open neighborhood Ux of x in E such that nE,j +1 = nE,j on Ux . Since E is compact, nE,∞ = nE,∞ on E. 2 nE,j +1 = nE,j , so nE,∞ = nE,j and Lemma 4.5. Let nE,1 be a bounded numbering function on E. Then NE,∞ is bounded on E if and only if nE,∞ is bounded on E. Proof. Suppose that NE,∞ is bounded. Let L1 = maxx∈E nE,1 (x) and L2 = maxx∈E NE,∞ (x). Then nE,j L1 NE,j and nE,∞ L1 L2 on E. The converse follows from NE,∞ nE,∞ on E. 2 Let nE,1 be a numbering function on E. For an open and closed subset E0 of E, let nE0 ,1 = nE,1 |E0 . By the definition of a tilde function, we have nE0 ,j = nE,j and nE0 ,∞ = nE,∞ on E0 . We call this fact as the locally stable property of numbering functions.


2107

Lemma 4.6. If NE,∞ is bounded, then for each x1 ∈ E there is an open and closed neighborhood Ex1 of x1 in E such that max

ξ ∈π(Ex1 )

NEx1 ,∞ (y): y ∈ Ex1 ∩ π −1 (ξ ) = NEx1 ,∞ (x1 ).

Proof. We write x1 = (ξ1 , z1 ). By (4.1), we put E ∩ π −1 (ξ1 ) = (ξ1 , z1 ), (ξ1 , z2 ), . . . , (ξ1 , zk ) ,

zi = zj

(i = j ).

Take r1 > 0 such that Dr1 (zi ) ∩ Dr1 (zi ) = ∅ for i = j . Next, take an open and closed neighborhood W1 of ξ1 in βN such that E ∩ π −1 (W1 ) ⊂

k

W1 × Dr1 (zi ).

i=1

Let Ex1 = E ∩ (W1 × Dr1 (z1 )). We have NEx1 ,∞ = NE,∞ on Ex1 . By Corollary 4.4, Ex ,∞ (x1 ) = NEx ,∞ (x1 ), so retaking smaller W1 we have N 1 1 NEx1 ,∞ (x1 ) = max

ξ ∈W1

NEx1 ,∞ (y): y ∈ Ex1 ∩ π −1 (ξ ) .

2

The following example will help to understand the argument in Sections 5–6. Example 4.7. We give an example of a compact subset E of X such that #(E ∩ π −1 (ξ )) 2 for every ξ ∈ π(E) and NE,∞ (x) = ∞ for some x ∈ E. Let {Ni }i be a family of subsets of N such that N = ∞ i=1 Ni , #(Ni ) = ∞ for every i ∈ N and Ni ∩ Nj = ∅ for i = j . Let {a1,j }j ∈N1 be a sequence in D1/2 with a1,j = 0 for every j ∈ N1 satisfying a1,j → 0 as j → ∞ in N1 . Let X X E1 = (j, 0): j ∈ N1 ∪ (j, a1,j ): j ∈ N1 ⊂ N1 βN × D. Then X E1 = (j, 0): j ∈ N1 ∪ (j, a1,j ): j ∈ N1 and #(E1 ∩ π −1 (ξ )) 2 for every ξ ∈ π(E1 ). Let x = (ξ, z) ∈ E1 . If ξ ∈ N1 , then NE1 ,2 (x) = 1 and #(E1 ∩ π −1 (ξ )) = 2, and if ξ ∈ N1 βN \ N1 , then NE1 ,2 (x) = 2 and #(E1 ∩ π −1 (ξ )) = 1. ∞ Let {N2,j }j be a family of subsets of N2 such that N2 = j =1 N2,j , #(N2,j ) = ∞ for every j ∈ N and N2,j ∩ N2, = ∅ for j = . For each j ∈ N, there are homeomorphisms η1,j : N1 βN → N2,j βN and τ1,j : N1 βN × D → N2,j βN × D

2108


such that τ1,j (ξ, z) = (η1,j (ξ ), z) for (ξ, z) ∈ N1 βN × D. Let E2,j = τ1,j (E1 ). Take ξ2,j ∈ N2,j βN \ N2,j . Then NE2,j ,2 (ξ2,j , 0) = 2. Let {a2,j }j be a sequence in D1/2 with a2,j = 0 for every j ∈ N satisfying a2,j → 0 as j → ∞. Let E2 =

∞

X

E2,j ∪ (ξ2,j , a2,j ): j ∈ N .

j =1

Then E2 is a compact subset of X and #(E2 ∩ π −1 (ξ )) 2 for every ξ ∈ π(E2 ). Let ξ2 be a (ξ2 )) = 1 and NE2 ,3 (ξ2 , 0) = 3. cluster point of {ξ2,j }j in βN. Then (ξ2 , 0) ∈ E2 , #(E2 ∩ π −1 Let {N3,j }j be a family of subsets of N3 such that N3 = ∞ j =1 N3,j , #(N3,j ) = ∞ for every j ∈ N and N3,j ∩ N3, = ∅ for j = . For each j ∈ N, there are homeomorphisms η2,j : N2 βN → N3,j βN and τ2,j : N2 βN × D → N3,j βN × D such that τ2,j (ξ, z) = (η2,j (ξ ), z) for (ξ, z) ∈ N2 βN × D. Let E3,j = τ2,j (E2 ). Take ξ3,j ∈ N3,j βN \ N3,j . Then NE3,j ,3 (ξ3,j , 0) = 3. Let {a3,j }j be a sequence in D1/2 with a3,j = 0 for every j ∈ N satisfying a3,j → 0 as j → ∞. Let E3 =

∞

X

E3,j ∪ (ξ3,j , a3,j ): j ∈ N .

j =1

Then E3 is a compact subset of X and #(E3 ∩ π −1 (ξ )) 2 for every ξ ∈ π(E3 ). Let ξ3 be a cluster point of {ξ3,j }j in βN. Then (ξ3 , 0) ∈ E3 , #(E3 ∩ π −1 (ξ3 )) = 1 and NE3 ,4 ((ξ3 , 0)) = 4. Repeating the same argument, we get a sequence {En }n of compact disjoint subsets of X such that En ⊂ Nn βN × D, #(En ∩ π −1 (ξ )) 2 for every ξ ∈ π(En ) and NEn ,n+1 (ξn , 0) = n + 1 for X −1 some ξn ∈ Nn βN \ Nn . Let E = ∞ n=1 En . Then #(E ∩ π (ξ )) 2 for every ξ ∈ π(E) and NE,∞ (x) = ∞ for some x ∈ E. 5. Associated primary ideals For a nonvoid compact subset E of X, let I (E) = f ∈ A: f (x) = 0, x ∈ E . Then I (E) is a closed ideal in A and E ⊂ Z(I (E)). We call I (E) the associated primary ideal of E. In this section, we study E for which Z(I (E)) = E, and characterize ord(I (E), x) for x ∈ E. For a closed ideal I in A satisfying Z(I ) ⊂ X, recall that ord(I, x): x ∈ Z(I ) ∩ π −1 (ξ ) , ξ ∈ βN Ordξ (I ) = and mI = max ord(I, x) < ∞. x∈Z(I )


2109

Let MI =

max

ξ ∈π(Z(I ))

Ordξ (I ).

By Theorem 3.3, I contains an inner function ϕ, Z(I ) is totally disconnected, ord(I, x) ord(ϕ, x) for x ∈ Z(I ) and by Corollary 2.5 we have MI

max

ξ ∈π(Z(ϕ))

Ordξ (ϕ) < ∞.

Lemma 5.1. Let I be a closed ideal in A and U be an open subset satisfying Z(I ) ⊂ U ⊂ X, and W be an open and closed subset of βN such that π(Z(I )) ⊂ W . Then there is an inner function ϕ of order mI in I such that Z(ϕ) ⊂ U ∩ π −1 (W ) and maxξ ∈π(Z(ϕ)) Ordξ (ϕ) = MI . Proof. By Corollary 3.6, for each ξ ∈ π(Z(I )) there is an inner function ψ(ξ ) of order mI in I such that Ordξ (ψ(ξ ) ) = Ordξ (I ) and Z(ψ(ξ ) ) ⊂ U ∩ π −1 (W ). By Corollary 2.6, there is an open and closed subset Wξ of βN such that ξ ∈ Wξ ⊂ W and Ordλ (ψ(ξ ) ) = Ordξ (ψ(ξ ) ) = Ordξ (I ),

λ ∈ Wξ .

Then there are ξ1 , ξ2 , . . . , ξk ∈ π(Z(I )) such that π(Z(I )) ⊂ ki=1 Wξi ⊂ W . Let V1 = Wξ1 and i−1 Vi = Wξi \ j =1 Wξj for 2 i k. Then {Vi : 1 i k} is a set of mutually disjoint open and closed subsets of βN and ki=1 Vi = ki=1 Wξi ⊂ W . Let V0 = βN \ ki=1 Vi . Then χπ −1 (V0 ) = 0 on Z(I ), and by Theorem 3.9 we have χπ −1 (V0 ) ∈ I . Hence ϕ := χπ −1 (V0 ) +

k

χπ −1 (Vi ) ψ(ξi ) ∈ I,

i=1

ϕ is an inner function and Z(ϕ) ⊂ U ∩ π −1 (W ). Since order(ψ(ξi ) ) = mI , we have order(ϕ) mI . Since ϕ ∈ I , again by Theorem 3.9 we have order(ϕ) mI . Hence order(ϕ) = mI . We have also MI

max

ξ ∈π(Z(ϕ))

Thus we get the assertion.

Ordξ (ϕ) = max Ordξi (ψ(ξi ) ) = max Ordξi (I ) MI . 1ik

1ik

2

If Z(I (E)) ⊂ X, then Z(I (E)) is totally disconnected, so E is totally disconnected. To study E satisfying Z(I (E)) ⊂ X, we may assume that E is totally disconnected. By Theorem 3.3, there is an inner function ϕ satisfying E ⊂ Z(ϕ). So also we may assume that

max # E ∩ π −1 (ξ ) < ∞.

ξ ∈π(E)

Lemma 5.2. Let E be a nonvoid compact and totally disconnected subset of X. Then Z(I (E)) ⊂ X if and only if Z(I (E)) = E.

2110


Proof. Suppose that Z(I (E)) ⊂ X. By Theorem 3.3, there is an inner function ϕ in I (E). Then E ⊂ Z(ϕ). Let U be an open subset of X such that E ⊂ U and Z(ϕ) ∩ U is open and closed in Z(ϕ). Let ψ be the subfactor of ϕ with zeros Z(ϕ) ∩ U . Then E ⊂ Z(ψ) ⊂ U . This shows that Z(I (E)) = E. 2 Lemma 5.3. Let E be a nonvoid compact and totally disconnected subset of X satisfying Z(I (E)) = E and E1 be a nonvoid open and closed subset of E. Then Z(I (E1 )) = E1 and ord(I (E1 ), x) = ord(I (E), x) for every x ∈ E1 . Proof. We have I (E) ⊂ I (E1 ). Then Z(I (E1 )) ⊂ Z(I (E)) = E ⊂ X. By Lemma 5.2, we have Z(I (E1 )) = E1 . We have also ord(I (E1 ), x) ord(I (E), x) for x ∈ E1 . Suppose that ord(I (E1 ), x1 ) < ord(I (E), x1 ) for some x1 ∈ E1 . Take open subsets U1 , U2 of X satisfying E1 ⊂ U1 , E \ E1 ⊂ U2 and U1 ∩ U2 = ∅. By Corollary 3.5, there is an inner function ϕ1 in I (E1 ) satisfying ord(ϕ1 , x1 ) = ord(I (E1 ), x1 ) and Z(ϕ1 ) ⊂ U1 . We have also Z(I (E \ E1 )) = E \ E1 . Let ϕ2 ∈ I (E \ E1 ) be an inner function satisfying Z(ϕ2 ) ⊂ U2 . Then ϕ1 ϕ2 ∈ I (E) and

ord(ϕ1 ϕ2 , x1 ) = ord(ϕ1 , x1 ) = ord I (E1 ), x1 < ord I (E), x1 . This is a contradiction.

2

We call the above fact the locally stable property of ord(I (E), x), x ∈ E. The following answers the A-version of Question 1 and is a generalized A-version of Theorem 7.6 in [19]. Theorem 5.4. Let E be a nonvoid compact and totally disconnected subset of X, and maxξ ∈π(E) #(E ∩ π −1 (ξ )) < ∞. Then the following conditions are equivalent. (i) Z(I (E)) = E. (ii) NE,∞ is bounded on E. (iii) ord(I (E), x) is bounded in x ∈ E. In this case, we have that ord(I (E), x) = NE,∞ (x) for every x ∈ E. Proof. (i) ⇒ (ii). Suppose that (i) holds. Since NE,1 (x) ord(I (E), x) for every x ∈ E, by Theorem 4.2 we have NE,∞ (x) ord(I (E), x) for every x ∈ E. By Lemma 3.1, we get (ii). (iii) ⇒ (i). Suppose that ord(I (E), x) < ∞ for every x ∈ E. For each x ∈ E, there exists fx ∈ I (E) such that ord(fx , x) < ∞. By Lemma 2.10, there are an open neighborhood Ux of x satisfies |hx | > 0 on Ux . By in X and a factorization fx = ϕx hx , where ϕx is inner and hx ∈ A the compactness of E, there are x1 , x2 , . . . , x ∈ E such that E ⊂ i=1 Uxi . Since fxi ∈ I (E), we have ϕxi = 0 on E ∩ Uxi for 1 i . Hence i=1 ϕxi = 0 on E, so i=1 ϕxi ∈ I (E). We have

Z I (E) ⊂ Z ϕxi ⊂ X. i=1

By Lemma 5.2, we get (i).


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(ii) ⇒ (iii). Suppose that (ii) holds. We shall prove that

ord I (E), x = NE,∞ (x),

x ∈ E.

(5.1)

As a result, we get (iii). For each positive integer k, let Ωk = E: max NE,∞ (x) k . x∈E

(5.2)

We have Ωk ⊂ Ωk+1 . We shall prove (5.1) by induction on k in Ωk . Let E ∈ Ω1 . Then 1 = NE,1 = NE,2 on E, so E is ρ-separated. By Theorem 2.19, there is an IBP b such that E ⊂ Z(b). Hence Z(I (E)) ⊂ Z(b) ⊂ X. Then by Lemma 5.2, we have Z(I (E)) = E. We have also ord(I (E), x) = 1 = NE,∞ (x) for every x ∈ E. Let k 2. Suppose that (5.1) holds for every E ∈ Ωk−1 . Let E ∈ Ωk . If E ∈ Ωk−1 , then by the assumption of induction we have (5.1). So by (5.2), we may assume that maxx∈E NE,∞ (x) = k. We have NE,∞ (x) ord(I (E), x) for x ∈ E. So we need to prove that ord(I (E), x) NE,∞ (x) for every x ∈ E. Let E∞ = x ∈ E: NE,∞ (x) = k . / E∞ , then NE,∞ (x1 ) k − 1. By the locally stable properties of ord(I (E), x) Let x1 ∈ E. If x1 ∈ of E such that x1 ∈ E, ord(I (E), x) = and NE,∞ , there is an open and closed subset E Hence E ∈ Ωk−1 . By the asord(I (E), x) and NE,∞ (x) = NE,∞ (x) k − 1 for every x ∈ E. x1 ) = NE,∞ sumption of induction, we have ord(I (E), (x1 ). Hence ord(I (E), x1 ) = NE,∞ (x1 ). We assume that x1 ∈ E∞ . If ord(I (E ), x1 ) = NE ,∞ (x1 ) for some open and closed neighborhood E of x1 in E, then by the locally stable properties of ord(I (E), x) and NE,∞ , we have that ord(I (E), x1 ) = NE,∞ (x1 ). So by Lemma 4.6, we may assume that max

ξ ∈π(E)

NE,∞ (y): y ∈ E ∩ π −1 (ξ ) = NE,∞ (x1 ) = k.

(5.3)

# E ∩ π −1 π(x) = 1,

(5.4)

This shows that x ∈ E∞ .

E,∞ = NE,∞ on E. So by Lemma 4.1, E∞ is a closed set. By (5.4) By Corollary 4.4, we have N and Lemma 2.14, E∞ is a simple interpolation set, so there is a simple IBP q such that E∞ ⊂ Z(q). Since k 2, we have E ⊂ Z(q). By (5.4) again, we have π(E∞ ) ∩ π(E \ Z(q)) = ∅. Since E \ Z(q) is an Fσ -set, so is π(E \ Z(q)). Hence there is a sequence {Wj }j of open and closed subsets of βN such that Wj ∩ W = ∅,

π E \ Z(q) ⊂

j = , ∞ j =1

and

Wj

(5.5) (5.6)

2112


π(E∞ ) ∩

∞

Wj = ∅.

(5.7)

j =1

Let Ej = E ∩ π −1 (Wj ). Then Ej is an open and closed subset of E. By the locally stable property of NE,∞ , we have NEj ,∞ (x) = NE,∞ (x) for x ∈ Ej . By (5.7), maxx∈Ej NEj ,∞ (x) < k, so we have Ej ∈ Ωk−1 . By the assumption of induction, we have ord(I (Ej ), x) = NEj ,∞ (x) for every x ∈ Ej . Since (iii) ⇒ (i) holds, we have Z(I (Ej )) = Ej ⊂ π −1 (Wj ). Hence by (5.3), kj := max ξ ∈Wj

ord I (Ej ), y : y ∈ Ej ∩ π −1 (ξ ) k.

Therefore kj = maxξ ∈π(Ej ) Ordξ (I (Ej )) k. By Lemma 5.1, there is an inner function ϕj ∈ I (Ej ) such that Z(ϕj ) ⊂ π −1 (Wj ) and maxξ ∈π(Z(ϕj )) Ordξ (ϕj ) = kj k. By Theorem 2.8(ii), kj there are simple IBPs ψj,1 , ψj,2 , . . . , ψj,kj such that ϕj = i=1 ψj,i . Let Ej,i =

Ej ∩ Z(ψj,i ), ∅,

1 i kj , kj + 1 i k.

Since Ej ⊂ Z(ϕj ), we have Ej = ki=1 Ej,i . Since Z(ϕj ) ⊂ π −1 (Wj ), we have Ej,i ⊂ π −1 (Wj ). Since ψj,i is a simple IBP, Ej,i is a simple interpolation set. We have E= E

∞

∞

∪

Ej

j =1

Ej .

j =1

∞ By (5.6), we have E \ ∞ j =1 Ej ⊂ Z(q), and since q is a simple IBP, E \ j =1 Ej is a simple interpolation set. By (5.5), ∞ ∞

X

Ej ⊂ E

∞

n=1 j =n

Ej .

(5.8)

j =1

Let Γi = E

∞

∪

Ej

j =1

∞

Ej,i ,

1 i k.

j =1

By (5.8), Γi is closed. Since π E

∞ j =1

Ej

∩π

∞

Ej,i = ∅,

j =1

interpoby (5.5) we have #(Γi ∩ π −1 (ξ )) 1 for every ξ ∈ βN. By Lemma 2.14, Γi is a simple lation set, so there is a simple IBP bi such that Γi ⊂ Z(bi ) for 1 i k. Since E = ki=1 Γi , we have E ⊂ Z( ki=1 bi ). Hence ord(I (E), x) k for every x ∈ E. Since x1 ∈ E∞ , we have


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ord(I (E), x1 ) k = NE,∞ (x1 ). Thus we get ord(I (E), x1 ) = NE,∞ (x1 ). As a result, we have ord(I (E), x) = NE,∞ (x) for every x ∈ E. 2 6. Zero’s order of closed ideals Let I be a closed ideal in A satisfying E := Z(I ) ⊂ X and nE (x) = ord(I, x) for x ∈ E. By Theorem 4.2, nE is a bounded numbering function satisfying nE = nE on E. In this section, we prove the converse of the above assertion. Let {ϕj }j be a sequence of inner functions in A and {Wj }j be a sequence of mutually disjoint open and closed subsets of βN satisfying Z(ϕj ) ⊂ π −1 (Wj ). Suppose that ϕj = 1 on π −1 (βN \ Wj ), ∞

X

Z(ϕj ) ⊂ X

j =1

and supj maxξ ∈Wj Ordξ (ϕj ) < ∞. Then we may define the function ψ on N × D by ψ(n, z) = ∞ j =1 ϕj (n, z) for every (n, z) ∈ N × D. By Theorem 2.8(i), there is an inner function ψ in A | satisfying ψ N×D = ψ . We write = ψ

∞

ϕj .

j =1

The infinite product ∞ j =1 ϕj is the usual infinite product on N × D and Wj , but it is not on X. ∞ −1 Any way j =1 ϕj is an inner function and ∞ j =1 ϕj = ϕj on π (Wj ). Let B be an inner function in A and {Vj }j be a sequence of mutually disjoint open and closed subsets of βN. For each j 1, if Z(B) ∩ π −1 (Vj ) = ∅, put Bj = 1, and if Z(B) ∩ π −1 (Vj ) = ∅, −1 let Bj be the subfactor of B with zeros Z(B) ∩ π −1 (Vj ). have Bj = 1 on We ∞π (βN \ Vj ). ∞ By the last paragraph, we may define the inner function j =1 Bj . We have j =1 Bj ≺ B and ∞ −1 |B/ ∞ j =1 Bj | > 0 on j =1 π (Vj ). Let {ψj }j be a sequence of inner functions in A such that ψj ≺ Bj and ψj =1 on π −1 (βN ∞\ Vj ) for every j 1. We may define the inner func∞ tion ∞ ψ . We have ψ ≺ j =1 j j =1 j j =1 Bj ≺ B. The following theorem answers the A-version of Question 2 in the introduction. Theorem 6.1. Let E be a nonvoid compact and totally disconnected subset of X, and maxξ ∈π(E) #(E ∩ π −1 (ξ )) < ∞. Let nE be a numbering function on E. If nE is bounded and nE = nE on E, then there is a closed ideal I in A such that Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E. Proof. We divide the proof into three steps. Step 1. For each positive integer k, let Ωk = (E, nE ): nE is bounded, nE = nE , max nE (x) k . x∈E

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We have Ωk ⊂ Ωk+1 . For each (E, nE ) ∈ Ωk , we shall prove the existence of a closed ideal I in A such that Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E. We shall prove the assertion by induction on k in Ωk . Suppose that k = 1 and (E, nE ) ∈ Ω1 . Then nE (x) = 1 for every x ∈ E. Since nE = 1, we have NE,∞ = 1 on E. By Theorem 5.4,

ord I (E), x = NE,∞ (x) = 1 = nE (x),

x ∈ E.

Step 2. Let k be a positive integer with k 2. Suppose that for every (E, nE ) ∈ Ωk−1 , there is a closed ideal I in A such that Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E. Let (E, nE ) ∈ Ωk . If (E, nE ) ∈ Ωk−1 , by the assumption of induction we have the assertion. So we assume that k = maxx∈E nE (x). Let I = f ∈ A: ord(f, x) nE (x), x ∈ E . Then I is a closed ideal of A satisfying Z(I ) ⊃ E and ord(I, x) nE (x) for x ∈ E. Since NE,∞ nE on E, NE,∞ is bounded on E. By Theorem 5.4, we have Z(I (E)) = E. We have f k ∈ I for every f ∈ I (E). Hence Z(I ) = E. To show that ord(I, x) = nE (x) for x ∈ E, for each x1 ∈ E it is sufficient to show the existence of inner function ϕx1 ∈ I such that ord(ϕx1 , x1 ) = nE (x1 ). We put x1 = (ξ1 , z1 ) ∈ E. Claim. If there are an inner function ψ and an open and closed neighborhood Ux1 of x1 in E such that ord(ψ, x1 ) = nE (x1 ) and ord(ψ, y) nE (y) for every y ∈ Ux1 , then there exists an inner function ϕx1 ∈ I such that ord(ϕx1 , x1 ) = nE (x1 ). Proof. By the locally stable property, NE\Ux1 ,∞ = NE,∞ on E \ Ux1 . By Theorem 5.4, we have Z(I (E \ Ux1 )) = E \ Ux1 . By Theorem 3.3, there is an inner function q ∈ I (E \ Ux1 ) satisfying q(x1 ) = 0. Since k = maxx∈E nE (x), we have ϕx1 := ψq k ∈ I and ord(ϕx1 , x1 ) = ord(ψ, x1 ) = nE (x1 ). 2 We continue the proof of Theorem 6.1. Since maxλ∈π(E) #(E ∩ π −1 (λ)) < ∞, we put E ∩ π −1 (ξ1 ) = {x1 , x2 , . . . , xt },

xi = xj

(i = j ).

We write xi = (ξ1 , zi ). Then there exists r > 0 such that Dr (zi ) ∩ Dr (zj ) = ∅ for i = j , and there is an open and closed neighborhood Wξ1 of ξ1 in βN such that E ∩ π −1 (Wξ1 ) =

t

E ∩ Wξ1 × Dr (zi )

i=1

and E ∩ (Wξ1 × Dr (zi )) is open and closed in E for 1 i t. Let

E1 = E ∩ Wξ1 × Dr (z1 ) . Then x1 ∈ E1 . Let nE1 = nE |E1 . By the locally stable property of a numbering function, we have nE1 (x1 ), taking smaller Wξ1 we may nE1 = nE1 on E1 and (E1 , nE1 ) ∈ Ωk . By the definition of assume that


nE1 (x1 )

nE1 (y): y ∈ E1 ∩ π −1 (λ) ,

λ ∈ Wξ1 .

2115

(6.1)

If nE1 (x1 ) k − 1, then (E1 , nE1 ) ∈ Ωk−1 , so by the assumption on induction there is a closed ideal J in A such that Z(J ) = E1 and ord(J, y) = nE1 (y) for every y ∈ E1 . By Corollary 3.5, there is an inner function ψ in J such that ord(ψ, x1 ) = nE1 (x1 ) = nE (x1 ). We have also ord(ψ, y) ord(J, y) = nE1 (y) = nE (y) for every y ∈ E1 . By Claim, there is an inner function ϕx1 ∈ I such that ord(ϕx1 , x1 ) = nE (x1 ). Step 3. Next, suppose that nE1 (x1 ) = k. Then (E1 , nE1 ) ∈ Ωk . Let E∞ = x ∈ E1 : nE1 (x) = k . By Lemma 4.1 and the definition of a tilde function, E∞ is a closed ρ-separated set, so by Theorem 2.19, E∞ is an interpolation set. By (6.1), #(E∞ ∩ π −1 (λ)) = 1 for every λ ∈ π(E∞ ). By Lemma 2.14, there is a simple IBP q such that E∞ ⊂ Z(q). Since E1 \ Z(q) is an Fσ -set, so is π(E1 \ Z(q)). By (6.1) again,

π(E∞ ) ∩ π E1 \ Z(q) = ∅. Then there is a sequence of open and closed subsets {Wj }j of βN such that Wj ∩ Wi = ∅, π(E∞ ) ∩

∞

j = i,

(6.2)

Wj = ∅,

(6.3)

j =1 ∞

π E1 \ Z(q) ⊂ Wj

(6.4)

j =1

and q(x) → 0,

max

x∈E1 ∩π −1 (Wj )

j → ∞.

(6.5)

For each j 1, let E1,j = E1 ∩ π −1 (Wj ). By (6.2), {E1,j }j is a set of mutually disjoint open and closed subsets of E1 , and E1 = E1

∞ j =1

E1,j

∪

∞

E1,j .

j =1

By (6.4), we have

E1

∞ j =1

E1,j ⊂ Z(q).

(6.6)

2116


By (6.3), E∞ ∩ E1,j = ∅. Let nE1,j = nE1 |E1,j . Then nE1,j = nE1,j = nE1 on E1,j and maxy∈E1,j nE1,j (y) k − 1, so (E1,j , nE1,j ) ∈ Ωk−1 . By the assumption of induction, there is a closed ideal Ij in A such that Z(Ij ) = E1,j ⊂ π −1 (Wj ) and ord(Ij , y) = nE1,j (y) = nE1 (y),

y ∈ E1,j .

(6.7)

Recall that MI j =

max

λ∈π(Z(Ij ))

ord(Ij , y): y ∈ Z(Ij ) ∩ π −1 (λ) .

By (6.1) and (6.7), MIj = max λ∈Wj

nE1 (y): y ∈ Z(Ij ) ∩ π −1 (λ) nE1 (x1 ) = k.

Let {Uj }j be a sequence of open subsets of X such that E1,j ⊂ Uj ⊂ U j ⊂ X and Uj ⊂ π −1 (Wj ) for every j . By (6.5), we may further assume that max q(x) → 0,

x∈Uj

j → ∞.

(6.8)

By Lemma 5.1, there is an inner function ψj in Ij such that Z(ψj ) ⊂ Uj ⊂ π −1 (Wj ),

max

λ∈π(Z(ψj ))

Ordλ (ψj ) = MIj k,

(6.9)

and ψj = 1 on π −1 (βN \ Wj ). By Theorem 2.8(ii), there are simple IBPs bj,1 , bj,1 , . . . , bj,MIj MI such that ψj = s=1j bj,s and bj,s = 1 on π −1 (βN \ Wj ) for 1 s MIj . Since MIj k, put bj,s = 1 for MIj + 1 s k, then ψj =

k

(6.10)

bj,s .

s=1

By (6.2), (6.8) and (6.9), we may define fixed 1 s k, we may also define

∞

j =1 ψj ,

qs =

∞

and

∞

j =1 ψj

is an inner function. For each

bj,s ,

(6.11)

j =1

and qs is a simple IBP. By (6.6) and (6.8), Γs := E1

∞

E1,j

∪ E1 ∩ Z(qs )

j =1

is a simple interpolation set. So there is a simple IBP ϕs such that Γs ⊂ Z(ϕs ). Let

(6.12)


ψx1 =

k

2117

(6.13)

ϕs .

s=1

Then ψx1 is an inner function. By (6.12), ord(ψx1 , x) = k,

x ∈ E1

∞

(6.14)

E1,j .

j =1

Since x1 ∈ E∞ , we have ord(ψx1 , x1 ) = k = nE1 (x1 ) = nE (x1 ). Let y ∈ E1 ∩ π

−1

∞

Wj

=

j =1

∞

E1,j .

j =1

By (6.2), there exists a unique j 1 satisfying y ∈ E1,j . Since Z(Ij ) = E1,j and ψj ∈ Ij , we have nE1 (y) = ord(Ij , y)

by (6.7)

ord(ψj , y) k = ord bj,s , y by (6.10) ord

s=1 k

qs , y

by (6.11)

s=1

ord(ψx1 , y)

by (6.12) and (6.13).

Thus we get ord(ψx1 , y) nE1 (y) = nE (y),

y∈

∞

E1,j .

j =1

By (6.14), ord(ψx1 , y) = k nE1 (y) = nE (y),

y ∈ E1

∞

E1,j .

j =1

By Claim, there is an inner function ϕx1 ∈ I such that ord(ϕx1 , x1 ) = nE (x1 ). This completes the proof. 2 Combining Theorem 6.1 with Theorem 4.2, we have the following.

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Corollary 6.2. Let E be a nonvoid compact and totally disconnected subset of X, and maxξ ∈π(E) #(E ∩ π −1 (ξ )) < ∞. Let nE be a numbering function on E. Then nE is bounded and nE = nE on E if and only if there is a closed ideal I in A such that Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E. Let I1 , I2 be closed ideals in A satisfying Z(Ii ) ⊂ X for i = 1, 2. Let I3 = I1 + I2 and E3 = Z(I3 ). Then I3 is a closed ideal and E3 = Z(I1 ) ∩ Z(I2 ). Then for x ∈ E3 , we have ord(I3 , x) = min ord(f1 + f2 , x): f1 ∈ I1 , f2 ∈ I2 = min min ord(f1 , x), ord(f2 , x) : f1 ∈ I1 , f2 ∈ I2 = min ord(I1 , x), ord(I2 , x) . Let I4 = I1 ∩ I2 and E4 = Z(I4 ). Then I4 is a closed ideal and E4 = Z(I1 ) ∪ Z(I2 ). Since I4 ⊂ Ii , we have ord(Ii , x) ord(I4 , x) for x ∈ E4 , i = 1, 2. Hence max ord(I1 , x), ord(I2 , x) ord(I4 , x),

x ∈ E4 .

Proposition 6.3. If nE4 is a numbering function such that max ord(I1 , x), ord(I2 , x) nE4 (x) ord(I4 , x),

x ∈ E4

and nE4 = nE4 on E4 , then nE4 (x) = ord(I4 , x) for every x ∈ E4 . Proof. By Theorem 6.1, there is a closed ideal I in A such that Z(I ) = E4 and ord(I, x) = nE4 (x) for x ∈ E4 . Since nE4 (x) ord(I4 , x) for x ∈ E4 , by Theorem 3.9 we have I4 ⊂ I . Since ord(Ii , x) nE4 (x), we have also I ⊂ I1 ∩ I2 = I4 . Thus we get I = I4 , so nE4 (x) = ord(I4 , x) for x ∈ E4 . 2 It is not difficult to give an example of I1 and I2 such that max ord(I1 , x), ord(I2 , x) = ord(I4 , x),

x ∈ E4 .

7. Tensor products of closed ideals Let I1 , I2 be closed ideals in A satisfying Z(Ii ) ⊂ X for i = 1, 2. Let I1 ⊗ I2 =

n 2

fi,j : fi,j ∈ Ii , i = 1, 2, n 1 .

j =1 i=1

Then I1 ⊗ I2 is an ideal in A (may not be closed) and is called the tensor product of I1 and I2 . We denote by I1 ⊗ I2 the closure of I1 ⊗ I2 in A. Then I1 ⊗ I2 is a closed ideal. We call I1 ⊗ I2 the closed tensor product. It is not difficult to see that Z(I1 ⊗ I2 ) = Z(I1 ⊗ I2 ) = Z(I1 ) ∪ Z(I2 )


2119

and ord(I1 ⊗ I2 , x) = ord(I1 ⊗ I2 , x) = ord(I1 , x) + ord(I2 , x) for every x ∈ Z(I1 ) ∪ Z(I2 ). The purpose of this section is to prove that I1 ⊗ I2 is a closed ideal in A. Let Ei = Z(Ii ),

i = 1, 2 and E = E1 ∪ E2 .

Then Z(I1 ⊗ I2 ) = E ⊂ X. We say that I1 ⊗ I2 has the factorization property if for every inner function ϕ in I1 ⊗ I2 , there are inner functions ϕ1 , ϕ2 such that ϕi ∈ Ii for i = 1, 2 and ϕ1 ϕ2 ≺ ϕ. For each x ∈ E, we may consider localizations of I1 ⊗ I2 . Since E is totally disconnected, there are many open and closed neighborhoods Ex of x in E. Let Ii,Ex = f ∈ A: ord(f, y) ord(Ii , y), y ∈ Ex ,

i = 1, 2.

Then Ii,Ex is a closed ideal, Ii ⊂ Ii,Ex , Z(Ii,Ex ) = Z(Ii ) ∩ Ex and ord(Ii,Ex , y) = ord(Ii , y),

y ∈ Z(Ii ) ∩ Ex .

When Z(Ii ) ∩ Ex = ∅, we note that Ii,Ex = A. Let IEx = I1,Ex ⊗ I2,Ex . We call IEx a localization of the closed tensor product I1 ⊗ I2 at x ∈ E. We have ord(IEx , y) = ord(I1 ⊗ I2 , y),

y ∈ Ex .

If IEx = I1,Ex ⊗ I2,Ex has the factorization property for some Ex , we say that I1 ⊗ I2 has the local factorization property at x ∈ E. Lemma 7.1. If I1 ⊗ I2 has the local factorization property at every point in E = Z(I1 ) ∪ Z(I2 ), then I1 ⊗ I2 has the factorization property. Proof. Let ϕ ∈ I1 ⊗ I2 be an inner function. By the assumption, there are points x1 , x2 , . . . , xn in E and their open and closed neighborhoods Ex1 , Ex2 , . . . , Exn of E such that E = nj=1 Exj and IExj = I1,Exj ⊗ I2,Exj has the factorization property for every 1 j n. Since ϕ ∈ IExj , there are inner functions ψ1,j , ψ2,j such that ψi,j ∈ Ii,Exj for i = 1, 2 and ψ1,j ψ2,j ≺ ϕ. Let j −1 n. Then {Vj : 1 j n} is a set of mutually V1 = Ex1 and Vj = Exj \ =1 Ex for 2 j disjoint open and closed subsets of E and E = nj=1 Vj . Take open subsets Uj of X, 1 j n, such that Vj ⊂ Uj , Uj ∩ U = ∅ for j = and Z(ψi,j ) ∩ Uj is open and closed in Z(ψi,j ) for i = 1, 2. For each i = 1, 2 and 1 j n, let ϕi,j be the subfactor of ψi,j with zeros Z(ψi,j )∩Uj . Then ord(ϕi,j , y) = ord(ψi,j , y) for y ∈ Z(ψ i,j ) ∩ Uj . We have ϕ1,j ϕ2,j ≺ ψ1,j ψ2,j ≺ ϕ for 1 j n. Since Z(ϕ1,j ϕ2,j ) ⊂ Uj , we have nj=1 ϕ1,j ϕ2,j ≺ ϕ. Let ϕi = nj=1 ϕi,j for i = 1, 2. Then ϕ1 ϕ2 = nj=1 ϕ1,j ϕ2,j ≺ ϕ. Since ψi,j ∈ Ii,Exj , for every y ∈ Vj we have

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ord(Ii , y) = ord(Ii,Exj , y) ord(ψi,j , y) = ord(ϕi,j , y) = ord(ϕi , y). Then ord(Ii , y) ord(ϕi , y) for every y ∈ E. By Theorem 3.9, we have ϕi ∈ Ii for i = 1, 2. Thus we get the assertion. 2 Let J be a closed ideal in A satisfying Z(J ) ⊂ X. Recall that mJ = max ord(J, x) x∈Z(J )

and MJ =

max

ξ ∈π(Z(J ))

Ordξ (J ),

where Ordξ (J ) =

ord(J, x): x ∈ Z(J ) ∩ π −1 (ξ ) .

If ϕ is an inner function in J , then we have MJ

max

ξ ∈π(Z(J ))

Ordξ (ϕ).

Lemma 7.2. Let J be a closed ideal in A satisfying Z(J ) ⊂ X and B be an inner function in J . Then there is an inner function ϕ of order mJ in J such that ϕ ≺ B and maxξ ∈π(Z(ϕ)) Ordξ (ϕ) = MJ . Proof. As in the proof of Corollary 3.14, for each ξ ∈ π(Z(J )) there is an inner function ψ(ξ ) of order mJ in J satisfying ψ(ξ ) ≺ B and is an open and closed neighborhood Wξ of ξ in βN such that Ordλ (J ) Ordλ (ψ(ξ ) ) = Ordξ (ψ(ξ ) ) = Ordξ (J ),

λ ∈ Wξ .

By the compactness, there are ξ1 , ξ1 , . . . , ξk in π(Z(J )) such that π(Z(J )) ⊂ ki=1 Wξi . Let i−1 V1 = Wξ1 and Vi = Wξi \ j =1 Wξj for 2 i k. Then {Vi : 1 i k} is a set of mutually disjoint open and closed subsets of βN and π(Z(J )) ⊂ ki=1 Wξi = ki=1 Vi . Let ϕi be the subfactor of ψ(ξi ) with zeros Z(ψ(ξi ) ) ∩ π −1 (Vi ). Then ϕi ≺ ψ(ξi ) ≺ B, order(ϕi ) mJ and Ordλ (ϕi ) = Ordλ (ψ(ξi ) ) = Ordξi (J ),

λ ∈ Vi .

Let ϕ = ki=1 ϕi . Since Vi ∩ Vj = ∅ for i = j , we have ϕ ≺ B and order(ϕ) mJ . For each λ ∈ π(Z(ϕ)), there is a unique 1 k such that λ ∈ V . Then Ordλ (ϕ) = Ordλ (ϕ ) = Ordξ (J ). Hence maxλ∈π(Z(ϕ)) Ordλ (ϕ) MJ . For x ∈ Z(J ), similarly we have x ∈ π −1 (V ) for some 1 k and ord(ϕ, x) = ord(ϕ , x) = ord(ψ(ξ ) , x) ord(J, x). By Theorem 3.9, we have maxλ∈π(Z(ϕ)) Ordλ (ϕ) = MJ . 2

ϕ ∈ J.

Therefore

we

get

order(ϕ) = mJ

and


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Lemma 7.3. Let ϕ, ψ be inner functions satisfying ψ ≺ ϕ. Let t be a positive integer. If max

ξ ∈π(Z(ψ))

Ordξ (ψ) t

min

ξ ∈π(Z(ϕ))

Ordξ (ϕ),

then there is an inner function q such that ψ ≺ q ≺ ϕ and Ordλ ( q ) = t for every λ ∈ π(Z(ϕ)). Proof. For each n ∈ π(Z(ϕ)) ∩ N, by the assumption we have ψn ≺ ϕn and Ordn (ψ) t Ordn (ϕ). Let bn (z) be a Blaschke subproduct of ϕn such that

ord(bn , w): w ∈ Z(bn ) = t

and ψn (z) ≺ bn (z). Let q(n, z) be the function on N × D defined by q(n, z) =

bn (z), 1,

n ∈ π(Z(ϕ)) ∩ N, n ∈ N \ π(Z(ϕ)).

By Theorem 2.8, there is an inner function q in A such that q |N×D = q. It is not difficult to see that q satisfies the desired conditions. 2 Lemma 7.4. Let I1 , I2 be closed ideals in A satisfying Z(Ii ) ⊂ X for i = 1, 2. Let 1 , 2 be positive integers satisfying MIi i for i = 1, 2. Suppose that I1 ⊗I2 has the factorization property. Let ϕ be an inner function in I1 ⊗ I2 such that 1 + 2 Ordξ (ϕ) for every ξ ∈ π(Z(ϕ)). Then there are inner functions ϕ1 , ϕ2 such that ϕi ∈ Ii for i = 1, 2, ϕ1 ϕ2 ≺ ϕ and Ordξ (ϕi ) = i for every ξ ∈ π(Z(ϕ)) and i = 1, 2. Proof. By the assumption, there are inner functions ψ1 , ψ2 such that ψi ∈ Ii for i = 1, 2 and ψ1 ψ2 ≺ ϕ. By Lemma 7.2, we may assume that max

ξ ∈π(Z(ψi ))

Ordξ (ψi ) = MIi ,

i = 1, 2.

For ξ ∈ π(Z(ϕ)), we have Ordξ

ϕ ψ2

= Ordξ (ϕ) − Ordξ (ψ2 ) 1 + 2 − MI2 1

and ψ1 ≺ ϕ/ψ2 . We have also that max

ξ ∈π(Z(ψ1 ))

Ordξ (ψ1 ) = MI1 1 .

By Lemma 7.3, there is an inner function ϕ1 such that ψ1 ≺ ϕ1 ≺ ϕ/ψ2 and Ordξ (ϕ1 ) = 1 for every ξ ∈ π(Z(ϕ/ψ2 )) = π(Z(ϕ)). Since ψ1 ∈ I1 , we have ord(I1 , x) ord(ψ1 , x) ord(ϕ1 , x), By Theorem 3.9, we have ϕ1 ∈ I1 .

x ∈ Z(I1 ).

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For ξ ∈ π(Z(ϕ)), we have Ordξ

ϕ ϕ1

= Ordξ (ϕ) − Ordξ (ϕ1 ) (1 + 2 ) − 1 = 2

and ψ2 ≺ ϕ/ϕ1 . By Lemma 7.3 again, there is an inner function ϕ2 such that ψ2 ≺ ϕ2 ≺ ϕ/ϕ1 and Ordξ (ϕ2 ) = 2 for every ξ ∈ π(Z(ϕ/ϕ1 )) = π(Z(ϕ)). Since ψ2 ∈ I2 , we have also ϕ2 ∈ I2 . 2 For each positive integer m, let Γm be the family of closed tensor products I1 ⊗ I2 of closed ideals I1 and I2 in A such that Z(Ii ) ⊂ X for i = 1, 2 and max

x∈Z(I1 ⊗I2 )

ord(I1 ⊗ I2 , x) m.

Theorem 7.5. Let I1 , I2 be closed ideals in A satisfying Z(Ii ) ⊂ X for i = 1, 2. Then I1 ⊗ I2 has the factorization property. Proof. We divide the proof into four steps. Step 1. We write Ei = Z(Ii ) for i = 1, 2 and E = E1 ∪E2 . We prove the assertion by induction on m in Γm . First, we consider the case m = 1. Take I1 ⊗I2 ∈ Γ1 arbitrary. Then ord(I1 , x)+ord(I2 , x) = 1 for every x ∈ E. So we have E1 ∩ E2 = ∅ and ord(Ii , x) = 1 for x ∈ Ei and i = 1, 2. By Theorem 3.3, E1 , E2 and E are all interpolation sets. Hence I1 ⊗ I2 = I (E). Let ϕ be an inner function in I1 ⊗ I2 . Then ϕ ∈ I (E), and by Lemma 2.17 it is not difficult to see the existence of IBPs ϕ1 , ϕ2 such that ϕi ∈ Ii and ϕ1 ϕ2 ≺ ϕ. Hence I1 ⊗ I2 has the factorization property. Step 2. Let m be a positive integer with m 2. Suppose that J1 ⊗ J2 has the factorization property for every J1 ⊗ J2 ∈ Γm−1 . Let I1 ⊗ I2 ∈ Γm . Applying Lemma 7.1, we shall show that I1 ⊗ I2 has the factorization property. Let x1 ∈ E. We shall prove that I1 ⊗ I2 has the local factorization property at x1 ∈ E. Since ord(I1 ⊗ I2 , y) is upper semicontinuous in y ∈ E, there is an open and closed neighborhood Ex1 of x1 in E such that ord(I1 ⊗ I2 , y) ord(I1 ⊗ I2 , x1 ) for every y ∈ Ex1 . Let IEx1 = I1,Ex1 ⊗ I2,Ex1 be a localization of the tensor product I1 ⊗ I2 at x1 ∈ E. Since ord(IEx1 , y) ord(I1 ⊗ I2 , x1 ) = ord(IEx1 , x1 ),

y ∈ Ex1 ,

if ord(I1 ⊗ I2 , x1 ) m − 1 we have IEx1 ∈ Γm−1 . By the assumption of induction, I1 ⊗ I2 has the local factorization property at x1 ∈ E. So we may assume that ord(I1 ⊗ I2 , x1 ) = m. We write x1 = (ξ1 , z1 ). Since #(E ∩ π −1 (ξ1 )) < ∞, we may take r > 0 as


2123

E ∩ {ξ1 } × D r (z1 ) = (ξ1 , z1 ) = {x1 }. Then there exists an open and closed neighborhood Wξ1 of ξ1 in βN such that E ∩ (Wξ1 × Dr (z1 )) is open and closed in E. Taking smaller r and Wξ1 , we may assume that

Ex1 = E ∩ Wξ1 × Dr (z1 ) . We have ord(IEx1 , y) =

2

ord(Ii,Ex1 , y) = ord(I1 ⊗ I2 , y) m,

y ∈ Ex1 ,

i=1

Ordξ1 (IEx1 ) = ord(IEx1 , x1 ) = ord(I1 ⊗ I2 , x1 ) = m and IEx1 ∈ Γm . By Corollary 3.14, Ordλ (IEx1 ) is upper semicontinuous in λ ∈ βN, so retaking smaller Wξ1 we may assume that Ordλ (IEx1 ) Ordξ1 (IEx1 ) = m,

λ ∈ π(Ex1 ).

Moreover we may assume that Ordλ (Ii,Ex1 ) ord(Ii,Ex1 , x1 ),

λ ∈ π(Ex1 ), i = 1, 2.

We shall prove that IEx1 has the factorization property. Step 3. To simplify the notations, we put Ji = Ii,Ex1 . If Ji = A, then there is nothing to prove, so we may assume that x1 ∈ Z(Ji ) for i = 1, 2. Let J = J1 ⊗ J 2 . Then we have x1 = (ξ1 , z1 ) ∈ Z(J ) = Z(J1 ) ∪ Z(J2 ). We have also J ∈ Γm , m = max ord(J, y) = ord(J, x1 ) = ord(J1 , x1 ) + ord(J2 , x1 )

(7.1)

Ordλ (Ji ) Ordξ1 (Ji ) = ord(Ji , x1 ) = 0

(7.2)

y∈Z(J )

and

for every i = 1, 2 and λ ∈ π(Z(J )). We shall prove that J has the factorization property.

2124


Let ϕ be an inner function in J . By (7.1) and (7.2), mJ = MJ = m. By Lemma 7.2, there is an inner function ψ of order m in J such that ψ ≺ ϕ and max

λ∈π(Z(ψ))

Ordλ (ψ) = Ordξ1 (ψ) = ord(ψ, x1 ) = m.

By Corollary 2.6,

V1 = λ ∈ π Z(ψ) : Ordλ (ψ) = m

and V2 = π Z(ψ) \ V1

are open and closed in βN. We have x1 ∈ Z(Ji ) ∩ π −1 (V1 ). Let ψ1 be the subfactor of ψ with zeros Z(ψ) ∩ π −1 (V1 ) and ψ2 = ψ/ψ1 . Then max

λ∈π(Z(ψ2 ))

Ordλ (ψ2 ) m − 1.

Let J2,i = f ∈ A: ord(f, x) ord(Ji , x), x ∈ Z(Ji ) ∩ π −1 (V2 ) for i = 1, 2. Then J2,i is a closed ideal such that Ji ⊂ J2,i , Z(J2,i ) = Z(Ji ) ∩ π −1 (V2 ) and ord(J2,i , y) = ord(Ji , y) for y ∈ Z(J2,i ). Since ψ ∈ J , we have ψ2 ∈ J2,1 ⊗ J2,2 . Hence J2,1 ⊗ J2,2 ∈ Γm−1 . By the assumption of induction, there are inner functions ψ2,1 , ψ2,2 such that ψ2,i ∈ J2,i for i = 1, 2 and ψ2,1 ψ2,2 ≺ ψ2 . Next, we study on π −1 (V1 ). Let J1,i = f ∈ A: ord(f, x) ord(Ji , x), x ∈ Z(Ji ) ∩ π −1 (V1 ) for i = 1, 2. Then J1,i is a closed ideal such that Ji ⊂ J1,i , Z(J1,i ) = Z(Ji ) ∩ π −1 (V1 ) and ord(J1,i , y) = ord(Ji , y) for y ∈ Z(J1,i ). We have ψ1 ∈ J1,1 ⊗ J1,2 and Ordλ (ψ1 ) = m for every λ ∈ V1 . In Step 4, we shall show the existence of inner functions ψ1,1 , ψ1,2 such that ψ1,i ∈ J1,i for i = 1, 2 and ψ1,1 ψ1,2 ≺ ψ1 . Let ϕi = ψ1,i ψ2,i for i = 1, 2. Since V1 ∩ V2 = ∅, we have ϕ1 ϕ2 = (ψ1,1 ψ1,2 )(ψ2,1 ψ2,2 ) ≺ ψ1 ψ2 = ψ ≺ ϕ. Let x ∈ Z(Ji ). Then either x ∈ π −1 (V1 ) or x ∈ π −1 (V2 ). If x ∈ π −1 (V1 ), then ord(ϕi , x) = ord(ψ1,i , x) ord(Ji , x). If x ∈ π −1 (V2 ), then similarly ord(ϕi , x) ord(Ji , x). By Theorem 3.9, we have ϕi ∈ Ji for i = 1, 2. Thus we get the assertion. Step 4. To simplify the notations again, we put Li = J1,i for i = 1, 2 and q = ψ1 . Let L = L1 ⊗ L 2 . Then q ∈ L and Ordλ (q) = m,

λ ∈ π Z(q) .

(7.3)


2125

We have L ∈ Γm , x1 ∈ Z(L) and ord(q, x1 ) = m. By Corollary 2.6, π(Z(q)) is an open and closed subset of βN. We have also

i = 1, 2, λ ∈ π Z(L)

Ordλ (Li ) Ordξ1 (Li ) = ord(Li , x1 ),

(7.4)

and ord(L1 , x1 ) + ord(L2 , x1 ) = m.

(7.5)

A1 = x ∈ Z(L): ord(L, x) = m .

(7.6)

Let

Then x1 ∈ A1 . To complete the proof, we need to show the existence of inner functions q1 , q2 such that qi ∈ Li for i = 1, 2 and q1 q2 = q. By (7.3) and Theorem 2.8(ii), there are simple IBPs b1 , b2 , . . . , bm such that q=

m

bj

and π Z(q) = π Z(bj ) ,

1 j m.

(7.7)

j =1

Let m bj (x),

x ∈ X.

(7.8)

A2 = x ∈ Z(q): F (x) = 0 .

(7.9)

F (x) =

j =1

Then F (x) is a continuous function on X. Let

Since q ∈ L, by (7.6) and (7.7) we have A1 ⊂ A2 . Since A2 is a closed Gδ -set, so is π(A2 ). Hence there is a sequence of mutually disjoint open and closed subsets {W } of βN such that ∞

W . π Z(q) \ π(A2 ) = =1

We have

Z(q) = A2 ∪ Z(q) ∩ π

−1

∞

W

(7.10)

=1

and A2 ∩ π

−1

∞ =1

W = ∅.

(7.11)

2126


Since bj is a simple IBP, by (7.7) we have

# Z(q) ∩ π −1 (ξ ) = 1,

∞

ξ ∈ π Z(q) W .

(7.12)

=1

Let fix a positive integer for a while. For each i = 1, 2, let Li, = f ∈ A: ord(f, x) ord(Li , x), x ∈ Z(Li ) ∩ π −1 (W ) . Then Li, is a closed ideal, Li ⊂ Li, , Z(Li, ) = Z(Li ) ∩ π −1 (W )

(7.13)

and ord(Li, , x) = ord(Li , x),

x ∈ Z(Li, ).

(7.14)

We put K = L1, ⊗ L2, . Let y ∈ Z(K ). Then y ∈ Z(L1, ) ∪ Z(L2, ). By (7.13), we have y ∈ π −1 (W ). By (7.11), A2 ∩ / A1 . So we have π −1 (W ) = ∅. Since A1 ⊂ A2 , y ∈ m > ord(L, y)

by (7.6)

= ord(L1 , y) + ord(L2 , y) = ord(L1, , y) + ord(L2, , y)

by (7.14)

= ord(K , y). Hence K ∈ Γm−1 . By the assumption of induction, K has the factorization property. Let B be the subfactor of q with zeros Z(q) ∩ π −1 (W ). Then Z(B ) ⊂ π −1 (W ), π(Z(B )) = W and B = 1 on π −1 (βN \ W ). Since q ∈ L and y ∈ π −1 (W ), we have ord(B , y) = ord(q, y) ord(L, y) =

2

ord(Li, , y) = ord(K , y).

i=1

By Theorem 3.9, we have B ∈ K . By (7.3), Ordξ (B ) = m for every ξ ∈ W . We have also MLi, MLi . By (7.4) and (7.5), ML1 + ML2 = ord(L1 , x1 ) + ord(L2 , x1 ) = m. Hence by Lemma 7.4, there are inner functions q1, , q2, such that qi, ∈ Li, ,

i = 1, 2,

q1, q2, = B

(7.15)


2127

and Ordλ (qi, ) = ord(Li , x1 ),

λ ∈ W , i = 1, 2.

(7.16)

Since B = 1 on π −1 (βN \ W ), moreover we may assume that qi, = 1 on π −1 (βN \ W ) for i = 1, 2. Next, we fix i = 1, 2 and move . As mentioned in Section 6, we may define the inner functions ∞

∞

i = 1, 2 and

qi, ,

=1

B .

=1

We have B ≺ q and Z(B ) ⊂ π −1 (W ). Since W ∩ Wj = ∅ for = j , by Corollary 2.9 and (7.15) we have 2 ∞

qi, =

i=1 =1

∞

∞

q1, q2, =

=1

B ≺ q.

(7.17)

=1

Let ∞

W0 =

W .

=1

Then W0 is an open and closed subset of βN and π Z

2 ∞

qi,

= W0 .

i=1 =1

By (7.16), we have Ordλ

∞

λ ∈ W0 , i = 1, 2

qi, = ord(Li , x1 ),

(7.18)

=1

and by (7.5) Ordλ

2 ∞

qi, =

i=1 =1

2

ord(Li , x1 ) = m,

i=1

Hence by (7.10),

Z(q)

π

−1

∞ =1

By (7.12) and (7.18), we have

W = A2 .

λ ∈ W0 .

(7.19)

2128


ord

∞

qi, , x = ord(Li , x1 ),

x ∈ Z(Li ) ∩ π

−1

W0

∞

=1

W .

(7.20)

=1

Let q ∞

q0 = 2

(7.21)

.

=1 qi,

i=1

By (7.17), q0 is an inner function. Hence by (7.3) and (7.19), Ordλ (q0 ) =

0, m,

λ ∈ W0 , λ ∈ π(Z(q)) \ W0 .

(7.22)

By (7.7), (7.8) and (7.9), we have ord(q, x) = m for every x ∈ A2 . We have Z(q0 ) ⊂ A2 , and by (7.3) and (7.12) ord(q0 , x) = m for every x ∈ Z(q0 ). Since A2 is a simple interpolation set, by (7.5) there are inner functions p1 , p2 such that ord(pi , x) = ord(Li , x1 ),

x ∈ Z(q0 ), i = 1, 2

(7.23)

and q 0 = p1 p2 .

(7.24)

Let q i = pi

∞

i = 1, 2.

qi, ,

(7.25)

=1

Then qi is an inner function. By (7.21) and (7.24), we have q = q1 q2 . We shall show that qi ∈ Li for i = 1, 2. Let x ∈ Z(Li ). Since q ∈ L, x ∈ Z(Li ) ⊂ Z(L) ⊂ Z(q). We have ∞ ∞

W ∪ W . π Z(q) = π Z(q) \ W0 ∪ W0 =1

=1

Then either

π(x) ∈ π Z(q) \ W0

(7.26)

or π(x) ∈ W0

∞ =1

W

(7.27)


2129

or π(x) ∈

∞

(7.28)

W .

=1

Suppose that (7.26) holds. By (7.22), x ∈ Z(q0 ). By (7.4), (7.23) and (7.25), we have ord(qi , x) = ord(pi , x) = ord(Li , x1 ) ord(Li , x). Suppose that (7.27) holds. By (7.4), (7.20) and (7.25), ord(qi , x) = ord

∞

qi, , x = ord(Li , x1 ) ord(Li , x).

=1

Suppose that (7.28) holds. Then there is a unique 1 satisfying π(x) ∈ W1 . Since x ∈ Z(Li ) ∩ π −1 (W1 ), by (7.13) we have x ∈ Z(Li,1 ), so ord(qi , x) = ord(qi,1 , x)

by (7.25)

ord(Li,1 , x)

by (7.15)

= ord(Li , x)

by (7.14).

Hence by Theorem 3.9, we have qi ∈ Li for i = 1, 2. Thus we get the assertion.

2

The following corollary answers the A-version of Question 3. Corollary 7.6. Let I1 , I2 be closed ideals in A satisfying Z(Ii ) ⊂ X for i = 1, 2. Then I1 ⊗ I2 is a closed ideal in A. Proof. We have I1 ⊗ I2 ⊂ I1 ⊗ I2 and Z(I1 ⊗ I2 ) = Z(I1 ) ∪ Z(I2 ) ⊂ X. Let ϕ be an inner function in I1 ⊗ I2 . By Theorem 7.5, there are inner functions ϕ1 , ϕ2 such that ϕ1 ∈ I1 , ϕ2 ∈ I2 and ϕ1 ϕ2 ≺ ϕ. We have ϕ1 ϕ2 ∈ I1 ⊗ I2 , so ϕ ∈ I1 ⊗ I2 . By Corollary 3.4, we have I1 ⊗ I2 ⊂ I1 ⊗ I2 . Thus we get I1 ⊗ I2 = I1 ⊗ I2 . 2 Let I1 , I2 , . . . , Ik be closed ideals in A satisfying Z(Ii ) ⊂ X for 1 i k. We may define the !k k !k i=1 Ii . We have Z( i=1 Ii ) = i=1 Z(Ii ) ⊂ X.

tensor product

Corollary 7.7. Let I1 , I2 , . . . , Ik be closed ideals in A satisfying Z(Ii ) ⊂ X for 1 i k. Then ! k i=1 Ii is a closed ideal in A. Let I be a closed ideal in A satisfying Z(I ) ⊂ X. Let E1 = Z(I ). By Lemma 5.2, Z(I (E1 )) = E1 so I ⊂ I (E1 ). Let NE1 ,∞ be the associated numbering function of E1 . By Theorem 5.4, NE1 ,∞ (x) = ord(I (E1 ), x) for x ∈ E1 . Hence NE1 ,∞ (x) ord(I, x) for every x ∈ E1 . If I = I (E1 ), there is nothing to say more, so we stop the argument. Suppose that I = I (E1 ).

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Let E2 = x ∈ E1 : NE1 ,∞ (x) = ord(I, x) . Then E2 = ∅. If E2 is not closed, then there are no closed ideals J satisfying I = I (E1 ) ⊗ J , and we stop the argument. Suppose that E2 is closed. As in Section 4, let NE2 ,1 (x) = 1 for x ∈ E2 . Then we have NE1 ,∞ (x) + NE2 ,1 (x) ord(I, x) for every x ∈ E1 . By Theorem 4.2 and the definition of a tilde function, we have NE1 ,∞ (x) + NE2 ,∞ (x) ord(I, x) for every x ∈ E1 . Hence by Theorem 3.9 and Corollary 7.6, we have I ⊂ I (E1 ) ⊗ I (E2 ). If I = I (E1 ) ⊗ I (E2 ), we stop the argument. Suppose that I = I (E1 ) ⊗ I (E2 ). Let E3 = x ∈ E2 : NE1 ,∞ (x) + NE2 ,∞ (x) = ord(I, x) . Then E3 = ∅. If E3 is not closed, then there are no closed ideals J satisfying I = I (E1 ) ⊗ I (E2 ) ⊗ J , and we stop the argument. If E3 is closed, then we have I ⊂ I (E1 ) ⊗ I (E2 ) ⊗ I (E3 ). We may repeat the same argument. Suppose that all E1 , E2 , . . . , Ek are closed and stop here. This means that I=

k "

I (Ei ),

E1 ⊃ E2 ⊃ · · · ⊃ Ek = ∅.

i=1

It is not difficult to give an example of I which does not have the above form. 8. Local ideal theory in H ∞ We shall study closed ideals I in H ∞ satisfying Z(I ) ⊂ G. Let E be a nonvoid compact and totally disconnected subset of G. Let nE : E → {1, 2, . . .} be a bounded numbering function on E. For each x ∈ E, let {Uα (x)}α be a net of fundamental open neighborhoods of x in G. We define the order α β by Uβ (x) ⊂ Uα (x). For each 0 < r < 1, the value of sup

ξ ∈Uα (x)

nE (ζ ): ζ ∈ Lξ (Dr ) ∩ E

decreases as α → ∞, where Lξ is the Hoffman map at ξ ∈ G. Then the value of sup

ξ ∈Uα (x)

nE (ζ ): ζ ∈ Lξ (Dr ) ∩ E

is eventually constant for sufficiently large α. Hence we may define lim

sup

α→∞ ξ ∈U (x) α

nE (ζ ): ζ ∈ Lξ (Dr ) ∩ E ∈ {1, 2, . . . , ∞}.

Also the value of lim

sup

α→∞ ξ ∈U (x) α

nE (ζ ): ζ ∈ Lξ (Dr ) ∩ E


2131

decreases as r decreases to 0. Hence the value of nE (ζ ): ζ ∈ Lξ (Dr ) ∩ E sup lim α→∞ ξ ∈U (x) α

is eventually constant for sufficiently small r > 0. Thus we may define nE (x) = lim

lim

sup

r→0 α→∞ ξ ∈Uα (x)

nE (ζ ): ζ ∈ Lξ (Dr ) ∩ E .

(8.1)

nE (x) for every x ∈ E. Moreover there exist α0 and r0 > 0 such that We have nE (x) nE (x) = sup

ξ ∈Uα (x)

nE (ζ ): ζ ∈ Lξ (Dr ) ∩ E

(8.2)

nE is upper semifor every α α0 and 0 < r r0 . It is not diffcult to show that the function continuous on E. Let E1 be an open and closed subset of E. We define nE1 = nE |E1 . By the n E1 = nE |E1 on E1 . This fact is called the locally stable property definition of nE (x), we have of a tilde function. Hoffman’s work in [14] gives us many informations on the local theory in H ∞ . Let δ, η and ε be numbers such that 0 < δ < 1,

0 0 such that nE (x) = sup

nE (ζ ): ζ ∈ E ∩ γ {λ} × Dr (z)

λ∈Wβ (ξ )

for every β β0 and 0 < r r0 . We define the numbering function nγ −1 (E) on γ −1 (E) by

nγ −1 (E) (y) = nE γ (y) ,

y ∈ γ −1 (E).

Condition (8.4) is converted into condition (4.1) for the set γ −1 (E). By the works in Section 4, we have


nE (x) = sup λ∈Wβ (ξ )

= sup λ∈Wβ (ξ )

nE (ζ ): ζ ∈ E ∩ γ {λ} × Dr (z)

nγ −1 (E) (y): y ∈ γ −1 (E) ∩ {λ} × Dr (z)

2133

= nγ −1 (E) (ξ, z). Hence we have the following. nE (x) for every x ∈ E. Lemma 8.1. nγ −1 (E) (γ −1 (x)) = nE,1 . Since nE,1 is also Let nE,1 be a bounded numbering function on E. We put nE,2 = bounded on E, for every positive integer j with j 2, we may define inductively nE,j = nE,j −1 on E. Since nE,j −1 nE,j , we may define nE,∞ (x) = lim nE,j (x), j →∞

x ∈ E.

As the special case, let NE,1 (x) = 1 for every x ∈ E. We may also define NE,j (x) = E,j −1 (x) on E for j 2. Since NE,j −1 (x) NE,j (x), we may define N NE,∞ (x) = lim NE,j (x), j →∞

x ∈ E.

We call NE,∞ : E → {1, 2, . . . , ∞} the associated numbering function of E. It is considered that NE,∞ represents a geometrically quantity of E, i.e. NE,∞ represents a generalized crossing number of lines in E at x ∈ E. We have the following. Lemma 8.2. Nγ −1 (E),∞ (γ −1 (x)) = NE,∞ (x) and nγ −1 (E),∞ (γ −1 (x)) = nE,∞ (x) for every x ∈ E. Corollary 8.3. Let nE,1 be a bounded numbering function on E. Then NE,∞ is bounded on E if E,∞ = NE,∞ and and only if nE,∞ is bounded on E. In this case, we have N nE,∞ = nE,∞ on E. Proof. Combining Lemmas 8.1 and 8.2 with Corollary 4.4 and Lemma 4.5, we get the assertion. 2 Let f ∈ H ∞ and x ∈ Z(f ) ∩ {|b| < ε}. Then we have ord(f, x) = ord(f ◦ γ , γ −1 (x)). Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ {|b| < ε}. Let x ∈ Z(I ). Then

ord(I, x) = min ord(f, x) = min ord g, γ −1 (x) . f ∈I

g∈I ◦γ

Generally, I ◦ γ is not a closed ideal in A, so let J be a closed ideal in A generated by I ◦ γ . Then we have the following.

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Lemma 8.4. (i) Z(J ) = γ −1 (Z(I )). (ii) ord(J, γ −1 (x)) = ord(I, x) for every x ∈ Z(I ). Let ψ be a simple IBP in A and N = N ∩ Z(ψ). We have #(Z(ψ) ∩ π −1 (n)) = 1 for every n ∈ N . Let {an } = Z(ψ) ∩ π −1 (n) for n ∈ N . Then {an : n ∈ N }X = Z(ψ). Let cn = γ (an ) for n ∈ N . Since γ (n, 0) = zn , we have cn ∈ Rn = z ∈ D: ρ(z, zn ) < η, b(z) < ε ,

n ∈ N.

It is known that {cn : n ∈ N} is an interpolating sequence in D (see [7, p. 405]). We denote by Ψ (ψ) the interpolating Blaschke product on D with zeros {cn : n ∈ N }. Then we have γ (Z(ψ)) = Z(Ψ (ψ)). Let ψ be an inner function in A. By Theorem 2.8(ii), m there are simple IBPs ψ1 , ψ2 , . . . , ψm in A such that ψ = m ψ . We define Ψ (ψ) = i=1 i i=1 Ψ (ψi ). We easily get the following lemma. Lemma 8.5. For an inner function ψ in A, Ψ (ψ) is a CN Blaschke product such that γ (Z(ψ)) = Z(Ψ (ψ)) ⊂ {|b| < ε} and ord(ψ, y) = ord(Ψ (ψ), γ (y)) for y ∈ Z(ψ). Conversely, let ϕ be a CN Blaschke product satisfying Z(ϕ) ⊂ {|b| < ε}. Then ϕ ◦ γ ∈ A and |ϕ ◦ γ | > 0 on ∂X. By Corollary 2.11, there are an inner function Φ(ϕ) in A and an invertible function h in A such that ϕ ◦ γ = Φ(ϕ)h. For the sake of simplicity, we ignore unimodular inner factors in A and unimodular constants in CN Blaschke products. We easily check the following. Lemma 8.6. (i) For every inner function ψ in A, we have (Φ ◦ Ψ )(ψ) = ψ. (ii) For every CN Blaschke product ϕ satisfying Z(ϕ) ⊂ {|b| < ε}, we have (Ψ ◦ Φ)(ϕ) = ϕ, Z(Φ(ϕ)) = γ −1 (Z(ϕ)) and ord(Φ(ϕ), γ −1 (x)) = ord(ϕ, x) for x ∈ Z(ϕ). By Lemmas 8.5 and 8.6, we have the following. Lemma 8.7. (i) Let ϕ1 , ϕ2 be CN Blaschke products satisfying Z(ϕi ) ⊂ {|b| < ε} for i = 1, 2. Then ϕ1 ≺ ϕ2 if and only if Φ(ϕ1 ) ≺ Φ(ϕ2 ). (ii) Let ψ1 , ψ2 be inner functions in A. Then ψ1 ≺ ψ2 if and only if Ψ (ψ1 ) ≺ Ψ (ψ2 ). Lemma 8.8. Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ {|b| < ε} and J be the closed ideal in A generated by I ◦ γ . Then we have the following. (i) If ϕ ∈ I is a CN Blaschke product satisfying Z(ϕ) ⊂ {|b| < ε}, then Φ(ϕ) ∈ J . (ii) If ψ ∈ J is an inner function, then Ψ (ψ) ∈ I .


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Proof. (i) Let ϕ ∈ I be a CN Blaschke product. Then ord(ϕ, x) ord(I, x) for every x ∈ Z(I ). By Lemma 8.4(i), we have Z(J ) = γ −1 (Z(I )). For y ∈ Z(J ), we have

ord Φ(ϕ), y = ord ϕ, γ (y) by Lemma 8.6(ii)

ord I, γ (y) = ord(J, y)

by Lemma 8.4(ii).

By Theorem 3.9, we have Φ(ϕ) ∈ J . (ii) Let ψ ∈ J be an inner function in A. Then ord(ψ, y) ord(J, y) for y ∈ Z(J ). We have Z(I ) = γ (Z(J )), and for x ∈ Z(I )

ord Ψ (ψ), x = ord ψ, γ −1 (x)

ord J, γ −1 (x) = ord(I, x) By Theorem A, we get Ψ (ψ) ∈ I .

by Lemma 8.5

by Lemma 8.4(ii).

2

Proposition 8.9. Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ {|b| < ε}, x ∈ Z(I ) and U be an open subset satisfying Z(I ) ⊂ U ⊂ {|b| < ε}. If B is a CN Blaschke product in I , then there is a CN Blaschke product ϕ of order mI in I such that Z(ϕ) ⊂ U , ϕ ≺ B and ord(ϕ, x) = ord(I, x). Proof. Let J be a closed ideal in A generated by I ◦γ . By Lemma 8.4(i), Z(J ) = γ −1 (Z(I )). By Lemma 8.8(i), we have Φ(B) ∈ J . Since γ : X → {|b| ε} is a homeomorphic map, γ −1 (U ) is an open subset of X such that Z(J ) ⊂ γ −1 (U ) ⊂ X. By Theorem 3.11, there is an inner function ψ of order mJ in J such that Z(ψ) ⊂ γ −1 (U ), ψ ≺ Φ(B) and ord(ψ, γ −1 (x)) = ord(J, γ −1 (x)). By Lemmas 8.4–8.7, we have Z(Ψ (ψ)) ⊂ U , Ψ (ψ) ≺ B and ord(Ψ (ψ), x) = ord(ψ, γ −1 (x)) = ord(I, x). Let ϕ = Ψ (ψ). Then Z(ϕ) ⊂ U and ord(ϕ, x) = ord(I, x). By Lemma 8.8(ii), we have ϕ ∈ I . 2 Corollary 8.10. Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ {|b| < ε} and B ∈ I be a CN Blaschke product. Then I is generated by CN Blaschke products ϕ in I such that ϕ ≺ B as a closed ideal. Proof. Let B be a CN Blaschke product in I . Let I1 be a closed ideal in H ∞ generated by CN Blaschke products ϕ in I satisfying ϕ ≺ B. Then I1 ⊂ I . Let {Uα }α be a set of open subsets of G such that Z(I ) = α Uα and Uα ⊂ {|b| < ε} for every α. For each x ∈ Z(I ) and α, by Proposition 8.9 there is a CN Blaschke product ϕx,α ∈ I such that Z(ϕx,α ) ⊂ Uα , ϕx,α ≺ B and ord(ϕx,α , x) = ord(I, x). Then Z(I1 ) = Z(I ) and ord(I1 , x) = ord(I, x) for every x ∈ Z(I ). By Theorem A, we get I1 = I . 2 Proposition 8.11. Let I be a closed ideal in H ∞ satisfying E := Z(I ) ⊂ {|b| < ε}. Let nE (x) = ord(I, x) for x ∈ E. Then nE is a bounded numbering function satisfying nE = nE on E.

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Proof. Let J be the closed ideal in A generated by I ◦ γ . By Lemma 8.4, we have Z(J ) = γ −1 (E). Let nγ −1 (E) (y) = nE (γ (y)) for y ∈ Z(J ). By Lemma 8.1, nγ −1 (E) (y) = nE (γ (y)) for every y ∈ Z(J ). By Lemma 8.4, we have

ord(J, y) = ord I, γ (y) = nE γ (y) = nγ −1 (E) (y),

y ∈ Z(J ).

Then by Theorem 4.2, nγ −1 (E) (y) is a bounded numbering function and nγ −1 (E) (y) = nγ −1 (E) (y) for every y ∈ Z(J ). Thus we get nE = nE on E. 2 Proposition 8.12. Let E be a nonvoid compact and totally disconnected subset of {|b| < ε}, and maxξ ∈Z(b) #(E ∩ Rξ ) < ∞. Then the following conditions are equivalent. (i) Z(I (E)) = E. (ii) NE,∞ is bounded. (iii) ord(I (E), x) is bounded in x ∈ E. In this case, we have that ord(I (E), x) = NE,∞ (x) for every x ∈ E. Proof. The condition maxξ ∈Z(b) #(E ∩ Rξ ) < ∞ is equivalent to max

ξ ∈π(γ −1 (E))

# γ −1 (E) ∩ π −1 (ξ ) < ∞.

Let J be the closed ideal in A generated by I (E) ◦ γ . Then J ⊂ I (γ −1 (E)) and Z(J ) = Z(I (γ −1 (E))) = γ −1 (E). Let ψ be an inner function in I (γ −1 (E)). By Lemma 8.5, we have Ψ (ψ) ∈ I (E) and Z(Ψ (ψ)) ⊂ {|b| < ε}. By Lemma 8.6(i), Φ(Ψ (ψ)) = ψ, and by Lemma 8.8(i) we have ψ ∈ J . By Corollary 3.4, we get J = I (γ −1 (E)). By Lemma 8.4(i), Z(I (E)) = E if and only if Z(J ) = γ −1 (E). Since J = I (γ −1 (E)), Z(I (E)) = E if and only if Z(I (γ −1 (E))) = γ −1 (E). By Lemma 8.2, NE,∞ (x) = Nγ −1 (E),∞ (γ −1 (x)), and by Lemma 8.4(ii) and J = I (γ −1 (E)), we have ord(I (E), x) = ord(I (γ −1 (E)), γ −1 (x)) for every x ∈ E. Therefore by Theorem 5.4, we get the assertion. 2 Proposition 8.13. Let E be a nonvoid compact and totally disconnected subset of {|b| < ε}, and nE = nE on E. maxξ ∈Z(b) #(E ∩ Rξ ) < ∞. Let nE be a bounded numbering function satisfying Then there is a closed ideal I in H ∞ such that Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E. Proof. Let nγ −1 (E) (y) = nE (γ (y)) for y ∈ γ −1 (E). By Lemma 8.1, we have

nE γ (y) = nE γ (y) = nγ −1 (E) (y), nγ −1 (E) (y) =

y ∈ γ −1 (E).

By Theorem 6.1, there is a closed ideal J in A such that Z(J ) = γ −1 (E) and ord(J, y) = nγ −1 (E) (y) for every y ∈ γ −1 (E). Let I be a closed ideal in H ∞ generated by Ψ (ψa ) for all inner functions ψa (α ∈ Λ) in J . We have γ −1 (E) =

a∈Λ

Z(ψa )

by Corollary 3.4


= γ −1

Z Ψ (ψa )

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by Lemma 8.5

a∈Λ

= γ −1 Z(I ) . Hence Z(I ) = E. For each x ∈ E, there exists α0 ∈ Λ such that

ord ψa0 , γ −1 (x) = ord J, γ −1 (x) = nγ −1 (E) γ −1 (x) = nE (x). Hence by Lemma 8.5, we get

ord Ψ (ψa0 ), x = ord ψa0 , γ −1 (x) = nE (x). Thus we get ord(I, x) nE (x). For each α ∈ Λ, we have

nE (x) = nγ −1 (E) γ −1 (x) = ord J, γ −1 (x)

ord ψα , γ −1 (x)

= ord Ψ (ψα ), x by Lemma 8.5. Hence we get nE (x) ord(I, x). Thus we get ord(I, x) = nE (x) for every x ∈ E.

2

Let I1 , I2 be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for i = 1, 2. Similarly as in Section 7, we may define the tensor product I1 ⊗ I2 and the closed tensor product I1 ⊗ I2 . We have Z(I1 ⊗ I2 ) = Z(I1 ⊗ I2 ) = Z(I1 ) ∪ Z(I2 ) and ord(I1 ⊗ I2 , x) = ord(I1 ⊗ I2 , x) = ord(I1 , x) + ord(I2 , x) for every x ∈ Z(I1 ⊗ I2 ). We say that I1 ⊗ I2 has the factorization property if for every CN Blaschke product ϕ in I1 ⊗ I2 , there are CN Blaschke products ϕ1 , ϕ2 such that ϕi ∈ Ii for i = 1, 2 and ϕ1 ϕ2 ≺ ϕ. Proposition 8.14. Let I1 , I2 be closed ideals in H ∞ satisfying Z(Ii ) ⊂ {|b| < ε} for i = 1, 2. Then I1 ⊗ I2 has the factorization property. Proof. Let ϕ ∈ I1 ⊗I2 be a CN Blaschke product. Considering a subproduct, we may assume that Z(ϕ) ⊂ {|b| < ε}. For i = 1, 2, let Ji be the closed ideal in A generated by Ii ◦ γ . By Lemma 8.4, Z(Ji ) = γ −1 (Z(Ii )) and ord(Ji , γ −1 (x)) = ord(Ii , x) for every x ∈ Z(Ii ). Let J be the closed ideal in A generated by (I1 ⊗ I2 ) ◦ γ . Then by Lemma 8.4 again, Z(J ) = γ −1 (Z(I1 ⊗ I2 )), and for y ∈ Z(J ) we have

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K.J. Izuchi, Y. Izuchi / Journal of Functional Analysis 260 (2011) 2086–2147 2

ord(J, y) = ord I1 ⊗ I2 , γ (y) = ord Ii , γ (y) i=1

=

2

ord(Ji , y) = (J1 ⊗ J2 , y).

i=1

By Theorem 3.9 and Lemma 8.8(i), we have Φ(ϕ) ∈ J = J1 ⊗ J2 . By Theorem 7.5, there are inner functions ψ1 , ψ2 in A such that ψi ∈ Ji for i = 1, 2 and ψ1 ψ2 ≺ Φ(ϕ). By Lemmas 8.6(ii) and 8.7(ii), Ψ (ψ1 )Ψ (ψ2 ) ≺ ϕ. By Lemma 8.8(ii), Ψ (ψi ) ∈ Ii for i = 1, 2. Hence I1 ⊗ I2 has the factorization property. 2 By the results in this section, A is a nice space to study the local ideal theory of H ∞ in G. But the local version of Theorem A may not be proved using Theorem 3.9. So Theorem A is a crucial theorem in ideal theory of H ∞ . 9. Ideal theory in H ∞ In this section, we shall answer Questions 1–4 given in the introduction. Let E be a compact and totally disconnected subset of G. For each x ∈ E, by Hoffman’s work there is an interpolating Blaschke product bx such that bx (x) = 0. Take δx , ηx , εx satisfying (8.3) and 0 < δx < δ(bx ). ∩ Ux is open and Let Ux be an open subset of M(H ∞ ) such that x ∈ Ux ⊂ {|bx | < εx } and E closed in E. By the compactness, there are x1 , x2 , . . . , xk ∈ E such that E = ki=1 E ∩ Uxi . Let j −1 E1 = E ∩ Ux1 and Ej = (E ∩ Uxj ) \ i=1 E ∩ Uxi for 2 i k. Then {Ej : 1 j k} is a set of mutually disjoint open and closed subsets of E. We have Ej ⊂ {|bxj | < εxj } for 1 j k. As a summary, we have the following. Lemma 9.1. Let E be a nonvoid compact and totally disconnected subset of G. Then there are interpolating Blaschke products b1 , b2 , . . . , b k and {Ei : 1 i k} a set of mutually disjoint open and closed subsets of E such that E = ki=1 Ei and Ei ⊂ {|bi | < εi }, where δi , ηi and εi satisfy (8.3) and 0 < δi < δ(bi ) for 1 i k. For E, we take the same notations given in Lemma 9.1. We are interested in the case that there is a closed ideal I in H ∞ satisfying Z(I ) = E and Z(I ) ⊂ G. Since I contains a CN Blaschke product, we have that maxx∈Z(bi ) #(Ei ∩ P (x) ∩ {|bi | < εi }) < ∞ for 1 i k. Therefore we assume that

max # Ei ∩ P (x) ∩ |bi | < εi < ∞,

x∈Z(bi )

1 i k.

(9.1)

If (9.1) holds for a partition E = ki=1 Ei given in Lemma 9.1, we say that E satisfies (9.1). Condition (9.1) corresponds to (8.4) for Ei and bi , 1 i k. nEi is a bounded For a bounded numbering function nE on E, let nEi = nE |Ei . By Section 8, n Ei = numbering function on Ei . By the locally stable property of a tilde function, we have nE is a bounded numbering function and nE nE on E. nE |Ei . Hence Let nE,1 be a bounded numbering function on E. For each positive integer j , we may define nE,j . Then nE,j nE,j +1 and we may define nE,∞ as before. We set also inductively nE,j +1 = NE,1 (x) = 1 for x ∈ E, and define NE,∞ as in Section 8.


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Theorem 9.2. Let E be a nonvoid compact and totally disconnected subset of G satisfying (9.1). Let nE,1 be a bounded numbering function on E. Then NE,∞ is bounded on E if and only if E,∞ = NE,∞ and nE,∞ = nE,∞ on E. nE,∞ is bounded on E. In this case, we have N Corollary 9.3. Let E be a nonvoid compact and totally disconnected subset of G satisfying (9.1). If there is a bounded numbering function nE satisfying nE = nE on E, then NE,∞ is bounded on E. Proof. Let E = ki=1 Ei be a partition of E given in Lemma 9.1. For each 1 i k, put nEi ,1 = nE,1 |Ei . By Corollary 8.3, NEi ,∞ is bounded on Ei if and only if nEi ,∞ is bounded on Ei . Ei ,∞ = NEi ,∞ and nEi ,∞ = nEi ,∞ on Ei . Since NEi ,∞ = NE,∞ and In this case, we have N nEi ,∞ = nE,∞ on Ei , we get the assertion. 2 Lemma 9.4. Let E be a nonvoid compact and totally disconnected subset of G satisfying (9.1). Then we have the following. (i) Z(I (E)) = E if and only if Z(I (Ei )) = Ei for every 1 i k. (ii) NE,∞ is bounded on E if and only if NEi ,∞ is bounded on Ei for every 1 i k. (iii) ord(I (E), x) is bounded on E if and only if ord(I (Ei ), x) is bounded on Ei for every 1 i k. (iv) ord(I (E), x) = NE,∞ (x) for every x ∈ E if and only if ord(I (Ei ), x) = NEi ,∞ (x) for every x ∈ Ei , 1 i k. Proof. Suppose that Z(I (E)) = E. For 1 i k, let Ui be an open subset such that Ei ⊂ Ui ⊂ U i ⊂ {|bi | < εi } and Ui ∩ Uj = ∅ for i = j . By [9], there is a CN Blaschke product ϕ in I (E). Let ϕi be a subproduct of ϕ with zeros Z(ϕ) ∩ Ui ∩ D counting multiplicities. Then ϕi ∈ I (Ei ) and Ei ⊂ Z(I (Ei )) ⊂ Z(ϕi ) ⊂ U i . Hence Z(I (Ei )) = Ei for 1 i k. ϕi ∈ I (Ei ). Suppose that Z(I (Ei )) = Ei for 1 i k. Then there is a CN Blaschke product Let U be an open subset of G such that E ⊂ U . Let ϕ be a subproduct of ki=1 ϕi with zeros Z( ki=1 ϕi ) ∩ U ∩ D counting multiplicities. Then ϕ ∈ I (E) and Z(ϕ) ⊂ U . This shows that Z(I (E)) = E. Thus we get (i). By the locally stable properties of NE,∞ and ord(I (E), x), NE,∞ = NEi ,∞ and ord(I (E), x) = ord(I (Ei ), x) on Ei for every 1 i k. By these facts, we get (ii), (iii) and (iv). 2 The following theorem answers Question 1. Theorem 9.5. Let E be a nonvoid compact and totally disconnected subset of G satisfying (9.1). Then the following conditions are equivalent. (i) Z(I (E)) = E. (ii) NE,∞ is bounded on E. (iii) ord(I (E), x) is bounded in x ∈ E. In this case, we have that ord(I (E), x) = NE,∞ (x) for every x ∈ E.

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Proof. Let E = ki=1 Ei be a partition of E given in Lemma 9.1. By Proposition 8.12 and Lemma 9.4, we get the assertion. 2 Since NE,∞ represents a geometrical quantity of E, Theorem 9.5(ii) gives a geometrical characterization of E in G satisfying Z(I (E)) = E. Corollary 9.6. Let I be a closed ideal in H ∞ satisfying E := Z(I ) ⊂ G. Then NE,∞ is bounded on E. Proof. We have I ⊂ I (E), so E ⊂ Z(I (E)) ⊂ Z(I ) = E. By Theorem 9.5, we get the assertion. 2 ∞ Let I be a closed k ideal in H satisfying E := Z(I ) ⊂ G. It is known that E is totally disconnected. Let E = i=1 Ei be a partition of E given in Lemma 9.1. For each 1 i k, let

Ii = f ∈ H ∞ : ord(f, x) ord(I, x), x ∈ Ei . Then Ii is a closed ideal in H ∞ , I ⊂ Ii and Z(Ii ) = Ei . ! For closed ideals I1 , I2 , . . . , Ik in H ∞ , we may define the tensor product ki=1 Ii and the !k closed tensor product i=1 Ii . Lemma 9.7. (i) ord(Ii , x) = ord(I, x) for x ∈ Ei , 1 i k. !k (ii) i=1 Ii = I . Proof. (i) We have ord(Ii , x) ord(I, x) for every x ∈ Ei . Since I ⊂ Ii , we have ord(Ii , x) ord(I, x) for x ∈ Ei . Thus we get (i). (ii) Let x ∈ E. Then there is a unique 1 i k satisfying x ∈ Ei . By (i), we have ord

k

"

Ii , x = ord(Ii , x) = ord(I, x).

i=1

Since x ∈ E is arbitrary, by Theorem A we get (ii).

2

Theorem 9.8. Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ G and B ∈ I be a CN Blaschke product. Then I is generated by CN Blaschke products ϕ in I such that ϕ ≺ B as a closed ideal. Proof. Let E = Z(I ). Let {Ei : 1 i k}, {bi : 1 i k} and {εi : 1 i k} be given in Lemma 9.1. Take open subsets {Vi : 1 i k} of G such that Ei ⊂ Vi ⊂ {|bi | < εi } for 1 i k and Vi ∩ Vj = ∅ for i = j . For each 1 i k, let ϕi be the subproduct of B with zeros Z(B) ∩ Vi ∩ D counting multiplicities. Then ϕi ≺ B, Z(ϕi ) ⊂ V i and ord(B, x) = ord(ϕi , x) for x ∈ Ei . Hence by Lemma 9.7(i), we have ord(ϕi , x) = ord(B, x) ord(I, x) = ord(Ii , x),

x ∈ Ei .


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By Theorem A, ϕi ∈ Ii . We note that ki=1 ϕi ≺ B. Since Z(Ii ) = Ei ⊂ {|bi | < εi }, by Corollary 8.10, Ii is generated by CN Blaschke products ψi ∈ Ii such that ψi ≺ ϕi as a closed ideal. By Lemma 9.7(ii), we get the assertion. 2 Theorem 9.9. Let I be a closed ideal in H ∞ satisfying E := Z(I ) ⊂ G. Let nE (x) = ord(I, x) nE = nE on E. for x ∈ E. Then nE is a bounded numbering function satisfying Proof. Since I contains a CN Blaschke product, nE is bounded on E. Let E = partition of E given in Lemma 9.1 and Ii = f ∈ H ∞ : ord(f, x) ord(I, x), x ∈ Ei ,

k

i=1 Ei

be a

1 i k.

Let nEi (x) = ord(Ii , x) for x ∈ Ei . By Lemma 9.7(i), we have nEi = nE on Ei . By Proposition 8.11, nEi = nEi on Ei . Since nE1 = nE |E1 , by the locally stable property, we have nE = nE on Ei for 1 i k. Thus we get the assertion. 2 nE1 = nE |E1 . Hence Theorem 9.10. Let E be a nonvoid compact and totally disconnected subset of G satisfying (9.1) nE = nE on E. Then there is a closed ideal I and nE be a bounded numbering function satisfying in H ∞ such that Z(I ) = E and ord(I, x) = nE (x) for every x ∈ E. Proof. Let E = ki=1 Ei be a partition of E given in Lemma 9.1. Let nEi = nE |Ei for 1 i k. By the assumption and the locally stable property, we have nEi = nEi . By Proposition 8.13, there is closed ideal Ii in H ∞ such that Z(Ii ) = Ei and ord(Ii , x) = nEi (x) for every x ∈ Ei . !k Let I = i=1 Ii . Then I is a closed ideal and Z(I ) = E. For x ∈ E = Z(I ), there is a unique 1 i k satisfying x ∈ Ei and we have ord(I, x) = ord(Ii , x) = nEi (x) = nE (x). Thus we get the assertion.

2

Combining Theorem 9.10 with Theorem 9.9, we have the following corollary, which answers Question 2. Corollary 9.11. Let E be a nonvoid compact and totally disconnected subset of G satisfying (9.1) and nE be a numbering function on E. Then there is a closed ideal I in H ∞ such that Z(I ) = E nE = nE on E. and ord(I, x) = nE (x) for every x ∈ E if and only if nE is bounded and Corollary 9.12. Let E be a nonvoid compact and totally disconnected subset of G satisfying (9.1). Suppose that NE,∞ is bounded on E. Let nE,1 be a bounded numbering function on E and I = f ∈ H ∞ : ord(f, x) nE,1 (x), x ∈ E . Then I is a closed ideal satisfying Z(I ) = E and ord(I, x) = nE,∞ . Proof. By the definition, I is a closed ideal in H ∞ . Since NE,∞ is bounded on E, by TheonE,∞ = nE,∞ on E. By Theorem 9.10, there is a closed ideal I1 rem 9.2 nE,∞ is bounded and in H ∞ such that Z(I1 ) = E and ord(I1 , x) = nE,∞ (x) for every x ∈ E. Since nE,1 nE,∞ , we

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have I1 ⊂ I . Hence E ⊂ Z(I ) ⊂ Z(I1 ) = E, so Z(I ) = E. Since nE,1 (x) ord(I, x) for x ∈ E, by Theorem 9.9 we have nE,∞ (x) ord(I, x) for x ∈ E, so nE,∞ (x) ord(I, x) ord(I1 , x) = nE,∞ (x),

x ∈ E.

Therefore ord(I, x) = ord(I1 , x) = nE,∞ (x) for every x ∈ E. By Theorem A, we get I = I1 .

2

Let I1 , I2 be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for i = 1, 2. We say that I1 ⊗ I2 has the factorization property if for every CN Blaschke product ϕ in I1 ⊗ I2 , there are CN Blaschke products ϕ1 , ϕ2 such that ϕi ∈ Ii for i = 1, 2 and ϕ1 ϕ2 ≺ ϕ. Theorem 9.13. Let I1 , I2 be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for i = 1, 2. Then I1 ⊗ I2 has the factorization property. Proof. We have Z(I1 ⊗ I2 )= Z(I1 ) ∪ Z(I2 ) ⊂ G. Let E =Z(I1 ) ∪ Z(I2 ). By Lemma 9.1, we have partitions Z(Ii ) = kj =1 Ei,j for i = 1, 2 such that { 2i=1 Ei,j : 1 j k} is a set of mutually disjoint open and closed subsets of E and E1,j ∪ E2,j ⊂ {|bj | < εj }. Let Ii,j = f ∈ H ∞ : ord(f, x) ord(Ii , x), x ∈ Ei,j for 1 j k and i = 1, 2. By Lemma 9.7(ii), Ii =

!k

j =1 Ii,j

for i = 1, 2. We have

k

I1 ⊗ I 2 =

" (I1,j ⊗ I2,j ),

Z(I1,j ⊗ I2,j ) = E1,j ∪ E2,j

j =1

and Z(I1,j ⊗ I2,j ) ∩ Z(I1, ⊗ I2, ) = ∅ for j = . Let ϕ ∈ I1 ⊗ I2 be a CN Blaschke product. Take {Uj : 1 j k} a set of open subsets of G such that E1,j ∪ E2,j ⊂ Uj for 1 j k and Uj ∩ U = ∅ for j = . Let ϕj be the subproduct of ϕ with zeros Z(ϕ) ∩ Uj ∩ D counting multiplicities. We have ord(ϕj , x) ord(I1 ⊗ I2 , x) = ord(I1,j ⊗ I2,j , x),

x ∈ E1,j ∪ E2,j

and kj =1 ϕj ≺ ϕ. By Theorem A, ϕj ∈ I1,j ⊗ I2,j . By Proposition 8.14, there are CN Blaschke products ψi,j ∈ Ii,j such that ψ1,j ψ2,j ≺ ϕj . We have qi :=

k j =1

k

ψi,j ∈

"

Ii,j = Ii ,

i = 1, 2

j =1

and q1 q2 ≺ ϕ. Hence I1 ⊗ I2 has the factorization property.

2

The following corollary answers Question 3. Corollary 9.14. Let I1 , I2 be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for i = 1, 2. Then I1 ⊗ I2 is a closed ideal in H ∞ , so I1 ⊗ I2 = I1 ⊗ I2 .


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Proof. Trivially we have I1 ⊗ I2 ⊂ I1 ⊗ I2 . Let ϕ ∈ I1 ⊗ I2 be a CN Blaschke product. By Theorem 9.13, there are CN Blaschke products ϕ1 ∈ I1 and ϕ2 ∈ I2 such that ϕ1 ϕ2 = ϕ. Then ϕ ∈ I1 ⊗ I2 . By Theorem B, we get I1 ⊗ I2 ⊂ I1 ⊗ I2 , so we have the assertion. 2 Corollary 9.15. Let Ii , 1 i n be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for 1 i n. ! !n Then ni=1 Ii = i=1 Ii . A compact subset E of G is called ρ-separated if there exists a positive number δ satisfying ρ(x, y) δ for every x, y ∈ E with x = y. In [20], the authors showed that if I is countably generated closed ideal in H ∞ satisfying Z(I !) ⊂ G, then there are closed Gδ and ρ-separated subsets E1 , E2 , . . . , Ek of G such that I = ni=1 I (Ei ). So by Corollary 9.15, I coincides with the tensor product of the associated primary ideals I (Ei ), 1 i n. Let I1 , I2 , . . . , In , be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for 1 i n. Let n i=1

Ii =

n

fi : fi ∈ Ii , 1 i n .

i=1

!n !n We have ni=1 Ii ⊂ i=1 Ii . In the last part of this paper, we shall prove that ni=1 Ii = i=1 Ii . The following is proved in Theorem 3.6 in [19]. Lemma 9.16. Let E be a nonvoid compact and ρ-separated subset of G. Let A ⊂ D satisfy E ⊂ A and A ∩ D = A. Then there is an interpolating Blaschke product b such that E ⊂ Z(b) and Z(b) ∩ D ⊂ A. For f ∈ H ∞ , we put

Z∞ (f ) = x ∈ M H ∞ : ord(f, x) = ∞ . Lemma 9.17. Let I be a closed ideal in H ∞ satisfying Z(I ) ⊂ G and f ∈ I with f = 0. Then there is a CN Blaschke product ϕ such that f/ϕ ∈ H ∞ and ord(ϕ, x) ord(I, x) for every x ∈ Z(I ) \ Z∞ (f ). x). By Theorem B, there are interpolating Blaschke products Proof. Let mI = maxx∈Z(I ) ord(I, I b1 , b2 , . . . , bmI such that m b ∈ I . Let f = Bh, where B is a Blaschke product and h ∈ H ∞ i i=1 satisfies |h| > 0 on D. Since Z(I ) ⊂ Z(f ), we have Z(I ) \ Z∞ (f ) ⊂ Z(B) ∩ D.

(9.2)

Since Z(bi ) is ρ-separated, by Lemma 9.16 there is an interpolating Blaschke product ϕ1 such that Z(I ) ∩ Z(B) ∩ D ∩ Z(b1 ) ⊂ Z(ϕ1 )

and ϕ1 ≺ B.

(9.3)

If Z(I ) ∩ Z(B) ∩ D ∩ Z(b1 ) = ∅, then we put ϕ1 = 1. Also there is an interpolating Blaschke product ϕ2 such that

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Z(I ) ∩ Z(B/ϕ1 ) ∩ D ∩ Z(b2 ) ⊂ Z(ϕ2 )

and ϕ2 ≺ B/ϕ1 .

If Z(I ) ∩ Z(B/ϕ1 ) ∩ D ∩ Z(b2 ) = ∅, then we put ϕ2 = 1. Repeating the same argument, there are interpolating Blaschke products ϕ3 , ϕ4 , . . . , ϕmI such that for each 3 j mI we have

B

Z(I ) ∩ Z j −1 i=1

ϕi

∩ D ∩ Z(bj ) ⊂ Z(ϕj )

(9.4)

and B ϕj ≺ j −1 i=1

. ϕi

I We note that ϕi = 1 for some 1 i mI . Let ϕ = m i=1 ϕi . Then ϕ is a CN Blaschke product and f/ϕ ∈ H ∞ . I Let x ∈ Z(I ) \ Z∞ (f ). Since f and m i=1 bi are contained in I , we have ord(I, x) ord(f, x)

ord(I, x) ord

and

m I

(9.5)

bi , x .

i=1

By (9.2), we have x ∈ Z(I ) ∩ Z(B) ∩ D, so by (9.3) we have ord(b1 , x) ord(ϕ1 , x).

(9.6)

Since ord(f, x) < ∞, we have 0 ord(f/ϕ, x) < ∞. If ord(f/ϕ, x) = 0, then ord(f, x) = ord(ϕ, x). Hence by (9.5), we have ord(I, x) ord(ϕ, x). Suppose that 1 ord(f/ϕ, x) < ∞. Then 1 ord(B/ϕ, x) < ∞, so we have B x ∈ Z j −1 i=1

∩ D,

ϕi

2 j mI .

Hence by (9.4), ord(bj , x) ord(ϕj , x) for 2 j mI . Therefore by (9.5) and (9.6), we have ord(I, x)

mI j =1

ord(bj , x)

mI

ord(ϕj , x) = ord(ϕ, x).

2

j =1

The following theorem answers Question 4. Theorem 9.18. Let I1 , I2 be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for i = 1, 2. Then I1 I2 = I1 ⊗ I2 , so I1 I2 is a closed ideal. Proof. Let J = I1 ⊗I2 . Since Z(J ) = Z(I1 )∪Z(I2 ) ⊂ G, Z(J ) is totally disconnected. Trivially we have I1 I2 ⊂ J . To show the reverse inclusion, let f ∈ J with f = 0. Let f = Bh, where B is a Blaschke product and h ∈ H ∞ satisfies |h| > 0 on D. Then Z(h) is a closed Gδ -set. By


2145

Corollary 3.1 in [16], Z∞ (B) is also a closed Gδ -set. Then Z∞ (f ) = Z∞ (B) ∪ Z(h) is a closed Gδ -set. There is a sequence of open subsets {Un }n of G such that Z(J ) \ Z∞ (f ) ⊂

∞

(9.7)

Un ,

n=1

Un ∩ U = ∅,

n = ,

(9.8)

Z(J ) ∩ Un is open and closed in Z(J )

(9.9)

and Z∞ (f ) ∩ Un = ∅

for n 1.

(9.10)

By Lemma 9.17, there is a CN Blaschke product ϕ such that f/ϕ ∈ H ∞ and ord(ϕ, x) ord(J, x),

x ∈ Z(J ) \ Z∞ (f ).

(9.11)

Then we have ϕ ≺ B and Z∞ (B) = Z∞ (B/ϕ), so f = (B/ϕ)hϕ

and Z∞ (f ) = Z∞ (B/ϕ) ∪ Z(h).

By Theorem 3.1 in [16], there are Blaschke products B1 , B2 such that B/ϕ = B1 B2 and Z∞ (B/ϕ) = Z∞ (B1 ) = Z∞ (B2 ). Since |h| > 0 on D, there is h1/2 ∈ H ∞ such that h = (h1/2 )2 . Hence we have

f = B1 h1/2 B2 h1/2 ϕ

(9.12)

Z∞ (f ) = Z∞ B1 h1/2 = Z∞ B2 h1/2 .

(9.13)

and

By (9.11), we have Z(J ) \ Z∞ (f ) ⊂ Z(ϕ). By (9.10), Z(J ) ∩ Un ⊂ Z(ϕ) for every n 1. Let ϕn be the subproduct of ϕ with zeros Z(ϕ) ∩ Un ∩ D counting multiplicities. By (9.8), we have ∞

ϕn ≺ ϕ

(9.14)

n=1

and by (9.11) we have ord(J, x) ord(ϕ, x) = ord(ϕn , x),

x ∈ Z(J ) ∩ Un .

(9.15)

Since J = I1 ⊗ I2 and Z(J ) = Z(I1 ) ∪ Z(I2 ), by (9.9) for each n 1 Z(Ii ) ∩ Un is open and closed in Z(Ii ) for i = 1, 2. Let Ii,n = f ∈ H ∞ : ord(f, x) ord(Ii , x), x ∈ Z(Ii ) ∩ Un ,

i = 1, 2

2146


and Jn = f ∈ H ∞ : ord(f, x) ord(J, x), x ∈ Z(J ) ∩ Un . Then Z(Ii,n ) = Z(Ii ) ∩ Un , ord(Ii,n , x) = ord(Ii , x) for x ∈ Z(Ii ) ∩ Un , Z(Jn ) = Z(J ) ∩ Un and ord(Jn , x) = ord(J, x) for x ∈ Z(J ) ∩ Un . We have also

Z(Jn ) = Z(I1 ) ∩ Un ∪ Z(I2 ) ∩ Un = Z(I1,n ) ∪ Z(I2,n ), and for x ∈ Z(Jn ) ord(Jn , x) = ord(J, x) = ord(I1 , x) + ord(I2 , x) = ord(I1,n , x) + ord(I2,n , x) = ord(I1,n ⊗ I2,n , x). Therefore by Theorem A, we have Jn = I1,n ⊗ I2,n . By (9.15), ord(Jn , x) ord(ϕn , x) for x ∈ Z(Jn ), so by Theorem A again ϕn ∈ Jn = I1,n ⊗ I2,n . By Theorem 9.13, there are Blaschke products ϕ1,n ∈ I1,n such that ϕn = ϕ1,n ϕ2,n . Let ψi =

∞

n=1 ϕi,n

ψ1 ψ2 =

∞

and ϕ2,n ∈ I2,n

(9.16)

for i = 1, 2. By (9.14),

ϕ1,n ϕ2,n =

n=1

∞

ϕn ≺ ϕ.

n=1

Let b1 = ψ1 and b2 = ψ2 (ϕ/(ψ1 ψ2 )). Then ϕ = b1 b2 . Let x ∈ Z(Ii ) \ Z∞ (f ) for i = 1, 2. Then x ∈ Z(J ) \ Z∞ (f ). By (9.7) and (9.8), there is a unique n such that x ∈ Z(Ii ) ∩ Un . Hence ord(bi , x) ord(ψi , x) = ord(ϕi,n , x) ord(Ii,n , x)

by (9.16)

= ord(Ii , x). Hence ord(bi , x) ord(Ii , x),

x ∈ Z(Ii ) \ Z∞ (f ), i = 1, 2.

(9.17)

Let f1 = b1 B1 h1/2 and f2 = b2 B2 h1/2 . By (9.12), we have f = f1 f2 . To show fi ∈ Ii , let x ∈ Z(Ii ). If x ∈ Z(Ii ) \ Z∞ (f ), then by (9.17) we have ord(fi , x) ord(Ii , x). If x ∈ Z∞ (f ), by (9.13) we have ord(fi , x) = ∞ > ord(Ii , x). By Theorem A, we get fi ∈ Ii for i = 1, 2. Hence f = f1 f2 ∈ I1 I2 . Thus I1 ⊗ I2 ⊂ I1 I2 , so we get I1 ⊗ I2 = I1 I2 . 2


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Corollary 9.19. Let I1 , I2 , . . . , Ik be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for 1 i k. !k Then ki=1 Ii = i=1 Ii and ki=1 Ii is a closed ideal in H ∞ . References [1] J. Bourgain, On finitely generated closed ideals in H ∞ (D), Ann. Inst. Fourier (Grenoble) 35 (1985) 163–174. [2] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958) 921–930. [3] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962) 547– 559. [4] S. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976) 81–89. [5] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. [6] T. Gamelin, Uniform Algebras, Prentice Hall, New Jersey, 1969. [7] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [8] P. Gorkin, Functions not vanishing on trivial Gleason parts of Douglas algebras, Proc. Amer. Math. Soc. 104 (1988) 1086–1090. [9] P. Gorkin, R. Mortini, Interpolating Blaschke products and factorization in Douglas algebras, Michigan Math. J. 38 (1991) 147–160. [10] P. Gorkin, R. Mortini, A. Nicolau, The generalized corona theorem, Math. Ann. 301 (1995) 135–154. [11] P. Gorkin, K. Izuchi, R. Mortini, Higher order hulls in H ∞ II, J. Funct. Anal. 177 (2000) 107–129. [12] C. Guillory, K. Izuchi, D. Sarason, Interpolating Blaschke products and division in Douglas algebras, Proc. Roy. Irish Acad. Sect. A 84 (1984) 1–7. [13] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, New Jersey, 1962. [14] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. 86 (1967) 74–111. [15] K.J. Izuchi, Countably generated Douglas algebras, Trans. Amer. Math. Soc. 299 (1987) 171–192. [16] K. Izuchi, Factorization of Blaschke products, Michigan Math. J. 40 (1993) 53–75. [17] K.J. Izuchi, Interpolating Blaschke products and factorization theorems, J. Lond. Math. Soc. (2) 50 (1994) 547–567. [18] K.J. Izuchi, The structure of the maximal ideal space of H ∞ , Sugaku Expositions 17 (2004) 171–184. [19] K.J. Izuchi, Y. Izuchi, Factorization of Blaschke products and primary ideals in H ∞ , J. Funct. Anal. 259 (2010) 975–1013. [20] K.J. Izuchi, Y. Izuchi, Gleason parts and countably generated closed ideals in H ∞ , preprint. [21] H.-M. Lingenberg, Interpolation sets in the maximal ideal space of H ∞ , Michigan Math. J. 39 (1992) 53–63. [22] D. Marshall, Subalgebras of L∞ containing H ∞ , Acta Math. 137 (1976) 91–98. [23] D. Sarason, Function Theory on the Unit Circle, Lectures Notes, Virginia Polytech. Inst. and State Univ., Blacksburg, VA, 1978. [24] V. Tolokonnikov, Blaschke products with the Carleson–Newman condition, and ideals of the algebra H ∞ , J. Soviet Math. 42 (1988) 1603–1610. ˇ [25] R.C. Walker, The Stone–Cech Compactification, Springer-Verlag, Berlin, 1974.


On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity Okihiro Sawada a , Ryo Takada b,∗ a Department of Mathematics, Darmstadt University of Technology, Schlossgartenstrasse 7,

D-64289 Darmstadt, Germany b Mathematical Institute, Tohoku University, 6-3 Aoba, Sendai 980-8578, Japan

Received 30 July 2010; accepted 11 December 2010 Available online 24 December 2010 Communicated by J. Coron

Abstract The Cauchy problem of the Euler equations in the whole space is considered with non-decaying initial 1 . It is proved that if the initial velocity is real analytic then the solution is velocity in the frame work of B∞,1 also real analytic in spatial variables. Furthermore, a new estimate for the size of the radius of convergence of Taylor’s expansion is established. The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms. It is also shown the propagation of the almost periodicity in spatial variables. © 2010 Elsevier Inc. All rights reserved. Keywords: The Euler equations; Analyticity; Almost periodicity; Non-decaying initial velocity


E-mail addresses: [email protected] (O. Sawada), [email protected] (R. Takada). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.011

O. Sawada, R. Takada / Journal of Functional Analysis 260 (2011) 2148–2162

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1. Introduction and main results Let us consider the Euler equations in Rn with n 2, describing the motion of perfect incompressible fluids, ⎧ ∂u ⎪ ⎪ + (u · ∇)u + ∇p = 0 in Rn × (0, T ), ⎨ ∂t (E) div u = 0 in Rn × (0, T ), ⎪ ⎪ ⎩ u(x, 0) = u0 (x) in Rn , where the unknown functions u = u(x, t) = (u1 (x, t), . . . , un (x, t)) and p = p(x, t) denote the velocity field and the pressure of the fluid, respectively, while u0 = u0 (x) = (u10 (x), . . . , un0 (x)) denotes the given initial velocity field. The purpose of this paper is to show the propagation properties of the analyticity and the almost periodicity in spatial variables for the solution of (E) with non-decaying initial velocity. For the local-in-time existence and uniqueness of solutions for (E), Kato [5] proved that for the given initial velocity field u0 ∈ H m (Rn )n with m > n/2 + 1, there exist T = T (u0 H m ) and a unique solution u of (E) in the class C([0, T ]; H m (Rn )n ). Kato and Ponce [6] extended this result to the fractional-ordered Sobolev spaces W s,p (Rn )n = (1 − )−s/2 Lp (Rn )n for s > n/p + 1, 1 < p < ∞. In order to treat the initial velocity with the minimal regularity, Pak and Park [8] 1 (Rn ) and obtained the following result. studied in the framework of B∞,1 1 (Rn )n with div u = 0, there exists Theorem 1.1. (See Pak and Park [8].) For every u0 ∈ B∞,1 0 1 n n a T > 0 such that (E) possesses a unique solution u ∈ C([0, T ]; B∞,1 (R ) ) with the pressure ∇p = ni,j =1 ∇(−)−1 ∂xi uj ∂xj ui . 1 (Rn ) and its properties will be explained in Section 2. The definition of the Besov space B∞,1 The reader can find the other results concerning the local-in-time existence and uniqueness of solutions to (E) in the reference of [8]. It has already been known that Kato’s solution is real analytic in spatial variables if u0 ∈ C ω (Rn )n ; see Alinhac and Métivier [2], Kukavica and Vicol [7] and the references therein. In this paper, we prove the propagation of analyticity of Pak–Park’s solutions. In particular, we give an improvement for the estimate for the size of the radius of convergence of Taylor’s expansion. Before stating our result about the analyticity, we set some notation. Let N0 := N ∪ {0}, where N is the set of all positive integers. For k ∈ N0 , put

mk := c

k! , (k + 1)2

where c is a positive constant such that one has α m|β| m|α−β| m|α| , α ∈ Nn0 , β 0βα α m|β|−1 m|α−β|+1 |α|m|α| , α ∈ Nn0 \ {0}n . β

0 7/2, and obtained the following estimate for uniform analyticity radius: t

α 1

∂x rot u(t)L∞ − |α| 2 −1 ρ 1+t exp −λ ∇u(τ ) L∞ dτ lim inf |α|→∞ α! 0

with some ρ := ρ(s, rot u0 ) and λ = λ(s). Hence our result is an improvement of the previous analyticity-rate in the sense that (1 + t 2 )−1 is replaced by (1 + t)−1 , and clarifies that ρ = ρ0 /L. The proof of Theorem 1.2 is based on the inductive argument with respect to |α|. The key of the proof is to derive the suitable estimates for the higher order derivatives of the nonlinear term of (E). To this end, we appeal to the technique due to [4], and use the commutator type estimates, the bilinear estimates (see Lemma 2.2 and Lemma 2.3 below) and the trajectory flow argument. We next consider the almost periodicity in spatial variables. We recall the definition of the almost periodicity in the sense of Bohr.


2151

Definition 1.4. Let f be a bounded continuous function on Rn . Put Σf := τξ f ξ ∈ Rn ⊂ L∞ Rn ,

τξ f := f (· + ξ ).

Then, f is called almost periodic in Rn if Σf is relatively compact in L∞ (Rn ). We now state the second result of this paper. 1 (Rn )n with div u = 0 and let u ∈ C([0, T ]; B 1 (Rn )n ) be the Theorem 1.5. Let u0 ∈ B∞,1 0 ∞,1 solution of (E). Suppose that u0 is almost-periodic in Rn , then the solution u(·, t) of (E) is almost-periodic in Rn for all t ∈ [0, T ].

The same assertion is known for the solutions to the Navier–Stokes equations by Giga, Mahalov and Nicolaenko [3]. Recently, Taniuchi, Tashiro and Yoneda [9] proved the almost periodicity of weak solutions to (E) in the whole plane R2 when u0 ∈ L∞ (R2 )2 . On the other hand, in the Theorem 1.5, we treat the classical solutions and all space-dimensions n 2. The proof of Theorem 1.5 is based on the argument given by [3]. The key of the proof is to use the estimate concerning the continuity with respect to the initial velocities, see Lemma 4.1 below. This paper is organized as follows. In Section 2, we introduce the notation that will be used throughout the paper, and recall the key lemmas which play important roles in our proof. In Sections 3 and 4, we present the proof of Theorems 1.2 and 1.5, respectively. 2. Preliminaries In this section, we introduce some notation and the function spaces. Let S (Rn ) be the Schwartz class of all rapidly decreasing functions, and let S (Rn ) be the space of all tempered distributions. We first recall the definition of the Littlewood–Paley operators. Let Φ and ϕ be the functions in S (Rn ) satisfying the following properties: ⊂ ξ ∈ Rn |ξ | 5/6 , supp Φ )+ Φ(ξ

∞

supp ϕ ⊂ ξ ∈ Rn 3/5 |ξ | 5/3 , ϕj (ξ ) = 1,

ξ ∈ Rn ,

j =0

where ϕj (x) = 2j n ϕ(2j x) and f denotes the Fourier transform of f ∈ S (Rn ) on Rn . Given f ∈ S (Rn ), we denote ⎧ ⎨ Φ ∗ f, j = −1, j f := ϕj ∗ f, j 0, ⎩ 0, j −2,

Sk f :=

j f,

k ∈ Z,

j k

s (Rn ) by the where ∗ denotes the convolution operator. Then, we define the Besov spaces Bp,q following definition. s (Rn ) is defined to be the set Definition 2.1. For s ∈ R and 1 p, q ∞, the Besov space Bp,q n of all tempered distributions f ∈ S (R ) such that the norm

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s f Bp,q = 2sj j f Lp j ∈Z q is finite. s (Rn ) is a Banach space with its norm · s . It is easy to see that Note that Bp,q Bp,q

f f L∞ =

j

j ∈Z

L∞

j f L∞ = f B 0 . ∞,1

j ∈Z

0 (Rn ) ⊂ L∞ (Rn ), and this embedding is continuous. It is also easily obtained Therefore B∞,1 0 that B∞,1 (Rn ) ⊂ BU C(Rn ), where BU C(Rn ) is the space of all bounded uniformly continu1 (Rn ) ⊂ W 1,∞ (Rn ), which is conous functions on Rn . Analogously, we can prove that B∞,1 1 tinuous embedding. Moreover, B∞,1 (R) contains some non-decaying functions, for example, −x

]. For more details, see Triebel [10]. [x → sin x], [x → cos x] and [x → tanh x = eex −e +e−x We now prepare the commutator type estimates and the bilinear estimates for nonlinear terms of (E). x

Lemma 2.2. (See Pak and Park [8].) There exists a positive constant C = C(n) such that j ∈Z

2j (Sj −2 u · ∇)j f − j (u · ∇)f L∞ CuB 1 f B 1 ∞,1

∞,1

1 (Rn )n+1 with div u = 0. holds for all (u, f ) ∈ B∞,1

Lemma 2.3. There exists a positive constant C = C(n) such that f gB 1

∞,1

C f L∞ gB 1

∞,1

+ gL∞ f B 1

∞,1

1 (Rn ). holds for all f, g ∈ B∞,1

The proof of Lemma 2.3 follows from the characterization by differences of Besov norm, easily; see [10]. Hence we skip the detail of the proof. Next, we give the estimate for the gradient of pressure π = ∇p. Lemma 2.4. (See Pak and Park [8].) There exists a positive constant C = C(n) such that

π(u, v)

1 B∞,1

CuB 1 vB 1 ∞,1

∞,1

1 (Rn )n with div u = div v = 0, where holds for all u, v ∈ B∞,1

π(u, v) =

n j,k=1

∇(−)−1 ∂xj uk ∂xk v j = ∇(−)−1 div (u · ∇)v .


2153

Finally, we recall the Gronwall inequality. Lemma 2.5 (The Gronwall inequality). Let A 0, and let f, g and h be non-negative, continuous functions on [0, T ] satisfying t

t g(s) ds +

f (t) A +

h(s)f (s) ds

0

0

for all t ∈ [0, T ]. Then it holds that f (t) Ae

t

t

0 h(τ ) dτ

+

e

t s

h(τ ) dτ

g(s) ds

0

for all t ∈ [0, T ]. 3. Proof of Theorem 1.2 s (Rn )n ) for all s 1, if u ∈ Proof of Theorem 1.2. We first notice that u ∈ C([0, T ]; B∞,1 0 s n n ∞ n n B∞,1 (R ) for all s 1. Hence u(·, t) ∈ C (R ) for all t ∈ [0, T ] and then u ∈ C ∞ (Rn × [0, T ])n , if u0 ∈ C ∞ (Rn )n . Moreover, the time-interval in which the solution exists does not depend on s. Indeed, T C/u0 B 1 with some constant C depending only on n, and the ∞,1 solution u satisfies

sup u(t) B 1

∞,1

t∈[0,T ]

C0 u0 B 1

(3.1)

∞,1

with some positive constant C0 depending only on n. Now let u0 satisfy the assumption of Theorem 1.2. We discuss with the induction argument. In the case α = 0, (1.1) follows from (3.1) with K = C0 K0 . Next, we consider the case |α| 1. We first introduce some notation. For l ∈ N and λ, L > 0, we put

Xl (t) := max ∂xα u(t) B 1 , |α|=l

Yl = Ylλ,L

∞,1

:= max sup

1kl t∈[0,T ]

t ∈ [0, T ],

Mk (t) Xk (t) , mk

where −λk Mk (t) = Mkλ,L (t) := ρ0k L−(k−1) (1 + t)−(k−1) e

t 0

u(τ )B 1

∞,1

dτ

.

The similar notation were used in [1] and [2]. In what follows, we shall show that Y|α| 2K0 for all α ∈ Nn0 with |α| 1 when λ and L are sufficiently large. We now consider the case |α| = 1. Let k be an integer with 1 k n. Taking the differential operation ∂xk to the first equation of (E), we have

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∂t (∂xk u) + (∂xk u · ∇)u + (u · ∇)∂xk u + ∂xk π(u, u) = 0,

(3.2)

where ∇p = π(u, u) =

n

∇(−)−1 ∂xj uk ∂xk uj = ∇(−)−1 div (u · ∇)u .

j,k=1

Applying the Littlewood–Paley operator j and adding the term (Sj −2 u · ∇)j (∂xk u) to the both sides of (3.2), we have ∂t j (∂xk u) + (Sj −2 u · ∇)j (∂xk u) = (Sj −2 u · ∇)j (∂xk u) − j (u · ∇)∂xk u − j (∂xk u · ∇)u − j ∂xk π(u, u) .

(3.3)

Here we consider the family of trajectory flows {Zj (y, t)} defined by the solution of the ordinary differential equations ⎧ ⎨ ∂ Zj (y, t) = Sj −2 u Zj (y, t), t , ∂t ⎩ Z (y, 0) = y. j

(3.4)

Note that Zj ∈ C 1 (Rn × [0, T ])n , and div Sj −2 u = 0 implies that each y → Zj (y, t) is a volume preserving mapping from Rn onto itself. From (3.3) and (3.4), we see that ∂t j (∂xk u) + (Sj −2 u · ∇)j (∂xk u)|(x,t)=(Zj (y,t),t) =

∂ j (∂xk u) Zj (y, t), t , ∂t

which yields that j (∂xk u) Zj (y, t), t = j (∂xk u0 )(y) −

t

j (∂xk u · ∇)u Zj (y, s), s ds

0

t +

(Sj −2 u · ∇)j (∂xk u) − j (u · ∇)∂xk u Zj (y, s), s ds

0

t −

j ∂xk π(u, u) Zj (y, s), s ds.

(3.5)

0

Since the map y → Zj (y, t) is bijective and volume-preserving for all t ∈ [0, T ], by taking the L∞ -norm with respect to y to both sides of (3.5), we have

j (∂x u)(t) ∞ j (∂x u0 ) ∞ + k k L L

t 0

j (∂x u · ∇)u (s) ∞ ds k L


t +

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(Sj −2 u · ∇)j (∂x u) − j (u · ∇)∂x u (s) ∞ ds k k L

0

t +

j ∂x π(u, u) (s) ∞ ds. k L

(3.6)

0

Multiplying both sides of (3.6) by 2j and then taking the 1 -norm in j , we obtain that

∂x u(t) 1 k B

t

∞,1

∂xk u0 B 1

+

∞,1

(∂x u · ∇)u(s) 1 ds + k B ∞,1

0

+

t 0 j ∈Z

t

∂x π(u, u)(s) 1 ds k B ∞,1

0

2j (Sj −2 u · ∇)j (∂xk u) − j (u · ∇)∂xk u (s) L∞ ds

=: I1 + I2 + I3 + I4 .

(3.7)

It follows from the assumption on u0 that I1 K0 ρ0−1 m1 .

(3.8)

From Lemma 2.3, we see that t I2 C

∇u(s)

∞ ∇u(s)

L

t 1 B∞,1

ds C

0

u(s)

1 B∞,1

X1 (s) ds,

(3.9)

0

where we used the continuous embedding ∇f L∞ Cf B 1 . For the pressure term I3 , it ∞,1 follow from Lemma 2.4 that t I3 2

π(∂x u, u)(s) 1 ds C k B

t

∞,1

0

u(s)

1 B∞,1

X1 (s) ds.

(3.10)

0

For the estimate of I4 , we have from Lemma 2.2 that t I4 C

u(s)

1 B∞,1

∂x u(s) 1 ds C k B

t

∞,1

0

u(s)

1 B∞,1

X1 (s) ds.

(3.11)

0

Substituting (3.8), (3.9), (3.10) and (3.11) into (3.7), we have

∂x u(t) 1 k B

∞,1

K0 ρ0−1 m1

t + C1 0

u(s)

1 B∞,1

X1 (s) ds

(3.12)

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with some positive constant C1 depending only on n. Since k ∈ {1, . . . , n} is arbitrary, it follows from (3.12) that

X1 (t) K0 ρ0−1 m1

t + C1

u(s)

1 B∞,1

X1 (s) ds,

0

which implies by Lemma 2.5 that X1 (t) K0 ρ0−1 m1 e

C1

t 0

u(τ )B 1

∞,1

dτ

.

(3.13)

By choosing λ C1 , we obtain from (3.13) that (C1 −λ) M1 (t) X1 (t) K0 e m1

t 0

u(τ )B 1

∞,1

dτ

K0 ,

which yields that Y1 K0 .

(3.14)

Next, we consider the case |α| 2. Let α be a multi-index with |α| 2. Taking the differential operation ∂xα to the first equation of (E), we have α α ∂xβ u · ∇ ∂xα−β u + ∂xα π(u, u) = 0. ∂t ∂x u + β

(3.15)

0βα

Applying the Littlewood–Paley operator j and adding the term (Sj −2 u · ∇)j (∂xα u) to the both sides of (3.15), we have ∂t j ∂xα u + (Sj −2 u · ∇)j ∂xα u = (Sj −2 u · ∇)j ∂xα u − j (u · ∇)∂xα u α j ∂xβ u · ∇ ∂xα−β u − j ∂xα π(u, u) . − β

(3.16)

00 s∈R t>0 The proof is complete.

2

Now we are in the position to check the required commutator condition and thus to prove Theorem 5.2. Proof of Theorem 5.2. Consider now the evolution semigroup corresponding to the nonautonomous equation (5). The corresponding generator is given formally as −

d + + V (t). ds

A. Bátkai et al. / Journal of Functional Analysis 260 (2011) 2163–2190

2185

Take now f ∈ BUC1 (R; H2 (Rd )), and notice that then f belongs to the domain D(A). We calculate the commutator of A and B. We have [A, B]f = −V (t)f (t) + V (t) f + 2∇V (t) · ∇f (t). Now, if we assume that V ∈ BUC1 (R; L∞ (Rd )) and V ∈ BUC(R; W 2,∞ (Rd )), then the first two terms can be estimated by c f , so we have only to deal with the term 2∇V · ∇f , for which it suffices to estimate ∂i f (t) for i = 1, . . . , d. We have

∂i f (t) c 1/2 f (t)

2 2

∂i is 1/2 -bounded on L2 .

By Proposition 5.4 this completes the proof of the commutator condition (1) in the form

[A, B]f (−A)α f for all f ∈ D(A) with some given α 1/2. Hence Theorem 1.2 yields the assertion.

2

6. Numerical examples illustrating the convergence In Section 5 we already introduced the non-autonomous parabolic equation (sometimes also called imaginary time Schrödinger equation) ∂t u(x, t) = u(x, t) + V (x, t)u(x, t) in Rd with appropriate initial conditions with V being a smooth and bounded function. In the following we will apply the sequential splitting introduced in Section 3 to the sub-operators A(t) := and B(t) := multiplication by V (x, t). In Theorem 2.2 we showed that the product formula describing the sequential splitting is convergent also in the case if we are able to solve the corresponding autonomous Cauchy problems (Eq. 1)–(Eq. 2) with operators A(r) and B(r) for every time level r ∈ R. We will use this result when constructing our numerical scheme. In order to illustrate numerically the convergence of the sequential splitting and give an estimate on its order, let us consider the following non-autonomous equation with boundary and initial conditions: ⎧ 2 ⎪ ⎨ ∂t u(x, t) = ∂x u(x, t) + V (x, t)u(x, t), u(0, t) = u(1, t) = 0, ⎪ ⎩ u(x, 0) = u0 (x),

t 0, x ∈ [0, 1], t 0, x ∈ [0, 1]

with functions V (x, t) and u0 (x) given later on in the example.

(7)

2186


6.1. Error analysis Let (uspl )ni denote the approximation of the exact solution u(iδ, nτ ) of problem (7) at time nτ and at the grid point iδ (with n = 0, . . . , N − 1 and i = 0, . . . , I − 1) using sequential splitting. 1 At this point the time-step τ = N 1−1 and the grid size δ = I −1 have certain given values. We call n n n n (uspl ) = ((uspl )0 , (uspl )1 , . . . , (uspl )I −1 ), n = 0, 1, . . . , N − 1, the split solution of problem (7). As already seen, the order of the splitting procedure can be estimated with the help of the splitting error defined by

n Espl := un − unspl

where un = (un0 , un1 , . . . , unI−1 ) with uni = u(iδ, nτ ), i = 0, 1, . . . , I − 1. With this notation the splitting procedure (or an arbitrary finite difference method) is of order p > 0 if for sufficiently smooth initial values there is a constant C > 0 such that for all t ∈ [0, t0 ] we have n Espl

C , np

or, if the method is stable, equivalently, 1 C τ p+1 . Espl 1 is In general, the exact solution of problem (7) is unknown, therefore, the local splitting error Espl n to be estimated as well. To this end we compute a so-called reference solution uref on a finer space grid using no splitting procedure. Then the order p of the splitting procedure can be determined 1 Cτ p+1 . Approximating un with un , we as follows. From the definition of p we have Espl ref 1 ≈E 1 := u1 − u1 Cτ p+1 . Thus, obtain Espl spl ref spl 1 log Espl (p + 1) log τ + log C. 1 for many different Then we can estimate p by computing the approximate local splitting error Espl values of the time-step τ , plotting the logarithm of the results, and fitting a line of form y(w) = aw + b to them. Hence, a ≈ p + 1 and b ≈ log C. Note, however, that the split solution contains not only the splitting error but also a certain amount of error originating from the spatial and temporal discretization. In what follows we show how to determine the numerical solutions u1ref and u1spl . We also note that it is reasonable to compute a relative local error defined as

Eloc =

1 Espl

u1ref

because this yields the ratio how the split solution differs from the reference solution.


2187

6.2. Numerical scheme In order to solve numerically the problem (7) we should discretize it in both space and time. For the temporal discretization we used the Crank–Nicholson method, and we chose the finite difference method for the spatial discretization. 6.2.1. Reference solution As mentioned above, we need a reference solution unref computed without using splitting procedures. After discretizing the equation, we obtain the following numerical scheme for determining (un+1 ref )i : −1 n+1 1 + (Href )ni unref i uref i = 1 − (Href )n+1 i

(8)

with (Href )ni

n+1 n+1 n+1 τ ui+1 − 2ui + ui−1 n = + Vi , 2 δ2

where Vin := V (iδ, nτ ). 6.2.2. Split solution Application of sequential splitting means that instead of the whole problem (7) two subproblems are solved. In our examples the first sub-problem corresponds to the diffusion equation ∂t uA (x, t) = ∂x2 uA (x, t). Its numerical solution unA can also be computed using Crank–Nicholson temporal and finite difference spatial discretization methods. Then we obtain the following numerical scheme similar to (8): n+1 −1 uA i = 1 − (HA )n+1 1 + (HA )ni unA i i

(9)

with (HA )ni =

n+1 n+1 n+1 τ ui+1 − 2ui + ui−1 . 2 δ2

The second sub-problem has the multiplication operator by V (x, t) on its right-hand side, i.e. ∂t uB (x, t) = V (x, t)uB (x, t). We refer again to Theorem 2.2 and take the function V only at time levels t = nτ , n = 0, 1, . . . , N − 1. In this (autonomous) case the exact solution uB (x, t) = etV (x,nτ ) u0 (x) is known. At the nth time level and on the space grid it has the form n uB i = uB (iδ, nτ ) = eτ V (iδ,nτ ) u0 (iδ). Due to the product formula (3), the split solution unspl is given by the following algorithm: for i = 0, . . . , I − 1 initial function: (u0A )i := u0 (iδ) end

(10)

2188


Fig. 1. Numerical solution of Eq. (7) at time levels t = 0, t = 10−3 , t = 5 · 10−3 , and t = 10−2 , respectively.

for n = 0, 1, . . . , N − 1 for i = 0, 1, . . . , I − 1 solve the first sub-problem using (9) ⇒ (unA )i end for i = 0, 1, . . . , I − 1 solve the second sub-problem using (10) ⇒ (unB )i end end −1 N −1 split solution: uN spl := uB

6.3. Numerical results Now we present some numerical results on the following example. Choose V (x, t) = t − 500x 2

and u0 (x) = e−50(x−0.4) . 2

Since the exact solution is unknown in this case, we should estimate the local splitting error using the reference solution instead of the exact one. Then the relative local splitting error Eloc and its order p can be measured. In Fig. 1 the time-behavior of the reference solution can be seen at the four time levels t = 0, t = 10−3 , t = 5 · 10−3 , and t = 10−2 , respectively. The effect of the diffusion can be clearly observed. Fig. 2 shows the result of the fitting. The dots correspond to log(Eloc ) for the various step sizes. The line fitted to these points has the form y(log(τ )) = a log(τ ) + b with a = 1.9470 and b = 3.25925. As mentioned above, the order of the splitting procedure p can be estimated by a − 1 ≈ 1, that is, the sequential splitting is of first order.


2189

Fig. 2. Results obtained by applying the sequential splitting with various time steps (dots), and the line y(w) = aw + b fitted to them with parameters a = 1.9470 ≈ p + 1 and b = 3.25925.

Acknowledgments A. Bátkai was supported by the Alexander von Humboldt-Stiftung. We thank Wolfgang Arendt (Ulm), Roland Schnaubelt (Karlsruhe) and Alexander Ostermann (Innsbruck) for interesting and useful discussions. The European Union and the European Social Fund have provided financial support to the project under the grant agreement no. TÁMOP-4.2.1/B-09/1/KMR. References [1] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H.P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, Lecture Notes in Math., vol. 1184, Springer-Verlag, Berlin, 1986. [2] A. Bátkai, P. Csomós, G. Nickel, Operators and spatial approximations for evolution equations, J. Evol. Equ. 9 (2009) 613–636. [3] M. Bjørhus, Operator splitting for abstract Cauchy problems, IMA J. Numer. Anal. 18 (1998) 419–443. [4] V. Cachia, V.A. Zagrebnov, Operator-norm approximation of semigroups by quasi-sectorial contractions, J. Funct. Anal. 180 (2001) 176–194. [5] P.R. Chernoff, Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators, Mem. Amer. Math. Soc., vol. 140, American Mathematical Society, Providence, RI, 1974. [6] P. Csomós, G. Nickel, Operator splitting for delay equations, Comput. Math. Appl. 55 (2008) 2234–2246. [7] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., vol. 194, Springer-Verlag, New York, 2000, with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli, R. Schnaubelt. [8] D.E. Evans, Time dependent perturbations and scattering of strongly continuous groups on Banach space, Math. Ann. 221 (1976) 275–290. [9] I. Faragó, Á. Havasi, Consistency analysis of operator splitting methods for C0 -semigroups, Semigroup Forum 74 (2007) 125–139. [10] I. Faragó, Á. Havasi, Operator Splittings and Their Applications, Math. Res. Dev., Nova Science Publishers, New York, 2009. [11] T. Graser, Operator multipliers generating strongly continuous semigroups, Semigroup Forum 55 (1997) 68–79. [12] E. Hansen, A. Ostermann, Exponential splitting for unbounded operators, Math. Comp. 78 (2009) 1485–1496. [13] E. Hansen, A. Ostermann, Dimension splitting for time dependent operators, in: X. Hou, et al. (Eds.), Dynamical Systems and Differential Equations, Proceedings of the 7th AIMS International Conference, Arlington, Texas, USA, DCDS Supplement 2009, American Institute of Mathematical Sciences, Springfield MO, 2009, pp. 322–332.

2190


[14] H. Holden, K.H. Karlsen, K.-A. Lie, N.H. Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions, European Mathematical Society, 2010. [15] J.S. Howland, Stationary scattering theory for time-dependent Hamiltonians, Math. Ann. 207 (1974) 315–335. [16] T. Ichinose, H. Neidhardt, V.A. Zagrebnov, Trotter–Kato product formula and fractional powers of self-adjoint generators, J. Funct. Anal. 207 (2004) 33–57. [17] T. Jahnke, C. Lubich, Error bounds for exponential operator splittings, BIT 40 (4) (2000) 735–744. [18] T. Kato, Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970) 241–258. [19] H. Kellermann, Linear evolution equations with time-dependent domain, Semesterberichte Funktionalanalysis, Tübingen, WS, 1985. [20] M. Kovács, On positivity, shape, and norm-bound preservation of time-stepping methods for semigroups, J. Math. Anal. Appl. 304 (2005) 115–136. [21] R. Nagel, G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems, in: Evolution equations, semigroups and functional analysis, Milano, 2000, in: Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2002, pp. 279–293. [22] R. Nagel, G. Nickel, S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaest. Math. 19 (1996) 83–100. [23] H. Neidhardt, On abstract linear evolution equations, I, Math. Nachr. 103 (1981) 283–298. [24] H. Neidhardt, On abstract linear evolution equations, II, Prepr., Akad. Wiss. DDR, Inst. Math. P-MATH-07/81, Berlin, 1981. [25] H. Neidhardt, On abstract linear evolution equations, III, Prepr., Akad. Wiss. DDR, Inst. Math. P-MATH-05/82, Berlin, 1982. [26] H. Neidhardt, V.A. Zagrebnov, Trotter–Kato product formula and symmetrically normed ideals, J. Funct. Anal. 167 (1999) 113–147. [27] H. Neidhardt, V.A. Zagrebnov, Linear non-autonomous Cauchy problems and evolution semigroups, Adv. Differential Equations 14 (2009) 289–340. [28] G. Nickel, Evolution semigroups for nonautonomous Cauchy problems, Abstr. Appl. Anal. 2 (1997) 73–95. [29] G. Nickel, Evolution semigroups and product formulas for nonautonomous Cauchy problems, Math. Nachr. 212 (2000) 101–116. [30] G. Nickel, R. Schnaubelt, An extension of Kato’s stability condition for nonautonomous Cauchy problems, Taiwanese J. Math. 2 (1998) 483–496. [31] R. Schnaubelt, Sufficient conditions for exponential stability and dichotomy of evolution equations, Forum Math. 11 (1999) 543–566. [32] 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. [33] P.-A. Vuillermot, A generalization of Chernoff’s product formula for time-dependent operators, J. Funct. Anal. 259 (2010) 2923–2938. [34] P.-A. Vuillermot, W.F. Wreszinski, V.A. Zagrebnov, A Trotter–Kato product formula for a class of non-autonomous evolution equations, Nonlinear Anal. 69 (2008) 1067–1072. [35] P.-A. Vuillermot, W.F. Wreszinski, V.A. Zagrebnov, A general Trotter–Kato formula for a class of evolution operators, J. Funct. Anal. 257 (2009) 2246–2290.


Loops in SU(2) and factorization Doug Pickrell Mathematics Department, University of Arizona, 617 N. Santa Rita, Tucson, AZ, United States Received 18 May 2009; accepted 3 January 2011 Available online 13 January 2011 Communicated by L. Gross

Abstract We discuss analytic issues associated with a refinement of triangular factorization for the loop group of SU(2). This factorization is of interest because (1) Toeplitz determinants factor in the associated coordi2 nates, and (2) the factorization is intimately related to the critical degree of smoothness for loops, W 1/2,L . © 2011 Elsevier Inc. All rights reserved. Keywords: Loop group; Factorization; Toeplitz operator; Determinant

0. Introduction The main purpose of this paper is to prove functional analytic generalizations of Theorems 0.1 and 0.2 below (which are basically algebraic). Let Lfin SU(2) (Lfin SL(2, C)) denote the group consisting of functions S 1 → SU(2) (SL(2, C), respectively) having finite Fourier series, with pointwise multiplication. For example, for ζ ∈ C and n ∈ Z, the function 1 ζ z−n , S 1 → SU(2) : z → a(ζ ) −ζ¯ zn 1 where a(ζ ) = (1 + |ζ |2 )−1/2 , is in Lfin SU(2). It is known is dense in that Lfin SU(2) C ∞ (S 1 , SU(2)) (Proposition 3.5.3 of [8]). Also, if f (z) = fn zn , let f ∗ (z) = f¯n z−n . If f ∈ H 0 (), then f ∗ ∈ H 0 (∗ ), where is the open unit disk, ∗ is the open unit disk at ∞, and H 0 (U ) denotes the space of holomorphic functions for a domain U . E-mail address: [email protected]. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.001

2192

D. Pickrell / Journal of Functional Analysis 260 (2011) 2191–2221

Theorem 0.1. Suppose that k1 ∈ Lfin SU(2). The following are equivalent: (I.1) k1 is of the form k1 (z) =

b(z) , a ∗ (z)

a(z) −b∗ (z)

z ∈ S1,

where a and b are polynomials in z, and a(0) > 0. (I.2) k1 has a factorization of the form k1 (z) = a(ηn )

−η¯ n zn 1

1 ηn z−n

. . . a(η0 )

1 η0

−η¯ 0 1

,

for some finite subset {η0 , . . . , ηn } ⊂ C. (I.3) k1 has triangular factorization of the form

n 1 −j j =0 y¯j z

0 1

0

a1 0

a1−1

α1 (z) γ1 (z)

β1 (z) , δ1 (z)

where a1 > 0, the third factor is a polynomial in z which is unipotent upper triangular at z = 0. Similarly, for k2 ∈ Lfin SU(2), the following are equivalent: (II.1) k2 is of the form k2 (z) =

−c∗ (z) , d(z)

d ∗ (z) c(z)

z ∈ S1,

where c and d are polynomials in z, c(0) = 0, and d(0) > 0. (II.2) k2 has a factorization of the form

1 k2 (z) = a(ζn ) ¯ − ζn z n

ζn z−n 1

for some finite subset {ζ1 , . . . , ζn } ⊂ C. (II.3) k2 has triangular factorization of the form n −j a 1 2 j =1 x¯ j z 0 0 1

1 . . . a(ζ1 ) −ζ¯1 z

0 a2−1

α2 (z) γ2 (z)

ζ1 z−1 1

,

β2 (z) , δ2 (z)

where a2 > 0 and the third factor is a polynomial in z which is unipotent upper triangular at z = 0. Remark. The two sets of conditions are equivalent; they are intertwined by the outer involution σ of LSL(2, C) given by σ

a c

b d

=

d bz

cz−1 a

.

(0.1)


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This theorem basically follows from results in [6], but it is possible to give a direct argument (not involving Lie theory). We will present this, and functional analytic generalizations, in Section 2. The terminology regarding triangular factorization and Toeplitz operators in the following theorem is reviewed in Section 1. Theorem 0.2. (a) If {ηi } and {ζj } are rapidly decreasing sequences of complex numbers, then the limits k1 (z) = lim a(ηn ) n→∞

1

−η¯ n zn

ηn z−n

1

. . . a(η0 )

1

−η¯ 0

η0

1

and k2 (z) = lim a(ζn ) n→∞

1 −ζ¯n zn

ζn z−n 1

. . . a(ζ1 )

1 −ζ¯1 z

ζ1 z−1 1

,

exist in C ∞ (S 1 , SU(2)). (b) Suppose g ∈ C ∞ (S 1 , SU(2)). The following are equivalent: (i) g has a triangular factorization g = lmau (see (1.1)), where l and u have C ∞ boundary values. (ii) g has a factorization of the form g(z) = k1∗ (z)

eχ(z) 0

0 e−χ(z)

k2 (z),

where χ ∈ C ∞ (S 1 , iR), and k1 and k2 are as in (a). (iii) The Toeplitz operator A(g) (see (1.3)) and the shifted Toeplitz operator A1 (g) (see the paragraph following (1.5)) are invertible. Remarks. (a) Suppose that g ∈ Lfin SU(2). The l and u factors in (i) are also in Lfin SL(2, C), but they are essentially never unitary on S 1 . On the other hand the factors kj in (ii) are unitary, but in general they are not in Lfin SU(2). [If k1 , k2 ∈ Lfin SU(2), then χ must be constant. Since Lfin SU(2) is dense in C ∞ (S 1 , SU(2)), the parameterization in (ii) implies that generically g will correspond to nonconstant χ .] (b) There is a generalization of this theorem with U (2) in place of SU(2), where one restricts to loops in the identity component. We will restrict our attention to SU(2), to simplify the exposition. (c) This factorization is of great interest because in particular (1) the Toeplitz determinant det(A(g)∗ A(g)) factors in the associated coordinates (see Theorem 2.2 below), (2) the invariant measures discussed in Part III of [5] factor in these coordinates, and conjecturally (3) the Evens–Lu homogeneous Poisson structure discussed in [6] factors in these coordinates.

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The outline of the paper is the following. Section 1 is a review of standard facts about triangular factorization. In Sections 2 and 3, we prove Theorems 0.1 and 0.2, respectively. In these two sections, the main point is to extend the equivalences above to other function spaces, especially the critical 2 Sobolev space W 1/2,L ; see Theorems 2.3 and 3.2. It seems possible that there are L2 generalizations of these theorems. This is briefly discussed in Section 4. In Appendix A we discuss the combinatorial relation between x ∗ and ζ in Theorem 0.1. This relation is central to the L2 question, and applications. Unfortunately this relation remains mysterious to me. The generalization of the algebraic aspects of this paper from SU(2) to general simply connected compact groups is known [6,7], but considerably more complicated. For SU(2) it suffices to consider one representation, the defining representation, which greatly simplifies everything. Notation. Sobolev spaces will be denoted W s , and will always be understood in the L2 by ∞ sense. The space of sequences satisfying n=1 n|ζn |2 < ∞ will be denoted by w 1/2 . We will write Meas(S 1 , SU(2)) for the group of (equivalence classes of) measurable maps. This group is usually equipped with the topology of convergence in measure, but this will not play a role in this paper. We will use [4] as a general reference for Hankel and Toeplitz operators. C) 1. Triangular factorization for LSL(2,C Suppose that g ∈ L1 (S 1 , SL(2, C)). A triangular factorization of g is a factorization of the form g = l(g)m(g)a(g)u(g),

(1.1)

where l=

l11 l21

l12 l22

l has a L2 radial limit, m = u=

u11 u21

∈ H ∗ , SL(2, C) , 0

0 , 0 m−1 0

m0

u12 u22

m0 ∈ S 1 , a(g) =

∈ H , SL(2, C) , 0

l(∞) = 0 , 0 a0−1

a0

1 l21 (∞)

0 1

,

a0 > 0,

u(0) =

1 u12 (0) 0 1

,

and u has a L2 radial limit. Note that (1.1) is an equality of measurable functions on S 1 . A Birkhoff (or Wiener–Hopf, or Riemann–Hilbert) factorization is a factorization of the form g = g− g0 g+ , where g− ∈ H 0 (∗ , ∞; SL(2, C), 1), g0 ∈ SL(2, C), g+ ∈ H 0 (, 0; SL(2, C), 1), and g± have L2 radial limits on S 1 . Clearly g has a triangular factorization if and only if g has a Birkhoff factorization and g0 has a triangular factorization, in the usual sense of matrices. Proposition 1. Birkhoff and triangular factorizations are unique.


2195

Proof. If g− g0 g+ = h− h0 h+ are two Birkhoff factorizations, then the function F equal to −1 1 1 h−1 − g− for |z| 1 and (h0 h+ ) g0 g+ for |z| 1 is holomorphic on C \ S and integrable on S . Integrability implies that the singularities along S 1 are removable. Therefore F is constant, and the normalization conditions force F = 1. This implies uniqueness. 2 Remark. In the definition of Birkhoff factorization, if the L2 condition is replaced by the weaker condition that g± have pointwise radial limits a.e. on S 1 , then factorization is not unique. For example

1 0 0 1

z+1 =

z−1

0

0

z−1 z+1

−1 0 0 −1

− z−1 z+1

0

0

− z+1 z−1

is a factorization in this weaker sense. At least for the purposes of this paper, L2 appears to be the natural regularity condition in the definitions of factorization. As in [8], consider the polarized Hilbert space H := L2 S 1 , C 2 = H+ ⊕ H− ,

(1.2)

where H+ = P+ H consists of L2 -boundary values of functions holomorphic in . If g ∈ L∞ (S 1 , SL(2, C)), we write the bounded multiplication operator defined by g on H as Mg =

A(g) C(g)

B(g) D(g)

(1.3)

where A(g) = P+ Mg P+ is the (block) Toeplitz operator associated to g and so on. If g has the an bn , then relative to the basis for H: Fourier expansion g = gn zn , gn = cn dn

. . . 1 z, 2 z, 1 , 2 , 1 z−1 , 2 z−1 , . . .

(1.4)

where {1 , 2 } is the standard basis for C2 , the matrix of Mg is block periodic of the form . .. a0 .. c0 .. a−1 .. c−1 − − .. a−2 .. c−2 .

. b0 d0 b−1 d−1 − b−2 d−2 .

. a1 c1 a0 c0 − a−1 c−1 .

. b1 d1 b0 d0 − b−1 d−1 .

. . | a2 | c2 | a1 | c1 − − | a0 | c0 . .

. b2 d2 b1 d1 − b0 d0 .

.. .. .. .. − .. ..

(1.5)

From this matrix form, it is clear that, up to equivalence, Mg has just two types of “principal minors”, the matrix representing A(g), and the matrix representing the shifted Toeplitz operator A1 (g), the compression of Mg to the subspace spanned by {i zj : i = 1, 2, j > 0} ∪ {1 }. Relative to the basis (1.4), the involution σ defined by (0.1) is equivalent to conjugation by the shift

2196


operator, i.e. the matrix of Mσ (g) is obtained from the matrix for Mg by shifting one unit along the diagonal (in either direction: the result is the same, because Mg commutes with Mz , the square of the shift operator). Consequently the shifted Toeplitz operator is equivalent to the operator A(σ (g)). Theorem 1.1. Suppose that g ∈ L∞ (S 1 , SL(2, C)). (a) If A(g) is invertible, then g has a Birkhoff factorization, where −1

(g0 g+ )

−1 1 −1 0 , A(g) . = A(g) 0 1

(1.6)

(b) If A(g) and A1 (g) are invertible, then g has a triangular factorization. Proof. For part (a), let M denote the 2 × 2 matrix valued loop on the right hand side of (1.6). The columns of this matrix are in H+ . We must check that det(M) = 1 on . Because the entries of M are in L2 (S 1 ), det(M) ∈ L1 (S 1 ). Because det(g) = 1 on S 1 , and gM = 1 + h, where the columns of h are in H− , det(M) is holomorphic on , and on S 1 equals a function which is holomorphic in ∗ and equal to 1 at ∞. Consequently det(M) has a holomorphic extension to ˆ and hence must be identically 1. We can now take g0 g+ = M −1 . This will have L2 all of C, entries, because M is unimodular. α β For part (b), suppose that g has Birkhoff factorization g = g− g0 g+ , and let g0 = γ δ . The matrix representing Mg0 g+ has the form

.. .. .. .. − .. ..

. α γ 0 0 − 0 0 .

. β δ 0 0 − 0 0 .

. ∗ ∗ α γ − 0 0 .

. ∗ ∗ β δ − 0 0 .

. | | | | − | | .

. ∗ ∗ ∗ ∗ − α γ .

. ∗ ∗ ∗ ∗ − β δ .

.. .. .. .. − .. ..

The matrix representing Mg− is unipotent and lower triangular. Consequently A1 (g) = A1 (g− )A1 (g0 g+ ), A1 (g− ) is unipotent lower triangular, and A1 (g) is invertible iff A1 (g0 g+ ) is invertible iff α = (g0 )11 = 0. This implies part (b). 2 In Theorem 1.1 we are assuming that g is bounded. It is not generally true that the factors g± are bounded. Recall (see [2]) that a Banach ∗-algebra A ⊂ L∞ (S 1 ) is said to be decomposing if A = A+ ⊕ A− , i.e. P+ : A → A+ is continuous. For example C s (S 1 ) is decomposing, provided s > 0 and nonintegral (see p. 60 of [2]), and W s is a decomposing algebra, provided s > 1/2. (Note: W 1/2 is not an algebra.)


2197

Corollary 1. Suppose that g ∈ L∞ (S 1 , SL(2, C)) belongs to a decomposing algebra A and has a Birkhoff factorization. Then the factors g± belong to A. This follows from the continuity of P+ on A and the formula in (a) of Theorem 1.1. Theorem 1.2. If g ∈ L∞ (S 1 , SL(2, C)), then B(g) and C(g) are compact operators if and only if g ∈ VMO, the space of functions with vanishing mean oscillation. If g ∈ QC := L∞ ∩ VMO, then A(g) and D(g) are Fredholm of index 0. The first statement is due to Hartmann, and the second to Douglas (see pp. 27 and 108 of [4], respectively). Remarks. (a) In the context of Theorem 1.1, if g has a Birkhoff factorization, then A(g) is 1–1: for if h ∈ H+ , then there is a Hardy decomposition of (not necessarily L2 ) C2 valued functions −1 −1 g− (Mg h)+ = g0 g+ h − g− (Mg h)− ;

thus if A(g)h = 0, then h = 0. A Birkhoff factorization for bounded g does not imply A(g) is invertible (see Theorem 5.1, p. 109 of [4]). (b) For g ∈ QC(S 1 , SL(2, C)), the converse in (a) (and also (b)) of Theorem 1.1 holds, because the Fredholm index of A(g) vanishes. Moreover there is a notion of generalized triangular factorization for all g (see [2] and Chapter 8 of [8]). (c) Theorem 1.2 implies that the Toeplitz operator defines a holomorphic map QC S 1 , SL(2, C) → Fred(H+ ) : g → A(g). There is a determinant line bundle Det → Fred(H+ ) with canonical section, A → det(A), which is nonvanishing precisely when A is invertible. In the notation of [6], σ0 = det(A(g)) ˜ is the pullback of the canonical section, and σ1 = det(A(σ (g))), ˜ viewed as holomorphic functions of g˜ in the universal C∗ extension of QC(S 1 , SL(2, C)). If g has a triangular factorization, then m(g)a(g) =

σ1 /σ0 0

0 σ0 /σ1

,

(1.7)

as the matrix manipulations above suggest (see (1.5)–(1.6) of [6]). 2. Proof of Theorem 0.1, and generalizations to other function spaces In the course of proving Theorem 0.1, we will also prove the following Theorem 2.1. Suppose that k1 ∈ C s (S 1 , SU(2)), where s > 0 and nonintegral. The following are equivalent:

2198


(I.1) k1 is of the form k1 (z) =

a(z)

b(z)

−b∗ (z)

a ∗ (z)

z ∈ S1,

,

where a, b ∈ H 0 () have C s boundary values, a(0) > 0, and a and b do not simultaneously vanish at a point in . (I.3) k1 has triangular factorization of the form

1 ∞ ∗ −j j =0 yj z

0 1

0

a1 0

a1−1

α1 (z) γ1 (z)

β1 (z) , δ1 (z)

where the factors have C s boundary values. Similarly, the following are equivalent: (II.1) k2 is of the form k2 (z) =

−c∗ (z) , d(z)

d ∗ (z) c(z)

z ∈ S1,

where c, d ∈ H 0 () have C s boundary values, c(0) = 0, d(0) > 0, and c and d do not simultaneously vanish at a point in . (II.3) k2 has triangular factorization of the form

1 0

∞

∗ −j j =1 xj z

1

a2 0

0

a2−1

α2 (z) γ2 (z)

β2 (z) , δ2 (z)

where the factors have C s boundary values. Remarks. (a) When k2 ∈ Lfin SU(2), the determinant condition c∗ c + dd ∗ = 1 can be interpreted as an equality of finite Laurent expansions in C∗ . Together with d(0) > 0, this implies that c and d do not simultaneously vanish. Thus the added hypotheses in (I.1) and (II.1) of Theorem 2.1 are superfluous in the finite case. (b) The kind of example we have to avoid in the C ∞ case is k2 =

d∗ 0

0 , d

d=

z−r rz − 1

where 0 < r < 1. (c) The factorizations in (I.2) and (II.2) of Theorem 0.1 are akin to nonabelian Fourier expansions. Consequently it is highly unlikely that one can characterize the coefficients for C s loops. For this purpose we consider a Sobolev completion at the end of this section.


2199

Proof. As we remarked in the Introduction, the two sets of conditions are intertwined by the outer involution σ . Also it is evident that (II.3) ⇒ (II.1): by multiplying the matrices in (II.3), we see that c = a2−1 γ2 and d = a2−1 δ2 , and these cannot simultaneously vanish at a point in . We will now prove, in reference to Theorem 0.1, that (II.2) ⇒ (II.1) ⇒ (II.3) ⇒ (II.2). The second step will also complete the proof of Theorem 2.1. It is straightforward to calculate that a loop as in (II.2) has the matrix form in (II.1): Proposition 2. The product in (II.2) equals

δ∗ 2 a(ζi ) γ2

−γ2∗ δ2

,

where γ2 (z) =

∞

γ2,n zn ,

n=1

(−ζ¯i1 )ζj1 . . . (−ζ¯ir )ζjr (−ζ¯ir+1 ), γ2,n = the sum over multiindices satisfying

0 < i1 < j1 < · · · < jr < ir+1 ,

i∗ −

j∗ = n,

and δ2 (z) = 1 + δ2,n =

∞

δ2,n zn ,

n=1

ζi1 (−ζ¯j1 ) . . . ζir (−ζ¯jr ),

the sum over multiindices satisfying 0 < i1 < j1 < · · · < jr ,

(j∗ − i∗ ) = n.

This is a straightforward induction, which we omit. Now suppose that we are given a loop k2 satisfying the conditions in (II.1), with one exception: for later convenience, we initially assume that k2 is merely measurable. Suppose that A(k2 )f = P+

d∗ c

−c∗ d

f1 f2

0 = . 0

Then cf1 + df2 = 0 ∈ H 0 (), and hence by the independence of c and d around S 1 , (f1 , f2 ) = λ(d, −c). Because c and d do not simultaneously vanish, this implies that λ is holomorphic in . We also have (d ∗ λd − c∗ λ(−c))+ = λ+ = 0. Thus λ = 0. Thus the Toeplitz operator is invertible. [Note: conversely, if c and d have a common zero z0 ∈ , then the Toeplitz operator is not invertible: take λ = 1/(z − z0 ).] The same argument shows that A1 (k2 ), and also D(k2 ), are invertible.

2200


We must now show that this loop has a triangular factorization as in (II.3), i.e. we must solve for a2 , x ∗ , and so on, in n ∗ −j a 1 0 α2 (z) β2 (z) d (z) −c∗ (z) 2 j =1 x¯ j z = . (2.1) k2 (z) = c(z) d(z) 0 a2−1 γ2 (z) δ2 (z) 0 1 The form of the second row implies that we must have a2 = d(0)−1 , and γ2 = a2 c,

and δ2 = a2 d,

(2.2)

because δ2 (0) = 1. This does define a2 > 0, γ2 and δ2 in a way which is consistent with (II.3), because c(0) = 0 and d(0) > 0. Using (2.2), the first row in (2.1) is equivalent to d ∗ = α2 + x ∗ c,

and −c∗ = β2 + x ∗ d.

(2.3)

In the finite case, by considering the second equation as an equality in C∗ , we can immediately obtain that x ∗ = −(c∗ /d)− . The C s case is more involved. Consider the Hardy space polarization H := L2 S 1 , dθ = H + ⊕ H − , and the operator T : H − → H − ⊕ H − : x∗ →

∗ ∗ cx − , dx − .

The operator T is the restriction of D(k2 )∗ = D(k2∗ ) to the subspace {(x ∗ , 0) ∈ H− }, consequently it is injective with closed image. The adjoint of T is given by T ∗ : H − ⊕ H − → H − : f ∗ , g ∗ → c∗ f ∗ + d ∗ g ∗ . If (f ∗ , g ∗ ) ∈ ker(T ∗ ), then c∗ f ∗ + d ∗ g ∗ vanishes in the closure of ∗ , and because |c|2 + |d|2 = 1 around S 1 , (f ∗ , g ∗ ) = λ∗ (d ∗ , −c∗ ), where λ∗ is holomorphic in ∗ and vanishes at ∞ ∗ , −c∗ ) ∈ ker(T ∗ )⊥ : because d ∗ (∞) = d(0) > 0. We now claim that (d−

∗ d− f + −c∗ g dθ =

λ d ∗ d + c∗ c dθ =

λ dθ = 0,

because λ(0) = 0. Because T has closed image, there exists x ∗ ∈ H − such that ∗ d− = x∗c −,

and −c∗ = x ∗ d − .

(2.4)

We can now solve for α2 and β2 in (2.3). This shows that k2 in (II.1) has a triangular factorization as in (II.3). When k2 ∈ C s , by Corollary 1, the factors are C s . This completes the proof of Theorem 2.1.


2201

We have now shown that (II.2) ⇒ (II.1) ⇒ (II.3). To prove that (II.3) implies (II.2), one method is to explicitly solve for x ∗ in terms of the ζ variables, then show that this relation can be inverted. The formula for x in terms of ζ is discussed in Appendix A. For our present purposes we only need to know that x∗ =

∞

x1∗ (ζj , . . .)z−j ,

j =1

where x1∗ (ζ1 , . . .) = ζ1

∞ ∞

1 + |ζk |2 + ζ2 1 + |ζk |2 s2 (ζ2 , ζ3 , . . .) k=2

+ ζ3

∞

k=3

1 + |ζk |2 s3 (ζ3 , ζ4 , . . .) + · · ·

k=4

(in the current context, these are finite sums). This structure implies that we can solve for the ζj in terms of the xi , and in fact ζn (x1 , x2 , . . .) = ζ1 (xn , xn+1 , . . .). (Note: the equivalence of (II.2) and (II.3) is implied by Theorem 5 of [6], which uses Lie theory; here we are emphasizing the elementary nature of the correspondence.) This completes the proof of Theorem 0.1. 2 It is obvious that for k2 in Theorem 0.1, there is a factorization a2 = a(ζj )−1 . By considering the Kac–Moody central extension of LSU(2), one can obtain a refinement of this factorization (recall (1.7), which suggests the existence of this refinement). Theorem 2.2. For ki as in Theorem 0.1, det(A∗ A(k1 )) equals −n −1 ˙ 1 + |ηn |2 = lim det AN (k1 ) = det 1 − C ∗ C(k1 ) = det 1 + B˙ ∗ B(y)

N →∞

n1

and det(A∗ A(k2 )) equals −n −1 ˙ 1 + |ζn |2 = , lim det AN (k2 ) = det 1 − C ∗ C(k2 ) = det 1 + B˙ ∗ B(x)

N →∞

n1

where AN denotes the finite dimensional compression of A to the span of {i zk : 0 k N }, and in the third expressions, x and y are viewed as multiplication operators on H = L2 (S 1 ), with Hardy space polarization. The first equalities are special cases of Theorem 6.1 of [9]; these are included for perspective: they demonstrate that finite dimensional approximations detect the magnitude of det A, not its

2202


phase. The second equalities follow from the unitarity of the Mki ; they explain why the determinants are well-defined, since C(ki ) is Hilbert–Schmidt if and only if ki ∈ W 1/2 (this follows immediately from the matrix expression for Mki in Section 1). The last two equalities follow from Theorem 5 of [6]. Lemma 1. Suppose that ζ = (ζn ) ∈ l 2 . As in Theorem 0.1, let (N ) k2

=

d (N )∗ c(N )

−c(N )∗ d (N )

:=

N

a(ζn )

ζN z−N 1

1 −ζ¯N zN

n=1

1 ... −ζ¯1 z

ζ1 z−1 1

.

Then c(N ) and d (N ) converge uniformly on compact subsets of to holomorphic functions c = c(ζ ) and d = d(ζ ), respectively, as N → ∞. The functions c and d have radial limits at a.e. point of S 1 , c and d are uniquely determined by these radial limits, k2 (z) = k2 (ζ )(z) :=

d(ζ )∗ (z) c(ζ )(z)

−c(ζ )∗ (z) d(ζ )(z)

∈ Meas S 1 , GL(2, C) ,

and det(k2 ) 1 on S 1 . A crucial lingering issue is the unitarity of k2 . In the course of proving Theorem 2.3, we will prove that k2 is unitary on S 1 when ζ ∈ w 1/2 . It is unclear whether this is true more generally for ζ ∈ l 2 (see Section 4). Proof. Because d (N ) d (N )∗ + c(N ) c(N )∗ = 1, both (c(N ) ) and (d (N ) ) are sequences of holomorphic functions on which are bounded by 1. By the Arzela–Ascoli Theorem, there exist subsequences which converge uniformly to holomorphic functions on , which will also be bounded by 1. We claim these limits are unique. As in Proposition 2, write k (N ) as

N

a(ζn )

(N )∗

(N )∗

−γ2

(N )

δ2

δ2

(N )

γ2

n=1

.

2 The ∞ n=1 a(ζn ) converges, because ζ ∈ l . Proposition 2 gives explicit expressions for the co(N ) (N ) efficients of γ2 and δ2 . Very crude estimates show that these expressions have well-defined limits as N → ∞. To see this, consider the formula for the nth coefficient of δ2 , and let P(n) denote the set of partitions of n (i.e. decreasing sequences n1 n2 · · · nl > 0, where nj = n is the magnitude and l = l(nj ) is the length of the partition). Then |δ2,n |

|ζi1 ||ζ¯j1 | · · · |ζir ||ζ¯jr |,

where the sum is over multiindices satisfying 0 < i1 < j1 < · · · < jr ,

(j∗ − i∗ ) = n.

(2.5)


2203

If nk = jk − ik , then nk = n, but this sequence is not necessarily decreasing. However if we eliminate the constraints i1 < · · · < ir , then we can permute the indices (1 k r) for the ik and nk . We can crudely estimate that (2.5) is

|ζi1 ||ζi1 +n1 | · · · |ζil ||ζil +nl | =

(ni )∈P (n) i1 ,...,il >0 2l((n )) |ζ |l 2 i . P (n)

l

|ζis ||ζis +ns |

(ni )∈P (n) s=1 is >0

This shows that the Taylor coefficients of any limiting function for the δ (N ) will be given by the formulas in Proposition 2. The same considerations apply to the γ (N ) . Thus the sequences (γ (N ) ) and (δ (N ) ) converge uniformly on compact sets of to unique limiting functions. This proves our claim about uniqueness of the limits c and d. Because c and d are bounded by 1 on , c and d have radial limits at a.e. point of S 1 , and these boundary values uniquely determine c and d. 1 2 ) on S . Since c and d are holomorphic in , and d(0) = Finally we consider det(k 2 2 a(ζj ) = 0, det(k2 ) = |d| + |c| is nonzero a.e. on S 1 . Thus k2 is invertible a.e. on S 1 . Clearly |d|2 + |c|2 2 on the closure of , since |d| and |c| are bounded by 1. This also holds for d (N ) and c(N ) . If ρ ∈ L1 (S 1 , dθ ) is positive, then

2 2 |d| + |c|2 ρ dθ = lim |d| + |c|2 reiθ ρ eiθ dθ, r↑1

S1

S1

(by dominated convergence)

= lim lim

r↑1 N →∞

(N ) 2 (N ) 2 iθ iθ d + c re ρ e dθ

S1

lim lim sup N →∞

r↑1

= lim

N →∞

(N ) 2 (N ) 2 iθ iθ d + c re ρ e dθ

S1

(N ) 2 (N ) 2 iθ iθ d + c e ρ e dθ =

S1

ρ eiθ dθ.

S1

Since ρ is a general positive integrable function, this implies that |d|2 + |c|2 1 on S 1 . This completes the proof. 2 Remark. To show that k2 has values in SU(2), it would suffice to show

2 1 |d| + |c|2 dθ = 1. 2π

(2.6)

S1

This would follow immediately (by dominated convergence) if we knew thatc(N ) (d (N ) ) converged to c (d, respectively) on S 1 . But we have not shown this. Since d(0) = a(ζj ), it is clear that (2.6) is bounded below by a(ζj )2 .

2204


Theorem 2.3. Suppose that k1 ∈ Meas(S 1 , SU(2)). The following are equivalent: (I.1) k1 is of the form k1 (z) =

a(z)

b(z)

−b∗ (z)

a ∗ (z)

z ∈ S1,

,

where a, b ∈ H 0 () have W 1/2 boundary values, a(0) > 0, and a and b do not simultaneously vanish at a point in . (I.2) k1 has a factorization of the form k1 (z) = lim a(ηn ) n→∞

−η¯ n zn 1

1 ηn z−n

. . . a(η0 )

1 η0

−η¯ 0 1

,

where η ∈ w 1/2 , and the limit is understood as in Lemma 1. (I.3) k1 has triangular factorization of the form

1 ∗ −j j =0 yj z

0 1

∞ where y =

∞

j =0 yj z

a1 0

0

a1−1

α1 (z) γ1 (z)

β1 (z) , δ1 (z)

has W 1/2 boundary values.

j

Moreover this defines a bijective correspondence between η ∈ w 1/2 and (yn ) ∈ w 1/2 . Similarly, the following are equivalent: (II.1) k2 is of the form k2 (z) =

−c∗ (z) , d(z)

d ∗ (z) c(z)

z ∈ S1,

where c, d ∈ H 0 () have W 1/2 boundary values, c(0) = 0, d(0) > 0, and c and d do not simultaneously vanish at a point in . (II.2) k2 has a factorization of the form k2 (z) = lim a(ζn ) n→∞

1 ¯ − ζn z n

ζn z−n 1

. . . a(ζ1 )

1 −ζ¯1 z

ζ1 z−1 1

,

where ζ ∈ w 1/2 , and the limit is understood as in Lemma 1. (II.3) k2 has triangular factorization of the form

where x =

∞

∞

∗ −j j =1 xj z

1 0

j =1 xj z

1 j

a2 0

0

a2−1

α2 (z) γ2 (z)

β2 (z) , δ2 (z)

has W 1/2 boundary values.

Moreover this defines a bijective correspondence between ζ ∈ w 1/2 and (xn ) ∈ w 1/2 .


2205

Remarks. (a) If |ζn | < ∞, then the products in (I.2) and (II.2) converge absolutely and uniformly in z ∈ S 1 , and the limits are C 0 . However n|ζn |2 < ∞ does not imply absolute convergence of the sum of the {ζn } and vice versa; similarly C 0 does not imply W 1/2 and vice versa. It is for this reason that the weak notion of convergence in Lemma 1 is used in (I.2) and (II.2). (b) In connection with (I.2) and (II.2), note that zn converges to zero uniformly on compact subsets of , but |zn | = 1, for all n, on S 1 . Thus it is not evident in (I.2) and (II.2) that k2 is unitary; this is the problem which we could not resolve in Lemma 1. Proof. The two sets of conditions are intertwined by σ . We will first show (II.1) is equivalent to (II.3); we will then show these conditions are equivalent to (II.2). Suppose that k2 satisfies the conditions in (II.1), except that at the outset we only assume k2 is measurable. In the course of proving Theorem 2.1, we showed that k2 has a triangular factorization as in (II.3), where

x∗ 0

−1 = D k2∗

(d ∗ )− −c∗

(2.7)

(and the other factors are given explicitly by (a) of Theorem 1.1). In particular x ∗ ∈ L2 . For the Birkhoff factorization of k2 , (k2 )− =

1 x∗ 0 1

.

Because Mk2 is unitary, −1 A(k2 )A(k2 )∗ = 1 + Z(k2 )∗ Z(k2 ) ,

(2.8)

where Z(k2 ) := C(k2 )A(k2 )−1 . A matrix calculation (see (5.13) and (5.14) of [6], and note that in [6], g = k2 , and x is written in place of x ∗ ) shows that Z(k2 ) = Z (k2 )− = C (k2 )− ,

(2.9)

and relative to the basis (1.4), C((k2 )− ) is represented by the matrix ⎛

.

0 xn

⎜. 0 ⎜ ⎜ ⎜. . ⎜ ⎜ ⎜ ⎜ ⎜. ⎜ ⎜. ⎜ ⎜ ⎝. 0 0

0 .

. .

0 0

. 0

x3 0 x4 0 . .

0 ..

0 x2 0 0 0 x3 0 0

0 0 0 0 0

. 0

0 0

⎞ x1 0⎟ ⎟ ⎟ x2 ⎟ ⎟ 0⎟ ⎟ ⎟. x3 ⎟ ⎟ ⎟ ⎟ ⎟ xn ⎠ 0

(2.10)

2206


Now suppose that k2 ∈ W 1/2 . In this case A(k2 )A(k2 )∗ is the identity plus trace class. By (2.8) and (2.9), C((k2 )− ) is Hilbert–Schmidt. By (2.10), x ∗ ∈ W 1/2 . Conversely, given x ∗ ∈ W 1/2 , by Lemma 4 of [6], we can explicitly compute k2 and the corresponding triangular factorization: ∗ −1 ∗ ∗ x , δ2∗ = 1 + C˙ x ∗ γ2 , γ2 = − 1 + C˙ zx ∗ C˙ zx ∗ β = −a2−2 A˙ x ∗ (δ2 ) α2 = a2−2 1 − A˙ x ∗ (γ2 ) ,

(2.11) (2.12)

and a22 =

˙ ∗ )) ˙ ∗ )∗ C(x det(1 + C(x . ∗ ∗ ∗ )) ˙ ˙ det(1 + C(zx ) C(zx

(2.13)

In the derivation of Eqs. (2.11) and (2.12) in Lemma 4 of [6], the fact that k2 is unimodular is not used explicitly; the derivation only uses (k2 )(1,1) = (k2 )∗(2,2) and (k2 )(1,2) = −(k2 )∗(2,1) . However, because α2 δ2 − β2 γ2 ∈ H 0 (), and has real values |c|2 + |d|2 on S 1 , α2 δ2 − β2 γ2 ˆ Since it equals 1 at z = 0, it is identically 1. This shows that extends holomorphically to C. unimodularity follows automatically. This determines a unitary k2 with measurable coefficients. The calculations (2.8), (2.9), and (2.10) imply that k2 ∈ W 1/2 . Thus (II.1) is equivalent to (II.3). Lemma 1 implies that if (ζn ) ∈ l 2 , then k2 defined as in (II.2) is in Meas(S 1 , GL(2, C)). Now suppose that ζ ∈ w 1/2 . By Theorem 2.2 N (N ) 2 ∗ −1

−n 1 + |ζn |2 = , detA k2 = det 1 + B˙ x (N ) B˙ x (N )

(2.14)

n=1

and this converges to a positive number as N → ∞. First suppose that ζn 0 for all n. Proposition 4 of Appendix A implies that the coefficients of x(ζ )(N ) are nonnegative and converge up to the coefficients of x(ζ ). This implies that the ma˙ (N ) )∗ will be nonnegative and converge in a monotone way to those ˙ (N ) )B(x trix entries of B(x ∗ ˙ (N ) )B(x ˙ (N ) )∗ ), which is bounded because (2.14) con˙ B(x) ˙ . Thus the sequence tr(B(x for B(x) ∗ ˙ B(x) ˙ ). This implies that (xn ) ∈ w 1/2 . For a general ζ ∈ w 1/2 , verges, will converge to tr(B(x) since the coefficients for x(|ζ |) dominate those for x(ζ ) we can conclude in the same way that (xn ) ∈ w 1/2 . We can now obtain a triangular factorization for k2 using (2.11)–(2.13). As we argued in the paragraph following (2.13), this automatically implies that k2 is unitary. The calculations (2.8), (2.9), and (2.10) imply that k2 ∈ W 1/2 and A(k2 ) is invertible. Since A(k2 ) is 1–1, this implies that c and d do not simultaneously vanish in (see the note in the second paragraph following Proposition 2). Thus (II.2) implies (II.1). n (N ) Suppose that we are given k2 and x as in (II.1) and (II.3). Let x (N ) = N n=1 xn z , and let ζ (N ) and k2 denote the corresponding objects. Theorem 2.2 implies that N 2 n ∗

det 1 + B˙ x (N ) B˙ x (N ) = 1 + ζn(N ) .

(2.15)

n=1

Because x ∈ W 1/2 , the sequence of numbers (2.15) has a limit. Therefore the sequence {ζ (N ) } is bounded in w 1/2 . Because the inclusion w 1/2 → l 2 is a compact operator, there are subsequences


2207

which converge in l 2 . By Lemma 1 these limiting sequences correspond to k2 . Thus there is a unique limiting sequence, {ζn } ∈ l 2 . Since (2.15) has a limit, ζ ∈ w 1/2 . Thus (II.1) and (II.3) imply (II.2). This completes the proof. 2 3. Proof of Theorem 0.2, and generalizations Part (a) of Theorem 0.2 is obvious. We will deduce the remaining parts of Theorem 0.2 from the following Theorem 3.1. Assume s > 0 and nonintegral, or s = ∞. For g ∈ C s (S 1 , SU(2)), the following are equivalent: (i) g has a triangular factorization g = lmau, where l and u have C s boundary values. (ii) g has a factorization g = k1∗ λk2 , where k1 , k2 ∈ C s (S 1 , SU(2)) satisfy the equivalent conditions (I.1) and (I.3) ((II.1) and (II.3), respectively) of Theorem 2.1, and λ ∈ C s (S 1 , T )0 . Proof. We will use the notation in (1.1) for g, and the notation in Theorem 2.1 for the entries of the ki and their triangular factorizations. Without much comment, we will use the fact that C s is a decomposing algebra, so that factors in various decompositions will remain in C s . We proved that (ii) implies (i) in [6] (see the proof of Theorem 7); we briefly recall the calλ 0 k2 , as in (ii). We can culation. Suppose that g ∈ C s (S 1 , SU(2)) can be factored as g = k1∗ −1 0λ

write λ = exp(−χ ∗ + χ0 + χ), where χ0 ∈ iR and χ ∈ H 0 (), χ(0) = 0, with C s boundary values. Then g has triangular factorization of the form g = l(g)

eχ0 a1 a2

0

0

(eχ0 a1 a2 )−1

(3.1)

u(g),

where m0 = eχ0 ∈ S 1 , a0 = a1 a2 > 0, l(g) :=

l11 l21

l12 l22

=

α1∗ β1∗

γ1∗ δ1∗

e−χ 0

∗

0 eχ

∗

∗

1 a12 e2χ0 P− (ye2χ + x ∗ e2χ ) 0 1

(3.2)

and u(g) :=

u11 u21

u12 u22

=

1 0

∗

a2−2 e−2χ0 P+ (ye2χ + x ∗ e2χ ) 1

eχ 0

0 e−χ

α2 γ2

β2 δ2

. (3.3)

Thus (i) is implied by (ii). Now suppose that g has triangular factorization g = lmau as in (i). We must solve for k1 , χ , and k2 . An elegant way to do this (discovered after this paper was completed) is presented in the proof of Theorem 4.1 of [7]. Here we will present a somewhat more explicit (if clumsy) calculation. Eq. (3.2) implies l11 = α1∗ exp −χ ∗ ,

l21 = β1∗ exp −χ ∗

(3.4)

2208


and (3.3) implies u21 = γ2 exp(−χ),

u22 = δ2 exp(−χ).

(3.5)

The special forms of k1 and k2 imply that on S 1 , |α1 |2 + |β1 |2 = a1−2 , |δ2 | + |γ2 | 2

2

= a22 .

(3.6) (3.7)

Therefore on S 1 |l11 |2 + |l21 |2 = a1−2 exp −2 Re(χ) , |u21 |2 + |u22 |2 = a22 exp −2 Re(χ) .

(3.8) (3.9)

This implies that on S 1 we must have −1/2 −1/2 Re(χ) = log a1−1 + log |l11 |2 + |l21 |2 = log(a2 ) + log |u21 |2 + |u22 |2 . (3.10) Assuming that the obvious consistency condition is satisfied, this pair of equations determines χ and the ai : because χ must be holomorphic in the disk and vanish at z = 0, the average of Re(χ) around S 1 must vanish, hence

1 log |l11 |2 + |l21 |2 dθ , a1 = exp − 4π

(3.11)

S1

1 2 2 a2 = exp log |u21 | + |u22 | dθ , 4π

(3.12)

S1

and Im(χ) = i Re(χ)− − i Re(χ)+ .

(3.13)

To see that χ and the ai are well-defined, we must check that |l11 |2 + |l21 |2 = (a1 a2 )−2 |u21 |2 + |u22 |2 ,

(3.14)

as functions on S 1 . Because g ∗ g = 1, l ∗ l = (a(g)u)−∗ (a(g)u)−1 , on S 1 . This implies three independent equations |l11 |2 + |l21 |2 = a0−2 |u22 |2 + |u21 |2 , ∗ ∗ l12 + l21 l22 = −m20 u∗22 u12 + u∗21 u11 , l11 |l12 |2 + |l22 |2 = a02 |u12 |2 + |u11 |2

(3.15) (3.16) (3.17)


2209

for the (1, 1), (1, 2) (or (2, 1)), and (2, 2) entries, respectively. The (1, 1) entry implies the consistency condition (3.14). Together with (3.4) and (3.5), this completely determines the ki : ∗ a(z) = a1 exp(χ)l11 ,

c(z) = a2−1 exp(χ)u21 ,

∗ b(z) = a1 exp(χ)l21 ,

(3.18)

d(z) = a2−1 exp(χ)u22 .

(3.19)

Because l ∗ is invertible at all points of , the entries a and b of k1 do not simultaneously vanish. Similarly, because u is invertible, the entries c and d do not simultaneously vanish. The fact that these are C s in the appropriate sense follows from the continuity of the projections P± on C s . Thus by Theorem 2.1 (and the ensuing Remark (b)) the ki have appropriate triangular factorizations. We have now solved for ki and χ . We have also observed that the diagonal term of g determines exp(χ0 ), so λ is determined as well. We now must show that g = k1−1 λk2 . From the definitions of ki and λ, both sides of this equation have the same m, a, l11 , l21 , u21 , and u22 coordinates. The proof is completed by the following explicit calculations, which I will need in a sequel to this paper. 2 Proposition 3. Suppose that g has a triangular factorization as in (1.1) and has values in SU(2). If l11 and u22 are nonvanishing, then

∗ / l ) + m2 (u∗ /u ) (l21 11 0 21 22 , l12 = −l11 P− 2 |l11 | + |l21 |2 ∗ (l21 / l11 ) + m20 (u∗21 /u22 ) 1 l22 = , − l21 P− l11 |l11 |2 + |l21 |2 ∗ (l21 / l11 ) + m20 (u∗21 /u22 ) −2 , u12 = −(m0 a0 ) u22 P+ |l11 |2 + |l21 |2 ∗ (l / l11 ) + m20 (u∗21 /u22 ) 1 . u11 = − (m0 a0 )−2 u21 P+ 21 u22 |l11 |2 + |l21 |2

In particular g is determined by m, a, l11 , l21 , u21 , and u22 . Proof. Because l11 and u22 are nonvanishing, we can use the unimodularity of l and u to solve for l22 and u11 in terms of l12 and u12 . Eq. (3.16) can be rewritten as ∗ ∗ l12 + l21 l22 + m20 u∗22 u12 + u∗21 u11 l11 1 + l12 l21 1 + u12 u21 ∗ ∗ 2 ∗ ∗ + m0 u22 u12 + u21 = 0. = l11 l12 + l21 l11 u22 Using (3.15) this can be rewritten as (l ∗ / l11 ) + m20 (u∗21 /u22 ) l12 u12 + m20 a02 = − 21 l11 u22 |l11 |2 + |l21 |2 by applying P± to this equation, and solving, we obtain the equations in the proposition.

2

2210


Suppose that g ∈ C s (S 1 , SU(2)), s > 1/2, and g has a triangular factorization. By Theorem 7 of [6], det A∗ A(g) = det A∗ A k1−1 det A∗ A(λ) det A∗ A(k2 ) ∞ ∞ ∞

−k 2 −i 2 = 1 + |ηi | 1 + |ζk |2 . exp −2 j |χj | j =1

i=1

(3.20)

k=1

These expressions make sense because C s ⊂ W 1/2 for s > 1/2. In the remainder of this section, our goal is to use these equalities to obtain a W 1/2 analogue of Theorem 3.1, which also incorporates the condition (bi ). This involves some subtleties, because W 1/2 functions are not necessarily continuous. Because SU(2) is compact, W 1/2 (S 1 , SU(2)) is a separable topological group. In contrast to the function spaces C s , s > 0, W s , s > 1/2, and L∞ ∩ W 1/2 , for the function space W 1/2 , the loop group W 1/2 (S 1 , SU(2)) is not a Lie group, because W 1/2 (S 1 , su(2)) is not a Lie algebra (whereas, e.g. L∞ ∩ W 1/2 (S 1 , su(2)) has a Lie algebra structure). Moreover the inclusion C ∞ (S 1 , SU(2)) ⊂ W 1/2 (S 1 , SU(2)) is dense and presumably a homotopy equivalence (whereas this is false for the L∞ ∩ W 1/2 topology). With respect to the W 1/2 topology, the operator-valued function g→

A(g) C(g)

B(g) D(g)

is continuous, provided the diagonal is equipped with the strong operator topology, and the offdiagonal with the Hilbert–Schmidt topology. In reference to the following lemma, we recall that the notion of degree (or winding number) can be extended from C 0 to VMO(S 1 , S 1 ), hence degree is well-defined for W 1/2 (S 1 , S 1 ) (see Section 3 of [1] for an amazing variety of formulas, and further references, or pp. 98–100 of [4]). Also given λ ∈ W 1/2 (S 1 , S 1 ), we view λ as a multiplication operator on H = L2 (S 1 ), with the ˙ Hardy polarization. We write A(λ) for the Toeplitz operator, and so on (with the dot), to avoid confusion with the matrix case. Lemma 2. There is an exact sequence of topological groups degree exp 0 → 2πiZ → W 1/2 S 1 , iR −−→ W 1/2 S 1 , S 1 −−−→ Z → 0. ˙ Moreover degree(λ) = −index(A(λ)). There is a more general version of this involving VMO, which is implicit on pp. 100–101 of [4]. Proof. Suppose that f ∈ W 1/2 (S 1 , iR). It is convenient to use the equivalent Besov form of the W 1/2 norm,

|f |2W 1/2 =

|f (θ1 ) − f (θ2 )|2 dθ1 dθ2 . |eiθ1 − eiθ2 |2


2211

Because |eiθ − 1| |θ |,

|ef (θ1 ) − ef (θ2 ) |2 dθ1 dθ2 |f |2W 1/2 . |eiθ1 − eiθ2 |2

Thus exp(f ) is also W 1/2 . This inequality also shows that exp is continuous at 0. Since exp is a homomorphism, this implies exp is globally continuous. Continuity implies that the image of exp is contained in the identity component. Conversely ˙ suppose that λ ∈ W 1/2 (S 1 , S 1 )0 . Then A(λ) is invertible. This implies the existence of a Birkhoff factorization λ = λ− λ0 λ+ , where for example λ+ ∈ H 0 (, 0; C∗ , 1) and has L2 boundary values. By taking logarithms on the disks, we can write λ = exp(−χ ∗ + χ0 + χ). By a formula of Szego and Widom (Theorem 7.1 of [9]), ∞ ∗ ∗ ˙ 2 ˙ ˙ ˙ j |χj | . det A A(λ) = det 1 − C C(λ) = exp −2

(3.21)

j =1

The determinant depends continuously on λ in the W 1/2 topology. Therefore χ ∈ W 1/2 . This shows the sequence is exact at W 1/2 (S 1 , S 1 ). A W 1/2 function cannot have jump discontinuities. This implies that the kernel of exp is 2πiZ. Thus the sequence in the statement of the lemma is continuous and exact. 2 Theorem 3.2. For g ∈ W 1/2 (S 1 , SU(2)), the following are equivalent: (i) g has a triangular factorization g = lmau. (ii) g has a factorization g = k1∗ λk2 , where the ki ∈ W 1/2 (S 1 , SU(2)) satisfy the equivalent conditions of Theorem 2.3, and λ ∈ W 1/2 (S 1 , T )0 . In both cases the factorization is unique. Proof. Assume (ii). Given Lemma 2, we can write λ = exp(χ). Since W 1/2 (S 1 , SU(2)) is a group, g will be in W 1/2 , and det(A(g)A(g)∗ ) will depend continuously on k1 , χ and k2 . The formula (3.20) now implies that A(g) is invertible, and hence g has a Birkhoff factorization. The triangular factorization is calculated exactly as in the proof of Theorem 3.1; see (3.2) and (3.3). (Note: we invoked (3.20), because it is not a priori clear that (3.2) and (3.3) are L2 .) Now assume (i). We can again solve for ki and χ , as in the proof of Theorem 3.1. The determi(N ) (N ) nant formulas (3.20) can be applied to g (N ) = k1 exp(χ (N ) )k2 , where the subscript indicates that ζn , χn , ηn are set equal to 0, for n > N . In (3.20), applied to g (N ) , all of the individual factors in (3.20) are bounded above by 1, and are tending monotonically down. Since g ∈ W 1/2 , det(A(g)A(g)∗ ) is positive, and det(A(g (N ) )A(g(N ))∗ will remain bounded away from zero. This implies that all of the factors in (3.20), applied to g(N ), will be bounded away from 0. Thus ζ , χ and η are in w 1/2 . By Theorem 2.3, ki ∈ W 1/2 . This implies (ii). 2 Corollary 2. The dense open set of g ∈ W 1/2 (S 1 , SU(2)) having triangular factorization is parameterized by y, χ0 ∈ iR mod 2πiZ, χ , and x, where y, χ and x are holomorphic functions in with W 1/2 boundary values, and x(0) = χ(0) = 0.

2212


Remark. This implies that an open neighborhood of 1 ∈ W 1/2 (S 1 , SU(2)) is parameterized by a Hilbert space, even though this group is not a Lie group and there does not exist an exponential map. In this respect this group is similar to the group of W s homeomorphisms of a compact d-manifold, where s > d2 + 1, although in this case right multiplication is smooth and there does exist an exponential map (see [3]). This contrasts with the finite dimensional situation, where a topological group locally homeomorphic to Rn is automatically a C ω Lie group. 4. A conjectural L2 generalization Suppose that ζ ∈ l 2 . By Lemma 1 there is a unique limit k2 ∈ Meas(S 1 , GL(2, C)) for the product in (4.1) below. When A(k2 ) is invertible, e.g. if ζ ∈ w 1/2 (by Theorem 2.3), there are three different expressions for k2 , 1 ζn z−n 1 ζ1 z−1 . . . a(ζ ) 1 n→∞ −ζ¯n zn 1 −ζ¯1 z 1

δ ∗ (z) −γ ∗ (z) 1 x ∗ (z) a 0 α2 (z) 2 2 2 = a(ζn ) = −1 γ2 (z) 0 1 0 a2 γ2 (z) δ2 (z)

k2 (z) = lim a(ζn )

β2 (z) , (4.1) δ2 (z)

where a2 = a(ζj )−1 , and γ2 and δ2 are determined by the formulas in Proposition 2. The existence of the triangular factorization implies that k2 has values in SU(2) on S 1 . Since the expression for a2 is convergent for all ζ ∈ l 2 , it is plausible that the triangular factorization in (4.1) is valid for all ζ ∈ l 2 . A further leap of faith suggests the following Conjecture. Suppose that k2 ∈ Meas(S 1 , SU(2)). The following are equivalent: (II.1) k2 is of the form k2 (z) =

−c∗ (z) , d(z)

d ∗ (z) c(z)

z ∈ S1,

where c, d ∈ H 0 (), c(0) = 0, d(0) > 0, and c and d do not simultaneously vanish at a point in . (II.2) k2 has a factorization of the form

1 k2 (z) = lim a(ζn ) n→∞ −ζ¯n zn

ζn z−n 1

1 . . . a(ζ1 ) −ζ¯1 z

ζ1 z−1 1

where ζ ∈ l 2 , and the limit is understood as in Lemma 1. (II.3) k2 has triangular factorization of the form

1 0

∞

∗ −j j =1 xj z

1

a2 0

0 a2−1

α2 (z) γ2 (z)

β2 (z) . δ2 (z)

Moreover this defines a bijective correspondence between ζ ∈ l 2 and (xn ) ∈ l 2 .

,


2213

In reference to this conjecture, recall that the condition (II.1) implies that A(k2 ) is 1–1. This entails invertibility when k2 ∈ QC (see Theorem 1.2), but not in general. When k2 is expressed as in (II.3), the third paragraph of the proof of Theorem 2.3, together with results of Nehari and Fefferman (pp. 3–5 of [4]), implies that A(k2 ) is invertible precisely when x has BMO boundary values. Thus the implications (II.2) ⇒ (II.1) ⇒ (II.3) hinge on the question of whether ζ ∈ l 2 ⇒ (xn ) ∈ l 2 , and this is different from the question of when A(k2 ) is invertible. The implication (II.3) ⇒ (II.1) hinges on the formulas (2.11)–(2.13) for k2 in terms of x. The first two formulas make sense for x ∈ BMO, as in the preceding paragraph, but it is not clear that this is the natural domain for x. Regarding the formula for a2 , which a priori depends on (xn ) ∈ w 1/2 , the second order term in the expansion at x = 0 is ∗ ∗ tr C x ∗ C x ∗ − tr C zx ∗ C zx ∗ = |xn |2 , the l 2 norm. This is at least consistent with the conjecture. Appendix A. The relation between x ∗ and ζ In this appendix, we consider the relation between x ∗ and (ζj ), in Theorem 0.1, at the level of combinatorial formulas. A.1. x ∗ as a function of ζ Proposition 4. x ∗ has the form ∗

x =

∞

x1∗ (ζj , . . .)z−j ,

j =1

where x1∗ (ζ1 , . . .) =

∞

ζn

n=1

∞

2 1 + |ζk | sn (ζn , ζn+1 , ζ¯n+1 , . . .), k=n+1

s1 = 1 and for n > 1, sn =

n−1

sn,r ,

sn,r =

ci,j ζi1 ζ¯j1 ζi2 ζ¯j2 . . . ζir ζ¯jr

r=1

where the sum is over multiindices satisfying the constraints

n and ci,j is a positive integer.

j1 ∨

···

i1

···

jr ∨,

r (jl − il ) = n − 1,

ir

l=1

(A.1)

2214


Remark. The main features of the formula for x1∗ are (i) the appearance of the infinite products, which isolates the part of the expression which has to be “renormalized” in probabilistic applications, and (ii) the positivity of the coefficients. For example (ii) implies that if ζ 0, then x(ζ1 , . . . , ζN , 0, . . .) converges monotonically up to x(ζ ) as N → ∞. Proof. The fact that x ∗ is completely determined by its residue x1∗ is (b) of Theorem 5 of [6]. We will show that x1∗ has the form claimed in the lemma (I stated this without proof in [6]). Clearly x1∗ (ζ1 ) = ζ1 . The proof hinges on the following recursion (see Lemma 2 and (5.12) of [6]) x1∗ (ζ1 , . . . , ζN +1 )

= 1 + |ζN +1 |2 x1 (ζ1 , . . . , ζN ) + +

x1 (ζi , . . . , ζN )x1 (ζj , . . . , ζN )ζ¯N +1

i+j =N +2

i+j +k=2N +3

+

x1 (ζi , . . . , ζN )x1 (ζj , . . . , ζN )x1 (ζk , . . . , ζN )ζ¯N2 +1

i+j +k+l=3N +4

x1 (ζi , . . . , ζN )x1 (ζj , . . . , ζN )x1 (ζk , . . . , ζN )x1 (ζl , . . . , ζN )ζ¯N3 +1 + · · · .

From this recursion one can immediately see that coefficients will be nonnegative. We assume that x1∗ (ζ1 , . . . , ζN ) =

N

N

1 + |ζk |2 sn (ζn , . . . , ζN ),

ζn

n=1

k=n+1

where s1 = 1 and for n > 1 sn (ζn , . . . , ζN ) =

ci,j ζi1 ζ¯j1 ζi2 ζ¯j2 . . . ζir ζ¯jr ,

the sum is over multiindices as in (A.1), with jr N , and ci,j is a positive integer (for N > 1, sN (ζN ) = 0). This implies x1∗ (ζI , . . . , ζN ) =

N −(I −1)

ζn+(I −1)

n=1

=

N m=I

ζm

N −(I

−1)

1 + |ζk+(I −1) |2 sn (ζn+(I −1) , . . .)

k=n+1 N

1 + |ζk |2 sm−(I −1) (ζm , . . . , ζN )

k=m+1

where sm−(I −1) (ζm , . . . , ζN ) = the sum is over multiindices satisfying

¯ ¯ ci−(I −1)1,j −(I −1)1 ζi1 ζj1 . . . ζiL ζjL ,


m

j1 ∨ i1

···

jL ∨ iL

···

N

L

,

2215

(jl − il ) = m − I,

l=1

and in the notation for the coefficient, i − (I − 1)1 means that we subtract I − 1 from each of the components of i. We now plug this into the recursion relation, and rewrite the expression so that it has the same form as the sum involving N variables: x1 (ζ1 , . . . , ζN +1 ) = 1 + |ζN +1 |2 s0

= 1 + |ζN +1 |2 s0

s+1

l=1 Il =s(N +1)+1

x1 (ζIl , . . . , ζN ) ζ¯Ns +1

Il

N

Il =s(N +1)+1 Il

ζm l

ml =Il

N

2 1 + |ζk | sml −(Il −1) (ζml , . . . , ζN ) ζ¯Ns +1 × k=ml +1

= 1 + |ζN +1 |2 s0

N

...

Il =s(N +1)+1 m1 =I1

N

ζm l

ms+1 =Is+1 Il

N

2 ¯s ¯ ¯ 1 + |ζk | c il −(Il −1)1, × jl −(Il −1)1 ζil,1 ζjl,1 . . . ζil,Ll ζjl,Ll ζN +1 k=ml +1

= 1 + |ζN +1 |2 s0

N

...

Il =s(N +1)+1 m1 =I1

N

ms+1 =Is+1

1

...

ζm l

s+1 Il

N

2 ¯s ¯ ¯ 1 + |ζk | c il −(Il −1)1, × jl −(Il −1)1 ζil,1 ζjl,1 . . . ζil,Ll ζjl,Ll ζN +1 ,

(A.2)

k=ml +1

where for each 1 l s + 1, the sum

ml

jl,1 ∨ il,1

··· ···

l

is over multiindices satisfying

jl,Ll ∨ il,Ll

N ,

Ll (jl,τ − il,τ ) = ml − Il . τ =1

Consider a term in this sum of the form N

1 + |ζk |2 ζil,1 ζ¯jl,1 . . . ζil,Ll ζ¯jl,Ll ζ¯Ns +1 , ζm l Il

k=ml +1

where ml il,1 for each l. Let n = min{ml : 1 l s + 1}, and factor out

(A.3)

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D. Pickrell / Journal of Functional Analysis 260 (2011) 2191–2221 N

1 + |ζk |2

ζn

k=n+1

in (A.3). What remains can be expressed as a positive integral combination of monomials ζi1 ζ¯j1 ζi2 ζ¯j2 . . . ζir ζ¯jL , where

n

j1 ∨ i1

···

jL ∨ iL

···

N +1 ,

L (jl − il ) = n − 1. l=1

Multiplicities arise when the factors with ml = m, N

1 + |ζk |2 k=ml +1

are expanded. Thus the entire sum can be written as N n=1

N +1

ζn

1 + |ζk |2 sn (ζn , . . . , ζN +1 )

k=n+1

with sn (ζn , . . . , ζN +1 ) =

(N +1) ζi1 ζ¯j1 ζi2 ζ¯j2 i,j

c

. . . ζir ζ¯jL ,

the sum is over multiindices satisfying

n

j1 ∨ i1

···

···

jL ∨ iL

N +1 ,

L

(jl − il ) = n − 1,

l=1

(N +1)

(N +1)

and c can be computed, in principle, recursively. If jL N , then ci,j i,j the index (i, j ) has the form

i0

j1 ∨ i1

· · · jr ∨ · · · ir

< N + 1 ··· . ∨ ir+1 · · ·

(N )

= ci,j . Otherwise

N +1 ∨ iL

where r + s = L. The corresponding terms will all originate from the term involving the index s in the last expression for (A.2). There are many ways that terms could arise, and at best we obtain a formula for c(N +1) in terms of coefficients c(N ) . So at this point we can only see that these coefficients are positive. 2


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Our aim now is to consider another approach which yields a closed formula for “generic” ci,j . This formula a priori involves signs, and we will make use of Proposition 4 to identify cancellations. The matrix n ∗ −j a 1 0 α(z) β(z) 2 j =1 xj z 0 a2−1 0 1 γ (z) δ(z) −1 a2 α + x ∗ a2 γ a2 β + x ∗ a2−1 δ = a2−1 δ a2−1 γ is special unitary, for all z ∈ S 1 . Therefore −γ ∗ = a22 β + x ∗ δ, and initially assuming δ is nonvanishing, this implies x ∗ = P− (−γ ∗ δ −1 ). In particular x1∗ = Residue −γ ∗ δ −1 = −γ1∗ + γ2∗ δ1 + γ3∗ δ2 + · · · − γ3∗ δ 2 2 + · · · =− γm∗ (−1)s δn1 . . . δns m1

where the second sum is over tuples n1 , . . . , ns 1 satisfying for γ ∗ and δ in Proposition 2, x1∗ =

nl = m − 1. Using the formulas

(−1)s+1 (−1)rm +1 ζim,1 ζ¯jm,1 . . . ζim,rm ζ¯jm,rm ζim,rm +1 × (−1)rn1 ζin1 ,1 ζ¯jn1 ,1 . . . ζin1 ,rn ζ¯jn1 ,rn . . . (−1)rns ζins ,1 ζ¯jns ,1 . . . ζins ,rns ζ¯jns ,rns 1

1

where the indexing can be described in the following way: the first sum is over m, n1 , . . . , ns 1 satisfying l nl = m − 1, the first internal sum, or cluster indexed by m, is over indices satisfying r m +1

0 < im,1 < jm,1 < · · · < jm,r < im,rm +1 ,

k=1

im,k −

rm

jm,k = m

k=1

and the cluster indexed by nl is over indices satisfying r

0 < inl ,1 < jnl ,1 < · · · < jnl ,rnl ,

nl (jnl ,k − inl ,k ) = nl .

k=1

We now write this as a single sum and consider one of the terms. We can put the i-indices (which are organized in clusters) im,1 , . . . , im,rm +1 ; in1 ,1 , . . . , in1 ,rn1 ; . . . ; ins ,1 , . . . , ins ,rns and the j -indices

2218


jm,1 , . . . , jm,rm ; jn1 ,1 , . . . , jn1 ,rn1 ; . . . ; jns ,1 , . . . , jns ,rns in nondecreasing order, which we write as i 0 i1 · · · iL

and j1 · · · jL ,

respectively. Lemma 3. In addition to being nondecreasing, the indices il , jl satisfy il−1 < jl , for l = 1, . . . , L. Proof. With the possible exception of im,r+1 , for any given i-index, it is possible to find a j-index with greater value, so that the map from these i-indices to j-indices is 1–1 (simply map in,l to jn,l ). One of iL−1 or iL must be strictly less than jL , hence iL−1 must be strictly less than jL . Similarly one of iL−2 or iL−1 or iL must be strictly less than jL−1 , hence iL−2 must be strictly less than jL−1 . Continuing in this way, this implies the strict inequalities in the lemma. 2 We claim that we can additionally assume that il jl ,

l = 1, . . . , L.

(A.4)

This is not implied by cluster decomposition considerations. For example the index set

1

2 2 1 3

violates (A.4), yet there are two cluster decompositions: 1 < 2 < 3; 1 < 2 (with (−1)s+L = (−1)1+2 = −1) and 3; 1 < 2; 1 < 2 (with (−1)s+L = (−1)2+2 = 1). This claim is justified by Proposition 4, which implies that terms corresponding to indices not satisfying (A.4) will cancel out. (It would clearly be desirable to see this cancellation directly, but I do not know how to do this.) This implies the following formula. Lemma 4. x1∗ =

ci,j ζi0 ζi1 ζ¯j1 . . . ζiL ζ¯jL , where the indices satisfy the constraints

0 < i0 i 1 · · · i L ,

j1 · · · jL , i1 j1 , . . . , iL jL , i0 < j1 , . . . , iL−1 < jL , i− j = 1,

(A.5)

and ci,j =

(−1)s+L ,

where the sum is over all possible ways in which the indices can be partitioned as im,1 , . . . , im,rm+1 ; in1 ,1 , . . . , in1 ,rn1 ; . . . ; ins ,1 , . . . , ins ,rns , jm,1 , . . . , jm,rm ; jn1 ,1 , . . . , jn1 ,rn1 ; . . . ; jns ,1 , . . . , jns ,rns so that the strict interlacing inequalities

(A.6)


0 < im,1 < jm,1 < · · · < jm,r < im,r+1 ,

im,k −

k

2219

jm,k = m

k

and 0 < inl ,1 < jnl ,1 < · · · < jnl ,r ,

(jnl ,k − inl ,k ) = nl k

hold for l = 1, . . . , s. To compare with the formula in Proposition 4, we first sum over n = i0 , and write x1∗ =

∞

ζn

c(n,i),j ζi1 ζ¯j1 . . . ζiL ζ¯jL

(A.7)

n=1

where (n, i) now stands for n i1 · · · iL . This implies

∞

ci,j ζi1 ζ¯j1 ζi2 ζ¯j2 . . . ζir ζ¯jr 1 + |ζk |2

c(n,i),j ζi1 ζ¯j1 . . . ζiL ζ¯jL =

(A.8)

k=n+1

where the indexing set for the latter sum satisfies the constraints in Proposition 4. To directly compare the coefficients we expand the product of factors (1 + |ζj |2 ) and distribute the pairs ζj and ζ¯j . This implies the following Lemma 5. Consider an index as in (A.5), with n = i0 . (a) If {il } ∩ {jl } is null, then c(n,i),j = ci,j . (b) In general c(n,i),j =

ci,j ,

where the sum is over all subindexing sets of (n, i, j), resulting from cancellation of pairs il = jl , which satisfy the constraints in Proposition 4. (c) In particular for any indexing set (i, j ) as in Proposition 4, ci,j c(n,i),j . Example. To clarify (b), given an indexing set such as 5 6 3 4 5

7 6

5 4

7 5

there are three proper subindexing sets,

3

6 4

7 6

3

7 3 4

2220


Part (a) of Lemma 5, and Lemma 4, yield an expression for a generic ci,j , where generic is defined by the null intersection condition in (a). Using this formula it is possible to write “most” of the terms in sn,r in Proposition 4 in terms of products of the Hermitian expressions bn (m) = ζn ζ¯n+m + ζn+1 ζ¯n+1+m + · · · . These expressions can be estimated using Cauchy–Schwarz, and they are also easy to understand in probabilistic contexts. Unfortunately I do not know how to systematically estimate nongeneric terms. Example. s2 = s2,1 = b2 (1) + b3 (1) and in general sn,1 = bn (n − 1) + bn+1 (n − 1). s3,2 is a quadratic expression in terms of the variables ζ3 ζ¯4 , ζ4 ζ¯5 , . . . . The matrix is 1

3 2 2 2 3 6 4 4 3 6 4 3 6

... 4 ... 4 4 4

4

... 4

...

Therefore s3,2 = b3 (1)2 + b4 (1)2 +

ζi ζ¯i+1 ζi ζ¯i+1 + ζ3 ζ¯4 ζ4 ζ¯5 + 2

i4

ζi ζ¯i+1 ζi+1 ζ¯i+2 .

i4

Thus “most” of s3,2 can be written in terms of powers of Hermitian expressions, and two “diagonal” sums near the boundary of the cone that we are adding over. A.2. ζ in terms of x We have ζn = ζ1 (xn , xn+1 , . . .), and for a finite number of variables, one can generate formulas for ζ1 . For example, if pn = j >n (1 + |ζj |2 ), then ζ1 (x1 , x2 , x3 , x4 ) =

1 1 1 1 x1 − x 2 x¯3 + 2 x2 x32 x¯3 x¯4 − 2 x2 x3 x¯4 p1 p1 p2 p3 2 p1 p3 p4 p1 p2 p32 p4 −

1

x34 x¯3 x¯42 p1 p2 p33 p42

+

1 x 3 x¯42 , p1 p32 p42 3

where the pi can be expressed in terms of x using the displayed line following (6.10) in [6]. But I have not made any progress toward finding a general formula.


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References [1] H. Brezis, New questions related to the topological degree, in: The Unity of Mathematics, in Honor of the Ninetieth Birthday of I.M. Gelfand, Birkhäuser, 2006, pp. 137–154. [2] K. Clancey, I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Birkhäuser, 1981. [3] D.G. Ebin, J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. 92 (1) (1970) 102–163. [4] V. Peller, Hankel Operators and Their Applications, Springer, 2003. [5] D. Pickrell, Invariant measures for unitary forms of Kac–Moody groups, Mem. Amer. Math. Soc. 693 (2000) 1–144. [6] D. Pickrell, Homogeneous Poisson structures on loop spaces of symmetric spaces, Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 069. [7] D. Pickrell, B. Pittmann-Polletta, Unitary loop groups and factorization, J. Lie Theory 20 (1) (2010) 93–112. [8] A. Pressley, G. Segal, Loop Groups, Oxford Math. Monogr., Oxford Science Publications, Oxford University Press, New York, 1986. [9] H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants. II, Adv. Math. 21 (1976) 1–29.


Direct sums and the Szlenk index ✩ Philip A.H. Brooker 1 Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia Received 1 April 2010; accepted 16 December 2010 Available online 20 January 2011 Communicated by Gilles Godefroy

Abstract For α an ordinal and 1 < p < ∞, we determine a necessary and sufficient condition for an p -direct sum of operators to have Szlenk index not exceeding ωα . It follows from our results that the Szlenk index of an p -direct sum of operators is determined in a natural way by the behaviour of the ε-Szlenk indices of its summands. Our methods give similar results for c0 -direct sums. © 2010 Elsevier Inc. All rights reserved. Keywords: Szlenk index; Asplund operators; Direct sums; Banach spaces

0. Introduction The Szlenk index was introduced by W. Szlenk in his influential paper [22], where an ordinal index was used to show that the class of all separable, reflexive Banach spaces contains no universal element. Since then, the Szlenk index and its variants have taken on an increasingly important role in the study of Banach spaces and their operators. We refer the reader to the surveys [14] and [19] for details on some of the main applications of the Szlenk index. A class of closed operator ideals naturally related to the Szlenk index has been introduced and systematically studied by the present author in [2]. These operator ideals are denoted SZ α , where α is an ordinal, and elements of SZ α are known as α-Szlenk operators. The operator ✩

Research supported by an ANU PhD Scholarship. E-mail address: [email protected]. 1 This work forms part of the author’s doctoral dissertation, written at the Australian National University under the supervision of Dr. Richard J. Loy. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.016

P.A.H. Brooker / Journal of Functional Analysis 260 (2011) 2222–2246

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ideals SZ α are studied in [2] with regard to their operator ideal properties and their relationship to other closed operator ideals, in particular the class of Asplund operators. The purpose of the present paper is to present a detailed analysis of the behaviour of the Szlenk index under the process of taking c0 and p -direct sums of operators. In particular, we give a precise formulation of the Szlenk index of a direct sum of operators in terms of the behaviour of the ε-Szlenk indices of the summands. Our motivation for this is as follows. Firstly, forming direct sums is a fundamental construction in Banach space theory, often being used to construct examples with a particular property, and so we feel it essential to understand precisely how the Szlenk index behaves under this procedure. Secondly, we are motivated by the following basic question of operator ideal theory: Question 0.1. Let I be a given operator ideal. Does I have the factorisation property? That is, does every element of I factor continuously and linearly through a Banach space whose identity operator belongs to I ? In [2], results and techniques developed in the current paper are applied to obtain both positive and negative answers to Question 0.1 for the case I = SZ α , with the answer depending upon ordinal properties of α. We now outline the structure of the current paper. In Section 1 we detail necessary notation and background results regarding the Szlenk index, including several relevant results from [2]. Our main results are presented in Section 2. Firstly, we consider the Szlenk index of 1 and ∞ -direct sums; this case is rather straightforward, but worth noting explicitly for the sake of completeness. We then move on to our main concern, providing a formulation of the Szlenk index of c0 and p -direct sums of operators, where 1 < p < ∞ (see, in particular, Theorem 2.10). This case is far more subtle than the case of 1 and ∞ -direct sums and, as such, requires substantially more effort to accomplish the desired formulation of the Szlenk index of the direct sum. Section 2 concludes with some applications of the earlier operator theoretic results to the Szlenk index of Banach spaces. The final section, Section 3, constitutes almost half of the paper and is devoted to proving the main technical lemma used in Section 2, namely Lemma 2.5. 1. Preliminaries Banach spaces are typically denoted by the letters E and F . For a Banach space E and nonempty bounded S ⊆ E, we define |S| := sup{x | x ∈ S}. By BE we denote the closed unit ball of E, and by IE the identity operator of E. The class of all bounded linear operators between arbitrary Banach spaces is denoted by B, and the class of all compact operators by K . We write O RD for the class of all ordinals, whose elements are typically denoted by the lower-case Greek letters α, β and γ . For Λ a set, Λ 0, there exists B ∈ Σ such that μ(B) > μ(Ω) − ε and {f χB | f ∈ ST (BE )} is relatively compact in L∞ (Ω, Σ, μ) (here χB denotes the characteristic function of B on Ω). We note that some authors, for example in [17] and [10], refer to Asplund operators as decomposing operators. Standard references for Asplund operators are [17] and [21], where it is shown that the Asplund operators form a closed operator ideal and that a Banach space is an Asplund space if and only if its identity operator is an Asplund operator. A further impressive result is that every Asplund operator factors through an Asplund space; this is due independently to O. Re˘ınov [18], S. Heinrich [10] and C. Stegall [21]. We now define the Szlenk index, noting that our definition varies from that given by W. Szlenk in [22]. However, the two definitions give the same index for operators acting on separable Banach spaces containing no isomorphic copy of 1 (see the proof of [12, Proposition 3.3] for details). Let E be a Banach space, K ⊆ E ∗ a w ∗ -compact set and ε > 0. Define sε (K) := x ∈ K diam(K ∩ V ) > ε for every w ∗ -open V x . We iterate sε transfinitely as follows: let sε0 (K) = K, sεα+1 (K) = sε (sεα (K)) for each ordinal α β and, if α is a limit ordinal, sεα (K) = β0 Szε (K). Note that Szε (K) (resp., Sz(K)) is either an ordinal or the class O RD of all ordinals. If Szε (K) (resp., Sz(K)) is an ordinal, then we write Szε (K) < ∞ (resp., Sz(K) < ∞), and otherwise we write Szε (K) = ∞ (resp., Sz(K) = ∞).


2225

For a Banach space E, the ε-Szlenk index of E is Szε (E) = Szε (BE ∗ ), and the Szlenk index of E is Sz(E) = Sz(BE ∗ ). If T : E −→ F is an operator, the ε-Szlenk index of T is Szε (T ) = Szε (T ∗ BF ∗ ), whilst the Szlenk index of T is Sz(T ) = Sz(T ∗ BF ∗ ). It is clear that the Szlenk index of a nonempty w ∗ -compact set cannot be 0. We note also that, by w ∗ -compactness, the ε-Szlenk index of a nonempty w ∗ -compact set K is never a limit ordinal. The following proposition states some known facts about the Szlenk index. Proposition 1.1. Let E and F be Banach spaces, T : E −→ F an operator and K ⊆ E ∗ a nonempty w ∗ -compact set. (i) If E is isomorphic to a quotient or subspace of F , then Sz(E) Sz(F ). In particular, the Szlenk index is an isomorphic invariant of a Banach space. (ii) Sz(E) < ∞ if and only if E is an Asplund space. Similarly, Sz(T ) < ∞ if and only if T is an Asplund operator. (iii) If K is absolutely convex and Sz(K) < ∞, then there exists an ordinal α such that Sz(K) = ωα . In particular, the Szlenk index of an Asplund space or Asplund operator is of the form ωα for some (unique) ordinal α. (iv) Sz(K) = 1 if and only if K is norm-compact. In particular, Sz(E) = 1 if and only if dim(E) < ∞, and Sz(T ) = 1 if and only if T is compact. (v) Sz(E ⊕ F ) = max{Sz(E), Sz(F )}. Part (i) of Proposition 1.1 is discussed in [8]. Part (ii) is discussed in [8] in the case of spaces, and the more general case of operators is established in [2, Proposition 2.10]. Part (iii) was proved for K = BE ∗ in [13]; see also p. 64 of [9]. As the proof of the case K = BE ∗ relies only upon the fact that BE ∗ is convex and symmetric (that is, absolutely convex), the proof applies also to arbitrary absolutely convex K. Part (iv) is a consequence of the fact that a w ∗ -compact set is norm-compact if and only if its relative w ∗ and norm topologies coincide (see, e.g., [4, Corollary 3.1.14]), with the final assertion regarding operators requiring the use of Schauder’s theorem. Part (v) is essentially Proposition 2.4 of [7] (see also [15, Proposition 14] for the separable case), and will be improved upon in Theorem 2.11 below. Definition 1.2. For each ordinal α, define SZ α := {T ∈ B | Sz(T ) ωα }. As noted in the introduction, elements of SZ α are known as α-Szlenk operators. We have the following: Theorem 1.3. (See [2, Theorem 2.2].) Let α be an ordinal. Then SZ α is a closed operator ideal. 2. Main results It is obvious that a direct sum of operators factors any of its summands. Thus, since {T ∈ B | Sz(T ) < ∞} is the operator ideal of Asplund operators (see Proposition 1.1(ii)), it is only interesting to consider the Szlenk index of a direct sum of operators in the case that all of the summands are Asplund. With this in mind, we henceforth consider direct sums of Asplund operators only.

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2.1. 1 -Direct sums and ∞ -direct sums The task of determining the Szlenk index of 1 -direct sums and ∞ -direct sums of operators is made considerably easier by the fact that the Banach spaces 1 and ∞ fail to be Asplund, for this ensures that the norms of the summand operators must exhibit c0 -like behaviour in order for the direct sum operator to be Asplund. More precisely, we have the following result. Proposition 2.1. Let Λ be a set, {Eλ | λ ∈ Λ} and {Fλ | λ ∈ Λ} families of Banach spaces, {Tλ ∈ B(Eλ , Fλ ) | λ ∈ Λ} a uniformly bounded family of Asplund operators and p = 1 or p = ∞. The following are equivalent: (i) Sz((λ∈Λ Tλ )p ) < ∞ (that is, ( λ∈Λ Tλ )p is Asplund). (ii) Sz(( λ∈Λ Tλ )p ) = sup{Sz(Tλ ) | λ ∈ Λ}. (iii) (Tλ )λ∈Λ ∈ c0 (Λ). Proof. We prove (iii) ⇒ (ii) ⇒ (i) ⇒ (iii). Suppose (iii) holds; we will show Sz(( λ∈Λ Tλ )p ) = sup{Sz(Tλ ) | λ ∈ Λ}. By Proposition 1.1(iii) there exist ordinals αλ , λ ∈ Λ, with Sz(Tλ ) =ωαλ for each λ. Let αΛ = sup{αλ | λ ∈ Λ}, so that sup{Sz(Tλ ) | λ ∈ Λ} = ωαΛ . To see that ( λ∈Λ Tλ )p ∈ SZ αΛ , for n ∈ N and λ ∈ Λ let Tλ if Tλ > 1/n, Tλ,n = 0 otherwise and Vn = ( λ∈Λ Tλ,n )p . Note that {Tλ,n | λ ∈ Λ, n ∈ N} ⊆ SZ αΛ , hence Vn ∈ SZ αΛ also that factor some element of {Tλ,n | since each Vn can be written as a (finite) sum of operators λ ∈ Λ, n ∈ N}. It follows from the definitions that Vn − ( λ∈Λ Tλ )p 1/n for each n ∈ N, hence Vn −→ ( λ∈Λ all n and SZ ααΛ is closed (TheoTλ )p as n −→ ∞. Since Vn ∈ SZ αΛ for Λ rem 1.3), we have ( λ∈Λ Tλ )p ∈ SZ αΛ . In particular, Sz(( λ∈Λ Tλ )p ) ω = sup{Sz(Tλ ) | λ ∈ Λ}. The reverse inequality follows by Theorem 1.3 and the fact that ( λ∈Λ Tλ )p factors each of the operators Tλ , λ ∈ Λ. We have now shown (iii) ⇒ (ii). It is trivial that (ii) ⇒ (i), so remains only to show that (i) ⇒ (iii). To this end, suppose that (iii) does not hold. Then there exists δ > 0 and an infinite set Λ ⊆ Λ such that Tλ > δ for all λ ∈ Λ , and so ( λ∈Λ T λ )p factors an isomorphic embedding of the non-Asplund space p . By Proposition 1.1(ii), Sz(( λ∈Λ Tλ )p ) = ∞. 2 2.2. c0 -Direct sums and p -direct sums (1 < p < ∞) In this section we consider the Szlenk index of a direct sum operator ( λ∈Λ Tλ )p , where p = 0 or 1 < p < ∞. As in the cases p = 1 and p = ∞, if (Tλ )λ∈Λ ∈ c0 (Λ) then / Sz(( λ∈Λ Tλ )p ) = sup{Sz(Tλ ) | λ ∈ Λ}. However, the situation is not so clear if (Tλ )λ∈Λ ∈ c0 (Λ), and we demonstrate this by way of an example. For an ordinal γ , we may equip the ordinal γ + 1 with its order topology, thereby making it a compact Hausdorff space. C. Samuel α has shown that for each α < ω1 , Sz(C(ωω + 1)) = ωα+1 (Samuel’s calculation is found in [20], however a more direct approach has been discovered by P. Hájek and G. Lancien [7]). By the Bessaga–Pełczyński linear isomorphic classification of C(K) spaces with K countable [1, Theorem 1], C(ωn + 1) is linearly isomorphic to C(ω + 1) for all 0 < n < ω. Thus, in particular,


2227

Sz(C(ωn + 1)) = Sz(C(ω + 1)) = ωfor all 0 < n < ω. For each 0 < n < ω, let Tn denote the identity operator on C(ωn + 1). As ( 0 0. Let d = max{diam(Ki ) | 1 i n} and let m and M be natural numbers such that M m 2 and (2q − 1)ε q M 8q d q (m − 1). Suppose α α is an ordinal such that sεω ·M (Bq (Ki | 1 i n)) = ∅. Then for every δ ∈ (0, ε/16) there is α i n such that sδω ·m (Ki ) = ∅. The proof of Lemma 2.5 is delayed until Section 3. To show (iii) ⇒ (i) we require the following discrete variant of [7, Lemma 3.3]: Lemma 2.6. Let Λ be a set, (Eλ )λ∈Λ a family of Banach spaces, 1 q < ∞, p predual to q and K ⊆ ( λ∈Λ Eλ )∗p nonempty and w ∗ -compact. Let α be an ordinal, R ⊆ Λ and ε > δ > 0. ∗ xq > |K|q − ( ε−δ )q , then U ∗ x ∈ s α (U ∗ K). If x ∈ sεα (K) and UR δ 2 R R Proof. We fix ε, δ and R and proceed by induction on α. The conclusion of the lemma is trivially true for α = 0. So suppose that β is an ordinal such that the conclusion of the lemma holds with α = β; we show that it holds then also for α = β + 1. To this end, let x ∈ K be such that β+1 β+1 ∗ xq > |K|q − ( ε−δ )q and U ∗ x ∈ ∗ K). Our goal is to show that x ∈ (UR / sε (K), so UR 2 R / sδ β ∗ x ∈ s β (U ∗ K) by the inductive hypothesis. It follows we may assume that x ∈ sε (K), hence UR δ R ∗ x ∈ V and d := diam(V ∩ s β (U ∗ K)) δ. that there is w ∗ -open V ⊆ ( λ∈R Eλ )∗p such that UR δ R ∗ x does not belong to the w ∗ -closed set (|K|q − ( ε−δ )q )1/q B ∗ As UR , we may assume ( λ∈R Eλ )p 2

ε−δ V ∩ |K| − 2 q

q 1/q

B(λ∈R Eλ )∗p = ∅.

∗ )−1 (V ) and let u ∈ W ∩ s (K). Then U ∗ uq > |K|q − ( ε−δ )q and u ∈ s (K), Let W = (UR ε ε 2 R ∗ u ∈ V ∩ s β (U ∗ K). So for u , u ∈ W ∩ s β (K) we have hence by the induction hypothesis UR 1 2 ε δ R ∗ u − U ∗ u q d q δ q . Moreover, since U ∗ u q > |K|q − ( ε−δ )q it follows that UR 1 2 R 2 R 1 β

u1 − P ∗ U ∗ u1 |K|q − P ∗ U ∗ u1 q 1/q = |K|q − U ∗ u1 q 1/q < ε − δ . R R R R R 2

β

P.A.H. Brooker / Journal of Functional Analysis 260 (2011) 2222–2246 ∗ U∗ u < Similarly, u2 − PR R 2

ε−δ 2 .

2229

We now deduce that

∗ ∗ q q ∗ ∗ ∗ ∗ ∗ ∗ u1 − u2 q = PR UR u1 − PR UR u2 + u1 − PR UR u1 − u2 − PR UR u2

∗ q ε−δ q ∗ UR u1 − UR u2 + 2 · 2 δ q + (ε − δ)q εq . β

β+1

In particular, diam(W ∩ sε (K)) ε. It follows that x ∈ / sε (K), as desired. The lemma passes easily to limit ordinals, so we are done. 2 In order to state the third (and final) lemma required in the proof of Proposition 2.4, we give the following definition. Definition 2.7. For real numbers a 0, b > c > 0 and 1 d < ∞, define

d

2a d b σ (a, b, c, d) := inf n ∈ N n − +1 . b−c b−c With regards to Definition 2.7, note that σ (a, b, c, d) = 1 whenever 2a b. Lemma 2.8. Let Λ be a set, {Eλ | λ ∈ Λ} a family of Banach spaces, 1 q < ∞, p predual to q, K ⊆ ( λ∈Λ Eλ )∗p a nonempty, w ∗ -compact set and ε > δ > 0. Suppose ηδ is a nonzero ordinal η ·σ (|K|,ε,δ,q)

such that sδ δ (UF∗ K) = ∅ for every F ∈ Λ 0 let 1 < mε < ω and βε < α be such that sup{Szε/32 (Kλ ) | λ ∈ Λ} < ωβε ·mε . Set d = sup{diam(Kλ ) | λ ∈ Λ} and for each ε ∈ (0, 1) let Mε ∈ N be such that (2q − 1)ε q Mε 8q d q (mε − 1). By Lemma 2.5, for F ∈ Λ 0. It follows that if ( λ∈Λ Tλ )p is noncompact, then Sz

λ∈Λ

Tλ

= inf ωα sup Szε (Tλ ) λ ∈ Λ < ωα for all ε > 0 .

p

Proof. For convenience we set T = ( λ∈Λ Tλ )p . The equivalence of (i) and (ii) is achieved by applying Proposition 2.4 with Kλ = Tλ∗ BFλ∗ for all λ ∈ Λ, for in this case T ∗ B(λ∈Λ Eλ )∗p = Bq (Tλ∗ BFλ∗ | λ ∈ Λ), where q ∈ [1, ∞) is dual to p. For each λ ∈ Λ let αλ denote the unique ordinal satisfying Sz(Tλ ) = ωαλ . Let αΛ = sup{αλ | λ ∈ Λ}. The set α ω sup Szε (Tλ ) λ ∈ Λ < ωα for all ε > 0 ωαΛ +1 is nonempty, hence Sz(T ) inf ωα sup Szε (Tλ ) λ ∈ Λ < ωα for all ε > 0 by the implication (ii) ⇒ (i) above. To complete the proof, we now suppose that T is noncompact. As Sz(T ) is a power of ω, it is enough to show that Sz(T ) > ωβ holds for β satisfying ωβ < inf{ωα | sup{Szε (Tλ ) | λ ∈ Λ} < ωα for all ε > 0}. Take such β. If β = 0, then Sz(T ) > ωβ by noncompactness of T . On the other hand, if β > 0 then there is ε > 0 so small that Szε (T ) sup{Szε (Tλ ) | λ ∈ Λ} ωβ . As Szε (T ) cannot be a limit ordinal, we conclude that Sz(T ) Szε (T ) > ωβ . 2

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2.3. Applications Our first result here is the following Banach space analogue of Theorem 2.10 which determines precisely the Szlenk index of a c0 -direct sum or p -direct sum of Banach spaces in terms of the behaviour of the ε-Szlenk indices of the summand spaces. Theorem 2.11. Let Λ be a set, {Eλ | λ ∈ Λ} a family of Asplund spaces, α > 0 an ordinal and p = 0 or 1 < p < ∞. The following are equivalent: (i) Sz(( λ∈Λ Eλ )p ) ωα . (ii) sup{Szε (Eλ ) | λ ∈ Λ} < ωα for all ε > 0. It follows that if ( λ∈Λ Eλ )p is infinite dimensional, then Sz

λ∈Λ

Eλ

= inf ωα sup Szε (Eλ ) λ ∈ Λ < ωα for all ε > 0 .

p

Proof. The conclusions of the theorem follow by taking Tλ to be the identity operator of Eλ for each λ ∈ Λ in the statement of Theorem 2.10. 2 Theorem 2.12. Let Λ be a set, E an infinite dimensional Banach space and 1 < p < ∞. Then

Sz(E) = Sz c0 (Λ, E) = Sz p (Λ, E) . Proof. Apply Theorem 2.11 with Eλ = E for all λ ∈ Λ.

2

The previous theorem, Theorem 2.12, allows us to add to the class of ordinals γ for which the Szlenk index of C(γ + 1) is known (here, γ + 1 is equipped with its order topology). The computation of the Szlenk index of C(ω1 + 1), in particular Sz(C(ω1 + 1)) = ω1 · ω, is due to Hájek and Lancien [7]. Essentially using the fact that Sz(C(ξ + 1)) = Sz(C(ζ + 1)) for ordinals ξ and ζ satisfying ξ ζ < ξ · ω (an easy consequence of Proposition 1.1(v)), Hájek and Lancien deduce that Sz(C(γ + 1)) = ω1 · ω whenever ω1 γ < ω1 · ω. We claim that Sz(C(γ + 1)) = ω1 · ω whenever ω1 γ < ω1 · ωω , a fact that will follow once we have shown that Sz(C(ξ + 1)) = Sz(C(ζ + 1)) whenever ξ and ζ are ordinals satisfying ω ξ ζ < ξ · ωω . If ξ and ζ are ordinals satisfying ω ξ ζ < ξ · ωω , then there exists n < ω such that C(ζ + 1) is isomorphic to a subspace of C(ξ · ωn + 1). Thus, by Proposition 1.1(i), it suffices to show that Sz(C(ξ + 1)) = Sz(C(ξ · ωn + 1)) for all n < ω. This is obviously true for n = 0, and if true for some n then, since C(ξ · ωn+1 + 1) is isomorphic to c0 (ω, C(ξ · ωn + 1)), Theorem 2.12 yields

Sz C ξ · ωn+1 + 1 = Sz c0 ω, C ξ · ωn + 1

= Sz C ξ · ωn + 1

= Sz C(ξ + 1) , which completes the proof.


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The following proposition asserts that the set of all countable values of the Szlenk index of Banach spaces is attained by the class of Banach spaces with a shrinking basis. A further consequence of this result is that if for α < ω1 there exists a Banach space of Szlenk index ωα , then Pełczyński’s complementably universal basis space (see [16]) has a complemented subspace of Szlenk index ωα . Proposition 2.13. Let 0 < α < ω1 . The following are equivalent: (i) There exists a Banach space E with Sz(E) = ωα . (ii) There exists a Banach space E with a shrinking basis and Sz(E) = ωα . To prove Proposition 2.13, we shall call on the following result regarding subspaces and quotients, due to G. Lancien [13] and [11, Theorem III.1]: Proposition 2.14. Let β < ω1 and let E be a Banach space such that Sz(E) > β. (i) There is a separable closed subspace F of E such that Sz(F ) > β. (ii) If E ∗ is norm separable, then for every δ > 0 there is a closed subspace F of E such that Sz(E/F ) > β and E/F has a shrinking basis with basis constant not exceeding 1 + δ. With the exception of the basis constant assertion of part (ii), Proposition 2.14 is proved in [13]. Lancien’s proof follows closely the proof of [11, Theorem III.1], and the extra assertion above regarding the basis constant is easily added to Lancien’s result using the observations regarding basis constants in the proof of [11, Theorem III.1]. Proposition 2.13 is an immediate consequence of the following: Proposition 2.15. Let α > 0 be a countable ordinal and E a Banach space with Sz(E) = ωα . Then there exist closed subspaces F ⊆ E and G ⊆ 2 (F ) such that 2 (F )/G has a shrinking basis and Sz(2 (F )/G) = ωα . Proof. For each n ∈ N, Proposition 2.14(i) yields a separable closed subspace Dn of E such that Sz(Dn ) > Sz1/n (E). Let F = span( n∈N Dn ). Then ωα = Sz(E) = sup Sz1/n (E) sup Sz(Dn ) Sz(F ) Sz(E) = ωα , n

n

hence equality holds throughout. In particular, Sz(F ) = ωα and, as F is a separable Asplund space (indeed, Sz(F ) < ∞), F ∗ is norm separable. For each n ∈ N let Fn = F . Then, by Proposition 2.14(ii), for each n ∈ N there is a closed subspace Gn of Fn such that Sz(Fn /Gn ) > Fn /Gn has a shrinking basis with basis constant 2. Let G denote Sz1/n (E) and not exceeding embedding into ( F ) . Then ( the image of ( n∈N Gn )2 under its natural n 2 n∈N n∈N Fn )2 /G is naturally isometrically isomorphic to ( n∈N Fn /Gn )2 . Note that ( n∈N Fn /Gn )2 has a shrinking basis since it is the 2 -direct sum of a countable family of Banach spaces with shrinking bases that have uniformly bounded basis constants. On the one hand, by Theorem 2.12 we have

= Sz(F ) = ωα . Fn /G Sz Fn Sz n∈N

2

n∈N

2

2234


On the other hand,

sup Sz1/n (En ) = Sz(E) = ωα . Sz Fn /G = Sz Fn /Gn n∈N

2

n∈N

n

2

Thus ( n∈N Fn )2 /G has a shrinking basis and Szlenk index ωα .

2

Proposition 2.16. Let α be an ordinal. Then there exists a Banach space of Szlenk index ωα+1 . Proof. Our proof is based on the construction of Szlenk in [22], by which we construct Banach spaces Eβ indexed by the class of ordinals β. Let E0 = {0}, Eβ+1 = Eβ ⊕1 2 and, if β is a limit ordinal, Eβ = ( γ β for all ordinals β. As the assertion of the proposition is known to be true for α = 0 (for example, Sz(2 ) = ω), we assume that α > 0 and let β denote the least ordinal such that Sz(Eβ ) > ωα . Then, by Proposition 1.1(iii), Sz(Eβ ) ωα+1 . By Proposition 1.1(v) and the definition of β , it must be that β is a limit ordinal, hence Eβ = ( β 0; (H2) f (x, u) = a(x)u + g(x, u) with a(x) ∈ L∞ (Ω) ∩ C ∞ (Ω), g(x, u) = o(u) as u → 0 uniformly in x and g(x, u) = o(|u|s−1 ) as u → ∞ uniformly in x. From (H1) to (H2), it follows f (x, 0) = 0 and that f is a lower-order perturbation of |u|s−2 u at infinite in the sense that limu→∞ f|u|(x,u) s−1 = 0 uniformly in x ∈ Ω. Moreover, we assume that f (x, u) satisfies: (H3) (H4) (H5)

∂f ∂u (x, u) is continuous on Ω × R; s−2 ), ∀u ∈ R uniformly in x ∈ Ω; | ∂f ∂u (x, u)| C(1 + |u| f (x,u) f1 (x, u) := u is non-decreasing in u > 0 and non-increasing

in u < 0 for a.e. x ∈ Ω.

For K = 1, f (x, u) = λu and λ ∈ (0, λ1 ) where λ1 is the first eigenvalue of − for Dirichlet boundary condition, the problem has a strong background from some variational problems in geometry and physics, such as the Yamabe’s problem with lack of compactness. This was considered by Brezis and Nirenberg for positive solutions in their pioneer work in [3]. Then it has been studied extensively in the last three decades. We recall briefly some results about the existence and multiplicity of sign-changing solutions to the problem (1) for K = 1 and f (x, u) = λu. For any fixed λ > 0, the first multiplicity result was due to Cerami, Fortunato and Struwe [5]. They obtained the number of the solutions of (1) is bounded below by the number of the eigenvalues of − lying in the open interval (λ, λ + S|Ω|−2/N ), where S is the best constant for

Y. Ge et al. / Journal of Functional Analysis 260 (2011) 2247–2282

2249

∗

the Sobolev embedding D 1,2 (RN ) → L2 (RN ) (see the definition below) and |Ω| denotes the Lebesgue measure of Ω. Capozzi, Fortunato and Palmieri in [4] established the existence of a nontrivial solution for λ > 0 which is not an eigenvalue of − when N 4 and for any λ > 0 when N 5 (see also [43]). In [8], Devillanova and Solimini proved that, if N 7, then (1) has infinitely many solutions for every λ > 0. They proved also in [9] that, if N 4 and λ ∈ (0, λ1 ), then there exist at least N2 + 1 pairs of nontrivial solutions. Clapp and Weth [6] have extended this last result to all λ > 0 with N 4. In the same paper they also obtained some extensions to critical biharmonic problems for N 8. When the domain Ω is a ball and N 4, Fortunato and Jannelli [12] proved there are infinitely many sign-changing solutions which are built using the symmetry of the domain Ω. Schechter and Zou in [35] showed the same result for any domain Ω when N 7. In particular, if λ λ1 , it has and only has infinitely many sign-changing solutions except zero. Their work is based on the estimates of Morse indices of nodal solutions. Concerning the polyharmonic case, Pucci and Serrin in [32] have studied the problem (1) for K = 2 and λ > 0 when Ω is a ball. They proved that it admits nontrivial radial symmetric solutions for all λ ∈ (0, λ1 ) if and only if N 8. If N = 5, 6, 7, then there exists λ∗ ∈ (0, λ1 ) such that the problem admits no nontrivial radial symmetric solutions whenever λ ∈ (0, λ∗ ]. Here λ1 is understood as the first eigenvalue of 2 for Dirichlet boundary conditions. This is the counterpart of the well-known result of [3] on the nonexistence for radial symmetric solutions for small λ in dimension N = 3 and K = 1 (where λ∗ = λ1 /4). They called these dimensions as critical dimensions. They conjectured that for general K 1, the critical dimensions are 2K + 1, . . . , 4K − 1. The conjecture is not completely solved for all K 1. Grunau [22] defined later the notion of weakly critical dimensions as the space dimensions for which a necessary condition for the existence of a positive radial solution of (1) in B1 is λ ∈ (λ∗ , λ1 ) for some λ∗ > 0. He proved that the conjecture is true in the weak sense. Gazzola, Grunau and Squassina [16] proved nonexistence of positive radial symmetric solutions for Navier boundary condition for small λ > 0. They established also some existence results for λ = 0. Their result strongly depends on the geometry of domains. For biharmonic operators, Bartsch, Weth and Willem in [1] and Ebobisse and Ahmedou in [10] have studied the problem (1) on domains with nontrivial topology under Dirichlet boundary condition and Navier boundary condition respectively. For related problems, we infer to [2,11,13,15,20,21,29] and the references therein. For general case K 1, Ge has studied in [19] the same type of Eq. (1) for Navier boundary condition when f (x, u) = λu with 0 λ < λ1 and λ1 the first eigenvalues of (−)K . He established the existence of positive solutions in some general domain under the suitable assumptions. In particular unstable solutions in higher level set are obtained by Coron’s topological method in domains perforated with the small holes. The purpose of this paper is to continue the study of the semilinear polyharmonic problem (1) to general K 1 with Dirichlet boundary condition for general domains. Let us denote the polyharmonic operator L := (−)K − a(x) and λ1 (Ω) λ2 (Ω) · · · λn (Ω) · · · the eigenvalues of L under the homogeneous Dirichlet boundary condition. It is well known that each eigenvalue λk (Ω), k 1, can be described as the minimax value

vLv λk (Ω) = min max Ω 2 . V ⊂H0K (Ω), dim V =k v∈V Ωv

2250


It follows that λk (Ω) is a non-increasing functional on the domains, that is, if Ω2 ⊂ Ω1 , then λk (Ω2 ) λk (Ω1 ). Moreover, from the unique continuation principle, we have λk (Ω2 ) > λk (Ω1 ) for any k 1, provided Ω1 is connected (see [24,31]). For the perforated domain Ω := Ω1 \ Ω2 with the smooth bounded domains Ω2 ⊂ Ω1 , with the help of the above description, we have lim λk (Ω1 \ Ω2 ) = λk (Ω1 ), where the limit is taken as the diameter of Ω2 goes to 0. To this aim, it suffices to consider Ω2 = B(x, ) balls with small radius > 0 in the sequel. Assume now λn (Ω) 0 and λn+1 (Ω) > 0 for some n 1. Under our assumptions, the energy functional E is not bounded from below. Thus, we could not use directly the minimization procedure. We split the tangent bundle into two parts: at any tangent space, Tu H0K = T1 ⊕ T2 , where T1 is a finite vector space and T2 is infinite one. The second differential d 2 E is non-positive on T1 and is definite positive on T2 . First, we solve the equation dE(u)|T1 = 0, which leads to consider an infinite-dimensional Finsler manifold. Then, we study the energy functional E on such manifold in order to solve dE(u)|T2 = 0. In such way, we get a solution of the initial problem. Compared to the classic Lyapunov reduction method, we follow the similar strategy, but inverse the procedure. More precisely, let ei (x) be an eigenfunction associated to λk (Ω) with ei K,2,Ω = 1 for any 1 i n. Define M := v ∈ H0K (Ω) \ {0} dE(v)(w) = 0, ∀w ∈ Span(v, e1 , . . . , en ) . We prove in Section 2 that under the hypothesis (H1) to (H5), M is then a complete C 1 Finsler manifold and it will be a C 1,1 Finsler manifold with additional assumptions (H6) to (H7) (see Section 2). This permits to consider the following minimization problem κ := inf E(v). v∈M

We prove then κ

N K 2K N (SK (Ω))

for any f satisfying (H1) to (H5), where we denote

SK (Ω) :=

inf

v∈H0K (Ω)\{0}

v2K,2,Ω v2Ls (Ω)

the best constant for the embedding H0K (Ω) → Ls (Ω). Here, as for K = 1, it is well known that SK (Ω) is independent of Ω and SK (Ω) = SK (RN ) := infv∈H K (RN )\{0}

v2

K,2,RN

v2 s

(see also [15,

L (RN )

18,37,40,41]). Therefore we denote it by SK in the sequel. Our first result concerns the nonN 2K critical dimension case, i.e., we prove that if we have the strict inequality κ < K N (SK (Ω)) , then the infimum for κ is achieved by some u ∈ M which is a solution of (1). Such √ situation is realized for example by either N 4K and λn (Ω) < 0, λn+1 (Ω) > 0 or N > 2( 2 + 1)K and λn (Ω) 0, λn+1 (Ω) > 0, see Proposition 2 below. Notice that this existence result in Proposition 2 is similar to the main result of [13] of Gazzola (see also [14]). The method we used is a reduction type method which is different from the method used by Gazzola. This reduction method can be seen as an alternative approach to the linking method (see [36]). The manifold M


2251

we defined here is a generalization of the Nehari manifold [28], which is equivalent to the manifold defined in [30,33,38,39]. They employed a similar variational approach-reduction method, for existence of solutions to the stationary Schödinger equation and some semilinear elliptic equations. We have even an improvement √ under some suitable conditions on the eigenfunctions when the dimension is less than 2( 2 + 1)K. This is stated in Proposition 3 and in Corollary 1. In particular, when K = 1 and N = 4, if λ > λ1 , with λ1 is a simple eigenvalue, there are ground state solutions for (1). The existence result in such cases is not new and is proved by Clapp and Weth in [6] even under some weaker assumptions on the eigenvalues. However, the new part is that the solutions we obtained here are ground state solutions. For the critical dimension 2K < N < 4K, the existence of solutions to (1) is a delicate issue. To our knowledge, there are few results on it, even for the case K = 1. The reason is that the minimizing method fails, for example, for K = 1, when Ω ⊂ R3 is a ball and when f (x, u) = λu with 0 < λ < λ41 . It is well known that there are no positive solutions. In Section 3, we study the existence of solutions for some perforated domains in such critical dimensions. We analyze the concentration phenomenon when the minimizing solutions do not exist. Then following Coron’s strategy of topological argument [7], we obtain the existence of unstable critical points in higher level sets for domains perforated with small holes. The approach of combining the variational method and Coron’s topological strategy is new for the existence of nontrivial solutions in the indefinite case. In all this paper, C, C and c denote generic positive constants independent of u, even their value could be changed from one line to another one. We give also some notations here. The N ∞ ∞ N space DK,2 (RN ) (resp. DK,2 (RN + )) is the completion of C0 (R ) (resp. C0 (R+ )) for the norm · K,2,RN (resp. · K,2,RN ). +

2. Study of the energy functional E on M We begin this section by studying some properties of the set M. Observe that v ∈ M is equivalent to say v = 0 and satisfying l0 (v) := v2K,2,Ω − vsLs (Ω) − f (x, v)v = 0, Ω

li (v) := (v, ei )Ω −

|v|s−2 vei − Ω

f (x, v)ei = 0,

∀1 i n.

(4)

Ω

Let us denote V0 := Span(e1 , . . . , en ) the n-dimensional vector space spanned by e1 , . . . , en . We prove now the following proposition. Proposition 1. Suppose (H1) to (H5) are satisfied. Then M is a complete C 1 Finsler manifold. Furthermore, suppose that (H6) (H7)

∂2f (x, u) is continuous on Ω × R and u → |u|s−2 u is C 2 on ∂u2 ∂2 s−3 , ∀u ∈ R uniformly in x ∈ Ω. | ∂u 2 f (x, u)| C(|u| + 1)

Then M is a complete C 1,1 Finsler manifold.

R;

2252


Proof. The proof is divided into several steps. Step 1. M is not empty. / V0 and By the assumptions (H1)–(H2), E is a continuous functional on H0K (Ω). Fixing v ∈ let V := Span(v, e1 , . . . , en ). Clearly, for all w ∈ V , we have 1 1 1 2 2 a(x)w − |w|s , (5) E(w) wK,2,Ω − 2 2 s Ω

Ω

0 and F (x, u) 12 a(x)u2 for all u ∈ R \ {0} since it follows from (H2) and (H5) that g(x,u) u and for a.e. x ∈ Ω. As V is a finite-dimensional vector space, all the norms on it are equivalent. In particular, the norms · K,2,Ω and · Ls (Ω) are equivalent on V . This implies lim

w∈V , w→∞

E(w) = −∞.

(6)

On the other hand, again from (H2), we infer for any given ε > 0, there exists C > 0 such that for all u ∈ R and for a.e. x ∈ Ω g(x, u) ε|u| + C|u|s−1 ,

F (x, u)

so that for all w ∈ V 1 1 E(w) w2K,2,Ω − 2 2

1 C a(x) + ε u2 + |u|s , 2 s

1+C a(x) + ε w 2 − s

Ω

(7)

|w|s . Ω

Since v ∈ / V0 , we can choose v ∈ V ∩ (V0 )⊥ such that 12 v 2K,2,Ω − 12 taking a sufficiently small ε > 0, we have 2 2 1 1 v 2 a(x) + ε v ε v K,2,Ω . − K,2,Ω 2 2

Ω

a(x)(v )2 > 0. By

(8)

Ω

As a consequence, we obtain sup E(w) > 0.

(9)

w∈V

Together with (6), there exists v˜ ∈ V such that E(v) ˜ = maxw∈V E(w) since V is a finitedimensional vector space. Clearly, v˜ ∈ M. Step 2. M is closed. We define the map L : H0K (Ω) → Rn+1 , v → l0 (v), . . . , ln (v) .


2253

In view of the assumptions (H1)–(H2), L is continuous on H0K (Ω). Let (vk ) ⊂ M be a sequence in M such that vk → v in H0K (Ω). Then we get L(v) = 0. Now it suffices to show v = 0. First, we note vk ∈ / V0 for all k ∈ N. Indeed, we have vk 2K,2,Ω

−

a(x)vk2

= vk sLs (Ω)

Ω

+

g(x, vk )vk .

(10)

Ω

If we have vk ∈ V0 for some k 1, the term on the left-hand is non-positive. But that one on the right-hand is non-negative. Thus, vk sLs (Ω) = 0 and the desired contradiction vk = 0 gives the result. Now, we claim there exists some positive number c > 0 such that vk K,2,Ω > c. We denote the orthogonal projection of vk on V0 by

vk :=

n (vk , ei )Ω ei i=1

and vk⊥ its orthogonal complementary

vk⊥ := vk − vk . As vk ∈ M, we obtain vk , vk Ω −

a(x)vk vk

Ω

g(x, vk ) s−2 |vk | vk vk . = + vk Ω

Together with (10), we have

⊥ 2 v k

K,2,Ω

−

2 ⊥ 2 2 a(x) vk − vk − a(x) vk

Ω

|vk |s−2 +

= Ω

Ω

g(x, vk ) ⊥ 2 2 vk − vk vk

which implies ⊥ 2 v k K,2,Ω −

2 a(x) vk⊥

Ω

g(x, vk ) ⊥ 2 s−2 |vk | vk , + vk

Ω

since 2 v

k K,2,Ω

− Ω

Gathering (5), (8) and (11), we get

2 a(x) vk 0.

(11)

2254


2 2ε v ⊥ k

K,2,Ω

2 v ⊥ −

2 a(x) + ε vk⊥ (1 + C)

k

Ω

⊥ 2 (1 + C)vk s−2 Ls (Ω) vk Ls (Ω)

2 |vk |s−2 vk⊥

Ω

⊥ 2 C (1 + C)vk s−2 Ls (Ω) vk K,2,Ω .

Finally, vk Ls (Ω) c > 0 and the desired claim follows. Step 3. dL(v) is surjective and its kernel splits for all v ∈ M. By (H3) and (H4), f (x, u)u and f (x, u) are C 1 on Ω × R and ∂(f (x, u)u) C 1 + |u|s−1 , (x, u) ∂u

uniformly in x ∈ Ω and ∀u ∈ R.

(12)

Therefore, L is C 1 on H0K (Ω) provided the assumptions (H1)–(H4) hold. A direct calculation leads to

dl0 (v)(w) = 2(v, w)Ω − s

f (x, v) + v

|v|s−2 vw − Ω

Ω

dli (v)(w) = (w, ei )Ω − (s − 1)

|v|s−2 wei − Ω

∂f (x, v) w, ∂v

∂f (x, v) wei , ∂v

∀1 i n.

(13)

Ω

We claim dL(v)|V , the restriction on V of dL(v), is a bijective endomorphism from V on Rn+1 . As V and Rn+1 have the same dimension, it suffices to prove Ker(dL(v)|V ) = {0}. Let w ∈ Ker(dL(v)|V ) and write w = μv + ni=1 μi ei where μ, μi ∈ R for each i. Combining (4) and (13), we get

dl0 (v)(w) = −(s − 2) Ω

dli (v)(w) = −

−f (x, v) + v

|v|s−2 vw −

∂f (x, v) w = 0, ∂v

Ω

n f (x, v) ∂f (x, v) − + μvei − (s − 2) |v|s−2 μvei + μ j ej , ei v ∂v

Ω

j =1

Ω

− (s − 1)

|v|s−2 ei Ω

n

μj ej −

j =1

Ω

∂f (x, v) ei μj ej = 0, ∂v n

j =1

for all 1 i n. On the other hand, we have μdl0 (v)(w) +

n i=1

Together with (14), we infer

Ω

μi dli (v)(w) = 0.

(14)


(s − 2)

|v| Ω

+

s−2

w +

|v|

2

s−2

n g(x, v) μj ej v

Ω

2

2 −

j =1

Ω

n

μj ej ,

j =1

n

μj ej ,

j =1

n

n

∂f (x,v) ∂v

− Ω

+ Ω

0,

a(x)

μi ei

μi ei

i=1

i=1

+ We know from (H2) and (H5) that − f (x,v) v

f (x, v) ∂f (x, v) − + w2 v ∂v

+

μj ej

j =1

Ω

n

2255

2 μj ej

= 0.

j =1

Ω

g(x,v) v n

a(x)

n

0 and 2 0.

μj ej

j =1

Ω

Finally, we deduce vw(x) = 0,

v

n

μj ej (x) = 0 for a.e. x ∈ Ω

(15)

j =1

and

n

μj ej ,

j =1

n

−

μi ei

i=1

Ω

a(x)

n

2 μj ej

= 0.

(16)

j =1

Ω

Thus we have μv 2 = vw − v

n

μj ej = 0

j =1

which yields μ = 0. Moreover, it follows from (16) that Lw = 0. By the unique continuation principle, we have either w ≡ 0 or w(x) = 0 for a.e. x ∈ Ω. Indeed, we state first w is regular. All the derivatives of w vanish a.e. on the set {x ∈ Ω; w(x) = 0} provided this set is not a negligible measurable set. Thus, w vanishes of infinite order at such points. By the strong unique continuation principle [24], w vanishes. Going back to (15), we have w ≡ 0 and the desired claim follows. As a consequence, for all v ∈ M, dL(v) is surjective and H0K (Ω) = ker(dL(v)) ⊕ V . M is thus a complete C 1 Finsler manifold (see [23]). Furthermore, M is a complete C 1,1 Finsler manifold provided (H6) and (H7) are satisfied. 2 For any v ∈ H0K (Ω) \ V0 , we denote by V

+

:= tv +

n

μi ei for all t > 0, μi ∈ R ,

i=1

the (n + 1)-dimensional half space spanned by v and {ei } for all 1 i n. We have the following

2256


Lemma 1. Under the assumptions (H1) to (H5), then there exists a unique v0 ∈ M such that M ∩ V + = {v0 }.

(17)

E(v0 ) = max E(w).

(18)

Moreover we have w∈V +

Proof. Given v ∈ H0K (Ω) \ V0 , we define for any t > 0 the n-dimensional affine vector space Vt := tv + V0 . We divide the proof into several steps. Step 1. For any t > 0 there exists a unique v(t) ∈ Vt such that E(v(t)) = maxVt E. Moreover, {v(t), t > 0} is a C 1 curve in V + . From (H1) to (H4), it is known that E is C 2 on V + . Thanks to (6), we have lim

w∈Vt , w→∞

E(w) = −∞.

Thus there exists some v(t) ∈ Vt such that E(v(t)) = maxw∈Vt E(w). A direct calculation leads to ∂g(x, v) 2 2 2 s−2 (s − 1)|v| w2 . + d E(v)(w, w) = wK,2,Ω − a(x)w − ∂v Ω

Ω

By (H5), we infer g(x, v) 0 v

and

∂g(x, v) g(x, v) 0. ∂v v

Hence, d 2 E(v) < 0 on Vt , that is, the functional E is strictly concave on Vt . This yields the uniqueness. We note {v(t), t > 0} = {w ∈ V + | dE(w)|V0 = 0}. As the second variation d 2 E of E is negative define on V0 , it follows from the Implicit Function Theorem that {v(t), t > 0} is a C 1 curve in V + which finishes the proof of Step 1. Step 2. For all w ∈ M ∩ V + , the restriction of E on V + has a strictly local maximum at w. nRecall V := Span(v, e1 , . . . , en ). Let v = 0 satisfying dE(v)|V = 0 and w = μv + i=1 μi ei ∈ V . As in the proof of Proposition 1, we have by (H2), d E(v)(w, w) = −(s − 2)

s−2

Ω

|v|

2

w −

|v|

2

s−2

Ω

n j =1

2 μj ej


2257

2 n f (x, v) ∂f (x, v) g(x, v) 2 − + w − − μj ej v ∂v v Ω

+

n

μj ej ,

j =1

n

Ω

−

μi ei

i=1

j =1

Ω

a(x) Ω

n

2 μj ej

j =1

which implies from (H1) to (H5) d 2 E(v)(w, w) < 0 provided w = 0. Therefore, the desired claim follows. Step 3. There exists a unique t0 > 0 such that v(t0 ) ∈ M. Moreover, dE(v(t))(v(t)) > 0 for any 0 < t < t0 and dE(v(t))(v(t)) < 0 for any t > t0 . With the same arguments as in the proof of Proposition 1, we have sup E(w) > 0.

(19)

w∈V +

On the other hand, it follows from (5) that ∀w ∈ V0 E(w) 0.

(20)

In particular, we obtain sup E(w) = sup E(w) = sup E v(t) ,

w∈V +

w∈V +

t>0

where V + is the closure of V + . Combining (6), (19) and (20) and using the continuity of E on V + , there exists some v0 ∈ M ∩ V + such that E(v0 ) = sup E(w). w∈V +

We know M ∩ V + ⊂ w ∈ V + dE(w)|V0 = 0 = v(t) t > 0

(21)

so that there exists t0 > 0 such that v(t0 ) = v0 . Set α(t) := E(v(t)) then α (t) = dE(v(t))(v (t)) = l0 (v(t)) since v (t) − v ∈ V0 and dE(v(t))|V0 = 0. We claim M ∩ V + = {v(t) | α (t) = 0}. t Obviously, M ∩ V + ⊂ {v(t) | α (t) = 0}. Conversely, for any v(t) with α (t) = 0, by the method of Lagrange multipliers, there exist μ1 , . . . , μn ∈ R such that dE(v(t))|V + n i=1 μi dli (v(t))|V = 0. Hence, we have on V0 , n i=1

μi d 2 E v(t) (·, ei ) = 0.

2258


By virtue of the fact d 2 E(v)|V0 < 0 for all v ∈ Vt , we infer μ1 = · · · = μn = 0 which proves the claim. Applying (6), we infer lim α(t) = −∞,

t→+∞

since inf wK,2,Ω = +∞.

lim

t→+∞ w∈Vt

It follows from Step 2 that there exists only strictly local maximum points for α(t). Hence, t0 is the only critical point of α(t). Moreover, α (t) > 0 for any 0 < t < t0 and α (t) < 0 for any t > t0 . The lemma is proved. 2 Now let us consider the minimization problem κ := inf E(v).

(22)

v∈M

We have then Lemma 2. Under assumptions (H1) to (H5), there holds κ

N K (SK ) 2K . N

(23)

Proof. Let B(x0 , R) ⊂ Ω for some x0 ∈ Ω and R > 0. We consider for some small number ν > 0 and for all ∈ (0, ν), the function u (x) := CN,K

( 2

(N −2K)/2 , + |x − x0 |2 )(N −2K)/2

where the constant CN,K independent of is chosen such that u sLs (RN ) = u 2K,2,RN = N

(SK ) 2K . Let ξ ∈ C0∞ (B(x0 , R)) be a fixed cut-off function satisfying 0 ξ 1 and ξ ≡ 1 on B(x0 , R/2). Putting w := ξ u ∈ C0∞ (Ω) as in [3] and [21], we obtain as → 0 N N and w 2K,2,Ω = (SK ) 2K + O N −2K . w sLs = (SK ) 2K + O N

(24)

It is clear that as → 0, we have w 0

weakly in H0K (Ω),

w 0 weakly in Ls (Ω), strongly in Lq (Ω) (∀q < s) and a.e. in Ω. Therefore, there holds f (x, w ) → 0

s

strongly in L s−1 (Ω).

(25)


2259

Indeed, for any M > 0, let fM (x, u) :=

if |u| M, if |u| > M.

f (x, u), 0,

From (H1) to (H2), it follows that ∀δ > 0, there exists M > 0 such that fM (x, u) − f (x, u) δ|u|s−1

for a.e. x ∈ Ω and ∀u ∈ R.

Therefore, we have f (x, w )

s

L s−1

s + fM (x, w ) s f (x, w ) − fM (x, w ) s−1 L L s−1 s . δw s−1 Ls + fM (x, w ) s−1

(26)

L

Using Lesbegue’s theorem, we infer that ∀β > 0 fM (x, w )

Lβ

→ 0.

Letting → 0 in (26), we obtain lim supf (x, w )

s

L s−1

→0

2δC.

Thus (25) is proved. Similarly, we have F (x, w ) = 0.

lim

→0 Ω

Set e0 = w . Clearly, e0 , e1 , . . . , en are linearly independent. Denote V the (n + 1)-dimensional vector space spanned by e0 , . . . , en and let w˜ ∈ V ∩ M. We claim lim w − w˜ K,2,Ω = 0.

→0

For this purpose, fix some small number r > 0. For all (γ¯0 , . . . , γ¯n ) ∈ Rn+1 with with the same arguments as above, we have the following expansions:

F x, w +

Ω

n i=0

γ¯i ei =

F x,

Ω

n

n

2 i=0 γ¯i

γ¯i ei + o(1),

i=1

2 2 n n γ¯i ei = (1 + γ¯0 )2 w 2K,2,Ω + γ¯i ei + o(1), w + i=0 i=1 K,2,Ω K,2,Ω s s n n s s γ¯i ei = |1 + γ¯0 | |w | + γ¯i ei + o(1) w + Ω

i=0

Ω

Ω

i=1

= r2,

2260


where o(1) tends to 0 uniformly with respect to (γ¯0 , . . . , γ¯n ). As a consequence, we infer E w +

n

1 1 γ¯i ei (1 + γ¯0 )2 w 2K,2,Ω − |1 + γ¯0 |s w sLs (Ω) 2 s i=0 n s n 1 2 1 + γ¯i λi (Ω) − γ¯i ei + o(1), s 2 s i=1

i=1

(27)

L (Ω)

since F (x, u) 12 a(x)u2 for a.e. x ∈ Ω. Gathering (24) and (27), we deduce E w +

n

γ¯i ei < E(w )

(28)

i=0

provided is sufficiently small. On the other hand, E(w˜ ) = supv∈V E(v). Hence, we have w˜ − w = ni=0 γi ei with Γ = (γ0 , . . . , γn ) ∈ Rn+1 satisfying |Γ |2 = ni=0 γi2 < r 2 , that is, the claim is proved. Now, applying (24) and (27), we infer lim E(w˜ ) = lim E(w ) =

→0

This yields the desired result.

→0

N K (SK ) 2K . N

2

Now we state our main result of this section. Theorem 1. Suppose (H1) to (H5) and κ
0. First we suppose that the case (i) occurs. In this case there exists t ∈ (0, 1) such that u(t) ∈ M because of Step 3 of Lemma 1. Set vk := uk − u as before and u˜ k := tuk + u(t) − tu = tvk + u(t). We define for all w ∈ H0K (Ω), 1 1 E∞ (w) := w2K,2,Ω − 2 s

|w|s . Ω

As vk 0 weakly in H0K (Ω), we obtain E(u˜ k ) = E∞ (tvk ) + E u(t) + o(1). Suppose E(u(t)) > κ, otherwise E(u(t)) = κ and then we finish the proof. By Lemma 1 and the fact u˜ k − tuk ∈ V0 , we have E(u˜ k ) E(uk ) = κ + o(1) which implies E∞ (tvk ) < 0 for sufficiently large k. In particular, vk = 0. Consequently, for sufficiently large k, s s tvk sLs (Ω) > tvk 2K,2,Ω SK tvk 2Ls (Ω) > SK tvk 2Ls (Ω) 2 2

(37)

so that N

vk sLs (Ω) > (SK ) 2K .

(38)

On the other hand, we have vk sLs (Ω) = uk sLs (Ω) − usLs (Ω) + o(1)

N κ − usLs (Ω) + o(1), K

which contradicts (38) by using Lemma 2. Thus case (i) is impossible.

(39)


Now we treat the case (ii). By the same arguments in Step 2, we have f (x, uk )uk = f (x, u)u + o(1) and F (x, uk ) = F (x, u) + o(1). Ω

Ω

Ω

2263

(40)

Ω

Thus, according to (31) and (40), we have 1 N N s vk Ls (Ω) = κ + F (x, u) − f (x, u)u − usLs (Ω) + o(1). K K 2

(41)

Ω

Similarly, we have vk 2K,2,Ω

N N = κ+ K K

N F (x, u) + 1 − f (x, u)u − u2K,2,Ω + o(1). 2K

Ω

(42)

Ω

Combining (40) and (42), we see that l0 (u) > 0 implies for sufficiently large k vk sLs (Ω) > vk 2K,2,Ω . N

Consequently, by the definition of SK , we obtain vk sLs (Ω) > (SK ) 2K for sufficiently large k. This is (38) and as before, we conclude that (ii) does not occur and thus u ∈ M. Moreover E(uk ) = E(u) + E∞ (vk ) + o(1) and vk sLs (Ω) = vk 2K,2,Ω + o(1). Thus E(u) = E(uk ) −

K vk 2K,2,Ω + o(1). N

Finally, we deduce vk 2K,2,Ω = o(1) and therefore E(u) = κ. Step 4. u is a solution to (1). In fact u is a critical point of E on M. By the method of Lagrange multipliers, there exists μ, μ1 , . . . , μn ∈ R such that dE(u) + μdl0 (u) +

n

μi dli (u) = 0.

i=1

We consider its restriction on V , this means μdl0 (u) +

n i=1

μi dli (u) = 0 V

since dE(u)|V = 0. On the other hand, we have seen from Proposition 1 that dL(u)|V is an isomorphism from V on Rn+1 . Consequently, μ = μ1 = · · · = μn = 0, that is, dE(u) = 0. Finally, u solves the problem (1) which finishes the proof. 2

2264


The following propositions concern the linear perturbation problem for the non-critical dimensions case where the assumption (29) is justified. Some similar existence results under various assumptions have been obtained by other approaches for example for the polyharmonic operators in [13] and for harmonic and biharmonic operators in [6]. Proposition 2. We suppose f (x, u) = μu for some √ μ > 0. Then (29) holds provided either N 4K and λn (Ω) < 0, λn+1 (Ω) > 0; or N > 2( 2 + 1)K and λn (Ω) 0, λn+1 (Ω) > 0. Proof. We keep the same notations as in the proof of Lemma 2. Direct calculations lead to w s−1 w L1 = O (N −2K)/2 and w 2L2 c1 2K = O (N −2K)/2 , (43) Ls−1 for some positive constant c1 > 0. Notice that when N = 4K, we have more precise estimate w 2L2 c1 2K |log |. On the other hand, we have for any i = 1, . . . , n (w , ei )Ω = w (−)K ei = O (N −2K)/2 . Ω

We prove the lemma in two cases. Case 1. N 4K and λn (Ω) < 0, λn+1 (Ω) > 0. Set K¯ = min{2K, (N − 2K)/2}. As in Lemma 2, we write w˜ − w = ni=0 γi ei with Γ = ¯ (γ0 , . . . , γn ) ∈ Rn+1 . We claim that |Γ |2 = ni=0 γi2 < 2K |log |, provided is sufficiently small. We want to prove for all sufficiently small and for all Γ¯ = (γ¯0 , . . . , γ¯n ) ∈ Rn+1 satisfying ¯ |Γ¯ |2 = ni=0 γ¯i2 = 2K |log |, (28) holds. As before, we have 2 n γ¯i ei w + i=0

= (1 + γ¯0 )2 w 2K,2,Ω

2 n + γ¯i ei

¯ + O 2K |log |1/2 ,

(44)

2 n + γ¯i ei 2

¯ + O 2K |log |1/2 .

(45)

i=1

K,2,Ω

K,2,Ω

and 2 n γ¯i ei w + 2 i=0

= (1 + γ¯0 )

2

w 2L2 (Ω)

i=1

L (Ω)

L (Ω)

From the fact that function · → | · |s is convex on R, we have s n γ¯i ei w + s i=0

L (Ω)

(1 + γ¯0 )s |e0 |s +

Ω

Ω

s(1 + γ¯0 )s−1 |e0 |s−2 e0 Ω

¯ (1 + γ¯0 )s |e0 |s + O 2K |log |1/2 .

n

γ¯i ei

i=1

(46)


2265

Gathering (44) to (46) and using (24), there holds E w +

n i=0

1 1 γ¯i ei (1 + γ¯0 )2 w 2K,2,Ω − |1 + γ¯0 |s w sLs (Ω) 2 s ¯ μ 1 2 (1 + γ¯0 )2 w 2L2 (Ω) + γ¯i λi (Ω) + O 2K |log |1/2 2 2 n

−

i=1

N 1 1 s−2 2 γ¯ (SK ) 2K w 2K,2,Ω − w sLs (Ω) − 2 s 4 0 n ¯ μ 1 2 − w 2L2 (Ω) + γ¯i λi (Ω) + O 2K |log |1/2 . 2 2

(47)

i=1

Here we use the facts 12 (1 + t)2 − 1s (1 + t)s ¯

K N

−

s−2 2 4 t

provided t small and w 2L2 (Ω) =

O( K ). Therefore we obtain (28) and the desired claim follows. By (24) and (47), there exists some positive constant c > 0 such that E(w˜ )

K ¯ N N K ¯ (SK ) 2K − c 2K |log | + O 2K |log |1/2 < (SK ) 2K , N N ¯

provided is sufficiently small since we have 2K 2K |log | for small when N > 4K and ¯ w 2L2 c1 2K |log | when N = 4K. This implies (29). √ Case 2. N > 2( 2 + 1)K and λn (Ω) = 0, λn+1 (Ω) > 0. Assume λn−l (Ω) = · · · = λn (Ω) = 0 and λn−l−1 (Ω) < 0. Set B = B1 × B2 where B1 := (γ¯0 , . . . , γ¯n−l−1 ) ∈ R

n−l

n−l−1 2 2K¯ γ¯i < |log | i=0

and B2 := (γ¯n−l , . . . , γ¯n ) ∈ R

l+1

n 2(N−2K)K¯ 1 2 γ¯i < N+2K |log | s . i=n−l

We claim for all sufficiently small there holds sup E w + ∂B

We write w +

n

i=0 γ¯i ei

n

γ¯i ei < sup E w +

i=0

= (w +

n−l−1 i=0

B

n

γ¯i ei .

i=0

γ¯i ei ) + ( ni=n−l γ¯i ei ) := w1 + w2 . We have

Lw2 = (−)K w2 − μw2 = 0,

(48)

2266


so that (w2 , w1 )Ω − μ

w2 w1 = (w2 , w2 )Ω − μ

w22

= (e0 , w2 )Ω − μ

e0 w2 = 0.

(49)

As a consequence, we deduce 1 μ 1 E(w1 + w2 ) = w1 2K,2,Ω − w1 2L2 (Ω) − 2 2 s

|w1 + w2 |s .

(50)

We want to prove the claim by two steps. Step 1. There exists some 0 > 0 independent of (γ¯n−l , . . . , γ¯n ) such that for all ∈ (0, 0 ) and for all (γ¯n−l , . . . , γ¯n ) ∈ B2 there holds n n sup E w + γ¯i ei < E w + γ¯i ei . (51) (γ¯0 ,...,γ¯n−l−1 )∈∂B1

i=n−l

i=0

With the same arguments as in Case 1 and by (49), we have for all (γ¯0 , . . . , γ¯n−l−1 ) ∈ ∂B1 1 μ 1 μ w1 2K,2,Ω − w1 2L2 (Ω) = (1 + γ¯0 )2 e0 + w2 2K,2,Ω − (1 + γ¯0 )2 e0 + w2 2L2 (Ω) 2 2 2 2 n−l−1 ¯ 1 2 γ¯i λi (Ω) + O 2K |log |1/2 2

+

(52)

i=1

and 1 s

1 |w1 + w2 | = s 1 s s

w1 + (1 + γ¯0 )w2 − γ¯0 w2 s (1 + γ¯0 )s |e0 + w2 |s

+ (1 + γ¯0 )

|e0 + w2 |

s−1

s−2

n−l−1 (e0 + w2 ) γ¯i ei − γ¯0 w2 i=1

¯ 1 s2 2 1 + s γ¯0 + γ¯0 |e0 + w2 |s + O 2K |log |1−1/2s . s 4

(53)

Here we use the facts |a + b|s−1 2s−1 (|a|s−1 + |b|s−1 ) for all a, b ∈ R and (1 + γ¯0 )s 1 + 2 s γ¯0 + s4 γ¯02 for small γ¯0 . Gathering (52) and (53), we obtain E w +

n

γ¯i ei

i=0

E w +

n i=n−l

γ¯i ei +

n−l−1 ¯ 1 2 γ¯i λi (Ω) + O 2K |log |1−1/2s 2 i=1


2267

+ γ¯0 e0 + w2 2K,2,Ω − μe0 + w2 2L2 (Ω) − |e0 + w2 |s s 1 2 2 2 s + γ¯0 e0 + w2 K,2,Ω − μe0 + w2 L2 (Ω) − |e0 + w2 | . 2 2 From the inequality |a + b|s − |a|s − |b|s C |a||b|s−1 + |b||a|s−1 ,

(54)

∀a, b ∈ R

and for some constant C > 0, we infer from (43) (N−2K)K¯ |e0 + w2 |s − |e0 |s − |w2 |s O N−2K 2 + N+2K |log |1/2s ,

(55)

which implies by (24),

¯ N |e0 + w2 |s = (SK ) 2K + o K .

(56)

On the other hand, again by (24), we have ¯ N e0 + w2 2K,2,Ω − μe0 + w2 2L2 (Ω) = e0 2K,2,Ω − μe0 2L2 (Ω) = (SK ) 2K + O K ,

(57)

¯

since e0 2L2 (Ω) = O( K ). Combining (54) to (57), we get finally E w +

n

γ¯i ei E w +

n

γ¯i ei +

i=n−l

i=0

−

n−l−1 1 2 γ¯i λi (Ω) 2 i=1

N s −2 2 ¯ γ¯0 (SK ) 2K + O 2K |log |1−1/2s . 4

(58)

This gives the desired result (51). Step 2. There exists some 1 > 0 independent of (γ¯0 , . . . , γ¯n−l−1 ) such that for all ∈ (0, 1 ) and for all (γ¯0 , . . . , γ¯n−l−1 ) ∈ B1 , there holds n sup E w + γ¯i ei < E(w ). (59) (γ¯n−l ,...,γ¯n )∈∂B2

i=0

Using (49), (55) and (58), we estimate E w +

n i=0

γ¯i ei E(w ) −

1 s

1 E(w ) − s

¯ |e0 + w2 |s − |e0 |s + O 2K |log |1−1/2s N−2K (N−2K)K¯ ¯ |w2 |s + O 2 + N+2K |log |1/2s + 2K |log |1−1/2s . (60)

2268


As all the norms on the finite dimension vector space are equivalent, we have

n

|w2 |s c

s/2 γ¯i2

(61)

i=n−l

which implies for all (γ¯n−l , . . . , γ¯n ) ∈ ∂B2 , E w +

n

2N K¯

γ¯i ei E(w ) − c N+2K |log |1/2

i=0

N−2K (N−2K)K¯ ¯ + O 2 + N+2K |log |1/2s + 2K |log |1−1/2s . K¯ K¯ K¯ N −2K Hence, we prove the desired result in Step 2 since N2N K¯ and N2N + (NN−2K) +2K < 2 +2K 2 +2K . Therefore, claim (48) follows. Now we write w˜ = w + ni=0 γi ei with Γ = (γ0 , . . . , γn ) ∈ B1 × B2 . Using (60), we have

N−2K (N−2K)K¯ ¯ E(w˜ ) E(w ) + O 2 + N+2K |log |1/2s + 2K |log |1−1/2s N−2K (N−2K)K¯ N K ¯ (SK ) 2K − c 2K + O 2 + N+2K |log |1/2s + 2K |log |1−1/2s N N K < (SK ) 2K N

√ provided sufficiently small since N > 2( 2 + 1)K implies 2K < the proof. 2

N −2K 2

(N −2K) + 2(N +2K) . We finish 2

Under more assumptions on the eigenfunctions, we have the following improved result. Proposition 3. Under the same assumptions as in Proposition 2, we suppose λn−l (Ω) = · · · = λn (Ω) = 0, λn+1 (Ω) > 0 and λn−l−1 (Ω) < 0. Moreover, we assume Σ0 := x ∈ Ω; en−l (x) = · · · = en (x) = 0 = ∅. Then (29) holds provided either N 4K and K 4; or N > 2(K − 1 + K 5.

(62) √ 2K 2 − 2K + 1 ) and

Proof. We √ keep the same notations as in Proposition 2. We need only to consider the case 4K ¯ (K¯ + 2)/(s − 1)}. N 2(1 + 2 )K so that K¯ = (N − 2K)/2. Let x0 ∈ Σ0 and α := min{2K/s,

Set B = B1 × B2 where B1 is defined as in the proof of Proposition 2 and B2

:= (γ¯n−l , . . . , γ¯n ) ∈ R

l+1

n 2 2 2α γ¯i < |log | s . i=n−l


2269

Similar to (48), we claim for all sufficiently small there holds sup E w + ∂B

We write again w + before.

n

γ¯i ei < sup E w + B

i=0

n

i=0 γ¯i ei

n

γ¯i ei .

(63)

i=0

= w1 + w2 . To prove (63), we divide the proof in two steps as

Step 1. With the same arguments as in Step 1 of the proof of Proposition 2, there exists some 2 > 0 independent of (γ¯n−l , . . . , γ¯n ) such that for all ∈ (0, 2 ) and for all (γ¯n−l , . . . , γ¯n ) ∈ B2 , (51) holds. In fact, we observe (s − 1)α > K¯ and we infer thus E w +

n

γ¯i ei E w +

n

γ¯i ei +

i=n−l

i=0

n−l−1 1 2 γ¯i λi (Ω) 2 i=1

¯ N s−2 2 − γ¯0 (SK ) 2K + O 2K |log |1/2 , 4

(64)

which proves the desired claim in Step 1. Step 2. There exists some 3 > 0 independent of (γ¯0 , . . . , γ¯n−l−1 ) such that for all ∈ (0, 3 ) and for all (γ¯0 , . . . , γ¯n−l−1 ) ∈ B1 , there holds sup

(γ¯n−l ,...,γ¯n )∈∂B2

E w +

n

γ¯i ei < E(w ).

(65)

i=0

First we have similarly, E w +

n

γ¯i ei

i=0

1 E(w ) − s

|e0 + w2 | − |e0 | s

s

¯ + O 2K |log |1/2 .

(66)

We will estimate carefully the second term on the right side. Observe the basic inequality |a + b|s − |a|s − |b|s − s|a|s−2 ab C |a||b|s−1 + |b|2 |a|s−2 ,

∀a, b ∈ R.

Using the facts

|w |s−2 =

O( 2K ) O( 2K |log |)

when N > 4K, when N = 4K

¯ we obtain from (43) and min{(s − 1)α, K + α} > K, |e0 + w2 |s − |e0 |s − |w2 |s − s|e0 |s−2 e0 w2 = O 2K¯ .

(67)

2270


To handle

|e0 |s−2 e0 w2 , we write w2 (x) = ∇w2 (x0 ), x − x0 + O α |log |1/s |x − x0 |2

since w2 (x0 ) = 0. Thus, we imply

|e0 |s−2 e0 w2 = O (N +2K)/2 + (N −2K+4)/2 α |log |1/s

when K > 1

(68)

|e0 |s−2 e0 w2 = O (N +2)/2 + (N +2)/2 |log | α |log |1/s

when K = 1,

(69)

and

by remarking ∇w2 (x0 ), x − x0 is an odd function with respect to x − x0 so that

|e0 |s−2 e0 ∇w2 (x0 ), x − x0 = 0.

B(x0 ,R/2)

Recalling (61) and gathering (66) to (69), we have for all (γ¯n−l , . . . , γ¯n ) ∈ ∂B2 , E w +

n

γ¯i ei

i=0 ¯

¯

E(w ) − c sα |log | + O( K+2+α |log |1/s + 2K |log |1/2 )

if K > 1,

E(w

if K = 1.

¯ ) − c sα |log | + O( K+2+α |log |(s+1)/s

¯ + 2K |log |1/2 )

¯ for all K and Hence, we prove the desired result in Step 2 since sα min{K¯ + 2 + α, 2K} ¯ ¯ sα = 2K < K + 2 + α for K = 1. Therefore, the claim follows. Now we write w˜ = w + ni=0 γi ei with Γ = (γ0 , . . . , γn ) ∈ B1 × B2 . Using (64), (67) to (69), we have ⎧ N ¯ ¯ ⎨ K (SK ) 2K − μ2 w 2L2 + O( K+2+α |log |1/s + 2K |log |1/2 ) if K > 1, N E(w˜ ) N ¯ ¯ μ 2 K+2+α (s+1)/s 2 K 1/2 ⎩ K (SK ) 2K − w + O( |log | + |log | ) if K = 1. N

2

L2

√ ¯ and 2K < min{K¯ + 2 + α, 2K}. ¯ When K 3 and 4K < N 2(1 + 2 )K, we have α = 2K/s ¯ = K , K = K¯ and 2K K¯ + 2 + α. When K 4 When K 4 and N = 4K, we have α = 2K/s 2 √ and N > 2(K − 1 + 2K 2 − 2K + 1 ), we have α = (K¯ + 2)/(s − 1) and 2K < min{K¯ + ¯ Recall w 2 2 c1 2K when N > 4K and w 2 2 c1 2K |log | when N = 4K. 2 + α, 2K}. L L In all above cases, there holds E(w˜ )
0}. From the Green’s formula, we have ∂en−1 ∂en−1 ∂en en = en − en−1 = en−1 en − en en−1 = 0. ∂ν ∂ν ∂ν ∂Ω1

Ω1

∂Ω1

n−1 > 0 a.e. on the boundary ∂Ω1 . Thus, there exists It follows from Maximum’s principle that ∂e∂ν some point x ∈ ∂Ω1 \ ∂Ω such that en (x) = 0 since ∂Ω1 \ ∂Ω is not empty. Therefore, the condition (62) is satisfied. For the general K, from the orthogonality condition Ω ei ej = 0 for i = j , there exists at most one eigenfunction which keeps√the sign. As a consequence, we have some ground state solutions for the dimension less than 2( 2 + 1)K provided 0 is a simple eigenvalue. In [6], by some different approach, the authors proved the existence of a solution under somewhat weaker assumptions. More precisely, when K = 1 and N 4 or K = 2 and N 8, if λ is an eigenvalue of multiplicity m < N + 2, then it has at least N +1−m pairs of nontrivial solutions. 2 However, we do not know whether these solutions are ground state ones or not. Comparing to their result, Corollary 1 gives some more information about some found solutions, that is, there are ground state solutions under appropriate assumptions on the eigenvalues.

3. Existence of solutions for some perforated domains In this section, we analyze first the concentration phenomenon for the problem (1). For this purpose, set FK (v) :=

((−)M v)2 |∇(−)M v|2

if K = 2M, if K = 2M + 1.

Similarly to Theorem 6 of [19], we have the following theorem and here we just give a sketch of the proof. Theorem 2. Suppose the assumptions (H1) to (H5) are satisfied. Moreover, suppose that κ=

N K (SK ) 2K N

(70)

and E(v) > κ,

∀v ∈ M.

(71)

2272


Let (uk ) ⊂ M be a minimizing sequence for κ, that is, limn→∞ E(uk ) = κ. Then there exists x0 ∈ Ω¯ such that μk := ζΩ FK (uk ) dx SK δx0

weakly in R RN

and νk := ζΩ |uk |s dx SK δx0

weakly in R RN ,

where R(RN ) denotes the space of non-negative Radon measures on RN with finite mass, δx0 denotes the Dirac measure concentrated at x0 with mass equal to 1 and ζΩ designates the characteristic function of Ω. Proof. As in the proof of Theorem 1, we see that (uk ) is bounded in H0K (Ω). Extracting a subsequence, there exists some u ∈ H0K (Ω) such that uk u weakly in H0K (Ω), uk u

weakly in Ls (Ω) and a.e. on Ω.

Moreover, for all 1 j n, we have lj (u) = 0. Furthermore, we have u = 0. Otherwise, with the same arguments as in Theorem 1, we infer u ∈ M and E(u) = κ which contradicts (71). Now the rest of proof is just a consequence of concentration compactness principle (for details cf. [25,17,19]). 2 In the following, we give some classification result. First we recall a basic fact for nonexistence result on the half space RN + . It can be stated as follows: Lemma 3. Let u ∈ DK,2 (RN + ) be a weak positive solution of the problem

(−)K u = |u|s−2 u

in RN +,

u = Du = · · · = D K−1 u = 0 on ∂RN +.

(72)

Then u ≡ 0. A stronger result have been obtained by Reichel and Weth in [34] very recently. Here we give a proof based on the Pohozaev formula (see [26]). Proof. It follows from the Pohozaev formula D K u = 0 on ∂RN + (see the details cf. [19] for the Navier boundary conditions). Now, (−)K−1 ((−)u) = us > 0 in RN + verifying Dirichlet . Thanks to the Boggio’s result, boundary condition (−)u = · · · = D K−2 (−)u = 0 on ∂RN + we know the Green function for the operator (−)K−1 on the half space with Dirichlet bound∂u ary condition is positive. Thus, (−)u > 0 in RN + . From Hopf’s Maximum principle, ∂n > 0 on ∂RN + . This contradiction finishes the proof of lemma. 2


2273

A similar problem in the whole space can be stated as follows: Lemma 4. Let u ∈ DK,2 (RN ) be a weak positive solution of the problem (−)K u = |u|s−2 u

in RN .

(73)

Then there exist a constant λ 0 and a point x0 ∈ RN such that u(x) =

2λ 2 1 + λ |x − x0 |2

N−2K 2

.

(74)

This result has been proved by Wei and Xu (Theorem 1.3 in [42]). K,2 (RN )) be a weak sign-changing solution of the Lemma 5. Let u ∈ DK,2 (RN + ) (resp. u ∈ D problem (72) (resp. (73)). Then

E∞ (u)

N 2K (SK ) 2K . N

(75)

Proof. Our proof is an adaptation of Gazzola–Grunau–Squassina’s approach [16]. We consider the closed convex cone v 0 a.e. in RN C1 = v ∈ DK,2 RN + + and its dual cone (w, v) N 0, ∀v ∈ C1 . C2 = w ∈ DK,2 RN + R +

We claim that C2 ⊂ −C1 . Given h ∈ C0∞ (RN + ) ∩ C1 , let v be the solution to the problem (−)K v = h

in RN +.

Again from the Boggio’s result, we have v 0 since the Green function for the operator (−)K on the half space with Dirichlet boundary condition is positive. Consequently, for all w ∈ C2 , we have hw = (−)K vw = (v, w)RN 0. +

RN +

RN +

This implies w 0 a.e. in RN + . Hence the claim is proved. Using a result of Moreau [27], for any ), there exists a unique pair (u1 , u2 ) ∈ C1 × C2 such that u ∈ DK,2 (RN + u = u1 + u2

with (u1 , u2 )RN = 0. +

Now let u be a sign-changing solution of the problem (72). Then ui = 0 for all i = 1, 2. From the above claim, we see u1 0 and u2 0 so that |u(x)|s−2 u(x)ui (x) |ui (x)|s for i = 1, 2. Applying the Sobolev inequality for ui (i = 1, 2), we obtain

2274


SK ui 2Ls

ui 2K,2,RN +

= (u, ui )RN = +

(−) uui K

RN +

ui (x)s = ui s s L

RN +

so that N

ui sLs (Ω) (SK ) 2K . Consequently, using the fact u2

K,2,RN +

= usLs , we infer

K K u2K,2,RN = u1 2K,2,RN + u2 2K,2,RN + + + N N N K 2K SK u1 2Ls + u2 2Ls (SK ) 2K . N N

E∞ (u) =

Similarly, we have the same result for u ∈ DK,2 (RN ).

2

Theorem 3. Assume (H1), (H2), (H5), (70) and (71) are satisfied. Let (uk ) ⊂ H0K (Ω) be a (P.S.)β sequence such that

N N K 2K (SK ) 2K , (SK ) 2K , N N K ∗ dE(uk ) → 0 in H0 (Ω) .

E(uk ) → β ∈

(76) (77)

Then (uk ) is precompact in H0K (Ω). Proof. The blow up analysis for (P.S.)β sequences is more or less standard. Its proof follows from the P. Lions’ concentration compactness principle and it is close to one in [19]. The only difference is that we need Lemma 6 to rule out sign-changing bubbles. We leave this part to interested readers. 2 As a consequence, we have Corollary 2. Under the assumptions (H1) to (H5), (70) and (71), assume moreover (H8) en (Ω) < 0. Let (uk ) ⊂ M be a (P.S.)β sequence for E on M such that N N K 2K E(uk ) → β ∈ (SK ) 2K , (SK ) 2K , N N dE(uk ) → 0. (T M)∗

uk

Then (uk ) is precompact in M.

(78) (79)


2275

Proof. As in the proof of Theorem 1, (uk ) is a bounded sequence in H0K (Ω). On the other hand, using (30), (31) and (H2), we infer that (uk ) is bounded from below by some positive constant in H0K (Ω) and also in Ls (Ω). Set V k the (n + 1)-dimensional vector space spanned by uk , e1 , . . . , en . If there is no confusion, we drop the index k. We claim there exists some positive constant c > 0 independent of k such that ∀k ∈ N, ∀w ∈ H0K (Ω), we can decompose w = w1 + w2

(80)

where w1 ∈ V k and w2 ∈ Tuk M satisfying w1 K,2,Ω cwK,2,Ω ,

w2 K,2,Ω cwK,2,Ω .

Set e0 = uk and θi = dli (uk )(w) ∈ R for all i = 0, . . . , n. Using (13) and the fact that (uk ) is a bounded sequence in H0K (Ω), the vector Θ = (θ0 , . . . , θn )T is bounded in Rn+1 with respect to k. Moreover, we can estimate |Θ| cwK,2,Ω . Define (n + 1) × (n + 1) symmetric matrix M(k) = (mij )0i,j n by mij = d 2 E(uk )(ei , ej ). We write w1 =

n

ψ i ei

i=0

where ψi ∈ R. Denote the vector Ψ = (ψ0 , . . . , ψn )T ∈ Rn+1 . Again from (13), the decomposition (80) is equivalent to solve d 2 E(uk )(w1 , ei ) = dli (uk )(w),

∀0 i n,

that is, M(k)Ψ = Θ. As in the proof of Lemma 1, the matrix is negative definite. Clearly, the matrix M(k) is uniformly bounded. We show there exists c > 0 independent of k such that M(k) −cI T n+1 where In is the identity matrix. For this purpose, for any vector Γ = (γ0 , . . . , γn ) ∈ R , denote ξ = i=0 γi ei we have

Γ T M(k)Γ = d 2 E(uk )(ξ, ξ ) n 2 s−2 2 s−2 −(s − 2) |uk | ξ − |uk | γj e j Ω

Ω

j =1

2276


+

n

γj e j ,

j =1

−

s −2 s −1

n

−

γi e i

i=1

Ω

|uk |s γ02 +

n

a(x) Ω

n

2 γj e j

j =1

γj2 λj (Ω).

j =1

Ω

Thus, the desired result follows. As a consequence, (Ψ = (M(k))−1 Θ)k is a bounded sequence. More precisely, we infer w1 K,2,Ω cwK,2,Ω . Therefore, w2 K,2,Ω wK,2,Ω + w1 K,2,Ω cwK,2,Ω , that is, the claim is proved. Hence, dE(uk )(w) = dE(uk )(w2 ) cdE(uk ) wK,2,Ω . (Tuk M)∗ Thus, there holds dE(uk )

(H0K (Ω))∗

cdE(uk )(T M)∗ uk

so that lim dE(uk )(H K (Ω))∗ = 0.

n→∞

0

Finally, applying Theorem 3, we finish the proof.

2

Now, we can prove the main result for domains with the small holes. Recall that Ω = Ω1 \ Ω2 is a bounded domain satisfying Ω2 ⊂ B(0, ) and Ω1 is fixed. To search solutions of (1) in such Ω, we minimize the energy functional E on the Finsler manifold M. We see that the concentration phenomenon occurs if E cannot reach the minimum. In this case, we will employ Coron’s strategy to search unstable critical points in higher level sets. Theorem 4. Let Ω be a bounded domain satisfying the above assumption. Assume (H1) to (H7) hold. Then there exists η > 0 such that for all < η, the problem (1) admits a nontrivial solution in Ω. Proof. Thanks to Lemma 2, we have κ

N K 2K N (SK )

. In the case κ < N K 2K N (SK )

N K 2K N (SK )

, the desired re-

sult follows from Theorem 1. So we suppose κ = . If there exists u ∈ M such that E(u) = κ, we finish the proof by Step 4 in the proof of Theorem 1. Hence, we assume ∀v ∈ M there holds E(v) > κ. From the properties of eigenvalues λi (Ω) described in the previous sections, (H8) is always satisfied for the perforated domain Ω, provided is sufficiently small. In fact, in case λi (Ω1 ) = 0 for all i ∈ N, it follows from the continuity of λi (Ω). In the case λn (Ω1 ) = · · · = λn+k (Ω1 ) = 0, we have λn (Ω) > 0.


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We divide the proof into several steps. Step 1. We choose a radially symmetric function ϕ ∈ C0∞ (RN ) such that 0 ϕ 1, ϕ ≡ 1 on the annulus {x ∈ RN | 1/2 < |x| < 1} and ϕ ≡ 0 outside the annulus {x ∈ RN | 1/4 < |x| < 2}. For any R 1, define ⎧ ⎨ ϕ(Rx) ϕR (x) = 1 ⎩ ϕ(x/R)

if 0 |x| < 1/R, if 1/R |x| < R, if |x| R.

Denote the unit sphere S N −1 = {x ∈ RN | |x| = 1}. For σ ∈ S N −1 , 0 t < 1, we set uσt (x) = CN,K

1−t (1 − t)2 + |x − tσ |2

N−2K 2

∈ H K RN , N

σ (x) = where the choice of CN,K is such that uσt 2K,2,RN = uσt sLs (RN ) = (SK ) 2K . Let w˜ t,R N−2K

σ (x) = (4R) 2 w σ (4Rx). Hence w σ ∈ H K (B(0, 1/2)\B(0, 1/16R 2 )), uσt (x)ϕR (x) and wt,R ˜ t,R t,R 0 ∀σ ∈ S N −1 and ∀t ∈ [0, 1). Clearly,

σ w˜ s N = w σ s N , t,R L (R ) t,R L (R ) σ σ w˜ t,R K,2,RN = wt,R K,2,RN .

(81) (82)

A direct computation leads to ∀R > 1 σ N −2K 2K−N w˜ − uσ 2 R t K,2,RN C(1 − t) t,R

(83)

σ w˜ − uσ s s N CR −N (1 − t)N . t L (R ) t,R

(84)

and

Consequently σ 2 σ s N s N = (SK ) 2K = lim w˜ t,R lim w˜ t,R K,2,RN L (R )

R→∞

R→∞

σ ∈ M ∩ Vect{e (Ω), . . . , e (Ω), w σ } where uniformly for t ∈ [0, 1) and σ ∈ S N −1 . Set w¯ t,R 1 n t,R Ω = Ω1 \ Ω2 , B(0, 1/2) ⊂ Ω1 and Ω2 ⊂ B(0, 1/16R 2 ). Thanks to the Implicit Function Theorem, the continuous map

wR : S N −1 × [0, 1) → H0K (Ω), σ (σ, t) → wt,R

yields a continuous map

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w¯ R : S N −1 × [0, 1) → M, σ (σ, t) → w¯ t,R .

Recall Ω1 is fixed. Without loss of generality, we assume en (Ω) < 0 and en+1 (Ω) > 0. A ba∞ (Ω \ {0}) away from 0 sic observation is that ei (Ω) → ei (Ω1 ) for all i = 1, . . . , n in Cloc 1 and strongly in H0K (Ω1 ) as R → +∞. For this purpose, we prolong ei (Ω) by setting 0 in Ω1 \ Ω and denote it by e¯i (Ω). We remark first from regularity theory of elliptic equation ∞ (Ω \ {0}) and is also bounded family in H K (Ω ). that {e¯i (Ω)} is bounded family in Cloc 1 1 0 Thus, the weak limit function v of e¯i (Ω) in H0K (Ω1 ) solves some linear elliptic equation of eigenvalue type in Ω1 \ {0}. As v ∈ H0K (Ω1 ), {0} is a removable singularity point. Thus, v is an eigenfunction in Ω1 . On the other hand, it follows from the fact λi (Ω) → λi (Ω1 ) that ei (Ω)K,2,Ω = e¯i (Ω)K,2,Ω1 → vK,2,Ω1 so that we have the strong convergence in ∞ (Ω \ {0}) comes from the compactness of this H0K (Ω1 ). Moreover, the convergence in Cloc 1 family in such space. Furthermore, the orthogonality of {ei (Ω)} with respect to i gives the orthogonality of the limit eigenfunctions and the desired claim follows. We remark that

1 1 1 E(u) u2K,2,Ω − usLs (Ω) − 2 s 2

a(x)u2 . Ω

In the following, we consider the simple case F (x, u) = 12 a(x)u2 (we can treat the general case with the same arguments). Fix some small number r > 0. As in the proof of Lemma 2, for all Γ = (γ0 , . . . , γn ) ∈ Rn+1 with ni=0 γi2 r 2 , we infer σ sup E wt,R +

t,σ,Ω2

n

N N 1 1 γi ei (1 + γ0 )2 (SK ) 2K − |1 + γ0 |s (SK ) 2K 2 s i=0 n s n 1 2 1 + γi λi (Ω1 ) − γi ei (Ω1 ) s 2 s

i=1

i=1

+ o(1),

L (Ω1 )

where o(1) is uniformly with respect to Γ as R → ∞. Consequently, we deduce sup E t,σ,Ω2

σ wt,R

+

n

σ γi ei < E wt,R

i=0

for

n

γi2 = r 2

i=0

provided R is sufficiently large. This implies σ σ w¯ t,R − wt,R =

n

γi ei (Ω)

for some |Γ | < r,

i=0

so that σ K N lim sup E w¯ t,R = (SK ) 2K . N

R→∞ t,σ,Ω2

(85)


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Hence, we can choose R0 > 0 such that for any R R0 σ 2K N (SK ) 2K . < E w¯ t,R N t∈[0,1), σ ∈S N−1 , Ω2 ⊂B(0,1/16R 2 ) sup

(86)

Thus we can define a map α : B(0, 1) → M, σ . (t, σ ) → w¯ t,R 0

Step 2. Set η := 1/16R02 and fix Ω2 ⊂ B(0, η). From (81) to (84), we infer that σ 2 σ s N lim w¯ t,R = lim w¯ t,R = (SK ) 2K 0 K,2,Ω 0 Ls (Ω)

t→1

t→1

uniformly for σ ∈ S N −1

which implies for any σ ∈ S N −1 K N lim E α(t, σ ) = (SK ) 2K . N

t→1

Step 3. For any v ∈ M, let γ (v) =

s x v(x) dx ∈ RN

Ω

denote its center mass. We claim there exists δ˜ > 0 such that for any v ∈ M satisfying E(v) N K ˜ we have 2K + δ, N (SK ) N γ (v) ∈ RN \ B 0, 2 (SK ) 2K /2

(87)

where B(0, 2 ) ⊂ Ω2 . Otherwise, we can find a sequence (vn ) ⊂ M satisfying N K (SK ) 2K , N N γ (vn ) ∈ B 0, 2 (SK ) 2K /2 .

lim E(vn ) =

n→∞

Applying Theorem 2, there exists x0 ∈ Ω¯ such that s N ζΩ vn (x) dx (SK ) 2K δx0 . Consequently, N N γ (vn ) → (SK ) 2K x0 ∈ / B 0, 2 (SK ) 2K

(88) (89)

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which contradicts (89). Thus, the desired claim yields. Choosing t0 ∈ [0, 1) such that ∀σ ∈ S N −1 N ˜ we set 2K + δ, and ∀t ∈ [t0 , 1), we have E(α(t, σ )) < K N (SK ) β := min

max

f ∈H (t,σ )∈(0,t0 ]×S N−1

E f (t, σ ) ,

where H is the set of any function homotopic to α on B(0, t0 ) with the fixed boundary data, that is, H = f f : B(0, t0 ) → M is continuous, f |∂B(0,t0 ) = α|∂B(0,t0 ) and f is homotopic to α . We see that ∀f ∈ H , γ ◦ f : B(0, t0 ) → RN is a contraction of the loop γ ◦ α|∂B(0,t0 ) ⊂ RN \ N B(0, 2 (SK ) 2K /2). On the other hand, it follows from Steps 1 and 2 N

lim γ ◦ α(t, σ ) = (SK ) 2K

t→1

σ 4R0

uniformly in σ ∈ S N −1 . N

Thus, γ ◦ α|∂B(0,t0 ) is a nontrivial loop in RN \ B(0, 2 (SK ) 2K /2). Using (89), we obtain sup (t,σ )∈B(0,t0

K N ˜ E f (t, σ ) (SK ) 2K + δ, N )

which implies β

N N K K (SK ) 2K + δ˜ > (SK ) 2K . N N

On the other hand, it follows from Step 1 β

sup (t,σ )∈B(0,t0

2K N (SK ) 2K . E α(t, σ ) < N )

Recalling Theorem 1 and Corollary 2 and using the deformation lemma, we infer β is a critical value. Finally, the problem (1) admits a nontrivial critical point u such that E(u) = β. 2 Remark 1. The condition a ∈ L∞ (Ω) ∩ C ∞ (Ω) could be weakened. Remark 2. We can use the above strategy to treat also the problem with Navier boundary conditions. Acknowledgment The authors would like to thank the referee for pointing out the references [30,33,38] that the equivalent Nehari type manifolds are defined therein and valuable comments on Propositions 2 and 3.


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References [1] T. Bartsch, T. Weth, M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differential Equations 18 (2003) 253–268. [2] F. Bernis, J. Garcia-Azorero, I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations 1 (1996) 219–240. [3] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477. [4] A. Capozzi, D. Fortunato, G. Palmieri, An existence results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (6) (1985) 463–470. [5] G. Cerami, D. Fortunato, M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (5) (1984) 341–350. [6] M. Clapp, T. Weth, Multiple solutions for the Brezis–Nirenberg problem, Adv. Differential Equations 10 (2005) 463–480. [7] J.M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I 299 (1984) 209–212. [8] G. Devillanova, S. Solimini, Concentration estimates and multiple solutions to elliptic problem at critical growth, Adv. Differential Equations 7 (2002) 1257–1280. [9] G. Devillanova, S. Solimini, A multiplicity result for elliptic equations at critical growth in low dimension, Commun. Contemp. Math. 5 (2003) 171–177. [10] F. Ebobisse, M. Ould Ahmedou, On a nonlinear fourth order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal. 52 (5) (2003) 1535–1552. [11] D.E. Edmunds, D. Fortunato, E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Ration. Mech. Anal. 112 (3) (1990) 269–289. [12] D. Fortunato, E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987) 205–213. [13] F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 251–263. √ [14] F. Gazzola, H.C. Grunau, On the role of space dimension n = 2 + 2 2 in the semilinear Brezis–Nirenberg eigenvalue problem, Analysis 20 (2000) 395–399. [15] F. Gazzola, H.C. Grunau, Critical dimensions and higher order Sobolev inequalities with remainder terms, NoDEA Nonlinear Differential Equations Appl. 8 (2001) 35–44. [16] F. Gazzola, H.C. Grunau, M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations 18 (2003) 117–143. [17] Y. Ge, Estimations of the best constant involving the L2 norm in Wente’s inequality and compact H -surfaces in Euclidean space, ESAIM Control Optim. Calc. Var. 3 (1998) 263–300. [18] Y. Ge, Sharp Sobolev inequalities in critical dimensions, Michigan Math. J. 51 (2003) 27–45. [19] Y. Ge, Positive solutions in semilinear critical problems for polyharmonic operators, J. Math. Pures Appl. 84 (2005) 199–245. [20] H.C. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators. II, Boll. Unione Mat. Ital. (7) 9-B (1995) 815–847. [21] H.C. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Differential Equations 3 (1995) 243–252. [22] H.C. Grunau, On a conjecture of P. Pucci–J. Serrin, Analysis 16 (1996) 399–403. [23] S. Lang, Introduction to Differentiable Manifolds, Interscience, New York, 1962. [24] C. Lin, Strong unique continuation for m-th powers of a Laplacian operator with singular coefficients, Proc. Amer. Math. Soc. 135 (2007) 569–578. [25] P.L. Lions, The concentration-compactness principle in the calculus of variations: The limit case. Part I and Part II, Rev. Mat. Iberoam. 1 (1) (1985) 145–201, Rev. Mat. Iberoam. 1 (2) (1985) 45–121. [26] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993) 125–151. [27] J.J. Moreau, Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris Sér. I 255 (1962) 238–240. [28] Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960) 101–123. [29] E.S. Noussair, C.A. Swanson, J. Yang, Critical semilinear biharmonic equations in R N , Proc. Roy. Soc. Edinburgh Sect. A 121 (1992) 139–148. [30] A. Pankov, Periodic nonlinear Schrödinger equation with application to Photonic crystals, Milan J. Math. 73 (2005) 259–287.

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[31] M.H. Protter, Unique continuation for elliptic equation, Trans. Amer. Math. Soc. 95 (1960) 81–91. [32] P. Pucci, J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. 69 (1990) 55–83. [33] M. Ramos, H. Tavares, Solutions with multiple spike patterns for an elliptic system, Calc. Var. Partial Differential Equations 31 (2008) 1–25. [34] W. Reichel, T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z. 261 (2009) 805–827. [35] M. Schechter, W.M. Zou, On the Brezis–Nirenberg problem, Arch. Ration. Mech. Anal. 197 (1) (2010) 337–356. [36] M. Struwe, Variational Methods, Springer, Berlin, Heidelberg, New York, Tokyo, 1990. [37] C.A. Swanson, The best Sobolev constant, Appl. Anal. 47 (4) (1992) 227–239. [38] A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009) 3802–3822. [39] A. Szulkin, T. Weth, M. Willem, Ground state solutions for a semilinear problem with critical exponent, Differential Integral Equations 22 (2009) 913–926. [40] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353–372. [41] R.C.A.M. Van der Vorst, Best constant for the embedding of the space H 2 ∩ H01 (Ω) into L2N/(N −4) Ω, Differential Integral Equations 6 (2) (1993) 259–276. [42] J. Wei, X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (2) (1999) 207–228. 4

[43] D. Zhang, On multiple solutions of u + λu + |u| N−2 u = 0, Nonlinear Anal. 13 (1989) 353–372.


Bounded mean oscillation and bandlimited interpolation in the presence of noise Gaurav Thakur Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA Received 4 August 2010; accepted 19 October 2010 Available online 28 October 2010 Communicated by Alain Connes

Abstract We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker–Shannon–Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties. © 2010 Elsevier Inc. All rights reserved. Keywords: Sampling theorem; Nonuniform sampling; Paley–Wiener spaces; Entire functions of exponential type; BMO; Sine-type functions

1. Introduction The classical Whittaker–Shannon–Kotelnikov (WSK) sampling theorem is a central result in signal processing and forms the basis of analog-to-digital and digital-to-analog conversion in a variety of contexts involving signal encoding, transmission and detection. If we normalize

E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.015

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G. Thakur / Journal of Functional Analysis 260 (2011) 2283–2299

∞ the Fourier transform as fˆ (ω) = −∞ f (t)e−2πiωt dt, then the sampling theorem states that a function f ∈ L2 (R) with supp(fˆ ) ⊂ [− b2 , b2 ] can be expressed as a series of the form f (t) =

∞ k=−∞

ak

sin(π(bt − k)) , π(bt − k)

(1)

where ak = f (k) are its samples. Conversely, for a given collection of data {ak } ∈ l 2 , the series (1) defines a function in L2 (R) with supp(fˆ ) ⊂ [− b2 , b2 ] called the bandlimited interpolation of {ak }. The calculation or approximation of this series is a standard procedure in many applications. For example, in audio processing it is used for resampling signals at a higher rate, typically by applying a lowpass filter to the piecewise-constant zero order hold function of the samples [6]. In this paper, we consider the situation of bounded noise in the samples ak . Building on recent work by Boche and Mönich on related problems [3–5], we study some properties of the effect of the noise on the bandlimited interpolation f . Before we discuss our problems, it will be convenient to define the Paley–Wiener spaces for 1 p ∞ by b b p PW b = f ∈ Lp : supp(fˆ ) ⊂ − , , 2 2 where fˆ is interpreted in the sense of tempered distributions. Our notation PW b essentially follows Seip [12], and is slightly different from the one used by Boche and Mönich. Without loss of generality, we will set b = 1 in what follows. Returning to the series (1), we consider corrupted samples of the form ak = Tk + Nk , where Tk are the true samples and Nk is some form of noise, and we correspondingly write f (t) = T (t) + N(t). One obstacle we face is that the noise {Nk } may not naturally decay in time alongside the signal, and even if {Tk } ∈ l 2 , it is often more physically meaningful to consider {Nk } ∈ l ∞ . The WSK sampling theorem shows that for any collection of samples {ak } ∈ l 2 , there exists a unique function f ∈ PW 21 with f (k) = ak . However, for bounded samples {ak } ∈ l ∞ , the series (1) does not necessarily converge. In fact, a given {ak } ∈ l ∞ may correspond to multiple functions f ∈ PW ∞ 1 , or to no such function [3]. A simple example of the former possibility (non-uniqueness) is given by ak ≡ 0, which corresponds to the functions f (t) ≡ 0 and f (t) = sin(πt). It turns out that adding one extra sample to the collection {ak } resolves this ambiguity, and allows us to consider the unique bandlimited interpolation of any bounded data {ak } ∈ l ∞ . We discuss the details of this procedure in Section 3. The latter possibility (non-existence) is less obvious, but in [3], Boche and Mönich presented an explicit example of this phenomenon. They showed that for the samples given by ak = 0, k < 1, and ak = (−1)k / log(k + 1), k 1, there is no f ∈ PW ∞ 1 with f (k) = ak . It is also possible to construct other, similar examples using standard special functions, and we describe one such sequence of {ak } in Section 3 and discuss its properties. The main observation of this paper is that such examples of {ak } are in a sense “highly oscillating.” By assuming that the noise Nk is statistically incoherent and defining N (t) carefully, we can rule out these examples and obtain sharper statements on the behavior of N (t). More precisely, we show in Section 4 that if Nk is a uniformly distributed, independent white noise r process, then supr>0 2r1 −r |N (t)| dt < ∞ almost surely. In other words, the average of |N (t)| is p


2285

globally bounded. We find that this result does not generally hold for {Nk } ∈ l ∞ that lack such a statistical condition, and we discuss examples that illustrate the differences. We also study a second topic motivated by further understanding N (t). As discussed in [7], the WSK series (1) can be interpreted as a discrete Hilbert transform operator H , mapping a space of samples into a space of bandlimited functions (see also [1] and [11]). The Plancherel p formula shows that H maps l 2 into PW 21 . In fact, H also maps l p into PW 1 for any 1 < p < ∞, and the series (1) converges for any {ak } ∈ l p [10]. This can be compared with the continuous Hilbert transform, and more generally any Calderon–Zygmund singular integral operator, which maps Lp into itself for any 1 < p < ∞. Such operators behave differently for p = ∞, mapping L∞ into the space BMO of functions with bounded mean oscillation [13]. It is thus reasonable to expect that if we consider samples {ak } ∈ l ∞ , the “right” target space for H may be one of bandlimited functions lying in the space BMO. However, this heuristic reasoning turns out to be incorrect. We consider bandlimited BMO functions in Section 5 and establish some of their properties. In particular, we find that such a function f is either in L∞ or that its samples {f ( ks )} are unbounded for any sampling rate s > 0. We exhibit a concrete example of such a function, and study it in the context of our other results. We review some existing theory on bandlimited functions and the space BMO in Section 2, and discuss some preliminary results in Section 3. The main results of the paper are presented in Sections 4 and 5. We also develop our results for a class of general, nonuniformly spaced interpolation points, given by zeros of sine-type functions. The above discussion for uniformly spaced points is a special case. 2. Background material We will write f1 f2 if the inequality f1 Cf2 holds for a constant C independent of f1 and f2 . We define f1 f2 similarly, and write f1 f2 if both f1 f2 and f1 f2 . For a set of points Y = {yk } and an extra element y, ˜ we denote the collection {yk } ∪ {y} ˜ by Y˜ , with Y˜ l p := (Y l p + |y| ˜ p )1/p and Y˜ l ∞ := max(Y l ∞ , |y|). ˜ These conventions will be used throughout the paper. p We first review a basic, alternative formulation of PW b , 1 p ∞. An entire function f is said to be of exponential type b if

b = inf β: f (z) eβ|z| , z ∈ C . We denote this by writing type(f ) = b, and by type(f ) = ∞ if b = ∞ or f is not entire. By p the Paley–Wiener–Schwartz theorem [9], PW b can be equivalently described as the space of all entire functions with type(f ) πb whose restrictions to R are in Lp . It also follows that p q p PW b ⊂ PW b for p < q. Functions f ∈ PW b satisfy the classical estimates f Lp πbf Lp and f (· + ic)Lp eπb|c| f Lp , respectively known as the Bernstein and Plancherel–Polya inequalities [10,12]. p There is a rich and well-developed theory of nonuniform sampling for functions in PW b . We only cover a few aspects of it that we will need in this paper, and refer to [12] and [15] for more details. We consider a sequence of points X = {xk } ⊂ R, indexed so that xk < xk+1 . The

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separation constant of X is defined by λ(X) = infk |xk+1 − xk |, and X is said to be separated if λ(X) > 0. The generating function of X is given by the product S(z) = z

δX

lim

r→∞

0 ε, C1 (ε) e−πb| Im(z)| S(z) C2 (ε).

(3)

It can be shown that condition (II) is equivalent to requiring that the bounds (3) only hold in some half-plane {z: | Im(z)| c}, c > 0. Furthermore, a sine-type function S also satisfies the bounds |S (xk )| 1 and forces X to satisfy supk |xk+1 − xk | < ∞ [10]. Now suppose the sequence X = {xk } has a sine-type generating function S with type(S) = πb. p Let 1 < p < ∞. Then any f ∈ PW b can be expressed in terms of its samples ak = f (xk ), f (z) =

∞

ak

k=−∞

S (x

S(z) , k )(z − xk )

(4)

with uniform convergence on compact subsets of C. Conversely, for any {ak } ∈ l p , the series p (4) converges uniformly on compact subsets of C and defines a function f ∈ PW b with ak = f (xk ) [10]. The simplest example of a sequence X with a sine-type generating function is the uniform sequence xk = bk , for which S(z) = sin(πbz) and the expansion (4) reduces to the WSK samπb pling theorem. More generally, any finite union of uniform sequences has a sine-type generating function. As a more interesting example, the Bessel function J0 has real, separated zeros, sat-

2 isfies J0 (z) = J0 (−z), and has the asymptotic formula J0 (z) = πz cos(z − π4 )(1 + O( 1z )) as |z| → ∞ and | arg z| < π (see [14]). This implies that for sufficiently small ε > 0, S(z) = π(z+ε) zJ0 ( πz 2 )J0 ( 2 ) is a sine-type function with type(S) = π . Sequences X with sine-type generating functions are not the most general class for which f has an expansion of the form (4), but they have several convenient properties and cover some important cases encountered in applications, such as that of periodic interpolation points. Such sequences X and various properties of the series (4) have recently been studied in [4] in a computational context. The above results do not directly carry over to bounded functions f ∈ PW ∞ b , but in this case we still have the following theorem [2].

Theorem (Beurling). For a sequence X = {xk }, let N (X, I ) be the number of xk in an interval I . Then f L∞ f (X)l ∞ for all f ∈ PW ∞ b if and only if D − (X) := lim sup inf r→∞

a

N (X, [a, a + r)) > b. r


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D − (X) is called the lower uniform density of X. For a uniform sequence xk = ks , D − (X) = s, and Beurling’s theorem implies that f ∈ PW ∞ b is uniquely determined by its samples if we oversample it beyond its Nyquist rate. We finally review a few properties of the Banach space BMO of functions with bounded mean oscillation, which has been studied extensively in connection with singular integral operators. It is defined by 1 f : f BMO = sup I |I | I

f (t) − 1 dt < ∞ , f (s) ds |I | I

where the supremum runs over all real intervals I . The quantity f BMO is technically a seminorm, since f BMO = f +cBMO for any constant c. Now for any g ∈ L1 , we denote its Hilbert ∞ g(t) transform by H g(z) := −∞ π(t−z) dt and its Riesz projections by P ± g := (g ± iH g)/2. We can then consider the “real” Hardy space H 1 (R), given by f : f H 1 (R) = f L1 + H f L1 < ∞ . Finally, it will also be useful to define the subspaces U1 = f ∈ C0∞ :

∞

f (t) dt = 0 , −∞

U2 = f ∈ H 1 (R): 1 + t 2 P + f (t) ∈ L∞ which are both norm dense in H 1 (R) [8,13]. These spaces are all closely related, as the following theorem shows. Theorem (Fefferman). BMO is the dual space of H 1 (R). More specifically, we have the inequality ∞ f (t)g(t) dt , f BMO sup g 1 g∈U H (R) 1

−∞

where U can be taken as U1 or U2 . Conversely, for any bounded linear functional L on H 1 (R), there is an f ∈ BMO with L f BMO . We write w = u + iv for the complex variable w in what follows. Let C± = {w: ±v > 0} be the upper and lower half-planes, and let P (w, t) = π1 (u−t)v2 +v 2 be the Poisson kernel on C+ . Now define the square Qa,r = {w: a < u < a + r, 0 < v < r}. A measure μ on C+ is said to μ(Q ) be a Carleson measure if we have N (μ) := sup( r a,r , a ∈ R, r > 0) < ∞. In other words, the measure μ of any square protruding from the real axis must be comparable to the length of its edge. The following theorem characterizes BMO in terms of such measures.

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∞ (t)| Theorem (Fefferman–Stein). Suppose −∞ |f dt < ∞, so that P (w, ·) f is well defined. t 2 +1 Then

2

1/2 f BMO N v ∇u,v P (w, ·) f du dv . (5) A detailed discussion of BMO and the significance of these theorems can be found in [8] or [13]. 3. Bandlimited interpolation of bounded data In this section, we establish a preliminary result showing how adding an extra sample allows us to treat the bandlimited interpolation of bounded data, such as the noise model discussed in Section 1. We define −2 −πb| Im(z)| |dz| < ∞ . z e (6) PW + = f entire: lim sup f (z) b r→∞

|z|=r

+ + The Plancherel–Polya inequality shows that PW ∞ b ⊂ PW b . Functions in PW b can be expanded in the following way.

Theorem 1. Suppose X = {xk } ⊂ R is separated and has a sine-type generating function S with ˜ ˜ type(S) = πb, and let x˜ ∈ / X. If f ∈ PW + b and A = f (X), then f (z) = a˜

∞ S(z) 1 S(z0 ) 1 + ak lim − , z0 →z S (xk ) z0 − xk S(x) ˜ x˜ − xk

(7)

k=−∞

with uniform convergence of compact subsets of C. Conversely, for any A˜ ∈ l ∞ , the series (7) converges uniformly on compact subsets of C and f ∈ PW + b. Proof. We use a standard complex variable argument. Assume z is in a closed ball B with z ∈ / X, and choose a real sequence {rn } with rn → ∞ and dist({rn }, X) > 0. We can then consider the integral

1 1 f (w)S(z) 1 − |dw|. J (rn ) := 2πi S(w) z − w x˜ − w |w|=rn

For sufficiently large n, it can be seen by calculating residues that

1 S(z) S(z) 1 . + J (rn ) = −f (z) + a˜ ak − S(x) ˜ S (xk ) z − xk x˜ − xk |xk | 0, induces growth on the negative real axis too. This property can be seen in the graph of G1 in Fig. 1. It is also present in the bandlimited interpolation of Boche and Mönich’s example ak = 0, k < 1, and ak = (−1)k / log(k + 1), k 1, where we take x˜ = 12 and a˜ = 0. 4. Bandlimited interpolation of random data We can now state the main result of this paper. Theorem 2. Suppose X ⊂ R is separated and has a sine-type generating function S with type(S) πb, and let x˜ ∈ / X. Suppose also that A˜ = {ak } ∪ a˜ is a collection of i.i.d. random ˜ variables uniformly distributed in [−α, α]. Let f be the bandlimited interpolation of A˜ at X. Then almost surely, 1 sup r>0 2r

r −r

f (t) dt < ∞.

(11)


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We make a few comments before proving Theorem 2. This result deals with the same situation discussed in Section 1, even though it has been formulated slightly differently. In the notation of Section 1, we can take Tk to be zero by linearity and only consider the noise Nk . As we saw in Section 3, the extra sample a˜ can be taken as deterministic and changed arbitrarily without affecting the result of Theorem 2. The exact probability distribution of A˜ is also of little significance here, and the result holds more generally for any symmetric, finitely supported distribution. We split the proof of Theorem 2 into three lemmas for clarity. Our approach is to write the function f as the sum of two parts, each with only zero samples in one direction along the real axis, and show that each one is almost surely bounded on that side. This shows directly that the nonlocal effect discussed in Section 3 does not occur. We then move to the deterministic setting and show that this one-sided boundedness forces a certain regularity upon the other side, resulting in the function having a bounded global average. For the rest of this section, we assume that X˜ and S are as given in Theorem 2, without repeating the conditions on them every time. Lemma 3. For k such that xk > 0, let {ak } be a collection of i.i.d. random variables uniformly distributed in [−α, α], let ak = 0 for all other k and let a˜ = 0. Suppose f is the bandlimited ˜ Then supt 0) and x˜ > 0, as the general case follows from the remarks after Theorem 1. Let bk = S (xk )(akx−x ˜ k ) . Then we have ∞

E(bk ) = 0

k=0

and the separation property shows that for some constant d, ∞

var(bk ) =

k=0

∞ 1 α2 2 3 S (xk ) (x˜ − xk )2 k=0

∞ 1 α2 3 (dist(x, ˜ X) + λ(X)|k − d|)2 k=0

< ∞. By Kolmogorov’s three-series theorem,

∞

k=0 bk

converges almost surely. Now let

∞ 1 f (t) ak 1 . = g(t) = − S(t) S (xk ) t − xk x˜ − xk k=0

∞ It is easy to check that if ∞ k=0 bk converges, then limt→−∞ g(t) = k=0 bk . Since |g(0)| < ∞, it follows by continuity that supt 12 . Recalling that T is the tent function (12), we have K(c, ·) − K c , · 1 L

∞ ∞ 2cT (2N t/c + n) 2c T (2N t/c + n) + dt π|t (t − c)| π|t (t − c )| n=−∞

−∞

16N + π

R\(I 0 ∪I c )

2c dt + π|t (t − c)|

3 2

R\(I 0 ∪I c )

c − c . Now suppose that 1 c that

2c dt π|t (t − c )|

and |c − c | 12 , so that N = 2. Some elementary estimates show

K(c, ·) − K c , · 1 L 1/2 5/2 4 4 − dt + max 4 − sin(4πc/c ) , 4 − sin(4πc /c) dt c c c π(c − c ) c π(c − c) −1/2

+

1/2

c − c |t|−3/2 dt

R\(−1/2,5/2)

c − c .

Following the same arguments, we can also obtain the bound H K(c, ·) − H K(c , ·)L1 |c − c | for the above choices of c and c . By Fefferman’s duality theorem and the fact that K(c, ·) − K(c , ·) ∈ U2 , we have ∞

1 f (t) K(c, t) − K c , t dt f BMO K(c, ·) − K(c , ·)H 1 (R) f (c) − f (c ) , c − c

−∞

(13)

where the constant in the inequality is independent of c and c . Since the BMO seminorm is translation-invariant, the inequality (13) actually holds for all c, c ∈ R. Combining this with Lemma 6 proves (I) and letting c → c gives (IV). If we fix R = |c − c | > 0, this also shows that f BMO + |f (c)| |f (c )|, where the implied constant depends on R, and we can interchange c and c to get (II). Finally, the statement (III) is just (II) phrased in a different way. 2


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Remark. The closure of the set of uniformly continuous BMO functions under the BMO seminorm is called VMO, for vanishing mean oscillation. Theorem 7(I) shows that PW b ⊂ VMO. Note that there are two non-equivalent definitions of VMO in the literature, and we use the one given in [8]. Remark. Theorem 7(IV) is a sharper form of the p = ∞ case of Bernstein’s inequality. We mention that the opposite inequality does not generally hold (even if f BMO is replaced by / PW b . f BMO,c ), and there are functions f such that f ∈ PW ∞ b but f ∈ Corollary 8. Let f ∈ PW b . Then either f ∈ PW ∞ b or there is no separated sequence X with D − (X) > 0 such that f (X) ∈ l ∞ . Proof. Suppose we have a separated X = {xk } with D − (X) > 0 and f (X) ∈ l ∞ . This means that for some large fixed r, every real interval I of length r contains a point xn ∈ X. Theorem 7(IV) then shows that for any t ∈ I , t f (t) = f (xn ) + f (u) du f (xn ) + rf BMO .

2

xn

Intuitively, Corollary 8 says that an unbounded PW b function is large in most places on the real line. It also shows that the bandlimited interpolation of bounded data A˜ ∈ l ∞ can never be in PW b unless it is actually in PW ∞ b . This occurs in spite of Lemma 4 and highlights a p basic difference between PW b and PW b , 1 < p < ∞. In Lemma 4, we generally cannot remove 1 from the inequality and conclude that f ∈ BMO. In contrast, for A ∈ l p , the the factor S(·+ic) (·+ic) series (4) can be used to find that fS(·+ic) ∈ Lp (see [10]), which clearly implies f (· + ic) ∈ Lp p and thus f ∈ L . We finally study an example of an unbounded PW b function that illustrates the “largeness” property described above.

Example. The function G3 (z) =

∞

πz k k=0 (−1) sin( 3·2k )

is in PW 1/3 \PW ∞ 1/3 .

To see this, we use the identity sin z = 2i1 (eiz − e−iz ) to write G3 = G3+ + G3− , where P (w, ·) G3± = G3± (w) for w ∈ C± , and then apply the Fefferman–Stein theorem (5) to each part. Let w = u + iv. We first note that by analyticity,

∇ P (w, ·) G3+ 2 = ∇G3+ (u + iv) 2 = 2 G (w) 2 . 3+ πz πz Since | sin 3·2 k | | 3·2k | for large k, the series defining G3 converges uniformly on compact sets, so we have 2 ∞ d 1 2

πi −k 2 w e3 N 2v G3+ (w) du dv = N 2v du dv dw 2i

N

2ve

− 2π 3 v

k=0

∞ π 3 · 2k k=0

2

du dv 2.

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Fig. 3. Left: The function G3 (z) on [−100, 100]. Right: The absolute value of G3 (z) on [0, 5000]. The peaks at powers of 2 are clearly visible, as well as a self-similarity effect at different scales.

Doing the same calculation with G3− , we find that G3 ∈ PW 1/3 . On the other hand, G3 satisfies the identity G3 (2z) = sin( 2πz 3 ) − G3 (z). This implies that for integer n 2, k n−1 n

π2 n n−k (−1) sin G3 2 = (−1) g3 (1) + 3 k=0 √

3 = (−1)n g3 (1) − (n − 2) , 2 / PW ∞ so G3 ∈ 1/3 . By Corollary 8, the samples G3 (X) are unbounded for any separated sequence X − with D (X) > 0. It is interesting to note that such a function can still be bounded on a sequence X that is “very sparse” in the sense that D − (X) = 0. It is easy to check that G3 (3 · 2n ) = (−1)n G3 (3) and G3 (−z) = −G3 (z), so G3 (X) ∈ l ∞ for the sequence xn = 3 · 2n sign(n). Some graphs of G3 are shown in Fig. 3. Acknowledgment The author would like to thank Professor Ingrid Daubechies for many valuable discussions in the course of this work. References [1] Y. Belov, T.Y. Mengestie, K. Seip, Unitary discrete Hilbert transforms, J. Anal. Math. (2009). [2] A. Beurling, Collected works of Arne Beurling, in: L. Carleson, P. Malliavin, J. Neuberger, J. Wermer (Eds.), Harmonic Analysis, in: Contemp. Math., vol. 2, Birkhäuser Boston, Boston, MA, 1989. [3] H. Boche, U.J. Mönich, On the behavior of Shannon’s sampling series for bounded signals with applications, Signal Process. 88 (2007) 492–501. [4] H. Boche, U.J. Mönich, Convergence behavior of non-equidistant sampling series, Signal Process. 90 (2009) 145– 156. [5] H. Boche, U.J. Mönich, Global and local approximation behavior of reconstruction processes for Paley–Wiener functions, Sampl. Theory Signal Image Process. 8 (1) (2009) 23–51. [6] R. Crochiere, L.R. Rabiner, Multirate Digital Signal Processing, Prentice–Hall, Englewood Cliffs, NJ, 1983. [7] C. Eoff, The discrete nature of the Paley–Wiener spaces, Proc. Amer. Math. Soc. 123 (2) (1995) 505–512. [8] J.B. Garnett, Bounded Analytic Functions, revised first ed., Grad. Texts in Math., Springer, New York, NY, 2007.


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[9] L. Hörmander, The Analysis of Linear Partial Differential Operators, Classics Math., vol. I, Springer, Berlin, Heidelberg, Germany, 2003. [10] B.Ya. Levin, Lectures on Entire Functions, Transl. Math. Monogr., vol. 150, Amer. Math. Soc., Providence, RI, 1996. [11] Y. Lyubarskii, K. Seip, Complete interpolating sequences and Muckenhoupt’s (Ap) condition, Rev. Mat. Iberoamericana 13 (2) (1997) 361–376. [12] K. Seip, Interpolation and Sampling in Spaces of Analytic Functions, Univ. Lecture Ser., vol. 33, Amer. Math. Soc., Providence, RI, 2004. [13] E.M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton Math. Ser., vol. 43, Princeton Univ. Press, Princeton, NJ, 1993. [14] E.T. Whittaker, G.M. Watson, A Course of Modern Analysis, Cambridge Math. Lib., fourth ed., Cambridge Univ. Press, Cambridge, UK, 1927. [15] R.M. Young, An Introduction to Nonharmonic Fourier Series, vol. 93, revised first ed., Academic Press, San Diego, CA, 2001.


Global well-posedness of the Maxwell–Dirac system in two space dimensions Piero D’Ancona a , Sigmund Selberg b,∗ a Department of Mathematics, University of Rome “La Sapienza”, Piazzale Aldo Moro 2, I-00185 Rome, Italy b Department of Mathematical Sciences, Norwegian University of Science and Technology, Alfred Getz’ vei 1,

N-7491 Trondheim, Norway Received 13 August 2010; accepted 12 December 2010 Available online 30 December 2010 Communicated by I. Rodnianski

Abstract In recent work, Grünrock and Pecher proved that the Dirac–Klein–Gordon system in 2d is globally wellposed in the charge class (data in L2 for the spinor and in a suitable Sobolev space for the scalar field). Here we obtain the analogous result for the full Maxwell–Dirac system in 2d. Making use of the null structure of the system, found in earlier joint work with Damiano Foschi, we first prove local well-posedness in the charge class. To extend the solutions globally we build on an idea due to Colliander, Holmer and Tzirakis. For this we rely on the fact that MD is charge subcritical in two space dimensions, and make use of the null structure of the Maxwell part. © 2010 Elsevier Inc. All rights reserved. Keywords: Maxwell–Dirac equations; Well-posedness

Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . Main results . . . . . . . . . . . . From local to global solutions Preliminaries . . . . . . . . . . . . Local well-posedness . . . . . .

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E-mail addresses: [email protected] (P. D’Ancona), [email protected] (S. Selberg). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.010

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2301 2305 2309 2312 2318

P. D’Ancona, S. Selberg / Journal of Functional Analysis 260 (2011) 2300–2365

6. The quadrilinear estimate . . . . . . . . . . . . . . . . 7. Bilinear and null form estimates . . . . . . . . . . . . 8. Proof of the dyadic quadrilinear estimate, Part I . 9. Proof of the dyadic quadrilinear estimate, Part II . 10. Proof of the trilinear estimate . . . . . . . . . . . . . . 11. Estimates for the electromagnetic field . . . . . . . 12. Proof of Lemma 11.2 . . . . . . . . . . . . . . . . . . . s,b;p 13. Proof of the linear estimates in Xφ(ξ ) . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2320 2325 2328 2334 2342 2349 2360 2361 2364

1. Introduction The Maxwell–Dirac system (MD) describes the motion of an electron interacting with an electromagnetic field. Here we study the 2d (two space dimensions) case, where the electron is restricted to move in the (x 1 , x 2 )-plane. Then the electric field E is constrained to the same plane, the magnetic field B is perpendicular to it, and all fields depend only on (t, x 1 , x 2 ) (not on x 3 ), so we write x = (x 1 , x 2 ), and occasionally t = x 0 . The partial derivative with respect to x μ is denoted ∂μ for μ = 0, 1, 2; we write ∂t = ∂0 , and ∇ denotes the spatial gradient. The summation convention is in effect: Roman indices j, k, . . . run over {1, 2}, Greek indices μ, ν, . . . over {0, 1, 2}, and repeated upper/lower indices are implicitly summed over these ranges. Indices are raised and lowered using the metric diag(−1, 1, 1) on R1+2 . In terms of a potential A = {Aμ }μ=0,1,2 with Aμ : R1+2 → R, B = ∇ × A = (0, 0, ∂1 A2 − ∂2 A1 ),

E = ∇A0 − ∂t A,

where A = (A1 , A2 , 0) denotes the spatial part of A. Expressing Maxwell’s equations in terms of A, and imposing the Lorenz gauge condition ∂ μ Aμ = 0 ( ⇐⇒ ∂t A0 = ∇ · A), the MD system reads (see e.g. [12]) −iα μ ∂μ + Mβ ψ = Aμ α μ ψ, Aμ = −α μ ψ, ψ,

(1.1)

where ψ : R1+2 → CN is the Dirac spinor, M ∈ R is a constant and = ∂μ ∂ μ = −∂t2 + x is the D’Alembertian on R1+2 . Since we work in 2d, the smallest possible dimension of the spinor space is N = 2, and then for the 2 × 2 Dirac matrices we can take the representation α 0 = I2×2 , α 1 = σ 1 , α 2 = σ 2 , β = σ 3 , where the σ j are the Pauli matrices. Finally, ·,· is the standard C2 inner product. Recently there has been significant progress in the regularity theory for MD and the simpler Dirac–Klein–Gordon system (DKG), −iα μ ∂μ + Mβ ψ = φβψ, (1.2) − + m2 φ = βψ, ψ, where φ is real-valued and m ∈ R is a constant.

2302


A key question for both systems is whether global regularity holds, i.e. starting from smooth initial data, does the solution exist for all time and stay smooth? For small data this has been answered affirmatively by Georgiev [16] in 3d, but for large data there was until quite recently only the 1d result of Chadam [7]. To make progress on the large data question in 2d and 3d, a natural strategy is to study local (in time) well-posedness for rough data and exploit conservation laws to extend the solutions globally. But for both DKG and MD, the energy lacks a definite sign (see [17]), so the only conserved quantity that appears to be immediately useful is the charge: ψ(t)2 2 = const. L This constant will be called the charge constant in what follows. The charge conservation was of course a key ingredient in Chadam’s global result for 1d MD [7], later improved for the 1d DKG case by Bournaveas [5], in the sense that the regularity requirements were lowered to the charge class (data in L2 for the spinor and in some Sobolev space for the scalar field). Since then a number of papers improving the local and global theory for 1d DKG have appeared, see [13,1,22,26–28,31,32,23]. As the space dimension increases, however, it becomes much more difficult to prove local existence in the charge class, and therefore correspondingly difficult to exploit the charge conservation. Indeed, it was to take more than thirty years from the 1d result of Chadam until the next major breakthrough in the global theory was achieved quite recently by Grünrock and Pecher [19], who proved global well-posedness for 2d DKG. At the same time, but independently, Ovcharov [25] proved a corresponding result under a spherical symmetry assumption. Decisive improvements in the local theory have been made possible through the discovery, by the authors in joint work with Damiano Foschi, of the complete null structure of first DKG, in [11], and then MD, in [12], permitting significant progress compared to earlier local results such as [18,4,6,24,14,2], where at most partial null structure was used. In [19], Grünrock and Pecher use the DKG null structure combined with bilinear estimates similar to those used in [10], where in particular it was shown that 2d DKG is locally well-posed for data ψ(0) ∈ L2 R2 ,

φ(0) ∈ H 1/2 R2 ,

∂t φ(0) ∈ H −1/2 R2 ,

(1.3)

with a time of existence depending only on the size of the data norm. Thus, to get a global result it suffices—in view of the conservation of charge—to show that φ(t)

H 1/2 (R2 )

+ ∂t φ(t)H −1/2 (R2 )

cannot blow up in finite time. In fact, Grünrock and Pecher prove this for an equivalent norm which we shall denote D(t). In our reformulation, they prove: Theorem 1.1. (See [19].) The local solution of 2d DKG exists up to a time T > 0 determined by T 1/2 1 + D(0) = ε,

(1.4)


2303

where ε > 0 depends on the charge constant. Moreover, if D(0) 1 then sup D(t) D(0) + CT 1/2 ,

(1.5)

0tT

where C depends on the charge constant. Both DKG and MD are charge subcritical in 2d (whereas the 3d problems are charge critical). To be precise, the critical regularity determined by scaling is half a derivative below the regularity of the charge class data (1.3), hence the half power of T in (1.4) is optimal, and in fact so is the half power in (1.5). The fact that the two exponents add up to 1 enabled Grünrock and Pecher to apply a scheme devised by Colliander, Holmer and Tzirakis [8] to extend solutions globally. We recall the argument here since a modified version of it will be used for MD. Since the only possible impediment to global existence is D(t) becoming large, one may assume D(t) 1 for all t 0 for which the solution exists. Now as long as D(t) 2D(0), Theorem 1.1 can be applied repeatedly with a uniform time increment T given by T 1/2 [1 + 2D(0)] = ε. In view of (1.5) the theorem can be applied n times, where n is the smallest integer such that nCT 1/2 > D(0). In this way one covers a total time interval of length nT = nCT 1/2

1 1/2 ε ε ε > D(0) T D(0) = > 0, C C[1 + 2D(0)] C[3D(0)] 3C

the crucial point being that ε/3C is independent of D(0). Repeating the whole argument one can therefore cover a time interval of arbitrary length. The purpose of the present paper is to extend the result of Grünrock and Pecher to the full MD system. This adds significant difficulties since MD has a far more complicated null structure than DKG, and since instead of a single scalar field φ we have to deal with the electromagnetic field (E, B). Because of these additional difficulties, we have to face the following two issues, affecting the above global existence argument: (i) For MD we are only able to prove the analog of (1.5) up to a logarithmic loss in the factor T 1/2 , i.e. the term CT 1/2 on the right-hand side is replaced by CT 1/2 log(1/T ), where D(t) is now a certain norm of (E, B)(t) such that local existence holds up to a time 0 < T 1 determined by (1.4). (ii) The norm D(t) actually depends implicitly on T . Because of these issues, we are not able to apply the scheme of Colliander, Holmer and Tzirakis in its original form, but with some extra work—exploiting in particular a crucial monotonicity property of our data norm with respect to T —we are nevertheless able to obtain a global existence result. The detailed argument is given in Section 3, but as a warm-up we sketch here the argument in the much simpler situation where we ignore the implicit dependence of D(t) on T . The local result can then be iterated until nCT 1/2 log(1/T ) > D(0), giving a total time ≡ nT >

D(0) ε 1 1 ∼ ∼ , log(1/T ) C[1 + 2D(0)] log(1/T ) log D(0)

2304


where (1.4) was used. Moreover, one can easily show D() 3D(0), so by a further iteration one covers successive time intervals of length 1 , 2 , . . . such that j +1

1 log(3j D(0))

∼

1 j +1

for j 0, hence ∞ j =1 j = ∞. Some notation: The Fourier transforms on R2 and R1+2 are defined by f (ξ ) =

e−ix·ξ f (x) dx,

e−i(tτ +x·ξ ) u(t, x) dt dx,

u(X) =

R2

R1+2

u. where ξ ∈ R2 , τ ∈ R and X = (τ, ξ ). We also write F u = If A is a subset of R1+2 , or a condition describing such a set, the multiplier PA is defined by P u(X), A u(X) = χA (X) where χA is the characteristic function of A, and similarly if A ⊂ R2 . We write D = −i∇, and given h : R2 → C we denote by h(D) the multiplier defined by (ξ ) = h(ξ )f (ξ ). h(D)f The notation · is reserved for the L2 -norms on both R2 and R1+2 (which one it is will be clear from the context): f =

f (x)2 dx

1/2

u =

,

R2

u(t, x)2 dt dx

1/2 ,

R1+2

and similarly in Fourier space. For s ∈ R, the Sobolev space H s = H s (R2 ) is defined as the completion of the Schwartz space S(R2 ) with respect to the norm f H s = Ds f , s =B ˙ s (R2 ) is the completion of S(R2 ) with where ξ = (1 + |ξ |2 )1/2 . The Besov space B˙ 2,1 2,1 respect to the norm

f B˙ s =

2,1

N s P|ξ |∼N f ,

N >0

where N is understood to be dyadic, i.e. of the form 2j with j ∈ Z. In estimates we use the shorthand X Y for X CY , where C 1 is either an absolute constant or depends only on quantities that are considered fixed; X = O(R) is short for |X| R; X ∼ Y means X Y X; X Y stands for X C −1 Y , with C as above. We write for equality up to multiplication by an absolute constant (typically factors involving 2π ).


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2. Main results 2.1. Local well-posedness We consider the initial value problem for 2d MD starting from data ψ(0, x) = ψ0 (x),

E(0, x) = E0 (x),

B(0, x) = B0 (x) = 0, 0, B03 ,

which by Maxwell’s equations [see (2.6) below] must satisfy ∇ · E0 = |ψ0 |2 and ∇ · B0 = 0. But the latter automatically holds in 2d, since B = (0, 0, B 3 ) does not depend on x 3 , whereas the constraint ∇ · E0 = |ψ0 |2 determines the curl-free part1 of E0 , so we only specify data Edf 0 for the divergence-free part Edf . Thus, −1 2 E0 = Edf 0 + ∇ |ψ0 | . The data for the potential A, Aμ (0, x) = aμ (x),

∂t Aμ (0, x) = a˙ μ (x)

(μ = 0, 1, 2),

are fixed by choosing a0 = a˙ 0 = 0. Then the spatial parts a = (a1 , a2 , 0) and a˙ = (a˙ 1 , a˙ 2 , 0) are given by, since ∇ · a = 0 by the Lorenz condition, a = −−1 ∂2 B03 , −∂1 B03 , 0 ,

a˙ = −E0 .

Solving the second equation in (1.1) and splitting Aμ into its homogeneous and inhomogeneous parts, we reduce MD to a nonlinear Dirac equation μ −iα μ ∂μ + Mβ ψ = Ahom. μ α ψ − N (ψ, ψ, ψ),

(2.1)

where = 0, Ahom. μ

Ahom. μ (0, x) = aμ (x),

∂t Ahom. ˙ μ (x), μ (0, x) = a

and N (ψ1 , ψ2 , ψ3 ) = −1 α μ ψ1 , ψ2 α μ ψ3 . Here −1 F denotes the solution of u = F on R1+2 with vanishing data at t = 0. 1 Recall the splitting of E (or indeed any vector field) into divergence-free and curl-free parts: E = −−1 ∇ × (∇ × E) + −1 ∇(∇ · E) ≡ Edf + Ecf .

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Assuming the following data regularity: 2 2 ⎧ 2 ⎪ ⎨ ψ0 ∈ L R , C , −1/2 R2 , R2 , P|ξ |1 Edf 0 ∈H ⎪ ⎩ P|ξ |1 B03 ∈ H −1/2 R2 , R ,

2 2 ˙0 P|ξ | 0 sufficiently small. 2.2. Growth estimate for the electromagnetic field Having obtained ψ , we reconstruct the full potential − −1 α μ ψ, ψ, Aμ = Ahom. μ


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which by the definition of the data (aμ , a˙ μ ) satisfies the Lorenz gauge condition ∂ μ Aμ = 0 (see [12]). Now define B = ∇ × A = (0, 0, ∂1 A2 − ∂2 A1 ),

E = ∇A0 − ∂t A.

Since Aμ = −α μ ψ, ψ, it follows that Maxwell’s equations hold: ∇ · E = ρ,

∇ × E + ∂t B = 0,

∇ · B = 0,

∇ × B − ∂t E = J,

(2.6)

where ρ = J 0 = |ψ|2 ,

J = J 1, J 2, 0 ,

J μ = α μ ψ, ψ .

The first equation in (2.6) determines the curl-free part of E and implies E = Edf + −1 ∇ |ψ|2 , where Edf = Pdf E is the divergence-free part of E. Here Pdf = −−1 ∇×∇× is the projection onto divergence-free fields. From Maxwell’s equations we know that E = ∇ρ + ∂t J and B = −∇ × J, hence

Edf = Pdf (−∇J0 + ∂t J), Edf (0) = Edf ∂t Edf (0) = ∇ × 0, 0, B03 − Pdf J(0), 0,

(2.7)

and

B 3 = ∂1 J2 − ∂2 J1 , B 3 (0) = B03 ,

3 ∂t B 3 (0) = − ∇ × Edf 0 .

(2.8)

We want to use these wave equations to prove an estimate analogous to (1.5) in Theorem 1.1 for our norm DT (t). To be precise, we aim to prove sup DT (t) DT (0) + CT 1/2 log(1/T ), 0tT

but in order to avoid a constant factor C > 1 in front of the first term on the right-hand side, we first split the wave equations into first order equations and modify DT (t) accordingly. Recall that the splitting u = u+ + u− given by u± =

1 u ± i|D|−1 ∂t u 2

transforms u = F into −1 −i∂t ± |D| u± = − ±2|D| F.

(2.9)

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The term |D|−1 ∂t u in (2.9) causes problems at low frequency if u = Edf , however. To avoid this we use a general trick going back at least as far as [27], and used also in [19]: Adding −Edf to both sides of (2.7) gives the Klein–Gordon equation

( − 1)Edf = Pdf (−∇J0 + ∂t J) − Edf , Edf (0) = Edf ∂t Edf (0) = ∇ × 0, 0, B03 − Pdf J(0). 0,

(2.10)

The extra term −Edf on the right-hand side is relatively easy to handle due to the gain in regularity, and the key advantage is that we can now use the analog of (2.9) for the Klein–Gordon equation: The splitting v = v+ + v− given by v± =

1 v ± iD−1 ∂t v 2

(2.11)

transforms ( − 1)v = G into −1 −i∂t ± D v± = − ±2D G. df 3 3 3 Applying (2.11) to Edf and (2.9) to B 3 , we now write Edf = Edf + + E− and B = B+ + B− , where

df −1 df df −1 2Edf ∇ × 0, 0, B 3 − Pdf J , ± = E ± iD ∂t E = E ± iD 3 3 2B± = B 3 ± i|D|−1 ∂t B 3 = B 3 ± i|D|−1 − ∇ × Edf

(2.12) (2.13)

satisfy −1 −i∂t ± D Edf Pdf (−∇J0 + ∂t J) − Edf , ± = − ±2D 3 −1 −i∂t ± |D| B± = − ±2|D| (∂1 J2 − ∂2 J1 ).

(2.14) (2.15)

Define the corresponding norm df 3 3 D˜ T (t) = Edf + (t) (T ) + E− (t) (T ) + B+ (t) (T ) + B− (t) (T ) ,

(2.16)

and note that D˜ T (0) < ∞. Indeed, D˜ T (0) C D˜ 1 (0) by Lemma 3.1 below, and D˜ 1 (0) < ∞ in view of the assumption (2.2) and some straightforward Sobolev estimates for J [see (4.14) and (4.15) below]. Since DT (t) D˜ T (t) by the triangle inequality, the iteration argument used to prove Theorem 2.1 will also immediately give us: Theorem 2.2. Theorem 2.1 still holds with DT (0) replaced by D˜ T (0) in (2.5): T 1/2 1 + D˜ T (0) ε, where ε > 0 depends only on the charge constant and |M|.

(2.17)


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We shall prove the following growth estimate for D˜ T (t). Theorem 2.3. Let ψ be the solution of 2d MD obtained in Theorem 2.2, with existence time T 3 satisfying (2.17), and reconstruct the electromagnetic field as above. Then Edf ± and B± , as functions of t ∈ [−T , T ], describe continuous curves in the data space (2.2), hence the same is true df 3 3 3 for Edf = Edf + + E− and B = B+ + B− . Moreover, we have sup D˜ T (t) D˜ T (0) + CT 1/2 log(1/T ),

(2.18)

0tT

where C depends only on the charge constant and |M|. Combining Theorems 2.2 and 2.3, we shall obtain the global well-posedness: Theorem 2.4. The solution of 2d MD obtained in Theorem 2.2 extends globally in time. In particular, for smooth data the solution is smooth on R1+2 , so global regularity holds for 2d MD. The rest of this paper is organized as follows: In the next section we prove Theorem 2.4, in Section 4 we introduce various notation and function spaces needed for the proof of Theorems 2.1 and 2.2, given in Sections 5–10. Finally, in Section 11 we prove Theorem 2.3. 3. From local to global solutions Here we prove that if the conclusions of Theorems 2.2 and 2.3 hold, then the solutions extend globally in time, hence we obtain Theorem 2.4. We follow as closely as possible the argument outlined at the end of Section 1, but the fact that our norm depends implicitly on T creates some difficulties. To resolve these we rely crucially on the following monotonicity property of the norm (2.4): Lemma 3.1. There exists C > 1 such that for all 0 < S < T 1 and f ∈ S(R2 ), f (S) Cf (T ) . Proof. By definition, f (S) = P|ξ |1/S f H −1/2 + S 1/2

P|ξ |∼N f ,

0 0. Thus w1 Xs,1/2;1 (S φ(ξ )

T)

ρT w1 Xs,1/2;1 φ(ξ )

∞ 1 n T (t/T )n ρ(t/T )S(t)fn Xs,1/2;1 n! φ(ξ ) n=1

∞ 1 n T t n ρ(t)B 1/2 T −n+1/2+b GXs,b;∞ φ(ξ ) 2,1 n! n=1 ∞ n2n−1 T 1/2+b GXs,b;∞ , φ(ξ ) n!

n=1

where we used the elementary estimate 1/2−s s T s ρT B2,1 ρB2,1

(0 < s 1/2)

with s = 1/2 and ρ(t) replaced by t n ρ(t). The splitting w2 = a − b is defined as in (13.4) and (13.5) but with the obvious modifications, and we have aXs,1/2;1 ∼ φ(ξ )

L 1/T

L 1/T

1 L1/2 Ds Pτ +φ(ξ )∼L G L L−1/2−b GXs,b;∞ ∼ T 1/2+b GXs,b;∞ , φ(ξ )

φ(ξ )

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provided that −1/2 − b < 0, i.e. b > −1/2. Since, by Cauchy–Schwarz, hH s

1 L1/2 Ds Pτ +φ(ξ )∼L G, L

L 1/T

we also have bXs,1/2;1 (S φ(ξ )

T)

ρbXs,1/2;1 hH s T 1/2+b GXs,b;∞ , φ(ξ )

φ(ξ )

completing the proof of (4.11). 13.3. Proof of (4.12) With w(t) =

"t 0

S(t − t )G(t ) dt , (13.1) gives w

(t, ξ ) e−itφ(ξ )

eit (λ+φ(ξ )) − 1 G(λ, ξ ) dλ, i(λ + φ(ξ ))

implying (4.12). References [1] Nikolaos Bournaveas, Dominic Gibbeson, Low regularity global solutions of the Dirac–Klein–Gordon equations in one space dimension, Differential Integral Equations 19 (2) (2006) 211–222. [2] Philippe Bechouche, Norbert J. Mauser, Sigmund Selberg, On the asymptotic analysis of the Dirac–Maxwell system in the nonrelativistic limit, J. Hyperbolic Differ. Equ. 2 (1) (2005) 129–182. [3] I. Bejenaru, S. Herr, J. Holmer, D. Tataru, On the 2D Zakharov system with L2 -Schrödinger data, Nonlinearity 22 (5) (2009) 1063–1089. [4] Nikolaos Bournaveas, Local existence for the Maxwell–Dirac equations in three space dimensions, Comm. Partial Differential Equations 21 (5–6) (1996) 693–720. [5] Nikolaos Bournaveas, A new proof of global existence for the Dirac–Klein–Gordon equations in one space dimension, J. Funct. Anal. 173 (1) (2000) 203–213. [6] Nikolaos Bournaveas, Low regularity solutions of the Dirac–Klein–Gordon equations in two space dimensions, Comm. Partial Differential Equations 26 (7–8) (2001) 1345–1366. [7] John Chadam, Global solutions of the Cauchy problem for the (classical) coupled Maxwell–Dirac equations in one space dimension, J. Funct. Anal. 13 (1973) 173–184. [8] James Colliander, Justin Holmer, Nikolaos Tzirakis, Low regularity global well-posedness for the Zakharov and Klein–Gordon–Schrödinger systems, Trans. Amer. Math. Soc. 360 (9) (2008) 4619–4638. [9] J. Colliander, C. Kenig, G. Staffilani, Local well-posedness for dispersion-generalized Benjamin–Ono equations, Differential Integral Equations 16 (12) (2003) 1441–1472. [10] Piero D’Ancona, Damiano Foschi, Sigmund Selberg, Local well-posedness below the charge norm for the Dirac– Klein–Gordon system in two space dimensions, J. Hyperbolic Differ. Equ. 4 (2) (2007) 295–330. [11] Piero D’Ancona, Damiano Foschi, Sigmund Selberg, Null structure and almost optimal local regularity of the Dirac– Klein–Gordon system, J. Eur. Math. Soc. (JEMS) 4 (2007) 877–898. [12] Piero D’Ancona, Damiano Foschi, Sigmund Selberg, Null structure and almost optimal local well-posedness of the Maxwell–Dirac system, Amer. J. Math. 132 (3) (2010) 771–839. [13] Y.F. Fang, On the Dirac–Klein–Gordon equations in one space dimension, Differential Integral Equations 17 (11– 12) (2004) 1321–1346. [14] Y.F. Fang, M. Grillakis, On the Dirac–Klein–Gordon equations in three space dimensions, Comm. Partial Differential Equations 30 (4–6) (2005) 783–812. [15] G.B. Folland, Real Analysis: Modern Techniques and Their Applications, second ed., John Wiley, New York, 1999.


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[16] Vladimir Georgiev, Small amplitude solutions of the Maxwell–Dirac equations, Indiana Univ. Math. J. 40 (3) (1991) 845–883. [17] Robert Glassey, Walter Strauss, Conservation laws for the classical Maxwell–Dirac and Klein–Gordon–Dirac equations, J. Math. Phys. 20 (3) (1979) 454–458. [18] Leonard Gross, The Cauchy problem for the coupled Maxwell and Dirac equations, Comm. Pure Appl. Math. 19 (1966) 1–15. [19] Axel Grünrock, Hartmut Pecher, Global solutions for the Dirac–Klein–Gordon system in two space dimensions, Comm. Partial Differential Equations 1 (2010) 89–112. [20] Carlo Kenig, Gustavo Ponce, Luis Vega, The Cauchy problem for the KdV equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1) (1994) 1–21. [21] Sergiu Klainerman, Matei Machedon, Smoothing estimates for null forms and applications, Duke Math. J. 81 (1) (1995) 99–133. [22] Shuji Machihara, The Cauchy problem for the 1-D Dirac–Klein–Gordon equation, NoDEA Nonlinear Differential Equations Appl. 14 (5–6) (2007) 625–641. [23] Shuji Machihara, Kenji Nakanishi, Kotaro Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math. 50 (2) (2010) 403–451. [24] Nader Masmoudi, Kenji Nakanishi, From Maxwell–Klein–Gordon and Maxwell–Dirac to Poisson–Schrödinger, Int. Math. Res. Not. IMRN 13 (2003) 697–734. [25] Evgeni Ovcharov, Inhomogeneous Strichartz estimates with spherical symmetry and applications to the Dirac– Klein–Gordon system in two space dimensions, arXiv:0903.5339. [26] Hartmut Pecher, Low regularity well-posedness for the one-dimensional Dirac–Klein–Gordon system, Electron. J. Differential Equations 150 (2006), 13 pp. (electronic). [27] Hartmut Pecher, Modified low regularity well-posedness for the one-dimensional Dirac–Klein–Gordon system, NoDEA Nonlinear Differential Equations Appl. 15 (3) (2008) 279–294. [28] Sigmund Selberg, Global well-posedness below the charge norm for the Dirac–Klein–Gordon system in one space dimension, Int. Math. Res. Not. IMRN 17 (2007), Art. ID rnm058, 25 pp. [29] Sigmund Selberg, Anisotropic bilinear L2 estimates related to the 3D wave equation, Int. Math. Res. Not. IMRN (2008), Art. ID rnn107, 63 pp. [30] Sigmund Selberg, Bilinear Fourier restriction estimates related to the 2d wave equation, preprint, 2010, available on http://arxiv.org/abs/1003.5978, Adv. Differential Equations, in press. [31] Sigmund Selberg, Achenef Tesfahun, Low regularity well-posedness of the Dirac–Klein–Gordon equations in one space dimension, Commun. Contemp. Math. 10 (2) (2008) 181–194. [32] Achenef Tesfahun, Global well-posedness of the 1D Dirac–Klein–Gordon system in Sobolev spaces of negative index, J. Hyperbolic Differ. Equ. 6 (3) (2009) 631–661.


Power boundedness in Fourier and Fourier–Stieltjes algebras and other commutative Banach algebras E. Kaniuth a,∗,1 , A.T. Lau b,2 , A. Ülger c,3 a Institut für Mathematik, Universität Paderborn, D-33095 Paderborn, Germany b Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1 c Department of Mathematics, Koc University, 34450 Sariyer, Istanbul, Turkey

Received 30 August 2010; accepted 19 November 2010 Available online 26 November 2010 Communicated by K. Ball

Abstract We study power boundedness in the Fourier and Fourier–Stieltjes algebras, A(G) and B(G), of a locally compact group G as well as in some other commutative Banach algebras. The main results concern the question of when all elements with spectral radius at most one in any of these algebras are power bounded, the characterization of power bounded elements in A(G) and B(G) and also the structure of the Gelfand transform of a single power bounded element. © 2010 Elsevier Inc. All rights reserved. Keywords: Commutative Banach algebra; Structure space; Power bounded element; Locally compact group; Fourier algebra; Figà–Talamanca–Herz algebra; Fourier–Stieltjes algebra; Segal algebra; Dual algebra; Coset ring

0. Introduction This research is motivated by the work of Schreiber [30] on power bounded elements in the measure algebra M(G) of a locally compact abelian group G and is to some extent a continuation * Corresponding author.

E-mail addresses: [email protected] (E. Kaniuth), [email protected] (A.T. Lau), [email protected] (A. Ülger). 1 Supported by the German Research Foundation. 2 Supported by NSERC grant MS 100. 3 Supported by the TUBA and Tubitak Isbap project No. 107T896. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.012

E. Kaniuth et al. / Journal of Functional Analysis 260 (2011) 2366–2386

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of our recent article [23]. Recall that an element a of an arbitrary Banach algebra A is called power bounded if supn∈N a n < ∞ and that the spectral radius r(a) of any power bounded element a is 1. The Banach algebra A is said to have the power boundedness property (pbproperty) if every a ∈ A with r(a) 1 is power bounded. Let G be a locally compact group and let A(G) and B(G) be the Fourier and the Fourier– Stieltjes algebra of G, as introduced by Eymard [7]. These algebras are natural generalizations of the measure algebras and the L1 -algebras of locally compact abelian groups and have since been a major object of investigation in abstract harmonic analysis. The purpose of this paper is twofold. On the one hand, our aim is to find criteria, in terms of the group structure, for algebras such as A(G) and B(G) to have the power boundedness property. On the other hand, we want to characterize the power bounded elements of these algebras. Even though many of our results concern Fourier and Fourier–Stieltjes algebras, some are of considerably more general nature. The contents can be briefly described as follows. In Section 2 we prove that an abstract Segal algebra in A(G) (hence A(G) itself, in particular) has the power boundedness property if and only if G is discrete. As a consequence we obtain that B(G) has the power boundedness property precisely when G is finite. We also discuss the so-called Figà–Talamanca–Herz algebras, the Lp -analogues of A(G). However, for them we are only able to show that power boundedness forces G to be discrete under the additional hypothesis that G is amenable. Section 3 is devoted to extend major results of [30, Section 6] to general locally compact groups. Theorem 3.2 gives a necessary condition, in terms of the closed coset ring and characters of subgroups of G, for a closed subset E of G to be of the form Eu = {x ∈ G: |u(x)| = 1} for some power bounded element of B(G) and for a continuous function on E to be the restriction of some power bounded element of B(G). This generalizes the corresponding result of [30]. Necessary and sufficient conditions are given when E is both open and closed. When G is connected and amenable, a somewhat deeper characterization of power bounded elements in B(G) can be obtained. They turn out to be precisely those functions in B(G) which are either constant of modulus one or for which the sequence of powers is w ∗ -convergent to zero (Theorem 4.6). The statement of Theorem 4.6 is even new for connected abelian groups, and this also applies to several other of our results. In the more general setting of a commutative dual Banach algebra A, connectedness of (A) turns out to be equivalent to certain w ∗ -convergence conditions placed on the power bounded elements of A (Theorem 4.2). In Section 5 we give a criterion for discreteness of a locally compact group in terms of convergence of Ishikawa sequences associated with power bounded elements in B(G), and in the final section we determine explicitly the power bounded elements of two function algebras. Power bounded elements in the Fourier–Stieltjes algebra of a locally compact abelian group were first studied by Beurling and Helson [2], and later by Andersson [1] and other authors. There is an extensive literature on power bounded operators in Banach spaces. As a sample we mention [24], one of the main results of which we shall use. 1. Preliminaries Let A be a Banach algebra. An element a of A is said to be power bounded if supn∈N a n < ∞. The set of all power bounded elements of A will be denoted by PB(A). By Theorem 1.2 of [30], PB(A) has the following properties: (1) Every element a of PB(A) has spectral radius r(a) at most one.

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(2) If r(a) < 1, then a ∈ PB(A). (3) If A is commutative, then PB(A) is convex. We say that A has the power boundedness property (pb-property) if every a ∈ A with r(a) 1 is power bounded. Note that every uniform algebra has the pb-property. For a commutative Banach algebra A, we shall always denote by (A) the Gelfand spectrum of A, equipped with the w ∗ -topology, and by a → a, ˆ where a(γ ˆ ) = γ (a) for γ ∈ (A), the Gelfand homomorphism. For a ∈ A, let ˆ ) = 1 and Fa = γ ∈ (A): a(γ Ea = γ ∈ (A): a(γ ˆ )=1 . Recall that A is said to be regular if given a closed subset F of (A) and γ ∈ (A) \ F , there exists a ∈ A such that a(γ ˆ ) = 0 and a| ˆ F = 0. Given a closed subset F of (A), there are two distinguished ideals of A with hull equal to F , namely j (F ) = {a ∈ A: aˆ has compact support disjoint from F } and k(F ) = a ∈ A: a(γ ˆ ) = 0 for all γ ∈ F . The set F is called a set of synthesis or spectral set if k(F ) is the only closed ideal with hull equal to F , and F is a Ditkin set if a ∈ aj (F ) for every a ∈ k(F ). If A is regular and I is any ideal with h(I ) = F , then j (F ) ⊆ I ⊆ k(F ), and hence in this case F is a set of synthesis if and only if j (F ) = k(F ). As general references to spectral synthesis, we mention [21] and [28]. For any group H , the coset ring R(H ) is the Boolean ring generated by all cosets of subgroups of H . If H is a topological group, then the closed coset ring Rc (H ) is defined to be Rc (H ) = E ∈ R(H ): E is closed in H . For a locally compact abelian group G, the elements of Rc (G) have been completely described by Gilbert [14] and Schreiber [31]. Forrest [9] verified that the analogous description is valid for arbitrary locally compact groups G. A subset E of G belongs to Rc (G) if and only if E is of the form E=

n i=1

xi Hi \

ni

yij Kij ,

j =1

where xi , yij ∈ G, Hi is a closed subgroup of G and Kij is an open subgroup of Hi , n, ni ∈ N0 , 1 i n, 1 j ni . In particular, we shall use the fact that if E ∈ R(G), then the closure E of E belongs to Rc (G), which in [14] is the key step to the structure theorem of elements in Rc (G). Moreover, every compact set in R(G) is a finite union of cosets. For examples, see [29]. Let G be a locally compact group. The Fourier–Stieltjes algebra and the Fourier algebra, B(G) and A(G), have been introduced and studied extensively by Eymard in his seminal article [7].


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The space B(G) is the linear span of the set P (G) of all continuous positive definite functions on G and can be identified with the dual space of the group C ∗ -algebra C ∗ (G). With pointwise multiplication and the dual norm, B(G) is a semisimple commutative Banach algebra. The Fourier algebra A(G) is the closed ideal of B(G) generated by all compactly supported functions in B(G). The spectrum of A(G) can be canonically identified with G. More precisely, the map x → ϕx , where ϕx (u) = u(x) for u ∈ A(G), is a homeomorphism from G onto (A(G)). The algebra A(G) is regular and, as shown in [25], admits a bounded approximate identity if and denotes the dual group of G, then only if G is amenable. Note that when G is abelian and G the Fourier–Stieltjes transform furnishes isometric isomorphisms between the measure algebra and the group algebra L1 (G) and A(G), respectively. For all this, compare [7]. M(G) and B(G) 2. Power bounded elements in Fourier algebras and Segal algebras on locally compact groups We start by recalling the definition of a Segal algebra from [4]. Let (B, · B ) be any Banach algebra. A Banach algebra (A, · A ) is called a Segal algebra in (B, · B ) if (1) A is a dense ideal in B; (2) there exists a constant α > 0 such that aB αaA for all a ∈ A; (3) there exists a constant β > 0 such that a1 a2 A βa1 B a2 A for all a1 , a2 ∈ A. Suppose that B is commutative. Then, by [4, Theorem 2.1], the map ϕ → ϕ|A is a homeomorphism from (B) onto (A). Moreover, A is semisimple if B is semisimple. For Segal algebras on locally compact abelian groups compare [27] and [28]. In the sequel we study the power boundedness property for Segal algebras in the Fourier algebra of a locally compact group. There are plenty of such Segal algebras. For instance, for any 1 p < ∞, we can take A(G) ∩ Lp (G), equipped with the norm f = f A(G) + f p ,

f ∈ A(G) ∩ Lp (G).

Segal algebras in Fourier algebras were recently studied in [12] under operator space aspects. We start with three lemmas, which are used to prove Theorem 2.4 below, but appear to be of independent interest. Lemma 2.1. Let G be a locally compact group with the property that each compact subset of G belongs to the coset ring Rc (G). Then G is discrete. Proof. For the sake of brevity we say that a locally compact group H has property (∗) if every compact subset of H lies in Rc (H ). We first show that if H is any σ -compact amenable locally compact group satisfying (∗), then H must be discrete. For that, by [22, Proposition 2.1] it suffices to show that spectral synthesis holds for A(H ). Thus let E be any closed subset of H . Since H is σ -compact, E = ∞ i=1 Ei where each Ei is compact. By hypothesis, Ei ∈ Rc (H ). Since H is amenable, Ei is a spectral set and the ideal k(Ei ) has a bounded approximate identity [11, Lemma 2.2]. In particular, each Ei is a Ditkin set. Since a closed countable union of Ditkin sets is again a Ditkin set (see [21, Theorem 5.2.2]), we conclude that E is a spectral set for A(H ), as was to be shown.

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Notice next that if H is a closed subgroup of a locally compact group G and G has property (∗), then so does H . Indeed, if K is any compact subset of H , then K ∈ Rc (G) and, using the structure of sets in Rc (G) and elementary group theory, it is easily verified that E ∩ H ∈ Rc (H ) for every E ∈ Rc (G). Now let G be the given group and let G0 denote the connected component of the identity of G. Since G/G0 is totally disconnected, we can choose an open subgroup H of G such that H /G0 is compact. We have to show that H is discrete. Every such group H is a projective limit of Lie groups. So there exists a compact normal subgroup C of H such that H /C is a Lie group. Since C has property (∗) and is compact, it must be finite, and hence H is a Lie group. Let R denote the radical of the connected Lie group H0 . We now further exploit the fact that if L is an amenable σ -compact group having property (∗), then L is discrete. Since R is solvable and connected, it follows that R is trivial. So H0 is a connected semisimple Lie group. If H0 is compact, it must be trivial. If H0 is noncompact then the Iwasawa decomposition shows that H0 contains a closed subgroup which is isomorphic to R and has property (∗). This is impossible and therefore H0 is trivial and hence H is discrete since H0 is open in H . This completes the proof. 2 Recall that for any locally compact group G and u ∈ B(G), Eu = x ∈ G: u(x) = 1 and Fu = x ∈ G: u(x) = 1 . Lemma 2.2. Let H be a totally disconnected compact group. If A(H ) has the power boundedness property, then H must be finite. Proof. Since H is totally disconnected it is a projective limit of finite groups. Suppose that H is infinite. Then we can find of closed normal subgroups Hn of H

a strictly decreasing ∞ sequence −n 1 1 of finite index. Let K = ∞ H and u = 2 Hn ∈ P (H ). Then K has infinite index n=1 n n=1 in H and Fu = K. By a well-known result due to Kakutani and Kodaira [20] there exists a closed normal subgroup N of H such that H /N is second countable and u is constant on cosets of N . Since Fu = K, N is contained in K. So N/K is second countable and therefore every closed subset of H /K equals Ev for some v ∈ A(H /K). Since A(H /K) has the pb-property, by Theorem 4.1 of [23] every closed subset of H /K is contained in Rc (H /K). Lemma 2.1 now implies that H /K is discrete and hence finite. This contradiction completes the proof. 2 Lemma 2.3. Let G be a locally compact group, H an open subgroup of G and K a compact normal subgroup of H such that H /K is second countable. Let A be a Segal algebra in A(G) and suppose that A has the power boundedness property. Then, for any compact subset C of H which is a union of K-cosets, we have C ∈ Rc (G). Proof. Since H /K is second countable, by [23, Lemma 4.3] there exists w ∈ B(H /K) such that w∞ = 1 and Fw = C/K. Let q denote the quotient homomorphism and let v denote the trivial extension of w ◦ q to all of G. Then v∞ = 1 and Fu = C. As C is compact, there exists u ∈ j (∅) such that 0 u 1 and u = 1 on C. Then uv ∈ j (∅) ⊆ A and it satisfies uv∞ = 1 and Fuv = C. Since A has the pb-property, there exists a constant C > 0 such that (uv)n C for all n ∈ N. Now · A(G) α · A for some constant α. It follows that uv is a power bounded element of A(G) and hence C = Fuv ∈ Rc (G) by [23, Theorem 4.1]. 2


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The following theorem is the first main result of this section. Theorem 2.4. Let G be a locally compact group and let A be a Segal algebra in A(G). Then A has the power boundedness property if and only if G is discrete. In particular, A(G) has the power boundedness property if and only if G is discrete. Proof. Since A is semisimple, A has the pb-property if G is discrete [30, Corollary 2.3]. Conversely, suppose that A has the pb-property. Since G/G0 is totally disconnected, we can choose an open subgroup H of G such that H /G0 is compact. It suffices to show that H is finite. Now H is a projective limit of Lie groups. Fix a compact normal subgroup K of H such that H /K is a Lie group, so second countable. Then, by Lemma 2.3 every compact subset C of H , which is a union of K-cosets, belongs to Rc (G). Thus every compact subset of H /K belongs to Rc (G/K) and hence to Rc (H /K). Lemma 2.1 now implies that H /K is discrete. Thus G0 ⊆ K, and since both H /G0 and K are compact, it follows that H is compact. We show next that A(H ) has the pb-property. Since H is open and compact, A(H ) embeds isometrically into Ac (G) = A(G) ∩ Cc (G) via the mapping i : u → i(u), where i(u) is the trivial extension of u to G. Let u ∈ A(H ) be such that u∞ = 1. Since (A) = G (point evaluations) and i(u) vanishes outside of H , rA (u) = u∞ = 1. Therefore, since A has the pb-property, i(u)n A C for some constant C and all n ∈ N. On the other hand, · A(G) α · A for some constant α. It follows that n u

A(H )

= i(u)n A(G) α i(u)n A αC

for all n ∈ N. This shows that A(H ) has the pb-property. Since H is a totally disconnected compact group, Lemma 2.2 implies that H is finite. 2 Next we consider the power boundedness property for certain ideals of A(G). Proposition 2.5. Let G be a locally compact group and E a closed subset of G. If the ideal k(E) has the power boundedness property, then every compact subset of G \ E belongs to the coset ring R(G). Proof. Let K be any compact subset of G \ E and choose a compactly generated open subgroup H of G containing K. Then H \ K is σ -compact and hence there exists v ∈ A(H ) with v∞ 1 and Fv = K and v = 0 on H ∩E (compare the proof of Theorem 2.4). Now let u denote the trivial extension of v to all of G. Then u ∈ k(E), u∞ 1 and Fu = K. By hypothesis, u is power bounded and hence Fu ∈ Rc (G). 2 Corollary 2.6. Let G be an amenable locally compact group and suppose that E is a countable subset of G and G \ E is σ -compact. If the ideal k(E) has the power boundedness property, then G is discrete. Proof. The statement follows once we have seen that every closed subset F of G is a set of synthesis for A(G). As G \ E is σ -compact, F = ∞ j =1 Fn ∪ F0 , where F0 is a closed subset of E and each Fn , n ∈ N, is a compact subset of G \ E. By Proposition 2.5, Fn ∈ Rc (G), n ∈ N, and hence Fn is a Ditkin set for A(G). On the other hand, F0 is a countable union of singletons and hence also is a Ditkin set. Being a countable union of Ditkin sets, E is a Ditkin set. 2

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Since k(∅) = A(G), the preceding corollary extends Theorem 2.4 when G is an amenable second countable group. In Corollary 6.5 of [30] it was shown that if G is a connected locally compact abelian group fˆ(γ ) = 1} is finite. Since conand f is a power bounded element of L1 (G), then the set {γ ∈ G: is compact-free (see [16, Theorem 24.17]), the following nectedness of G is equivalent to that G generalizes [30, Corollary 6.5]. Corollary 2.7. Let G be a locally compact group such that G contains no nontrivial compact subgroup. Let u be a power bounded element of B0 (G) = u ∈ B(G): u ∈ C0 (G) . Then Eu is finite. Proof. Since u vanishes at infinity, the set Eu , which belongs to Rc (G), must be compact. Therefore Eu is a finite union of cosets of compact subgroups of G. By the hypothesis on G, this means that Eu is finite. 2 We now turn to the problem of when the Fourier–Stieltjes algebra B(G) has the power boundedness property. The result will be an easy consequence of Theorem 2.4 and the following lemma. Lemma 2.8. Let G be a discrete group. If every subset of G is contained in R(G), then G is finite. Proof. Let WAP(G) denote the space of weakly almost periodic functions on G. The hypothesis implies that 1E ∈ B(G) ⊆ WAP(G) for every subset E of G. This in turn implies that ∞ (G) ⊆ WAP(G) and hence these two spaces are equal. Since WAP(G) has a unique invariant mean [3], so does ∞ (G). However, this forces G to be finite (see [26, Chapter VI]). 2 Corollary 2.9. Let G be any locally compact group. Then B(G) has the power boundedness property if and only if G is finite. Proof. We only have to show that if B(G) has the pb-property, then G is finite. If B(G) has the pb-property, so does A(G) and hence G is discrete by Theorem 2.4. By the standard argument, we can assume that G is countable. Then, given any subset E of G, by [23, Lemma 4.3], there exists u ∈ B(G) with u∞ = 1 and Fu = E. Since u is power bounded, Fu ∈ R(G) by [23, Theorem 4.1]. Lemma 2.8 now shows that G is finite. 2 For a locally compact group G, Herz has introduced Lp -versions, Ap (G), of the Fourier algebra for all 1 < p < ∞. These algebras are nowadays usually referred to as the Figà– Talamanca–Herz algebras. We refer to [15] and [8] for the definition and basic properties of Ap (G). In particular, Ap (G) is a semisimple regular commutative Banach algebra with spectrum G. It is expected that in the following theorem, which is the second main result of this section, the hypothesis of amenability can be dropped. However, we have been unable to show this. Theorem 2.10. Let G be a first countable amenable locally compact group and 1 < p < ∞. Then Ap (G) has the power boundedness property if and only if G is discrete.


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Proof. We only have to show the necessity of the condition and for that we can assume that G is second countable. Indeed, if H is a compactly generated open subgroup of G, then H is second countable and Ap (H ) can be considered as a closed subalgebra of Ap (G) and hence has the power boundedness property. Let E be any compact subset of G. Let Bp (G) denote the algebra of all bounded continuous functions v on G which are multipliers of Ap (G), that is, vAp (G) ⊆ Ap (G). Then, as in the proof of [23, Lemma 4.3], we find v ∈ Bp (G) such that Fv = E and v∞ = 1. Since Ap (G) is regular, there exists w ∈ Ap (G) such that w = 1 on E and w∞ = 1. Now the element u = vw of Ap (G) satisfies Fu = E and u∞ = 1. Because G is amenable, the algebra Ap (G) has a bounded approximate identity [15] and therefore the closed ideal I = (1G − u)Ap (G) of Ap (G) has a bounded approximate identity by [23, Theorem 1.7]. Since h(I ) = E, [10, Proposition 3.13] implies that E ◦ , the interior of E, is closed in G. It is now easy to see that if E ◦ is closed in G for every compact subset E of G, then G must be discrete. In fact, if G0 , the connected component of the identity, is nontrivial, then take E = U , where U is any nonempty, relatively compact subset of G0 , whereas, if G is totally disconnected and H is any infinite compact open subgroup of G, one can take for E the intersection of a strictly decreasing sequence of subgroups of finite index in H (see the proof of Lemma 2.2). 2 With somewhat more effort, similar arguments as those in the proof of Theorem 2.4 can be used to show that in Theorem 2.10 the hypothesis that G be first countable can be dropped. 3. On power bounded elements in Fourier–Stieltjes algebras The main purpose of this section is to describe, for power bounded elements u of the Fourier– Stieltjes algebra B(G), the restriction of u to Eu , u|Eu , in terms of the coset ring of G and affine maps. The main result, Theorem 3.2 below, gives a necessary condition for a subset E of G to be of the form Eu for some u ∈ PB(B(G)) and for a continuous function on E to be the restriction of some u ∈ PB(G). We start by recalling the notions of affine and piecewise affine maps. Let G and H be groups. A map α : C ⊆ G → H is called affine if C is a coset and for any r, s, t ∈ C,

α rs −1 t = α(r)α(s)−1 α(t). A map α : Y ⊆ G → H is called piecewise affine if (i) there exist pairwise disjoint sets Yi ∈ R(G), i = 1, . . . , n, such that Y = ni=1 Yi , (ii) each Yi is contained in a coset Ci on which there is an affine map αi : Ci → H such that αi |Yi = α|Yi . The proof of the following lemma is patterned after that of [30, Lemma 6.1]. Lemma 3.1. Let G be a locally compact group and u a power bounded element of B(G) such that Eu is open in G. Then u|Eu is a piecewise affine map from Eu into T. Proof. For f ∈ B(T), define a function φ(f ) on G by φ(f )(x) = f (u(x)) for x ∈ Eu and φ(f )(x) = 0 otherwise. Then φ(f ) is continuous since Eu is open and closed in G. Because 1 (Z), we have B(T) =

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fˇ(n)u n ∈ B(G),

n∈Z

where fˇ denotes the inverse Fourier transform of f , and φ(f )(x) =

fˇ(n)u(x) n

n∈Z

for all x ∈ Eu . Since Eu ∈ Rc (G), 1Eu ∈ B(G), and therefore φ(f ) = 1Eu ·

fˇ(n)u n ∈ B(G).

n∈Z

Since f g is the inverse Fourier transform of fˇ ∗ g, ˇ it is straightforward to check that φ is a homomorphism from B(T) into B(G). Since φ is bounded and B(T) = 1 (Z) carries the MAX operator space structure [6, p. 316], φ is actually completely bounded [6, p. 49]. It now follows from [18, Theorem 3.7] that there exists an affine map α : Y ⊆ G → T such that, for each f ∈ B(T) and x ∈ G, φ(f )(x) = f (α(x)) whenever x ∈ Y and φ(f )(x) = 0 otherwise. Here Y = x ∈ G: φ(f )(x) = 0 for some f ∈ B(T) . It is then obvious that Y = Eu and α = u|Eu . So u|Eu is piecewise affine.

2

Theorem 3.2. Let G be a locally compact group and let u be a power bounded element of B(G). Then there exist closed subsets F1 , . . . , Fn of G with the following properties: (1) Fj ∈ Rc (G), 1 j n, and Eu = nj=1 Fj . (2) For each j = 1, . . . , n, there exist a closed subgroup Hj of G, aj ∈ G, αj ∈ T and a continuous character γj of Hj such that Fj ⊆ aj Hj and

u(x) = αj γj aj−1 x for all x ∈ Fj . Proof. We apply Lemma 3.1 to Gd , the group G equipped with the discrete topology. Let i : Gd → G denote the identity map. Then u ◦ i ∈ B(Gd ) and u ◦ iB(Gd ) = uB(G) [7, Théorème 2.20], and hence u ◦ i is power bounded. Therefore, by Lemma 3.1 there exist subsets Si of G, subgroups Li of G, ci ∈ G and affine maps βi : ci Li → T, i = 1, . . . , n, with the following properties: (1) Si ∈ R(Gd ) and Eu = ni=1 Si . (2) For each i = 1, . . . , n, Si ⊆ ci Li and βi |Si = u|Si . Now each Si is of the form q l=1

d l Ml \

ql k=1

elk Nlk ,


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where dl , elk ∈ G, the Ml are subgroups of G and the Nlk are subgroups of Ml , 1 l q, 1 k ql . Thus, by a further reduction step, we can assume that we only have to consider a set S of the form m S=a H \ bj Kj ⊆ bT , j =1

where bj ∈ H and the Kj are subgroups of H , and that there exists an affine map β : bT → T such that β|S = u|S . Furthermore, we can assume that each Kj has infinite index in H because otherwise, for some j , H is a finite union of Kj -cosets and therefore can be assumed to be simply a coset. Now m

bj Kj H = H ∩ a −1 bT ∪

and H ∩ a −1 bT = ∅,

j =1

because otherwise at least one of the Kj has finite index in H . It follows that H ∩ a −1 bT = h(H ∩ T ) for some h ∈ H and H ∩ T has finite index in H . So S is contained in a finite union of cosets of T ∩ H and consequently we can assume that S ⊆ c(T ∩ H ) for some c ∈ G. Since also S ⊆ bT , we have bT = cT . Hence δ = β|c(T ∩H ) is an affine map satisfying δ|S = u|S . Now S ⊆ c(T ∩ H ) implies that a = ch for some h ∈ H and therefore S=c H \

m

= c (T ∩ H ) \

hbj Kj

j =1

m

hbj Kj .

j =1

If hbj Kj ∩ (T ∩ H ) = ∅, then hbj = tk for some t ∈ (T ∩ H ) and k ∈ Kj and hence hbj Kj ∩ (T ∩ H ) = tKj ∩ (T ∩ H ) = t (Kj ∩ T ∩ H ). Thus, setting A = T ∩ H and Bj = hbj Kj ∩ (T ∩ H ), we have S =c A\

m

Bj ,

j =1

where Bj is either empty or a coset in A. In addition, since Kj has infinite index in H and A has finite index in H , the subgroup corresponding to Bj has infinite index in A. When G is abelian, in precisely the above setting it was shown by Cohen [5] that, since u ∈ B(G) is uniformly continuous, the affine map δ : cA → T is uniformly continuous as well and hence extends to a continuous affine map δ : cA → T. We briefly indicate the proof in the current nonabelian situation. The first observation is that there are elements a1 , . . . , am+1 of S / Kj whenever k = l, 1 k, l m + 1. To see this, let Bj such that, for all j = 1, . . . , n, ak−1 al ∈ be a coset of the subgroup Hj of H , pick an element a1 of S, and define inductively the sequence of ak s by choosing ak+1 ∈ S \

k i=1

ai (H1 ∪ · · · ∪ Hn )

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arbitrarily. This is possible because H cannot be covered by finitely many cosets of subgroups of infinite index in H [18, Proposition 2.2]. Next observe that if x, y ∈ cH , then xy −1 ak ∈ S for some 1 k m + 1. Indeed, otherwise there exist k, l ∈ {1, . . . , m + 1}, k = l, such that −1 −1

ak−1 al = xy −1 ak xy al ∈ Kj for some j , contradicting the choice of the ak . The remainder of the proof is now entirely analogous to the one on page 223 of [5], using the uniform continuity of the function u ∈ B(G). Then δ agrees with u on S since u is continuous. Let γ denote the continuous character of A associated with δ. Then u(x) = αγ (c−1 x) for all x ∈ S. Finally, since Eu is closed in G, Eu is a finite union of such sets S and on each such set S, u is of the form stated in (2). This completes the proof of the theorem. 2 Corollary 3.3. Let u be a power bounded element of A(G). Then in the description of Eu and u|Eu in Theorem 3.2 each Fj can be chosen to be a compact coset in G. Proof. We only have to note that Eu is compact and that every compact set in R(G) is a finite union of cosets of compact subgroups of G. 2 When G is abelian, every character of a closed subgroup H of G extends to a character of G. The preceding theorem therefore generalizes Theorem 6.2 of [30]. Even for abelian G and u ∈ B(G), there seems to be no necessary and sufficient criterion in terms of Eu , the closed coset ring and piecewise affine maps for u to be power bounded. However, if Eu is open in G, we have the following result, which generalizes [30, Theorem 6.7]. Theorem 3.4. Let G be an arbitrary locally compact group and let u ∈ B(G) be such that Eu is open in G. Then u is power bounded if and only if there exist (i) pairwise disjoint open sets F1 , . . . , Fn in R(G) such that Eu = nj=1 Fj and open subgroups Hj of G and aj ∈ G such that Fj ⊆ aj Hj , j = 1, . . . , n, and (ii) characters γj of Hj and αj ∈ T, j = 1, . . . , n, such that

u(x) = αj γj aj−1 x for all x ∈ Fj . Proof. Suppose first that u is power bounded. By Lemma 3.1, u|Eu is a piecewise affine map from Eu into T. Let R0 (G) denote the open coset ring of G, the smallest ring of subsets of G containing all open cosets. Using [18, Lemma 1.3(ii)] and its proof, we can write Eu as a disjoint union Eu = nj=1 Fj of open sets in R0 (G) such that for each j , there are an open subgroup Hj , an element aj of G and a continuous affine map βj : aj Hj → T such that βj |Fj = u|Fj . Now define γj : Hj → T by γj (h) = βj (aj )−1 βj (aj h),

h ∈ H.

Then it is known and easily verified that γj is a continuous character of Hj , and of course γj satisfies u(x) = γj (aj−1 x) for all x ∈ Fj .


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Conversely, let (i) and (ii) be satisfied and let γ˜j denote the trivial extension of γj to all of G. Since Fj is an open and closed set in R(G), 1Fj is an idempotent in B(G). Now u can be written as u=

n

αj 1Fj · Laj γ˜j .

j =1

Since the sets Fj are pairwise disjoint, it follows that, for all q ∈ N, uq =

n

q q αj 1Fj Laj γ˜j ,

j =1

and therefore, since |αj | = 1 = 1Fj B(G) , q u

B(G) =

n q La γ˜ j j B(G) j =1

n q γ˜

n q γ

j =1

j =1

j

n j =1

whence u is power bounded.

= B(G)

j

B(G)

q

γj B(Hj ) = n,

2

Corollary 3.5. Let u ∈ A(G) be such that u∞ 1 and Eu is open in G. Then u is power bounded if and only if there exist a compact open subgroup K of G, characters χ1 , . . . , χn of K, elements a1 , . . . , an of G and α1 , . . . , αn ∈ T such that (i) Eu = nj=1 aj K; (ii) u(x) = αj χj (aj−1 x) for x ∈ aj K, 1 j n. Proof. By Theorem 3.4 we only have to show that if u is power bounded, then the sets Fj in (i) of that theorem can be chosen to be cosets of a single compact open subgroup of G. Fix j and p note that, since Eu is compact, F = Fj can be written as a finite union F = i=1 ci Ci , were all Ci are compact subgroups. Since F is open in G, some of the Ci , say precisely C1 , . . . , Cq , q q p, are open in G. Then the open subset F \ i=1 ci Ci has to be empty. It follows that F is a

q finite union of cosets of the compact open subgroup i=1 Ci of G. Thus each Fj is a finite union of cosets

of some compact open subgroup Kj and hence also of the compact open subgroup K = nj=1 Kj . 2 The following corollary, which will be used in Section 4, characterizes the power bounded elements u of B(G) satisfying Eu = G.

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Corollary 3.6. Let G be a connected locally compact group and let u ∈ B(G) such that |u(x)| = 1 for all x ∈ G. Then u is power bounded if and only if there exist α ∈ T and a character γ of G such that u(x) = αγ (x) for all x ∈ G. In particular, such a u is constant on cosets of the commutator subgroup of G. Finally, we determine the extreme points of the convex set PB(A) for certain subalgebras A of B(G). Proposition 3.7. Let A be any closed subalgebra of B(G) containing A(G). Then u ∈ A is an extreme point of PB(A) if and only if |u(x)| = 1 for all x ∈ G. In particular, if G is noncompact and A ⊆ C0 (G), then PB(A) has no extreme points. Proof. If Eu = G, then u is an extreme point of the unit ball of C b (G). If u fails to be an extreme point of PB(A), then there exists v ∈ A, v = 0, such that the elements 12 (u + v) and 12 (u − v) are power bounded. Thus 12 (u + v)∞ 1 and 12 (u − v)∞ 1. This contradicts the fact that u is an extreme point of the unit ball of C b (G). Conversely, let u be an extreme point of PB(A) and, towards a contradiction, assume that |u(x0 )| < 1 for some x0 ∈ G. Choose an open, relatively compact neighbourhood V of x0 such that V ∩ Eu = ∅. Then u|V ∞ < 1. Since A(G) ⊆ A and A(G) is regular, there exists v ∈ A such that v = 0 on G \ V , v(x0 ) = 0 and v∞ < 1 − u|V ∞ . Then, by construction of v, u = u ± v = 1, u = u ± v on a neighbourhood of Eu and Eu±v = Eu . This implies that u ± v ∈ A is power bounded [30, Corollary 3.4], contradicting the fact that u is an extreme point of PB(A). 2 Corollary 3.8. Let G be a connected group. Then the extreme points of PB(B(G)) are precisely the functions u of the form u(x) = αγ (x), x ∈ G, where α ∈ T and γ is a continuous character of G. 4. Connected Gelfand spectrum and power bounded elements Recall that a semisimple commutative Banach algebra A is said to be a dual Banach algebra if there exists a Banach space X such that A = X ∗ and the multiplication in A is separately w ∗ continuous. Connectedness of the spectrum of a dual Banach algebra A turns out to be closely related to convergence of sequences (a n )n∈N for power bounded elements a ∈ A. To accomplish the corresponding results, we are going to employ, apart from harmonic analysis tools, various other resources, such as the Ishikawa iteration process [19] (as in [23, Corollary 1.3]), a theorem of Katznelson and Tzafriri [24] concerning the perispheral spectrum of power bounded operators on a Banach space and a Toeplitz summation theorem (see [32]). Suppose that A is unital with identity e and let a ∈ A. In the sequel, the Ishikawa sequence associated to a is always understood n to be sequence of elements ( e+a 2 ) , n ∈ N. Proposition 4.1. Let A = X ∗ be a dual semisimple commutative Banach algebra with identity e, and let a ∈ A be power bounded. n (i) If u ∈ A is a w ∗ -cluster point of the sequence ( e+a 2 ) , n ∈ N, then u is an idempotent and satisfies au = u. n (ii) If w ∗ -limn→∞ a n = 0, then w ∗ -limn→∞ ( e+a 2 ) = 0.


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Proof. (i) The mapping La : A → A, b → ab is power bounded since La = a and Lna = La n . Therefore, by [23, Corollary 1.3], e + a n+1 e+a n b − b →0 2 2 for every b ∈ A. In particular, taking b = e, e + a n+1 e + a n → 0. − 2 2 n Now, let u ∈ A be a w ∗ -cluster point of the sequence ( e+a 2 ) , n ∈ N. Then 2 u = u and also au = u.

(ii) For n, k ∈ N0 , let ck (n) = 2−n nk . Then

e+a 2 u = u. This implies

∞ n (1) k=0 ck (n) = k=0 ck (n) = 1 for all n ∈ N0 ; (2) limn→∞ ck (n) = 0 for each k ∈ N0 . Since the sequence (a n )n∈N is w ∗ -convergent and

e+a 2

n

n ∞ 1 n k a = = n ck (n)a k , k 2 k=0

k=0

it follows that, for all x ∈ X, n (3) limn→∞ ( e+a 2 ) , x = limn→∞

∞

k=0 ck (n)a

k , x.

Now (1) and (2) show that the summation method defined by the doubly infinite matrix with entries ck (n) is ‘regular’ in the sense of summation theory. It then follows from (3), w ∗ limn→∞ a n = 0 and the Toeplitz summation theorem (see [32]) that lim

n→∞

for each x ∈ X, as was to be shown.

e+a 2

n

,x = 0

2

The main results of this section are the following theorem and Theorem 4.6 below. Theorem 4.2. Let A = X ∗ be a semisimple commutative dual Banach algebra with identity e. Then the following are equivalent. (i) (A) is connected. (ii) For each power bounded element a of A, a = e, with Ea = Fa , we have w ∗ -limn→∞ a n = 0. n (iii) For each power bounded element a of A, a = e, w ∗ -limn→∞ ( e+a 2 ) = 0.

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Proof. (i) ⇒ (ii) Let a ∈ PB(A) be such that a = e and Ea = Fa . As in the proof of Proposition 4.1(i), consider the power bounded linear operator La : b → ab of A. Then

σ (La ) = aˆ (A)

and aˆ (A) ∩ T = Ea = Fa .

Thus σ (La ) ∩ T ⊆ {1}, and it follows from [24, Theorem 1 and the Remark on page 317] that lim a n+1 − a n = lim Ln+1 − Lna = 0. a

n→∞

n→∞

Let u be a w ∗ -cluster point of the sequence (a n )n∈N . Then au = u and this in turn implies that u2 = u. Hence, since (A) is connected, either u = 0 or u = e. In the latter case it follows that a = e. This contradiction shows that 0 is the only w ∗ -cluster point of the sequence (a n )n∈N . Consequently, w ∗ -limn→∞ a n = 0. (ii) ⇒ (iii) is immediate from Proposition 4.1(ii). (iii) ⇒ (i) Towards a contradiction, assume that (A) is not connected. Then, by Shilov’s idempotent theorem, there exists an idempotent a in A with 0 = a = e. Clearly, a is power e+a n bounded and ( e+a 2 ) = 2 = 0 for all n ∈ N, contradicting (iii). 2 We now present the Hardy algebra as an example to which the preceding theorem applies. The result can also easily be deduced from the fact that w ∗ -convergence in H ∞ (D) is equivalent to pointwise convergence plus uniform boundedness (see Theorem 5.3 and its proof of Garnett’s book [13]). Example 4.3. Let H ∞ (D) be the algebra of all bounded analytic functions on the open unit disk D = {z ∈ C: |z| < 1}, equipped with the uniform norm. By Carleson’s Corona Theorem, the spectrum of H ∞ (D) is connected (see [13, Chapter VIII] or [17, Chapter 10]). Moreover, H ∞ (D) is a dual Banach algebra. In fact, its unique predual is X = L1 (T)/H0 (T), where H0 (T) is the closure in L1 (T) of the set of all complex polynomials without constant term [13, Chapter V, Section 5]. n Let f ∈ H ∞ (D) with f = 1 and f ∞ 1. Then, by Theorem 4.2, w ∗ -limn→∞ ( 1+f 2 ) =0 ∗ n and, if in addition Ef = Ff , then w -limn→∞ f = 0. Turning to locally compact groups and A = B(G), note that if u ∈ B(G) = C ∗ (G)∗ has the property that w ∗ -limn→∞ un = 0, then u is power bounded. In fact, this follows readily from the uniform boundedness principle. We now attack the problem of whether conversely power boundedness of u entails w ∗ -limn→∞ un = 0. The first partial answer is a consequence of Theorem 4.2. Corollary 4.4. Let G be a connected locally compact group and let u ∈ B(G) be such that u = 1G and Eu = Fu . Then u is power bounded if and only if w ∗ -limn→∞ un = 0. In particular, if u ∈ B(G), then |u| is power bounded if and only if w ∗ -limn→∞ |u|n = 0. Proof. If G is connected, so is (B(G)). Indeed, otherwise there exists an idempotent v ∈ B(G) such that 0 = v = 1G , and since G is determining for B(G), we conclude that {x ∈ G: u(x) = 1} is a proper nonempty open and closed subset of G. Thus, if u is power bounded, the implication (i) ⇒ (ii) of Theorem 4.2 shows that w ∗ -limn→∞ un = 0. 2 The next proposition, which will be used in Theorem 4.6 below, shows that if Eu = G, then in Corollary 4.4 we can drop the absolute value.


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Proposition 4.5. Let G be a connected locally compact group and let u ∈ B(G) be such that |u(x)| = 1 for all x ∈ G and u is nonconstant. Then u is power bounded if and only if w ∗ limn→∞ un = 0. Proof. We only have to show that the limit condition is necessary. By Lemma 3.6 there exist α ∈ T and a character γ of G such that u(x) = αγ (x) for all x ∈ G. Of course we can assume that α = 1. Let N = {x ∈ G: γ (x) = 1} and define β on G/N by β(xN ) = γ (x). Then β is a faithful character of G/N . Since G/N is connected, β(G/N ) = T and β is a topological isomorphism between G/N and T. Thus we can identify G/N with T, and then β is of the form β(z) = zm for all z ∈ T and some m ∈ Z, m = 0. Let φ : L1 (G) → L1 (G/N ) = L1 (T) be the surjective homomorphism defined by φ(f )(xN ) = N f (xt) dt, x ∈ G. Then φ extends uniquely to a continuous homomorphism from C ∗ (G) onto C ∗ (G/N ). It suffices to verify that γ n , f → 0 for all f in some subalgebra A of L1 (G) which is dense in C ∗ (G). Choose A to consist of all f ∈ L1 (G) such that φ(f ) is a trigonometric polynomial on G/N = T. Then A is dense in L1 (G) since the trigonometric polynomials are dense in L1 (T). Fix f ∈ A and let φ(f )(z) =

r

cj znj ,

z ∈ T,

j =1

where c1 , . . . , cr ∈ C and n1 , . . . , nr ∈ Z. Then, normalizing Haar measures on G, N and T appropriately and using Weil’s formula, γ n, f =

γ (xt)n f (xt) dt d(xN )

G/N N

=

β(z)n φ(f )(z) dz T

=

r j =1

Since

Tz

q

znm+nj dz.

cj T

dz = 0 whenever q = −1, we get that γ n , f = 0 for sufficiently large n ∈ N.

2

Of course, it is desirable to drop the hypothesis that Eu = Fu . As we are now going to show, using entirely different tools, this can be done under the assumption that G is amenable. Theorem 4.6. Let G be an amenable and connected locally compact group and let u be a nonconstant function in B(G). Then u is power bounded if and only if w ∗ -limn→∞ un = 0. Proof. By the remark preceding Corollary 4.4, we have to show that if u ∈ B(G) is power bounded, then w ∗ -limn→∞ un = 0. By Proposition 4.5 we can assume that Eu = G. Consider the closed ideal

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A0 (u) = v ∈ A(G): lim un v A(G) = 0 n→∞

of A(G). Then, as proved in [23, Theorem 2.6], j (Eu ) ⊆ A0 (u) ⊆ k(Eu ). Moreover, by [23, Theorem 4.1], Eu ∈ Rc (G). Since G is amenable, it follows that Eu is a set of synthesis and k(Eu ) has a bounded approximate identity [11, Lemma 2.2]. Therefore A0 (u) = k(Eu ) and the ideal A0 (u) has a bounded approximate identity, (uα )α say. Now, because Eu = G and hence A0 (u) = {0}, any w ∗ -cluster point of (uα )α in B(G) is a nonzero idempotent. Since G is connected, we conclude that 1G is the only such w ∗ -cluster point. Hence uα → 1G in the w ∗ -topology of B(G). Now let v ∈ B(G) be a w ∗ -cluster point of the bounded sequence (un )n∈N . Then v = w ∗ limι unι for some subnet (unι )ι∈I of (un )n∈N . Since the multiplication in B(G) is separately w ∗ -continuous, for any f ∈ C ∗ (G), v, uα · f = uα v, f → v, f . Since (uα )α ⊆ A0 (u), it follows that v, f = limv, uα · f = lim lim unι , uα · f α α ι = lim lim unι uα , f = 0. α

ι

Hence v = 0, and this shows that 0 is the only w ∗ -cluster point of the sequence (un )n∈N . Consequently, w ∗ -limn→∞ un = 0. 2 yields Theorem 4.6, applied to an abelian locally compact group G and μ ∈ M(G) = B(G), the following corollary. and let Corollary 4.7. Let G be a locally compact abelian group with connected dual group G μ ∈ M(G) = C0 (G)∗ be such that μ is not a multiple of the Dirac measure δe . Then μ is power bounded if and only if w ∗ -limn→∞ μn = 0. 5. A criterion for discreteness in terms of Ishikawa sequences In this section we first give a criterion for a locally compact group G to be discrete in terms of norm convergence of the Ishikawa sequences associated to power bounded elements of B(G). Lemma 5.1. Let G be a locally compact group. If Fu is open in G for every u ∈ P 1 (G) = {v ∈ P (G): v(e) = 1}, then G is discrete. Proof. Suppose first that G is first countable and let (Vn )n∈N be a neighbourhood basis of the identity e. For each n ∈ N, we can choose vn ∈ P 1 (G) such that vn = 0 on G \ Vn . Then v = ∞ −n 1 n=1 2 vn ∈ P (G) and Fv = {e}. So G is discrete.


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Notice next that the condition presumed in the lemma passes to open subgroups and to quotient groups. Let G0 be the connected component of the identity, and choose an open subgroup H of G such that H /G0 is compact. Then H is a projective limit of first countable groups (actually, Lie groups) H /Kα . By the first paragraph and the preceding remark, it follows that each H /Kα is discrete and hence finite since it is almost connected. Thus H is a compact totally disconnected group. We have to show that H is finite. Assuming that H is infinite, we find a strictly decreasing ∞ −n 1 sequence (Hn )n of open subgroups of H . Let u = n=1 2 1Hn ; then u ∈ P (H ) and therefore

∞ Fu is open in H . But Fu = n=1 Hn , which is of infinite index in H and hence not open in H . This contradiction finishes the proof. 2 Theorem 5.2. Let G be a locally compact group. Then G is discrete if and only if for every power n bounded element u ∈ A(G), the sequence ( 1+u 2 ) , n ∈ N, converges in norm in B(G). Proof. Suppose first that G is discrete and let u be a power bounded element of A(G). By n ∗ Proposition 4.1, the sequence ( 1+u 2 ) , n ∈ N, has a subsequence which converges in the w ∗ topology to some idempotent v in B(G). Since G is discrete, w -convergence implies pointwise convergence. Hence v(x) = 1 if u(x) = 1 and v(x) = 0 otherwise. n Now let w ∈ B(G) be another w ∗ -cluster point of the sequence ( 1+u 2 ) , n ∈ N. Then also w(x) = 1 if u(x) = 1 and w(x) = 0 otherwise, and hence w = v. Thus the whole sequence n ∗ ( 1+u 2 ) , n ∈ N, converges to v in the w -topology. Since G is discrete, the mapping A(G) → A(G), v → uv, is a compact operator. As the 1 n sequence vn = ( 1+u 2 ) − 2n , n ∈ N, is in A(G) and bounded, (uvn )n converges in norm. Since n 2−n → 0, it follows that u( 1+u 2 ) converges in norm to uv = v. Since

1+u 2

n+1

−

1+u 2

n =

u 1+u n 1 1+u n − , 2 2 2 2

n u( 1+u 2 ) → v in norm and

1 + u n+1 1 + u n → 0, − 2 2 n it follows that ( 1+u 2 ) → v in norm. n Conversely, suppose that whenever u ∈ A(G) is power bounded, the sequence ( 1+u 2 ) , n ∈ N, converges in norm to some element of B(G), say v. By Proposition 4.1, v is an idempotent and vu = u. Moreover, since the convergence is in norm, for x ∈ G, considered as a linear functional of B(G), we get that v(x) = 1 if u(x) = 1 and v(x) = 0 otherwise. This implies that for any such u the set Fu is open in G. In particular, Fu is open in G for every u ∈ P 1 (G). Now, Lemma 5.1 implies that G is discrete. 2

Suppose that G is discrete and u is a power bounded element of A(G). Then, by the preceding n theorem, the sequence ( 1+u 2 ) , n ∈ N, converges in norm in B(G). One might wonder whether 1+u n n limn→∞ ( 2 ) = 0. Note that, by [23, Corollary 2.8], limn→∞ ( 1+u 2 ) = 0 if and only if 1 − u is invertible in B(G).

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As an interesting consequence of Theorem 5.2 we obtain the following corollary. Corollary 5.3. Let G be a locally compact abelian group. Then G is compact if and only if for n every power bounded element f of L1 (G), the sequence ( δe +f 2 ) , n ∈ N, converges in M(G). In concluding this section, we present an application of power boundedness to the existence of a weakly compact homomorphism between two commutative Banach algebras. The reader will observe that we do not assume the algebra A to have a bounded approximate identity. Theorem 5.4. Let A and B be semisimple commutative Banach algebras and let φ : A → B be a weakly compact homomorphism with dense range. Suppose that given any ϕ ∈ (A), there exists a power bounded element u of A such that Eu = {ϕ}. Then (B) is discrete, and hence B has the power boundedness property. Proof. Let γ ∈ (B) be arbitrary and let ϕ = φ ∗ (γ ). By hypothesis, there exists u ∈ PB(A) such that Eu = {ϕ}. Let m be a w ∗ -cluster point of the sequence (un )n in A∗∗ . Then m, ϕ = 1 and m, δ = 0,

δ ∈ (A), δ = ϕ.

Let b = φ ∗∗ (m). Then b ∈ B since φ is weakly compact. As φ ∗ is one-to-one, φ ∗ (ρ) = ϕ for ρ ∈ (B), ρ = γ . It follows that b, ϕ = φ ∗∗ (m), ρ = m, φ ∗ (ρ) = 0 for ρ ∈ (B) \ {γ } and |b, γ | = 1. Since b ∈ B, this shows that the singleton {ψ} is open in (B). Finally, since B is semisimple and (B) is discrete, B has the pb-property. 2 Let G be a first countable locally compact group. Then, given any x ∈ G, there exists u ∈ A(G), actually a translate of a positive definite function, such that Eu = {x} and uA(G) = 1. Therefore the following corollary is an immediate consequence of the preceding theorem. Corollary 5.5. Let G be a first countable locally compact group and B a semisimple commutative Banach algebra whose spectrum is not discrete. Then there does not exist a weakly compact homomorphism from A(G) into B with dense range. 6. Power bounded elements of two function algebras We finish the paper by determining explicitly the power bounded elements of two commutative Banach algebras which are neither uniform algebras nor algebras associated with locally compact groups. Example 6.1. Let X be a compact metric space with metric d and let Lip(X) denote the space of all Lipschitz functions of order one on X, that is, all continuous complex-valued functions f on X for which |f (x) − f (y)| : x, y ∈ X, x = y p(f ) = sup d(x, y)


2385

is finite. With pointwise multiplication and the norm f = f ∞ + p(f ), the set Lip(X) is a commutative Banach algebra, and the map x → ϕx , where ϕx (f ) = f (x) for f ∈ Lip(X), is a homeomorphism from X onto (Lip(X)). In particular, σLip(X) (f ) = f (X) and hence rLip(X) (f ) = f ∞ for each f ∈ Lip(X). We claim that f ∈ Lip(X) is power bounded if and only if X is the disjoint union of open sets U and V such that f |U ∞ < 1 and V = {x ∈ X: |f (x)| = 1} and f is locally constant on V . In particular, if X is connected then f ∈ Lip(X) is power bounded if and only if either f ∞ < 1 or f (X) = {z} for some z ∈ T. Suppose first that f is power bounded and let U = {x ∈ X: |f (x)| < 1}. Towards a contradiction, assume that there exists x ∈ U such that |f (x)| = 1. Of course, we can assume that f (x) = 1. Since f is power bounded, there exists C > 0 such that |f (y)n − 1| C d(y, x) for all y ∈ X and n ∈ N. As x ∈ U , there exists y ∈ U with d(y, x) < 1/2C and hence |f (y)n −1| < 1/2 for all n ∈ N, which is impossible since f (y)n → 0. This contradiction shows that U is closed in X and V = {x ∈ X: |f (x)| = 1} is open. √ To show that f is locally constant on V , fix x ∈ V and put W = {y ∈ V √ : d(y, x) < 1/ 3}. Again we can assume that f (x) = 1. Then, for y ∈ W , |f (y)n − 1| < 1/ 3 for all n ∈ N and hence for all n ∈ Z since |f (y)| √ = 1. So the set {f (y)n : n ∈ Z} is a subgroup of T, which is contained in {z ∈ T: |z − 1| < 1/ 3}. However, every such subgroup must be trivial. This shows that f (y) = 1 for all y ∈ W . Conversely, suppose that f satisfies the above conditions on U and V . Since V is compact, V is a disjoint union of open sets V1 , . . . , Vm such that f is constant on each Vj . Let δ = min d(Vj , Vk ): 1 j, k m, j = k . Then δ > 0 and for x ∈ Vj and y ∈ Vk , j = k, we have δ d(x, y) d(Vj , Vk ) δ f (x)n − f (y)n 2 for all n ∈ N. Since f is constant on each Vj , we conclude that f |V ∈ Lip(V ) is power bounded. Moreover, since f |U ∞ < 1, f |U ∈ Lip(U ) is power bounded. Using these facts and d(U, V ) > 0, it follows that f is power bounded. Example 6.2. Let C 1 [a, b] be the algebra of all continuously differentiable, complex-valued functions on the interval [a, b]. Equipped with the norm f = f ∞ + f ∞ , C 1 [a, b] is a semisimple commutative Banach algebra, the spectrum of which can be canonically identified with [a, b] in the sense that the map t → ϕt , where ϕt (f ) = f (t) for f ∈ C 1 [a, b] and t ∈ [a, b], is a homeomorphism between [a, b] and (C 1 [a, b]). We claim that apart from functions f ∈ C 1 [a, b] with rC 1 [a,b] (f ) = f ∞ < 1, the constant functions of absolute value one are the only power bounded elements of C 1 [a, b]. Of course, this is reminiscent of the description of the power bounded elements of Lip(X) in Example 6.1, and the following arguments are indeed similar. Let f ∈ C 1 [a, b] be power bounded with f ∞ = 1. If t0 ∈ [a, b] is such that |f (t0 )| = 1 then |f (t)| = 1 in a neighbourhood of t0 . To verify this, we can assume that f (t0 ) = 1. By hypothesis, (f n ) ∞ C < ∞ for all n ∈ N and hence, by the mean value theorem, f (t)n − 1 = f n (t) − f n (t0 ) C|t − t0 |

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for all t ∈ [a, b] and all n ∈ N. It follows that for t ∈ [a, b] with |t − t0 | < 1/C, we cannot have |f (t)| < 1. This shows that the set {t ∈ [a, b]: |f (t)| = 1} is open (and closed) in [a, b] and hence equal to [a, b]. Using this, it is now seen as in Example 6.1 that f is locally constant, and hence constant, on the interval [a, b]. Acknowledgments The authors are grateful to the reviewer for carefully checking the manuscript and for drawing their attention to Garnett’s book [13] in connection with Example 4.3. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

R. Andersson, Power bounded restrictions of Fourier–Stieltjes transforms, Math. Scand. 46 (1980) 129–153. A. Beurling, H. Helson, Fourier–Stieltjes transforms with bounded powers, Math. Scand. 1 (1953) 120–126. R.B. Burckel, Weakly Almost Periodic Functions on Semigroups, Gordon and Breach, New York, 1970. J.T. Burnham, Closed ideals in subalgebras of Banach algebras. I, Proc. Amer. Math. Soc. 32 (1972) 551–555. P.J. Cohen, On homomorphisms of group algebras, Amer. J. Math. 82 (1960) 213–226. E.G. Effros, Z.-J. Ruan, Operator Spaces, London Math. Soc. Monogr. Ser., vol. 23, Clarendon Press, Oxford, 2000. P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964) 181–236. P. Eymard, Algèbre Ap et convoluteurs de Lp , in: Sèminaire Bourbaki No. 367, 1969/70, pp. 55–72. B.E. Forrest, Amenability and ideals in A(G), Austral. J. Math. Ser. A. 53 (1992) 143–155. B.E. Forrest, Amenability and the structure of Ap (G), Trans. Amer. Math. Soc. 343 (1994) 233–243. B.E. Forrest, E. Kaniuth, A.T. Lau, N. Spronk, Ideals with bounded approximate identities in Fourier algebras, J. Funct. Anal. 203 (2003) 286–304. B.E. Forrest, N. Spronk, P. Wood, Operator Segal algebras in Fourier algebras, Studia Math. 179 (2007) 277–295. J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. J.E. Gilbert, On projections of L∞ (G) and translation invariant subspaces, Proc. Lond. Math. Soc. 19 (1969) 69–88. C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973) 91–123. E. Hewitt, K.A. Ross, Abstract Harmonic Analysis. I, Springer, New York, 1963. K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, NJ, 1962. M. Ilie, N. Spronk, Completely bounded homomorphisms of the Fourier algebras, J. Funct. Anal. 225 (2005) 480– 499. S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976) 65–71. S. Kakutani, K. Kodaira, Über das Haarsche Maß in der lokal bikompakten Gruppe, Proc. Imp. Acad. Tokyo 20 (1944) 444–450. E. Kaniuth, A Course in Commutative Banach Algebras, Grad. Texts in Math., vol. 246, Springer, New York, 2009. E. Kaniuth, A.T. Lau, Spectral synthesis for A(G) and subspaces of VN(G), Proc. Amer. Math. Soc. 129 (2001) 3253–3263. E. Kaniuth, A.T. Lau, A. Ülger, Multipliers of commutative Banach algebras, power boundedness and Fourier– Stieltjes algebras, J. Lond. Math. Soc. (2) 81 (2010) 255–275. Y. Katznelson, L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986) 313–328. H. Leptin, Sur l’algèbre de Fourier d’un groupe localement compact, C. R. Math. Acad. Sci. Paris Ser. A 266 (1968) 1180–1182. A.L.T. Paterson, Amenability, Math. Surveys Monogr., vol. 29, American Mathematical Society, Providence, RI, 1988. H. Reiter, L1 -Algebras and Segal Algebras, Lecture Notes in Math., vol. 231, Springer, New York, 1971. H. Reiter, J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, Oxford, 2000. W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1960. B. Schreiber, Measures with bounded convolution powers, Trans. Amer. Math. Soc. 151 (1970) 405–431. B. Schreiber, On the coset ring and strong Ditkin sets, Pacific J. Math. 33 (1970) 805–812. A. Wilansky, Summability through Functional Analysis, Math. Stud., vol. 85, North-Holland, Amsterdam, 1984.


Non-existence of vortices in the small density region of a condensate ✩ Amandine Aftalion a,∗ , Robert L. Jerrard b , Jimena Royo-Letelier c,d a CNRS et Université Versailles-Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles,

CNRS UMR 8100, 45 avenue des États-Unis, 78035 Versailles Cédex, France b Dept. of Mathematics University of Toronto, Toronto, Canada M5S2E4 c CMAP, Ecole Polytechnique, 91128 Palaiseau Cédex, France d Université Versailles-Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, 45 avenue des États-Unis, 78035 Versailles Cédex, France Received 31 August 2010; accepted 5 December 2010 Available online 30 December 2010 Communicated by H. Brezis

Abstract In this paper, we answer a question raised by Lev Pitaevskii and prove that the ground state of the Gross– Pitaevskii energy describing a Bose–Einstein condensate in a rotationally symmetric trap at low rotation does not have vortices in the low density region. Therefore, the first ground state with vortices has its vortices in the bulk. In fact we prove something stronger, which is that the ground state for the model at low and moderate rotations is equal to the ground state in a condensate with no rotation. This is obtained by proving that for small rotational velocities, the ground state is multiple of the ground state with zero rotation. We rely on sharp bounds of the decay of the wave function combined with weighted Jacobian estimates. © 2010 Elsevier Inc. All rights reserved. Keywords: Bose–Einstein condensates; Jacobian estimates; Ground state; Radial symmetry

✩

This work was partially supported by French ministry grant ANR-0238 VoLQuan and by the National Science and Engineering Research Council of Canada, under operating Grant 261955. * Corresponding author. E-mail address: [email protected] (A. Aftalion). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.003

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1. Introduction Among the many experiments on Bose–Einstein condensates, one consists in rotating the trap holding the atoms in order to observe a superfluid behavior: the appearance of quantized vortices [1,23,18–20,2]. This takes place for sufficiently large rotational velocities. On the contrary, at low rotation, no vortex is detected in the bulk of the condensate. The system can be described by a complex valued wave function minimizing a Gross–Pitaevskii type energy. A vortex corresponds to zeroes of the wave function with phase around it. The density of the condensate is significant in a region which is either a disk or an annulus, and gets exponentially small outside this domain. Vortices are experimentally visible in the bulk of the condensate. A question raised by Lev Pitaevskii is whether for small rotational velocity, when there are no vortices in the bulk, vortices could exist in the low density region. For very large rotational velocities, when bulk vortices are arranged on a triangular lattice, it has been shown [5] that in a simplified model, obtained by formally projecting the Gross–Pitaevskii energy onto the lower Landau level, the vortex distribution extends to infinity. This suggests that in this case, there are many vortices in the low density region. It is then very natural to wonder whether vortices first appear in the bulk or at infinity. It is experimentally and numerically difficult to observe a vortex, which is a zero, in a low density region. Mathematically this could not be achieved through energy estimates or expansion since the contribution of a vortex in a low density region is very small. In this paper, we introduce new ideas to answer Pitaevskii’s question and prove that at low velocity, there are indeed no vortices in the condensate, even in the low density region. Therefore, the first ground state with vortices has its vortices in the bulk. Since a condensate is a trapped object, the geometry of the trap plays a role. An important special case is a radial harmonic trapping potential V (r) = r 2 . The space can then be split into two regions, a region of the form D = {λ0 > V (r)} (for a suitable constant λ0 ), where the wave function is significant and the condensate is mainly located, and a region R2 \ D where the modulus of the wave function is exponentially small [2]. In this latter region, it is very difficult to determine mathematically the contribution of a vortex to the energy. Ignat and Millot [11,12] following ideas from [7], have determined the critical rotational velocity Ωc for the nucleation of the first vortex inside D. This theorem does not describe the behavior in R2 \ D. A natural question is whether for Ω < Ωc , the minimizer of the energy has zeroes in this region, whether there is a smaller critical velocity than Ωc where the minimizer is unique and vortex free. At very high velocity, it has been proved in [5] that vortices exist up to infinity in a reduced model so it seems reasonable that at smaller velocity, vortices may exist in the exponentially small region, far away from the bulk and could arrange themselves on disks or arrays close to infinity. In fact, we prove that this is not the case before Ωc , namely that the minimizer is unique and does not vanish. It means that for a large range of rotational velocities Ω, the minimizer exactly equals the ground state of a condensate at rest. We consider here a two-dimensional setting and define the energy for the complex-valued wave function u, such that R2 |u|2 = 1, as Eε (u) =

1 1 1 |∇u|2 + 2 |u|4 + 2 V (x)|u|2 − Ωx ⊥ · (iu, ∇u) dx, 2 4ε 2ε

(1.1)

R2

where Ω is the angular velocity, x = (x1 , x2 ), x ⊥ = (−x2 , x1 ), ε > 0 is a small parameter, V (x) is the trapping potential and (iu, ∇u) = iu∇u∗ − iu∗ ∇u. The class of potentials includes


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the model case V = x12 + x22 . Then, the critical angular velocity for nucleation of vortices is of order | log ε| (see [11]). An upper bound on the rotational velocity is given by Ω < 1/ε when the confinement breaks down. The condensate is mostly concentrated in the region D := x ∈ R2 : V < λ0

(1.2)

where λ0 is chosen so that

+ λ0 − V (x) dx = 1.

(1.3)

R2

We refer to [2] for more details on how this is derived from the physical experiments. In recent experiments in which a laser beam is superimposed upon the magnetic trap holding the atoms, the trapping potential V (x) is of a different type [21,23,24,3]: V (r) = r 2 + V0 e−r

2 /w 0

(1.4)

.

When the gaussian is expanded around the origin, this leads to a harmonic plus quartic potential [16,23] k V (r) = (1 − b)r 2 + r 4 . 4

(1.5)

If b is small (b < 1 + (3k 2 /4)1/3 ), the domain D given by (1.2) is a disc, while if b > 1 + (3k 2 /4)1/3 , it is an annulus. According to the values of V0 and w0 in the case of (1.4), the domain D can also be a disk or an annulus. In this paper, we consider potentials V including r 2 and of the type (1.4) or (1.5) when the bulk D is a disk. In the case where D is a disk, the potential V is not necessarily required to be increasing. 1.1. Assumptions Throughout this paper, we make the following assumptions about the potential V . First, V is nonnegative and radial,

V ∈ C1,

(1.6)

and there exist c0 > 0, p 2 such that

1 p r V (r) c0 r p c0

if r c0 .

(1.7)

This assumption is easily seen to imply that Eε is bounded below for |Ω| 1ε and that the angular momentum term x ⊥ · (iu, ∇u) is integrable as long as u has finite energy. We will also use (1.7) to obtain decay estimates that justify for example the integration by parts leading to a decoupling of the energy. We fix λ0 ∈ R such that (1.2)–(1.3) hold. Such a λ0 exists due to the growth of V . We further assume that the bulk D is a disk and not an annulus, that is V is such that

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D = BR (0)

for some R > 0

(1.8)

and that there exist δ0 > 0 and a C 1 function R : (−2δ0 , 2δ0 ) → R also denoted Rδ = R(δ), such that R0 = R, 0
0 such that V (r) − λ0 c1 r 2 − R 2 for all r R.

(1.10)

Our assumptions include indeed potentials like r 2 or (1.5) for a disk case, and do not require V to be increasing. 1.2. Main result Our main result is Theorem 1.1. Assume that uε minimizes Eε (·) with rotation Ω, and let ηε denote the minimizer of Eε (·) for Ω = 0. There exists ε0 , ω0 , ω1 > 0 such that if 0 < ε < ε0 and Ω ω0 | log ε| − ω1 log | log ε| then uε = eiα ηε in R2 for some constant α. In the pure quadratic case V = r 2 , Ignat and Millot [11,12] have shown that the bulk of the condensate (that is any domain contained in D) is vortex-free for |Ω| ω0 | log ε|−ω1 log | log ε|, for some ω1 > 0 and the same constant ω0 that we find in Theorem 1.1. They have no information on what happens in R2 \ D. Our theorem proves that vortices do not lie in R2 \ D. They have also shown that there exists δ > 0 such that the ground state has at least one vortex in the bulk if Ω ω0 | log ε| + δ log | log ε|. In this sense, our estimate |Ω| ω0 | log ε| − ω1 log | log ε| captures the sharp leading-order term, and the correct scaling of the next-order term, of the critical velocity for vortex formation. We point out that our arguments also deal with more general potentials. The arguments used in [11] to prove the existence of interior vortices for rotations greater than ω0 | log ε|+δ log | log ε| should extend with few changes to the more general potentials considered here, using results about auxiliary functions that we establish in Section 3 in place of parallel results from [11]. Thus the constant ω0 should also be sharp for these more general potentials. We split the proof into two independent results. The first main result of this paper asserts roughly speaking that symmetry breaking occurs first in the interior of D: if Ω is small enough that there are no vortices in D, then there are no vortices anywhere, and in fact the rotation has absolutely no effect on the ground state. Theorem 1.2. Assume that uε minimizes Eε (·) with rotation Ω, and let ηε denote the minimizer of Eε (·) for Ω = 0. Assume also that Ω C| log ε| for some C.


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There exists ε0 > 0 such that if 0 < ε < ε0 and Ω is subcritical in the sense that 1 |uε | ηε in D1 := x ∈ D: dist(x, ∂D) | log ε|−3/2 2

(1.11)

then uε = eiα ηε in R2 for some constant α. Our second main theorem gives an estimate for the critical value of Ω. The statement of the theorem refers to an auxiliary function f0 : let a(x) = λ0 − V (x), η0 :=

√

a+,

∞ ξ0 (r) =

sη02 (s) ds, r

f0 (r) =

0

if r R,

ξ0 (r)/η02 (r)

if r R.

(1.12)

Theorem 1.3. Let ω0 = 2 f10 ∞ . There exist ω1 > 0 and ε1 > 0 such that if |Ω| ω0 | log ε| − ω1 log | log ε| and 0 < ε < ε1 , then Ω is subcritical in the sense of (1.11), and the conclusion of Theorem 1.2 thus holds. In our proof of Theorem 1.3, as in estimates of the critical rotation in works such as [11] and [4], a main point is to obtain sharp energy lower bounds. In all earlier works that we know of, this is done using the vortex ball construction originally introduced by [13] and [22]. In our proof of Theorem 1.3, we avoid any explicit1 mention of vortex balls by instead appealing to a result from [14], stated here as Lemma 4.1. This makes our argument considerably shorter than those in [4,11] and other references. We point out that the results of [11,12] do not directly imply that Theorem 1.3 holds in the case V = r 2 , although it is possible that this conclusion can be extracted with relatively little effort from arguments in these references. 1.3. Main ideas of the proof The energy minimizers with Ω = 0 provide real solutions to the Euler–Lagrange equations when Ω = 0, Eε (η) = Gε (η), where Gε (η) =

1 1 1 |∇η|2 + 2 |η|4 + 2 V (x)|η|2 dx. 2 4ε 2ε

(1.13)

R2

Our main goal consists in proving that up to the critical velocity of nucleation of bulk vortices, the minimizer of Eε with velocity Ω is in fact equal to ηε . The minimizer ηε of Gε under the L2 constraint of norm 1, is (up to a complex multiplier of modulus one) the unique positive solution of 1 However, the proof of Lemma 4.1, see Lemma 8 in [14], ultimately relies on a vortex ball construction appearing in [15].

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−ηε +

1 1 ηε V (x) + ηε2 = 2 λε ηε 2 ε ε

(1.14)

where ε12 λε is the Lagrange multiplier, which is also necessarily unique. Moreover, λε → λ0 , and ηε2 converges to a + in L2 (D) and uniformly on any compact set of D. We will need some estimates on the decay of ηε at infinity that we prove in Section 2. By a remarkable identity (see Lassoued and Mironescu [17]), for any u, the energy Eε for any Ω splits into two parts, the energy Gε (ηε ) of the density profile and a reduced energy of the complex phase v = u/ηε : Eε (u) = Gε (ηε ) + Fε (v),

(1.15)

where Fε (v) =

2 ηε2 ηε4 2 2 2 ⊥ |∇v| + 2 |v| − 1 − ηε Ωx · (iv, ∇v) dx. 2 4ε

(1.16)

R2

In particular the potential V (x) only appears in Gε . We will recall the proof of (1.15), as well as that of (1.18) below, in Section 3. This kind of splitting of the energy is by now standard in the rigorous analyzes of functionals such as Eε . Next, define ∞ ξε (r) =

sηε2 (s) ds,

(1.17)

r

so that ∇ ⊥ ξε = x ⊥ ηε2 . An integration by parts yields Fε (v) = R2

2 ηε2 4Ωξε ηε4 2 2 dx |∇v| − 2 J v + 2 |v| − 1 2 ηε 4ε

(1.18)

where J v = 12 ∇ × (iv, ∇v) = (ivx1 , vx2 ) is the Jacobian. We recall that the function fε := ξε /ηε2 appearing in Fε is important since it is well known that vortices in the interior of D first appear near where this function attains a local maximum [2,4,11,12]; its importance is also clear from (1.18), since it controls the relative strength of the positive and negative contributions to Fε . The proofs of Theorems 1.2 and 1.3 rest on new bounds for fε in R2 \ D and near ∂D, which in turn rely on decay estimates for ηε . In particular, we show in Lemma 2.4 that fε Cε 2/3 in R2 \ D. The other part of the proof consists essentially of bounds of 2Ω ηε2 fε J v by the positive terms in Fε . Away from the bulk, we use our estimates of fε to find that 2Ωfε J v is bounded pointwise by 12 |∇v|2 . In the bulk, where ηε2 is not too small, we have

2 2 1 1 2 η4 1 ηε |∇v|2 + ε2 |v|2 − 1 ηε2 |∇v|2 + 2 |v|2 − 1 2 2 4ε 4˜ε


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for some ε˜ such that | log ε˜ | = | log ε|(1 + o(1)). We obtain the desired bounds by combining this with a weighted Jacobian estimate mentioned above, Lemma 4.1, which directly implies that

2Ω

χηε2 fε J v Ω

2 fε ∞ | log ε˜ |

χηε2

2 1 1 + small error terms |∇v|2 + 2 |v|2 − 1 2 4˜ε

where χ is a cutoff function supported in the bulk. Note that the leading-order critical rotation ω0 ε ∞ is such that Ω( 2 f log ε˜ | ) ≈ Ω/ω0 | log ε|. The proof of Theorem 1.3 is completed by assembling these ingredients and controlling error terms. The proof of Theorem 1.2 relies on an additional inv gredient, which is that if |v| 12 in an open set U , then J v is extremely close in U to J ( |v| ) = 0. Theorem 1.1 follows immediately from combining Theorems 1.2 and 1.3. An interesting open problem is to see to what extent this analysis continues to hold if the assumption of radial symmetry is dropped. In our arguments, this symmetry is used heavily in our analysis of the behavior of fε away from the bulk, and near the boundary of the bulk. We briefly remark on the assumption (1.7) of quadratic growth. Our proofs show that the absence of vortices in the low density region is a consequence of the fact that the auxiliary function fε = ξε /ηε2 is very small in R2 \ D. The proof of this fact (see Lemma 2.4) can be modified to show that if for example (1.7) holds with p < 2, then fε (r) Cεr 1−p/2 → ∞ as r → ∞. However, in this situation Eε is unbounded below for any Ω = 0. This reflects the fact that a subquadratic trapping potential is not strong enough to contain a rotated condensate. 2. Properties of auxiliary functions In this section we study the real-valued minimizer ηε and the auxiliary functions ξε and fε = ξε /ηε2 defined as ∞ ξε (r) =

sηε2 (s) ds,

fε (r) = ξε (r)/ηε2 (r).

(2.19)

r

Theorem 2.1. Assume that V satisfies (1.6), (1.9). Then for every ε > 0, there exists a unique positive minimizer ηε of Gε in H := u ∈ H 1 R2 : |u|2 V (x) < ∞, |u|2 = 1 . R2

R2

Every minimizer of Gε in H has the form eiα ηε , for α constant. Moreover, ηε is a radial smooth positive function and satisfies (1.14) with |λε − λ0 | Cε| log ε|1/2

(2.20)

where λ0 is defined by (1.3). Finally, recall the notations Rδ from (1.9) and a = λ0 − V , the following estimates are satisfied:

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A. Aftalion et al. / Journal of Functional Analysis 260 (2011) 2387–2406 −1/3 (R−r)

ηε (r) Cε 1/6 ecε √ √ ηε − a + Cε 1/3 a +

in R2 \ D,

in BR−ε1/3 ,

(2.21) (2.22)

∇ηε L∞ (R2 ) Cε −1 ,

(2.23)

ηε (r) 0 for all r ∈ (R−δ0 , Rδ0 ), C η (r) ηε (r) V (r) for all sufficiently large r ε ε

(2.24) (2.25)

if ε < ε0 . Certain parts of the proof follow quite closely arguments given in [4] and in the pure quadratic case in [11]. Note that some arguments in [11] rely strongly on the special shape of the potential and cannot be generalized to other functions. Since V is not necessarily increasing, we have property (2.24) only in the neighborhood of ∂D. Proof. Step 1: existence of minimizers. This follows from standard arguments once we notice that R2 |un |2 V dx is uniformly bounded for any sequence (un ) minimizing Gε , and the set of functions in H satisfying such a uniform bound is precompact with respect to weak convergence in H 1 (R2 ). This last fact is proved by straightforward and well-known arguments, such as are explained in the proof in [11], Lemma 2.1, for V quadratic, the point being that the bound on |u|2 V prevents mass escaping to ∞. Standard theory then implies that any minimizer is smooth. If η is any minimizer, then |η| is as well, since G(|ζ |) G(ζ ) for all ζ . The strong maximum principle then implies that |η| (and hence η) never vanishes, and since G(η) G(|η|), it is easy to see that η/|η| = eiα for some constant α. We henceforth let ηε denote a fixed positive minimizer. Step 2: uniqueness of ηε . This follows from ideas in [9]. Multiplying (1.14) by ηε and integrating by parts we find that με is positive. Suppose that there are two couples (η0 , μ0 ) and (η1 , μ1 ) satisfying (1.14) such that η0 L2 = 1 = η1 L2 and μ0 > μ1 , and define w = ηη10 . This function verify

η02 (w − 1)2 dx = 2 R2

R2

2 η1 − η0 η1 dx = 2

η02 w(w − 1) dx

R2

and 1 −∇ · η02 ∇w + 2 η04 w w 2 − 1 = (μ1 − μ0 )η02 w. ε Multiplying the second equality by (w − 1), integrating by parts and then using the first equality we find 2 1 4 1 2 2 2 2 η0 ∇(w − 1) + 2 η0 w(w − 1) (w + 1) + (μ0 − μ1 )η0 (w − 1) dx = 0. 2 ε R2

The integration by parts is justified in view of (2.21), (2.25), which apply to both η0 and η1 , and the proofs of which do not rely on the uniqueness of the minimizer. Hence w ≡ 1 and μ0 = μ1 .


2395

From (1.6) is easy to see that the compose of ηε with any rotation has the same energy, so it is also a minimizer of Gε . The unicity implies then that ηε is a radial function. Step 3: estimate of λε − λ0 . We next note, following standard arguments, that Gε can be rewritten 2 1 1 1 1 + 2 1 |∇η|2 + 2 η2 − a + + 2 a − η2 dx + 2 λ0 − Gε (η) = a 2 2 4ε 2ε 2ε R2

if η 2 = 1. Let G1ε (η) denote the first integral above. We claim that G1ε (ηε ) C| log ε|. Since ηε is a minimizer, to prove this it suffices to construct a competitor for which G1ε is suitably small. To do this, define

gε (s) :=

s ε √ s

if s ε 2 , if s

and η˜ ε :=

ε2 ,

gε (a + ) .

gε (a + ) L2

Note that 1=

a+

gε2 a + =

a+ −

a + ε 2

a+ a + 1 − 2 1 − Cε 2 . ε

Using this and explicit calculations such as those in [14], Lemma 12, the claim is easily verified. We now multiply (1.14) by ηε , integrate by parts and rewrite, recalling the L2 constraint, to find that 1 1 (λε − λ0 ) = |∇ηε |2 + 2 ηε2 + (V − λ0 ) ηε2 dx (2.26) ε2 ε 1 = |∇ηε |2 + 2 ηε2 − a + + a − ηε2 dx ε 2 1 = |∇ηε |2 + 2 a − ηε2 + ηε2 − a + + ηε2 − a + a + dx (2.27) ε

1 1 4G1ε (ηε ) + 2 ηε2 − a + L2 a + L2 C G1ε (ηε ) + G1ε (ηε ) . ε ε Thus we have proved (2.20). Step 4: estimates of ηε . We claim that ηε2 max(λε − V ) =: A. D

(2.28)

√ To see this define w = 1ε (ηε − A). We have that ηε ∈ L3loc , so after (1.14) w, w ∈ L1loc . Kato’s inequality gives w + sgn+ (w)w. Using (1.14) again we find

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sgn+ (w) 2 ηε ηε − A 3 ε √ 2 2 √ + 3 sgn+ (w) = (εw + A ) ε w + 2εw A w ε3

w +

in D .

Hence we have −w + + (w + )3 0 in D (R2 ) and w ∈ L3loc , so using Lemma 2 in [8], ≡ 0. We remark that the properties of the potential V at the boundary (1.9) implies that the maximum of λε − V is attained at an interior point x0 of D such that dist(x0 , ∂D) > cδ0 . The minimizer being a solution of (1.14) in L∞ , by elliptic regularity we derive that it is a smooth function. w+

Proof of (2.21). We construct a supersolution of (1.14) of the form ⎧√ λ0 − V (x) + 8δ ⎪ ⎪ ⎨ λ −δ−V (x) √ 0 √ +3 δ η(x) ¯ := 6 δ ⎪ ⎪ ⎩ − |x| γe σ

if |x| R−δ , if R−δ |x| Rδ , if Rδ |x|,

where 0 < δ < δ0 is small parameter that will be determined later and γ , σ are chosen such that η¯ ∈ C 1 (R2 ), i.e., √ 8 δ Rδ /σ e γ= 3

and σ =

16δ . |∇V (Rδ )|

A straightforward computation shows that for δ = Cε 1/3 , η¯ is a supersolution of (1.14) and we also have σ = O ε 1/3

−1/3 and γ = O ε 1/6 eε R .

Moreover, with this choice of δ, η¯ 2 > λε − V for every |x| R−δ , so using (2.28) ηε2 (x0 ) A = λε − V (x0 ) < η(x ¯ 0 ). Because ηε and η¯ are going to zero at infinity, if the function ηε − η¯ is positive somewhere in (r0 , ∞), for r0 := |x0 |, then it attains a positive maximum at r˜ ∈ (r0 , ∞), i.e. ηε (˜r ) = η¯ (˜r ) and ηε

(˜r ) < η¯

(˜r ). Given the structure of (1.14) and because η¯ is a supersolution and ηε a solution, if V (˜r ) − λε 0 we ¯ r ). In another hand, if V (˜r ) − λε < 0 then we would have that ηε (˜r ) η(˜ would have η(˜ ¯ r ) < λε − V (˜r ), which for ε small enough, contradicts the definition of η. ¯ Hence ηε (r) η(r) ¯

in (r0 , ∞).

2

Proof of (2.22). Using assumption (1.9), by exactly following [4], one finds that |ηε − Cε 1/3 aε+ , for aε := λε − V = a + λε − λ0 . In view of (2.20), this implies (2.22). 2

aε+ |


2397

Proof of (2.23). For x ∈ R2 define η(y) ˜ = ηε (ε(y − x)) in B2L (x). This function satisfies η˜ = η˜ V ε(y − x) + η˜ 2 − λε =: hε . After estimates (2.21) and (2.22) |hε | C, so using a Hölder estimate for the first derivative of η˜ (see Theorem 8.32 in [10]) we have that ∇ η

˜ L∞ (BL (x)) C for a constant C independent of x and hence the result. 2 Step 5: proof of (2.24). We denote L the elliptic operator obtained by linearizing equation (1.14) L := − +

1 V (x) + 3ηε2 − λε , ε2

and λj , j = 1, 2, . . . , its eigenvalues in R2 . Let μ be the first Dirichlet eigenvalue of L in the half space Ω = {x1 > 0} and ψ the corresponding eigenfunction (which exists because of the compact embedding of H in L2 ). Since V and ηε are radial, is clear that the odd extension of ψ to R2 is an eigenfunction for L in R2 with corresponding eigenvalue μ = λj . Note that j 2 because the odd extension change sign in R2 . We have that Lηε = 2ηε4 > 0 and ηε > 0. Using the maximum principle due to Berestycki, Nirenberg and Varadhan [6], this implies that the first eigenvalue of L is positive. We will prove that if (2.24) does not hold, then μ < 0, which contradicts the fact that λ1 > 0. Assume that ηε (r) > 0 at some r ∈ (R−δ0 , Rδ0 ). Then there exists α < r < β such that

ηε (α) = ηε (β) = 0 and ηε > 0 in (α, β). If α R−2δ0 , then ηε is increasing on (R−2δ0 , R√ −δ0 ), so that ηε (R−2δ0 ) ηε (R−δ0 ). This is impossible for all sufficiently small ε, since ηε → a + uniformly for r < R−ε1/3 , by (2.22), and a + (R−2δ0 ) > a + (R−δ0 ). Thus α R−2δ0 . The same argument, but using (2.21) instead of (2.22), shows that β R2δ0 . Now let D := {x ∈ R2 : x1 > 0, α < |x| < β}. Then ∂ηε > 0 in D, ∂x1

∂ηε = 0 in ∂D ∂x1

∂ηε and L ∂x1

=−

∂V ηε 0 in D. ∂x1

The last inequality come from the differentiation of (1.14) and hypothesis (1.9), which implies that ∂V /∂R > 0 for r ∈ (R−2δ0 , R2δ0 ). Using the monotonicity of Dirichlet eigenvalues with respect to the domain, this implies that μ < 0. Step 6: proof of (2.25). For any r R, define a function η˜ : (r, ∞) → R by p+2 2α p+2 s 2 −r 2 η(s) ˜ := ηε (r) exp − p+2

where c0 and p are the constants in (1.7). It follows from (2.20) and (1.7) that if s r and r is sufficiently large, then V (s) − λε + η˜ 2 (s) V (s) c0 s p , so that if r is sufficiently large, then −η˜ +

1 c0 V − λε + η˜ 2 η˜ −η(s) ˜ + 2 s p η˜ = 2 ε ε

−α 2 +

p p c0 p 2 −1 η. + 1 s + α s ˜ 2 ε2

2398

A. Aftalion et al. / Journal of Functional Analysis 260 (2011) 2387–2406 1/2

Choosing α = (2c0ε) , it follows that η˜ is a subsolution of (1.14) in (r, ∞) if r is sufficiently large. For such r, noting that η(r) ˜ = ηε (r), we can argue as in the proof of (2.21) to deduce that ηε − η˜ is nonnegative in (r, ∞). ˜ ηε (s) for s r, we again use (1.7) to conclude that Then since η(r) ˜ = ηε (r) and η(s) ηε (r) η˜ (r) = −

√ c0 (2c0 )1/2 p r 2 ηε (r) − 2 V (r)ηε (r) ε ε √

c

˜ we obtain for sufficiently large r. On the other hand, by choosing α = 2ε0 in the definition of η, a decreasing supersolution (still denoted η) ˜ such that η(r) ˜ = ηε (r). A similar application of the maximum principle shows that ηε is bounded above by (the new) η˜ on (r, ∞), and in particular this implies that ηε (r) 0. These facts combine to establish (2.25). 2 We next prove: Lemma 2.2. Assume that V satisfies (1.6) and (1.9) and the quadratic growth condition (1.10). Let ηε be the positive minimizer found in Theorem 2.1. Let fε (r) := ξε (r)/ηε2 (r), where ξε was defined in (1.17). Then there exists a constant C independent of ε ∈ (0, ε1 ] such that fε |x|

C dist(x, ∂D) + Cε 2/3

if x ∈ D,

Cε 2/3

if not.

(2.29)

In addition, for all sufficiently small ε,

∇ξε ∞ C

(2.30)

fε − f0 ∞ Cε 1/3 .

(2.31)

and

Proof. For every s r Rδ (where 0 < δ δ0 will be chosen later), we define η(s) ˜ = ηε (r)e−μδ (s

2 −r 2 )/2

and

μ2δ =

c1 (Rδ2 − R 2 ) + (λε − λ0 ) Rδ2 ε 2

.

(2.32)

Using (1.10), where the constant c1 is defined, and arguing as in the proof of (2.25), we find that η˜ − ηε is nonnegative in (r, ∞). We use the previous estimate and the definition of ξε to compute 1 fε (r) = 2 ηε (r)

∞

∞ sηε2 (s) ds

r

e−μδ (s

2 −r 2 )

s ds =

r η (r)

1 2μδ

for r Rδ .

The definition of fε implies that fε (r) = −r − 2fε (r) ηεε (r) , and from the monotonicity (2.24) of ηε , we infer that fε (r) −r in (R−δ0 , Rδ0 ). Thus for any R−δ0 r Rδ ,


fε (r)

2399

Rδ2 − r 2 1 + . 2 2μδ

We now fix δ = ε 2/3 , and we conclude from (1.9) and (2.20) that (2.29) holds as long as r R−δ0 . For 0 r R−δ0 , we write 1 fε (r) = ηε (r)2

R−δ0

sηε2 (s) ds + r

ηε2 (R−δ0 ) f (R−δ0 ). ηε2 (r)

From (2.22) and (1.9), we see that if 0 r s R−ε1/3 , then ηε2 (s) (1 + Cε 1/3 )2 a + (s) C ηε2 (r) (1 − Cε 1/3 )2 a + (r)

for sufficiently small ε,

(2.33)

and by using the and the fact that fε (R−δ0 ) Cε 2/3 + Cδ0 , one easily deduces that (2.29) holds for r ∈ [0, R−δ0 ). Next, the definition of ξε implies that |∇ξε (x)| = |x|ηε2 (x), so that (2.30) follows from (2.28) and (2.21). For r R−ε1/3 , we see from (2.29) that |fε (r) − f0 (r)| Cε 1/3 + |f0 (r)|. This is trivially bounded by Cε 1/3 if r R. If R−ε−1/3 r R then (1.9) implies that c(R − r) a(r) C(R − r), and thus f0 (r) = f0 (r)

C r −R

R s(R − s) ds C(R − r) Cε 1/3 . r

For 0 r R−ε1/3 we write

fε (r) − f0 (r) =

1 ηε2 (r)

+

R−ε1/3

sηε2 (s) ds

1 − a(r)

r

R−ε1/3

sa(s) ds r

ηε2 (R−ε1/3 ) fε (R−ε1/3 ) − ηε2 (r)

a(R−ε1/3 ) f0 (R−ε1/3 ) a(r)

= I + II − III. Using (2.33) and our earlier estimates of fε , f0 for r R−ε1/3 , we see that |II| Cfε (R−ε1/3 ) Cε 1/3

and |III| Cf0 (R−ε1/3 ) Cε 1/3 .

We further decompose the remaining term as I=

1 1 − ηε2 (r) a(r)

R−ε1/3 sηε2 (s) ds r

1 + a(r)

R−ε1/3

r

s ηε2 (s) − a(s) ds.

2400


Using (2.22), it follows that R−ε1/3

|I | Cε

1/3 r

Due to (2.24), ηε2 (s) ηε2 (r)

ηε2 (s) ηε2 (r)

η2 (s) ds + Cε 1/3 s ε2 ηε (r)

R−ε1/3

s

a(s) ds. a(r)

r

1 if R−δ0 r s R−ε1/3 . And if 0 r R−δ0 then ηε2 (r) C −1 and so

C. Thus the first integral is bounded by Cε 1/3 . The second integral is similarly estimated, using (1.9) in place of (2.24). 2 Remark 2.3. In the case of a potential V for which (1.8) fails, so that for example D has the form BR \ BR , one expects that instead of being small, fε is large, namely, fε cec/ε in the interior of BR . This is related to the formation at very low rotations of a giant vortex in the interior of BR . The arguments used to prove Lemma 2.4 show in this situation that if V grows quadratically in the complement of BR , as in (1.10), then fε is very small in R2 \ BR . This suggests that at low rotations there should be no vortices in R2 \ BR , but this cannot be deduced from the arguments we use to prove Theorems 1.2 and 1.3. The last lemma in this section examines the case when V has subquadratic growth and fε is also large so that in principle vortices could exist in the low density region. Lemma 2.4. Assume that V satisfies (1.6), (1.9) and there exist c2 > 0 and p < 2 such that V (r) c2 r p + 1 for all r R.

(2.34)

Then fε (x) → +∞ as |x| → ∞. Note that with these assumptions on V , there is a sequence of functions ζα in H such that infα Gε (ζα ) = −∞. Physically this happens because the centrifugal force due to rotation is bigger than the subquadratic trapping potential. This indicates that, although one can prove that in this situation, fε → ∞ as r → ∞ (compare Lemma 2.4), this is not expected to give any information about the physical behavior of condensates. Proof. Let q > 2. For every r max{1, R}, we claim that ηε (s) ηε (r)e−νε,r (s

q −r q )/q

for all s r. Where νε,r is the positive root of the polynomial ν 2 − small satisfy

(2.35) q rq ν

−

c , ε 2 r 2q−2−p

which for ε

νε,r < C ε −1 r −β with β = q − 1 − p/2. Indeed, the right-hand side of (2.35) is a subsolution in (r, ∞) of (1.14) while ηε is a solution. Boths functions are going to zero at infinity and they are equal at s = r, so the result come arguing as in the proof of (2.21).


2401

We use the previous estimates and the definition of ξε to compute

fε (r) =

1 ξε (r) = ηε2 (r) ηε2 (r)

and hence the result.

∞

∞ sηε2 (s) ds

r

e−νr (s

q −r q )

r

s ds

r 2−q > Cεr 1−p/2 νr

2

3. Splitting the energy In this section we recall the proofs of (1.15) and (1.18). For U ⊂ R2 , we will write Eε (w; U ) etc to denote the integrals over U of the energy density appearing in the definition of Eε (u) = Eε (u; R2 ), and similarly Gε (·, U ), Fε (·, U ). Note that v = u/ηε is well defined since ηε > 0. Since ηε satisfies (1.14), we multiply it by ηε (1 − |v|2 ) and integrate over a ball Br to find that

2 1 λε 2 1 1 |v| − 1 − ηε2 + 2 ηε2 V (x) + ηε2 + |∇ηε |2 = 2 |u| − ηε2 . 4 2 2ε ε

Br

Br

Note that the Lagrange multiplier term tends to 0 as r → ∞, since both the L2 norms of u and ηε are 1. Moreover, Eε (vηε ; Br ) = Gε (ηε ; Br ) + Fε (v; Br ) +

1 1 |∇ηε |2 |v|2 − 1 + ηε ∇ηε · ∇|v|2 2 2

Br

2 1 1 1 1 1 − 2 ηε4 1 − |v|2 + 2 η4 |v|4 + 2 V (x)η2 |v|2 − 2 η4 − 2 V (x)η2 . 4ε 4ε 2ε 4ε 2ε We integrate by parts to obtain

1 ηε ∇ηε · ∇|v|2 = − 2

Br

Br

1 2 |v| ηε2 + 4

1 2 |v| ηε ν · ∇η. 2

∂Br

We use (2.25) to estimate C 1 2 1 2 2√ C −1/2 |v| η V (r) η ν · ∇η |v| V = V |u|2 . ε ε 2 2 ε ∂Br

∂Br

∂Br

Since R2 V |u|2 < ∞, we can easily find a sequence rk → ∞ such that the above integral tends to 0. Combining the above and letting rk → ∞ along this sequence, we obtain (1.15). The only property of V that the above argument used (implicitly) was (1.7), which will be used in the proof of (2.25).

2402


The integration by parts that leads to (1.18) is justified in a similar fashion. One must estimate boundary terms of the form ∂Br ξ ν · (iv, ∇v). To do this we note that ξ ν · (iv, ∇v) = fε (r)ηε2 (iv, ∇v) = fε (r)(iu, ∇u) fε ∞ |u|2 + |∇u|2 . We prove in (2.29) that fε is bounded as long as V satisfies (1.10) (in fact we show that fε Cε 2/3 for large r) and since u ∈ H 1 (R2 ), we can again find a sequence rk → ∞ such that the boundary terms vanish. Note also that the fact that fε ∈ L∞ , or equivalently that |ξε | Cηε2 , implies that the term ξε J v appearing in (1.18) is integrable on R2 for v = u/ηε , whenever u has finite energy. 4. Proofs of Theorems 1.2 and 1.3 In this section we use the estimates we have already established to complete the proofs of our main theorems. Proof of Theorem 1.2. We assume that uε minimizes Eε and that Ω C| log ε| is such that (1.11) holds. Let χ be a smooth function such that χ ≡ 1 in {x ∈ D: dist(x, ∂D) 2| log ε|−3/2 }, and with support in D1 . We also assume that ∇χ ∞ 2| log ε|3/2 . Let v = uε /ηε , so that Eε (u) = Gε (ηε ) + Fε (v) = Eε (ηε ) + Fε (v). Thus Fε (v) 0. We write Fε (v) = A1 − A2 + B where A1 =

2 ηε2 ηε4 2 2 dx, |∇v| + 2 |v| − 1 χ 2 4ε

A2 = 2Ω

R2

χξε J v dx R2

and B=

2 2 η4 η dx. (1 − χ) ε |∇v|2 − 4Ωfε J v + ε2 |v|2 − 1 2 4ε

R2

It follows directly from our estimates on fε that 0 < fε C(ε 2/3 + | log ε|−3/2 ) in the support of 1 − χ , for small enough ε. Since Ω C| log ε|, it follows that Ωfε 14 for all sufficiently small ε and (recalling that |J v| 12 |∇v|2 ) we deduce that 1 |∇v|2 − 4Ωfε J v |∇v|2 2 in the support of 1 − χ . It follows immediately that

2 2 η η4 dx 0 B (1 − χ) ε |∇v|2 + ε2 |v|2 − 1 4 4ε R2

and hence that B = 0 if and only if v is a constant of modulus 1 in the support of 1 − χ .

(4.36)


2403

Since Fε (v) 0, it is clear that A1 + B A2 . Next, define ε˜ = ε/(infD1 ηε ), so that (in view of (2.22) and the definition of D1 ) 1 ηε2 ε˜ 2 ε2

ε˜ Cε| log ε|3/4 ,

in D1 .

Then (4.36) and (2.22) imply that, D1

−2 2 1 1 |∇v|2 + 2 |v|2 − 1 inf ηε (A1 + 2B) C| log ε|3/2 A2 . 2 4˜ε D1

(4.37)

v = w 1 + iw 2 . From (1.11) we see that |v| 12 in D1 , and hence it is To continue, let w = |v| clear that w ∈ H 1 (D1 ), and |w|2 ≡ 1. It follows that J w = 0; we will recall a standard proof of this fact in a moment. Thus A2 = 2Ω χξε (J v − J w) dx = 2Ω ∇ ⊥ (χξε ) · (iv, ∇v) − (iw, ∇w) dx.

D1

D1

If we write v = ρeiφ in D1 , then a calculation shows that (iv, ∇v) = ρ 2 ∇φ,

(iw, ∇w) = ∇φ.

From the latter fact we see that J w = 12 ∇ × (iw, ∇w) = 0, as we asserted above. Also, from this and the fact that ρ 12 in D1 we estimate 2 (iv, ∇v) − (iw, ∇w) = |ρ − 1| |ρ∇φ| 2|v|2 − 1|∇v|. ρ

Using (4.37), we deduce that A2 2Ω ∇(χξε )∞

D1

2 ε˜ 1 2 dx |∇v|2 + |v| − 1 2 2˜ε

CΩ ∇(χξε )∞ ε| log ε|9/4 A2 . One checks easily from the definitions and from (2.30) that ∇(χξε )

∞

∇χ ∞ ξε ∞ + ∇ξε ∞ C| log ε|3/2

(4.38)

so we conclude that A2 Cε| log ε|15/4 A2 12 A2 for all sufficiently small ε. We know from (4.37) that A2 0, and it follows that A2 = 0, and hence (again appealing to (4.37)) that A1 = B = 0. Thus ∇v L2 = 1 − |v|2 L2 = 0, and so v is a constant of modulus 1 as required. 2 The proof of Theorem 1.3 will use the following result, which is Lemma 8 in [14].

2404


Lemma 4.1. There exists a universal constant C > 0 such that for any κ ∈ (1, 2), open set U ⊂ R2 and u ∈ H 1 (U ; R2 ), and ε ∈ (0, 1), φJ u κ |φ| eε (u) | log ε| U

|φ| + 1 eε (u) dx + Cε (κ−1)/50 1 + φ W 1,∞ φ ∞ + 1 +

(4.39)

supp φ

for all φ ∈ Cc0,1 (U ). Here eε (u) = 12 |∇u|2 +

1 (|u|2 4ε 2

− 1)2 .

The lemma as stated in [14] does not explicitly specify the exponent (κ − 1)/50 appearing on the right-hand side of (4.39). By inspection of the proof, however, one sees that this exponent can be taken to have the form 12 α, where α = (κ − 1)/12κ as in Theorem 2.1 of [15]. Proof of Theorem 1.3. We continue to use notation from the proof of Theorem 1.2, such as A1 , A2 , B, ε˜ , and so on. We first invoke the lemma, with ε˜ in place of ε and χξε in place of φ, and with κ > 1 to be chosen. This yields |A2 | 2Ωκ

χξε R2

eε˜ (v) dx + E, | log ε˜ |

where E denotes the error terms in (4.39). We note that for all sufficiently small ε > 0, the error term satisfies the bound E Cε β (1 + |A2 |), for β = (κ − 1)/100, for all sufficiently small ε. This is a consequence of (4.37) and the estimates

χξε W 1,∞ C| log ε|3/2 ,

χξε L∞ C.

These in turn follow from (4.38) together with (2.30). Now the choice of ε˜ implies that eε˜ (v) 12 |∇v|2 + ξε = fε ηε2 , we obtain

fε ∞ 1 − Cε β |A2 | 2Ωκ | log ε˜ | = 2Ωκ

χ

ηε2 (|v|2 4ε 2

− 1)2 in D1 , and recalling that

2 η4 ηε2 + Cε β |∇v|2 + ε2 |v|2 − 1 2 4ε

fε ∞ A1 + Cε β . | log ε˜ |

We know from (2.31) that fε ∞ (1 + Cε 1/3 ) f0 ∞ (1 + Cε β ) f0 ∞ , and from the choice of ε˜ , for any K > 0 there exists ε0 > 0 such that | log ε˜ | (| log ε| − log | log ε|)(1 + Kε β ) if 0 < ε < ε0 . Thus |A2 | Ω

2 f ∞ κA1 + Cε β | log ε| − log | log ε|


2405

for all sufficiently small ε. Assume that Ω 2 f1 ∞ (| log ε| − (c1 + 1) log | log ε|), for c1 to be chosen below. Then log | log ε| log | log ε| |A2 | 1 − c1 κA1 + Cε β 1 − c1 κA1 + Cε β . (4.40) | log ε| − log | log ε| | log ε| | log ε| We now take κ := 1 + c1 log| log ε| , so that β = (κ − 1)/100 = B A2 and that B 0, clearly A1 A2 , so we deduce that

c12

log | log ε| | log ε|

2

c1 log | log ε| 100 | log ε| .

Recalling that A1 +

A1 Cε β = C| log ε|−c1 /100 .

If c1 = 400 then we conclude that A1 C| log ε|−2 . Then (4.40) implies that A2 C| log ε|−2 , and it follows that B C| log ε|−2 . In view of (4.37), this implies that |∇v|2 + D1

The estimate ∇v ∞

C ε

2 1 2 |v| − 1 C| log ε|−2 . 2 4ε

(4.41)

(see (2.23)) and (4.41) are easily seen to imply that |v| 1 − C| log ε|−1

in D1

for all sufficiently small ε. Thus Ω is subcritical for small enough ε.

(4.42) 2

References [1] J.R. Abo-Shaeer, C. Raman, J.M. Vogels, W. Ketterle, Observation of vortex lattices in Bose–Einstein condensates, Science 292 (2001) 476. [2] A. Aftalion, Vortices in Bose Einstein Condensates, Progr. Nonlinear Differential Equations Appl., vol. 67, Birkhäuser Boston, Inc., Boston, MA, 2006. [3] A. Aftalion, P. Mason, Rotation of a Bose Einstein condensate held under a toroidal trap, Phys. Rev. A 81 (2010) 023607. [4] A. Aftalion, S. Alama, Lia Bronsard, Giant vortex and the breakdown of strong pinning in a rotating Bose–Einstein condensate, Arch. Ration. Mech. Anal. 178 (2005) 247–286. [5] A. Aftalion, X. Blanc, F. Nier, Lowest Landau level for Bose Einstein condensates and Bargmann transform, J. Funct. Anal. 241 (2006) 661–702. [6] H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47–92. [7] F. Bethuel, H. Brezis, F. Hélein, Ginzburg–Landau Vortices, Progr. Nonlinear Differential Equations Appl., vol. 13, Birkhäuser Boston, Inc., Boston, MA, 1994. [8] H. Brezis, Semilinear equations in RN without conditions at infinity, Appl. Math. Optim. 12 (1984) 271–282. [9] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986) 55–64. [10] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, ISBN 978-3540-41160-4, 1998. [11] R. Ignat, V. Millot, The critical velocity for vortex existence in a two-dimensional rotating Bose–Einstein condensate, J. Funct. Anal. 233 (2006) 260–306. [12] R. Ignat, V. Millot, Energy expansion and vortex location for a two-dimensional rotating Bose–Einstein condensate, Rev. Math. Phys. 18 (2006) 119–162. [13] R.L. Jerrard, Lower bounds for generalized Ginzburg–Landau functionals, SIAM J. Math. Anal. 30 (1999) 721–746.

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[14] R.L. Jerrard, Local minimizers with vortex filaments for a Gross–Pitaevsky functional, ESAIM Control Optim. Calc. Var. 13 (2007) 35–71. [15] R. Jerrard, H.M. Soner, The Jacobian and the Ginzburg–Landau energy, Calc. Var. Partial Differential Equations 14 (2002) 151–191. [16] K. Kasamatsu, M. Tsubota, M. Ueda, Giant hole and circular superflow in a fast rotating Bose–Einstein condensate, Phys. Rev. B 66 (2002) 053606. [17] L. Lassoued, P. Mironescu, Ginzburg–Landau type energy with discontinuous constraint, J. Anal. Math. 77 (1999) 1–26. [18] K.W. Madison, F. Chevy, V. Bretin, J. Dalibard, Stationary states of a rotating Bose–Einstein condensate: Routes to vortex nucleation, Phys. Rev. Lett. 86 (2001) 4443–4446. [19] C.J. Pethick, H. Smith, Bose Einstein Condensation in Dilute Gases, Cambridge University Press, 2002. [20] L. Pitaevskii, S. Stringari, Bose Einstein Condensation, Internat. Ser. Monogr. Phys., vol. 116, Oxford Science Publications, 2003. [21] C. Ryu, et al., Observation of persistent flow of a Bose Einstein condensate in a toroidal trap, Phys. Rev. Lett. 99 (2007) 260401. [22] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998) 379–403; J. Funct. Anal. 171 (2000) 233 (erratum). [23] S. Stock, V. Bretin, F. Chevy, J. Dalibard, Shape oscillation of a rotating Bose–Einstein condensate, Europhys. Lett. 65 (2004) 594. [24] C.N. Weiler, et al., Spontaneous vortices in the formation of Bose Einstein condensates, Nature 455 (2008) 948.


Minimal and maximal operator spaces and operator systems in entanglement theory Nathaniel Johnston a,∗ , David W. Kribs a,b , Vern I. Paulsen c , Rajesh Pereira a a Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada b Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada c Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA

Received 2 September 2010; accepted 6 October 2010 Available online 15 October 2010 Communicated by D. Voiculescu

Abstract We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps. © 2010 Elsevier Inc. All rights reserved. Keywords: Operator space; Operator system; Quantum information theory; Entanglement


E-mail addresses: [email protected] (N. Johnston), [email protected] (D.W. Kribs), [email protected] (V.I. Paulsen), [email protected] (R. Pereira). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.003

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N. Johnston et al. / Journal of Functional Analysis 260 (2011) 2407–2423

1. Introduction A primary goal of this paper is to formally link central areas of study in operator theory and quantum information theory. More specifically, we connect recent investigations in operator space and operator system theory [18,21] on the one hand and the theory of entanglement [9,1] on the other. As benefits of this combined perspective, we obtain new results and new elementary proofs in both areas. We give further details below before proceeding. Given a (classical description of a) quantum state ρ, one of the most basic open questions in quantum information theory asks for an operational criterion for determining whether ρ is separable or entangled. Much progress has been made on this front over the past two decades. For instance, a revealing connection between the separability problem and operator theory was established in [7], where it was shown that ρ is separable if and only if it remains positive under the application of any positive map to one half of the state. Another more recent approach characterizes separability via maps that are contractive in the trace norm on Hermitian operators [8]. In this work we show that these two approaches to the separability problem can be seen as arising from the theory of minimal and maximal operator systems and operator spaces, respectively. Additionally, this work can be seen as demonstrating how to rephrase certain positivity questions that are relevant in quantum information theory in terms of norms that are relevant in operator theory instead. For example, instead of using positive maps to detect separability of quantum states, we can construct a natural operator system into which positive maps are completely positive. Then the completely bounded norm on that operator system serves as a tool for detecting separability of quantum states as well. A natural generalization of the characterization of separable states in terms of positive linear maps was implicit in [25] and proved in [22] – a state has Schmidt number no greater than k if and only if it remains positive under the application of any k-positive map to one half of the state. Recently, a further connection was made between operator theory and quantum information: a map is completely positive on what is known as the maximal (resp. minimal) operator system on Mn , the space of n × n complex matrices, if and only if it is a positive (resp. entanglementbreaking [10]) map [19]. Thus, the maps that serve to detect quantum entanglement are the completely positive maps on the maximal operator system on Mn . Similarly, completely positive maps on “k-super maximal” and “k-super minimal” operator systems on Mn [26] have been studied and shown to be the same as k-positive and k-partially entanglement breaking maps [2], respectively. We will reprove these statements via an elementary proof that shows that the cones of positive elements that define the k-super maximal (resp. k-super minimal) operator systems are exactly the cones of (unnormalized) states with Schmidt number at most k (resp. k-block positive operators). Analogous to the minimal and maximal operator systems, there are minimal and maximal operator spaces (and appropriate k-minimal and k-maximal generalizations). We will show the norms that define the k-minimal operator spaces on Mn coincide with a family of norms that have recently been studied in quantum information theory [13,12,11,17,3–5] for their applications to the problems of detecting k-positive linear maps and NPPT bound entangled states. Furthermore, we will connect the dual of a version of the completely bounded minimal operator space norm to the separability problem and extend recent results about how trace-contractive maps can be used to detect entanglement. We will see that the maps that serve to detect quantum entanglement via norms are roughly the completely contractive maps on the minimal operator space on Mn . The natural generalization to norms that detect states with Schmidt number k is proved via a


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stabilization result for the completely bounded norm from Mr to the k-minimal operator space (or system) of Mn . In Section 2 we introduce the reader to the various relevant notions from quantum information theory such as separability and Schmidt rank. In Section 3 we introduce (abstract) operator spaces and the k-minimal and k-maximal operator space structures, and investigate their relationship with norms that have been used in quantum information theory. In Section 4 we give a similar treatment to abstract operator systems and the k-super minimal and k-super maximal operator system structures. We then investigate some norms on the k-super minimal operator system structures in Section 5. We close in Section 6 by considering the completely bounded version of some of the norms that have been presented and establish a relationship with the Schmidt number of quantum states. 2. Quantum information theory preliminaries Given a vector space V , we will use Mm,p (V ) to denote the space of m × p matrices with elements from V . For brevity we will write Mm (V ) := Mm,m (V ) and Mm := Mm (C). It will occasionally be convenient to use tensor product notation and identify Mm ⊗ V ∼ = Mm (V ) in the standard way, especially when V = Mn . We will make use of bra-ket notation from quantum mechanics as follows: we will use “kets” |v ∈ Cn to represent unit (column) vectors and “bras” v| := |v∗ to represent the dual (row) vectors, where (·)∗ represents the conjugate transpose. Unit vectors represent pure quantum states (or more specifically, the projection |vv| onto the vector |v represents a pure quantum state) and thus we will sometimes refer to unit vectors as states. Mixed quantum states are represented by density operators ρ ∈ Mm ⊗ Mn that are positive semidefinite with Tr(ρ) = 1. A state |v ∈ Cm ⊗ Cn is called separable if there exist |v1 ∈ Cm , |v2 ∈ Cn such that |v = |v1 ⊗ |v2 ; otherwise it is said to be entangled. The Schmidt rank of a state |v, denoted SR(|v), is the least number separable states |vi needed to write |v = i αi |vi , where αi are some (real) coefficients. The analogue of Schmidt rank for a bipartite mixed state ρ ∈ Mm ⊗ Mn is Schmidt number [25], denoted SN(ρ), which is defined to be the least integer k such that ρ can be written in the form ρ = i pi |vi vi | with {pi } forming a probability distribution and SR(|vi ) k for all i. An operator X = X ∗ ∈ Mm ⊗ Mn is said to be k-block positive (or a k-entanglement witness) if v|X|v 0 for all vectors |v with SR(|v) k. In the extreme case when k = min{m, n}, we see that the k-block positive operators are exactly the positive semidefinite operators (since SR(|v) min{m, n} for all vectors |v), and for smaller k the set of k-block positive operators is strictly larger. In [13,12], a family of operator norms that have several connections in quantum information theory was investigated. Arising from the Schmidt rank of bipartite pure states, they are defined for operators X ∈ Mm ⊗ Mn as follows XS(k) = sup v|X|w: SR |v , SR |w k . |v,|w

(1)

These norms were shown to be useful for determining whether or not an operator is k-block positive, and also have applications to the problem of determining whether or not there exist bound entangled non-positive partial transpose states [13,17]. The problem of computing these norms was investigated in [12].

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The completely bounded norm of a linear map Φ : Mr → Mn is defined to be Φcb := sup (idm ⊗ Φ)(X): X ∈ Mm (Mr ) with X 1 . m1

It was shown by Smith [24] (and independently later by Kitaev [15] from the dual perspective) that it suffices to fix m = n so that Φcb = idn ⊗ Φ. We will see in Section 6 a connection between the norms idk ⊗ Φ for 1 k n and the norms (1). 3. k-Minimal and k-maximal operator spaces We will now present (abstract) operator spaces and the k-minimal and k-maximal operator space structures. An operator space is a vector space V together with a family of L∞ matrix norms · Mm (V ) on Mm (V ) that make V into a matrix normed space. That is, we require that if A = (aij ), B = (bij ) ∈ Mp,m and X = (xij ) ∈ Mm (V ) then A · X · B ∗ Mp (V ) AXMm (V ) B, where ∗

A · X · B :=

m

aik xk bj ∈ Mp (V )

k,=1

and A, B represent the operator norm on Mp,m . The L∞ requirement is that X ⊕ Y Mm+p (V ) = max{XMm (V ) , Y Mp (V ) } for all X ∈ Mm (V ), Y ∈ Mp (V ). When the particular operator space structure (i.e., the family of L∞ matrix norms) on V is not important, we will denote the operator space simply by V . We will use Mn itself to denote the “standard” operator space structure on Mn that is obtained by associating Mm (Mn ) with Mmn in the natural way and using the operator norm. For a more detailed introduction to abstract operator spaces, the interested reader is directed to [18, Chapter 13]. Given an operator space V and a natural number k, one can define a new family of norms on Mm (V ) that coincide with the matrix norms on Mm (V ) for 1 m k and are minimal (or maximal) for m > k. We will use MIN k (V ) and MAX k (V ) to denote what are called the k-minimal operator space of V and the k-maximal operator space of V , respectively. For X ∈ Mm (V ) we will use XMm (MIN k (V )) and XMm (MAX k (V )) to denote the norms that define the k-minimal and k-maximal operator spaces of V , respectively. For X ∈ Mm (V ) one can define the k-minimal and k-maximal operator space norms via XMm (MIN k (V )) := sup Φ(Xij ) : Φ : V → Mk with Φcb 1

(2)

XMm (MAX k (V )) := sup Φ(Xij ) : Φ : V → B(H) with idk ⊗ Φ 1 .

(3)

and

Indeed, the names of these operator space structures come from the facts that if O(V ) is any operator space structure on V such that · Mm (V ) = · Mm (O(V )) for 1 m k then · Mm (MIN k (V )) | · |Mm (O(V )) · Mm (MAX k (V )) for all m > k. In the k = 1 case, these operator spaces are exactly the minimal and maximal operator space structures that are fundamental in operator space theory [18, Chapter 14]. The interested reader is directed to [16] and the references therein for further properties of MIN k (V ) and MAX k (V ) when k 2.


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One of the primary reasons for our interest in the k-minimal operator spaces is the following result, which says that the k-minimal norm on Mm (Mn ) is exactly equal to the S(k)-norm (1) from quantum information theory. Theorem 1. Let X ∈ Mm (Mn ). Then XMm (MIN k (Mn )) = XS(k) . Proof. A fundamental result about completely bounded maps says (see [14, Theorem 19], for example) that any completely bounded map Φ : Mn → Mk has a representation of the form Φ(Y ) =

nk

Ai Y Bi∗

with Ai , Bi ∈ Mk,n

i=1

nk nk

∗ ∗ and Ai Ai Bi Bi = Φ2cb . i=1

(4)

i=1

By using the fact that Φ is completely contractive (so Φcb = 1) and a rescaling of the operators {Ai } and {Bi } we have nk

∗ XMm (MIN k (Mn )) = sup (Im ⊗ Ai )X Im ⊗ Bi : i=1

nk nk

∗ ∗ Ai Ai = Bi Bi = 1 , i=1

i=1

where the supremum is taken over all families of operators {Ai }, {Bi } ⊂ Mk,n satisfying the normalization condition. Now define αij |aij := A∗i |j and βij |bij := Bi∗ |j , and let |v = k k m k j =1 γj |cj ⊗ |j , |w = j =1 δj |dj ⊗ |j ∈ C ⊗ C be arbitrary unit vectors. Then simple algebra reveals k

∗ νi |vi := Im ⊗ Ai |v = αij γj |cj ⊗ |aij j =1

and k

μi |wi := Im ⊗ Bi∗ |w = βij δj |dj ⊗ |bij . j =1

In particular, SR(|vi ), SR(|wi ) k for all i. Furthermore, by the normalization condition on {Ai } and {Bi } we have that v| Im ⊗

nk

Ai A∗i

i=1

|v =

nk

νi2

1 and w| Im ⊗

i=1

nk

Bi Bi∗

|w =

i=1

nk

μ2i 1.

(5)

i=1

Thus we can write nk nk nk

∗ νi μi vi |X|wi νi μi vi |X|wi . v|(Im ⊗ Ai )(X) Im ⊗ Bi |w = i=1

i=1

i=1

(6)

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The normalization condition (5) and the Cauchy–Schwarz inequality tell us that there is a particular i such that the sum (6) |vi |X|wi |. Taking the supremum over all vectors |v and |w gives the “” inequality. The “” inequality can be seen by noting that if we have two vectors in their Schmidt decom positions |v = ki=1 αi |ci ⊗ |ai and |w = ki=1 βi |di ⊗ |bi , then we can define operators A, B ∈ Mk,n by setting their ith row in the standard basis to be ai | and bi |, respectively. Because the orthonormal sets, A = B = 1. Additionally, if we define rows of A and B form k−1

= α |c ⊗ |i and |w β |d |v = k−1 i=0 i i i=0 i i ⊗ |i, then (Im ⊗ A)(X) Im ⊗ B ∗ v (Im ⊗ A)(X) Im ⊗ B ∗ w = v|X|w. Taking the supremum over all vectors |v, |w with SR(|v), SR(|w) k gives the result.

2

Remark 2. When working with an operator system (instead of an operator space) V , it is more natural to define the norm (2) by taking the supremum over all completely positive unital maps Φ : V → Mk rather than all complete contractions (similarly, to define the norm (3) one would take the supremum over all k-positive unital maps rather than k-contractive maps). In this case, the k-minimal norm no longer coincides with the S(k)-norm on Mm (Mn ) but rather has the following slightly different form: XMm (OMIN k (Mn )) = sup v|X|w: SR |v , SR |w k and |v,|w

∃P ∈ Mm s.t. (P ⊗ In )|v = |w ,

(7)

where the notation OMIN k (Mn ) refers to a new operator system structure that is being assigned to Mn , which we discuss in detail in the next section. Intuitively, this norm has the same interpretation as the norm (1) except with the added restriction that the vectors |v and |w look the same on the second subsystems. We will examine this norm in more detail in Section 5. In particular, we will see in Theorem 8 that the norm (7) is a natural norm on the k-super minimal operator system structure (to be defined in Section 4), which plays an analogous role to the k-minimal operator space structure. Now that we have characterized the k-minimal norm in a fairly concrete way, we turn our attention to the k-maximal norm. The following result is a direct generalization of a corresponding known characterization of the MAX(V ) norm [18, Theorem 14.2]. Theorem 3. Let V be an operator space and let X ∈ Mm (V ). Then XMm (MAX k (V )) = inf AB: A, B ∈ Mm,rk , xi ∈ Mk (V ), xi Mk (V ) 1 with X = A · diag(x1 , . . . , xr ) · B ∗ , where diag(x1 , . . . , xr ) ∈ Mrk (V ) is the r × r block diagonal matrix with entries x1 , . . . , xr down its diagonal, and the infimum is taken over all such decompositions of X. Proof. The “” inequality follows simply from the axioms of an operator space: if X = (Xij ) = A · diag(x1 , . . . , xr ) · B ∗ ∈ Mm (V ) then


2413

Φ(Xij ) = A · diag (idk ⊗ Φ)(x1 ), . . . , (idk ⊗ Φ)(xr ) · B ∗ . Thus Φ(Xij ) AB max (idk ⊗ Φ)(x1 ), . . . , (idk ⊗ Φ)(xr ) . By taking the supremum over maps Φ with idk ⊗ Φ 1, the “” inequality follows. We will now show that the infimum on the right is an L∞ matrix norm that coincides with · Mm (V ) for 1 m k. The “” inequality will then follow from the fact that · Mm (MAX k (V )) is the maximal such norm. First, denote the infimum on the right by Xm,inf and fix some 1 m k. Then the inequality XMm (V ) Xm,inf follows immediately by picking any particular decomposition X = A · diag(x1 , . . . , xr ) · B ∗ and using the axioms of an operator space to see that A · diag(x1 , . . . , xr ) · B ∗ M

m (V )

AB max x1 , . . . , xr AB Xm,inf .

The fact that equality is attained by some decomposition of X comes simply from writing X = (XMm (V ) I ) · (X ⊕ 0k−m ) · I . It follows that · Mm (V ) = · m,inf for 1 m k. All that remains to be proved is that · m,inf is an L∞ matrix norm, which we omit as it is directly analogous to the proof of [18, Theorem 14.2]. 2 As one final note, observe that we can obtain lower bounds of the k-minimal and k-maximal operator space norms simply by choosing particular maps Φ that satisfy the normalization condition of their definition. Upper bounds of the k-maximal norms can be obtained from Theorem 3. The problem of computing upper bounds for the k-minimal norms was investigated in [12]. 4. k-Super minimal and k-super maximal operator systems We will now introduce (abstract) operator systems, and in particular the minimal and maximal operator systems that were explored in [19] and the k-super minimal and k-super maximal operator systems that were explored in [26]. Our introduction to general operator systems will be brief, and the interested reader is directed to [18, Chapter 13] for a more thorough treatment. Let V be a complex (not necessarily normed) vector space as before, with a conjugate linear involution that will be denoted by ∗ (such a space is called a ∗-vector space). Define Vh := {v ∈ V : v = v ∗ } to be the set of Hermitian elements of V . We will say that (V , V + ) is an ordered ∗-vector space if V + ⊆ Vh is a convex cone satisfying V + ∩ −V + = {0}. Here V + plays the role of the “positive” elements of V – in the most familiar ordering on square matrices, V + is the set of positive semidefinite matrices. Much as was the case with operator spaces, an operator system is constructed by considering the spaces Mm (V ), but instead of considering various norms on these spaces that behave well with the norm on V , we will consider various cones of positive elements on Mm (V ) that behave well with the cone of positive elements V+ . To this end, given a ∗-vector space V we let Mm (V )h

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denote the set of Hermitian elements in Mm (V ). It is said that a family of cones Cm ⊆ Mm (V )h (m 1) is a matrix ordering on V if C1 = V + and they satisfy the following three properties: • each Cm is a cone in Mm (V )h ; • Cm ∩ −Cm = {0} for each m; and • for each n, m ∈ N and X ∈ Mm,n we have X ∗ Cm X ⊆ Cn . A final technical restriction on V is that we will require an element e ∈ Vh such that, for any v ∈ V , there exists r > 0 such that re − v ∈ V + (such an element e is called an order unit). It is said that e is an Archimedean order unit if re + v ∈ V + for all r > 0 implies that v ∈ V + . A triple (V , C1 , e), where (V , C1 ) is an ordered ∗-vector space and e is an Archimedean order unit, will be referred to as an Archimedean ordered ∗-vector space or an AOU space for short. Furthermore, if e ∈ Vh is an Archimedean order unit then we say that it is an Archimedean matrix order unit if the operator em := Im ⊗ e ∈ Mm (V ) is an Archimedean order unit in Cm for all m 1. We are now able to define abstract operator systems: Definition 4. An (abstract) operator system is a triple (V , {Cm }∞ m=1 , e), where V is a ∗-vector is a matrix ordering on V , and e ∈ V is an Archimedean matrix order unit. space, {Cm }∞ h m=1 For brevity, we may simply say that V is an operator system, with the understanding that there is an associated matrix ordering {Cm }∞ m=1 and Archimedean matrix order unit e. Recall from [19] that for any AOU space (V , V + , e) there exists minimal and maximal operator system min }∞ structure OMIN(V ) and OMAX(V ) – that is, there exist particular families of cones {Cm m=1 ∞ max }∞ + , e) then C max ⊆ and {Cm such that if {D } is any other matrix ordering on (V , V m m=1 m m=1 min for all m 1. In [26] a generalization of these operator system structures, analogous D m ⊆ Cm to the k-minimal and k-maximal operator spaces presented in Section 3, was introduced. Given an operator system V , the k-super minimal operator system of V and the k-super maximal operator system of V , denoted OMIN k (V ) and OMAX k (V ) respectively, are defined via the following families of cones: min,k + Cm := (Xij ) ∈ Mm (V ): Φ(Xij ) ∈ Mm ∀ unital CP maps Φ : V → Mk , max,k Cm := A · D · A∗ ∈ Mm (V ): A ∈ Mm,rk , D = diag(D1 , . . . , Dr ), D ∈ Mk (V )+ ∀, r ∈ N . max,k need not define an operator system due to If V is infinite-dimensional then the cones Cm Im ⊗ e perhaps not always being an Archimedean order unit, though it was shown in [26] how to Archimedeanize the space to correct this problem. We will avoid this technicality by working explicitly in the V = Mn case from now on. Observe that the interpretation of the k-super minimal and k-super maximal operator systems is completely analogous to the interpretation of k-minimal and k-maximal operator spaces. min,k max,k and Cm coincide with the families of positive cones of The families of positive cones Cm V for 1 m k, and out of all operator system structures on V with this property they are the largest (smallest, respectively) for m > k. min,k ⊆ Mm ⊗ Mn are exactly the cones In terms of quantum information theory, the cones Cm max,k ⊆ Mm ⊗ Mn are exactly the cones of (unnorof k-block positive operators, and the cones Cm


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malized) density operators ρ with SN(ρ) k. These facts have appeared implicitly in the past, but their importance merits making the details explicit: Theorem 5. Let X, ρ ∈ Mm ⊗ Mn . Then min,k if and only if X is k-block positive; and (a) X ∈ Cm max,k (b) ρ ∈ Cm if and only if SN(ρ) k.

Proof. To see (a), we will use techniques similar to those used in the proof of Theorem 1. Use min,k if and only if the Choi–Kraus representation of completely positive maps so that X ∈ Cm nk

(Im ⊗ Ai )X Im ⊗ A∗i ∈ (Mm ⊗ Mk )+

for all {Ai } ⊂ Mk,n with

i=1

nk

Ai A∗i = Ik .

i=1

Now define αij |aij := A∗i |j and let |v = vector. Then some algebra reveals

k−1

j =0 γj |cj ⊗ |j ∈ C

m ⊗ Ck

be an arbitrary unit

k−1

νi |vi := Im ⊗ A∗i |v = αij γj |cj ⊗ |aij . j =0

In particular, SR(|vi ) k for all i. Thus we can write nk nk

v|(Im ⊗ Ai )(X) Im ⊗ A∗i |v = νi2 vi |X|vi 0. i=1

(8)

i=1

Part (a) follows by noting that we can choose |v and a CP map with one Kraus operator A1 so that (Im ⊗ A∗1 )|v is any particular vector of our choosing with Schmidt rank no larger than k. To see the “only if” implication of (b), we could invoke various known duality results from operator theory and quantum information theory so that the result would follow from (a), but for completeness we will instead prove it using elementary means. To this end, suppose max,k . Thus we can write ρ = A · D · A∗ for some A ∈ Mm,rk and D = diag(D ρ ∈ Cm 1 , . . . , Dr ) = r + with D ∈ Mk (Mn ) for all . Furthermore, write D = =1 || ⊗ D h d,h |v,h v,h | where |v,h = ki=1 |i ⊗ |d,h,i . Then if we define α,i |a , i := A(| ⊗ |i), we have A · D · A∗ =

kn

r

d,h

ij =1

h=1 =1

=

kn

r

h=1 =1

=

kn

r

h=1 =1

k

A || ⊗ |ij | A∗ ⊗ |d,h,i d,h,j |

d,h

k

α,i α,j |a,i a,j | ⊗ |d,h,i d,h,j |

ij =1

d,h |w,h w,h |,

2416


where |w,h :=

k

α,i |a,i ⊗ |d,h,i .

i=1

Since SR(|w,h ) k for all , h, it follows that SN(ρ) k as well. For the “if” implication, we note that the above argument can easily be reversed.

2

One of the useful consequences of Theorem 5 is that we can now easily characterize completely positive maps between these various operator system structures. Recall that a map Φ ∞ between operator systems (V , {Cm }∞ m=1 , e) and (V , {Dm }m=1 , e) is said to be completely positive if (Φ(Xij )) ∈ Dm whenever (Xij ) ∈ Cm . We then have the following result that characterizes k-positive maps, entanglement-breaking maps, and k-partially entanglement breaking maps as completely positive maps between these k-super minimal and k-super maximal operator systems. Corollary 6. Let Φ : Mn → Mn and let k n. Then (a) Φ : OMIN k (Mn ) → Mn is completely positive if and only if Φ is k-partially entanglement breaking; (b) Φ : Mn → OMAX k (Mn ) is completely positive if and only if Φ is k-partially entanglement breaking; (c) Φ : OMAX k (Mn ) → Mn is completely positive if and only if Φ is k-positive; (d) Φ : Mn → OMIN k (Mn ) is completely positive if and only if Φ is k-positive. Proof. Fact (a) follows from [2, Theorem 2] and fact (b) follows from the fact that Φ is kpartially entanglement breaking (by definition) if and only if SN((idm ⊗Φ)(ρ)) k for all m 1. Fact (c) follows from [25, Theorem 1] and (d) follows from (c) and the fact that the cone of unnormalized states with Schmidt number at most k and the cone of k-block positive operators are dual to each other [23]. 2 Remark 7. Corollary 6 was originally proved in the k = 1 case in [19] and for arbitrary k in [26]. min,k max,k and Cm Both of those proofs prove the result directly, without characterizing the cones Cm as in Theorem 5. 5. Norms on operator systems Given an operator system V , the matrix norm induced by the matrix order {Cm }∞ m=1 is defined for X ∈ Mm (V ) to be rem X ∈ C2m . (9) XMm (V ) := inf r: X ∗ rem In the particular case of X ∈ Mm (OMIN k (V )) or X ∈ Mm (OMAX k (V )), we will denote the norm (9) by XMm (OMIN k (V )) and XMm (OMAX k (V )) , respectively. Our first result characterizes XMm (OMIN k (Mn )) in terms of the Schmidt rank of pure states, much like Theorem 1 characterized XMm (MIN k (Mn )) .


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Theorem 8. Let X ∈ Mm (OMIN k (Mn )). Then XMm (OMIN k (Mn )) = sup v|X|w: SR |v , SR |w k and |v,|w

∃P ∈ Mm s.t. (P ⊗ In )|v = |w .

Proof. Given X ∈ Mm (OMIN k (Mn )), consider the operator X˜ :=

rIn X∗

X rIn

∈ (M2 ⊗ Mm ) ⊗ Mn ∼ = M2m ⊗ Mn .

min,k ˜ (Mn ) if and only if v|X|v 0 for all |v ∈ C2m ⊗ Cn with SR(|v) k. If we Then X˜ ∈ C2m multiply on the left and the right by a Schmidt-rank k vector |v := ki=1 βi |ai ⊗ |bi , where |ai = αi1 |1 ⊗ |ai1 + αi2 |2 ⊗ |ai2 ∈ C2 ⊗ Cm and |bi ∈ Cn , we get

˜ v|X|v =

k k

2 2 + r βi2 αi1 + βi2 αi2 2αi1 αj 2 βi βj Re ai1 | ⊗ bi | X |aj 2 ⊗ |bj ij =1

i=1

=r +

k

2αi1 αj 2 βi βj Re ai1 | ⊗ bi | X |aj 2 ⊗ |bj

ij =1

= r + 2c1 c2 Re v1 |X|v2 , where c1 |v1 := ki=1 αi1 βi |ai1 ⊗ |bi , c2 |v2 := ki=1 αi2 βi |ai2 ⊗ |bi ∈ Cm ⊗ Cn . Notice that the normalization of the Schmidt coefficients tells us that c12 + c22 = 1. Also notice that |v1 and |v2 can be written in this way using the same vectors |bi on the second subsystem if and only if there exists P ∈ Mm such that (P ⊗ In )|v1 = |v2 . Now taking the infimum over r and requiring that the result be non-negative tells us that the quantity we are interested in is XMm (OMIN k (Mn )) = sup 2c1 c2 Re v1 |X|v2 : SR |v1 , SR |v2 k, c12 + c22 = 1, ∃P ∈ Mm s.t. (P ⊗ In )|v1 = |v2 = sup v1 |X|v2 : SR |v1 , SR |v2 k and ∃P ∈ Mm s.t. (P ⊗ In )|v1 = |v2 , where the final equality comes applying a complex phase to |v1 so that Re(v1 |X|v2 ) = |v1 |X|v√2 |, and from Hölder’s inequality telling us that the supremum is attained when c1 = c2 = 1/ 2. 2 Of course, the matrix norm induced by the matrix order is not the only way to define a norm on the various levels of the operator system V . What is referred to as the order norm of v ∈ Vh [20] is defined via vor := inf{t ∈ R: −te v te}.

(10)

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It is not difficult to see that for a Hermitian element X = X ∗ ∈ Mm (V ), the matrix norm induced by the matrix order (9) coincides with the order norm (10). It was shown in [20] how the order norm on Mm (V )h can be extended (non-uniquely) to a norm on all of Mm (V ). Furthermore, there exists a minimal order norm · m and a maximal order norm · M satisfying · m · M 2 · m . We will now examine properties of these two norms as well as some other norms (all of which coincide with the order norm on Hermitian elements) on the k-super minimal operator system structures. We will consider an operator X ∈ Mm (OMIN k (Mn )), where recall by this we mean X ∈ Mm (Mn ), where the operator system structure on the space is OMIN k (Mn ). Then we recall the minimal order norm, decomposition norm · dec , and maximal order norm from [20]: Xm := sup f (X): f : Mm OMIN k (Mn ) → C a pos. linear functional s.t. f (I ) = 1 , r

r

min,k |λi |Pi : X = λi Pi with Pi ∈ Cm (Mn ) and λi ∈ C , Xdec := inf i=1 i=1 or r

r

∗ |λi |Hi or : X = λi Hi with Hi = Hi and λi ∈ C . XM := inf i=1

i=1

Our next result shows that the minimal order norm can be thought of in terms of vectors with Schmidt rank no greater than k, much like the norms · S(k) and · Mm (OMIN k (Mn )) introduced earlier. Theorem 9. Let X ∈ Mm (OMIN k (Mn )). Then Xm = sup v|X|v: SR |v k . |v

Proof. Note that if we define a linear functional f : Mm (OMIN k (Mn )) → C by f (X) = v|X|v min,k (Mn ) (by for some fixed |v with SR(|v) k then it is clear that f (X) 0 whenever X ∈ Cm definition of k-block positivity) and f (I ) = 1. The “” inequality follows immediately. To see the other inequality, note that if X = X ∗ = (Xij ) and |v, |w can be written |v = k k r=1 αr |ar ⊗ |br and |w = r=1 γr |cr ⊗ |br , then v|X|w =

k

αr γs ar | br |Xij |bs ij |cs

rs=1

⎛ (b |X |b ) 1 ij 1 ij .. = α1 a1 |, . . . , αk ak | ⎝ . (bk |Xij |b1 )ij

··· .. . ···

(b1 |Xij |bk )ij ⎞ ⎛ γ1 |c1 ⎞ .. ⎠ ⎝ .. ⎠ . . . γk |ck (bk |Xij |bk )ij

Because X is Hermitian, so is the operator in the last line above, so if we take the supremum over all |v, |w of this form, we may choose αi |ai = γi |ci for all i. It follows that sup v|X|v: SR |v k = sup v|X|w: SR |v , SR |w k and ∃P ∈ Mm s.t. (P ⊗ In )|v = |w .


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The “” inequality follows from Theorem 8, the fact that · Mm (OMIN k (Mn )) is an order norm, and the minimality of · m among order norms. 2 The characterization of · m given by Theorem 9 can be thought of as in the same vein as [20, Proposition 5.8], where it was shown that for a unital C∗ -algebra, · m coincides with the numerical radius. In our setting, · m can be thought of as a bipartite analogue of the numerical radius, which has been studied in quantum information theory in the k = 1 case [6]. In the case where X is not Hermitian, equality need not hold between any of the order norms that have been introduced. We now briefly investigate how they compare to each other in general. Proposition 10. Let X ∈ Mm (OMIN k (Mn )). Then Xm XMm (OMIN k (Mn )) Xdec XM . Proof. The first and last inequalities follow from the fact that · m and · M are the minimal and maximal order norms, respectively. Thus, all that needs to be shown is that XMm (OMIN k (Mn )) Xdec . To this end, let |v, |w ∈ Cm ⊗ Cn with SR(|v), SR(|w) k be such that there exists some P ∈ Mm such that (P ⊗ In )|v = |w. Then for any decomposition min,k (Mn ) and λi ∈ C we can use a similar argument to that used in X = ri=1 λi Pi with Pi ∈ Cm the proof of Theorem 9 to see that v|Pi |v |v|Pi |w| 0 because each Pi is k-block positive (by Theorem 5). Thus r r r r

v|X|w = λi v|Pi |w |λi |v|Pi |w |λi |v|Pi |v |λi |Pi . i=1

i=1

i=1

i=1

or

Taking the supremum over all such vectors |v and |w and the infimum over all such decompositions of X gives the result. 2 We know in general that · m and · M can differ by at most a factor of two. We now present an example some of these norms and to demonstrate that in fact even · m and · Mm (OMIN k (Mn )) can differ by a factor of two. Example 11. Consider the rank-1 operator X := |φψ| ∈ OMIN kn (Mn ), where 1

|i ⊗ |i, |φ := √ n n−1 i=0

It is easily verified that if |v =

n−1 1

|ψ := √ |i ⊗ i + 1 (mod n) . n i=0

k

i=1 αi |ai ⊗ |bi

then

k n−1

1

v|X|v = αr αs ar |ibr |ij |as j + 1 (mod n)bs n rs=1 ij =0

k n−1 k

1 = Tr αr |ar br | · j | αr |ar br | j + 1 (mod n) . n r=1

j =0

r=1

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In the final line above we have the trace of an operator with rank at most k, multiplied by the sum of the elements on the superdiagonal of the same operator, subject to the constraint that the k k and so Xm = 2n Frobenius norm of that operator is equal to 1. It follows that |v|X|v| 2n k−1 1 (equality can be seen by taking |v = i=0 √ |i ⊗ (|i + |i + 1 (mod n))). 2k k−1 k−1 √1 To see that Xop is twice as large, consider |v = √1 i=0 |i⊗|i and |w = i=0 |i⊗ k

k

|i + 1 (mod n). Then it is easily verified that v|X|w = nk . Moreover, if P ∈ Mm is the cyclic permutation matrix such that P |i = |i − 1 (mod n) for all i then (P ⊗ In )|v = |w, showing that Xop nk . 6. Contractive maps as separability criteria We now investigate the completely bounded version of the k-minimal operator space norms and k-super minimal operator system norms that have been introduced. We will see that these completely bounded norms can be used to provide a characterization of Schmidt number analogous to its characterization in terms of k-positive maps. Given operator spaces V and W , the completely bounded (CB) norm from V to W is defined by ΦCB(V ,W ) := sup (idm ⊗ Φ)(X)M m1

m (W )

: X ∈ Mm (V ) with XMm (V ) 1 .

Clearly this reduces to the standard completely bounded norm of Φ in the case when V = Mr and W = Mn . We will now characterize this norm in the case when V = Mr and W = MIN k (Mn ). In particular, we will see that the k-minimal completely bounded norm of Φ is equal to the perhaps more familiar operator norm idk ⊗ Φ – that is, the CB norm in this case stabilizes in much the same way that the standard CB norm stabilizes (indeed, in the k = n case we get exactly the standard CB norm). This result was originally proved in [16], but we prove it here using elementary means for completeness and clarity, and also because we will subsequently need the operator system version of the result, which can be proved in the same way. Theorem 12. Let Φ : Mr → Mn be a linear map and let 1 k n. Then idk ⊗ Φ = ΦCB(Mr ,MIN k (Mn )) . Proof. To see the “” inequality, simply notice that Y Mk (MIN k (Mn )) = Y Mk (Mn ) for all Y ∈ Mk (Mn ). We thus just need to show the “” inequality, which we do in much the same manner as Smith’s original proof that the standard CB norm stabilizes. First, use Theorem 1 to write ΦCB(Mr ,MIN k (Mn )) = sup (idm ⊗ Φ)(X)S(k) : X 1 .

(11)

m1

Now fix m k and a pure state |v ∈ Cm ⊗ Cn with SR(|v) k. We begin by showing that ˜ ∈ Ck ⊗ Cn such that (V ⊗ In )|v ˜ = |v. there exists an isometry V : Ck → Cm and a state |v To this end, write |v in its Schmidt Decomposition |v = ki=1 αi |ai ⊗ |bi . Because k m, k m ˜ := we k may define an isometry V : C → C by V |i = |ai for 1 i k. If we define |v α |i ⊗ |b then (V ⊗ I )| v ˜ = |v, as desired. i i n i=1


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˜ 1 and the supremum (11) (holding m fixed) is Now choose X˜ ∈ Mm (Mr ) such that X ˜ Then choose vectors |v, |w ∈ Cm ⊗ Cn with SR(|v), SR(|w) k such that attained by X. (idm ⊗ Φ)(X) ˜

v|(idm ⊗ Φ)(X)|w . ˜ = S(k)

As we saw earlier, there exist isometries V , W : Ck → Cm and unit vectors |v, ˜ |w ˜ ∈ Ck ⊗ C n such that (V ⊗ In )|v ˜ = |v and (W ⊗ In )|w ˜ = |w. Thus (idm ⊗ Φ)(X) ˜

S(k)

∗ ˜ ˜ V ⊗ In (idm ⊗ Φ)(X)(W = v| ⊗ In )|w ˜ ∗ ˜ ˜ = v|(id ⊗ Ir ) |w ˜ k ⊗ Φ) V ⊗ Ir X(W ∗ ˜ (idk ⊗ Φ) V ⊗ Ir X(W ⊗ Ir ) sup (idk ⊗ Φ)(X): X ∈ Mk (Mr ) with X 1 ,

˜ ⊗ Ir ) 1. The desired where the final inequality comes from the fact that (V ∗ ⊗ Ir )X(W inequality follows, completing the proof. 2 We will now show that the operator system versions of these norms have applications to testing separability of quantum states. To this end, notice that if we instead consider the completely bounded norm from Mr to the k-super minimal operator systems on Mn , then a statement that is analogous to Theorem 12 holds. Its proof can be trivially modified to show that if Φ : Mr → Mn and 1 k n then sup v|(idk ⊗ Φ)(X)|v: X 1, X = X ∗ = sup v|(idm ⊗ Φ)(X)|v: X 1, X = X ∗ , SR |v k .

(12)

m1

Eq. (12) can be thought of as a stabilization result for the completely bounded version of the norm described by Theorem 9. We could also have picked one of the other order norms on the k-super minimal operator systems to work with, but from now on we will be working exclusively with Hermiticity-preserving maps Φ. So by the fact that all of the operator system order norms are equal on Hermitian operators, it follows that these versions of their completely bounded norms are all equal as well. Before proceeding, we will need to define some more notation. If Φ : Mn → Mr is a linear map, then we define a Hermitian version of the induced trace norm of Φ: ∗ ΦH tr := sup Φ(X) tr : Xtr 1, X = X . Because of convexity of the trace norm, it is clear that the above norm is unchanged if instead of being restricted to Hermitian operators, the supremum is restricted to positive operators or even just projections. Now by taking the dual of the left and right norms described by Eq. (12), and using the fact that the operator norm is dual to the trace norm, we arrive at the following corollary:

2422


Corollary 13. Let Φ : Mn → Mr be a Hermiticity-preserving linear map and let 1 k n. Then idk ⊗ ΦH tr = sup (id m ⊗ Φ)(ρ) tr : ρ ∈ Mm ⊗ Mn with SN(ρ) k . m1

We will now characterize the Schmidt number of a state ρ in terms of maps that are contractive in the norm described by Corollary 13. Our result generalizes the separability test of [8]. We begin with a simple lemma that will get us most of the way to the linear contraction characterization of Schmidt number. The k = 1 version of this lemma appeared as [8, Lemma 1], though our proof is more straightforward. Lemma 14. Let ρ ∈ Mm ⊗ Mn be a density operator. Then SN(ρ) k if and only if (idm ⊗ Φ)(ρ) 0 for all trace-preserving k-positive maps Φ : Mn → M2n . Proof. The “only if” implication of the proof is clear, so we only need to establish that if SN(ρ) > k then there is a trace-preserving k-positive map Φ : Mn → M2n such that (idm ⊗ Φ)(ρ) 0. To this end, let Ψ : Mn → Mn be a k-positive map such that (idm ⊗ Ψ )(ρ) 0 (which we know exists by [25,22]). Without loss of generality, Ψ can be scaled so that Ψ tr n1 . Then if Ω : Mn → Mn is the completely depolarizing channel defined by Ω(ρ) = n1 In for all ρ ∈ Mn , it follows that (Ω − Ψ )(ρ) 0 for all ρ 0 and so the map Φ := Ψ ⊕ (Ω − Ψ ) : Mn → M2n is k-positive (and easily seen to be trace-preserving). Because (idm ⊗ Ψ )(ρ) 0, we have (idm ⊗ Φ)(ρ) 0 as well, completing the proof. 2 We are now in a position to prove the main result of this section. Note that in the k = 1 case of the following theorem it is not necessary to restrict attention to Hermiticity-preserving linear maps Φ (and indeed this restriction was not made in [8]), but our proof for arbitrary k does make use of Hermiticity-preservation. Theorem 15. Let ρ ∈ Mm ⊗ Mn be a density operator. Then SN(ρ) k if and only if (idm ⊗ Φ)(ρ)tr 1 for all Hermiticity-preserving linear maps Φ : Mn → M2n with idk ⊗ ΦH tr 1. Proof. To see the “only if” implication, simply use Corollary 13 with r = 2n. For the “if” implication, observe that any positive trace-preserving map Ψ is necessarily Hermiticity-preserving and has Ψ H tr 1. Letting Ψ = id k ⊗ Φ then shows that any k-positive trace-preserving map Φ has idk ⊗ ΦH tr 1. Thus the set of Hermiticity-preserving linear maps Φ with idk ⊗ ΦH 1 contains the set of k-positive trace-preserving maps, so the “if” implitr cation follows from Lemma 14. 2 Acknowledgments Thanks are extended to Marius Junge for drawing our attention to the k-minimal and kmaximal operator space structures. N.J. was supported by an NSERC Canada Graduate Scholarship and the University of Guelph Brock Scholarship. D.W.K. was supported by Ontario Early Researcher Award 048142, NSERC Discovery Grant 400160 and NSERC Discovery Accelerator Supplement 400233. R.P. was supported by NSERC Discovery Grant 400096.


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References ˙ [1] I. Bengtsson, K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, Cambridge, 2006. [2] D. Chruscinski, A. Kossakowski, On partially entanglement breaking channels, Open Syst. Inf. Dyn. 13 (2006) 17–26. [3] D. Chru´sciński, A. Kossakowski, Spectral conditions for positive maps, Comm. Math. Phys. 290 (2009) 1051–1064. [4] D. Chru´sciński, A. Kossakowski, G. Sarbicki, Spectral conditions for entanglement witnesses vs. bound entanglement, preprint, 2009, arXiv:0908.1846v1 [quant-ph]. [5] D.P. DiVincenzo, P.W. Shor, J.A. Smolin, B.M. Terhal, A.V. Thapliyal, Evidence for bound entangled states with negative partial transpose, Phys. Rev. A 61 (2000) 062312, arXiv:quant-ph/9910026v3. ˙ [6] P. Gawron, Z. Puchala, J.A. Miszczak, L. Skowronek, M.-D. Choi, K. Zyczkowski, Local numerical range: a versatile tool in the theory of quantum information, E-print: arXiv:0905.3646v1 [quant-ph]. [7] M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Lett. A 223 (1996) 1–8. [8] M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed quantum states: linear contractions approach, Open Syst. Inf. Dyn. 13 (2006) 103. [9] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement, Rev. Modern Phys. 81 (2009) 865–942. [10] M. Horodecki, P.W. Shor, M.B. Ruskai, General entanglement breaking channels, Rev. Math. Phys. 15 (2003) 629– 641. [11] N. Johnston, Characterizing operations preserving separability measures via linear preserver problems, preprint, 2010, arXiv:1008.3633v1 [quant-ph]. [12] N. Johnston, D.W. Kribs, A family of norms with applications in quantum information theory II, preprint, 2010, arXiv:1006.0898v1 [quant-ph]. [13] N. Johnston, D.W. Kribs, A family of norms with applications in quantum information theory, J. Math. Phys. 51 (2010) 082202. [14] N. Johnston, D.W. Kribs, V. Paulsen, Computing stabilized norms for quantum operations, Quantum Inf. Comput. 9 (1–2) (2009) 16–35. [15] A.Yu. Kitaev, Quantum computations: algorithms and error correction, Russian Math. Surveys 52 (1997) 1191– 1249. [16] T. Oikhburg, E. Ricard, Operator spaces with few completely bounded maps, Math. Ann. 328 (2004) 229–259. [17] L. Pankowski, M. Piani, M. Horodecki, P. Horodecki, A few steps more towards NPT bound entanglement, IEEE Trans. Inform. Theory 56 (2010) 4085–4100. [18] V.I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, Cambridge, 2003. [19] V. Paulsen, I. Todorov, M. Tomforde, Operator system structures on ordered spaces, Proc. Lond. Math. Soc. (2010), doi:10.1112/plms/pdq011. [20] V. Paulsen, M. Tomforde, Vector spaces with an order unit , Indiana Univ. Math. J. 58 (3) (2009) 1319–1359. [21] G. Pisier, Introduction to Operator Space Theory, Cambridge University Press, Cambridge, 2003. [22] K.S. Ranade, M. Ali, The Jamiołkowski isomorphism and a simplified proof for the correspondence between vectors having Schmidt number k and k-positive maps, Open Syst. Inf. Dyn. 14 (2007) 371–378. ˙ [23] Ł. Skowronek, E. Størmer, K. Zyczkowski, Cones of positive maps and their duality relations, J. Math. Phys. 50 (2009) 062106. [24] R.R. Smith, Completely bounded maps between C∗ -algebras, J. London Math. Soc. 27 (1983) 157–166. [25] B.M. Terhal, P. Horodecki, Schmidt number for density matrices, Phys. Rev. A 61 (2000) 040301R. [26] B. Xhabli, Universal operator system structures on ordered spaces and their applications, PhD thesis, 2009.


Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system Fatiha Alabau-Boussouira a,∗,1 , Kaïs Ammari b a Université Paul Verlaine-Metz Metz. LMAM UMR 7122, 57045 Metz Cedex 1, France b Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia

Received 5 September 2010; accepted 4 January 2011 Available online 15 January 2011 Communicated by J. Coron

Abstract We consider the problem of sharp energy decay rates for nonlinearly damped abstract infinitedimensional systems. Direct methods for nonlinear stabilization generally rely on multiplier techniques, and thus are valid under restrictive geometric conditions compared to the optimal geometric optics condition of Bardos et al. (1992) [10]. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely an observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine optimal geometric conditions, as for instance that of Bardos et al. (1992) [10] and the optimal-weight convexity method of the first author (AlabauBoussouira, 2010 [6], Alabau-Boussouira, 2005 [2]) to deduce very simple and quasi-optimal energy decay rates for nonlinearly locally damped systems. We also show that using arguments based on Russell’s principle (Russell, 1978 [24]), one can deduce sharp energy decay rates from the exponential stabilization of the linearly damped system. Our results extend to nonlinearly damped systems, those of Haraux (1989) [14] and Ammari and Tucsnak (2001) [9] which concern linearly damped systems. © 2011 Elsevier Inc. All rights reserved. Keywords: Nonlinear stabilization; Dissipative systems; Observability; Energy decay rates; Wave equation; Hyperbolic equation


E-mail addresses: [email protected] (F. Alabau-Boussouira), [email protected] (K. Ammari). 1 Present position Délégation CNRS at MAPMO, UMR 6628.


F. Alabau-Boussouira, K. Ammari / Journal of Functional Analysis 260 (2011) 2424–2450

2425

1. Introduction and main results In this paper we characterize the stabilization for some nonlinear infinite-dimensional systems. These results have been partially announced in [7]. We show that if the linear system is observable through a locally distributed observation, then any dissipative nonlinear feedback locally distributed stabilize the system and we give a general easily computable energy decay formula. We show by this way that for the locally distributed case, one can combine the optimal geometric optics conditions of Bardos, Lebeau and Rauch [10] (see also [11,12]) and the optimalweight convexity method by the first author [1,2,6] (see also [3,4]) based on nonlinear Gronwall inequalities with optimal weight to deduce sharp easily computable energy decay rates for nonlinear damped systems. Using recent results of the first author [6], a very simple, upper estimate is given for feedbacks with general growth close to the origin (not close to a linear behavior) and linear at infinity. Optimality of these estimates has been proved in the finite-dimensional case in [6] and in certain infinite-dimensional situations [2] using optimality results by Vancostenoble and Martinez [26] (see also [25]). Our results extend to nonlinear feedbacks, previous results by Haraux [14] and Ammari and Tucsnak [9,8] valid for linear feedbacks. A result using this indirect approach has been obtained for boundary and localized dampings for wave-type equations by Daoulatli, Lasiecka and Toundykov in [13], using the ODE approach of [16] for nonlinear boundary and localized stabilization. Theorem 2.2 of [13] can be compared to our main result Theorem 1.1. Let us denote by w the solution of the nonlinearly damped system, by z the solution of the linearly damped system and by φ the solution of the conservative system. The proof of Theorem 2.2 [13] relies on an observability estimate for the corresponding linearly damped system solved by z, estimates of the mixed products of the form a(x)zt wt , a(x)zt ρ(., wt ) Ω

Ω

and the ODE-convexity approach of [16] (see also [20,28]) which consists in estimating the energy decay rate of the nonlinear stabilization system by the solution S of a nonlinear separable ODE of the form S (t) + q S(t) = 0, S(0) = Ew (0) where q = I − (I + h−1 ◦ (K.I ))−1 . Here I stands for the identity map on R, K depends on the minimal time T (above which the observability inequality holds), on the observability constant and on the damping region. Moreover h is a strictly increasing concave function on [0, ∞), such that h(0) = 0 and related to the damping ρ (assumed to depend only on the second variable) as follows h ρ(s)s s 2 + ρ 2 (s), ∀|s| 1. The above nonlinear ODE can be replaced by a simplified one under further hypotheses. Our approach here relies rather on an observability inequality for the conservative system, on two comparison properties – namely a comparison property (see later in Lemma 2.3) between the localized observation for the conservative system and the time integral of the localized kinetic energy of the solution of the linearly damped system, and a comparison property (see

2426


later in Lemma 2.2) between this last quantity and the time integral of the localized linear and nonlinear kinetic energies of the nonlinearly damped system – and on the optimal-weight convexity method [2,6] we above mentioned. It consists in determining an optimal-weight thanks to convexity properties of the function H introduced later in (1.9) and to prove a nonlinear Gronwall type inequality relative to this weight. The optimal-weight convexity method generalizes the power-like integral method (see [15] and references therein). Optimality of the sharp upper estimate given in Theorem 1.1 is proved in [6] in the finite-dimensional case for dampings which are not close to a linear behavior close to the origin (see later in Theorem 1.1 the condition lim supx→0+ ΛH (x) < 1). Moreover, we show in [6] that the upper estimate given in Theorem 1.1, can be estimated from above by the energy of an associated ODE of first order which involves only the function g of Assumption (A1), this holding in the finite as well as in the infinite-dimensional case. It should be noted that the dampings considered in [13] have more general growth behaviors at infinity (they can be sublinear or superlinear at infinity) than in the present paper. Both the ODE-convexity method [16] and the optimal-weight convexity method [2,6] provide sharp energy decay rates, but use somehow different ways to measure the decay of the energy of solutions. Our purpose here is indeed to provide a self-contained, easy and explicit approach based on a general methodology initiated in [2,6], pursued through lower energy estimates and further comparison properties in [5] and to combine it with and stress the importance of quasi-optimal geometric conditions on the observation region derived thanks to micro-local analysis [10]. Our study in [2,6] is valid only under the less general multiplier geometric conditions. On the other hand, the upper estimates derived thanks to this approach are obtained through a very simple formula in the case of dampings which are not close to a linear behavior at the origin. Therefore, it is important to show that it is possible to combine this approach for capturing optimal and quasi-optimal energy decay rates and the geometric optics approach of [10] which allows optimal and quasi-optimal geometric conditions on the support of the damping region. In this, we extend the results of [14,9] and give a different but related method compared to [13] and a different expression for sharp upper energy decay rates of the solutions of nonlinearly locally damped PDEs. We also give an explicit dependence of the parameters which are involved in our estimate with respect to the observability constant, the initial energy and the minimal time for observability. This is also important for numerical purposes. We present now the general set-up for our results. We consider the following second order differential equation w(t) ¨ + Aw(t) + a(.)ρ(., w) ˙ = 0, t ∈ (0, ∞), x ∈ Ω, (1.1) 0 w(0) ˙ = w1 , w(0) = w , where Ω is a bounded open set in RN , with a boundary Γ . We assume that Ω is either convex or of class C 1,1 . We set H = L2 (Ω), with its usual scalar product denoted by ·,· H and the associated norm · H and where A : D(A) ⊂ H → H is a densely defined self-adjoint linear operator satisfying Au, u H Cu2H ,

∀u ∈ D(A)

(1.2)

for some C > 0. We also introduce the scale of Hilbert spaces Hα , as follows: for every α 0, Hα = D(Aα ), with the norm zα = Aα zH . The space H−α , is defined by duality with respect to the pivot space H as follows: H−α = Hα∗ , for α > 0. The operator A can be extended (or


2427

restricted) to each Hα , such that it becomes a bounded operator A : Hα → Hα−1 ,

∀α ∈ R.

(1.3)

Eq. (1.1) is understood as an equation in H−1/2 , i.e., all the terms are in H−1/2 . The energy of a solution is defined by Ew (t) =

2 1 w(t), w(t) ˙ . H ×H 1/2 2

(1.4)

Most of the coupled linear equations modelling the damped vibrations of elastic structures can be written in the form (1.1), where w stands for the displacement field and the term B w(t) ˙ = a(.)ρ(., w), ˙ represents a viscous feedback damping. The system (1.1) is well-posed. More precisely, the following holds: Suppose that (w 0 , w 1 ) ∈ H1/2 × H . Then the problem (1.1) admits a unique solution w ∈ C [0, ∞); H1/2 ∩ C 1 [0, ∞); H . Moreover w satisfies, for all t 0, the energy identity 0 1 2 w ,w H

1/2 ×H

2 ˙ − w(t), w(t)

H1/2 ×H

t =2

a(.)ρ ., w(s) ˙ w(s) ˙ dx ds.

(1.5)

0 Ω

The aim of this paper is to deduce sharp simple computable energy decay rates for the damped system (1.1) from observability estimates for the associated undamped system, that is

¨ + Aφ(t) = 0, φ(t) ˙ φ(0) = φ 0 , φ(0) = φ1.

(1.6)

Before stating our main results, let us specify some hypotheses on the feedback and give some preliminary definitions. We make the following assumptions on the feedback ρ and on a: Assumption (A1). ρ ∈ C(Ω × R; R) is a continuous monotone nondecreasing function with respect to the second variable on Ω such that ρ(., 0) = 0 on Ω and there exists a continuous strictly increasing odd function g ∈ C([−1, 1]; R), continuously differentiable in a neighbourhood of 0 and satisfying g(0) = g (0) = 0, with

c1 g |v| ρ(., v) c2 g −1 |v| , c1 |v| ρ(., v) c2 |v|,

|v| 1, a.e. on Ω, |v| 1, a.e. on Ω,

(1.7)

where ci > 0 for i = 1, 2. Moreover a ∈ C(Ω), with a 0 on Ω and ∃a− > 0 such that a a− on ω.

(1.8)

2428


Here ω stands for the subregion of Ω on which the feedback ρ is active and U = L2 (ω). We define a function H (see [2]) by H (x) =

√ √ xg( x ),

x ∈ 0, r02 .

(1.9)

Thanks to Assumption (A1), H is of class C 1 and is strictly convex on [0, r02 ], where r0 > 0 is the extension of H to R where H (x) = +∞ for a sufficiently small number. We denote by H x ∈ R\[0, r02 ]. We also define a function L by L(y) =

(y) H y ,

if y ∈ (0, +∞), if y = 0,

0,

(1.10)

stands for the convex conjugate function of H , i.e.: H (y) = supx∈R {xy − H (x)}. where H We prove in [2] that L is strictly increasing continuous and onto from [0, +∞) on [0, r02 ). We define a function ΛH on (0, r02 ] by H (x) . xH (x)

ΛH (x) =

(1.11)

We also define 1 + ψr (x) = H (r02 )

H (r02 )

1/x

v 2 (1 − Λ

1 dv, −1 H ((H ) (v)))

x

1 H (r02 )

.

(1.12)

Let us state our main results: Theorem 1.1. Assume that ρ and a satisfy Assumption (A1) and that there exists r0 > 0 sufficiently small so that the function H defined by (1.9) is strictly convex on [0, r02 ]. Assume that lim

x→0+

H (x) =0 ΛH (x)

(1.13)

where ΛH is defined by (1.11). Moreover assume that there exists T > 0 such that the following observability inequality is satisfied for the linear conservative system (1.6) T cT Eφ (0)

√ 2 ˙ H dt, | a φ|

∀(φ0 , φ1 ) ∈ H1/2 × H

(1.14)

0

with a certain cT > 0. Then, the energy of the solution of (1.1) satisfies Ew (t) βT L

1 ψr−1 ( t−T T0 )

,

for t sufficiently large.

(1.15)


2429

If further, lim supx→0+ ΛH (x) < 1 then we have the simplified decay rate −1 Ew (t) βT H

DT0 , t −T

(1.16)

for t sufficiently large. Here D is a positive constant which is independent of Ew (0) and T , whereas T0 depends on T and is defined by (3.10), β is a positive constant chosen so that

Ew (0) 2αT Ew (0) , , β > max , CT L(H (r02 )) δ

(1.17)

where the constants CT > 0, α and δ > 0 are respectively defined by (2.25), (2.26) and (2.34). Remark 1.2. If 0 < lim inf ΛH (x) x→0+

(1.18)

holds, then since limx→0+ H (x) = 0, (1.13) holds. Moreover, under the above hypotheses, we have L

1

ψr−1 ( t−T T0 )

→ 0 as t → ∞.

We refer to [6,5] for lower energy estimates for the nonlinearly damped wave equation with locally distributed or boundary dampings. For several examples of PDEs, exponential decay for the linear damped case, has been proved under geometric conditions. We now give an important result showing that sharp energy decay rates for the case of arbitrary nonlinear damping is a consequence of exponential stabilization for the case of linear damping. This corollary is deduced from Theorem 1.1 and from Russell’s principle [24] as generalized by K. Liu [18]. Let us formulate this result. For this, we consider the case of the linearly damped system:

z¨ (t) + Az(t) + a(.)˙z = 0, z(0) = z , 0

t ∈ (0, ∞), x ∈ Ω,

z˙ (0) = w . 1

(1.19)

We define the energy of a solution z of (1.19) by Ez as in (1.4) replacing w by z and for initial date (z0 , z1 ) ∈ H1/2 × H . Corollary 1.3. Assume that ρ and a satisfy Assumption (A1) and that there exists r0 > 0 sufficiently small so that the function H defined by (1.9) is strictly convex on [0, r02 ]. Assume also that (1.13) holds. We moreover assume that the system (1.19) is exponentially stable, that is there exist μ > 0 and C > 0 such that Ez (t) CEz (0)e−μt ,

∀(z0 , z1 ) ∈ H1/2 × H.

(1.20)

2430


Then there exists T > 0 such that the energy of the solution of (1.1) satisfies (1.15). If further lim supx→0+ ΛH (x) < 1, then Ew satisfies the simplified decay rate (1.16). The proof of Theorem 1.1 relies on the next theorem. This second result is interesting in itself since it allows to compare in full generality, discrete energy inequalities (valid for sequences of time converging to infinity) to continuous ones. For this, we consider the following assumption. Assumption (A2). H is a continuously differentiable strictly convex function on [0, r02 ] with H (0) = H (0) = 0. The function M defined by M(x) = xL−1 (x),

x ∈ 0, r02

(1.21)

is such that limx→0+ M (x) = 0, where L is defined by (1.10). Remark 1.4. Thanks to Assumption (A2), for all positive constant κ, there exists δ ∈ (0, r02 ] such that the function x → x − κM(x) is strictly increasing on [0, δ]. Theorem 1.5. Assume that Assumption (A2) holds and let T > 0 and ρT > 0 be given. Let δ > 0 be such that the function defined by x → x − ρT M(x) is strictly increasing on [0, δ]. Assume is a nonnegative, nonincreasing function defined on [0, ∞) with E(0) < δ and satisfying that E (k + 1)T E(kT E ) 1 − ρT L−1 E(kT ) ,

∀k ∈ N.

(1.22)


(1.23)

satisfies the upper estimate Then E TL E(t)

1 ψr−1 ( (t−TT )ρT )

,

If moreover lim supx→0+ ΛH (x) < 1, then we have the simplified decay rate T H −1 E(t)

DT , ρT (t − T )

(1.24)

and of T . for t sufficiently large and where D is a positive constant independent of E(0) The paper is organized as follows. In the second section, we establish preliminary technical results. In Section 3, we give the proof of our three main results, that is Theorem 1.1, Corollary 1.3 and Theorem 1.5. We give examples of applications of our results to various examples of feedbacks growth and to examples of PDEs, namely the wave and Bernoulli–Euler plate equations.


2431

2. Preliminary intermediate results In all this section the initial data (w(0), w(0)) ˙ will be kept fixed. We extend H by +∞ on R\[0, r02 ] and still denote this extension by H . We define the convex conjugate of H and denote it by H . Moreover we define a weight function f such that sf (s) , H f (s) = β where β > max( αT CT ,

Ew (0) , Ewδ(0) ) L(H (r02 ))

s ∈ 0, βr02 ,

(2.1)

where the constants CT > 0, α and δ > 0 are respectively

defined by (2.25), (2.26) and (2.34). We recall that f is defined by f (s) = L

−1

s , β

∀s ∈ 0, βr02 ,

where L is the continuous strictly increasing function defined from [0, +∞) onto [0, r02 ) by (1.10). One can show [2] that f is a strictly increasing function from [0, βr02 ) onto [0, ∞). We start by a key lemma which relies on the optimal-weight convexity method of [2]. Lemma 2.1. Assume that ρ and a satisfy Assumption (A1) and that there exists r0 > 0 sufficiently small so that the function H defined by (1.9) is strictly convex on [0, r02 ]. Let (w 0 , w 1 ) ∈ H1/2 ×H be given and (φ 0 , φ 1 ) = (w 0 , w 1 ) and w and φ be the respective solutions of (1.1) and of (1.6). Then the following inequality holds T

f Eφ (0)

2 ˙ dx dt a(x)|w| ˙ 2 + a(x)ρ(x, w)

Ω

0

c5 T H f Eφ (0) + c6 f Eφ (0) + 1

T a(x)ρ(x, w) ˙ w˙ dx dt,

(2.2)

0 Ω

where c5 = |Ω| 1 + c22 , and |Ω| =

Ω

c6 =

1 + c2 , c1

dσ , with dσ = a(.) dx.

Proof. Define ε0 = g(r0 ) < 1. We can easily check that there exist c1 > 0 and c2 > 0 such that c1 g |v| ρ(x, v) c2 g −1 |v| ,

x ∈ Ω, |v| ε0 ,

(2.3)

and c1 |v| ρ(x, v) c2 |v|,

x ∈ Ω, |v| ε0 .

(2.4)

2432


Define now r12 = H −1 ( cc12 H (r02 )) and ε1 = min(r0 , g(r1 )). We can assume, without loss of generality that c1 < c2 , so that ε1 ε0 holds. Moreover, one can easily prove that there exist constants, that we still denote by c1 > 0 and c2 > 0 such that c1 |v| ρ(x, v) c2 |v|,

x ∈ Ω, |v| ε1

(2.5)

and c1 g |v| ρ(x, v) c2 g −1 |v| ,

x ∈ Ω, |v| ε1 .

(2.6)

T Step 1. Estimate of 0 f (Eφ (0)) Ω a(x)|ρ(x, w)| ˙ 2 dx dt. t ˙ x)| ε0 }. We also set We set for all fixed t 0, Ω1 = {x ∈ Ω, |w(t, cg =

1 . c2

(2.7)

Thus, by definition of cg and thanks to (2.3), we have 2 ˙ x) r02 , cg2 ρ x, w(t, We set dσ = a(x) dx and |Ω1t | = σ (Ω1t ) = 1 |Ω1t |

Ω1t

∀x ∈ Ω1t .

dσ . Since

2

˙ x) dσ ∈ 0, r02 , cg2 ρ x, w(t,

Ω1t

which is the domain of convexity of H , and thanks to Jensen’s inequality, we have H

1 |Ω1t |

2 ˙ x) dσ cg2 ρ x, w(t,

Ω1t

1 |Ω1t | 1 |Ω1t |

2 ˙ x) dσ H cg2 ρ x, w(t,

Ω1t

˙ x) g cg ρ x, w(t, ˙ x) a(x) dx. cg ρ x, w(t,

(2.8)

Ω1t

But thanks to (2.3), and since g is increasing, we have on Ω1t : w(t) ˙ ˙ g cg ρ x, w(t)

on Ω1t .

Using this last inequality in (2.8), we deduce that H

1 |Ω1t |

Ω1t

2 1 ˙ x) a(x) dx cg2 ρ x, w(t, cg w(t)(x)ρ ˙ x, w(t, ˙ x) a(x) dx. t |Ω1 | Ω1t

(2.9)


2433

On the other hand, thanks to (2.3), we obtain 1 |Ω1t |

Ω1t

1 cg w(t, ˙ x)ρ x, w(t, ˙ x) dσ |Ω1t |

ε0 g −1 (ε0 )a(x) dx = H r02 .

(2.10)

Ω1t

Hence, we have H −1

1 |Ω1t |

cg w(t, ˙ x)ρ x, w(t, ˙ x) a(x) dx ∈ 0, r02 .

Ω1t

Thanks (2.8) and to Young’s inequality, we have T

f Eφ (t)

2 ˙ x) dx dt a(x)ρ x, w(t,

Ω1t

0

T 0

T 0

−1 cg |Ω1t | f Eφ (t) H a(x)w(t, ˙ x)ρ x, w(t, ˙ x) a(x) dx dt |Ω1t | cg2 Ω1t

1 |Ω1t | H f Eφ (t) + 2 cg cg

T

w(t, ˙ x)ρ x, w(t, ˙ x) a(x) dx dt.

0 Ω1t

On the complementary set of Ω1t in Ω, since ρ has a linear growth, and since ˙ 2 2Eφ (t) = (φ(t), φ(t) H

1/2 ×H

= 2Eφ (0),

∀t 0,

we have T 0

f Eφ (t)


Ω\Ω1t

f (Eφ (0)) cg

T

a(x)w(t, ˙ x)ρ x, w(t, ˙ x) dx dt.

0 Ω\Ω1t

Hence, thanks to the above two inequalities, we have

(2.11)

2434


T

f Eφ (t)


Ω

0

f (Eφ (0)) + 1 |Ω| 2 T H f Eφ (0) + cg cg

T


(2.12)

0 Ω

T ˙ 2 dx dt. Step 2. Estimate of 0 f (Eφ (0)) Ω a(x)|w| t ˙ x)| ε1 } Thus, we have We set Ω2 = {x ∈ Ω, |w(t, 1 w(t, ˙ x)g w(t, ˙ x) w(t, ˙ x)ρ x, w(t, ˙ x) , c1

∀x ∈ Ω2t ,

and 1 |Ω2t |

2

w(t, ˙ x)) dσ ∈ 0, r02 ,

Ω2t

which is the domain of convexity of H . Therefore thanks to Jensen’s inequality and since H is nondecreasing, we have

2 w(t, ˙ x) dσ Ω2t H −1

Ω2t

Ω2t H −1

1 |Ω2t |

2 ˙ x) dσ H w(t,

Ω2t

1 |Ω2t |c1

a(x)w(t, ˙ x)ρ x, w(t, ˙ x) dx .

Ω2t

Thanks (2.13) and to Young’s inequality, we have T

f Eφ (t)

2 ˙ x) dx dt a(x)w(t,

Ω2t

0

T

t Ω f Eφ (t) H −1 2

0

T 0

1 |Ω2t |c1

t Ω H f Eφ (t) + 1 2 c1

a(x)w(t, ˙ x)ρ x, w(t, ˙ x) dx dt

Ω2t

T


0 Ω2t

On the complementary set of Ω2t in Ω, since ρ has a linear growth, and since 2 ˙ 2Eφ (t) = φ(t), φ(t) H

1/2 ×H

= 2Eφ (0),

∀t 0,

(2.13)


2435

we have T

f Eφ (t)

2 ˙ x) dx dt a(x)w(t,

Ω\Ω2t

0

f (Eφ (0)) c1

T


0 Ω\Ω2t

Hence, thanks to the above two inequalities, we have f Eφ (0)

T a(x)|w| ˙ 2 dx dt 0 Ω

f (Eφ (0)) + 1 |Ω|T H f Eφ (0) + c1

T

a(x)wρ ˙ x, w(t, ˙ x) dx dt.

(2.14)

0 Ω

Now thanks to the definition of the weight function f in (2.1), we have f Eφ (0)

T |w| ˙ 2 dx dt

(2.15)

0 ω

|ω|αT Eφ (0)f Eφ (0) f (Eφ (0)) + 1 + c1 a−

T

w(t, ˙ x)ρ x, w(t, ˙ x) dx dt.

(2.16)

0 Ω

Inequalities (2.12) and (2.14) lead to the desired result.

2

The next lemma compares the localized kinetic damping of the linearly damped equation with the localized linear and nonlinear kinetic energies of the nonlinearly damped equation. Lemma 2.2. Assume that ρ ∈ C(Ω × R; R) is a continuous monotone nondecreasing function with respect to the second variable on Ω such that ρ(., 0) = 0 on Ω. Let w be the solution of (1.1) with initial data (w 0 , w 1 ) ∈ H1/2 × H . Let us introduce z solution of the linear locally damped problem

z¨ + Az + a(x)˙z = 0, z(0) = w 0 , z˙ (0) = w 1 .

2436


Then the following inequality holds T

T a(x)|˙z| dx dt 2 2

0 Ω

2 a(x)|w| ˙ 2 + a(x)ρ x, w˙ dx dt.

(2.17)

0 Ω

Proof. Set ψ = w − z. Then ψ is solution of ψ¨ + Aψ + a(x)ρ(., w) ˙ − a(x)˙z = 0, ψ(0) = 0,

˙ ψ(0) = 0.

(2.18)

Therefore, we have T

(ψ¨ + Aψ)ψ˙ +

0 Ω

T

a(x)ρ(., w) ˙ − a(x)˙z ψ˙ dx dt = 0.

0 Ω

Thus, we have T Eψ (T ) +

T a|˙z| dx dt = 2

0 Ω

−a(x)ρ(., w) ˙ w˙ + a z˙ w˙ + a(x)˙z ρ(., w) ˙ dx dt.

0 Ω

Since ρ(., v) is monotone increasing with respect to v and vanishes at v = 0, we deduce from the above equality that T

T a|˙z| dx dt 2

0 Ω

a z˙ w˙ + a(x)˙zρ(., w) ˙ dx dt

0 Ω

T η

a|˙z|2 dx dt +

2 1 ˙ dx dt, a|w| ˙ 2 + a(x)ρ(., w) 2η

∀δ > 0.

0 Ω

We choose η = 12 . Thus T

T a|˙z| dx dt 2 2

0 Ω

2 ˙ dx dt. a|w| ˙ 2 + a(x)ρ(., w)

2

0 Ω

The next lemma compares the localized observation for the conservative undamped equation with the localized damping of the linearly damped equation. Lemma 2.3. Assume that a ∈ C(Ω), with a 0 on Ω. Let T > 0 be given, then there exists kT > 0 such that for all (w 0 , w 1 ) ∈ H1/2 × H


T

˙ 2 dx dt kT a|φ|

0 Ω

2437

T a|˙z|2 dx dt

(2.19)

0 Ω

where φ is the solution of the conservative equation (1.6) with (φ 0 , φ 1 ) = (w 0 , w 1 ) and z is the solution of (2.17). Proof. We set θ = φ − z. Then θ satisfies

θ¨ + Aθ = a(x)˙z, θ (0) = 0, θ˙ (0) = 0.

Let t 0 be given. Then we have t Eθ (t) =

a z˙ θ˙ dx ds.

0 Ω

We integrate both sides with respect to t on [0, T ]. This gives T

T Eθ (t) dt =

(T − t)a z˙ θ˙ dx dt.

0 Ω

0

Thus, bounding appropriately the right-hand side of the above relation we obtain T

|θ˙ | dx dt 4T aL∞ (Ω) 2

T a|˙z|2 dx dt.

2

0 Ω

(2.20)

0 Ω

Since φ = θ + z and thanks to (2.20), we obtain (2.19) with kT = 8T 2 a2L∞ (Ω) + 2.

2

Theorem 2.4. We assume the hypotheses of Lemma 2.1 and denote by w and φ the respective w = Ew . Then, the solutions of (1.1) and (1.6) where (w 0 , w 1 ) = (φ 0 , φ 1 ) ∈ H1/2 × H . We set E β following inequality holds w (T ) E w (0) w (0) 1 − ρT L−1 E E

(2.21)

where ρT =

cT . 4kT (c6 H (r02 ) + 1)

(2.22)

2438


Proof. Thanks to our assumptions and to (2.19), we know that there exist cT > 0 and kT > 0 such that T cT Eφ (0)

˙ dx dt kT a|φ|

T a|˙z|2 dx dt.

2

0 Ω

(2.23)

0 Ω

Thanks to the choice of β, we have f Eφ (0) H r02 . This together with (2.17), (2.2) and the definition of the weight function f lead to αT CT Eφ (0)f Eφ (0) Eφ (0)f Eφ (0) + β

T a(x)ρ(x, w) ˙ w˙ dx dt,

(2.24)

0 Ω

where CT =

cT , 2kT (c6 H (r02 ) + 1)

(2.25)

and α=

c5 . (c6 H (r02 ) + 1)

(2.26)

Now the dissipation relation for w gives T a(x)ρ(x, w) ˙ w˙ dx dt = Ew (0) − Ew (T ). 0 Ω

Since Eφ (0) = Ew (0), we obtain

αT f Ew (0) . Ew (T ) Ew (0) 1 − CT − β Thanks to our choice of β in (1.17), we have CT − αT β >

CT 2

(2.27)

= ρT > 0. Thus, we have (2.21).

2

Corollary 2.5. Assume the hypotheses of Lemma 2.1. We set w (kT ), Ek = E

∀k ∈ N.

(2.28)

We define M as in (1.21). Then the following inequalities hold Ek+1 − Ek + ρT M(Ek ) 0,

∀k ∈ N,

(2.29)


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with w (0). E0 = E

(2.30)

Proof. Due to the invariance by translation of (1.1) and (1.6), so that working on the interval [kT , (k + 1)T ] and making the time translation t − kT , we deduce that w (k + 1)T E w (kT ) 1 − CT L−1 E w (kT ) , E

∀k ∈ N.

(2.31)

2

We then easily deduce (2.29).

Proposition 2.6. Assume the hypotheses of Theorem 1.1 and define ψ by ψ(x) = x − ρT M(x),

x ∈ 0, r02

(2.32)

where ρT is defined by (2.22). Then, there exists δ > 0 such that ψ is strictly increasing on [0, δ]. Proof. Thanks to the definition of L and M, we have x(H )2 (x) H (x) = , M L ◦ H (x) = H (x) ΛH (x)

x ∈ 0, r02 .

Since L and H are vanishing at 0 and are invertible in a neighbourhood of 0 and thanks to (1.13), we deduce that lim M (y) = 0.

y→0+

(2.33)

Hence, there exists δ > 0 such that M (y) < Thus ψ is strictly increasing on [0, δ].

1 , ρT

∀y ∈ [0, δ].

(2.34)

2

Proposition 2.7. We assume that (A1) holds. Then lim inf x→0+

H (x) =0 ΛH (x)

(2.35)

where ΛH is defined by (1.11). Proof. We remark that (A1) implies that H (0) = 0. Moreover H and ΛH are nonnegative in a right neighbourhood of 0, so that lim infx→0+ ΛHH(x) (x) = γ exists and is nonnegative. Assume to the contrary that γ > 0. Then, there exist η0 > 0 and δ1 > 0 such that η0

H (x) , ΛH (x)

∀x ∈ (0, δ1 ).

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Hence, we have √ η0 H (s) , √ √ 2 s 2 H (s)

∀s ∈ (0, δ1 ).

Since H (0) = 0, this implies that H (0) > 0 which contradicts Assumption (A1). Hence (2.35) holds. 2 Remark 2.8. Hence the only situation where (1.13) can be violated occurs if lim inf ΛH (x) = 0 x→0+

and limx→0+

H (x) ΛH (x)

does not exist.

3. Comparison with an Euler scheme and proof of Theorem 1.1, Corollary 1.3 and Theorem 1.5 We start by a first comparison result between the energy evaluated at time kT and a sequence yk which is a numerical approximation obtained by a standard Euler scheme applied to an appropriate ordinary differential equation as will be seen later on. Lemma 3.1. Assume the hypotheses of Theorem 1.5. We set ), Ek = E(kT

∀k ∈ N.

(3.1)

We consider the sequence ( yk )k defined by induction as follows

y k + ρT M( yk ) = 0, k+1 − y y0 = E0 .

k ∈ N,

(3.2)

Then the following inequality holds Ek yk ,

(3.3)

for all k ∈ N. Proof. Thanks to the hypotheses of Theorem 1.5, we know that (2.29) holds. On the other hand, the sequence ( yk )k satisfies (3.2). Hence, we have yk ), Ek+1 − y k+1 ψ(Ek ) − ψ(

∀k ∈ N

(3.4)

where ψ is defined by (2.32). We prove (3.3) par induction on k. Since E0 y0 , (3.3) holds for is nonincreasing and k = 0. Assume that (3.3) holds at the order k. First, we remark that since E thanks to our assumption E0 < δ, we have Ek < δ,

∀k ∈ N.


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Moreover, it is easy to check that the sequence ( yk )k is nonincreasing, so that yk y0 = E0 < δ,

∀k ∈ N.

Thanks to our choice of δ, and since we make the assumption that Ek yk , we deduce from Proposition 2.6 that ψ(Ek ) − ψ( yk ) 0. Using this last estimate in (3.4), we deduce that (3.3) holds at the order k + 1.

2

We now compare the sequence ( yk ) obtained using an Euler scheme to the solution of the associated ordinary differential equation at time kT . Lemma 3.2. Assume the hypotheses of Theorem 1.5. We define Ek as in (3.1). We consider the ordinary differential equation

ρT M y(s) = 0, T y(0) = E0 y (s) +

s 0,

(3.5)

and set sk = kT ,

yk = y(sk ),

∀k ∈ N.

(3.6)

Then we have for all k in N yk yk ,

(3.7)

where ( yk )k is defined by (3.2). Remark 3.3. As mentioned before, the sequence ( yk )k is a numerical approximation of the sequence (y(sk ))k thanks to the Euler scheme applied to (3.5). Proof of Lemma 3.2. We integrate (3.5) between sk and sk+1 and compare with the equation satisfied by yk . Thus we have ρT yk+1 − y k ) + k+1 − (yk − y T

sk+1 M y(s) − M( yk ) ds = 0,

∀k ∈ N.

(3.8)

sk

We prove (3.7) by induction on k. The property clearly holds for k = 0. Assume that it holds at the order k. Since y is nonincreasing, we deduce that yk = y(sk ) y0 = E0 < δ. Thus y(s) yk < δ,

∀s ∈ [sk , sk+1 ].

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Since M is nondecreasing, we deduce from (3.8) that yk ) yk+1 − y ψ(yk ) − ψ( k+1 . Since we assume that (3.7) holds at the order k and since ψ is nondecreasing on [0, δ], we deduce yk ) . 0 ψ(yk ) − ψ( Using this last inequality in the above one, we prove (3.7) at the order k + 1.

2

We deduce from Lemma 3.1 and Lemma 3.2 the following result. Corollary 3.4. Assume the hypotheses of Theorem 1.5. Then we have Ek y(sk ),

∀k ∈ N.

(3.9)

We can now give the proof of Theorem 1.5 and Theorem 1.1. Proof of Theorem 1.5. We set T0 =

T , ρT

r = E(0).

(3.10)

We also define r Kr (τ ) =

1 dv. M(v)

(3.11)

τ

Thus the solution y of (3.5) is characterized as y(t) = Kr−1

t , T0

t 0.

(3.12)

On the other hand, we define Ek by (3.1). Then, thanks to (1.22), Ek satisfies (2.29) for all k ∈ N. Let l ∈ N be an arbitrary fixed integer. We have in particular Ek+1+i − Ek+i + ρT M(Ek+i ) 0,

for i = 0, . . . , i = l.

Summing these inequalities from i = 0 to i = l, and using the fact that (Ek )k is a nonincreasing sequence whereas M is a nondecreasing function, we obtain Ek+l+1 − Ek +

1 (l + 1)T M(Ek+l ) 0 T0

so that (l + 1)T M(Ek+l ) T0 Ek ,

∀k, l ∈ N.

(3.13)


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In particular, we have for any arbitrary p ∈ N M(Ep )

T0 T

Ep−l . l+1

inf

l∈{0,...,p}

(3.14)

Now thanks to Corollary 3.4 and to (3.12), we have Ei yi = Kr−1

iT , T0

∀i ∈ N.

Using this last relation in (3.14), we deduce that M(Ep )

T0 T

inf

Kr−1 ( (p−l)T T0 )

l+1

l∈{0,...,p}

(3.15)

.

Let now t T be given and p ∈ N be the unique integer so that t ∈ [pT , (p + 1)T ). Let θ ∈ (0, t − T ] be arbitrary and l ∈ N be the unique integer so that θ ∈ [lT , (l + 1)T ). Then, thanks to (3.15) and by construction, we have

T0 M E(t) M(Ep ) T

inf

Kr−1 ( (p−l)T T0 ) l+1

l∈{0,...,p}

,

and Kr−1

(p − l)T T0

Kr−1

t −θ −T . T0

We deduce that

T −1 t − T − θ M E(t) Kr , θ T0

∀θ ∈ (0, t − T ].

Since M is strictly increasing, we deduce that T M −1 E(t)

inf

θ∈(0,(t−T )]

1 −1 (t − T − θ ) . K θ r T0

Using now the proof of Theorem 2.1 of [2], we deduce that TL E(t)

1 ψr−1 ( t−T T0 )

,

∀t T .

So that (1.23) is proved. If we further assume that lim supx→0+ ΛH (x) < 1, then using Theorem 2.3 of [6] we obtain (1.24). 2 We can now give the proof of our two main results. We start by

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Proof of Theorem 1.1. Since (1.13) holds, we have that limx→0+ M (x) = 0. This, together with = Ew /β. Then the assumptions of Theorem 1.1 imply that Assumption (A2) holds. We set E 0 = Ew (0)/β < δ. Thus, thanks to our assumptions thanks to our choice of β in (1.17) we have E we can apply Corollary 2.5, so that the sequence (Ek )k defined by (2.28) satisfies (2.29). This im satisfies (1.22). We can therefore apply Theorem 1.5 to E, so that E satisfies (1.23). plies that E If additionally lim supx→0+ ΛH (x) < 1 we obtain (1.24). Going back to the definition of E we conclude. 2 Proof of Corollary 1.3. Thanks to Theorem 3.2 in [18], exponential stabilization for system (1.19) implies that there exist T > 0 and cT > 0 such that (1.14) holds for (1.6). We can thus apply Theorem 1.1 to conclude. 2 Remark 3.5. The fact that exponential stabilization implies controllability in Theorem 3.2 in [18] is the generalization of Russell’s principle. 4. Applications to examples of PDEs and dampings Now, we give applications of Theorem 1.1 and Corollary 1.3. In the next result, we denote by CT (E(0)) a positive (explicit) constant depending on E(0) and T whereas KT is a positive constant depending on T . We also only give the expression of g in a right neighbourhood of 0, since as long as g has a linear growth at infinity, the asymptotic behavior of the energy depends only on the behavior of g close to 0. 4.1. Examples of dampings Theorem 4.1. We assume that ρ ∈ C(Ω × R; R) is a continuous monotone nondecreasing function with respect to the second variable on Ω such that ρ(., 0) = 0 on Ω and satisfying (1.7). We assume that a ∈ C(Ω) satisfies (1.8) with a 0 on Ω. We assume that there exists T > 0 such that the solution of (1.6) satisfies the observability inequality (1.14). Then, we have the following results: Example 1. Let g be given by g(x) = x p where p > 1 on (0, r0 ]. Then the energy of solution of (1.1) satisfies the estimate −2 E(t) CT E(0) t p−1 , for t sufficiently large and for all (u0 , u1 ) ∈ H1/2 × L2 (Ω). Example 2. Let g be given by g(x) = x p (ln( x1 ))q where p > 2 and q > 1 on (0, r0 ]. Then the energy of solution of (1.1) satisfies the estimate −2q/(p−1) E(t) CT E(0) t −2/(p−1) ln(t) , for t sufficiently large and for all (u0 , u1 ) ∈ H1/2 × L2 (Ω). Example 3. Let g be given by g(x) = e

−

1 x2

on (0, r0 ].

(4.1)


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Then the energy of solution of (1.1) satisfies the estimate −1 E(t) CT E(0) ln(t) ,

(4.2)

for t sufficiently large and for all (u0 , u1 ) ∈ H1/2 × L2 (Ω). 1

Example 4. Let g be given by g(x) = e−(ln( x )) where 1 < p < 2 on (0, r0 ]. Then the energy of solution of (1.1) satisfies the estimate p

1/p E(t) CT E(0) e−2(ln(KT t)) , for t sufficiently large and for all (u0 , u1 ) ∈ H1/2 × L2 (Ω). Example 5. Let g be given by g(x) = x(ln( x1 ))−p where p > 0. Then the energy of solution of (1.1) satisfies the estimate 1/(p+1) 1/(p+1) 1 E(t) CT E(0) e−KT t , t

(4.3)

for t sufficiently large and for all (u0 , u1 ) ∈ H1/2 × L2 (Ω). Proof. For all examples, g satisfies the assumptions in (A1) and H satisfies the assumption of Theorem 1.1. For Examples 1 and 2, limx→0+ ΛH (x) exists and is in (0, 1). Hence (1.13) holds so that the energy satisfies the simplified upper estimate (1.16). Using [2,5], we deduce the desired upper estimates for both examples. For Examples 3 and 4, limx→0+ ΛH (x) = 0. For Example 3, we have e−1/x H (x) = √ , ΛH (x) x thus (1.13) holds. For Example 4, we find that e H (x) = ΛH (x)

−(ln( √1x ))p

√ x

,

thus (1.13) holds. Therefore, the assumptions of Theorem 1.1 are satisfied. Moreover, since limx→0+ ΛH (x) = 0, the energy satisfies the simplified upper estimate (1.16). Using [2,5], we deduce the desired upper estimates for both examples. For Example 5, limx→0+ ΛH (x) = 1, hence (1.13) holds and the energy satisfies the general upper estimate (1.15). We refer to [2,6] for the computation of the desired estimate. 2

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4.2. First example: stabilization of the nonlinear damped wave equation We consider the following initial and boundary problem: ⎧ ⎨ utt − u + a(x)ρ(x, ut ) = 0, (x, t) ∈ Ω × (0, +∞), u = 0, on ∂Ω × (0, +∞), ⎩ u(x, 0) = u0 (x), ut (x, 0) = u1 (x), on Ω,

(4.4)

where ρ and a satisfy (A1). Hence u satisfies an equation of the form (1.1) with: A = − : D(A) ⊂ H = L2 (Ω) → L2 (Ω), D(A) = u ∈ L2 (Ω), u ∈ L2 (Ω), u|∂Ω = 0 , H1/2 = H01 (Ω). It is well known that A is a self-adjoint operator satisfying (1.2). The conservative equation (1.6) becomes in this case: ⎧ ⎨ φtt − φ = 0, Ω × (0, +∞), φ = 0, ∂Ω × (0, +∞), ⎩ φ(x, 0) = u0 (x), φt (x, 0) = u1 (x),

(4.5) Ω.

We consider the control geometric condition, also called the condition of geometric optics of Bardos, Lebeau and Rauch [10,17] (see also [11,12]): (G.C.C.) The generalized ray of Ω has a finite order contact with the boundary ∂Ω and there exists T0 > 0 such that every generalized ray of Ω with length greater than T0 hits the open set ω. The stability result can now be stated as follows. Theorem 4.2. We assume that Ω is a C ∞ bounded open set with a boundary of class C ∞ . We assume that ρ and a satisfy Assumption (A1) with a ∈ C ∞ (Ω; [0, ∞)). Assume that there exists r0 > 0 sufficiently small so that the function H defined by (1.9) is strictly convex on [0, r02 ] and that (1.13) is satisfied. Moreover assume that the geometric condition (G.C.C.) is valid. Then, there exists T > 0 such that the energy of the solution of (4.4) satisfies Eu (t) βT L

1 ψr−1 ( t−T T0 )

,


(4.6)

If further lim supx→0+ ΛH (x) < 1, then we have the simplified decay rate −1 Eu (t) βT H

DT0 , t −T

(4.7)

for t sufficiently large. Here D is a positive constant which is independent of Eu (0) and T , whereas T0 depends on T and is defined by (3.10), β is a positive constant chosen such as


Eu (0) 2αT Eu (0) , , β > max , CT L(H (r02 )) δ

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(4.8)

where the constants CT > 0, α and δ > 0 are respectively defined by (2.25), (2.26) and (2.34). Proof. Thanks to Theorem 0 in Lebeau [17], exponential stabilization holds for the associated linear damped system (1.19). Hence applying Corollary 1.3, we conclude. 2 For the sake of completeness, let us now describe another geometric condition, namely the piecewise multiplier geometric condition (HG) given below (see K. Liu [18]). It is less general than the condition (G.C.C.) but requires light smoothness assumptions on Ω and a (the smoothness assumptions required in [10] have been strongly weakened in [11,12]). To state this condition, we need some notation. If Ωj ⊂ Ω is a Lipschitz domain, we denote by Γj its boundary and by νj the outward unit normal to Γj . Moreover, if U is a subset of RN and x ∈ R N , we set d(x, U ) = infy∈U |x − y|, and Nε (U ) = {x ∈ RN , d(x, U ) ε}. We make the following geometric assumptions on Ω and ω as in [18] and [21] (for use of the piecewise multiplier method, see [22] for the bondary damped case): ⎧ ⎪ ⎨ ∃ε > 0, domains Ωj ⊂ Ω with Lipschitz boundary Γj for 1 j J, and points xj in RN such that Ωi ∩ Ωj = ∅ if i = j, (HG) ⎪ ⎩ Ω ∩ N [ γ (x ) ∪ (Ω\ Ω )] ⊂ ω, ε j j j j j where γj (xj ) = {x ∈ Γj , (x − xj ) · νj (x) > 0}. These assumptions generalize Zuazua’s assumptions [27] (see also [29]), valid to a single domain Ω1 = Ω and to a single observation point. It allows to treat situations for which for instance Ω is a ball and the damping coefficient a vanishes at the two poles of this ball, so that two observation points at least are requested. Theorem 4.3. We assume that ρ and a satisfy Assumption (A1) where Ω is a bounded open set which is either convex or of class C 1,1 . We also assume that there exists r0 > 0 sufficiently small so that the function H defined by (1.9) is strictly convex on [0, r02 ]. Assume also that (1.13) holds. Then, under the geometric condition (HG), the energy of the solution of the nonlinearly damped equation (4.4) satisfies the estimates given in Theorem 4.2. Our result is also valid for the more general PDE considered by Lebeau [17,10]. Thanks to Theorem 0 in [17] and to [10], and applying Corollary 1.3, we deduce that Theorem 4.4. We assume that (Ω, g) is a C ∞ Riemannian compact and connex manifold, with a boundary of class ∞, whereas −A is the Laplacian on Ω for the metrics g. We assume that ρ and a satisfy Assumption (A1) with a ∈ C ∞ (Ω; [0, ∞)). We assume that there exists r0 > 0 sufficiently small so that the function H defined by (1.9) is strictly convex on [0, r02 ]. Assume also either that (1.18) or (1.13) holds. Then, under the geometric condition (G.C.C.), the energy of the solution of the nonlinearly damped equation (1.1) satisfies the estimates given in Theorem 4.2. We now consider a third example studied in [23] ⎧ ⎨ utt − u + aqu + a(x)ρ(x, ut ) = 0, (x, t) ∈ Ω × (0, +∞), ∂ u = 0, on ∂Ω × (0, +∞), ⎩ ν u(x, 0) = u0 (x), ut (x, 0) = u1 (x), on Ω,

(4.9)

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where q ∈ C(Ω) is a nonnegative and nonzero function and ν represents the outward unit normal vector to the boundary ∂Ω. We define the energy of a solution u by 1 Eu (t) = 2

2 2 2 ut + |∇u| + aqu .

Ω

Theorem 4.5. We assume that Ω is a bounded open set which is either convex or of class C 1,1 , and that ρ and a satisfy Assumption (A1). We further assume that there exists r0 > 0 sufficiently small so that the function H defined by (1.9) is strictly convex on [0, r02 ]. Assume also that (1.13) holds. Then, under the geometric hypothesis (HG), the energy Eu of the solution of (4.9) satisfies the estimates given in Theorem 4.2. Proof. Thanks to Martinez’s result [23], exponential stabilization holds for Eq. (4.9) in case of a linear damping. Applying our Corollary 1.3, we conclude. 2 Remark 4.6. A similar result can be deduced under the geometric condition (G.C.C.) for smoother domains Ω and coefficients a and q. 4.3. Second example: stabilization of a nonlinear Bernoulli–Euler plate equation We consider the following initial and boundary value problem: ⎧ 2 ⎪ ⎨ utt + u + a(x)ρ(x, ut ) = 0, Ω × (0, +∞), u = 0, u = 0, ∂Ω × (0, +∞), ⎪ ⎩ u(x, 0) = u0 (x), ut (x, 0) = u1 (x), Ω,

(4.10)

where ρ and a satisfy (A1) and Ω is a bounded open set which is either convex or of class C 1,1 of RN . In this case: A = 2 ,

D(A) = {u ∈ L2 (Ω), 2 u ∈ L2 (Ω), u|∂Ω = 0, u|∂Ω = 0}.

(4.11)

Moreover the conservative equation (1.6) becomes in this case ⎧ 2 ⎪ ⎨ φtt + φ = 0, Ω × (0, +∞), φ = 0, φ = 0, ∂Ω × (0, +∞), ⎪ ⎩ 0 φ(x, 0) = u (x), φt (x, 0) = u1 (x),

(4.12) Ω.

The stability result can now be stated as follows. Theorem 4.7. Assume that there exists r0 > 0 sufficiently small so that the function H defined by (1.9) is strictly convex on [0, r02 ] and that (1.13) is satisfied. Moreover assume that the geometric


2449

condition (HG) is valid. Then, the energy of the solution of (4.10) satisfies Eu (t) βT L

1 ψr−1 ( t−T T0 )

,


(4.13)

If further lim supx→0+ ΛH (x) < 1, then we have the simplified decay rate −1 Eu (t) βT H

DT0 , t −T

(4.14)

for t sufficiently large. Here D is a positive constant which is independent of Eu (0) and T , whereas T0 depends on T and is defined by (3.10), β is a positive constant chosen such as

2αT Eu (0) Eu (0) β > max , , , CT L(H (r02 )) δ

(4.15)

where the constants CT > 0, α and δ > 0 are respectively defined by (2.25), (2.26) and (2.34). Remark 4.8. 1. In the case where a ∈ C ∞ (Ω) and under a geometric condition like (G.C.C.) we obtain the same stability result (like Theorem 4.7) (by decomposing the plate-like operator in two Schrödinger-like operators ∂t2 + 2 = (i∂t + )(−i∂t + )). 2. By using the equivalence between exact internal controllability of the Kirchhoff plate-like equation (4.16) and the wave equation (see [19] for more details), we obtain a stability result as Theorem 4.7 for the following system, under the same geometric condition (G.C.C.) in the case where a ∈ C ∞ (Ω) and under condition (HG) in the case where a ∈ C(Ω). ⎧ 2 ⎪ ⎨ utt − γ utt + u + a(x)ρ(x, ut ) = 0, (x, t) ∈ Ω × (0, +∞), (4.16) u = 0, u = 0, on ∂Ω × (0, +∞), ⎪ ⎩ 0 1 u(x, 0) = u (x), ut (x, 0) = u (x), on Ω, where ρ and a satisfy (A1), γ > 0 is a constant and Ω is a bounded smooth domain of RN , N 2 (the smoothness assumptions on Ω being adapted if one considers (G.C.C.) or (HG)). Acknowledgments We are grateful to the referees for their valuable comments and suggestions. References [1] F. Alabau-Boussouira, Une formule générale pour le taux de décroissance des systèmes dissipatifs non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 338 (2004) 35–40. [2] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim. 51 (2005) 61–105. [3] F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equations with nonlinear dissipation, J. Evol. Equ. 6 (2006) 95–112.

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[4] F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA 14 (2007) 643–669. [5] F. Alabau-Boussouira, New trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems, J. Differential Equations 249 (2010) 1145–1178. [6] F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differential Equations 248 (2010) 1473–1517. [7] F. Alabau-Boussouira, K. Ammari, Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE’s, C. R. Acad. Sci. Paris Sér. I Math. 348 (2010) 165–170. [8] K. Ammari, M. Tucsnak, Stabilization of Bernoulli–Euler beams by means of a pointwise feedback force, SIAM J. Control Optim. 39 (2000) 1160–1181. [9] K. Ammari, M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var. 6 (2001) 361–386. [10] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992) 1024–1065. [11] N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal. 14 (1997) 157–191. [12] N. Burq, P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 749–752. [13] M. Daoulatli, I. Lasiecka, D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth conditions, Discrete Contin. Dyn. Syst. Ser. S 2 (2009) 67–94. [14] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math. 46 (1989) 245–258. [15] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Collection RMA, vol. 36, Masson/John Wiley, Paris/Chicester, 1994. [16] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations 8 (1993) 507–533. [17] G. Lebeau, Equation des ondes amorties, in: Algebraic and Geometric Methods in Mathematical Physics, Kaciveli, 1993, in: Math. Phys. Stud., vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, pp. 73–109. [18] K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim. 35 (1997) 1574–1590. [19] K. Liu, X. Yu, Equivalence between exact internal controllability of the Kirchhoff plate-like equation and the wave equation, Chin. Math. Ann. 218 (2000) 71–76. [20] W.-J. Liu, E. Zuazua, Decay rates for dissipative wave equations, Ric. Mat. 48 (1999) 61–75. [21] P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut. 12 (1999) 251–283. [22] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var. 4 (1999) 419–444. [23] P. Martinez, Stabilization for the wave equation with Neumann boundary condition by a locally distributed damping, ESAIM Proc. 8 (2000) 119–136. [24] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978) 639–739. [25] J. Vancostenoble, Optimalité d’estimation d’énergie pour une équation des ondes amortie, C. R. Acad. Sci. Paris Sér. I 328 (1999) 777–782. [26] J. Vancostenoble, P. Martinez, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim. 39 (2000) 776–797. [27] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations 15 (1990) 205–235. [28] E. Zuazua, Uniform stabilization of the wave equation by nonlinear feedbacks, SIAM J. Control Optim. 28 (1990) 466–477. [29] E. Zuazua, Propagation, observation and control of wave approximation by finite difference methods, SIAM Rev. 47 (2005) 197–243.


ζ -function and heat kernel formulae Fedor Sukochev a,∗ , Dmitriy Zanin b,1 a School of Mathematics and Statistics, University of New South Wales, Sydney, 2052, Australia b School of Computer Science, Engineering and Mathematics, Flinders University, Bedford Park, 5042, Australia

Received 14 September 2010; accepted 11 October 2010 Available online 9 November 2010 Communicated by Alain Connes

Abstract We present a systematic study of asymptotic behaviour of (generalised) ζ -functions and heat kernels used in noncommutative geometry and clarify their connections with Dixmier traces. We strengthen and complete a number of results from the recent literature and answer (in the affirmative) the question raised by M. Benameur and T. Fack (2006) [1]. © 2010 Elsevier Inc. All rights reserved. Keywords: Zeta function; Heat kernel formulae; Dixmier trace; Noncommutative geometry

1. Introduction The interplay between Dixmier traces, ζ -functions and heat kernel formulae is a cornerstone of noncommutative geometry [8]. These formulae are widely used in physical applications. To define these objects, let us fix a Hilbert space H and let B(H ) be the algebra of all bounded operators on H with its standard trace Tr. Let A and B be positive operators from B(H ). Consider the following [0, ∞]-valued functions t→

1 1+1/t Tr A , t

t→

1 1+1/t Tr A B t


E-mail addresses: [email protected] (F. Sukochev), [email protected] (D. Zanin). 1 Research supported by the Australian Research Council.


(1)

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F. Sukochev, D. Zanin / Journal of Functional Analysis 260 (2011) 2451–2482

and, for fixed 0 < q < ∞, t→

1 Tr exp −(tA)−q , t

t→

1 Tr exp −(tA)−q B . t

(2)

When these functions are finitely-valued, they are frequently referred to as ζ -functions and heat kernel functions associated with the operators A and B. When these functions are bounded, a particular interest is attached to their asymptotic behaviour when t → ∞, which is usually measured with the help of some generalised limit γ : L∞ (0, ∞) → R yielding the following functionals 1 1+1/t 1 1+1/t Tr A Tr A ζγ (A) := γ , ζγ ,B (A) := γ B (3) t t and ϕγ (A) := γ

1 Tr exp −(tA)−q , t

ϕγ ,B (A) := γ

1 Tr exp −(tA)−q B . t

(4)

A natural class of operators for which the formulae (1) and (3) are well defined (respectively, (2) and (4)) is given by the set M1,∞ (respectively, L1,∞ ) of compact operators from B(H ). More precisely, denote by μn (T ), n ∈ N, the singular values of a compact operator T (the singular values are the eigenvalues of the operator |T | = (T ∗ T )1/2 arranged with multiplicity in decreasing order [24, §1]). Then

n 1 M1,∞ := M1,∞ (H ) = T : sup μk (T ) < ∞ n∈N log(n + 1)

(5)

k=1

defines a Banach ideal of compact operators. We set

L1,∞ := T ∈ M1,∞ : ∃C > 0 such that μn (A) C/n, n 1 . It is important to observe that the subset L1,∞ is not dense in M1,∞ (see e.g. [17]). It should also be pointed out that our notation here differs from that used in [8]. It follows from [6, Theorem 4.5] that the functions defined in (1) are bounded if and only if A ∈ M1,∞ . It also follows from [6] and [4] that the functions defined in (2) are bounded if and only if A ∈ L1,∞ . In fact the last result is a strong motivation to consider the following modification of formulae (2). Let us consider a Cesaro operator on L∞ (0, ∞) given by 1 (Mx)(t) = log(t)

t x(s)

ds , s

t ∈ (0, ∞).

1

It follows from [6] and [4] that the functions 1 M t → Tr exp −(tA)−q , t

1 M t → Tr exp −(tA)−q B t

(6)


2453

are bounded if and only if A ∈ M1,∞ . Therefore, for a given generalised limit ω, let us set ω := ω ◦ M

(7)

and instead of the functions given in (4) consider the functions ξω (A) := ω

1 −q Tr exp −(tA) , t

ξω,B (A) := ω

1 −q Tr exp −(tA) B . t

(8)

The class of dilation invariant states ω as above was introduced by A. Connes (see [8]) and it is natural to refer to this class as “Connes states”. We prove in Section 5 that if ω in (7) is dilation invariant, then ξω is a linear functional on M1,∞ . In fact, we also show in Proposition 18 that if ω in (7) is such that ξω is linear on M1,∞ , then necessarily there exists a dilation invariant generalised limit ω0 such that ξω = ξω0 . There is a deep reason to require that the functionals ξω and ζγ be defined on M1,∞ and be linear (and thus, by implication, to consider Connes states). Important formulae in noncommutative geometry [8] and its semi-finite counterpart [5,7,1,6,4] then connect these functionals with Dixmier traces on M1,∞ . Recall that in [9], J. Dixmier constructed a non-normal semi-finite trace (a Dixmier trace) on B(H ) using the weight Trω (T ) := ω

∞ n 1 μk (T ) , log(1 + n) k=1

T > 0,

(9)

n=1

where ω is a dilation invariant state on L∞ (0, ∞). The interplay between positive functionals Trω , ζγ and ξω on M1,∞ makes an important chapter in noncommutative geometry and has been treated (among many other papers) in [8,5, 7,1,6,22,4,23]. We now list a few most important known results concerning this interplay and explain our contribution to this topic. In [5], the equality 1 1+1/t B = ζω◦log,B (A), Trω (AB) = (ω ◦ log) τ A t

0 A ∈ M1,∞ ,

(10)

was established for every B ∈ B(H ) under very restrictive conditions on ω. These conditions are dilation invariance for both ω and ω ◦ log and M-invariance of ω. In [6], for the special case B = 1, the assumption that ω is M-invariant has been removed. However, the case of an arbitrary B appears to be inaccessible by the methods in that article. In Section 4, we prove the general result which implies, in particular, that the equality (10) holds without requiring M-invariance of ω. In [5], the equality 1 1 −q τω (AB) ω τ exp −(tA) B = Γ 1 + t q

(11)

was established under the same conditions on ω and ω ◦ log as above. In [23], in the special case B = 1 the equality (11) was established under the assumption that ω is M-invariant. However,

2454


again the case of an arbitrary B appears to be inaccessible by the methods in that article. Here, we are able to treat the case of a general operator B. In [1] a more general approach to the heat kernel formulae is suggested. It consists of replacing the function t → exp(t −q ) with an arbitrary function f from the Schwartz class. The following equality was proved in [1] ∞ 1 1 ds · τω (AB) ω τ f (tA)B = f t s

(12)

0

for A ∈ L1,∞ and M-invariant ω. In [1, p. 51], M. Benameur and T. Fack have asked whether the result above continues to stand without the M-invariance assumption on ω. In Theorem 49 below, we answer this question affirmatively for a much larger class of functions than the Schwartz class and for any A ∈ M1,∞ . Finally, it is important to emphasise the connection between our results with the theory of fully symmetric functionals. Recall that a linear positive functional ϕ : M1,∞ → C is called fully symmetric if ϕ(B) ϕ(A) for every positive A, B ∈ M1,∞ such that B ≺≺ A. The latter symbol means that n k=1

μk (B)

n

μk (A),

∀n ∈ N.

k=1

It is obvious that every Dixmier trace Trω is a fully symmetric functional. However, the fact that every fully symmetric functional coincides with a Dixmier trace is far from being trivial (see [16] and Theorem 1 below). It is therefore quite natural to ask whether a similar result holds for the sets of all linear positive functionals on M1,∞ formed by the ξω and ζγ respectively. To this end, we establish results somewhat similar to those of [16]. Firstly, in Theorem 22 we prove that if ω in (7) is dilation invariant, then the functional ξω extends to a fully symmetric functional on M1,∞ . Secondly, in Theorem 31 we show that in fact every normalised fully symmetric functional on M1,∞ coincides with some ξω , where ω is dilation invariant. Thus, in view of [16], we can conclude that the set {Trω : ω is a dilation invariant generalised limit} coincides with the set {ξω : ω is a dilation invariant generalised limit} (up to a norming constant). At the same time, a natural question, namely, whether the equality 1 Trω ξω = Γ 1 + q holds for every dilation invariant generalised limit ω is answered in the negative in Theorem 37. Finally, we note that the question on the relationship between the sets {Trω : ω is a dilation invariant generalised limit}, {ζγ : γ is a generalised limit} and {ζω : ω is a dilation invariant generalised limit} remains open. 2. Definitions and notations The theory of singular traces on operator ideals rests on some classical analysis which we now review for completeness.


2455

As usual, L∞ (0, ∞) is the set of all bounded Lebesgue measurable functions on the semi-axis equipped with the uniform norm · . Given a function x ∈ L∞ (0, ∞), one defines its decreasing rearrangement μ(x) = μ(·, x) by the formula (see e.g. [19])

μ(t, x) = inf s 0: m |x| > s t . Let H be a Hilbert space and let B(H ) be the algebra of all bounded operators on H equipped with the uniform norm · . Let N ⊂ B(H ) be a semi-finite von Neumann algebra with a fixed faithful and normal semi-finite trace τ . For every A ∈ N , the generalised singular value function μ(A) = μ(·, A) is defined by the formula (see e.g. [14])

μ(t, A) := inf Ap: τ (1 − p) t . If, in particular, N = B(H ), then μ(A) is a step function and, therefore, can be identified with the sequence {μ(n, A)}n0 of singular numbers of the operators A (the singular values are the eigenvalues of the operator |A| = (A∗ A)1/2 arranged with multiplicity in decreasing order). Equivalently, μ(A) can be defined in terms of the distribution function dA of A. That is, setting dA (s) := τ E|A| (s, ∞) ,

s 0,

we obtain

μ(t, A) = inf s: dA (s) t ,

t > 0.

Here, E|A| denotes the spectral measure of the operator |A|. The following formula follows directly from the von Neumann definition of trace (see the definition at [20, Definition 15.1.1]) τ f (A) = −

∞ f (λ) ddA (λ).

(13)

0

Using the Jordan decomposition, every operator A ∈ B(H ) can be uniquely written as A = (A)+ − (A)− + i (A)+ − (A)− . Here, (A) := 12 (A + A∗ ) (respectively, (A) := 2i1 (A − A∗ )) for any operator A ∈ B(H ) and B+ = BEB (0, ∞) (respectively, B− = BEB (−∞, 0)) for any self-adjoint operator B ∈ B(H ). Recall that A, A ∈ N for every A ∈ N and B+ , B− ∈ N for every self-adjoint B ∈ N . Let ψ : R+ → R+ be an increasing concave function such that ψ(t) = O(t) as t → 0. The Marcinkiewicz function space Mψ (see e.g. [19]) consists of all x ∈ L∞ (0, ∞) satisfying

xMψ

1 := sup t>0 ψ(t)

t μ(s, x) ds < ∞. 0

2456


The Marcinkiewicz operator space Mψ := Mψ (N , τ ) (see e.g. [7,6]) consists of all A ∈ N satisfying 1 AMψ := sup t>0 ψ(t)

t μ(s, A) ds < ∞. 0

We are especially interested in Marcinkiewicz spaces M1,∞ and M1,∞ that arise when ψ(t) = log(1 + t), t 0. In the literature, the ideal M1,∞ is sometimes referred to as the Dixmier ideal. We recommend the recent paper of A. Pietsch, [21], discussing the origin of M1,∞ in mathematics. For s > 0, dilation operators σs : L∞ → L∞ are defined by the formula (σs x)(t) = x(t/s). Clearly, σs : M1,∞ → M1,∞ (see also [19, Theorem II.4.4]). Further, we need to recall the important notion of Hardy–Littlewood majorization. Let A, B ∈ N . B is said to be majorized by A and written B ≺≺ A if and only if t

t μ(s, B) ds

0

μ(s, A) ds,

t 0.

(14)

0

We have (see [14]) A + B ≺≺ μ(A) + μ(B) ≺≺ 2σ1/2 μ(A + B).

(15)

One of the most widely used ideals in von Neumann algebras is 1/p

Lp := Lp (N , τ ) = A ∈ N : Ap := τ |A|p 1. A linear functional ϕ : M1,∞ → C is said to be symmetric if ϕ(B) = ϕ(A) for every positive A, B ∈ M1,∞ such that μ(B) = μ(A). A linear functional ϕ : M1,∞ → C is said to be fully symmetric if ϕ(B) ϕ(A) for all A, B ∈ M+ 1,∞ such that B ≺≺ A [10–12]. Every fully symmetric functional is symmetric and bounded. The converse fails [17]. A positive normalised linear functional γ : L∞ (0, ∞) → R is called a generalised limit if γ (z) = 0 for every z ∈ L∞ (0, ∞) such that limt→∞ z(t) = 0. A linear functional γ : L∞ (0, ∞) → R is called dilation invariant if γ (σs z) = γ (z) for every z ∈ L∞ (0, ∞) and every s > 0. Let S ⊆ B(H ). We denote by S + the set of all positive operators from S. Let ω : L∞ (0, ∞) → R be a dilation invariant generalised limit. Define a functional τω on M+ 1,∞ by the formula

1 τω (A) = ω log(1 + t)

t

μ(s, A) ds .

0


2457

The functional τω is additive and unitarily invariant on M+ 1,∞ . Thus, τω extends to a fully symmetric functional on M1,∞ . One usually refers to it as to a Dixmier trace. We refer the reader to [9,8,5,7,6,16] for details. Further, we use the following properties of Dixmier traces. Let A ∈ M1,∞ and let B ∈ N . We have (see [8,5]) τω (AB) = τω (BA).

(16)

Suppose that B > 0. It follows from (16) that τω (AB) = τω B 1/2 AB 1/2 .

(17)

Suppose that the trace τ on the von Neumann algebra N is infinite and the algebra N is either diffuse (that is with no minimal projections) or else is B(H ). Given any finite sequence {An } of operators, we can construct a sequence of operators {Bn } such that μ(An ) = μ(Bn ) for all n’s and Bn Bm = 0 for all n = m. Further, we refer to any such sequence {Bn } as a “sequence of disjoint copies of {An }”. Cesaro operator M is defined on L∞ (0, ∞) by the formula 1 (Mx)(t) = log(t)

t x(s)

ds , s

t ∈ (0, ∞).

1

3. Preliminary important results In this section, for the reader’s convenience, we collect a number of key known results, which will be used throughout this paper. The following important theorem is proved in [16, Theorem 11] for general Marcinkiewicz spaces. Theorem 1. Every fully symmetric functional on M1,∞ is a Dixmier trace. The following theorem is an analog of Lidskii formula (see [24]) for Dixmier traces. It is proved in [23, Theorem 33] for a large subclass of Marcinkiewicz spaces which contains M1,∞ . Theorem 2. Let A ∈ M1,∞ and let τω be an arbitrary Dixmier trace on M1,∞ . We have

1 τω (A) = ω log(t)

λ .

|λ|>log(t)/t, λ∈σ (A)

The following ω-variant of the classical Karamata theorem is established in [5]. Theorem 3. Let β be a continuous increasing function. Set ∞ h(t) = 0

e−(u/t) dβ(u). q

2458


We have

h(t) ω t

1 β(t) =Γ 1+ ω q t

for any dilation invariant generalised limit ω. Consider the ideal KN of τ -compact operators in N (that is the norm closed ideal generated by the projections E ∈ N with τ (E) < ∞). The following result is not new (see [15, Chapter II, Lemma 3.4]). We present a short proof for convenience of the reader. Theorem 4. Let A, B ∈ N be positive τ -compact operators. We have B ≺≺ A if and only if τ (B − t)EB (t, ∞) τ (A − t)EA (t, ∞) ,

∀t > 0.

(18)

Proof. Fix t > 0. It follows from the definition of generalised singular value function that μ(AEA (t, ∞)) = μ(A)χ[0,dA (t)] . Applying [14, Proposition 2.7] to the operator AEA (t, ∞), we have τ AEA (t, ∞) =

dA (t)

μ(s, A) ds, 0

and hence τ (A − t)EA (t, ∞) =

dA (t)

μ(s, A) − t ds.

0

The function u u→

μ(s, A) − t ds

0

attains its maximum at u = dA (t). If B ≺≺ A, then dB (t)

dB (t)

dA (t)

0

0

0

μ(s, B) − t ds

μ(s, A) − t ds

Inequality (18) follows now from (19).

μ(s, A) − t ds.

(19)


2459

Suppose now that (18) holds. Fix u > 0 and set t = μ(u, A). It follows that u

dB (t)

μ(s, B) − t ds = τ (B − t)EB (t, ∞)

μ(s, B) − t ds

0

0

τ (A − t)EA (t, ∞) =

u

μ(s, A) − t ds.

0

Hence, u

u μ(s, B) ds

0

Since u is arbitrary, we have B ≺≺ A.

μ(s, A) ds. 0

2

4. ζ -function formulae We begin by showing that the functionals given in (3) are well defined on M+ 1,∞ . Lemma 5. If γ : L∞ (0, ∞) → R is a generalised limit, then ζγ (A) < ∞ and ζγ ,B (A) < ∞ for any A ∈ M+ 1,∞ . Proof. It is clear that μ(s, A) ≺≺ (1 + s)−1 A1,∞ . Therefore, 1+1/t τ A1+1/t A1,∞

∞ 0

dt 1+1/t = tA1,∞ . (1 + s)1+1/t

Hence, ζγ (A) A1,∞ . It follows from τ A1+1/t B Bτ A1+1/t that ζγ ,B (A) Bζγ (A).

2

Remark 6. Let x, y ∈ L∞ (0, ∞). For any generalised limit γ such that γ (|x − 1|) = 0, we have γ (xy) = γ (y). Indeed, |γ (xy − y)| γ (|x − 1|)y = 0. Lemma 7. For any A, C ∈ M+ 1,∞ we have τ A1+s + C 1+s τ (A + C)1+s 2s τ A1+s + C 1+s ,

s > 0.

2460


Proof. In the special case when N = B(H ), the first inequality can be found in [18, (2.9)]. In the general case, it follows directly from Proposition 4.6(ii) of [14] when f (u) = u1+s , u > 0. The second inequality follows from the same proposition by setting there a = a ∗ = b = b∗ = 2−1/2 . 2 Let A ∈ M1,∞ . For a functional ζγ defined on M+ 1,∞ by (3) (see Lemma 5), we set ζγ (A) := ζγ (A)+ − ζγ (A)− + i ζγ (A)+ − ζγ (A)− .

(20)

The following theorem shows that functionals ζγ defined by (20) are fully symmetric on M1,∞ . Theorem 8. If γ : L∞ (0, ∞) → R is a generalised limit, then ζγ is a fully symmetric linear functional on M1,∞ . Proof. To verify that ζγ is linear, it is sufficient to check that ζγ (A + C) = ζγ (A) + ζγ (C) for any A, C ∈ M+ 1,∞ . It follows from the left-hand side inequality of Lemma 7 that ζγ (A + C) ζγ (A) + ζγ (C). Noting that γ (|21/t − 1|) = 0, it follows from the right-hand side inequality of Lemma 7 and Remark 6 that ζγ (A + C) ζγ (A) + ζγ (C). Therefore, we have ζγ (A + C) = ζγ (A) + ζγ (C). The homogeneity of ζγ follows from Remark 6. Finally, if 0 C ≺≺ A ∈ M+ 1,∞ , then C, A ∈ 1 1 1+s 1+s 1+1/t 1+1/t ) t τ (A ) and so ζγ (C) ζγ (A). 2 L1+s and τ (C ) τ (A ). Hence, t τ (C Let B ∈ N . We extend the functional ζγ ,B on M1,∞ , similarly to (20). Observe that ζγ ,B1 +B2 (A) = ζγ ,B1 (A) + ζγ ,B2 (A),

B1 , B2 ∈ N , A ∈ M1,∞ .

Lemma 9. If A ∈ M1,∞ and Bn → B in N , then ζγ ,Bn (A) → ζγ ,B (A). Proof. It is sufficient to prove the assertion for A ∈ M+ 1,∞ . Since 1+s τ A B − τ A1+s Bn τ A1+s B − Bn , we obtain ζγ ,B (A) − ζγ ,B (A) ζγ (A)B − Bn . n

2


2461

The following lemma follows immediately from [5, Lemma 3.3]. Lemma 10. Let A, B ∈ B + (H ) and let s > 0. We have (i) (B 1/2 AB 1/2 )1+s B 1/2 A1+s B 1/2 if 0 B 1. (ii) (B 1/2 AB 1/2 )1+s B 1/2 A1+s B 1/2 if B 1. The result below significantly strengthens [5, Proposition 3.6] by removing all extra assumptions on the generalised limit γ . Proposition 11. If γ : L∞ (0, ∞) → R is a generalised limit, then ζγ ,B (A) = ζγ B 1/2 AB 1/2 ,

∀A ∈ M1,∞ , B ∈ N + .

Proof. It is sufficient to prove the assertion for A ∈ M+ 1,∞ . Suppose first that there are constants 0 < m M < ∞ such that m B M. Applying Lemma 10 to the operators A and M −1 B (respectively, m−1 B), we have 1+s M s B 1/2 A1+s B 1/2 . ms B 1/2 A1+s B 1/2 B 1/2 AB 1/2 Therefore, 1+1/t 1 1/t 1+1/t 1 1/t 1+1/t 1 1/2 m τ A M τ A B τ B AB 1/2 B . t t t Since γ (|m1/t − 1|) = 0 and γ (|M 1/t − 1|) = 0, it follows from Remark 6 that ζγ ,B (A) = ζγ (B 1/2 AB 1/2 ). For an arbitrary B ∈ N + , we set Bn := BEB (1/n, ∞) + 1/nEB [0, 1/n], n 1. From the first part of the proof, we have 1/2 1/2 ζγ ,Bn (A) = ζγ Bn ABn . 1/2

1/2

Since Bn ABn

→ B 1/2 AB 1/2 in M1,∞ , we have by Theorem 8 1/2 1/2 ζγ Bn ABn → ζγ B 1/2 AB 1/2 .

On the other hand, by Lemma 9 we have ζγ ,Bn (A) → ζγ ,B (A).

2

The following is our main result on the ζ -function. Theorem 12. If γ : L∞ (0, ∞) → R is a generalised limit, then ζγ ,B (A) = ζγ (AB),

∀A ∈ M1,∞ , B ∈ N .

2462


Proof. It is sufficient to prove the assertion for B ∈ N + . By Theorems 8 and 1, we know that ζγ is a Dixmier trace on M1,∞ . Hence, by (17), we have ζγ (B 1/2 AB 1/2 ) = ζγ (AB). The assertion follows now from Proposition 11. 2 Our remaining objective in this section is to provide strengthening of several formulae linking Dixmier traces and ζ -functions from [5,6]. −1 Lemma 13. Let A ∈ M+ 1,∞ . The mapping s → s ζγ ◦σs (A) is convex and, therefore, continuous.

Proof. For all t, s > 0, we have s

−1

σs

1 1+1/t 1 τ A = τ A1+s/t . t t

Therefore, for every s > 0 s −1 ζγ ◦σs = γ

1 1+s/t . τ A t

Let λi > 0 and let λ1 + λ2 = 1. Since the mapping t → a 1+t is convex for every a > 0, it follows from the spectral theorem that the map s → As is also convex. Therefore, for all positive real numbers s1 , s2 and t, we have A1+(λ1 s1 +λ2 s2 )/t λ1 A1+s1 /t + λ2 A1+s2 /t . The assertion follows immediately.

2

Let γ be a generalised limit on L∞ (0, ∞). Below, we will formally apply the notation ζγ ,B (A) introduced in (3) to some unbounded positive operators B on H . Lemma 14. Let A ∈ N be a positive τ -compact operator and let B 1 be an unbounded operator commuting with A. If (the closure of ) the product AB ∈ M1,∞ and AB n ∈ N for every n ∈ N, then ζγ (AB) = ζγ ,B (A). Proof. It follows from AB = BA and B 1 that A1+s B (AB)1+s . The inequality ζγ ,B (A) ζγ (AB) follows immediately. 1/2n Set cn := AB 2n , n 1 and observe that BA1/2n cn . Setting Bn = BEA [0, cn−1 ], we obtain Bn A1/n = BA1/2n · A1/2n EA 0, cn−1 (cn A)1/2n EA 0, cn−1 1. It follows from (21) that A1+1/t Bn (ABn )1+n/t (n−1) . Thus, γ

1 1+1/t 1 n−1 τ A τ (ABn )1+n/t (n−1) = ζγ ◦σn/(n−1) (ABn ). Bn γ t t n

(21)


2463

Since A is τ -compact, then B −Bn is bounded operator with finite support. Due to the linearity with respect to B, we have ζγ ,B (A) = ζγ ,Bn (A)

n−1 n−1 ζγ ◦σn/(n−1) (ABn ) = ζγ ◦σn/(n−1) (AB). n n 2

The assertion follows now from Lemma 13.

The following result is mainly known (see [5,6]). Our proof is however much simpler than the arguments used there. Theorem 15. If ω is a dilation invariant generalised limit such that the generalised limit ω ◦ log is still dilation invariant, then τω = ζω◦log . Proof. It is sufficient to verify the equality τω = ζω◦log on positive operators A ∈ M+ 1,∞ such that A e−1 . Define a continuously increasing function β : (0, ∞) → (0, ∞) by ∞ β(u) := −

λ ddA (λ).

ue−u

Let h be as in Theorem 3 as applied to the above β. Define an operator B 1 by the formula A = Be−B and set C = e−B . We have ∞ h(t) =

e

−u/t

∞ dβ(u) = −

0

(13) e−u(1+1/t) u ddA ue−u = τ C 1+1/t B .

(22)

0

The conditions of Lemma 14 are valid for B and C. Indeed, B commutes with C, BC = A ∈ M1,∞ and B n e−B ∈ N for every n ∈ N. By Lemma 14, we have h(t) . ζω◦log (A) = ζω◦log,B (C) = (ω ◦ log) t By Theorem 2, we have

−1 τω (A) = ω log(t)

∞

β(t) . λ ddA (λ) = (ω ◦ log) t

(23)

log(t)/t

We can now conclude h(t) (Thm. 3) β(t) (23) ζω◦log (A) = (ω ◦ log) = (ω ◦ log) = τω (A). t t (22)

2

The following corollary strengthens and extends the results of [6, Theorem 4.11] and [5, Theorem 3.8]. It follows immediately from Theorems 15 and 12.

2464


Corollary 16. If ω is a dilation invariant generalised limit such that the generalised limit ω ◦ log is still dilation invariant, then 1 1+1/t B , ∀A ∈ M+ τω (AB) = (ω ◦ log) τ A 1,∞ , B ∈ N . t 5. The linearity criterion for functionals ξγ In this section we focus on functionals ξγ (·) defined in (8). It follows from the proof of [6, Theorem 5.2] that 1 M t → τ exp −(tA)−q ∈ L∞ (0, ∞), ∀A ∈ M+ 1,∞ , t and therefore, 1 ξγ (A) := (γ ◦ M) t → τ exp −(tA)−q t

(24)

is finite for every A ∈ M+ 1,∞ and every generalised limit γ on L∞ (0, ∞). We note, in passing that a stronger result than [6, Theorem 5.2] is established in Theorem 40 below. Let A ∈ M1,∞ . For a functional ξγ , we set ξγ (A) := ξγ (A)+ − ξγ (A)− + i ξγ (A)+ − ξγ (A)− .

(25)

It is an open question how to describe the set of all generalised limits γ for which (25) yields a linear functional ξγ . However, the class of linear functionals ξγ is an easier object. Below in Proposition 18, we show that the sets of linear functionals {ξγ : γ is a generalised limit} and linear functionals {ξω : ω is a dilation invariant generalised limit} coincide. Lemma 17. For every locally integrable z with Mz ∈ L∞ (0, ∞), we have (M ◦ σs −1 − σs −1 ◦ M)(z) ∈ C0b (0, ∞),

∀s > 0.

Here, C0b (0, ∞) is the space of all bounded continuous functions tending to 0 at ∞. Proof. Fix s > 0. The assertion follows by writing 1 (M ◦ σs −1 − σs −1 ◦ M)(z) = log(t)

st

1 du − z(u) u log(st)

s

st z(u) 1

and noting that the assumption Mz ∈ L∞ (0, ∞) easily implies that 1 log(st)

st 1

1 du − z(u) u log(t)

st z(u) 1

du ∈ Cb0 (0, ∞). u

2

du u


2465

Proposition 18. Suppose that a generalised limit γ on L∞ (0, ∞) is such that ξγ is a linear functional on M1,∞ . Then, there exists a dilation invariant generalised limit ω on L∞ (0, ∞) such that ξγ = ξω . Proof. Fix s > 0 and observe that

−q 1 1 −q t → τ exp −(tsA) = sσs −1 t → τ exp −(tA) . t t

(26)

Therefore, 1 −q ξγ (sA) = s(γ ◦ M ◦ σs −1 ) τ exp −(tA) . t By the assumption, we have ξγ (sA) = sξγ (A) and appealing to Lemma 17, we obtain 1 ξγ (A) = (γ ◦ σs −1 ◦ M) τ exp −(tA)−q , t

∀s > 0.

(27)

Let E be the linear span of the functions t →M

1 τ exp −(tA)−q , t

A ∈ M+ 1,∞ ,

and let F := E + C0b (0, ∞). We claim that the space F is dilation invariant. Indeed, it follows from Lemma 17 and (26) that every function −q 1 σs −1 t → M τ exp −(tA) t belongs to the set 1 + C0b (0, ∞). s −1 t → M τ exp −(tsA)−q t It follows from (27) that γ ◦ σs −1 = γ on F . By the invariant form of the Hahn–Banach theorem (see [13, p. 157]) applied to the group of dilations {σs }s>0 , we see that γ |F can be extended to a dilation invariant generalised limit ω on L∞ (0, ∞). 2 The following lemma can be found in [23]. We present a shorter proof for convenience of the reader. Lemma 19. If ω is a dilation invariant generalised limit on L∞ (0, ∞), then 1 1 1 ξω (A) = Γ 1 + (ω ◦ M) dA , q t t

∀A ∈ M+ 1,∞ .

(28)

2466


Proof. It follows from (13) that τ exp −(tA)−q =

∞

e−(u/t) ddA q

1 . u

(29)

0

Setting β(u) = dA (1/u), multiplying both sides of (29) by 1/t and applying Theorem 3 to ω ◦ M (which is dilation invariant, see [8]), we obtain (28). 2 Lemma 20. Let A ∈ M+ 1,∞ and let ω be a dilation invariant generalised limit on L∞ (0, ∞). We have 1 1 1 1 ξω (A) = Γ 1 + ω τ A− EA , ∞ . (30) q log(1 + t) t t Proof. In view of Lemma 19, it is sufficient to show that right-hand sides of (28) and (30) coincide. This easily follows from the following computation, where we use integration by parts t 1 1 1 ds 1 1 1 = M dA dA = dA (u) du t t log(t) s s2 log(t)

1

1/t

1 1 udA (u)|11/t − = log(t) log(t)

1 u ddA (u) 1/t

1 1 1 τ A− EA , ∞ + o(1). = log(t) t t

2

Lemma 21. Let ω be a dilation invariant generalised limit on L∞ (0, ∞) and let A, B ∈ M+ 1,∞ be such that B ≺≺ A. We have ξω (B) ξω (A). Proof. The assertion follows from Lemma 20 and Theorem 4.

2

The following is the main result of this section. Theorem 22. For any dilation invariant generalised limit ω on L∞ (0, ∞), the functional ξω given by (25) is linear and fully symmetric on M1,∞ . Proof. The assertion follows from Lemma 21 provided we have shown that ξω (A + B) = ξω (A) + ξω (B),

∀A, B ∈ M+ 1,∞ .

(31)

To this end, we observe first that since ω and ω ◦ M are dilation invariant, it follows from Lemma 21 and (15) that ξω (A + B) = ξω μ(A) + μ(B) ,

∀A, B ∈ M+ 1,∞ .


2467

Now, let C and D be disjoint copies of A and B (see Section 2). Thus, we have ξω (C + D) = ξω μ(C) + μ(D) = ξω μ(A) + μ(B) = ξω (A + B). However, the equality ξω (C + D) = ξω (C) + ξω (D) for positive operators C and D such that CD = 0 follows immediately from the definition (24). Since the equalities ξω (A) = ξω (C), ξω (B) = ξω (D) are obvious, we arrive at (31). 2 6. Every fully symmetric functional has form ξω It follows from Theorems 22 and 1 that the functional ξω is a fully symmetric functional on M1,∞ whenever ω is a dilation invariant generalised limit ω on L∞ (0, ∞). In this section, we show the converse. Define a (non-linear) operator T : M+ 1,∞ → L∞ (0, ∞) by the formula (T A)(t) =

1 τ log(1 + t)

A−

1 1 EA , ∞ , t t

t > 0.

(32)

We need some properties of the operator T . Firstly, we show that it is additive on certain pairs of A, B ∈ M+ 1,∞ . Lemma 23. Let A, B ∈ M+ 1,∞ be such that AB = BA = 0. It follows that T (A+B) = T A+T B. Proof. It follows immediately from the assumption that 1 1 1 1 1 1 A+B − EA+B ,∞ = A − EA , ∞ + B − EB ,∞ . t t t t t t

2

Next, we explain the connection of the operator T with fully symmetric functionals on M1,∞ . Lemma 24. Let the operators A, B ∈ M+ 1,∞ be such that T B T A. For every fully symmetric functional ϕ on M1,∞ , we have ϕ(B) ϕ(A). Proof. It follows immediately from the definition (32) that τ

1 1 1 1 EB ,∞ τ A− EA , ∞ , B− t t t t

Applying Theorem 4 we obtain B ≺≺ A and so ϕ(B) ϕ(A).

∀t > 0.

2

Lemma 25. Let A, B ∈ M+ 1,∞ . For every fully symmetric functional ϕ on M1,∞ , we have ϕ(B) − ϕ(A) ϕM∗1,∞ lim sup(T B − T A)(t). t→∞

2468


Proof. Without loss of generality, ϕM∗1,∞ = 1. Denote the right-hand side by c and suppose that c 0 (the case when c < 0 is treated similarly). Fix ε > 0. We have (T B − T A)(t) c + ε for all sufficiently large t. Let C be an operator with μ(t, C) = (c + 2ε)/(1 + t). We have T B T A + T C for all sufficiently large t. Let A1 and C1 be disjoint copies of A and C, respectively. It follows from Lemma 23 that T B(t) T (A1 + C1 )(t) for all sufficiently large t. Choose 0 < δ small enough to guarantee T B1 (t) T (A1 + C1 )(t) for all t > 0, where B1 := min{B, δ}. By Lemma 24, we have ϕ(B1 ) ϕ(A1 ) + ϕ(C1 ), or equivalently ϕ(B) ϕ(A) + c + 2ε. Since ε is arbitrarily small, we are done. 2 Lemma 26. Let A1 , . . . , An ∈ M+ 1,∞ and let λ1 , . . . , λn ∈ R for some n 1. For every fully symmetric functional ϕ on M1,∞ we have n

λk ϕ(Ak ) lim sup

n

t→∞

k=1

λk (T Ak )(t).

(33)

k=1

Proof. Both sides of the inequality (33) depend continuously on the λk ’s. Without loss of generality, we may assume that all λk ∈ Q. Multiplying both sides by the common denominator, we may assume that all λk ∈ Z. Writing

λk Ak =

|λk |

sgn(λk )Ak

k=1

we see that it is sufficient to prove (33) only for the case when λk = ±1 for every k. Let {Bk } be a disjoint copy sequence of {Ak }. Both sides of the inequality (33) do not change if we replace Ak with Bk . Without loss of generality, the operators Ak Aj = 0, k = j . By Lemma 25 we have n

λk ϕ(Ak ) = ϕ

Ak − ϕ

λk =1

k=1

Ak

λk =−1

lim sup T Ak − T Ak (t). t→∞

λk =1

λk =−1

Since Ak Aj = 0 for all k = j , we have by Lemma 23 that

T

λk =1

and the assertion follows.

n Ak − T Ak = λk T Ak λk =−1

k=1

2

b Lemma 27. Let E be the linear span of T M+ 1,∞ and C0 (0, ∞). For every s > 0 we have σs E = E.


2469

Proof. It follows from the definition (32) that for every s > 0, we have σs T A ∈ sT s −1 A + C0b (0, ∞),

∀A ∈ M+ 1,∞ .

2

(34)

Let ϕ be a normalised fully symmetric functional on M1,∞ . We need the following linear functional on E. Definition 28. For every z ∈ E such that z∈

n

λk T Ak + C0∞ (0, ∞)

k=1

we set ρ(z) =

n

λk ϕ(Ak ).

k=1

That ρ is well defined is proved below. Lemma 29. The linear functional ρ : E → R is well defined. For every z ∈ E, we have ρ(z) lim sup z(t). t→∞

Proof. Let z ∈ E be such that z∈

n

λk T Ak + C0b (0, ∞),

z∈

k=1

m

μk T Bk + C0b (0, ∞).

k=1

We have n

λk T Ak −

k=1

m

μk T Bk ∈ C0b (0, ∞).

k=1

It follows from Lemma 26 that n k=1

λk ϕ(Ak ) =

m

μk ϕ(Bk ),

k=1

so that ρ is well defined. The second assertion directly follows from Lemma 26.

2

Lemma 30. Let ϕ be a normalised fully symmetric functional on M1,∞ . There exists a dilation invariant generalised limit ω on L∞ (0, ∞) such that ϕ(A) = ω(T A) for every A ∈ M+ 1,∞ .

2470


Proof. For every A ∈ M+ 1,∞ , we have Def. 28 −1 (34) ρ(σs T A) = ρ sT s −1 A = sϕ s A = ρ(T A). Therefore, ρ is σs -invariant on E. It follows from Lemma 29 that ρ(z) lim sup z(t), t→∞

z ∈ E.

By the invariant form of the Hahn–Banach theorem (see [13, p. 157]) applied to the group of dilations {σs }s>0 , we can extend ρ to a dilation invariant generalised limit on L∞ (0, ∞). 2 The following assertion is the main result of this section. It permits representation of a fully symmetric functional ϕ via heat kernel formulae. Theorem 31. Let ϕ be a fully symmetric functional on M1,∞ . There exists a dilation invariant generalised limit ω on L∞ (0, ∞) such that ϕ = const · ξω . Proof. It follows from Lemma 30 that there exists a dilation invariant generalised limit ω such that 1 1 1 τ A− . ϕ(A) = ω EA , ∞ log(1 + t) t t The assertion follows now from Lemma 20.

2

7. A counterexample It is known (see [23, Theorem 33] and the more general result in Corollary 51 below) that the equality 1 ξω (A) = Γ 1 + τω (A), q

A ∈ M+ 1,∞ ,

holds for every M-invariant generalised limit ω on L∞ (0, ∞) (see also earlier results with more restrictive assumptions on ω in [5, Theorem 4.1] and [6, Theorem 5.2]). In view of Theorems 31 and 1, it is quite natural to ask whether the equality above holds for every dilation invariant generalised limit ω. In this section we prove that this is not the case. Lemma 32. Let ω be a dilation invariant generalised limit on L∞ (0, ∞). For every s > 1, we have (35) χ[eek ,seek ) = 0, ω ω

k

k

χ(ek+ek /s,ek+ek ] = 0.

(36)


2471

Proof. Denote the left-hand side of (35) by f (s). Due to the dilation invariance of ω, we have f (s) = ω

χ[teek ,steek ) = f (st) − f (t),

s, t > 1.

k

Since f is monotone and bounded, we have f = 0. Denote the left-hand side of (36) by g(s). Due to the dilation invariance of ω, we have g(s) = ω

χ(ek+ek /st,ek+ek /t] = g(st) − g(t),

s, t > 1.

k

Since g is monotone and bounded, we have g = 0.

2

Lemma 33. Let ω be a dilation invariant generalised limit on L∞ (0, ∞). We have (i) ω

k

t −ek e χ[ek−1+ek−1 ,ek+ek ] (t) = 0. log(t)

(ii) ω

k

1 k ek+e χ[eek ,eek+1 ] (t) = 0. t log(t)

Proof. We only prove the first assertion. Proof of the second one is similar. Fix s > 1. We have t 2 k k e−e + 2e−e /2 , log(t) s

k

∀t ek+e /s, ∀k 1

and, therefore, k

t 2 k e−e χ[ek−1+ek−1 ,ek+ek ] (t) + χ[ek+ek /s,ek+ek ] (t) log(t) s k k e−e /2 χ[ek−1+ek−1 ,ek+ek ] (t). +2 k

Clearly, ω

k

e−e

k /2

χ[ek−1+ek−1 ,ek+ek ] (t) = 0.

2472


It follows from Lemma 32 that ω

k

t 2 −ek e χ[ek−1+ek−1 ,ek+ek ] (t) . log(t) s

Since s is arbitrarily large, we have

ω

k

t k e−e χ[ek−1+ek−1 ,ek+ek ] (t) = 0. log(t)

2

Lemma 34. There exists a dilation invariant generalised limit ω on L∞ (0, ∞) such that ω

χ[eek ,ek+ek ) = 1,

ω

k

χ[ek+ek ,eek+1 ) = 0.

k

Proof. Define a positive, homogeneous functional π on L∞ (0, ∞) by the formula 1 π(x) = lim sup log log(N ) N →∞

N log(N )

x(s)

ds . s

N

It is verified in [23, Lemma 4] that every ω ∈ L∞ (0, ∞)∗ satisfying ω π is dilation invariant. Observing that

π

χ[eek ,ek+ek ) = 1,

k

let us select ω ∈ L∞ (0, ∞)∗ satisfying ω π and such that ω

χ[eek ,ek+ek ) = 1.

k

Therefore,

ω

χ[ek+ek ,eek+1 ) = 1 − ω χ[eek ,ek+ek ) = 0.

k

2

k

Define a function x by the formula x = sup e−e χ[0,ek+ek ] . k

k∈N

(37)


Fix k 1. For every t ∈ [ek−1+e 1 log(1 + t)

k−1

2473

k

, ek+e ], we have

t

ek+e

x(s) ds e

x(s) ds e

1−k

0

k

1−k

k

e−e · en+e n

n

n=1

0

e2 , e−1

which guarantees x ∈ M1,∞ . Lemma 35. Let x be as in (37) and let ω be as in Lemma 34. We have τω (x) = (e − 1)−1 . Proof. Fix t ∈ [ek−1+e

k−1

k

, ek+e ]. We have t x(u) du = 0

ek k + te−e + O(1). e−1

It follows that τω (x) = (e − 1)−1 ω +ω

k

k

t log(t)

ek χ[ek−1+ek−1 ,ek+ek ] (t) log(t) −ek e χ[ek−1+ek−1 ,ek+ek ] (t) .

By Lemma 33, the second generalised limit above vanishes. We claim that the first generalised limit above is 1. Indeed, k

ek χ[ek−1+ek−1 ,ek+ek ] (t) 1 + o(1) χ[eek ,ek+ek ] (t) log(t) k

and k

ek χ[ek−1+ek−1 ,ek+ek ] (t) χ[eek ,ek+ek ] (t) + e χ[ek−1+ek−1 ,eek ] . log(t) k

The claim follows from Lemma 34.

k

2

Lemma 36. Let x be as in (37) and let ω be as in Lemma 34. We have 1 e ξω (x) = Γ 1+ . e−1 q k

Proof. Fix t ∈ [ee , ee

k+1

). We have

x>1/t

1 1 ek+1 k − ek+e + O(1). x(u) − du = t e−1 t

2474


This estimate and Lemma 20 yield k 1 e e ξω (x) = ω χ[eek ,eek+1 ] (t) Γ (1 + 1/q) e−1 log(t) k 1 k+ek χ[eek ,eek+1 ] (t) . −ω e t log(t) k

It follows from Lemma 33 that the second generalised limit is 0. We claim that the first generalised limit is 1. Indeed, k

ek χ[eek ,eek+1 ] (t) 1 + o(1) χ[eek ,ek+ek ] log(t) k

and k

The claim follows from Lemma 34.

ek χ k k+1 (t) 1. log(t) [ee ,ee ] 2

The following theorem delivers the promised counterexample. Theorem 37. There exist A ∈ M1,∞ and dilation invariant generalised limit ω on L∞ (0, ∞) such that 1 τω (A) < ξω (A). Γ 1+ q Proof. For brevity, we assume that the von Neumann algebra N is of type II (the argument can be easily adjusted when N is of type I ). Let x be as in (37) and let A ∈ M+ 1,∞ be such that x = μ(A). The assertion follows from Lemmas 35 and 36. 2 8. Correctness of the definition for generalised heat kernel formulae Let ω be a dilation invariant generalised limit on L∞ (0, ∞) and let B ∈ N . Following [1], we consider the functionals on M+ 1,∞ defined by the formula 1 ξω,B,f (A) = (ω ◦ M) t → τ f (tA)B . t The main result of this section, Theorem 40, shows that the function 1 M t → τ f (tA)B t is bounded, and so the formula (38) is well defined.

(38)


2475

2 −1 log(t)) as t → ∞. Lemma 38. Let A ∈ M+ 1,∞ . We have τ (A EA [0, 1/t]) = O(t

Proof. Let c := A1,∞ . We have μ(s, A) ≺≺ c(1 + s)−1 . Fix t > 0. Define decreasing function xt ∈ M1,∞ (0, ∞) by setting xt (s) =

log(1+ct log(t)) , t log(t) c 1+s ,

0 s ct log(t), s > ct log(t).

Define a decreasing function yt ∈ M1,∞ (0, ∞) by setting 1 yt (s) = μ(A)χ{μ(A)1/t} (s) + χ{μ(A)1/t} (s), t

s > 0.

We claim that yt ≺≺ xt . Indeed, yt (s) 1/t xt (s) for s ct log(t) and s

s yt (u) du c

0

0

du = 1+u

s xt (u) du 0

for s > ct log(t). It follows that

2

τ A EA

1 0, t

∞

∞ yt2 (s) ds

0

xt2 (s) ds. 0

We have ∞ xt2 (s) ds

c log2 (1 + ct log(t)) + = t log(t)

0

∞

ct log(t)

c2 log(t) . ds 5c t (1 + s)2

2

Lemma 39. Let f (t) = t 2 χ[0,1] (t) and let A ∈ M+ 1,∞ . We have t →M

1 τ f (tA) ∈ L∞ (0, ∞). t

Proof. For fixed t > 0, we have t t 1 1 1 1 1 2 2 M τ f (tA) = τ A EA 0, EA 0, ds = τ A ds . t log(t) s log(t) s

1

1

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Integrating by parts, we obtain

t EA 1

t 1 1 1 t 1 1 t 1 −1 ds = sEA 0, − s dEA 0, = sEA 0, + u dEA , u 0, s s 1 s s 1 t 1

= O(1) + A−1 EA

1 1 , ∞ + tEA 0, . t t

1/t

Therefore,

1 1 1 1 t 1 2 M τ f (tA) = τ AEA , ∞ + τ A EA 0, +O . t log(t) t log(t) t log(t) It follows from the definitions of · 1,∞ and dA (·) that for every A ∈ M1,∞ and every t > 0, we have

1 max 1, A1,∞ log(1 + t). dA t Clearly, 1 1 1 τ AEA 0, = log(t) t log(t)

dA(1/t)

μ(s, A) ds

log(dA (1/t)) A1,∞ ∈ L∞ . log(t)

0

The assertion follows now from Lemma 38.

2

Theorem 40. Let a bounded function f ∈ C 2 [0, ∞) be such that f (0) = f (0) = 0. Let A ∈ M+ 1,∞ and let B ∈ N . We have 1 M t → τ f (tA)B ∈ L∞ (0, ∞). t Proof. Due to the well-known inequality τ (CB) τ (|C|)B, it suffices to prove the theorem only when B = 1. In this case, for the function f (t) := t 2 χ[0,1] (t), the assertion follows from Lemma 39. If f (t) := χ(1,∞) (t) then it holds trivially. Thus, it holds for the function f (t) := min{1, t 2 }. Finally, observe that the assumptions on f guarantee that there exists a constant c > 0 such that |f (t)| c min{1, t 2 }. 2 Since the function t → exp(−t −q ) satisfies the assumptions of Theorem 40 we obtain the following corollary, which was implicitly proved in [6, Theorem 5.2]. Corollary 41. For every q > 0 and every A ∈ M+ 1,∞ , we have 1 −q ∈ L∞ (0, ∞). M t → τ exp −(tA) t


2477

9. Reduction theorem for generalised heat kernel formulae The results of this section extend and generalise those of [5, Theorem 4.1] and [6, Theorem 5.2]. We also give an answer to the question asked in [1, p. 52]. We explicitly prove that the functional ξω,B,f (extended to M1,∞ as in (25)) is linear on M1,∞ . Lemma 42. Let f ∈ C 2 [0, ∞) be such that f (0) = f (0) = 0. Let A ∈ M+ 1,∞ and let B ∈ N . For every dilation invariant generalised limit ω on L∞ (0, ∞), we have ε 1 B = 0. lim (ω ◦ M) τ f tAEA 0, ε→0 t t Proof. Since |f (t)| const · t 2 for t ∈ [0, 1], it is sufficient to prove the assertion for f (t) = t 2 . As in the proof of Theorem 40, it is sufficient to assume that B = 1. By Theorem 40, for every ε > 0 we have ε 2 1 ∈ L∞ (0, ∞). M t → τ tAEA 0, t t Since ω is dilation invariant, we conclude 1 ε 2 1 1 2 (ω ◦ M) τ tAEA 0, = ε(ω ◦ M) τ tAEA 0, . t t t t The assertion follows immediately.

2

Lemma 43. Let f ∈ L∞ (0, ∞) be such that f (0) = 0. Let A ∈ M+ 1,∞ and let B ∈ N . For every dilation invariant generalised limit ω on L∞ (0, ∞), we have 1 1 ,∞ B = 0. lim (ω ◦ M) τ f tAEA ε→0 t εt Proof. As before, we may assume that B = 1. It is clear that 1 1 f tAEA ,∞ f EA ,∞ . εt εt Since ω ◦ M is dilation invariant, we obtain 1 1 1 1 (ω ◦ M) τ EA ,∞ = ε(ω ◦ M) dA . t εt t t The assertion follows immediately.

2

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Lemma 44. Let f : R+ → R be monotone on [a, b] and such that f (0) = 0. Let A ∈ M+ 1,∞ and let B ∈ N . For every dilation invariant generalised limit ω on L∞ (0, ∞) we have

b 1 1 a b 1 ds (ω ◦ M) τ f tAEA , f (s) 2 · (ω ◦ M) τ EA , ∞ B . B = t t t t t s a

Proof. Without loss of generality, we may assume that f is increasing on [a, b] and that B 0. Let a = a0 a1 a2 · · · an = b. For every given t > 0, we have EA

a b , t t

=

n−1

EA

k=0

ak ak+1 , . t t

Since f is increasing on [a, b] and f (0) = 0, we have f (ak )EA

ak ak+1 , t t

ak ak+1 ak ak+1 f tAEA , f (ak+1 )EA , . t t t t

Therefore, n−1 1 a b ak ak+1 1 (ω ◦ M) τ f tAEA , B B f (ak+1 )(ω ◦ M) τ EA , t t t t t t k=0

and n−1 1 1 a b ak ak+1 , (ω ◦ M) τ f tAEA , f (ak )(ω ◦ M) τ EA B B . t t t t t t k=0

We have EA

ak ak+1 , t t

= EA

ak ak+1 , ∞ − EA ,∞ . t t

For all c > 0, we have 1 c 1 1 −1 (ω ◦ M) τ EA , ∞ B = c (ω ◦ M) τ EA , ∞ B . t t t t Therefore, 1 ak ak+1 1 1 1 1 (ω ◦ M) τ EA , − (ω ◦ M) τ EA , ∞ B . B = t t t ak ak+1 t t


2479

Hence, n−1

1 1 1 1 f (ak ) − (ω ◦ M) τ EA , ∞ B ak ak+1 t t k=0 a b 1 B (ω ◦ M) τ f tAEA , t t t n−1 1 1 1 1 f (ak+1 ) − (ω ◦ M) τ EA , ∞ B . ak ak+1 t t k=0

Both coefficients in the latter formula tend to

b a

f (s)s −2 ds.

2

Lemma 45. Let a bounded function f ∈ C 2 [0, ∞) be such that f (0) = f (0) = 0. Let A ∈ M+ 1,∞ and let B ∈ N . For every dilation invariant generalised limit ω on L∞ (0, ∞) we have ∞ ξω,B,f (A) =

1 ds 1 f (s) 2 (ω ◦ M) τ EA , ∞ B . t t s

0

Proof. Let f satisfy the assumptions above. Observe that the assertion of Lemma 44 holds for the function f |[a,b] , where 0 < a < b < ∞. Indeed, every such function is a function of bounded variation and therefore may be written as a difference of two monotone functions. Now the assertion follows from Lemmas 42, 43, 44 by setting a := ε and b := ε −1 and letting ε → 0. 2 Corollary 46. Let a bounded function f ∈ C 2 [0, ∞) be such that f (0) = f (0) = 0. Let A ∈ + M+ 1,∞ and let B ∈ N . For every dilation invariant generalised limit ω on L∞ (0, ∞) we have ∞ ξω,B,f (A) = 0

1 1 ds 1 τ A− EA , ∞ B . f (s) 2 ω log(1 + t) t t s

Proof. It follows from the definition of Cesaro operator M that t 1 1 1 1 ds M t → τ EA , ∞ B = τ EA , ∞ B 2 . t t log(t) s s 1

Integrating by parts, we obtain 1 log(t)

t 1 ds τ EA , ∞ B 2 s s 1

1 = log(t)

1 1/t

τ EA (u, ∞)B du

2480


1

1 1 1 = · uτ EA (u, ∞)B 1/t − log(t) log(t)

udτ EA (u, ∞)B

1/t

∞

1 −1 −1 · τ EA , ∞ B + τ u dEA (u, ∞)B + o(1). = t log(t) t log(t) 1/t

Evidently, ∞ −τ

1 u dEA (u, ∞)B = τ AEA , ∞ B . t

1/t

Therefore, 1 1 1 1 1 M t → τ EA , ∞ B = τ A− EA , ∞ B + o(1). t t log(t) t t The assertion follows now from Lemma 45.

2

The first assertion in lemma below can be found in [3, Theorem 11]. For the second assertion we refer to [2, Theorem 3.5]. Lemma 47. Let A, B ∈ B + (H ) and let f be convex continuous function such that f (0) = 0. We have (i) τ (B 1/2 f (A)B 1/2 ) τ (f (B 1/2 AB 1/2 )) if B 1. (ii) τ (B 1/2 f (A)B 1/2 ) τ (f (B 1/2 AB 1/2 )) if B 1. We show in the following lemma that ξω,B,f depends continuously on B. Lemma 48. If A ∈ M+ 1,∞ and let Bn , B ∈ N , n 1, then ξω,B (A) − ξω,B (A) ξω (A) · Bn − B. n Proof. The assertion follows from the inequality τ f (tA)Bn − τ f (tA)B τ f (tA) · Bn − B.

2

The following theorem extends the results of [5,6] and gives an affirmative answer to the question stated in [1]. It also shows that the functionals ξω,B,f (·) are linear functionals on M1,∞ for a wide class of functions f .


2481

Theorem 49. Let a bounded function f ∈ C 2 [0, ∞) be such that f (0) = f (0) = 0. Let A ∈ M1,∞ and let B ∈ N . For every dilation invariant generalised limit ω on L∞ (0, ∞) we have 1 ξω,B,f (A) = Γ (1 + 1/q)

∞

ds f (s) 2 ξω (AB). s

(39)

0

Proof. It follows from Theorem 22 that ξω is linear and fully symmetric. By Theorem 1 and (17), we have ξω (B 1/2 AB 1/2 ) = ξω (AB). Recall that function u → (u − 1/t)+ is convex. It follows from Lemma 47 that (i) τ ((A − 1t )+ B) τ ((B 1/2 AB 1/2 − 1t )+ ) if B 1. (ii) τ ((A − 1t )+ B) τ ((B 1/2 AB 1/2 − 1t )+ ) if B 1. It follows from Corollary 46 that for 0 B 1 we have 1 ξω,B,f (A) Γ (1 + 1/q)

∞

ds f (s) 2 ξω B 1/2 AB 1/2 . s

(40)

0

Since both sides are homogeneous, the inequality (40) is valid for every B. It follows from 46 that for B 1 we have 1 ξω,B,f (A) Γ (1 + 1/q)

∞

ds f (s) 2 ξω B 1/2 AB 1/2 . s

(41)

0

Since both sides are homogeneous, the inequality (41) is valid if B is bounded from below by a strictly positive constant. Thus, we have the equality (39) valid for every B bounded from below by a strictly positive constant. Set Bn = BEB (1/n, ∞) + 1/nEB [0, 1/n]. It follows that equality (39) holds with B replaced with Bn throughout. By Lemma 48, we have ξω,Bn ,f (A) → ξω,B,f (A). Since ABn → AB in M1,∞ and since ξω is bounded on M1,∞ , we have ξω (ABn ) → ξω (AB). The assertion follows immediately. 2 The following corollary treats the case of classical heat kernel formulae. We use the notation 1 ξω,B (A) = (ω ◦ M) τ exp −(tA)−q B . t Corollary 50. Let A ∈ M+ 1,∞ and let B ∈ N . For every dilation invariant generalised limit ω on L∞ (0, ∞) we have ξω,B (A) = ξω (AB). Proof. Use f (t) = exp(−t −q ) in Theorem 49 and observe that ∞ f (s) 0

ds 1 = Γ 1 + . q s2

2

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The following assertion extends [23, Theorem 33]. Corollary 51. Let A ∈ M+ 1,∞ and let B ∈ N . For every dilation invariant generalised limit ω on L∞ (0, ∞) such that ω = ω ◦ M, we have 1 ξω,B (A) = Γ 1 + τω (AB). q References [1] M. Benameur, T. Fack, Type II noncommutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math. 199 (2006) 29–87. [2] J. Bourin, Convexity or concavity inequalities for Hermitian operators, Math. Inequal. Appl. 7 (4) (2004) 607–620. [3] L. Brown, H. Kosaki, Jensen’s inequality in semi-finite von Neumann algebras, J. Operator Theory 23 (1) (1990) 3–19. [4] A.L. Carey, V. Gayral, A. Rennie, F. Sukochev, Integration on locally compact noncommutative spaces, arXiv: 0912.2817v1. [5] A. Carey, J. Phillips, F. Sukochev, Spectral flow and Dixmier traces, Adv. Math. 173 (1) (2003) 68–113. [6] A. Carey, A. Rennie, A. Sedaev, F. Sukochev, The Dixmier trace and asymptotics of zeta functions, J. Funct. Anal. 249 (2) (2007) 253–283. [7] A. Carey, F. Sukochev, Dixmier traces and some applications to noncommutative geometry, Uspekhi Mat. Nauk 61 (6(372)) (2006) 45–110 (in Russian); English translation in: Russian Math. Surveys 61 (6) (2006) 1039– 1099. [8] A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994. [9] J. Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris 262 (1966) A1107–A1108. [10] P. Dodds, B. de Pagter, A. Sedaev, E. Semenov, F. Sukochev, Singular symmetric functionals, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 290 (2002), Issled. po Linein. Oper. i Teor. Funkts. 30, 42–71 (in Russian); English translation in: J. Math. Sci. (N. Y.) 124 (2) (2004) 4867–4885. [11] P. Dodds, B. de Pagter, A. Sedaev, E. Semenov, F. Sukochev, Singular symmetric functionals with additional invariance properties, Izv. Ross. Akad. Nauk Ser. Mat. 67 (6) (2003) 111–136 (in Russian); English translation in: Izv. Math. 67 (2003) 1187–1213. [12] P. Dodds, B. de Pagter, E. Semenov, F. Sukochev, Symmetric functionals and singular traces, Positivity 2 (1) (1998) 47–75. [13] R. Edwards, Functional Analysis, Holt, Rinehart and Winston, New York, 1965. [14] T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable operators, Pacific J. Math. 123 (2) (1986) 269–300. [15] I. Gohberg, M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18, American Mathematical Society, Providence, RI, 1969. [16] N. Kalton, A. Sedaev, F. Sukochev, Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces, Adv. Math., in press. [17] N. Kalton, F. Sukochev, Rearrangement-invariant functionals with applications to traces on symmetrically normed ideals, Canad. Math. Bull. 51 (2008) 67–80. [18] L.S. Koplienko, Trace formula for nontrace-class perturbations, Sibirsk. Mat. Zh. 25 (5) (1984) 62–71 (in Russian); English translation in: Sib. Math. J. 25 (5) (1984) 735–743. [19] S. Krein, Ju. Petunin, E. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English translation in: Transl. Math. Monogr., vol. 54, American Mathematical Society, 1982. [20] F.J. Murray, J. von Neumann, On rings of operators, Ann. Math. 37 (1) (1936) 116–229. [21] A. Pietsch, About the Banach Envelope of l1,∞ , Rev. Mat. Complut. 22 (1) (2009) 209–226. [22] A. Sedaev, Generalized limits and related asymptotic formulas, Math. Notes 86 (4) (2009) 612–627. [23] A. Sedaev, F. Sukochev, D. Zanin, Lidskii-type formulae for Dixmier traces, Integral Equations Operator Theory, doi:10.1007/s00020-010-1828-1, in press, http://arxiv.org/pdf/1003.1817. [24] B. Simon, Trace Ideals and Their Applications, American Mathematical Society, 2005.


Note

Perturbations and operator trace functions Walter D. van Suijlekom Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands Received 4 August 2010; accepted 13 December 2010 Available online 22 December 2010 Communicated by Alain Connes

Abstract We study the spectral functional A → Tr f (D + A) for a suitable function f , a self-adjoint operator D having compact resolvent, and a certain class of bounded self-adjoint operators A. Such functionals were introduce by Chamseddine and Connes in the context of noncommutative geometry. Motivated by the physical applications of these functionals, we derive a Taylor expansion of them in terms of Gâteaux derivatives. This involves divided differences of f evaluated on the spectrum of D, as well as the matrix coefficients of A in an eigenbasis of D. This generalizes earlier results to infinite dimensions and to any number of derivatives. © 2010 Elsevier Inc. All rights reserved. Keywords: Noncommutative geometry; Perturbation theory

1. Introduction The spectral action in noncommutative geometry [4] is given as the trace Tr f (D) of a suitable function f (D) of an unbounded self-adjoint operator D, which is assumed to have compact resolvent. One is interested in this trace function as D is perturbed to D + A where A is a certain self-adjoint bounded operator. For instance, the so-called inner fluctuations of a spectral triple are of this type; they are central in the applications of noncommutative geometry to high-energy physics [1–3] (cf. also [6]). A natural question that arises is what happens to the trace function when D is perturbed to D + A. It is the goal of this paper to address this question. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.012

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W.D. van Suijlekom / Journal of Functional Analysis 260 (2011) 2483–2496

We aim for a Taylor expansion of the spectral action by Gâteaux deriving it with respect to A. As we will see, the context of finite-dimensional noncommutative manifolds (i.e. spectral triples) allows for a derivation of results previously obtained only for finite-dimensional (matrix) algebras [13]. Our main result is the expansion: SD [A] =

∞ 1 n=0

n

Ai1 i2 · · · Ain i1 f [λi1 , . . . , λin ]

i1 ,...,in

where f [λi1 , . . . , λin ] is the divided difference of order n of f (cf. Definition 14 below) evaluated on the spectrum of D, and Aij are the matrix coefficients with respect to an eigenbasis of D. This paper is organized as follows. First, we recall in Section 2 some results on perturbations of operators, in the setting of noncommutative geometry. Then, we give a precise definition of the spectral action functional in Section 3. In that section, we also recall the definition of divided differences and derive our main result on the Taylor expansion of the spectral action. We end with some conclusions and an appendix recalling a theorem by Getzler and Szenes. 2. Perturbations and spectral triples Recall that a spectral triple consists of an algebra A of bounded operators on a Hilbert space H, together with a self-adjoint operator D with compact resolvent such that the commutator [D, a] is a bounded operator for all a ∈ A. The key example is associated to a compact Riemannian spin manifold M: ∞ ∂ , C (M), L2 (M, S), / ∂ is a Dirac operator on the spinor bundle S → M. Indeed, / ∂ is an elliptic differential where / operator of degree one and smooth functions satisfy [/ ∂ , f ] = f Lip < ∞ in the Lipschitz norm of f . In general, a spectral triple (A, H, D) is said to be of finite summability if there exists an n 0 such that (1 + D 2 )−n/2 is a traceclass operator on H. Let us start with a basic and well-known result. Lemma 1. Let p be a polynomial on R. Then for any t > 0 the operator p(D)e−tD is traceclass. 2

Proof. By finite summability and Hölder’s inequality (1 + D 2 )−n/2 is traceclass for some n. Thus, −n/2 2 p(D)e−tD = ϕ(D) 1 + D 2 with ϕ defined by functional calculus for the function −n/2 −tx 2 e . ϕ(x) = p(x) 1 + x 2


2485

For t > 0, this is a bounded function on R so that ϕ(D)(1 + D 2 )−n/2 is in the ideal L1 (H) of traceclass operators as required. 2 In particular, this applies to p(x) = 1, i.e. finite summability implies so-called θ -summability: 2 Tr e−tD < ∞ (t ∈ R+ ).

(1)

2.1. Fréchet algebra of smooth operators Given the derivation δ(·) = [|D|, ·] on B(H), there is a natural structure of a Fréchet algebra on the smooth domain of δ. Proposition 2. The following define a multiplicative family of semi-norms on B(H): n δ (T )

T ∈ B(H)

indexed by n ∈ Z0 . Proof. The derivation property of δ yields n n n n k n n−k δ (T1 T2 ) = δk (T1 )δn−k (T2 ). δ (T1 )δ (T2 ) k k k=0

2

k=0

We will denote B n (H) = T ∈ B(H): δ k (T ) < ∞ for all k n . Evidently, we have B ∞ (H) ⊂ · · · ⊂ B 2 (H) ⊂ B 1 (H) ⊂ B(H) where by definition B ∞ (H) =

n∈Z0

B n (H).

Remark 3. Recall that a spectral triple (A, H, D) is called regular if both the algebra A and [D, A] are in the smooth domain of δ. This can thus be reformulated as: the algebra generated by a and [D, b] (a, b ∈ A) is a subalgebra of B ∞ (H). In particular, the A-bimodule of Connes’ differential one-forms [4, Sect. VI.1], 1 (A) = ΩD

j

is a subspace of B ∞ (H).

aj [D, bj ]

2486


2.2. Perturbations of heat operators In this subsection, we take a closer look at the heat operator e−tD and its perturbations. First, recall that the standard m-simplex is given by an m-tuple (t1 , . . . , tm ) satisfying 0 t1 · · · tm 1. Equivalently, it can be given by an m + 1-tuple (s0 , s1 , . . . , sm ) such that s0 + · · · + sm = 1 and 0 si 1 for any i = 0, . . . , m. Indeed, we have s0 = t1 , si = ti+1 − ti and sm = 1 − tm and, vice versa, tk = s0 + s1 + · · · + sk−1 . For later use, we prove the following bound, which already appeared in a slightly different form in [10]. 2

Proposition 4. For any m 0 and 0 k m + 1 we have the bound

d m s(s0 · · · sk−1 )−1/2

m

πk . (m − k)!

Proof. In terms of the parameters ti for the m-simplex, we have to find an upper bound for

1

tm

dt1

dtm−1 · · ·

dtm 0

t2

0

0

1 t1 (t2 − t1 ) · · · (tk − tk−1 )

,

where tm+1 ≡ 1. First, note that by a standard substitution

t2 0

1 = π. dt1 √ t1 (t2 − t1 )

For the subsequent integral over t2 :

t3 0

1 dt2 √ t3 − t2

t3 dt2 √ 0

1 =π t2 (t3 − t2 )

since t2 1. This we can repeat k times, leaving us with the integral

1

tm dtm

0

tk+1 dtm−1 · · · dtk =

0

0

1 . (m − k)!

2

Lemma 5. Let A be a bounded operator and denote DA = D + A. Then

e

−t (DA )2

=e

−tD 2

1 −t 0

with P (A) = DA + AD + A2 .

ds e−st (DA ) P (A)e−(1−s)tD 2

2


2487

Proof. Note that e−tDA is the unique solution of the Cauchy problem 2

(dt + DA )u(t) = 0, u(0) = 1

with dt = d/dt. Using the fundamental theorem of calculus, we find that dt e

−tD 2

t −

2 −(t−t )DA

dt e

P (A)e

−t D 2

2 = −DA

e

−tD 2

t −

0

2 −(t−t )DA

dt e

P (A)e

−t D 2

0

showing that the bounded operator e−tD − Cauchy problem. 2 2

t 0

dt e−(t−t )DA P (A)e−t D also solves the above 2

2

The following estimates were proved in a slightly different form in [10]. Lemma 6. If the operators A, Ai are bounded, and αi ∈ {0, 1} are such that

i

αi = k, then

Tr A0 |DA |α0 e−s0 tDA2 A1 |D|α1 e−s1 tD 2 · · · An |D|αn e−sn tD 2 d n s n

A0 · · · An Tr e−(1−)tD (n − k)!(π −2 t)k/2

2

for any 0 < < 1. Proof. Recall Hölder’s inequality: Tr(T0 · · · Tn ) T0

s0−1

· · · Tn s −1

(2)

n

when s0 + · · · + sn = 1. Also, we estimate for some arbitrary 0 < < 1: Ai e−si tD 2

si−1

2 s 2 s Ai Tr e−tD i Ai Tr e−(1−)tD i ,

Ai |D|e−si tD 2 −1 Ai |D|e−si tD 2 Tr e−(1−)tD 2 si s i

2 s (si t)−1/2 Ai Tr e−(1−)tD i

writing e−stD = e−stD e−(1−)stD . We have used Lemma 1 and the fact that 2

2

−stD 2 e 1,

2

|D|e−stD 2 sup xe−stx 2 = (2est)−1/2 . x∈R+

Moreover, Theorem C in [10] (cf. Appendix A) gives Tr e−t (1−/2)(DA ) e(1+2/)tA Tr e−t (1−)D . 2

2

2

(∗)

2488


This further yields A0 |DA |e−s0 tDA2

s0−1

2 2 s A0 |DA |e−/2si tDA Tr e−(1−/2)tDA i 2 2 s (es0 t)−1/2 e(1+2/)tA A0 Tr e−(1−)tD 0 .

Combining these estimates with (2), we obtain for instance in the case that the first k αi are nonzero (i.e. α0 = · · · = αk−1 = 1): Tr A0 |DA |α0 e−s0 tDA2 A1 |D|α1 e−s1 tD 2 · · · An |D|αn e−sn tD 2

A0 · · · An 2 Tr e−(1−)tD k/2 s0 · · · sk (t)

making use of the fact that s0 + s1 + · · · + sn = 1. The bounds of Proposition 4 complete the proof. 2 Let us introduce the following convenient notation (cf. [10]). If A0 , . . . , An are operators, we define a t-dependent quantity by

A0 , . . . , An n := t Tr n

A0 e−s0 tD A1 e−s1 tD · · · An e−sn tD d n s. 2

2

2

(3)

n

Note the difference in notation with [10], for which the same symbol is used for the supertrace of the same expression, rather than the trace. Also, we are integrating over the ‘inflated’ n-simplex tn , yielding the factor t n . The forms A0 , . . . , An satisfy, mutatis mutandis, the following properties. Lemma 7. (See [10].) In each of the following cases, we assume that the operators Ai are such that each term is well defined: 1. 2. 3. 4.

Ai , . . . , An , . . . , Ai−1 n ; A0 , . . . , An n = n , . . . , A

= A 0 n n i=0 1, . . . , Ai , . . . , An , A0 , . . . , Ai−1 n ; n i=0 A0 , . . . , [D, Ai ], . . . , An n = 0; A0 , . . . , [D 2 , Ai ], . . . , An n = A0 , . . . , Ai−1 Ai , , . . . , An n−1 − A0 , . . . , Ai Ai+1 , . . . , An n−1 .

2.3. Gâteaux derivatives As a preparation for the next section, we recall the notion of Gâteaux derivatives, referring to the excellent treatment [12] for more details. Definition 8. The Gâteaux derivative at x ∈ X of a map F : X → Y between locally convex topological vector spaces is defined for h ∈ X by F (x + uh) − F (x) . u→0 u

F (x)(h) = lim


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In general, the map F (x)(·) is not linear, in contrast with the Fréchet derivative. However, if X and Y are Fréchet spaces, then the Gâteaux derivatives actually defines a linear map F (x)(·) for any x ∈ X [12, Theorem 3.2.5]. In this case, higher order derivatives are denoted as F , F et cetera, or more conveniently as F (k) for the k-th order derivative. The latter will be understood as a linear bounded operator from X × · · · × X (k + 1 copies) to Y . Theorem 9 (Taylor’s formula with integral remainder). For a Gâteaux k + 1-differentiable map F : X → Y between Fréchet spaces X and Y it holds for x, a ∈ X that F (x) = F (a) + F (a)(x − a) + +

1 F (a)(x − a, x − a) + · · · 2!

1 (k) F (a)(x − a, . . . , x − a) + Rk (x) n!

with integral remainder given by 1 Rk (x) = k!

1

F (k+1) a + t (x − a) (1 − t)h, . . . , (1 − t)h, h dt.

0

3. Trace functionals In this section, we consider trace functionals of the form A → Tr f (D + A). Here D is the self-adjoint operator forming a finitely summable spectral triple (A, H, D), and A is a bounded operator. We derive a Taylor expansion of this functional in A. Our main motivation comes from the spectral action principle introduced by Chamseddine and Connes [1,2] and we define accordingly Definition 10. (See Chamseddine and Connes [2].) The spectral action functional SD [A] is defined by A ∈ B(H) .

SD [A] = Tr f (D + A)

The square brackets indicate that SD [A] is considered as a functional of A ∈ B(H). Remark 11. Actually, Chamseddine and Connes considered SD [A] for so-called gauge fields 1 (A) which associated to the spectral triple (A, H, D). These are self-adjoint elements A in ΩD 2 by Remark 3 is a subset of B (H). For the function f we assume that it is a Laplace–Stieltjes transform:

f (x) = t>0

for which we make the additional:

e−tx dμ(t) 2

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Assumption 1. For all α > 0, β > 0, γ > 0 and 0 < 1, there exist constants Cαβγ such that

Tr t α |D|β e−t (D

2 −β)

dμ(t) < Cαβγ .

t>0

In view of Theorem 9, we have the following Taylor expansion (around 0) in A ∈ B 2 (H) for the spectral action SD [A]: ∞ 1 (n) S (0)(A, . . . , A). SD [A] = n! D

(4)

n=0

Indeed, SD is Fréchet differentiable on B 2 (H) as the following Proposition establishes. (n)

Proposition 12. If n = 0, 1, . . . and A ∈ B 2 (H), then SD (0)(A, . . . , A) exists and (n) (0)(A, . . . , A) = n! SD

n

1, (1 − ε1 ){D, A} + ε1 A2 , . . . ,

(−1)k

ε1 ,...,εk

k=0

(1 − εk ){D, A} + εk A2 k dμ(t), where the sum is over multi-indices (ε1 , . . . , εk ) ∈ {0, 1}k such that

k

i=1 (1 + εi ) = n.

Proof. We will prove this by induction on n; the case n = 0 being trivial. By definition of the Gâteaux derivative and using Lemma 5, (n+1) SD (0)(A, . . . , A) = n!

n k=0 ε1 ,...,εk

k (−1)k+1 1, (1 − ε1 ){D, A} + ε1 A2 , . . . , i=1

{D, A}, . . . , (1 − εk ){D, A} + εk A2 i

+

k+1

k (−1)k 1, (1 − ε1 ){D, A} + ε1 A2 , . . . , 2(1 − εi )A2 , . . . , i=1

(1 − εk ){D, A} + εk A2 k dμ(t). The first sum corresponds to a multi-index ε = (ε1 , . . . , εi−1 , 0, εi , . . . , εk ), the second sum corresponds to ε = (ε1 , . . . , εi + 1, . . . , εk ) if εi = 0, counted with a factor of 2. In both cases, we compute that j (1 + εj ) = n + 1. In other words, the induction step from n to n + 1 corresponds to inserting in a sequence of 0’s and 1’s (of, say, length k) either a zero at any of the k + 1 places, or replace a 0 by a 1 (with the latter counted twice). In orderto arrive at the right combinatorial coefficient (n + 1)!, we have to show that any ε satisfying i (1 + εi ) = n + 1 appears in precisely n + 1 ways from ε that satisfy i (1 + εi ) = n. If ε has length k, it contains n + 1 − k


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times 1 as an entry and, consequently, 2k − n − 1 a 0. This gives (with the double counting for the 1’s) for the number of possible ε: 2(n + 1 − k) + 2k − n − 1 = n + 1 2

as claimed. This completes the proof. Example 13. (1) SD (0)(A) =

(2) SD (0)(A, A) = 2

− 1, {D, A} 1 dμ(t),

− 1, A2 1 + 1, {D, A}, {D, A} 2 dμ(t),

(3)

SD (0)(A, A, A) = 3!

1, A2 , {D, A} 2 + 1, {D, A}, A2 2

− 1, {D, A}, {D, A}, {D, A} 3 dμ(t). 3.1. Divided differences Recall the definition of and some basic results on divided differences. Definition 14. Let g : R → R and x0 , x1 , . . . , xn be distinct points on R. The divided difference of order n is defined by the recursive relations g[x0 ] = g(x0 ), g[x0 , x1 , . . . , xn ] =

g[x1 , . . . , xn ] − g[x0 , x1 , . . . , xn−1 ] . xn − x0

On coinciding points we extend this definition as the usual derivative: g[x0 , . . . , x, . . . , x, . . . , xn ] := lim g[x0 , . . . , x + u, . . . , x, . . . , xn ]. u→0

Finally, as a shorthand notation, we write for an index set I = {i1 , . . . , in }: g[xI ] = g[xi1 , . . . , xin ]. Also note the following useful representation due to Hermite [14]. Proposition 15. For any x0 , . . . , xn ∈ R:

f [x0 , x1 , . . . , xn ] = n

f (n) (s0 x0 + s1 x1 + · · · + sn xn ) d n s.

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As an easy consequence, we derive n

f [x0 , . . . , xi , xi , . . . , xn ] = f [x0 , x1 , . . . , xn ].

i=0

Proposition 16. For any x1 , . . . , xn ∈ R we have for f (x) = g(x 2 ): f [x0 , . . . , xn ] =

(xi + xi+1 ) g xI2

{i−1,i}⊂I

I

where the sum is over all ordered index sets I = {0 = i0 < i1 < · · · < ik = n} such that ij − ij −1 2 for all 1 j k (i.e. there are no gaps in I of length greater than 1). Proof. This follows from the chain rule for divided difference: if f = g ◦ ϕ, then [9] f [x0 , . . . , xn ] =

n

k−1 g ϕ(xi0 ), . . . , ϕ(xik ) ϕ[xij , . . . , xij +1 ].

k=1 0=i0

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