# Analyticity of the Schrodinger propagator on the Heisenberg group

## Full text

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DOI 10.1007/s00605-012-0424-7

### Analyticity of the Schrödinger propagatoron the Heisenberg group

S. Parui · P. K. Ratnakumar · S. Thangavelu

Received: 17 May 2011 / Accepted: 22 June 2012

Abstract We discuss the analytic extension property of the Schrödinger propaga- tor for the Heisenberg sublaplacian and some related operators. The result for the sublaplacian is proved by interpreting the sublaplacian as a direct integral of an one parameter family of dilated special Hermite operators.

Keywords Schrödinger equation·Oscillatory group·Special Hermite expansion· Heisenberg group·Sublaplacian

Mathematics Subject Classification (1991) Primary 22E30;

Secondary 35G10·47A63

1 Introduction

It is well known that the heat equation is infinitely smoothing in the sense that the solution u(x,t)of the heat equationtu=u with initial condition u(x,0)= f(x)

Communicated by K. Gröchenig.

S. Parui (

### B

)

School of Mathematics, National Institute of Science Education and Research, IOP Campus, Bhubaneswar 751005, India

e-mail: parui@niser.ac.in P. K. Ratnakumar

S. Thangavelu

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India e-mail: veluma@math.iisc.ernet.in

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is real analytic even if f is only in L2(Rn). Actually much more is true: if qt stands for the heat kernel,

qt(x)=(4πt)n2e4t1|x|2

then for fL2(Rn), the solution u(x,t) = fqt(x)extends toCnas an entire function u(x+i y,t)which satisfies

R2n

|u(x+i y,t)|2qt/2(y)d x d y=cn

Rn

|f(x)|2d x.

Conversely, any entire function F(x+i y)onCnfor which

R2n

|F(x+i y)|2qt/2(y)d x d y<

can be factored as F(x) = fqt(x)for some fL2(Rn).This is a celebrated theorem attributed to Bargmann .

In contrast to this, the Schrödinger equation cannot be expected to have such a smoothness property. In fact, the solutionv(x,t)of the equation

i∂tv=v, v(x,0)= f(x)

is given by the unitary group ei t, which is an isometry onto L2, so without any improvement in the smoothing. In view of this, the following result of Hayashi and Saitoh  looks surprising at the first sight. In 1990, they have shown that the solution of the one dimensional Schrödinger equation

i∂tu+1

2x2u=0, u(x,0)= f(x), t,x∈R

extends toCas an entire function, provided the initial condition f has enough decay at infinity. More precisely, they have shown that when f is square integrable with respect to ex2d x, then u(x,t)extends toCas an entire function and

√1 πt2

−∞

−∞

e

y2

t 2|e2tiz2u(z,t)|2d x d y=

−∞

|f(x)|2ex2 d x

for all t =0. Their proof is based on the identity

√1πα

−∞

−∞

ey2α|f(x+i y)|2d x d y =

j=0

αj j!

−∞

|∂xj f(x)|2d x.

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However, the result can be directly proved in all dimensions without any recourse to the above identity. Indeed, the solution of the Schrödinger equation

i∂tu=u, u(x,0)= f(x) is given by

u(x,t)=(4πi t)n2

Rn

f(y)e4i t1 |xy|2d y.

Therefore, u(x,t)e4i t1|x|2 is up to a constant multiple, the Fourier transform (at 2t1x) of the function f(y)e4i t1|y|2.If we assume that f has a Gaussian decay, then it is well known that its Fourier transform extends toCnas a holomorphic function. There- fore, it is not surprising that the solution of the Schrödinger equation also has some smoothing properties, provided we start with initial conditions having enough decay.

If we assume that f is square integrable with respect to e|y|2d y, then a simple calculation shows that

u(x,t)e4ti x2j =(4πi t)n2

Rn

f(y)e4ti y2j

e2tix·yd y,

which can be written as

u(x,t)e4ti x2j =(2i t)n/2gˆ∗qs

1 2tx

where g(y)= f(y)e4ti|y|2es|y|2. As gL2(Rn), it follows from the theorem of Segal and Bargmann that F(x,t)=u(x,t)e4ti x2j extends toCnas an entire function and

Cn

|F(z,t)|2qs/2

y 2t

d x d y =(2t)n

Cn

| ˆgqs(z)|2qs/2(y)d x d y

=(2t)n

Rn

| ˆg(y)|2d y,

which leads to the identity, with s =12,

Cn

|u(z,t)e4ti z2j|2e

1 4t2|y|2

d x d y =(4t2π)n/2

Rn

|f(y)|2e|y|2d y.

Actually, the converse is also true.

In this article we consider the Schrödinger equation associated to the Hermite and special Hermite operators and also to the sublaplacian on the Heisenberg group.

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The Hermite and Special Hermite cases are similar to the Euclidean Laplacian and straightforward, whereas the sublaplacian case is very different. We obtain an iso- metric isomorphism between certain subspaces of L2(Hn)and a direct integral of weighted Bergman spaces related to special Hermite operator.

The paper is organised as follows. In Sect. 2, we treat the Hermite and special Hermite cases. In Sect.3, we consider the sublaplacian on the Heisenberg group. In Sect.4, the generalized sublaplacian has been treated.

2 Schrödinger equation for the Hermite and the special Hermite operators First we consider the regularising properties of the unitary group ei t H, t ∈ Rgen- erated by the Hermite operator H = −+ |x|2onRn. As is well known, ei t H is a fractional Fourier transform and hence the holomorphic extendability of ei t Hf under suitable decay assumption on f is only but expected. The spectral decomposition of H is given by

H = k=0

(2k+n)Pk

where Pkf =

|α|=kf, αα, αbeing the normalised Hermite functions onRn. Using Mehler’s formula (see ) we obtain

ei t Hf(x)=

Rn

Ki t(x,y)f(y)d y,

where

Ki t(x,y)=(2πi sin 2t)n2ei4(cot 2t)(|x|2+|y|2)−i(csc 2t)x·y.

Note that Ki t(x,y)is well defined as long as 2t is not an integral multiple ofπ.

The one parameter group ei t H is periodic in t and for 2t = kπ, k ∈ Z,ei t H is given by the above integral. For such values of t,

e4i(cot 2t)|x|2ei t Hf(x)=(2πi sin 2t)n2

Rn

f(y)ei4(cot 2t)|y|2ei(csc 2t)x·yd y.

If we assume that f(y)es|y|2L2(Rn),then defining g(y)= f(y)es|y|2ei4(cot 2t)|y|2, we obtain

e4i(cot 2t)|x|2ei t Hf(x)=(2πi sin 2t)n2gˆ∗qs((csc 2t)x)

where qs is the Euclidean heat kernel onRn.Thus we see that ei t H is related to the Segal–Bargmann transform or the heat kernel transform associated to the standard Laplacian onRn.

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We briefly recall the main theorem on Segal–Bargmann transform for the conve- nience of the reader. The operator which takes fL2(Rn)into the entire function fqt(z) onCn is called the Segal–Bargmann transform. This is a variant of the Bargmann transform studied by Bargmann  and independently by Segal. The main theorem proved in  can be restated as follows.

Theorem 2.1 The Segal–Bargmann transform takes L2(Rn)onto the space of entire functions F on Cn which are square integrable with respect to the measure qt/2(y)d xd y.Moreover, for any f ∈Ł2(Rn)

Cn

|fqt(x+i y)|2qt/2(y)d xd y=

Rn

|f(x)|2d x.

Using the above theorem of Segal and Bargmann we can easily prove the following result for the operators ei t H.

Theorem 2.2 For any t = k2π,k∈Z, ei t Hdefines an isomorphism between L2(Rn, e|x|2d x)and the Hilbert space of entire functions onCn that are square integrable with respect to the weight function(2π|sin 2t|)ne−(csc22t)|y|2e(cot 2t)x·y.

Proof As gL2(Rn),the function F(x,t)= e4i(cot 2t)|x|2ei t Hf(x)has an entire extension toCnand according to the theorem of Bargmann, we have

Cn

|F(x+i y,t)|2qs/2((csc 2t)y)d x d y

=(2πn)|sin 2t|n

Cn

| ˆgqs(x+i y)|2qs/2(y)d x d y

=(2π)n|sin 2t|n

Rn

| ˆg(y)|2d y

=(2π)n|sin 2t|n

Rn

|f(y)|2e2s|y|2d y.

Taking s=12 we observe that

Cn

|F(x+i y)|2e−(csc22t)|y|2 d x d y

=cn(2π)n|sin 2t|n

Rn

|f(y)|2e|y|2d y.

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This can be rewritten as

Cn

|ei t Hf(x+i y)|2e(cot 2t)x·y|2πsin 2t|ne−((csc22t)|y|2)d x d y

=cn

Rn

|f(y)|2e|y|2d y.

The converse follows from the fact that the Segal–Bargmann transform is onto. The proof of this fact is involved which uses the action of Bargmann transform on Hermite functions. See also Section 6, Chapter 1 of Folland  for details about Bargmann transform.

Remark 2.3 We remark that when t = π4, ei t Hf is a constant multiple of f and theˆ above result reduces to

Rn

Rn

| ˆf(x+i y)|2e−|y|2 d x d y=cn

Rn

|f(y)|2e|y|2d y.

Therefore, we can view Theorem 2.1 as a result for the fractional Fourier transform.

We now turn our attention to the special Hermite operator Lλdefined by

Lλ= −+λ2

4 (|x|2+ |u|2) n

j=1

(xjujujxj)

whereis the Laplacian onR2nandλ∈R\{0}. The spectral decomposition of Lλis given by

Lλf =(2π)n

k=0

(2k+n)|λ|fλϕkλ

where theλtwisted convolution between two functions is defined as fλg(x,u)=

Cn

f(xx,uu)g(x,u )eiλ2(u·xu·x)d x du,

so that the Schrödinger equation is solved by the function

ei t Lλf(x,u)=cλ k=0

e(2k+n)|λ|i t fλϕλk(x,u). (2.1)

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Again the kernel of ei t Lλ is explicitly known in terms of the heat kernel ptλ(x,u) associated to the special Hermite operator Lλ(see ): ei t Lλf = fλ pλi t,where

pλi t(x,u)=cn

λ sinλt

n

eiλ4(cot tλ)(|x|2+|u|2).

As in the case of H , ei t Lλis also periodic in t and the above kernel exists as long as tλ is not an integral multiple ofπ. Also note that for such t, pi tλ(x,u)extends toCn×Cn as an entire function.

For t ∈R, λ∈R\{0}, andξ, η∈R2n, setωtλ(ξ, η)=ecotλt)ξ·ηqsin2 tλ

λ2 (η)where qs(η)is the Euclidean heat kernel onR2n. Consider the Hilbert space Stλ(C2n)of all entire functions for which

F2Sλ

t =

C2n

|F(ξ+iη)|2ωλt(ξ, η)dξ dη <∞.

We now prove

Theorem 2.4 The operator ei t Lλ is an isometric isomorphism between L2(R2n, e|ξ|2dξ)and the Hilbert space Stλ(C2n):

C2n

|ei t Lλf(ξ+iη)|2ωλt(ξ, η)dξdη=

R2n

|f(ξ)|2e|ξ|2dξ. (2.2)

Moreover, the weight functionωtλis unique.

Proof For fL2(R2n,e|ξ|2dξ), consider

ei t Lλf(ξ)= fλ pi tλ(ξ), ξ=(x,u) which is given by the integral

cn

λ sinλt

n

R2n

f(a,b)eiλ4(cot tλ)[|xa|2+|ub|2]eiλ2(a·ub·x)da db.

Definingvt(ξ)=ei t Lλf(ξ)pλi t(ξ), we can write the above as vt(ξ)=

λ sinλt

2n

R2n

gλ(η)eiλ2(Aη·ξ)

where gλ(η)= f(η)eiλ4(cot tλ)|η|2 and A is the 2n×2n matrix A=A(λ)=

cot(tλ)In −In

In cot(tλ)In

.

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We would like to expressvt as the convolution of a function hλL2(R2n)with the Euclidean heat kernel onR2n, still denoted by qs, for suitable s. Let B =(At)1and C= 2λB. We then have

vt(ξ)= λ

sinλt 2n

R2n

gλ(Cη)|det C|eiξ·η

=

R2n

hˆλ(η)es|η|2eiξ·η

wherehˆλ(η)= sinλλt2n

|det C|gλ(Cη)es|η|2 for a suitable s to be chosen later.

Therefore,vt(ξ)=hλqs(ξ),the convolution with the Euclidean heat kernel on R2n. If hλL2(R2n), then it follows from the Bargmann–Segal theorem thatvt(ξ) extends toC2n as an entire function and

C2n

|vt+iη)|2qs/2(η)dξdη=

R2n

|hλ(η)|2dη.

Now,

R2n

|hλ(η)|2=

R2n

| ˆhλ(η)|2

= λ

sinλt 4n

|det C|2

R2n

|gλ(Cη)|2e2s|η|2

= λ

sinλt 4n

|det C|

R2n

|gλ(η)|2e2s|C−1η|2dη.

After computing the determinant of C and|C1η|we end up with

R2n

|hλ(η)|2= λ4n (sin tλ)4n

22n

λ2n(sin tλ)2n

R2n

|gλ(η)|2eλ

2

2 |η|2s csc2tλ

=

2nλn (sin tλ)n

2n

R2n

|f(η)|2e2|

by choosing s=2sinλ22tλ. Thus we have proved the isometry

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C2n

|ei t Lλf(ξ+iη)|2|pλi t+iη)|2qsin2 tλ λ2

(η)dξdη

=cλ

R2n

|f(η)|2e|η|2 dη,

where cλ= sin t2λλ2n

. Note that

|pλi t+iη)|2= λ

sin tλ 2n

|eiλ4(cot tλ)(ξ+iη)2|2= λ

sin tλ 2n

eλ2(cot tλ)ξ·η and henceωλt(ξ, η)=eλ(cot tλ)ξ·ηqsin2 tλ

λ2

(η).

Observe that the equation

vt(ξ)=ei t Lλf(ξ)pλi t(ξ)=hλqs(ξ)

with s chosen as above sets up a one to one correspondence between the image of L2(R2n,e|ξ|2dξ)under ei t Lλand the image of L2(R2n)under the heat kernel transform viz. hhqs+iη). Indeed, if

F(ξ+iη)=hλqs+iη), hλL2(R2n) define f in terms of hλby the equations

hˆλ(η)= |det C|gλ(Cη)es|η|2, f(η)=gλ(η)eiλ4(cot tλ)|η|2. In other words,

f(η)= ˆhλ(C1η)es|C1η|2|det C|1eiλ4(cot tλ)|η|2. Since 2s|C1η|2= |η|2,we see that

R2n

|f(η)|2e|η|2=cλ

R2n

| ˆhλ(C1η)|2dη <∞.

Clearly,

ei t Lλf(ξ)pλi t(ξ)=hλqs(ξ)=F(ξ) as the above calculation shows, and hence ei t Lλis onto.

We now prove the uniqueness for the weight function. LetFs(C2n)be the image of L2(R2n)under the transform hhqs(ζ )where s=2sinλ22λt.In view of the above remarks it is enough to prove the weight function characterising the spaceFs(C2n)

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is unique. We know from  that this space is a weighted Bergman space with the weight function qa/2(η), η∈R2n.The spaceFsis invariant under the action ofR2n. Consequently, if there exists another nonnegative weight function Ws(ξ, η)such that

C2n

|G(ξ+iη)|2Ws(ξ, η)dξdη=

C2n

|G(ξ+iη)|2qs/2(η)dξdη

then we should have Ws(ξ, η)=Ws(η)([14, p. 308]). Using the Plancherel formula in theξ variable we have

R2n

|f(ξ)|2 =

C2n

|fqs+iη)|2Ws(η)dξdη

=

R2n

R2n

|f(ξ)|2e2ξ·ηe2s|ξ|2 Ws(η)dξdη.

Taking f to be the Gaussian e12|x|2 we see that

Rn

Ws(η)e(1+2s)1|η|2dη <∞.

Moreover, Ws(η)is given by the equation e2s|ξ|2 =

R2n

e2ξ·ηWs(η)dη.

In view of the above integrability of Ws,both sides of this equation extend toCnas entire functions. Consequently we see that

e2s|ξ|2 =

R2n

e2iξ·ηWs(η)dη.

and hence Wsis equal to qs/2(η). Thus the weightωλt is unique for Stλ(C2n).

3 Schrödinger equation for the sublaplacian Let Xj = xj12uj

∂ξ, Uj =uj +12xj

∂ξ and Z= ∂ξ be the left invariant vector fields on Hnforming a basis for the Heisenberg Lie algebra. Then the sublaplacian on Hnis defined by

L= − n

j=1

(X2j+U2j)

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and it plays the role of Laplacian onRn. It is well known thatL is a nonnegative essentially self adjoint operator which generates a semigroup etL of convolution with the smooth kernel pt:

pt(x,u, ξ)=

−∞

eiλξpλt(x,u)dλ

where pλt is the kernel associated to the special Hermite operator Lλstudied in Sect.2.

Note that etLand et Lλare related via

(etLf)λ(x,u)=et Lλfλ(x,u).

Here fλ(x,u)=

R f(x,u, ξ)eiλξ . At the convolution level(fpt)λ(x,u)= fλλ ptλ(x,u).In , the authors have characterised the image of L2(Hn)under the semigroup et(L+Z2). There it has been proved that the image is not a weighted Bergman space as expected, but can be expressed as a direct integral of weighted (twisted) Bergman spaces. Here we consider a similar problem for the Schrödinger equation and try to characterise the image of L2(Hn)under ei tL.

Formallyvt(x,u, ξ)=ei tLf(x,u, ξ)solving the Schrödinger equation i∂tvt(x,u, ξ)=Lvt(x,u, ξ), v0(x,u, ξ)= f(x,u, ξ) is given by fpi t(x,u, ξ)which can also be written as

fpi t(x,u, ξ)=

−∞

eiλξfλλ pλi t(x,u)dλ.

Note that the kernel pi t has singularities at the points whereλt is an integral multiple ofπ. Therefore, the above formula is not valid even for fixed t, unless the function λfλis compactly supported so that the singularities are avoided. Observe that

pλt(x,u)=cn

λ sin tλ

n

eλ4(cot tλ)(|x|2+|u|2)

so that there is no problem atλ =0. Therefore, if we assume thatλfλ is sup- ported in, say|λ| ≤ R, the above integral representation for fpi t makes sense for all t,|t|< πR.If we further assume that fλ(x,u), has enough decay as a function of (x,u), then fpi t(x,u, ξ)can be holomorphically extended to the whole ofC2n+1 which is the complexification of Hn.

For R>0 letHRstand for the Hilbert subspace of L2(Hn)consisting of functions f having the following two properties:

(i) fλis supported in|λ| ≤R (ii)

R2n+1|f(x,u, ξ)|2e(|x|2+|u|2)d x du dξ <∞.

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DefinefHRto be the square root of the quantity in (ii). ThenHRbecomes a Hilbert space with the inner product

(f,g)=

R2n+1

f(x,u, ξ)g(x,u, ξ)e(|x|2+|u|2)d x du dξ.

We have the following direct integral characterisation of the image ofHRunder ei tL. Let Tt stand for the map which takes fL2(Hn)into(ei t Lλfλ)λ∈[−R,R].

Theorem 3.1 Let R > 0 and |t| < πR. Then Tt : HRR

R⊕Stλ is, up to a constant multiple, an isometric isomorphism.

Proof The theorem follows from Theorem 2.4. Indeed, consider the map f(ei tLf)λ fromHRR

R⊕Stλdλ.As(ei tLf)λ=ei t Lλfλ, we have ei t Lλfλ2Sλ

t =cn

R2n

|fλ(x,u)|2e(|x|2+|u|2)d x du.

Integrating with respect toλand using Plancherel in theξvariable, we obtain R

R

(ei tLf)λ2Sλ

t=cn

R2n+1

|f(x,u, ξ)|2e(|x|2+|u|2)d x du dξ.

To prove the converse, suppose (Fλ)λ∈[−R,R] belong to R

RStλdλ. Then using Theorem 2.4 for eachλ∈ [−R,R]there exists unique fλL2(R2n,e|x|2+|u2|d x du) such that ei t Lλfλ=Fλand the equality

R2n

|fλ(x,u)|2e|x|2+|u2|d x du= Fλ2Sλ t

holds. We define f on Hnby

f(x,u,t)= R

R

fλ(x,u)eiλtdλ.

We see that fHRand Ttf is(Fλ)λ∈[−R,R].Now the function f is unique as fλis so. This proves the surjectivity.

We now address the problem of characterising the image ofHR under ei tLas a weighted Bergman space. Note that the operator ei tLcommutes with left translations by elements of the Heisenberg group (being a right convolution operator) but the space

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HRis not invariant under left translations. However, it is invariant under translation in the last variable. This means thatHRis invariant under translation by elements of the centre of the Heisenberg group. This shows that the image ofHRunder ei tLis also invariant under the action of centre of Heisenberg group. Suppose now that there is a nonnegative weight function Wt(z, w,s)onCn×Cn×Csuch that

C2n+1

|ei tLf(z, w, ζ )|2Wt(z, w, ζ )d z dwdζ (3.1)

=cn

R2n+1

|f(x,u, ξ)|2e(|x|2+|u|2)d x du dξ

for all fHR.Replacing f byτaf(x,u, ξ)= f(x,u, ξa),a∈Rand using the fact that ei tLcommutes withτawe see that

C2n+1

|ei tLf(z, w, ζ )|2Wt(z, w, ζ+a)d z dwdζ (3.2)

=cn

R2n+1

|f(x,u, ξ)|2e(|x|2+|u|2)d x du dξ

for all fHRand aR.Since the right hand side is independent of a we see that Wt(z, w, ζ+a)=Wt(z, w, ζ )for all a which means that Wt(z, w, ζ )depends only on the imaginary part ofζ.Thus if we letζ =ξ +then

C2n+1

|ei tLf(z, w, ζ )|2Wt(z, w,iη)d z dwdζ (3.3)

=cn

R2n+1

|f(x,u, ξ)|2e(|x|2+|u|2)d x du dξ

Using the Euclidean Plancherel theorem in theξ variable, the above leads to

R

R

C2n

|ei t Lλfλ(z, w)|2e2ληWt(z, w,iη)d z dwdηdλ

=cn

R

R2n

|fλ(x,u)|2e(|x|2+|u|2)d x du dλ.

Letϕk be a sequence of Schwartz functions onRsuch thatˆk)2converges to the Dirac mass atλ0.Replacing f by f(x,u, ξ)ϕk(ξ)in the above equation and letting k

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tend to infinity we obtain

R

C2n

|ei t Lλ0 fλ0(z, w)|2e2λ0ηWt(z, w,iη)d z dwdη (3.4)

=cn

R2n

|fλ0(x,u)|2e(|x|2+|u|2)d x du.

This is true for anyλ0,0| ≤R.Since the weight function for the image of L2(R2n) under ei t Lλis unique, the above leads to the condition

−∞

e2ληWt(z, w,iη)dη (3.5)

=cn

λ2 sin2

n

e

1 4

λ2 sin2 tλ

(|y|2+|v|2)

eλ(cot tλ)(x·y+u·v) which should be valid for all|λ| ≤R.

We now show that the above condition cannot be satisfied by a non negative weight function leading to the following negative result.

Theorem 3.2 There does not exist a non negative weight function Wt(z, w,iη)such that

C2n+1

|ei tLf(z, w, ζ )|2Wt(z, w,iη)d z dwdζ

=cn

R2n+1

|fλ(x,u)|2e(|x|2+|u|2)d x du dλ

for all fHR.

Proof As we have observed, the weight function has to satisfy the condition (3.5) for all|λ| ≤ R.We fix z and wand consider Wt(z, w,iη)as a function ofη.In view of (3.5) it is integrable as a function ofηas we are assuming that Wt is nonnegative.

Moreover,

+i s)

R

e2(λ+i sWt(z, w,iη)dη

is holomorphic on the strip|λ|<R. To see this, we need only to verify the convergence of the integral

R

|η|e2ληWt(z, w,iη)dη.

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