Department of Mathematics Indian Institute of Science Bangalore...

Hermite and Laguerre semigroups

some recent developments

S. Thangavelu

Department of Mathematics

Indian Institute of Science

Bangalore

Technical Report No. 2006/7

March 26, 2006

HERMITE AND LAGUERRE SEMIGROUPS

SOME RECENT DEVELOPMENTS

SUNDARAM THANGAVELU

Abstract. We introduce and study some properties of Hermite, special Her-mite and Laguerre semigroups. The image of L

2 under these semigroups areshown to be certain weighted Bergman spaces of entire functions. The readeris expected to have some basic knowledge of Fourier Analysis but otherwisethis notes is self-contained.

1. Introduction

It is no exaggeration to say that Hermite functions are ubiquitous in Mathemat-ics. They appear in such diverse fields as harmonic analysis, differential equations,mathematical physics and probability theory. They are eigenfunctions of the simpleharmonic oscillator and hence play an important role in quantum mechanics. Theyare also eigenfunctions of the Fourier transform, a fact exploited by Norbert Wienerin his treatment of the Fourier transform. Hermite functions can be expressed asLaguerre functions of type 1

2 and − 12 . In this sense Laguerre functions are gener-

alisations of the Hermite functions. But there is a deeper relation between thesetwo families of functions which arises in connection with analysis on the Heisenberggroup Hn.

The modern theory of Hermite and Laguerre expansions makes use of this con-nection. Heisenberg group Hn is the most well known example from the realm ofnilpotent Lie groups. No detailed harmonic analysis can be done on H

n withoutusing Hermite functions. The most beautiful relation between Hermite and La-guerre functions is experessed by the formula W (ϕn−1

k ) = (2π)nPk where ϕn−1k are

Laguerre functions of type (n−1), Pk are the projections associated to the Hermiteoperator and W is the Weyl transform related to the Schrodinger representationπ1 of Hn.

Our aim in these lectures is to introduce the Hermite and Laguerre semigroupsvia the Heisenberg group. Both semigroups are related to the semigroup generatedby the sublaplacian on H

n. The group Fourier transform on Hn takes this latter

semigroup into the Hermite semigroup whereas the Fourier decomposition in thecentral variable in Hn leads to the socalled special Hermite semigroup. This lastsemigroup, generated by the special Hermite operator, encompasses all Laguerresemigroups of integer type.

Date: January 16, 2006.1991 Mathematics Subject Classification. 42A38, 42B08, 42B15.Key words and phrases. Hermite, special Hermite and Laguerre functions, Heisenberg group,

twisted convolution, heat kernels, Bergman spaces.ST wishes to thank the organisers for inviting him to deliver these lectures.

1

2 SUNDARAM THANGAVELU

After introducing the Hermite, special Hermite and Laguerre semigroups weproceed to the description of the image of L2 under these semigroups. It is a classicalresult of Bargmann and Fock that the image of L2(Rn) under the Gauss-Weierstrasssemigroup can be defined as a weighted Bergman space of entire functions. Thisspace was associated to the realisation of the creation and annihilation operatorsfor Bosons in quantum physics. Similar results are known from the works of Hall[5] and Stenzel [10] for the semigroups generated by the Laplace-Beltrami operatoron compact symmetric spaces. Recently it has been shown that the situation isquite different in the case of Heisenberg groups ( see Krotz-Thangavelu-Xu [8] )and noncompact symmetric spaces (see Krotz-Olafsson-Stanton [9]).

The plan of the notes is as follows. We introduce Hermite and Laguerre semi-groups in Section 2. In the process we introduce Segal-Bargmann transform onRn and study the Fock-Bergman space associated to the standard Laplacian on

Rn. We also consider the Bessel semigroup and the associated Bergman spaces. InSection 3 we study the Hermite-Bergman and twisted Bergman spaces. The resultsfor Laguerre semigroups are deduced from the corresponding results for Hermiteand special Hermite semigroups.

2. Hermite and Laguerre semigroups

2.1. Hermite functions and Bargmann transform. We begin with the defi-nition of Hermite polynomials Hk(x) where k is a nonnegative integer and x ∈ R.

These are defined by

Hk(x) = (−1)kex2 dk

dxk(e−x

2

).

It is then easy to see that the functions hk(x) = Hk(x)e− 1

2x2

are eigenfunctions of

the Hermite operator H = − d2

dx2 + x2. More precisely,

(− d2

dx2+ x2)hk(x) = (2k + 1)hk(x).

From this one can easily conclude that {hk : k = 0, 1, 2, ...} forms an orthogonalsystem in the Hilbert space L2(R). The operator H can be factorised as H =12 (AA∗ +A∗A) where A = d

dx+x and A∗ = − d

dx+x is its formal adjoint. Another

easy calculation shows that

A∗hk(x) = hk+1(x), Ahk(x) = 2k ˜hk−1(x).

This shows that hk(x) = A∗k(e−12x

2

). Using this and the relation A∗A = H + 1along with an induction argument we can show that

∫

R

(hk(x))2dx = 2kk!π

12 .

Therefore, we conclude that the functions

hk(x) = (2kk!π12 )−

12 hk(x).

form an orthonormal family of functions in L2(R). We can actually show that theyform an orthonormal basis ( see below).

From the definition of the Hermite polynomials it follows that Hk are given bythe generating function

∞∑

k=0

Hk(x)

k!wk = e2xw−w2

HERMITE AND LAGUERRE SEMIGROUPS 3

for any w ∈ C. This can be checked by Taylor expanding the right hand side about

w = 0 and using the definition of Hk. Defining ζk(w) = (2kk!π12 )−

12wk we can

rewrite the above as

(2.1)

∞∑

k=0

hk(x)ζk(w) = π− 12 e−

12 (x−w)2e

14w

2

.

The series converges uniformly over compact subsets of C. From the above we caneasily deduce

Theorem 2.1. The family {hk : k = 0, 1, 2, ...} is an orthonormal basis for L2(R).

Proof. As {hk : k = 0, 1, 2, ...} is an orthonormal system Bessel’s inequality showsthat

∞∑

k=0

|(f, hk)|2 ≤ ‖f‖22.

Therefore, the series∑∞k=0(f, hk)ζk(w) converges absolutely and equals

Bf(w) = π− 12 e

14w

2

∫

R

f(x)e−12 (x−w)2dx.

If now f ∈ L2(R) is orthogonal to all hk then the integral defining Bf will be zero

for all w. This means convolution of f with the Gaussian h0(x) = (π)−14 e−

x2

2 is

identically zero. Taking Fourier transform we get f = 0 and consequently f = 0.This proves the theorem. �

The operator B is called the Bargmann transform in the literature and hasinteresting properties. It takes functions f ∈ L2(R) into entire fuctions Bf(w) onC. These are not merely entire functions but also square integrable with respect to

the Gaussian measure dµ(w) = (4π)−12 e−

12 |w|2dw. An easy calculation using polar

coordinates show that the fuctions ζk form an orthonormal system in L2(C, dµ(w)).Let F be the subspace of L2(C, dµ(w)) consisting of entire functions. Then theequation

∞∑

k=0

(f, hk)ζk(w) = Bf(w)

shows that∫

C

|Bf(w)|2dµ(w) =

∫

R

|f(x)|2dx.

This leads to the interesting result

Theorem 2.2. The Bargmann transform B is an isometric isomorphism between

L2(R) and F .Proof. We only need to check that B is onto but this will follow once we observe thatB takes the Hermite basis into {ζk : k = 0, 1, 2, ...} and this family is an orthonormalbasis for F . This last claim is justified since the expansion of a function F from Fin terms of ζk is nothing but its Taylor expansion. �

Before proceeding further let us introduce multi-dimensional Hermite functions.For each multi-index α ∈ Nn and x ∈ Rn we define

Φα(x) = hα1(x1).....hαn(xn).


Then it is clear that {Φα : α ∈ Nn} is an orthonormal basis for L2(Rn). Wedefine the functions ζα and the space F(Cn) in a similar way. The functions ζαform an orthonormal basis for F(Cn) and the Bargmann transform is an isometricisomorphism between L2(Rn) and F(Cn).

We conclude this subsection with another useful formula known as Mehler’sformula for the Hermite functions.

Proposition 2.3. For all w ∈ C with |w| < 1 we have

∑

α∈Nn

Φα(x)Φα(y)w|α| = π−n2 (1 − w2)−

n2 e

− 12

1+w2

1−w2 (x2+y2)+ 2w

1−w2 x·y.

Proof. It is enough to prove the formula in one dimension. As the Bargmanntransform B takes the Hermite basis into the basis ζk it is unitary and hence itsinverse is given by the adjoint B∗. Hence

∞∑

k=0

hk(x)hk(y)wk =

∞∑

k=0

hk(x)B∗ζk(y)w

k

and so in view of (2.1) we need to caculate B∗Fw(y) when

Fw(z) = π− 12 e−

12 (x−wz)2e

14w

2z2 .

Since

(B∗F, f) = (4π)−12

∫

C

F (z)Bf(z)e−12 |z|

2

dz

an easy calculation shows that

B∗F (y) = (4π)−12

∫

C

F (z)e−12 (y−z)2e

14 z

2

e−12 |z|

2

dz.

Taking F (z) = Fw(z) = π− 12 e−

12 (x−wz)2e

14w

2z2 in this formula and evaluating theGaussian Fourier transform we complete the proof. �

2.2. Gauss-Weierstrass kernel and Bergman spaces. The space F(Cn) isknown as the Fock space in the literature. As a motivation for what we plan todo with the Hermite and Laguerre semigroups let us look at F(Cn) more closely.Consider the Gauss-Weierstrass kernel or simply the heat kernel associated to thestandard Laplacian ∆ on Rn defined by

gt(x) = (4πt)−n2 e−

x2

4t .

(Here and later in these notes we will be writing x2 in place of |x|2 =∑n

j=1 x2j . By

the same convention for z ∈ Cn we let z2 stand for∑n

j=1 z2j .) The name heat kernel

is justified since the function

Gtf(x) = f ∗ gt(x) =

∫

Rn

f(y)gt(x− y)dy

satisfies the heat equation for the Laplacian with initial condition f . Here f canbe from any of the Lp spaces over Rn.

For w ∈ Cn the heat kernel gt(w − y) makes sense as an entire function and sois Gtf(x). That is to say the function Gtf(x) extends to Cn as an entire functionof w = u+ iv. Note that when t = 1

2 we have the relation

G 12f(w) = Bf(w)e−

14w

2


for all f ∈ L2(Rn). The isometry between L2(Rn) and F(Cn) takes the form∫

Cn

|G 12f(u+ iv)|2e−v2dudv =

∫

Rn

|f(x)|2dx.

More generally, for any t > 0 we have∫

Cn

|Gtf(u+ iv)|2e− v2

2t dudv = ct

∫

Rn

|f(x)|2dx

for an appropriate constant ct. If we define Bt(Cn) to be the space of all entire

functions on Cn which are square integrable with respect to the measure e−

v2

2t dudv

we have the following result.

Theorem 2.4. The heat kernel transform defined by the Gauss-Weierstrass kernel

is an isometric isomorphism between L2(Rn) and Bt(Cn).

The spaces Bt(Cn) are called weighted Bergman spaces asociated to the stan-

dard Laplacian. Thus the image of L2(Rn) under the heat kernel transform is theweighted Bergman space Bt(C

n). We are mainly interested in studying analoguesof such spaces when the Laplacian is replaced by other operators such as Hermiteand special Hermite operators.

2.3. Laguerre functions. For any α > −1 Laguerre polynomials of type α aredefined by

Lαk (x)e−xxα =1

k!

dk

dxk(e−xxk+α).

Here x ∈ R+ = [0, ∞) and k ∈ N. Explicitly, the Laguerre polynomials are givenby

Lαk (x) =

k∑

j=0

Γ(k + α+ 1)

Γ(j + α+ 1)Γ(k − j + 1)j!(−1)jxj .

From the definition we easily get the generating function

∞∑

k=0

Lαk (x)wk = (1 − w)−α−1e−w

1−wx

for all x ∈ R+ and |w| < 1. We also have another generating function identity whichwill be useful later. Recall that the Bessel function Jα of order α > −1 is given bythe power series

Jα(t) = (t

2)α

∞∑

k=0

(−1)k

k!Γ(k + α+ 1)(t

2)2k

which converges uniformly over compact subsets of C. By a direct calculation usingthe above explicit formula for Lαk we can easily prove

(2.2)

∞∑

k=0

Lαk (x)

Γ(k + α+ 1)wk = ew(xw)−

α2 Jα(2(xw)

12 ).

It also follows from the definition that Lαk forms an orthogonal system withrespect to the measure xαe−xdx on R+. Indeed, if p(x) is any polynomial

∫ ∞

0

p(x)Lαk (x)xαe−xdx =1

k!

∫ ∞

0

p(x)dk

dxk(e−xxk+α)dx.


If p is of degree less than k then an integration by parts shows that∫ ∞

0

p(x)Lαk (x)xαe−xdx = 0.

In particular∫ ∞

0

Lαj (x)Lαk (x)xαe−xdx = 0

for all j 6= k. When k = j we get∫ ∞

0

Lαk (x)Lαk (x)xαe−xdx =1

k!

∫ ∞

0

xk+αe−xdx.

The last integral can be evaluated and equals Γ(k+α+1)Γ(k+1) . We slightly change the

notation and consider the functions

ψαk (r) =

(

2−αk!

Γ(k + α+ 1)

)12

Lαk (1

2r2)e−

14 r

2

.

These form an orthonormal system in L2(R+, r2α+1dr). Actually, more is true.

Theorem 2.5. The above system {ψαk : k ∈ N} is an orthonormal basis for

L2(R+, r2α+1dr).

Proof. If f ∈ L2(R+, r2α+1dr) is orthogonal to all ψαk then the generating function

identity (2.2) shows that∫ ∞

0

f(r)Jα(λr)

(λr)α2r2α+1dr = 0

for all λ ∈ R+. This means the Hankel transform of f is identically zero and henceby the inversion formula for the Hankel transform we get f = 0. �

We remark that when α = (n2 − 1) where n is a nonnegative integer the Hankeltransform of a function f is a constant multiple of the Fourier transform of theradial function F (x) = f(|x|) on Rn. Thus for these special values of α the aboveresult follows from the Fourier inversion theorem.

The Hermite and Laguerre polynomials are related to each other. On the

one hand we have the formulas H2k(x) = (−1)k22kk!L− 1

2

k (x2) and H2k+1(x) =

(−1)k22kk!xL12

k (x2). On the other hand Laguerre polynomials can also be expressed

in terms of Hermite polynomials. For α > − 12 one has the formula

Lαk (x) =(−1)kπ− 1

2

Γ(α+ 12 )

Γ(k + α+ 1)

(2k)!

∫ 1

−1

(1 − t2)α−12H2k((xt)

12 )dt.

There is another deeper connection between these two families of functions. Inorder to bring that out we have to analyse some aspects of the Heisenberg group.

2.4. Heisenberg group and Weyl transform. As a set the Heisenberg groupHn is just Cn × R but the group operation is defined by

(z, t)(w, s) = (z + w, t+ s+1

2=(z · w))

for z, w ∈ Cn and t, s ∈ R. Here z · w =∑n

j=1 zjwj . This group is nonabelian butunimodular and the Lebesgue measure dzdt on Cn × R is the Haar measure onHn. The group Fourier transform on Hn is defined in terms of certain irreducibleunitary representations which we now proceed to describe.


For each nonzero real λ and (z, t) ∈ Hn consider the operator πλ(z, t) defined onL2(Rn) by

πλ(z, t)ϕ(ξ) = eiλteiλ(x·ξ+ 12x·y)ϕ(ξ + y)

where ϕ ∈ L2(Rn) and z = x+iy. It is easy to see that πλ(z, t) is a unitary operatorand

πλ(z, t)πλ(w, s) = πλ((z, t)(w, s)).

We say that πλ is a unitary representation of Hn. It can be shown that πλ isirreducible in the sense that L2(Rn) does not have any nontrivial subspace that isinvariant under πλ(z, t) for all (z, t) ∈ H

n.

The group Fourier transform of a function f ∈ L1(Hn) is the operator valuedfunction

f(λ) =

∫

Hn

f(z, t)πλ(z, t)dzdt.

As πλ(z, t) are unitary it is clear that f(λ) is bounded on L2(Rn) and the operatornorm is bounded by ‖f‖1. From the definition πλ(z, t) = eiλtπλ(z, 0) and thereforeif we define

fλ(z) =

∫

R

f(z, t)eiλtdt

then it follows that f(λ) = Wλ(fλ) where

Wλ(fλ) =

∫

Cn

fλ(z)πλ(z, 0)dz.

This suggests that in order to study the group Fourier transform we have to lookat the operators Wλ(f

λ) more closely.Let us take λ = 1 and consider W (g) = W1(g) for functions g ∈ L1(Cn) defined

on Cn. The operator W (g) is called the Weyl transform of g. It follows from thedefinition that

W (g)ϕ(ξ) =

∫

Rn

Kg(ξ, η)ϕ(η)dη

where the kernel is explicitly given by

Kg(ξ, η) =

∫

Rn

g(x+ i(η − ξ))ei2x·(ξ+η)dx.

Thus W (g) is an integral operator and if g also belongs to L2(Cn) then Kg ∈L2(Rn × Rn). Therefore, W (g) is a Hilbert-Schmidt operator.

Theorem 2.6. The Weyl transform initially defined on L1 ∩ L2 extends to the

whole of L2(Cn) as an isometric isomorphism onto the space of all Hilbert-Schmidt

operators on L2(Rn).

Proof. As W (g) is an integral operator with kernel Kg we know that

‖W (g)‖2HS =

∫

Rn×Rn

|Kg(ξ, η)|2dξdη.

The explicit formula for the kernelKg along with Plancherel theorem for the Fouriertransform on R

n yields ‖W (g)‖2HS = (2π)n‖g‖2

2. This completes the proof. �


The bilinear form (W (g)ϕ, ψ) on L2(Rn) takes the form

(W (g)ϕ, ψ) =

∫

Cn

g(z)(π(z, 0)ϕ, ψ)dz

and this suggests that we look at functions of the form

Vϕ,ψ(z) = (2π)−n2 (π(z, 0)ϕ, ψ).

This is called the Fourier-Wigner transform of ϕ and ψ and explicitly given by

(2.3) Vϕ,ψ(z) = (2π)−n2

∫

Rn

eix·ξϕ(ξ +1

2y)ψ(ξ − 1

2y)dξ.

This formula leads to the following interesting result.

Proposition 2.7. For f, g, ϕ, ψ ∈ L2(Rn) we have

(Vf,g , Vϕ,ψ) = (f, ϕ)(ψ, g)

where the inner product on the left is on Cn whereas those on the right are on Rn.

Proof. From the explicit formula (2.3) we obtain∫

Cn

|Vϕ,ψ(z)|2dz =

∫

Rn×Rn

|ϕ(ξ +1

2y)|2|ψ(ξ − 1

2y)|2dξdy.

By a change of variables we get

‖Vϕ,ψ‖22 = ‖ϕ‖2

2‖ψ‖22.

By polarising this identity we obtain the proposition. �

Corollary 2.8. For f, g, ϕ, ψ ∈ L2(Rn) we have

(W (Vϕ,ψ)f, g) = (2π)n2 (f, ϕ)(ψ, g).

2.5. Special Hermite functions. In view of Proposition 2.7 we see that we canstart with an orthonormal system in L2(Rn) and construct an orthonormal systemfor L2(Cn). If we take the Hermite basis for L2(Rn) we get what we call the or-thonormal basis of special Hermite functions. These functions are defined for eachpair of multi-indices by

Φα,β(z) = VΦα,Φβ(z) = (2π)−

n2 (π(z, 0)Φα,Φβ).

We then have

Theorem 2.9. The family {Φα,β : α, β ∈ Nn} is an orthonormal basis for L2(Cn).

Proof. The orthonormality follows from Proposition 2.7. To prove the completenesswe make use of the result of Theorem 2.6. If f ∈ L2(Cn) is orthogonal to all Φα,βthen it follows that (W (f)Φα,Φβ) = 0 for all α, β ∈ Nn and hence W (f) = 0. Anappeal to Theorem 2.6 completes the proof. �

The name special Hermite functions is suggested by the fact that Φα,β are eigen-functions of the Hermite operator −∆ + 1

4 |z|2 on Cn. Indeed, by direct calculation

we can check that

(−∆ +1

4|z|2)Φα,β(z) = (|α| + |β| + n)Φα,β(z).


These functions are also eigenfunctions of the special Hermite operator, also calledtwisted Laplacian, which is defined by

L = −∆ +1

4|z|2 − i

n∑

j=1

(xj∂

∂yj− yj

∂

∂xj).

More precisely, we have

LΦα,β = (2|β| + n)Φα,β

which means the eigenspaces of L corresponding to the eigenvalues (2k+n) are in-finite diemensional, an interesting new feature not shared by the Hermite operator.

The special Hermite operatorL is related to the sublaplacian L on the Heisenberggroup. On Hn consider the first order differential operators

Xj = (∂

∂xj− 1

2yj∂

∂t), Yj = (

∂

∂yj+

1

2xj∂

∂t)

for j = 1, 2, ..., n. These ’vector fields’ along with T = ∂∂t

generate the Heisenberg

Lie algebra and they play the role of ∂∂xj

on Rn. Analogous to the Laplacian on R

n

the sublaplacian on Hn is defined by

L = −n∑

j=1

(X2j + Y 2

j )

and is given explicitly by the expression

L = −∆ − 1

4|z|2 ∂

2

∂t2+

n∑

j=1

(xj∂

∂yj− yj

∂

∂xj)∂

∂t.

Then it follows that

L(e−itf(z)) = e−itLf(z)

and this is the reason why L is sometimes called the twisted Laplacian.Another interesting property of special Hermite functions is the fact that they are

expressible in terms of Laguerre functions. Moreover, multiple Laguerre expansionsof certain kind can be studied by looking at special Hermite expansions of functionshaving special homogeneity properties. Here we restrict ourselves in calculatingΦα,α and show that they are nothing but multiple Laguerre functions.

Proposition 2.10.

Φα,α(z) = (2π)−n2

n∏

j=1

L0αj

(1

2|zj |2)e−

14 |zj |

2

.

Proof. It is enough to consider the one dimensional case as Φα,α(z) is the product ofone dimensional functions. We begin with Mehler’s formula proved in Proposition2.3, namely

∞∑

k=0

hk(ξ +1

2y)hk(ξ −

1

2y)rk = π− 1

2 (1 − r2)−12 e−

1+r1−r

y2

4 e−1−r1+r

ξ2 .

If we take the Fourier transform in the ξ variable then the left hand side becomes∑∞k=0 Φk,k(z)r

k . On the other hand, the right hand side becomes

1√2π

(1 − r2)−12 e−

1+r1−r

y2

4

∫

R

eixξe−1−r1+r

ξ2dξ =1√2π

(1 − r)−1e−14

1+r1−r

(x2+y2).


But this is nothing but

1√2π

∞∑

k=0

L0k(

1

2(x2 + y2))e−

14 (x2+y2)rk

in view of the generating function identity for the Laguerre functions. A comparisonof these two expressions proves the proposition. �

2.6. The Hermite semigroup. The Hermite operator H = −∆ + |x|2 which isself-adjoint admits a spectral decomposition which is explicitly given in terms ofHermite functions. Given f ∈ L2(Rn) we have the Hermite expansion

f =∑

α∈Nn

(f,Φα)Φα

the series being convergent to f in L2(Rn). For each k ∈ N let Pk stand for theorthogonal projection of L2(Rn) onto the eigenspace spannned by {Φα : |α| = k}.Then the spectral decomposition of H is explicitly given as

Hf =

∞∑

k=0

(2k + n)Pkf.

The operator H defines a semigroup, called the Hermite semigroup and denoted bye−tH , t > 0 by the expansion

e−tHf =

∞∑

k=0

e−(2k+n)tPkf

for f ∈ L2(Rn). On a dense subspace, say the space of all Schwartz functions, theabove can be written as

e−tHf(x) =

∫

Rn

f(y)Kt(x, y)dy

where the kernel Kt(x, y) is given by the expansion

Kt(x, y) =∑

α∈Nn

e−(2|α|+n)tΦα(x)Φα(y).

In view of Mehler’s formula the above series can be summed up and we obtain

Kt(x, y) = (2π)−n2 (sinh(2t))−

n2 e

− 12 coth(2t)(x2+y2)+ 1

sinh(2t)x·y.

The integral representation of the Hermite semigroup leads to the following result.

Theorem 2.11. The Hermite semigroup e−tH initially defined on L2 ∩ Lp(Rn)extends to the whole of Lp(Rn) and we have ‖e−tHf‖p ≤ Ct‖f‖p for all 1 ≤ p ≤ ∞.

To prove the theorem we only need to check that the integral of Kt(x, y) withrespect to y is bounded by a constant Ct independent of x. For then it follows thate−tH is bounded on L1 and as it is already bounded on L2 interpolation proves thetheorem when 1 ≤ p ≤ 2. Duality then takes care of the other values of p. We canactually show that e−tH is a contraction on every Lp.

Let us take a digression and bring out the relation between e−tH and the semi-group generated by the sublaplacian L on Hn. Though the operator L is not ellipticit is hypoelliptic, thanks to a theorem of Hormander. As it is self-adjoint and non-negative it generates a semigroup e−aL. According to a theorem of G. Hunt this


semigroup is given by convolution with a measure which is absolutely continuouswith respct to the Haar measure on Hn. Thus

e−aLf(z, t) =

∫

Hn

f((z, t)(w, s)−1)qa(w, s)dwds.

The function qa which is a positive Schwartz function is called the heat kernelassociated to the sublaplacian. The Fourier transform of qa in the t variable isexplicitly known (see below).

As in the Euclidean case the group Fourier transform on Hn transforms convolu-

tions into products. That is to say (f∗g)(λ) = f(λ)g(λ) for functions f, g ∈ L1(Hn).

Therefore, (e−aLf )(λ) = f(λ)qa(λ) and we can show that qa(λ) = e−aH(λ) whereH(λ) = −∆ + λ2|x|2 is the scaled Hermite operator. Since the kernel of e−aH(λ)

is explicitly known the heat kernel qa(z, t) can be calculated. In fact, we will showlater that

(2.4) qa(z, t) = cn

∫

R

eiλt(λ

sinh(aλ))ne−

λ4 coth(aλ)|z|2dλ.

In order to do this we have to use the connection between the Hermite and specialHermite semigroups.

2.7. The special Hermite semigroup. Consider the special Hermite expansionof a function f ∈ L2(Cn) written as

f =∑

α

∑

β

(f,Φα,β)Φα,β

the series being convergent in L2(Cn). We now show that this series can be put ina compact form using the notion of twisted convolution. If f and g are functionson Hn then we can easily check that

(f ∗ g)λ(z) =

∫

Cn

fλ(z − w)gλ(w)ei2λ=(z·w)dw.

This suggests that we define a new convolution ( called twisted convolution) on Cn

by

F ∗λ G(z) =

∫

Cn

F (z − w)G(w)ei2λ=(z·w)dw

so that (f ∗ g)λ(z) = fλ ∗λ gλ(z). When λ = 1 we use the notation F ×G in placeof F ∗1 G. It then follows that W (F × G) = W (F )W (G) where W is the Weyltransform.

The Hermite projections Pk are given by

Pkϕ =∑

|α|=k

(ϕ,Φα)Φα

for ϕ ∈ L2(Rn). Combining the results of Proposition 2.10 and Corollary 2.8 wecan write the above as

Pkϕ = (2π)−n2

∑

|α|=k

W (Φα,α)ϕ.

This leads to the following result which connects the projections Pk with Laguerrefunctions. We define

ϕn−1k (z) = Ln−1

k (1

2|z|2)e− 1

4 |z|2

.


Proposition 2.12. We have W (ϕn−1k ) = (2π)nPk and consequently ϕk × ϕj =

(2π)2nδkj for every j, k ∈ N.

Proof. In view of the above observations we only need to show that

(2.5)∑

|α|=k

Φα,α(z) = (2π)−n2 ϕn−1

k (z).

The Laguerre functions ϕn−1k (z) satisfy the generating function

(2.6)

∞∑

k=0

ϕn−1k (z)rk = (1 − r)−ne−

14

1+r1−r

|z|2 .

On the other hand Proposition 2.10 gives

Φα,α(z) = (2π)−n2

n∏

j=1

L0αj

(1

2|zj |2)e−

14 |zj |

2

and each L0αj

( 12 |zj |2)e−

14 |zj |

2

satisfies

∞∑

k=0

L0k(

1

2|zj |2)e−

14 |zj |

2

rk = (1 − r)−1e−14

1+r1−r

|zj |2

.

From the last two equations it is clear that

∞∑

k=0

∑

|α|=k

Φα,α(z)

rk = (2π)−n2 (1 − r)−ne−

14

1+r1−r

|z|2 .

Comparing this with (2.6) we get the proposition. �

We are now in a position to rewrite the special Hermite expansion in a compactform as follows.

Theorem 2.13. For every f ∈ L2(Cn) its special Hermite expansion takes the

form

f = (2π)−n∞∑

k=0

f × ϕk

where the series converges to f in L2(Cn).

Proof. In order to prove the theorem we have to show that

f × ϕk = (2π)n∑

|β|=k

∑

α

(f,Φα,β)Φα,β .

From the definition of twisted convolution and the relation (2.5) we only need toshow

Φβ,β(z − w)e−i2=(z·w) = (2π)

n2

∑

α

Φα,β(z)Φα,β(w).

We can now expand the function on the left hand side in terms of the orthonormalbasis {Φα,γ : α, γ ∈ Nn}. We note that

∫

Cn

Φβ,β(z − w)e−i2=(z·w)Φα,γ(w)dw = Φα,γ × Φβ,β.

By taking the Weyl transform one easily checks that Φα,γ × Φβ,β vanishes unlessγ = β in which case it is equal to (2π)

n2 Φα,β(z). This proves the theorem. �


We are now ready to define the special Hermite semigroup by the relation

e−tLf = (2π)−n∞∑

k=0

e−(2k+n)tf × ϕk

for functions f ∈ L2(Cn). For Schwartz class functions it is clear that e−tLf(z) =f × pt(z) where

pt(z) = (2π)−n∞∑

k=0

e−(2k+n)tϕk(z).

The generating function for Laguerre functions of type (n− 1) leads to the explicitformula

pt(z) = (2π)−n(sinh t)−ne−14 coth(t)|z|2 .

This is the heat kernel associated to the special Hermite operator. As in the caseof Hermite semigroup we have the following result.

Theorem 2.14. The special Hermite semigroup initially defined on L2 ∩ Lp(Cn)extends to the whole of Lp(Cn) and satisfies ‖e−tLf‖p ≤ Ct‖f‖p for all 1 ≤ p ≤ ∞.

We conclude this subsection by relating all the three semigroups. Let us definethe scaled special Hermite operatorLλ by the equationL(e−iλtf(z)) = e−iλtLλf(z).Then it follows that e−tLλf(z) = f ∗λ pλt (z) where

pλt (z) = (2π)−n(λ

sinh(λt))ne−

14λ coth(λt)|z|2 .

It is also clear that Wλ(pλt ) = e−tH(λ). Moreover, for f ∈ L2(Hn) the function

fλ ∗λ qλa satisfies the heat equation for Lλ and hence qλa should be equal to pλaproving our formula (2.4) for the heat kernel qa(z, t) on the Heisenberg group.

2.8. Laguerre semigroups. In constructing the Laguerre semigroups we restrictourselves to the cases where α = n

2 −1 is a half integer. This is because in these casesthe Laguerre semigroups are related to Hermite and special Hermite semigroups.We begin with the following observation.

Proposition 2.15. Denote by µr the normalised surface measure on the sphere

{w : |w| = r} in Cn. Then for any k ∈ N we have∫

|w|=r

ϕk(z − w)ei2=(z·w)dµr(w) =

k!(n− 1)!

(k + n− 1)!ϕn−1k (r)ϕk(z).

Proof. Consider the equation ϕn−1k ×ϕn−1

j = (2π)nδkjϕn−1k which written explicitly

gives∫ ∞

0

(

∫

|w|=r

ϕk(z − w)ei2=(z·w)dµr(w)

)

ϕn−1j (r)r2n−1dr = (2π)nδkjϕ

n−1k (z).

This simply means that the Laguerre expansion of the radial function∫

|w|=r

ϕk(z − w)ei2=(z·w)dµr(w)

reduces to a single term and we get the proposition. �

An immediate corollary of the above proposition is the fact that the specialHermite expansion of a radial function reduces to a Laguerre expansion.


Corollary 2.16. Let f ∈ L2(Cn) be radial,i.e. f(z) depends only on |z|. Then

f × ϕn−1k (z) =

k!(n− 1)!

(k + n− 1)!

(∫ ∞

0

f(r)ϕn−1k (r)r2n−1dr

)

ϕn−1k (z).

Thus the Laguerre expansion of type (n− 1) of a function f ∈ L2(R+, r2n−1) is

nothing but the special Hermite expansion of the radial function f(|z|) on Cn. Thefunctions ψn−1

k are eigenfunctions of the Laguerre operator

L(n− 1) = − d2

dr2+

2n− 1

r

d

dr+

1

4r2

with eigenvalues (2k + n). The Laguerre semigroup of type (n − 1) generated byL(n− 1) is given by

e−tL(n−1)f(r) =

∞∑

k=0

e−(2k+n)t(f, ψn−1k )ψn−1

k (r).

In view of the above remarks we also have

e−tL(n−1)f(r) = e−tLf(z), z ∈ Cn, |z| = r.

( By abuse of notation we have identified f(|z|) with f.)The Laguerre semigroup is an integral operator of the form

e−tL(n−1)f(r) =

∫ ∞

0

f(s)Kt(r, s)s2n−1ds

where the kernel is given by the expansion

Kt(r, s) =

∞∑

k=0

21−nk!

(k + n− 1)!e−(2k+n)tϕn−1

k (r)ϕn−1k (s).

This series can be summmed up to obtain

Theorem 2.17. The kernel of the Laguerre semigroup is explicitly given by

Kt(r, s) = cn(sinh t)−1e−14 coth(t)(r2+s2) Jn−1(

irs2 sinh t )

(rs)n−1

where Jn−1 is the Bessel function of type (n− 1) and cn is a constant.

The connection between Laguerre and special Hermite semigroups show that

Kt(r, s) =

∫

|w|=r

pt(z − w)e−i2=(z·w)dµr(w).

Therefore, the theorem follows once we have

Proposition 2.18.

∫

|w|=r

e−14 coth(t)|z−w|2e−

i2=(z·w)dµr(w) = (sinh(t))n−1e−

14 coth(t)(|z|2+|w|2)

Jn−1(i|z||w|

2 sinh(t) )

(|z||w|)n−1.

It is possible to prove the proposition by evaluating the integral using polarcoordinates. But there is another easy way of proving the result which uses therelation between Hermite and special Hermite semigroups. Recall that L = H( 1

2 )−iN whereN =

∑nj=1(xj

∂∂yj

−yj ∂∂xj

). The operatorN is called the rotation operator

since it is nothing but∑nj=1

∂∂θj

in terms of the polar coordiantes zj = rjeiθj .

Therefore, N kills all radial functions and hence Lf = H( 12 )f whenever f is radial.


Consequently, for such functions e−tLf = e−tH( 12 )f. This also shows that for radial

functions the Hermite semigroup on R2n reduces to the Laguerre semigroup of type(n−1). The same is true on Rn which connects Laguerre semigroups of type (n2 −1)with Hermite semigroups.

The Hermite functions Φα(2−12 z) are eigenfunctions of H( 1

2 ) on R2n with eigen-

values 12 (2|α| + 2n). Therefore, the kernel of e−tH( 1

2 ) reduces to

(2π)−n(sinh t)−ne−14 coth(t)(|z|2+|w|2)e−

12 sinh t

<(z·w).

In order to prove the proposition we only need to evaluate the integral∫

|w|=r

e−1

2 sinh t<(z·w)dµr(w);

But it is a standard result in Fourier analysis that the above is a constant multipleof

(sinh t)n−1 Jn−1(i|z||w|2 sinh t )

(|z||w|)n−1.

This completes the proof of the proposition and hence the theorem is proved.

3. The image of L2 under the semigroups

3.1. Hermite-Bergman spaces. We begin by investigating the image of L2(Rn)under the Hermite semigroup. Note that the Hermite functions Φα have exten-

sions to Cn as entire functions. Indeed, as Φα(x) = Hα(x)e−12x

2

where Hα(x) are

polynomials the extension is given by Φα(z) = Hα(z)e−12 z

2

. The same is true ofe−tHf(x) for f ∈ L2(Rn). To see this recall that

e−tHf(x) =

∫

Rn

Kt(x, u)f(u)du

where the kernel Kt(x, u) is explicitly given by (see Proposition 2.3 )

Kt(x, u) = (2π)−n2 (sinh(2t))−

n2 e−

12 coth(2t)(x2+u2)e

1sinh(2t)x·u.

Since the kernel Kt(x, u) extends as an entire function Kt(z, u) the same is true ofe−tHf(x). This can be verified by using Morera’s theorem.

We are interested in characterising the space of all functions e−tHf(z) as f variesover L2(Rn) as a weighted Bergman space. Let us consider the weight function

Ut(x+ iy) = 2n(sinh(4t))−n2 etanh(2t)x2−coth(2t)y2

.

Let Ht(Cn) be the space of all entire functions on Cn for which the norms

‖F‖2Ht

=

∫

R2n

|F (x + iy)|2Ut(x + iy)dxdy

are finite. Let us call this space the Hermite-Bergman space. The following is ourmain result which is the analogue of Theorem 2.4 in the context of the Hermitesemigroup.

Theorem 3.1. An entire function F on Cn belongs to Ht(Cn) if and only if F (x) =

e−tHf(x) for some f ∈ L2(Rn). Moreover, we have the equality of norms: ‖F‖Ht=

‖f‖2 whenever F (x) = e−tHf(x).


We break up the theorem into two parts: first we show that the image of L2

under e−tH is contained in Ht(Cn). Then we prove the surjectivity of the heat

kernel transform: f → e−tHf.

Before proceeding with the proof let us recall some properties of the EuclideanFourier transform which we define as

f(ξ) = (2π)−n2

∫

Rn

f(x)e−ix·ξdx

for functions, say f ∈ L1 ∩ L2(Rn). The inversion formula reads as

f(x) = (2π)−n2

∫

Rn

f(ξ)eix·ξdξ

and we also have the Plancherel formula: ‖f‖2 = ‖f‖2 which allows us to extend

the definition of Fourier transform for all functions in L2(Rn). When f has enoughdecay the inversion formula shows that we can extend f as an entire function onCn

f(x+ iy) = (2π)−n2

∫

Rn

f(ξ)e−y·ξeix·ξdξ

and Plancherel formula applied to the function f(x+ iy) leads to∫

Rn

|f(x+ iy)|2dx =

∫

Rn

|f(ξ)|2e−2y·ξdξ.

This is called the Gutzmer’s formula for the Euclidean Fourier transform. We nowprove

Proposition 3.2. F (z) = e−tHf(z) belongs to Ht(Cn) for every f ∈ L2(Rn) and

‖f‖2 = ‖F‖Ht.

Proof. For 0 < r < 1, let Mr(x, u) be the Mehler’s kernel given explicitly by

Mr(x, u) = π− n2 (1 − r2)−

n2 e

− 12

1+r2

1−r2 (x2+u2)e

2r

1−r2 x·u.

When r = e−2t we haveMr(x, u) = entKt(x, u). Therefore, we consider the functionFr(x) =

∫

Rn f(u)Mr(x, u)du. Note that Fr extends to Cn as an entire function. Weneed to show that

rn∫

R2n

|Fr(x+ iy)|2Ut(x+ iy)dxdy =

∫

Rn

|f(x)|2dx.

To prove this, we first observe that

Fr(x+ iy) = 2n2 (1 − r2)−

n2 e

− 12

1+r2

1−r2 (x+iy)2g(

2r

1 − r2(−y + ix))

where g(y) = f(y)e− 1

21+r2

1−r2 y2

which leads to∫

Rn

e1+r2

1−r2 (x2−y2)|Fr(x+ iy)|2dy = 2n(1 − r2)−n∫

Rn

|g( 2r

1 − r2(−y + ix))|2dy.

Applying Gutzmer’s formula and making a change of variables and simplifying weget

∫

Rn

e1+r2

1−r2 (x2−y2)|Fr(x + iy)|2dy = r−n∫

Rn

|f(y)|2e−1+r2

1−r2 y2

e4r

1−r2 x·ydy.


Mutiplying both sides by rn(2π)−n2 e

− 4r2

1−r4 x2

, integrating with respect to x andnoting that

(2π)−n2

∫

Rn

e4r

1−r2 x·ye− 4r2

1−r4 x2

dx = (8r2)−n2 (1 − r4)

n2 e

1+r2

1−r2 y2

we get∫

R2n

e− 1+r2

1−r2 y2

e1−r2

1+r2 x2

|Fr(x + iy)|2dxdy = (8r4)−n2 (1 − r4)

n2

∫

Rn

|f(y)|2dy.

Taking r = e−2t and simplifying we get the proposition.�

As a corollary of the proposition we get the following orthogonality property ofthe complexified Hermite functions.

Corollary 3.3.∫

Cn

Φα(z)Φβ(z)Ut(z)dz = e2(2|α|+n)tδα,β .

As e−tHΦα = e−(2|α|+n)tΦα the corollary follows by polarising the identity∫

Cn

|e−tHf(z)|2Ut(z)dz =

∫

Rn

|f(y)|2dy.

From the corollary it follows that the functions Φα(z) = e−(2|α|+n)tΦα(z) forman orthonormal system in Ht(C

n). We prove our theorem by showing that thesefunctions form an orthonormal basis for Ht(C

n).

Proposition 3.4. The system {Φα : α ∈ Nn} is an orthonormal basis for Ht(Cn).

Proof. We show that any F ∈ Ht(Cn) which is orthogonal to all Φα vanishes

identically. Since the monomials zα can be written as a finite linear combinationof the Hermite polynomials it follows that F is orthogonal to all the functions

ψα(z) = zαe−12 z

2

. Recall the Fock spaces Fs(Cn) defined as the space of all entirefunctions G for which

‖G‖2Fs

=

∫

Cn

|G(z)|2e−s|z|2dz

are finite. It is easy to see that F ∈ Ht if and only if G(z) = F (z)e12 coth(4t)z2 ∈ Fs(t)

where s(t) = (cosh 2t − cosh 4t). To check this one uses the identity tanh 2t +coth 2t = 2 coth 4t.

The assumption that F is orthogonal to all ψα(z) leads to the condition that

G(z) = F (z)e12 coth(4t)z2 is orthogonal to all ψα(z)e

12 coth(4t)z2 in Ft. The Taylor

expansion of the function G(z)e12 (1−coth(4t))z2 leads to

G(z) =∑

α

aαψα(z)e12 coth(4t)z2 .

Since G is orthogonal to all ψα(z)e12 coth(4t)z2 we get aα = 0 for all α and hence

G = 0. This proves our claim that F vanishes identically. �


3.2. Bergman spaces for Bessel semigroups. In our study of Bergman spacesassociated to the Hermite semigroup we have made use of the weighted Bergmanspaces defined by the Gauss-Weierstrass semigroup. In the same way in orderto study Laguerre-Bergman spaces we require Bergman spaces defined by Besselsemigroups. We mainly treat the case of semigroups defined by Bessel functionsof type (n − 1) though the general case is not very different. We impose thisrestriction for the simple reason that the Bessel semigroup is then related to theGauss-Weierstrass semigroup on R2n.

Consider the Gauss-Weierstrass semigroup on R2n defined by

Gtf(x) = (4πt)−n∫

R2n

f(y)e−14t

(x−y)2dy.

When f is a radial function Gtf(x) is also a radial function and we have

Gtf(x) = (4πt)−n∫ ∞

0

f(r)

(∫

S2n−1

e−14t

(x−rω)2dµ(ω)

)

r2n−1dr

where dµ(ω) is the normalised surface measure on the unit sphere S2n−1. By ex-panding the Gaussian in the inner integral and making use of the formula

(2π)−n∫

S2n−1

e−iλx·ωdµ(ω) = cn(λ|x|)1−nJn−1(λ|x|)

we see that

Gtf(r) = cn

∫ ∞

0

f(s)e−14t

(r2+s2)(2t)−1(rs)1−nJn−1(irs

2t)s2n−1dr.

Here cn is a normalising constant which doesn’t play any important role. Therefore,we will be sloppy about such constants and will not make any attempt to write downtheir exact values.

We call the above Bessel semigroup of type (n−1) and denote it by e−tBf wheref ∈ L2(R+, r

2n−1dr). Note that

e−tBf(r) =

∫ ∞

0

f(s)bt(r, s)s2n−1dr

where the kernel bt is given by

bt(r, s) = cn(2t)−1(rs)1−nJn−1(

irs

2t)e−

14t

(r2+s2).

From the above formula for the Bessel function as the Fourier transform of thesurface measure ( or from the power series expansion) it is clear that Jn−1(r)r

1−n

extends to C an an even entire function. Hence for f ∈ L2(R+, r2n−1dr) the same

is true of e−tBf(r). We are interested in characterising the image of L2 under theBessel semigroup as a weighted Bergman space.

Consider the generating function for the Laguerre polynomials, namely,

∞∑

k=0

Lαk (x)

Γ(k + α+ 1)wk = ew(xw)−

α2 Jα(2(xw)

12 ).

Replacing w by − 18w

2, x by 12r

2 and taking α = n− 1 we get

∞∑

k=0

2−kϕn−1k (r)

(k + n− 1)!(−1)k(

1

2w)2k = 4n−1e

18w

2

e−14 (r2+w2)(irw)1−nJn−1(

i

2rw).


Recalling the definition of the Bessel semigroup we see that

e−Bf(w)e18w

2

= cn

∞∑

k=0

2−k(f, ϕn−1

k )

(k + n− 1)!(−1)k(

1

2w)2k .

The functions ( 12w)2k form an orthogonal system in L2(C, h(|w|)dw) for any

radial weight function h. If we can choose the weight function in such a way that∫

C

2−4k|w|4kh(|w|)dw = 22kk!(k + n− 1)!

∫ ∞

0

|ϕn−1k (r)|2r2n−1dr

then we get∫

C

|e−Bf(w)e18w

2 |2h(|w|)dw = cn

∞∑

k=0

k!

(k + n− 1)!|(f, ϕn−1

k )|2.

Since we have∫ ∞

0

|ϕn−1k (r)|2r2n−1dr =

21−nk!

(k + n− 1)!

we get∫

C

|e−Bf(w)e18w

2 |2h(|w|)dw = cn

∫ ∞

0

|f(r)|2r2n−1dr.

Fortunately, there is such a radial weight function. Let Kν be the modifiedBessel function of the third kind, also called the McDonald function defined by

Kν(z) = (π

2z)

12

e−z

Γ(ν + 12 )

∫ ∞

0

e−ttν−12 (1 +

t

(2z))ν−

12 dt

Then we can read out from Erdelyi et al [4, p.51] that∫ ∞

0

r4(k+n−12 )Kn−1(r

2)2rdr = 2n−122kk!(k + n− 1)!.

Therefore, the function h(|w|) = cnKn−1(14 |w|2) has the required property for a

suitable constant cn. More generally the following result is true. Define

ht(u+ iv) = cnKn−1(1

4t(u2 + v2))e

14t

(u2−v2).

We then have

Theorem 3.5. For f ∈ L2(R+, r2n−1dr) the function e−tBf extends to C as an

even entire function and∫

C

|e−tBf(w)|2ht(w)dw =

∫ ∞

0

|f(r)|2r2n−1dr.

Moreover, the map f(r) → e−tBf(w) is onto the space of even entire functions F

for which∫

C|F (w)|2ht(w)dw <∞.

Proof. Only the surjectivity of the map remains to be proved. Note that the heatkernel transform takes the Laguerre functions ϕn−1

k into w2k. So we only need toshow that these monomials form a complete orthogonal system for the space underconsideration. This can be done as in the case of the Bergman space for the Gauss-Weierstrass semigroup going through the corresponding Fock spaces. The detailsare left to the reader. �


3.3. Laguerre-Bergman spaces. This subsection is devoted to a characterisationof the image of L2(R+, r

2n−1dr) under the Laguerre semigroup e−tL(n−1). Let usremark that we do not have to restrict to Laguerre semigroups of type (n− 1) andwhatever we say about this case holds good for the general case of e−tL(α). Resultsfor the semigroup e−tL(n−1) will be deduced from the corresponding results for theBessel semigroup. The situation is very much similar to the case of e−tH .

Let us recall that e−tL(n−1)f(r) =∫∞

0f(s)Kt(r, s)s

2n−1ds where

Kt(r, s) = cn(sinh t)−1e−14 coth(t)(r2+s2)

Jn−1(irs

2 sinh(t) )

(irs)n−1.

It is clear that for f ∈ L2(R+, r2n−1dr) the function e−tL(n−1)f extends to C as

an even entire function. Note that the kernel Kt is related to the kernel bt of theBessel semigroup. From this an easy calculation shows that

e−tL(n−1)f(w) = e−12 tanh( t

2 )w2

e− sinh(t)Bg(w)

where we have written g(s) = f(s)e−12 tanh( t

2 )s2 . From the characterisation of theimage of L2(R+, r

2n−1dr) under the Bessel semigroup we get the following result.

Let us define the measure dµt(s) = e− tanh( t2 )s2s2n−1ds.

Theorem 3.6. The image of L2(R+, dµt) under the Laguerre semigroup e−tL(n−1)

is the Bergman space of all even entire functions on C that are square integrable with

respect to Kn−1(1

4 sinh(t) (u2 + v2))etanh( t

2 )(u2−v2)e1

4 sinh(t)(u2−v2)

. Moreover, equality

of norms holds good.

The above theorem characterises the image of L2(R+, dµt) whereas we are moreinterested in characterising the image of L2(R+, r

2n−1dr) under the semigroup. Atpresent we don’t know which weighted Bergman space on C can be identified withthe image of L2(R+, r

2n−1dr) under the Laguerre semigroup. There is a result ofKarp [6] which gives a characterisation for a slightly different Laguerre semigroup (semigroup defined in terms of Laguerre polynomials Ln−1

k (r)) and it may be possibleto deduce a result for our semigroup. By using the relation between Laguerre andHermite semigroups we get a different characterisation of the image.

Given f ∈ L2(R+, r2n−1dr) we think of it as a radial function on R2n. Then

we have seen that e−tL(n−1)f = e−tH( 12 )f and hence e−tL(n−1)f extends to C

2n

as an entire function which is radial in the sense that e−tL(n−1)f(z) depends only

on z2 =∑2n

j=1 z2j , where z = (z1, z2, ..., z2n). This follows from the explicit formula

for the Hermite semigroup. We also remark that e−tH( 12 )f is radial if and only if

f is radial. For every nonzero λ ∈ R let Hλt (C

2n) be the Hermite-Bergman spaceassociated to H(λ) on R2n. We then have

Theorem 3.7. The image of L2(R+, r2n−1dr) under the semigroup e−tL(n−1) is

the radial subspace of H12t (C2n) with equality of norms.

Recall that for radial functions the special Hermite expansions on Cn also re-duces to Laguerre expansions of type (n− 1) and hence we can deduce yet anothercharacterisation once we have a result for e−tL. This is our goal in the next twosubsections.


3.4. The image of L2 under e−tL. Let us begin with a slight change of notationwhich will be used for the rest of the notes. We identify Cn with R2n and z =x + iu ∈ Cn with (x, u) ∈ R2n. Thus functions on Cn will be denoted by f(x, u)etc. The definition of the twisted convolution of f with g then takes the form

f × g(x, u) =

∫

R2n

f(x′, u′)g(x− x′, u− u′)e−i2 (x′·u−x·u′)dx′du′.

The complexification of R2n is C2n and we write (z, w) = (x+ iy, u+ iv) to denotethe elements of C2n.

With the above notation we observe that the special Hermite functions Φα,β(x, u)extend to C2n as entire functions. The extensions are simply given by Φα,β(z, w) as

these are products of polynomials with the Gaussian e−14 (x2+u2). For f ∈ L2(Cn)

the function e−tLf(x, u) is given by

f × pt(x, u) =

∫

R2n

f(x′, u′)pt(x− x′, u− u′)e−i2 (x′·u−x·u′)dx′du′.

Recalling that

pt(x, u) = (2π sinh(t))−ne−14 coth(t)(x2+u2)

it is clear that f × pt(x, u) extends as an entire function f × pt(z, w). Let us denotethe space of all functions as f runs through L2(Cn) by imL2(Cn). We are interestedin realising this as a weighted Bergman space.

We endow imL2(Cn) with Hilbert topology induced from L2(Cn). By this wemean the Hilbert space structure on imL2(Cn) defined by (e−tLf, e−tLg) = (f, g).We consider e−tL as a map from L2(Cn) onto imL2(Cn) which is clearly injective.Define the twisted translation τ(a, b), (a, b) ∈ R2n by

τ(a, b)f(x, u) = f(x− a, u− b)e−i2 (a·u−b·x)

for all f ∈ L2(Cn). It has a natural extension to entire functions F on C2n:

τ(a, b)F (z, w) = F (z − a, w − b)e−i2 (a·w−b·z).

The transform e−tL is equivariant with respect to twisted translations. By thiswe mean τ(a, b)(e−tLf)(z, w) = e−tL(τ(a, b)f)(z, w). Thus the image is invariantunder the twisted translation. Moreover, for any compact subset K of C2n we have

sup(z,w)∈K

∫

R2n

|τ(a, b)pt(z, w)|2dadb <∞

and hence e−tL is continuous as a map from L2(Cn) onto imL2(Cn).Thus imL2(Cn) is a Hilbert space of entire functions and hence has continuous

point evaluations. That is, for all (z, w) ∈ C2n the map imL2(Cn) → C, F →F (z, w) is continuous. Hence F (z, w) = (F,Kt

(z,w)) for a unique element Kt(z,w) ∈

imL2(Cn). We thus obtain a positive definite kernel

Kt : C2n × C

2n → C, Kt((z, w), (z′, w′)) = (Kt(z′,w′),K

t(z,w))

which is holomorphic in (z, w) and antiholomorphic in (z′, w′). Moreover, the in-variance of imL2(Cn) under τ(a, b) translates into the property

Kt(τ(a, b)(z, w), τ(a, b)(z′, w′)) = Kt((z, w), (z′, w′)).

The kernel Kt((z, w), (z′, w′)) is called the reproducing kernel for imL2(Cn) sincefor any F ∈ imL2(Cn) we have F (z, w) = (F,Kt

(z,w)).


We can easily calculate the reproducing kernel in terms of the heat kernelpt(z, w). Fix (z, w) and consider F (z, w) = f × pt(z, w), f ∈ L2(Cn).

(F,Kt(z,w)) =

∫

R2n

f(a, b)pt(z − a, w − b)e−i2 (a·w−b·z)dadb.

As this is true for all F ∈ imL2(Cn) we conclude that Kt(z,w) belongs to imL2(Cn);

more precisely, Kt(z,w) = e−tLg(z,w) where g(z,w)(a, b) = pt(z − a, w − b)e

i2 (a·w−b·z).

From this we get

Kt(z,w)(z

′, w′) =

∫

R2n

pt(a, b)g(z,w)(z′ − a, w′ − b)e

i2 (a·w′−b·z′)dadb.

This simplifies to give the relation

Kt(z,w)(z

′, w′) = ei2 (z′·w−w′·z)p2t(z − z′, w − w′)

where we have used the fact that pt × pt = p2t. In particular, for (a, b) ∈ R2n wehave

Kt(a,b)(z, w) = p2t(z − a, w − b)e−

i2 (a·w−b·z).

The inner product on imL2(Cn) is uniquely determined by the requirement that

Kt((a, b), (a′, b′)) = (Kt(a,b),K

t(a′,b′))

for all (a, b), (a′, b′) ∈ R2n. The invariance of imL2(Cn) under twisted translationallows us to assume (a′, b′) = (0, 0). If the image is a weighted Bergman space givenby a weight function Wt(z, w) then the above condition translates into

p2t(a, b) =

∫

Cn

∫

Cn

p2t(z − a, w − b)e−i2 (a·w−b·z)p2t(z, w)Wt(z, w)dzdw.

We now determine a weight function Wt(z, w) satisfying this property.

Proposition 3.8. A weight function Wt(z, w) satisfying the above condition is

given by

Wt(x+ iy, u+ iv) = 4ne(u·y−v·x)p2t(2y, 2v).

It is easy to see that the weight function satisfies

Wt(x+ iy, u+ iv) = e(u·y−v·x)Wt(iy, iv).

This follows from the fact that imL2(Cn) is invariant under twisted translationswhich leads to the invariance of the weight function. In the next lemma we provethat the above weight function has the required property.

Lemma 3.9. For a, b ∈ Rn we have

p2t(a, b) =

∫

Cn

∫

Cn

p2t(z + a, w + b)ei2 (a·w−b·z)p2t(z, w)Wt(z, w)dzdw.

Proof. In view of the product nature of the functions involved, we may assume thatn = 1. Expanding out and simplifying we have

p2t(x+ a+ iy, u+ b+ iv)p2t(x+ iy, u+ iv) = (4π)−2(sinh 2t)−2e−12 (coth 2t)(x2+u2)

· e− 14 (coth 2t)(a2+b2)e

12 (coth 2t)(y2+v2)e−

12 (coth 2t)(a(x+iy)+b(u+iv)).

We can combine the terms e−12 (coth 2t)(x2+u2) and

e(uy−vx) = e(coth 2t)(uy tanh(2t)−xv tanh(2t))


to get

p2t(x + a+ iy, u+ b+ iv)p2t(x+ iy, u+ iv)e(uy−vx)

= (4π)−2(sinh 2t)−2e−14 (coth 2t)(a2+b2)e

12 (coth 2t+tanh 2t)(y2+v2)

· e− 12 (coth 2t)((x+v tanh(2t))2+(u−y tanh(2t))2)e−

12 (coth 2t)(a(x+iy)+b(u+iv)).

Using the identity tanh 2t+ coth 2t = 2 coth 4t and simplifying further we get

p2t(z + a, w + b)ei2 (aw−bz)p2t(z, w)Wt(z, w)

= 4−2π−3(sinh 2t)−3e−18 coth 2t(a2+b2)e(coth 4t−coth 2t)(y2+v2)

· e− 12 (coth 2t)((x+ a

2 +tanh(2t)v)2+(u+ b2−y tanh(2t))2)e−

i2 (coth 2t)(ay+bv)e

i2 (au−bx),

where z = x+ iy and w = u+ iv. First consider the integral∫

R2

ei2 (au−bx)e−

12 (coth 2t)((x+ a

2 +v tanh(2t))2+(u+ b2−y tanh(2t))2) dx du

= ei2 (tanh 2t)(ay+bv)

∫

R2

ei2 (au−bx)e−

12 (coth 2t)(x2+u2) dx du

= 2π(tanh 2t)ei2 (tanh 2t)(ay+bv)e−

18 (tanh 2t)(a2+b2).

Up to an explicit factor the remaining integral is∫

R2

e−i2 (coth 2t−tanh 2t)(ay+bv)e−(coth 2t−coth 4t)(y2+v2) dy dv.

As coth 2t − tanh 2t = 2(sinh 4t)−1 and coth 2t − coth 4t = (sinh 4t)−1 the aboveintegral reduces to∫

R2

e−i(sinh 4t)−1(ay+bv)e−(sinh 4t)−1(y2+v2) dy dv = π(sinh 4t)e−14 (sinh 4t)−1(a2+b2).

Combining results yields∫

C2

p2t(z + a, w + b)ei2 (aw−bz)p2t(z, w)Wt(z, w) dz dw

= 8−1π−1(sinh 2t)−3(tanh 2t)(sinh 4t)e−18 (coth 2t+tanh2t)(a2+b2)e−

14 (sinh 4t)−1(a2+b2).

Finally using the identities coth 2t+ tanh 2t = 2 coth 4t and coth 4t+ (sinh 4t)−1 =coth 2t and simplifying we get

∫

C2

p2t(z + a, w + b)ei2 (aw−bz)p2t(z, w)Wt(z, w) dz dw

=1

4π(sinh 2t)−1e−

14 coth 2t(a2+b2) = p2t(a, b) .

This proves the lemma. �

3.5. Twisted Bergman spaces and the surjectivity of e−tL. Let O(C2n) bethe space of all entire functions on C2n. We define the twisted Bergman spaces by

B∗t (C

2n) = {f ∈ O(C2n) : ‖f‖2 =

∫

Cn×Cn

|f(z, w)|2Wt(z, w) dz dw <∞} .

Clearly B∗t (C

2n) is a Hilbert space of holomorphic functions on C2n. It follows fromProposition 3.8 that e−tL : L2(R2n) → B∗

t (C2n) is an isometric embedding.


Our goal for this subsection is to show that e−tL is onto. We begin with adescription of a useful orthonormal basis for imL2(Cn) in terms of the specialHermite functions Φα,β(x, u). For each α, β ∈ Nn, let us consider

Φα,β(z, w) = (2π)−ne−(2|β|+n)tΦα,β(z, w)

where Φα,β(z, w) is the extension of Φα,β(x, u) to Cn×Cn. The functions Φα,β(x, u)satisfy the orthogonality relation

(Φα,β × Φµ,ν)(x, u) = δβ,µΦα,ν(x, u) .

Lemma 3.10. The set {Φα,β : α, β ∈ Nn0} is an orthonormal basis for im(L2(Cn)).

Proof. As the heat kernel pt(x, u) is given by

pt(x, u) = (2π)−n∑

µ

e−(2|µ|+n)tΦµ,µ(x, u) ,

we obtain the relation

(Φα,β × pt)(x, u) = (2π)−ne−(2|β|+n)tΦα,β(x, u) .

Thus e−tL(Φα,β)(z, w) = Φα,β(z, w) and, therefore, using Proposition 3.8 we obtain∫

C2n

Φα,β(z, w)Φµ,ν(z, w)Wt(z, w) dz dw

=

∫

C2n

e−tL(Φα,β)(z, w)e−tL(Φµ,ν)(z, w)Wt(z, w) dz dw

=

∫

R2n

Φα,β(x, u)Φµ,ν(x, u) dx du .

Hence {Φα,β : α, β ∈ Nn} is an orthonormal system in im(L2(Cn)).To show that it is an orthonormal basis for im(L2(Cn)), we only need to show

that∫

C2n

e−tLf(z, w)Φα,β(z, w)Wt(z, w) dz dw = 0

for all α, β implies f = 0. But the above simply means, by Proposition 3.8, that∫

R2n

f(x, u)Φα,β(x, u) dx du = 0

for all α, β and we know that {Φα,β : α, β ∈ Nn} is an orthonormal basis for

L2(R2n). Hence f = 0 and the proof is complete. �

We will show that {Φα,β : α, β ∈ Nn} is also an orthonormal basis for B∗t (C

2n).Clearly this implies that e−tL : L2(R2n) → B∗

t (C2n) is onto.

Note that Φα,β ∈ B∗t (C

2n) for any t > 0 and {Φα,β : α, β ∈ Nn} will be anorthonormal basis for any B∗

t (C2n).

As z = x+iy and w = u+iv, we note that u ·y−v ·x = =(z · w) is the symplecticform on R2n. Thus =(σz · σw) = =(z · w) for σ ∈ U(n).

We introduce the twisted Fock space F∗t (C2n) by

F∗t (C2n) = {G ∈ O(C2n) :

‖G‖2 =

∫

Cn×Cn

|G(z, w)|2e=(z·w)e−12 (coth 2t)(|z|2+|w|2) dz dw <∞} .


Clearly, the prescription

U(n) ×F∗t (C2n) → F∗

t (C2n), (σ,G) → Gσ ; Gσ(z, w) = G(σz, σw)

defines a unitary representation of U(n) on F∗t (C2n).

The Hilbert spaces B∗t (C

2n) and F∗t (C2n) are related through

(3.1) F (z, w) ∈ B∗t (C

2n) if and only if F (z, w)e14 (coth 2t)(z·z+w·w) ∈ F∗

t (C2n) .

Let T ' (S1)n be the diagonal subgroup of U(n). We write the elements of T asσ = (eiϕ1 , . . . , eiϕn). For each n−tuple of integers m = (m1,m2, . . . ,mn) let χm(σ)

be the character of T defined by χm(σ) = eiP

nj=1 mjϕj . For each G ∈ F∗

t (C2n)define

Gm(z, w) =

∫

T

G(σz, σw)χm(σ) dσ .

As G is holomorphic it is clear that Gm = 0 unless m is a multi-index in Nn. Bythe Fourier expansion

G(σz, σw) =∑

m∈Nn0

Gm(z, w)χm(σ)

and by the Plancherel theorem we have

(3.2)

∫

T

|G(σz, σw)|2dσ =∑

m∈Nn

|Gm(z, w)|2 .

Note that the functions Gm satisfy the homogeneity condition

Gm(σz, σw) = χm(σ)Gm(z, w) .

For any G ∈ F∗t (C2n) we observe that, as =(z · w) = =(σz · σw),

∫

C2n

G(z, w)e=(z·w)e−12 (coth 2t)(|z|2+|w|2)dz dw

=

∫

T

∫

C2n

G(σz, σw)e=(z·w)e−12 (coth 2t)(|z|2+|w|2)dz dw dσ.

In view of this and the homogeneity condition we arrive at the orthogonality rela-tions

∫

C2n

Gm(z, w)Gm′(z, w)e=(z·w)e−12 (coth 2t)(|z|2+|w|2)dz dw = 0 ,

whenever m and m′ are different. We also note that each Gm has an expansion ofthe form

Gm(z, w) =∑

α+β=m

cα,βzαwβ .

Hence each Gm is a polynomial.

Lemma 3.11. The linear span of Pα,β(z, w) = zαwβ, α, β ∈ Nn, is dense in

F∗t (C2n) .

Proof. If G ∈ F∗t (C2n) is orthogonal to all Pα,β then

∫

C2n

G(z, w)Gm(z, w)e=(z·w)e−12 (coth 2t)(|z|2+|w|2)dz dw = 0

for any m ∈ Nn. In view of the homogeneity property of Gm this means that∫

C2n

|Gm(z, w)|2e=(z·w)e−12 (coth 2t)(|z|2+|w|2)dz dw = 0 .


Hence Gm(z, w) = 0 for every m and so G = 0 in view of (3.2). �

It follows from Lemma 3.11 and equation (3.1) that every F ∈ B∗t (C

2n) has theorthonormal expansion

F (z, w) =∑

m

∑

α+β=m

cα,βPα,β(z, w)e−14 (coth 2t)(z·z+w·w) .(3.3)

The functions

Ψm(z, w) =∑

α+β=m

cα,βPα,β(z, w)e−14 (coth 2t)(z·z+w·w)

are orthogonal in B∗t (C

2n) but not orthogonal in any other B∗s(C

2n) when s 6= t.Another crucial property of these functions is proved in the next lemma.

Lemma 3.12. All the functions Ψmα,β(z, w) = Pα,β(z, w)e−

14 (coth 2t)(z·z+w·w) belong

to the image im(L2(Cn)) of L2(Cn) under the heat kernel transform e−tL.

Proof. It will suffice to show that for each pair α, β ∈ Nn there exists a function

fα,β ∈ L2(R2n) such that

e−tL(fα,β)(z, w) = (fα,β × pt)∼(z, w) = zαwβe−

14 (coth 2t)(z2+w2) .

As both sides are holomorphic it is enough to prove this for z = x and w = u wherex, u ∈ Rn. Thus we need to solve the equation

(3.4) (fα,β × pt)(x, u) = xαuβp2t(x, u) .

In the sequel it will be convenient to identify R2n with Cn via z = x + iu. Thenxαuβ = 2−|α|(2i)−|β|(z + z)α(z − z)β . It is then sufficient to solve the equation

(3.5) (fα,β × pt)(z) = zαzβp2t(z)

where pt(z) = pt(x, u). We solve this equation using properties of the Weyl trans-form.

Recall that the Weyl transform W (f) of a function f ∈ L1(Cn), is defined to bethe bounded operator on L2(Rn) given by

W (f)ϕ(ξ) =

∫

Cn

f(z)π(z)ϕ(ξ) dz (ξ ∈ Rn)

where π(z) = π1(z, 0) and π1 is the Schrodinger representation of the Heisenberggroup Hn with parameter λ = 1. Then for f ∈ L1

⋂

L2(Cn), W (f) is a Hilbert-Schmidt operator and W extends to L2(Cn) as an isometry onto the space ofHilbert-Schmidt operators. Moreover, W (f × g) = W (f)W (g) and W (pt) = e−tH .Here H denotes the Hermite operator

H = (−∆ + |ξ|2) =1

2

n∑

j=1

(AjA∗j +A∗

jAj),

in which Aj = − ∂∂ξj

+ ξj and A∗j = ∂

∂ξj+ ξj are the creation and annihilation

operators. The eigenfunctions of H are the Hermite functions Φα. They satisfy

AjΦα = (2αj + 2)12 Φα+ej

, A∗jΦα = (2αj)

12 Φα−ej

where ej are the coordinate vectors. Given a bounded linear operator T on L2(Rn),define the derivations

δjT = [A∗j , T ] = A∗

jT − TA∗j , δjT = [T,Aj ] = TAj −AjT .


Then it can be shown that (see [13])

W (zjf) = δjW (f), and W (zjf) = δjW (f) .

By iteration we obtain

W (zαzβf) = δαδβW (f)

where δαδβ

are defined in an obvious way.Returning to our equation (3.5), we take the Weyl transform on both sides and

obtain that

W (fα,β)e−tH = δαδ

βe−2tH .

Testing against the Hermite basis it is easy to see that the densely defined operator

T = (δαδβe−2tH)etH

extends to the whole L2(Rn) as a Hilbert-Schmidt operator. Hence, T = W (fα,β)for some fα,β ∈ L2(Cn). This completes the proof of the lemma. �

Theorem 3.13. The twisted heat kernel transform, i.e., the special Hermite semi-

group e−tL : L2(R2n) → B∗t (C

2n) is an isometric isomorphism. Moreover, {Φα,β :α, β ∈ Nn} is an orthonormal basis for B∗

t (C2n).

Proof. All what is left to show is that e−tL is onto. Suppose that F ∈ B∗t (C

2n) is

orthogonal to all Φα,β. We have to verify that F = 0. The function

G(z, w) = F (z, w)e14 (coth 2t)(z·z+w·w)

is orthogonal in F∗t to all functions of the form

f × pt(z, w)e−14 (coth 2t)(z·z+w·w).

In view of Lemma 3.12, G is orthogonal to all Pα,β . Hence by Lemma 3.11 we getG = 0 and so F = 0 as desired. �

We conclude this subsection with a proof of the uniqueness of the weight functionWt.

Lemma 3.14. Wt is the unique non-negative measurable weight function for the

twisted Bergman space B∗t (C

2n).

Proof. In view of (3.1), the statement is equivalent to the assertion that

(3.6) Wt(w, z) = e=(z·w)e−12 (coth 2t)(|z|2+|w|2)

is the unique weight function for the twisted Fock space F∗t (C2n). This will be

verified in the sequel.We may restrict ourselves to the notationally convenient case n = 1. Let Ut :

C2 → R+ be a measurable function such that

(3.7)

∫

C2

f(z, w)g(z, w) Wt(z, w) dz dw =

∫

C2

f(z, w)g(z, w) Ut(z, w) dz dw

holds for all f, g ∈ F∗t (C2). We have to show that Wt = Ut almost everywhere.

Recall from Lemma 3.12 that all polynomials zmwn lie in F∗t (C2). In particular

the constant function belongs to F∗t (C2) and (3.6)implies that Ut is integrable.


Let us introduce polar coordinates on C2 by (z, w) = (reiφ, seiθ). Consider theFourier expansions of Wt and Ut given by

Wt(reiφ, seiθ) =

∑

m,n∈Z

am,n(r, s)eimφeinθ

and

Ut(reiφ, seiθ) =∑

m,n∈Z

bm,n(r, s)eimφeinθ .

The isometry relation applied to f = zkwl yields the estimates∫ ∞

0

∫ ∞

0

r2k+1s2l+1|am,n(r, s)| dr ds ≤ ‖zkwl‖2F∗

t (C2)(3.8)

∫ ∞

0

∫ ∞

0

r2k+1s2l+1|bm,n(r, s)| dr ds ≤ ‖zkwl‖2F∗

t (C2)(3.9)

for all m,n ∈ Z.We finish the proof and show cm,n = am,n − bm,n = 0 for all m,n ∈ Z. In fact

for f = zm1wn1 and g = zm2wn2 for m1,m2, n1, n2 ∈ N we obtain from (3.6) that

(3.10)

∫ ∞

0

∫ ∞

0

rm1+m2+1sn1+n2+1cm2−m1,n2−n1(r, s) dr ds = 0 .

Note that the integral on the left is absolutely convergent by (3.7-3.8). Fix nowm,n ∈ Z. Reformulating (3.9) reads

(3.11)

∫ ∞

0

∫ ∞

0

r|m|+2k+1s|n|+2l+1cm,n(r, s) dr ds = 0

for all k, l ∈ N. In view of (3.7-3.8), we have the estimate

(3.12)

∫ ∞

0

∫ ∞

0

r|m|+2k+1s|n|+2l+1|cm,n(r, s)| dr ds ≤ 2‖z|m|+kw|n|+l‖2F∗

t (C2) + C

with C =∫

|z|<1,|w|<1(Wt(z, w) + Ut(z, w)) dz dw > 0 a constant independent ofm,n.

Denote by C+ = {ζ ∈ C : <ζ > 0} the right halfplane. Let us recall theelementary fact that a bounded holomorphic function f : C+ → C which vanisheson α + βN for some α ≥ 0, β > 0 is identically zero (see , [7] Lemma A.1 for aproof).

The explicit formula for Wt in Proposition 3.8 yields a crude but sufficient esti-mate for the norm of monomials: there exists constants c, γ > 0 such that for allk, l ∈ N one has

(3.13) ‖zkwl‖2 ≤ c · eγ(k+l) .

Now define the function

Fm,n :C+ × C+ → C,

(ζ1, ζ2) → e−3γ(ζ1+ζ2)

∫ ∞

0

∫ ∞

0

r|m|+2ζ1+1s|n|+2ζ2+1cm,n(r, s) dr ds .

It is a consequence of (3.11) and (3.12) that Fm,n is bounded and holomorphicon C+ × C+. As Fm,n|N×N = 0 by (3.10), we conclude that Fm,n = 0. But thencm,n = 0 by the properties of the Mellin transform. �


We conclude this subsection by proving a formula for the inverse map of thetwisted heat kernel transform e−tL : L2(R2n) → B∗

t (C2n). It is in the nature of the

problem that (e−tL)−1 can only be defined nicely on a dense subspace of B∗t (C

2n).The precise statement is as follows:

Theorem 3.15. The inverse of e−tL : L2(R2n) → B∗t (C

2n) is given by

(e−tL)−1(F ) = lims→0+

Fs (F ∈ B∗t (C

2n)) ,

where

Fs(a, b) =

∫

C2n

F (z + a, w + b)ei2 (a·w−b·z)pt+s(z, w)Wt(z, w) dz dw .

Proof. Let F ∈ B∗t (C

2n). Since the space B∗t (C

2n) is twisted-translation invariant,it is clear that the function

(τ(−a,−b)F )(z, w) = F (z + a, w + b)ei2 (a·w−b·z)

belongs to B∗t (C

2n). Hence, by Cauchy-Schwarz inequality, the integral defining Fsconverges. According to Theorem 3.13 we have F = e−tLf = (f × pt) for somef ∈ L2(R2n). It is easy to see that Fs ∈ L2(R2n) and that Fs converges to f . Infact, we have

Fs(a, b) =

∫

C2n

(τ(−a,−b)e−tLf)(z, w)e−tL(ps)(z, w)Wt(z, w) dz dw

=

∫

C2n

e−tL(τ(−a,−b)f)(z, w)e−tL(ps)(z, w)Wt(z, w) dz dw

=

∫

R2n

(τ(−a,−b)f)(x, u)ps(x, u) dx du .

As (ps)s>0 is a Dirac sequence, it therefore follows that

Fs(a, b) → (τ(−a,−b)f)(0, 0) = f(a, b)

for s→ 0+. This proves the theorem. �

References

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177-202[4] A. Erdelyi et al, Higher transcendental functions, Vol. 2, McGraw-Hill, New York, 1953.[5] B. Hall, The Segal-Bargmann “coherent state” transform for compact Lie groups, J. Funct.

Anal. 122 (1994), no. 1, 103-151[6] D. Karp, Square summability with geometric weights for classical orthogonal expansions,

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Anal. 165 (1999), no. 1, 44-58[11] G. Szego, Orthogonal polynomials, AMS Colloq. Public., 1991.


[12] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Math. Notes, No.42, PrincetonUniv. Press, Princeton, 1993.

[13] S. Thangavelu, Harmonic analysis on the Heisenberg group, Progress in Math. Vol. 159,Birkhauser, Boston, 1998.

[14] S. Thangavelu, An Introduction to the Uncertainty Principle, Progress in Math. Vol.217,Birkhauser, Boston, 2004.

[15] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge Uni-versity Press, London, 1962.

Department of mathematics, Indian Institute of Science, Bangalore-560 012, India.

E-mail address: [email protected]

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