under the Hermite Semigroup

International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 287218,13pages

doi:10.1155/2008/287218

Research Article

Image of L

R

under the Hermite Semigroup

R. Radha and D. Venku Naidu

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

Correspondence should be addressed to R. Radha,[email protected] Received 9 June 2008; Revised 8 October 2008; Accepted 9 December 2008 Recommended by Misha Rudnev

It is shown that the Hermitepolynomialsemigroup{e^−tH:t >0}mapsL^pRⁿ, ρinto the space of holomorphic functions inL^rCⁿ, V_t,p/2^r/2for each > 0, whereρis the Gaussian measure, V_t,p/2^r/2is a scaled version of Gaussian measure withr pif 1< p < 2 andr pif 2< p <∞ with 1/p1/p 1. Conversely ifFis a holomorphic function which is in a “slightly” smaller space, namelyL^rCⁿ, V_t,p/2^r/2, then it is shown that there is a function f ∈ L^pRⁿ, ρ such that e^−tHf F. However, a single necessary and sufficient condition is obtained for the image of L²Rⁿ, ρp/2undere^−tH, 1< p <∞. Further it is shown that ifFis a holomorphic function such that F∈L¹Cⁿ, V_t,p/2^1/2orF∈L^1,p_mR²ⁿ, then there exists a functionf ∈L^pRⁿ, ρsuch thate^−tHfF, wheremx, y e^{−x2/p−1e4t1}e^{−y2/e4t−1}and 1< p <∞.

Copyrightq2008 R. Radha and D. V. Naidu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that the Bargmann transform is an isometric isomorphism ofL²Rⁿonto the Fock spaceFCⁿ, which is associated with the realization of the creation and annihilation operators for Bosons in quantum mechanics. We refer to1,2for further results. Similar type of results are shown for semigroups generated by the Laplace-Beltrami operator on compact spacessee3,4. The image ofL²Hⁿunder the heat kernel transform is studied for the Heisenberg group in5. Such type of results are also known for Hermite, special Hermite, Bessel, and Laguerre semigroups, see6.

Hall in7studied the problem of characterizing the image ofL^pRⁿunder the Segal- Bargmann transform. In this paper, we want to study this problem for Hermite semigroup instead of Segal-Bargmann transform. In fact, the idea of extending the classical results involving the standard Laplacian onRⁿor Fourier transform onRⁿto Hermite expansions is not new: to cite a few, summability theorems8,9, multipliers10, Sobolev spaces11, and Hardy’s inequalities12,13.

In order to prove that a holomorphic function to be in the image ofL^pRⁿ, ρunder the Segal-Bargmann transform Hall in7obtained a necessary condition for the range 1< p≤2 and the sufficient condition for 2≤p <∞. In this paper, we tried to obtain a single necessary and sufficient condition for a holomorphic function to be in the image ofL^pRⁿ, ρ under Hermite polynomial semigroup. Though we were not completely successful, we could prove the following.

If f ∈ L^pRⁿ, ρ, then e^−tHf is holomorphic and e^−tHf ∈ L^rCⁿ, V_t,p/2^r/2 for every >0, but we are able to prove the converse only for the holomorphic functions which are in L^rCⁿ, V_t,p/2^r/2 withr p, if 1< p <2 andr p, if 2< p <∞with 1/p1/p 1, where V_t,p/2z π²/4^−n/2e^2ntp−1e^4t1e^4t−1^−n/2e^{−2x2/p−1e4t1}e^{−2y2/e4t−1}andV_t,p/2^s ,the sth power ofV_t,p/2.However, we are able to obtain a single necessary and sufficient condition for the image ofL²Rⁿ, ρ_p/2undere^−tH,1< p <∞,whereρ_p/2u pπ/2^−n/2e^−2/pu2.

Notice thatHL^rCⁿ, V_t,p/2^r/2 ⊂

>0HL^rCⁿ, V_t,p/2^r/2 asV_t,p/2^r/2 ≤ CV_t,p/2^r/2 ,where the constant C depends on , t, p, andn. We remark that the Gaussian-type density V_t,p/2z defines a finite measure with total mass e^2nt when n ≥ 1 and t > 0.Let m be a weight function on R²ⁿ and let 1 ≤ p, q ≤ ∞. Then the weighted mixed-norm space L^p,q_m R²ⁿ consists of all Lebesgue measurable functions on R²ⁿ, such that the norm F_L^p,q_m

Rⁿ

Rⁿ|Fx, w|^pmx, w^pdx^q/pdw^1/qis finite. Ifp ∞orq∞,then the corresponding p-norm is replaced by the essential supremum. In this paper, we also prove that if F is holomorphic and ifF ∈L¹Cⁿ, V_t,p/2^1/2orL^1,p_m R²ⁿ,then there exists a functionf ∈L^pRⁿ, ρ such thatFis the image off.Heremx, y e^{−x2/p−1e4t1}e^{−y2/e4t−1.}

The advantage of takingL^pRⁿ, ρinstead ofL^pRⁿhas been nicely explained by Hall in7. We also wish to point out the following interesting fact, namely, iff∈L^pRⁿ,then the pointwise estimate ofe^−tHfis given by|e^−tHfxiy| ≤C_t,p,ne^{−1/2tanh 2t x2}e^{1/2coth 2t y2}which is independent ofpexcept the constant factor, which does not help in the current problem.

Here the constantCt,p,n e^−nt2π^−n/2sinh 2t^−n/22π/qcoth 2t^n/2q,with 1/p1/q 1.

However, iff ∈ L^pRⁿ, ρ, we get pointwise bounds as in Theorem 3.1. Further, we found the semigroup associated with Hermite polynomials to be more suitable for this problem rather than the semigroup associated with the Hermite functions. This is mainly because questions aboutL^pstructure forp /2 do depend on the measures used in the particular setup, while questions aboutL²structure do not. InSection 2, we discuss the Hermitepolynomial semigroup and discuss the image ofL²Rⁿ, ρunder the Hermite semigroup. InSection 3, we prove our main results.

2. Hermite (polynomial) semigroup

Let Hkx −1^ke^x2d^k/dx^ke^−x2 denote the Hermite polynomial. For a multi-index α ∈ Nⁿ, x x1, x₂, . . . , x_n ∈ Rⁿ, define Hαx Hα1x1Hα2x2· · ·Hαnxn and Hαx 2^αα!^−n/2Hαx. This collection {Hαx : α ∈ Nⁿ}forms an orthonormal basis for L²Rⁿ, ρ, whereρxdx π^−n/2e^−x2dx. Further, the functionsHα are eigenfunctions of the operatorH−Δ 2_n

j1x_j∂/∂xj nIwith eigenvalues 2|α|n. Forf ∈L²Rⁿ, ρ,consider

f

α∈Nⁿ f,HαHα, where the sum converges tofinL²Rⁿ, ρ. For eachk∈N, letQkdenote the orthogonal projection ofL²Rⁿ, ρonto the eigenspace spanned by{Hα :|α| k}. Then the spectral decomposition ofHcan be written as Hf _∞

k02knQkf.The operatorH defines a semigroup, called the Hermite polynomial semigroup, denoted bye^−tH,for each

t >0 using the expansion

e^−tHf^∞

k0

e^−2kntQkf. 2.1

Before stating our theorem, we will state the following identity which will be repeatedly used in this paper:

1

coth 2t1−2/psinh²2t1−coth 2t 2

p−1e^4t1. 2.2

In fact, the left-hand side of2.2can be rewritten as 1

cosh 2tsinh 2t−2/psinh 2tsinh 2tsinh 2t−cosh 2t

sinh 2t . 2.3

By using the exponential formula for sinh 2tand cosh 2t,the right-hand side of2.2can be obtained by straightforward simplification.

We state here Mehler’s formula for Hermite functions

Φαx ⁿ

i1

2^αiαi!π^1/2−1/2−1^αi d^αi

dx^α_iⁱe^−x2ie^x2i/2, 2.4

where α α1, α2, . . . , αn ∈ Nⁿ, x x1, x2, . . . , xn ∈ Rⁿ which will lead to a formula involving Hermite polynomialsfor the proof, we refer to14or6.

Mehler’s formula

For allw∈Cwith|w|<1, one has

α∈Nⁿ

ΦαxΦαyw^|α|π^−n/21−w^2−n/2e^{−1/21w2/1−w2x2y22w/1−w2x·y}, 2.5

for allx, y∈Rⁿ.Herex² _n

i1x_i²andx·y_n

i1x_i·y_i.But

α∈Nⁿ

ΦαxΦαyw^|α|π^−n/2e^−x2/2e^−y2/2

α∈Nⁿ

HαxHαyw^|α| 2.6

from which it follows that

α∈Nⁿ

HαxHαyw^|α|π^n/2e^x2/2e^y2/2

α∈Nⁿ

ΦαxΦαyw^|α|. 2.7

Iff∈L²Rⁿ, ρ, then

Qkfu

|α|k

f,HαHαu

|α|k RⁿfxHαxρxdx

Hαu. 2.8

Thus, it follows that

e^−tHfx

RⁿKtx, ufuρudu

RⁿLtx, ufudu,

2.9

where

Ltx, u π^−n/2

α∈Nⁿ

e^−2|α|ntHαxHαue^−u2 π^−n/2e^−u2e^−nt

α∈Nⁿ

e^−2t_|α|

HαxHαu. 2.10

Notice that,Ktis the kernel ofe^−tH.However,Ltis used for computational purpose. Then by using2.7, we can write

Ltx, u 2π^−n/2sinh 2t^−n/2e−1/2coth 2t−1x²e−1/2coth 2t1u²e^{x·u/sinh 2t}. 2.11

SinceKtx, uextends to an entire functionKtz, uforz ∈Cⁿ,Fz e^−tHfzcan also be extended toCⁿas an entire function, wherezxiy.This can be verified by using Morera’s theorem.

Remark 2.1. The mape^−tHis one-one . Lete^−tHf 0 forf∈L²∩L^pRⁿ, ρ.Thene^−tHf−iξ 0∀ξ∈Rⁿ.

Letgξ 2π ^−n/2

Rⁿgxe^−ix·ξdxdenote the Fourier transform ofg.Then it follows from2.11that

e^−tHf−iξ sinh 2t^−n/2e1/2coth 2t−1ξ²G ξ

sinh 2t

0, 2.12

whereGu fue−1/2coth 2t−1u²forcingGξ/ sinh 2t 0.Then by uniqueness of Fourier transformG0,which in turn impliesf0,showing thate^−tHis one-one.In fact, the above proof shows thate^−tHis injective onL²Rⁿ, ρ.However, in general, one can show thate^−tHis injective on theL^pspace1 < p <∞using the fact that the Fourier transform is injective on theL^pspace.

We should call e^−tHf Hermite Bargmann transform. Hereafter, we should write HL^rCⁿ, αzfor the class of holomorphic functions inL^rCⁿ, αz.

Theorem 2.2. Fixt > 0 and let 1 < p < ∞.Then the Hermite polynomial semigroupe^−tH is an isometric isomorphism ofL²Rⁿ, ρ_p/2onto the space of holomorphic functionsHL²Cⁿ, V_t,p/2. Proof. Letf∈L²Rⁿ, ρ_p/2.ThenFz e^−tHfzis given by

Fz 2π^−n/2sinh 2t^−n/2e−1/2coth 2t−1z²

Rⁿe−1/2coth 2t1u²e^{z·u/sinh 2t}fudu 2π^−n/2sinh 2t^−n/2e−1/2coth 2t−1z²

Rⁿe−1/2coth 2t1u²e^{−i−yix·u/sinh 2t}fudu.

2.13

Putgu fue−1/2coth 2t1u²e^{x·u/sinh 2t}.Then

Rⁿ|Fz|²e^{coth 2t−1x2−y2}dy sinh 2t⁻ⁿ

Rⁿ

g −y

sinh 2t ²dy

Rⁿ|gy|²dy by applying change of variables

Rⁿ|gy|²du using Plancherel formula

Rⁿ|fu|²e^{−coth 2t1u2}e^{2x·u/sinh 2t}du

pπ 2

n/2

Rⁿ|fu|²e−coth 2t1−2/pu²e^{2x·u/sinh 2t}ρ_p/2udu.

2.14

Multiplying both sides bye^−x2/coth 2t1−2/psinh²2tand integrating with respect tox, we get

Cⁿ|Fz|²e^{coth 2t−1x2−y2}e^−x2/coth 2t1−2/psinh²2tdy dx

pπ 2

_n/2

Rⁿ|fu|²e−coth 2t1−2/pu²

Rⁿe^−x2/coth 2t1−2/psinh²t2x·y/sinh 2tdx ρ_p/2udu

pπ 2

_n/2

Rⁿ|fu|²

Rⁿe^−x/√

coth 2t1−2/psinh 2t−u√

coth 2t1−2/p²dx ρ_p/2udu.

2.15

LetKt,p,n pπ²/2^−n/2coth 2t1−2/psinh²2t^−n/2.Then it follows that

Kt,p,n

Cⁿ|Fz|²e^{coth 2t−1x2−y2}e^−x2/coth 2t1−2/psinh²2tdx dyf²_L2Rⁿ,ρp/2. 2.16

By using2.2, the left-hand side of the above equation can be written as pπ²

2

−n/2

coth 2t1− 2 p

sinh²2t

_−n/2

Cⁿ|Fz|²e^{−2x2/p−1e4t1}e^{−2y2/e4t−1} dz

π² 4

_−n/2 e^2nt

p−1e^4t1e^4t−1−n/2

Cⁿ|Fz|²e^{−2x2/p−1e4t1}e^{−2y2/e4t−1} dz, 2.17

which implies thate^−tHis an isometry from the spaceL²Rⁿ, ρ_p/2intoHL²Cⁿ, V_t,p/2.In the above,dzdx dy, zxiy.

It remains to show thate^−tHdefines an onto map. Sincee^−tHis an isometry, the range is closed inHL²Cⁿ, Vt,p/2.It is enough to show that the range is dense inHL²Cⁿ, Vt,p/2.By using exponential formula for sinh 2t,cosh 2t,and coth 2t, e^−tHcan be written as

e^−tHfz 2π^−n/2sinh 2t^−n/2

Rⁿe^{−e−tz−etu2/2 sinh 2t}fudu. 2.18

It can be easily seen that e^−tH will take real variable polynomials in L²Rⁿ, ρp/2 to holomorphic polynomials inHL²Cⁿ, V_t,p/2.On the other hand, if we take a holomorphic polynomial inHL²Cⁿ, V_t,p/2,it can be expressed as an image of a real variable polynomial inL²Rⁿ, ρp/2undere^−tH.In fact, suppose, for instance, ifn1,the first one can show that

e^−tHu^mz C m k0

−1^kmcke^−tz^m−kwk, 2.19

for fixedm≥0,whereC−2π^−n/2sinh 2t^−n/2e^−m1t,wk

Rⁿe−1/2 sinh 2ty²y^kdy.Now if we takeFz _m

i0c_izⁱ,anmth degree holomorphic polynomial inHL²C, Vt,p/2,we wish to choose anmth degree real variable polynomialfx _m

i0aiuⁱ inL²R, ρp/2such that e^−tHf F. This leads to the determination of the coefficientsai such thate^−tH_m

i0aiuⁱ _m

i0c_izⁱ.Using 2.19 and comparing the coefficients of z^k on both sides for 0 ≤ k ≤ m, one ends up with a matrix equation UX Y with U an upper triangular matrix with U_iic₀e⁻ⁱ¹w₀,wherec₀−2π^−n/2sinh 2t^−n/2, X a0, a₁, . . . , a_m^t, Y c0, c₁, . . . , c_m^t. Sincew0/0, detU /0 which in turn gives a unique solution fora0, a1, . . . , am.Thus every holomorphic polynomial in HL²Cⁿ, V_t,p/2 is an image of a real variable polynomial in L²Rⁿ, ρ_p/2.This idea can be appropriately extended for higher dimensions also. It remains to show that the set of all holomorphic polynomials are dense inHL²Cⁿ, Vt,p/2,which will force the image ofe^−tH to be dense inHL²Cⁿ, Vt,p/2.Toward this end, we show that any F∈ HL²Cⁿ, V_t,p/2which is orthogonal to all holomorphic polynomials vanishes identically.

In particular,Fis orthogonal to all monomialsz^α, α∈Nⁿ.Now consider the following Fock spacesF_stCⁿ,defined as the space of all entire functionsGfor which

G²_Fst

Cⁿ|Gz|²e^−st|z|2dz 2.20

are finite. It is easy to see that

F∈ HL²Cⁿ, V_t,p/2⇐⇒Gz Fze^wtz2 ∈ F_st, 2.21

wherest 1/p−1e^4t1 1/e^4t−1>0,wt 1/21/e^4t−1−1/p−1e^4t1 The assumption thatFis orthogonal to allz^αleads to the condition thatGz Fze^wtz2is orthogonal to allz^αe^wtz2inF_st.The Taylor expansionFz

α∈Nⁿaαz^αleads to

Gz

α∈Nⁿ

a_αz^αe^wtz2. 2.22

SinceGis orthogonal to allz^αe^wtz2, we geta_α 0 for allαand soF 0,thus proving our assertion.

In particular, whenp 2,we obtain the following result in which the functionf ∈ L²Rⁿ, ρand the measureρ are independent oft, bute^−tHf ∈ L²Cⁿ, V_t,whereV_t V_t,1, which depends ontsee also15.

Corollary 2.3. A holomorphic function F on Cⁿ belongs to HL²Cⁿ, Vt if and only if Fz e^−tHfzfor somef ∈ L²Rⁿ, ρ, whereV_tz π²/2^−n/2sinh 4t^−n/2e^−2x2/e4t1e^{−2y2/e4t−1}. Moreover, one has the equality of norms

Fz_HL2Cⁿ,Vtf_L2Rⁿ,ρ, 2.23

wheneverFe^−tHf.

3. The main results

First, we will obtain a pointwise bound for Hermite Bargmann transform of a functionf ∈ L^pRⁿ, ρ.From here onward, in order to definee^−tH onL^pRⁿ, ρ,1 < p < ∞,we will first definee^−tH on the class of functionsL²∩L^pRⁿ, ρ.Then using standard density argument, we will definee^−tHonL^pRⁿ, ρ.

Theorem 3.1. Fixt >0 and let 1< p <∞.Then for allf∈L^pRⁿ, ρ,one has

e^−tHfxiy≤Kt,p,nf_LpRⁿ,ρe^x2/p−1e4t1e^y2/e4t−1. 3.1 Proof. We have e^−tHfz 2π^−n/2sinh 2t⁻ⁿe−1/2coth 2t−1z²

Rⁿe−1/2coth 2t−1u²e^{z·u/sinh 2t} fue^−u2du, for f ∈ L² ∩ L^pRⁿ, ρ. Let hu e−1/2coth 2t−1u²e^{z·u/sinh 2t}, Ctx, y 2 sinh 2t^−n/2e−1/2coth 2t−1x²−y².Asf ∈L^pRⁿ, ρandh∈L^pRⁿ, ρby applying H ¨older’s inequality, it can be shown that

e^−tHfz≤C_tx, yf_LpRⁿ,ρe−1/2coth 2t−1u²e^{z·u/sinh 2t}

L^pRⁿ,ρ. 3.2

Consider

e−1/2coth 2t−1u²e^{z·u/sinh 2tp}

L^pRⁿ,ρ

Rⁿe^−p/2coth 2t−1u²e^{px·u/sinh 2t}π^−n/2e^−u2du π^−n/2

Rⁿe^−p/2√

coth 2t−12/pu−x/sinh 2t√

coth 2t−12/p²e^px2/2sinh22tcoth 2t−12/pdu

2

p _n/2

e^px2/2sinh22tcoth 2t1−2/p

coth 2t−1 2 p

_−n/2 .

3.3

Thus

|e^−tHfz| ≤K_t,p,nf_LpRⁿ,ρe1/2coth 2t−1y²e^x2/21/sinh22tcoth 2t1−2/p1−coth 2t, 3.4

whereK_t,p,nis a constant depending ont, p, n.By using2.2, it follows that

|e^−tHfz| ≤Kt,p,nf_LpRⁿ,ρe^x2/p−1e4t1e^y2/e4t−1, zxiy. 3.5

SinceL²∩L^pRⁿ, ρis dense inL^pRⁿ, ρ,1< p <∞,the result follows.

The next theorem follows fromTheorem 3.1, by a straightforward computation.

Theorem 3.2. Fixt > 0 and let 1 < p≤2. Iff ∈L^pRⁿ, ρ,thene^−tHf ∈ HL^pCⁿ, V_t,p/2^p/2for every fixed >0. In particulare^−tHf ∈

>0HL^pCⁿ, V_t,p/2^p/2.

Remark 3.3. The above theorem is valid for 1 < p < ∞. We will see in Theorem 3.8 that Theorems 3.2 and 3.7can be put together in a general form. At this point, we thank one of the referees for suggesting us this general form, namely,Theorem 3.8.

Theorem 3.4. IfFis holomorphic andF ∈L^pCⁿ, V_t,p/2^p/2, where 1< p≤2 and t is a fixed number greater than zero, then there exists a unique functionf∈L^pRⁿ, ρsuch thate^−tHfF.

Proof. Let Gf e^−tHf. Then it follows from Theorem 2.2, that G is an isometric isomorphism fromL²Rⁿ, ρp/2ontoHL²Cⁿ, Vt,p/2.Gcan be explicitly written asGfz

RⁿLtz, u/ρp/2ufuρ_p/2udu, whereLtis given in2.11. LetG^∗,pdenote the adjoint of G,whereGis an operator from Hilbert spaceL²Rⁿ, ρp/2into the Hilbert spaceL²Cⁿ, Vt,p/2. Note that,G^∗,pFwill coincide withG⁻¹FifF is a holomorphic function onCⁿ.Thus we can write

G^∗,pFu

Cⁿ

Ltz, u

ρ_p/2 FzVt,p/2dx dy. 3.6

In order to change the measuresVt,p/2andρp/2into Lebesgue measure, construct a mapG^∗,p: L²Cⁿ, dxdy → L²Rⁿ, dudefined by

G^∗,pFu ρ^1/2_p/2G^∗,pV_t,p/2^−1/2Fz. 3.7

An explicit computation shows thatG^∗,pcan be written as

G^∗,pFu C

Cⁿeix·ycoth 2t−1e^{−iy·u/sinh 2t}

×e{−1/2coth 2t1−2/pu−x/sinh 2tcoth 2t1−2/p²}Fzdx dy.

3.8

It can be easily verified that G^∗,p defines a bounded linear map of L¹Cⁿ, dx dy into L¹Rⁿ, du.By applying interpolation theorem16, M. Riesz convexity theorem, it follows thatG^∗,p is a bounded map ofL^qCⁿ, dx dy intoL^qRⁿ, du forqsatisfying 1 ≤ q ≤ 2.In particular if we take p q, thenG^∗,p will be bounded fromL^pCⁿ, dx dyintoL^pRⁿ, du.

Again, we wish to change Lebesgue measure onRⁿ to ρ, and Lebesgue measure onCⁿ to V_t,p/2^p/2. Toward this end, we define

G^∗,p:L^p

Cⁿ, V_t,p/2^p/2

−→L^pRⁿ, ρ by G^∗,pFu ρ^−1/puG^∗,p

V_t,p/2^1/2Fz

. 3.9

Note that,ρu^−1/p ce^u2/p cρ^−1/2_p/2 for some constantscandc. Then it follows from3.7 thatG^∗,pFu CG^∗,pFu.This shows thatG^∗,pis bounded fromL^pCⁿ, V_t,p/2^p/2intoL^pRⁿ, ρ.

We claim that ifFis in the holomorphic subspace ofL^pCⁿ, V_t,p/2^p/2,thenGG^∗,pFF.

Let Psz be the “polydisk” of radius s, centered atz, namely, Psz {w ∈ Cⁿ :

|wk−z_k|< s, k 1,2, . . . , n,andw_k u_kiv_k, z_k x_kiy_k}see17. ThenFzcan be written as

Fz πs²⁻ⁿ

Cⁿχ_Psz 1

αwFwαwdu dv, 3.10

whereχ_Pszdenotes the characteristic function onP_sz.

Defineαw e^{−pu2/p−1e4t1}e^{−pv2/e4t−1}, wuiv, β p−1/p.By using H ¨older’s inequality,

Cⁿ|Fmz|²e^{−2x2/p−1e4t1}e^{−2y2/e4t−1}dx dy

Cⁿ|Fλmz|²e^{−2x2/p−1e4t1}e^{−2y2/e4t−1}dx dy C

Cⁿ|Fz|²e^{−2x2/p−1e4t1}e^{−2y2/λ2me4t−1}dx dy.

3.11

From this we can see that

|Fz| ≤Ce^{xs2/p−1e4t1}e^ys2/e4t−1. 3.12

Now defineFmz Fλmz,whereλmis an increasing sequence of numbers tending to 1.

ThenF_m∈L^pCⁿ, V_t,p/2^p/2andF_mwill converge toFin the norm ofL^pCⁿ, V_t,p/2^p/2.Consider

|Fz| ≤C

χ_Psz 1

|αw|

L^pCⁿ,αwF_L^p_Cn,αw

≤C sup

w∈Psz

1

|αw|^βmPsz^1/pF_L^p_Cⁿ_,αw C sup

w∈Psz

1

|αw|^β C sup

w∈Psz

e^u2/p−1e4t1e^v2/e4t−1 sincepβ1.

3.13

Then by using3.12, we get

3.11≤C

Cⁿe^{xs2/p−1e4t1}e^ys2/e4t−1e^{−2x2/λ2mp−1e4t1}e^{−2y2/λ2me4t−1}dx dy

<∞

as 2< 2

λ²_m for eachm

.

3.14

This shows thatF_m ∈ HL²Cⁿ, V_t,p/2for eachmwhich in turn implies thatGG^∗,pF_m F_m for each m. Since G^∗,p is bounded from L^pCⁿ, V_t,p/2^p/2 into L^pRⁿ, ρ, G^∗,pFm converges to G^∗,pF.ThenGG^∗,pFm Fmwill converge uniformly toGG^∗,pFon compact sets. SinceFmalso converges toFin the norm ofL^pCⁿ, V_t,p/2^p/2, the pointwise limit andL^plimit must coincide, showingGG^∗,pF F.Then takingf G^∗,pFproves our existence assertion. The uniqueness follows fromRemark 2.1.

Remark 3.5. As mentioned in the introduction,

>0HL^pCⁿ, V_t,p/2^p/2 is larger than HL^pCⁿ, V_t,p/2^p/2. In fact, if fz e^z2/p−1e4t1, then f ∈

>0HL^pCⁿ, V_t,p/2^p/2 but f/∈HL^pCⁿ, V_t,p/2^p/2.But we are able to show that the transforme^−tHis only onto the functions inHL^pCⁿ, V_t,p/2^p/2.

The following theorem shows that the image ofL^pRⁿ, ρunder Hermite polynomial semigroup will be contained inHL^pCⁿ, V_t,p/2^p/2also.

Theorem 3.6. Fixt >0 and let 1< p≤2. Iff ∈L^pRⁿ, ρ,thene^−tHf ∈ HL^pCⁿ, V_t,p/2^p/2,where pis such that 1/p1/p1.

Proof. LetGf e^−tHf.Then it follows fromTheorem 2.2thatGis an isometric isomorphism from L²Rⁿ, ρ_p/2 onto HL²Cⁿ, V_t,p/2. In order to change the weighted measure into Lebesgue measures, construct a mapG_p :L²Rⁿ, du → L²Cⁿ, dx dydefined byG_pfz V_t,p/2^1/2 Gρp/2u^−1/2fu.An explicit computation shows thatG_pcan be written as

Gpfxiy C

Rⁿe−icoth 2t−1x·ye^{iy·u/sinh 2t}e{−1/2coth 2t1−2/pu−x/sinh 2tcoth 2t1−2/p²}fudu, 3.15

wherez xiy andCis a constant depending ont, p, n. It can be easily verified thatG_p defines a bounded operator fromL¹Rⁿ intoL^∞Cⁿ.By the interpolation theorem, G_p is also bounded fromL^qRⁿ, duintoL^qCⁿ, dx dy,forqsatisfying 1 ≤q≤2.In particular we takep q,thenG_pwill be bounded fromL^pRⁿ, duintoL^pCⁿ, dx dy.Again, to change measures, we defineGp :L^pRⁿ, ρu → L^pCⁿ, V_t,p/2^p/2byGpfz V_t,p/2^−1/2Gpρu^1/pfu we see that the operatorsGandG_pturn out to be the same up to a constant multiple. ThusG is bounded fromL^pRⁿ, ρuintoL^pCⁿ, V_t,p/2^p/2,proving our assertion.

Using pointwise estimate inTheorem 3.1, one can obtain the following result.

Theorem 3.7. Fixt >0 and let 2≤p <∞.Iff ∈L^pRⁿ, ρ,thene^−tHf ∈ HL^pCⁿ, V_t,p/2^p/2for any fixed >0.In particulare^−tHf ∈

>0HL^pCⁿ, V_t,p/2^p/2.

As mentioned earlier, Theorems3.2and3.7are special cases of the following theorem.

Theorem 3.8. Suppose thatf ∈ L^pRⁿ, ρand that 1< p < ∞, t > 0 and >0.Thene^−tHf ∈ HL^sCⁿ, V_t,p/2^s/2for any 1≤s <∞.

The proof is simply an application of the pointwise estimate proved inTheorem 3.1.

Then one getsTheorem 3.2by takingspandTheorem 3.7by takingsp. As in the case ofTheorem 3.4, we prove the following result.

Theorem 3.9. IfFis holomorphic andF∈L^pCⁿ, V_t,p/2^p/2, where 2≤p <∞andtis a fixed number greater than zero, then there exists a unique functionf ∈L^pRⁿ, ρsuch thate^−tHf F, wherepis such that 1/p1/p1.

Proof. In the proof ofTheorem 3.4, we have noticed thatG^∗,p is bounded fromL¹Cⁿ, dx dy into L¹Rⁿ, du. Instead, one can also verify that G^∗,p is bounded from L¹Cⁿ, dx dy into L^∞Rⁿ, du.In this case the interpolation theorem will show thatG^∗,pwill be bounded from L^pCⁿ, dx dyintoL^pRⁿ, du.SoG^∗,pwill also be bounded fromL^pCⁿ, V_t,p/2^p/2intoL^pRⁿ, ρ.

In this case also, we can show thatGG^∗,pF F, forF ∈ HL^pCⁿ, V_t,p/2^p/2.Now, letf G^∗,pF.

The uniqueness follows fromRemark 2.1.

Remark 3.10. As mentioned earlier, we are able to show that the transforme^−tH is only onto the functions inHL^pCⁿ, V_t,p/2^p/2,instead of

>0HL^pCⁿ, V_t,p/2^p/2.