JJ II

volume 6, issue 1, article 12, 2005.

Received 13 October, 2004;

accepted 04 December, 2004.

Communicated by:H.M. Srivastava

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Journal of Inequalities in Pure and Applied Mathematics

HARDY-TYPE INEQUALITIES FOR HERMITE EXPANSIONS

R. BALASUBRAMANIAN AND R. RADHA

The Institute of Mathematical Sciences C.I.T. Campus

Tharamani

Chennai - 600 113 INDIA EMail:[email protected] Department of Mathematics

Indian Institute of Technology Madras Chennai - 600 036 INDIA

EMail:[email protected]

2000c Victoria University ISSN (electronic): 1443-5756 191-04

Hardy-Type Inequalities For Hermite Expansions

R. Balasubramanian and R. Radha

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Abstract

Hardy-type inequalities are established for Hermite expansions forf ∈H^p(R),0<

p≤1.

2000 Mathematics Subject Classification: Primary: 42C10; Secondary: 42B30, 33C45

Key words: Atomic decomposition, Fourier-Hermite coefficient, Hardy spaces, Her- mite functions.

One of the authors (R.R.) wishes to thank Prof. S. Thangavelu for initiating her into this work.

Contents

1 Introduction. . . 3 2 The Main Result . . . 5

References

Hardy-Type Inequalities For Hermite Expansions

R. Balasubramanian and R. Radha

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1. Introduction

Hardy’s inequality for a Fourier transformF is stated as Z

R

|Ff(ξ)|^p

|ξ|^2−p dξ ≤Ckfk^p_ReHp 0< p ≤1,

whereReH^pdenotes the real Hardy space consisting of the boundary values of real parts of functions in the Hardy spaceH^pon the unit disc in the plane. Kan- jin in [1] has proved Hardy’s inequalities for Hermite and Laguerre expansions for functions inH¹. In [4] Satake has obtained Hardy’s inequalities for Laguerre expansions forH^p where0 < p ≤ 1.In connection with regularity properties of spherical means on Cⁿ, Thangavelu [6] has proved a Hardy’s inequality for special Hermite functions. These type of inequalities for higher dimensional expansions are studied in [2], [3]. In this short note we obtain such inequalities for Hermite expansions for one dimension, namely forf ∈H^p(R), 0< p≤1.

In fact, it is to be noted from Theorem2.1 that the resulting inequality for Her- mite expansions (0 < p ≤ 1)is sharper than the inequalities for the classical Fourier transform as well as the Laguerre function expansion.

AnH^patom,0< p≤1is defined to be a functionasatisfying the following conditions:

i. ais supported in an interval[b, b+h]

ii. |a(x)| ≤h^−1/palmost everywhere and iii. R

Rx^ka(x)dx= 0for allk = 0,1,2, . . . ,h

1 p −1i

.

Hardy-Type Inequalities For Hermite Expansions

R. Balasubramanian and R. Radha

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Making use of the atomic decomposition we define the Hardy spaceH^p to be the collection of all functionsf satisfying f = P∞

k=0λkak,where aj is an H^p - atom,λ_k is a sequence of complex numbers withP|λ_k|^p <∞and

Ckfk_H^p ≤X

|λ_k|^p1_p

≤C⁰kfk_H^p.

For various other definitions ofH^p-spaces we refer to Stein [5].

Hardy-Type Inequalities For Hermite Expansions

R. Balasubramanian and R. Radha

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2. The Main Result

LetH_kdenote the Hermite polynomials H_k(x) = (−1)^k d^k

dx^k

e^−x2

e^x2, k= 0,1,2, . . . .

Then the Hermite functionsh˜_kare defined by

h˜k(x) =Hk(x)e^−12x2, k = 0,1,2, . . . . The normalized Hermite functionsh_kare defined as

h_k(x) = (2^kk!√

π)⁻¹²h˜_k(x).

These functions {h_k} form an orthonormal basis forL²(R). They are eigen- functions for the Hermite operatorH =−∆ +x² with eigenvalues2k+ 1. For more results concerning Hermite expansions, we refer to [7].

The following inequalities for Hermite functions are well known:

|h_k(x)| ≤Ck⁻¹²¹ and

d dxh_k(x)

≤Ck¹²⁵.

Using these inequalities and the relation d

dxh_k(x) = k

2 ¹₂

h_k−1(x) +

k+ 1 2

¹₂

h_k+1(x)

we obtain the estimate

d^m dx^mh_k(x)

≤Ck^−121+m2 for m= 0,1,2, . . . , which can be verified easily by applying induction onm.

Hardy-Type Inequalities For Hermite Expansions

R. Balasubramanian and R. Radha

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Theorem 2.1. Let{h_k}be the normalized Hermite functions onR. Let0< p≤ 1andm = h

1 p

i

. Then for everyf ∈ H^p(R), the Fourier - Hermite coefficient off, namely,

fˆ(k) = Z

R

f(x)hk(x)dx, k = 0,1,2,3, . . .

satisfies the inequality

∞

X

k=0

|f(k)|ˆ ^p

(k+ 1)^σ ≤Ckfk_H^p,

whereCis a constant andσ = ^2−p_{12 18m+11}

2m+1 = ³₄ + _12m+6¹

(2−p).

Proof. In order to prove the theorem, it is enough to prove that

∞

X

k=0

|f(k)|ˆ ^p (k+ 1)^σ ≤C

for anH^p-atomf. Letf be anH^p atom. By considering the remainder term of the Taylor series expansion forh_k(x), we write the Fourier-Hermite coefficient off as

f(k) =ˆ 1 m!

Z b+h

b

f(x)d^m

dt^mh_k(t)(x−b)^mdx, wheret∈[b, x]andm=h

1 p

i .

Hardy-Type Inequalities For Hermite Expansions

R. Balasubramanian and R. Radha

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Then

|fˆ(k)| ≤Ch^m Z b+h

b

|f(x)|

d^m dt^mh_k(t)

dx

≤Ch^mk^−121+m2 Z b+h

b

|f(x)|dx

≤Ch^mk^−121+m2h^−1p+1. Consider

∞

X

k=0

|fˆ(k)|^p

(k+ 1)^σ =X

k≤γ

|f(k)|ˆ ^p

(k+ 1)^σ +X

k>γ

|fˆ(k)|^p (k+ 1)^σ

=S₁+S₂.

We chooseγ =h^{−6(2m+1)6m+5} . Then

S₁ ≤Ch^mp−1+pX

k≤γ

k

−p 12 + ^mp₂ (k+ 1)^σ.

Sinceσ= ^2−p_{12 18m+11}

2m+1 andm=h

1 p

i

, we get m

2 − 1 12

p−σ+ 1 = (6m+ 5){(m+ 1)p−1}

2m+ 1 >0.

Thus

S1 ≤Ch^mp−1+pγ(^m₂⁻₁₂¹)^p−σ+1 ≤C

Hardy-Type Inequalities For Hermite Expansions

R. Balasubramanian and R. Radha

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by the choice ofγ.

On the other hand, applying Hölder’s inequality withP = ²_p, we get,

S₂ =X

k>γ

|f(k)|ˆ ^p (k+ 1)^σ

≤ X

k>γ

|f(k)|ˆ ²

!^p₂ X

k>γ

1 (k+ 1)^2−p2σ

!^2−p₂

≤ kfk^p₂γ(⁻2−p^2σ +1)

2−p 2 .

Using property (ii) of anH^p-atom, we getkfk^p₂ ≤h^−1+p2 and thus S2 ≤h^−1+p2γ^−σ+(^2−p₂ ) ≤C

again by the choice ofγ, thus proving our assertion.

Remark 1. In the case of higher dimensions, the result has been proved with σ = ⁿ₄ + ¹₂

(2−p) (see [3]). However, here, we need an additional factor

1

12m+6 which approaches0 asp → 0. But when p = 1, the value ofσ = ²⁹₃₆, which coincides with the result of Kanjin in [1].

Hardy-Type Inequalities For Hermite Expansions

R. Balasubramanian and R. Radha

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References

[1] Y. KANJIN, Hardy’s inequalities for Hermite and Laguerre expansions, Bull. London Math. Soc., 29 (1997), 331–337.

[2] R. RADHA, Hardy type inequalities, Taiwanese J. Math., 4 (2000), 447–

456.

[3] R. RADHAANDS. THANGAVELU, Hardy’s inequalities for Hermite and Laguerre expansions, Proc. Amer. Math. Soc., 132(12) (2004), 3525–3536.

[4] M. SATAKE, Hardy’s inequalities for Laguerre expansions, J. Math. Soc., 52(1) (2000), 17–24.

[5] E.M. STEIN, Harmonic Analysis: Real variable methods, Orthogonality and Oscillatory integrals, Princeton Univ. Press, 1993.

[6] S. THANGAVELU, On regularity of twisted spherical means and special Hermite expansion, Proc. Ind. Acad. Sci., 103 (1993), 303–320.

[7] S. THANGAVELU, Lectures on Hermite and Laguerre expansions, Math- ematical Notes, 42, Princeton Univ. Press, 1993.