Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics

http://jipam.vu.edu.au/

Volume 6, Issue 1, Article 12, 2005

HARDY-TYPE INEQUALITIES FOR HERMITE EXPANSIONS

R. BALASUBRAMANIAN AND R. RADHA THEINSTITUTE OFMATHEMATICALSCIENCES

C.I.T. CAMPUS

THARAMANI

CHENNAI- 600 113 INDIA [email protected] DEPARTMENT OFMATHEMATICS

INDIANINSTITUTE OFTECHNOLOGYMADRAS

CHENNAI- 600 036 INDIA [email protected]

Received 13 October, 2004; accepted 04 December, 2004 Communicated by H.M. Srivastava

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

p≤1.

Key words and phrases: Atomic decomposition, Fourier-Hermite coefficient, Hardy spaces, Hermite functions.

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

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,

where ReH^p denotes the real Hardy space consisting of the boundary values of real parts of functions in the Hardy spaceH^p on the unit disc in the plane. Kanjin 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 for H^p where 0 < p ≤ 1. In connection with regularity properties of spherical means onCⁿ, 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 Theorem 2.1 that the resulting inequality for Hermite expansions(0< p≤1)is sharper than the inequalities for the classical Fourier transform as well as the Laguerre function expansion.

ISSN (electronic): 1443-5756

c 2005 Victoria University. All rights reserved.

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

191-04

2 R. BALASUBRAMANIAN ANDR. RADHA

AnH^p atom,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

.

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

k=0λ_ka_k, wherea_j is anH^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].

2. THEMAINRESULT

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) =H_k(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 for L²(R). They are eigenfunctions for the Hermite operatorH =−∆ +x²with 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

dxhk(x) = k

2 ¹₂

hk−1(x) +

k+ 1 2

¹₂

hk+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.

Theorem 2.1. Let{h_k}be the normalized Hermite functions onR. Let0 < p ≤ 1and m = h1

p

i

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

f(k) =ˆ Z

R

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

satisfies the inequality

∞

X

k=0

|fˆ(k)|^p

(k+ 1)^σ ≤Ckfk_H^p, whereCis a constant andσ= ^2−p_{12 18m+11}

2m+1 = ³₄ +_12m+6¹

(2−p).

J. Inequal. Pure and Appl. Math., 6(1) Art. 12, 2005 http://jipam.vu.edu.au/

HARDY-TYPEINEQUALITIESFORHERMITEEXPANSIONS 3

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

∞

X

k=0

|fˆ(k)|^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 = h1

p

i . Then

|f(k)| ≤ˆ 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+m2 h^−1p+1. Consider

∞

X

k=0

|f(k)|ˆ ^p

(k+ 1)^σ =X

k≤γ

|fˆ(k)|^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 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−p2 .

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

J. Inequal. Pure and Appl. Math., 6(1) Art. 12, 2005 http://jipam.vu.edu.au/

4 R. BALASUBRAMANIAN ANDR. RADHA

again by the choice ofγ, thus proving our assertion.

Remark 2.2. In the case of higher dimensions, the result has been proved withσ = ⁿ₄ + ¹₂ (2−

p)(see [3]). However, here, we need an additional factor _12m+6¹ which approaches0asp →0.

But whenp= 1, the value ofσ= ²⁹₃₆, which coincides with the result of Kanjin in [1].

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. RADHA AND S. 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, Mathematical Notes, 42, Prince- ton Univ. Press, 1993.

J. Inequal. Pure and Appl. Math., 6(1) Art. 12, 2005 http://jipam.vu.edu.au/