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:balu@imsc.ernet.in Department of Mathematics
Indian Institute of Technology Madras Chennai - 600 036 INDIA
EMail:radharam@iitm.ac.in
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 ∈Hp(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
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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 ≤CkfkpReHp 0< p ≤1,
whereReHpdenotes the real Hardy space consisting of the boundary values of real parts of functions in the Hardy spaceHpon 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 forHp 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 ∈Hp(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.
AnHpatom,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
Rxka(x)dx= 0for allk = 0,1,2, . . . ,h
1 p −1i
.
Hardy-Type Inequalities For Hermite Expansions
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Making use of the atomic decomposition we define the Hardy spaceHp to be the collection of all functionsf satisfying f = P∞
k=0λkak,where aj is an Hp - atom,λk is a sequence of complex numbers withP|λk|p <∞and
CkfkHp ≤X
|λk|p1p
≤C0kfkHp.
For various other definitions ofHp-spaces we refer to Stein [5].
Hardy-Type Inequalities For Hermite Expansions
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2. The Main Result
LetHkdenote the Hermite polynomials Hk(x) = (−1)k dk
dxk
e−x2
ex2, 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 functionshkare defined as
hk(x) = (2kk!√
π)−12h˜k(x).
These functions {hk} 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:
|hk(x)| ≤Ck−121 and
d dxhk(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
dm dxmhk(x)
≤Ck−121+m2 for m= 0,1,2, . . . , which can be verified easily by applying induction onm.
Hardy-Type Inequalities For Hermite Expansions
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Theorem 2.1. Let{hk}be the normalized Hermite functions onR. Let0< p≤ 1andm = h
1 p
i
. Then for everyf ∈ Hp(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 ≤CkfkHp,
whereCis a constant andσ = 2−p12 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 anHp-atomf. Letf be anHp atom. By considering the remainder term of the Taylor series expansion forhk(x), we write the Fourier-Hermite coefficient off as
f(k) =ˆ 1 m!
Z b+h
b
f(x)dm
dtmhk(t)(x−b)mdx, wheret∈[b, x]andm=h
1 p
i .
Hardy-Type Inequalities For Hermite Expansions
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Then
|fˆ(k)| ≤Chm Z b+h
b
|f(x)|
dm dtmhk(t)
dx
≤Chmk−121+m2 Z b+h
b
|f(x)|dx
≤Chmk−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 ≤Chmp−1+pX
k≤γ
k
−p 12 + mp2 (k+ 1)σ.
Sinceσ= 2−p12 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 ≤Chmp−1+pγ(m2−121)p−σ+1 ≤C
Hardy-Type Inequalities For Hermite Expansions
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by the choice ofγ.
On the other hand, applying Hölder’s inequality withP = 2p, 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
≤ kfkp2γ(−2−p2σ +1)
2−p 2 .
Using property (ii) of anHp-atom, we getkfkp2 ≤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
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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.