On the theorems of Hardy and Miyachi for the Jacobi–Dunkl transform

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On the theorems of Hardy and Miyachi for the Jacobi–Dunkl transform

R. Daher ^a

a Department of Mathematics , University of Hassan 2. Faculty of Sciences of Ain Chock. Laboratory T. A. G. A , Casablanca, Morocco

Published online: 24 Apr 2007.

To cite this article: R. Daher (2007) On the theorems of Hardy and Miyachi for the Jacobi–Dunkl transform, Integral Transforms and Special Functions, 18:5, 305-311, DOI:

10.1080/10652460701318244

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Vol. 18, No. 5, May 2007, 305–311

On the theorems of Hardy and Miyachi for the Jacobi–Dunkl transform

R. DAHER*

University of Hassan 2. Faculty of Sciences of Ain Chock.

Department of Mathematics. Laboratory T. A. G. A. Casablanca. Morocco.

(Received 9 February 2005)

We generalize theorems of Hardy and Miyachi for the Fourier transform on real line to the Jacobi–

Dunkl transform. More precisely in the first part of this paper we give another proof of the heart of Hardy’s theorem (we meanαβ=1/4), which has been obtained recently by Chouchane, Mili, and Trimèche. In the second part, we get some analog of Miyachi’s theorem for the Jacobi–Dunkl transform.

Keywords: Hardy’s theorem; Jacobi-Dunkl transform; Heat kernel

1. Introduction

A famous theorem of Hardy ([1], [2]) asserts that a measurable function f on Rand its Fourier transformfcannot both be very rapidly decreasing. More precisely, assume that

|f (t )| ≤Ce⁻^α^|^t^|² and|f (λ) | ≤Ce⁻^β^|^λ^|² whereC,αandβ are positive constants. Hardy’s theorem [2] states that if

i) αβ >1/4, then f (t )=0,

ii) αβ=1/4, then f (t )=const e⁻^αt²,

iii) αβ <1/4,then there are infinitely many suchf that are lineary independent and satisfy the above conditions.

We note that (ii) is the heart of the theorem because it implies (i) and (iii). This central part of Hardy’s theorem can be reformulated in terms of heat kernelht(x)=(4π t )⁻^(1/2)e⁻^(x²^{/4t )}, t >0. We note thath_t(x)=e⁻^{t x}², and thus the only functions satisfying(ii)are constant mul- tiples ofh_β, withβ=(1/4α). An analog of this celebrated result for the Fourier transform on semi-simple Lie groups and Riemannian symmetric spaces of the non-compact type have been the object of interest in several recent papers [3–10]. The interested reader may consult the Thangavelu’s book [10] and the references therein. AnL^pversion of this theorem was, further- more, given by Cowling and Price [11]. Gallardo and Trimèche have obtained anL^pversion

*Email: ra_daher@yahoo.fr

Integral Transforms and Special Functions

DOI: 10.1080/10652460701318244

Downloaded by [UQ Library] at 14:19 15 October 2014

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306 R. Daher

of Hardy’s theorem for the Dunkl transform onRⁿ[12]. Chouchaneet al.have formulated in [13] the Hardy’s theorem in terms of the heat kernel of the Jacobi–Dunkl operator.

In the first part of this paper, applying the same method as in [14], we give another proof of the generalization of the analogue of the Hardy’s theorem for the Jacobi–Dunkl transform obtained already by Chouchaneet al.[13]. Their approach is different from ours. Their proof is based on Hardy’s theorem for the usual Fourier transform and the properties of the Jacobi–

Dunkl intertwing operators and its dual. In the second part, we show that Miyachi’s theorem can also be reformulated in terms of the heat kernel of the Jacobi–Dunkl operator.

The contents of this paper are as follows:

In section 2, we review some main results concerning the harmonic analysis associated with the Jacobi–Dunkl operator. In section 3, we recall two crucial lemmas of the complex variables theory, which are a version of the Phragmen–Lindelöff theorem. Section 4 is devoted to giving another proof of Hardy’s theorem associated with the Jacobi–Dunkl transform.

In the last section, an analog of Miyachi’s theorem is obtained for the Jacobi–Dunkl transform.

The proof of these results requires many tools introduced in ref. [13, 15].

2. The harmonic analysis associated with the Jacobi–Dunkl operator

In this section, we collect relevant material from the harmonic analysis associated with the Jacobi–Dunkl operator, which was developped recently in [13] and [15].

2.1 The Jacobi–Dunkl kernel

In this subsection, we consider the differential-difference operatorα,βonR, given by α,βf (x)= d

dxf (x)+[(2α+1)coth(x)+(2β+1)tanh(x)]f (x)−f (−x) 2 whereα≥β ≥(−1/2);α=(−1/2).

The Jacobi–Dunkl kernel_λ^(α,β)is the uniqueC^∞-solution onRof the differential-difference equation

α,βu= −iλu

u(0)=1, λ∈C. (1)

It has the the Laplace integral representation

∀λ∈C, ∀x ∈R\{0}, _λ^(α,β)(x)= _|x|

−|x|K(x, y)e⁻^iλydy, (2) whereK(x,·)is a positive function onR, continuous on(−|x|,|x|)and satisfies

∀x ∈R\{0},

RK(x, y)dy=1. (3)

From (2) and (3) we deduce that the function _λ^(α,β) satisfies the following elementary estimate

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LEMMA1 Assume thatα≥β ≥(−1/2);α=(−1/2),λ∈C. Then

_λ^(α,β)≤_iIm(λ)^(α,β) (4)

and

∀x∈R, _λ^(α,β)(x)≤e^|Im(λ)^||^x^|. (5)

2.2 The Jacobi–Dunkl transform

We denote byρ=α+β+1, whereα≥β ≥(−1/2);α=(−1/2),D(R)the space of com- pactly supported C^∞-functions on R, S(R)the space of C^∞-functionsg onR which are rapidly decreasing together with their derivatives and byS^r(R), 0< r≤1, the generalized Schwartz space defined byS^r(R)=(cosh(x))⁽^−2ρ/r)S(R).

The Jacobi–Dunkl transform of functionf ∈D(R)is defined by

∀λ∈C, Ff (λ)=

R

f (x)_λ^(α,β)(x) α,β(x)dx, (6) where

α,β(x)=(2 sinh(x))^2α⁺¹(2 cosh(x))^2β⁺¹ This transform has the following inversion formula [13, 15].

LEMMA2 Forα≥β ≥(−1/2)andα=(−1/2),f ∈S^r(R),(0< r≤1),x∈R, f (x)=

RFf (λ)₋^(α,β)_λ (x)dσ (λ), (7) wheredσis the measure given by

dσ (λ)= |λ|

8π(λ²−ρ²)^(1/2)c((λ²−ρ²)^(1/2))²χ_R\(−ρ,ρ)(λ)dλ, (8) whereχ_R\(−ρ,ρ)(λ)is the characteristic function ofR\(−ρ, ρ)and

c(μ)= 2^ρ⁻^iμ(α+1)(iμ)

(((1/2)(ρ+iμ))(1/2)(α−β+1+iμ)), μ∈C\ {iN}. (9) 2.3 The heat kernel

DEFINITION3 Lett >0. The heat kernelEt associated with the Jacobi-Dunkl operator is defined by

∀x ∈R, Et(x)=F⁻¹(e⁻^{t λ}²)(x). (10) This heat kernelEt has the following properties [13] and [15]

(P₁) For allt >0,Et is an even positiveC^∞-function onR. (P₂) ∀t >0,∀λ∈R,FEt(λ)=e⁻^{t λ}².

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308 R. Daher

2.4 Version of the Phragmen-Lindelöff theorem

The proof of the main results of the following section depends on the two complex variable lemmas, which will be presented in this section.

L^EMMA4 Lethbe an entire function onCsuch that:

∀z∈C, |h(z)| ≤Ce⁻^a^|^z^|², (11) and

∀t ∈R, |h(t )| ≤Ce^at², (12) for some positive constantsaandC. Thenh(z)=conste⁻^az²,z∈C.

Proof (See ref. 7, Lemma 2.1)

As usual, let us define log⁺(x)=log(x)ifx >1, and log⁺(x)=0 otherwise. We also need the following lemma

LEMMA5 Supposegis an entire function and suppose there exist constantsA, B >0such that

For allz∈C, |g(z)| ≤Ae^B(Re(z))². (13) Also suppose

_+∞

−∞ log⁺|g(z)|<∞. (14) Thengis a constant function.

Proof (See [16, Lemma 4])

3. Main results

In this section we state and prove analogues of Hardy’s and Miyachi’s theorems.

3.1 An analog of Hardy’s theorem

THEOREM6 Letf be a measurable function onRsuch that

∀x∈R, |f (x)| ≤MEa(x) (15)

and

∀λ∈R, |Ff (λ)| ≤Me⁻^aλ² (16) for some constanta >0,andM >0. Then the functionf is a constant multiple of the heat kernelEa.

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Proof First, since by the condition (15),Ff (λ)is well defined for allλ, andFf is an entire function onC. Moreover, using the estimate (4) and (15) for allλ∈C, we have

|Ff (λ)| ≤

R|f (x)|_λ^(α,β)(x) ^α,β(x)dx

≤M

R

E_a(x)_iIm(λ)^(α,β)(x) _α,β(x)dx

=MF(E_a)(iIm(λ)).

From (10) we obtain

|Ff (λ)| ≤Me^a(Im(λ))². (17)

Or(Im(λ))²≤ |λ|²then we have

|Ff (λ)| ≤Me^a^|^λ^|², for allλ∈C. (18) We also have by assumption

|Ff (λ)| ≤Me⁻^a^|^λ^|², for allλ∈R. So by Lemma 4, we have

Ff (λ)=const·e⁻^aλ², forλ∈C. Using (10) we deduce that

f (x)=const·E_a(x).

This proves the theorem.

3.2 An analogue of Miyachi’s theorem We denote by

• L^pα,β(R),p∈(1,∞), the space of measurable functionf onRsuch that f1,α,β =

R|f (x)| α,β(x)dx <∞ and

f_∞,α,β=ess sup

x∈R|f (x)|<∞.

• L^p(R),p∈(1,∞)is defined in the obvious way.

Our principal result reads as follows:

THEOREM7 Leta >0. Supposef is a function onRsuch that E_a⁻¹(x)f (x)∈(L¹α,β+L^∞α,β)(Rx)

and _+∞

−∞ log⁺

Ff (λ)e^aλ² ξ

dλ <∞

for someξ, 0< ξ <∞. Thenf is a constant multiple of the heat kernelE_a.

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310 R. Daher

Proof We consider

Ff (λ)=

R

f (x)_λ^(α,β)(x) α,β(x)dx, forλ∈C

Using estimate (5) in Lemma 1, we get the following estimate on Ff (λ)(for different positive constantsC)

|Ff (λ)| ≤

R|f (x)|_iIm(λ)^(α,β)(x) α,β(x)dx.

The integrand of the above integral can be written as follows:

E_a⁻¹(x)f (x)E_a(x)_iIm(λ)^(α,β)(x) _α,β(x).

The first factor of which belongs to(L¹α,β+L^∞α,β)(Rx), by assumption and the second belongs to(L¹α,β∩L^∞α,β)(Rx).

Hence,Ff (λ)is well defined for allλ∈C, andFf is an entire function onC.

Moreover,

|Ff (λ)| ≤

R

E_a⁻¹(x)f (x)·Ea(x)_iIm(λ)^(α,β)(x) α,β(x)dx

≤C

R

E_a(x)_iIm(λ)^(α,β)(x) _α,β(x)dx

≤CF(Ea)(iIm(λ))

≤Ce^a(Im(λ))² with a constantC, independent ofλ.

Letg(λ)=e^aλ²Ff (λ), this is also an entire function.

Using (18), we have

|g(λ)| ≤Ce^a(Re(λ))², forλ∈C We also have, by assumption,

_+∞

−∞ log⁺

|g(λ)| ξ

dλ <∞.

Hence, applying the crucial Lemma 5 to the functiong(λ)/ξ, we see thatg(λ)=constant= K, or equivalently

Ff (λ)=Ke⁻^aλ².

From (10), by Fourier inversion we deduce that the functionf satisfies f (x)=KEa(x).

The theorem is then proved.

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References

[1] Hardy, G.H., 1933, A theorem concerning Fourier transform.Journal of the London Mathematical Society,8, 227–231.

[2] Dym, H. and McKean, H.P., 1972,Fourier Series and Integrals(Probability and Mathematical Statistics) (New York: Academic Press, Inc.).

[3] Narayanan, E.K. and Ray, S.S., 2002, The heat kernel and Hardy’s theorem for semi simple Lie groups.

Proceeding of the American Mathematical Society,130, 1859–1866.

[4] Ray, S.K. and Sarkar, R., 2001, AnL^pversion of Hardy’s theorem and charaterization of heat kernels on symmetric spaces. Preprint.

[5] Sarkar, R., 2002, A complete analogue of Hardy’s theorem onSL2(R)and characterization of the heat kernel.

Proceedings of the Indian Academy of Science,112, 579–594.

[6] Shimeno, N., 2001, An analogue of Hardy’s theorem for the Harish-Chandra transform.Hiroshima Mathematical Journal,31, 383–390.

[7] Sitaram, A. and Sundari, M., 1997, An analogue of Hardy’s theorem for very rapidly decreasing functions on semi simple Lie groups.Pacific Journal of Mathematics177, 187–200.

[8] Thangavelu, S., 2001, An analogue of Hardy’s theorem for the Heisenberg groups. Colloquium Mathematicum, 87, 137–145.

[9] Thangavelu, S., 2002, Hardy’s theorem for the Helgason-Fourier transform on non-compact rank one symmetric spaces.Colloquium Mathematicum,94(2), 263–280.

[10] Thangavelu, S., 2002, An introduction to uncertainty principle.Progress in Mathematics(Boston: Birkhauser).

[11] Cowling, M.G. and Price, J.F., 1983, Generalizations of Heisenberg inequality. Lecture Notes in Mathematics, 992 (Berlin: Springer), pp. 443–449.

[12] Gallardo, L. and Trimèche, K., 2002, Un analogue d’un théorème de Hardy pour la transformation de Dunkl.

Comptes Rendus Académie Science Paris,334(I), 849–854.

[13] Chouchane, F., Mili M., and Trimèche, K., 2004, AnL^pversion of Hardy’s theorem for the Jacobi–Dunkl transform.Integral Transforms and Special Functions,15(3), 225–237.

[14] Kawazoe, T. and Liu, J., 2003, Heat kernel and Hardy’s theorem for Jacobi transform.Chinese Annals on Mathematics,24B(3), 359–366.

[15] Chouchane, F., Mili, M. and Trimèche, K., 2003, Positivity of the Jacobi-Dunkl intertwining operator and its dual and applications.Journal of Analysis Application,1(4), 387–412.

[16] Miyachi, A., 1997, A generalization of a theorem of Hardy. Harmonic analysis seminar (Japan, Izunagaoka) 97.