International Journal of Mathematics and Mathematical Sciences

Volume 2010 (2010), Article ID 204509, 17 pages

http://dx.doi.org/10.1155/2010/204509

## Convolution Operators and Bochner-Riesz Means on Herz-Type Hardy Spaces in the Dunkl Setting

Department of Mathematics, Faculty of Sciences of Tunis, University of EL-Manar, Tunis 2092, Tunisia

Received 12 February 2010; Accepted 20 October 2010

Academic Editor: Ricardo Estrada

Copyright © 2010 A. Gasmi and F. Soltani. 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.

#### Abstract

We study the Dunkl convolution operators on Herz-type Hardy spaces and we establish a version of multiplier theorem for the maximal Bochner-Riesz operators on the Herz-type Hardy spaces .

#### 1. Introduction

The classical theory of Hardy spaces on has received an important impetus from the work of Fefferman and Stein, Lu and Yang [1, 2]. Their work resulted in many applications involving sharp estimates for convolution and multiplier operators.

By using the technique of Herz-type Hardy spaces for the Dunkl operator , we are attempting in this paper to study the Dunkl convolution operators, and we establish a version of multiplier theorem for the maximal Bochner-Riesz operators on these spaces.

The Dunkl operator , , associated with the reflection group on : is the operator devised by Dunkl [3] in connection with a generalization of the classical theory of spherical harmonics. The Dunkl analysis with respect to concerns the Dunkl operator , the Dunkl transform , the Dunkl convolution , and a certain measure on .

In this paper we define a Herz-type Hardy spaces , , in the Dunkl setting. Next, we consider the Dunkl convolution operators , where is a locally integrable function on . We use the atomic decomposition of the Herz-type Hardy spaces to study the -bounded and the -bounded of the operators . Finally, we establish a version of multiplier theorem for the maximal Bochner-Riesz operators , , : where on the Herz-type Hardy spaces . In this version we prove the -bounded of the operators , for .

The content of this work is the following. In Section 2, we recall some results about harmonic analysis and we define a Herz-type Hardy spaces , for the Dunkl operator . In Section 3, we study the -bounded and the -bounded of the convolution operators . In Section 4, we prove the -bounded of the maximal Bochner-Riesz operators .

Throughout the paper we use the classic notation. Thus and are the Schwartz space on and the space of tempered distributions on , respectively. Finally, will denote a positive constant not necessary the same in each occurrence.

#### 2. The Dunkl Harmonic Analysis on

For and , the initial problem
has a unique analytic solution called Dunkl kernel [4–6] given by
where
is the modified spherical Bessel function of order *α*.

We notice that, the Dunkl kernel can be expanded in a power series [7] in the form where

*Note. *Let be the measure on given by
We denote by , , the space of measurable functions on , such that

The Dunkl kernel gives rise to an integral transform, called Dunkl transform on , which was introduced by Dunkl in [8], where already many basic properties were established. Dunkl's results were completed and extended later on by de Jeu in [5].

The Dunkl transform of a function , is given by

For , we define the Dunkl transform of , by

*Note. *For all , we put
where
We denote by the following signed measure:
The Dunkl translation operators , (see [6]) are defined for (the space of continuous functions on ), by

Let and be two functions in . We define the Dunkl convolution product of and by

For and , we define the Dunkl convolution product by

We begin by recalling the definition of the Herz-type Hardy space in the Dunkl setting. Firstly we introduce a class of fundamental functions that we will call atoms.

Let . A measurable function on is called a atom, if satisfies the following conditions:

(i)there exists such that ;(ii), where is given in (i);(iii), for all , where , (the integer part of ).

Let . Our Herz-type Hardy space is constituted by all those that can be represented by
where and is a atom, for all , such that and the series in (2.16) converges in .

We define on the norm by
where the infimum is taken over all those sequences such that is given by (2.16) for certain atoms , .

As the same in [7], we prove the following theorem.

Theorem 2.1. *Let and . Then
*

#### 3. Dunkl Convolution Operators on

In the following, we study on , the Dunkl convolution operators defined by , where is a locally integrable function on .

Theorem 3.1. *Let . Assume that for every we are given and a function such that *(i)*,*(ii)*,*(iii)*. **Suppose also that and define . Then defines a bounded linear mapping from into .**To prove this theorem the following lemma is needed.*

Lemma 3.2. *For and , there are a constants , such that
**
where are the constants given by (2.4).*

*Proof. *Let , . By dominated convergence theorem, we can write
and by derivation under the integral sign, we get
Then, from Theorem 2.4, [6] we obtain
Let for , where . According to (2.11), [9],
where , , and are polynomials of degree at most with respect to each variable. Moreover,
Then, from (2.4) we deduce
Therefore,
where
This finishes the proof of the lemma.

*Proof of Theorem 3.1. *Firstly, notice that , . Hence, the series defining converges in and .

Let be a atom. Suppose that and that , where . We can write
*Step 1. *Let . From (2.13), for , we have
Hence, for , we deduce
*Step 2. *Firstly, let us consider that . From (Proposition 3(i), [10]) and condition (ii) of the theorem, we have

Assume now that . Since , , with , we have
where are the constants given by (2.4).

Using the properties of the Dunkl transform established by de Jeu [5] (see also [7, 10]), we deduce
According to page 302, [7]
Using condition (iii) of the theorem and Hölder's inequality, we get
where
Using the fact that and , we obtain
*Step 3. *We now prove for all ; , that
Fubini's theorem and [6] (see also page 20, [10]) lead to
Hence, by Lemma 3.2 and by taking into account that , , with , we get

According to the previous three steps, we conclude that is a atom. Then, , and

Let now be in . Assume that , where and is a atom, for every , and such that . The series defining converges in . In fact, it is sufficient to note that . Hence . Moreover, . Then by (Proposition 3(i), [10]), the operator is bounded from into itself, and from this, we deduce that . Using the fact that , we obtain
This completes the proof of the Theorem 3.1.

We now study the Dunkl convolution operators on the Herz-type Hardy spaces .

Theorem 3.3. *Let be a locally integrable function on . Assume that the following three conditions are satisfied:*(i)* defines a bounded linear operator from into itself.*(ii)* defines a bounded linear operator from into .*(iii)*There exist and such that **
Then defines a bounded linear mapping from into .*

*Proof. *Let be a atom. We choose such that and . We can write
Here is the one given in (iii).

From condition (i) of the theorem and Hölder's inequality, we deduce that
Also, by taking into account that , the condition (iii) of the theorem allows us to write
Hence, it concludes that
Note that the positive constant is not depending on the atom .

Let now be in . Then and , where and is a atom, for every and .

The series defining converges in . In fact, it is sufficient to note that , for every atom . Hence . Then the condition (ii) of the theorem implies that
By (3.29) the series in (3.30) converges in and . Hence .

#### 4. Maximal Bochner-Riesz Operators on

The Bochner-Riesz mean , for and associated to the Dunkl transform is defined by The maximal operator , associated to the Bochner-Riesz means , , is defined by

Lemma 4.1. *For and , one has*(i)*, where
Here is the modified spherical Bessel function given by (2.3).*(ii)*The operator is bounded from , into itself.**Proof. * Let and .(i)* By taking into account that the functions and are bounded on it is not hard to see that *
On the other hand, from [11], we have
Thus,
Applying Inversion Theorem[5], we obtain
where is the characteristic function of the set . Thus,
and from (Proposition 3 (ii), [10]), we deduce that
(ii)*Using (i) and (Proposition 3 (i), [10]), we obtain *
This clearly yields the result.*Theorem 4.2. Let and . For and , the operator extended to a bounded operator from into .*

*Proof. *According to Theorem 2.1, if , then is in and it is defined by
Moreover,
Hence is a bounded operator from into .

We now study the behavior of the maximal Bochner-Riesz operator on .

Theorem 4.3. *Let . Then the maximal Bochner-Riesz operator , is bounded from into .**To prove this theorem we need the following lemma.*

Lemma 4.4. *
(i) For and ,
**
(ii) For and ,*

*Proof. *
(i) Since the function is bounded on , it follows that
According to [6] and the fact that is even, we obtain
where
By (4.15) and using the fact that , we deduce that
(ii)From [11], we have
Then, similarly to the proof in (i), we have
which can be written as
Then from (4.20), it follows
Hence, if , we obtain
which completes the proof of the lemma.

*Proof of Theorem 4.3. *Let us first show that , exists such that
for every atom .

Let be an atom. Suppose that , and . We choose such that , we write
where
being , where will be specified later.

According to Lemma 4.4 (i), for , , we get

Then, using the fact that , we obtain
and hence, it concludes that
Note that the last series is convergent provide that .

On the other hand, since , from Lemma 4.4 (ii) we have

Then, for , we obtain
and in fact, we deduce that
The last series converges provided that .

Note that we can find such that the series in (4.31) and (4.34) converge if and only if . By combining (4.31) and (4.34) we show that
for a certain that is not depending on .

From Lemma 4.1 (ii) and (4.35) we deduce that
that is, (4.25) holds. Let now be in . Assume that , where the series converges in , and for every , is a atom and , such that . From Theorem 4.2, for , we can write
Hence, from (4.25) it follows . Thus we conclude that .

#### Acknowledgment

A. Gasmi and F. Soltani partially supported by DGRST project 04/UR/15-02 and CMCU program 10G 1503.

#### References

- C. Fefferman and E. M. Stein, “${H}^{p}$ spaces of several variables,”
*Acta Mathematica*, vol. 129, no. 3-4, pp. 137–193, 1972. View at Google Scholar - S. Lu and D. Yang, “Multiplier theorems for Herz type Hardy spaces,”
*Proceedings of the American Mathematical Society*, vol. 126, no. 11, pp. 3337–3346, 1998. View at Publisher · View at Google Scholar - C. F. Dunkl, “Differential-difference operators associated to reflection groups,”
*Transactions of the American Mathematical Society*, vol. 311, no. 1, pp. 167–183, 1989. View at Publisher · View at Google Scholar - C. F. Dunkl, “Integral kernels with reflection group invariance,”
*Canadian Journal of Mathematics*, vol. 43, no. 6, pp. 1213–1227, 1991. View at Google Scholar - M. F. E. de Jeu, “The Dunkl transform,”
*Inventiones Mathematicae*, vol. 113, no. 1, pp. 147–162, 1993. View at Publisher · View at Google Scholar - M. Rösler, “Bessel-type signed hypergroups on $\mathbb{R}$,” in
*Probability Measures on Groups and Related Structures, XI (Oberwolfach, 1994)*, H. Heyer and A. Mukherjea, Eds., pp. 292–304, World Scientific, River Edge, NJ, USA, 1995. View at Google Scholar - A. Gasmi, M. Sifi, and F. Soltani, “Herz-type Hardy spaces for the Dunkl operator on the real line,”
*Fractional Calculus & Applied Analysis*, vol. 9, no. 3, pp. 287–306, 2006. View at Google Scholar - C. F. Dunkl, “Hankel transforms associated to finite reflection groups,” in
*Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications*, vol. 138 of*Contemporary Mathematics*, pp. 123–138, American Mathematical Society, Providence, RI, USA, 1992. View at Google Scholar - H. Ben Mohamed and K. Trimèche, “Dunkl transform on $\mathbb{R}$ and convolution product on new spaces of distributions,”
*Integral Transforms and Special Functions*, vol. 14, no. 5, pp. 437–458, 2003. View at Publisher · View at Google Scholar - F. Soltani, “${L}^{p}$-Fourier multipliers for the Dunkl operator on the real line,”
*Journal of Functional Analysis*, vol. 209, no. 1, pp. 16–35, 2004. View at Publisher · View at Google Scholar - G. N. Watson,
*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, Cambridge, Mass, USA, 1966.