#### Abstract

The main purpose of this paper is to establish a Hörmander multiplier theorem for Herz-type Hardy spaces associated with the Dunkl transform.

#### 1. Introduction

Let be a multiplier operator defined in terms of Fourier transforms by for suitable functions . The multiplier theorem of Hörmander [1] gives a sufficient condition on for the operator to be bounded on whenever , namely, that is a bounded -function on satisfying the Hörmander condition as follows: where is the least integer greater than and . In [2], the authors proved that if satisfies the Hörmander condition with , then is bounded on the Hardy spaces with .

In [3], the authors considered the following multiplier operator which is associated with the Dunkl transform: where designs the Dunkl transform and using Hörmander’s technique proved the following theorem.

Theorem 1. *Let be the least integer greater than and let be a bounded -function on which satisfies the Hörmander condition as follows:
**
where is a constant independent of and . Then, the multiplier operator associated with the Dunkl transform can be extended to a bounded operator from into itself for , where is the Lebesgue space on with respect to the following measure:
*

The Hardy spaces associated with Herz spaces can be regarded as the local version at the origin of the classical Hardy spaces and they are good substitutes for when we study the boundedness of nontranslation invariant operators. To establish the boundedness of operators in hardy-type spaces on , one usually appeals to the atomic decomposition characterization of these spaces. In [4, 5], the authors studied the Herz-type Hardy spaces for the Dunkl operator in one-dimension and gave an atomic decomposition characterization of these spaces. The aim of this work is to prove the following Hörmander multiplier theorem on the spaces .

Theorem 2. *Let , , and be an integer greater than . If satisfies the Hörmander condition , then the operator is bounded on . *

The paper is organized as follows. In Section 2, we recall some results about harmonic analysis and Herz-type Hardy spaces associated with the Dunkl operator on . In Section 3, we give the proof of the main result of this work. Then, as an application, we obtain the boundedness of the generalized Hilbert transform on .

Throughout this paper, let be the usual Schwartz space and let be the space of -functions on . We always use to denote a positive constant that is independent of the main parameters involved but whose value may differ from line to line. We use the shorter notation instead of .

#### 2. Preliminaries

In this section, we recapitulate some results about harmonic analysis on Dunkl hypergroups and the Herz-type Hardy space and its atomic decomposition which will be used later. For details, the reader is referred to [6–8].

Let . We consider the differential-difference operator introduced in [9] as follows:
and call it the *Dunkl operator*.

For , the following initial value problem:
has a unique solution (called the *Dunkl kernel*) given by
where is the normalized Bessel function of the first kind (with order ) defined on by
The integral representation of is given by
From which, we get

The * Dunkl transform *, which was introduced by [10] and studied in [11], is defined for by
This transform satisfies the following properties.(i)For all , we have
(ii)For all such that , we have the following inversion formula:
(iii)For all ,
(iv) is a topological isomorphism from into itself. (v) is an isometric isomorphism of , and we have the following Parseval formula:

The following lemma can be proved, similar to Lemma 7.25, page 343, in [12].

Lemma 3. *Let be the least integer greater than . If satisfies the Hörmander condition , then there is a constant independent of , such that if or , the following inequality holds:
**
Furthermore, in case , then and is continuous on . *

*Notation. *For all , we put
where
The Dunkl translation operator , is defined for a continuous function on by
where is the signed measures given by
The operator has the following properties. (i)For and a continuous function on , we have
(ii)For all , the operator can be extended to , and for , we have
(iii)For all and , we have

Let such that . The convolution product of and is defined by
and we have
If , then

Now, let us recall the definition of the Herz-type Hardy space and its atomic decomposition. For being sufficiently large, we denote by the subset of constituted by all those such that and for all such that , we have Moreover, the system of seminorms generates the topology of .

Let . We define the -grand maximal function of by where is the dilation of given by

*Definition 4. *Let , , and .(i) The homogeneous weighted Herz space is the space constituted by all functions , such that
where is the characteristic function of .(ii) The nonhomogeneous weighted Herz space is defined, as usual, by . Moreover, .

Note that .

*Definition 5. *Let , , and . The Herz-type Hardy space is the space of distributions such that . Moreover, we define
In the same way, we define the space for the non-homogeneous case.

*Definition 6. *Let and . A measurable function on is called a (central) -atom if it satisfies the following: (i), for some ,(ii), (iii), , where and denotes the integer part function.

The following theorem is shown in [4].

Theorem 7. *Let and . Then, if and only if, for all , there exist a -atom and , such that and . Moreover,
**
where the infimum is taking over all atomic decompositions of . *

In the sequel, fix and .

*Definition 8. *For . Set , , , and . A central -molecule is a function satisfying the following: (i), (ii), (iii), .

Proposition 9. *Let be the triple cited in the previous definition. Every central -molecule belongs to and , where the constant is independent of . *

* Proof. *Let be a central -molecule and suppose that . In the general case, letting , we have .

Let , , and , , where is the characteristic function of . For each , there exists a unique polynomial , of degree at most , such that if ; then

Using some ideas in [2], we can show that each is a multiple of a central -atom with a sequence of coefficients in . We also show that the sum can be written as an infinite linear combination of central -atom with a sequence of coefficients in . Since a -atom is also -atom, hence,
where is a central -atom and . It follows from Theorem 7 that and .

The following Lemma plays an important role in the proof of the main result of this work.

Lemma 10. *Let be a -atom. For all integer and every , there exists a constant independent of , such that
*

*Proof. *(i) Let be a -atom. Consider that such that and that . From (9), (iii) of Definition (19), and the estimate for the remainder in Taylors’ formula, it follows that
From (ii) of Definition (19), we obtain
(ii) For ,
Using (10), we get the following for all :
From (ii) of Definition (19), we obtain the following for all :
For ,
For ,
Finally, we get the following for all :

#### 3. Proof of Theorem 2

Let and be an integer greater than . Set , , , and .

We have ; then, according to Proposition 9 to prove Theorem 2 it suffices to prove that, for any -atom , is a central -molecule with for some constant independent of . In other words, we need to check that Firstly, we prove (i) and (ii).

satisfies the Hörmander condition ; then, by Theorem 1, there exists a constant independent of , such that From (14) and (13), we have Then, by Plancherel theorem to estimate , it suffices to estimate , which turns out to prove that

By induction, we have where and are constants.

But, using Leibniz formula, we have the following for : So, to establish (47), it suffices to claim that For the case , we use Lemma 10 (ii) with and Lemma 3 to get the following: For , we have where and is the integer, such that Firstly, we estimate .

Using (i) of Lemma 10 and the fact that satisfies the Hörmander condition , we get By (54), we obtain Now, we estimate . By Holder’s inequality, we have Using (ii) of Lemmas 10 and 3, we get To guarantee the convergence of this summation, we choose the pair as follows:(a)if , we choose ;(b)if and , we choose ;(c)if and , we choose such that .

Furthermore, by (54), we get

Finally, combining (56) and (59), we obtain (47). (i) and (ii) are hence proved.

To prove (iii), it suffices to prove that for all integer and : indeed if according to (14), which we have is continuous, and hence .

Now, we check . We write , where Using the fact that and Schwarz’s inequality, we get For , we have Using the fact that , we get .

Finally, we check We have where and are constants. Then, to prove (63), it suffices to prove that By (i) of Lemma 10, we have According to Lemma 3, we have ; indeed ; then, we obtain where (63) is hence proved. This finishes the proof of Theorem 2.

Corollary 11. *Let . Then, the generalized Hilbert transform defined by
**
where is given by (19), is bounded on . *

*Proof. *From Proposition 3.6 in [3], the generalized Hilbert transform is a multiplier operator with ; then the proof of the corollary follows from Theorem 2.

#### Acknowledgment

This paper was supported by a generous grant from Taibah University Research Project.