Generalized Hausdorff Operators on and in the Dunkl Settings

Abstract

In the present paper, we obtain some new results, and we generalize some known results for the Hausdorff operators. We have studied the generalized Hausdorff operators on the Dunkl-type homogeneous weighted Herz spaces and Dunkl Herz-type Hardy spaces . We have determined simple sufficient conditions for these operators to be bounded on these spaces. As applications, we provide necessary and sufficient conditions for generalized Cesàro operator to be bounded on and Hardy inequality for .

1. Introduction and Preliminaries

We recall that the Fourier transform of a (complex-valued) function in is defined as

The Hausdorff operator generated by a function in as introduced in [1] can be defined both directly and via the Fourier transform. The latter reads as follows: where is also in . The existence of such a function in is established in [1]. The theory of Hausdorff operators, while dating in a sense back to Hurwitz and Silverman [2] in 1917 with summability of number series, now becomes a notable ingredient in modern harmonic analysis and has received an extensive attention in recent years. To save the length of this article, we refer the reader to the survey article [3] for its background and historical developments.

Dunkl’s theory generalizes classical Fourier analysis on . This theory began twenty years ago with Dunkl’s seminal work in [4]. It was later developed by many mathematicians. On the real line, the Dunkl operators are differential-difference operators associated with the reflection group on . An important motivation to study Dunkl operators originates from their relevance for the analysis of quantum many-body systems of the Calogero-Moser-Sutherland type. These describe algebraically integrable systems in one dimension and have gained considerable interest in mathematical physics (see [5]).

Let be the measure on , given by

We denote by , , the space of measurable functions on such that

The Dunkl-Hausdorff operator (see [6–9]) acting on generated by a function belonging to is defined directly as: and for all function in , the Dunkl-Hausdorff operator verifies where is the Dunkl transform. When , the operator is the direct definition of the Hausdorff operator associated with Fourier transform defined in (2) from which several well-known operators can be deduced for suitable choices of , e.g., for , the operator reduces to the standard Hardy averaging operator while for , it reduces to the adjoint of Hardy averaging operator

Chen et al. [10] established boundedness of the classical Hausdorff operators in Herz type spaces, which are a natural generalization of the Lebesgue spaces . Gasmi et al. [11] introduced a new weighted Herz space associated with the Dunkl operators on . They also characterize the corresponding Herz-type Hardy spaces by atomic decomposition. Motivated by this result concerning Herz spaces (see also [12–14] and reference therein), this paper is aimed at extending these results to the context of Dunkl theory. We investigate the Dunkl-Hausdorff operators on the Dunkl-type homogeneous weighted Herz spaces and Dunkl Herz-type Hardy space in the spirit of those in [10]. As applications, we provide necessary and sufficient conditions for Dunkl-Cesàro operator and sufficient conditions for Dunkl-Hardy operator to be bounded on the homogeneous weighted Herz space .

This paper is organized as follows: in Section 1, we have presented some definitions and fundamental results from Dunkl’s analysis. In Sections 2 and 3, we have presented and proven our main results.

For , the Dunkl differential-difference operator is defined as (see [4])

For , the initial 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

The Dunkl transform is defined for by:

This transform satisfies the following properties: (i)For all such that , we have the inversion formula(ii)For all (the usual Schwartz space)

For any , we consider: where and for ,

We further have (see [15])

In the sequel, we consider the signed measure on given by

For and a continuous function on , we put which is called the Dunkl translation operator. The Dunkl translation operator has the following properties: (i)For and a continuous function on , we have(ii)(Product formula) For all (iii)For all and , we have

The Dunkl convolution of two functions , on is defined by the relation

Let and such that , we have where is the dilation of given by

For all , 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 (see [16]).

Let and . The -grand maximal function of -order of is defined by

The -grand maximal function is a bounded continuous operator from into itself, for every , provided that (see [11]).

2. Boundedness of on the Homogeneous Weighted Herz Space

Let , , and . The homogeneous weighted Herz space is the space constituted by all the functions , such that where is the characteristic function of the set and is the space

Note that . The main result of this subsection is the following theorem.

Theorem 1. Let , , , , and a measurable function on such that Then, the Dunkl-Hausdorff operator is bounded from to itself, i.e.,

Proof. Note that for any , there exists an integer number satisfying . The Minkowski inequality for guarantees that for any , By applying the Minkowski inequality for , we obtain Since and the definition of the Herz space , we estimate Therefore, we obtain which implies that is bounded from to itself.

Remark 2. When , Theorem 1 reduce to ([10], Theorem 2.4).

2.1. Hardy Inequality for

If , then (6) is of the following form:

In this case, reduces to the Hardy-type averaging operator for which we deduce the following result.

Corollary 3. Let , , , , and . Then, the Dunkl-Hardy operator is bounded from to itself and we have

Proof. From Theorem 1, we have

2.2. Generalized Cesàro Operator

If is supported in the interval , then reduces to the generalized Cesàro operator defined by (see [6, 17]).

Corollary 4. Let , , , , and a nonnegative measurable function defined on . Then, the generalized Cesàro operator is bounded from to itself if and only if

Proof. By Theorem 1, we need just to prove the necessity part. For any , we set then for and for , we have where Hence, it yields thus .
Now, it is easy to see that It follows from (49) that which implies when , This completes the proof.

3. Boundedness of on the Dunkl Herz-Type Hardy Space

Definition 5. Let , , , , and . The Herz-type Hardy space is the space of distributions such that . Moreover, we have In the sequel, we are interested in the spaces , when . Now, we turn to the atomic characterization of the space .

Definition 6. Let , , and . A measurable function on is called a (central) -atom if it satisfies: (1), for a certain (2),(3), where is the integer part of .

Theorem 7. Let , , , , and ; . Then, if and only if there exist, for all , an -atomand, such that and . Moreover, where the infimum is taken over all atomic decompositions of .

The main result of this subsection is the following theorem.

Theorem 8. Let , , , and ; . (i)For , letIf for some , then (ii)For , letIf , then

Proof. By the central atomic decomposition, for , we write where Then, we have Let us show that where each again is a central -atom and We write So, Now, we check that each satisfies the same cancellation condition as .
For , where is the integer part of , we have Also, the size of is then If , we have which means for all This tells us that Now, we write where It is easy to check that is a central atom and we have Let using Holder inequality yields the following since , then . It follows from (75) that This shows By the atomic decomposition, we obtain and this end the proof of .

The argument of part can be proved in an analogous way.

Remark 9. When , Theorem 8 reduce to ([10], Theorem 2.5).

We now return to the example of the generalized Cesàro operator .

Corollary 10. Let , , , and ; . (i)For , letIf for some , then (ii)For , letIf , then

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.