#### Abstract

We consider the generalized shift operator, associated with the Dunkl operator , . We study some embeddings into the Morrey space (-Morrey space) , and modified Morrey space (modified -Morrey space) associated with the Dunkl operator on . As applications we get boundedness of the fractional maximal operator , , associated with the Dunkl operator (fractional -maximal operator) from the spaces to for and from the spaces to for .

#### 1. Introduction

On the real line, the Dunkl operators are differential-difference operators introduced in 1989 by Dunkl  and are denoted by , where is a real parameter . These operators are associated with the reflection group on . Rösler in  shows that the Dunkl kernel verifies a product formula. This allows us to define the Dunkl translations , .

In the theory of partial differential equations, together with weighted spaces, Morrey spaces play an important role. Morrey spaces were introduced by Morrey in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations (see ). Later, Morrey spaces found important applications to Navier-Stokes [4, 5] and Schrödinger  equations, elliptic problems with discontinuous coefficients [9, 10], and potential theory . An exposition of the Morrey spaces can be found in the book .

In the present work, we study some embeddings into the -Morrey and modified -Morrey spaces. As applications we give boundedness of the fractional -maximal operator in the -Morrey and modified -Morrey spaces.

The paper is organized as follows. In Section 2, we present some definitions and auxiliary results. In Section 3, we give some embeddings into the -Morrey and modified -Morrey spaces. In Section 4, we prove the boundedness of the fractional -maximal operator from the spaces to for and from the spaces to for .

#### 2. Preliminaries

On the real line, we consider the first-order differential-difference operator defined by

which is called the Dunkl operator. For , the Dunkl kernel on was introduced by Dunkl in  (see also ) and is given by

where is the normalized Bessel function of the first kind and order , defined by

The Dunkl kernel is the unique analytic solution on of the initial problem for the Dunkl operator (see ).

Let be the weighted Lebesgue measure on given by

For every , we denote by the spaces of complex-valued functions , measurable on such that

For we denote by the weak spaces defined as the set of locally integrable functions , with the finite norm

Note that

For all , we put

where

and is the Bessel kernel given by

where and .

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

For and being a continuous function on , the Dunkl translation operator is given by

Using the change of variable , we have also (see )

where and .

Proposition 2.1 (see Soltani ). For all the operator extends to , and we have for ,

Let , and , where . For (the space of locally integrable functions on ), we consider

Theorem 2.2 (see ). () If , then for every where is independent of .
(2) If , , then and
where is independent of .

Corollary 2.3. If , then for a.e. .

#### 3. Some Embeddings into the -Morrey and Modified -Morrey Spaces

Definition 3.1 (see ). Let , , and , . We denote by Morrey space ( -Morrey space) and by the modified Morrey space ( modified -Morrey space), associated with the Dunkl operator as the set of locally integrable functions , with the finite norms respectively.

If or , then , where is the set of all functions equivalent to on .

Note that

Definition 3.2 (see ). Let , . We denote by the weak -Morrey space and by the modified weak -Morrey space as the set of locally integrable functions , with finite norms respectively.

We note that

Lemma 3.3 (see ). Let . Then

Lemma 3.4. Let , . Then

Proof. Let . Then by (3.3) we have
Let . Then
Therefore, and the embedding is valid.
Thus .

From Lemmas 3.3 and 3.4 for , we have

Lemma 3.5. Let . Then

Proof. The embedding is a consequence of Hölder's inequality and Proposition 2.1. Indeed,

On the -Morrey spaces, the following embedding is valid.

Lemma 3.6 (see ). Let and . Then for where .

On the modified -Morrey spaces, the following embedding is valid.

Lemma 3.7. Let and . Then for

Proof. Let , , and . By the Hölder's inequality, we have
Note that

Therefore, and

#### 4. Some Applications

In this section, using the results of Section 3, we get the boundedness of the fractional -maximal operator in the -Morrey and modified -Morrey spaces.

For we define the fractional maximal functions

In the case , we denote by . Note that .

Lemma 4.1. Let , and . Then and the following equality is valid.

Proof.

Taking in Lemma 4.1 and using Lemma 3.3, we get for the following result.

Corollary 4.2. Let . Then

Lemma 4.3. Let , and . Then and the following equality is valid.

Corollary 4.4. Let and . Then the operator is bounded from to for . Moreover,

Corollary 4.5. . Then the operator is bounded from to for . Moreover,

#### Acknowledgment

The authors express their thank to the referees for careful reading, helpful comments and suggestions on the manuscript of this paper.