Research Article | Open Access

# Boundedness of Marcinkiewicz Integrals and Their Commutators on Generalized Weighted Morrey Spaces

**Academic Editor:**Lars E. Persson

#### Abstract

We will study the boundedness of Marcinkiewicz integrals with rough kernel on the generalized weighted Morrey spaces. We will also prove that the commutator operators formed by a function and Marcinkiewicz integrals are also bounded on the generalized weighted Morrey spaces.

#### 1. Introduction and Results

Suppose that is the unit sphere in equipped with the normalized Lebesgue measure . Let with be homogeneous of degree zero and satisfy the cancellation condition where for any . The Marcinkiewicz integral of higher dimension is defined by where We will also consider the commutator generated by Marcinkiewicz integral and is defined as follows:

The following results concerning the boundedness of Marcinkiewicz integrals and their commutators on weighted space are known.

Theorem 1 (see [1]). *Suppose that with . Then, for every and , there is a constant independent of such that
*

Theorem 2 (see [2]). *Suppose that with and . Then, for every and , there is a constant independent of such that
*

The classical Morrey spaces, introduced by Morrey [3] in 1938, have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces. See [4â€“7] for details.

Given , and . Morrey spaces are defined by

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

In [8] Chiarenza and Frasca show the boundedness of the Hardy-Littlewood maximal operator in that allows them to prove continuity of fractional and classical CalderÃ³n-Zygmund operators in these spaces. Recall that integral operators of that kind appear in the representation formulae of the solutions of elliptic/parabolic equations and systems. Thus the continuity of the CalderÃ³n-Zygmund integrals implies regularity of the solutions in the corresponding spaces.

Let , , be a growth function, that is, a positive increasing function in , and satisfy doubling condition where is a doubling constant independent of . In [9] Mizuhara gave generalization Morrey spaces considering instead of in (7). He studied also a continuity in of some classical integral operators.

Komori and Shirai [10] introduced a version of the weighted Morrey space , which is a natural generalization of the weighted Lebesgue space . Let , and let be a weight function. Then the spaces are defined by

Recently, Aliev and Guliev in [11] introduced another generalization of the Morrey spaces. Let and let be a positive measurable function on . We denote by the generalized Morrey space, the space of all functions with finite norm They also studied in [11] the boundedness of and their commutator on generalized Morrey space when .

Let and let be a positive measurable function on and let be a nonnegative measurable function on . Following [12], we denote by the generalized weighted Morrey space, the space of all functions with finite norm where

If and with , then is the classical Morrey space; if , , then is the weighted Morrey space; if , , then . If , then .

The purpose of this paper is to discuss the boundedness properties of Marcinkiewicz integrals with rough kernel and their commutators on the generalized weighted Morrey spaces . Our main results can be formulated as follows.

Theorem 3. *Suppose that with . Let , , and satisfy the condition
**
where does not depend on or . Then there is a constant independent of such that
*

Theorem 4. *Suppose that with . Let , , and satisfy the condition
**
If , then there is a constant independent of such that
*

Corollary 5. *Let with and . *(i)*Suppose satisfy the condition
Then there is a constant independent of such that
*(ii)*Suppose satisfy the condition
and . Then there is a constant independent of such that
*

*Remark 6. *(i) Let , and ; then satisfy conditions (13) and (15).

(ii) Let . If , then also satisfy conditions (13) and (15).

We verify only (i). (ii) can be verified similarly. In fact, from (36) in Section 2 we have constant such that Then

Corollary 7. *Suppose that with and . If , , then there is a constant independent of such that
**
If , , and , then there is a constant independent of such that
*

Corollary 8. *Suppose that with . If , , then there is a constant independent of such that
**
If , , and , then there is a constant independent of such that
*

#### 2. Some Preliminaries

We begin with some properties of weights which play a great role in the proofs of our main results.

A weight is a nonnegative, locally integrable function on . Let denote the ball with the center and radius . For a given weight function and a measurable set , we also denote the Lebesgue measure of by and set weighted measure . For any given weight function on , and , denote by the space of all function satisfying

A weight is said to belong to for , if there exists a constant where is the dual of such that . The class is defined by A weight is said to belong to if there are positive numbers and so that for all balls and all measurable . It is well known that By (28), we have for . Note that is true for any real-valued nonnegative function and measurable on (see [13] page 143) and (29); we get

The classical weight theory was first introduced by Muckenhoupt in the study of weighted -boundedness of Hardy-Littlewood maximal function in [14].

Lemma 9 (see [14, 15]). *Suppose and the following statements hold. *(i)*For any , there is a positive number such that
*(ii)*For any , there are positive numbers and such that
*(iii)*For any , one has .*

Following [16], a locally integrable function is said to be in if where

Lemma 10 (see [12]). *Suppose and . Then, for any and , one has
*

#### 3. Proof of Theorem 3

We first prove the following conclusions.

Theorem 11. *Suppose that , . Then, for every , , there is a constant independent of such that
*

*Proof. *We represent as , where denotes the characteristic function of . Then
Since and from the boundedness of on (Theorem 1) it follows that
By HÃ¶lder inequality,
Then, for ,
By (32), we get

To estimate , observe that when and ,
Therefore, by Minkowskiâ€™s inequality, we have

When , then by assumption we have , . It follows from the HÃ¶lder inequality and (32) that
Then

When , . It follows from the HÃ¶lder inequality that
When and , then by a direct calculation we can see that . Hence
We also note that if , , then . Consequently
It follows from (47), (50), (51), and (52) that
Since , . Thus from the HÃ¶lder inequality we get that
Note that . It follows from the condition that
That is,
Then
Therefore
Combining (45), (49), and (58), the proof of Theorem 11 is completed.

*Proof of Theorem 3. *Since , from (33) and the fact is a nondecreasing function of , we get
For , since satisfy (13), we have
Then by (40), we get
This completes the proof of Theorem 3.

#### 4. Proof of Theorem 4

As in the proof of Theorem 3, it suffices to prove the following result.

Theorem 12. *Assume that with and . Then, for every , , there is a constant independent of such that
*

*Proof. *We represent as
Then
By Theorem 2, we have
As the proof of (45), we get

We now turn to deal with the term . For any given , we have
By (48) and (57), for any , , we have
Then from Lemma 10 we get

When , by assumption, we have , . It follows from the HÃ¶lder inequality that
By Lemma 9, we have . Then by Lemma 10 and (32),
Then
Therefore,