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 [47] 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 Calderón-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 Calderón-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 Hölder inequality, Then, for , By (32), we get
To estimate , observe that when and , Therefore, by Minkowski’s inequality, we have
When , then by assumption we have , . It follows from the Hölder inequality and (32) that Then
When , . It follows from the Hölder 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 Hölder 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 Hölder inequality that By Lemma 9, we have . Then by Lemma 10 and (32), Then Therefore,
When , it follows from (50), (51), and (52) that Set . Since , from Lemma 9, we know . By Hölder inequality Consequently, Since , from (32), we know Using Lemma 10 and the fact that , we thus obtain