Boundedness of Marcinkiewicz Integrals and Their Commutators on Generalized Weighted Morrey Spaces
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 ). Suppose that with . Then, for every and , there is a constant independent of such that
Theorem 2 (see ). Suppose that with and . Then, for every and , there is a constant independent of such that
The classical Morrey spaces, introduced by Morrey  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 [4–7] 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  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  Mizuhara gave generalization Morrey spaces considering instead of in (7). He studied also a continuity in of some classical integral operators.
Komori and Shirai  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  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  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 , 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
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  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 .
Following , a locally integrable function is said to be in if where
Lemma 10 (see ). 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.
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
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