Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces

Abstract

The boundedness of multilinear commutators of Calderón-Zygmund operator on generalized weighted Morrey spaces with the weight function belonging to Muckenhoupt's class is studied. When and , , , the sufficient conditions on the pair which ensure the boundedness of the operator from to are found. In all cases the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in .

1. Introduction

Let be a Calderón-Zygmund singular integral operator and . A well known result of Coifman et al. [1] states that if and is a Calderón-Zygmund operator, then the commutator operator is bounded on for . The commutators of Calderón-Zygmund operator play an important role in studying the regularity of solutions of elliptic, parabolic and ultraparabolic partial differential equations of second order (see, [2–7]).

The classical Morrey spaces were originally introduced by Morrey in [8] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [2–4, 8, 9].

Let , and be locally integrable functions when we consider multilinear commutators as defined by where is Calderón-Zygmund kernel. That is, for all distinct , and all with , there exist positive constant and such that(i)(ii); and(iii)when , it is the classical commutator which was introduced by Coifman et al. in [1]. It is well known that Calderón-Zygmund operators play an important role in harmonic analysis (see [10–12]).

We define the generalized weighed Morrey spaces as follows.

Definition 1. Let , be a positive measurable function on and be non-negative measurable function on . We denote by the generalized weighted Morrey space, the space of all functions with finite norm where denotes the weighted -space of measurable functions for which
Furthermore, by we denote the weak generalized weighted Morrey space of all functions for which where denotes the weak -space of measurable functions for which

Remark 2. (1) If , then is the generalized Morrey space.
(2) If , then is the weighted Morrey space.
(3) If , then is the two weighted Morrey space.
(4) If and with , then is the classical Morrey space and is the weak Morrey space.
(5) If , then is the weighted Lebesgue space.

The commutators are useful in many nondivergence elliptic equations with discontinuous coefficients, [2–5]. In the recent development of commutators, Pérez and Trujillo-González [13] generalized these multilinear commutators and proved the weighted Lebesgue estimates. The weighted Morrey spaces was introduced by Komori and Shirai [14]. Moreover, they showed that some classical integral operators and corresponding commutators are bounded in weighted Morrey spaces. Feng in [15] obtained the boundedness of the multilinear commutators in weighted Morrey spaces for and , where the symbol belongs to bounded mean oscillation . Furthermore, was given the weighted weak type estimate of these operators in weighted Morrey spaces of for and .

Recently, the generalized weighted Morrey spaces introduced by Guliyev [16, 17]. Moreover, in [16, 17] he studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [18–20]).

The following statement was proved in [18].

Theorem A. Let , and satisfies the condition where does not depend on and . Then the operator is bounded from to for and from to .

Remark 3. Note that, Theorem A was proved in the case in [21] and in the case and in [14].

Definition 4. is the Banach space modulo constants with the norm defined by where and

In this paper, we prove the boundedness of the multilinear commutators of Calderón-Zygmund operator from one generalized weighted Morrey space to another for and , , .

By we mean that with some positive constant independent of appropriate quantities. If and , we write and say that and are equivalent.

2. Main Results

In the following, main results are given. First, we present some estimates which are the main tools to prove our theorems, for the boundedness of the multilinear commutator operators on the generalized weighted Morrey spaces.

Theorem 5. Let , , , , , and be a multilinear commutators defined as (6). Then holds for any ball and for all , where does not depend on , and .

Theorem 6. Let , , , , and be a multilinear commutators defined as (6). Then holds for any ball and for all , where does not depend on , and , where and .

Now we give a theorem about the boundedness of the multilinear commutator operator on the generalized weighted Morrey spaces.

Theorem 7. Let , , and satisfies the condition where does not depend on and . Let , , . Then the operator is bounded from to . Moreover,

Theorem 8. Let , and satisfies the condition where does not depend on and . Let , , . Then the operator is bounded from to . Moreover, where and .

When , from Theorem 7 we also get the following new result.

Corollary 9. Let , , , , , . Then the operator is bounded on for and from to for , where and .

Proof. Let , , and , , . Then the pair satisfies the condition (11) for and the condition (13) for . Indeed, by Lemma 10 there exists and such that for all and : Then

Note that from Corollary 9 was proved in [15].

3. Some Lemmas

Let be the -dimensional Euclidean space of points with norm . For and , denote the open ball centered at of radius . Let be the complement of the ball , and be the Lebesgue measure of .

A weight function is a locally integrable function on which takes values in almost everywhere. For a weight and a measurable set , we define , the Lebesgue measure of by , and the characteristic function of by . Given a weight , we say that satisfies the doubling condition if there is a constant such that for any ball . When satisfies the doubling condition, we denote , for short.

If is a weight function, then we denote the weighted Lebesgue space by with the norm and when .

We recall that a weight function is in the Muckenhoupt’s class , , if where the sup is taken with respect to all the balls and . Note that, for all balls we have by Hölder’s inequality. For , the class is defined by the condition with , and for we define .

Lemma 10 (see [22]). We have the following:(1)If for some , then . Moreover, for all we have (2)If , then . Moreover, for all we have (3)If for some , then there exist and such that for any ball and a measurable set ,

The following results are proved by Pérez and Trujillo-González [13].

Lemma 11. Let and and suppose that , , , then there exists a constant such that

Although the commutators with function are not of weak type , they have the following inequality.

Lemma 12. Let and suppose that , , , then there exists a constant such that where .

Lemma 13. Let and suppose that , , , then there exists a constant such that where .

In this paper, we need the following statement on the boundedness of the Hardy type operator where be a non-negative Borel measure on .

Theorem 14. The inequality holds for all non-negative and non-increasing on if and only if and .

Note that, Theorem 14 is proved analogously to Theorem  4.3 in [21].

Lemma 15 (see [23, Theorem 5, page 236]). Let . Then the norm of is equivalent to the norm of , where

Remark 16. (1) The John-Nirenberg inequality: there are constants , , such that for all and (2)For the John-Nirenberg inequality implies that and for and

Indeed, it follows from the John-Nirenberg inequality and using Lemma 10  (3), we get for some . Hence, this inequality implies that

To prove the requested equivalence we also need to have the right inequality, that is easily obtained using Hölder inequality, then we get (32). Note that (31) follows from (32) in the case .

The following lemma was proved in [24].

Lemma 17. Let be a function in . Let also , , and . Then where is independent of , , and .

The following lemma was proved in [17].

Lemma 18. Let and be a function in . Let also , , and . Then where is independent of , , and .
Let and be a function in . Let also , , and . Then where is independent of , , and .

4. Proof of the Theorems

Proof of Theorem 5. Let . For arbitrary and , set . Write with and . Hence From the boundedness of in (see Lemma 11) it follows that:
For the term , without loss of generality, we can assume . Thus, the operator can be divided into four parts
For we have
Then Let us estimate Applying Hölder’s inequality and by Lemma 18, we get
Let us estimate
Applying Hölder’s inequality and by Lemma 18, we get
In the same way, we shall get the result of
In order to estimate note that By Lemma 18, we get Applying Hölder’s inequality, we get Thus, by (50) Summing up and , for all we get
On the other hand,
Finally, and the statement of Theorem 5 follows by (53).

Proof of Theorem 6. Let . To deal with this result, we split as above by , which yields From the boundedness of from to (see Lemma 13) it follows that:
For the last term , without loss of generality, we still assume . By homogeneity it is enough to assume and hence, we only need to prove that for all . In fact, by Lemma 12, we get where . We use the Fefferman-Stein maximal inequality for any functions and . This yields
Then

Proof of Theorem 7. By Theorem 5 and Theorem 14 we have for

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for careful reading the paper and useful comments. The research of Vagif S. Guliyev was supported by the grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003.13.003) and (PYO.FEN.4003-2.13.007).