Rough Multilinear Fractional Integrals on Weighted Morrey Spaces

Abstract

It is showed that a class of multilinear fractional operators with rough kernels, which are similar to the higher-order commutators for the rough fractional integrals, are bounded on the weighted Morrey spaces.

1. Introduction and Results

Suppose that denotes the unit sphere of equipped with the usual Lebesgue measure, is homogeneous of degree zero, and , , are functions defined on . Consider the following multilinear fractional integral with rough kernel defined by where ,  , and

When , , and , then is just the commutator of the rough fractional integral with the function : If , then is nontrivial generalization of the above commutator. The weighted -boundedness of the operator was given by Wu and Yang in [1].

When , for , and , then is the higher-order commutator of the fractional integral with the function : The weighted -boundedness of was given by Ding and Lu in [2].

Ding and Lu in [3] proved the following result.

Theorem 1. Let ,  , let , and let be homogeneous of degree zero with Moreover, for , , , and . If , then there exists a constant , independent of and , such that

Here and in the sequel, always denotes the conjugate index of any ; that is, , and stands for a constant which is independent of the main parameters, but it may vary from line to line.

The purpose of this paper is to discuss the boundedness properties of the rough fractional multilinear integral operators on appropriate weighted Morrey spaces.

The classical Morrey spaces were introduced by Morrey [4] in 1938 and have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations [5–7].

Let and ; Morrey spaces are defined by where Note that and If or , then , where is the set of all functions equivalent to on

Let ,  , be a growth function, that is, a positive increasing function in , which satisfies doubling condition where is a doubling constant independent of . Mizuhara in [8] gave generalization Morrey spaces considering instead of in (7).

Komori and Shirai [9] introduced a version of the weighted Morrey space , which is a natural generalization of the weighted Lebesgue spaces .

Let ,  , and a weight function. The spaces are defined by where

In order to deal with the fractional order case, we need to consider the weighted Morrey spaces with two weight functions; they were also introduced by Komori and Shirai in [9].

Let , For two weight functions and , the spaces are defined by where

Let ,   a positive measurable function on , and a nonnegative measurable function on . We denote by other weighted Morrey spaces, the spaces of all functions with finite norm where

Remark 2. (1) If and with , then .
(2) If , then .
(3) If , then .
(4) If , then .

Our main results can be formulated as follows.

Theorem 3. Let , , let , and let be homogeneous of degree zero with , . Suppose and satisfies the condition where does not depend on and . If have derivatives of order in , , then there is a constant , independent of and , such that

Remark 4. Let , let , and let . If , it is easy to prove that satisfies condition (15).

Remark 5. Let , , , and . By [10], we know that satisfies condition (15).

Corollary 6. Let , , let , and let be homogeneous of degree zero with . Suppose , and have derivatives of order in , ; then is bounded from to and

Remark 7. In Corollary 6, let and ; then we obtain the main result in [11].

Corollary 8. Let , , let , and let be homogeneous of degree zero with , Suppose , , and have derivatives of order in , ; then is bounded from to and

Remark 9. In Corollary 8, if and , then we obtain the main result in [12]; if and , then we obtain the main result in [13].

2. Some Preliminaries

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

Weight is a nonnegative, locally integrable function on . Let denote the ball with the center and radius and let for any . 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 functions satisfying

Weight is said to belong to for , if there exists a constant such that denotes the set consisting of all weight functions : Weight is said to belong to if there are positive numbers and such that for all balls and all measurable . It is well known that

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

Lemma 10 (see [14, 15]). Suppose . The following statements hold: (i)For any , there is a positive number such that (ii)For any , there is a positive number and such that (iii)For any , one has .

We also need another weight class introduced by Muckenhoupt and Wheeden in [16] to study weighted boundedness of fractional integral operators.

Given , we say that if there exists a constant such that, for every ball , the inequality holds when , and for every ball the inequality holds when .

By (26), we have

We summarize some properties about weights ; see [15, 16].

Lemma 11. Given (i) if and only if ;(ii) if and only if ;(iii)if and , then .

John and Nirenberg introduced the function space of BMO in [17]. A locally integrable function is said to be in if where

Lemma 12 (see [10]). Suppose and Then for any and , we have

Below we recall some conclusions about

Lemma 13 (see [9]). Let be a function on with the th derivatives in , . Then

Lemma 14 (see [18]). For fixed , let Then .

Lemma 15. Let , and let . Then

Proof. From Lemma 14, we have By Lemma 13, If , , we can easily see that . We get Then Thus Note that Then Thus Combining with (35), (36), and (42), then (34) is proved.

Finally, we recall a relationship between essential supremum and essential infimum.

Lemma 16 (see [19]). Let be a real-valued nonnegative function and measurable on Then

3. A Local Estimate

To prove Theorem 3, we first investigate the following local estimate.

Theorem 17. Let , , let , and let be homogeneous of degree zero with , . Suppose and have derivatives of order in , ; then, for any , there is a constant independent of such that

Proof. To simplify the proof process of Theorem 17 in the following discussion we consider only the case The method can be used to deal with the case without any essential difficulty.
We write as , where , and denotes the characteristic function of . Then Since , by the boundedness of from to (Theorem 1) we get Noting that and , then by Hölder’s inequality This means Thus Since , by (28), we get holds for all Then Let , and let Suppose By Lemma 15, By Hölder’s inequalities, When and , then by a direct calculation, we can see that . Then We also note that if , then . Consequently Then Since , it follows from Hölder’s inequality that Then From (50) we know Then By Hölder’s inequality and (54), (55) we have Applying Hölder’s inequality again, we get Consequently, By and (ii) of Lemma 11 we know Then it follows from (31) and (59) that Hence Similar to the estimates for , we have Finally, we come to estimate .
Similar to (59), we get By Hölder’s inequality we get Then Since , similar to (64), we can deduce Then by (70) we get Thus Combining the estimates for , , , and , we get Then This completes the proof of Theorem 17.

4. Proof of Theorem 3

Since , then by Lemma 16 and the fact that is a nondecreasing function of , we get Since satisfies (15), we have Then by Theorem 17 we get

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors are very grateful to the anonymous referees and the editor for their insightful comments and suggestions.