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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 280401, 10 pages
Multilinear Singular Integrals and their Commutators with Nonsmooth Kernels on Weighted Morrey Spaces
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
Received 21 October 2013; Accepted 4 December 2013
Academic Editor: Alberto Fiorenza
Copyright © 2013 Songbai Wang and Yinsheng Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Some multilinear maximal functions and the generalized Calderón-Zygmund operators and their commutators with nonsmooth kernels are studied. The purpose of this paper is to establish that these operators are bounded on certain product Morrey spaces . Based on the boundedness of these operators from to , we obtained that they are also bounded from to with , , and , .
Let and be the Schwartz spaces of all rapidly decreasing functions and tempered distributions, respectively. Let be a multilinear operator initially defined on the -fold product of Schwartz spaces and taking values into the space of tempered distributions, Following , the -multilinear Calderón-Zygmund operator satisfies the following conditions:(S1)there exist , it extends to such that a bounded multilinear operator from to , where (S2)there exists a function , defined off the diagonal in , satisfying for all and , where for some and all , whenever .
We also take some notation following . Given a locally integrable vector function . The commutator of and the -linear Calderón-Zygmund operator , denoted here by , was introduced by Pérez and Torres in  and is defined via where
And the iterated commutators are defined by
To clarify the notations, if is associated in the usual way with a Calderón-Zygmund kernel , then at a formal level
Let be a class of integral operators, which play the role of the approximation to the identity. We always assume that the operators are given by kernels in the sense that for all and , and the kernels satisfy the following conditions: where is a positive fixed constant and is a positive, bounded, decreasing function satisfying that for some
Recall that the th transpose of the -linear operator is defined via for all in . It is seen that the kernel of is related to the kernel of via the identity
If an -linear operator maps a product of Banach spaces into another Banach space , then the transpose maps to . Moreover, the norms of and are equal. To maintain uniform notation, we may occasionally denote by and by .
Assumption 1. Assume that for each there exist operators with kernels that satisfy conditions (11) and (12) with constants and and that, for every , there exist kernel such that for all in with . Also assume that there exist a function with and constants and so that for every and every , we have whenever .
If satisfies Assumption 1 we will say that is an -linear operator with generalized Calderón-Zygmund kernel . The collection of function satisfying (15) and (16) with parameters , and will be denoted by -linear . We say that is of class - if has an associated kernel in -. Throughout this paper, we always assume that the -linear operator satisfies the following assumption.
Assumption 2. Assume that there exist some and some with , such that maps to .
Theorem 3 (see ). Assume that is a multilinear operator in -. Let with , all the following statement are valid:(i)when all , then can be extended to be a bounded operator from the -fold product to ;(ii)when some , then can be extended to be a bounded operator from the -fold product to .
Moreover, there exists a constant such that
Assumption 4. Assume that there exist operators with kernels that satisfy condition (11) and (12) with constants and . Let We assume that the kernels satisfy the following estimates; there exist a function with and constants and such that whenever , and whenever .
It is known that condition (16) is weaker than, and indeed a consequence of, the Calderón-Zygmund kernel condition (5) from the proof of Proposition 2.1 in . And also it is pointed out that Assumption 4 is weaker than the condition (5) for in .
For be an -linear Calderón-Zygmund operator, and with and , Lerner et al.  proved that and bounded from to and Pérez et al.  extended the result to when all , in the case of the endpoint, that is, some , weak type estimates have been established; for some details refer [2, 7]. To obtain the same results for the multilinear singular integral operators in - with kernel satisfying Assumption 4, some authors have done so much work. Duong et al.  obtained that maps to , where with . Grafakos et al.  proved that maps to where all and , and maps to with some . For with , Anh and Duong  established that are of boundedness from to ; after that, Chen and Wu  extended the results of Lerner et al.  and Pérez et al.  to the multilinear singular integral operators in - without the endpoint case.
The following relationship with the above three maximal functions is easy to check:
Let , and . We define the following multilinear maximal functions:
We have that
The following statements are our main results.
Theorem 6. Let , and . Let , , and for some ( depending only on ), if all , then and are bounded from to , and or else, bounded from to .
Corollary 7. Under the same assumptions as in Theorem 6. are bounded from to or .
Theorem 8. Assume that is a multilinear operator in - with kernel satisfying Assumption 4. Let , , with , and . Then we have the following:(i)when all , there exists a constant such that (ii)when some , there exists a constant such that where .
Theorem 9. Assume that is a multilinear operator in - with kernel satisfying Assumption 4. Let , , and with = and and . Then, there exists a constant such that
Following , for positive integers and with , we denote by the family of all finite subsets of of different elements, where we always take if . For any , the associated complementary sequence is given by with the convention . Given an -tuple of functions and , we also use the notation for the -tuple obtained from given by . Similar to , we define for in - and in , the th order iterated commutator that is, formally Clearly, when , and when . We have the following general forms of Theorem 9 without the proof.
Theorem 10. Assume that is a multilinear operator in - with kernel satisfying Assumption 4. Let , , , and with and and . Then, there exists a constant such that
2. Some Definitions and Results
In this section, we introduce some definitions and results used be later on.
Definition 11 ( weights). A weight is a nonnegative, locally integrable function on . Let ; we call that a weight function that belongs to the class , if there is a constant such that, for any cube , and belongs to the class , if there is a constant such that, for any cube , We denote .
Definition 12 (see ). For exponents , we often write for the number given by and denote by the vector . A multiple weight is said to satisfy the condition if for it holds that when , is understood as .
As remarked in , is strictly contained in ; moreover, in general does not imply for any , but instead where the condition in the case is understood as .
Definition 13 (see ). Let , , and be a weight function on . The weighted Morrey space is define by where The weighted weak Morrey space is defined by where
We say that a weight satisfies the doubling condition, denoting , if there is a constant such that holds for any cube . If with , we know that for all , then .
Lemma 14 (see ). Suppose , then there exists a constant such that for any cube.
Lemma 15 (see ). If , then for any cube , we have where .
Lemma 16 (see ). Suppose , then . Here
From the fact and Lemma 16, we can deduce that .
Lemma 17 (see ). Assume that is a multilinear operator in - with kernel satisfying Assumption 4. Let , with and . Then we have the following:(i) extends to a bounded operators from to if all the exponents are strictly greater than 1;(ii) extends to a bounded operators from to if some exponents are equal to 1.
In either case, the norm of is bounded by , where is a positive constant depending on , and .
Lemma 18 (see ). Assume that is a multilinear operator in - with kernel satisfying Assumption 4. Let with and with , . Then we have the following:(i)there exists a constant such that (ii)if , then there exists a constant such that where .
3. Proof of Theorems
Proof of Theorem 6. Here, we only prove the boundedness of . From , there exists some only depend on such that where is the weighted centered maximal operator. Then by the Hölder inequality, The weak version is a very similar process by the Hölder inequality for the weak spaces. We omit the details.
Proof of Theorem 8. For any , we split where , and ; then
where each term of contains at least one . Write then
From Definition 12, Lemma 17, we can get
The last inequality holds by Lemma 15. For , we first consider the case when . Taking , since and , we get
hence, . By Assumption 4, we have that
For any , then by Assumption 4,
Since , then there is a positive such that
It remains to estimate the terms with for some and . We have
Therefore, we also have
Combining the above estimates and then taking the supermum over all balls in , we have proved the previous part of Theorem 8.
Next, we turn to complete the proof of the weak inequality. For any , we can write By Lemmas 17 and 15, we can easily check that From the proof of (53) and (56), we have the following pointwise estimate: Since at least one , we can assume that such that and others greater than 1. Then, Suppose that ; then we have that therefore, Taking the supremum over all balls and all , we complete the proof of Theorem 6.
Proof of Theorem 9. We will show the proof for because the proof for is very similar but easier. Moreover, for simplicity of the expansion, we only present the case .
For any cube , we also split as with and . Then it remains only to verify the following inequalities: From Lemma 19, Lemma 15, and Hölder's inequality, we can get Since and are symmetric, we only estimate . Taking , can be divided into four part: From the proof of Theorem 8 we know that, for any , Applying (67), Hölder's inequality and Lemma 16, we have The last inequality is obtained by the property of : there is a constant such that For , by the Assumption 4, Lemma 15, and Lemma 16, it follows that Hölder's inequality and Lemma 16 tell us that Similarly, we also have that