Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 128520, 20 pages

doi:10.1155/2012/128520

Research Article

Weighted Estimates for Maximal Commutators of Multilinear Singular Integrals

Dongxiang Chen and Suzhen Mao

Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China

Received 11 July 2012; Accepted 1 September 2012

Academic Editor: Ti- Xiao

Copyright © 2012 Dongxiang Chen and Suzhen Mao. 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.

Abstract.

This paper is concerned with the pointwise estimates for the sharp function of the maximal multilinear commutators and maximal iterated commutator , generalized by -linear operator and a weighted Lipschitz function . The boundedness and the boundedness are obtained for maximal multilinear commutator and maximal iterated commutator , respectively.

1. Introduction and Notation

The theory of multilinear Calderón-Zygmund singular integral operators,originated from the work of Coifman and Meyers’, has an important role in harmonic analysis. Its study has been attracting a lot of attention in the last few decades. So far, a number of properties for multilinear operators are parallel to those of the classical linear Calderón-Zygmund operators but new interesting phenomena have also been observed. A systematic analysis of many basic properties of such multilinear operators can be found in the articles by Coifman and Meyer [1], Grafakos and Torres [2–4], and Lerner et al. [5]. So we first recall the definition and results of multilinear Calderón-Zygmund operators as well as the corresponding maximal multilinear operators.

Definition 1.1. Let be a multilinear operator initially defined on the m-fold product of Schwartz space and taking values into the space of tempered distributions:

Following [2], we say that is an m-linear Calderón-Zygmund operator if for some , it extends to a bounded multilinear operator from to , where , and if there exists a function , defined off the diagonal in , satisfying

for all supp.

And

for some and all , where .

The maximal multilinear singular integral operator was defined by

where is the smooth truncation of given by

As pointed in [4], is pointwise well defined when with .

The study for the multilinear singular integral operator and its maximal operators attracts many authors’ attention. For maximal multilinear operator , one can see [4] for details. We list some results for as follows.

Theorem A (see [4]). Let and q such that and . Let be an m-linear Calderón-Zygmund operator. Then there exists a constant , such that for all satisfying

where W is the norm of in the mapping : .

Theorem B (see [4]). Let be an m -linear Calderón-Zygmund operator. Then, for all exponents , satisfying , one has

when , one also has

when at least one is equal one. In either cases the norm of is controlled by a constant multiple of .

Definition 1.2 (see [5] (commutators in the th entry)). Given a collection of locally integrable function , we define the commutators of the -linear Calderón-Zygmund operator to be

where each term is the commutator of and in the th entry of , that is

In [6], the following more general iterated commutators of multilinear Calderón-Zygmund operators and pointwise multiplication with functions in BMO were defined and studied in products of Lebesgue spaces, including strong type and weak end-point estimates with multiple weights. That is,

For the operator , when is the Calderón-Zygmund singular integral operator and (the homogeneous Lipschitz spaces), Paluszyński [7] established the boundedness with and . Hu and Gu [8] extended this results to the case: with .

Now we present the definitions of two classes of maximal commutators of multilinear singular integral operators. One is

the other is

where . It is obvious to see that

The main purpose of this paper is to extend the results in [8] to the maximal commutators generated by multilinear singular integrals and functions .

We can formulate our result as following.

Theorem 1.3. Assume that the kernel satisfies (1.3) and (1.4). Let be given numbers satisfying . And assume that maps to . For and let , , , and   with , . Given such that and  , then one has

From (1.15) and (1.16), one can get

If , one can get the following.

Theorem 1.4. Assume that the kernel satisfies (1.3) and (1.4). Let be given numbers satisfying . And assume that maps to . For and let , , and   with , . Set , then one has

From (1.15), one can get

The following theorem states the weighted estimates with two different weights for maximal iterated commutator of multilinear singular integrals.

Theorem 1.5. Assume that the kernel satisfies (1.3) and (1.4). Let be given numbers satisfying . And assume that maps to . Let , , , with  , , and , . Given such that and , then one has

Similarly as Theorem 1.4, one also obtains the unweighted estimates of maximal iterated commutators.

Theorem 1.6. Assume that the kernel satisfies (1.3) and (1.4). Let be given numbers satisfying . And assume that maps to . Let , , , with , , and , . Set , then one has

The rest of this paper is organized as follows. In Section 2, we recall some standard definitions and lemmas. Section 3 is devoted to the proof of our theorems. Throughout this paper, we use the letter to denote a positive constant that varies line to line, but it is independent of the essential variable. For any , the is always used to denote the dual index such that .

2. Preliminaries

A nonnegative function defined on is called weight if it is locally integrable. A weight is said to belong to the Muckenhoupt class , , if there exists a constant such that

for every ball . A weight is said to belong to class if

for every ball . The class can be characterized as .

Many properties of weights can be found in the book [9], we only collect some of them in the following lemma which will be used bellow.

Lemma 2.1.   for ;

(ii) if , then for ;(iii)for , if and only if .

A locally integrable function belongs to the weighted Lipschitz space for , and if

The smallest bound satisfying (1.19) is then taken to be the norm of denoted by . Put .

If , from the definition of , it is obvious to see

where .

The important properties of the weights are the weighted estimates for the maximal function, the sharp maximal function and their variants. One first recalls the maximal function defined by

It is well known that for ,   maps into itself if and only if , see [10].

The sharp maximal function is defined by

One also recalls the variants , and . We denote the weighted fractional maximal operators by

Recall that is the weighted fractional maximal operators, that is

The following lemmas are all from [11].

Lemma 2.2 (Kolmogorov’s inequality). Let be a probability measure space and let then there exists a constant such that for any measurable function

Lemma 2.3. Let and , there exists depending on the constant of such that

for any function for which the left side of the above inequality is finite.

Lemma 2.4. Suppose that , , . If , then there exists a constant such that for any measurable function

Lemma 2.5. Suppose that , , . If , then there exists a constant such that for any measurable function

3. Two Estimates for Maximal Multilinear Commutators

We will prove our theorems in this section. To begin, we prepare another two iterated operators to control the commutators.

Let such that , and satisfying

We define the maximal operators

For simplicity, we denote , and

The kernels of and satisfy conditions (1.3) and (1.4) uniformly in , respectively. And by the same argument in [4], both and have the same weighted estimates to that appeared in Theorems A and B.

It is easy to see that . Moreover,

where

For simplicity, we will only prove for the case . The arguments for the case are similar. For the similarity to the two commutators and , we might as well consider the former. We only consider the former. And we establish the following crucial lemma.

Lemma 3.1. Let and , with , . Let . Then one has

Proof. Without loss of generality, we only consider the case and denote by for convenience. Fix and let , be the average of on , where . To proceed, we decompose , where , . Let be a constant to be fixed along the proof.

Since , we have

For , since , and , by Hölder’ inequality, we have

To estimate the second term . Since , using Kolmogorov’s inequality with , , , and the -boundedness of , we derive that

where we have used the analogous technique in to get the last inequality.

For the term , using the fact for any , , and note that satisfies (1.3) uniformly in , we obtain

For the term , using the fact for any , , and note that satisfies (1.3) uniformly in , and using (2.4), we obtain

For , fix the value of by taking , recall that satisfies (1.4) uniformly in , then we can obtain

where in the last inequality, we use the same computation in the term.

Consequently, combining the estimates of , and , we conclude the proof of Lemma 3.1.

Now we are ready to return to prove Theorem 1.3.

Proof. First, by Lemma 2.1, we have that , and hence . Then by Lemma 2.3, we obtain

For , by Lemma 3.1, we reduce to bound the norm of the right-hand side of (3.6). For the first term, since and taking such that , by Lemma 2.4, and Theorem B(ii), we have

For the second term, we let , and . Then by Lemma 2.4 again, together with Hölder’s inequality, we obtain

We can obtain that

Similarly, we have

Consequently, by the above arguments, we conclude the proof of Theorem 1.3.

Similarly as the proof of Lemma 3.1 and that of Theorem 1.3, we only consider the case and establish the following sharp maximal function for .

Lemma 3.2. Let and ; , and . And let , . Then one has

Proof. Fix and let with . Taking , the average of on , , where . Let be a constant to be fixed along the proof. We split in the following way:

Since , then we have

For the term , since , and , then by Hölder’s inequality, we have

For the term , noting that , we use the facts and , then by Hölder’s inequality and Komolgorov’s inequality (Lemma 2.2) and Theorem B, we have

Similarly, for the term , we have

Now we turn to estimate the last term . To proceed, we denote that , where , . Let , where

We split in the following way:

where

For the term , we choose and use Kolmogorov’s inequality and Theorem B, then we use the same computation as to deduce that

For , by the fact , for any , , and note that satisfies (1.4) uniformly in , then we get that

where we have used the same computation of to gain the last inequality.

Similarly as , we can get the estimates for ,

Now we turn to , by the fact , for any , , and recalling that satisfies (1.4) uniformly in , then we can obtain

Therefore,

Consequently, the estimate for , together with those of , and , can conclude the proof of Lemma 3.2.

Now we return to prove Theorem 1.5.

Proof. Similarly as the proof of Theorem 1.3,

We reduce to bound the norm of the right-hand side of (3.19). We estimate each term as follows. For the first term, since and choosing such that , by Lemma 2.4 and Theorem A and observe that , we obtain

For the second term, since , by Hölder’s inequality and Lemma 2.4, we get

Similarly, we also have

This estimate together with that for finishes the proof of Theorem 1.5.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (1096015, 10871173, and 11261023) and the Natural Science Foundation of Jiangxiang Teacher’s Division (GJJ10397).

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