Abstract

Suppose , is the iterated commutator of and the -linear Calderón-Zygmund operator . The purpose of this paper is to discuss the boundedness properties of on weighted Herz spaces with general Muckenhoupt weights.

1. Introduction and Results

In recent years, the study of multilinear integrals has received more attention, which is motivated by not only the generalization of the linear theory but also the natural appearance of singular integral theory. The initial work on the class of multilinear Calderón-Zygmund operators was done by Coifman and Meyer in [1] and was later systematically studied by Grafakos and Torres in [2–4]. More results on the multilinear Calderón-Zygmund operators and their commutators can be found in [5–10].

Let be the -dimensional Euclidean space, and let be the -fold product space . We denote by the space of all Schwartz functions on and by its dual space, the set of all tempered distributions on .

We say that a locally integrable function defined away from the diagonal in is a kernel in the class - if it satisfies the size estimatefor some and all with for some . Moreover, assume that for some and all it satisfies the smoothness estimateswhenever

Let be a multilinear operator initially defined on the -fold product of Schwartz spaces and taking values into the space of tempered distributions,We say that is a -linear Calderón-Zygmund operator if, for some and with , it extends to a bounded multilinear operator from into and if there exists a kernel function in the class -, defined away from the diagonal in , satisfyingfor all and are functions with compact support.

Given a locally integrable vector function , the iterated commutator of and the -linear Calderón-Zygmund operator , denoted by , was introduced in [6] and is defined by

To clarify the notation, if is associated in the usual way with a Calderón-Zygmund kernel , then at a formal level

Suppose is an -linear Calderón-Zygmund operator. Let , let with , and let In 2014, Pérez et al. [6] proved that the commutator is bounded from to .

Let and for any Denote for , where is the characteristic function of the set . The following weighted Herz space is introduced by Lu and Yang in [11].

Let , and be two weight functions on . The homogeneous weighted Herz space is defined by where

Obviously, if , then for any Thus, the weighted Herz spaces are generalizations of the weighted Lebesgue spaces.

In 2000, Lu et al. in [12] proved boundedness results for sublinear operators on weighted Herz spaces with general Muckenhoupt weights. Recently, many authors considered the boundedness of operators and their commutators on weighted Herz spaces. Wang in [13] proved that the intrinsic square functions are bounded on weighted Herz spaces and in [14] he also obtained the boundedness of the intrinsic square functions on weighted Herz type Hardy spaces. In [15], Kuang considered the boundedness of generalized Hausdorff operators on weighted Herz spaces; Hu et al. in [16] established the weighted boundedness for the commutator of fractional integral operators on Herz spaces. The authors also have studied the boundedness of the multilinear integrals on weighted (unweighted) Herz spaces in [17–19]. In this paper, we focus on the boundedness of the iterated commutator of and the -linear Calderón-Zygmund operator on weighted Herz spaces. Our main results in this paper are formulated as follows.

Theorem 1. Suppose , is an -linear Calderón-Zygmund operator, , and For , let , let , and let If satisfy then is bounded from into provided that for

2. Some Preliminaries

A weight is a nonnegative, locally integrable function on . The symbol denotes the ball with center , radius and for any , will stand for the ball . For a given weight function and a measurable set , we also denote the Lebesgue measure of by and set weighted measure .

A weight is said to belong to for , if there exists a constant such that, for every ball ,where is the dual of such that The class is defined by replacing the above inequality with

A weight is said to belong to if there are positive numbers and so thatfor all balls and all measurable . It is well known that

By (12), we havefor

The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal function in [20].

Lemma 2 (see [20, 21]). Suppose and the following statements hold.(i)For any , there are positive numbers and such that(ii) for any .(iii)For any , one has .

A locally integrable function is said to be in if where .

Lemma 3 (see [22]). Suppose and . Then, for any and , we have

Now let us recall the definition of multiple weights. For exponents , we write Let , and let with Given , set We say that satisfies the condition if it satisfiesWhen , is understood as

Let , and let By Hölder’s inequality,Supposing , then by (22) and the definition of multiple weights we have

Lemma 4 (see [6]). Let , and let . Then if and only ifwhere and the condition in the case is understood as .

3. Proof of Theorem 1

Without loss of generality, we only prove the theorem for the case . The method can be extended for any -linear case without any essential difficulty.

Let , and let . If , then from (17) we haveIf , then from (18) we have

We have the following conclusions.

Theorem 5. Suppose , is an -linear Calderón-Zygmund operator, and . For , let , and let . If satisfy and , thenwhere , andfor .

Proof. When , then, by the boundedness of from to and , we obtainThis means if .
In the other case, we see thatfor .
TakingthenThusNow, we will estimate each , separately. Applying (29), By Hölder’s inequality and (16),we haveThenBy Lemma 4 we know . If , then by Hölder’s inequality, Lemma 3, and (22) we haveIf , then by Hölder’s and Lemma 3 we haveThusWhen . Then by (38) and (24), we getWhen , by (38) and (25), we getWhen Then by (38) and (24)-(25) we get By symmetry, when , we haveWhen Noting that , thenSimilar to the estimates (43), we can verify (26) in the following case: and Thus we obtainApplying (29) and Hölder’s inequality, By Lemma 2 we know . Then, by Lemma 3, we haveThen, by (46) and (16), we haveNote that, for ,Thus,Similar to the estimates of (44), we getBy symmetry, we also getFinally, it remains to estimate . Applying (29) and Hölder’s inequality,By (46) and (16),ThenThus Now, we give the proof of Theorem 1.
If , by Minkowski’s inequality and Theorem 5, we havefor any , since whenever .
If , note the fact that and then by Theorem 5, the inequality , and Hölder’s inequality, we have Since , by (56), (58) and Hölder’s inequality,It is enough to show that and By the symmetry, we only give the estimate for ConsiderIf , then, by the inequality and the fact that we getNote thatThen, for , we haveSince for , thenThus, we have obtained that holds for any
Let us now turn to estimate the last term . If , thenButThenNote thatThen, for , we haveThis completes the proof of Theorem 1.