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Journal of Function Spaces and Applications
Volume 2013, Article ID 394732, 8 pages
http://dx.doi.org/10.1155/2013/394732
Research Article

Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Nondoubling Measures

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Received 22 August 2013; Accepted 5 October 2013

Academic Editor: Yongsheng S. Han

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.

Abstract

Let be a nonnegative Radon measure on satisfying for all , and some fixed and The authors obtain some estimates for the multilinear commutators generated by Marcinkiewicz integral and functions on certain Hardy-type spaces, where space was introduced by Tolsa in (Mathematische Annalen, 2001).

1. Introduction

We will work on the -dimensional Euclidean space with a nonnegative Radon measure which only satisfies the following growth condition that there exists a consist such that where is the open ball centered at some point and having radius . The measure in (1) is not assumed to satisfy the doubling condition which is a key assumption in the analysis on spaces of homogeneous type. In recent years, considerable attention has been paid to the study of function spaces and the boundedness of Calderón-Zygmund operators with nondoubling measures and many classical results have been proved still valid if the underlying measure is substituted by a nondoubling Radon measure as in (1); see [17] and their references. It is worth pointing out that the analysis with nondoubling measures plays an essential role in solving the long-standing Painlevé open problem; see [1].

By a cube we mean a closed cube whose sides are paralleled to the axes and we denote its side length by and its center by . Let and be positive constants such that and ; for a cube , we say that is -doubling if , where denotes the cube concentric with , having side length . In what follows, for definiteness, if and are not specified, by a doubling cube we mean a -doubling cube. Especially, for any give cube , we denote by the smallest doubling cube in the family . For two cubes , set where is the first positive integer such that .

Definition 1. Let be some fixed constant, we say that a function belongs to the space if there is a constant such that and if are doubling cubes, where the supremum is taken over all cubes concentered at some point of and is the mean value of on ; namely, The minimal constant in (3) and (4) is defined to be norm of and is defined by .
In [1] Tolsa introduced the space and showed that the definition of is independent of the choice of numbers . In the sequel we will choose .
To state our results, we need to recall some necessary notation and definitions. Let be a locally integrable function on . Assume that there exists a constant such that for with , and for any , , and , with , for any . The Marcinkiewicz integral associated with the above kernel and the measure as in (1) is defined by In what follows, we will always assume that is bounded on . For and , , we formally define the multilinear commutator by where not only satisfies the size condition (6) but also satisfies a strong Hömander type condition: for all and For , , we denote by .

Remark 2. With the assumption that is bounded on , Hu et al. [8] established that the integral as in (8) with kernel satisfying (6) and (7) is bounded from to , from to , and from to , respectively, where the function space is the subset of function space . At the same time, the authors obtain the boundedness of the commutator , respectively, from to itself for and from the space to , where the kernel satisfies (6) and (10). Such type of multilinear commutators when is a nondoubling measure was introduced by Zhang [9] and its -boundedness has been established.
In this sequel, for , we denote by the family of all finite subsets of with different elements. For any , the complementary sequence is given by . Let be a finite family of locally integrable functions. For all and , we define where and are cubes in and , . With this notation, we write If , we simply write

In this paper, we will deal with the multilinear commutators for Marcinkiewicz integrals on the Hardy-type spaces with nondoubling measures. Now we recall the definition of the atomic block Hardy spaces.

Definition 3. Let , , . Suppose for . A function is called a -atomic block if (a)there exists some cube such that ;(b) for all and ;(c)for , there is function supported on cubes and numbers such that and Then, we denote We say that , if there exist -atomic blocks such that with . The norm of is defined by where the infimum is taken over all the possible decomposition of in -atomic blocks.
From the definition, we have the following properties that for any , and with , and for any and , and for any , and , For some more details about this Hardy-type space, see [4] and its references. It is valuable to point out that, if , the space is just the Hardy space introduced by Tolsa in [1] with equivalent norm, which was proved by the authors in [10].

Let us state our main results as follows.

Theorem 4. Let , , and , for , and let be as in (9) with satisfying (6) and (10). Then is bounded from to weak .

Theorem 5. Let , , , for , and be as in (9) with satisfying (6) and (10). Then is bounded from to .

Remark 6. For , Theorem 4 is just Theorem 3.6 in [8], so we extend the result significantly.

Throughout this paper, denotes a constant that is independent of the main parameters involved but whose value may differ from line to line. We use the constant with subscripts to indicate its dependence on the parameters in the subscripts.

2. Proof of Theorems

To prove our Theorems, we need the following lemma; see [1].

Lemma 7. Let and , for , and . Then there exists a constant such that for any cube ,

Proof of Theorem 4. For each fixed , one has the decomposition where are -atomic blocks as in Definition 3, such that Let be a cube such that ; for each fixed , decompose as where for and is supported on some cube such that and it satisfies the following condition: With the aid of the formulate that for , we can write With weak -boundedness of , Hölder’s inequality, Lemma 7, and (26), it states that where we have used the face derived from [1], Now we turn to estimate the term . Write
To estimate , further decompose Choose ; the Hölder inequality, -boundedness of (see [9]), Lemma 7, (26), and (30) yield that On the other hand, , and ; it is easy to get that is equivalent to and . By (6), the Hölder inequality, the Minkowski inequality, Lemma 7, (26) and (30), one has where we have used the fact that for ,
Now we turn to estimate :
Note that for ,
By (6), the Minkowski inequality, Lemma 7, the Hölder inequality, (26), and (37), we obtain that For , by the generalization of the Hölder inequality, John-Nirenberg’s inequality, (10), and a similar statement in [4, pages 12-13], we have Then, we can get the estimate of : Combining the estimates for and then yields the desired estimate for , which together with the estimate for indicates that
This finishes the proof of Theorem 4.

Proof of Theorem 5. By a standard argument, we only need to verify that for any -atomic block as in Definition 3 with , where is a constant independent of . Let all the notation be the same as in Definition 3. By our choices, , now they satisfy satisfies the following condition:
Write To estimate , we further decompose The Hölder inequality, -boundedness of (see [9]), and (43) tell us From (6), the Hölder inequality, Lemma 7, and (43), it follows that Now we turn to estimate . Involving the vanishing moment of , we can write An argument similar to estimate together with the Hölder inequality, Lemma 7, and (43) tell us that
The estimates for and lead to (42). The proof of Theorem 5 is completed.

Then, we can get the following corollaries directly.

Corollary 8. Let , , and , and let be as in (9) with satisfying (6) and (10). Then is bounded from to weak .

Corollary 9. Let ,  , and , and let be as in (9) with satisfying (6) and (10). Then is bounded from to .

We remark that Corollary 9 is a generalization version of Theorem 1 in [11].

As a special example of Theorem 3.4 in [8], we conclude that the commutator version for Marcinkiewicz integral are bounded from to using a different method.

Theorem HL (see [12]). Let , and , and let T be a sublinear operator that is bounded from to weak and is bounded from to weak ; then there is a positive constant such that for any , measurable function with  ,

From the above Theorem  HL and Theorem 4, one has the following corollary.

Corollary 10. Let ,  ,   and satisfy (6) and (10). Then is bounded from to ; that is,

Conflict of Interests

The authors declare that they have no conflict of interests.

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