Journal of Function Spaces
Volume 2015 (2015), Article ID 548165, 12 pages
http://dx.doi.org/10.1155/2015/548165

Research Article

Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

Guanghui Lu and Shuangping Tao

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received 10 October 2015; Accepted 30 November 2015

Academic Editor: Henryk Hudzik

Copyright © 2015 Guanghui Lu and Shuangping Tao. 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 metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutator generated by the Marcinkiewicz integral and Lipschitz function . The authors prove that is bounded from the Lebesgue spaces to weak Lebesgue spaces for , from the Lebesgue spaces to the spaces for , and from the Lebesgue spaces to the Lipschitz spaces for . Moreover, some results in Morrey spaces and Hardy spaces are also discussed.

1. Introduction

As we know, the Littlewood-Paley operators are playing an important role in harmonic analysis and PDE. The Marcinkiewicz integral is an essential Littlewood-Paley -function. It is firstly introduced by Marcinkiewicz on and it is conjectured that it is bounded on for any (see [1]). In 1958, Stein gave the higher-dimensional Marcinkiewicz integral (see [2]). Suppose that is homogeneous of degree zero on , for , and has mean value zero on the unit sphere ; Marcinkiewicz is defined by

In [2], Stein proved that if for some , then was bounded on for any and also bounded from to . In 1990, Torchinsky and Wang established boundedness for the commutator generated by and function (see [3]). In 2007, Mo and Lu obtained boundedness of the commutator generated by and function in [4]. For more results about this operator, we refer the reader to see [5–8].

Let be a nonnegative Radon measure on which satisfies the polynomial growth condition; that is, there exist positive constants and such that, for all and , , where . The analysis associated with nondoubling measures is proved to play a striking role in solving the long-standing open Painlevé problem by Tolsa in [9]. Obviously, the nondoubling measure with the polynomial growth condition may not satisfy the well-known doubling condition, which is a key assumption in harmonic analysis on spaces of homogeneous type. Since then, many results from real analysis and harmonic analysis on the classical Euclidean spaces have been extended to the spaces with nondoubling measures satisfying the polynomial growth condition (see [10–14]). The Marcinkiewicz integral operators and commutators have also been discussed widely on the spaces with nondoubling measure (see [15–17]). In 2010, Hytönen introduced a new class of metric measure spaces satisfying both the so-called geometrically doubling and the upper doubling conditions, which are called nonhomogeneous metric measure space in [18]. In particular, in recent years, a lot of classical results have been proved to be still valid if the underlying spaces are replaced by the nonhomogeneous spaces of Hytönen et al. (see [19, 20]). For example, Lin and Yang in [21] obtained the boundedness of Marcinkiewicz integral on nonhomogeneous metric measure spaces. In 2013, Cao and Zhou considered some operators on Morrey spaces over nonhomogeneous metric measure spaces in [22].

In this paper, we will give some estimates for the commutator of Marcinkiewicz integral on the Lebesgue spaces, Lipschitz spaces, spaces, Morrey spaces, and Hardy spaces on nonhomogeneous metric measure spaces.

To state our main results, we first recall some necessary notions and remarks. The notion of upper doubling metric measure spaces was originally introduced by Hytönen [18] (see also [19]) as follows.

Definition 1. A metric measure space is said to be upper doubling, if is Borel measure on and there exist a dominating function and a positive constant such that for each is nondecreasing and, for all and ,

Remark 2. (1) Obviously, a space of homogeneous type is a special case of upper doubling spaces, where one can take the dominating function . Moreover, let be a nonnegative Radon measure on which only satisfies the polynomial growth condition. By taking , we see that is also an upper doubling measure space.
(2) It was proved in [20] that there exists another dominating function related to satisfying the property that there exists a positive constant such that and, for all with ,

Based on this, we always assume that is a nonhomogeneous metric measure spaces with the dominating function that satisfies (3).

The following notion of the geometrically doubling condition is well-known in analysis on metric spaces, which was firstly introduced by Coifman and Weiss in [23, pp. 66-67].

Definition 3. A metric space is said to be geometrically doubling, if there exists some such that, for any ball , there exists a finite ball covering of such that the cardinality of this covering is at most

Remark 4. Let be a metric space. Hytönen showed that the following statements are mutually equivalent (see [18]):

(1) is geometrically doubling.(2)For any and ball , there exists a finite ball covering of such that the cardinality of this covering is at most . Here and in what follows, is as Definition 3 and (3)For every , any ball can contain at most centers of disjoint balls with radius (4)There exists such that any ball can contain at most centers of disjoint balls .

Now we recall the notion of the coefficient introduced by Hytönen (see [18]), which is analogous to the quantity introduced by Tolsa (see [13, 14]).

Definition 5. For any two balls , define

where above and in that follows, for a ball and , is the center of ball .

Remark 6. The following discrete version, , of defined in Definition 5, was first introduced by Bui and Duong in nonhomogeneous metric measure spaces (see [24]), which is more close to the quantity introduced by Tolsa [12] in the setting of nondoubling measures. For any two balls , let be defined by

where and , respectively, denote the radius of the balls and and denote the smallest integer satisfying . Obviously, . As was pointed by Bui and Duong in [24], in general, it is not true that .

In [25], Zhou and Wang gave the notion of Lipschitz function as follows.

Definition 7. Given , the function satisfies a Lipschitz condition of order provided

and we claim that .

Remark 8. Lipschitz condition can also be defined by

See [25].

Let . A ball is called -doubling if . It was proved in [18] that if a metric measure space is upper doubling and , then, for every ball , there exists some such that is -doubling. Moreover, let be geometrically doubling and with and is Borel measure on which is finite on bounded sets. In [18] Hytönen also showed that for -almost every there exist arbitrarily small -doubling balls centered at . Furthermore, the radii of their balls may be chosen to be of the form for and any preassigned number . Throughout this paper, for any and ball , denotes the smallest -doubling ball of the form with , where

In what follows, by a doubling ball we mean a -doubling ball and is simply denoted by

Now we give the definition of Marcinkiewicz integral (see [21]).

Let be a -locally integrable function on . Assume that there exists a positive constant such that, for any with ,

and, for any ,

The Marcinkiewicz integral associated with the above kernel is defined by

Definition 9. Let and satisfy (8) and (9). The commutator of Marcinkiewicz is formally defined by

Obviously, by taking , we see that, in the classical Euclidean space , if

with homogeneous of degree zero and for some , then satisfies (8) and (9) and as in (10) is just the Marcinkiewicz integral introduced by Stein in [2].

In 2007, Hu et al. introduced a Hörmander-type condition in [8], defined as follows:

According to this, we will consider the following condition to replace (13).

Definition 10. Let . The kernel is said to satisfy a Hörmander-type condition if there exist and such that, for any and ,

We denote by the class of kernels satisfying this condition. It is obvious that these classes are nested:

Now we recall the notion of (see [18]) as follows.

Definition 11. Let . A function is said to be in the if there exist a positive constant and a complex number for any balls ,

and that for any balls ,

And is defined to be the infimum of the positive constants in the above inequalities. From [18], it follows that the definition is independent of the choice of .

We begin with recalling some useful properties of in Definition 5 (see [18]).

Lemma 12. (1) For all balls , it holds true that
(2) For any , there exists a positive constant , depending on , such that, for all balls with
(3) For any , there exists a positive constant , depending on , such that, for all balls ,
(4) There exists a positive constant such that, for all balls , In particular, if and are concentric, then .
(5) There exists a positive constant such that, for all balls ,; moreover, if and are concentric, then .

The following characterizations of from [25] play a key role in the proofs of theorems.

Lemma 13. For a function , the following conditions (i), (ii), and (iii) are equivalent.
There exist some constant and a collection of numbers of , one for each ball , such that these two properties hold: For any ball with radius

and, for any ball such that and ,

There is a constant such that

for -almost every and in the support of .
For any given , , there is a constant , such that, for every ball of radius , we have

where and also for any ball such that and

In addition, the quantities , , and with a fixed are equivalent and denoted by .

Remark 14. For , (20) is equivalent to

for any two balls with (see [25]).

The organization of this paper is as follows. In Sections 2 and 3, we study the commutator in the case of and establish that is bounded from the Lebesgue spaces to the Lebesgue spaces for , from the Lebesgue spaces to the spaces for , and from the Lebesgue spaces to the Lipschitz spaces for . In Section 4, we establish the boundedness of commutator of the Marcinkiewicz integral from the Morrey space to the Morrey spaces , from the Morrey spaces to the spaces , and from the Morrey spaces to the Lipschitz spaces . Finally, we establish the boundedness of in for .

Throughout this paper, we use the constant with subscripts to indicate its dependence on the parameters. For a -measurable set , denotes its characteristic function. For any , we denote by its conjugate index; namely, .

2. Boundedness of in Lebesgue Spaces

In this section, we investigate the boundedness of commutator as in (11) in the Lebesgue spaces. The main results are listed as follows.

Theorem 15. Let , and . Suppose that satisfies (8) and (9), is bounded on , and is defined as (11). Then there exists a positive constant such that, for all , one has

Proof. By Minkowski inequality and the kernel condition, we deduce that

where denotes the fractional integral operator defined by

By applying (2) and Theorem   in [26], it is easy to get that there exists a positive constant , such that, for all , one has

This finishes the proof of Theorem 15.

By applying Marcinkiewicz interpolation theorem, it is easy to get the following result.

Corollary 16. Let , and . Suppose that satisfies (8) and (9), is bounded on , and is defined as (11). Then there exists a positive constant such that, for all ,

3. Boundedness of in Lipschitz Spaces

In this section, we investigate the boundedness of commutator as in (11) in the Lipschitz space.

Theorem 17. Let Suppose that and satisfies (8) and condition. Then is bounded from the Lebesgue spaces to the Lipschitz spaces ; namely, there exists a positive constant , such that

Proof. For any balls and in such that satisfies , let

It is easy to see that and are real numbers. By Lemma 13, we need to show that there exists a constant such that

Firstly, let us estimate (31). For a fixed ball and , decompose , where and . Write that

Thus, for and , by Hölder inequality and Lemma 13, we can get

Secondly, let us prove . For any , one has

Now let us estimate . In order to do this, we write

For , we have

Noticing that, for any , , it holds true that . By Hölder inequality and Lemma 13, we can deduce that

As , we have

By a similar argument as in the estimate of , it follows that

As , noting that satisfies the condition , we have by applying Minkowski inequality

Finally, let us estimate . For any , we have

By a similar argument for , we can get that

Combining these estimates, we can conclude

Thus, (31) is proved.
Now, let us proceed to show (32). For any balls with , where is an arbitrary ball and is a doubling ball, denote simply by . Write

By a similar argument for , we have

As ,, it follows that

Therefore, we obtain that

With a similar argument to that in the estimate of , we can get

Thus, (32) is proved and this finishes the proof of Theorem 17.

Theorem 18. Let , and . Suppose that satisfies (9) and condition, is bounded on , and is defined as in (11). Then there exists a positive constant such that, for all bounded functions with compact support, one has

Proof. With a slight change in the proof of Theorem 17, by applying Lemma 13, it is not difficult to get the proof of Theorem 18. We omit the details here.

4. Boundedness of in Morrey Spaces

In this section, we investigate the boundedness for the commutator defined as (11) in the Morrey space . Before stating our main results, we need to recall the definition of the Morrey space.

Definition 19. Let and :

where

It is easy to see that and for If the underlying spaces are replaced by the nonhomogeneous spaces of Tolsa or Euclidean spaces, the definition of Morrey spaces can be seen in [27]. Cao and Zhou proved that the Morrey space is independent of the choice of (see [22]).

Suppose that . Let

In [22], it was proved that

Theorem 20. Let , and . Suppose that satisfies (8) and (9), is bounded on , and is defined as in (11). Then is bounded from the space to the space ; namely, there exists a positive constant such that

Proof. By Minkowski inequality and the kernel conditions, we can get

from the following fact proved in [22]:

so we can easily get the proof of Theorem 20.

The following theorem is adapted from [28].

Theorem 21. Let , and . Suppose that satisfies (9) and condition, is bounded on , and is defined as in (11). Then there exists a positive constant such that, for all ,

Theorem 22. Let , and . Suppose that satisfies (9) and condition, is bounded on , and is defined as in (11). Then there exists a positive constant such that, for all with compact support,

Theorem 23. Let . Suppose that satisfies (9) and condition, is bounded on , and is defined as in (11). Then there exists a positive constant such that, for all ,

Remark 24. Since , the proof of Theorem 21 is similar to Theorem 17. Theorem 22 can be immediately deduced as a conclusion of Theorem 21 in the case of . By applying Lemma 13, with a slight change in the proof of Theorem 21, it is not difficult to show Theorem 23. Thus, we omit the proofs of Theorems 21–23.

5. Boundedness of in Hardy Spaces

This section is devoted to the behavior of in Hardy spaces. In order to define the Hardy space , we need to recall the grand maximal operator introduced in [14].

Definition 25. Consider , defined by

where the notation means that and satisfies

(1),(2) for all ,(3) for all

Based on Theorem  1.2 in [14], we give the following definition of the Hardy space from [12].

Definition 26. The Hardy space is the set of all functions satisfying that and . Moreover, the norm of is defined by

Now we recall the definition of the atomic Hardy space from [20].

Definition 27. Let . A function is called an atomic block if

(1)there exists some ball such that ,(2),(3)for , there are functions supported on cubes and numbers such that and

Then define

Define and as follows:

where the infimum is taken over all possible decompositions of in atomic blocks and is the set of all finite linear combinations of -atoms.

Remark 28. It was proved in [12, 20] that, for each , the atomic Hardy space is independent of the choice of and the spaces and coincide with equivalent norms.

Then we state the main theorem which was adapted from [28] of this section.

Theorem 29. Let , and . Suppose that satisfies (9) and condition. Then is bounded from the Hardy spaces into the Lebesgue spaces ; namely, there exists a positive constant such that for all

Proof. Without loss of generality, we may assume that and as a finite of atomic blocks defined in Definition 27. It is easy to see that we only need to prove the theorem for one atomic block . Let be a ball such that ,, and

where , for , is a real number, , and , for , is a bounded function supported on some balls and it satisfies

Write

By (67), we have

To estimate , write

We choose and such that , and . By Hölder inequality, the fact that , and -boundedness of (see Theorem 15), we have

Denote simply by . Invoking the fact that , therefore, we get

and here we have used the fact that

So we have

With an argument similar to that used in the proof of , we have

Combining the estimates for and yields
For , we have . By Minkowski inequality, we get

For any , we have . It follows that

For , by Minkowski inequality, we conclude that

and here we have used the fact that and .
Now we turn to estimate . Noting that, for any , we have and by Minkowski inequality,

So we have

Combining the estimates for , and yields that

This finishes the proof of Theorem 29.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally to the writing of this paper. All authors read and approved the final paper.

Acknowledgment

This paper is supported by National Natural Foundation of China (Grant no. 11561062).

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