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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 196459, 8 pages
http://dx.doi.org/10.1155/2013/196459
Research Article

Morrey Spaces for Nonhomogeneous Metric Measure Spaces

Department of Mathematics, Xinjiang University, Urumqi 830046, China

Received 6 May 2013; Accepted 14 August 2013

Academic Editor: Vakhtang M. Kokilashvili

Copyright © 2013 Cao Yonghui and Zhou 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

The authors give a definition of Morrey spaces for nonhomogeneous metric measure spaces and investigate the boundedness of some classical operators including maximal operator, fractional integral operator, and Marcinkiewicz integral operators.

1. Introduction

During the past fifteen years, many results from real and harmonic analysis on the classical Euclidean spaces have been extended to the spaces with nondoubling measures only satisfying the polynomial growth condition (see [19]). The Radon measure on is said to only satisfy the polynomial growth condition, if there exists a positive constant such that for all and , , where is some fixed number in and . The analysis associated with such nondoubling measures is proved to play a striking role in solving the long-standing open Painlevé's problem by Tolsa [6]. 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. To unify both spaces of homogeneous type and due to the fact that the metric spaces endow with measures only satisfying the polynomial growth condition, Hytönen [10] introduced a new class of metric measure spaces satisfying both the so-called geometrically doubling and the upper doubling conditions (see Definition 3), which are called nonhomogeneous spaces. Recently, many classical results have been proved still valid if the underlying spaces are replaced by the nonhomogeneous spaces of Hytönen (see [1117]).

In this paper, we give a natural definition of Morrey spaces associated with the nonhomogeneous spaces of Hytönen and investigate the boundedness of some classical operators including maximal operator, fractional integral operator and Marcinkiewicz integrals operator. To state the main results of this paper, we first recall some necessary notion, and notations. The following notions of geometrically doubling and upper doubling metric measure spaces were originally introduced by Hytönen [10].

Definition 1. 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 2. Let be a metric space. In [10], Hytönen showed that the following statements are mutually equivalent.(1) is geometrically doubling.(2)For any and any ball , there exists a finite ball covering of such that the cardinality of this covering is at most , where .(3)For any and any ball contains at most centers of disjoint balls .(4)There exists such that any ball contains at most centers of disjoint balls .

Definition 3. A metric measure space is said to be upper doubling if is a Borel measure on and there exist a dominating function and a positive constant such that, for each , is nondecreasing and

It was proved in [14] that there exists a dominating function related to satisfying the property that there exists a positive constant such that , , and, for all , with , . Based on this, in this paper, we always assume that the dominating function also satisfies it.

The following coefficients for all ball and were introduced in [10] as analogues of Tolsa’s number in [5].

Definition 4. For all balls , let where, as in the above mentioned, and in what follows, for a ball and , .

Definition 5. Let . A ball is called -doubling if .

It was proved in [10] that if a metric measure space is upper doubling and satisfying , then, for any ball , there exists some such that is -doubling. Moreover, let be geometrically doubling, with and a Borel measure on which is finite on bounded sets. Hytönen [10] also showed that, for -almost every , there exist arbitrary small -doubling balls centered at . Furthermore, the radii of these balls may be chosen to be from for and any preassigned number . Throughout this paper, for any and ball , the smallest -doubling ball of the form with is denoted by , where

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

Let and . We define the Morrey space associated with the nonhomogeneous spaces of Hytönen. This is an analogy of [1820].

Definition 6. Let and , as where

Clearly we have and , . If the underlying spaces are replaced by the nonhomogeneous spaces of Tolsa or Euclidean spaces, the definition of Morrey spaces can been seen in [18]. We will prove in Section 2 that the Morrey space is independent of choice of .

In [21], Chiarenza and Frasca showed that the Hardy-Littlewood maximal operator is bounded on the Morrey space. By establishing a pointwise estimate of fractional integrals in terms of the maximal function, they also showed the boundedness of fractional integral operator on Morrey space. If the underlying spaces are replaced by the nonhomogeneous spaces of Tolsa, Sawano and Tanaka also obtained these results in [18]. When the underlying spaces are the nonhomogeneous spaces of Hytönen, these operators have been discussed in Lebesgue space and RBMO space (see [22, 23]).

Main theorems of this paper are stated in each section. The definition of Morrey space and its equivalent definition are shown in Section 2. Section 3 is devoted to the study of maximal operator and fractional maximal operator. Section 4 deals with the fractional integral operator for the nonhomogeneous spaces of Hytönen. In Section 5, we investigate the behavior of the Marcinkiewicz integrals operator. In what follows the letter will be used to denote constants that may change from one occurrence to another.

2. Morrey Space and Its Equivalent Definition

We firstly prove that the definition of Morrey space is independent of the choice of the parameter (see [18, Proposition 1.1]).

Theorem 7. Let ; then .

Proof. This result is a special case of the results in [24, Theorem 1.2]. For the sake of convenience, we provide the details. Let . By the definition of Morrey space, we have where . So the inclusion is obvious.
Let and ball . Exploiting Remark 2(2), where , we have that there exists ball with the same radius such that Using this covering, we obtain That is, . We complete the proof of the theorem.

With this theorem in mind, we sometimes omit parameter in .

Let is -doubling ball}. Now we give an equivalent definition of Morrey space.

Definition 8. Let ; as where This definition and Theorem 9 are analogy of [20].

Theorem 9. Let and ; then .

Proof. We only need to prove that .
For every ball and , let be the largest doubling ball centered at , having radius , . So . By Besicovitch covering lemma, there is a subcollection that covers so that no point belongs to more than of , where only depends on space . We write . Using Remark 2(2), we know, cardinal number of set . For all , we have So

3. Maximal Inequalities

In this section we will investigate some maximal inequalities. Now we give the definitions of some maximal operators.

Definition 10. Let , , , as

In [11, 22, 2527], the boundedness of these maximal operators has been proven in Lebesgue spaces.

Lemma 11. Let , . Then the maximal operators and are bounded on space.

Lemma 12. Let , , , and . Then the maximal operator is bounded from space to space.

Remark 13. When , Lemma 11 also is right.
Now we extend these results to the Morrey spaces.

Theorem 14. If and , then the maximal operators and are bounded on space.

Proof. The proof of the boundedness of has been obtained in [24, 28]. We only prove the boundedness of . For simplicity, we take . Let and , where . Then for every we have From the definitions of and it follows that For , , the simple calculus yields . Thus we have It follows that

We obtain the conclusion of the theorem.

Lemma 15. If , , , and , then

Proof. This Proof is an analogy of [18, 29]. For every , we write . So
For , we have If , there exists a such that . It follows that We complete the proof of Lemma 15.

Using Lemma 15 and Theorem 14, we have the following theorem.

Theorem 16. If , , , and , then operator is bounded from to .

4. Fractional Integral Operator

In this section, we prove the boundedness of fractional integral operator on Morrey space. The definition of fractional integral operator can be seen in [22]. The investigation of fractional integrals on quasimetric measure spaces with nondoubling measure (nonhomogeneous spaces) in Lebesgue spaces was researched in [30, chapter 6].

Definition 17. Let , for all with bounded support, as

In what follows, we assume that the dominating function satisfies where is the dominating function of the measure of in Definition 3. The condition about was first introduced by Bui and Duong in [11] to study the boundedness of commutators of Calderón-Zygmund operators. In [22], the authors obtain the boundedness of . The boundedness of fractional integral operators of other type can be seen in [31, 32].

Lemma 18. Let , , and . Then is bounded from space to space.

Lemma 19. Let , , and . Then

Proof. Let . We write For , we have Similarly, we have
For every , we take that satisfies . Then So we have

Using this lemma and the boundedness of maximal operator, we obtain the following result.

The following proof of Theorem 20 is similar to that of [33].

Theorem 20. Let , , , and , . Then is bounded from space to space.

Proof. For all ball , we have Thus we have proved the theorem.

5. Marcinkiewicz Integral Operator

Firstly, we introduce the definition of Marcinkiewicz integral operator (see [23]).

Definition 21. Let be a locally integrable function on . Assume that there exists a positive constant such that, for all with , The Marcinkiewicz integral associated with the above kernel is defined by setting

The boundedness on has been proved in [23].

Lemma 22. Suppose that is bounded on space for some . Then is bounded on spaces for all .

Now we extend this result to the Morrey spaces .

Theorem 23. Let . If is bounded on space for some , then is bounded on space.

Proof. For every ball , , let , where .
We can estimate For the first term , we have For , we firstly estimate , as For any ball , , , and , we have Similarly, we obtain
That is to say, for all ball and .
Using it we have The proof of Theorem 23 is completed.

Acknowledgments

Cao Yonghui is supported by the National Natural Science Foundation of China (Grant no. 11261055) and by the National Natural Science Foundation of Xinjiang (Grant no. 2011211A005). Zhou Jiang is supported by the National Science Foundation of China (Grant nos. 11261055 and 11161044) and the National Natural Science Foundation of Xinjiang (Grant nos. 2011211A005 and BS120104).

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