Morrey Spaces for Nonhomogeneous Metric Measure Spaces

and Applied Analysis 3 Using this covering, we obtain μ(kB) 1/p−1/q (∫ B 󵄨󵄨󵄨󵄨f 󵄨󵄨󵄨󵄨 q dμ) 1/q


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 [1][2][3][4][5][6][7][8][9]).The Radon measure  on R  is said to only satisfy the polynomial growth condition, if there exists a positive constant  such that for all  ∈ R  and  > 0, ((, )) ≤   , where  is some fixed number in (0, ] and (, ) = { ∈ R  : | − | < }.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 wellknown 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 [11][12][13][14][15][16][17]).
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 (X, ) is said to be geometrically doubling if there exists some  0 ∈ N such that, for any ball (, ) ⊂ X, there exists a finite ball covering {(  , /2)}  of (, ) such that the cardinality of this covering is at most  0 .Remark 2. Let (X, ) be a metric space.In [10], Hytönen showed that the following statements are mutually equivalent.
Definition 3. A metric measure space (X, , ) is said to be upper doubling if  is a Borel measure on X and there exist a dominating function  : X × (0, ∞) → (0, ∞) and a positive constant   such that, for each  ∈ X,  → (, ) is nondecreasing and It was proved in [14] that there exists a dominating function λ related to  satisfying the property that there exists a positive constant  λ such that λ ≤ ,  λ ≤   , and, for all ,  ∈ X,  > 0 with (, ) ≤ , λ(, ) ≤  λ λ(, ).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  > 0,  = (  ,   ).
It was proved in [10] that if a metric measure space (X, , ) is upper doubling and ,  ∈ (0, ∞) satisfying  >  log 2   =  V , then, for any ball , there exists some  ∈ N ∪ {0} such that    is (, )-doubling.Moreover, let (X, , ) be geometrically doubling,  >   with  = log 2  0 and  a Borel measure on X which is finite on bounded sets.Hytönen [10] also showed that, for -almost every  ∈ X, there exist arbitrary small (, )-doubling balls centered at .Furthermore, the radii of these balls may be chosen to be from  −  for  ∈ N and any preassigned number  > 0. Throughout this paper, for any  ∈ (1, ∞) and ball , the smallest (,   )-doubling ball of the form    with  ∈ N is denoted by B , where In what follows, by a doubling ball we mean a (6,  6 )doubling ball and B6 is simply denoted by B.
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.

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]).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 1/ − 1/ < 0. So the inclusion    (, ) ⊂    (, ) is obvious.
With this theorem in mind, we sometimes omit parameter  in    (, ).Let  = { ⊂ X :  is (6,  6 )-doubling ball}.Now we give an equivalent definition of Morrey space. where This definition and Theorem 9 are analogy of [20].

Maximal Inequalities
In this section we will investigate some maximal inequalities.Now we give the definitions of some maximal operators.
Proof.This Proof is an analogy of [18,29].For every  ∈ X, we write If () >   , there exists a  ∈ N such that We complete the proof of Lemma 15.
Using Lemma 15 and Theorem 14, we have the following theorem.

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 0 <  < 1, 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].
Then   is bounded from   () space to   () space.
For , we have Similarly, we have For every  ∈ X, we take  that satisfies (, 1) 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].
Proof.For all ball (, ), we have Thus we have proved the theorem.

Marcinkiewicz Integral Operator
Firstly, we introduce the definition of Marcinkiewicz integral operator (see [23]).
The Marcinkiewicz integral M() associated with the above kernel  is defined by setting The boundedness on   () has been proved in [23].
Now we extend this result to the Morrey spaces    ().
Using it we have  ≤ (4) The proof of Theorem 23 is completed.