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.

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 [

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 [

A metric space

Let

For any

For any

There exists

A metric measure space

It was proved in [

The following coefficients

For all balls

Let

It was proved in [

In what follows, by a doubling ball we mean a

Let

Let

Clearly we have

In [

Main theorems of this paper are stated in each section. The definition of Morrey space and its equivalent definition are shown in Section

We firstly prove that the definition of Morrey space is independent of the choice of the parameter

Let

This result is a special case of the results in [

Let

With this theorem in mind, we sometimes omit parameter

Let

Let

Let

We only need to prove that

For every ball

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

Let

In [

Let

Let

When

Now we extend these results to the Morrey spaces.

If

The proof of the boundedness of

We obtain the conclusion of the theorem.

If

This Proof is an analogy of [

For

Using Lemma

If

In this section, we prove the boundedness of fractional integral operator on Morrey space. The definition of fractional integral operator can be seen in [

Let

In what follows, we assume that the dominating function

Let

Let

Let

For every

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

The following proof of Theorem

Let

For all ball

Firstly, we introduce the definition of Marcinkiewicz integral operator (see [

Let

The boundedness on

Suppose that

Now we extend this result to the Morrey spaces

Let

For every ball

We can estimate

That is to say,

Using it we have

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).