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We define and investigate generalized local Morrey spaces and generalized local Campanato spaces, within a context of a general quasimetric measure space. The locality is manifested here by a restriction to a subfamily of involved balls. The structural properties of these spaces and the maximal operators associated to them are studied. In numerous remarks, we relate the developed theory, mostly in the “global” case, to the cases existing in the literature. We also suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets of

A quasimetric on a nonempty set

for every

for every

there is a constant

Given

Two quasimetrics

In what follows, if

Let

Given a function

By a

If

Parallel to the main theory, we shall also develop an alternative theory in the framework of closed balls

The general notion of local maximal operators was introduced in [

Throughout the paper, we use a standard notation. While writing estimates, we use the notation

By defining and investigating generalized local Morrey and Campanato spaces on quasimetric measure spaces, we adapt the general approach to these spaces presented by Nakai [

Let

Clearly, working with a general

Let the system

An alternative way of defining the local sharp maximal operator is

For

Another property to be immediately noted is

Finally, in case of considering maximal operators based on closed balls, we shall use the notations

It may be worth mentioning that the following (local) variant of the Hardy-Littlewood maximal operator,

An interesting discussion of mapping properties of (global) fractional maximal operators in Sobolev and Campanato spaces in measure metric spaces equipped with a doubling measure

The following lemma enhances [

For any admissible

In the noncentered case no assumption on

In the centered case, we use the assumptions imposed on

Exactly, the same argument works for the level set

To relate maximal operators based on closed balls with these based on open balls, we must assume something more on the function

We then have the following.

Assume that (

For every

The proof of (

Given

The following lemma enhances [

Suppose that

The assumption

It is probably worth pointing out that in the setting of

The generalized local Morrey and Campanato spaces in the setting of the given system

Other properties to be observed are the inequality

When

Since

In what follows we shall abuse slightly the language (in fact, we already did it) using in several places the term norm instead of (the proper term) seminorm.

The definition of the generalized local Morrey and Campanato spaces based on closed balls requires using in (

Assume that (

Consider the global case; that is,

If

Recall that a quasimetric measure space

In the framework of a space of homogeneous type

Of course it may happen that

See also [

In the Euclidean setting of

The result that follows compares generalized local Morrey and Campanato spaces for the given system

Let

To prove the first claim, take

Under the assumptions of Proposition

In the case when, in the system

The following example generalizes the situation of equivalency of theories based on the Euclidean balls or cubes mentioned above.

Let

Sawano and Tanaka [

For a parameter

Sawano [

Recently, Liu et al. [

In [

In the final example of this section, we analyse a specific case that shows that, in general, things may occur unexpected.

Take

Consider first

Consider now the case of

In this subsection we suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets of

As it was mentioned in [

Similar indecisions accompany the process of choosing a suitable definition of Morrey and Campanato spaces for a general open proper subset

An alternative way of defining generalized Morrey and Campanato spaces on open proper (not necessarily bounded) subset

Given a parameter

The analogous definitions (and comments associated to them) obey

In what follows, rather than considering a general

The following propositions partially contain [

Let

Let

Consider first the case of (

Considering (

Let

The present proof mimics the one of Proposition

For the sake of completeness, we include an outline of the proof of the first aforementioned property. We shall use the following simple geometrical fact: given

Now, take any

The results of Propositions

Let

We focus on proving the statement concerning the Campanato spaces; the argument for the Morrey spaces is analogous (and slightly simpler). Given a cube

Fix

Clearly, the concept of Morrey and Campanato spaces on open proper subsets of

Boundedness of classical operators of harmonic analysis on Morrey spaces was investigated in a vast number of papers; see, for instance, [

In this section, we assume the system

Let

For the notational convention, let

Take

Consider the global case,

Similarly, if

In the literature, several variants of

Let

Since

García-Cuerva and Gatto proved that [

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

The research was initiated when Krzysztof Stempak visited the Department of Mathematics of Zhejiang University of Science and Technology, China, in April 2013. He is thankful for the warm hospitality he received. The authors would like to thank the referees for their careful comments. The research of Krzysztof Stempak is supported by NCN of Poland under Grant 2013/09/B/ST1/02057. The research of Xiangxing Tao is supported by NNSF of China under Grants nos. 11171306 and 11071065.