A version of one-sided weighted Morrey space is introduced. The boundedness of some classical one-sided operators in harmonic analysis and PDE on these spaces are discussed, including the Riemann-Liouville fractional integral.

The reasons to study one-sided operators involve not only the generalization of the theory of two-sided operators but also the requirements of ergodic theory [

The study of one-sided spaces emerged naturally alongside the study of one-sided operators. In one previous study, the authors studied one-sided BMO spaces associated with one-sided sharp functions and their relationship to good weights for the one-sided Hardy-Littlewood maximal functions [

A version of one-sided weighted Morrey spaces and Campanato spaces is introduced in this paper. The boundedness of some one-sided operators and its effects on these spaces are investigated. First recall some definitions of the classical Campanato spaces and Morrey spaces.

Let

The study of weighted estimates and their effects on these spaces is important to harmonic analysis. Weighted inequalities arise naturally in Fourier analysis, but their use is best justified by the variety of applications in which they appear. For example, the theory of weights plays an important role in the study of boundary value problems inherent in Laplace's equations on Lipschitz domains. Many authors are interested in the study of the events that occur when the weight function belongs to one of the Muckenhoupt classes. Let

The study of weights for one-sided operators is motivated by their natural emergence in harmonic analysis. For example, certain measures are required when the one-sided Hardy-Littlewood maximal operators [

Let

Similar results can be obtained for the left-hand-side operator by changing the condition

A function

The one-sided

Let

Also, a result concerning the converse of Theorem

In addition to singular integral operators, fractional integral operators also play an important role in harmonic analysis. The problem of fractional derivation was an early impetus to study fractional integrals [

Let

The weight conditions

The one-sided Campanato space and one-sided Morrey space can now be introduced.

Let

When

When

Case

Let

A standard calculation shows that

Section

Throughout this paper, for

In this section, the boundedness of the one-sided operators mentioned in Section

Let

Theorem

For the fractional case, the following is true.

Let

First, some basic propositions of one-sided weight classes are selected for use in the analysis.

(a) If

(b)

(c) If

(d)

According to the definitions of

Suppose

Proof is given only for

If

Let

Like the one-sided doubling condition, the following proposition also plays an important role in the present arguments.

Let

For the proof of (a), we first claim that

The proof of (b) is a byproduct of (a) and the fact that

The proof of (a) is given first. Because

Theorem

For the term

We begin with the proof for

By Theorem

The proof of (b) is a reprise of the argument given in the proof of Theorem

The method used in the proof of Theorem

A class of more general one-sided operators that do not necessarily have convolution kernels can now be studied. Let

For the fractional case, the corresponding size condition can be introduced:

It is easy to confirm that the condition (

Let

Let

Theorems

It is sufficient to show that there exists

In view of (

An argument similar to that used in the proof of Theorem

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

The authors cordially thank the referees for their careful reading and helpful comments. This work was partially supported by NSF of China (Grants nos. 11301249, 11271175, and 1171345), NSF of Shandong Province (Grant no. ZR2012AQ026), and AMEP of Linyi University.