Fractional operators on weighted Morrey spaces

TL;DR

This paper establishes necessary and sufficient conditions for one-weight inequalities involving fractional maximal and integral operators on weighted Morrey spaces, highlighting differences in their behaviors.

Contribution

It provides new characterizations for these inequalities and clarifies the distinct behaviors of fractional maximal and integral operators in Morrey spaces.

Findings

Conditions are established for fractional maximal operator inequalities.

Conditions are established for fractional integral operator inequalities.

Differences between the operators' behaviors are clarified.

Abstract

A necessary condition and a sufficient condition for one weight norm inequalities on Morrey spaces to hold are given for the fractional maximal operator and the fractional integral operator. We clarify the difference between the behavior of the fractional maximal operator and the one of the fractional integral operator which is originated from the structure of Morrey spaces. Both the necessary condition and the sufficient condition are also verified for the power weights.