The singular integral operator and its commutator on weighted Morrey spaces

TL;DR

This paper investigates the boundedness of singular integral operators and their commutators on weighted Morrey spaces, revealing unique phenomena not explained by classical $A_q$-theory and exploring properties of related maximal operators.

Contribution

It provides necessary and sufficient conditions for boundedness on weighted Morrey spaces and analyzes the behavior of commutators and maximal operators in this context.

Findings

Classical $A_q$-theory does not fully explain boundedness in Morrey spaces.

Necessary and sufficient conditions for boundedness are established.

Differences between maximal and singular integral operators are clarified.

Abstract

In this paper, we give necessary conditions and sufficient conditions respectively for the boundedness of the singular integral operator on the weighted Morrey spaces. We observe the phenomenon unique to the case of Morrey spaces; the $A_q$-theory by Muckenhoupt and Wheeden does not suffice. We also discuss the boundedness of the commutators. The difference between the maximal operator and the singular integral operators will be clarified. Further, the property of the sharp maximal operator is investigated.