Muckenhoupt-type conditions on weighted Morrey spaces

TL;DR

This paper introduces a Muckenhoupt-type condition on weighted Morrey spaces, establishing its necessity and sufficiency for the boundedness of key operators like the maximal and Calderón operators, with implications for weighted inequalities.

Contribution

It defines a new Muckenhoupt-type condition on weighted Morrey spaces and proves its equivalence to boundedness of important operators, extending classical weighted inequality results.

Findings

The condition characterizes boundedness of the maximal operator on weighted Morrey spaces.

It provides a simplified criterion for the Hardy-Littlewood maximal operator on local Morrey spaces.

The condition extends to global Morrey spaces with an additional local $A_p$ condition.

Abstract

We define a Muckenhoup-type condition on weighted Morrey spaces using the K\"othe dual of the space. We show that the condition is necessary and sufficient for the boundedness of the maximal operator defined with balls centered at the origin on weighted Morrey spaces. A modified condition characterizes the weighted inequalities for the Calder\'on operator. We also show that the Muckenhoup-type condition is necessary and sufficient for the boundedness on weighted local Morrey spaces of the usual Hardy-Littlewood maximal operator, simplifying the previous characterization of Nakamura-Sawano-Tanaka. For the same operator, in the case of global Morrey spaces the condition is necessary and for the sufficiency we add a local $A_p$ condition. We can extrapolate from Lebesgue $A_p$-weighted inequalities to weighted global and local Morrey spaces in a very general setting, with applications to many operators.