A note on the maximal operator on weighted Morrey spaces

TL;DR

This paper investigates the boundedness of the Hardy--Littlewood maximal operator on weighted Morrey spaces with specific cube families, extending previous results to lacunary sequences of cube centers.

Contribution

It extends the characterization of maximal operator boundedness to Morrey spaces with cube families centered at lacunary sequences, generalizing prior local space results.

Findings

Positive characterization for lacunary-centered cube families

Extension of previous local Morrey space results

Addresses open problem for global Morrey spaces

Abstract

In this paper we consider weighted Morrey spaces ${\mathcal M}_{\lambda, {\mathcal F}}^p(w)$ adapted to a family of cubes ${\mathcal F}$, with norm $$\|f\|_{{\mathcal M}_{\lambda, {\mathcal F}}^p(w)}:=\sup_{Q\in {\mathcal F}}\left(\frac{1}{|Q|^{\lambda}}\int_Q|f|^pw\right)^{1/p},$$ and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy--Littlewood maximal operator on ${\mathcal M}_{\lambda, {\mathcal F}}^p(w)$. In the case of the global Morrey spaces (when ${\mathcal F}$ is the family of all cubes in ${\mathbb R}^n$) this question is still open. In the case of the local Morrey spaces (when ${\mathcal F}$ is the family of all cubes centered at the origin) this question was answered positively in a recent work of Duoandikoetxea--Rosenthal \cite{DR21}. We obtain an extension of \cite{DR21} by showing that the answer is positive when ${\mathcal F}$ is the family of all cubes centered at a sequence of points in ${\mathbb R}^n$ satisfying a certain lacunary-type condition.