Hardy-Littlewood maximal operator on reflexive variable Lebesgue spaces over spaces of homogeneous type

TL;DR

This paper extends the characterization of the boundedness of the Hardy-Littlewood maximal operator from Euclidean spaces to spaces of homogeneous type, establishing a duality criterion in reflexive variable Lebesgue spaces.

Contribution

It generalizes a known Euclidean result to a broader setting of spaces of homogeneous type, providing a duality condition for boundedness.

Findings

Boundedness of the maximal operator is equivalent on a reflexive variable Lebesgue space and its dual.

Extends Lars Diening's Euclidean result to spaces of homogeneous type.

Provides a necessary and sufficient condition for boundedness in this setting.

Abstract

We show that the Hardy-Littlewood maximal operator is bounded on a reflexive variable Lebesgue space $L^{p(\cdot)}$ over a space of homogeneous type $(X,d,\mu)$ if and only if it is bounded on its dual space $L^{p'(\cdot)}$, where $1/p(x)+1/p'(x)=1$ for $x\in X$. This result extends the corresponding result of Lars Diening from the Euclidean setting of $\mathbb{R}^n$ to the setting of spaces of homogeneous type $(X,d,\mu)$.