Self-improving boundedness of the maximal operator on quasi-Banach lattices over spaces of homogeneous type

TL;DR

This paper establishes a self-improvement property of the Hardy--Littlewood maximal operator on quasi-Banach lattices in spaces of homogeneous type, extending previous Euclidean space results to more general metric measure spaces.

Contribution

It generalizes the boundedness criterion for maximal operators from Euclidean spaces to spaces of homogeneous type using dyadic cube techniques.

Findings

Proves self-improvement of maximal operator boundedness in this setting.

Extends results to variable Lebesgue spaces over spaces of homogeneous type.

Utilizes dyadic cube structures for the proof.

Abstract

We prove the self-improvement property of the Hardy--Littlewood maximal operator on quasi-Banach lattices with the Fatou property in the setting of spaces of homogeneous type. Our result is a generalization of the boundedness criterion obtained in 2010 by Lerner and Ombrosi for maximal operators on quasi-Banach function spaces over Euclidean spaces. The specialty of the proof for spaces of homogeneous type lies in using adjacent grids of Hyt\"onen--Kairema dyadic cubes and studying the maximal operator alongside its dyadic version. Then we apply the obtained result to variable Lebesgue spaces over spaces of homogeneous type.