Hardy-Littlewood maximal operator on the associate space of a Banach function space

TL;DR

This paper extends a theorem by Lerner to spaces of homogeneous type, showing that the boundedness of the Hardy-Littlewood maximal operator on a Banach function space implies a specific condition on its associate space.

Contribution

It generalizes Lerner's Euclidean result to spaces of homogeneous type, establishing equivalence conditions for maximal operator boundedness on associate spaces.

Findings

Boundedness of M on  implies condition _ on '

Extension of Lerner's theorem to non-Euclidean spaces

Characterization of associate space boundedness in homogeneous type

Abstract

Let $\mathcal{E}(X,d,\mu)$ be a Banach function space over a space of homogeneous type $(X,d,\mu)$. We show that if the Hardy-Littlewood maximal operator $M$ is bounded on the space $\mathcal{E}(X,d,\mu)$, then its boundedness on the associate space $\mathcal{E}'(X,d,\mu)$ is equivalent to a certain condition $\mathcal{A}_\infty$. This result extends a theorem by Andrei Lerner from the Euclidean setting of $\mathbb{R}^n$ to the setting of spaces of homogeneous type.