Lower bounds for uncentered maximal functions on metric measure space

TL;DR

This paper establishes that uncentered Hardy-Littlewood maximal operators on certain metric measure spaces have uniform lower bounds greater than one in $L^p$, under mild conditions, highlighting the importance of measure continuity.

Contribution

It proves lower bounds for uncentered maximal functions in metric measure spaces with the Besicovitch property, extending classical results and emphasizing the necessity of measure continuity.

Findings

Lower bounds for maximal operators are strictly greater than 1.

The bounds are independent of the measure under mild conditions.

Counterexamples show the importance of measure continuity.

Abstract

We show that the uncentered Hardy-Littlewood maximal operators associated with the Radon measure $\mu$ on $\mathbb{R}^d$ have the uniform lower $L^p$-bounds (independent of $\mu$) that are strictly greater than $1$, if $\mu$ satisfies a mild continuity assumption and $\mu(\mathbb{R}^d)=\infty$. We actually do that in the more general context of metric measure space $(X,d,\mu)$ satisfying the Besicovitch covering property. In addition, we also illustrate that the continuity condition can not be ignored by constructing counterexamples.