Weighted norm inequalities for the maximal operator on $\Lpp$ over spaces of homogeneous type

TL;DR

This paper establishes necessary and sufficient weighted norm inequalities for the Hardy-Littlewood maximal operator on variable exponent Lebesgue spaces over spaces of homogeneous type, extending Euclidean space results.

Contribution

It generalizes Euclidean space results to spaces of homogeneous type, proving the variable Muckenhoupt condition is both necessary and sufficient under certain regularity conditions.

Findings

Weighted norm inequalities hold under the $ ext{A}_{oldsymbol{p}(oldsymbol{x})}$ condition.

The $ ext{A}_{oldsymbol{p}(oldsymbol{x})}$ condition is necessary and sufficient for strong-type inequalities.

Results extend classical Euclidean space theorems to more general metric measure spaces.

Abstract

Given a space of homogeneous type $(X,\mu,d)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^\pp$. We prove that the variable Muckenhoupt condition $\App$ is necessary and sufficient for the strong type inequality if $\pp$ satisfies log-H\"older continuity conditions and $1 < p_- \leq p_+ < \infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved by Cruz-Uribe, Fiorenza and Neugebuaer (2012).