Weighted norm inequalities for the maximal operator on L^p(·) over spaces of homogeneous type

Abstract

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