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 \(1 < p_- \leq p_+ < \infty\). Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved by Cruz-Uribe, Fiorenza and Neugebauer (2012).