Weighted Norm Inequalities for the Maximal Operator on Variable Lebesgue Spaces Over Spaces of Homogeneous Type

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 $L\backslash pp$. We prove that the variable Muckenhoupt condition $\backslash App$ is necessary and sufficient for the strong type inequality if $\backslash pp$ satisfies log-H"older continuity conditions and . Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved in [14].