Weighted modular estimates for a generalized maximal operator on spaces of homogeneous type

Abstract

We study weighted modular inequalities for a generalized maximal operator associated to a Young function in the context of spaces of homogeneous type. We prove the equivalence between these inequalities and a Dini-type condition, which involves the function associated to the operator and the functions related to the modular estimates. Particularly we obtain a generalization of a result of Perez and Wheeden (J Funct Anal 181(1):146–188, 2001). In addition we prove a characterization of the A 1-Muckenhoupt class, that extends and improves the corresponding results proved by Kita (Acta Math Hungar 72(4):291–305, 1996; Math Nachr 178:1180–1189, 2005).