Potential operators, maximal functions, and generalizations of A∞

Abstract

We derive weighted norm estimates which relate integral operators of potential type (fractional integrals) to corresponding maximal operators (fractional maximal operators). We also derive norm estimates for the maximal operators. The conditions that we impose on the weights involve A ∞ conditions of “content type” which are weaker than the usual A ∞ condition. The analysis is carried out in the context of spaces of homogeneous type.