Some Characterizations of the Preimage of A∞ for the Hardy-Littlewood Maximal Operator and Consequences

Abstract

The purpose of this paper is to give some characterizations of the weight functions w such that Mw∈A∞(Rn). We show that, for these Mw weights, being in A∞ ensures being in A1. We give a criterion in terms of the local maximal functions and we present a pair of applications, one of them similar to the Coifman-Rochberg characterization of A1 but using functions of the form (f#)δ and (mλu)δ instead of (Mf)δ.