Some Characterizations of the Preimage of \(A_{\infty }\) 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 \(Mw\in A_{\infty }\left( \mathbb{R}^{n}\right) \). We show that, for these \(Mw\) weights, being in \(A_{\infty }\) ensures being in \(A_{1}\). We give a criterion in terms of the local maximal functions \(% m_{\lambda }\) and we present a pair of applications, one of them similar to the Coifman-Rochberg characterization of \(A_{1}\) but using functions of the form \(\left( f^{\#}\right) ^{\delta }\) and \(\left( m_{\lambda }u\right) ^{\delta }\) instead of \(\left( Mf\right) ^{\delta }\).