Some characterizations of the preimage of $A_{\mathrm{\infty }}$ for the Hardy-Littlewood maximal operator and consequences

The purpose of this paper is to give some characterizations of the weight functions $w$ such that $Mw$ is in $A_{\mathrm{\infty }}$. We show that for those weights to be in $A_{\mathrm{\infty }}$ ensures to be in $A_1$. We give a criterion in terms of the local maximal functions $m_{\lambda }$ and we present a pair of applications, among them someone similar to the Coifman-Rochberg characterization of $A_1$ but using functions of the form $(f^{\mathrm{\#}}{)}^{\delta }$ and $(m_{\lambda }u{)}^{\delta }$ instead of $(Mf{)}^{\delta }$.