  The purpose of this paper is to give some characterizations of the weight
functions $w$ such that $Mw$ is in $A_{\infty}$. We show that for those weights
to be in $A_{\infty}$ 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 $(f^{\#})^{\delta}$
and $(m_{\lambda}u)^{\delta}$ instead of $(Mf)^{\delta}$.
