On the embedding of $A_1$ into $A_\infty$

Abstract

We give a quantitative embedding of the Muckenhoupt class $A_1$ into $A_\infty$. In particular, we show how $\epsilon$ depends on $[w]_{A_1}$ in the inequality which characterizes $A_\infty$ weights: \[ \frac{w(E)}{w(Q)} \leq \biggl( \frac{|E|}{|Q|} \biggr)^\epsilon, \] where $Q$ is any dyadic cube and $E$ is any subset of $Q$. This embedding yields a sharp reverse-H\"older inequality as an easy corollary.