On the embedding of 𝐴₁ into 𝐴_{∞}

Rey, Guillermo

doi:10.1090/proc/13087

On the embedding of $A_1$ into $A_\infty$
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by Guillermo Rey PDF

Proc. Amer. Math. Soc. 144 (2016), 4455-4470 Request permission

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-Hölder inequality as an easy corollary.

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