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: \[ \frac{w(E)}{w(Q)} \leq \biggl( \frac{|E|}{|Q|} \biggr)^ε, \] 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.

Minor changes following the suggestions of the anonymous referee

References in corpus (3)

• Sharp Reverse Hölder property for A_\infty weights on spaces of homogeneous type
• Borderline Weak Type Estimates for Singular Integrals and Square Functions
• Extremizers and sharp weak-type estimates for positive dyadic shifts