Sharp Reverse Hölder property for A_\infty weights on spaces of homogeneous type

Abstract

In this article we present a new proof of a sharp Reverse Hölder Inequality for $A_{\infty }$ weights that is valid in the context of spaces of homogeneous type. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy-Littlewood maximal function involving $A_{\infty }$ constants: |M|_{L^p(w)} \leq c (\frac{1}{p-1} [w]_{A_p}[σ]_{A_\infty})^{1/p}, where $1<p<\infty$, $\sigma =w^{\frac{1}{1-p}}$ and $c$ depends only on the doubling constant of the measure $\mu$ and the geometric constant $\kappa$ of the quasimetric.

Corrected version. Theorem 1.1 has changed and now is a weak sharp reverse Holder property in the case of spaces of homogeneous type. However, the main results derived from that Theorem remain valid as in the previous version, with minor modifications. We thank Andrei Lerner for his observations and corrections