Sharp Reverse H\"older property for A_\infty weights on spaces of   homogeneous type
  In this article we present a new proof of a sharp Reverse H\"older 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}[\sigma]_{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. 
Tuomas Hyt\"onen, Carlos P\'erez, Ezequiel Rela
math.CA