Reverse Hölder Property for Strong Weights and General Measures

Abstract

We present dimension-free reverse Hölder inequalities for strong $A^{\ast}_p$ weights, $1 \le p < \infty$. We also provide a proof for the full range of local integrability of $A^{\ast}_1$ weights. The common ingredient is a multidimensional version of Riesz’s “rising sun” lemma. Our results are valid for any nonnegative Radon measure with no atoms. For $p = \infty$, we also provide a reverse Ho ̈lder inequality for certain product measures. As a corollary we derive mixed $A^{\ast}_p − A^{\ast}_{\infty}$ weighted estimates.