Reverse H\"older Property for strong weights and general measures

TL;DR

This paper establishes dimension-free reverse H"older inequalities for strong weights and measures, extending the understanding of weighted inequalities in harmonic analysis with broad applicability.

Contribution

It introduces new reverse H"older inequalities for strong $A^*_p$ weights and measures, including a multidimensional Riesz's lemma-based proof and results for product measures.

Findings

Dimension-free reverse H"older inequalities for strong weights

Full range local integrability of $A_1^*$ weights proved

Reverse H"older inequality for certain product measures

Abstract

We present dimension-free reverse H\"older inequalities for strong $A^*_p$ weights, $1\le p < \infty$. We also provide a proof for the full range of local integrability of $A_1^*$ 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 H\"older inequality for certain product measures. As a corollary we derive mixed $A_p^*-A_\infty^*$ weighted estimates.