Characterizations of $A_\infty$ Weights in Ergodic Theory

TL;DR

This paper develops a discrete Calderón-Zygmund decomposition in ergodic theory to characterize $A_ abla$ weights, establishing their equivalence with reverse Hölder inequalities and exploring their properties under doubling conditions.

Contribution

It introduces a discrete Calderón-Zygmund decomposition in ergodic theory and provides new characterizations of $A_ abla$ weights, linking them with reverse Hölder inequalities.

Findings

Characterizations of reverse Hölder's inequality for $A_ abla$ weights.

$A_ abla$ implies reverse Hölder's inequality.

Under doubling condition, reverse Hölder's inequality implies $A_ abla$.

Abstract

We establish a discrete weighted version of Calder\'{o}n-Zygmund decomposition from the perspective of dyadic grid in ergodic theory. Based on the decomposition, we study discrete $A_\infty$ weights. First, characterizations of the reverse H\"{o}lder's inequality and their extensions are obtained. Second, the properties of $A_\infty$ are given, specifically $A_\infty$ implies the reverse H\"{o}lder's inequality. Finally, under a doubling condition on weights, $A_\infty$ follows from the reverse H\"{o}lder's inequality. This means that we obtain equivalent characterizations of $A_{\infty}$. Because $A_{\infty}$ implies the doubling condition, it seems reasonable to assume the condition.