Reverse H\"{o}lder Inequalities for log-Lipschitz Functions

TL;DR

This paper explores reverse H"{o}lder inequalities for log-Lipschitz functions, introducing weaker conditions based on Transport-Entropy inequalities and concentration, expanding the understanding of these inequalities in probability spaces.

Contribution

It presents a novel approach using Transport-Entropy inequalities to derive reverse H"{o}lder inequalities for log-Lipschitz functions, including approximately log-Lipschitz functions, under weaker assumptions.

Findings

Weaker conditions than log-Sobolev inequalities can yield reverse H"{o}lder inequalities.

Transport-Entropy inequalities are effective in handling approximately log-Lipschitz functions.

Comparison with Poincaré inequality scenarios highlights different assumptions needed.

Abstract

Reverse H\"{o}lder inequalities for a class of functions on a probability space constitute an important tool in Analysis in Probability. After revisiting how a (modified) log-Sobolev inequality can be used to derive reverse H\"{o}lder inequalities for the class of log-Lipschitz functions, we obtain a weaker condition using general Transport-Entropy inequalities, which can also handle approximately log-Lipschitz functions. In its weakest form, the condition degenerates to the assumption of satisfying a concentration inequality. We compare this with a scenario in which the underlying space only satisfies a Poincar\'e inequality.