Reverse H\"{o}lder Inequalities for log-Lipschitz Functions
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.
Keywords
Cite
@article{arxiv.2007.14149,
title = {Reverse H\"{o}lder Inequalities for log-Lipschitz Functions},
author = {Emanuel Milman},
journal= {arXiv preprint arXiv:2007.14149},
year = {2020}
}
Comments
13 pages. Added a paragraph in the Introduction on reverse Holder inequalities on the discrete cube. To appear in special issue of Pure Appl. Funct. Anal. dedicated to Louis Nirenberg