Reverse Hölder Inequalities for log-Lipschitz Functions
arXiv:2007.14149
Abstract
Reverse Hölder 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 Hölder 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 Poincaré inequality.
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