A journey with the integrated $\gamma$2 criterion and its weak forms

TL;DR

This paper explores extensions and applications of the $\Gamma$2 integrated criterion, introducing weak forms and demonstrating their equivalence to weak Poincaré inequalities, with relevance to log-concave measures.

Contribution

It introduces general weak versions of the $\Gamma$2$ criterion and establishes their equivalence to weak Poincaré inequalities, expanding the theoretical framework.

Findings

Weak versions are equivalent to weak Poincaré inequalities

Extensions applicable to log-concave measures

Provides new tools for studying measure concentration

Abstract

As the title indicates this paper will describe several extensions and applications of the $\Gamma$2 integrated criterion introduced by M. Ledoux following ideas of B. Hellffer. We introduce general weak versions and show that they are equivalent to the weak Poincar{\'e} inequalities introduced by M. R{\"o}ckner and F. Y. Wang. We also discuss special weak versions appropriate to the study of log-concave measures and log-concave perturbations of product measures.