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

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.