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

As the title indicates this paper will describe several extensions and applications of the $Γ$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{é} inequalities introduced by M. R{ö}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.

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