  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.
