From super Poincaré to weighted log-Sobolev and entropy-cost inequalities

Abstract

We derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, Talagrand inequalities with super quadratic cost functions are obtained. In particular, on a complete connected Riemannian manifold, we prove that the log δ -Sobolev inequality with δ ∈ ( 1 , 2 ) implies the L 2 / ( 2 − δ ) -transportation cost inequality: W 2 / ( 2 − δ ) ρ ( f μ , μ ) 2 / ( 2 − δ ) ⩽ C μ ( f log f ) , μ ( f ) = 1 , f ⩾ 0 for some constant C > 0 , and they are equivalent if the curvature of the corresponding generator is bounded below. Weighted log-Sobolev and entropy-cost inequalities are also derived for a large class of probability measures on R d .