From Super Poincar\'e to Weighted Log-Sobolev and Entropy-Cost Inequalities

We derive weighted log-Sobolev inequalities from a class of super Poincar'e inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the ${\log }^{\text{dd}}$-Sobolev inequality with $\text{dd}\in (1,2)$ implies the $L^{2\mathrm{/}(2-\text{dd})}$-transportation cost inequality $$W_{2\mathrm{/}(2-\text{dd})}^{\text{rr}}(f\mu ,\mu {)}^{2\mathrm{/}(2-\text{dd})}\le C\mu (f\log f),\mu (f)=1,f\ge 0$$ 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 ${\mathbb{R}}^d$.