Generalized transportation-cost inequalities and applications

For µ: = eV (x)dx a probability measure on a complete connected Riemannian manifold, we establish a correspondence between the Entropy-Information inequality Ψ(µ(f 2 log f 2)) ≤ µ(|∇f |2) and the transportation-cost inequalityW2(f 2µ, µ) ≤ Φ(µ(f 2 log f 2)) for µ(f 2) = 1, where Φ and Ψ are increasing functions. Moreover, under the curvature-dimension condition, a Sobolev type HWI (Entropy-Cost-Information) inequality is established. As applications, explicit estimates are obtained for the Sobolev constant and the diameter of a compact manifold, which either extend or improve some corresponding known results