Generalized Transportation-Cost Inequalities and Applications

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For μ: = e V(x)dx a probability measure on a complete connected Riemannian manifold, we establish a correspondence between the Entropy-Information inequality $\Psi(\mu(f^2{\rm{log}} f^2))\le \mu(|{\nabla}\!\!f |^2)$ and the transportation-cost inequality $W_2(f^2\mu,\mu)\le \Phi(\mu(f^2{\rm{log}} f^2))$ 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.