Sobolev-Kantorovich inequalities under $CD(0,\infty )$ condition

Abstract

We refine and generalize several interpolation inequalities bounding the $L^p$ norm of a probability density with respect to the reference measure $\mu$ by its Sobolev norm and the Kantorovich distance to $\mu$ on a smooth weighted Riemannian manifold satisfying $CD(0,\infty )$ condition.