Sobolev-Kantorovich inequalities under $CD(0, \infty)$ condition
  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. 
V. I. Bogachev, A.V. Shaposhnikov, F.-Yu. Wang
math.PR math.DG math.FA