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

TL;DR

This paper develops refined inequalities connecting probability density norms, Sobolev norms, and Kantorovich distances on weighted Riemannian manifolds satisfying the $CD(0, \, \infty)$ condition, advancing the mathematical understanding of these relationships.

Contribution

It generalizes and refines interpolation inequalities involving $L^p$ norms, Sobolev norms, and Kantorovich distances under the $CD(0, \, \infty)$ curvature-dimension condition.

Findings

New interpolation inequalities relating $L^p$ norms and Kantorovich distances.

Extensions of inequalities to weighted Riemannian manifolds with $CD(0, \infty)$ condition.

Enhanced bounds for probability densities in geometric analysis.

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.