Sobolev's inequality under a curvature-dimension condition
Analysis of PDEs
2021-01-21 v3 Probability
Abstract
In this note we present a new proof of Sobolev's inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous inequality and to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In particular, we elucidate computations used in numerous previous works, starting with Bidaut-V{\'e}ron and V{\'e}ron's 1991 classical work.
Cite
@article{arxiv.2011.07840,
title = {Sobolev's inequality under a curvature-dimension condition},
author = {Louis Dupaigne and Ivan Gentil and Simon Zugmeyer},
journal= {arXiv preprint arXiv:2011.07840},
year = {2021}
}