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arXiv · 2011.07840

Sobolev's inequality under a curvature-dimension condition

Louis Dupaigne·Ivan Gentil·Simon Zugmeyer
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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.

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Louis Dupaigne, Ivan Gentil, Simon Zugmeyer. 2020-11-16. Sobolev's inequality under a curvature-dimension condition. https://arxiv.org/abs/2011.07840

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