Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion
Analysis of PDEs
2012-06-08 v1
Abstract
In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension . The main consequence is an improvement of Sobolev's inequality when , which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension , Onofri's inequality plays the role of Sobolev's inequality and can also be related to its dual inequality, the logarithmic Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equation.
Keywords
Cite
@article{arxiv.1103.1145,
title = {Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion},
author = {Jean Dolbeault},
journal= {arXiv preprint arXiv:1103.1145},
year = {2012}
}