Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion

Details Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion

2011-03-06

arxiv.org/abs/1103.1145v1

Mathematics

Analysis of PDEs

Scientific paper

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 $d\ge3$. The main consequence is an improvement of Sobolev's inequality when $d\ge5$, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension $d=2$, 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.

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