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Article Dans Une Revue Math. Res. Lett. Année : 2011

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

Résumé

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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hal-00573943 , version 1 (05-03-2011)

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Jean Dolbeault. Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion. Math. Res. Lett., 2011, 18 (6), pp.1037-1050. ⟨hal-00573943⟩
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