Sobolev and hardy-littlewood-sobolev inequalities: Duality and fast diffusion

Jean Dolbeault
In the euclidean space of dimension d ≥ 3, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. We investigate how to relate these inequalities using the flow of a fast diffusion equation. Up to a term which is needed for homogeneity reasons, the difference of the two terms in Sobolev's inequality can be seen as the derivative with respect to time along the flow of an entropy functional based on the Hardy-Littlewood-Sobolev inequality. A...
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