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

In the euclidean space of dimension $d\ge3$, 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 similar result also holds in dimension $d=2$ with Sobolev and Hardy-Littlewood-Sobolev inequalities replaced respectively by a variant of Onofri’s inequality and by the logarithmic Hardy-Littlewood-Sobolev inequality, while the flow is determined by a super-fast diffusion equation.

By considering second derivatives in time of the entropy functional along the flow of the fast diffusion equation, we obtain an improvement of Sobolev’s inequality in terms of the entropy. However, for integrability reasons, the method is restricted to $d\ge5$.

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Published 20 February 2012