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Elsevier, Journal of Differential Equations, 6(257), p. 1689-1720, 2014

DOI: 10.1016/j.jde.2014.04.021

International Press, Mathematical Research Letters, 6(18), p. 1037-1050

DOI: 10.4310/mrl.2011.v18.n6.a1

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Sobolev and Hardy–Littlewood–Sobolev inequalities

Journal article published in 2014 by Jean Dolbeault ORCID, Gaspard Jankowiak
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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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 $d\ge3$. 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.