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

Dolbeault, Jean (2011), Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion, Mathematical Research Letters, 18, 6, p. 1037-1050. http://dx.doi.org/10.4310/MRL.2011.v18.n6.a1

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00573943/fr/
Date
2011-03
Journal name
Mathematical Research Letters
Volume
18
Number
6
Publisher
International Press
Pages
1037-1050
Publication identifier
Metadata
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Author(s)
Dolbeault, Jean cc
Abstract (EN)
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.
Subjects / Keywords
extinction; fast diffusion equation; stereographic projection; best constants; duality; extremal functions; Gagliardo-Nirenberg inequality; Onofri's inequality; Sobolev's inequality; logarithmic Hardy-Littlewood-Sobolev inequality; Hardy-Littlewood-Sobolev inequality; Sobolev spaces