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| HAL : hal-00573943, version 1 |
| arXiv : 1103.1145 |
| Fiche détaillée |  Récupérer au format |
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| Math. Res. Lett. (2012) to appear |
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| Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion |
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| Jean Dolbeault 1 |
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| (2012) |
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| 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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| 1 : | CEntre de REcherches en MAthématiques de la DEcision (CEREMADE) |
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CNRS : UMR7534 – Université Paris IX - Paris Dauphine |
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| Domaine | : | Mathématiques/Equations aux dérivées partielles |
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| Sobolev spaces – Hardy-Littlewood-Sobolev inequality – logarithmic Hardy-Littlewood-Sobolev inequality – Sobolev's inequality – Onofri's inequality – Gagliardo-Nirenberg inequality – extremal functions – duality – best constants – stereographic projection – fast diffusion equation – extinction |
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| hal-00573943, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00573943 | |
| oai:hal.archives-ouvertes.fr:hal-00573943 | |
| Contributeur : Jean Dolbeault | |
| Soumis le : Samedi 5 Mars 2011, 22:37:48 | |
| Dernière modification le : Jeudi 2 Février 2012, 17:41:34 | |
