Sobolev and Hardy-Littlewood-Sobolev inequalities

Jean Dolbeault (1) , Gaspard Jankowiak (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin-Talenti functions.

Mots clés: Sobolev spaces; Sobolev inequality; Hardy-Littlewood-Sobolev inequality; logarithmic Hardy-Littlewood-Sobolev inequality; Onofri's inequality; Caffarelli-Kohn-Nirenberg inequalities; extremal functions; duality; best constants; stereographic projection; fast diffusion equation

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Gaspard Jankowiak. Sobolev and Hardy-Littlewood-Sobolev inequalities. Journal of Differential Equations, 2014, 257 (6), pp.1689-1720. ⟨hal-00915998v3⟩