Sharp interpolation inequalities on the sphere : new methods and consequences

Jean Dolbeault (1) , Maria J. Esteban (1) , Michal Kowalczyk (2) , Michael Loss (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DIM - Depto Ingeniería Matemática
3 School of Mathematics - Georgia Institute of Technology

These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere. We shall address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.

Mots clés: Sobolev inequality; interpolation; Gagliardo-Nirenberg inequalities; logarithmic Sobolev inequality; heat equation

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Maria J. Esteban, Michal Kowalczyk, Michael Loss. Sharp interpolation inequalities on the sphere : new methods and consequences. Chinese Annals of Mathematics - Series B, 2013, 34 (1), pp.99-112. ⟨10.1007/s11401-012-0756-6⟩. ⟨hal-00739140⟩