Optimal functional inequalities for fractional operators on the sphere and applications

Jean Dolbeault (1) , An Zhang (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is devoted to optimal functional inequalities for fractional Laplace operators on the sphere. Based on spectral properties, subcritical inequalities are established. Their consequences for fractional heat flows are considered. These subcritical inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities. Their optimal constants are determined by a spectral gap. In the subcritical range, the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. We also consider inequalities which interpolate between fractional logarithmic Sobolev and fractional Poincaré inequalities. Finally, weighted inequalities of Caffarelli-Kohn-Nirenberg type involving the fractional Laplacian are obtained in the Euclidean space, using a stereographic projection and scaling properties.

Mots clés: Hardy-Littlewood-Sobolev inequality; fractional Sobolev inequality; fractional logarithmic Sobolev inequality; spectral gap; fractional Poincaré inequality; fractional heat flow; subcritical interpolation inequalities on the sphere; stereographic projection; Euclidean fractional Caffarelli-Kohn-Nirenberg inequalities

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, An Zhang. Optimal functional inequalities for fractional operators on the sphere and applications. 2016. ⟨hal-01358059v1⟩