Abstract
This paper is devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.
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In honor of the scientific heritage of Jacques-Louis Lions
Project supported by ANR grants CBDif and NoNAP, the ECOS project (No. C11E07), the Chilean research grants Fondecyt (No. 1090103), Fondo Basal CMM-Chile, Project Anillo ACT-125 CAPDE and the National Science Foundation (No. DMS-0901304).
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Dolbeault, J., Esteban, M.J., Kowalczyk, M. et al. Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences. Chin. Ann. Math. Ser. B 34, 99–112 (2013). https://doi.org/10.1007/s11401-012-0756-6
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DOI: https://doi.org/10.1007/s11401-012-0756-6
Keywords
- Sobolev inequality
- Interpolation
- Gagliardo-Nirenberg inequality
- Logarithmic Sobolev inequality
- Heat equation