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Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

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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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Authors and Affiliations

  1. Place de Lattre de Tassigny, Université Paris-Dauphine, 75775, Paris Cédex 16, France

    Jean Dolbeault & Maria J. Esteban

  2. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Michal Kowalczyk

  3. Skiles Building, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Michael Loss

Authors
  1. Jean Dolbeault
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  2. Maria J. Esteban
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  3. Michal Kowalczyk
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  4. Michael Loss
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Correspondence to Jean Dolbeault.

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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

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