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Partial Differential Equations: Theory, Control and Approximation pp 225–242Cite as

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.

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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Author information

Authors and Affiliations

  1. Ceremade, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris Cedex 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. Georgia Institute of Technology, Skiles Building, 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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Corresponding author

Correspondence to Jean Dolbeault .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, City University of Hong Kong, Hong Kong, Hong Kong SAR

    Philippe G. Ciarlet

  2. School of Mathematical Sciences, Fudan University, Shanghai, People's Republic of China

    Tatsien Li

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie(Paris VI), Paris, France

    Yvon Maday

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Dolbeault, J., Esteban, M.J., Kowalczyk, M., Loss, M. (2014). Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences. In: Ciarlet, P., Li, T., Maday, Y. (eds) Partial Differential Equations: Theory, Control and Approximation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41401-5_9

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