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Extremal functions for Caffarelli—Kohn—Nirenberg and logarithmic Hardy inequalities

Published online by Cambridge University Press:  10 August 2012

Jean Dolbeault  and
Maria J. Esteban
Jean Dolbeault
Affiliation:
Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France (dolbeaul@ceremade.dauphine.fr; esteban@ceremade.dauphine.fr)
Maria J. Esteban
Affiliation:
Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France (dolbeaul@ceremade.dauphine.fr; esteban@ceremade.dauphine.fr)
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Abstract

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We consider a family of Caffarelli–Kohn–Nirenberg interpolation inequalities and weighted logarithmic Hardy inequalities that were obtained recently as a limit case of the Caffarelli–Kohn–Nirenberg inequalities. We discuss the ranges of the parameters for which the optimal constants are achieved by extremal functions. The comparison of these optimal constants with the optimal constants of Gagliardo–Nirenberg interpolation inequalities and Gross's logarithmic Sobolev inequality, both without weights, gives a general criterion for such an existence result in some particular cases.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 142 , Issue 4 , August 2012 , pp. 745 - 767
Copyright
Copyright © Royal Society of Edinburgh 2012