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Article Dans Une Revue Proceedings of the Royal Society of Edinburgh: Section A, Mathematics Année : 2012

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

Résumé

We consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities and weighted logarithmic Hardy inequalities which have been obtained recently as a limit case of the first ones. 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' logarithmic Sobolev inequality, both without weights, gives a general criterion for such an existence result in some particular cases.
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Dates et versions

hal-00496936 , version 1 (01-07-2010)
hal-00496936 , version 2 (27-01-2012)

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Jean Dolbeault, Maria J. Esteban. Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 2012, 142A, pp.745-767. ⟨hal-00496936v2⟩
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