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

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

2010-07-02

arxiv.org/abs/1007.0309v2

Mathematics

Analysis of PDEs

Proc. Edinburgh A (2012) To appear

Scientific paper

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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