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

TL;DR

This paper investigates extremal functions for a class of weighted inequalities, establishing conditions for the existence of optimal functions and comparing constants with classical inequalities.

Contribution

It provides new insights into the existence of extremal functions for weighted Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities, including parameter range analysis.

Findings

Optimal constants are achieved by extremal functions in certain parameter ranges.

Comparison criteria are established between weighted and unweighted inequalities.

Conditions for the existence of extremal functions are identified.

Abstract

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.