On Hardy and Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper develops enhanced Hardy and Caffarelli-Kohn-Nirenberg inequalities by incorporating nonlocal, nonconvex functionals, advancing the mathematical understanding of these inequalities and their applications in topology and Sobolev space characterizations.

Contribution

It introduces improved inequalities using nonlocal nonconvex functionals, extending classical results with novel analytical tools.

Findings

Established new inequalities with nonlocal functionals

Connected inequalities to topological degree estimates

Provided characterizations of Sobolev spaces

Abstract

We establish improved versions of the Hardy and Caffarelli-Kohn-Nirenberg inequalities by replacing the standard Dirichlet energy with some nonlocal nonconvex functionals which have been involved in estimates for the topological degree of continuous maps from a sphere into itself and characterizations of Sobolev spaces.