Anisotropic $L^{2}$-weighted Hardy and $L^{2}$-Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper derives sharp weighted Hardy and Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups and Euclidean spaces with quasi-norms, providing best constants and explicit examples.

Contribution

It introduces new sharp inequalities with optimal constants on homogeneous groups and Euclidean spaces with arbitrary quasi-norms, expanding the theoretical framework.

Findings

Established sharp remainder terms for inequalities on homogeneous groups

Derived new sharp inequalities in Euclidean spaces with quasi-norms

Provided explicit examples illustrating the results

Abstract

We establish sharp remainder terms of the $L^{2}$-Caffarelli-Kohn-Niren\-berg inequalities on homogeneous groups, yielding the inequalities with best constants. Our methods also give new sharp Caffarelli-Kohn-Nirenberg type inequalities in $\mathbb{R}^{n}$ with arbitrary quasi-norms. We also present explicit examples to illustrate our results for different weights and in abelian cases.