Unified weighted Hardy-type inequalities

TL;DR

This paper introduces a unified method to derive sharp Hardy-type inequalities on nilpotent Lie groups, improving classical results and extending to new Caffarelli-Kohn-Nirenberg inequalities in Euclidean and Lie group contexts.

Contribution

It provides a novel unified framework for Hardy inequalities on Lie groups, yielding sharper constants and new inequalities in both Euclidean and non-Euclidean settings.

Findings

Sharp Hardy inequalities with optimal constants on nilpotent Lie groups

Extension of Hardy inequalities to Caffarelli-Kohn-Nirenberg type inequalities

Improvement of classical Euclidean Hardy inequalities

Abstract

We present a unified approach to obtain Hardy-type inequalities in the context of nilpotent Lie groups with sharp constants. The unified methodology employed herein allows for exploration of the sharp Hardy inequalities on various Lie group structures and improves previously known inequalities even in the classical Euclidean setting. By leveraging this framework, as an application, we derive new Caffarelli-Kohn-Nirenberg type inequalities both in the classical Euclidean setting and on general homogeneous Lie groups.