Caffarelli-Kohn-Nirenberg inequalities on Lie groups of polynomial growth

TL;DR

This paper establishes Caffarelli-Kohn-Nirenberg inequalities on Lie groups with polynomial growth, utilizing Lorentz spaces and Carnot-Caratheodory distances to refine classical inequalities in this geometric setting.

Contribution

It extends Caffarelli-Kohn-Nirenberg inequalities to Lie groups of polynomial growth using Lorentz spaces and H"ormander vector fields, providing a refined framework for Sobolev and Hardy inequalities.

Findings

Derived inequalities involving weights as powers of Carnot-Caratheodory distance.

Utilized weak L^p spaces and Lorentz spaces for proofs.

Refined classical Sobolev and Hardy-Sobolev inequalities.

Abstract

In the setting of a Lie group of polynomial volume growth, we derive inequalities of Caffarelli-Kohn-Nirenberg type, where the weights involved are powers of the Carnot-Caratheodory distance associated with a fixed system of vector fields which satisfy the H\"ormander condition. The use of weak $L^p$ spaces is crucial in our proofs and we formulate these inequalities within the framework of $L^{p,q}$ Lorentz spaces (a scale of (quasi)-Banach spaces which extend the more classical $L^p$ Lebesgue spaces) thereby obtaining a refinement of, for instance, Sobolev and Hardy-Sobolev inequalities.