A simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fields

Hardy, Rellich and Uncertainty principle inequalities on Carnot Groups

In this paper we prove sharp weighted Hardy-type inequalities on Carnot groups with the homogeneous norm $N=u^{1/(2-Q)}$ associated to Folland's fundamental solution $u$ for the sub-Laplacian $Δ_{\mathbb{G}}$. We also prove uncertainty principle, Caffarelli-Kohn-Nirenberg and Rellich inequalities on Carnot groups.

刻画Carnot群中各类函数空间可移除集的容量

Capacities Characterizing Removable Sets for Various Function Spaces in Carnot Groups

本文研究了Carnot群中各类函数空间的可移除集，通过不同容量和算子刻画其可移除性及Lipschitz函数特性。

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

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 Hörmander 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.