Abstract
In this chapter, we demonstrate how some ideas originating in the analysis on groups can be applied in related settings without the group structure. In particular, in Chapter 7 we showed a number of Hardy and Rellich inequalities with weights expressed in terms of the so-called $$\mathcal{L}$$ -gauge. There, the $$\mathcal{L}$$ -gauge is a homogeneous quasi-norm on a stratified group which is obtained from the fundamental solution to the sub-Laplacian. At the same time, in Chapter 11 we used the fundamental solutions of the sub-Laplacian for the advancement of the potential theory on stratified groups, and in Section 7.3 fundamental solutions for the p-sub-Laplacian and their properties were used on polarizable Carnot groups for the derivation of further Hardy estimates in that setting.