Hardy and Rellich Inequalities for Sums of Squares of Vector Fields

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 L-gauge. There, the 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.