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.

Highlights

  • In this chapter, we demonstrate how some ideas originating in the analysis on groups can be applied in related settings without the group structure

  • The aim of this chapter is to show that given the existence of a fundamental solution one can use the ideas from the analysis on groups to establish a number of Hardy inequalities on spaces without group structure

  • By the well-known Hormander sums of the squares theorem from [Hor67], the operator L is locally hypoelliptic if the iterated commutators of the vector fields {Xk}Nk=1 generate the tangent space at each point

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Summary

Chapter 12

We demonstrate how some ideas originating in the analysis on groups can be applied in related settings without the group structure. The aim of this chapter is to show that given the existence of a fundamental solution one can use the ideas from the analysis on groups to establish a number of Hardy inequalities on spaces without group structure. By the well-known Hormander sums of the squares theorem from [Hor67], the operator L is locally hypoelliptic if the iterated commutators of the vector fields {Xk}Nk=1 generate the tangent space at each point. Such operators have been investigated under weaker conditions or without the hypoellipticity property.

Chapter 12. Hardy and Rellich Inequalities for Sums of Squares
12.1 Assumptions
12.1. Assumptions
12.1.1 Examples
12.2. Divergence formula
12.2 Divergence formula
12.3 Green’s identities for sums of squares
Green’s second identity
12.3.1 Consequences of Green’s identities
12.4 Local Hardy inequalities
12.5 Anisotropic Hardy inequalities via Picone identities
12.6 Local uncertainty principles
12.6. Local uncertainty principles and also
12.7 Local Rellich inequalities
12.8 Rellich inequalities via Picone identities
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