Chapter 12 Hardy and Rellich Inequalities for Sums of Squares of Vector Fields

  • Michael Ruzhansky
  • Durvudkhan Suragan
Open Access
Part of the Progress in Mathematics book series (PM, volume 327)


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.

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Authors and Affiliations

  • Michael Ruzhansky
    • 1
    • 2
    • 3
  • Durvudkhan Suragan
    • 4
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Department of Mathematics: Analysis, Logic and Discrete MathematicsGhent UniversityGhentBelgium
  3. 3.School of Mathematical SciencesQueen Mary University of LondonLondonUK
  4. 4.Department of MathematicsNazarbayev UniversityAstanaKazakhstan

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