Layer potentials, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups

Michael Ruzhansky, Durvudkhan Suragan

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35 Citations (Scopus)

Abstract

We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.

Original languageEnglish
Pages (from-to)483-528
Number of pages46
JournalAdvances in Mathematics
Volume308
DOIs
Publication statusPublished - Feb 21 2017

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Keywords

  • Hardy inequality
  • Homogeneous Carnot group
  • Integral boundary condition
  • Layer potentials
  • Newton potential
  • Stratified group
  • Sub-Laplacian

ASJC Scopus subject areas

  • Mathematics(all)

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