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

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Title: Layer potentials, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups
Authors: Ruzhansky, M
Suragan, D
Item Type: Journal Article
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.
Issue Date: 2-Jan-2017
Date of Acceptance: 20-Dec-2016
ISSN: 1090-2082
Publisher: Elsevier
Start Page: 483
End Page: 528
Journal / Book Title: Advances in Mathematics
Volume: 308
Copyright Statement: © 2017 The Authors. Published by Elsevier. Open Access funded by Engineering and Physical Sciences Research Council. Under a Creative Commons license (
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
The Leverhulme Trust
Funder's Grant Number: EP/K039407/1
Keywords: math.AP
35R03, 35S15
General Mathematics
0101 Pure Mathematics
Notes: 35 pages; a revised version with some (small) changes to the last section
Publication Status: Published
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences

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