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

(2017) ADVANCES IN MATHEMATICS. 308. p.483-528
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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.
Keywords
Sub-Laplacian, Integral boundary condition, Homogeneous Carnot group, Stratified group, Newton potential, Layer potentials, Hardy inequality, NILPOTENT LIE-GROUPS, HEISENBERG-GROUP, DIFFERENTIAL-OPERATORS, LAPLACIAN, PRINCIPLE, CALCULUS, EQUATION

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Chicago
Ruzhansky, Michael, and Durvudkhan Suragan. 2017. “Layer Potentials, Kac’s Problem, and Refined Hardy Inequality on Homogeneous Carnot Groups.” Advances in Mathematics 308: 483–528.
APA
Ruzhansky, M., & Suragan, D. (2017). Layer potentials, Kac’s problem, and refined Hardy inequality on homogeneous Carnot groups. ADVANCES IN MATHEMATICS, 308, 483–528.
Vancouver
1.
Ruzhansky M, Suragan D. Layer potentials, Kac’s problem, and refined Hardy inequality on homogeneous Carnot groups. ADVANCES IN MATHEMATICS. 2017;308:483–528.
MLA
Ruzhansky, Michael, and Durvudkhan Suragan. “Layer Potentials, Kac’s Problem, and Refined Hardy Inequality on Homogeneous Carnot Groups.” ADVANCES IN MATHEMATICS 308 (2017): 483–528. Print.
@article{8585402,
  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.},
  author       = {Ruzhansky, Michael and Suragan, Durvudkhan},
  issn         = {0001-8708},
  journal      = {ADVANCES IN MATHEMATICS},
  keywords     = {Sub-Laplacian,Integral boundary condition,Homogeneous Carnot group,Stratified group,Newton potential,Layer potentials,Hardy inequality,NILPOTENT LIE-GROUPS,HEISENBERG-GROUP,DIFFERENTIAL-OPERATORS,LAPLACIAN,PRINCIPLE,CALCULUS,EQUATION},
  language     = {eng},
  pages        = {483--528},
  title        = {Layer potentials, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups},
  url          = {http://dx.doi.org/10.1016/j.aim.2016.12.013},
  volume       = {308},
  year         = {2017},
}

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