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

Michael Ruzhansky,Durvudkhan Suragan

doi:10.1016/j.aim.2016.12.013

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.

Highlights

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
We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green’s first and second formulae on homogeneous Carnot groups
The sub-Laplacian L is a left invariant homogeneous hypoelliptic differential operator and it is known that L is elliptic if and only if the step of G is equal to 1

Summary

Preliminaries

There are several equivalent definitions of homogeneous Carnot groups. The sub-Laplacian L is a left invariant homogeneous hypoelliptic differential operator and it is known that L is elliptic if and only if the step of G is equal to 1. Consider the following second order hypoelliptic differential operator based on the matrix A and the vector fields {X1, ..., XN1 }, given by ak,j XkXj. For instance, in the Euclidean case (N1 = N ), that is, for G = (RN , +), the constant coefficient second order operator of elliptic type is transformed into. Let G be a free homogeneous Carnot group, and let A be a given positive-definite symmetric matrix. Let X = {X1, ..., XN1 } be left-invariant vector fields in the first stratum of the free homogeneous Carnot group G with the corresponding sub-Laplacian (2.1). See [44] for analogues of such constructions on compact Lie groups

Sub-Laplacian Green’s formulae and their consequences

Traces and Kac’s problem for the sub-Laplacian

Powers of the sub-Laplacian

Refined Hardy inequality and uncertainty principles

Background