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We propose the analogues of boundary layer potentials for the sub-Laplacianon homogeneous Carnot groups/stratified Lie groups and prove continuity resultsfor them. In particular, we show continuity of the single layer potential andestablish the Plemelj type jump relations for the double layer potential. Weprove sub-Laplacian adapted versions of the Stokes theorem as well as ofGreen's first and second formulae on homogeneous Carnot groups. Severalapplications to boundary value problems are given. As another consequence, wederive formulae for traces of the Newton potential for the sub-Laplacian topiecewise smooth surfaces. Using this we construct and study a nonlocalboundary value problem for the sub-Laplacian extending to the setting of thehomogeneous Carnot groups M. Kac's "principle of not feeling the boundary". Wealso obtain similar results for higher powers of the sub-Laplacian. Finally, asanother application, we prove refined versions of Hardy's inequality and of theuncertainty principle.

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