Hölder continuity of the traces of Sobolev functions to hypersurfaces in Carnot groups and the $\mathcal{P}$-differentiability of Sobolev mappings

We study the behavior of Sobolev functions and mappings on the Carnot groups with the left invariant sub-Riemannian metric. We obtain some sufficient conditions for a Sobolev function to be locally Hölder continuous (in the Carnot–Carathéodory metric) on almost every hypersurface of a given foliation. As an application of these results we show that a quasimonotone contact mapping of class $W^{1,\nu}$ of Carnot groups is continuous, $\mathcal{P}$-differentiable almost everywhere, and has the $\mathcal{N}$-Luzin property.