Schauder estimates up to the boundary on H-type groups: an approach via the double layer potential

Abstract

We establish the Schauder estimates at the boundary out of the characteristic points for the Dirichlet problem by means of the double layer potential in first Heisenberg H 1 group. Despite its singularity we manage to invert the double layer potential restricted to the boundary thanks to a reflection technique for an approximate operator in H 1 . This is the first instance where a reflection-type argument appears to be useful in the sub-Riemannian setting. k i=1 X 2 i is the sub-Laplacian in a generic Carnot