  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 $\mathbb{H}^1$ 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 $\mathbb{H}^1$. This is the
first instance where a reflection-type argument appears to be useful in the
sub-Riemannian setting.
