We
present a new approach to Schauder estimates at the boundary for
sub-Laplacian operators
in Carnot groups. While internal Schauder estimates have been deeply
studied, sub-Riemannian estimates at the boundary were known only in the
Heisenberg group. The proof in the Heisenberg setting, due to Jerison
(1981), is based on the Fourier transform technique
which can not be repeated in general Lie groups. In collaboration with
Baldi and Cupini we introduced a new approach,
and build a Poisson kernel starting from the fundamental solution,
under a
geometric assumption on the boundary. In collaboration with G.
Giovannardi and Y. Sire we removed this assumption in H-type groups.