Giovanna Citti: Schauder estimates at the boundary in Carnot groups

Abstract:

We prove Schauder estimates at the boundary for sub-Laplacians in Carnot groups. While sub-Riemannian internal Schauder estimates are well known, the estimates at the boundary were known only in the Heisenberg group. Here we face the problem in general Carnot groups at non characteristic points, building a Poisson kernel in term of the fundamental solution. In collaboration with Baldi and Cupini we provide a first result, related to the Caffarelli Silvestre approach, which can be applied if the boundary is an Hörmander structure. In a second result, obtained in collaboration with Giovannardi and Sire, we remove this assumption in case of H-type groups, using a double potential method.