Schauder estimates at the boundary for sub-laplacians in Carnot groups

TL;DR

This paper establishes boundary Schauder estimates for sub-Laplacian operators in Carnot groups, extending known results beyond the Heisenberg group using a novel approach that constructs a Poisson kernel from the fundamental solution.

Contribution

It introduces a new method to obtain boundary Schauder estimates in general Carnot groups, overcoming limitations of previous Fourier transform techniques.

Findings

Proves boundary Schauder estimates in Carnot groups

Develops a new approach to construct Poisson kernels

Extends boundary regularity results beyond Heisenberg groups

Abstract

In this paper we prove Schauder estimates at the boundary for sub-Laplacian type operators in Carnot groups. While internal Schauder estimates have been deeply studied, up to now subriemannian estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison, is based on the Fourier transform technique and can not be repeated in general Lie groups. After the result of Jerison no new contribution to the boundary problem has been provided. In this paper we introduce a new approach, which allows to built a Poisson kernel starting from the fundamental solution, from which we deduce the Schauder estimates at non characteristic boundary points.