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

TL;DR

This paper develops boundary Schauder estimates for the Dirichlet problem on H-type groups using a novel reflection technique to invert the double layer potential, overcoming singularities in the sub-Riemannian setting.

Contribution

It introduces a new reflection-based method to invert the double layer potential, enabling Schauder estimates up to the boundary in sub-Riemannian geometries.

Findings

Successfully inverts the double layer potential on H-type groups.

Establishes Schauder estimates at the boundary away from characteristic points.

First application of reflection techniques in sub-Riemannian analysis.

Abstract

We establish the Schauder estimates at the boundary away from the characteristic points for the Dirichlet problem by means of the double layer potential in a Heisenberg-type group $\mathbb{G}$. 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 $\mathbb{G}$. This is the first instance where a reflection-type argument appears to be useful in the sub-Riemannian setting.