The method of double layer potential for the Schauder estimates at the boundary in $\mathbb{H}^1$
Giovanna Citti, Gianmarco Giovannardi, Yannick Sire · arXiv (Cornell University) · 2021
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.