Higher order Boundary Schauder Estimates in Carnot Groups

TL;DR

This paper extends Schauder boundary estimates to Carnot groups, filling a gap in sub-Riemannian analysis by establishing optimal regularity results near non-characteristic boundaries.

Contribution

It provides the first Schauder estimates near boundaries in Carnot groups for higher-order operators, generalizing Jerison's results from the Heisenberg group.

Findings

Proves optimal $ abla^{k,eta}$ Schauder estimates in Carnot groups.

Establishes boundary regularity for perturbations of horizontal Laplacians.

Fills a significant gap in sub-Riemannian boundary regularity theory.

Abstract

In his seminal 1981 study D. Jerison showed the remarkable negative phenomenon that there exist, in general, no Schauder estimates near the characteristic boundary in the Heisenberg group $\mathbb H^n$. On the positive side, by adapting tools from Fourier and microlocal analysis, he developed a Schauder theory at a non-characteristic portion of the boundary, based on the non-isotropic Folland-Stein H\"older classes. On the other hand, the 1976 celebrated work of Rothschild and Stein on their lifting theorem established the central position of stratified nilpotent Lie groups (nowadays known as Carnot groups) in the analysis of H\"ormander operators but, to present date, there exists no known counterpart of Jerison's results in these sub-Riemannian ambients. In this paper we fill this gap. We prove optimal $\Gamma^{k,\alpha}$ ($k\geq 2$) Schauder estimates near a $C^{k,\alpha}$ non-characteristic portion of the boundary for $\Gamma^{k-2, \alpha}$ perturbations of horizontal Laplacians in Carnot groups.