Higher order boundary Schauder estimates in Carnot groups

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 Hölder 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 Hörmander 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 \(\Gamma ^{k,\alpha }\) \((k\ge 2)\) 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.