论文标题
Carnot组的高阶边界Schauder估算
Higher order Boundary Schauder Estimates in Carnot Groups
论文作者
论文摘要
D. Jerison在他的1981年开创性研究中表明，在海森伯格集团$ \ mathbb h^n $的特征边界附近，Schauder总体上没有估算出了显着的负面现象。从积极的一面来看，通过适应傅立叶和微局部分析的工具，他基于非异端Folland-SteinHölder类，在边界的非特征部分开发了Schauder理论。另一方面，1976年的罗斯柴尔德（Rothschild）和斯坦因（Stein）在举重定理上的著名作品确立了分层的nilpotent Lie群体（如今被称为Carnot群体）的核心地位，但在霍曼德运营商的分析中，但到现在，目前，杰里森（Jerison）在这些次级居民中尚无这些以下的杰里森（Jerison）在这些次级里曼尼亚（Riemannian）的结果中的结果。在本文中，我们填补了这个空白。我们证明了最佳$γ^{k，α} $（$ k \ geq 2 $）Schauder在$ c^{k，α} $非特征部分的$ c^{k，α}附近，对于$γ^{k-2，α} $扰动carnot群中的水平laplacians。
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 $Γ^{k,α}$ ($k\geq 2$) Schauder estimates near a $C^{k,α}$ non-characteristic portion of the boundary for $Γ^{k-2, α}$ perturbations of horizontal Laplacians in Carnot groups.
