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引用次数: 0
摘要
在 1981 年的开创性研究中,D.Jerison 在他 1981 年的开创性研究中展示了一个显著的负面现象,即在海森堡群 \(\mathbb {H}^{n}.\) 的特征边界附近一般不存在肖德尔估计。从正面看,通过改编傅里叶分析和微局域分析的工具,他基于非各向同性的福兰-斯坦霍尔德类,在边界的非特征部分发展了肖德尔理论。另一方面,1976 年罗斯柴尔德和斯坦因关于其提升定理的著名研究确立了分层零potent Lie 群(现称为卡诺群)在赫曼德算子分析中的核心地位,但迄今为止,杰里逊的结果在这些亚黎曼环境中还没有已知的对应物。本文将填补这一空白。我们证明了最优的 \(\Gamma ^{k,\α }\) \((k\ge 2)\)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.
Higher order boundary Schauder estimates in Carnot groups
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.
期刊介绍:
Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin.
The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin.
Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.