Problems in harmonic analysis and several complex variables

调和分析中的问题和几个复变量

• 批准号: 0457500
• 负责人: Kenneth Koenig
• 金额: $ 1.86万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0457500&HistoricalAwards=false
• 关键词: Problems; harmonic; analysis; several; complex

DMS - 0400505PI: Kenneth KoenigUniversity of ChicagoProblems in harmonic analysis and several complex variablesAbstract Many of the fundamental questions in complex analysis are closely related to the regularity properties of solutions to the Cauchy-Riemann equations. The principal investigator plans to study Lp-Sobolev and Holder regularity for the interior and tangential Cauchy-Riemann complexes on smoothly bounded, pseudoconvex domains in Cn when there is at least some gain of regularity in the standard Sobolev spaces (e. g. for finite-type domains, in any dimension). In particular, he will focus on the following basic problems: (1) Determination of the precise relation, up to smoothing operators, between the Bergman and Szego projections; (2) Transference of Sobolev and Holder estimates from the interior to the boundary of such domains (and vice versa), and connections to nonisotropic maximal hypoellipticity; (3) Kernel estimates for the Neumann operator and for the canonical solutions to the Cauchy-Riemann operator, based on properties of certain generalized singular integrals on the boundary. The principal investigator anticipates that his work will also provide insight into questions concerning global and exact regularity for arbitrary smoothly bounded, pseudoconvex domains. The proposed research will lead to a better understanding of the broader question: how are the regularity properties of solutions to a system of partial differential equations (with prescribed boundary conditions) on a given domain related to the ones for an associated system on the boundary? Some of the methods introduced by the principal investigator should have applications to other PDE that arise in the physical sciences. He expects to pursue this possibility along with independent interests in nonlinear dispersive wave equations, and he will also contribute to the wider dissemination (among graduate students and other researchers) of modern methods in harmonic analysis and several complex variables.

DMS - 0400505PI：Kenneth Koenig 芝加哥大学调和分析中的问题和几个复变量摘要复分析中的许多基本问题与柯西-黎曼方程解的正则性质密切相关。首席研究员计划研究 Cn 中平滑有界伪凸域上的内部和切向柯西-黎曼复形的 Lp-Sobolev 和 Holder 正则性，当标准 Sobolev 空间中至少有一些正则性增益时（例如，对于有限型域） ，在任何维度）。他将特别关注以下基本问题：（1）确定Bergman和Szego投影之间的精确关系，直至平滑算子； (2) Sobolev 和 Holder 估计从这些域的内部到边界的转移（反之亦然），以及与非各向同性最大亚椭圆性的联系； (3) 基于边界上某些广义奇异积分的性质，对 Neumann 算子和 Cauchy-Riemann 算子的规范解进行核估计。首席研究员预计他的工作还将深入了解有关任意光滑边界伪凸域的全局和精确规律性的问题。 拟议的研究将有助于更好地理解更广泛的问题：给定域上偏微分方程组（具有规定边界条件）的解的正则性性质与边界上相关系统的解的正则性性质有何关系？ 主要研究者引入的一些方法应该适用于物理科学中出现的其他偏微分方程。他希望追求这种可能性以及对非线性色散波动方程的独立兴趣，并且他还将为谐波分析和几个复杂变量的现代方法的更广泛传播（在研究生和其他研究人员中）做出贡献。