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：谐波分析中的Chicagopbroblems的Kenneth Koeniguniversity和复杂分析中的几个复杂变量消除许多基本问题与Cauchy -Riemann方程的解决方案的规律性属性密切相关。首席研究者计划研究LP-Sobolev和持有人的规律性，以实现CN平滑界限的，pseudoconvex结构域的内部和切向Cauchy-Riemann络合物，当时在标准SOBOLEV空间中至少有一些规律性的增益（例如，对于有限型型域中，至少有一些规律性的增益）。特别是，他将重点关注以下基本问题：（1）确定伯格曼和塞戈预测之间的确切关系，直到平滑运营商； （2）Sobolev和Holder估计从内部到边界的转移（反之亦然），以及与非偶然最大性低纤维化的连接； （3）基于边界上某些广义奇异积分的特性，Neumann运算符和Cauchy-Riemann操作员的规范解决方案的内核估计值。首席调查员预计，他的工作还将提供有关全球和确切规律性的问题，以实现任意平滑限制的伪内电图域的问题。 拟议的研究将使人们对更广泛的问题有更好的了解：在与边界上关联系统相关的给定域中，解决方程组（带有规定边界条件）的解决方案的规律性特性如何？ 主要研究者引入的一些方法应在物理科学中对其他PDE应用。他希望在非线性分散波方程中追求这种可能性以及独立的利益，他还将为谐波分析和几个复杂变量的现代方法的更广泛的传播（在研究生和其他研究人员中）做出贡献。