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: 0400505
• 负责人: Kenneth Koenig
• 金额: $ 1.86万
• 依托单位: University of Chicago
• 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: 2004-07-01 至 2005-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400505&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 -0400505 PI：Kenneth Koenig芝加哥大学调和分析和多复变量问题摘要复分析中的许多基本问题与Cauchy-Riemann方程解的正则性密切相关。主要研究者计划研究Lp Sobolev和保持器正则性的内部和切Cauchy-Riemann复形的光滑有界，伪凸域在Cn时，至少有一些增益的正则性标准Sobolev空间（e。G.对于有限型域，在任何维度上）。特别地，他将集中于以下基本问题：（1）确定Bergman和Szego投影之间的精确关系，直到平滑算子;（2）Sobolev和保持器估计从这种区域的内部到边界的转移（反之亦然），以及与各向异性最大亚椭圆率的联系;（3）基于边界上某些广义奇异积分的性质，给出了Neumann算子和Cauchy-Riemann算子的正则解的核估计。首席研究员预计，他的工作也将提供洞察问题的全球和精确的规则性任意光滑有界，伪凸域。 拟议的研究将导致更好地理解更广泛的问题：如何正则性的解决方案的偏微分方程系统（规定的边界条件）在一个给定的域相关的边界上的一个相关系统？ 主要研究者介绍的一些方法应该适用于物理科学中出现的其他PDE。他希望追求这种可能性沿着独立的利益在非线性色散波方程，他也将有助于更广泛的传播（研究生和其他研究人员）的现代方法在谐波分析和几个复杂的变量。