Problems in harmonic analysis and several complex variables

谐波分析与若干复变量问题摘要复分析中的许多基本问题都与Cauchy-Riemann方程解的正则性密切相关。当在标准Sobolev空间（例如，在任何维度的有限型域）中至少有一些正则性增益时，主要研究者计划研究Cn中光滑有界伪凸域上的内切Cauchy-Riemann复合体的Lp-Sobolev和Holder正则性。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估计从这些区域的内部转移到边界（反之亦然），以及与非各向同性极大次椭圆的联系；(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算子正则解的核估计。首席研究员预计，他的工作还将为有关任意光滑有界伪凸域的全局和精确规律性问题提供见解。提出的研究将导致更好地理解更广泛的问题：在给定域上的偏微分方程系统（具有规定的边界条件）的解的正则性如何与边界上的相关系统的解的正则性相关？首席研究员介绍的一些方法应该适用于物理科学中出现的其他PDE。他希望在非线性色散波动方程的独立兴趣中追求这种可能性，他也将为谐波分析和一些复杂变量的现代方法（在研究生和其他研究人员中）的更广泛传播做出贡献。

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.