Singular Integral Operators in Several Complex Variables

多个复数变量中的奇异积分算子

• 批准号: 0654195
• 负责人: Jennifer Brooks
• 金额: $ 7.42万
• 依托单位: University of Montana
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0654195&HistoricalAwards=false
• 关键词: Singular; Integral; Operators; Several; Complex

The most important partial differential operators in the theory of functions of several complex variables are the Cauchy-Riemann (CR) operator and the tangential CR- operator, obtained by restricting the CR-operator to a surface in complex n-dimensional space. This project aims to extend in a significant way known results concerning the tangential CR-operator and the associated system of partial differential equations. In general, this system does not have a unique solution, so the best that one can do is to find the unique square-integrable solution orthogonal to the null-space of the operator. For this reason a fundamental object of study is the Szego projection operator, which is the orthogonal projection of the space of square-integrable functions onto the null space of the tangential CR-operator. This operator is well understood in the case of hypersurfaces in two-dimensional complex space that are of finite type (which means that they are not "too flat"). Part of this project examines the Szego projection operator for infinite-type surfaces in two-space. The approach is to think of the operator in terms of integration against a kernel and then to estimate the kernel function. The methods used in the finite-type case do not extend to this setting, so new ideas are needed. The principal investigator also intends to study the kernel for the Szego projection operator for convex surfaces in higher dimensions. The operator is reasonably well understood in this situation, though previous approaches have used knowledge of the Bergman kernel rather than direct analysis of the integral kernel. An examination of the latter should give additional insight into the nonisotropic metric that arises in this case. Finally, the CR- operator itself will be studied for a model non-(pseudo)convex surface in complex two-space for which estimates on the Szego kernel exist, but for which little is known about the CR-operator itself.Partial differential equations provide a powerful language for describing relationships between changing quantities. Some model heat flow, others are concerned with wave propagation, and yet others are just objects of study in their own right. For any partial differential one is interested in solving for unknown functions in terms of given initial data. The objective is to determine conditions under which a solution exists, conditions under which a solution is unique, and relationships between the properties of the given data and properties of the solution. Although this project studies partial differential equations that are of fundamental interest in pure rather than applied mathematics, the new techniques developed will be much more broadly applicable.

在复变函数理论中最重要的偏微分算子是Cauchy-Riemann （CR）算子和切向CR-算子，它们是通过将CR-算子限定在复n维空间的曲面上而得到的。本项目旨在以一种有意义的方式推广关于切向cr -算子和相关偏微分方程组的已知结果。一般来说，这个系统没有唯一解，所以我们能做的最好的就是找到唯一的平方可积解正交于算子的零空间。因此，一个基本的研究对象是Szego投影算子，它是平方可积函数空间在切向cr -算子的零空间上的正交投影。这个算子在二维复杂空间的有限型超曲面（这意味着它们不是“太平坦”）的情况下是很容易理解的。这个项目的一部分研究了二维空间中无限型曲面的Szego投影算子。方法是考虑算子对核函数的积分，然后估计核函数。在有限类型情况下使用的方法不能扩展到这种情况，因此需要新的想法。研究了高维凸曲面的Szego投影算子的核。在这种情况下，虽然以前的方法使用的是伯格曼核的知识，而不是对积分核的直接分析，但算子是相当容易理解的。对后者的考察应该对在这种情况下产生的非各向同性度量提供额外的见解。最后，我们将研究复二空间中模型非（伪）凸曲面的CR-算子本身，该曲面存在Szego核的估计，但对CR-算子本身知之甚少。偏微分方程为描述变化量之间的关系提供了一种强大的语言。有些模拟热流，有些与波的传播有关，还有一些本身就是研究对象。对于任何一种偏微分，人们都对用给定的初始数据求解未知函数感兴趣。目标是确定解决方案存在的条件，解决方案唯一的条件，以及给定数据的属性和解决方案的属性之间的关系。虽然这个项目研究的偏微分方程是纯数学而不是应用数学的基本兴趣，但所开发的新技术将有更广泛的应用。