CR Manifolds and Singular Integrals

CR 流形和奇异积分

• 批准号: 1200815
• 负责人: Jennifer Brooks
• 金额: $ 9.66万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200815&HistoricalAwards=false
• 关键词: CR; Manifolds; Singular; Integrals

Much of modern complex analysis is concerned with the study of the tangential Cauchy-Riemann (CR) operator. This operator arises by restricting the classical Cauchy-Riemann operator in complex Euclidean space to a hypersurface or to an appropriate class of manifolds called CR manifolds. This operator is not invertible. Thus to understand it, we must understand a related singular integral operator, the Szego projection operator, which is the orthogonal projection of the space of square-integrable functions on a manifold onto the null space of the associated tangential CR operator. The Szego projection is relatively well-understood for boundaries of pseudoconvex domains of finite type. Comparatively little is known if one relaxes the finite-type hypothesis, and even less is known if one relaxes the pseudoconvexity hypothesis. Exploring these contexts systematically is the goal of the current project. More specifically, Halfpap and her collaborators will analyze explicit expressions for the integral kernel associated with the Szego projection for several classes of hypersurfaces and CR manifolds. The goals are 1) to determine the locations and sizes of the singularities of the Szego kernel, 2) to determine how these are connected to the geometry of the underlying manifold, and 3) to understand the mapping properties of the projection operator itself. In general, the study of partial differential equations (PDEs) is important because PDEs give a language for describing relationships among changing quantities; some model heat flow, some wave propagation, and some are simply objects of study in their own right. For a general PDE, several fundamental questions arise: Can one solve for the unknown function in terms of the given data? Is the solution unique? If the data are known to have certain special properties (e.g., smoothness or square-integrability) what can be said about the solution? Often one seeks a so-called ``fundamental solution" -- a kernel function against which one may integrate the data to obtain a solution of the original PDE. Theorems relating properties of the solution to those of the data thus require detailed information about the associated integral kernel function. Frequently, the kernel function has singularities. In these cases one obtains a singular integral operator. This is the larger context in which the current project is situated, and thus the results of the current project have broader implications for other areas in which PDEs and singular integral operators arise. This mathematics research project also has the potential to increase diversity within mathematics; Halfpap is an active advisor of graduate students (as well as an effective teacher and mentor of undergraduates) at a university serving a largely-rural state.

现代复变分析的很多内容都是关于切向Cauchy-Riemann(CR)算子的研究。这个算子是通过将复欧氏空间中的经典柯西-黎曼算子限制在一个超曲面或一类称为CR流形的适当流形上而产生的。这个运算符是不可逆的。因此，要理解它，我们必须了解一个相关的奇异积分算子，Szego投影算子，它是流形上平方可积函数空间到相关切向CR算子的零空间的正交投影。Szego投影对于有限类型的伪凸域的边界是相对容易理解的。如果放松有限类型假设，我们知道的相对较少，如果放松伪凸性假设，我们知道的就更少了。系统地探索这些背景是本项目的目标。更具体地说，Halfpap和她的合作者将分析几类超曲面和CR流形上与Szego投影相关的积分核的显式表达式。目标是1)确定Szego核的奇点的位置和大小，2)确定这些奇点如何与底层流形的几何联系，以及3)理解投影算子本身的映射性质。一般而言，偏微分方程组(PDE)的研究很重要，因为偏微分方程组提供了一种描述变量之间关系的语言；一些模型热流，一些波传播，还有一些本身就是研究的对象。对于一般的偏微分方程组，出现了几个基本问题：能否根据给定的数据来求解未知函数？该解决方案是否独一无二？如果已知数据具有某些特殊性质(例如，光滑性或平方可积性)，关于解可以说什么？通常，人们寻求所谓的“基本解”--一种核函数，可以根据它对数据进行积分，以获得原始偏微分方程组的解。因此，将解的性质与数据的性质联系起来的定理需要有关相关积分核函数的详细信息。通常，核函数有奇点。在这些情况下，我们得到一个奇异积分算子。这是当前项目所处的更大背景，因此当前项目的结果对出现偏微分方程组和奇异积分算子的其他领域具有更广泛的影响。这个数学研究项目也有可能增加数学中的多样性；Halfpap是一所大学的研究生的积极顾问(也是一名有效的教师和本科生的导师)，该大学服务于一个以农村为主的州。