Mathematical Sciences: Solvability, Regularity and Embeddability of the Tangential Cauchy-Riemann Operators

数学科学：切向柯西-黎曼算子的可解性、正则性和可嵌入性

• 批准号: 8901455
• 负责人: Mei-Chi Shaw
• 金额: $ 3.47万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-09-01 至 1992-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8901455&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Solvability; Regularity; Embeddability

This project continues mathematical research into basic problems at the interface of the theories of several complex variables and partial differential equations. The analytic object of concern is the tangential Cauchy-Riemann operator, a differential operator acting on functions defined on the boundary of a domain in the space of several complex variables (or on more general complex manifolds). These operators agree with CR- operators defined on neighborhoods of boundary points. The object of much current activity in this area today addresses the issue of when can a function satisfying the homogeneous CR- equations on the boundary be the restriction of a holomorphic function defined on some surrounding open set? This is the extension problem and it involves solving systems of complex partial differential equations. The equations themselves are interesting because they serve as prototypes of overdetermined systems of differential equations which are not elliptic complexes. Work will be done investigating the regularity properties of the tangential operators on weakly pseudo-convex boundaries. Specifically, one would like to make estimates of how the functions in certain classes are (or are not) preserved under the action of these operators. The case where the functions are square-integrable has been worked out in detail. For the other Lebesgue spaces and for functions which are Holder continuous, good progress has recently been made in two dimensions. Efforts will be made to extend these results to higher dimension. Additional work will be done in seeking a resolution of the problem of embedding abstract strongly pseudo-convex CR-structures in complex space. For all odd real dimensions this problem has been solved by Kuranishi's embedding theorem except for the dimension three. This research will seek to exploit a new homotopy approach to determine if the final case can be settled.

这个项目继续对几个复变量理论和偏微分方程界面上的基本问题进行数学研究。所关注的分析对象是切向柯西-黎曼算子，它是作用于定义在多个复变量空间(或更一般的复流形)的区域边界上的函数的一种微分算子。这些算子与定义在边界点邻域上的CR算子一致。目前这一领域的研究主要集中在满足边界上齐次CR方程的函数何时能成为定义在周围开集上的全纯函数的限制这一问题上。这就是延拓问题，它涉及到求解复杂的偏微分方程组。这些方程本身很有趣，因为它们是超定微分方程组的原型，这些微分方程组不是椭圆形复形。我们将研究弱伪凸边界上切向算子的正则性。具体地说，人们想要估计某些类中的函数在这些运算符的作用下是如何保留(或不保留)的。对函数平方可积的情形进行了详细的推导。对于其他勒贝格空间和Holder连续的函数，最近在二维方面取得了很好的进展。将努力将这些成果扩展到更高的维度。进一步的工作将是寻求在复空间中嵌入抽象强伪凸CR-结构问题的解决方案。对于所有奇实数维，这个问题都已用Kuranishi的嵌入定理解决了，除了三维。这项研究将寻求利用一种新的同伦方法来确定最终案件是否可以解决。