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

这个项目继续在几个复杂变量理论和偏微分方程的界面上对基本问题进行数学研究。所关注的分析对象是切向Cauchy-Riemann算子，这是一种作用于在几个复变量空间（或更一般的复流形）的域边界上定义的函数的微分算子。这些算子与边界点邻域上定义的CR算子一致。当前在这一领域的许多活动的目标是解决这样一个问题：在边界上满足齐次CR-方程的函数何时可以成为在某些周围开集中定义的全纯函数的限制？这是扩展问题，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系统的原型，它们不是椭圆复合体。研究了弱伪凸边界上切算子的正则性。具体来说，我们希望估计在这些操作符的作用下，某些类中的函数是如何（或不）保留的。对函数平方可积的情况作了详细的计算。对于其他勒贝格空间和Holder连续的函数，最近在二维空间中取得了很好的进展。将努力将这些结果扩展到更高的维度。在复杂空间中寻找嵌入抽象强伪凸cr结构问题的解决方案方面将做进一步的工作。对于所有奇实维，这个问题都可以用Kuranishi的嵌入定理来解决，除了第三维。这项研究将寻求利用一种新的同伦方法来确定最终的情况是否可以解决。

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.