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

该项目继续进行数学研究， 几个复杂的理论之间的界面问题 变量和偏微分方程。 的解析 关注的对象是切向柯西-黎曼算子， 微分算子作用于定义在边界上的函数 多个复变量空间中的域（或多个复变量空间） 一般复流形）。 这些运营商同意CR- 定义在边界点邻域上的算子。 的 目前在这一领域的许多活动的目的是解决 一个函数何时可以满足齐次CR- 边界上的方程是一个全纯 定义在某个周围开集上的函数？ 这是 它涉及到解决复杂的系统， 偏微分方程 方程本身是 因为它们是超定的原型 非椭圆型微分方程组 配合物 工作将做调查的规律性性质 弱伪凸边界上的切算子 具体来说，人们想估计 在某些类中的函数被（或不被）保留在 这些运营商的行动。 函数为 的平方可积性。 对于其他 Lebesgue空间和函数是保持器连续的， 最近在两个方面取得了良好进展。 努力 将这些结果推广到更高的维度。 将开展更多的工作，寻求解决 嵌入抽象强伪凸问题 复空间中的CR结构 对于所有奇真实的维， 这个问题已经解决了Kuranishi的嵌入定理，除了 第三维度。 这项研究将寻求利用一个 一种新的同伦方法来确定最终情况是否可以 定了

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.