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

点击翻译按钮获取中文摘要

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.