The most important partial differential operators in the theory of functions of several complex variables are the Cauchy-Riemann (CR) operator and the tangential CR- operator, obtained by restricting the CR-operator to a surface in complex n-dimensional space. This project aims to extend in a significant way known results concerning the tangential CR-operator and the associated system of partial differential equations. In general, this system does not have a unique solution, so the best that one can do is to find the unique square-integrable solution orthogonal to the null-space of the operator. For this reason a fundamental object of study is the Szego projection operator, which is the orthogonal projection of the space of square-integrable functions onto the null space of the tangential CR-operator. This operator is well understood in the case of hypersurfaces in two-dimensional complex space that are of finite type (which means that they are not "too flat"). Part of this project examines the Szego projection operator for infinite-type surfaces in two-space. The approach is to think of the operator in terms of integration against a kernel and then to estimate the kernel function. The methods used in the finite-type case do not extend to this setting, so new ideas are needed. The principal investigator also intends to study the kernel for the Szego projection operator for convex surfaces in higher dimensions. The operator is reasonably well understood in this situation, though previous approaches have used knowledge of the Bergman kernel rather than direct analysis of the integral kernel. An examination of the latter should give additional insight into the nonisotropic metric that arises in this case. Finally, the CR- operator itself will be studied for a model non-(pseudo)convex surface in complex two-space for which estimates on the Szego kernel exist, but for which little is known about the CR-operator itself.
Partial differential equations provide a powerful language for describing relationships between changing quantities. Some model heat flow, others are concerned with wave propagation, and yet others are just objects of study in their own right. For any partial differential one is interested in solving for unknown functions in terms of given initial data. The objective is to determine conditions under which a solution exists, conditions under which a solution is unique, and relationships between the properties of the given data and properties of the solution. Although this project studies partial differential equations that are of fundamental interest in pure rather than applied mathematics, the new techniques developed will be much more broadly applicable.
