The aims of this project concern fundamental questions arising in the mathematical theory of several complex variables. Work to be done involves the relationship of function theory on domains of finite type to the geometry of the boundaries of the domains. Other work will focus on problems of classical function theory and Euclidean harmonic analysis. One important effort, of a continuing nature, is that of extending to domains in three or more complex dimensions, recent results on pseudoconvex domains of finite type. At this time, they have only been extablished in two dimensions. The results concern the construction of a nonisotropic metric which estimates the two most important integral kernels, the Bergman and Szego kernels. As a consequence, sharp regularity properties of the Bergman and Szego projections are obtained. This result is one of the foremost discoveries in the field in many years. Describing the corresponding result in higher dimensions is a natural and challenging goal. Work on harmonic analysis will concentrate on estimates for Fourier transforms defined along closed hypersurfaces in Euclidean space. When the mainifold bounds a convex region, estimates of the transform are known. This research will seek to obtain decay estimates for the transform when the manifold is not convex. A second objective will be to ask the same questions for manifolds with higher codimension. Work of this nature derives from problems in number theory and partial differential equations. Other investigations will be carried out on finding proper conditions for the existence of Cauchy-Riemann extensions from submanifolds of finite type and on analyzing Bergman projections in weighted spaces of entire functions in several complex variables.
