Many of the fundamental questions in complex analysis are closely related to the regularity properties of solutions to the Cauchy-Riemann equations. The principal investigator plans to study Lp-Sobolev and Holder regularity for the interior and tangential Cauchy-Riemann complexes on smoothly bounded, pseudoconvex domains in Cn when there is at least some gain of regularity in the standard Sobolev spaces (e. g. for finite-type domains, in any dimension). In particular, he will focus on the following basic problems: (1) Determination of the precise relation, up to smoothing operators, between the Bergman and Szego projections; (2) Transference of Sobolev and Holder estimates from the interior to the boundary of such domains (and vice versa), and connections to nonisotropic maximal hypoellipticity; (3) Kernel estimates for the Neumann operator and for the canonical solutions to the Cauchy-Riemann operator, based on properties of certain generalized singular integrals on the boundary. The principal investigator anticipates that his work will also provide insight into questions concerning global and exact regularity for arbitrary smoothly bounded, pseudoconvex domains.
The proposed research will lead to a better understanding of the broader question: how are the regularity properties of solutions to a system of partial differential equations (with prescribed boundary conditions) on a given domain related to the ones for an associated system on the boundary? Some of the methods introduced by the principal investigator should have applications to other PDE that arise in the physical sciences. He expects to pursue this possibility along with independent interests in nonlinear dispersive wave equations, and he will also contribute to the wider dissemination (among graduate students and other researchers) of modern methods in harmonic analysis and several complex variables.
