Variations on the twisted dbar-complex

As functions of several complex variables occur in many branches of mathematics, the field of several complex variables is an important part of analysis. Research in several complex variables involves using tools and techniques from various mathematical areas like partial differential equations, functional and harmonic analysis as well as differential and algebraic geometry. In fact, the connection to these other fields is so strong that methods developed in complex analysis of several variables have been instrumental in some of these other branches. For instance, the theory of pseudo-differential operators, which was developed to solve a certain problem in several complex variables, is integral to the developments in linear partial differential equations. My main interest is complex analysis in several variables and its connection to partial differential equations. The primary form of this connection is through the Cauchy-Riemann (CR) differential equations; holomorphic functions are homogeneous solutions to these equations, and solving the inhomogeneous system is a powerful tool for obtaining results in complex function theory of several variables. A fertile approach to solving the inhomogeneous CR equations is to analyze perturbations of the dbar-complex, the complex associated to the CR equations. I propose to study a particular kind of perturbation, the so-called twisted dbar-complex, in regards to questions of existence and regularity of solutions to the CR equations. In particular, I seek to analyze what range of twist factors is "negligible" in certain analytic estimates which are of importance to the analysis of solutions to the CR equations. This includes projects regarding: subellipticity of the dbar-Neumann operator, global regularity and a smoothing property of the Bergman projection, compactness of the dbar-Neumann operator, the closed range property of the dbar-operator on certain non-pseudoconvex domains, the construction of smooth solutions to the inhomogeneous CR equations on the worm domain, and convexity-like conditions in the several complex variables setting.

The most significant, scientific result of the project was the discovery of smoothing properties of the holomorphic Bergman projection (in joint work with J. D. McNeal, E. J. Straube).Holomorphic functions are one of the main objects of interest in the field of several complex variables. A way to analyze these functions (and their properties) on a given domain is to understand the so-called holomorphic Bergman projection (and its properties) associated to the domain in consideration. The Bergman projection is an operator which acts on square-integrable functions and reproduces holomorphic, square-integrable functions in a particular manner. An operator is called smoothing if the output of the operator is, in some sense, more smooth than the input data. The Bergman projection is the identity on square-integrable, holomorphic functions, that is, holomorphic functions are mapped to themselves. As holomorphic functions are not necessarily smooth (up to the boundary of the domain in consideration) a possible smoothing property of the operator is somewhat unexpected. An initial analysis, using the methods from the study of the twisted dbar-complex, revealed that there is some "space" for smoothing. Using methods from functional analysis led us to proving the smoothing phenomena in rather general circumstances. This approach also let us clarify that this phenomena holds for a more general class of projection operators as well.