Operator Properties for the d-bar Neumann Problem

The d-bar equation on a weighted L^2 space on C can be connected with a Schrödinger operator with magnetic field on R^2. It can be shown that the canonical solution operator to d-bar is compact as an operator on a weighted L^2 space if and only if the corresponding Schrödinger operator with magnetic field has compact resolvent. A complete characterization of all weight functions such the canonical solution operator on the weighted L^2 space is compact is still not known, even in the case of complex dimension one. Various methods can be used to attack this problem: complex analytic, Brownian motion and potential theory and spectral theory of Schrödinger operators. In the polynomial case results of Helffer-Nourrigat can be used, in the non-polynomial case techniques from Helffer-Mohamed and Kondratiev-Shubin will give at least partial answers. For a bounded pseudoconvex domain compactness of the d-bar Neumann operator implies global regularity in the sense of preservation of Sobolev spaces. It is not clear what compactness of the canonical solution operator to d- bar means for the weighted L^2. A certain gain in the estimate for the solution seems to be a consequence of compactness. In the case of several complex variables the connection between the d-bar Neumann problem and Schrödinger operators with magnetic field is much more complicated. A first attempt for so-called decoupled weights has already been carried out. A promising approach is to find appropriate conditions on the Levi matrix of the weight function, which imply compactness of the solution operator to d-bar mainly using methods from the spectral theory of Schrödinger operators with magnetic fields. It is of interest to clarify the situation how compactness of the restriction to forms with holomorphic coefficients already implies compactness of the original solution operator to d-bar. This is the case for convex domains. There are many examples for non-compactness, where the obstruction already occurs for forms with holomorphic coefficients. The commutators between the Bergman projection and multiplication operators by z_j-bar, for j=1,... ,n are compact on L^2 if and only if the restriction of the canonical solution operator to d-bar to (0,1)-forms with holomorphic coefficients is compact. An interesting question is, under which additional conditions on the commutators one gets compactness of the solution operator on the whole L^2. Especially in the case of unbounded domains almost nothing is known for the question of compactness of the canonical solution operator to d-bar, with a few exceptions. In this connection there are also many interesting questions concerning the tangential Cauchy-Riemann complex and operator-theoretic properties of the canonical solution operator to d-bar-b. Here one is concerned with solving the tangential Cauchy-Riemann equations, such that the solution belongs to the orthogonal complement of the Hardy space on the boundary of the domain considered. If the domain is unbounded the range of d-bar-b is not closed in L^2 and so the solution operator is not bounded on L^2. Therefore one has to find appropriate weights to describe operator theoretic properties of the solution to d-bar-b. The topics of this proposal fit perfectly into the framework of the program "Complex Analysis, Operator Theory, and Applications to Mathematical Physics" which took place at the Erwin Schrödinger International Institute of Mathematical Physics ( ESI ) in fall 2005 and which was organized by the author of the present proposal in cooperation with Emil Straube (Texas A & M University) and Harald Upmeier (University of Marburg), with a follow-up program in October 2006.

This project is located at the intersection of complex analysis in several variables with functional analysis, partial differential equations and potential theory. The main issue was to investigate operator theoretic properties of the solution of inhomogeneous Cauchy-Riemann differential equations, which are of special interest in complex analysis. Starting point of the project was a result on the canonical solution operator of the inhomogeneous Cauchy-Riemann differential equations which was proved by means of methods from real analysis, such as spectral theory of Schrödinger operators and the Witten Laplacian. During the project it turned out how to understand these results only by means of complex analysis. For this purpose it was necessary to develop a suitable concept of a weighted Sobolev space and an appropriate Rellich lemma in order to handle the question of compactness of the canonical solution operator. Finally we found out that an abstract characterization of precompact subsets in L^2 spaces was the right tool to describe compactness of the d-bar Neumann operator and the canonical solution operator of the inhomogeneous Cauchy-Riemann equations. In this connection there are two points of importance: Garding`s inequality for elliptic differential operators and the behaviour at infinity of the eigenvalues of the Levi matrix of the weight function. Klaus Gansberger finished his PhD thesis during the time of the project, where he described interesting applications of complex analysis to spectral analysis of Schrödinger operators with magnetic fields and Pauli- and Dirac operators. He also explained the interplay between boundary behaviour and behaviour at infinity for the d-bar Neumann operator on weighted L^2 spaces of unbounded domains. Anne-Katrin Herbig investigated the consequences if the domains in C^n have defining functions which are plurisubharmonic on the boundary of the domain, especially for the so-called Diederich-Fornaess exponent. During the project there were three international conferences and workshops on the theme of the project organized by the leader of the project: at the Banach center in Warsaw (40 participants), at Erwin Schrödinger Institute at Vienna (100 participants) and at CIRM, Luminy , Marseille (30 participants).