Proposal: DMS-9801539 Principal Investigators: Harold P. Boas, Emil Straube Abstract: The L-2 Sobolev theory of the d-bar Neumann problem on pseudoconvex domains has seen dramatic progress in the 1980s and 1990s, notably through the work of Catlin and D'Angelo on subellipticity and finite type, of the principal investigators on global regularity on large classes of (weakly) pseudoconvex domains, and of Barrett, Christ, and Siu on failure of regularity on the so called "worm" domains. However, there is currently not even a conjecture about what combination of geometric and potential theoretic conditions might be used to characterize global regularity in the d-bar Neumann problem. The study of such conditions is one of the directions of this project. A related (and quite possibly more tractable) question is that of characterizing domains with compact d-bar Neumann operators. (This question also has repercussions in operator theory.) In addition, the principal investigators will study L-p estimates with gain for the d-bar Neumann problem on domains of finite type. Here, they will build on new insight gained from recent work concerning nonsmooth domains of finite type. These investigations are also of interest from the point of view of the Kaehler geometry associated with the Bergman kernel of a domain. (In addition to their intrinsic significance, Bergman-like kernels play a prominent role in a quantization scheme --Berezin quantization -- that has recently attracted much attention from both mathematicians and physicists). During this project, the principal investigators will train three graduate students, and they will supervise a young mathematician at the post-doctoral level (supported through a Group Infrastructure Grant at Texas A&M University) in his study of problems concerning polynomial hulls. (While intrinsically motivated, these questions have intimate connections to control engineering). The study of analysis in several complex variables can be motivated by the centrality of the subject within mathematics (thus making it essential, in the long term, for the well-being of the scientific and technological enterprise) or through a more direct appeal to its usefulness. For example, one of the basic laws of nature, causality, when transcribed via a mathematical device called the Fourier transform, immediately gives rise to analytic functions of several (in this case four) complex variables. Likewise, as indicated above, the work in this project will impact not only the core areas of several complex variables and partial differential equations, but also other areas of science through connections to operator theory, to mathematical physics, and to control engineering. Finally, the project will contribute significantly to human resources development in mathematics.
