Research and Education in Several Complex Variables

摘要：伪凸域上D -bar Neumann问题的L-2 Sobolev理论在20世纪80年代和90年代取得了巨大的进展，特别是通过Catlin和D'Angelo关于亚椭圆性和有限型的研究，主要研究人员关于大（弱）伪凸域上的整体正则性的研究，以及Barrett、Christ和Siu关于所谓“蠕虫”域上正则性失效的研究。然而，目前甚至没有一个关于几何和潜在理论条件的组合可以用来表征d-bar Neumann问题的全局正则性的猜想。对这些条件的研究是本课题的研究方向之一。一个相关的（很可能更容易处理的）问题是用紧致d-bar诺伊曼算子来表征域。（这个问题在算子理论中也有影响。）此外，主要研究者将研究有限型域上d-bar Neumann问题的L-p估计。在这里，他们将建立在最近关于有限类型的非光滑域的工作中获得的新见解。从与域的Bergman核相关的Kaehler几何的角度来看，这些研究也很有趣。（除了它们的内在意义之外，类伯格曼核在量子化方案——Berezin量子化——中发挥着突出的作用，该方案最近引起了数学家和物理学家的广泛关注）。在这个项目中，主要研究人员将培养三名研究生，他们将指导一名博士后水平的年轻数学家（由德克萨斯农工大学集团基础设施资助）研究多项式船体问题。（虽然这些问题具有内在的动机，但它们与控制工程有着密切的联系）。对几个复杂变量的分析的研究可以被数学学科的中心地位所激励（因此，从长远来看，对科学和技术企业的福祉至关重要），或者通过对其有用性的更直接的呼吁。例如，自然的基本定律之一，因果关系，当通过称为傅里叶变换的数学装置转录时，立即产生几个（在这种情况下是四个）复杂变量的解析函数。同样，如上所述，本项目的工作不仅会影响几个复杂变量和偏微分方程的核心领域，还会通过与算子理论、数学物理和控制工程的联系影响其他科学领域。最后，该项目将为数学领域的人力资源开发做出重大贡献。

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.