Mathematical Sciences: Boundary Behavior of Holomorphic Functions and Mappings

要做的工作是继续一个程序，在n维复空间的域上发展函数理论，使用度量几何的语言。Hardy空间理论、Nevanlinna理论、沿曲线的边界极限以及Lipschitz和Bloch空间理论的最新进展为这一数学研究的方向提供了相当大的推动力。特别是对Lusin区域积分和谐波分析的其他方面的研究将使用度量几何方法进行。这项工作的长期目标是发展一个统一的理论，在所有维度全纯函数的边界行为，其中统一的语言是黎曼几何。具体的课题包括使用新定义的逼近区域研究强伪凸区域的Lusin区域积分的映射性质，以及研究d-bar问题（一阶偏微分方程）的正则性。后者涉及在常规功能空间中无法实现的极其微妙的范数估计。必须开发新的空间以取得明显的效果。将进行额外的工作来确定在与图像域具有高阶接触的全纯矢量值映射中是否必须存在一定的刚性。这样的结果在一个复杂的维度上是错误的，但有相当多的证据表明，在更高的维度上情况是不同的。我们还将继续努力了解域间生物全纯映射的自同构群的结构。强有力的结果显示了这些映射如何在强伪凸区域的边界上全纯扩展。下一个逻辑步骤将是考虑弱伪凸区域，特别是有限类型的伪凸区域，以确定这种解析延拓的范围。

Work to be done continues a program developing the function theory on domains in n-dimensional complex space using the language of metric geometry. Recent progress made in the theories of Hardy spaces, Nevanlinna theory, boundary limits along curves and the theories of Lipschitz and Bloch spaces has provided considerable impetus to the directions this mathematical research will take. In particular studies of the Lusin area integral and other aspects of harmonic analysis will be carried out using the metric geometry approach. The long-term goal of this work is to develop a unified theory for boundary behavior of holomorphic functions in all dimensions in which the unifying language is that of Riemannian geometry. Specific topics to be undertaken include investigations into mapping properties of the Lusin area integral for strongly pseudoconvex regions using newly defined approach regions and a study of the regularity for the d-bar problem - a first order partial differential equation. The latter concerns extremely delicate norm estimates which cannot be achieved in conventional function spaces. New spaces will have to be developed to obtain sharp results. Additional work will be carried out to establish whether or not certain rigidity must occur in holomorphic vector-valued maps which exhibit higher order contact with the image domain. Such results are false in one complex dimension but considerable evidence exists suggesting that the situation is different in higher dimensions. Continuing efforts will also be made to understand the structure of automorphism groups of biholomorphic maps between domains. Powerful results show how these maps extend holomorphically across boundaries of strongly pseudoconvex regions. The next logical step will be to consider weakly pseudoconvex regions, especially those of finite type, to determine the extent of such analytic continuations.