Mathematical Sciences: Boundary Behavior of Holomorphic Functions and Mappings

要做的工作继续一个程序开发的功能 论域在n维复空间中使用的 度量几何的语言。 最近的进展 哈代空间理论，Nevanlinna理论，边界极限 沿着曲线和Lipschitz和Bloch空间的理论 提供了相当大的推动力的方向，这个数学 研究将采取。 特别是对卢辛地区的研究 积分和其他方面的谐波分析将进行 使用度量几何方法。 的长期目标 本文的工作是发展一个统一的理论， 全纯函数在所有维度，其中统一 语言就是黎曼几何。 将开展的具体专题包括： 强Lusin面积积分的映射性质 伪凸区域使用新定义的方法区域和一个 一阶d杆问题的正则性研究 偏微分方程 后者涉及到极大的 在传统的标准中无法实现的微妙的规范估计 函数空间 必须开发新的空间， 尖锐的结果。 将开展更多的工作，以确定是否或 全纯向量值映射不一定有刚性 其表现出与图像域的高阶接触。 等 结果在一个复杂的维度上是错误的， 有证据表明，情况有所不同， 更高的维度 还将继续努力， 了解双全纯的自同构群的结构 域之间的映射。 强有力的结果表明，这些地图 强伪凸的边界上全纯扩张 地区 下一个合乎逻辑的步骤将是考虑弱 伪凸区域，特别是有限类型的区域， 确定这种分析延续的程度。

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.