Mathematical Sciences: Poisson Integrals and Cayley Transforms on Bounded Homogeneous Domains in Cn

数学科学：Cn 有界齐次域上的泊松积分和凯莱变换

• 批准号: 9306222
• 负责人: Richard Penney
• 金额: $ 6.04万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1996-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9306222&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Poisson; Integrals; Cayley

Penney will continue his investigation to determine what aspects of the harmonic analysis of Hermitian symmetric spaces generalize to non-symmetric situations. Particular questions yet to be resolved are (a) how to characterize the Poisson integrals of hyperfunctions on the boundary and (b) how to treat unbounded domains. The first step will be to determine if there exist on a bounded homogeneous domain a system of differential operators which annihilates the Gindikin-Poisson kernel. The basic idea behind Penney's project can be seen already in the motion group of the line, which consists of translations and reflection about zero. The quotient of the motion group by the reflection subgroup is naturally isomorphic to the line. Similarly, more general geometric objects can be realized as the quotient of a transformation group by a certain subgroup. This introduces natural classes of group invariant operators on the geometric space, and analysis is developed using eigenfunctions of the group invariant operators.

彭尼（Penney）将继续他的调查，以确定赫尔米尼对称空间的谐波分析的哪些方面概括为非对称情况。 尚未解决的特定问题是（a）如何表征边界上超功能的泊松积分以及（b）如何处理无界域。 第一步将是确定在有界均匀域上是否存在鉴定算子的系统，该系统将歼灭Gindikin-Poisson内核。 Penney项目背后的基本思想可以在该行的运动组中看到，该行动包括翻译和反思零。 反射亚组的运动组的商自然是该线的同构。 同样，更通用的几何对象可以通过某个子组实现为转换组的商。 这是在几何空间上介绍了组不变运算符的天然类别，并使用组不变运算符的本征函数进行了分析。