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

Penney将继续他的研究，以确定厄米对称空间的谐波分析的哪些方面可以推广到非对称情况。有待解决的具体问题是(a)如何刻画边界上超函数的泊松积分(b)如何处理无界域。第一步是确定在有界齐次域上是否存在一个能湮灭Gindikin-Poisson核的微分算子系统。Penney项目背后的基本思想已经可以在线的运动组中看到，它由关于零的平移和反思组成。运动群与反射子群的商自然与直线同构。类似地，更一般的几何对象可以被实现为一个变换群与某一子群的商。引入了几何空间上群不变算子的自然类别，并利用群不变算子的特征函数进行了分析。

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.