Mathematical Sciences: Function Theory on Symmetric Spaces

科拉尼 8901496 这项工作试图利用之间的联系， 微分几何的数学理论，表示法 理论和经典分析。 这是一个持续的 结合经典课题的研究计划，如贝塞尔 功能协调发展的 Hankel变换和Luzin面积定理 最近的主题及其概括，如 拟对称空间和更一般的幂零连续 组 研究中一个相对较新的重点是 上拟共形映射理论的发展 柯西-黎曼流形，特别是海森堡群。 此外，还将在 对称锥上函数论的研究 大部分 广义超几何函数的早期研究 真实的和复矩阵的情况已被扩展到 任意对称锥 现在必须做的是 结合现有的各种理论， 一般适当的设置。 一个新的想法正在考虑中仍然开放的问题 找到一个推广的卢津面积积分， 黎曼对称空间 它是基于建筑 一个跳跃过程，它集中在 明显的边界比布朗运动慢得多 但其调和函数仍是鞅。 解的边界性质有许多应用 这些微分方程直到 成功的Fatou型定理可以在这种情况下开发。

Koranyi 8901496 This work seeks to exploit the connections between the mathematical theories of differential geometry, representation theory and classsical analysis. It is part of a continuing program of research combining classical topics such as Bessel functions. Hankel transforms and Luzin's area theorem with more recent topics and their generalizations such as quasi-symmetric spaces and more general nilpotent continuous groups. A relatively new thrust in the research concerns the development of a theory of quasiconformal mapping on Cauchy-Riemann manifolds, especially the Heisenberg group. Work of a continuing nature will also be carried out in the study of function theory on symmetric cones. Most of the earlier research on generalized hypergeometric functions in the case of real and complex matrices has been extended to arbitrary symmetric cones. What must be done now is to tie together the various existing theories and find the most general appropriate setting. A new idea is being considered in the still open problem of finding a generalization of Luzin's area integral for Riemannian symmetric spaces. It is based on the construction of a jump process which concentrates in neighborhoods of the distinguished boundary much less rapidly than Brownian motion but for which the harmonic functions are still martingales. There are many applicatons to boundary properties of solutions of differential equations which will not be resolved until successful Fatou-type theorems can be developed in this setting.