Mathematical Sciences: Function Theory on Symmetric Spaces

数学科学：对称空间函数论

• 批准号: 8901496
• 负责人: Adam Koranyi
• 金额: $ 13.01万
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1992-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8901496&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Function; Theory; Symmetric

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.

Koranyi 8901496 这项工作旨在探索微分几何、表示论和经典分析的数学理论之间的联系。 它是结合贝塞尔函数等经典主题的持续研究计划的一部分。 Hankel 变换和 Luzin 面积定理具有更新的主题及其概括，例如拟对称空间和更一般的幂零连续群。 研究中一个相对较新的重点涉及柯西-黎曼流形（尤其是海森堡群）上的拟共形映射理论的发展。 对称锥函数论的研究也将进行持续性的工作。 大多数关于实数和复数矩阵情况下的广义超几何函数的早期研究已扩展到任意对称锥体。 现在要做的就是将现有的各种理论结合起来，找到最普遍的合适设置。 在仍然悬而未决的问题中正在考虑一种新的想法，即寻找黎曼对称空间的 Luzin 面积积分的推广。 它基于跳跃过程的构造，该过程集中在区分边界的邻域，速度远低于布朗运动，但调和函数仍然是鞅。 微分方程解的边界性质有许多应用，只有在这种情况下成功发展出法图型定理，这些应用才能得到解决。