Mathematical Sciences: Function Theory on Symmetric Spaces

该奖项将支持数学研究，重点是 三个问题领域。 第一部分是拟共形问题的继续 定义在流形上的映射。 这些地图在以下方面发挥着重要作用： 了解欧几里得空间的变换，其局部 失真在固定的范围内。 流形的扩张 海森堡群是新的，提供了一个潜在的 在半单或多谐分析中有价值的工具 幂零李群 此外，流形下 强伪凸域的边界 几个复杂的变量 然后，自然会问（如 是在真实的变量情况下）的条件时，这些 拟共形映射是可以扩展的。 许多实变量 方法不会延续。 然而，一些主要的 结果就是。 工作将集中在隔离条件， 将给出扩展结果。 涉及对称空间的第二类研究 涉及由对称锥体和管域形成的域。在 问题是是否可以通过一个 Jordan代数结构 如果能做到这一点， 很可能，目前可用于凸锥的结果可以是 传递到非凸的。 这将有特别的 对表征理论的影响。 第三个调查领域将考虑谐波 黎曼曲面和离散结构上的函数。 将努力继承一些经典的- 变函数理论来完善黎曼空间。 特殊 感兴趣的是使用广义面积积分， 定义在这些空间上。 使用这个积分功将完成 在确定那些面积积分 给出了与传统极大函数相同的哈代类。

This award will support mathematical research focusing on three problem areas. The first continues work on quasiconformal maps defined on manifolds. These maps play an essential role in understanding transformations of Euclidean space whose local distortions lie within fixed limits. The extension to manifolds such as the Heisenberg group is new, providing a potentially valuable tool in the harmonic analysis of semi-simple or nilpotent Lie groups. In addition, the manifolds under consideration form boundaries of strongly pseudoconvex domains in several complex variables. It then becomes natural to ask (as was in the real variable case) for conditions on when these quasiconformal maps can be extended. Many of the real-variable methods do not carry over. Nevertheless, some of the main results do. Work will concentrate on isolating conditions which will give the extension result. A second line of research involving symmetric spaces concerns domains formed by symmetric cones and tube domains. At issue is whether or not on can obtain all symmetric tubes by a Jordan algebra construction. If this can be done, then it is likely that current results available for convex cones can be carried over to the non-convex. This will have particular implications for representation theory. The third area of investigation will consider harmonic functions on Riemann surfaces and on discrete structures. Efforts will be made to carry over some of the classical one- variable function theory to complete Riemann spaces. Of special interest is the use of a generalized area integral which can be defined on these spaces. Using this integral work will be done in determining those function spaces for which the area integral gives the same Hardy class as the traditional maximal function.