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Boundary behavior of analytic functions and harmonic functions

解析函数和调和函数的边界行为

基本信息

批准号：
11640187
负责人：
SEGAWA Shigeo
金额：
$ 1.22万
依托单位：
Daido Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

1. Segawa showed that every positive harmonic function on a finitely sheeted unlimited covering surface of an open Riemann surface of positive boundary is a pullback of a positive harmonic function on the base surface by the projection map if and only if the Martin compactification of the covering surface is isomorphic to that of the base surface via the projection map. Segawa proved an analogous result of the above for bounded harmonic functions in terms of Martin boundary. Segawa determined the Martin boundaries of m-sheeted cyclic unlimited covering surfaces of the complex plane. Nakai showed that Royden p-compactifications for 1<p<d of two d-dimensional Riemannian manifolds (d【greater than or equal】2) are homeomorphic if and only if there exists a almost quasiisometric homeomorphism between these Riemannian manifolds. 2. Nakai and Tada determined the maximal growth of a rotation free density which is an exceptional perturbation for Picard principle. Tada and Nakai showed that if the Picard principle is valid for a rotation free density P, then there exists an essential set of P which is arbitrarily small and rare in a sense. 3. Nakai and Tada proved an extension of the Liouville theorem for a class of functions which properly contains polyharmonic functions. 4. Ueda showed that for a family of entire functions, the zeros of each function in the family are of odd order. Ueda generalized Nevanlinna's three-function theorem. 5. Narita gave a sufficient condition for bounded domains in order that a harmonic interpolating sequence is also an interpolating sequence. Narita gave a sufficient condition for bounded domains without irregular boundary points in order that there exists a harmonic interpolating sequence which is not interpolating. Nakai showed that the uniqueness theorem is sufficient but not necessary for the occurrence of the Myrberg phenomenon.

1. Segawa证明了正边界的开Riemann曲面的无限复盖曲面上的每个正调和函数都是投影映射对基曲面上的正调和函数的拉回当且仅当复盖曲面的Martin紧化通过投影映射同构于基曲面的Martin紧化。Segawa证明了一个类似的结果，上面的有界调和函数的马丁边界。Segawa确定了复平面的m-单圈无限覆盖曲面的Martin边界。Nakai证明了两个d维黎曼流形（d[大于或等于]2）的1<p<d的Royden p-紧化是同胚的当且仅当这些黎曼流形之间存在几乎拟等距同胚。2. Nakai和Tada确定了旋转自由密度的最大增长，这是Picard原理的一个例外扰动。多田和中井表明，如果皮卡德原理对旋转自由密度P有效，那么存在P的本质集，该本质集在某种意义上任意小且稀有。3. Nakai和Tada证明了Liouville定理的一个扩展，它适用于一类包含多调和函数的函数。4.上田表明，对于一个家庭的整个职能，零点的每个职能的家庭是奇数秩序。上田推广了Nevanlinna三函数定理。5. Narita给出了有界域上调和插值序列也是插值序列的一个充分条件。Narita给出了有界区域上不存在非规则边界点的调和插值序列的一个充分条件。Nakai证明了唯一性定理对于Myrberg现象的发生是充分的，但不是必要的。