Boundary behavior of anlytic functions and harmonic functions

(1)Segawa(与Masaoka(京都尚约大学)合作)研究了给定的开Riemann曲面R的有限薄片无限覆盖曲面R的Martin边界，证明了对于R的Martin边界的一点p，存在一个位于p上的Martin边界的极小点p当且仅当p是极小点.他们还用精细拓扑刻画了R的极小Martin边界点的个数，这些极小Martin边界点位于给定的R的极小Martin边界点上。应用这些结果，对具有格林函数的R，证明了R上的每一个正调和函数都是R上的一个正调和函数的投影拉回当且仅当对R的每个极小Martin边界点p，R上位于p上的极小Martin边界点的个数为1。(2)Tada和Nakai研究了关于R^n上具有给定符号测值的薛定谔方程的区域D上格林函数的存在性或不存在性，他们证明了如果D是一个连续的区域且D上存在格林函数，(3)Ueda推广了Nevanlinna关于三个亚纯函数及其零极点集的一个结果。他还证明了某个亚纯函数的零一极点集在某种意义上是稀疏的。(4)Narita证明了如果一个解析函数的代数A在开Riemaun曲面R上分离R的某个小子集的点，则A弱分离R的点。此外，他构造了两个例子，证明了结果在某种意义上是尖锐的。

(1) Segawa (with Masaoka (Kyoto Sangyo Univ.)) studied Martin boundary of finitely sheeted unlimited covering surfaces R of a given open Riemann surface R and showed that for a point p of the Martin boundary of R there exists a minimal point p of the Martin boundary of R which lies over p if and only if p is a minimal point. They also characterized the number of minimal Martin boundary points of R which lie over a given minimal Martin boundary point of R, in terms of fine topology. Applying these results, for R with Green's functions, they showed that every positive harmonic function on R is a pullback of a positive harmonic function on R by the projection if and only if for every minimal Martin boundary point p of R the number of minimal Martin boundary points of R which lie over p is one.(2) Tada and Nakai studied existence or nonexistence of Green's functions on a domain D in the Euclidian space R^n with respect to Schrodinger equation with a given signed measure potential on R^n They showed that if D is a continuous domain and there exist Green's functions on D, then there is a domain F which contains D and on which there exist Green's functions.(3) Ueda extended a result by Nevanlinna for three meromorphic functions and their zero-one-pole sets. He also showed that the zero-one-pole set of a certain meromorphic function is thin in a sense.(4) Narita showed that if an algebra A of analytic functions on an open Riemaun surface R separates the points of a certain small subset of R, then A weakly separates the points of R.Moreover, he constructed two examples showing that the result is sharp in a sense.