Study of boundary behavior of holomorphic functions and harmonic functions

1.(1)给出了复平面无限层覆盖曲面不存在格林函数的充分条件。(2)对于双曲（即具有格林函数）黎曼曲面R，证明了当且仅当R的Martin边界由有限多个极小点组成且每个极小点具有正调和量时，R上不存在无界正调和函数。此外，对于双曲Riemann曲面R，证明了当且仅当R的Martin边界由有限多个极小点组成，且每个极小点具有正调和量，且每个极小函数为Dirichlet有限时，R上不存在Dirichlet无穷正调和函数。(3)证明了复平面的2张或3张无限覆盖曲面R和S是拟共形等价的，则R的调和维数等于S的调和维数。(1)证明了自由旋转双曲带符号Radon测度dm的Picard维数等于在dm上加一个基本测度得到的测度的Picard维数。(2)对于以Radon测度为势的平稳不变Schoredinger方程的Green函数，给出了Ohtsuka定理、Herve定理和Lahtinen定理的推广。(1)在满足一定几何条件的平面域上，如果序列的两个部分的闭包互不相交，则证明序列是内插的。(2)证明了如果平面域的正则边界点上的最大理想空间的光纤不是峰集，则存在一个收敛于该边界点的非插值而是调和插值序列。

1.Riemann surfaces(1)A sufficient condition was given for certain infinitely sheeted covering surfaces of the complex plane in order to possess no Green's functions.(2)For hyperbolic (i.e.possessing Green's function) Riemann surfaces R, it was shown that there exists no unbounded positive harmonic functions on R if and only if the Martin boundary of R consists of finitely many minimal points and each minimal point has positive harmonic measure. Moreover, for hyperbolic Riemann surfaces R, it was shown that there exists no Dirichlet infinite positive harmonic functions on R if and only if the Martin boundary of R consists of finitely many minimal points, each minimal point has positive harmonic measure and each minimal function is Dirichlet finite.(3)It was shown that, if 2 or 3-shetted unlimited covering surfaces R and S of the complex plane are quasiconformal equivalent to each other, then the harmonic dimension of R is equal to that of S.2.Picard Principle(1)It was shown that the Picard dimension of a rotation free hyperbolic signed Radon measure dm is equal to that of a measure obtained by adding a fundamental measure to dm.(2)For Green's function with respect to a staionary invariant Schoredinger equation with a Radon measure as potential, generalizations of Ohtsuka's theorem, Herve's theorem and Lahtinen' theorem are given.3.Bounded holomorphic functions(1)It was shown that a sequence in a plane domain satisfying a certain geometric condition is interpolating if the closures of two parts of the sequence is mutually disjoint.(2)It was shown that if the fiber of the maximal ideal space over a regular boundary point of a plane domain is not a peak set, then there exists a sequence converging to the boundary point which is not interpolating but harmonic interpolating.