The Algebras of Bounded Analytic Functions on a Riemann Surface and the isomorphic problem

黎曼曲面上有界解析函数的代数与同构问题

• 批准号: 12640147
• 负责人: HAYASHI Mikihiro
• 金额: $ 2.56万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640147/
• 关键词: Rieman surface; bounded analytic function; maximal ideal space; point separation problem; isomorphic problem; Hardy class; Myrberg phenomena; uniqueness theorem; 葉被覆面; 有界正則関数環; Hardy族; Myrberg現象; 2葉被覆面

The algebra of all bounded analytic functions on a Riemann surface is of course determined by the Riemann surface. This converse is no longer obvious, and we call it the isomorphic problem. If a Riemann surface admits a meromorphic function bounded at the boundary of the surface and if bounded analytic functions separate the points of the surface, then the answer is yes. However, the investigator found a counter example when a surface admits no such meromorphic function.When we ask the isomorphic problem, we assume that bounded analytic functions (weakly) separate the points of a surface, for otherwise isomorphic problem fails always. Thus, it is an interesting problem to determine whether a given surface is separating or not with respect to bounded analytic functions. This problem is also hard to solve in general. The investigator, together with Prof. Mitsuru Nakai and Dc. Yasuyuki Kobayashi, consider the case of a two-sheeted disc, where the projection of the sequence of ramification points converging to the center of the disc. We considered the case that a sequence of small discs centered at ramification points are removed, and found several sufficient conditions for separating and also not separating.

当然，所有有界分析函数的代数当然由黎曼表面确定。这种相反不再是显而易见的，我们称之为同构问题。如果riemann表面允许在表面边界上有界面的meromorthic函数，如果有界的分析函数将表面的点分开，则答案是肯定的。但是，研究者在表面不承认这种同构功能时发现了一个反示例。当我们提出同构问题时，我们假设有界的分析函数（弱）将表面的点分开，因为否则同构问题总是会失败。因此，确定给定表面是否相对于有限的分析功能是一个有趣的问题。这个问题通常也很难解决。调查员与Mitsuru Nakai教授和DC一起。 Yasuyuki Kobayashi，考虑一个两片圆盘的情况，其中分支点的序列的投影融合到了圆盘中心。我们认为，删除了一个集中在分离点的小盘子的序列，并发现了几个足够的分离条件，也没有分离。

项目成果

Toshiya JIMBO: "Polynomial hulls of graphs of antiholomorphic functions"Scientiae Mathematicae Japonicae. 57. 121-127 (2003)

Toshiya JIMBO：“反全纯函数图的多项式壳”Scientiae Mathematicae Japonicae。

K. Izuchi: "Trivial points in the maximal ideal space of H^∞. III."Houston J. Math.. (in press).

K. Izuchi：“H^∞ 的最大理想空间中的琐碎点。III.”Houston J. Math..（出版中）。

K.Izuchi: "Trivial points in the maximal ideal space of H^∞.III"Houston J. Math.. 27. 873-881 (2001)

K. Izuchi：“H^∞.III 的最大理想空间中的平凡点”Houston J. Math.. 27. 873-881 (2001)

T.Nakazi: "Compact Toeplitz operators with continuous symbols on weighted Bergman spaces"Glasgow J.Math.. 42. 31-35 (2000)

T.Nakazi：“加权 Bergman 空间上具有连续符号的紧凑 Toeplitz 算子”Glasgow J.Math.. 42. 31-35 (2000)

神保敏弥: "多重調和関数のグラフの多項式凸包"数理解析研究所講究録. 1277. 136-140 (2002)

Toshiya Jimbo：“多重调和函数图的多项式凸包”数学研究所 Kokyuroku。1277. 136-140 (2002)