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Functional Analyistic Studies On The Algebra Of Bounded Analytic Functions On A Riemann Surface

黎曼曲面上有界解析函数代数的泛函分析研究

基本信息

批准号：
16540132
负责人：
HAYASHI Mikihiro
金额：
$ 2.48万
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：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540132/
关键词：
bounded analytic function maximal ideal space Riemann surface pole set Shilov boundary Hardy class composition operator invariant subspace 補間問題同型問題 pole集合

项目摘要

1. Studies on Riemann surfaces and bounded analytic functions and harmonic functions on them:In the case that a given Riemann surface is not included openly and homeomorphically in the maximal ideal space of the algebra of all bounded analytic functions, we find a condition on a Riemann surface whose Shilov boundary is totally disconeccted. This result can be applied to the uniqueness theorem of linear extremal problem. Also, we studied the structure of the fibre in the maximal ideal space. In addition, we show that Royden's resolution of a Riemann surface can be constructed by the method of simultaneous analytic continuation of a give family of meromorphic functions.We studied on relations between the Martin boundary and harmonic functions. We succeeded in a characterization for the existence of Green functions on a infinite-sheeted covering Riemann surface over the Riemann sphere, that is obtained by pasting a pair of sheets along each curve in a given sequence of curves, in terms of the sequence of capacities of curves.2. Stuies on Hardy classes and the function theory of several variables:We give the best possible estimate of the norm of cross-commutators of two subnormal operators by means of spectral area, and generalized a former result to the case of p-quasi normal operators.We give a characterization of weighted Herz space by means of wavelets. Our result correct an error in the result published by Tang-Yang in 2000, and further showed that our system of wavelets form an unconditional basis.We gave a new type of factorization of an inner function in relation of connected components of the zero of the inner function, which solves a problem posed in a paper published in 2004. We also studies when a linear combination of composition operators is compact for an analytic function space.

1.关于Riemann曲面及其上的有界解析函数和调和函数的研究：在给定的Riemann曲面不同胚地包含在所有有界解析函数代数的极大理想空间中的情况下，得到了Riemann曲面的Shilov边界完全不连通的一个条件。这一结果可应用于线性极值问题的唯一性定理。同时，我们还研究了极大理想空间中纤维的结构。此外，我们还证明了Riemann曲面的Royden分解可以通过亚纯函数族的同时解析延拓的方法来构造，并研究了Martin边界与调和函数的关系。我们成功地刻画了黎曼球面上无限片覆盖黎曼曲面上绿色函数的存在性，这是通过在给定的曲线序列中的每一条曲线上沿着一对片来获得的，它是根据曲线的容量序列来刻画的.关于哈代类和多元函数论的研究：利用谱面积给出了两个次正规算子的交叉算子的范数的最佳可能估计，并将以前的结果推广到p-拟正规算子的情形，利用小波给出了加权Herz空间的一个特征。我们的结果纠正了Tang-Yang在2000年发表的结果中的一个错误，并进一步证明了我们的小波系统是一个无条件基，给出了一种新的关于内函数零点连通分量的分解，解决了2004年发表的一篇论文中的一个问题.我们还研究了复合算子的线性组合对于解析函数空间何时是紧的。