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

1. 黎曼曲面及其上有界解析函数和调和函数的研究：在给定黎曼曲面不公开同纯地包含在所有有界解析函数代数的极大理想空间的情况下，我们发现黎曼曲面上希洛夫边界完全不连通的一个条件。该结果可推广到线性极值问题的唯一性定理。同时，我们还研究了纤维在最大理想空间中的结构。此外，我们还证明了Riemann曲面的Royden分辨率可以用给定亚纯函数族的同时解析延拓的方法来构造。研究了马丁边界与调和函数之间的关系。我们成功地描述了在Riemann球面上覆盖Riemann曲面的无限片上Green函数的存在性，这是通过在给定曲线序列中的每条曲线上粘贴一对片来获得的，根据曲线的容量序列。Hardy类与多变量函数理论的研究：利用谱面积给出了两个次正规算子的交叉换易子的范数的最佳估计，并将先前的结果推广到p-拟正规算子的情况。利用小波给出了加权赫兹空间的一个表征。我们的结果纠正了唐扬在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无条件基。给出了一种新型的内函数的零连通分量分解方法，解决了2004年发表的一篇论文中提出的问题。我们还研究了对于解析函数空间，复合算子的线性组合在什么情况下是紧的。

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.