管状域上的Hardy空间及其算子理论的研究

The Hardy space, as a classical and important space of analytic functions, has had a profound influence on pure mathematical theory and applied analysis, therefore, it has been an active research object of function spaces and operator theory. However, since tube is unbounded and the higher-dimensional structure is complex, there is far from adequate studies on the theory of Hardy spaces over the higher-dimensional tubes and of operators on them, except for some classical basic results obtained by Stein and Upmeier about function spaces and operator theory, respectively. For a long time, the theory of Hardy spaces on tubes has been under-developed, lacking further and thorough investigation. Based on our some preliminary research of the theory of Hardy spaces on tubes, in order to develop the theory better, using three tools of functional analysis, complex method of several complex variables and real method of harmonic analysis, the project intends to study the Fourier spectrum characterizations, integral formulas, Hardy decomposition theorems, characterization of the backward shift invariant subspace, the properties of the composition operators and other operators. By seeking some new and more effective research methods, the project aims to enrich and develops the theory of analytic function spaces over the higher-dimensional unbounded domains and of operators on them.

Hardy空间是一类经典而重要的解析函数空间，在纯数学理论和应用分析领域都有着深刻的影响，因此一直是函数空间与算子理论领域中活跃的研究对象。然而，关于高维管状域上的Hardy空间及其算子理论的研究，由于管状域的无界性与高维结构的复杂性等困难，除了Stein与Upmeier等人分别在函数论与算子理论方面取得的少量经典结果之外，很长时间以来发展较缓慢，仍缺乏深入而系统的研究。基于申请者近几年来对管状域上Hardy空间的研究基础，为了更好地发展该空间理论及其算子理论，本项目将应用泛函分析、多复变函数论的复方法与调和分析的实方法三大分析工具，主要研究Fourier谱的特征刻画、积分表示、分解定理、后移不变子空间的特征刻画以及复合算子与其他算子的性质。通过本项目的研究，寻求新的更有效的研究途径，期望丰富与发展高维无界区域上的解析函数空间及其算子理论。

本项目的主要研究解析函数空间及其算子理论，在项目执行过程中，考虑了Hardy空间的Fourier谱的特征刻画问题及函数的积分表示。建立ardy空间分解定理及讨论该分解的唯一性。Dirichlet空间上的复合算子的有界性、Bergman 空间上的再生核的最佳逼近问题以及凸域上的Carleson测度问题。