高维及无限维区域上函数空间的算子论及相关调和分析

本项目立足于算子理论和函数论交叉领域，拟研究多变量、可数无限变量解析函数空间上的算子理论及相关的调和分析。近年来，单变量解析函数空间上的很多研究成果表明算子论、函数论、几何、群论等数学分支之间存在密切深入的联系，这些工作激励人们在高维区域上作进一步的探讨，也启发人们在可数无限变量的Dirichlet-Hardy空间上展开。我们将着重研究一些基本课题，如高维区域上算子组的联合约化子空间及其函数符号几何的联系、Cowen-Thomson换位定理在高维的推广、Dirichlet-Hardy空间上的Carleson测度问题、嵌入问题等。这些问题是算子论和函数论研究的主流课题，问题的突破对函数论、复分析、调和分析、算子论等具有重要意义。

Operator theory and operator algebra on function spaces have been one of the hot topics in recent years. This project aims to investigate operator theory and harmonic analysis on multi-variable and infinite-variable function spaces. Recent years have seen fruitful progresses that reveal a lot strong interplays between operator theory, function theory, geometry and even group theory, and that inspire people do more on spaces over high dimensional domains or Dirichlet-Hardy spaces of countably many variables. The investigator will focus on several important topics such as the relationship between joint reducing subspaces of a tuple of operators and the defining functions of these operators, the Carleson problem and embedding problem on Dirichlet-Hardy spaces. These objects are main stream in the study of operator theory and function theory. The breakthrough of these problems is definitely of significant importance to various aspects in mathematics, such as function theory, complex analysis, harmonic analysis and operator theory.