解析Sobolev空间上的算子理论

本项目主要研究解析Sobolev空间上的Toeplitz算子、Hankel算子、积分算子及复合算子的性质，这类空间将复解析函数空间结构Sobolev空间结构相结合，在复分析与调和分析之间架设了一座桥梁，使得调和分析的一些方法可以运用到复解析函数空间。众所周知复解析函数空间比实函数空间具有有更好的结构，所以这些空间上的算子与算子代数的性质与结构相对于实变情形更加精细。调和分析中的一些经典积分变换通过Bargman变换可以转换到解析Sobolev空间上来，为复解析函数空间提供新型的算子类。与经典的解析函数空间相比，解析Sobolev空间的结构则要复杂很多(它涉及分数阶导数)，复解析函数空间中的经典技巧与方法几乎不再适用于这类空间，需要发展新的技巧与方法。本项目将继续探索这些空间上乘子、Toeplitz算子、新型积分型算子以及复合算子的性质。

This project mainly studies the properties of Toeplitz operator, Hankel operator, integral operator and composition operator on analytic Sobolev space. This kind of space combines the Sobolev space structure and the complex analytic function space structure, and builds a bridge between complex analysis and harmonic analysis, so that some methods of harmonic analysis can be applied to complex analytic function space. It is well known that complex analytic function spaces have better structures than real function spaces, so the properties and structures of operators and operator algebras in these spaces are more refined than real variable cases. Some classical integral transforms in harmonic analysis can be transformed into analytic Sobolev space by bargman transform, which provides new operator classes for complex analytic function space. Compared with the classical analytic function space, the structure of the analytic Sobolev space is much more complex (it involves fractional derivative). The classical techniques and methods in the complex analytic function space are almost no longer suitable for this kind of space, and new techniques and methods need to be developed. This project will continue to explore the properties and structures of multipliers, Toeplitz Operators, new integral operators and composite operators on these spaces.