解析Sobolev型空间上的算子理论

The produce of Sobolev space marked the birth of modern partial differential equation. Sobolev-type space, which originates in the estimation of the boundedness of singular integral operators with non-smooth kernel, is a product of the combination of the structure of classical Sobolev space and that of some other classical function spaces in real variables. However, the structure of this type space and the operators and operator algebras on them seems very complex for lack of analytic structure, essential difficlties reflected in the estimation about some relevant inequalities and integral operators. After adding analytic structure to this kind space, the corresponding theory of analytic Sobolev-type space can be built by the two valid mathematical tools consist of the theory of analytic functions and real analysis. Hardy-Sobolev space and Fock-Sobolev space contain almost all classical analytic function spaces, not only be more general than any classical analytic function space, but also be more special than the classical Sobolev space. In this project, we mainly research the operators and their related problems on these two kinds of analytic Sobolev-type spaces. Specifically, include investigating the space structure of themselves and characterizing the structure and properties of the multipliers, Toeplitz operators and composition operators on them.

Sobolev空间的产生标志着现代偏微分方程的诞生，Sobolev型空间是经典Sobolev空间结构与实变量的经典函数空间结构相结合的产物，它源于调和分析中对具有非光滑核的奇异积分算子有界性问题的估计。由于缺乏解析结构，对这类空间的结构及其上算子与算子代数的研究极其复杂，本质的困难在于相关不等式的估计与积分估计。而通过对Sobolev型空间赋予解析结构，则可凭借解析函数理论与实分析两大强有力的数学工具建立相应的解析Sobolev型空间理论。Hardy-Sobolev空间和Fock-Sobolev空间包含了几乎所有的经典解析函数空间，既比任何经典解析函数空间广泛，又比经典Sobolev空间特殊。本项目主要研究这两类解析Sobolev型空间上的算子及其相关问题，包括对空间自身结构的探索以及对定义在该两类空间上的乘法算子、Toeplitz算子、复合算子结构与性质的刻画。

本项目主要研究了Hardy-Sobolev空间和Fock-Sobolev空间这两类解析Sobolev型空间上的算子及其相关问题，包括对空间自身结构的探索以及对定义在该类空间上的乘法算子、Toeplitz算子、复合算子及其所生成代数的结构和性质的研究，取得了一系列深刻的结果。目前，围绕本课题在国内外重要期刊上发表科研学术论文13篇，并完成另外3篇科研学术论文的撰写。