解析函数空间上的Toeplitz型奇异积分算子

Singular integral operator is the central research object in harmonic analysis and exists in operator theory naturally and widely. This project is devoted to the study of singular integral operators induced by the important analytic reproducing kernels in operator theory, such as Szego kernel, Dirichlet kernel and Drury-Arveson kernel. Firstly, from the viewpoint of harmonic analysis, some classical problems are taken into account, such as boundedness, compactness, space lifting, and so on. Then we consider the boundedness of Toeplitz operator on Bergman space with local integral symbol by some boundedness criterions in harmonic analysis. Based on these research and the Calderon-Zygmund singular integral operator theory in harmonic analysis, we are intend to refine some abstract conclusions about singular integral operators induced by analytic reproducing kernels in operator theory, which contains boundedness criterions, inequalities of weak type, etc. We expect to promote the development of the application of harmonic analysis methods in operator theory, which will enrich the research techniques in operator theory and provide new viewpoints for the classical remained problems in functional analysis.

奇异积分算子是调和分析的核心研究对象，同时在函数空间上的算子理论中广泛且自然存在。本项目将围绕由算子理论中重要的解析再生核(Szego核， Dirichlet核，Drury-Arveson核)诱导的奇异积分算子展开研究。首先从调和分析的角度，在具体函数空间上研究这类算子的有界性、紧性、空间提升等经典问题。然后结合调和分析中各类有界性估计条件，考虑Bergman空间上局部可积符号的Toeplitz算子有界性问题。在此基础上，参照调和分析中的Calderon-Zygmund奇异积分算子理论，在算子理论框架内提炼由解析再生核诱导的奇异积分算子的一般性结论，包括有界性判别条件、弱型不等式等。此项研究有望进一步促进调和分析方法在算子理论中的发展，丰富算子理论的研究手段，为泛函分析中经典问题的研究带来新的动力。

奇异积分算子是调和分析的核心研究对象，同时在函数空间上的算子理论中广泛且自然存在。本项目围绕由算子理论中重要的解析再生核(Szego 核， Dirichlet 核，Drury-Arveson 核)诱导的奇异积分算子展开研究。首先，研究了Szego核， Dirichlet 核，Drury-Arveson核诱导的积分算子 有界性及相关基本性质。其次，我们建立了一般解析再生核诱导的积分算子有界性与某类Carleson测度的等价性。最后，利用B(pq)权解决了Szego核诱导的积分算子的Mukehoupt问题。此项目进一步促进调和分析方法在算子理论中的发展，丰富算子理论的研究工具。