Bergman 空间和Hardy空间上的复合算子差分

The aim of this project is to study the difference of composition operators between weighted Bergman spaces on the unit ball or half-plane as well as the difference of composition operators on Hardy spaces on the unit disc. Firstly, in order to obtain some estimates for the norm and essential norm of the difference of composition operators, We will apply the Carleson type measure, r-lattice, test functions and techniques of weighted Bergman spaces to derive the connection of the difference of composition operators between weighted Bergman spaces and the corresponding joint pull-back measure. Secondly, based on the necessary and sufficient conditions for boundedness and compactness of the difference operator, we will also study the bounded or compact difference of the linear fractional composition operators between weighted Bergman spaces on the half-plane, to understand the discrepancies of the difference of composition operators between the unit ball setting and the half-plane setting. Finally, we aim to solve the problem of the compact difference of composition operators on Hardy spaces on the unit disc. Through our research, we hope to establish the basic theory and the method of the difference of composition operators between weighted Bergman spaces on unit ball or half-plane, and work out the problem of the compact difference of composition operator on Hardy spaces on unit disc.

本项目旨在研究单位球及上半平面上加权Bergman空间，以及单位圆盘上Hardy空间的复合算子差分问题。具体包括：通过Carleson型测度、r-网和测试函数等研究工具，建立单位球及上半平面上加权Bergman空间复合算子差分，与其上两个解析自映照的联合拉回测度之间的联系，得到单位球及上半平面上加权Bergman空间复合算子差分的算子范数和本性范数估计，以及复合算子差分的有界性和紧性刻画；研究上半平面上加权Bergman空间的分式线性复合算子差分，并构造合适的例子来揭示单位球与上半平面上加权Bergman空间复合算子差分存在的差异性；给出单位圆盘上Hardy空间上复合算子紧差分的刻画。通过本项目建立单位球及上半平面上加权Bergman空间复合算子差分的基本理论与研究方法，并解决单位圆盘上Hardy空间的复合算子紧差分的刻画问题。

本项目给出了单位圆盘上加权Bergman空间复合算子差分有界性、紧性刻画的正确证明，纠正了Saukko先期文章的重要错误。并把相关结论推广到单位球上加权Bergman空间，另外我们研究了双倍权Bergman空间的复合算子的复合算子差分紧性问题，把 Moorhouse关于标准权Bergman空间复合算子差分紧性刻画的结论推广到双倍权Bergman空间，给出双倍权Bergman空间的复合算子差分紧性刻画。进一步，我们给出单位球上双倍权Bergman空间的Carleson测度和Valterra 积分算子的有界性、紧性刻画。我们还研究了从有界解析函数空间到Bloch空间的加权复合算子差分的有界性、紧性刻画。我们还利用调和分析中的稀疏控制估计方法来研究上半平面上的Bergman空间加权复合算子，并给出该算子有界性、紧性的新刻画。通过本项目建立了加权Bergman空间复合算子差分的基本理论与研究方法，完善了Bergman空间复合算子差分的理论，也建立了上半复平面上Bergman空间的加权复合算子的稀疏控制估计理论。