本项目主要研究对象是与Bergman核相关的几类积分算子,注重泛函分析、微分几何、PDE、调和分析等现代数学分支在函数空间与算子理论方向的应用与融合.首先,研究Siegel上半空间上的Forelli-Rudin算子理论,建立与Bergman投影相关的一类积分算子从经典权函数空间L^p到L^q的有界性,给出Bergman投影在一般权函数空间L^p(u)上的精确范数.其次,研究Berezin型算子从Bergman空间A^p到L^q空间的有界性、紧性、Schatten类.最后,给出Toeolitz算子在一般径向权Bergman空间上的本性范数,开展Hankel算子在一般径向权Bergman空间上的理论研究.

The project deals with several classes of integral operators related to Bergman kernel with paying attention to the application and the combination of Functional Analysis、Differential Geometry、PDE and Harmonic Analysis in the research of function spaces and operator theory. First, we will study the theory of Forelli-Rudin operators over Siegel upper half-space. We are going to establish the boundedness of a class of integral operator related to Bergman projection from L^p to L^q space under classical weights and give the accurate L^p norm of Bergman projection on L^p space under general weights. Second, we will study the boundedness、the compactness and Schatten class of Berezin-type operator from Bergman space A^p to L^q space. At last, we will calculate the essential norm of Toeplitz operator on weighted Bergman space under general radial weights, and develop the theory of Hankel operator on weighted Bergman space under general radial weights.