函数空间上的算子理论与算子代数是现代数学的重要分支之一，而加权复合算子与内函数是其中两个非常重要的研究对象。这是因为它不仅为一般算子理论研究提供实例，而且加权复合算子与内函数的性质强烈依赖于相应的函数空间，从而在算子理论与经典函数理论之间建立起了内在联系。Dirichlet型空间D_p与莫比乌斯不变Dirichlet型空间Q_p是两类重要的解析函数空间，其上的加权复合算子与内函数的研究还不完善。本项目我们将利用解析自映射的高价刻画，对数导数，Carleson曲线，对偶及伪解析延拓等研究方法，研究D_p空间，Q_p空间及其在Bloch空间闭包上的加权复合算子与内函数的各种性质。

Operator theory and operator algebra on function spaces is one of very important branchs of mordern mathematics, weighted composition operator and inner function are two important subject in this field. Because they provide some examples for general operator theory, the properties of weighted composition operator and inner function strongly depend on the corresponding function space. Thus, operator theory is related with the classcial function theory. Dirichlet type spaces D_p and Mobius invariant Dirichlet type spaces Q_p are two important analytic function spaces, and the theory of weighted composition operator and inner function on these spaces are not plentiful. In this project, using higher order of analytic self-map, logarithmic derivatives, Carleson curve, duality and pseudoanalytic extension, we will characterize various properties of weighted composition operator and inner functions on D_p spaces, Q_p spaces and the closures of these spaces in Bloch spaces.