Bloch-Type空间$B_{\alpha }$与$Z_p$空间上的超复合算子

Superposition Operators between Bloch-Type Spaces $B_{\alpha }$ and $Z_p$ Spaces

摘要: 复合算子在泛函分析与算子理论当中有丰富的应用与深厚的研究背景，在泛函分析中，人们关注各种函数空间的性质与结构，如$L^p$ 空间和连续函数空间$C\left(X\right)$ 等。复合函数作为一种将函数空间进行变换的工具，有助于深入理解函数空间之间的关系与结构，例如通过研究复合算子在不同函数空间上的作用，可以揭示函数空间的嵌入性、紧性以及有界性等等。不仅如此，复合算子在动力系统与遍历理论、量子力学、算子代数等数学分支当中也作为算子论的工具，推动其他数学分支的发展。而超复合算子是指以整函数作为复合子的特殊复合算子，推动着算子论的发展：超复合算子的有界性是研究复合算子当中比较重要的部分。近年来，有许多学者研究了超复合算子作用在一些经典的解析函数空间上的有界性。本文通过利用$Z_p$ 空间的定义与性质，讨论了超复合算子上Bloch-type空间$B_{\alpha }$ 到$Z_p$ 空间的有界性问题以及Bloch-type空间$B_{\alpha }$ 到Morrey空间上的有界性问题。

Abstract: Composite operators have extensive applications and a profound research background in functional analysis and operator theory. In functional analysis, researchers focus on the properties and structures of various function spaces, such as $L^p$ spaces and the continuous function space $C\left(X\right)$ . As a tool for transforming function spaces, composite operators contribute to a deeper understanding of the relationships and structures among function spaces. For instance, by studying the actions of composite operators on different function spaces, one can reveal properties like the embedding, compactness, and boundedness of function spaces. Moreover, composite operators as tools in operator theory, also play a role in promoting the development of other mathematical branches, such as dynamical systems and ergodic theory, quantum mechanics, and operator algebras. Superposite operators are special composite operators with entire functions as the composition elements, and they also drive the development of operator theory. The boundedness of Superposite operators is a crucial part of the study of composite operators. In recent years, many scholars have investigated the boundedness of Superposite operators acting on some classical analytic function spaces. This paper, by making use of the definitions and properties of $Z_p$ spaces, derives the boundedness problem of Superposite operators from the Bloch-type space $B_{\alpha }$ to the $Z_p$ space and Bloch-type space $B_{\alpha }$ to Morrey space.