Czechoslovak Mathematical Journal, first online, pp. 1-18
Composition operators on variable exponent Bloch spaces
Xin He, Cezhong Tong, Zicong Yang, Zehua Zhou
Received September 18, 2024. Published online March 11, 2025.
Abstract: We consider the composition operator $C_{\varphi}$ on the variable exponent Bloch space $\mathcal{B}^{\alpha({\cdot})}$, which consists of all analytic functions $f$ on the unit disk $\mathbb{D}$ such that $\sup\{(1-|z|^2)^{\alpha(z)}|f'(z)| \colon z\in\mathbb{D} \}<\infty.$ Here, $\alpha(z)$ is a log-Hölder continuous function on $\mathbb{D}$. The boundedness and compactness of $C_{\varphi}$ are characterized. Besides, we show that $(1-|z|^2)^{\alpha(z)}f'(z)$ is Lipschitz continuous in terms of the pseudo-hyperbolic metric under the Lipschitz continuity of $\alpha(z)$. By using this result, we study the bounded and compact difference $C_{\varphi}-C_{\psi}$ of two composition operators on $\mathcal{B}^{\alpha({\cdot})}$, and the boundedness from below of $C_{\varphi}$ is partially described.
Affiliations: Xin He, Cezhong Tong, Zicong Yang (corresponding author), Department of Mathematics, Hebei University of Technology, No. 5340, Xiping Road, Beichen District, Tianjin 300401, P. R.China, e-mail: hexinsx2000@163.com, ctong@hebut.edu.cn, zicongyang@126.com, zc25@hebut.edu.cn; Zehua Zhou, School of Mathematics, Tianjin University, No. 135, Ya Guan Road, Jinnan District, Tianjin 300354, P. R. China, e-mail: zehuazhoumath@aliyun.com
