Weighted Composition Operators from H ∞ to α , m -Bloch Space on Cartan-Hartogs Domain of the First Type

Abstract

Let $Y_I$ be nonhomogeneous Cartan-Hartogs domain of the first type, $\varphi$ a holomorphic self-map, and $\psi$ a fixed holomorphic function on $Y_I$ . We study the weighted composition operator $\psi C_{\varphi }f=\psi \left(f\circ \varphi \right)$ for a function $f$ holomorphic on $Y_I$ . Our main results generalize both cases of the unit ploydisc and the unit ball obtained by Li and Stević (Li 2007 and Li 2008). Firstly, we obtain two crucial inequalities on $Y_I$ ; furthermore, the boundedness and compactness of operator $\psi C_{\varphi }$ from the space $H^{\infty }$ of all bounded holomorphic functions to the $\left(\alpha ,m\right)$ -Bloch space ${\mathcal{B}}^{\left(\alpha ,m\right)}$ on $Y_I$ are investigated.