Weighted Composition Operators from Bloch to Zygmund Spaces: An Alternative Characterization

Abstract

Let $\psi$ be a holomorphic function and $\varphi$ a holomorphic self-map of the unit ball $\mathbb{B} \subset \mathbb{C}^N$. 
Denote by $\mathcal{B}_{\nu}$ and $\mathcal{Z}_{\mu}$ the weighted Bloch-type and Zygmund-type spaces with normal weights $\nu$ and $\mu$. 
The weighted composition operator
\[
W_{\psi,\varphi} :
\mathcal{B}_{\nu}\,(\text{or } \mathcal{B}_{\nu,0}) \to
\mathcal{Z}_{\mu}\,(\text{or } \mathcal{Z}_{\mu,0}), \quad f \mapsto \psi \cdot (f \circ \varphi),
\]
is studied in terms of boundedness, compactness, and asymptotic norm behavior. 
The obtained characterizations are expressed via analytic properties of the symbols $\psi$ and $\varphi$, particularly a component function $\varphi_k$ of $\varphi$.