Characterizing Weighted Composition Operators on Weighted-Type High-Order Growth Spaces via the Component Function $\varphi_p$

Let $\psi$ be a holomorphic function on the open unit ball $\BB \subset \C^N$, and let $\varphi$ be a holomorphic self-map of $\BB$, associated with normal weights $\nu$ and $\mu$. We consider the weighted composition operator $W_{\psi,\varphi} : \mathcal H_\nu^{(n)} \to \mathcal H_\mu^{(m)}, \quad n,m \in \N,$ acting between weighted-type high-order growth spaces. Unlike previous studies that involve the full symbol $\varphi$, this paper establishes characterizations of the boundedness, compactness, and asymptotic norm estimates of $W_{\psi,\varphi}$ \emph{solely in terms of the symbol $\psi$ and a single component function $\varphi_p$ of $\varphi$}, offering a new approach to the analysis of such operators.