Characterizing Weighted Composition Operators on Weighted-Type High-Order Growth Spaces via the Component Function ${\phi }_p$

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