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

TL;DR

The paper addresses the boundedness and compactness of weighted composition operators $W_{\\psi,\\varphi}$ between weighted-type high-order growth spaces on the unit ball, showing that these properties can be fully characterized using only the symbol $\\psi$ and a single component $\\varphi_p$ of the map $\\varphi$. It introduces the $(n,\\mu)$-condition and a family of functionals $\\mathfrak B_{n,j}$ and $\\mathscr B_{j}^n$ to quantify operator action, proving equivalences that tie boundedness to finiteness of these functionals and the inclusion $\\psi, \\psi\\cdot\\varphi_p^i \\in \\mathcal{H}^{(n)}_{\\mu}$, with the norm controlled by $|\\psi(0)|\\|\\delta_{\\varphi(0)}^{\\mathcal{H}^{(n+m)}_{\\nu}}\\| + \sum_{j=0}^n \\mathscr B_{j,p}^n$. The results extend existing Bloch- and Zygmund-type analyses to high-order growth spaces and provide precise compactness criteria via limit conditions and integral finiteness $I^{k}_\\nu(1)$. By leveraging ball automorphisms and the geometry of the sets $\\widetilde{S}_p({\Bbb B})$, this work offers a new framework for reducing the study of weighted composition operators to component-symbol data with clear operator-norm and asymptotic behavior implications.

Abstract

Let $ψ$ 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 $ν$ and $μ$. We consider the weighted composition operator $ W_{ψ,\varphi} : \mathcal H_ν^{(n)} \to \mathcal H_μ^{(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 $W_{ψ,\varphi}$ \emph{solely in terms of the symbol $ψ$ and a single component function $\varphi_p$ of $\varphi$}, offering a new approach to the analysis of such operators.

Theorems & Definitions (21)

• Proposition 1: Qu1, Proposition 3.3
• Example 1
• Example 2
• Example 3
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof
• Example 4