Composition Operators on $\bf H^{p,q,s}(B_{n})$ of $\bf \mathbb{C}^{n}$

TL;DR

The paper investigates composition operators $C_\varphi$ on the general Hardy type space $H^{p,q,s}(B_n)$ for holomorphic mappings $\varphi: B_n\to B_n$, aiming to characterize when $C_\varphi$ is bounded or compact. It develops a framework built on growth estimates (Lemma 2.1), Bergman-type integral representations (Lemmas 2.2–2.3), and kernel bounds (Lemmas 2.4–2.5) to analyze the action of composition operators. The main contributions are Theorems 3.1 and 3.2, which provide necessary and sufficient conditions for boundedness and compactness, including the automorphism case, a limit ratio criterion for compactness, and a connection to boundedness on $H^{pn/(q+n)}(B_n)$. This work extends classical results for Hardy and weighted Hardy spaces to the full family of general Hardy type spaces, offering a unified approach to operator theory on holomorphic function spaces in the unit ball.

Abstract

Let $B_{n}$ be the unit ball in the complex vector space $\mathbb{C}^{n}$, and let $\varphi: B_{n}\rightarrow B_{n}$ be a holomorphic mapping. In this paper, we characterize those symbols $\varphi$ such that composition operators $C_{\varphi}$ are bounded or compact on the general Hardy type space $H^{p,q,s}(B_{n})$. These results extend the relevant results on Hardy space and some other classical function spaces.