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

TL;DR

This paper characterizes when composition operators induced by holomorphic maps are bounded or compact on a generalized Hardy space in several complex variables, extending classical results to more general function spaces.

Contribution

It provides a comprehensive characterization of boundedness and compactness of composition operators on $H^{p,q,s}(B_{n})$, broadening the understanding beyond classical Hardy spaces.

Findings

Characterization of symbols $$ for bounded composition operators.

Criteria for compactness of composition operators.

Extension of classical Hardy space results to generalized spaces.

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.