Composition operators on weighted Bergman spaces of the polydisc

TL;DR

This paper investigates the properties of composition operators on weighted Bergman spaces of the polydisc, focusing on continuity criteria influenced by the symbol's behavior on the polytorus, with detailed analysis for the tridisc.

Contribution

It provides a general continuity result based on symbol behavior on the polytorus and explores the complexity of characterizing continuity solely through derivatives.

Findings

Continuity depends on the symbol's behavior on the polytorus.

Automatic continuity results for induced operators.

Examples show derivative-based characterization is unlikely.

Abstract

We study composition operators between weighted Bergman spaces of the polydisc induced by smooth symbols. We prove a general result of continuity which only involves the behaviour of the symbol on the polytorus. We deduce from this several consequences about the automatic continuity of the induced operator. We study in depth the case of the tridisc and exhibit several examples showing that a characterization of continuity using only derivatives seems impossible.