Compactness of composition operators on the Bergman spaces of convex domains and analytic discs

TL;DR

This paper investigates when composition operators are compact on Bergman spaces of convex domains in complex space, especially those with boundary analytic discs, and characterizes this compactness for polydiscs.

Contribution

It provides a characterization of the compactness of composition operators with continuous symbols on Bergman spaces of convex domains, including polydiscs, considering boundary analytic discs.

Findings

Characterization of compactness for composition operators on convex domains.

Analysis of boundary analytic discs' impact on operator compactness.

Specific results for the polydisc case.

Abstract

We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in $\mathbb{C}^n$ with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the composition operator with a continuous symbol (up to the closure) on the Bergman space of the polydisc.