# Compactness and Singular Points of Composition Operators on Bergman spaces

## TL;DR

This paper investigates the conditions under which composition operators on Bergman spaces of convex domains are compact, focusing on boundary Jacobian behavior and providing counterexamples to previous assumptions.

## Contribution

It offers a partial characterization of compactness for composition operators with regular symbols and demonstrates that the converse of this characterization does not hold.

## Key findings

- Boundary Jacobian behavior influences compactness
- Partial characterization of compactness established
- Counterexample disproves the converse implication

## Abstract

Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain with a $C^2$-smooth boundary. We study the compactness of composition operators on the Bergman spaces of smoothly bounded convex domains. We give a partial characterization of compactness of the composition operator (with sufficient regularity of the symbol) in terms of the behavior of the Jacobian on the boundary. We then construct a counterexample to show the converse of the theorem is false.