Compactness and Singular Points of Composition Operators on Bergman spaces

Abstract

Let $\Omega \subset {\mathbb{C}}^n$ for $n\ge 2$ 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.

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