Compactness and Singular Points of Composition Operators on Bergman spaces

Abstract

Let Ω ⊂ C n for n ≥ 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.