ABSTRACT
Thhis chapter discusses the regularity property Condition R of the Bergman projection operator and its connection to the Neumann operator. Kiselman constructs additional examples, including a smoothly bounded pseudoconvex domain in a two-dimensional complex manifold, for which Condition R fails. For the class of domains of finite type, the proof that Condition R holds requires at present the machinery of the ∂-Neumann problem and Catlin's theorem on subelliptic estimates. The extension will be a finite holomorphic mapping in some neighborhood of any boundary point. By the Baouendi-Rothschild result a proper mapping f between them extends holo- morphically past the origin, and is a finite mapping there.
