Complete Hartogs domains in \({\mathbb{C}}^ 2\) have regular Bergman and Szegö projections (Q1107679)

The Bergman and Szegö projections do not preserve regularity of functions in all smooth bounded domains. The known counterexamples, constructed by Barrett and by Barrett and Fornæss, are certain (non- pseudoconvex) domains in \({\mathbb{C}}^ 2\) that are Hartogs domains. This paper delimits the nature of the obstruction to regularity exhibited by these domains. It is shown that the Bergman and Szegö projections exactly preserve differentiability of functions as measured by Sobolev norms in smooth bounded Hartogs domains in \({\mathbb{C}}^ 2\) that are complete (but not necessarily pseudoconvex). Consequently, biholomorphic mappings between such domains extend smoothly to the boundaries.