Exact regularity of the Bergman and the Szegö projections on domains with partially transverse symmetries (Q1117363)

If D is a smooth bounded pseudoconvex domain in \({\mathbb{C}}^ n\) that has symmetries transverse on the complement of a compact subset of the boundary consisting of points of finite type, then the Bergman projection for D maps the Sobolev space \(W^ r(D)\) continuously into itself and the Szegö projection maps the Sobolev space \(W^ r(bD)\) continuously into itself. If D has symmetries, coming from a group of rotations, that are transverse on the complement of a B-regular subset of the boundary, then the Bergman projection, the Szegö projection, and the \({\bar \partial}\)-Neumann operator on (0,1)-forms all exactly preserve differentiability measured in Sobolev norms. The results hold, in particular, for all smooth bounded strictly complete pseudoconvex Hartogs domains in \({\mathbb{C}}^ 2\), as well as for Sibony's counterexample domain that fails to have sup-norm estimates for solutions of the \({\bar \partial}\)-equation.