Exact regularity of the Bergman and Szeg� projections on domains with partially transverse symmetries

Abstract

IfD is a smooth bounded pseudoconvex domain in Cn that has symmetries transverse on the complement of a compact subset of the boundary consisting of points of finite type, then the Bergman projection forD maps the Sobolev spaceWr(D) continuously into itself and the Szego projection maps the Sobolev spaceWsur(bD) continuously into itself. IfD has symmetries, coming from a group of rotations, that are transverse on the complement of aB-regular subset of the boundary, then the Bergman projection, the Szego 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 C2, as well as for Sibony's counterexample domain that fails to have sup-norm estimates for solutions of the\(\bar \partial \)-equation.