Small sets of infinite type are benign for the \overline∂-Neumann problem

Abstract

An explicit construction shows that the ∂ ¯ \bar \partial -Neumann operator and the Bergman and Szegő projections are globally regular in every smooth bounded pseudoconvex domain whose set of boundary points of infinite type has Hausdorff two-dimensional measure equal to zero. On the other hand there are examples of domains with globally regular ∂ ¯ \bar \partial -Neumann operator but whose infinite-type points fill out an open subset of the boundary.