Small Sets of Infinite Type are Benign for the $\overline\partial$-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.