Good Stein neighborhood bases and regularity of the $\overline\partial$-Neumann problem

Abstract

We show that the $\bar\partial$-Neumann problem is globally regular on a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$ whose closure admits a sufficiently nice Stein neighborhood basis. We also discuss (what turns out to be) a generalization: global regularity holds as soon as the weakly pseudoconvex directions at boundary points are limits, from inside, of weakly pseudoconvex directions of level sets of the boundary distance.