The Diederich--Fornæss index and the regularities on the $\bar{\partial}$-Neumann problem

Abstract

We show, under an assumption on the weakly pseudoconvex points, the trivial Diederich-Fornaess index directly implies the global regularities of the ∂-Neumann operators.

Maximal Estimates for the $\bar\partial$-Neumann Problem on Non-pseudoconvex domains

“…Maximal estimates have many applications. In addition to those given by Derridj and Ben Moussa, see also [6], [22], or [23].…”

The $\bar{\partial}$-Neumann problem and boundary integral equations

“…There are some recent results on sufficient conditions of global regularity, see e.g., Straube [14], Staube-Zeytuncu [16], Harrington [7], Harrington-Liu [6], Liu-Raich [11] and Liu [10]. However, the complete answer, to the cut of when the regularity holds or not, is still unknown.…”