The Diederich-Fornaess index and the global regularity of the di-bar-Neumann problem

TL;DR

This paper investigates the relationship between the Diederich-Fornaess index and the regularity of the Bergman projection, providing new estimates and extending known results to forms of general degree.

Contribution

It refines pseudodifferential calculus at the boundary and relates the Diederich-Fornaess index to Sobolev regularity of the Bergman projection for all form degrees.

Findings

Established a condition linking the Diederich-Fornaess index to Sobolev regularity.

Extended Kohn's results from functions to forms of any degree.

Refined boundary pseudodifferential calculus for better operator estimates.

Abstract

We describe along the guidelines of Kohn "Quantitative estimates..." (1999), the constant E_s which is needed to control the commutator of a totally real vector field T with di-bar* in order to have Sobolev s-regularity of the Bergman projection in any degree of forms, on a smooth pseudoconvex domain D of the complex space. This statement, not explicit in Kohn's paper, yields Straube's Theorem in "A sufficient condition..." (2008). Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function r of D, the operators (T^+)^{-delta/2} and (-r)^{delta/2}. We are thus able to extend to general degree of forms the conclusion of Kohn which only holds for functions: if for the Diederich-Fornaess index delta of D, we have that (1-\delta)^{1/2} < E_s, then the Bergman projection is s-regular.