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

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.