On plurisubharmonic defining functions for pseudoconvex domains in $\mathbb{C}^2$

Abstract

We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in C 2 . In particular, we construct a family of simple counterexamples to the existence of plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains.

Diederich-Fornæss index and global regularity in the $\overline{\partial}$-Neumann problem: domains with comparable Levi eigenvalues

“…The supremum over all such η has come to be known as the Diederich-Fornaess index of Ω. The relevance of the index stems from the fact that in general, Ω does not admit a defining function that is plurisubharmonic on Ω near the boundary, not even locally ( [18], [2], [20]).…”

“…First, index one does not imply that there is a defining function that is plurisubharmonic on Ω (near bΩ). Indeed, domains with real analytic boundaries are of finite type, so satisfy property(P), yet need not admit even local plurisubharmonic defining functions ( [2,20]). Second, and perhaps more strikingly, all domains in the list with index one are known to have globally regular Bergman projections and ∂-Neumann operators ( [8,9,10,33,22]), while on the worm domains, these operators are regular only up to a finite Sobolev level that is closely related to their index (see [6,1,13,29]).…”