The Diederich–Fornæss index I: For domains of non-trivial index

Abstract

We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich-Fornaess index defined in 1977. This connects the Diederich-Fornaess index to boundary conditions and refines the Levi pseudoconvexity. We also prove the β-worm domain is of index π/(2β). It is the first time that a precise non-trivial Diederich-Fornaess index in Euclidean spaces is obtained. This finding also indicates that the Diederich-Fornaess index is a continuum in (0, 1], not a discrete set. The ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations.global regularity for many other pseudoconvex domains (with smooth boundary) is still lacking a complete answer (see ). In particular, the∂-Neumann operator may or may not preserve the sobolev space W s (p,q) (Ω) of (p, q)-forms for all s > 0 (see Barrett [3], Kiselman [19] and Boas-Straube [5]) even under the category of smoothly bounded pseudoconvexity. This demands a refinement of pseudoconvexity to further classify the (weakly) pseudoconvex domains.In 1977, Diederich-Fornaess proved a celebrated theorem in [10]. The idea of the theorem is to replace the classical log composition in the notion of Hartogs pseudoconvexity with a power function composition. This classifies the notion of pseudoconvexity in the detail.Definition 0.2. Let Ω be a bounded pseudoconvex domain with C 2 boundary in C n . The function ρ is called a defining function if following conditions are satisfied:1. the function ρ is C 2 on a neighborhood of Ω, 2. the domain Ω = {z ∈ C n : ρ(z) < 0}, and 3. the gradient ∇ρ = 0 on ∂Ω.where the supremum is taken over all defining functions of Ω, is called the Diederich-Fornaess index of the domain Ω (see ).The index does a fundamental job to refine the notion of Hartogs pseudoconvexity in terms of the Sobolev regularity for the∂-Neumann operator and Bergman projection. For example, Berndtsson-Charpentier showed in [4], that the∂-Neumann operator and the Bergman projection preserves W k (Ω) for k < η 0 /2 when the Diederich-Fornaess index is η 0 . However, the Diederich-Fornaess index has not been understood thoroughly because the verifications and computations are rather difficult. In this paper, we understand the index by refining the notion of Levi pseudoconvexity. In other words, we define a natural index of the boundary and show this new index is equal to the Diederich-Fornaess index. This is given by Theorem 2.9. This theorem can be thought of as a refinement of the equivalence of Hartogs pseudoconvexity and Levi pseudoconvexity.To prove the theorem, we first simplify some common quantities and establish several useful identities. Many of the simplifications are motivated by the geometric analysis for Riemannian and Kähler manifolds. We also pass the plurisubharmonicity to the study of quadratic equations. Using the useful identities and the discriminant for quadratic equations, we are able to obtain the ...

On the Steinness Index

“…The author ( [10]), exploiting the idea of [7], characterized the Steinness index by way of a differential inequality on the set of weakly pseudoconvex boundary points (in fact, on the set of infinite type boundary points in the sense of D'Angelo). Also, we may induce a similar description for the Diederich-Fornaess index by the same argument.…”

“…We give here an explaination about the difference of Theorem 4.1 and the formula of Liu (Thoerem 2.9 in[7]). Liu dealt with a defining function of the form re ψ , where r is a defining function and ψ is a smooth function near the boundary.…”

The Diederich–Fornæss index II: For domains of trivial index

“…Just a while ago, an index associated to boundary was found by the author [22]. He showed this new index is equivalent to the Diederich-Fornaess index.…”

The Diederich-Fornaess index and good vector fields

“…The outline for the remainder of the paper is as follows: in Section 2, we will carefully define the structures that we need to provide a precise statement of our main result. In Section 3, we adapt ideas of Boas and Straube [4] to show that the existence of a family of good vector fields implies that a certain 1-form is exact (see [20] for a similar analysis along these lines). In Section 4 we combine this with a family of weight functions with large hessians in order to build a weight function that is well-suited to the study of the Diederich-Fornaess Index (see [21] for a more careful analysis of such weight functions).…”