Necessary geometric and analytic conditions for general estimates in the $\bar{\partial}$ -Neumann problem

Abstract

We show the geometric and analytic consequences of a general estimate in the∂-Neumann problem: a "gain" in the estimate yields a bound in the "type" of the boundary, that is, in its order of contact with an analytic curve as well as in the rate of the Bergman metric. We also discuss the potential-theoretical consequence: a gain implies a lower bound for the Levi form of a bounded weight.

Boundary regularity of the solution to the complex Monge-Ampère equation on pseudoconvex domains of infinite type

“…The f -Property consists in the existence of a bounded family of weights in the spirit of [Cat87] and it is sufficient for an f -estimate for the∂-Neumann problem [Cat87,KZ10]. We also notice that when lim t→∞ f (t) log t = ∞ the solution of the∂-Neumann problem is regular [Koh02,KZ12]. with f (t) t 1/2 decreasing, we say that Ω has an f -Property if there exist a neigborhood U of bΩ and a family of functions {φ δ } such that (i) the functions φ δ are plurisubharmonic, C 2 on U, and satisfy −1 ≤ φ δ ≤ 0, and…”

Hypoellipticity of the $\overline{\partial}$-Neumann problem at a point of infinite type

“…Similar is the notion of hypoellipticity of the Laplacian or the regularity of the inverse Neumann operator N = −1 . It has been proved by Kohn in [17] and by the two last authors in [14] that superlogarithmic estimates suffice for local hypoellipticity of the problem in the boundary and in the domain. (Note that hypoellipticity for the domain, [17] Theorem 8.3, is deduced from microlocal hypoellipticity for the boundary, [17] Theorem 7.1, but a direct proof is also available, [10] Theorem 5.4.)…”

On the Bergman metric on bounded pseudoconvex domains an approach without the Neumann operator

“…In [7,Theorem 1.4 (ii)], the Bergman metric of a smoothly bounded pseudoconvex domain D is estimated near a boundary point z 0 ∈ ∂D in a more general manner as follows. If f : R + → R + is a smooth increasing function and an f -estimate for the∂ Neumann problem at the level of (0, 1)-forms on D is satisfied on a neighborhood of z 0 , (for the notion "f -estimate" see [7, Definition 1.2]) then, given a positive number η > 0, there exists a constant c η , such that for any X ∈ C n and z ∈ D close to z 0 one has…”