Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in CPn\mathbb{CP}^n

In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $\Omega$ in $\mathbb{CP}^n$ there exists a Lipschitz defining function $\rho$ and an exponent $0<\eta<1$ such that $-(-\rho)^\eta$ is strictly plurisubharmonic on $\Omega$. This generalizes a result of Ohsawa and Sibony for $C^2$ domains. In contrast to the Ohsawa-Sibony result, we provide a counterexample demonstrating that we may not assume $\rho=-\delta$, where $\delta$ is the geodesic distance function for the boundary of $\Omega$

The Diederich-Fornaess Index and Good Vector Fields

We consider the relationship between two sufficient conditions for regularity of the Bergman Projection on smooth, bounded, pseudoconvex domains. We show that if the set of infinite type points is reasonably well-behaved, then the existence of a family of good vector fields in the sense of Boas and Straube implies that the Diederich-Fornaess Index of the domain is equal to one.Comment: This version contains some additional corrections to the previous version

The ∂ˉ\bar{\partial}-Neumann operator with the Sobolev norm of integer orders

Let $\Omega\subset\mathbb{C}^m$ be a bounded pseudoconvex domain with smooth boundary. For each $k\in\mathbb{N}$, we give a sufficient condition to estimate the $\bar\partial$-Neumann operator in the Sobolev space $W^k(\Omega)$. The key feature of our results is a precise formula for $k$ in terms of the geometry of the boundary of $\Omega$

Closed Range for ∂ˉ\bar\partial and ∂ˉb\bar\partial_b on Bounded Hypersurfaces in Stein Manifolds

We define weak $Z(q)$, a generalization of $Z(q)$ on bounded domains $\Omega$ in a Stein manifold $M^n$ that suffices to prove closed range of $\bar\partial$. Under the hypothesis of weak $Z(q)$, we also show (i) that harmonic $(0,q)$-forms are trivial and (ii) if $\partial\Omega$ satisfies weak $Z(q)$ and weak $Z(n-1-q)$, then \dbar_b has closed range on $(0,q)$-forms on $\partial\Omega$. We provide examples to show that our condition contains examples that are excluded from $(q-1)$-pseudoconvexity and the authors' previous notion of weak $Z(q)$.Comment: 29 page

Hartogs Domains and the Diederich Forn{\ae}ss Index

We study a geometric property of the boundary on Hartogs domains which can be used to find upper and lower bounds for the Diederich-Forn{\ae}ss Index. Using this, we are able to show that under some reasonable hypotheses on the set of weakly pseudoconvex points, the Diederich-Forn{\ae}ss Index for a Hartogs domain is equal to one if and only if the domain admits a family of good vector fields in the sense of Boas and Straube. We also study the analogous problem for a Stein neighborhood basis, and show that under the same hypotheses if the Diederich-Forn{\ae}ss Index for a Hartogs domain is equal to one then the domain admits a Stein neighborhood basis.Comment: 28 page

Closed range of ∂ˉ\bar\partial on unbounded domains in Cn\mathbb C^n

In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$ and $L^2$-Sobolev spaces on $(0,q)$-forms for a fixed $q$ on domains in $\mathbb{C}^n$. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical $Z(q)$ condition that we call weak $Z(q)$. We provide examples that explain the necessity of working in weighted spaces both for closed range in $L^2$ and even more critically, in $L^2$-Sobolev spaces.Comment: 19 pages. We split the original paper into two, smaller paper

Strong Closed Range Estimates: Necessary Conditions and Applications

The $L^2$ theory of the $\bar\partial$ operator on domains in $\mathbb{C}^n$ is predicated on establishing a good basic estimate. Typically, one proves not a single basic estimate but a family of basic estimates that we call a family of strong closed range estimates. Using this family of estimates on $(0,q)$-forms as our starting point, we establish necessary geometric and potential theoretic conditions. The paper concludes with several applications. We investigate the consequences for compactness estimates for the $\bar\partial$-Neumann problem, and we also establish a generalization of Kohn's weighted theory via elliptic regularization. Since our domains are not necessarily pseudoconvex, we must take extra care with the regularization.Comment: 26 page

A remark on boundary estimates on unbounded Z(q)Z(q) domains in Cn\mathbb{C}^n

The goal of this note is to explore the relationship between the Folland-Kohn basic estimate and the $Z(q)$-condition. In particular, on unbounded pseudoconvex (resp., pseudoconcave) domains, we prove that the Folland-Kohn basic estimate is equivalent to uniform strict pseudoconvexity (resp., pseudoconcavity). As a corollary, we observe that despite the Siegel upper half space being strictly pseudoconvex and biholomorphic to a the unit ball, it fails to satisfy uniform strict pseudoconvexity and hence the Folland-Kohn basic estimate fails. On unbounded non-pseudoconvex domains, we show that the Folland-Kohn basic estimate on $(0,q)$-forms implies a uniform $Z(q)$ condition, and conversely, that a uniform $Z(q)$ condition with some additional hypotheses implies the Folland-Kohn basic estimate for $(0,q)$-forms.Comment: 6 page

Sobolev Spaces and Elliptic Theory on Unbounded Domains in Rn\mathbb R^n

In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb R^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case.Comment: 47 page

A Modified Morrey-Kohn-H\"ormander Identity and Applications

We prove a modified form of the classical Morrey-Kohn-H\"ormander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in $\mathbb{C}^n$, where the inner domain has $\mathcal{C}^{1,1}$ boundary, we show that the $L^2$ Dolbeault cohomology group in bidegree $(p,q)$ vanishes if $1\leq q\leq n-2$ and is Hausdorff and infinite-dimensional if $q=n-1$, so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the $L^2$ Sobolev space $W^1$ on any pseudoconvex domain with $\mathcal{C}^{1,1}$ boundary. We also generalize our results to annuli between domains which are weakly $q$-convex in the sense of Ho for appropriate values of $q$.Comment: Version 2: some minor typos have been fixe