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Phillip S. Harrington

Publications and source records attributed to Phillip S. Harrington.

18 recordsLinked to original sources
arXiv2024

Sobolev Regularity of the Bergman Projection on a Smoothly Bounded Stein Domain that is not Hyperconvex

For every $0<r<\frac{1}{2}$, we will construct a flat Kähler manifold $M$ and a relatively compact domain with smooth boundary $Ω\subset M$ that is Stein but not hyperconvex such that the Bergman projection $P$ on $Ω$ is regular in the $L^2$ Sobolev space $W^s(Ω)$ for all $0\leq s<r$ but irregular in $W^r(Ω)$. On these domains, we will also construct $f\in C^\infty(\overlineΩ)$ such that $Pf\notin C^\infty(\overlineΩ)$. We will prove the same result for the invariant Bergman projection on $(2,0)$-forms. These domains are modelled on a construction of Diederich and Ohsawa.

math.CV↗
arXiv2024

Sobolev Regularity for the Bergman Projection on Relatively Compact Domains in Hermitian manifolds

Generalizing a result of Berndtsson and Charpentier, we provide sufficient conditions for $L^2$ Sobolev regularity of the Bergman projection acting on $L^2$ sections of a holomorphic line bundle restricted to a relatively compact domain with Lipschitz boundary in a Hermitian manifold. We provide examples to show that our methods work for domains in Hopf manifolds endowed with a suitable Hermitian metric.

math.CV↗
arXiv2024

The $\bar\partial$-problem on $Z(q)$-domains

Given a complex manifold containing a relatively compact $Z(q)$ domain, we give sufficient geometric conditions on the domain so that its $L^2$-cohomology in degree $(p,q)$ (known to be finite dimensional) vanishes. The condition consists of the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms.

math.CV↗
arXiv2023

The Strong Diederich-Fornæss Index on $C^2$ Domains in Hermitian Manifolds

For a relatively compact Stein domain $Ω$ with $C^2$ boundary in a Hermitian manifold $M$, we consider the strong Diederich-Fornæss index, denoted $DF(Ω)$: the supremum of all exponents $0<η<1$ such that eigenvalues of the complex Hessian of $-(-ρ)^η$ are bounded below by some positive multiple of $(-ρ)^η$ on $Ω$ for some $C^2$ defining function $ρ$. We will show that $DF(Ω)$ is completely characterized by the existence of a Hermitian metric with curvature terms satisfying a certain inequality when restricted to the null-space of the Levi-form.

math.CV↗
arXiv2022

Maximal Estimates for the $\bar\partial$-Neumann Problem on Non-pseudoconvex domains

It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$). In this paper, we explore necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. We also discuss when the necessary conditions and sufficient conditions agree and provide examples. Our results subsume the earlier known results from the pseudoconvex case.

math.CV↗
arXiv2019

On Competing Definitions for the Diederich-Fornæss Index

Let $Ω\subset\mathbb{C}^n$ be a bounded pseudoconvex domain. We define the Diederich-Fornæss index with respect to a family of functions to be the supremum over the set of all exponents $0<η<1$ such that there exists a function $ρ_η$ in this family such that $-ρ_η$ is comparable to the distance to the boundary of $Ω$ on $Ω$ and such that $-(-ρ_η)^η$ is plurisubharmonic on $Ω$. We first prove that computing the Diederich-Fornæss index with respect to the family of upper semi-continuous functions is the same as computing the Diederich-Fornæss index with respect to the family of Lipschitz functions. When the boundary of $Ω$ is $C^k$, $k\geq 2$, we prove that the Diederich-Fornæss index with respect to the family of $C^k$ functions is the same as the Diederich-Fornæss index with respect to the family of $C^2$ functions.

math.CV↗
arXiv2019

Hartogs Domains and the Diederich Fornæss 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-Fornæss Index. Using this, we are able to show that under some reasonable hypotheses on the set of weakly pseudoconvex points, the Diederich-Fornæss 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-Fornæss Index for a Hartogs domain is equal to one then the domain admits a Stein neighborhood basis.

math.CV↗
arXiv2019

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.

math.CV↗
arXiv2018

A Modified Morrey-Kohn-Hörmander Identity and Applications

We prove a modified form of the classical Morrey-Kohn-Hörmander 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 $q$.

math.CV↗
arXiv2018

Boundary invariants and the closed range property for $\bar\partial$

This paper provides a connection between two distinct branches of research in CR geometry -- namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR manifolds. Specifically, we work on not necessarily pseudoconvex domains $Ω\subset\mathbb{C}^n$ and define third and fourth order CR invariants on $\bd\Om$ and show that these invariants provide enough information to establish closed range for the $\bar\partial$-Laplacian in $L^2_{(0,q)}(Ω)$ for a given, fixed $q$. The closed range estimates follow from our previously defined weak $Z(q)$ condition. We also develop powerful linear algebra machinery to translate the information from the invariants into information about the Levi form and its eigenvalues. We conclude with several examples that demonstrate the usefulness and ease of use of the new conditions.

math.CV↗
arXiv2018

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.

math.CV↗
arXiv2017

Closed range of $\bar\partial$ on unbounded domains in $\mathbb C^n$

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.

math.CV↗
arXiv2017

Closed range of $\bar\partial$ in $L^2$-Sobolev spaces on unbounded domains in $\mathbb{C}^n$

Let $Ω\subset\mathbb{C}^n$ be a domain and $1 \leq q \leq n-1$ fixed. Our purpose in this article is to establish a general sufficient condition for the closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$-Sobolev spaces on $(0,q)$-forms. 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.

math.CV↗
arXiv2015

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

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.

math.CV↗
arXiv2015

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

In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $Ω$ in $\mathbb{CP}^n$ there exists a Lipschitz defining function $ρ$ and an exponent $0<η<1$ such that $-(-ρ)^η$ is strictly plurisubharmonic on $Ω$. 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 $ρ=-δ$, where $δ$ is the geodesic distance function for the boundary of $Ω$.

math.CV↗
arXiv2013

Regularity equivalence of the Szegö projection and the complex Green operator

In this paper we prove that on a CR manifold of hypersurface type that satisfies the weak $Y(q)$ condition, the complex Green operator $G_q$ is exactly (globally) regular if and only if the Szegö projections $S_{q-1}, S_q$ and a third orthogonal projection $S'_{q+1}$ are exactly (globally) regular. The projection $S'_{q+1}$ is closely related to the Szegö projection $S_{q+1}$ and actually coincides with it if the space of harmonic $(0,q+1)$-forms is trivial. This result extends the important and by now classical result by H. Boas and E. Straube on the equivalence of the regularity of the $\bar\partial$-Neumann operator and the Bergman projections on a smoothly bounded pseudoconvex domain. We also prove an extension of this result to the case of bounded smooth domains satisfying the weak $Z(q)$ condition on a Stein manifold.

math.CV↗