Phillip S. Harrington

activity

20092024

\partial ˉ

L^{2}

-Sobolev spaces on unbounded domains in

C^{n}

3 citations · 15 across the 17 of their papers we have counts for

collaborators

22 papers

math.CV2024

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

Debraj Chakrabarti, Phillip S. Harrington, Andrew Raich

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…

math.CV2024

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

Phillip S. Harrington

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 t…

math.CV2022

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

Phillip S. Harrington

For a relatively compact Stein domain $Ω$ with $C^{2}$ boundary in a Hermitian manifold $M$ , we consider the strong Diederich-Fornæss index, denoted $D F (Ω)$ : the supremum of all expo…

math.CV2022

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

Phillip S. Harrington, Andrew Raich

It is well known that elliptic estimates fail for the $\partial ˉ$ -Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be…

math.CV2020

Exact sequences and estimates for the $\partial$ -problem

Debraj Chakrabarti, Phil Harrington

We study Sobolev estimates for solutions of the inhomogenous Cauchy-Riemann equations on annuli in $\cx^n$, by constructing exact sequences relating the Dolbeault cohomology of the…

math.CV2019

On Competing Definitions for the Diederich-Fornæss Index

Phillip S. Harrington

Let $Ω \subset C^{n}$ 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 expo…

math.CV2019★ 2 cited

The $\partial ˉ$ -Neumann operator with the Sobolev norm of integer orders

Phillip Harrington, Bingyuan Liu

Let $Ω \subset C^{m}$ be a bounded pseudoconvex domain with smooth boundary. For each $k \in N$ , we give a sufficient condition to estimate the $\partial ˉ$ -Neumann o…

math.CV2019

Strong Closed Range Estimates: Necessary Conditions and Applications

Phillip S. Harrington, Andrew Raich

The $L^{2}$ theory of the $\partial ˉ$ operator on domains in $C^{n}$ is predicated on establishing a good basic estimate. Typically, one proves not a single basic estimate…

math.CV2018★ 2 cited

A Modified Morrey-Kohn-Hörmander Identity and Applications

Debraj Chakrabarti, Phillip S. Harrington

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 d…

math.CV2018

Hartogs Domains and the Diederich Fornæss Index

Muhenned Abdulsahib, Phillip S. Harrington

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 t…

math.CV2018

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

Phillip S. Harrington, Andrew Raich

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…

math.CV2017★ 2 cited

$L^{p}$ Mapping Properties for the Cauchy-Riemann Equations on Lipschitz Domains Admitting Subelliptic Estimates

Phillip S. Harrington, Yunus E. Zeytuncu

We show that on bounded Lipschitz pseudoconvex domains that admit good weight functions the $\partial$ -Neumann operators $N_{q}, \partial^{*} N_{q}$ , and $\overli…

math.CV2017

The Diederich-Fornaess Index and Good Vector Fields

Phillip S. Harrington

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 infinit…

math.CV2017★ 3 cited

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

Phillip S. Harrington, Andrew S. Raich

Let $Ω \subset C^{n}$ 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-…

math.CV2015

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

Phillip S. Harrington, Andrew Raich

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., pseudoconcav…

math.CV2015★ 2 cited

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

Phillip S. Harrington

In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $Ω$ in $CP^{n}$ there exists a Lipschitz defining function $ρ$ and an exponent…

math.CV2015★ 1 cited

Closed range of $\partial ˉ$ on unbounded domains in $C^{n}$

Phillip S. Harrington, Andrew Raich

In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator $\partial ˉ$ in appropriately weighted $L^{2}$ and $L^{2}$ -Sobolev spaces…

math.CV2013

Regularity equivalence of the Szegö projection and the complex Green operator

Phillip S. Harrington, Marco M. Peloso, Andrew S. Raich

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 on…

math.AP2012

Sobolev Spaces and Elliptic Theory on Unbounded Domains in $R^{n}$

Phillip S. Harrington, Andrew Raich

In this article, we develop the theory of weighted $L^{2}$ Sobolev spaces on unbounded domains in $R^{n}$ . As an application, we establish the elliptic theory for elliptic oper…

math.DG2011

Defining Functions for Unbounded $C^{m}$ Domains

Phillip Harrington, Andrew Raich

For a domain $Ω \subset R^{n}$ , we introduce the concept of a uniformly $C^{m}$ defining function. We characterize uniformly $C^{m}$ defining functions in terms of the signed dista…

math.CV2011★ 3 cited

Closed Range for $\partial ˉ$ and $\partial ˉ_{b}$ on Bounded Hypersurfaces in Stein Manifolds

Phillip Harrington, Andrew Raich

We define weak $Z (q)$ , a generalization of $Z (q)$ on bounded domains $Ω$ in a Stein manifold $M^{n}$ that suffices to prove closed range of $\partial ˉ$ . Under the hypothesis of we…

math.CV2009

Regularity results for $\partial ˉ_{b}$ on CR-manifolds of hypersurface type

Phillip Harrington, Andrew Raich

We introduce a class of embedded CR manifolds satisfying a geometric condition that we call weak $Y (q)$ . For such manifolds, we show that dbar-b has closed range on $L^{2}$ and that…

The ∂ˉ-problem on Z(q)-domains

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

The Strong Diederich-Fornæss Index on C2 Domains in Hermitian Manifolds

Maximal Estimates for the ∂ˉ-Neumann Problem on Non-pseudoconvex domains

Exact sequences and estimates for the ∂-problem

On Competing Definitions for the Diederich-Fornæss Index

The ∂ˉ-Neumann operator with the Sobolev norm of integer orders

Strong Closed Range Estimates: Necessary Conditions and Applications

A Modified Morrey-Kohn-Hörmander Identity and Applications

Hartogs Domains and the Diederich Fornæss Index

Boundary invariants and the closed range property for ∂ˉ

Lp Mapping Properties for the Cauchy-Riemann Equations on Lipschitz Domains Admitting Subelliptic Estimates

The Diederich-Fornaess Index and Good Vector Fields

Closed range of ∂ˉ in L2-Sobolev spaces on unbounded domains in Cn

A remark on boundary estimates on unbounded Z(q) domains in Cn

Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in CPn

Closed range of ∂ˉ on unbounded domains in Cn

Regularity equivalence of the Szegö projection and the complex Green operator

Sobolev Spaces and Elliptic Theory on Unbounded Domains in Rn

Defining Functions for Unbounded Cm Domains

Closed Range for ∂ˉ and ∂ˉb​ on Bounded Hypersurfaces in Stein Manifolds

Regularity results for ∂ˉb​ on CR-manifolds of hypersurface type

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

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

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

Exact sequences and estimates for the $\partial$ -problem

The $\partial ˉ$ -Neumann operator with the Sobolev norm of integer orders

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

$L^{p}$ Mapping Properties for the Cauchy-Riemann Equations on Lipschitz Domains Admitting Subelliptic Estimates

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

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

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

Closed range of $\partial ˉ$ on unbounded domains in $C^{n}$

Sobolev Spaces and Elliptic Theory on Unbounded Domains in $R^{n}$

Defining Functions for Unbounded $C^{m}$ Domains

Closed Range for $\partial ˉ$ and $\partial ˉ_{b}$ on Bounded Hypersurfaces in Stein Manifolds

Regularity results for $\partial ˉ_{b}$ on CR-manifolds of hypersurface type