arXiv subjects

Emil J. Straube

Publications and source records attributed to Emil J. Straube.

At least 19 recordsLinked to original sources

arXiv2026

Diederich-Fornæss index and global regularity of the complex Green operator: domains with comparable Levi eigenvalues

Let $Ω\subset \mathbb{C}^{n}$, with $n \geq 3$, be a smooth bounded pseudoconvex domain satisfying the symmetric eigenvalue comparability condition $D(q_0)$ for some $1\le q_0\le n-2$. We show that if the Diederich-Fornaess-index of $Ω$ is one, then the complex Green operator $G_q$, associated with $Ω$, is globally regular for $q$ in the range $\min\{q_0,\, n - 1 - q_0\} \leq q \leq \max\{q_0,\, n - 1 - q_0\}$.

math.CV↗

arXiv2025

Regularity in the $\overline{\partial}$--Neumann problem, D'Angelo forms, and Diederich--Fornæss index

This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\overline{\partial}$--Neumann operators on $Ω$ preserve the Sobolev spaces $W^{s}_{(0,q)}(Ω)$, $s\geq 0$. The same authors then proved a further regularity result and made explicit the role of D'Angelo forms for regularity (\cite{BoasStraube93}). A few years later, Kohn (\cite{Kohn99}) initiated a quantitative study of the results in \cite{BoasStraube91} by relating the Sobolev level up to which regularity holds to the Diederich--Fornæss index of the domain. Many of these ideas were synthesized and developed further by Harrington (\cite{Harrington11,Harrington19,Harrington22}). Then, around 2020, Liu (\cite{Liu19b, Liu19}) and Yum (\cite{Yum21}) discovered that the DF--index is closely related to certain differential inequalities involving D'Angelo forms. This relationship in turn led to a recent new result which supports the conjecture that DF--index one should imply global regularity in the $\overline{\partial}$--Neumann problem (\cite{LiuStraube22}). Much of the work described above relies heavily on Kohn's groundbreaking contributions to the regularity theory of the $\overline{\partial}$--Neumann problem.

math.CV↗

arXiv2025

Diederich--Fornæss index and global regularity in the $\overline{\partial}$--Neumann problem: domains with comparable Levi eigenvalues

Let $Ω$ be a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$. Let $1\leq q_{0}\leq (n-1)$. We show that if $q_{0}$--sums of eigenvalues of the Levi form are comparable, then if the Diederich--Fornæss index of $Ω$ is $1$, the $\overline{\partial}$--Neumann operators $N_{q}$ and the Bergman projections $P_{q-1}$ are regular in Sobolev norms for $q_{0}\leq q\leq n$. In particular, for domains in $\mathbb{C}^{2}$, Diederich--Fornæss index $1$ implies global regularity in the $\overline{\partial}$--Neumann problem.

math.CV↗

arXiv2024

Sobolev regularity of the Bergman and Szegö projections in terms of $\overline{\partial}\oplus\overline{\partial}^{}$ and $\overline{\partial}_{b}\oplus\overline{\partial}_{b}^{}$

Let $Ω$ be a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$. It is shown that for $0\leq q\leq n$, $s\geq 0$, the embedding $j_{q}: dom(\overline{\partial})\cap dom(\overline{\partial}^{*}) \hookrightarrow L^{2}_{(0,q)}(Ω)$ is continuous in $W^{s}(Ω)$--norms if and only if the Bergman projection $P_{q}$ is (see below for the modification needed for $j_{0}$). The analogous result for the operators on the boundary is also proved (for $n\geq 3$). In particular, $j_{1}$ is always regular in Sobolev norms in $\mathbb{C}^{2}$, notwithstanding the fact that $N_{1}$ need not be.

math.CV↗

arXiv2023

Modifications of the Levi core

We construct a family of subdistributions of the Levi core $\mathfrak{C}(\mathcal{N})$ called modified Levi cores $\{\mathcal{M}\mathfrak{C}_{\mathcal{A}}\}_{\mathcal{A}}$ indexed over closed distributions $\mathcal{A}$ that contain the Levi null distribution $\mathcal{N}$ and are contained in the complex tangent bundle $T^{1, 0}bΩ$ of a smooth bounded pseudoconvex domain $Ω$. We show that Catlin's Property ($P$) holds on $bΩ$ if and only if Property ($P$) holds on the support of one, and hence all, of the modified Levi cores. In $\mathbb{C}^2$, all of the modified Levi cores coincide. For a smooth bounded pseudoconvex complete Hartogs domain in $\mathbb{C}^2$ that satisfies Property ($P$), we show that its modified Levi core is trivial. This contrasts with $\mathfrak{C}(\mathcal{N})$, which can be nontrivial for such domains.

math.CV↗

arXiv2021

A Sufficient condition for compactness of Hankel operators

Let $Ω$ be a bounded convex domain in $\mathbb{C}^{n}$. We show that if $φ\in C^{1}(\overlineΩ)$ is holomorphic along analytic varieties in $bΩ$, then $H^{q}_φ$, the Hankel operator with symbol $φ$, is compact. We have shown the converse earlier, so that we obtain a characterization of compactness of these operators in terms of the behavior of the symbol relative to analytic structure in the boundary. A corollary is that Toeplitz operators with these symbols are Fredholm (of index zero).

math.CV↗

arXiv2020

Compactness of Hankel operators with continuous symbols on convex domains

Let $Ω$ be a bounded convex domain in $\mathbb{C}^{n}$, $n\geq 2$, $1\leq q\leq (n-1)$, and $ϕ\in C(\barΩ)$. If the Hankel operator $H^{q-1}_ϕ$ on $(0,q-1)$--forms with symbol $ϕ$ is compact, then $ϕ$ is holomorphic along $q$--dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only `finitely many' varieties, $1\leq q\leq n$, and $ϕ\in C(\barΩ)$ is analytic along the ones of dimension $q$ (or higher), then $H^{q-1}_ϕ$ is compact.

math.CV↗

arXiv2019

Convex domains, Hankel operators, and maximal estimates

Let $1\leq q\leq (n-1)$. We first show that a necessary condition for a Hankel operator on $(0,q-1)$-forms on a convex domain to be compact is that its symbol is holomorphic along $q$-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension $(n-1)$ (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level $q$ if and only if they hold at all levels.

math.CV↗

arXiv2017

Estimates for the complex Green operator: symmetry, percolation, and interpolation

Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees, that is, they hold for $(p,q)$--forms if and only if they hold for $(m-p,m-1-q)$--forms. Here $m$ equals the CR dimension of $M$ plus one. Symmetries of this type are known to hold for compactness estimates. We further show that with the usual microlocalization, compactness estimates for the positive part percolate up the complex, i.e. if they hold for $(p,q)$--forms, they also hold for $(p,q+1)$--forms. Similarly, compactness estimates for the negative part percolate down the complex. As a result, if the complex Green operator is compact on $(p,q_{1})$--forms and on $(p,q_{2})$--forms ($q_{1}\leq q_{2}$), then it is compact on $(p,q)$--forms for $q_{1}\leq q\leq q_{2}$. It is interesting to contrast this behavior of the complex Green operator with that of the $\overline{\partial}$--Neumann operator on a pseudoconvex domain.

math.CV↗

arXiv2017

$L^{2}$-Sobolev theory for the complex Green operator

These notes are concerned with the $L^{2}$-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed CR--submanifolds of $\mathbb{C}^{n}$ of hypersurface type. This class of submanifolds generalizes that of boundaries of pseudoconvex domains. We first discuss briefly the CR--geometry of general CR--submanifolds and then specialize to this class. Next, we review the basic $L^{2}$-theory of the tangential Cauchy-Riemann operator and the associated complex Green operator(s) on these submanifolds. After these preparations, we discuss recent results on compactness and regularity in Sobolev spaces of the complex Green operator(s).

math.CV↗

arXiv2014

Sobolev estimates for the complex Green operator on CR submanifolds of hypersurface type

Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. Our main result says that when a certain $1$-form on $M$ is exact on the null space of the Levi form, then the complex Green operator on $M$ satisfies Sobolev estimates. This happens in particular when $M$ admits a set of plurisubharmonic defining functions or when $M$ is strictly pseudoconvex except for the points on a simply connected complex submanifold.

math.CV↗

arXiv2014

Compactness of the $\overline{\partial}$-Neumann operator on the intersection of two domains

Assume that $Ω_{1}$ and $Ω_{2}$ are two smooth bounded pseudoconvex domains in $\mathbb{C}^{2}$ that intersect (real) transversely, and that $Ω_{1} \cap Ω_{2}$ is a domain (i.e. is connected). If the $\overline{\partial}$-Neumann operators on $Ω_{1}$ and on $Ω_{2}$ are compact, then so is the $\overline{\partial}$-Neumann operator on $Ω_{1} \cap Ω_{2}$. The corresponding result holds for the $\overline{\partial}$-Neumann operators on $(0,n-1)$-forms on domains in $\mathbb{C}^{n}$.

math.CV↗

arXiv2011

Duality of holomorphic functions spaces und smoothing properties of the Bergman projection

Let $Ω\subset\mathbb{C}^n$ be a bounded domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_{1}}(Ω)$ (continuously) into $H^{k_{2}}(Ω)$. We establish two smoothing results: (i) the full Sobolev norm $\|Bf\|_{k_{2}}$ is controlled by $L^2$ derivatives of $f$ taken along a single, distinguished direction (of order $\leq k_{1}$), and (ii) the projection of a conjugate holomorphic function in $L^{2}(Ω)$ is automatically in $H^{k_{2}}(Ω)$. There are obvious corollaries for when $B$ is globally regular.

math.CV↗

arXiv2011

On extending $L^{2}$ holomorphic functions from complex hyperplanes

The key to the proof of the Ohsawa-Takegoshi Extension Theorem is a certain $\bar{\partial}$-estimate. The purpose of this note is to show that the 'curvature term' that arises in the Kohn-Morrey-Hörmander inequality (or the Bochner-Kodaira technique) is sufficient to produce such an estimate. We exploit self boundedness of the gradients of the weight functions to change the weight with respect to which the adjoint is taken. The weights, on the other hand, are the usual ones used in this context.

math.CV↗

arXiv2010

Geometric sufficient conditions for compactness of the complex Green operator

We establish compactness estimates for $\bar{\partial}_{M}$ on a compact pseudoconvex CR-submanifold $M$ of $\mathbb{C}^{n}$ of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the $\bar{\partial}$-Neumann operator given by the authors earlier. These conditions are formulated in terms of certain short time flows in complex tangential directions.

math.CV↗

arXiv2010

The complex Green operator on CR-submanifolds of $\mathbb{C}^{n}$ of hypersurface type: compactness

We establish compactness estimates for $\overline{\partial}_{b}$ on a compact pseudoconvex CR-submanifold of $\mathbb{C}^{n}$ of hypersurface type that satisfies property(P). When the submanifold is orientable, these estimates were proved by A.~Raich via microlocal methods. Our proof deduces the estimates from (a slight extension, when $q>1$, of) those known on hypersurfaces via the fact that locally, CR-submanifolds of hypersurface type are CR-equivalent to a hypersurface. The relationship between two potential theoretic conditions is also clarified.

math.CV↗

arXiv2008

Observations regarding compactness in the $\overline{\partial}$-Neumann problem

We show that compactness of the $\overline{\partial}$-Neumann operator is independent of the metric, and we give a new proof of this independence for subellipticity. We define an abstract obstruction to compactness, namely the common zero set of all the compactness multipliers, and we identify this subset of the boundary for convex domains in $\mathbb{C}^{n}$ and for complete Hartogs domains in $\mathbb{C}^{2}$.

math.CV↗

arXiv2007

Compactness of the Complex Green Operator

Let $Ω\subset\C^n$ be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_{q}$ on $(0,q)$-forms on $bΩ$ implies compactness of the $\bar{\partial}$-Neumann operator $N_{q}$ on $Ω$. We prove that if $1 \leq q \leq n-2$ and $bΩ$ satisfies $(P_q)$ and $(P_{n-q-1})$, then $G_{q}$ is a compact operator (and so is $G_{n-1-q}$). Our method relies on a jump type formula to represent forms on the boundary, and we prove an auxiliary compactness result for an `annulus' between two pseudoconvex domains. Our results, combined with the known characterization of compactness in the $\bar{\partial}$-Neumann problem on locally convexifiable domains, yield the corresponding characterization of compactness of the complex Green operator(s) on these domains.

math.CV↗