Smooth Bounded Pseudoconvex Domain Research Articles

Spectral Stability of the {overline{partial }}-Neumann Laplacian: Domain Perturbations

We study spectral stability of the {bar{partial }}-Neumann Laplacian on a bounded domain in {mathbb {C}}^n when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the {bar{partial }}-Neumann Laplacian on bounded pseudoconvex domains in {mathbb {C}}^n, lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in {mathbb {C}}^n.

Sufficient conditions for compactness of the ∂-Neumann operator on high level forms

By establishing a unified estimate of the twisted Kohn-Morrey-H\"{o}rmander estimate and the $q$-pseudoconvex Ahn-Zampieri estimate, we discuss variants of Property $(P_q)$ of Catlin and Property $(\widetilde{P_q})$ of McNeal on the boundary of a smooth pseudoconvex domain in $\mathbb{C}^n$ for certain high level forms. These variant conditions on the one side, imply $L^2$-compactness of the $\overline{\partial}$-Neumann operator on the associated domain, on the other side, are different from the classical Property $(P_q)$ and Property $(\widetilde{P_q})$. As an application of our result, we show that if the Hausdorff $(2n-2)$-dimensional measure of the weakly pseudoconvex points on the boundary of a smooth bounded pseudoconvex domain is zero, then the $\overline{\partial}$-Neumann operator $N_{n-1}$ is $L^2$-compact on $(0,n-1)$-level forms. This result generalizes Boas and Sibony's results on $(0,1)$-level forms.

Projected Composition Operators on Pseudoconvex Domains

Let \(\Omega \subset {\mathbb {C}}^n\) be a smooth bounded pseudoconvex domain and \(A^2 (\Omega )\) denote its Bergman space. Let \(P:L^2(\Omega )\longrightarrow A^2(\Omega )\) be the Bergman projection. For a measurable \(\varphi :\Omega \longrightarrow \Omega \), the projected composition operator is defined by \((K_\varphi f)(z) = P(f \circ \varphi )(z), z \in \Omega , f\in A^2 (\Omega ).\) In 1994, Rochberg studied boundedness of \(K_\varphi \) on the Hardy space of the unit disk and obtained different necessary or sufficient conditions for boundedness of \(K_\varphi \). In this paper we are interested in projected composition operators on Bergman spaces on pseudoconvex domains. We study boundedness of this operator under the smoothness assumptions on the symbol \(\varphi \) on \({{\overline{\Omega }}}\).

On compactness and Lp‐regularity in the ∂¯‐Neumann problem

Let Ω be a C 4 -smooth bounded pseudoconvex domain in C 2 . We show that if the ∂ ¯ -Neumann operator N 1 is compact on L ( 0 , 1 ) 2 ( Ω ) , then the embedding operator J : D o m ( ∂ ¯ ) ∩ D o m ( ∂ ¯ ∗ ) → L ( 0 , 1 ) 2 ( Ω ) is L p -regular for all 2 ⩽ p < ∞ .

A local weighted Axler–Zheng theorem in ℂn

The well-known Axler-Zheng theorem characterizes compactness of finite sums of finite products of Toeplitz operators on the unit disk in terms of the Berezin transform of these operators. Subsequently this theorem was generalized to other domains and appeared in different forms, including domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-Neumann operator $N$ is compact. In this work we remove the assumption on $N$, and we study weighted Bergman spaces on smooth bounded pseudoconvex domains. We prove a local version of the Axler-Zheng theorem characterizing compactness of Toeplitz operators in the algebra generated by symbols continuous up to the boundary in terms of the behavior of the Berezin transform at strongly pseudoconvex points. We employ a Forelli-Rudin type inflation method to handle the weights.

Asymptotic boundary behavior of the Bergman curvatures of a pseudoconvex domain

We present a method of obtaining a lower bound estimate of the curvatures of the Bergman metric without using the regularity of the kernel function on the boundary. As an application, we prove the existence of an uniform lower bound of the bisectional curvatures of the Bergman metric of a smooth bounded pseudoconvex domain near the boundary with constant Levi rank.

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt.

On noncompactness of the [formula omitted]-Neumann problem on pseudoconvex domains in [formula omitted

We prove that a smooth bounded pseudoconvex domain Ω⊆C3 with a one-dimensional complex manifold M in its boundary has a noncompact ∂‾-Neumann operator on (0,1)-forms, under the additional assumption that bΩ has finite regular D'Angelo 2-type at a point of M.

On Hardy spaces on worm domains

AbstractIn this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth worm domains.

Axler–Zheng Type Theorem on a Class of Domains in $${\mathbb{C}^n}$$ C n

We prove a version of Axler-Zheng's Theorem on smooth bounded pseudoconvex domains in C^n on which the dbar-Neumann operator is compact.

CONTINUITY OF PLURISUBHARMONIC ENVELOPES IN ℂ2

We show that in ℂ2 if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up to the boundary are continuous on the closure of the domain. Using this result we prove that smooth bounded pseudoconvex Reinhardt domains in ℂ2 are c-regular.

Strong Stein neighbourhood bases

Let Ω be a smooth bounded pseudoconvex domain in ℂ n . We give several characterizations for the closure of Ω to have a Stein neighbourhood basis in the sense that Ω has a defining function ρ such that {z ∈ ℂ n : ρ(z) < ϵ} is pseudoconvex for sufficiently small ϵ > 0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly up to the boundary.

On compactness of the \overline{∂}-Neumann problem and Hankel operators

Let Ω = Ω 1 ∖ Ω ¯ 2 \Omega =\Omega _1\setminus \overline {\Omega }_2 , where Ω 1 \Omega _1 and Ω 2 \Omega _2 are two smooth bounded pseudoconvex domains in C n , n ≥ 3 , \mathbb {C}^n, n\geq 3, such that Ω ¯ 2 ⊂ Ω 1 . \overline {\Omega }_2\subset \Omega _1. Assume that the ∂ ¯ \overline {\partial } -Neumann operator of Ω 1 \Omega _1 is compact and the interior of the Levi-flat points in the boundary of Ω 2 \Omega _2 is not empty (in the relative topology). Then we show that the Hankel operator on Ω \Omega with symbol ϕ , H ϕ Ω , \phi , H^{\Omega }_{\phi }, is compact for every ϕ ∈ C ( Ω ¯ ) \phi \in C(\overline {\Omega }) but the ∂ ¯ \overline {\partial } -Neumann operator on Ω \Omega is not compact.

Comparison of the Bergman and Szegö kernels

The quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domains in C n is bounded from above by a constant multiple of δ | log δ | p for any p > n , where δ is the distance to the boundary. For a class of domains that includes those of DʼAngelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by a constant multiple of δ | log δ | p for any p < − 1 . Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of δ.

Boundary behavior of the Kobayashi-Royden metric in smooth pseudoconvex domains

We show some lower estimates for the Kobayashi-Royden metric on a class of smooth bounded pseudoconvex domains.

Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let β be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Ω ⊂ C n . We show that, if Ω is convex or the Levi form of the boundary of Ω is of rank at least n − 2 , then compactness of the Hankel operator H β implies that β is holomorphic “along” analytic discs in the boundary. Furthermore, when Ω is convex in C 2 we show that the condition on β is necessary and sufficient for compactness of H β .

Hearing the type of a domain in [formula omitted] with the [formula omitted]-Neumann Laplacian

A smooth bounded pseudoconvex domain in C 2 is of finite type if and only if the number of eigenvalues of its ∂ ¯ -Neumann Laplacian that are less than or equal to λ has at most polynomial growth as λ goes to infinity.

A remark on irregularity of the $\overline {\partial }$-Neumann problem on non-smooth domains

It is an observation due to J.J. Kohn that for a smooth bounded pseudoconvex domain D in $C^n$ there exists s>0 such that the dbar-Neumann operator on D maps $W^s_{(0,1)}(D)$ (the space of $(0,1)$-forms with coefficient functions in $L^2$-Sobolev space of order s) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain D in $C^2$, smooth except at one point, whose dbar-Neumann operator is not bounded on $W^s_{(0,1)}(D)$ for any s>0.

Analytic discs, plurisubharmonic hulls, and non-compactness of the -Neumann operator

We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain Ω in is an obstruction to compactness of the -Neumann operator on Ω, provided that at some point of M, the Levi form of bΩ has the maximal possible rank n−1−dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary.

Geometric conditions which imply compactness of the ${\overline\partial}$-Neumann operator

For smooth bounded pseudoconvex domains in ℂ 2 , we provide geometric conditions on the boundary which imply compactness of the ∂ ¯-Neumann operator. It is noteworthy that the proof of compactness does not proceed via verifying the known potential theoretic sufficient conditions.