Bounded Pseudoconvex Domain Research Articles

On New Decomposition Theorems in some Analytic Function Spaces in Bounded Pseudoconvex Domains

We provide new sharp decomposition theorems for multifunctional Bergman spaces in the unit ball and bounded pseudoconvex domains with smooth boundary expanding known results from the unit ball. Namely we prove that mΠ j=1 jjfj jjXj ≍ jjf1 : : : fmjj Ap for various (Xj) spaces of analytic functions in bounded pseudoconvex domains with smooth boundary where f; fj ; j = 1; : : : ;m are analytic functions and where Ap ; 0 < p < 1; > �����1 is a Bergman space. This in particular also extend in various directions a known theorem on atomic decomposition of Bergman Ap spaces.

The -Neumann operator with Sobolev estimates up to a finite order

Let be a bounded pseudoconvex domain with smooth boundary. For each we give a sufficient condition to estimate the -Neumann operator in the Sobolev space The key feature of our results is a precise formula for k in terms of the geometry of the boundary of Ω. It also shows the bound in the precise formula for Sobolev estimates of order k is comparably reciprocal of the Sobolev order k.

Localization of the Kobayashi metric and applications

In this paper we introduce a new class of domains— log-type convex domains, which have no boundary regularity assumptions. Then we will localize the Kobayashi metric in log-type convex subdomains. As an application, we prove a local version of continuous extension of rough isometric maps between two bounded domains with log-type convex Dini-smooth boundary points. Moreover we prove that the Teichmuller space $${\mathcal {T}}_{g,n}$$ is not biholomorphic to any bounded pseudoconvex domain in $$\mathbb C^{3g-3+n}$$ which is locally log-type convex near some boundary point.

Lp and weighted estimates for barred derivatives

We generalize the basic L2 inequality for barred derivatives that holds on bounded pseudoconvex domains. The analogue in Lp-Sobolev norms is established for smooth, bounded pseudoconvex domains of finite type with comparable Levi eigenvalues. A weighted L2 version and Lp estimates with loss are obtained on any smooth bounded weakly q-convex domain, where the weights are fractional powers of the distance to the boundary.

The automorphism group and limit set of a bounded domain I: The finite type case

For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and connected Lie group locally isomorphic to Aut(Bk). Further, the limit set is a smooth submanifold diffeomorphic to the sphere of dimension 2k−1. As applications we prove a new finite jet determination theorem and a Tits alternative theorem. The geometry of the Kobayashi metric plays an important role in the paper.

Some properties of h-extendible domains in [formula omitted

The purpose of this article is twofold. The first aim is to characterize h-extendibility of smoothly bounded pseudoconvex domains in Cn+1 by their noncompact automorphism groups. Our second goal is to show that if the squeezing function tends to 1 at an h-extendible boundary point of a smooth pseudoconvex domain in Cn+1, then this point must be strongly pseudoconvex.

The Diederich–Fornæss index II: For domains of trivial index

We study bounded pseudoconvex domains in complex Euclidean spaces. We find analytical necessary conditions and geometric sufficient conditions for a domain being of trivial Diederich–Fornæss index (i.e. the index equals to 1). We also connect a differential equation to the index. This reveals how a topological condition affects the solution of the associated differential equation and consequently obstructs the index being trivial. The proofs rely on a new method of study of the complex geometry of the boundary. The method was motivated by geometric analysis of Riemannian manifolds. We also generalize our main theorems under the context of de Rham cohomology.

Estimates on the Bergman kernels on pseudoconvex domains with comparable Levi-forms

ABSTRACTLet Ω be a smoothly bounded pseudoconvex domain in , and assume that is a point of regular finite 1-type, assume that the -eigenvalues of the Levi-form are comparable in a neighborhood of , and . Then we get optimal estimates of the Bergman kernel function along some ‘almost tangential curve’ .

The Diederich–Fornæss index I: For domains of non-trivial index

We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich–Fornæss index defined in 1977. This connects the Diederich–Fornæss index to boundary conditions and refines the Levi pseudoconvexity. We also prove the β-worm domain is of index π/(2β). It is the first time that a precise non-trivial Diederich–Fornæss index in Euclidean spaces is obtained. This finding also indicates that the Diederich–Fornæss index is a continuum in (0,1], not a discrete set. The ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations.

Compactness of Hankel Operators with Symbols Continuous on the Closure of Pseudoconvex Domains

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $\phi\in C(\overline{\Omega})$ such that $H_{\phi}$ is compact on $A^2(\Omega)$. Then $\phi\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow b\Omega$.

Estimates on the Bergman Kernels in a Tangential Direction on Pseudoconvex Domains in C3

Let Ω be a smoothly bounded pseudoconvex domain in C3 and assume that TΩreg(z0)<∞ where z0∈bΩ, the boundary of Ω. Then we get optimal estimates of the Bergman kernel function along some “almost tangential curve” Cb(z0,δ0)⊂Ω∪z0.

On the Steinness Index

We introduce the concept of Steinness index related to the Stein neighborhood basis. We then show several results: (1) The existence of Steinness index is equivalent to that of strong Stein neighborhood basis. (2) On the Diederich–Fornaess worm domains in particular, we present an explicit formula relating the Steinness index to the well-known Diederich–Fornaess index. (3) The Steinness index is 1 if a smoothly bounded pseudoconvex domain admits finitely many boundary points of infinite type.

Behavior of the squeezing function near h-extendible boundary points

It is shown that if the squeezing function tends to one at an h-extendible boundary point of a $\mathcal C^\infty$-smooth, bounded pseudoconvex domain, then the point is strictly pseudoconvex.

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.

On the $L^2$-Dolbeault cohomology of annuli

For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli.

Smoothing properties of the Friedrichs operator on Lp spaces

We show that the Friedrichs operator exhibits smoothing properties in the [Formula: see text] scale. In particular, we prove that on any smoothly bounded pseudoconvex domain the Friedrichs operator maps [Formula: see text] to [Formula: see text] for some [Formula: see text].

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 Boundary Points at Which the Squeezing Function Tends to One

J. E. Fornaess posed the question whether the boundary point of smoothly bounded pseudoconvex domain is strictly pseudoconvex, when the asymptotic limit value of the squeezing function is 1. The purpose of this paper is to give an affirmative answer if the domain is in $$\mathbb {C}^{2}$$ with smooth boundary of finite type in the sense of D’Angelo (Ann Math (2) 115(3):615–637, 1982).

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.