Bounded Pseudoconvex Domain Research Articles

Splitting type results for pseudoconvex domains and remarks on their Nebenhülle

We give a very general splitting type theorem for biholomorphic maps close to identity in the context of smoothly bounded pseudoconvex domains (Theorem 1.4). As a particular case, in the context of worm domains, we essentially reprove the splitting type result (Theorem 1.3) from Bracci et al. (Math Z 292:879–893, 2019) (by a different method). We also discuss some properties of the Nebenhülle of worm domains.

Capacities, Green function, and Bergman functions

AbstractUsing the logarithmic capacity, we give quantitative estimates of the Green function, as well as precise lower bounds of the Bergman kernel for bounded pseudoconvex domains in and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies the sharp estimate if is a bounded pseudoconvex domain with ‐boundary.

Bi-Hölder Extensions of Quasi-isometries on Pseudoconvex Domains of Finite Type in $${\mathbb {C}}^2$$

In this paper, we prove that the identity map for the smoothly bounded pseudoconvex domain of finite type in $${\mathbb {C}}^2$$ extends to a bi-Hölder map between the Euclidean boundary and Gromov boundary. As an application, we show the bi-Hölder boundary extensions for quasi-isometries between these domains. Moreover, we get a more accurate index of the Gehring–Hayman type theorem for the bounded m-convex domains with Dini-smooth boundary.

Curvature positivity of invariant direct images of Hermitian vector bundles

We prove that the invariant part, with respect to a compact group action satisfying certain condition, of the direct image of a Nakano positive Hermitian holomorphic vector bundle over a bounded pseudoconvex domain is Nakano positive. We also consider the action of the noncompact group \({\mathbb {R}}^m\) and get the same result for a family of tube domains, which leads to a new method to the matrix-valued Prekopa’s theorem originally proved by Raufi.

On competing definitions for the Diederich–Fornæss index

Let \(\Omega \subset {\mathbb {C}}^n\) 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 \(0<\eta <1\) such that there exists a function \(\rho _\eta \) in this family that is comparable to \(-{{\,\mathrm{dist}\,}}(z,b\Omega )\) on \(\Omega \) and such that \(-(-\rho _\eta )^\eta \) is plurisubharmonic on \(\Omega \). 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 \(\Omega \) is \(C^k\), \(k\ge 2\), 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.

Solvability of the Gleason Problem on a Class of Bounded Pseudoconvex Domains

We show that if a bounded pseudoconvex domain satisfies the solvability of the bounded $\bar{\partial}$ problem, then the ideal of bounded holomorphic functions vanishing at a point in the domain is finitely generated. We also prove a smooth analog of the main result for bounded pseudoconvex domains with a sufficiently smooth boundary and also consider the Bergman space case.

Boundary behavior of the Szegö kernel

We give a H\"ormander-type localization principle for the Szeg\"o kernel $S_\Omega(z)$. We also show that for each boundary point $z_0$, $S_\Omega(z)\gtrsim|z-z_0|^{-\frac{1}{3}}$ holds non-tangentially for any bounded pseudoconvex domain with smooth boundary in ${\mathbb C}^2$.

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.

Rooftop envelopes and residual plurisubharmonic functions

Given a negative plurisubharmonic function $\phi $ in a bounded pseudoconvex domain of $\mathbb C^n$, we introduce and study its <em>residual function</em> $g_\phi $ determined by the asymptotic behavior of $\phi $ near its singularity points, both inside

Weak pre-orthogonal adaptive Fourier decomposition in Bergman spaces of pseudoconvex domains

We study a weak greedy type algorithm called the weak pre-orthogonal adaptive Fourier decomposition (WPOAFD) for the Bergman space on a bounded pseudoconvex domain Ω with smooth boundary. We show that any function can be approximated by linear combinations of kernel functions in weak greedy sense.

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.

Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

A solution to the effectiveness problem in Kohn's algorithm for generating holomorphic subelliptic multipliers is provided for general classes of domains of finite type in Cn, that include the so-called special domains given by finite and infinite sums of squares of absolute values of holomorphic functions. Also included is a more general class of domains recently discovered by M. Fassina [23]. More generally, for any smoothly bounded pseudoconvex domain we introduce an invariantly defined associated sheaf S of C-subalgebras of holomorphic function germs, that combined with a result of Fassina, reduces the existence of effective subelliptic estimates at p to a purely algebraic geometric question of controlling the multiplicity of S.Our main new tool, a triangular resolution, is the construction of subelliptic multipliers decomposable as Q∘Γ, where Γ is constructed from pre-multipliers and Q is part of a triangular system. The effectiveness is proved via a sequence of newly proposed procedures, called here meta-procedures, built on top of the holomorphic Kohn's procedures, where the order of subellipticity can be effectively tracked. Important sources of inspiration are algebraic geometric techniques by Y.-T. Siu [54,55] and procedures for triangular systems by D.W. Catlin and J.P. D'Angelo [16,8].The proposed procedures are purely algebraic and as such can be of wider interest for geometric and computational problems involving Jacobian determinants, such as resolving singularities of holomorphic maps.

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 Modified Morrey-Kohn-Hörmander Identity and Applications to the $$\overline{\partial }$$-Problem

We prove a modified form of the classical Morrey-Kohn-Hormander 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 $$1\le q\le n-2$$ 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.

Every bounded pseudo‐convex domain with Hölder boundary is hyperconvex

We show that every bounded pseudo-convex domain with Hölder boundary in C n is hyperconvex.

Growth of Sibony metric and Bergman kernel for domains with low regularity

It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with C1 boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general C1 boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the Blocki-Zwonek Suita-type theorem implies growth to infinity of the Bergman kernel; the fact that the Bergman kernel grows as the square of the reciprocal of the distance to the boundary, proved by S. Fu in the C2 case, is extended to bounded pseudoconvex domains with Lipschitz boundaries.

A method of potential scaling in the study of pseudoconvex domains with noncompact automorphism group

The affine scaling method has been a typical approach to study complex domains with noncompact automorphism group. In this article, we will introduce an alternative approach, so called, the method of potential scaling to construct a certain class of potential functions of the Kähler-Einstein metric. We will also prove that if a bounded pseudoconvex domain admits a potential function of the Kähler-Einstein metric whose differential has constant length, then there is an 1-parameter family of automorphisms.

On J-Holomorphic Variational Vector Fields and Extremal Discs

We prove that every J-holomorphic variational vector field can be realized as derivation \(\frac{d}{dt}_{|t=0}f_t\) where \((f_t)\) is a one parametric family of J-holomorphic discs. Furthermore, we discuss properness of an extremal J-holomorphic disc in a bounded pseudoconvex domain.

On some new estimates for integrals of the Lusin’s square function in the unit polydisk

The purpose of the note is to obtain new estimates for the quasinorm of Hardy’s analytic classes of in the polydisk. We extend some classical onedimensional assertions to the case of several complex variables. Our results more precisely provide direct new extention of some known one variable theorems concerning area integral to the case of simplest product domains namely the unit polydisk in $\mathbb{C}^n$. Let further $D$ be a bounded or unbounded domain in $\mathbb{C}^n$. For example, tubular domain over symmetic cone or bounded pseudoconvex domain with smooth boundary. Our results can be probably extended to the case of products of such type complicated domains, namely even to $D\times\dots\times D$. This can be probably done based on some approaches we suggested and used in this paper. On the other hand our results in simpler case namely in the unit polydisk may also have various interesting applications in complex function theory in the unit polydisk.