Pseudoconvex Domains Research Articles

A lower bound for the Kähler-Einstein distance from the Diederich-Fornæss index

In this paper we establish a lower bound for the distance induced by the Kähler-Einstein metric on pseudoconvex domains with positive hyperconvexity index (e.g. positive Diederich-Fornæss index). A key step is proving an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.

Commutators of Cauchy–Fantappiè Type Integrals on Generalized Morrey Spaces on Complex Ellipsoids

Let $$\Omega $$ be a domain which belongs to a class of bounded weakly pseudoconvex domains of finite type in $${\mathbb {C}}^n$$ , let $$d\lambda $$ be the Monge–Ampere boundary measure on $$b\Omega $$ and $$\varrho \ge 0$$ be a non-decreasing function. The aim of this paper is to establish the characterizations of boundedness and compactness for the commutator operators of Cauchy–Fantappie type integrals with $$L^1(b\Omega ,d\lambda )$$ functions on the generalized Morrey spaces $$L^{p}_\varrho (b\Omega ,d\lambda )$$ , with $$p\in (1, \infty )$$ .

Subharmonic variation of Azukawa pseudometrics for balanced domains

We consider the subharmonicity property of the logarithm of Azukawa pseudometrics of pseudoconvex domains under pseudoconvex variations. We prove that such a property holds for the variation of balanced domains. We also give a non-balanced example. The relation of the volume of Azukawa indicatrix and the estimate in the Ohsawa–Takegoshi [Formula: see text]-extension theorem is also discussed.

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.

Berezin regularity of domains in $\mathbb {C}^n$ and the essential norms of Toeplitz operators

For the open unit disc $\mathbb{D}$ in the complex plane, it is well known that if $\phi \in C(\overline{\mathbb{D}})$ then its Berezin transform $\widetilde{\phi}$ also belongs to $C(\overline{\mathbb{D}})$. We say that $\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb{C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.

The pluricomplex Poisson kernel for strongly pseudoconvex domains

In this paper we introduce, via a Phragmén-Lindelöf type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the pluricomplex Poisson kernel because it shares many properties with the classical Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it is zero on the boundary except at one boundary point where it has a non-tangential simple pole, and reproduces pluriharmonic functions. We also use such a function to obtain a new “intrinsic” version of the classical Julia's Lemma and Julia-Wolff-Carathéodory's Theorem.

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.

The Pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Ampere equations, modelled on the Kahler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.

Bounded strictly pseudoconvex domains in C2 with obstruction flat boundary

On a bounded strictly pseudoconvex domain in $\Bbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\Bbb{C}^2$ which are diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. In particular we observe that, up to biholomorphism, the unit ball in $\Bbb{C}^2$ is rigid with respect to deformations in the class of strictly pseudoconvex domains with obstruction flat boundary. We further show that for more general deformations of the unit ball, the order of vanishing of the obstruction equals the order of vanishing of the CR curvature. Finally, we give a generalization of the recent result of the second author that for an abstract CR manifold with transverse symmetry, obstruction flatness implies local equivalence to the CR $3$-sphere.

Corona theorem for strictly pseudoconvex domains

Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that M. Kosiek and K. Rudol obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved "abstract" Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries.

CONNECTION AND CURVATURE ON BUNDLES OF BERGMAN AND HARDY SPACES

We consider a complex domain D x V in the space C-m x C-n and a family of weighted Bergman spaces on V defined by a weight e(-k phi(z , w)) for a pluri-subharmonic function phi(z, w) with a quantization parameter k. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain D. We consider the natural covariant differentiation del(z) on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures R-(k)(Z,Z) for large k and for the induced connection [del((k))(Z), T-f((k))] on Toeplitz operators T-f. In the special case when the domain D is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.

On smoothing properties of the Bergman projection

We study smoothing properties of the Bergman projection and weighted Bergman projections. In particular, we relate these properties to the hyperconvexity index of a pseudoconvex domain in Cn. The notion of a hyperconvexity index was first introduced by B.Y. Chen, which provides a flexible criterion for studying geometric properties of hyperconvex domains. We also obtain a new estimate of weighted Bergman projections, which extends a well-known estimate of Berndtsson and Charpentier. We then give some applications of this estimate.

On the Dimension of the Bergman Space of Some Hartogs Domains with Higher Dimensional Bases

Let D be a Hartogs domain of the form $$D=D_{\varphi }(G)=\{(z,w)\in G\times \mathbb {C}^{N}\,:\,\Vert w\Vert <e^{-\varphi (z)}\}$$ , where $$\varphi $$ is a plurisubharmonic function on G and $$G\subseteq \mathbb {C}^{M}$$ is a pseudoconvex domain. We expand on the results of Jucha (J Geom Anal 22(1):23–37, 2012) and prove the infinite-dimensionality or triviality of the space of square integrable holomorphic functions on $$D_{\varphi }(G)$$ for various choices of $$\varphi $$ and G.

Holomorphic mappings between domains with low boundary regularity

We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with boundaries of low regularity.

Semi-continuity of the Diederich–Fornaess and Steinness indices

In this paper, we prove the semi-continuity theorem of Diederich–Fornaess index and Steinness index under a smooth deformation of pseudoconvex domains in Stein manifolds.

Renormalized characteristic forms of the Cheng–Yau metric and global CR invariants

For each invariant polynomial Φ, we construct a global CR invariant via the renormalized characteristic form of the Cheng–Yau metric on a strictly pseudoconvex domain. When the degree of Φ is 0, the invariant agrees with the total Q′-curvature. When the degree is equal to the CR dimension, we construct a primed pseudo-hermitian invariant IΦ′ which integrates to the corresponding CR invariant. These are generalizations of the I′-curvature on CR five-manifolds, introduced by Case–Gover.

Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type

We study the Lorentz boundedness properties of Calderón-type commutators and relate those properties to the spaces of bounded mean oscillation and vanishing mean oscillation. In application, we obtain Lorentz boundedness and compactness characterization of integral commutators with singular kernel of Calderón–Zygmund type on strictly pseudoconvex domains in ℂn, or convex domain of finite type in ℂn.

Estimates of Kähler-Einstein metrics on pseudoconvex domains of finite type with locally diagonalizable Levi form

In this short note, we use the weights introduced by Ph. Charpentier and Y. Dupain on pseudoconvex domains of finite type with locally diagonalizable Levi form to prove that on these domains, the Kähler-Einstein metrics are comparable with the Carathéodory, Kobayashi and Bergman metrics.

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.