On some extremal problems in Bergman spaces in weakly pseudoconvex domains

Romi F. Shamoyan; Olivera R. Mihić

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Displaying similar documents to “On some extremal problems in Bergman spaces in weakly pseudoconvex domains”

On boundary behaviour of the Bergman projection on pseudoconvex domains

M. Jasiczak (2005)

Studia Mathematica

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It is shown that on strongly pseudoconvex domains the Bergman projection maps a space $L v_{k}$ of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space $E \supset L^{\infty} (Ω)$ defined by weighted-sup seminorms and equipped...

$L ²_{h}$ -domains of holomorphy and the Bergman kernel

Peter Pflug, Włodzimierz Zwonek (2002)

Studia Mathematica

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We give a characterization of $L ²_{h}$ -domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.

The pluricomplex Green function on some regular pseudoconvex domains

Gregor Herbort (2014)

Annales Polonici Mathematici

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Let D be a smooth bounded pseudoconvex domain in ℂⁿ of finite type. We prove an estimate on the pluricomplex Green function $_{D} (z, w)$ of D that gives quantitative information on how fast the Green function vanishes if the pole w approaches the boundary. Also the Hölder continuity of the Green function is discussed.

Peak functions on convex domains

Kolář, Martin

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Let $Ω \subset ℂ^{n}$ be a domain with smooth boundary and $p \in \partial Ω$ . A holomorphic function $f$ on $Ω$ is called a $C^{k}$ ( $k = 0, 1, 2, \dots$ ) peak function at $p$ if $f \in C^{k} (\bar{Ω})$ , $f (p) = 1$ , and $| f (q) | < 1$ for all $q \in \bar{Ω} ∖ {p}$ . If $Ω$ is strongly pseudoconvex, then $C^{\infty}$ peak functions exist. On the other hand, J. E. Fornaess constructed an example in $ℂ^{2}$ to show that this result fails, even for $C^{1}$ functions, on a weakly pseudoconvex domain [Math. Ann. 227, 173-175 (1977; Zbl 0346.32026)]. Subsequently, E. Bedford and J. E. Fornaess showed that there is always a continuous peak function...

On some new sharp embedding theorems in minimal and pseudoconvex domains

Romi F. Shamoyan, Olivera R. Mihić (2016)

Czechoslovak Mathematical Journal

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We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in $ℂ^{n} .$ Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our...

Quantitative estimates for the Green function and an application to the Bergman metric

Klas Diederich, Gregor Herbort (2000)

Annales de l'institut Fourier

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Let $D \subset ℂ^{n}$ be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by $G_{D} (., .)$ . In this article we give for a compact subset $K \subset D$ a quantitative upper bound for the supremum ${sup}_{z \in K} | G_{D} (z, w) |$ in terms of the boundary distance of $K$ and $w$ . This enables us to prove that, on a smooth bounded regular domain $D$ (in the sense of Diederich-Fornaess), the Bergman differential metric $B_{D} (w; X)$ tends to infinity, for $X \in ℂ^{n} / {O}$ , when $w \in D$ tends to a boundary...

The Bergman kernel functions of certain unbounded domains

Friedrich Haslinger (1998)

Annales Polonici Mathematici

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We compute the Bergman kernel functions of the unbounded domains $Ω_{p} = (z^{'}, z) \in ℂ ² : z > p (z^{'})$ , where $p (z^{'}) = {| z^{'} |}^{α} / α$ . It is also shown that these kernel functions have no zeros in $Ω_{p}$ . We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.

The $\bar{\partial}$ -Neumann operator on Lipschitz $q$ -pseudoconvex domains

Sayed Saber (2011)

Czechoslovak Mathematical Journal

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On a bounded $q$ -pseudoconvex domain $Ω$ in $ℂ^{n}$ with a Lipschitz boundary, we prove that the $\bar{\partial}$ -Neumann operator $N$ satisfies a subelliptic $(1 / 2)$ -estimate on $Ω$ and $N$ can be extended as a bounded operator from Sobolev $(- 1 / 2)$ -spaces to Sobolev $(1 / 2)$ -spaces.

A natural localization of Hardy spaces in several complex variables

Mihai Putinar, Roland Wolff (1997)

Annales Polonici Mathematici

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Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in $ℂ^{n}$ . The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.