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

Klas Diederich; Gregor Herbort

Displaying similar documents to “Quantitative estimates for the Green function and an application to the Bergman metric”

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...

Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

Bo Berndtsson (2006)

Annales de l’institut Fourier

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Let $D$ be a pseudoconvex domain in $ℂ_{t}^{k} \times ℂ_{z}^{n}$ and let $φ$ be a plurisubharmonic function in $D$ . For each $t$ we consider the $n$ -dimensional slice of $D$ , $D_{t} = {z; (t, z) \in D}$ , let $φ^{t}$ be the restriction of $φ$ to $D_{t}$ and denote by $K_{t} (z, ζ)$ the Bergman kernel of $D_{t}$ with the weight function $φ^{t}$ . Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n = 1$ and $φ = 0$ ) we prove that $log K_{t} (z, z)$ is a plurisubharmonic function in $D$ . We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in...

On the Green function on a certain class of hyperconvex domains

Gregor Herbort (2008)

Annales Polonici Mathematici

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We study the behavior of the pluricomplex Green function on a bounded hyperconvex domain D that admits a smooth plurisubharmonic exhaustion function ψ such that 1/|ψ| is integrable near the boundary of D, and moreover satisfies the estimate $| ψ | \leq C e x p (- C^{'} {(l o g (1 / δ_{D}))}^{α})$ at points close enough to the boundary with constants C,C’ > 0 and 0 < α < 1. Furthermore, we obtain a Hopf lemma for such a function ψ. Finally, we prove a lower bound on the Bergman distance on D.

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.

Global boundary regularity for the $\bar{p a r t i a l}$ -equation on $q$ -pseudo-convex domains

Heungju Ahn (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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For a bounded domain $D$ of $C^{n}$ , we introduce a notion of « $q$ -pseudoconvexity» of new type and prove that for a given $\bar{\partial}$ -closed $(p, r)$ -form $f$ that is smooth up to the boundary on $D$ , and for $r \geq q$ , there exists a $(p, r - 1)$ -form $u$ smooth up to the boundary on $D$ which is a solution of the equation $\bar{\partial} u = f$

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...

Two-sided $L^{1}$ -estimates for Szegö kernels on classical domains

Josephine Mitchell, G. Sampson (1990)

Annales Polonici Mathematici

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