Boundary behaviour of holomorphic functions in Hardy-Sobolev spaces on convex domains in ℂⁿ

Marco M. Peloso; Hercule Valencourt

Displaying similar documents to “Boundary behaviour of holomorphic functions in Hardy-Sobolev spaces on convex domains in ℂⁿ”

Some properties of Reinhardt domains

Le Mau Hai, Nguyen Quang Dieu, Nguyen Huu Tuyen (2003)

Annales Polonici Mathematici

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We first establish the equivalence between hyperconvexity of a fat bounded Reinhardt domain and the existence of a Stein neighbourhood basis of its closure. Next, we give a necessary and sufficient condition on a bounded Reinhardt domain D so that every holomorphic mapping from the punctured disk $Δ_{*}$ into D can be extended holomorphically to a map from Δ into D.

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...

Proper holomorphic liftings and new formulas for the Bergman and Szegő kernels

E. H. Youssfi (2002)

Studia Mathematica

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We consider a large class of convex circular domains in $M_{m ₁, n ₁} (ℂ) \times . . . \times M_{m_{d}, n_{d}} (ℂ)$ which contains the oval domains and minimal balls. We compute their Bergman and Szegő kernels. Our approach relies on the analysis of some proper holomorphic liftings of our domains to some suitable manifolds.

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...

On the algebra of $A^{k}$ -functions

Ulf Backlund, Anders Fällström (2006)

Mathematica Bohemica

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For a domain $Ω \subset ℂ^{n}$ let $H (Ω)$ be the holomorphic functions on $Ω$ and for any $k \in ℕ$ let $A^{k} (Ω) = H (Ω) \cap C^{k} (\bar{Ω})$ . Denote by $𝒜_{D}^{k} (Ω)$ the set of functions $f Ω \to [0, \infty)$ with the property that there exists a sequence of functions $f_{j} \in A^{k} (Ω)$ such that ${| f_{j} |}$ is a nonincreasing sequence and such that $f (z) = {lim}_{j \to \infty} | f_{j} (z) |$ . By $𝒜_{I}^{k} (Ω)$ denote the set of functions $f Ω \to (0, \infty)$ with the property that there exists a sequence of functions $f_{j} \in A^{k} (Ω)$ such that ${| f_{j} |}$ is a nondecreasing sequence and such that $f (z) = {lim}_{j \to \infty} | f_{j} (z) |$ . Let $k \in ℕ$ and let $Ω_{1}$ and $Ω_{2}$ be bounded $A^{k}$ -domains of holomorphy in $ℂ^{m_{1}}$ and $ℂ^{m_{2}}$ respectively. Let $g_{1} \in 𝒜_{D}^{k} (Ω_{1})$ , $g_{2} \in 𝒜_{I}^{k} (Ω_{1})$ and $h \in 𝒜_{D}^{k} (Ω_{2}) \cap 𝒜_{I}^{k} (Ω_{2})$ . We prove...

The Quantitative Isoperimetric Inequality for Planar Convex Domains

Carlo Nitsch (2008)

Bollettino dell'Unione Matematica Italiana

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We prove that among all the convex bounded domains in $\mathbb{R}^{2}$ having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domains.

$L ²_{h}$ -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.

Lipschitz spaces of holomorphic and pluriharmonic functions on bounded symmetric domains in $C^{N}$ (N>1)

Josephine Mitchell (1981)

Annales Polonici Mathematici

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Balls for the Kobayashi distance and extension of the automorphisms of strictly convex domains in $C^{n}$ with real analytic boundary

Andrea Iannuzzi (1994)

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

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It is shown that given a bounded strictly convex domain $Ω$ in $C^{n}$ with real analitic boundary and a point $x_{0}$ in $Ω$ , there exists a larger bounded strictly convex domain $Ω^{'}$ with real analitic boundary, close as wished to $Ω$ , such that $Ω$ is a ball for the Kobayashi distance of $Ω^{'}$ with center $x_{0}$ . The result is applied to prove that if $Ω$ is not biholomorphic to the ball then any automorphism of $Ω$ extends to an automorphism of $Ω^{'}$ .

Duality theorems for Hardy and Bergman spaces on convex domains of finite type in $ℂ^{n}$

Steven G. Krantz, Song-Ying Li (1995)

Annales de l'institut Fourier

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We study Hardy, Bergman, Bloch, and BMO spaces on convex domains of finite type in $n$ -dimensional complex space. Duals of these spaces are computed. The essential features of complex domains of finite type, that make these theorems possible, are isolated.