A result on extension of C.R. functions

Makhlouf Derridj; John Erik Fornaess

Displaying similar documents to “A result on extension of C.R. functions”

An extension theorem for separately holomorphic functions with analytic singularities

Marek Jarnicki, Peter Pflug (2003)

Annales Polonici Mathematici

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Let $D_{j} \subset ℂ^{k_{j}}$ be a pseudoconvex domain and let $A_{j} \subset D_{j}$ be a locally pluriregular set, j = 1,...,N. Put $X : = ⋃_{j = 1}^{N} A ₁ \times . . . \times A_{j - 1} \times D_{j} \times A_{j + 1} \times . . . \times A_{N} \subset ℂ^{k ₁ + . . . + k_{N}}$ . Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with ${f ̂ |}_{X ∖ M} = f$ . The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001]. ...

The non-pluripolarity of compact sets in complex spaces and the property $(L B^{\infty})$ for the space of germs of holomorphic functions

Le Mau Hai, Tang Van Long (2002)

Studia Mathematica

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The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $(L B^{\infty})$ for the dual space of the space of germs of holomorphic functions on that compact set.

On spaces of holomorphic functions in ℂⁿ

Diana D. Jiménez S., Lino F. Reséndis O., Luis M. Tovar S. (2014)

Banach Center Publications

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Following the line of Ouyang et al. (1998) to study the $_{p}$ spaces of holomorphic functions in the unit ball of ℂⁿ, we present in this paper several results and relations among $_{p} (ₙ)$ , the α-Bloch, the Dirichlet $_{p}$ and the little $_{p, 0}$ spaces.

Biholomorphic maps determined on the boundary

Nozomu Mochizuki (1977)

Annales de l'institut Fourier

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Let $D$ be a bounded domain in $C^{n}$ such that the boundary $b D$ is topologically $S^{2 n - 1}$ in $R^{2 n}$ with a regular point; let $f : \tilde{D} \to C^{n}$ be a holomorphic map where $\tilde{D}$ is a neighborhood of $\overline{D}$ . If $f$ is one-to-one when restricted to $b D$ , then $f : D \to f (D)$ is biholomorphic.

Proper holomorphic self-mappings of the minimal ball

Nabil Ourimi (2002)

Annales Polonici Mathematici

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The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping f: D → Ω with branch locus $V_{f}$ is factored by automorphisms if and only if $f_{*} (π ₁ (D ∖ f^{- 1} (f (V_{f})), x))$ is a normal subgroup of $π ₁ (Ω ∖ f (V_{f}), b)$ for some $b \in Ω ∖ f (V_{f})$ and $x \in f^{- 1} (b)$ .

The exceptional sets for functions of the Bergman space in the unit ball

Piotr Jakóbczak (1993)

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

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Let $D$ be a domain in $C^{2}$ . Given $w \in C$ , set $D_{w} = \{z \in C ∣ (z, w) \in D\}$ . If $f$ is a holomorphic and square-integrable function in $D$ , then the set $E (D, f)$ of all $w$ such that $f (\cdot, w)$ is not square-integrable in $D_{w}$ has measure zero. We call this set the exceptional set for $f$ . In this Note we prove that whenever $0 < r < 1$ there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in $C^{2}$ such that $E (B, f)$ is the circle $C (0, r) = \{z \in C ∣ |z| = r\}$ .

On L₁-subspaces of holomorphic functions

Anahit Harutyunyan, Wolfgang Lusky (2010)

Studia Mathematica

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We study the spaces $H_{μ} (Ω) = f : Ω \to ℂ h o l o m o r p h i c : \int_{0}^{R} \int_{0}^{2 π} | f (r e^{i φ}) | d φ d μ (r) < \infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{μ} (Ω)$ is either isomorphic to l₁ or to ${(\sum \oplus A ₙ)}_{(1)}$ . Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

On the geometry of the space of m-pairs of holomorphic subspaces of the n-dimensional projective space $P_{n}$ ove r an albebra $𝒜$

Е.М. Кузнецова (1986)

Trudy Geometriceskogo Seminara

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Approximation of sets defined by polynomials with holomorphic coefficients

Marcin Bilski (2012)

Annales Polonici Mathematici

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Let X be an analytic set defined by polynomials whose coefficients $a ₁, . . ., a_{s}$ are holomorphic functions. We formulate conditions on sequences $a_{1, ν}, . . ., a_{s, ν}$ of holomorphic functions converging locally uniformly to $a ₁, . . ., a_{s}$ , respectively, such that the sequence $X_{ν}$ of sets obtained by replacing $a_{j}$ ’s by $a_{j, ν}$ ’s in the polynomials converges to X.

Shilov boundary for holomorphic functions on some classical Banach spaces

María D. Acosta, Mary Lilian Lourenço (2007)

Studia Mathematica

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Let $_{\infty} (B_{X})$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_{X}$ of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let $_{u} (B_{X})$ be the subspace of $_{\infty} (B_{X})$ of those functions which are uniformly continuous on $B_{X}$ . A subset $B \subset B_{X}$ is a boundary for $_{\infty} (B_{X})$ if $∥ f ∥ = s u p_{x \in B} | f (x) |$ for every $f \in_{\infty} (B_{X})$ . We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for $_{\infty} (B_{X})$ . On the other hand, for X = , the Schreier...

The space $N_{*}$ of holomorphic functions on bounded symmetric domains

Manfred Stoll (1976)

Annales Polonici Mathematici

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Linearization of germs: regular dependence on the multiplier

Stefano Marmi, Carlo Carminati (2008)

Bulletin de la Société Mathématique de France

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We prove that the linearization of a germ of holomorphic map of the type $F_{λ} (z) = λ (z + O (z^{2}))$ has a $𝒞^{1}$ -holomorphic dependence on the multiplier $λ$ . $𝒞^{1}$ -holomorphic functions are $𝒞^{1}$ -Whitney smooth functions, defined on compact subsets and which belong to the kernel of the $\bar{\partial}$ operator. The linearization is analytic for $| λ | \neq 1$ and the unit circle $𝕊^{1}$ appears as a natural boundary (because of resonances,roots of unity). However the linearization is still defined at most points of $𝕊^{1}$ , namely those points which lie “far enough...

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

Ngaiming Mok (2012)

Journal of the European Mathematical Society

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We study the extension problem for germs of holomorphic isometries $f : (D; x_{0}) \to (Ω; f (x_{0}))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d s_{D}^{2}$ on $D$ and $d s_{Ω}^{2}$ on $Ω$ . Our main focus is on boundary extension for pairs of bounded domains $(D, Ω)$ such that the Bergman kernel $K_{D} (z, w)$ extends meromorphically in $(z, \bar{w})$ to a neighborhood of $\bar{D} \times D$ , and such that the analogous statement holds true for the Bergman kernel $K_{Ω} (ς, ξ)$ on $Ω$ . Assuming that $(D; d s_{D}^{2})$ and $(Ω; d s_{Ω}^{2})$ are complete Kähler manifolds, we prove that...

On some extremal problems in classes of holomorphic functions $S_{p}^{} (ρ)$ , $S_{p}^{)} (ρ)$ , $Σ_{p}^{} (ρ)$ , $M C (ρ)$ at additional condition

B. Platynowicz (1980)

Matematički Vesnik

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Complete pluripolar graphs in $ℂ^{N}$

Nguyen Quang Dieu, Phung Van Manh (2014)

Annales Polonici Mathematici

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Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that $Γ_{g} (F) : = (z, g (z)) : z \in F$ is complete pluripolar in $ℂ^{N + 1}$ . Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that $Γ_{g} (D)$ is complete pluripolar in $ℂ^{N + 1}$ . These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math....