On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in ${\bf C}^n$

Jean Erik Björk

Displaying similar documents to “On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in $𝐂^{n}$ ”

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]. ...

A result on extension of C.R. functions

Makhlouf Derridj, John Erik Fornaess (1983)

Annales de l'institut Fourier

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Let $Ω$ an open set in $C^{4}$ near $z_{0} \in \partial Ω$ , $λ$ a suitable holomorphic function near $z_{0}$ . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\overline{\partial} u = λ f$ , ( $f$ is a $(0, 1)$ form, $\overline{\partial}$ closed in $U (z_{0})$ in $U (z_{0})$ with supp $(u) \subset \overline{Ω} \cap U (z_{0})$ , then we deduce an extension result for $C . R .$ functions on $\partial Ω \cap U (z_{0})$ , as holomorphic fonctions in $Ω \cap V (z_{0})$ .

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.

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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Equilibrium measures for holomorphic endomorphisms of complex projective spaces

Mariusz Urbański, Anna Zdunik (2013)

Fundamenta Mathematicae

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Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space $ℙ^{k}$ , k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number $κ_{f} > 0$ such that if ϕ: J → ℝ is a Hölder continuous function with $s u p (ϕ) - i n f (ϕ) < κ_{f}$ , then ϕ admits a unique equilibrium state $μ_{ϕ}$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f, μ_{ϕ})$ is K-mixing, whence...

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.

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.

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....

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 $x \in f^{- 1} (b)$ .

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...

Equidistribution towards the Green current for holomorphic maps

Tien-Cuong Dinh, Nessim Sibony (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^{n}$ its iterate of order $n$ . We prove that the pull-back by $f^{n}$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1, 1)$ -currents and a similar result for regular polynomial automorphisms of $ℂ^{k}$ .

$J$ -holomorphic discs and real analytic hypersurfaces

William Alexandre, Emmanuel Mazzilli (2014)

Annales de l’institut Fourier

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We give in $ℝ^{6}$ a real analytic almost complex structure $J$ , a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$ , such that there is no germ of $J$ -holomorphic disc $γ$ included in $M$ with $γ (0) = 0$ and $\frac{\partial γ}{\partial x} (0) = v$ , although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$ , we give sufficient conditions under which there exists such a germ of disc.

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.

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 the isomorphism classes of weighted spaces of harmonic and holomorphic functions

Wolfgang Lusky (2006)

Studia Mathematica

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Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of $H v = f : Ω \to ℂ h o l o m o r p h i c : s u p_{z \in Ω} | f (z) | v (z) < \infty$ and investigate some isomorphism classes of $h v = f : Ω \to ℂ h a r m o n i c : s u p_{z \in Ω} | f (z) | v (z) < \infty$ where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either $H v \sim l_{\infty}$ or $H v \sim H_{\infty}$ , and at least two possibilities for hv, again $h v \sim l_{\infty}$ and $h v \sim H_{\infty}$ . We also discuss many new examples of weights.

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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Displaying similar documents to “On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in 𝐂 n ”

Displaying similar documents to “On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in $𝐂^{n}$ ”