On L₁-subspaces of holomorphic functions

Anahit Harutyunyan; Wolfgang Lusky

Displaying similar documents to “On L₁-subspaces of holomorphic functions”

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.

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 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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$J$ -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.

Spaces of geometrically generic configurations

Yoel Feler (2008)

Journal of the European Mathematical Society

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Let $X$ denote either ${ℂℙ}^{m}$ or $ℂ^{m}$ . We study certain analytic properties of the space $ℰ^{n} (X, g p)$ of ordered geometrically generic $n$ -point configurations in $X$ . This space consists of all $q = (q_{1}, \dots, q_{n}) \in X^{n}$ such that no $m + 1$ of the points $q_{1}, \dots, q_{n}$ belong to a hyperplane in $X$ . In particular, we show that for a big enough $n$ any holomorphic map $f : ℰ^{n} ({ℂℙ}^{m}, g p) \to ℰ^{n} ({ℂℙ}^{m}, g p)$ commuting with the natural action of the symmetric group $𝐒 (n)$ in $ℰ^{n} ({ℂℙ}^{m}, g p)$ is of the form $f (q) = τ (q) q = (τ (q) q_{1}, \dots, τ (q) q_{n})$ , $q \in ℰ^{n} ({ℂℙ}^{m}, g p)$ , where $τ : ℰ^{n} ({ℂℙ}^{m}, g p) \to 𝐏𝐒𝐋 (m + 1, ℂ)$ is an $𝐒 (n)$ -invariant holomorphic map. A similar result holds true for mappings of the configuration...

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...

On highly nonintegrable functions and homogeneous polynomials

P. Wojtaszczyk (1997)

Annales Polonici Mathematici

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We construct a sequence of homogeneous polynomials on the unit ball $_{d}$ in $ℂ^{d}$ which are big at each point of the unit sphere . As an application we construct a holomorphic function on $_{d}$ which is not integrable with any power on the intersection of $_{d}$ with any complex subspace.

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 $ℂ^{k}$ .

Holomorphic line bundles and divisors on a domain of a Stein manifold

Makoto Abe (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^{k} (D, 𝒪) = 0$ for $2 \leq k \leq n - 1$ . We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$ .

Grauert's line bundle convexity, reduction and Riemann domains

Viorel Vâjâitu (2016)

Czechoslovak Mathematical Journal

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We consider a convexity notion for complex spaces $X$ with respect to a holomorphic line bundle $L$ over $X$ . This definition has been introduced by Grauert and, when $L$ is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if $H^{0} (X, L)$ separates each point of $X$ , then $X$ can be realized as a Riemann domain over the complex projective...

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...

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....

Pluriharmonic extension in proper image domains

Rafał Czyż (2009)

Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s ≥ 3. Also let $Ω_{π}$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f : \partial Ω_{π} \to ℝ$ that can be extended to a real-valued pluriharmonic function in $Ω_{π}$ .

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...

New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces

Xin-Cui Guo, Ze-Hua Zhou (2015)

Czechoslovak Mathematical Journal

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Let $u$ be a holomorphic function and $ϕ$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u C_{ϕ}$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$ , $ϕ$ , their derivatives, and $ϕ^{n}$ , the $n$ -th power of $ϕ$ . Moreover, we obtain some similar estimates for the essential norms of the operators $u C_{ϕ}$ , from which sufficient and necessary conditions of compactness of $u C_{ϕ}$ follows immediately. ...

A finiteness theorem for holomorphic Banach bundles

Jürgen Leiterer (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $id + K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0 \leq q \leq n$ . Then: If $V \to X$ is a holomorphic vector bundle (of finite rank) with $H^{q} (X, V) = 0$ , then $dim H^{q} (X, V \otimes E) < \infty$ . In particular, if $dim H^{q} (X, 𝒪) = 0$ , then $dim H^{q} (X, E) < \infty$ .

Robin functions and extremal functions

T. Bloom, N. Levenberg, S. Ma'u (2003)

Annales Polonici Mathematici

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Given a compact set $K \subset ℂ^{N}$ , for each positive integer n, let $V^{(n)} (z)$ = $V_{K}^{(n)} (z)$ := sup $1 / (d e g p) V_{p (K)} (p (z))$ : p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V_{K}^{(n)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_{K} (z)$ := max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${| | p | |}_{K} \leq 1$ ]. Our main result is that if K is regular, then all of the functions $V_{K}^{(n)}$ are continuous; and their associated Robin functions $ϱ_{V_{K}^{(n)}} (z) : = l i m s u p_{| λ | \to \infty} [V_{K}^{(n)} (λ z) - l o g (| λ |)]$ increase to $ϱ_{K} : = ϱ_{V_{K}}$ for all z outside a pluripolar...

Steinness of bundles with fiber a Reinhardt bounded domain

Karl Oeljeklaus, Dan Zaffran (2006)

Bulletin de la Société Mathématique de France

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Let $E$ denote a holomorphic bundle with fiber $D$ and with basis $B$ . Both $D$ and $B$ are assumed to be Stein. For $D$ a Reinhardt bounded domain of dimension $d = 2$ or $3$ , we give a necessary and sufficient condition on $D$ for the existence of a non-Stein such $E$ (Theorem $1$ ); for $d = 2$ , we give necessary and sufficient criteria for $E$ to be Stein (Theorem $2$ ). For $D$ a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for $E$ to be Stein (Theorem...

Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ

Kuzman Adzievski (2006)

Annales Polonici Mathematici

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We study questions related to exceptional sets of pluri-Green potentials $V_{μ}$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{μ}$ are defined by $V_{μ} (z) = \int_{B} l o g (1 / | ϕ_{z} (w) |) d μ (w)$ , where for a fixed z ∈ B, $ϕ_{z}$ denotes the holomorphic automorphism of B satisfying $ϕ_{z} (0) = z$ , $ϕ_{z} (z) = 0$ and $(ϕ_{z} \circ ϕ_{z}) (w) = w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals

Shuzo Izumi (2015)

Annales Polonici Mathematici

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Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection $_{b}$ from the local ring $_{n, b}$ onto the space $Z_{b}$ of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and $_{b}$ induces the structure of an Artinian algebra on $Z_{b}$ . In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈...

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 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.

Equidistribution towards the Green current

Vincent Guedj (2003)

Bulletin de la Société Mathématique de France

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Let $f : ℙ^{k} \to ℙ^{k}$ be a dominating rational mapping of first algebraic degree $λ \geq 2$ . If $S$ is a positive closed current of bidegree $(1, 1)$ on $ℙ^{k}$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks $λ^{- n} {(f^{n})}^{*} S$ converge to the Green current $T_{f}$ . For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$ -invariant currents.