Boundary functions on a bounded balanced domain

Piotr Kot

Displaying similar documents to “Boundary functions on a bounded balanced domain”

Description of simple exceptional sets in the unit ball

Piotr Kot (2004)

Czechoslovak Mathematical Journal

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For $z \in \partial B^{n}$ , the boundary of the unit ball in $ℂ^{n}$ , let $Λ (z) = {λ | λ | \leq 1}$ . If $f \in 𝕆 (B^{n})$ then we call $E (f) = {z \in \partial B^{n} \int_{Λ (z)} | f (z) |^{2} d Λ (z) = \infty}$ the exceptional set for $f$ . In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a $G_{δ}$ and $F_{σ}$ subset of the projective $(n - 1)$ -dimensional space $ℙ^{n - 1} = ℙ (ℂ^{n})$ then there exists a holomorphic function $f$ in the unit ball $B^{n}$ so that $E (f) = E$ .

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 $C (0, r) = \{z \in C ∣ |z| = r\}$ .

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 $Ω \cap V (z_{0})$ .

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.

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 an integral-type operator from Privalov spaces to Bloch-type spaces

Xiangling Zhu (2011)

Annales Polonici Mathematici

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Let H(B) denote the space of all holomorphic functions on the unit ball B of ℂⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. We study the integral-type operator $C_{φ}^{g} f (z) = \int_{0}^{1} ℜ f (φ (t z)) g (t z) d t / t$ , f ∈ H(B). The boundedness and compactness of $C_{φ}^{g}$ from Privalov spaces to Bloch-type spaces and little Bloch-type spaces are studied

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]. ...

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.

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.

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.

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

Manfred Stoll (1976)

Annales Polonici Mathematici

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