# An extension theorem for separately holomorphic functions with analytic singularities

• Volume: 80, Issue: 1, page 143-161
• ISSN: 0066-2216

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## Abstract

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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 $X:={\bigcup }_{j=1}^{N}A₁×...×{A}_{j-1}×{D}_{j}×{A}_{j+1}×...×{A}_{N}\subset {ℂ}^{k₁+...+{k}_{N}}$. 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 ${f̂|}_{X\setminus M}=f$. The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001].

## How to cite

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Marek Jarnicki, and Peter Pflug. "An extension theorem for separately holomorphic functions with analytic singularities." Annales Polonici Mathematici 80.1 (2003): 143-161. <http://eudml.org/doc/281010>.

@article{MarekJarnicki2003,
abstract = {Let $D_j ⊂ ℂ^\{k_j\}$ be a pseudoconvex domain and let $A_j ⊂ D_j$ be a locally pluriregular set, j = 1,...,N. Put $X: = ⋃_\{j=1\}^N A₁ ×. .. × A_\{j-1\} × D_j × A_\{j+1\} ×. .. × A_N ⊂ ℂ^\{k₁+...+k_N\}$. 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 $f̂|_\{X∖M\} = f$. The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001].},
author = {Marek Jarnicki, Peter Pflug},
journal = {Annales Polonici Mathematici},
keywords = {separately holomorphic; pluriregular; holomorphic extension},
language = {eng},
number = {1},
pages = {143-161},
title = {An extension theorem for separately holomorphic functions with analytic singularities},
url = {http://eudml.org/doc/281010},
volume = {80},
year = {2003},
}

TY - JOUR
AU - Marek Jarnicki
AU - Peter Pflug
TI - An extension theorem for separately holomorphic functions with analytic singularities
JO - Annales Polonici Mathematici
PY - 2003
VL - 80
IS - 1
SP - 143
EP - 161
AB - Let $D_j ⊂ ℂ^{k_j}$ be a pseudoconvex domain and let $A_j ⊂ D_j$ be a locally pluriregular set, j = 1,...,N. Put $X: = ⋃_{j=1}^N A₁ ×. .. × A_{j-1} × D_j × A_{j+1} ×. .. × A_N ⊂ ℂ^{k₁+...+k_N}$. 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 $f̂|_{X∖M} = f$. The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001].
LA - eng
KW - separately holomorphic; pluriregular; holomorphic extension
UR - http://eudml.org/doc/281010
ER -

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