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An extension theorem for separately holomorphic functions with pluripolar singularities


Authors: Marek Jarnicki and Peter Pflug
Journal: Trans. Amer. Math. Soc. 355 (2003), 1251-1267
MSC (2000): Primary 32D15, 32D10
Published electronically: November 5, 2002
MathSciNet review: 1938756
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Abstract: Let $D_j\subset\mathbb{C} ^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,\dots,N$. Put

\begin{displaymath}X:=\bigcup_{j=1}^N A_1\times\dots\times A_{j-1}\times D_j\tim... ...thbb{C} ^{n_1}\times\dots\times\mathbb{C} ^{n_N}=\mathbb{C} ^n.\end{displaymath}

Let $U\subset\mathbb{C} ^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,\dots,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times\dots\times A_{j-1}) \times(A_{j+1}\times\dots\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb{C} ^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,\dots,\Sigma_N$ are pluripolar. Put
\begin{multline*}X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times\dots\times A_{j-... ...imes(A_{j+1}\times\dots\times A_N):\\ (z',z'')\notin\Sigma_j\}. \end{multline*}
Then there exists a relatively closed pluripolar subset $\widehat{M}\subset\widehat X$ of the ``envelope of holomorphy'' $\widehat{X}\subset\mathbb{C} ^n$ of $X$ such that:

$\bullet$ $\widehat M\cap X'\subset M$,

$\bullet$ for every function $f$ separately holomorphic on $X\setminus M$ there exists exactly one function $\widehat f$ holomorphic on $\widehat X\setminus\widehat M$ with $\widehat f=f$ on $X'\setminus M$, and

$\bullet$ $\widehat M$ is singular with respect to the family of all functions $\widehat f$.


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Additional Information

Marek Jarnicki
Affiliation: Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland
Email: jarnicki@im.uj.edu.pl

Peter Pflug
Affiliation: Carl von Ossietzky Universität Oldenburg, Fachbereich Mathematik, Postfach 2503, D-26111 Oldenburg, Germany
Email: pflug@mathematik.uni-oldenburg.de

DOI: https://doi.org/10.1090/S0002-9947-02-03193-8
Received by editor(s): February 12, 2002
Received by editor(s) in revised form: June 3, 2002
Published electronically: November 5, 2002
Additional Notes: The first author was supported in part by KBN grant no. 5 P03A 033 21.
Both authors were supported in part by the Niedersächsisches Ministerium für Wissenschaft und Kultur, Az. 15.3 – 50 113(55) PL
Article copyright: © Copyright 2002 American Mathematical Society