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An extension theorem for separately holomorphic functions with pluripolar singularities
Authors:Marek Jarnicki   Peter Pflug
Affiliation:Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland ; Carl von Ossietzky Universität Oldenburg, Fachbereich Mathematik, Postfach 2503, D-26111 Oldenburg, Germany
Abstract:Let $D_jsubsetmathbb{C} ^{n_j}$ be a pseudoconvex domain and let $A_jsubset D_j$ be a locally pluriregular set, $j=1,dots,N$. Put

begin{displaymath}X:=bigcup_{j=1}^N A_1timesdotstimes A_{j-1}times D_jtim... ...thbb{C} ^{n_1}timesdotstimesmathbb{C} ^{n_N}=mathbb{C} ^n.end{displaymath}

Let $Usubsetmathbb{C} ^n$ be an open neighborhood of $X$ and let $Msubset U$ be a relatively closed subset of $U$. For $jin{1,dots,N}$ let $Sigma_j$ be the set of all $(z',z')in(A_1timesdotstimes A_{j-1}) times(A_{j+1}timesdotstimes A_N)$ for which the fiber $M_{(z',cdot,z')}:={z_jinmathbb{C} ^{n_j}: (z',z_j,z')in M}$ 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_1timesdotstimes A_{j-... ...imes(A_{j+1}timesdotstimes A_N): (z',z')notinSigma_j}. end{multline*}
Then there exists a relatively closed pluripolar subset $widehat{M}subsetwidehat X$ of the ``envelope of holomorphy' $widehat{X}subsetmathbb{C} ^n$ of $X$ such that:

$bullet$ $widehat Mcap X'subset M$,

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

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

Keywords:
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