An extension theorem for separately holomorphic functions with pluripolar singularities

Let $D_j\subset {\mathbb{C}}^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$bealocallypluriregularset,$j=1,…,N$\mathrm{.}Put$$ X:=\bigcup_{j=1}^N A_1\times…\times A_{j-1}\times D_j\times A_{j+1}\times …\times A_N\subset\Bbb C^{n_1}\times…\times\Bbb C^{n_N}=\Bbb C^n. Let $U\subset\Bbb C^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,...,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1}) \times(A_{j+1}\times...\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\Bbb C^{n_j}\: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,...,\Sigma_N$ are pluripolar. Put X’:=\bigcup_{j=1}^N{(z’,z_j,z’‘)\in(A_1\times…\times A_{j-1})\times D_j \times(A_{j+1}\times…\times A_N): (z’,z’‘)\notin\Sigma_j}. $$ 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: $\hat M\cap X’\subset M$,foreveryfunction$f$separatelyholomorphicon$X\setminus M$thereexistsexactlyonefunction$\hat f$holomorphicon$\hat X\setminus\hat M$with$\hat f=f$on$X’\setminus M$,and$\hat M$ is singular with respect to the family of all functions $\widehat{f}$. Some special cases were previously studied in \cite{Jar-Pfl 2001c}.