An extension theorem for separately holomorphic functions with pluripolar singularities

Let $D_j\subset\mathbb{C} ^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,\dots,N$. Put

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

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 $\widehat f$.