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We continue to discuss the example presented in \cite{JarPfl2015}. In particular, we clarify some gaps and complete the description of the Shilov boundary.

Let $D, G\subset{\Bbb C}$ be domains, let $A\subset D$, $B\subset G$ be locally regular sets, and let $X:=(D\times B)\cup(A\times G)$. Assume that $A$ is a Borel set. Let $M$ be a proper analytic subset of an open neighborhood of $X$. Then there exists a pure 1-dimensional analytic subset $\hat M$ of the envelope of holomorphy $\hat X$ of $X$ such that any function separately holomorphic on $X\setminus M$ extends to a holomorphic function on $\hat X\setminus\hat M$. The result generalizes special cases which were studied in \cite{\"Okt 1998}, \cite{\"Okt 1999a}, and \cite{Sic 2000}., Comment: 8 pages

We prove that in the extension theorem for separately holomorphic functions on an $N$-fold cross with singularities the case of analytic singularities follows from the case of 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, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset X$ be relatively closed. For any $j\in\{1,...,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1})\times(A_{j+1}\times...\times A_N)$ such that the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb C^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,...,\Sigma_N$ are pluripolar. Put ${multline*} X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times...\times A_{j-1})\times D_j \times(A_{j+1}\times...\times A_N): (z',z'')\notin\Sigma_j\}$. Then there exists a relatively closed pluripolar subset $\widetilde M\subset\widetilde X$ of the `envelope of holomorphy' $\widetilde X$ of $X$ such that: $\bullet$ $\widetilde M\cap X'\subset M$, $\bullet$ every function $f$ separately meromorphic on $X\setminus M$ extends to a (uniquely determined) function $\widetilde f$ meromorphic on $\widetilde X\setminus\widetilde M$, $\bullet$ if $f$ is separately holomorphic on $X\setminus M$, then $\widetilde f$ is holomorphic on $\widetilde X\setminus\widetilde M$, and $\bullet$ $\widetilde M$ is singular with respect to the family of all functions $\widetilde f$. \noindent In the case where N=2, $M=\varnothing$, the above result may be strengthened., Comment: 11 pages

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb C^{k_1+...+k_N}.$$Let $U$ be an open connected neighborhood of $X$ and let $M\varsubsetneq U$ be an analytic subset. Then there exists an analytic subset $\hat M$ of the `envelope of holomorphy' $\hat X$ of $X$ with $\hat M\cap X\subset M$ such that for every function $f$ separately holomorphic on $X\setminus M$ there exists an $\hat f$ holomorphic on $\hat X\setminus\hat M$ with $\hat f|_{X\setminus M}=f$. The result generalizes special cases which were studied in \cite{\"Okt 1998}, \cite{\"Okt 1999}, \cite{Sic 2000}, and \cite{Jar-Pfl 2001}., Comment: 20 pages; This a new version of the paper (including "An extension theorem for separately holomorphic functions with singularities, II")

This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089., Comment: This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089

We present various characterizations of $n$-circled domains of holomorphy $G\subset\CC^n$ with respect to some subspaces of $\Cal H^\infty(G)$.