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We study the universal overconvergence and its relations with Ostrowski gaps for holomorphic functions of several complex variables.

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_j subset mathbb{C}^{n_j} $ be a pseudoconvex domain and let $ A_j subset D_j $ be a locally pluriregular set, $ j = 1, ldots ,N $. Put $ X colon = cup_{j=1}^{N} {A_1\times cdots \times A_{j-1} \times D_j \times A_{j+1} \times cdots \times A_N $. Let $ M subset X $ be relatively closed. For any $ j in {1, ldots ,N } $ let $ {sum}_j $ be the set of all $ (z',z") in (A_1\times cdots \times A_{j-1}) \times (A_{j+1} \times cdots \times A_N) $ such that the fiber $ M_(z', cdotp ,z") colon = {z_j in mathbb{C}^{n_j} colon (z',z_j,z") in M } $ is not pluripolar. Assume that $ {sum}_1, ldots , {sum}_N $ are pluripolar. Put $ X' colon = cup_{j=1}^{N} {(z',z_j,z") in (A_1 \times cdots \times A_{j-1}) \times D_j \times (A_{j+1} \times cdots \times A_N) colon (z',z") \otin {sum}_j } $. Then (Theorem 1.3) there exists a relatively closed pluripolar subset $ hut{M} subset hut{X} $ of the 'envelope of holomorphy' $ hut{X} $ of $ X $ such that: $ sullet $ $ hut{M} cap X' subset M $; $ \bullet $ every function $ f $ separately meromorphic on $ X \backslash M $ (Definition 1.2) extends to a (uniquely determined) function $ hut{f} $ meromorphic on $ hut{X} \backslash hut{M} $; $ \bullet $ if $ f $ is separately holomorphic on $ X \backslash M $, then $ f $ is holomorphic on $ hut{X} \backslash hut{M} $; and $ \bullet $ $ hut{M} $ is singular with respect to the family of all functions $ hut{f} $. The case of separately holomorphic functions was solved by Jarnicki and Pflug in an earlier paper. In the case where $ N = 2, M = empty $, the above result will be strengthened in Theorem 1.4.

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")

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