Let D ⊂ ℂⁿ and $G\subset {ℂ}^m$ be pseudoconvex domains, let A (resp. B) be an open subset of the boundary ∂D (resp. ∂G) and let X be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Suppose in addition that the domain D (resp. G) is locally ² smooth on A (resp. B). We shall determine the “envelope of holomorphy” X̂ of X in the sense that any function continuous on X and separately holomorphic on (A×G)∪(D×B) extends to a function continuous on X̂ and holomorphic on the interior of X̂. A generalization of this result to N-fold...

We construct a domain of holomorphy in ${ℂ}^N$, N≥ 2, whose envelope of holomorphy is not diffeomorphic to a domain in ${ℂ}^N$.

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$$Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f:W:=(A\times Y)\cup (X\times B)\to Z$ extends to a holomorphic mapping $\widehat{f}$ on $\widehat{W}:=\left\{(z,w)\in X\times Y:\tilde{\omega }(z,A,X)+\tilde{\omega }(w,B,Y)\<1\right\}$ such that $\widehat{f}=f$...

We extend the theory of separately holomorphic mappings between complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay theorem on holomorphic discs and our recent joint-work with Pflug on boundary cross theorems in dimension $1.$ It also relies on our new technique of conformal mappings and a generalization of Siciak’s relative extremal function. Our approach illustrates the unified character: “From local information to global extensions”. Moreover, it avoids systematically...

Let $D_j\subset {ℂ}^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, j = 1,...,N. Put $X:={\bigcup }_{j=1}^{N}A₁\times ...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset {ℂ}^{k₁+...+k_N}$. Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with ${f̂|}_{X\setminus M}=f$. The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001].

The main result of the paper is a new Hartogs type extension theorem for generalized (N,k)-crosses with analytic singularities for separately holomorphic functions and for separately meromorphic functions. Our result is a simultaneous generalization of several known results, from the classical cross theorem, through the extension theorem with analytic singularities for generalized crosses, to the cross theorem with analytic singularities for meromorphic functions.

Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.