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.

Let X, Y be two complex manifolds of dimension 1 which are countable at infinity, let D ⊂ X, G ⊂ Y be two open sets, let A (resp. B) be a subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D∪A)×B) ∪ (A×(B∪G)). Suppose in addition that D (resp. G) is Jordan-curve-like on A (resp. B) and that A and B are of positive length. We determine the "envelope of holomorphy" Ŵ of W in the sense that any function locally bounded on W, measurable on A × B, and separately holomorphic on (A × G) ∪ (D × B)...

We show that the restriction operator of the space of holomorphic functions on a complex Lie group to an analytic subset V has a continuous linear right inverse if it is surjective and if V is a finite branched cover over a connected closed subgroup Γ of G. Moreover, we show that if Γ and G are complex Lie groups and V ⊂ Γ × G is an analytic set such that the canonical projection ${\pi }_1:V\to \Gamma$ is finite and proper, then $R_V:O\left(\Gamma \times G\right)\to Im{R}_V\subset O\left(V\right)$ has a right inverse

We prove that a Cousin-I open set D of an irreducible projective surface X is locally Stein at every boundary point which lies in $X_{reg}$. In particular, Cousin-I proper open sets of ℙ² are Stein. We also study K-envelopes of holomorphy of K-complete spaces.

Let D,G ⊂ ℂ be domains, let A ⊂ D, B ⊂ G be locally regular sets, and let X:= (D×B)∪(A×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 M̂ of the envelope of holomorphy X̂ of X such that any function separately holomorphic on X∖M extends to a holomorphic function on X̂ ∖M̂. The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], and [Sic 2000].

Analytic continuation and domains of holomorphy for solution to the complex Laplace and Dirac equations in ${𝐂}^n$ are studied. First, geometric description of envelopes of holomorphy over domains in ${𝐄}^n$ is given. In more general case, solutions can be continued by integral formulas using values on a real $n-1$ dimensional cycle in ${𝐂}^n$. Sufficient conditions for this being possible are formulated.