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### A boundary cross theorem for separately holomorphic functions

Annales Polonici Mathematici

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...

### A domain whose envelope of holomorphy is not a domain

Annales Polonici Mathematici

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

### A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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 $f:\phantom{\rule{4pt}{0ex}}W:=\left(A×Y\right)\cup \left(X×B\right)\to Z$ extends to a holomorphic mapping $\stackrel{^}{f}$ on $\stackrel{^}{W}:=\left\{\left(z,w\right)\in X×Y:\phantom{\rule{4pt}{0ex}}\stackrel{˜}{\omega }\left(z,A,X\right)+\stackrel{˜}{\omega }\left(w,B,Y\right)<1\right\}$ such that $\stackrel{^}{f}=f$...

### A unified approach to the theory of separately holomorphic mappings

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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...

### An Envelope of Holomorphy for Certain Normal Complex Spaces.

Mathematische Annalen

### An extension theorem for separately holomorphic functions with analytic singularities

Annales Polonici Mathematici

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 $X:={\bigcup }_{j=1}^{N}A₁×...×{A}_{j-1}×{D}_{j}×{A}_{j+1}×...×{A}_{N}\subset {ℂ}^{k₁+...+{k}_{N}}$. 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].

### An extension theorem with analytic singularities for generalized (N,k)-crosses

Annales Polonici Mathematici

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.

### Analytic extension from non-pseudoconvex boundaries and $A\left(D\right)$-convexity

Annales de l’institut Fourier

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.

### Balanced domains of holomorphy of type ${H}^{\alpha }$

Matematički Vesnik

### Bases communes dans certains espaces de fonctions harmoniques et fonctions séparément harmoniques sur certains ensembles de ${𝐂}^{n}$

Annales de la Faculté des sciences de Toulouse : Mathématiques

### Biholomorphic Mappings Between Certain Real Analytic Domains in C2.

Mathematische Annalen

### Boundary cross theorem in dimension 1

Annales Polonici Mathematici

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)...

### Cartan-Thullen theorem for domains spread over ...-spaces.

Journal für die reine und angewandte Mathematik

### Central morphisms and envelopes of holomorphy.

Portugaliae Mathematica

### Construction d'enveloppes d'holomorphie par l'méthode de H. Cartan et P.Thullen.

Mathematische Annalen

### Continuous linear extension operators on spaces of holomorphic functions on closed subgroups of a complex Lie group

Annales Polonici Mathematici

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 ${R}_{V}:O\left(\Gamma ×G\right)\to Im{R}_{V}\subset O\left(V\right)$ has a right inverse

### Cousin-I spaces and domains of holomorphy

Annales Polonici Mathematici

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.

### Cross theorem

Annales Polonici Mathematici

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].

### Envelopes of Holomorphy and Polynomial Hulls.

Mathematische Annalen

### Envelopes of holomorphy for solutions of the Laplace and Dirac equations

Commentationes Mathematicae Universitatis Carolinae

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.

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