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1
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Extension of separately holomorphic functions-a survey 1899-2001

100%
Pflug P.
Annales Polonici Mathematici
|
2003
|
tom 80
|
nr 1
21-36
EN
This note is an attempt to describe a part of the historical development of the research on separately holomorphic functions.
2
Content available remote

Invariant distances and metrics in complex analysis-revisited

69%
Jarnicki M., Pflug P.
Instytut Matematyczny Polskiej Akademii Nauk
3
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Extension theorems of Sakai type for separately holomorphic and meromorphic functions

64%
Pflug P., Nguyên V.
Annales Polonici Mathematici
|
2003
|
tom 82
|
nr 2
171-187
EN
We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions.
4
Content available remote

Estimates for the Bergman kernel and metric of convex domains in ℂⁿ

64%
Nikolov N., Pflug P.
Annales Polonici Mathematici
|
2003
|
tom 81
|
nr 1
73-78
EN
Sharp geometrical lower and upper estimates are obtained for the Bergman kernel on the diagonal of a convex domain D ⊂ ℂⁿ which does not contain complex lines. It is also proved that the ratio of the Bergman and Carathéodory metrics of D does not exceed a constant depending only on n.
5
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An extension theorem for separately holomorphic functions with analytic singularities

64%
Jarnicki M., Pflug P.
Annales Polonici Mathematici
|
2003
|
tom 80
|
nr 1
143-161
EN
Let $D_j ⊂ ℂ^{k_j}$ be a pseudoconvex domain and let $A_j ⊂ D_j$ be a locally pluriregular set, j = 1,...,N. Put $X: = ⋃_{j=1}^N A₁ ×. .. × A_{j-1} × D_j × A_{j+1} ×. .. × A_N ⊂ ℂ^{k₁+...+k_N}$. 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 $f̂|_{X∖M} = f$. The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001].
6
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Invariant pseudodistances and pseudometrics - completeness and product property

64%
Jarnicki M., Pflug P.
Annales Polonici Mathematici
|
1991
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tom 55
|
nr 1
169-189
EN
A survey of properties of invariant pseudodistances and pseudometrics is given with special stress put on completeness and product property.
7
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New examples of effective formulas for holomorphically contractible functions

64%
Jarnicki M., Pflug P.
Studia Mathematica
|
1999
|
tom 135
|
nr 3
219-230
EN
Let $G ⊂ ℂ^n$ and $B ⊂ ℂ^m$ be domains and let Φ:G → B be a surjective holomorphic mapping. We characterize some cases in which invariant functions and pseudometrics on G can be effectively expressed in terms of the corresponding functions and pseudometrics on B.
8
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A remark on separate holomorphy

64%
Jarnicki M., Pflug P.
Studia Mathematica
|
2006
|
tom 174
|
nr 3
309-317
EN
Let X be a Riemann domain over $ℂ^{k} × ℂ^{ℓ}$. If X is a domain of holomorphy with respect to a family ℱ ⊂𝓞(X), then there exists a pluripolar set $P ⊂ ℂ^{k}$ such that every slice $X_{a}$ of X with a∉ P is a region of holomorphy with respect to the family ${f|_{X_{a}}: f ∈ ℱ}$.
9
Content available remote

On balanced L²-domains of holomorphy

64%
Jarnicki M., Pflug P.
Annales Polonici Mathematici
|
1996
|
tom 63
|
nr 1
101-102
EN
We show that any bounded balanced domain of holomorphy is an $L²_h$-domain of holomorphy.
10
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A boundary cross theorem for separately holomorphic functions

64%
Pflug P., Nguyên V.-A.
Annales Polonici Mathematici
|
2004
|
tom 84
|
nr 3
237-271
EN
Let D ⊂ ℂⁿ and $G ⊂ ℂ^m$ 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 𝓒² 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 crosses is also given.
11
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$L²_{h}$-domains of holomorphy and the Bergman kernel

64%
Pflug P., Zwonek W.
Studia Mathematica
|
2002
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tom 151
|
nr 2
99-108
EN
We give a characterization of $L²_{h}$-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.
12
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Cross theorem

64%
Jarnicki M., Pflug P.
Annales Polonici Mathematici
|
2001
|
tom 77
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nr 3
295-302
EN
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].
13
Content available remote

Effective formulas for invariant functions - case of elementary Reinhardt domains

64%
Pflug P., Zwonek W.
Annales Polonici Mathematici
|
1998
|
tom 69
|
nr 2
175-196
EN
We find effective formulas for the invariant functions, appearing in the theory of several complex variables, of the elementary Reinhardt domains. This gives us the first example of a large family of domains for which the functions are calculated explicitly.
14
Content available remote

On n-circled $ℋ^{∞}$-domains of holomorphy

64%
Jarnicki M., Pflug P.
Annales Polonici Mathematici
|
1996-1997
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tom 65
|
nr 3
253-264
EN
We present various characterizations of n-circled domains of holomorphy $G ⊂ ℂ^n$ with respect to some subspaces of $ℋ^∞(G)$.
15
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A remark on the identity principle for analytic sets

64%
Jarnicki M., Pflug P.
Colloquium Mathematicum
|
2011
|
tom 123
|
nr 1
21-26
EN
We present a version of the identity principle for analytic sets, which shows that the extension theorem for separately holomorphic functions with analytic singularities follows from the case of pluripolar singularities.
16
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Erratum to "Effective formulas for invariant functions - case of elementary Reinhardt domains" (Ann. Polon. Math. 69 (1998), 175-196)

64%
Pflug P., Zwonek W.
Annales Polonici Mathematici
|
1998
|
tom 69
|
nr 3
301
17
Content available remote

Boundary cross theorem in dimension 1

64%
Pflug P., Nguyên V.-A.
Annales Polonici Mathematici
|
2007
|
tom 90
|
nr 2
149-192
EN
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) "extends" to a function holomorphic on the interior of Ŵ.
18
Content available remote

On extremal holomorphically contractible families

52%
Jarnicki M., Jarnicki W., Pflug P.
Annales Polonici Mathematici
|
2003
|
tom 81
|
nr 2
183-199
EN
We prove (Theorem 1.2) that the category of generalized holomorphically contractible families (Definition 1.1) has maximal and minimal objects. Moreover, we present basic properties of these extremal families.
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