Marek Jarnicki

Extension of holomorphic functions

Riemann domains - Riemann domains over Cn holomorphic functions examples of Riemann regions holomorphic extension of Riemann domains the boundary of a Riemann domain union, intersection, and direct limit of Riemann domains domains of existence maximal holomorphic extensions liftings of holomorphic mappings I holomorphic convexity Riemann surfaces pseudocenvexity - Plurisubharmonic functions pseudoconvexity the Kiselman minimum principle d-operator solution of the Levi problem regular solutions approximation the Remmert embedding theorem the Docquier-Grauert criteria the division theorem spectrum liftings of holomorphic mappings II envelopes of holomorphy for special domains - Univalent envelopes of holomorphy k-tubular domains matrix Reinhardt domains the envelope of holomorphy of X/M separately holomorphic functions extension of meromorphic functions existence domains of special families of holomorphic functions - special domains the Ohsawa-Takegoshi extension theorem the Skoda division theorem the Catlin-Hakim-Sibony theorem structure of envelopes of holomorphy.

A general cross theorem with singularities

We present a general cross theorem for separately holomorphic and meromorphic functions with singularities.

Effective formulas for the Carathéodory distance

For a class of Reinhardt domains we give formulas for the Carathéodory distance. As an application we discuss the product property of the Carathéodory distance when one factor domain is a Reinhardt domain of special type.

A new cross theorem for separately holomorphic functions

Theorem 1.1 ([Sic 1969a], [Sic 1969b], [Zah 1976], [Sic 1981a], [Ngu-Sic 1991], [Ngu-Zer 1991], [Ngu-Zer 1995], [NTV 1997], [Ale-Zer 2001], [Zer 2002]). For each f ∈ Os(X) there exists exactly one f ∈ O(X) such that f = f on X and sup b X |f | = supX |f |. The aim of this note is to extend the above theorem to a class of more general objects, namely (N, k)– crosses XN,k defined for k ∈ {1, . . . , N} as follows: XN,k = XN,k((Aj , Dj)j=1) := ⋃

On the upper semicontinuity of the Wu metric

We discuss continuity and upper semicontinuity of the Wu pseudometric. The Wu pseudometric was introduced by H. Wu in [Wu 1993] (and [Wu]). Various properties of the Wu metric may be found for instance in [Che-Kim 1996], [Che-Kim 1997], [Kim 1998], [Che-Kim 2003], [Juc 2002]. Nevertheless, it seems that even quite elementary properties of this metric are not completely understood, e.g. its upper semicontinuity. First, let us formulate the definition of the Wu pseudometric in an abstract setting. Let h : C −→ R+ be a C–seminorm. Put: I = I(h) := {X ∈ C : h(X) < 1} (I is convex), V = V (h) := {X ∈ C : h(X) = 0} ⊂ I (V is a vector subspace of C), U = U(h) := the orthogonal complement of V with respect to the standard Hermitian scalar product 〈z, w〉 := ∑n j=1 zjwj in C, I0 := I ∩ U , h0 := h|U (h0 is a norm, I = I0 + V ). For any pseudo–Hermitian scalar product s : C × C −→ C, let qs(X) := √ s(X,X), X ∈ C, E(s) := {X ∈ C : qs(X) < 1}. Consider the family F of all pseudo–Hermitian scalar products s : C × C −→ C such that I ⊂ E(s), equivalently, qs ≤ h. In particular, V ⊂ I = I0 + V ⊂ E(s) = E(s0) + V, where s0 := s|U×U (note that E(s0) = E(s)∩U). Let Vol(s0) denote the volume of E(s0) with respect to the Lebesgue measure of U . Since I0 is bounded, there exists an s ∈ F with Vol(s0) < +∞. Observe that for any basis e = (e1, . . . , em) of U (m := dimC U) we have Vol(s0) = C(e) detS , where C(e) > 0 is a constant (independent of s) and S = S(s0) denotes the matrix representation of s0 in the basis e, i.e. Sj,k := s(ej , ek), j, k = 1, . . . ,m. In particular, if U = C × {0}n−m and e = (e1, . . . , em) is the canonical basis, then C(e) is the volume of the open unit Euclidean ball Bm ⊂ C. We are interested in Received by the editors October 3, 2003. 2000 Mathematics Subject Classification. Primary 32F45.

A counterexample to a theorem of Bremermann on Shilov boundaries

We give a counterexample to the following theorem of Bremermann on Shilov boundaries: if

An example of a Carathéodory complete but not finitely compact analytic space

D

On the extended tube conjecture

is a bounded domain in

A counterexample to a theorem of Bremermann on Shilov boundaries - revisited

\mathbb C^n

Baire Category Approach

having a univalent envelope of holomorphy, say