An extension theorem for separately holomorphic functions with singularities.pdf
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AN EXTENSION THEOREM FOR SEPARATELY
HOLOMORPHIC FUNCTIONS WITH SINGULARITIES
Marek Jarnicki (Krako?w)(?) Peter Pflug (Oldenburg)(?)
Abstract. Let Dj ? C
kj be a pseudoconvex domain and let Aj ? Dj be a locally
pluripolar set, j = 1, . . . , N . Put
X :=
N?
j=1
A1 × · · · × Aj?1 × Dj × Aj+1 × · · · × AN ? C
k1+···+kN .
Let U be an open connected neighborhood of X and let M U be an analytic
subset. Then there exists an analytic subset M? of the ‘envelope of holomorphy’
X? of X with M? ∩ X ? M such that for every function f separately holomorphic
on X \ M there exists an f? holomorphic on X? \ M? with f? |X\M = f . The result
generalizes special cases which were studied in [O?kt 1998], [O?kt 1999], [Sic 2000],
and [Jar-Pfl 2001].
1. Introduction. Main Theorem. Let N ∈ N, N ≥ 2, and let
? 6= Aj ? Dj ? C
kj ,
where Dj is a domain, j = 1, . . . , N . We define an N–fold cross
X := X(A1, . . . , AN ;D1, . . . , DN)
:=
N?
j=1
A1 × · · · ×Aj?1 ×Dj ×Aj+1 × · · · ×AN ? C
k1+···+kN . (1)
Observe that X is connected.
Let ? ? Cn be an open set and let A ? ?. Put
hA,? := sup{u : u ∈ PSH(?), u ≤ 1 on ?, u ≤ 0 on A},
1991 Mathematics Subject Classification. 32D15, 32D10.
(?) Research partially supported by the KBN grant No. 5 P03A 033 21.
(?) Research partially supported by the Niedersa?chsisches Ministerium fu?r Wissenschaft und
Kultur, Az. 15.3 – 50 113(55) PL.
Typeset by AMS-TEX
1
where PSH(?) denotes the set of all functions plurisubharmonic on ?. Define
ωA,? := lim
k→+∞
h?A∩?k,?k ,
where (?k)
∞
k=1 is a sequence of relatively compact open sets ?k ? ?k+1 ?? ? with?∞
k=1 ?k = ? (h
? denotes the upper semicontinuous regularization of h). Observe
that the definition is independent of the chosen exhausting sequence (?k)
∞
k=1.
Moreover, ωA,? ∈ PSH(?).
For an N–fold cross X = X(A1, . . . , AN ;D1, . . . , DN ) put
X? := {(z1, . . . , zN) ∈ D1 × · · · ×DN :
N∑
j=1
ωAj,Dj (zj) 1};
notice that X? may be
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