Let
Dj ⊂ C
nj be a pseudoconvex domain and let
Aj ⊂
Dj be a locally pluriregular set,
j = 1, . . . ,
N. Put
X :=?
A1 ×· · ·×
Aj-1 ×
Dj ×
Aj+1 ×· · ·×
AN.
Let
M ⊂
X be relatively closed. For any
j ∈ {1, . . . ,
N} let Σ
j be the set of all (
z',
z'') ∈ (
A1 × · · · ×
Aj-1) × (
Aj+1 × · · · ×
AN) such that the fiber
M(z',·,z'') := {
zj ∈ C
nj: (
z', zj , z'') ∈
M} is not pluripolar. Assume that Σ
1, . . . ,Σ
N are pluripolar. Put
X' :=?{(
z', zj , z'') ∈ (
A1×· · ·×
Aj-1)×
Dj ×(
Aj+1×· · ·×
AN) : (
z',
z'') ∉ Σ
j}.
Then (Theorem 1.3) there exists a relatively closed pluripolar subset
M ⊂
X of the ‘envelope of holomorphy’
X of
X such that:
•
M &cao;
X' ⊂
M;
• every function
f separately meromorphic on
X\\
M (Definition 1.2) extends to a (uniquely determined) function
f meromorphic on
X\\
M;
• if f is separately holomorphic on
X\\
M, then
f is holomorphic on
X\\
M; and
•
M is singular with respect to the family of all functions
f.
The case of separately holomorphic functions was solved by Jarnicki and Pflug in an earlier paper. In the case where
N = 2,
M = ∅, the above result will be strengthened in Theorem 1.4.
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