Kyushu Journal of Mathematics
Online ISSN : 1883-2032
Print ISSN : 1340-6116
ISSN-L : 1340-6116
AN EXTENSION THEOREM FOR SEPARATELY MEROMORPHIC FUNCTIONS WITH PLURIPOLAR SINGULARITIESINEQUALITIES
Marek JARNICKIPeter PFLUG
Author information
JOURNALS FREE ACCESS

2003 Volume 57 Issue 2 Pages 291-302

Details
Abstract

Let Dj ⊂ Cnj be a pseudoconvex domain and let AjDj be a locally pluriregular set, j = 1, . . . ,N. Put
X :=?A1 ×· · ·×Aj-1 × Dj × Aj+1 ×· · ·×AN.
Let MX be relatively closed. For any j ∈ {1, . . . ,N} let Σj be the set of all (z', z'') ∈ (A1 × · · · × Aj-1) × (Aj+1 × · · · × AN) such that the fiber M(z',·,z'') := {zj ∈ Cnj: (z', zj , z'') ∈ M} is not pluripolar. Assume that Σ1, . . . ,ΣN are pluripolar. Put
X' :=?{(z', zj , z'') ∈ (A1×· · ·×Aj-1)×Dj ×(Aj+1×· · ·×AN) : (z', z'') ∉ Σj}.
Then (Theorem 1.3) there exists a relatively closed pluripolar subset MX 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.

Information related to the author
© 2003 by Faculty of Mathematics, Kyushu University
Previous article Next article
feedback
Top