AN EXTENSION THEOREM FOR SEPARATELY MEROMORPHIC FUNCTIONS WITH PLURIPOLAR SINGULARITIESINEQUALITIES

Abstract

Let D_j ⊂ C^n_j be a pseudoconvex domain and let A_j ⊂ D_j be a locally pluriregular set, j = 1, . . . ,N. Put
X :=?A₁ ×· · ·×A_j-1 × D_j × A_j+1 ×· · ·×A_N.
Let M ⊂ X be relatively closed. For any j ∈ {1, . . . ,N} let Σ_j be the set of all (z', z'') ∈ (A₁ × · · · × A_j-1) × (A_j+1 × · · · × A_N) such that the fiber M_(z',·,z'') := {z_j ∈ C^n_j: (z', z_j , z'') ∈ M} is not pluripolar. Assume that Σ₁, . . . ,Σ_N are pluripolar. Put
X' :=?{(z', z_j , z'') ∈ (A₁×· · ·×A_j-1)×D_j ×(A_j+1×· · ·×A_N) : (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.