An extension theorem for separately meromorphic functions with pluripolar singularities

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Let Dⱼⁿʲ be a pseudoconvex domain and let Aⱼ⊂ Dⱼ be a locally pluriregular set, j=1,...,N. Put X:=ⱼ₌₁ⁿ A₁×...× Aⱼ₋₁× Dⱼ× Aⱼ₊₁×...× Aₙ. Let M⊂ X 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''₎:={zⱼⁿʲ: (z',zⱼ,z'')∈ M} is not pluripolar. Assume that Σ₁,...,Σₙ are pluripolar. Put multline* X':=ⱼ₌₁ⁿ{(z',zⱼ,z'')∈(A₁×...× Aⱼ₋₁)× Dⱼ ×(Aⱼ₊₁×...× Aₙ): (z',z'')∉Σⱼ}. Then there exists a relatively closed pluripolar subset M⊂ X of the `envelope of holomorphy' X of X such that: M∩ X'⊂ M, every function f separately meromorphic on X M 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. In the case where N=2, M=, the above result may be strengthened.