An extension theorem for separately holomorphic functions with pluripolar singularities

2001/12/31 by Marek Jarnicki, Peter Pflug
Mathematics · #math.CV #msc:32D15 #msc:32D10

paper · pdf

published as Trans. Amer. Math. Soc. 355 (2003), 1251-1267. · 19 pages

arxiv created 2002/02/13 · arxiv updated 2009/11/30

Abstract

Let D_j⊂\Bbb C^n_j be a pseudoconvex domain and let A_j⊂ D_j be a locally pluriregular set, j=1,...,N. Put X:=\bigcup_j=1^N A₁×...× A_j-1× D_j× A_j+1× ...× A_N⊂\Bbb C^n₁×...×\Bbb C^n_N=\Bbb Cⁿ. Let U⊂\Bbb Cⁿ be an open neighborhood of X and let M⊂ U be a relatively closed subset of U. For j∈\1,...,N\ let Σ_j be the set of all (z',z'')∈(A₁×...× A_j-1) ×(A_j+1×...× A_N) for which the fiber M_(z',⋅,z''):=\z_j∈\Bbb C^n_j (z',z_j,z'')∈ M\ is not pluripolar. Assume that Σ₁,...,Σ_N are pluripolar. Put X':=\bigcup_j=1^N\(z',z_j,z'')∈(A₁×...× A_j-1)× D_j ×(A_j+1×...× A_N) (z',z'')∉Σ_j\. Then there exists a relatively closed pluripolar subset M⊂ X of the `envelope of holomorphy' X⊂\Bbb Cⁿ of X such that: M∩ X'⊂ M, for every function f separately holomorphic on X∖ M there exists exactly one function f holomorphic on X∖ M with f=f on X'∖ M, and M is singular with respect to the family of all functions f. Some special cases were previously studied in \citeJar-Pfl 2001c.

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