An extension theorem for separately holomorphic functions with singularities.pdf

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An extension theorem for separately holomorphic functions with singularities.pdf

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An extension theorem for separately holomorphic functions with singularities

a r X i v : m a t h / 0 1 0 4 0 8 9 v 3 [ m a t h .C V ] 1 0 O c t 2 0 0 1 AN EXTENSION THEOREM FOR SEPARATELY HOLOMORPHIC FUNCTIONS WITH SINGULARITIES Marek Jarnicki (Krako?w)(?) Peter Pflug (Oldenburg)(?) Abstract. Let Dj ? C kj be a pseudoconvex domain and let Aj ? Dj be a locally pluripolar set, j = 1, . . . , N . Put X := N? j=1 A1 × · · · × Aj?1 × Dj × Aj+1 × · · · × AN ? C k1+···+kN . Let U be an open connected neighborhood of X and let M U be an analytic subset. Then there exists an analytic subset M? of the ‘envelope of holomorphy’ X? of X with M? ∩ X ? M such that for every function f separately holomorphic on X \ M there exists an f? holomorphic on X? \ M? with f? |X\M = f . The result generalizes special cases which were studied in [O?kt 1998], [O?kt 1999], [Sic 2000], and [Jar-Pfl 2001]. 1. Introduction. Main Theorem. Let N ∈ N, N ≥ 2, and let ? 6= Aj ? Dj ? C kj , where Dj is a domain, j = 1, . . . , N . We define an N–fold cross X := X(A1, . . . , AN ;D1, . . . , DN) := N? j=1 A1 × · · · ×Aj?1 ×Dj ×Aj+1 × · · · ×AN ? C k1+···+kN . (1) Observe that X is connected. Let ? ? Cn be an open set and let A ? ?. Put hA,? := sup{u : u ∈ PSH(?), u ≤ 1 on ?, u ≤ 0 on A}, 1991 Mathematics Subject Classification. 32D15, 32D10. (?) Research partially supported by the KBN grant No. 5 P03A 033 21. (?) Research partially supported by the Niedersa?chsisches Ministerium fu?r Wissenschaft und Kultur, Az. 15.3 – 50 113(55) PL. Typeset by AMS-TEX 1 where PSH(?) denotes the set of all functions plurisubharmonic on ?. Define ωA,? := lim k→+∞ h?A∩?k,?k , where (?k) ∞ k=1 is a sequence of relatively compact open sets ?k ? ?k+1 ?? ? with?∞ k=1 ?k = ? (h ? denotes the upper semicontinuous regularization of h). Observe that the definition is independent of the chosen exhausting sequence (?k) ∞ k=1. Moreover, ωA,? ∈ PSH(?). For an N–fold cross X = X(A1, . . . , AN ;D1, . . . , DN ) put X? := {(z1, . . . , zN) ∈ D1 × · · · ×DN : N∑ j=1 ωAj,Dj (zj) 1}; notice that X? may be