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Generalization of a theorem of Gonchar. (English) Zbl 1161.31005
Let $$X$$ and $$Y$$ be two complex manifolds, let $$D\subset X$$ and $$G\subset Y$$ be two nonempty open sets, let $$A$$ (resp. $$B$$) be an open subset of $$\partial D$$ (resp. $$\partial G$$), and let $$W$$ be the $$2$$-fold cross $$((D\cup A)\times B)\cup(A\times (B\cup G))$$. Under a geometric condition on the boundary sets $$A$$ and $$B$$, the authors show that every function locally bounded, separately continuous on $$W$$, continuous on $$A\times B$$, and separately holomorphic on $$(A\times G)\cup(D\times B)$$ “extends” to a function continuous on a “domain of holomorphy” $$\widehat W$$ and holomorphic on the interior of $$\widehat W$$.
The proof is based on Gonchar’s theorem, the techniques introduced by the authors in [Ann. Pol. Math. 84, No. 3, 237–271 (2004; Zbl 1068.32010)], Poletsky theory of holomorphic discs and Rosay’s theorem developed in a recent article of the second author [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 2, 219–254 (2005; Zbl 1170.32306)] and a thorough geometric study of the plurisubharmonic measure.

##### MSC:
 31C10 Pluriharmonic and plurisubharmonic functions 32D15 Continuation of analytic objects in several complex variables 32U05 Plurisubharmonic functions and generalizations
##### Citations:
Zbl 1068.32010; Zbl 1170.32306
Full Text:
##### References:
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