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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
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