zbMATH — the first resource for mathematics

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.

31C10 Pluriharmonic and plurisubharmonic functions
32D15 Continuation of analytic objects in several complex variables
32U05 Plurisubharmonic functions and generalizations
Full Text: DOI arXiv
[1] Airapetyan, R. A. and Henkin, G. M., Analytic continuation of CR-functions across the ”edge of the wedge”, Dokl. Akad. Nauk SSSR 259 (1981), 777–781 (Russian). English transl.: Soviet Math. Dokl. 24 (1981), 128–132.
[2] Airapetyan, R. A. and Henkin, G. M., Integral representations of differential forms on Cauchy–Riemann manifolds and the theory of CR-functions. II, Mat. Sb. 127(169) (1985), 92–112, 143 (Russian). English transl.: Math. USSR-Sb. 55 (1986), 91–111. · Zbl 0589.32036
[3] Dru\.zkowski, L. M., A generalization of the Malgrange–Zerner theorem, Ann. Polon. Math. 38 (1980), 181–186. · Zbl 0461.32004
[4] Edigarian, A., Analytic discs method in complex analysis, Dissertationes Math. (Rozprawy Mat.) 402 (2002). · Zbl 0993.31003
[5] Goluzin, G. M., Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs 26, Amer. Math. Soc., Providence, RI, 1969. · Zbl 0183.07502
[6] Gonchar, A. A., On analytic continuation from the ”edge of the wedge”, Ann. Acad. Sci. Fenn. Math. 10 (1985), 221–225. · Zbl 0603.32008
[7] Gonchar, A. A., On N. N. Bogolyubov’s ”edge of the wedge” theorem, Tr. Mat. Inst. Steklova 228 (2000), Probl. Sovrem. Mat. Fiz. 24–31 (Russian). English transl.: Proc. Steklov Inst. Math. 228 (2000), 18–24. · Zbl 0988.32009
[8] Komatsu, H., A local version of Bochner’s tube theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 (1972), 201–214. · Zbl 0239.32012
[9] Lárusson, F. and Sigurdsson, R., Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math. 501 (1998), 1–39. · Zbl 0901.31004
[10] Nguyên, V.-A., A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005), 219–254. · Zbl 1170.32306
[11] Pflug, P. and Nguyên, V.-A., A boundary cross theorem for separately holomorphic functions, Ann. Polon. Math. 84 (2004), 237–271. · Zbl 1068.32010
[12] Poletsky, E. A., Plurisubharmonic functions as solutions of variational problems, in Several Complex Variables and Complex Geometry (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math. 52, Part 1, pp. 163–171, Amer. Math. Soc., Providence, RI, 1991. · Zbl 0739.32015
[13] Poletsky, E. A., Holomorphic currents, Indiana Univ. Math. J. 42 (1993), 85–144. · Zbl 0811.32010
[14] Rosay, J.-P., Poletsky theory of disks on holomorphic manifolds, Indiana Univ. Math. J. 52 (2003), 157–169. · Zbl 1033.31006
[15] Zerner, M., Quelques résultats sur le prolongement analytique des fonctions de variables complexes, Séminaire de Physique Mathématique.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.