Suppose that D ⊂ ℂn is a domain with smooth boundary ∂D, E ⊂ ∂D is a boundary subset of positive Lebesgue measure mes(E) > 0, and F ⊂ G is a nonpluripolar compact set in a strongly pseudoconvex domain G ⊂ ℂm. We prove that, under some additional conditions, each function separately analytic on the set X = (D×F)∪(E× G) can be holomorphically continued into the domain
F. Hartogs, “Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veranderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiben,” Math. Ann., 62 (1906), 1–88.MATHMathSciNetCrossRefGoogle Scholar
2.
B. V. Shabat, Introduction to Complex Analysis, vol. 2 [in Russian], Nauka, Moscow, 1985.Google Scholar
3.
M. Hukuhara, L’extensions du théorème d’Osgood et de Hartogs, Kansu-hoteisiki ogobi Oyo-Kaiseki, vol. 48, 1930.Google Scholar
4.
J. Siciak, “Separately analytic functions and envelopes of holomorphy of some lower-dimensional subsets of ℂn,” Ann. Pol. Math., 22 (1966), no. 1, 145–171.MathSciNetGoogle Scholar
5.
A. Sadullaev, “Plurisubharmonic functions,” in: Current Problems in Mathematics: Fundamental Directions [in Russian], vol. 8, Itogi Nauki i Tekhniki [Progress in Science and Technology], Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1985, pp. 65–113.Google Scholar
6.
Nguyen Thanh Van and A. Zeriahi, “Une extension du théorème de Hartogs sur les fonctions séparément analytiques,” in: Analyse complexe multivariable: recents developpements (Guadeloupe, 1988) (A. Meril, ed.), EditEl, Rende, 1991, pp. 183–194.Google Scholar
7.
A. S. Sadullaev, “Rational approximations and pluripolar sets,” Mat. Sb. [Math. USSR-Sb.], 119(161) (1982), no. 1, 96–118.MATHMathSciNetGoogle Scholar
8.
A. S. Sadullaev, “Plurisubharmonic measures and capacities on complex manifolds,” Uspekhi Mat. Nauk [Russian Math. Surveys], 36 (1981), no. 4, 53–105.MATHMathSciNetGoogle Scholar
9.
E. Bedford and B. A. Taylor, “A new capacity for plurisubharmonic functions,” Acta. Math., 149 (1982), 1–40.MATHMathSciNetCrossRefGoogle Scholar
10.
A. A. Gonchar, “On analytic continuation from the “edge of the wedge,”” Ann. Acad. Sci. Fin. Ser. A I Math., 10 (1985), 221–225.MATHMathSciNetGoogle Scholar
11.
A. A. Gonchar, “On N. N. Bogolyubov’s ‘edge of the wedge’ theorem” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 228 (2000), 24–31.MATHMathSciNetGoogle Scholar
12.
V. P. Zakharyuta, “Extremal plurisubharmonic functions, Hilbert scales, and isomorphisms of spaces of analytic functions in several variables, I, II,” Theory of Functions, Functional Analysis, and Their Applications (1974), no. 19, 133–157 (1974), no. 21, 65–83.Google Scholar
13.
B. S. Mityagin, “Approximative dimension and bases in nuclear spaces,” Uspekhi Mat. Nauk [Russian Math. Surveys], 16 (1961), no. 4, 63–132.MATHGoogle Scholar
