Abstract
It is shown that the set \( \mathfrak{L}_\Gamma \) of all complex lines passing through a germ of a generating manifold Γ is sufficient for any continuous function f defined on the boundary of a bounded domain D ⊂ ℂn with connected smooth boundary and having the holomorphic one-dimensional extension property along all lines from \( \mathfrak{L}_\Gamma \) to admit a holomorphic extension to D as a function of many complex variables.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. L. Agranovskii and R. E. Val’skii, “Maximality of invariant algebras of functions,” Sibirsk. Mat. Zh. 12(1), 3–12 (1971).
E. L. Stout, “The boundary values of holomorphic functions of several complex variables,” Duke Math. J. 44(1), 105–108 (1977).
L. A. Aizenberg and A. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis (Nauka, Novosibirsk, 1979) [in Russian].
A. M. Kytmanov, The Bochner-Martinelli Integral and Its Applications (Nauka, Novosibirsk, 1992; Birkhäuser, Basel, 1995).
A. M. Kytmanov and S. G. Myslivets, “On a boundary analogue of the Morera theorem,” Sibirsk. Mat. Zh. 36(6), 1350–1353 (1995) [Siberian Math. J. 36 (6), 1171–1174 (1995)].
A. M. Kytmanov and S. G. Myslivets, “On an application of the Bochner-Martinelli operator,” in Contemp. Math., Vol. 212: Operator Theory for Complex and Hypercomplex Analysis: Proceedings of a Conference, Mexico City, Mexico, 1994 (Amer. Math. Soc., Providence, RI, 1998), pp. 133–136.
A. M. Kytmanov and S. G. Myslivets, “Higher-dimensional boundary analogs of the Morera theorem in problems of analytic continuation of functions,” J. Math. Sci. (N. Y.) 120(6), 1842–1867 (2004).
M. L. Agranovskii and A. M. Semenov, “Boundary analogues of the Hartogs theorem,” Sibirsk. Mat. Zh. 32(1), 168–170 (1991) [Siberian Math. J. 32 (1), 137–139 (1991)].
J. Globevnik and E. I. Stout, “Boundary Morera theorems for holomorphic functions of several complex variables,” Duke Math. J. 64(3), 571–615 (1991).
M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Real Submanifolds in Complex Space and Their Mappings, in Princeton Mathematical Series (Princeton Univ. Press, Princeton, NJ, 1999), Vol. 47.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A. M. Kytmanov, S. G. Myslivets, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 4, pp. 545–551.
Rights and permissions
About this article
Cite this article
Kytmanov, A.M., Myslivets, S.G. On families of complex lines sufficient for holomorphic extension. Math Notes 83, 500–505 (2008). https://doi.org/10.1134/S0001434608030231
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434608030231