Abstract
We consider the continuous functions on the boundary of a bounded n-circular domain D in ℂn, n > 1, which admit one-dimensional holomorphic extension along a family of complex straight lines passing through finitely many points of D. The question is addressed of the existence of a holomorphic extension of these functions to D.
References
Agranovsky M. L. and Val’skiĭ R. E., “Maximality of invariant algebras of functions,” Sib. Math. J., 12, No. 1, 1–7 (1971).
Stout E. L., “The boundary values of holomorphic functions of several complex variables,” Duke Math. J., 44, No. 1, 105–108 (1977).
Aizenberg L. A. and Yuzhakov A. P., Integral Representations and Residues in Multidimensional Complex Analysis, Amer. Math. Soc., Providence (1983).
Kytmanov A. M. and Myslivets S. G., “Higher-dimensional boundary analogs of the Morera theorem in problems of analytic continuation of functions,” J. Math. Sci., 120, No. 6, 1842–1867 (2004).
Kytmanov A. M. and Myslivets S. G., Multidimensional Integral Representations. Problems of Analytic Continuation, Springer-Verlag, Heidelberg; New York; Dordrecht; London (2015).
Globevnik J. and Stout E. L., “Boundary Morera theorems for holomorphic functions of several complex variables,” Duke Math. J., 64, No. 3, 571–615 (1991).
Kytmanov A. M. and Myslivets S. G., “On families of complex lines sufficient for holomorphic extension,” Math. Notes, 83, No. 4, 500–505 (2008).
Kytmanov A. M., Myslivets S. G., and Kuzovatov V. I., “Minimal dimension families of complex lines sufficient for holomorphic extension of functions,” Sib. Math. J., 52, No. 2, 256–266 (2011).
Agranovsky M. L., “Propagation of boundary CR-foliations and Morera type theorems for manifolds with attached analytic discs,” Adv. Math., 211, No. 1, 284–326 (2007).
Agranovsky M., “Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of Cn,” J. Anal. Math., 113, No. 1, 293–304 (2011).
Baracco L., “Holomorphic extension from the sphere to the ball,” J. Math. Anal. Appl., 388, No. 2, 760–762 (2012).
Globevnik J., “Small families of complex lines for testing holomorphic extendibility,” Amer. J. Math., No. 6, 1473–1490 (2012).
Baracco L., “Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball,” Amer. J. Math., 135, No. 2, 493–497 (2013).
Globevnik J., “Meromorphic extensions from small families of circles and holomorphic extensions from spheres,” Trans. Amer. Math. Soc., 364, No. 11, 5857–5880 (2012).
Kytmanov A. M. and Myslivets S. G., “Holomorphic extension of functions along finite families of complex lines in the ball,” J. Sib. Fed. Univ. Math. Phys., 5, No. 4, 547–557 (2012).
Kytmanov A. M. and Myslivets S. G., “An analog of the Hartogs theorem in a ball of Cn,” Math. Nahr., 288, No. 2–3, 224–234 (2015).
Aizenberg L. A., “Integral representations of functions holomorphic in n-circular domains (“Continuation” of Szegö kernels),” Mat. Sb., 65, No. 1, 104–143 (1964).
Khenkin G. M., “The method of integral representations in complex analysis,” in: Contemporary Problems of Mathematics. Fundamental Trends [in Russian], VINITI, Moscow, 1985, 7, pp. 23–124 (Itogi Nauki i Tekhniki).
Stein E. and Weiss G., Introduction to Harmonic Analysis on Euclidean Spaces, Princeton University Press, Princeton (1970).
Egorychev G. P., Integral Representation and Calculation of Combinatorial Sums [in Russian], Nauka, Novosibirsk (1977).
Shabat B. V., An Introduction to Complex Analysis. Part 1: Functions of a One Variable [in Russian], Nauka, Moscow (1976).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2016 Kytmanov A.M. and Myslivets S.G.
Krasnoyarsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 4, pp. 792–808, July–August, 2016; DOI: 10.17377/smzh.2016.57.406. Original article submitted September 18, 2015.
The authors were supported by the Russian Foundation for Basic Research (Grant 14–01–00544), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh–9149.2016.1), and the Government of the Russian Federation for the State Maintenance Program for the Leading Scientific Schools at Siberian Federal University (Grant 14.Y26.31.0006).
Rights and permissions
About this article
Cite this article
Kytmanov, A.M., Myslivets, S.G. Holomorphic extension of functions along finite families of complex straight lines in an n-circular domain. Sib Math J 57, 618–631 (2016). https://doi.org/10.1134/S0037446616040066
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446616040066