Abstract
In the present paper we consider integrable functions given on the boundary of \(n\)-circular domain \(D\subset \mathbb C^n\), \(n>1\) and having one-dimensional property of holomorphic extension along the families of complex lines, passing through finite number of points of \(D.\) We prove the existence of holomorphic extension of such functions in \(D.\)
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Agranovskii, M.L., Val’skii, R.E.: Maxiniality of invariant algebras of functions. Sibirsk. Mat. Zh. 12(1), 3–12 (1971); English translation in Siberian Math. J. 12 (1971)
Agranovsky, M.L.: Propagation of boundary \(CR\)-foliations and Morera type theorems for manifolds with attached analytic discs. Adv. Math. 211, 284–326 (2007)
Agranovsky, M.L.: Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of \(\mathbb{C}^n\). J. d’Analyse Mathematique 113, 293–304 (2011)
Aizenberg, L.A.: Integral representations of functions holomorphic in n-circular domains (Continuation of Szego kernels). Mat. Sb. 65, 104–143 (1964)
Aizenberg, L.A., Yuzhakov, A.P.: Integral Representations and Residues in Multidimensional Complex Analysis. American Mathematical Society, Providence (1983). (translated from the Russian)
Aytuna, A., Sadullaev, A.: \(S^\ast \)-parabolic manifolds. TWMS J. Pure Appl. Math. 2, 6–9 (2011)
Baracco, L.: Holomorphic extension from the sphere to the ball. J. Math. Anal. Appl. 388, 760–762 (2012)
Baracco, L.: Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball. Amer. J. Math. 135, 493–497 (2013)
Egorychev, G.P.: Integral Representation and the Computation of Combinatorial Sums, Nauka, Novosibirsk, 1977 (in Russian). English transl. Transl. Math. Monographs 59, Amer. Math. Soc., Providence. RI 1984; 2nd ed. (1989)
Globevnik, J.: Meromorphic extension from small families of circles and holomorphic textension from spheres Trans. Amer. Math. Soc. 364, 5867–5880 (2012)
Globevnik, J.: Small families of complex lines for testing holomorphic extendibility. Amer. J. Math. 134, 1473–1490 (2012)
Globevnik, J., Stout, E.L.: Boundary Morera theorems for holomorphic functions of several complex variables. Duke Math. J. 64, 571–615 (1991)
Henkin, G.M.: The method of integral representations in complex analysis, Complex analysis-several variables 1, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 7, VINITI, Moscow, pp. 23–124 (1985)
Kytmanov, A.M.: The Bochner-Martinelli Integral and Its Applications. Birkhauser, Basel (1995)
Kytmanov, A.M., Myslivets, S.G.: Higher-dimensional boundary analogs of the Morera theorem in problems of analytic continuation of functions. J. Math. Sci. 120, 1842–1867 (2004)
Kytmanov, A.M., Myslivets, G.: On families of complex lines sufficient for holomorphic extension. Math. Notes 83, 500–505 (2008)
Kytmanov, A.M., Myslivets, S.G., Kuzovatov, V.I.: Minimal dimension families of complex lines sufficient for holomorphic extension of functions. Siberian Math. J. 52, 256–266 (2011)
Kytmanov, A.M., Myslivets, S.G.: Holomorphic continuation of functions along finite families of complex lines in the ball. J. Sib. Fed. Univ. Math. Phys. 5(4), 547–557 (2012)
Kytmanov, A.M., Myslivets, S.G.: On the families of complex lines which are sufficient for holomorphic continuation of functions given on the boundary of the domain. J. Sib. Fed. Univ. Math. Phys. 5(2), 213–222 (2012)
Kytmanov, A.M., Myslivets, S.G.: Holomorphic extension of continuous functions along finite families of complex lines in a ball. J. Sib. Fed. Univ. Math. Phys. 8(3), 291–302 (2015)
Kytmanov, A.M., Myslivets, S.G.: Holomorphic extension of functions along the finite families of complex lines in a ball of \(\mathbb{C}^n\). Math. Nahr 288(2–3), 224–234 (2015)
Otemuratov, B.P.: Functions of class \(L^p\) with the one-dimensional holomorphic extension property. Vestnik KrasGU 9, 95–100 (2006)
Otemuratov, B.P.: A multidimensional Morera’s theorem for integrable functions. Uzbek Math. J. 2, 112–119 (2009)
Otemuratov, B.P.: Some families of complex lines of minimal dimension which are sufficient for holomorphic continuation of integrable functions. J. Sib. Fed. Univ. Math. Phys. 5(1), 97–105 (2012)
Shabat, B.V.: Introduction to Complex Analysis, Part I, 2nd edn. Nauka, Moscow (1976). (in Russian)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1975)
Stout, E.L.: The boundary values of holomorphic functions of several complex variables. Duke Math. J. 44(1), 105–108 (1977)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Otemuratov, B. (2018). A Multidimensional Boundary Analogue of Hartogs’s Theorem on \(\mathbf n\)-Circular Domains for Integrable Functions. In: Ibragimov, Z., Levenberg, N., Rozikov, U., Sadullaev, A. (eds) Algebra, Complex Analysis, and Pluripotential Theory. USUZCAMP 2017. Springer Proceedings in Mathematics & Statistics, vol 264. Springer, Cham. https://doi.org/10.1007/978-3-030-01144-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-01144-4_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-01143-7
Online ISBN: 978-3-030-01144-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)