A MULTIDIMENSIONAL BOUNDARY ANALOGUE OF HARTOGS’S THEOREM FOR INTEGRABLE FUNCTIONS ON n-CIRCULAR DOMAINS
Keywords:
integrable functions, complex analysis, dimensional holomorphic extension, logarithmic residue, theorem.
Abstract
In the present paper we consider integrable functions given on the boundary of n-circular domain D⊂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.
References
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L. A. Aizenberg and A. P. Yuzhakov, Integral representations and residues in multidimensional complex analysis. American Mathematical Society, Providence, RI, 1983. (translated from the Russian).
M. L. Agranovskii and R. E. Val’skii, Maxiniality of invariant algebras of functions, Sibirsk. Mat. Zh. 12 (1971), no. 1, 3-12; English translation in Siberian Math. J. 12 (1971).
M. L. Agranovsky, Propagation of boundary CR-foliations and Morera type theorems for manifolds with attached analytic discs, Advances in Math., 211 284–326. (2007).
M. L. Agranovsky, Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of C^n, Journal d’Analyse Mathematique, 113 293-304 (2011).
A. Aytuna, A. Sadullaev, S^*-Parabolic manifolds, TWMS J. Pure Appl. Math. 2 6-9 (2011).
L. Baracco, Holomorphic extension from the sphere to the ball, Journal of Mathematical Analysis and Applications, 388 760–762 (2012).
L. Baracco, Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball, Amer. J. of Math. 135 493-497 (2013).
L. Baracco, Holomorphic Extension from a convex hypersurface, Asian J. Math., 20 (2) 263–266 (2016).
G.P. Egorychev, 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. in 1989.
G. M. Henkin, 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, 1985, 23-124.
J. Globevnik, E. L. Stout, Boundary Morera theorems for holomorphic functions of several complex variables, Duke Math. J., 64 571–615 (1991).
J. Globevnik, Small families of complex lines for testing holomorphic extendibility, Amer. J. of Math. 134 1473-1490 (2012).
J. Globevnik, Meromorphic extension from small families of circles and holomorphic textension from spheres Trans. Amer. Math. Soc., 364 5867-5880 (2012).
A. M. Kytmanov, The Bochner-Martinelli Integral and Its Applications, Birkhauser Verlag, Basel 1995.
A. M. Kytmanov, S. G. Myslivets, Higher-dimensional boundary analogs of the Morera theorem in problems of analytic continuation of functions, J. Math. Sci., 120 1842-1867 (2004).
A. M. Kytmanov, G. Myslivets, On families of complex lines sufficient for holomorphic extension, Mathematical Notes, 83 500-505 (2008).
A. M. Kytmanov, S. G. Myslivets, V. I. Kuzovatov, Minimal dimension families of complex lines sufficient for holomorphic extension of functions, Siberian Mathematical Journal, 52 256-266 (2011).
A. M. Kytmanov, S. G. Myslivets, Holomorphic continuation of functions along finite families of complex lines in the ball, J. Sib. Fed. Univ. Math. Phys., 5:4 (2012), 547-557.
A.M.Kytmanov, S.G.Myslivets, Holomorphic extension of functions along the finite families of complex lines in a ball of C^n, Math. Nahr, 288, no. 2-3, 224-234 (2015).
A. M. Kytmanov, S. G. Myslivets, 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).
A. M. Kytmanov, S. G. Myslivets, 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).
M.G. Lawrence. Hartogs’ separate analyticity theorem for CR functions. Internat. J. Math., 18 (3) 219–229 (2007).
B. P. Otemuratov, Functions of class L^p with the one-dimensional holomorphic extension property, Vestnik KrasGU, no 9, 95-100 (2006).
B. P. Otemuratov, A multidimensional Morera’s theorem for integrable function, Uzbek Math. Journ., no 2, 112-119 (2009).
B. P. Otemuratov, 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).
B. V. Shabat, Introduction to complex analysis, 2nd ed., Part I, Nauka, Moscow, 1976. (in Russian).
E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1975.
E. L. Stout, The boundary values of holomorphic functions of several complex variables, Duke Math. J., 44 no. 1, 105–108 (1977).
L. A. Aizenberg and A. P. Yuzhakov, Integral representations and residues in multidimensional complex analysis. American Mathematical Society, Providence, RI, 1983. (translated from the Russian).
M. L. Agranovskii and R. E. Val’skii, Maxiniality of invariant algebras of functions, Sibirsk. Mat. Zh. 12 (1971), no. 1, 3-12; English translation in Siberian Math. J. 12 (1971).
M. L. Agranovsky, Propagation of boundary CR-foliations and Morera type theorems for manifolds with attached analytic discs, Advances in Math., 211 284–326. (2007).
M. L. Agranovsky, Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of C^n, Journal d’Analyse Mathematique, 113 293-304 (2011).
A. Aytuna, A. Sadullaev, S^*-Parabolic manifolds, TWMS J. Pure Appl. Math. 2 6-9 (2011).
L. Baracco, Holomorphic extension from the sphere to the ball, Journal of Mathematical Analysis and Applications, 388 760–762 (2012).
L. Baracco, Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball, Amer. J. of Math. 135 493-497 (2013).
L. Baracco, Holomorphic Extension from a convex hypersurface, Asian J. Math., 20 (2) 263–266 (2016).
G.P. Egorychev, 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. in 1989.
G. M. Henkin, 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, 1985, 23-124.
J. Globevnik, E. L. Stout, Boundary Morera theorems for holomorphic functions of several complex variables, Duke Math. J., 64 571–615 (1991).
J. Globevnik, Small families of complex lines for testing holomorphic extendibility, Amer. J. of Math. 134 1473-1490 (2012).
J. Globevnik, Meromorphic extension from small families of circles and holomorphic textension from spheres Trans. Amer. Math. Soc., 364 5867-5880 (2012).
A. M. Kytmanov, The Bochner-Martinelli Integral and Its Applications, Birkhauser Verlag, Basel 1995.
A. M. Kytmanov, S. G. Myslivets, Higher-dimensional boundary analogs of the Morera theorem in problems of analytic continuation of functions, J. Math. Sci., 120 1842-1867 (2004).
A. M. Kytmanov, G. Myslivets, On families of complex lines sufficient for holomorphic extension, Mathematical Notes, 83 500-505 (2008).
A. M. Kytmanov, S. G. Myslivets, V. I. Kuzovatov, Minimal dimension families of complex lines sufficient for holomorphic extension of functions, Siberian Mathematical Journal, 52 256-266 (2011).
A. M. Kytmanov, S. G. Myslivets, Holomorphic continuation of functions along finite families of complex lines in the ball, J. Sib. Fed. Univ. Math. Phys., 5:4 (2012), 547-557.
A.M.Kytmanov, S.G.Myslivets, Holomorphic extension of functions along the finite families of complex lines in a ball of C^n, Math. Nahr, 288, no. 2-3, 224-234 (2015).
A. M. Kytmanov, S. G. Myslivets, 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).
A. M. Kytmanov, S. G. Myslivets, 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).
M.G. Lawrence. Hartogs’ separate analyticity theorem for CR functions. Internat. J. Math., 18 (3) 219–229 (2007).
B. P. Otemuratov, Functions of class L^p with the one-dimensional holomorphic extension property, Vestnik KrasGU, no 9, 95-100 (2006).
B. P. Otemuratov, A multidimensional Morera’s theorem for integrable function, Uzbek Math. Journ., no 2, 112-119 (2009).
B. P. Otemuratov, 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).
B. V. Shabat, Introduction to complex analysis, 2nd ed., Part I, Nauka, Moscow, 1976. (in Russian).
E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1975.
E. L. Stout, The boundary values of holomorphic functions of several complex variables, Duke Math. J., 44 no. 1, 105–108 (1977).
Published
2023-07-01
How to Cite
Bayrambay Otemuratov. (2023). A MULTIDIMENSIONAL BOUNDARY ANALOGUE OF HARTOGS’S THEOREM FOR INTEGRABLE FUNCTIONS ON n-CIRCULAR DOMAINS. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 4(7), 17-26. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/478
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