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Uzbekistan Tashkent; Urgench
Year
2023
Volume
33
Issue
1
Pages
17-31
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Section Mathematics
Title On Shimoda's Theorem
Author(-s) Atamuratov A.A.ab, Rasulov K.K.b
Affiliations Institute of Mathematics, National Academy of Sciences of Uzbekistana, Urgench State Universityb
Abstract The present work is devoted to Shimoda's Theorem on the holomorphicity of a function $f(z,w)$ which is holomorphic by $w\in V$ for each fixed $z\in U$ and is holomorphic by $z\in U$ for each fixed $w\in E$, where $E\subset V$ is a countable set with at least one limit point in $V$. Shimoda proves that in this case $f(z,w)$ is holomorphic in $U\times V$ except for a nowhere dense closed subset of $U\times V$. We prove the converse of this result, that is for an arbitrary given nowhere dense closed subset of $U$, $S\subset U$, there exists a holomorphic function, satisfying Shimoda's Theorem on $U\times V\subset {\mathbb C}^{2}$, that is not holomorphic on $S\times V$. Moreover, we observe conditions which imply empty exception sets on Shimoda's Theorem and prove generalizations of Shimoda's Theorem.
Keywords Hartogs's phenomena, Shimoda's Theorem, separately holomorphic functions, power series
UDC 517.55
MSC 32A05, 32A10
DOI 10.35634/vm230102
Received 15 November 2022
Language English
Citation Atamuratov A.A., Rasulov K.K. On Shimoda's Theorem, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 1, pp. 17-31.
References
  1. Shabat B.V. Introduction to complex analysis. Part II. Functions of several variables, Providence, RI: AMS, 1992. https://doi.org/10.1090/mmono/110
  2. Hukuhara M. L'extensions du théorème d'Osgood et de Hartogs, Kansû-hòteisiki ogobi Ôyô-Kaiseki, 1930, vol. 48, no. 8, pp. 48-49.
  3. Shimoda I. Notes on the functions of two complex variable, Journal of Gakugei, Tokushima University. Mathematics, 1957, vol. 8, no. 8, pp. 1-3.
  4. Terada T. Sur une certaine condition sous laquelle une fonction de plusieurs variables complexes est holomorphe. Diminution de la condition dans le théorème de Hartogs, Publications of the Research Institute for Mathematical Sciences, 1966, vol. 2, no. 3, pp. 383-396. https://doi.org/10.2977/prims/1195195767
  5. Siciak J. Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of ${\mathbb C}$${n}$ , Annales Polonici Mathematici, 1969, vol. 22, no. 2, pp. 145-171. https://doi.org/10.4064/ap-22-2-145-171
  6. Zaharjuta V.P. Separately analytic functions, generalizations of Hartogs' theorem, and envelopes of holomorphy, Mathematics of the USSR-Sbornik, 1976, vol. 30, no. 1, pp. 51-67. https://doi.org/10.1070/SM1976v030n01ABEH001898
  7. Gonchar A.A. On analytic continuation from the “edge of the wedge”, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, 1985, vol. 10, pp. 221-225. https://zbmath.org/an:0603.32008
  8. Sadullaev A.S., Chirka E.M. On continuation of functions with polar singularities, Mathematics of the USSR-Sbornik, 1988, vol. 60, no. 2, pp. 377-384. https://doi.org/10.1070/SM1988v060n02ABEH003175
  9. Jarnicki M., Pflug P. An extension theorem for separately holomorphic functions with analytic singularities, Annales Polonici Mathematici, 2003, vol. 80, pp. 143-161. https://doi.org/10.4064/ap80-0-12
  10. Jarnicki M., Pflug P. An extension theorem for separately holomorphic functions with pluripolar singularities, Transactions of the American Mathematical Society, 2003, vol. 355, no. 3, pp. 1251-1267. https://doi.org/10.1090/S0002-9947-02-03193-8
  11. Jarnicki M., Pflug P. A new cross theorem for separately holomorphic functions, Proceedings of the American Mathematical Society, 2010, vol. 138, no. 11, pp. 3923-3932. https://doi.org/10.1090/S0002-9939-2010-10552-X
  12. Sadullaev A.S., Imomkulov S.A. Extension of holomorphic and pluriharmonic functions with thin singularities on parallel sections, Proceedings of the Steklov Institute of Mathematics, 2006, vol. 253, issue 1, pp. 144-159. https://doi.org/10.1134/S0081543806020131
  13. Bang P.H. Separately locally holomorphic functions and their singular sets, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie. Nouvelle Série, 2008, vol. 51 (99), no. 2, pp. 103-108. https://www.jstor.org/stable/43679098
  14. Sadullaev A.S., Tuychiev T.T. On continuation of Hartogs series that admit holomorphic extension to parallel sections, Uzbekskii Matematicheskii Zhurnal, 2009, no. 1, pp. 148-157 (in Russian).
  15. Baracco L. Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball, American Journal of Mathematics, 2013, vol. 135, no. 2, pp. 493-497. https://doi.org/10.1353/ajm.2013.0015
  16. Boman J. Siciak's theorem on separate analyticity, Analysis meets geometry, Cham: Birkhäuser, 2017. P. 135-145. https://doi.org/10.1007/978-3-319-52471-9_10
  17. Krantz S.G. On a theorem of F. Forelli and a result of Hartogs, Complex Variables and Elliptic Equations, 2018, vol. 63, issue 4, pp. 591-597. https://doi.org/10.1080/17476933.2017.1345890
  18. Bochnak J., Kucharz W. Global variants of Hartogs' theorem, Archiv der Mathematik, 2019, vol. 113, issue 3, pp. 281-290. https://doi.org/10.1007/s00013-019-01340-7
  19. Cho Y.-W.L., Kim K.-T. Functions holomorphic along a $C$${1}$ pencil of holomorphic discs, The Journal of Geometric Analysis, 2021, vol. 31, issue 11, pp. 10634-10647. https://doi.org/10.1007/s12220-021-00660-x
  20. Boggess A., Dwilewicz R., Porten E. On the Hartogs extension theorem for unbounded domains in ${\mathbb C}$${n}$ , Annales de l'Institut Fourier, 2022, vol. 72, no. 3, pp. 1185-1206. https://doi.org/10.5802/aif.3514
  21. Lawrence M.G. The $L$${p}$ CR Hartogs separate analyticity theorem for convex domains, Mathematische Zeitschrift, 2018, vol. 288, issues 1-2, pp. 401-414. https://doi.org/10.1007/s00209-017-1894-z
  22. Lewandowski A. A new Hartogs-type extension result for the cross-like objects, Kyushu Journal of Mathematics, 2015, vol. 69, issue 1, pp. 77-94. https://doi.org/10.2206/kyushujm.69.77
  23. Thai Thuan Quang, Nguyen Van Dai. On Hartogs extension theorems for separately $(\cdot,W)$-holomorphic functions, International Journal of Mathematics, 2014, vol. 25, no. 12, 1450112. https://doi.org/10.1142/s0129167x14501122
  24. Thai Thuan Quang, Nguyen Van Dai. On the holomorphic extension of vector valued functions, Complex Analysis and Operator Theory, 2015, vol. 9, issue 3, pp. 567-591. https://doi.org/10.1007/s11785-014-0382-2
  25. Thai Thuan Quang, Lien Vuong Lam. Cross theorems for separately $(\cdot,W)$-meromorphic functions, Taiwanese Journal of Mathematics, 2016, vol. 20, issue 5, pp. 1009-1039. https://doi.org/10.11650/tjm.20.2016.7363
  26. Imomkulov S.A., Abdikadirov S.M. Removable singularities of separately harmonic functions, Journal of Siberian Federal University. Mathematics and Physics, 2021, vol. 14, issue 3, pp. 369-375. https://doi.org/10.17516/1997-1397-2021-14-3-369-375
  27. Palamodov V.P. Hartogs phenomenon for systems of differential equations, The Journal of Geometric Analysis, 2014, vol. 24, issue 2, pp. 667-686. https://doi.org/10.1007/s12220-012-9350-0
  28. Goluzin G.M. Geometricheskaya teoriya funktsii kompleksnogo peremennogo (Geometric theory of functions of a complex variable), Moscow: Nauka, 1966.
  29. Gamelin T.W. Uniform algebras, AMS, 1969.
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