Separation of singularities of analytic functions with preservation of boundedness
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V. P. Khavin
Translated by: S. V. Kislyakov - St. Petersburg Math. J. 16 (2005), 259-283
- DOI: https://doi.org/10.1090/S1061-0022-04-00850-7
- Published electronically: December 17, 2004
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Abstract:
For which pairs $(O_1,O_2)$ of open sets on the complex plane is it true that the operator \begin{equation*}J:(f_1,f_2)\mapsto (f_1+f_2)|(O_1\cap O_2) \end{equation*} from $H^{\infty }(O_1)\times H^{\infty }(O_2)$ to $H^{\infty }(O_1\cap O_2)$ is a surjection? In the first part of the paper, a method is indicated for constructing pairs without this property. In the second part, for some classes of pairs $(O_1,O_2)$ a right inverse for $J$ is constructed explicitly. The paper continues the previous studies of the author jointly with A. H. Nersessian and J. Ortega Cedrá.References
- N. Aronszajn, Sur les décompositions des fonctions analytiques uniformes et sur leurs applications, Acta Math. 65 (1935), 1–156.
- Carlos A. Berenstein and Roger Gay, Complex variables, Graduate Texts in Mathematics, vol. 125, Springer-Verlag, New York, 1991. An introduction. MR 1107514, DOI 10.1007/978-1-4612-3024-3
- M. Fréchet, Sur certaines décompositions de la fonction complexe uniforme la plus générale, Acta Math. 54 (1930), 37–79.
- Dieter Gaier, Lectures on complex approximation, Birkhäuser Boston, Inc., Boston, MA, 1987. Translated from the German by Renate McLaughlin. MR 894920, DOI 10.1007/978-1-4612-4814-9
- Dieter Gaier, Remarks on Alice Roth’s fusion lemma, J. Approx. Theory 37 (1983), no. 3, 246–250. MR 693011, DOI 10.1016/0021-9045(83)90050-3
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- Paul M. Gauthier, Mittag-Leffler theorems on Riemann surfaces and Riemannian manifolds, Canad. J. Math. 50 (1998), no. 3, 547–562. MR 1629823, DOI 10.4153/CJM-1998-030-1
- V. P. Havin, The separation of the singularities of analytic functions, Dokl. Akad. Nauk SSSR 121 (1958), 239–242 (Russian). MR 0098168
- V. P. Havin, S. V. Hruščëv, and N. K. Nikol′skiĭ (eds.), Linear and complex analysis problem book, Lecture Notes in Mathematics, vol. 1043, Springer-Verlag, Berlin, 1984. 199 research problems. MR 734178, DOI 10.1007/BFb0072183
- V. P. Havin and A. H. Nersessian, Bounded separation of singularities of analytic functions, Entire functions in modern analysis (Tel-Aviv, 1997) Israel Math. Conf. Proc., vol. 15, Bar-Ilan Univ., Ramat Gan, 2001, pp. 149–171. MR 1890536
- V. P. Havin, A. H. Nersessian, and J. Ortega Cerdà, Uniform estimates in the Poincaré–Aronszajn theorem on the separation of singularities of analytic functions, Preprint, 2003.
- B. S. Mitjagin and G. M. Henkin, Linear problems of complex analysis, Uspehi Mat. Nauk 26 (1971), no. 4 (160), 93–152 (Russian). MR 0287297
- H. Poincaré, Sur les fonctions à espaces lacunaires, Amer. J. Math. 14 (1892), 201–221.
- A. V. Bicadze, Mathematics during 40 years in the USSR (brief survey), Advancement in Math. 4 (1958), 583–585 (Chinese). MR 97995
- P. L. Polyakov, Continuation of bounded holomorphic functions from an analytic curve in general position into the polydisc, Funktsional. Anal. i Prilozhen. 17 (1983), no. 3, 87–88 (Russian). MR 714234
- Georges Valiron, Fonctions analytiques, Presses Universitaires de France, Paris, 1954 (French). MR 0061658
- I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765
- Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
- A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199 (Russian). MR 0229838
- A. V. Kozlov and V. P. Khavin, Separation of singularities of analytic functions with the preservation of continuity up to the boundary (in preparation). (Russian)
Bibliographic Information
- V. P. Khavin
- Affiliation: St. Petersburg State University, Department of Mathematics and Mechanics, Petrodvorets, Bibliotechnaya Pl. 2, St. Petersburg 198504, Russia
- Received by editor(s): September 23, 2003
- Published electronically: December 17, 2004
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 259-283
- MSC (2000): Primary 30E99
- DOI: https://doi.org/10.1090/S1061-0022-04-00850-7
- MathSciNet review: 2068355
Dedicated: Dedicated to Mikhail Shlemovich Birman on the occasion of his 75th birthday