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Separation of singularities of analytic functions with preservation of boundedness

Author(s): V. P. Khavin
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 16 (2004), vypusk 1.
Journal: St. Petersburg Math. J. 16 (2005), 259-283.
MSC (2000): Primary 30E99
Posted: December 17, 2004
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Abstract | References | Similar articles | Additional information

Abstract: For which pairs $(O_1,O_2)$ of open sets on the complex plane is it true that the operator

\begin{displaymath}J:(f_1,f_2)\mapsto (f_1+f_2)\vert(O_1\cap O_2) \end{displaymath}

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:

1.
N. Aronszajn, Sur les décompositions des fonctions analytiques uniformes et sur leurs applications, Acta Math. 65 (1935), 1-156.

2.
C. A. Berenstein and R. Gay, Complex variables. An introduction, Grad. Texts in Math., vol. 125, Springer-Verlag, New York, 1991. MR 1107514 (92f:30001)

3.
M. Fréchet, Sur certaines décompositions de la fonction complexe uniforme la plus générale, Acta Math. 54 (1930), 37-79.

4.
D. Gaier, Lectures on complex approximation, Birkhäuser Boston, Inc., Boston, MA, 1987. MR 0894920 (88i:30059b)

5.
-, Remarks on Alice Roth's fusion lemma, J. Approx. Theory 37 (1983), 246-250. MR 0693011 (84h:41032)

6.
T. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1969. MR 0410387 (53:14137)

7.
P. M. Gauthier, Mittag-Leffler theorems on Riemann surfaces and Riemannian manifolds, Canad. J. Math. 50 (1998), 547-562. MR 1629823 (99h:30046)

8.
V. P. Khavin, The separation of singularities of analytic functions, Dokl. Akad. Nauk SSSR 121 (1958), no. 2, 239-242. (Russian) MR 0098168 (20:4630)

9.
-, Golubev series and the analyticity on a continuum, Linear and Complex Analysis Problem Book. 199 Research Problems (V. P. Khavin, S. V. Khrushchev, N. K. Nikolskii, eds.), Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin etc., 1984, pp. 670-673. MR 0734178 (85k:46001)

10.
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 (2002m:31018)

11.
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.

12.
B. S. Mityagin and G. M. Khenkin, Linear problems of complex analysis, Uspekhi Mat. Nauk 26 (1971), no. 4, 93-152; English transl. in Russian Math. Surveys 26 (1971), no. 4. MR 0287297 (44:4504)

13.
H. Poincaré, Sur les fonctions à espaces lacunaires, Amer. J. Math. 14 (1892), 201-221.

14.
P. L. Polyakov and G. M. Khenkin, Integral formulas for solution of the $\overline{\partial}$-equation, and interpolation problems in analytic polyhedra, Trudy Moskov. Mat. Obshch. 53 (1990), 130-170; English transl., Trans. Moscow Math. Soc. 1991, 135-175. MR 097995 (92b:32008)

15.
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; English transl., Funct. Anal. Appl. 17 (1983), no. 3, 237-239. MR 0714234 (84k:32019)

16.
G. Valiron, Fonctions analytiques, Presses Univ. France, Paris, 1954. MR 0061658 (15:861a)

17.
I. I. Privalov, Boundary properties of analytic functions, 2nd ed., GITTL, Moscow-Leningrad, 1950 (Russian); German transl., Randeigenschaften analytischer Funktionen, Deutscher Verlag des Wissenschaften, Berlin, 1956. MR 0047765 (13:926h)

18.
L. Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, NJ etc., 1966. MR 0203075 (34:2933)

19.
A. G. Vitushkin, Analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), no. 6, 141-199; English transl., Russian Math. Surveys 22 (1967), 130-200. MR 0229838 (37:5404)

20.
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)

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Additional Information:

V. P. Khavin
Affiliation: St. Petersburg State University, Department of Mathematics and Mechanics, Petrodvorets, Bibliotechnaya Pl. 2, St. Petersburg 198504, Russia

DOI: 10.1090/S1061-0022-04-00850-7
PII: S 1061-0022(04)00850-7
Keywords: Bounded analytic function, Cauchy potential, plane continuum, separation of singularities
Received by editor(s): 23/SEP/2003
Posted: December 17, 2004
Dedicated: Dedicated to Mikhail Shlemovich Birman on the occasion of his 75th birthday
Copyright of article: Copyright 2004, American Mathematical Society

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