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Subsequences of zeros for classes of holomorphic functions, their stability, and the entropy of arcwise connectedness. I


Authors: B. N. Khabibullin, F. B. Khabibullin and L. Yu. Cherednikova
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 20 (2008), nomer 1.
Journal: St. Petersburg Math. J. 20 (2009), 101-129
MSC (2000): Primary 30C15.
DOI: https://doi.org/10.1090/S1061-0022-08-01040-6
Published electronically: November 14, 2008
MathSciNet review: 2411972
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Abstract: For a domain $ \Omega$ in the complex plane $ \mathbb{C}$, let $ H(\Omega)$ denote the space of functions holomorphic in $ \Omega$, and let $ \mathscr{P}$ be a family of functions subharmonic in $ \Omega$. Denote by $ H_{\mathscr{P}}(\Omega )$ the class of $ f\in H(\Omega)$ satisfying $ \vert f(z)\vert\leq C_f\exp p_f(z)$, $ z\in \Omega$, where $ p_f \in \mathscr{P}$ and $ C_f$ is a constant. The paper is aimed at conditions for a set $ \Lambda \subset \Omega$ to be included in the zero set of some nonzero function in $ H_{\mathscr{P}}(\Omega )$. In the first part, certain preparatory theorems are established concerning ``quenching'' the growth of a subharmonic function by adding to it a function of the form $ \log \vert h\vert$, where $ h$ is a nonzero function in $ H(\Omega)$. The method is based on the balayage of measures and subharmonic functions.


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  • 1. S. V. Shvedenko, Hardy classes and related spaces of analytic functions in the unit disc, polydisc and ball, Itogi Nauki i Tekhniki. Mat. Anal., vol. 23, VINITI, Moscow, 1985, pp. 3-124; English transl. in J. Soviet Math. 39 (1987), no. 6. MR 0824267 (87h:30075)
  • 2. A. B. Aleksandrov, Function theory in the ball, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fundam. Napravleniya, vol. 8, VINITI, Moscow, 1985, pp. 115-190; English transl., Encyclopaedia Math. Sci., vol. 8, Springer-Verlag, Berlin, 1994, pp. 107-178. MR 0850487 (88b:32002)
  • 3. H. Hedenmalm, Recent progress in the function theory of the Bergman space, Holomorphic Spaces (Berkeley, CA, 1995), Math. Sci. Res. Inst. Publ., No. 33, Cambridge Univ. Press, Cambridge, 1998, pp. 35-50. MR 1630644 (99e:46035)
  • 4. P. Colwell, Blaschke products. Bounded analytic functions, Univ. Michigan Press, Ann Arbor, 1985. MR 0779463 (86f:30033)
  • 5. A. Djrbashian and F. A. Shamoian, Topics in the theory of $ A_{\alpha}^p$ spaces, Teubner-Texte Math., Bd. 105, Teubner, Leipzig, 1988. MR 1021691 (91k:46019)
  • 6. F. A. Shamoyan, A factorization theorem of M. M. Dzhrbashyan and the characteristic of zeros of functions analytic in the circle with a majorant of finite growth, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 13 (1978), no. 5-6, 405-422; English transl. in Soviet J. Contemporary Math. Anal. 13 (1978), no. 5-6. MR 0541789 (80i:30054)
  • 7. -, Zeros of functions analytic in the disk and growing near the boundary, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 18 (1983), no. 1, 15-27; English transl., Soviet J. Contemporary Math. Anal. 18 (1983), no. 1, 13-25. MR 0705983 (84g:30032)
  • 8. C. Horowitz, Zero sets and radial zero sets in function spaces, J. Anal. Math. 65 (1995), 145-159. MR 1335372 (96h:30066)
  • 9. B. Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), 187-219. MR 0425124 (54:13081)
  • 10. E. Beller, Factorization for non-Nevanlinna classes of analytic functions, Israel J. Math. 27 (1977), no. 3-4, 320-330. MR 0442234 (56:620)
  • 11. E. Beller and C. Horowitz, Zero sets and random zero sets in certain function spaces, J. Anal. Math. 64 (1994), 203-217. MR 1303512 (95j:30005)
  • 12. K. Seip, On a theorem of Korenblum, Ark. Mat. 32 (1994), 237-243. MR 1277927 (95f:30054)
  • 13. -, On Korenblum's density condition for the zero sequences of $ A^{-\alpha}$, J. Anal. Math. 67 (1995), 307-322. MR 1383499 (97c:30044)
  • 14. J. Bruna and X. Massaneda, Zero sets of holomorphic functions in the unit ball with slow growth, J. Anal. Math. 66 (1995), 217-252. MR 1370351 (97f:32006)
  • 15. D. Luecking, Zero sequences for Bergman spaces, Complex Variables Theory Appl. 30 (1996), 345-362. MR 1413164 (97g:30007)
  • 16. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Grad. Texts in Math., vol. 199, Springer-Verlag, New York, 2000. MR 1758653 (2001c:46043)
  • 17. O. Blasco, A. Kukuryka, and M. Nowak, Luecking's condition for zeros of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 58 (2004), 1-15. MR 2199585 (2006k:30010)
  • 18. V. V. Napalkov, Spaces of analytic functions of given growth near the boundary, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 2, 287-305; English transl., Math. USSR-Izv. 30 (1988), no. 2, 263-281. MR 0896998 (88g:46036)
  • 19. B. N. Khabibullin, Zero (sub)sets for spaces of holomorphic functions and (sub)harmonic minorants, Electronic Archive at LANL, 18 Dec. 2004, 42 pp., http://arxiv.org/abs/math.CV/0412359.
  • 20. -, Zero sequences of holomorphic functions, representation of meromorphic functions, and harmonic minorants, Mat. Sb. 198 (2007), no. 2, 121-160; English transl., Sb. Math. 198 (2007), no. 1-2, 261-298. MR 2355445 (2008h:30054)
  • 21. -, Sets of uniqueness in spaces of entire functions of one variable, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 5, 1101-1123; English transl., Math. USSR-Izv. 39 (1992), no. 2, 1063-1084. MR 1149889 (93e:30062)
  • 22. -, A theorem on the least majorant and its applications. I. Entire and meromorphic functions, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 1, 129-146; English transl., Russian Acad. Sci. Izv. Math. 42 (1994), no. 1, 115-131. MR 1220584 (94f:31003)
  • 23. -, Nonconstructive proofs of the Beurling-Malliavin theorem on the radius of completeness, and nonuniqueness theorems for entire functions, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 4, 125-148; English transl., Russian Acad. Sci. Izv. Math. 45 (1995), no. 1, 125-149. MR 1307059 (96g:30005)
  • 24. P. Koosis, Leçons sur le théorème de Beurling et Malliavin, Univ. Montréal, Publ. CRM, Montreal, QC, 1996. MR 1430571 (99e:42023)
  • 25. T. J. Ransford, Jensen measures, Approximation, Complex Analysis, and Potential Theory (Montreal, QC, 2000), NATO Sci. Ser. II Math. Phys. Chem., vol. 37, Kluwer Acad. Publ., Dordrecht, 2001, pp. 221-237. MR 1873590 (2002m:31019)
  • 26. B. N. Khabibullin, Dual approach to certain questions for the weighted spaces of holomorphic functions, Entire Functions in Modern Analysis (Tel-Aviv, 1997), Israel Math. Conf. Proc., vol. 15, Bar-Ilan Univ., Ramat Gan, 2001, pp. 207-219. MR 1890538 (2003e:30009)
  • 27. -, Dual representation of superlinear functionals and its applications in function theory. II, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 5, 167-190; English transl., Izv. Math. 65 (2001), no. 5, 1017-1039. MR 1874358 (2003b:46024)
  • 28. -, Completeness of systems of entire functions in spaces of holomorphic functions, Mat. Zametki 66 (1999), no. 4, 603-616; English transl., Math. Notes 66 (1999), no. 3-4, 495-506 (2000). MR 1747088 (2001d:32002)
  • 29. -, Estimates for the volume of null sets of holomorphic functions, Izv. Vyssh. Uchebn. Zaved. Mat. 1992, no. 3, 58-63. (Russian) MR 1204818 (93k:32003)
  • 30. -, Uniqueness theorems for holomorphic functions, and balayage, Complex Analysis. Operator Theory. Mathematical Modelling, Vychisl. Nauchn. Tsentr. Ross. Akad. Nauk, Vladikavkaz, 2006, pp. 118-132. (Russian)
  • 31. L. Yu. Cherednikova and B. N. Khabibullin, Nonuniqueness sets for weighted algebras of functions holomorphic in the disk, Complex Analysis, Differential Equations and Related Problems. I. Complex Analysis, Trans. Internat. Conf., Ross. Akad. Nauk Ural. Nauchn. Tsentr. Inst. Mat. Vychisl. Tsentr, Ufa, 2000, pp. 195-200. (Russian)
  • 32. -, Nonuniqueness sequences for weighted algebras of functions holomorphic in the disk, Trudy Mat. Tsentr. Lobachevsk. 19 (2003), 221-223. (Russian)
  • 33. L. Yu. Cherednikova, Nonuniqueness sequences for weighted algebras of holomorphic functions in the unit disk, Mat. Zametki 77 (2005), no. 5, 775-787; English transl., Math. Notes 77 (2005), no. 5-6, 715-725. MR 2178847 (2006i:30074)
  • 34. L. Yu. Cherednikova and B. N. Khabibullin, Stability of nonuniqueness sequences for weighted algebras of functions holomorphic in the disk, Scientific Conf. on Scientific-Technical Programs of Education Ministry of the Russian Federation: Collected Works. Part I, Bashkir. Univ., Ufa, 2000, pp. 25-28. (Russian)
  • 35. B. N. Khabibullin and F. B. Khabibullin, Zero subsets for spaces of functions and the entropy of arcwise connectedness, Geometric Analysis and its Applications: Thesis of Internat. School-Conf., Volgograd. Univ., Volgograd, 2004, pp. 193-195.
  • 36. B. N. Khabibullin, Zero sets for weighted classes of holomorphic functions, Vestnik Bashkir. Univ. 2004, no. 2, 59-63. (Russian)
  • 37. L. Yu. Cherednikova, On ``quenching'' of the growth of subharmonic functions, Regional School-Conf. for Students, Post Graduate Students, and Young Scientists in Mathematics and Physics. Vol. 1. Mathematics, Bashkir. Univ., Ufa, 2001, pp. 239-245. (Russian)
  • 38. W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. 1, London Math. Soc. Monogr., No. 9, Acad. Press, London-New York, 1976. MR 0460672 (57:665)
  • 39. T. W. Gamelin, Uniform algebras and Jensen measures, London Math. Soc. Lecture Note Ser., vol. 32, Cambridge Univ. Press, Cambridge-New York, 1978. MR 0521440 (81a:46058)
  • 40. B. J. Cole and T. J. Ransford, Subharmonicity without upper semicontinuity, J. Funct. Anal. 147 (1997), 420-442. MR 1454488 (98h:31005)
  • 41. -, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29-53. MR 1876284 (2003c:31002)
  • 42. B. N. Khabibullin, Criteria for (sub-)harmonicity and the continuation of (sub-)harmonic functions, Sibirsk. Mat. Zh. 44 (2003), no. 4, 905-925; English transl., Siberian Math. J. 44 (2003), no. 4, 713-728. MR 2010135 (2004g:31004)
  • 43. O. V. Epifanov, Solvability of the inhomogeneous Cauchy-Riemann equation in classes of functions that are bounded with weight and a system of weights, Mat. Zametki 51 (1992), no. 1, 83-92; English transl., Math. Notes 51 (1992), no. 1-2, 54-60. MR 1165283 (93h:30060)
  • 44. L. Hörmander, Notions of convexity, Progr. Math., vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301332 (95k:00002)
  • 45. T. J. Ransford, Potential theory in the complex plane, London Math. Soc. Student Texts, vol. 28, Cambridge Univ. Press, Cambridge, 1995. MR 1334766 (96e:31001)
  • 46. J. B. Garnett, Bounded analytic functions, Pure Appl. Math., vol. 96, Acad. Press, Inc., New York-London, 1981. MR 0628971 (83g:30037)
  • 47. L. Yu. Cherednikova, Elementary estimates in terms of Harnack distances, Regional School-Conf. for Students, Post Graduate Students, and Young Scientists in Mathematics and Physics. Vol. II. Mathematics, Bashkir. Univ., Ufa, 2002, pp. 87-90. (Russian)
  • 48. M. Brelot, Eléments de la théorie classique du potentiel, Centre Doc. Univ., Paris, 1959. MR 0106366 (21:5099)
  • 49. -, On topologies and boundaries in potential theory, Lecture Notes in Math., vol. 175, Springer-Verlag, Berlin-New York, 1971. MR 0281940 (43:7654)
  • 50. A. V. Arkhangel'skiĭ and V. I. Ponomarev, Fundamentals of general topology. Problems and exercises, ``Nauka'', Moscow, 1974; English transl., D. Reidel Publ. Co., Dordrecht, 1984. MR 0445439 (56:3781); MR 0785749 (87i:54001)
  • 51. W. K. Hayman, Subharmonic functions. Vol. 2, London Math. Soc. Monogr., vol. 20, Acad. Press, London, 1989. MR 1049148 (91f:31001)

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Additional Information

B. N. Khabibullin
Affiliation: Bashkir State University, Institute of Mathematics with Computer Center, Urals Scientific Center, Russian Academy of Sciences, Ufa, Bashkortostan, Russia
Email: khabib-bulat@mail.ru

F. B. Khabibullin
Affiliation: Bashkir State University, Institute of Mathematics with Computer Center, Urals Scientific Center, Russian Academy of Sciences, Ufa, Bashkortostan, Russia

L. Yu. Cherednikova
Affiliation: Bashkir State University, Institute of Mathematics with Computer Center, Urals Scientific Center, Russian Academy of Sciences, Ufa, Bashkortostan, Russia

DOI: https://doi.org/10.1090/S1061-0022-08-01040-6
Keywords: Holomorphic function, algebra of functions, weighted spaces, nonuniqueness sequence.
Received by editor(s): November 6, 2006
Published electronically: November 14, 2008
Additional Notes: Supported by RFBR, grant no. 06-01-00067, and by the Program of state subventions for leading scientific schools, grant NSh-10052.2006.1
Article copyright: © Copyright 2008 American Mathematical Society

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