Subsequences of zeros for classes of holomorphic functions, their stability, and the entropy of arcwise connectedness. I. (English. Russian original) Zbl 1206.30074

Khabibullin, B. N.; Khabibullin, F. B.; Cherednikova, L. Yu.

St. Petersbg. Math. J. 20, No. 1, 101-129 (2009); translation from Algebra Anal. 20, No. 1, 146-189 (2008).

Summary: For a domain \( \Omega\) in the complex plane \( \mathbb{C}\), let \( H(\Omega)\) denote the space of functions holomorphic in \( \Omega\), and let \( \mathcal P\) be a family of functions subharmonic in \( \Omega\). Denote by \( H_{\mathcal P}(\Omega )\) the class of \(f\in H(\Omega)\) satisfying \( | f(z)|\leq C_f\exp p_f(z), z\in \Omega\), where \(p_f \in \mathcal P\) 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 \( H_{\mathcal P}(\Omega)\). 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 \( \log | h|\), where \( h\) is a nonzero function in \( H(\Omega)\). The method is based on the balayage of measures and subharmonic functions.

MSC:

30H05

Spaces of bounded analytic functions of one complex variable

Keywords:

holomorphic function; algebra of functions; weighted spaces; nonuniqueness sequence

References:

[1]	S. V. Shvedenko, Hardy classes and related spaces of analytic functions in the unit disc, polydisc and ball, Mathematical analysis, Vol. 23, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 3 – 124, 288 (Russian). · Zbl 0603.32004
[2]	A. B. Aleksandrov, Function theory in the ball, Current problems in mathematics. Fundamental directions, Vol. 8, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 115 – 190, 274 (Russian). · Zbl 0614.32005
[3]	Håkan Hedenmalm, Recent progress in the function theory of the Bergman space, Holomorphic spaces (Berkeley, CA, 1995) Math. Sci. Res. Inst. Publ., vol. 33, Cambridge Univ. Press, Cambridge, 1998, pp. 35 – 50. · Zbl 1126.46303
[4]	Peter Colwell, Blaschke products, University of Michigan Press, Ann Arbor, MI, 1985. Bounded analytic functions. · Zbl 0315.30035
[5]	Ashot E. Djrbashian and Faizo A. Shamoian, Topics in the theory of \?^{\?}_{\?} spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 105, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1988. With German, French and Russian summaries.
[6]	F. A. Šamojan, A factorization theorem of M. M. Džrbašjan 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, 541 (Russian, with English and Armenian summaries).
[7]	F. A. Shamoyan, 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 (Russian, with English and Armenian summaries).
[8]	Charles Horowitz, Zero sets and radial zero sets in function spaces, J. Anal. Math. 65 (1995), 145 – 159. · Zbl 0837.30008 · doi:10.1007/BF02788769
[9]	Boris Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), no. 3-4, 187 – 219. · Zbl 0323.30030 · doi:10.1007/BF02392019
[10]	Eliyahu Beller, Factorization for non-Nevanlinna classes of analytic functions, Israel J. Math. 27 (1977), no. 3 – 4, 320 – 330. · Zbl 0361.30012 · doi:10.1007/BF02756490
[11]	E. Beller and C. Horowitz, Zero sets and random zero sets in certain function spaces, J. Anal. Math. 64 (1994), 203 – 217. · Zbl 0819.30003 · doi:10.1007/BF03008409
[12]	Kristian Seip, On a theorem of Korenblum, Ark. Mat. 32 (1994), no. 1, 237 – 243. · Zbl 0812.30003 · doi:10.1007/BF02559530
[13]	Kristian Seip, On Korenblum’s density condition for the zero sequences of \?^{-\?}, J. Anal. Math. 67 (1995), 307 – 322. · Zbl 0845.30014 · doi:10.1007/BF02787795
[14]	Joaquim Bruna and Xavier Massaneda, Zero sets of holomorphic functions in the unit ball with slow growth, J. Anal. Math. 66 (1995), 217 – 252. · Zbl 0858.32009 · doi:10.1007/BF02788823
[15]	Daniel H. Luecking, Zero sequences for Bergman spaces, Complex Variables Theory Appl. 30 (1996), no. 4, 345 – 362. · Zbl 0871.30004
[16]	Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. · Zbl 0955.32003
[17]	Oscar Blasco, Artur Kukuryka, and Maria Nowak, Luecking’s condition for zeros of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 58 (2004), 1 – 15. · Zbl 1102.30006
[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, 446 (Russian); English transl., Math. USSR-Izv. 30 (1988), no. 2, 263 – 281. · Zbl 0623.30054
[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]	B. N. Khabibullin, Zero sequences of holomorphic functions, representation of meromorphic functions, and harmonic minorants, Mat. Sb. 198 (2007), no. 2, 121 – 160 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 1-2, 261 – 298. · Zbl 1145.30005 · doi:10.1070/SM2007v198n02ABEH003837
[21]	B. N. Khabibullin, Sets of uniqueness in spaces of entire functions of one variable, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 5, 1101 – 1123 (Russian); English transl., Math. USSR-Izv. 39 (1992), no. 2, 1063 – 1084. · Zbl 0745.30028
[22]	B. N. Khabibullin, 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 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 42 (1994), no. 1, 115 – 131. · Zbl 0798.31003 · doi:10.1070/IM1994v042n01ABEH001526
[23]	B. N. Khabibullin, 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 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 45 (1995), no. 1, 125 – 149. · Zbl 0847.30020 · doi:10.1070/IM1995v045n01ABEH001622
[24]	Paul Koosis, Leçons sur le théorème de Beurling et Malliavin, Université de Montréal, Les Publications CRM, Montreal, QC, 1996 (French). · Zbl 0869.30023
[25]	Thomas 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. · Zbl 0992.31007 · doi:10.1007/978-94-010-0979-9_8
[26]	B. N. Khabibullin, Dual approach to certain questions for 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. · Zbl 0995.30006
[27]	B. N. Khabibullin, Dual representation of superlinear functionals and its applications in the theory of functions. II, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 5, 167 – 190 (Russian, with Russian summary); English transl., Izv. Math. 65 (2001), no. 5, 1017 – 1039. · Zbl 1052.32004 · doi:10.1070/IM2001v065n05ABEH000361
[28]	B. N. Khabibullin, Completeness of systems of entire functions in spaces of holomorphic functions, Mat. Zametki 66 (1999), no. 4, 603 – 616 (Russian, with Russian summary); English transl., Math. Notes 66 (1999), no. 3-4, 495 – 506 (2000). · Zbl 1395.32002 · doi:10.1007/BF02679100
[29]	B. N. Khabibullin, Estimates for the volume of null sets of holomorphic functions, Izv. Vyssh. Uchebn. Zaved. Mat. 3 (1992), 58 – 63 (Russian).
[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 (Russian, with Russian summary); English transl., Math. Notes 77 (2005), no. 5-6, 715 – 725. · Zbl 1076.30053 · doi:10.1007/s11006-005-0072-5
[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. I, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. London Mathematical Society Monographs, No. 9. · Zbl 0419.31001
[39]	T. W. Gamelin, Uniform algebras and Jensen measures, London Mathematical Society Lecture Note Series, vol. 32, Cambridge University Press, Cambridge-New York, 1978. · Zbl 0418.46042
[40]	B. J. Cole and T. J. Ransford, Subharmonicity without upper semicontinuity, J. Funct. Anal. 147 (1997), no. 2, 420 – 442. · Zbl 0873.31007 · doi:10.1006/jfan.1996.3070
[41]	Brian J. Cole and Thomas J. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29 – 53. · Zbl 0988.31002 · doi:10.1515/crll.2001.094
[42]	B. N. Khabibullin, Criteria for (sub-)harmonicity and the continuation of (sub-)harmonic functions, Sibirsk. Mat. Zh. 44 (2003), no. 4, 905 – 925 (Russian, with Russian summary); English transl., Siberian Math. J. 44 (2003), no. 4, 713 – 728. · Zbl 1030.31004 · doi:10.1023/A:1024796809786
[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 (Russian); English transl., Math. Notes 51 (1992), no. 1-2, 54 – 60. · Zbl 0851.35026 · doi:10.1007/BF01229435
[44]	Lars Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. · Zbl 0835.32001
[45]	Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. · Zbl 0828.31001
[46]	John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. · Zbl 0469.30024
[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, Éléments de la théorie classique du potentiel, Les Cours de Sorbonne. 3e cycle, Centre de Documentation Universitaire, Paris, 1959 (French). · Zbl 0084.30903
[49]	Marcel Brelot, On topologies and boundaries in potential theory, Enlarged edition of a course of lectures delivered in 1966. Lecture Notes in Mathematics, Vol. 175, Springer-Verlag, Berlin-New York, 1971. · Zbl 0222.31014
[50]	Основы общей топологии в задачах и упражнениях., Издат. ”Наука”, Мосцощ, 1974 (Руссиан). А. В. Архангел\(^{\приме}\)ский анд В. И. Пономарев, Фундаменталс оф генерал топологы, Матхематицс анд иц Апплицатионс, Д. Реидел Публишинг Цо., Дордречт, 1984. Проблемс анд ешерцисес; Транслатед фром тхе Руссиан бы В. К. Јаин; Щитх а форещорд бы П. Алешандрофф [П. С. Александров].
[51]	W. K. Hayman, Subharmonic functions. Vol. 2, London Mathematical Society Monographs, vol. 20, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989. · Zbl 0699.31001