Functional Analysis and Its Applications

, Volume 52, Issue 1, pp 21–34 | Cite as

On the Distribution of Zero Sets of Holomorphic Functions

Article
  • 12 Downloads

Abstract

Let M be a subharmonic function with Riesz measure ν M in a domain D in the n-dimensional complex Euclidean space ℂ n , and let f be a nonzero function that is holomorphic in D, vanishes on a set ZD, and satisfies |f| ⩽ expM on D. Then restrictions on the growth of ν M near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.

Key words

holomorphic function zero set subharmonic function Riesz measure Jensen measure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Ph. Griffiths and J. King, “Nevanlinna theory and holomorphic mappings between algebraic varieties,” Acta Math., 130 (1973), 145–220.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Sh. A. Dautov and G. M. Khenkin, “Zeros of holomorphic functions of finite order and weighted estimates for solutions of the ¯?-equations,” Mat. Sb., 107(149):2(10) (1978), 163–174; English transl.: USSR Sb. Math., 35:4 (1979), 449–459.MathSciNetMATHGoogle Scholar
  3. [3]
    J. Bruna and X. Massaneda, “Zero sets of holomorphic functions in the unit ball with slow growth,” J. Anal. Math., 66 (1995), 217–252.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    B. N. Khabibullin, “Dual representation of superlinear functionals and its application in function theory. II,” Izv. Ross. Akad. Nauk Ser. Mat., 65:5 (2001), 167–190; English transl.: Russian Acad. Sci. Izv. Math., 65:5 (2001), 1017–1039.MathSciNetCrossRefGoogle Scholar
  5. [5]
    G. M. Khenkin, “The method of integral representations in complex analysis,” in: Complex analysis — several variables — 1, Itogi Nauki i Tekhniki. Sovrem. Probl. Mat. Fund. Napr., vol. 7, VINITI, Moscow, 1985, 23–124.MathSciNetMATHGoogle Scholar
  6. [6]
    S. V. Shvedenko, “Hardy classes and related spaces of analytic functions in the unit disc, polydisc, and ball,” in: Itogi Nauki i Tekhniki. Mat. Analiz, vol. 23, VINITI, Moscow, 1985, 3–124.MathSciNetMATHGoogle Scholar
  7. [7]
    L. I. Ronkin, Elements of the Theory of Analytic Functions of Several Variables [in Russian], Naukova dumka, Kiev, 1977.MATHGoogle Scholar
  8. [8]
    L. I. Ronkin, “Entire functions,” in: Complex analysis—several variables—3, Itogi Nauki i Tekhniki. Sovrem. Probl. Mat. Fund. Napr., vol. 9, VINITI, Moscow, 1986, 5–36.MathSciNetGoogle Scholar
  9. [9]
    P. Lelong and L. Gruman, Entire Functions of Several Comples Variables, Springer-Verlag, Berlin–Heidelberg, 1986.CrossRefMATHGoogle Scholar
  10. [10]
    L. I. Ronkin, Functions of Completely Regular Growth, Mathematics and Its Applications (Soviet Series), Kluver Academic Publishers Group, Dordrecht–Boston–London, 1992.CrossRefGoogle Scholar
  11. [11]
    B. N. Khabibullin, “Completeness of systems of entire functions in spaces of holomorphic functions,” Mat. Zametki, 66:4 (1999), 603–616; English transl.: Math. Notes, 66:4 (1999), 495–506.MathSciNetCrossRefGoogle Scholar
  12. [12]
    B. N. Khabibullin, Completeness of systems of exponentials and sets of uniqueness, 4th ed., completed, RITs BashGU, Ufa, 2012.Google Scholar
  13. [13]
    S. Yu. Favorov and L. D. Radchenko, “The Riesz measure of functions subharmonic in the exterior of a compact set,” Matematichni Studii, 40:2 (2013), 149–158.MathSciNetMATHGoogle Scholar
  14. [14]
    B. N. Khabibullin and N. R. Tamindarova, “Distribution of zeros and masses for holomorphic and subharmonic functions. I, II. Hadamard- and Blaschke-type conditions,” Mat. Sb, 2018 (to appear); http://arxiv.org/abs/1512.04610v2.Google Scholar
  15. [15]
    B. Khabibullin and N. Tamindarova, “Distribution of zeros for holomorphic functions: Hadamard- and Blaschke-type conditions,” in: Abstracts of International Workshop on “Nonharmonic Analysis and Differential Operators” (May 25-27, 2016), Institute of Mathematics and Mechanics of Azerbaijan National Academy of Sciences, Azerbaijan, Baku, 2016, 63.Google Scholar
  16. [16]
    B. Khabibullin and N. Tamindarova, “Uniqueness theorems for subharmonic and holomorphic functions of several variables on a domain,” Azerb. J. Math., 7:1 (2017), 70–79.MathSciNetMATHGoogle Scholar
  17. [17]
    H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000.Google Scholar
  18. [18]
    E. G. Kudasheva and B. N. Khabibullin, “The distribution of zeros of holomorphic functions of moderate growth in the unit disc and the representation of meromorphic functions there,” Mat. Sb, 200:9 (2009), 95–126; English transl.: Russian Acad. Sci. Sb. Math., 200:9 (2009), 1353–1382.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    W. Hayman and P. Kennedy, Subharmonic Functions, Academic Press, London–New York, 1976.MATHGoogle Scholar
  20. [20]
    Th. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.CrossRefMATHGoogle Scholar
  21. [21]
    E. M. Chirka, Complex Analytic Sets, Kluwer Academic Publishers, Dordrecht, 1985.MATHGoogle Scholar
  22. [22]
    M. Klimek, Pluripotential Theory, Clarendon Press, Oxford University Press, New York, 1991.MATHGoogle Scholar
  23. [23]
    P. Lelong, “Propriétés métriques des variétés analytiques complexes définies par une équation,” Ann. Sci. Ecole Norm. Sup., 67 (1950), 393–419.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    M. G. Arsove, “Functions representable as differences of subharmonic functions,” Trans. Amer. Math. Soc., 75 (1953), 327–365.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    A. F. Grishin, Nguen Van Quynh, and I. V. Poedintseva, “Representation theorems of d-subharmonic functions [in Russian],” Vestnik Kharkov Nat. Univ. Ser. Mat., Prikl. Math. i Mekh., 1133:70 (2014), 56–75; http://vestnik-math.univer.kharkov.ua/ Vestnik-KhNU-1133-2014-grish.pdf.MATHGoogle Scholar
  26. [26]
    B. J. Cole and T. J. Ransford, “Subharmonicity without upper semicontinuity,” J. Func. Anal., 147 (1997), 420–442.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    B. N. Khabibullin, “Criteria for (sub)harmonicity and continuation of (sub)harmonic functions,” Sibirsk. Mat. Zh., 44:4 (2003), 905–925; English transl.: Siberian. Math. J., 44:4 (2003), 713–728.MathSciNetMATHGoogle Scholar
  28. [28]
    B. N. Khabibullin and N. R. Tamindarova, “Subharmonic test functions and the distribution of zero sets of holomorphic functions,” Lobachevskii J. Math., 38:1 (2017), 38–43; http://arxiv.org/abs/1606.06714v1.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    E. A. Poletsky, “Disk envelopes of functions II,” J. Funct. Anal., 163:1 (1999), 111–132.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Bashkir State UniversityUfaRussia

Personalised recommendations