Advertisement

Siberian Mathematical Journal

, Volume 44, Issue 4, pp 713–728 | Cite as

Criteria for (Sub-)Harmonicity and Continuation of (Sub-)Harmonic Functions

  • B. N. Khabibullin
Article

Abstract

We obtain criteria for harmonicity and subharmonicity of a function in a domain in R d , d≥2, in terms of special Arens–Singer and Jensen measures. We also establish a criterion for (sub-)harmonicity of a δ-subharmonic function in terms of the associated Riesz charge and special Arens–Singer and Jensen functions. To this end, we use the theorem of this article on continuation of (sub-)harmonic functions to polar sets.

harmonic function subharmonic function Jensen measure Arens–Singer measure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gamelin T. W., Uniform Algebras [Russian translation], Mir, Moscow (1973).Google Scholar
  2. 2.
    Gamelin T. W., Uniform Algebras and Jensen Measures, Cambridge Univ. Press, Cambridge (1978).Google Scholar
  3. 3.
    Function Algebras, Proc. Intern. Sympos. Function Algebras, Birtel F. (ed.), Scott, Foresman and Co., Chicago (1966).Google Scholar
  4. 4.
    Koosis P., Lecons sur le Théorème de Beurling et Malliavin, Les Publications CRM, Montréal (1996).Google Scholar
  5. 5.
    Cole B. J. and Ransford T. J., “Subharmonicity without upper semicontinuity,” J. Funct. Anal., 147, 420–442 (1997).Google Scholar
  6. 6.
    Khabibullin B. N., “Sets of uniqueness in spaces of entire functions of a single variable,” Izv. Akad. Nauk SSSR Ser. Mat., 55, No. 5, 1101–1123 (1991).Google Scholar
  7. 7.
    Khabibullin B. N., “Estimations of the volume of the zero sets of holomorphic functions,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 58–63 (1992).Google Scholar
  8. 8.
    Khabibullin B. N., “On zero sets of entire functions and a representation of meromorphic functions,” Mat. Zametki, 59, No. 4, 611–617 (1996).Google Scholar
  9. 9.
    Hansen W. and Nadirashvili N., “Mean values and harmonic functions,” Math. Ann., 297, 157–170 (1993).Google Scholar
  10. 10.
    Bu S. and Schachermayer W., “Approximation of Jensen measures by image measures under holomorphic functions and applications,” Trans. Amer. Math. Soc., 331, No. 2, 585–608 (1992).Google Scholar
  11. 11.
    Poletsky E. A., “Holomorphic currents,” Indiana Univ. Math. J., 42, No. 1, 85–144 (1993).Google Scholar
  12. 12.
    Netuka I. and Vesely J., “Mean value property and harmonic functions,” in: Classical and Modern Potential Theory and Applications, Kluwer Acad. Publ., Dordrecht, 1994, pp. 359–398.Google Scholar
  13. 13.
    Khabibullin B. N., “On the type of entire and meromorphic functions,” Mat. Sb., 183, No. 11, 35–44 (1992).Google Scholar
  14. 14.
    Khabibullin B. N., “Nonconstructive proofs of the Beurling-Malliavin theorem and the nonuniqueness theorem for entire functions,” Izv. Ross. Akad. Nauk Ser. Mat., 58, No. 4, 125–148 (1994).Google Scholar
  15. 15.
    Arsove M. G., “Functions representable as differences of subharmonic functions,” Trans. Amer. Math. Soc., 75, 327–365 (1953).Google Scholar
  16. 16.
    Hayman W. K. and Kennedy P. B., Subharmonic Functions [Russian translation], Mir, Moscow (1980).Google Scholar
  17. 17.
    Doob J. L., Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, New York (1984).Google Scholar
  18. 18.
    Watson N. A., “Superharmonic extensions, mean values and Riesz measures,” Potential Anal., 2, 269–294 (1993).Google Scholar
  19. 19.
    Khabibullin B. N., “Jensen measures on open sets,” Vestnik Bashkir. Univ., 2, 3–6 (1999).Google Scholar
  20. 20.
    Khabibullin B. N., “Subharmonicity criteria in terms of measures and Jensen functions,” in: Proceedings of the School-Conference “The Theory of Functions, Its Applications, and Related Problems,” Kazan', Kazansk. Mat. Obshch., 1999, pp. 236–237.Google Scholar
  21. 21.
    Gardiner S. J., Harmonic Approximation, Cambridge Univ. Press, Cambridge (1995).Google Scholar
  22. 22.
    Gauthier P. M., “Uniform approximation,” in: Complex Potential Theory, Kluwer Acad. Publ., Dordrecht; Boston; London, 1994, 235–271.Google Scholar
  23. 23.
    Gaier D., Lectures on Complex Approximation [Russian translation], Mir, Moscow (1986).Google Scholar
  24. 24.
    Gauthier P. M., “Subharmonic extensions and approximations,” Canad. Math. Bull., 37, No. 1, 46–53 (1994).Google Scholar
  25. 25.
    Brelot M., Fundamentals of Classical Potential Theory [Russian translation], Mir, Moscow (1964).Google Scholar
  26. 26.
    Landkof N. S., Fundamentals of Modern Potential Theory [in Russian], Nauka, Moscow (1966).Google Scholar
  27. 27.
    Hayman W. K., Subharmonic Functions. Vol. 2, Acad. Press, London (1989).Google Scholar
  28. 28.
    Azarin V. S., “On asymptotic behavior of subharmonic functions of finite order,” Mat. Sb., 108, No. 2, 147–167 (1979).Google Scholar
  29. 29.
    Azarin V. S., On the Theory of Growth of Subharmonic Functions: Lectures [in Russian], Khar'kov Univ., Khar'kov (1978).Google Scholar
  30. 30.
    Hörmander L., The Analysis of Linear Partial Differential Operators. Vol. 1 and 2 [Russian translation], Mir, Moscow (1986).Google Scholar
  31. 31.
    Brelot M., On Topologies and Boundaries in Potential Theory, Mir, Moscow (1974).Google Scholar
  32. 32.
    Ransford T. J., Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge (1995).Google Scholar
  33. 33.
    Newman M. H. A., Elements of the Topology of Plane of Points, Cambridge Univ. Press, Cambridge (1964).Google Scholar
  34. 34.
    Bagby T. and Gauthier P. M., “Harmonic approximation on closed subsets of Riemannian manifolds,” in: Complex Potential Theory, Kluwer Acad. Publ., Dordrecht; Boston; London, 1994, pp. 75–89.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • B. N. Khabibullin
    • 1
  1. 1.Bashkir State UniversityUfa

Personalised recommendations