Journal of Mathematical Sciences

, Volume 234, Issue 5, pp 608–615 | Cite as

Modules of Families of Vector Measures on a Riemann Surface

  • Yu. V. DymchenkoEmail author
  • V. A. Shlyk

The paper considers a Riemann surface (in a broad sense of the term in the Hurwitz–Courant terminology) and an open set with a compact closure on this surface. It is proved that, following Aikawa–Ohtsuka, with a condenser on a given open set one can associate a family of vector measures, whose modules are computed directly with the aid of the weighted capacity (with Muchenhoupt weight) of the condenser.


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  1. 1.
    A. Hurwit and R. Courant, Function Theory [Russian translation], Nauka, Moscow (1968).Google Scholar
  2. 2.
    K. Ioshida, Functional Analysis [Russian translation], Mir, Moscow (1967).Google Scholar
  3. 3.
    D. Milman, “On some criteria for the regularity of spaces of type (B),” Dokl. AN SSSR, 20, 243–246 (1938).zbMATHGoogle Scholar
  4. 4.
    P. A. Pugach and V. A. Shlyk, “Weighted modules and capacities on a Riemann surface,” Zap. Nauchn. Semin. POMI, 458, 164–217 (2017).MathSciNetzbMATHGoogle Scholar
  5. 5.
    S. Stoilov, Lectures on Topological Principles in the Theory of Analytic Functions [in Russian], Nauka, Moscow (1964).Google Scholar
  6. 6.
    D. Adams and L. Hedberg, Function Spaces and Potential Theory, Springer-Verlag (1996).Google Scholar
  7. 7.
    H. Aikawa and M. Ohtsuka, “Extremal length of vector measures,” Ann. Acad. Sci. Fenn. Ser. A., 24, 61–88 (1999).MathSciNetzbMATHGoogle Scholar
  8. 8.
    K. Fan, “Minimax theorems,” Proc. Nat. Acad. Sci. USA, 39, No. 1, 42–47 (1953).MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Ohtsuka, Extremal Length and Precise Functions, Gakkōtosho (2003).Google Scholar
  10. 10.
    P. Pugach and V. Shlyk, “Moduli, capacity, BV-functions on the Riemann surfaces,” Lobachevskii J. Math., 38, No. 2, 338–351 (2017).MathSciNetCrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.Far Eastern Federal UniversityVladivostokRussia
  2. 2.Vladivostok Branch of the Russian Customs AcademyVladivostokRussia

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