Skip to main content
Download PDF

Generalized capacities, compound curves, and removable sets

Journal of Mathematical Sciences volume 193, pages 55–65 (2013)Cite this article

Relations between the generalized capacity of a condenser in the sense of Aikawa-Ohtsuka and the module of the family of compound curves connecting the condenser plates through a given set are established. Conditions of the removability of a compact set for the generalized capacity of a condenser are obtained. Properties of the extremal length of vector measures are used. Bibliography: 9 titles.

References

  1. K. Kuratowski, Topology, Vol. II [Russian translation], Mir, Moscow (1969).

  2. P. A. Pugach and V. A. Shlyk, “Generalized capacities and polyhedral surfaces,” Zap. Nauchn. Semin. POMI, 383, 148–179 (2010).

    Google Scholar 

  3. P. A. Pugach and V. A. Shlyk, “Removable sets for the generalized module of a surface family,” Zap. Nauchn. Semin. POMI, 392, 163–190 (2011).

    Google Scholar 

  4. V. A. Shlyk, “Normal domains and removable singularities,” Izv. RAN, Ser. Mat., 57, 93–117 (1993).

    Google Scholar 

  5. H. Aikawa and M. Ohtsuka, “Extremal length of vector measures,” Ann. Acad. Sci. Fenn. Math., 24, 61–88 (1999).

    MathSciNet  MATH  Google Scholar 

  6. B. Fuglede, “Extremal length and functional completion,” Acta Math., 126, 171–219 (1957).

    MathSciNet  Article  Google Scholar 

  7. L. I. Hedberg, “Removable singularities and condenser capacities,” Ark. Mat., 12, 181–201 (1974).

    MathSciNet  MATH  Article  Google Scholar 

  8. 8 B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal functions,” Trans. Amer. Math. Soc., 192, 207–226 (1972).

    MathSciNet  Article  Google Scholar 

  9. M. Ohtsuka, Extremal Length and Precise Functions (GAKUTO Int. Ser. Math. Sci. Appl., 19), Gakkōtosho, Tokyo (2003).

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics of the Far Eastern Branch of the RAS, Vladivostok, Russia

    Yu. V. Dymchenko

  2. Far Eastern Federal University, Vladivostok, Russia

    V. A. Shlyk

Authors
  1. Yu. V. Dymchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. A. Shlyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. V. Dymchenko.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 404, 2012, pp. 100–119.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dymchenko, Y.V., Shlyk, V.A. Generalized capacities, compound curves, and removable sets. J Math Sci 193, 55–65 (2013). https://doi.org/10.1007/s10958-013-1433-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1433-3

Keywords

  • Vector Measure
  • Generalize Capacity
  • Extremal Length
  • Compound Curf