Siberian Mathematical Journal

, 50:760 | Cite as

Ned sets on a hyperplane

  • V. V. AseevEmail author


Under study are the sets in ℝ n (NED sets) each of which does not affect the conformal capacity of any condenser with connected plates disjoint from this set. These sets are removable singularities of quasiconformal mappings, which explains our interest in them. For compact sets on a hyperplane we obtain a geometric criterion of the NED property; we point out a simple sufficient condition for an NED set in terms of the connected attainability of its points from its complement in the hyperplane. For compact sets on a hypersphere we obtain a criterion for an NED set in terms of the reduced module at a pair of points in its complement. We establish that a compact set on a hypersphere S, removable for the capacity in at least one spherical ring concentric with S and containing S, is an NED set.


module of a family of curves NED set quasiconformal mapping removable singularity capacity of a condenser reduced generalized module capacity defect attainable boundary point 


  1. 1.
    Ahlfors L. and Beurling A., “Conformal invariants and functional-theoretic null-sets,” Acta Math., 83, 100–129 (1950).MathSciNetGoogle Scholar
  2. 2.
    Pesin I. N., “Metric properties for Q-quasiconformal mappings,” Mat. Sb., 40, No. 3, 281–294 (1956).MathSciNetGoogle Scholar
  3. 3.
    Väisälä J., “On the null-sets for extremal distances,” Ann. Acad. Sci. Fenn. Ser. A I Math., 322, 1–12 (1962).Google Scholar
  4. 4.
    Aseev V. V. and Sychëv A. V., “On removable sets of quasiconformal space mappings,” Sibirsk. Mat. Zh., 15, No. 6, 1213–1227 (1974).zbMATHGoogle Scholar
  5. 5.
    Vodop’ yanov S. K. and Gol’dshtĭn V. M., “Criteria for the removability of sets in spaces of L p1 quasiconformal and quasi-isometric mappings,” Siberian Math. J., 18, No. 1, 35–50 (1977).zbMATHCrossRefGoogle Scholar
  6. 6.
    Kopylov A. P., “Removability of plane sets in the class of three-dimensional quasiconformal mappings,” in: Metric Questions of the Theory of Functions and Mappings. Vol. 1 [in Russian], Naukova Dumka, Kiev, 1969, pp. 21–23.Google Scholar
  7. 7.
    Kopylov A. P. and Pesin I. N., “Removability of certain sets in the class of three-dimensional quasiconformal mappings,” Math. Notes, 7, 432–434 (1970).zbMATHMathSciNetGoogle Scholar
  8. 8.
    Väisälä J., “Removable sets for quasiconformal mappings,” J. Math. Mech., 19, No. 1, 49–51 (1969).zbMATHMathSciNetGoogle Scholar
  9. 9.
    Miklyukov V. M., “Removable singularities of quasiconformal mappings in space,” Dokl. Akad. Nauk SSSR, 188, No. 3, 525–527 (1969).zbMATHMathSciNetGoogle Scholar
  10. 10.
    Aseev V. V., “An example of an NED-set in n-dimensional Euclidean space, having positive (n-1)-dimensional Hausdorff measure,” Soviet Math., Dokl., 15, 855–858 (1974).zbMATHMathSciNetGoogle Scholar
  11. 11.
    Hedberg L. I., “Removable singularities and condenser capacities,” Ark. Mat., 12, No. 2, 181–201 (1974).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Shlyk V. A., “The geometry of removable sets for the space FD p, p ∈ (1,+∞), and Hedberg normal domains,” Soviet Math., Dokl., 41, No. 3, 471–474 (1990).zbMATHMathSciNetGoogle Scholar
  13. 13.
    Shlyk V. A., “The structure of compact sets generating normal domains, and removable singularities for the space L p1(D),” Math. USSR-Sb., 71, No. 2, 405–418 (1992).CrossRefMathSciNetGoogle Scholar
  14. 14.
    Shlyk V. A., “Normal domains and removable singularities,” Izv. Russ. Acad. Sci. Math., 43, No. 1, 83–104 (1994).CrossRefMathSciNetGoogle Scholar
  15. 15.
    Caraman P., “About the equality between the p-module and the p-capacity in ℝn,” in: Analytic Functions. Proc. of Conf. Blazejewko, 1982, Springer-Verlag, Berlin; Heidelberg; New York, 1983, pp. 32–83 (Lecture Notes in Math.; 1039).Google Scholar
  16. 16.
    Caraman P., “Relations between p-capacity and p-module,” Rev. Roumaine Math. Pures Appl., 39, No. 6, 509–577 (1994).zbMATHMathSciNetGoogle Scholar
  17. 17.
    Aseev V. V., “Reduced generalized module in the space problems of capacity tomography,” Dal’nevostochn. Mat. Zh., 7, No. 1–2, 17–29 (2007).Google Scholar
  18. 18.
    Dubinin V. N., Capacities of Condensers in Geometric Function Theory [in Russian], Izdat. Dal’nevostochn. Univ., Vladivostok (2003).Google Scholar
  19. 19.
    Dubinin V. N., “Symmetrization in the geometric theory of functions of a complex variable,” Russian Math. Surveys, 49, No. 1, 1–79 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Dubinin V. N. and Eĭrikh N. V., “Reduced generalized module,” Dal’nevostochn. Mat. Zh., 3, No. 2, 150–164 (2002).Google Scholar
  21. 21.
    Aseev V. V., “Description of NED-sets lying on hyperspheres,” in: Modern Methods of the Theory of Boundary Value Problems [in Russian], Proceedings of the Voronezh Spring Mathematical School “Pontryagin Readings-17,” Voronezh, Tsentr.-Chernozem. Knizh. Izdat., 2006, p. 9.Google Scholar
  22. 22.
    Hesse J., “A p-extremal length and p-capacity equality,” Ark. Mat., 13, No. 1, 131–144 (1975).zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Vuorinen M., Conformal Geometry and Quasiregular Mappings, Springer-Verlag, Berlin etc. (1988) (Lecture Notes in Math.; 1319).zbMATHGoogle Scholar
  24. 24.
    Aseev V. V., “On a property for the module,” Dokl. Akad. Nauk SSSR, 200, No. 3, 513–514 (1971).MathSciNetGoogle Scholar
  25. 25.
    Fuglede B., “Extremal length and functional completion,” Acta Math., 98, No. 3–4, 171–219 (1957).zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Fikhtengol’ts G. M., A Course of Differential and Integral Calculus. Vol. 3 [in Russian], Nauka, Moscow (1969).Google Scholar
  27. 27.
    Caraman P., n-Dimensional Quasiconformal (QCf) Mappings, Editura Acad. Bucurečsti, Romania; Abacus Press Tunbridge Wells, Kent (1974).zbMATHGoogle Scholar
  28. 28.
    Väisälä J., Lectures on n-Dimensional Quasiconformal Mappings, Springer-Verlag, Berlin etc. (1971) (Lecture Notes in Math.; 229).Google Scholar
  29. 29.
    Goluzin G. M., Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence (1969).zbMATHGoogle Scholar
  30. 30.
    Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton (1970).zbMATHGoogle Scholar
  31. 31.
    Kuratowski K., Topology. Vol. 2 [Russian translation], Mir, Moscow (1969).Google Scholar
  32. 32.
    Miklyukov V. M., “On one multiplicative inequality for the space mappings with bounded distortion,” in: Metric Questions of the Theory of Functions and Mappings. Vol. 1 [in Russian], Naukova Dumka, Kiev, 1969, pp. 162–184.Google Scholar
  33. 33.
    Gol’dshteĭn V. M. and Reshetnyak Yu. G., An Introduction to the Theory of Functions with Generalized Derivatives and the Quasiconformal Mappings [in Russian], Nauka, Moscow (1983).Google Scholar
  34. 34.
    Sychëv A. V., Modules and Quasiconformal Space Mappings [in Russian], Nauka, Novosibirsk (1983).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

Personalised recommendations