Simple sufficient conditions for a compact set on a hyperplane to be a NED-set in terms of the spherical accessibility of its points from the complement of this set to the hyperplane are derived. Bibliography: 14 titles.
Similar content being viewed by others
References
V. V. Asseev, “NED-sets lying in a hyperplane,” Sib. Mat. Zh., 50, 760–775 (2009).
V. V. Aseev and A. V. Sychev, “On sets removable for space quasiconformal maps,” Sib. Mat. Zh., 15, 1213–1227 (1974).
S. K. Vodop’yanov and V. M. Gol’dstein, “A criterion of removability of sets for the spaces \(L^1_p\), quasiconformal and quasiisometrical maps,” Sib. Mat. Zh., 18, 48–68 (1977).
B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis [Russian translation], Mir, Moscow (1967).
V. Golubev, Univalent Analytic Functions: Automorphic Functions [in Russian], Moscow (1961).
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable [in Russian], Moscow (1966).
Yu. V. Dymchenko and V. A. Shlyk, “Sufficiency of a family of broken lines in the module method and removable sets,” Sib. Mat. Zh., 51, 1028–1042 (2010).
A. P. Kopylov, “On the removability of plane sets in the class of three-dimensional quasiconformal maps,” Metric Questions of the Theory of Functions and Mappings, Vyp. 1, 21–23 (1969).
V. A. Shlyk, “Weight capacities, condenser capacities, and sets exceptional in the sense of Fuglede,” Dokl. RAN, 332, 428–431 (1993).
L. Ahlfors and A. Beurling, “Conformal invariants and function-theoretic null-sets,” Acta Math., 83, 101–129 (1950).
B. Fuglede, “Extremal length and functional completion,” Acta Math., 126, 171–219 (1957).
M. Ohtsuka, Extremal Length and Precise Functions (GAKUTO Int. Ser. Math. Sci. Appl., 19), Gakkōtosho, Tokyo (2003).
J. Väisälä, “On the null-sets for extremal distances,” Ann. Acad. Sci. Fenn. Ser. A., 322, 1–12 (1962).
M. Vuorinen, Conformal Geometry and Quasiregular Mappings (Lect. Notes Math., 1319), Springer-Verlag (1988).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 404, 2012, pp. 248–258.
Rights and permissions
About this article
Cite this article
Shlyk, V.A. Spherical symmetrization and NED-sets on a hyperplane. J Math Sci 193, 145–150 (2013). https://doi.org/10.1007/s10958-013-1443-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-013-1443-1