We consider the so-called ring Q-mappings, which are natural generalizations of quasiregular mappings in a sense of Väisälä’s geometric definition of moduli. It is shown that, under the condition of nondegeneracy of these mappings, their inner dilatation is majorized by a function Q(x) to within a constant depending solely on the dimension of the space.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York (2009).
O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York (1973).
Yu. F. Strugov, “Compactness of the families of mappings quasiconformal in the mean,” Dokl. Acad. Nauk SSSR, 243, No. 4, 859–861 (1978).
C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, “On conformal dilatation in space,” Int. J. Math. Math. Sci., 22, 1397–1420 (2003).
V. M. Miklyukov, Conformal Mappings of Irregular Surfaces and Their Applications [in Russian], Volgograd University, Volgograd (2005).
F. W. Gehring, “Rings and quasiconformal mappings in space,” Trans. Amer. Math. Soc., 103, 353–393 (1962).
Yu. G. Reshetnyak, Space Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982).
S. Rickman, “Quasiregular mappings,” Results Math. Relat. Areas, 26, No. 3 (1993).
R. Salimov, “Absolute continuity on lines and the differentiability of one generalization of quasiconformal mappings,” Izv. Ross. Acad. Nauk, Ser. Mat., 72, No. 5, 141–148 (2008).
R. R. Salimov and E. A. Sevost’yanov, “Theory of ring Q-mappings in geometric function theory,” Mat. Sb., 201, No. 6, 131–158 (2010).
V. I. Kruglikov, “Capacities of capacitors and space mappings quasiconformal in the mean,” Mat. Sb. 130, No. 2, 185–206 (1986).
G. T. Whyburn, Analytic Topology, American Mathematical Society, Providence, RI (1942).
S. Saks, Theory of the Integral, Dover, New York (1937).
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 11, pp. 1531–1537, November, 2010.
About this article
Cite this article
Salimov, R.R., Sevost’yanov, E.A. Estimation of dilatations for mappings more general than quasiregular mappings. Ukr Math J 62, 1775–1782 (2011). https://doi.org/10.1007/s11253-011-0467-2
- Quasiconformal Mapping
- Maximum Elevation
- Quasiregular Mapping
- Lebesgue Measurable Function
- Nonnegative Measurable Function