This work is devoted to the investigation of ring Q-homeomorphisms. We formulate conditions for a function Q(x) and the boundary of a domain under which every ring Q-homeomorphism admits a homeomorphic extension to the boundary. For an arbitrary ring Q-homeomorphism f: D → D’ with Q ∈ L 1(D); we study the problem of the extension of inverse mappings to the boundary. It is proved that an isolated singularity is removable for ring Q-homeomorphisms if Q has finite mean oscillation at a point.
Similar content being viewed by others
References
B. Bojarski, V. Gutlyanskii, and V. Ryazanov, “General Beltrami equations and BMO,” Ukr. Math. Bull., 5, No. 3, 305–326 (2008).
V. Ryazanov, U. Srebro, and E. Yakubov, “On ring solutions of Beltrami equations,” J. Anal. Math., 96, 117–150 (2005).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn. Ser. A1, 30, No. 1, 49–69 (2005).
E. A. Sevost’yanov, “Theory of modules, capacities, and normal families of mappings admitting branching,” Ukr. Mat. Vestn., 4, No. 4, 582–604 (2007).
E. A. Sevost’yanov, “Liouville, Picard, and Sokhotskii theorems for ring mappings,” Ukr. Mat. Vestn., 5, No. 3, 366–381 (2008).
V. I. Ryazanov and R. R. Salimov, “Weakly flat spaces and boundaries in the theory of mappings,” Ukr. Mat. Vestn., 4, No. 2, 199–234 (2007).
A. Ignat’ev and V. Ryazanov, “Finite mean oscillation in the theory of mappings,” Ukr. Mat. Vestn., 2, No. 3, 395–417 (2005).
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. Ryazanov, U. Srebro, and E. Yakubov, “On convergence theory for Beltrami equations,” Ukr. Math. Bull., 5, No. 4, 524–535 (2008).
E. A. Sevost’yanov, “Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for Q-mappings,” Ukr. Mat. Zh., 61, No. 1, 116–126 (2009).
V. Gutlyanski, O. Martio, T. Sugava, and M. Vuorinen, “On the degenerate Beltrami equation,” Trans. Amer. Math. Soc., 357, No. 3, 875–900 (2005).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. Anal. Math., 93, 215–236 (2004).
K. Kuratowski, Topology [Russian translation], Vol. 2, Mir, Moscow (1969).
F. W. Gehring, ”Quasiconformal mappings,” in: Complex Analysis and Its Applications, Vol. 2, International Atomic Energy Agency, Vienna (1976), pp. 213–268.
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 10, pp. 1329–1337, October, 2009.
Rights and permissions
About this article
Cite this article
Lomako, T.V. On extension of some generalizations of quasiconformal mappings to a boundary. Ukr Math J 61, 1568–1577 (2009). https://doi.org/10.1007/s11253-010-0298-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-010-0298-6