Abstract
We investigate the boundary behavior of so-called Q-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function Q(x) and the boundary of the domain under which any Q-homeomorphism with respect to a measure admits a continuous extension to a boundary point.
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References
K. Astala, T. Iwaniec, P. Koskela, and G. Martin, “Mappings of BMO-bounded distortion,” Math. Ann., 317, 703–726 (2000).
F. W. Gehring and T. Iwaniec, “The limit of mappings with finite distortion,” Ann. Acad, Sci. Fenn. Math., 23, 253–264 (1999).
J. Heinonen and P. Koskela, “Sobolev mappings with integrable dilations,” Arch. Ration. Mech. Anal., 125, 81–97 (1993).
I. Holopainen and P. Pankka, “Mappings of finite distortion: global homeomorphism theorem,” Ann. Acad. Sci. Fenn. Math., 29, No. 1, 59–80 (2004).
T. Iwaniec and G. Martin, Geometrical Function Theory and Non-Linear Analysis, Clarendon Press, Oxford (2001).
T. Iwaniec, P. Koskela, and J. Onninen, “Mappings of finite distortion: monotonicity and continuity,” Invent. Math., 144, No. 3, 507–531 (2001).
T. Iwaniec, P. Koskela, and J. Onninen, “Mappings of finite distortion: compactness,” Ann. Acad. Sci. Fenn. Math., 27, No. 2, 349–417 (2002).
J. J. Manfredi and E. Villamor, “An extension of Reshetnyak’s theorem,” Indiana Univ. Math. J., 47, No. 3, 1131–1145 (1998).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. Anal. Math., 93, 215–236 (2004).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn. Math., 30, 49–69 (2005).
A. A. Ignat’ev and V. I. Ryazanov, “Finite mean oscillation in the theory of mappings,” Ukr. Mat. Vestn., 2, No. 3, 395–417 (2005).
J. Heinonen and P. Koskela, “Quasiconformal maps in metric spaces with controlled geometry,” Acta Math., 181, 1–61 (1998).
O. Martio, “Modern tools in the theory of quasiconformal mappings,” Texts Math., Ser. B, 27, 1–43 (2000).
J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York (2001).
J. Vaisala, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).
B. Fuglede, “Extremal length and functional completion,” Acta Math., 908, 171–219 (1957).
G. T. Whyburn, Analytic Topology, American Mathematical Society, Phode Island (1942).
P. Buser, “A note on the isoperimetric constant,” Ann. Sci. Ecole Norm Supér., 4, No. 15, 213–230 (1982).
I. Chavel, Riemannian Geometry—a Modern Introduction, Cambridge University, Cambridge (1993).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1068–1074, August, 2007.
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Salimov, R.R. On the boundary behavior of imbeddings of metric spaces into a Euclidean space. Ukr Math J 59, 1184–1191 (2007). https://doi.org/10.1007/s11253-007-0079-z
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DOI: https://doi.org/10.1007/s11253-007-0079-z