On boundary extension of one class of mappings in terms of prime ends

  • E.A. Sevost'yanov Zhytomyr Ivan Franko State University, Zhytomyr, Ukraine
  • S. A. Skvortsov Zhytomyr Ivan Franko State University, Zhytomyr, Ukraine
  • I.A. Sverchevska Zhytomyr Ivan Franko State University, Zhytomyr, Ukraine
Keywords: metric spaces, quasiconformal mappings, mappings with bounded and nite distortion, equiconti- nuity, moduli of families of paths


Here we consider the classes of mappings of metric spaces that distort the modulus of families of paths similarly to Poletsky inequality. For domains, which are not locally connected at the boundaries, we obtain results on the boundary extension of the indicated mappings. We also investigate the local and global behavior
of mappings in the context of the equicontinuity of their families. The main statements of the article are proved under the condition that the majorant responsible for the distortion of the modulus of the families of paths has a finite mean oscillation at the corresponding points. The results are applicable to well-known classes of conformal and quasiconformal mappings as well as mappings with a finite distortion.


E. Sevost'yanov, On boundary extension of mappings in metric spaces in terms of prime ends, Ann. Acad. Sci. Fenn. Math. 44 (2019), №1, 65-90.

E. Sevost'yanov, A. Markysh, On Sokhotski-Casorati-Weierstrass theorem on metric spaces, Complex Variables and Elliptic Equations 64 (2019), №12, 1973-1993.

T. Adamowicz, Prime ends in metric spaces and quasiconformal-type mappings, Analysis and Mathematical Physics, https://doi.org/10.1007/s13324-019-00292-z.

T. Adamowicz, A. Bjorn, J. Bjorn, N. Shanmugalingam, Prime ends for domains in metric spaces, Adv. Math. 238 (2013), 459-505.

V. Ya. Gutlyanskii, V. I. Ryazanov, E. Yakubov, The Beltrami equations and prime ends, Ukr. Mat. Visn. 12 (2015), №1, 27-66; transl. J. Math. Sci. (N.Y.) 210 (2015), №1, 22-51.

D.A. Kovtonyuk, V.I. Ryazanov, On the theory of prime ends for space mappings , Ukr. Mat. Zh. 67 (2015), №4, 467-479; transl. Ukrainian Math. J. 67 (2015), №4, 528-541.

A. Bjorn, J. Bjorn, Nonlinear Potential Theory on Metric Spaces, in EMS Tracts in Mathematics 17, European Math. Soc., Zurich, 2011.

B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171-219.

O. Martio, V. Ryazanov, U. Srebro and E. Yakubov, Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, New York etc., 2009.

A. Ignat'ev and V. Ryazanov, Finite mean oscillation in mapping theory (Russian), Ukr. Mat. Visn. 2 (2005), №3, 395-417, translation Ukr. Math. Bull. 2 (2005), №3, 403-424.

K. Kuratowski, Topology, v. 2, Academic Press, New York-London, 1968.

E.S. Smolovaya, Boundary behavior of ring Q-homeomorphisms in metric spaces, Ukrainian Math. J. 62 (2010), №5, 785-793.

E.S. Afanas'eva, On boundary behavior of one class of mappings in metric spaces , Ukrainian Math. Journ. 66 (2014), №1, 16-29.

R. Nakki and B. Palka, Uniform equicontinuity of quasiconformal mappings, Proc. Amer. Math. Soc. 37 (1973), №2, 427-433.

E. Sevost'yanov, Local and boundary behavior of maps in metric spaces , Algebra i analiz 28 (2016), №6, 118-146 (in Russian); translation St. Petersburg Math. J. 28 (2017), №6, 807-824.

J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517.

How to Cite
Sevost’yanov E, Skvortsov SA, Sverchevska I. On boundary extension of one class of mappings in terms of prime ends. Mat. Stud. [Internet]. 2020Mar.17 [cited 2020Apr.5];53(1):29-0. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/11