On the behavior of one class of homeomorphisms at infinity

  • R. R. Salimov Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv
  • B. A. Klishchuk Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv
Keywords: p-modulus of a family of curves, p-capacity of a condenser, ring homeomorphisms with respect to p-modulus

Abstract

UDC 517.5

We study the behavior of ring $Q$-homeomorphisms with respect to the $p$-modulus with $p>n$  at infinity.

References

J. V"{a}is"{a}l"{a, Lectures on n-dimensional quasiconformal mappings, Lect. Notes Math., vol. 229, Springer-Verlag, Berlin (1971).

V. I. Ryazanov, E. A. Sevost'yanov, Equicontinuous classes of ring $Q$-homeomorphisms, Sib. Math. J., 48, № 6, 1093 – 1105 (2007).

O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, $Q$-homeomorphisms, Complex Analysis and Dynamical Systems, Contemp. Math., 364, 193 – 203 (2004).

O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, On $Q$-homeomorphisms, Ann. Acad. Sci. Fenn. Math., 30, № 1, 49 – 69 (2005).

O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Math. Monogr., New York (2009).

R. Salimov, ACL and differentiability of a generalization of quasiconformal maps, Izv. Math., 72, № 5, 977 – 984 (2008).

A. Golberg, Differential properties of $(alpha, Q)$-homeomorphisms, Further Progress in Analysis, Proc. 6th ISAAC Congr. (2009), p.~218 – 228.

A. Golberg, Integrally quasiconformal mappings in space, Trans. Inst. Math. NAS Ukraine, 7, № 2, 53 – 64 (2010).

A. Golberg, R. Salimov, Logarithmic H{"o}lder continuity of ring homeomorphisms with controlled p-module, Complex Var. and Elliptic Equat., 59, № 1, 91 – 98 (2014).

A. Golberg, R. Salimov, E. Sevost'yanov, Distortion estimates under mappings with controlled $p$-module, Ann. Univ. Bucharest, Math. Ser., 63, № 5, 95 – 114 (2014).

R. Salimov, On finitely Lipschitz space mappings, Sib. Elecron. Math. Rep., 8, 284 – 295 (2011).

R. Salimov, Estimation of the measure of the image of the ball, Sib. Math. J., 53, № 4, 920 – 930 (2012).

R. Salimov, To a theory of ring $Q$-homeomorphisms with respect to a $p$-modulus, Ukr. Math. Bull., 10, № 3, 379 – 396 (2013).

R. Salimov, Оne property of ring $Q$-homeomorphisms with respect to a $p$-module, Ukr. Math. J., 65, № 5, 728 – 733 (2013).

R. Salimov, B. Klishchuk, The extremal problem for the area of an image of a disc, Rep. NAS Ukraine, 10, 22 – 27 (2016).

B. Klishchuk, R. Salimov, Lower bounds for the area of the image of a circle, Ufa Math. J., 9, № 2, 55 – 61 (2017).

R. Salimov, B. Klishchuk, Extremal problem for the area of the image of a disk, Zap. Nauchn. Sem. POMI, 456, 160 – 171 (2017).

R. Salimov, B. Klishchuk, An extremal problem for the volume functional, Mat. Stud., 50, № 1, 36 – 43 (2018).

B. Klishchuk, R. Salimov, Lower bounds for the volume of the image of a ball, Ukr. Math. J., 71, № 6, 774 – 785 (2019).

M. Cristea, Local homeomorphisms satisfying generalized modular inequalities, Complex Var. and Eliptic Equat., 59, № 2, 232 – 246 (2014).

M. Cristea, Some properties of open discrete generalized ring mappings, Complex Var. and Eliptic Equat., 61, № 5, 623 – 643 (2016).

M. Cristea, Eliminability results for mappings satisfying generalized modular inequalities, Complex Var. and Eliptic Equat., 64, № 4, 676 – 684 (2019).

А. А. Маркиш, Р. Р. Салимов, Е. А. Севостьянов, Об оценке искажения расстояния снизу для одного класса отображений, Укр. мат. журн, 70, № 11, 1553 – 1562 (2018).

A. Golberg, R. Salimov, E. Sevost'yanov, Singularities of discrete open mappings with controlled p-module, J. Anal. Math., 127, 303 – 328 (2015).

A. Golberg, R. Salimov, E. Sevost'yanov, Poletskii type inequality for mappings from the Orlicz – Sobolev classes, Complex Anal. and Oper. Theory, 10, 881 – 901 (2016).

A. Golberg, R. Salimov, E. Sevost'yanov, Estimates for Jacobian and dilatation coefficients of open discrete mappings with controlled p-module, Complex Anal. and Oper. Theory, 11, № 7, 1521 – 1542 (2017).

R. Salimov, E. Smolovaya, On the order of growth of ring Q-homeomorphisms at infinity, Ukr. Math. J., 62, № 6, 829 – 836 (2010).

Ю. Г. Решетняк, Пространственные отображения с ограниченным искажением, Новосибирск, Наука (1982).

O. Martio, S. Rickman, J. V"{a}is"{a}l"{a}, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Math.,448, 1 – 40 (1969).

V. A. Shlyk, The equality between $p$-capacity and $p$-modulus, Sib. Math. J., 34, № 6, 216 – 221 (1993).

V. Maz'ya, Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces, Contemp. Math., 338, 307 – 340 (2003).

F. W. Gehring, Lipschitz mappings and the $p$-capacity of ring in $n$-space, Adv. Theory Riemann Surfaces (Proc. Conf. Stonybrook, New York, 1969), Ann. Math. Stud., 66, 175 – 193 (1971).

Published
27.11.2022
How to Cite
SalimovR. R., and KlishchukB. A. “On the Behavior of One Class of Homeomorphisms at Infinity”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 10, Nov. 2022, pp. 1416 -26, doi:10.37863/umzh.v74i10.7158.
Section
Research articles