# On prime end distortion estimates of mappings with the Poletsky condition in domains with the Poincar´e inequality

### Abstract

This article is devoted to the study of mappings with bounded and

finite distortion defined in some domain of the Euclidean space. We

consider mappings that satisfy some upper estimates for the

distortion of the modulus of families of paths, where the order of

the modulus equals to $p,$ $n-1<p\leqslant n.$ The main problem

studied in the manuscript is the investigation of the boundary

behavior of such mappings, more precisely, the distortion of the

distance under mappings near boundary points. The publication is

primarily devoted to definition domains with ``bad boundaries'', in

which the mappings not even have a continuous extension to the

boundary in the Euclidean sense. However, we introduce the concept

of a quasiconformal regular domain in which the specified continuous

extension is valid and the corresponding distance distortion

estimates are satisfied; however, both must be understood in the

sense of the so-called prime ends. More precisely, such estimates

hold in the case when the mapping acts from a quasiconformal regular

domain to an Ahlfors regular domain with the Poincar\'e inequality.

The consideration of domains that are Ahlfors regular and satisfy

the Poincar\'e inequality is due to the fact that, lower estimates for

the modulus of families of paths through the diameter of the

corresponding sets hold in these domains. (There are the so-called

Loewner-type estimates). We consider homeomorphisms and mappings

with branching separately. The main analytical condition under which

the results of the paper were obtained is the finiteness of the

integral averages of some majorant involved in the defining modulus

inequality under infinitesimal balls. This condition includes the

situation of quasiconformal and quasiregular mappings, because for

them the specified majorant is itself bounded in a definition

domain. Also, the results of the article are valid for more general

classes for which Poletsky-type upper moduli inequalities are

satisfied, for example, for mappings with finite length distortion.

### References

O. Martio, S. Rickman, J. Vaisala, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 465 (1970), 1–13.

O. Martio, R. Nakki, Boundary Holder continuity and quasiconformal mappings, J. London Math. Soc. (2), 44 (1991), 339–350.

V.I. Ryazanov, E.A. Sevost’yanov, Toward the theory of ring Q-homeomorphisms, Israel J. Math., 68 (2008), 101–118.

O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Monographs in Mathematics, Springer, New York etc., 2009.

M. Arsenovic, M. Mateljevic, On the Holder continuity of ring Q-homeomorphisms, Georgian Math. J., 29 (2022), №6, 805—811.

V. Ryazanov, R. Salimov, E. Sevost’yanov, On the H¨older property of mappings in domains and on boundaries, Journal of Mathematical Sciences, 246 (2020), №1, 60–74.

M. Mateljevic, R. Salimov, E. Sevost’yanov, Holder and Lipschitz continuity in Orlicz-Sobolev classes, distortion and harmonic mappings, Filomat, 36 (2022), №16, 5359–5390.

M. Mateljevic, E. Sevost’yanov, On the behavior of Orlicz-Sobolev mappings with branching on the unit sphere, Journal of Mathematical Sciences, 270 (2023), №3, 467–499.

P. Hajlasz, P. Koskela, Sobolev met Poincare, Mem. Amer. Math. Soc., 145 (2000), №688, 1–101.

J. Vaisala, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., V.229, Springer–Verlag, Berlin etc., 1971.

J. Heinonen, Lectures on Analysis on metric spaces, Springer Science+Business Media, New York, 2001.

D.A. Kovtonyuk, V.I. Ryazanov, On the theory of prime ends for space mappings, Ukrainain Mathematical Journal, 67 (2015), №4, 528—541.

R. Nakki, Prime ends and quasiconformal mappings, Journal d’Analyse Mathematique, 35 (1979), 13–40.

O. Martio, J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A1 Math., 4 (1978/1979), 384–401.

F.W. Gehring, O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, Journal d’Analyse Math´ematique, 45 (1985), 181–206.

T. Adamowicz, N. Shanmugalingam, Non-conformal Loewner type estimates for modulus of curve families, Ann. Acad. Sci. Fenn. Math., 35 (2010), 609—626.

M. Vuorinen, Exceptional sets and boundary behavior of quasiregular mappings in n−space, Ann. Acad. Sci. Fenn. Ser. AI Math. Dissertattiones, 11 (1976), 1–44.

K. Kuratowski, Topology, V.2, Academic Press, New York–London, 1968.

R. Nakki, Continuous boundary extension of quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I. Math., 511 (1972), 1–10.

R.R. Salimov, E.A. Sevost’yanov, ACL and differentiability of open discrete ring (p,Q)-mappings, Mat. Stud., 35 (2011), №1, 28–36.

V.I. Ryazanov, E.A. Sevost’yanov, On the convergence of spatial homeomorphisms, Mat. Stud., 39 (2013), №1, 34–44.

E.A. Sevost’yanov, S.A. Skvortsov, N.S. Ilkevych, On boundary behavior of mappings with two normalized conditions, Mat. Stud., 49 (2018), №2, 150–157.

E.A. Sevost’yanov, S.A. Skvortsov, N.S. Ilkevych, On removable singularities of mappings in uniform spaces, Mat. Stud., 52 (2019), №1, 24–31.

E.A. Sevost’yanov, S.A. Skvortsov, I.A. Sverchevska, On boundary extension of one class of mappings in terms of prime ends, Mat. Stud., 53 (2020), №1, 29–40.

O.P. Dovhopiatyi, E.A. Sevost’yanov, On compact classes of solutions of Dirichlet problem in simply connected domains, Mat. Stud., 58 (2022), №2, 159–173.

R.R. Salimov, E.O. Sevost’yanov, V.A. Targonskii, On modulus inequality of the order p for the inner dilatation, Mat. Stud., 59 (2023), №2, 141–155.

*Matematychni Studii*,

*61*(2), 148-159. https://doi.org/10.30970/ms.61.2.148-159

Copyright (c) 2024 O. P. Dovhopiatyi, N. S. Ilkevych, E. O. Sevost'yanov, A. L. Targonskii

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.