On distortion under mappings satisfying the inverse Poletsky inequality

E. O. Sevost'yanov; R. R. Salimov; V. S. Desyatka; N. S. Ilkevych

doi:10.30970/ms.63.1.31-47

E. O. Sevost'yanov Zhytomyr Ivan Franko State University; Institute of Applied Mathematics and Mechanics of NAS of Ukraine https://orcid.org/0000-0001-7892-6186
R. R. Salimov Institute of Mathematics of NAS of Ukraine Kyiv, Ukraine
V. S. Desyatka Zhytomyr Ivan Franko State University Zhytomyr, Ukraine
N. S. Ilkevych Zhytomyr Ivan Franko State University Zhytomyr, Ukraine

DOI: https://doi.org/10.30970/ms.63.1.31-47

Keywords: quasiconformal mappings, mappings with bounded and finite distortion, boundary behavior, moduli of families of paths

Abstract

As it is known, conformal mappings are locally Lipschitz at inner points of a domain, and quasiconformal (quasiregular) mappings are locally H ̈older continuous. As for estimates of the distortion of mappings at boundary points of the domain, this problem has not been studied sufficiently even for these classes. We partially fill this gap by considering in this manuscript not even local behavior at the boundary points, but global behavior in the domain of one class of mappings. The paper is devoted to studying mappings with finite distortion. The goal of our investigation is obtaining the distance distortion for mappings at inner and boundary points. Here we study mappings satisfying Poletsky’s inequality in the inverse direction. We obtain conditions under which these mappings are either logarithmic H ̈older continuous or H ̈older continuous in the closure of a domain. We consider several important cases in the manuscript, studying separately bounded convex domains and domains with locally quasiconformal boundaries, as well as domains of more complex structure in which the corresponding distortion estimates must be understood in terms of prime ends. In all the above situations we show that the maps are logarithmically H ̈older continuous, which is somewhat weaker than the usual H ̈older continuity. However, in the last section we consider the case where the maps are still H ̈older continuous in the usual sense. The research technique is associated with the use of the method of moduli and the method of paths liftings. A key role is also played by the lower bounds of the Loewner type for the modulus of families of paths, which are valid only in domains with a special geometry, in particular, bounded convex domains. Another important fact which is also valid for domains of the indicated type, is the possibility of joining pairs of different points in a domain by paths lying (up to a constant) at a distance no closer than a distance between above points.

Author Biographies

E. O. Sevost'yanov, Zhytomyr Ivan Franko State University; Institute of Applied Mathematics and Mechanics of NAS of Ukraine

Profeesor of Math. Analysis Dept. of Zhytomyr Ivan Franko State University; Leading Researcher of Function Theory Dept., Institute of Applied Mathematics and Mechanics of NAS of Ukraine

R. R. Salimov, Institute of Mathematics of NAS of Ukraine Kyiv, Ukraine

Institute of Mathematics of NAS of Ukraine
Kyiv, Ukraine

V. S. Desyatka, Zhytomyr Ivan Franko State University Zhytomyr, Ukraine

Zhytomyr Ivan Franko State University Zhytomyr, Ukraine

N. S. Ilkevych, Zhytomyr Ivan Franko State University Zhytomyr, Ukraine

Zhytomyr Ivan Franko State University Zhytomyr, Ukraine

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