On distortion under mappings satisfying the inverse Poletsky inequality

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.