Skvortsov S. Local behavior of mappings with unbounded characteristics.

The dissertation is devoted to the development of mapping theory, namely the study of local and global behavior of mappings in Euclidean space, as well as in metric spaces. The families of mappings, their continuous extension to the boundary, equicontinuity at the boundary and in the inner points of the domain are considered in the manuscript. The mappings under consideration satisfy some modular condition (the so-called Poletsky inequality or the "inverse" Poletsky inequality, wherein the class of mappings considered in the dissertation includes the main types of mappings with finite distortion. The main results that determine the scientific novelty of the dissertation: -there are proved different theorems on the equicontinuous families of mappings of the Euclidean space at inner and boundary points of a domain, inverse of which are Q-mappings or ring Q-mappings provided that a function Q is integrable; -it is proved the logarithmic Hölder continuity of open discrete mappings satisfying the inverse Poletsky inequality; -there are established sufficient conditions for the removability of isolated singularities of ring Q-mappings of the Euclidean space and metric spaces with sphericalization; -there are established sufficient conditions for the continuous extension to the boundary of the domain of ring Q-homeomorphisms and mappings inverse to which are Q-homeomorphisms and for discrete closed Q-mappings; -in the case of complex boundaries, there are established sufficient conditions under which homeomorphisms with inverse Poletsky inequality have a continuous extension to the boundary of the domain in terms of prime ends; -there are established sufficient conditions for the continuous extension of open discrete ring Q-mapping of the metric space to the boundary of the domain, filled with its prime ends, defined in terms of (p, q) –modulus.