Isolated singularities of mappings with the inverse Poletsky inequality

Abstract

The manuscript is devoted to the study of mappingswith finite distortion, which have been actively studied recently.We consider mappings satisfying the inverse Poletsky inequality,which can have branch points. Note that mappings with the reversePoletsky inequality include the classes of con\-for\-mal,quasiconformal, and quasiregular mappings. The subject of thisarticle is the question of removability an isolated singularity of amapping. The main result is as follows. Suppose that $f$ is an opendiscrete mapping between domains of a Euclidean $n$-dimensionalspace satisfying the inverse Poletsky inequality with someintegrable majorant $Q.$ If the cluster set of $f$ at some isolatedboundary point $x_0$ is a subset of the boundary of the image of thedomain, and, in addition, the function $Q$ is integrable, then $f$has a continuous extension to $x_0.$ Moreover, if $f$ is finite at$x_0,$ then $f$ is logarithmic H\"{o}lder continuous at $x_0$ withthe exponent $1/n.$