Isolated singularities of mappings with the inverse Poletsky inequality

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 $1/n.$

Hardy Spaces for Quasiregular Mappings and Composition Operators

We define Hardy spaces $${\mathcal {H}}^p$$ H p for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper $${\mathcal {H}}^p$$ H p -theory for Quasiconformal Mappings (Pure Appl Math Q 7(1):19–50, 2011).

On slow escaping and non-escaping points of quasimeromorphic mappings

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.