On behavior of inverse homeomorphisms in terms of prime ends
Authors: Sevost'yanov Evgeny , Skvortsov Sergei , Ilkevych N. 
Published: May 2020

As is known, even conformal mappings of plane simply connected domains do not, generally speaking, have continuous boundary extension in the Euclidean sense. One of the minimum requirements necessary for such an extension is the local connectedness of the definition domain of the corresponding map on its boundary. Of course, quite a lot of simply connected domains do not have this property. For example, the unit disk with a cut along the positive part of the real axis is not locally connected at the boundary. In the same way, the mapped domain must also satisfy certain conditions necessary for the continuous extension of a mapping. The situation changes significantly if we are not talking about the Euclidean boundary behavior of mappings, but about extension in terms of the so-called prime ends. In this case, the domain of definition of mappings should be only regular, that is, this domain should be the image of a domain with a locally quasiconformal boundary under some quasiconformal mapping. A similar requirement also applies to the mapped domain. In this article, we study the equicontinuous families of maps at inner and boundary points in the case where the prime ends of the domain serve as boundary points. Relatively speaking, the paper consists of two parts, one of which contains a number of auxiliary statements, and the second, the final part of the work, contains the formulation of the main theorem and its proof. We consider a class of homeomorphisms of Euclidean space, inverse of which distort moduli of families of paths by the Poletsky type inequality. Note that these classes include most well-known mappings, such as conformal mappings, quasiconformal mappings, mappings with finite length and area distortion, and so on. It should be noted that under conformal mappings, distortion of the modulus of families of paths does not occur, therefore, when passing to inverse mappings, we remain in the class under study. A similar situation is in the case of quasiconformal mappings, since, as is known, the inverse mapping to a quasiconformal is also quasiconformal. In more general situations, the studied configurations can turn out to be much more complicated, in particular, the transition to inverse mappings can significantly change their properties (this is confirmed by specific examples of mappings, which are rather easy to construct in this case). This article is actually devoted to the study of this particular case, that is, when we are dealing with a certain family of homeomorphisms with an unbounded characteristic, in addition, mappings inverse to them are studied. In more detail, we consider mappings whose inverse satisfy the upper distortion estimate of the modulus of families of paths with integrable majorant. In the article, we proved that the families of the indicated mappings are equicontinuous both at the inner and boundary points of the domain, provided that the majorant responsible for the distortion of the modulus of the families of paths is integrable, besides that, the definition and mapped domains are regular, and the boundary points are prime ends of the definition domain. The results obtained in the paper are applicable to well-known classes of mappings, such as mappings with bounded and finite distortion, as well as to the Sobolev and Orlicz-Sobolev classes.