Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

Abstract

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio–Ryazanov–Srebro–Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk–Petkov–Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries.

We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Carathéodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Carathéodory prime ends are obtained.