On local behavior of one class of inverse mappings

Abstract

As is known, the local behavior of maps is one of the most important problems of analysis. This, in particular, relates to the study of mappings with bounded and finite distortion, which have been actively studied recently. As for this work, here we solve the problem of the behavior of maps, the inverse of which satisfies the Poletsky inequality. The main result is the statement about the equicontinuity of the indicated mappings inside the domain in the case when the majorant corresponding to the distortion of the module under the mapping is integrable in the original domain. It should be emphasized that the proof of this result is largely geometric, at the same time, it uses only the conditions of boundedness of the direct and mapped domains and does not involve any requirements on their boundaries. The study of families of mappings inverse to a given class may turn out to be trivial if we are talking about quasiconformal mappings. In the latter case, we do not go beyond the limits of the class under study in the transition to inverse maps. Nevertheless, when studying mappings with unbounded characteristic, this question is quite substantial, as simple examples of the corresponding classes show. The idea of the proof of the main result is based on the fact that the inner points of an arbitrary domain are weakly flat. The last statement can be called the Väisälä lemma, which was established in his monograph and related to families of curves joining two continua between the plates of a spherical condenser. The proof is also based on the fact that the module of families of curves joining two converging continua in a good domain must tend to infinity. In this case, the neighborhood of some inner point of the mapped domain serves as ''good'' region, in which we check the equicontinuity of the inverse family of maps. The results of this article are applicable to many other classes of mappings such as mappings with a finite distortion in the sense of Iwaniec, Sobolev classes on the plane and in space, and so on.