On the global behavior of inverse mappings in terms of prime ends

Abstract

The paper is devoted to the study of mappings with finite distortion, actively studied recently. For mappings whose inverse satisfy the Poletsky inequality, the results on boundary behavior in terms of prime ends are obtained. In particular, it was proved that the families of the indicated mappings are equicontinuous at the points of the boundary if a certain function determining the distortion of the modulus of families of paths under the mappings is integrable.

On discrete boundary extension of mappings in terms of prime ends

“…One of the versions of the following statement is established in [SevSkv 1 , item v, Lemma 2] for homeomorphisms and "good" boundaries, see also [SevSkv 2 , Lemma 4.1]. Let us also point out the case relating to mappings with branching and good boundaries, see [SSD,Lemma 6.1], as well as the case of bad boundaries and homeomorphisms, see [ISS,Lemma 2.13]. The statement below seems to refer to the most general situation when some function Q participating in inequality (1.2), generally speaking, is not integrable; however, it has finite averages over spheres.…”

“…The following lemma was established in [Sev,Lemma 2] for the case of integrable functions Q; see also the case of homeomorphisms established in [ISS,Lemma 2.2].…”

On boundary Hölder logarithmic continuity of mappings in some domains

“…The possibility of a continuous extension of the mapping f to the boundary of D follows by Theorem 3.1 in [10]. In particular, locally quasiconformal boundaries of domains are weakly flat (see [15,Proposition 2.2], see also [9,Theorem 17.10]), and convex domains are obviously locally connected at its boundary. Put x 0 ∈ ∂D.…”