On nonhomeomorphic mappings with the inverse Poletsky inequality

On the inverse Poletskii inequality in metric spaces and prime ends

“…Now let us talk about the equicontinuity of families of mappings in the closure of a domain. Statements like the ones below have been established in various situations in [SSD,Theorem 1.2] and [Sev 2 , Theorem 2]. To such a great degree of generality, this statement is proved for the first time.…”

“…The following statement was proved in [SSD,Theorem 7.1] in the case of a fixed function Q and for the Euclidean space.…”

“…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 a mapping acts between domains of metric spaces.…”

On logarithmic Hölder continuity of mappings on the boundary

“…which was required to prove. Now, the possibility of a continuous extension of f to the boundary of B n is established in [SSD,Theorem 3.1]. In particular, the weakly flatness of ∂B n = S n−1 follows by [Va,Theorems 17.10 and 17.12].…”

“…Without loss of generality, we may assume that x m → x 0 as m → ∞. By [SSD,Theorem 3.1] f m has a continuous extension to x 0 , moreover, {f m } ∞ m=1 is equicontinuous at x 0 (see, e.g., [SSD,Theorem 1.2]). Now, for any ε > 0 there is…”