On the prime ends extension of unclosed inverse mappings

TL;DR

This paper investigates the boundary extension and equicontinuity of certain non-closed mappings that distort path families, using prime ends theory, under specific conditions.

Contribution

It extends the theory of boundary behavior of inverse mappings to non-closed cases using prime ends, providing new continuity and equicontinuity results.

Findings

Boundary extension results in the prime ends sense

Equicontinuity of mapping families established

Conditions for continuous boundary extension derived

Abstract

We consider mappings that distort the modulus of families of paths in the opposite direction in the manner of Poletsky's inequality. Here we study the case when the mappings are not closed, in particular, they do not preserve the boundary of the domain under the mapping. Under certain conditions, we obtain results on the continuous boundary extension of such mappings in the sense of prime ends. In addition, we obtain corresponding results on the equicontinuity of families of such mappings in terms of prime ends.