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On prime ends equicontinuity of unclosed inverse mappings

Zarina Kovba, Evgeny Sevost'yanov

TL;DR

This work extends equicontinuity results for families of open discrete mappings that satisfy a Poletsky-type inverse inequality to settings where the image domain boundary is not locally connected and the mappings need not be boundary-preserving or closed. By building a prime-end framework with a boundary metric ρ on the prime-end closure, the authors prove boundary extensions and equicontinuity of the extended family, under conditions on the distortion function Q and boundary regularity (including weak flatness). They further establish that, in the finitely connected case, equicontinuity holds in the prime-end closure and maps extend to the full prime-end boundary, broadening Näkki–Palka type results to non-closed, boundary-nonpreserving mappings. The results unify and generalize previous work in quasiconformal and quasiregular mapping theory, providing robust boundary control tools with potential applications in geometric function theory and related areas.

Abstract

We study mappings satisfying the inverse Poletsky-type inequality in a domain of Euclidean space. Such inequalities are well known and play an important role in the study of quasiconformal and quasiregular mappings. We consider the case when the mapped domain, generally speaking, is not locally connected on its boundary. At the same time, we consider the situation when the mapping is open and discrete, but may not preserve the boundary of the domain. In terms of prime ends, we obtain results on the equicontinuity of families of such mappings in the closure of the definition domain.

On prime ends equicontinuity of unclosed inverse mappings

TL;DR

This work extends equicontinuity results for families of open discrete mappings that satisfy a Poletsky-type inverse inequality to settings where the image domain boundary is not locally connected and the mappings need not be boundary-preserving or closed. By building a prime-end framework with a boundary metric ρ on the prime-end closure, the authors prove boundary extensions and equicontinuity of the extended family, under conditions on the distortion function Q and boundary regularity (including weak flatness). They further establish that, in the finitely connected case, equicontinuity holds in the prime-end closure and maps extend to the full prime-end boundary, broadening Näkki–Palka type results to non-closed, boundary-nonpreserving mappings. The results unify and generalize previous work in quasiconformal and quasiregular mapping theory, providing robust boundary control tools with potential applications in geometric function theory and related areas.

Abstract

We study mappings satisfying the inverse Poletsky-type inequality in a domain of Euclidean space. Such inequalities are well known and play an important role in the study of quasiconformal and quasiregular mappings. We consider the case when the mapped domain, generally speaking, is not locally connected on its boundary. At the same time, we consider the situation when the mapping is open and discrete, but may not preserve the boundary of the domain. In terms of prime ends, we obtain results on the equicontinuity of families of such mappings in the closure of the definition domain.

Paper Structure

This paper contains 4 sections, 7 theorems, 79 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th2']}
  4. The case of a finitely connectedness of a domain on $E$

Key Result

theorem 1.1

Let $D$ and $D^{\,\prime}$ be domains in ${\Bbb R}^n,$$n\geqslant 2,$ let $D$ be a domain with a weakly flat boundary and let $E$ be a set in $\overline{D^{\,\prime}},$ which is closed in $\overline{{\Bbb R}^n}$ and such that $\partial D^{\,\prime}\subset E.$ Suppose that the following conditions ar

Figures (5)

  • Figure 1: To the statement of Lemma \ref{['lem1']}
  • Figure 2: The formulation of Lemma \ref{['lem2']}
  • Figure 3: To the proof of theorem \ref{['th2']}.
  • Figure 4: To the proof of Theorem \ref{['th2']}, case III. a)
  • Figure 5: To the proof of Theorem \ref{['th2']}, case III. b)

Theorems & Definitions (12)

  • remark 1.1
  • theorem 1.1
  • remark 1.2
  • lemma 2.1
  • lemma 2.2
  • proof
  • lemma 2.3
  • proof
  • proposition 2.1
  • lemma 2.4
  • ...and 2 more