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On lower distance estimates of mappings in metric spaces

Evgeny Sevost'yanov, Denys Romash, Nataliya Ilkevych

Abstract

We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit mapping of a sequence of mappings converging locally uniformly. We separately consider cases when mappings are defined in Euclidean $n$-dimensional space and in a metric space.

On lower distance estimates of mappings in metric spaces

Abstract

We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit mapping of a sequence of mappings converging locally uniformly. We separately consider cases when mappings are defined in Euclidean -dimensional space and in a metric space.

Paper Structure

This paper contains 4 sections, 5 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th1']}
  3. Proof of Theorem \ref{['th2']}
  4. Convergence theorems

Key Result

theorem 1.1

If $Q\in L^1(D),$ then there exist a constant $C>0$ depending only on $n$ such that for any $x, y\in K$ and any $f\in \frak{F}^{A, \delta}_{K, Q}(D).$

Figures (1)

  • Figure 1: To proof of Theorem \ref{['th2']}

Theorems & Definitions (6)

  • theorem 1.1
  • theorem 1.2
  • theorem 1.3
  • theorem 1.4
  • remark 1.1
  • proposition 2.1