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Sharpness of uniform continuity of quasiconformal mappings onto s-John domains

Chang-Yu Guo, Pekka Koskela

Abstract

We construct examples to show the sharpness of uniform continuity of quasiconformal mappings onto $s$-John domains. Our examples also give a negative answer to a prediction in [7].

Sharpness of uniform continuity of quasiconformal mappings onto s-John domains

Abstract

We construct examples to show the sharpness of uniform continuity of quasiconformal mappings onto -John domains. Our examples also give a negative answer to a prediction in [7].

Paper Structure

This paper contains 2 sections, 6 theorems, 40 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Proofs of the main results

Key Result

Theorem 1.1

Let $\Omega'\subset\mathbb{R}^n$ be a domain and $\Omega\subset\mathbb{R}^n$ be an $s$-John domain with $s\in (1,1+\frac{1}{n-1})$. Then each quasiconformal mapping $f:\Omega'\to\Omega$ satisfies for every pair $x',y'$ of distinct points in $\Omega'$, where $D_I$ is defined by taking the infimum of the diameters over all rectifiable curves in $\Omega$ joining the desired pair of points.

Figures (7)

  • Figure 1: the 2-John domain $\Omega$
  • Figure 2: $\Omega$ and $\Omega'$ in the step $j$
  • Figure 3: The quasiconformal mapping from $\tilde{ Q}_i$ to $Q_i'$
  • Figure 4: The first part of our domain $\Omega$
  • Figure 5: The new "legs" at step $j$
  • ...and 2 more figures

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • proof : Proof of Theorem \ref{['example:sharpness']}
  • proof : Proof of Corollary \ref{['coro:co-example to hk']}
  • proof : Proof of Theorem \ref{['example:sharpness2']}
  • proof : Proof of Corollary \ref{['coro:co-example to hk2']}
  • ...and 1 more