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Refined geometric characterizations of weak $p$-quasiconformal mappings

Ruslan Salimov, Alexander Ukhlov

Abstract

In this paper we consider refined geometric characterizations of weak $p$-quasiconformal mappings $\varphi:Ω\to\widetildeΩ$, where $Ω$ and $\widetildeΩ$ are domains in $\mathbb R^n$. We prove that mappings with the bounded on the set $Ω\setminus S$, where a set $S$ has $σ$-finite $(n-1)$-measure, geometric $p$-dilatation, are $W^1_{p,\loc}$-- mappings and generate bounded composition operators on Sobolev spaces.

Refined geometric characterizations of weak $p$-quasiconformal mappings

Abstract

In this paper we consider refined geometric characterizations of weak -quasiconformal mappings , where and are domains in . We prove that mappings with the bounded on the set , where a set has -finite -measure, geometric -dilatation, are -- mappings and generate bounded composition operators on Sobolev spaces.
Paper Structure (5 sections, 11 theorems, 70 equations)

This paper contains 5 sections, 11 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Composition operators on Sobolev spaces
  3. Sobolev spaces
  4. Composition operators
  5. Refined geometric characterizations of mappings

Key Result

Theorem 2.1

Let $\varphi:\Omega\to\widetilde{\Omega}$ be a homeomorphic mapping between two domains $\Omega$ and $\widetilde{\Omega}$. Then $\varphi$ generates a bounded composition operator if and only if $\varphi\in W^1_{q,\mathop{\mathrm{loc}}\nolimits}(\Omega)$ and The norm of the operator $\varphi^\ast$ is estimated as $\|\varphi^\ast\| \leq K_{p,q}(\varphi;\Omega)$.

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • Theorem 3.6
  • ...and 6 more