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Boundary Extensions for mappings between metric spaces

Yao-Lan Tian, Yi Xuan

Abstract

In this paper, we consider boundary extensions of two classes of mappings between metric measure spaces. These two mapping classes extend in particular the well-studied geometric mappings such as quasiregular mappings with integrable Jacobian determinant and mappings of exponentially integrable distortion with integrable Jacobian determinant. Our main results extend the corresponding results of Äkkinen and Guo [Ann. Mat. Pure. Appl. 2017] to the setting of metric measure spaces.

Boundary Extensions for mappings between metric spaces

Abstract

In this paper, we consider boundary extensions of two classes of mappings between metric measure spaces. These two mapping classes extend in particular the well-studied geometric mappings such as quasiregular mappings with integrable Jacobian determinant and mappings of exponentially integrable distortion with integrable Jacobian determinant. Our main results extend the corresponding results of Äkkinen and Guo [Ann. Mat. Pure. Appl. 2017] to the setting of metric measure spaces.
Paper Structure (5 sections, 11 theorems, 76 equations)

This paper contains 5 sections, 11 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and auxiliary results
  3. Proof of the main results
  4. Existence of limits along curves
  5. Uniqueness of limits along John curves

Key Result

Theorem 1.3

Let $\Omega\subset X_{1}$ be a $\varphi$-length John domain with center $x_{0}$. Suppose $\Omega$ is Ahlfors $q$-regular. Given a map $f:\Omega\rightarrow X_{2}$, let $E_{f}$ be the set of points $\omega\in\partial\Omega$ for which there exists a curve $\gamma\in I^{(\varphi,c)}(\omega,x_{0})$ so th

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Ahlfors $q$-regularity
  • Definition 2.2: Doubling metric space
  • Proposition 2.3: hk12
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 18 more