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Quasiregular values from generalized manifold with controlled geometry

Deguang Zhong

Abstract

The main aim of this paper is to establish the Reshetnyak's theorem for quasiregualr values from generalized $n$-manifold with suitable controlled geometry to Euclidean space $\mathbb{R}^{n}.$ This generalizes a previous result due to Kangasniemi and Onninen on the setting of Euclidean space [A single-point Reshetnyak's theorem, Trans. Amer. Math. Soc., 378(2025): 3105-3128].

Quasiregular values from generalized manifold with controlled geometry

Abstract

The main aim of this paper is to establish the Reshetnyak's theorem for quasiregualr values from generalized -manifold with suitable controlled geometry to Euclidean space This generalizes a previous result due to Kangasniemi and Onninen on the setting of Euclidean space [A single-point Reshetnyak's theorem, Trans. Amer. Math. Soc., 378(2025): 3105-3128].
Paper Structure (27 sections, 23 theorems, 116 equations)

This paper contains 27 sections, 23 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Newtonian spaces
  4. Generalized $n$-manifolds with controlled geometry
  5. Ahlfors $Q$-regular
  6. Rectifiable sets
  7. Generalized $n$-manifolds
  8. Orientable and metrically orientable
  9. Linearly locally contractible
  10. Local degree and local index
  11. Generalized $n$-manifolds with controlled geometry
  12. Quasiregular values and values of finite distortion
  13. Hölder regularity for generalized finite distortion and its applications
  14. Hölder regularity for generalized finite distortion mappings
  15. Corollary I: Hölder regularity for values of finite distortion
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $K\geq1$ be a constant, $y_{0}\in\mathbb{R}^{n},$$\mathcal{S}\subset \mathbb{R}^{m}$ be a generalized $n$-manifolds with controlled geometry in the sense of subsection adjhaka and $\Omega\subset\mathcal{S}$ be a domain. Suppose that $f\in N_{loc}^{1,n}(\Omega,\mathbb{R}^{n})$ is an nonconstant m for almost every $x\in\Omega.$ If $\Sigma\in L_{loc}^{p}(\Omega)$ for some $p>1,$ then $f^{-1}\{y_{

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.1
  • Definition 2.7
  • ...and 46 more