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Mappings of finite distortion of polynomial type

Changyu Guo

Abstract

Suppose that $f: \bR^n\to\bR^n$ is a mapping of $K$-bounded $p$-mean distortion for some $p>n-1$. We prove the equivalence of the following properties of $f$: doubling condition for $J(x,f)$ over big balls centered at origin, boundedness of multiplicity function $N(f,\bR^n)$, polynomial type of $f$ and polynomial growth condition for $f$.

Mappings of finite distortion of polynomial type

Abstract

Suppose that is a mapping of -bounded -mean distortion for some . We prove the equivalence of the following properties of : doubling condition for over big balls centered at origin, boundedness of multiplicity function , polynomial type of and polynomial growth condition for .

Paper Structure

This paper contains 6 sections, 9 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Auxiliary results
  3. Area formula
  4. Basic modulus estimates
  5. Growth estimate
  6. Proof of theorem A

Key Result

Theorem A

Let $f: \mathbb{R}^n\to\mathbb{R}^n$ be a mapping of $K$-bounded $p$-mean distortion for some $p>n-1$. Then the following statements are equivalent Moreover, each of the above condition implies that $A(r)$ is doubling for $r\geq r_0$, where $r_0>0$ is a constant. In particular, $f$ has finite lower order.

Theorems & Definitions (17)

  • Theorem A
  • Lemma 1: Lemma 4.9, r93
  • Lemma 2
  • Lemma 3: r04, Theorem 2.1
  • Lemma 4: ko06, Theorem 4.1
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 7 more