Mappings of finite distortion of polynomial type

TL;DR

This paper establishes the equivalence of several properties for mappings of finite distortion of polynomial type, including doubling conditions, bounded multiplicity, and polynomial growth, under certain distortion bounds.

Contribution

It proves the equivalence of key geometric and analytic properties for mappings with bounded p-mean distortion, extending understanding of their structure.

Findings

Doubling condition for Jacobian over large balls is equivalent to polynomial growth.

Bounded multiplicity function characterizes polynomial type mappings.

Mappings of finite distortion with bounded p-mean distortion exhibit polynomial behavior.

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 $f$.