Local properties of quasihyperbolic and freely quasiconformal mappings

TL;DR

This paper proves that local quasihyperbolic and freely quasiconformal properties imply global properties for homeomorphisms between Banach space domains, providing conditions for FQC mappings based on the $j_D$ metric.

Contribution

It establishes that local quasihyperbolic and freely quasiconformal conditions lead to global properties, with explicit bounds and conditions in Banach spaces.

Findings

Local M-QH or $$-FQC maps are globally M_1-QH or $_1$-FQC.

Provides a sufficient $j_D$ metric condition for a homeomorphism to be FQC.

Results apply to Banach spaces of dimension at least 2.

Abstract

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\subset E$ and $D'\subset E'$ are domains, and that $f: D\to D'$ is a homeomorphism. In this paper, we prove that if there exists some constant $M>1$ (resp. some homeomorphism $\phi$) such that for all $x\in D$, $f: B(x,d_D(x))\to f(B(x,d_D(x)))$ is $M$-QH (resp. $\phi$-FQC), then $f$ is $M_1$-QH with $M_1=M_1(M)$ (resp. $\phi_1$-FQC with $\phi_1=\phi_1(\phi)$). We apply our results to establish, in terms of the $j_D$ metric, a sufficient condition for a homeomorphism to be FQC.