Local properties of quasihyperbolic and freely quasiconformal mappings

Abstract

Suppose that $E$ and $E^{\prime }$ denote real Banach spaces with dimension at least 2, that $D\subset E$ and $D^{\prime }\subset E^{\prime }$ are domains, and that $f:D\to D^{\prime }$ is a homeomorphism. In this paper, we prove that if there exists some constant $M>1$ (resp. some homeomorphism $\varphi$) such that for all $x\in D$, $f:B(x,d_D(x))\to f(B(x,d_D(x)))$ is $M$-QH (resp. $\varphi$-FQC), then $f$ is $M_1$-QH with $M_1=M_1(M)$ (resp. ${\varphi }_1$-FQC with ${\varphi }_1={\varphi }_1(\varphi )$). We apply our results to establish, in terms of the $j_D$ metric, a sufficient condition for a homeomorphism to be FQC.

arXiv comment13 pages

References in corpus (1)

• Local Convexity Properties of Quasihyperbolic Balls in Punctured Space