On the subinvariance of uniform domains in Banach spaces

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 the following subinvariance property for the class of uniform domains: Suppose that $f$ is a freely quasiconformal mapping and that $D^{\prime }$ is uniform. Then the image $f(D_1)$ of every uniform subdomain $D_1$ in $D$ under $f$ is still uniform. This result answers an open problem of Väisälä in the affirmative.

arXiv comment20 pages, 4 figures. arXiv admin note: text overlap with arXiv:1105.4684

References in corpus (1)

• Local Convexity Properties of Quasihyperbolic Balls in Punctured Space