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arXiv · 1209.4541

On the subinvariance of uniform domains in Banach spaces

Manzi Huang·Xiantao Wang·Matti Vuorinen
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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 the following subinvariance property for the class of uniform domains: Suppose that $f$ is a freely quasiconformal mapping and that $D'$ 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 V\"ais\"al\"a in the affirmative.

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Manzi Huang, Xiantao Wang, Matti Vuorinen. 2012-09-20. On the subinvariance of uniform domains in Banach spaces. https://arxiv.org/abs/1209.4541

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