On the subinvariance of uniform domains in Banach spaces

TL;DR

This paper proves that under freely quasiconformal mappings, the image of a uniform subdomain remains uniform in Banach spaces, confirming a conjecture and advancing understanding of domain invariance properties.

Contribution

It establishes the subinvariance of uniform domains under freely quasiconformal maps in Banach spaces, solving an open problem posed by V"ais"al"a.

Findings

The image of a uniform subdomain under a freely quasiconformal map is uniform.

The result holds in Banach spaces of dimension at least 2.

It confirms the subinvariance property for uniform domains.

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.