On quasim\"obius maps in real Banach spaces
Manzi Huang, Yaxiang Li, Matti Vuorinen, Xiantao Wang

TL;DR
This paper proves that under certain conditions, a homeomorphism between domains in real Banach spaces extends to a quasim"obius map on the boundary, affirmatively resolving an open problem from 1991.
Contribution
It establishes a characterization of uniform domains via boundary extension and quasim"obius maps in real Banach spaces, solving an open problem from V"ais"al"a's 1991 work.
Findings
D' is uniform iff f extends to a boundary homeomorphism that is η-quasim"obius
Confirms the boundary extension property characterizes uniform domains in this setting
Addresses and resolves an open problem from V"ais"al"a (1991)
Abstract
Suppose that and denote real Banach spaces with dimension at least 2, that and are domains, that is an -CQH homeomorphism, and that is uniform. The main aim of this paper is to prove that is a uniform domain if and only if extends to a homeomorphism and is -QM relative to . This result shows that the answer to one of the open problems raised by V\"ais\"al\"a from 1991 is affirmative.
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Taxonomy
TopicsAnalytic and geometric function theory · Holomorphic and Operator Theory · Geometric and Algebraic Topology
