Quasimöbius invariance of uniform domains

Qingshan Zhou; Antti Rasila

doi:10.4064/SM191215-22-10

Quasimöbius invariance of uniform domains

Qingshan Zhou^*, Antti Rasila

^*Corresponding author for this work

Mathematics with Computer Science

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We study quasimöbius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as (diameter) uniformity, δ-uniformity and the min-max property. Then we show that all of these conditions are equivalent if the domain is ψ-natural. As applications, we partially answer an open question proposed by Väisälä, and provide a new method to prove a recent result of Huang et al. (2013), which also gives an answer to another question raised by Väisälä.

Original language	English
Pages (from-to)	1-24
Number of pages	24
Journal	Studia Mathematica
Volume	261
Issue number	1
DOIs	https://doi.org/10.4064/SM191215-22-10
State	Published - 2021

Keywords

(relative) quasimöbius mapping
Min-max property
Natural condition
Quasisymmetric mapping
Uniform domain

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AU - Rasila, Antti

N1 - Publisher Copyright: © Instytut Matematyczny PAN, 2021

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N2 - We study quasimöbius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as (diameter) uniformity, δ-uniformity and the min-max property. Then we show that all of these conditions are equivalent if the domain is ψ-natural. As applications, we partially answer an open question proposed by Väisälä, and provide a new method to prove a recent result of Huang et al. (2013), which also gives an answer to another question raised by Väisälä.

AB - We study quasimöbius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as (diameter) uniformity, δ-uniformity and the min-max property. Then we show that all of these conditions are equivalent if the domain is ψ-natural. As applications, we partially answer an open question proposed by Väisälä, and provide a new method to prove a recent result of Huang et al. (2013), which also gives an answer to another question raised by Väisälä.

KW - (relative) quasimöbius mapping

KW - Min-max property

KW - Natural condition

KW - Quasisymmetric mapping

KW - Uniform domain

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Quasimöbius invariance of uniform domains

Abstract

Keywords

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