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

10 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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JO - Studia Mathematica

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Quasimöbius invariance of uniform domains

Abstract

Keywords

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