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October 2019 A reverse quasiconformal composition problem for $Q_\alpha(\mathbb{R}^n)$
Jie Xiao, Yuan Zhou
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Ark. Mat. 57(2): 451-469 (October 2019). DOI: 10.4310/ARKIV.2019.v57.n2.a11

Abstract

We give a partial converse to [8, Theorem 1.3] (as a resolution of [2, Problem 8.4] for the quasiconformal $Q$-composition) for $Q_{0 \lt \alpha \lt 2^{-1}} (\mathbb{R}^{n \geq 2})$, and yet demonstrate that if $f : \mathbb{R}^2 \to \mathbb{R}^2$ is a homeomorphism then the boundedness of $u \mapsto u \circ f$ on $Q_{2^{-1} \lt \alpha \lt 1} (\mathbb{R}^2) \subset BMO (\mathbb{R}^2)$ yields the quasiconformality of $f$.

Funding Statement

J.X. is supported by NSERC of Canada (# 202979463102000). Y.Z. is supported by AvH-foundation, and by the National Natural Science Foundation of China (# 11522102 & 11871088), respectively.

Citation

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Jie Xiao. Yuan Zhou. "A reverse quasiconformal composition problem for $Q_\alpha(\mathbb{R}^n)$." Ark. Mat. 57 (2) 451 - 469, October 2019. https://doi.org/10.4310/ARKIV.2019.v57.n2.a11

Information

Received: 6 March 2018; Revised: 4 April 2019; Published: October 2019
First available in Project Euclid: 16 April 2020

zbMATH: 07114515
MathSciNet: MR4018763
Digital Object Identifier: 10.4310/ARKIV.2019.v57.n2.a11

Subjects:
Primary: 30H25, 42B35, 46E30, 47B38

Rights: Copyright © 2019 Institut Mittag-Leffler

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