A reverse quasiconformal composition problem for $Q_\alpha(\mathbb{R}^n)$

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 $f$.

Keywords

quasi-conformality, composition, Essén–Janson–Peng–Xiao’s space, reverse

2010 Mathematics Subject Classification

30H25, 42B35, 46E30, 47B38

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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.

Received 6 March 2018

Received revised 4 April 2019

Accepted 17 April 2019

Published 7 October 2019