Asian Journal of Mathematics

Volume 23 (2019)

Number 5

The $Q_{ \alpha}$-restriction problem

Pages: 837 – 876



Z. Wang (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

J. Xiao (Department of Mathematics and Statistics, Memorial University, St. Johns, Newfoundland, Canada)

Y. Zhou (School of Mathematical Science, Beijing Normal University, Beijing, China)


Let $\alpha \in [0, 1)$ and $\Omega$ be an open connected subset of $\mathbb{R}^{n \geq 2}$. This paper shows that the $Q_{\alpha}$-restriction problem $Q_{\alpha} \vert {}_{\Omega} = \mathscr{Q}_{\alpha} (\Omega)$ is solvable if and only if $\Omega$ is an Ahlfors $n$-regular domain; i.e., $\operatorname{vol} \bigl ( B(x, r) \cap \Omega \bigr ) \gtrsim r^n$ for any Euclidean ball $B(x, r)$ with center $x \in \Omega$ and radius $r \in \bigl ( 0, \operatorname{diam} (\Omega) \bigr ) $ , thereby not only yielding an exponential $Q_{\alpha}$-integrability as a proper adjustment of the John–Nirenberg type inequality for $Q_{\alpha}$ conjectured in [3, Problem 8.1, (8.2)] but also resolving the quasiconformal extension problem for $Q_{\alpha}$ posed in [3, Problem 8.5].


$Q$ space, restriction, Ahlfors regular domain, uniform domain, Minkowski type dimension

2010 Mathematics Subject Classification

42B35, 46E35

Z. Wang and Y. Zhou were supported by the National Natural Science Foundation of China (#11871088). J. Xiao was supported by the NSERC of Canada.

Received 24 May 2018

Accepted 5 July 2018

Published 30 April 2020