A quasiconformal composition problem for the Q-spaces

TL;DR

This paper investigates how quasiconformal mappings affect the boundedness of composition operators on Q-spaces, revealing new phenomena and establishing sharp conditions based on the Jacobian's degeneracy set and Minkowski dimension.

Contribution

It provides sharp criteria for the boundedness of composition operators on Q-spaces under quasiconformal maps, solving a known problem and uncovering phenomena different from classical function spaces.

Findings

Boundedness depends on the index α and the Jacobian's degeneracy set.

Tukia-V"ais"al"a's extension preserves Q-spaces for certain parameters.

Q-spaces are invariant under inversions for all 0<α<1.

Abstract

Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\bf C}_f$ on the spaces $Q_{\alpha}(\mathbb R^n)$ depends on the index $\alpha$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $\alpha$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently, Tukia-V\"ais\"al\"a's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of an arbitrary quasisymmetric mapping $g:\mathbb R^{n-p}\to \mathbb R^{n-p}$ is shown to preserve $Q_{\alpha} (\mathbb R^n)$ for any $(\alpha,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}$. Moreover, $Q_{\alpha}(\mathbb R^n)$ is shown to be invariant under inversions for all $0<\alpha<1$.