A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on $\mathcal{Q}_\alpha(\mathbb R^n)$

Abstract

The spaces $\mathcal Q_\alpha(\mathbb R^n)$ for the critical index range $\alpha\in(0,\min\{1,n/2\})$ form a scale-invariant family lying strictly between the space of constant functions and $\mathrm{BMO}(\mathbb R^n)$. A long-standing open problem, posed by Ess\'en, Janson, Peng and Xiao in 2000, asks to characterize the quasisymmetric mappings $\varphi:\ \mathbb R^n\to\mathbb R^n$ for which the composition operator $\mathcal C_\varphi(f)=f\circ\varphi^{-1}$ is bounded on $\mathcal Q_\alpha(\mathbb R^n)$. This paper establishes the full intrinsic characterization by introducing a new geometric quantity, the capacity gauge $\|\varphi\|_{\mathscr G_\beta}$, which measures the distortion of $\varphi$ through pullback volume ratios along the dyadic tree. We prove that for any quasisymmetric mapping $\varphi$ (with a mild Muckenhoupt \(A_\infty(\mathbb R)\) assumption on the Jacobian determinants of \(\varphi\) and \(\varphi^{-1}\) when \(n=1\)), the composition operator $\mathcal C_\varphi$ is bounded on $\mathcal Q_\alpha(\mathbb R^n)$ if and only if $\|\varphi\|_{\mathscr G_{1-\frac{2\alpha}{n}}}<\infty,$ with quantitative equivalence $$ \|\mathcal C_\varphi\|_{\mathcal Q_\alpha(\mathbb R^n)\to\mathcal Q_\alpha(\mathbb R^n)}^2 \simeq \|\varphi\|_{\mathscr G_{1-\frac{2\alpha}{n}}}. $$ This sharp, necessary and sufficient characterization refines earlier sufficient criteria of Koskela, Xiao, Zhang and Zhou in 2017, which were formulated in terms of local or global self-similar Minkowski dimension of the exceptional sets of the Jacobian. As applications, we obtain the composition stability of the finite capacity gauge, the invariance of $\mathcal Q_\alpha$-removability under quasisymmetric mappings with finite capacity gauge, and the propagation of $\mathcal Q_\alpha(\mathbb R^n)$-regularity and initial-data stability for transport equations driven by quasisymmetric flows.