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 Essén, Janson, Peng and Xiao in 2000, asks to characterize the quasisymmetric mappings $\phi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ for which the composition operator ${\mathcal{C}}_{\phi }(f)=f\circ {\phi }^{-1}$ 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 $\Vert \phi {\Vert }_{{\mathcal{G}}_{\beta }}$, which measures the distortion of $\phi$ through pullback volume ratios along the dyadic tree. We prove that for any quasisymmetric mapping $\phi$ (with a mild Muckenhoupt \(A_\infty(\mathbb R)\) assumption on the Jacobian determinants of \(φ\) and \(φ^{-1}\) when \(n=1\)), the composition operator ${\mathcal{C}}_{\phi }$ is bounded on ${\mathcal{Q}}_{\alpha }({\mathbb{R}}^n)$ if and only if $\Vert \phi {\Vert }_{{\mathcal{G}}_{1-\frac{2\alpha }{n}}}<\infty ,$ with quantitative equivalence $$\Vert {\mathcal{C}}_{\phi }{\Vert }_{{\mathcal{Q}}_{\alpha }({\mathbb{R}}^n)\to {\mathcal{Q}}_{\alpha }({\mathbb{R}}^n)}^2\simeq \Vert \phi {\Vert }_{{\mathcal{G}}_{1-\frac{2\alpha }{n}}}.$$ 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.

64 pages