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

The spaces $\mathcal Q_α(\mathbb R^n)$ for the critical index range $α\in(0,\min\{1,n/2\})$ 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 chara