关于$\u003cspan style=\"font-style: italic;\"\u003eQ\u003c/span\u003e_α(\u003cspan style=\"font-style: italic;\"\u003eR\u003c/span\u003e^n)$上拟对称复合开问题的一个精确容量规解
A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on $\mathcal{Q}_α(\mathbb R^n)$
AI总结：
本文引入容量规这一新几何量，完整刻画了使复合算子在$\mathcal Q_α(\mathbb R^n)$上有界的拟对称映射，解决了2000年提出的长期开问题，并给出若干应用。
AI中文摘要：
临界指数范围$α\in(0,\min\{1,n/2\})$下的空间$\mathcal Q_α(\mathbb R^n)$构成一个严格介于常函数空间与$\mathrm{BMO}(\mathbb R^n)$之间的尺度不变族。由Essén、Janson、Peng和Xiao于2000年提出的一个长期悬而未决的问题，是刻画使得复合算子$\mathcal C_φ(f)=f\circφ^{-1}$在$\mathcal Q_α(\mathbb R^n)$上有界的拟对称映射$φ:\mathbb R^n\to\mathbb R^n$。本文通过引入一个新的几何量——容量规$\\|φ\\|_{\mathscr G_β}$，建立了完整的内蕴刻画，该容量规通过二进树上的拉回体积比来衡量$φ$的畸变。我们证明，对于任意拟对称映射$φ$（当$n=1$时，对$φ$和$φ^{-1}$的雅可比行列式施加温和的Muckenhoupt $A_\infty(\mathbb R)$假设），复合算子$\mathcal C_φ$在$\mathcal Q_α(\mathbb R^n)$上有界当且仅当$\\|φ\\|_{\mathscr G_{1-\frac{2α}{n}}}\u003c\infty$，且满足定量等价关系$\\|\mathcal C_φ\\|_{\mathcal Q_α(\mathbb R^n)\to\mathcal Q_α(\mathbb R^n)}^2 \simeq \\|φ\\|_{\mathscr G_{1-\frac{2α}{n}}}$。这一精确的充要刻画改进了Koskela、Xiao、Zhang和Zhou于2017年提出的早期充分性准则，后者是用雅可比行列式例外集的局部或全局自相似Minkowski维数来表述的。作为应用，我们得到了有限容量规的复合稳定性、$\mathcal Q_α$-可去性在具有有限容量规的拟对称映射下的不变性，以及由拟对称流驱动的输运方程的$\mathcal Q_α(\mathbb R^n)$-正则性传播和初值稳定性。
英文摘要：
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 characterize the quasisymmetric mappings $φ:\ \mathbb R^n\to\mathbb R^n$ for which the composition operator $\mathcal C_φ(f)=f\circφ^{-1}$ is bounded on $\mathcal Q_α(\mathbb R^n)$. This paper establishes the full intrinsic characterization by introducing a new geometric quantity, the capacity gauge $\|φ\|_{\mathscr G_β}$, which measures the distortion of $φ$ through pullback volume ratios along the dyadic tree. We prove that for any quasisymmetric mapping $φ$ (with a mild Muckenhoupt \(A_\infty(\mathbb R)\) assumption on the Jacobian determinants of \(φ\) and \(φ^{-1}\) when \(n=1\)), the composition operator $\mathcal C_φ$ is bounded on $\mathcal Q_α(\mathbb R^n)$ if and only if $\|φ\|_{\mathscr G_{1-\frac{2α}{n}}}<\infty,$ with quantitative equivalence $$ \|\mathcal C_φ\|_{\mathcal Q_α(\mathbb R^n)\to\mathcal Q_α(\mathbb R^n)}^2 \simeq \|φ\|_{\mathscr G_{1-\frac{2α}{n}}}. $$ 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 $\mathcal Q_α$-removability under quasisymmetric mappings with finite capacity gauge, and the propagation of $\mathcal Q_α(\mathbb R^n)$-regularity and initial-data stability for transport equations driven by quasisymmetric flows.
