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

[2608.27915] A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on $\mathcal{Q}_α(\mathbb R^n)$ Skip to main content Search Submit Donate Log in Mathematics > Classical Analysis and ODEs arXiv:2608.27915 (math) [Submitted on 28 Aug 2026] Title: A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on $\mathcal{Q}_α(\mathbb R^n)$ Authors: Liu Liguang , Xiao Jie View PDF HTML (experimental) 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 $\varphi:\ \mathbb R^n\to\mathbb R^n$ 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. Comments: 64 pages Subjects: Classical Analysis and ODEs (math.CA) Cite as: arXiv:2608.27915 [math.CA] (or arXiv:2608.27915v1 [math.CA] for this version) https://doi.org/10.48550/arXiv.2608.27915 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Liguang Liu [ view email ] [v1] Fri, 28 Aug 2026 04:42:56 UTC (53 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CA < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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