Skip to content
DraconiticResearch
Search Sources API
197,357 studies31 sourcesUpdated just now
PreprintarXiv licencearXiv

A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on 𝒬α(ℝn)

  • Liu Liguang
  • Xiao Jie
Open at sourcePDFDOICite

Abstract

The spaces 𝒬α(ℝn) for the critical index range α∈(0,min{1,n/2}) form a scale-invariant family lying strictly between the space of constant functions and BMO(ℝn). A long-standing open problem, posed by EssΓ©n, Janson, Peng and Xiao in 2000, asks to characterize the quasisymmetric mappings Ο†: ℝn→ℝn for which the composition operator π’žΟ†(f)=fβˆ˜Ο†βˆ’1 is bounded on 𝒬α(ℝn). This paper establishes the full intrinsic characterization by introducing a new geometric quantity, the capacity gauge |Ο†|𝒒β, which measures the distortion of Ο† through pullback volume ratios along the dyadic tree. We prove that for any quasisymmetric mapping Ο† (with a mild Muckenhoupt A∞(ℝ) assumption on the Jacobian determinants of Ο† and Ο†βˆ’1 when n=1), the composition operator π’žΟ† is bounded on 𝒬α(ℝn) if and only if |Ο†|𝒒1βˆ’2Ξ±/n<∞, with quantitative equivalence |π’žΟ†|𝒬α(ℝn)→𝒬α(ℝn)2 ≃ |Ο†|𝒒1βˆ’2Ξ±/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 𝒬α-removability under quasisymmetric mappings with finite capacity gauge, and the propagation of 𝒬α(ℝn)-regularity and initial-data stability for transport equations driven by quasisymmetric flows.

Related

  1. A Complete Solution to the Conjecture on the Inverse Collatz–Sinogowitz Irregularity Problem
    Xue Ji, Zhiwen Wang, Ji-Ming Guo
    Discrete Mathematics LettersPeer-reviewedCC BYFull textGeometry & topologyDOAJ
  2. A solution to Haagerup's Problem on decomposable maps
    Rui Okayasu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. A solution to Berndtsson's problem and uniqueness of twisted KE currents
    Yinji Li, Haoyuan Sun, Zhiwei Wang +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv
  4. Solution of Laczkovich's Lipschitz mapping problem
    Attila GΓ‘spΓ‘r
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  5. A solution to Michael's problem for FrΓ©chet algebras
    Antonio Acuaviva
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{liguang2026sharp,
  title = {{A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on $\mathcalQ_Ξ±(\mathbb R^n)$}},
  author = {Liu Liguang and Xiao Jie},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27915},
  eprint = {2608.27915},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27915},
}
RIS
TY  - UNPB
TI  - A Sharp Capacity Gauge Solution to the Open Problem of Quasisymmetric Composition on 𝒬_Ξ±(ℝ^n)
AU  - Liu Liguang
AU  - Xiao Jie
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27915
UR  - https://arxiv.org/abs/2608.27915
AB  - The spaces 𝒬_Ξ±(ℝ^n) for the critical index range α∈(0,min{1,n/2}) form a scale-invariant family lying strictly between the space of constant functions and BMO(ℝ^n). A long-standing open problem, posed by EssΓ©n, Janson, Peng and Xiao in 2000, asks to characterize the quasisymmetric mappings Ο†: ℝ^n→ℝ^n for which the composition operator π’ž_Ο†(f)=fβˆ˜Ο†^βˆ’1 is bounded on 𝒬_Ξ±(ℝ^n). This paper establishes the full intrinsic characterization by introducing a new geometric quantity, the capacity gauge |Ο†|_𝒒_Ξ², which measures the distortion of Ο† through pullback volume ratios along the dyadic tree. We prove that for any quasisymmetric mapping Ο† (with a mild Muckenhoupt A_∞(ℝ) assumption on the Jacobian determinants of Ο† and Ο†^βˆ’1 when n=1), the composition operator π’ž_Ο† is bounded on 𝒬_Ξ±(ℝ^n) if and only if |Ο†|_𝒒_1βˆ’2Ξ±/n<∞, with quantitative equivalence |π’ž_Ο†|_𝒬_Ξ±(ℝ^n)→𝒬_Ξ±(ℝ^n)^2 ≃ |Ο†|_𝒒_1βˆ’2Ξ±/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 𝒬_Ξ±-removability under quasisymmetric mappings with finite capacity gauge, and the propagation of 𝒬_Ξ±(ℝ^n)-regularity and initial-data stability for transport equations driven by quasisymmetric flows.
ER  -