{
  "abstract": "We study endpoint Schatten class properties of commutators $[I_θ^ε,M_x]$ on quantum tori of dimension $d\\geq2$, where $I_θ^ε$ is the Riesz potential of order $ε$ and $M_x$ denotes the left multiplication operator by an operator $x$ on quantum tori $\\mathbb T_θ^d$. For $-\\frac{d}{2}<ε<1$, $ε\\neq0$, and $x\\in L_1(\\mathbb T_θ^d)$, we prove that $[I_θ^ε,M_x]\\in \\mathcal L_{\\frac{d}{1-ε},\\infty}$ if and only if $x\\in\\dot W_{\\frac{d}{1-ε}}^1(\\mathbb T_θ^d)$. We also determine the singular value asymptotics of this commutator for $x\\in\\dot W_{\\frac{d}{1-ε}}^1(\\mathbb T_θ^d)$. The necessary Sobolev condition holds throughout $1-d<ε<1$, $ε\\neq0$. For $ε\\leq-\\frac{d}{2}$, we characterise Hilbert--Schmidt membership, with a logarithmically weighted Fourier condition at $ε=-\\frac{d}{2}$. In order to get the singular value asymptotics of commutators, we prove singular value asymptotics for pseudodifferential operators of arbitrary negative order on quantum tori using $C^*$-algebraic principal symbols. This asymptotic result confirms Conjecture~7.1 of McDonald and Ponge (\\emph{Adv. Math.} 412 (2023), 108815).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16747",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinyu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33atre6yyn462.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16747",
  "fulltext_url": null,
  "id": "33atre6yyn462",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16747",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7221775873",
  "pdf_url": "https://arxiv.org/pdf/2610.16747",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33atre6yyn462.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16747",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16747"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Endpoint Schatten class properties of commutators on quantum tori",
  "title_text": "Endpoint Schatten class properties of commutators on quantum tori",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/33atre6yyn462",
  "venue": "arXiv",
  "versions": []
}