{
  "abstract": "Let $U,V$ be matrix $\\mathcal A_p$ weights, $1<p<\\infty$, and let $B$ be a matrix-valued function. For Calderón--Zygmund operators $T$ with scalar standard kernels, we characterize boundedness and compactness of $[M_B,T\\otimes I_m]:L^p(U)\\to L^p(V)$ by two-matrix BMO and vanishing oscillation conditions. The converse statements require non-degeneracy condition. For matrix \\(\\mathcal A_2\\) weights \\(U,V\\) and a fixed Riesz transform \\(R_j\\) on \\(\\mathbb R^d\\), \\(d\\geq2\\), we characterize the Schatten class membership of \\(C_{B,R_j}:L^2(U)\\to L^2(V)\\). We characterize $\\mathcal S^r$ membership for $r>d$ and $\\mathcal S^{d,\\infty}$ membership by $\\ell^r$ and weak $\\ell^d$ conditions on the two Hilbert--Schmidt cube oscillations. For $0<r\\leq d$, $\\mathcal S^r$ membership is equivalent to $B$ being constant almost everywhere. The weak endpoint condition implies a weighted first-order Sobolev estimate for $C^1$ symbols, with a converse for constant or uniformly elliptic weights.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17054",
  "author_total": 4,
  "authors": [
    {
      "name": "Ji Li"
    },
    {
      "name": "Chong-Wei Liang"
    },
    {
      "name": "Chun-Yen Shen"
    },
    {
      "name": "Brett D. Wick"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6h2nffyxgnbq7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17054",
  "fulltext_url": null,
  "id": "6h2nffyxgnbq7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17054",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.17054",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6h2nffyxgnbq7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17054",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17054"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Commutators with two matrix weights",
  "title_text": "Commutators with two matrix weights",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/6h2nffyxgnbq7",
  "venue": "arXiv",
  "versions": []
}