{
  "abstract": "We establish sharp weak-type bounds for commutators of Calderón--Zygmund operators and $\\text{BMO}(\\mathbb{R}^n)$ symbols with respect to $A_p$ weights. We prove that the bound \\looseness=-1 $$ \\|[b,T]\\|_{L^p(w)\\rightarrow L^{p,\\infty}(w)}\\lesssim [w]_{A_p}^{\\max(2,p')} $$ for $p \\in (1,\\infty)$ and $w \\in A_p$ is sharp for general $T \\in \\text{CZO}(\\mathbb{R}^n)$ and $b \\in \\text{BMO}(\\mathbb{R}^n)$. In the case of the standard logarithmic symbol, we show that the estimate improves to $$ \\|[\\log|\\cdot|,T]\\|_{L^p(w)\\rightarrow L^{p,\\infty}(w)}\\lesssim [w]_{A_p}^{\\max\\big(2,\\frac{1}{p}+\\frac{p'}{p}\\big)} $$ and that this dependence is optimal for this symbol. Our upper bound for the logarithmic symbol follows from the pointwise control of the commutator by a sum of the Hardy--Littlewood maximal operator and the squares of the radial Hardy operator and its adjoint, while our lower bounds are consequences of a general result for singular integral operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31493",
  "author_total": 3,
  "authors": [
    {
      "name": "Adam Mair"
    },
    {
      "name": "Óscar Andrés Ramírez"
    },
    {
      "name": "Cody B. Stockdale"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/52v685kbav3f4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31493",
  "fulltext_url": null,
  "id": "52v685kbav3f4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31493",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214615363",
  "pdf_url": "https://arxiv.org/pdf/2609.31493",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/52v685kbav3f4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31493",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31493"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp weighted weak-type bounds for commutators of singular integrals",
  "title_text": "Sharp weighted weak-type bounds for commutators of singular integrals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/52v685kbav3f4",
  "venue": "arXiv",
  "versions": []
}