{
  "abstract": "Let $d\\geq 2$ and $T^*_{Ω,b}$ be the commutator of the rough maximal singular integral on $\\mathbb{R}^d$ defined by $$T^*_{Ω,b}f(x)=\\sup_{\\varepsilon>0}\\left|\\int_{|x-y|>\\varepsilon}\\bigl(b(x)-b(y)\\bigr)\\frac{Ω(x-y)}{|x-y|^d}f(y)\\,dy\\right|.$$ Assume that $Ω\\in L(\\log L)^2(\\mathbb{S}^{d-1})$ has mean zero and that $b\\in\\operatorname{BMO}(\\mathbb{R}^d)$. We prove that, for every $λ>0$, \\[\\bigl|\\{x\\in\\mathbb{R}^d:T^*_{Ω,b}f(x)>λ\\}\\bigr|\\lesssim_{Ω,b}\\int_{\\mathbb{R}^d}\\frac{|f(x)|}λ\\log\\!\\left(e+\\frac{|f(x)|}λ\\right)\\,dx.\\] The argument starts from a Calderón-Zygmund decomposition of $f$ and a dyadic linearization of the maximal truncation. A decomposition of $Ω$ by size, followed by regularization of the spatial cutoffs and a microlocal decomposition, reduces the problem to a family of estimates with quantitative decay. The required decay follows by interpolating localized $L^1$ and $L^3$ bounds and using an analytic-family argument for the commutators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34311",
  "author_total": 2,
  "authors": [
    {
      "name": "Shenglong Lin"
    },
    {
      "name": "Qingying Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bzqffs4tk5d09.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34311",
  "fulltext_url": null,
  "id": "bzqffs4tk5d09",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34311",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214780115",
  "pdf_url": "https://arxiv.org/pdf/2609.34311",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bzqffs4tk5d09.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34311",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34311"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators",
  "title_text": "Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bzqffs4tk5d09",
  "venue": "arXiv",
  "versions": []
}