{
  "abstract": "Let $Ω\\in L^1(S^{d-1})$ have vanishing average, and let $T_Ω^\\ast$ be the maximal truncation of the associated rough homogeneous singular integral. We prove quantitative sparse bounds for $T_Ω^\\ast$. If $Ω\\in L^\\infty(S^{d-1})$, then, for every $1<p<\\infty$, \\[ \\|T_Ω^\\ast\\|_{(1,p)\\text{-}\\mathrm{sparse}} \\lesssim_d p'\\|Ω\\|_{L^\\infty(S^{d-1})}. \\] For unbounded angular kernels, if $1<q<\\infty$ and $Ω\\in L^{q,1}\\log L(S^{d-1})$, then the same estimate holds for $q'\\leq p<\\infty$, with the right-hand side replaced by \\[ C_{d,q}p' \\|Ω\\|_{L^{q,1}\\log L(S^{d-1})}. \\] These estimates retain a genuine $L^1$ average in the first entry of the sparse form. In the bounded-kernel case, the upper bound has the same linear growth in $p'$ as the known sparse bound for the nonmaximal operator. The unbounded-kernel estimate includes the critical exponent $p=q'$. As a consequence, we obtain weighted weak-type $(1,1)$ estimates for all $A_1$ weights in the bounded-kernel case and for weights in $A_1\\cap RH_{q'}$ in the unbounded-kernel case. The proof combines physical-space linearization and microlocal decomposition with localized sparse testing. An amplitude decomposition of the second input, together with the Rademacher--Menshov inequality, yields the quantitative dependence on $p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03751",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuhao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpzdwbc3nnr9y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03751",
  "fulltext_url": null,
  "id": "fpzdwbc3nnr9y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03751",
  "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": "W7208720544",
  "pdf_url": "https://arxiv.org/pdf/2609.03751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpzdwbc3nnr9y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.03751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03751"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sparse bounds for maximal rough singular integrals",
  "title_text": "Sparse bounds for maximal rough singular integrals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fpzdwbc3nnr9y",
  "venue": "arXiv",
  "versions": []
}