{
  "abstract": "In this present paper, we study the weighted weak type $(1,\\,1)$ bounds for rough maximal singular integral operators as well as the the corresponding variation and jump operators. Our first result is the weighted weak type $(1,\\,1)$ bound for the maximal singular integral $$T_Ω^{*}f(x)=\\sup\\limits_{\\varepsilon>0}|T_{\\varepsilon,Ω}f(x)|=\\sup\\limits_{\\varepsilon>0}\\bigg|\\int_{|x-y|>\\varepsilon} \\frac{Ω(x-y)}{|x-y|^{d}}f(y)dy\\bigg|,$$ where $Ω\\in L^\\infty(\\mathbb{S}^{d-1})$, homogeneous of degree zero, and satisfies the cancellation condition. We show that $$\\|T_Ω^{*}\\|_{L^1(w)\\rightarrow L^{1,\\infty}(w)}\\lesssim[w]_{A_1}[w]_{A_\\infty}\\log([w]_{A_\\infty}+1),$$ where $w$ belongs to the Muckenhoupt class $A_1(\\mathbb{R}^d)$. This represents an essential improvement of a result (Honz\\'ık, Inter.Math. Res. Not. 2020) and a result (Bhojak and Mohanty, J. Funct. Anal.2023). Our second one is the weighted weak type $(1,\\,1)$ bounds for variation and jump operators corresponding to $\\{T_{\\varepsilon,Ω}\\}_{\\varepsilon\\in2^{\\mathbb{Z}}}$ and $\\{T_{\\varepsilon, Ω}^ϕ\\}_{\\varepsilon\\in\\mathbb{R}^{+}}$ under the condition $Ω\\in L^\\infty(\\mathbb{S}^{d-1})$, where $T_{ε,Ω}^ϕ$ represents a smooth truncation of rough singular integral operator.These results of this part are the first weighted weak type $(1,\\,1)$ variation inequalities and jump inequalities for rough singular integrals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12361",
  "author_total": 3,
  "authors": [
    {
      "name": "Ting Chen"
    },
    {
      "name": "Feng Liu"
    },
    {
      "name": "Zhou Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dmdpvy07s2x0e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12361",
  "fulltext_url": null,
  "id": "dmdpvy07s2x0e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12361",
  "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": "W7213150426",
  "pdf_url": "https://arxiv.org/pdf/2609.12361",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dmdpvy07s2x0e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12361",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12361"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Weighted Endpoint Variational and Jump Inequalites for Rough Singular Integrals",
  "title_text": "Weighted Endpoint Variational and Jump Inequalites for Rough Singular Integrals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/dmdpvy07s2x0e",
  "venue": "arXiv",
  "versions": []
}