{
  "abstract": "In this article, we establish sharp quantitative weighted estimates for the lifted Hardy--Littlewood (resp Sobolev sharp) maximal operators $\\{\\mathcal M_θ\\}_{θ\\in(0,\\infty)}$ (resp $\\{\\mathcal M_θ^\\sharp\\}_{θ\\in(0,\\infty)}$). Specifically, for $p\\in(1,\\infty)$, $γ\\in\\mathbb R\\setminus\\{0\\}$, $\\upsilon\\in A_p$, and $f\\in L^p_\\upsilon$, we prove that \\begin{align*} \\sup_{θ,λ\\in(0,\\infty)}λ^p \\underset{\\mathcal M_θ(f)(x,t)>λt^γ} {\\int_{\\mathbb R^n}\\int_0^\\infty} t^{γp-1}\\,dt\\,\\upsilon(x)\\,dx \\lesssim [\\upsilon]_{A_p}^{α_{p,γ}} \\|f\\|_{L^p_\\upsilon}^p, \\end{align*} where $α_{p,γ}:=\\frac{p}{p-1}$ for $γ\\in[-n,0)$ and $α_{p,γ}:=1$ otherwise and where the implicit positive constant depends only on $n$, $p$, and $γ$. For $\\mathcal M_θ^\\sharp$, the corresponding estimate holds with $\\|f\\|_{L^p_\\upsilon}^p$ replaced by $\\|\\nabla f\\|_{L^p_\\upsilon}^p$ for any $f\\in\\dot W^{1,p}_\\upsilon$, and with $α_{p,γ}:=\\frac{p}{p-1}$ for $γ\\in[-1,0)$ and $α_{p,γ}:=1$ otherwise. We show that, for an $A_1$ weight $\\upsilon$, the weight constant in the above estimates becomes $[\\upsilon]_{A_1}$ for the sharp ranges $p\\in(1,\\infty)$ and $γ\\in\\mathbb R\\setminus\\{0\\}$, or $p=1$ and $γ\\in\\mathbb R\\setminus[-n,0]$ (resp. $γ\\in\\mathbb R\\setminus[-1,0]$). We also prove that $α_{p,γ}$ is sharp, and we establish exact limiting formulae and characterize Muckenhoupt weights via weak-type bounds. As applications, we characterize the weighted Hardy(--Sobolev) space $H^1_\\upsilon$ with $\\upsilon\\in A_1$ in terms of the truncated Riesz transforms with optimal range $γ\\in \\mathbb R\\setminus\\{0\\}$. More applications include the sharp weighted Caffarelli--Silvestre extension and the generalization to ball Banach function spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33998",
  "author_total": 4,
  "authors": [
    {
      "name": "Meng Geng"
    },
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Yirui Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cvvx5z43t83w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33998",
  "fulltext_url": null,
  "id": "1cvvx5z43t83w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33998",
  "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": "W7223833170",
  "pdf_url": "https://arxiv.org/pdf/2610.33998",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cvvx5z43t83w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33998",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33998"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Quantitative Weighted Estimates for Lifted Maximal Operators with Applications",
  "title_text": "Sharp Quantitative Weighted Estimates for Lifted Maximal Operators with Applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/1cvvx5z43t83w",
  "venue": "arXiv",
  "versions": []
}