{
  "abstract": "Let $n\\ge2$, $α\\in(0,n)$, and $μ$ be a locally finite nonnegative Borel measure on $\\mathbb{R}^n$. We prove that, for any $f\\in C_{\\rm c}^\\infty(\\mathbb{R}^n)$, \\begin{align*} \\|I_αf\\|_{L^{\\frac{n}{n-α},1}(μ)} \\le C_{n,α} \\int_{\\mathbb{R}^n} |Rf(x)|[Mμ(x)]^{1-\\fracα{n}}\\,dx, \\end{align*} where $I_α$ is the Riesz potential, $R$ is the Riesz transform, $Mμ$ is the Hardy--Littlewood maximal function of $μ$, and $C_{n,α}$ is a positive constant depending only on $n$ and $α$. This answers a question posed by Bortz, Moen, Olivo, Pérez, and Rela, with the case $α=1$ already known, and extends Spector's optimal unweighted Lorentz inequality to general measures. As applications, we obtain a Lorentz estimate for curl-free vector fields, Sobolev--Lorentz and Hardy inequalities with respect to general measures, and an $A_1$-weighted Lorentz estimate with a sharp exponent of the $A_1$ characteristic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23154",
  "author_total": 3,
  "authors": [
    {
      "name": "Pingxu Hu"
    },
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Wen Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/825m18dz21f48.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23154",
  "fulltext_url": null,
  "id": "825m18dz21f48",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23154",
  "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": "W7222987616",
  "pdf_url": "https://arxiv.org/pdf/2610.23154",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/825m18dz21f48.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23154",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23154"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Endpoint Lorentz Inequality for Riesz Potentials with Respect to General Measures",
  "title_text": "Endpoint Lorentz Inequality for Riesz Potentials with Respect to General Measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/825m18dz21f48",
  "venue": "arXiv",
  "versions": []
}