{
  "abstract": "Let $d \\ge 2$, let $α_1,\\dots, α_n > 0$, and put $A = \\sum_k α_k$. We determine, up to constants depending only on $d$, the smallest possible $L^1$ norm in the unit ball $\\mathbb{B}^d\\subset \\mathbb{R}^d$ of the Coulomb field generated by charges of strengths $α_k$ placed on the unit sphere $\\mathbb{S}^{d-1}$. More precisely, we prove \\[ \\inf_{x_1,\\dots,x_n \\in \\mathbb{S}^{d-1}} \\int_{\\mathbb{B}^d} \\left| \\sum_{k=1}^n α_k \\frac{x_k-x}{|x_k - x|^d} \\right| dx \\asymp_d A^{-1/(d-1)} \\sum_{k=1}^n α_k^{d/(d-1)}. \\] The lower bound follows from an $L^1$ Lipschitz trace estimate and an explicit weighted tent function on the sphere. For the upper bound we use a partition of the sphere into cells of masses $α_k/A$ and diameters $O_d \\left((α_k/A)^{1/(d-1)} \\right)$. For the unit weights the sharp order is $n^{(d-2)/(d-1)}$, and in dimension three it is $\\sqrt{n}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35123",
  "author_total": 3,
  "authors": [
    {
      "name": "Evgueni Doubtsov"
    },
    {
      "name": "Ioann Vasilyev"
    },
    {
      "name": "Grigory Voinov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8zb8en4zxmj2z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35123",
  "fulltext_url": null,
  "id": "8zb8en4zxmj2z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35123",
  "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": "W7214776313",
  "pdf_url": "https://arxiv.org/pdf/2609.35123",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8zb8en4zxmj2z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35123",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35123"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp weighted $L^1$ bounds for Coulomb fields on the sphere",
  "title_text": "Sharp weighted L^1 bounds for Coulomb fields on the sphere",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/8zb8en4zxmj2z",
  "venue": "arXiv",
  "versions": []
}