{
  "abstract": "We establish the endpoint gradient estimate for centered homothetic maximal averages in every dimension $d\\ge2$ over each fixed convex body with $C^{1,θ}$ boundary, $0<θ\\le1$. In particular, the centered ball maximal operator satisfies the $W^{1,1}$ gradient bound in every dimension. Neither strict convexity nor a lower bound on the support gap is required. The geometric argument combines concentration in gradient values with nonhomogeneous calibrated rows and a summable radial correction. It extends the scalar refinement, direction-history construction and global assembly of the planar centered-disk proof of OpenAI. A separate transfer theorem gives sharp singular-derivative identities for bounded compact probability kernels once local BV output is known, and gradient continuity under an additional spatial-tail condition. Quantitative contact and translation estimates yield further profile and kernel examples. The estimates retain the fixed body's regularity data; the endpoint for arbitrary $C^1$ and nonsmooth convex bodies remains open here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35760",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhouyu Long"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1bw98z640xjzx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35760",
  "fulltext_url": null,
  "id": "1bw98z640xjzx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35760",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7224001511",
  "pdf_url": "https://arxiv.org/pdf/2610.35760",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1bw98z640xjzx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:43Z",
      "native_id": "2610.35760",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35760"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Endpoint gradient bounds for centered convex maximal operators and probability kernels",
  "title_text": "Endpoint gradient bounds for centered convex maximal operators and probability kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/1bw98z640xjzx",
  "venue": "arXiv",
  "versions": []
}