{
  "abstract": "Let $1<p<2$ and $D_j$ be the discrete partial derivative on the Hamming cube $Ω_n = \\{ -1, 1\\}^n$. Let $Δ=\\sum_{j=1}^nD_j$ be the discrete Laplacian. We prove the endpoint inequality \\[ \\left\\|\\left(\\sum_{j=1}^n|D_jf|^2\\right)^{1/2}\\right\\|_{p} \\lesssim_p\\|Δ^{1/p}f\\|_{p}, \\quad \\forall f:Ω_n \\to \\mathbb C. \\] This result answers the conjecture proposed by Naor, Eskenazis and Ivanisvili (see [BenEfraimLustPiquard, IvanisviliVolberg] or [Remark 45, EskenazisIvanisvili]). The proof relies heavily on the noncommutative semigroup BMO theory [JungeMei].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03993",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhendong Xu"
    },
    {
      "name": "Hao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b8pzw4m2mn16d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03993",
  "fulltext_url": null,
  "id": "b8pzw4m2mn16d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03993",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7208791183",
  "pdf_url": "https://arxiv.org/pdf/2609.03993",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b8pzw4m2mn16d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2609.03993",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03993"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Endpoint Fractional Riesz Estimate on the Hamming Cube",
  "title_text": "The Endpoint Fractional Riesz Estimate on the Hamming Cube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/b8pzw4m2mn16d",
  "venue": "arXiv",
  "versions": []
}