{
  "abstract": "Let $Ω_{n}=\\{-1,1\\}^n$ be the $n$-dimensional hypercube equipped with the normalized uniform measure, let $\\nabla$ be the Walsh gradient and let $Δ$ be the Walsh Laplacian. For every $1<p\\leq 2$ we prove the following estimate \\[ \\|\\nabla f\\|_{L_p(Ω_n;\\ell_2^n)} \\leq c_{\\rm abs}(p-1)^{-2}\\|Δ^{1/p}f\\|_{L_p(Ω_n)}. \\] The exponent $\\frac1p$ is optimal, thus this settles the open problem on the sharp fractional Riesz estimate by Efraim and Lust-Piquard \\cite{E-LP2008} which was subsequently highlighted by Ivanisvili and Volberg \\cite{I-V2022}. We also establish the higher-order counterpart. As applications of our results, we obtain simpler proofs of the optimal short-time estimate for $\\nabla e^{-tΔ}$, and the Bernstein-Markov type inequality for $d$-bounded degree functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09040",
  "author_total": 4,
  "authors": [
    {
      "name": "Yong Jiao"
    },
    {
      "name": "Sijie Luo"
    },
    {
      "name": "Dmitriy Zanin"
    },
    {
      "name": "Dejian Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7y1xpjsfqfd6d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09040",
  "fulltext_url": null,
  "id": "7y1xpjsfqfd6d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09040",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7212034150",
  "pdf_url": "https://arxiv.org/pdf/2609.09040",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7y1xpjsfqfd6d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.09040",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09040"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Fractional Riesz Estimates on the Hypercube",
  "title_text": "Sharp Fractional Riesz Estimates on the Hypercube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/7y1xpjsfqfd6d",
  "venue": "arXiv",
  "versions": []
}