{
  "abstract": "Eldan's level-$1{:}3$ inequality is a central tool in his second-order correlation inequalities for increasing Boolean functions. His proof is going to Gaussian space and uses an entropy estimate for second Hermite tensors. We give a direct proof on the discrete hypercube. For increasing real-valued functions $f,g:\\{0,1\\}^n\\to\\mathbb R$, we prove $$ W_3[f,g]\\ge -2W_1[f,g]\\log\\!\\left(\\frac{\\sqrt{\\mathrm{Var}(f)\\mathrm{Var}(g)}}{W_1[f,g]}\\right). $$ The proof combines complex hypercontractivity with the discrete Riesz transforms. We also give a short proof of a level-$1:d$ inequality obtained by Keller and Kindler. As an application, we recover Eldan's correlation inequality with a slight improvement. If $f,g:\\{0,1\\}^n\\to\\{0,1\\}$ are increasing and $g$ is antipodal, then $$ \\mathrm{Cov}(f,g)\\ge \\frac1{24}\\,\\frac{\\sum_i\\mathrm{Inf}_i[f]\\mathrm{Inf}_i[g]}{\\sqrt{\\log\\!\\left(4e\\sqrt{\\mathrm{Var}(f)\\mathrm{Var}(g)}/(\\sum_i\\mathrm{Inf}_i[f]\\mathrm{Inf}_i[g])\\right)}}, $$ where the square-root logarithm cannot be improved as shown in the OR--majority example.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08289",
  "author_total": 4,
  "authors": [
    {
      "name": "Fan Chang"
    },
    {
      "name": "Zhenyu Li"
    },
    {
      "name": "Guanghui Wang"
    },
    {
      "name": "Mingrui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7qse5c88ma3q6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08289",
  "fulltext_url": null,
  "id": "7qse5c88ma3q6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08289",
  "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.CO"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08289",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7qse5c88ma3q6.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08289",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08289"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Fourier level inequalities on the discrete cube via complex hypercontractivity and Riesz transforms",
  "title_text": "Fourier level inequalities on the discrete cube via complex hypercontractivity and Riesz transforms",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:49Z",
  "url": "https://research.draconitic.com/paper/7qse5c88ma3q6",
  "venue": "arXiv",
  "versions": []
}