{
  "abstract": "We study sparse Kahane--Salem--Zygmund constructions and weighted Hardy--Littlewood inequalities for homogeneous polynomials. For supports of cardinality $n^{d+o(1)}$, we determine the sharp power of $n$ governing the smallest norm of a unimodular $m$-linear form on $\\ell_{p_1}^n\\times\\cdots\\times\\ell_{p_m}^n$; in the diagonal case, this yields the missing polynomial growth exponent in the coefficient-versus-supremum norm problem for $2\\le p\\le m$ and $2\\le r\\le\\infty$. We then introduce a diagonal weighted Hardy--Littlewood functional which, on $s=p\\ge m$, agrees exactly with the classical Hardy--Littlewood coefficient norm with the same optimal constant. We determine the optimal diagonal weight exponent for $2\\le p\\le m$ and $1\\le q\\le2$, on the full critical line $p=m$, and on a sharp part of the region $q>2$; at $q=\\infty$ the optimal weight exponent is obtained for every $2\\le p\\le m$. The sparse coefficient estimates provide the matching dimensional obstructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01500",
  "author_total": 4,
  "authors": [
    {
      "name": "Anderson Barbosa"
    },
    {
      "name": "Daniel Núñez-Alarcón"
    },
    {
      "name": "Anselmo Raposo"
    },
    {
      "name": "Joedson Santos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/be6j1mnbq7gsx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01500",
  "fulltext_url": null,
  "id": "be6j1mnbq7gsx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01500",
  "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": "W7216177484",
  "pdf_url": "https://arxiv.org/pdf/2610.01500",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/be6j1mnbq7gsx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01500",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01500"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sparse Kahane--Salem--Zygmund Forms and Weighted Hardy--Littlewood Inequalities Across the Critical Endpoint",
  "title_text": "Sparse Kahane--Salem--Zygmund Forms and Weighted Hardy--Littlewood Inequalities Across the Critical Endpoint",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/be6j1mnbq7gsx",
  "venue": "arXiv",
  "versions": []
}