{
  "abstract": "In this paper, we resolve the convolution inequality problem posed by Naor and Schechtman in \\cite[Question 6.1]{N-S2016} (see also \\cite{Na2016}). More precisely, we establish the following strong convolution inequality on products of finite cyclic groups: For each $1\\leq p<\\infty$ there exists $c_{p}>0$ such that for every $n$, $m\\in \\mathbb{N}$ the following holds \\[ \\begin{split} &\\sum_{\\varepsilon\\in \\{-1,1\\}^{n}}\\sum_{x\\in \\Z}\\left|E_{\\{1,\\cdots,n\\}}f(x+\\varepsilon)-E_{\\{1,\\cdots,n\\}}f(x-\\varepsilon)\\right|^{p}\\\\ \\leq& c^{p}_{p}\\sum_{\\varepsilon\\in \\{-1,1\\}^{n}}\\sum_{x\\in \\Z}\\left|\\varepsilon_{j}\\left[E_{\\{1,\\cdots,n\\}\\setminus\\{j\\}}f(x+e_{j})-E_{\\{1,\\cdots,n\\}\\setminus\\{j\\}}f(x-e_{j})\\right]\\right|^{p}, \\end{split} \\] $E_{A}f(x)=\\frac{1}{2^{n}}\\sum_{δ\\in \\{-1,1\\}^{n}}f\\left(x+\\sum_{j\\in A}δ_{j}e_{j}\\right)$ for every $A\\subseteq \\{1,\\cdots,n\\}$. The growth order we obtain here $c_{p}=O(p)$ as $p\\to \\infty$. For the endpoint $p=\\infty$, we show by a counterexample that the strong convolution inequality fails for the $L_{\\infty}$, and we prove that the appropriate endpoint for $p=\\infty$ is the martingale $BMO$ space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15768",
  "author_total": 4,
  "authors": [
    {
      "name": "Yong Jiao"
    },
    {
      "name": "Sijie Luo"
    },
    {
      "name": "Dmitriy Zanin"
    },
    {
      "name": "Dejian Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/brp1a28z774xm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15768",
  "fulltext_url": null,
  "id": "brp1a28z774xm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15768",
  "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": "W7213351964",
  "pdf_url": "https://arxiv.org/pdf/2609.15768",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/brp1a28z774xm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.15768",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15768"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Strong Naor--Schechtman Convolution Inequality on the Product of Cyclic Groups",
  "title_text": "The Strong Naor--Schechtman Convolution Inequality on the Product of Cyclic Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/brp1a28z774xm",
  "venue": "arXiv",
  "versions": []
}