{
  "abstract": "In this paper we investigate the optimal dimension-dependent estimates of the second-order discrete Riesz transforms \\[ R_{\\mathrm{dis}}^{(jk)}f(n) = c_d\\sum_{m\\in\\mathbb Z^d\\setminus\\{0\\}} \\frac{m_jm_k}{|m|^{d+2}}f(n-m), \\qquad c_d=\\frac{Γ\\left(\\frac{d+2}{2}\\right)}{π^{d/2}}. \\] For $j\\neq k$ and every fixed $1<p<\\infty$, we prove that \\[ \\|R_{\\mathrm{dis}}^{(jk)}\\|_{\\ell^p\\to\\ell^p} = c_d\\left[ \\frac{2}{2^{d/2}} + \\left(\\frac83+o(1)\\right)\\frac{d}{3^{d/2}} \\right] \\] and \\[ \\frac{2c_d}{2^{d/2}} \\leq \\|R_{\\mathrm{dis}}^{(jk)}\\|_{\\ell^1\\to\\ell^{1,\\infty}} \\leq c_d\\left[ \\frac{2}{2^{d/2}} + \\left(\\frac83+o(1)\\right)\\frac{d}{3^{d/2}} \\right]. \\] Since $ c_d\\sim \\sqrt{πd}\\left(\\frac{d}{2πe}\\right)^{d/2} $ by Stirling's formula, thl $\\ell^p$ estimates give a negative answer to the conjecture proposed by Bañuelos and Kim in \\cite{BK2}. The diagonal case exhibits quite a different phenomenon: for every $j$ and every $1<p<\\infty$, $R_{\\mathrm{dis}}^{(jj)}$ is neither bounded from $\\ell^p(\\mathbb Z^d)$ to $\\ell^p(\\mathbb Z^d)$ nor of weak type $(1,1)$. Cancellation is restored for the operators $R_{\\mathrm{dis}}^{(jj-kk)} =R_{\\mathrm{dis}}^{(jj)}-R_{\\mathrm{dis}}^{(kk)}$. For every fixed $1<p<\\infty$, \\[ \\|R_{\\mathrm{dis}}^{(jj-kk)}\\|_{\\ell^p\\to\\ell^p} = 4c_d\\left[ 1+(1+o(1))\\frac{d}{2^{d/2}} \\right], \\] and \\[ 4c_d \\leq \\|R_{\\mathrm{dis}}^{(jj-kk)}\\|_{\\ell^1\\to\\ell^{1,\\infty}} \\leq 4c_d\\left[ 1+(1+o(1))\\frac{d}{2^{d/2}} \\right]. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09770",
  "author_total": 2,
  "authors": [
    {
      "name": "Hanli Tang"
    },
    {
      "name": "Zewei Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5yw92495sshpg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09770",
  "fulltext_url": null,
  "id": "5yw92495sshpg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09770",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7212164164",
  "pdf_url": "https://arxiv.org/pdf/2609.09770",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5yw92495sshpg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09770",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09770"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal dimension-dependent $\\ell^p$ and $\\ell^{1,\\infty}$ estimates of the second-order discrete Riesz transforms",
  "title_text": "Optimal dimension-dependent ℓ^p and ℓ^1,∞ estimates of the second-order discrete Riesz transforms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/5yw92495sshpg",
  "venue": "arXiv",
  "versions": []
}