{
  "abstract": "We prove a spectral multiplier theorem for the standard sub-Laplacian on the Engel group under a scale-invariant local $H^s$ condition with $s>2$. The multiplier operators $F(\\mathcal L)$ are of weak type $(1,1)$ and bounded on $L^p$ for every $1<p<\\infty$. Together with the known lower bound, this identifies the sharp threshold as $2$, half the topological dimension. The proof combines a weighted Plancherel estimate for quartic oscillators with a decomposition into rank-two spectral subspaces, which controls parameter derivatives uniformly as one well escapes to infinity. For fixed smooth multipliers compactly supported in $(0,\\infty)$, we also determine the leading term and first correction for the localized kernels. Whether the weak type $(1,1)$ estimate holds under the local $H^2$ condition remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08976",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhicheng Han"
    },
    {
      "name": "Zhipeng Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bh8h1pma4b4q6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08976",
  "fulltext_url": null,
  "id": "bh8h1pma4b4q6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08976",
  "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"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
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  "openalex": "W7220918212",
  "pdf_url": "https://arxiv.org/pdf/2610.08976",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bh8h1pma4b4q6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.08976",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08976"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A sharp spectral multiplier theorem on the Engel group",
  "title_text": "A sharp spectral multiplier theorem on the Engel group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/bh8h1pma4b4q6",
  "venue": "arXiv",
  "versions": []
}