{
  "abstract": "Let $Δ_{\\rm d}$ be the nonnegative discrete Laplacian on $\\mathbb Z^d$. We prove that, for $0<p\\le1$, \\[ \\|Δ_{\\rm d}^{iγ}\\|_{H^p_{Δ_{\\rm d}}\\to H^p_{Δ_{\\rm d}}} \\simeq (1+|γ|)^{d(1/p-1/2)}, \\] and, for $1<p<\\infty$, \\[ \\|Δ_{\\rm d}^{iγ}\\|_{\\ell^p\\to\\ell^p} \\simeq (1+|γ|)^{d|1/p-1/2|}. \\] The estimates are sharp with respect to the exponent of $1+|γ|$. We also explain why the general spectral multiplier theorem and the usual Calderón--Zygmund approach do not give the sharp power.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27385",
  "author_total": 3,
  "authors": [
    {
      "name": "The Anh Bui"
    },
    {
      "name": "The Quan Bui"
    },
    {
      "name": "Thanh Tinh Cao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eb2j5h9yjfbh3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27385",
  "fulltext_url": null,
  "id": "eb2j5h9yjfbh3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27385",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7223153936",
  "pdf_url": "https://arxiv.org/pdf/2610.27385",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eb2j5h9yjfbh3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27385",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27385"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp estimates for imaginary powers of discrete Laplacians",
  "title_text": "Sharp estimates for imaginary powers of discrete Laplacians",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/eb2j5h9yjfbh3",
  "venue": "arXiv",
  "versions": []
}