{
  "abstract": "We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global $\\mathscr{H}^p$-norm controls local $L^p$-mass on the critical line $\\operatorname{Re}s=1/2$. More precisely, for every $2<p<\\infty$, there exists a constant $C_p<\\infty$ such that every Dirichlet polynomial $P$ satisfies $$ \\sup_{θ\\in\\mathbb{R}}\\int_θ^{θ+1}\\left|P\\left(\\frac12+it\\right)\\right|^p\\,\\mathrm{d}t\\le C_p\\left\\lVert P\\right\\rVert_{\\mathscr{H}^p}^{p}, $$ with $C_p$ independent of the number and choice of prime variables on which $P$ depends. Before the present work, the embedding was known at $p=2$ and, by taking integer powers, at the even exponents $p=2k$; it had been conjectured that these exhaust the finite positive cases above $2$. Together with the known failure for $0<p<2$, our theorem gives the sharp finite-exponent classification: the local embedding property holds exactly for $p\\ge2$. Thus, the true threshold is $p=2$, rather than even integrality. The proof passes to the dual exponent $q=p/(p-1)\\in(1,2)$, where an exact frequency decomposition isolates a single resonant Euler-product term. A covariance-preserving replacement of the shared prime factors reduces the resulting fractional-moment estimate to a log-correlated Gaussian field, and a critical branching-random-walk bound supplies the required multiscale decay. A finite-cyclic square-function estimate assembles the resonant scales, and Hardy-quotient duality converts the resulting vector-valued bound into the critical-line trace. For $1\\le p<\\infty$, known equivalences give the same sharp threshold in several classical problems, including the conformally invariant half-plane embedding, the reverse local Carleson-measure transfer, and boundedness of all characteristic-zero Gordon--Hedenmalm composition operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11560",
  "author_total": 6,
  "authors": [
    {
      "name": "Bonan Chen"
    },
    {
      "name": "Xiang Fang"
    },
    {
      "name": "Feng Guo"
    },
    {
      "name": "Shengzhao Hou"
    },
    {
      "name": "Yizhou Shao"
    },
    {
      "name": "Qi Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/advdxs0j3037q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11560",
  "fulltext_url": null,
  "id": "advdxs0j3037q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11560",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212274901",
  "pdf_url": "https://arxiv.org/pdf/2609.11560",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/advdxs0j3037q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.11560",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11560"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Local Embedding Problem for Hardy Spaces of Dirichlet Series",
  "title_text": "The Local Embedding Problem for Hardy Spaces of Dirichlet Series",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/advdxs0j3037q",
  "venue": "arXiv",
  "versions": []
}