{
  "abstract": "We prove lower bounds for Fourier transforms of surface measures $$|\\widehat{ψ\\,dσ}(ξ)| \\gtrsim_{ψ, K, σ, R_0} σ(B(p, |ξ|^{-1})), \\text{ given }p\\in K\\Subset\\{ψ>0\\} ,\\ \\vec{n}_{p}\\parallel ξ,\\ |ξ|\\ge R_0,$$ on convex hypersurfaces of finite type with at most one vanishing principal curvature at each point. These bounds are uniform in the choice of $p\\in K\\Subset \\{ψ>0\\}$ and match the classical upper bound of Bruna-Nagel-Wainger. The key new analytic ingredient is a uniform lower bound for parameter-dependent oscillatory integrals on $\\mathbb{R}$ with convex phases. We also give convex examples of finite type with two vanishing principal curvatures for which $$\\inf_{p\\in K}|\\widehat{ψ\\,dσ}(R \\vec{n}_p)|\\lesssim_N R^{-N},\\ \\text{ for arbitrary fixed } \\{0\\}\\Subset K\\Subset\\{ψ>0\\}.$$ Though the estimate above has its own interest, our original motivation is to study Fourier frames for surface measures. In particular, we show that, if $Γ$ is a closed convex smooth surface in $\\mathbb{R}^d, d\\geq 2$, of finite type and at most one principal curvature vanishes at each point, then the surface measure on $Γ$ does not admit any Fourier frame. This weakens the non-vanishing curvature condition in a previous result of Iosevich, Lai, the second author, and Wyman. On centrally symmetric surfaces of non-vanishing Gaussian curvature, we obtain more domains without Fourier frames. For example, we show that $\\{x\\in S^{d-1}:0<x_d<δ\\}, d\\geq 3$, does not admit any Fourier frame for all $δ>0$, which generalizes the result of Chen and the second author on the hemisphere.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04799",
  "author_total": 2,
  "authors": [
    {
      "name": "Shilei Fan"
    },
    {
      "name": "Bochen Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d106va3w9rbq7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04799",
  "fulltext_url": null,
  "id": "d106va3w9rbq7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04799",
  "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.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7220671393",
  "pdf_url": "https://arxiv.org/pdf/2610.04799",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d106va3w9rbq7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04799",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04799"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Lower bounds for parameter-dependent oscillatory integrals and Fourier frames on hypersurfaces",
  "title_text": "Lower bounds for parameter-dependent oscillatory integrals and Fourier frames on hypersurfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/d106va3w9rbq7",
  "venue": "arXiv",
  "versions": []
}