{
  "abstract": "We study Fourier extension from compact convex $C^2$ planar curves by organizing the mass $|f|^2\\,\\mathrm dσ$ through the Gauss map. The pushforward measure \\[ ν_f= N_\\#\\bigl(|f|^2\\,\\mathrm dσ\\bigr) \\] records its distribution in normal directions. The turning measure $\\mathrm dμ_κ=κ\\,\\mathrm dσ$ determines an intrinsic terminal tangential scale $r_R(ξ)$ through \\[ r_R(ξ)\\, μ_κ\\bigl(B_Γ(ξ,r_R(ξ))\\bigr) \\asymp R^{-1}, \\] and hence a position-dependent angular resolution $ρ_R(ξ)=R^{-1}/r_R(ξ)$. Under a doubling hypothesis on $μ_κ$, $r_R$ is, up to structural constants, the largest scale on which the curve can be linearized to precision $R^{-1}$. These scales define a normal-direction energy, and the associated terminal decomposition and transverse geometry yield local $L^4$ Fourier extension estimates and weighted variants. For the monomial curves $γ_k(t)=(t,t^k)$, $k\\geq3$ real, the terminal scales are explicit and the energy admits a multiscale representation in terms of angular correlations of $ν_f$. Under an $s$-dimensional Frostman condition on $ν_f$, this yields a growth diagram with critical threshold \\[ s_c(k)=\\frac{k-2}{3k-4}, \\] separating the flat-point and nondegenerate regimes. The resulting rates, including the critical logarithmic correction, are sharp at the energy level.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20643",
  "author_total": 1,
  "authors": [
    {
      "name": "Vicente Vergara"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1vrypdzn3en1b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20643",
  "fulltext_url": null,
  "id": "1vrypdzn3en1b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20643",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7213611495",
  "pdf_url": "https://arxiv.org/pdf/2609.20643",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1vrypdzn3en1b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.20643",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20643"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Normal-Direction Energy and Fourier Restriction for Convex Planar Curves",
  "title_text": "Normal-Direction Energy and Fourier Restriction for Convex Planar Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/1vrypdzn3en1b",
  "venue": "arXiv",
  "versions": []
}