{
  "abstract": "We study the sharp Fourier extension inequality from $L^2(\\mathbb S^{d-1})$ to the radial--angular mixed space $L^q_{\\mathrm{rad}}L^2_{\\mathrm{ang}}$. Our work is motivated by and builds upon the foundational analysis of Carneiro, Oliveira e Silva and Sousa \\cite{COS}, who initiated the sharp mixed-norm spherical restriction problem from the viewpoint of spherical-harmonic decomposition. Their reduction revealed that the extremal problem is governed by a competition among the individual harmonic modes, and their results established degree-zero extremizers in several important ranges of exponents, including even integer exponents and the endpoint case. The present paper addresses this question by establishing a complete modal ordering result. Building on their spherical-harmonic reduction, the sharp-constant problem is reduced to the weighted Bessel moments \\[ I_{ν,k}(q) := \\int_0^\\infty |J_{ν+k}(r)|^q r^{2ν+1-νq}\\,dr, \\qquad ν:=\\frac d2-1. \\] For every integer $d\\ge2$ and every finite admissible exponent $q>2d/(d-1)$, we prove \\[ I_{ν,k}(q)<I_{ν,0}(q), \\qquad k\\ge1, \\] and hence constants are the unique nonzero extremizers of the sharp mixed-norm extension inequality. In dimensions $d\\ge3$ we prove the stronger strict ordering \\[ I_{ν,0}(q)>I_{ν,1}(q)>I_{ν,2}(q)>\\cdots. \\] We resolve this difficulty by combining exact Hankel asymptotics with a finite remainder comparison. The modal ordering also yields the optimal quadratic global stability inequality. For every finite admissible $q$, \\[ C_{d,q}^2\\|f\\|_2^2 -\\|\\widehat{fσ}\\|_{L^q_{\\mathrm{rad}}L^2_{\\mathrm{ang}}}^2 \\ge Γ_{d,q}\\|f-P_0f\\|_2^2, \\] with the best possible coefficient. When $d\\ge3$, this coefficient is explicitly \\[ Γ_{d,q} =(2π)^d \\left[ I_{ν,0}(q)^{2/q}-I_{ν,1}(q)^{2/q} \\right]. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19648",
  "author_total": 1,
  "authors": [
    {
      "name": "Zewei Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3g64yjp9zy3hz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19648",
  "fulltext_url": null,
  "id": "3g64yjp9zy3hz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19648",
  "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"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7222021906",
  "pdf_url": "https://arxiv.org/pdf/2610.19648",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3g64yjp9zy3hz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19648",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19648"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp radial--angular mixed norm Fourier extension and global stability on spheres",
  "title_text": "Sharp radial--angular mixed norm Fourier extension and global stability on spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/3g64yjp9zy3hz",
  "venue": "arXiv",
  "versions": []
}