{
  "abstract": "We investigate the optimal endpoint $L^2$ restriction estimates of Laplace eigenfunctions on submanifolds of codimension 2 in a smooth closed Riemannian manifold $M$. For every fixed smooth codimension-two submanifold $Σ$, we prove a little-o improvement $o(λ^{1/2}\\sqrt{\\logλ})$ on the classical estimate $O(λ^{1/2}\\sqrt{\\logλ})$ of Burq--Gérard--Tzvetkov and Hu. Our proof uses the Bargmann transform and Tataru's phase-space representation to reduce the problem to Stein--Street's estimate for singular Radon transforms. The three-dimensional case can be handled directly by Ricci--Stein's estimate. Moreover, we construct explicit examples to show that the little-o improvement is optimal in general. These are closely related to earlier counterexamples for endpoint Strichartz estimates by Montgomery--Smith and Carbery--Hofmann. In particular, we establish the log-free estimate $O(λ^{1/2})$ when $(M,g)$ and $Σ$ are real analytic, whereas this estimate fails in general in the smooth setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18290",
  "author_total": 2,
  "authors": [
    {
      "name": "Xing Wang"
    },
    {
      "name": "Cheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bnns1c3vsvt61.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18290",
  "fulltext_url": null,
  "id": "bnns1c3vsvt61",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18290",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7213498288",
  "pdf_url": "https://arxiv.org/pdf/2609.18290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bnns1c3vsvt61.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.18290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Endpoint eigenfunction restriction estimates in codimension two",
  "title_text": "Endpoint eigenfunction restriction estimates in codimension two",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/bnns1c3vsvt61",
  "venue": "arXiv",
  "versions": []
}