{
  "abstract": "For a bounded domain $Ω\\subset\\mathbb R^d$, the spherical average $E_Ω(R)=\\int_{\\mathbb S^{d-1}}|\\widehat{\\mathbb{1}_Ω}(Rω)|^2dω$ is never $O(R^{-a})$ with $a>d+1$, and for smooth boundaries it is $O(R^{-d-1})$, as it is for the ball, whether or not the boundary is curved. In 2003, Brandolini, Hofmann and Iosevich extended this to every convex domain and to every domain with $C^{1,\\frac{1}{2}}$ boundary, and this regularity condition has not been improved since. How much regularity does the decay $R^{-d-1}$ require? We show that among Hölder classes the answer is $C^{1,\\frac{1}{4}}$. Every bounded domain with $C^{1,\\frac{1}{4}}$ boundary satisfies $E_Ω(R)\\leq C_ΩR^{-d-1}$, while for every $0<s<\\frac{1}{4}$ any smooth domain can be altered near a single boundary point to produce a $C^{1,s}$ domain for which this bound fails along a sequence of radii. More generally, when the boundary has fractional Sobolev regularity $W^{1+s,p}$, we determine the optimal decay exponent under a Dini condition on the normal, and in a large range of $s$ and $p$ without it. The upper bounds follow from $L^2$ estimates for nondegenerate oscillatory integral operators whose phases are distances between points of two graphs with Dini or fractional Sobolev gradients, and hence barely better than $C^1$. These are far too rough for the integrations by parts techniques from the smooth theory. We replace derivatives by finite differences adapted to the phase, a device that we expect to be useful in a variety of related problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04426",
  "author_total": 3,
  "authors": [
    {
      "name": "A. Iosevich"
    },
    {
      "name": "A. Merlo"
    },
    {
      "name": "M. Mourgoglou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2my0y7xspra4n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04426",
  "fulltext_url": null,
  "id": "2my0y7xspra4n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04426",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04426",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2my0y7xspra4n.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04426",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04426"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp decay of the spherical averages of the Fourier transform of open sets",
  "title_text": "Sharp decay of the spherical averages of the Fourier transform of open sets",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/2my0y7xspra4n",
  "venue": "arXiv",
  "versions": []
}