{
  "abstract": "Let $p_d=(d^2+1)/(d(d-1))$ and $q_d=(d^2+1)/(d-1)$. We prove that the local spherical maximal operator $\\sup_{1\\leq t\\leq 2}|f*dσ_t|$ is bounded from $L^{p_d}(\\mathbb{R}^d)$ to $L^{q_d}(\\mathbb{R}^d)$ for every $d\\geq 2$. Previously only the restricted weak type estimate was known at this endpoint. Together with known results, this completes the characterization of the strong type $L^p\\to L^q$ bounds for the local spherical maximal operator. We also prove the maximal estimate \\[ \\Big\\|\\sup_{t\\in\\mathbb{R}}|e^{it\\sqrt{-Δ}}f|\\Big\\|_{L^{2(d+1)/(d-1)}(\\mathbb{R}^d)} \\lesssim\\|f\\|_{\\dot H^{d/(d+1)}(\\mathbb{R}^d)} \\] for the wave equation at the critical exponent. Both results are obtained by exploiting the gain in bilinear estimates that comes from the interaction between high and low frequencies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27803",
  "author_total": 2,
  "authors": [
    {
      "name": "Sanghyuk Lee"
    },
    {
      "name": "Shuijiang Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ezd4chsw25kqe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27803",
  "fulltext_url": null,
  "id": "ezd4chsw25kqe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27803",
  "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": "W7223141727",
  "pdf_url": "https://arxiv.org/pdf/2610.27803",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ezd4chsw25kqe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27803",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27803"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The strong type endpoint estimate for the local spherical maximal function",
  "title_text": "The strong type endpoint estimate for the local spherical maximal function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/ezd4chsw25kqe",
  "venue": "arXiv",
  "versions": []
}