{
  "abstract": "In contrast to Christ's theorem that the lacunary spherical maximal operator is bounded from $H^1(\\mathbb R^d)$ to $L^{1,\\infty}(\\mathbb R^d)$ ($d\\geq2$), we show that the lacunary bilinear spherical maximal operator $$ \\mathcal B_{\\mathrm{lac}}(f,g)(x) =\\sup_{\\ell\\in\\mathbb Z} \\Bigl|\\int_{\\mathbb S^{2d-1}}f(x-2^\\ell y)\\,g(x-2^\\ell z)\\,dσ(y,z)\\Bigr| $$ is not bounded from $H^1(\\mathbb R^d)\\times H^1(\\mathbb R^d)$ to $L^{1/2,\\infty}(\\mathbb R^d)$ for any $d\\geq1$. In particular, it is not bounded from $L^1(\\mathbb R^d)\\times L^1(\\mathbb R^d)$ to $L^{1/2,\\infty}(\\mathbb R^d)$. More precisely, for every sufficiently large integer $N$, we construct two finite linear combinations of smooth, compactly supported, mean-zero $H^1(\\mathbb R^d)$-atoms for which the ratio of the weak $L^{1/2}$ quasinorm to the product of the two Hardy norms is at least $c_d\\log(N+1)$. This lower bound holds already for the supremum over a suitable set of $N$ dyadic radii. For $d=1$ this answers the case $p=q=1$ of a question in Borges's survey. For every $k\\ge3$ and $d\\ge1$, a modification of the construction proves that the lacunary $k$-linear spherical maximal operator does not map $(H^1(\\mathbb R^d))^k$ into $L^{1/k,\\infty}(\\mathbb R^d)$. We also indicate extensions to certain hypersurfaces and to horizontal spherical averages on the Heisenberg group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40100",
  "author_total": 4,
  "authors": [
    {
      "name": "Sanghyuk Lee"
    },
    {
      "name": "Ji Li"
    },
    {
      "name": "Chaojie Wen"
    },
    {
      "name": "Liangchuan Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5smdbzq0r7fcp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40100",
  "fulltext_url": null,
  "id": "5smdbzq0r7fcp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40100",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40100",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5smdbzq0r7fcp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40100",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40100"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Failure of the $H^1\\times H^1\\longrightarrow L^{1/2,\\infty}$ estimate for the lacunary bilinear spherical maximal function and applications",
  "title_text": "Failure of the H^1× H^1⟶ L^1/2,∞ estimate for the lacunary bilinear spherical maximal function and applications",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5smdbzq0r7fcp",
  "venue": "arXiv",
  "versions": []
}