{
  "abstract": "We study the parabolic maximal operator $M_{\\textrm{par}}$ along the moment curve $(t,t^2)$. In 1988, Christ proved that $M_{\\textrm{par}}$ maps the parabolic Hardy space $H_{\\textrm{par}}^1(\\mathbb R^2)$ into $L^{1,\\infty}(\\mathbb R^2)$. We show that this result is sharp at $p=1$: for every $0<p<1$, the natural extension from $H_{\\textrm{par}}^p(\\mathbb R^2)$ to $L^{p,\\infty}(\\mathbb R^2)$ fails even for the corresponding single-scale operator. We then introduce a curvature-adapted modified Hardy space $H_{\\textrm{par}}^{p,*}(\\mathbb R^2)$ and a weak tendril space $\\mathcal T^{p,\\infty}(\\mathbb R^2)$, and prove that $$ M_{\\textrm{par}}: H_{\\textrm{par}}^{p,*}(\\mathbb R^2) \\longrightarrow \\mathcal T^{p,\\infty}(\\mathbb R^2), \\qquad 0<p<1, $$ is bounded. At $p=1$, these spaces recover those in Christ's theorem: $H_{\\textrm{par}}^{1,*}(\\mathbb R^2)=H_{\\textrm{par}}^1(\\mathbb R^2)$ and $\\mathcal T^{1,\\infty}(\\mathbb R^2)=L^{1,\\infty}(\\mathbb R^2)$. Thus, our result provides a natural extension of Christ's work to the range $0<p<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.15427",
  "author_total": 2,
  "authors": [
    {
      "name": "Yongsheng Han"
    },
    {
      "name": "Bingyang Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dezhdc7gfzn1e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.15427",
  "fulltext_url": null,
  "id": "dezhdc7gfzn1e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.15427",
  "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": "W7169752328",
  "pdf_url": "https://arxiv.org/pdf/2607.15427",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dezhdc7gfzn1e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2607.15427",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.15427"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hardy space estimates for the parabolic maximal operator below $p=1$",
  "title_text": "Hardy space estimates for the parabolic maximal operator below p=1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/dezhdc7gfzn1e",
  "venue": "arXiv",
  "versions": []
}