{
  "abstract": "We establish noncommutative maximal inequalities for averages over submanifolds in the $L^p$-spaces associated with semifinite von Neumann algebras. For every $1<p\\leq\\infty$, we prove local maximal bounds for smooth submanifolds of finite type at the parameter origin, and global bounds over all positive scales for polynomial parametrizations, without a finite-type assumption and with constants depending only on $p$, the degree and the dimensions. For variable hypersurfaces in $\\mathbb{R}^n$, $n\\geq3$, satisfying a uniform rotational curvature condition, we obtain local maximal bounds for $p>n/(n-1)$. To our knowledge, these are the first such noncommutative inequalities for general lower-dimensional submanifolds and polynomial images. The key step is a weak type $(1,1)$ estimate for a regularized auxiliary family adapted to non-isotropic dilations. The geometric kernels need not satisfy the pointwise smoothness or $L^2$-Dini conditions required by existing noncommutative Calderón--Zygmund theories. We combine a Calderón--Zygmund decomposition based on Cuculescu projections with integral regularity estimates, a non-isotropic Littlewood--Paley decomposition and Fourier decay to prove this endpoint estimate. Marcinkiewicz interpolation yields maximal $L^p$ bounds for the regularized averages, and interpolation with Fourier-based $L^2$ bounds recovers the original averages. The variable-hypersurface result uses oscillatory $TT^*$ estimates and localization. As applications, we obtain maximal ergodic inequalities for trace-preserving actions of $\\mathbb{R}^n$ and, under additional assumptions, bilaterally almost uniform convergence results for the associated normalized averages.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34906",
  "author_total": 2,
  "authors": [
    {
      "name": "Xudong Lai"
    },
    {
      "name": "Siyu Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eyhjmf9w3z1yv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34906",
  "fulltext_url": null,
  "id": "eyhjmf9w3z1yv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34906",
  "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.OA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214747065",
  "pdf_url": "https://arxiv.org/pdf/2609.34906",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eyhjmf9w3z1yv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34906",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34906"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces",
  "title_text": "Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/eyhjmf9w3z1yv",
  "venue": "arXiv",
  "versions": []
}