{
  "abstract": "Let $\\mathbb{G}$ be a stratified Lie group and $\\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\\nabla_{H} \\mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step, and the underlying group structure of $\\mathbb{G}$. Our result provides a noncommutative generalization of the dimension-free Euclidean theorem of Ouyang, Spector, and Stockdale arXiv:2608.18068, with the same universal constant. Our proof relies upon a fractional obstacle problem adapted to stratified Lie groups by using the functional calculus of $\\mathcal L$ instead of the Fourier transform. As a consequence, by interpolation we obtain uniform $L^p$ bounds for the full horizontal Riesz transform $\\nabla_{H} \\mathcal L^{-1/2}$ for $p \\in (1,2]$. By contrast, for every $p > 2$, we construct a sequence of stratified Lie groups with fixed horizontal dimension $5$ and steps tending to infinity for which the $L^p$ norms of the horizontal Riesz transforms diverge.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.20267",
  "author_total": 3,
  "authors": [
    {
      "name": "Sheng-Chen Mao"
    },
    {
      "name": "Yaojun Wang"
    },
    {
      "name": "Ye Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3tht9jwgahaxh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.20267",
  "fulltext_url": null,
  "id": "3tht9jwgahaxh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.20267",
  "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": "W7203896776",
  "pdf_url": "https://arxiv.org/pdf/2608.20267",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3tht9jwgahaxh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2608.20267",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.20267"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A $p = 2$ dichotomy for uniform Riesz transform bounds on stratified Lie groups",
  "title_text": "A p = 2 dichotomy for uniform Riesz transform bounds on stratified Lie groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/3tht9jwgahaxh",
  "venue": "arXiv",
  "versions": []
}