{
  "abstract": "Let $(M,\\mathcal{D},g)$ be a smooth equiregular sub-Riemannian manifold equipped with a smooth positive measure $μ$. We study the small-scale limit of the metric-ball mean-value operator \\[ A_hf(x)=\\frac{1}{h^{2}μ(B(x,h)) }\\int_{B(x,h)}(f(q)-f(x))\\, dμ(q). \\] Exact homogeneity yields the pointwise limit on Carnot groups. On a general equiregular manifold, convergence for every smooth test function is equivalent to convergence of the rescaled horizontal first moments of metric balls in first-kind privileged coordinates. This criterion is independent of $μ$; when it holds, the principal symbol is determined by the normalized second-moment tensor of the tangent unit ball, and the drift satisfies an explicit change-of-measure formula. In step at most two, we verify the criterion by combining a real-analytic finite-jet reduction with tame integration, which rules out oscillation of the normalized moments. We also compute the limit on Lie groups and prove unconditional distributional convergence of the volume-weighted operators on every equiregular manifold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02710",
  "author_total": 3,
  "authors": [
    {
      "name": "Fabrice Baudoin"
    },
    {
      "name": "Jonathan Junné"
    },
    {
      "name": "David Tewodrose"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4mgjhgv8exffc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02710",
  "fulltext_url": null,
  "id": "4mgjhgv8exffc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02710",
  "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.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7207774217",
  "pdf_url": "https://arxiv.org/pdf/2609.02710",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4mgjhgv8exffc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02710",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02710"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Asymptotic mean value Laplacian on equiregular sub-Riemannian manifolds",
  "title_text": "Asymptotic mean value Laplacian on equiregular sub-Riemannian manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/4mgjhgv8exffc",
  "venue": "arXiv",
  "versions": []
}