{
  "abstract": "We study the tangent bundle $T_μ$ to a Radon measure $μ$ in Euclidean space, introduced by Bouchitté, Champion and Jimenez (2005). Its construction, inspired by Monge-Kantorovich optimal transport theory, involves the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures. Precisely, a velocity field $σ$ is $μ$-tangent when the divergence of $σμ$ lies in the Arens-Eells space. The tangent bundle $T_μ$ defined $μ$-almost everywhere provides a local projection that allows to construct a $μ$-tangential gradient operator on Lipschitz functions that is weakly continuous and enjoys integration by parts. In this paper, we introduce a new quantitative estimate involving the tangential and normal components of a given velocity vector field $σ\\in L^1_μ(\\mathbb{R}^d)$. Specifically, we show that the tangential condition $σ\\in T_μ$ holding $μ$ a.e. is equivalent to each of the following two conditions: the convergence in the Arens-Eells space of $h^{-1} \\left((id+h σ)_{\\#} μ- μ\\right)$ as $h \\to 0$, and the differentiability of Lipschitz functions along $σ$. Moreover, given a field of non-tangent directions, we construct a Lipschitz function that is not differentiable on a large set, confirming that $T_μ$ agrees with the decomposability bundle of Alberti and Marchese. We finally relate $T_μ$ to the tangent measures of Preiss and survey further properties of the tangential differential calculus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29401",
  "author_total": 3,
  "authors": [
    {
      "name": "Adolfo Arroyo-Rabasa"
    },
    {
      "name": "Averil Aussedat"
    },
    {
      "name": "Guy Bouchitté"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4pztwqcyrng8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29401",
  "fulltext_url": null,
  "id": "f4pztwqcyrng8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29401",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214293341",
  "pdf_url": "https://arxiv.org/pdf/2609.29401",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4pztwqcyrng8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29401",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29401"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Monge-Kantorovich tangent bundle to a measure",
  "title_text": "The Monge-Kantorovich tangent bundle to a measure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/f4pztwqcyrng8",
  "venue": "arXiv",
  "versions": []
}