{
  "id": "2505.02053",
  "version": "v1",
  "published": "2025-05-04T10:10:19.000Z",
  "updated": "2025-05-04T10:10:19.000Z",
  "title": "An atomic decomposition for functions of bounded variation",
  "authors": [
    "Daniel Spector",
    "Cody B. Stockdale",
    "Dmitriy Stolyarov"
  ],
  "comment": "13 pages",
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "In this paper, we give a decomposition of the gradient measure $Du$ of an arbitrary function of bounded variation $u$ into a sum of atoms $\\mu=D\\chi_{F}$, where $F$ is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each $\\mu$, there exists a cube $Q$ such that $\\operatorname*{supp}\\mu\\subset Q$, $\\mu(Q)=0$, $|\\mu|(Q)\\leq 1$, and, denoting by $p_t$ the heat kernel in $\\mathbb{R}^d$, \\[ \\sup_{x \\in \\mathbb{R}^d, t>0} |t^{1/2} p_t \\ast \\mu (x)| \\leq \\frac{1}{l(Q)^{d-1}}. \\] Our proof relies on a sampling of the coarea formula and a new boxing identity. We present several consequences of this result, including Sobolev inequalities, dimension estimates, and trace inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-04T10:10:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bounded variation",
      "atomic decomposition",
      "gradient measure",
      "sobolev inequalities",
      "coarea formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}