{
  "id": "2505.08010",
  "version": "v1",
  "published": "2025-05-12T19:22:37.000Z",
  "updated": "2025-05-12T19:22:37.000Z",
  "title": "On equality of the $L^\\infty$ norm of the gradient under the Hausdorff and Lebesgue measure",
  "authors": [
    "Ze-An Ng"
  ],
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $\\Omega$ be an open subset of $\\mathbb R^n$, and let $f: \\Omega \\to \\mathbb R$ be differentiable $\\mathcal H^k$-almost everywhere, for some nonnegative integer $k < n$, where $\\mathcal H^k$ denotes the $k$=dimensional Hausdorff measure. We show that $\\|\\nabla f\\|_{L^\\infty (\\mathcal H^k)} = \\|\\nabla f\\|_{L^\\infty(\\mathcal H^n)}.$ We deduce that convergence in the Sobolev space $W^{1, \\infty}$ preserves everywhere differentiability. As a further corollary, we deduce that the class $C^1 (\\Omega)$ of continuously differentiable functions is closed in $W^{1, \\infty}(\\Omega)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-12T19:22:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A16"
    ],
    "keywords": [
      "lebesgue measure",
      "dimensional hausdorff measure",
      "open subset",
      "sobolev space",
      "nonnegative integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}