{
  "id": "1803.10855",
  "version": "v2",
  "published": "2018-03-28T21:09:21.000Z",
  "updated": "2019-08-17T15:39:13.000Z",
  "title": "Pointwise differentiability of higher order for distributions",
  "authors": [
    "Ulrich Menne"
  ],
  "comment": "33 pages, no figures. Additions and changes in version 2: (1) description of the relation to asymptotic expansions; (2) alternative proof of Theorem E; (3) minor corrections in 2.2, 2.16, 2.23, and 3.7; (4) updates of acknowledgements, references, and affiliations; (5) minor expository improvements. Comments of R. Estrada and a referee induced (1)+(2) and (5), respectively",
  "categories": [
    "math.FA",
    "math.AP",
    "math.CA"
  ],
  "abstract": "For distributions, we build a theory of higher order pointwise differentiability comprising, for order zero, {\\L}ojasiewicz's notion of point value. Results include Borel regularity of differentials, higher order rectifiability of the associated jets, a Rademacher-Stepanov type differentiability theorem, and a Lusin type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincar\\'e inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2019-08-17T15:39:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46F10",
      "26B05",
      "41A58"
    ],
    "keywords": [
      "distributions",
      "order pointwise differentiability comprising",
      "rademacher-stepanov type differentiability theorem",
      "point value",
      "distributional partial derivatives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}