{
  "id": "2207.08738",
  "version": "v1",
  "published": "2022-07-18T16:30:19.000Z",
  "updated": "2022-07-18T16:30:19.000Z",
  "title": "On fine differentiability properties of Sobolev functions",
  "authors": [
    "Paz Hashash",
    "Vladimir Gol'dshtein",
    "Alexander Ukhlov"
  ],
  "comment": "24 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study fine differentiability properties of functions in Sobolev spaces. We prove that the difference quotient of $f\\in W^{1}_{p}(\\mathbb R^n)$ converges to the formal differential of this function in the $W^{1}_{p,\\loc}$-topology $\\cp_p$-a.~e. under an additional assumption of existence of a refined weak gradient. This result is extended to convergence of remainders in the corresponding Taylor formula for functions in $W^{k}_{p}$ spaces. In addition we prove $L_p-$approximately differentiability $\\cp_p-$a.e for functions $f\\in W^1_p(\\mathbb{R}^n)$ with a refined weak gradient.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-18T16:30:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "31B15"
    ],
    "keywords": [
      "sobolev functions",
      "refined weak gradient",
      "study fine differentiability properties",
      "formal differential",
      "additional assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}