{
  "id": "1808.07190",
  "version": "v1",
  "published": "2018-08-22T02:29:53.000Z",
  "updated": "2018-08-22T02:29:53.000Z",
  "title": "The distributional hyper-Jacobian determinants in fractional Sobolev spaces",
  "authors": [
    "Qiang Tu",
    "Chuanxi Wu",
    "Xueting Qiu"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we give a positive answer to a question raised by Baer-Jerison in connection with hyper-Jacobian determinants and associated minors in fractional Sobolev spaces. Inspired by recent works of Brezis-Nguyen and Baer-Jerison on the Jacobian and Hessian determinants, we show that the distributional $m$th-Jacobian minors of degree $r$ are weak continuous in fractional Sobolev spaces $W^{m-\\frac{m}{r},r}$, and the result is optimal, satisfying the necessary conditions, in the frame work of fractional Sobolev spaces. In particular, the conditions can be removed in case $m=1,2$, i.e., the $m$th-Jacobian minors of degree $r$ are well defined in $W^{s,p}$ if and only if $W^{s,p} \\subseteq W^{m-\\frac{m}{r},m}$ in case $m=1,2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-22T02:29:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "46F10",
      "42B35"
    ],
    "keywords": [
      "fractional sobolev spaces",
      "distributional hyper-jacobian determinants",
      "th-jacobian minors",
      "necessary conditions",
      "baer-jerison"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}