{
  "id": "1306.2662",
  "version": "v3",
  "published": "2013-06-11T21:33:57.000Z",
  "updated": "2014-04-03T13:36:29.000Z",
  "title": "Densely Defined Multiplication on the Sobolev Space",
  "authors": [
    "Joel A. Rosenfeld"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Following Sarason's classification of the densely defined multiplication operators over the Hardy space, we classify the densely defined multipliers over the Sobolev space, $W^{1,2}[0,1]$. In this paper we find that the collection of such multipliers for the Sobolev space is exactly the Sobolev space itself. This sharpens a result of Shields concerning bounded multipliers. The densely defined multiplication operators over the subspace $W_0 = \\{f \\in W^{1,2}[0,1] : f(0)=f(1)=0 \\}$ are also classified. In this case the densely defined multiplication operators can be written as a ratio of functions in $W_0$ where the denominator is non-vanishing. This is proved using a contructive argument.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-03T13:36:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sobolev space",
      "densely defined multiplication operators",
      "shields concerning bounded multipliers",
      "hardy space",
      "sarasons classification"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.2662R"
    }
  }
}