{
  "id": "1504.07778",
  "version": "v1",
  "published": "2015-04-29T09:03:34.000Z",
  "updated": "2015-04-29T09:03:34.000Z",
  "title": "A new approach to Sobolev spaces in metric measure spaces",
  "authors": [
    "Tomas Sjödin"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $(X,d_X,\\mu)$ be a metric measure space where $X$ is locally compact and separable and $\\mu$ is a Borel regular measure such that $0 <\\mu(B(x,r)) <\\infty$ for every ball $B(x,r)$ with center $x \\in X$ and radius $r>0$. We define $\\mathcal{X}$ to be the set of all positive, finite non-zero regular Borel measures with compact support in $X$ which are dominated by $\\mu$, and $\\mathcal{M}=\\mathcal{X} \\cup \\{0\\}$. By introducing a kind of mass transport metric $d_{\\mathcal{M}}$ on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for real valued functions $F$ on $\\mathcal{X}$, and then for real valued functions $f$ on $X$ by identifying them with the unique function $F_f$ on $\\mathcal{X}$ defined by the mean-value integral: $$F_f(\\eta)= \\frac{1}{\\|\\eta\\|} \\int f d\\eta.$$ In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space $\\mathbb{R}^n$ with Lebesgue measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-29T09:03:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "30L99",
      "31E05"
    ],
    "keywords": [
      "metric measure space",
      "finite non-zero regular borel measures",
      "real valued functions",
      "first order sobolev spaces",
      "mass transport metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150407778S"
    }
  }
}