{
  "id": "0804.2833",
  "version": "v1",
  "published": "2008-04-17T15:20:30.000Z",
  "updated": "2008-04-17T15:20:30.000Z",
  "title": "Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces",
  "authors": [
    "Donatella Danielli",
    "Nicola Garofalo",
    "Nguyen Cong Phuc"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider various types of Hardy-Sobolev inequalities on a Carnot-Carath\\'eodory space $(\\Om, d)$ associated to a system of smooth vector fields $X=\\{X_1, X_2,...,X_m\\}$ on $\\RR^n$ satisfying the H\\\"ormander's finite rank condition $rank Lie[X_1,...,X_m] \\equiv n$. One of our main concerns is the trace inequality \\int_{\\Om}|\\phi(x)|^{p}V(x)dx\\leq C\\int_{\\Om}|X\\phi|^{p}dx,\\qquad \\phi\\in C^{\\infty}_{0}(\\Om), where $V$ is a general weight, i.e., a nonnegative locally integrable function on $\\Om$, and $1<p<+\\infty$. Under sharp geometric assumptions on the domain $\\Om\\subset \\Rn$ that can be measured equivalently in terms of subelliptic capacities or Hausdorff contents, we establish various forms of Hardy-Sobolev type inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-17T15:20:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35H20",
      "26D10"
    ],
    "keywords": [
      "carnot-carathéodory spaces",
      "hardy-sobolev type inequalities",
      "smooth vector fields",
      "sharp geometric assumptions",
      "finite rank condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.2833D"
    }
  }
}