{
  "id": "1402.2030",
  "version": "v3",
  "published": "2014-02-10T03:57:26.000Z",
  "updated": "2014-03-11T03:42:53.000Z",
  "title": "Rough metrics on manifolds and quadratic estimates",
  "authors": [
    "Lashi Bandara"
  ],
  "comment": "1 figure, v2: Slight change to Section 4.4 v3: Submitted version, corrected a few typos, changed Prop 5.5 from \"geodesic\" to \"minimising geodesic\", changed Thm 6.10 from inj(M,g) to inj(M\\{x_0},g), changed remarks following Thm 6.10",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations of the space and obtain solutions to this problem on functions and forms on compact manifolds with a continuous metric. Furthermore, we show that a lower bound on the injectivity radius is not a necessary condition to solve the Kato square root problem.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-03-11T03:42:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J05",
      "58J60",
      "47B44",
      "46E35"
    ],
    "keywords": [
      "kato square root problem",
      "rough metrics",
      "transmit quadratic estimates",
      "necessary condition",
      "measure zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.2030B"
    }
  }
}