{
  "id": "math/0701515",
  "version": "v1",
  "published": "2007-01-18T18:58:43.000Z",
  "updated": "2007-01-18T18:58:43.000Z",
  "title": "Resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, I",
  "authors": [
    "Colin Guillarmou",
    "Andrew Hassell"
  ],
  "comment": "28 pages, 1 figure",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We analyze the resolvent $R(k)=(P+k^2)^{-1}$ of Schr\\\"odinger operators $P=\\Delta+V$ with short range potential $V$ on asymptotically conic manifolds $(M,g)$ (this setting includes asymptotically Euclidean manifolds) near $k=0$. We make the assumption that the dimension is greater or equal to 3 and that $P$ has no $L^2$ null space and no resonance at 0. In particular, we show that the Schwartz kernel of $R(k)$ is a conormal polyhomogeneous distribution on a desingularized version of $M\\times M\\times [0,1]$. Using this, we show that the Riesz transform of $P$ is bounded on $L^p$ for $1<p<n$ and that this range is optimal if $V$ is not identically zero or if $M$ has more than one end. We also analyze the case V=0 with one end. In a follow-up paper, we shall deal with the same problem in the presence of zero modes and zero-resonances.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-01-18T18:58:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "58J37"
    ],
    "keywords": [
      "asymptotically conic manifolds",
      "riesz transform",
      "schrodinger operators",
      "low energy",
      "short range potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1515G"
    }
  }
}