{
  "id": "math/0703316",
  "version": "v1",
  "published": "2007-03-12T01:38:17.000Z",
  "updated": "2007-03-12T01:38:17.000Z",
  "title": "Resolvent at low energy and Riesz transform for Schroedinger operators on asymptotically conic manifolds. II",
  "authors": [
    "Colin Guillarmou",
    "Andrew Hassell"
  ],
  "comment": "41 pages, 1 figure",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Let $(M^\\circ, g)$ be an asymptotically conic manifold, in the sense that $M^\\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. A special case of particular interest is that of asymptotically Euclidean manifolds, where $\\partial M = S^{n-1}$ and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel $(P + k^2)^{-1}$ and Riesz transform of the operator $P = \\Delta_g + V$, where $\\Delta_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function $V$ that is smooth on $M$ and vanishes to some finite order at the boundary. In the first paper in this series we made the assumption that $n \\geq 3$ and that $P$ has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary on a blown up version of $M^2 \\times [0, k_0]$, and (ii) the Riesz transform of $P$ is bounded on $L^p(M^\\circ)$ for $1 < p < n$, and that this range is optimal unless $V \\equiv 0$ and $M^\\circ$ has only one end. In the present paper, we perform a similar analysis assuming again $n \\geq 3$ but allowing zero modes and zero-resonances. We find the precise range of $p$ for which the Riesz transform (suitably defined) of $P$ is bounded on $L^p(M)$ when zero modes (but not resonances, which make the Riesz transform undefined) are present. Generically the Riesz transform is bounded for $p$ precisely in the range $(n/(n-2), n/3)$, with a bigger range possible if the zero modes have extra decay at infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-12T01:38:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "58J37"
    ],
    "keywords": [
      "riesz transform",
      "asymptotically conic manifold",
      "zero modes",
      "schroedinger operators",
      "low energy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3316G"
    }
  }
}