{
  "id": "math/0411648",
  "version": "v1",
  "published": "2004-11-30T09:28:53.000Z",
  "updated": "2004-11-30T09:28:53.000Z",
  "title": "Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends",
  "authors": [
    "Gilles Carron",
    "Thierry Coulhon",
    "Andrew Hassell"
  ],
  "categories": [
    "math.AP",
    "math.DG",
    "math.FA"
  ],
  "abstract": "Let $M$ be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, $\\RR^n \\setminus B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \\to L^p(M; T^*M)$ for $1 < p < n$ and unbounded for $p \\geq n$ if there is more than one end. It follows from known results that in such a case the Riesz transform on $M$ is bounded for $1 < p \\leq 2$ and unbounded for $p > n$; the result is new for $2 < p \\leq n$. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in $L^p$ for some $p > 2$ for a more general class of manifolds. Assume that $M$ is a $n$-dimensional complete manifold satisfying the Nash inequality and with an $O(r^n)$ upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on $L^p$ for some $p > 2$ implies a Hodge-de Rham interpretation of the $L^p$ cohomology in degree 1, and that the map from $L^2$ to $L^p$ cohomology in this degree is injective.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-11-30T09:28:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J37",
      "58J35",
      "42B20"
    ],
    "keywords": [
      "riesz transform",
      "euclidean ends",
      "cohomology",
      "heat kernel estimates",
      "smooth riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11648C"
    }
  }
}