{
  "id": "1109.6396",
  "version": "v1",
  "published": "2011-09-29T04:31:42.000Z",
  "updated": "2011-09-29T04:31:42.000Z",
  "title": "$L^p$ estimates for the Hilbert transforms along a one-variable vector field",
  "authors": [
    "Michael Bateman",
    "Christoph Thiele"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, $L^2$ whenever $v$ is Lipschitz. We establish a wide range of $L^p$ estimates for this operator when $v$ is a measurable, non-vanishing, one-variable vector field in $\\bbr ^2$. Aside from an $L^2$ estimate following from a simple trick with Carleson's theorem, these estimates were unknown previously. This paper is closely related to a recent paper of the first author (\\cite{B2}).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-09-29T04:31:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25"
    ],
    "keywords": [
      "one-variable vector field",
      "hilbert transform",
      "wide range",
      "simple trick",
      "carlesons theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.6396B"
    }
  }
}