{
  "id": "0704.0808",
  "version": "v3",
  "published": "2007-04-05T21:33:13.000Z",
  "updated": "2008-01-11T00:26:28.000Z",
  "title": "On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields",
  "authors": [
    "Michael Lacey",
    "Xiaochun Li"
  ],
  "comment": "92 pages, 20+ figures. Final version of the paper. To appear in Memoirs AMS",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $ v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform \\operatorname H_{v, \\epsilon}f(x) := \\text{p.v.}\\int_{-\\epsilon}^ \\epsilon f(x-yv(x)) \\frac{dy}y where $ \\epsilon $ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.\\thinspace M.\\thinspace Stein, that if $ v$ is Lipschitz, there is a positive $ \\epsilon $ for which the transform above is bounded on $ L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $ v$. This sufficient condition is that this new maximal function be bounded on some $ L ^{p}$, for some $ 1<p<2$. We show that the maximal function is bounded from $ L ^{2}$ to weak $ L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-01-11T00:26:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert transform",
      "em stein",
      "conjecture",
      "lipschitz maximal function",
      "study sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 92,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0704.0808L"
    }
  }
}