{
  "id": "1701.00477",
  "version": "v1",
  "published": "2017-01-02T18:49:02.000Z",
  "updated": "2017-01-02T18:49:02.000Z",
  "title": "Restriction of the Fourier transform to some oscillating curves",
  "authors": [
    "Xianghong Chen",
    "Dashan Fan",
    "Lifeng Wang"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\phi$ be a smooth function on a compact interval $I$. Let $$\\gamma(t)=\\left (t,t^2,\\cdots,t^{n-1},\\phi(t)\\right).$$ In this paper, we show that $$\\left(\\int_I \\big|\\hat f(\\gamma(t))\\big|^q \\big|\\phi^{(n)}(t)\\big|^{\\frac{2}{n(n+1)}} dt\\right)^{1/q}\\le C\\|f\\|_{L^p(\\mathbb R^n)}$$ holds in the range $$1\\le p<\\frac{n^2+n+2}{n^2+n},\\quad 1\\le q<\\frac{2}{n^2+n}p'.$$ This generalizes an affine restriction theorem of Sj\\\"olin (1974) for $n=2$. Our proof relies on ideas of Sj\\\"olin (1974) and Drury (1985), and more recently Bak-Oberlin-Seeger (2008) and Stovall (2016), as well as a variation bound for smooth functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-02T18:49:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10",
      "42B99"
    ],
    "keywords": [
      "fourier transform",
      "oscillating curves",
      "smooth function",
      "affine restriction theorem",
      "compact interval"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}