{
  "id": "1512.09356",
  "version": "v1",
  "published": "2015-12-31T19:58:38.000Z",
  "updated": "2015-12-31T19:58:38.000Z",
  "title": "On the Boundedness of The Bilinear Hilbert Transform along \"non-flat\" smooth curves. The Banach triangle case ($L^r,\\: 1\\leq r<\\infty$)",
  "authors": [
    "Victor Lie"
  ],
  "comment": "25 pages, one figure. Submitted in Fall 2015",
  "categories": [
    "math.CA"
  ],
  "abstract": "We show that the bilinear Hilbert transform $H_{\\Gamma}$ along curves $\\Gamma=(t,-\\gamma(t))$ with $\\gamma\\in\\n\\f$ is bounded from $L^{p}(\\R)\\times L^{q}(\\R)\\,\\rightarrow\\,L^{r}(\\R)$ where $p,\\,q,\\,r$ are H\\\"older indices, i.e. $\\frac{1}{p}+\\frac{1}{q}=\\frac{1}{r}$, with $1<p<\\infty$, $1<q\\leq\\infty$ and $1\\leq r<\\infty$. Here $\\n\\f$ stands for a wide class of smooth \"non-flat\" curves near zero and infinity whose precise definition is given in Section 2. This continues author's earlier work on this topic, extending the boundedness range of $H_{\\Gamma}$ to any triple of indices $(\\frac{1}{p},\\,\\frac{1}{q},\\,\\frac{1}{r'})$ within the Banach triangle. Our result is optimal up to end-points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-31T19:58:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A45"
    ],
    "keywords": [
      "bilinear hilbert transform",
      "banach triangle case",
      "smooth curves",
      "boundedness",
      "continues authors earlier work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}