{
  "id": "1903.11002",
  "version": "v1",
  "published": "2019-03-26T16:42:23.000Z",
  "updated": "2019-03-26T16:42:23.000Z",
  "title": "The Boundedness of the (Sub)Bilinear Maximal Function along \"non-flat\" smooth curves",
  "authors": [
    "Alejandra Gaitan",
    "Victor Lie"
  ],
  "comment": "22 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\mathcal{N}\\mathcal{F}$ be the class of smooth non-flat curves near the origin and near infinity previously introduced by the second author and let $\\gamma\\in\\mathcal{N}\\mathcal{F}$. We show - via a unifying approach relative to the correspondent bilinear Hilbert transform $H_{\\Gamma}$ - that the (sub)bilinear maximal function along curves $\\Gamma=(t,-\\gamma(t))$ defined as $$M_{\\Gamma}(f,g)(x):=\\sup\\limits_{\\epsilon>0} \\frac{1}{2\\epsilon} \\int_{-\\epsilon}^{\\epsilon} |f(x-t)\\,g(x+\\gamma(t))|\\,dt$$ is bounded from $L^p(\\mathbb{R})\\times L^{q}(\\mathbb{R})\\to L^r(\\mathbb{R})$ for all $p, q$ and $r$ H\\\"older indices, that is $\\frac{1}{p}+\\frac{1}{q}=\\frac{1}{r}$, with $1<p,\\,q\\leq\\infty$ and $1\\le r\\leq\\infty$. This is the maximal boundedness range for $M_{\\Gamma}$, that is, our result is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-26T16:42:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A45",
      "42B20"
    ],
    "keywords": [
      "bilinear maximal function",
      "smooth curves",
      "correspondent bilinear hilbert transform",
      "maximal boundedness range",
      "smooth non-flat curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}