{
  "id": "2006.04346",
  "version": "v1",
  "published": "2020-06-08T04:08:00.000Z",
  "updated": "2020-06-08T04:08:00.000Z",
  "title": "Bilinear Hilbert Transforms and (Sub)Bilinear Maximal Functions along Convex Curves",
  "authors": [
    "Junfeng Li",
    "Haixia Yu"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we determine the $L^p(\\mathbb{R})\\times L^q(\\mathbb{R})\\rightarrow L^r(\\mathbb{R})$ boundedness of the bilinear Hilbert transform $H_{\\gamma}(f,g)$ along a convex curve $\\gamma$ $$H_{\\gamma}(f,g)(x):=\\mathrm{p.\\,v.}\\int_{-\\infty}^{\\infty}f(x-t)g(x-\\gamma(t)) \\,\\frac{\\textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $\\frac{1}{p}+\\frac{1}{q}=\\frac{1}{r}$, and $r>\\frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p(\\mathbb{R})\\times L^q(\\mathbb{R})\\rightarrow L^r(\\mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{\\gamma}(f,g)$ along a convex curve $\\gamma$ $$M_{\\gamma}(f,g)(x):=\\sup_{\\varepsilon>0}\\frac{1}{2\\varepsilon}\\int_{-\\varepsilon}^{\\varepsilon}|f(x-t)g(x-\\gamma(t))| \\,\\textrm{d}t.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-08T04:08:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bilinear maximal function",
      "bilinear hilbert transform",
      "convex curve",
      "boundedness property holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}