{
  "id": "1808.01447",
  "version": "v1",
  "published": "2018-08-04T08:14:05.000Z",
  "updated": "2018-08-04T08:14:05.000Z",
  "title": "$L^p$ Boundedness of Hilbert Transforms and Maximal Functions along Variable Curves",
  "authors": [
    "Junfeng Li",
    "Haixia Yu"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, the $L^p$ boundedness of the Hilbert transform along variable curve $(t, P(x_1)\\gamma(t))$ $$H_{P,\\gamma}f(x_1,x_2):=\\mathrm{p.\\,v.}\\int_{-\\infty}^{\\infty}f(x_1-t,x_2-P(x_1)\\gamma(t))\\,\\frac{\\textrm{d}t}{t}$$ and the corresponding maximal function $$M_{P,\\gamma}f(x_1,x_2):=\\sup_{\\varepsilon>0}\\frac{1}{2\\varepsilon}\\int_{-\\varepsilon}^{\\varepsilon}|f(x_1-t,x_2-P(x_1)\\gamma(t))|\\,\\textrm{d}t$$ are obtained, where $p\\in (1,\\infty)$, $P$ is a real polynomial on $\\mathbb{R}$. It gives a positive answer to a concern proposed by Bennett in \\cite{BJ}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-04T08:14:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert transform",
      "variable curve",
      "boundedness",
      "corresponding maximal function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}