{
  "id": "1610.05203",
  "version": "v1",
  "published": "2016-10-17T16:51:53.000Z",
  "updated": "2016-10-17T16:51:53.000Z",
  "title": "Maximal operators and Hilbert transforms along variable non-flat homogeneous curves",
  "authors": [
    "Shaoming Guo",
    "Jonathan Hickman",
    "Victor Lie",
    "Joris Roos"
  ],
  "comment": "38 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove that the maximal operator associated with variable homogeneous planar curves $(t, u t^{\\alpha})_{t\\in \\mathbb{R}}$, $\\alpha\\not=1$ positive, is bounded on $L^p(\\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\\mathbb{R}^2 \\to \\mathbb{R}$ is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with $(t, ut^{\\alpha})_{t\\in \\mathbb{R}}$, $\\alpha\\not=1$ positive, is bounded on $L^p(\\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\\mathbb{R}^2\\to \\mathbb{R}$ is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, $TT^*$ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-17T16:51:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25",
      "44A12"
    ],
    "keywords": [
      "variable non-flat homogeneous curves",
      "maximal operator",
      "hilbert transform",
      "stationary phase methods",
      "lipschitz function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}