{
  "id": "1601.07677",
  "version": "v1",
  "published": "2016-01-28T07:55:01.000Z",
  "updated": "2016-01-28T07:55:01.000Z",
  "title": "Restriction of averaging operators to algebraic varieties over finite fields",
  "authors": [
    "Doowon Koh",
    "Seongjun Yeom"
  ],
  "comment": "15 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "We study the mapping properties of restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\\mathbb F_q$ with $q$ elements. Given an algebraic variety $V \\subset \\mathbb F_q^d,$ let us denote by $dx$ and $\\sigma$ the normalized counting measure on $\\mathbb F_q^d$ and the normalized surface measure on $V$, respectively. For functions $f: \\mathbb F_q^d \\to \\mathbb C$, a restricted averaging operator $A_V$ is defined by restricting $f\\ast \\sigma$ to the variety $V$, that is $A_Vf=f\\ast \\sigma|_V.$ The main purpose of this paper is to investigate $L^p\\to L^r$ estimates of the restricted averaging operator $A_V.$ We relate this problem to both the Fourier restriction problem and the averaging problem over $V\\subset \\mathbb F_q^d.$ As a consequence, we obtain the optimal results on the restricted averaging problems for regular varieties such as the spheres, the paraboloids for dimensions $d\\ge2,$ and the cones for odd dimensions $d\\ge 3.$ In addition, when the variety $V$ is the cone lying in an even dimensional vector space over $\\mathbb F_q$, we obtain the sharp weak-type estimates for restricted averaging operators $A_V.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-28T07:55:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B05",
      "11T23"
    ],
    "keywords": [
      "algebraic variety",
      "restricted averaging operator",
      "finite fields",
      "dimensional vector space",
      "averaging problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160107677K"
    }
  }
}