{
  "id": "1804.09845",
  "version": "v1",
  "published": "2018-04-26T01:10:28.000Z",
  "updated": "2018-04-26T01:10:28.000Z",
  "title": "$ \\ell ^{p}$-improving inequalities for Discrete Spherical Averages",
  "authors": [
    "Michael T. Lacey",
    "Robert Kesler"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $ \\lambda ^2 \\in \\mathbb N $, and in dimensions $ d\\geq 5$, let $ A _{\\lambda } f (x)$ denote the average of $ f \\;:\\; \\mathbb Z ^{d} \\to \\mathbb R $ over the lattice points on the sphere of radius $\\lambda$ centered at $x$. We prove $ \\ell ^{p}$ improving properties of $ A _{\\lambda }$. \\begin{equation*} \\lVert A _{\\lambda }\\rVert _{\\ell ^{p} \\to \\ell ^{p'}} \\lesssim \\lambda ^{-d (1 - \\frac{2}p)}, \\qquad \\tfrac{d} {d-2} < p \\leq 2. \\end{equation*} The inequalities hold for the extended range $ \\frac{d+1} {d-1} < p \\leq \\frac{d} {d-2}$ under the restriction that $ \\lambda ^2 $ has a bounded number of distinct prime factors. We show that these inequalities cannot hold for $ p < \\frac{d+1} {d-1}$. These inequalities are discrete versions of a classical inequality of Littman and Strichartz on the $ L ^{p}$ improving property of spherical averages on $ \\mathbb R ^{d}$. The proof uses the decomposition of the corresponding multiplier whose properties were established by Magyar-Stein-Wainger, Magyar, and Hunt. Various endpoint estimates are combined with interpolation estimates of different types.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-26T01:10:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "discrete spherical averages",
      "improving inequalities",
      "distinct prime factors",
      "improving property",
      "inequalities hold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}