{
  "id": "1907.05734",
  "version": "v1",
  "published": "2019-07-12T13:22:17.000Z",
  "updated": "2019-07-12T13:22:17.000Z",
  "title": "Averages along the Square Integers: $\\ell^p$ improving and Sparse Inequalities",
  "authors": [
    "Rui Han",
    "Michael T Lacey",
    "Fan Yang"
  ],
  "comment": "27 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $f\\in \\ell^2(\\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\\frac{1}{N}\\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \\ell ^{p}$-improving estimate, for $ 3/2 < p \\leq 2$: \\begin{equation*} N ^{-2/p'} \\lVert A_N f \\rVert _{ p'} \\lesssim N ^{-2/p} \\lVert f\\rVert _{\\ell ^{p}}, \\end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =\\frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = \\sup _{N\\geq 1} |A_Nf| $ satisfies a similar sparse bound. Novel weighted and vector valued inequalities for $ A$ follow. A critical step in the proof requires the control of a logarithmic average over $ q$ of a function $G(q,x)$ counting the number of square roots of $x$ mod $q$. One requires an estimate uniform in $x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-12T13:22:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inequality",
      "square integers",
      "sparse inequalities",
      "similar sparse bound",
      "square roots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}