{
  "id": "1907.04753",
  "version": "v1",
  "published": "2019-07-10T14:31:20.000Z",
  "updated": "2019-07-10T14:31:20.000Z",
  "title": "Endpoint estimates for the maximal function over prime numbers",
  "authors": [
    "Bartosz Trojan"
  ],
  "comment": "to appear in Journal of Fourier Analysis And Applications",
  "categories": [
    "math.DS",
    "math.CA",
    "math.NT"
  ],
  "abstract": "Given an ergodic dynamical system $(X, \\mathcal{B}, \\mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\\log L)^2(\\log \\log L)(X, \\mu)$, the ergodic averages \\[ \\frac{1}{\\pi(N)} \\sum_{p \\in \\mathbb{P}_N} f\\big(T^p x\\big), \\] converge for $\\mu$-almost all $x \\in X$, where $\\mathbb{P}_N$ is the set of prime numbers not larger that $N$ and $\\pi(N) = \\# \\mathbb{P}_N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-10T14:31:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A45",
      "46E30",
      "42B25"
    ],
    "keywords": [
      "prime numbers",
      "maximal function",
      "endpoint estimates",
      "ergodic dynamical system",
      "ergodic averages"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}