{
  "id": "1908.08487",
  "version": "v1",
  "published": "2019-08-22T16:47:18.000Z",
  "updated": "2019-08-22T16:47:18.000Z",
  "title": "Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator",
  "authors": [
    "Samuel Zbarsky"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For all $p>1$ and all centrally symmetric convex bodies $K\\subset \\mathbb{R}^d$ define $Mf$ as the centered maximal function associated to $K$. We show that when $d=1$ or $d=2$, we have $||Mf||_p\\ge (1+\\epsilon(p,K))||f||_p$. For $d\\ge 3$, let $q_0(K)$ be the infimum value of $p$ for which $M$ has a fixed point. We show that for generic shapes $K$, we have $q_0(K)>q_0(B(0,1))$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-22T16:47:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "centered hardy-littlewood maximal operator",
      "fixed point",
      "lower bounds",
      "centrally symmetric convex bodies",
      "centered maximal function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}