{
  "id": "1410.1574",
  "version": "v1",
  "published": "2014-10-06T21:49:14.000Z",
  "updated": "2014-10-06T21:49:14.000Z",
  "title": "The behaviour of square functions from ergodic theory in $L^{\\infty}$",
  "authors": [
    "Guixiang Hong"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we analyze carefully the behaviour in $L^\\infty(\\mathbb R)$ of the square functions $S$ and $S_\\mathcal I$'s, originating from ergodic theory. Firstly, we show that we can find some function $f\\in L^\\infty(\\mathbb{R})$, such that $Sf$ equals infinity on a nonzero measure set. Secondly, we can find compact supported function $f\\in L^\\infty(\\mathbb{R})$ and $\\mathcal I$ such that $S_\\mathcal{I} f$ does not belong to $BMO$ space. Finally, we show that $S$ is bounded from $L^{\\infty}_c$ to $BMO$ space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl in \\cite{JKRW98}. That is, $S_\\mathcal I$ are uniformly bounded in $L^p(\\mathbb R)$ with respect to $\\mathcal I$ for $2<p<\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-06T21:49:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "47G10"
    ],
    "keywords": [
      "square functions",
      "ergodic theory",
      "nonzero measure set",
      "compact supported function",
      "equals infinity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.1574H"
    }
  }
}