{
  "id": "1512.00569",
  "version": "v1",
  "published": "2015-12-02T04:01:11.000Z",
  "updated": "2015-12-02T04:01:11.000Z",
  "title": "On The Boundedness of Bi-parameter Littlewood-Paley $g_λ^{*}$-function",
  "authors": [
    "Mingling Cao",
    "Qingying Xue"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Let $m,n\\ge 1$ and $g_{\\lambda_1,\\lambda_2}^*$ be the bi-parameter Littlewood-Paley $g_{\\lambda}^{*}$-function defined by $$ g_{\\lambda_1,\\lambda_2}^*(f)(x)= \\bigg(\\iint_{\\R^{m+1}_{+}} \\big(\\frac{t_2}{t_2 + |x_2 - y_2|}\\big)^{m \\lambda_2} \\iint_{\\R^{n+1}_{+}} \\big(\\frac{t_1}{t_1 + |x_1 - y_1|}\\big)^{n \\lambda_1}|\\theta_{t_1,t_2} f(y_1,y_2)|^2 \\frac{dy_1 dt_1}{t_1^{n+1}} \\frac{dy_2 dt_2}{t_2^{m+1}} \\bigg)^{1/2}, \\lambda_1>1,\\quad \\lambda_2>1 $$ where $\\theta_{t_1,t_2} f$ is a non-convolution kernel defined on $\\mathbb{R}^{m+n}$. In this paper, we showed that the bi-parameter Littlewood-Paley function $g_{\\lambda_1,\\lambda_2}^*$ was bounded from $L^2(\\R^{n+m})$ to $L^2(\\R^{n+m})$. This was done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-02T04:01:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "47G10"
    ],
    "keywords": [
      "boundedness",
      "bi-parameter littlewood-paley function",
      "non-convolution kernel",
      "probabilistic methods",
      "double whitney regions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151200569C"
    }
  }
}