{
  "id": "1309.6512",
  "version": "v1",
  "published": "2013-09-25T14:13:10.000Z",
  "updated": "2013-09-25T14:13:10.000Z",
  "title": "Boundedness of Intrinsic Littlewood-Paley Functions on Musielak-Orlicz Morrey and Campanato Spaces",
  "authors": [
    "Yiyu Liang",
    "Eiichi Nakai",
    "Dachun Yang",
    "Junqiang Zhang"
  ],
  "comment": "47pages, Banach J. Math. Anal. (to appear)",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $\\varphi: {\\mathbb R^n}\\times [0,\\infty)\\to[0,\\infty)$ be such that $\\vz(x,\\cdot)$ is nondecreasing, $\\varphi(x,0)=0$, $\\varphi(x,t)>0$ when $t>0$, $\\lim_{t\\to\\infty}\\varphi(x,t)=\\infty$ and $\\vz(\\cdot,t)$ is a Muckenhoupt $A_\\infty({\\mathbb R^n})$ weight uniformly in $t$. Let $\\phi: [0,\\infty)\\to[0,\\infty)$ be nondecreasing. In this article, the authors introduce the Musielak-Orlicz Morrey space $\\mathcal M^{\\varphi,\\phi}(\\mathbb R^n)$ and obtain the boundedness on $\\mathcal M^{\\varphi,\\phi}(\\mathbb R^n)$ of the intrinsic Lusin area function $S_{\\alpha}$, the intrinsic $g$-function $g_{\\alpha}$, the intrinsic $g_{\\lambda}^*$-function $g^\\ast_{\\lambda, \\alpha}$ and their commutators with ${\\rm BMO}(\\rn)$ functions, where $\\alpha\\in(0,1]$, $\\lambda\\in(\\min\\{\\max\\{3,\\,p_1\\},3+2\\az/n\\},\\infty)$ and $p_1$ denotes the uniformly upper type index of $\\vz$. Let $\\Phi: [0,\\infty)\\to[0,\\infty)$ be nondecreasing, $\\Phi(0)=0$, $\\Phi(t)>0$ when $t>0$, and $\\lim_{t\\to\\infty}\\Phi(t)=\\infty$, $w\\in A_\\infty(\\mathbb R^n)$ and $\\phi: (0,\\infty)\\to(0,\\infty)$ be nonincreasing. The authors also introduce the weighted Orlicz-Morrey space $M_w^{\\Phi,\\phi}(\\mathbb R^n)$ and obtain the boundedness on $M_w^{\\Phi,\\phi}(\\mathbb R^n)$ of the aforementioned intrinsic Littlewood-Paley functions and their commutators with ${\\rm BMO}(\\rn)$ functions. Finally, for $q\\in[1,\\fz)$, the boundedness of the aforementioned intrinsic Littlewood-Paley functions on the Musielak Orlicz Campanato space $\\mathcal L^{\\varphi,q}(\\mathbb R^n)$ is also established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-25T14:13:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B35",
      "46E30",
      "46E35"
    ],
    "keywords": [
      "boundedness",
      "aforementioned intrinsic littlewood-paley functions",
      "musielak orlicz campanato space",
      "intrinsic lusin area function",
      "musielak-orlicz morrey space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.6512L"
    }
  }
}