{
  "id": "1808.10168",
  "version": "v1",
  "published": "2018-08-30T08:10:16.000Z",
  "updated": "2018-08-30T08:10:16.000Z",
  "title": "Atomic and Maximal Function Characterizations of Musielak-Orlicz-Hardy Spaces Associated to Non-negative Self-adjoint Operators on Spaces of Homogeneous Type",
  "authors": [
    "Sibei Yang",
    "Dachun Yang"
  ],
  "comment": "52 pages; Submitted",
  "categories": [
    "math.CA",
    "math.AP",
    "math.FA"
  ],
  "abstract": "Let $\\mathcal{X}$ be a metric space with doubling measure and $L$ a non-negative self-adjoint operator on $L^2(\\mathcal{X})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $\\varphi:\\ \\mathcal{X}\\times[0,\\infty) \\to[0,\\infty)$ satisfies that $\\varphi(x,\\cdot)$ is an Orlicz function and $\\varphi(\\cdot,t)\\in {\\mathbb A}_{\\infty}(\\mathcal{X})$ (the class of uniformly Muckenhoupt weights). Let $H_{\\varphi,\\,L}(\\mathcal{X})$ be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors characterize the space $H_{\\varphi,\\,L}(\\mathcal{X})$ by means of atoms, non-tangential and radial maximal functions associated with $L$. In particular, when $\\mu(\\mathcal{X})<\\infty$, the local non-tangential and radial maximal function characterizations of $H_{\\varphi,\\,L}(\\mathcal{X})$ are obtained. As applications, the authors obtain various maximal function and the atomic characterizations of the \"geometric\" Musielak-Orlicz-Hardy spaces $H_{\\varphi,\\,r}(\\Omega)$ and $H_{\\varphi,\\,z}(\\Omega)$ on the strongly Lipschitz domain $\\Omega$ in $\\mathbb{R}^n$ associated with second-order self-adjoint elliptic operators with the Dirichlet and the Neumann boundary conditions; even when $\\varphi(x,t):=t$ for any $x\\in\\mathbb{R}^n$ and $t\\in[0,\\infty)$, the equivalent characterizations of $H_{\\varphi,\\,z}(\\Omega)$ given in this article improve the known results via removing the assumption that $\\Omega$ is unbounded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-30T08:10:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B35",
      "46E30",
      "30L99"
    ],
    "keywords": [
      "non-negative self-adjoint operator",
      "musielak-orlicz-hardy space",
      "homogeneous type",
      "radial maximal function characterizations",
      "gaussian upper bound estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}