{
  "id": "1803.10302",
  "version": "v2",
  "published": "2018-03-27T20:22:48.000Z",
  "updated": "2018-05-13T10:31:27.000Z",
  "title": "Remark on atomic decompositions for Hardy space $H^1$ in the rational Dunkl setting",
  "authors": [
    "Jacek Dziubański",
    "Agnieszka Hejna"
  ],
  "comment": "2 figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Delta$ be the Dunkl Laplacian on $\\mathbb R^N$ associated with a normalized root system $R$ and a multiplicity function $k(\\alpha)\\geq 0$. We say that a function $f$ belongs to the Hardy space $H^1_{\\Delta}$ if the nontangential maximal function $\\mathcal M_H f(\\mathbf x)=\\sup_{\\| \\mathbf x-\\mathbf y\\|<t} |\\exp(t^2\\Delta )f(\\mathbf x)|$ belongs to $L^1(w(\\mathbf x)\\, d\\mathbf x)$, where $w(\\mathbf x)=\\prod_{\\alpha\\in R} |\\langle \\alpha,\\mathbf x\\rangle|^{k(\\alpha)}$. We prove that $H^1_\\Delta$ coincides with the space $H^1_{\\rm atom}(\\mathbb R^N, \\| \\mathbf x-\\mathbf y\\|, w(\\mathbf x)d\\mathbf x)$ understood as the atomic Hardy space on the space of homogeneous type in the sense of Coifman--Weiss. To this end we improve estimates for the heat kernel of $e^{t\\Delta}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-05-13T10:31:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B30",
      "33C52",
      "35J05",
      "35K08",
      "42B25",
      "42B35",
      "42B37",
      "42C05"
    ],
    "keywords": [
      "rational dunkl setting",
      "atomic decompositions",
      "nontangential maximal function",
      "atomic hardy space",
      "normalized root system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}