{
  "id": "1512.04790",
  "version": "v1",
  "published": "2015-12-15T14:07:18.000Z",
  "updated": "2015-12-15T14:07:18.000Z",
  "title": "Absolutely summing operators and atomic decomposition in bi-parameter Hardy spaces",
  "authors": [
    "Paul F. X. Müller",
    "Johanna Penteker"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For $f \\in H^p(\\delta^2)$, $0<p\\leq 2$, with Haar expansion $f=\\sum f_{I \\times J}h_{I\\times J}$ we constructively determine the Pietsch measure of the $2$-summing multiplication operator \\[\\mathcal{M}_f:\\ell^{\\infty} \\rightarrow H^p(\\delta^2), \\quad (\\varphi_{I\\times J}) \\mapsto \\sum \\varphi_{I\\times J}f_{I \\times J}h_{I \\times J}. \\] Our method yields a constructive proof of Pisier's decomposition of $f \\in H^p(\\delta^2)$ \\[|f|=|x|^{1-\\theta}|y|^{\\theta}\\quad\\quad \\text{ and }\\quad\\quad \\|x\\|_{X_0}^{1-\\theta}\\|y\\|^{\\theta}_{H^2(\\delta^2)}\\leq C\\|f\\|_{H^p(\\delta^2)}, \\] where $X_0$ is Pisier's extrapolation lattice associated to $H^p(\\delta^2)$ and $H^2(\\delta^2)$. Our construction of the Pietsch measure for the multiplication operator $\\mathcal{M}_f$ involves the Haar coefficients of $f$ and its atomic decomposition. We treated the one-parameter $H^p$-spaces in [P.F.X M\\\"uller, J.Penteker, $p$-summing multiplication operators, dyadic Hardy spaces and atomic decomposition, Houston Journal Math.,41(2):639-668,2015.].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-15T14:07:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B30",
      "46B25",
      "46B09",
      "46B42",
      "46E40",
      "47B10",
      "60G42"
    ],
    "keywords": [
      "atomic decomposition",
      "bi-parameter hardy spaces",
      "absolutely summing operators",
      "summing multiplication operator",
      "pietsch measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}