{
  "id": "1412.0470",
  "version": "v1",
  "published": "2014-12-01T13:31:35.000Z",
  "updated": "2014-12-01T13:31:35.000Z",
  "title": "Operator-valued dyadic shifts and the T(1) theorem",
  "authors": [
    "Timo S. Hänninen",
    "Tuomas P. Hytönen"
  ],
  "comment": "29 pages, 1 figure",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "In this paper we extend dyadic shifts and the dyadic representation theorem to an operator-valued setting: We first define operator-valued dyadic shifts and prove that they are bounded. We then extend the dyadic representation theorem, which states that every scalar-valued Calder\\'on-Zygmund operator can be represented as a series of dyadic shifts and paraproducts averaged over randomized dyadic systems, to operator-valued Calder\\'on-Zygmund operators. As a corollary, we obtain another proof of the operator-valued, global T(1) theorem. We work in the setting of integral operators that have R-bounded operator-valued kernels and act on functions taking values in UMD-spaces. The domain of the functions is the Euclidean space equipped with the Lebesgue measure. In addition, we give new proofs for the following known theorems: Boundedness of the dyadic (operator-valued) paraproduct, a variant of Pythagoras' theorem for (vector-valued) functions adapted to a sparse collection of dyadic cubes, and a decoupling inequality for (UMD-valued) martingale differences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-01T13:31:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "46E40"
    ],
    "keywords": [
      "dyadic representation theorem",
      "first define operator-valued dyadic shifts",
      "extend dyadic shifts",
      "operator-valued calderon-zygmund operators",
      "martingale differences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.0470H"
    }
  }
}