{
  "id": "1107.4348",
  "version": "v1",
  "published": "2011-07-21T19:42:40.000Z",
  "updated": "2011-07-21T19:42:40.000Z",
  "title": "Paraproducts via $H^\\infty$-functional calculus",
  "authors": [
    "Dorothee Frey"
  ],
  "comment": "26 pages",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\\{e^{-tL}\\}_{t>0}$ satisfies Davies-Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated to $L$. We show various boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO spaces $H^p_L(X)$ and $BMO_L(X)$. In generalization of standard paraproducts constructed via convolution operators, we show $L^2(X)$ off-diagonal estimates as a substitute for Calder\\'on-Zygmund kernel estimates. As an application, we study differentiability properties of paraproducts in terms of fractional powers of the operator $L$. The results of this paper are fundamental for the proof of a T(1)-Theorem for operators beyond Calder\\'on-Zygmund theory, which will be the subject of a forthcoming paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-07-21T19:42:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "study differentiability properties",
      "bounded holomorphic functional calculus",
      "satisfies davies-gaffney estimates",
      "calderon-zygmund kernel estimates",
      "calderon-zygmund theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.4348F"
    }
  }
}