{
  "id": "math/0305072",
  "version": "v1",
  "published": "2003-05-05T10:02:34.000Z",
  "updated": "2003-05-05T10:02:34.000Z",
  "title": "Littlewood-Paley decompositions and Besov spaces related to symmetric cones",
  "authors": [
    "D. Bekolle",
    "A. Bonami",
    "G. Garrigos",
    "F. Ricci"
  ],
  "comment": "48 pages, 1 figure",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Starting from a Whitney decomposition of a symmetric cone $\\Omega$, analog to the dyadic partition $[2^j, 2^{j+1})$ of the positive real line, in this paper we develop an adapted Littlewood-Paley theory for functions with spectrum in $\\Omega$. In particular, we define a natural class of Besov spaces of such functions, $B^{p,q}_\\nu$, where the role of usual derivation is now played by the generalized wave operator of the cone $\\Delta(\\frac{\\partial}{\\partial x})$. Our main result shows that $B^{p,q}_\\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p,q}_\\nu(T_\\Omega)$, at least in a ``good range'' of indices $1\\leq q<q_{\\nu,p}$. We obtain the sharp $q_{\\nu,p}$ when $p\\leq 2$, and conjecture a critical index for $p>2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_\\nu\\colon L^{p,q}_\\nu\\to A^{p,q}_\\nu$, for which our result implies a positive answer when $q_{\\nu,p}'<q<q_{\\nu,p}$. This extends to general cones previous work of the authors in the light-cone. Finally, we conclude the paper with a finer analysis in light-cones, for which we establish a link between our conjecture and the cone multiplier problem. Moreover, using recent work by Tao, Vargas and Wolff, we improve in dimension 3 the range of $q$'s for which the Bergman projection is bounded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-05-05T10:02:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B35",
      "32M15"
    ],
    "keywords": [
      "besov spaces",
      "symmetric cone",
      "littlewood-paley decompositions",
      "distributional boundary values",
      "cone multiplier problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......5072B"
    }
  }
}