{
  "id": "1409.4612",
  "version": "v1",
  "published": "2014-09-16T12:46:38.000Z",
  "updated": "2014-09-16T12:46:38.000Z",
  "title": "Atomic decompositions for Hardy spaces related to Schrödinger operators",
  "authors": [
    "Marcin Preisner"
  ],
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "Let L_U = -Delta+U be a Schr\\\"odinger operator on R^d, where U\\in L^1_{loc}(R^d) is a non-negative potential and d\\geq 3. The Hardy space H^1(L_U) is defined in terms of the maximal function for the semigroup K_{t,U} = exp(-t L_U), namely H^1(L_U) = {f\\in L^1(R^d): \\|f\\|_{H^1(L_U)}:= \\|sup_{t>0} |K_{t,U} f| \\|_{L^1(R^d)} < \\infty. Assume that U=V+W, where V\\geq 0 satisfies the global Kato condition sup_{x\\in R^d} \\int_{R^d} V(y)|x-y|^{2-d} < \\infty. We prove that, under certain assumptions on W\\geq 0, the space H^1(L_U) admits an atomic decomposition of local type. An atom a for H^1(L_U) is either of the form a(x)=|Q|^{-1}\\chi_Q(x), where Q are special cubes determined by W, or a satisfies the cancellation condition \\int a(x)w(x) dx = 0, where w is an (-Delta+V)-harmonic function given by w(x) = lim_{t\\to \\infty} K_{t,V} 1(x). Furthermore, we show that, in some cases, the cancellation condition \\int_{R^d} a(x)w(x) dx = 0 can be replaced by the classical one \\int_{R^d} a(x) dx = 0. However, we construct another example, such that the atomic spaces with these two cancellation conditions are not equivalent as Banach spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-16T12:46:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B30",
      "35J10",
      "42B25",
      "42B35"
    ],
    "keywords": [
      "atomic decomposition",
      "hardy spaces",
      "schrödinger operators",
      "cancellation condition",
      "global kato condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.4612P"
    }
  }
}