{
  "id": "1310.8531",
  "version": "v1",
  "published": "2013-10-31T14:48:11.000Z",
  "updated": "2013-10-31T14:48:11.000Z",
  "title": "Local $Tb$ theorem with $L^2$ testing conditions and general measures: Calderón-Zygmund operators",
  "authors": [
    "Michael T. Lacey",
    "Henri Martikainen"
  ],
  "comment": "31 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Local $Tb$ theorems with $L^p$ type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. Until very recently, local $Tb$ theorems in the non-homogeneous case had only been proved assuming scale invariant ($L^{\\infty}$ or BMO) testing conditions. The combination of non-scale-invariance and general measures is a delicate issue. In a previous paper we overcame this obstacle in the model case of square functions defined using general measures. In this paper we finally tackle the very demanding case of Calder\\'on-Zygmund operators. That is, we prove a non-homogeneous local $Tb$ theorem with $L^2$ type testing conditions for all Calder\\'on-Zygmund operators. In doing so we prove general twisted martingale transform inequalities which turn out to be subtle in our general framework.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-31T14:48:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "general measures",
      "calderón-zygmund operators",
      "type testing conditions",
      "scale invariant",
      "calderon-zygmund operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.8531L"
    }
  }
}