{
  "id": "2006.05628",
  "version": "v1",
  "published": "2020-06-10T03:02:55.000Z",
  "updated": "2020-06-10T03:02:55.000Z",
  "title": "A two weight inequality for Calderón-Zygmund operators on spaces of homogeneous type with applications",
  "authors": [
    "Xuan Thinh Duong",
    "Ji Li",
    "Eric T. Sawyer",
    "Manasa N. Vempati",
    "Brett D. Wick",
    "Dongyong Yang"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $(X,d,\\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $\\mu $ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Borel measures on $(X,d,\\mu )$. Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calder\\'{o}n--Zygmund operator $T$ from $L^{2}(u)$ to $L^{2}(v)$ in terms of the $A_{2}$ condition and two testing conditions. For every cube $B\\subset X$% , we have the following testing conditions, with $\\mathbf{1}_{B}$ taken as the indicator of $B$ \\begin{equation*} \\Vert T(u\\mathbf{1}_{B})\\Vert _{L^{2}(B, v)}\\leq \\mathcal{T}\\Vert 1_{B}\\Vert _{L^{2}(u)}, \\end{equation*}% \\begin{equation*} \\Vert T^{\\ast }(v\\mathbf{1}_{B})\\Vert _{L^{2}(B, u)}\\leq \\mathcal{T}\\Vert 1_{B}\\Vert _{L^{2}(v)}. \\end{equation*}% The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-10T03:02:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homogeneous type",
      "calderón-zygmund operators",
      "weight inequality",
      "applications",
      "pivotal side condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}