{
  "id": "1309.5839",
  "version": "v3",
  "published": "2013-09-23T15:20:09.000Z",
  "updated": "2014-02-12T20:14:46.000Z",
  "title": "Two weight norm inequalities for the $g$ function",
  "authors": [
    "Michael T Lacey",
    "Kangwei Li"
  ],
  "comment": "15 pages. Reflects the report from referee",
  "categories": [
    "math.CA"
  ],
  "abstract": "Given two weights $\\sigma, w$ on $\\mathbb R ^{n}$, the classical $g$-function satisfies the norm inequality $\\lVert g (f\\sigma)\\rVert_{L ^2 (w)} \\lesssim \\lVert f\\rVert_{L ^2 (\\sigma)}$ if and only if the two weight Muckenhoupt $A_2$ condition holds, and a family of testing conditions holds, namely \\begin{equation*} \\iint_{Q (I)} (\\nabla P_t (\\sigma \\mathbf 1_I)(x, t))^2 \\; dw \\, t dt \\lesssim \\sigma (I) \\end{equation*} uniformly over all cubes $I \\subset \\mathbb R ^{n}$, and $Q (I)$ is the Carleson box over $I$. A corresponding characterization for the intrinsic square function of Wilson also holds.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-02-12T20:14:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weight norm inequalities",
      "norm inequality",
      "intrinsic square function",
      "weight muckenhoupt",
      "condition holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.5839L"
    }
  }
}