{
  "id": "1412.2593",
  "version": "v1",
  "published": "2014-12-08T14:52:11.000Z",
  "updated": "2014-12-08T14:52:11.000Z",
  "title": "Two-weight $L^p$-$L^q$ bounds for positive dyadic operators: unified approach to $p\\leq q$ and $p>q$",
  "authors": [
    "Timo S. Hänninen",
    "Tuomas P. Hytönen",
    "Kangwei Li"
  ],
  "comment": "27 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We characterize the $L^p(\\sigma)\\to L^q(\\omega)$ boundedness of positive dyadic operators of the form $ T(f\\sigma)=\\sum_{Q\\in\\mathscr{D}}\\lambda_Q\\int_Q f\\,\\mathrm{d}\\sigma\\cdot 1_Q, $ and the $L^{p_1}(\\sigma_1)\\times L^{p_2}(\\sigma_2)\\to L^q(\\omega)$ boundedness of their bilinear analogues, for arbitrary locally finite measures $\\sigma,\\sigma_1,\\sigma_2,\\omega$. In the linear case, we unify the existing \"Sawyer testing\" (for $p\\leq q$) and \"Wolff potential\" (for $p>q$) characterizations into a new \"sequential testing\" characterization valid in all cases. We extend these ideas to the bilinear case, obtaining both sequential testing and potential type characterizations for the bilinear operator and all $p_1,p_2,q\\in(1,\\infty)$. Our characterization covers the previously unknown case $q<\\frac{p_1p_2}{p_1+p_2}$, where we introduce a new two-measure Wolff potential.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-08T14:52:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "47G40"
    ],
    "keywords": [
      "positive dyadic operators",
      "unified approach",
      "two-weight",
      "two-measure wolff potential",
      "potential type characterizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.2593H"
    }
  }
}