{
  "id": "1812.04952",
  "version": "v1",
  "published": "2018-12-12T14:04:03.000Z",
  "updated": "2018-12-12T14:04:03.000Z",
  "title": "Two Weight Inequalities for Positive Operators: Doubling Cubes",
  "authors": [
    "Wei Chen",
    "Michael T. Lacey"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "For the maximal operator $ M $ on $ \\mathbb R ^{d}$, and $ 1< p , \\rho < \\infty $, there is a finite constant $ D = D _{p, \\rho }$ so that this holds. For all weights $ w, \\sigma $ on $ \\mathbb R ^{d}$, the operator $ M (\\sigma \\cdot )$ is bounded from $ L ^{p} (\\sigma ) \\to L ^{p} (w)$ if and only the pair of weights $ (w, \\sigma )$ satisfy the two weight $ A _{p}$ condition, and this testing inequality holds: \\begin{equation*} \\int _{Q} M (\\sigma \\mathbf 1_{Q} ) ^{p} \\; d w \\lesssim \\sigma ( Q), \\end{equation*} for all cubes $ Q$ for which there is a cube $ P \\supset Q$ satisfying $ \\sigma (P) < D \\sigma (Q)$, and $ \\ell P = \\rho \\ell Q$. This was recently proved by Kangwei Li and Eric Sawyer. We give a short proof, which is easily seen to hold for several closely related operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-12T14:04:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weight inequalities",
      "positive operators",
      "doubling cubes",
      "short proof",
      "finite constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}