{
  "id": "1010.0755",
  "version": "v2",
  "published": "2010-10-05T02:54:56.000Z",
  "updated": "2010-12-08T14:14:40.000Z",
  "title": "Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture",
  "authors": [
    "Tuomas Hytönen",
    "Carlos Pérez",
    "Sergei Treil",
    "Alexander Volberg"
  ],
  "comment": "38 pages; v2: the weighted bound for shifts is now quadratic in shift complexity (as opposed to cubic in v1)",
  "categories": [
    "math.CA"
  ],
  "abstract": "We give a self-contained proof of the $A_2$ conjecture, which claims that the norm of any Calderon-Zygmund operator is bounded by the first degree of the $A_2$ norm of the weight. The original proof of this result by the first author relied on a subtle and rather difficult reduction to a testing condition by the last three authors. Here we replace this reduction by a new weighted norm bound for dyadic shifts - linear in the $A_2$ norm of the weight and quadratic in the complexity of the shift -, which is based on a new quantitative two-weight inequality for the shifts. These sharp one- and two-weight bounds for dyadic shifts are the main new results of this paper. They are obtained by rethinking the corresponding previous results of Lacey-Petermichl-Reguera and Nazarov-Treil-Volberg. To complete the proof of the $A_2$ conjecture, we also provide a simple variant of the representation, already in the original proof, of an arbitrary Calderon-Zygmund operator as an average of random dyadic shifts and random dyadic paraproducts. This method of the representation amounts to the refinement of the techniques from nonhomogeneous Harmonic Analysis.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-12-08T14:14:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B35",
      "47A30"
    ],
    "keywords": [
      "sharp weighted estimates",
      "conjecture",
      "original proof",
      "arbitrary calderon-zygmund operator",
      "random dyadic shifts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.0755H"
    }
  }
}