{
  "id": "1007.4330",
  "version": "v1",
  "published": "2010-07-25T16:12:33.000Z",
  "updated": "2010-07-25T16:12:33.000Z",
  "title": "The sharp weighted bound for general Calderon-Zygmund operators",
  "authors": [
    "Tuomas P. Hytönen"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "For a general Calderon-Zygmund operator $T$ on $R^N$, it is shown that $\\|Tf\\|_{L^2(w)}\\leq C(T)\\|w\\|_{A_2}\\|f\\|_{L^2(w)}$ for all Muckenhoupt weights $w\\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Perez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the Nazarov-Treil-Volberg method of random dyadic systems with just one random system and completely without bad parts; (ii) a resulting representation of a general Calderon-Zygmund operator as an average of dyadic shifts; and (iii) improvements of the Lacey-Petermichl-Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-25T16:12:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25"
    ],
    "keywords": [
      "general calderon-zygmund operator",
      "sharp weighted bound",
      "dyadic shifts",
      "random dyadic systems",
      "lacey-petermichl-reguera estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.4330H"
    }
  }
}