{
  "id": "1312.0843",
  "version": "v3",
  "published": "2013-12-03T15:04:30.000Z",
  "updated": "2019-11-18T15:47:17.000Z",
  "title": "The two-weight inequality for the Hilbert transform with general measures",
  "authors": [
    "Tuomas P. Hytönen"
  ],
  "comment": "The last version of the article submitted to the publisher (but without journal style file)",
  "journal": "Proc. Lond. Math. Soc. (3) 117 (2018), no. 3, 483-526",
  "doi": "10.1112/plms.12136",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "The two-weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures $\\sigma$ and $w$ on $\\mathbb R$. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two-weight problem by Lacey, Sawyer, Shen and Uriarte-Tuero. Our characterization is in terms of Sawyer-type testing conditions and a variant of the two-weight $A_2$ condition, where $\\sigma$ and $w$ are integrated over complementary intervals only. A key novely of the proof is a two-weight inequality for the Poisson integral with `holes'.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-02T20:52:15.000Z",
      "abstract": "The two-weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures on the line. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two-weight problem by Lacey, Sawyer, Shen and Uriarte-Tuero. Our characterization is in terms of Sawyer-type testing conditions and a variant of the two-weight $A_2$ condition, where the two measures are integrated over complementary intervals only. One of the key novelties of the proof is a two-weight inequality for the Poisson integral with `holes'.",
      "comment": "45 pages. v2: small changes, mainly typo-fix. This version has been submitted",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2019-11-18T15:47:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25"
    ],
    "keywords": [
      "two-weight inequality",
      "hilbert transform",
      "general measures",
      "common point masses",
      "poisson integral"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.0843H"
    }
  }
}