{
  "id": "2003.00208",
  "version": "v1",
  "published": "2020-02-29T08:15:25.000Z",
  "updated": "2020-02-29T08:15:25.000Z",
  "title": "An integral arising from dyadic average of Riesz transforms",
  "authors": [
    "Chih-Chieh Hung",
    "Chun-Yen Shen"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In the work of S. Petermichl, S. Treil and A. Volberg it was explicitly constructed that the Riesz transforms in any dimension $n \\geq 2$ can be obtained as an average of dyadic Haar shifts provided that an integral is nonzero. It was shown in the paper that when $n=2$, the integral is indeed nonzero (negative) but for $n \\geq 3$ the nonzero property remains unsolved. In this paper we show that the integral is nonzero (negative) for $n=3$. The novelty in our proof is the delicate decompositions of the integral for which we can either find their closed forms or prove an upper bound.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-29T08:15:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riesz transforms",
      "dyadic average",
      "integral arising",
      "dyadic haar shifts",
      "nonzero property remains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}