{
  "id": "2204.01886",
  "version": "v1",
  "published": "2022-04-04T23:21:36.000Z",
  "updated": "2022-04-04T23:21:36.000Z",
  "title": "Singular integral operators, $T1$ theorem, Littlewood-Paley theory and Hardy spaces in Dunkl Setting",
  "authors": [
    "Chaoqian Tan",
    "Yanchang Han",
    "Yongsheng Han",
    "Ming-Yi Lee",
    "Ji Li"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "The purpose of this paper is to introduce a new class of singular integral operators in the Dunkl setting involving both the Euclidean metric and the Dunkl metric. Then we provide the $T1$ theorem, the criterion for the boundedness on $L^2$ for these operators. Applying this singular integral operator theory, we establish the Littlewood-Paley theory and the Dunkl-Hardy spaces. As applications, the boundedness of singular integral operators, particularly, the Dunkl-Rieze transforms, on the Dunkl-Hardy spaces is given. The $L^2$ theory and the singular integral operator theory play crucial roles. New tools developed in this paper include the weak-type discrete Calder\\'on reproducing formulae, new test functions, and distributions, the Littlewood-Paley, the wavelet-type decomposition, and molecule characterizations of the Dunkl-Hardy space, Coifman's approximation to the identity and the decomposition of the identity operator on $L^2$, Meyer's commutation Lemma, and new almost orthogonal estimates in the Dunkl setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-04T23:21:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "littlewood-paley theory",
      "dunkl setting",
      "theory play crucial roles",
      "operator theory play crucial",
      "hardy spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}