{
  "id": "1907.00690",
  "version": "v1",
  "published": "2019-07-01T12:15:10.000Z",
  "updated": "2019-07-01T12:15:10.000Z",
  "title": "On pointwise $\\ell^r$-sparse domination in a space of homogeneous type",
  "authors": [
    "Emiel Lorist"
  ],
  "comment": "35 pages",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "We prove a general sparse domination theorem in spaces of homogeneous type, in which we control a vector-valued operator pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the $\\ell^1$-sum in the sparse operator is replaced by an $\\ell^r$-sum. This sparse domination theorem is applicable to many operators from both harmonic analysis and PDE and yields simple and unified proofs for the (sharp) weighted $L^p$-boundedness of these operators. As an illustration of the versatility of our theorem we prove the $A_2$-theorem for vector-valued Calder\\'on--Zygmund operators in a space of homogeneous type and sharp weighted norm inequalities for Littlewood--Paley operators and both the lattice Hardy--Littlewood and the Rademacher maximal operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-01T12:15:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25",
      "46E40"
    ],
    "keywords": [
      "homogeneous type",
      "general sparse domination theorem",
      "sparse operator",
      "rademacher maximal operator",
      "sharp weighted norm inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}