{
  "id": "1606.04309",
  "version": "v1",
  "published": "2016-06-14T11:04:49.000Z",
  "updated": "2016-06-14T11:04:49.000Z",
  "title": "Non-homogeneous square functions on general sets: suppression and big pieces methods",
  "authors": [
    "Henri Martikainen",
    "Mihalis Mourgoglou",
    "Emil Vuorinen"
  ],
  "comment": "48 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local $Tb$ theorems. The setting is new: we consider conical square functions with cones $\\{x \\in \\mathbb{R}^n \\setminus E: |x-y| < 2 \\operatorname{dist}(x,E)\\}$, $y \\in E$, defined on general closed subsets $E \\subset \\mathbb{R}^n$ supporting a non-homogeneous measure $\\mu$. We obtain boundedness criteria in this generality in terms of weak type testing of measures on regular balls $B \\subset E$, which are doubling and of small boundary. Due to the general set $E$ we use metric space methods. Therefore, we also demonstrate the recent techniques from the metric space point of view, and show that they yield the most general known local $Tb$ theorems even with assumptions formulated using balls rather than the abstract dyadic metric cubes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-14T11:04:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "big pieces methods",
      "non-homogeneous square functions",
      "general set",
      "suppression",
      "abstract dyadic metric cubes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}