{
  "id": "2007.15928",
  "version": "v1",
  "published": "2020-07-31T09:45:36.000Z",
  "updated": "2020-07-31T09:45:36.000Z",
  "title": "Quadratic sparse domination and Weighted Estimates for non-integral Square Functions",
  "authors": [
    "Julian Bailey",
    "Gianmarco Brocchi",
    "Maria Carmen Reguera"
  ],
  "comment": "31 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove a quadratic sparse domination result for general non-integral square functions $S$. That is, we prove an estimate of the form \\begin{equation*} \\int_{M} (S f)^{2} g \\, \\mathrm{d}\\mu \\le c \\sum_{P \\in \\mathcal{S}} \\left(\\frac{1}{\\lvert 5P \\rvert}\\int_{5 P} \\lvert f\\rvert^{p_{0}} \\, \\mathrm{d}\\mu\\right)^{2/p_{0}} \\left(\\frac{1}{\\lvert 5P \\rvert} \\int_{5 P} \\lvert g\\rvert^{q_{0}^*}\\,\\mathrm{d}\\mu\\right)^{1/q_{0}^*} \\lvert P\\rvert, \\end{equation*} where $q_{0}^{*}$ is the H\\\"{o}lder conjugate of $q_{0}/2$, $M$ is the underlying doubling space and $\\mathcal{S}$ is a sparse collection of cubes on $M$. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace-Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space $L^{p}(w)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-31T09:45:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B37"
    ],
    "keywords": [
      "weighted estimates",
      "general non-integral square functions",
      "quadratic sparse domination result",
      "divergence form elliptic operators",
      "derive optimal norm estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}