{
  "id": "1703.02935",
  "version": "v1",
  "published": "2017-03-08T17:46:49.000Z",
  "updated": "2017-03-08T17:46:49.000Z",
  "title": "Absolute continuity and $α$-numbers on the real line",
  "authors": [
    "Tuomas Orponen"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $\\mu,\\nu$ be Radon measures on $\\mathbb{R}$. For an interval $I \\subset \\mathbb{R}$, define $\\alpha_{\\mu,\\nu}(I) := \\mathbb{W}_{1}(\\mu_{I},\\nu_{I})$, the Wasserstein distance of normalised blow-ups of $\\mu$ and $\\nu$ restricted to $I$. Let $\\mathcal{S}_{\\nu}$ be either one of the square functions $$\\mathcal{S}^{2}_{\\nu}(\\mu)(x) = \\sum_{x \\in I \\in \\mathcal{D}} \\alpha_{\\mu,\\nu}^{2}(I) \\quad \\text{or} \\quad \\mathcal{S}_{\\nu}^{2}(\\mu)(x) = \\int_{0}^{1} \\alpha_{\\mu,\\nu}^{2}(B(x,r)) \\, \\frac{dr}{r},$$ where $\\mathcal{D}$ is the family of dyadic intervals of side-length at most one. Assuming that $\\nu$ is doubling, and $\\mu$ does not charge the boundaries of $\\mathcal{D}$ (in the case of the first square function), I prove that $\\mu|_{G} \\ll \\nu$, where $$G = \\{x : \\mathcal{S}_{\\nu}(\\mu)(x) < \\infty\\}.$$ If $\\mu$ is also doubling, and the square functions satisfy suitable Carleson conditions, then absolute continuity can be improved to $\\mu \\in A_{\\infty}(\\nu)$. The results answer the simplest \"$n = d = 1\"$ case of a problem of J. Azzam, G. David and T. Toro.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-08T17:46:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A99"
    ],
    "keywords": [
      "absolute continuity",
      "real line",
      "square functions satisfy suitable carleson",
      "functions satisfy suitable carleson conditions",
      "first square function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}