{
  "id": "1710.01346",
  "version": "v1",
  "published": "2017-10-03T18:48:28.000Z",
  "updated": "2017-10-03T18:48:28.000Z",
  "title": "The Sharp Constant in the Weak (1,1) Inequality for the Square Function: A New Proof",
  "authors": [
    "Irina Holmes",
    "Paata Ivanisvili",
    "Alexander Volberg"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "In this note we give a new proof of the sharp constant $C = e^{-1/2} + \\int_0^1 e^{-x^2/2}\\,dx$ in the weak (1, 1) inequality for the dyadic square function. The proof makes use of two Bellman functions $\\mathbb{L}$ and $\\mathbb{M}$ related to the problem, and relies on certain relationships between $\\mathbb{L}$ and $\\mathbb{M}$, as well as the boundary values of these functions, which we find explicitly. Moreover, these Bellman functions exhibit an interesting behavior: the boundary solution for $\\mathbb{M}$ yields the optimal obstacle condition for $\\mathbb{L}$, and vice versa.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-03T18:48:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp constant",
      "inequality",
      "bellman functions",
      "dyadic square function",
      "optimal obstacle condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}