{
  "id": "1711.10578",
  "version": "v1",
  "published": "2017-11-28T21:54:39.000Z",
  "updated": "2017-11-28T21:54:39.000Z",
  "title": "Martingale transform and Square function: some end-point weak weighted estimates",
  "authors": [
    "Paata Ivanisvili",
    "Alexander Volberg"
  ],
  "categories": [
    "math.AP",
    "math.CA",
    "math.PR"
  ],
  "abstract": "Following the ideas of [8] we prove that there is a sequence of weights $w\\in A^d_1$ such that $[w]^d_{A_1}\\to \\infty$, and martingale transforms $T$ such that $\\|T: L^1(w) \\to L^{1, \\infty}(w)\\| \\ge c [w]^d_{A_1}\\log [w]^d_{A_1}$ with an absolute positive $c$. We also show the existence of the sequence of weights such that $\\|S\\|_{w} \\ge c \\|M^d\\|_{w}\\sqrt{\\log \\|M^d\\|_{w}}$. Finally we show that for any weight $w\\in A^d_2$ and any characteristic function of a dyadic interval $\\|S_w\\chi_I\\|_{L^{2, \\infty}(w^{-1})} \\le C \\sqrt{[w]_{A^d_2}}\\, \\|\\chi_I\\|_w$. So on characteristic functions at least, no logarithmic correction is needed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-28T21:54:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B35",
      "47A30"
    ],
    "keywords": [
      "end-point weak weighted estimates",
      "martingale transform",
      "square function",
      "characteristic function",
      "dyadic interval"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}