{
  "id": "1805.06060",
  "version": "v1",
  "published": "2018-05-15T22:49:15.000Z",
  "updated": "2018-05-15T22:49:15.000Z",
  "title": "Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type",
  "authors": [
    "Amalia Culiuc",
    "Francesco Di Plinio",
    "Michael Lacey",
    "Yumeng Ou"
  ],
  "comment": "24 pages, 4 figures. Submitted",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove sharp endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain a sharp range of weighted norm inequalities for these operators. In particular, we obtain the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<\\infty$ for both Marcinkiewicz multipliers and Walsh-Littlewood-Paley square functions. Zygmund's $L{(\\log L)^{{\\frac12}}}$ inequality is the core of our lacunary multi-frequency projection proof.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-15T22:49:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "endpoint sparse bounds",
      "marcinkiewicz type",
      "walsh-fourier multipliers",
      "sharp endpoint-type sparse bounds",
      "lacunary multi-frequency projection proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}