{
  "id": "1702.04848",
  "version": "v1",
  "published": "2017-02-16T03:37:33.000Z",
  "updated": "2017-02-16T03:37:33.000Z",
  "title": "On the norm of the operator $aI+bH$ on $L^p(\\mathbb R)$",
  "authors": [
    "Yong Ding",
    "Loukas Grafakos",
    "Kai Zhu"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky \\cite{HKV}: let $H$ be the Hilbert transform and let $a,b$ be real constants. Then for $1<p<\\infty$ the norm of the operator $aI+bH$ from $L^p(\\mathbb R)$ to $L^p(\\mathbb R)$ is equal to $$ \\bigg(\\max_{x\\in \\Bbb R}\\frac{|ax-b+(bx+a)\\tan \\frac{\\pi}{2p}|^p+|ax-b-(bx+a)\\tan \\frac{\\pi}{2p}|^p}{|x+\\tan \\frac{\\pi}{2p}|^p+|x-\\tan \\frac{\\pi}{2p}|^p} \\bigg)^{\\frac 1p}. $$ Our proof avoids passing through the analogous result for the conjugate function on the circle, as in \\cite{HKV}, and is given directly on the line. We also provide new approximate extremals for $aI+bH$ in the case $p>2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-16T03:37:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A45",
      "42A50",
      "42A99"
    ],
    "keywords": [
      "direct proof",
      "hilbert transform",
      "real constants",
      "conjugate function",
      "approximate extremals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}