{
  "id": "2408.08366",
  "version": "v1",
  "published": "2024-08-15T18:18:55.000Z",
  "updated": "2024-08-15T18:18:55.000Z",
  "title": "The Operator Norm of Paraproducts on Bi-parameter Hardy spaces",
  "authors": [
    "Shahaboddin Shaabani"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "It is shown that for all positive values of $p$, $q$, and $r$ with $\\frac{1}{q} = \\frac{1}{p} + \\frac{1}{r}$, the operator norm of the dyadic paraproduct of the form \\[ \\pi_g(f) := \\sum_{R \\in \\Dtwo} g_R \\avr{f}{R} h_R, \\] from the bi-parameter dyadic Hardy space $\\dyprodhp$ to $\\dotdyprodhq$ is comparable to $\\dotdyprodhrn{g}$. We also prove that for all $0 < p < \\infty$, there holds \\[ \\dyprodbmon{g} \\simeq \\|\\pi_g\\|_{\\dyprodhp \\to \\dotdyprodhp}. \\] Similar results are obtained for bi-parameter Fourier paraproducts of the same form.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-15T18:18:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bi-parameter hardy spaces",
      "operator norm",
      "bi-parameter dyadic hardy space",
      "bi-parameter fourier paraproducts",
      "dyadic paraproduct"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}