{
  "id": "1103.1822",
  "version": "v1",
  "published": "2011-03-09T16:17:14.000Z",
  "updated": "2011-03-09T16:17:14.000Z",
  "title": "Paraproducts and Products of functions in $BMO(\\mathbb R^n)$ and $H^1(\\mathbb R^n)$ through wavelets",
  "authors": [
    "Aline Bonami",
    "Sandrine Grellier",
    "Luong Dang Ky"
  ],
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \\BMO(\\bR^n)$ and $\\H^1(\\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\\H^1(\\bR^n)\\times \\BMO(\\bR^n) $ into $L^1(\\bR^n)$, the other one from $\\H^1(\\bR^n)\\times \\BMO(\\bR^n) $ into a new kind of Hardy-Orlicz space denoted by $\\H^{\\log}(\\bR^n)$. More precisely, the space $\\H^{\\log}(\\bR^n)$ is the set of distributions $f$ whose grand maximal function $\\mathcal Mf$ satisfies $$\\int_{\\mathbb R^n} \\frac {|\\mathcal M f(x)|}{\\log(e+|x|) +\\log (e+ |\\mathcal Mf(x)|)}dx <\\infty.$$ The two bilinear operators can be defined in terms of paraproducts. As a consequence, we find an endpoint estimate involving the space $\\H^{\\log}(\\bR^n)$ for the $\\div$-$\\curl$ lemma.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-09T16:17:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "paraproducts",
      "grand maximal function",
      "continuous bilinear operators",
      "distribution sense",
      "endpoint estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.1822B"
    }
  }
}