{
  "id": "1506.05910",
  "version": "v1",
  "published": "2015-06-19T08:35:31.000Z",
  "updated": "2015-06-19T08:35:31.000Z",
  "title": "Products of Functions in ${\\mathop\\mathrm{BMO}}({\\mathcal X})$ and $H^1_{\\rm at}({\\mathcal X})$ via Wavelets over Spaces of Homogeneous Type",
  "authors": [
    "Xing Fu",
    "Dachun Yang",
    "Yiyu Liang"
  ],
  "comment": "80 pages, Submitted",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $({\\mathcal X},d,\\mu)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\\rm at}({\\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hyt\\\"onen, the authors prove that the product $f\\times g$ of $f\\in H^1_{\\rm at}({\\mathcal X})$ and $g\\in\\mathop\\mathrm{BMO}({\\mathcal X})$, viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $H^1_{\\rm at}({\\mathcal X})\\times\\mathop\\mathrm{BMO}({\\mathcal X})$ into $L^1({\\mathcal X})$ and from $H^1_{\\rm at}({\\mathcal X})\\times\\mathop\\mathrm{BMO}({\\mathcal X})$ into $H^{\\log}({\\mathcal X})$, which affirmatively confirms the conjecture suggested by A. Bonami and F. Bernicot (This conjecture was presented by L. D. Ky in [J. Math. Anal. Appl. 425 (2015), 807-817]). As byproducts, the authors obtain an unconditional basis of $H^1_{\\rm at}({\\mathcal X})$ and several equivalent characterizations of $H^1_{\\rm at}({\\mathcal X})$ in terms of wavelets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-19T08:35:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B30",
      "47A07",
      "42C40",
      "30L99"
    ],
    "keywords": [
      "homogeneous type",
      "metric measure space",
      "atomic hardy space",
      "equivalent characterizations",
      "unconditional basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 80,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150605910F"
    }
  }
}