{
  "id": "1309.7576",
  "version": "v2",
  "published": "2013-09-29T11:24:44.000Z",
  "updated": "2014-02-06T13:08:04.000Z",
  "title": "Towards spaces of harmonic functions with traces in square Campanato space and its scaling invariant",
  "authors": [
    "Renjin Jiang",
    "Jie Xiao",
    "Dachun Yang"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "For $n\\ge 1$ and $\\alpha\\in (-1,1)$, let $H^{\\alpha,2}$ be the space of harmonic functions $u$ on the upper half space $\\mathbb{R}^{n+1}_+$ satisfying $$\\displaystyle\\sup_{(x_0,r)\\in \\mathbb R^{n+1}_+}r^{-(2\\alpha+n)}\\int_{B(x_0,r)}\\int_0^r|\\nabla_{x,t} u(x,t)|^2t\\,dt\\,dx<\\infty,$$ and $\\mathcal{L}_{2,n+2\\alpha}$ be the Campanato space on $\\mathbb R^n$. We show that $H^{\\alpha,2}$ coincide with $e^{-t\\sqrt{-\\Delta}}\\mathcal{L}_{2,n+2\\alpha}$ for all $\\alpha\\in (-1,1)$, where the case $\\alpha\\in [0,1)$ was originally discovered by Fabes, Johnson and Neri [Indiana Univ. Math. J. 25 (1976), 159-170] and yet the case $\\alpha\\in (-1,0)$ was left open. Moreover, for the scaling invariant version of $H^{\\alpha,2}$, $\\mathcal{H}^{\\alpha,2}$, which comprises all harmonic functions $u$ on $\\mathbb R^{n+1}_+$ satisfying $$\\sup_{(x_0,r)\\in\\mathbb R^{n+1}_+}r^{-(2\\alpha+n)}\\int_{B(x_0,r)} \\int_0^r|\\nabla_{x,t} u(x,t)|^2\\,t^{1+2\\alpha} \\,dt\\,dx<\\infty,$$ we show that $\\mathcal{H}^{\\alpha,2}=e^{-t\\sqrt{-\\Delta}}(-\\Delta)^\\frac{\\alpha}{2}\\mathcal{L}_{2,n+2\\alpha}$, where $(-\\Delta)^{\\frac{\\alpha}{2}}\\mathcal{L}_{2,n+2\\alpha}$ is the collection of all functions $f$ such that $(-\\Delta)^{-\\frac{\\alpha}{2}}f$ are in $\\mathcal{L}_{2,n+2\\alpha}$. Analogues for solutions to the heat equation are also established. As an application, we show that the spaces $\\big((-\\Delta)^{\\frac{\\alpha}{2}}\\mathcal{L}_{2,n+2\\alpha}\\big)^{-1}$ unify $Q_{\\alpha}^{-1}$, ${\\mathrm{BMO}}^{-1}$ and $\\dot{B}^{-1,\\infty}_\\infty$ naturally.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-06T13:08:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B05",
      "35Q30",
      "42B37",
      "46E35"
    ],
    "keywords": [
      "harmonic functions",
      "square campanato space",
      "upper half space",
      "heat equation",
      "indiana univ"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.7576J"
    }
  }
}