{
  "id": "1103.0616",
  "version": "v1",
  "published": "2011-03-03T08:15:22.000Z",
  "updated": "2011-03-03T08:15:22.000Z",
  "title": "Estimates at or beyond endpoint in harmonic analysis: Bochner-Riesz means and spherical means",
  "authors": [
    "Shunchao Long"
  ],
  "comment": "50 pages",
  "categories": [
    "math.CA",
    "math.AP",
    "math.FA"
  ],
  "abstract": "We introduce some new functions spaces to investigate some problems at or beyond endpoint. First, we prove that Bochner-Riesz means $B_R^\\lambda$ are bounded from some subspaces of $L^p_{|x|^\\alpha}$ to $L^p_{|x|^\\alpha}$ for $ \\frac{n-1}{2(n+1)}<\\lambda \\leq \\frac{n-1}{2}, 0 < p\\leq p'_\\lambda=\\frac{2n}{n+1+2\\lambda}, n(\\frac{p}{p_\\lambda}-1)< \\alpha<n(\\frac{p}{p'_\\lambda}-1)$, and $0<R<\\infty,$ and so are the maximal Bochner-Riesz means $B_*^\\lambda$ for $ \\frac{n-1}{2}\\leq \\lambda < \\infty, 0 < p\\leq 1$ and $-n< \\alpha<n(p-1)$. From these we obtain the $L^p_{|x|^\\alpha}$-norm convergent property of $B_R^\\lambda $ for these $\\lambda,p,$ and $\\alpha$. Second, let $n\\geq 3,$ we prove that the maximal spherical means are bounded from some subspaces of $L^p_{|x|^\\alpha}$ to $L^p_{|x|^\\alpha}$ for $0<p\\leq \\frac{n}{n-1}$ and $ -n(1-\\frac{p}{2})<\\alpha<n(p-1)-n$. We also obtain a $L^p_{|x|^\\alpha}$-norm convergent property of the spherical means for such $p$ and $\\alpha$. Finally, we prove that some new types of $|x|^\\alpha$-weighted estimates hold at or beyond endpoint for many operators, such as Hardy-Littlewood maximal operator, some maximal and truncated singular integral operators, the maximal Carleson operator, etc. The new estimates can be regarded as some substitutes for the $(H^p,H^p)$ and $(H^p,L^p)$ estimates for the operators which fail to be of types $(H^p,H^p)$ and $(H^p,L^p)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-03T08:15:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10",
      "42B20",
      "42B25",
      "42B35",
      "42B37"
    ],
    "keywords": [
      "harmonic analysis",
      "norm convergent property",
      "hardy-littlewood maximal operator",
      "maximal bochner-riesz means",
      "truncated singular integral operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.0616L"
    }
  }
}