{
  "id": "2011.12655",
  "version": "v1",
  "published": "2020-11-25T11:32:11.000Z",
  "updated": "2020-11-25T11:32:11.000Z",
  "title": "$L^{p}$ estimates and weighted estimates of fractional maximal rough singular integrals on homogeneous groups",
  "authors": [
    "Yanping Chen",
    "Zhijie Fan",
    "Ji Li"
  ],
  "comment": "45 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the $L^{p}$ boundedness and $L^{p}(w)$ boundedness ($1<p<\\infty$ and $w$ a Muckenhoupt $A_{p}$ weight) of fractional maximal singular integral operators $T_{\\Omega,\\alpha}^{\\#}$ with homogeneous convolution kernel $\\Omega(x)$ on an arbitrary homogeneous group $\\mathbb H$ of dimension $\\mathbb{Q}$. We show that if $0<\\alpha<\\mathbb{Q}$, $\\Omega\\in L^{1}(\\Sigma)$ and satisfies the cancellation condition of order $[\\alpha]$, then for any $1<p<\\infty$, \\begin{align*} \\|T_{\\Omega,\\alpha}^{\\#}f\\|_{L^{p}(\\mathbb{H})}\\lesssim\\|\\Omega\\|_{L^{1}(\\Sigma)}\\|f\\|_{L_{\\alpha}^{p}(\\mathbb{H})}, \\end{align*} where for the case $\\alpha=0$, the $L^p$ boundedness of rough singular integral operator and its maximal operator were studied by Tao (\\cite{Tao}) and Sato (\\cite{sato}), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if $0\\leq\\alpha<\\mathbb{Q}$ and $\\Omega$ satisfies the same cancellation condition but a stronger condition that $\\Omega\\in L^{q}(\\Sigma)$ for some $q>\\mathbb{Q}/\\alpha$, then for any $1<p<\\infty$ and $w\\in A_{p}$, \\begin{align*} \\|T_{\\Omega,\\alpha}^{\\#}f\\|_{L^{p}(w)}\\lesssim\\|\\Omega\\|_{L^{q}(\\Sigma)}\\{w\\}_{A_p}(w)_{A_p}\\|f\\|_{L_{\\alpha}^{p}(w)},\\ \\ 1<p<\\infty. \\end{align*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-25T11:32:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B20",
      "43A85"
    ],
    "keywords": [
      "fractional maximal rough singular integrals",
      "homogeneous group",
      "weighted estimates",
      "fractional maximal singular integral operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}