{
  "id": "2103.08799",
  "version": "v1",
  "published": "2021-03-16T01:43:30.000Z",
  "updated": "2021-03-16T01:43:30.000Z",
  "title": "Weighted estimates of commutators for $0<p<\\infty$",
  "authors": [
    "Shunchao Long"
  ],
  "comment": "26 pages,0 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "$BMO$ commutators of some sublinear operators such as singular integral operators and Hardy-Littelwood maximal operator are well known to be bounded from $L_w^p$ to itself for all $1<p<\\infty$ and all $w\\in A_p$ (the classical Muckenhopt class), but for these commutators, it has been an open question how to extend the estimate to all $0<p<\\infty$. In addition, there are many classical operators, especially some maximal operators such as Carleson maximal operator, for which the $L^p$-boundedness holds for $1<p<\\infty$, but for the $BMO$ commutators of these operators, it has also been an open question whether there are similar unweighted or weighted estimates for each $0<p<\\infty$. In this paper, giving a weights class $A_p^+$ with $0<p\\leq \\infty$ which is an extension of $A_p$ when $1\\leq p\\leq \\infty$ and is a refinement of $A_1$ when $0< p\\leq 1$, for $BMO$ commutators of some sublinear operators, we obtain weighted estimates from some subspaces of $L^p_w$ to $L^p_w$ (and to themselves) for all $0<p<\\infty$ and $w\\in A_q^+$ with $0<q<p$. These results are applied the commutators in above two questions. In particular, these imply that the $BMO$ commutators of many classical operators including those mentioned above are bounded from $H^p_w$ to $L^p_w$ and from $H^p_w$ to itself for all $0<p\\leq 1$ and $w\\in A_q^+$ with $0<q<p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-16T01:43:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weighted estimates",
      "commutators",
      "sublinear operators",
      "open question",
      "hardy-littelwood maximal operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}