{
  "id": "1203.6335",
  "version": "v3",
  "published": "2012-03-28T18:51:10.000Z",
  "updated": "2015-04-09T05:07:56.000Z",
  "title": "Endpoint estimates for commutators of singular integrals related to Schrödinger operators",
  "authors": [
    "Luong Dang Ky"
  ],
  "comment": "Rev. Mat. Iberoam. (to appear)",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $L= -\\Delta+ V$ be a Schr\\\"odinger operator on $\\mathbb R^d$, $d\\geq 3$, where $V$ is a nonnegative potential, $V\\ne 0$, and belongs to the reverse H\\\"older class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a class $\\mathcal K_L$ of sublinear operators containing the fundamental operators in harmonic analysis related to $L$. More precisely, when $T\\in \\mathcal K_L$, we prove that there exists a bounded subbilinear operator $\\mathfrak R= \\mathfrak R_T: H^1_L(\\mathbb R^d)\\times BMO(\\mathbb R^d)\\to L^1(\\mathbb R^d)$ such that $|T(\\mathfrak S(f,b))|- \\mathfrak R(f,b)\\leq |[b,T](f)|\\leq \\mathfrak R(f,b) + |T(\\mathfrak S(f,b))|$, where $\\mathfrak S$ is a bounded bilinear operator from $H^1_L(\\mathbb R^d)\\times BMO(\\mathbb R^d)$ into $L^1(\\mathbb R^d)$ which does not depend on $T$. The subbilinear decomposition (\\ref{abstract 1}) explains why commutators with the fundamental operators are of weak type $(H^1_L,L^1)$, and when a commutator $[b,T]$ is of strong type $(H^1_L,L^1)$. Also, we discuss the $H^1_L$-estimates for commutators of the Riesz transforms associated with the Schr\\\"odinger operator $L$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-04-13T16:19:49.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-09T05:07:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B35",
      "35J10",
      "42B20"
    ],
    "keywords": [
      "singular integrals",
      "endpoint estimates",
      "schrödinger operators",
      "commutator",
      "fundamental operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.6335D"
    }
  }
}