{
  "id": "1804.11167",
  "version": "v1",
  "published": "2018-04-30T12:59:39.000Z",
  "updated": "2018-04-30T12:59:39.000Z",
  "title": "The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator",
  "authors": [
    "Tuomas P. Hytönen"
  ],
  "comment": "34 pages",
  "categories": [
    "math.CA",
    "math.AP",
    "math.FA"
  ],
  "abstract": "Supplying the missing necessary conditions, we complete the characterisation of the $L^p\\to L^q$ boundedness of commutators $[b,T]$ of pointwise multiplication and Calder\\'on-Zygmund operators, for arbitrary pairs of $1<p,q<\\infty$ and under minimal non-degeneracy hypotheses on $T$. For $p\\leq q$ (and especially $p=q$), this extends a long line of results under more restrictive assumptions on $T$. In particular, we answer a recent question of Lerner, Ombrosi, and Rivera-R\\'ios by showing that $b\\in BMO$ is necessary for the $L^p$-boundedness of $[b,T]$ for any non-zero homogeneous singular integral $T$. We also deal with iterated commutators and weighted spaces. For $p>q$, our results are new even for special classical operators with smooth kernels. As an application, we show that every $f\\in L^p(R^d)$ can be represented as a convergent series of normalised Jacobians $Ju=\\det\\nabla u$ of $u\\in \\dot W^{1,dp}(R^d)^d$. This extends, from $p=1$ to $p>1$, a result of Coifman, Lions, Meyer and Semmes about $J:\\dot W^{1,d}(R^d)^d\\to H^1(R^d)$, and supports a conjecture of Iwaniec about the solvability of the equation $Ju=f\\in L^p(R^d)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-30T12:59:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25",
      "42B37",
      "35F20",
      "47B47"
    ],
    "keywords": [
      "jacobian operator",
      "boundedness",
      "commutators",
      "application",
      "non-zero homogeneous singular integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}