{
  "id": "2303.14855",
  "version": "v1",
  "published": "2023-03-26T23:42:01.000Z",
  "updated": "2023-03-26T23:42:01.000Z",
  "title": "Weighted $L^p\\to L^q$-boundedness of commutators and paraproducts in the Bloom setting",
  "authors": [
    "Timo S. Hänninen",
    "Emiel Lorist",
    "Jaakko Sinko"
  ],
  "comment": "37 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We characterize the $L^p(\\mu)\\to L^q(\\lambda)$ boundedness of commutators of Calder\\'on--Zygmund operators and pointwise multiplication in the upper triangular Bloom setting: Let $1<q<p<\\infty$, $\\mu\\in A_p$, $\\lambda\\in A_q$ and define the Bloom weight $\\nu$ by $\\nu^{1/p+1/q'}:= \\mu^{1/p} \\lambda^{-1/q}$. For a locally integrable function $b$ and a non-degenerate Calder\\'on--Zygmund operator $T$, the commutator $[b,T]:L^p(\\mu)\\to L^q(\\lambda)$ is bounded if and only if $M^\\#_\\nu b\\in L^{r}(\\nu)$, where $\\frac1r := \\frac1q-\\frac1p$ and $M^\\#_\\nu b$ is the weighted sharp maximal function defined by $$ M^\\#_\\nu b:=\\sup_{Q} \\frac{\\mathbf{1}_Q}{\\nu(Q)} \\int |b-\\langle b\\rangle_Q |\\,\\mathrm{d}x.$$ In parallel to commutators, we also characterize the boundedness of dyadic paraproducts $\\Pi_b$ in the Bloom setting for all $p,q \\in (1,\\infty)$. This completes the characterisation of the $L^p(\\mu) \\to L^q(\\lambda)$-boundedness of both commutators of Calder\\'on--Zygmund operators and pointwise multiplication and of paraproducts in the Bloom setting for all $p,q \\in (1,\\infty)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-26T23:42:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "47B47"
    ],
    "keywords": [
      "commutator",
      "boundedness",
      "paraproducts",
      "weighted sharp maximal function",
      "non-degenerate calderon-zygmund operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}