{
  "id": "2208.10016",
  "version": "v1",
  "published": "2022-08-22T02:02:03.000Z",
  "updated": "2022-08-22T02:02:03.000Z",
  "title": "The $L^p$-to-$L^q$ Compactness of Commutators with $p>q$",
  "authors": [
    "Tuomas Hytönen",
    "Kangwei Li",
    "Jin Tao",
    "Dachun Yang"
  ],
  "categories": [
    "math.FA",
    "math.AP",
    "math.CA"
  ],
  "abstract": "Let $1<q<p<\\infty$, $\\frac1r:=\\frac1q-\\frac1p$, and $T$ be a non-degenerate Calder\\'on--Zygmund operator. We show that the commutator $[b,T]$ is compact from $L^p({\\mathbb R}^n)$ to $L^q({\\mathbb R}^n)$ if and only if the symbol $b=a+c$ with $a\\in L^r({\\mathbb R}^n)$ and $c$ being any constant. Since both the corresponding Hardy--Littlewood maximal operator and the corresponding Calder\\'on--Zygmund maximal operator are not bounded from $L^p({\\mathbb R}^n)$ to $L^q({\\mathbb R}^n)$, we take the full advantage of the compact support of the approximation element in $C_{\\rm c}^\\infty({\\mathbb R}^n)$, which seems to be redundant for many corresponding estimates when $p\\leq q$ but to be crucial when $p>q$. We also extend the results to the multilinear case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-22T02:02:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B47",
      "42B20",
      "42B25",
      "42B35",
      "46E30"
    ],
    "keywords": [
      "commutator",
      "compactness",
      "non-degenerate calderon-zygmund operator",
      "corresponding hardy-littlewood maximal operator",
      "corresponding calderon-zygmund maximal operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}