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The $L^p$-to-$L^q$ compactness of commutators with $p \gt q$

Volume 271 / 2023

Tuomas Hytönen, Kangwei Li, Jin Tao, Dachun Yang Studia Mathematica 271 (2023), 85-105 MSC: Primary 47B47; Secondary 42B20, 42B25, 42B35, 46E30. DOI: 10.4064/sm220910-10-1 Published online: 1 March 2023


Let $1 \lt q \lt p \lt \infty $, $1/r:=1/q-1/p$, and $T$ be a non-degenerate Calderón–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 $b=a+c$ with $a\in L^r(\mathbb R^n)$ and $c$ a constant. Since neither the corresponding Hardy–Littlewood maximal operator nor the corresponding Calderón–Zygmund maximal operator is 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 is crucial when $p \gt q$. We also extend the results to the multilinear case.


  • Tuomas HytönenDepartment of Mathematics and Statistics
    University of Helsinki
    00014 Helsinki, Finland
  • Kangwei LiCenter for Applied Mathematics
    Tianjin University
    300072 Tianjin, People’s Republic of China
  • Jin TaoHubei Key Laboratory
    of Applied Mathematics
    Faculty of Mathematics and Statistics
    Hubei University
    430062 Wuhan, People’s Republic of China
  • Dachun YangLaboratory of Mathematics and
    Complex Systems
    (Ministry of Education of China)
    School of Mathematical Sciences
    Beijing Normal University
    100875 Beijing, People’s Republic of China

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