The $L^p$-to-$L^q$ compactness of commutators with $p \gt q$
Volume 271 / 2023
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
Abstract
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.
