The $L^p$-to-$L^q$ Compactness of Commutators with $p>q$

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.