  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.

Functional Analysis;Analysis of PDEs;Classical Analysis and ODEs