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

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