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

Let $1<q<p<\mathrm{\infty }$, $\frac{1}{r}:=\frac{1}{q}-\frac{1}{p}$, 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 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 Calderón–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 $C_{\mathrm{c}}^{\mathrm{\infty }}({\mathbb{R}}^n)$, which seems to be redundant for many corresponding estimates when $p\le q$ but to be crucial when $p>q$. We also extend the results to the multilinear case.