Note on the compactness of commutators

The optimal sufficient conditions for the $L^p$-to-$L^q$ compactness of commutators of singular integral operators of both Calderón-Zygmund and of rough type are shown in the different exponent ranges $\mathrm{‘}\mathrm{‘}q>p\mathrm{"}$, $\mathrm{‘}\mathrm{‘}q=p\mathrm{"}$ and $\mathrm{‘}\mathrm{‘}q<p\mathrm{"}$ to quickly follow from each other. The approach is through classical compactness interpolation methods. We also present a new elementary off-diagonal to diagonal extrapolation principle for the compactness of commutators of linear operators, which is of independent interest.