Note on the compactness of commutators

Abstract

The optimal sufficient conditions for the $L^p$-to-$L^q$ compactness of commutators of singular integral operators of both Calder\'on-Zygmund and of rough type are shown in the different exponent ranges $``q>p"$, $``q=p"$ and $``q<p"$ 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.