Compactness of commutators of rough singular integrals

Abstract

We study the two-weighted off-diagonal compactness of commutators of rough singular integral operators \(T_\Omega\) that are associated with a kernel \(\Omega\in L^q(\mathbb{S}^{d-1})\). We establish a characterisation of compactness of the commutator \([b,T_\Omega]\) in terms of the function \(b\) belonging to a suitable space of functions with vanishing mean oscillation. Our results expand upon certain previous compactness characterisations in that the results do not require smoothness from the kernel of the singular integral operator. Additionally, we prove a matrix-weighted compactness result for \([b,T_\Omega]\) by applying the so-called matrix-weighted Kolmogorov–Riesz theorem.