Compactness of commutators of rough singular integrals

We study the two-weighted off-diagonal compactness of commutators of rough singular integral operators $T_{\mathrm{\Omega }}$ that are associated with a kernel $\mathrm{\Omega }\in L^q({\mathbb{S}}^{d-1})$. We establish a characterisation of compactness of the commutator $[b,T_{\mathrm{\Omega }}]$ in terms of the function $b$ belonging to a suitable space of functions with vanishing mean oscillation. Our results expand upon the previous compactness characterisations for Calderón-Zygmund operators. Additionally, we prove a matrix-weighted compactness result for $[b,T_{\mathrm{\Omega }}]$ by applying the so-called matrix-weighted Kolmogorov-Riesz theorem.