Commutators with two matrix weights

Let $U,V$ be matrix $\mathcal A_p$ weights, $1<p<\infty$, and let $B$ be a matrix-valued function. For Calderón--Zygmund operators $T$ with scalar standard kernels, we characterize boundedness and compactness of $[M_B,T\otimes I_m]:L^p(U)\to L^p(V)$ by two-matrix BMO and vanishing oscillation conditions. The converse statements require non-degeneracy condition. For matrix \(\mathcal A_2\) weights \(U,V\) and a fixed Riesz transform \(R_j\) on \(\mathbb R^d\), \(d\geq2\), we characterize the Schatten class membership of \(C_{B,R_j}:L^2(U)\to L^2(V)\). We characterize $\mathcal S^r$ membership for $r>d$ and $\mathcal S^{d,\infty}$ membership by $\ell^r$ and weak $\ell^d$ conditions on the two Hilbert--Schmidt cube oscillations. For $0<r\leq d$, $\mathcal S^r$ membership is equivalent to $B$ being constant almost everywhere. The weak endpoint condition implies a weighted first-order Sobolev estimate for $C^1$ symbols, with a converse for constant or uniformly elliptic weights.