Commutators with two matrix weights

Let $U,V$ be matrix ${\mathcal{A}}_p$ weights, $1<p<\mathrm{\infty }$, 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 ${\mathbb{R}}^d$, $d\ge 2$, 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 ${\mathcal{S}}^{d,\mathrm{\infty }}$ membership by ${\mathrm{\ell }}^r$ and weak ${\mathrm{\ell }}^d$ conditions on the two Hilbert–Schmidt cube oscillations. For $0<r\le d$, ${\mathcal{S}}^r$ 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.