Commutators with two matrix weights

Abstract

Let $U,V$ be matrix $\mathcal A_p$ weights, $1 d$ 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.