Schatten--Lorentz properties of commutators with matrix weights

Abstract

We study the membership of commutators $[b,T]$ of Calderón--Zygmund operators in the Schatten--Lorentz classes in the matrix-weighted setting. Working on a space of homogeneous type $(X,ρ,μ)$ with a matrix $A_2$ weight $W$, we characterise, for $p\in(1,\infty)$ and $q\in[1,\infty)$, the Schatten--Lorentz norm $\|[b,T]\|_{S^{p,q}(L^2(W))}$ in terms of an oscillation (Besov-type) norm $\|b\|_{\mathrm{Osc}^{p,q}}$ of the symbol: for all $p\in(1,\infty)$, finiteness of $\|b\|_{\mathrm{Osc}^{p,q}}$ is necessary; and for $p>p(η)$, where $p(η)$ depends on the regularity of the kernel of $T$ and a dimensional parameter of the space, the two-sided estimate $\|[b,T]\|_{S^{p,q}(L^2(W))}\approx\|b\|_{\mathrm{Osc}^{p,q}}$ holds. Extending a recent unified framework for unweighted and scalar-weighted versions of such results by one of us to the matrix-weighted setting, our approach combines the dyadic representation of singular integral operators with the nearly weakly orthogonal (NWO) sequences of Rochberg and Semmes.