Commutators in the two scalar and matrix weighted setting

Abstract

In this paper, we approach the two weighted boundedness of commutators via matrix weights. This approach provides both a sufficient and a necessary condition for the two weighted boundedness of commutators with an arbitrary linear operator in terms of one matrix weighted norm inequalities for this operator. Furthermore, using this approach, we surprisingly provide conditions that almost characterize the two matrix weighted boundedness of commutators with CZOs and completely arbitrary matrix weights, which is even new in the fully scalar one weighted setting. Finally, our method allows us to extend the two weighted Holmes/Lacey/Wick results to the fully matrix setting (two matrix weights and a matrix symbol), completing a line of research initiated by the first two authors.

Dyadic product BMO in the Bloom setting

“…Their proof of these equivalences employed a duality result between dyadic $$\text{BMO}(\nu )$$ and a certain dyadic weighted $$H^1$$ space that they established in the same work, as well as characterizations of two‐weight BMO spaces in terms of two‐weight boundedness of certain paraproducts. It should be noted that the results of [18] were very recently extended to the matrix‐valued setting by J. Isralowitz, S. Pott and S. Treil [22]. In fact, the authors of [22] proved there several results for the case of completely arbitrary (not necessarily $$A_p$$) matrix‐valued weights, that are new even if one specializes to the fully scalar setting.…”

“…There are as well some lower bounds that do not rely on (), like [18, 19], [22]. However, all these employ variants of the original argument by Coifman–Rochberg–Weiss [9].…”

“…It should be noted that the results of [18] were very recently extended to the matrix‐valued setting by J. Isralowitz, S. Pott and S. Treil [22]. In fact, the authors of [22] proved there several results for the case of completely arbitrary (not necessarily $$A_p$$) matrix‐valued weights, that are new even if one specializes to the fully scalar setting.…”