We characterize the boundedness of the commutators with biparameter Journé operators in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little bmo norm of the symbol . Specifically, if and are biparameter weights, is the Bloom weight, and is in , then we prove a lower bound and testing condition , where and are Riesz transforms acting in each variable. Further, we prove that for such symbols and any biparameter Journé operators , the commutator is bounded. Previous results in the Bloom setting do not include the biparameter case and are restricted to Calderón–Zygmund operators. Even in the unweighted, case, the upper bound fills a gap that remained open in the multiparameter literature for iterated commutators with Journé operators. As a by-product we also obtain a much simplified proof for a one-weight bound for Journé operators originally due to R. Fefferman.
"Weighted little bmo and two-weight inequalities for Journé commutators." Anal. PDE 11 (7) 1693 - 1740, 2018. https://doi.org/10.2140/apde.2018.11.1693