Weighted little bmo and two-weight inequalities for Journé commutators

We characterize the boundedness of the commutators [b, T] with biparameter Journé operators T 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 b. Specifically, if μ and λ are biparameter A_p weights, υ:=μ^1/pλ^-1/p is the Bloom weight, and b is in bmo.(υ), then we prove a lower bound and testing condition ||b||_bmo(υ)≲ sup ||[b,R_k¹R_l²]:L^p(μ)→L^p(λ)||, where R_k¹ and R_l² are Riesz transforms acting in each variable. Further, we prove that for such symbols b and any biparameter Journé operators T, the commutator [b,T]:L^p(μ)→L^p(λ)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, p = 2 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.