Weighted little bmo and two-weight inequalities for Journ\'e commutators

We characterize the boundedness of the commutators $[b, T]$ with biparameter Journ\'{e} 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 $\mu$ and $\lambda$ are biparameter $A_p$ weights, $\nu := \mu^{1/p}\lambda^{-1/p}$ is the Bloom weight, and $b$ is in $bmo(\nu)$, then we prove a lower bound and testing condition $\|b\|_{bmo(\nu)} \lesssim \sup \| [b, R_k^1 R_l^2]: L^p(\mu) \rightarrow L^p(\lambda)\|$, where $R_k^1$ and $R_l^2$ are Riesz transforms acting in each variable. Further, we prove that for such symbols $b$ and any biparameter Journ\'{e} operators $T$ the commutator $[b, T]:L^p(\mu) \rightarrow L^p(\lambda)$ is bounded. Previous results in the Bloom setting do not include the biparameter case and are restricted to Calder\'{o}n-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 Journ\'e operators. As a by-product we also obtain a much simplified proof for a one-weight bound for Journ\'{e} operators originally due to R. Fefferman.