Weighted little bmo and two-weight inequalities for Journé commutators

Abstract

We characterize the boundedness of the commutators $\left[b,T\right]$ 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 $\mu$ and $\lambda$ are biparameter $A_p$ weights, $\nu :={\mu }^{1∕p}{\lambda }^{-1∕p}$ is the Bloom weight, and $b$ is in $bmo\left(\nu \right)$, then we prove a lower bound and testing condition $\parallel b{\parallel }_{bmo\left(\nu \right)}\lesssim sup\parallel \left[b,R_k^1{R}_l^2\right]:L^p\left(\mu \right)\to L^p\left(\lambda \right)\parallel$, 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 Journé operators $T$, the commutator $\left[b,T\right]:L^p\left(\mu \right)\to L^p\left(\lambda \right)$ 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.