Weighted little bmo and two-weight inequalities for Journé commutators

Abstract

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 $\mu$ 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 $\Vert b{\Vert }_{bmo(\nu )}\lesssim \sup \Vert [b,R_k^1{R}_l^2]:L^p(\mu )\to L^p(\lambda )\Vert$, 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 $[b,T]:L^p(\mu )\to L^p(\lambda )$ 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.

Final journal (Analysis and PDE) version, with a new introduction and a few new Propositions/Lemmas

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