#### Vol. 11, No. 7, 2018

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Weighted little bmo and two-weight inequalities for Journé commutators

### Irina Holmes, Stefanie Petermichl and Brett D. Wick

Vol. 11 (2018), No. 7, 1693–1740
##### 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 $\parallel b{\parallel }_{bmo\left(\nu \right)}\lesssim sup\parallel \left[b,\underset{k}{\overset{1}{R}}{R}_{l}^{2}\right]:{L}^{p}\left(\mu \right)\to {L}^{p}\left(\lambda \right)\parallel$, 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.

##### Keywords
commutators, Calderón–Zygmund operators, bounded mean oscillation, weights, Journé operators, little BMO, multiparameter harmonic analysis, singular integrals, weighted inequalities
##### Mathematical Subject Classification 2010
Primary: 42B20, 42B25, 42A50
##### Milestones
Revised: 10 September 2017
Accepted: 24 October 2017
Published: 20 May 2018
##### Authors
 Irina Holmes Department of Mathematics Michigan State University East Lansing, MI United States Stefanie Petermichl Institut de Mathématiques de Toulouse Université Paul Sabatier Toulouse France Brett D. Wick Department of Mathematics Washington University - St. Louis St. Louis, MO United States