Weighted little bmo and two-weight inequalities for Journé commutators
Résumé
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 Ap weights, ν := µ 1/p λ −1/p is the Bloom weight, and b is in bmo(ν), then we prove a lower bound and testing condition b bmo(ν) sup [b, R 1 k R 2 l ] : L p (µ) → L p (λ), where R 1 k and R 2 l 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.
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