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2018 Weighted little bmo and two-weight inequalities for Journé commutators
Irina Holmes, Stefanie Petermichl, Brett D. Wick
Anal. PDE 11(7): 1693-1740 (2018). DOI: 10.2140/apde.2018.11.1693


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 A p 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 k 1 R l 2 ] : L p ( μ ) L p ( λ ) , 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 ( 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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Irina Holmes. Stefanie Petermichl. Brett D. Wick. "Weighted little bmo and two-weight inequalities for Journé commutators." Anal. PDE 11 (7) 1693 - 1740, 2018.


Received: 14 June 2017; Revised: 10 September 2017; Accepted: 24 October 2017; Published: 2018
First available in Project Euclid: 15 January 2019

zbMATH: 1395.42064
MathSciNet: MR3810470
Digital Object Identifier: 10.2140/apde.2018.11.1693

Primary: 42A50, 42B20, 42B25

Rights: Copyright © 2018 Mathematical Sciences Publishers


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Vol.11 • No. 7 • 2018
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