Weighted little bmo and two-weight inequalities for Journé commutators

By I. Holmes, S. Petermichl, Brett D. Wick. 2017

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We characterize the boundedness of the commutators $[b, T]$ with biparameter Journe 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 $\|b\|_{bmo(\nu)} \lesssim \sup \| [b, R_k^1 R_l^2]: L^p(\mu) \rightarrow L^p(\lambda)\|$, 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 Journe operators $T$ the commutator $[b, T]:L^p(\mu) \rightarrow L^p(\lambda)$ is bounded. Previous results in the Bloom setting do not include the biparameter case and are restricted to Calderon-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 Journe operators. As a by-product we also obtain a much simplified proof for a one-weight bound for Journe operators originally due to R. Fefferman.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/070bad96274dacdba15343231f356b629453b38a

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In Applied Mathematics

Citations in the corpus

Two Weight Commutators on Spaces of Homogeneous Type and Applications (2018, methodology): “Note that in [22] the authors only considered the lower bound of commutator with respect to double Riesz transforms, and their proof relies on Fourier transform and hence can not be adapted to spaces of homogeneous type.”