Weighted little bmo and two-weight inequalities for Journé commutators Academic Article uri icon

abstract

  • We characterize the boundedness of the commutators $[b, T]$ with biparameter Journ'{e} 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, $
    u := mu^{1/p}lambda^{-1/p}$ is the Bloom weight, and $b$ is in $bmo(
    u)$, then we prove a lower bound and testing condition $|b|_{bmo(
    u)} lesssim sup | [b, R_k^1 R_l^2]: L^p(mu)
    ightarrow L^p(lambda)|$, 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 Journ'{e} operators $T$ the commutator $[b, T]:L^p(mu)
    ightarrow L^p(lambda)$ is bounded. Previous results in the Bloom setting do not include the biparameter case and are restricted to Calder'{o}n-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 Journ'e operators. As a by-product we also obtain a much simplified proof for a one-weight bound for Journ'{e} operators originally due to R. Fefferman.

altmetric score

  • 0.75

author list (cited authors)

  • Holmes, I., Petermichl, S., & Wick, B. D.

citation count

  • 8

publication date

  • May 2018