Commutators, little bmo and weak factorization

Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang

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    Abstract

    In this paper, we provide a direct and constructive proof of weak factorization of h1 (R × R) (the predual of little BMO space bmo(R × R) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f ∈ h1 (R × R) there exist sequences {αk j } ∈ and functions gj k, hk j ∈ L2 (R2 ) such that ∞ ∞ f = αk j k j H1 H2 gj k − gj kH1 H2 hk k=1 j=1 in the sense of h1 (R × R), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm fh1(R×R) is given in terms of gj k L2(R2) and hk j L2(R2). By duality, this directly implies a lower bound on the norm of the commutator [b, H1 H2 ] in terms of bbmo(R×R). Our method bypasses the use of analyticity and the Fourier transform, and hence can be extended to the higher dimension case in an arbitrary n-parameter setting for the Riesz transforms.

    Original languageEnglish
    Pages (from-to)109-129
    Number of pages21
    JournalAnnales de l'Institut Fourier
    Volume68
    Issue number1
    DOIs
    Publication statusPublished - 1 Jan 2018

    Bibliographical note

    Copyright Association des Annales de l’institut Fourier, 2018. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

    Keywords

    • Bmo(R × R)
    • Commutator
    • H1(R × R)
    • Hilbert transform
    • Weak factorization

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