Author's School

Arts & Sciences

Author's Department

Mathematics

Document Type

Article

Publication Date

2018

Originally Published In

Annales de l'institut Fourier, 68 no. 1 (2018), p. 109-129, doi: 10.5802/aif.3153

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.

Comments

© Association des Annales de l’institut Fourier, 2018,

CC-BY-ND

Article is available at Annales de l'institut Fourier, 68 no. 1 (2018), p. 109-129, doi: 10.5802/aif.3153

DOI

10.5802/aif.3153

Creative Commons License

Creative Commons Attribution-No Derivative Works 3.0 License
This work is licensed under a Creative Commons Attribution-No Derivative Works 3.0 License.

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Analysis Commons

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