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 Є h1 (ℝ×ℝ) there exist sequences {αkj} Є l and functions gjk, hkj Є L2 (ℝ2 ) such that [Equation Unavailable] 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 ║gjk║ L2(ℝ2) and ║hkj║ L2(ℝ2). By duality, this directly implies a lower bound on the norm of the commutator [b, H1 H2 ] in terms of ║b║bmo(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

ORCID

https://orcid.org/0000-0003-1890-0608 [Wick]

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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