Commutators, Little BMO and Weak Factorization
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 109-129.

In this paper, we provide a direct and constructive proof of weak factorization of h 1 (×) (the predual of little BMO space bmo(×) studied by Cotlar–Sadosky and Ferguson–Sadosky), i.e., for every fh 1 (×) there exist sequences {α j k } 1 and functions g j k ,h j k L 2 ( 2 ) such that

f=k=1j=1αjkhjkH1H2gjk-gjkH1H2hjk

in the sense of h 1 (), where H 1 and H 2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm f h 1 (×) is given in terms of g j k L 2 ( 2 ) and h j k L 2 ( 2 ) . By duality, this directly implies a lower bound on the norm of the commutator [b,H 1 H 2 ] in terms of b bmo(×) .

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.

Dans ce papier, nous donnons une preuve directe et constructive de la factorisation faible de h 1 (×) (le prédual de l’espace little BMO bmo(×) étudié par Cotlar–Sadosky et Ferguson–Sadosky), i.e., pour chaque fh 1 (×) il existe des suites {α j k } 1 et des fonctions g j k ,h j k L 2 ( 2 ) telles que

f=k=1j=1αjkhjkH1H2gjk-gjkH1H2hjk

au sens de h 1 (×), où H 1 et H 2 sont les transformées de Hilbert dans la première et la seconde variable, respectivement. De plus, la norme f h 1 (×) est donnée en termes de g j k L 2 ( 2 ) et h j k L 2 ( 2 ) . Par dualité, ceci implique directement une borne inférieure de la norme du commutateur [b,H 1 H 2 ] en termes de b bmo(×) .

Notre méthode contourne l’utilisation de l’analyticité et de la transformée de Fourier, et peut donc être étendue en dimension supérieure dans le cadre de n-paramètres arbitraires, pour les transformées de Riesz.

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DOI: 10.5802/aif.3153
Classification: 42B30,  42B20,  42B35
Keywords: bmo(×), h 1 (×), commutator, weak factorization, Hilbert transform
Duong, Xuan Thinh 1; Li, Ji 1; Wick, Brett D. 2; Yang, Dongyong 3

1 Macquarie University Department of Mathematics NSW, 2109 (Australia)
2 Department of Mathematics Washington University – St. Louis One Brookings Drive St. Louis, MO 63130 (USA)
3 School of Mathematical Sciences Xiamen University Xiamen 361005 (China)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Duong, Xuan Thinh; Li, Ji; Wick, Brett D.; Yang, Dongyong. Commutators, Little BMO and Weak Factorization. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 109-129. doi : 10.5802/aif.3153. https://aif.centre-mersenne.org/articles/10.5802/aif.3153/

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