Commutators, Little BMO and Weak Factorization
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, p. 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 thatf=k=1j=1αjkhjkH1H2gjk-gjkH1H2hjkin 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 quef=k=1j=1αjkhjkH1H2gjk-gjkH1H2hjkau 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.
Received : 2016-05-06
Revised : 2016-12-15
Accepted : 2017-06-15
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3153
Classification:  42B30,  42B20,  42B35
Keywords: bmo(×), h 1 (×), commutator, weak factorization, Hilbert transform
@article{AIF_2018__68_1_109_0,
     author = {Duong, Xuan Thinh and Li, Ji and Wick, Brett D. and Yang, Dongyong},
     title = {Commutators, Little BMO and Weak Factorization},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     pages = {109-129},
     doi = {10.5802/aif.3153},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_1_109_0}
}
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/item/AIF_2018__68_1_109_0/

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