In this paper, we provide a direct and constructive proof of weak factorization of (the predual of little BMO space studied by Cotlar–Sadosky and Ferguson–Sadosky), i.e., for every there exist sequences and functions such that
in the sense of , where and are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm is given in terms of and . By duality, this directly implies a lower bound on the norm of the commutator in terms of .
Our method bypasses the use of analyticity and the Fourier transform, and hence can be extended to the higher dimension case in an arbitrary -parameter setting for the Riesz transforms.
Dans ce papier, nous donnons une preuve directe et constructive de la factorisation faible de (le prédual de l’espace little BMO étudié par Cotlar–Sadosky et Ferguson–Sadosky), i.e., pour chaque il existe des suites et des fonctions telles que
au sens de , où et sont les transformées de Hilbert dans la première et la seconde variable, respectivement. De plus, la norme est donnée en termes de et . Par dualité, ceci implique directement une borne inférieure de la norme du commutateur en termes de .
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 -paramètres arbitraires, pour les transformées de Riesz.
Revised:
Accepted:
Published online:
Keywords: $\protect \operatorname{bmo}(\protect \mathbb{R}\times \protect \mathbb{R})$, $h^1(\protect \mathbb{R}\times \protect \mathbb{R})$, commutator, weak factorization, Hilbert transform
Mot clés : $\protect \operatorname{bmo}(\protect \mathbb{R}\times \protect \mathbb{R})$, $h^1(\protect \mathbb{R}\times \protect \mathbb{R})$, commutateur, factorisation faible, transformée de Hilbert
@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}, pages = {109--129}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {1}, year = {2018}, doi = {10.5802/aif.3153}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3153/} }
TY - JOUR AU - Duong, Xuan Thinh AU - Li, Ji AU - Wick, Brett D. AU - Yang, Dongyong TI - Commutators, Little BMO and Weak Factorization JO - Annales de l'Institut Fourier PY - 2018 SP - 109 EP - 129 VL - 68 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3153/ DO - 10.5802/aif.3153 LA - en ID - AIF_2018__68_1_109_0 ER -
%0 Journal Article %A Duong, Xuan Thinh %A Li, Ji %A Wick, Brett D. %A Yang, Dongyong %T Commutators, Little BMO and Weak Factorization %J Annales de l'Institut Fourier %D 2018 %P 109-129 %V 68 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3153/ %R 10.5802/aif.3153 %G en %F AIF_2018__68_1_109_0
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. http://archive.numdam.org/articles/10.5802/aif.3153/
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