no. 1
Commutators, Little BMO and Weak Factorization
[Commutateurs, Little BMO et Factorisation Faible]
Duong, Xuan Thinh1 ; Li, Ji1 ; Wick, Brett D.2 ; Yang, Dongyong3
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)
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 109-129.

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

f=∑k=1∞∑j=1∞αjkhjkH1H2gjk-gjkH1H2hjk

au sens de h 1 (ℝ×ℝ), 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 (ℝ×ℝ) est donnée en termes de ∥g j k ∥ L 2 (ℝ 2 ) et ∥h j k ∥ L 2 (ℝ 2 ) . Par dualité, ceci implique directement une borne inférieure de la norme du commutateur [b,H 1 H 2 ] en termes de ∥b∥ bmo(ℝ×ℝ) .

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.

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

f=∑k=1∞∑j=1∞αjkhjkH1H2gjk-gjkH1H2hjk

in the sense of h 1 (ℝ), where H 1 and H 2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm ∥f∥ h 1 (ℝ×ℝ) is given in terms of ∥g j k ∥ L 2 (ℝ 2 ) and ∥h j k ∥ L 2 (ℝ 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(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.

Reçu le :
Révisé le :
Accepté le :
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DOI : 10.5802/aif.3153
Classification : 42B30, 42B20, 42B35
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
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)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Commutators, {Little} {BMO} and {Weak} {Factorization}},
     journal = {Annales de l'Institut Fourier},
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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, Tome 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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