Commutators, Little BMO and Weak Factorization

Duong, Xuan Thinh; Li, Ji; Wick, Brett D.; Yang, Dongyong

doi:10.5802/aif.3153

Commutators, Little BMO and Weak Factorization
[Commutateurs, Little BMO et Factorisation Faible]

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)

Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 109-129.

Résumé
Abstract

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

f = \sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} α_{j}^{k} (h_{j}^{k} H_{1} H_{2} g_{j}^{k} - g_{j}^{k} H_{1} H_{2} h_{j}^{k})

au sens de $h^{1} (ℝ \times ℝ)$ , où $H_{1}$ et $H_{2}$ sont les transformées de Hilbert dans la première et la seconde variable, respectivement. De plus, la norme ${∥f∥}_{h^{1} (ℝ \times ℝ)}$ 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 ${∥ b ∥}_{bmo (ℝ \times ℝ)}$ .

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 $h^{1} (ℝ \times ℝ)$ (the predual of little BMO space $bmo (ℝ \times ℝ)$ studied by Cotlar–Sadosky and Ferguson–Sadosky), i.e., for every $f \in h^{1} (ℝ \times ℝ)$ there exist sequences ${α_{j}^{k}} \in ℓ^{1}$ and functions $g_{j}^{k}, h_{j}^{k} \in L^{2} (ℝ^{2})$ such that

f = \sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} α_{j}^{k} (h_{j}^{k} H_{1} H_{2} g_{j}^{k} - g_{j}^{k} H_{1} H_{2} h_{j}^{k})

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 ${∥ f ∥}_{h^{1} (ℝ \times ℝ)}$ 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 ${∥ b ∥}_{bmo (ℝ \times ℝ)}$ .

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 : 2016-05-06
Révisé le : 2016-12-15
Accepté le : 2017-06-15
Publié le : 2018-04-18

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

Affiliations des auteurs :

Duong, Xuan Thinh ¹ ; Li, Ji ¹ ; Wick, Brett D. ² ; Yang, Dongyong ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.3153/}
}

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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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