Commutators, Little BMO and Weak Factorization

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

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 $h^1(ℝ\times ℝ)$ (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 $\left\{{\alpha }_j^k\right\}\in {\ell }^1$ et des fonctions $g_j^k,h_j^k\in L^2\left({ℝ}^2\right)$ telles que

$$f=\sum _{k=1}^{\infty }\sum _{j=1}^{\infty }{\alpha }_j^k\left(h_j^k{H}_1{H}_2{g}_j^k-g_j^k{H}_1{H}_2{h}_j^k\right)$$

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 $\parallel g_j^k{\parallel }_{L^2\left({ℝ}^2\right)}$ et $\parallel h_j^k{\parallel }_{L^2\left({ℝ}^2\right)}$. Par dualité, ceci implique directement une borne inférieure de la norme du commutateur $[b,H_1{H}_2]$ en termes de ${\parallel b\parallel }_{bmo(ℝ\times ℝ)}$.

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 $\left\{{\alpha }_j^k\right\}\in {\ell }^1$ and functions $g_j^k,h_j^k\in L^2\left({ℝ}^2\right)$ such that

$$f=\sum _{k=1}^{\infty }\sum _{j=1}^{\infty }{\alpha }_j^k\left(h_j^k{H}_1{H}_2{g}_j^k-g_j^k{H}_1{H}_2{h}_j^k\right)$$

in the sense of $h^1\left(ℝ\right)$, where $H_1$ and $H_2$ are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm ${\parallel f\parallel }_{h^1(ℝ\times ℝ)}$ is given in terms of $\parallel g_j^k{\parallel }_{L^2\left({ℝ}^2\right)}$ and $\parallel h_j^k{\parallel }_{L^2\left({ℝ}^2\right)}$. By duality, this directly implies a lower bound on the norm of the commutator $[b,H_1{H}_2]$ in terms of ${\parallel b\parallel }_{bmo(ℝ\times ℝ)}$.

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.

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

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