Commutators, Little BMO and Weak Factorization

Abstract

In this paper, we provide a direct and constructive proof of weak factorization of $h^1(\mathbb{R})$ (the predual of little BMO space bmo$(\mathbb{R}\times \mathbb{R})$ studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every $f\in h^1(\mathbb{R}\times \mathbb{R})$ there exist sequences $\{{\alpha }_j^k\}\in {\ell }^1$ and functions $g_j^k,h_j^k\in L^2({\mathbb{R}}^2)$ such that \begin{align*} f=\sum_{k=1}^\infty\sum_{j=1}^\inftyα^k_j\Big(\, h^k_j H_1H_2 g^k_j - g^k_j H_1H_2 h^k_j\Big) \end{align*} in the sense of $h^1(\mathbb{R})$, where $H_1$ and $H_2$ are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm $\Vert f{\Vert }_{h^1(\mathbb{R}\times \mathbb{R})}$ is given in terms of $\Vert g_j^k{\Vert }_{L^2({\mathbb{R}}^2)}$ and $\Vert h_j^k{\Vert }_{L^2({\mathbb{R}}^2)}$. By duality, this directly implies a lower bound on the norm of the commutator $[b,H_1{H}_2]$ in terms of $\Vert b{\Vert }_{\mathrm{bmo}(\mathbb{R}\times \mathbb{R})}$. 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.

arXiv comment17 pages. To appear in Annales de l'Institut Fourier