Commutators, Little BMO and Weak Factorization

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 $g_j^k,h^k_j\in L^2(\mathbb{R}^2)$ such that \begin{align*} f=\sum_{k=1}^\infty\sum_{j=1}^\infty\alpha^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 $\|f\|_{h^1(\mathbb{R}\times\mathbb{R})}$ is given in terms of $\|g^k_j\|_{L^2(\mathbb{R}^2)}$ and $\|h^k_j\|_{L^2(\mathbb{R}^2)}$. By duality, this directly implies a lower bound on the norm of the commutator $[b,H_1H_2]$ in terms of $\|b\|_{{\rm bmo}(\mathbb{R}\times\mathbb{R})}$. 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.