A characterization of product BMO by commutators.
Let \({\mathbb R}^2_+\) be the upper half plane and \(\text{BMO} ({\mathbb R}^2_+\times{\mathbb R}^2_+)\) be the dual of the real-variable Hardy space \(H^1\) on the product domain \({\mathbb R}^2_+\times {\mathbb R}^2_+\), which is also called the product BMO space and was introduced by \textit{S.-Y. A. Chang} and \textit{R. Fefferman} in [Bull. Am. Math. Soc., New Ser. 12, 1--43 (1985; Zbl 0557.42007)]. Moreover, let \(H_j\) denote the one-dimensional Hilbert transform in the \(j\)th variable for \(j=1,\,2\), and \([\,[M_b,\,H_1],\,H_2]\) be the nested commutator determined by the function \(b\) and the operator \(H_j\), where, for a function \(b\) on the plane, \(M_bf=bf\). The main result of this paper is that there is a constant \(C>0\) such that \[ \| b\| _{\text{BMO}}\leq C\| [\,[M_b,\,H_1],\,H_2]\| _{L^2\to L^2} \] for all functions \(b\in BMO({\mathbb R}^2_+\times {\mathbb R}^2_+)\). It was proved by \textit{S. H. Ferguson} and \textit{C. Sadosky} in [J. Anal. Math. 81, 239--267 (2000; Zbl 0988.32002)] that \[ \| [\,[M_b,\,H_1],\,H_2]\| _{L^2\to L^2}\leq C\| b\| _{\text{BMO}}. \] Combining both results allows one to concretely identify \(\text{BMO}({\mathbb R}^2_+\times {\mathbb R}^2_+)\) as an operator space on \(L^2({\mathbb R}^2)\). The proof of the authors relies on three key ideas: the dyadic characterization of the BMO norm given by Chang and Fefferman in the above mentioned paper, a variant of Journé's covering lemma whose proof is included in the appendix, and the estimates of Ferguson and Sadosky in the aforementioned paper. Let \(H^1({\mathbb R}^2_+\times{\mathbb R}^2_+)\) be the two-variable biholomorphic Hardy space, which can be defined in terms of the boundary values of biholomorphic function on \({\mathbb R}^2_+\times {\mathbb R}^2_+\). Applying the above result, the authors obtain a weak factorization for the (biholomorphic) space \(H^1({\mathbb R}^2_+\times {\mathbb R}^2_+)\), namely, the authors prove that if \(h\in H^1({\mathbb R}^2_+\times {\mathbb R}^2_+)\) with \(\| h\| _1=1\), then there exist functions \(\{f_j\},\,\{g_j\}\subseteq H^2({\mathbb R}^2_+\times {\mathbb R}^2_+)\) such that \(h=\sum_{j=1}^\infty f_jg_j\) and \(\sum_{j=1}^\infty\| f_j\| _2\| g_j\| _2\leq C\). The authors point out that the weak factorization above implies the analogous factorization for \(H^1\) of the bidisk.
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