Dyadic BMO on the bidisk

Abstract

We give several new characterizations of the dual of the dyadic Hardy space $H^{1,d}(\mathbb{T}^2)$, the so-called dyadic BMO space in two variables and denoted ${\mathrm{BMO}}^{\mathit d}_{prod}}$. These include characterizations in terms of Haar multipliers, in terms of the ``symmetrised paraproduct'' $\Lambda_b$, in terms of the rectangular BMO norms of the iterated ``sweeps'', and in terms of nested commutators with dyadic martingale transforms. We further explore the connection between ${\mathrm{BMO}}^{\mathit d}_{prod}}$ and John-Nirenberg type inequalities, and study a scale of rectangular BMO spaces.