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 ${\mathrm{BMO}}_{\mathrm{prod}}^d$. These include characterizations in terms of Haar multipliers, in terms of the “ymmetrised paraproduct” ${\Lambda }_b$, in terms of the rectangular BMO norms of the iterated “weeps”, and in terms of nested commutators with dyadic martingale transforms. We further explore the connection between ${\mathrm{BMO}}_{\mathrm{prod}}^d$ and John–Nirenberg type inequalities, and study a scale of rectangular BMO spaces.