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}_{\mathrm{prod}} . 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}^{\mathit d}_{\mathrm{prod}} and John–Nirenberg type inequalities, and study a scale of rectangular BMO spaces.