Over the last decade, Bonami, Grellier, Ky, Iwaniec, Jones, Ky, Zinsmeister and others have investigated products of $H^1$ and BMO functions, and associated bilinear forms. Here, we consider natural dyadic decompositions of such products. We also give an outlook on the biparameter setting.

(This is joint work with Odysseas Bakas, Salvador Rodriguez, and Alan Sola).