Dyadic product BMO in the Bloom setting

Ó. Blasco and S. Pott showed that the supremum of operator norms over L 2 $L^2$ all bicommutators (with same symbol) one-parameter Haar multipliers dominates biparameter dyadic product BMO norm symbol itself. In present work we extend this result to Bloom setting, any exponent 1 < p ? $1<p<\infty$ . The main tool is a new characterization in terms paraproducts two-weight John–Nirenberg inequalities for setting. We also our results whole scale indexed spaces between little bmo general multiparameter with appropriate iterated commutator each case.