Dyadic product BMO in the Bloom setting

Abstract

Ó. Blasco and S. Pott showed that the supremum of operator norms over L2$L^2$ of all bicommutators (with the same symbol) of one‐parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1<p<∞$1&lt;p&lt;\infty$. The main tool is a new characterization in terms of paraproducts and two‐weight John–Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multiparameter setting, with the appropriate iterated commutator in each case.