Dyadic product weighted BMO spaces and commutators with Haar multipliers

In this talk, based on joint work with Spyridon Kakaroumpas (Universitäa Würzburg), we will discuss some results about dyadic multiparameter weighted $\mathrm{BMO}$ spaces. We will recall a result due to O. Blasco and S. Pott in the unweighted setting which asserts that the supremum of operator norms over $L^2$ of all bicommutators (with the same symbol $b$) of one-parameter Haar multipliers dominates the biparameter dyadic product $BMO$ norm of the symbol $b$ itself. In the current work, we extend this result to the weighted setting, and considering operator norms over $L^p$ spaces for any $1 < p < \infty.$ The main tool is a new characterisation in terms of paraproducts and two-weight John--Nirenberg inequalities for dyadic product $\mathrm{BMO}$ in the weighted setting.