Paraproducts, Bloom BMO and sparse BMO functions

We address $L^p(\mu)\to L^p(\lambda)$ bounds for paraproducts in the Bloom setting. introduce certain “sparse BMO” functions associated with sparse collections no infinitely increasing chains, and use these to express operators as sums of martingale transforms – essentially, Haar multipliers well obtain an equivalence norms between $\mathcal{A}\_{\mathcal{S}}$ compositions $\Pi^\*\_a\Pi\_b$.