Paraproducts, Bloom BMO and sparse BMO functions

A. We address Lp(μ) → Lp(λ) bounds for paraproducts in the Bloom setting. We introduce certain “sparse BMO” functions associated with sparse collections with no infinitely increasing chains, and use these to express sparse operators as sums of paraproducts and martingale transforms – essentially, as Haar multipliers – as well as to obtain an equivalence of norms between sparse operators AS and compositions of paraproducts ΠaΠb.