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Given a functionb, and using adapted Haar wavelets, we define a BMO-type norm which is dependent onb. In both global and local cases, we find the dependence of the bounds on ∥f∥BMO by the bounds on theb-weighted BMO norm off. We show that the dependence is sharp in the global case. Multiscale analysis is used in the local case. We formulate as corollaries global and local dyadicT(b) theorems whose hypotheses include a bound on theb-weighted BMO-norm ofT *(1).

b-weighted dyadic BMO from dyadic BMO and associatedT(b) theorems