$b$-weighted dyadic BMO from dyadic BMO and associated $T(b)$ theorems

Given a function $b$, and using adapted Haar wavelets, we define a $\BMO$-type norm which is dependent on $b$. In both global and local cases, we find the dependence of the bounds on $\|f\|_{\BMO}$ by the bounds on the $b$-weighted $\BMO$ norm of $f$. 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 dyadic $T(b)$ theorems whose hypotheses include a bound on the $b$-weighted $\BMO$-norm of $T^*(1)$.