\(b\)-weighted dyadic BMO from dyadic BMO and associated \(T(b)\) theorems (Q972116)

Denote by \(\Delta\) a finite set of dyadic intervals on the half-line. Let \(b\) be a function with \(|[b]_I|\geq 1\) for all \(I \in \Delta\), and let \(\psi_I^b\) and \(\rho_I^b\) be the associated \(b-\)adapted Haar wavelets. The (global) BMO\((b)\) semi-norm of a function \(f\) if defined by \[ \|f\|_{\text{BMO}(b)}=\sup_{J}\frac{1}{|J|^{1/2}}\bigg(\sum_{I\subseteq J} |\langle f,\rho_I^b\rangle|^2\bigg)^{1/2}. \] In this paper, a theorem comparing the BMO\((b)\) norm of a local \(L^2\) function and its dyadic BMO norm is proved. A dyadic global \(b-\)output \(T(b)\) theorems follows as an corollary. For the local case, by using the multiscale analysis, corresponding theorems are also established.