Applications of commutator theory to weighted BMO and matrix analogs of \(A_ 2\) (Q1113033)

For the maximal operator commutator \(T_ bf(x)=\sup_{x\in I}| \int_{I}[b(x)f(t)-b(t)f(t)]dt|,\) we obtain the Theorem: Let \(\mu\) and \(\lambda \in A_ p\), \(1<p<\infty\). Put \(\nu =(\mu \lambda^{-1})^{1/p}\). Then \(b\in BMO_{\nu}\) if and only if \(T_ b: L^ p(\mu)\to L^ p(\lambda)\) is a bounded operator. Applications of this theorem include several unusual characterizations of BMO and weighted BMO, and a vector-valued weighted norm inequality.