A commutator theorem and weighted BMO

The main result of this paper is a commutator theorem: If $\mu$ and $\lambda$ are ${A_p}$ weights, then the commutator $H$, ${M_b}$ is a bounded operator from ${L^p}(\mu )$ into ${L^p}(\lambda )$ if and only if $b \in {\operatorname {BMO} _{{{(\mu {\lambda ^{ - 1}})}^{1/p}}}}$. The proof relies heavily on a weighted sharp function theorem. Along the way, several other applications of this theorem are derived, including a doubly-weighted ${L^p}$ estimate for BMO. Finally, the commutator theorem is used to obtain vector-valued weighted norm inequalities for the Hilbert transform.