A commutator theorem and weighted BMO

Abstract

The main result of this paper is a commutator theorem: If μ \mu and λ \lambda are A p {A_p} weights, then the commutator H H , M b {M_b} is a bounded operator from L p ( μ ) {L^p}(\mu ) into L p ( λ ) {L^p}(\lambda ) if and only if b ∈ BMO ( μ λ − 1 ) 1 / p 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 {L^p} estimate for BMO. Finally, the commutator theorem is used to obtain vector-valued weighted norm inequalities for the Hilbert transform.