A commutator theorem and weighted BMO

The main result of this paper is a commutator theorem: If μ μ and λ λ 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}(μ ) into L p ( λ ) {L^p}(λ ) if and only if b ∈ BMO ( μ λ − 1 ) 1 / p b ∈ {BMO _{{{(μ {λ - 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.