Commutators with fractional integral operators

We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for mu, lambda is an element of A(p,q) and alpha/n+1/q = 1/p, the norm parallel to[b, I-alpha] : L-p (mu(p)) -> L-q(lambda(q))parallel to is equivalent to the norm of b in the weighted BMO space BMO(nu), where nu = mu lambda(-1). This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.