Two-Weight Inequalities for Commutators with Fractional Integral Operators
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In this paper we investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $mu,lambdain A_{p,q}$ and $alpha/n+1/q=1/p$, the norm $| [b,I_alpha]:L^p(mu^p) o L^q(lambda^q) |$ is equivalent to the norm of $b$ in the weighted BMO space $BMO( u)$, where $ u=mulambda^{-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.