Two-Weight Inequalities for Commutators with Fractional Integral Operators

Abstract: 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 $\mu,\lambda\in A_{p,q}$ and $\alpha/n+1/q=1/p$, the norm $| [b,I_\alpha]:L^p(\mu^p)\to L^q(\lambda^q) |$ 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.