Weighted Estimates for Bilinear Fractional Integral Operators and their Commutators

In this paper, we prove several weighted estimates for bilinear fractional integral operators and their commutators with BMO functions. We also prove maximal function control theorem for these operators; that is, we prove the weighted L-p norm of the operator is bounded by the weighted Lp norm of a natural maximal operator when the weight belongs to A(infinity). As a corollary, we are able to obtain new weighted estimates for the bilinear maximal function associated with the bilinear Hilbert transform. Finally, as an application of our results, we prove a bilinear Stein-Weiss inequality.