Bilinear Fractional Integral Operators

We study the bilinear fractional integral considered by Kenig and Stein, where linear combinations of variables with matrix coefficients are involved. Under more general settings, we give a complete characterization of the corresponding parameters for which the bilinear fractional integral is bounded from $L^{p_1}({\mathbb{R}}^{n_1})\times L^{p_2}({\mathbb{R}}^{n_2})$ to $L^q(\mathbb R^m)$.