  In this paper, we study a class of multilinear fractional integral operators
which have correlation kernels $\prod_{1\leq i<j \leq
k}|x_i-x_j|^{-\alpha_{ij}}$. The necessary and sufficient conditions are
obtained under which these oprators are bounded from $L^{p_1}\times \cdots
\times L^{p_k}$ into $L^q$. As a consequence, we also get the endpoint
estimates from $L^{p_1}\times \cdots \times L^{p_k}$ to $BMO$ of these
operators.
