$L^p$ improving multilinear Radon-like transforms

We characterize (up to endpoints) the $k$-tuples $(p_1,\dots ,p_k)$ for which certain $k$-linear generalized Radon transforms map $L^{p_1} \times \cdots \times L^{p_k}$boundedlyinto$\mathbb R$. This generalizes a result of Tao and Wright.