Transversal multilinear Radon-like transforms: Local and global estimates

We prove local "L-improving" estimates for a class of multilinear Radon-like transforms satisfying a strong transversality hypothesis. As a consequence, we obtain sharp multilinear convolution estimates for measures supported on fully transversal submanifolds of Euclidean space of arbitrary dimension. Motivated by potential applications in diffraction tomography, we also prove global estimates for the same class of Radon-like transforms under a natural homogeneity assumption.