Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers

We prove that the singularities of a complex valued potential q in the Schrodinger hamiltonian Delta + q can be reconstructed from the linear Born approximation for full scattering data by averaging in the extra variables. We prove that, with this procedure, the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data. In particular, for q is an element of W-alpha,W- 2 for alpha >= 0, in 2D we recover the main singularity of q with an accuracy of one derivative; in 3D the accuracy is epsilon > 1/2, increasing with alpha. This gives a mathematical basis for diffraction tomography. The proof is based on some new estimates for multidimensional bilinear Fourier multipliers of independent interest.