Optimal estimates for the double dispersion operator in backscattering

We obtain optimal results in the problem of recovering the singularities of a potential from backscattering data. To do this we prove new estimates for the double dispersion operator of backscattering, the first nonlinear term in the Born series. In particular, by measuring the regularity in the H\"older scale, we show that there is a one derivative gain in the integrablity sense for suitably decaying potentials $q\in W^{\beta,2}(\mathbb{R}^n)$ with $\beta \ge (n-2)/2$. In the case of radial potentials, we are able to give stronger optimal results in the Sobolev scale.