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We obtain optimal results in the problem of recovering the singularities of apotential from backscattering data. To do this we prove new estimates for thedouble dispersion operator of backscattering, the first nonlinear term in theBorn 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 forsuitably 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 strongeroptimal results in the Sobolev scale.

Optimal estimates for the double dispersion operator in backscattering