The double dispersion operator in backscattering: Hölder estimates and optimal Sobolev estimates for radial potentials

Abstract

We study the problem of recovering the singularities of a potential $q$ from backscattering data. In particular, we prove two new different estimates for the double dispersion operator $Q_2$ of backscattering, the first nonlinear term in the Born series. In the first, by measuring the regularity in the Hölder 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 $\beta \ge (n-2)/2$ and $n\ge 3$. In the second, we give optimal estimates in the Sobolev scale for $Q_2(q)$ when $n\ge 2$ and $q$ is radial. In dimensions 2 and 3 this result implies an optimal result of recovery of singularities from the Born approximation.