The double dispersion operator in backscattering: Hölder estimates and optimal Sobolev estimates for radial potentials (Q2031510)

Summary: 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.