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We study the problem of recovering the singularities of a potential from backscattering data. In particular, we prove two new different estimates for the double dispersion operator 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 with and . In the second, we give optimal estimates in the Sobolev scale for when and is radial. In dimensions 2 and 3 this result implies an optimal result of recovery of singularities from the Born approximation.
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Cristóbal J. Meroño, The double dispersion operator in backscattering: Hölder estimates and optimal Sobolev estimates for radial potentials. Rev. Mat. Iberoam. 37 (2021), no. 3, pp. 1175–1205DOI 10.4171/RMI/1223