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

### Cristóbal J. Meroño

Universidad Politécnica de Madrid, Spain

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## 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.

## Cite this article

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. 1175–1205

DOI 10.4171/RMI/1223