JournalsrmiVol. 37, No. 3pp. 1175–1205

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

  • Cristóbal J. Meroño

    Universidad Politécnica de Madrid, Spain
The double dispersion operator in backscattering: Hölder estimates and optimal Sobolev estimates for radial potentials cover
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Abstract

We study the problem of recovering the singularities of a potential qq from backscattering data. In particular, we prove two new different estimates for the double dispersion operator Q2Q_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 qWβ,2(Rn)q\in W^{\beta,2}(\mathbb{R}^n) with β(n2)/2\beta \ge (n-2)/2 and n3n \ge 3. In the second, we give optimal estimates in the Sobolev scale for Q2(q)Q_2(q) when n2n\ge 2 and qq 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