Optimal estimates for the double dispersion operator in backscattering

TL;DR

This paper establishes optimal regularity estimates for the double dispersion operator in backscattering, improving potential recovery accuracy by demonstrating a derivative gain in specific function spaces.

Contribution

It provides new nonlinear estimates for the double dispersion operator, achieving optimal regularity results in both Hölder and Sobolev scales for potential recovery.

Findings

One derivative gain in Hölder scale for decaying potentials.

Stronger results in Sobolev scale for radial potentials.

Optimal estimates improve the understanding of potential singularity recovery.

Abstract

We obtain optimal results in the problem of recovering the singularities of a potential from backscattering data. To do this we prove new estimates for the double dispersion operator of backscattering, the first nonlinear term in the Born series. In particular, by measuring the regularity in the H\"older 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 $\beta \ge (n-2)/2$. In the case of radial potentials, we are able to give stronger optimal results in the Sobolev scale.