The fractional Calder\'on problem: low regularity and stability

TL;DR

This paper advances the fractional Calderón problem by establishing logarithmic stability and extending results to potentials in low regularity spaces, using quantitative approximation techniques.

Contribution

It provides the first stability estimates for the fractional Calderón problem with low regularity potentials, improving upon previous uniqueness results.

Findings

Logarithmic stability for the fractional Calderón problem.

Extension of results to potentials in $L^p$ and negative Sobolev spaces.

Quantitative approximation properties for solutions of fractional equations.

Abstract

The Calder\'on problem for the fractional Schr\"odinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant $L^p$ or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli-Silvestre extension and a duality argument.