On Instability Properties of the Fractional Calder\'{o}n Problem

We prove exponential instability properties for the fractional Calder\'on problem and the conductivity formulation of the fractional Calder\'on problem in the regime of fractional powers s∈ (0,1). We particularly focus on two settings: First, we discuss instability properties in general domain geometries with scaling critical Ln/2s potentials and constant background metrics. Secondly, we investigate instability properties in general geometries with Ln/2s potentials and low regularity, variable coefficient, possibly anisotropic background metrics. In both settings we make use of the methods introduced in {KRS21} and we deduce strong compression estimates for the forward problem. In the first setting this is based on analytic smoothing estimates for a suitable comparison operator while in the second setting involving low regularity metrics this is based on an iterated compression gain. We thus generalize the results from {RS18} to generic geometries and variable coefficients and further also discuss the setting of fractional conductivity equations. In particular, this proves that the logarithmic stability estimates for the fractional Calder\'on problem from {RS20} are optimal.