Roughly speaking, a unique continuation property states that a solution of certain partial differential equation is determined by its behaviour in a subset. In this talk we will see this kind of properties, including their strong and quantitative versions, for some classes of nonlocal operators like the Hilbert transform, which arise in medical imaging, or the (higher order) fractional Laplacian. These results rely on commonly used tools as Carleman estimates and the Caffareli-Silvestre extension, but also on two alternative mechanisms. As an application we will see Runge approximation results.
This is joint work with Angkana Rüland.