Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

Abstract

We prove a unique continuation property for the fractional Laplacian $(-\Delta {)}^s$ when $s\in (-n/2,\infty )\setminus \mathbb{Z}$. In addition, we study Poincaré-type inequalities for the operator $(-\Delta {)}^s$ when $s\ge 0$. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $d$-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

35 pages, final version