Entanglement principle for fractional Laplacian on hyperbolic spaces and applications to inverse problem

Abstract

We establish an entanglement principle for fractional powers of the Laplace-Beltrami operator on hyperbolic space ${\mathbb{H}}^n$, $n\ge 2$. More precisely, we prove that if finitely many distinct noninteger powers of $-{\Delta }_{{\mathbb{H}}^n}$, acting on functions that vanish on a common nonempty open set, satisfy a linear dependence relation on that set, then each of these functions must vanish identically on ${\mathbb{H}}^n$. This extends the recently developed entanglement principle for the fractional Laplacian on ${\mathbb{R}}^n$ to the negatively curved setting of hyperbolic space. As an application, we derive global uniqueness results for inverse problems associated with fractional polyharmonic equations on ${\mathbb{H}}^n$, including a fractional Calderón problem. The proof relies on the heat semigroup representation of fractional powers together with sharp global heat kernel estimates on hyperbolic space.

22 pages. All comments are welcome