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

TL;DR

This paper extends the entanglement principle for fractional Laplacians from Euclidean space to hyperbolic space, establishing unique continuation results and applications to inverse problems like the fractional Calderón problem.

Contribution

It introduces an entanglement principle for fractional Laplacians on hyperbolic spaces and applies it to prove uniqueness in inverse fractional polyharmonic problems.

Findings

Proved an entanglement principle for fractional Laplacians on hyperbolic space.

Derived global uniqueness results for inverse fractional polyharmonic problems.

Extended the entanglement principle from Euclidean to hyperbolic geometry.

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 Calder\'on problem. The proof relies on the heat semigroup representation of fractional powers together with sharp global heat kernel estimates on hyperbolic space.