Harnack inequality for fractional Laplacian-type operators on hyperbolic spaces

Abstract

We establish the Krylov--Safonov theory for a large class of nonlocal operators of order $2s \in (0,2)$ on hyperbolic spaces $\mathbb{H}^{n}_{\kappa}$ with curvature $-\kappa<0$. We prove the Alexandrov--Bakelman--Pucci (ABP) estimates, Krylov--Safonov Harnack inequality, and H\"older estimates. Notably, the Harnack inequality is new even for the fractional Laplacian. The novelty of the results lies in the robustness of the regularity estimates as $s \to 1$ and $\kappa \to 0$: they recover the classical regularity estimates for second-order operators on $\mathbb{H}^{n}_{\kappa}$ as $s \to 1$, and for fractional-order operators on Euclidean spaces as $\kappa \to 0$. Since the operators on hyperbolic spaces exhibit qualitatively different behavior compared to their Euclidean counterparts, we introduce new scale functions which take the effect of negative curvatures into account.