Harnack inequality for fractional Laplacian-type operators on hyperbolic spaces

TL;DR

This paper develops a comprehensive regularity theory for nonlocal fractional operators on hyperbolic spaces, including new Harnack inequalities, with results that unify and extend classical estimates to curved geometries.

Contribution

It introduces a robust regularity framework for fractional Laplacian-type operators on hyperbolic spaces, including new Harnack inequalities and scale functions accounting for curvature effects.

Findings

Established ABP estimates, Krylov--Safonov Harnack inequality, and Hölder estimates on hyperbolic spaces.

Proved the Harnack inequality is new even for the fractional Laplacian.

Demonstrated the estimates recover classical results as curvature or fractional order approaches Euclidean limits.

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.