Harnack inequality for fractional Laplacian-type operators on hyperbolic spaces
Jongmyeong Kim, Minhyun Kim, Ki-Ahm Lee
TL;DR
<p>The work addresses regularity for fully nonlinear nonlocal operators of order $2s$ on negatively curved manifolds, specifically hyperbolic spaces $\mathbb{H}^n_\kappa$, by proving ABP-type estimates, a Krylov–Safonov Harnack inequality, and Hölder continuity. It introduces a novel scale-function framework $\mathcal{I}_{0,\kappa}$, $\mathcal{I}_{\infty,\kappa}$ to account for non-homogeneous hyperbolic geometry and employs hyperbolic dyadic rings and barrier methods to develop a unified regularity theory. The results are robust as $s\to1$ and $\kappa\to0$, recovering classical second-order hyperbolic estimates and Euclidean fractional estimates in the respective limits. This work advances the understanding of nonlocal, fully nonlinear equations on curved spaces and provides tools potentially applicable to heat kernel estimates and stochastic control on manifolds with negative curvature.</p>
Abstract
We establish the Krylov--Safonov theory for a large class of nonlocal operators of order $2s \in (0,2)$ on hyperbolic spaces $\mathbb{H}^{n}_κ$ with curvature $-κ<0$. We prove the Alexandrov--Bakelman--Pucci (ABP) estimates, Krylov--Safonov Harnack inequality, and Hölder 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 $κ\to 0$: they recover the classical regularity estimates for second-order operators on $\mathbb{H}^{n}_κ$ as $s \to 1$, and for fractional-order operators on Euclidean spaces as $κ\to 0$. 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.