Harnack inequality for Nonlocal operators on Manifolds with nonnegative curvature

TL;DR

This paper proves Harnack inequalities and regularity estimates for fully nonlinear nonlocal operators on manifolds with nonnegative curvature, extending classical PDE results to a geometric nonlocal setting.

Contribution

It introduces nonlocal Pucci operators on manifolds and establishes uniform regularity results, bridging local and nonlocal PDE theories in geometric contexts.

Findings

Harnack inequalities hold for nonlocal operators on curved manifolds.

Uniform Hölder estimates are established for these operators.

Results recover classical local operator results as a limit.

Abstract

We establish the Krylov Safonov Harnack inequalities and Holder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity results for these operators which recover the classical results for second order local operators as limits.