Summary: In this paper we obtain some extensions of the classical Krylov-Safonov Harnack inequality. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. We require that at scale larger than some \(r_0> 0\) (small) the functions satisfy the comparison principle with a standard family of quadratic polynomials, while at scale \(r_0\) they satisfy a Weak Harnack type estimate.
We also give several applications of the main result in very different settings such as discrete difference equations, nonlocal equations, homogenization and the quasi-minimal surfaces of Almgren.