We need ABP.

The goal of the Krylov–Safonov theory is to establish quantitative regularity for PDEs in nondivergence form with merely measurable coefficients. The elliptic result is the following. Let \(A=(a_{ij})\) satisfy \(\lambda \mathrm{Id}\le A(x)\le\Lambda \mathrm{Id}\) for constants \(0<\lambda\le\Lambda\). We write

Theorem: Every nonnegative solution of \(Lu=0\) in \(B_{2r}\) satisfies

where \(C=C(n,\Lambda/\lambda)>0\). Moreover, every solution of \(Lu=0\) in \(B_{2r}\) satisfies

where \(\alpha=\alpha(n,\Lambda/\lambda)>0\).

The role of ABP

The key ingredient was the Aleksandrov–Bakelman–Pucci estimate, developed in works of Aleksandrov (Dokl. Akad. Nauk SSSR, 1960, archive), Bakelman (Sibirsk. Mat. Zh., 1961, archive), and Pucci (Ann. Mat. Pura Appl., 1966, DOI). For \(Lu=f\) in a bounded domain \(\Omega\), with \(u=0\) on \(\partial\Omega\),

Krylov–Safonov (Dokl. Akad. Nauk SSSR, 1979, archive) used a probabilistic method to establish quantitative propagation of positivity from sets of positive measure. A growth lemma and a covering argument yield the Harnack inequality and Hölder continuity. The parabolic result appeared in Krylov–Safonov (Math. USSR-Izvestiya, 1981; Russian original 1980, DOI), alongside the elliptic result of Safonov (J. Soviet Math., 1983; Russian original 1980, DOI).