On certain degenerate and singular elliptic PDEs. III: Nondivergence form operators and \(RH_\infty\)-weights (Q2656303)

The author studies the regularity properties of the strong solutions to certain second-order elliptic PDEs with rough coefficients whose degeneracies/singularities are shaped by weights in the reverse-Hölder class \(RH_\infty\). In the seminal papers by \textit{L. A. Caffarelli} and \textit{C. E. Gutierrez} [Trans. Am. Math. Soc. 348, No. 3, 1075--1092 (1996; Zbl 0858.35034); Am. J. Math. 119, No. 2, 423--465 (1997; Zbl 0878.35039)], the proof of Harnack inequalities for nonnegative classical solutions \(u\) of suitable elliptic degenerate PDEs is based on a reverse-Hölder inequality with respect to the Monge-Ampère sections. In order to extend this result to more general operators (such as Grushin operators) another condition is necessary, i.e., the existence of a positive constant \(\sigma\) such that \[\sigma \delta \phi (x_0, x) \leq D\phi(x_0,x).\] This paper is devoted to investigate the deep connections between the above inequality and the reverse Hölder inequality with respect to the Monge-Ampère sections.