Harnack Inequality for a class of Degenerate Elliptic Equations in Non-Divergence Form

Abstract

We provide two proofs of an invariant Harnack inequality in small balls for a class of second order elliptic operators in non-divergence form, structured on Heisenberg vector fields. We assume that the coefficient matrix is uniformly positive definite, continuous, and symplectic. The first proof emulates a method of E. M. Landis, and is based on the so-called growth lemma, which establishes a quantitative decay of oscillation for subsolutions. The second proof consists in establishing a critical density property for non-negative supersolutions, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez and Lanconelli to obtain Harnack’s inequality.