An invariant Harnack inequality for a class of subelliptic operators under global doubling and Poincaré assumptions, and applications

The aim of this paper is to prove an invariant, non-homogeneous Harnack inequality for a class of subelliptic operators L in divergence form, with low-regular coefficients. The main assumption, whose geometric meaning is well known in the literature on Harnack inequalities, is the requirement that L be naturally associated with a Carnot-Carathéodory doubling metric space, where a Poincaré inequality also holds. Both doubling and Poincaré conditions are assumed to hold globally for every CC-ball: accordingly, the Harnack inequality will hold true on every CC-ball. Applications to inner and boundary Hölder estimates are provided, together with pertinent results on the Green function for L. An explicit example of a class of operators for which our results are fulfilled is also given. Via the Green function for L, the global nature of the Harnack inequality can be applied to the study of the existence of a fundamental solution Γ for L, globally defined out of the diagonal of RN×RN.