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–Caratheodory doubling metric space, where a Poincare inequality also holds. Both doubling and Poincare 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 Holder 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 R N × R N .