Harnack inequality for non-divergence form operators on stratified groups

Abstract

We prove lower bounds for the fundamental solutions of the non-divergence form operators\[∑i,jai,j(x,t)XiXj−∂tand∑i,jai,j(x)XiXj,{\textstyle \sum _{i,j}} a_{i,j}(x,t)\,X_iX_j-\partial _t \quad \text {and}\quad {\textstyle \sum _{i,j}}a_{i,j}(x)\,X_iX_j,\]where theXiX_i’s are Hörmander vector fields generating a stratified groupG\mathbb {G}and(ai,j)i,j(a_{i,j})_{i,j}is a positive-definite matrix with Hölder continuous entries. We then prove an invariant Harnack inequality for such operators. As a byproduct we also study some relevant properties of the Green functions on bounded domains.