Levi’s parametrix for some sub-elliptic non-divergence form operators

Abstract

We construct the fundamental solutions for the sub-elliptic operators in non-divergence form∑i,jai,j(x,t)XiXj−∂t{\textstyle \sum _{i,j}} a_{i,j}(x,t)\,X_iX_j-\partial _tand∑i,jai,j(x)XiXj{\textstyle \sum _{i,j}}a_{i,j}(x)\,X_iX_j, where theXiX_i’s form a stratified system of Hörmander vector fields andai,ja_{i,j}are Hölder continuous functions belonging to a suitable class of ellipticity.