Fundamental solutions for non-divergence form operators on stratified groups

Abstract

We construct the fundamental solutionsΓ\Gammaandγ\gammafor the non-divergence form operators∑i,jai,j(x,t)XiXj−∂t\,{\textstyle \sum _{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial _t\,and∑i,jai,j(x)XiXj{\,\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 also provide Gaussian estimates ofΓ\Gammaand its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates ofΓ\Gammaallow us to constructγ\gammausing bothtt-saturation and approximation arguments.