Geometric Foundations and Nonsmooth Extensions for Harnack Inequalities on Carnot Groups

This paper supplies the geometric and approximation results needed to extend the smooth Harnack inequalities proved in the companion paper {GutierrezMohammedSmooth} to weak solutions of divergence-form subelliptic equations on Carnot groups. We collect horizontal perimeter, Gauss--Green, coarea, and regular-level-set tools, then approximate measurable coefficients and continuous monotone nonlinearities while preserving the structural bounds and pass the Harnack estimates to the weak limit.