Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities

In this work we deal with linear second order partial differential operators of the following type: \[ H = ∂ t − L = ∂ t − ∑ i , j = 1 q a i j ( t , x ) X i X j − ∑ k = 1 q a k ( t , x ) X k − a 0 ( t , x ) H=∂ ₜ-L=∂ ₜ-∑ i,j=1qaᵢⱼ ( t,x ) XᵢXⱼ-∑ ₖ₌₁qaₖ ( t,x ) Xₖ-a₀ ( t,x ) \] where X 1 , X 2 , … , X q X₁,X₂,… ,Xq is a system of real Hörmander’s vector fields in some bounded domain Ω ⊆ R n <mml:annotation encoding="application/x-te