Weighted L^p-Liouville theorems for hypoelliptic partial differential operators on Lie groups

We prove weighted L^p-Liouville theorems for a class of second-order hypoelliptic partial differential operators L on Lie groups G whose underlying manifold is n-dimensional space. We show that a natural weight is the right-invariant measure ˇH of G. We also prove Liouville-type theorems for C^2 subsolutions in L p(G, ˇH ). We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator L − ∂t .