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 ({mathcal{L}}) on Lie groups ({mathbb{G}}) whose underlying manifold is ({n})-dimensional space. We show that a natural weight is the right-invariant measure (check{H}) of ({mathbb{G}}). We also prove Liouville-type theorems for ({C^{2}}) subsolutions in ({L^{p}(mathbb{G},check{H})}). 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 ({mathcal{L}-partial_{t}}).