Search

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 \({\mathcal{L}-\partial_{t}}\).

Aim of this paper is to provide new examples of Hormander operators \({\mathcal{L}}\) to which a Lie group structure can be attached making \({\mathcal{L}}\) left invariant. Our class of examples contains several subclasses of operators appearing in literature and arising both in theoretical and in applied fields: evolution Kolmogorov operators, degenerate Ornstein–Uhlenbeck operators, Mumford and Fokker–Planck operators, Ornstein–Uhlenbeck operators with time-dependent periodic coefficients. Our examples basically come from exponential of matrices, as well as from linear constant-coefficient ODE’s, in \({\mathbb{R}}\) or in \({\mathbb{C}}\) . Furthermore, we describe how these groups can be combined together to obtain new structures and new operators, also having an interest in the applied field.