A Liouville-type Theorem on half-spaces for sub-Laplacians

Abstract

Let L be a sub-Laplacian on LN and let G = (LN , ◦, δλ) be its related homogeneous Lie group. Let E be a Euclidean subgroup of LN such that the orthonormal projection π : G −→ E is a homomorphism of homogeneous groups, and let 〈 , 〉 be an inner product in E. Given α ∈ E, α = 0, define Ω(α) := {x ∈ G : 〈α, π(x)〉 > 0}. We prove the following Liouville-type theorem. If u is a nonnegative L-superharmonic function in Ω(α) such that u ∈ L1(Ω(α)), then u ≡ 0 in Ω(α).