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

Abstract:

Let $\mathcal {L}$ be a sub-Laplacian on $\mathcal {L}^N$ and let $\mathbb {G}=(\mathcal {L}^N,\circ ,\delta _\lambda )$ be its related homogeneous Lie group. Let $\mathbb {E}$ be a Euclidean subgroup of $\mathcal {L}^N$ such that the orthonormal projection $\pi :\mathbb {G} \longrightarrow \mathbb {E}$ is a homomorphism of homogeneous groups, and let $\langle \ ,\ \rangle$ be an inner product in $\mathbb {E}$. Given $\alpha \in \mathbb {E}$, $\alpha \neq 0$, define $\Omega (\alpha ):= \{ x\in \mathbb {G} \ :\ \langle \alpha , \pi (x) \rangle >0\}$. We prove the following Liouville-type theorem.

If $u$ is a nonnegative $\mathcal {L}$-superharmonic function in $\Omega (\alpha )$ such that $u\in L^1(\Omega (\alpha ))$, then $u\equiv 0$ in $\Omega (\alpha )$.