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


Author: Alessia E. Kogoj
Journal: Proc. Amer. Math. Soc. 143 (2015), 239-248
MSC (2010): Primary 35B53, 35R03, 31C05, 31B05; Secondary 35H20, 35H10
DOI: https://doi.org/10.1090/S0002-9939-2014-12210-6
Published electronically: August 28, 2014
MathSciNet review: 3272749
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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 )$.


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Additional Information

Alessia E. Kogoj
Affiliation: Basque Center for Applied Mathematics (BCAM), Mazzaredo, 14, E48009 Bilbao, Basque Country, Spain
Address at time of publication: Dipartimento di Matematica, Alma Mater Studiorum - Università di Bologna Piazza di Porta, S. Donato, 5, 40126 Bologna, Italy
Email: alessia.kogoj@unibo.it

DOI: https://doi.org/10.1090/S0002-9939-2014-12210-6
Keywords: Liouville-type theorems, half-spaces, superharmonic functions, subLaplacians, homogeneous Lie groups
Received by editor(s): September 11, 2012
Received by editor(s) in revised form: March 10, 2013, and March 24, 2013
Published electronically: August 28, 2014
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.