A Liouville-type Theorem on half-spaces for sub-Laplacians
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- by Alessia E. Kogoj
- Proc. Amer. Math. Soc. 143 (2015), 239-248
- DOI: https://doi.org/10.1090/S0002-9939-2014-12210-6
- Published electronically: August 28, 2014
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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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Bibliographic 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
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3272749