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Weighted sub-Laplacians on Métivier groups: Essential self-adjointness and spectrum


Authors: Tommaso Bruno and Mattia Calzi
Journal: Proc. Amer. Math. Soc. 145 (2017), 3579-3594
MSC (2010): Primary 22E30, 58J50, 35R03
DOI: https://doi.org/10.1090/proc/13551
Published electronically: January 25, 2017
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Abstract: Let $ G$ be a Métivier group and let $ N$ be any homogeneous norm on $ G$. For $ \alpha >0$ denote by $ w_\alpha $ the function $ e^{-N^\alpha }$ and consider the weighted sub-Laplacian $ \mathcal {L}^{w_\alpha }$ associated with the Dirichlet form $ \phi \!\mapsto \!\int _{G}\Vert\nabla _\mathcal {H}\phi (y)\Vert^2 w_\alpha (y)\,dy$, where $ \nabla _\mathcal {H}$ is the horizontal gradient on $ G$. Consider $ \mathcal {L}^{w_\alpha }$ with domain $ C_c^\infty $. We prove that $ \mathcal {L}^{w_\alpha }$ is essentially self-adjoint when $ \alpha \geq 1$. For a particular $ N$, which is the norm appearing in $ \mathcal {L}$'s fundamental solution when $ G$ is an H-type group, we prove that $ \mathcal {L}^{w_\alpha }$ has purely discrete spectrum if and only if $ \alpha >2$, thus proving a conjecture of J. Inglis.


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Additional Information

Tommaso Bruno
Affiliation: Dipartimento di Matematica, Università degli Studi di Genova, Via Dodecaneso, 35 16146 Genova, Italy
Email: brunot@dima.unige.it

Mattia Calzi
Affiliation: Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri, 7 56126 Pisa, Italy
Email: mattia.calzi@sns.it

DOI: https://doi.org/10.1090/proc/13551
Received by editor(s): September 8, 2016
Published electronically: January 25, 2017
Communicated by: Michael Hitrik
Article copyright: © Copyright 2017 American Mathematical Society