Weighted sub-Laplacians on Métivier groups: Essential self-adjointness and spectrum
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- by Tommaso Bruno and Mattia Calzi PDF
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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 $\phi \!\mapsto \!\int _{G}\|\nabla _\mathcal {H}\phi (y)\|^2 w_\alpha (y) dy$, 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.References
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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
- Received by editor(s): September 8, 2016
- Published electronically: January 25, 2017
- Communicated by: Michael Hitrik
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3579-3594
- MSC (2010): Primary 22E30, 58J50, 35R03
- DOI: https://doi.org/10.1090/proc/13551
- MathSciNet review: 3652809