Let G be a Métivier group and let N be any homogeneous norm on G. For α > 0 denote by wα the function e−N α and consider∫ the weighted sub-Laplacian Lwα associated with the Dirichlet form φ→G ‖∇Hφ(y)‖2 wα (y) dy, where ∇H is the horizontal gradient on G. Consider Lwα with domain Cc∞. We prove that Lwα is essentially self-adjoint when α ≥ 1. For a particular N, which is the norm appearing in L’s fundamental solution when G is an H-type group, we prove that Lwα has purely discrete spectrum if and only if α > 2, thus proving a conjecture of J. Inglis.
Weighted sub-laplacians on métivier groups: Essential self-adjointness and spectrum / T. Bruno, M. Calzi. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9939. - 145:8(2017 Aug), pp. 3579-3594. [10.1090/proc/13551]
Weighted sub-laplacians on métivier groups: Essential self-adjointness and spectrum
M. Calzi
2017
Abstract
Let G be a Métivier group and let N be any homogeneous norm on G. For α > 0 denote by wα the function e−N α and consider∫ the weighted sub-Laplacian Lwα associated with the Dirichlet form φ→G ‖∇Hφ(y)‖2 wα (y) dy, where ∇H is the horizontal gradient on G. Consider Lwα with domain Cc∞. We prove that Lwα is essentially self-adjoint when α ≥ 1. For a particular N, which is the norm appearing in L’s fundamental solution when G is an H-type group, we prove that Lwα has purely discrete spectrum if and only if α > 2, thus proving a conjecture of J. Inglis.Pubblicazioni consigliate
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