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On the Essential Self-Adjointness of Singular Sub-Laplacians

Franceschi, Valentina; Prandi, Dario; Rizzi, Luca (2019), On the Essential Self-Adjointness of Singular Sub-Laplacians, Potential Analysis, 53, p. 89-112. 10.1007/s11118-018-09760-w

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1708.09626.pdf (299.6Kb)
Type
Article accepté pour publication ou publié
Date
2019
Journal name
Potential Analysis
Volume
53
Publisher
Step Communications
Pages
89-112
Publication identifier
10.1007/s11118-018-09760-w
Metadata
Show full item record
Author(s)
Franceschi, Valentina
Laboratoire de Mathématiques d'Orsay [LM-Orsay]
Centre de Mathématiques Appliquées - Ecole Polytechnique [CMAP]
Prandi, Dario cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laboratoire des signaux et systèmes [L2S]
Rizzi, Luca cc
Institut Fourier [IF ]
Abstract (EN)
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. We also show that, in the compact case, this criterion implies discreteness of the sub-Laplacian spectrum even though the total volume of the manifold is infinite. As a consequence of our result, the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp’s measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This settles a conjecture formulated by Boscain and Laurent (Ann. Inst. Fourier (Grenoble) 63(5), 1739–1770, 2013), under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.
Subjects / Keywords
Sub-Laplacian; Hörmander-type operators; Singular measure; Popp’s measure; Quantum confinement

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