Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

In this article, we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity...

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Bibliographic Details
Published in:Communications in Partial Differential Equations
Main Authors: Cassano B., Franceschi V., Krejcirik D., Prandi D.
Other Authors: Cassano, B., Franceschi, V., Krejcirik, D., Prandi, D.
Format: Article in Journal/Newspaper
Language:English
Published: Taylor and Francis Ltd. 2023
Subjects:
laptev
Online Access:https://hdl.handle.net/11577/3481542
https://doi.org/10.1080/03605302.2023.2191326
https://www.tandfonline.com/doi/full/10.1080/03605302.2023.2191326
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Summary:In this article, we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov–Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland–Stein operator, that we prove in this article and has an interest on its own.

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