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...
Published in: | Communications in Partial Differential Equations |
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Main Authors: | , , , |
Other Authors: | , , , |
Format: | Article in Journal/Newspaper |
Language: | English |
Published: |
Taylor and Francis Ltd.
2023
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Subjects: | |
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 |
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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