Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866908318683889664 |
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| author | Cassano, Biagio Franceschi, Valentina Krejcirik, David Prandi, Dario |
| author_facet | Cassano, Biagio Franceschi, Valentina Krejcirik, David Prandi, Dario |
| contents | In this paper 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 paper and has an interest on its own. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_13775 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group Cassano, Biagio Franceschi, Valentina Krejcirik, David Prandi, Dario Spectral Theory Analysis of PDEs In this paper 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 paper and has an interest on its own. |
| title | Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group |
| topic | Spectral Theory Analysis of PDEs |
| url | https://arxiv.org/abs/2110.13775 |