Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

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, i...

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Main Authors: Cassano, Biagio, Franceschi, Valentina, Krejcirik, David, Prandi, Dario
Format: Article in Journal/Newspaper
Language:unknown
Published: arXiv 2021
Subjects:
Spectral Theory math.SP
Analysis of PDEs math.AP
FOS Mathematics
Folland
laptev
Online Access:https://dx.doi.org/10.48550/arxiv.2110.13775
https://arxiv.org/abs/2110.13775
id ftdatacite:10.48550/arxiv.2110.13775
record_format openpolar
spelling ftdatacite:10.48550/arxiv.2110.13775 2023-05-15T17:07:16+02:00 Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group Cassano, Biagio Franceschi, Valentina Krejcirik, David Prandi, Dario 2021 https://dx.doi.org/10.48550/arxiv.2110.13775 https://arxiv.org/abs/2110.13775 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Spectral Theory math.SP Analysis of PDEs math.AP FOS Mathematics Article CreativeWork article Preprint 2021 ftdatacite https://doi.org/10.48550/arxiv.2110.13775 2022-03-10T14:02:54Z 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. : 44 pages Article in Journal/Newspaper laptev DataCite Metadata Store (German National Library of Science and Technology) Folland ENVELOPE(7.471,7.471,63.004,63.004)
institution Open Polar
collection DataCite Metadata Store (German National Library of Science and Technology)
op_collection_id ftdatacite
language unknown
topic Spectral Theory math.SP
Analysis of PDEs math.AP
FOS Mathematics
spellingShingle Spectral Theory math.SP
Analysis of PDEs math.AP
FOS Mathematics
Cassano, Biagio
Franceschi, Valentina
Krejcirik, David
Prandi, Dario
Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group
topic_facet Spectral Theory math.SP
Analysis of PDEs math.AP
FOS Mathematics
description 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. : 44 pages
format Article in Journal/Newspaper
author Cassano, Biagio
Franceschi, Valentina
Krejcirik, David
Prandi, Dario
author_facet Cassano, Biagio
Franceschi, Valentina
Krejcirik, David
Prandi, Dario
author_sort Cassano, Biagio
title Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group
title_short Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group
title_full Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group
title_fullStr Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group
title_full_unstemmed Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group
title_sort horizontal magnetic fields and improved hardy inequalities in the heisenberg group
publisher arXiv
publishDate 2021
url https://dx.doi.org/10.48550/arxiv.2110.13775
https://arxiv.org/abs/2110.13775
long_lat ENVELOPE(7.471,7.471,63.004,63.004)
geographic Folland
geographic_facet Folland
genre laptev
genre_facet laptev
op_rights arXiv.org perpetual, non-exclusive license
http://arxiv.org/licenses/nonexclusive-distrib/1.0/
op_doi https://doi.org/10.48550/arxiv.2110.13775
_version_ 1766062643841335296

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