Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

Biagio Cassano; Valentina Franceschi; David Krejcirik; Dario Prandi

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arXiv · 2110.13775

Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

Biagio Cassano·Valentina Franceschi·David Krejcirik·Dario Prandi

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Abstract

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.

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Biagio Cassano, Valentina Franceschi, David Krejcirik, Dario Prandi. 2021-10-26. Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group. https://arxiv.org/abs/2110.13775

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