Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group
Biagio Cassano, Valentina Franceschi, David Krejcirik, Dario Prandi
TL;DR
The paper develops a magnetic framework for the Heisenberg group and shows that magnetic fields can substantially alter the spectral and functional-inequality landscape of the sub-Laplacian. Through the Rumin complex and horizontal-differential tools, it derives a sub-Riemannian analogue of Laptev–Weidl subcriticality: uniform fields raise the bottom of the spectrum with a bound $\inf\sigma(-\Delta_A) \ge c|B|^{2/3}$, while AB potentials induce sharp Hardy improvements $-\Delta_{A_\alpha} \ge d(\alpha,\mathbb{Z})^2/r^2$ and related center- and cylinder-based inequalities. The results extend the Euclidean diamagnetic picture to sub-Riemannian geometry, and they establish Hardy-type improvements for the Garofalo–Lanconelli inequality via the Folland–Stein operator, including quantitative and localized enhancements. Collectively, these findings deepen our understanding of magnetic effects in subelliptic settings and hint at consequences for long-time behavior of magnetic heat semigroups and quantum dynamics in the Heisenberg group.
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.