Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

• Main Authors: Cassano, Biagio, Franceschi, Valentina, Krejcirik, David, Prandi, Dario
• Other Authors: Università degli studi di Bari Aldo Moro = University of Bari Aldo Moro (UNIBA), Dipartimento di Matematica Padova, Università degli Studi di Padova = University of Padua (Unipd), Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Université Paris-Saclay, Centre National de la Recherche Scientifique (CNRS), Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
• Format: Report
• Language: English
• Published: HAL CCSD 2021
• Subjects: [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]; Folland; laptev
• Online Access: https://hal.science/hal-03434702

44 pages 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.