Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

IF 2.1 2区数学 Q1 MATHEMATICS

Communications in Partial Differential Equations Pub Date : 2021-10-26 DOI:10.1080/03605302.2023.2191326

B. Cassano, Valentina Franceschi, D. Krejčiřík, D. Prandi

引用次数: 1

Abstract

Abstract In this article, 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 article and has an interest on its own.

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Heisenberg群中的水平磁场和改进的Hardy不等式

摘要在本文中，我们在海森堡群中引入了磁场的概念，并研究了它对相应的磁性（亚椭圆）拉普拉斯算子的光谱性质的影响。我们发现，均匀的磁场提升了光谱的底部。对于在无穷远处消失的磁场，包括Aharonov–Bohm势，我们对海森堡次拉普拉斯算子的各种Hardy型不等式进行了磁性改进。特别地，我们建立了平面中磁性拉普拉斯算子的Laptev和Weidl亚临界结果的亚黎曼类似物。我们的论点的工具是Folland–Stein算子的Hardy型不等式的有效性，我们在本文中证明了这一点，并对其本身感兴趣。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications in Partial Differential Equations 数学-数学

CiteScore

3.60

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： This journal aims to publish high quality papers concerning any theoretical aspect of partial differential equations, as well as its applications to other areas of mathematics. Suitability of any paper is at the discretion of the editors. We seek to present the most significant advances in this central field to a wide readership which includes researchers and graduate students in mathematics and the more mathematical aspects of physics and engineering.