This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

中文翻译：

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涉及维度 2 和维度 3 中 Aharonov-Bohm 磁势的不等式

本文致力于收集与涉及 Aharonov-Bohm 磁势的薛定谔算子相关的非线性插值不等式的结果，以及一些结果。由于对称性对于确定最优性结果具有重要作用，我们将考虑对应于圆、二维球体或二维环面的各种情况，以及 2 维和 3 维欧几里得空间。大多数结果是新的我们将重点放在方法上，因为在存在磁场的情况下对对称性、刚性和最优性知之甚少。最引人注目的应用是 2 维和 3 维中的新磁哈代不等式。