We prove the rigidity of the Heisenberg–Pauli–Weyl uncertainty principle and the Caffarelli–Kohn–Nirenberg interpolation inequality, on metric measure spaces satisfying measure contraction property. Non-trivial examples fitting our setting include Finsler manifolds with non-negative Ricci curvature and many ideal sub-Riemannian manifolds, such as Heisenberg groups, the Grushin plane and Sasakian structures.

中文翻译：

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度量测度空间上的尖锐不确定性原理

我们在满足测度收缩性质的度量测度空间上证明了 Heisenberg-Pauli-Weyl 不确定性原理和 Caffarelli-Kohn-Nirenberg 插值不等式的刚性。适合我们设置的重要示例包括具有非负 Ricci 曲率的 Finsler 流形和许多理想的亚黎曼流形，例如海森堡群、Grushin 平面和 Sasakian 结构。