Sharp uncertainty principles on general Finsler manifolds

Huang, Libing; Kristály, Alexandru; Zhao, Wei

doi:10.1090/tran/8178

Sharp uncertainty principles on general Finsler manifolds
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by Libing Huang, Alexandru Kristály and Wei Zhao;

Trans. Amer. Math. Soc. 373 (2020), 8127-8161

DOI: https://doi.org/10.1090/tran/8178

Published electronically: September 9, 2020

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Uncorrected version: Original version posted September 9, 2020
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Abstract:

The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg, and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fully characterizes the nature of the Finsler manifold in terms of three non- Riemannian quantities, namely, its reversibility and the vanishing of the flag curvature and $S$-curvature induced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem.

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Sharp uncertainty principles on general Finsler manifolds
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Abstract: