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Transactions of the American Mathematical Society

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Sharp uncertainty principles on general Finsler manifolds


Authors: Libing Huang, Alexandru Kristály and Wei Zhao
Journal: Trans. Amer. Math. Soc. 373 (2020), 8127-8161
MSC (2010): Primary 26D10, 53C60, 53C23
DOI: https://doi.org/10.1090/tran/8178
Published electronically: September 9, 2020
MathSciNet review: 4169684
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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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Additional Information

Libing Huang
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, 300071 Tianjin, People’s Republic of China
Email: huanglb@nankai.edu.cn

Alexandru Kristály
Affiliation: Department of Economics, Babeş-Bolyai University, 400591 Cluj-Napoca, Romania; and Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary
Email: alex.kristaly@econ.ubbcluj.ro; kristaly.alexandru@nik.uni-obuda.hu

Wei Zhao
Affiliation: Department of Mathematics, East China University of Science and Technology, 200237 Shanghai, People’s Republic of China
ORCID: 0000-0001-9319-6070
Email: szhao_wei@yahoo.com

DOI: https://doi.org/10.1090/tran/8178
Keywords: Uncertainty principles, Caffarelli-Kohn-Nirenberg interpolation inequality, Heisenberg-Pauli-Weyl inequality, Hardy inequality, Finsler manifold, reversibility, sharp constant, rigidity
Received by editor(s): December 6, 2018
Received by editor(s) in revised form: April 7, 2020
Published electronically: September 9, 2020
Additional Notes: The research of the second author was supported by the National Research, Development and Innovation Fund of Hungary, financed under the K_18 funding scheme, Project no. 127926.
The third author was supported by the National Natural Science Foundation of China (No. 11501202, No. 11761058) and the grant of China Scholarship Council (No. 201706745006).
Article copyright: © Copyright 2020 American Mathematical Society