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Improved Caffarelli–Kohn–Nirenberg Inequalities and Uncertainty Principle

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Abstract

In this paper we prove some improved Caffarelli–Kohn–Nirenberg inequalities and uncertainty principle for complex- and vector-valued functions on \({\mathbb {R}}^n\), which is a further study of the results in Dang et al. (J Funct Anal 265:2239-2266, 2013). In particular, we introduce an analogue of “phase derivative" for vector-valued functions. Moreover, using the introduced “phase derivative", we extend the extra-strong uncertainty principle to cases for complex- and vector-valued functions defined on \(\mathbb S^n,n\ge 2.\)

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Acknowledgements

W.-X. Mai was supported by NSFC Grant No.11901594, FRG Program of the Macau University of Science and Technology, No. FRG-22-076-MCMS, and the Science and Technology Development Fund, Macau SAR (File no. 0133/2022/A, 0022/2023/ITP1). P. Dang was supported by FRG Program of the Macau University of Science and Technology, No. FRG-23-033-FIE, and the Science and Technology Development Fund, Macau SAR (File No. 0030/2023/ITP1).

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Authors and Affiliations

  1. Department of Engneering Science, Faculty of Innovation Engineering, Macau University of Science and Technology, Macao, China

    Pei Dang

  2. Macao Center for Mathematical Sciences, Macau University of Science and Technology, Macao, China

    Weixiong Mai

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  1. Pei Dang
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  2. Weixiong Mai
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Correspondence to Weixiong Mai.

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Dang, P., Mai, W. Improved Caffarelli–Kohn–Nirenberg Inequalities and Uncertainty Principle. J Geom Anal 34, 70 (2024). https://doi.org/10.1007/s12220-023-01524-2

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