On the extremal functions of second order uncertainty principles: symmetry and symmetry breaking

2025/08/21 by Xiao‐Ping Chen, Chun‐Lei Tang, Chen, Xiao-Ping +1 · 1 citation
Decision Sciences · Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probabilistic and Robust Engineering Design #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2508.15221

openalex publication_date 2025/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper focus on the symmetry and symmetry breaking about the second order Hydrogen Uncertainty Principle. Firstly, by choosing a suitable test function, we give a negative answer to the conjecture presented by Cazacu, Flynn and Lam in [J. Funct. Anal. 283 (2022), Paper No. 109659, 37 pp] for N∈\2,3\, and emphasizing the symmetry breaking phenomenon. Secondly, we obtain a family of sharp weighted second order Hydrogen Uncertainty Principle, and prove the extremal functions are radial, which extends the work of Duong and Nguyen [The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle, arXiv:2102.01425].

Citations

• Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities
• Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and\n second order derivatives
• The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle
• Sharp second order uncertainty principles
• Sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities and their extremal functions
• Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on half space via mass transport and consequences
• Weighted inequalities and Stein-Weiss potentials
• Best Constants in the Hardy-Rellich Inequalities and Related Improvements

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• On Sharp Heisenberg Uncertainty Principle and the stability

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