标题： 关于二阶不确定性原理的极值函数：对称性与对称性破缺 显示英文标题

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

摘要： 本文关注二级氢不确定性原理的对称性和对称性破缺。 \emph{首先}，通过选择合适的测试函数，我们对 Cazacu、Flynn 和 Lam 在 [\emph{J. Funct. Anal.} \textbf{283} (2022)，论文编号 109659，37 页] 中提出的猜想给出了否定回答，针对$N\in\{2,3\}$，并强调了对称性破缺现象。 \emph{其次}，我们获得了一族精确的加权二级氢不确定性原理，并证明了极值函数是径向的，这扩展了 Duong 和 Nguyen 的工作 [精确的二级 Caffareli-Kohn-Nirenberg 不等式和精确的二级不确定性原理的稳定性估计，arXiv:2102.01425]。 显示英文摘要

摘要： This paper focus on the symmetry and symmetry breaking about the second order Hydrogen Uncertainty Principle. \emph{Firstly}, by choosing a suitable test function, we give a negative answer to the conjecture presented by Cazacu, Flynn and Lam in [\emph{J. Funct. Anal.} \textbf{283} (2022), Paper No. 109659, 37 pp] for $N\in\{2,3\}$, and emphasizing the symmetry breaking phenomenon. \emph{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].