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

TL;DR

This paper investigates the symmetry properties of the second order Hydrogen Uncertainty Principle, disproving a conjecture for dimensions 2 and 3 and establishing the radial nature of extremal functions for a family of sharp inequalities.

Contribution

It provides a negative answer to a conjecture on symmetry breaking and extends the understanding of extremal functions in second order uncertainty principles.

Findings

Disproved the conjecture for N=2,3, showing symmetry breaking.

Established that extremal functions are radial for a family of sharp inequalities.

Extended previous work on second order Caffarelli-Kohn-Nirenberg inequalities.

Abstract

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].