螺线管场的海森堡不确定性原理及二阶Caffarelli--Kohn--Nirenberg不等式的最优稳定性层级

中文摘要 AI 辅助

2018年，V. Maz'ya提出了分析学和偏微分方程中的75个开放问题。其中之一是关于螺线管矢量场的海森堡不确定性原理的最佳常数，该问题源于流体力学中的疑问。Cazacu、Flynn和Lam在二维情形下解决了该问题，随后Hamamoto通过建立具有显式常数的尖锐不等式，在所有维度$N\ge3$中解决了该问题。然而，难度大得多的稳定性问题在所有维度中仍然开放。本文的第一个主要结果是建立一个尖锐的稳定性层级，通过证明其亏量控制到极值集合的距离，对于$N\ge4$，尖锐稳定性常数为$\frac12(N-\sqrt{N^2-4N+12})$，对于$N=3$，常数为$1$，同时给出衡量到显式更大的极向场和环向场族距离的余项链。第二个主要结果是直线$1+a+b=0$上带权重$|x|^{-2a}$和$|x|^{-2b}$的二阶$L^2$-Caffarelli--Kohn--Nirenberg不等式，我们获得其稳定性常数为$\frac12\min\{4(1+a),\sqrt{(N+2a)^2+4N-4}-(N+2a)\}$。当第二个数较小时，该CKN不等式是尖锐的。我们发展了一种新方法：与球面模态相关的四阶一维问题，通过实数阶的Fourier--Hankel变换，转化为一阶不等式，其尖锐稳定性是加权高斯Poincaré不等式的稳定性。逆变换以Kummer函数形式产生极值。作为我们新方法的副产品，我们还获得了Hamamoto的一维不等式及其螺线管场尖锐不确定性原理的显著更简单的证明，并带有尖锐余项和等式情形。

英文摘要

In 2018, V. Maz'ya proposed 75 open problems in analysis and PDEs. One of them is about the best constant for the Heisenberg Uncertainty Principle for solenoidal vector fields motivated by questions in hydrodynamics. This was answered by Cazacu, Flynn and Lam in dimension two and subsequently by Hamamoto in all dimensions $N\ge3$ by establishing sharp inequalities with explicit constants. Nevertheless, the much harder stability problem remains open in all dimensions. The first main result in this paper is to establish a sharp stability hierarchy by proving that its deficit controls the distance to the set of extremals, with the sharp stability constant $\frac12(N-\sqrt{N^2-4N+12})$ for $N\ge4$ and $1$ for $N=3$, together with chains of remainder terms measuring the distance to explicit larger families of poloidal and toroidal fields. The second main result is the second order $L^2$-Caffarelli--Kohn--Nirenberg inequality with the weights $|x|^{-2a}$ and $|x|^{-2b}$ on the line $1+a+b=0$, for which we obtain the stability constant $\frac12\min\{4(1+a),\sqrt{(N+2a)^2+4N-4}-(N+2a)\}$. This CKN inequality is sharp whenever the second number is the smaller one. We develop a novel method: the fourth order one-dimensional problems attached to the spherical modes are transformed, by a Fourier--Hankel transform of real order, into first order inequalities whose sharp stability is that of weighted Gaussian Poincaré inequalities. The inverse transform produces the extremals in terms of Kummer's function. As a byproduct of our new approach, we also obtain substantially simpler proofs, with sharp remainder terms and equality cases, of Hamamoto's one-dimensional inequality and of his sharp uncertainty principle for solenoidal fields.

发表机构

• University of Connecticut（康涅狄格大学）
• Hanoi University of Science and Technology（河内科学技术大学）
• Memorial University of Newfoundland（纽芬兰纪念大学）
• FPT University（FPT大学）

机构由 AI 辅助整理，请以论文原文为准。

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39 pages