论文标题
尖锐的二阶不确定性原理
Sharp second order uncertainty principles
论文作者
论文摘要
我们研究了欧几里得空间中的caffarelli-kohn-nirenberg类型的尖锐的二阶不等式，$ \ mathbb {r}^{n} $，其中$ n $表示维度。该分析等同于研究特殊类矢量场的不确定性原理的研究。特别是，我们表明，当从标量字段切换$ u：\ rr^n \ rightarrow \ mathbb {c} $转换为表格$ \ vec {u}的向量字段：= \ nabla u $（$ u $（$ u $）是标量的标量范围）在Heisenberg noceptionty interiply（Hup）中的最佳常数（ $ \ frac {n^{2}} {4} $ to $ \ frac {（n+2）^{2}} {4} {4} $，氢不确定性原理（HYUP）的最佳常数从$ \ frac {\ frac {\ frac {\ weft（n-1 \ right）^{n-1 \右） $ \ frac {（n+1）^{2}} {4} $。由于结果，我们回答了Maz'ya（整体方程式操作员理论2018）的开放问题，$ n = 2 $涉及divergence Free Vector Fields的HUP。
We study sharp second order inequalities of Caffarelli-Kohn-Nirenberg type in the euclidian space $\mathbb{R}^{N}$, where $N$ denotes the dimension. This analysis is equivalent to the study of uncertainty principles for special classes of vector fields. In particular, we show that when switching from scalar fields $u: \rr^n\rightarrow \mathbb{C}$ to vector fields of the form $\vec{u}:=\nabla U$ ($U$ being a scalar field) the best constant in the Heisenberg Uncertainty Principle (HUP) increases from $\frac{N^{2}}{4}$ to $\frac{(N+2)^{2}}{4}$, and the optimal constant in the Hydrogen Uncertainty Principle (HyUP) improves from $\frac{\left( N-1\right)^{2}}{4}$ to $\frac{(N+1)^{2}}{4}$. As a consequence of our results we answer to the open question of Maz'ya (Integral Equations Operator Theory 2018) in the case $N=2$ regarding the HUP for divergence free vector fields.
