Sharp second order uncertainty principles

TL;DR

This paper establishes sharp second order inequalities related to uncertainty principles in Euclidean space, revealing how the best constants change for vector fields derived from scalar potentials and resolving an open question for divergence-free fields in two dimensions.

Contribution

The paper introduces sharp second order inequalities of Caffarelli-Kohn-Nirenberg type for vector fields, extending uncertainty principles and solving an open problem for divergence-free fields in two dimensions.

Findings

Best constant in HUP increases from N^2/4 to (N+2)^2/4 for gradient vector fields.

Optimal constant in HyUP improves from (N-1)^2/4 to (N+1)^2/4 for gradient vector fields.

Resolved an open question by Maz'ya for N=2 regarding HUP for divergence-free vector fields.

Abstract

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.