Sharp second order uncertainty principles

摘要

We study sharp second order inequalities of Caffarelli-Kohn-Nirenberg type in the euclidian space 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 : R-n -> C to vector fields of the form (u) over bar:=del U(Ubeing a scalar field) the best constant in the Heisenberg Uncertainty Principle (HUP) increases from N2/4 to (N+2)(2)/4, and the optimal constant in the Hydrogen Uncertainty Principle (HyUP) improves from (N-1)(2)/4to (N+1)(2)/4. As a consequence of our results we answer to the open question of Maz'ya [21, Section 3.9] in the case N = 2 regarding the HUP for divergence free vector fields. (c) 2022 Elsevier Inc. All rights reserved.