Sharp second order uncertainty principles

Abstract

We study sharp second order inequalities of Caffarelli-Kohn-Nirenberg type in the euclidian space RN, 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:Rn→C to vector fields of the form u→:=∇U (U being a scalar field) the best constant in the Heisenberg Uncertainty Principle (HUP) increases from N24 to (N+2)24, and the optimal constant in the Hydrogen Uncertainty Principle (HyUP) improves from (N−1)24 to (N+1)24. 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.