Sharp second order uncertainty principles

2020/12/23 by Cazacu, Cristian, Flynn, Joshua, Lam, Nguyen · 4 citations
#26D10 #26D15 #46E35 #58A10 #81S07 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2012.12667

Abstract

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: \rrⁿ→ 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 \fracN²4 to \frac(N+2)²4, and the optimal constant in the Hydrogen Uncertainty Principle (HyUP) improves from \frac( N-1)²4 to \frac(N+1)²4. 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.

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