The Uncertainty Principle in Fourier Analysis Oleg Kovrijkine

The Uncertainty Principle is the following vague statement: a function and its Fourier transform can not both be concentrated on small sets. The famous Heysenberg Uncertainty Principle in Quantum Mechanics is a particular version of this general principle. We consider the following type of the Uncertainty Principle: ∫ |f|^{2} ≤ C(∫_{E^{c}} |f|^{2} + ∫_{Σ^{c}} |f̂|^{2}) where C does not depend

on f ∈ L^{2}(ℝ^{d}) and E and Σ are ”small” sets in ℝ^{d} and, in particular, it implies that if f is supported on E and f̂ is supported on Σ then f ≡ 0. The challenging part is to find such pair of sets E and Σ. There are three main results of this type: the Amrein-Berthier theorem where E and Σ are sets of finite Lebesgue meausre, the Logvinenko-Sereda theorem where Σ is a ball and E^{c} is relatively dense and Wolff’s theorem where E and Σ are so called ϵ-thin sets. The last two theorems have numerous applications for PDE. We obtain a new version of the Uncertainty Principle which links the last two theorems by introducing a new notion of density for sets E and Σ. The obtained result is sharp. We apply the new result to estimate certain operators on L^{2}(ℝ^{d}).