The Uncertainty Principle for certain densities

Abstract

We prove a new version of the Uncertainty Principle of the form $\int |f|^2 \lesssim \int_{E^c} |f|^2 + \int_{Σ^c}|\hat f|^2 $ where the sets $E$ and $Σ$ are $ε$-thin in the following sense: $|E \cap D(x, ρ_1(x))| \le ε|D(x, ρ_1(x))|$ and $|Σ\cap D(x, ρ_2(x))| \le ε|D(x, ρ_2(x))|$. This is an intermediate result between Logvinenko-Sereda's and Wolff's versions of the Uncertainty Principle.