Fractional uncertainty

Abstract

We use techniques of dyadic analysis in order to prove that, for every $0<s<\tfrac{1}{2}$, there exists a positive constant $\gamma(s)$ such that the inequality $$\left(\iint_{\mathbb{R}^2}|x-y|^{2s-1}|\varphi(x)||\varphi(y)|dx dy\right)\left(\iint_{\mathbb{R}^2}|x-y|^{-2s-1}|\varphi(x)-\varphi(y)|^2 dx dy\right)\geq \gamma(s)$$ holds for every $\varphi$ with $||\varphi||_{L^2(\mathbb{R})}=1$. The second integral on the left hand side is the energy quadratic form of order $s$, which for the limit case $s=1$ gives the local form $Var|\hat{\varphi}|^2$ or $\int|\nabla\varphi|^2$. The first is a natural substitution of the position form, which on the Haar system shows the same behavior of the classical $Var|\varphi|^2$.