Fractional uncertainty

Abstract

We use techniques of dyadic analysis in order to prove that, for every $0<s<\frac{1}{2}$, there exists a positive constant $\gamma (s)$ such that the inequality $$\left({\iint }_{{\mathbb{R}}^2}|x-y{|}^{2s-1}|\phi (x)||\phi (y)|dxdy\right)\left({\iint }_{{\mathbb{R}}^2}|x-y{|}^{-2s-1}|\phi (x)-\phi (y){|}^{2}dxdy\right)\ge \gamma (s)$$ holds for every $\phi$ with $||\phi |{|}_{L^2(\mathbb{R})}=1$. 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 $Var|\hat{\phi }{|}^2$ or $\int |\nabla \phi {|}^2$. The first is a natural substitution of the position form, which on the Haar system shows the same behavior of the classical $Var|\phi {|}^2$.

12 pages, 2 figures