Fractional uncertainty

TL;DR

This paper proves a fractional uncertainty inequality involving dyadic analysis, establishing a lower bound for a product of two integrals related to function localization and energy, valid for all functions with unit L2 norm.

Contribution

It introduces a novel fractional uncertainty inequality using dyadic analysis, connecting nonlocal energy forms with position measures for functions in L2 space.

Findings

Established a lower bound for the product of two integral expressions involving fractional powers.

Connected fractional energy forms with classical variance and gradient measures.

Extended uncertainty principles to fractional and nonlocal contexts.

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 $Var|\varphi|^2$.