Uncertainty Principle for distributions with Fourier transform in $L_{p,q}(\mathbb{R}^d)$

TL;DR

This paper refines the Uncertainty Principle for functions in Lorentz spaces with Fourier transforms supported on sets of specific Hausdorff measure, identifying the precise conditions at the critical endpoint.

Contribution

It establishes the sharp form of the Uncertainty Principle at the endpoint case for Lorentz spaces, characterizing support conditions via Netrusov--Hausdorff capacity.

Findings

Existence of non-zero functions in Lorentz spaces with Fourier support on sets of zero Netrusov--Hausdorff capacity.

The UP does not hold at the critical Hausdorff measure threshold.

Precise capacity conditions determine the support of Fourier transforms in Lorentz spaces.

Abstract

A version of the Uncertainty Principle says: There does not exist a non zero function in $L_p(\mathbb{R}^d)$ if its Fourier transform is supported by a set of finite $\alpha$-Hausdorff measure with $\alpha<2d/p$. This UP does not hold at the endpoint $\alpha=2d/p$. We find the sharp form of the UP in the limit case. We prove that there exists a non-zero function in the Lorentz space $L_{p,q}(\mathbb{R}^d)$ such that its Fourier transform is supported by a set of zero $(\frac{2d}{p},\beta)$-Netrusov--Hausdorff capacity if and only if $\beta>\frac{q}{2(q-1)}$.