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

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 $\beta>\frac{q}{2(q-1)}$.