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arXiv · 2604.26096

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

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

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BibTeXRIS

Nikita Dobronravov. 2026-04-28. Uncertainty Principle for distributions with Fourier transform in $L_{p,q}(\mathbb{R}^d)$. https://arxiv.org/abs/2604.26096

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