Uncertainty Principle for distributions with Fourier transform in $L_{p,q}(\mathbb{R}^d)$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910175918555136 |
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| author | Dobronravov, Nikita |
| author_facet | Dobronravov, Nikita |
| contents | 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 $α$-Hausdorff measure with $α<2d/p$. This UP does not hold at the endpoint $α=2d/p$. 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 $(\frac{2d}{p},β)$-Netrusov--Hausdorff capacity if and only if $β>\frac{q}{2(q-1)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26096 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uncertainty Principle for distributions with Fourier transform in $L_{p,q}(\mathbb{R}^d)$ Dobronravov, Nikita Classical Analysis and ODEs 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 $α$-Hausdorff measure with $α<2d/p$. This UP does not hold at the endpoint $α=2d/p$. 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 $(\frac{2d}{p},β)$-Netrusov--Hausdorff capacity if and only if $β>\frac{q}{2(q-1)}$. |
| title | Uncertainty Principle for distributions with Fourier transform in $L_{p,q}(\mathbb{R}^d)$ |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.26096 |