Subcritical Fourier uncertainty principles
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913420200116224 |
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| author | Saucedo, Miquel Tikhonov, Sergey |
| author_facet | Saucedo, Miquel Tikhonov, Sergey |
| contents | It is well known that if a function $f$ satisfies $$\|f(x) e^{πα|x|^2}\|_p + \| \widehat{f}(ξ) e^{πα|ξ|^2} \|_q<\infty \qquad\qquad\qquad(*)$$
with $α=1$ and $1\le p,q<\infty$, then $f\equiv 0.$
We prove that if $f$ satisfies $(*)$ with some $0<α<1$ and $1\le p,q\leq \infty$, then $$ |f(y)|\le C
(1+|y|)^{\frac{d}{p}}
e^{- πα|y|^2}, \quad y\in \mathbb{R}^d, $$ with $ C=C(α,d,p,q)$ and this bound is sharp for $p\neq 1$. We also study a related uncertainty principle for functions satisfying $\;\;\displaystyle\|f(x)|x|^m\|_p+ \|\widehat{f}(ξ)|ξ|^n\|_q <\infty.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subcritical Fourier uncertainty principles Saucedo, Miquel Tikhonov, Sergey Classical Analysis and ODEs Functional Analysis 42A38, 42B35 It is well known that if a function $f$ satisfies $$\|f(x) e^{πα|x|^2}\|_p + \| \widehat{f}(ξ) e^{πα|ξ|^2} \|_q<\infty \qquad\qquad\qquad(*)$$ with $α=1$ and $1\le p,q<\infty$, then $f\equiv 0.$ We prove that if $f$ satisfies $(*)$ with some $0<α<1$ and $1\le p,q\leq \infty$, then $$ |f(y)|\le C (1+|y|)^{\frac{d}{p}} e^{- πα|y|^2}, \quad y\in \mathbb{R}^d, $$ with $ C=C(α,d,p,q)$ and this bound is sharp for $p\neq 1$. We also study a related uncertainty principle for functions satisfying $\;\;\displaystyle\|f(x)|x|^m\|_p+ \|\widehat{f}(ξ)|ξ|^n\|_q <\infty.$ |
| title | Subcritical Fourier uncertainty principles |
| topic | Classical Analysis and ODEs Functional Analysis 42A38, 42B35 |
| url | https://arxiv.org/abs/2404.07375 |