$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912570867187712 |
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| author | Duan, Shengze Pham, Minh-Quy Ryou, Donggeun |
| author_facet | Duan, Shengze Pham, Minh-Quy Ryou, Donggeun |
| contents | For $0 < α\leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\mathbb{R}^d$ such that $N(E,\varepsilon) \lesssim \varepsilon^{-α}$ for $0 <\varepsilon <1$ where $N(E,\varepsilon)$ is the smallest number of $\varepsilon$-balls needed to cover $E$. We proved that if $f \in L^p(\mathbb{R}^d)$ with \begin{align*}
1 \leq p\leq p_α:=
\begin{cases}
\frac{d^2+d+2α}{2α} & d \geq 3,
\frac{4}α &d =2,
\end{cases}
\end{align*} and $\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p$ is optimal by considering random Cantor sets on the moment curve. We extended the result of Guo, Iosevich, Zhang and Zorin-Kranich, including the endpoint. We also considered applications of our results to the failure of the restriction estimates and Wiener Tauberian Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve Duan, Shengze Pham, Minh-Quy Ryou, Donggeun Classical Analysis and ODEs primary: 42B10, secondary: 42B20, 28A75 For $0 < α\leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\mathbb{R}^d$ such that $N(E,\varepsilon) \lesssim \varepsilon^{-α}$ for $0 <\varepsilon <1$ where $N(E,\varepsilon)$ is the smallest number of $\varepsilon$-balls needed to cover $E$. We proved that if $f \in L^p(\mathbb{R}^d)$ with \begin{align*} 1 \leq p\leq p_α:= \begin{cases} \frac{d^2+d+2α}{2α} & d \geq 3, \frac{4}α &d =2, \end{cases} \end{align*} and $\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p$ is optimal by considering random Cantor sets on the moment curve. We extended the result of Guo, Iosevich, Zhang and Zorin-Kranich, including the endpoint. We also considered applications of our results to the failure of the restriction estimates and Wiener Tauberian Theorem. |
| title | $L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve |
| topic | Classical Analysis and ODEs primary: 42B10, secondary: 42B20, 28A75 |
| url | https://arxiv.org/abs/2412.19956 |