Dimension-free Fourier restriction inequalities
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915292952657920 |
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| author | Silva, Diogo Oliveira e Wróbel, Błażej |
| author_facet | Silva, Diogo Oliveira e Wróbel, Błażej |
| contents | Let ${{\bf R}_{\mathbb{S}^{d-1}}}(p\to q)$ denote the best constant for the $L^p(\mathbb{R}^d)\to L^q(\mathbb{S}^{d-1})$ Fourier restriction inequality to the unit sphere $\mathbb{S}^{d-1}$, and let ${\bf R}_{\mathbb{S}^{d-1}} (p\to q;\textrm{rad})$ denote the corresponding constant for radial functions. We investigate the asymptotic behavior of the operator norms ${{\bf R}_{\mathbb{S}^{d-1}}}(p\to q)$ and ${\bf R}_{\mathbb{S}^{d-1}} (p\to q;\textrm{rad})$ as the dimension $d$ tends to infinity. We further establish a dimension-free endpoint Stein-Tomas inequality for radial functions, together with the corresponding estimate for general functions which we prove with an $O(d^{1/2})$ dependence. Our methods rely on a uniform two-sided refinement of Stempak's asymptotic $L^p$ estimate of Bessel functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03942 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimension-free Fourier restriction inequalities Silva, Diogo Oliveira e Wróbel, Błażej Classical Analysis and ODEs 42B10 Let ${{\bf R}_{\mathbb{S}^{d-1}}}(p\to q)$ denote the best constant for the $L^p(\mathbb{R}^d)\to L^q(\mathbb{S}^{d-1})$ Fourier restriction inequality to the unit sphere $\mathbb{S}^{d-1}$, and let ${\bf R}_{\mathbb{S}^{d-1}} (p\to q;\textrm{rad})$ denote the corresponding constant for radial functions. We investigate the asymptotic behavior of the operator norms ${{\bf R}_{\mathbb{S}^{d-1}}}(p\to q)$ and ${\bf R}_{\mathbb{S}^{d-1}} (p\to q;\textrm{rad})$ as the dimension $d$ tends to infinity. We further establish a dimension-free endpoint Stein-Tomas inequality for radial functions, together with the corresponding estimate for general functions which we prove with an $O(d^{1/2})$ dependence. Our methods rely on a uniform two-sided refinement of Stempak's asymptotic $L^p$ estimate of Bessel functions. |
| title | Dimension-free Fourier restriction inequalities |
| topic | Classical Analysis and ODEs 42B10 |
| url | https://arxiv.org/abs/2412.03942 |