Bounds for spectral projectors on the three-dimensional torus
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| Format: | Preprint |
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2025
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| _version_ | 1866909782950019072 |
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| author | Germain, Pierre Myerson, Simon L. Rydin Pezzi, Daniel |
| author_facet | Germain, Pierre Myerson, Simon L. Rydin Pezzi, Daniel |
| contents | We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window limit we are left with $L^p$ norms of eigenfunctions of the Laplacian, as considered for instance by Bourgain. For the three-dimensional torus we prove new cases of a previous conjecture of the first two authors concerning the size of these norms; we also refine certain prior results to remove $ε$-losses in all dimensions. We use methods from number theory: the geometry of numbers, the circle method and exponential sum bounds due to Guo. We complement these techniques with height splitting and a bilinear argument to prove sharp results.
We exposit on the various techniques used and their limitations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds for spectral projectors on the three-dimensional torus Germain, Pierre Myerson, Simon L. Rydin Pezzi, Daniel Analysis of PDEs Classical Analysis and ODEs 42A45, 42B15 (Primary), 11L07, 11H06, 11P21 (Secondary) We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window limit we are left with $L^p$ norms of eigenfunctions of the Laplacian, as considered for instance by Bourgain. For the three-dimensional torus we prove new cases of a previous conjecture of the first two authors concerning the size of these norms; we also refine certain prior results to remove $ε$-losses in all dimensions. We use methods from number theory: the geometry of numbers, the circle method and exponential sum bounds due to Guo. We complement these techniques with height splitting and a bilinear argument to prove sharp results. We exposit on the various techniques used and their limitations. |
| title | Bounds for spectral projectors on the three-dimensional torus |
| topic | Analysis of PDEs Classical Analysis and ODEs 42A45, 42B15 (Primary), 11L07, 11H06, 11P21 (Secondary) |
| url | https://arxiv.org/abs/2508.05573 |