Sharp Spectral Projection Estimates for the Torus at $p_c=\frac{2(n+1)}{n-1}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918116871634944 |
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| author | Pezzi, Daniel |
| author_facet | Pezzi, Daniel |
| contents | We prove sharp spectral projection estimates for tori in all dimensions at the exponent $p_c=\frac{2(n+1)}{n-1}$ for shrinking windows of width $1$ down to windows of length $λ^{-1+κ}$ for fixed $κ>0$. This improves and slightly generalizes the work of Blair-Huang-Sogge who proved sharp results for windows of width $λ^{-\frac{1}{n+3}}$, and the work of Hickman, Germain-Myerson, and Demeter-Germain who proved results for windows of all widths but incurred a sub-polynomial loss. Our work uses the approaches of these two groups of authors, combining the bilinear decomposition and microlocal techniques of Blair-Huang-Sogge with the decoupling theory and explicit lattice point lemmas used by Hickman, Germain-Myerson, and Demeter-Germain to remove these losses. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_02746 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp Spectral Projection Estimates for the Torus at $p_c=\frac{2(n+1)}{n-1}$ Pezzi, Daniel Analysis of PDEs Classical Analysis and ODEs 42A45, 42B15 (Primary) 11L07, 11H06, 11P21 (Secondary) We prove sharp spectral projection estimates for tori in all dimensions at the exponent $p_c=\frac{2(n+1)}{n-1}$ for shrinking windows of width $1$ down to windows of length $λ^{-1+κ}$ for fixed $κ>0$. This improves and slightly generalizes the work of Blair-Huang-Sogge who proved sharp results for windows of width $λ^{-\frac{1}{n+3}}$, and the work of Hickman, Germain-Myerson, and Demeter-Germain who proved results for windows of all widths but incurred a sub-polynomial loss. Our work uses the approaches of these two groups of authors, combining the bilinear decomposition and microlocal techniques of Blair-Huang-Sogge with the decoupling theory and explicit lattice point lemmas used by Hickman, Germain-Myerson, and Demeter-Germain to remove these losses. |
| title | Sharp Spectral Projection Estimates for the Torus at $p_c=\frac{2(n+1)}{n-1}$ |
| topic | Analysis of PDEs Classical Analysis and ODEs 42A45, 42B15 (Primary) 11L07, 11H06, 11P21 (Secondary) |
| url | https://arxiv.org/abs/2405.02746 |