The $L^p$ restriction bounds for Neumann data on surface
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911812075651072 |
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| author | Wu, Xianchao |
| author_facet | Wu, Xianchao |
| contents | Let $\{u_λ\}$ be a sequence of $L^2$-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. We seek to get an $L^p$ restriction bounds of the Neumann data $ λ^{-1} \partial_νu_λ\,\vline_γ$ along a unit geodesic $γ$. Using the $T$-$T^*$ argument one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is $O(λ^{-\frac{1}p+\frac{3}2})$. The Van De Corput theorem (Lemma 2.1) plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $L^p$ restriction bounds for Neumann data on surface Wu, Xianchao Analysis of PDEs Let $\{u_λ\}$ be a sequence of $L^2$-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. We seek to get an $L^p$ restriction bounds of the Neumann data $ λ^{-1} \partial_νu_λ\,\vline_γ$ along a unit geodesic $γ$. Using the $T$-$T^*$ argument one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is $O(λ^{-\frac{1}p+\frac{3}2})$. The Van De Corput theorem (Lemma 2.1) plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal. |
| title | The $L^p$ restriction bounds for Neumann data on surface |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.16445 |