Fourier Restriction: From Linear Restriction to Multilinear Restriction
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915672575967232 |
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| author | Zhang, Sicheng |
| author_facet | Zhang, Sicheng |
| contents | This dissertation studies the Fourier restriction, which is to find the range of the constants p, q such that the L^q norm on a chosen subset of the Fourier domain is bounded above by the L^p norm in a spacial domain, up to some constant that is independent of the function. We discuss linear restriction, including Hausdorff-Young's inequality, A proof of the restriction estimate on curves, and further discussions on the restriction problem on the sphere and paraboloid via the Stein-Tomas argument. We then discuss bilinear restriction, where the estimate on 2-dimensional case is proved by the reverse square function estimate and the bilinear interaction of transverse wave packets. The result is further used to verify the restriction conjecture on the 2-dimensional paraboloid. We discuss about multi-linear restriction in the final section, focusing on a short proof of a close result of the multilinear restriction estimate from I. Bejenaru. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11888 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier Restriction: From Linear Restriction to Multilinear Restriction Zhang, Sicheng History and Overview This dissertation studies the Fourier restriction, which is to find the range of the constants p, q such that the L^q norm on a chosen subset of the Fourier domain is bounded above by the L^p norm in a spacial domain, up to some constant that is independent of the function. We discuss linear restriction, including Hausdorff-Young's inequality, A proof of the restriction estimate on curves, and further discussions on the restriction problem on the sphere and paraboloid via the Stein-Tomas argument. We then discuss bilinear restriction, where the estimate on 2-dimensional case is proved by the reverse square function estimate and the bilinear interaction of transverse wave packets. The result is further used to verify the restriction conjecture on the 2-dimensional paraboloid. We discuss about multi-linear restriction in the final section, focusing on a short proof of a close result of the multilinear restriction estimate from I. Bejenaru. |
| title | Fourier Restriction: From Linear Restriction to Multilinear Restriction |
| topic | History and Overview |
| url | https://arxiv.org/abs/2512.11888 |