The Fourier extension conjecture for the paraboloid
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913135406874624 |
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| author | Rios, Cristian Sawyer, Eric T. |
| author_facet | Rios, Cristian Sawyer, Eric T. |
| contents | We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized. This is then used to establish a local inequality that is well known to be equivalent to the Fourier extension conjecture, and is accomplished by using a variant of the bilinear equivalence of the Fourier extension conjecture given by Tao, Vargas and Vega in [TaVaVe]. A key aspect of our proof is that the bilinear inequality, when taken over smooth Alpert projections, only requires an averaging over grids of functions mollified by discrete multipliers, which converts a difficult exponential sum into an oscillatory integral with periodic amplitude. After extracting Dirichlet kernels in yet another averaging over lattices, this is then controlled using a stationary phase estimate with periodic amplitude, and altogether we then obtain the desired localization on the Fourier side. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fourier extension conjecture for the paraboloid Rios, Cristian Sawyer, Eric T. Classical Analysis and ODEs 42B05, 42B08, 42B20, 42C40 We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized. This is then used to establish a local inequality that is well known to be equivalent to the Fourier extension conjecture, and is accomplished by using a variant of the bilinear equivalence of the Fourier extension conjecture given by Tao, Vargas and Vega in [TaVaVe]. A key aspect of our proof is that the bilinear inequality, when taken over smooth Alpert projections, only requires an averaging over grids of functions mollified by discrete multipliers, which converts a difficult exponential sum into an oscillatory integral with periodic amplitude. After extracting Dirichlet kernels in yet another averaging over lattices, this is then controlled using a stationary phase estimate with periodic amplitude, and altogether we then obtain the desired localization on the Fourier side. |
| title | The Fourier extension conjecture for the paraboloid |
| topic | Classical Analysis and ODEs 42B05, 42B08, 42B20, 42C40 |
| url | https://arxiv.org/abs/2512.24990 |